%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LATEX FILE OF THE PAPER:
% "Expanding and contracting universes in
% third quantized string cosmology"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[12pt,titlepage,epsfig]{article}
%\input epsf
\def\baselinestretch{1.4}
\setlength{\oddsidemargin}{0.0cm}
\setlength{\textwidth}{16.5cm}
\setlength{\topmargin}{-.9cm}
\setlength{\textheight}{22.5cm}%
%\renewcommand{\thesection}{\arabic{section}}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
\font\small=cmr8 scaled \magstep0
\font\grande=cmr10 scaled \magstep4
\font\medio=cmr10 scaled \magstep2
\outer\def\beginsection#1\par{\medbreak\bigskip
\message{#1}\leftline{\bf#1}\nobreak\medskip
\vskip-\parskip
\noindent}
\def\obdot{\hskip-8pt \vbox to 11pt{\hbox{..}\vfill}}
\def\obbdot{\hskip-8pt \vbox to 14pt{\hbox{..}\vfill}}
\def\odot{\hskip-6pt \vbox to 6pt{\hbox{..}\vfill}}
%minore o circa uguale
\def\laq{\raise 0.4ex\hbox{$<$}\kern -0.8em\lower 0.62
ex\hbox{$\sim$}}
%maggiore o circa uguale
\def\gaq{\raise 0.4ex\hbox{$>$}\kern -0.7em\lower 0.62
ex\hbox{$\sim$}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def \me {\buildrel <\over \sim}
\def \Me {\buildrel >\over \sim}
\def \pa {\partial}
\def \ra {\rightarrow}
\def \big {\bigtriangledown}
\def \fb {\overline \phi}
\def \fbp {\dot{\fb}}
\def \bp {\dot{\beta}}
\def \rb {\overline \rho}
\def \pb {\overline p}
\def \pr {\prime}
\def \se {\prime \prime}
\def \H {{a^\prime \over a}}
\def \fp {{\phi^\prime}}
\def \ti {\tilde}
\def \la {\lambda}
\def \ls {\lambda_s}
\def \La {\Lambda}
\def \Da {\Delta}
\def \b {\beta}
\def \a {\alpha}
\def \ap {\alpha^{\prime}}
\def \ka {\kappa}
\def \Ga {\Gamma}
\def \ga {\gamma}
\def \sg {\sigma}
\def \da {\delta}
\def \ep {\epsilon}
\def \r {\rho}
\def \om {\omega}
\def \Om {\Omega}
\def \noi {\noindent}
\def \pfb {\Pi_{\fb}}
\def \pM {\Pi_{M}}
\def \pbe {\Pi_{\b}}
\begin{document}
\bibliographystyle {unsrt}
\titlepage
\begin{flushright}
CERN-TH/96-322 \\
IFUP-TH/96-65 \\
hep-th/9701146\\
\end{flushright}
\vspace{6mm}
\begin{center}
{\bf EXPANDING AND CONTRACTING UNIVERSES\\
IN THIRD QUANTIZED STRING COSMOLOGY}\\
\vspace{10mm}
A. Buonanno${}^{(a) (b)}$, M. Gasperini${}^{(c) (d)}$,
M. Maggiore${}^{(a) (b)}$ and C. Ungarelli${}^{(a) (b)}$\\
\vspace{6mm}
${}^{(a)}$
{\sl Dipartimento di Fisica, Universit\`a di Pisa, \\
Piazza Torricelli 2, I-56100 Pisa, Italy} \\
%
${}^{(b)}$
{\sl Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy} \\
%
${}^{(c)}$
{\sl Theory Division, CERN, CH-1211 Geneva 23, Switzerland} \\
%
${}^{(d)}$
{\sl Dipartimento di Fisica Teorica, Universit\`a di Torino, \\
Via P. Giuria 1, 10125 Turin, Italy }\\
\end{center}
\vskip 1cm
\centerline{\medio Abstract}
\noindent
We discuss the possibility of quantum transitions from the string
perturbative vacuum to cosmological configurations
characterized by isotropic contraction and decreasing
dilaton. When the dilaton potential preserves the sign of the Hubble
factor throughout the evolution, such transitions
can be represented as an anti-tunnelling of the
Wheeler--De Witt wave function in minisuperspace or, in a
third-quantization language, as the production of pairs of universes
out of the vacuum.
