Unimodular f(G) gravity

In this paper we study a modified version of unimodular general relativity in the context of f(G), G denoting the Gauss–Bonnet invariant. We focus on Bianchi-type I and Friendmann–Robertson–Walker universes and search for unimodular f(G) models according to the de Sitter and power-law solutions. Assuming unimodular f(G) gravity as a perfect fluid and making use of the slow-roll parameters, the inflationary model has been reconstructed in concordance with the Planck observational data. Moreover, we investigate the realization of the bounce and loop quantum cosmological ekpyrotic paradigms. Assuming suitable and appropriate scale factors, unimodular f(G) models able to reproduce superbounce and ekpyrotic scenarios have been reconstructed.


Introduction
It is well known nowadays that interest in gravity theories derived from Lagrangians has extended beyond the general relativity (GR). The first step is introducing the cosmological constant, originating from the vacuum expectation value of the quantum field [1,2]. However, there is no intrinsic mechanism in the theory that can dynamically induce the cosmological constant [3][4][5]. On the other hand, unimodular gravity  is an interesting gravitational theory that can be considered as a specific case of the GR, and in which the cosmological constant appears as the trace-free part of the gravitational field equations, fixing the determinant of the metric tensor as a number or a function [2,5]. In other words, it would be more appropriate to state that unimodular gravity, in a sense, can offer a proposal to solve the cosmological problem. It is important to mention that in the context of unimodular gravity the problem of late-time acceleration of the universe can be developed, where the metric can be decomposed in the unimodular metric part and a scalar field a e-mail: sthoundjo@yahoo.fr part [23][24][25]. Instead of considering GR, different kinds of modified gravity based on the curvature scalar have been performed in the recent years, as f (R) [31][32][33][34][35][36][37][38][39] where R is the curvature scalar, f (R, T ), T being the trace of the energy-momentum tensor [40][41][42][43][44][45][46][47][48][49], and f (G), G denoting the Gauss-Bonnet (GB) invariant .
In this paper we focus our attention to the f (G) and the purpose here is to realize some cosmological evolutions with specific and realistic Hubble rate, and to investigate which unimodular f (G) model can yield such cosmological evolutions of the universe. We fundamentally adopt two kinds of metric in this paper; the Bianchi type-I (BI) metric and the Friedman-Robertson-Walker (FRW) metric. It is obvious that for both metrics there is no compatibility with the unimodular constraint, and so we fix suitably the metrics in order to satisfy this constraint. With the BI metric we search for f (G) models able to reproduce inflation through the de Sitter and power-law form of the scale factor. Through the use of the FRW metric, we still search for f (G) models that agree with inflation. As we are addressing inflationary models we find it important to formulate the related observables, like the spectral index, the tensor-to-tensor ratio and the running of the spectral index. By considering the f (G) model coming from power-law solutions of the scale factor, with non-specified integration constants in the perfect fluid description, we determine the slow-roll indices and confront them with the observational recent Planck data [85,86] in order to correctly calculate the constants, and then obtaining the effective unimodular f (G) model able to realize the inflationary epoch of our universe.
An alternative to the standard description of the acceleration of inflation is provided by bouncing cosmologies [87,118], in which the appearance of the initial singularity is prevented. This feature can be represented by the use of scalar fields , by the models of modified gravity [93][94][95][96][97][98][99][100][101][102][103][104][105], and also by the use of the Loop Quantum Cosmology (LQC) [106][107][108][109][110][111][112][113][114]. In the general case, the realization of the bounce is related to the coupling of matter fields with an equation of state in order to violate the null energy condition [87][88][89]118]. Bouncing cosmologies have been developed in several works [115][116][117]119], but essentially it has been shown that bouncing cosmologies may present primordial instabilities [90,91], which can be solved by the ekpyrotic scenario. In this paper we also pay attention to the superbounce and the LQC ekpyrotic scenarios in the context of unimodular f (G) gravity by reconstructing a suitable model characteristic of each scenario. This kind of work has been performed in [119], but not in the unimodular context. For other reviews as regards bouncing cosmologies see [120,121] for f (R) theory, [122][123][124][125] for f (G) theory and  for f (T ) theory, where T denotes the torsion scalar.
The manuscript is organized as follows: in Sect. 2 we present the general description of unimodular gravity. Section 3 is devoted to the formalism of f (G) and unimodular equations of motion. The reconstruction of f (G) gravity in an inhomogeneous universe is performed in Sect. 4, for both de Sitter solutions in Sect. 4.1 and power-law solutions in Sect. 4.2. In the FRW metric context, the reconstructions of unimodular f (G) models have been developed within the de Sitter solutions in Sect. 5.1 and powerlaw solutions in Sect. 5.2, and the latter has been achieved according to Planck results in Sect. 6, finding the integration constants accordingly. The ekpyrotyc scenario reconstruction and superbounce reconstruction from unimodular f (G) gravity is performed in Sect. 7. We present our conclusion in Sect. 8.

