Bounce universe from string-inspired Gauss-Bonnet gravity

We explore cosmology with a bounce in Gauss-Bonnet gravity where the Gauss-Bonnet invariant couples to a dynamical scalar field. In particular, the potential and and Gauss-Bonnet coupling function of the scalar field are reconstructed so that the cosmological bounce can be realized in the case that the scale factor has hyperbolic and exponential forms. Furthermore, we examine the relation between the bounce in the string (Jordan) and Einstein frames by using the conformal transformation between these conformal frames. It is shown that in general, the property of the bounce point in the string frame changes after the frame is moved to the Einstein frame. Moreover, it is found that at the point in the Einstein frame corresponding to the point of the cosmological bounce in the string frame, the second derivative of the scale factor has an extreme value. In addition, it is demonstrated that at the time of the cosmological bounce in the Einstein frame, there is the Gauss-Bonnet coupling function of the scalar field, although it does not exist in the string frame.


I. INTRODUCTION
Cosmological observations have suggested that the current cosmic expansion is accelerating. If the current universe is homogeneous, isotropic, and spatially flat, dark energy with negative pressure or the modification of gravity at a large distance is necessary to explain the observations (there are recent reviews on the dark energy problem and modified gravity, e.g., in Refs. [1][2][3]).
On the other hand, inflation can explain the homogeneity, isotropy, and flatness of the universe, and include the mechanism to generate the primordial density perturbations. Therefore, inflation is the most promising scenario to describe the early universe. As an viable alternative scenario to inflation, there has been proposed the matter bounce cosmology [4,5], where the initial singularity in the beginning of the universe can be avoided. In this scenario, matter dominates the universe at the bounce point, and the density fluctuations compatible with observations are generated (see, for instance, Ref. [6] for a review on bounce cosmology). In cosmology with a bounce, there have been various discussions [7] on the BKL instability [8], the bounce phenomena [9] in the Ekpyrotic scenario [10], and the density perturbations [11]. Moreover, observational implications of the cosmological bounce have been argued in Ref. [12].
In the matter bounce scenario, the primordial density perturbations with a nearly scaleinvariant and adiabatic spectrum of can be generated [5]. Especially, the perturbations of the quantum vacuum, whose original scale is smaller than that of the Hubble horizon, are produced. Its scale becomes larger than the Hubble horizon in the epoch of the contraction where matter dominates the universe, and eventually it evolves as the curvature perturbations with the (almost) scale-invariant spectrum. Similarly, it is known that in the Ekpyrotic scenario in the framework of brane world models, the primordial density perturbations with such a spectrum can also been produced. One of the most important aims in this scenario is to connect cosmology in the early universe to more fundamental theories such as superstring theories and M-theories [10].
Cosmology with a bounce has been examined in various gravity theories including F (R) gravity [13], modified Gauss-Bonnet gravity [14], f (T ) gravity [15], where T is the torsion scalar in teleparallelism, non-linear massive gravity with its extension [16], and loop quantum gravity [17,18] (for references on loop quantum cosmology, see, for example, Ref. [19]). The comparison of the bounce cosmology with the BICEP2 experimental data [20] 1 has been executed in Ref. [25]. The parameters of the bounce cosmology with the quasi-matter domination have been introduced in Ref. [26]. The theories leading to the cosmological bounce may be represented as a kind of a non-minimal Brans-Dicke-like theory [27], in which anti-gravity behaviours could be realized.
In this paper, we investigate the cosmological bounce in scalar Gauss-Bonnet gravity, where a dynamical scalar field non-minimally couples to the Ricci scalar and/or the Gauss-Bonnet invariant. It is known that the Gauss-Bonnet term appears in string theories through the approach to derive the low-energy effective action. Furthermore, we compare the bounce phenomenon in the string (Jordan) frame with that in the Einstein frame by making the conformal transformation and explore the relations between these conformal frames. We note that the cosmological perturbations [28] and a cosmological scenario for the structure formation [29] in a scalar field theory coupling to the Gauss-Bonnet invariant have been examined. Moreover, cosmological non-singular solutions in superstring theories have also been analyzed in Ref. [30]. We use units of k B = c l = = 1, where c is the speed of light, and denote the gravitational constant 8πG by κ 2 ≡ 8π/M Pl 2 with the Planck mass of The organization of the paper is as follows. In Sec. II, we explain a scalar field theory with non-minimal coupling to gravity and derive the equations of motions. In the Einstein frame, we reconstruct scalar Gauss-Bonnet gravity in Sec. III the Hubble parameter and the scalar field around the cosmological bounce in Sec. IV. In Sec. V, the reconstruction of scalar Gauss-Bonnet gravity is performed in the string frame. In Sec. VI, we make the conformal transformation of bounce solutions from the string frame to the Einstein frame, and vice versa. We demonstrate that in general, the bounce in the string frame does not correspond to that in the Einstein frame, and vice versa. It is also shown that the bounce universe can be transformed to the accelerating universe in several cases. Conclusions are described in Sec. VII.
1 Very recently, the new joint analysis by BICEP2/Keck Array and Planck [21] on B-mode polarization and Planck 2015 data [22][23][24] on various cosmological aspects have been released.

