Off--Diagonal Ekpyrotic Scenarios and Equivalence of Modified, Massive and/or Einstein Gravity
{\qquad }
Theory Division
CERN
CH-1211
Geneva 23
Switzerland \footnote{associated visiting research}
and
Rector's Office
Alexandru Lapu\c sneanu street
nr. 14
UAIC -- Corpus R
office 323;
Ia\c si
\ Romania
700057
We show how generic off--diagonal cosmological solutions depending, in general, on all spacetime coordinates can be constructed in massive gravity using the anholonomic frame deformation method. There are found new classes of locally anisotropic and (in) homogeneous cosmological metrics with open and closed spatial geometries. Such solutions describe the late time acceleration due to effective cosmological terms induced by nonlinear off--diagonal interactions and graviton mass. The cosmological metrics and related St\" uckelberg fields are constructed in explicit form up to nonholonomic frame transforms of the Friedmann--Lama\^{i}tre--Robertson--Walker (FLRW) coordinates. The solutions include matter, graviton mass and other effective sources modelling nonlinear gravitational and matter fields interactions with polarization of physical constants and deformations of metrics, which may explain certain dark energy and dark matter effects. There are stated the conditions when such configurations mimic interesting solutions in general relativity and modifications and recast the general Painlev\' e--Gullstrand and FLRW metrics. Finally, we sketch a reconstruction procedure for a subclass of off--diagonal cosmological solutions which describe cyclic and ekpyrotic universes. \vskip5pt Keywords:\ massive gravity, modified gravity, off--diagonal cosmological solutions; ekpyrotic and little rip universe. \vskip5pt {PACS:\ 04.50.Kd, 04.90.+e, 98.80.Jk, 98.80.Cq, 95.30.Sf, 95.36.+x, 95.35.+d}
1304.1080.tex

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\begin{document}

\title{Off--Diagonal  Ekpyrotic Scenarios and Equivalence of  Modified,  Massive and/or Einstein Gravity}
\date{November 10, 2014}

\author{{\Large \textbf{Sergiu I. Vacaru}}\thanks{sergiu.vacaru@uaic.ro}\\
{\qquad } \\
 Theory Division, CERN, CH-1211, Geneva 23, Switzerland \footnote{associated visiting research}\\
 and \\
    Rector's Office, Alexandru Ioan Cuza University, \\
 Alexandru Lapu\c sneanu street, nr. 14, UAIC -- Corpus R, office 323;\\
 Ia\c si,\ Romania, 700057
  }


\renewcommand\Authands{ and }

\maketitle

\begin{abstract}
We show how generic off--diagonal cosmological solutions depending, in general, on all spacetime coordinates can be constructed in massive gravity using the anholonomic frame deformation method. There are found new classes of locally anisotropic and (in) homogeneous cosmological metrics with open and closed spatial geometries. Such solutions describe the late time acceleration due to effective cosmological terms induced by nonlinear off--diagonal interactions and graviton mass. The cosmological metrics and related St\" uckelberg fields are constructed in explicit form up to nonholonomic frame transforms of the Friedmann--Lama\^{i}tre--Robertson--Walker (FLRW) coordinates. The solutions include matter, graviton mass and other effective sources modelling nonlinear gravitational and matter fields interactions with polarization of physical constants and deformations of metrics, which may explain certain dark energy and dark matter effects. There are stated the conditions when such configurations mimic interesting solutions in general relativity and modifications and recast the general Painlev\' e--Gullstrand and FLRW metrics. Finally, we sketch a reconstruction procedure for a subclass of off--diagonal cosmological solutions which describe cyclic and ekpyrotic universes.

\vskip5pt

\textbf{Keywords:}\ massive gravity, modified gravity, off--diagonal cosmological solutions; ekpyrotic and little rip universe.
\vskip5pt
{\small PACS:\ 04.50.Kd, 04.90.+e, 98.80.Jk, 98.80.Cq, 95.30.Sf, 95.36.+x, 95.35.+d}
\end{abstract}


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%\section{Introduction}

%\vskip5pt

%\newpage

The idea that graviton may have a nontrivial mass was proposed by Fierz and Pauli work \cite{fierz} (for recent reviews and related f(R) modifications, see \cite{odints}). The key steps in elaborating a modern version of a ghost free (bimetric) massive gravity theory were made in a series of papers: The so--called vDVZ discontinuity problem was solved using the Vainshtein mechanism \cite{dvz} (avoiding discontinuity by going beyond the linear theory), or following more recent approaches based on DGP model \cite{dgp}. But none solution was found for another problem with ghosts because at nonlinear order in massive gravity appears a sixth scalar degree of freedom as a ghost (see the Boulware and Deser paper and similar issues related to the effective field theory approach in Refs. \cite{boul}). That stoppled for almost two decades the research on formulating a consistent theory of massive gravity.

Recently, a substantial progress was made when de Rham and co--authors had
shown how to eliminate the scalar mode and Hassan and Rosen established a
complete proof for a class of bigravity / bimetric gravity theories, see
\cite{drg}. The second metric describes an effective exotic matter related
to massive gravitons and does not suffer from ghost instability to all
orders in a perturbation theory and away from the decoupling limit.

The possibility that the graviton has a nonzero mass $\mathring{\mu}$
results not only in fundamental theoretical implications but give rise to
straightforward phenomenological consequences. For instance, a gravitational
potential of Yuka\-wa form $\sim e^{-\mathring{\mu}r}/r$ results in decay of
gravitational interactions at scales $r\geq \mathring{\mu}^{-1}$ and this
can result in the accelerated expansion of the Universe. This way, a theory
of massive gravity provides alternatives to dark energy and, via effective
polarizations of fundamental physical constants (in result of generic
off--diagonal nonlinear interactions), may explain certain dark matter
effects. Recently, various cosmological models derived for ghost free
(modified) massive gravity and bigravity theories have been elaborated and
studied intensively (see, for instance, Refs. \cite{mgcosm,odints,kobayashi}%
).

The goal of this work is to construct generic off--diagonal cosmological
solutions in massive gravity theory, MGT, and state the conditions when such
configurations are modelled equivalently in general relativity (GR). As a
first step, we consider off--diagonal deformations of a "prime" cosmological
solution taken in general Painlev\'{e}--Gullstrand (PG) form, when the
Friedman--Lama\^{\i}tre--Robertson--Walker (FLRW) can be recast for
well--defined geometric conditions. At the second step, the "target" metrics
will be generated to possess one Killing symmetry (or other none Killing
symmetries) and depend on timelike and certain (all) spacelike coordinates.
In general, such off--diagonal solutions are with local anisotropy and
inhomogeneities for effective cosmological constants and polarizations of
other physical constants and coefficients of cosmological metrics which can
be modelled both in MGT and GR. Finally (the third step), we shall emphasize
and speculate on importance of off--diagonal nonlinear gravitational
interactions for elaborating cosmological scenarios when dark matter and
dark energy effects can be explained by anisotropic polarizations of vacuum
and/or de Sitter like configurations.

