# Bounce cosmology in generalized modified gravities

G. Minas<EMAIL_ADDRESS>Section of Astrophysics, Astronomy and
Mechanics, Department of Physics, National and Kapodistrian University of
Athens, Panepistimiopolis 15783, Athens, Greece  E. N. Saridakis
<EMAIL_ADDRESS>Department of Physics, National Technical University of
Athens, Zografou Campus GR 157 73, Athens, Greece  Department of Astronomy,
School of Physical Sciences, University of Science and Technology of China,
Hefei 230026, China  P. C. Stavrinos<EMAIL_ADDRESS>Department of
Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis
15784, Athens, Greece  A. Triantafyllopoulos<EMAIL_ADDRESS>Section of
Astrophysics, Astronomy and Mechanics, Department of Physics, National and
Kapodistrian University of Athens, Panepistimiopolis 15783, Athens, Greece

###### Abstract

We investigate the bounce realization in the framework of generalized modified
gravities arising from Finsler and Finsler-like geometries. In particular, the
richer intrinsic geometrical structure is reflected in the appearance of extra
degrees of freedom in the Friedmann equations that can drive the bounce. We
examine various Finsler and Finsler-like constructions. In the cases of
general very special relativity as well as of Finsler-like gravity on the
tangent bundle we show that a bounce cannot be easily obtained. However, in
the Finsler-Randers space the induced scalar anisotropy can fulfill the bounce
conditions and bouncing solutions are easily obtained. Finally, for the
general class of theories that include a nonlinear connection a new scalar
field is induced, leading to a scalar-tensor structure that can easily drive a
bounce. These features reveal the capabilities of Finsler and Finsler-like
geometries.

bounce, cosmology, Finsler geometry, modified gravities

###### pacs:

98.80.-k, 95.36.+x, 04.50.Kd

##  I Introduction

Bounce cosmologies offer an alternative view of the early universe
Biswas:2005qr ; Cai:2007qw ; Cai:2008qw ; Cai:2012va ; Cai:2014bea ;
Brandenberger:2016vhg  (for a review see Novello:2008ra ). Historically, this
idea belongs to Tolman who in 1930’s first suggested the possibility of a re-
expansion of a closed universe which has already collapsed to an extremely
dense state Tolman:1931fei . Since then, various bouncing models have been
proposed within an effort for a systematic explanation of the origin of our
universe.

The main advantage of bouncing cosmology is that it provides a way of solving
the singularity problem which appears in the standard cosmological paradigm.
The singularity (Big Bang) is replaced with a smooth transition from
contraction to expansion (Big Bounce). In this sense, bounce cosmology offers
an opportunity of obtaining a more continuous picture of the early universe.
The efficiency of bouncing models in solving basic cosmological problems in
comparison with inflationary scenarios is visualized via the wedge diagram
introduced in Ijjas:2018qbo .

In general, the realization of a bounce requires a violation of the null
energy condition. This can be achieved with the introduction of extra degrees
of freedom which are added ad hoc into the Lagrangian Singh:2015bca ;
Cai:2012va . The violation of null energy condition needs to be handled with
care, in order not to spoil the usual thermal history and the sequence of
epochs after the bounce. Nevertheless, such violations can easily be acquired
from modified Novello:2008ra  or quantum gravity Barrau:2016fcg . In
particular, they can be easily acquired in the Pre-Big-Bang Gasperini:2003pb ;
Gasperini:2004ss  and the Ekpyrotic Khoury:2001wf ; Khoury:2001bz  models, in
gravity actions with higher order corrections Tirtho1 ; Nojiri:2013ru , in
$f(R)$ gravity Bamba:2013fha ; Nojiri:2014zqa , in $f(T)$ gravity Cai:2011tc ,
in braneworld scenarios Shtanov:2002mb ; Saridakis:2007cf , in non-
relativistic gravity Brandenberger:2009yt ; Cai:2009in ; Saridakis:2009bv , in
Galileon theory Easson:2011zy ; Qiu:2013eoa , in massive gravity Cai:2012ag ,
in Lagrange modified gravity Cai:2010zma , in loop quantum cosmology
Bojowald:2001xe ; Odintsov:2014gea ; Odintsov:2015uca , etc. Moreover, a
nonsingular bounce model which supports magnetogenesis at the inflationary
epoch is presented in Membiela:2013cea .

Among modified gravity theories, an interesting class is that of gravitational
models based on Finsler and Finsler-like geometries. These are natural
extensions of Riemannian geometry in which the physical quantities may
directly depend on observer 4-velocity, and this velocity-dependence reflects
the Lorentz-violating character of the kinematics. Such a property is called
_dynamic anisotropy_ Bogoslovsky:1999pp ; Chang:2007vq ; Kouretsis:2010vs ;
Vacaru:2010fi ; Mavromatos:2010jt ; Vacaru:2010rd ; Mavromatos:2010nk ;
Torrome:2012kt ; Basilakos:2013hua ; Basilakos:2013ij ; Hohmann:2016pyt ;
Hohmann:2018rpp . Additionally, Finsler and Finsler-like geometries are
strongly connected to the effective geometry within anisotropic media Born ;
Perlick  and naturally enter the analogue gravity program Barcelo:2005fc .
These features suggest that Finsler and Finsler-like geometries may play an
important role within quantum gravity physics. The dependence of the metric
tensor and other quantities on the position coordinates of the base-manifold
and the directional/velocity variables of the tangent space suggest that the
natural geometrical framework for the description of these models is the
tangent bundle of a smooth manifold. Finally, in the case where there is no
velocity-dependence, Finsler geometry becomes Riemannian.

The intrinsic geometrical space-time dynamical anisotropy of Finsler geometry
(not to be confused with the spatial anisotropy that may exist also in
Riemannian geometry, as for instance in Bianchi cases) is included in the
geometry of space-time as an intrinsic field (variable) which influences its
geometrical and physical concepts. Hence, it can give us the form of
anisotropy as a hypothetical field, the _anisotropion_ , which produces this
deviation from isotropy. This appears in the Friedmann equations and Lorentz
violations Koivisto:2008ig ; Stavrinos:2012ty ; kour-stath-st 2012 ;
Triantafyllopoulos:2018bli , and thus the anisotropy arises as a property of
Finslerian spacetime stavrinos-ikeda 1999 ; Stavrinos:2016xyg ;
Stavrinos:2012ty ; Triantafyllopoulos:2018bli .

In the present work we are interested in investigating the bounce realization
in the framework of modified gravity related to Finsler and Finsler-like
geometries. In particular, we desire to see how the new features of Finsler
geometry can drive bouncing solutions, and to examine the evolution of the
intrinsic anisotropy during the bounce. In some bouncing scenarios the
anisotropy decreases in the contracting phase and remains quite small during
the bounce, in agreement with the current observational data Cai:2012va . On
the other hand, there are also scenarios where the reduction of anisotropy in
the contracting phase is followed by its exponential growth during the bounce,
mainly due to quantum fluctuations of curvature Xue:2011nw . Finally, we
mention that nonsingular bounces are also possible to be generated in models
which spontaneously violate Lorentz symmetry kour-stath-st 2012 ;
Gasperini:1985aw ; Gasperini:1998eb . In this framework Lorentz symmetry
violations lead to interactions with anisotropies kour-stath-st 2010 . Hence,
we can establish a connection between anisotropic fields and nonsingular
bounce. In summary, we can depict all the above form of connections in the
diagram of Fig. 1.

anisotropyLorentz violationnonsingular bounce Figure 1:  Connections of
anisotropy, nonsingular bounce and Lorentz violation.

The plan of this work is the following: In Section II we first describe the
basic conditions for a bounce realization and we briefly review Finsler
geometry and gravity. Then we examine the bounce realization in general very
special relativity and Finsler-Randers models. In Section III we study the
case of Finsler-like gravity on a tangent bundle, while in Section IV we
analyze bouncing solutions from scalar-tensor theory on the fiber bundle.
Finally, in Section V we present the summary and the conclusions.

##  II Bounce from Finsler gravity

In this section we are interested in studying the bounce realization in the
framework of Finsler gravity. We start by describing the conditions for a
bounce realization, and we provide the basics of Finsler geometry and gravity.
Then we proceed to the examination of the bounce realization in specific
models, such as general very special relativity and Finsler-Randers ones.

