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\begin{flushright}
BA-TH/99-367\\
hep-th/0004149
\end{flushright}
\vspace*{0.8truein}
{\Large\bf\centering\ignorespaces
String Cosmology\\
\bigskip
versus Standard and Inflationary Cosmology
\vskip2.5pt}
{\dimen0=-\prevdepth \advance\dimen0 by23pt
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\vspace{0.8 cm}
M. Gasperini
\par}
\vspace{0.5 cm}
{\small\it\centering\ignorespaces
Dipartimento di Fisica, Universit\`a di Bari,\\
Via G. Amendola 173, 70126 Bari, Italy \\
\vspace{0.2 cm}
and\\
\vspace{0.2 cm}
Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy\\
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\centerline{\bf Abstract}
\noi
This paper presents a review of the basic, model-independent
differences between the pre-big bang scenario, arising naturally in a
string cosmology context, and the standard inflationary scenario. We use
an unconventional approach in which the introduction of technical details
is avoided as much as possible, trying to focus the reader's attention on
the main conceptual aspects of both scenarios. The aim of the paper is
not to conclude in favour either of one or of the other scenario, but to
raise questions that are left to the reader's meditation. Warnings: the
paper does not contain equations, and is not intended as a
complete review of all aspects of string cosmology.
\vspace{0.8cm}
\begin{center}
------------------------------
\vspace{0.8cm}
To appear in {\bf Classical and Quantum Gravity}\\
(Topical Review Section)
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\title[String cosmology]{String cosmology versus standard and
inflationary cosmology }
\author{M. Gasperini\dag\ddag
\footnote[3]{E-mail: gasperini@ba.infn.it.}}
\address{\dag Dipartimento di Fisica , Universit\`a di Bari,
Via G. Amendola 173, 70126 Bari, Italy}
\address{\ddag Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
Bari, Italy}
\begin{abstract}
This paper presents a review of the basic, model-independent
differences between the pre-big bang scenario, arising naturally in a
string cosmology context, and the standard inflationary scenario. We use
an unconventional approach in which the introduction of technical details
is avoided as much as possible, trying to focus the reader's attention on
the main conceptual aspects of both scenarios. The aim of the paper is
not to conclude in favour either of one or of the other scenario, but to
raise questions that are left to the reader's meditation. Warnings: the
paper does not contain equations, and is not intended as a
complete review of all aspects of string cosmology.
\end{abstract}
~~~~~~~~~~~~~Preprint BA-TH/99-367,
~~~~~~~~~~~~~E-print Archives: hep-th/0004149
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\section{Introduction}
\label{I}
The standard cosmological scenario \cite{Wein,0}, rightfully one of the best
celebrated conquests of the physics of the XX century, cannot be
extrapolated to arbitrarily high energy and curvature scales without
clashing with the singularity problem.
A singularity, on the other hand, often represents a signal that
the physical laws we are applying have been extrapolated outside
their domain of validity. As a well known example, we may quote here the
case of the spectral energy distribution of radiation in thermal
equilibrium. By applying the laws of classical physics one finds indeed
the Rayleigh-Jeans spectrum, that diverges like $\om^3$ at high
frequency. By taking into account instead the appropriate quantum
corrections, this classical singularity is regularized by the bell-like
Planck distribution, $\om^4\left(e^{\om/T}-1\right)^{-1}$, as
illustrated in Fig. 1.
\begin{figure}
\centerline{\epsfxsize=9.0cm
\epsffile{f1rev.ps}}
\caption{\sl The Planck distribution (full line) regularizes the classical
Rayleigh-Jeans prediction (dashed line), for the spectral energy density
of radiation in thermal equilibrium. The classical distribution is only
valid at low enough frequency scales.}
\label{fig1}
\end{figure}
String theory suggests that the initial curvature singularity of the
standard cosmological scenario could be similarly regularized by a
bell-shaped curve, as qualitatively illustrated in Fig. 2. As we go back
in time the curvature, instead of blowing up, could reach a maximum
controlled by the string scale, and then decrease towards a state of
very low curvature and weak coupling, approaching asymptotically the
so-called string perturbative vacuum. This behaviour, indirectly suggested
by duality and thermodynamical arguments \cite{1}, as well as,
independently, by the motion of strings in rolling backgrounds \cite{2}, is
naturally grounded on the duality symmetry of the cosmological string
effective action \cite{3}, and its possible implementation in the context of
a realistic model of the early Universe has a lot of dynamical and
phenomenological consequences \cite{4}.
In the string comology scenario of Fig. 2, the big bang singularity
is replaced by a phase of high but finite curvature. It comes thus natural,
in such a context, to call ``pre-big bang" the initial phase in which the
curvature is growing, in contrast to the subsequent ``post-big bang"
phase, with decreasing curvature and standard decelerated evolution.
\begin{figure}[b]
\centerline{\epsfxsize=9.0cm
\epsffile{f2rev.ps}}
\caption{\sl Curvature scale versus time for standard, inflationary (de
Sitter), and string cosmology models of the Universe. In the pre-big bang
scenario the classical curvature singularity is regularized, and the
standard cosmological evolution is only valid at late enough time scales.}
\label{fig2}
\end{figure}
The conventional inflationary scenario \cite{0,4a}, on the other hand,
suggests a picture of the early Universe which is approximately
intermediate between the standard and the string cosmology one. In that
context the curvature, instead of blowing up, is
expected to approach a nearly constant value, typical of a de Sitter-like
geometry (see Fig. 2). However, a de Sitter phase with
exponential expansion at constant curvature, implemented in the context
of the conventional, potential-dominated inflation, cannot be extended
back in time for ever, as first discussed in \cite{5}. Indeed, quoting Alan
Guth's recent survey of inflationary cosmology \cite{6}:
\bigskip
{\sl ``... Nevertheless, since inflation appears to be eternal
only into the future,
but not to the past, an important question remains open. How did
all start? Although eternal inflation pushes this question far into the
past, and well beyond the range of observational tests, the question
does not disappear."}
\bigskip
A possible answer to the above question, suggested by string cosmology
and represented graphically in Fig. 2, is that all started from a state
approaching, asymptotically, the flat, cold and empty
string perturbative vacuum. Even if this starting point is infinitely
far into the past, however, the initial state of our
Universe might be non completely beyond the range of present
observations, in contrast to the sentence quoted above, because the
starting point may affect the dynamics of the subsequent inflationary
evolution. The initial curvature scale, for instance, is constant or
decreasing according to the conventional inflationary scenario,
while it is growing according to the pre-big bang scenario, and this may
lead to important phenomenological consequences.
