Research ArticleInteracting Quintessence Dark Energy Models in Lyra Manifold

M Khurshudyan1 J Sadeghi2 R Myrzakulov3 Antonio Pasqua4 and H Farahani5

1 Department of Theoretical Physics Yerevan State University 1 Alex Manookian 0025 Yerevan Armenia2Department of Physics Islamic Azad University Ayatollah Amoli Branch PO Box 678 Amol Iran3Department of Physics Eurasian International Center for Theoretical Physics Eurasian National UniversityAstana 010008 Kazakhstan

4Department of Physics University of Trieste Via Valerio No 2 34127 Trieste Italy5 Department of Physics Mazandaran University PO Box 47416-95447 Babolsar Iran

Correspondence should be addressed to H Farahani hfarahaniumzacir

Received 8 April 2014 Revised 8 August 2014 Accepted 13 August 2014 Published 1 September 2014

Academic Editor Sally Seidel

Copyright copy 2014 M Khurshudyan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

We consider two-component dark energy models in Lyra manifoldThe first component is assumed to be a quintessence field whilethe second component may be a viscous polytropic gas a viscous Van der Waals gas or a viscous modified Chaplygin gas Wealso consider the possibility of interaction between components By using the numerical analysis we study some cosmologicalparameters of the models and compare them with observational data

1 Introduction

Observations of high redshift type Supernovae Ia (SNeIa) [1ndash3] reveal the accelerated expansion of our universe whosenature is not exactly clear until now It is found that thedensity of matter is very much less than critical density [4]Moreover cosmic microwave background (CMB) radiationanisotropies observations indicate that the universe can beconsidered flat and the total energy density is very close tothe critical value (Ωtot ≃ 1) [5] Based on the experimentaldata a component of the energy dubbed as dark energy isthought to be responsible for the physics of the acceleratedexpansion but it seems that it is not alone in the universe sothemysteriousmatter component which is called darkmattershould also exist Dark energy can be described by a pressuresufficiently negative in order to drive the acceleration of theuniverse and by positive energy density There are severaldifferent models proposed to explain the nature of darkenergy The cosmological constant Λ is the simplest modelwhich can be considered but in presence of many researchpapers in these fields the origin of dark energy and darkmatter is still unknown and the possible connection betweenthem is also unknown as well as real role of the componentsin the history of the universe This situation gives a lot of

freedom to researchers and possibility of some simulationsThe cosmological constant faced two main problems thatis the absence of a fundamental mechanism which sets thecosmological constant zero or very small value (which isknown as fine-tuning problem) and the problem known ascosmological coincidence problem which asks why we areliving in an epoch in which the densities of dark energy andmatter are comparable One of the interesting ways to solvethe above mentioned problems is to consider interactionsbetween components [6] From observational point of viewno piece of evidence has been so far presented againstsuch interactions Indeed possible interactions between thecomponents of universe have been discussed in recent yearsIt is found that a suitable interaction can help to alleviate thecoincidence problem Different interacting models of darkenergy have been investigated [7ndash14]

Alternative models of dark energy suggest a dynamicalform of dark energy which at least in an effective levelcan originate from a variable cosmological constant [15 16]or from various fields such as a canonical scalar field [17ndash21] (quintessence) a phantom field [20ndash22] or quintom[23ndash36] Finally interesting attempts to probe the nature ofdark energy according to some basic quantum gravitationalprinciples are the holographic dark energy paradigm [37ndash45]

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 878092 13 pageshttpdxdoiorg1011552014878092

2 Advances in High Energy Physics

and agegraphic dark energy models [46 47] Among thema quintessence model is interesting in this paper as a com-ponent of dark energy In that case the dark energy may bedynamical approaching zero with time or it may be slowlyincreasing It is now dominating the universe because thereduction of mass and radiation energy density with thescale factor (which gives some information about the sizeof the universe) is greater than the decrease in dark energydensity in the present epoch In general we would like thequintessence field to be decreasing with the scale factor andtime at a smaller rate than the mass energy so that it willbecome dominant at redshifts less than oneThe quintessencefield has the property of being veryweakly coupled to baryonsbut contributing a negative pressure to the equation of stateIn the past it had a small contribution but with time it hasdecreased less quickly with the scale factor than the matterand radiation densities and is dominant now

For the dark energy component we consider severalmod-els in this paper including viscosity Indeed bulk viscosityis added to obtain more realistic models However viscouspressure can itself play the role of an agent that drives thepresent acceleration of the universe [48]

One of interesting dark energy models is the polytropicgas which was proposed to explain the accelerated expansionof the universe [49] It was shown that the polytropic gasmodel in the presence of interaction can behave as phantomfield [50] It was pointed out that a polytropic scalar field canbe reconstructed according to the evolutionary behaviors ofthe holographic and new agegraphic dark energy densitiesThe validity of the generalized second lawof thermodynamicswas also examined for the polytropic gas model in [51]Another interesting model of dark energy may be Vander Waals gas which could be accounted as a fluid withunusual EoS or could be thought of as a fluid satisfyingmore general form of EoS that is 119865(120588 119875) = 0 [52] Thereare also some important models to describe dark energybased on Chaplygin gas equation of state which were recentlyconsidered by several papers such as [53ndash56] and yield goodagreement with observational data

On the other hand the Lyra geometry provides one ofthe possible alternatives in modification of the cosmologicalmodels As we know the modification of the gravitationaltheory has long been famous but the late-time cosmologicalacceleration caused more research in this field [57] Nowwe like to consider a universe filled with a two-componentdark energy in Lyra manifold with possibility of interactionbetween components The first component is assumed to bequintessence while we have several choices for the secondcomponent such as viscous polytropic gas viscous Van derWaals gas or viscous Chaplygin gas We suggest these as toymodels to describe universe and compare our results withobservational data to choose one of them as the best model

This paper is organized as follows In Section 2 we intro-duce our models In Section 3 we recall the main propertiesof field equations In Section 4 we give numerical resultscorresponding to constantΛ In Section 5 we give numericalresults corresponding to varying Λ In Section 6 we obtainsome observational constraints Finally in Section 7 wewritethe conclusions of this paper

2 The Models

One of the well studied dark energy models is thequintessence model [17 18] which is a scalar field modeldescribed by a field 120601 and a 119881(120601) potential It representsthe simplest scalar field scenario without having theoreticalproblems such as the appearance of ghosts and LaplacianinstabilitiesThe energy density 120588119876 and the pressure 119875119876 of thequintessence scalar field model are given respectively by

120588119876 =1

2

1206012+ 119881 (120601)

119875119876 =1

2

1206012minus 119881 (120601)

(1)

Canonical scalar field is not the unique solution We cangeneralize it as follows [58]

120588119876 =120596

2120601119896 1206012+ 119881 (120601)

119875119876 =120596

2120601119896 1206012minus 119881 (120601)

(2)

In the case of 119896 = 0 (2) transform to the canonical scalarfield model with rescaling of the field Below we would liketo consider an interaction term 119876 between dark energy anddark matter described by

119876 = 3119867119887120588119876 + 120574 (120588119887 minus 120588119876)

120601

120601 (3)

where 119887 and 120574 are positive constants with a typical value of001ndash003 Nature of the interaction between dark energy anddark matter is not clear If we believe that it has a quantumorigin then an absence of the final theory of quantum gravityleaves this question as an open problem However if webelieve that the link existing between components is due tothe same origin of the dark energy and dark matter thenthis approach does not give any exact solution because thenature of the two components is not formulated and it isanother open problem Therefore only phenomenologicalassumption is an appropriate approach For the dark mattermodel we will consider once a viscous modified Chaplygingas with the following equation of state (EoS)

119875 = 119860120588 minus119861

120588120572minus 3120585119867 (4)

where 119860 119861 and 120572 are constants (with 0 le 120572 le 1 in generalrelativity)

For the second model we will use viscous polytropic fluidwith EoS given by

119875 = 1198701205881+1119899

minus 3120585119867 (5)

where 119870 is the polytropic index and 120585 represents the viscouscoefficient

In the third model we would like to consider interactionbetween quintessence dark energy and a viscous Van derWaals gas of the general form

119875 =119860120588

119861 minus 120588minus 1198611205882minus 3120585119867 (6)

Advances in High Energy Physics 3

where 119860 and 119861 are constants Furthermore we will considertwo regimes (1)Λ is a numerical constant (2)Λ is a functionof the cosmic time 119905 therefore it is a varying quantity Inparticular we choose the following form for the time-varyingΛ

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (7)

where 120575 is a positive constant and 119881(120601) is the potential of thefield which we consider as follows

119881 (120601) = 1198810119890(minus1206010120601) (8)

where 1206010 is a constant parameterWe investigate the behavior of the cosmological param-

eters like the Hubble parameter 119867 deceleration parameter119902 and EoS parameters of the quintessence dark energy andan effective two-component fluid Moreover we performstability analysis via the squared speed of the sound1198622119878 whichis defined as follows

1198622

119904 =120597119875

120597120588 (9)

where 119875 and 120588 are respectively the pressure and the energydensity of the effective fluid given by

119875 = 119875119876 + 119875119894

120588 = 120588119876 + 120588119894

(10)

where 119894 refers to one of the viscous fluids described aboveWe will finish our paper with the results obtained fromobservational constraints Consideration of the statefinderanalysis different forms of interaction terms and differentΛ(119905) and varying viscosity is possible and an interestingresearch We hope to approach this question in future withforthcoming articles

3 The Field Equations

The field equations governing our model are given by

119877120583] minus1

2119892120583]119877 minus Λ119892120583] +

3

2120601120583120601] minus

3

4119892120583]120601120572120601120572 = 119879120583] (11)

Considering the content of the universe to be a perfect fluidwe have

119879120583] = (120588 + 119875) 119906120583119906] minus 119875119892120583] (12)

where 119906120583 = (1 0 0 0) is the 4-velocity of the comovingobserver satisfying the relation 119906120583119906

120583= 1 Let 120601120583 be a time-

like vector field of displacement then

120601120583 = (2

radic3120573 0 0 0) (13)

where120573 = 120573(119905) is a function of time alone and the factor 2radic3is inserted in order to simplify the writing of all the followingequations By using FRWmetric for a flat universe given by

1198891199042= minus119889119905

2+ 119886(119905)

2(1198891199032+ 1199032119889Ω2) (14)

field equations can be reduced to the following Friedmannequations

31198672minus 1205732= 120588 + Λ

2 + 31198672+ 1205732= minus119875 + Λ

(15)

where 119867 = 119886119886 is the Hubble parameter the dot standsfor differentiation with respect to the cosmic time 119905 119889Ω2 =1198891205792+ sin21205791198891206012 and 119886(119905) represents the scale factor The 120579

and 120601 parameters are the usual azimuthal and polar anglesof spherical coordinates with 0 le 120579 le 120587 and 0 le 120601 lt 2120587 Thecoordinates (119905 119903 120579 120601) are called comoving coordinates

The continuity equation is given by

120588 + Λ + 2120573 120573 + 3119867 (120588 + 119875 + 21205732) = 0 (16)

The continuity equation given in (16) can be also rewritten inthe compact form

120588 + 3119867 (120588 + 119875) = 0 (17)

