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\title{Brane Worlds and the Cosmic Coincidence Problem}
\author{Massimo Pietroni}
\address{{\it
INFN -- Sezione di Padova,\\
Via F. Marzolo 8, I-35131 Padova, Italy}}
\maketitle
\begin{abstract}{
Brane world models with `large' extra dimensions with radii in the 
$r_l \sim 100\;\mu{\rm m}$ range and smaller ones at  
$r_s \leq (1 \,{\rm TeV})^{-1}$ have the potential to solve the cosmic 
coincidence problem, {\it i.e.} the apparently fortuitous equality between
dark matter and dark energy components today.
The main ingredient is the assumption of a stabilization mechanism fixing
the total volume of the compact submanifold, but allowing for
shape deformations.
The latter are associated with phenomenologically safe 
ultra-light scalar fields.
Bulk fields Casimir energy  naturally plays the role of dark 
energy, which decreases in time because
of expanding $r_l$. Stable Kaluza Klein states may play the role
of dark matter with increasing, $O(1/r_s)$, mass. 
The cosmological equations exhibit attractor solutions in which 
the global equation of state is 
negative, the ratio between dark energy and dark matter is constant and 
the observed value of the ratio is obtained for two large extra 
dimensions.

Experimental searches of large extra dimensions should take into account that,
 due to the strong coupling between dark matter and radii dynamics, the size 
of the large extra dimensions inside the galactic halo may be smaller than
 the average value  of $O(100\;\mu{\rm m})$.

}
\end{abstract}
%\pacs{PACS numbers: 11.10.Gh 11.10.Hi 11.25.-w\\
DFPD/02/TH/06,
%UPRF-2001-02}

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The energy content of the Universe is nowadays, more than ever, the
subject of intense research. The picture emerging is the
following \cite{parameters}; 
the total energy density of the  Universe equals the
critical value  $\rho_c \simeq (2 \cdot 10^{-3}\, {\rm eV})^4 \simeq
(100\, \mic)^{-4}$  ({\it i.e.} $\Omega \simeq 1$),
 non-relativistic dark matter (DM) accounts for at 
most forty percent of it ( $\Omega_{DM}\simeq 0.2 - 0.4$), 
and the rest is in the form of a smooth - {\it i.e.}
non-clusterized - component, named Dark Energy (DE) (plus a small amount
 of baryons, $\Omega_b \simeq 0.05$).
Luminosity-redshift measurements of supernovae Ia \cite{supernovae}, 
as well as the
resolution of the so called age  problem \cite{age},
agree with this picture and point towards a DE with negative  equation
of state $W=p/\rho$, where $p$ and $\rho$ are pressure and energy
density, respectively.  A viable DE candidate is then Einstein's
cosmological constant ($W=-1$), but more general fluids  with $W<0$
have been considered,
emerging, for instance,  from the dynamics of scalar fields with
appropriate potentials.  The old cosmological constant problem
\cite{Wein} now becomes double-faced.  On one hand we still have the
old puzzle of the cosmological constant -- or the  scalar field
potential-- being so tiny compared to any common particle physics
scale. On the other hand -- and this is the new face of the problem-- 
the presence of DE today
in roughly the same amount as DM energy density is 
embarassing. 
Indeed, the ratio between the two energy  densities
scales as
\[
\frac{\rho_{DM}}{\rho_{DE}}\sim a^{3 W}\;,
\] 
where $a$ is the scale factor of the Universe. So, the approximate
equality between the two  components today looks quite an amazing
coincidence in the cosmic history if DE has $W\neq 0$  as required
by observations.

In principle, brane world scenarios offer a suggestive contact with the DE
problem. Indeed, extra spatial  dimensions compactified to a size as
large as $100 \mic - 1 \mm$ have been shown to be a  viable
possibility, provided that no Standard Model (SM) field propagates through
them \cite{arkani}. 
It might then  be tempting to attribute the observed value of
the DE energy density to the Casimir energy  associated to some field
propagating in these large extra dimensions of size $r$, {\it i.e.}
$\rho_{DE} \simeq 1/r^4$. 