In spite of the existence of a few particular examples \cite{6a}, an
unambiguous regularization of the curvature singularity (to all orders in
the string effective action), toghether with a complete description of the
transition from the pre- to the post-big bang regime, is not an easy
achievement (see however \cite{6b} for recent encouraging results).
Also, it is fair to say that the pre-big bang models, at present, are not
free from other (more or less important) difficulties, that some aspects of
the pre-big bang scenario are still unclear, and that further work is
certainly needed for a final answer to all the difficulties.
Assuming that all the problems can (and will) be solved in a
satisfactory way, string cosmology will provide eventually a model of the
early Universe somewhat different, however, from the standard
inflationary picture. The aim of this paper, therefore, is to present in a
compact form a comparison (and a short discussion) of the way in which a
phase of inflation could be implemented within a cosmology based on the
string effective action, with respect to the standard cosmology based on
the Einstein equations. Basically all the differences arise, as we will see,
from the fact that in string cosmology the Universe starts evolving from
an initial state at very low curvature and weak coupling, while in
conventional inflation the initial state is assumed to approach the
Planckian, quantum gravity regime.
The paper is organized as follows. We will discuss kinematical and
dynamical differences in Sect. \ref{II},
quantum cosmology differences in Sect. \ref{III},
and phenomenological differences in Sect. \ref{IV}.
Sect. \ref{V} is devoted to some concluding remarks. We will avoid as
much as possible the introduction of technical details -- all contained in
the papers quoted in the bibliography -- trying to emphasize the ideas and
the main physical aspects of the different inflationary scenarios.
As already stressed in the Abstract, it seems appropriate to recall that
this paper is {\em not} intended as a complete review of all aspects of
string cosmology. Rather, the paper is narrowly focused on those aspects
where there is an important overlapping of methods and objectives and,
simultaneously, a strong contrast of basic assumptions, for the string
cosmology and the standard scenario. For a more exhaustive and
systematic approach, the interested reader is referred to the excellent
review paper devoted to a presentation and a technical discussion of all
presently existing (super)string cosmology models \cite{10a}, as well as
to two recent introductory lectures on the pre-big bang scenario
\cite{lec}.
\section {Kinematical and dynamical differences}
\label{II}
The idea of inflation hystorically was born \cite{7} to solve the
problem of monopoles that could be largely produced in the early
Universe, at the energy scale of grand unified theories ($GUT$). More
generally, inflation is now understood as a period of accelerated
evolution that can explain why the present Universe is so flat and smooth
over a so large scale of distances \cite{0,4a}.
For hystorical reasons, i.e. because inflation was first implemented as
a period of supercooling of the Universe trapped in a ``false" vacuum
state, in the context of $GUT$ phase transitions, inflation was first
associated \cite{7} to a phase of exponential
expansion (in cosmic time)
of the scale factor $a(t)$, corresponding to a de Sitter (or
``quasi-de Sitter") geometrical state. But it was soon realized that any
type of accelerated expansion ($\dot a >0$, $\ddot a >0$, where the dots
denote differentiation with respect to cosmic time), can in principle
solve the kinematic problems of the standard scenario \cite{8}.
Actually, even accelerated contraction ($\dot a <0$, $\ddot a <0$) is
effective to this purpose
\cite{9}, as expected from the fact that accelerated expansion can
be transformed into accelerated contraction through
an appropriate field redefinition, like the one connecting the String and
the Einstein frame. Indeed, physical effects such as the dilution of
inhomogeneities should be independent from the choice of the frame.
So, inflation can be generally identified, in a frame-independent way,
as a period of accelerated evolution of the scale factor (sign $\dot a=$
sign $\ddot a$).
There are, however, two classes of accelerated evolution very
different from a dynamical point of view, and depending on the
behaviour -- growing or decreasing in time -- of the curvature of the
space-time manifold. A first important point to be stressed is thus the
fact that the pre-big bang scenario corresponds to a phase of
accelerated evolution characterized by growing (or non-decreasing)
curvature \cite{3,4}, while the phase that we shall call
``standard inflation " \cite{0,4a} is characterized by a curvature scale
which tends to be constant -- in the limiting case of a de Sitter metric --
or slightly decreasing in time.
Before proceeding further, two remarks are in order. The first is that a
phase of accelerated evolution and growing curvature, also called
superinflation (or pole-inflation) \cite{12}, is not a peculiarity of string
cosmology, but is possible even in general relativity: in
higher-dimensional backgrounds, for instance, in the context of dynamical
dimensional reduction. The important difference is that, in string
cosmology, superinflation does not necessarily requires neither the
shrinking of the internal dimensions, nor some exotic matter source
\cite{11a} or symmetric breaking mechanism \cite{11b}: it can be simply
driven, even in three spatial dimensions, by the kinetic energy of the
rolling dilaton field $\phi$ \cite{3,4}, parametrizing the growth of the
string coupling $g=\exp (\phi/2)$ from zero (the string perturbative
vacuum), to the strong coupling regime, $g \sim 1$.
The second remark is that the words ``pre-big bang" should be referred
to the complete cosmological evolution from the initial
state approaching the string perturbative vacuum, up to the beginning of
the hot, radiation-dominated phase of the standard
scenario. Altough in most of this paper we shall restrict our discussion to
the low-energy part of the pre-big bang phase, appropriately described in
terms of the low-energy string effective action, the complete pre-big
bang history necessarily includes a high-curvature ``stringy" phase
\cite{13}, which may also be of the inflationary type \cite{17b}, and
which has a curvature expected to be, on the average, non
decreasing.
For a power-law, accelerated, conformally flat background, the
time-behaviour of the curvature scale follows the behaviour of the
absolute value of the Hubble parameter, $|H| =| \dot a /a|$. In such a
background, on the other hand, the inverse of the Hubble parameter (i.e.
the Hubble horizon $|H|^{-1}$) also controls the (finite) proper
distance between the surface of the event horizon and the wordline of a
comoving observer. Such a distance is shrinking for pre-big bang inflation
(where $|H|$ is growing), while it is non-decreasing in standard inflation.