Comparing (16) and (17)we obtain the following link betweenΛ and 120573

Λ + 2120573 120573 + 61198671205732= 0 (18)

In order to introduce an interaction between dark energy anddark matter we should mathematically split (17) into the twofollowing equations

120588119894 + 3119867 (120588119894 + 119875119894) = 119876 (19)

120588119876 + 3119867 (120588119876 + 119875119876) = minus119876 (20)

The cosmological parameters of our interest are the EoSparameter of each fluid component 120596119894 = 119875119894120588119894 the EoSparameter of composed fluid

120596tot =119875119876 + 119875119894

120588119876 + 120588119894

(21)

and the deceleration parameter 119902 which can be written asfollows

119902 =1

2(1 + 3

119875

120588) (22)

where index 119894 refers to the first components which is viscousmodified Chaplygin gas or viscous polytropic fluid and index119876 refers to the quintessence scalar field A differential equa-tion describing dynamics of the DE after some mathematicscan be rewritten as

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601 (23)

Taking into account the form of the varyingΛ(119905) from (7) forthe Hubble parameter119867 we will have

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (24)

Hereafter we will consider 1206010 = 1 for mathematical simplic-ity

4 Advances in High Energy Physics

1 2 3 4 5

04

05

06

07

08

09

10

11

t

H

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(a)

minus05

00

05

10

15

1 2 3 4 5

t

q

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 1 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 1

4 Case of Constant Λ

We found it reasonable to start our analysis from the modelswith constant Λ Without loss of generality we would liketo describe equations allowing us to find dynamics of themodels According to the assumption with constant Λ (16)will be modified as follows

120588 + 2120573 120573 + 3119867 (120588 + 119875 + 21205732) = 0 (25)

and using the expression 120588 + 3119867(120588 + 119875) = 0 we will obtainthat

120573 + 3119867120573 = 0 (26)

The last equation can be integrated very easily and the resultis the following

120573 = 1205730119886minus3 (27)

where 119886(119905) is the scale factor and 1205730 is the integrationconstant In our future calculations we will use 1205730 = 1 asinitial condition For the Hubble parameter119867 we will obtain

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (28)

Concerning the form of the field equations we need only toassume the form of 119876 and we will obtain the cosmologicalsolutions Concerning the mathematical hardness of theproblem we will analyze models numerically and investi-gate graphical behavior of various important cosmologicalparameters with respect to the cosmic time In the followingsubsections we consider three models with the particularform of 119876 with given forms of the EoS for the viscous darkmatter fluids considered in the Introduction

41Model 1 Thefirst toymodel describes the dynamics of theuniverse within an effective fluid in case of the cosmologicalconstant The dynamics of the energy density of the viscousmodified Chaplygin gas which will model dark matter inour universe and the differential equation describing thedynamics of it can be found to be

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(29)

From (23) for the dynamics of the dark energy we have

120588119876 + 3119867(1 + 119887 + 120596119876 minus120574

3119867

120601

120601)120588119876 = minus120574

120601

120601120588Ch (30)

Thebest fit for the theoreticalmodel of our considerationwiththe observational data we obtained for1198670 = 14 ΩCh = 03Λ = 07 119860 = 25 119861 = 09 120574 = 002 119887 = 001 and 120585 = 01

In Figures 1ndash3 we have chosen to plot some of thequantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01120572 = 05 and119860 = 25 andwehave chosen five differentvalues of Λ that is 0 02 03 05 and 07

Figure 1 shows that theHubble parameter119867 is decreasingwith time to a constant at the late universe as expected andits value increased by Λ On the other hand the value of thedeceleration parameter decreased with Λ It is illustrated bythe right plot of Figure 1 For Λ = 0 we can see 119902 rarr minus04while for Λ = 07 we can see 119902 rarr minus07 Also acceleration todeceleration phase transition is seen in this model At the latetime we have 119902 sim minus05 being in agreement with observationaldata As we know recent observations of type SNIa indicatethat universe is accelerating with the deceleration parameterlying somewhere in the range minus1 lt 119902 le 0

Advances in High Energy Physics 5

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596tot

120574 = 002 b = 001 and 120585 = 01

(a)

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596Q

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 2 Behavior of EoS parameters 120596tot and 120596119876 against 119905 for the constant Λ Model 1

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

minus1

0

1

2

3

C2 s

120574 = 002 b = 001 and 120585 = 01

Figure 3 Squared sound speed against 119905 for the constant Λ Model1

Moreover Figure 2 shows that the EoS parameter tendsto minus1 at the late time with 120596tot ge minus1 corresponding to aquintessence-like universe

Unfortunately analysis of squared sound speed (seeFigure 3) shows that this model is not stable at the late timeand will be considered only for the early universe

42 Model 2 In the second model after some mathematicalcalculations we obtain the following differential equation tostudy dynamics of the model

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(31)

where 120596119875 is given as

120596119875 = 1198701205881119899

119875 minus3120585119867

120588119875

(32)

The best fitted values of parameters are1198670 = 12 Ω119875 = 025Λ = 15 119870 = 25 119899 = 10 120574 = 002 119887 = 001 and 120585 = 02

In Figures 4 and 5 we have chosen to plot some ofthe quantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01 119899 = 1 and 119870 = 25 and we have chosen five differentvalues of Λ that is 0 05 10 12 and 15

Numerical results of the Hubble expansion parameterand the deceleration parameter 119902 show good behavior (seeFigure 4) but the stability analysis illustrated in Figure 5shows that this model also has instability in the late timeand is only useful for the early universe However the effectof constant Λ in this parameter is similar to the first modeland the EoS parameter tends to minus1 as before In the case ofΛ = 0 we can see that 119902 sim minus02 which is not coincident withobservational data It tells that presence ofΛmaybenecessaryto obtain agreement with observations

Figure 5 shows that instability of model may solve for thelarge value of the cosmological constant Therefore model 2is stable at the late time for the large value of the Λ

43 Model 3 In the third model we have the followingexpression for the pressure 119875

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (33)

The constant Λ assumption will lead us to the followingexpression for the Hubble parameter119867

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (34)

6 Advances in High Energy Physics

02

04

06

08

10

12

1 2 3 4 5

t

H

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(a)

04

02

00

minus02

minus04

minus06

minus08

q

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(b)

Figure 4 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 2

minus1

0

1

2

3

C2 s

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

Figure 5 Squared sound speed against 119905 for the constant Λ Model2

The differential equations describing the dynamics of theenergy densities of both components are given by the follow-ing equations

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119882

120601

120601

(35)

where 120596119876 is the EoS parameter of dark energy The bestfit for the theoretical model of our consideration with the

5 10 15 20

t

10

05

00

minus05

minus10

C2 s

Λ = 0 A = 15 B = 12

Λ = 03 A = 15 B = 12Λ = 05 A = 15 B = 12

Λ = 07 A = 15 B = 12

Λ = 12 A = 15 B = 12

120574 = 002 b = 0 and 120585 = 04

Figure 6 Squared sound speed against 119905 for the constant Λ Model3

observational data is obtained for 1198670 = 13 Ω119875 = 0217Λ = 12 119860 = 15 119861 = 12 120574 = 002 119887 = 000 and 120585 = 04

The behaviors of the cosmological parameters are similarto the previous models and we can see late time instability ofthis model in Figure 6 This suggests considering varying Λto obtain more appropriate models

5 The Case of Varying Λ

In this section we will consider three interacting fluidmodels and will investigate cosmological parameters like theHubble parameter 119867 deceleration parameter 119902 and EoSparameters of the total fluid and dark energy 120596119876 Based onnumerical solutions we will discuss graphical behaviors of

Advances in High Energy Physics 7

04

05

06

07

08

09

10

11

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

H

120574 = 002 b = 001 and 120585 = 05

(a)

04

06

08

10

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

H

120575 = 1120574 = 002 b = 001 and

(b)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

120574 = 002 b = 001 and 120585 = 05

(c)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

120575 = 1120574 = 002 b = 001 and

(d)

Figure 7 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 4

the cosmological parameters For the varying Λ we take aphenomenological form which was considered by us recently[59] The formula of Λ is given as the following expression

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (36)

which is a function of the Hubble parameter potential of thescalar field and time derivative of the scalar field For thepotential we take a simple form 119881(120601) = 119890

[minus120601] therefore theform of Λ can be written also in the following way as only afunction of the filed 120601

Λ (119905) = 1198672120601minus2+ 120575119890minus120601 (37)

Therefore the dynamics of 120573 can be obtained from thefollowing differential equation

2120573 120573 + 61198671205732+ 2119867120601

minus2minus 21198672120601minus3 120601 minus 120575119890

[minus120601] 120601 = 0 (38)

In forthcoming subsections within three different forms of119876wewill investigate the dynamics of the universeThe question

of the dynamics for the energy densities of the dark energyand dark matter is already discussed in a previous sectionthereforewewill not consider themhere andwewill startwiththe comments on the graphical behaviors of the cosmologicalparameters of themodelsWe will start with themodel where

119876 = 3119867119887120588119876 + 120574 (120588119894 minus 120588119876)

120601

120601 (39)

51 Model 4 Interacting viscous modified Chaplygin gaswith the quintessence dark energy in the case of varying Λgives the following differential equation

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(40)

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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AstronomyAdvances in

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Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

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AstrophysicsJournal of

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Physics Research International

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Solid State PhysicsJournal of

 Computational  Methods in Physics

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Soft MatterJournal of

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ThermodynamicsJournal of

2 Advances in High Energy Physics

and agegraphic dark energy models [46 47] Among thema quintessence model is interesting in this paper as a com-ponent of dark energy In that case the dark energy may bedynamical approaching zero with time or it may be slowlyincreasing It is now dominating the universe because thereduction of mass and radiation energy density with thescale factor (which gives some information about the sizeof the universe) is greater than the decrease in dark energydensity in the present epoch In general we would like thequintessence field to be decreasing with the scale factor andtime at a smaller rate than the mass energy so that it willbecome dominant at redshifts less than oneThe quintessencefield has the property of being veryweakly coupled to baryonsbut contributing a negative pressure to the equation of stateIn the past it had a small contribution but with time it hasdecreased less quickly with the scale factor than the matterand radiation densities and is dominant now

For the dark energy component we consider severalmod-els in this paper including viscosity Indeed bulk viscosityis added to obtain more realistic models However viscouspressure can itself play the role of an agent that drives thepresent acceleration of the universe [48]

One of interesting dark energy models is the polytropicgas which was proposed to explain the accelerated expansionof the universe [49] It was shown that the polytropic gasmodel in the presence of interaction can behave as phantomfield [50] It was pointed out that a polytropic scalar field canbe reconstructed according to the evolutionary behaviors ofthe holographic and new agegraphic dark energy densitiesThe validity of the generalized second lawof thermodynamicswas also examined for the polytropic gas model in [51]Another interesting model of dark energy may be Vander Waals gas which could be accounted as a fluid withunusual EoS or could be thought of as a fluid satisfyingmore general form of EoS that is 119865(120588 119875) = 0 [52] Thereare also some important models to describe dark energybased on Chaplygin gas equation of state which were recentlyconsidered by several papers such as [53ndash56] and yield goodagreement with observational data

