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CURVATURE MATTER COUPLING: SOME
COSMIC ASPECTS
By
Muhammad Zubair
A THESIS
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
MATHEMATICS
Supervised By
Prof. Dr. Muhammad Sharif
UNIVERSITY OF THE PUNJAB
LAHORE-PAKISTAN
DECEMBER, 2013
CERTIFICATE
I certify that the research work presented in this thesis is
the original work of Mr. Muhammad Zubair S/O Mehboob
Ahmed and is carried out under my supervision. I endorse its
evaluation for the award of Ph.D. degree through the official
procedure of University of the Punjab.
Prof. Dr. Muhammad Sharif(Supervisor)
ii
DECLARATION
I, Mr. Muhammad Zubair S/O Mehboob Ahmed,
hereby declare that the matter printed in this thesis is my
original work. This thesis does not contain any material that
has been submitted for the award of any other degree in any
university and to the best of my knowledge, neither does this
thesis contain any material published or written previously by
any other person, except due reference is made in the text of
this thesis.
Muhammad Zubair
iii
Table of Contents
Table of Contents v
List of Figures vii
Abstract ix
Acknowledgements xi
Notations xiii
Introduction 1
1 Modified Gravities and Their Implications 6
1.1 Modified Gravitational Theories . . . . . . . . . . . . . . . . 7
1.1.1 Theories Involving Non-Minimal Coupling . . . . . . 8
1.1.2 f(R, T ) Gravity . . . . . . . . . . . . . . . . . . . . . 8
1.1.3 f(R, T,Q) Gravity . . . . . . . . . . . . . . . . . . . 11
1.2 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Zeroth Law . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 First Law . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Second Law . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4 Third Law . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Laws of BH Dynamics or Thermodynamics . . . . . . . . . . 17
1.4.1 Zeroth Law . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 First Law . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Second Law . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 Generalized Second Law . . . . . . . . . . . . . . . . 19
1.4.5 Thermal Equilibrium . . . . . . . . . . . . . . . . . . 21
1.4.6 Third Law . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Anisotropic Cosmologies . . . . . . . . . . . . . . . . . . . . 24
v
1.7 Cosmological Parameters . . . . . . . . . . . . . . . . . . . . 26
1.7.1 Hubble’s Law and Hubble Parameter . . . . . . . . . 26
1.7.2 Mean and Directional Hubble Parameters . . . . . . 27
1.7.3 Anisotropy Parameter of Expansion . . . . . . . . . . 28
1.7.4 Deceleration Parameter . . . . . . . . . . . . . . . . . 28
1.8 The Expansion and Shear Scalar . . . . . . . . . . . . . . . . 29
2 Thermodynamics Laws in f(R, T ) and f(R, T,Q) Modified
Theories 30
2.1 Thermodynamics in f(R, T ) Gravity . . . . . . . . . . . . . 31
2.1.1 First Law of Thermodynamics . . . . . . . . . . . . . 33
2.1.2 Generalized Second Law of Thermodynamics . . . . . 35
2.2 Redefining the Dark Components . . . . . . . . . . . . . . . 37
2.2.1 First Law of Thermodynamics . . . . . . . . . . . . . 39
2.2.2 Generalized Second Law of Thermodynamics . . . . . 41
2.3 Thermodynamics in f(R, T,Q) Gravity . . . . . . . . . . . . 42
2.3.1 First Law of Thermodynamics . . . . . . . . . . . . . 44
2.3.2 Generalized Second Law of Thermodynamics . . . . . 48
3 Energy Conditions Constraints and Stability of f(R, T ) and
f(R, T,Q) Modified Theories 56
3.1 Energy Conditions in f(R, T,Q) Gravity . . . . . . . . . . . 57
3.2 Constraints on Class of f(R, T,Q) Models . . . . . . . . . . 63
3.2.1 f(R, T,Q) = R + αQ . . . . . . . . . . . . . . . . . . 63
3.2.2 f(R, T,Q) = R(1 + αQ) . . . . . . . . . . . . . . . . 65
3.3 Energy Conditions in f(R, T ) Gravity . . . . . . . . . . . . . 67
3.3.1 Power Law Solutions . . . . . . . . . . . . . . . . . . 72
3.4 Stability of Power Law Solutions . . . . . . . . . . . . . . . 80
3.4.1 f(R, T ) = f(R) + λT . . . . . . . . . . . . . . . . . . 80
3.4.2 f(R, T ) = R + 2f(T ) . . . . . . . . . . . . . . . . . . 82
4 Anisotropic Universe Models in f(R, T ) Gravity 84
4.1 f(R, T ) Gravity and Bianchi I Universe . . . . . . . . . . . . 85
4.2 Solution of the Field Equations . . . . . . . . . . . . . . . . 86
4.2.1 Exponential Expansion Model . . . . . . . . . . . . . 88
4.2.2 Power Law Expansion Model . . . . . . . . . . . . . 91
4.3 Massless Scalar Field Models . . . . . . . . . . . . . . . . . . 95
4.4 Solutions for Fixed Anisotropy Parameter . . . . . . . . . . 97
5 Discussion and Conclusion 104
Bibliography 117
vi
List of Figures
2.1 Evolution of GSLT for the Lagrangian f(R, T,Q) = R+αQ,
the left panel shows the bound on m for α = −2 whereas in
the right panel we set m = 10. It is evident that GSLT is
valid only if α < 0, m > 1. . . . . . . . . . . . . . . . . . . . 53
2.2 Evolution of GSLT for the Lagrangian f(R, T,Q) = R(1 +
αQ), the left panel shows constraint on m for α = 10 whereas
in the right panel, we set m = 10. It is evident that GSLT
is valid only if α > 0, m > 1. . . . . . . . . . . . . . . . . . . 54
2.3 Evolution of GSLT for the Lagrangian f(R, T,Q) = R +
f(Q) + g(T ), the left panel shows constraints on parameters
C and D for z = 0 whereas in the right panel, we set Ci and
Di in terms of unique parameter Υ and constrain the values
of Υ. We choose H0 = 67.3, Ωm0 = 0.315 from the recent
Planck results [33]. . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Evolution of ∆ versus t for different values of n. We set
l = 0.1, k = 3, and α = 0.05. . . . . . . . . . . . . . . . . . . 89
4.2 Evolution of NEC for n = 2. The left graph shows that NEC
is satisfied for α < 0 and it is violated for α > 0 at the right
side. We set l = λ = 0.1 and k = 3. . . . . . . . . . . . . . . 90
4.3 The left graph shows the behavior of ρ for −6 < n ≤ 0 and
n < −6, while the right graph presents the evolution of ρ for
n > 0. We set l = λ = 0.1, k = 3 and α = 0.05. . . . . . . . 91
vii
4.4 Plot of ∆ versus t for different values of n. We set l = 0.1,
k = c2 = 3, m = 0.9 and α = 0.05. . . . . . . . . . . . . . . . 92
4.5 Behavior of NEC versus α for n = 3. The left part shows
that NEC is satisfied for α < 0, while it is violated for α > 0
shown on the right side. We set l = λ = 0.1, k = c2 = 3 and
m = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 This figure is plotted for n = 2. The left part shows that
NEC is violated for α < 0, whereas NEC is satisfied for
α > 0 shown on right side. . . . . . . . . . . . . . . . . . . . 94
4.7 Evolution of ρ versus t for n ≥ 0. We set l = λ = 0.1,
k = c2 = 3, m = 0.9 and α = 0.05. . . . . . . . . . . . . . . . 95
4.8 Evolution of φ versus t for m = 0 and different values of
n: solid(black) n = 1; dashed(red), n = 0; dahsed(blue),
n = −1. We set l = λ = 0.1, k = 3 and α = 0.05. . . . . . . 96
4.9 Evolution of φ versus t for m 6= 0 and different values of
n: solid(black) n = 1; dashed(red), n = 0; dahsed(blue),
n = −1. We set l = λ = 0.1, k = c2 = 3, m = 0.9 and
α = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.10 Evolution of NEC for exponential model. The NEC is vio-
lated for (a) λ > 0 whereas it can be satisfied for (b) λ < −25.
We set χ = 3, l = c1 = 0.1 and γ = 0.001. . . . . . . . . . . 99
4.11 Plot of energy density corresponding to different values of
coupling parameter. . . . . . . . . . . . . . . . . . . . . . . . 99
4.12 Evolution of NEC for power law model. Plot (a) shows that
NEC is violated for λ > −25 which favors the accelerated
expansion. It can be met for λ < −25 as shown in plot (b). . 101
4.13 Evolution of NEC for 0 < m < 1 with λ = 10. For λ < −25,
we have ρ + p > 0. . . . . . . . . . . . . . . . . . . . . . . . 102
4.14 Evolution of ρ for power law model. In plot (a) we set m =
0.9 and −10 < λ < 10 whereas in plot (b) we vary m in the
range of phantom evolution and set λ = 0.1 . . . . . . . . . 102
viii
Abstract
This thesis studies some cosmic aspects in modified theories involving cur-
vature matter coupling. In this setting, we concentrate on f(R, T ) and
f(R, T, RµνTµν) theories to discuss the thermodynamic laws with the non-
equilibrium description at the apparent horizon of FRW universe. It is
shown that Friedmann equations can be transformed to the form of Clau-
sius relation ThSeff = δQ, Seff is the entropy which contains contributions
both from horizon entropy as well as additional entropy term introduced
due to the non-equilibrating description and δQ is the energy flux across
the horizon. The generalized second law of thermodynamics is also estab-
lished in a more comprehensive form and one can recover the corresponding
results in Einstein as well as f(R) theories. We remark that equilibrium
description in such theories needs more study to follow.
Moreover, we discuss the validity of energy conditions in f(R, T,RµνTµν)
gravity. The corresponding energy conditions are presented in terms of re-
cent values of Hubble, deceleration, jerk and snap parameters. In particular,
we use two specific models recently developed in literature to study concrete
application of these conditions as well as Dolgov-Kawasaki instability. We
explore f(R, T ) gravity as a specific case to this modified theory for expo-
nential and power law models. The exact power law solutions are obtained
for two particular cases in homogeneous and isotropic f(R, T ) cosmology.
Finally, we find certain constraints which have to be satisfied to ensure that
power law solutions may be stable and match the bounds prescribed by the
energy conditions.
We also explore the locally rotationally symmetric Bianchi type I model
ix
x
with perfect fluid as matter content in f(R, T ) gravity. The exact solutions
of the field equations are obtained for two expansion laws namely exponen-
tial and power law expansions. The physical and kinematical quantities are
examined for both cases in future evolution of the universe. We investigate
the validity of null energy condition and conclude that our solutions are
consistent with the current observations.
Acknowledgements
All praises and thanks to Almighty Allah, Who (Alone) created the heavens
and the earth, and prevailed the darkness and sparked the light. I owe my
deep gratitude to Him, Who endowed me with opportunity, knowledge,
patience and potential to impart a drop, in the sea of knowledge. All
praise and best regards to the Holy Prophet Hazrat Muhammad (PBUH),
the mercy on mankind, who is the greatest inspiration for all knowledge
seekers.
One of the joys of this completion is to look over the past journey and
remember all the friends and family who have helped and supported me
along this long but fulfilling road. This thesis has been kept on track and
seen through to completion with the support and encouragement of numer-
ous people. I would like to thank all those who made this thesis possible
and an unforgettable experience for me.
First and foremost, I feel great pleasure to express my heartiest gratitude
and deep sense of obligation to my distinguished supervisor and chairman
Prof. Dr. Muhammad Sharif for every bit of guidance, assistance,
expertise, enthusiasm and constructive criticism. Sir! you have been a
tremendous mentor for me and I feel extremely privileged to have worked
under your supervision. I am grateful to Department of Mathematics for
providing the research facilities and Higher Education Commission, Islam-
abad for its financial support through the Indigenous Ph.D. 5000 Fellowship
Program Batch-VII.
I want to express my deeply felt thanks to all faculty members and wor-
thy school as well as college teachers especially Assoc. Prof. Aslam Malik,
Mr. Daud Ahmed, Mr. M. Riaz, Dr. Aziz Ullah, Prof. Shahid Siddiqi, Dr.
Ghazala Akram, Mr. Zafar Islam, Prof. Hameed Siddiqi and Prof. Hamid
Shah for their guidance and helping nature. I am obliged to Prof. Asghar
xi
xii
Qadir for his valuable suggestions and comments to improve this thesis. I
would also like to acknowledge him with much appreciation for suggesting a
new direction regarding modified theories of gravity. I would like to thank
Prof. Dr. Jin Lin Han and his group members for their warm hospitality
at National Astronomical Observatories, Chinese Academy of Sciences Bei-
jing, during my three months stay. I really enjoyed that period of my life
and it was unforgettable tour with my senior Dr. Abbas. I acknowledge
him for his support in this tour and in my PhD.
It is my pleasure to acknowledge all my PhD fellows especially Mr.
Jawad, Mr. Hamood, Mr. Younis, Mr. Muzammal and my cabin fellow Mr.
M. Azam for their help, cooperative behavior and providing a stimulating
and conducive environment. I would also like to thank my colleague Miss
Saira Waheed for cooperative and supporting attitude. I am obliged to
many people especially Zahid bahi, Advoc. Hafiz Sami, Mr. Shakeel and
Mr. Tariq who in some way contributed to my educational career. I would
particularly thank neurophysician Dr. Mazhar Badshah for his medication
to recover from my brain disease and encouragement to continue my PhD.
I am grateful to all my friends especially Ihtesham Zafar, Haseeb Muzaf-
far, Jawad Ali, Imran Sarwar, ShahRukh, Irtaza, Muqaddar, Zahid and
Quyum for their unflagging love, care and encouragement throughout this
period. I give credit to Ihtesham who always tried to keep us connected and
wanted me to be there in every gathering. I apologize for ignoring them
most of the times because of my occupied schedules. I am ever indebted to
Imran bahi for his valuable support in my whole hostel life.
This acknowledgement will be incomplete without mentioning my feel-
ings with tearful eyes for my loving parents who taught me to take the first
step, to speak the first word and inspired me throughout of my life. My
deepest gratitude goes to my family for their unwearying love and support;
this thesis would have been simply impossible without them. Many special
thanks are due to my sister Rakhshanda for her support in completing my
educational career. My father always encouraged and supported me and
my mother whose hands always arose in prayers for me, she is everything
for me.
Lahore Muhammad Zubair
December, 2013
Notations
In this thesis, the convention to be used for the metric signatures will be
(+,−,−,−) and Greek indices will vary from 0 to 3, if different it will be
mentioned. Also, we shall use the following list of notations and abbrevia-
tions.
AH: Apparent Horizon
BAO: Baryon Acoustic Oscillations
BH: Black Hole
BI: Bianchi Type I
CMB: Cosmic Microwave Background
CMC: Curvature Matter Coupling
DE: Dark Energy
DEC: Dominant Energy Condition
DM: Dark Matter
EoS: Equation of State
FLT: First Law of Thermodynamics
FRW: Friedmann-Roberston-Walker
GR: General Relativity
GSLT: Generalized Second Law of Thermodynamics
ΛCDM: Λ-Cold Dark Matter
LRS: Locally Rotationally Symmetric
MGT: Modified Gravitational Theory
NEC: Null Energy Condition
SEC: Strong Energy Condition
SNeIa: Supernovae Type Ia
WEC: Weak Energy Condition
xiii
Introduction
Humans have been speculating about the Universe in search of reasons to
questions like, how did it come into being and how will it evolve in future?
What is its matter energy contents and how these are structured? Cosmol-
ogy, the study of the cosmos, explains the origin and evolution of entire
cosmic contents, tries to understand the underlying physical processes and
comprehend the laws of physics assumed to hold throughout the cosmos.
Cosmology is ranked among the modern and dynamic physical sciences due
to its progress both in theory and observations. The development of cos-
mology and gravitation can be seen as one of the scientific triumphs of the
twentieth century. In 1998, observations of SNeIa accumulated by the high-
redshift SN team [1] and SN cosmology project team [2] appeared as illumi-
nating candles in disclosing the expansion of the Universe. The source for
this observed cosmic acceleration may be an anonymous energy component
entitled as DE. In spite of tremendous efforts, late cosmic acceleration is
certainly a major challenge for cosmologists. The direct evidence for cosmic
acceleration has strengthened over time with measurements from tempera-
ture anisotropies in CMB [3] and BAO [4] which confirm the existence of
DE.
Contemporary Planck results [5] acquired by ESA’s Planck space tele-
scope predict that cosmos is made of 4.9% baryons, 26.8% DM and 68.3%
1
2
DE which confirms that cosmic energy budget is dominated by DE. Dark
energy is recognized by its distinctive nature from ordinary matter sources
having negative pressure which may lead to cosmic expansion counter strik-
ing the gravitational pull. To explore the properties of DE, one needs to
clarify whether it is Λ or it originates from other dynamical sources. If the
origin of DE is not Λ then one may seek for some other possibilities to count
the cosmic expansion. A useful way to categorize the candidates of DE is
according to how they modify Einstein equations which relate geometry
with energy and matter in cosmic contents. The first proposal is to modify
the matter part in Einstein equations by considering exotic matter source
with a negative pressure [6]. In this proposal, the representative models are
quintessence [7], K-essence [8], phantom field [9] and Chaplygin gas [10].
These dynamical DE models can be distinguished from Λ by defining the
evolution for EoS parameter.
The other proposal for the construction of DE models is the modification
of Einstein-Hilbert action which leads to modified gravity models. Of the
many proposals for modified gravity, we will be interested in f(R) gravity
[11], f(R, T ) gravity [12] and f(R, T,Q) gravity [13, 14], where R is the
scalar curvature, T is the trace of energy-momentum tensor and Q is the
contraction of Ricci tensor and energy momentum tensor. Harko et al. [12]
introduced a matter geometry coupled system in the setting of Lagrangian
f(R, T ), a generic function of R and T . The field equations were formu-
lated for general and some specific forms of Lagrangian in metric formalism.
Alvarenga et al. [15] explored the scalar cosmological perturbations for a
specific model in this theory to guarantee the standard continuity equa-
tion and obtained the matter density perturbed equations. Shabani and
3
Farhoudi [16] discussed the cosmological solutions for three specific cate-
gories in this theory through phase space analysis. Recently, an extension of
f(R, T ) theory is proposed by assuming the non-minimal coupling through
the contraction of Ricci tensor Rαβ as well as Tαβ and resulting action is
refereed as f(R, T,Q) [13, 14]. Haghani et al. [13] developed the field equa-
tions in metric formalism and investigated the cosmological implications for
conserved as well as nonconserved Tαβ. Odinstov and Saez-Gomez [14] re-
constructed this theory for some well-known solutions like de Sitter, power
law and ΛCDM cosmology and discussed the issue of matter instability.
The discovery of black hole thermodynamics set up a significant connec-
tion between gravity and thermodynamics [17, 18]. The Hawking tempera-
ture T = |κsg |2π
, where κsg is the surface gravity, and horizon entropy S = A4G
satisfy FLT. The association of FLT with Einstein equations has been ex-
plored extensively in settings of GR and MGTs. Cai and Kim [19] developed
a connection between FLT at AH with the field equations for the FRW Uni-
verse model. The FRW equations for any spatial curvature are derived using
the relation ThdSh = −dE (Th = 1/2πrA and Sh = πrA/G), where E is the
heat flow across the horizon. Eling et al. [20] realized that thermodynamic
derivation of Einstein equations in f(R) gravity needs a modification to
non-equilibrium setting. In order to get the right equations, it is necessary
to add an extra entropy production term in the Clausius relation to balance
the energy conservation. This corresponds to non-equilibrium description
of thermodynamics. In spite of of these studies, reinterpretation of non-
equilibrium correction has also been explored and alternative treatments
have been suggested. Bamba and Geng [21, 23] suggested that equilibrium
picture of thermodynamics can be established in MGTs by incorporating
4
the extra degrees of freedom in effective Tαβ. They discussed the FLT and
GSLT in both equilibrium and non-equilibrium descriptions.
In GR, the theory of matter is specified on the basis of classical energy
conditions namely, weak, null, strong and dominant conditions which make
certain constraints such as positivity of energy density and dominance of
energy density over pressure [24]. Santos [28] explored the energy conditions
bounds in f(R) gravity and constrained two known models in terms of
recent figured values of deceleration, jerk and snap parameters. This scheme
has been implemented in other MGTs including f(T ) gravity [29], f(G)
gravity [30], scalar-tensor theories [31] and modified gravities with CMC
[32].
This thesis is devoted to look into the cosmological implications of
f(R, T ) and f(R, T,Q) theories of gravity. We address the thermodynamic-
gravity relation and energy conditions bounds on these theories. The exact
solutions are discussed for power law cosmology and anisotropic cosmic
models which assist in reconstructing the corresponding Lagrangian. The
thesis is outlined in the following format.
Chapter One presents an overview of the current results concerning
the dynamics of cosmos and indications for the modification of GR. We
briefly introduce modified theories involving CMC and their corresponding
formalisms.
Chapter Two deals with the study of thermodynamic laws in the frame-
work of f(R, T ) and f(R, T,Q) gravities. We establish the FLT and GSLT
at the AH of FRW spacetime in non-equilibrium picture of thermodynamics.
The validity of GSLT is examined for two particular models by constraining
the coupling parameters. We also investigate the existence of equilibrium
5
description of thermodynamics for these theories.
Chapter Three presents the picture of energy conditions constraints in
the configuration of FRW spacetime for f(R, T ) and f(R, T,Q) gravities.
The corresponding inequalities are obtained in terms of recent values of
Hubble, deceleration, jerk and snap parameters which can reduce to well-
known results in GR and f(R) gravity. The exact power law solutions in
f(R) gravity are constrained against the energy conditions and linear ho-
mogeneous perturbations. We also consider two specific forms of f(R, T,Q)
gravity to develop concrete application of these conditions as well as Dolgov-
Kawasaki instability.
Chapter Four is devoted to discuss the LRS BI model with matter con-
tent as perfect fluid in f(R, T ) gravity. The exact solutions of the field equa-
tions are obtained for two expansion laws namely exponential and power
law expansions. We check the validity of NEC and conclude that these
solutions favor the phantom model. We also establish the functional forms
of Lagrangian for both dynamical and constant anisotropy parameters.
Chapter Five comprises of concluding remarks and suggests some issues
requiring further consideration.
Chapter 1
Modified Gravities and TheirImplications
The most significant characteristic of our cosmos is its large scale homogene-
ity and isotropy, the so called Cosmological Principle which is considered as
the cornerstone of modern cosmology. According to this principle at each
epoch, the Universe represents the same aspect from every point, except
for local irregularities. In fact, there is no privileged direction or position
in the Universe. Following this idea, the line element of homogeneous and
isotropic FRW spacetime is given by
ds2 = dt2 − a2(t)[dr2
1− kr2+ r2dθ2 + r2 sin2 θdφ2], (1.0.1)
where k = +1,−1, 0 corresponds to closed, open and flat geometries, re-
spectively.
In this chapter, we present the candidates of MGTs involving matter
geometry coupling and overview thermodynamic laws, energy conditions
and some other cosmological components.
6
7
1.1 Modified Gravitational Theories
Modified gravitational theories have been the subject of great interest in
cosmology and provide a convincing way for settling the issue of late-time
acceleration. The concept that gravity is not described precisely by GR
but rather by some alternative theories has been viewed under different cir-
cumstances. There are various ways to modify GR incorporating quadratic
Lagrangian, consisting of second order curvature invariants such as R2,
RαβRαβ, RαβγδRαβγδ, CαβγδC
αβγδ. Therefore, the general modification of
GR action is of the form
I =1
2κ2
∫dx4
√−gf(R, RαβRαβ, RαβγδRαβγδ, ..) +
∫dx4
√−gLm(gαβ,Ψm),
(1.1.1)
where κ2 = 8πG and Lm is the matter Lagrangian with matter field Ψm.
Such theories involve the higher order derivatives and allow the dynami-
cal equations to be higher than second order. In this respect, a particularly
interesting modification is to replace the linear dependence of scalar curva-
ture with the more generic function and resulting action is named as f(R)
gravity. There are three different approaches to formulate the field equa-
tions in this modified gravity namely, metric, Palatini and metric-affine
formalism. In metric formalism, the metric tensor variation of the f(R)
action yields
RαβfR(R)− 1
2gαβf(R) + (gαβ2−∇α∇β)fR(R) = κ2Tαβ, (1.1.2)
where fR = ∂f/∂R. Recently, f(R) theory and its subclass have been
presented in many writings. In the next section, we overview the subclass
of these theories, in particular, the ones involving dependence of T .
