Physics Letters B 732 (2014) 349–355
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Physics Letters B
www.elsevier.com/locate/physletb
Bouncing cosmology in modified Gauss–Bonnet gravity
Kazuharu Bamba
a,b,∗
, Andrey N. Makarenko
c,d
, Alexandr N. Myagky
e
,
Sergei D. Odintsov
f,g,h
a
Leading Graduate School Promotion Center, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan
b
Kobayashi–Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 464-8602, Japan
c
Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia
d
National Research Tomsk State University, Lenin Avenue, 36, 634050 Tomsk, Russia
e
National Research Tomsk Polytechnic University, Lenin Avenue, 30, 634050 Tomsk, Russia
f
Instituciò C atalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
g
Institut de Ciencies de l’Espai (CSIC–IEEC), Campus UAB, Facultat de Ciencies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain
h
King Abdulaziz University (KAU), Jeddah, Saudi Arabia
article info abstract
Article history:
Received 13 March 2014
Received in revised form 3 April 2014
Accepted 3 April 2014
Available online 12 April 2014
Editor: J. Hisano
We explore bounce cosmology in F(G) gravity with the Gauss–Bonnet invariant G.WereconstructF (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 are explicitly analyzed.
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
1. 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
areview,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 inves-
tigated [7,8]. Moreover, the density perturbations in the matter
bounce scenario with two scalar fields have recently been exam-
ined [9].
On the other hand, various cosmological observations support
the
current cosmic accelerated expansion. To explain this phe-
nomenon in the homogeneous and isotropic universe, it is nec-
essary to assume the existence of dark energy, which has nega-
tive pressure, or propose that gravity is modified on large scales
(for recent reviews on issues of dark energy and modified grav-
ity theories, see, e.g., [10,11]). Regarding the latter approach, there
have been proposed a number of modified gravity theories such
*
Corresponding author.
as F (R) gravity. The bouncing behavior has been investigated
in F
(R) gravity [12–15], string-inspired gravitational theories [16],
non-local gravity [17]. A relation between the bouncing behavior
and the anomalies on the cosmic microwave background (CMB)
radiation has 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 ten-
sor 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 stud-
ied [21].Weuseunitsofk
B
= c
l
=
¯
h = 1, where c is the speed of
light, and denote the gravitational constant 8
π G by κ
2
≡ 8π/M
2
Pl
with the Planck mass of M
Pl
= G
−1/2
= 1.2 ×10
19
GeV.
In the following, we first explain F
(G) gravity and its recon-
struction method. We also investigate the stability of the solutions
in the reconstructed F
(G) gravity model. As more concrete ex-
amples, we study an exponential model and a power-law model,
in which the bouncing behavior happens. In addition, the F
(G)
gravity model to make the ekpyrotic scenario [22] is built. Next,
we examine a sum of two exponential functions model of the
scale factor. We explicitly show that in this model the bouncing
http://dx.doi.org/10.1016/j.physletb.2014.04.004
0370-2693/
© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by
SCOAP
3
.