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\preprint{DCPT-03/15}
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\title{Comment on ``Global monopole in Asymptotically dS/ AdS Spacetime''}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\author{Bruno Bertrand \  \ and \  \ Yves Brihaye\footnote{Yves.Brihaye@umh.ac.be}}
\address{Facult\'e des Sciences, Universit\'e de Mons-Hainaut,
 B-7000 Mons, Belgium}
\author{Betti Hartmann\footnote{Betti.Hartmann@durham.ac.uk}}
\address{Department of Mathematical Sciences, University
of Durham, Durham DH1 3LE, U.K.}
\date{\today}
\setlength{\footnotesep}{0.5\footnotesep}

\maketitle
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\begin{abstract}
In this comment on the recently published
paper ```Global monopole in Asymptotically dS/ AdS Spacetime'' [Phys. Rev. {\bf
D 66} (2002), 107701], we reconsider global monopoles in asymptotic de Sitter/ Anti- de Sitter
space-time. In comparison to the mentioned publication, we find that the introduction
of a cosmological constant can {\bf not} render a positive
mass of the global monopole. Moreover, we find (confirmed by our numerical
analysis) a different asymptotic behaviour of the metric functions.
\end{abstract}

\pacs{PACS numbers: 04.20Jb, 04.40.Nr, 14.80.Hv }

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\section{Introduction}
In recent years topological defects \cite{vilenkin} in asymptotic de Sitter (dS)/
Anti- de Sitter (AdS) space-time have gained renewed interest.
This is mainly due to the proposed dS/CFT \cite{strominger}, resp.
AdS/CFT \cite{adscft} correspondences.  These correspondences 
suggest a holographic duality between gravity in a $d$-dimensional
dS, resp. AdS space and a conformal field theory (CFT) ``living''
on the boundary of the dS, resp. AdS spacetime and thus being $d-1$-dimensional. 
However, dS space-time is also interesting from a cosmological point of
view since it seems to be confirmed by observational data \cite{super}
that we live in a universe with positive cosmological constant.

Consequently, global monopoles have been studied in dS/AdS space-time \cite{li}.
Gravitating global monopoles in asymptotically flat space-time 
were first discussed in \cite{vile,harari}. These  topological defects
were found to have a negative mass and a deficit angle depending on the
vacuum expectation value (vev) of the scalar Goldstone field and the gravitational
coupling. For sufficiently high enough values of the vev the solutions
have an horizon \cite{lieb}. These solutions were named (after their string counterparts 
\cite{laguna}) ``supermassive monopoles''. 

In  \cite{li}, it was found that the introduction of a cosmological constant
can render the mass of the monopole positive. This was demonstrated by a figure showing the mass
function as function of the radial coordinate for different choices of the cosmological
constant. In our recent work, we are mainly interested in composite monopole
defects \cite{com} in dS/ AdS space-time \cite{ybe}. This is why we cross-checked the
results of \cite{li} and found discrepancies between our results and those in \cite{li}.

The paper is organised as follows: we give the model in Section II and discuss the
asymptotic behaviour, which should be compared to that in \cite{li}, in Section III.
We give our numerical results in Section IV and conclude in Section V.
 

\section{The Model}
We consider the following action~:
\begin{equation}
\label{action}
S =\int \left( \frac{1}{16\pi G}(R- 2 \Lambda)
- \frac{1}{2}\partial_{\mu} \xi^a \partial^{\mu} \xi^a  - 
\frac{ \lambda}{4}(\xi^a\xi^a- \eta^2)^2   \right) \sqrt{-g} d^4 x
\end{equation}
which describes a Goldstone triplet $\xi^a$, $a=1,2,3$, interacting with
gravity in an asymptotically de Sitter (dS) (for the cosmological constant
$\Lambda > 0$), resp. Anti-de Sitter (AdS) ($\Lambda < 0$) space-time.
$G$ is Newton's constant,
$\lambda$ is the self-coupling constant of the Goldstone field
 and $\eta$ the vacuum expectation value (vev) of the Goldstone field.

For the metric, the spherically symmetric Ansatz
in Schwarzschild-like coordinates reads~:
\begin{equation}
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=
-A^{2}(r)N(r)dt^2+N^{-1}(r)dr^2+r^2 (d\theta^2+\sin^2\theta
d^2\varphi)
\label{metric}
\ , \end{equation}
while for the Goldstone field, we choose the hedgehog Ansatz \cite{vile}~:
\begin{equation} 
 {\xi}^a = \eta  h(r) {e_r}^a 
\ . 
\end{equation}
We introduce the following dimensionless variable and coupling constants~:
\begin{equation}
     x = \eta r \quad , \quad 
     \alpha^2 = 4 \pi G \eta^2 \quad , \quad  
     \gamma = \frac{\Lambda}{\eta^2} \ .
\end{equation}


