What did we learn from studying \\acoustic black holes ?
%\footnotesize Renaud Parentani\footnote{ %\footnotesize %Typeset names in 8 pt roman, uppercase. Use the footnote to indicate the %present or permanent address of the author. %\address Laboratoire de Math\'ematique et Physique Th\'eorique, CNRS-UMR 6083, %Universit\'e de Tours, Parc de Grammont
\footnotesize Universit\'e de Tours, Parc de Grammont, 37200 Tours, France.}
The study of acoustic black holes has been undertaken to provide new insights about the role of high frequencies in black hole evaporation. Because of the infinite gravitational redshift from the event horizon, Hawking quanta emerge from configurations which possessed ultra high (trans-Planckian) frequencies. Therefore Hawking radiation cannot be derived within the framework of a low energy effective theory; and in all derivations there are some assumptions concerning Planck scale physics. The analogy with condensed matter physics was thus introduced to see if the asymptotic properties of the Hawking phonons emitted by an acoustic black hole, namely stationarity and thermality, are sensitive to the high frequency physics which stems from the granular character of matter and which is governed by a non-linear dispersion relation. In 1995 Unruh showed that they are not sensitive in this respect, in spite of the fact that phonon propagation near the (acoustic) horizon drastically differs from that of photons. In 2000 the same analogy was used to establish the robustness of the spectrum of primordial density fluctuations in inflationary models. This analogy is currently stimulating research for experimenting Hawking radiation. Finally it could also be a useful guide for %indicating the appropriate techniques for going beyond the semi-classical description of black hole evaporation. % Perhaps the new question is: %instaed of asking if Hawking radiation is robust, one now ask %whether the linear dispersion relation of photons is robust against %radiative corrections. %pave the way of quantum gravity beyond the %semi-classical description of black hole evaporation. %The abstract should summarize the context, content %and conclusions of the paper in less than 200 words. It should %not contain any references or displayed equations. Typeset the %abstract in 8 pt roman with baselineskip of 10 pt, making %an indentation of 1.5 pica on the left and right margins. %\vskip .1 truecm %\keywords{Hawking radiation; acoustic black hole; quantum gravity.}
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\begin{document}

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\title{What did we learn from studying \\acoustic black holes ?}
%INSTRUCTIONS FOR TYPESETTING MANUSCRIPTS\\
%USING \TeX\ OR \LaTeX\footnote{For the title, try not to use more than 3 lines.
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\author{
%\footnotesize
Renaud Parentani\footnote{
%\footnotesize
%Typeset names in 8 pt roman, uppercase. Use the footnote to indicate the
%present or permanent address of the author.
%\address
Laboratoire de Math\'ematique et Physique Th\'eorique, CNRS-UMR 6083, 
%Universit\'e de Tours, Parc de Grammont\\ \footnotesize
Universit\'e de Tours, Parc de Grammont,
37200 Tours, France.}}
%\footnote{State completely without abbreviations, the
%affiliation and mailing address, including country. Typeset in 8 pt
%italic.}


%\author{SECOND AUTHOR}
%\address{Group, Laboratory, Address\\
%City, State ZIP/Zone, Country}

\maketitle                            
%\pub{Received (Day Month Year)}
%{Revised (Day Month Year)}

\begin{abstract}
The study of acoustic black holes has been undertaken to provide new 
insights about the role of high frequencies in black hole evaporation.
Because of the infinite gravitational redshift from the event horizon,
Hawking quanta emerge from configurations which possessed ultra high 
(trans-Planckian) frequencies. Therefore Hawking radiation cannot be 
derived within the framework of a low energy effective theory; 
and in all derivations there are some assumptions concerning Planck scale
physics. The analogy with condensed matter physics was thus introduced
to see if the asymptotic properties of the Hawking phonons emitted by 
an acoustic black hole, namely stationarity and thermality, 
are sensitive to the high frequency physics 
which stems from the granular character of matter
and which is governed by a non-linear dispersion relation. 
In 1995 Unruh showed that they are not sensitive in this respect, 
in spite of the fact that phonon propagation
near the (acoustic) horizon drastically differs from that of photons. 
In 2000 the same analogy was used to establish the robustness of 
the spectrum of primordial density fluctuations in inflationary models. 
This analogy is currently stimulating research for experimenting 
Hawking radiation. Finally it could also be a useful guide for 
%indicating the appropriate techniques for 
going beyond the semi-classical description of black hole evaporation.
% Perhaps the new question is:
%instaed of asking if Hawking radiation is robust, one now ask 
%whether the linear dispersion relation of photons is robust against 
%radiative corrections.