\vspace{7mm}
\centerline{\sl To appear in {\bf Class. Quantum Grav.}}
\vspace{5mm}
\vfill
\begin{flushleft}
CERN-TH/96-322 \\
November 1996
\end{flushleft}
\newpage
At very early times, according to the standard cosmological scenario,
the Universe is expected to approach a Planckian, quantum gravity
regime where a classical description of the spacetime manifold is no
longer appropriate. A possible quantum description of the Universe, in
that regime, is based on the Wheeler--De Witt (WDW) wave function
\cite{1,2}, generally defined on the superspace spanned by all
three-dimensional geometric configurations. In that context it becomes
possible to compute, with an appropriate model of (mini)superspace, the
probability distribution of a given cosmological configuration versus an
appropriate ``state" parameter (for instance the cosmological constant
$\La$). The results, however, are in general affected by
operator-ordering ambiguities, and are also strongly dependent on the
boundary conditions \cite{3}--\cite{5}
imposed on the solutions of the WDW equation.
String theory has recently motivated the study of a cosmological
scenario in which the Universe starts from the string perturbative
vacuum and evolves through an initial, ``pre-big bang" phase \cite{6},
characterized by an accelerated growth of the curvature and of the
gauge coupling $g=e^{\phi/2}$ ($\phi$ is the dilaton field).
In such a context, the WDW equation
is obtained from the low-energy string effective
action \cite{7}--\cite{9}, and has no
operator ordering ambiguities \cite{7}
since the ordering is uniquely fixed by the
duality symmetries of the action. Also, the boundary conditions are
determined by the choice of the perturbative vacuum as the initial
state for the cosmological evolution.
According to the lowest-order effective action, the classical evolution
from the perturbative vacuum necessarily leads the background
to a singularity, and
the transition to the present decelerated ``post-big bang"
configuration is impossible, for any realistic type of (local) dilaton
potential \cite{10}. With an appropriate potential, however, the
transition may become allowed at the quantum level even if, for the same
potential, it remains classically forbidden.
This effect was discussed in previous papers \cite{7}, in which the WDW
equation was applied to compute the transition probability between
two duality-related pre- and post-big bang cosmological phases.
The string perturbative vacuum is, in general, a higher-dimensional
state, and the initial growth of the dilatonic coupling $g$ requires,
according to the lowest-order action, a large enough number of expanding
dimensions. For instance, in a Bianchi-type I background with $d$
expanding and $n$ contracting isotropic spatial dimensions, the growth
of $g$ requires \cite{6}
$d+\sqrt{d+n}>n$, which cannot be satisfied by $d=3$, in
particular, in the ten-dimensional superstring vacuum. With a monotonic
evolution of the scale factor, this represents another obstruction to
a smooth transition to our present,
dimensionally reduced Universe.
The aim of this paper is to show that the initial perturbative vacuum
is not inconsistent, at the quantum level, with a final contracting
cosmological configuration,
when we add to the lowest-order action an
appropriate dilaton potential (such as the simple one induced by an
effective cosmological constant). In particular,
for a WDW potential which is
translationally invariant in minisuperspace, along the direction
parametrized by the scale factor, and for which the sign of the Hubble
factor is classically conserved during the whole evolution,
the cosmological contraction corresponds to a pure quantum effect. It can be
described as an ``anti-tunnelling" of the WDW wave function
from the string perturbative vacuum or, in a
third quantization \cite{10a} language,
as a production of ``pairs of universes"
(one expanding, the other contracting) out of the third quantized
vacuum. Such a process requires
the identification of the time-like coordinate in minisuperspace with
the direction parametrized by the shifted dilaton $\fb$ (see below), and
is complementary to the process of spatial reflection of the wave
function, which describes transitions from pre- to post-big bang
configurations \cite{7}.
We shall adopt, in this paper, the
minisuperspace model
already discussed in \cite{7}, based on the tree-level,
lowest-order in $\ap$, string effective action \cite{11}. Working in the
simplifying assumption that only the metric and the dilaton contribute
non-trivially to the background, in $d$ isotropic spatial dimensions, the
corresponding action can be written as
\beq
S = -\frac{1}{2\,\lambda_s^{d-1}}\,\int\,d^{d+1}x\,\sqrt{|g|}\,e^{-\phi}
\,\left(R+\partial_{\mu}\phi\partial^{\mu}\phi +V \right ).
