%Date: Fri, 22 Oct 1993 16:24:02 -0600
%From:     kevin%cahill.unm.edu@Sdsc.BITnet (Kevin Eric Cahill)
%To:       podolsky%cspuni12.BITNET@Sdsc.BITnet
%Subject: good news
%cc:       kevin%cahill.unm.edu@Sdsc.BITnet

%Dear Jiri,

%   Congratulations on your thesis.  Dan Finley tells me you did very well.

%   J Phys G may well publish our paper.They faxed me today two referee reports.
%One was favorable and the other asked us to clarify a point of physics.

%   Basically the point was that in our eq.(13) rho is related to V as internal
%energy is related to free energy at finite temperature.
%But V is T_{00}; and at finite temperature, T_{00} is the same as internal energy.
%So how can V be both free energy and internal energy? These phrases are the best
%I can do with what the referee wrote.

%   I think I can look at Kolb & Turner and puzzle out the matter.
%But do e-mail me your thoughts on it.%

%                              Best wishes,
%                              Kevin

%=============================================================================

%Date: Tue, 26 Oct 1993 23:48:11 -0600
%From:     kevin%cahill.unm.edu@Sdsc.BITnet (Kevin Eric Cahill)
%To:       podolsky%cspuni12.BITNET@Sdsc.BITnet
%Subject: G/47039/PAP revised

%                            26 October 1993
%
%   Dear Ms. Grimes:
%
%      Thanks you for your kind letter of 21 October
%   about the paper G/47039/PAP by Podolsky and me.
%
%      You asked me to respond to the comments of the
%   first referee.  Let me do so now:
%
%      The referee points out that we have treated
%   the scalar potential $V$ as a free energy.  He
%   has pointed out a confusion due to our condensed
%   notation.  In fact the zero-temperature effective
%   potential $V_0(\phi)$ to lowest order is the scalar
%   potential, but the finite-temperature effective
%   potential $V(\phi,T)$ is the Gibbs free energy
%   of the particle theory.  The relation (13) that
%   expresses the energy density $\rho$ in terms of
%   the finite-temperature effective potential $V(\phi,T)$,
%   a.k.a. Gibbs free energy, is quite standard, being
%   used, as it is, by Kolb \& Turner (The Early Universe), p.198;
%   by S.~Weinberg (PRD 9 (1974) 3357, 3372, \& 3374); and
%   by Sher (Phys. Rep. 179 (1989) 326 and 327.
%
%      Since our notation is confusing even to someone
%   as learned as the referee, we have added language that
%   makes it clear that $V(\phi,T)$ is the Gibbs free energy
%   and not the scalar potential.
%
%      The explicit changes (apart from punctuation) are as follows:
%
%................................................................
%
%251,253c245,246
%< the Higgs field $\phi$ lies at a local minimum $\phi_0$
%< of the effective potential with free energy
%< $V(\phi_0,T) \approx V_0 - \pi^2 N T^4 /90$.
%---
%> the Higgs field lies at a local minimum $V_0$
%> of the effective potential with free energy
%>$A(T) = V_0 - \pi^2 N T^4 /90$.
%272,274c265,268
%< The effective potential $V(\phi,T)$
%< is the Gibbs free energy of the particle theory;
%< it is a function of the Higgs field and the temperature.
%---
%> The effective potential
%> $V(\phi,T)$ is a function
%> of the Higgs field
%> and the temperature.
%309c303
%< mean value $\phi_c \equiv \phi$ of the Higgs field.
%---
%> magnitude $\phi$ of the Higgs field.
%373d366
%< which is the Gibbs free energy,
%798c791,792
%< \item{[9]} K.~Cahill, hep-ph/9301294.
%---
%> \item{[9]} K.~Cahill, {\sl Proceedings of the XXVIth International
%>   Conference on High-Energy Physics\/} (AIP, in press),
%> hep-ph/9210273.
%
%.................................................................
%
%      What follows is the revised paper.
%
%      Please let me know if you have received this TeX
%   file in reasonable shape.
%
%      The paper is written in plain TeX.