As a consequence of the fact that the horizon is shrinking, and the
curvature is growing, it turns out that the initial state of the phase
of inflation, in the pre-big bang scenario, is characterized by a curvature
which is very small in Planck (or string) units, and by a Hubble horizon very
large in the same units (for the sake of simplicity we may identify,
at the end of inflation, the present value of the string length $L_s$ and of
the Planck lenght $L_p$; indeed, at tree-level, they are related by
\cite{12a} $L_s=\langle g\rangle L_p= \langle \exp \phi/2 \rangle L_p$,
with a present dilaton expectation value $\langle g \rangle \sim
0.3-0.03$). The initial state is also characterized by another very small
dimensionless number, the initial string coupling $g \ll 1$,
corresponding to a dilaton approaching the perturbative vacuum
value, $\phi \ra -\infty$.
By contrast, the standard inflationary scenario is characterized by a
dilaton already settled to its present vacuum expectation value;
the coupling is always strong ($g$ is of order one), and the size of
the initial horizon may be of order one in Planck units, if the initial
curvature approaches the Planck scale. These kinematical and dynamical
differences between standard and pre-big bang inflation are summarized
in Table I.
As evident from the pre-big bang curve of Fig. 2, the smaller is the
value of the initial curvature, the longer is the duration of the phase
of pre-big bang inflation. The request that the inflation phase be long
enough to solve the horizon and flatness problems \cite{7} thus imposes
bounds on the dimensionless parameters of the initial state, controlled
by the curvature and by the string coupling. Parameters such as the
typical size of an initial homogeneous domain, in Planck units, must be by
far greater than one for inflation to be successful. This aspect of string
cosmology was pointed out already in the first papers on the pre-big
bang scenario \cite{3,4} and, even before, also in the context of
string-driven superinflation \cite{2}.
\begin{table}
\tabcolsep .07cm
\renewcommand{\arraystretch}{2.0}
\begin{center}
\begin{tabular}{|c||c||c|}
\hline
& {\bf Standard Inflation} & {\bf Pre-big bang Inflation} \\ \hline
%\\[-0.75cm] \hline
Time evolution & {\sl accelerated} & {\sl accelerated} \\ \hline
Driving energy & {\sl inflaton potential} &
{\sl dilaton kinetic energy} \\ \hline
Curvature & {\sl constant or decreasing} & {\sl growing} \\ \hline
Event horizon & {\sl constant or growing} & {\sl shrinking} \\ \hline
Initial curvature scale & {\sl model-dependent} & {\sl arbitrarily small}
\\ \hline
Initial coupling & {\sl strong, non-perturbative} &
{\sl arbitrarily weak, perturbative} \\ \hline
\end{tabular}
\bigskip
\caption{ Kinematical and dynamical differences
between standard and pre-big bang inflation.}
\end{center}
\end{table}
The fact that the initial curvature and coupling are
small, and that the initial state is characterized by very large
dimensionless numbers, may be interpreted however as a possible
fine-tuning of the pre-big bang models \cite{14}, or even as a serious
drawback, preventing the solution of the flatness and homogeneity
problems, and supporting the conclusion that ``the current version of the
pre-big bang scenario cannot replace usual inflation" \cite{15}.
Consider, for instance, the horizon/homogeneity problem. At the onset of
pre-big bang inflation, as usual in an inflationary context, the
maximal allowed homogeneity scale is bounded by the size of the horizon.
The point is that the initial horizon size $H^{-1}$ is very large, instead of
being of Planckian order. The basic question thus becomes: is an initial
homogeneity scale of the order of the maximal scale $H^{-1}$ necessarily
unnatural if the initial curvature is small and, consequently, the
initial horizon is large in Planck (or string) units? In other words, which
basic length scale has to be used to measure the naturalness of the
initial homogeneous domain, which subsequently inflates to reproduce the
presently observed Universe? The Planck length or the radius of the causal
horizon?
The choice of the Planck length, emphasized in \cite{15}, is certainly
appropriate when the initial conditions are imposed in a state approaching
the high-curvature, quantum gravity regime, like in models of chaotic
inflation \cite{16}, for instance. In the pre-big bang scenario, on the
contrary, the initial conditions are to be imposed when the Universe is
deeply inside the low-curvature, weak coupling, classical regime. In
that regime the Universe does not know about the Planck length, and the
only available classical scale of distance, the horizon, should not be
discarded ``a priori" as unnatural \cite{21a}.
This does not mean, of course,
that the horizon should be always {\sl assumed} as the natural scale of
homogeneity. This suggests, however, that the naturalness of
homogeneity over a large horizon scale should be discussed on the
grounds of some quantitative and objective criterium, as attempted for
instance in \cite{17}, taking into account also the effects of quantum
fluctuations \cite{18} that could destroy the initial, classical homogeneity
(see also \cite{17a} for a discussion of ``generic" initial conditions in a
string cosmology context).
Another point concerns the flatness problem. In order to explain the
precise fine-tuning of the present density to the critical one, the
initial state of pre-big bang inflation must be characterized by large
dimensionless parameters, thus reintroducing from the beginning the
large numbers that one would like to explain.
This may seem to be quite unsatisfactory, as emphasized in
\cite{15}. It should be pointed out, however, that if one accepts the point
of view that large numbers are always to be avoided at the onset of
inflation, then should also accept the fact that natural initial
conditions are only possible in the context of models in which inflation
starts at the Planck scale, in order to have, for the curvature, an initial
dimensionless ratio of order one. This rules out, as a satisfactory
explanation of our present cosmological state, not only the pre-big bang
scenario, but any model in which inflation starts at scales smaller than
Planckian (unless we imagine a scenario with different stages of inflation,
each of them responsible for solving different problems, and occurring at
different scales: but, again, is this a natural cosmological configuration?)
