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\begin{flushright}
CERN-TH/95-85 \\
DFTT-26/95 \\
April 1995
\end{flushright}
\vspace{15mm}
\begin{center}
{\grande Primordial Magnetic Fields}\\
\vspace{5mm}
{\grande from String Cosmology}
\vspace{10mm}
M. Gasperini \\
{\em Dipartimento di Fisica Teorica, Via P. Giuria 1, 10125 Turin,
Italy} \\
M. Giovannini and G. Veneziano \\
{\em Theory Division, CERN, CH-1211 Geneva 23, Switzerland} \\
\end{center}
\vspace{10mm}
\centerline{\medio Abstract}
\noindent
Sufficiently large seeds for generating the observed
(inter)galactic magnetic fields emerge naturally in string
cosmology from the amplification of electromagnetic
vacuum fluctuations due to a dynamical dilaton background.
The success of the mechanism
depends crucially on two features of the so-called
pre-big-bang scenario, an early epoch
of dilaton-driven inflation at very small coupling, and a
sufficiently long intermediate stringy era preceding
the standard radiation-dominated evolution.
%\vspace{5mm}
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%CERN-TH/95-85 \\
%April 1995
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\vspace{10mm}
\noindent
---------------------------------------------
\vspace{10mm}
To appear in {\bf Phys. Rev. Lett.}
\newpage
\setcounter{equation}{0}
It is widely believed that the observed galactic (and intergalactic)
magnetic fields, of microgauss strength, are generated and
maintained by
the action of a cosmic dynamo \cite{Parker}. The dynamo model,
as well as any other model, requires however
a primordial seed field; in spite of many
attempts \cite{Turner}-\cite{Harrison},
it is fair to say that no compelling mechanism has yet been
suggested, which would be able to
generate the required seed
field coherent over the Mpc scale, and with an
energy density to radiation density ratio
$\rho_{B}/\rho_{\gamma}~~\gaq~~~ 10^{-34}$ (possibly much greater,
according to a careful analysis of the turbolence of the
interstellar medium \cite{Kulsrud}).
A priori, an appealing mechanism for the origin
of the seed field is the cosmological amplification of the
vacuum quantum fluctuations
of the electromagnetic field, the same kind of mechanism
as is believed to generate primordial metric and energy
density perturbations. The minimal coupling of
photons
to the metric background is, however, conformally invariant
(in $d=3$ spatial dimensions). As a consequence, a
cosmological evolution involving a conformally flat metric
(as is effectively the case in inflation)
cannot amplify magnetic fluctuations, unless conformal
invariance is broken.
Possible attempts to generate large enough seeds thus include
considering exotic higher-dimensional scenarios, or coupling
non-minimally the electromagnetic field to the
background curvature
\cite{Turner} with some ``ad hoc" prescription, or breaking
conformal invariance at the quantum level through the so-called
trace anomaly \cite{Dolgov}.
In critical superstring theory the
electromagnetic field $F_{\mu\nu}$ is coupled not only
to the metric ($g_{\mu\nu}$), but also to the dilaton background
($\phi$).
In the low energy limit such an interaction is represented by the
string effective action \cite{Lovelace}, which reads, after
reduction from ten to four external dimensions,
\begin{equation}
S=- \int d^4x\sqrt{-g}e^{-\phi}\left( R +
\partial_{\mu} \phi \partial^{\mu} \phi
+ \frac{1}{4} F_{\mu\nu}F^{\mu\nu}\right) + ...
\label{action4}
\end{equation}
where $\phi = \Phi - \ln{V_6} \equiv \ln (g^2)$ controls the
tree-level four-dimensional gauge coupling ($\Phi$ being the
ten-dimensional dilaton field, and $V_6$ the volume of the
six-dimensional compact internal space) and the dots refer to
other moduli originating from the compactification.
In the inflationary models based on the above
effective action \cite{Veneziano,Gasperini}
the dilaton background is not at all constant, but
undergoes an accelerated evolution
from the string perturbative vacuum ($\phi= -\infty$) towards the
strong
coupling regime, where it is expected to remain frozen at its
present value. In this
context, the quantum fluctuations of
the electromagnetic field can
thus
be amplified $\it{directly}$ through their coupling to the dilaton,
according
to eq. (\ref{action4}). In the following we will discuss the
conditions
under which such a mechanism is able to produce
large enough primordial magnetic fields to seed the galactic
dynamo (a scalar-vector coupling similar to that of
eq. (\ref{action4}) was previously discussed in \cite{Ratra}, but
$\phi$ was there identified with the conventional inflaton
undergoing a dynamical evolution much different from the
dilaton evolution considered here).