%
%                            Sincerely yours,
%                            Kevin Cahill
%
%
%

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\centerline{  }
\bigskip
\bigskip
\cl{\bf Inflation Pressures}
\bigskip
\bigskip
\cl{Kevin Cahill and Ji\v r\'\i\ Podolsk\'y \footnote*
{\tenpoint \baselineskip=0.17 in
Present address:
Department of Theoretical Physics,
Faculty of Mathematics and Physics,
Charles University, V Hole\v sovi\v ck\'ach 2,
180 00 Praha~8, Czech Republic.}}
\bigskip
\cl{Department of Physics and Astronomy, University of New Mexico}
\cl{Albuquerque, New Mexico 87131--1156}
\medskip
\cl{kevin@cahill.unm.edu and podolsky@cspuni12.bitnet}
\bigskip
\bigskip
\bigskip
\cl{14 April 1993}
\bigskip
\bigskip
\bigskip
\bigskip
\cl{\bf Abstract}
\medskip
\midinsert
\narrower\narrower
If the sum of the energy density $\rho$
and thrice the pressure $p$ is negative,
then a Robertson--Walker universe can inflate
or even oscillate.
But we show at the one-loop level
that in a generic grandly unified theory
$\rho + 3 p$ cannot be negative
at the absolute minimum
of the finite-temperature effective potential.
We discuss the implications of this result
for inflation in two grandly unified theories
and speculate on the effects of gravitational friction
and anti-friction.
\endinsert
\goodbreak
\vfill\eject
\bigskip
\centerline{\bf The Inflation Pressure}
\par
In the early universe,
the sum of the energy density $\rho$
and three times the pressure $p$
is a crucial quantity.
In a homogeneous and isotropic universe,
the sum $\rho + 3 p$ determines
the acceleration $\ddot R$ of the scale factor $R$
$$ {\ddot R \over R} = - {4 \pi G \over 3} (\rho + 3 p). \n $$
Inflation, which is characterized by an exponentially increasing
scale factor, can occur only when $\rho + 3 p$ is negative.
If $\rho + 3p$, which might be called the {\it inflation pressure\/},
is negative and constant,
then the scale factor $R$ may inflate~[1--4] as
$$R(t) = e^{Ht} R_0 \n $$
with $H^2 = -4\pi G(\rho+3p)/3$.
But when the inflation pressure $\rho + 3 p$ is positive,
the universe cannot inflate.
\par
The inflation pressure
also determines whether a uniform universe
has had an initial singularity.
If $\rho + 3 p$ has always been positive,
then $\ddot R$ has always been negative
and the velocity $\dot R$ of the scale factor
must always have been greater than at present.
Thus at some finite time in the past
the scale factor $R$ must have vanished: there was an
initial singularity.
\par
The opposite scenario can occur
in a contracting universe
if the inflation pressure is negative and
the Higgs field $\phi$ lies at a local minimum $\phi_0$
of the effective potential with free energy
$V(\phi_0,T) \approx V_0 - \pi^2 N T^4 /90$.
%$$A(T) = V_0 - {\pi^2 N \over 90} T^4 . \n $$
In this case the scale factor may vary inversely with
the temperature, $R = a/T$;
and the universe may exhibit deflation followed by inflation
according to the nonsingular solution~[5]
$$ R(t) =  R_1 \left[ 1 + (1 - x^2)^{1 \over 2}
\cosh {\sqrt{2} t \over R_1} \right]^{1 \over 2}. \n  $$
For typical values of $V_0$, the constants
$R_1 = \sqrt{3/(16\pi G V_0)}$
and $x = 2 (2\pi a/3)^2 \sqrt{6 N V_0 / 5}$
are such that the minimum radius
of this bounce solution
exceeds the Planck length by a factor of a million or more.
\par
In this paper we shall discuss
the inflation pressure $\rho + 3p$ in terms of
the one-loop, finite-temperature effective potential~[6]
for a generic grandly unified model~[7].
The effective potential $V(\phi,T)$
is the Gibbs free energy of the particle theory;
it is a function of the Higgs field and the temperature.
We shall show that whenever the Higgs field
is at the absolute minimum of $V(\phi,T)$,
then the inflation pressure $\rho + 3p$ cannot be negative,
which implies that $\ddot R < 0$.
This result is a constraint upon
inflation and particularly upon oscillation.
\par
We then numerically describe the behavior of the inflation pressure
in two simple, grandly unified models
and the implications of that behavior for inflation.
For the SU(5) model of Albrecht and Steinhardt~[4,8],
inflation begins when the temperature has dropped
below $T_i = 0.0755\sigma$, where $\sigma$
is the scale of grand unification.
In a generic supersymmetric grandly unified model,
inflation cannot start until the temperature is below
$T_i = 0.123\sigma$ and then not unless the Higgs
field lies in the region of negative inflation pressure.
