Physics Letters B 712 (2012) 6–9

Contents lists available at SciVerse ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Some cosmological solutions of 5D Einstein equations with dark spinorcondensate

Tae Hoon Lee

Department of Physics, Soongsil University, Seoul 156-743, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 February 2012Received in revised form 14 April 2012Accepted 21 April 2012Available online 25 April 2012Editor: M. Trodden

Keywords:CosmologyHigher dimensional gravityDark spinor

We study the 5D Einstein gravity equations with dark spinor condensate, and under the cylindercondition we find an exponentially expanding cosmological solution for the scale factor of our universe,even without a cosmological constant. The stability condition for the solution is given. Some power-law cosmological solutions are also derived when bulk matter sources in the form of a perfect fluid areadditionally introduced.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The standard model of electroweak and strong interactions havebeen very successful in explaining almost all kinds of high energyphenomena, while a complete theory of quantum gravity at Planckscale is anticipated to be constructed in higher dimensional theo-ries such as string theory. However there still remain some prob-lems to be understood, and among them it is important to accountfor the existence of a big gap between the electroweak and thePlanck scale. To solve the hierarchy problem Arkani-Hamed et al.[1] proposed, instead of assuming supersymmetry, that the stan-dard model fields are confined to a 3-brane in extra dimensions,and when studying such a 3-brane embedded in a 5-dimensional(5D) space–time Randall and Sundrum [2] made the suggestionthat the weak scale is generated from the Planck scale throughan exponential hierarchy. Various kinds of higher dimensional Ein-stein gravity theory have been intensively studied in relation to thehierarchy problem and the cosmological constant problem. In theSpace–Time–Matter theory [3] Einstein’s four-dimensional theoryof gravity with induced matter is embedded in 5D vacuum gen-eral relativity, which is considered for the purpose of explainingthe cosmological problems and particle physics phenomena.

Even if the original Space–Time–Matter theory do not includeother matter than induced matter obtained from 5D vacuum Ein-stein equations, we think it is also important to extend the theoryto include other matter for the possible relations to brane models[2,4]. Such extensions, up to now, were restricted to special cases

E-mail address: thlee@ssu.ac.kr.

0370-2693/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physletb.2012.04.051

as the 5D cosmological constant [5], bulk scalar [6], and Brans–Dicke fields [7]. In this Letter we consider (bulk) dark spinor field[8] in the 5D model for the purpose of better understanding ofcosmological problems including the cosmological constant prob-lem.

We study cosmological solutions of the field equations in thetheory of 5D gravity coupled with only dark spinor, considered asa generalized version of Space–Time–Matter theory [3,5,7]. Darkspinor, a new spin half field with mass dimension one, was theo-retically found in Ref. [8] and has been recently investigated as acandidate for dark energy [9] and dark matter [8,10,11]. Assum-ing the cylinder condition [3], we find exponentially expandingsolutions for the scale factors of both our universe and the extraspace and determine conditions on which the solutions are sta-ble. Some power-law cosmological solutions are also derived whenbulk matter sources in the form of a perfect fluid are additionallyintroduced.

2. 5D gravity with dark spinor

Without the projection term we consider the following actionof the 5D gravity theory with a dark spinor field Ψ (and its dualspinor Ψ ) [10].

S =∫

d5x√

g

[R

2κ+ 1

2gMN∇MΨ ∇NΨ − V (Ψ Ψ )

]+ Sm, (1)

where Sm is the action for other matter than dark spinor and Vis the potential of the spinor. ∇MΨ = ∂MΨ + Ψ ΓM and ∇NΨ =∂NΨ − ΓNΨ are covariant derivative of them. κ = 8πG , xM =(xμ, y) with the four-dimensional coordinate xμ and x5 = y, and

T.H. Lee / Physics Letters B 712 (2012) 6–9 7

ΓM = −1

8ωAB

M [γA, γB ] (2)

is the spin connection in the 5D space–time with ωABM =

e AN(∂M eN B + Γ N

MLeLB) and γA -matrices satisfying the Clifford al-gebra, {γA, γB} = ηAB ≡ diag(1,−1,−1,−1,−1), in a locally flatinertial coordinate (A, B = 0, 1, 2, 3, 5).