Let us first define a few important parameters of the inflationary
scenario (also called ``pre-big-bang" scenario)
discussed in \cite{Gasperini}. The phase
of growing curvature and dilaton coupling
($\dot H>0$, $\dot\phi>0$), driven by the kinetic energy of the
dilaton field, is correctly described in terms of
the lowest order string effective action only
up to the conformal time $\eta=\eta_{s}$ at which the curvature
reaches the string scale $H_{s}=\lambda_{s}^{-1}$ ($
\lambda_{s}\equiv
\sqrt{\alpha^{\prime}}$ is the fundamental length of string theory).
A first important parameter of this cosmological model is thus the
value $\phi_s$ attained by the dilaton at $\eta=\eta_{s}$.
Provided such a
value is sufficiently negative, it is also arbitrary, since
there
is no perturbative potential to break invariance under shifts of
$\phi$.
For $\eta >\eta_{s}$ high-derivatives terms (higher
orders in $\alpha^{\prime}$)
become important in the string effective action,
and the background enters a genuinely ``stringy" phase of
unknown duration. It was shown in
\cite{Brustein} that it is impossible to have a graceful
exit to standard cosmology without such an intermediate
stringy phase. Such
stringy phase eventually ends at some conformal time
$\eta_1$,
in the strong coupling regime. At this time, the dilaton,
feeling a non-trivial potential, freezes to its present constant
value
$\phi=\phi_{1}$, and the standard radiation-dominated era starts.
The total duration $\eta_1/\eta_s$, or the total red-shift $z_s$
encountered during
the stringy epoch (i.e. between $\eta_s$ and $\eta_1$),
will be the second crucial parameter (besides $\phi_{s}$)
entering our discussion. For the purpose of this paper, two
parameters are enough to specify completely our model of
background, if we accept that during the string phase the
curvature freezes at the string scale, that is $H\simeq
\lambda_{s}^{-1}$
for
$\eta_s<\eta <\eta_1$.
We will work all the time in the String (also
called Brans-Dicke) frame, in which test strings move along
geodesic surfaces. In this frame the string scale $\lambda_{s}$
is constant, while the Planck scale
$\lambda_{P}= e^{\frac{\phi}{2}} \lambda_{s}$ grows from zero
(at the initial vacuum) to its present value,
reached at the end of the
string phase. We have explicitly checked, however, that all our results
also follow in the more commonly used (but less natural
in a string context) Einstein frame.
We shall now consider, in the above background, the amplification
of the quantum fluctuations of the electromagnetic field,
assuming that, at the very
beginning, it was in its vacuum state. In a four-dimensional,
conformally flat background,
the Fourier modes $A_{k}^{\mu}$ of the (correctly
normalized) variable
corresponding to the standard
electromagnetic field, and obeying canonical commutation
relations,
satisfy the equation
\begin{equation}
A_{k}^{\prime\prime}+
[k^2-V(\eta)]A_{k}=0~~~~,~~~~V(\eta)=
g(g^{-1})^{\prime\prime}~~~~,~~~~
g(\eta)\equiv e^{\frac{\phi}{2}}~~~~~~~.
\label{equazione}
\end{equation}
where a prime denotes
differentiation with respect to the conformal time $\eta$.
This equation is valid for each polarization component, and
is obtained from the action (\ref{action4}) with the gauge
condition
$\partial_{\nu}[e^{-\phi}\partial^{\mu}(e^{\frac{\phi}{2}}
A^{\nu})]=0$, which, for backgounds depending just on time, is
equivalent to the conventional
radiation gauge for electromagnetic waves in the vacuum.
The effective potential $V(\eta)$ grows from zero like
$\eta^{-2}$, for $\eta \rightarrow 0_-$, in the phase of
dilaton-driven inflation, is expected to reach some maximum
value during the string phase, and then goes rapidly to zero at
the beginning of the radiation-dominated era (where $\phi =$
const).
The approximate solution of eq.