\par
Finally we argue that the effects of gravitational
friction and anti-friction may allow some portions
of a locally nonuniform, contracting universe
to avoid a singularity and reexpand.
\goodbreak
\bigskip
\medskip
\centerline{\bf The Finite-Temperature Effective Potential}
\par
We shall assume that the breaking and
restoration of symmetry can be described
by a finite-temperature Higgs mechanism~[6]
in a generic model of particle physics
with grand unification~[7] at an energy scale $\sigma$.
For simplicity we shall consider only the
mean value $\phi_c \equiv \phi$ of the Higgs field.
We shall also restrict ourselves
to the case of weak Higgs self coupling so that we can ignore
the contributions of the scalar loops,
which require special treatment~[9].
If the Higgs field is described at the one-loop
level by the zero-temperature effective potential
$V_1(\phi)$, then for finite temperatures the
one-loop effective potential $V(\phi,T)$ is of the form
$$V(\phi,T) = V_1(\phi) + T^4 I(\phi^2/T^2).\n$$
Here the one-loop, zero-temperature
effective potential $V_1(\phi)$
is the sum of the classical potential $V_0(\phi)$
and the one-loop contributions
$$V_1(\phi) = V_0(\phi) +
\sum_i {(-1)^{2j_i}(2j_i+1) \over 64\pi^2}
m_i^4(\phi)\log(\phi^2/\sigma^2), \n $$
one for each fermion and gauge boson of spin $j_i$
and mass $m_i(\phi) = c_i \phi$,
with Dirac fermions being counted as two two-component
spinors.
The function $I(\phi^2/T^2)$ is a similar sum of integrals
$$I(\phi^2/T^2) = \sum_i {(-1)^{2j_i}(2j_i+1) \over 2\pi^2}
\int_0^\infty \! x^2
\log\left( 1 - (-1)^{2j_i} e^{-(x^2+(m_i(\phi)/T)^2)^{1/2}}
\right)dx . \n $$
It is strictly negative for all $\phi$ and $T$,
$$I(\phi^2/T^2) < 0 . \n $$
The product $T^4 I(\phi^2/T^2)$ contains all the temperature dependence
of the one-loop, finite-temperature
effective potential $V(\phi,T)$.
\par
If $b$ is the number of gauge bosons and $f$ the number
of fermions (with electrons and positrons counted together as one),
then at very high temperatures, $T \gg m_i(\phi) $
for all particles $i$,
we may expand the effective potential as
$$V(\phi,T) \approx V_1(\phi) - {\pi^2 N \over 90} T^4
+ { n {\bar m(\phi)}^2 T^2 \over 24} \n $$
where  $N$ is the effective number of
degrees of freedom  $N=3 b + 7 f/2$,
$n=3b+2f$, and $\bar m(\phi) = {\bar c} \phi$
is an average mass.
At such high temperatures, the absolute minimum
of the effective potential is at $\phi = 0$,
and the symmetry of the lagrangian is restored.
\par
At low temperatures, the effective potential is approximately
$$ V(\phi,T) \approx V_1(\phi) - \sum_i (2j_i+1) T^4
\left({ m_i(\phi) \over 2\pi T}\right)^{3/2} e^{-m_i(\phi)/T} , \n $$
and its minima are those of $V_1(\phi)$
which spoil the symmetry of the lagrangian.
We shall take these minima to lie at $\phi = \sigma$,
the scale of grand unification.
\goodbreak
\par
\bigskip
\medskip
\centerline{\bf The Minimum of the Effective Potential}
\par
We shall now show
that at the absolute minimum of the one-loop, finite-temperature
effective potential
$$V(\phi,T) = V_1(\phi) + T^4 I(\phi^2/T^2),\n$$
which is the Gibbs free energy,
the inflation pressure $\rho + 3 p$ cannot be negative.
Since in a homogeneous isotropic universe,
a positive inflation pressure rules out inflation,
this result is a constraint upon inflationary models.
We shall assume that in our generic grandly unified model,
the ground-state or vacuum energy vanishes
so that the one-loop, zero-temperature effective
potential $V_1(\phi)$
is positive except at its global minimum $\phi=\sigma$
where it is zero.
\par
Now the function $I(\phi^2/T^2)$
is negative as noted in eq.(7), and so
$$V(\phi,T) \leq V(\phi,0) . \n$$
Also as its argument $\phi^2/T^2$ goes to infinity,
the function $I(\phi^2/T^2)$ tends monotonically to zero
from below.
