text:
\section{\normalsize\bf{Introduction}} Accelerated phenomenon of
the universe is now well established observable fact but the
physical origin of this acceleration is unknown till now. There
are so many observational studies like the different luminosity
type Ia supernovae (SNIa) \cite{Gold2009,
Nolta2009,Bachall1999,Perlmutter1999}, in associated with Large
scale Structure \cite{Abazajian2004,Abazajian2005} and Cosmic
Microwave Background (CMB) \cite{Perlmutter1998,Riess,Riess1,
Bennet,Sperge} radiation which strongly support that our universe
is currently expanding with an acceleration. Many candidates are
considered as the main responsible for this observational facts of
expanding scenario which triggered out a new type of matter which
violates the strong energy condition i.e., $\rho+p<0$. The energy
density of such mysterious matters which have negative pressure
thats leads to cosmic acceleration, are known as Dark Energy (DE)
\cite{Sahni2006,Sahni2006b,Seikel,Clarkson,Liu}. Recent
observational studies indicate that Dark energy occupies about
$73\%$ of the energy of our universe, while dark matter
contribution is $\sim$$23\%$ and the usual baryonic matter is
$\sim$$4\%$. Dark energies are separated depending on their
equation of state (EOS). In the case of quintessence type DE, the
EOS is $-1 <\omega<1/3$, whereas for phantom type DE, EOS is $
\omega<-1$ but in the case of quintom type DE, the EOS can cross
the phantom divide $ \omega=-1$ from both sides. The most charming
and simplest candidate for DE is the cosmological constant
$\Lambda$. However, it is known to all that the cosmological
constant suffers two serious theoretical problems, i.e. the
cosmological constant problem and the coincidence problem. Hence,
for seeking the most reliable model for describing the DE
phenomenon, different methods have been adopted and for that
different dynamical DE models have been
developed.\\

There are another point of view to describe that acceleration
scenario. Recently many researchers consider Modified Gravity as
the alternating candidate of dark energy in support of
accelerating phenomenon. The basic idea of Modified Gravity is
that a concept of modification of gravitational theory and as a
result it provides a very natural gravitational alternative for
exotic matter. Without commencing negative kinetic term of dark
energies, Modify Gravity models can explain the phantom or
non-phantom or quintom phase of the universe, not only that, it
also illustrates about the early time inflation then transition
from deceleration to late time acceleration. The most popular
String/M-theory also supports Modify Gravity models. In recent day
scenario, it is treated as the rival of General Relativity.
Although there are some rigid constraints in modify gravity
theory, some particular form of modified gravity obtains a great
attention of current researchers. Ricci curvature $R$ of
Lagrangian is one of them. Different form of Ricci curvature have
been considered \cite{Vollick,Briscese, Carroll,Abdalla,Linder} to
explain the accelerating phase of the Universe. In this regards,
some remarkable work was done by Nojiri et al
\cite{Nojiri2003,Nojiri2007} where they considered the modified
Lagrangian as $R+ R^m + R^{-m}$ and as a result they obtained an
inflation at an early stage and also a late time of accelerated
expansion. These result lead to open new window of gravity
dependent modern research. The most reliable modified gravity
models incorporate $f(R)$ gravity (where $R$ represent the Ricci
Scalar Curvature) \cite{Nojiri2007a,Li,Nojiri2006}, $f(T)$ gravity
(where $T$ is the torsion scalar) \cite{Li2,Myrzakulov,Wu}, $f(G)$
gravity (where $G$ represents the Gauss-Bonnet invariant)
\cite{Rastkar,Nojiri2005}, $f(R,T)$ gravity \cite{Harko2011,
Jamil2012}, $f(R,G)$ gravity \cite{Bamba2010,Myrzakulov2013} and
so on. Recently, reconstruction between different dark energy
models become a very appealing scenario in modern cosmological
study. In 2008, Setare et al. \cite{Setare2008} studied the
cosmological proposition of the correspondence of Holographic dark
energy model and Gauss-Bonnet dark energy model. That showed the
way to explanation of the accelerated expansion of the universe by
imposing specific constraints. In \cite{Liu2009} Liu et al. talked
about the New Agegraphic Dark Energy (NADE) model in the framework
of the Brans-Dicke theory with the help of EoS and deceleration
parameters and finally they showed that the EoS parameter has a
quintom-like behavior for that model and it indicates the
accelerated expansion of the universe. The NADE in $f(R)$ gravity
model has been discussed in \cite{Setare2010} and as a result it
found that there may exist a phantom-like universe. Also in
\cite{Jamil2010} the NADE model is correspond with Horava-Lifshitz
gravity which specified that the accelerated expanding universe is
consistent with cosmological observations. The reconstruction of
Entropy-Corrected Holographic Dark Energy (ECHDE) in the $f(G)$
gravity also been investigated to explain the expanding universe
with acceleration in \cite{Setare2010b}. Recently the HDE model in
the framework of the $f(G)$ gravity has been discussed in
\cite{Jawad2013} and the different phenomenon for the accelerating
universe are explained.\\

The main purpose of this work is to analyzed some cosmological
parameters for interacting new Holographic dark energy model in
$f(T)$ gravity model. For this reason, homogeneous, isotropic
modified FRW for $f(T)$ gravity has been considered and after that
conservation equations for interacting new Holographic dark energy
and dark matter have been solved in Section 2. Then density and
pressure are converted into dimensionless quantity. After that
Deceleration parameter, EoS parameter, $\omega $--$ \omega'$ plane
analysis, State-finder parameters, Cosmographic parameters and
{\it Om} parameter are converted in terms of current density,
pressure and Hubble expansion rate parameter and graphically
analyzed in Section 3 respectively. In Section 4, stability of
this model have been analyzed. Finally, the results of the paper
are summarized in Section 5.

\section{\bf{Field equations and their solutions}}

The metric of a spatially flat homogeneous and isotropic universe
in FRW model is given by
\begin{equation}
ds^{2}=dt^{2}-a^{2}(t)[dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})]
\end{equation}
where $a(t)$ is the scale factor.

The Einstein field equation are given by
\begin{equation}
H^{2}=\frac{1}{3}\rho
\end{equation}
and
\begin{equation}
2\dot{H}+3H^{2}=-p
\end{equation}
where $\rho$ and $p$ are energy density and isotropic pressure
respectively (choosing $8\pi G=c=1$)and $H=\frac{\dot{a}}{a}$ is
the Hubble parameter.

Let us we start with $f(T)$ gravity models where the action is
given by
\begin{equation}
S=\frac{1}{2\kappa^{2}}\int dx^{4}[\sqrt{-g}f(T)+L_{m}]
\end{equation}
where $T$ is the torsion scalar, $f(T)$ is general differentiable
function of the torsion and $L_{m}$ corresponds to the matter
Lagrangian, $\kappa^{2}=8\pi G=1$. Therefore the field equations
for modified FRW model are given by
\begin{equation}\label{2.5}
3H^{2}=\rho_{MG}=\rho+\rho_{T}
\end{equation}
and
\begin{equation}\label{2.6}
2\dot{H}+3H^{2}=-p_{MG}=-(p+p_{T})
\end{equation}
where
\begin{equation}
\rho_{T}=\frac{1}{2}(2Tf_{T}-f+6H^{2}),~~~p_{T}=-\frac{1}{2}[-8\dot{H}Tf_{TT}+(2T-4\dot{H})f_{T}-f+4\dot{H}+6H^{2}]
\end{equation}
and $T=-6H^{2}$.

We consider the power law model $f(T)=T^{n}$, where $n$ is a
constant. Restricting ourselves to $n=2$ for simplicity, we obtain
\begin{equation}\label{2.8}
\rho_{T}=3H^{2}(18H^{2}+1),~~~~p_{T}=-2\dot{H}-3H^{2}(24\dot{H}+18H^{2}+1)
\end{equation}

 Also we consider that our universe is filled with new Holographic
dark energy and dark matter. So we assume,
$\rho=\rho_{x}+\rho_{m}$ and $p=p_{x}+p_{m}$. Here $\rho_{x}$,
$p_{x}$ and $\rho_{m}$, $p_{m}$ are respectively the energy
density and pressure for new holographic dark energy and dark
matter. Although the dark matter has negligible pressure, i.e.,
$p_{m}\approx 0$ but here we taken into account the non-zero value
of $p_{m}$(which is very small value).

The energy conservation equation is given by
\begin{equation}\label{2.9}
\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0
\end{equation}
Now we consider the model of interaction between dark matter and
the new holographic dark energy model, through a phenomenological
interaction term Q. Keeping into consideration the fact that the
Supernovae and CMB data determine that decay rate should be
proportional to the present value of the Hubble parameter. The
interaction term describes the energy flow between the two fluids.
Therefore the conservation equation (\ref{2.9}) becomes
\begin{equation}\label{2.10}
\dot{\rho}_{x}+3H(\rho_{x}+p_{x})=-Q
\end{equation}
and
\begin{equation}\label{2.11}
\dot{\rho}_{m}+3H(\rho_{m}+p_{m})=Q
\end{equation}
There are many different forms of Q in the literature, we choose
the following form of Q as
\begin{equation}
Q=3\delta H\rho_{m}
\end{equation}
where $\delta$ (may positive or negative)is the interaction
parameter. In our present study we take $\delta\geq 0$.

 The new Holographic dark energy density is given by
\begin{equation}\label{2.13}
\rho_{x}=3(\alpha H^{2}+\beta \dot{H})
\end{equation}
where $\alpha$ and $\beta$ are constants.
 Now the EOS of dark matter is given by
$p_{m}=\omega_{m}\rho_{m}$, where $\omega_{m}$ is very small and
so the equation (\ref{2.11}) becomes
\begin{equation}
\rho_{m}=\rho_{m0}a^{-3(1+\omega_{m}-\delta)}
\end{equation}
 and consequently from the equation (\ref{2.5}), we have
\begin{equation}
H^{2}=\frac{\sqrt{-\frac{\beta}{a^{3(1+\omega_{m})}}}\left[\alpha
\sqrt{-\frac{a^{3(1+\omega_{m})}}{\beta}}
J_{-U}(V)C+\sqrt{-\frac{6a^{3\delta}\rho_{m0}}{\beta}}\left(2J_{-1+U}(V)+\left(J_{-1-U}(V)-
J_{1-U}(V)\right)C\right)\right]}{36\left(J_{U}(V)+J_{-U}(V)C\right)}
\end{equation}
where $J_{U}(V)$ is a Bessel function of first kind,
$U=\frac{2\alpha}{3\beta(1-\delta+\omega_{m})}$ and
$V=-\frac{\sqrt{\frac{32}{3}\rho_{m0}}
a^{-\frac{3}{2}(1-\delta+\omega_{m})}}{\beta(1-\delta+\omega_{m})}$
and $\rho_{m0}$ is the present value of matter density (at $a=1$)
and $C$ is the arbitrary integration constant.

