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#  Non-singular Bounce from a Phase transition

C. C. Wong  Department of Electrical and Electronic Engineering, University of
Hong Kong. H.K.

###### Abstract

The single field Ekpyrosis of Khoury and Steinhardt khoury -khoury1  admits a
Jordan frame scalar field with a double well potential. After the end of the
ekpyrotic phase, as energy density increases we find a phase transition
scenario to a symmetric phase at the conformal coupling fixed point leads to a
non-singular bounce in Einstein frame.

###### pacs:

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##  I Introduction

The density fluctuations in the observed Cosmic Microwave Background ade1
-ade2  requires a nearly scale invariant curvature power spectrum within an
early Hubble radius, see brandenberger . The current paradigm to provide these
fluctuations, as well as a flat, homogeneous and isotropic universe, is
inflation [4]. Inflation can be realised using effective models of canonical
quantum scalar fields with a plateau potential in Einstein frame brandenberger
-martin . From a similar scalar field plateau potential, one can also obtain
nearly invariant curvature perturbations in slowly evolving (contracting or
expanding) solutions, piao -lyth . For the inflation solution, conceptual
challenges such as super-Planckian problem, multiverse problem aside
brandenberger -steinhardt , the isotropic power loss of $5\sim 10\%$ in the
small multipole $l<40$ range, suppression of quadruple and dipolar asymmetry
ade2 , are notable concerns though they are arguably within statistical
variance. Attempts to address these issues usually require pre-”slow-roll
inflation” physics such as a constant roll phase, starobinsky  and a fast roll
phase linde , a ”slowly expanding” phase or a ”slowly contraction” phase piao
. The slowly contracting phase we focus on in this work is the ”Ekpyrotic”
phase, which requires a bounce to connect the contracting phase to the
subsequent expanding phase, whether it be a radiation dominant phase or an
inflation phase. A bounce can be non-singular if it occurs at energy scales
sufficiently lower than the Planck scale, where the evolution of the classical
background is more trustable.

To obtain a bounce mechanism presents a major challenge. After the Ekpyrotic
phase, the Einstein frame Hubble constant is $H_{E}<0$ and the scale factor of
the universe is in a contracting phase. For an effective bounce to occur, one
requires $\dot{H}_{E}>0$ and after the bounce a subsequent field roll in a non
negative potential $V\geq 0$, see xue . In the scalar field kinetic energy
dominant epoch, the field kinetic energy has the largest power $\epsilon$ in
$a^{-2\epsilon}$ amongst cosmological fluids, spatial curvature and static
field potential, a contracting universe will keep contracting unless some
”additional” conditions exist that leads to $\dot{H}_{E}>0$. A common approach
is by allowing, for a short time, the kinetic or gradient term to take the
wrong sign with its cocomitant violation of Null Energy Conditon (NEC) ljjas
where the instabilities are controllable. This is implemented using non-
canonical kinetic terms. Some successes are found at Galileon theories ljjas1
-ljjas2  despite a no-go theorem for the cubic Galileon theories libanov .
Other notable solution involves higher derivatives operator piao2019 .
Attempts to obtain a bounce from Canonical scalar fields include using a
generic positive curvature matsui  and adding string inspired negative energy
density term brandenberger2 . In Jordan frame, one can implement a bounce
without Null Energy Condition (NEC) violation, li -piao2 . Yet these solutions
are shown to be dual to inflation in Einstein frame. A Jordan frame model with
a conformal coupling involving the Higgs field is also considered in bars
-bars2 , in which the Einstein frame scaler field plateau potential is also
described as a double well potential in Jordan frame. A phase transition is
effected when the quartic coupling of the scalar field passes between a
positive value region and a negative value region. In this work, we consider a
Jordan frame double well potential which goes through a phase transition at
conformal coupling.

As we consider a bounce after the end of Ekpyrotic phase in khoury -khoury1 ,
we recall their results for completeness. Here one considers a canoncial
scalar field with a lifted plateau potential in Einstein frame

$V=V_{1}-V_{0}e^{-c\varphi/M_{pl}},$ (1)

where $M_{pl}$ is the Planck mass, $V=-V_{0}e^{-c\varphi/M_{p}}$ is the
negative potential and $V_{1}$, usually taken as $V_{1}=V_{0}$, is the lift to
take the overall potential to positive density at large $\varphi$. This
potential provides an effective model for Inflation martin . However, the same
model also admits two other solution branches, namely adiabatic Ekpyrosis and
a slow expansion followed by a slow contraction lyth .

