%The entropy stored in inflationary spectra
David Campo
astro-ph0404021.tex

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\vskip 1truecm

\title{
%The entropy stored in inflationary spectra}
Inflationary 
%primordial 
spectra, decoherence, 
%entropy, 
and two-mode coherent states}
\vskip 1truecm
\author{David Campo}
\email[]{campo@phys.univ-tours.fr}
\affiliation{Laboratoire de Math\'{e}matiques et Physique
Th\'{e}orique, CNRS UMR 6083,
Universit\'{e} de Tours, 37200 Tours, France}
\author{Renaud Parentani}
%\email[]{parenta@celfi.phys.univ-tours.fr}
\email[]{Renaud.Parentani@th.u-psud.fr}
%\affiliation{Laboratoire de Math\'{e}matiques et Physique
%Th\'{e}orique, CNRS UMR 6083,
%Universit\'{e} de Tours, 37200 Tours, France}
\affiliation{Laboratoire de Physique Th\'{e}orique, CNRS UMR 8627,
B\^atiment 210,
Universit\'{e} Paris XI, 91405 Orsay Cedex, France}
%\date{D-28 Novembre, 2003}
\maketitle


\vskip 1truecm
\centerline{{\bf Abstract }}
\vskip 0.5truecm

%It is well understood that the temporal coherence of primordial 
%density fluctuations
%, which is a prediction of inflation,
%can be implemented by neglecting the decaying mode.
%when the mode wave length is still much larger than the Hubble radius.
% when computing the power spectrum.
%However this simplified treatment fails when posing questions concerning
%for instance, 
%decoherence processes occuring
%, and their associated entropy increase,
%which could have taken place 
%in the early universe.
% and modify the distribution.
%For similar reasons, the generic deviations from the standard distribution
%cannot be described in these simplified settings.
We re-examine the question of the entropy stored in the 
distribution of primordial density fluctuations. To this end
%To address these questions it is appropriate to use 
we make use of two-mode coherent states. These states
%escribe decoherence processes
%, and their associated entropy increase,
%which could have taken place in the early universe
%or to parametrize the generic deviations from the standard 
%distribution of primordial density fluctuations,
%e.address these questions it is 
%instead 
%appropriate to use two-mode coherent states. These
% they simply 
incorporate the isotropy of the distribution
as well as the temporal coherence and the
semi-classical character of highly amplified modes. 
%More importantly t
They also provide a lower bound for the 
%decoherence 
entropy 
%in the inflationary scenario 
if, as one expects, decoherence processes erase the squeezing
which originally characterized the distribution in inflationary models.
%as one expects.
% decoherence processes have erased.
This lower bound is one half the maximal (thermal) value.
%since they constitute the 
%appropriate basis to describe
%basis in which the density matrix rapidly becomes
%diagonal when taking into account weak interactions.
By considering backreaction effects, we also provide an upper bound
for this entropy at the onset of the adiabatic era.
%We provide an exact expression for 
%The entropy of these Gaussian distributions
%is exactly calculated. 
%by considering backreaction effects.
%they aptly describe the most general 
%homogeneous and they constitute the r
%appropriate basis to describe
%basis in which the density matrix becomes
%diagonal when taking into account weak interactions
%amongst modes.
% small non-linearities inducing some decoherence.
%isotropic Gaussian distribution in the high occupation number limit.

\vskip 1truecm 

\section{Introduction}

Inflation tells us that the primordial
density fluctuations arise from the amplification 
of vacuum fluctuations \cite{Mukhanov81,Staro79}. As a result
of this amplification,
the initial vacuum state becomes a product of 
highly squeezed two-mode states \cite{Grishchuk90}.
%e described by a stochastic Gaussian ensemble.
%The quantum distribution which results from this amplification process
%specifies this ensemble
In spite of the complexity of this 
%quantum 
%(Gaussian) 
state, 
%This quantum state is such that, 
when computing expectation values, i.e. the two-point function,
the modes exhibit a temporal coherence upon horizon re-entry.
%whereas their amplitude is governed by a Gaussian ensemble.
%To extract the predictions 
%study the behaviour of these modes is then conveniant to 
%Therefore, w
When considering the physics which took place near the recombination, 
%late time behaviour of the two-point function
it is therefore convenient and sufficient to enforce the 
temporal coherence by 
%completely neglecting 
putting to zero
the decaying mode.  
%In the same vein, because of the 
%high occupation number (typically of the order of $10^100$)
Then the residual random properties consist in 
%reduce to 
%amount in 
treating the amplitude of the growing mode as a stochastic
variable, thereby ignoring the quantum properties
of the original Gaussian distribution.
%one can safely replace the quantum distribution by 
%a classical stochastic distribution.
% of his coherence is simply and effectively enforced by 

