Compact static stars in minimal dilatonic gravity
P. P. Fiziev \footnote{fizev@phys.uni-sofia.bg\
\
\
and\
\
\
fizev@theor.jinr.ru
We present the basic equations and relations for the relativistic static spherically symmetric stars (SSSS) in the model of minimal dilatonic gravity (MDG) which is {\em locally equivalent to the f(R) theories of gravity and gives an alternative description of the effects of dark matter and dark energy. The results for the simplest form of the relativistic equation of state (EOS) of neutron matter are represented. Our approach overcomes the well-known difficulties of the physics of SSSS in the f(R) theories of gravity introducing two novel EOS for cosmological energy-pressure densities and dilaton energy-pressure densities, as well as proper boundary conditions. %PACS numbers:
mdg_starsPRL.tex
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\title{Compact static stars in minimal dilatonic gravity}

\author{P. P.  Fiziev
\footnote{fizev@phys.uni-sofia.bg\,\,\,and\,\,\,
fizev@theor.jinr.ru}}
\affiliation{Sofia University Foundation of Theoretical and Computational Physics and Astrophysics, Boulevard
5 James Bourchier, Sofia 1164, Bulgaria\\
and\\
BLTF, JINR, Dubna, 141980 Moscow Region, Rusia}


 \begin{abstract}

 We present the basic equations and relations
 for the relativistic static spherically symmetric  stars (SSSS)
 in the  model of minimal dilatonic  gravity  (MDG)
 which is {\em locally} equivalent to the f(R) theories of gravity and gives an alternative
 description of the effects of dark matter and dark energy.
 The results for the simplest form of the relativistic equation of state (EOS) of neutron matter are represented.
 Our approach overcomes the well-known difficulties of the physics of SSSS in the f(R) theories of gravity
 introducing two novel EOS for cosmological energy-pressure densities
 and dilaton energy-pressure densities, as well as  proper boundary conditions.

 %PACS numbers:
\end{abstract}
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\maketitle
\section{Introduction}\label{Intro}
The MDG model was proposed and studied in \cite{OHanlon72,Fiziev00a,Fiziev02,Fiziev13}.
It describes a proper generalization of the Einstein general relativity (GR), being
based on the following action of the gravi-dilaton sector
\ben
{\cal A}_{g,\Phi}={\frac c {2\kappa}}\int d^4 x\sqrt{|g|}
 \bigl( \Phi R - 2 \Lambda U(\Phi) \bigr).
\la{A_MDG}
\een
%
Without any relation with astrophysics and cosmology,
it was studied by O'Hanlon, as early as in \cite{OHanlon72}.
His goal was to justify the Fujii idea about the "fifth force".
There the term "dilaton" for the field $\Phi$ was introduced.
Hence, the name MDG appeared.
It corresponds to the Branse-Dicke theory with the identically vanishing parameter $\omega$.
This property is also used as the name of the model in publications on the $f(R)$
theories of gravity, see \cite{Starobinsky80,Starobinsky07,Faraoni06,Frolov08,Felice10,Sotiriou10,Clifton12} and the huge amount of references therein.
The $f(R)$ theories are only {\em locally} equivalent to MDG \cite{Fiziev13}.

In Eq. \eqref{A_MDG}, $\kappa=8\pi G_{N}/c^2$ is the Einstein constant, $G_N$ is the Newton gravitational constant,
$\Lambda$ is the cosmological constant, and $\Phi\in (0,\infty)$ is the dilaton field. The values $\Phi$ must be positive
since a change of the sign of $\Phi$ entails a change of the sign of the gravitational factor $G_N/\Phi$
and leads to antigravity instead of gravity.
Such a change is physically unacceptable. Besides, the value $\Phi=0$
must be excluded since it leads to an infinite gravitational factor and makes the Cauchy problem
in MDG not well posed \cite{EFarese}.
The value $\Phi=\infty$ turns off the gravity and is also physically unacceptable.

