We reconstruct F(R) gravity models with exponential and power-law forms of the scale factor in which bounce cosmology can be realized. We explore the stability of the reconstructed models with analyzing the perturbations from the background solutions. Furthermore, we study an F(R) gravity model with a sum of exponentials form of the scale factor, where the bounce in the early universe as well as the late-time cosmic acceleration can be realized in a unified manner. As a result, we build a second order polynomial type model in terms of R and show that it could be stable. Moreover, when the scale factor is expressed by an exponential form, we derive F(R) gravity models of a polynomial type in case of the non-zero spatial curvature and that of a generic type in that of the zero spatial curvature. In addition, for an exponential form of the scale factor, an F(R) bigravity model realizing the bouncing behavior is reconstructed. It is found that in both the physical and reference metrics the bouncing phenomenon can occur, although in general the contraction and expansion rates are different each other.

We explore bounce cosmology in F (G) gravity with the Gauss-Bonnet invariant G. We reconstruct F (G) gravity theory to realize the bouncing behavior in the early universe and examine the stability conditions for its cosmological solutions. It is demonstrated that the bouncing behavior with an exponential as well as a power-law scale factor naturally occurs in modified Gauss-Bonnet gravity. We also derive the F (G) gravity model to produce the ekpyrotic scenario. Furthermore, we construct the bounce with the scale factor composed of a sum of two exponential functions and show that not only the early-time bounce but also the late-time cosmic acceleration can occur in the corresponding modified Gauss-Bonnet gravity. Also, the bounce and late-time solutions in this unified model is explicitly analyzed.Introduction-As a cosmological model to describe the early universe, the matter bounce scenario [1] is known. In this scenario, in the contraction phase the universe is dominated by matter, and a non-singular bounce occurs. Also, the density perturbations whose spectrum is consistent with the observations can be produced (for a review, see [2]). In addition, after the contracting phase, the so-called BKL instability [3] happens, so that the universe will be anisotropic. The way of avoiding this instability [4] and issues of the bounce [5] in the Ekpyrotic scenario [6] has been investigated [7,8]. Moreover, the density perturbations in the matter bounce scenario with two scalar fields has recently been examined [9].On the other hand, various cosmological observations support the current cosmic accelerated expansion. To explain this phenomenon in the homogeneous and isotropic universe, it is necessary to assume the existence of dark energy, which has negative pressure, or propose that gravity is modified on large scales (for recent reviews on issues of dark energy and modified gravity theories, see, e.g., [10,11]). Regarding the latter approach, there have been proposed a number of modified gravity theories such as F (R) gravity. The bouncing behavior has been investigated in F (R) gravity [12][13][14][15], string-inspired gravitational theories [16], non-local gravity [17]. A relation between the bouncing behavior and the anomalies on the cosmic microwave background (CMB) radiationhas also been discussed [18].In this Letter, we explore bounce cosmology in F (G) gravity with F (G) an arbitrary function of the Gauss-Bonnet invariant G = R 2 −4R µν R µν +R µνρσ R µνρσ , where R µν is the Ricci tensor and R µνρσ is the Riemann tensor. Such F (G) theory has been proposed as gravitational alternative for dark energy and inflation in Ref. [19] and its application to the late-time cosmology [20] has been studied. Moreover, cosmology in a theory with a dynamical dilaton coupling to the Gauss-Bonnet invariant has also been studied [21]. 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 M Pl = G −1/2 = 1.2 × 10 19 GeV. In the followin...

We investigate dynamics of (4 + 1) and (5 + 1) dimensional flat anisotropic Universe filled with a perfect fluid in the Gauss-Bonnet gravity. An analytical solutions valid for particular values of the equation of state parameter w = 1/3 have been found. For other values of w structure of cosmological singularity have been studied numerically. We found that for w > 1/3 the singularity is isotropic. Several important differences between (4 + 1) and (5 + 1) dimensional cases are discussed.

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.PACS numbers: 04.50. Kd, 95.36.+x, 98.80.Cq