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\begin{document}
\title{\vskip-1.7cm \bf  Nonlocal action for 
long-distance modifications of gravity theory}
\date{}
\author{A.O.Barvinsky}
\maketitle \hspace{-8mm}{\em Theory Department, Lebedev 
Physics
Institute, Leninsky Prospect 53, Moscow 117924, Russia}
\begin{abstract}
We construct the covariant nonlocal action for recently 
suggested
long-distance modifications of gravity theory motivated 
by the
cosmological constant and cosmological acceleration 
problems. This
construction is based on the special nonlocal form of 
the
Einstein-Hilbert action explicitly revealing the fact 
that this
action within the covariant curvature expansion begins 
with
curvature-squared terms.
\end{abstract}

\section{Introduction}
\hspace{\parindent} The purpose of this paper is to 
suggest the
class of nonlocal actions for covariantly consistent 
infrared
modifications of Einstein theory discussed in 
\cite{AHDDG}. The
modified equations of motion were suggested to have the 
form of
Einstein equations
    \begin{eqnarray}
    M_P^2\,\Big(1+{\cal F}(L^2\Box\,)\Big)
    \left(R_{\mu\nu}-\frac12
    g_{\mu\nu}R\right)=\frac12\,T_{\mu\nu}  \label{1.1}
    \end{eqnarray}
with "nonlocal" inverse gravitational constant or 
Planck mass,
$M_P^2(\Box)=M_P^2\,\big(1+{\cal F}(L^2\Box\,)\big)$, 
being some
function of the dimensionless combination of the 
covariant
d'Alem\-bertian $\Box=g^{\alpha\beta}
\nabla_\alpha\nabla_\beta$
and the additional scale $L$ -- the length at which 
infrared
modification becomes important, $1/\sqrt{-\Box}\sim L$. 
If the
function of this dimensionless combination $z=L^2\Box$ 
satisfies
the conditions
    \begin{eqnarray}
    &&{\cal F}(z)\to 0,\,\,\,\,\,\,\,\,
    z\gg 1\nonumber\\
    &&{\cal F}(z)\to {\cal F}(0)\gg 1,
    \,\,\,\, z\to 0,                           \label
{1.2}
    \end{eqnarray}
then the long-distance modification is inessential for 
processes
varying in spacetime faster than $1/L$ and is large for 
slower
phenomena at wavelengthes $\sim L$ and larger. This 
opens the
prospects for resolving the cosmological constant 
problem,
provided one identifies the scale $L$ with the horizon 
size of the
present Universe $L\sim 1/H_0\sim 10^{28} cm$. Indeed, 
equations
(\ref{1.2}) then interpolate between the Planck scale 
of the
gravitational coupling constant $G_P=16\pi/M^2_P$ for 
local matter
sources of size $\ll L$ and the long distance 
gravitational
constant $G_{LD}=16\pi/M^2_P(1+{\cal F}(0))\ll G_P$ 
with which the
sources nearly homogeneous at the horizon scale $L$ are
gravitating. Therefore, the vacuum energy ${\cal E}$,
$T_{\mu\nu}={\cal E}g_{\mu\nu}$, of TeV or even 
Planckian scale
(necessarily arising in all conceivable models with 
spontaneously
broken SUSY or in quantum gravity) will not generate a
catastrophically big spacetime curvature incompatible 
with the
tiny observable $H_0^2$. This mechanism is drastically 
different
from the old suggestions of supersymmetric cancellation 
of ${\cal
E}$ \cite{Weinberg}, because it relies on the fact that 
the nearly
homogeneous vacuum energy gravitates very little, 
rather than it
is itself very small. It will generate the curvature 
$H^2\sim
G_{LD}{\cal E}\sim G_P\,{\cal E}/{\cal F}(0)$ which can 
be very
small due to large ${\cal F}(0)$.

