%to switch ON running title
%\markboth{B. Feng, X. Gong \& X. Wang}
%{Reheating Energy Scale on Primordial Spectrum and CMB} %condensed title

\wstoc{Sample Chapter: Assessing the Effects of the Uncertainty in
Reheating Energy Scale on Primordial Spectrum and CMB}{B. Feng, X.
Gong and X. Wang}

\title{SAMPLE CHAPTER: ASSESSING THE EFFECTS OF THE UNCERTAINTY IN
REHEATING ENERGY SCALE ON PRIMORDIAL SPECTRUM
AND CMB\footnote{Supported by a FPPI grant.}}

\author{BO FENG, XIAOBO GONG\footnote{Also at
Department of Physics, Peking University, Beijing 100871, P. R. China.} and
XIULIAN WANG$^\ddag$}
\aindx{Feng, B.}\aindx{Gong, X.}\aindx{Wang, X.}

\address{Institute of Physics, Chinese Academy of Science,
P. O. Box 98-4,\\
Beijing 100039, P. R. China\\
\email{$^\ddag\!$E-mail: wang@mail.ihep.ac.cn}}

\begin{abstract}
We search for regular tachyon kinks in an extended model,
which includes the tachyon action recently proposed to describe the
tachyon field. The extended model that we propose adds a new contribution
to the tachyon action, which allows on to obtain stable tachyon kinks
of regular profile, which may appropriately lead to the singular kink
found by Sen sometime ago. Also, under specific conditions we may
find periodic array of kink--antikink configurations.
\end{abstract}

\bodymatter

\section{Introduction}
String theory is perhaps the most plausible candidate for a
relativistic theory to describe the electromagnetic, weak, strong, and
gravitational interactions altogether. It engenders a very rich
structure which includes stable or BPS and unstable or non-BPS
branes.\footnote{The theory of electromagnetic.}

In the last case, instabilities of non-BPS branes are marked
by the presence of tachyon fields, whose dynamics are directly related
to the process in which non-BPS branes decay into BPS branes --- see
e.g.~Refs.~\refcite{t1}--\refcite{t8}.

These recent investigations on tachyons have suggested that the
tachyon dynamics is described by the action
\begin{equation}
S=-\int d^{p+1}xV(T)
\sqrt{1+\eta^{\mu\nu}\partial_\mu T\partial_\nu T}\,,
\label{a}
\end{equation}
where $T=T({\bf x},t)$ is the tachyon field, real, and
$x^\mu=(t,{\bf x})$ is the position vector. Also, the Minkowski metric
has signature $(-,+,+,+,\ldots)$, and $V(T)$ is the potential, which
is non-negative, obeys $V(T\to\pm\infty)\to0$ and attains its global
maximum at $T=0.$ We consider $0\leq V(T)\leq 1$.

In the present work, in the above action we modify the Lagrange
density to the new form
\begin{equation}
\tilde{\mathcal{L}}=-V(T)
\left(\sqrt{1+\eta^{\mu\nu}\partial_\mu T\partial_\nu T}
-\frac1{\sqrt{1+r^2F^2}}\right)\,,
\label{ma}
\end{equation}
where $r$ is a parameter, real and positive, and
$F=F(T,\eta^{\mu\nu}\partial_\mu T\partial_\nu T)$. The real parameter
$r$ is introduced to control the way one goes beyond former
investigations: for $F$ that behaves appropriately, the limit
$r\to\infty$ gives the former action (\ref{a}) and leads to the
problem investigated by Sen in Ref.~\refcite{S}; for $r\gg 1$ one may
get to the recent investigation,\cite{css} in which one modifies the
action (\ref{a}) by adding a term which depends on the derivative of
the tachyon field.\footnote{The action which modifies the derivative
of the fields.} In Ref.~\refcite{css}, the term included in the
action is controlled by a very small parameter, and admits a
perturbative investigation which very nicely leads to regular kink,
and gives the singular kink of Ref.~\refcite{S} in the appropriate
limit.

