# Bouncing Cosmological Isotropic Solutions in
Scalar-Tensor Gravity

D. Polarski1, A. A. Starobinsky2,3, Y. Verbin4
david.polarski@umontpellier.fralstar<EMAIL_ADDRESS>
###### Abstract

Bouncing non-singular isotropic cosmological solutions are investigated in a
simple model of scalar-tensor gravity. New families of such solutions are
found and their properties are presented and analyzed using an effective
potential as the main tool. Bouncing solutions are shown to exist for a Higgs-
like self-interaction potential which is bounded from below, in contrast to
previous solutions that appeared in the literature based on potentials which
were unbounded from below. In the simplest version of a scalar field with the
quartic potential and conformal coupling to gravity, bouncing spatially flat
solutions either have the Hubble function diverging in the past before the
bounce, but with a well-behaved future, or are globally regular but unstable
with respect to anisotropic or inhomogeneous perturbations at some finite
values of the scalar field and curvature. Regular solutions can only exist in
the part of the parameter space where the maximum of the effective potential
is larger than the first zero of the potential, and gravity becomes repulsive
at the bounce.

1Université de Montpellier & CNRS, Laboratoire Charles Coulomb,

UMR 5221, F-34095 Montpellier, France

2L. D. Landau Institute for Theoretical Physics RAS,

Moscow 119334, Russian Federation

3Kazan Federal University, Kazan 420008, Republic of Tatarstan, Russian
Federation

4Astrophysics Research Center & Department of Natural Sciences,

The Open University of Israel, Raanana 4353701, Israel

##  1 Introduction

Inflation as the stage of the early Universe evolution preceding the hot Big
Bang (the radiation dominated stage) is widely considered to be the most
elegant and promising idea to provide definite initial conditions for the
evolution of the Universe up to the present time. And indeed, a narrow
subclass of inflationary models, the simplest of them depending on one free
parameter to be fixed by observations only, quantitatively predicted the form
of the observed spectrum of primordial scalar (matter density) inhomogeneities
in the Universe at large scales, while many other models have been already
excluded mainly by the measured value of the slope of the power spectrum of
primordial curvature perturbations $n_{s}(k)-1$ [1] and by the upper value of
the tensor-to-scalar ratio $r(k)$ (see [2] for the most recent strong upper
limit). However, the duration of the inflationary stage inside our past light
cone is finite in all these viable inflationary models. Thus, even if we
accept the existence of the inflationary stage in the past of our Universe, it
is natural to think what was before it. One possibility is that inflation was
preceded by a generic inhomogeneous and anisotropic stage with curvature much
exceeded that during inflation (even up to singular values). Indeed in this
case it is straightforward to show that inflation is an intermediate attractor
and it can be realized for a set of initial pre-inflationary conditions with a
nonzero measure111In this case the fact that inflationary models do not
’solve’ the singularity problem (that they did not intend to), so they permit
more general solutions with much higher curvature, actually serves as an
advantage, not a drawback.. For the idealized cases of inflation in General
Relativity (GR) with a cosmological constant and power-law inflation where
inflation is the generic late-time attractor, this was proved in [3] and [4]
respectively, see also the recent papers [5, 6] for anisotropic cosmological
solutions in the more realistic $R+R^{2}$ and Higgs inflationary models.

However, another possibility considered in this paper is presented by bouncing
cosmologies which have a regular minimum of the scale factor at some finite
value of curvature much less than the Planck one. It is easy to obtain such
solutions even in the Friedmann-Lemaître-Robertson-Walker (FLRW) case in the
presence of spatial curvature. For example, in [7] the bouncing FLRW model
with a massive scalar field minimally coupled to gravity and positive spatial
curvature was constructed which, in fact, had both a contracting quasi-de
Sitter stage before the bounce and an expanding slow-roll quasi-de Sitter
stage after it. However, though this model permits solutions with an arbitrary
large number of bounces, completely regular solutions with an infinite number
of bounces exist in it, too, but they are not generic [8, 9, 10, 11]. So, this
bouncing model does not solve the singularity problem. Many other bouncing
models in theories of modified gravity including scalar-tensor gravity were
proposed later, see in particular [12, 13, 14, 15] and the recent papers [16,
17, 18]. Such theories are an ideal framework for the realization of
nonsingular bouncing cosmologies, since they allow for the violation of the
null energy condition that is necessary for the existence of a bounce in the
absence of spatial curvature [19, 20, 21, 22]. In particular, one can obtain
bouncing solutions in pre-big-bang [23] and ekpyrotic [24, 25] models (though
with some additional assumptions on matching of solutions before and after the
bounce).

It is just the case of FLRW bouncing models with zero spatial curvature on
which we will concentrate in this paper. In a series of three recent papers
[26, 27, 28] by Boisseau, Giacomini, Polarski and Starobinsky (BGPS), bouncing
cosmological solutions were found in a simple field theoretical model: BGPS
took the simple potential $U(\Phi)=U_{0}+\lambda\Phi^{4}/4$ and found that in
order to obtain bouncing cosmological solutions, the cosmological term
$(U_{0})$ should be positive and the self-coupling ($\lambda$) should be
negative, thus rendering the potential unbounded from below. The case of
$\xi=1/6$ which corresponds to conformally coupled scalar, results a
considerable simplifications in the field equations and allows for explicit
(analytic) solutions [26]. The BGPS bouncing solutions start at a point in the
past where the scalar field has a maximal value, from which it decreases while
the scale factor $a(t)$ decreases as well and passes through a minimum. The
scalar field keeps decreasing and vanishes asymptotically, while the scale
factor starts increasing exponentially with an asymptotically constant Hubble
function $H=\dot{a}/a$. As usual the bounce point is defined in terms of the
Hubble function as the time where $H=0$ and $\ddot{a}>0$. Some further
progress in this direction was made more recently [29, 30] still using
unbounded from below potentials. Finally, we must mention the problem of the
negative effective gravitational constant [31] (see also [32]) which is
obvious if we notice that the coefficient of the Einstein-Hilbert term in the
action (2.1) is the field dependent factor $1/\kappa^{2}-\xi\Phi^{2}$. This
factor vanishes for a definite field value, but as we will see, homogeneous
and isotropic cosmological solutions cross this field value without any
noticeable difficulty. However, small deviations from isotropy [31, 33, 34] or
from homogeneity [35] diverge as the scalar field tends to
$\Phi^{2}\rightarrow 1/\xi\kappa^{2}$. For the sake of completeness of the
discussion, we choose first to present the homogeneous and isotropic
cosmological solutions in the entire domain where they exist without imposing
the further restriction on inhomogeneous and anisotropic instabilities. The
paper is organized in the following way. In Section 2 we present the scalar-
tensor model that we study in this paper written in the Jordan frame in a
general context which will allow further extensions. In Section 3 we
specialize to the case of conformal coupling which results in considerable
simplifications of the equations. We obtain the effective potential of the
scalar field and the conditions on the (quartic) potential parameters
$m^{2},\lambda$ are derived in order to obtain bouncing solutions. In Section
4 the relevant regions in this parameter space are found, in particular that
region where a negative effective gravitational constant around the bounce can
be avoided, and we discuss the properties of bounce solutions found there. The
region where all quantities remain regular at all times, corresponds to that
region where the effective gravitational constant necessarily becomes
negative. In section 5 we analyze the problem in the Einstein frame and we
recover the region where bounce solutions avoiding a negative effective
gravitational constant around the bounce are possible. Our results are
summarized and discussed briefly in our Conclusion.