\label{21}
\eeq
Here $\lambda_s=(\ap)^{1/2}$ is the
fundamental string length parameter governing the
higher-derivative expansion of the action, and $V$ is a (possibly
non-perturbative) dilaton potential. By using the parametrization
appropriate to an isotropic, spatially flat cosmological background:
\beq
g_{\mu\nu} ={\rm diag} \left(N^2(t), -a^2(t) \da_{ij}\right), ~~~~~~~~
a= \exp\left[\b (t)/\sqrt{d}\right], ~~~~~~~~ \phi=\phi(t),
\label{22}
\eeq
and assuming spatial sections of finite volume, the action can be
expressed in the convenient form
\beq
S=\frac{\lambda_s}{2}\,\int\,dt\,{e^{-\fb}\over N}\,
\left(\dot{\beta}^2-\dot{\fb}^2-
N\,V \right)\,,
\label{23}
\eeq
where $\fb$ is the shifted dilaton:
\beq
\fb=\phi-\log\,\int\,d^dx/\lambda_s^d -\sqrt{d}\,\beta \,.
\label{24}
\eeq
The variation with respect to $N$ then leads to the Hamiltonian
constraint
\beq
\Pi^2_{\beta}-\Pi^2_{\fb}
+\lambda_s^2\,V(\b,\fb)\,e^{-2\,\fb}=0~,
\label{25}
\eeq
where $\pbe,\pfb$ are the (dimensionless) canonical momenta (in the
gauge $N=1$):
\beq
\Pi_{\beta}={\da S\over \da\dot{\beta}}=
\lambda_s\,\dot{\beta}\,e^{-\fb} , ~~~~~~~~~~~~
\Pi_{\fb}={\da S\over \da\dot{\fb}}=
-\lambda_s\,\dot{\fb}\,e^{-\fb} .
\label{26}
\eeq
When $V=0$, the classical solutions of the action
(\ref{23}) describing the phase of accelerated pre-big bang
evolution are characterized by two duality-related branches \cite{6},
defined in the negative time range:
\beq
t<0, ~~~~a=a_0(-t)^{\mp 1/\sqrt d}, ~~~~\fb-\phi_0=-\ln(-t)=\pm \b,
~~~~\pbe=\pm k={\rm const}, ~~~~\pfb=\mp \pbe <0
\label{27}
\eeq
($k$, $a_0$ and $\phi_0$ are integration constants).
For the upper-sign branch the metric is expanding ($\pbe>0$), and the
curvature scale $\dot\b ^2$ and the string coupling $g(t)$ are
growing, starting asymptotically from the perturbative vacuum,
the state with flat metric
($\bp=0=\fbp$) and vanishing coupling constant ($\phi=-\infty$,
$g=0$). The lower-sign branch corresponds instead to a contracting
configuration ($\pbe<0$), in which the coupling $g(t)$ is decreasing.
In the presence of a constant dilaton potential,
$V=\La= {\rm const}$, the accelerated pre-big bang solutions
are again characterized by two branches \cite{12}:
\beq
t<0, ~~~~a=a_0\left[\tanh(-t\sqrt{\Lambda}/{2})
\right]^{\mp 1/\sqrt{d}}, ~~~~\fb-\phi_0=-\ln \sinh \left(-t\sqrt{\Lambda}
\right), ~~~~\pfb <0,
\label{29}
\eeq
which are respectively expanding with growing dilaton
(upper sign, $\pbe>0$) and contracting with decreasing dilaton
(lower sign, $\pbe<0$). In this case
both branches are
dominated, in the low-curvature regime, by the contribution
of a positive cosmological constant $\La$. The initial
perturbative vacuum is replaced by a
configuration with flat metric and linearly evolving dilaton ($\bp=0$,
$\dot\phi={\rm const}$), another well-known string theory
background \cite{13} (exact solution to all orders
in the $\ap$ expansion). Near the singularity ($t\ra 0_-$), however, the
contribution of $\La$ becomes negligible, and the solution
(\ref{29}) asymptotically approaches that of eq. (\ref{27}).