Now we define:
\begin{equation}\label{2.16}
\begin{array}{l}
\widetilde{H}^{2}=\frac{H^{2}}{H_{0}^{2}};~~~~~~\\\\
\widetilde{\rho}_{m}=\frac{\rho_{m}}{3H_{0}^{2}};~~~~
\widetilde{\rho}_{x}=\frac{\rho_{x}}{3H_{0}^{2}};~~~~
\widetilde{\rho}_{T}=\frac{\rho_{T}}{3H_{0}^{2}};~~~~~~\\\\
\widetilde{p}_{m}=\frac{p_{m}}{3H_{0}^{2}};~~~~
\widetilde{p}_{x}=\frac{p_{x}}{3H_{0}^{2}};~~~~
\widetilde{p}_{T}=\frac{p_{T}}{3H_{0}^{2}};~~~~~~\\\\
\widetilde{\Omega}_{m0}=\frac{\rho_{m0}}{3H_{0}^{2}};~~~~
\widetilde{\Omega}_{x0}=\frac{\rho_{x0}}{3H_{0}^{2}};~~~~
\widetilde{\Omega}_{T0}=\frac{\rho_{T0}}{3H_{0}^{2}};~~~~
\end{array}
\end{equation}

where $H_{0}$ is the present value of the Hubble parameter,
$\widetilde{H}$ is the Hubble expansion rate, current density
parameters are $\Omega_{m0}$ and $\Omega_{x0}$ for matter and dark
energy respectively. The scale factor $a$ is expressed in terms of
redshift z as
\begin{equation}
a=\frac{1}{1+z}
\end{equation}
Therefore from the equations
(\ref{2.5})-(\ref{2.8}),(\ref{2.10})-(\ref{2.16}) we have

\begin{equation}
\widetilde{\rho}_{m}=\widetilde{\Omega}_{m0}(1+z)^{3(1-\delta+\omega_{m})}
\end{equation}

\begin{equation}
\widetilde{\rho}_{T}=(18H^{2}+1)\widetilde{H}^{2}
\end{equation}

\begin{equation}
\widetilde{\rho}_{x}=\alpha \widetilde{H}^{2}-
\frac{\beta(1+z)}{2}\frac{d\widetilde{H}^{2} }{dz}
\end{equation}

\begin{equation}
\widetilde{p}_{m}=\omega_{m}\widetilde{\Omega}_{m0}(1+z)^{3(1-\delta+\omega_{m})}
\end{equation}

\begin{equation}
\widetilde{p}_{T}=\frac{1}{3\beta}\left[\left((2\alpha-3\beta)
+18H^{2}(4\alpha-3\beta)
\right)\widetilde{H}^{2}-2(36H^{2}+1)\widetilde{\rho}_{x}\right]
\end{equation}
and
\begin{equation}
\widetilde{p}_{x}=-\delta\widetilde{\rho}_{m}-\alpha
\widetilde{H}^{2}+\frac{1}{3}(\alpha+\beta)(1+z)\frac{d\widetilde{H}^{2}
}{dz}-\frac{1}{6}\beta(1+z)^{2}\frac{d^2\widetilde{H}^{2} }{dz^2}
\end{equation}
where we use the relations
$\frac{\dot{H}}{H^{2}}=-\frac{(1+z)}{2\widetilde{H}^{2}}
\frac{d\widetilde{H}^{2} }{dz}$ \&
$\frac{\ddot{H}}{H^{3}}=\frac{(1+z)}{2\widetilde{H}^{2}}
\frac{d\widetilde{H}^{2}
}{dz}+\frac{(1+z)^{2}}{2\widetilde{H}^{2}}\frac{d^{2}\widetilde{H}^{2}
}{dz^{2}}$.

\section{{\normalsize\bf{Some Well-Known Cosmological
Parameters}}}
\subsection{\normalsize\bf{Deceleration Parameter}}