Even for a single stage of inflation, occurring very near to the Planck
scale, we are not free from problems, however, as we are led eventually
to the following question: can we trust the naturalness of inflation
models in which classical general relativity is applied to set up initial
conditions at Planckian curvature scales, i.e. deeply inside the
non-perturbative, quantum gravity regime? (in string cosmology, the
Planckian regime directly affects the exit from the inflationary phase, and
only indirectly set constraints on the initial conditions, through the finite
duration of the low energy phase).
Assuming that the answer be positive, we are led to a situation that can
be graphically summarized, in a qualitative way, as in Fig. 3.
\begin{itemize}
\item{}
Case (a) represents a standard inflationary model of the Universe in
which inflation starts at the Planck scale. The time arrow points from
bottom to top, and the shaded area at the time $t_0$ represents a
spatial section of our present homogeneous Universe, of size fixed by
the present Hubble radius $H_0^{-1}$. As we go back in time, according
to the solutions of the standard cosmological model, the horizon shrinks
linearly in cosmic time, while the proper size of the present
homogeneous region, controlled by the scale factor $a(t)$, shrinks
slowly. When we reach the Planck scale, at the time $t_f$, the causal
horizon is smaller than the homogeneous region, roughly by the factor
$10^{-30}$.
To solve this problem, the phase of standard evolution is
preceeded by a phase of exponential de Sitter inflation, long enough
in time from $t_i$ to $t_f$, during which the curvature and the horizon
stay frozen at the Planck scale, and our present portion of the Universe
may ``re-enter" inside the causal horizon.
It should be stressed that, in a realistic inflationary model,
the horizon has to be slightly increasing from $t_i$ to $t_f$, because
the scale corresponding to our present Hubble radius has to cross the
horizon, during inflation, at a curvature scale $H_1$ smaller than
Planckian. We must require, in particular, that $H_1/M_p~ \laq~
10^{-5}$ in order to avoid too much amplification of gravitational
perturbations, that would contradict the present degree of
homogeneity observed by COBE \cite{19} at large angular scales.
However, for the sake of graphical simplicity, we shall ignore this
complication that is not essential for our present discussion.
\item{}
Case (b) represents a string cosmology model of the Universe, in which
the inflationary pre-big bang phase, from $t_i$ to $t_f$, is represented
in terms of the contracting metric of the Einstein frame \cite{9}, in
order to emphasize (graphically)
the underlying duality and time-reversal
symmetry of the scenario (there is no need, of course, that in the
Einstein frame the pre-big bang scale factor exactly coincide with the
time-reversal of the post-big bang solution).
The main difference from
case (a) is that in the pre-big bang epoch the curvature is growing, and
the event horizon shrinks linearly in cosmic time, from $t_i$ to $t_f$,
instead of being constant. Since the scale factor shrinks at a slower
rate, however, it is still possible for the initial homogeneous domain
to be ``pushed out" of the horizon, and for the Universe to emerge at
the Planck scale, at the time $t_f$, in the same configuration as in
case (a). The subsequent evolution from $t_f$ to $t_0$ is the same as in
the standard scenario.
\item{}
Case (c), finally, represents a string cosmology model in which the period
of pre-big bang inflation corresponds in part to a phase of growing
curvature, growing dilaton and shrinking horizon (from $t_i$ to $t_s$), and
in part to a phase in which the curvature, the horizon, and eventually the
dilaton, are frozen at the Planck scale (from $t_s$ to $t_f$). The initial
horizon is still large in Planck units, but it is no longer
reflection-symmetric to $t_0$, depending on the duration of the high
curvature phase from $t_s$ to $t_f$.
\end{itemize}
\begin{figure}
\centerline{\epsfxsize=9.0cm
\epsffile{f3rev.ps}}
\caption{\sl Qualitative evolution of the horizon scale and of the
proper size of a homogeneous region for (a) standard de Sitter
inflation with constant Hubble horizon, (b) pre-big bang inflation
(in the Einstein frame) with
shrinking horizon, and (c) pre-big bang including a
phase of high curvature inflation at the string scale.
The time
direction coincides with the vertical axis. The three horizontal
spatial sections $t_0, t_f$ and $t_i$
corresponds, from top to bottom, to the present
time, to the end, and to the beginning of inflation. The shaded
area represents the horizon, and the dashed lines its time
evolution. The full curves represent the time evolution of the proper size
of the homogeneous region, controlled by the scale factor.}
\label{fig3}
\end{figure}
According to \cite{15}, the model (a) represents an acceptable solution
to the horizon and flatness problem because inflation starts at the
Planck scale, and all the dimensionless parameters characterizing the
initial configuration are of order one. When the phase at constant
curvature of the model (c) extends in time like in case (a) the two
models pratically coincide for what concerns the naturalness of the
initial conditions, as in both cases our Universe emerges at the Planck
scale from a single domain of Planckian size, and we loose any
observational tracks of what happened before.
The aim of the pre-big bang scenario, on the other hand, is to attempt a
description of the possible cosmological
evolution {\sl before} the Planck epoch. The main difference between
case (a) and (c) is that, if the duration of the phase of inflation at
constant curvature is shorter than the minimal duration required
for a solution of all the standard kinematic problems, what happened
before the Planck scale may then become visible. In other words, there
are phenomenological consequences that can be ascribed to the phase of
pre-Planckian evolution, and that can be tested (at least in principle) even
today (see Section \ref{IV}). The model of case (b), in particular, is the
limiting case in which the duration of the high curvature phase shrinks to
a point, and the Universe emerges at the Planck curvature scale with a
homogeneous domain large enough to fill our present Hubble radius
through the subsequent standard evolution.
The above discussion
refers to the curvature in Planck units, but the same arguments can also
be applied to the initial value of the string coupling. If the
coupling is of
order one at the beginning of inflation then it is natural, in the sense
of \cite{15}, if it is much smaller than one, then the inflationary growth of
the coupling may have observable consequences.
To conclude this discussion it seems difficult, in our
opinion, to discard a model of pre-Planckian evolution, like those
illustrated in case (b), (c), only on the grounds of the large parameters
characterizing its initial conditions. Such an argument could be applied in
the impossibility of observational tests, namely in the absence of any
phenomenological evidence about the cosmological evolution before the
Planck era. But, as pointed out before, the pre-Planckian epoch becomes
invisible only in the limiting case (a), in which the effective models
reduces to standard inflation starting at the Planck scale, with no
unnatural initial conditions.