The scalar field $\Phi$ is introduced to consider a variable gravitational factor $G(\Phi)=G_N/\Phi=G_N g(\Phi)$ instead
of the gravitational constant $G_N$.
The cosmological potential $U(\Phi)$ is introduced to consider  a variable cosmological factor $\Lambda U(\Phi)$
instead of the cosmological constant $\Lambda$. In GR with cosmological constant $\Lambda$
we have $\Phi\equiv 1$, $g(\Phi)\equiv 1$, and $U(1)\equiv 1$.
Due to its specific physical meaning,
the field $\Phi$ has quite unusual properties.

The function $U(\Phi)$ defines the cosmological potential
which must be a positive {\em single valued} function of the dilaton field $\Phi$ by astrophysical reasons\footnote{The
astrophysical observations show that in the observable Universe the cosmological term in Eq. \eqref{A_MDG} has a unique negative value,
$\Lambda$ being a positive constant.}.
See \cite{Fiziev13} for all physical requirements on the cosmological potential $U(\Phi)$,
necessary for a sound MDG model.
There the class of {\em withholding} potentials was introduced.
These confine dynamically the values of the dilaton $\Phi$ in the physical domain.
It is hard to formulate such a property for the function $f(R)$ in a simple intuitive way.

Some more physical and astrophysical consequences of MDG are described in
\cite{OHanlon72,Fiziev00a,Fiziev00b,Fiziev02,Fiziev03,Fiziev13,EFarese}.
There one can find a comparison of MDG with observations,
observational restrictions on the mass of the dilaton  $m_\Phi$,
discussions of the MDG-cosmology, and some study of the structure of boson stars in MDG.
This model provides an alternative explanation of the observed astrophysical effects
without introduction of dark energy and dark matter.

According to the widespread opinion, the f(R) gravity does not admit the existence of stable star
configurations and thus may be considered unrealistic outside cosmology, see for example
\cite{Frolov08,Felice10,Sotiriou10,Clifton12,Kobayashi08}.

On the other hand, this conclusion seems to be hasty and misleading,
since by representing $f(R)=R+\Delta f(R)$ with small perturbation $\Delta f(R)$
one obtains models of SSSS which deviate not very much
from the corresponding GR models \cite{Arapoglu11,Deliduman12,Artyom14}.

The goal of the present article is to solve this problem in MDG and to give the simplest
examples of physically consistent families of SSSS for an admissible cosmological potential $U(\Phi)$.

We explicitly show how the dilatonic field $\Phi$ changes the structure of the compact stars
and creates a specific dilatonic sphere around them.

\section{Basic equations and boundary conditions for SSSS in MDG}
In units $G_N=c=1$  the field equations of MDG can be written in the form:
\begin{subequations}\label{DGE:ab}
\ben
\Phi \hat{R}_\alpha^\beta +\widehat{\nabla_\alpha\nabla^\beta}\Phi+ 8\pi\hat{T}_\alpha^\beta&=&0, \label{DGE:a}\\
\Box\Phi+ \Lambda V^\prime(\Phi) &=& {\frac {8\pi} 3} T. \label{DGE:b}
\een
\end{subequations}
Here ${T}_\alpha^\beta$ is the standard energy-momentum tensor of the matter, $\hat{X}_\alpha^\beta={X}_\alpha^\beta - {\frac 1 4}X\delta_\alpha^\beta$
is the traceless part of any tensor ${X}_\alpha^\beta$ in four dimensions,
$X=X_\alpha^\alpha$ is its trace, the relation $V^\prime(\Phi)={\frac 2 3}\Big(\Phi U^\prime(\Phi) -2U(\Phi)\Big)$ introduces the dilatonic potential $V(\Phi)$, and
the prime denotes differentiation with respect to the variable $\Phi$. See for conventions and notation \cite{Fiziev13}.