8
1.1.1 Theories Involving Non-Minimal Coupling
There are various approaches to identify the DE problem and other cosmic
aspects, and one can classify most of them as (i) MGTs or (ii) inserting
exotic matter components to GR action. MGTs are constructed by in-
corporating the geometric part whereas matter contribution is considered
as an additional term in Lagrangian. Nevertheless one can put further
modification by introducing direct coupling between matter and curvature
components; such theory is named as non-minimally coupled gravity. Such
couplings were initially proposed in [34, 35] which were formulated in the
context of f(R) theories by considering explicit and also arbitrary couplings
with Lm. These types of Lagrangian are listed as follows:
• L = f1(R) + (1 + λf2(R))Lm;
• L = f1(R) + G(Lm)f2(R);
• L = f(R,Lm);
• L = f(R, T );
• L = f(R, T,Q).
Here, fi’s and G involve arbitrary dependence on their respective argu-
ments.
1.1.2 f(R, T ) Gravity
The issue of accelerated cosmic expansion can be explained by taking into
account the MGTs involving CMC such as f(R, T ) gravity. In these the-
ories, one can explore the present cosmic issues without resorting exotic
energy component or additional spatial dimension. The f(R, T ) theory can
9
be reckoned as a useful candidate of DE components which may help to re-
alize the accelerated expansion. In this theory, cosmic expansion can result
not just from the scalar-curvature part of the entire cosmic energy density,
but can include a matter component as well. In [12], f(R) theory is mod-
ified by inserting an arbitrary dependence of the function f on T yielding
the action
I =1
2κ2
∫ √−gdx4f(R, T ) +
∫ √−gdx4Lm. (1.1.3)
The matter energy-momentum tensor is given by [36]
Tαβ = − 2√−g
δ(√−gLm)
δgαβ. (1.1.4)
If Lm depends only upon the components of gαβ rather than its derivatives
then Eq.(1.1.4) yields
Tαβ = gαβLm − 2∂Lm
∂gαβ. (1.1.5)
The field equations corresponding to the action (1.1.3) are
κ2Tαβ − fT (R, T )Tαβ − fT (R, T )Θαβ −RαβfR(R, T ) +1
2gαβf(R, T )
+ (∇α∇β − gαβ2)fR(R, T ) = 0, (1.1.6)
where subscripts mark the derivatives with respect to R and T , 2 =
∇α∇α, ∇α denotes covariant derivative and Θαβ is defined by
Θαβ =gµνδTµν
δgαβ= −2Tαβ + gαβLm − 2gµν ∂2Lm
∂gαβ∂gµν. (1.1.7)
The trace of Eq.(1.1.6) is
κ2T − fT (R, T )T − fT (R, T )Θ−RfR(R, T )− 2f(R, T ) + 32fR(R, T ) = 0,
or equivalently
f(R, T ) =1
2
[RfR(R, T ) + 32fR(R, T )− κ2T + fT (R, T )T + fT (R, T )Θ
].
(1.1.8)
10
where Θ = Θαα. As the dynamical equations in this theory depends upon
contribution from matter contents, therefore one can obtain particular scheme
of equations corresponding to every selection of Lm.
We consider matter part as perfect fluid whose energy-momentum tensor
is
Tαβ = (ρ + p)uαuβ − pgαβ, (1.1.9)
where ρ and p indicate the energy density and pressure, respectively, and
uα is the four-velocity. Here, we take Lm = −p [12] which leads the second
derivative of matter Lagrangian to zero and hence Θαβ becomes
Θαβ = −2Tαβ − pgαβ.
Consequently, the field equations take the form
κ2Tαβ + fT (R, T )Tαβ + fT (R, T )pgαβ −RαβfR(R, T ) +1
2gαβf(R, T )
+ (∇α∇β − gαβ2)fR(R, T ) = 0. (1.1.10)
One can cast the above equation as effective Einstein equations
Gαβ = Rαβ − 1
2Rgαβ = 8πGeffT
(m)αβ + T
(DC)αβ , (1.1.11)
where effective matter dependent gravitational coupling Geff and energy-
momentum tensor of dark components corresponding to matter geometry
coupling are defined as
Geff =1
fR(R, T )
(G +
fT (R, T )
8π
), (1.1.12)
T(DC)αβ =
1
fR(R, T )
[1
2gαβ(f(R, T )−RfR(R, T )) + fT (R, T )pgαβ + (∇α∇β
− gαβ2)fR(R, T )] . (1.1.13)
11
In f(R, T ) gravity, the divergence of energy-momentum tensor is non-zero
and is obtained as
∇αTαβ =fT
κ2 − fT
[(Tαβ + Θαβ)∇α ln fT +∇αΘαβ − 1
2gαβ∇αT
](1.1.14)
The particular class of models can be listed through the following three
choices.
• f(R, T ) = R + 2f(T ): This corresponds to gravitational Lagrangian
with time dependent cosmological constant being function of T and
hence represents the ΛCDM model.
• f(R, T ) = f1(R) + f2(T ): This choice does not imply the direct non-
minimal CMC nevertheless it can be considered as correction to f(R)
gravity. We shall use the linear form of f2 and distinct results can
be obtained on the basis of non-trivial coupling as compared to f(R)
gravity.
• f(R, T ) = f1(R) + f2(T )f3(R): This model involves the explicit non-
minimal CMC and consequences of this type of theory would be dif-
ferent from other models.
1.1.3 f(R, T,Q) Gravity
This theory is also an interesting candidate among the modified theories
which are based on non-minimal CMC. The action of this modified theory
is of the form [13, 14]
I =1
2κ2
∫ √−gdx4f(R, T,Q) +
∫ √−gdx4Lm. (1.1.15)
The function f(R, T,Q) necessities an arbitrary dependence on R, T and
contraction of Rαβ and Tαβ. The metric tensor variation of this action
12
implies that
RαβfR − 1
2f − LmfT − 1
2∇µ∇ν(fQT µν)gαβ + (gαβ2−∇α∇β)fR
+1
22(fQTαβ) + 2fQRµ(αT µ
β) −∇µ∇(α[T µβ)fQ]−GαβLmfQ − 2 (fT gµν
+ fQRµν)∂2Lm
∂gαβ∂gµν= (1 + fT +
1
2RfQ)Tαβ. (1.1.16)
This equation can be reduced to well-known forms of the field equations
in f(R) and f(R, T ) theories by setting some particular choices of the La-
grangian. It can be rearranged as that of Eq.(1.1.11) with
Geff =1
fR − fQLm
(G +
1
8π
[fT +
1
2(R−2) fQ
]), (1.1.17)
T(DC)αβ =
[1
2(f −RfR)− LmfT − 1
2∇µ∇ν(fQT µν)
gαβ + (∇α∇β
− gαβ2) fR − 1
2(fQ2Tαβ +∇µfQ∇µTαβ)− 2fQRµ(αT µ
β)
+ ∇µ∇(α[T µβ)fQ] + 2 (fT gµν + fQRµν)
∂2Lm
∂gµν∂gαβ
]. (1.1.18)
1.2 Stability Criteria
The study of stability criteria is a significant aspect in modified theories
for the viability of such modification to GR. In fact, any MGT needs to
possess exact cosmological dynamics and avoids the instabilities, such as
ghosts degrees of freedom endorsed in Ostrogradski’s instability, tachyon
and Dolgov-Kawasaki instability [37].
Ghost is referred as a field having kinetic term with wrong sign. Ghost
appears as common property of any MGT that informs the DE as a source
behind current cosmic acceleration. This may be induced due to a myste-
rious force which is repulsive in nature acting between the massive objects
at significant distances. In fact, higher derivative MGTs such as presented
13
in action (1.1.1) give rise to ghosts and Ostrogradski’s instability. Ac-
cording to Ostrogradski’s theorem, Lagrangians that contain higher than
second order time derivatives imply the ghost instability which limits the
modification of gravity to a function of R. Thus, theories of the type
f(R, RαβRαβ, RαβγδRαβγδ) are plagued by ghosts that can be avoided in
f(R) and f(R, R2 − 4RαβRαβ + RαβγδRαβγδ) (where second term is named
as Gauus-Bonnet term) theories. The condition of effective gravitational
coupling to be positive is also important to keep the attractive nature of
gravity. In f(R) gravity, this condition requires fR > 0 which is also neces-
sary to avoid the appearance of ghost [38].
A tachyon is any hypothetical particle that travels faster than the speed
of light. For such particles, the moving mass would be imaginary and one
could assume the imaginary rest mass so that moving mass would now
be real now. However, such solutions are generally discarded on physical
grounds and overcome such instability criterion, one needs to have m2 > 0.
Tachyon instability is appeared in massive modes, it can appear for scalar
field and spin 2 modes. In f(R) and f(R, G) gravities, the condition of
stability is equivalent to fRR > 0 and fGG > 0, respectively. Dolgov and
Kawasaki [39] explored this instability in R − µ4/R model which becomes
unstable if fRR < 0 and sets the stability condition for viable f(R) models
as fRR > 0.
Thus viable f(R) models require to satisfy the following stability con-
straints
fR(R) > 0, fRR(R) > 0, R≥R0,
where R0 is the the Ricci scalar today. This instability criterion is also
generalized to f(R) gravity involving matter geometry coupling [32]. In [13,
14
14], the authors suggested that the Dolgov-Kawasaki instability in f(R, T )
gravity requires similar sort of constraints as in f(R) gravity and Eq.(1.1.12)
implies additional constraint 1 + fT (R, T ) > 0 for Geff > 0. Thus for
f(R, T ) gravity, we require
fR(R, T ) > 0, 1 + fT (R, T ) > 0, fRR(R, T ) > 0, R≥R0. (1.2.1)
The instability analysis for f(R, T,Q) gravity yields the conditions of Dolgov-
Kawasaki instability and effective gravitational coupling as
3fRR +
(1
2T − T 00
)fQR > 0,
1 + fT + 12RfQ
fR − fQLm
> 0. (1.2.2)
1.3 Thermodynamics
Thermodynamics (a word coined from two Greek words, thermos means
heat and dunamiz means power) is the study of the relationship between
heat and mechanical energy and conversion of one into other [40]. Classical
thermodynamics is restricted to a consideration of macroscopic properties
of the system independent of its constituents. Quantities like pressure, vol-
ume, internal energy, temperature, heat capacity and entropy are discussed
in this branch of thermodynamics. Since a typical thermodynamic system
is composed of an assembly of atoms or molecules, we can surely presume
that its macroscopic behavior can be expressed in terms of the microscopic
properties of its constituent particles. This basic concept provides the foun-
dation for the subject of statistical thermodynamics. Here, we present the
overview of four laws of classical thermodynamics as follows [40].
1.3.1 Zeroth Law
Zeroth law or law of thermal equilibrium is an important principle of ther-
modynamics which provides the operational definition of temperature. It
15
states that “objects in thermal equilibrium with a third object are in ther-
mal equilibrium with each other”. It is based on the fact that systems
in thermal contact are not in complete equilibrium until they have same
temperature.
1.3.2 First Law
First law is more or less based on the principle of energy conservation and
tells that “Entire quantity of energy in a system remains constant but can
change from one form to another”. The first law says that there is a gen-
eralized amount of energy possessed by a thermodynamic system, called its
internal energy U , which can be changed by adding or subtracting energy
of any form and that the algebraic sum of these amounts is equal to the
net, dU , of the internal energy of the system. In thermodynamic process,
the change in a system’s internal energy dU is the difference between the
heat added dQ and the work done by the system dW . The differential form
of this law is
dU = TdS + dQ− dW = TdS + dQ− PdV + JdL + ...
where dQ is the heat added to the system and dW is the work done by the
system. If the system has uniform pressure then a small increase in volume
dV imply that system did the work. If the system is a rubber band having
tension J then it would require a work to be done on it to increase its length
by an amount dL.
1.3.3 Second Law
The second law deals with entropy also recognized as law of increase of en-
tropy. According to this law “For a thermally isolated system, the system’s
16
entire entropy remains constant for reversible process and increases for the
irreversible processes or entropy of an isolated system can never diminish”
i.e., dS > 0.
Entropy S is a state variable which measures the extent of disorder of the
system. The change in entropy dS occurs when a given quantity of energy
is transferred as heat, if heat enters the system its entropy increases, dS is
positive and vice-versa if heat leaves the system. For system interacting in
any way, the change in entropy is
dS = diS + deS,
diS represents the entropy change as a result of modifications occurring
inside the system and deS is produced on account of interaction with the
surroundings. Here, deS = dQ/Tsys, Tsys being the temperature of the
surroundings and dQ is the heat absorbed by the system from surround-
ings. For irreversible process, we have diS > 0 and hence dS > dQ/Tsys.
In fact, natural processes are irreversible and involve spontaneous changes
such as transfer of heat from hot to cold body. For reversible process, en-
tropy depends upon initial and final states of the system and it remains
constant, dS = drQ/Tsys, where the subscript r signifies that the transfer
must be carried out reversibly (without entropy production other than in
the system).
1.3.4 Third Law
It is presented in three different ways: two different Nernst’s statements and
one Planck’s statement. Planck’s statement is more effective from which one
can produce the Nernst’s statements. Walther Nernst (1906) articulated
a principle “As absolute zero is approached, all chemical and/or physical
17
transformations in thermodynamic systems that are in internal equilibrium
occur with zero change in entropy”. In 1912, Nernst gave another argument
(often cited as unattainability statement of third law) according to which
“Temperature cannot be limited to zero in a finite series of steps”. Following
the Nernst’s initial thought, Max Planck hypothesized that “The entropy
of all thermodynamic systems in the state of inner equilibrium tends to zero
as the temperature goes to zero”.
1.4 Laws of BH Dynamics or Thermodynam-
ics
There are two intuitive routes to BH thermodynamics, namely the laws of
BH dynamics and classical thermodynamics. In GR, BHs obey certain laws
which have mathematical resemblance with ordinary laws of thermodynam-
ics. GR describes BHs as massive objects with such a strong gravitational
field that even light cannot escape their surface (the black hole horizon).
Classically, these are perfect absorbers but do not radiate, however, quan-
tum theory predicts that BHs emit particles moving away from the horizon.
In fact, the theory of BH enabled us to develop a relation between gravita-
tion and thermodynamics. We present the overview of laws of BH dynamics
and thermodynamics as follows [41].
1.4.1 Zeroth Law
Zeroth law of BH dynamics suggests that “The surface gravity κ of a sta-
tionary BH is uniform across the horizon”. This property is reminiscent
of zeroth law in classical thermodynamics, according to which temperature
is uniform everywhere in a system in thermal equilibrium. According to
18
Hawking, ~κ/2π is the physical temperature of BH (Th ∝ κ), so that the
constancy of κ on the horizon translates to constancy of temperature be-
tween systems in thermal equilibrium. Thus the temperature of a BH is
constant over the horizon.
1.4.2 First Law
It relates the energy difference of two nearby stationary BH equilibrium
states to the difference in the area of event horizon A in the angular mo-
mentum J and in the charge Q
dM =κ
8πdA + ΩdJ + ΦdQ,
where Ω and Φ denote the angular velocity and electric potential at the
horizon. This relation is for the rotating charged BH. If stationary matter
is present outside the BH then there are additional terms on the right side
of the above result. The term ΩdJ + ΦdQ represents the work done on
the BH by an external agent which increases BH’s angular momentum and
charge by dJ and dQ. This law has striking resemblances with its counter
part in classical thermodynamics, according to which the change in energy
E, entropy S and other state parameters satisfy the following relation
dE = TdS + “workterm”.
Thus the first law of BH dynamics is also the FLT by taking Sh ∝ A and
Th ∝ κ.
1.4.3 Second Law
Hawking proved a remarkable theorem about BHs “In any interaction, the
surface area of a BH can never decrease assuming cosmic censorship and
19
positive energy condition”, i.e., dA ≥ 0. The area law endures a resem-
blance to the second law in classical thermodynamics that entropy in a
closed system can never decrease. The analogy is uniform to the extent
that it follows the first law where entropy of a BH is identified with its
area. The direct translation of area theorem in GR would be that entropy
of BH can never decrease.
1.4.4 Generalized Second Law
We present some arguments related to second law which helps to formulate
the GSL. In classical thermodynamics, it is postulated that entire matter
entropy in cosmos can never decrease, nevertheless some serious trouble
arises with the presence of BH. For a BH, one needs to pay attention to
matter and radiation outside it. As BH accretes matter falls into a singu-
larity, in any case, loss of information occurs which cannot be measured
since events beyond the horizon are not visible to external observer. How-
ever, in this process the entropy of external contents of BH decreases which
is not compensated through any means. Bekenstein proposed BH entropy
as some multiple of BH area measured in units of squared Planck length
L2p = ~G/c3. He defined the generalized entropy S as consisting of BH en-
tropy SBH as well as entropy associated with radiation and matter outside
the BH Sm. Thus the second law is replaced by GSL, i.e., the total entropy
can never decrease
dS = d(SBH + Sm) > 0.
The proposal of GSL was presented prior to the discovery of quantum ef-
fects. In 1974, Hawking presented that all BHs behave as black bodies and
radiate with a thermal spectrum. Hawking radiations emitted by a BH
20
leads to a decrease in horizon area.
Black hole evaporation can be understood as the pair creation in the
gravitational field of a BH, one member of pair is created beneath the hori-
zon while other is created outside the horizon. Hawking radiations carry
away energy resulting in decrease of BH mass. Following the energy conser-
vation principle, there must be a flux of negative energy through the horizon
into BH to balance the outgoing flux of Hawking radiation at infinity. This
can happen only if expectation value of the energy-momentum tensor does
not satisfy NEC, violation of one of the postulates in area theorem. If
energy conservation holds, an isolated BH must lose mass to compensate
the energy flux at infinity. This will evaporate entirely heading towards
decrease in mass and hence the area. Consequently, the area theorem is
violated under the quantum effects.
We have seen that the presence of BH and quantum effects leads to
the violation of second law and area theorem. Initially, when Bekenstein
proposed the GSL, he did not consider the possibility of decrease in area.
According to Bekenstein, loss of matter outside BH is compensated by the
increase in horizon area. Since the quantum effects violate the condition
for applicability of area theorem, one counts this issue as “BH evaporation
is accompanied by a rise in entropy in the surroundings space through the
emitted thermal radiations.” Hawking showed that coefficient of propor-
tionality between BH entropy and A/~G is 1/4 so that SBH = A/4~G. The
GSL thus takes the form “Entire cosmic entropy including that of BH can
never decrease”, i.e., dS = d(Sext+S) > 0, where Sext is the cosmic entropy
excluding BH.
21
1.4.5 Thermal Equilibrium
Thermodynamics does not permit equilibrium when different parts of a
system are at different temperatures. The existence of a state of ther-
modynamic equilibrium and temperature is postulated by the zeroth law
of thermodynamics. In GR, there is no equilibrium state involving BHs.
If a BH is placed in a radiation bath, it continuously absorbs radiations
without ever coming to the equilibrium. Likewise, considering the quan-
tum effects, if there is no matter outside the BH, Hawking radiation is the
only process that changes the state of a stationary BH. If there is matter
or radiation outside the BH, Hawking evaporation is accompanied by the
process of accretion of this matter and radiation onto the BH. It emerges
that a particular matching of parameters of the matter distribution to the
BH parameters produces an equilibrium situation in which the loss of par-
ticles through accretion in each mode is exactly compensated by the BH
radiation in this mode.
1.4.6 Third Law
In thermodynamics, the third law is formulated in variety of ways as pre-
sented in section 1.3.4. The most acceptable statement for third law in BHs
is of the form “It is inconceivable by any mean to reduce the BH temperature
to zero by a finite sequence of operations.”
1.5 Energy Conditions
In GR, matter and energy distribution are defined by the energy-momentum
tensor Tαβ. It is no more universal depending upon particular type of
matter and interactions which you involve in your model. As the cosmos
22
is composed of large number of various matter fields, it would be much
complicated to signify exact Tαβ even if one knows the contribution of each
field and governing dynamical equations. In this case, it is convenient to
impose conditions on Tαβ to limit the arbitrariness so that it represents a
realistic matter source. However, there are certain inequalities which appear
to be physically relevant for Tαβ and adequate to explore the occurrence of
singularities independent of the exact form of Tαβ. Such inequalities are
named as energy conditions which provide certain constraints on energy
density and pressure [25].
We first present these conditions in GR and search a way to express them
in modified theories. The SEC and NEC are originated from geometric
principle namely, Raychaudhuri equation together with the requirement
of attractive gravity. In fact, Raychaudhuri equation plays a key role to
prove singularity theorems and explain the congruence of timelike and null
geodesics. Raychaudhuri’s equation for the congruence of timelike geodesics
is defined as
dθ
dτ= −1
3θ2 − σαβσαβ + ωαβωαβ −Rαβuαuβ, (1.5.1)
where θ denotes the expansion parameter (if θ > 0 then congruence will be
diverging and for θ < 0, it will be converging), σαβ and ωαβ measure the
distortion of volume and rotation of curves linked to the congruence set by
the vector field uα. In case of null geodesics characterized by the vector
field κα, the temporal variation of expansion is given by
dθ
dτ= −1
2θ2 − σαβσαβ + ωαβωαβ −Rαβκακβ. (1.5.2)
It is significant to remark that Raychaudhuri equation is exclusively
geometric and hence develops no deal with any theory of gravity under
23
discussion. Actually, the energy-momentum tensor can have contribution
from different sources and it is convenient to set some constraints to deal
with it on physical grounds. There are certain inequalities which may limit
the arbitrariness in the energy-momentum tensor based on Raychaudhuri
equation with attractiveness property of gravity. The association of Ray-
chaudhuri equation can be set from the fact that the variation of expan-
sion parameter is related to Tαβ if one finds Rαβ from the respective field
equations. Hence, one can develop the physical constraints on the energy-
momentum tensor through the connection between Raychaudhuri equation
and the field equations.
As σαβσαβ > 0 (shear tensor is purely spatial), so the condition of
attractive gravity ( dθdτ
< 0) along with hypersurface orthogonal (ωαβ = 0)
congruence of timelike and null geodesics, takes the form
SEC : Rαβuαuβ > 0, NEC : Rαβκακβ > 0. (1.5.3)
One can use the field equations to relate Rαβ to the energy-momentum ten-
sor Tαβ. Thus, the connection between Raychaudhuri and Einstein equa-
tions can set the physical conditions for Tαβ. In the framework of GR, the
conditions (1.5.3) can be written as
Rαβuαuβ =
(Tαβ − T
2gαβ
)uαuβ > 0, Rαβκακβ = Tαβκακβ > 0. (1.5.4)
If the matter part is considered as perfect fluid then these conditions reduce
to the most familiar form of strong and null energy conditions in GR as
ρ + 3p > 0 and ρ + p > 0.
The WEC represents the physically reasonable requirement that for any
matter contribution, the energy density must be non-negative as measured
by observer, i.e., Tαβuαuβ > 0 for all timelike vector, or equivalently that
24
ρ > 0 and ρ + p > 0. The DEC includes WEC as well as the requirement
that Tαβuα is a non-spacelike vector. It may be interpreted as for any
observer, energy density must be non-negative and local energy flow vector
is timelike or null. In terms of components of Tαβ, it implies that ρ > 0 and
ρ±p > 0. Thus the DEC is the WEC with the additional requirement that
pressure should not exceed the energy density.
In modified theories, one can employ an approach analogous to that in
GR and define the effective energy-momentum tensor so that the conditions
in Raychaudhuri equations are represented as
(T eff
αβ − T eff
2gαβ
)uαuβ > 0, T eff
αβ κακβ > 0. (1.5.5)
In determining the WEC and DEC, the modified form of these conditions
in GR can be used under the transformations ρ → ρeff and p → peff . Thus
the WEC and DEC are obtained as
WEC : ρeff > 0 ρeff+peff > 0,
DEC : ρeff > 0 ρeff±peff > 0. (1.5.6)
Since the Raychaudhuri equation is a geometrical principle which agrees to
any MGT, one can keep the physical motivation of focussing of geodesic
congruences along with attractive nature of gravity to formulate the energy
constraints in modified theories.
1.6 Anisotropic Cosmologies
Despite the success of FRW model, the concept of inhomogeneous and
anisotropic cosmos cannot be neglected at least on certain scales and to
a certain range. In this perspective, the candidates having more degrees of
25
freedom than FRW can be useful to investigate the cosmic evolution. These
models can represent the anisotropic modes, including rotation and global
magnetic field. Bianchi models are spatially homogeneous but not necessar-
ily isotropic. A spacetime is said to be spatially homogeneous if there exists
a one-parameter set of spacelike hypersurfaces foliating the spacetime such
that given any two points p and q there is an isometry that takes p into q.