We study modified massive gravity theories determined on a
pseudo--Riemannian spacetime $\mathbf{V}$ with physical metric $\mathbf{g=\{g%
}_{\mu \nu }\}$ and certain fiducial metric as we shall explain below. The
action for our model is
\begin{eqnarray}
S &=&\frac{1}{16\pi }\int \delta u^{4}\sqrt{|\mathbf{g}_{\alpha \beta }|}[%
\widehat{f}(\widehat{R})-\frac{\mathring{\mu}^{2}}{4}\mathcal{U}(\mathbf{g}%
_{\mu \nu },\mathbf{K}_{\alpha \beta })+\ ^{m}L]  \label{act} \\
&=&\frac{1}{16\pi }\int \delta u^{4}\sqrt{|\mathbf{g}_{\alpha \beta }|}%
[f(R)+\ ^{m}L].  \label{act1}
\end{eqnarray}%
In this formula, $\widehat{R}$ is the scalar curvature for an auxiliary
(canonical) connection $\widehat{\mathbf{D}}$ uniquely determined by two
conditions 1) it is metric compatible, $\widehat{\mathbf{D}}\mathbf{g}=0%
\mathbf{,}$ and 2) its $h$- and $v$-torsions are zero (but there are nonzero
$h-v$ components of torsion $\widehat{\mathcal{T}}$ completely determined by
$\mathbf{g)}$ for a conventional splitting $\mathbf{N}:\ T\mathbf{V=}h\mathbf{%
V\oplus }v\mathbf{V,}$ see details in \cite{vadm1}.\footnote{%
We consider a conventional $2+2$ splitting when coordinates are labelled in
the form $u^{\alpha }=(x^{i},y^{a}),$ or $u=(x,y),$ with indices $%
i,j,k...=1,2$ and $a,b,...=3,4.$ There will be used boldface symbols in
order to emphasize that certain geometric/physical objects and/or formulas
are written with respect to N--adapted bases (\ref{nframes}). There will be
considered also left up/low indices as labels for some geometric/physical
objects.} The "priority" of the connection $\widehat{\mathbf{D}}$ is that it allows to
decouple the field equations in various gravity theories and construct exact
solutions in very general forms. We shall work with generic off--diagonal
metrics and generalized connections depending on all spacetime coordinates
when, for instance, of type $\widehat{\mathbf{D}}=\nabla +\widehat{\mathbf{Z}%
}[\widehat{\mathcal{T}}].$ Such distortion relations from the Levi--Civita
(LC) connection $\nabla $ are uniquely determined by a distorting tensor $%
\widehat{\mathbf{Z}}$ completely defined by $\widehat{\mathcal{T}}$ and (as
a consequence for such models) by $(\mathbf{g},\mathbf{N}).$ Having
constructed integral varieties (for instance, locally anisotropic and/or
inhomogeneous cosmological ones), we can impose additional nonholonomic
(non--integrable constraints) when $\widehat{\mathbf{D}}_{\mid \widehat{%
\mathcal{T}}=0}\rightarrow \nabla $ and $\widehat{R}\rightarrow R,$ where $R$
is the scalar curvature of $\nabla ,$ and it is possible to extract exact
solutions in GR.

The theories with actions of type (\ref{act}) generalize the so--called
modified $f(R)$ gravity, see reviews and original results in \cite{odints},
and the ghost--free massive gravity (by de Rham, Gabadadze and Tolley, dRGT)
\cite{drg}. We use the units when $\hbar =c=1$ and the Planck mass $M_{Pl}$
is defined via $M_{Pl}^{2}=1/8\pi G$ with 4--d Newton constant $G.$ We write
$\delta u^{4}$ instead of $d^{4}u$ because there are used N--elongated
differentials (see below the formulas (\ref{nframes})) and consider $%
\mathring{\mu}=const$ as the mass of graviton. For LC--configurations, we
can fix (as a particular case) condition of type
\begin{equation}
\widehat{f}(\widehat{R})-\frac{\mathring{\mu}^{2}}{4}\mathcal{U}(\mathbf{g}%
_{\mu \nu },\mathbf{K}_{\alpha \beta })=f(\widehat{R}),\mbox{ \ or \ }%
\widehat{f}(\widehat{R})=f(R),\mbox{ \ or \ }\widehat{f}(\widehat{R})=R,
\label{mgrfunct}
\end{equation}%
which depend on the type of models we elaborate and what classes of solutions
we wont to construct. It will be possible to find solutions in explicit form
if we fix the coefficients $\{N_{i}^{a}\}$ of $\mathbf{N}$ and local frames
for $\widehat{\mathbf{D}}$ when $\widehat{R}=const$ and $\partial _{\alpha }%
\widehat{f}(\widehat{R})=(\partial _{\widehat{R}}\widehat{f})\times \partial
_{\alpha }\widehat{R}=0$ but, in general, $\partial _{\alpha }f(R)\neq 0.$
The equations of motion for such modified massive gravity theory can be
written\footnote{%
see details on action and variational methods in \cite{drg}; we shall follow
some conventions from \cite{kobayashi}; the Einstein summation rule on
repeating indices will be applied if the contrary is not stated}%
\begin{equation}
(\partial _{\widehat{R}}\widehat{f})\widehat{\mathbf{R}}_{\mu \nu }-\frac{1}{%
2}\widehat{f}(\widehat{R})\mathbf{g}_{\mu \nu }+\mathring{\mu}^{2}\mathbf{X}%
_{\mu \nu }=M_{Pl}^{-2}\mathbf{T}_{\mu \nu },  \label{mgrfe}
\end{equation}%
where $M_{Pl}$ is the Plank mass, $\widehat{\mathbf{R}}_{\mu \nu }$ is the
Einstein tensor for a pseudo--Riemannian metric $\mathbf{g}_{\mu \nu }$ and $%
\widehat{\mathbf{D}},$ $\mathbf{T}_{\mu \nu }$ is the standard matter
energy--momentum tensor. \ For $\widehat{\mathbf{D}}\rightarrow \nabla ,$ we
get $\widehat{\mathbf{R}}_{\mu \nu }\rightarrow R_{\mu \nu }$ with a
standard Ricci tensor $R_{\mu \nu }$ for $\nabla .$ The effective
energy--momentum tensor $\mathbf{X}_{\mu \nu }$ is defined in a
"sophisticate" form by the potential of graviton $\mathcal{U}=\mathcal{U}%
_{2}+\alpha _{3}\mathcal{U}_{3}+\alpha _{4}\mathcal{U}_{4}$, where $\alpha
_{3}$ and $\alpha _{4}$ are free parameters. The values $\mathcal{U}_{2},%
\mathcal{U}_{3}$ and $\mathcal{U}_{4}$ are certain polynomials on traces of
some other polynomials of a matrix $\mathcal{K}_{\mu }^{\nu }=\delta _{\mu
}^{\nu }-\left( \sqrt{g^{-1}\Sigma }\right) _{\mu }^{\nu }$ for a tensor
determined by four St\"{u}ckelberg fields $\phi ^{\underline{\mu }}$ as
\begin{equation}
\Sigma _{\mu \nu }=\partial _{\mu }\phi ^{\underline{\mu }}\partial _{\nu
}\phi ^{\underline{\nu }}\eta _{\underline{\mu }\underline{\nu }},
\label{bm}
\end{equation}%
when $\eta _{\underline{\mu }\underline{\nu }}=(1,1,1,-1).$ Following a
series of arguments presented in \cite{kobayashi}, when the parameter choice
 $\alpha _{3}=(\alpha -1)/3,\alpha _{4}=(\alpha ^{2}-\alpha +1)/12$
is useful for avoiding potential ghost instabilities, we can fix
\begin{equation}
\mathbf{X}_{\mu \nu }=\alpha ^{-1}\mathbf{g}_{\mu \nu }.  \label{cosmconst}
\end{equation}%
De Sitter solutions for an effective cosmological constant are possible, for
instance, for ansatz of PG type,%
\begin{equation}
ds^{2}=U^{2}(r,t)[dr+\epsilon \sqrt{f(r,t)}dt]^{2}+\widetilde{\alpha }%
^{2}r^{2}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2})-V^{2}(r,t)dt^{2}.
\label{pgm}
\end{equation}%
In above formula, there are used spherical coordinates labelled in the form $%
u^{\beta }=(x^{1}=r,x^{2}=\theta ,y^{3}=\varphi ,y^{4}=t),$ the function $f$%
\ takes non--negative values and the constant $\widetilde{\alpha }=\alpha
/(\alpha +1)$ and $\epsilon =\pm 1.$ For such bimetric configurations, the St%
\"{u}ckelberg fields are parameterized in the unitary gauge as $\phi ^{%
\underline{4}}=t$ and $\phi ^{\underline{1}}=r\widehat{n}^{\underline{1}%
},\phi ^{\underline{2}}=r\widehat{n}^{\underline{2}},\phi ^{\underline{3}}=r%
\widehat{n}^{\underline{3}},$ where a three dimensional (3--d) unit vector
is defined as $\widehat{n}=(\widehat{n}^{\underline{1}}=\sin \theta \cos
\varphi ,\widehat{n}^{\underline{2}}=\sin \theta \sin \varphi ,\widehat{n}^{%
\underline{3}}=\cos \theta ).$ Any PG metric of type (\ref{pgm}) defines
solutions both in GR and in MGT. It allows us to extract the de Sitter
solution, in the absence of matter, and to obtain standard cosmological
equations with FLRW metric, for a perfect fluid source%
\begin{equation}
T_{\mu \nu }=\left[ \rho (t)+p(t)\right] u_{\mu }u_{\nu }+p(t)g_{\mu \nu },
\label{memt}
\end{equation}%
where $u_{\mu }=(0,0,0,-V)$ can be reproduced for the effective cosmological
constant $\ ^{eff}\lambda =\mathring{\mu}^{2}/\alpha .$ It is also possible
to express metrics of type (\ref{pgm}) in a familiar cosmological FLRW form
(see formulas (23), (24) and (27) in \cite{kobayashi}).