Thus its derivative $I^\prime=I^\prime(\phi^2/T^2)$ with respect
to its argument $\phi^2/T^2$ is positive,
$$I^\prime \ge 0. \n$$
In terms of $V_1=V_1(\phi)$, $I$,  and $I^\prime$,
the energy density
$$\rho = V(\phi,T) - T \partial V(\phi,T)/\partial T\n$$
is
$$\rho = V_1 - 3 T^4 I + 2 \phi^2 T^2 I^\prime \n $$
and the pressure $p = - V(\phi,T)$ is
$$p = - V_1 - T^4 I.\n  $$
So the inflation pressure $\rho + 3p$
is the sum of three terms
$$ \rho + 3p = -2 V_1 - 6T^4 I + 2 \phi^2 T^2 I^\prime . \n $$
Of these terms only the first is negative.
\par
Now if the field $\phi$ is at the
absolute minimum of the effective potential $V(\phi,T)$,
then we must have
$$ V(\phi,T) = V_1(\phi) + T^4 I(\phi^2/T^2) \leq V(\sigma,T)
\leq V(\sigma,0) = 0 , \n $$
where the second inequality follows from eq.(11).
Hence
$$-T^4 I \geq V_1 \geq 0, \n $$
and so the inflation pressure $\rho + 3p$ exceeds
the sum of two non-negative terms
$$ \rho + 3p \geq 4 V_1 + 2 \phi^2 T^2 I^\prime \geq 0 \n $$
and therefore is non-negative.
\par
Thus if the Higgs field
remains in equilibrium at the absolute minimum
of the effective potential,
then
$$\ddot R = - 4\pi G(\rho + 3p) R/3 \leq 0. \n $$
The acceleration $\ddot R$
of the scale factor is then negative or zero,
and neither inflation nor a bounce is possible.
\goodbreak
\par
\bigskip
\medskip
\centerline{\bf Implications for Inflation}
\par
Obviously this result is a constraint upon inflation.
At the very high temperatures
of an early, post-big-bang, Robertson--Walker universe,
the absolute minimum of the effective potential is
at $\phi =0$,
and the symmetry of the lagrangian is restored.
At such high temperatures, the inflation pressure
$\rho + 3p$ is typically positive
for many models of particle physics,
and inflation cannot begin
until the temperature has dropped considerably.
For two generic, grandly unified models,
we have numerically integrated the equations (4--6) and (14)
that determine the one-loop inflation pressure.
In the SU(5) model of Albrecht and Steinhardt~[4,8],
inflation begins when the temperature has dropped below
$T_i = 0.0755 \sigma$, where $\sigma = 2\times 10^{15}$ GeV
is the scale of grand unification.
In a supersymmetric model, we found that inflation
cannot start until the temperature has dropped below
$T_i = 0.123 \sigma$, where $\sigma = 10^{16}$ GeV
is the scale of grand unification, and then not unless
the Higgs field is in the region of negative inflation pressure
when $T < T_i$.
\par
The first model is the SU(5) model
of ``new inflation''
discussed by Albrecht and Steinhardt~[4,8]
in which there are twelve heavy gauge bosons.
The one-loop zero-temperature
effective potential $V_1(\phi)$ is
of the Coleman - Weinberg form
$$V_1(\phi) = {25 \alpha_{GUT}^2\over16}
\left[{\sigma^4\over2} + \phi^4
\left(\log{\phi^2\over\sigma^2} - {1\over2}\right)
\right]  \n $$
with $\alpha_{GUT} = 1/45$
at $\sigma = 2\times 10^{15}$ GeV.
The heavy gauge bosons have a common
mass $m(\phi) = \sqrt{5\pi\alpha_{GUT}/3}\,\phi$, and
the function $I(\phi^2/T^2)$ of eq.(6) is
$$I(\phi^2/T^2) = {18\over\pi^2}
\int_0^\infty \! x^2
\log\left( 1 - e^{-(x^2+ (5\pi\alpha_{GUT}/3)(\phi/T)^2)^{1/2}}
\right)dx . \n $$
In this model twelve gauge bosons
and all the fermions are light;
these light particles have a negligible
effect upon the effective potential.
We ignore the effects of these light particles
as well as those of loops of Higgs bosons.