%pave the way of quantum gravity beyond the 
%semi-classical description of black hole evaporation. 



%The abstract should summarize the context, content
%and conclusions of the paper in less than 200 words. It should
%not contain any references or displayed equations. Typeset the
%abstract in 8 pt roman with baselineskip of 10 pt, making
%an indentation of 1.5 pica on the left and right margins.
%\vskip .1 truecm
%\keywords{Hawking radiation; acoustic black hole; quantum gravity.}
\end{abstract}

%\newpage

%\section*
%{The motivation for the analogy}
%\section{The motivation for the analogy}	%) A SECTION HEADING
\subsection*{The motivation for the analogy}

The semi-classical description of black hole evaporation
is based on the hypothesis that
the dynamics of the system (black hole + radiation) 
can be well approximated by treating the gravitational field classically. 
In this case the quantized radiation field $\phi$
obeys the d'Alembert equation 
\begin{equation}
%\box \phi = 
{1 \over \sqrt{-g} }\partial_\mu   \sqrt{-g}g^{\mu \nu }\partial_\nu
 \phi = 0\, ,
\end{equation}
where $g^{\mu \nu }$ is the metric of the collapsing body.  
%When adopting this scheme, 
Following Hawking\cite{hawk},
one obtains that an isolated black hole formed by gravitational
collapse radiates away its mass by emitting a thermal flux 
with a temperature given by $T= \kappa / 2\pi$ where $\kappa$ is 
its surface gravity.

%At first sight, t
The hypothesis of having treated $g^{\mu \nu }$
%the gravitational field 
classically finds support in the fact that $\langle T_{\mu \nu } \rangle$,
the expectation value of the energy-momentum tensor of $\phi$,
%the radiation field,
is regular. 
%This guarantees that the geometry will stay regular:
% during the whole evaporation process:
%will proceed adiabatically with a temperature fixed by the surface gravity 
%of the residual black hole. 
By solving Einstein equations driven by $\langle T_{\mu \nu } \rangle$,
one then finds 
%indeed
that the evaporation stays adiabatic as long as the residual mass 
of the hole is much larger than the Planck mass, see e.g. \cite{GO}. 