In this paper we shall assume that an effective cosmological constant
$\La$ is
generated non-perturbatively in the strong coupling, Planckian regime,
and we shall use the WDW equation to discuss the possibility of
transitions,
induced by $\La$, from the perturbative vacuum to a final configuration
with contracting metric and decreasing dilaton. We shall consider, in
particular,
the case in which the effective dilaton potential can be approximated
by the Heaviside step function $\theta$ as
$V(\b, \fb)= \La ~\theta (\fb)$. The corresponding WDW equation, in the
minisuperspace spanned by $\b$ and $\fb$, is obtained from the
Hamiltonian constraint (\ref{25}) through the differential
representation $\Pi= -i \nabla$:
\beq
\left [ \partial^2_{\fb} -
\partial^2_{ \beta}
+\lambda_s^2\,\Lambda\,\theta(\fb)\,e^{-2\fb} \right ]\, \Psi= 0 \,.
\label{31}
\eeq
The momentum along the $\b$ axis is conserved,
\beq
[\pbe ,H] =0, ~~~~~~~~~~~~\pbe = \ls \dot{\b} e^{-\fb} = k =
{\rm const} ,
\label{212a}
\eeq
and the general solution of the WDW equation can be factorized as
$\Psi_k(\fb,\b) = \psi_k(\fb) e^{ik\b}$.
Note that we have assumed a potential $V$ depending explicitly only on
$\fb$ because the classical evolution of the
scale factor, in that case, is monotonic, and no contracting
configuration can be eventually obtained, classically,
if we start from the isotropic perturbative vacuum.
From a quantum-mechanic point of view, however, the situation is
different. Indeed, if we assign to $\fb$ the role of time-like
coordinate, eq. (\ref{31}) is formally equivalent to a
Klein--Gordon equation with time-dependent mass term. The solution
$\psi_k$ is a linear combination of plane waves for $\fb <0$,
and of Bessel functions \cite{14} $J_{\pm\nu}(z)$, of imaginary index
$\nu=ik$ and argument $z=\la_s\sqrt\La e^{-\fb}$, for $\fb>0$. In
particular, the functions
\bea
\Psi^{(\pm)}_{k}=\frac{e^{ik\b}}{\sqrt{4\pi k}}\,e^{\mp ik\fb}\,,
~~~~~~~~~~~~~~\fb&<&0 , \\
\Psi^{(\pm)}_{k}=\frac{e^{ik\b}}{\sqrt{4\pi k}}
\,\left(\frac{z_0}{2}\right)^{\mp \nu}\,\Gamma(1\pm \nu)\,
J_{\pm \nu}(z)\,,
~~~~~~~~~~~~~~\fb&>&0 ,
\label{33}
\eea
where $z_0=\la_s\sqrt\La$ and $\Ga$ is the Euler function,
provide orthonormal sets of solutions with respect to the
Klein-Gordon scalar product
\beq
\label{scal}
(\Psi^1,\Psi^2)=-i\,\int d \beta \,
\Psi^1(\beta,\fb)\stackrel{\leftrightarrow}{\partial}_{\fb}
{\Psi^2}^*(\beta,\fb)\,\,.
\label{34}
\eeq
We shall fix the boundary conditions by imposing that, for $\fb<0$, the
Universe is represented by the wave function
\beq
\Psi_{Ik}(\b, \fb<0)=\frac{1}{\sqrt{4\pi k}}\,e^{ ik(\b-\fb )},
\label{35}
\eeq
corresponding to a state of growing dilaton and
accelerated pre-big bang expansion from the perturbative vacuum, with
$\pbe=-\pfb=k>0$ according to eq. (\ref{27}). The eigenvalue $k$ of $\pbe$
parametrizes the initial state in the space of all classical configurations
(\ref{27}). For $\fb>0$ the wave function is uniquely determined by
the matching conditions for $\Psi$ and $\partial_{\fb} \Psi$
at $\fb=0$, in terms of the functions (\ref{33}), as
\beq
\Psi_{IIk}(\b, \fb>0)= A_k^+\,\Psi_{k}^{(+)} +
A_k^-\,\Psi_{k}^{(-)} ,
\label{36}
\eeq
where
\beq
A^{\pm}_{k}=\frac{i\,z_0}{2\,k}\,\left
(\frac{z_0}{2}\right)^{\pm ik}\,\Gamma(1\mp ik)
\,\left[\pm{J}_{\mp ik}^{\prime}(z_0)\mp
\frac{i\,k}{z_0}\,{J}_{\mp ik}(z_0)\right]
\label{37}
\eeq
(a prime denotes differentiation of the Bessel functions with respect to
their argument).