Let us consider an ansatz
\begin{eqnarray}
ds^{2} &=&\eta _{1}(r,\theta )\mathring{g}_{1}(r)dr^{2}+\eta _{2}(r,\theta )%
\mathring{g}_{2}(r)d\theta ^{2}  \label{offdans} \\
&&+\omega ^{2}(r,\theta ,\varphi ,t)\{\eta _{3}(r,\theta ,t)\mathring{h}%
_{3}(r,\theta )[d\varphi +n_{i}(r,\theta )dx^{i}]^{2}  \notag \\
&&+\eta _{4}(r,\theta ,t)\mathring{h}_{4}(r,\theta ,t)[dt+(w_{i}(x^{k},t)+%
\mathring{w}_{i}(x^{k}))dx^{i}]^{2}\},  \notag
\end{eqnarray}%
with Killing symmetry on $\partial _{3}=\partial _{\varphi },$ which (in
general) can not be diagonalized by coordinate transforms. The values $\eta
_{\alpha }$ are called "polarization" functions; $\omega $ is the so--called
"vertical", v, conformal factor. The off--diagonal, N--coefficients, are
labelled $N_{i}^{a}(x^{k},y^{4}),$ where (for this ansatz) $%
N_{i}^{3}=n_{i}(r,\theta )$ and $N_{i}^{4}=w_{i}(x^{k},t)+\mathring{w}%
_{i}(x^{k}).$ The  data for the "primary" metric
are
\begin{eqnarray}
\mathring{g}_{1}(r) &=&U^{2}-\mathring{h}_{4}(\mathring{w}_{1})^{2},%
\mathring{g}_{2}(r)=\widetilde{\alpha }^{2}r^{2},\mathring{h}_{3}=\widetilde{%
\alpha }^{2}r^{2}\sin ^{2}\theta ,\mathring{h}_{4}=\sqrt{|fU^{2}-V^{2}|},
\notag \\
\mathring{w}_{1} &=&\epsilon \sqrt{f}U^{2}/\mathring{h}_{4},\mathring{w}%
_{2}=0,\mathring{n}_{i}=0,  \label{pmdata}
\end{eqnarray}%
when the coordinate system is such way fixed that the values $f,U,V$ in (\ref%
{pgm}) result in a coefficient $\mathring{g}_{1}$ depending only on $r$.

We shall work with respect to a class of N--adapted (dual) bases
\begin{eqnarray}
\mathbf{e}_{\alpha } &=&(\mathbf{e}_{i}=\partial _{i}-N_{i}^{b}\partial
_{b},e_{a}=\partial _{a}=\partial /\partial y^{a})\mbox{ and }  \notag \\
\mathbf{e}^{\beta } &=&(e^{j}=dx^{i},\mathbf{e}^{b}=dy^{b}+N_{c}^{b}dy^{c}),
\label{nframes}
\end{eqnarray}%
which are nonholonomic (equivalently, anholonomic) because, in general,
there are satisfied relations of type $\mathbf{e}_{\alpha }\mathbf{e}_{\beta
}-\mathbf{e}_{\beta }\mathbf{e}_{\alpha }=W_{\alpha \beta }^{\gamma }\mathbf{%
e}_{\gamma },$ for certain nontrivial anholonomy coefficients $W_{\alpha
\beta }^{\gamma }(u).$ For simplicity, we shall consider energy momentum
sources (\ref{memt}) and effective (\ref{cosmconst}) which up to
frame/coordinate transforms can be parameterized in the form
{\small
\begin{equation}
\Upsilon _{\beta }^{\alpha }=\frac{1}{M_{Pl}^{2}(\partial _{\widehat{R}}%
\widehat{f})}(\mathbf{T}_{\beta }^{\alpha }+\alpha ^{-1}\mathbf{X}_{\beta
}^{\alpha })=\frac{1}{M_{Pl}^{2}(\partial _{\widehat{R}}\widehat{f})}(\
^{m}T+\alpha ^{-1})\delta _{\beta }^{\alpha }=(\ ^{m}\Upsilon +\ ^{\alpha
}\Upsilon )\delta _{\beta }^{\alpha }  \label{effectsourc}
\end{equation}%
}
for constant values $\ ^{m}\Upsilon :=M_{Pl}^{-2}(\partial _{\widehat{R}}%
\widehat{f})^{-1}\ ^{m}T$ and $\ ^{\alpha }\Upsilon =M_{Pl}^{-2}(\partial _{%
\widehat{R}}\widehat{f})^{-1}\alpha ^{-1},$ with respect to N--adapted
frames (\ref{nframes}).\footnote{%
In general, such sources are not diagonal and may depend on spacetime
coordinates. We fix such N--adapted parameterizations which will allow to
construct exact solutions in explicit form.}

Let us explain an important decoupling property of the gravitational field
equations in GR and various generalizations/ modifications studied in
details in Refs. \cite{vadm1}. That anholonomic frame deformation method
(AFDM) can be applied for decoupling, and constructing solutions of the MGT
field equations (\ref{mgrfe}) with any effective source (\ref{effectsourc}).
We consider target off--diagonal metrics $\mathbf{g}=(g_{i}=\eta _{i}%
\mathring{g}_{i},h_{a}=\eta _{a}\mathring{h}_{a},N_{j}^{a})$ [there is not
summation on repeating indices in this formula] with coefficients determined
by ansatz (\ref{offdans}). For convenience, we shall use brief denotations
for partial derivatives:  $\partial _{1}\psi =\psi ^{\bullet
},\partial _{2}\psi =\psi ^{\prime },\partial _{3}\psi =\psi ^{\diamond }$
and $\partial _{4}\psi =\psi ^{\ast }.$ Computing the N--adapted
coefficients of the Ricci and Einstein tensors (see  details in Refs.
\cite{vadm1}), we transform (\ref{mgrfe}) into a system of nonlinear
partial differential equations (PDE):%
\begin{eqnarray}
\widehat{R}_{1}^{1} &=&\widehat{R}_{2}^{2}\Longrightarrow \psi ^{\bullet
\bullet }+\psi ^{\prime \prime }=2(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon ),
\label{mgfeq1} \\
\widehat{R}_{3}^{3} &=&\widehat{R}_{4}^{4}\Longrightarrow \phi ^{\ast
}h_{3}^{\ast }=2h_{3}h_{4}(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon ),  \notag \\
\widehat{R}_{3k} &\Longrightarrow &n_{i}^{\ast \ast }+\gamma n_{i}^{\ast
}=0,\ \widehat{R}_{4k}\Longrightarrow \beta w_{i}-\alpha _{i}=0,\   \notag \\
\partial _{k}\omega &=&n_{k}\omega ^{\diamond }+w_{k}\omega ^{\ast },
\label{conf2}
\end{eqnarray}%
\begin{equation}
\mbox{for \ }\phi =\ln \left\vert \frac{h_{3}^{\ast }}{\sqrt{|h_{3}h_{4}|}}%
\right\vert ,\gamma :=(\ln \frac{|h_{3}|^{3/2}}{|h_{4}|})^{\ast },\ \ \alpha
_{i}=\frac{h_{3}^{\ast }}{2h_{3}}\partial _{i}\phi ,\ \beta =\frac{%
h_{3}^{\ast }}{2h_{3}}\phi ^{\ast },  \label{c1}
\end{equation}%
where $\Longrightarrow $ is used in order to show that certain equations
follow from respective coefficients of the Ricci tensor $\widehat{\mathbf{R}}%
_{\mu \nu }.$ In these formulas, the system of coordinates and polarization
functions are fixed for configurations with $g_{1}=g_{2}=e^{\psi (x^{k})}$
and nonzero values $\phi ^{\ast }$ and $h_{a}^{\ast }.$ The equations result
in solutions for the Levi--Civita configurations (with zero torsion) if the
coefficients of metrics are subjected to the conditions$\ $%
\begin{equation}
w_{i}^{\ast }=\mathbf{e}_{i}\ln \sqrt{|\ h_{4}|},\mathbf{e}_{i}\ln \sqrt{|\
h_{3}|}=0,\partial _{i}w_{j}=\partial _{j}w_{i}\mbox{ and }n_{i}^{\ast }=0.
\label{lccond}
\end{equation}