However, the validity of the semi-classical description becomes dubious when 
inquiring about the origin of Hawking quanta.
%\footnote{Another 
%dubious prediction is that Hawking radiation is strictly 
%incoherent (because 
%%. This follows from the fact that 
%the ``partners'' of Hawking quanta are all on the 
%other side of the horizon, inaccessible to outside observers). 
%This leads to the famous ``information loss paradox''.
%Let us simply say here that in the context of super string theory, 
%%no information loss was found (for near extremal holes), 
%and that we shall not further discuss these aspects
%since the analogy with acoustic black holes does not seem appropriate 
%(to the author) to confront them in relevant terms.}
Eq. (1) predicts that they emerge from configurations 
localized extremely close to the horizon and characterized by 
ultra-high frequencies (when measured by an infalling 
observer). This is specific and
%This feature is specific 
intrinsic to horizon quantum physics 
as it follows from free propagation near an event horizon 
and from second quantization. 
%In view of the importance of this observation,
%let us pay some attention on it.
%the nature of the question.
%It should be clear that the appeal to ultra-high frequencies
%which is specific of quantum horizon physics as it follows from
%propagation near an event horizon, and from second quantization.
The infinite gravitational redshift from the event horizon 
%conditions the structure of the outgoing null geodesics,
%and this in turn determines 
%the properties of outgoing configurations of the radiation field, 
%the solutions of Eq. (1).
% which propagate along these geodesics.
%These are clearly seen by using a regular coordinate system, 
%e.g. the advanced Eddington-Finkelstein $v,r$. 
% since they are the characteristics of the d'Alembert equation. 
%Classically, it 
implies that during the last stage of the 
collapse the radiation emitted by the body is redshifted according to
\begin{equation}
\lambda(t) \propto \Omega \ e^{-\kappa t} \, ,
\end{equation}
where $\Omega$ is the emitted frequency 
and $\lambda$ that received on ${\cal J}^+$.
% the characteristic time being $\kappa$.
%which governs this exponential law is 
%, the surface gravity of the hole. 
%In classical terms 
Classically, there is thus no appeal to high frequencies
since the initial frequencies $\Omega$ are bounded.
%%%%%Quantum mechanically one gets the opposite result:
%The appeal to high frequencies follows from quantum mechanics:
%It arises from the fact that 
%vacuum fluctuations
%contain, as usual, all frequencies, but more specifically that 
On the contrary, Hawking quanta emerge from vacuum fluctuations 
with very high $\Omega$
%frequencies 
precisely because these are redshifted as in Eq. (2). 
%%%%%A closer analysis confirms that 
Indeed, the steady character of Hawking radiation 
is achieved at time $t$ by referring to vacuum frequencies which grow 
according to $\Omega(t) \propto \kappa e^{\kappa t}$,
see Eq. (3.34) in \cite{GO}. 
% non-vanishing character of the Bogoliubov coefficients
%directly follows from this exponential redshift.
% of vacuum frequencies.
%the non-trivial frequency mixing engendered by Eq. (2). 
%between modes specified before the collapse and modes specified
%on ${\cal J}^+$. Technically, this mixing is expressed by the non-vanishing
%character of the Bogoliubov coefficients. 
%The consequence of the exponential redshift of vacuum frequencies is
%Hawking radiation: pairs of quanta are spontaneously emitted, 
%one of each member on each side of the horizon.
%The spectrum of the outside quanta is thermal 
%The surprising fact is that 
%a {\it constant} production rate of asymptotic quanta.
%with a temperature given 
%by $T= \kappa / 2\pi$ and the production rate is constant.
%himportantly for our purposes, 
%the production process is {\it stationary}. 
%By a closer examination, see e.g. Eq. (3.34) in \cite{GO}, 
%one can show that the stationarity of the flux at time $t$ is achieved by 
%refering to vacuum configurations with frequencies which grow 
%according Eq.(2), i.e. $\Omega(t) \propto \kappa e^{+\kappa t}$. 

Hawking radiation therefore cannot be derived within the framework of a 
low energy effective theory. Moreover since Planck scale physics
%the ultra-violet behaviour of quantum gravity theory 
is unknown,
% (since we do not know how to deal with 
%quantum gravitational interactions), 
it is important to determine if/why the asymptotic properties 
of Hawking radiation could be insensitive 
%to this unknown physics.
% influence the asymptotic properties of Hawking radiation.
%and that these grow This is nicely illustrated by the analysis
%of \cite{ted93}
%wherein it is shown that one {\it can} obtain Hawking radiation 
%without appeal to high frequencies
%but only by introducing a {\it time-like} boundary at some
%distance from the horizon from which one starts the forward evolution,
%or, equivalently, one stops the backward in time propagation 
%needed to inquire about the origin of Hawking quanta.
to it. From the outset\cite{unruh81}
%,ted91}, 
this question has been the principle 
motivation for studying acoustic black holes.
%Let us first explain the nature of the analogy.