However this simplified description has several drawbacks.
In particular the settings are too restrictive
%cannot be used 
%to address the following 
% when askingseveral 
%questions. 
%which concern first 
%the  of non-linearities, second those of 
to describe the distribution which 
would result from some decoherence process
which would have taken place during the early universe.
%inflation. 
%Therefore it is not possible to evaluate the entropy associated with
%this decoherence.
% considerations,third
%fourth 
More generally, the simplified settings are unable
%degenerate in that they cannot be used 
to 
%phenomenologically 
parameterize 
%phenomenological description of 
 deviations from the standard results
%which results from some unknow process but 
which preserve
%maintain 
the isotropy and the Gaussianity of the distribution.

In this paper, we show that the appropriate basis to investigate 
these questions is provided by two-mode coherent states. 
The reasons are the following. 
First, a two-mode coherent state provides the quantum counterpart
of a particular classical realization of the ensemble
of metric fluctuations. This correspondence is well defined
%these states naturally incorporate 
%the temporal coherence and 
%in the semi-classical character of 
for the highly excited modes we are dealing with. 
(Remember that the observed temperature anisotropies of relative amplitude 
$10^{-5}$ fix the occupation number $n$
to be of the order of $10^{100}$.)  
Second, ensembles of two-mode coherent states
%they can
%easily 
can characterize 
{\it any} Gaussian isotropic distribution,
thereby allowing to describe 
%an 
%ensembles with 
arbitrary levels of coherence. 
This follows from the requirement of isotropy and homogeneity
which
%drastically 
restricts the non-vanishing matrix elements
of the distribution: only two-mode elements with opposite wave vectors
could be different from zero.
%In particular the usual description which
%consists in negelecting the decaying mode is found to correspond
%to a {\it minimal} decoherence. 
Third, they allow to make contact with the general remark
of \cite{Zurekbook,Zurek93}
%,clarified by Zurek et al \cite{Zurekbook,Zurek93}, that 
%highly 
according to which squeezed states rapidly decay
into a statistical mixture of coherent states when small non-linearities
are no longer neglected\cite{Maldacena03}. 
%i, General Relativity .
%not being a quadratic theory. 
In fact when applying this general theorem to 
highly squeezed two-mode states,
the resulting 
%decohered 
distribution is precisely a 
diagonal density matrix of two-mode coherent states. 

%We claim that t
This distribution defines the {\it minimal} entropy 
stored in the primordial spectrum
% that decoherence processes could induce 
(or in other words, the minimal coarse graining) 
when no squeezing remains,
that is, when there is no 
%element of the density matrix which 
direction in phase space in which the spread 
%smaller
of the non-diagonal elements of the density matrix 
is 
%a quantum spread 
smaller than that of the vacuum.
%FIn this case, f
For each 
%relevant 
mode the entropy bound is $1/2$ the maximal (thermal) value, 
given the occupation number.
We shall see that
%As a final benefit, since we have at our disposal the most general settings, 
%we can determine the status of 
this entropy coincides with that associated  with
the simplified prescription which consists in neglecting the 
decaying mode. It is to be stressed that this identification can be
reached because in the large occupation number limit, the entropy
is extremely sensitive to the level of decoherence whereas the power spectrum is 
instead extremely robust. Typically the relative modifications 
of the latter are $1/n$ whereas the changes of the former are in $\ln n$.
  
The open question concerns the efficiency of 
decoherence processes in nature: 
are they powerful enough to suppress
the squeezing that the initial density matrix possessed?
This question is currently under investigation. 
Preliminary results indicate that the squeezing is indeed erased,
thereby implying that the resulting entropy is larger than (or equal to)
the above mentionned bound.

% $1/2$ the thermal entropy.
 %  is indeed a lower bound.