Various aspects of this idea have been discussed in 
much detail in
\cite{AHDDG}. One formal difficulty with this 
construction was
particularly emphasized by the authors of \cite{AHDDG}. 
Point is
that for any nontrivial form factor ${\cal F}(L^2\Box\,)
$ the left
hand side of (\ref{1.1}) does not satisfy the Bianchi 
identity
and, therefore, cannot be generated by generally 
covariant action,
which undermines the consistency of the whole scheme.

In this paper we suggest to circumvent this problem by 
the
following simple observation. Point is that the long-
distance
infrared regime, which is crucial for the resolution of 
the
cosmological constant problem, implies not only the
long-wavelength but also the {\em weak field} 
approximation. This
means that the equation (\ref{1.1}) is literally valid 
only as a
first term of the perturbation expansion in powers of 
the
curvature. Therefore, the left hand side of (\ref{1.1}) 
should be
modified by higher than linear terms in the curvature, 
and the
modified nonlocal action $S_{NL}[\,g\,]$ generating 
these
equations should be found from the variational equation
    \begin{eqnarray}
    \frac{\delta S_{NL}[\,g\,]}{\delta g_{\mu\nu}(x)}=
    M_P^2\,g^{1/2}\Big(1+{\cal F}(L^2\Box\,)\Big)
    \left(R^{\mu\nu}-\frac12
    g^{\mu\nu}R\right)+{\rm O}\,[\,R_{\mu\nu}^2\,]. 
\label{1.3}
    \end{eqnarray}
Flexibility in higher orders of the curvature allows 
one to
guarantee the integrability of this equation and to 
construct the
nonlocal action as a generally covariant (but nonlocal) 
curvature
expansion. Here we explicitly present this construction 
along the
lines of covariant curvature expansion developed in
\cite{CPTII,basis} a number of years ago. As a starting 
point we
consider a special nonlocal form of the conventional
Einstein-Hilbert action revealing its basic property -- 
the
absence of a linear in metric perturbation part (on 
flat-space
background), which is apparently the classical analogue 
of tadpole
elimination technique in non-SUSY string models \cite
{tadpole}.
Then we introduce a needed long-distance modification 
by a simple
replacement of the nonlocal form factor in the 
curvature-squared
term of the obtained action. The paper is accomplished 
by a brief
discussion of the obtained result in brane induced 
model of
\cite{DGP} and its generalization to asymptotically 
deSitter
spacetimes.

\section{Nonlocal form of the Einstein action}
\hspace{\parindent} For simplicity, we start with the 
Euclidean
(positive-signature) asymptotically-flat spacetime in 
$d$
dimensions. The action of Einstein theory
    \begin{eqnarray}
    S_E[\,g\,]=-M_P^2
    \int dx\,g^{1/2}\,R(\,g\,)
    -2\,M_P^2\int_\infty\!
    d^{d-1}\sigma\,\Big(g^{(d-1)}\Big)^{1/2}\,
    \Big(K-K_0\Big)                 \label{2.1}
    \end{eqnarray}
includes the bulk integral of the $d$-dimensional 
scalar curvature
and the surface integral over spacetime infinity, 
$|x|\to\infty$,
with induced metric $g^{d-1}$. The latter is usually 
called the
Gibbons-Hawking action which in the covariant form 
contains the
trace of the extrinsic curvature of the boundary $K$ 
(with the
subtraction of the flat space background $K_0$). This 
surface term
guarantees the consistency of the variational problem 
for this
action which yields as a metric variational derivative 
the
Einstein tensor
    \begin{eqnarray}
    \frac{\delta S_E[\,g\,]}{\delta g_{\mu\nu}(x)}
    =M_P^2\,g^{1/2}
    \left(R^{\mu\nu}-\frac12
    g^{\mu\nu}R\right).           \label{2.2}
    \end{eqnarray}

The action is local and manifestly covariant, but it 
contains
together with the spacetime metric auxiliary 
structures -- as a
part of boundary conditions it involves at spacetime 
infinity the
vector field normal to the boundary and the 
corresponding
extrinsic curvature. As we will now see these 
structures can be
identically excluded from the action without loosing 
covariance,
but by the price of locality -- the local action will be
transformed to the manifestly nonlocal form which will 
serve as a
hint for constructing {\em covariant} long-distance 
modifications.