The system of nonlinear PDE (\ref{mgfeq1})--(\ref{lccond}) can be integrated in general forms for any $\omega $ constrained by a system of linear first order equations (\ref{conf2}). The explicit solutions are given by quadratic elements%
\begin{eqnarray}
ds^{2} &=&e^{\psi (x^{k})}[(dx^{1})^{2}+(dx^{2})^{2}]+\frac{\Phi ^{2}\omega
^{2}}{4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )}\mathring{h}_{3}[d\varphi
+\left( \partial _{k}\ n\right) dx^{k}]^{2}  \notag \\
&&-\frac{(\Phi ^{\ast })^{2}\omega ^{2}}{(\ ^{m}\Upsilon +\ ^{\alpha
}\Upsilon )\Phi ^{2}}\mathring{h}_{4}[dt+(\partial _{i}\ \widetilde{A}%
)dx^{i}]^{2}.  \label{nvlcmgs}
\end{eqnarray}%
for any $\Phi =\check{\Phi},$ $(\partial _{i}\check{\Phi})^{\ast }=\partial
_{i}\check{\Phi}^{\ast }$ and $w_{i}+\mathring{w}_{i}=\partial _{i}\check{%
\Phi}/\check{\Phi}^{\ast }=\partial _{i}\ \widetilde{A}.$\footnote{%
We can construct exact solutions even such conditions are not satisfied, i.e. the zero torsion conditions are not stated or there are given in non--explicit form; this way, it is possible to generate off--diagonal metrics and nonholonomically induced torsions etc, see details in \cite{vadm1}. There will pe presented physical arguments for what type of generating/ integration functions and sources we have to chose in order to construct realistic scenarios for Universe acceleration and observable dark energy/ matter effects.} To generate new solutions we can consider arbitrary nontrivial sources, $\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon \neq 0,$ and generating functions, $\Phi (x^{k},t):=e^{\phi }$ and $n_{k}=\partial _{k}n(x^{i}).$ Such metrics are generic off--diagonal and can not be diagonalized via coordinate transforms in a finite spacetime region because, in general, the anholonomy coefficients $W_{\alpha \beta }^{\gamma }$ for (\ref{nframes}) are not zero (we can check by explicit computations). The polarization $\eta $--functions for (\ref{nvlcmgs}) are computed in the form
{\small
\begin{equation}
\eta _{1}=e^{\psi }/\mathring{g}_{1},\eta _{2}=e^{\psi }/\mathring{g}%
_{2},\eta _{3}=\Phi ^{2}/4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon ),\eta
_{4}=(\Phi ^{\ast })^{2}/(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )\Phi ^{2}.
\label{polarf}
\end{equation}%
} So, prescribing any generating functions $\check{\Phi}(r,\theta ,t),$ $%
n(r,\theta ),$ $\omega (r,\theta ,\varphi ,t)$ and sources $\ ^{m}\Upsilon
,\ ^{\alpha }\Upsilon $ and then computing $\check{A}(r,\theta ,t),$ we can
transform any PG (and FLRW) metric $\mathbf{\mathring{g}}=(\mathring{g}_{i},%
\mathring{h}_{a},\mathring{w}_{i},\mathring{n}_{i})$ in MGT and/or GR into
new classes of generic off--diagonal exact solutions depending on all
spacetime coordinates. Such metrics define Einstein manifolds in GR with
effective cosmological constants $\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon .$
With respect to N--adapted frames (\ref{nframes}) the coefficients of metric
encode contributions from massive gravity, determined by $\ ^{\alpha
}\Upsilon ,$ and matter fields, included in $\ ^{m}\Upsilon .$

It is possible to provide an "alternative" treatment of (\ref{nvlcmgs}) as
exact solutions in MGT. Following such an approach, we have to define and
analyze the properties of fiducial St\"{u}ckelberg fields $\phi ^{\underline{%
\mu }}$ and the corresponding bimetric structure resulting in target
solutions $\mathbf{g}=(g_{i},h_{a},N_{j}^{a})$: Let us analyze the primary
configurations related to $\mathring{\phi}^{\underline{\mu }}=(\mathring{\phi%
}^{\underline{i}}=a(\tau )\rho \widetilde{\alpha }^{-1}\widehat{n}^{%
\underline{i}},\mathring{\phi}^{\underline{3}}=a(\tau )\rho \widetilde{%
\alpha }^{-1}\widehat{n}^{\underline{3}},\mathring{\phi}^{\underline{4}%
}=\tau \kappa ^{-1}),$ when the corresponding prime PG--metric $\mathbf{%
\mathring{g}}$ is taken in FLRW form $ds^{2}=a^{2}(d\rho ^{2}/(1-K\rho
^{2})+\rho ^{2}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}))-d\tau ^{2}.$
The related fiducial tensor (\ref{bm}) is computed
\begin{equation*}
\mathring{\Sigma}_{\underline{\mu }\underline{\nu }}du^{\underline{\mu }}du^{%
\underline{\nu }}=\frac{a^{2}}{\tilde{\alpha}^{2}}[d\rho ^{2}+\rho
^{2}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2})+2H\rho d\rho d\tau -(\frac{%
\tilde{\alpha}^{2}}{\kappa ^{2}a^{2}}-H^{2}\rho ^{2})d\tau ^{2}],
\end{equation*}%
where the coefficients and coordinates are re--defined in the form $%
r\rightarrow \rho =\widetilde{\alpha }r/a(\tau )$ and $t\rightarrow \tau
=\kappa t,$ for $K=0,\pm 1;\kappa $ is an integration constant; $H:=d\ln
a/d\tau $ and the local coordinates are parameterized in the form $x^{%
\underline{1}}=\rho ,x^{\underline{2}}=\theta ,y^{\underline{3}}=\varphi ,y^{%
\underline{4}}=\tau .$