%\section{Acoustic black holes}

\subsection*{The acoustic black hole geometry}
%The analogy}

The concept of acoustic black hole arises from 
the analogy between light propagation in a curved background 
and sound propagation in a moving fluid\cite{unruh81,visser}.
%This analogy is rather simple\cite{unruh81,visser}. 
Consider a vorticity-free flow of a barotropic fluid. The 
%%first condition means that the 
velocity field can be written as 
$v= \nabla \Psi$ and the pressure 
$P$ is related to the density $\rho$ by an 
equation of state $P=P(\rho)$. 
Consider also the equation of continuity and Euler equation. 
Then linearize these three equations about a background solution, 
 $v_B= \nabla \Psi_B $,  $P_B $,  $\rho_B$, 
and consider the equation for 
%%%%%the potential fluctuation 
$\psi=\Psi- \Psi_B $, see \cite{visser} for details.
Verify that $\psi$ 
%can be rewritten in the form of
obeys Eq. (1) 
when introducing the $4$D acoustic metric $g_{\mu \nu}^{ac}$ 
 given (up to a conformal factor) 
by $ g_{tt}^{ac}=-(c^2_s -v_B^2), \, 
g_{ti}= -v_{B\, i}, \, g_{ij}= \delta_{ij}$. The velocity of sound $c_s$
is given by $c_s^2 = dP/d\rho$.

%\subsection*{The acoustic black hole geometry}

To obtain an acoustic geometry which behaves like that of
a black hole, one must consider a flow which becomes supersonic 
without developing any shock wave. A well-known device 
(by hydrodynamicians) which realizes this is the Laval nozzle:
 a pipe wherein $\vert v_B \vert$ reaches $c_s$ at its narrowest radius. 
When considering
a flow to the left in the $x$-direction, the velocity behaves as 
$v_B(x)= - c_s + x \kappa_{ac}$ and the acoustic horizon is located at $x=0$. 
In the near horizon region, for $x \ll c_s/\kappa_{ac}$, 
 null (retarded) geodesics behave like 
\begin{equation}
x(t)=x_0 \, e^{\kappa_{ac}(t-t_0)}\,\, .
\label{neweq}
\end{equation}
Hence, for $x < 0$, right moving sound waves 
are carried away to the left by the supersonic flow.
For $x > 0$ instead, they propagate to the right and 
their frequencies are redshifted
%thereby giving rise to a redshift of phonon frequencies obeying 
according to Eq. (2) with $\kappa=\kappa_{ac}$.

Thus, if the phonon field $\psi$ is second quantized and 
if its state is regular across the horizon, the standard
treatment\cite{hawk,GO} applies. It tells us that a constant flux
of phonons at temperature $T_{ac}= \kappa_{ac}/2 \pi$
will be emitted against the flow, to the right. 
The quantization of the phonon field is physically pertinent in ${\rm He}^4$
and in Bose-Einstein condensates (BEC). The regularity of the state
of $\psi$
requires that the flow $v_B(x)$ be regular across the horizon. 
%The regularity of the state
%can be meaningfully addressed only after having taken into account 
%the dispersion-relation.
%follows from that of the flow $v_B(x)$. 

\subsection*{The usefulness of the analogy}

So far we have not learned much because 
%the analogy is too good. However, we can learn something by 
we have not yet taken into account the fact
%it is imperfect at high frequency. Indeed,
that 
%%%%%%%in condensed matter 
the dispersion relation of phonons 
is not linear when the wave length approaches the inter-atomic 
distance, $d_c$. 
Three possibilities can be obtained. Two can be modeled by 
\begin{equation}
\Omega^2  = c_s^2 k^2 [ 1 + \xi \, k^2 d_c^2 + O(k^4)] \, ,
\end{equation}
 with $ \xi = \pm 1$. $ \xi = 1$ is 
%The first case, 
realized in BEC and leads to supersonic propagation. $ \xi = - 1$
is
%, and the second,
realized  in ${\rm He}^4$ and leads to subsonic propagation.
The third case describes propagation in an absorptive medium
 such as a viscous fluid\cite{visser}. It is governed by a correction term,
the bracket in Eq. (4), given by $[1-i \vert k \vert d_c]$.
%, see \cite{visser}.
%%%%%(found in ${\rm He}^4$), 
%and the third describes propagation in an absorptive medium
% such as a viscous fluid.