Another property of the action (\ref{2.1}) is that it is
explicitly linear in the curvature. However, this 
linearity is in
essence misleading, because the variational derivative 
(\ref{2.2})
is also linear in curvature and, therefore, it is at 
least linear
in metric perturbation on flat-space background $R_
{\mu\nu}\sim
h_{\mu\nu}$. Thus the flat-space perturbation theory 
for the
Einstein action should start with the quadratic order, 
${\rm
O}\,[\,h_{\mu\nu}^2\,]\sim {\rm O}\,[\,R_{\mu\nu}^2\,]
$. This is a
well known fact from the theory of free massless spin-2 
field. Our
goal is to make this $h_{\mu\nu}$-expansion manifestly 
covariant,
that is to convert it to the covariant (but generally 
nonlocal)
expansion in powers of the curvature. A systematic way 
to do this
is to use the technique of covariant perturbation 
theory of
\cite{CPTII,basis}. This technique begins with the 
derivation of
the expression for the metric perturbation in terms of 
the
curvature and in our context looks as follows.

Expand the Ricci curvature in metric perturbations on 
flat-space
background
    \begin{eqnarray}
    &&R_{\mu\nu}=-\frac12\,\Box\,h_{\mu\nu}+\frac12\,
    \nabla_\mu\!
    \left(\nabla^\lambda h_{\nu\lambda}
    -\frac12\,\nabla_\nu h\right)\nonumber\\
    &&\qquad\qquad\qquad\qquad\qquad\qquad
    +\frac12\,\nabla_\nu\!
    \left(\nabla^\lambda h_{\mu\lambda}
    -\frac12\,\nabla_\mu h\right)
    +{\rm O}\,[\,h_{\mu\nu}^2\,]         \label{2.3}
    \end{eqnarray}
and solve it by iterations as a nonlocal expansion in 
powers of
the curvature. This expansion starts with the following 
terms
    \begin{eqnarray}
    h_{\mu\nu}=-\frac2{\Box}R_{\mu\nu}
    +\nabla_\mu f_\nu+\nabla_\nu f_\mu
    +{\rm O}\,[\,R_{\mu\nu}^2\,].        \label{2.4}
    \end{eqnarray}
Here $1/\Box$ acting on $R_{\mu\nu}$ denotes the action 
of the
Green's function $G_{\mu\nu}^{\;\;\;\;\alpha\beta}(x,y)
$ of the
{\em covariant metric-dependent} d'Alembertian $\Box
\delta^{\alpha\beta}_{\mu\nu}\equiv
g^{\lambda\sigma}\nabla_\lambda\nabla_\sigma
\delta^{\alpha\beta}_{\mu\nu}$ on the space of symmetric
second-rank tensors with natural zero boundary 
conditions at
infinity
    \begin{eqnarray}
    &&\frac1{\Box}R_{\mu\nu}(x)\equiv
    \frac{\delta^{\alpha\beta}_{\mu\nu}}{\Box}R_
{\alpha\beta}(x)
    =\int dy\,
    G_{\mu\nu}^{\;\;\;\;\alpha\beta}(x,y)\,R_
{\alpha\beta}(y),
    \nonumber\\
    &&\Box_x
    G_{\mu\nu}^{\;\;\;\;\alpha\beta}(x,y)=
    \delta^{\alpha\beta}_{\mu\nu}\delta(x,y),\,\,\,\,
    G_{\mu\nu}^{\;\;\;\;\alpha\beta}(x,y)\,\Big|_
{\,|x|\to\infty}=0.
    \end{eqnarray}
In what follows we will not specify the tensor 
structure of the
Green's functions of $\Box$ implicitly assuming that it 
is always
determined by the nature of the quantity acted upon by 
$1/\Box$.