If, on the contrary, the high curvature phase
is shorter than in case (a), then the model requires an initial horizon larger
than Planckian, and a small initial coupling -- which are possibly unnatural
according to standard criteria \cite{15}. In that case, however, such initial
conditions are in principle accessible to present observations, so why do
not try to test the scenario observationally, and try to analyze the
naturalness in terms of a Bayesan approach, as attempted in \cite{17}? In
that case, the computation of ``a posteriori" probabilities suggests that
the observation of a large initial horizon and a small initial coupling may
become ``a posteriori" natural \cite{17}, because of the duality symmetry
intrinsic to the pre-big bang scenario.
Finally, even if the initial state should require a certain degree of
fine-tuning, this does not necessarily implies that the pre-big bang
cosmology described by string models is to be discarded (after all, the
description of our late-time Universe given by the standard cosmological
model is rather satisfactory, in spite of the fine-tuning required in such a
model if the initial state is extrapolated back in time until the Planck
epoch).
Usually, the need for fine-tuning in the initial conditions means
that the model is incomplete, and that a more general dynamical
mechanism is required, to explain the particular initial conditions.
Thus, it might well be that the pre-big bang picture provided by string
cosmology does not represents the whole story of our Universe, and that
only an earlier evolution can explain why, at a certain instant of time, the
Universe is lead to a state so similar to the string perturbative vacuum.
\section {Quantum cosmology differences}
\label{III}
In the standard inflationary scenario the phase of exponential,
de Sitter-like expansion at constant curvature cannot be infinitely
extended towards the past, for a well known reason of geodesic
completeness. A complete manifold requires an earlier contracting phase:
in that case, however, it seems impossible, in models dominated by the
inflaton potential, to stop the collapse and to bounce towards the
expanding phase \cite{5}. One has thus the problem of explaining how the
Universe could emerge at the Planck scale in the initial state appropriate
to exponential expansion.
At curvature scales of order one in Planck units we are in the full
quantum gravity regime, and the use of the quantum cosmology approach
seems to be appropriate. In this approach the Universe is represented by
a wave function satisfying the Wheeler-De Witt (WDW) equation \cite{20},
and evolving in the so-called superspace, whose points represent
all possible spatial geometric configurations. For pratical applications,
however, the evolution of the WDW wave function is usually studied in a
``minisuperspace" context, where only a finite numbers of coordinates
is chosen to parametrize the different geometrical configurations.
With an appropriate choice of the boundary conditions it is then possible
to obtain cosmological solutions of the WDW equation describing the
``birth of the Universe" as an effect of quantum tunnelling \cite{21,22}.
In that case, if the geometric state of the Universe is characterized by a
cosmological constant $\La$ (due, for instance, to the vacuum
energy-density induced by a scalar field potential), the tunnelling
probability is found
to be proportional to $\exp \left(-\La^{-1}\right)$, where $\La$
is measured in Planck units. The Universe tends thus to emerge in a
state of big vacuum energy, just appropriate to the onset of inflation.
The quantum cosmology approach seems thus to provide a natural
mechanism to explain the formation of ``baby universes", emerging at
the Planck scale, and ready to inflate according to the standard
inflationary scenario \cite{23}.
The minisuperspace approach to quantum cosmology is known to be
affected by various problems of technical nature: the probabilistic
interpretation, the unambiguous determination of an appropriate time
parameter, the semiclassical limit, the ordering of quantum operators,
and so on. The most unsatisfactory aspect of this approach, in our
opinion, is however the fact that the boundary conditions for the
tunnelling process are to be chosen ``ad-hoc". They are by no means
compelling, and it is possible indeed to impose different boundary
conditions, for instance according to the ``no boundary" criterium
\cite{24}, leading to a completely different result for the probability
of creation of universes -- results that are not always appropriate
to inflationary initial conditions.
The source of this problem is the fact that, in a quantum description of
the birth of the Universe, the final cosmological state (i.e. the
Universe that we want to obtain) is well known, while the initial
cosmological state (before the quantum transition) is completely
unknown, at least in the context of the standard inflationary scenario.
Indeed, the cosmological tunnelling is usually referred as a proces of
tunnelling ``from nothing" \cite{21}, just to stress the ignorance about
the initial vacuum state. The classical theory of the standard
cosmological scenario cannot help, because the initial
state, in that context,
is the big bang singularity, i.e. just what the quantum approach
would like to avoid.
In a string cosmology context, the quantum approach based on
minisuperspace can be implemented in a straightforward way, with the
only difference that the differential WDW equation represents the
Hamiltonian constraint following not from the Einstein action, but from the
low-energy string effective action \cite{25,26,27}. As a consequence, the
``minimal" minisuperspace is at least two-dimensional, because the action
always contains the dilaton, besides the metric. The formal problems
related to the minisuperspace approach remains, with the possible
exception of the operator-ordering problem, as the quantum ordering is
unambiguosly fixed by the global, pseudo-orthogonal $O(d,d)$ symmetry of
the low-energy string effective action \cite{25,27}. Another possible
exception is the identification of the time-like coordinate in
minisuperspace \cite{33a}.
There is, however, a radical difference for what concerns boundary
conditions. In the context of the pre-big bang scenario the initial,
asymptotic state of the Universe is unambiguosly prescribed -- the
string perturbative vacuum -- and cannot be chosen ``ad-hoc". Such
initial state is perfectly appropriate to a low-energy normalization of
the WDW wave function, and the transition probability of string
cosmology only depends on the dynamics, i.e. on the effective potential
appearing in the WDW equation.
It is now interesting to observe that if we compute, in the context of
the pre-big bang scenario, the transition probability from the
perturbative vacuum to a final, post-big bang configuration
characterized by a non-vanishing cosmological constant, we obtain
\cite{26} a probability distribution $P(\La)$ very similar to the one of the
conventional quantum cosmology, computed with tunneling boundary
conditions. The reason is that, by imposing the perturbative vacuum as
the boundary condition to the WDW equation, the WDW solutions contain
only outgoing waves at the singular spacetime boundary, just like in
the case of tunnelling boundary conditions \cite{21}. In this sense, we can
say that the ``ad-hoc" prescription of tunnelling boundary conditions
simulates, in a phenomenological way, the birth of the Universe from the
string perturbative vacuum. This suggests that, instead of ``tunnelling
from nothing", we should speak of ``tunnelling from the string
perturbative vacuum" or, even better, of quantum instability and ``decay"
of the perturbative vacuum \cite{28}.