\section{Study of Some Special Orbits}
In the loop space formalism for gauge theories the fields depend more
on the paths rather than on spacetime points. The fundamental quantity
that arises from this path-dependent approach is the non-integrable
phase factor that represents a gauge field more adequately than the
field strength does.

\subsection{The curves and angle}
The spin connection $\Gamma_\phi$ corresponds to closed curves with
the time and the $\theta$ coordinate constants. The spin connection
$\Gamma_\theta$ corresponds to closed curves with time and azimuthal
angle constants.

\subsubsection{Single site events data}
The peak detection procedure is carried out within an energy interval
that seems to not contain (according to the left-hand side) lines
other than the one at $Q_{\beta\beta}$. This interval is broad enough
(about $\pm 5$ standard deviations of the Gaussian line) for a
meaningful analysis.

\section{Generalities \label{geni}}
We use the modified model to examine the energy corresponding
to static configuration $T=T({\bf x})$. We get
\begin{equation}
E=\int d^px\,V(T)\left(\sqrt{1+\partial_iT\partial^iT}
-\frac1{\sqrt{1+r^2F^2}}\right)\,. \label{2.1}
\end{equation}
We investigate stability of static solutions with the Derrick--Hobart
theorem\cite{H} --- see also Ref.~\refcite{css}. We
change $T({\bf x})\to T^\lambda({\bf x})=T(\lambda{\bf x})$ to get
\begin{equation}
\frac{p+(p-1)\partial_i T\partial^i T}
{\sqrt{1+\partial_i T\partial^i T}}=\frac{p}{\sqrt{1+r^2F^2}}\,.
\label{bps}
\end{equation}
The case of a single spatial dimension is special. Here the above
condition reduces to the simpler form $T^{\prime 2}(x)=r^2 F^2(T)$, or
better
\[
T^{\prime}(x)=\pm r F(T)\,,
\]
where the prime stands for derivative with respect to $x$. We note
that the above Eq.~(\ref{bps}) reproduces first-order differential
equations that appear in the bosonic sector of supersymmetric field
theory described a single chiral superfield; see,
e.g.~Ref.~\refcite{bb00}.

Let us now investigate the equation of motion that follows from the
modified action. It can be written in the form
\begin{eqnarray}
&&\frac1{\sqrt{1+\partial_\mu T\partial^\mu T}}
\bigg(\frac{dV}{dT}-V\partial_\mu\partial^\mu T
+V\frac{\partial^\mu T\partial^\nu T\partial_\mu\partial_\nu T}
{1+\partial_\mu T\partial^\mu T}\bigg)\nonumber\\[5pt]
&&\qquad =\frac1{\sqrt{1+r^2F^2}}
\bigg(\frac{dV}{dT}-\frac{r^2 FV}{1+r^2F^2}\frac{dF}{dT}\bigg)\,.
\label{2.4}
\end{eqnarray}
In the case $p=1$, the static field obeys
\begin{equation}
\frac1{T^\prime}\frac{d}{dx}
\bigg(\frac{V}{\sqrt{1+T^{\prime2}}}
-\frac{V}{\sqrt{1+r^2F^2}}\bigg)=0\,.
\label{em1}
\end{equation}
This equation is solved by $T^\prime\to\infty$, which are stable
solutions of Eq.~(\ref{bps}) in the limit $r\to\infty$, which leads to
the case first investigated by Sen in Ref.~\refcite{S}, giving rise to
the stable but singular tachyon kinks
\begin{equation}
T^{\pm}_S(x)=\begin{cases}
\pm\infty \qquad& {\rm for}\ x>0\,,\cr
0 \qquad & {\rm for}\ x=0\,,\cr
\mp\infty \qquad & {\rm for} \ x<0\,.\end{cases}
\label{sk}
\end{equation}
There are other solutions, which obey
\begin{equation}
T^{\prime 2}=\frac{1}{\left(\frac{V_0}{V}+\frac{1}
{\sqrt{1+r^2F^2}}\right)^2}-1\,,
\label{s}
\end{equation}
where $V_0$ is a real constant, constrained to obey
\begin{equation}
0\leq{\frac1{\sqrt{1+r^2F^2}}+\frac{V_0}{V}}\leq 1\,.
\label{cons}
\end{equation}
The case $V_0=0$ reproduces the former Eq.~(\ref{bps}), leading to the
conclusion that the solutions of the equation of motion are stable for
$V_0=0$.