In the problems under consideration, the space-time-interval is
$ds^2=e^{\nu(r)}dt^2-e^{\lambda(r)}dr^2 - r^2 d\Omega^2 $ \cite{Landau},
where $r$ is the luminosity distance to the center of symmetry,
and $d\Omega^2$ describes the space-interval on the unite sphere.
Then, after some algebra one obtains  the following basic results
for a SSSS of the luminosity radius $r^*$.

In the inner domain   $r\in [0,r^*]$  the SSSS structure is determined by the system:
\begin{subequations}\label{DE:abcd}
\ben
{\frac {dm}{dr}}&=&4\pi r^2\epsilon_{eff}/\Phi, \label{DE:a}\\
{\frac {d\Phi}{dr}}&=&-4\pi r^2 p_{{}_\Phi}/\Delta,  \label{DE:b}\\
{\frac {dp_{{}_\Phi}}{dr}}&=&- {\frac{ p_{{}_\Phi}}{r\Delta}}\left(3r -7 m-{\frac 2 3}\Lambda r^3+4\pi r^3\epsilon_{eff}/\Phi\right)\nonumber \\
&-&{\frac{2}{r}}\epsilon_{{}_\Phi}, \label{DE:c}\\
{\frac {dp}{dr}}&=&- {\frac {p+\epsilon}{r}}\,{\frac{m+4\pi r^3 p_{eff}/\Phi}{\Delta-2\pi r^3 p_{{}_\Phi}/\Phi} }. \label{DE:d}
\een
\end{subequations}
The four unknown functions are $m(r)$, $\Phi(r)$, $p_{{}_\Phi}(r)$, and $p(r)$. In Eqs. \eqref{DE:abcd}
$\Delta=r-2 m-{\frac 1 3}\Lambda r^3$,
$\epsilon_{eff}=\epsilon+\epsilon_{{}_\Lambda}+\epsilon_{{}_\Phi}$, $p_{eff}=p+p_{{}_\Lambda}+p_{{}_\Phi}$.
%
In addition, we obtain two novel EOS, which are specific for MDG:
\begin{subequations}\label{NewEOS:abc}
\ben
\epsilon_{{}_\Lambda}&=&- p_{{}_\Lambda}-{\frac \Lambda {12\pi}}\Phi \,;\la{NewEOS:a}\\
\epsilon_{{}_\Phi}&=&p-{\frac 1 3}\epsilon +
{\frac \Lambda {8\pi}}V^\prime(\Phi)\nonumber \\
&+&{\frac {p_{{}_\Phi}} 2}\,{\frac{m+4\pi r^3 p_{eff}/\Phi}{\Delta-2\pi r^3 p_{{}_\Phi}/\Phi}}\,; \la{NewEOS:b}\\
\epsilon &=& \epsilon(p)\la{NewEOS:c}
\een
\end{subequations}


Equation \eqref{NewEOS:a} is the EOS for the cosmological energy density
$\epsilon_{{}_\Lambda}={\frac \Lambda {8\pi}} \Big(U(\Phi)-\Phi\Big)$
and the cosmological pressure
$p_{{}_\Lambda}=-{\frac \Lambda {8\pi}} \Big(U(\Phi)-{\frac 1 3}\Phi\Big)$,
i.e.  the cosmological EOS (CEOS).

Equation \eqref{NewEOS:b} replaces the field equation \eqref{DGE:b} and presents the dilatonic EOS (DEOS).
It relates the dilatonic energy density
$\epsilon_{{}_\Phi}={\frac 1 {8\pi r^2}}\left(\Delta/r\right)^{1/2}{\frac d {dr}}\left(r^2 \left(\Delta/r\right)^{1/2}{\frac d {dr}}\Phi\right)$
and  the dilatonic pressure   $p_{{}_\Phi}=-{\frac \Delta {4\pi r^2}}  {\frac d {dr}}\Phi$, which
measures the gradient of gravitational factor.

Equation \eqref{NewEOS:c} presents the usual EOS of star matter (MEOS), see \cite{Luciano} for a modern detailed survey.