###  II.1 Bounce conditions

Let us start by discussing the basic requirements for a bouncing solution. For
the moment we consider the ordinary Friedmann-Robertson-Walker (FRW) geometry
with metric

$[g_{\mu\nu}(x)]=\mathrm{diag}\left(-1,\frac{a^{2}(t)}{1-kr^{2}},a^{2}(t)r^{2},a^{2}(t)r^{2}\sin^{2}\theta\right),$
(1)

with $a(t)$ the scale factor and $k=-1,0,+1$ corresponding to open, flat and
closed spatial geometry respectively. As usual, in such a geometry the general
field equations of any theory give rise to the Friedmann and Raychaudhuri
equations, which can be written in a compact form as

$\displaystyle H^{2}=\frac{8\pi G}{3}\rho_{tot}-\frac{k}{a^{2}}$ (2)
$\displaystyle\dot{H}=-4\pi G(\rho_{tot}+P_{tot})+\frac{k}{a^{2}},$ (3)

where $G$ is the Newton’s constant, $H={\dot{a}}/{a}$ is the Hubble function,
and with dots denoting derivatives with respect to the cosmic time $t$. In the
above expressions $\rho_{tot}$ and $P_{tot}$ are respectively the total energy
density and pressure of the universe, which include matter, radiation, dark
energy and any other gravitational or geometrical contribution that a theory
or scenario may have.

In order to obtain a bounce realization we need a contracting universe, namely
with $H<0$, succeeded by an expanding universe, namely with $H>0$, and hence
from continuity we deduce that at the bounce point we must have $H=0$.
Furthermore, one can see that at the bounce point and around it we must have
$\dot{H}>0$. Observing the form of the general Friedmann and Raychaudhuri
equations (2),(3), and focusing on the physically more interesting flat case,
we deduce that the above requirements can be fulfilled if

$\displaystyle\rho_{tot}=0$ (4)

exactly at the bounce point, and if additionally the null energy condition is
violated around the bounce point, namely if

$\displaystyle\rho_{tot}+P_{tot}<0$ (5)

(in the case of a non-flat universe the bounce can be driven by the curvature
term without null energy condition violation Novello:2008ra ). Therefore, in
order to obtain a bounce one needs to construct theories in which the extra
contributions to the total energy density and pressure are such that the null
energy condition is violated around the bounce point and requirement (5)
holds, and moreover the total energy becomes zero exactly at the bounce point
and condition (4) holds. As we see in the following, scenarios based on
Finsler gravity can fulfill these necessary conditions.

###  II.2 Finsler gravity

We first briefly review the basics of Finsler gravity, since this lies in the
center of the investigation of the present work. Finsler gravity is a
geometrical extension of general relativity, where the role of the metric is
played by the real-valued fundamental function $F(x,y)$, defined on the
tangent bundle $TM$ over a smooth spacetime manifold $M$. Variable $y$ is an
element of the tangent space of $M$ at a point $x$ (we have suppressed indices
for convenience). The distance of two neighboring points on $M$ is defined as
$ds=F(x,dx)$. We consider the following properties to hold:

  1. 1.

$F$ is continuous on $TM$ and smooth on $\widetilde{TM}\equiv
TM\setminus\\{0\\}$ i.e. the tangent bundle minus the null section.

  2. 2.

$F$ is positively homogeneous of first degree on its second argument:

$F(x,ky)=kF(x,y),\qquad k>0.$ (6)

  3. 3.

The form

$f_{\mu\nu}(x,y)=\dfrac{1}{2}\dfrac{\partial^{2}F^{2}}{\partial
y^{\mu}\partial y^{\nu}}$ (7)

defines a non-degenerate matrix on $TM$ minus the null set $\\{(x,y)\in
TM|F(x,y)=0\\}$:

$\det\left[f_{\mu\nu}\right]\neq 0.$ (8)

Using homogeneity condition (6) it can be shown that:

$F^{2}(x,y)=|f_{\mu\nu}(x,y)y^{\mu}y^{\nu}|,$ (9)

and therefore $f_{\mu\nu}(x,y)$ can play the role of the metric for the vector
space spanned by $y$. When studying gravity, the metric $f_{\mu\nu}(x,y)$ is
considered to be of Lorentzian signature $(-,+,+,+)$.

###  II.3 General very special relativity on cosmology

A particularly interesting Finslerian cosmological model is elaborated in the
framework of the so-called general very special relativity on cosmology kour-
stath-st 2010 . The metric function takes the form

$F(x,y)=\big{(}g_{\mu\nu}(x)y^{\mu}y^{\nu}\big{)}^{(1-b)/2}\big{(}n_{\kappa}y^{\kappa}\big{)}^{b},$
(10)

where $g_{\mu\nu}(x)$ is the ordinary FRW metric (1). Expression (10) is a
direct cosmological generalization of the general very special relativity
description, where the line-element is

$ds=\big{(}\eta_{\mu\nu}dx^{\mu}dx^{\nu}\big{)}^{(1-b)/2}\big{(}n_{\kappa}dx^{\kappa}\big{)}^{b},$
(11)

with $[\eta_{\mu\nu}]=\mathrm{diag}\big{(}-1,1,1,1\big{)}$, which is invariant
under transformations generated by the deformation $DISIM_{b}(2)$ of the
Lorentz subgroup $ISIM(2)$ Cohen:2006ky ; Gibbons:2007iu . The one-form
$n_{\kappa}$ is called “spurionic field”. We mention that the parameter $b$
quantifies the deviation from Riemannian geometry, i.e. the Lorentz violation
in the gravitational sector. Parametrized post-Newtonian (PPN) analysis
Will:2005va  and use of solar system data provides the most stringent
constraints on it, and thus Gravity Probe B put an upper bound at $10^{-7}$
Bailey:2013oda .

The Riemannian osculating approach is followed, namely
$g_{\mu\nu}(x)=f_{\mu\nu}\big{(}x,y(x)\big{)}$, where $y(x)$ is the tangent
vector to the cosmological fluid’s (matter fluid) flow lines. As usual the
matter fluid is described by the energy-momentum tensor of the perfect fluid:

$T_{\mu\nu}=P_{m}g_{\mu\nu}+(\rho_{m}+P_{m})y_{\mu}y_{\nu},$ (12)

where $\rho_{m}$ is the energy density and $P_{m}$ the pressure. The field
equations for this construction are then:

$L_{\mu\nu}-\frac{1}{2}Lg_{\mu\nu}=-8\pi GT_{\mu\nu},$ (13)

where $L_{\mu\nu}$ is the Ricci tensor for the metric $g_{\mu\nu}(x)$ and
$L=g^{\mu\nu}L_{\mu\nu}$.

Applying the above geometrical construction in a cosmological framework we
consider the spurionic field to be parallel to the velocity of the comoving
observer, namely

$n^{\kappa}=\big{(}n(t),0,0,0\big{)}.$ (14)

As a simple model, in kour-stath-st 2010  the following approximations we
imposed

$\displaystyle n(t)\approx At+B$ $\displaystyle A\rightarrow 0$ $\displaystyle
B\rightarrow 0,$ (15)

since $n(t)$, parametrized by $A$,$B$, needs to be suitably small in order to
be consistent with the observational small bound on $b$. For these choices,
the Ricci tensor components for the metric function (10) are calculated as
kour-stath-st 2010

$\displaystyle\\!\\!\\!\\!\\!\\!\\!L_{00}=3\,\dfrac{\ddot{a}}{a}+3\dfrac{Ab}{B}\dfrac{\dot{a}}{a}+O\left({A}^{2}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!L_{11}=-\dfrac{a\dot{a}+2\dot{a}^{2}+2k}{1-kr^{2}}+\dfrac{5A}{B}b\dfrac{a\dot{a}}{1-kr^{2}}+O(A^{2})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!L_{22}=-r^{2}(a\ddot{a}+2\dot{a}^{2}+2k)-\dfrac{5A}{B}br^{2}a\ddot{a}+O(A^{2})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!L_{33}=-r^{2}(a\ddot{a}+2\dot{a}^{2}+2k)\sin^{2}\theta-\dfrac{5A}{B}br^{2}a\ddot{a}\sin^{2}\theta$
$\displaystyle\ \ \ \ \ \ +O(A^{2}).$ (16)

Therefore, using the above, we result to the following generalization of the
Friedmann equations:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!H^{2}\\!+\\!\frac{k}{a^{2}}\\!+\\!2\frac{A}{B}bH=\frac{8\pi
G}{3}\\!\left[\rho_{m}\\!-\\!2\frac{A}{B}bP_{m}\\!\left(t\\!+\\!\frac{B}{A}\ln
B\right)\right]$ (17)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\dot{H}+H^{2}+\frac{Ab}{B}H=-\frac{4\pi
G}{3}\big{[}(\rho_{m}+3P_{m})$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \,\ \ \ \ \ \ \ +4\ln(At+B)b(\rho_{m}+P_{m})\big{]}.$ (18)

Unfortunately, as one can see, the above Friedmann equations do not accept a
bounce solution. One could still try to construct a model with a different
approximation than (15) of kour-stath-st 2010 , however such a detailed
investigation of a new construction lies beyond the scope of the present work.
Hence, in the following subsection we examine the case of another Finslerian
construction, where bounce realization is possible.