Theorem~\ref{theg} in Sec.~\ref{geni} will discuss the geometric
property of the averaged Hall conductance
$\overline{\sigma_{xy}(\mbox{\boldmath $\phi$})}$ as a geometric
invariant for non-Abelian gauge fields on the gauge torus
${\mathcal T}_{\rm g}$.

\begin{theorem}
\label{theg}
Let $\mathcal L$ be a Banach space with a norm $|| \cdot ||_{{\mathcal
L}}$ such that ${\mathcal L} \subset {\mathcal H}$ is a dense
subset of a Hilbert space $\mathcal H,$ and for any
$f\in {\mathcal L}$.
\end{theorem}

Since both $U^{(N)}(t,s)$ and $U_0^{(N)}(t,s)$ are bounded, one has
the following Lemma~\ref{ufi}:

\begin{lemma}
\label{ufi}
In the strong sense\/$,$ $U^{(N)}(t,s)\rightarrow U_0^{(N)}(t,s)$ as
$F\rightarrow 0,$ uniformly in $t,s$ in any finite interval.
\end{lemma}

\section{Establish Equilibrium Mergers}
Regular clusters may have gone some time since their last
mergers. However, since the timescale to re-establish equilibrium
after a major merger event is of the order of several
Gyr, and since projection effects can hide
evidence for mergers, the fraction of clusters that appear regular
most likely provides an underestimate of the fraction that have
recently undergone mergers.

\begin{proposition}
\label{pro-n}
There exist orthonormal vectors
${\hat\Phi}_{0,\mu}^{(N)}(\mbox{\boldmath $\phi$}),$ $\mu=1,2,\ldots,q$
such that the sector of the $($quasi\/$)$ degenerate ground state is
spanned by the vectors.
\end{proposition}

In Proposition~\ref{ve} the vector $E=0$.  It is easy to see that $b=1$
is actually a minimum and that these integrals are bounded by a factor
$\exp{-c N}$.

\begin{proposition}
\label{ve}
The ${\hat\Phi}_{0,\mu}^{(N)}(\mbox{\boldmath $\phi$}),$
$\mu=1,2,\ldots,q,$ and
that all the vectors ${E=0+\hat \Phi}_{0,\mu}^{(N)}(\mbox{\boldmath
$\phi$}),$ $\mu=1,2,\ldots,q,$ are infinitely differentiable with
respect to the gauge parameters $\mbox{\boldmath $\phi$}$ on the gauge
torus ${\mathcal T}_{\rm g}$.
\end{proposition}

Typical X-ray luminosities of clusters range between about $10^{44}$
and $10^{45}$~erg~s$^{-1}$.  The X-ray emission of the ICM is due to
thermal bremsstrahlung for $T>3\cdot 10^7~{\rm K}$ while line cooling
becomes very important at lower temperatures.

\begin{corollary}
\label{corla}
There exists a recursively closed, strongly polynomial-time approximable
set $S=R$ is a nonrecurve real number between $0$ and $1$.
\end{corollary}

We construct a simple fractal curve whose dimension is equal to $3/2$.
An important group of fractals is generated by dynamical systems.