General description of unimodular gravity
In this section we point out the generalization of the GR gravity formalism in order to provide the unimodular f (G) gravity formalism. The unimodular gravity approach is based on the assumption that the determinant of the metric tensor is fixed, expressed by the relation g μν δg μν = 0. Throughout this paper the components of the metric are chosen in such a way that [2] √ −g = 1. (1) Let us start our study by using the Bianchi type-I metric, from which the usual and practical FRW metric can be recovered, It is easy to check that the unimodular constraint, expressed by Eq. (1) is not satisfied by the Bianchi-type I metric (2). In order to meet this condition we need to redefine the cosmic time coordinate as follows: in such a way that the metric (2) becomes This metric obviously satisfies the constraint (1) and we shall refer to the latter as the unimodular Bianchi-type I metric from which the FRW metric can be recovered.

Formalism of f (G) gravity and unimodular equations of motion
Let us introduce the general R + f (G) action as follows: where R represents the Ricci scalar, and the modification function f (G) corresponds to a generic globally differentiable Gauss-Bonnet topological invariant G function. The matter action S m is the one which induces the energymomentum tensor T μν . We focus our attention on the metric formalism where the variation of the action (5) with respect to the metric tensor yields within the minimum principle the following general equation of motion [50][51][52][53][54]: Here f G = d f (G) dG , and the Gauss-Bonnet term is defined by G = R 2 − R μν R μν + R μνλσ R μνλσ , R μν and R μνλσ being the Ricci tensor and Riemann tensor, respectively. We also point out some adopted definitions as follows: the signature of the Riemannian metric is ( ων for the covariant derivative of a covariant vector and the Riemann tensor, respectively. Making use of the metric (4) the field equations read where we undertook the matter content as an anisotropic fluid described by the following energy-momentum tensor: with ρ being the energy density, u μ the 4-velocity and v μ the space-like vector in the radial direction; p r (the radial pressure) is the pressure in the direction of v μ and p t (the tangential pressure) the pressure in the direction orthogonal to v μ . Due to the fact that one considers an anisotropic spherically symmetric matter one has p r = p t (their equality corresponds to isotropic spherically symmetric matter).

Reconstructing f (G) gravity in inhomogeneous universe
In this section we search for the f (G) action through the reconstruction scheme for some particular solution of the class of the metric explored in the previous section. We will just consider the de Sitter and power-law solutions. In the general case, using (4) and (11) the conservation equation for the energy-momentum tensor can easily be obtained bẏ where the following definitions have been made:

Searching for de Sitter solutions
The de Sitter solutions are well known in the context of cosmology due to the fact that they may approximately describe the early and current epochs of the universe where its expansion is accelerated. Here, for the three directions the scale factors present an exponential expansion yielding constant Hubble parameters in each direction. Then we assume the scale factors as follows: where H x0 , H y0 and H z0 are positive constants expressing the instantaneous rates of the expansion at t = 0. By simple derivations one can see that the rates of the expansion for each directions read Then, in the context of the cosmic time t the Gauss-Bonnet invariant reads = 8H x0 H y0 H z0 H x0 + H y0 + H z0 (16) and the field equations become Since we address de Sitter solutions, by assuming p r = p t = p, one has p = −ρ, and the combination of any two of the field equations yields where K is a constant depending on H x0 , H y0 and H z0 . The general solution of Eq. (21) reads where K is an integration constant. Making use of (3) and (13) the expressions of the scale factors can be written as such that the metric (4) in the unimodular context becomes with

Power-law solutions
In this subsection we devote our attention to the cosmological evolution described by a power-law functions of cosmic time in each direction of the space. We assume the following expressions for the scale factors in terms of cosmic time: where a, b, c and A 0 , B 0 , C 0 are constants that should be determined from the initial conditions. In this case the rate of the expansion for each direction is given by and the expression of Gauss-Bonnet invariant in the context of cosmic time reads Then the field equations take the following forms: Here we also assume for simplicity that p r = p t = p = ωρ, ω being the parameter of the equation of state. Considering the first two field equations one gets with By making use of (33) the previous equations take the following form: where The general solution of the previous equation reads with β a positive constant to be determined using the cosmological data. In this case the metric (4) in the unimodular context becomes