##  2 Scalar-Tensor Gravity: Action and Field Equations

We consider a scalar-tensor gravity model whose action is given by

$S=\int
d^{4}x\sqrt{-g}\left(\frac{R}{2\kappa^{2}}+\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-U(\Phi)-\frac{\xi}{2}R\Phi^{2}\right)~{},$
(2.1)

where $R$ is the Ricci scalar222We will use the $(+,-,-,-)$ signature, while
our definition of $R$ has a minus sign compared to Landau-Lifshitz. and $\Phi$
is a real scalar field non-minimally coupled to gravity through the coupling
constant $\xi$. Here $\kappa^{2}=8\pi G$, with $G$ the usual gravitational
(Newton’s) constant. The action (2.1) is actually a special case of the
general family which can be written in the notation of [26] as

$S=\frac{1}{2}\int
d^{4}x\sqrt{-g}~{}\left(F(\Phi)R+Z(\Phi)\partial_{\mu}\Phi\partial^{\mu}\Phi-2U(\Phi)\right)~{},$
(2.2)

where we choose here

$F(\Phi)=\kappa^{-2}-\xi\Phi^{2}~{},$ (2.3)

and

$Z(\Phi)=1~{}.$ (2.4)

It is obvious that by a field redefinition the function $Z(\Phi)$ can be
reduced to be $\pm 1$. We note further that $Z(\Phi)=-1$ is another
possibility which is perfectly acceptable provided the corresponding Brans-
Dicke parameter (function) $\omega_{\rm BD}$ satisfies
$-\frac{3}{2}<\omega_{\rm BD}<0$. We will consider however models with
$Z(\Phi)=1$.

The corresponding field equations of the general theory (2.2) with $Z(\Phi)=1$
read

$\nabla_{\mu}\nabla^{\mu}\Phi+U^{\prime}(\Phi)-\frac{R}{2}F^{\prime}(\Phi)=0\;\;\
,\;\;\;\
F(\Phi)G_{\mu\nu}+T^{(min)}_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}F(\Phi)-g_{\mu\nu}\nabla_{\rho}\nabla^{\rho}F(\Phi)=0$
(2.5)

where $T^{(min)}_{\mu\nu}$ is the energy-momentum tensor of the minimally
coupled scalar field:

$T^{(min)}_{\mu\nu}=\partial_{\mu}\Phi\partial_{\nu}\Phi-\left(\frac{1}{2}\partial_{\rho}\Phi\partial^{\rho}\Phi-U(\Phi)\right)g_{\mu\nu}~{}.$
(2.6)

In this work we would like to concentrate on the scalar field dynamics in the
simplest case (2.3). Writing the field equations for the scalar field, we find

$\nabla_{\mu}\nabla^{\mu}\Phi+U^{\prime}(\Phi)+\xi R\Phi=0~{},$ (2.7)

while we obtain for the gravitational field

$(1-\xi\kappa^{2}\Phi^{2})G_{\mu\nu}+\kappa^{2}T^{(min)}_{\mu\nu}+2\xi\kappa^{2}\left(\partial_{\rho}\Phi\partial^{\rho}\Phi+\Phi\nabla_{\rho}\nabla^{\rho}\Phi\right)g_{\mu\nu}-2\xi\kappa^{2}\left(\partial_{\mu}\Phi\partial_{\nu}\Phi+\Phi\nabla_{\mu}\nabla_{\nu}\Phi\right)=0~{},$
(2.8)

where $T^{(min)}_{\mu\nu}$ is given by (2.6). By tracing Eq. (2.8) we obtain
an expression for $R$ in terms of the scalar field

$R=\kappa^{2}\frac{4U(\Phi)-6\xi\Phi
U^{\prime}(\Phi)-(1-6\xi)(\partial\Phi)^{2}}{1-\xi\kappa^{2}(1-6\xi)\Phi^{2}}$
(2.9)

where we use the notation
$(\partial\Phi)^{2}=\partial_{\rho}\Phi\partial^{\rho}\Phi$ . So the equation
of the scalar field (2.7) may be simplified to be written in terms of an
effective potential $U_{\rm eff}(\Phi)$ as

$\nabla_{\mu}\nabla^{\mu}\Phi-\frac{\xi\kappa^{2}(1-6\xi)\Phi}{1-\xi\kappa^{2}(1-6\xi)\Phi^{2}}(\partial\Phi)^{2}+U_{eff}^{\prime}(\Phi)=0$
(2.10)

where the effective potential is defined in terms of its derivative333This
effective potential is different from the one used by Pozdeeva et al. [30] in
a similar context.:

$U_{eff}^{\prime}(\Phi)=\frac{4\xi\kappa^{2}\Phi
U(\Phi)+(1-\xi\kappa^{2}\Phi^{2})U^{\prime}(\Phi)}{1-\xi\kappa^{2}(1-6\xi)\Phi^{2}}$
(2.11)

##  3 General Considerations for conformal coupling $\xi=\frac{1}{6}$

We will further specialize in this work to a conformally coupled scalar i.e.
$\xi=1/6$ which obviously results considerable simplifications. In that case
the Ricci scalar may be expressed in terms of the scalar field through the
potential only, while the field derivative term drops:

$R=\kappa^{2}\left(4U(\Phi)-\Phi U^{\prime}(\Phi)\right)~{}.$ (3.1)

So the equation of the scalar field is written in terms of the effective
potential

$\nabla_{\mu}\nabla^{\mu}\Phi+U_{\rm eff}^{\prime}(\Phi)=0~{},$ (3.2)

where the effective potential is now defined in terms of its derivative by a
much simpler relation:

$U_{\rm
eff}^{\prime}(\Phi)=(1-\frac{\kappa^{2}}{6}\Phi^{2})U^{\prime}(\Phi)+\frac{2\kappa^{2}}{3}\Phi
U(\Phi)$ (3.3)

At this point we note that since it is $U_{\rm eff}^{\prime}(\Phi)$ which
appears in the equation of motion for the scalar field $\Phi$, we can think of
the field $\Phi$ as if it were minimally coupled to gravity with (effective)
potential given by $U_{\rm eff}(\Phi)$ (instead of $U(\Phi)$).

In the BGPS models, the effective potential has a shape similar to those shown
in Figure 1(a). Hence one may ask whether it is possible to start with a
potential which is bounded from below still yielding an effective potential
which has the same shape. For example, for an even polynomial of the sixth
order we have

$\displaystyle U(\Phi)$ $\displaystyle=$ $\displaystyle
U_{0}+\frac{m^{2}}{2}\Phi^{2}+\frac{\lambda}{4}\Phi^{4}+\frac{\nu}{6}\Phi^{6}\;\;,$
(3.4) $\displaystyle U_{\rm eff}(\Phi)$ $\displaystyle=$
$\displaystyle\left(\frac{m^{2}}{2}+\frac{\kappa^{2}U_{0}}{3}\right)\Phi^{2}+\frac{1}{4}\left(\lambda+\frac{\kappa^{2}m^{2}}{6}\right)\Phi^{4}+\frac{\nu}{6}\Phi^{6}-\frac{\kappa^{2}\nu}{144}\Phi^{8}.$
(3.5)

Figure 1:  The rescaled effective potential $\bar{U}_{\rm eff}(\bar{\Phi})$ is
shown for $\xi=1/6$, $\nu=0$ and various parameters $\bar{m}^{2}$ . (a) the
parameter $\bar{\lambda}$ is fixed to $\bar{\lambda}=2$, while the mass
parameter $\bar{m}^{2}$ (see the definition in Eq.(3.9)) takes the values:
$-\bar{m}^{2}=2,\,2.35,\,2.5,\,2.7,\,3,\,4$. In order to identify the
corresponding curves, note that $\bar{U}_{eff}(\bar{\Phi})$ decreases with
$-\bar{m}^{2}$. These effective potentials can be obtained for potentials
_bounded from below_ due to the non-zero parameter $\bar{m}^{2}<0$ satisfying
the inequalities (4.8) or equivalently (3.8). Our bouncing solutions are
constructed with the effective potentials displayed on the left panel; (b)
$\bar{\lambda}=6$ and $-\bar{m}^{2}=4,\,4.75,\,5.25,\,5.5,\,5.65,\,6$ so these
values violate the inequalities (4.8) or equivalently (3.8).