On the other hand the Lyra geometry provides one ofthe possible alternatives in modification of the cosmologicalmodels As we know the modification of the gravitationaltheory has long been famous but the late-time cosmologicalacceleration caused more research in this field [57] Nowwe like to consider a universe filled with a two-componentdark energy in Lyra manifold with possibility of interactionbetween components The first component is assumed to bequintessence while we have several choices for the secondcomponent such as viscous polytropic gas viscous Van derWaals gas or viscous Chaplygin gas We suggest these as toymodels to describe universe and compare our results withobservational data to choose one of them as the best model

This paper is organized as follows In Section 2 we intro-duce our models In Section 3 we recall the main propertiesof field equations In Section 4 we give numerical resultscorresponding to constantΛ In Section 5 we give numericalresults corresponding to varying Λ In Section 6 we obtainsome observational constraints Finally in Section 7 wewritethe conclusions of this paper

2 The Models

One of the well studied dark energy models is thequintessence model [17 18] which is a scalar field modeldescribed by a field 120601 and a 119881(120601) potential It representsthe simplest scalar field scenario without having theoreticalproblems such as the appearance of ghosts and LaplacianinstabilitiesThe energy density 120588119876 and the pressure 119875119876 of thequintessence scalar field model are given respectively by

120588119876 =1

2

1206012+ 119881 (120601)

119875119876 =1

2

1206012minus 119881 (120601)

(1)

Canonical scalar field is not the unique solution We cangeneralize it as follows [58]

120588119876 =120596

2120601119896 1206012+ 119881 (120601)

119875119876 =120596

2120601119896 1206012minus 119881 (120601)

(2)

In the case of 119896 = 0 (2) transform to the canonical scalarfield model with rescaling of the field Below we would liketo consider an interaction term 119876 between dark energy anddark matter described by

119876 = 3119867119887120588119876 + 120574 (120588119887 minus 120588119876)

120601

120601 (3)

where 119887 and 120574 are positive constants with a typical value of001ndash003 Nature of the interaction between dark energy anddark matter is not clear If we believe that it has a quantumorigin then an absence of the final theory of quantum gravityleaves this question as an open problem However if webelieve that the link existing between components is due tothe same origin of the dark energy and dark matter thenthis approach does not give any exact solution because thenature of the two components is not formulated and it isanother open problem Therefore only phenomenologicalassumption is an appropriate approach For the dark mattermodel we will consider once a viscous modified Chaplygingas with the following equation of state (EoS)

119875 = 119860120588 minus119861

120588120572minus 3120585119867 (4)

where 119860 119861 and 120572 are constants (with 0 le 120572 le 1 in generalrelativity)

For the second model we will use viscous polytropic fluidwith EoS given by

119875 = 1198701205881+1119899

minus 3120585119867 (5)

where 119870 is the polytropic index and 120585 represents the viscouscoefficient

In the third model we would like to consider interactionbetween quintessence dark energy and a viscous Van derWaals gas of the general form

119875 =119860120588

119861 minus 120588minus 1198611205882minus 3120585119867 (6)

Advances in High Energy Physics 3

where 119860 and 119861 are constants Furthermore we will considertwo regimes (1)Λ is a numerical constant (2)Λ is a functionof the cosmic time 119905 therefore it is a varying quantity Inparticular we choose the following form for the time-varyingΛ

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (7)

where 120575 is a positive constant and 119881(120601) is the potential of thefield which we consider as follows

119881 (120601) = 1198810119890(minus1206010120601) (8)

where 1206010 is a constant parameterWe investigate the behavior of the cosmological param-

eters like the Hubble parameter 119867 deceleration parameter119902 and EoS parameters of the quintessence dark energy andan effective two-component fluid Moreover we performstability analysis via the squared speed of the sound1198622119878 whichis defined as follows

1198622

119904 =120597119875

120597120588 (9)

where 119875 and 120588 are respectively the pressure and the energydensity of the effective fluid given by

119875 = 119875119876 + 119875119894

120588 = 120588119876 + 120588119894

(10)

where 119894 refers to one of the viscous fluids described aboveWe will finish our paper with the results obtained fromobservational constraints Consideration of the statefinderanalysis different forms of interaction terms and differentΛ(119905) and varying viscosity is possible and an interestingresearch We hope to approach this question in future withforthcoming articles

3 The Field Equations

The field equations governing our model are given by

119877120583] minus1

2119892120583]119877 minus Λ119892120583] +

3

2120601120583120601] minus

3

4119892120583]120601120572120601120572 = 119879120583] (11)

Considering the content of the universe to be a perfect fluidwe have

119879120583] = (120588 + 119875) 119906120583119906] minus 119875119892120583] (12)

where 119906120583 = (1 0 0 0) is the 4-velocity of the comovingobserver satisfying the relation 119906120583119906

120583= 1 Let 120601120583 be a time-

like vector field of displacement then

120601120583 = (2

radic3120573 0 0 0) (13)

where120573 = 120573(119905) is a function of time alone and the factor 2radic3is inserted in order to simplify the writing of all the followingequations By using FRWmetric for a flat universe given by

1198891199042= minus119889119905

2+ 119886(119905)

2(1198891199032+ 1199032119889Ω2) (14)

field equations can be reduced to the following Friedmannequations

31198672minus 1205732= 120588 + Λ

2 + 31198672+ 1205732= minus119875 + Λ

(15)

where 119867 = 119886119886 is the Hubble parameter the dot standsfor differentiation with respect to the cosmic time 119905 119889Ω2 =1198891205792+ sin21205791198891206012 and 119886(119905) represents the scale factor The 120579

and 120601 parameters are the usual azimuthal and polar anglesof spherical coordinates with 0 le 120579 le 120587 and 0 le 120601 lt 2120587 Thecoordinates (119905 119903 120579 120601) are called comoving coordinates

The continuity equation is given by

120588 + Λ + 2120573 120573 + 3119867 (120588 + 119875 + 21205732) = 0 (16)

The continuity equation given in (16) can be also rewritten inthe compact form

120588 + 3119867 (120588 + 119875) = 0 (17)

Comparing (16) and (17)we obtain the following link betweenΛ and 120573

Λ + 2120573 120573 + 61198671205732= 0 (18)

In order to introduce an interaction between dark energy anddark matter we should mathematically split (17) into the twofollowing equations

120588119894 + 3119867 (120588119894 + 119875119894) = 119876 (19)

120588119876 + 3119867 (120588119876 + 119875119876) = minus119876 (20)

The cosmological parameters of our interest are the EoSparameter of each fluid component 120596119894 = 119875119894120588119894 the EoSparameter of composed fluid

120596tot =119875119876 + 119875119894

120588119876 + 120588119894

(21)

and the deceleration parameter 119902 which can be written asfollows

119902 =1

2(1 + 3

119875

120588) (22)

where index 119894 refers to the first components which is viscousmodified Chaplygin gas or viscous polytropic fluid and index119876 refers to the quintessence scalar field A differential equa-tion describing dynamics of the DE after some mathematicscan be rewritten as

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601 (23)

Taking into account the form of the varyingΛ(119905) from (7) forthe Hubble parameter119867 we will have

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (24)

Hereafter we will consider 1206010 = 1 for mathematical simplic-ity

4 Advances in High Energy Physics

1 2 3 4 5

04

05

06

07

08

09

10

11

t

H

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(a)

minus05

00

05

10

15

1 2 3 4 5

t

q

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 1 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 1

4 Case of Constant Λ

We found it reasonable to start our analysis from the modelswith constant Λ Without loss of generality we would liketo describe equations allowing us to find dynamics of themodels According to the assumption with constant Λ (16)will be modified as follows

120588 + 2120573 120573 + 3119867 (120588 + 119875 + 21205732) = 0 (25)

and using the expression 120588 + 3119867(120588 + 119875) = 0 we will obtainthat

120573 + 3119867120573 = 0 (26)

The last equation can be integrated very easily and the resultis the following

120573 = 1205730119886minus3 (27)

where 119886(119905) is the scale factor and 1205730 is the integrationconstant In our future calculations we will use 1205730 = 1 asinitial condition For the Hubble parameter119867 we will obtain

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (28)

Concerning the form of the field equations we need only toassume the form of 119876 and we will obtain the cosmologicalsolutions Concerning the mathematical hardness of theproblem we will analyze models numerically and investi-gate graphical behavior of various important cosmologicalparameters with respect to the cosmic time In the followingsubsections we consider three models with the particularform of 119876 with given forms of the EoS for the viscous darkmatter fluids considered in the Introduction

41Model 1 Thefirst toymodel describes the dynamics of theuniverse within an effective fluid in case of the cosmologicalconstant The dynamics of the energy density of the viscousmodified Chaplygin gas which will model dark matter inour universe and the differential equation describing thedynamics of it can be found to be

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(29)

From (23) for the dynamics of the dark energy we have

120588119876 + 3119867(1 + 119887 + 120596119876 minus120574

3119867

120601

120601)120588119876 = minus120574

120601

120601120588Ch (30)

Thebest fit for the theoreticalmodel of our considerationwiththe observational data we obtained for1198670 = 14 ΩCh = 03Λ = 07 119860 = 25 119861 = 09 120574 = 002 119887 = 001 and 120585 = 01

In Figures 1ndash3 we have chosen to plot some of thequantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01120572 = 05 and119860 = 25 andwehave chosen five differentvalues of Λ that is 0 02 03 05 and 07

Figure 1 shows that theHubble parameter119867 is decreasingwith time to a constant at the late universe as expected andits value increased by Λ On the other hand the value of thedeceleration parameter decreased with Λ It is illustrated bythe right plot of Figure 1 For Λ = 0 we can see 119902 rarr minus04while for Λ = 07 we can see 119902 rarr minus07 Also acceleration todeceleration phase transition is seen in this model At the latetime we have 119902 sim minus05 being in agreement with observationaldata As we know recent observations of type SNIa indicatethat universe is accelerating with the deceleration parameterlying somewhere in the range minus1 lt 119902 le 0

Advances in High Energy Physics 5

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596tot

120574 = 002 b = 001 and 120585 = 01

(a)

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596Q

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 2 Behavior of EoS parameters 120596tot and 120596119876 against 119905 for the constant Λ Model 1

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

minus1

0

1

2

3

C2 s

120574 = 002 b = 001 and 120585 = 01

Figure 3 Squared sound speed against 119905 for the constant Λ Model1

Moreover Figure 2 shows that the EoS parameter tendsto minus1 at the late time with 120596tot ge minus1 corresponding to aquintessence-like universe

Unfortunately analysis of squared sound speed (seeFigure 3) shows that this model is not stable at the late timeand will be considered only for the early universe

42 Model 2 In the second model after some mathematicalcalculations we obtain the following differential equation tostudy dynamics of the model

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(31)

where 120596119875 is given as

120596119875 = 1198701205881119899

119875 minus3120585119867

120588119875

(32)

The best fitted values of parameters are1198670 = 12 Ω119875 = 025Λ = 15 119870 = 25 119899 = 10 120574 = 002 119887 = 001 and 120585 = 02

In Figures 4 and 5 we have chosen to plot some ofthe quantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01 119899 = 1 and 119870 = 25 and we have chosen five differentvalues of Λ that is 0 05 10 12 and 15

