Slow expansion in the scalar-tensor theory of gravity

Annals of Physics 409 (2019) 167921

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Slow expansion in the scalar-tensor theory of gravity ∗

Qihong Huang a , , He Huang b , Feiquan Tu a , Lu Zhang a , Jun Chen c a

School of Physics and Electronic Science, Zunyi Normal University, Zunyi 563006, China College of Physics and Electronic Engineering, Guangxi Teachers education university, Nanning 530001, China c School of Science, Kaili University, Kaili, Guizhou 556011, China b

article

info

Article history: Received 11 December 2018 Accepted 25 May 2019 Available online 6 August 2019

a b s t r a c t In this paper, we analyze the slow expansion scenario in the scalar-tensor theory of gravity. After considering the slow expansion solution H ≃ 0, we discuss the slow expansion in the scalar-tensor theory by terms of the nearly scale-invariant primordial power spectrum and find that the primordial power spectrum can be determined by the coupling function f (ϕ ). After the slow expansion phase ends, general relativity can be recovered under certain conditions, and then the universe evolves in accordance with the standard cosmology. In the scalar-tensor theory, the slow expansion scenario provides a possible way to avoid the big bang singularity. © 2019 Elsevier Inc. All rights reserved.

1. Introduction General relativity is one of the most fundamental theories in physics. With the advent of modern cosmology, the hot big bang model based on general relativity produces many remarkable achievements. However, there are still several puzzling drawbacks in this model, such as the horizon problem, the singularity problem. Fortunately, the inflationary scenario conceived by Guth and Linde et al. [1–4] gives a reasonable solution to most of these problems and provides the nearly scaleinvariant primordial perturbation responsible for the observable universe [5], leaving the singularity problem unsolved. To avoid this singularity, the pre-big bang [6] and the cyclic scenario [7] have ∗ Corresponding author. E-mail address: [email protected] (Q. Huang). https://doi.org/10.1016/j.aop.2019.167921 0003-4916/© 2019 Elsevier Inc. All rights reserved.

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been proposed. Another scenario named as emergent scenario which is a model without singularity was first proposed by Ellis et al. [8,9]. In this scenario, the universe stays in an Einstein static state past eternally and then evolves to an inflationary era naturally after exiting the static state. Recently, an emergent scenario with slow expansion in the spatially flat universe was proposed by Liu et al. [10,11], in which the universe originates from a finite static state and expands very slowly. During this era, it provides the nearly scale-invariant primordial power spectrum. After this era, the universe will evolve into the hot big-bang era. The slow expansion scenario which provides the nearly scale-invariant primordial perturbation was first constructed by Piao [12] and re-clarified in Ref. [13]. It has been found that the scale-invariant curvature perturbation in the slow expansion scenario was induced by the adiabatic perturbation [10–12,14,15] or the entropy perturbation [16], and there is no ghost instability for the generalized Galileon gravity [10,11]. In Ref. [10], the squared 2 sound speed cs2 ∼ 1.4, and the squared sound speed decreases rapidly from ∞ to e 5 during the slow evolution in Ref. [11]. Recently, the slow expansion scenario was analyzed in the gravity theory with nonminimal derivative coupling [17], the squared sound speed also decreases rapidly from ∞ to a constant. Thus, the sound speed in these theories becomes superluminal (cs2 > 1) during the slow evolution. Furthermore, in the slow expansion scenario, the flatness and the horizon problems 1 can be solved naturally because the comoving Hubble length aH is decreasing with time during the slow expansion era [10,11]. The scalar-tensor theory was conceived originally by Jordan who embeds a four dimensional curved manifold in the five dimensional flat spacetime [18]. The idea of the scalar-tensor theory reached full maturity with the work of Brans and Dicke [19]. Scalar-tensor theory provides a simple but versatile generalization of Einstein’s general relativity theory and is one of the simplest alternative gravitational theories. It introduces a scalar field which is nonminimally coupled to curvature tensor and plays an important role in explaining the accelerated expansion of the universe [20]. This theory has received various attention since it can produce the early cosmic inflation [21] and Higgs inflation [22]. It can also help to avoid the big bang singularity through Einstein static universe [23] or cyclic cosmology [24]. Moreover, cosmological observations of dark energy are often explained by a scalar field [25]. Although the slow expansion scenarios have been realized in the generalized Galileon gravity [10,11] and the gravity theory with nonminimal derivative coupling [17] without the ghost instability, the sound speed becomes superluminal. In this paper, we analyze the slow expansion scenario in the scalar-tensor theory, in which there is no ghost and the sound speed does not exceed the speed of light. The paper is organized as follows. In Section 2 we review the action and field equations in the scalar-tensor theory of gravity. In Section 3 we discuss the slow expansion in the scalartensor theory. Finally, our main conclusions are presented in Section 4. Throughout this paper, unless specified, we set 8π G = 1 and adopt the metric signature (−, +, +, +). Latin indices run from 0 to 3 and the Einstein convention is assumed for repeated indices. 2. Field equations of scalar-tensor theory In this paper, we begin with the following action [26,27]