(\ref{equazione}), for a mode $k$ ``hitting" the effective
potential barrier at $\eta=\eta_{ex}$, and
with initial conditions corresponding to vacuum
fluctuations, is given by:
\begin{eqnarray}
A_{k} &=&{e^{-ik\eta}\over \sqrt{k}}\;\;~~~,~~~~~
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\eta < \eta_{ex} \;
\nonumber \\
A_{k} &=& g^{-1}(\eta)
[C_k + D_k \int^{\eta} d\eta'~~ g^2(\eta^{\prime})] \; \;
{}~~~~,~~~~~\eta_{ex} < \eta < \eta_{re} \; \nonumber \\
A_{k} &=& {1\over \sqrt{k}}[ c_+(k) e^{-ik\eta} + c_-(k)
e^{ik\eta}]
\;\;~~~~~,~~~~~~~~ \eta > \eta_{re}
\label{soluzione}
\end{eqnarray}
where $\eta_{ex}$ and $\eta_{re}$ are the times of exit and
reentry of
the comoving scale associated with $k$, defined by the
conditions $k^2 = |V(\eta_{ex})| = |V(\eta_{re})|$ ($C, D,
c_{\pm}$ are integration constants). We are following here the
usual convention for which a mode in the underbarrier region is
referred to, somewhat improperly, as being ``outside the
horizon". Moreover, we
are considering a background in which the potential $V(\eta)$
keeps growing in the string phase until the final time $\eta_1$,
so that a mode crossing the horizon during dilaton-driven
inflation remains outside the horizon during the whole string
phase, i.e. $\eta_{re}\geq \eta_1$.
The Bogoliubov coefficients $c_{\pm}(k)$, determining the
parametric amplification of a mode $k<|V(\eta_1)|$, are easily
determined by matching these various solutions. One finds:
\begin{eqnarray}
{2ik}e^{ik(\eta_{ex} \mp \eta_{re})} c_\pm =
&\mp &
\frac{g_{ex}}{g_{re}}\left(-\frac{{g_{re}}^{\prime}}{g_{re}} \mp
ik\right)\pm
\frac{g_{re}}{g_{ex}}\left(-\frac{{g_{ex}}^ {\prime}}{g_{ex}}
+ik\right) \pm
\nonumber \\
&\pm &\frac{1}{g_{ex} g_{re}} \left(-\frac{{g_{ex}}^{\prime}}{g_{ex}}
+ik\right) \left(-\frac{{g_{re}}^{\prime}}{g_{re}} \mp ik\right)
\int_{\eta_{ex}}^{\eta_{re}} g^2 d\eta
\label{Bog}
\end{eqnarray}
Remembering that reentry occurs during the radiation epoch
in which the dilaton freezes to a constant value
($g^{\prime}_{re}\simeq 0, g_{re}\simeq 1$), it is easy to
estimate the
complicated-looking expression (\ref{Bog})
and to obtain, for the leading contribution,
the amazingly simple and intuitive result:
$|c_{-}|\simeq {g_{re}}/{g_{ex}}\equiv
\exp [-(\phi_{ex}-\phi_{re})/2]$.
An important feature of this result is that, for perturbations
which went out of horizon during the dilaton-driven phase,
the final result does not depend upon
the details of the background during the high curvature stringy
phase. This is because the perturbation evolves in a purely
kinematical way while outside the horizon.
We are thus trusting the large wavelength
part of our spectrum in spite of the present lack of
understanding of the stringy phase.
The coefficient $|c_-|$ defines
the energy density distribution ($\rho_{B}(\omega)$) over
the amplified fluctuation spectrum,
$ d\rho_{B}/d\ln\omega \simeq \omega^4
|c_{-}(\omega)|^2$ , where $\omega= k/a$ is the red-shifted,
present value of the
amplified proper frequency. We are interested in the ratio
\begin{equation}
r(\omega)=\frac{\omega}{\rho_{\gamma}} \frac{d\rho_{B}}
{d\omega}
\simeq
\frac{\omega ^{4}}{\rho_{\gamma}} |c_{-}(\omega)|^2 \simeq
\frac{\omega^{4}}{\rho_{\gamma}}
\left(g_{re}\over g_{ex}\right)^2~~~~~~,
\label{r}
\end{equation}
which measures the fraction of electromagnetic energy stored in the
mode
$\omega$ (in particular, for the intergalactic scale,
$\omega_{G}\simeq (1 {\rm Mpc})^{-1}\simeq 10^{-14}$Hz),
relative to the background radiation energy $\rho_{\gamma}$.
The ratio $r(\omega)$ stays
constant
during the phase of matter-dominated as well as
radiation-dominated
evolution, in which the universe behaves like a good
electromagnetic
conductor
\cite{Turner}.
In terms of $r(\omega)$ the condition for a large enough
magnetic field to seed the galactic dynamo is \cite{Turner}
$r(\omega_{G})\gaq 10^{-34}$.
Using the known value of $\rho_{\gamma}$ and $e^{\phi_{re}}$
we thus find, from eqs.(\ref{r}), the condition
$ g_{ex}(\omega_G) \laq 10^{-33}$.