Furthermore, since $V_0$ must obey the constraint (\ref{cons}),
we can also have two other distinct possibilities: one for $V_0$
positive, representing the case of negative pressure, and the other
for $V_0$ negative, representing the case of positive pressure. We
shall show below that in the case of nonvanishing $V_0$, we must
compactify the real line in order to have finite energy, and this will
give rise to periodic tachyon kinks. Thus, in the modified model we
shall find stable and regular tachyon kinks in an environment with
vanishing pressure. And also, we shall find periodic kink--antikink
array in another environment, with negative pressure. We notice from
the above Eq.~(\ref{cons}) that the limit $r\to\infty$, which leads to
the standard tachyon action, gives rise to the case of negative
pressure, and nothing more --- see Refs.~\refcite{kkl} and
\refcite{bms}.

\section{Specific Models}
We investigate the case with $V_0=0$, which corresponds to vanishing
pressure.  This case gives rise to models that support stable tachyon
kinks of regular profile. The energy depends on the tachyon potential,
$V(T)$. Thus, we choose the tachyon potential such that $E_S=1$. In
this case the energy of the tachyon configurations is restricted to be
in the interval $0<E<1$. We may choose $V_{I}(T)=\exp(-\pi T^2)$,
$V_{IIa}(T)=1/2\cosh^2(T)$, $V_{IIb}(T)=1/\pi\cosh(T)$, and
$V_{III}(T)=1/\pi(1+T^2)$ which identify type-I, type-II and type-III
models, all leading to unit energy. We will study these models to have
a better understanding of the role of the potential for the modified
tachyon action that we propose in (\ref{ma}). Specific models
involving the choices $F(T)=1$ and $F(T)=1/\cosh(T)$ will be
investigated below.

\subsection{Type-I models}
We consider the case $F(T)=1$, which implies that $T^\prime=\pm r$,
giving rise to the solutions $T_\pm(x)=\pm r x$, which lead to the
singular kinks (\ref{sk}) in the limit $r\to\infty$.  This case
reproduces the solutions of Ref.~\refcite{css}. The energy
corresponding to these solutions has the form
\begin{equation}
E_I(r)=\frac{r^2}{\sqrt{1+r^2}}\int_{-\infty}^{\infty} dx\,V(T)\,.
\label{3.1}
\end{equation}
Thus, for $V(T)=\exp(-\pi T^2)$ and for $T_\pm=\pm r x$ we get
\begin{equation}
E_I(r)=\frac{r}{\sqrt{1+r^2}}\,.
\label{e1}
\end{equation}
We use this result to get $E(r=0)=0$. This is interesting, since the
limit $r\to0$ leads to a constant tachyon configuration. Our model
gives vanishing energy for trivial constant tachyon configurations
at $r=0$, and unit energy for the singular kink (\ref{sk}) at the
limit $r\to\infty$. The stable kink solutions $T^{\pm}(x)=\pm r x$
are parametrized by $r$, and have energy as in Eq.~(\ref{e1}), which
is well-defined in the entire interval $0\leq r\leq1$. In
Fig.~\ref{fig4.1} we plot $E_I(r)$ in the whole interval
$r\in[0,\infty)$.

\begin{figure}[ht]
\centerline{\epsfxsize=7.5cm\epsfbox{pr-f1.eps}} \caption{The energy
as a function of the real parameter $r$. The thick line corresponds
to $F(T)=1$, for all the models. The dashed, dash-dotted, and dotted
lines correspond to $F(T)=1/\cosh(T)$ for the type-I, type-IIa, and
type-IIb models, respectively. The thin line corresponds to the
type-III model, for $F(T)=1/\cosh(T)$.} \label{fig4.1}
\end{figure}