From the above action we have the 5D Einstein equations

RMN − 1

2gMN R = −κ

(T (e)

MN + T (m)MN

), (3)

where the former of the right-hand side is the 5D energy–momentum tensor of the dark spinor,

T (e)MN = ∇MΨ ∇NΨ − 1

2gMN

(∇ LΨ ∇LΨ − 2V) + 1

2∇L J MN

L, (4)

whose last term is derived from the spin-action in Eq. (1) [10]. Thelatter is the energy–momentum tensor derived from the matteraction,

√gT (m)

MN = 2δSm/δgMN .

3. Cosmology

To discuss Friedman cosmology on the four-dimensional hyper-surface embedded in a 5D space–time, we use the following metricincluding the flat (k = 0) Robertson–Walker (RW) space–time.

ds2 = gMN dxM dxN = gμν dxμ dxν − b2(t)dy2 (5)

with

gμν = diag(1,−a2(t),−a2(t),−a2(t)

), (6)

where a(t) is the scale factor of our (1 + 3)-dimensional universeand the scale factor of the extra dimension b(t) is defined also asa function of only the cosmic time t under the cylinder condition(∂yb = 0) as in the Space–Time–Matter-like theory [3]. We assumeboth such a cylinder condition (by dropping all derivatives withrespect to the external coordinate y) and a scalar condensate ofthe dark spinor field [10]

Ψ(xM) = ϕ(t)ξ (7)

with a constant spinor ξ such that ξξ = 1 but ∇Mξ �= 0 because ofpresence of its spin connection.

By means of vierbeins the metric (5) can be rewritten asηAB e AeB with

e A = (dt,a(t)dxi,b(t)dy

) = e AM dxM . (8)

Using such related physical quantities as (e AN eN

B = δAB )

eNB = diag(1,1/a,1/a,1/a,1/b) (9)

and

eN B ≡ eNC ηC B = diag(1,−1/a,−1/a,−1/a,−1/b), (10)

we have spin connections;

Γt = 0, Γi = −γiγ0a

2, Γy = −γ5γ0

b

2(11)

and non-vanishing components of the 5D Christoffel symbol;

Γ ti j = aaδi j, Γ t

yy = bb,

Γ kt j = a−1aδk

j , Γy

ty = b−1b. (12)

3.1. Field equations

From the formula (4) of the previous section, we calculatethe energy density ρcos , the pressure of our space–time pcos , andthe pressure in the external space p(5)

cos , which are effected byboth the spin connections [10] and the extra space in the space–time given in Eq. (5), and we write down them in the following,

where 12 ∇L Jμν

L = 12 ∇ρ Jμν

ρ + b2b Jμν

t − Γy

y(μ Jν)yy and so on are

used.Exploiting the above equations in the case Sm = 0, we have

four differential equations from the 5D Einstein equations andthe dark spinor field equation with definitions H = a/a and K =b/b;

3H2 + 3H K = κρcos, (13)

−(

2a

a+ H2 + b

b+ 2H K

)δ

ji = κ pcosδ

ji , (14)

ϕ + (3H + K )ϕ + V ,ϕ −1

4

(3H2 + K 2)ϕ = 0, (15)

3

(a

a+ H2

)= −κ p(5)

cos = κ

[− ϕ2

2+ V − 3

8H2ϕ2 + 1

8K 2ϕ2

+ 1

4Kϕ2 + 1

2Kϕϕ + 3

4H Kϕ2

], (16)

where

ρcos = ϕ2

2+ V +

(3

8H2 + 1

8K 2

)ϕ2, (17)

pcos = ϕ2

2− V − 3

8H2ϕ2 − 1

4Hϕ2 − 1

2Hϕϕ

+ 1

8K 2ϕ2 − 1

4H Kϕ2, (18)

and a four-dimensional (4D) scalar curvature 6( aa + H2) = R(4) .