For a target metric $\mathbf{g}={\mathbf{g}_{\alpha \beta }}$ and frames $%
\mathbf{e}_{\alpha }^{\ }=\mathbf{e}_{\alpha }^{\ \underline{\alpha }%
}\partial _{\underline{\alpha }},$ we can write
\begin{equation*}
\mathbf{g}_{\alpha \beta }=\mathbf{e}_{\alpha }^{\ \underline{\alpha }}%
\mathbf{e}_{\beta }^{\ \underline{\beta }}\eta _{\underline{\alpha }%
\underline{\beta }}=\left[
\begin{array}{cc}
g_{ij}+N_{i}^{a}N_{j}^{b}g_{ab} & N_{i}^{a}h_{ab} \\
N_{i}^{a}h_{ab} & h_{ab}%
\end{array}%
\right] ,\mbox{ for }\mathbf{e}_{\alpha }^{\ \underline{\alpha }}=\left[
\begin{array}{cc}
\mathbf{e}_{i}^{\ \underline{i}} & N_{i}^{b}\mathbf{e}_{b}^{\ \underline{a}}
\\
0 & \mathbf{e}_{a}^{\ \underline{a}}%
\end{array}%
\right] .
\end{equation*}%
The values $g_{ij}=\mathbf{e}_{i}^{\ \underline{\alpha }}\mathbf{e}_{j}^{\
\underline{\beta }}\eta _{\underline{\alpha }\underline{\beta }}=e^{\psi
}\delta _{ij}=diag[\eta _{i}\mathring{g}_{i}]$, $h_{ab}=\mathbf{e}_{i}^{\
\underline{\alpha }}\mathbf{e}_{j}^{\ \underline{\beta }}\eta _{\underline{%
\alpha }\underline{\beta }}=diag[\eta _{a}\mathring{h}_{a}]$, $N_{i}^{3}=%
\partial _{k}\ n$ and $N_{i}^{4}=\partial _{k}\ \widetilde{A}$ are related
algebraically to data (\ref{polarf}) resulting in off--diagonal solutions (%
\ref{nvlcmgs}). We can compute the "target" St\"{u}ckelberg fields as $\phi
^{\mu ^{\prime }}=\mathbf{e}_{\ \underline{\mu }}^{\mu ^{\prime }\ }\phi ^{%
\underline{\mu }}$ with $\mathbf{e}_{\ \underline{\mu }}^{\mu ^{\prime }\ }$
being inverse to $\mathbf{e}_{\alpha }^{\ \underline{\alpha }},$ and the
fiducial tensor $\Sigma _{\alpha \beta }=(\mathbf{e}_{\alpha }^{\ }\phi ^{%
\underline{\mu }})(\mathbf{e}_{\beta }^{\ }\phi ^{\underline{\nu }})\eta _{%
\underline{\mu }\underline{\nu }}=\mathbf{e}_{\alpha }^{\ \underline{\alpha }%
}\mathbf{e}_{\beta }^{\ \underline{\beta }}\Sigma _{\underline{\alpha }%
\underline{\beta }}.$ If the value $\mathring{\Sigma}_{\underline{\mu }%
\underline{\nu }}$ carries information about two constants $\kappa $ and $%
\widetilde{\alpha },$ a tensor $\Sigma _{\mu \nu }$ associated to
off--diagonal solutions encodes data about generating and integration
functions and via superpositions on possible Killing symmetries, on various
integration constants \cite{geroch}. In the framework of MGT, two
cosmological solutions $\mathbf{\mathring{g}}$ and $\mathbf{g}$ related by
nonholonomic deformations\footnote{%
involving not only frame transforms but also deformation of the linear
connection structure when at the end there are imposed additional
constraints for zero torsion} are characterised respectively by two
invariants $\mathring{I}^{\underline{\alpha }\underline{\beta }}=\mathring{g}%
^{\alpha \beta }\partial _{\alpha }\mathring{\phi}^{\underline{\alpha }%
}\partial _{\beta }\mathring{\phi}^{\underline{\beta }}$ and $\mathbf{I}%
^{\alpha \beta }=\mathbf{g}^{\alpha \beta }\mathbf{e}_{\alpha }\phi ^{%
\underline{\alpha }}\mathbf{e}_{\beta }\phi ^{\underline{\beta }}.$ The
tensor $\mathring{I}^{\underline{\alpha }\underline{\beta }}$ does not
contain singularities because there are not coordinate singularities on
horizon for PG metrics. The symmetry of $\Sigma _{\mu \nu }$ is not the same
as that of $\mathring{\Sigma}_{\underline{\mu }\underline{\nu }}$ and the
singular behaviour of $\mathbf{I}^{\alpha \beta }$ depends on the class of
generating and integration functions we chose of constructing a target
solution $\mathbf{g.}$

In GR and/or Einstein--Finsler gravity theories \cite{vadm1},
off--cosmological solutions of type (\ref{nvlcmgs}) were found to
generalized various models of Biachi, Kasner, G\"{o}del and other universes.
For instance, Bianchi type anisotropic cosmological metrics are generated if
we impose corresponding Lie algebra symmetries on metrics. It was emphasized
in \cite{kobayashi} that "any PG--type solution in general relativity (with
a cosmological constant) is also a solution to massive gravity." Such a
conclusion can be extended to a large class of generic off--diagonal
cosmological solutions generated by effective cosmological constants but it
is not true, for instance, if we consider nonholonomic deformations with
nonholonomically induced torsion like in metric compatible Finsler theories.

We note that the analysis of cosmological perturbations around an
off--diagonal cosmological background is not trivial because the fiducial
and reference metrics do not respect the same symmetries. Nevertheless,
fluctuations around de Sitter backgrounds seem to have a decoupling limit
which implies that one can avoid potential ghost instabilities if the
parameter choice  is considered both for diagonal and
off--diagonal cosmological solutions, see details in \cite{fluct}. This
special choice also allows us to have a structure $X_{\mu \nu }\sim g_{\mu
\nu }$ at list in N--adapted frames when the massive gravity effects can be
approximated by effective cosmological constants and exact solutions in MGT
which are also solutions in GR.

Let us consider three examples of off--diagonal cosmological solutons with
solitonic modifications in MGT and (with alternative interpretation) GR. Two
and three dimensional solitonic waves are typical nonlinear wave
configurations which can be used for generating spacetime metrics with
Killing, or non--Killing, symmetries and can be characterised by additional
parametric dependencies and solitonic symmetries.

\vskip5pt

\textbf{Example 1:} Taking a nonlinear radial (solitonic, with left $s$%
-label) generating function
\begin{equation}
\Phi =\ ^{s}\check{\Phi}(r,t)=4\arctan e^{q\sigma (r-vt)+q_{0}}
\label{1solgf}
\end{equation}%
and $\omega =1,$ we construct a metric
\begin{eqnarray}
\mathbf{ds}^{2} &=&e^{\psi (r,\theta )}(dr^{2}+\ d\theta ^{2})+~\frac{\ ^{s}%
\check{\Phi}^{2}}{4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )}\mathring{h}%
_{3}(r,\theta )d\varphi ^{2}  \notag \\
&&-\frac{(\partial _{t}\ ^{s}\check{\Phi})^{2}}{(\ ^{m}\Upsilon +\ ^{\alpha
}\Upsilon )\ ^{s}\check{\Phi}^{2}}\mathring{h}_{4}(r,t)[dt+(\partial _{r}\
\widetilde{A})dr]^{2},  \label{offdsol1}
\end{eqnarray}%
where, for simplicity, we consider $n(r,\theta )=0,$\ $\widetilde{A}(r,t)$
is defined as a solution of $\ ^{s}\check{\Phi}^{\bullet }/\ ^{s}\check{\Phi}%
^{\ast }=\partial _{r}\ \widetilde{A}$ and $\mathring{h}_{a}$ are given by
PG--data (\ref{pmdata}). The generating function (\ref{1solgf}), where $%
\sigma ^{2}=(1-v^{2})^{-1}$ for constants $q,q_{0},v,$ is just a 1--soliton
solution of the sine--Gordon equation
 $\ ^{s}\check{\Phi}^{\ast \ast }-\ ^{s}\check{\Phi}^{\bullet \bullet }+\sin \
^{s}\check{\Phi}=0$.
 For any class of small polarizations with $\eta _{a}\sim 1),$ we can
consider that the source $(\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )$ is
polarized by $\ ^{s}\check{\Phi}^{-2}$ when $h_{3}\sim \mathring{h}_{3}$ and
$h_{4}\sim \mathring{h}_{4}(\ ^{s}\check{\Phi}^{\ast })^{2}/\ ^{s}\check{\Phi%
}^{-4}$ with an off--diagonal term $\partial _{r}\ \widetilde{A}$ resulting
in a stationary solitonic universe. If we consider that $(\partial _{%
\widehat{R}}\widehat{f})^{-1}=\ ^{s}\check{\Phi}^{-2}$ in (\ref{effectsourc}%
), we can model $\widehat{f}$--interactions of type (\ref{act}) via
off--diagonal interactions and "gravitational polarizations". In absence of
matter, $\ ^{m}\Upsilon =0,$ the off--diagonal cosmology is completely
determined by $\ ^{\alpha }\Upsilon $ when $\ ^{s}\check{\Phi}$ transforms $%
\mathring{\mu}$ into an anisotropically polarized/variable mass of solitonic
waves. Such configurations can be modelled if $\ ^{m}\Upsilon \ll \ ^{\alpha
}\Upsilon .$ If $\ ^{m}\Upsilon \gg \ ^{\alpha }\Upsilon ,$ we generate
cosmological models determined by distribution off matter fields when
contributions from massive gravity are with small anisotropic polarization.
For a class of nonholonomic constraints on $\Phi $ and $\psi $ (which may be
not of solitonic type), when solutions (\ref{offdsol1}) are of type (\ref%
{offdans}) with $\eta _{\alpha }\sim 1$ and $n_{i},w_{i}\sim 0,$ we
approximate PG--metrics of type (\ref{pgm}). Hence, by an appropriate choice
of generating functions and sources, we can model equivalently modified
gravity effects, massive gravity contributions or matter field
configurations in GR and MGT interactions. For well defined conditions, such
configurations can be studied in the framework of some classes of
off--diagonal solutions in Einstein gravity with effective cosmological
constants.

