Using developed earlier our methods for multidimensional models \cite{M1,M2,M3} a family of cosmological-type solutions in $D$-dimensional model with two sets of scalar fields $\vec{\varphi}$ and $\vec{\psi}$ and exponential potential depending upon $\vec{\varphi}$ is considered. The solutions are defined on a product of $n$ Ricci-flat spaces. The fields from $\vec{\varphi}$ have positive kinetic terms and $\vec{\psi}$ are "phantom" fields with negative kinetic terms. For vector coupling constant obeying $0< \vec{\lambda}^2 < (D-1)/(D-2)$ a subclass of non-singular solutions is singled out. The solutions from this subclass are regular for all values of synchronous "time" $\tau \in ( - \infty, + \infty )$. For $\vec{\lambda}^2 < 1/(D-2)$ we get an asymptotically accelerated and isotropic expansion for large values of $\tau$.
gr-qc0504044.tex


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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{document}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{center} \large \bf

 Non-singular solutions in multidimensional
    model with scalar fields and exponential potential

\end{center}

\vspace{0.63truecm}

\bigskip


\begin{center}

 \normalsize

   J.-M. Alimi\fnm[1]\fnt[1]{jean-michel.alimi@obspm.fr},

 \bigskip

  Laboratoire de l'Univers et de ses Theories
  CNRS UMR8102,  Observatoire de Paris
  92195, Meudon Cedex, France,

 \bigskip


   V.D. Ivashchuk\fnm[2]\fnt[2]{rusgs@phys.msu.ru}
   and
   V.N. Melnikov\fnm[3]\fnt[3]{melnikov@phys.msu.ru},

 \bigskip

  Center for Gravitation and Fundamental Metrology,
  VNIIMS and  Institute of Gravitation and Cosmology,
  Peoples' Friendship University of Russia,
  3-1 M. Ulyanovoy Str., Moscow, 119313, Russia


 \end{center}

 \bigskip


 \begin{abstract}

 Using developed earlier our  methods for multidimensional models
  \cite{M1,M2,M3} a  family of  cosmological-type solutions
  in $D$-dimensional model with  two
  sets of scalar  fields  $\vec{\varphi}$ and $\vec{\psi}$
  and exponential potential depending  upon $\vec{\varphi}$ is considered.
  The solutions are defined on a product of $n$ Ricci-flat spaces.
  The fields from $\vec{\varphi}$ have  positive kinetic terms and
  $\vec{\psi}$ are "phantom" fields with  negative kinetic terms.
  For  vector coupling constant obeying $0< \vec{\lambda}^2 < (D-1)/(D-2)$
  a subclass of non-singular solutions is singled out. The solutions
  from this subclass are regular for all values of
  synchronous  "time"  $\tau \in ( -  \infty, + \infty )$.
  For  $\vec{\lambda}^2 < 1/(D-2)$ we get an asymptotically accelerated and
  isotropic expansion for large values of $\tau$.

 \end{abstract}

  \np

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \section{Introduction}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 Recently,  the discovery of the cosmic acceleration
 \cite{Riess,Perl} stimulated a lot of papers, on multidimensional
 cosmology in particular, with the aim to explain this phenomenon
 using certain
 multidimensional models, e.g. those of superstring or supergravity
 origin (see \cite{Town,IMS,Chen1,Ohta,IMS1,IMK}, and refs. therein).


 At present rather popular models are those with  multiple
 exponential potential of the scalar fields
 (see \cite{RP,CopLW,BCopN,StarSah} and refs. therein).
 %They are of current interest mainly due to
 %"quintessence" approach \cite{CDS}.
 Potentials of such type  arise naturally in certain
 supergravitational models \cite{Town}
 and in sigma-models \cite{IMsigma1,IMC},
 related to configurations with $p$-branes (for a review see also
  \cite{IMSam,IMtop,IMerice}).

  Here we are interested in non-singular solutions
  (e.g. with a bounce) that appear in certain cosmological models, see
  \cite{AWands,BrF} and refs. therein. Such non-singular solutions
  are not new ones.
  For pioneering papers with non-singular solutions based on scalar fields
  see \cite{bm69,SM} - for conformal massless scalar field and cosmological
  constant, \cite{ZKol} - for phantom scalar fields, \cite{MelOrl} - for
  conformal Higgs-type scalar field with spontaneous symmetry breaking
  (SSB) back reaction, \cite{Starpol} - for a scalar field with
  polarization of vacuum (PV) effect, \cite{bmedd} - for a scalar field
  with SSB and PV etc. See all these results also in \cite{SM}.


 In this paper we consider the $D$-dimensional model governed by the
 action

 \bear{1.1}
     S =  \int_{M} d^{D}z \sqrt{|g|} \{
     {R}[g] - g^{MN} \partial_{M} \vec{\varphi} \partial_{N}
              \vec{\varphi}
            \\ \nonumber
             + g^{MN} \partial_{M} \vec{\psi} \partial_{N}
            \vec{\psi}   - 2 V(\varphi) \},
 \ear
     with the scalar potential

 \beq{1.2}
     V(\varphi) =  \Lambda \exp(2 \vec{\lambda} \vec{\varphi} ).
 \eeq

  Here   $\vec{\varphi} = (\varphi^1, \dots, \varphi^k)$
         is a set (vector) of scalar fields,
         $\Lambda $ is a constant,
         $\vec{\lambda} = (\lambda_1, \dots, \lambda_k) \in  \R^k$
         is a vector of dilatonic couplings,
         $\vec{\psi}= (\psi^1, \dots, \psi^m) $  is a
         set (vector) of "phantom" scalar fields,
         $g = g_{M N} dx^{M} \otimes dx^{N}$ is metric,
         $|g| = |\det(g_{M N})|$.