A 4-dimensional manifold M with metric tensor is called a Bianchi
cosmology model if it involves a 3-dimensional group of isometries acting on
spacelike hypersurfaces (i.e., any point on one of these surfaces is equivalent
to any other point on the same surface) named as surfaces of homogeneity
[42]. The classification is based on commutation laws of Killing vector fields
which gives the basic identity
[ξα, ξβ] = Cµαβξµ,
where Cµαβ are called structure constants. Cµ
αβ can be decomposed in terms
of symmetric contravariant tensor nαβ = diag(n1, n2, n3) and covariant vec-
tor aα = (a, 0, 0) (satisfying the condition nαβaα = 0) as
Cµαβ = εαβγn
µγ + aαδµβ − aβδµ
α, (1.6.1)
εαβγ is the antisymmetric tensor and δµβ is the Kronecker delta.
One can define Bianchi models into two classes A and B according to
aα is or not zero. In defining the class B, one may introduce a scalar h
which satisfies the relation a2 = hn2n3. The classification of Bianchi types
is shown in Table 1.1 indicating that h < 0 in type V Ih and h > 0 in type
V IIh. Bianchi groups allow higher symmetry subcases such as isotropic or
locally rotationally symmetric (LRS) models. The FRW models appear as
a limited subclass of Bianchi models because of their isotropy. The Bianchi
26
Table 1.1: Classification of Bianchi models into two groups and ten types.Bianchi type III ia specific case of V Ih with h = −1.
Class Type a n1 n2 n3 FRW as specific case
A I 0 0 0 0 κ = 0
II 0 + 0 0 -
VI0 0 0 + − -
VII0 0 0 + + κ = 0
VIII 0 − + + -
IX 0 + + + κ = +1
B V + 0 0 0 κ = −1
IV + 0 0 + -
VIh + 0 + − -
VIIh + 0 + + κ = −1
type I model is the generalization of the flat FRW and is given by
ds2 = dt2 − A2(t)dx2 −B2(t)dy2 − C2(t)dz2, (1.6.2)
The scale factors A, B and C in different directions are allowed to vary
independently of each other.
1.7 Cosmological Parameters
In this section, we discuss different parameters that can be used to explore
the cosmic expansion history and its future evolution.
1.7.1 Hubble’s Law and Hubble Parameter
American astronomer Slipher measured shifts in the spectra of more than
20 galaxies between 1912 and 1925. He found that almost all the galaxies
showed red shifts. Later, in 1929, Hubble noticed that our Universe is
27
expanding with the passage of time distant galaxies are moving away from
each other [43]. He determined the distances for a number of galaxies and
found that galaxies at larger distances also showed larger red shifts. He
constructed a linear relation between distances of galaxies from the Earth
and recessional velocity as determined by the red shifts. It can be stated
as [43] v = cz = HD, where v is the recessional velocity, H is the Hubble
constant and D is the distance from the Earth to the galaxy and z is its
redshift. This relation is called Hubble’s law. The Hubble constant or more
appropriately Hubble parameter, since it depends on time, is defined as
H =a(t)
a(t), (1.7.1)
where a(t) is the scale factor which represents cosmic expansion and dot
indicates differentiation with respect to time. a(t) is an increasing function
of time in an expanding cosmos and would be zero at the time of big-bang.
Hubble parameter represents the expansion rate that changes with time.
1.7.2 Mean and Directional Hubble Parameters
Isotropic expansion rate is specified by the Hubble parameter H as given
in Eq.(1.7.1) but in case of anisotropic expansion, we use the mean Hubble
parameter. This is the average of H in each spatial direction. If the value of
Hubble parameter varies in each spatial direction with the passage of time
then the mean Hubble parameter can be defined as
H =1
3(lnV ) = (lna) =
1
3(Hx + Hy + Hz) , (1.7.2)
where Hx, Hy and Hz represent the expansion rate with time in x, y and
z axes, respectively and known as directional Hubble parameters.
28
1.7.3 Anisotropy Parameter of Expansion
The anisotropy parameter of expansion is characterized by the mean and
directional Hubble parameters defined as
∆ =1
3
3∑i=1
(Hi −H
H
)2
, (1.7.3)
which can be represented in the form of expansion and shear scalars as
∆ = 6(σ
θ
)2
. (1.7.4)
The anisotropy of expansion results in isotropic cosmic expansion in the
limit of ∆ −→ 0.
1.7.4 Deceleration Parameter
The deceleration parameter q measures the deceleration of cosmic expansion
and is defined in terms of scale factor a(t) as well as its derivatives as
q = −a(t)a(t)
a2(t). (1.7.5)
Current observational data provided conclusive evidence for cosmic deceler-
ation that preceded the present epoch of cosmic acceleration. q can explain
the transition from past deceleration to the present accelerating epoch. The
sign of deceleration parameter indicates whether the cosmic expansion is ac-
celerating or decelerating. A positive value of q corresponds to deceleration
while the negative value indicates the accelerating behavior of cosmos. The
deceleration parameter can be expressed in the form of H as follows
q =d
dt
(1
H
)− 1 = −
(H2 + H
H2
). (1.7.6)
29
1.8 The Expansion and Shear Scalar
Let O be an open region in spacetime. A congruence in O is a family of
curves such that through each point in O there passes only one curve from
this family. Congruences generated by timelike, null and spacelike curves
are called timelike, null and spacelike congruences, respectively. Consider
the congruence of timelike geodesics (each curve in the family is a timelike
geodesic) and associated timelike vector field uα.
The expansion scalar measures the fractional rate of change of volume
per unit time and is defined as [24]
θ = uα;α = uα
,α + Γααβuβ. (1.8.1)
For θ > 0, congruence will be diverging (geodesic flying apart) which shows
that the Universe is expanding whereas for θ < 0, congruence will be con-
verging (geodesics coming closer) representing the decelerating behavior of
the Universe. The shear tensor measures the distortion in timelike curves
keeping the volume constant. It represents the possibility of initial sphere
of geodesics to become distorted into an ellipsoidal shape. It can be written
as
σαβ = θαβ − 1
3θhαβ = u(α;β) − u(αuβ) − 1
3θhαβ. (1.8.2)
The shear tensor is symmetric in its indices and shear scalar σ is given by
σ2 =1
2σαβσαβ. (1.8.3)
Chapter 2
Thermodynamics Laws inf (R, T ) and f (R, T,Q) ModifiedTheories
In this chapter, we explore the thermodynamic properties at the AH of
FRW cosmos in MGTs involving matter geometry coupling. Eling et al.
[20] suggested that non-equilibrium picture of thermodynamics is required
in non-linear MGTs such as f(R) and scalar-tensor theories. However, it has
been shown that equilibrium thermodynamics can be achieved in f(R) and
f(T ) theories by incorporating the curvature/torsion contribution terms
to the effective energy-momentum tensor [21, 23]. In our discussion, we
consider the non-equilibrium description of thermodynamics to establish
the first and second laws in f(R, T ) and f(R, T,Q) theories. We take two
forms of the energy-momentum tensor of dark components in f(R, T ) grav-
ity and demonstrate that equilibrium description of thermodynamics is not
achievable in such kind of theories. Therefore, we opt the non-equilibrium
approach and show that the field equations for these theories can be ex-
pressed in the form of FLT, ThSeff = δQ, where Seff contains contribu-
tions both from horizon entropy and an additional component introduced
due to the non-equilibrium description. The validity of GSLT is also tested
30
31
in these circumstances.
The chapter is organized in the following format. In section 2.1, we
formulate dynamical equations in f(R, T ) gravity and investigate the va-
lidity of first and second laws in this theory. Section 2.2 redefines the
contributions from exotic components in f(R, T ) gravity and explores the
thermodynamic properties in this context. Section 2.3 is devoted to the
FLT and GSLT in f(R, T,Q) and also restricts the specific forms of La-
grangian (1.1.15) for the validity of GSLT. The results presented in this
chapter have been published in [44, 45].
2.1 Thermodynamics in f (R, T ) Gravity
In this section, we first present the general formulation of dynamical equa-
tions in f(R, T ) gravity. The action of f(R, T ) gravity is given by Eq.(1.1.3)
whose variation with respect to the metric tensor yields the field equations
(1.1.6) that depend upon the source term and each choice of Lm results in
particular set of equations.
We consider the matter part as perfect fluid with Lm = −p. Conse-
quently, the field equations can be rewritten as effective Einstein equations
(1.1.11). The corresponding field equations for FRW model are
3
(H2 +
k
a2
)= 8πGeffρm +
1
fR
[1
2(RfR − f)− 3H(RfRR
+ T fRT )], (2.1.1)
−(
2H + 3H2 +k
a2
)=
1
fR
[1
2(f −RfR) + 2H(RfRR + T fRT ) + RfRR
+ R2fRRR + 2RT fRRT + T fRT + T 2fRTT
].(2.1.2)
32
These can be rewritten as
3
(H2 +
k
a2
)= 8πGeff (ρm + ρDC), (2.1.3)
−2
(H − k
a2
)= 8πGeff (ρm + ρDC + pDC), (2.1.4)
where we have assumed pressureless matter and ρDC , pDC are the energy
density and pressure of dark components
ρDC =1
8πGF[1
2(RfR − f)− 3H(RfRR + T fRT )
], (2.1.5)
pDC =1
8πGF[−1
2(RfR − f) + 2H(RfRR + T fRT ) + RfRR + R2fRRR
+ 2RT fRRT + T fRT + T 2fRTT
]. (2.1.6)
Here F = 1 + fT (R,T )8πG
. The EoS parameter of dark fluid ωDC is given by
(pDC = ωDCρDC)
ωDC = −1 + RfRR + R2fRRR + 2RT fRRT + T fRT + T 2fRTT −H(RfRR
+ T fRT )/1
2(RfR − f)− 3H(RfRR + T fRT ). (2.1.7)
The ordinary matter continuity equation involving interaction term is
of the form
ρ + 3Hρ = q. (2.1.8)
Assuming that TDCαβ behaves as perfect fluid which satisfies the following
equations
ρDC + 3H(ρDC + pDC) = qDC , (2.1.9)
ρtot + 3H(ρtot + ptot) = qtot, (2.1.10)
where ρtot = ρm+ρDC , ptot = pDC and qtot = q+qDC denote the entire energy
transfer term and qDC is the energy transfer of the fluid generated from the
33
modification to gravity. Replacing Eqs.(2.1.3) and (2.1.4) in (2.1.10), it
follows that
qtot =3
8πG(H2 +
k
a2)∂t
(fR
F)
. (2.1.11)
Clearly, this reduces to the energy transfer relation for f(R) theory if F = 1.
If the effective gravitational coupling is purely a constant, we obtain qtot = 0
implying the standard conservation law in GR.
Now, we investigate the validity of first and second laws of thermody-
namics at the AH of FRW universe.
2.1.1 First Law of Thermodynamics
The relation hαβ∂αr∂β r = 0 implies the radius of dynamical AH and for
FRW geometry it becomes
rA =
(H2 +
k
a2
)−1/2
, (2.1.12)
yielding the Hubble horizon rA = 1/H for flat case. Differentiating this
equation with respect to cosmic time, it follows that
1
Hr3A
drA
dt=
(H − k
a2
). (2.1.13)
The temperature associated with the AH is defined as [19]
Th =|κsg|2π
, (2.1.14)
where
κsg =1
2√−h
∂α(√−hhαβ∂β rA) = − 1
rA
(1−
˙rA
2HrA
)
= − rA
2
(2H2 + H +
k
a2
). (2.1.15)
34
is the surface gravity. The Bekenstein-Hawking relation Sh = A/4G [17, 18]
defines the horizon entropy in GR, while in MGTs Wald [46] suggested that
horizon entropy is associated with Noether charge and in f(R) theory it is
defined as Sh = AfR/4G. Bamba et al. [21] remarked that this entropy
relation is analogous in both metric and Palatini formalisms of f(R) gravity.
Brustein et al. [47] showed that Wald’s entropy can be represented in terms
of effective gravitational coupling as Sh = A/4Geff . Thus, one can define
the horizon entropy in f(R, T ) gravity as
Sh =AfR
4GF . (2.1.16)
This implies the corresponding results in GR and f(R) gravity depending
upon the variation of f . Employing Eqs.(2.1.13) and (2.1.16), we get
1
2πrA
dSh = 4πr3A(ρtot + ptot)Hdt +
rA
2GF dfR +rAfR
2Gd
(1
F)
. (2.1.17)
Multiplying (1− ˙rA/2HrA) on both sides of the above equation, it follows
that
ThdSh = 4πr3A(ρtot + ptot)Hdt− 2πr2
A(ρtot + ptot)drA +πr2
AThdfR
GF+
πr2AThfR
Gd
(1
F)
. (2.1.18)
In GR, the Misner-Sharp energy is defined as [48] E = rA
2Gwhich can
be extended to the form E = rA
2Geffin MGTs [49]. In terms of volume
V = 43πr3
A, we have
E =3V
8πGeff
(H2 +
k
a2
)= V ρtot, (2.1.19)
which represents the matter energy inside the sphere of radius rA. For
E > 0, one needs to set the positive effective gravitational coupling so that
35
Geff = GFfR
> 0. It follows from Eqs.(2.1.3) and (2.1.19) that
dE = −4πr3A(ρtot+ptot)Hdt+4πr2
AρtotdrA+rAdfR
2GF +rAfR
2Gd
(1
F)
. (2.1.20)
Substituting Eq.(2.1.20) in (2.1.18), we have
ThdSh = −dE + WdV +(1 + 2πrATh)rAdfR
2GF +(1 + 2πrATh)rAfR
2G
× d
(1
F)
, (2.1.21)
where W = −12T (tot)αβhαβ = 1
2(ρtot − ptot) is the work density [50]. Thus
FLT in f(R, T ) gravity can be represented as
ThdSh + ThdSh = −dE + WdV, (2.1.22)
where
dSh = − rA
2GTh
(1 + 2πrATh)d
(fR
F)
= −F(E + ShTh)
ThfR
d
(fR
F)
, (2.1.23)
is the entropy production term developed due to the non-equilibrium set-
tings in this theory as compared to GR, Gauss-Bonnet, braneworld and
Lovelock gravities [51]-[54]. dSh marks to the non-equilibrium represen-
tation of thermodynamics resulting from the effects of matter geometry
coupling. The FLT in f(R) gravity [55] and its traditional form in GR can
be recovered for f(R, T ) = f(R) and f(R, T ) = R, respectively.
2.1.2 Generalized Second Law of Thermodynamics
In literature, it has been shown that GSLT can be met in the framework
of MGTs [21, 23, 57]-[59]. It would be interesting to test the validity of
GSLT in f(R, T ) gravity. This states that the sum of entropy associated
with horizon and that of matter fluid components inside the horizon is not
36
decreasing with time. Thus one needs to show that [57]
Sh + dSh + Sin ≥ 0, (2.1.24)
where Sh is the entropy associated with AH in f(R, T ) gravity, dSh =
∂t(dSh) and Sin is the entropy of entire matter and energy sources within
horizon. The Gibb’s equation relating the entropy Sin and temperature Tin
of matter and energy sources within the horizon to the density and pressure
is given by [60]
TindSin = d(ρtotV ) + ptotdV. (2.1.25)
The temperature of matter and energy sources within the horizon is as-
sumed in proportion to the temperature of AH [56, 57, 60], i.e., Tin = bTh,
where 0 < b < 1 to ensure Tin > 0 and smaller than Th. In fact, it is natural
to assume that such proportionality relation between the temperatures of
AH and entire contents inside the horizon which results in local equilibrium
by setting the proportionality constant b as unity. In general, the horizon
temperature does not match to that of fluid components within the horizon
which makes the spontaneous flow of energy between the horizon and fluid
contents so that local thermal equilibrium is no longer preserved [60]. Fur-
thermore, mutual matter curvature coupling in these modified theories may
also play its role in energy flow and systems must experience interaction for
some period of time ahead of achieving the thermal equilibrium.
Substituting Eqs.(2.1.22) and (2.1.25) in Eq.(2.1.24), we obtain
Sh + dSh + Sin =24πΞ
rAbR≥ 0, (2.1.26)
where
Ξ = (1− b)ρtotV + (1− b
2)(ρtot + ptot)V
37
is the comprehensive constraint to meet the GSLT in MGTs [57]. Using
Eqs.(2.1.3) and (2.1.4), condition (2.1.26) becomes
12πXbRGF(H2 + k
a2 )2≥ 0, (2.1.27)
where
X = 2(1− b)H
(H − k
a2
)(H2 +
k
a2
)fR + (2− b)H
(H − k
a2
)2
fR
+ (1− b)
(H2 +
k
a2
)2
F∂t
(fR
F)
.
Therefore, the constraint to meet the GSLT is equivalent to X ≥ 0. For
flat FRW geometry, the validity of GSLT requires the conditions ∂t(fR
F ) ≥0, H > 0 and H ≥ 0. Also, F and fR are positive in order to keep E > 0.
If b = 1, i.e., temperature on either side of horizon boundary stays identical
then validity of GSLT requires
J =
(H − k
a2
)2fR
F ≥ 0. (2.1.28)
The effective EoS is given by ωeff = −1 − 2(H − ka2 )/3(H2 + k
a2 ), where
H < ka2 represents the quintessence era and H > k
a2 constitutes the phantom
regime of cosmos. Thus GSLT in f(R, T ) gravity is met in both phantom
and non-phantom eras. The validity of GSLT has also been established in
f(R) and f(T ) theories [21, 23].
2.2 Redefining the Dark Components
In the previous section, it has been found that an additional entropy term
dSh is raised in thermodynamic laws which can be regarded as the conse-
quence of non-equilibrium statement of the field equations. One can specify
the components of dark fluid so that resulting equations eliminate auxiliary
38
entropy element. Such approach is classified as an equilibrium treatment
which has been developed in MGTs [21, 23], where it is shown that one can
get rid of additional entropy production term.
We would like to explore the existence of equilibrium description in
f(R, T ) gravity. As a matter of fact, following [21, 23] we may be able to
limit the entropy production term but it cannot be eliminated completely
in this theory. We redefine the components of dark fluid so that the field
equations (2.1.3) and (2.1.4) can be rearranged with Geff =(G + fT (R,T )
8π
)
and
ρDC =1
8πGF[1
2(RfR − f)− 3H(RfRR + T fRT ) + 3(1− fR)(H2
+k
a2)
], (2.2.1)
pDC =1
8πGF[−1
2(RfR − f) + 2H(RfRR + T fRT ) + RfRR + R2fRRR
+ 2RT fRRT + T fRT + T 2fRTT − (1− fR)(2H + 3H2 +k
a2)
],(2.2.2)
being the energy density and pressure of redefined dark fluid. The EoS
parameter ωDC in this description becomes
ωDC = −1 + RfRR + R2fRRR + 2RT fRRT + T fRT + T 2fRTT −H(RfRR
+ T fRT )− 2(1− fR)(H − k
a2)/1
2(RfR − f)− 3H(RfRR + T fRT )
+ 3(1− fR)(H2 +k
a2). (2.2.3)
It is evident from Eqs.(2.1.7) and (2.2.3) that the EoS parameter is not
unique in both cases. Thus one should regard the two formulations of dark
fluid in discussions on cosmic issues.
The entire energy exchange term for this case turns out to be
qtot =3
8πG(H2 +
k
a2)∂t
(1
F)
. (2.2.4)
39
Since in general ∂t(fT (R, T )) 6= 0 in this theory, so qtot is non-zero. However,
it may disappear in some specific cases which involve the linear dependence
on T such as f(R, T ) = f(R)+cT and results would be very similar to that
in f(R) gravity. Therefore, we may not be able to develop the equilibrium
picture of thermodynamics in f(R, T ) gravity. Consequently, we again need
to consider the non-equilibrium treatment of thermodynamics. This result
differentiates f(R, T ) gravity from other MGTs due to the matter depen-
dence of the Lagrangian density. In f(R) and f(T ) theories, the redefinition
of dark fluid results in local conservation of the energy-momentum tensor
of dark components [21, 23].
Now we explore the validity of the first and second laws of thermody-
namics in this setting.
2.2.1 First Law of Thermodynamics
In this particular representation of dark fluid, the time derivative of radius
rA at the AH is given by
drA = 4πr3AGF(ρtot + ptot)Hdt. (2.2.5)
In f(R, T ) gravity, the equilibrium description is not executable as com-
pared to other MGTs such as f(R), f(T ) and scalar-tensor theories etc.
Thus, we employ the Wald entropy relation Sh = A/(4Geff ) rather than
introducing Bekenstein-Hawking entropy. For this case, the differential of
horizon entropy is given by
1
2πrA
dSh = 4πr3A(ρtot + ptot)Hdt +
rA
2Gd
(1
F)
. (2.2.6)
40
The evolution of entropy can be represented in terms of temperature as
ThdSh = 4πr3A(ρtot + ptot)Hdt− 2πr2
A(ρtot + ptot)drA +πr2
ATh
G
×(
1
F)
. (2.2.7)
Introducing the Misner-Sharp energy
E =rA
2GF = V ρtot, (2.2.8)
we obtain
dE = −4πr3A(ρtot + ptot)Hdt + 4πr2
AρtotdrA +rA
2Gd
(1
F)
. (2.2.9)
Combining Eqs.(2.2.7) and (2.2.9), it gives the FLT
ThdSh + ThdSh = −dE + WdV, (2.2.10)
where
dSh = − rA
2ThG(1 + 2πrATh)d
(1
F)
= −F(
E
Th
+ Sh
)d
(1
F)
= − π(4H2 + H + 3k/a2)
G(H2 + k/a2)(2H2 + H + k/a2)d
(1
F)
, (2.2.11)
is an additional entropy term produced due to matter contents of the cos-
mos. It involves derivative of f(R, T ) with respect to T . Notice that the
FLT, ThdSh = −dE + WdV holds at the AH of FRW universe in equilib-
rium description of MGTs [21, 23, 56]. However, in f(R, T ) gravity, this
law does not hold due to the presence of an additional term dSh. This
term vanishes if we take f(R, T ) = f(R) which leads to the equilibrium
description of thermodynamics in f(R) gravity.
41
2.2.2 Generalized Second Law of Thermodynamics
To formulate the GSLT in this description of f(R, T ) gravity, we consider
the Gibbs equation in terms of all matter fields and energy contents
TindSin = d(ρtotV ) + pindV, (2.2.12)
where Tin and Sin denote the temperature and entropy of all the matter
and energy sources within the horizon. For this representation, the GSLT
can be represented as
˙Sh + d
˙Sh +
˙Stot ≥ 0, (2.2.13)
which implies that
12πYbRGF(H2 + k
a2 )2≥ 0, (2.2.14)
where
Y = 2(1− b)H
(H − k
a2
)(H2 +
k
a2
)+ (2− b)H
(H − k
a2
)2
+ (1− b)
(H2 +
k
a2
)2
F∂t
(1
F)
.
Thus the GSLT is met only if Y ≥ 0. For flat FRW geometry, the GSLT
is satisfied with the constraints ∂t(1F ) ≥ 0, H > 0 and H ≥ 0. In thermal
equilibrium b = 1, the above constraint is reduced to
B =12πH
(H − k
a2
)2
G(H2 + k
a2
)2R
1
F ≥ 0, (2.2.15)
for V = 43πr3
A and R = 6(H + 2H2 + k/a2). B ≥ 0 clearly holds when
the Hubble parameter and scalar curvature have the same signatures. It
can be seen that significant conflict of results of f(R, T ) with f(R) gravity
is the factor F = 1 + fT (R,T )8πG
. We remark that in both definitions of dark
components, the GSLT is met both in phantom and non-phantom cosmic
eras.