###  II.4 Bounce in Finsler-Randers Space

Let us now consider a different Finslerian construction, namely the Finsler-
Randers (FR) space Stavrinos:2015 ; Stavrinos:2006rf . In this space a
Lagrangian metric function is given by

$F(x,y)=\upalpha(x,y)+u_{\mu}y^{\mu},\,\|u_{\mu}\|\ll 1,$ (19)

where $\upalpha(x,y)=\sqrt{g_{\kappa\lambda}(x)y^{\kappa}y^{\lambda}}$ and
$g_{\kappa\lambda}(x)$ is the FRW metric (1), with
$\kappa,\lambda,\mu\in\\{0,1,2,3\\}$.

In this cosmological model an important role is played by the variation of
anisotropy $Z_{t}$. In the case of the FRW geometry (1) the modified Friedmann
equations of the generalized form of the FR-type cosmology have been studied
in Stavrinos:2006rf , and are written as

$\displaystyle H^{2}=\frac{8\pi G}{3}\rho_{m}-HZ_{t}-\frac{k}{a^{2}},$ (20)
$\displaystyle\dot{H}=-4\pi
G\left(\rho_{m}+P_{m}\right)+\frac{1}{4}HZ_{t}+\frac{k}{a^{2}}.$ (21)

In these expressions, we have defined the variation of anisotropy $Z_{t}$ as
$Z_{t}=\dot{u}_{0}$, namely as the derivative of the time component of the
unit vector $\hat{u}_{a}$ Stavrinos:2006rf . This variation affects the form
of geometry as can be seen from relations (20),(21), and at the limit
$Z_{t}\rightarrow 0$ we recover the ordinary Friedmann equations of general
relativity. Finally, we have considered the matter sector to correspond to a
perfect fluid with energy density and pressure $\rho_{m}$ and $P_{m}$
respectively.

Observing the form of two Friedmann equation (20),(21) we can define the
effective energy density and pressure of geometrical origin as

$\displaystyle\rho_{FR}\equiv-\frac{3}{8\pi G}HZ_{t}$ (22) $\displaystyle
P_{FR}\equiv\frac{5}{16\pi G}HZ_{t}.$ (23)

Therefore, the total energy density and pressure respectively become
$\rho_{tot}=\rho_{m}+\rho_{FR}$ and $P_{tot}=P_{m}+P_{FR}$, and the Friedmann
equations take the usual form of (2),(3). Hence, we can now easily examine
what are the conditions in order to fulfill the bounce requirements (4) and
(5).

Firstly, from (4) we deduce that for the flat universe exactly at the bounce
point $\rho_{m}$ must be zero ($\rho_{FR}$ becomes also zero exactly at the
bounce point, since $H=0$). This is a usual assumption in many bouncing models
and it is expected to be fulfilled in the early universe. Taking this into
account, we moreover see that condition (5) implies that around the bouncing
point $\rho_{FR}+P_{FR}<0$ and thus that $HZ_{t}>0$. Hence, we deduce that the
above requirements can be fulfilled if we suitably choose the variation of
anisotropy $Z_{t}$.

In order to provide a specific example we focus on a flat FRW geometry ($k=0$)
and we consider a bouncing scale factor of the form

$a(t)=a_{b}(1+Bt^{2})^{1/3},$ (24)

where $a_{b}$ is the scale factor value at the bounce, while $B$ is a positive
parameter which determines how fast the bounce takes place. In this case time
varies between $-\infty$ and $+\infty$, with $t=0$ the bouncing point, and
where away from the bounce one obtains the usual expansion behavior. Moreover,
we consider that the matter sector is absent in the early universe. Inserting
these into (20) we immediately find that

$Z_{t}=-\frac{2Bt}{3(1+Bt^{2})}.$ (25)

Hence, it is this $Z_{t}$, that comes from the Finslerian modification of the
geometry, that generates the bouncing scale factor (24). Moreover, we remark
that the variation of anisotropy $Z_{t}$ actually determines the physically
important quantity $B$ in (24).

##  III Finsler-like gravity on a tangent bundle

In this Section we are interested in examining whether a bounce can be
realized from Finsler-like gravity on a tangent bundle. Generally, we will use
the term Finsler-like for any metric theory in which the various structures
may depend on a set of _internal_ variables ($y,\phi,$ etc) apart from the
position or _external_ ones which we denote as $x^{\mu}$ through this work.
Finsler-like extensions of general relativity on the tangent bundle have been
presented in the bibliography bucataru-miron 2007 ; Vacaru:2005ht ; miron-
anastasiei 2012 ; pfeifer-wohlfarth 2012 ; Caianiello:1989wm  and bouncing
cosmological scenarios have been studied on them Triantafyllopoulos:2018bli ;
Caianiello:1991 ; Gasperini:1991pp . In the following we focus our interest on
a tangent bundle $TM$ equipped with a Finslerian Sasaki-type metric:

$\mathcal{G}=g_{\mu\nu}(x,y)\,\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}+v_{\alpha\beta}(x,y)\,\delta
y^{\alpha}\otimes\delta y^{\beta},$ (26)

where $x^{\mu}$ are the coordinates on the base manifold, with
$\kappa,\lambda,\mu,\nu,\ldots=0,1,2,3$, and $y^{\alpha}$ are the fiber
coordinates, with $\alpha,\beta,\ldots,\theta=0,1,2,3$. On the total space
$TTM$ of $TM$, the adapted basis is
$\\{\delta_{\mu},\dot{\partial}_{\alpha}\\}$ and its dual is given by
$\\{\mathrm{d}x^{\mu},\delta y^{\alpha}\\}$, and the following definitions
hold:

$\displaystyle\delta_{\mu}=\dfrac{\delta}{\delta
x^{\mu}}=\dfrac{\partial}{\partial
x^{\mu}}-N^{\alpha}_{\mu}(x,y)\dfrac{\partial}{\partial y^{\alpha}}$
$\displaystyle\dot{\partial}_{\alpha}=\dfrac{\partial}{\partial y^{\alpha}}$
$\displaystyle\delta
y^{\alpha}=\mathrm{d}y^{\alpha}+N^{\alpha}_{\nu}\mathrm{d}x^{\nu},$ (27)

where $N^{\alpha}_{\mu}(x,y)$ are the coefficients of a nonlinear connection
on $TM$. This connection is defined by a splitting of the total space $TTM$ of
$TM$ into an h-subspace $HTM$ spanned by $\\{\delta_{\mu}\\}$ and a v-subspace
$VTM$ spanned by $\\{\dot{\partial}_{\alpha}\\}$ Vacaru:2005ht . The tangent
space of $TM$ is thus a Whitney sum of the h-subspace and v-subspace, namely

$TTM=HTM\oplus VTM.$ (28)

One can now introduce the $d-$connection $\mathcal{D}$ as a covariant linear
differentiation rule that preserves h-space and v-space:

$\displaystyle\mathcal{D}_{\delta_{\kappa}}\delta_{\nu}=L^{\mu}_{\nu\kappa}(x,y)\delta_{\mu}\qquad\mathcal{D}_{\dot{\partial}_{\gamma}}\delta_{\nu}=C^{\mu}_{\nu\gamma}(x,y)\delta_{\mu}$
(29)
$\displaystyle\mathcal{D}_{\delta_{\kappa}}\dot{\partial}_{\beta}=L^{\alpha}_{\beta\kappa}(x,y)\dot{\partial}_{\alpha}\qquad\mathcal{D}_{\dot{\partial}_{\gamma}}\dot{\partial}_{\beta}=C^{\alpha}_{\beta\gamma}(x,y)\dot{\partial}_{\alpha}.$
(30)

A canonical $d-$connection is a linear connection that is compatible with the
metric (26) and it preserves under parallel translation the horizontal and
vertical subspaces $HTM$ and $VTM$ Vacaru:2005ht . It can be uniquely defined
if one demands that it only depends on $g_{\mu\nu},v_{\alpha\beta}$ and
$N^{\alpha}_{\mu}$, and moreover that the connection coefficients
$L^{\mu}_{\nu\kappa}$ and $C^{\alpha}_{\beta\gamma}$ are symmetric on the
lower indices. In this case, its coefficients turn out to be miron-watanabe-
ikeda 1987 :

$\displaystyle\\!\\!\\!\\!L^{\mu}_{\nu\kappa}=\frac{1}{2}g^{\mu\rho}\left(\delta_{k}g_{\rho\nu}+\delta_{\nu}g_{\rho\kappa}-\delta_{\rho}g_{\nu\kappa}\right)$
$\displaystyle\\!\\!\\!\\!L^{\alpha}_{\beta\kappa}=\dot{\partial}_{\beta}N^{\alpha}_{\kappa}+\frac{1}{2}v^{\alpha\gamma}\left(\delta_{\kappa}v_{\beta\gamma}-v_{\delta\gamma}\,\dot{\partial}_{\beta}N^{\delta}_{\kappa}-v_{\beta\delta}\,\dot{\partial}_{\gamma}N^{\delta}_{\kappa}\right)$
$\displaystyle\\!\\!\\!\\!C^{\mu}_{\nu\gamma}=\frac{1}{2}g^{\mu\rho}\dot{\partial}_{\gamma}g_{\rho\nu}$
$\displaystyle\\!\\!\\!\\!C^{\alpha}_{\beta\gamma}=\frac{1}{2}v^{\alpha\delta}\left(\dot{\partial}_{\gamma}h_{\delta\beta}+\dot{\partial}_{\beta}h_{\delta\gamma}-\dot{\partial}_{\delta}v_{\beta\gamma}\right).$
(31)

Now, the curvature of the nonlinear connection is defined as

$\Omega^{\alpha}_{\nu\kappa}=\dfrac{\delta N^{\alpha}_{\nu}}{\delta
x^{\kappa}}-\dfrac{\delta N^{\alpha}_{\kappa}}{\delta x^{\nu}},$ (32)

and the space at hand is equipped with various Ricci curvature tensors such
as:

$\displaystyle\overline{R}_{\mu\nu}$
$\displaystyle=\delta_{\kappa}L^{\kappa}_{\mu\nu}-\delta_{\nu}L^{\kappa}_{\mu\kappa}+L^{\rho}_{\mu\nu}L^{\kappa}_{\rho\kappa}-L^{\rho}_{\mu\kappa}L^{\kappa}_{\rho\nu}$
(33) $\displaystyle S_{\alpha\beta}$
$\displaystyle=\dot{\partial}_{\gamma}C^{\gamma}_{\alpha\beta}-\dot{\partial}_{\beta}C^{\gamma}_{\alpha\gamma}+C^{\epsilon}_{\alpha\beta}C^{\gamma}_{\epsilon\gamma}-C^{\epsilon}_{\alpha\gamma}C^{\gamma}_{\epsilon\beta}.$
(34)

Hence, the generalized Ricci scalar curvature reads as

$\mathcal{R}=g^{\mu\nu}\overline{R}_{\mu\nu}+v^{\alpha\beta}S_{\alpha\beta}\equiv\overline{R}+S.$
(35)

One can now write a Hilbert-like action, namely bucataru-miron 2007 ;
Vacaru:2005ht ; miron-anastasiei 2012 ; pfeifer-wohlfarth 2012

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathcal{S}_{TM}=\dfrac{1}{16\pi
G}\mathcal{S}_{H}+\mathcal{S}_{M}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\equiv\dfrac{1}{16\pi
G}\\!\int\\!d^{8}\mathcal{U}\sqrt{\det\mathcal{G}}\,\mathcal{L}_{H}+\\!\int\\!d^{8}\mathcal{U}\sqrt{\det\mathcal{G}}\,\mathcal{L}_{M},$
(36)

with

$d^{8}\mathcal{U}=\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}\wedge\mathrm{d}y^{0}\wedge\mathrm{d}y^{1}\wedge\mathrm{d}y^{2}\wedge\mathrm{d}y^{3},$
(37)

where the gravitational part of the action $\mathcal{S}_{H}$ is constructed by
the gravitational Lagrangian

$\mathcal{L}_{H}=\mathcal{R}=(\overline{R}+S),$ (38)

and the matter action $\mathcal{S}_{M}$ by the matter Lagrangian
$\mathcal{L}_{M}$.

Extremization of the total action $\mathcal{S}_{TM}$ with respect to the
metric components $g_{\mu\nu}$ and $v_{\alpha\beta}$ leads to the following
field equations Triantafyllopoulos:2018bli :

$\displaystyle\overline{R}_{(\mu\nu)}-\frac{1}{2}(\overline{R}+S)g_{\mu\nu}=8\pi
GT_{\mu\nu}$ (39) $\displaystyle
S_{\alpha\beta}-\frac{1}{2}(\overline{R}+S)v_{\alpha\beta}=8\pi
GY_{\alpha\beta},$ (40)

where we have defined
$T_{\mu\nu}=-\frac{2}{\sqrt{\det\mathcal{G}}}\frac{\delta{\left(\mathcal{L}_{M}\sqrt{\det\mathcal{G}}\right)}}{\delta
g^{\mu\nu}}$ and
$Y_{\alpha\beta}=-\frac{2}{\sqrt{\det\mathcal{G}}}\frac{\delta{\left(\mathcal{L}_{M}\sqrt{\det\mathcal{G}}\right)}}{\delta
v^{\alpha\beta}}$. Applying these field equations in the FRW metric (1),
focusing on the flat case, and assuming the usual matter perfect fluid (12),
one obtains the following modified Friedmann equations
Triantafyllopoulos:2018bli :

$\displaystyle H^{2}=\frac{8\pi G}{3}\rho_{m}-\frac{1}{6}S$ (41)
$\displaystyle\dot{H}+H^{2}=-\frac{4\pi
G}{3}\left(\rho_{m}+3P_{m}\right)-\frac{1}{6}S,$ (42)

where due to the imposed symmetries all quantities depend only on time.

From the form of the two Friedmann equations (41),(42) we can see that we
obtain extra contributions that reflect the Finsler-like structure of the
tangent bundle. In particular, these induce an effective energy density and
pressure of geometrical origin as

$\displaystyle\rho_{S}\equiv-\frac{1}{16\pi G}S$ (43) $\displaystyle
P_{S}\equiv\frac{1}{16\pi G}S.$ (44)

Hence, the total energy density and pressure respectively become
$\rho_{tot}=\rho_{m}+\rho_{S}$ and $P_{tot}=P_{m}+P_{S}$, and the Friedmann
equations acquire the usual form of (2),(3). Thus, we can examine what are the
conditions in order to fulfill the bounce requirements (4) and (5). Concerning
(4) we deduce that for the flat universe exactly at the bounce point we must
have $S=16\pi G\rho_{m}$, while (5) requires $\rho_{m}+P_{m}<0$ (since
according to (43),(44) $P_{S}+\rho_{S}=0$). Therefore, we conclude that in the
case of a flat universe and for standard matter a bounce cannot be obtained in
the scenario at hand.