Adopting the widespread assumption that the SSSS-center C ($\Rightarrow$ index "c") is at $r_c=0$ we obtain the boundary conditions
\ben
m(0)=m_c=0,\,\,\,\,\,\Phi(0)=\Phi_c,\,\,\,\,\,p(0)=p_c,\nonumber \\
p_{{}_\Phi}(0)=p_{\Phi c}={\frac 2 3}\left({\frac {\epsilon(p_c)} 3}- p_c\right)-{\frac{\Lambda}{12\pi}}V^\prime(\Phi_c).
\la{center}
\een
Requiring $m_c=0$ we ensure finiteness of pressure $p_c$ simultaneously for the Newton-, GR- and MDG-SSSS.
The condition on $p_{\Phi c}\,(=-{\frac 2 3}\epsilon_{\Phi c})$  ensures its finiteness,
being a specific MDG-centre-values-relation: $F_\Phi(p_{\Phi c},p_c,\Phi_c)=0$.


The SSSS-edge ($\Rightarrow$ index "*")  is defined by the condition $p^*=p(r^*;p_c,\Phi_c)=0$ (and typically $\epsilon^* =0$). Then
\ben
m^*=m(r^*;p_c,\Phi_c),\,\,\,\Phi^*=\Phi(r^*;p_c,\Phi_c),\nonumber \\
p_{{}_\Phi}^*=p_{{}_\Phi}(r^*;p_c,\Phi_c).
\la{edge}
\een
The luminosity radius of a compact star for different physically sound MEOS varies: $r^*\sim 5\div 20$ km.

Outside the star $p\equiv 0$ and $\epsilon\equiv 0$, and we have a dilaton-sphere,
in brief -- {\em a disphere}. Its structure is determined
by the shortened system \eqref{DE:abcd}: Eq. \eqref{DE:d} has to be omitted.
%
For the exterior domain $r\in [r^*,r_{{}_{\!U}}]$ we use Eqs. \eqref{edge} as left boundary conditions.
The right boundary is defined by the cosmological horizon $r_{{}_{\!U}}$:  $\Delta(r_{{}_{\!U}}; p_c,\Phi_c)=0$,
where the de Sitter vacuum is reached:
$\Phi(r_{{}_{\!U}}; p_c,\Phi_c)=1$.
Thus we obtain a new MDG-centre-values-relation:   $F_\Lambda(p_c,\Phi_c)=0$.

The two MEOS-dependent-relations
\ben
F_\Phi(p_{\Phi c},p_c,\Phi_c)=0,\,\,\,F_\Lambda(p_c,\Phi_c)=0,
\la{F}
\een
show that in MDG, as well as in the Newton gravity and GR, we have a one-parameter-family of SSSS.

The observable  value $\Lambda\approx 1.27\times 10^{-44}$ $km^{-2}$ is very small.
As a result, the luminosity radius of the Universe $r_{{}_{\!U}} \sim 1/\sqrt{\Lambda}\sim 10^{22}\,\text{km}$
is very large in comparison with the luminosity radius of the compact star $r^*\sim 10$ km.


Further on, we use the cosmological potential
$U(\Phi)=\Phi^2+{\frac{3}{16\, \mathfrak{p}^2}}\left(\Phi-1/ \Phi\right)^2$. Hence,
$V^\prime(\Phi)={\frac 1 {2 \mathfrak{p}^2}}\left(1-1/\Phi^2\right)$.
For useful comments and a more general form of the admissible cosmological potentials $U(\Phi)$ see \cite{Fiziev02,Fiziev13}.
%
The parameter $\mathfrak{p}=\sqrt{\Lambda}\,\hbar/c\,m_\Phi$ is the dimensionless
Compton length (measured in cosmological units) of the dilaton $\Phi$.
MDG is consistent with observation if $\mathfrak{p}\lesssim 10^{-30}$ ($m_\Phi> 10^{-3}\, \text{eV}/c^2)$ \cite{Fiziev00a,Fiziev02}.
If one takes as a dilaton the only currently known fundamental scalar particle,
the Higgs boson with mass $m_H \approx 125\, GeV/c^2$, then $\mathfrak{p} \approx 1.8\times 10^{-43}$.
Further on, we  use a non-realistic big value $\mathfrak{p} =  10^{-21}$ (the Compton length $\sim 9$ km) just for a more transparent
graphical representation of the results.