The term $\nabla_\mu f_\nu+\nabla_\nu f_\mu$ in (\ref
{2.4})
reflects the gauge ambiguity in the solution of (\ref
{2.3}) for
$h_{\mu\nu}$ (originating from the harmonic-gauge type 
terms in
the right-hand side of (\ref{2.3})), but its explicit 
form is not
important for our purposes here\footnote{The only 
important
property of this term is that this is a gauge 
transformation with
some gauge parameter $f_\mu\sim \nabla^\nu h_{\mu\nu}-
\nabla_\mu
h/2+{\rm O}\,[h_{\mu\nu}^2]$. Explicit gauge fixing 
procedure for
the equation (\ref{2.3}) becomes important in higher 
orders of
curvature expansion and it is presented in much detail 
in
\cite{CPTII,basis}.}.

Now restrict ourselves with the approximation quadratic 
in
$R_{\mu\nu}$ (or equivalently, $h_{\mu\nu}$) and 
integrate the
variational equation (\ref{2.2}) for $S_E[\,g\,]$. 
Since the
variational derivative is at least linear in $h_{\mu\nu}
$, $\delta
S_E/\delta g_{\mu\nu}\sim h_{\alpha\beta}$, the 
quadratic part of
the action in view of this equation is given by the 
integral
    \begin{eqnarray}
    S_E[\,g\,]=\frac12\int dx \,h_{\mu\nu}(x)
    \frac{\delta S_E[\,g\,]}{\delta g_{\mu\nu}(x)}
    +{\rm O}\,[\,R_{\mu\nu}^3\,].           \label{2.5}
    \end{eqnarray}
Substituting (\ref{2.2}) and (\ref{2.4}) and 
integrating by parts
one finds that the contribution of the gauge parameters 
$f_\mu$
vanishes in view of the Bianchi identity for the 
Einstein tensor,
and the final result reads
    \begin{eqnarray}
    S_E[\,g\,]=
    M_P^2\int dx\,g^{1/2}\,\left\{\,
    -\Big(R^{\mu\nu}
    -\frac12\,g^{\mu\nu}R\Big)\,
    \frac1{\Box}R_{\mu\nu}
    +{\rm O}\,[\,R_{\mu\nu}^3\,]\,\right\}.  \label{2.6}
    \end{eqnarray}
This is the covariant {\em nonlocal} form of the {\em 
local}
Einstein action which was originally observed in our 
previous
papers on braneworld scenarios with two repulsive branes
\cite{brane,nlbwa}. This nonlocal incarnation of (\ref
{2.1})
explicitly features: i) the absence of linear in 
curvature term
and ii) the absence of auxiliary structures associated 
with
spacetime infinity. Before we go over to the 
construction of
long-distance modifications of the theory, let us 
briefly dwell on
higher-order curvature terms. This, in particular, will 
clarify
the role played by the Gibbons-Hawking action in the 
subtraction
of the linear term.