Ekpyrosis scenario turok -turok1  and steinhardt2 , where a period of ultra
slow scale factor contraction, with equation of state parameter $\epsilon\gg
1$, can be shown to produce scale invariant power spectrum fluctuations. For
potential in Eq.(1), we take $M_{pl}=1$, with scale factor $a(t)$ and Einstein
frame Hubble constant $H_{E}$, where the cosmic time $t$ is negative and
increasing toward $0_{-}$. At large c, the field has a tachyonic roll given by
the equation

$\ddot{\varphi}+3H_{E}\dot{\varphi}+V_{,\varphi}=0.$ (2)

Given that $-t_{ek-beg}$ and $-t_{ek-end}$ are the points where Ekpyrotic
phase begins and ends, Eq.(2) has the following scaling solution

$a(t)=\bigg{(}\frac{-t}{-t_{ek-
end}}\bigg{)}^{1/\epsilon};\>\>\>H_{E}=\frac{2}{c^{2}t},\>\>\>\>\dot{H}_{E}=-\frac{2}{c^{2}t^{2}};\>\>\>\epsilon=\frac{3}{2}(1+\omega)=-\frac{\dot{H}_{E}}{H_{E}^{2}}=\frac{c^{2}}{2}$
(3)
$\varphi=\frac{2}{c}\ln\bigg{(}-\sqrt{\frac{V_{0}}{2(1-6/c^{2})}}ct\bigg{)}.$
(4)

Given the density $\rho$ and pressure $P$ of the cosmological fluid, the
Einstein frame Hubble constant is given by

$3H_{E}^{2}=\rho=\frac{1}{2}\dot{\varphi}^{2}+V(\varphi);\>\;\>P=\frac{1}{2}\dot{\varphi}^{2}-V(\varphi),$
(5)

with $\rho=a^{-2\epsilon}$ and at nearly constant $\epsilon$, we obtain

$6H_{E}\dot{H}_{E}=\dot{\rho}=-3H_{E}(\rho+P);\>\;\;\;\dot{H}_{E}=-\frac{1}{2}\dot{\varphi}^{2}=-\frac{1}{2}(\rho+P).$
(6)

where the Null Energy condition (NEC) is $\rho+P\geq 0$. The Ekpyrotic phase
describes a contracting scale factor with $H_{E}<0$ and $\dot{H}_{E}<0$.
However, it is argued that the curvature spectrum of this canonical scalar
field is necessarily blue-tiled creminelli0

By extrapolating the ekpyrotic scaling solution to much earlier time, the
authors of khoury -khoury1  find a ”Pre-ekpyrotic ” phase in which a scale
invariant power spectrum can be made red-tilted.

Specifically, by integrating $\dot{H}_{E}=-\frac{2}{c^{2}t^{2}}$ one obtains

$H_{E}=\frac{2}{c^{2}t}-\sqrt{\frac{V_{1}}{3}};$ (7)

where the equation of state paramter $\epsilon=\frac{3}{2}(1+\frac{p}{\rho})$
is given by

$\epsilon=\frac{6}{c^{2}V_{0}}\frac{1}{(t+t_{ek-beg})^{2}}\sim\frac{1}{t^{2}}$
(8)

for larger $|t|\gg|t_{ek-beg}|$ where $t_{ek-beg}$ is the time at which the
pre-Ekpyrotic phase ends and the Ekpyrotic phase begins. The equation of state
parameter $\epsilon$ changes from a small value at $t^{2}\gg 1$ towards a
large constant $\frac{c^{2}}{2}$ as time grows towards $t_{ek-beg}$.

In this ”Pre-Ekpyrotic” phase, the Hubble radius is large and the small
momentum perturbations grow outside of a curvature scale (a reminiscence of
the correlation length) that is smaller than the Hubble radius. These scale
invariant perturbations which is red-tilted stay inside the Hubble radius and
can only exit the Hubble radius during a subsequent Ekpyrotic (contracting
Hubble radius) phase. During the ekpyrotic contraction, the red tilt spectrum
will exit the Hubble radius first to be followed by the blue tilt ”ekpyrotic”
spectrum.

Cosmological observations of non-gaussianity and a small red-tilt requires
that $c(\varphi)$ in Eq.(1) to be very large ($\sim 10^{28})$ khoury  in both
the ”pre-Ekpyrotic” phase and at the beginning of the Ekpyrotic phase. Also
$c^{{}^{\prime}}(\varphi)>0$ (that is $c$ decreases as $\varphi$ decreases) is
required to reduce $c(\varphi)$ to $O(1)$ where the Ekpyrotic phase ends.
After $c(\varphi)$ takes $O(1)$ value, we have
$\dot{H}_{E}=-\frac{1}{2}\dot{\varphi}^{2}$. To match observations that the
Hubble radius is expanding at late time. $\dot{H}_{E}$ needs to change from a
negative value to positive value ($\dot{H}_{E}>0$). Typically the connection
between the contracting phase to expanding phase (the bounce) is described
schematically as $a\rightarrow a$, $\dot{a}\rightarrow-\dot{a}$,
$\varphi\rightarrow\varphi$, $\dot{\varphi}\rightarrow\dot{\varphi}$.