Nevertheless, a bounce could still be possible with the addition of extra
fields, e.g. Triantafyllopoulos:2018bli , but still one has to be careful with
the constraints imposed to $S$ via (40). For example, if we consider the
trivial case where $Y_{\alpha\beta}=0$ then the trace of (40) gives

$S=-2\overline{R}.$ (45)

We assume that the extra field can be modeled to a perfect fluid as in (12),
with energy density and pressure $\rho_{eff}$ and $P_{eff}$ respectively, and
thus the Friedmann equation (41) takes the form

$H^{2}=\frac{8\pi G}{3}(\rho_{m}+\rho_{eff})-\frac{1}{6}S.$ (46)

Substituting (45) to (46)111In our case $\overline{R}$ reduces to the ordinary
flat FRW Ricci scalar curvature of general relativity, due to the fact that
the metric components $g_{\mu\nu}(x)$ do not depend on the $y$ variables, as
was shown in Triantafyllopoulos:2018bli . gives $3H^{2}+2\dot{H}+8\pi
G(\rho_{m}+\rho_{eff})/3=0$. This relation implies that in order for an extra
field with trivial $Y_{\alpha\beta}$ to induce a bounce solution for our
spatially flat metric it would need to have $\rho_{eff}<0$, which is
undesirable from a physical point of view.

##  IV Bounce from scalar-tensor theory on the fiber bundle

In this section we investigate the bounce generation in theories which include
scalar-tensor sectors on the fiber bundle. These constructions are very
general, with very rich structure and behavior, which reveals the significant
capabilities of Finsler-like geometry. We first present the basics of this
construction and then we proceed to the investigation of two explicit
scenarios.

###  IV.1 The model

We consider a fibered space over a pseudo-Riemannian spacetime manifold $M$ of
the form $M\times\\{\phi^{(1)}\\}\times\\{\phi^{(2)}\\}$, where
$\phi^{(1)},\phi^{(2)}$ stand for the fiber coordinates. Under coordinate
transformations on the base manifold, fiber coordinates behave like scalars.
Moreover, the space is equipped with a nonlinear connection with coefficients
$N^{(\alpha)}_{\mu}(x^{\nu},\phi^{(\beta)})$, where $\mu,\nu$ take the values
from $0$ to $3$ and $\alpha,\beta$ take the values $1$ and $2$ stavrinos-ikeda
1999 . Its adapted bases for the tangent and cotangent spaces are
$\\{\delta_{\mu}=\partial_{\mu}-N^{(\beta)}_{\mu}\partial_{\phi^{(\beta)}},\partial_{\phi^{(\alpha)}}\\}$,
where a summation is implied over the possible values of $\beta$, and
$\\{\mathrm{d}x^{\mu},\delta\phi^{(\alpha)}=\mathrm{d}\phi^{(\alpha)}+N^{(\alpha)}_{\mu}\mathrm{d}x^{\mu}\\}$
with a summation implied over the possible values of $\mu$. The metric
structure of the space is defined as stavrinos-ikeda 1999

$\mathbf{G}=g_{\mu\nu}(x)\,\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}+v_{(\alpha)(\beta)}(x)\,\delta\phi^{(\alpha)}\otimes\delta\phi^{(\beta)}.$
(47)

The metric coefficients for the fiber coordinates are set as
$v_{(0)(0)}=v_{(1)(1)}=\phi(x^{\mu})$ and $v_{(0)(1)}=v_{(1)(0)}=0$. Note that
the function $\phi$ is clearly a scalar under coordinate transformations. The
detailed investigation of the above construction has been performed in
stavrinos-ikeda 1999 , where a metrical d-connection has been introduced and
its curvature and torsion tensor coefficients have been calculated.
Additionally, the Raychaudhuri equations for the model have been derived in
Stavrinos:2016xyg .

We can now write an action as Stavrinos:2016xyg

$\mathcal{S}_{G}=\frac{1}{16\pi
G}\int\sqrt{|\det\mathbf{G}|}\,\mathcal{L}_{G}dx^{(N)},$ (48)

where $\mathcal{L}_{G}$ is taken equal to the scalar curvature of the
d-connection, and
$dx^{(N)}=d^{4}x\wedge\mathrm{d}\phi^{(1)}\wedge\mathrm{d}\phi^{(2)}$. In the
special case of a holonomic basis, i.e. $[\delta_{\mu},\delta_{\nu}]=0$, the
scalar curvature of the d-connection is

$\mathcal{R}=R-\frac{2}{\phi}\square\phi+\frac{1}{4\phi^{2}}\partial^{\mu}\phi\partial_{\mu}\phi,$
(49)

where $R$ is the scalar curvature of Levi-Civita connection and $\square$ is
the d’Alembert operator with respect to it. On the other hand, in the general
case one obtains the scalar curvature as

$\mathcal{\tilde{R}}=R-\frac{2}{\phi}\square\phi+\frac{1}{4\phi^{2}}\partial^{\mu}\phi\partial_{\mu}\phi+\frac{1}{\phi}\partial^{\mu}\phi\,\partial_{\phi^{(\alpha)}}N^{(\alpha)}_{\mu}.$
(50)

Additionally, we can add the matter sector too, considering the total action

$\mathcal{S}=\frac{1}{16\pi
G}\int\sqrt{|\det\mathbf{G}|}\,\mathcal{L}_{G}dx^{(N)}+\int\sqrt{|\det\mathbf{G}|}\,\mathcal{L}_{M}dx^{(N)}.$
(51)

Since for the determinants $\det\mathbf{G}$ and $\det g$ we have the relation
$\det\mathbf{G}=\phi^{2}\det g$, the above total action can be re-written as

$\mathcal{S}=\frac{1}{16\pi G}\int\sqrt{|\det
g|}\,\phi\mathcal{L}_{G}dx^{(N)}+\int\sqrt{|\det
g|}\,\phi\mathcal{L}_{M}dx^{(N)}.$ (52)

In the following two subsections we study the bounce realization in the
holonomic ($\mathcal{L}_{G}=\mathcal{R}$) and nonholonomic
($\mathcal{L}_{G}=\mathcal{\tilde{R}}$) basis separately.

###  IV.2 Bounce in holonomic basis

Let us consider the total action (52) in the case of holonomic basis, allowing
also for a potential for the scalar field, namely Stavrinos:2016xyg

$\displaystyle\mathcal{S}=$ $\displaystyle\\!\\!\\!\\!\\!\\!\frac{1}{16\pi
G}\int\\!\\!\sqrt{|\det g|}\,\big{[}\phi\mathcal{R}-V(\phi)\big{]}dx^{(N)}$
(53) $\displaystyle\\!\\!\\!\\!\\!\\!\\!+\int\\!\\!\sqrt{|\det
g|}\,\phi\mathcal{L}_{M}dx^{(N)},$

where $\mathcal{R}$ is the holonomic scalar curvature (49). We mention here
that the above action belongs to the Horndeski class, and hence the resulting
equations of motion are guaranteed to have up to second order derivatives
DeFelice:2011bh . In particular, the field equations for the metric are
extracted as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!E_{\mu\nu}=8\pi
GT_{\mu\nu}+\frac{1}{\phi}\big{(}\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\square\phi\big{)}$ $\displaystyle\
+\frac{1}{4\phi^{2}}\left[\frac{1}{2}g_{\mu\nu}(\nabla\phi)^{2}-\nabla_{\mu}\phi\nabla_{\nu}\phi\right]-\frac{1}{2\phi}g_{\mu\nu}V,$
(54)

where $E_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor,
$T_{\mu\nu}=-\frac{2}{\sqrt{|g|}}\frac{\delta(\sqrt{|g|}\mathcal{L}_{M})}{\delta
g^{\mu\nu}}$ is the energy-momentum tensor, and $\nabla_{\mu}$ is the Levi-
Civita covariant derivative, while the scalar field (extension of Klein-Gordon
) equation reads as

$\square\phi=2\phi\big{(}R-V^{\prime}\big{)}+\frac{1}{2\phi}(\nabla\phi)^{2}+32\pi
G\mathcal{L}_{M}\phi,$ (55)

with $V^{\prime}=dV/d\phi$. Note the interesting fact that in the scenario at
hand we obtain an effective interaction between the scalar field and the
matter sector due to the transformation from $\mathbf{G}$-metric to
$g$-metric.