II. MODEL
We explore a model of a homogeneous scalar field φ = φ(t) non-minimally coupling to gravity. Our model action is given by [31] Here, g is the determinant of the metric g µν , (∇φ) 2 ≡ g µν ∇ µ φ∇ ν φ, where ∇ µ is the covariant derivative associated with g µν . Moreover, G is the Gauss-Bonnet invariant with R the scalar curvature, R µν the Ricci tensor, and R µνρσ the Riemann tensor. In addition, f (φ, R) is an arbitrary function of φ and R, ω(φ) and ξ(φ) are arbitrary functions of φ, V (φ) is the potential of φ, and α 1 and α 2 are constants. In the following, for simplicity, we set κ 2 = 1.
Here, we mention that in the framework of string theories (for a detailed review, see, e.g., [32]), the most general expression of the last term in the brackets { } of the action in Eq. (II.1) is represented as [33] whereᾱ ′ ≡ l 2 string with l string the fundamental length scale of strings is an expansion parameter,λ is an additional parameter, d i (i = 1, . . . , 4) are constants, G µν ≡ R µν − (1/2) g µν R is the Einstein tensor, and ≡ g µν ∇ µ ∇ ν is the covariant d'Almbertian for a scalar field φ.
Non-singular cosmology [34,35] and the cosmological perturbations [31] in a theory including such higher-order correction terms have been investigated. By comparing our action in Eq. (II.1) with the expression in Eq. (II.3), we see that in our action in Eq. (II.1), we have taken d 2 = 0 and d 3 = 0. Furthermore, in the action in Eq. (III.1) as is shown in the next section, we further set α 1 = 1 and α 2 = 0. Namely, we take − (1/2)ᾱ ′λ d 1 = 1 and d 4 = 0 in the full action in Eq. (II.3). This means that our action corresponds to a special model of the general theory studied in Ref. [31]. The coefficients d i have to be determined so that that the full action can agree with the three-graviton scattering amplitude [31].
The explanations for the reasons why we have neglected several terms in our model action are as follows. For the action in Eq. (II.1), this could be regarded as a different version of string-inspired actions from the general action including the expression in Eq. (II.3). In other words, this action is a kind of a toy model built in a phenomenological approach. On the other hand, for the action in Eq. (III.1), we consider the case that the non-linear terms of derivatives of the scalar field φ, i.e., the last three terms in the action in Eq. (III.1), are much smaller than the Gauss-Bonnet term. Therefore, we only take the first term of the action in Eq. (II.3) with − (1/2)ᾱ ′λ d 1 = 1 and neglect the other last three terms by setting We suppose the flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric where a(t) is the scale factor.
It is known that the system of Eqs. (II.5) and (II.6) is an overdetermined set of equations.
We see that Eq. (II.6) is a consequence of Eqs. (II.5) and (II.7). By combining Eqs. (II.5) and (II.6), we find If the scalar field φ(t) and the scale factor a(t) are given, the coupling function ξ(φ) may be obtained by solving the differential equation (II.12). Hence, the potential of the scalar field V (φ) can be acquired from Eq. (II.5).