Deceleration parameter plays a very important role for any
cosmological model to determine the cosmic acceleration. A
negative value of q represents cosmic acceleration, whereas a
positive q gives a decelerating universe. The deceleration
parameter is given by
\begin{equation}
q=-1-\frac{\dot{H}}{H^{2}}=-1+\frac{(1+z)}{2\widetilde{H}^{2}}
\frac{d\widetilde{H}^{2} }{dz}
\end{equation}
and in this present scenario, the expression for $q$ is obtained
as
\begin{equation}
\begin{array}{ll}
q=-1+\left(18 \left(\frac{1}{1+z}\right)^{-1+\frac{3}{2}
\left(1+\omega _m\right)} \left(J_{U}(V)+J_{-U}(V) C\right) H_0^2
\right. \left(\left(\left(\frac{1}{1+z}\right)^{1-\frac{3}{2}
\left(1+\omega _m\right)} \sqrt{-\frac{1}{\beta }} \beta
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }}
J_{-U}(V) C+\right.\right.\right.\\
\left.\left.\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3 \delta
/2} \left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V)) C\right)
\sqrt{-\frac{\rho_{m0}}{\beta }}\right) \left(1+\omega
_m\right)\right)\right/\left(24 \left(J_U(V)+J_{-U}(V) C\right)
H_0^2\right)+\\ \left(\left(\frac{1}{1+z}\right)^{-\frac{3}{2}
\left(1+\omega _m\right)} \sqrt{-\frac{1}{\beta }} \beta  \right.
\left(-3 \sqrt{\frac{3}{2}} \left(\frac{1}{1+z}\right)^{1+\frac{3
\delta }{2}} \delta  \left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V))
C\right) \sqrt{-\frac{\rho _{m0}}{\beta }}-\right.\\
\left.\left.\left.\frac{3}{2}
\left(\frac{1}{1+z}\right)^{1+\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C
\left(1+\omega _m\right)\right)\right)\right/
\left.\left.\left.\left(36 \left(J_U(V)+J_{-U}(V) C\right)
H_0^2\right)\right)\right)\right/\\
\left(\sqrt{-\frac{1}{\beta }} \beta  \right.
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C+\right.
\left.\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3 \delta /2}
\left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V)) C\right)
\sqrt{-\frac{\rho _{m0}}{\beta }}\right)\right)
\end{array}
\end{equation}
The deceleration parameter is plotted against the redshift
parameter in Fig 1.
\begin{figure}
~~~~~~~~~~~~~~~~~~~~~~~~~~~\includegraphics[scale=0.85]{fig1.eps}\\\label{4.1}
\vspace{2mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.1~~\\
\vspace{4mm} Fig.1 shows the variation of deceleration parameter
(q) against the redshift (z). \vspace{6mm}
\end{figure}
\subsection{\normalsize\bf{EoS Parameter}}
The nature of matter content of the universe is depicted by the
Equation of State (EoS) parameter $\omega$. Basically its gives an
idea about the era of the universe. When $\omega=0,1/3$ and $1$,
it can be predicted that the universe is in the dust era, the
radiation era and stiff fluid respectively whereas $\omega=-1/3,
-1$ and $\omega<-1$ represent the quintessence DE, $\Lambda$CDM
era and Phantom era respectively. The EoS parameter can be
obtained as

\begin{equation}
\omega=\frac{2q-1}{3}
\end{equation}

\begin{figure}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~\includegraphics[scale=0.85]{fig2.eps}\\
\vspace{2mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.2~~\\
\vspace{4mm} Fig.2 depicts the change of EoS parameter against the
redshift. \vspace{6mm}
\end{figure}
Here we have obtained it for the given model as:
\begin{equation}
\begin{array}{ll}\label{4.2}
\omega=-\frac{1}{3}\\+\frac{2}{3}\left[-1+\left(18
\left(\frac{1}{1+z}\right)^{-1+\frac{3}{2} \left(1+\omega
_m\right)} \left(J_{U}(V)+J_{-U}(V) C\right) H_0^2 \right.
\left(\left(\left(\frac{1}{1+z}\right)^{1-\frac{3}{2}
\left(1+\omega _m\right)} \sqrt{-\frac{1}{\beta }} \beta
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }}
J_{-U}(V) C+\right.\right.\right.\right.\\
\left.\left.\left.\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3
\delta /2} \left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V)) C\right)
\sqrt{-\frac{\rho_{m0}}{\beta }}\right) \left(1+\omega
_m\right)\right)\right/\left(24 \left(J_U(V)+J_{-U}(V) C\right)
H_0^2\right)+\right.\\
\left.\left(\left(\frac{1}{1+z}\right)^{-\frac{3}{2}
\left(1+\omega _m\right)} \sqrt{-\frac{1}{\beta }} \beta  \right.
\left(-3 \sqrt{\frac{3}{2}} \left(\frac{1}{1+z}\right)^{1+\frac{3
\delta }{2}} \delta  \left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V))
C\right) \sqrt{-\frac{\rho _{m0}}{\beta }}-\right.\right.\\
\left.\left.\left.\left.\frac{3}{2}
\left(\frac{1}{1+z}\right)^{1+\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C
\left(1+\omega _m\right)\right)\right)\right/
\left.\left.\left.\left(36 \left(J_U(V)+J_{-U}(V) C\right)
H_0^2\right)\right)\right)\right/\right.\\
\left.\left(\sqrt{-\frac{1}{\beta }} \beta  \right.
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C+\right.
\left.\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3 \delta /2}
\left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V)) C\right)
\sqrt{-\frac{\rho _{m0}}{\beta }}\right)\right)\right]
\end{array}
\end{equation}
The plot of the EoS parameter against the redshift parameter is
obtained in Fig 2.

\subsection{\normalsize\bf{\Large{$\omega$--$ \omega'$} Plane
Analysis}}

The $\omega$--$ \omega'$(where prime denotes derivatives with
respect to $x = ln~a$) plane analysis is very significant tool in
our modern days cosmological analysis. Basically, it has been used
to distinguish different DE models through trajectories on its
plane. At first this approach was proposed by Caldwell and Linder
in 2005 \cite{Caldwell2005}. Initially, this method has been
applied on quintessence DE model which leads to two classes of its
plane, one of which has been known as thawing region where the
area occupied by the region as $\omega>0$ for $\omega'<0$ on
$\omega$--$ \omega'$ plane while the other known as freezing
region where $\omega<0$ for $\omega'<0$ and it is also notable
that the expansion of the universe is comparatively more
accelerating in freezing region. Recently, this tool has been
widely applied to other well-known dynamical DE models such as
more general form of quintessence \cite{Scherrer}, Pilgrim DE
\cite{Sharif2013a,Sharif2013b,Sharif2014}, quintom \cite{Guo},
phantom \cite{Chiba}, polytropic DE \cite{Malekjani} and others.
Here we try to analyze and emphasize such regions with the help of
this method. Now $\omega'$ can be obtained by differentiating Eq.
(\ref{4.2}) with respect to $x$. The plot for $\omega$ versus $
\omega'$ for our predicted model with different value of
$\delta=1, 0, 0.5$ are given in Figs. 3-5 respectively. Also it
can be observed from those figures that the freezing regions are
possible for non-interacting scenario i.e., $\delta=0$ and for
interacting scenario with $\delta=0.5$, whereas for high
interaction scenario, $\delta=1.0$ the thawing region is a
possibility with the assumption that the other parameters are
fixed at same value as the previous cases. Again it can be
observed that $\omega'\rightarrow 0$ when $\omega\rightarrow -1$
i.e., $\Lambda$CDM limit may be possible only the in case when $\delta=0$.\\

\begin{figure}
\includegraphics[scale=0.75]{fig3.eps}~~
\includegraphics[scale=0.75]{fig4.eps}~~\\
\vspace{1mm} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.3
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.4~~\\\\
\vspace{1mm}~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\includegraphics[scale=0.85]{fig5.eps}~~\\
\vspace{1mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.5~~\\
\vspace{1mm} Fig. 3-5 show the plot of $\omega$--$\omega'$ for
$\delta=1.0,0,0.5$ respectively with other parameters fixed at
$C=0.005$, $\omega_{m}=0.3$, $\rho_{m0}=0.01$, $\alpha=0.95$,
$\beta=-2.1$ and $H_{0}=73$.
\end{figure}

\subsection{\normalsize\bf{Statefinder Parameters}} Statefinder
Parameters solved the problem of discriminating between the
various candidates for dark energy model. In 2003 Sahni et
al\cite{Sahni2003a} proposed the trajectories in the \{$r,s$\}
plane corresponding to different cosmological models to depict
qualitatively different behavior. The statefinder diagnostic along
with future SNAP observations may perhaps be used to discriminate
between different dark energy models. The above statefinder
diagnostic pair are defined as follows:

\begin{equation}
r=1+3\frac{\dot{H}}{H^{2}}+\frac{\ddot{H}}{H^{3}} ~~\text{and} ~~
s=\frac{r-1}{3(q-\frac{1}{2})}
\end{equation}
where $q$ is the deceleration parameter defined by
$q=-1-\frac{\dot{H}}{H^{2}}$ and $H$ is the Hubble parameter.
 These parameters

Now r and s can be written in terms of Hubble parameter H in the
following forms:
\begin{equation}
r=1+3\frac{\dot{H}}{H^{2}}+\frac{\ddot{H}}{H^{3}}
\end{equation}
and
\begin{equation}
s=-\frac{3H\dot{H}+\ddot{H}}{3H(2\dot{H}+3H^{2})}
\end{equation}
Or, equivalently in terms of $\widetilde{H}$ as follows:

\begin{equation}
r=1-\frac{(1+z)}{\widetilde{H}^{2}}\frac{d\widetilde{H}^{2} }{dz}
+\frac{(1+z)^{2}}{2\widetilde{H}^{2}}\frac{d^{2}\widetilde{H}^{2}
}{dz^{2}}
\end{equation}

\begin{equation}
s=\frac{-\frac{(1+z)}{\widetilde{H}^{2}}\frac{d\widetilde{H}^{2}
}{dz}+\frac{(1+z)^{2}}{2\widetilde{H}^{2}}\frac{d^{2}\widetilde{H}^{2}
}{dz^{2}}}{3\left(\frac{(1+z)}{\widetilde{H}^{2}}\frac{d\widetilde{H}^{2}
}{dz}-3\right)}
\end{equation}
where the explicit form of $r$ is as
\begin{equation}
\begin{array}{ll}
r=1-\left(36 \left(\frac{1}{1+z}\right)^{-1+\frac{3}{2}
\left(1+\omega _m\right)} \left(J_{U}(V)+J_{-U}(V) C\right) H_0^2
\right. \left(\left(\left(\frac{1}{1+z}\right)^{1-\frac{3}{2}
\left(1+\omega _m\right)} \sqrt{-\frac{1}{\beta }} \beta
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V)
C+\right.\right.\right.\\
\left.\left.\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3 \delta
/2} \left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V)) C\right)
\sqrt{-\frac{\rho_{m0}}{\beta }}\right) \left(1+\omega
_m\right)\right)\right/\left(24 \left(J_{U}(V)+J_{-U}(V)
C\right) H_0^2\right)+\\
\left(\left(\frac{1}{1+z}\right)^{-\frac{3}{2} \left(1+\omega
_m\right)} \sqrt{-\frac{1}{\beta }} \beta  \right. \left(-3
\sqrt{\frac{3}{2}} \left(\frac{1}{1+z}\right)^{1+\frac{3 \delta
}{2}} \delta  \left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V))
C\right) \sqrt{-\frac{\rho_{m0}}{\beta }}-\right.\\
\left.\left.\left.\frac{3}{2}
\left(\frac{1}{1+z}\right)^{1+\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C
\left(1+\omega _m\right)\right)\right)\right/
\left.\left.\left.\left(36 \left(J_{U}(V)+J_{-U}(V) C\right)
H_0^2\right)\right)\right)\right/
\left(\sqrt{-\frac{1}{\beta }} \beta  \right.\\
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C+\right.
\left.\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3 \delta /2}
\left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V))
C\right) \sqrt{-\frac{\rho_{m0}}{\beta }}\right)\right)+\\
\left(18 \left(\frac{1}{1+z}\right)^{-2+\frac{3}{2} \left(1+\omega
_m\right)} \left(J_{U}(V)+J_{-U}(V) C\right) H_0^2 \right.
\left(-\left(\left(\frac{1}{1+z}\right)^{2-\frac{3}{2}
\left(1+\omega _m\right)} \sqrt{-\frac{1}{\beta }} \beta
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V)
C+\right.\right.\right.\\
\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3 \delta /2} \left(2
J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V)) C\right)
\sqrt{-\frac{\rho_{m0}}{\beta }}\right) \left.\left(1+\omega
_m\right) \left(1-\frac{3}{2} \left(1+\omega
_m\right)\right)\right)/\left(24 \left(J_{U}(V)+J_{-U}(V) C\right)
H_0^2\right)\\
+\left(\left(\frac{1}{1+z}\right)^{1-\frac{3}{2} \left(1+\omega
_m\right)} \sqrt{-\frac{1}{\beta }} \beta  \left(1+\omega
_m\right) \right. \left(-3 \sqrt{\frac{3}{2}}
\left(\frac{1}{1+z}\right)^{1+\frac{3 \delta }{2}} \delta  \left(2
J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V))
C\right) \sqrt{-\frac{\rho_{m0}}{\beta }}-\right.\\
\left.\left.\frac{3}{2} \left(\frac{1}{1+z}\right)^{1+\frac{3}{2}
\left(1+\omega _m\right)} \alpha  \sqrt{-\frac{1}{\beta }}
J_{-U}(V) C \left(1+\omega _m\right)\right)\right)/\left(12
\left(J_{U}(V)+J_{-U}(V) C\right) H_0^2\right)+
\left(\left(\frac{1}{1+z}\right)^{-\frac{3}{2} \left(1+\omega
_m\right)} \sqrt{-\frac{1}{\beta }} \beta  \right.\\
\left(3 \sqrt{\frac{3}{2}} \left(\frac{1}{1+z}\right)^{2+\frac{3
\delta }{2}} \delta \left(1+\frac{3 \delta }{2}\right) \left(2
J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V))
C\right) \sqrt{-\frac{\rho_{m0}}{\beta }}+\right.\\
\left.\left.\left.\frac{3}{2}
\left(\frac{1}{1+z}\right)^{2+\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C
\left(1+\omega _m\right) \left(1+\frac{3}{2} \left(1+\omega
_m\right)\right)\right)\right)\right/\left.\left.\left.\left(36
\left(J_{U}(V)+J_{-U}(V) C\right)
H_0^2\right)\right)\right)\right/\\
\left(\sqrt{-\frac{1}{\beta }} \beta  \right.
\left(\left(\frac{1}{1+z}\right)^{\frac{3}{2} \left(1+\omega
_m\right)} \alpha  \sqrt{-\frac{1}{\beta }} J_{-U}(V) C+\right.
\left.\left.\sqrt{6} \left(\frac{1}{1+z}\right)^{3 \delta /2}
\left(2 J_{-1+U}(V)+(J_{-1-U}(V)-J_{1-U}(V)) C\right)
\sqrt{-\frac{\rho_{m0}}{\beta }}\right)\right)
\end{array}
\end{equation}

\begin{figure}
\includegraphics[scale=0.75]{fig6.eps}~~
\includegraphics[scale=0.75]{fig7.eps}\\
\vspace{2mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.7\\
\vspace{6mm}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\includegraphics[scale=0.7]{fig8.eps}~~\\
\vspace{2mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\vspace{1mm} Fig. 6 shows the variation of the state statefinder
parameter r with redshift z. \\
Fig. 7 shows the variation of the state statefinder parameter s
with redshift z. \\
Fig. 8 shows the variation of the state
statefinder parameter s against r.\vspace{1mm}
\end{figure}

\subsection{\normalsize\bf{Cosmographic parameters}}

Standard candle is an astronomical ideal object that has a known
absolute magnitude. It is extremely important since by measuring
the apparent magnitude of the object, the distance can also be
determine. It helps to reconstruct the Hubble diagram, i.e, the
redshift - distance relation up to high redshift values and for
that it is commonly used to parameterized model where it can
verify the viability of the data and characterizing parameters.
There were some uncertainty as the method is model dependent.
Therefore some doubts come to mind when we examine the validity of
derived quantities depending on the present day values of the
deceleration parameters and the age of the universe. In order to
conquer such problem we need a model independent approach and for
that the alternative choice is cosmography where the scale factor
is expanded in Taylor series with respect to the cosmic time. Such
expansions not only show the way to distance-redshift relation, it
also supports the assumption of the Robertson-Walker metric. It
does not depend on the particular form of the solution of cosmic
equations. For that aim, the Cosmographic parameters that are
proportional to the coefficients of Taylor series expansion of the
scale factor with respect to the cosmic time defined as
\begin{equation}
q=-\frac{1}{aH^{2}}\frac{d^{2}a}{dt^{2}}=-\frac{1}{H}\frac{d}{da}(aH)=-1+\frac{(1+z)}{\widetilde{H}^{2}}
\frac{d\widetilde{H}^{2} }{dz}
\end{equation}
\begin{equation}
J=r=\frac{1}{aH^{3}}\frac{d^{3}a}{dt^{3}}=-\frac{1}{H^{2}}\frac{d}{da}(aH^{2}q)=1-\frac{(1+z)}{\widetilde{H}^{2}}\frac{d\widetilde{H}^{2}
}{dz}
+\frac{(1+z)^{2}}{2\widetilde{H}^{2}}\frac{d^{2}\widetilde{H}^{2}
}{dz^{2}}
\end{equation}
\begin{equation}
S=\frac{1}{aH^{4}}\frac{d^{4}a}{dt^{4}}=\frac{1}{H^{3}}\frac{d}{da}(aH^{3}J)=-\frac{(1+z)^{2}}
{\widetilde{H}^{3}}\frac{d}{dz}\left(\frac{\widetilde{H}^{3}}{1+z}J\right)
\end{equation}
\begin{equation}
L=\frac{1}{aH^{5}}\frac{d^{5}a}{dt^{5}}=\frac{1}{H^{4}}\frac{d}{da}(aH^{4}S)=-\frac{(1+z)^{2}}
{\widetilde{H}^{4}}\frac{d}{dz}\left(\frac{\widetilde{H}^{4}}{1+z}S\right)
\end{equation}

which are usually referred to as the deceleration ($q$), jerk
($J$), snap ($S$) and lerk ($L$) parameters, respectively. The
Cosmographic Parameters's present day values (which are denote
with a subscript 0) may be used to characterize the evolutionary
status of the universe. For instance, $q_0 < 0$ denotes an
accelerated expansion, while $J_0$ allows to discriminate among
different accelerating models.

\begin{figure}
\includegraphics[scale=0.75]{fig9.eps}~~
\includegraphics[scale=0.75]{fig10.eps}\\
\vspace{2mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.9~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.10\\
\vspace{1mm} Fig. 9 shows the variation of the Snap parameter S
with redshift z. \\
Fig. 10 shows the variation of the Lerk
parameter L with redshift z.
\end{figure}

\subsection{$Om$ \normalsize\bf{Parameter}}

Recently, Sahni et al \cite{Sahni2003a,Sahni2008} proposed a new
cosmological parameter named {\it Om} which was introduced to
differentiate $\Lambda$CDM from other dark energy models  {\it Om}
diagnostic has been discussed together with statefinder for
generalized Chaplygin gas model from cosmic observations in
\cite{Lu2009,Tong2009}. Generally, it is a combination of Hubble
parameter and the cosmological redshift and provides a null test
of dark energy. For $\Lambda$CDM model, ${\it Om} = \Omega_{m0}$
is a constant, independent of redshift $z$. Also it helps to
distinguish the present matter density constant $\Omega_{m0}$ in
different models more effectively. The main utility for {\it Om}
diagnostic is that the quantity of {\it Om} can distinguish dark
energy models with less dependence on matter density
${\Omega_{m0}}$ relative to the EoS of dark energy. Our starting
point for {\it Om} diagnostics in the Hubble parameter and it is
defined as:

\begin{equation}
Om(z)=\frac{\widetilde{H}^{2}(z)-1}{(1+z)^{3}-1}
\end{equation}

Using the above result we have

\begin{equation}
Om(z)=\frac{\frac{\sqrt{-\frac{\beta}{a^{3(1+\omega_{m})}}}\left[\alpha
\sqrt{-\frac{a^{3(1+\omega_{m})}}{\beta}}
J_{-U}(V)C+\sqrt{-\frac{6a^{3\delta}\rho_{m0}}{\beta}}\left(2J_{-1+U}(V)+\left(J_{-1-U}(V)-
J_{1-U}(V)\right)C\right)\right]}{36H_{0}^{2}\left(J_{U}(V)+J_{-U}(V)C\right)}-1}{(1+z)^{3}-1}
\end{equation}

\begin{figure}
~~~~~~~~~~~~~~~~~\includegraphics[scale=0.85]{fig11.eps}~~\\
\vspace{2mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.11~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\vspace{4mm}Fig.11 gives the plot of \textit{Om}-diagnostic
against redshift parameter $z$ \vspace{6mm}
\end{figure}