 We note that phantom scalar fields (with negative
 kinetic terms)  appear in various multidimensional
 models, e.g. in a field model for $F$-theory
 \cite{KKLP}, in a chain of $B_D$-models
 in dimensions $D \geq 11$  \cite{IMJ}
 ($B_{11}$ corresponds to  $M$-theory \cite{M-th}
  and $B_{12}$ to $F$-theory \cite{Vafa}).

 The article is devoted mainly to cosmological type
 solutions with the vector coupling constant obeying

 \beq{1.3}
      \vec{\lambda}^2 \neq b \equiv \frac{D-1}{D-2},
 \eeq
 which follow from condition of non-zero scalar product
 for certain $U$-vector \cite{IMS}.

 The paper is organized as follows. In Section 2
 a class of solutions on a product of $n$ Ricci-flat
 spaces  for the model  under consideration is described.
 In Section 3 non-singular   solutions
 (e.g. with simple bounce of volume scale factor)
 are singled out.


 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{\bf  Solutions on product of Ricci-flat spaces}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  We deal with  the cosmological solution
  to field equations in the model (\ref{1.1})-(\ref{1.3})
  that is  the special case of solutions obtained in \cite{IMS}.
  Our main interest here will be the role of phantom scalar fields
  in combination with the usual matter fields.

  These solutions are defined on the manifold
   \beq{2.1}
     M = (u_{-},u_{+})  \times M_{1} \times \ldots \times M_{n}
   \eeq
   and are described by the metric
   \beq{2.2}
      g =  f^{2b h} [ - \e^{2c^0 t+ 2\bar c^0} w du \otimes du
           +   f^{-2 h} \sum_{i = 1}^n
               \e^{2c^i t+ 2 \bar c^i} g^i ].
    \eeq
     and scalar fields: material
    \beq{2.3}
     \vec{\varphi} = -  h \vec{\lambda} \ln |f| +
                        \vec{c}_{\varphi} t + \vec{{\bar c}},
    \eeq
    and phantom
   \beq{2.3a}
     \vec{\psi} =  \vec{c}_{\psi} t  + \vec{{\bar c}}_{\psi}.
    \eeq

  Here and in what follows we  denote
   \beq{2.4}
     h \equiv (\vec{\lambda}^2 -  b)^{-1}
   \eeq
   and $w = \pm 1$.
   The manifolds $(M_i,g^i)$ are Ricci-flat,
   \beq{02.Ric}
    {\rm Ric}[g^i ] = 0,
   \eeq
   and $d_i = {\rm dim} M_i$,
   $i =1, \dots, n$.

      The  function $f$ reads
   \bear{2.5}
     f = R \sinh(\sqrt{C}(u- u_0)), \;  C>0, \; \epsilon<0;
      \\ \label{2.5a}
      R \sin(\sqrt{|C|}(u- u_0)), \; C<0, \; \epsilon<0;
      \\ \label{2.5b}
      R \cosh(\sqrt{C}(u- u_0)), \; C>0, \; \epsilon>0;
      \\ \label{2.5c}
      |2 \Lambda/h|^{1/2}(u- u_0), \; C=0, \; \epsilon < 0,
    \ear
     where $u_0$ and $C$ are constants and
   \bear{2.6}
        R = |2 \Lambda/(h C)|^{1/2},
             \\ \label{2.6a}
                \epsilon = - \sign(\Lambda w h).
    \ear
   satisfy the following constraint relations

 \bear{2.7}
     \sum_{i = 1}^{n} d_i c^i +  \vec{\lambda} \vec{c}_{\varphi} =0,
     \qquad
     \sum_{i = 1}^{n} d_i \bar c^i +
     \vec{\lambda} \vec{\bar c}_{\varphi} = 0,
 \ear
 and
 \beq{2.8}
     C h + (\vec{c}_{\varphi})^2 -  (\vec{c}_{\psi})^2
          + \sum_{i= 1}^n d_i (c^i)^2
          - \left(\sum_{i=1}^n d_i c^i \right)^2 = 0.
 \eeq

 Relations (\ref{2.7}) means that
 integration constants are orthogonal
 to a certain U-vector
 and relation (\ref{2.8}) is equivalent to the  zero-energy
 constraint, see \cite{GIM,BGrIM,GrIM,IMJ}.

 In (\ref{2.2})
    \beq{2.9}
     c^0 = \sum_{i = 1}^{n} d_i c^i, \qquad
    \bar c^0 = \sum_{i = 1}^{n} d_i \bar c^i.
 \eeq

 The solutions under consideration are general
 $D$-dimensional cosmological type solutions
 defined on product of $n$ Ricci-flat spaces
 $D = \sum_{i = 1}^{n} d_i$ with generalized
 (i.e. multidimensional) Bianchi-I
 metrics in the model (\ref{1.1})-(\ref{1.3}). Here
 $u$ is the harmonic time variable. In what follows we put $u > u_0$.