The scalar field dynamics is best understood from the shape of $U_{\rm eff}$,
rather than from the “bare” potential $U(\Phi)$ itself. Actually the potential
in the Einstein frame (EF) is similar to the effective potential in the Jordan
frame (JF) [27] so this is another way to understand the scalar field
dynamics. We will return to this point later.

In order that generic bounce solutions exist, it is the effective potential
which should have a maximum at some non-zero $\Phi$, and may be unbounded from
below as happens in BGPS. The “bare” potential itself may be completely
“normal” and still produce an effective potential which will be unbounded from
below as seen in Fig.1(a) which corresponds to the choice $\lambda>0$ and
$\nu=0$ in Eqs.(3.4)–(3.5).

Restricting ourselves in the following to polynomial potentials of order four,
therefore setting $\nu=0$ in (3.5), inspection of (3.5) gives immediately an
elegant answer: we must have a positive factor in front of $\Phi^{2}$, and a
negative one in front of $\Phi^{4}$. In other words, we get the condition

$\frac{m^{2}}{2}+\frac{\kappa^{2}U_{0}}{3}>0~{},$ (3.6)

and

$\lambda+\frac{\kappa^{2}m^{2}}{6}<0~{}.$ (3.7)

These inequalities can be summarized as follows

$-\frac{2\kappa^{4}U_{0}}{3}<\kappa^{2}m^{2}<-6\lambda~{}.$ (3.8)

The inequalities (3.8) constitute the key result of this section, they allow
to generalize the bouncing solutions based on potentials unbounded from below,
as in BGPS, to quartic potentials which are _bounded from below_ when
$\lambda>0$. It will be convenient to reformulate these inequalities later
using dimensionless variables, see Eq.(4.8).

The effective potentials shown on Fig.1(a) (left panel) will be used to
construct bouncing solutions. They correspond to potentials $U$ bounded from
below though these effective potentials have a shape similar to that of the
BGPS effective potential. The latter is obtained by substituting $m=0$ (and
$\nu=0$), $U_{0}>0$ and $\lambda<0$ in Eqs. (3.4)–(3.5). Indeed, if $m^{2}=0$,
this is the only way to satisfy the conditions $\eqref{condUeff}$ and we are
led to the inverted (unbounded from below) potential considered in BGPS. From
this wider perspective, the BGPS potential is just a simple special member in
a large family of potentials which may give rise to bouncing cosmologies.

This is the clearest way to see why the BGPS potential can be modified easily
to become bounded from below and more natural. The simplest way to turn the
BGPS potential itself (with $\lambda<0$) bounded from below is to add an
increasing higher power which will turn the potential function $U(\Phi)$
bounded from below. If the turning point is for large enough field values, the
bouncing solutions will not change their general behavior, since the effective
potential $U_{eff}(\Phi)$ will be still unbounded. The two functions of
Eqs.(3.4)-(3.5) for $\lambda<0$ and $\nu>0$ (take for simplicity $m=0$) are a
simple example for this. We note incidentally that the potential of Eq. (3.4)
is in much use for $Q$-ball construction [38, 39, 40].

Viewed from a different point of view, if the potential $U(\Phi)$ contains a
positive contribution with a power higher than 4, the effective potential will
necessarily become unbounded from below as demonstrated in Eq. (3.5) for a
contribution of power 6. We will thus assume from now on that our “bare”
potential is always bounded from below, i.e. in the polynomial potential of
Eq. (3.4) the parameter $\nu$ must be non-negative, and if it vanishes, then
$\lambda>0$. In what follows we will often keep $\nu$ in the equations, but we
will not go into extensive study of its role which is straightforward but
cumbersome. In other words, in most of our concrete calculations and
demonstrations we will take $\nu=0$.

It turns out to be much more transparent to analyze the situation in terms of
rescaled dimensionless variables. We will just substitute $\xi=1/6$ in the
general relation for arbitrary $\xi$

$\bar{\Phi}=\sqrt{\xi\kappa^{2}}\Phi\;\;,\;\;\bar{x}^{\mu}=\sqrt{\xi\kappa^{2}U_{0}}x^{\mu}\;\;,\;\;\bar{m}=\frac{m}{\sqrt{\xi\kappa^{2}U_{0}}}\;\;,\;\;\bar{\lambda}=\frac{\lambda}{\xi^{2}\kappa^{4}U_{0}}\;\;,\;\;\bar{\nu}=\frac{\nu}{\xi^{3}\kappa^{6}U_{0}}~{},$
(3.9)

in terms of which the “bare” and the effective potential become (for
$\xi=1/6$)

$\displaystyle\bar{U}(\bar{\Phi})$ $\displaystyle=$
$\displaystyle\frac{U(\bar{\Phi})}{U_{0}}=1+\frac{\bar{m}^{2}}{2}\bar{\Phi}^{2}+\frac{\bar{\lambda}}{4}\bar{\Phi}^{4}+\frac{\bar{\nu}}{6}\bar{\Phi}^{6}$
(3.10) $\displaystyle\bar{U}_{\rm eff}(\bar{\Phi})$ $\displaystyle=$
$\displaystyle\frac{U_{\rm
eff}(\bar{\Phi})}{U_{0}}=\left(\frac{\bar{m}^{2}}{2}+2\right)\bar{\Phi}^{2}+\frac{1}{4}\left(\bar{\lambda}+\bar{m}^{2}\right)\bar{\Phi}^{4}+\frac{\bar{\nu}}{6}\bar{\Phi}^{6}-\frac{\bar{\nu}}{24}\bar{\Phi}^{8}$

One may get the dimensionless version of all other equations for the conformal
coupling by the formal substitutions $\kappa=6$ and $U_{0}=1$ and putting
“bars” over all other quantities. We will use alternatively dimensionful and
dimensionless variables according to convenience.