In order to see whether or not the previous condition
can be fulfilled, we go back to our two-parameter
cosmological model.
The discussion is greatly helped by looking at {\bf Fig.1} where we
plot, on a double-logarithmic scale against the scale factor $a$,
the evolution of the
coupling strength (i.e. of $e^{\phi/2}$) and that of the
``horizon" size (defined here by $a |V|^{-1/2}$, whose behaviour
coincides with that of the Hubble radius $H^{-1}$
during the dilaton driven-epoch).
The horizon curve has an inverted trapezoidal shape,
corresponding to the fact that
$V=0$ during the radiation era, that $\dot\phi$ and $H$ are
approximately constant during the string era,
and that, during the dilaton-driven era
\cite{Veneziano,Gasperini},
\begin{equation}
a=(-t)^{\alpha},~~~~~ \alpha = - \frac {1}{\sqrt{3}}
\sqrt{1- \Sigma}, \,\,\,\,\,\,\,\,
a|V|^{-\frac{1}{2}}\simeq
a(t)\int_{t}^{0} dt^{\prime} a^{-1}(t^{\prime})
\simeq a^{{1\over \alpha}}~~~~~~~.
\end{equation}
Here
$\Sigma \equiv \sum_{i}{\beta_i^2}$ represents the
possible effect of internal dimensions, whose radii $b_i$ shrink
like $(-t)^{\beta_i}$ for $t \rightarrow 0_-$ (for the sake of
definiteness we show in the figure the case $\Sigma=0$).
The shape of the coupling curve corresponds to the fact that the
dilaton
is constant during the radiation era, that $\dot\phi$ is
approximately constant during the string era, and that it evolves
like
\begin{equation}
g(\eta) = a^{\lambda}, \,\,\,\,\,\,\,\,\,\,
\lambda =
\frac{1}{2}\left(3+\frac{\sqrt{3}}{\sqrt{1-\Sigma}}\right)
\end{equation}
during the dilaton-driven era \cite{Veneziano,Gasperini}
($\Sigma=0$ is the case shown in the picture). Notice that,
during the stringy phase, the dilaton keeps growing (at an
approximately constant and large rate) so that, ultimately,
one is lead into the strong coupling regime
in which the dilaton potential becomes important.
%\begin{figure}
%\centerline{\psfig{file=f1.eps,width=3.5in}}
%\caption{Evolution of the horizon scale $H^{-1}$ (thick lines), of
%the galactic scale ${\omega_{G}}^{-1}$ (thin solid line) and of the
%coupling $g=e^{{\phi}/{2}}$ (dashed lines). Dots on the latter
%lines show the values of $g_{ex}(\omega_{G})$ for three cases
%corresponding to different values of $z_{s}$, showing that, for a
%sufficiently fast variation of the dilaton during the string era,
%larger values of $z_{s}$ give a lower $g_{ex}(\omega_{G})$.}
%\end{figure}
We can now easily see when a sufficient amplification is achieved.
The galactic scale of length ${\omega_G}^{-1}$ was about $10^{25}$
in string (or Planck) units at
the beginning of the radiation era. By definition, at earlier times
it evolves as a straight line of slope 1 on our plot and
thus inevitably
hits the horizon curve sometimes during the string or the
dilaton-driven
era. At that time, the value of $g$ should have been smaller than
$10^{-33}$. One can easily convince her/himself that this is all but
impossible provided:
i) $z_s=\eta_s/\eta_1=
a_1/a_s$ is sufficiently large, and ii) the dilaton
evolution during the
string era is sufficiently fast. For the first condition a minimal
red-shift
$z_{s}$ of
$10^{10}$ is necessary; for the average
ratio ${\dot \phi / H}$ during the string era, a value
below but not too far from the one just before $\eta_s$ is
sufficient.
The combination of (i) and (ii) also implies that the coupling
at the onset of the string era has to be smaller
that $10^{-20}$ or so,
which thus supports the scenario advocated in \cite{Brustein} for
the gracious exit problem.
We can express our results more quantitatively
by showing the allowed region in the
$g_s$-$z_s$ plane in order to have sufficiently large seeds.