A further, important difference should be mentioned. The
transition from the pre- to the post-big bang phase induced by the
cosmological constant (or, more generally, by an appropriate dilaton
potential), is represented, in the minisuperspace of string cosmology, not
like a tunnelling effet, but like a quantum reflection of the WDW wave
function, over an effective potential barrier. The correct description that
we obtain in string cosmology for the birth of the Universe, therefore, is
that of a ``quantum scattering" effect \cite{25,26,33a}.
The various differences between quantum inflationary cosmology and
quantum string cosmology are summarized in Table II. Besides the formal
aspects (such as tunnelling versus reflection), the basic difference is
that in the standard inflationary scenario the Universe, because of
quantum cosmology effects, is expected {\em to enter} in the
inflationary regime, while in the pre-big bang scenario the Universe is
expected {\em to exit} from the inflationary regime (or at least from
the phase of growing curvature). So, in standard inflation, quantum
effects at the Planck scale are expected to be responsible for
inflationary initial conditions. In string cosmology, on the contrary,
initial conditions are to be imposed in the opposite, low energy quantum
regime, where quantum effects are negligible.
\begin{table}
\tabcolsep .07cm
\renewcommand{\arraystretch}{2.0}
\begin{center}
\begin{tabular}{|c||c||c|}
\hline
& {\bf Quantum cosmology} & {\bf Quantum string cosmology}
\\ \hline
%\\[-0.75cm] \hline
Formal approach & {\sl WDW, minisuperspace} &
{\sl WDW, minisuperspace} \\ \hline
Quantum ordering & {\sl arbitrary } &
{\sl fixed by duality} \\ \hline
Boundary conditions & {\sl arbitrary } & {\sl string perturbative vacuum}
\\ \hline
Outgoing waves & {\sl tunnelling from nothing } & {\sl
reflection from pre-big bang} \\ \hline
Quantum transition & {\sl beginning of inflation} & {\sl
exit from inflation} \\ \hline
\end{tabular}
\bigskip
\caption{ Quantum cosmology differences
between standard and pre-big bang inflation.}
\end{center}
\end{table}
It seems appropriate, at this point, to comment on the fact that the
initial state of pre-big bang inflation seems to be characterized by a
large entropy $S$, if one assumes, as in \cite{15}, that the de
Sitter relation between entropy and horizon area remains valid also when
the horizon is not constant in time (or, in other words, if one assumes a
saturation of the bound provided by the holographic principle \cite{34a},
applied however to the Hubble horizon \cite{34b}). If $S$ is large in Planck
units, the probability that such a configuration be obtained through a
process of quantum tunnelling, $\exp (-S)$, is exponentially small, as
emphasized in \cite{15}. However, as stressed above, in string cosmology
quantum effects such as tunnelling or reflection are expected to be
effective {\em at the end} of inflation, and not {\em at the
beginning}, i.e. {\em not} to explain the origin of the initial state. A large
entropy of the initial state, in the weakly-coupled, highly-classical regime,
can only correspond to a large probability of such configuration, which is
proportional to $\exp (+S)$, like for every classical and macroscopic
configurations (not arising from quantum tunnelling).
Let us stress, finally, another important difference between
conventional quantum cosmology and quantum string cosmology. In string
cosmology quantum geometrical effects cannot be fully accounted for,
as fas as we limit ourself to a WDW equation obtained from the
low-energy string effective action. Indeed, when approaching the Planck
scale, the string theory action acquires (even at small coupling, i.e. at
tree-level in the quantum loop expansion) higher curvature correction
\cite{29}, weighed by the inverse string tension $\alpha'$. They are to be
included into the Hamiltonian constraint, and lead in general to a
higher-derivative WDW equation. This problem has been discussed in
\cite{30}, and it has been shown in \cite{31} that when the higher
curvature corrections appear in the form of an Eulero density, then the
WDW approach can only be applied to a dimensionally reduced version of
the theory.
The quantum cosmology results reported in this section refer, in
this sense, only to a model of ``low-energy" quantum string cosmology
\cite{28}. In the full quantum gravity regime, in order to include all the
higher-derivative contributions, the correct WDW equation should follow
not from the effective action, but possibly from a conformal, sigma-model
action \cite{43a}, which automatically takes into account all orders in
$\alpha'$. We note, however, that duality transformations in toroidal
moduli space have recently suggested \cite{30a} that the Lorenztian
structure of the low-energy minisuperspace may have an exact meaning
also in an $M$-theory context, even if the exact $M$-theory equations are
expected to be in general different.
In fact, when the curvature is large in string units ($\ap H^2 >1$), and also
the string coupling is large ($g^2=e^\phi >1$), we necessarily enter the
$M$-theory regime where new quantum effects are possible, such as a
copious production of higher-dimensional $D$-branes \cite{M1}. If the
curvature is small enough, the strong coupling regime of string
cosmology is then expected to be described by $11$-dimensional
supergravity theory, and the dilaton to be interpreted as the radius (i.e.,
the modulus field) of the $11$-th dimension \cite{M2}. In this context
string cosmology becomes $U$-duality covariant \cite{10a, M3,M4},
and the presence in the action of Ramond-Ramond fields may be helpful to
evade the problem of the curvature singularity \cite{M5,M6,M7}. The
singularity, in addition, could also disappear as a result of the embedding
of the low-energy solutions of string theory into a higher-dimensional
($d=11$) manifold \cite{M4,M8}.
These results seem to suggests that an appropriate quantum description
of the birth of the Universe will be probably achieved only within a full
$M$-theory approach to the strong coupling regime, in which the pre-big
bang acceleration is damped, the curvature is regularized, and the Universe
bounces back to the phase of standard evolution.
\section {Phenomenological differences}
\label{IV}
One of the most important (and probably also most spectacular)
phenomenological predictions of inflation is the
parametric amplification of metric (and of other different types of)
perturbations \cite{32}, and the corresponding generation of primordial
inhomogeneity spectra, directly from the quantum fluctuations of the
background fields in their vacuum state (see \cite{33} for a review).