42
2.3 Thermodynamics in f (R, T,Q) Gravity
In this section, we consider more general modified gravity involving the
dependence on matter energy-momentum tensor whose action is presented
in Eq.(1.1.15). The variation of action (1.1.15) with respect to metric im-
plies field equations in f(R, T,Q) gravity which are shown in Eqs.(1.1.16)-
(1.1.18). For perfect fluid as the matter energy-momentum tensor the (0−0)
and (i− i) components of TDCαβ can be determined as
T(DC)00 =
1
fR − fQLm
[1
2(f −RfR)− LmfT − 3H∂tfR +
3
2
(H2 + 3H
)
× ρfQ − 3
2
(3H2 + H
)pfQ +
3
2H∂t [(p− ρ)fQ] +
3
2Hρ∂tfQ
+1
2ρ∂ttfQ
], (2.3.1)
T(DC)ii =
1
fR − fQLm
[1
2(RfR − f) + LmfT +
1
2
(H + 3H2
)ρfQ
+1
2
(5(3H2 + H) +
8κ
a2
)pfQ + 2H∂t fR + (ρ + p) fQ
+ ∂tt
fR +
1
2(ρ− p)fQ
+
3
2Hp∂tfQ +
1
2p∂ttfQ
]gii, (2.3.2)
where ρ, p represent the energy density and pressure of matter fluid. Using
Eqs.(2.3.1) and (2.3.2), the field equations in f(R, T,Q) gravity can be
rearranged as
−2
(H − k
a2
)= 8πGeff (ρ + p) +
1
fR − fQLm
[(3H2 + 5H)ρfQ
+ (3H2 + H +4κ
a2)pfQ −H∂tfR − 1
2(ρ + 7p)fQ
+3
2H(ρ + p)∂tfQ + ∂ttfR +
1
2(ρ− p)fQ+
1
2(ρ + p)
× ∂ttfQ] . (2.3.3)
The covariant divergence of the energy-momentum tensor can be set by
43
taking the divergence of the field equation (1.1.16), yielding
∇µTµν =2
2(1 + fT ) + RfQ
[∇µ(fQRαµTαν) +∇ν(LmfT )− 1
2(fQRσζ
+ fT gσζ)∇νTσζ −Gµν∇µ(fQLm)− 1
2[∇µ(RfQ) + 2∇µfT ] Tµν
].
(2.3.4)
For FRW spacetime, the matter energy density ρ satisfies
ρ + 3H(ρ + p) =1
2 + 3(fT − (H + 3H2)fQ)
[3
H(H − 3H2)(ρ + p)
− (H + 2H2)∂tp− 2H2∂tLm
fQ + 6H2(ρ− Lm)∂tfQ + fT ∂t(2Lm + 3p)
+ 2(Lm − ρ)∂tfT ] . (2.3.5)
It is significant to see that ideal continuity equation does not agree in this
modified theory which is also true in other modified theories involving non-
minimal matter geometry coupling [12, 13, 14]. This equation represents
the standard continuity equation when the left side vanishes which is only
possible if Lagrangian has null variation with respect to T and Q.
Now, we analyze thermodynamic properties in f(R, T,Q) gravity and
discuss the first and second laws of thermodynamics at the AH of FRW
universe. In section 2.2, it is shown that equilibrium description may not
be achievable in f(R, T ) gravity. As this theory is more general to f(R, T )
gravity, therefore we adopt the non-equilibrium description of thermody-
namics.
44
2.3.1 First Law of Thermodynamics
Let us proceed to establish FLT in the above modified theory. Substituting
Eq.(2.3.3) in (2.1.13) and multiplying with the factor 4πrA, we obtain
1
2πrA
16π2rA(fR − fQLm)drA
κ2 + fT + 12(R−2)fQ
=4πr3
AH
κ2 + fT + 12(R−2)fQ
[(3H2 + 5H)
× ρfQ + (3H2 + H +4κ
a2)pfQ −H∂tfR − 1
2(ρ + 7p)fQ+
3
2H(ρ + p)
× ∂tfQ + ∂ttfR +1
2(ρ− p)fQ+
1
2(ρ + p)∂ttfQ
]dt + 4πr3
AH(ρ + p)dt.
(2.3.6)
We define the following differential
d
(8πA(fR − fQLm)
4(κ2 + fT + 12(R−2)fQ)
)=
8πrA
κ2 + fT + 12(R−2)fQ)
[2π(fR
− fQLm)drA + rA
∂t(fR − fQLm)− (fR − fQLm)∂t
(ln[κ2 + fT
− 1
2(R−2)fQ]
)dt
],
so that Eq.(2.3.6) is modified to the form
1
2πrA
d
(8πA(fR − fQLm)
4(κ2 + fT + 12(R−2)fQ)
)= 4πr3
AH[(ρ + p) +
(3H2
+ 5H)ρfQ + (3H2 + H +4κ
a2)pfQ −H∂tfR − 1
2(ρ + 7p)fQ+
1
2(ρ + p)
× (3H∂tfQ + ∂ttfQ) + ∂ttfR +1
2(ρ− p)fQ
/
(κ2 + fT +
1
2(R−2)
× fQ)] dt +rA
2
8π
κ2 + fT + 12(R−2)fQ
∂t(fR − fQLm)− (fR − fQLm)
× ∂t
(ln[κ2 + fT − 1
2(R−2)fQ]
)dt. (2.3.7)
Moreover, if we multiply the above equation by the term (1− ˙rA/2HrA),
45
one can obtain
|κsg|2π
d
(8πA(fR − fQLm)
4(κ2 + fT + 12(R−2)fQ)
)=
(1−
˙rA
2HrA
)4πr3
AH [(ρ + p)
+
(3H2 + 5H)ρfQ + (3H2 + H +
4κ
a2)pfQ −H∂tfR − 1
2(ρ + 7p)fQ
+1
2(ρ + p)(3H∂tfQ + ∂ttfQ) + ∂ttfR +
1
2(ρ− p)fQ
/
(κ2 + fT +
1
2
× (R−2)fQ)] dt +
(1−
˙rA
2HrA
)rA
2
8π
κ2 + fT + 12(R−2)fQ
∂t(fR
− fQLm)− (fR − fQLm)∂t
(ln[κ2 + fT − 1
2(R−2)fQ]
)dt, (2.3.8)
where κsg = 1rA
(1 − ˙rA
2HrA) is the surface gravity and |κsg|
2πis identified as
temperature of AH. The term inside the differential on left side of the
above equation is the entropy [55, 61], S = 8πA(fR−fQLm)
4(κ2+fT + 12(R−2)fQ)
|rAof AH
in f(R, T,Q) gravity. The entropy relation in f(R, T,Q) gravity repro-
duces the corresponding results in f(R) [55, 21] and f(R, T ) [44] theories.
Consequently, Eq.(2.3.8) can be rewritten as
ThdS = 4πr3AH(ρ + p)dt− 2πr2
A(ρ + p)drA +2πAHr2
A
κ2 + fT + 12(R−2)fQ
× T
(3H2 + 5H)ρfQ + (3H2 + H +
4κ
a2)pfQ −H∂tfR − 1
2(ρ
+ 7p)fQ+1
2(ρ + p)(3H∂tfQ + ∂ttfQ) + ∂ttfR +
1
2(ρ− p)fQ
dt
+TA
4
8π
κ2 + fT + 12(R−2)fQ
∂t(fR − fQLm)− (fR − fQLm)
× ∂t
(ln[κ2 + fT − 1
2(R−2)fQ]
)dt. (2.3.9)
Taking differential of energy relation E = V ρ, we get
dE = 4πr2AρdrA +
4
3πr3
Aρdt. (2.3.10)
Using the expression of time derivative of energy density (2.3.5), the
46
above result takes the following form
dE = 4πr2AρdrA − 4πr3
A(ρ + p)Hdt +V
2 + 3(fT − (H + 3H2)fQ))[3 H
× (H − 3H2)(ρ + p)− (H + 2H2)∂tp− 2H2∂tLm
fQ + 6H2(ρ− Lm)
× ∂tfQ + fT ∂t(2Lm + 3p) + 2(Lm − ρ)∂tfT ] dt. (2.3.11)
Incorporating dE in Eq.(2.3.9), it follows that
ThdSh = −dE +1
2(ρ− p)d(4/3πr3
A) +TA
4[G + (fT + 12(R−2)fQ)/8π]
×[Hr2
A
(3H2 + 5H)ρ + (3H2 + H +
4κ
a2)p
fQ + (1−H2r2
A)∂tfR +1
2
× H2r2A∂t[(ρ + p)fQ] +
3
2H2r2
A(ρ + p)∂tfQ − ∂t(fQLm) + Hr2A ∂tt (fR
+1
2(ρ− p)fQ
)+
1
2(ρ + p)∂ttfQ
− (fR − fQLm)∂t(ln[κ2 + fT +
1
2(R
+ 2)fQ])] dt +2πV
|κ|[2 + 3(fT − (H + 3H2)fQ)]
3
H(H − 3H3)(ρ + p)
− (H + 2H2)∂tp− 2H2∂tLm
fQ + fT ∂t(2Lm + 3p) + 2(ρ− Lm)(3H2
× ∂tfQ − ∂tfT ) dt. (2.3.12)
As the work density is defined as W = 12(ρ − p), so the above equation
implies FLT in this modified theory as follows
ThdSh + ThdSh = −dE + WdV, (2.3.13)
where
dSh =−TA
4[G + (fT + 12(R−2)fQ)/8π]
[Hr2
A
(3H2 + 5H)ρ + (3H2
+ H +4κ
a2)p
fQ + (1−H2r2
A)∂tfR +1
2H2r2
A∂t[(ρ + 7p)fQ] +3
2H2r2
A(ρ
+ p)∂tfQ − ∂t(fQLm) + H2r2A
∂tt
(fR +
1
2(ρ− p)fQ
)+
1
2(ρ + p)∂ttfQ
− (fR − fQLm)∂t(ln[κ2 + fT +1
2(R + 2)fQ])
]dt+
47
2πV
3
H(H − 3H2)(ρ + p)− (H + 2H2)∂tp− 2H2∂tLm
fQ
+ fT ∂t(2Lm + 3p) + 2(ρ− Lm)(3H2∂tfQ − ∂tfT )
dt/|κ|[2 + 3(fT
− (H + 3H2)fQ)],
is the entropy production term developed in this modified theory. Such
additional term marks to the non-equilibrium treatment of thermodynamics
and is produced internally due to matter curvature coupling.
In gravitational theories such as Einstein, Gauss-Bonnet, braneworld
and Lovelock gravities, the usual FLT is satisfied by the respective field
equations. In fact, these theories do not involve any surplus term in univer-
sal form of FLT, i.e., TdS = −dE +WdV . It is worth mentioning here that
FLT for non-equilibrium treatment in f(R, T ) gravity can be retrieved from
this result if Lagrangian (1.1.15) has null variation with respect to Q so that
f(R, T,Q) = f(R, T ). One can also obtain FLT exclusively in f(R) theory
by setting f(R, T,Q) = f(R) which is similar to that in [55] for (3+1)-
dimensional FRW spacetime while it does correspond to GR if fT = fQ = 0
and fR = 1 so that dSh vanishes. Thus FLT (2.3.13) established in this
modified theory is more comprehensive and ensures the results in f(R) and
f(R, T ) theories. One can define the effective entropy term being the sum
of horizon entropy and entropy production term as Seff = Sh + S so that
Eq.(2.3.13) can be rewritten as
ThdSeff = −dE + WdV,
where Seff is the effective entropy related to the CMC in this modified
theory at the AH of FRW spacetime.
48
2.3.2 Generalized Second Law of Thermodynamics
Here, we explore the validity of GSLT in the framework of f(R, T,Q) grav-
ity. Let us proceed with the Gibb’s equation which relates the entropy of
matter and energy sources to the pressure in the horizon given by
TindSin = (ρ + p)dV + V dρ. (2.3.14)
Using the divergence of energy-momentum tensor, we can evaluate the evo-
lution of entropy inside the horizon as
TinSin = 4πr2A(ρ + p)( ˙rA −HrA) +
V
2 + 3(fT − (H + 3H2)fQ)3 H
× (H − 3H3)(ρ + p)− (H + 2H2)∂tp− 2H2∂tLm
fQ + fT ∂t(2Lm
+ 3p) + 2(ρ− Lm)(3H2∂tfQ − ∂tfT )
. (2.3.15)
One can find the relations of matter energy density and pressure for the
FRW universe using the field equation (1.1.16) as follows
ρ =1
κ2 + fT + 12(R−2)fQ
[1
2f − 3(H + H2)fR + Lm(fT − 3(H2 +
κ
a2)
× fQ)− 3
2
(H2 + 3H
)ρfQ +
3
2
(3H2 + H
)pfQ + 3H∂t
(fR +
1
2[(ρ
− p)fQ])− 3
2Hρ∂tfQ − 1
2ρ∂ttfQ
], (2.3.16)
p =1
κ2 + fT + 12(R−2)fQ
[1
2f + Lm
((2H + 3H2 +
κ
a2)− fT
)+ (H
+ 3H2 +2κ
a2)fR − 1
2
(H + 3H2
)ρfQ − 1
2
(5(3H2 + H) +
8κ
a2
)pfQ
− 2H∂t fR + (ρ + p) fQ − ∂tt
fR +
1
2(ρ− p)fQ
− 3
2Hp∂tfQ
− 1
2p∂ttfQ
]. (2.3.17)
49
Substituting Eqs.(2.3.16) and (2.3.17) in (2.3.15), it follows that
TinSin = 4πr2A( ˙rA −HrA)
1
κ2 + fT + 12(R−2)fQ
[2(
κ
a2− H)(fR
− fQLm)− (3H2 + 5H)ρfQ − (3H2 + H +4κ
a2)pfQ + H∂tfR
− 1
2(ρ + 7p)fQ − 1
2(ρ + p)(3H∂tfQ + ∂ttfQ)− ∂ttfR +
1
2(ρ
− p)fQ] V
2 + 3(fT − (H + 3H2)fQ)
3
H(H − 3H3)(ρ + p)
− (H + 2H2)∂tp− 2H2∂tLm
fQ + fT ∂t(2Lm + 3p)
+ 2(ρ− Lm)(3H2∂tfQ − ∂tfT )
. (2.3.18)
The entropy of AH in f(R, T,Q) gravity is defined as
Sh =A(fR − fQLm)
4[G + 18π
(fT + 12(R−2)fQ)]
. (2.3.19)
The evolution of Sh multiplied with the horizon temperature implies that
ThSh =2π(2HrA − ˙rA)
κ2 + fT + 12(R−2)fQ
[2
˙rA
rA
(fR − fQLm) + ∂t(fR − fQLm)−
× (fR − fQLm)∂tκ2 + fT +1
2(R−2)fQ
]. (2.3.20)
Eventually, the validity of GSLT requires the condition (ThSh+ TinSin) > 0.
In this setting, we assume a relation between Tin and Th as Tin = bTh.
After some manipulations, Eqs.(2.3.18) and (2.3.20) can be summed to the
following form
ThStot =2π
(H2 + κ
a2
)−5/2
κ2 + fT + 12(R−2)fQ
[2H
( κ
a2− H
) κ
a2+ (1− 2b)H + 2(1
− b)H2
(fR − fQLm) + ∂tfR
κ
a2
( κ
a2+ H + 3H2
)+ (1− 2b)H
× H2 + 2(1− b)H4
+ 2bH(H + H2)∂ttfR − (κ
a2+ H2)
( κ
a2+ H
+ 2H2)
(fR − fQLm)∂t
[ln
(κ2 + fT +
1
2(R−2)fQ
)]+ ∂t(fQ
× Lm)]
50
−4bπ(H2 + κ
a2
)−5/2H(H + H2)
κ2 + fT + 12(R−2)fQ
[−(3H2 + 5H)ρfQ − (3H2 + H
+4κ
a2)pfQ − 1
2H∂t(ρ + 7p)fQ − 1
2(ρ + p)(3H∂tfQ + ∂ttfQ)− ∂ttfR
+1
2(ρ− p)fQ
]+
43bπr3
A
2 + 3[fT − (H + 3H2)fQ]
3
H(H − 3H2)(ρ
+ p)− (H + 2H2)∂tp− 2H2∂tLm
fQ + fT ∂t(2Lm + 3p) + 2(ρ
− Lm)(3H2∂tfQ − ∂tfT )
> 0, (2.3.21)
where Stot = Sh +Sin and condition to meet the GSLT counts the choice of
action in this modified theory. The above result appears to be more general
and one can deduce the expressions of GSLT in Einstein, f(R) and f(R, T )
gravities.
If we set f(R, T,Q) = f(R, T ), so that the variation of f with respect
to Q is null then inequality to fulfill the GSLT is given by
ThStot =2π
(H2 + κ
a2
)−5/2
κ2 + fT
[2H
( κ
a2− H
) κ
a2+ (1− 2b)H + 2(1− b)
× H2
fR + ∂tfR
κ
a2
( κ
a2+ H + 3H2
)+ (1− 2b)HH2 + 2(1− b)H4
+ 2bH(H + H2)∂ttfR −( κ
a2+ H2
)( κ
a2+ H + 2H2
)fR∂t[ln(κ2 + fT )]
+4
3πr3
A
b
2 + 3fT
fT ∂t(2Lm + 3p) + 2(Lm − ρ)∂tfT]
> 0. (2.3.22)
When fT = fQ = 0 (purely f(R) gravity), the GSLT takes the form
ThStot =2π
(H2 + κ
a2
)−5/2
κ2
[2H
( κ
a2− H
) κ
a2+ (1− 2b)H + 2(1− b)
× H2
fR + κ
a2
( κ
a2+ H + 3H2
)+ (1− 2b)HH2 + 2(1− b)H4
∂tfR
+ 2bH(H + H2)∂ttfR
]> 0. (2.3.23)
The GSLT in Einstein gravity can be recovered by replacing fR = 1 in
Eq.(2.3.23)
ThStot =1
2GH
(H − κ
a2
)2 (H2 +
κ
a2
)−5/2
.
51
We limit our discussion of GSLT by taking the system (includes matter
and energy contents bounded by the horizon) in equilibrium position so
that energy would not flow in the system and temperature within the hori-
zon more or less matches with the horizon temperature, i.e., b = 1. This
situation would correspond to the case of late times where the universe
components and horizon would have interacted for long time while its ex-
istence for early or intermediate times would be ambiguous. Though the
assumption of thermal equilibrium is limiting in some sense to avoid the
non-equilibrium complexities but it has widely been accepted to study the
GSLT [21, 23, 57, 58, 59, 61].
To illustrate the validity of GSLT in f(R, T,Q) gravity, we take some
concrete models in this modified theory namely [13]
(i) f(R, T,Q) = R + αQ, (ii) f(R, T,Q) = R(1 + αQ),
where α is a coupling parameter. These models are proposed in [13] where
authors explored the evolution of scale factor and deceleration parameter in
this scenario. Recently, we have examined the validity of energy conditions
for the above particular forms of Lagrangian (1.1.15) [62]. Here we are
interested to develop constraints on the validity of GSLT.
(i) f(R, T,Q) = R + αQ
For this model, the GSLT becomes
ThStot =2π
(H2 + κ
a2
)−5/2
κ2 + α2R
[2H
( κ
a2− H
)2
(1− αLm)−( κ
a2+ H2
)( κ
a2
+ H + 2H2)
(1− αLm)∂t
[ln
(κ2 +
α
2R
)]+ α∂tLm
− 2H(H
+ H2)
−(3H2 + 5H)αρ− (3H2 + H +
4κ
a2)αp− α
2H(ρ + 7p)
− α
2(ρ− p)
]
52
+4πα
(κa2 + H2
)−3/2
2− 3α(H + 3H2)
H(H − 3H2)(ρ + p)− (H + 2H2)∂tp
− 2H2∂tLm
> 0.
Cosmic expansion history is thought to have experienced the decelerated
phase and hence transition to accelerating epoch. Thus, power law solutions
can play vital role to connect the matter dominated phase with accelerating
paradigm. The existence of power law solutions in FRW setting is partic-
ularly relevant to intimate all possible cosmic evolutions. The scale factor
for power law cosmology is defined as
a(t) = a0tm,
where m is a positive constant. If 0 < m < 1, then the resulting power
law solution favors decelerating expansion whereas for m > 1, it exhibits
accelerating behavior. To be more explicit for the above constraint, we set
the power law cosmology for accelerated cosmic expansion (m > 1) with
ρ = ρ0t−3m and choose Lm = −p. For the flat FRW geometry, the plots of
GSLT are shown in Figure 2.1. We find that the GSLT holds if one sets
the parameters as α < 0 and m > 1 while in the case of Lm = ρ, it can be
easily examined that validity of GSLT requires m > 8.
(ii) f(R, T,Q) = R(1 + αQ)
In this case, Eq.(2.3.21) takes the form
ThStot =2π
(H2 + κ
a2
)−5/2
κ2 + α2(R−2)R
[2H
( κ
a2− H
)2
(1−RαLm)−( κ
a2+ H2
)
×( κ
a2+ H + 2H2
)(1−RαLm)∂t
[ln
(κ2 +
α
2(R−2)R
)]+ α
× ∂t(RLm) − 2H(H + H2)−αρ(3H2 + 5H)R− αp(3H2 + H
+4κ
a2)R− α
2H∂[(ρ + 7p)R]− 3α
2H(ρ + p)R− α
2∂tt[(ρ− p)R]
]
53
Figure 2.1: Evolution of GSLT for the Lagrangian f(R, T,Q) = R + αQ,the left panel shows the bound on m for α = −2 whereas in the right panelwe set m = 10. It is evident that GSLT is valid only if α < 0, m > 1.
+4πα
(κa2 + H2
)−3/2
2− 3α(H + 3H2)
H(H − 3H2)(ρ + p)− (H + 2H2)∂tp− 2H2
× ∂tLm + 6αH2(ρ− Lm)R
> 0.
We set Lm = −p so that in flat FRW power law cosmology, the above
inequality depends on the parameters m and α. Figure 2.2 depicts the
evolution of GSLT for this model which shows that GSLT can be met only
if α > 0 with m > 1. Similarly, one can explore the validity of GSLT for
the choice Lm = ρ which requires m > 3 and α > 0.
Recently, Odintsov and Saez-Gomez [14] discussed the FRW cosmolog-
ical dynamics in f(R, T,Q) gravity and reconstructed the Lagrangian for
ΛCDM and de Sitter universe models. They proposed the model f(R, T,Q) =
R + f(Q) + g(T ) in terms of redshift by considering de Sitter cosmology.
The functions f(Q) and g(T ) are defined as
f(Q) = H20F
( QQ0
), g(T ) = H2
0G
(T
T0
), (2.3.24)
54
Figure 2.2: Evolution of GSLT for the Lagrangian f(R, T,Q) = R(1+αQ),the left panel shows constraint on m for α = 10 whereas in the right panel,we set m = 10. It is evident that GSLT is valid only if α > 0, m > 1.
where
F (Q) = C1
( QQ0
)α/3
+
C2 cos
(w
3lnQQ0
)+ C3 cos
(w
3lnQQ0
)
×( QQ0
)β/3
+ C4 + 3Ωm0QQ0
,
G(T ) = D1
(T
T0
)α/3
+
D2 cos
(w
3ln
T
T0
)+ D3 cos
(w
3ln
T
T0
)
×(
T
T0
)β/3
+ D4 − 3Ωm0T
T0
,
Ci’s, Di’s are integration constants, Q/Q0 = T/T0 = (1 + z)3 and α =
−1.327, β = 3.414, ω = 1.38. This model is constructed for Lm = −p by
considering dust matter so that ρ ∝ (1 + z)3. We are interested to explore
the validity of GSLT in the background of de Sitter universe for the model
(2.3.24). Substituting Eq.(2.3.24) in (2.3.21), one can obtain the condition
for the validity of GSLT. We plot the GSLT for two cases.
• In first case, we assume the constants Ci = C, Di = D and present
the evolution of GSLT for current value of redshift z = 0 in the left
panel of Figure 2.3. The value of parameter C is very critical which
is set in the range 0 < C 6 0.6 and parameter D can be assigned any
value.
55
0.0
0.2
0.4
0.6
C
0
10
20D
0
10
20
30
40
Th S
tot
0
5
10z
10
15
20U
0
2´106
4´106
Figure 2.3: Evolution of GSLT for the Lagrangian f(R, T,Q) = R+f(Q)+g(T ), the left panel shows constraints on parameters C and D for z = 0whereas in the right panel, we set Ci and Di in terms of unique parameterΥ and constrain the values of Υ. We choose H0 = 67.3, Ωm0 = 0.315 fromthe recent Planck results [33].
• In second case, we set Ci = Υ and Di are also represented in terms of
Υ so that one can constrain parameters in the evolution −1 < z 6 10.
We find that GSLT is satisfied for the range Υ > 8.