\par
At temperatures $T > \sigma$ above the scale of unification,
the global minimum of the effective potential
$V(\phi,T)$ is at $\phi=0$
where $V(0,T) \ll 0$,
and the inflation pressure $\rho + 3p$
is large and positive.
At $T = \sigma$ the global minimum
of $V(\phi,\sigma)$ is still at $\phi=0$
where $V(0,\sigma) = - 10,232 V_1(0)$
as compared with its value at $\phi=\sigma$
of $V(\sigma,\sigma) = - 9,864 V_1(0)$.
(Here and throughout our discussion of these models,
the number of significant figures quoted
is subjective and does not reflect
the errors due to the use of the one-loop
approximation.)
The inflation pressure is hugely positive
with a value of $\rho + 3p = 61,395 V_1(0)$ at $\phi = 0$.
When the temperature $T$ has dropped
to $T = 0.127 \sigma$,
a second minimum begins to develop
near $\phi = 0.80\sigma$.
The critical temperature $T_c$
is $0.1065 \sigma$.
The two local minima are at $\phi=0$
and at $\phi = 0.95 \sigma$
and have the equal values
$V(0,T_c) = V(0.95\sigma,T_c)
= -0.31 \sigma$.
These minima are separated by
a maximum of $V(0.5 \sigma, T_c) = -0.016 \sigma$
at $\phi = 0.5 \, \sigma$,
and so the phase transition is first order.
\par
The inflation pressure initially turns negative
when the temperature falls to $T = 0.082 \sigma$
attaining the value of $\rho + 3p = - 0.01 V_1(0)$
near $\phi = 0.5 \sigma$.
But inflation probably doesn't start until the temperature
has dropped further to $T_i = 0.0755 \sigma$.
At this temperature the inflation pressure
is negative for $\phi \le 0.75 \sigma$.
In particular the inflation pressure is negative,
$\rho + 3p = -0.005 V_1(0)$, near $\phi = 0$
which is now a high local minimum
of the effective potential,
$V(0,T_i) = 0.67 V_1(0)$.
At $T \le T_i$,
the global minimum is
at $\phi = \sigma$ where
$V(\sigma,T_i) = -0.029 V_1(0)$.
In this theory the point $\phi = 0$ is always
a local minimum of the one-loop
effective potential $V(\phi,T)$.
Thus the Higgs field remains
trapped at $\phi = 0$
until either the one-loop approximation
looses its validity or the field $\phi$
tunnels through the potential barrier,
which becomes ever thinner and shallower
as the temperature drops.
Below $T = 0.05 \sigma$, the inflation pressure
is negative for $|\phi - \sigma| \ge 0.03 \sigma$.
(Throughout this paper,
we are using $\phi$ to mean the length $|\phi|$
of the vector of Higgs fields.)
\par
In this model inflation begins at $T = T_i$ and continues
until the Higgs field enters a region
in which the inflation pressure is positive
or until this semi-classical picture is no
longer adequate.
\par
The second model is a nearly supersymmetric,
grandly unified model with $b=100$ kinds of gauge bosons
and $f=75$ kinds of fermions.
For simplicity, we set $c_i=0.5$ for all $i$.
In this case all the gauge bosons and fermions
have the same mass $m_i(\phi)=\phi$,
and the one-loop, zero-temperature
effective potential reduces to the tree-level
potential which we took to be
$V_0(\phi)=(\lambda/4!)(\phi^2-\sigma^2)^2$
with $\lambda = 1$ and $\sigma = 10^{16}$ GeV.
\par
At a temperature ten times the unification scale
$T = 10\sigma$,
the minimum of the effective potential $V(\phi,T)$
is at $\phi=0$, but this minimum is very shallow:
$V(0,10\sigma)$ is lower than $V(\sigma,10\sigma)$
by less than one part in a thousand;
both are large and negative,
about $ - 1.5\times10^7 V_0(0)$.
At this temperature the inflation pressure (13)
is enormously positive, being $8.9\times10^7 V_0(0)$
at $\phi = 0$.
At a temperature equal to the unification scale, $T = \sigma$,
the inflation pressure is still huge:
$\rho + 3p = 8,881 V_0(0)$ at $\phi = 0$,
which is the absolute minimum
of the effective potential.
At this temperature, the effective potential at $\phi=0$
is $- 1,479 V_0(0)$ which
is 7\% lower than at $\phi=\sigma$.
By $T = 0.2 \sigma$,
the inflation pressure
has dropped to $12 V_0(0)$ at $\phi = 0$.