Applying the above equations in the FRW metric (1), focusing on the flat case,
and neglecting the matter sector, since we are interesting in the early-time
bounce realization, we obtain the following modified Friedmann equations:

$\displaystyle
3H^{2}=-3H\frac{\dot{\phi}}{\phi}-\frac{\dot{\phi}^{2}}{8\phi^{2}}+\frac{1}{2\phi}V$
(56)
$\displaystyle\dot{H}+H^{2}=-\frac{1}{2\phi}\big{(}\ddot{\phi}+H\dot{\phi}\big{)}+\frac{\dot{\phi}^{2}}{12\phi^{2}}+\frac{V}{6\phi}$
(57)
$\displaystyle\ddot{\phi}+3H\dot{\phi}=-12\phi\left(2H^{2}+\dot{H}\right)+\frac{\dot{\phi}^{2}}{2\phi}+2\phi
V^{\prime},$ (58)

out of which two are independent.

We now proceed to show how it is possible to obtain a specific bounce in this
construction. As we observe from the above equations, we may choose a specific
scalar-field potential that can satisfy the general bounce conditions (4) and
(5) and thus induce the bounce realization. We follow the procedure of
Cai:2011tc ; Cai:2012ag ; Qiu:2013eoa ; Cai:2010zma ; Cai:2011bs  and we first
start from the desired result, that is we impose a known form of the scale
factor $a(t)$ possessing a bouncing behavior. Thus, $H(t)$ is known too.
Eliminating $V$ from (56),(57) gives the simple differential equation

$4\phi(t)\ddot{\phi}(t)-\dot{\phi(t)}[\dot{\phi}(t)+4H(t)\phi(t)]+8\dot{H}(t)\phi(t)^{2}=0,$
(59)

which can be solved to provide $\phi(t)$. Then this $\phi(t)$ can be inserted
into (56) and provide $V(t)$ as

$\displaystyle
V(t)=6H(t)[\dot{\phi}(t)+\phi(t)H(t)]+\frac{\dot{\phi}(t)^{2}}{4\phi(t)}.$
(60)

Figure 2:  The solution for the scalar field $\phi(t)$ (upper graph) and of
the potential $V(t)$ (lower graph), for the holonomic basis, under the imposed
bouncing scale factor (24) with $B=1$, in units where $8\pi G=1$.

Finally, knowing both $\phi(t)$ and $V(t)$, eliminating time we can extract
the explicit form of the potential $V(\phi)$. Hence, it is this potential that
generates the initially given desired bouncing scale factor $a(t)$.

Let us provide an explicit example of the bounce realization. We start by
inserting the desired bouncing scale factor (24) and we apply the above steps.
Since analytical solutions cannot be obtained, we numerically solve (59) and
find $\phi(t)$, and then we use (60) to find $V(t)$. These two functions are
shown in Fig. 2. Hence, from these $\phi(t)$ and $V(t)$ we reconstruct the
potential $V(\phi)$, which is depicted in Fig. 3.

Figure 3:  The reconstructed scalar potential $V(\phi)$ using Fig. 2, under
the imposed bouncing scale factor (24) with $B=1$, in units where $8\pi G=1$.

Therefore, if this $V(\phi)$ is imposed as an input, one acquires the bounce
realization, and in particular the bouncing scale factor (24).

###  IV.3 Bounce in nonholonomic basis

We now proceed to the investigation of the nonholonomic case, namely we
consider the total action (52) with $\mathcal{L}_{G}=\mathcal{\tilde{R}}$,
i.e.

$\mathcal{S}=\frac{1}{16\pi
G}\int\sqrt{|g|}\,\phi\mathcal{\tilde{R}}dx^{(N)}+\int\sqrt{|g|}\,\phi\mathcal{L}_{M}dx^{(N)},$
(61)

where $\mathcal{\tilde{R}}$ is the nonholonomic scalar curvature (50). This
action leads to the following equations of motion for the metric and the
scalar field:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!E_{\mu\nu}=8\pi
GT_{\mu\nu}+\frac{1}{\phi}\big{(}\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\square\phi\big{)}$
$\displaystyle+\frac{1}{4\phi^{2}}\left[\frac{1}{2}g_{\mu\nu}(\nabla\phi)^{2}-\nabla_{\mu}\phi\nabla_{\nu}\phi\right]$
$\displaystyle-\left(\delta^{\lambda}_{\mu}\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\partial^{\lambda}\phi\right)N_{\lambda}$
(62)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\square\phi=2\phi
R+\frac{1}{2\phi}(\nabla\phi)^{2}+32\pi G\mathcal{L}_{M}\phi-\phi
D^{\mu}N_{\mu},$ (63)

where $N_{\mu}\equiv\partial_{\phi^{(\alpha)}}N^{(\alpha)}_{\mu}$, and with
$D_{\mu}N^{\lambda}=\delta_{\mu}N^{\lambda}+\Gamma^{\lambda}_{\kappa\mu}N^{\kappa}$
the d-covariant differentiation on the fiber bundle where
$\Gamma^{\lambda}_{\kappa\mu}$ are the Christoffel symbols. We mention that
the last term in (63), which reflects the internal structure of Finsler-like
geometry, can be seen to act as an effective potential for the scalar field
$\phi$. Since every other quantity in (62),(63) depends on $x^{\mu}$
coordinates only, this should also be the case for $N_{\lambda}$ for
consistency (equivalently
$\partial_{\phi^{(\beta)}}\partial_{\phi^{(\alpha)}}N^{(\alpha)}_{\mu}=0$ on
shell).

Applying the above equations of motion in the FRW metric (1), focusing on the
flat case, and neglecting the matter sector, since we are interesting in the
early-time bounce realization, leads to the modified Friedmann equations

$\displaystyle
3H^{2}=-3H\frac{\dot{\phi}}{\phi}-\frac{\dot{\phi}^{2}}{8\phi^{2}}-\frac{1}{2}\dot{\phi}N_{0}$
(64)
$\displaystyle\dot{H}+H^{2}=-\frac{1}{2\phi}\big{(}\ddot{\phi}+H\dot{\phi}\big{)}+\frac{\dot{\phi}^{2}}{12\phi^{2}}+\frac{1}{3}\dot{\phi}N_{0}$
(65)
$\ddot{\phi}+3H\dot{\phi}=-12\phi\left(2H^{2}+\dot{H}\right)+\frac{\dot{\phi}^{2}}{2\phi}+\phi\left(\dot{N}^{0}+3HN^{0}\right),$
(66)

out of which two are independent, where as we mentioned, due to symmetries,
all quantities depend only on time. Thus, in the Friedmann equations we
acquire a modification reflecting the nonholonomicity of the fiber bundle of
the underlying Finsler-like geometry.

Let us now show how this construction may give rise to the bounce realization.
From the form of the Friedmann equations (64),(65) we deduce that we may
choose a specific nonholonomic function $N^{0}(t)$ that can satisfy the
general bounce conditions (4) and (5) and thus induce the bounce. We first
start from the desired result, that is we impose as an input a scale factor
form $a(t)$ that possesses bouncing behavior. Therefore, $H(t)$ is known too.
Eliminating $N^{0}$ from (64),(65) gives the simple differential equation

$\displaystyle\ddot{\phi}(t)+5H(t)\dot{\phi(t)}+2\phi(t)[\dot{H}(t)+3H(t)^{2}]=0,$
(67)

which can be solved to provide $\phi(t)$. Then this $\phi(t)$ can be
substituted into (64) and provide $N^{0}(t)$ as

$\displaystyle
N_{0}(t)=-6\left[\frac{H(t)}{\phi(t)}+\frac{\dot{\phi(t)}}{24\phi(t)^{2}}+\frac{H(t)^{2}}{\dot{\phi}(t)}\right].$
(68)

Hence, it is this $N_{0}(t)$, induced by the nonlinear connection of the
Finsler-like geometry, that generates the initially given desired bouncing
scale factor $a(t)$.

We close this subsection by providing an explicit example of the bounce
realization. We use as input the bouncing scale factor (24) and we apply the
above steps. We numerically solve (67) and find $\phi(t)$, and then we use
(68) to find $N_{0}(t)$. In Fig. 4 we depict the solution for $N_{0}(t)$.
Hence, if this $N_{0}$ is imposed as an input, one obtains the bounce
realization, and in particular the bouncing scale factor (24).

Figure 4:  The reconstructed time-dependent part $N_{0}(t)$ related to the
nonlinear connection, for the nonholonomic basis, under the imposed bouncing
scale factor (24) with $B=1$, in units where $8\pi G=1$.