Varying (\ref{action}) with respect to the metric fields gives 
the Einstein equations which can be combined to give two first order
differential equations for $A$ and $\mu$:~
\begin{equation}
\label{a}
A' = \alpha^2 A x  (h')^2  
\end{equation}
\begin{equation}
\label{mu}
\mu' = \alpha^2  \left( h^2-1 + x^2\frac{\lambda}{4}(h^2-1)^2 +\frac{1}{2}
x^2N (h')^2 \right) 
\end{equation}
and $N$ and $\mu$ are related as follows:~
\begin{equation} 
\label{nmu}
N(x) = 1 - 2\alpha^2 - 2 \frac{\mu(x)}{x} 
                - \frac{\gamma}{3} x^2  \ . \
\end{equation}
Note that for $\gamma=0$, the existence of solutions without horizon
is restricted by $\alpha < \sqrt{\frac{1}{2}}$ \cite{lieb}.

Variation with respect to the matter fields yields the Euler-Lagrange
equations for the Goldstone field~:
\begin{equation}
(x^2 A N h')' = A( 2 h + \lambda x^2 h(h^2-1)) \ , \\
\label{feq}
\end{equation}
The prime denotes the derivative with respect to $x$.
Note that the equations have the same structure as  for the 
asymptotically flat space-time \cite{vile,harari}. The cosmological
constant just appears in the relation defining $\mu(x)$ and $N(x)$.


In order to solve the system of equations uniquely,
we have to introduce $4$ boundary conditions, which we choose to be~: 
\begin{equation}
     \mu(0) = 0 \quad ,  \quad h(0) = 0 \quad , \quad  A(\infty) = 1 \quad  , 
     \quad h(\infty) = 1 \quad .
\end{equation}


The dimensionless mass of the solution is determined by the asymptotic 
value $\mu(\infty)=\mu_{\infty}$  of the function $\mu(x)$ and is given
by $\mu_{\infty}/\alpha^2$.




\section{Asymptotic behaviour}
Expanding the functions
around the origin gives~:
\begin{equation}
f(x\rightarrow 0) = c_1 x + O(x^3) \ \ , \ \
\mu(x\rightarrow 0) = - \alpha^2 x + O(x^2) \ \ , \ \
A(x\rightarrow 0) = A(0)(1 + O(x^2))
\end{equation}
where $c_1$ and $A(0)$ are free parameters to be determined numerically.
The asymptotic behaviour ($x\rightarrow\infty$) is given by~:
\begin{eqnarray}
\label{asp}
f(x\rightarrow\infty) &=&  1 - \frac{K}{x^2} +O(\frac{1}{x^3}) \ \  , \ \  
\mu(x\rightarrow\infty) = \mu_{\infty} + \frac{\alpha^2 \lambda K^2}{x} + O(\frac{1}{x^2}) \ ,  \nonumber \\
& & 
A(x\rightarrow\infty) =   1 - \frac {\alpha^2 K^2}{x^4}+O(\frac{1}{x^5})
\label{ainfty}
\end{eqnarray}
with the constant:
\begin{equation} 
K \equiv (\lambda - \frac{\gamma}{3})^{-1} \ .
\end{equation}
It is worthwhile to contrast the coefficient of the $1/x$ correction for the
mass function $\mu(x)$ appearing in  the above equation (\ref{asp}) 
with its counterpart in \cite{li}, eq.(18). While in the latter
the coefficient is independent on both the cosmological constant and the
self-coupling of the Goldstone field, we find here a non-trivial dependence
on these parameters (which is indeed confirmed by our numerical analysis).
We also remark that the expansion presented in \cite{li} is in contradiction
with the figure presented in that paper. The figure of \cite{li} seems incompatible
with the fact that the first asymptotic correction to the mass is
supposed to be independent of the cosmological constant.
 
\section{Numerical results}
We remark that without loosing generality, we can choose $\lambda=1.0$.

Following the investigation in \cite{li}, we have studied the dependence of the mass
$\mu_{\infty}/\alpha^2$ on the cosmological constant $\gamma$.
It is well know that for $\gamma=0$ 
global monopoles have a negative mass \cite{vile,harari}. Thus, the global monopole
has a repulsive effect on a test particle in its neighbourhood. The authors of \cite{li}
addressed the question whether the mass can become positive for specific choices of the
cosmological constant and found that for positive cosmological
constants \footnote{Note that the sign convention in \cite{li} is opposite to the
one used by us, i.e. dS space has a negative cosmological constant in \cite{li}.} 
this is possible. As a check for a future publication \cite{ybe} on composite 
monopole defects in dS/AdS space-time, we have tried to obtain the results given in \cite{li}
and found contradictions.