∫ S=

√

d4 x −g

[1 2

f (ϕ )R −

1 2

] ω(ϕ )g αβ ∇α ϕ∇β ϕ − V (ϕ ) ,

(1)

where R is the Ricci curvature scalar, ϕ is the scalar field of the scalar-tensor theory, f (ϕ ) and ω(ϕ ) are the coupling functions of the scalar field, and V (ϕ ) is the potential. Here, the coupling function f (ϕ ) is positive for the gravitons to carry positive energy. Varying the action (1) with respect to the metric tensor g αβ and the scalar field ϕ , we obtain f (Rαβ −

1 2

(

gαβ R) − ∇α ∇β f + gαβ ∇σ ∇ σ f = ω ∇α ϕ∇β ϕ −

1 2

)

gαβ ∇ σ ϕ∇σ ϕ − gαβ V ,

(2)

and f,ϕ R + ω,ϕ ∇ σ ϕ∇σ ϕ + 2ω∇σ ∇ σ ϕ − 2V,ϕ = 0 , where, ϕ denotes the derivative with respect to ϕ .

(3)

Q. Huang, H. Huang, F. Tu et al. / Annals of Physics 409 (2019) 167921

3 eff

The field equations (2) can be expressed as the standard form of general relativity Gαβ = Tαβ with a modification in the energy–momentum tensor eff

Tαβ =

1[ f

( )] 1 ∇α ∇β f − gαβ (∇σ ∇ σ f + V ) + ω ∇α ϕ∇β ϕ − gαβ ∇ σ ϕ∇σ ϕ . 2

(4)

For the flat FRW universe, the perturbed metric in the longitudinal gauge is [28,29] ds2 = −(1 + 2Φ )dt 2 + (1 − 2Ψ )a2 (t)δij dxi dxj

(5)

where, the perturbative quantities Φ and Ψ characterize the scalar perturbation. The background equation can be obtained by substituting this metric into the field equations (2) and ignoring the perturbation quantities Φ and Ψ . The 0 − 0 components of the field equation (2) and the scalar field equation (3) can be expressed as 3H 2 + 3H

f˙ f

=

1(1 f 2

) ωϕ˙ 2 + V ,

(6)

and

ϕ¨ + 3H ϕ˙ +

1 (

) ω,ϕ ϕ˙ 2 + 2V,ϕ − f,ϕ R = 0 .

2ω where a dot denotes a derivative with respect to the cosmic time t.