Lagrange multiplier formulation and reconstruction of unimodular f (G) gravity
In this section we will present the Lagrange multiplier formulation of f (G) gravity, according to the method performed in context of f (R) in [152][153][154]. This procedure is just used to ensure that the unimodular condition is satisfied. To do so, we introduce the Lagrangian multiplier λ and the unimodular f (G) gravity with matter can be expressed as From now we will assume the FRW metric and consider the matter energy-momentum tensor as corresponding to a perfect fluid with the energy density and pressure ρ and p, respectively. Thereby the GB invariant expressions in terms of t and τ read where the parameter H ≡ 1 dτ is the relative Hubble parameter in this context. Hence the field equations become By directly summing Eqs. (49) and (85), the term ( f − λ) is eliminated, giving rise to the following equations: In order to perform a consistent analysis, it is important to make use of the continuity equation for the matter, namely Let us adopt the equation of state p = wρ. Thus solving the previous equation, one gets Now, in order to obtain cosmological f (G) models in the unimodular context, one just needs a suitable expression of the scale depending on τ , solving Eq. (51).

Reconstruction of unimodular f (G) model describing de Sitter universe
In this subsection we consider the FRW metric that describes a de Sitter expanding universe with the scale factor reading where H 0 is an arbitrary positive constant. By using Eq. (3), setting B = C = A, one can extract the t in terms of τ as such that the scale factor in terms of the τ reads In this case, addressing the de Sitter universe, ω = −1 and the differential equation (51) becomes whose general solutions are where C 1 and C 2 are integration constants. On the other hand, one can express the GB invariant (48) in terms of τ , which obviously is a constant. The task to be done now is to invert the function G(τ ) in order to obtain τ = τ (G), but since G(τ ) = Const it is common to fix C 1 = 0, such that f G = d f (G)/dG = C 2 , whose general solution takes the following form: This result is consistent with the one obtained in (22), confusing (α, K ) with (C 1 , C 3 ), characteristic of the de Sitter universe.

Reconstruction of unimodular f (G) model according to power-law solution
In this case we assume the scale factor in terms of the cosmic time to be where t 0 and A 0 are the initial time and the initial value of the scale factor, respectively. According to the relation dτ = A(t) 3

dt one gets
It is easy to see that, for σ = 1/3, the de Sitter universe is recovered, while the power-law inflation is obtained for q > 1 (1/4 < σ < 1/3). The accelerated expansion is realized for 0 < q ≤ 1 (0 < σ ≤ 1/4), where inflation is possible. The GB invariant in this case is Thus the differential equation (51) becomes with The previous equation can be rewritten by dividing the whole equation by J 01 = 0, where whose general solution reads K 1 and K 2 being integration constants. From Eq. (63), one can express τ and τ τ 0 in terms of G, Hence, Eq. (68) can now be written in terms of G as By integrating the expression in the right hand side of the (70) with respect to G, one gets

Unimodular inflationary cosmological f (G) model
In the previous section we performed the reconstruction of the specific f (G) form that generates scale-factor evolution.
The obtained model will be explored in the present section to investigate inflation realization in unimodular f (G) gravity. Moreover, we will focus our attention on the investigation of inflationary observables such as scalar and tensor spectral indices, the tensor-to-tensor ratio and the running spectral index confronting them with the cosmological observational data.