By solving the three equations numerically we constructed
solutions for positive and negative values of $\gamma$.
First, we checked, whether the asymptotic behaviour found in (\ref{ainfty})
is correct. We indeed confirmed numerically that the coefficients have the
given dependence on the coupling constants. We have then fixed $\alpha=0.1$
and determined the mass $\mu_{\infty}/\alpha^2$ in dependence on $\gamma$.
Our results are illustrated in Fig.~1. First, we remark that
for $\gamma=0$, the mass of the solution is close to
$-\frac{\pi}{2}$. Since the mass of the global monopole
in flat space is just $=-\frac{\pi}{2}$ \cite{harari} (of course in rescaled units in comparison
to here), the value of $\frac{\mu_{\infty}}{\alpha^2} \lesssim -\frac{\pi}{2}$
is in good agreement with these results. 

Since solutions in AdS space ($\gamma < 0$) are
not restricted by the appearance of horizons, we can choose $\gamma\rightarrow -\infty$.
Clearly, we find that the mass increases, but stays negative for {\bf all}
values of the cosmological constant. Choosing $\gamma$ positive, solutions
exist only for small values of $\gamma$ since a cosmological horizon appears in dS space.
Thus, we have not plotted the mass dependence here. However, we remark that
$\left( \mu_{\infty}(\gamma > 0)-\mu_{\infty}(\gamma=0)\right)/\alpha^2 < 0$ for all $\gamma >0$.
Since the mass curve does only alter its shape very little when choosing different $\alpha$,
we conclude that in contrast to what is claimed in \cite{li}, the appearance of
a cosmological constant (of either sign) can {\bf not} alter the sign of the mass of the global monopole.
Rather, we find that the mass gets more negative for increasing $\gamma >0$ which can be related to the
fact that the core of the monopole increases due to increased cosmological expansion. For $\gamma < 0$, we find
that the mass becomes less negative for $|\gamma|$ increasing, which is due to the decreased cosmological
expansion, but however stays negative for all values of $\gamma$ in AdS space.


We contrast the behaviour of the mass-function $\mu(x)$ for different values of $\gamma$ with that in Fig.1
of \cite{li}. Since the authors choose the vev of the Goldstone field to be $=0.01$, while we choose it to
be $1$, the values of their cosmological constant in their plot corresponds to choosing
$\gamma=-10$, $-3$ and $5$ here. Moreover, their choice of $G$ leads to $\alpha=0.0355$.
First, we remark that we are surprised that the authors of \cite{li} have managed to find
solutions which correspond to our $\gamma=5$. We find that increasing $\gamma$ from zero to positive values,
we can construct solutions only for $\gamma \lesssim 0.073$. The reason is that with increasing
cosmological constant the horizon which appears for the dS solutions decreases to lie closer and closer to
the core of the monopole. We thus present $\mu(x)$ for $\gamma=-10$, $-3$ and $0.073$ in Fig.~2. 
Clearly, the mass function $\mu(x)$ is a constantly decreasing function of the coordinate $x$ and
doesn't have local extrema like in \cite{li}. Moreover, the asymptotic values of $\mu(x)$ are always negative.



\section{Conclusions}
Topological defects \cite{vilenkin} are believed to be relevant 
for structure formation in the universe. 
Global defects, i.e. defects which don't involve gauge fields
are of special interest in this context since they have
a long-range scalar field. This leads to the infiniteness of energy
in flat space, but however renders a strong gravitational
effect when the topological defects are studied in curved space.
Moreover, in the case of the global monopole, the coupling to gravity
can remove the singularity present in flat space. The space-time then
has a deficit angle and is not locally flat. Moreover, the mass
of the monopole is negative, which was interpreted as a repulsive effect
of the monopole. 

The authors of \cite{li} have studied global monopoles in a dS/AdS space-time and
found that the inclusion of the cosmological constant can render the mass of the 
monopole positive. Reconsidering these solutions with a highly accurate numerical routine (see 
\cite{bhk} for a short description) and
studying the asymptotic behaviour  we come to a different conclusion~:
global monopoles do {\bf not} acquire a positive mass in AdS or dS space-time.



\begin{acknowledgments}
BH was supported by an EPSRC grant.
\end{acknowledgments}

\begin{thebibliography}{99}
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%gravitational field of a monopole
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%%%repulsive gravitational effects of global monopoles
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\end{thebibliography}


\newpage
\begin{figure}
\centering
\epsfysize=10cm
\mbox{\epsffile{bbh1.eps}}
\caption{The value of the mass $\mu_{\infty}/\alpha^2$ is given for the AdS monopoles
($\gamma < 0$) as function of $-\gamma$. We have chosen $\alpha=0.1$, $\lambda=1.0$. }
\end{figure}
\newpage
\begin{figure}
\centering
\epsfysize=10cm
\mbox{\epsffile{bbh2.eps}}
\caption{The mass function $\mu(x)$ is shown as function of $x$ for
$\gamma=-10$, $-3$ and $0.073$. We have chosen $\alpha=0.0355$, $\lambda=1.0$. }
\end{figure}
\end{document}