\section{\normalsize\bf{Stability Analysis}}

In this section, we have examined the stability of this model. In
this connection, the most important quantity is the squared speed
of sound which is denoted as $v_s^2$ and defined as a ratio of the
effective pressure and energy densities i.e.,
$v_s^2=\frac{\dot{p}}{\dot{\rho}}$. The sign of $v_s^2$ plays a
vital role for stability analysis of a background evolution of
cosmic models. It is well known that the model is stable if
$v_s^2>0$ and if $v_s^2<0$ implies that the model is classically
unstable\cite{Myung}. In 2008, Kim et al. \cite{Kim} found that
$v_s^2$ for agegraphic DE is always negative which leads to
classically instability of that model. Recently many researchers
are using this methods to analyzed the stability of that models on
which they worked and of which some authors
\cite{Jawad,Ebrahimi,Sharif,Setare} have reached to a conclusion
that HDE, ADE, NADE, Chaplygin gas, holographic Chaplygin,
holographic $f(T)$, holographic $f(G)$, new agegraphic $f(T)$, new
agegraphic $f(G)$ models are classically unstable because squared
speed of sound is negative i.e., $v_s^2<0$  throughout the
evolution of the universe. Here we consider
\begin{equation}
 v_s^2=\frac{\dot{p}_{MG}}{\dot{\rho}_{MG}}
\end{equation}
and plot $v_s^2$ versus $t$ by taking the power-law scale factor
in this model. Fig. 12 shows the variation of $v_s^2$ with $t$ for
the above mentioned cases.
\begin{figure}
~~~~~~~~~~~~~~~~~\includegraphics[scale=0.85]{fig12.eps}~~\\
\vspace{2mm}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fig.12~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\vspace{4mm} Fig 12 shows the variation of the squared speed of
sound $v_s^2$ against the cosmic time $t$.\vspace{6mm}
\end{figure}
In this model we observed that the square speed of sound remains
negative for the present and future epoch, which implies that our
predicted model for the above mentioned scenarios with power-law
form of scale factor is classically unstable.

\section{\normalsize\bf{Discussions}}

In this work we have studied the behaviour of some cosmological
parameters like Deceleration parameter, EoS parameter, {\it Om}
parameter, State-finder parameters and Cosmographic parameters in
an universe described by interacting new holographic dark energy
model in $f(T)$ gravity. We have considered a suitable interaction
between holographic dark energy and dark matter to account for the
coincidence scenario. We have derived the expressions for the
above mentioned cosmological parameters and plotted them against
the cosmological redshift parameter $z$.

In fig.1, we have plotted the deceleration parameter against the
redshift. From the plot we see that at the current epoch $(z=0)$,
the value of $q$ is around $-0.8$, which shows that the universe
is going through an accelerated expansion. From the curve it is
also evident that the universe is evolving from an early
decelerating phase to a late accelerating phase. In fig.2, the EoS
parameter is plotted against the redshift parameter. It is seen
that currently the value of $\omega$ is around $-0.9$, which lies
in the quintessence range $(-1<\omega<-1/3)$. Moreover it is seen
that near the $\omega=-1$ axis the curve assumes an asymptotic
behaviour, which shows that the universe never enters the phantom
regime, although the trajectory is infinitesimally close to the
phantom divide. In the figs. 3, 4 and 5, the trajectories in the
$\omega$--$ \omega'$ plane are obtained for different values of
interactions. It is seen that for no or smaller interaction models
we get freezing regions which are best suited for the current
accelerating scenario. But for high interactions we get thawing
models. In the figs. 6 and 7, the statefinder parameters $r$ and
$s$ are respectively plotted against the redshift parameter. From
the trajectories it is seen that in the present time $r=1, s=0$,
which corresponds to the standard $\Lambda$CDM cosmological model
of accelerating universe. In fig. 8, trajectories in the $r-s$
plane are obtained for the given model. This figure also confirms
the fact that when $r=1$, we have $s=0$, which is quite
characteristic for an accelerating model of the universe. In figs.
9 and 10, the Snap and the Lerk parameters are respectively
plotted against the redshift parameter. From fig.9, we see that
after passing through a series of negative values in the early
universe, it assumes constancy around the zero level in the
present universe. Almost similar is the case for the Lerk
parameter in fig 10, the only difference being its values lying in
the positive region in the early universe. In fig. 11, we have
obtained the trajectories for the $Om-$diagnostic. The
trajectories obtained are unique for the given model and can be
used to differentiate the model from others. Finally in fig. 12,
the squared speed of sound $v_s^{2}$ is plotted against cosmic
time $t$. From the figure it is seen that the trajectory for the
present model remains at the negative level thus exhibiting the
unstable nature of the model. Nevertheless the study reveals that
the model is perfectly consistent with the notion of the
accelerating universe, at least for low interaction cases, and the
derived values of all the parameters comply to this fact.\\

{\bf Acknowledgement:}\\

The authors are thankful to IUCAA, Pune, India for warm
hospitality where a part of the work was carried out.\\


text:
\section{Introduction}

   After the paradigm of the Hilbert's Lagrangian formulation of
 Einstein's theory of gravity it was clear how one
 could consistently formulate other,
 higher derivative gravitational theories (that is theories in
 which the field equations have higher than second metric derivatives).
 And such theories where indeed proposed and used as alternatives to
 Einstein's theory in attempts to unify other fields with
 gravity  \cite{Weyl}  and to remedy some of its seemingly undesirable
 consequences  as, for example, at the classical level, the unavoidance
 of cosmological singularities  \cite{Kerner} and, at the quantum level,
 the non renormalizability of the quantized version of general
 relativity \cite{Stelle}.

 One of the main motivations for studying higher derivative theories
 comes from the  semiclassical general relativity.
 There,  it seems to be  a matter of self-consistency to consider higher
 derivative terms in the gravitational Lagrangian  since such terms
 arise generically in one-loop calculations \cite{Birrell}.
 Certainly this notion of self-consistency is a delicate issue and, as
 Simon has recently suggested \cite{Simon},  it needs to be reconsidered
 if one wants to avoid undesirable semiclassical predictions such  as
 unstable Minkowski spacetime.
 Another recent motivation for considering higher derivative
 gravitational theories is that such theories have arisen as low energy
 limit of several superstring theories \cite{Superstring}.

 Higher derivative theories are of interest to cosmology mainly because,
 even vacuum theories admit cosmological models which give rise to the,
 so called, Starobinsky inflation \cite{Starobinsky} (see however
 Ref. \cite{Simon} for a  critisism on its consistency in the semiclassical
 limit), without fine tuning of the initial conditions \cite{Feldman}.

 In this paper we want to look at another topic of cosmological
 relevance namely, the effects of higher derivative theories on the
 gravitational field of topological defects as monopoles, cosmic strings
 and domain walls.
 These are objects that may have formed during phase transitions in the
 cooling down of the Early-Universe and    may have played a key role in
 the formation of the large scale structure of the Universe mainly
 through their gravitational interactions \cite{Vilenkin85,Brandenberger}.
 Since their main interaction is gravitational, it is important to have
 an idea of what modifications one should expect in their gravitational
 field when the relevant gravitational theory has higher derivatives.
 Some work has recently been done in this direction, but only for gauge
 cosmic strings \cite{Linet}.
 This work shows that  in the weak  field limit only short range
 corrections to the Einstein theory arise which
 are associated with the presence of additional massive fields in the
 spectrum of higher derivative theories.
 However, this is not expected to be in general true, especially for
 global topological defects which are extended field configurations and
 not localized as the gauge cosmic strings.

 In this work we have in mind  theories that can be separated in a part
 ${\cal L}_G$ for the gravitational field $g_{\mu\nu}$ constructed with
 geometrical scalars of the Ricci tensor $R_{ab}$, and another  part
 ${\cal L}_M$ containing  matter fields with standard coupling to the
 gravitational field $g_{\mu\nu}$
\begin{equation}
 {\cal L} = {1\over 2\kappa}{\cal L}_G(R_{ab}) + {\cal L}_M(g_{ab}).
\label{FullLagrangian}\end{equation}
 Hereafter $\kappa:=8\pi G$ where $G$ is the gravitational constant.
 For theories of this type it has been  noted that  they can be recasted
 into an equivalent theory  of Einstein gravity  interacting with
 additional matter fields \cite{Whitt,Magnano,Jakubiec}. However, as it
 was stressed by Brans \cite{Brans} and we shall explain in the next
 section, this  equivalence is in general only at a mathematical level
 and not at a physical one. Nevertheless, based on such an equivalent
 system, Whitt \cite{Whitt} was able to show that the black holes of
 general relativity are the only black hole solutions of  $R+R^2$
 theories (no hair theorem).

 For the  discussion of this paper we will deal  with theories that have
 as gravitational part  the following, often appearing in the
 literature, Lagrangians
\begin{equation}
    {\cal L}_G = \sqrt{-g}
    ( R + \alpha R^2 + \beta R_{\mu\nu} R^{\mu\nu} )
\label{QuadraticLagrangian}\end{equation}
 and
\begin{equation}
    {\cal L}_G = \sqrt{-g} F(R) ,
\label{ScalarQuadraticLagrangian}\end{equation}
 where  $\alpha, \beta$ are some coupling constants,
 $g:={\rm det}g_{\mu\nu}$, and $R=g^{\mu\nu}R_{\mu\nu}$. Finally the $F$
 in Eq.\ (\ref{ScalarQuadraticLagrangian}) is in principle an arbitrary
 function of the curvature scalar $R$. However, later on we will
 take $F$ to differ only slightly from the Einstein value $R$, that is
 $F=R+\alpha R^2$ with $\alpha |R|\ll 1$.
 See Ref. \cite{Whitt} for a treatment of the  $F=R+\alpha R^2$ theory in
 vacuum and the Ref. \cite{Maeda} together with references therein for
 generalizations to arbitrary $F(R)$ in the presence of particular forms
 of matter.

 The structure of the paper is as follows. Section II contains a brief
 review of higher derivarive theories to the extent needed in
 this paper. The field equations for the
 theories in (\ref{FullLagrangian}),(\ref{QuadraticLagrangian}) and
 (\ref{FullLagrangian}),(\ref{ScalarQuadraticLagrangian}),
 are written down, and
 their spectrum is explained. With  the procedure that enables the
 recasting of these field equations  into  Einstein type ones we obtain
 the basic result that is used  in the Sec. III for the comparison of