If the radius $r$ is stabilized, then the
energy density behaves  exactly as a cosmological constant, and we
have made no substantial progress with respect to any of the two problems
mentioned above. Indeed, we have no clue about why the Casimir energy
would be the only non-vanishing contribution to the cosmological
constant and, moreover, the equality between the latter and the matter
energy density would appear as fortuitous as before.

In order to gain some insight at least on the cosmic coincidence
problem it seems then necessary to make the radius
$r$ dynamical, so that it may evolve on a cosmological time-scale. 
But this turns out to be quite dangerous. 
The `radion' field, whose expectation value fixes $r$,
couples to the trace of the SM energy momentum tensor  with gravitational
strength. Moreover, in order to
evolve with $\dot{r}/r \sim H$ ($H$ being the Hubble parameter), it must be 
extremely light, $m \simeq H_0 \simeq 10^{-33}\;{\rm eV}$. It then behaves
as a massless Brans-Dicke scalar with $O(1)$ couplings to matter,
whereas present bounds are $O(10^{-3})$ \cite{will}.
Thus, the geometric explanation of a dynamical DE looks quite unlikely 
(see however \cite{albrecht}).

The purpose of this letter is to show that the above conclusion is
indeed premature, and that standard brane world scenarios with large extra 
dimensions may quite naturally solve the cosmic coincidence problem.
Unfortunately, we will add nothing
new with respect to the first problem, that of the vanishing of all 
present contributions to the
cosmological constant larger than $O(100\, \mic)^{-4}$.

The key point in our discussion is the realization that  besides 
large extra-dimensions in the $100 \,\mic$ range there could also be
smaller ones, compactified on a radius that is usually assumed to
be smaller than $(1 {\rm TeV})^{-1}$. Considering for simplicity 
only two compact subspaces, each caracterized by a single radius, we note that
the trace of the 
four-dimensional
energy-momentum tensor couples with the total {\it volume} of the compact 
space, that
is with the combination $r_s^{n_s} r_l^{n_l}$, where $r_{s,l}$ and $n_{s,l}$ 
are the radii and dimensionalities of the two subvolumes, 
respectively. 
If we assume a
stabilization mechanism that fixes the {\it total volume} ${\cal V}$
 of the compactified
manifold but {\it not its shape}, the potentially dangerous combination of 
radion fields associated
with volume fluctuations is made harmless, whereas the orthogonal one, 
associated with
shape deformation, is decoupled from normal matter and may then 
be ultra-light.
As a consequence, $r_l$ can grow on a cosmological
time-scale (such that the associated Casimir energy $1/r_l^4$
decreases) and at the same time $r_s$ shrinks so that ${\cal V}$ keeps a fixed
value.

The coincidence problem is solved if we
associate the dark matter with some stable state living on the
$4+n_s$-dimensional brane, because the resulting
cosmological  equations exhibit an {\it attractor solution} in which the
Casimir energy and $\rho_{DM}$ redshift at the
same rate with their ratios fixed at a $O(1)$ value. 

If the DM candidate has a mass
independent on $r_s$ then the attractor corresponds to the equation of
state of non-relativistic  matter, {\it i.e.} $W=0$, which is
disfavoured by cosmological observations.

On the other hand, if the mass of the DM particle is inversely proportional 
to some power of $r_s$, as for a Kaluza-Klein (KK)
state ($m_{DM} = m_{KK} \sim r_s^{-1}$), then
a more interesting behavior is obtained. Indeed, on  the attractor now we have
\[
\rho_{DM} \sim 1/(r_s a^3) \sim \rho_{DE} ;
\;,
\]
which, due to the shrinking of $r_s$, decreases slower than $a^{-3}$
during the cosmic expansion.  Remembering that a fluid with equation
of state $W$ scales as $\rho\sim a^{-3(W+1)}$ we see that the energy densities
scale now with a  {\it negative} effective
equation of state. Moreover, 
the values of $W$, $\Omega_{DM}$, and $\Omega_{DE}$ depend  only
on the two parameters, $n_s$ and $n_l$. 