In this work, by noting that the Einstein frame canonical scalar field with a
plateau potential in Eq.(1) admits a corresponding Jordan frame scalar field
with a double well potential, we consider that the system is in a broken
symmetric phase. As energy density increases we study the scenario that this
Jordan frame scalar field goes through a phase transition where the non-
minimal coupling bounces off its conformal value with zero mass. We find that
$\dot{H}_{E}>0$ appears naturally in the broken symmetric phase leading to a
bounce into the symmetric phase. In the next section we recapitulate the
correspondence of Eq.(1) in Jordan frame. We consider the scaling region in
the broken symmetric and the symmetric phase, the bounce and the subsequent
field roll in the effective potential in the symmetric phase. We end with a
conclusion and discussion.

###  I.1 The Jordan frame double well potential

Starting with a Jordan frame action

$S_{J}=\int
d^{4}x\sqrt{-g_{J}}\bigg{(}\frac{1}{2}F(\phi)M_{p}^{2}R_{J}+\frac{K}{2}g_{J}^{\mu\nu}\partial^{\mu}\phi\partial_{\mu}\phi-
V_{J}(\phi^{2})\bigg{)},$ (9)

the Einstein frame action is given by (see for example darabi )

$S_{E}=\int
d^{4}x\sqrt{-g_{E}}\bigg{(}\frac{1}{2}M_{p}^{2}R_{E}-\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi-\frac{V_{J}(\phi)}{F^{2}}\bigg{)}$
(10)

where

$F=\phi^{2}=e^{c\varphi};\>\>\>\frac{\phi_{,\mu}}{\phi}=\frac{c}{2}\varphi_{,\mu},$
(11)

for $K\leq 6$ (where $K=6$ is the conformal coupling value), where the
positive kinetic energy corresponds to $K<0$. The canonical field is related
to Jordan frame field by

$\varphi_{,\mu}=\phi_{,\mu}\sqrt{\frac{3}{2}\bigg{(}\frac{F_{,\phi}}{F}\bigg{)}^{2}-\frac{K}{F}}=\bigg{(}\sqrt{6-K}\bigg{)}\frac{\phi_{,\mu}}{\phi}\>\>\>.$
(12)

Here $"c"$ is no longer a free parameter but is related to the non-conformal
coupling $"K"$ by the equation

$c=\frac{2}{\sqrt{6-K}}.$ (13)

such that for near conformal coupling value $K\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}6$, the parameter
$c$ takes on a very large value.

In Scalar-Tensor theory modelling of cosmological evolution from high energy
density, as energy density $\rho$ reduces, $K(\rho)<6$ is usually taken as a
fine-tunable parameter, while the parameters in potential $V_{J}$ is either
taken to be fixed or to vary with $K$, see for example wetterich2014 .

We consider a a double well potential in Jordan frame,

$V_{J}=-\frac{1}{2}m^{2}\phi^{2}+\lambda\phi^{4}=F(\phi)^{2}V(\varphi)$ (14)

its Einstein frame action is given by

$S_{E}=\int
d^{4}x\sqrt{-g_{E}}\bigg{(}\frac{1}{2}M_{p}^{2}R_{E}-\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi+V_{E}\bigg{)},\>\>V_{E}=\frac{V_{J}}{F^{2}}=-\frac{1}{2}m^{2}e^{-c\frac{\varphi}{M_{p}}}+\lambda$
(15)

where the potential $V_{E}$ matches the plateau potential of Eq.(1) if we set
$\lambda=V_{1}$ and $\frac{1}{2}m^{2}=V_{0}$. The positive potential with
$V_{0}=-\frac{1}{2}m^{2}$ is the simplest choice in many scalar-tensor
theories. wetterich2014

In the Ekpyrotic phase where $a(\tau)\sim\>constant$, the Einstein frame
equation of state parameter is given by

$\epsilon=\epsilon_{E}=\frac{c^{2}}{2}=\frac{2}{6-K}.$ (16)

Although our primary attention is in the Einstein frame dynamics, the Jordan
frame dynamics, especially the scaling behaviour of the parameters play a
critical role. The equation between $H_{J}$ and $H_{E}$ is given as

$a=\sqrt{F}a_{J};\>\>\>dt=\sqrt{F}dt_{J};\>\>H_{E}=\frac{a_{,t}}{a}=\frac{1}{\sqrt{F}}\bigg{(}H_{J}+\frac{F_{,t_{J}}}{2F}\bigg{)},\>\dot{\phi}=\frac{d\phi}{dt}.$
(17)
$H_{J}+\frac{c}{2}\dot{\varphi}\phi=H_{J}+\frac{\dot{\phi}}{\phi}=H_{E}\phi.$
(18)
$H_{E}=\pm\sqrt{\bigg{(}1-\frac{K}{6}\bigg{)}\frac{\dot{\phi}^{2}}{\phi^{2}}+\frac{1}{3\phi^{4}}\bigg{(}-\frac{1}{2}m^{2}\phi^{2}+\lambda\phi^{4}\bigg{)}}=\pm\sqrt{\frac{1}{6}\dot{\varphi}^{2}-\frac{1}{6}m^{2}e^{-c\varphi}+\frac{1}{3}\lambda}.$
(19)