 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{\bf Non-singular solutions}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    Here we restrict our consideration to
    small dilatonic couplings when
    \beq{3.1}
        w \Lambda < 0, \qquad   0 < \vec{\lambda}^2 < b
                                     = \frac{D-1}{D-2},
    \eeq
  see (\ref{1.2}). Thus, in (\ref{2.6a}) $\epsilon = -1$.
  As we shall see below the inequalities (\ref{3.1}) guarantee
  the existence of non-singular solutions.

  In what follows we investigate the
  behaviour of solutions from Section 2
  for parameters from (\ref{3.1}) using
  the so-called "synchronous" variable.

  We introduce scale factors
  \beq{3.10}
      a_i = f^{h/(D - 2)} \e^{c^i t + \bar c^i},
  \eeq
  $i = 1,\dots, n$. The relation between
  synchronous and harmonic variables
  reads
  \beq{3.11}
      \tau  = \int_{u}^{u_1} du' v(u'),
  \eeq
   where $u_1 > u_0$ and
  \beq{3.12}
       v = a_1 \dots a_n = f^{bh} \e^{c^0 u + \bar c^0},
   \eeq
   is a volume scale factor.

  From   (\ref{3.11})  for
   $u \to u_0 + 0$  we get $\tau \to +\infty$
   for all values of parameters   and
  \beq{3.12a}
      v \sim \tau^{(D-1) \nu}.
  \eeq


   It follows from (\ref{3.11}) that the solution
   is a non-singular one for $\tau \in ( -  \infty, + \infty )$
   only in such three cases
   \bear{3.13}
      (A_{+}) \quad
              c^0  > b|h| \sqrt{C}, \quad  \  C  \geq 0;
              \\ \label{3.14a}
      (A_{0}) \quad
              c^0  = b|h| \sqrt{C}, \quad  \  C  > 0;
              \\ \label{3.14}
      (B) \qquad \qquad \qquad \qquad   \ C < 0.
    \ear

    In the cases $(A_{+})$ and  $(B)$
    we get a bouncing behaviour of volume scale factor $v(\tau)$
    at some point $\tau_{b}$, i.e.
    the function $v(\tau)$ is monotonically
    decreasing in the interval
    $( -  \infty,  \tau_b )$ and monotonically
    increasing in the interval $( \tau_b, + \infty )$.

    In the case $(A_{0})$  the solution is
    non-singular, the function  $v(\tau)$ is monotonically
    increasing to infinity from a non-zero  value.

    For other values of parameters the solution is
    singular in general position (excepting certain
    special cases),  the function $v(\tau)$
    is monotonically increasing to infinity
    from  zero value. This   takes place when either
          (i) $c^0  < b |h| \sqrt{C}$, $C  > 0$
       or
          (ii) $C = 0$,  $c^0 \leq 0$.
    The solution for
    these values of parameters is defined in the
    semi-infinite interval $( \tau_0, + \infty )$.

    The relations (\ref{3.1}) lead to isotropisation of scale factors
    for large values of synchronous variable $\tau$.
    Namely, for $\vec{\lambda}^2 \neq 0$
    we get the following asymptotical relations
    in the limit $\tau \to +\infty$ \cite{IMS}
     \bear{3.2}
      g_{as}=   w d \tau \otimes d\tau
           + \sum_{i = 1}^{n} A_i \tau^{2 \nu} g^i, \\
                 \label{3.3}
      \vec{\varphi}_{as} = - \frac{\vec{\lambda} }{\vec{\lambda}^2}
                        \ln \tau  + \vec{\varphi}_0,
      \\   \label{3.4}   \vec{\psi}_{as} = \vec{\psi}_0,
     \ear
    where
    \beq{3.5}
      \nu = \frac{1}{(D - 2) \vec{\lambda}^2},
    \eeq
  $A_i > 0$ are constant, and $\vec{\varphi}_0^i$,
  $\vec{\psi}_0$ are  constant
  vectors  obeying
   \beq{3.6}
    2 |\Lambda| \exp( 2 \vec{\lambda} \vec{\varphi}_0) =
    |\vec{\lambda}^2 -  b| / (\vec{\lambda}^2)^2.
   \eeq


 For $\vec{\lambda} = 0$ we get for $\tau \to +\infty$ \cite{IMS}

  \bear{3.7}
    g_{as}=  w d\tau \otimes d \tau +
             \sum_{i = 1}^{n} A_i \exp(2 M \tau) g^i,
      \\ \label{3.8}
             \vec{\varphi}_{as}=  \vec{\varphi}_0,
             \qquad \vec{\psi}_{as}=  \vec{\psi}_0
  \ear
   where  $|2 \Lambda| =  (D-2)(D-1) M^2$
   (in agreement with the case of one scalar field
    from \cite{IM-tmf,BIMZ}.)

 Accelerated expansion in the limit
 $\tau \to +\infty$  takes place for
  \beq{3.9}
      \lambda^2 < \frac{1}{D-2}
  \eeq
  (In (\ref{3.7}) should be $M > 0$ for $\lambda = 0$.)

 \section{\bf The role of scalar charges}

    Here we  show that non-singular solutions
    defined for all $\tau \in \R$ exist only for
    large enough value of (vector) phantom charge squared
    $(\vec{c}_{\psi})^2$.