III. RECONSTRUCTION OF SCALAR GAUSS-BONNET GRAVITY IN THE EINSTEIN FRAME
In this section, we study the action in Eq. (II.1) with f (φ, R) = R, ω(φ) = γ ≡ ±1, α 1 = 1, and α 2 = 0, expressed as This is an action for string-inspired Gauss-Bonnet gravity. We reconstruct several models of scalar Gauss-Bonnet gravity. We assume that the time dependence of the scalar field has the following form: where φ 0 is a constant.

B. Exponential model
Next, we study the case that the scale factor has an exponential form as where α (> 0) is a positive constant. In this case, we have There occurs only one cosmological bounce at t = t b = 0. At this time, we obtain It follows from Eqs. (II.14) and (II.15) that where 2 F 2 (σ 1 , σ 2 , σ 3 ; χ) with σ i (i = 1, 2, 3) constants and χ a variable is a hyper geometric function, erf(χ) and erfi(χ) are the Gauss's error function, and γ E is the Euler's constant.
In the end of this section, we explicitly state the physical motivation of our procedure to reconstruct the potential V (φ) of the scalar field φ and its coupling function ξ(φ) to the Gauss-Bonnet invariant for specific forms of the evolutions of φ and the scale factor a(t) by using the gravitational field equations and the equation of motion for φ. Basically, in string theories, the functions of V (φ) and ξ(φ) are known only in some approximations with keeping the leading terms. Therefore, it is considered to be very interesting and significant subject to study which forms of these functions are predicted by non-singular cosmology such as cosmology with a bounce to avoid the initial singularity in the early universe.
In addition, we explain the justification of the reconstructed potentials V (φ) of the scalar field φ in Eqs. (III.7) and (III.18) and its coupling function ξ(φ) to the Gauss-Bonnet invariant. First, we examine the physical behaviours of the resultant potential V (φ). In the reconstructed potentials V (φ) in Eqs. (III.7) for the hyperbolic form of the scale factor Fig. 1, the behaviour of V (φ) for c 1 = 0, c 2 = 0, λ = 1, γ = 1, and φ 0 = 1 is drawn.
It follows from this graph that the time of the cosmological bounce is t b = 0, and hence around the bounce point, the value of the potential V (φ) changes from positive to negative.
While, in the reconstructed potentials V (φ) in Eq. (III.18) for the exponential form of the scale factor exp (αt 2 ) in Eq. (III.15) and φ = φ 0 t in Eq. (III.2), on the left panel in Fig. 2, the behaviour of V (φ) for α = 1, c 1 = 0, c 2 = 0, and φ 0 = 1 is shown. From this plot, we can see that the time of the cosmological bounce is t b = 0, and therefore around the bounce point, the absolute value of the potential V (φ) becomes a minimum.
When we consider inflation in the early universe in the theory whose action is given by  (1), the slow-roll inflation could occur because the slope of the potential is sufficiently flat.
This form resembles a kind of the inflaton potential in the so-called new inflation models [36,37]. Moreover, for the potential shown on the left panel in Fig. 2, if the initial value of φ at the inflationary stage is φ i ≃ 0, the potential form is similar to that in the so-called natural (or axion) inflation [38]. Consequently, it is considered that the reconstructed potential form V (φ) includes the terms which could have physical and cosmological meanings.
Next, we discuss the justification of the coupling function ξ(φ) of φ to the Gauss-Bonnet invariant. If we consider the effective action with the loop correction [30,39,40], in which non-singular cosmological solutions have been derived [41,42], it is known that the coupling function ξ(φ) includes a constant term, a linear term, an exponential term, and a logarithmic term in terms of the scalar field φ [28,29]. Indeed, there exist these terms in the expressions of ξ(φ) in Eqs. (III.8) and (III.19). For ξ(φ) in Eq. (III.8), the first term is a constant term, the third term is a linear term, the second and fourth terms are exponential terms, and the fifth term is a logarithmic term. Moreover, for ξ(φ) in Eq. (III.19), the first term and the first and second terms within the round brackets in the last term on the last line are constant terms, the third term is a linear term, the second term is an exponential term, and the last term within the round brackets in the last term on the last line is a logarithmic term. Thus, it is interpreted that the resultant coupling function ξ(φ) could consist of the terms whose existence is justified based on the considerations in terms of the loop-corrected effective action.
We also describe the behaviour of ξ(φ) when the scalar field φ increases. On the right panel in Fig. 1, we see that in the limit that φ → +∞, ξ(φ) → −∞ . Moreover, on the right panel in Fig. 2, we find that φ → ±∞, ξ(φ) → −∞. Thus, when φ grows, ξ(φ) diverges. We can also be obtained for scalar Gauss-Bonnet gravity in the string frame as is shown in Sec. V. This phenomena can be seen in Figs. 3 and 4).