We also consider the case of $F(T)=1/\cosh(T)$ to get
\begin{equation}
T^\prime=\pm \frac{r}{\cosh(T)}\,.
\label{rk}
\end{equation}
This equation was already solved in Ref.~\refcite{b}. The solutions
are $T(x)=\pm\,{\rm arcsinh}(r\,x)$, and we realize that the
singular kink (\ref{sk}) is now very naturally recovered in the
limit $r\to\infty$.  The energy of the regular kinks can be written
as
\begin{equation}
\tilde E_I(r)=r\int_{-\infty}^{\infty}dx
\frac{e^{-\pi\,{\rm arcsinh}^2(x)}}{(1+x^2)\sqrt{1+\frac{r^2}{1+x^2}}}
\label{3.4}
\end{equation}
and depends on the parameter $r$. It vanishes for $r=0$, and
converges to unit in the limit $r\to\infty$. In Fig.~\ref{fig4.1} we
plot the energy density for $r\in[0,\infty)$. We note that $\tilde
E_I(r)$ is very close to $E_I(r)$, indicating that the choice of
$F(T)$ determine no qualitative behavior.

\begin{sidewaysfigure}[pt]
\centerline{\psfig{file=pr-f2.eps,width=15cm}}
\caption{Erosion depth at the moment of maximum velocity and around
flow reversal (All dimensions are in mm). $T =3$~s, $U_{\max}
=1.4$~m/s, $D_{50} =0.3$~mm.\label{fig2} }
\end{sidewaysfigure}

To make the difference between the unmovable bed and mobile layer
become clearer, the brightness values are enlarged 20~times during our
image analysis. Figure~\ref{fig2} shows typical visualized images for
determining erosion depth under the experimental condition $T =3$~s,
$U_{\max} =1.4$~m/s, $D_{50} =0.3$~mm at the moment of maximum
velocity and around flow reversal (zero velocity).

\subsection{Type-II models}
We first consider type-IIa models, and use $F(T)=1$. The investigation
is similar to the former case. The kink solutions and the energy give
the very same results already obtained in the corresponding type-I
model. Thus, we consider the next case: $F(T)=1/\cosh(T)$. Here the
energy changes to
\begin{equation}
\tilde E_{IIa}(r)=\frac{r}{2}\int_{-\infty}^{\infty}dx
\frac{1}{(1+x^2)^2}\frac{1}{\sqrt{1+\frac{r^2}{1+x^2}}}\,.
\label{3.5}
\end{equation}
It vanishes for $r=0$, and converges to unit in the limit
$r\to\infty$. In Fig.~\ref{fig4.1} we plot the energy $\tilde
E_{IIa}(r)$ in the entire interval $r\in[0,\infty)$. We note that
$\tilde E_{IIa}(r)$ is very close to $\tilde E_{I}(r)$, suggesting
that the specific choice of the tachyonic potential seems to
determine no qualitative behavior.

An imaging experiment with OH$^+$ in CRYRING led to discovery of a
very small kinetic energy release when the electron energy was
sufficient for the ${\rm O}(^3P) + {\rm H}(n=2)$ dissociation limit
(see Table~\ref{table1}) to become energetically allowed.