Chapter 3
Energy Conditions Constraintsand Stability of f (R, T ) andf (R, T,Q) Modified Theories
In this chapter, we present the energy conditions constraints in the context
of f(R, T ) and f(R, T,Q) theories. The corresponding energy conditions
appear to be more general and can reduce to the familiar forms of these
conditions in GR and f(R) theory. The generic inequalities set by these
energy constraints are presented in terms of recent values of Hubble, de-
celeration, jerk and snap parameters. In particular, some specific models
are considered in these theories to study the concrete application of energy
bounds as well as the Dolgov-Kawasaki instability. Moreover, we obtain
the exact power law solutions for two particular cases namely, f(R) + λT
and R + 2f(T ) in homogeneous and isotropic f(R, T ) cosmology. We also
establish certain conditions to ensure the viability of these solutions.
The present chapter has the following format. In section 3.1, we derive
the energy conditions in f(R, T,Q) gravity and respective inequalities are
presented in terms of cosmographic parameters. Section 3.2 is devoted to
study the energy conditions bounds for some specific forms of f(R, T,Q)
gravity and the Dolgov-Kawasaki instability. In section 3.3, the energy
56
57
constraints are also shown for specific functional forms of f(R, T ) involv-
ing an exponential function and the coupling between R and T . We also
explore the existence of power law solutions and corresponding constraints
from energy conditions. Section 3.4 is devoted to examine whether these
solutions are stable against linear homogeneous perturbations in f(R, T )
gravity. This chapter is based upon the results presented in [62, 63].
3.1 Energy Conditions in f (R, T,Q) Gravity
In this section, we derive the energy conditions in the context of f(R, T,Q)
gravity and express in terms of well-known cosmographic parameters. The
field equation (1.1.16) can be rearranged in the following form
Gαβ = Rαβ − 1
2Rgαβ = T eff
αβ , (3.1.1)
which is analogous to the standard field equations in GR. Here T effαβ , the
effective energy-momentum tensor in f(R, T,Q) gravity is defined as
T effαβ =
1
fR − fQLm
[(1 + fT +
1
2RfQ)Tαβ + 1
2(f −RfR)− LmfT
− 1
2∇α∇β(fQTαβ)gαβ − (gαβ2−∇α∇β)fR − 1
22(fQTαβ)
− 2fQRα(αTαβ) +∇α∇(α[T α
β)fQ] + 2(fT gαβ + fQRαβ
)
× ∂2Lm
∂gαβ∂gαβ
]. (3.1.2)
We take the homogeneous and isotropic flat FRW metric defined as
ds2 = dt2 − a2(t)dx2,
where dx2 is the spatial part of the metric. The corresponding effective
energy density and effective pressure can be taken such that T effαβ assumes
58
the form of perfect fluid. In FRW background, ρeff and peff can be obtained
in this modified theory as
ρeff =1
fR − fQLm
[ρ + (ρ− Lm)fT +
1
2(f −RfR)− 3H∂tfR − 3
2(3H2
− H)ρfQ − 3
2(3H2 + H)pfQ +
3
2H∂t[(p− ρ)fQ]
], (3.1.3)
peff =1
fR − fQLm
[p + (p + Lm)fT +
1
2(RfR − f) +
1
2(H + 3H2)(ρ
− p)fQ + ∂ttfR + 2H∂tfR +1
2∂tt[(ρ− p)fQ] + 2H∂t[(ρ + p)
× fQ]] , (3.1.4)
where R = −6(H + 2H2), H = aa
being Hubble parameter and over dot
refers to time derivative. Here, we neglect the terms involving second deriv-
ative of matter Lagrangian with respect to the metric tensor. As we are
dealing with perfect fluid, so matter Lagrangian can either be Lm = p or
Lm = −ρ which makes it obvious to ignore such term.
We adopt the procedure developed in [28, 32] for f(R), f(R,Lm) and
f(R) gravity with arbitrary and non-minimal matter geometry coupling
to extend it to a more general f(R, T,Q) grvaity. The Ricci tensor in
Eq.(3.1.1) can be represented in terms of T effαβ and its trace T eff as
Rαβ = T effαβ − 1
2gαβT eff , (3.1.5)
where the contraction of Eq.(3.1.2) yields the trace of the energy-momentum
tensor
T eff =1
fR − fQLm
[(1 + fT +
1
2RfQ)T + 2(f −RfR)− 4LmfT
− ∇α∇β(fQTαβ)− 32fR − 1
22(fQT )− 2fQRαβTαβ + 2gαβ(fT gαβ
+ fQRαβ)∂2Lm
∂gαβ∂gαβ
]. (3.1.6)
59
The attractive nature of gravity needs to satisfy the following additional
constraint
1 + fT + 12RfQ
fR − fQLm
> 0, (3.1.7)
which does not depend on the conditions (1.5.3) derived from the Ray-
chaudhuri equation. In fact, this condition corresponds to the effective
gravitational coupling in f(R, T,Q) gravity.
In section 1.5, it is shown that Raychaudhuri equations with attractive
behavior of gravitational interaction give rise to SEC and NEC which hold
for any theory of gravitation. In this modified theory, we can employ an
approach analogous to that in GR to develop the energy conditions. We
also assume that standard matter obeys the energy conditions. We combine
Eqs.(1.5.4) and (3.1.6) so that NEC is of the form
T effαβ κακβ > 0.
Inserting Eq.(3.1.2) in the above relation leads to the following inequality
ρeff + peff =1
fR − fQLm
[(1 + fT )(ρ + p)− 3H2(ρ + 2p)fQ + 2H(ρ
− p)fQ −H∂tfR − 1
2(ρ + 7p)fQ+ ∂ttfR +
1
2(ρ− p)fQ
]> 0, (3.1.8)
which is the NEC in f(R, T,Q) gravity. One can represent the SEC in
f(R, T,Q) gravity in the form
T effαβ uαuβ − 1
2T eff > 0, (3.1.9)
where gαβuαuβ = 1. Using Eqs.(3.1.2) and (3.1.6), it follows that
ρeff + 3peff =1
fR − fQLm
[(ρ + 3p) + (ρ + 3p + 2Lm)fT + RfR − f
+ 3[H(ρ− p)− 3H2p]fQ + 3H∂t[fR +1
2(3ρ + 5p)fQ]
+ 3∂tt[fR +1
2(ρ− p)fQ]
]> 0. (3.1.10)
60
It is remarked that one can obtain the NEC and SEC in f(R) and f(R, T )
theories by taking f(R, T,Q) = f(R) and f(R, T,Q) = f(R, T ), respec-
tively. Moreover, the traditional structures for the NEC and SEC can be
found in the framework of GR as a specific case with f(R, T,Q) = R.
To derive the WEC and DEC, we can extend the GR approach by in-
troducing an effective energy-momentum tensor. We consider the modified
form of energy conditions in GR which are obtained under the transforma-
tions ρ → ρeff and p → peff . We would like to mention here that the null
and strong energy conditions given by Eqs.(3.1.8) and (3.1.10) are derived
from the Raychaudhuri equation. One can obtain equivalent results follow-
ing the same procedure as that in GR with conditions ρeff + peff > 0 and
ρeff + 3peff > 0. We extend this approach to develop the constraints for
WEC and DEC so that these conditions for f(R, T,Q) gravity are given by
ρeff > 0 and ρeff − peff > 0. Using Eqs.(3.1.3) and (3.1.4), one can obtain
the constraints on WEC and DEC. The WEC requires the condition (3.1.8)
and the following inequality
ρeff =1
fR − fQLm
[ρ + (ρ− Lm)fT +
1
2(f −RfR)− 3H∂tfR − 3
2(3H2
− H)ρfQ − 3
2(3H2 + H)pfQ +
3
2H∂t[(p− ρ)fQ]
]> 0, (3.1.11)
whereas the DEC is satisfied by meeting the inequalities (3.1.8), (3.1.11)
and the condition
ρeff − peff =1
fR − fQLm
[(ρ− p) + (ρ− p− 2Lm)fT + f −RfR
+ H(ρ− p)− 3H2(3ρ + p)fQ −H∂t[1
2(7ρ + p)fQ + 5fR]
− ∂tt[fR +1
2(ρ− p)fQ]
]> 0. (3.1.12)
When we assume f(R, T,Q) = f(R, T ), the above expressions reduce to
61
the WEC and DEC in f(R, T ) gravity which are similar to that in [63].
Also, by neglecting the dependence on Tαβ and its trace, we can have the
energy conditions in f(R) gravity consistent with the results in [28]. If the
variation of Lagrangian with respect to T and Q is null and fR = 1 then
such conditions constitute ρ > 0 and ρ + p > 0, i.e., the WEC and DEC in
GR.
One can utilize the energy conditions constraints (3.1.8)-(3.1.12) to re-
strict some specific models in f(R, T,Q) gravity in the framework of FRW
model. To be more definite about these energy constraints, we define de-
celeration, jerk and snap parameters as [64]
q = − 1
H2
a
a, j =
1
H3
...a
a, and s =
1
H4
....a
a,
and express the Hubble parameter as well as its time derivatives in terms
of these parameters
H = −H2(1 + q), H = H3(j + 3q + 2),...H = −H4(5q + 2j − s + 3).
Since R, R and R are represented in terms of the above relations, so using
these parameters the energy conditions (3.1.8)-(3.1.12) can be constituted
as
(ρ + p)(1 + fT ) +1
2ρ− p + H(ρ + 7p)− 4(1 + q)H2(ρ− p)− 6H2
× (ρ + 2p)fQ − 6H2(s− j + (q + 1)(q + 8))fRR + (T −HT )fRT + Q
− HQ − 3H3(j − q − 2)(2(ρ− p) + H(ρ + 7p))− 3H4(ρ− p)(s + q2 + 8q
+ 6)fRQ +1
22T + (2(ρ− p) + H(ρ + 7p))TfTQ +
1
22Q+ (2(ρ− p)
+ H(ρ + 7p))QfQQ + [6H3(j − q − 2)]2fRRR − 12H3(j − q − 2)T fRRT
+ 18(ρ− p)[H3(j − q − 2)]2 − 12H3(j − q − 2)QfRRQ + 2T [Q − 3H3(ρ
62
−p)(j − q − 2)]fRTQ + T 2fRTT + Q[Q − 6H3(ρ− p)(j − q − 2)]fRQQ
+1
2(ρ− p)T [QfTQQ + T fTTQ] +
1
2(ρ− p)Q[T fTQQ + QfQQQ] > 0, (NEC)
(3.1.13)
ρ(1 + fT )− LmfT +1
2f + 3H2(1− q)fR +
3
2Hp− ρ− 2H(2ρ + p)
− H(ρ− p)qfQ − 3HQ+ 3H3(p− ρ)(j − q − 2)fRQ +3
2H(p− ρ)(Q
× fQQ + T fTQ) + 18H4(j − q − 2)fRR − 3HTfRT ) > 0, (WEC) (3.1.14)
(ρ + 3p)(1 + fT ) + 2LmfT − f − 6H2(1− q)fR +3
2ρ− p + H(3ρ + 5p)
+ 2H2[(p− ρ)(1 + q)− 3p]fQ − 18H4(s + j + q2 + 7q + 4)fRR + 3(T
+ HT )fRT + 3Q+ HQ − 3H3(j − q − 2)[2(ρ− p) + H(3ρ + 5p)]− 3H4
× (ρ− p)(s + q2 + 8q + 6)fRQ +3
2(ρ− p)T + [2(ρ− p) + H(3ρ + 5p)T ]
× fTQ +3
2(ρ− p)Q+ [2(ρ− p)−H(ρ + 3p)Q]fQQ + 3[6H3(j − q − 2)]2
× fRRR − 36H3(j − q − 2)T fRRT + 3−12H3(j − q − 2)Q+ 18(ρ− p)[6
× H3(j − q − 2)]2fRRQ + 6TQ − 3(ρ− p)H3(j − q − 2)fRTQ + 3T 2
× fRTT + 3QQ − 6H3(ρ− p)(j − q − 2)fRQQ +3
2(ρ− p)T2QfTQQ
+ T fTTQ+3
2(ρ− p)Q2fQQQ > 0, (SEC) (3.1.15)
(ρ− p)(1 + fT )− 2LmfT + f + 6H2(1− q)fR +1
2p− ρ−H(7p + ρ)
− 6H2(3ρ + p)− 2H2(ρ− p)(1 + q)fQ − Q+ 5HQ − 6H3(j − q − 2)
× (ρ− p)− 3H4(j − q − 2)(7ρ + p)− 3H4(s + q2 + 8q + 6)(ρ− p)
− 1
2(ρ− p)T + [2(ρ− p) + H(7ρ + p)]TfTQ − 1
2(ρ− p)Q+ [2(ρ− p)
+ H(7ρ + p)]QfQQ + 6H4[s + 5j + (q − 1)(q + 4)]fRR − (T + 5TH)fRT
− [6H3(j − q − 2)]2fRRR + 12H3(j − q − 2)T fRRT + 12H3(j − q − 2)Q
− 18[H3(j − q − 2)]2(ρ− p)fRRQ − 2TQ − 6H3(j − q − 2)(ρ− p)fRTQ
− T 2fRTT − QQ − 6H3(j − q − 2)(ρ− p)fRQQ − 1
2(ρ− p)TQfTQQ+
63
T fTTQ1
2(ρ− p)QT fTQQ + QfQQQ > 0. (DEC) (3.1.16)
The results of energy conditions in terms of cosmographic parameters for
f(R) and f(R, T ) theories can be achieved from the constraints (3.1.13)-
(3.1.16).
3.2 Constraints on Class of f (R, T,Q) Models
To illustrate how these energy conditions put limits on f(R, T,Q) grav-
ity, we consider some specific functional forms for the Lagrangian (1.1.15)
namely [13],
• f(R, T,Q) = R + αQ,
• f(R, T,Q) = R(1 + αQ),
where α is a coupling parameter. Recently, these models have been studied
in [13] which suggest that exponential and de Sitter type solutions exist
for these forms of f(R, T,Q) gravity. Thus one can deduce that matter
geometry coupling may cause the current cosmic acceleration.
3.2.1 f(R, T,Q) = R + αQ
In the first place, we consider the Lagrangian given by R + αQ. In FRW
background, the energy conditions for such model can be represented as
αA1 + H∂tA2 > A3, (3.2.1)
64
where Ai’s purely depend upon the energy conditions under discussion. For
NEC, one can have
ANEC1 = (2H − 3H2)ρ− 2(H + 3H2)p + ∂tt[α
−1 +1
2(ρ− p)],
ANEC2 = −(1− α
2(ρ + 7p)), ANEC
3 = −(ρ + p). (3.2.2)
For WEC, this yields
AWEC1 = −3H2ρ, AWEC
2 =3α
2(p− ρ)− 3, AWEC
3 = −ρ.(3.2.3)
For SEC, one can find
ASEC1 = 3ρ(2H + H2)− 6p(H + 3H2) + 3∂tt[α
−1 +1
2(ρ− p)],
ASEC2 = 3(1 +
α
2(3ρ + 5p)), ASEC
3 = −(ρ + 3p). (3.2.4)
For DEC, it implies that
ADEC1 = 2H(p− ρ) + 6H2(p− 2ρ)− ∂tt[α
−1 +1
2(ρ− p)],
ADEC2 = −α
2(7ρ + p)− 5, ADEC
3 = −(ρ− p). (3.2.5)
We can also find the condition of attractive gravity for this model from
inequality (3.2.1) so that
AAG1 = (1− αLm)
(1
α+
R
2
)−1
, AAG2 = constant, AAG
3 = 0.
The energy conditions (3.2.1)-(3.2.5) can be expressed in terms of deceler-
ation parameter (see appendix A1). It can be seen that these conditions
depend only upon the parameters H, q and α. In our discussion, we set the
present day values of cosmographic parameters as q0 = −0.81+0.14−0.14, j0 =
2.16+0.81−0.75 [65] and H0 = 73.8 [66], while matter is assumed to be pressure-
less. To exemplify how these conditions can constrain the above model, we
consider the WEC given by the relation
ρ− 6H2 − 3αHρ > 0. (3.2.6)
65
For the given H and q, one can see that the above inequality relies on the
measures of parameter α and time derivative of energy density. Here, ρ can
be evaluated using Eq.(2.3.4) which takes the form
ρ = −6Hρ1 + αH2(2 + 5q − 3H(1 + q))/2− 3α(2− q)H2,
which shows that ρ is always negative. Using this value of ρ in Eq.(3.2.6),
we find that WEC for the model f(R, T,Q) = R + αQ is satisfied if α > 0
for present day values of q and H.
3.2.2 f(R, T,Q) = R(1 + αQ)
In this example, we consider the function f given by R(1+αQ) and energy
conditions for such model can be written as
αB1 + H∂tB2 > B3, (3.2.7)
where α = (−1)α and Bi’s purely depend upon the energy conditions under
discussion. For NEC, one can have
BNEC1 = [−3H2(ρ + 2p) + 2H(ρ− p)]R + ∂tt[α
−1 +Q+1
2(ρ− p)R],
BNEC2 = −(1 + αQ− α
2(ρ + 7p)R), BNEC
3 = −(ρ + p). (3.2.8)
For WEC, we have
BWEC1 = −3
2[(3H2 − H)ρ + (3H2 + H)p]R,
BWEC2 = 3[
α
2(p− ρ)R− (1 + αQ)], BWEC
3 = −ρ. (3.2.9)
For SEC, it follows that
BSEC1 = 3[H(ρ− p)− 3H2p]R + ∂tt[α
−1 +Q+1
2(ρ− p)R],
BSEC2 = 3[1 + αQ+
α
2(3ρ + 5p)R], BSEC
3 = −(ρ + 3p). (3.2.10)
66
For DEC, this yields
BDEC1 = −3H2(3ρ + p)R− (ρ + p)HR− ∂tt[α
−1 +Q+1
2(ρ− p)R],
BDEC2 = −(5(1 + αQ) +
α
2(7ρ + p)R), BDEC
3 = −(ρ− p). (3.2.11)
The condition of attractive gravity can be obtained from the inequality
(3.2.7) and relevant components are
BAG1 = (1 + αQ−RLm)
(1
α+
R2
2
)−1
, BAG2 = constant, BAG
3 = 0.
The viability of modified theories is under debate to develop the criteria
for different modifications to the Einstein-Hilbert action. In this perspec-
tive, one of the important criterion is Dolgov-Kawasaki instability which
has been developed to constrain the f(R) and f(R) with curvature matter
coupling gravities [32, 37, 39]. Recently, the authors [13, 14] have executed
this instability analysis for f(R, T,Q) gravity which yields the condition of
Dolgov-Kawasaki instability as
3fRR +
(1
2T − T 00
)fQR > 0. (3.2.12)
For the model f = R(1 + αQ), the inequality (3.2.12) takes the form
α(ρ− 3p) + 6αH(ρ + p)∂t
(H
R
)> 0,
where
α =
(−1)α, if R,Q < 0,
α, if R,Q > 0.
One can derive the above inequality using the relation (3.2.7) so that Bi’s
are given by
BAG1 =
ρ− 3p
ρ + p, BAG
2 =6αH
R, BAG
3 = 0.
67
We check the validity of constraints (3.2.8)-(3.2.11) for this model. The
constraint to ensure WEC is given by
ρ[1 + 9αH4(2j − q2 − 3q + 2)] + 9αH3(1− 2q)ρ > 0.
As in the previous case, we evaluate ρ
ρ = −3Hρ1 + 6αH4(j − 4q + 2q2 − 1)/1 + 9αH4(1− q)(2− q).
Here, ρ < 0 for any value of α and hence the WEC is satisfied only if
parameter α is positive.
3.3 Energy Conditions in f (R, T ) Gravity
Now, we present f(R, T,Q) gravity models which involve null variation
with respect to Q and correspond to f(R, T ) gravity. The effective field
equations in f(R, T ) gravity can be expressed as that of Eq.(3.1.1) with
T effαβ =
1
fR(R, T )
[(1 + fT (R, T ))Tαβ + pgαβfT (R, T ) +
1
2(f(R, T )
− RfR(R, T ))gαβ + (∇µ∇ν − gµν2)fR(R, T )] . (3.3.1)
For this modified theory, the conditions (3.1.8)-(3.1.12) reduce to the fol-
lowing form
NEC :
ρeff + peff =1
fR
[(ρ + p)(1 + fT ) + (R− RH)fRR + R2fRRR
+ 2RT fRRT + (T − TH)fRT + T 2fRTT
]> 0, (3.3.2)
WEC :
ρeff =1
fR
[ρ + (ρ + p)fT +
1
2(f −RfR)− 3H(RfRR
+ T fRT )]
> 0, ρeff + peff > 0, (3.3.3)
68
SEC :
ρeff + 3peff =1
fR
[(ρ + 3p) + (ρ + p)fT − f + RfR + 3R2fRRR
+ 3(R + RH)fRR + 6RT fRRT + 3(T + TH)fRT
+ 3T 2fRTT
]> 0, ρeff + peff > 0, (3.3.4)
DEC :
ρeff − peff =1
fR
[(ρ− p) + (ρ + p)fT + f −RfR − R2fRRR
− (R + 5RH)fRR − 2RT fRRT − (T + 5TH)fRT
− T 2fRTT
]> 0, ρeff + peff > 0, ρeff > 0. (3.3.5)
The inequalities (3.3.2)-(3.3.5) represent the null, weak, strong and domi-
nant energy constraints in f(R, T ) theory for FRW spacetime. The above
conditions can also imply the respective constraints in f(R) gravity similar
to that in [28] for vanishing fT . To illustrate how above conditions can
be exercised to place bounds on f(R, T ) gravity, we consider two particular
forms of f(R, T ) gravity. We are interested in more general functional forms
of f(R, T ) involving an exponential function and also the coupling between
R and T . We present the energy conditions constraints for the following
two models
f(R, T ) = αexp
(R
α+ λT
), f(R, T ) = R + ηRmT n,
where α, λ, η, m and n are arbitrary constants.
• f(R, T ) = αexp(
Rα
+ λT)
If Rα
+λT ¿ 1 then f(R, T ) ≈ α+R + λαT + ... representing the ΛCDM
model. The energy constraints in f(R, T ) gravity can be achieved by placing
null variation of f with respect to Q in the results (3.1.8)-(3.1.12). For this
69
model, these conditions take the form
exp(
Rα
+ λT)
1 + αλ exp(
Rα
+ λT) (C1 + C2) > C3, (3.3.6)
where Ci’s depend upon the energy conditions given in Appendix A2.
The condition of attractive gravity in f(R, T ) gravity is (1 + fT )/fR > 0
which becomes (1 + αλ exp(
Rα
+ λT))/ exp
(Rα
+ λT)
> 0 for the expo-
nential model. We can obtain this inequality from Eq.(3.3.6) for C1 = 1,
C2 = C3 = 0. It is suggested [13] that Dolgov-Kawasaki instability in
f(R, T ) gravity would be identical to that in f(R) gravity so that one can
check the viability of f(R, T ) models on similar steps as in f(R) theory.
Thus for f(R, T ) theory, we have
fR(R, T ) > 0, fRR(R, T ) > 0, R≥R0.
The instability conditions are exp(
Rα
+ λT)
> 0 and 1α
exp(
Rα
+ λT)
>
0 which can be derived from relation (3.3.6) by taking C1 = 1, C2 =
αλ exp(
Rα
+ λT), C3 = 0 and C1 + C2 = 1
α
(1 + αλ exp
(Rα
+ λT))
, C3 = 0,
respectively. One can represent energy conditions (A2) in the form of
cosmographic parameters.
The inequality to fulfill the WEC is
ρ
(1 + αλ exp
(Rα
+ λT)
exp(
Rα
+ λT)
)+ α(0.5− λLm) + 3H2(1− q) + 6α−1H2
× (j − q − 2) − 3λHT > 0.
Using the WEC results in GR, i.e., ρ > 0 and also the condition of attractive
gravity (1 + αλ exp(
Rα
+ λT))/ exp
(Rα
+ λT)
> 0, the above inequality is
reduced to
α(0.5− λLm) + 3H2(1− q) + 6α−1H2(j − q − 2) − 3λHT > 0.
70
We take Lm = p and assume the pressureless matter so that
0.5α + 3H2(1− q) + 6H2(j − q − 2)α−1 − 3λHρ > 0. (3.3.7)
If we consider the present day values of the parameters like Hubble, de-
celeration and jerk then the above inequality depends upon ρ and values
of constants (α, λ). We find ρ from the energy conservation equation in
f(R, T ) gravity as
ρ + 3H(ρ + p) =−1
1 + f2T
[(ρ− Lm)fT − LmfT +
1
2T fT
], (3.3.8)
which takes the following form for exponential model
ρ = −3Hρ1 + λ(α− 2H2(j − q − 2)) exp(
Rα
+ λT)
1 + (1.5 + λρ)αλ exp(
Rα
+ λT) .