Such fluctuations, when decomposed in Fourier modes, satisfy a
canonical Schrodinger-like equation, whose effective potential is
determined by the so-called`` pump field", which depends in its
turn on the background geometry.
It is then evident that different backgrounds lead to different pump
fields, to a different evolution of perturbations, and thus to different
spectra. In string cosmology, in particular, there are two main
properties of the background that can affect the final form of the
perturbation spectra. They are:
\begin{itemize}
\item{}
(A) the growth of the curvature scale;
\item{}
(B) the scalar-tensor (i.e. gravi-dilaton) nature of the background.
\end{itemize}
Property (A) has two important consequences. The first, that we will call
(A1), is that the pre-big bang scenario leads to metric perturbation
spectra growing with frequency \cite{34} (instead of being
flat, or decreasing,
like in standard inflation), because the spectral distribution of metric
perturbations tends to follow the behaviour of the curvature scale
at the time of the first horizon crossing. The second, that we will call
(A2), is that the growth of the curvature can also force the comoving
amplitude of perturbations to grow (instead of being frozen) outside the
horizon. This effect, implicitly contained in the earlier, pioneer studies
\cite{41a}, was first explicitly pointed out in \cite{35}, and only later
independently re-discovered in a string cosmology context \cite{36}. As
a further consequence of property (A) we should mention, finally, the fact
that perturbations are amplified in a final ``squeezed vacuum" state, and
not in a ``squeezed thermal vacuum" \cite{36a}.
Let us first discuss the second effect (A2). This effect, on one hand, is
interesting because it may lead to an amplification of
perturbations more efficient than in the standard inflationary scenario. On
the other hand, however, it is dangerous, because the perturbation
amplitude could grow too much, during the pre-big bang phase, so as to
prevent the application of the standard linearized approach, which
neglects effects of back-reaction \cite{33}.
Such an ``anomalous" growth of perturbations cannot be eliminated
by a change of frame, because the associated
physical (i.e. observable) energy density spectrum is obviously
frame-independent. However, the breakdown of the linear approximation
{\em is}, in general, {\em gauge-dependent}. For the particular case of
scalar metric perturbations, in three isotropic dimensions, the linear
approximation breaks down in the standard longitudinal gauge, but is
restored in a more appropriate ``off-diagonal" gauge \cite{37}, also
called ``uniform-curvature" gauge \cite{38}. Moreover, as a consequence
of a particolar form of duality invariance that appears explicitly in
the Hamiltonian approach to perturbation theory \cite{39}, the final
energy-density spectrum can always be correctly estimated by
neglecting the growing mode, provided one includes in the full Hamiltonian
both the contribution of the amplitude and of its conjugate momentum.
There are backgrounds, however, in which the growth of perturbations
remains too strong even after the elimination of all unphysical gauge
effects, and we have to limit ourselves to a restricted portion of
parameter space for the linear approximation to be valid. Even in this
case, however, the effect (A1) has two interesting consequences.
The first is that a growing spectrum leads to the
formation of relic backgrounds whose amplitude is higher at higher
frequency, where in general the backgrounds are also more easily
detectable. A typical example is the formation of a relic background of
cosmic gravitons which, in the frequency range of present resonant-mass
and interferometric detectors ($10^2-10^3$ Hz), could be up to eight
orders of magnitude stronger than expected in the context of standard
inflation \cite{34,40}. Thus in principle detectable, in a not-so-far future,
by the (planned) advanced version of the interferometric gravitational
antennas, or by spherical resonant detectors.
The second consequence is that the normalization of the peak of the
spectrum, at high frequency, is automatically controlled by the string
scale \cite{41}. The peak amplitude may be high enough to support a
picture in which all the radiation, that becomes dominant at the beginning
of the standard era, is produced through a process of parametric
amplification, directly from the quantum fluctuations of the vacuum
during the pre-big bang epoch \cite{41,42,50a}.
Indeed, for the background fields that interact more strongly
than gravitationally, the amplified fluctuations are expected to
thermalize, and their energy-density is expected to grow in time with
respect to the dilaton kinetic energy that was driving the background
during the phase of pre-big bang inflation. This possibility is absent in
the standard inflationary scenario, where the spectrum of perturbations
is decreasing, and the normalization of the spectrum is determined at low
frequency by the observation of the large scale CMB anisotropy
\cite{19}: the resulting energy-density of the fluctuations, in that case, is
by far too low to dominate, eventually, the post-inflationary background.
Up to now we have reported some phenomenological consequences of the
property (A).
For what concerns the property (B), i.e. the gravi-dilaton nature of the
background, we will quote here a peculiar string cosmology effect, the
amplification of the quantum fluctuations of the electromagnetic field
due to their direct coupling to the dilaton, according to the effective
Lagrangian density $ \sqrt{-g} e^{-\phi} F_{\mu\nu}F^{\mu\nu}$.
In general relativity the dilaton is absent, the Lagrangian is
invariant under a conformal rescaling of the metric, and the coupling of
the electromagnetic field to a conformally flat metric, typical of inflation,
can always be eliminated (unless the coupling is non-minimal and/or
violates $U(1)$ gauge invariance \cite{54a}). As a consequence, the
inflationary evolution of the metric background is unable to amplify the
electromagnetic fluctuations. In string cosmology, on the contrary, such
fluctuations are amplified by the accelerated growth of the dilaton (acting
as the pump field) during the pre-big bang phase. If the high-curvature
string phase is long enough, it is then possible to produce in this way the
``seeds", required for instance by the galactic dynamo, for the generation
of cosmic magnetic fields on a large scale \cite{43}. String cosmology thus
provides a possible solution to a longstanding astrophysical ``puzzle", i.e.
the generation of the primordial seed fields, through a mechanism which
is uneffective in the standard inflationary scenario.
This certainly represents an advantage with respect to the standard
scenario. The different amplification of perturbations, however, is also
asssociated to possible drawbacks. In particular, the fact that the
metric perturbation spectrum, in string cosmology, grows with a very
steep slope, and it is rigidly normalized at high frequency, makes
problematic the matching to the anisotropy observed at the present
horizon scale \cite{19}. The generation of the observed CMB anisotropy,
with the right spectrum, is instead one of the most celebrated results
of standard inflation \cite{0,4a,33}.