When taking into account the non-linearities of Eq. (4)
%%%%%%the dispersion relation 
while deriving the production of Hawking phonons, one can {\it test} if
the asymptotic properties of the flux
are sensitive to these non-linearities.
%behavior governed by $d_c$ and $\xi$. 
In 1995 Unruh\cite{unruh81} showed (numerically and for $\xi=-1$) 
that they are not sensitive when $T_{ac} \ll c_s/ d_c$ 
and when the initial state is vacuum for $\Omega \gg T_{ac}$. 
These conclusions are now understood 
analytically, see \cite{tedriver} for a review.
%This follows from an adiabatic decoupling of the two scales. 
%In spite of this, the configurations giving rise Hawking quanta crucially 
%Nevertheless, 
In spite of this absence of asymptotic modification,
%depend on the dispersion relation governed by $d_c$ and $\xi$.
%Indeed 
when propagating backwards in time a mode (or better a wave packet), 
%to determine the origin of Hawking quanta, 
one finds that its near horizon propagation is dramatically modified
%, in a $\xi$ dependent manner, 
% in a backward in time extrapolation
when its blue-shifted frequency reaches the critical frequency 
$\Omega_c = c_s /d_c$, see \cite{bmps} for schematic representations
of the modified trajectories. 
%These conclusions have now been extended, 
%see \cite{tedriver} for a nice review. 

Being insensitive to high energy physics, 
%The lesson is that 
the asymptotic Hawking radiation is truly a low
energy $O(T)$ phenomenon.
% since it is insensitive to the high energy physics.
% which is determined by the value of $\xi$: $0,1,-1$ or $i$.
%as pointed out in citeGO, the same time (see Sect. 3.7 in \cite{GO}),
This reinforces our confidence in the low energy predictions
of the semi-classical treatment but does not take us any further
concerning its dubious high energy predictions.
%namely that Hawking quanta should emerge from vacuum 
%fluctuations with $\Omega \propto \kappa e^{+\kappa t}$.

\newpage

\subsection*{Inflationary models}

In inflationary scenarios, the primordial density fluctuations 
find their origin in the non-adiabatic evolution of the 
non-homogeneous modes of the inflaton which were in their vacuum
state at the onset of inflation.
% which results in copious pair creation. 
Given the exponentially growing redshift during inflation,
their initial frequency $\Omega$ was proportional to $e^{H \Delta t}$,
where $H$ is the Hubble constant and
$H \Delta t$ is the number of e-folds during inflation.
% they suffer from that time to present observation,
%More precisely the scale-invariant spectrum originates from
%a steady conversion of vacuum configurations whose frequency, 
%at the begining of the inflation
%grows like $\Omega \propto e^{H t}$ where $H$ is the Hubble constant
%during inflation. Therefore,
If the latter is taken to be larger than $\sim 70$ (which is the minimal 
number for inflation to work when the adiabatic era starts just below
GUT scale, near $10^{15}\,$Gev) there is an appeal to Planckian physics.
Therefore the doubts concerning Hawking radiation apply here as 
well\cite{tedriver}.
Thus the appeal to modified dispersion relations can also be 
called upon to see if the scale-invariance of the primordial fluctuations
%(Harisson-Zeldovich)
spectrum is affected by modifications of Planck 
scale physics\cite{cosm}. 

As in black hole physics, the properties of the spectrum are robust.
%not modified in all cases where Hawking properties were 
%also recovered\cite{cosm2}.
The two conditions guaranteeing this robustness are
the adiabaticity of the evolution from Planck scale to that of $H$, 
and the regularity of the initial configurations.
The adiabaticity essentially follows from scale separation, 
$M_{Planck} \gg H$, whereas the second condition
is severely constrained by self-consistency in the following
sense\cite{cosm2}. The very existence of inflation implies 
that the dominant 
term in the Friedman equation is the inflaton potential energy.
Thus the contribution of the other modes must be negligible.
% during the whole inflationary era. 
When $H \ll M_{Planck}$
and when inflation lasts more than $70$ e-folds,
% starts several e-folds before the minimum numbe, 
this condition requires that the initial state of the modes 
observed today be vacuum. 