(7)

3. Slow expansion in the scalar-tensor theory of gravity In this section, we will construct a slow expansion scenario in the scalar-tensor theory which can provide scale-invariant primordial perturbation power spectrum. The parameterization of the slow expansion was presented in Refs. [10,11], a reasonable selection for the scale factor a(t) is 1 b (t −t)b 0

b

, (8) β b (t0 − t)b+1 where t < t0 and b > 0. a → a0 and H → 0 are set in the infinite past while a ≃ a0 e and H ≃ b for te ≃ O(1)t0 . te corresponds to the time in which the slow expansion ends. Thus, the universe stems initially from a finite size and then expands slowly in the regime −∞ < t <[ te . During the ] slow evolution, the scale factor satisfies aa ≃ 1 which can be written as a = a0 1 + β b (t 1−t)b , 0 0 i.e. β (t0 − t) ≫ 1 is required. When the change of a is not negligible, the exit is assumed to occur. This period of slow expansion ( ) can be defined as emergence [11]. One can define ϵ = dtd H1 , and ϵ < 0 in the slow expansion since H is rapidly increasing. When initially β (t0 − t) ≫ 1, the evolution of the slow expansion corresponds to ϵ ≪ −1. The evolution ends at | ϵ |∼ 1, when β (t0 − t) ≃ 1. a = a0 e β

,H =

In this scenario, the big bang singularity can be avoided because the universe stems from a finite size. Furthermore, the slow expansion scenario can also solve the flatness and the horizon problems 1 because the comoving Hubble length aH is decreasing with time. As is pointed out in Ref. [30], the flatness problem is simply that the combination aH is a decreasing function of time, and the horizon problem can be solved because of the dramatic reduction in the comoving Hubble length during inflation. Thus, the origin of the flatness and horizon problems is that the comoving Hubble length 1 grows with time. If we had an early period in the history of the universe where aH decreases with time, the flatness and the horizon problems can be solved [31,32]. 3.1. The background of slow expansion To analyze the slow expansion in the scalar-tensor theory, we consider a special coupling between scalar field and gravity f (ϕ ) = 1 − ξ ϕ + λϕ 2 ,

ω(ϕ ) = ω0 ϕ −1 ,

with ξ , λ, ω0 and V0 being the coupling constant.

3

V (ϕ ) = −V0 ϕ 2

(9)

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Q. Huang, H. Huang, F. Tu et al. / Annals of Physics 409 (2019) 167921

Following e.g. [11,17], we consider the slow expansion solution H ≃ 0 [12], the background equation (6) reduces to 1

3

ω0 ϕ −1 ϕ˙ 2 − V0 ϕ 2 ≃ 0,

(10)

2 which gives

ϕ=

64ω02

1

V02 (t0 − t)4

.

(11)

f˙

Thus, H ≃ − f gives H=

256ξ ω02

1

V02

(t0 − t)5

,

(12)

and ∫

a = a0 e

Hdt

= a0 e

For b = 4 and β =

64ξ ω2 1 0 V 2 (t0 −t)4 0

V02

(

[ 64ξ ω02 = a0 1 + 2 V0

) 41

64ξ ω02

1 (t0 − t)4

]

.

(13)

, Eq. (8) is obtained, which is the required evolution. Because a hardly

changes and H increases with t, the comoving Hubble length is decreasing with time during the whole slow expansion era. Thus, the flatness and horizon problem can be solved naturally during the slow expansion era. Since H ∼ (t −1 t)5 , we have H ϕ˙ ≪ ϕ¨ and Eq. (7) can be approximated as 0

ω0 ω0 1 2 ϕ¨ − 2 ϕ˙ 2 − 3V0 ϕ 2 ≃ 0, ϕ ϕ

(14)

which is consistent with the solution (11). From Eq. (13), one can see that the condition of slow expansion is V02 64ξ ω02

(t0 − t)4 ≫ 1,

which shows ϵ ≃ −

5V02 256ξ ω02

(15)

(t0 − t)4 ≪ −1. When the condition (15) is broken, the slow expansion

era closes, and the condition t0 − te ≃

( 64ξ ω02 ) 14 V02

64ξ ω02 V02

1 (t0−t)4

≃ 1 gives the ending time te

.

(16)

When the conditions λ = ξ 2 and ξ ω0 = 1 are satisfied, general relativity is recovered, the universe will evolve in accordance with the standard cosmology. 3.2. The curvature perturbation In this section, we will calculate the primordial perturbations generated during the slow expansion. The quadratic action of the curvature perturbation is [33]

δS =

1

∫

2

{

a3 Z δ φ˙ 2 −

where, δφ = δϕ +

ω+

Z = ( 1+

ϕ H

1 a2

∇ α δφ∇α δφ +

1 H [ 3 ( ϕ˙ ). ]. 2 } 3 a Z δφ dtd x, a3 Z ϕ˙ H

Ψ is a gauge-invariant variable, and Z is defined as

3 f˙ 2 2 f ϕ˙ 2 f˙ 2Hf

(17)

)2 .