The above results rely on naive
dimensional reduction, without assuming drastic modifications to the
radion  kinetic terms as was done for instance in \cite{albrecht}.  

For sake of clarity, we summarize our assumptions here. They are;

1) the vanishing of any contributions to the cosmological constant
larger than $O(100 \,\mic)^{-4}$ today;

2) a stabilization mechanism for the total volume (rather than for each single
 radius separately)
of the compactified space effective at low  energies.

We now show explicitly how the above mentioned conclusions follow.

The starting point is the action
\beq
S=S_{\rm bulk} + S_{4+n_s} + S_4\;.
\eeq
The bulk action is given by
\beq
S_{\rm bulk} = \frac{1}{16\pi G_{4+{\cal N}}}\int d^{4+{\cal{N}}} X 
\sqrt{-\cal G}
\left[R({\cal G_{A B}}) + \ldots\right]\;,
\label{sbulk}
\eeq
where ${\cal N}=n_s+n_l$ and the $4+{\cal N}$- dimensional metric is given by
\beqra
&&\ds
d s^2 = {\cal G}_{A B} d X^A d X^B \nonumber \\
&&= g_{\mu\nu} dx^\mu dx^\nu + \sum_{i=s,l} b_i(x)^2 
\gamma^{(i)}_{\alpha_i \beta_i} dy^{\alpha_i} dy^{\beta_i}\;,
%+ b_1(x)^2 \gamma^{(1)}_{a b} dy^a dy^b+ b_l(x)^2 \gamma^{(2)}_{i j} dz^i dz^j
\eeqra 
where $\mu,\nu =0,\ldots,3$, $\alpha_i, \beta_i = 1,\ldots, n_i$, and 
$b_i(x)=r_i(x)/r_i^0$, with $r_i^0$'s the average values of the radii today. 
The dots in eq. (\ref{sbulk}) represent
contributions from fields living in the bulk and from the bulk
cosmological constant, which we  do not need to specify here. Apart from the
overall $b_i$'s factors, 
the metrics $\gamma^{(i)}$'s are assumed to be non-dynamical.