\section{A simple EOS for neutron matter}

We do not present here a realistic model of a neutron star.
The most idealized relativistic MEOS of neutron matter is the well-known one
used in the Tolman-Oppenheimer-Volkov (TOV) model  \cite{TOV}: $\epsilon={\frac 1 {4\pi}x\left( \sinh t -t\right)}$,
$p={\frac 1 {12\pi}\left( \sinh t -8\sinh(t/2)+3t\right)}$.
It describes the ideal Fermi neutron gas at zero temperature (IFNG0T) and is free from
the difficulties related with unknown properties of neutron matter.
This facilitates our study of the new pure-MDG-effects in SSSS, shown in Figs. \ref{FLambda}--\ref{gr}.
Note that we plot in Figs. \ref{pPhiePhi} and \ref{epPhi_st_r_st} the functions $2\times\epsilon_\Phi$ and $p_\Phi$.
When applicable, we use different colors for star's interior, and for star's exterior.


%
\begin{figure}[!ht]
%\centering
\begin{minipage}{8.cm}
\vskip -.truecm
\hskip -1.truecm
\includegraphics[width=5.truecm]{FLambda.eps}
\vskip -.3truecm
\caption{\small The specific MDG-curve\\ $F_\Lambda(p_c,\Phi_c)=0$}
\label{FLambda}
\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
%\centering
\begin{minipage}{8.cm}
\vskip -.truecm
\hskip -.5truecm
\includegraphics[width=5.truecm]{mep.eps}
\vskip -.3truecm
\caption{\small The MDG-SSSS interior in accord with MEOS}
\label{mep}
\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
%\centering
\begin{minipage}{8.cm}
\vskip -.truecm
\hskip -.5truecm
\includegraphics[width=5.truecm]{mmdi.eps}
\vskip -.3truecm
\caption{\small The SSSS-disphere-mass-dependence on $r$}
\label{mmdi}
\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
\centering
\begin{minipage}{8.cm}
\vskip .truecm
\hskip -.8truecm
\includegraphics[width=5.5truecm]{eLpL_st_r_st.eps}
\vskip -.3truecm
\caption{\small The dependencies $\epsilon_\Lambda(r)$ and $p_\Lambda(r)$ in accord with CEOS}
\label{eLpL_st_r_st}
\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
\centering
\begin{minipage}{8.cm}
\vskip -.truecm
\hskip -.5truecm
\includegraphics[width=5.truecm]{pPhiePhi.eps}
\vskip -.3truecm
\caption{\small The dependencies $2\epsilon_\Phi(r)$ and $p_\Phi(r)$ in accord with DEOS}
\label{pPhiePhi}
\end{minipage}
\end{figure}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[!ht]
\centering
\begin{minipage}{9.cm}
\vskip -.truecm
\hskip -1.truecm
\includegraphics[width=5.truecm]{m_centerD.eps}
\vskip -.3truecm
\caption{\small The m* -- $\epsilon_c$ dependencies}
\label{m centerD}
\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
\centering
%\begin{minipage}{20.cm}
\vskip -.truecm
\hskip -1.9truecm
\includegraphics[width=5.truecm]{m_r.eps}
\vskip -.3truecm
\caption{\small Mass - radius relations for IFNG0T in \\
\hskip 1.truecm GR ($m_{max}^*\approx .7051\,m_{\odot}$, $r_{max}^*\approx 9.209$ km) and\\
\hskip 1.truecm in MDG ($m_{max}^*\approx .5073\,m_{\odot}$, $r_{max}^*\approx 7.092$ km)}
\label{m_r}
%\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
\centering
%\begin{minipage}{20.cm}
\vskip -.truecm
\hskip -1.truecm
\includegraphics[width=5.truecm]{Phi_st_r_st.eps}
\vskip -.3truecm
\caption{\small The specific MDG-dependence $\Phi^*(r*)$}
\label{Phi_st_r_st}
%\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
\centering
%\begin{minipage}{20.cm}
\vskip -.truecm
\hskip -1.truecm
\includegraphics[width=5.truecm]{epPhi_st_r_st.eps}
\vskip -.3truecm
\caption{\small The specific MDG-dependences $2 \epsilon_\Phi^*(r*)$, $p_\Phi^*(r*)$.\\
Maximal values are reached for smaller $r*$ than in Fig. \ref{m_r}}
\label{epPhi_st_r_st}
%\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
\centering
\begin{minipage}{8.cm}
\vskip -.truecm
\hskip -1.5truecm
\includegraphics[width=6.truecm]{eLpL.eps}
\vskip -.3truecm
\caption{\small The specific MDG-dependences \\ $\epsilon_\Lambda^*(r*)$ and $p_\Lambda^*(r*)$ in accord with CEOS}
\label{eLpL}
\end{minipage}
\end{figure}
%