Finally, we shall also provide an 
upper bound for this entropy by considering
backreaction effects at the end of the inflationary period.
This upper bound is given by $3/4$ of the maximal entropy.
% can find the particular case which corresponds to 
%the usual description which consists in neglecting the decaying mode. 
%We shall see that it corresponds to a {\it minimal} decoherence scheme. 
%The ex
%above mentionned minimal decoherence scheme has a simple interpretation
%along these lines.
%character of 
The evaluation of these entropies is exactly performed
by exploiting the fact that any
Gaussian isotropic distribution can be expressed in terms of
thermal distributions, see Appendix C. 

%It should be said that these 
Related questions have been already analyzed
in several papers, see 
\cite{MukhaBrandyProkopecg,Albrecht94,Giovanninig,Matacz94,PolStar96,KiefPolStar98a,KiefPolStar00}. 
%We agree with the most conclusions. 
What we add in the present paper is 
a further clarification of the matters,
%the generality of the discussion, 
the usefulness and the relevance of two-mode coherent states,
 the lower and upper bounds on the entropy, and relationships
between various elements which have been some how separately discussed.
Notice that 
we shall not discuss the physical relevance of this entropy for structure
formation. For this interesting question we refer to \cite{MukhaBrandyProkopecg}.
% for the physical relevance of this entropy

%in the past.
%former work.




%The paper is organized as follows. 
%In section II we review the basic elements which determine
%the properties of the distribution of primordial fluctuations.
%leads to  the conclusion that the amplification that modes
%experience when their wave length is larger than the Hubble radius
%leads in quantum settings to the squeezing of two-modes states.  
%In Section III we 



\section{A review of the standard derivation of primordial spectra}

\subsection{Quantum distribution of two-mode states}

In this subsection we recall how the amplification of modes 
of a massless field propagating in a FRW universe translates 
in quantum settings in the fact that 
%propagation of a quantum field in a FRW universe initially in its 
the initial ground state
%homogeneous, non-stationary background 
evolves into a product of
%to a quantum distribution of 
highly squeezed two-mode states. 
%i.e. entangled, squeezed states. 
%The properties of the spectrum of primordial
%fluctuations follows. Indeed, 
Before proceeding, we remind the reader that 
it has been shown that the evolution of linearized cosmological perturbations
(metric and density perturbations) reduces to the
propagation of 
%real, 
massless
%, minimally coupled 
scalar fields in 
a FRW spacetime \cite{MBF92}.
In this article, we shall only consider the massless scalar test field
%work with this simplified version.  
since the transposition
%The translation 
of the results 
to physical fields represents no difficulty.
Indeed, when preserving
the linearity of the evolution,
the only modification concerns the late time dependence of the modes.
%  see Appendix A.



Let us work in a flat FRW universe. 
The line element is:
%with flat spacial section is :
\ba \label{metric}
 ds^2 &=& a(\eta)^2 \left[ -d\eta^2 +  \delta_{ij}dx^{i}dx^{j} \right] \, .
\ea
For definiteness and simplicity, we consider a 
cosmological evolution which starts with an inflationary de Sitter
phase and ends
with a %matter dominated period after having experienced a 
radiation dominated period.
%In these two periods,
When using the conformal time $\eta$ to parametrize the
evolution, the scale factor is respectively given by 
\sba \label{aeta}
 a(\eta) &=& - \frac{1}{H(\eta-2 \eta_r)} \, , \qquad 
 {\mbox{for\, \, }} -\infty < \eta < \eta_r  \, , \\
 a(\eta) &=& \frac{1}{H\eta_r^2} \, \eta  \, , \qquad 
  {\mbox{for\, \, }}  \eta > \eta_r \, , 
 % \\
 % a(\eta) &=&  \frac{1}{4H\eta_r^2} 
 % \frac{(\eta-4\eta_r+\eta_{eq})^2}{\eta_{eq}-2\eta_r} \, , \qquad 
 %{\mbox{for\, \, }} \eta_{eq} < \eta \, ,
\sea
where $\eta_r> 0 $ %and $\eta_{eq}$ 
designates  % respectively 
the end of inflation. 
%The shift in the parenthesis in the second %and third lines are 
%line is necessary to parametrize the two periods by a single conformal 
%time $\eta$. 
The transition
%pasting of these periods 
is such that
the scale factor and the Hubble parameter are
continuous functions.
%and the time of equilibrium between radiation and matter. 
This approximation based on an instantaneous transition is 
perfectly justified for modes 
relevant for CMB physics. Indeed, 
%these
%or relevant modes, 
%, i.e. modes which contribute to visible anisotropies in the CMB,
their wave vector $k$ 
%of these modes 
obeys 
%$k/Ha(\eta_r) = 
$k \eta_r  \sim 10^{-25}\sim e^{-60}$ when inflation
lasts for $60$ e-folds.  
Hence, the 
phase shift they could accumulate in a more realistic smoothed out transition
is completely negligible. 