A possible solution to this problem, in string cosmology, comes from the
amplification of the fluctuations of the Kalb-Ramond axion, which is one
of the fundamental fields appearing in the string effective action, already
at low energy. Indeed, unlike metric perturbations, the axion
perturbations can be amplified with a rather flat spectrum \cite{51a} and,
through the (integrated) Sachs-Wolfe effect \cite{44}, they can induce the
temperature anisotropy observed in the CMB radiation on a large scale,
both in case of massless \cite{45} and massive \cite{46} axion fluctuations.
It is important to observe, in that case, that the slope
of the spectrum is no longer arbitrary, but rigidly determined by the
COBE normalization (imposed at low frequency),
and by the string normalization
(imposed at the opposite, high-frequency end of the spectrum). The
resulting slope turns out to be slightly increasing \cite{45,46}, but still in
agreement with the observational limits at the horizon scale \cite{47}. At
higher frequency scales, however, important differences from standard
inflation may appear in the peak structure of the spectrum \cite{55a}.
The possible axionic origin of the fluctuations of the CMB temperature is
thus expected to be confirmed (or disproved) in a very near future, by the
planned satellite observations.
\begin{table}
\tabcolsep .07cm
\renewcommand{\arraystretch}{2.0}
\begin{center}
\begin{tabular}{|c||c||c|}
\hline
& {\bf Standard Inflation} & {\bf Pre-big bang Inflation} \\ \hline
%\\[-0.75cm] \hline
Pump field & {\sl metric} &
{\sl metric and dilaton} \\ \hline
Spectrum (vs frequency) & {\sl flat or decreasing } &
{\sl increasing} \\ \hline
Amplitude outside horizon & {\sl frozen } & {\sl increasing} \\ \hline
Normalization & {\sl low frequency (COBE) } & {\sl
high frequency (string scale) } \\ \hline
Electromagnetic fields & {\sl unaffected } & {\sl
amplified by the dilaton} \\ \hline
CMB anisotropy & {\sl metric fluctuations} &
{\sl axion fluctations} \\ \hline
Dilaton fluctuations & {\sl absent} &
{\sl dilaton productions} \\ \hline
\end{tabular}
\bigskip
\caption{Amplification of vacuum fluctuations in the inflationary and
pre-big bang scenario.}
\end{center}
\end{table}
The main differences between standard and string cosmology inflation,
for what concerns cosmological perturbations, are summarized in
Table III. The last entry of the Table refers to the amplification of the
fluctuations of the dilaton background \cite{48}, another peculiar effect
of string cosmology, because the dilaton is absent in the standard
scenario. This effect, analogous to the amplification of tensor metric
perturbations, can be interpreted (in a second quantization approach) as a
process of dilaton production, which leads to the formation of a relic
background of cosmic dilatons. The background is subject to various
constraints, depending on the slope of the spectrum and on the mass of
the dilaton, but if dilatons are light enough \cite{49} (namely $m ~\laq
~10$ KeV), they are not yet decayed and could represent today a
significant fraction of the dark matter, that seems required to match
various astrophysical observations.
Detecting such a background, through the gravity-like
interactions of the dilatons at low-energy scales, is however a challenge
that seems beyond the possibilities of present technology \cite{57a},
unless the dilaton couples universally to macroscopic bodies, represents
a significant fraction of dark matter, and its mass lies within the
sensitivity band of gravitational detectors \cite{64a}.
\section {Conclusion}
\label{V}
The pre-big bang scenario provides a model, suggested and
supported by string theory, of the possible cosmological evolution before
our Universe emerged at the Planck scale.
If the subsequent post-Planckian evolution follows the standard
inflationary scenario, then initial conditions can be imposed at the Planck
scale, and are probably natural in the sense of \cite{15}, but any track of
what happened before disappears from our observational range. If, on the
contrary, inflation at the Planck scale is not too long, and inside our
present Hubble radius there are comoving length scales that crossed the
horizon during the low-curvature pre-big bang phase, then pre-Plackian
initial conditions are in principle accessible to present observations, and
their naturaleness can be discussed in terms of a Bayesan analysis
\cite{17}, based on ``a posteriori" probabilities. Quantum cosmology
methods can also be applied, taking into account however that quantum
effects are possibly important at the end, and not
at the beginning, of inflation.
We believe that
the possibility of looking back in the past before the Planck
era is the most distinctive aspect of string cosmology, with respect to
the standard inflationary cosmology. Concerning the possible tracks of
the pre-Planckian Universe, we have emphasized , in particular, three
effects, referring to observations to be performed $1)$ in a
not-so-far future, $2)$ in a near future, and $3)$ to observations already
(in part) performed. These effects are, respectively: $1)$ the production of
a cosmic gravity wave background, $2)$ the axion-induced anisotropy of
the CMB radiation, and $3)$ the production of seeds for the cosmic
magnetic fields.
These effects are not necessarily compatible among them (at least in the
``minimal" version of the pre-big bang models), and it seems thus possible
to test, and eventually exclude (or confirm) the pre-big bang scenario on
the grounds of its phenomenological consequences. Even if, as recently
stressed \cite{50}, the idea itself of inflation ``cannot be falsified", the
particular models can (and must) be tested, and the pre-big bang scenario,
after all, can be regarded as a particular, unconventional model of
primordial inflation.
It seems thus appropriate to stress, in conclusion, that the pre-big bang
scenario is not alternative to the idea of inflation, but only alternative to
a more conventional realization of inflation which, for hystorical reason,
is still deeply anchored to the standard big bang picture, where the
initial state must necessarily represent a very small, curve and dense
Universe. The effort of this paper aims at stimulating the reader's
meditation on the fact that this standard picture is a possibility, not a
necessity, and that quite different initial conditions are possible, and not
necessarily unlikely.
\bigskip
\ack
It is a pleasure to thank Gabriele Veneziano, for the long and fruitful
collaboration on many aspects of the string
cosmology scenario discussed in this paper.
\bigskip
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\end{document}