We note that from Eq.(19) that the ”Pre-Ekpyrotic” phase of khoury -khoury1
corresponds to a stable $3H_{E}^{2}=\lambda$ where the sum of the negative
potential and the kinetic term is small comparing to $\lambda$

$\frac{m^{2}}{2}e^{-c\varphi}\sim\frac{1}{2}\dot{\varphi}^{2}\ll\lambda.$ (20)

In Jordan frame, this corresponds to a nearly conformal coupling $K\simeq 6$
and a large $\phi^{2}$ for some fixed $m^{2}$. As $\varphi$ (and $\phi^{2}$)
decreases, when the negative potential dominates over $\lambda$, one enters
the ”Ekpyrotic phase” .

At the beginning of the Ekpyrotic phase, starting with a large positive field
($\varphi>0$), a negative field speed ($\dot{\varphi}<0$) and large negative
time $t$, the evolution of system in Einstein frame follows the work of khoury
-khoury1  with solution in Eq.(3). From Eq.(16), the unnaturally large value
$c\sim 10^{28}$ can be easily understood in Eq.(13) as $K$ having a near
conformal coupling value. As $K$ moves away from the conformal coupling value
to $6>K>16/3$, we have $\epsilon>3$ towards the end of Ekpyrotic phase. From
Eq.(11) we take the corresponding starting $\phi$ at $\phi>0$, with
$\dot{\phi}=(c\phi)\dot{\varphi}<0$. In this scenario the Jordan frame field
$\phi$ rolls down from a large positive field to its bottom (corresponding to
the Einstein frame pre-Ekpyrotic) and rolls up the potential towards the
$\phi=0$ (corresponding to the Ekpyrotic and Kinetic energy phase). At the end
of Ekpyrotic phase, $K$ is still not far away from the conformal coupling
value. We note that scale invariant curvature spectrum is also obtained in li
-piao2  using Jordan frame model with nearly scale invariant potentials,
though this spectrum are inflation related.

After the Ekpyrotic phase, the scale factor continues to shrink in the
Einstein frame (which is also true in Jordan frame for slow varying $\phi>0$).
The Hubble constant increases towards the Planck scale. As the canonical
kinetic term dominates over the potential with decreasing scale factor, the
energy density which scales as $\sim a^{-6}$ can only continue to increase
since none of the known cosmological components can slow it down. To avoid
reaching the Planck energy density, one way is to consider the existence of a
high energy density fixed point below the Planck scale. Scalar-Tensor theories
in the form of Eq. (9) are used to describe the universe’s evolution from very
high energy densities to low energy densities over a large energy density
scales. Conformally invariant fixed point $K=6$ and $m^{2}=0$ provides an
important solution. In wetterich2 , for a positive potential

$V_{J}(\phi)=+\frac{1}{2}m^{2}\phi^{2}+\lambda\phi^{4},$ (21)

it is found that as $\frac{m^{2}}{\phi^{2}}\rightarrow 0$, the system under
functional renormalisation group flow will reach the conformally invariant
fixed point described above. In principle, the equilibrium renormalised field
$\phi$ remains finite, however, at finite $-\varphi$, for $c$ in
$c\varphi=\ln\phi^{2}$ to reach the conformal value ($c\rightarrow\infty$) it
is also required that $\phi^{2}\rightarrow 0$ as $m^{2}\rightarrow 0$. This
conformally invariant fixed point can be conformally transformed in Einstein
frame to the Einstein-Hilbert action, which is also to be an UV fixed point
for gravitational constant and cosmological constant flows wetterich2
-litim2004 .

With an unimpeded energy density rise and a double well potential, it is not
hard to imagine that the system could go through a phase transition at the
conformal point into a symmetric phase with potential described by Eq.(21). A
second order phase transition in quantum gravity around some critical field
value $\phi^{2}_{c}$ has also been proposed in falls . In a flat space
$R_{J}=0$ scenario, this is the most common mechanism used for phase
transition studies luscher . In cosmological evolution, as the energy scale
varies across a large range that a scalar field with a given symmetry is
expected to go through phase transitions, which could lead to inflation guth2
as well as in the production of cosmological defects kibble .

More specifically, we consider the scenario that as the energy density
increases the system enters into the scaling (Ginsberg) region of a phase
transition, in which system parameters follow scaling laws. Its broken
symmetric phase (renormalised) mass term $-m^{2}\phi^{2}<0$ in Eq.(14) will go
through zero into the symmetric phase scaling region with $+m^{2}\phi^{2}>0$.
If the phase transition fixed point is effectively at the conformal invariant
fixed point, as the renormalised mass goes through zero, the coupling $K$
approaches the conformal coupling limit and then moves away. The end result is
that in Einstein frame, the mass term $m^{2}(\rho)$ has an energy density
dependence, with a deriative w.r.t to $t$ and $\rho$ given by

$m^{2}_{,t}=m^{2}_{,\rho}\dot{\rho}(a).$ (22)

When $\rho$ increases over time, we have $\dot{\rho}>0$ and the renormalised
mass flow is $m^{2}_{,\rho}<0$. As $c^{2}\propto(6-K)^{-1}$, the corresponding
rate of change for $c$ is $c_{,\rho}>0$ towards the critical point.