$S_{4+n_s}$ is the action for the fields living in all the
 $4+n_s$ dimensions. In the following, we will only assume
that there is at least one stable KK state, which we will associate to DM
. 
The 4-dimensional action is
\beq
S_4=\int d^4x\sqrt{-g}\left[{\cal L}_{SM}(g_{\mu\nu}, \psi) +  V_{\rm
stab}(b_s^{n_s} b_l^{n_l})\right]\;,
\eeq
where $\psi$ represent SM fields  and $V_{\rm stab}$ is a phenomenological 
potential accounting for
the stabilization mechanism of 
the ${\cal N}$-dimensional compact volume. Close to its
minimum, it can be expanded as
\beq
V_{\rm stab} = \mu^4
\left(b_s^{n_s} b_l^{n_l}
-1\right)^2\;,
\eeq
with $\mu$ some mass
scale typically $O((G_{4+{\cal N}})^{-1/(2+{\cal N})})$.  
The dimensional reduction of $S_{\rm bulk}$ is a straightforward 
procedure (see for instance \cite{carroll}). We define new
`radion' variables, $\phi_a(x)$, as
$\ln b_i = A_{ia}\phi_a$
%\left(\frac{\cal N}{2 (2+{\cal N})}\right)^{1/2} \phi_j$
with $i= s,l$, $a=1,2$, and
\beq
A = \frac{1}{\sqrt{\cal N}} 
\left[
\begin{array}{cc}
\sqrt{\frac{2}{2+{\cal N}}} & \sqrt{\frac{n_l}{n_s}}\\
\sqrt{\frac{2}{2+{\cal N}}} & -\sqrt{\frac{n_s}{n_l}}
\end{array}
\right]\,.
\eeq
By also rescaling 
the 4-dimensional metric as 
\beq
g_{\mu\nu} = \e^{-C_{\cal N} \phi_1} \tilde{g}_{\mu\nu}\,,
\label{einstein}
\eeq
where $C_{\cal N}=\sqrt{2 {\cal N}/(2+{\cal N})}$,
we obtain the `Einstein frame' action,
\beqra
&& \ds S_{\rm bulk} = \int d^4x \sqrt{-\tilde{g}} \left[ \frac{1}{16 \pi G_4}
\tilde{R}(\tilde{g}_{\mu\nu}) \right. \;\;\;\;\;\;\;\;\;\;\nonumber \\ 
%+ \sum_{i=1}^2 \e^{-2 \alpha_i} R(\gamma^{(i)})
%\right.
%\nonumber\\
&& \ds \left. - \frac{M_p^2}{2} \sum_{a=1}^2 \tilde{g}^{\mu\nu}\partial_\mu
\phi_a \partial_\nu \phi_a +\ldots \right]\;, \;\;\;\;\;\;\;\;\;\;
\eeqra
where we assumed that both $R(\gamma^{(i)})=0$, as in toroidal 
compactifications.
%\[
%\eta_{ij}=\frac{n_i n_j}{2+{\cal N}} - n_i \delta_{ij} \,.
%\]
In the same frame, $S_4$ takes the form
\beqra
S_4= && \ds
\int d^4x \sqrt{-\tilde{g}} \e^{-2 C_{\cal N} \phi_1} 
\left[{\cal L}(\e^{- C_{\cal N} \phi_1} \tilde{g}_{\mu\nu}, \psi) 
\right.\nonumber\\
&&\ds \left.+ \mu^4 \left(\e^{C_{\cal N} \phi_1}-1\right)^2 \right]\;,
\eeqra
from  which we see that only $\phi_1$, which we might call `the volumon', 
couples to matter. The stabilizing potential $V_{\rm stab}$ gives
a mass $O(\mu^2/M_p)$ to it, which is phenomenologically safe for
$\mu \gta {\rm TeV}$ (no forces of range $\gta 1\, \mm$ are induced).
Moreover, in a cosmological setting, $\phi_1$ may be considered frozen at the 
minimum of 
its potential ({\it i.e.} $\phi_1 = 0$) as long as its mass is bigger than the 
Hubble parameter, that is, for $T \lta \mu$. In this situation, the `physical'
 and the Einstein frames are equivalent (see eq.~(\ref{einstein})). 

The orthogonal combination $\phi_2$ does not couple to SM fields directly. 
On the other hand, fields living in the bulk, or on the $4+n_s$-dimensional
brane, are sensitive to the shape of the compact manifold, and as a 
consequence they couple to the light $\phi_2$ fluctuations. In the 
4-dimensional
language, $\phi_2$ couples -- besides the graviton-- 
to {\it non-zero} KK modes, since their mass terms are proportional 
to $1/r_i \sim \exp(-A_{i2}\phi_2)$  ($i=s,l$).
Integrating the non-zero KK modes out leaves $O(r_{s,l}^{-4})$ 
contributions to the free 
energy. Consistently with our assumptions, we will neglect the 
$O(r_s^{-4})$ term (as well as the
the UV-divergent ones), since they must be cancelled by
the (unknown) mechanism solving the cosmological constant problem.
Thus, we are left with a potential for $\phi_2$ given by
\beq V(\phi_2)= \frac{B}{r_l^4}= \frac{B}{(r_l^0)^4} 
\exp\left(4 \sqrt{\frac{n_s}{n_l {\cal N}}} \phi_2\right)\,,
\label{potential}
\eeq
where we have fixed $\phi_1=0$ and
$B$ is a $O(1)$ coefficient depending on the particle content of the
$4+{\cal N}$-dimensional bulk, which we require to be such that $B>0$.