\begin{sidewaystable}
\tbl{Sample statistics for A-share premia for Shanghai companies that
issued A- and B-shares before April 1994 (sample period: April 1,
1994--October 31, 1998). \label{table1}}
{\begin{tabular}{@{}ccccccccc@{}}
\hline
\multicolumn{9}{c}{}\\[-2ex]
Year &Taiwan &Dow Jones &Nikkei &South &London's &Hong Seng &Thailand's
&Singapore's\\
{} &Index &Index &Index &Korea's &FTSE &Index &Set Index &Strait Times\\
{} &{} &{} &{} &Kospi &Index &{} &{} &{}\\
{} &{} &{} &{} &Index &{} &{} &{} &{}\\[0.25ex]
\hline
\multicolumn{9}{c}{}\\[-2ex]
1988 &5,119.11 &\phantom{0}2,168.57 &30,159.00 &907.20 &1,455.30
&\phantom{0}2,687.44 &386.73 &1,038.62\\
1989 &9,624.18 &\phantom{0}2,753.20 &38,915.87 &909.72 &1,916.60
&\phantom{0}2,836.57 &879.19 &1,481.33\\
1990 &4,530.16 &\phantom{0}2,633.70 &23,848.71 &696.11 &1,673.40
&\phantom{0}3,243.30 &612.86 &1,154.48\\
1991 &4,600.67 &\phantom{0}3,168.83 &22,983.77 &610.92 &1,891.30
&\phantom{0}4,297.33 &711.36 &1,490.70\\
1992 &3,377.06 &\phantom{0}3,301.11 &16,924.95 &678.44 &2,185.20
&\phantom{0}5,512.39 &893.42 &1,524.40\\
1993 &6,070.56 &\phantom{0}3,754.09 &17,417.24 &866.18 &2,559.50
&11,888.39 &1,565.12\phantom{0.} &2,425.68\\
1994 &7,124.66 &\phantom{0}3,834.44 &19,723.06 &1,027.37\phantom{0.}
&3,065.50 &\phantom{0}8,191.04 &1,360.09\phantom{0.} &2,239.56\\
1995 &5,173.73 &\phantom{0}5,117.12 &19,868.15 &882.94 &3,689.30
&10,073.39 &1,280.81\phantom{0.} &2,266.54\\
1996 &6,933.94 &\phantom{0}6,448.27 &19,361.35 &651.22 &4,118.50
&13,451.45 &831.57 &2,216.79\\
1997 &8,187.27 &\phantom{0}7,905.25 &15,258.74 &376.31 &5,135.50
&10,722.76 &372.69 &1,529.84\\
1998 &6,418.43 &\phantom{0}9,181.43 &13,842.17 &562.46 &5,882.60
&10,048.58 &355.81 &1,392.73\\
1999 &8,448.84 &11,497.12 &18,934.34 &1,028.07\phantom{0.} &6,930.20
&16,962.10 &481.92 &2,479.58\\
2000 &5,544.18 &10,971.14 &14,539.60 &514.48 &6,438.40 &14,895.34
&271.84 &1,976.54\\
1 &9,744.89 &10,940.53 &19,539.70 &943.88 &6,268.50 &15,532.34 &477.57
&2,230.28\\
2 &9,435.94 &10,128.31 &19,959.52 &828.38 &6,232.60 &17,169.44 &374.32
&2,120.50\\
3 &9,854.95 &10,921.92 &20,337.32 &860.94 &6,540.20 &17,406.54 &400.32
&2,132.59\\
4 &8,777.35 &10,733.91 &17,973.70 &725.39 &6,327.40 &15,519.30 &390.40
&2,164.11\\
5 &8,939.52 &10,522.33 &16,332.45 &731.88 &6,359.30 &14,713.86 &323.29
&1,795.13\\
6 &8,265.09 &10,447.89 &17,411.05 &821.22 &6,312.70 &16,155.78 &325.69
&2,037.97\\
7 &8,114.92 &10,521.98 &15,727.49 &705.97 &6,365.30 &16,840.98 &284.67
&2,051.21\\
\hline
\end{tabular}}
\begin{tabnote}
The state-budget funds as a dominant financial source of investment
of SOEs, regardless of investment decisions being made by governments
or enterprises. The retained profits and non-bank debts are also used by
SOEs to finance their investment and operation.
\end{tabnote}
\end{sidewaystable}

\section{Ending Comments}
In this work we have modified the tachyon action (\ref{a}) by changing
the square root contribution as in (\ref{ma}). We have found stable,
finite energy tachyon kinks in the case of vanishing pressure in
several different models. We have also found a network of
kink--antikink configurations in the case of negative
pressure. Although the scenario one finds in the case of negative
pressure is similar to other extensions that appeared recently, the
case of vanishing pressure is new, and it supports regular, stable and
finite energy tachyon kinks which nicely lend themselves to Sen's
singular solutions in the appropriate limit.