    Let us introduce anisotropy parameters
      $\tilde c^i$   by the following relations
    \beq{3.19}
     c^i = - \frac{\vec{\lambda}\vec{c}_{\varphi}}{D-1} + \tilde c^i,
    \eeq
    $i = 1,\dots,n$. We get from (\ref{2.9})
    \beq{3.20}
     \sum_{i = 1}^{n} d_i \tilde c^i = 0.
    \eeq

  In terms of these parameters  the main relations
  on parameters (\ref{2.8})
  may be rewritten as following
   \bear{3.21}
     C h +  (\vec{c}_{\varphi}^{~par})^2
      - \frac{1}{b} (\vec{\lambda}\vec{c}_{\varphi})^2
      \\ \nonumber
      = (\vec{c}_{\psi})^2  - (\vec{c}_{\varphi}^{~ort})^2
       -  \sum_{i= 1}^n d_i (\tilde c^i)^2 \equiv Q,
   \ear
  It follows also from (\ref{3.13})  and  (\ref{3.14})
  that
    \beq{3.22}
     c^0 = -  \vec{\lambda} \vec{c}_{\varphi}  \geq
      b h \sqrt{C}, \quad
        {\rm for}  \  C  \geq 0
   \eeq
  and, hence,
    \beq{3.22a}
     (c^0)^2  \geq b^2 h^2 C
    \eeq
  for all non-singular cases $(A_{+})$, $(A_{0})$ and $(B)$.

  Here we use decomposition of the vector of scalar charges
  into a sum of two terms orthogonal and parallel to
  $\vec{\lambda}$, respectively,
     \bear{3.21a}
     \vec{c}_{\varphi} =
     \vec{c}_{\varphi}^{~ort} + \vec{c}_{\varphi}^{~par},
      \\          \label{3.21b}
       \vec{c}_{\varphi}^{~par} =
       (\vec{c}_{\varphi} \vec{\lambda})
       \frac{\vec{\lambda}}{\vec{\lambda}^2},
      \\          \label{3.21c}
       \vec{c}_{\varphi}^{~ort} = \vec{c}_{\varphi}
       - (\vec{c}_{\varphi} \vec{\lambda})
       \frac{\vec{\lambda}}{\vec{\lambda}^2}.
   \ear

   It is clear that
   \bear{3.21d}
     \vec{c}_{\varphi}^{~2} =
     (\vec{c}_{\varphi}^{~ort}) + (\vec{c}_{\varphi}^{~par})^2.
   \ear

   We call $Q$ the "main" parameter of the model.
   We show that the necessary
   condition for the existence of non-singular solutions
   (defined for $\tau \in ( -  \infty, + \infty )$ ) is
   the following one
   \beq{3.Q}
      Q \geq 0,
   \eeq

     that means

   \beq{3.M}
     (\vec{c}_{\psi})^2  \geq (\vec{c}_{\varphi}^{~ort})^2
       + \sum_{i= 1}^n d_i (\tilde c^i)^2.
   \eeq


  Indeed, relations   (\ref{3.21}) and  (\ref{3.22}) imply
    either
    \beq{3.23}
     (A) \quad  Q_{-} \leq Q \leq Q_{+},
         \quad  0 \leq C \leq \vec{\lambda}^2 Q_{-},
    \eeq
    or
    \beq{3.25}
     (B) \qquad  Q > Q_{+},  \qquad  C < 0,
    \eeq
    where
    \beq{3.24}
     Q_{-} = \frac{(b - \vec{\lambda}^2)^2}{b^2 \vec{\lambda}^2}
     (\vec{c}_{\varphi} \vec{\lambda})^2,
     \quad
     Q_{+} = \frac{(b - \vec{\lambda}^2)}{b \vec{\lambda}^2}
     (\vec{c}_{\varphi} \vec{\lambda})^2
      \eeq
     ($\vec{\lambda} \neq 0$). Relation $Q_{-} \geq 0$
     implies  $Q \geq 0$.

   Subclasses (A) and (B) differ by  asymptotical relations
   for    $\tau \to  -  \infty$:
   \beq{3.26}
     (A) \qquad v \sim |\tau|.
   \eeq
   and
  \beq{3.27}
     (B) \qquad v \sim |\tau|^{(D-1) \nu}.
  \eeq
   with $\nu$ defined in (\ref{3.5}).

    The first asymptotics corresponds to Kasner-like behaviour
    of scale factors \cite{IMS} in agreement with the
    billiard approach of \cite{IMsbil,IMpbil}. The second
    one is just the same as in the limit $\tau \to  +  \infty$
    (see \ref{3.12a}).  This  follows from  the relations
    $\sin x = \sin (\pi -x) \sim x$ for small $x = (u - u_0)\sqrt{|C|}$
    and  (\ref{3.11}).

   Non-singular solutions
   defined for $\tau \in ( -  \infty, + \infty )$
   are absent for zero phantom charges $\vec{c}_{\psi} = 0$
   even for isotropic case  with  $\tilde c^i = 0$
   (in this case $\vec{c}_{\varphi} = C = 0$ and
   we get a solution with a power-law
   dependent scale factors, see also \cite{IMS}).

