%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % IOP Publishing Ltd and Adam Hilger Ltd % % Documentation for Latex style file ecssm.sty % % for producing CRC for Conference Proceedings % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Last updated 3 September 1997 % Comments/amendments to Electronic Production % E-mail prod2@IOPPUBLISHING.co.uk % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Character check % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ! exclamation mark " double quote % # hash ` opening quote (grave) % & ampersand ' closing quote (acute) % $ dollar % percent % ( open parenthesis ) close paren. % - hyphen = equals sign % | vertical bar ~ tilde % @ at sign _ underscore % { open curly brace } close curly % [ open square ] close square bracket % + plus sign ; semi-colon % * asterisk : colon % < open angle bracket > close angle % , comma . full stop % ? question mark / forward slash % \ backslash ^ caret (circumflex) % % ABCDEFGHIJKLMNOPQRSTUVWXYZ % abcdefghijklmnopqrstuvwxyz % 1234567890 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[ecssm]{article} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} %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 \pa {\partial} \def \ti {\widetilde} \def \se {{\prime\prime}} \def \ra {\rightarrow} \def \la {\lambda} \def \La {\Lambda} \def \Da {\Delta} \def \b {\beta} \def \a {\alpha} \def \ap {\alpha^{\prime}} \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 \bp {\dot{\beta}} \def \bpp {\ddot{\beta}} \def \fpu {\dot{\phi}} \def \fpp {\ddot{\phi}} \def \hp {\dot{h}} \def \hpp {\ddot{h}} \def \fb {\overline \phi} \def \rb {\overline \rho} \def \pb {\overline p} \def \fbp {\dot{\fb}} \def \ls {\lambda_{\rm s}} \begin{document} %%%%%%%start PREPRINT page %%%%%%%%%%%%%% \begin{flushright} BA-TH/02-454\\ gr-qc/0301032 \end{flushright} \vspace*{1.5 truein} {\Large\bf\centering\ignorespaces Phenomenology of the relic dilaton background \vskip2.8pt} {\dimen0=-\prevdepth \advance\dimen0 by23pt \nointerlineskip \rm\centering \vrule height\dimen0 width0pt\relax\ignorespaces 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\\ and \\ Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy\\ \par} \par \bgroup \leftskip=0.10753\textwidth \rightskip\leftskip \dimen0=-\prevdepth \advance\dimen0 by17.5pt \nointerlineskip \small\vrule width 0pt height\dimen0 \relax \vspace*{1truein} \centerline{\bf Abstract} \noindent We discuss the expected amplitude of a cosmic background of massive, non-relativistic dilatons, and we report recent results about its possible detection. This paper is a contracted version of a talk given at the 15th SIGRAV Conference on ``General Relativity and Gravitational Physics" (Villa Mondragone, Roma, September 2002). \vspace{2.5cm} \begin{center} ------------------------------ \vspace{0.5cm} To appear in Proc. of the \\ {\sl ``15th SIGRAV Conference on General Relativity and Gravitational Physics"},\\ Villa Mondragone, Monte Porzio Catone (Roma), September 9-12, 2002\\ Eds. V. Gorini et al. (IOP Publishing, Bristol, 2003) \end{center} \thispagestyle{plain} \par\egroup \vfill \newpage %%%%%%%%%%end PREPRINT page%%%%%%%%%%%%%% \setcounter{page}{1} \title{Phenomenology of the relic dilaton background} \author{Maurizio Gasperini\dag \ddag\footnote{E-mail: gasperini@ba.infn.it}} \affil{\dag\ Dipartimento di Fisica, Universit\`a di Bari, Via G. Amendola 173, 70126 Bari, Italy} \affil{\ddag\ Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy} \beginabstract We discuss the expected amplitude of a cosmic background of massive, non-relativistic dilatons, and we report recent results about its possible detection. \endabstract \section{The amplitude of the dilaton background and the dilaton coupling-strength} The aim of this talk is to discuss the production and the possible detection of a cosmic background of relic dilatons. The production is a well known string cosmology effect \cite{1}, so I will mainly concentrate here on the interaction of the dilaton background with a pair of realistic gravitational detectors \cite{2,3}. I will consider, in particular, the case of {\em massive} and {\em non-relativistic} dilatons, where some new result has recently been obtained \cite{4}. Let me start by recalling that the string effective action contains, already at lowest order, at least two fundamental fields, the metric and the dilaton, \beq S= -{1\over 2 \ls^2} \int d^4x \sqrt{-g} e^{-\phi} \left[ R+ \left(\nabla \phi\right)^2 + \dots\right]. \label{1} \eeq The dilaton $\phi$ controls the strength of all gauge interactions \cite{5} and, from a geometrical point of view, it may represent the radius of the $11$-th dimension \cite{6} in the context of M-theory and brane-world models of the Universe. What is important, for our discussion, is that during the accelerated evolution of the Universe the parametric amplification of the quantum fluctuations of the dilaton field may lead to the formation of a stochastic background of relic scalar waves \cite{1}, just like the amplification of (the tensor part of) metric fluctuations may lead to the formation of a relic stochastic background of gravitational waves \cite{7}. There are, however, two important differences between the graviton and dilaton case. The first is that the dilaton fluctuations are gravitationally coupled to the scalar part of the metric and matter fluctuations. Such a coupling may lead to a final spectral distribution different from that of gravitons. The second difference is that dilatons (unlike gravitons) could be massive. Since the proper momentum is redshifted with respect to the mass, then all modes tend to become non-relativistic as time goes on, and a typical dilaton spectrum should contain in general three branches: $\Om_1$ for relativistic modes, $\Om_2$ for modes becoming non-relativistic inside the horizon, and $\Om_3$ for modes becoming non-relativistic outside the horizon. The present energy-density of the background, per logarithmic momentum-interval and in critical units, can thus be written as follows \cite{1}: \bea && \Om_1(p,t_0)=\left(H_1/ M_{\rm P}\right)^2 \Om_r(t_0)\left(p/ p_1\right)^{\da}, ~~~~~~~~~~~~~~~~~~~~~~ m
p_1\sim (H_1/M_{\rm
P})^{1/2}10^{-4}$ eV, then all modes today are non-relativistic, and the
$\Om_1$ branch of the spectrum disappears. Note also that the
relativistic branch of the spectrum is typically growing \cite{1} in
minimal string cosmology models ($0<\da \leq 3$), while the
non-relativistic spectrum may have a flat or decreasing branch if $\da
\leq 1$.