Considering the possibility of galactic scale exit during the string
or the dilaton-driven phase, we find from eq. (\ref{r}) that
$r(\omega_G)$ can be expressed, in the two cases, respectively
as:
\begin{equation}
r(\omega_{G})\simeq \left(\frac{\omega_{G}}
{\omega_{1}}\right)^{4}
e^{-\phi_{ex}(\omega_G)}\simeq
\left(\frac{\omega_{G}}{\omega_{1}}\right)
^{4 +\frac{\phi_{s}}{\ln{z_{s}}}}
{}~~~~~~,~~~\omega_{s}<\omega_{G}<\omega_{1}
\label{romega}
\end{equation}
and
\begin{equation}
r(\omega_{G})\simeq\left
(\frac{\omega_{G}}{\omega_{1}}\right)^{4-2\gamma}
z_{s}^{-2\gamma} e^{-\phi_{s}}
{}~~~~\,\,\,\,\,\,\,,~~~\omega_{G}<\omega_{s} \label{romega2}
\end{equation}
where $\gamma =\lambda\alpha/(\alpha -1)$,
$\omega_1=H_1a_1/a\simeq 10^{11}$Hz is the maximal
amplified frequency,
and
$\omega_{s}=\omega_{1}/z_{s}$. In the previous formulae
(\ref{romega}), (\ref{romega2}) we used the fact that, according
to our model of background, the transition scale $H_{1}$ has to
be of the order of the Planck mass $M_p$, so that
$\rho_{\gamma}(t) \simeq
H_{1}^4[a_1/a(t)]^4 ={\omega_1}^4$.
The resulting limits obtained by imposing $r(\omega_G)>10^{-34}$
are plotted in {\bf Fig. 2}, where they provide the right-side
border of the allowed region (the shaded area). The
previous spectrum, however, has been obtained using a homogeneous model
of background. It is thus valid provided the fluctuations remain, at
all
times, small perturbations of a nearly homogeneous configuration, with
negligible back-reaction on the metric (see also \cite{BG}), namely for
$r(\omega)<1$ at all $\omega$. This provides, according to
eqs.(\ref{romega}) and
(\ref{romega2}), the condition $\log_{10} g_s >- 2\log_{10}z_s$, which
determines the left border of the allowed region.
It should be mentioned that such allowed region is compatible with the
bounds following from the presence of strong magnetic fields at
nucleosynthesis time
\cite{Cheng}. Moreover, in the part of the allowed region in which
$r \gaq 10^{-8}$ the primordial fields can even seed directly the
galactic magnetic field \cite{Turner}, thus avoiding the necessity
of a dynamo and the related difficulties discussed in \cite{Kulsrud}.
%\begin{figure}
%\centerline{\psfig{file=f2.ps,width=3.5in}}
%\caption{The shaded area represents the allowed region determined by
%the conditions $r(\omega_G)>10^{-34}$ and $r(\omega)<1$, and defines
%the
%values of $z_s$, $g_s$ compatible with a large enough amplification
%of the
%electromagnetic vacuum fluctuations to seed the
%galactic magnetic field.}
%\end{figure}
We want to recall, finally, that our results were
obtained in the framework
of the tree-level, string effective Lagrangian.
We know that we could have corrections coming either from
higher loops
(expansion in $e^{\phi}$) or from higher curvature terms
($\alpha^{\prime}$
corrections ).
Since we work in a range of parameters where the dilaton is
deeply in his perturbative
regime ($\log_{10}g_s<-20$),
we expect our results to be stable against loop
corrections, at least for scales leaving the horizon during the
dilaton driven phase.
As to the $\alpha^{\prime}$ corrections, they are instead invoked
in the basic assumption that the dilaton-driven era ends when
the curvature reaches the string scale $\lambda_{s}^{-2}$, and
leads to a quasi-de Sitter epoch.
It should be clear however that, once such an assumption is made,
the detailed way in which it is implemented will not affect the
behaviour
of perturbations which stay outside of the horizon throughout the
high-curvature phase. Such perturbations are frozen during that phase
and their evolution is merely kinematical.
In conclusion, our predictions for the large-wavelength part of the
spectrum should be regarded as more rubust than those
pertaining to shorter scales.
\section*{Acknowledgements}
It is a pleasure to thank R. Brustein and V. Mukhanov for a fruitful
collaboration
on the spectral
properties of metric perturbations in string cosmology, which
inspired part of this work.
\section*{Note added}
While this paper was being written, we received a paper by D.
Lemoine and M. Lemoine, ``Primordial magnetic fields in string
cosmology", whose content overlaps with ours where
the effects of dilaton-driven inflation on the amplification of
electromagnetic perturbations are concerned.
Their model of background does
not include, however, a sufficiently long, intermediate stringy
era
whose presence is crucial to produce the large
amplification discussed here.
\newpage
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\end{document}