Let $\xi(\eta,{\bf x})$ be a massless test scalar field 
propagating in this background metric.
%It obeys the d'Alembertian equation:
%\ba \label{PropPhi}
% \partial_{\eta}^2 \xi + 2 {\cal{H}} \partial_{\eta} \xi - 
% \inv{a^2}\bold{\nabla}^2 \xi = 0  \, , \ea
%where ${\cal{H}} ={\partial_{\eta} a}/{a}$ is the conformal Hubble parameter and
%$\bold{\nabla}$ is the gradient with respect to the comoving coordinates
%$\bf x$.
It is convenient to work with the rescaled field $\phi = a \xi $ and
to decompose it into Fourier modes
\ba
\phi(\eta, \bold x)
&=& \int\!\!d^3k \, \frac{e^{i \bold k \bold x}}{(2\pi)^{3/2}} 
  \phi_{\bf k}(\eta) \, .
\ea
The time dependent mode $\phi_{\bold k}$ obeys 
\ba \label{PropPhi_k}
 \partial_{\eta}^2 \phi_{\bf k}  + 
 \left(k^2 - \frac{\partial_\eta^2 a}{a} \right) \phi_{\bf k} = 0 \, ,
\ea
where $k=\abs{\bold k}$ is the norm of the conformal wave vector. 
%In Appendix A we recall how this equation gets modified when considering 
%density perturbations. The rescaled $\bold k$-modes of gravitational waves
%obey \Eqref{PropPhi_k}\cite{??}.
%and where the time-dependent frequency is given 
%b sound velocity $c_s^2=\delta p / \delta\rho$ \cite{MBF92}, e.g. $1/3$ 
%and $0$ in purely radiation and purely matter eras respectively. }
%\ba \label{omega_k}
% \omega_{k}^2(\eta) = {\bold k}^2 - \frac{a''}{a} \, .
%\ea
  
In our background solution,
%non-stationary backgrounds, 
${k}^2 - {\partial_\eta^2 a}/{a}$ 
%the squared frequency in Eq. (\ref{PropPhi_k}) 
is negative during the de Sitter period when the wavelength
is larger that the Hubble radius. 
%last on time, leading 
This leads to a large amplification of $\phi_k$. 
In quantum terms this mode amplification translates 
into spontaneous pair production characterized by 
correspondingly large occupation numbers.
% fluctuations. 
%Equivalently, one can say that the modes are spontaneously excited through
%non-adiabatic transitions. We will use this language in the following.

To obtain the final distribution of particles,
%ese transitions, 
%it is appropriate to 
one should introduce two sets of 
%asymptotic 
positive frequency solutions of Eq. \Eqref{PropPhi_k}. 
The in modes are defined at 
asymptotic early time, and the out ones at late time. Both have 
%normalized with 
unit positive Wronskian in conformity with the usual
particle interpretation \cite{B&D}. One gets  
\sba \label{modes}
 \phi_k^{in}(\eta) &=& \frac{1}{\sqrt{2k}} 
 	\left(1-\frac{i}{k(\eta - 2 \eta_r)}\right)
	e^{-ik(\eta - 2 \eta_r)} \, , \quad 
 \mbox{for\, \,} \eta < \eta_r   \, , \\
 \phi_k^{out}(\eta) &=& 
 	\frac{1}{\sqrt{2k}} e^{-ik\eta} \, ,\quad 
  \mbox{for\, \, }  \eta > \eta_r  \, .
%  \\
% \phi_k^{mat}(\eta) &=& \frac{1}
%%e^{i\phi_{mat}}}
%{\sqrt{2k}} 
% 	\left(1-\frac{i}{k(\eta - \eta_m)}\right)e^{-ik\eta} \, , \quad 
% \mbox{for\, \,} \eta_{eq} < \eta \, ,
\sea
In spite of the time dependence of the background,
these two positive frequency modes are unambiguously 
defined (up to an arbitrary constant
phase which drops in all expectation values and which has here been chosen
so as to simplify the forthcoming expressions).
In the radiation dominated era there is no ambiguity 
since the conformal frequency is constant because $\partial^2_\eta a = 0$. 
In the de Sitter epoch, there is no ambiguity either for relevant modes 
if inflation lasts more than $70$ e-folds, see \cite{NPC02} for the evaluation
of the small corrections one obtains when imposing vacuum at some finite 
early time. 
Similarly, in quantum settings, there is no ambiguity
for the initial state of relevant modes: at the onset of inflation
they must be in their ground state\cite{MBF92,cmbParenta}.
%. However, if the period of inflation lasts at least $70$ e-folds, 
%the asymptotic positive frequency
%modes are well defined. Thus, we can work with the in and out modes given at 
%Eq. \Eqref{modes}. 
%he modes depend on
%the norm of $\bf{k}$ only since we work in an isotropic background. 