Numerical results of the Hubble expansion parameterand the deceleration parameter 119902 show good behavior (seeFigure 4) but the stability analysis illustrated in Figure 5shows that this model also has instability in the late timeand is only useful for the early universe However the effectof constant Λ in this parameter is similar to the first modeland the EoS parameter tends to minus1 as before In the case ofΛ = 0 we can see that 119902 sim minus02 which is not coincident withobservational data It tells that presence ofΛmaybenecessaryto obtain agreement with observations

Figure 5 shows that instability of model may solve for thelarge value of the cosmological constant Therefore model 2is stable at the late time for the large value of the Λ

43 Model 3 In the third model we have the followingexpression for the pressure 119875

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (33)

The constant Λ assumption will lead us to the followingexpression for the Hubble parameter119867

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (34)

6 Advances in High Energy Physics

02

04

06

08

10

12

1 2 3 4 5

t

H

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(a)

04

02

00

minus02

minus04

minus06

minus08

q

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(b)

Figure 4 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 2

minus1

0

1

2

3

C2 s

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

Figure 5 Squared sound speed against 119905 for the constant Λ Model2

The differential equations describing the dynamics of theenergy densities of both components are given by the follow-ing equations

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119882

120601

120601

(35)

where 120596119876 is the EoS parameter of dark energy The bestfit for the theoretical model of our consideration with the

5 10 15 20

t

10

05

00

minus05

minus10

C2 s

Λ = 0 A = 15 B = 12

Λ = 03 A = 15 B = 12Λ = 05 A = 15 B = 12

Λ = 07 A = 15 B = 12

Λ = 12 A = 15 B = 12

120574 = 002 b = 0 and 120585 = 04

Figure 6 Squared sound speed against 119905 for the constant Λ Model3

observational data is obtained for 1198670 = 13 Ω119875 = 0217Λ = 12 119860 = 15 119861 = 12 120574 = 002 119887 = 000 and 120585 = 04

The behaviors of the cosmological parameters are similarto the previous models and we can see late time instability ofthis model in Figure 6 This suggests considering varying Λto obtain more appropriate models

5 The Case of Varying Λ

In this section we will consider three interacting fluidmodels and will investigate cosmological parameters like theHubble parameter 119867 deceleration parameter 119902 and EoSparameters of the total fluid and dark energy 120596119876 Based onnumerical solutions we will discuss graphical behaviors of

Advances in High Energy Physics 7

04

05

06

07

08

09

10

11

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

H

120574 = 002 b = 001 and 120585 = 05

(a)

04

06

08

10

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

H

120575 = 1120574 = 002 b = 001 and

(b)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

120574 = 002 b = 001 and 120585 = 05

(c)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

120575 = 1120574 = 002 b = 001 and

(d)

Figure 7 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 4

the cosmological parameters For the varying Λ we take aphenomenological form which was considered by us recently[59] The formula of Λ is given as the following expression

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (36)

which is a function of the Hubble parameter potential of thescalar field and time derivative of the scalar field For thepotential we take a simple form 119881(120601) = 119890

[minus120601] therefore theform of Λ can be written also in the following way as only afunction of the filed 120601

Λ (119905) = 1198672120601minus2+ 120575119890minus120601 (37)

Therefore the dynamics of 120573 can be obtained from thefollowing differential equation

2120573 120573 + 61198671205732+ 2119867120601

minus2minus 21198672120601minus3 120601 minus 120575119890

[minus120601] 120601 = 0 (38)

In forthcoming subsections within three different forms of119876wewill investigate the dynamics of the universeThe question

of the dynamics for the energy densities of the dark energyand dark matter is already discussed in a previous sectionthereforewewill not consider themhere andwewill startwiththe comments on the graphical behaviors of the cosmologicalparameters of themodelsWe will start with themodel where

119876 = 3119867119887120588119876 + 120574 (120588119894 minus 120588119876)

120601

120601 (39)

51 Model 4 Interacting viscous modified Chaplygin gaswith the quintessence dark energy in the case of varying Λgives the following differential equation

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(40)

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

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PhotonicsJournal of

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ThermodynamicsJournal of

Advances in High Energy Physics 3

where 119860 and 119861 are constants Furthermore we will considertwo regimes (1)Λ is a numerical constant (2)Λ is a functionof the cosmic time 119905 therefore it is a varying quantity Inparticular we choose the following form for the time-varyingΛ

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (7)

where 120575 is a positive constant and 119881(120601) is the potential of thefield which we consider as follows

119881 (120601) = 1198810119890(minus1206010120601) (8)

where 1206010 is a constant parameterWe investigate the behavior of the cosmological param-

eters like the Hubble parameter 119867 deceleration parameter119902 and EoS parameters of the quintessence dark energy andan effective two-component fluid Moreover we performstability analysis via the squared speed of the sound1198622119878 whichis defined as follows

1198622

119904 =120597119875

120597120588 (9)

where 119875 and 120588 are respectively the pressure and the energydensity of the effective fluid given by

119875 = 119875119876 + 119875119894

120588 = 120588119876 + 120588119894

(10)

where 119894 refers to one of the viscous fluids described aboveWe will finish our paper with the results obtained fromobservational constraints Consideration of the statefinderanalysis different forms of interaction terms and differentΛ(119905) and varying viscosity is possible and an interestingresearch We hope to approach this question in future withforthcoming articles

3 The Field Equations

The field equations governing our model are given by

119877120583] minus1

2119892120583]119877 minus Λ119892120583] +

3

2120601120583120601] minus

3

4119892120583]120601120572120601120572 = 119879120583] (11)

Considering the content of the universe to be a perfect fluidwe have

119879120583] = (120588 + 119875) 119906120583119906] minus 119875119892120583] (12)

where 119906120583 = (1 0 0 0) is the 4-velocity of the comovingobserver satisfying the relation 119906120583119906

120583= 1 Let 120601120583 be a time-

like vector field of displacement then

120601120583 = (2

radic3120573 0 0 0) (13)

where120573 = 120573(119905) is a function of time alone and the factor 2radic3is inserted in order to simplify the writing of all the followingequations By using FRWmetric for a flat universe given by

1198891199042= minus119889119905

2+ 119886(119905)

2(1198891199032+ 1199032119889Ω2) (14)

field equations can be reduced to the following Friedmannequations

31198672minus 1205732= 120588 + Λ

2 + 31198672+ 1205732= minus119875 + Λ

(15)

where 119867 = 119886119886 is the Hubble parameter the dot standsfor differentiation with respect to the cosmic time 119905 119889Ω2 =1198891205792+ sin21205791198891206012 and 119886(119905) represents the scale factor The 120579

and 120601 parameters are the usual azimuthal and polar anglesof spherical coordinates with 0 le 120579 le 120587 and 0 le 120601 lt 2120587 Thecoordinates (119905 119903 120579 120601) are called comoving coordinates

The continuity equation is given by

120588 + Λ + 2120573 120573 + 3119867 (120588 + 119875 + 21205732) = 0 (16)

The continuity equation given in (16) can be also rewritten inthe compact form

120588 + 3119867 (120588 + 119875) = 0 (17)

Comparing (16) and (17)we obtain the following link betweenΛ and 120573

Λ + 2120573 120573 + 61198671205732= 0 (18)

In order to introduce an interaction between dark energy anddark matter we should mathematically split (17) into the twofollowing equations

120588119894 + 3119867 (120588119894 + 119875119894) = 119876 (19)

120588119876 + 3119867 (120588119876 + 119875119876) = minus119876 (20)

The cosmological parameters of our interest are the EoSparameter of each fluid component 120596119894 = 119875119894120588119894 the EoSparameter of composed fluid

120596tot =119875119876 + 119875119894

120588119876 + 120588119894

(21)

and the deceleration parameter 119902 which can be written asfollows

119902 =1

2(1 + 3

119875

120588) (22)

where index 119894 refers to the first components which is viscousmodified Chaplygin gas or viscous polytropic fluid and index119876 refers to the quintessence scalar field A differential equa-tion describing dynamics of the DE after some mathematicscan be rewritten as

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601 (23)

Taking into account the form of the varyingΛ(119905) from (7) forthe Hubble parameter119867 we will have

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (24)

Hereafter we will consider 1206010 = 1 for mathematical simplic-ity

4 Advances in High Energy Physics

1 2 3 4 5

04

05

06

07

08

09

10

11

t

H

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(a)

minus05

00

05

10

15

1 2 3 4 5

t

q

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 1 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 1

4 Case of Constant Λ

We found it reasonable to start our analysis from the modelswith constant Λ Without loss of generality we would liketo describe equations allowing us to find dynamics of themodels According to the assumption with constant Λ (16)will be modified as follows

120588 + 2120573 120573 + 3119867 (120588 + 119875 + 21205732) = 0 (25)

and using the expression 120588 + 3119867(120588 + 119875) = 0 we will obtainthat

120573 + 3119867120573 = 0 (26)

The last equation can be integrated very easily and the resultis the following

120573 = 1205730119886minus3 (27)

where 119886(119905) is the scale factor and 1205730 is the integrationconstant In our future calculations we will use 1205730 = 1 asinitial condition For the Hubble parameter119867 we will obtain

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (28)

Concerning the form of the field equations we need only toassume the form of 119876 and we will obtain the cosmologicalsolutions Concerning the mathematical hardness of theproblem we will analyze models numerically and investi-gate graphical behavior of various important cosmologicalparameters with respect to the cosmic time In the followingsubsections we consider three models with the particularform of 119876 with given forms of the EoS for the viscous darkmatter fluids considered in the Introduction

41Model 1 Thefirst toymodel describes the dynamics of theuniverse within an effective fluid in case of the cosmologicalconstant The dynamics of the energy density of the viscousmodified Chaplygin gas which will model dark matter inour universe and the differential equation describing thedynamics of it can be found to be

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(29)

From (23) for the dynamics of the dark energy we have

120588119876 + 3119867(1 + 119887 + 120596119876 minus120574

3119867

120601

120601)120588119876 = minus120574

120601

120601120588Ch (30)

Thebest fit for the theoreticalmodel of our considerationwiththe observational data we obtained for1198670 = 14 ΩCh = 03Λ = 07 119860 = 25 119861 = 09 120574 = 002 119887 = 001 and 120585 = 01

In Figures 1ndash3 we have chosen to plot some of thequantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01120572 = 05 and119860 = 25 andwehave chosen five differentvalues of Λ that is 0 02 03 05 and 07

Figure 1 shows that theHubble parameter119867 is decreasingwith time to a constant at the late universe as expected andits value increased by Λ On the other hand the value of thedeceleration parameter decreased with Λ It is illustrated bythe right plot of Figure 1 For Λ = 0 we can see 119902 rarr minus04while for Λ = 07 we can see 119902 rarr minus07 Also acceleration todeceleration phase transition is seen in this model At the latetime we have 119902 sim minus05 being in agreement with observationaldata As we know recent observations of type SNIa indicatethat universe is accelerating with the deceleration parameterlying somewhere in the range minus1 lt 119902 le 0

Advances in High Energy Physics 5

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596tot

120574 = 002 b = 001 and 120585 = 01

(a)