##  4 Bounce Solutions for $\xi=1/6$

In the previous section we were mainly concerned with the scalar field
dynamics which could lead to cosmological bouncing solutions. We have to see
how the larger class of effective potentials found earlier will affect the
space-time dynamics when bouncing solutions are produced. For this
cosmological simple framework we assume a homogeneous time dependent scalar
field $\Phi(t)$ evolving in a standard flat FLRW Universe with metric

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dt^{2}-a^{2}(t)\left(dr^{2}+r^{2}d\Omega_{2}^{2}\right)~{}.$
(4.1)

Using the potential function of Eq. (3.4), the scalar field equation becomes

$\ddot{\Phi}+\frac{3\dot{a}}{a}\dot{\Phi}+\left(m^{2}+\frac{2\kappa^{2}U_{0}}{3}\right)\Phi+\left(\lambda+\frac{\kappa^{2}m^{2}}{6}\right)\Phi^{3}+\nu\Phi^{5}-\frac{\kappa^{2}\nu}{18}\Phi^{7}=0$
(4.2)

and the two Einstein equations (for $G^{1}_{1}=G^{2}_{2}=G^{3}_{3}$ and for
$G^{0}_{0}$) are given in explicit form by

$\displaystyle\left(\frac{3}{\kappa^{2}}-\frac{\Phi^{2}}{2}\right)\left(\frac{2\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\right)+\frac{\dot{a}}{a}\Phi\dot{\Phi}+\frac{\dot{\Phi}^{2}}{2}\hskip
142.26378pt$
$\displaystyle-3U_{0}+\left(\frac{2\kappa^{2}U_{0}}{3}-\frac{m^{2}}{2}\right)\Phi^{2}+\left(\frac{\lambda}{4}+\frac{\kappa^{2}m^{2}}{6}\right)\Phi^{4}+\frac{\nu}{2}\Phi^{6}-\frac{\kappa^{2}\nu}{18}\Phi^{8}=0$
(4.3)
$\displaystyle\left(\frac{3}{\kappa^{2}}-\frac{\Phi^{2}}{2}\right)\frac{\dot{a}^{2}}{a^{2}}-\frac{\dot{a}}{a}\Phi\dot{\Phi}-\frac{\dot{\Phi}^{2}}{2}-U_{0}-\frac{m^{2}}{2}\Phi^{2}-\frac{\lambda}{4}\Phi^{4}-\frac{\nu}{6}\Phi^{6}=0$
(4.4)

Another useful relation is the explicit form of Eq. (3.1) (or just the sum
(4.3)+(4.4)),

$R=6\left(\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\right)=\kappa^{2}\left(4U_{0}+m^{2}\Phi^{2}-\frac{\nu}{3}\Phi^{6}\right)~{},$
(4.5)

which may be used alternatively as a more convenient field equation. Actually,
for the case $m=\nu=0$ (while $\lambda$ may be non-zero) the scalar field
decouples as the expression for $R$, Eq.(4.5), yields a constant and it is
possible to completely integrate the equations of motion and get already a
bouncing universe tending in the asymptotic future to General Relativity (GR)
with a cosmological constant [26, 27, 28]. More accurately, in this particular
case $R$ is a constant at all times, but this feature of $R$ approaching a
positive constant, or $H$ tending to a constant, is characteristic to all the
bouncing solutions we will find here. So our aim is to find the conditions for
the existence of solutions to the field equations which pass through a bounce
point which we take to occur at $t=0$ by exploiting the freedom to redefine
the origin of the time coordinate. The solutions we are after, are therefore
solutions that obey the “initial conditions” $H(0)=0$ and $\dot{H}(0)>0$. From
this we notice that the bounce solutions are determined by the value of the
scalar field at $t=0$, $\Phi(0)$, since the value of $\dot{\Phi}(0)$ is
determined (up to a sign) by the first order constraint (4.4) to obey

$\dot{\Phi}(0)^{2}+U(\Phi(0))=0~{}.$ (4.6)

If we want to solve directly for $a(t)$, we may take an arbitrary value for
$a(0)$ due to the symmetry under rescaling of $a(t)$. We take naturally
$a(0)=1$.

One may also solve directly for $H(t)=\dot{a}/a$ using the following equation
which is a combination of the field equations (4.2)-(4.4) (compare also
Eq.(4.21) in what follows):

$\left(\frac{6}{\kappa^{2}}-\Phi^{2}\right)\dot{H}+H\Phi\dot{\Phi}+2\dot{\Phi}^{2}-\Phi\ddot{\Phi}=0$
(4.7)

Equation (4.7) will be very useful for some of our derivations. From the
scalar field equation Eq. (4.2), we easily see that the main properties of the
scalar field solutions are determined by the effective potential $U_{\rm
eff}(\Phi)$ defined in Eq. (3.5) whose derivative appears in Eq. (4.2). In
analogy with Newtonian mechanics we note that the scalar field equation is
similar to that of a point particle moving under the potential $U_{\rm
eff}(\Phi)$ with an additional damping term with the Hubble function in the
role of the viscosity coefficient which may also be negative.

In particular, by inspection of the effective potential $U_{\rm eff}(\Phi)$ we
find the possible asymptotic values of $\Phi$. If $\Phi$ tends to a constant
there are only two possibilities: (i) $\Phi(t)\rightarrow 0$, (ii) $\Phi(t)$
goes at late times to the second critical point where $U^{\prime}_{\rm
eff}(\Phi_{\ast})=0$, The third possibility is that $\Phi(t)$ diverges in a
finite time after the bounce but these are not interesting solutions. The
regular cases correspond to the following late time behavior of the Hubble
function: (i) $H(t)\rightarrow H_{0}=\sqrt{\kappa^{2}U_{0}/3}$ , (ii)
$H(t)\rightarrow
H_{\ast}=\sqrt{2\kappa^{2}U(\Phi_{\ast})/(6-\kappa^{2}\Phi^{2}_{\ast})}$ ,

By time reflection symmetry, the analogous possibilities exist for $t<0$ as
well as in the future if we change $t\rightarrow-t$; we however break “by
hand” the symmetry by the requirement that $\dot{H}(0)>0$ which selects the
realistic types of solutions with $H(t)>0$ for $t>0$.

In spite of the existence of the BGPS explicit (analytic) solution, it is
unrealistic to expect that analytic solutions can be found in general444See
however [29, 36] for some further analytical results in this theory and [37]
for Induced Gravity.. So the solutions of the field equations which will be
presented in this report were found numerically.

Figure 2:  Several representative “time-profiles” ($\bar{\Phi}(t)$ and $H(t)$)
of cosmological bounce solutions for the conformally coupled case ($\xi=1/6$).
The parameter values are $\bar{m}^{2}=-2.25$, $\bar{\lambda}=1$ ($\nu=0$)
which corresponds to the upper (red) part of the allowed region of parameter
space (see Fig.3). These models satisfy the inequalities (4.18) and _all_
bounce solutions violate the condition $F>0$ at the bounce, with
$\bar{\Phi}_{-}=1.104>1$. The solutions displayed have the following
$\bar{\Phi}(0)$ values: 1.104 = $\bar{\Phi}_{-}$, 1.114, 1.11437, 1.120,
1.145, 1.330, 1.560, 1.56025, 1.810. Note that the symmetric $\bar{\Phi}$
solution has a maximum at the bounce. Note also a symmetric $\bar{\Phi}$
solution with a minimum at $t=0$ with $\bar{\Phi}(0)>\bar{\Phi}_{*}$. Bounce
solutions are those that satisfy $H(0)=0$, $\dot{H}(0)>0$ and have no future
singularities.