To avoid the ghost and gradient instabilities, Z > 0 should be satisfied.

(18)

Q. Huang, H. Huang, F. Tu et al. / Annals of Physics 409 (2019) 167921

√ aϕ˙

Introducing v = z ζ = z Hϕ˙ δφ and z = following mode expansion

vˆ =

d3 x

∫ √

(

(2π )3

H

5

Z , and expanding the quantum operator vˆ in the

) vk eikx aˆ k + vk∗ e−ikx aˆ †k ,

(19)

Eq. (17) leads to an equation of motion

( z ′′ ) vk′′ + k2 − vk = 0,

(20)

z

where ′ represents the derivative with respect to the conformal time η, dt = adη. Using Eqs. (9), (11) and (12), we can obtain the expression for Z and z Z ≃ 4ω =

a60 V02 16ω0

(η 0 − η ) 4 ,

(21)

and z≃

a30 V0

√ (η 0 − η ) 2 . ω0

4ξ

(22)

To avoid the ghost and gradient instability, ω0 > 0 is required. ′′ For k2 = zz , the perturbation mode of ζ leaves its horizon which is named as ζ horizon

√

1

=

Hfreeze

|

z

1

| ≃ √ (η 0 − η ) ∼

z ′′

( 1 ) 15 H

2

Thus, the physical ζ horizon is

a Hfreeze

√1

≃

2

.

(23)

(t0 − t), while the Hubble horizon

1 H

can be given by

Eq. (12). This relation shows that both the ζ horizon and the Hubble horizon decrease with time t. During the slow expansion evolution, the perturbation mode of ζ first leaves the ζ horizon and then it is freezed out, but the perturbation mode does not leave the Hubble horizon. After a while, the perturbation mode inevitably leaves the Hubble horizon and becomes the observable primordial perturbation. 1 H

V02

When initially

64ξ ω02

(t0 − t)4 ≫ 1, the physical wavelength of the perturbation

a k

is much smaller

z ′′ . z

than the ζ horizon, i.e. k ≫ The normalized positive frequency modes corresponding to the minimal quantum fluctuations take the form 2

1

vk = √

2k

e−ik

∫

dη

.

(24)

Thus, the power spectrum for ζ can be written as Pζ =

k3 2π

2

|

vk

|2 ≃ k2

z

4ξ 2 ω0

1

π 2 a60 V02 (η0 − η)4

→ 0.

(25)

This insures that the initial background of static state is not broken by the perturbations in the infinite past. So, in the slow expansion scenario, the primordial perturbation must be generated by the increasing mode [10–14]. When

V02 64ξ ω02

(t0 − t)4 → 1, the ζ horizon decreases with time while the physical wavelength of

the perturbation ak hardly changes and is much larger than the ζ horizon, i.e. k2 ≪ of ζ given by Eq. (20) is

ζ ∼ C1 is constant mode, or ζ ∼ C2

∫

dη z2

is changed mode,

z ′′ , z

the solution

(26)

where C2 mode is increasing or decaying dependent on different evolutions. The scale invariance ′′ 2 of ζ requires zz ≃ (η −η , and z needs to satisfy z ∼ (η0 − η)2 for the increasing mode [10]. From )2 0

Eq. (22), one can see that

z ′′ z

≃

2 (η0 −η)2

is satisfied.