%Since Eq. \Eqref{PropPhi_k} is homogeneous and  linear, 
The in and out modes are related by a Bogoliubov transformation 
\ba \label{inoutmodes}
 \phi_k^{in}(\eta) = \alpha_k \phi_k^{out}(\eta) + 
 		     \beta_k^* \phi_k^{out\, *}(\eta) \, ,
\ea
where the Bogoliubov coefficients are given by the Wronskians
\ba \label{abcoef}
  \alpha_{k} = \left( \phi_{k}^{out} , \phi_{k}^{in} \right)
  \mbox{, \ \ }
  \beta_{k}^{*} = -  \left( \phi_{k}^{out\, *} , \phi_{k}^{in} \right) \, .
\ea 
These overlaps should be evaluated at transition time $\eta_r$ 
since modes satisfy different equations in each era.
One gets
\sba \label{bogolincosmo}
 \alpha_k &=& - \frac{ e^{2i k \eta_r}}{2k^2\eta_r^2} 
 \left( 1 - 2 i{k\eta_r}- 2k^2\eta_r^2 \right) 
%\left( 1 + \frac{i}{k\eta_r}-\frac{1}{2k^2\eta_r^2} \right) 
 = \frac{-1}{2k^2\eta_r^2} \, ( 1 + O(k\eta_r)^3) \, , \\
 \beta_k^{*} &=& 
 %\frac{e^{-2i k \eta_r}}
 \frac{1}{2k^2\eta_r^2} \, .
\sea
%For relevant modes, i.e. modes which contribute to visible anisotropies in the CMB,
%their wave length obeys 
%$k/Ha(\eta_r) = k \vert \eta_r \vert \sim 10^{-25}$. 
Thus, for relevant modes, $k\eta_r \sim 10^{-25}$,
the in modes are enormously amplified. Concomitantly, they are 
dominated by the sine during the radiation dominated era:
%read
\ba \label{neglectdecay}
 \phi_k^{in}(\eta) 
 &=& \frac{i}{k^{2}\eta_r^2} \left[ \frac{\sin k\eta}{\sqrt{2 k}} \, 
      + \, O(k\eta_r)^3 \,  \cos k\eta   \right]
      \, , \qquad  \eta \geq \eta_r \, .
\ea
%For relevant modes, the relative amplitude is of order 
%$(k\eta_r)^3 \sim 10^{-75}$.   
Once the cosine in Eq. \Eqref{neglectdecay} is neglected, the physical modes 
$\phi^{in}_k/a$ show the same temporal behaviour, e.g. they are constant until
they start oscillating as they re-enter the Hubble radius when $k \eta \simeq 1$.
%$\frac{1}{a}\phi \propto \sin x /x$.
%This temporal coherence leads to well defined
%is a necessary condition to obtain 
%peaks in the CMB ansotropies spectrum\cite{Albrecht00}.