The minimum of $V(\phi,0.2 \sigma)$
is still at $\phi=0$,
where it is now $-1.37 V_0(0)$
which is 54\% lower than
its value
$V(\sigma,0.2 \sigma) = -0.89 V_0(0)$
at $\phi=\sigma$.
At $T = .189\sigma$ a second
minimum develops near $\phi=0.65\sigma$.
The critical temperature is $T_c =
0.18\sigma$, and the phase transition is
first order since the minima
$V(0,T_c) = V(0.8\sigma,T_c)
= -0.55 V_0(0)$
are separated by a potential barrier
with a maximum of $-0.49\sigma$
at $\phi=.4\sigma$.
By $T = 0.133\sigma$,
the local minimum at $\phi=0$
has disappeared, and
the value there of the effective potential
has risen to $V(0,0.133\sigma)= 0.54 V_0(0)$
which is considerably greater than its minimum value
of $-0.08 V_0(0)$ at $\phi=0.97\sigma$.
The Higgs field can roll down to $\phi = \sigma$.
But the inflation pressure is positive for $\phi < 1.2\sigma$.
\par
Inflation cannot start until
the inflation pressure becomes negative
which happens for $0.15 \sigma \le \phi \le 0.40\sigma$
when the temperature has fallen to $T = T_i = 0.123\sigma$.
If the Higgs field is still in this region
of negative inflation pressure when $T \le T_i$,
then inflation will occur in this model.
At $T = 0.122 \sigma$, the inflation pressure
is negative for $\phi \le 0.45 \sigma$.
At $T = 0.122\sigma$,
the symmetric point $\phi = 0$ is a local maximum
of the effective potential
$V(0,0.12\sigma) = 0.67  V_0(0)$,
the absolute minimum being
at $\phi = \sigma$ where it is slightly negative,
$V(\sigma,0.12\sigma) = -0.04 V_0(0)$.
At $T = \sigma/10$, the inflation pressure
runs from $-1.11 V_0(0)$
at $\phi = 0$, where $V(0,0.1\sigma) = 0.85 V_0(0)$,
through zero at $\phi = 0.85 \sigma$
to $0.09 V_0(0)$ at $\phi=\sigma$,
where $V(\sigma, 0.1\sigma) = - 0.01 V_0(0)$.
At temperatures lower than $T = 0.05 \sigma$,
the absolute minima of the effective potential
are near $\phi = \sigma$,
and the inflation pressure is negative
for $|\phi - \sigma| > 0.005 \sigma$.
\par
In this model, inflation may not occur at all.
But if as the temperature drops below $T_i = 0.123 \sigma$,
the region of negative $\rho +3p$
grows faster than $\phi$,
which is slowed by gravitational friction,
then there could be substantial inflation.
\goodbreak
\par
\bigskip
\medskip
\centerline{\bf Gravitational Friction and Anti-Friction}
\par
In a Robertson--Walker universe,
the field equation for
the evolution of the mean value $\phi$ of the Higgs field is
$$\ddot \phi = - 3 \dot R \dot \phi / R - V^\prime(\phi,T) \n $$
in which spatial derivatives have been suppressed
and $V^\prime(\phi,T) = \partial V(\phi,T)/\partial \phi$.
The term $- 3 \dot R \dot \phi$
arises from the coupling of the Higgs field $\phi$
to the gravitational field and
can force $\phi$ from its equilibrium position
when $\dot R$ and $\dot \phi$ are appreciable.
As the universe contracts,
$\dot R$ is negative, and so
$\ddot \phi \propto \dot \phi$.
Thus for $\dot R < 0$,
any motion of the Higgs field $\phi$ is accelerated:
there is {\it anti-friction\/} or {\it negative\/} friction.
As the universe expands,
$\dot R$ is positive, and so
$\ddot \phi \propto - \dot \phi$:
any motion of $\phi$ is slowed by gravitational friction.
\par
At the one-loop level,
since $I^\prime >0$
for all $\phi$ and $T$,
the force $-V^\prime(\phi,T)$
due to the temperature-dependent part
of the effective potential
is negative where $V_0^\prime(\phi)\approx 0$,
which is true near $\phi = 0$ and $\phi = \sigma$.
These regions of negative $-V^\prime(\phi,T)$
grow with the temperature $T$ and
merge when symmetry is restored.