%
\begin{figure}[!ht]
\centering
\begin{minipage}{8.cm}
\vskip -.truecm
\hskip -1.truecm
\includegraphics[width=5.5truecm]{fg_r.eps}
\vskip -.3truecm
\caption{\small The specific variations of the  gravitational
factor $g(r)=1/\Phi(r)$ for MDG-SSSS}
\label{gr}
\end{minipage}
\end{figure}
%
\section{Comments and Concluding remarks}

The interior structure of MDG-SSSS is shown in Fig.\ref{m centerD}.
It differs from the corresponding structure of GR-SSSS only quantitatively.
As seen in  Fig. \ref{m_r}, MDG-SSSS are lighter and more compact than GR-SSSS.

Precisely as in GR, MDG-SSSS may be stable only until maximal mass is reached, see Figs. \ref{m centerD} and \ref{m_r}.
As well as in GR, the full investigation of the stability problem of MDG-SSSS requires a study of their oscillations.


The obvious similarity of Figs. \ref{FLambda} and  \ref{m centerD}
shows that both they follow the behaviour of the function $m^*(r^*)$, shown in Fig. \ref{m_r}.

The most interesting novel result is shown in Fig. \ref{mmdi}:
The mass of the disphere $m_{disp}(r)$ outside MDG-SSSS as a function of the
luminosity radius $r\in[r^*, r_{{}_U}]$. As expected,
it exponentially goes to a constant. The total mass of the object exceeds the mass of the
very star. In our illustrative example the MDG-SSSS mass $ m^* \approx  .5073\, M_\odot$
is close to the extremal one.  We found the total mass of the disphere $m_{disphere} \approx .1638\, M_\odot$, i.e. $\approx 32$ per cent of mass of the star.
In the case of consideration, the total mass of the object
$m_{total} \approx 0.6710\, M_\odot$ is reached with good precision at the distance 
about several hundredths of star radii from the center. The total mass is quite close to the mass of GR-SSSS.

It will be quite interesting to work out a model of a moving and rotating star in MDG.
In this case, one can expect not only an asymmetric configuration,
but also the appearance of different centers of the star and its disphere,
or even detachment of the parts of disphere.
This seem to confirm a possible interpretation of the dilaton field $\Phi$ as a dark matter.
For similar effects at cosmological scales see \cite{Clowe}.

The basic property of the MDG dilaton $\Phi$  is that it does not interact directly with matter
\cite{Fiziev00a,Fiziev02,Fiziev03,Fiziev13}. Such interaction is possible only in quantum field
theory as a perturbation effect of the second order with respect to the small gravitational constant $G_N$.
Hence, the corresponding cross-sections will be extremely small
in accordance with the latest observational data,
see, for example, \cite{LUX} and the references therein.