   For $\vec{c}_{\psi} \neq 0$
   the anisotropy parameters $\tilde c^i$ and
   orthogonal component  $\vec{c}_{\varphi}^{~ort}$
   should be small enough (in comparison
   with absolute value of phantom vector charge,
   see (\ref{3.M})) for the existence of
   non-singular solutions defined for all values
   of synchronous variable $\tau \in \R$.


   \section{Conclusions}

  In this paper we have considered
  a family of generalized Bianchi-I
  cosmological type solutions defined
  on product of Ricci-flat spaces
  in $D$-dimensional model  (\ref{1.1})-(\ref{1.3})
  with  $k$ ordinary scalar  fields  $\varphi^i$ and
  $m$ phantom scalar fields $\psi^j$. These
  solutions with phantom scalar fields
  are the special case of the solutions obtained
  earlier in \cite{IMS}.

  For the coupling constant obeying $0 < \vec{\lambda}^2 < (D-1)/(D-2)$
  we have singled out a subclass of
  non-singular solutions (e.g. with
  a simple bounce of the volume scale factor).
  Solutions from this subclass are regular for all values of
  synchronous  time $\tau \in ( -  \infty, + \infty )$
  and have an isotropization behaviour for $\tau \to  + \infty $.

  We have found a necessary condition on scalar charges
  and anisotropy parameters  for the existence of such
  non-singular solutions, see (\ref{3.M}).

  For  $\vec{\lambda}^2 < 1/(D-2)$ this expansion is asymptotically
  accelerated one for large values of $\tau$.

   \begin{center}
   {\bf Acknowledgments}
   \end{center}

   This work was supported in part by the Russian Foundation for
   Basic Researchs  and DFG Project  (436 RUS
   113/807/0-1(R)) for V.D.I. and V.N.M. and Universite Paris 7
   for V.N.M.

 \small
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 \bibitem{Chen1}
    C.-M. Chen, P.-M. Ho, I. P. Neupane, N. Ohta and J. E. Wang,
   Hyperbolic Space Cosmologies,
   {\it  JHEP} {\bf 0310} (2003) 058; hep-th/0306291.

  \bibitem{Ohta}
   N. Ohta, Accelerating Cosmologies and Inflation from M/superstring Theories,
   {\it Int. J. Mod. Phys.} {\bf A 20}, 1 (2005); hep-th/0411230.

  \bibitem{IMS1}
   V.D. Ivashchuk, V.N. Melnikov and A.B. Selivanov,
   Composite S-brane solutions on product of Ricci-flat spaces,
   {\it GRG} {\bf 36},  1593-1602 (2004); hep-th/0404113.

  \bibitem{IMK}
    V.D. Ivashchuk, V.N. Melnikov and S.-W. Kim,
    S-brane solutions with acceleration,
    in models with forms and multiple exponential
    potential,
   {\it Grav. Cosmol. } {\bf  10}, 141-148 (2004);  hep-th/0405009.


  \bibitem{IMsigma1}
    V.D. Ivashchuk and V.N. Melnikov,
    Intersecting p-brane Solutions in
    Multidimensional Gravity and M-Theory,
    {\it Grav. Cosmol.}, {\bf 2}, No 4 (8),
    297-305 (1996), hep-th/9612089; \\
    Generalized Intersecting P-brane Solutions from
    Sigma-model, {\it Phys. Lett. }, {\bf B 403},
    23-30 (1997).

  \bibitem{IMC}
   V.D. Ivashchuk and V.N. Melnikov, Sigma-model for the
   Generalized  Composite p-branes,
   {\it Class. Quantum Grav.}, {\bf 14}, 3001-3029, (1997);
   hep-th/9705036.

  \bibitem{KKLP}
   N. Khviengia, Z. Khviengia, H. L\"u and  C.N. Pope,
   Toward Field Theory of F-Theory,
   {\it Class. Quantum Grav.}, {\bf 15}, 759-773, (1998);
   hep-th/9703012.

  \bibitem{M-th}
  C. Hull and P. Townsend, Unity of Superstring Dualities,
  {\it Nucl. Phys.}, {\bf  B 438}, 109 (1995); \\
  P. Horava and E. Witten, {\it Nucl. Phys.} {\bf B 460}, 506 (1996).

  \bibitem{Vafa}
  C. Vafa, Evidence for F-Theory, Nucl. Phys. B 469, 403 (1996).

   \bibitem{GIM}
   V.R. Gavrilov, V.D. Ivashchuk and V.N. Melnikov,
   Integrable pseudo-Euclidean Toda-like systems in
   multidimemsional cosmology  with multicomponent perfect
   fluid., {\it J. Math. Phys.}, {\bf 36}, No 10, 5829-5847
   (1995).

  \bibitem{BGrIM}
  K.A. Bronnikov, M.A. Grebeniuk, V.D. Ivashchuk, V.N. Melnikov,
  Integrable multidimensional cosmology for intersecting
  $p$-branes, {\it Grav. Cosmol. } {\bf  3}, 2 (10), 105-112 (1997).

  \bibitem{GrIM}
   M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov,
   Integrable multidimensional quantum cosmology
   for intersecting p-branes,
   {\it Grav.  Cosmol.\/} {\bf 3}, No 3 (11), 243-249 (1997);
   gr-qc/9708031.