FIELD AROUND THE COSMOLOGICAL BOUNCE IN THE EINSTEIN FRAME
In this section, we explore the forms of the Hubble parameter and scalar field around the cosmological bounce point in the Einstein frame. If the cosmological bounce occurs at the time t = t b , the following conditions have to be satisfied We investigate the Cauchy problem for Eq. (II.5) with α 1 = 1 and α 2 = 0: The Cauchy problem formulated above takes place in the case that We solve the Cauchy problem in the form of the Taylor series in the powers of the deviation between the time and the cosmological bounce point (t − t b ): Similarly, from Eq. (II.12), we get the values of derivatives of H(t) at the bounce time Consequently, if we have the potential V (φ) of the scalar field φ and the coupling function of φ to the Gauss-Bonnet invariant ξ(φ), the expansion of the function H(t) around the bounce time t = t b can be written as (IV.14) The conditionḢ b > 0 (< 0) leads to the following expression V (φ b ) > 0 (< 0), but the expansion in Eq. (IV.14) is available only if γ = −1 (+1). Through the combination of Eqs. (IV.8)-(IV.13), we acquirė Therefore, the interaction of a scalar field with the Gauss-Bonnet invariant appears in the expansion near the point t = t b only from the fifth order. If the potential V (φ) and function ξ(φ) are represented as with V 0 , ξ 0 , and φ 0 constants, a scalar field can be expanded near the point of the cosmological bounce as The expansion of the function H(t) around the bouncing time t = t b becomes When the expressions of φ(t) and H(t) are known, it is possible to reconstruct the functions ξ(φ) and V (φ) around t = t b . However, only if the form of the term ξ(φ)(∇φ) 2 in the action is taken into account, the function ξ(φ) can completely be reconstructed.
Equation (II.12) is consistently differentiated with respect to the variable t. Accordingly, we find the coefficients of expansion of ξ(φ) in the Taylor series around the cosmological bounce at t = t b . The Taylor expansion of ξ(φ) is given by Here, for α 1 = 0 and α 2 = 0, we have , · · · , (IV. 25) whereas for α 1 = 1 and α 2 = 0, we obtain where c 1 is a constant.

V. RECONSTRUCTION OF SCALAR GAUSS-BONNET GRAVITY IN THE STRING FRAME
In Sec. III, we have examined scalar Gauss-Bonnet gravity in the Einstein frame, while in this section, we study scalar Gauss-Bonnet gravity in the string frame. In the string (Jordan) frame, the action has the form We consider the case that the scalar field is expressed as φ(t) = φ 0 t in Eq. (III.2). It follows from the action in Eq. (V.1) that in the FLRW space-time in Eq. (II.4), the gravitational field equations and the equation of motion for φ read For the string frame, from Eqs. (II.14) and (II.15) with f (φ, R) = e −φ R, ω(φ) = −e −φ , α 1 = 1, and α 2 = 0, we have .