Figures \ref{eLpL_st_r_st} and \ref{eLpL} confirm our expectation that the $\Lambda$-term in action \eqref{A_MDG} yields variable
positive $\epsilon_\Lambda$ and variable negative pressure $p_\Lambda$, inside, as well as outside MDG-SSSS.
The changes of  $\epsilon_\Lambda(r)$ and $p_\Lambda(r)$ are slightly asymmetric in accord to CEOS \eqref{NewEOS:a}.

These results remain valid also in the $f(R)$ theories of gravity,
which correspond to withholding MDG-potentials \cite{Fiziev13}.

As a result of the above statements and the previous work on MDG
a novel possibility arises: To look not only for realistic MEOS
but also for a withholding cosmological potentials able
to describe simultaneously the variety of cosmological, astrophysical and star phenomena.

The author is deeply indebted to the Directorate of the
Laboratory of Theoretical Physics, JINR, Dubna, for the
good working conditions. This
research was supported in part by the Foundation for
Theoretical and Computational Physics and Astrophysics
and Grant of the Bulgarian Nuclear Regulatory
Agency for 2014.

%\vskip .truecm

\begin{thebibliography}{28}
%
\bibitem{OHanlon72} O'Hanlon, Phys. Rev. Lett. {\bf 29} 137 (1972).

\bibitem{Fiziev00a} P. P. Fiziev, Mod. Phys. Lett. A, {\bf 15} 1077 (2000).

\bibitem{Fiziev00b} Fiziev P. P., Yazadjiev S., Boyadjiev T., Todorov M., Phys. Rev. D {\bf 61} 124018 (2000).

\bibitem{Fiziev02} Plamen P. Fiziev,  arXiv:gr-qc/0202074.

\bibitem{Fiziev03} P. Fiziev, Georgieva D., Phys. Rev. D {\bf 67} 064016 (2003).

\bibitem{Fiziev13} P. P. Fiziev, Phys. Rev. D {\bf 87} 0044053 (2013).


%%%%%%%%%%%%%%%%%%
% f(R)

\bibitem{Starobinsky80} A. A. Starobinsky, Phys. Lett. B {\bf 91}  99 (1980).

\bibitem{Faraoni06} V. Faraoni, Phis Rev. D {\bf 74} 104017 (2006).

\bibitem{Starobinsky07} A. A. Starobinsky, JETP Lett. {\bf 86} 157 (2007).

\bibitem{Frolov08} A. V. Frolov, PRL {\bf 101} 061103 (2008).

\bibitem{Felice10} A. De Felice, S. Tsujikawa, Living Rev. Rel. {\bf 13} 3 (2010).

\bibitem{Sotiriou10} T. P. Sotiriou, V. Faraoni, Rev. Mod. Phys. {\bf 82} 451 (2010).

\bibitem{Clifton12} T. Clifton, P. G. Ferreira, A. Padill, C. Skordis, Physics Reports {\bf 513} 1 (2012).


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

\bibitem{EFarese} G. Esposito-Farese, D. Polarski, Phys. Rev. D {\bf 63} 063504  (2001).


\bibitem{Kobayashi08} T. Kobayashi and K. Maeda, Phys. Rev. D {\bf 78} 064019 (2008).


\bibitem{Arapoglu11} A. S. Arapoglu, C. Deliduman, K. Y. Eksi,  J. Cosmology and Astroparticle Physics, {\bf 07} 020 (2011).

\bibitem{Deliduman12} C. Deliduman, K. Y. Eksi, V. Keles,  	J. Cosmology and Astroparticle Physics, {\bf 05} 036 (2012).

\bibitem{Artyom14} A. V., Astashenok, S. Capozziello, S. D. Odintsov, arXiv:1401.4546 [gr-qc].

\bibitem{Landau}  Landau, L.D., Lifshitz, E.M.  {\em The Classical Theory of Fields}, New York: Pergamon Press (1975).

\bibitem{Luciano} L. Rezzola, O. Zanotti, {em Relativistic Hydrodynamics}, Oxford University Press (2013).