  \bibitem{IMJ}
   V.D. Ivashchuk and V.N. Melnikov,
   Multidimensional Classical and Quantum Cosmology with
   Intersecting p-Branes,
   {\it J. Math. Phys.}, {\bf 39}, 2866-2889 (1998); hep-th/9708157.

   \bibitem{IMSam}
   V.D. Ivashchuk and V.N. Melnikov,
   Multidimensional cosmological and spherically symmetric solutions with
   intersecting p-branes. In: Lecture Notes in Physics, Vol. 537,
   ``Mathematical and Quantum Aspects of Relativity and Cosmology.
   Proceedings of the Second Samos Meeting on Cosmology, Geometry and
   Relativity held at Pythagoreon, Samos, Greece, 1998, eds: S. Cotsakis,
   G.W. Gibbons., Berlin, Springer, 2000, pp. 214-148.

  \bibitem{IMtop}
   V.D. Ivashchuk and V.N. Melnikov,  Exact solutions in
   multidimensional gravity with antisymmetric forms,
   topical review, {\it Class. Quantum Grav.} {\bf 18},
   R82-R157 (2001); hep-th/0110274.

  \bibitem{IMerice}
      V.D. Ivashchuk, On exact solutions in
    multidimensional gravity with antisymmetric forms,
    In: Proceedings of the 18th Course of the School on
    Cosmology and Gravitation: The Gravitational Constant. Generalized
    Gravitational Theories and Experiments (30 April-10 May 2003, Erice).
    Ed. by G.T. Gillies, V.N. Melnikov and V. de Sabbata, Kluwer
    Academic Publishers, Dordrecht, 2004, pp. 39-64;  gr-qc/0310114.

  \bibitem{AWands}
   L.E. Allen and D. Wands, Cosmological perturbations through
   a simple bounce, {\it Phys. Rev.} {\bf D 70} (2004) 063515; astro-ph/0404441.

\bibitem{BrF}
  K.A. Bronnikov and J.C. Fabris, Nonsingular multidimensional
  cosmologies without fine tuning, {\it  JHEP} {\bf 0209} (2002) 062;
  hep-th/0207213.


  \bibitem{bm69}
  K.A. Bronnikov and V.N. Melnikov, Abstr. IV Russian Geom. Conf., 1969,
  p. 167-168 (in Russian).

  \bibitem{ZKol}
  N.A. Zaitsev, S.M. Kolesnikov and A.G. Radynov. Preprint
  ITP-72-21P, Kiev, 1972.

  \bibitem{MelOrl}
  V.N. Melnikov, {\it Doklady Acad. Nauk},
  {\bf 246}, No 6,  1351 (1979) (in Russian);\\
  V.N. Melnikov and S.V. Orlov, {\it Phys. Lett.},
   {\bf  A 70}, No 4,  263 (1979).

  \bibitem{Starpol}
  V.Ts. Gurovich and A.A. Starobinsky, {\it JETP},
  {\bf 77},  1689 (1979) (in Russian).

  \bibitem{bmedd}
  K.A. Bronnikov and V.N. Melnikov, A non-singular model of
  preinflationary Universe, Abstr. of USSR Conf. GR-6,
  MGPI Press, Moscow, 1984, p. 46-47 (in Russian).

  K.A. Bronnikov and V.N. Melnikov, A nonsingular vacuum model for
  pre-inflationary universe. In: Proc. A. Eddington Cent. Symp.,
  V.3, Gravitational Radiation and Relativity, Eds. J. Weber and
  T.M. Karade,  WS, Singapore, 1986, p. 386.


  \bibitem{SM}
  K.P. Stanyukovich and V.N. Melnikov, Hydrodynamics, Fields and
  Constants in the Theory of Gravitation,
  Energoatomizdat, Moscow, 1983 (in Rissian).
  English translation of first 5 chapters see in:\\
  V.N. Melnikov,   Fields and Constants in the Theory of Gravitation.
  CBPF-MO-02/02, Rio de Janeiro, 2002, 134pp.

  \bibitem{IM-tmf}
  V.D. Ivashchuk and V.N. Melnikov, Exact Solutions in
  Multidimensional Cosmology with Cosmological Constant,
  {\it Teor. Mat. Fiz.} {\bf 98}, 312 (1994) (in  Russian);
  {\it Theor. Mat. Phys.} {\bf 98}, 212 (1994).

  \bibitem{BIMZ}
  U. Bleyer, V.D. Ivashchuk, V.N. Melnikov and A.I. Zhuk,
  Multidimensional Classical and Quantum Wormholes in Models with
  Cosmological Constant, {\it Nucl. Phys.}, {\bf B 429}, 177-204 (1994).

  \bibitem{IMpbil}
   V.D. Ivashchuk and V.N. Melnikov, Billiard representation for multi-
    dimensional cosmology with intersecting p-branes near the singularity.
    {\it J. Math. Phys.}, {\bf 41}, No 9, 6341-6363 (2000);
    hep-th/9904077.

  \bibitem{IMsbil}
    H. Dehnen, V.D. Ivashchuk and V.N. Melnikov,
    Billiard representation for multidimensional multi-scalar
    cosmological model with exponential potentials,
    {\it GRG}, {\bf 36}, N 7, 1563-1578 (2004); hep-th/0312317.