Given a pure initial state $\Psi_{I}^{(+)}$ of ``positive frequency" $k$,
the final state is thus a
mixture of ``positive" and ``negative" frequency modes,
$\Psi_{II}^{(+)}$ and $\Psi_{II}^{(-)}$, satisfying asymptotically the
conditions
\bea
&&\lim_{\fb \ra \infty} \Psi_{II}^{(\pm)}(\b,\fb) =
\Psi_{\infty}^{(\pm)}(\b,\fb)\sim e^{ik(\b \mp \fb)} ,
\nonumber\\
&&\pbe\Psi_{\infty}^{(\pm)}= -i \pa_\b\Psi_{\infty}^{(\pm)}=
k\Psi_{\infty}^{(\pm)},~~~~~ ~
\pfb\Psi_{\infty}^{(\pm)}= -i \pa_{\fb}\Psi_{\infty}^{(\pm)}=
\mp \pbe\Psi_{\infty}^{(\pm)} .
\label{213}
\eea
The mixing is determined by the
coefficients $A^{\pm}_{k}$, satisfying the standard Bogoliubov
normalization condition
$|A_k^+|^2-|A_{k}^-|^2=1$. In a second quantization context, it is well
known that such a mixing describes a process of pair production
\cite{15}, the negative energy mode being associated to an
antiparticle state of positive energy and opposite spatial momentum. It
thus seems correct to interpret the above splitting of the WDW wave
function, in a third quantization context \cite{10a}, as the production
of a pair of universes, with quantum numbers $\{\pbe,
\pfb \}$, corresponding to positive energy ($\pfb<0$) and opposite
momentum along the spacelike direction $\b$. One of the two universes is
isotropically expanding ($\pbe>0$), with growing dilaton; the
``anti-universe" is isotropically contracting ($\pbe<0$), with decreasing
dilaton. Both configurations evolve towards the curvature singularity of
the classical pre-big bang solution (\ref{29}). However, while the
growing dilaton state corresponds to a continuous classical evolution
from the perturbative vacuum, no smooth connection to such vacuum is
possible, classically, for the state with decreasing dilaton.
It is important to stress that, as long as $V=V(\fb)$ and,
consequently, $\pbe$ is
conserved, a third-quantized production of universes is
only possible provided we assign the role of time-like coordinate to
$\fb$, and the potential satisfies $V(\fb)e^{-2\fb}$ $\ra 0$ for $\fb \ra
+\infty$ (in order to identify, asymptotically, positive and negative
frequency modes). The pairs of universes are produced in the limit of
large positive $\fb$, so that we cannot describe in this context a
transition to post-big bang cosmological configurations, which are
instead characterized by $\fb<0$. A quantum description of the
transition from pre- to post-big bang requires in fact the
interpretation of $\b$ as the time-like axis, as discussed in \cite{7}. In
that case, a third quantized production of pairs becomes possible only
if $\pbe$ is not conserved, namely if $V$ depends also on $\b$.
For the process considered in this paper, the probability
is controlled by
$|A_{k}^-|^2$, which determines the expectation number pairs of universes
produced in the final state. The production probability is negligible when
$|A_{k}^-|\ll 1$; it has the typical probability of a vacuum fluctuation
effect when $|A_{k}^-|\sim |A_{k}^+|\sim 1$; finally, when
$|A_{k}^-|\sim |A_{k}^+|\gg1$, the initial wave function is parametrically
amplified \cite{15a} and the probability is large.
In our case, the interesting parameter characterizing the
process, besides $\La$, is the portion of proper spatial volume
$\Om=a^d\int d^dx$ undergoing the transition. Considering, in
particular, $d=3$ spatial dimensions, and using the
definitions of $k$ and $\fb$, the initial momentum $k$ can be
conveniently expressed as $k=\sqrt 3 \Om_s g_s^{-2}\la_s^{-3}$,
where $g_s=\exp(\phi_s/2)$ and $\Om_s$ are, respectively,
the value of the coupling and of the proper spatial
volume evaluated at the string scale $t=t_s$,
when $H\equiv \bp/\sqrt 3=\la_s^{-1}$.