If α > 0 then the first two terms in inequality (3.3.7) are positive whereas
for the last term we need to have −ρ > 0. From the above expression, we
see that −ρ > 0 if λ > 0 and α > 2H20 (j0 − q0 − 2). Thus, the WEC for
exponential f(R, T ) model is satisfied if λ > 0 and α > 2H20 (j0 − q0 − 2).
We consider another form of matter Lagrangian Lm = −ρ for which the
continuity equation and constraint to fulfill the WEC are given by
ρ = −3Hρ1 + λ(α− 4H2(j − q − 2)) exp(
Rα
+ λρ)
1 + (1.5 + 2λρ)αλ exp(
Rα
+ λρ) ,
α(0.5 + λρ) + 3H2(1− q) + 6α−1H2(j − q − 2) − 3λHρ > 0.
As in the previous case, we find a constraint for which ρ < 0 which is only
possible if α > 4H2(j − q − 2). It is to be noted that we set the present
values of H and other parameters so that the WEC is satisfied if λ > 0 and
α > 4H2(j − q − 2).
• f(R, T ) = R + ηRmT n
71
Here, we consider the power law type f(R, T ) model which involves
coupling between R and T . Such functional form of f(R, T ) matches to the
form of Lagrangian f(R, T ) = f1(R)+f2(R)f3(T ) with f1(R) = R, f2(R) =
Rm and f3(T ) = T n which involves the explicit non-minimal gravitational
matter geometry coupling. In a recent work [67], we have reconstructed
such type of f(R, T ) models corresponding to power law solutions. The
attractiveness of gravity implies that 1+ ηnRmT n−1/1+ ηmRm−1T n >
0. The energy condition constraints for this model can be represented as
η|R|m|T |n1− ηn|R|m|T |n−1
[D1 + LmT−1D2] > D3, (3.3.9)
where Di’s can have particular relations depending on the energy condi-
tions which are shown in appendix A3. We study the WEC inequality for
this model and develop the constraints as for the exponential model. The
condition to meet the WEC is given by
ρ + ηRmρn
[n(1− Lmρ−1) +
1
2(1−m)− 3m(m− 1)HRR−2
− 3mnHρR−1ρ−1]
> 0.
For Lm = p, the above inequality can be represented in the form of decel-
eration and jerk parameters as
2ρ + 2η[6H2(1− q)]mρn
[2n + 1−m + m(m− 1)
j − q − 2
(1− q)2
+mnρρ−1
2(1− q)H
]> 0. (3.3.10)
The coupling parameter η is assumed to be positive so that the above
constraint is satisfied if one can meet the condition in square bracket. For
this purpose, ρ can be obtained using Eq.(3.3.8) in the form
ρ = −3Hρ1 + nη[6H2(1− q)]mρn−1
(1 + m(j−q−2)
3(1−q)
)
1 + n(n + 0.5)η[6H2(1− q)]mρn−1.
72
Substituting ρ in Eq.(3.3.10), it is found that WEC is satisfied if both the
constants m and n are positive. One can also examine the WEC constraint
for Lm = −ρ for which WEC can be met if m,n > 0 with coupling para-
meter being positive.
3.3.1 Power Law Solutions
It is important to study the existence of exact power law solutions corre-
sponding to different phases of cosmic evolution. Such solutions are par-
ticularly relevant because in FRW background they represent all possible
cosmological evolutions such as radiation dominated, matter dominated and
dark energy eras. We discuss power law solutions for two particular models
of f(R, T ) gravity. Alvarenga et al. [68] studied the energy conditions for
some models of the type f(R, T ) = R + 2f(T ) and analyze their stability
under matter perturbations. We establish the energy conditions constraints
for those f(R, T ) models which confirm the existence of power law solutions
in this modified theory. We assume two specific forms of f(R, T ) gravity,
f(R, T ) = f(R) + λT, f(R, T ) = R + 2f(T ).
We shall obtain the power law solutions for each case and hence the con-
straints set by the respective energy conditions.
• f(R, T ) = f(R) + λT
For f(R, T ) = f(R) + λT [69], the effective Einstein field equations are
given by Eq.(3.1.1) with
T effµν =
1
fR
[(1 + λ)Tαβ + (λp +
1
2λT )gαβ +
1
2(f −RfR)gαβ + (∇µ∇ν
− gµν2)fR] . (3.3.11)
73
The Friedmann equation and the trace of the field equations are
θ2 =3
fR
[ρ + λ(ρ + p) +
λT
2+
1
2(f −RfR)− θRfRR
], (3.3.12)
RfR + 32fR(R, T )− 2f = (1 + 3λ)T + 4λp, (3.3.13)
where θ = 3a/a is the expansion scalar. The standard matter satisfies the
following energy conservation equation
ρ = −θ(ρ + p). (3.3.14)
The field equations can be represented by Raychaudhuri equation as
θ +1
3θ2 = − 1
2fR
[ρ + 3p + 4λp− f + RfR + (3R + θR)fRR
+ 3R2fRRR
]. (3.3.15)
Combination of Raychaudhuri and Friedmann equations yields
R = −2(θ +2
3θ2). (3.3.16)
We assume that there exists an exact power law solution to the modified
field equations
a(t) = a0tm, (3.3.17)
where m > 0. If 0 < m < 1, then the resulting power law solution favors
decelerating expansion whereas for m > 1 it exhibits accelerating behavior.
For EoS p = ωρ, the energy conservation equation leads to
ρ(t) = ρ0t−3m(1+ω). (3.3.18)
Using Eq.(3.3.17) in (3.3.16), the scalar curvature becomes
R = −6m(2m− 1)t−2 = −ηmt−2, (3.3.19)
74
where ηm = 6m(2m − 1). We see that the sign of R depends on the value
of m, R > 0 if 0 < m < 12
and R < 0 for m > 12. Since m = 1
2leads to
vanishing of R, so we exclude this value of m in our discussion.
Using Eqs.(3.3.18) and (3.3.19), the Friedmann equation (3.3.12) can be
written in terms of Ricci scalar R, f and fR as
fRRR2 +m− 1
2RfR +
1− 2m
2f− (2m−1)Aρ0
(−R
ηm
) 3m(1+ω)2
= 0, (3.3.20)
where A = 1 + λ2(3− ω). This represents second order differential equation
for f(R) whose general solution is
f(R) = Xmω
(−R
ηm
) 3m(1+ω)2
+ C1R14(3−m−√δm) + C2R
14(3−m+
√δm), (3.3.21)
where
Xmω =4A(2m− 1)ρ0
3m2(3ω + 4)(ω + 1)−m(9ω + 13) + 2, δm = m2 + 10m + 1,
and C1, C2 are arbitrary integration constants. Since m > 0, so δm > 0
for cosmologically viable solutions. Xmω is found to be real-valued but it
diverges for 3m2(3ω +4)(ω +1)−m(9ω + 13) + 2 = 0, i.e., m and ω satisfy
any of the relations ω = 3−7m±√δm
6mor m = 13+9ω±√9ω2+66ω+73
6(ω+1)(3ω+4). Since R < 0,
so (−R/ηm) > 0 for all R, thus we have real-valued solution f(R, T ) =
f(R) + λT showing that the power law solution exists for this model. For
λ = 0, we obtain solution as in f(R) gravity [70]. To check whether the
f(R, T ) gravity reduces to GR, we need to put C1 = C2 = λ = 0. When
m = 23(1+ω)
and ρ0 = 43(1+ω)2
, this theory reduces to GR. We are interested
to construct the f(R, T ) model of the form αRn+λT . If we put m = 2n3(1+ω)
,
then f(R) is given by
f(R) = αnω(−R)n, (3.3.22)
75
where
αnω =23−2n3n−1nA(n(4n− 3(1 + ω))1−n(1 + ω)2n−2
(n2(6ω + 8)− n(9ω + 13) + 3(ω + 1)),
and hence f(R, T ) = αnω(−R)n + λT . This model represents the exact
Friedmann-like power law solution a ∝ t2n
3(1+ω) and the limit n → 1 with
λ = 0 leads to GR. For n = 1, our solutions represent ΛCDM model of the
form f(R, T ) = R + λT .
We can construct the phantom phase power law solution which leads to
big rip singularity. For this case, the scale factor and Hubble parameter are
expressed as
a(t) = a0(ts − t)−m, H(t) =m
ts − t.
The scale factor diverges within finite time (t → ts) leading to big rip
singularity for m > 1. The results for this case can be recovered just by
replacing m by −m in the previous section. Hence, the phantom phase
power law solution exists for f(R) + λT gravity.
The energy conditions (3.3.2)-(3.3.5) can be applied to constrain the
given f(R) model in the context of f(R, T ) gravity. We assume that fR > 0
to keep the effective gravitational constant positive. For f(R) + λT model,
the energy constraints in terms of present day values of H, q, j and s are
given by
NEC : (1 + λ)(ρ0 + p0)− 6H4(s0 − j0 + (q0 + 1)(q0 + 8))f0RR + H40 [6
× H0(j0 − q0 − 2)]2f0RRR > 0,
WEC : ρ0 +λ
2(3ρ0 − p0) +
1
2f0 + 3H2
0 (1− q0)f0R + 18H40 [j0 − q0 − 2]
× f0RR > 0, ρeff + peff > 0,
SEC : (ρ0 + 3p0) + 4p0λ− f0 − 6H20 (1− q0)f0R + 3[6H3
0 (j0 − q0 − 2)]2
× f0RRR − 18H40 [s0 + j0 + q2
0 + 7q0 + 4]f0RR > 0, ρeff + peff > 0,
76
DEC : (ρ0 − p0) + 2λ(ρ0 − p0) + f0 + 6H20 (1− q0)f0R − [6H3
0 (j0 − q0
− 2)]2f0RRR − 6H4[s0 + 5j0 + (q0 − 1)(q0 + 4)]f0RR > 0,
ρeff + peff > 0, ρeff > 0.
In order to present the concrete application of the above energy conditions,
we employ the exact power law solution of f(R) + λT gravity. The present
day values of q, j and s parameters are taken as [65] q0 = −0.81+0.14−0.14, j0 =
2.16+0.81−0.75 and s0 = −0.22+0.21
−0.19. We shall discuss the WEC requirement to
illustrate how the above conditions place bounds on f(R, T ) gravity. One
can see that the above conditions depend upon the present day value of
pressure p0, so for simplicity we assume p = 0.
Now, we take the power law solution as an objective model which is
given by
f(R, T ) = αn(−R)n + λT, (3.3.23)
where n is an integer and αn = 23−2n3n−1nA(4n2−3n)1−n
(8n2−13n+3). The constraints to
fulfill the WEC, i.e., ρeff > 0, ρeff + peff > 0, are respectively obtained as
(2 + 3λ)ρ0 + αn[6H20 (1− q0)]
n[B1(n2 − n)− n + 1] > 0, (3.3.24)
(1 + λ)ρ0 + αnn(n− 1)6H40 [6H2
0 (1− q0)]n−2[−(s0 − j0 + (q0 + 1)
× (q0 + 8))−B2(n− 2)] > 0, (3.3.25)
where B1 = (j0− q0− 2)/(1− q0)2 and B2 = (j0− q0− 2)2/(1− q0). As the
standard matter is assumed to satisfy the necessary energy conditions and
λ > 0, so (2 + 3λ)ρ0 > 0 and (1 + λ)ρ0 > 0. Hence, the inequality (3.3.24)
is reduced to
αn(3.3H0)2nβn > 0, where βn = B1(n
2 − n)− n + 1.
77
It is clear from the above expression that the result is trivial for n = 0, 1.
We consider the following two cases:
(i) αn > 0, the allowed values for n are n = 2, 3, 4, .... Notice that βn > 0
in the range n = 4, 5, 6, ... and βn < 0 for n = 2, 3.
(ii) αn < 0, the acceptable values of n are n = −1,−2, ... and in this
particular range we have βn < 0. Thus, the inequality ρeff > 0 is satisfied
for n = ...,−2,−1, 4, 5, ....For the validity of Eq.(3.3.25) except n = 0, 1 as the result is trivial for
this choice, the inequality is transformed to the following form
αn(3.3H0)2n−2µn > 0, where µn = (n2 − n)(2.054− 0.52n).
The results of the above inequality can be interpreted as:
(i) µn > 0, if n = 2, 3,−1,−2, ... and for µn < 0, the acceptable values of
n are n = 4, 5, 6, ....(ii) αn > 0 with acceptable range n = 2, 3, 4, ... and αn < 0, when
n = −1,−2,−3, .... Hence, the condition ρeff + peff > 0 is satisfied for
n = 2, 3.
• f(R, T ) = R + 2f(T )
Now, we construct the power law solutions for R+2f(T ) gravity, where
f(T ) is an arbitrary function of T . The effective Einstein field equations
are given by Eq.(3.1.1) with
T effµν = (1 + 2fT )Tαβ + (2pfT + f)gαβ,
The Friedmann equation and the trace equation can be obtained as
θ2 = 3[ρ + 2(ρ + p)fT + f ], R = −(ρ− 3p)− 2(ρ + p)fT − 4f. (3.3.26)
78
The field equations can be represented as the Raychaudhuri equation
θ +1
3θ2 = −1
2[(ρ + 3p) + 2(ρ + p)fT − 2f ] . (3.3.27)
Combining Eqs.(3.3.26) and (3.3.27), we can get the Ricci scalar R given
in Eq.(3.3.16). Using Eq.(3.3.18), the above Friedmann equation can be
written in terms of T, f(T ) and its derivative with respect to T as
TfT +T
2(1 + ω)+
(1− 3ω)f
2(1 + ω)− K(1− 3ω)T
23m(1+ω)
2(1 + ω)= 0, (3.3.28)
where K = 3m2(ρ0(1− 3ω))−2
3m(1+ω) . This is the first order differential equa-
tion in f(T ) whose solution is
f(T ) =T
ω − 3+ LmωT
23m(1+ω) + C1T
−(1−3ω)2(1+ω) , (3.3.29)
where Lmω = 9m3(1−3ω)(ρ0(1−3ω))−2
3m(1+ω)
4+3m(1−3ω)and C1 is an arbitrary constant of
integration, Lmω is finite and real-valued unless 4 + 3m(1 − 3ω) = 0. In
general, the function f(T ) is real-valued if m and ω do not satisfy the
relation m = −43(1−3ω)
and if ω > 3. Therefore, the power law solutions exist
for R + 2f(T ) gravity.
For m = 0, we have a = a0 so that H = R = 0, it represents the Einstein
static universe and the corresponding solution is
f(T ) =T
ω − 3+ C1T
−(1−3ω)2(1+ω) . (3.3.30)
The standard Einstein gravity can be recovered for the choice C1 = 0,
m = 23(1+ω)
and ρ0 = 43(1+ω)2
. In order to develop a more general form of
function f(T ), we put m = 2n3(1+ω)
, so that
f(T ) =T
ω − 3+ anωT n, (3.3.31)
where anω = 23−2n3n−1n3−2n(1−3ω)1−n(1+ω)2n−2
4(1+ω)+2n(1−3ω). We can ensure that this theory
reduces to GR for n = 1. It is remarked that phantom power law solutions
79
exist for this model too, which can be obtained in a similar fashion as in
the 1st case.
The effective energy density ρeff and effective pressure peff for this
particular f(R, T ) gravity are defined as
ρeff = ρ + 2(ρ + p)fT + f, peff = p− f. (3.3.32)
Using Eq.(3.3.32) in energy conditions (3.3.2)-(3.3.5), the following form is
obtained [68]
NEC : (ρ + p)[1 + 2fT ] > 0,
WEC : ρ + 2(ρ + p)fT + f > 0, ρeff + peff > 0,
SEC : ρ + 3p + 2(ρ + p)fT − 2f > 0, ρeff + peff > 0,
DEC : ρ− p + 2(ρ + p)fT + 2f > 0, ρeff > 0, ρeff + peff > 0.
To check how these conditions place bounds on power law solution (3.3.31)
in R + λT gravity, we put p = 0 so that T = ρ. Hence, the function f(ρ) is
of the form
f(ρ) = −ρ
3+ anρ
n, (3.3.33)
where an = 22(1−n)3n−1n3−2n
n+2. The constraints to accomplish the above energy
conditions are obtained as follows:
NEC :ρ
3+ 2nanρn > 0,
WEC :ρ
3+ 5nanρn > 0,
SEC :4ρ
3+ 2(2n + 1)anρn > 0,
DEC : ρ + (7n + 2)anρn > 0.
These conditions are trivially satisfied for n = 0, 1. The quantities nan, (2n+
1)an and (7n + 2)an are negative when n = −3,−4,−5, ... and positive
80
for n = −1, 2, 3, .... Since ρ is assumed to be positive, so it is obvious
that these conditions are satisfied within the range of n = −1, 2, 3, ....
3.4 Stability of Power Law Solutions
In this section, we are interested to study the stability of power law solutions
against linear perturbations in f(R, T ) gravity. First, we assume a general
solution H(t) = Hh(t) for the cosmological background of FRW universe
that satisfies Eqs.(3.3.12) and (3.3.26). The matter fluid is assumed to be
dust which can be represented in terms of Hh(t) as
ρh(t) = ρ0e−3R
Hh(t)dt, (3.4.1)
where ρ0 is an integration constant. Since the matter perturbations also
contribute to the stability, so we introduce perturbations in Hubble para-
meter and energy density to study the perturbation around the arbitrary
solution Hh(t) as follows [68]
H(t) = Hh(t)(1 + δ(t)), ρ(t) = ρh(1 + δm(t)). (3.4.2)
In the following, we develop perturbation equations for two specific cases
f(R, T ) = f(R) + λT and f(R, T ) = R + 2f(T ).
3.4.1 f(R, T ) = f(R) + λT
To study the linear perturbations, f(R) is expanded in powers of Rh eval-
uated at H(t) = Hh(t) as
f(R) = fh + fhR(R−Rh) +O2, (3.4.3)
81
where f(R) and its derivative are evaluated at Rh. The term O2 includes
all the terms proportional to the square or higher powers of R. The Ricci
scalar R at H(t) = Hh(t) is given by
Rh = −6(Hh + 2H2h). (3.4.4)
By introducing the expressions (3.4.2) and (3.4.3) in the FRW equation
(3.3.12), the equation for the perturbation δ(t) becomes
δ(t) + c(t)δ(t) =Aρh
3HhRhfhRR
δm, (3.4.5)
where
c(t) =d
dt
[ln
(H−1
h R2hf
hRR
)]+ Hh[2(
d
dt[ln(fh
R)])−1 − 1].
The conservation equation (3.3.14) implies the second perturbation equa-
tion as
δm(t) + 3Hh(t)δ(t) = 0. (3.4.6)
We can eliminate δ(t) from Eqs.(3.4.5) and (3.4.6) and arrive at the follow-
ing second-order perturbation equation
δm(t) + c1(t)δm(t) +Aρh
3HhRhfhRR
δm = 0, (3.4.7)
where
c1(t) =d
dt
[ln
(H−2
h R2hf
hRR
)]+ Hh[2(
d
dt[ln(fh
R)])−1 − 1].
Here, we consider the f(R, T ) model proposed in section 3.3.1 for the
dust case which is defined as f(R, T ) = αn(−R)n+λT . We evaluate f(R, T )
and its derivatives at H(t) = Hh(t) and hence the perturbation δm(t) is
given by
δm(t) = C+tµ+ + C−tµ− , (3.4.8)
82
where C± are arbitrary constants and
µ± =8n2 − 15n + 13
6(n− 1)±
√n2(8n2 − 15n + 3)2 + 18(8n3 − 21n2 + 16n− 3)ρ0
6n(n− 1).
To study the stability of perturbation given by Eq.(3.4.8), one needs to
check the signs of exponents µ±. The exponents are found to be negative
provided that n 6 −2, otherwise µ± would be positive and the perturbation
is unstable. The perturbation δ(t) is found to be
δ(t) =−1
3Hh
(C+µ+tν+ + C−µ−tν−), (3.4.9)
where ν± = µ±− 1. It can be seen that exponent ν+ is negative for n 6 −2
and ν− is always negative. Hence, as the time evolves the condition n 6 −2
ensures the decay of perturbations δ(t) and δm(t) which implies the stability
of power law solution for this f(R, T ) gravity.
3.4.2 f(R, T ) = R + 2f(T )
We explore the behavior of perturbations (3.4.2) for this f(R, T ) model and
expand the function f(T ) in powers of Th(= ρh) as
f(T ) = fh + fhT (T − Th) +O2. (3.4.10)
The function f(T ) and its derivatives are evaluated at T = Th. Using
Eqs.(3.4.2) and (3.4.10) in FRW equation (3.3.26), it follows that
(Th + 3ThfhT + 2T 2
hfhTT )δm(t) = 6H2
hδ(t). (3.4.11)
Combining Eqs.(3.4.6) and (3.4.11), the first order matter perturbation
equation is
δm(t) +1
2Hh
(Th + 3ThfhT + 2T 2
hfhTT )δm(t) = 0, (3.4.12)
83
which leads to
δm(t) = C4 exp
−1
2
∫CT dt
, CT =
Th
Hh
(1 + 3fhT + 2Thf
hTT ). (3.4.13)
The behavior of perturbation δ(t) can be seen from the relation
δ(t) =C4CT
6Hh
exp
−1
2
∫CT dt
. (3.4.14)
We explore the stability of power law model (proposed in section 3.3.1)
of the form
f(T ) = a1T + a2Tn, (3.4.15)
where a1 and a2 are parameters. One can evaluate the expression CT and
integral −12
∫CT dt for the model (3.4.15) as
CT =3
2n
[ρ0(3a1 + 1)t−2n+1 + a2ρ
n0n(2n + 1)t−2n2+1
], (3.4.16)
−1
2
∫CT dt =
3
8n(n− 1)
[ρ0(3a1 + 1)t−2(n−1) +
a2ρn0n(2n + 1)
n + 1
× t−2(n2−1)]. (3.4.17)
As the time evolves, we need to set the conditions for decay of perturbations.
It is obvious that expressions (3.4.16) and (3.4.17) decay as time increases
for the choice n > 1 which results in decay of δ(t) and δm(t). Hence, for
large values of t, perturbation decays which corresponds to the stability of
power law solutions for R + 2f(T ) gravity. We find that the conditions
developed for stability are compatible with some constraints to fulfil the
energy conditions. Hence, we may remark that power law solutions are
acceptable regarding to the stability, energy conditions and late time cosmic
acceleration.
Chapter 4
Anisotropic Universe Modelsin f (R, T ) Gravity
This chapter studies the anisotropic LRS BI model with perfect fluid as
matter content in f(R, T ) gravity. The field equations are presented for
specific model f(R) + λT in the background of LRS BI universe. The prin-
ciple of mean Hubble parameter variation is assumed which results in two
different laws for cosmic expansion. We find solutions of the field equations
for both cases and examine the future evolution of the corresponding phys-
ical and kinematical quantities. We also explore the validity of NEC and
determine that our solutions are consistent with recent observations.
The layout of the chapter is as follows. In section 4.1, we formulate the
field equations for LRS BI model in f(R, T ) gravity. Section 4.2 provides
solutions of the field equations and investigates physical behavior of the
model and kinematical parameters. Section 4.3 contains solutions for the
massless scalar field. In section 4.4, we employ the anisotropic feature
of spacetime and discuss the exponential and power law expansions. The
results of this chapter have been published in the form of two research
papers [71, 72].