  \end{thebibliography}
  \end{document}

gr-qc0504044.extracted_bib
\begin{thebibliography}{99}

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 \bibitem{M2}
 V.N. Melnikov, Multidimensional Cosmology and  Gravitation,
 CBPF-MO-002/95, Rio de Janeiro, 1995, 210 pp.; \\
 V.N. Melnikov. In: {\it Cosmology and Gravitation. II}, ed. M.
 Novello, Editions Frontieres, Singapore, 1996, p. 465.

 \bibitem{M3}
 V.N. Melnikov,
 Exact Solutions in Multidimensional Gravity and Cosmology III.
 CBPF-MO-03/02,  Rio de Janeiro, 2002, 297 pp.

 \bibitem{Riess}
 A.G. Riess
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 S. Perlmutter {\it et al}, {\it ApJ}, {\bf 517}, 565 (1999).

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 Cosmological consequences of a rolling homogeneous scalar field,
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 E.J. Copeland, A. R. Liddle and D. Wands,
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 \bibitem{BCopN}
 T. Bareiro, E.I. Copeland and N.J. Nunes,
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 \bibitem{Town}
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 {\it JHEP} {\bf 0111} (2001) 042;  hep-th/0110072.

 \bibitem{DGM}
  H. Dehnen, V.R. Gavrilov and V.N. Melnikov,
  General solutions for flat  Friedmann universe
  filled by perfect fluid and scalar field with exponential
  potential, {\it Grav. Cosmol.}, {\bf 9}, No 3 (35), 189 (2003);
  gr-qc/0212107.

  \bibitem{IMS}
    V.D. Ivashchuk, V.N. Melnikov and A.B. Selivanov,
    Cosmological solutions in multidimensional model with multiple
    exponential potential,  {\it JHEP}  {\bf 0309}  (2003) 059;
    hep-th/0309027.

 \bibitem{Chen1}
    C.-M. Chen, P.-M. Ho, I. P. Neupane, N. Ohta and J. E. Wang,
   Hyperbolic Space Cosmologies,
   {\it  JHEP} {\bf 0310} (2003) 058; hep-th/0306291.

  \bibitem{Ohta}
   N. Ohta, Accelerating Cosmologies and Inflation from M/superstring Theories,
   {\it Int. J. Mod. Phys.} {\bf A 20}, 1 (2005); hep-th/0411230.

  \bibitem{IMS1}
   V.D. Ivashchuk, V.N. Melnikov and A.B. Selivanov,
   Composite S-brane solutions on product of Ricci-flat spaces,
   {\it GRG} {\bf 36},  1593-1602 (2004); hep-th/0404113.

  \bibitem{IMK}
    V.D. Ivashchuk, V.N. Melnikov and S.-W. Kim,
    S-brane solutions with acceleration,
    in models with forms and multiple exponential
    potential,
   {\it Grav. Cosmol. } {\bf  10}, 141-148 (2004);  hep-th/0405009.


  \bibitem{IMsigma1}
    V.D. Ivashchuk and V.N. Melnikov,
    Intersecting p-brane Solutions in
    Multidimensional Gravity and M-Theory,
    {\it Grav. Cosmol.}, {\bf 2}, No 4 (8),
    297-305 (1996), hep-th/9612089; \\
    Generalized Intersecting P-brane Solutions from
    Sigma-model, {\it Phys. Lett. }, {\bf B 403},
    23-30 (1997).

  \bibitem{IMC}
   V.D. Ivashchuk and V.N. Melnikov, Sigma-model for the
   Generalized  Composite p-branes,
   {\it Class. Quantum Grav.}, {\bf 14}, 3001-3029, (1997);
   hep-th/9705036.

  \bibitem{KKLP}
   N. Khviengia, Z. Khviengia, H. L\"u and  C.N. Pope,
   Toward Field Theory of F-Theory,
   {\it Class. Quantum Grav.}, {\bf 15}, 759-773, (1998);
   hep-th/9703012.

  \bibitem{M-th}
  C. Hull and P. Townsend, Unity of Superstring Dualities,
  {\it Nucl. Phys.}, {\bf  B 438}, 109 (1995); \\
  P. Horava and E. Witten, {\it Nucl. Phys.} {\bf B 460}, 506 (1996).

  \bibitem{Vafa}
  C. Vafa, Evidence for F-Theory, Nucl. Phys. B 469, 403 (1996).

   \bibitem{GIM}
   V.R. Gavrilov, V.D. Ivashchuk and V.N. Melnikov,
   Integrable pseudo-Euclidean Toda-like systems in
   multidimemsional cosmology  with multicomponent perfect
   fluid., {\it J. Math. Phys.}, {\bf 36}, No 10, 5829-5847
   (1995).

  \bibitem{BGrIM}
  K.A. Bronnikov, M.A. Grebeniuk, V.D. Ivashchuk, V.N. Melnikov,
  Integrable multidimensional cosmology for intersecting
  $p$-branes, {\it Grav. Cosmol. } {\bf  3}, 2 (10), 105-112 (1997).

  \bibitem{GrIM}
   M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov,
   Integrable multidimensional quantum cosmology
   for intersecting p-branes,
   {\it Grav.  Cosmol.\/} {\bf 3}, No 3 (11), 243-249 (1997);
   gr-qc/9708031.