As we have anticipated, the other ingredient of our model is dark matter. 
We associate it to a stable KK state, whose mass scales as $O(1/r_s)$.

The important point is that the cosmological abundance of this 
non-relativistic relic will scale as
\beq
\rho_{DM} \sim r_s^{-1} a^{-3} \sim 
\exp\left(- \sqrt{\frac{n_l}{n_s {\cal N}}}
 \phi_2\right) a^{-3}\,.
\label{dm}
\eeq
Since the runaway potential, eq.~(\ref{potential}), pushes the field 
$\phi_2$ to
$-\infty$, the mass of the  dark-matter particle increases during the 
cosmological expansion, and its energy density redshifts less than for common 
dark matter. It is an example of varying mass dark matter, a possibility
that was considered in refs.~\cite{vamp}. 

The fact that $\phi_2$ is the canonically normalized version of a radion field
is the reason for the exponential dependences in (\ref{potential}) and 
(\ref{dm}). This is of crucial importance for what follows, because 
exponentials, once inserted in the cosmological equations, allow {\it scaling}
solutions, in which $\rho_{DE}$ and $\rho_{DM}$ redshift at the same rate.

Defining
\beq
\beta= 4 \sqrt{\frac{n_s}{n_l {\cal N}}} \,,\;\;\;\;\;
\lambda= \sqrt{\frac{n_l}{n_s {\cal N}}}\;,
\eeq
the cosmological equations are,
\beqra
&&\ds \ddot{\phi_2} + 3 H \dot{\phi_2} = - \beta \,V + \lambda \,\rho_{DM}
\nonumber\\
&&\ds 
H^2 = \frac{1}{3 M_p^2} \left(\rho_{DM} +\frac{M_p^2}{2}\dot{\phi_2}^2 +V
\right)
\nonumber\\
&&\frac{\ddot{a}}{a}= - \frac{1}{6 M_p^2}
\left( \rho_{DM} +2 M_p^2\dot{\phi_2}^2 -2 V \right)\;.
\label{friedmann}
 \eeqra
They admit a solution of the form 
\[ \phi_2 = \frac{-3}{\lambda+\beta} \log a \;, \]  
($a_0=1$)
which is such that $\rho_{\phi_2} \sim \rho_{DM} \sim a^{-3(W+1)}$, with the 
equation of state
\beq
W= \frac{-\lambda}{\lambda+\beta} =  
- \left(1 + 4 \frac{n_s}{n_l} \right)^{-1}\,,\;
\label{eos}
\eeq
and fixed ratio,
\beqra
\Omega_{DE} &&=\ds \frac{\rho_{\phi_2}}{ \rho_{DM} +\rho_{\phi_2}} = 
\frac{3 + \lambda(\beta +\lambda)}{(\beta+\lambda)^2}\,,\nonumber\\
&&=\frac{n_l ( 3 n_s {\cal N} + 4 n_s  + n_l )}{(4 n_s + n_l )^2}\;,
\eeqra
independent on the scale factor $a$. 
The solution above is an attractor over solution space if 
\beq
n_s > \frac{n_l (3 n_l - 4 )}{16 - 3 n_l}\;.
\label{attractor}
\eeq
Differently from the case of a 
cosmological constant, or even from quintessence models with inverse power 
law potentials, once the attractor is reached $\Omega_{DE}$ and $\Omega_{DM}$
 become 
independent on the cosmological era, thus solving the cosmic coincidence 
problem. Moreover, the equation 
of state (\ref{eos}) is negative, as required by observations.