In asymptotically-flat (Euclidean) spacetime with the 
asymptotic
behavior of the metric
    \begin{eqnarray}
    g_{\mu\nu}=\delta_{\mu\nu}+h_{\mu\nu},\,\,\,\,
    h_{\mu\nu}={\rm
    O}\,\left(\frac1{|x|^{d-2}}\right),\,\,|x|\to\infty,
    \end{eqnarray}
the the noncovariant form of the Gibbons-Hawking term 
in Cartesian
coordinates reads as
    \begin{eqnarray}
    &&S_{\rm GH}[\,g\,]\equiv-2\,M_P^2\int_\infty\!
    d^{d-1}\sigma\,\Big(g^{(d-1)}\Big)^{1/2}\,
    \Big(K-K_0\Big)\nonumber\\
    &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
    =M_P^2\int\limits_{|x|\to\infty} d\sigma^\mu\,
    \big(\partial^\nu
    h_{\mu\nu}-\partial_\mu h\Big).        \label{2.7}
    \end{eqnarray}
This surface integral can be transformed to the bulk 
integral of
the integrand $\partial^\mu\big(\partial^\nu
h_{\mu\nu}-\partial_\mu h\Big)$ -- the linear in $h_
{\mu\nu}$ part
of the scalar curvature. From the viewpoint of the 
metric in the
interior of spacetime this is a topological invariant 
depending
only on the asymptotic behavior $g^\infty_{\mu\nu}
=\delta_{mu\nu}+
h_{\mu\nu}(x)\,\big|_{\,|x|\to\infty}$. Similarly to 
the above
procedure this integral can be covariantly expanded in 
powers of
the curvature. Up to cubic terms inclusive this 
expansion
reads\footnote{Validity of this result can be checked 
either by
the direct $h_{\mu\nu}$-expansion of the right-hand 
side or by
systematically expanding $h_{\mu\nu}$ on the left hand 
side as
covariant series in the curvature, starting with (\ref
{2.4})
\cite{CPTII,basis}.}
    \begin{eqnarray}
    \int_\infty\! d\sigma^\mu\,
    \big(\partial^\nu
    h_{\mu\nu}-\partial_\mu h\Big)&=&
    \int dx\,g^{1/2}\left\{\,R
    -\Big(R^{\mu\nu}
    -\frac12\,g^{\mu\nu}R\Big)\,\frac1{\Box}R_{\mu\nu}
    \right.\nonumber\\
       &&+\frac12\,R\left(\frac1{\Box}
    R^{\mu\nu}\right)\frac1{\Box} R_{\mu\nu}
       -R^{\mu\nu}\left(\frac1{\Box}
    R_{\mu\nu}\right)\frac1{\Box} R\nonumber\\
       &&
       +\left(\frac1{\Box} R^{\alpha\beta}\right)
    \left(\nabla_\alpha\frac1{\Box} R\right)
    \nabla_\beta\frac1{\Box} R\nonumber\\
    &&-2\,\left(\nabla^\mu\frac1{\Box} R^{\nu\alpha}
\right)
    \left(\nabla_\nu\frac1{\Box}
    R_{\mu\alpha}\right)\frac1{\Box} R \nonumber\\
    &&
    -2\,\left(\frac1{\Box} R^{\mu\nu}\right)
    \left(\nabla_\mu\frac1{\Box}
    R^{\alpha\beta}\right)\nabla_\nu\frac1{\Box}
    R_{\alpha\beta}
    \left.+{\rm O}\,[\,R_{\mu\nu}^4\,]\,\right\}.     
\label{2.8}
    \end{eqnarray}
As we see, when substituting to (\ref{2.1}) its linear 
term
cancels the Ricci scalar part, the quadratic terms 
reproduce those
of (\ref{2.6}) and the cubic terms recover ${\rm
O}\,[\,R_{\mu\nu}^3\,]$. Obviously, this type of 
expansion can be
extended to arbitrary order in curvature.

\section{Long-distance modification of the Einstein 
action}
\hspace{\parindent} Long distance modification of the 
Einstein
action that would generate (\ref{1.3}) as the left-hand 
side of
the gravitational equations of motion now can be simply 
obtained
from the nonlocal form of the Einstein action (\ref
{2.6}). It is
just enough to make the following replacement in the 
quadratic
part of (\ref{2.6})
    \begin{eqnarray}
    \frac1{\Box}\rightarrow
    \frac{1+{\cal F}(L^2\Box\,)}{\Box}.  \label{3.1}
    \end{eqnarray}
Indeed, the subsequent variation of the Ricci tensor, 
$\delta_g
R_{\mu\nu}=-\frac12\Box\,\delta g_{\mu\nu}+\nabla_\mu
f_\nu+\nabla_\nu f_\mu$, in
    \begin{eqnarray}
    &&\delta_g\int dx\, g^{1/2}
    \Big(R^{\mu\nu}
    -\frac12\,g^{\mu\nu}R\Big)
    \frac{1+{\cal F}(L^2\Box\,)}{\Box}\,
    R_{\mu\nu}=\nonumber\\
    &&\qquad\qquad\qquad\quad 2\int dx\, g^{1/2}
    \Big(R^{\mu\nu}
    -\frac12\,g^{\mu\nu}R\Big)
    \frac{1+{\cal F}(L^2\Box\,)}{\Box}\;
    \delta_g R_{\mu\nu}+\,{\rm O}\,[\,R_{\mu\nu}^2\,]
    \end{eqnarray}
and integration by parts "cancel" the denominator of 
(\ref{3.1}),
whereas the contribution of gauge parameters $f_\mu$ 
vanishes, as
above, in view of the Bianchi identity. All commutators 
of
covariant derivatives with the $\Box$ in the form factor
(\ref{3.1}) give rise to the curvature-squared order 
which is
beyond our control. This recovers the Einstein tensor 
term of
(\ref{1.3}) with the needed "nonlocal" Planckian mass
$M_P^2\Big(1+{\cal F}(L^2\Box\,)\Big)$.