\vskip5pt

\textbf{Example 2:} Three dimensional solitonic anistoropic waves can be
generated, for instance, if we take instead of (\ref{1solgf}) a generating
functions $\ ^{s}\check{\Phi}(r,\theta ,t)$ which is a solution of the
Kadomtsev--Petivashvili, KdP, equations \cite{kadom},%
\begin{equation*}
\pm \ ^{s}\check{\Phi}^{\prime \prime }+(\ ^{s}\check{\Phi}^{\ast }+\ ^{s}%
\check{\Phi}\ \ ^{s}\check{\Phi}^{\bullet }+\epsilon \ ^{s}\check{\Phi}%
^{\bullet \bullet \bullet })^{\bullet }=0,
\end{equation*}%
when solutions induce certain anisotropy on $\theta .$\footnote{%
In a similar form, we can construct various types of vacuum gravitational
2-d and 3-d configurations characterized by solitonic hierarchies and
related bi--Hamilton structures, for instance, of KdP equations with
possible mixtures with solutions for 2-d and 3-d sine--Gordon equations etc,
see details in Ref. \cite{vacarsolitonhier}.} In the dispersionless limit $%
\epsilon \rightarrow 0,$ we can consider that the solutions are independent
on $\theta $ and determined by Burgers' equation $\ ^{s}\check{\Phi}^{\ast
}+\ ^{s}\check{\Phi}\ \ ^{s}\check{\Phi}^{\bullet }=0.$ The solutions can be
parameterized and treated similarly to (\ref{offdsol1}) but with, in
general, a nontrivial term $(\partial _{\theta }\ \widetilde{A})d\theta $
after $\mathring{h}_{4},$ when $\ ^{s}\check{\Phi}^{\bullet }/\ ^{s}\check{%
\Phi}^{\ast }=\ \widetilde{A}^{\bullet }$ and $\ ^{s}\check{\Phi}^{\prime
}/\ ^{s}\check{\Phi}^{\ast }=\ \widetilde{A}^{\prime }.$

\vskip5pt

\textbf{Example 3:} Solitonic waves can be considered for a nontrivial
vertical conformal $v$--factor as in (\ref{offdans}), for instance, of KdP
type, when $\omega =\check{\omega}(r,\varphi ,t),$ when $x^{1}=r,$ $%
x^{2}=\theta ,y^{3}=\varphi ,y^{4}=t,$ for
\begin{equation}
\pm \check{\omega}^{\diamond \diamond }+(\partial _{t}\check{\omega}+\check{%
\omega}\ \check{\omega}^{\bullet }+\epsilon \check{\omega}^{\bullet \bullet
\bullet })^{\bullet }=0,  \label{kdp1}
\end{equation}%
In the dispersionless limit $\epsilon \rightarrow 0,$ we can consider that
the solutions are independent on angle $\varphi $ and determined by Burgers'
equation $\check{\omega}^{\ast }+\check{\omega}\ \check{\omega}^{\bullet
}=0. $ The conditions (\ref{conf2}) impose an additional constraint $\mathbf{%
e}_{1}\check{\omega}=\check{\omega}^{\bullet }+w_{1}(r,\theta ,\varphi )%
\check{\omega}^{\ast }+n_{1}(r,\theta )\check{\omega}^{\diamond }=0.$ In the
system of coordinates when $\check{\omega}^{\prime }=0,$ we can fix $w_{2}=0$
and $n_{2}=0.$ For any arbitrary generating function with LC--configuration,
$\check{\Phi}(r,\theta ,t),$ we construct exact solutions
\begin{eqnarray}
\mathbf{ds}^{2} &=&e^{\psi (r,\theta )}(dr^{2}+\ d\theta ^{2})+~\frac{\check{%
\Phi}^{2}\check{\omega}^{2}}{4\ (\ ^{m}\Upsilon +\ ^{\alpha }\Upsilon )}%
\mathring{h}_{3}(r,\theta )d\varphi ^{2}  \notag \\
&&-\frac{(\partial _{t}\ \check{\Phi})^{2}\check{\omega}^{2}}{(\
^{m}\Upsilon +\ ^{\alpha }\Upsilon )\ \check{\Phi}^{2}}\mathring{h}%
_{4}(r,t)[dt+(\partial _{r}\ \widetilde{A})dr]^{2},  \label{offdsol3}
\end{eqnarray}%
which are generic off--diagonal and depend on all spacetime coordinates.
Such stationary cosmological solutions are with polarizations on two angles $%
\theta $ and $\varphi .$ Nevertheless, the character of anisotropies is
different for metrics of type (\ref{offdsol1}) and (\ref{offdsol3}). In the
third class of metrics, we obtain a Killing symmetry on $\partial _{\varphi
} $ only in the limit $\check{\omega}\rightarrow 1,$ but in the first two
ones, such a symmetry exists generically. For (\ref{offdsol3}), the value $%
\check{\Phi}$ is not obligatory a solitonic one which can be used for
additional off--diagonal modifications of solutions and various types of
polarizations. We can provide an interpretation similar to that in Example
1, if the generating and integration functions are chosen to satisfy the
conditions $\eta _{\alpha }\sim 1$ and $n_{i},w_{i}\sim 0,$ we approximate
PG--metrics of type (\ref{pgm}). In a particular case, we can use a
conformal $v$--factor which is a 1--solitonic one, i.e. $\check{\omega}%
\rightarrow \omega (r,t)=4\arctan e^{q\sigma (r-vt)+q_{0}},$ where $\sigma
^{2}=(1-v^{2})^{-1}$ and constants $q,q_{0},v,$ defines a 1--soliton
solution of the sine--Gordon equation $\omega ^{\ast \ast }-\omega ^{\bullet
\bullet }+\sin \omega =0.$ Such a soliton propagates in time along the
radial coordinate.