The present amplitude of the non-relativistic spectrum is controlled by
two basic parameters, $H_1$ and $m$. In minimal string cosmology
models $H_1$ is typically fixed at the string scale, $H_1 \simeq M_{\rm
s}$, and the question {\em ``How strong is today the relic dilaton
background?"} thus becomes {\em ``How large is the dilaton mass?"}
We would like to have a mass small enough to avoid the decay and to
resonate with the present gravitational antennas, but large enough to
correspond to a detectable amplitude. It is possible?
The answer depends on the value of the dilaton mass. From the
theoretical side we have no compelling prediction, at present. From the
phenomenological side, however, we know that the allowed values of
mass (and thus of the range of the dilatonic forces) are strictly
correlated to the strength of the dilaton coupling to macroscopic
matter. We are thus lead to the related question: {\em ``How large is
the dilaton coupling?"}
For a better formulation of such a question we should recall that the
motion of a macroscopic body, in a gravi-dilaton background, is in
general non-geodesic, because of the direct coupling to the gradients
of the dilaton field \cite{2},
\beq
{du^\mu\over d\tau}+ \Ga_{\a\b}^\mu u^\a u^\b+q \nabla^\mu \phi=0,
\label{5}
\eeq
induced by the effective ``dilatonic charge" $q$ of the body, which
controls the relative strength of scalar-to-tensor forces. So, how large
is $q$?
The experimental value of $q$, as is well known, is directly constrained
by tests of the equivalence principle and of macroscopic gravity,
which provide exclusion plots in the $\{m, q^2\}$ plane
(see for instance \cite{11}). From a theoretical point of view, on the
other hand, there are two possible (alternative) scenarios. A first
possibility is that, by including all required loop corrections into the
effective action, the dilatonic charge $q$ becomes large and
composition-dependent \cite{12}. In that case one has to impose $m
\gaq 10^{-4}$ eV, to avoid contradiction with the present gravitational
experiments, which exclude testable deviations from the standard
Newton's law down to the millimeter scale \cite{13}. The alternative
possibility is that, when including loop corrections to all orders, the
resulting effective charge of ordinary macroscopic matter turns out to
be very small ($q \ll1$)and universal \cite{14}. In that case the dilaton
mass can be very small, or even zero. In the following discussion we
will accept this second (phenomenologically more interesting)
possibility, relaxing however the assumption of universal dilaton
interactions (which is not very natural, and which seems to require
fine-tuning).
We shall thus assume that the relic dilatons are weakly coupled to
matter, and light enough to have not yet decayed ($m \laq 100$ MeV),
so as to be available to present observations. The amplitude of the
background, as already noticed, depends however on $m$, and a larger
mass corresponds in principle to a stronger signal. In this range of
parameters, on the other hand, there is an upper limit on the mass
following from the critical density bound,
\beq
\int^{p_1} d(\ln p) \Om(p) \laq 1,
\label{6}
\eeq
which has to be imposed to avoid a Universe over-dominated by
dilatons. So, which values of mass may correspond to the stronger
signal, i.e. to an (almost) critical dilaton background?
The answer depends on the shape of the spectrum.
If $m>p_1$ and $\da \geq 1$, so that all modes are non-relativistic and
the spectrum is peaked at $p=p_1$, the critical bound reduces to
$\Om_2(p_1)\laq 1$, and implies
\beq
m<\left(H_{\rm eq}M_{\rm
P}^4/H_1^3\right)^{1/2}, \label{7}
\eeq
which, for $H_1 \sim M_{\rm s}\sim 10^{-1} M_{\rm P}$, is saturated by
$m \sim 10^2$ eV. This means that the density of the relic dilaton
background may approach the critical upper limit for values of mass
which are too large (as we shall see later) to fall within the resonant
band of present gravitational detectors. If $\da<1$ the above bound is
relaxed, but the mass is still constrained to be in the range $m>p_1$.
Let us thus
consider the case $m