The result of the replacement (\ref{3.1}) can be 
rewritten so that
the contribution of $1$ in the denominator of the new 
form factor
is again represented in the usual local form of the 
Einstein
action (\ref{2.1}). Then, the long-distance 
modification takes the
form of the additional nonlocal term
    \begin{eqnarray}
    S_{NL}[\,g_{\mu\nu}\,]\,
    &=&\,S_E[\,g_{\mu\nu}\,]\nonumber\\
    &&-M_P^2\int dx\,g^{1/2}\,\left\{\,
    \Big(R^{\mu\nu}
    -\frac12\,g^{\mu\nu}R\Big)
    \frac{{\cal F}(L^2\Box\,)}{\Box}\,
    R_{\mu\nu}
    +\,{\rm O}\,[\,R_{\mu\nu}^3\,]\,\right\}.  \label
{3.2}
    \end{eqnarray}
This term is not unique though, because it is defined 
by a given
form factor ${\cal F}(L^2\Box\,)$ only in quadratic 
order, while
we do not have good principles to fix its higher-order 
terms thus
far.

This action is manifestly generally covariant. 
Therefore, its
variational derivative (the left hand side of the 
modified
Einstein equations) exactly satisfies the Bianchi 
identity,
    \begin{eqnarray}
    &&\nabla_\mu\frac{\delta S_{NL}[\,g_{\mu\nu}\,]}
    {\delta g_{\mu\nu}(x)}=\nonumber\\
    &&\qquad\qquad
    -M_P^2\,g^{1/2}\nabla_\mu\left[\,\Big(1+{\cal F}(L^2
\Box\,)\Big)
    \left(R_{\mu\nu}-\frac12
    g_{\mu\nu}R\right)+{\rm O}\,[\,R_{\mu\nu}^2\,]
\,\right]=0,
    \end{eqnarray}
and thus does not suffer from the concerns of \cite
{AHDDG}. The
commutator of the covariant derivative with the form 
factor
$\Big(1+{\cal F}(L^2\Box\,)\Big)$ gives rise to 
curvature squared
terms and cancels against ${\rm O}\,[\,R_{\mu\nu}^2\,]$.

\section{Discussion}
\hspace{\parindent} Obviously, due to nonlocality the 
action
(\ref{3.2}) inherits all the problems associated with 
acausality,
possible instabilities caused by ghost states, etc., 
intensively
discussed in \cite{AHDDG}. The construction of the 
above type was
done in Euclidean spacetime and, thus, requires analytic
continuation to the physical Lorentzian signature. The 
principles
of this continuation are far from being obvious, and 
might require
currently developing paradigms like holographic dS/CFT-
conjecture
\cite{dS/CFT}, the concept of time as a holographically 
generated
dimension in which the RG-flow from the ultra-violet to 
infrared
fixed points takes place \cite{AHDDG}. Hopefully, these 
new
developments can shed light on the physically reasonable
boundary-value problem for nonlocal form factors in the 
action
(\ref{3.2}).