 the gravitational field of global monopoles, cosmic strings and domain
 walls in Enstein's theory, and  in the quadratic  $R+\alpha R^2$
 theory with $\alpha |R|\ll 1$. Section III also contains at the
 beginning a brief introduction to the topological defects.
 Finally in Sec. IV we conclude with a brief summary and comments.

 Throughout this paper we use the conventions $\hbar=c=1$, metric
 signature $(-+++$), Riemann tensor $R{}^a{}_{bcd}:=- \partial{}_d
 \Gamma{}^a{}_{bc}+ \dots$, and Ricci tensor
 $R{}_{ab}:=R{}^c{}_{acb}$.

\section{Theories with higher derivatives}

\subsection{Field equations}

 We shall give now the gravitational field equations for the higher
 derivative theories given by   the
 Eqs.\ (\ref{FullLagrangian}), (\ref{QuadraticLagrangian})
 and  Eqs.\ (\ref{FullLagrangian}), (\ref{ScalarQuadraticLagrangian}).
 The field equations  for $g_{\mu\nu}$ are obtained by varying the
 action corresponding to Eq.\ (\ref {FullLagrangian})
 with respect to $g^{\mu\nu}$
 and contain derivatives of the metric up to the fourth order. For the
 case of ${\cal L}_G$ of  Eq.\ (\ref{QuadraticLagrangian}) they read
\begin{eqnarray}
     (1 &+&2\alpha R)(R_{\mu\nu} - {1\over 2}g_{\mu\nu}R) +
          {\alpha \over 2}R^2  g_{\mu\nu}    \cr
     &+&(2\alpha + {\beta\over 2})g_{\mu\nu} R_{;p}{}^{;p}
         -(2\alpha+\beta) R_{;\mu\nu}         \cr
   &+&\beta  R_{\mu\nu;p}{}^{;p} -
   {\beta\over 2} R_{pq}R^{pq}g_{\mu\nu} +
  2\beta R_{pq}R{}_\mu{}^p{}_\nu{}^q  \cr
  && \qquad\qquad ={-2\kappa\over\sqrt{-g}}
         {\delta  S_M \over \delta (g^{\mu\nu})} :=
  \kappa T^{(M)}_{\mu\nu} .
\label{FieldEqs}\end{eqnarray}
 Notice that the trace of this equation is an inhomogeneous massive
 Klein-Gordon equation for the curvature scalar $R$
\begin{equation}
     (6\alpha+2\beta)R{}_{;p}{}^{;p} - R= \kappa T^{(M)}.
\label{TraceOfFieldEqs}\end{equation}

 Finally, the field equations  for the theory
 (\ref{FullLagrangian}) and (\ref{ScalarQuadraticLagrangian}) can be
 written as
\FL
\begin{equation}
    F^\prime G_{\mu\nu} = \kappa T^{(M)}_{\mu\nu} + {1\over 2}
  g_{\mu\nu}(F-F^\prime R- 2 F^\prime_{;p}{}^{;p}) +
  F^\prime_{;\mu\nu} ,
\label{ScalarFieldEqs}\end{equation}
 where $F^\prime= \partial F/  \partial R$ and $G_{\mu\nu}= R_{\mu\nu}
 -{1\over 2}g_{\mu\nu} R$ is the Einstein tensor. The trace of this
 equation  is
\begin{equation}
 3 F^\prime_{;p}{}^{p} + F^\prime R-2F = \kappa T^{(M)}.
\label{TraceOfScalarFieldEqs}\end{equation}

\subsection{Spectrum of quadratic theories - \\ Weak gravitational limit}

 We would like  to stress here the fact that quadratic theories  do not
 contain only the usual massless (long-range) spin-2 graviton field but
 also, in general,  two massive (short-range) fields with spin-0 and
 spin-2.

 This spectrum  can be  easily recognized in the case of ${\cal L}_G$ of
 Eq.\ (\ref{QuadraticLagrangian}) when one writes the field equations  in
 the linearized weak field  limit using a convenient gauge (coordinate
 system). Indeed following Teyssandier \cite{Teyssandier} we have that
 $g_{\mu\nu}$ can be decomposed in the weak gravitational limit (where
 $g_{\mu\nu}=\eta_{\mu\nu} + h_{\mu\nu}$ with $|h_{\mu\nu}| <<1$) as
\begin{equation}
   g_{\mu\nu}= \eta_{\mu\nu} + h^{\rm (E)}_{\mu\nu} +
             \chi\eta_{\mu\nu} +\psi_{\mu\nu} ,
\label{LinearizedEqsa}\end{equation}
 with the field equations
\FL
\begin{eqnarray}
          &\Box & h^{\rm (E)}_{\mu\nu} =
      -2\kappa( T_{\mu\nu}-{1\over 2}T \eta_{\mu\nu} ) ,
      \cr
        (&\Box& - m_0^2 )\chi = -{1\over 3} \kappa T ,
       \quad m_0^{-2}:=6\alpha+2\beta ,
      \cr
       (&\Box & - m_1^2 )\psi_{\mu\nu} =
                2\kappa( T_{\mu\nu}-{1\over 3}T \eta_{\mu\nu} ),
         \quad m_1^{-2}:=-\beta ,
\label{LinearizedEqsb}\end{eqnarray}
 and the gauge conditions
\begin{eqnarray}
    \partial^a( h^{\rm (E)}_{\mu a} - {1\over 2}
    h^{{\rm (E)}\lambda}_\lambda \eta_{\mu a})=0 ,
      \cr
     ( \psi^{ab} - \psi^\lambda_\lambda \eta^{ab})_{,ab} = 0 .
\label{LinearizedEqsc}\end{eqnarray}
 Here  indices are raised and lowered with the Minkowski metric tensor
 and the operator $\Box$ is  the Minkowskian one.  One recognizes in
 Eqs.\ (\ref{LinearizedEqsa})-(\ref{LinearizedEqsc})
 the usual Einstein contribution
 $h^{\rm (E)}_{\mu\nu}$, that is the graviton field which has 2 degrees
 of freedom.
 Then, a scalar field $\chi$ with mass $m_0$, which  obviously has one
 degree of freedom and appears as an overall conformal factor (in the
 considered approximation). Finally the massive tensorial field
 $\psi_{\mu\nu}$ with mass $m_1$ which turns out to have five degrees of
 freedom (note that in contrast to $h^{\rm (E)}_{\mu\nu}$ its components
 satisfy only one gauge condition) and thus it posseses the stucture of
 a massive spin-2 field.
 In order to keep the  ``mass'' parameters $m_0, m_1$  real  we shall
 demand the {\sl no-tachyon} constraint
\begin{equation}
 3\alpha + \beta \geq 0, \qquad \beta\leq 0.
\label{NoTachyon}\end{equation}

 We leave for the next subsection the case of the Lagrangian
 in Eq.\ (\ref{ScalarQuadraticLagrangian})
 where we will go beyond the weak
 gravitational limit and we will see that the spectrum of this theory
 consists of a graviton and a massive interacting scalar  field.

\subsection{Reformulation of quadratic theories}

 Interestingly enough, besides $g_{\mu\nu}$ there is an alternative
 canditate for  the metric field of the
 spacetime \cite{Magnano,Jakubiec}, namely the $\gamma_{\mu\nu}$ which
 is the inverse of $\gamma^{\mu\nu}$ where
\begin{equation}
    \sqrt{-\gamma}\gamma^{\mu\nu}:=
 { \partial {\cal L}_G \over  \partial  R_{\mu\nu}} ,
\label{NewMetricDefinition}\end{equation}
 and $\gamma:={\rm det}\gamma_{\mu\nu}$.

 In particular for the ${\cal L}_G$ of Eq.\ (\ref{QuadraticLagrangian})
 we have
\FL
\begin{equation}  \sqrt{-\gamma}\gamma^{\mu\nu}:=
            \sqrt{-g} \Bigl[ (1+2\alpha R)g^{\mu\nu} +
             2\beta R_{\alpha\beta}g^{\alpha\mu}g^{\beta\nu}\Bigr ] .
\label{NewMetric}\end{equation}
 Expressing the theory in terms of $\gamma_{\mu\nu}$ via a Legendre
 transformation, one can reduce the order of the derivatives that appear
 in the field equations  from fourth to second. But what is also
 important is, that the resulting theory takes the form of Einstein
 gravity for the metric $\gamma^{\mu\nu}$ plus some additional massive
 fields interpreted as matter fields.
 Thus the equation (\ref{NewMetric}) can be considered as a non linear
 decomposition of $g^{\mu\nu}$ in the physical spectum of the full
 theory:  a spin-2 massless field $\gamma^{\mu\nu}$, a scalar field
 that appears as a conformal factor and is a linear function of $R$,
 and finally, a tensor
 field related to $R^{ab}$ with 5 degrees of freedom ($R^{ab}$ is
 symmetric satisfying the 4 contracted Bianchi equations and its trace
 is essentially the previously mentioned scalar field).

 We can be more explicit in the  interesting case  of the theory in
 Eq.\ (\ref{FullLagrangian}) where ${\cal L}_G$ is given by
 Eq.\ (\ref{ScalarQuadraticLagrangian}).
 Here, only an additional scalar field appears since $\gamma^{\mu\nu}$
 and $g^{\mu\nu}$ are conformally related. Indeed
 Eq.\ (\ref{NewMetricDefinition}) gives
\begin{equation}  \sqrt{-\gamma}\gamma^{\mu\nu}:=
               \sqrt{-g} F^\prime g^{\mu\nu}  ,
\label{NewConformalMetric}\end{equation}
 which implies that
\begin{equation} \gamma_{\mu\nu}= F^\prime{}^{-1} g_{\mu\nu} .
\label{ConformalMetric}\end{equation}
 Defining a scalar field $\psi$ (not to be confused with the tensorial
 field $\psi_{\mu\nu}$ of Eqs.\ (\ref{LinearizedEqsa}),
 (\ref{LinearizedEqsb}) via
\begin{equation}
 F^\prime= \exp\left ( \sqrt{2\kappa\over 3}\psi \right ) ,
\label{psiField}\end{equation}
 the field equations (\ref{ScalarFieldEqs}) are written in the system
 $\gamma_{\mu\nu}$ as
\begin{equation}
 \widehat G_{\mu\nu} = {\kappa\over F^\prime} T^{(M)}_{\mu\nu}(g_{ab}) +
         \kappa T_{\mu\nu}^{(\psi)}
\label{TransformedFieldEqs}\end{equation}
 where, hereafter, hats denote quantities with respect to the metric
 $\gamma_{\mu\nu}$ and
\begin{equation}
    T_{\mu\nu}^{(\psi)}=  \widehat\nabla_\mu\psi \widehat\nabla_\nu\psi
     -{1\over 2} \gamma_{\mu\nu}\widehat\nabla_\lambda\psi
       \widehat\nabla^\lambda\psi -{1\over 2} \gamma_{\mu\nu} U(\psi).
\label{psiSEMtensor}\end{equation}
 The potential $U(\psi)$ is given by
\begin{equation}
 U(\psi)={1\over 2\kappa} F^\prime{}^{-2}(RF^\prime-F) ,
\label{Potential}\end{equation}
 which can be written as a function of $\psi$ alone in regions where
 Eq.\ (\ref{psiField}) is invertible.
 Finally  the scalar field $\psi$ satisfies the equation
\FL
\begin{equation}
 \widehat{\Box\vbox{\vskip 6pt}}\psi = \biggl (
 {\kappa\over 6}\biggr)^{1/2}
     F^\prime{}^{-2} \biggl [ {2F-RF^\prime\over \kappa}
      + g^{\mu\nu} T^{(M)}_{\mu\nu}(g_{ab})  \biggr ],
\label{psiFieldEq}\end{equation}
 which can be checked to be equivalent to the trace
 (\ref{TraceOfScalarFieldEqs}) of the initial field equations
 (\ref{ScalarFieldEqs}).

 These field equations (\ref{TransformedFieldEqs}) and (\ref{psiFieldEq})
 follow from the Lagrangian
\begin{eqnarray}
 {\cal L}^\prime = &&{1\over 2\kappa}\sqrt{-\gamma}
  \widehat{R}(\gamma_{ab})  \cr
  &+&\sqrt{-\gamma}[-{1\over 2}
  \gamma^{\mu\nu}\psi_{,\mu}\psi_{,\nu} -U(\psi)] +
  {\cal L}_M(g_{ab}) ,
\label{NewLagrangian}\end{eqnarray}
 which shows that  the quadratic theory (\ref{FullLagrangian}) is, loosely
 speaking, equivalent to ``Einstein's'' gravitational theory for the
 metric $\gamma_{ab}$ plus an interacting massive scalar field $\psi$
 plus the ``peculiar'' (not anymore usual) matter fields of
 ${\cal L}_M(g_{ab})$. They are indeed peculiar if $\gamma_{ab}$ is
 considered as the metric of spacetime since then  the dependence of
 ${\cal L}_M$ on $g_{ab}$ implies, via Eq.\ (\ref{ConformalMetric}), a non
 standard interaction of the metric field $\gamma_{ab}$ and of the field
 $\psi$ with the matter fields of ${\cal L}_M$. More on this issue of
 equivalence will be said at the end of this section.

\noindent\underbar {Case: $F(R) \approx R +\alpha R^2 $ with
                    $\alpha |R|\ll  1$.}

 \noindent We will now consider the interesting case where $F(R)$ can be
 expanded as a Taylor series around $R=0$ and deviates only slightly
 from Einstein's theory
\FL
\begin{equation}
   F(r)= R + \alpha R^2 + O(R^3),\quad \alpha |R|\ll 1,
   \quad
   \alpha:=
{F^{\prime\prime}\over 2}\bigr \vert_{R=0}.
\label{ApproxF}\end{equation}
 Assuming that the $O(R^3)$ terms can be ignored in this expression then
 the field equations (\ref{TransformedFieldEqs})
 and (\ref{psiFieldEq}) simplify
 considerably. Indeed, in this approximation  Eqs.\ (\ref{psiField}),
 (\ref{ApproxF}) imply
\begin{equation}
   \psi \approx \left (6\over \kappa \right)^{1/2}\alpha  R,
\label{psiApprox}\end{equation}
 while the leading term in the potential $U(\psi)$
 of Eq.\ (\ref{Potential})
 is,  assuming the no-tachyon constraint $\alpha>0$, a mass term
\begin{equation}
 U(\psi) = {1\over 2}m_0^2\psi^2 + O(\psi^3),