VI. CONFORMAL TRANSFORMATION OF BOUNCE SOLUTIONS IN THE TRANSITION FROM THE STRING FRAME TO THE EINSTEIN FRAME
In this section, we investigate the transition from the string frame (g S µν , φ) to the Einstein frame (g E µν , ψ), where ψ is the scalar field in the Einstein frame corresponding to the scalar field φ in the string frame. We begin with the action of the heterotic string theory in the string frame Here and in the following, the superscription (subscription) "S" denotes the quantities in the string frame, whereas the superscription (subscription) "E" shows the quantities in the Einstein frame.
We make a conformal transformation [43] g S µν → g E µν = e −φ g S µν . (VI. 2) The FLRW metric in the string frame is written as ds 2 = e −φ (−dt 2 S + a 2 S dx 2 S ). Hence, the relation between time in the Einstein frame and that in the string frame becomes In the further considerations, we choose the positive sign in Eq. (VI.3). This means that the direction of motion along the time axis in the string frame is the same as that in the Einstein frame. There are also the following relations of various quantities between in the Einstein and string frames a E = e −φ/2 a S , (VI.4) When the string frame moves to the Einstein frame, the potential of the scalar field and its coupling function to the Gauss-Bonnet invariant are changed as follows The action in the Einstein frame is given by where g E is the determinant of the metric g E µν in the Einstein frame, R E is the Ricci scalar, and G E are the Gauss-Bonnet invariant. In this action, there is the Gauss-Bonnet term.
Hence, if we start from the effective action in the string frame, the additional term F appears in the Einstein frame [44].
In the Einstein frame, the conditions for the existence of the cosmological bounce at t These relations have to be fulfilled at the bounce point. The corresponding conditions in the string frame to these relations in the Einstein frame are represented as If the scalar field linearly dependents on time as φ(t S ) = φ 0 t S , by taking Eq. (VI.3) into consideration, we get Under the conformal transformation, the time axis t S converts into a positive or negative time semi-axis t E . Furthermore, if φ 0 > 0 (< 0), the mapping occurs on the negative (positive) semi-axis t E .
We analyze the behaviours of the scale factor a E (t E ) and its second derivativeä E (t E ) around t ⋆ E , which corresponds to the point of the cosmological bounce t S b in the string frame. It is known thatȧ S (t S b ) = 0, and therefore the scale factor at t S b has an extreme value. Using the relation φ(t S ) = φ 0 t S and Eq. (VI.4), we can determine the values of higher derivatives of the scale factor at the point of t ⋆ E in the Einstein frame as (VI. 16) It follows that the sign of second and third derivatives of the scale factor around t S b will be maintained during the transition from the string frame to the Einstein frame. However, if the functionä S (t S ) has an extreme value at t S b , the functionä E (t E ) will also have an extreme value, but it has already had an extreme value at the point t ⋆ E .

A. Hyperbolic model in the string frame
We study the case that the scale factor in the string frame has the hyperbolic form a S (t S ) = σe λt S + τ e −λt S , λ > 0 . (VI.17) In this model, the conditions in (VI.12) for the existence of the cosmological bounce read We define the point of the cosmological bounce in the Einstein frame as For the model in the string frame, under the conformal transformation, the point of the cosmological bounce t S b moves to a point t ⋆ E in the Einstein frame We examine the behaviour of second derivative of the scale factor in the Einstein frame.
Accordingly, it is seen that at the point t It can be shown that the function has only this extreme value.
(VI. 25) In Fig. 5, we show the evolutions of a E (t E ) and H E (t E ) as functions of t E for a S (t S ) = cosh (λt S ) with φ(t S ) = λt S and λ = 1. The functions V E (ψ) and ξ E (ψ) are reconstructed as follows In Fig. 6, we illustrate the evolutions of V E (t E ) and ξ E (t E ) as functions of t E for a S (t S ) = cosh (λt S ) with φ(t S ) = λt S , c 1 = 0, c 2 = 0, and λ = 1.