###  I.2 Scaling region in the Broken symmetric phase

From the above discussion, in the scaling region of the broken symmetric
phase, we write that the mass term is dependent on the energy density of the
Einstein frame $\rho$, as

$V_{J}=-\frac{1}{2}m^{2}(\rho)\phi^{2}+\lambda\phi^{4}$ (23)

so that

$3H_{E}^{2}=\frac{1}{2}\dot{\varphi}^{2}-\frac{1}{2}m^{2}(\rho)e^{-c\varphi}+\lambda.$
(24)

Keeping in mind that it is the parameters in Jordan frame that scale as energy
density increases, we differentiate Eq. (24) w.r.t the time in Einstein frame
$t$,

$3\frac{d}{dt}(H_{E}^{2})=\dot{\varphi}\ddot{\varphi}+V_{,\varphi}\dot{\varphi}-\frac{1}{2}\frac{m^{2}_{,t}}{\phi^{2}}-\frac{1}{2}\frac{m^{2}}{\phi^{2}}\bigg{(}-\dot{c}\varphi\bigg{)}+\dot{\lambda}$
(25)
$\dot{H}_{E}=-\frac{1}{2}\dot{\varphi}^{2}-\frac{1}{12H_{E}}\bigg{[}\frac{m^{2}}{\phi^{2}}\bigg{(}\frac{m^{2}_{,t}}{m^{2}}-\dot{c}\varphi\bigg{)}-2\dot{\lambda}\bigg{]}.$
(26)

We note that

$m^{2}_{,\rho}=\frac{dm^{2}}{d\rho},\>\>\>c_{,\rho}=\frac{dc}{d\rho},\>\>\>\lambda_{,\rho}=\frac{d\lambda}{d\rho}.$
(27)

where respectively $m^{2}_{,\rho}$ and $\lambda_{,\rho}$ are the renormalised
mass flow and renormalised coupling flow due to energy density increases,
while $c_{,\rho}$ is a result of the non-conformal coupling flow towards the
conformal value. We can rewrite Eq.(26) as

$\dot{H_{E}}=-\frac{1}{2}\dot{\varphi}^{2}-\frac{\dot{\rho}}{12H_{E}}\bigg{[}\bigg{(}\frac{m^{2}_{,\rho}}{m^{2}}-c_{,\rho}\varphi\bigg{)}\frac{m^{2}}{\phi^{2}}-2\lambda_{,\rho}\bigg{]}$
(28)

We assume that scaling laws here is qualitatively similar to the flat space
canonical $\lambda\phi^{4}$ theory luscher , we have

$m^{2}=a_{0}(\rho_{c}-\rho)^{l};\>\;\>\>\;\>\lambda=\lambda_{0}(\rho_{c}-\rho)^{\gamma}.$
(29)

where $a_{0}>0$ and $\lambda_{0}>0$ are constants, and the exponent $l>0$ and
$\gamma>0$ describes the effect of mass renormalisation and quartic coupling
renormalisation respectively. (We note that in 4-dimensions, the mean field
value for $l$ is $l_{MF}=1$ for the flat space $\lambda\phi^{4}$ theory, and
it is also possible that $\lambda$ goes to zero logarthimically.

$\lambda=-\frac{\lambda_{0}}{\ln(\rho_{c}-\rho)}.$ (30)

The possibility that $\lambda$ is an non-trivial does not alter the result of
the analysis below.)

In Einstein frame, for the potential to rise as energy density increases
towards the phase transition, the mass term $m^{2}/\phi^{2}$ will need to
reduce towards zero. Also at finite $\varphi<0$ with $c\rightarrow\infty$
towards the phase transition. A consistent choice for $\phi$ and $c$ will be

$\phi^{2}=b_{0}(\rho_{c}-\rho)^{q},\>\>\>\>\;\>c=\frac{1}{\varphi}\ln\phi^{2}\propto
q\ln(\rho_{c}-\rho).$ (31)

(We note that $\phi^{2}$ can also dimishes logarithmically as
$\phi^{2}\propto-(\ln(\rho_{c}-\rho))^{-1}$, but this also does not change the
result of the following analysis.)