By exploiting the properties of the Bessel functions, we can then
express the asymptotic limits of
the Bogoliubov coefficients (\ref{37}) in terms of the physical
parameters $\Om_s$ and $\La$. We obtain, at fixed
$\Om_s/(g_s^2\la_s^3)=1$,
\bea
&&|A^+|^2-1 \simeq |A^-|^2 \simeq \frac{1}{48}\,\Lambda^{2}\,\lambda_s^4\,
, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\La\ll \la_s^{-2},
\label{310}\\
&&|A^+|^2 \simeq |A^-|^2 \simeq
\sqrt{\Lambda\,\lambda_s^2}\,
\,\frac{\cosh(\sqrt{3}\,\pi) - \sin(2\,\sqrt{\Lambda\,\lambda_s^2})}
{4\,\sqrt{3}\,\sinh(\sqrt{3}\,\pi)}\,,
~~~~~~~\La\gg \la_s^{-2},
\label{311}
\eea
and, at fixed $\La \la_s^2=1$,
\beq
|A^+|^2 \simeq |A^-|^2 \simeq
\frac{g_s^4\,\lambda_s^6}{12\,\Omega_s^2}\,|J^{\prime}_0(1)|^2\,,
~~~~~~~ |J^{\prime}_0(1)| \simeq 0.44, ~~~~~~~
\Om_s\ll g_s^2 \la_s^3
\label{312}
\eeq
(the limit $\Om_s\gg g_s^2 \la_s^3$ cannot be performed because the
quantum process is confined to the region of large $\fb$).
The quantum production of universes in a state with non-vanishing
cosmological constant $\La$ is thus strongly suppressed
for small values of $\La$,
while it is favoured in the opposite limit
of large $\La$ and of proper volumes that are
small in string units, in qualitative
agreement with previous results \cite{7}, and
also with the general approach to quantum cosmology
based on tunnelling boundary
conditions \cite{4,5}. Instead of a ``tunnelling from nothing", however,
this quantum production of expanding and contracting universes can be
seen as an ``anti-tunnelling from the string perturbative vacuum" of
the WDW wave function. Indeed, the asymptotic expansion of the solution
(\ref{35}), (\ref{36}),
\bea
\fb \ra +\infty &,& ~~~~~~~~~~~ \psi \sim A_{in} e^{-ik\fb}+
A_{ref} e^{ik\fb},\nonumber \\
\fb \ra -\infty &,& ~~~~~~~~~~~ \psi \sim A_{tr} e^{-ik\fb},
\label{313}
\eea
describes formally a scattering process along $\fb$, in which the
expanding universe corresponds to the incident part of the wave function,
the contracting anti-universe to the reflected part, and the initial
vacuum to the transmitted part. In the parametric amplification regime
of eqs. (\ref{311}) and (\ref{312}), where $|A^+|\sim |A^-|\gg 1$, the
reflection
coefficient $R=|A_{ref}|^2/|A_{in}|^2$ is approximately $1$, and
the Bogoliubov coefficient $|A^-|$, which controls the probability of
pair production,
becomes the inverse of the
tunnelling coefficient $T=|A_{tr}|^2/|A_{in}|^2$:
\beq
|A^-|^2= {|A_{ref}|^2\over|A_{tr}|^2}={R\over T}\simeq {1\over T}.
\label{314}
\eeq
In view of future applications,
we have also computed numerically the Bogoliubov coefficients $A^{\pm}$
by discretizing the
WDW equation with the explicit method \cite{16}, and using the routine Fast
Fourier Transform \cite{17}. A computer simulation, in which the pair
production process is graphically represented by the scattering and
reflection of an initial wave packet, has given results
in complete agreement with
the analytic computation (\ref{37}).
In conclusion, we have shown in this paper that it is not impossible,
in a quantum cosmology context, to nucleate universes
in a state characterized by isotropic contraction
and decreasing dilaton. The process can be described as the production
from the vacuum of
universe--anti-universe pairs in the strong coupling regime, triggered by
the presence of an effective cosmological constant.
When $V=V(\fb)$ and $\pbe$ is conserved, the pair-production
process requires the
identification of $\fb$ as time-like coordinate in minisuperspace,
while the transition from
pre- to post-big bang configurations requires the complementary choice of
$\b$ as the time-like axis.
The validity of our
analysis is limited by the very crude approximation (the step potential)
adopted to modellize the time-evolution of the non-perturbative dilaton
potential. Also, an appropriate
potential should depend on $\phi$ (not on $\fb$ as assumed in
this paper); in that case, however, the transition from expansion to
contraction may be allowed also classically (in an appropriate
limit), and is represented in minisuperspace as a reflection \cite{18}
(instead of an anti-tunnelling) of the wave function. In spite of these
limitations, the analysis of this paper confirms that the WDW approach
provides an adequate framework for a consistent formulation
of quantum string cosmology, with
the boundary conditions uniquely prescribed by the choice of
the initial perturbative vacuum.