84
85
4.1 f (R, T ) Gravity and Bianchi I Universe
Here, we take the generic dynamical equations developed in section 1.1.2
for matter part as perfect fluid in f(R, T ) gravity. We employ the trace
equation to formulate the specific form of the field equations. Substituting
f(R, T ) from Eq.(1.1.8) in (1.1.10), it follows that
κ2(Tαβ − 1
4Tgαβ) + fT (R, T )(Tαβ + pgαβ)− 1
4fT (R, T )(ρ + p)gαβ
− (Rαβ − 1
4Rgαβ)fR(R, T )− (
1
4gαβ2−∇α∇β)fR(R, T ) = 0. (4.1.1)
For fT = 0, this implies the field equations in f(R) gravity. We consider
the function f(R, T ) of the form [69]
f(R, T ) = f(R) + λT, (4.1.2)
where λ is a coupling parameter and λT represents correction to f(R) grav-
ity. The main reason behind the dier- ence on cosmology in ordinary f(R)
gravity and in the above f(R, T ) model is the non-trivial coupling between
matter and geometry. This choice involves explicit matter geometry cou-
pling which can produce significant results. For perfect fluid as matter
contents with Lm = −p, the corresponding field equations are obtained as(
A
A+ 2
B
B− 2
AB
AB− 2
B2
B2
)fR − 1
2
(A
A+ 2
B
B
)d
dtfR +
3
2
d2
dt2fR
= −3
2(κ2 + λ)(ρ + p), (4.1.3)
(A
A− 2
B
B+ 2
AB
AB− B2
B2
)fR +
(3
2
A
A− B
B
)d
dtfR − 1
2
d2
dt2fR
=1
2(κ2 + λ)(ρ + p), (4.1.4)
(B2
B2− A
A
)fR − 1
2
(A
A− 2
B
B
)d
dtfR − 1
2
d2
dt2fR =
1
2(κ2 + λ)
× (ρ + p), (4.1.5)
86
where over dot represents derivative with respect to cosmic time. The Ricci
scalar is
R = −2
(A
A+ 2
B
B+ 2
AB
AB+
B2
B2
). (4.1.6)
Equations (4.1.3)-(4.1.6) can be represented in the form of mean and
directional Hubble parameters as
(3H − 2HxHy + Hx
2)
fR − 3
2
(H
d
dtfR − d2
dt2fR
)= −3
2(κ2 + λ)
× (ρ + p), (4.1.7)(3H − 4Hy + 2HxHy + H2
x − 3H2y
)fR +
(9
2H − 4Hy
)d
dtfR − 1
2
d2
dt2fR
=1
2(κ2 + λ)(ρ + p), (4.1.8)
(−Hx −H2
x + H2y
)fR − 1
2(3H − 4Hy)
d
dtfR − 1
2
d2
dt2fR =
1
2(κ2 + λ)
× (ρ + p), (4.1.9)
R = −2(3H + 2HxHy + H2
x + 3H2y
), (4.1.10)
where H = (ln a) = 13(Hx + 2Hy) is the Hubble parameter and Hx = A
A,
Hy = Hz = BB
represent the directional Hubble parameters along x, y
and z axes, respectively. The corresponding average scale factor, volume,
expansion and shear scalars become
V = a3 = AB2, θ = ua;a =
A
A+ 2
B
B, σ2 =
1
2σabσ
ab =1
3
[A
A− B
B
]2
.
(4.1.11)
4.2 Solution of the Field Equations
To solve the field equation, we assume the variation law of mean Hubble
parameter defined by the relation
H = lV −m/3 = l(AB2)−m/3, l > 0, m > 0. (4.2.1)
87
This law has been used to formulate the exact solutions for anisotropic and
homogeneous Bianchi models in Einstein and MGTs [73]-[75, 76]. Berman
[77] proposed this law for FRW model which specifies constant value of q
and generates two discrete expansion laws. Using H and V for the LRS BI
model in Eq.(4.2.1), these laws are defined as
V =
c2e3lt, m = 0,
(mlt + c3)3/m, m 6= 0,
(4.2.2)
where ci are positive constants. The first law corresponds to de Sitter
expansion with the scale factor being an increasing function of cosmic time
as a(t) = a0eHt, H = l, a constant. The de Sitter model is convenient
tool to explain the present cosmic scenario which yields q = −1. The
second volumetric expansion law represents power law model with scale
factor a(t) = a0t1/m and q = m− 1. If m > 1 then such model develops the
decelerating behavior with q > 0 and for 0 < m < 1, we have accelerating
model of the universe. Subtracting Eq.(4.1.8) from (4.1.9) and after some
manipulation, it follows that
Hx −Hy =k
V F, (4.2.3)
where k > 0. The anisotropy parameter of expansion is
∆ =1
3
3∑j=1
(Hj −H
H
)2
=2
9
(Hx −Hy
H
)2
,
and using (4.2.3), we have
∆ = 6(σ
θ
)2
=
(k√
3V F
)2
. (4.2.4)
In the following, we discuss the above two cases separately.
88
4.2.1 Exponential Expansion Model
For the exponential expansion model with spatial volume V = c2e3lt, one
can find relations of A and B in the following form
A = c1/31 c
2/33 elt+ 2k
3
R1
V Fdt, B = c
1/31 c
−1/33 elt− k
3
R1
V Fdt. (4.2.5)
To find an explicit solution of the field equations, we assume a relation
between F and a as F ∝ an [74], which implies that
F = αenlt,
where α is the proportionality constant and n is any arbitrary constant. As
we are interested to discuss the exponential and power law expansions, so
it would be useful to assume unknown F in terms of these expansion laws.
This assists to reconstruct the f(R, T ) gravity depending upon the choice
of the scale factor. Using this value of F in Eq.(4.2.5), we obtain
A = c1/31 c
2/33 elt− 2k
3αl(n+3)e−(n+3)lt
, B = c1/31 c
−1/33 elt+ k
3αl(n+3)e−(n+3)lt
. (4.2.6)
For n > −3, we observe that the scale factors A(t) and B(t) are finite at
initial era which reveals that such model experiences no initial singularity,
while these diverge in future cosmic evolution. When n < −3, the scale
factors increase with time and approach to very large values as t →∞. For
n = −3, the model represents similar behavior in every direction. The di-
rectional, mean Hubble parameters and anisotropy parameter of expansion
turn out to be
Hx = l +2k
3αe−(n+3)lt, Hy = Hz = l − k
3αe−(n+3)lt, H = l,
∆ =2k2
9l2α2e−2(n+3)lt.
89
0 2 4 6 8 100
20 000
40 000
60 000
80 000
t
D
n=-2 n=0 n=2
Figure 4.1: Evolution of ∆ versus t for different values of n. We set l = 0.1,k = 3, and α = 0.05.
The parameter H is found to be constant whereas Hx and Hy are dynamical.
For n > −3, Hx and Hy become constant at t = 0 as well as for t → ∞.
These parameters vary from H by some constant at t = 0 and match for late
time comic evolution. As the constant being positive (negative), it would
increase (decrease) expansion on the x-axis and it decreases (increases)
expansion on y and z axes. For n = −3, Hx will increase from H by
a constant factor 2k3α
, while parameters Hy, Hz will decrease by a factor
k3α
. The anisotropy parameter of expansion results in finite values for early
cosmic times and vanishes as t →∞ for n > −3 as shown in Figure 4.1.
The deceleration parameter, expansion and shear scalars are given by
q = −1, θ = 3l = 3H, σ2 =k2
3α2e−2(n+3)lt. (4.2.7)
Cosmic volume V is an exponential function which expands with the in-
crease in time and becomes infinitely large for late times. Also, the expan-
sion scalar is generally uniform and hence the model would favor the uni-
form expansion. The deceleration parameter (q = −1) allows the existence
90
Ρ+p
0
2
4t
-6
-4
-2
Α
0.0
0.1
0.2
0.3
0.4
Ρ+p
0
2
4t
2
4
6
Α
-0.4
-0.3
-0.2
-0.1
0.0
Figure 4.2: Evolution of NEC for n = 2. The left graph shows that NECis satisfied for α < 0 and it is violated for α > 0 at the right side. We setl = λ = 0.1 and k = 3.
of accelerating model for this case which is in agreement with the current ob-
servations of SNeIa and CMB [1, 2]. Using Eq.(4.2.6) in Eqs.(4.1.7)-(4.1.9),
we obtain the following relation of energy density and pressure
ρ + p =−1
3α(8π + λ)
[2k2enlt−2(n+3)lt + 3b1e
nlt], (4.2.8)
where b1 = n(n−1)l2α2. This shows that the NEC is violated, i.e., ρ+p < 0
which implies that ω < −1. Matter component with ω < −1 is named as
“phantom energy” and is a possible candidate of the present accelerated
expansion. The phantom regime favors recent observational cosmology of
accelerated cosmic expansion. The behavior of NEC for different values of
α is displayed in Figure 4.2, which shows that NEC is violated for positive
values of α. Thus, we assume α > 0 for phantom universe. Equation (4.2.8)
implies the following dynamical variables of the perfect fluid
ρ =−1
3α(1 + ω)(8π + λ)
[2k2enlt−2(n+3)lt + 3b1e
nlt], (4.2.9)
p =−ω
3α(1 + ω)(8π + λ)
[2k2enlt−2(n+3)lt + 3b1e
nlt]. (4.2.10)
91
0 1 2 3 4 50
1
2
3
4
t
Ρ
n=-7
n=-6
n=-4
n=-1
0
5
10
t
0
5
10
n0
5
10
Ρ
Figure 4.3: The left graph shows the behavior of ρ for −6 < n ≤ 0 andn < −6, while the right graph presents the evolution of ρ for n > 0. We setl = λ = 0.1, k = 3 and α = 0.05.
For the phantom evolution of the universe, ρ decreases with cosmic
time and approaches to zero as t →∞ in the range of −6 < n ≤ 0. When
n < −6, ρ increases as time goes from zero to infinity and hence diverges.
Figure 4.3 shows that ρ decreases for n = −1,−4 and becomes uniform for
n = −6. However, the value of n = −7 shows increasing ρ for the future
evolution of the universe. If n > 0, ρ decreases with time but for large
values of n, it shows bouncing behavior as shown in right panel of Figure
4.3. For this model, the scalar curvature R and f(R, T ) are given by
R = − 2
3α2
[18l2α2 + k2e−2(n+3)lt
],
f(R, T ) =α
2(R + 3l(n2l + 3))enlt +
8π(1− 3ω) + λ(1− ω)
6α(8π + λ)(1 + ω)
× (2k2e−(n+6)lt + 3b1enlt).
4.2.2 Power Law Expansion Model
For m 6= 0, the spatial volume is given by Eq.(4.2.2) and the corresponding
deceleration parameter is q = m − 1. To obtain the accelerated expansion
model, we take m < 1. Solving the field equations (4.1.7)-(4.1.9), the scale
92
0 5 10 15 200
10
20
30
40
t
D
n=-0.5 n=0 n=0.5
Figure 4.4: Plot of ∆ versus t for different values of n. We set l = 0.1,k = c2 = 3, m = 0.9 and α = 0.05.
factors are found to be
A = c2/34 (mlt + c2)
1/me2k
3αl(m−n−3)(mlt+c2)1−
n+3m
,
B = c−1/34 (mlt + c2)
1/mek
3αl(n−m+3)(mlt+c2)1−
n+3m
. (4.2.11)
We discuss the evolution of the scale factors for two cases, i.e., m > n + 3
and m < n + 3 along with 0 < m < 1. If m > n + 3, the scale factor A
increases with time whereas B tends to zero. For m < n+3, the behavior of
scale factors is almost identical provided that n is always greater than −3 to
keep m positive. For scale factors (4.2.11), we get the following parameters
Hx = l(mlt + c2)−1 +
2k
3α(mlt + c2)
−n+3m , (4.2.12)
Hy = Hz = l(mlt + c2)−1 − k
3α(mlt + c2)
−n+3m , (4.2.13)
H = l(mlt + c2)−1, ∆ =
2k2
9l2α2(mlt + c2)
−2(n+3)
m . (4.2.14)
The Hubble parameters H, Hx, Hy and Hz become constant at the initial
epoch. As t → ∞, the values of these parameters tend to zero for n > −3
and become infinite for n < −3.
93
Ρ+p
0
2
4t
-6
-4
-2
Α
0.005
0.010
0.015 Ρ+p
0
2
4t
2
4
6
Α
-0.015
-0.010
-0.005
Figure 4.5: Behavior of NEC versus α for n = 3. The left part shows thatNEC is satisfied for α < 0, while it is violated for α > 0 shown on the rightside. We set l = λ = 0.1, k = c2 = 3 and m = 0.9.
If n < −3, 4 increases with cosmic time whereas for n > −3, its value
decreases and may result to isotropic expansion in future evolution of the
universe (see Figure 4.4). The expansion and shear scalars are
θ = 3l(mlt + c2)−1, σ2 =
k2
3α2(mlt + c2)
−2(n+3)
m . (4.2.15)
If we replace Eq.(4.2.11) in Eqs.(4.1.7)-(4.1.9), we obtain
ρ + p =−1
3α(8π + λ)
[2k2(mlt + c2)
− (n+6)m
+ 3b2(mlt + c2)(n−2m)
m
], (4.2.16)
where b2 = (n(n− 1)−m(n + 2))l2α2, ρ and p are obtained as follows
ρ =−1
3α(1 + ω)(8π + λ)
[2k2(mlt + c2)
− (n+6)m
+ 3b2(mlt + c2)(n−2m)
m
], (4.2.17)
p =−ω
3α(1 + ω)(8π + λ)
[2k2(mlt + c2)
− (n+6)m
+ 3b2(mlt + c2)(n−2m)
m
]. (4.2.18)
94
Ρ+p
0
2
4t
-6
-4
-2
Α
-0.005
-0.004
-0.003
-0.002
-0.001
Ρ+p
0
2
4t
2
4
6
Α
0.001
0.002
0.003
0.004
0.005
Figure 4.6: This figure is plotted for n = 2. The left part shows that NECis violated for α < 0, whereas NEC is satisfied for α > 0 shown on rightside.
Equation (4.2.16) shows that NEC is violated for the power law expan-
sion model. The behavior of NEC is shown in Figures 4.5-4.6 which de-
pends on the choice of α as well as n. For each value of n except 2 ≤ n ≤ 0,
NEC can be satisfied for α < 0 but the choice α > 0 does not support it.
If 2 ≤ n ≤ 0, the constraints to satisfy and violate NEC are interchanged
(see Figure 4.6). For m < 1, energy density decreases in the range of
−6 < n ≤ 0 and increases with cosmic time t for n > −7. For n > 0,
the behavior of ρ is shown in Figure 4.7. From Eqs.(1.1.8) and (4.1.6), the
Ricci scalar R and f(R, T ) are given by
R =2
3α2
[9l2α2(m− 2)(mlt + c2)
−2 − k2(mlt + c2)−2(n+3
m)],
f(R, T ) =α
2
[R(mlt + c2)
nm + 3n(n−m)l2(mlt + c2)
nm−2
+ 9l(mlt + c2)nm−1
]+
8π(1− 3ω) + λ(1− ω)
6α(8π + λ)(1 + ω)(2k2(mlt + c2)
−n+6m
+ 3b2(mlt + c2)nm−2).
95
0
2
4
6
8
t
0
2
4
6
8
n
0
5
10
15
20
Ρ
Figure 4.7: Evolution of ρ versus t for n ≥ 0. We set l = λ = 0.1, k = c2 = 3,m = 0.9 and α = 0.05.
4.3 Massless Scalar Field Models
The Lagrangian for massless scalar field φ is given by [78]
Lm = −1
2gµν∂µφ∂νφ, (4.3.1)
and the corresponding energy-momentum tensor is
Tµν = ∂µφ∂νφ− 1
2gµν∂γφ∂γφ. (4.3.2)
Here, Tµν represents stiff matter with EoS ωφ = 1. Using Eqs.(4.1.1) and
(4.3.2), we obtain the following field equations for massless scalar field(
A
A+ 2
B
B− 2
AB
AB− 2
B2
B2
)F +
3
2F − 1
2
(A
A+ 2
B
B
)F
= −3
2(8π + λ)φ2, (4.3.3)
(A
A− 2
B
B+ 2
AB
AB− 2
B2
B2
)F − 1
2F +
(3
2
A
A− B
B
)F
= −1
2(8π + λ)φ2, (4.3.4)
(B2
B2− A
A
)F − 1
2F − 1
2
(A
A− 2
B
B
)F =
1
2(8π + λ)φ2.(4.3.5)
96
Φ+
Φ-
0 2 4 6 8
-5
0
5
t
Φ
Figure 4.8: Evolution of φ versus t for m = 0 and different values of n:solid(black) n = 1; dashed(red), n = 0; dahsed(blue), n = −1. We setl = λ = 0.1, k = 3 and α = 0.05.
The field equations with massless scalar field are similar to the perfect
fluid case with ρφ + pφ = φ2, hence we obtain the same results for the scale
factors and other physical parameters. Substituting Eq.(4.2.6) in (4.3.3)-
(4.3.5), the time derivative of scalar field φ is
φ = ±√
−1
3α(8π + λ)[2k2enlt−2(n+3)lt + 3b1enlt]. (4.3.6)
Using Eqs.(4.2.6) and (4.3.6) in (1.1.8), it follows that
f(R, T ) =α
2(R + 3l(n2l + 3))enlt − 4π
3α(8π + λ)(2k2e−(n+6)lt + 3b1e
nlt).
(4.3.7)
The behavior of φ for exponential expansion is shown in Figure 4.8. If we
solve Eqs.(4.3.3)-(4.3.5) for the case m 6= 0, we get the similar solutions as
given in section 4.2.2. The expression of φ is obtained as follows
φ = ±√
−1
3α(8π + λ)
[2k2(mlt + c2)
− (n+6)m + 3b2(mlt + c2)
(n−2m)m
]. (4.3.8)
Evolution of φ versus cosmic time t for different values of n is shown in
97
Φ+
Φ-0 2 4 6 8 10
-1.0
-0.5
0.0
0.5
1.0
t
Φ
Figure 4.9: Evolution of φ versus t for m 6= 0 and different values of n:solid(black) n = 1; dashed(red), n = 0; dahsed(blue), n = −1. We setl = λ = 0.1, k = c2 = 3, m = 0.9 and α = 0.05.
Figure 4.9. Substituting Eqs.(4.2.11) and (4.3.8) in Eq.(1.1.8), we have
f(R, T ) =α
2
[R(mlt + c2)
nm + 3n(n−m)l2(mlt + c2)
nm−2
+ 9l(mlt + c2)nm−1
]− 4π
3α(8π + λ)
[2k2(mlt + c2)
−n+6m
+ 3b2(mlt + c2)nm−2
]. (4.3.9)
4.4 Solutions for Fixed Anisotropy Parame-
ter
Here, we are interested to solve the field equations by assuming known
anisotropy parameter in terms of some small constant. In section 4.2, we
have used f(R, T ) = f(R)+λT model to reconstruct some functions corre-
sponding to LRS BI model and considered an ansatz for f(R) function to
find the exact solutions. Now, we reconstruct the function f(R, T ) without
taking ansatz about f(R).
98
It has been suggested that normal congruence to the homogeneous ex-
pansion for spatially homogeneous metric yields σ/θ ≈ 0.3 [79]. Bunn et al.
[80] performed statistical analysis on 4-yr data from CMB and set a limit
for primordial anisotropy to be less than 10−3 in Planck era. As the Bianchi
models represent the anisotropic universe, so one can choose the anisotropy
parameter of expansion to be constant say γ. In this setting, Harko and
Mak [81] studied the anisotropy issue for Bianchi I and V universe models
in braneworld cosmology. Yilmaz et al. [82] employed this condition to
explore the behavior of quark matter and strange quark matter in f(R)
gravity for anisotropic Bianchi models. Several authors [?, 76] used this
relation to address different issues in Einstein and modified gravities. From
Eq.(4.2.4), we have
σ
θ=
χ√3V F
= γ, (4.4.1)
which implies that
F =χ√3γV
. (4.4.2)
Now, we assume the exponential and power law expansion models to
find the exact solutions and explore the corresponding behavior of LRS BI
model.
• Exponential Expansion Model
For this model, the explicit relations of A and B are obtained as
A = c2/31 c
1/32 el(1+2
√3γ)t, B = c
−1/31 c
1/32 el(1−√3γ)t. (4.4.3)
The scale factors A and B are found to be finite at initial era which reveals
that such model experiences no initial singularity. If γ > 0 then A(t)
would increase exponentially and carry indefinite values as t →∞ whereas
99
Ρ+p
0
5
10t
0
5
10
Λ
-400
-300
-200
-100
0
(a)
Ρ+p
0
5
10t
-30
-29
-28
-27
-26Λ
0
2000
4000
(b)
Figure 4.10: Evolution of NEC for exponential model. The NEC is violatedfor (a) λ > 0 whereas it can be satisfied for (b) λ < −25. We set χ = 3, l =c1 = 0.1 and γ = 0.001.
0 2 4 6 8 100
100
200
300
400
500
t
Ρ
Λ=-5 Λ=0 Λ=5
Figure 4.11: Plot of energy density corresponding to different values ofcoupling parameter.
B(t) approaches to some constant and vice-versa for γ < 0. The mean
and directional Hubble parameters are finite for this model. Substituting
Eq.(4.4.3) in (4.1.7), we obtain
ρ + p =−2χl(6γ4 − 2γ2 + 3)e−3lt
√3γc1(κ2 + λ)
. (4.4.4)
Now we explore the behavior of NEC for this expansion model. It is
clear from the above expression that the validity of NEC depends upon
100
the constants χ, l, c1, γ and coupling parameter λ. Since constants other
than λ are selected as positive, so we examine the NEC depending upon the
values of parameter λ. If λ > 0 then NEC is violated so that EoS parameter
< −1 which is shown in Figure 4.10(a). The candidate of exotic matter
corresponding to ω < −1 is termed as phantom energy and is responsible
for the present accelerating phase which favors the current observational
data [1, 2]. The NEC can be met only if the denominator of the above
relation is negative which is shown in Figure 4.10(b). The energy density is
plotted for different values of coupling parameter λ in the spirit of phantom
regime. Figure 4.11 shows that energy density is a decreasing function of
cosmic time. Substituting Eqs.(4.4.2) and (4.4.3) in (1.1.8), one can get the
expression of f(R, T ) for the above model
f(R, T ) =χe−3lt
6√
3γc1l(9l(3l + 1)−R) +
χe−3lt
√3γ(κ2 + λ)(1 + ω)c1
l(6γ4
− 2γ2 + 3)(κ2(1− 3ω) + λ(1 + ω)). (4.4.5)
• Power Law Expansion Model
We take power law expansion model defined by the scale factor a(t) =
(mlt + c3)1/m and deceleration parameter q = m− 1. This universe model
favors the accelerated expansion for 0 < m < 1 whereas for m > 1, it
describes the decelerating phase. Solving the field equations, we get the
scale factors as
A = c2/31 (mlt + c3)
1+2√
3γm , B = c
−1/31 (mlt + c3)
1−√3γm . (4.4.6)
At the present era, these scale factors are finite while in future epoch the
evolution of A and B depends upon constant γ. Using Eq.(4.4.6), we obtain
101
Ρ+p
0
5
10
t
-20
0
20
Λ
-0.002
-0.001
0.000
(a)
Ρ+p
0
5
10
t
-50
-40
-30Λ
0.001
0.002
0.003
0.004
(b)
Figure 4.12: Evolution of NEC for power law model. Plot (a) shows thatNEC is violated for λ > −25 which favors the accelerated expansion. It canbe met for λ < −25 as shown in plot (b).
H, Hx and Hy as
H = l(mlt + c3)−1, Hx = l(1 + 2
√3γ)(mlt + c3)
−1,
Hy = l(1−√
3γ)(mlt + c3)−1.
These parameters result in constant values for the present epoch and ap-
proach to zero for future epoch. Substituting Eq.(4.4.6) in (4.1.7), we obtain
the relation of NEC for the above model
ρ + p =χl(36γ4 − 36γ2 + 6m− 5)e−3lt
3√
3γ(κ2 + λ)(mlt + c3)(m+3)/m. (4.4.7)
This shows that ρ+p depends upon the choice of different constants. We
examine its dependence on the coupling parameter λ with other parameters
being positive. It is evident from Figure 4.12 that ρ + p < 0, i.e., NEC is
violated for λ > −25 whereas it can be satisfied if λ < −25. The dependence
of NEC on parameter m can be seen from Figure 4.13 where we have set
0 < m < 1. This choice of m favors the phantom regime which would
102
Ρ+p 0
5
10
t
0.2
0.4
0.6
0.8
1.0
m
-0.0006
-0.0004
-0.0002
0.0000
Figure 4.13: Evolution of NEC for 0 < m < 1 with λ = 10. For λ < −25,we have ρ + p > 0.
0
5
10
t
-10-5
05
10Λ
0.0004
0.0006
0.0008
Ρ
(a)
0
5
10
15
20t
0.2
0.4
0.6
0.8m
0.00000
0.00005
0.00010
0.00015
0.00020
Ρ
(b)
Figure 4.14: Evolution of ρ for power law model. In plot (a) we set m = 0.9and −10 < λ < 10 whereas in plot (b) we vary m in the range of phantomevolution and set λ = 0.1
103
violate the NEC as depicted in this plot. We also explore the behavior of
energy density for power law model and its evolution is presented for both
cases depending on the choice of parameters m and λ. In plot 4.14(a), we
fix m and vary λ and vice-versa for plot 4.14(b). The values of m and λ
are selected on the basis of phantom cosmology. The Ricci scalar R and
function f(R, T ) for power law model are found as
R = 6S2(m− 2− 3γ2)(mlt + c3),
f(R, T ) =(mlt + c3)
2m−3m R
4√
3γ(m− 2− 3γ2)l2+
lχ(mlt + c3)−(m+3)
m
2√
3γ
[l(m− 5 + 6γ2)
− (κ2(1− 3ω) + λ(1 + ω))(36γ4 − 36γ2 + 6m− 10)
3(κ2 + λ)(1 + ω)
]. (4.4.8)
Chapter 5
Discussion and Conclusion
The late time accelerated cosmic expansion is a major issue in cosmology.