\bibitem{TOV} Openheimer J. R., Volkoff G. M., Phys. Rev.  {\bf 55} 374 (1939).

\bibitem{Clowe} D. Clowe, M. Bradac,A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, D. Zaritsky, The Astrophysical Journal, {\bf 648} L109 (2006).

 \bibitem{LUX} LUX Collaboration, Accepted by Phys. Rev. Lett., arXiv:1310.8214.


\end{thebibliography}



\end{document}


1402.2813.extracted_bib
{OHanlon72 O'Hanlon, Phys. Rev. Lett. {29 137 (1972).
1402.2813.extracted_bib
{Fiziev00a P. P. Fiziev, Mod. Phys. Lett. A, {15 1077 (2000).
1402.2813.extracted_bib
{Fiziev00b Fiziev P. P., Yazadjiev S., Boyadjiev T., Todorov M., Phys. Rev. D {61 124018 (2000).
1402.2813.extracted_bib
{Fiziev02 Plamen P. Fiziev, arXiv:gr-qc/0202074.
1402.2813.extracted_bib
{Fiziev03 P. Fiziev, Georgieva D., Phys. Rev. D {67 064016 (2003).
1402.2813.extracted_bib
{Fiziev13 P. P. Fiziev, Phys. Rev. D {87 0044053 (2013). %%%%%%%%%%%%%%%%%% % f(R)
1402.2813.extracted_bib
{Starobinsky80 A. A. Starobinsky, Phys. Lett. B {91 99 (1980).
1402.2813.extracted_bib
{Faraoni06 V. Faraoni, Phis Rev. D {74 104017 (2006).
1402.2813.extracted_bib
{Starobinsky07 A. A. Starobinsky, JETP Lett. {86 157 (2007).
1402.2813.extracted_bib
{Frolov08 A. V. Frolov, PRL {101 061103 (2008).
1402.2813.extracted_bib
{Felice10 A. De Felice, S. Tsujikawa, Living Rev. Rel. {13 3 (2010).
1402.2813.extracted_bib
{Sotiriou10 T. P. Sotiriou, V. Faraoni, Rev. Mod. Phys. {82 451 (2010).
1402.2813.extracted_bib
{Clifton12 T. Clifton, P. G. Ferreira, A. Padill, C. Skordis, Physics Reports {513 1 (2012). %%%%%%%%%%%%%%%%%%%%%%%%
1402.2813.extracted_bib
{EFarese G. Esposito-Farese, D. Polarski, Phys. Rev. D {63 063504 (2001).
1402.2813.extracted_bib
{Kobayashi08 T. Kobayashi and K. Maeda, Phys. Rev. D {78 064019 (2008).
1402.2813.extracted_bib
{Arapoglu11 A. S. Arapoglu, C. Deliduman, K. Y. Eksi, J. Cosmology and Astroparticle Physics, {07 020 (2011).
1402.2813.extracted_bib
{Deliduman12 C. Deliduman, K. Y. Eksi, V. Keles, J. Cosmology and Astroparticle Physics, {05 036 (2012).
1402.2813.extracted_bib
{Artyom14 A. V., Astashenok, S. Capozziello, S. D. Odintsov, arXiv:1401.4546 [gr-qc].
1402.2813.extracted_bib
{Landau Landau, L.D., Lifshitz, E.M. {\em The Classical Theory of Fields, New York: Pergamon Press (1975).
1402.2813.extracted_bib
{Luciano L. Rezzola, O. Zanotti, {em Relativistic Hydrodynamics, Oxford University Press (2013).
1402.2813.extracted_bib
{TOV Openheimer J. R., Volkoff G. M., Phys. Rev. {55 374 (1939).
1402.2813.extracted_bib
{Clowe D. Clowe, M. Bradac,A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, D. Zaritsky, The Astrophysical Journal, {648 L109 (2006).
1402.2813.extracted_bib
{LUX LUX Collaboration, Accepted by Phys. Rev. Lett., arXiv:1310.8214.