3.2. Cosmological solutions

3.2.1. Maximally symmetric solutionAssuming the potential of a scalar condensate ϕ of the dark

spinor Ψ = ϕξ [10] as

V (ϕ) = V 0 + m2

2ϕ2 (19)

with a constant V 0 and a mass m, we first seek cosmological so-lutions for a (eternal) accelerated era of our universe by requiringthat H = 0 and H > 0 in Eqs. (13)–(16).

When ϕ = ϕ = 0 for a static condensate ϕ = ϕ0 and K = 0, wehave two solutions, K = H and K = −3H , of which the latter doesnot satisfy all equations. The former is given by H = K = m withκV 0 = m2(6 − κϕ2

0 ), and it is stable because there exist α < 0,β < 0, and γ < 0 which satisfy the stability condition

κϕ20

(β + 4 + 3(γ + 4)

)+ (

κϕ20 − 12

)(α + 4)(β + 4)(γ + 4) = 0. (20)

Eq. (20) is induced by three equations; 2(α + 4)ca = 3cb + cc ,(β + 4)cb = (γ + 4)cc , and (β + 4)cb = κϕ2/12(2ca + (γ + 4)cc),which are obtained by using linearized stability equations derivedfrom Eqs. (13)–(16) with such perturbations near the solution asϕ/ϕ = cameαmt , (H − m) = cbmeβmt , (K − m) = ccmeγ mt .

Note that V 0 = 0 in the special case when κϕ20 = 6, which is

also stable because there exist α < 0, β < 0, and γ < 0 such that

8 T.H. Lee / Physics Letters B 712 (2012) 6–9

β + 4 + 3(γ + 4) − (α + 4)(β + 4)(γ + 4) = 0, and we have a solu-tion to describe the accelerated era of our universe even without acosmological constant V 0.

3.2.2. Power-law solutions with additional bulk matterNext, we find that simple power-law cosmological solutions of

the 5D gravity theory with dark spinor condensate, by consideringadditional bulk matter sources in the form of a perfect fluid, T N

M ≡2δSdm/(

√gδgM

N ) = diag(ρ,−p,−p,−p,−p5).The matter action Sm in Eq. (1) of the previous section is re-

placed by∫d5x

√g

(u

MΨ Ψ Φ∗Φ

)+ Sdm

(Φ∗Φ

), (21)

where the 5D Planck mass M [2] (with a dimensionless constant u)is introduced because a complex scalar field Φ as well as the darkspinor have mass dimension 3/2 in the 5D space–time. Sdm is abulk matter action of the complex scalar field like the Higgs boson,a unique particle which can interact with dark spinor [8]. Assum-ing further that uΦ∗Φ/M = v2/(2t2) with a constant v and thatρ ∝ t−2, we have two following solutions when a ∝ th and b ∝ tk

with a static condensate ϕ = ϕ0 (h and k are constants).

3.2.2.1. Symmetric solution The first solution is given by h = k withh = v(> 0) and κϕ2 = 12 − 2κ(ρt2)/v2, and it seems to be natu-ral due to our setting of an isotropic and homogeneous, (1 + 4)-dimensional space–time. In this case we see the equation of statefor the pressure in the external space ω5 ≡ p5

ρ = −1 + 12v as well

as for the pressure in our space

ω ≡ p

ρ= −1 + 1

2v. (22)

From the 5D energy conservation and their equations of state, weget ρ ∝ a−4(1+ω) = a−2/v . Since a ∝ tv , ρ is proportional to t−2,which is consistent with our assumption given below Eq. (21).The power-law cosmological solution with a ∝ tv (0 < v < ∞) candescribe any period in the whole history of our universe fromradiation-dominated era to the era of recent acceleration. Whenspecially v = 1, a = 0 and ρ ∝ a−2, with ω = −1/2 which is dif-ferent from the ordinary case without being embedded in higherdimensional space–time.