Slow-roll parameters and inflationary observables
The exploration of any inflationary scenario is essentially based on the values of related observables such as the scalar spectral index of the curvature perturbations n s , its running α s ≡ dn s /d ln k, k being the absolute value of the wave number k, the tensor spectral index n T and the tensor-to-tensor ratio r . The determination of the values of the observables would be by detailed perturbation analysis and we propose to escape from these complicated procedures by making use of the Einstein frame, where all the inflationary information is driven by the so-called effective scalar potential V (φ). To do so it is common to define the slow-roll parameters , η and ξ in terms of this potential and its derivatives [155][156][157]161], We point out that the calculation of the spectral index of the primordial curvature perturbations and of the scalar-to-tensor ratio relies upon the technique of the perfect fluid approximation in modified gravity, and this was developed in [158,159]. It is important to mention that a more direct approach is to calculate the spectral index directly via the direct calculation of the power spectrum, as was done in [160], in the context of standard f (G) gravity; this is, however, very tedious. Also it has been shown in [5] that the inflation ends when = 1. The approximated expressions for the observables are presented in [161]: Since we are dealing with modified gravity, its obvious that the conformal transformation to the Einstein frame is not possible and one cannot define a scalar potential nor the potential slow-roll parameters. To do so, one can introduce the Hubble slow-roll parameters n by where the initial value of this parameter is 0 ≡ H ini /H and the parameter N , known as the e-folding number, is defined as N ≡ ln (a/a ini ), a ini being the scale factor at the beginning of the inflation and H ini the corresponding Hubble parameter. Hence one can express the first three as in such a way that the inflationary-related observables can now be written as According to (61) and (62), one gets yielding where we fix for simplicity K 1 = K 2 = 1 and K 3 = 0.

Superbounce and loop quantum ekpyrotic cosmology from unimodular f (G) gravity
In this section we propose to reconstruct the unimodular f (G) model able to realize the superbounce and loop quantum ekpyrotic cosmology. The first attempts to realize these cosmologies in modified gravity were performed in [163].

Ekpyrotic scenario reconstruction from unimodular f (G) gravity
The task here is to find the f (G) model in the unimodular context which, in the large cosmic time limit, corresponds to the late-time era of the ekpyrotic scenario. We assume the well-known gravitational action as in [124,125], By making use of the auxiliary scalar field φ the previous action becomes The variation of the action (80) with respect to φ yields whose solution, if it exists, will be a functional dependence of G, i.e. φ = φ(G) and now substituting it into (80) leads to and it is obvious that if we find φ(G), the algebraic f (G) will be obtained straightforwardly. We have By combining the above equations through a straightforward summation one gets the following equation: Since we are searching for an ekpyrotic model we have to take the large cosmic time limit of the scale factor assumed as corresponding, in the unimodular context, to According to the previous expression of the scale factor in the unimodular context, Eq. (86) takes the following form: with γ 1 (τ ) = 8Ālτ l−1 + 2(l − 1)τ −1 , and From this, according to the large cosmic time, we arrive at the following expression of ξ(τ ) and V (τ ): Making use of (48) and (87), one can express τ in terms of G as such that, from (83), the algebraic f (G) ekpyrotic model reads

Superbounce reconstruction from unimodular f (G) gravity
In this case the scale factor is given by [88] where t * denotes the big crunch time, and c is a parameter constrained to c > √ 6 [88]. In the context of unimodular gravity, one gets In this case, for simplicity and for large cosmic time t, i.e. near the bounce, the τ dependent functions ξ(τ ) and V (τ ) behave as On the other hand, from (97) and (48), one expresses τ in terms of G as such that the algebraic f (G) model near the bounce reads where Z 1 and Z 2 are integration constants depending on the parameter c.

Conclusion
In this paper we explored a type of modified GR theory, namely f (G) theory of gravity in the unimodular context, where G denotes the GB invariant. In such a theory an important condition is applied to the metric tensor constraining its determinant to a number or a specific function. Here we work fixing it to 1 and consider both appropriate BI and FRW universes. In a first step the task is to reconstruct the unimodular f (G) models considering de Sitter and power-law solutions for the scale factors. The resulting models are different from the standard f (G) model (without considering unimodular formalism). In the second step we focused our attention to the model provided by the power-law solutions in the FRW universe and searched for the input constants that can constrain it to an inflationary model. To do so, we proceeded to the determination of the slow-roll parameters and the corresponding observational indices from the unimodular f (G) field equations, which appear as functions of the input parameters. According to the recent Planck results we calculated accordingly the input parameters, obtaining an unimodular f (G) inflationary model. On the other hand we investigated, still in the unimodular f (G) context, the superbounce and the loop quantum cosmology ekpyrotic paradigms. As is well known, bouncing solutions can be an alternative to inflation and thus it appears interesting to search for a related f (G) model. To do so, we considered the well-known standard cosmic time depending scale factors able to lead to the bouncing cosmology in the one case and the ekpyrotic cosmology in the other. The corresponding scale factors in the unimodular context have been determined, namely in terms of the auxiliary time τ . Through the unimodular formalism and the related field equations, the bouncing and ekpyrotic f (G) models have been reconstructed.