The values of $W$ and $\Omega_{DE}$ on the attractor are 
functions of $n_s$ and $n_l$ only.
In Tab.~1 we list the possible values
of $\Omega_{DE}$, $W$ and of 
$H_0 t_0$, $t_0$ being the present age of the Universe. 
We limited the dimensionality of
the compact space according to the theoretical prejudice coming from string 
theory, {\it i.e.} ${\cal N} \le 6$.

\begin{quote}
\begin{tabular}{||c|c|c|c|c||}
\hline
$n_s$ & $n_l$ & $\Omega_{DE}$ & $W$ & $H_0 t_0$\\
\hline
\hline
1 - 5 &   1 &      $\le 0.44$  &  --  & -- \\
\hline
1     &   2 &  0.83            & $-1/3$ & 1 \\
\hline
2     &   2 &  0.68            &  -0.20& 0.83\\
\hline
3     &   2 &  0.60            &  -0.14& 0.78\\
\hline
4     &   2 &  0.56            & -0.11 & 0.75\\
\hline
1-3   &   3 &  $\ge 0.92$      &  --     &   --  \\
\hline
1-2   &   4 & no attractor& -- & -- \\
\hline
1    &5& no attractor & -- & -- \\
\hline
\hline
\end{tabular}  
\end{quote}
\begin{itemize}
\item[Tab.1] The values of $\Omega_{DE}$, $W$, and $H_0 t_0$ for 
different values of $n_s$ and $n_l$ such that ${\cal N}=n_s+n_l\le 6$.
\end{itemize}
%\begin{quote}
%\begin{tabular}{||c|c|c|c|c||}
%\hline
%$n_s$ & $n_l$ & $\Omega_{DE}$ & $W$ & $H_0 t_0$\\
%\hline
%\hline
%1 - 5 &   1 &      $\le 0.41$  &  --  & -- \\
%\hline
%1     &   2 &  0.92            & -0.20 & 0.83 \\
%\hline
%2     &   2 &  0.68            &  -0.20& 0.83\\
%\hline
%3     &   2 &  0.60            &  -0.20& 0.83\\
%\hline
%4     &   2 &  0.56            & -0.20 & 0.83\\
%\hline
%1-2   &   3 &  no attractor      &  --     &   --  \\
%\hline
%3   &   3 & 0.87 & -0.27  & 0.91 \\
%\hline
%1-2    & 4-5& no attractor & -- & -- \\
%\hline
%\hline
%\end{tabular}  
%\end{quote}
%\begin{itemize}
%\item[Tab. 2]
% Same as Tab.~1, but for $p=n_s/2$.
%\end{itemize}
The first noticeable fact about Tab.~1 is that the observed range for
the dark energy density, $0.6 \lta \Omega_{DE} \lta 0.8$ 
uniquely selects the number of `large' extra dimensions to be $n_l=2$, the
same value that is required by the totally unrelated issue of solving the
hierarchy problem with `millimeter' size extra dimensions \cite{arkani}.
 
Indeed, at the level of Tab.~1, we have not yet inserted any information
about absolute scales, such as $100 \,\mic$, TeV, $H_0$ and so on. We find it
a remarkable and inspiring `coincidence' that the observed balance between
$\Omega_{DM}$ and $\Omega_{DE}$ points to the same value,
$n_l=2$, obtained from scale dependent considerations, such
 as reproducing $H_0$ or
solving the hierarchy problem.
For  $n_l\neq 2$ we either find that the attractor corresponds to energy
densities outside the observed range or that the couplings $\lambda$ and 
$\beta$ lie outside the limit of eq.~(\ref{attractor}) and correspond to a 
different --unphysical --  attractor.

The values for $H_0 t_0$ listed in the table are obtained from
\[
H_0 (t_0 -t_{\rm att})  = \frac{2}{3(W+1)} 
\left( 1- a_{\rm att}^{\frac{3}{2}(W+1)}\right)\;,
\]
where $t_{\rm att}$ and $a_{\rm att}$ are the time and scale factor when the 
attractor is reached, assuming $a_{\rm att} < 10^{-2}$.
Taking  the $95\%$ 
confidence level lower limit on $t_0$ from globular cluster age estimates, 
$t_0 > 11 {\rm Gyr}$ \cite{age} we obtain the lower bound 
\[
H_0 t_0 > (0.71 - 0.79)
\]
 where we have varied $H_0$ in the range $(63 - 70)\,{\rm km \,s^{-1} 
Mpc^{-1}}$. The above bound 
is inconsistent with a flat, matter-dominated universe (for which 
$W=0$ and $H_0 t_0 = 2/3$) while it is satisfied by all the 
$n_l=2$ models in Tab.~1.