While the field $\phi$ is in these regions,
it is forced toward $\phi=0$
and its speed $-\dot \phi$ increases.
Thus although between these regions
(and before symmetry restoration)
the force $-V^\prime(\phi,T)$ is positive,
it may be possible during a collapse
for negative gravitational friction
to push the Higgs field into a region
of negative inflation pressure
(closer to the symmetric point $\phi = 0$),
and then for gravitational friction to keep it there
during a subsequent expansion.
By this mechanism it may be possible for some parts
of a collapsing universe to avoid an initial singularity.
\par
Our numerical computations of
the effective potential $V(\phi,T)$
and of the inflation pressure $\rho + 3p$
for the two models studied in the last section
suggest various possibilities.
One scenario applies only to
the SU(5) model of Albrecht and Steinhardt~[4,8]
and other theories
in which the symmetric point $\phi = 0$ can be
a local minimum of the effective potential
with negative inflation pressure.
In that model the inflation pressure
is negative near $\phi = 0$
when $T \le T_i = 0.0755 \sigma$
and also negative for $|\phi - \sigma| > 0.03 \sigma$
when $T \leq \sigma/20$.
The regions of negative $-V^\prime(\phi,T)$
grow with $T$ and merge at $ T = 0.128 \sigma$
when the symmetry is restored.
In this scenario
the Higgs field arrives at the lip of the cup
of the local minimum at $\phi=0$
where $\rho + 3p <0$
with so slow a speed $\dot\phi$
and at such a time
that it is still within the cup
as the universe begins to expand.
The chances of such lucky timing
may be very small~[10].
But in a locally non-uniform universe,
there are many somewhat different sets
of initial conditions, and so although
the timing required for a bounce may
be exquisitely precise, it is plausible
for the required timing to occur
in some small regions.
These regions then inflate.
\par
In another scenario, which applies to both
the SU(5) model and the {\tenpoint SUSY-GUT} model,
the Higgs field is driven by negative friction and
so oscillates about the minimum of the effective potential
with increasing amplitude.
In such models the field $\phi$ may spend most of its time
in regions of negative inflation pressure
when $T \le \sigma/20$,
since at such temperatures $\rho + 3p$ is negative
for $|\phi - \sigma| > 0.03 \sigma$ in the SU(5) model
and for $|\phi - \sigma| > 0.005\sigma$ in the susy-gut model.
If the contraction of the universe reverses
while the Higgs field
is in such a region, then gravitational friction
may hold it there long enough for the universe
to inflate substantially.
\goodbreak
\par
\bigskip
\medskip
\centerline{\bf Acknowledgements}
\par
The authors have benefited from discussions
with M.~J. Duff, B.~Kemmell, E.~Kolb, R.~Matzner, E.~Mottola,
R.~Reeder, P.~Steinhardt, and M.~Turner.
This research was supported by the U.S. Department of Energy
under contract DE-FG04-84ER40166.
\goodbreak
\bigskip
\medskip
\centerline{\bf References}
\par
\frenchspacing
\noindent
\vskip 1pt
\noindent
\item{[1]} A.~H. Guth, {\sl Phys.\ Rev.\ D\/} 23 (1981) 347.
\item{[2]} A.~D. Linde, {\sl Phys.\ Lett.\/} 108B (1982) 389
and 129B (1983) 177.
\item{[3]} A.~D. Linde, {\sl Inflation and Quantum Cosmology\/}
(Academic Press, New York, 1990).
\item{[4]} A.~Albrecht and P.~Steinhardt,  {\sl Phys.\ Rev.\ Lett.\/}
48 (1982) 1220.
\item{[5]} E.~R. Harrison, {\sl Mon. Not. R. astr. Soc.\/}
137 (1967) 69.
\item{[6]} L.~Dolan and R.~Jackiw, {\sl Phys.\ Rev.\ D\/} 9 (1974) 3320;
S.~Weinberg, {\it ibid\/} p.3357.
\item{[7]} H.~Georgi and S.~L. Glashow,
{\sl Phys.\ Rev.\ Lett.\/} 32 (1974) 438.
\item{[8]} E.~W. Kolb and M.~S. Turner,
{\sl The Early Universe\/}
(Addison-Wesley Pub.\ Co., 1990), p.\ 291.
\item{[9]} K.~Cahill, hep-ph/9301294.
\item{[10]} J.~D. Barrow and R.~Matzner,
{\sl Phys.\ Rev.\ D\/} 21 (1980) 336.
\bye