We now consider a reconstruction mechanism with distinguished off--diagonal
cosmological effects \cite{vadm1} by generalizing some methods elaborated for $%
f(R)$ gravity in \cite{odints}. Any cosmological solution in massive, MGT
and/or GR parameterized in a form (\ref{offdans}) (in particular, as (\ref%
{offdsol1}) and (\ref{offdsol3})) can be encoded into an effective
functional $\widehat{f}-\frac{\mathring{\mu}^{2}}{4}\mathcal{U}=f(\widehat{R}%
),\widehat{R}_{\mid \widehat{\mathbf{D}}\rightarrow \nabla }=R$ (\ref%
{mgrfunct}). This allows us to work as in MGT;  the conditions $\partial
_{\alpha }f(\widehat{R})=0$ if $\widehat{R}=const$  simplify
substantially the computations. The starting point is to consider a prime
flat FLRW like metric
\begin{equation*}
ds^{2}=a^{2}(t)[(dx^{1})^{2}+(dx^{2})^{2}+(dy^{3})^{2}]-dt^{2},
\end{equation*}%
where $t$ is the cosmological time. In order to extract a monotonically
expanding and periodic cosmological scenario, we parameterize $\ln
|a(t)|=H_{0}t+\tilde{a}(t)$ for a periodic function $\tilde{a}(t+\tau )=\
^{1}a\cos (2\pi t/\tau ),$ where $0<\ ^{1}a<H_{0}.$ Our goal is to prove
that such a behavior is encoded into off--diagonal solutions of type (\ref%
{offdsol1})--(\ref{offdsol3}).

We  write FLRW like equations with respect to N--adapted (moving) frames (%
\ref{nframes}) for a generalized Hubble function $H,$%
\begin{equation*}
3H^{2}=8\pi \rho \mbox{ and }3H^{2}+2\mathbf{e}_{4}H=-8\pi p.
\end{equation*}%
Using variables with $\partial _{\alpha }f(\widehat{R})_{|\widehat{R}%
=const}=0$, we can consider a function $H(t)$ when $\mathbf{e}_{4}H=\partial
_{t}H=H^{\ast }.$ The energy--density and pressure of an effective
perfect fluid are computed
\begin{eqnarray}
\rho  &=&(8\pi )^{-1}[(\partial _{R}f)^{-1}(\frac{1}{2}f(R)+3H\mathbf{e}%
_{4}(\partial _{R}f))-3\mathbf{e}_{4}H]  \label{rhomg} \\
&=&(8\pi )^{-1}[\partial _{\widehat{R}}\ln \sqrt{|\widehat{f}|}-3H^{\ast
}]=(8\pi )^{-1}[\partial _{\widehat{R}}\ln \sqrt{|\frac{\mathring{\mu}^{2}}{4%
}\mathcal{U}+f(\widehat{R})|}-3H^{\ast }],  \notag \\
p &=&-(8\pi )^{-1}[(\partial _{R}f)^{-1}(\frac{1}{2}f(R)+2H\mathbf{e}%
_{4}(\partial _{R}f)+\mathbf{e}_{4}\mathbf{e}_{4}(\partial _{R}f))+\mathbf{e}%
_{4}H]  \notag \\
&=&(8\pi )^{-1}[\partial _{\widehat{R}}\ln \sqrt{|\widehat{f}|}+H^{\ast
}]=-(8\pi )^{-1}[\partial _{\widehat{R}}\ln \sqrt{|\frac{\mathring{\mu}^{2}}{%
4}\mathcal{U}+f(\widehat{R})|}+H^{\ast }].  \notag
\end{eqnarray}%
In N--adapted variables, the equation of state, EoS, parameter for the
effective dark fluid is defined by
\begin{equation}
w=\frac{p}{\rho }=\frac{\widehat{f}+2H^{\ast }\partial _{\widehat{R}}%
\widehat{f}}{\widehat{f}-6H^{\ast }\partial _{\widehat{R}}\widehat{f}}=\frac{%
\frac{\mathring{\mu}^{2}}{4}\mathcal{U}+f(\widehat{R})+2H^{\ast }\partial _{%
\widehat{R}}\widehat{f}}{\frac{\mathring{\mu}^{2}}{4}\mathcal{U}+f(\widehat{R%
})-6H^{\ast }\partial _{\widehat{R}}\widehat{f}},  \label{wcosm}
\end{equation}%
when the corresponding EoS is $p=-\rho -(2\pi )^{-1}H^{\ast }$ and $\mathcal{%
U}(t)$ is computed, for simplicity, for a configuration of "target" St\"{u}%
ckelberg fields  $\phi ^{\mu ^{\prime }}=\mathbf{e}_{\ \underline{\mu }%
}^{\mu ^{\prime }\ }\phi ^{\underline{\mu }}$ when a found solution is finally modelled by generating functions with  dependencies
on $t.$

Taking a generating Hubble parameter $H(t)=H_{0}t+H_{1}\sin \omega t,$ for $%
\omega =2\pi /\tau ,$ we can recover the modified action for oscillations of
off--diagonal (massive) universe (see similar details in \cite{odints}),%
\begin{equation}
f(R(t))=6\omega H_{1}\int dt[\omega \sin \omega t-4\cos \omega
t(H_{0}+H_{1}\sin \omega t)]\exp [H_{0}t+\frac{H_{1}}{\omega}\sin \omega t].
\label{recovf}
\end{equation}%
We can not invert analytically to find in explicit form $R.$ Nevertheless,
we can prescribe any values of constants $H_{0}$ and $H_{1}$ and of $\omega $
and compute effective dark energy and dark matter oscillating cosmology
effects for any off--diagonal solution in massive gravity and/or effective
MGT, GR. To extract contributions of $\mathring{\mu}$ we can fix, for
instance, $\widehat{f}(\widehat{R})=$ $\widehat{R}=R$ and using (\ref{act})
and (\ref{act1}) we can relate $f(R(t))$ and respective constants to certain
observable data in cosmology.

The MGT theories studied in this work  encode, for respective nonholonomic
constraints, the ekpyrotic scenario which can be modelled similarly to  $f(R)
$ gravity. A scalar field is introduced into usual ekpyrotic models in order
to reproduce a cyclic universe and such a property exists if we consider
off--diagonal solutions with massive gravity terms and/or $f$%
--modifications.  Let us consider a prime configuration with energy--density
for pressureless matter $\mathring{\rho}_{m},$ for radiation and
anisotropies  we take respectively  $\mathring{\rho}_{r}$ and $\mathring{\rho%
}_{\sigma }$ for radiation and anisotropies, $\kappa $ is the spatial
curvature of the universe and a target effective energy--density $\rho $ (%
\ref{rhomg}). A FLRW model can be described by
\begin{equation*}
3H^{2}=8\pi \lbrack \frac{\mathring{\rho}_{m}}{a^{3}}+\frac{\mathring{\rho}%
_{r}}{a^{4}}+\frac{\mathring{\rho}_{\sigma }}{a^{6}}-\frac{\kappa }{a^{2}}%
+\rho ].
\end{equation*}%
We generate an off--diagonal/massive gravity cosmological \ cyclic  scenario
containing a contracting phase by solving the initial problems if $w>1,$ see
(\ref{wcosm}). A homogeneous and isotripic spatially flat universe is
obtained when the scale factor tends to zero and the effective $f$--terms
(massive gravity and off--diagonal contributions) dominate over the rests.
In such cases, the results are similar to those in the inflationary
scenario. For recovering (\ref{recovf}), the ekpyrotic scenario takes place
and mimic the observable universe for $t\sim \pi /2\omega $ in the
effective EoS parameter
 $w\approx -1+\sin \omega t/3\omega H_{1}\cos ^{2}\omega t\gg 1$.
 This allows us to conclude that in massive gravity and/or using
off--diagonal interactions in GR cyclic universes can be reconstructed in
such forms that the initial,  flatness and/or horizon problems can be solved.

In the diversity of off--diagonal cosmological solutions which can constructed using above presented methods, there are cyclic ones with singularities of the type of big bang/ crunch behaviour. Choosing necessary types generating and integration functions, we can avoid singularities and elaborate models with smooth transition. Using the possibility to generate nonholonomically constrained $f$--models with equivalence to certain classes of solutions in massive gravity and/or off--diagonal configurations in GR, we can study in this context, following methods in  \cite{odints,vadm1}, , big and/or little rip cosmology models, when the phantom energy--density is modelled by off-diagonal interactions. We omit such considerations in this letter.