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596Q

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 2 Behavior of EoS parameters 120596tot and 120596119876 against 119905 for the constant Λ Model 1

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

minus1

0

1

2

3

C2 s

120574 = 002 b = 001 and 120585 = 01

Figure 3 Squared sound speed against 119905 for the constant Λ Model1

Moreover Figure 2 shows that the EoS parameter tendsto minus1 at the late time with 120596tot ge minus1 corresponding to aquintessence-like universe

Unfortunately analysis of squared sound speed (seeFigure 3) shows that this model is not stable at the late timeand will be considered only for the early universe

42 Model 2 In the second model after some mathematicalcalculations we obtain the following differential equation tostudy dynamics of the model

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(31)

where 120596119875 is given as

120596119875 = 1198701205881119899

119875 minus3120585119867

120588119875

(32)

The best fitted values of parameters are1198670 = 12 Ω119875 = 025Λ = 15 119870 = 25 119899 = 10 120574 = 002 119887 = 001 and 120585 = 02

In Figures 4 and 5 we have chosen to plot some ofthe quantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01 119899 = 1 and 119870 = 25 and we have chosen five differentvalues of Λ that is 0 05 10 12 and 15

Numerical results of the Hubble expansion parameterand the deceleration parameter 119902 show good behavior (seeFigure 4) but the stability analysis illustrated in Figure 5shows that this model also has instability in the late timeand is only useful for the early universe However the effectof constant Λ in this parameter is similar to the first modeland the EoS parameter tends to minus1 as before In the case ofΛ = 0 we can see that 119902 sim minus02 which is not coincident withobservational data It tells that presence ofΛmaybenecessaryto obtain agreement with observations

Figure 5 shows that instability of model may solve for thelarge value of the cosmological constant Therefore model 2is stable at the late time for the large value of the Λ

43 Model 3 In the third model we have the followingexpression for the pressure 119875

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (33)

The constant Λ assumption will lead us to the followingexpression for the Hubble parameter119867

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (34)

6 Advances in High Energy Physics

02

04

06

08

10

12

1 2 3 4 5

t

H

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(a)

04

02

00

minus02

minus04

minus06

minus08

q

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(b)

Figure 4 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 2

minus1

0

1

2

3

C2 s

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

Figure 5 Squared sound speed against 119905 for the constant Λ Model2

The differential equations describing the dynamics of theenergy densities of both components are given by the follow-ing equations

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119882

120601

120601

(35)

where 120596119876 is the EoS parameter of dark energy The bestfit for the theoretical model of our consideration with the

5 10 15 20

t

10

05

00

minus05

minus10

C2 s

Λ = 0 A = 15 B = 12

Λ = 03 A = 15 B = 12Λ = 05 A = 15 B = 12

Λ = 07 A = 15 B = 12

Λ = 12 A = 15 B = 12

120574 = 002 b = 0 and 120585 = 04

Figure 6 Squared sound speed against 119905 for the constant Λ Model3

observational data is obtained for 1198670 = 13 Ω119875 = 0217Λ = 12 119860 = 15 119861 = 12 120574 = 002 119887 = 000 and 120585 = 04

The behaviors of the cosmological parameters are similarto the previous models and we can see late time instability ofthis model in Figure 6 This suggests considering varying Λto obtain more appropriate models

5 The Case of Varying Λ

In this section we will consider three interacting fluidmodels and will investigate cosmological parameters like theHubble parameter 119867 deceleration parameter 119902 and EoSparameters of the total fluid and dark energy 120596119876 Based onnumerical solutions we will discuss graphical behaviors of

Advances in High Energy Physics 7

04

05

06

07

08

09

10

11

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

H

120574 = 002 b = 001 and 120585 = 05

(a)

04

06

08

10

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

H

120575 = 1120574 = 002 b = 001 and

(b)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

120574 = 002 b = 001 and 120585 = 05

(c)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

120575 = 1120574 = 002 b = 001 and

(d)

Figure 7 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 4

the cosmological parameters For the varying Λ we take aphenomenological form which was considered by us recently[59] The formula of Λ is given as the following expression

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (36)

which is a function of the Hubble parameter potential of thescalar field and time derivative of the scalar field For thepotential we take a simple form 119881(120601) = 119890

[minus120601] therefore theform of Λ can be written also in the following way as only afunction of the filed 120601

Λ (119905) = 1198672120601minus2+ 120575119890minus120601 (37)

Therefore the dynamics of 120573 can be obtained from thefollowing differential equation

2120573 120573 + 61198671205732+ 2119867120601

minus2minus 21198672120601minus3 120601 minus 120575119890

[minus120601] 120601 = 0 (38)

In forthcoming subsections within three different forms of119876wewill investigate the dynamics of the universeThe question

of the dynamics for the energy densities of the dark energyand dark matter is already discussed in a previous sectionthereforewewill not consider themhere andwewill startwiththe comments on the graphical behaviors of the cosmologicalparameters of themodelsWe will start with themodel where

119876 = 3119867119887120588119876 + 120574 (120588119894 minus 120588119876)

120601

120601 (39)

51 Model 4 Interacting viscous modified Chaplygin gaswith the quintessence dark energy in the case of varying Λgives the following differential equation

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(40)

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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 Computational  Methods in Physics

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PhotonicsJournal of

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ThermodynamicsJournal of

4 Advances in High Energy Physics

1 2 3 4 5

04

05

06

07

08

09

10

11

t

H

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(a)

minus05

00

05

10

15

1 2 3 4 5

t

q

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 1 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 1

4 Case of Constant Λ

We found it reasonable to start our analysis from the modelswith constant Λ Without loss of generality we would liketo describe equations allowing us to find dynamics of themodels According to the assumption with constant Λ (16)will be modified as follows

120588 + 2120573 120573 + 3119867 (120588 + 119875 + 21205732) = 0 (25)

and using the expression 120588 + 3119867(120588 + 119875) = 0 we will obtainthat

120573 + 3119867120573 = 0 (26)

The last equation can be integrated very easily and the resultis the following

120573 = 1205730119886minus3 (27)

where 119886(119905) is the scale factor and 1205730 is the integrationconstant In our future calculations we will use 1205730 = 1 asinitial condition For the Hubble parameter119867 we will obtain

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (28)

Concerning the form of the field equations we need only toassume the form of 119876 and we will obtain the cosmologicalsolutions Concerning the mathematical hardness of theproblem we will analyze models numerically and investi-gate graphical behavior of various important cosmologicalparameters with respect to the cosmic time In the followingsubsections we consider three models with the particularform of 119876 with given forms of the EoS for the viscous darkmatter fluids considered in the Introduction

41Model 1 Thefirst toymodel describes the dynamics of theuniverse within an effective fluid in case of the cosmologicalconstant The dynamics of the energy density of the viscousmodified Chaplygin gas which will model dark matter inour universe and the differential equation describing thedynamics of it can be found to be

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(29)

From (23) for the dynamics of the dark energy we have

120588119876 + 3119867(1 + 119887 + 120596119876 minus120574

3119867

120601

120601)120588119876 = minus120574

120601

120601120588Ch (30)

Thebest fit for the theoreticalmodel of our considerationwiththe observational data we obtained for1198670 = 14 ΩCh = 03Λ = 07 119860 = 25 119861 = 09 120574 = 002 119887 = 001 and 120585 = 01

In Figures 1ndash3 we have chosen to plot some of thequantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01120572 = 05 and119860 = 25 andwehave chosen five differentvalues of Λ that is 0 02 03 05 and 07

Figure 1 shows that theHubble parameter119867 is decreasingwith time to a constant at the late universe as expected andits value increased by Λ On the other hand the value of thedeceleration parameter decreased with Λ It is illustrated bythe right plot of Figure 1 For Λ = 0 we can see 119902 rarr minus04while for Λ = 07 we can see 119902 rarr minus07 Also acceleration todeceleration phase transition is seen in this model At the latetime we have 119902 sim minus05 being in agreement with observationaldata As we know recent observations of type SNIa indicatethat universe is accelerating with the deceleration parameterlying somewhere in the range minus1 lt 119902 le 0

Advances in High Energy Physics 5

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596tot

120574 = 002 b = 001 and 120585 = 01

(a)

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596Q

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 2 Behavior of EoS parameters 120596tot and 120596119876 against 119905 for the constant Λ Model 1

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

minus1

0

1

2

3

C2 s

120574 = 002 b = 001 and 120585 = 01

Figure 3 Squared sound speed against 119905 for the constant Λ Model1

Moreover Figure 2 shows that the EoS parameter tendsto minus1 at the late time with 120596tot ge minus1 corresponding to aquintessence-like universe

Unfortunately analysis of squared sound speed (seeFigure 3) shows that this model is not stable at the late timeand will be considered only for the early universe

42 Model 2 In the second model after some mathematicalcalculations we obtain the following differential equation tostudy dynamics of the model

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(31)

where 120596119875 is given as

120596119875 = 1198701205881119899

119875 minus3120585119867

120588119875

(32)

The best fitted values of parameters are1198670 = 12 Ω119875 = 025Λ = 15 119870 = 25 119899 = 10 120574 = 002 119887 = 001 and 120585 = 02

In Figures 4 and 5 we have chosen to plot some ofthe quantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01 119899 = 1 and 119870 = 25 and we have chosen five differentvalues of Λ that is 0 05 10 12 and 15

Numerical results of the Hubble expansion parameterand the deceleration parameter 119902 show good behavior (seeFigure 4) but the stability analysis illustrated in Figure 5shows that this model also has instability in the late timeand is only useful for the early universe However the effectof constant Λ in this parameter is similar to the first modeland the EoS parameter tends to minus1 as before In the case ofΛ = 0 we can see that 119902 sim minus02 which is not coincident withobservational data It tells that presence ofΛmaybenecessaryto obtain agreement with observations

Figure 5 shows that instability of model may solve for thelarge value of the cosmological constant Therefore model 2is stable at the late time for the large value of the Λ

43 Model 3 In the third model we have the followingexpression for the pressure 119875

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (33)

The constant Λ assumption will lead us to the followingexpression for the Hubble parameter119867

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (34)

6 Advances in High Energy Physics

02

04

06

08

10

12

1 2 3 4 5

t

H

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(a)

04

02

00

minus02

minus04

minus06

minus08

q

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(b)

Figure 4 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 2

minus1

0

1

2

3

C2 s

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

Figure 5 Squared sound speed against 119905 for the constant Λ Model2

The differential equations describing the dynamics of theenergy densities of both components are given by the follow-ing equations

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119882

120601

120601

(35)

where 120596119876 is the EoS parameter of dark energy The bestfit for the theoretical model of our consideration with the

5 10 15 20

t

10

05

00

minus05

minus10

C2 s

Λ = 0 A = 15 B = 12

Λ = 03 A = 15 B = 12Λ = 05 A = 15 B = 12

Λ = 07 A = 15 B = 12

Λ = 12 A = 15 B = 12

120574 = 002 b = 0 and 120585 = 04

Figure 6 Squared sound speed against 119905 for the constant Λ Model3

observational data is obtained for 1198670 = 13 Ω119875 = 0217Λ = 12 119860 = 15 119861 = 12 120574 = 002 119887 = 000 and 120585 = 04