Nonlocalities of the type (\ref{3.2}) are 
characteristic of a
certain class of braneworld models. They cannot arise in
Randall-Sundrum type of models with strictly localized 
zero modes,
because in these models nontrivial form factors 
basically arise in
the transverse-traceless sector of the action (as 
kernels of
nonlocal quadratic forms in {\em Weyl} tensor, see \cite
{nlbwa},
where apparently the nonlocal form of the Einstein 
action was
originally observed). It is important that, in contrast 
to these
models, the nonlocal part of (\ref{3.2}) is not 
exhausted by the
square of the Weyl tensor $(R_{\mu\nu}^2-\frac13 R^2)$ 
(in
contrast to the structure $(R_{\mu\nu}^2-\frac12 R^2)$ 
of
(\ref{3.2}) and contains the {\em conformal} sector. It 
is the
sector of the conformal (trace) mode which is 
responsible for the
potential resolution of the cosmological constant 
problem. This
sector gets dynamically involved in models with 
metastable
graviton, like the Gregory-Rubakov-Sibiryakov model 
\cite{GRS},
and Dvali-Gabadaze-Porrati model (DGP) \cite{DGP} (the 
latter
apparently also suffering from instabilities due to 
tachion ghost
modes \cite{DR} in more than one extra dimensions).

In particular, for the (4+1)-dimensional DGP model the 
(Euclidean)
form factor ${\cal F}(L^2\Box)$ is actually singular at 
$\Box\to
0$ and has the form \cite{DGP,DefDG,DGZ}
    \begin{eqnarray}
    M_P^2\,{\cal F}(L^2\Box)=
    \frac{M^3}{\sqrt{-\Box}},\,\,\,\,\,
    M^3=\frac{M_P^2}L,
    \end{eqnarray}
where $M\sim 10^{-21}M_P\sim 100$ MeV is a mass scale 
of the bulk
gravity as opposed to the Planckian scale of the 
Einstein term on
the brane $M_P\sim 10^{19}$ GeV. The branch point of 
this form
factor (which is well defined only in the Euclidean 
space with
Dirichlet boundary conditions at infinity, such that the
d'Alembertian $\Box$ is negative semi-definite) raises 
the issue
of its continuation to the physical spacetime and is 
apparently
related to different branches of cosmological solutions 
including
the scenario of cosmological acceleration \cite{DefDG}.

Thus, in contrast to tentative models of \cite{AHDDG} 
with finite
${\cal F}(0)\gg 1$, which only interpolate between two 
Einstein
theories with different gravitational constants $G_{LD}
\sim
G_P/{{\cal F}(0)}\ll G_P$, this model is anticipated to 
suggest
the mechanism of the cosmological acceleration. This 
implies the
replacement of the asymptotically-flat spacetime by the
asymptotically-deSitter one. For small values of 
asymptotic
curvature (as is the case of the observable horizon 
scale
$H_0^2/M_P^2\sim 10^{-120}$) the curvature expansion of 
the above
type seems plausible, although the effect of the 
asymptotic
curvature might be in essence nonperturbative. 
Therefore, the
above construction might have to be modified. In 
particular, the
Gibbons-Hawking term should be replaced by its
asymptotically-deSitter analogue and the expansion in 
powers of
the curvature should be replaced by the expansion in 
powers of its
deviation from the asymptotic value $R_{\mu\nu}-\frac1d
g_{\mu\nu}R_\infty$. This would introduce in the 
formalism as a
free parameter the asymptotic value of the curvature,
$R_\infty\sim H_0^2$ that might be related to the CFT 
central
charge, $c_{UV}=M_P^2/R_\infty\sim 10^{120}$ \cite
{AHDDG} -- the
number of holographic degrees of freedom in dS/CFT 
conjecture. The
resulting modifications in the above construction are 
currently
under study and will be presented elsewhere.

\section*{Acknowledgements}

The author would like to thank V.Mukhanov for helpful 
stimulating
discussions and is grateful for hospitality of the 
Physics
Department of LMU, Munich, where a part of this work 
has been done
under the grant SFB375. This work was also supported by 
the
Russian Foundation for Basic Research under the grant No
02-02-17054 and the grant for Leading scientific 
schools No
1578.2003.2.

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\end{document}