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Q. Huang, H. Huang, F. Tu et al. / Annals of Physics 409 (2019) 167921

Because z

z ′′ z

can be written as

′′

z

=

where ν =

2 (η 0 − η ) 2 3 2

vk = e

=

1(

τ2

ν2 −

1) 4

,

(27)

and τ = η0 − η, the solution of Eq. (20) is given as (

i ν− 12

)

π

2

3

2ν− 2

Γ (ν ) 1 1 π 1 √ (−kτ ) 2 −ν = √ ei 2 [−k(η0 − η)]−1 . 3 Γ ( 2 ) 2k 2k

(28)

Since the perturbation is given by the increasing mode, the spectrum of ζ should be calculated around the time in which the slow expansion ends spectrum for ζ is

(t0 − t)4 ≃ 1. Using Eq. (22), the power

2

k3 1 1

1

4ξ 2 ω0

5

ξ 2 V0 Pζ = | | = = 2 6 2 ≃ , 2π 2 z 2π 2 z 2 2k k2 (η0 − ηe )2 128π 2 π a0 V0 (η0 − ηe )6 k3

vk

V02

64ξ ω02

1

(29)

where, the conditions Eq. (16) and ξ ω0 = 1 are used. Thus, the spectrum of ζ is scale invariant, 1

− 52

and Pζ2 ∼ 10−5 requires ξ ∼ 10−2 V0

. It implies that only one parameter needs to be adjusted − 52

in this model. In this manner, if we consider ξ = 10−2 V0 −2

1

1

, then we have Pζ2 ≃ 8.897 × 10−5 . As

another example, for ξ = 0.9 × 10−2 V0 5 , we have Pζ2 ≃ 7.799 × 10−5 When the slow expansion phase ends, the general relativity will recover and the universe will evolve in accordance with the standard cosmology. 4. Conclusion The scalar-tensor theory is an extension of general relativity by coupling a scalar field ϕ to the Ricci scalar R with terms f (ϕ )R. This theory can be expressed as general relativity with a modified energy–momentum tensor. Recently, the slow expansion scenario has been discussed in [10,11,17], and it has been found that the scale invariant primordial perturbation is obtained after the slow evolution ends and then the universe reheats and the evolution of hot big bang model begins. This suggests that the universe can stem from a static state in the infinite past with the slow expansion and then evolve into the hot big bang era. In this paper, we analyze the slow expansion scenario in the scalar-tensor theory by terms of the observed results, i.e. nearly scale-invariant primordial perturbation. We find that the primordial power spectrum can be determined by the coupling function f (ϕ ) and only one parameter requires to be adjusted. When the slow expansion phase ends, general relativity can be recovered under certain conditions, and then the universe evolves in accordance with the standard cosmology. Therefore, in the scalar-tensor theory, the slow expansion scenario can be used to avoid the big bang singularity. Different from the Einstein static universe [23], in which the initial static state requires to satisfy the stability conditions to avoid this state being broken by the perturbations, this problem can be solved naturally in the slow expansion scenario since the primordial perturbation is generated by the increasing mode. In addition, the Einstein static universe generally requires the existence of the spatial curvature, while the slow expansion scenario needs a flat spacetime. Finally, there still exist some comments about the slow expansion scenario proposed to avoid the big bang singularity. The equation of state of the dominant energy component violates the null energy condition during the slow expanding phase since the Hubble parameter increases. Fortunately, Z > 0 shows that there is no ghost and gradient instability, and this theory is healthy and does not suffer from pathologies in the fluctuation dynamics. Furthermore, in the scalar-tensor theory, the convergence towards Einstein gravity forces the energy density below the characteristic energy density of nucleosynthesis. So, there remains a question about how to obtain the appropriate characteristic energy density after the slow expansion phase ends, and in this regard, one possible explanation is that the universe will evolve in accordance with the standard cosmology since general

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7

relativity will recover when the slow expansion phase ends. It is also interesting to discuss the nucleosynthesis in the case in which general relativity is not recovered when the slow expansion phase ends. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants Nos. 11865018, 11865019, 11847031, 11847085, 11465011, the Foundation of the Guizhou Provincial Education Department of China under Grants Nos. KY[2018]312, KY[2018]028, KY[2017]247, KY[2016]104, the Doctoral Foundation of Zunyi Normal University of China under Grants No. BS[2017]07. References [1] [2] [3] [4] [5] [6] [7]

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[22]

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