In summary, we have found new cosmological off--diagonal solutions in massive gravity with flat, open and closed spatial geometries. We applied a geometric techniques for decoupling the field equations and constructing exact solutions in $f(R)$ gravity, theories with nontrivial torsion and noholonomic constraints to GR and possible extensions  on (co) tangent Lorentz bundles. A very important property of such generalized classes of solutions is that they depend, in principle, on all spacetime coordinates via generating and integration functions and constants. After some classes of solutions were constructed in general form, we can impose at the end nonholonomic constraints, cosmological approximations, extract configurations with a prescribed spacetime symmetry, consider asymptotic conditions etc. Thus, our solutions can be used not only for elaborating homogeneous and isotropic cosmological models with arbitrary spatial curvature, but also for study "non--spherical" collapse models of the formation of cosmic structure such as stars and galaxies (see also  \cite{kobayashi}).

   Deriving cosmological scenarios only for diagonalizable metrics, there are possibilities to discriminate the massive gravity theory form the $f$--gravity and/or GR. Mathematically, we work with certain nonlinear systems of ordinary differential equations with general solutions depending on integration constants. Following such an approach, we positively have to modify the GR theory in order to explain observational data in modern cosmology and elaborate realistic quantum models of massive gravity. Fundamental  field equations in gravity consists from nonlinear off--diagonal systems of PDE. It is very difficult to construct off--diagonal solutions and to provide certain physical meaning for such spacetimes. Perhaps, such solutions are important at large cosmological scales because of the possibility to models polarization of constants and nonlinear interactions, anisotropies and generalized sources in modified gravity theories. A new and important feature is that the off--diagonal anisotropic configurations allows us to model cosmic accelerations and massive gravity and/or dark energy and dark matter effects as certain effective Einstein spaces. For a large class of such solutions, we can put the question:\  May be we do not need to consider modifications of the Einstein gravity but only to extend the constructions to off--diagonal solutions and nonholonomic systems and try to apply this in modern cosmology? This is a quite complicated theoretical and experimental problem and the main goal of this paper was to analyze such constructions from the viewpoint of massive gravity theory when off--diagonal effects can be alternatively explained to other types of gravity theories.


We also developed the reconstruction  method for the massive gravity theory  which admits and effective off--diagonal interpretation in GR and $f$--modified gravity with cyclic and
ekpyrotic universe solution.  The expansion is around the GR action if we admit a nontrivial effective torsion. For zero torsion  constraints, it is also possible to perform off--diagonal cosmological models keeping the constructions in the framework of the GR theory. Our results indicate that theories with massive gravitons and off--diagonal interactions may lead to more complicated cyclic universe. Following such an approach,  the ekpyrotic (little rip) scenario can be realized with no need to introduce an additional field (or modifying gravity) but only in terms of massive gravity or GR. Further constructions can be related to   reconstruction scenarios of $f(R)$ and massive gravity theories leading to little rip universe. The dark energy for little rip models presents an example of non--singular phantom cosmology. Finally, we note  that other types of non--singular super--accelerating universe may be also reconstructed in $f(R)$ gravity.


\vskip5pt

\textbf{Acknowledgments:\ } The work is partially supported by the Program
IDEI, PN-II-ID-PCE-2011-3-0256. The author thanks S. Capozziello, P. Stavrinos, D. Singleton and  S. Rajpoot  for important discussions, critical remarks, or substantial support for seminars.

%\newpage

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110} (2010) 73
\end{thebibliography}

\end{document}

1304.1080.extracted_bib
\begin{thebibliography}{99}
\bibitem{fierz} M. Fierz and W. Pauli, Proc. Roy. Soc.\ Lond. A 173 (1939)
211

\bibitem{odints} K. Hinterbichler, Rev. Mod. Phys. 84 (2012) 671;\ S.
Nojiri, S. Odintsov and N. Shirai,  arXiv: 1212.2079; S. Nojiri and S. Odintsov, Phys.
Lett. B716 (2012) 377;\ S. Nojiri, S. Odintsov and D. Saez-Gomez, AIP Conf.
Proc. 1458 (2011) 207, arXiv: 1108.0768

\bibitem{dvz} H. van Dam and J. J. G. Veltman, Nucl. Phys B 22 (1970) 397;
V.\ Zakharov, JETP Letters (Sov. Phys.) 12 (1970) 312;\ A. I. Vainshtein,
Phys. Lett. B 39 (1972) 393

\bibitem{dgp} G. R. Dvali, G.\ Gabadadze and M. Porrati, Phys. Lett. B 485
(2000) 2008; C. Deffayet, G. R. Dvali and G. Babadaze, Phys. Rev. D 65
(2002) 044023; M. A. Luty, M. Porrati and R.\ Rattazzi, JHEP 0309 (2003)
029; A. Nicolis and R. Rattazzi, JHEP 0406 (2004) 059

\bibitem{boul} D. G. Boulware and S. Deser, Phys. Rev. D 6 (1972) 3368; N.
Arkani--Hamed, H. Gergi and M. D. Schwarz, Annals Phys. 305 (2003) 96

\bibitem{drg} C. de Rham, Phys. Lett. B 681 (2009) 89; C. de Rham and G.
Gabadadze, Phys.\ Rev. D 82 (2010) 044020; C. de Rham, G. Gabadadze, and A.
J. Tolley, Phys. Rev. Lett. \textbf{106} (2011) 231101;\ S. Hassan and R. A.
Rosen, Phys. Rev. Lett. 108 (2012) 041101; JHEP 1204 (2012) 2012

\bibitem{mgcosm} G. D'Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D.
Pirtskhalava and A. J. Tolley, Phys. Rev. D 84 (2011) 124046;\ A. E.
Gumrukcuoglu, C. Lin, and S.\ Mukoyama, JCAP 1203 (2012) 006;\ M. von
Strauss, A. Schmidt--May, J. Enander, E. Mortsell and S. F. Hassan, JCAP
1203 (2012) 042;\ M. S. Volkov, JHEP 1201 (2012) 035;\ S. Koyama, G. Niz,
and G.\ Tasinato, JHEP 1112 (2011) 065;\ P. Gratia, W. Hu and M. Wyman,
Phys. Rev. D 86 (2012) 061504

\bibitem{kobayashi} T. Kobayashi, S. Siino, M. Yamaguchi and D. Yoshida,
Phys. Rev. D 86 (2012) 061505

\bibitem{vadm1} P. Stavrinos and S. Vacaru, Class. Quant. Grav. 30 (2013)
055012;\ S. Vacaru, Gen. Relat. Grav. 44 (2012) 1015;\ Europhys. Letters
(EPL) 96 (2011) 5001;\ Class. Quant. Grav. 28 (2011) 215991; Int. J. Geom.
Meth. Mod. Phys. 8 (2011) 9;\ J. Phys.: Conf. Ser. 543 (2013) 012021;\ Int. J.
Theor. Phys. 49 (2010) 1788;\ Int. J. Mod. Phys. D 21 (2012) 1250072;\ EPJC 74 (2014) 3132

\bibitem{geroch} R. Geroch, J. Math. Phys. 12 (1971) 918; R. Geroch, J.
Math. Phys. 12 (1972) 394; S. Vacaru, Int. J. Geom. Meth. Mod. Phys. 4
(2008) 1285

\bibitem{fluct} L. Berezhiani, G. Chkareuli, C. de Rham, G. Gababadze and A.
J. Tolley, Phys. Rev. D 85 (2012) 044024; K. Koyama, G. Niz and G. Tasinato,
JHEP 1112 (2011) 065

\bibitem{kadom} B. B. Kadomtsev and V. I. Petviashvili, Doklady Akademii
Nauk SSS \textbf{192} (1970) 753;\ Sov. Phys. Dokl. \textbf{15}
(1970) 539 [English translation]

\bibitem{vacarsolitonhier} S. Vacaru, Acta Applicandae Mathematicae \textbf{%
110} (2010) 73
\end{thebibliography}