The behaviors of the cosmological parameters are similarto the previous models and we can see late time instability ofthis model in Figure 6 This suggests considering varying Λto obtain more appropriate models

5 The Case of Varying Λ

In this section we will consider three interacting fluidmodels and will investigate cosmological parameters like theHubble parameter 119867 deceleration parameter 119902 and EoSparameters of the total fluid and dark energy 120596119876 Based onnumerical solutions we will discuss graphical behaviors of

Advances in High Energy Physics 7

04

05

06

07

08

09

10

11

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

H

120574 = 002 b = 001 and 120585 = 05

(a)

04

06

08

10

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

H

120575 = 1120574 = 002 b = 001 and

(b)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

120574 = 002 b = 001 and 120585 = 05

(c)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

120575 = 1120574 = 002 b = 001 and

(d)

Figure 7 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 4

the cosmological parameters For the varying Λ we take aphenomenological form which was considered by us recently[59] The formula of Λ is given as the following expression

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (36)

which is a function of the Hubble parameter potential of thescalar field and time derivative of the scalar field For thepotential we take a simple form 119881(120601) = 119890

[minus120601] therefore theform of Λ can be written also in the following way as only afunction of the filed 120601

Λ (119905) = 1198672120601minus2+ 120575119890minus120601 (37)

Therefore the dynamics of 120573 can be obtained from thefollowing differential equation

2120573 120573 + 61198671205732+ 2119867120601

minus2minus 21198672120601minus3 120601 minus 120575119890

[minus120601] 120601 = 0 (38)

In forthcoming subsections within three different forms of119876wewill investigate the dynamics of the universeThe question

of the dynamics for the energy densities of the dark energyand dark matter is already discussed in a previous sectionthereforewewill not consider themhere andwewill startwiththe comments on the graphical behaviors of the cosmologicalparameters of themodelsWe will start with themodel where

119876 = 3119867119887120588119876 + 120574 (120588119894 minus 120588119876)

120601

120601 (39)

51 Model 4 Interacting viscous modified Chaplygin gaswith the quintessence dark energy in the case of varying Λgives the following differential equation

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(40)

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

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AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

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Soft MatterJournal of

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PhotonicsJournal of

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ThermodynamicsJournal of

Advances in High Energy Physics 5

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596tot

120574 = 002 b = 001 and 120585 = 01

(a)

minus10

minus08

minus06

minus04

minus02

00

02

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

120596Q

120574 = 002 b = 001 and 120585 = 01

(b)

Figure 2 Behavior of EoS parameters 120596tot and 120596119876 against 119905 for the constant Λ Model 1

1 2 3 4 5

t

Λ = 0 120572 = 05 A = 25

Λ = 02 120572 = 05 A = 25Λ = 03 120572 = 05 A = 25

Λ = 05 120572 = 05 A = 25

Λ = 07 120572 = 05 A = 25

minus1

0

1

2

3

C2 s

120574 = 002 b = 001 and 120585 = 01

Figure 3 Squared sound speed against 119905 for the constant Λ Model1

Moreover Figure 2 shows that the EoS parameter tendsto minus1 at the late time with 120596tot ge minus1 corresponding to aquintessence-like universe

Unfortunately analysis of squared sound speed (seeFigure 3) shows that this model is not stable at the late timeand will be considered only for the early universe

42 Model 2 In the second model after some mathematicalcalculations we obtain the following differential equation tostudy dynamics of the model

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(31)

where 120596119875 is given as

120596119875 = 1198701205881119899

119875 minus3120585119867

120588119875

(32)

The best fitted values of parameters are1198670 = 12 Ω119875 = 025Λ = 15 119870 = 25 119899 = 10 120574 = 002 119887 = 001 and 120585 = 02

In Figures 4 and 5 we have chosen to plot some ofthe quantities derived for different values of the parametersinvolved In particular we have chosen 120574 = 002 119887 = 001120585 = 01 119899 = 1 and 119870 = 25 and we have chosen five differentvalues of Λ that is 0 05 10 12 and 15

Numerical results of the Hubble expansion parameterand the deceleration parameter 119902 show good behavior (seeFigure 4) but the stability analysis illustrated in Figure 5shows that this model also has instability in the late timeand is only useful for the early universe However the effectof constant Λ in this parameter is similar to the first modeland the EoS parameter tends to minus1 as before In the case ofΛ = 0 we can see that 119902 sim minus02 which is not coincident withobservational data It tells that presence ofΛmaybenecessaryto obtain agreement with observations

Figure 5 shows that instability of model may solve for thelarge value of the cosmological constant Therefore model 2is stable at the late time for the large value of the Λ

43 Model 3 In the third model we have the followingexpression for the pressure 119875

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (33)

The constant Λ assumption will lead us to the followingexpression for the Hubble parameter119867

119867 =1

radic3

radic120588 + Λ + 1205730119886minus6 (34)

6 Advances in High Energy Physics

02

04

06

08

10

12

1 2 3 4 5

t

H

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(a)

04

02

00

minus02

minus04

minus06

minus08

q

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(b)

Figure 4 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 2

minus1

0

1

2

3

C2 s

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

Figure 5 Squared sound speed against 119905 for the constant Λ Model2

The differential equations describing the dynamics of theenergy densities of both components are given by the follow-ing equations

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119882

120601

120601

(35)

where 120596119876 is the EoS parameter of dark energy The bestfit for the theoretical model of our consideration with the

5 10 15 20

t

10

05

00

minus05

minus10

C2 s

Λ = 0 A = 15 B = 12

Λ = 03 A = 15 B = 12Λ = 05 A = 15 B = 12

Λ = 07 A = 15 B = 12

Λ = 12 A = 15 B = 12

120574 = 002 b = 0 and 120585 = 04

Figure 6 Squared sound speed against 119905 for the constant Λ Model3

observational data is obtained for 1198670 = 13 Ω119875 = 0217Λ = 12 119860 = 15 119861 = 12 120574 = 002 119887 = 000 and 120585 = 04

The behaviors of the cosmological parameters are similarto the previous models and we can see late time instability ofthis model in Figure 6 This suggests considering varying Λto obtain more appropriate models

5 The Case of Varying Λ

In this section we will consider three interacting fluidmodels and will investigate cosmological parameters like theHubble parameter 119867 deceleration parameter 119902 and EoSparameters of the total fluid and dark energy 120596119876 Based onnumerical solutions we will discuss graphical behaviors of

Advances in High Energy Physics 7

04

05

06

07

08

09

10

11

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

H

120574 = 002 b = 001 and 120585 = 05

(a)

04

06

08

10

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

H

120575 = 1120574 = 002 b = 001 and

(b)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

120574 = 002 b = 001 and 120585 = 05

(c)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

120575 = 1120574 = 002 b = 001 and

(d)

Figure 7 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 4

the cosmological parameters For the varying Λ we take aphenomenological form which was considered by us recently[59] The formula of Λ is given as the following expression

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (36)

which is a function of the Hubble parameter potential of thescalar field and time derivative of the scalar field For thepotential we take a simple form 119881(120601) = 119890

[minus120601] therefore theform of Λ can be written also in the following way as only afunction of the filed 120601

Λ (119905) = 1198672120601minus2+ 120575119890minus120601 (37)

Therefore the dynamics of 120573 can be obtained from thefollowing differential equation

2120573 120573 + 61198671205732+ 2119867120601

minus2minus 21198672120601minus3 120601 minus 120575119890

[minus120601] 120601 = 0 (38)

In forthcoming subsections within three different forms of119876wewill investigate the dynamics of the universeThe question

of the dynamics for the energy densities of the dark energyand dark matter is already discussed in a previous sectionthereforewewill not consider themhere andwewill startwiththe comments on the graphical behaviors of the cosmologicalparameters of themodelsWe will start with themodel where

119876 = 3119867119887120588119876 + 120574 (120588119894 minus 120588119876)

120601

120601 (39)

51 Model 4 Interacting viscous modified Chaplygin gaswith the quintessence dark energy in the case of varying Λgives the following differential equation

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(40)

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

02

04

06

08

10

12

1 2 3 4 5

t

H

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(a)

04

02

00

minus02

minus04

minus06

minus08

q

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

(b)

Figure 4 Behavior of Hubble parameter119867 and 119902 against 119905 for the constant Λ Model 2

minus1

0

1

2

3

C2 s

1 2 3 4 5

t

Λ = 0 n = 1K = 25

Λ = 05 n = 1K = 25Λ = 1 n = 1K = 25

Λ = 12 n = 1K = 25

Λ = 15 n = 1K = 25

120574 = 002 b = 001 and 120585 = 02

Figure 5 Squared sound speed against 119905 for the constant Λ Model2

The differential equations describing the dynamics of theenergy densities of both components are given by the follow-ing equations

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119882

120601

120601

(35)

where 120596119876 is the EoS parameter of dark energy The bestfit for the theoretical model of our consideration with the

5 10 15 20

t

10

05

00

minus05

minus10

C2 s

Λ = 0 A = 15 B = 12

Λ = 03 A = 15 B = 12Λ = 05 A = 15 B = 12

Λ = 07 A = 15 B = 12

Λ = 12 A = 15 B = 12

120574 = 002 b = 0 and 120585 = 04

Figure 6 Squared sound speed against 119905 for the constant Λ Model3

observational data is obtained for 1198670 = 13 Ω119875 = 0217Λ = 12 119860 = 15 119861 = 12 120574 = 002 119887 = 000 and 120585 = 04

The behaviors of the cosmological parameters are similarto the previous models and we can see late time instability ofthis model in Figure 6 This suggests considering varying Λto obtain more appropriate models

5 The Case of Varying Λ

In this section we will consider three interacting fluidmodels and will investigate cosmological parameters like theHubble parameter 119867 deceleration parameter 119902 and EoSparameters of the total fluid and dark energy 120596119876 Based onnumerical solutions we will discuss graphical behaviors of

Advances in High Energy Physics 7

04

05

06

07

08

09

10

11

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

H

120574 = 002 b = 001 and 120585 = 05

(a)

04

06

08

10

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

H

120575 = 1120574 = 002 b = 001 and

(b)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

120574 = 002 b = 001 and 120585 = 05

(c)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

120575 = 1120574 = 002 b = 001 and

(d)

Figure 7 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 4

the cosmological parameters For the varying Λ we take aphenomenological form which was considered by us recently[59] The formula of Λ is given as the following expression

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (36)

which is a function of the Hubble parameter potential of thescalar field and time derivative of the scalar field For thepotential we take a simple form 119881(120601) = 119890

[minus120601] therefore theform of Λ can be written also in the following way as only afunction of the filed 120601

Λ (119905) = 1198672120601minus2+ 120575119890minus120601 (37)

Therefore the dynamics of 120573 can be obtained from thefollowing differential equation

2120573 120573 + 61198671205732+ 2119867120601

minus2minus 21198672120601minus3 120601 minus 120575119890

[minus120601] 120601 = 0 (38)

In forthcoming subsections within three different forms of119876wewill investigate the dynamics of the universeThe question

of the dynamics for the energy densities of the dark energyand dark matter is already discussed in a previous sectionthereforewewill not consider themhere andwewill startwiththe comments on the graphical behaviors of the cosmologicalparameters of themodelsWe will start with themodel where