Supernovae Ia data, taken at face value, point towards a more negative equation
of state than those listed in Tab.~1, typically in the $W\lta -0.6$ range 
\cite{supernovae}. 
However, the analyses have been done assuming two fluids with
different equations of state, {\it i.e.}
matter ($W=0$) and `quintessence' ($W=W_x$), whereas in the present case the
two fluids scale with the same, negative, equation of state. 
 Since quintessence begins to dominate
the energy density quite recently, the negative equation of state takes over 
later than in our model, where it has been negative since much previous 
epochs. As a consequence, the supernovae bounds on the equation
of  state of the universe should be somehow relaxed in our model compared to 
quintessence.

Structure formation in a generic model of the type of eq.~(\ref{friedmann}) 
was studied in \cite{amendola} (where also the baryonic component was 
considered). There, it was shown that the non-zero -- and constant-- 
$\Omega_{DM}$ allows 
the growth of perturbations even if the expansion is accelerated. This behavior
is to be contrasted with the usual quintessence or cosmological constant case,
where the perturbations freeze out soon after the takeover because in that 
case $\Omega_{DM}$ drops quickly to zero.

The model we have presented in this letter is just the simplest version of a 
large family. In order to discuss its phenomenological implications thoroughly,
one should specify it in detail. Different dependences of the dark matter mass
on the small radius could also be envisaged, possibly leading to different 
predictions for the equation of state. Also, the couplings of the
KK sector to the SM are crucial. If the dark matter candidate is not a SM 
singlet, then its mass variation would induce variations in the gauge coupling
constants. Imposing the existing strong constraints on the latter
would imply that the $\phi_2$ field was frozen up to a not too distant 
past and that the 
attractor became effective quite recently.

It is, anyway, not trivial at all that the 
simplest choices, that is, tree-level dimensional reduction
 and the assumption of a 
$O(1/r_l^4)$ Casimir energy, lead to a model which solves the cosmic 
coincidence problem and is --if not fully compatible with-- at least very
close to the observations. 

A model-independent prediction of the framework outlined
in this letter is the presence of extra 
dimensions in the $100\, \mic$ range. Whereas for the solution of the hierarchy
problem discussed in \cite{arkani} this value was {\it allowed}, the link 
with the cosmological expansion discussed in this paper makes it a true
{\it prediction}. The expected value 
for $r_l$ is quite close to the present bound of
$200 \,\mic$, 
obtained from measurements of the Newton's law at small distances 
\cite{eotwash}. However, one cautionary remark is due on this point. 
The predicted value is determined by global 
observables, {\it i.e.} $H_0$ and $\ddot{a_0}/a_0$. Since the radius and DM 
are strongly coupled in this model, it is plausible that the local value of 
$r_l$, inside the galactic halo, would be somehow different. 
It is difficult to estimate the size of this effects, which would require a 
dedicated numerical study of the halo formation in this model. However, the 
sign would plausibly go in the direction of shrinking $r_l$ inside the halo. 
This is because the system would use the additional degree of freedom 
represented by the varying mass, to decrease the gravitational potential in 
overdense regions, thus expanding $r_s$ --and then shrinking $r_l$ -- with
respect to the average universe.

\vspace{0.5 cm}
It is a pleasure to thank Tony Riotto for useful discussions. This work was 
partially supported by European Contracts HPRN-CT-2000-00148 and
HPRN-CT-2000-00149.

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\end{document}