For the exponent $l>q>0$, as $\rho\rightarrow\rho_{c}$ from below, the mass
term

$\frac{m^{2}}{\phi^{2}}=\frac{a_{0}}{b_{0}}(\rho_{c}-\rho)^{(l-q)}\rightarrow
0.$ (32)

Also we have

$\frac{m^{2}_{,\rho}}{m^{2}}=\frac{-l}{(\rho_{c}-\rho)};\>\;\>\frac{2\phi_{,\rho}}{\phi}=\frac{-q}{(\rho_{c}-\rho)};\>\>\>\lambda_{,\rho}=\frac{-\gamma\lambda}{\rho_{c}-\rho}.$
(33)

As $c$ diverges logarthimically, we have near the critical point

$c_{,\rho}\varphi=2\frac{\phi_{,\rho}}{\phi}.$ (34)

We note that the energy density of the patch with scale factor $a(t)$ is
$\rho(a)=a^{-2\epsilon}$. Its time derivative is given by

$\dot{\rho}=\dot{\epsilon}\frac{d}{d\epsilon}a^{-2\epsilon}=-2\epsilon\rho
H_{E}-2\dot{\epsilon}\rho\ln\bigg{(}\frac{a}{a_{p}}\bigg{)},$ (35)

where $a_{p}$ is some reference scale that $a>a_{p}$. The equation of state
parameter $\epsilon$ from Eq.(5) in Einstein Frame is given by in the broken
symmetric phase

$\epsilon=\frac{3}{2}\bigg{(}1+\frac{P}{\rho}\bigg{)}=3\frac{\dot{\varphi}^{2}}{\dot{\varphi}^{2}+2V(\varphi)}=3\frac{\dot{\varphi}^{2}}{\dot{\varphi}^{2}-\frac{m^{2}}{\phi^{2}}+2\lambda}.$
(36)

Starting with large field speeds and a decreasing mass term $m^{2}/\phi^{2}$,
in the scaling region $\epsilon$ will remain at $\epsilon>3$ and we can assume
that $\dot{\epsilon}\sim 0$. For a constant $\epsilon$, from Eq.( 28) we
obtain

$\dot{H}_{E}=-\frac{1}{2}\dot{\varphi}^{2}+\frac{1}{6}\frac{\epsilon\rho}{(\rho_{c}-\rho)}\bigg{(}(q-l)\frac{m^{2}}{\phi^{2}}+2\gamma\lambda\bigg{)}.$
(37)

(If $\phi^{2}\rightarrow 0$ logarithmically, we have $q=\phi^{2}<1$.) In flat
space $\lambda\phi^{4}$ theory, the exponent $\gamma$ of the renormalised
coupling $\lambda$ is found to be very small. It is reasonable to assume that
$\gamma$ is smaller than the exponent of the mass term in Eq.(32), so that
$\gamma\ll(l-q)$. In this case, in the scaling region far away from the
critical point, the $m^{2}/\phi^{2}$ term dominates over the $\lambda$ term,
$\dot{H_{E}}$ remains negative. As the $m^{2}/\phi^{2}$ term diminishes
rapidly near the critical point, at some point the $\lambda$ term in Eq.(37)
dominates, (taking initially $\dot{\varphi}^{2}\sim\rho$,) where $\dot{H}_{E}$
becomes positive. A small $\lambda$ and $\gamma$ would requires
$\dot{H}_{E}>0$ to occur very close to the critical point.

###  I.3 Scaling region in the Symmetric phase

Kibble kibble -kibble2  considers that in a large physcial system going
through a phase transition, the correlation length does not have enough time
to reach infinity before the phase transition is completed. In this scenario,
phase transition occurs when the mass term $-\frac{1}{2}m^{2}\varphi^{2}$ is
frozen and simply flips to $+\frac{1}{2}m^{2}\varphi^{2}$ after a finite time.
For a small Hubble radius this may be problematic. If the cut-off scale factor
$a_{p}$ occurs at Planck scale ($\sim(10^{19}GeV)^{4}$), a phase transition at
energy scale $\sim(10^{14}GeV)^{4}$ may involve finite size effects or
modification of scaling laws. For our purpose, as in flat space the quartic
coupling exponent $\gamma$ is much smaller than the exponent of the mass term
$l-q$, we assume that the modification of scaling exponents does not affect
our result. We also note that when the Hubble radius becomes nearly infinite
around the phase transition point, the Kibble mechanism is perfectly valid
around this region.

In the symmetric phase, the Einstein frame Hubble constant turns positive,
$H_{E}>0$. We consider a scenario that for a short time the energy density can
continue to grow such that $\dot{\rho}>0$. In the symmetric phase the
potential is given by

$V(\varphi)=\frac{1}{2}m^{2}(\rho)e^{-c\varphi}+\lambda$ (38)

with Hubble parameter

$3H_{E}^{2}=\rho(a)=\frac{1}{2}\dot{\varphi}^{2}+\frac{1}{2}m^{2}e^{-c\varphi}+\lambda;\>\;\>\varphi<0.$
(39)

We assume for convenience that the scaling laws on both sides of the phase
transition are similar (which is true if the model is in flat space), $m^{2}$
and $\lambda$ scales as