3.2.2.2. Stiff solution The second solution is given as k = 1 − 3h,when a bulk stiff (dark) matter field is considered with the equa-tion of state p = p5 = ρ .

The constant h satisfy both equations 2h2 − h + 1−4v2

6 = 0 and

2h2 − h + κϕ2

843 v2 + κρ t2

3 = 0, which give us the condition κρt2 =12 − 2v2(1 + 4κ ϕ2

8 ) and

1

4� h <

1

2. (23)

The latter condition corresponds to the effective (1+3)-dimension-al equation of state 5/3 �ωeff < 1/3, which lies in the range fromthe radiation-dominated era of the universe to even earlier thanthe stiff era (with ωeff = 1).

4. Summary and discussions

Considering the gravity theory with dark spinor, in the 5Dspace–time containing a flat RW space–time as a subspace, wehave calculated additional contribution to the energy density andthe pressure of both our space and the extra space, in Eqs. (16)–(18) of Section 3, compared with previous results calculated in the

ordinary (1 + 3)-dimensional space–time [10]. Assuming the cylin-der condition [3] and a static condensate of dark spinor, we havefound a stable cosmological solution corresponding to an (eternal)accelerated era of the universe [12]. If the mass of dark spinor mis very small, it can be related with the late-time acceleration ofthe universe [9] and the relation is possible even without the cos-mological constant when κϕ2 = 6.

Some power-law cosmological solutions of our theory with thestatic dark spinor condensate are also obtained by consideringadditional bulk matter fields in the form of a perfect fluid. InSection 3.2.2.1 where the 5D space–time has such a maximal sym-metry as Section 3.2.1, arbitrary simple power-law solutions a ∝ tv

including the zero acceleration case with v = 1 are obtained. InSection 3.2.2.2, with a 5D stiff bulk matter field we have somepower-law solutions of a ∝ th with 1/4 � h < 1/2, which can de-scribe any period from the radiation-dominated era of the universeto much earlier than stiff era.

Our model can describe both (dark) matter-dominated and darkenergy-dominated era of the universe in the following two waysdepending on global properties of the 5D space–time: In the firstcase where we have spatial isotropy and homogeneity in the whole5D space–time with K = H , the universe makes a smooth phasetransition from matter-dominated era of Section 3.2.2.1 (throughthe v = 1, zero-acceleration epoch) to the (5D isometric and homo-geneous) dark energy-dominated era of Section 3.2.1, since the firstterm in Eq. (21) decreases in magnitude, as the temperature of theuniverse goes down much lower than the 5D Planck mass scale M .In the second case, the universe makes an abrupt phase transitionfrom the power-law expansion era of a ∝ th with h = (1 − k)/3 inSection 3.2.2.2 (with contracting extra dimension for 1/3 < h < 1/2or with slowly expanding one for 1/4 < h < 1/3) to the darkenergy-dominated era of Section 3.2.1, even if detailed dynamicsof the phase transition has not been studied in this Letter.

In this Letter we have assumed a static condensate of darkspinor for simple cosmological solutions, and they crucially dependalso on the cylinder condition. It would be interesting to checkwhether relaxing these conditions could lead to some more inter-esting solutions.

Note added

In order to clarify our standpoint on which our model of the universe is de-scribed by (a generalized) Space–Time–Matter theory, we append a brief explana-tion about it. Without any matter, effective dynamics in Space–Time–Matter theory[3] are governed by the vacuum 5D Einstein equation,

R MN − 1

2gMN R = 0, (24)

which has been extended in this Letter to include only dark spinor matter (andHiggs scalar fields to which the dark spinor couples uniquely, in Section 3.2.2). Wecan rewrite Eqs. (13)–(14), derived from Eq. (3) which are 5D Einstein equationswith the unique matter source, as

3H2 = ρind + κρcos, (25)

−(

2a

a+ H2

)= pind + κ pcos. (26)

Here ρind = −3H K and pind = b/b + 2H K are the induced energy and pressuredensity in Space–Time–Matter theory [3] with the cylinder condition.