 \qquad    m_0^2 = {1\over 6\alpha}.
\label{PotentialApprox}\end{equation}
  The metric $\gamma_{\mu\nu}$ and the field $\psi$ are obtained from the
 field equations (\ref{TransformedFieldEqs}) and (\ref{psiFieldEq}) which,
 to lowest order in the approximation (\ref{ApproxF}), read
\begin{eqnarray}
        \widehat G_{\mu\nu}&\approx & \kappa T^{(M)}_{\mu\nu}(\gamma_{ab}),
    \cr
         (\widehat{\Box\vbox{\vskip 6pt}} - m_0^2)\psi &\approx &
          \left ( {\kappa\over 6}\right)^{1/2} T^{(M)}(\gamma_{ab}).
\label{ApproxFieldEqs}\end{eqnarray}
 Making use of Eq.\ (\ref{ConformalMetric}) one can finally obtain the
 metric $g_{\mu\nu}$.
 Notice that, to the considered approximation,  it does not matter which
 metric we actually use in  $T^{(M)}_{\mu\nu}$. It is more convenient,
 however, from the technical point of view to use $\gamma_{\mu\nu}$.

 Concluding we arrive at the following result:

 \noindent
 For a given matter source $T^{(M)}_{\mu\nu}(g_{ab})$, the metric
 $g_{\mu\nu}$ in the quadratic theory
 (\ref{FullLagrangian}) and (\ref{ScalarQuadraticLagrangian})
 with $F(R)\approx R +\alpha R^2$ and $\alpha |R|\ll 1$ is given by
\begin{equation}
 g_{\mu\nu}=[1+\chi]\gamma_{\mu\nu} ,
\label{QuadraticFromEinsteinMetric}\end{equation}
 where $\gamma_{\mu\nu}$ is the metric in the Einstein's theory with
 source $T^{(M)}_{\mu\nu}(\gamma_{ab})$ while the field $\chi$
 satisfies the equation
\begin{equation}
 ({\Box} - m_0^2)\chi =
    -{\kappa\over 3} T^{(M)}(\gamma_{ab}), \qquad m_0^2:={1\over 6\alpha}
\label{chiFieldEq}\end{equation}
 with the ${\Box}$ operator taken with respect to the $\gamma_{ab}$
 metric.

 This result follows directly from Eqs.\ (\ref{ApproxFieldEqs}),
 (\ref{ConformalMetric})
 using the variable $\chi$ related to the field $\psi$ via
 $\chi:=-2[\kappa/6]^{1/2}\psi$.
 Note that from (\ref{psiApprox}) follows $\alpha R=-\chi/2$ and therefore
 the condition $\alpha |R| \ll 1$ for our approximation is equivalent to
 $|\chi/2|\ll 1$.
 Finally, let us notice that  in the weak gravitational limit  the
 Eqs.\ (\ref{QuadraticFromEinsteinMetric}) and (\ref{chiFieldEq})
 are consistent with the
 $\beta = 0$ limit of the linearized equations (\ref{LinearizedEqsa}) and
 (\ref{LinearizedEqsb}).

\subsection{Some remarks}

 Based on the decomposition (\ref{NewMetricDefinition}) several
 authors \cite{Magnano,Ferraris88,Jakubiec,Schmidt,Maeda} have dealt
 with the question of whether quadratic theories are equivalent to
 Einstein's theory plus some additional fields. It seems that this may
 well be true for vacuum  theories. However, as was pointed out by
 Brans \cite{Brans}, (see also Refs.\cite{Sokolowski,Ferraris90}), a
 subtlety appears in the case where usual matter is present.
 The problem is  that the equivalence principle, a basic guide that one
 may use in constructing theories coupled to gravity and in particular
 to Einstein's theory, cannot be valid in both the original and the
 reformulated theories.
 If it is valid in the original theory, then a test matter  in
 ${\cal L}_M$ of the Lagrangian (\ref{FullLagrangian}) will follow
 geodesics of the spacetime with metric $g_{\mu\nu}$ but,
 in general, will fail
 to do the same in the spacetime with the metric $\gamma_{\mu\nu}$. In
 this sense we are not entitled to consider the reformulated theory as
 Einstein's theory in the presence of some interacting fields.
 In the case that one is philosophically inclined to consider the
 $\gamma_{\mu\nu}$ as the physical metric, while the $g_{\mu\nu}$ as
 some sort of unifying field, then the equivalence principle should be
 implemented in the matter part ${\cal L}_M$ of
 Eq.\ (\ref{FullLagrangian}) using
 the metric $\gamma_{\mu\nu}$ in the place of  $g_{\mu\nu}$. Of course
 then, this ${\cal L}_M$ will be non standard with respect to
 $g_{\mu\nu}$.

 Whether or not nature chooses  to couple usual matter universally only
 to a spin-2 field (as the $\gamma_{\mu\nu}$) and not to a more
 composite one (as the $g_{\mu\nu}$)  is far from being experimentally
 testable. Trying to find an answer one may, however, employ some
 criteria of principle, as positivity of energy \cite{Sokolowski}. In
 any case, the use of new variables, as those of Eq.\ (\ref{NewMetric}) and
 Eq.\ (\ref{ConformalMetric}), which  turn out to simplify technically a
 physical problem, is undoubtfully very useful even if it is not clear
 whether one can attribute to these variables a foundamental character.

\section{Topological defects \\ in higher derivative theories}

Cosmological defects are formed during phase transitions in the
 evolving Early Universe whenever the symmetry group $G$ of the relevant
 field theory breaks down to a subgroup $H$ so that the vacuum manifold
 $M=G/H$ has some non trivial homotopy group \cite{Kibble}.
 Such a symmetry breakdown  at an energy scale $\eta$ can be realized,
 e.g., with an $n$-component scalar field $\phi^{(i)}$ having a
 Mexican-hat type of potential
\begin{equation}
 V(\phi) = -{\lambda\over 4}(\sum_{i=1}^n\phi^{(i)}\phi^{(i)}- \eta^2)^2 .
\label{MexicanHatPot}\end{equation}
 The  homotopic structure of the vacuum manifold depends on the number
 $n$ of  components of the scalar field and, thus, we may have the
 formation of domain walls for $n=1$, cosmic strings for $n=2$,
 monopoles for $n=3$.
 These defects are  respectively  surface-, line-,  and point-like
 configurations. Sufficiently away from these configurations, at
 distances  $d\gg \delta$, the scalar field $\phi^{(i)}$ approaches
 quickly its vacuum value $\sum_i\phi^{(i)}\phi^{(i)} \approx \eta^2$.
 Here $\delta$ is the width of the core of these defects, of the order
 of $m_{\phi}^{-1}$ where $m_{\phi}=\eta\sqrt{\lambda}$ is the mass of
 the scalar field's massive mode. Typically, for symmetry breaking at
 grand unification scale, $\delta\approx 10^{-30}{\rm cm}$ and
 $\kappa\eta^2\approx 10^{-6}$.

 Depending on whether the symmetry that breaks down is a gauge (local)
 or a global one we have respectively the formation of {\sl gauge} or
 {\sl global} topological defects.
 In the case of gauge symmetry there exists a well defined core, with
 width $\delta$,  where most of the energy of the topological defect
 configuration is localized.
 On the other hand, for global topological defects the components of the
 respective stress-energy-momentum tensor have, outside the ``core'', a
 relatively slow fall off due to the gradients of the Goldstone modes of
 the scalar field $\phi^{(i)}$. Thus, global defects are extended
 configurations. The reason for this difference between gauge and global
 defects is that in the case of gauge symmetry the presence of gauge
 fields can compensate the gradients of the scalar field.
 Finally, in the case of discrete symmetry breaking, which gives rise to
 domain walls, there are no Goldstone modes and thus domain walls are
 localized configurations.

 Based on the above properties, we will make in what follows the
 following approximations:
 \begin{itemize}
\item[(i)] Gauge topological defects and domain walls will be considered
    in the  zero core-thickness approximation
    and thus their stress-energy-momentum tensors will have components
    with appropriate Dirac $\delta$-fuctions.
\item[(ii)] For global defects, we will make the $\sigma$-model
    approximation
    where the scalar field is fixed to its asymptotic vacuum value
    everywhere outside the defect. This  is a sensible approximation at
    distances from the defect sufficiently larger than the ``core''
    width $\delta$.
 \end{itemize}

 In the following subsections we will obtain the gravitational field of
 cosmic strings, monopoles and domain walls in the quadratic theory $R+
 \alpha R^2$ with $\alpha |R| \ll 1$. For this we will make use of the
 result of the previous section (see Eqs.
 (\ref{QuadraticFromEinsteinMetric}), (\ref{chiFieldEq})), stating that
 the metric in the quadratic theory, $ds^2_{\rm (Q)}$,
 is conformally related
 with the metric in Einstein's theory, $ds^2_{\rm (E)}$,
\begin{equation}
    ds^2_{\rm (Q)} = (1+\chi) ds^2_{\rm (E)} , \qquad |\chi/2| \ll 1,
\label{MetricRelationship}\end{equation}
 with $\chi$ satisfying the massive Klein-Gordon equation
 (\ref{chiFieldEq})  in the $ds^2_{\rm (E)}$ metric.
 A consequence of Eq.\ (\ref{MetricRelationship}) is that there will be a
 modification of the  ``Newtonian'' potential equal to $\chi/2$. We will
 have below the oportunity to study its nature and its range in the case
 of topological defects, be them localized or extended sources.

 In general we shall restrict our attention to sufficiently large
 distances, $d$, away from the core, ($d\gg \delta$), but we will keep
 in mind that a proper treatment at short distances requires a proper
 model for the core of the defect itself. In this way we will be able to
 use the existing results in General Relativity for the gravitational
 field  of cosmic strings, monopoles and domain walls which were obtained
 by making use of the above approximations in model Lagrangians with
 symmetry breaking potential of the form (\ref{MexicanHatPot}).

\subsection{Global monopoles}

 The stress-energy-momentum tensor of a global monopole configuration,
 in regions far away from the core, can be approximated
 by \cite{Barriola,Harari}
\begin{equation} T^t_t= T^r_r \approx - {\eta^2\over r^2},  \qquad
          T^\theta_\theta=T^\varphi_\varphi=0 ,
\label{GlMonopoleSEM}\end{equation}
 while the respective metric in Einstein's theory of gravity is
 (approximately) given by \cite{Barriola,Harari}
\begin{eqnarray}
 ds_{\rm (E)}^2 =-(1&-&\Delta)dt^2 + (1-\Delta)^{-1}dr^2 +d\Omega^2  \cr
     d\Omega^2 &:=&r^2(d\theta^2+\sin^2\theta d\varphi^2 ), \cr
     \Delta &:=&8\pi G\eta^2=\kappa\eta^2.
\label{GlMonopoleEinsteinMetric}\end{eqnarray}
 This metric corresponds to a spacetime with a solid deficit angle:
 test particles are deflected by an angle $\pi\Delta/2$  irrespective of
 their velocity and their impact parameter.
 Here it should be added that a more careful treatment \cite{Harari}
 that takes into account the actual behaviour of the field at the
 monopole core, shows that the metric (\ref{GlMonopoleEinsteinMetric}) gets
 modified by terms which at distances $r\gg \delta=
 (\sqrt{\lambda}\eta)^{-1}$ correspond effectively to a negative
 mass term $M_{\rm eff}$, that is e.g. $g_{tt}\approx (1-\Delta -
 2GM_{\rm eff}/r)$. According to numerical analysis \cite{Harari}
 $M_{\rm eff} \approx -6\pi \sqrt{\lambda}\eta$. Thus, besides the
 topological deflection caused by the solid deficit angle, test
 particles experience also a repulsive radial force $- GM_{\rm eff}/r^2$
 away from the monopole.

 The metric in the quadratic theory is given by
 Eq.\ (\ref{MetricRelationship}) with $\chi$ satisfying
 Eq.\ (\ref{chiFieldEq}).
 Looking for spherically symmetric solutions we find that this equation
 for  $\chi=\chi(r)$ reads
\begin{eqnarray}
 \left\lbrace {1\over r^2}{d\over dr}\left [r^2{d\over dr}\right ] -
 \widehat{m}^2 \right\rbrace \chi(r)&=& {2\Delta \over 3(1-\Delta) r^2},\cr
 \widehat{m}^2&:=&m_0^2/ (1-\Delta).
\label{chiGlMonEq}\end{eqnarray}