119876 = 3119867119887120588119876 + 120574 (120588119894 minus 120588119876)

120601

120601 (39)

51 Model 4 Interacting viscous modified Chaplygin gaswith the quintessence dark energy in the case of varying Λgives the following differential equation

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(40)

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

04

05

06

07

08

09

10

11

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

H

120574 = 002 b = 001 and 120585 = 05

(a)

04

06

08

10

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

H

120575 = 1120574 = 002 b = 001 and

(b)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

120574 = 002 b = 001 and 120585 = 05

(c)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

120575 = 1120574 = 002 b = 001 and

(d)

Figure 7 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 4

the cosmological parameters For the varying Λ we take aphenomenological form which was considered by us recently[59] The formula of Λ is given as the following expression

Λ (119905) = 1198672120601minus2+ 120575119881 (120601) (36)

which is a function of the Hubble parameter potential of thescalar field and time derivative of the scalar field For thepotential we take a simple form 119881(120601) = 119890

[minus120601] therefore theform of Λ can be written also in the following way as only afunction of the filed 120601

Λ (119905) = 1198672120601minus2+ 120575119890minus120601 (37)

Therefore the dynamics of 120573 can be obtained from thefollowing differential equation

2120573 120573 + 61198671205732+ 2119867120601

minus2minus 21198672120601minus3 120601 minus 120575119890

[minus120601] 120601 = 0 (38)

In forthcoming subsections within three different forms of119876wewill investigate the dynamics of the universeThe question

of the dynamics for the energy densities of the dark energyand dark matter is already discussed in a previous sectionthereforewewill not consider themhere andwewill startwiththe comments on the graphical behaviors of the cosmologicalparameters of themodelsWe will start with themodel where

119876 = 3119867119887120588119876 + 120574 (120588119894 minus 120588119876)

120601

120601 (39)

51 Model 4 Interacting viscous modified Chaplygin gaswith the quintessence dark energy in the case of varying Λgives the following differential equation

120588Ch + 3119867(1 + 119860 minus119861

120588120572+1Ch

minus120574

3119867

120601

120601)120588Ch

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(40)

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

2 4 6 8 10 12 14t

120575 = 01 120572 = 05 A = 15

120575 = 03 120572 = 05 A = 15120575 = 05 120572 = 05 A = 15

120575 = 1 120572 = 05 A = 15

120575 = 15 120572 = 05 A = 15

minus05

minus06

minus07

minus08

minus09

minus10

120596tot

120574 = 002 b = 001 and 120585 = 05

(a)

120596Q

t

120585 = 0 120572 = 05 A = 15

120585 = 02 120572 = 05 A = 15120585 = 04 120572 = 05 A = 15

120585 = 05 120572 = 05 A = 15

120585 = 07 120572 = 05 A = 15

2 4 6 8 10 12 14

minus05

minus06

minus07

minus08

minus09

minus10

120575 = 1120574 = 002 b = 001 and

(b)

Figure 8 Behavior of the EoS parameters 120596tot and 120596119876 against 119905 for the case corresponding to varying Λ Model 4

00 05 10 15 2008

10

12

14

16

18

H(z)

z

Figure 9 Hubble parameter 119867 against redshift 119911 for varying ΛModel 4

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (41)

In Figure 7 we can see that the Hubble expansionparameter is a decreasing function of time 119905 which yields aconstant at the late time as expected It is clear that 120575 and120585 increase the value of the Hubble expansion parameter butdecrease the value of the deceleration parameter In order toobtain the deceleration parameter being in agreement withobservational data we should choose larger values of 120575 and120585 Also acceleration to deceleration phase transition happensin this model We find an instability at the initial time but themodel is completely stable at the late time

Figure 8 shows that the EoS parameters yield minus1 at the latetime in agreement with observational data Also effects of 120575and 120585 are illustrated in the plots of Figure 8

In Figure 9 we can see behavior of the Hubble expansionparameter with the redshift which is also in agreement withobservational data since it is increasing function

52 Model 5 A polytropic fluid interacting with thequintessence dark energy yields the following differentialequation

120588119875 + 3119867(1 + 120596119875 minus120574

3119867

120601

120601)120588119875

= 3119867(119887 minus120574

3119867

120601

120601)120588119876 + 9120585119867

(42)

with the Hubble parameter obtained as

119867 =1

radic3

radic120588 + 120575119890

[minus1206010120601] + 1205732

1 minus 120601minus23 (43)

In Figure 10 we can see behavior of the Hubble expansionparameter and the deceleration parameter with time andfind that the value of 119870 decreases the value of the Hubbleexpansion parameter while it increases the value of thedeceleration parameter Also acceleration to decelerationphase transition is seen in this model

However squared sound speed plotted in Figure 11 showsthat this model similar to the constant Λ is completelyinstable at the late time and will be useful only for the earlyuniverse

53 Model 6 Finally in the third model for the case ofvarying Λ where a viscous Van der Waals gas with EoS of thegeneral form

119875 =119860120588119882

119861 minus 120588119882

minus 1198611205882

119882 minus 3120585119867 (44)

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

02

04

06

08

10

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

H

120575 = 1120574 = 002 b = 001 and

(a)

04

02

00

minus02

minus04

minus06

q

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

120575 = 1120574 = 002 b = 001 and

(b)

Figure 10 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 5

5 10 15 20

t

120585 = 03 n = 05K = 05

120585 = 03 n = 1K = 1120585 = 03 n = 15K = 15

120585 = 03 n = 2K = 25

120585 = 03 n = 25K = 35

10

05

00

minus05

minus10

C2 s

120575 = 1120574 = 002 b = 001 and

Figure 11 Squared sound speed against 119905 for varying Λ Model 5

interacts with the quintessence dark energy the differentialequations describing the dynamics of the energy densities ofboth components are given

120588119882 + 3119867120588119882(1 +119860

119861 minus 120588119882

minus 119861120588119882 minus120574

3119867

120601

120601)

= 3119867120588119876(119887 minus120574

3119867

120601

120601) + 9120585119867

120588119876 + 3119867120588119876(1 + 119887 + 120596119876 minus120574

3119867

120601

120601) = minus120574120588119894

120601

120601

(45)

where 120596119876 is the EoS parameter of the dark energy Weobtain the behavior of important cosmological parameters as

illustrated in Figure 12 Moreover Figure 13 shows that thismodel is also instable at the late time

Therefore we can choose the first model (model 4) as anappropriate model for the late time cosmology Apart fromthe instabilities discussed above for the second and thirdmodels we can constructmodels by usingmore observationaldata discussed in the next section

6 Observational Constraints on InteractingModels with Varying Λ

The SNIa test is based on the distance modulus 120583 which isrelated to the luminosity distance119863119871 by

120583 = 119898 minus119872 = 5log10119863119871 (46)

where119863119871 is defined as

119863119871 = (1 + 119911)119888

1198670

int

119911

0

1198891199111015840

radic119867(1199111015840)

(47)

The quantities119898 and119872 denote the apparent and the absolutemagnitudes respectively Baryonic acoustic oscillations havetheir origin in oscillations in the photon-baryon plasma at themoment of the decoupling at about 119911 = 1090 They can becharacterized by the distance scale

119860 =radicΩ1198980

119867(119911119887)13[1

119911119887

int

119911119887

0

119889119911

119867 (119911)]

23

(48)

The WiggleZ data [58] indicates the following informationabout 119860 and 119911119887 119860 = 0474 plusmn 0034 0442 plusmn 0020 and0424 plusmn 0021 for the redshifts 119911119887 = 044 060 and 073

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in High Energy Physics

02

04

06

08

10

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

H

120575 = 05120574 = 002 b = 001 and

(a)

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

04

02

00

minus02

minus04

minus06

q

120575 = 05120574 = 002 b = 001 and

(b)

Figure 12 Behavior of Hubble parameter119867 and deceleration parameter 119902 against 119905 for varying Λ Model 6

10

05

00

minus05

minus10

C2 s

5 10 15 20

t

120585 = 0 A = 1 B = 12

120585 = 01 A = 1 B = 12120585 = 03 A = 1 B = 12

120585 = 04 A = 1 B = 12120585 = 05 A = 1 B = 12

120575 = 05120574 = 002 b = 001 and

Figure 13 Squared sound speed against 119905 for varying Λ Model 6

respectively The key quantity of a statistical analysis is the 1205942parameter

1205942(119909119895) =

119899

sum

119894

(119891(119909119895)119905

119894minus 119891(119909

119895)0

119894)

2

120590119894

(49)

where 119891(119909119895)119905119894 is the theoretical evaluation of a given observ-able depending on 119909

119895 free parameters 119891(119909119895)0119894 is the cor-responding observational value and 119899 is the total numberof observational data for the given test There are manydifferent SNIa data sets obtained with different techniquesIn some cases these different samples may give very differentresults The second point is the existence of two differentcalibrationmethodsOne of themuses cosmological relations

Table 1

119872 120575 120574 119887 120585 1198670 Ω1198980

4 13+02

minus02002+003

minus001001+002

minus00105+025

minus04511+01

minus0203+015

minus015

5 14+02

minus03002+002

minus001001+002

minus00103+035

minus01508+02

minus03025+03

minus01

6 075+035

minus015002+003

minus002003+001

minus002035+015

minus0114+005

minus01023+003

minus003

Table 2

119872 119860 119861 120572 119870 119899

4 17+02

minus0303+005

minus01505+02

minus04mdash mdash

5 mdash mdash mdash 15+03

minus0512+05

minus03

6 12+03

minus0111+04

minus02mdash mdash mdash

and takes into account SNIa with high 119911 and the otherone using astrophysical methods is suitable for small 119911(MLCS2k2) Our observational analysis of the backgrounddynamics uses the following three tests the differential ageof old objects based on the 119867(119911) dependence as well as thedata fromSNIa and fromBAOA fourth test could potentiallybe added the position of the first peak of the anisotropyspectrum of the cosmic microwave background radiation(CMB) However the CMB test implies integration of thebackground equations until 119911 asymp 1000 which requires theintroduction of the radiative component But the inclusion ofsuch radiative component considerably changes the structureof the equations and no analytic expression for 119867(119911) isavailable Hence we will limit ourselves to the mentionedthree tests for which a reliable estimation is possible

In Tables 1 and 2 we fix parameters of three models byusing mentioned observational data

In Figure 14 we can see the behavior of 120583 in all modelswhich is approximately in agreement with observational data

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 11

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(a)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(b)

00 01 02 03 04 05 06 07

35

40

45

50

55

120583

z

(c)

Figure 14 Observational data (SNeIa+BAO+CMB) for distance modulus versus our theoretical results for varying Λ in models 4 5 and 6