\begin{table}
\tbl{Number of spectra with a calculated confidence limit above a
given value.
\label{table2}
\vspace*{-2pt}}
{\tabcolsep8pt
$$\begin{tabular}{@{}lcccc@{}}
\hline
&&&&\\[-5pt]
C.L. &90.0\% &95.0\% &99.0\% &99.9\%\\[2pt]
Expected &$100\pm 31$ &$50\pm 7$ &$10\pm 3$ &$1\pm 1$\\[2pt]
Found &96 &42 &12 &0\\[4pt]
\hline
\end{tabular}$$}
\end{table}

Table~\ref{table2} presents the results: the first
column contains the corresponding confidence limit $c$ (precisely the
parameter $K_E$ defined earlier), the second column contains the
{\it expected\/} number of spectra indicating the existence of a
line with a confidence limit above the value $c$.

\section*{Acknowledgments}
We thank F. A. Brito for discussions, and CAPES, CNPq, PROCAD and
PRONEX for partial support.

\appendix{Bach-flat Spaces \label{app-a}}   %S1
An interesting question is the characterization of those
spaces (Appendix~\ref{app-a}) for which there is no obstruction.
In $d=2$ dimensions Eq.~(\ref{apd1}), this is equivalent to find
Bach-flat spaces
\begin{equation}
ds^2=dt^2 - a^2_i
\label{apd1}
\end{equation}
with Hubble paramers, $h_i-a^2_i$ and anisotropy parameters
$m_i=h_i-h/3$
\begin{equation}
3d^2(pr)/dt^2=8pr^3_4c\,.
\label{eqp2}
\end{equation}

\subappendix{The Bianchi identities\label{app-b}}   %S2
There are two variables ($\Lambda$ and $r_c$) in Eq.~(\ref{app4}).
$r_c$ can be regarded as the total energy within the cosmological
horizon, (Appendix~\ref{app-b}) on a phenomenological level,

\noindent
\begin{equation}
\frac{dE}{dt} \sim \frac{\Delta E}{\Delta t}
\sim G^{-1}\,.
\end{equation}

When $r_c$ is fixed, the entropy varies with $\Lambda$. That is to
say, there are fluctuations in entropy due to different $\Lambda$.

\eject

\subsubappendix{A fundamental principle\label{app-3}}   %S3
There is a cosmological horizon $r_c$ determined by
$f(r_c, Q,\Lambda)=0$
\begin{eqnarray}
\kappa_c &=& -\frac{1}{2} f_r^\prime (r_c, Q, \Lambda)\nonumber\\[5pt]
&=& r_c^{-1} \left (\frac{2\Lambda r_c^2}{3} -1\right )\,,
\label{app4}
\end{eqnarray}
where the second equality, (Appendix~\ref{app-3})
is derived from $f (r_c, Q, \Lambda) =0$.

\appendix{Higher Order Equations of Motion\label{app-nn}}  %S1
In this Appendix~\ref{app-nn} we present the equations of motion
from order $p^2$ to order $p^4$
\begin{equation}
3(d-6)h^3+(3h-3h^2+h_k)\,.
\end{equation}

The studies to date have considered bubbles in
idealized cluster atmospheres, clusters are known to undergo mergers
which dramatically affect their structure.

\begin{thebibliography}{00}

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\bibitem{t2} P. Horava, {\it Adv. Theor. Math. Phys.} {\bf2}, 1373 (1999).

\bibitem{t3} A. Sen, {\it JHEP} {\bf10}, 008 (1999).

\bibitem{t4} M. Garousi, {\it Nucl. Phys.} {\bf B584}, 284 (2000).

\bibitem{t5} E. Bergshoeff {\it et al.}, {\it JHEP} {\bf05}, 009 (2000).

\bibitem{t6} M. Garousi, {\it Nucl. Phys.} {\bf B647}, 117 (2002).

\bibitem{t7} D. Kutasov and V. Niarchos, {\it Nucl. Phys.} {\bf B666}, 56 (2003).

\bibitem{t8} A. Sen, {\it Phys. Rev.} {\bf D68}, 106003 (2003).

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