To compare explicitly our model that is described in terms of equations in theprevious section (and Eqs. (25) and (26)) with the 4D universe we observe, weneed to assume that the 5D space–time can be foliated by a family of hypersurfacesdefined by y = y0 with a constant y0 as in Refs. [13] and [14]. Without the cylindercondition [3], the (4D) dynamics of the metric fields obtained from a foliation of 5DEinstein equations in Eq. (3) are expressed by the effective 4D Einstein equations

[G(4)

μν

]y0

= −T indμν − κ

[T (e)μν + T (m)

μν

]y0

, (27)

where T indμν ≡ [Gμν − G(4)

μν ]y0 = diag(ρind, pind, pind, pind) is the induced energy–

momentum tensor, whereas T (e)μν and T (m)

μν are the additional energy–momentum

T.H. Lee / Physics Letters B 712 (2012) 6–9 9

tensor of the dark spinor and of the Higgs scalar fields on the hypersurface of theform y = y0. Gμν = Rμν − 1

2 gμν R are (μν)-components of the 5D Einstein tensor

and G(4)μν = R(4)

μν − 12 gμν R(4) the 4D Einstein tensor. Even when the cylinder condi-

tion is satisfied, the foliation is implicitly possible.We anticipate this Letter to be useful for future works beyond the cylinder con-

dition in the framework of the generalized Space–Time–Matter theory with darkspinor, and we also hope that it may serve as both a first step toward the devel-opment of a new one-brane model with a warp factor [2] and one of studies [4]on the relationship between Space–Time–Matter and brane theories, where b(t) inEq. (5) may play the role of a radion [15].

Acknowledgements

The author would like to thank colleagues including Dr. P. Ohand Dr. J. Lee, and also to an anonymous referee for their usefulcomments.

He is always grateful to H.Y. Park and G.R. Lee for their help.This research was supported by the Basic Science Research Pro-

gram through the National Research Foundation of Korea (NRF)funded by Ministry of Education, Science and Technology (2011-0005276).

References

[1] N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Lett. B 429 (1998) 263;N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, N. Kaloper, Phys. Rev. Lett. 84(2000) 586.

[2] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370;L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690.

[3] J.M. Overduin, P.S. Wesson, Phys. Rep. 283 (1997) 303.[4] P. De Leon, Mod. Phys. Lett. A 16 (2001) 2291, gr-qc/0111011.[5] T.H. Lee, Mod. Phys. Lett. A 26 (2011) 1673.[6] J.E.M. Aguilar, M. Bellini, Phys. Lett. B 596 (2004) 116, gr-qc/0405024.[7] J.E.M. Aguilar, C. Romero, A. Barros, Gen. Rel. Grav. 40 (2008) 117;

T.H. Lee, Class. Quant. Grav. 26 (2009) 137001.[8] D.V. Ahluwalia-Khalilova, D. Grumiller, JCAP 0507 (2005) 012;

D.V. Ahluwalia-Khalilova, D. Grumiller, Phys. Rev. D 72 (2005) 067701.[9] S.J. Perlmutter, et al., Astrophys. J. 517 (1999) 565;

A.G. Riess, et al., Astrophys. J. 659 (2007) 98.[10] C.G. Boehmer, J. Burnett, D.F. Mota, D.J. Shaw, JHEP 1007 (2010) 053, and ref-

erences therein.[11] Y.-X. Liu, X.-N. Zhou, K. Yang, F.-W. Chen, arXiv:1107.2506 [hep-th], 2011.[12] A. Vilenkin, Phys. Rev. D 27 (1983) 2848.[13] J.E.M. Aguilar, Eur. Phys. J. C 53 (2008) 133.[14] M. Bellini, Phys. Lett. B 709 (2012) 309.[15] S. Kobayashi, K. Koyama, JHEP 0212 (2002) 056.