7 Conclusion

In this paper we considered three different cosmologicalmodels for the universe based on Lyra geometry First of allwe introduced our models and then obtained field equationssolved numerically to find behavior of cosmological param-eters In order to find effect of varying Λ we also studied thecase of constant Λ and found that presence of Λ is necessaryto obtain results being in agreement with observational dataWe assumed viscous modified Chaplygin gas (models 1 and4) viscous polytropic gas (models 2 and 5) or viscousVan derWaals gas (models 3 and 6) to be a component which at theearly universe plays the role of dark matter with 120596 = 0 but atlate times it tends to a cosmological constant Moreover wehave a quintessence field which will contribute to the darkenergy sector including possibility of interaction betweencomponents Easily we can check thatΩDE andΩDM are of thesame order Also we considered case of varyingΛ and studiedbehavior of cosmological parameters numerically We usedobservational data to fix parameters of the models and seenagreement with observational data by investigation of 119867(119911)By using stability analysis we concluded that the model is thebest model considered in this paper to describe universe

For the future work we can extend the present paper toinclude shear viscosity or varying bulk viscosity [53] andwe can also consider cosmic Chaplygin gas versions [54] toobtain more general model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[2] S Perlmutter G Aldering1 G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999

[3] R Amanullah C Lidman D Rubin et al ldquoSpectra and hubblespace telescope light curves of six type Ia supernovae at 0511 lt119911 lt 112 and the union2 compilationrdquo The AstrophysicalJournal vol 716 p 712 2010

[4] A C Pope T Matsubara A S Szalay et al ldquoCosmologicalparameters from eigenmode analysis of sloan digital sky surveygalaxy redshiftsrdquo The Astrophysical Journal vol 607 no 2 pp655ndash660 2004

[5] D N Spergel L Verde H V Peiris et al ldquoFirst year WilkinsonMicrowave Anisotropy Probe (WMAP) observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement vol 148 p 175 2003

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

12 Advances in High Energy Physics

[6] M Jamil K Yesmakhanova D Momeni and R MyrzakulovldquoPhase space analysis of interacting dark energy in f(T) cosmol-ogyrdquoCentral European Journal of Physics vol 10 no 5 pp 1065ndash1071 2012

[7] W Zimdahl ldquoModels of interacting dark energyrdquo AIP Confer-ence Proceedings vol 147 p 51 2012

[8] M Khurshudyan B Pourhassan and E Chubaryan ldquoA Uni-verse with a generalized ghost dark energy and Van der Waalsfluid interacting with a fluidrdquo httparxivorgabs13021819

[9] Y L Bolotin A Kostenko O A Lemets and D A YerokhinldquoCosmological evolution with interaction between dark energyand dark matterrdquo httparxivorgabs13100085

[10] M Khurshudyan B Pourhassan and E O Kahya ldquoInteractingtwo-component fluid models with varying EoS parameterrdquoInternational Journal of Geometric Methods in Modern Physicsvol 11 no 6 Article ID 1450061 14 pages 2014

[11] P P Avelino and H M R da Silva ldquoEffective dark energyequation of state in interacting dark energy modelsrdquo PhysicsLetters B vol 714 no 1 pp 6ndash10 2012

[12] J Sadeghi M Khurshudyan A Movsisyan and H FarahanildquoInteracting ghost dark energy models in the higher dimen-sional cosmologyrdquo httparxivorgabs14016649

[13] M Khurshudyan J Sadeghi A Pasqua S ChattopadhyayR Myrzakulov and H Farahani ldquoInteracting Ricci darkenergy models with an effective Λ-term in Lyra manifoldrdquohttparxivorgabs14025678

[14] M Khurshudyan J Sadeghi M Hakobyan H Farahani andR Myrzakulov ldquoInteraction between modified Chaplygin gasand ghost dark energy in the presence of extra dimensionsrdquoTheEuropean Physical Journal Plus vol 129 article 119 2014

[15] J Sola and H Stefancic ldquoEffective equation of state for darkenergy Mimicking quintessence and phantom energy througha variable Λrdquo Physics Letters B vol 624 no 3-4 pp 147ndash1572005

[16] I L Shapiro and J Sola ldquoOn the possible running of thecosmological lsquoconstantrsquordquo Physics Letters B vol 682 no 1 pp105ndash113 2009

[17] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[18] M Khurshudyan E Chubaryan and B Pourhassan ldquoInteract-ing quintessence models of dark energyrdquo International Journalof Theoretical Physics vol 53 no 7 pp 2370ndash2378 2014

[19] Z K Guo N Ohta and Y Z Zhang ldquoParametrizations ofthe dark energy density and scalar potentialsrdquo Modern PhysicsLetters A vol 22 no 12 p 883 2007

[20] S Dutta E N Saridakis and R J Scherrer ldquoDark energy from aquintessence (phantom) field rolling near a potential minimum(maximum)rdquo Physical Review D vol 79 Article ID 1030052009

[21] E N Saridakis and S V Sushkov ldquoQuintessence and phantomcosmology with nonminimal derivative couplingrdquo PhysicalReview D vol 81 Article ID 083510 2010

[22] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 Article ID 0713012003

[23] B Feng X LWang and XM Zhang ldquoDark energy constraintsfrom the cosmic age and supernovardquo Physics Letters B vol 607no 1-2 pp 35ndash41 2005

[24] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[25] Z K Guo ldquoCosmological evolution of a quintommodel of darkenergyrdquo Physics Letters B vol 608 pp 177ndash182 2005

[26] M-Z Li B Feng and X-M Zhang ldquoA single scalar field modelof dark energy with equation of state crossing minus1rdquo Journal ofCosmology and Astroparticle Physics vol 2005 p 002 2005

[27] B Feng M Li Y-S Piao and X Zhang ldquoOscillating quintomand the recurrent universerdquo Physics Letters B vol 634 no 2-3pp 101ndash105 2006

[28] S Capozziello S Nojiri and S D Odintsov ldquoUnified phantomcosmology inflation dark energy and dark matter under thesame standardrdquo Physics Letters Section B vol 632 no 5-6 pp597ndash604 2006

[29] W Zhao and Y Zhang ldquoQuintom models with an equation ofstate crossing minus1rdquo Physical Review D vol 73 Article ID 1235092006

[30] Y F Cai T Qiu Y S Piao M Li and X Zhang ldquoBouncinguniverse with quintom matterrdquo Journal of High Energy Physicsvol 710 article 71 7 pages 2007

[31] E N Saridakis and J M Weller ldquoA Quintom scenario withmixed kinetic termsrdquo Physical Review D vol 81 no 12 ArticleID 123523 11 pages 2010

[32] Y F Cai T Qiu R Brandenberger Y S Piao and X ZhangldquoOnperturbations of quintombouncerdquo Joint Center forArtificialPhotosynthesis vol 2008 no 03 article 013 2008

[33] M R Setare and E N Saridakis ldquoCoupled oscillators as modelsof quintom dark energyrdquo Physics Letters B vol 668 no 3 pp177ndash181 2008

[34] M R Setare and E N Saridakis ldquoQuintom cosmology withgeneral potentialsrdquo International Journal of Modern Physics Dvol 18 no 4 pp 549ndash557 2009

[35] Y F Cai E N Saridakis M R Setare and J Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[36] T Qiu ldquoTheoretical aspects of Quintom modelsrdquo ModernPhysics Letters A vol 25 no 11-12 pp 909ndash921 2010

[37] Q G Huang andM Li ldquoThe holographic dark energy in a non-flat universerdquo Journal of Cosmology and Astroparticle Physicsvol 8 2004

[38] M Ito ldquoHolographic-dark-energy model with non-minimalcouplingrdquo Europhysics Letters vol 71 no 5 pp 712ndash716 2005

[39] X Zhang and F Q Wu ldquoConstraints on holographic darkenergy from type Ia supernova observationsrdquo Physical ReviewD vol 72 Article ID 043524 2005

[40] S Nojiri and S D Odintsov ldquoUnifying phantom inflationwith late-time acceleration scalar phantommdashnon-phantomtransition model and generalized holographic dark energyrdquoGeneral Relativity and Gravitation vol 38 pp 1285ndash1304 2006

[41] E Elizalde S Nojiri S D Odintsov and P Wang ldquoDarkenergy vacuum fluctuations the effective phantom phase andholographyrdquo Physical Review D vol 71 Article ID 103504 2005

[42] H Li Z K Guo and Y Z Zhang ldquoA tracker solution fora holographic dark energy modelrdquo International Journal ofModern Physics D vol 15 p 869 2006

[43] E N Saridakis ldquoRestoring holographic dark energy in branecosmologyrdquo Physics Letters B vol 660 no 3 pp 138ndash143 2008

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 13

[44] E N Saridakis ldquoHolographic dark energy in braneworldmodels with moving branes and the 119908 = minus1 crossingrdquo Journalof Cosmology and Astroparticle Physics vol 2008 no 4 article020 2008

[45] E N Saridakis ldquoHolographic dark energy in braneworld mod-els with a Gauss-Bonnet term in the bulk Interacting behaviorand the 119908 = minus1 crossingrdquo Physics Letters B vol 661 no 5 pp335ndash341 2008

[46] R-G Cai ldquoA dark energy model characterized by the age of theuniverserdquo Physics Letters B vol 657 no 4-5 pp 228ndash231 2007

[47] HWei and R G Cai ldquoInteracting agegraphic dark energyrdquoTheEuropean Physical Journal C vol 59 pp 99ndash105 2009

[48] O F Piattella J C Fabris and W Zimdahl ldquoBulk viscouscosmology with causal transport theoryrdquo Journal of Cosmologyand Astroparticle Physics vol 2011 no 5 article 029 2011

[49] K Karami Z Safari and S Asadzadeh ldquoCosmological con-straints on polytropic gas modelrdquo httparxivorgabs12096374

[50] K Karami S Ghaffari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquo EuropeanPhysical Journal C vol 64 no 1 pp 85ndash88 2009

[51] K Karami and S Ghaffari ldquoThe generalized second law ofthermodynamics for the interacting polytropic dark energyin non-flat FRW universe enclosed by the apparent horizonrdquoPhysics Letters B vol 688 p 125 2010

[52] G M Kremer ldquoCosmological models described by a mixture ofvan derWaals fluid and dark energyrdquo Physical ReviewD vol 68no 12 Article ID 123507 2003

[53] H Saadat and B Pourhassan ldquoEffect of varying bulk viscosityon generalized chaplygin gasrdquo International Journal of Theoret-ical Physics vol 53 no 4 pp 1168ndash1173 2014

[54] B Pourhassan ldquoViscous modified cosmic chaplygin gas cos-mologyrdquo International Journal of Modern Physics D vol 22 no9 Article ID 1350061 2013

[55] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified Chaplygin gas in flat spacerdquo Astrophysics andSpace Science vol 343 no 2 pp 783ndash786 2013

[56] H Saadat and B Pourhassan ldquoFRW bulk viscous cosmologywith modified cosmic Chaplygin gasrdquo Astrophysics and SpaceScience vol 344 no 1 pp 237ndash241 2013

[57] V K Shchigolev ldquoCosmology with an effective Λ-term in LyraManifoldrdquo Chinese Physics Letters vol 30 Article ID 1198012013

[58] M Jamil S Ali D Momeni and R Myrzakulov ldquoBianchi typeI cosmology in generalized Saez-Ballester theory via Noethergauge symmetryrdquo European Physical Journal C vol 72 no 4article 1998 pp 1ndash6 2012

[59] M Khurshudyan ldquoToy models of Universe with an effectivevarying Λ-termrdquo httparxivorgabs14030109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of