$m^{2}\propto(\rho-\rho_{c})^{l};\>\>\>\>\lambda\propto(\rho-\rho_{c})^{\gamma};\>\>l>0,\>\gamma>0.$
(40)

Assume also that

$\phi^{2}\propto(\rho-\rho_{c})^{q},\>\>\>c=\frac{1}{\varphi}\ln\phi^{2}.$
(41)

We have

$\dot{H}_{E}=-\frac{1}{2}\dot{\varphi}^{2}+\frac{\dot{\rho}}{12H_{E}}\bigg{[}\bigg{(}\frac{m^{2}_{,\rho}}{m^{2}}-\frac{2\dot{\phi}}{\phi}\bigg{)}\frac{m^{2}}{\phi^{2}}+2\gamma\lambda\bigg{]}.$
(42)
$\frac{m^{2}_{,\rho}}{m^{2}}=\frac{l}{\rho-\rho_{c}},\>\>\frac{2\dot{\phi}}{\phi}=\frac{q}{\rho-\rho_{c}},\>\>\lambda_{,\rho}=\frac{\gamma\lambda}{\rho-\rho_{c}}.$
(43)
$\dot{H}_{E}=-\frac{1}{2}\dot{\varphi}^{2}+\frac{\dot{\rho}}{12H_{E}}\frac{1}{(\rho-\rho_{c})}\bigg{(}(l-q)\frac{m^{2}}{\phi^{2}}+2\gamma\lambda\bigg{)}$
(44)

We see that for $H_{E}>0$ and $\dot{\rho}>0$, $\dot{H}_{E}>0$ is maintained
for a very small $\dot{\varphi}^{2}$.

###  I.4 The bounce

We recall the Eq.(36)

$\dot{\rho}=\dot{\epsilon}\frac{d}{d\epsilon}a^{-2\epsilon}=-2\epsilon\rho
H_{E}-2\dot{\epsilon}\rho\ln\bigg{(}\frac{a}{a_{p}}\bigg{)}.$ (45)

The Friedmann equation $3H_{E}^{2}=\frac{1}{2}\dot{\varphi}^{2}+V(\varphi)$
suggests that if $H_{E}$ goes to zero continuously, $H_{E}=0$ will be reached
in the broken symmetric phase where $V(\varphi)<0$.

If this occurs in a region where $\epsilon>3$ is still nearly constant, as
$H_{E}$ reaches $H_{E}=0_{-}$, we have

$\dot{\rho}=0;\>\>\dot{H}_{E}=0.$ (46)

The $\dot{H}_{E}>0$ term in Eq.(37) is switched off completely. At this point
$H_{E}$ goes back to the background energy density

$H_{E}=+\sqrt{\frac{\rho}{3}}.$ (47)

(We do not consider the solution that $H_{E}$ bounces back to $H_{E}<0$.)

More consistently, at $H_{E}=0$, we have the equation of state parameter
$\epsilon\rightarrow\infty$. The system can continue to move into the
symmetric phase, with $H_{E}>0$, as long as Eq.(45) remains valid. That is

$\frac{\dot{\rho}}{H_{E}}=-2\epsilon\rho\bigg{(}1+\frac{1}{H_{E}}\frac{\dot{\epsilon}}{\epsilon}\ln\bigg{(}\frac{a}{a_{p}}\bigg{)}\bigg{)}>0.$
(48)

That is the energy density can continue to increase $\dot{\rho}>0$ if
$\dot{\epsilon}<0$ and $a>a_{p}$. This is possible due to the fact that the
value of $\epsilon$ comes down from a very large value to $\epsilon<3$ as
$H_{E}$ passes through $H_{E}=0$. At some point where

$H_{E}=-\frac{\dot{\epsilon}}{\epsilon}\ln\bigg{(}\frac{a}{a_{p}}\bigg{)},$
(49)

the system reaches $H_{E}>0$ and $\dot{H}_{E}=0$ in the symmetric phase. The
bounce from this point leads naturally to an expanding phase. When the exit
potential is $V_{E}<\frac{1}{2}\dot{\varphi}^{2}$, we have $\epsilon>2$ the
system enters the radiation epoch directly. If $V_{E}>\dot{\varphi}^{2}$, we
have $\epsilon<1$ and the system will go through a short inflationary epoch.

We also notice that near the bounce where $H_{E}\sim 0$ and $\dot{H}_{E}\sim
0$, the Einstein frame Scalar curvature $R_{E}=12H_{E}^{2}+6\dot{H}_{E}$ is
small. One can also show that as the field roll $\dot{\phi}$ nearly stops that
$R_{J}\sim 0$. At this point, the background metric is nearly flat and for a
$\lambda\phi^{4}$ theory the phase transition structure around a critical
point is well understood luscher .