 


\subsection*{Experiments ?}

Can we conceive and realize experiments in which Hawking
quasi-particles could be measured ? 
We used the term ``quasi-particles'' rather than phonons to
emphasize that in condensed matter there is a large variety 
of collective excitations which can propagate in effective metrics.
At present it is not clear which material is the most suitable
for our purpose.
%One should therefore carefully choose the right material
%for the properties of its collective excitations.
Another challenge is to
build a regular horizon enclosing a trapped region. 
But probably the main difficulty is to get a sufficiently high
temperature. Indeed it does not seem 
easy to get $T_{ac} > 10^{-6}$K. 

To overcome the detection of these soft quanta,
one should perhaps consider a pair of acoustic holes.
A natural feature of acoustic holes is to come in pairs: 
it suffices to reduce the velocity flow $v_B$ so as to
re-cross the sound velocity. Moreover this second crossing 
leads to a white hole. And white holes are unstable. 
This instability would show up differently
according the sign of $\xi$. For positive  $\xi$, one would obtain 
a self-amplifying flux of soft phonons, a laser effect, 
see \cite{bhlaser}. On the other hand,
 for $\xi$ negative, the ``partners'' of Hawking quanta 
would leak out of the white horizon with a frequency close to the cutoff 
$\Omega_c = c_s/ d_c$ and would thus be easier to detect. 

\subsection*{Orienting research on Quantum Gravity ?}

Even though this is clearly a delicate enterprise and one 
which might turn out to be misleading, it
%to obtain insight for orienting research on quantum gravity 
has been one of the major reasons to study acoustic holes. 
Let us take some risk and present our opinion as it stands today. 

Besides the insensitivity of asymptotic properties, 
we have also learned that for {\it all} dispersion relations
but the scale invariant one $d_c = 0$, there is no appeal to 
unbounded frequencies.
%\footnote{
[When the gradient $dv_B/dx$ vanishes 
for $x-x_{horizon} \gg d_c$, the frequency of a backwards in time 
propagated wave packet stays of the order of $\Omega_c$ 
for both $\xi= \pm 1$, without any further blueshift. One learns from this 
that the stationarity of Hawking radiation now 
results, as in the Golden Rule, by a steady increase 
of the density of modes at some fixed frequency. 
This is not what is found when there is no dispersion, 
see Eq. (3.34) in \cite{GO}.]
%$\Omega$. 
This suggests that the quantum 
gravitational interactions neglected in the semi-classical treatment
might introduce a {\bf new} scale $d_{QG}$ which could act on light 
propagation near the event horizon as $d_c$ did on phonons. 
In order words, instead of questioning  the robustness of asymptotic 
properties, we are lead to question {\bf is $\Omega=k$ robust} ? 
%More on this delicate conjecture in the last paragraphs.
%focussing of back   has been learned 

I would claim probably not. Because $\Omega=k$ follows from Eq. (1) 
which results from a linearisation procedure: when starting from
a quantum formulation of black hole evaporation, Eq. (1)
is obtained by discarding all fluctuations of $g_{\mu \nu}$ 
around the mean metric, 
a solution of the semi-classical Einstein equations.
A first order analysis shows that the effects of 
%near horizon influence of 
metric fluctuations on
%of frequency $\lambda$ and 
configurations giving rise to Hawking quanta 
grow like $1/(r-2M)$.
%$G \omega \lambda e^{\kappa t}$, where $G$ is Newton's constant. 
Therefore a taming mechanism is bound to occur. 
This has been substantiated in \cite{hawkmf}
where it is shown that vacuum energy fluctuations
engender metric fluctuations whose random properties 
induce a damping similar to that obtained
by putting $\xi = - i$ in Eq. (4).
%, see \cite{hawkmf} for more details.
That near horizon quantum gravitational interactions 
should induce dissipative effects was anticipated in \cite{GO}.

\subsection*{Acknowledgements}

I wish to thank the organizers of the {\it Journ\'ees Relativistes} 
in Dublin,
as well as those of the  {\it ESF-COSLAB} meeting in London for 
the invitation.  
I also wish to thank Ted Jacobson for many discussions. 
This work has been
supported by the NATO Grant CLG.976417.

%This section should come before the References. Dedications and funding 
%information may also be included here.

\begin{thebibliography}{0}
\bibitem{hawk}
S. W. Hawking, Commun. Math. Phys. {\bf  43} (1975) 199.