Modified theories of gravity have appeared as convenient candidates to ad-
dress such issues and predict the destiny of the universe. Theories involving
CMC have attained significant importance to explore the enigma of cosmic
evolution and other cosmological aspects. In this setting, f(R, T ) gravity
can be reckoned as an effective campaigner of dark components with no
need of introducing either the existence of extra spatial dimension or an
exotic component of DE. Such theory is of great importance as the source
of DE components can be seen from an integrated contribution of both cur-
vature and matter Lagrangian parts. The matter geometry coupling results
in existence of extra force due to non-geodesic motion of test particles.
f(R, T,Q) is another more general modified gravity formulated on the
basis of CMC. This theory involves the contraction of Ricci tensor and mater
energy-momentum tensor Q = RµνTµν and can be regarded as an extended
form of f(R, T ) gravity. However, there is a significant difference in the
results of this theory with the rest of modifications. For instance, if one
considers the role of electromagnetic field or radiation dominated fluid (i.e.,
trace free energy-momentum tensor), the field equations in f(R, T ) gravity
104
105
reduce to that in f(R) gravity. Thus, the contribution of non-minimal
coupling would disappear in f(R, T ) gravity whereas in f(R, T,Q) gravity,
the effect of non-minimal coupling can be assured due to the contribution of
contraction term Q. Indeed, in Lagrangian (1.1.15) the interaction between
matter and geometry can be seen through the coupling of energy-momentum
and Ricci tensors, Q is the generic term responsible for the non-minimal
coupling as compared to other modified theories. Such behavior becomes
more explicit for the models of the type R + αQ and R(1 + αQ). However,
still one can formulate more complex models and discuss the cosmological
features.
In fact, the fundamental characteristic of theories involving non-minimal
matter geometry coupling is the non-conserved energy-momentum tensor
produced from the divergence of the field equations. As a result, motion of
test particles is non-geodesic and an extra force orthogonal to four-velocity
of the particle is present due to matter geometry coupling. This is consistent
with the interpretation of four force which states that component of force
orthogonal to particle’s four-velocity can influence its trajectory. It has
been shown that the extra force vanishes if one uses the matter Lagrangian
of the form Lm = p [83] for non-minimally coupled f(R) theories (as given
in [84, 85]). However, the extra force generated by the matter geometry
coupling does not vanish in this modified theory even for Lm = p. It also
involves the contribution from the Ricci tensor and may lead to significant
deviation from the geodesic paths. The extra force can be useful to explain
the DM properties and Pioneer anomaly. One can count the additional
curvature obtained from the CMC to inform the galactic rotation curves.
This theory can also present novel views about the early stages of cosmic
106
evolution specifically the inflationary paradigm.
In this thesis, we have discussed the thermodynamic laws at the AH of
FRW universe for f(R, T ) and f(R, T,Q) theories. In particular, we have
presented the non-equilibrium description of thermodynamics and explored
the existence of equilibrium picture. We have also addressed the question
how to constrain the various forms of Lagrangian in these theories on phys-
ical grounds employing the energy conditions and test the stability criteria.
The anisotropic universe models are also discussed to present the cosmic
evolution and reconstruct the corresponding form of Lagrangian. The sig-
nificant findings have been listed below.
In Chapter 2, it is shown that the representation of equilibrium thermo-
dynamics is not executable in f(R, T ) theory. Thus, the non-equilibrium
treatment of thermodynamics is used to discuss the laws of thermodynam-
ics. The FLT is formulated by using the Wald’s entropy relation. It is
found that additional entropy term is produced as a result of CMC. The
gravitational coupling between matter and higher derivative terms of cur-
vature describes a transfer of energy and momentum across the horizon
which can be the explanation for non-equilibrium picture. It is worth
mentioning here that no such term is present in Einstein, Gauss-Bonnet,
Lovelock and braneworld modified theories [19]-[54]. The equilibrium and
non-equilibrium descriptions of thermodynamics in f(R) gravity can be re-
produced from these results.
We have also explored the validity of GSLT at the AH of FRW uni-
verse in this modified theory. The time evolution of entropy Stot (contribu-
tions from horizon entropy and entropy associated with the matter contents
within the horizon) is presented in a comprehensive way. We have assumed
107
the proportionality relation between the temperatures related to AH and
matter components inside the horizon. The condition Tin = bTh relates the
temperature of ingredients inside the horizon to the temperature of AH.
We adopt the proposal of thermal equilibrium and set b = 1 to explore the
validity of GSLT. It is concluded that GSLT is satisfied in both phantom
and quintessence regimes of the universe.
In f(R, T,Q) gravity, the general formalism of the field equations for
FRW spacetime with any spatial curvature is presented. It is shown that
these equations can be cast to the form of FLT, ThdSh + ThdS = δQ, in
non-equilibrium description of thermodynamics. In this structure of FLT,
we have found that entropy Seff = Sh + S involves contribution from two
factors, the first corresponds to horizon entropy in terms of area and sec-
ond represents the entropy production term dS which is produced due to
the non-equilibrium description in f(R, T,Q) gravity. This shows that one
may need the non-equilibrium treatment of thermodynamics in this the-
ory. The entropy production term in f(R, T,Q) gravity is more general
and can reproduce the corresponding factor in f(R) and f(R, T ) theories.
This modified theory involves strong coupling resulting from the contrac-
tion of Ricci tensor and matter energy-momentum tensor. Due to this
interaction, the entropy production term would be ultimate in f(R, T,Q)
gravity. Different schemes have been suggested to avoid the auxiliary term
in FLT [21, 23, ?, 86] in f(R) and scalar-tensor theories. Bamba et al.
[21, 23] showed that one can redefine the energy-momentum tensor contri-
bution from the modified theories so that the conservation equation is truly
satisfied and hence leads to the omission of entropy production term. How-
ever, we have seen that such procedure is not fruitful in f(R, T ) theory and
108
equilibrium thermodynamics needs more study to follow. Since f(R, T,Q)
gravity is a more general theory, so the non-equilibrium representation of
thermodynamics is shown in the present work.
In order to get insights of GSLT in thermal equilibrium, we have taken
two specific gravitational models and developed the constraints for the ac-
celerated cosmic expansion with m > 1. We have found that for flat FRW
universe in power law cosmology, the validity of GSLT depends upon the
coupling parameter α as shown in Figures 2.1-2.2. The viability constraints
are given as follows:
• f(R, T,Q) = R + αQ,
If Lm = −p then GSLT is satisfied for α < 0, m > 1.
If Lm = ρ then GSLT is satisfied for α < 0, m > 8.
• f(R, T,Q) = R(1 + αQ),
For Lm = −p, GSLT is satisfied for α > 0, m > 1.
For Lm = ρ, GSLT is satisfied for α > 0, m > 3.
We have also considered the Lagrangian f(R, T,Q) = R + f(Q) + G(T )
[14] to validate the GSLT in de Sitter background and results are shown
in Figure 2.3. Thus, we have presented the validity of GSLT in expanding
universe with the assumption of thermal equilibrium which may be achieved
in later times. Finally, we conclude that our study on non-equilibrium
thermodynamics in f(R, T ) and f(R, T,Q) gravity is consistent with the
statements given in [20, 55, 21, 23].
Chapter 3 presents the energy conditions constraints and stability of
f(R, T ) and f(R, T,Q) theories. Lagrangian of f(R, T,Q) gravity is more
109
comprehensive implying that different functional forms of f can be sug-
gested. The versatility in Lagrangian raises the question that how to con-
strain such a theory on physical grounds. We have developed some con-
straints for general as well as specific forms of f(R, T,Q) gravity by exam-
ining the respective energy conditions. The NEC and SEC are derived using
the Raychaudhuri equation along with the condition that gravity is attrac-
tive. Moreover, these inequalities are equivalent to the results found from
conditions ρ + 3p > 0 and ρ + p > 0 under the transformations ρ → ρeff
and p → peff , respectively. One can employ the similar procedure to derive
the WEC and DEC by translating their counterpart in GR for effective
energy-momentum tensor. The conditions of positive effective gravitational
coupling and attractive nature of gravity are also obtained in this theory.
To illustrate how these conditions can constrain the f(R, T,Q) gravity,
we have taken two functional forms of f namely, f = R + αQ and f =
R(1 + αQ). It is shown that WEC for these models depends upon the
coupling parameter α which is satisfied only if α is negative. We have also
set the Dolgov-Kawasaki criterion in this discussion. The f(R, T ) gravity
is addressed as a specific case to this modified theory. We have taken two
interesting choices for the Lagrangian, one involving an exponential function
and other having explicit coupling between R and T . The validity of WEC
for both choices of matter Lagrangian Lm = p and Lm = −ρ have been
explored. The WEC for f(R, T ) = αexp(
Rα
+ λT)
is met in both cases if
coupling parameter λ > 0 and α > 2H20 (j0−q0−2)(or α > 4H2
0 (j0−q0−2)),
respectively. For the model f(R, T ) = R + ηRmT n, the WEC is satisfied if
both the constants m and n are positive.
We have discussed the power law solutions for two particular cases and
110
derived the corresponding energy bounds. We summarize the results of
these two models as follows.
• f(R) + λT
It is shown that exact power law solution exists for this form of f(R, T )
gravity given in Eq.(3.3.21). In the limit of λ = 0, the corresponding result
can be recovered in f(R) gravity. To ensure that this theory reduces to GR,
we need to set C1 = C2 = λ = 0 with m = 23(1+ω)
and ρ0 = 43(1+ω)2
. We have
constructed the general form of f(R) + λT model which corresponds to Rn
gravity. For this particular model, we have examined the WEC bounds in
terms of present day observational values H0, q0, j0 and s0.
• R + 2f(T )
A general form of f(T ) model (3.3.31) is obtained which corresponds
to GR in the limit n = 1. We have applied the energy conditions to set
the possible constraints on this f(R, T ) model. It is found that energy
conditions are globally satisfied within the range of n = −1, 2, 3, .... It
is worth mentioning here that results of power law solutions and energy
conditions found here are quite general which correspond to GR and f(R)
gravity.
We have also analyzed the stability of power law solutions under linear
homogeneous perturbations in the FRW background for f(R, T ) gravity. In
particular, perturbations for energy density and Hubble parameter are in-
troduced which produce linearized perturbed field equations. The stability
conditions are found to be compatible with energy conditions bounds to
some extent. Hence, power law solutions in f(R, T ) gravity can be consid-
ered as viable models to explain the cosmic expansion history.
111
Chapter 4 investigates the homogeneous but not necessarily isotropic
models in the context of f(R, T ) gravity. We have employed the f(R, T ) =
f(R) + λT choice to reconstruct some explicit models of f(R, T ) gravity
for LRS BI universe. The exact solutions of the modified field equations
are obtained for the LRS BI universe with perfect fluid and massless scalar
field. The law of variation of mean Hubble parameter is assumed implying
two cosmological models for m = 0 and m 6= 0 which support the recent
observations about the accelerated cosmic expansion. We have presented
physical properties as well as kinematical parameters of the models. In the
following, we summarize the results for these two models.
(i) Model for V = c1e3lt
For exponential expansion model, the accelerated expansion of the universe
may occur as q = −1. The kinematical parameters have been discussed for
two cases n > −3 and n < −3. The expansion scalar is constant, while the
Ricci scalar approaches to constant value as t → ∞ for n > −3 and take
infinitely large values for n < −3. The anisotropy parameter of expansion
depends upon time and vanishes in future evolution for n > −3. If α > 0,
NEC is violated. We are not able to find the explicit function of f(R, T )
by using Eq.(1.1.8). For λ = 0, we develop f(R) in terms of R and hence
the function f(R, T ) as
f(R) =
[2αω
1 + ω(R + 12H2) +
1
2α3α2H((n2 − 4)H + 3) +
1− 3ω
1 + ωb1
]enlt,
which can be expressed as
f(R1) = const1 ×R1m1 + const2 ×R1
m2 ,
112
where R1 = R+12H2, m1 = n+62(n+3)
and m2 = m1−1. The models of f(R1)
depending on n are shown in Table 5.1.
Table 5.1: Models of f(R1) corresponding to n
n f(R1)
n = 0 R1 + R01, R0
1 = const
n = −2 R1 + R21
n = −6 R01 + 1
R1
n = −4 R−11 + R−2
1
n = −125 R3
1 + R21
n = −185 R−2
1 + R−31
n = −32 R
321 + R
121
n = −92 R
−12
1 + R−32
1
For n = 0, f(R) represents the ΛCDM model, i.e., f(R) = R + Λ. If we
put constant = 0, then f(R, T ) is of the form f(R, T ) = R + T . The most
famous Starobinsky’s model [87], f(R) = R + αR2 is achieved for n = −2
and the corresponding f(R, T ) function is f(R, T ) = R + αR2 + T . For
n = −6, f(R, T ) can be presented as f(R, T ) = 1R
+ T . For massless scalar
field (m = 0), we have found similar results for scale factors as in perfect
fluid. The expression of f(R) is
f(R) =
[α(R + 12H2) +
1
2α3α2H((n2 − 4)H + 3)− b1
]enlt.
i.e., f(R1) = const3 ×R1m1 + const4.×R1
m2 .
(ii) Model for V = (mlt + c2)3/m
For m 6= 0, the deceleration parameter is q = m − 1, which leads to the
accelerating universe model for 0 < m < 1 and if m > 1(q > 0), the model
113
represents decelerating phase of the universe. The evolution of the scale
factors is discussed for two cases m > n+3 and m < n+3 with 0 < m < 1.
The anisotropy parameter of expansion increases for n < −3, whereas it
may result in isotropic expansion in future evolution of the universe for
n > −3. The Hubble parameter, expansion scalar and shear scalar approach
to constant at earlier times of the universe and tend to zero as t →∞. The
scalar curvature R becomes constant as t → ∞ for n > −3, whereas it
diverges for n < −3. When λ = 0, we have
f(R) = (mlt + c2)n/m
[2αω
1 + ω(R− 6(m− 2)H2) +
α
2H3H(n2
+ m− 4) +1− 3ω
α2l2(1 + ω)b2H + 9
],
which leads to
f(R2) = const5 ×R2m1 + const6 ×R2
m2 ,
where R2 = R− 6(m− 2)H2. In case of massless scalar field, f(R) is
f(R) = (mlt + c2)n/m
[α(R− 6(m− 2)H2) +
α
2H3H(n2 + m
− 4) +b2H
α2l2+ 9
],
i.e., f(R2) = const7 × R2m1 + const8 × R2
m2 . We have seen that all f(R)
represent identical behavior with different constraints. The NEC is found
to be violated for both models m = 0 and m 6= 0 which results in phantom
evolution. For ω < −1, energy density is found to be positive and pressure
is negative. Thus, our solutions for perfect fluid represent the phantom era
of DE. The isotropic behavior of models is observed for future evolution.
Although the anisotropy of CMB is restricted on cosmological scales
114
which can favor the FRW universe but allowing local anisotropy while pre-
serving isotropic expansion dynamics is possible at least on phenomenolog-
ical level. However, in section 4.4 we have considered that the deviations
from isotropy are small, and we have restricted our analysis to ansatz that
anisotropy parameter of expansion ∆ being small constant. We have dis-
cussed the above models in this setting. We have examined the evolution
of NEC counting the value of coupling parameter λ for model (i). For
λ > 0, NEC is violated resulting in EoS parameter ω < −1 which can be
met if λ < −25. The energy density turns out to be constant at present
era and gets smaller values depending on time. For exponential model, the
expression of f(R, T ) is given
f(R, T ) =χe−3lt
6√
3γc1l(9l(3l + 1)−R) +
χe−3lt
√3γ(κ2 + λ)(1 + ω)c1
l(6γ4
− 2γ2 + 3)(κ2(1− 3ω) + λ(1 + ω)).
The above expression depends upon R as well as exponential function of
time t which is due to scale factors but still one can represent the above
expression in terms of R and T .
From Eq.(4.4.4), we represent e−3lt in terms of T as
e−3lt = C1T, C1 =(1 + ω)
√3γc1(κ
2 + λ)
2lχ(1− 3ω)(2γ2 − 6γ4 − 3),
so that f(R, T ) is given by
f(R, T ) = α1T + α2RT.
For case (ii), we have explored the validity of NEC for both cases depending
on coupling parameter λ and m. For λ, the situation is similar to that for
exponential model and NEC is violated for 0 < m < 1 with λ > −25. In
this case, we can construct some explicit f(R, T ) models. For this purpose,
115
we choose λ = 0 to formulate f(R) in terms of R and hence the Lagrangian
f(R, T ). For λ = 0, Eq.(4.4.8) implies that
f(R, T ) =(mlt + c3)
2m−3m R
4√
3γ(m− 2− 3γ2)l2+
lχ(mlt + c3)−(m+3)
m
2√
3γ
[l(m− 5 + 6γ2)
− (1− 3ω)
3(1 + ω)(36γ4 − 36γ2 + 6m− 10)
],
which can be represented as
f(R2) = const1 ×R3/2m + const2 ×R(m+3)/2m.
We choose m in the range 0 < m < 1 so that the reconstructed models
may favor the accelerated expansion of the universe and the corresponding
results are shown in Table 5.2. Using these expressions of f(R), one can
Table 5.2: Models of f(R) corresponding to m
m f(R)
m = 1 const1 ×R + const2 ×R3/2
m = 34
const1 ×R2 + const2 ×R5/2
m = 12
const1 ×R3 + const2 ×R7/2
m = 38
const1 ×R4 + const2 ×R9/2
m = 14
const1 ×R6 + const2 ×R13/2
m = 13
const1 ×R9/2 + const2 ×R5
m = 23
const1 ×R9/4 + const2 ×R11/4
develop the relevant forms of f(R, T ) through f(R, T ) = f(R) + λT . In
this case, we can get f(R, T ) model which involves non-minimal matter
geometry coupling. Equation (4.4.7) implies the trace of energy-momentum
tensor T and using in Eq.(4.4.8), we get the Lagrangian (1.1.3) of the form
f(R, T ) = α1R−1/2T + α2T.
116
Thus we can find that NEC is violated in both cases m = 0 and m 6= 0
which implies the phantom DE with EoS parameter ω < −1. In this setting,
the energy density is positive and decreasing whereas pressure is negative.
To conclude, modified theories presented in this thesis appear as com-
pelling candidates to describe the properties of the gravitational interaction.
However, still one needs to establish the validity and viability through theo-
retical and experimental tests. The non-equilibrium description of thermo-
dynamics is presented and it would be interesting to establish the general
description of non-equilibrium picture which may apply to any MGTs. One
can also explore the issues like finite-future singularities, stability analysis
of Einstein static universe and other cosmological solutions, different cosmic
eras corresponding to anisotropic solutions other than the LRS BI.
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125
Appendix A
Energy conditions for some specific models.
f(R, T, Q) = R + αQ:
NEC : ρ + p +α
2ρ− p + H(ρ + 7p)−H2[ρ(5 + q) + 2p(2− q)] > 0,
WEC : ρ +3α
2H(p− ρ)− 2H2 > 0,
SEC : ρ + 3p +3α
2ρ− p + H(3ρ + 5p)− 2H2(1 + 2q)ρ− 4(2− q)
× H2p > 0,
DEC : ρ− p +α
2p− ρ−H(7ρ + p) + 4H2ρ(q − 5)ρ + 4H2(2− q)
× p > 0. (A1)
f(R, T ) = αexp(
Rα
+ λT):
NEC : CNEC1 =
1
α2
(R2 + α(R−HR) + 2αλRT
)+ λT −HT + λT 2,
CNEC2 = 0, CNEC
3 = −(ρ + p),
WEC : CWEC1 = 3(H + 2H2)− 3
αHR− 3λHT , CWEC
2 = α
(1
2− λLm
),
CWEC3 = −ρ,
SEC : CSEC1 = R +
3
α2
(R2 + α(R + HR) + 2αλRT
)+ 3λT + T (H
+ λT ), CSEC2 = α(2λLm − 1), CNEC
3 = −(ρ + 3p),
DEC : CDEC1 = −R− 1
α2
(R2 + α(R + 5HR) + 2αλRT
)− λT + T (5H
+ λT ), CDEC2 = α(1− 2λLm), CDEC
3 = −(ρ− p). (A2)
f(R, T ) = R + ηRmT n:
NEC : DNEC1 = m(m− 1)R−2R−HR + (m− 2)R2R−1 + 2nRTT−1
+ nR−1T−1T −HT + (n− 1)T 2T−1, DNEC2 = 0,
DNEC3 = −(ρ + p),
WEC : DWEC1 = (1−m)0.5 + 3mHRR−2 − 3mnHTR−1T−1,
DWEC2 = −n, DWEC
3 = −ρ,
SEC : DSEC1 = (m− 1)1 + 3mR−2R + HR + (m− 2)R2R−1 + 2nRT
× T−1+ 3mnR−1T−1T + HT + (n− 1)T 2T−1, DSEC2 = 2n,
DSEC3 = −(ρ + 3p),
126
DEC : DDEC1 = (1−m)1 + mR−2R + 5HR + (m− 2)R2R−1 + 2nRT
× T−1 −mnR−1T−1T + 5HT + (n− 1)T−1, DDEC2 = −2n,
DDEC3 = −(ρ− p). (A3)
127
B: List of Publications
The contents of this thesis are based on the following research papers pub-
lished in journals of international repute. These papers are also attached
herewith.
1. Sharif, M. and Zubair, M.: Thermodynamics in f(R, T ) Theory of
Gravity, J. Cosmology Astroparticle Phys. 03(2012)028.
2. Sharif, M. and Zubair, M.: Study of Thermodynamic Laws in f(R, T, RµνTµν)
Gravity, J. Cosmology Astroparticle Phys. 11(2013)042.
3. Sharif, M. and Zubair, M.: Energy Conditions in f(R, T,RµνTµν)
Gravity, J. High Energy Phys. (to appear 2013).
4. Sharif, M. and Zubair, M.: Energy Conditions Constraints and Sta-
bility of Power Law Solutions in f(R, T ) Gravity, J. Phys. Soc. Jpn.
82(2013)014002.
5. Sharif, M. and Zubair, M.: Anisotropic Universe Models with Per-
fect Fluid and Scalar Field in f(R, T ) Gravity, J. Phys. Soc. Jpn.
81(2012)114005.
6. Sharif, M. and Zubair, M.: Study of Bianchi I Anisotropic Model in
f(R, T ) Gravity, Astrophys. Space Sci. 349(2014)457.
We have also published/submitted the following papers related to this the-
sis.
1. Sharif, M. and Zubair, M.: Evolution of the Universe in Inverse and
lnf(R) Gravity, Astrophys. Space Sci. 342(2012)511.
2. Sharif, M. and Zubair, M.: Cosmology of Holographic and New Age-
graphic f(R, T ) Models, J. Phys. Soc. Jpn. 82(2013)064001.
3. Sharif, M. and Zubair, M.: Analysis of f(R) Theory Corresponding
to NADE and NHDE, Adv. High Energy Phys. 2013(2013)790967.
4. Sharif, M. and Zubair, M.: Thermodynamic Behavior of Particular
f(R, T ) Gravity Models, J. Exp. Theor. Phys. 117(2013)248.
5. Sharif, M. and Zubair, M.: Reconstruction and Stability of f(R, T )
Gravity with Ricci and Modified Ricci Dark Energy, Astrophys. Space
Sci. 349(2014)529.
128
6. Sharif, M. and Zubair, M.: Thermodynamics in Modified Gravity with
Curvature Matter Coupling, Adv. High Energy Phys. 2013(2013)947898.
7. Sharif, M. and Zubair, M.: Cosmological Reconstruction and Stability
in f(R, T ) Gravity Gen. Relativ. Grav. 46(2014)1723.
8. Sharif, M. and Zubair, M.: Cosmological Evolution of Pilgrim Dark
Energy, Astrophys. Space Sci. DOI 10.1007/s10509-014-1889-8.
9. Sharif, M. and Zubair, M.: Reconstructing f(R) Theory from Pilgrim
Dark Energy, (Submitted).
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