Solving the field equations we find several kinds of solutions which are shown
in Fig.2 for the representative choice $(\bar{\lambda}=1,\bar{m}^{2}=-2.25)$.
They indeed fit the general framework described above. Note that this figure
demonstrates solutions which are not all bounce solutions. The one with
$\bar{\Phi}(0)=1.810$ has only a finite lifetime, starting from a Big Bang and
ending in a big crunch. The solution with $\bar{\Phi}(0)=1.56025$ is the
limiting solution which tends asymptotically to the maximum of the effective
potential at $\bar{\Phi}=\bar{\Phi}_{\ast}=1.18322$ (see Eq. (4.9) below). The
corresponding spacetime starts from a Big Bang, crosses a maximal expansion
and contracts exponentially. The third solution in Fig.2 which does not
satisfy the initial conditions imposed above for a bounce, is actually a
bounce solution with the only “fault” of having the bounce point not at $t=0$,
but at some later time. At $t=0$ it is still contracting. However, there is no
need to include it as an independent bounce solution, since it is related by
time translation to one of the bounce solutions with smaller $\bar{\Phi}(0)$
which also have $\dot{H}(0)>0$. So, the maximal value of $\bar{\Phi}(0)$ that
we should include in the $\bar{\Phi}(0)$ domain which produces bounce solution
is the one which corresponds to $H(0)=\dot{H}(0)=0$.

These solutions exist in different regions of parameter space, i.e. the
$(\bar{\lambda}\,,\bar{m}^{2})$ plane, which correspond to different relations
between the maximum $\bar{\Phi}_{*}$ of $\bar{U}_{\rm eff}(\bar{\Phi})$ and
the zeroes $\bar{\Phi}_{-},~{}\bar{\Phi}_{+}$ of $\bar{U}(\bar{\Phi})$. This
analysis will be crucial for the assessment of possible bouncing solutions.

The inequalities (3.8) which are required in order to reproduce an effective
potential similar to that of the BGPS model, become in terms of our
dimensionless variables

$-4<\bar{m}^{2}<-\bar{\lambda}~{}.$ (4.8)

The maximum itself is located at555This corresponds to two values because of
the symmetry of the potential. We will concentrate on $\Phi>0$.

$\bar{\Phi}^{2}_{\ast}=-\frac{\bar{m}^{2}+4}{\bar{m}^{2}+\bar{\lambda}}~{}.$
(4.9)

Figure 3: The various regions of the $(\bar{\lambda}\,,\bar{m}^{2})$ plane
where bounce solutions exist for a Higgs-like potential function
$\bar{U}(\bar{\Phi})$, i.e. a potential bounded from below, and an effective
potential $\bar{U}_{\rm eff}(\bar{\Phi})$ which has a maximum (see Fig.1). We
take $\xi=1/6$. The upper part of the allowed region between $L1$ and $L3$
(red) satisfies the inequalities (4.18). Fig.2 corresponds to this region. All
bouncing solutions in this region violate the condition $F>0$ at the bounce,
but completely regular ($H$ and $\bar{\Phi}$ finite everywhere) bounce
solutions do exist. The lower part of the allowed region between $L2$ and $L3$
(black) satisfies the inequalities (4.19). Fig.4 corresponds to this region.
Bounce solutions exist in this region satisfying $F>0$ at the bounce till the
asymptotic future, but $H$ and $\bar{\Phi}$ become singular in the past for
all these solutions, regular bounce solutions do not exist. The grey area at
the bottom contains solutions with “wrong” asymptotic behavior. For more
details, see text.

A second necessary condition is inferred from the first order constraint - Eq.
(4.4) taken at $t=0$. The scalar field value $\Phi(0)$ at the bounce belongs
to the domain where the potential $U$ is negative

$U(\Phi(0))<0\;\;\;\;\
\Rightarrow\;\;\;\;\bar{\Phi}_{-}<\bar{\Phi}(0)<\bar{\Phi}_{+}~{},$ (4.10)

where $\bar{\Phi}_{\pm}$ are the zeroes of $\bar{U}(\bar{\Phi})$:

$\bar{\Phi}^{2}_{\pm}=-\frac{\bar{m}^{2}}{\bar{\lambda}}\pm\sqrt{\left(\frac{\bar{m}^{2}}{\bar{\lambda}}\right)^{2}-\frac{4}{\bar{\lambda}}}~{}.$
(4.11)

We find therefore, that in addition to the condition (4.8), the parameters of
the potential should also satisfy

$0<\bar{\lambda}<\bar{m}^{4}/4~{}.$ (4.12)

Finally, $F>0$ is guaranteed whenever $\bar{\Phi}<1$. Obviously, all field
values $\bar{\Phi}(0)$ at the bounce would satisfy $F>0$ for a given choice of
parameters yielding $\bar{\Phi}_{+}<1$. However, in view of (4.11), (4.8), we
always have

$\bar{\Phi}_{+}>1~{}.$ (4.13)

Therefore, if $\bar{\Phi}_{-}<1$, only part of the bouncing solutions will
satisfy $F>0$ namely those solutions obeying

$\bar{\Phi}_{-}\leq\bar{\Phi}(0)<1~{}.$ (4.14)

Hence while it looks naively as if $\bar{\Phi}(0)$ is bound from above by
$\bar{\Phi}_{+}$, the upper zero of $\bar{U}$, it is actually bound by $1$ if
we insist on $F>0$. So in this respect the situation is similar to the BGPS
solutions where the (inverted) potential $U$ has only one zero
($\bar{\Phi}_{-}$). Obviously, when $\bar{\Phi}_{-}>1$, all existing bouncing
solutions necessarily violate the condition $F>0$. So the condition $F>0$
brings additional important constraint on the model parameters.

Let us return to the $(\bar{\lambda}\,,\bar{m}^{2})$ parameter plane, see
Fig.3. If we insist on a potential bounded from below (i.e.
$\bar{\lambda}>0$), the region in which bounce solutions are found is the
right triangular domain with the “parabolic hypotenuse” (which we will denote
$L1$) shown in Fig.3. The curve $L1$ expresses the constraint (4.12). There is
however an additional condition which originates from the asymptotic behavior
of the solutions which should correspond to a universe without future
singularities. All the solutions below the line $L2$ in Fig.3 have the “wrong”
asymptotic behavior, so this line is the lower edge of the domain of existence
of bounce solutions: to each point in this domain corresponds a family of
bounce solutions characterized by the value of $\Phi(0)$ which lies in the
interval $(\Phi_{-},\Phi_{+})$. Remember that by bounce solutions we mean
solutions which satisfy $H(0)=0$, $\dot{H}(0)>0$ and no future singularities.
The line $L2$ can be found only numerically and it is well approximated by

$\bar{m}^{2}=-4+0.42(4-\bar{\lambda})^{2/3}$ (4.15)

This domain of bounce solutions is further divided by the line $L3$ described
by

$\bar{m}^{2}=-2-\bar{\lambda}/2$ (4.16)

which separates the region where $\bar{\Phi}_{\ast}$ is located between
$\bar{\Phi}_{-}$ and $\bar{\Phi}_{+}$ and the region where both
$\bar{\Phi}_{-}$ and $\bar{\Phi}_{+}$ are to the right of $\bar{\Phi}_{\ast}$.
These are the only two possibilities. Actually $L3$ plays a crucial role.
Indeed, it is easily shown that points on the line $L3$ satisfy

$\bar{\Phi}_{-}=\bar{\Phi}_{*}=1~{}.$ (4.17)

In the upper region between the line $L3$ and the curve $L1$, defined by
$\bar{m}^{2}>-2-\bar{\lambda}/2$, one has the following inequalities

$1<\bar{\Phi}_{-}<\bar{\Phi}_{*}<\bar{\Phi}_{+}~{}.$ (4.18)

Hence in this region _all_ quantities $\bar{\Phi}_{-}$, $\bar{\Phi}_{*}$, and
not only $\bar{\Phi}_{+}$, are larger than one. Therefore, if we impose the
condition $F>0$, this region of the parameter plane is excluded as all
bouncing solutions necessarily violate it (at least) at the bounce. This is a
key result of this section and we will return below to it.