 Making use of the Green function for this equation,
\begin{eqnarray}
 G(r,r^\prime)=-{1\over \widehat{m} r r^\prime}
 [&e^{-\widehat{m}r^\prime}&\sinh(\widehat{m}r)\Theta(r^\prime-r) \cr
 + &e^{-\widehat{m}r}&\sinh(\widehat{m}r^\prime)\Theta(r-r^\prime) ],
\label{GlMonGreenFun}\end{eqnarray}
 where the step function $\Theta(z):=\{0,1,1/2\}$ for
 $\{z<0, z>0, z=0\}$ respectively,
 we can write down the solution for $\chi(r)$ in terms of the
 Exponential-Integral $({\rm Ei})$ and Hyperbolic-Sine-Integral
 $({\rm shi})$ functions \cite{Gradshteyn} as
\FL
\begin{equation}
 \chi(r)= {2\Delta\over 3(1-\Delta)}{1\over \widehat{m}r}
 [{\rm Ei}(-\widehat{m}r)\sinh(\widehat{m}r)-e^{-\widehat{m}r}
 {\rm shi}(\widehat{m}r) ] .
\label{chiGlMon}\end{equation}
 Checking numerically the behavior of this function we find that its
 contribution to the `Newtonian' potential $\chi/2$ is an attractive
 one.  In particular, using the asymptotic behavior of the Ei and  shi
 functions \cite{Gradshteyn} we find that at large radial distances
 $r\to\infty$
\begin{equation}
 \chi(r) \approx -{2\over 3} {\Delta\over(m_0r)^2} ,
\label{chiGlMonAs}\end{equation}
 which implies a long range potential, exerting on test particles an
 attractive force $-(2\Delta/3m_0^2)r^{-3}$. Comparing this force to the
 repulsive force due to the core of the monopole we see that the former
 falls off faster by one power of $r$ and thus is negligible at very
 large distances. It overcomes, however, the effect of the latter at a
 distance $r\approx m_0^{-2}/(\lambda\delta)$ and, thus, it can be the
 dominant force within the region
 $\delta\ll r \ll m_0^{-2}/(\lambda\delta)$ which will exist provided
 that $m_0^{-1}\gg \delta$.

 Finally, let us note that the expression (\ref{chiGlMon}) diverges as
 $r\to 0$. This is due to the form of the energy-momentum-tensor in
 Eq.\ (\ref{GlMonopoleSEM}) which is not valid
 at distances comparable to the core of the monopole.

\subsection{Gauge Monopoles}

 A gauge monopole is a spherically symmetric configuration with mass $M$
 and a magnetic charge $g$. Its stress energy momentum tensor can be
 approximated by
\begin{eqnarray}
 T^t_t &=& -{M\over 4\pi} {\delta(r) \over r^2}-{(g/4\pi)^2\over r^4},
 \cr
 T^r_r&=&-T^\theta_\theta=-T^\varphi_\varphi=-{(g/4\pi)^2\over r^4} .
\label{GaugeMonSEM}\end{eqnarray}
 We consider the case where the metric outside the core of the monopole
 matches to a Reissner-Nordstrom one, (see Ref. \cite{Gibbons} for a recent
 review and new results on the gravitational field of monopoles)
\begin{eqnarray}
 ds^2=&-&\left (1-{2GM\over r}+ {G g^2\over 4\pi r^2}\right ) dt^2 \cr
        &+& \left (1-{2G M\over r}+ {G g^2\over 4\pi r^2}\right )^{-1} dr^2
	+ d\Omega^2 .
\label{GaugeMonEinsteinMetric}\end{eqnarray}

 Since the source for the $\chi$ field is the trace of the
 stress-energy-momentum tensor, only the mass term in
 Eq.\ (\ref{GaugeMonSEM}) will contribute.
 Furhermore, if we consider distances sufficiently far from the monopole
 $r\gg \delta \gg GM$, the equation for $\chi$ approximately reads
\begin{equation}
 \left\lbrace {1\over r^2}{d\over dr}\left [r^2{d\over dr}\right ] -
  m_0^2 \right\rbrace \chi(r)= {\kappa M\over 12\pi}
  {\delta(r) \over r^2}.
\label{chiGaugeMonEq}\end{equation}
 Demanding finiteness at radial infinity, this equation has as solution
 the Yukawa fuction
\begin{equation}
 \chi(r)= -{\kappa M\over 12\pi} {e^{-m_0 r}\over r}.
\label{chiGaugeMon}\end{equation}
 Notice that the Newtonian potential of the monopole will be  modified
 by the ammount $\chi/2$ corresponding to an attractive potential
 exponentially decreasing with an $e$-folding term characteristic of a
 massive scalar field with mass $m_0$.

 It worths remarking that the short range
  corrections of Eq.\ (\ref{chiGaugeMon})
 apply also to the external metric of  spherically symmetric mass
 distributions \cite{Teyssandier} such as neutron strars,  giving thus
 rise to ``fifth force'' terms. However, when one deals with black
 holes, the no-hair theorem for $R+R^2$ theories \cite{Whitt} implies
 that corrections of the type (\ref{chiGaugeMon}) are absent.

\subsection{ Global Cosmic Strings}

 As we explained in the introductory part of this section global cosmic
 strings are extended line configurations. The stress-energy-momentum
 tensor for a straight, static, cylindrically symmetric global string
 lying along the $z$-axis is approximately given for $r\gg \delta$ by
\begin{equation}
   T_t^t=T_z^z=T_r^r=-T_\theta^\theta \approx -{\eta^2 \over 2r^2} .
\label{GlStrSEM}\end{equation}
 The respective exact solution for the metric in Einstein's theory has
 been found in  \cite{Cohen}. However, it is quite complicated for the
 purpose of solving the equation (\ref{chiFieldEq}) for the field $\chi$.
 Furthermore, besides this technical problem, the spacetime of a global
 string has true spacetime singularities \cite{Harari88,Gregory}, a fact
 that demands carefull checking of the range of validity of the
 approximation ($\alpha |R| \ll 1$) on which our treatment is based.
 Instead, we prefer to work here in the weak field limit where the
 equation for the $\chi$ field is in Minkowski background metric.

 In the weak field limit of general relativity the metric of the global
 string reads \cite{Harari88}
\begin{eqnarray}
   ds^2_{\rm (E)} = \biggl [1-4G\mu \ln({r\over \delta})
   \biggr ] (-dt^2+dz^2)
      + &dr^2& \cr
      + r^2 \biggl [1-8G\mu\ln({r\over \delta}+c) \biggr ]&d\theta^2&,
\label{GlStringEinsteinMetric}\end{eqnarray}
 Here $\mu:=\pi\eta^2$, $\delta$ is the core width and $c$ is a constant
 of order unity that may partially take into account a global effect of
 the string core.
 Studying the motion of test particles it is seen that the static global
 string exerts a repulsive force $2G\mu/r$ \cite{Harari88}. It is
 interesting to explore how this force is modified in the quadratic
 theory that we are currently considering.

 The equation that $\chi$ statisfies in the weak field limit is
\begin{equation}
 \left\lbrace {1\over r}{d\over dr}\left [r{d\over dr}\right ] -
     m_0^2\right \rbrace \chi(r) = {\kappa\mu\over 3\pi r^2}.
\label{chiGlStringEq}\end{equation}
 The Green function for the homogeneous part of this differential
 equation, with the boundary conditions of finitness at the origin and
 at infinity, is easily found to be
\begin{eqnarray}
 G(r,r^\prime)= &-& K_0(m_0r)I_0(m_0r^\prime)\Theta(r-r^\prime) \cr
                &-& I_0(m_0r)K_0(m_0r^\prime)\Theta(r^\prime -r)
\label{GreenFunction}\end{eqnarray}
 where $\Theta$ is the step function.
 Thus the solution of Eq.\ (\ref{chiGlStringEq}) can be written as
\begin{eqnarray}
    \chi(r)=-{\kappa\mu\over 3\pi} \Bigl [&K_0(m_0r)&\int_\delta^r
      I_0(m_0r^\prime){dr^\prime\over r^\prime}  \cr
      + &I_0(m_0r)&\int_r^\infty
       K_0(m_0r^\prime){dr^\prime\over r^\prime} \Bigr] ,
\label{chiGlString}\end{eqnarray}
 where we have introduced a lower cutoff at $r=\delta$ to cope
 effectively with the divergence that  appears in the first integral if
 we let  $r\to 0$. This divergence is only due to the approximate form
 of the stress-energy-momentum tensor which as we have already stressed
 is not valid near the core of the string.

 The leading term in an asymptotic expansion of Eq.\ (\ref{chiGlString}) at
 large radial distances is
\begin{equation}
 \chi(r) \approx -{\kappa\mu\over 3\pi m_0^2 r^2}, \qquad r\to\infty .
\label{chiGlStringLargeAppr}\end{equation}
 From this expression we conclude that the additional `Newtonian'
 potential in the quadratic theory implies at large distances an
 attractive force $-(\kappa\mu/3m_0^2)r^{-3}$. Due to the slower fall
 off of the original repulsive force, the total force on test particles
 remains repulsive at large distances from the string. At around $r\sim
 m_0^{-1}$ the total force is expected to change sign.

\subsection{Gauge Cosmic Strings}

 Gauge cosmic strings are, in contrast to global ones, localized line
 configurations. The stress-energy-momentum tensor for a static,
 straight along the $z$-axis, gauge cosmic string with line energy
 density $\mu$ is
\FL
\begin{equation}
      T_t^t=T_z^z= -{\mu\over 2\pi(1- \kappa\mu/2)} {\delta(r)\over r},
      \quad T_r^r=T_\theta^\theta=0
\label{GaugeStringSEM}\end{equation}
 with corresponding metric in Einstein's theory \cite{Vilenkin85}
\FL
\begin{equation}
     ds^2_{\rm (E)}= -dt^2+ dz^2 + dr^2 + (1- \kappa\mu/2)^2r^2d\theta^2.
\label{GaugeStringEinsteinMetric}\end{equation}
 Here the polar coordinates $r,\theta$ have the usual range.
 This spacetime is everywhere flat except along the $z$-axis where the
 string is located.
 As one goes around the string one notices an  angle deficit. This
 topological property has  the consequence that test particles which
 localy do not feel any gravitational forces  are, however, deflected by
 the string.

 Let us now turn our attention to  the field $\chi$. It satisfies the
 equation
\FL
\begin{equation}
 \left\lbrace {1\over r}{d\over dr}\left [r{d\over dr}\right ] -
       m_0^2\right \rbrace \chi(r) = {\kappa\mu\over 3\pi
       (1- \kappa\mu/2)}{\delta(r)\over r}.
\label{chiGaugeStringEq}\end{equation}
 which can be easily solved by demanding for the field $\chi$ finiteness
 at infinity and  correct discontinuity at the origin. The solution
 reads
\begin{equation}
     \chi(r)= -{\kappa\mu\over 3\pi(1- \kappa\mu/2)} K_0(m_0 r),
\label{chiGaugeString}\end{equation}
 which can be  easily checked that satisfies  Eq.\ (\ref{chiGaugeStringEq})
 using the small argument asymptotic behavior of the modified Bessel
 function $K_0(z) \approx -\ln(z/2)$.   Finally notice that the field
 $\chi$ decays exponentially fast since at large distances
 $K_0(m_0r)\approx (\pi/2m_0r)^{1/2}\exp(-m_0r)$.
 Very close to the string the expression (\ref{chiGaugeString}) diverges
 logarithmically. Again, as the physical cosmic string has a finite core
 this divergence should not appear in a more proper  treatment near the
 core.

 From Eq.\ (\ref{chiGaugeString}) follows that the `Newtonian' potential
 of the cosmic string spacetime in an attractive, short range one. The
 respective force that the string will exert on test particles is
 $-[\kappa\mu m_0 / 6\pi(1- \kappa\mu/2)] K_1(m_0 r)$. Because of the
 large distance exponential fall off behavior of $K_1(m_0r)\propto
 (m_0r)^{-1/2}\exp(-m_0r)$ it follows that this force is significant
 only close to the string up to  distances $r\sim m_0^{-1}$.

 In closing this subsection let us remark that the result obtained here
 is in agreement with the recent result of Linet and
 Teyssandier \cite{Linet} in the weak field limit where
 $\kappa\mu\ll 1$. These authors have also obtained the cosmic string
 metric in the weak field limit of the  quadratic theory
 (\ref{FullLagrangian}) and (\ref{QuadraticLagrangian}) which
 contains also the massive tensorial field $\psi_{\mu\nu}$ of
 Eqs.\ (\ref{LinearizedEqsa})-(\ref{LinearizedEqsc}).
 In particular they find that the effect of this field on the Newtonian
 potential is a repulsive one with a range set by the inverse mass $m_1$
 of the field $\psi_{\mu\nu}$.