###  I.5 After the bounce

After the bounce, the field speed relation between Einstein frame and Jordan
frame is given by

$\dot{\varphi}=\frac{2}{c}\frac{\dot{\phi}}{\phi}+\frac{\dot{c}}{c}\varphi.$
(50)

Consider the point where $-\varphi\sim O(1)$. We take that $c$ remains
positive, and that the Jordan frame field lands at $0<\phi<1$ in a symmetric
potential. For $\phi$ to roll down the symmetric potential, we need the
kinetic energy term to have the right sign $K<0$ (and $c^{2}<1/3$), one has
$\frac{\dot{c}}{c}\ll 1$, the negative Einstein frame field speed
$\dot{\varphi}<0$ corresponds to $\dot{\phi}<0$ (and
$\frac{d\phi}{dt_{J}}<0$), so that $\varphi$ will continue its roll to the
$\varphi\rightarrow-\infty$ while $\phi$ rolls towards $\phi\sim 0$. It is
important to note that as $\varphi\rightarrow-\infty$, the Einstein frame
potential

$V(\varphi)=\frac{1}{2}m^{2}(\rho)e^{-c\varphi}+\lambda$ (51)

reduces to $\lambda$ asymptotically, only when the $m^{2}$ term has an energy
density dependence, so that as $\rho\rightarrow 0$, $m^{2}(\rho)\rightarrow 0$
faster than $e^{-c\varphi}=\phi^{-2}\rightarrow\infty$. This is possible due
to the fast energy density reduction in the expanding phase, and the Jordan
frame system parameters flow into a conformal infra-red fixed point according
to some scaling laws similar to that in the symmetric phase above. We recall
that a conformal Infra-red fixed point for the positive potential in Eq.(51)
is found in wetterich2 . That a non-minimal coupling model with $K\sim 6$ at
late time universe has recently found observational support hrycyna .

The field acceleration

$\ddot{\varphi}=-3H_{E}\dot{\varphi}-V_{,\varphi}=-3H_{E}\dot{\varphi}+\frac{1}{2}cm^{2}e^{-c\varphi}$
(52)

shows that both terms on the R.H.S. provide friction on the field speed
$\dot{\varphi}<0$. A sufficently slow $\dot{\phi}$ allows for a choice of the
Newtonian constant $G=\phi^{-2}$. If the bare theory is scale invariant, the
Jordan frame field potential in principle should change gradually into a scale
invarant effective potential as energy density reduces. A symmetry breaking
induced fixed minimum $\phi=\phi_{0}$ can be provided by the Coleman-Weinberg
mechanism ferreira -ferreira3  or more fundamentally by non-canonical
quantisation fubini -fubini2 .

###  I.6 Summary and Discussion

A lifted plateau potential has observational support in the inflationary
paradigm. Khoury and Steinhardt show that from the same plateau potential, one
can obtain similar red-tilted curvature spectrum from a contracting universe
scenario. Once inside the kinetic dominant phase, some new physics is required
to effect a connection into the familiar expanding universe phase. Although
there are now some potentially viable mechanisms, they are highly non-trivial
in comparision to that of a slow roll inflation. Here we start from the post-
Ekpyrotic phase of Khuory and Steinhardt. In Jordan frame, the plateau
potential resembles a double well potential. A key observation is that
phenomenologically the parameter $c$ takes an unnaturally large value at pre-
ekpyrotic phase and crosses many orders of magnitude to reach $O(1)$ at the
end of Ekpyrotic phase, which in Jordan frame simply means the non-minimal
coupling is initially very close to the conformal coupling value but moves
away mildly towards the end of Ekpyrotic phase. This indicates that conformal
coupling plays a crucial role in this Ekpyrotic model. The kinetic energy
dominance provides an unimpeded source of energy density increase, so that in
principle if there exists a phase transition point below the Planck energy, it
will eventually be reached. We further note from the work of Henz et al. that
as energy density decreases this Jordan frame model with a positive mass term
in the potential admits a conformal invariant fixed point at zero mass. A
cosmological phase transition for a scalar field is a much studied system, in
which there are regions at both sides of phase transition in which system
parameters scale adiabatically with scaling laws. As energy density passes the
critical energy density, the parameters will freeze and the mass parameter
changes sign. Within this paradigm, we show that we can obtain $\dot{H}_{E}>0$
naturally near the phase transition point. As the equation of state parameter
goes to infinity at $H_{E}=0$, its rapidly decline leads to a continuation of
energy density increase into the symmetric phase where $H_{E}>0$. As the
equation of state parameter decline finishes, the energy density increase
stops and the bounce occurs which takes $H_{E}>0$ to its expanding background
energy density. The subsequent field roll to minus infinity in Einstein frame
resembles that in the standard Ekpyrotic paradigm provided that the low energy
state is also a conformal invariant Infra-red fixed point in Jordan frame,
which is consistent with the solution found in Henz et al. In this case, the
Einstein frame asymptotic potential value is $\lambda$ which is the Jordan
frame renormalised quartic coupling which is naturally very small near a
conformal invariant fixed point, regardless whether it is trivial or non-
trivial.

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