In the lower region (between $L2$ and $L3$) defined by
$\bar{m}^{2}<-2-\bar{\lambda}/2$, one has

$\bar{\Phi}_{*}<\bar{\Phi}_{-}<1<\bar{\Phi}_{+}~{}.$ (4.19)

So in this region of the parameter plane part of the existing bounce solutions
do satisfy the condition $F>0$ at (and after) the bounce. However, as we will
show below $H$ cannot be globally regular, this region allows only for partial
bounce solutions with $H$ diverging, and becoming positive, before the bounce.
We are now in a position to confirm analytically the curves displayed on
Figures 2, 4 and to exhibit the differences with the BGPS solutions. While the
scalar field dynamics is best understood in terms of the effective potential
$U_{\rm eff}$ one must actually return to the potential $U$ itself in order to
analyze the space-time dynamics. To see this, it is convenient to consider the
Friedmann equations in a more compact form666Eq. (4.20) is actually the time
component of the Einstein equations of Eq. (2.5) for the flat FLRW metric
(4.1), while Eq. (4.21) is a combination of the time and space components.
Compare also Eq. (4.7).

$\displaystyle 3FH^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\Phi}^{2}-3H\dot{F}+U~{},$ (4.20)
$\displaystyle-2F\dot{H}$ $\displaystyle=$
$\displaystyle\dot{\Phi}^{2}+\ddot{F}-H\dot{F}~{}.$ (4.21)

and apply them to the present case with the choice (2.3) and (2.4). Let us
consider a bouncing solution where $\Phi$ is very large before the bounce so
that $F$ becomes negative. In order to be able to pass the bounce, the scalar
field must obviously decrease in this region so $\dot{\phi}$ must be negative.
We can immediately find a striking difference between an inverted and non-
inverted (polynomial) potential considered in this work. Indeed, looking more
closely at (4.20), it is seen that while the last term is positive for a non-
inverted potential, it is clear that Eq.(4.20) is satisfied provided we have

$-3H\dot{F}<0~{},$ (4.22)

which is our case forces $H$ to be _positive_. In other words, we cannot have
a globally contracting universe before the bounce for a non-inverted
(polynomial) potential. This shows also the impossibility for a bounce
solution to satisfy globally $\dot{H}>0$. In the BGPS model this was possible
precisely because $U$ was negative for large $\Phi$. So while the scalar field
dynamics is essentially similar with the potentials considered here, the
space-time dynamics on the other hand can show crucial differences. From this
discussion, it follows that the only way one can hope to have a globally
growing Hubble function $H$, also before the bounce when the field $\Phi$
becomes large, is to have a potential $U$ which becomes negative rather than
positive for large $\Phi$.

Of course, there is still the option that the field $\Phi$ itself does not
become large before the bounce. Such a solution would be regular before as
well as after the bounce. As we have seen from the scalar field dynamics, this
is possible if $\Phi$ tends either to zero or to $\Phi_{*}$ before and after
the bounce.

Using our previous analysis of the $(\bar{\lambda}\,,\bar{m}^{2})$ parameter
plane, we note that several situations can arise. In the full allowed region
between $L1$ and $L2$, we get from (4.11) the inequality

$\bar{\Phi}_{-}>0~{}.$ (4.23)

Therefore a solution tending to zero in the past and in the future must have a
local maximum at some finite time in order to satisfy
$\bar{\Phi}(0)\geq\bar{\Phi}_{-}>0$. Using (2.3),(4.21) or (4.7) in its
explicit form, the following equality is easily obtained at any extremum of
$\Phi$

$6F\dot{H}-\Phi\ddot{\Phi}=0~{}.$ (4.24)

Assuming globally the inequality $\dot{H}>0$, and not just at the bounce, we
see that a maximum is possible only when $F<0$. Several solutions of this kind
are displayed on Fig.2. There is still the possibility that the solution tends
asymptotically to $\bar{\Phi}_{*}$ in the past and/or in the future. This does
not modify our conclusions. Indeed, in view of the inequalities (4.18),(4.19),
either one has a local maximum at some finite time when the parameters are in
the lower part of the allowed region between lines $L2$ and $L3$ (see Fig.3);
either a maximum can be avoided in principle for
$\bar{\Phi}_{*}>\bar{\Phi}_{-}$ which is possible only when the parameters lie
in the upper part of the allowed region between lines $L3$ and $L1$ and both
$\bar{\Phi}_{*},~{}\bar{\Phi}_{-}$ violate $F>0$, see (4.18).

Note the existence of a symmetric solution with respect to time reversal
satisfying $\dot{\Phi}_{0}=0$ and $\ddot{\Phi}_{0}<0$ at the bounce, see
Fig.2. The corresponding Hubble function has a “step-like” shape tending to a
positive constant as $t\rightarrow\infty$ and a negative constant as
$t\rightarrow-\infty$. That is exponential contraction until a minimal value
of the scale factor followed by an exponential expansion. Such a realization
implies in particular $\Phi(0)=\Phi_{-}$ from (4.20). Hence for such a
solution to exist, the model parameters must be chosen such that
$\bar{\Phi}_{-}>1$. In other words, this is only possible in the upper part of
the allowed region, between lines $L1$ and $L3$. The parameter values of Fig.2
correspond indeed to this region. Such a symmetric solution cannot exist in
the lower part between lines $L2$ and $L3$, see e.g. the parameters chosen for
Fig. 4.

Figure 4: Several representative “time-profiles” ($\Phi(t)$ and $H(t)$) of
cosmological solutions of the conformally coupled case ($\xi=1/6$) in the
lower part of the allowed region of parameter space (black), see Fig.3. These
models satisfy the inequalities (4.19). The parameters values are
$\bar{m}^{2}=-2.7225$ and $\bar{\lambda}=1$ ($\nu=0$). The $\bar{\Phi}(0)$
values are: 0.94672, 0.9500, 0.9750, 1.025, 1.210, 1.53142. Bounce solutions
do exist which do not violate the condition $F>0$ at the bounce. However _all_
bounce solutions become singular in the past, regular bounce solutions cannot
exist. Note that the largest $\bar{\Phi}(0)$ solution (dashed curve) does not
fulfill the $\dot{H}(0)>0$ condition and is equivalent up to a time
translation to the solution with the smallest $\bar{\Phi}(0)$.

In our derivation of the existence of regular bounce solution in the upper
region between $L3$ and $L1$ however violating the condition $F>0$, we have
used the fact that $\Phi_{-}$ is either larger or smaller than one together
with the assumption that the background evolution obeys $\dot{H}>0$. Actually
it is possible to generalize these results. Indeed, from the dynamics of the
scalar field $\bar{\Phi}$ and the shape of the effective potential, it is
clear that any extremum of $\bar{\Phi}$ corresponds either to a maximum
$\bar{\Phi}_{\rm max}$ when

$\bar{\Phi}_{\rm max}<\bar{\Phi}_{*}~{},$ (4.25)

or to a minimum $\bar{\Phi}_{\rm min}$ when

$\bar{\Phi}_{\rm min}>\bar{\Phi}_{*}~{}.$ (4.26)

It is then immediately seen that the maximum of a regular bounce solution in
the upper region between $L3$ and $L1$ (satisfying (4.18)), will necessarily
violate the condition $F>0$ as we have in this case

$1<\bar{\Phi}_{-}\leq\bar{\Phi}_{\rm max}<\bar{\Phi}_{*}~{}.$ (4.27)

We stress that this result is now obtained without any assumption concerning
the behaviour of $H$. Of course at the maximum $\bar{\Phi}_{\rm max}$, we
necessarily have $\dot{H}>0$ from (4.24). Such solutions can and do exist.