##  V Conclusions

In this work we investigated the bounce realization in the framework of
Finsler and Finsler-like gravity. Finsler and Finsler-like geometries are
natural extensions of Riemannian one, where one allows that the physical
quantities may directly depend on observer 4-velocity. Hence, the
gravitational theory based on Finsler and Finsler-like gravity provides a
gravitational modification, since it induces extra terms in the field
equations. When applied in a cosmological framework, the richer intrinsic
structure of Finsler and Finsler-like geometries is reflected in extra terms
in the resulting modified Friedmann equations. Thus, these terms can lead to
the bounce realizations.

In our analysis we considered various Finsler and Finsler-like constructions
and we examined whether bouncing solutions can be obtained. As a first model
we considered the so-called general very special relativity, which presents a
slight Lorentz violation quantified by a single parameter and the “spurionic”
one-form. As we showed, under the linear approximation this scenario cannot
lead to a bounce. However, considering the Finsler-Randers space, in which the
intrinsic Finslerian structure is reflected to the appearance of a new
function in the Friedmann equations (the variation of anisotropy), we saw that
the bounce conditions can be easily fulfilled and thus the bounce can be
realized.

As a next construction we examined the Finsler-like gravity on the tangent
bundle. Performing the analysis and considering the two involved curvature
tensors, we extracted the Friedmann equations which contain a modification
resulting from the tangent-bundle related $S$-curvature. Nevertheless, for
simple models and standard matter, these extra terms cannot drive a bouncing
solution since they cannot lead to the violation of the null energy condition.

As a last construction we considered theories which include scalar-tensor
sectors on the fiber bundle. These theories present a very rich structure
revealing the capabilities of Finsler-like geometry. In particular, the
nonlinear connection induces a new degree of freedom that behaves as a scalar
under coordinate transformations. In a cosmological framework this scalar
field appears in the Friedmann equations, and therefore its dynamics may
trigger a bounce. In the case of holonomic basis we showed that the bounce can
be easily obtained, and we provided the way of the reconstruction of the
potential that gives rise to a desired bouncing scale factor. Similarly, in
the case of nonholonomic basis we saw that the bounce can be easily realized,
and we presented the reconstruction procedure of the time-coefficient related
to the nonlinear connection that induces the desired bounce.

In summary, we saw that Finsler and Finsler-like geometries is a natural
framework for the realization of bounce cosmology. Apart from the background
evolution one should additionally investigate the various scenarios at the
perturbation levels, since the process of perturbations through the bounce
phase is strongly related to the subsequent development of the large scale
structure and hence to observations. Such a detailed perturbation analysis
lies beyond the scope of the present work and it is left for a future
investigation.

###### Acknowledgements.

This research is co-financed by Greece and the European Union (European Social
Fund- ESF) through the Operational Programme “Human Resources Development,
Education and Lifelong Learning” in the context of the project “Strengthening
Human Resources Research Potential via Doctorate Research” (MIS-5000432),
implemented by the State Scholarships Foundation (IKY). This article is based
upon work from COST (European Cooperation in Science and Technology) Action
CA15117 “Cosmology and Astrophysics Network for Theoretical Advances and
Training Actions” (CANTATA).