B. Exponential model in the string frame
We discuss the case that the scale factor in the string frame is written as a S (t S ) = exp αt 2 S , α > 0 , (VI.28) We define the point of the cosmological bounce in the Einstein frame as For this model in Eq. (VI.28), through the conformal transformation, the point of the cosmological bounce t S b in the string frame moves to a point t ⋆ E in the Einstein frame We also study the behaviour of the functionä E (t E ) for this model. It follows from Eqs. (VI.15) and (VI.16) that at the point t ⋆ E , the function has an extreme value. Hence, when |φ 0 | < √ 24α (|φ 0 | > √ 24α), it has a minimum (maximum). Moreover, from Fig. 7, it is found that in the latter case of |φ 0 | > √ 24α, the functionä E (t E ) has an additional extreme value at the points In this case, we get Let us consider the case that φ(t S ) = φ 0 t S with φ 0 = √ 4α. In Fig. 8, we depict the evolutions of a E (t E ) and H E (t E ) as functions of t E for a S (t S ) = e αt 2 S with φ(t S ) = √ 4αt S and α = 1. Furthermore, the functions V E (ψ) and ξ E (ψ) are expressed as where t E < 0. In Fig. 9, we plot the evolutions of V E (t E ) and ξ E (t E ) as functions of t E for a S (t S ) = e αt 2 S with φ(t S ) = √ 4αt S , c 1 = 0, c 2 = 0, and α = 1.

VII. CONCLUSIONS
In the present paper, we have studied the bounce universe in the framework of scalar Gauss-Bonnet gravity. The existence of the Gauss-Bonnet invariant as a higher derivative quantum correction is strongly supported by string theories. Particularly, when the scale factor has the hyperbolic form or exponential form leading to cosmology with a bounce, we have explicitly reconstructed the potential form and Gauss-Bonnet coupling function of a dynamical scalar field.
In addition, we have explored the bounce behaviours in both the string and Einstein frames in detail by performing the conformal transformation and derived the relation of the bounce cosmology between these conformal frames. Through the conformal transformation, it has been seen that the difference of potential form of the scalar field between the two frames is the exponential function of the scalar field, whereas the coupling function of the scalar field to the Gauss-Bonnet invariant are the same in the two frames.
As a consequence, we have found the following three points. (i) In the case that the point of the cosmological bounce in the string frame is transformed into that in the Einstein frame, it does not retain its character. When the conformal transformation from the string frame to the Einstein frame is made, the bounce point in the string frame changes its qualitative natures and ceases to be bounce point in the Einstein frame. However, new bounce point(s) appears in the Einstein frame. (ii) If the second derivative of the scale factor takes an extreme value in the string frame, the second derivative of the scale factor in the Einstein frame has an extreme value at the point corresponding to the one of the cosmological bounce in the string frame. Especially, there are cosmological models in which at this point, the universe expands with its minimal acceleration, namely, the second derivative of the scale factor becomes its minimum value. However, in principle, the parameters of the theory may be chosen so that the second derivative of the scale factor at this point can take its maximum value. (iii) Third, in the Einstein frame, at the point of the cosmological bounce t E b , the gap interaction function ξ(φ) is missed unlike in the string frame.
In inflation paradigm, the spatially flat, homogeneous, and isotropic universe, which are suggested by quite precise cosmological observations, can be realized successfully. The primordial density perturbations with its spectrum consistent with the observations can also be generated during inflation. It is significant to discuss other possible scenarios for the early universe to explain the observations so that physics in the early universe can further be proved. As an attempt for this issue, the idea of the bounce universe has been examined. For instance, also in the matter bounce scenario and the Ekpyrotic cosmology, the primordial density perturbations with its almost scale-invariant spectrum can be generated.
Another additional merit of these scenarios is that they are motivated by fundamental theories including superstring/M-theories, which are hopeful candidates to describe the quantum aspects of gravity.
Finally, we remark that according to the investigations in F (R) gravity, when the cosmological bounce happens in the Einstein frame, the cosmic acceleration (inflation) may occur in the corresponding string frame via the conformal transformation [18]. Thus, if there exists a kind of duality between the bounce phenomenon in the Einstein frame and inflation in the string frame, and vice versa, by comparing the theoretical results on the spectral index of the curvature perturbations and the tensor-to-scalar ratio in the Einstein frame with their observational values and using such a duality, we can judge whether the corresponding bounce cosmology is realistic or not (see also Ref. [26]).