Looking at the lower region between $L2$ and $L3$ (satisfying (4.19)), it is
seen that regular bouncing solutions cannot exist at all. It is interesting
that in this case bounce solutions which do not violate the condition $F>0$ do
exist, and some examples are displayed on Fig.4, but regular solutions do not
exist. One could summarize the situation in the following way: either we can
avoid violation of the condition $F>0$ but no regular solution exists, or we
can construct regular bounce solutions but we necessarily violate the
condition $F>0$.

Fig.4 depicts the behavior of a representative family of solutions
corresponding to $(\bar{\lambda}=1,\bar{m}^{2}=-2.7225)$ (lower region between
$L2$ and $L3$). The asymptotic values are given as before by Eqs(4.28)–(4.29)
which give in the particular case of Fig.4 the two possibilities:
($\bar{\Phi}(t)\rightarrow 0$ , $\bar{H}(t)\rightarrow\sqrt{2}$) and
($\bar{\Phi}(t)\rightarrow 0.86119$, $\bar{H}(t)\rightarrow 0.9952$).

For the sake of completeness we note that monotonically expanding solutions
exist all over the place: they are the only kind where bounces cannot exist
(i.e. no $\dot{a}=0$) and coexist with bounce solutions where they are
possible. These expanding solutions start at a Big Bang and become typically
exponentially growing because of the $U_{0}>0$ parameter. The “time-profiles”
of Fig.2 correspond to the point $(\bar{\lambda}=1,\bar{m}^{2}=-2.25)$ which
lies in the upper region between $L1$ and $L3$. The solutions satisfy the
appropriate asymptotic conditions of the above mentioned types: type (i):

$\bar{\Phi}(t)\rightarrow
0\;\;,\;\;\;\;\bar{H}^{2}(t)\rightarrow\bar{H}^{2}_{0}=2$ (4.28)

and type (ii):

$\bar{\Phi}(t)\rightarrow\bar{\Phi}_{\ast}=\sqrt{-\frac{\bar{m}^{2}+4}{\bar{m}^{2}+\bar{\lambda}}}\;\;,\;\;\;\;\bar{H}^{2}(t)\rightarrow\bar{H}^{2}_{\ast}=\frac{4\bar{\lambda}-\bar{m}^{4}}{2(\bar{\lambda}+\bar{m}^{2})}.$
(4.29)

For the particular case of Fig.2 we have for type (ii) the numerical values:
$\bar{\Phi}(t)\rightarrow 1.18322$, $\bar{H}(t)\rightarrow-0.65192$. Fig.2
contains also one representative of type (iii) which “lives” for a finite time
between a Big Bang and a big crunch.

To summarize, it is possible for part of the parameter region to allow for
globally regular bouncing solutions, however such solutions necessarily
violate the condition $F>0$. While this is not problematic at the homogeneous
background level, it was shown in [31] that anisotropies would become infinite
when $F\to 0$. Hence in such a scenario, primordial fluctuations can be
generated only after the bounce, and more precisely after the quantity $F$
crosses zero (upwards) for the second time. In this respect, the situation is
quite similar to the standard inflationary picture in a Big Bang universe.

Finally we note the correspondence with the BGPS solutions. The crucial
difference is that since $m=0$, there is only one (positive) root to the
equation $U(\Phi)=0$, which is $\Phi^{2}_{1}=\sqrt{-4U_{0}/\lambda}$. It is
clear that we have $\lambda<0$, and hence also $\bar{\lambda}<0$, for
$U_{0}>0$. We must impose $\bar{\Phi}^{2}_{1}<1$ to ensure that the condition
$F>0$ is not violated at the bounce, or $\bar{\lambda}<-4$, which as expected
from our results is only possible for $\Phi^{2}_{\ast}<\Phi^{2}_{1}$ . In that
case regular bouncing solutions do not exist. The parameters were chosen in
[26] in this way so this corresponds to what happens in the lower region
between $L3$ and $L1$.

##  5 Einstein frame

It is interesting to recover our results and to see how these are expressed in
the Einstein frame. It is well-known that it is possible to transform the
Lagrangian (2.2) written in the Jordan frame (JF), to a new one written in the
Einstein frame (EF) as follows

$L=\frac{R_{*}}{2\kappa^{2}}+\frac{1}{2}g_{*}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)~{},$
(5.30)

where the asterisk denotes the geometrical quantities in the EF; $\phi$ and
$V$ are respectively the Einstein frame scalar field and potential. One can go
from the Jordan frame to the Einstein frame by means of the following
transformations (for $Z=1$)

$\displaystyle\left(\frac{d\phi}{d\Phi}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\kappa^{2}}\left(\frac{3}{4}\left(\frac{dF/d\Phi}{F}\right)^{2}+\frac{1}{2F}\right)~{},$
$\displaystyle g^{*}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\kappa^{2}F(\Phi)~{}g_{\mu\nu}=A^{-2}(\phi)~{}g_{\mu\nu}$
$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\kappa^{-4}U(\Phi)F^{-2}(\Phi)$ (5.31)

These transformations are obviously ill-defined when $F\leq 0$. We easily find
for our system

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\frac{\sqrt{6}}{\kappa}\tanh\left(\frac{\kappa}{\sqrt{6}}\phi\right)$
(5.32) $\displaystyle A(\phi)$ $\displaystyle=$
$\displaystyle\cosh\left(\frac{\kappa\phi}{\sqrt{6}}\right)~{}.$ (5.33)

We have in particular that $\Phi\geq 0$ is a monotonically growing function of
$\phi$ with $\bar{\Phi}\to 1$ for $\phi\to\infty$. Using (5.31) we easily
obtain the potential $V(\phi)$, viz.

$V(\phi)=U_{0}\cosh^{4}\left(\frac{\kappa}{\sqrt{6}}\phi\right)+\frac{3m^{2}}{\kappa^{2}}\cosh^{2}\left(\frac{\kappa}{\sqrt{6}}\phi\right)\sinh^{2}\left(\frac{\kappa}{\sqrt{6}}\phi\right)+\frac{9\lambda}{\kappa^{4}}\sinh^{4}\left(\frac{\kappa}{\sqrt{6}}\phi\right)~{}.$
(5.34)

This potential is even and its behaviour depends crucially on the parameter
values. We can now introduce dimensionless quantities $\bar{\phi},~{}\bar{V}$
in the same way as we did earlier in the JF, see (3.9), and write (5.34) in
the appealing form (see Fig. 5)

Figure 5: The rescaled potential function in the Einstein frame
$\bar{V}(\bar{\phi})$ is shown for $\xi=1/6$, $\nu=0$ and $\bar{\lambda}=2$.
The mass parameter $\bar{m}^{2}$ takes the values:
$-\bar{m}^{2}=2,\,3,\,3.1,\,3.2,\,3.5,\,4$. In order to identify the
corresponding curves, note that $\bar{V}(\bar{\phi})$ decreases as
$-\bar{m}^{2}$ increases. From top to bottom: the first curve corresponds to
the (red) upper region in parameter space, $V$ never becomes negative and no
bounce satisfying $F>0$ is possible for these parameters; the second curve is
on L3; the other curves correspond to the lower (black) region for which $V$
can be negative allowing for a bounce satisfying $F>0$.