\bibitem{GO}
R. Brout {\it et al.},
% S. Massar, P. Spindel and R. Parentani,
Phys. Rep. {\bf 260} (1995) 329, and refs. therein.

\bibitem{unruh81}
W. G. Unruh, Phys. Rev. Lett. {\bf 46} (1981) 1351;
T. Jacobson, Phys. Rev. D {\bf 44} (1991) 1731; W. G. Unruh,
Phys. Rev. D {\bf 51} (1995) 2827.

%\bibitem{ted91}
%T. Jacobson, Phys. Rev. D {\bf 44} (1991) 1731.

\bibitem{visser} M. Visser, Class. Quant. Grav. {\bf 15} (1998) 1767.

%\bibitem{unruh94}
%W. G. Unruh, Phys. Rev. D{\bf 51} (1995) 2827.

\bibitem{tedriver} 
T. Jacobson, Prog. Theor. Phys. Suppl. {\bf 136} (1999) 1, and refs. therein.

\bibitem{bmps}
 R. Brout {\it et al.}, Phys. Rev. D {\bf 52} (1995) 4559.
%S. Massar, R. Parentani, Ph. Spindel

\bibitem{cosm} J. Martin and R. Brandenberger, Phys. Rev. D {\bf 63} (2001)
 123501; 
J. Niemeyer, Phys. Rev. D {\bf 63} (2001) 123502.  

\bibitem{cosm2} J. Niemeyer and R. Parentani, Phys. Rev. D {\bf 64} (2001) 
101301. 

\bibitem{bhlaser}
S. Corley and T. Jacobson, Phys. Rev. D {\bf 59} (1999) 124011.

\bibitem{hawkmf} 
R. Parentani, Phys. Rev. D {\bf 63} (2001) 041503;
Int. J. of Theor. Phys. 40 (2001) 2201-2216, and refs. therein.
%mf, and refs. therein



\end{thebibliography}

\end{document}













gr-qc0204079.extracted_bib
\begin{thebibliography}{0}
\bibitem{hawk}
S. W. Hawking, Commun. Math. Phys. {\bf  43} (1975) 199.

\bibitem{GO}
R. Brout {\it et al.},
% S. Massar, P. Spindel and R. Parentani,
Phys. Rep. {\bf 260} (1995) 329, and refs. therein.

\bibitem{unruh81}
W. G. Unruh, Phys. Rev. Lett. {\bf 46} (1981) 1351;
T. Jacobson, Phys. Rev. D {\bf 44} (1991) 1731; W. G. Unruh,
Phys. Rev. D {\bf 51} (1995) 2827.

%\bibitem{ted91}
%T. Jacobson, Phys. Rev. D {\bf 44} (1991) 1731.

\bibitem{visser} M. Visser, Class. Quant. Grav. {\bf 15} (1998) 1767.

%\bibitem{unruh94}
%W. G. Unruh, Phys. Rev. D{\bf 51} (1995) 2827.

\bibitem{tedriver} 
T. Jacobson, Prog. Theor. Phys. Suppl. {\bf 136} (1999) 1, and refs. therein.

\bibitem{bmps}
 R. Brout {\it et al.}, Phys. Rev. D {\bf 52} (1995) 4559.
%S. Massar, R. Parentani, Ph. Spindel

\bibitem{cosm} J. Martin and R. Brandenberger, Phys. Rev. D {\bf 63} (2001)
 123501; 
J. Niemeyer, Phys. Rev. D {\bf 63} (2001) 123502.  

\bibitem{cosm2} J. Niemeyer and R. Parentani, Phys. Rev. D {\bf 64} (2001) 
101301. 

\bibitem{bhlaser}
S. Corley and T. Jacobson, Phys. Rev. D {\bf 59} (1999) 124011.

\bibitem{hawkmf} 
R. Parentani, Phys. Rev. D {\bf 63} (2001) 041503;
Int. J. of Theor. Phys. 40 (2001) 2201-2216, and refs. therein.
%mf, and refs. therein



\end{thebibliography}