\vskip 2 cm
\section*{Acknowledgements}
We are grateful to Vittorio de Alfaro and Roberto Ricci for
discussions and clarifying comments. Special thanks are due to Gabriele
Veneziano for a careful reading of the manuscript and for helpful
suggestions.
\newpage
\begin{thebibliography}{999}
\newcommand{\pl}{{ Phys. Lett.}\ }
\newcommand{\prl}{{Phys. Rev. Lett.}\ }
\newcommand{\prd}{{ Phys. Rev.}\ }
\newcommand{\np}{{ Nucl. Phys.}\ }
\newcommand{\cmp}{{ Com. Math. Phys.}\ }
\newcommand{\bb}{\bibitem}
\bibitem{1} De Witt B S 1967 Phys. Rev. 160 1113;
Wheeler J A 1968 {\sl Battelle Rencontres} ed De Witt C and
Wheeler J A (Benjamin, New York).
\bb{2}For a recent review on quantum cosmology see Vilenkin A 1996
{\sl String gravity and physics at the Planck energy scale} ed
Sanchez N and Zichichi A (Kluwer Acad. Pub., Dordrecht)
p 345.
\bb{3}Hartle J B and Hawking S W 1983 Phys. Rev. D28 2960;
Hawking S W 1984 \np B239 257;
Hawking S W and Page D N 1986 \np B264 185.
\bb{4}Linde A D 1984 Sov. Phys. JETPT 60 211; Lett. Nuovo Cimento 39 401;
Zel'dovich Y B and Starobinski A A 1984 Sov. Astron. Lett. 10
135; Rubakov V A 1984 \pl B148 280.
\bb{5}Vilenkin A 1984 Phys. Rev. D30 509, 1986
D33 3560, 1988 D37 888.
\bibitem{6}Gasperini M and Veneziano G 1993 Astropart. Phys. 1 317;
Mod. Phys. Lett. A8 3701; 1994 Phys. Rev. D50 2519. An updated
collection of papers on the pre-big bang scenario is available at {\tt
http://www.to.infn.it/teorici/gasperini/}.
\bb{7} Gasperini M Maharana J and Veneziano G 1996 \np B472 349;
Gasperini M and Veneziano G 1996 Gen. Rel. Grav. 28 1301.
\bb{8}Kehagias A A and Lukas A 1996 Nucl. Phys. B477 549.
\bb{9}Lidsey J E 1996 {\sl Inflationary and deflationary branches in
extended pre-big bang cosmology} (gr-qc/9605017) Phys. Rev. D (in
press).
\bibitem{10} Brustein R and G. Veneziano G 1994
Phys. Lett. B329 429;
Kaloper N Madden R and Olive K A 1995 Nucl. Phys. B452
677; 1996 Phys. Lett. B371 34;
Easther R Maeda K and Wands D 1996 Phys.
Rev. D53 4247.
\bb{10a}Rubakov V A 1988 \pl B214 503;
Kozimirov N and Tkachev I I 1988 Mod. Phys. Lett. A4 2377;
McGuigan M 1988 \prd D38 3031, 1989 D39 2229; 1990 \prd D41 418.
\bibitem{11}See for instance
Metsaev R R and Tseytlin A A 1987 Nucl. Phys. B293 385.
\bb{12} Veneziano G 1991 \pl B265 387.
\bb{13} Myers R C 1987 \pl B199 371.
\bibitem{14}Abramowicz M and Stegun I A 1972 {\sl Handbook of
Mathematical Functions} (Dover, New York).
\bibitem{15}See for instance Birrel N and Davies P 1982 {\sl
Quantum fields in
curved space} (Cambridge University Press, Cambridge).
\bb{15a}Grishchuk L P 1975 Sov. Phys. JEPT 40 409;
Starobinski A A 1979 JEPT Letters 30 682.
\bibitem{16}Mitchell A R and Griffiths D F 1980 {\sl The finite difference
method in partial differential equations} (John Wiley and Sons Ltd.,
Chichester).
\bibitem{17}Press W H et al 1992
{\sl Numerical recipes in Fortran: the art of scientific
computing} (Cambridge University Press, Cambridge).
\bibitem{18}Ricci R et al 1996 in preparation.
\end{thebibliography}
\end{document}