\subsection{Domain Walls}

 The energy content and the gravitational field of domain wall
 configurations in Einstein's theory  has been studied  extensively in
 the literature, see  Ref. \cite{Wang} and references therein.
 We will consider here Vilenkin's vacuum plain domain wall solution
 discussed in   \cite{Vilenkin83}.
 For a domain wall with surface energy density $\sigma$, lying on the
 $|z|=0$ plane,  the stress-energy-momentum tensor is
\begin{equation}
 T^t_t=T^x_x=T^y_y=-\sigma \delta(z),\qquad T^z_z=0,
\label{DWSEM}\end{equation}
 while the respective domain wall spacetime is described by the
 metric \cite{Vilenkin83}
\FL
\begin{eqnarray}
 ds^2_{\rm (E)}=&&(1-\nu|z|)^2\bigl [-dt^2 + e^{2\nu t}(dx^2 + dy^2)
            \bigl ]+ dz^2 , \cr
 &&\nu:=2\pi G\sigma=\kappa\sigma/4.
\label{DWEinsteinMetric}\end{eqnarray}
 Note that some of the metric components are time dependent (no static
 solutions can be found). Test particles in this spacetime are repelled
 with a proper acceleration $\nu$ away from the wall \cite{Vilenkin83},
 a property that we may deduce just by looking at the `Newtonian'
 potential term in the $g_{tt}$ component of the metric
 (\ref{DWEinsteinMetric}).
 Finally we should mention  that at $|z|=\nu^{-1}$ an event horizon
 appears \cite{Vilenkin83}.
 In what follows we restrict our attention to spacetime regions with
 $|z|\leq \nu^{-1}$.

 Although the metric in (\ref{DWEinsteinMetric}) is time dependent we can
 find, however, static solutions to the equation (\ref{chiFieldEq})
 for the field $\chi$ depending only on $|z|$. For such solutions, equation
 (\ref{chiFieldEq})  reduces to the ordinary differential equation
\FL
\begin{equation}
    \left\lbrace (1-\nu|z|)^{-3}{d\over dz}\left
       [(1-\nu|z|)^3{d\over dz}\right]
            - m_0^2 \right\rbrace \chi(z) = 4\nu \delta(z).
\label{chiDWEq}\end{equation}
 The homogeneous part of this equation can be easily transformed into a
 Bessel differential equation for $\widehat{\chi}$ where
 $\chi(|z|):=\widehat{\chi}(\widehat{z})/\widehat{z}$ using the new
 variable $\widehat{z}=\nu^{-1} -|z|$.
 In this way we find that the solution for $\chi(z)$ is, in regions with
 $z\neq 0$, a linear combination of the terms
 $I_1(m_0\widehat{z})/\widehat{z}$ and
 $K_1(m_0\widehat{z})/\widehat{z}$ where $K_1, I_1$ denote modified
 Bessel functions. The coefficients of this solution are determined by
 demanding finiteness at the horizon $\widehat{z}=0$, while on the
 domain wall, $z=0$, the field $\chi$ should be continous  and  have the
 appropriate discontinuity in its first derivative  which, according to
 Eq.\ (\ref{chiDWEq}), is $[{d\over dz}\chi]_{z=0}=4\nu$. Thus we finally
 obtain

\begin{equation}
    \chi(z)= -2\left[{m_0\over\nu} I_2({m_0\over \nu})\right ]^{-1}
    {I_1({m_0\over \nu}[1-\nu|z|]) \over 1-\nu|z|} .
\label{chiDW}\end{equation}

 It is easy to check that this implies  an attractive and  short range
 contribution to the Newtonian potential. This cannot overwhelm the
 original repulsive potential of the domain wall except very close to
 the wall  for $|z|\alt m_0^{-1}$. This should be obvious in writing
 Eq.\ (\ref{chiDW}) in the sensible
 approximation $m_0/\nu \gg 1$ and near the
 domain wall where $\chi$ takes its  largest values
\begin{equation}
     \chi(z)\approx -{2\nu\over m_0}\exp(-m_0|z|) .
\label{chiDWApprox}\end{equation}
 Away from the wall the field $|\chi|$ decreases exponentially
 and  at the horizon
 $\chi$ attains the small value $-1/I_2({m_0\over \nu})$.

\section{Conclusions and remarks}

 We have dealt in this paper mainly with the higher derivative  theory
 (\ref {FullLagrangian}), (\ref{ScalarQuadraticLagrangian})
 with $F=R+\alpha R^2$ in the approximation $\alpha |R|\ll 1$.
 We showed that one can simplify the problem of solving the
 corresponding fourth order field equations using the conformal
 transformation (\ref{ConformalMetric}) which leaves us with  the system
 of field equations (\ref{TransformedFieldEqs}), (\ref{psiFieldEq})
 having only  second order derivatives.
 This  is formally a system of Einstein type  equations
 plus the field equations for a massive scalar field  with mass
 $m_0^2=1/(6\alpha)$ interacting non minimally with gravity.
 We then found that in the approximation $\alpha |R|\ll 1$ the
 gravitational fields in the $R+\alpha R^2$ theory and in the Einstein
 theory are conformally related according to the
 Eqs.\ (\ref{QuadraticFromEinsteinMetric}), (\ref{chiFieldEq}).
 Using this result we looked in Sec. III for solutions representing the
 gravitational field of monopoles, cosmic strings and domain walls

 For localized topological defect configurations as gauge strings, gauge
 monopoles and domain walls we have found  {\sl short range}, {\sl
 attractive} corrections. Their range  $\sim 1/m_0$ is characteristic of
 the pressence of the  massive field.

 For extended sources as global monopoles and global strings we have
 again found {\sl attractive} corrections but with a {\sl long range}.
 Their  fall off rate  depends on the corresponding
 stress-energy-momentum of these defect configurations. In particular,
 for distances $r\gg m_0^{-1}$ is found that the attractive correction
 to the Newtonian potential is $\approx \kappa T/( 6m_0^2)$ where $T$ is
 the trace of the corresponding stress-energy-momentum tensor.

 A more detailed investigation of the gravitational effects of
 topological defects in more general higher order derivative theories is
 in progress and we hope to present the results elsewhere.
 We can however already here reestablish the above given conclusions in
 the $R+\alpha R^2$ theory and at the same time extend them to the case
 of the theory (\ref{FullLagrangian}),(\ref{QuadraticLagrangian}) which,
 for $\beta\neq 0$, contains also a massive tensorial field.
 This will be done by making use of the linearized field equations
 (\ref{LinearizedEqsa}), (\ref{LinearizedEqsb}).
 First  observe that the Newtonian potential $\Phi_N$ will have, besides
 the Einstein term $\Phi_N^{(E)}$, also the contributions
 $\Phi_N^{(\chi)}$ and $\Phi_N^{(\psi_{\mu\nu})}$ from the scalar field
 $\chi$ and massive field $\psi_{\mu\nu}$ respectively
\FL
\begin{equation}
 \Phi_N = \Phi_N^{(E)} + \Phi_N^{(\chi)} + \Phi_N^{(\psi_{\mu\nu})}=
      {1\over 2}[-h^{(E)}_{00} + \chi - \psi_{00}] .
\label{NewtonianPotential}\end{equation}
 For static spacetimes we have from Eqs.\ (\ref{LinearizedEqsa}) and
 (\ref{LinearizedEqsb}) that
\begin{eqnarray}
    \nabla^2 \Phi_N^{(E)} &=& {\kappa\over 2}(\rho + P_1 + P_2 + P_3),
    \cr
   (\nabla^2- m_0^2)\Phi_N^{(\chi)}&=&{\kappa\over 6}(\rho -P_1 -P_2 -P_3),
    \cr
   (\nabla^2- m_1^2)\Phi_N^{(\psi_{\mu\nu})}&=&-{\kappa\over 3}
                                   (2\rho + P_1 + P_2 + P_3) \cr
    m_0^{-2}:=6\alpha +2\beta, && \quad m_1^{-2}:=-\beta,
\label{NewtonianPotentialEqs}\end{eqnarray}
 where $\rho$ denotes the mass density and $P_1,P_2,P_3$ the principal
 pressures of the matter.

{\sl Sign of forces:}
 Look at the r.h.s of these equations. For the topological defect
 configurations  discussed in the present paper $\rho + \sum_iP_i$ is
 $>0$ for gauge monopoles, $0$ for gauge strings and global monopoles,
 and $<0$ for global strings and domain walls. On the other hand
 $\rho -\sum_iP_i > 0$, while $-(2\rho + \sum_iP_i) \leq 0$ with equality
 holding for domain walls. Thus the Einstein contribution in
 Eq.\ (\ref{NewtonianPotential}) is attractive for gauge monopoles,
 zero for gauge strings and global monopoles,
 and repulsive for global strings and
 domain walls. The $\chi$-contribution is always attractive, while the
 $\psi_{\mu\nu}$-contribution is in general repulsive except for domain
 walls where it is  zero.

{\sl Range of forces:}
 From Eqs.\ (\ref {NewtonianPotentialEqs})
 it is clear that the Einstein term
 provides in general a long range interaction due to an effective mass
 density $\rho + \sum_iP_i$. It gives, however, a zero effect for gauge
 cosmic strings and global monopoles.
 For the  contributions of $\chi$ and $\psi_{\mu\nu}$ terms we have
 that:
 \begin{itemize}
\item[(i)] If the stress energy momentum tensor vanishes (or falls off
    sufficiently  rapidly) outside a localized source then at distances
    $d$ from the source, these contributions are of short range $\propto
    \exp(-md) /d^p$ where $m$ stands for the mass $m_0$ (or $m_1$) and
    $p$ is a parameter depending on the symmetry of the spacetime and is
    equal to $0, {1\over 2}$ and $1$ for plain domain walls, strings and
    monopoles respectively.
\item[(ii)] If the sources are not localized then for the $\chi$,
    $\psi_{\mu\nu}$ contributions there are two characteristic regimes:

(a) At distances $d\gg {\rm max}(m_0^{-1}, m_1^{-1})$ the mass terms
          dominate over the derivative terms in the two last equations of
       (\ref{NewtonianPotentialEqs}). Thus, asymptotically at large
       distances we have long range contributions
           $\Phi_{N}^{(\chi)}\approx -\kappa (\rho -\sum_iP_i)/(6m_0^2)=
           \kappa T/(6m_0^2)$, as we found in this paper, and
           $\Phi_{N}^{(\psi_{\mu\nu})}\approx
           \kappa (2\rho + \sum_iP_i)/(3m_1^2)$.

(b) At distances $d\ll{\rm min}(m_0^{-1}, m_1^{-1})$
               the derivative terms dominate over the mass terms.
               The interesting thing to note here is
               that at such distances the total `Newtonian' potential
               in Eq.\ (\ref{NewtonianPotential}) satisfies
               $\nabla^2 \Phi_N \approx 0$. This has implications for the
               differentiability of the spacetime metric and
               implies drastic changes in the
               singularity  structure of gravity at short distances. For
               example, the gravitational potential of a point massive
	       particle is finite at the origin in contrast to the $1/r$
	       Coulomb behavior in the Newtonian theory.
 \end{itemize}

 These considerations are in agreement with the results of the previous
 sections and the results of  \cite{Linet}  for gauge cosmic strings. They
 may be particularly relevant to the study of the evolution of
 topological defects in the very early universe: (a) for structure
 formation scenarios based on global defects where the long range
 modifications of the quadratic theories may play an important role; (b)
 for collisions of cosmic strings where the drastic short range
 modifications may change significantly the predictions of these
 simulations for the evolution parameters of a string network.  Thus it
 is interesting to study further topological defects and  collisions of
 cosmic strings in quadratic gravitational theories and implement
 appropriate modifications in future numerical simulations. The outcome
 of such an investigation confronted with observation, may, among other
 things, allow one to put constraints on the $m_0, m_1$ parameters of
 quadratic gravitational theories.

\acknowledgments
 This work was supported by the European Community DG XII. C.O.L was
 also supported by the Alexander von Humboldt foundation.

redpajama_set_name: RedPajamaArXiv