  \bibitem{IMJ}
   V.D. Ivashchuk and V.N. Melnikov,
   Multidimensional Classical and Quantum Cosmology with
   Intersecting p-Branes,
   {\it J. Math. Phys.}, {\bf 39}, 2866-2889 (1998); hep-th/9708157.

   \bibitem{IMSam}
   V.D. Ivashchuk and V.N. Melnikov,
   Multidimensional cosmological and spherically symmetric solutions with
   intersecting p-branes. In: Lecture Notes in Physics, Vol. 537,
   ``Mathematical and Quantum Aspects of Relativity and Cosmology.
   Proceedings of the Second Samos Meeting on Cosmology, Geometry and
   Relativity held at Pythagoreon, Samos, Greece, 1998, eds: S. Cotsakis,
   G.W. Gibbons., Berlin, Springer, 2000, pp. 214-148.

  \bibitem{IMtop}
   V.D. Ivashchuk and V.N. Melnikov,  Exact solutions in
   multidimensional gravity with antisymmetric forms,
   topical review, {\it Class. Quantum Grav.} {\bf 18},
   R82-R157 (2001); hep-th/0110274.

  \bibitem{IMerice}
      V.D. Ivashchuk, On exact solutions in
    multidimensional gravity with antisymmetric forms,
    In: Proceedings of the 18th Course of the School on
    Cosmology and Gravitation: The Gravitational Constant. Generalized
    Gravitational Theories and Experiments (30 April-10 May 2003, Erice).
    Ed. by G.T. Gillies, V.N. Melnikov and V. de Sabbata, Kluwer
    Academic Publishers, Dordrecht, 2004, pp. 39-64;  gr-qc/0310114.

  \bibitem{AWands}
   L.E. Allen and D. Wands, Cosmological perturbations through
   a simple bounce, {\it Phys. Rev.} {\bf D 70} (2004) 063515; astro-ph/0404441.

\bibitem{BrF}
  K.A. Bronnikov and J.C. Fabris, Nonsingular multidimensional
  cosmologies without fine tuning, {\it  JHEP} {\bf 0209} (2002) 062;
  hep-th/0207213.


  \bibitem{bm69}
  K.A. Bronnikov and V.N. Melnikov, Abstr. IV Russian Geom. Conf., 1969,
  p. 167-168 (in Russian).

  \bibitem{ZKol}
  N.A. Zaitsev, S.M. Kolesnikov and A.G. Radynov. Preprint
  ITP-72-21P, Kiev, 1972.

  \bibitem{MelOrl}
  V.N. Melnikov, {\it Doklady Acad. Nauk},
  {\bf 246}, No 6,  1351 (1979) (in Russian);\\
  V.N. Melnikov and S.V. Orlov, {\it Phys. Lett.},
   {\bf  A 70}, No 4,  263 (1979).

  \bibitem{Starpol}
  V.Ts. Gurovich and A.A. Starobinsky, {\it JETP},
  {\bf 77},  1689 (1979) (in Russian).

  \bibitem{bmedd}
  K.A. Bronnikov and V.N. Melnikov, A non-singular model of
  preinflationary Universe, Abstr. of USSR Conf. GR-6,
  MGPI Press, Moscow, 1984, p. 46-47 (in Russian).

  K.A. Bronnikov and V.N. Melnikov, A nonsingular vacuum model for
  pre-inflationary universe. In: Proc. A. Eddington Cent. Symp.,
  V.3, Gravitational Radiation and Relativity, Eds. J. Weber and
  T.M. Karade,  WS, Singapore, 1986, p. 386.


  \bibitem{SM}
  K.P. Stanyukovich and V.N. Melnikov, Hydrodynamics, Fields and
  Constants in the Theory of Gravitation,
  Energoatomizdat, Moscow, 1983 (in Rissian).
  English translation of first 5 chapters see in:\\
  V.N. Melnikov,   Fields and Constants in the Theory of Gravitation.
  CBPF-MO-02/02, Rio de Janeiro, 2002, 134pp.

  \bibitem{IM-tmf}
  V.D. Ivashchuk and V.N. Melnikov, Exact Solutions in
  Multidimensional Cosmology with Cosmological Constant,
  {\it Teor. Mat. Fiz.} {\bf 98}, 312 (1994) (in  Russian);
  {\it Theor. Mat. Phys.} {\bf 98}, 212 (1994).

  \bibitem{BIMZ}
  U. Bleyer, V.D. Ivashchuk, V.N. Melnikov and A.I. Zhuk,
  Multidimensional Classical and Quantum Wormholes in Models with
  Cosmological Constant, {\it Nucl. Phys.}, {\bf B 429}, 177-204 (1994).

  \bibitem{IMpbil}
   V.D. Ivashchuk and V.N. Melnikov, Billiard representation for multi-
    dimensional cosmology with intersecting p-branes near the singularity.
    {\it J. Math. Phys.}, {\bf 41}, No 9, 6341-6363 (2000);
    hep-th/9904077.

  \bibitem{IMsbil}
    H. Dehnen, V.D. Ivashchuk and V.N. Melnikov,
    Billiard representation for multidimensional multi-scalar
    cosmological model with exponential potentials,
    {\it GRG}, {\bf 36}, N 7, 1563-1578 (2004); hep-th/0312317.

  \end{thebibliography}