## References

  * (1) T. Biswas, A. Mazumdar and W. Siegel, _Bouncing universes in string-inspired gravity_ , JCAP 0603, 009 (2006).
  * (2) Y. F. Cai, T. Qiu, Y. S. Piao, M. Li and X. Zhang, _Bouncing universe with quintom matter_ , JHEP 0710, 071 (2007).
  * (3) Y. F. Cai, T. t. Qiu, R. Brandenberger and X. m. Zhang, _A Nonsingular Cosmology with a Scale-Invariant Spectrum of Cosmological Perturbations from Lee-Wick Theory_ , Phys. Rev. D 80, 023511 (2009).
  * (4) Y. F. Cai, D. A. Easson and R. Brandenberger, _Towards a Nonsingular Bouncing Cosmology_ , JCAP 1208, 020 (2012).
  * (5) Y. F. Cai, _Exploring Bouncing Cosmologies with Cosmological Surveys_ , Sci. China Phys. Mech. Astron. 57, 1414 (2014).
  * (6) R. Brandenberger and P. Peter, _Bouncing Cosmologies: Progress and Problems_ , Found. Phys. 47, no. 6, 797 (2017).
  * (7) M. Novello and S. E. P. Bergliaffa, Bouncing Cosmologies, Phys. Rept. 463, 127 (2008).
  * (8) R. C. Tolman, _On the Theoretical Requirements for a Periodic Behaviour of the Universe_ , Phys. Rev. 38, no. 9, 1758 (1931).
  * (9) A. Ijjas and P. J. Steinhardt, _Bouncing Cosmology made simple_ , Class. Quant. Grav. 35, no. 13, 135004 (2018).
  * (10) T. Singh, R. Chaubey and A. Singh, _Bounce conditions for FRW models in modified gravity theories_ , Eur. Phys. J. Plus 130, no. 2, 31 (2015).
  * (11) A. De Felice and S. Tsujikawa, _Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models_ , JCAP 1202, 007 (2012).
  * (12) A. Barrau, B. Bolliet, M. Schutten and F. Vidotto, _Bouncing black holes in quantum gravity and the Fermi gamma-ray excess_ , Phys. Lett. B 772, 58 (2017).
  * (13) M. Gasperini, M. Giovannini and G. Veneziano, Perturbations in a nonsingular bouncing universe, Phys. Lett. B 569, 113 (2003)
  * (14) M. Gasperini, M. Giovannini and G. Veneziano, Cosmological perturbations across a curvature bounce, Nucl. Phys. B 694, 206 (2004)
  * (15) J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, The ekpyrotic universe: Colliding branes and the origin of the hot big bang, Phys. Rev. D 64, 123522 (2001).
  * (16) J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok, From big crunch to big bang, Phys. Rev. D 65, 086007 (2002).
  * (17) T. Biswas, A. Mazumdar and W. Siegel, Bouncing universes in string-inspired gravity, JCAP 0603, 009 (2006).
  * (18) S. Nojiri and E. N. Saridakis, Phantom without ghost, Astrophys. Space Sci. 347, 221 (2013).
  * (19) K. Bamba, A. N. Makarenko, A. N. Myagky, S. Nojiri and S. D. Odintsov, Bounce cosmology from $F(R)$ gravity and $F(R)$ bigravity, JCAP 1401 (2014) 008.
  * (20) S. Nojiri and S. D. Odintsov, Mimetic $F(R)$ gravity: inflation, dark energy and bounce, Mod. Phys. Lett. A 29, no. 40, 1450211 (2014).
  * (21) Y. -F. Cai, S. -H. Chen, J. B. Dent, S. Dutta and E. N. Saridakis, Matter Bounce Cosmology with the f(T) Gravity, Class. Quant. Grav. 28, 215011 (2011).
  * (22) Y. Shtanov and V. Sahni, Bouncing braneworlds, Phys. Lett. B 557, 1 (2003).
  * (23) E. N. Saridakis, Cyclic Universes from General Collisionless Braneworld Models, Nucl. Phys. B 808, 224 (2009).
  * (24) R. Brandenberger, Matter Bounce in Horava-Lifshitz Cosmology, Phys. Rev. D 80, 043516 (2009).
  * (25) Y. F. Cai and E. N. Saridakis, Non-singular cosmology in a model of non-relativistic gravity, JCAP 0910, 020 (2009).
  * (26) E. N. Saridakis, Horava-Lifshitz Dark Energy, Eur. Phys. J. C 67, 229 (2010).
  * (27) D. A. Easson, I. Sawicki and A. Vikman, G-Bounce, JCAP 1111, 021 (2011).
  * (28) T. Qiu, X. Gao and E. N. Saridakis, Towards anisotropy-free and nonsingular bounce cosmology with scale-invariant perturbations, Phys. Rev. D 88, no. 4, 043525 (2013).
  * (29) Y. F. Cai, C. Gao and E. N. Saridakis, Bounce and cyclic cosmology in extended nonlinear massive gravity, JCAP 1210, 048 (2012).
  * (30) Y. -F. Cai and E. N. Saridakis, Cyclic cosmology from Lagrange-multiplier modified gravity, Class. Quant. Grav. 28, 035010 (2011).
  * (31) M. Bojowald, Absence of singularity in loop quantum cosmology, Phys. Rev. Lett. 86, 5227 (2001).
  * (32) S. D. Odintsov and V. K. Oikonomou, Matter Bounce Loop Quantum Cosmology from $F(R)$ Gravity, Phys. Rev. D 90, no. 12, 124083 (2014).
  * (33) S. D. Odintsov, V. K. Oikonomou and E. N. Saridakis, Superbounce and Loop Quantum Ekpyrotic Cosmologies from Modified Gravity: $F(R)$, $F(G)$ and $F(T)$ Theories, Annals Phys. 363, 141 (2015).
  * (34) F. A. Membiela, Primordial magnetic fields from a non-singular bouncing cosmology, Nucl. Phys. B 885, 196 (2014)
  * (35) G. Y. Bogoslovsky and H. F. Goenner, Finslerian spaces possessing local relativistic symmetry, Gen. Rel. Grav. 31, 1565 (1999).
  * (36) Z. Chang and X. Li, Lorentz invariance violation and symmetry in Randers-Finsler spaces, Phys. Lett. B 663, 103 (2008).
  * (37) A. P. Kouretsis, M. Stathakopoulos and P. C. Stavrinos, Imperfect fluids, Lorentz violations and Finsler Cosmology, Phys. Rev. D 82, 064035 (2010).
  * (38) S. I. Vacaru, Principles of Einstein-Finsler Gravity and Perspectives in Modern Cosmology, Int. J. Mod. Phys. D 21, 1250072 (2012).
  * (39) N. E. Mavromatos, S. Sarkar and A. Vergou, Stringy Space-Time Foam, Finsler-like Metrics and Dark Matter Relics, Phys. Lett. B 696, 300 (2011).
  * (40) S. I. Vacaru, Modified Dispersion Relations in Horava-Lifshitz Gravity and Finsler Brane Models, Gen. Rel. Grav. 44, 1015 (2012).
  * (41) N. E. Mavromatos, V. A. Mitsou, S. Sarkar and A. Vergou, Implications of a Stochastic Microscopic Finsler Cosmology, Eur. Phys. J. C 72, 1956 (2012).
  * (42) R. Gallego Torrome, P. Piccione and H. Vitorio, On Fermat’s principle for causal curves in time oriented Finsler spacetimes, J. Math. Phys. 53, 123511 (2012).
  * (43) S. Basilakos, A. P. Kouretsis, E. N. Saridakis and P. Stavrinos, Resembling dark energy and modified gravity with Finsler-Randers cosmology, Phys. Rev. D 88, 123510 (2013).
  * (44) S. Basilakos and P. Stavrinos, Cosmological equivalence between the Finsler-Randers space-time and the DGP gravity model, Phys. Rev. D 87, no. 4, 043506 (2013).
  * (45) M. Hohmann and C. Pfeifer, Geodesics and the magnitude-redshift relation on cosmologically symmetric Finsler spacetimes, Phys. Rev. D 95, no. 10, 104021 (2017).
  * (46) M. Hohmann, C. Pfeifer and N. Voicu, Finsler gravity action from variational completion, arXiv:1812.11161 [gr-qc].
  * (47) M. Born and E. Wolf, Principles of Optics, Cambridge University Press, England (1999).
  * (48) V. Perlick,Ray Optics, Fermats’s principle and applications to General Relativity, Springer, Heidelberg (2000).
  * (49) C. Barcelo, S. Liberati and M. Visser, _Analogue gravity_ , Living Rev. Rel. 8, 12 (2005) [Living Rev. Rel. 14, 3 (2011)].
  * (50) P. C. Stavrinos and S. I. Vacaru, _Cyclic and Ekpyrotic Universes in Modified Finsler Osculating Gravity on Tangent Lorentz Bundles_ , Class. Quant. Grav. 30, 055012 (2013).
  * (51) A. Triantafyllopoulos and P. C. Stavrinos, _Weak field equations and generalized FRW cosmology on the tangent Lorentz bundle_ , Class. Quant. Grav. 35, no. 8, 085011 (2018).
  * (52) A.P. Kouretsis, M. Stathakopoulos, and P.C. Stavrinos, _Covariant kinematics and gravitational bounce in Finsler space-times_ , Phys. Rev. D 86 124025 (2012).
  * (53) T. Koivisto and D. F. Mota, _Anisotropic Dark Energy: Dynamics of Background and Perturbations_ , JCAP 0806, 018 (2008).
  * (54) P.C. Stavrinos and S. Ikeda, _Some connections and variational principle to the Finslerian scalar-tensor theory of gravitation_ , Rep. Math. Phys. 44 221-230 (1999)
  * (55) P. C. Stavrinos and M. Alexiou, _Raychaudhuri equation in the Finsler-Randers space-time and generalized scalar-tensor theories_ , Int. J. Geom. Meth. Mod. Phys. 15, no. 03, 1850039 (2017).
  * (56) B. Xue and P. J. Steinhardt, _Evolution of curvature and anisotropy near a nonsingular bounce_ , Phys. Rev. D 84, 083520 (2011).
  * (57) M. Gasperini, Inflation and Broken Lorentz Symmetry in the Very Early Universe, Phys. Lett. 163B, 84 (1985).
  * (58) M. Gasperini, Repulsive gravity in the very early universe, Gen. Rel. Grav. 30, 1703 (1998)
  * (59) A.P. Kouretsis, M. Stathakopoulos and P.C. Stavrinos, _General very special relativity in Finsler cosmology_ , Phys. Rev. D 79 104011 (2010).
  * (60) A. G. Cohen and S. L. Glashow, _Very special relativity_ , Phys. Rev. Lett. 97, 021601 (2006).
  * (61) G. W. Gibbons, J. Gomis and C. N. Pope, _General very special relativity is Finsler geometry_ , Phys. Rev. D 76, 081701 (2007).
  * (62) C. M. Will, _The Confrontation between general relativity and experiment_ , Living Rev. Rel. 9, 3 (2006).
  * (63) Q. G. Bailey, R. D. Everett and J. M. Overduin, _Limits on violations of Lorentz Symmetry from Gravity Probe B_ , Phys. Rev. D 88, 102001 (2013).
  * (64) P. C. Stavrinos, _Congruences of Fluids in a Finslerian Anisotropic Space-Time._ Int. J. Theor. Phys 44, 245 (2005)
  * (65) P. C. Stavrinos, A. P. Kouretsis and M. Stathakopoulos, _Friedmann Robertson-Walker model in generalised metric space-time with weak anisotropy_ , Gen. Rel. Grav. 40, 1403 (2008).
  * (66) S. Vacaru, P. Stavrinos, E. Gaburov and D. Gonţa, _Clifford and Riemann-Finsler structures in geometric mechanics and gravity_ , Geometry Balkan Press, Bucharest, Romania (2006), [gr-qc/0508023].
  * (67) I. Bucataru, R. Miron, _Finsler-Lagrange geometry_ , Editura Academiei Romane (2007).
  * (68) R. Miron and M. Anastasiei, _The geometry of Lagrange spaces: theory and applications_ , Springer Science & Business Media (2012).
  * (69) C. Pfeifer and M.N.R. Wohlfarth, _Finsler geometric extension of Einstein gravity_ , Phys. Rev. D 85 064009 (2012).
  * (70) E. R. Caianiello, A. Feoli, M. Gasperini and G. Scarpetta, Quantum Corrections to the Space-time Metric From Geometric Phase Space Quantization, Int. J. Theor. Phys. 29, 131 (1990).
  * (71) E. R. Caianiello, M. Gasperini and G. Scarpetta, Inflation and singularity prevention in a model for extended-object-dominated cosmology, Class. Quantum Grav. 8, 659 (1991)
  * (72) M. Gasperini, A Geometric regularization procedure for the curvature of cosmological background, DFTT-4-91.
  * (73) R. Miron, S. Watanabe, S. Ikeda, _Some connections on tangent bundle and their applications to general relativity_ , Tensor, N. S. 46, (1987).
  * (74) Y. F. Cai and E. N. Saridakis, _Non-singular Cyclic Cosmology without Phantom Menace_ , J. Cosmol. 17, 7238 (2011).