$\bar{V}(\bar{\phi})=\cosh^{4}(\bar{\phi})+\frac{\bar{m}^{2}}{2}\cosh^{2}(\bar{\phi})\sinh^{2}(\bar{\phi})+\frac{\bar{\lambda}}{4}\sinh^{4}(\bar{\phi})~{}.$
(5.35)

Remembering the different regions in parameter space, see Fig.3, we see that
$\bar{V}$ tends asymptotically to $+\infty$ in the (upper red) region between
$L3$ and $L1$ and on the line $L3$, while $V$ tends to $-\infty$ in the (lower
black) region between $L2$ and $L3$.

The crucial fact is that only in the last case does the potential become
negative. It is straightforward to check from (5.31) that when the EF is well-
defined, the EF potential $V$ is negative when the JF potential $U$ is
negative. As the existence of a bounce solution in the JF requires that $U$
becomes negative – in our case between $\Phi_{-}$ and $\Phi_{+}$ – around the
bounce, it is clear that only the (lower black) region between $L2$ and $L3$
can produce bounce solutions satisfying $F>0$ around the bounce. The limit is
the line $L3$ on which $\bar{\Phi}_{*}=\bar{\Phi}_{-}=1$ or
$\bar{\phi}_{*}=\bar{\phi}_{-}=\infty$, and
$V=\left(1-\frac{\lambda}{4}\right)\cosh^{2}(\bar{\phi})+\frac{\lambda}{4}>0$.
In the upper region including the line $L3$, bounce solutions satisfying $F>0$
cannot exist at all because $V$ is non negative there. This confirms the
results found in section 3, namely that while bounce solutions do exist in the
JF when the parameters lie in the upper region between $L3$ and $L1$, these
solutions violate the condition $F>0$ around the bounce. Our analysis in the
EF confirms that this violation is unavoidable, which is the key result of
this section.

We add now some more material which helps in understanding the correspondence
between both frames. We know that a spatially flat Robertson-Walker (RW)
metric in the JF yieds a spatially flat RW metric in the EF, viz.

$ds_{*}^{2}=dt_{*}^{2}-a_{*}^{2}(t_{*})d\vec{x}^{2}~{},$ (5.36)

where we have defined $dt=A(\phi)dt_{*}$ and $a=A(\phi)a_{*}$. Note that
$t_{*}$ is a monotonically growing function of $t$ in the region $F>0$. We
need of course the Hubble function in the EF which is obviously
$H_{*}=(da_{*}/dt_{*})/a_{*}$. So, the Friedmann equations in the EF are given
by

$\displaystyle\frac{3}{\kappa^{2}}H_{*}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{d\phi}{dt_{*}}\right)^{2}+V$ (5.37)
$\displaystyle\frac{1}{\kappa^{2}}\frac{dH_{*}}{dt_{*}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{d\phi}{dt_{*}}\right)^{2}$ (5.38)

while the Klein-Gordon equation for the scalar field is

$\frac{d^{2}\phi}{dt_{*}^{2}}+3H_{*}\frac{d\phi}{dt_{*}}+\frac{dV}{d\phi}=0~{}.$
(5.39)

Note that from (5.38) we cannot get a bounce in the EF because
$\frac{dH_{*}}{dt_{*}}\leq 0$. We can further easily derive the following
equalities

$\frac{d\phi}{dt_{*}}=\cosh^{3}\left(\frac{\kappa}{\sqrt{6}}\phi\right)~{}\frac{d\Phi}{dt}~{},$
(5.40)

and

$H=A^{-1}\left[H_{*}+\frac{\kappa}{\sqrt{6}}\tanh\left(\frac{\kappa}{\sqrt{6}}\phi\right)\frac{d\phi}{dt_{*}}\right]~{}.$
(5.41)

Eq.(5.40) shows that an extremum of $\Phi$ in the JF corresponds to an
extremum of $\phi$ in the EF. One can show through Eq.(5.32) that the maximum
of $V(\phi)$ corresponds to the maximum of $U_{\rm eff}(\Phi)$. To illustrate
the correspondence between both frames in a specific case, we note that in
particular when $U(\Phi_{-})=0$, a bouncing solution is possible with
$\dot{\Phi}\big{|}_{0}=0$, $\Phi_{0}=\Phi_{-}$ at the bounce. It is easily
checked from (5.31),(5.40) and (5.41), that this corresponds in the EF to
$V(\phi_{-})=0$, $\frac{d\phi}{dt_{*}}=0$ and $H_{*}=0$.

##  6 Conclusion

Cosmological bouncing universes without spatial curvature require violation of
the null energy condition. Effective violation of this condition and the
construction of bouncing solutions is possible in scalar-tensor models
considered in this work. We have presented bouncing solutions which are
extensions of the BGPS solutions found earlier. The latter were based on
potentials unbounded from below. As a first important result, considering
quartic potentials, we show that if the potential parameters satisfy the
inequalities (4.8) or equivalently (3.8), an arbitrarily large family of
potentials _which are bounded from below_ is obtained which can produce
bouncing solutions.

Like in the BGPS solutions, the scalar field $\Phi(t)$ diverges generically in
the past (and hence the condition $F>0$ is violated in the past) in the new
models considered in this work too. However, unlike the BGPS solutions, we
have shown the impossibility to have a regular Hubble function $H(t)$ from the
asymptotic past to the asymptotic future, $H(t)$ will necessarily diverge in
the past before the bounce, see eqs. (4.20) - (4.22). This is due precisely to
the fact that the potential $U$ is unbounded from below in the BGPS model
while it is positive for large $\Phi$ for the models considered in this work.
Actually the only way to avoid this divergence of $H(t)$ is to avoid
altogether the divergence of $\Phi(t)$. In that case, regular bouncing
solutions with a step-like Hubble function $H$ can be constructed in part of
the parameter space. We have shown however that such solutions necessarily
violate the condition $F>0$ at the bounce, see the discussion in section 3 and
in particular the crucial inequalities (4.18), (4.19). We have found the
regions in parameter space, see Fig.3, where any of these solutions exist. All
these possibilities are illustrated in Fig.2 and Fig.4. These results were
confirmed by looking at the scalar field potential in the Einstein frame, see
Eq. (5.34) or (5.35). As a bouncing solution requires $V<0$ (at the bounce),
we have found indeed that $V<0$ is only possible in the (lower black) region
between L2 and L3 of the parameter space. This corresponds indeed to the
region found in the Jordan frame where bounce solutions with $F>0$ exist.

To summarize, either we have bouncing solutions which are regular in the
future and which do not violate the condition $F>0$ around the bounce, but the
scalar field $\Phi$ _and also the Hubble function $H$_ will diverge before the
bounce; or else we can have bouncing solutions for which both $\Phi$ and $H$
are globally regular, but these will violate the condition $F>0$ around the
bounce. If such a scenario is realized in our Universe, it implies that the
globally regular bouncing universes should be perfectly isotropic around the
bounce, in particular there can be no perturbations before the bounce. The
introduction of positive spatial curvature can provide a way to avoid the
problem of $F>0$ violation around the bounce, that we leave for future
investigation.

Acknowledgement

AAS was supported by the RSF grant 21-12-00130. YV acknowledges support by the
OUI Research Authority.

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