Lets now see how these considerations translate in second quantized settings.
%ation, t
Each mode operator is decomposed twice 
\ba \label{hatphik}
 \hat \phi_{\bold k}(\eta) = 
 \hat a_{\bold k}^j  \phi_{k}^j(\eta) + 
 \hat a_{-\bold k}^{j\, \dagger} \phi_{k}^{j \, *}(\eta) \, ,
\ea
%where the 'hat' characterizes operators and $\phi_{k}$ are given at 
where $j$ stands for both the in and out basis.
%Eq. \Eqref{inoutmodes}. A familly of 'in' and 'out' 
%annihilation and creation operators is 
%defined for each set of modes respectively :
%\ba
% \hat a_{\bold k}^{in} = \left(\phi_{k}^{in}\, , \hat \phi \right) \, , \qquad
%\hat a_{-\bold k}^{in \, \dagger} = - \left(\phi_{k}^{in\, *}\, , \hat \phi
%\right) \,  , 
%\ea
%and similar expressions hold for the out operators. In and out 
The operators so defined are related by the 
transformation 
\ba \label{inoutoperator}
 \hat a_{\bold k}^{in} = \alpha_k^{*} \, \hat a_{\bold k}^{out} -
		      \beta_k \, \hat a_{-\bold k}^{out \, \dagger} \, .
\ea 
%Because of the homogeneity of the background,
This transformation 
%is $2\times 2$ block-diagonal as it 
couples ${\bf k}$ to $-{\bf k}$ only.
Hence, when starting from the in vacuum (the state annihilated by the
$\hat a_{\bold k}^{in}$ operators), 
every out particle 
of momentum ${\bf k}$ will be accompanied by a partner 
of momentum $-{\bf k}$. Moreover, pairs characterized 
by different momenta are incoherent 
%in the in vacuum
(in the sense that in expectation values any
product of annihilation and creation operators of different momenta 
will factorize).

These two properties are explicit when  
writing the in vacuum in terms of out states (i.e.
states with a definite out particle content). From Eq. \Eqref{inoutoperator}, 
%using 
%the notations of Eq. \Eqref{z}, 
one gets (see \cite{MukhaBrandyProkopecg,PhysRep95})
\ba \label{inoutvac}
 \ket{0,\, in} 
&=&  
 \widetilde{\prod\limits_{\bold k}} \otimes  
 \kett{0,\, \bold k, in} 
\nonumber\\ 
&=& 
\widetilde{\prod_{\bold k}}  \otimes 
\left(  \inv{\vert \alpha_{k} \vert} 
   \exp \left( z_{k}\, 
  \hat  a_{\bold k}^{out \, \dagger}  \, 
  \hat a_{-\bold k}^{out \, \dagger} \right)
  \ket{0,\, \bold k, out}\otimes \ket{0,\, -\bold k, out} \right) \, .
\ea
The tilde tensorial product takes into account only half the modes.
It must be introduced 
because the squeezing operator acts both on the $\bf k$ and the $-\bf k$
sectors. 
%For later conveniance, we introduce t
%Notice that 
%the division of 
The definition of this product 
%over only half 
requires the introduction of an arbitrary wave vector 
to divide the modes into two sets. The sign of $k_z$ can be used. 
Notice that a rigorous definition of $\widetilde{\prod_{\bold k}}$
%this product 
requires to consider a
discrete set of modes normalized with Kronekers (that is, to normalize the 
modes in a finite conformal 3-volume).
To be explicit, the two-mode state $\kett{0,\, \bold k, in} $
is given by
%he in vacuum factorizes into a product over half the momenta
%of sums of 'two-mode' out states. 
%In the following, we note such a two-mode state
\ba
\label{in2mode}
 \kett{0,\, \bold k, in} = 
 \ket{0,\, \bold k, in} \otimes \ket{0,\, -\bold k, in} \, ,
\ea 
where $\ket{0,\, \bold k, in}$ is the ground state of the 
$\bold k$-th mode at the onset of inflation.
%In Eq. \Eqref{inoutvac} t
The complex parameter 
$z_k$ appearing in the squeezing operator
in Eq. \Eqref{inoutvac}
is given by the ratio of the Bogoliubov coefficients
%, since it will play an important role in the following. 
%In the long-wavelength limit, it is given by
\ba \label{z}
 z_k &=& \frac{\beta_k}{\alpha_k^*} = - e^{-2i k \eta_r} 
 \frac{1}{\left( 1 + i2{k\eta_r}- 2k^2\eta_r^2 \right)} 
 \nonumber \\
 &=& -  1 +  O(k\eta_r)^3  \, .
\ea
%Hence, taking t
The high occupation number limit corresponds to $\abs{z_k} \to 1^-$.
%This limit should be taken with care.PETITE PHRASE TOUTE SEULE.
%is equivalent to neglect 
%means that the decaying mode.

It has to be emphasized that none of the out states in Eq. \Eqref{inoutvac} 
carry
3-momentum. Hence, the distribution is homogeneous in a strong sense:
at late time the 3-momentum operator is still
annihilated by the state of Eq. \Eqref{inoutvac}. 
(This property is not satisfied 
by incoherent distributions such as thermal baths. In those cases, 
the 3-momentum fluctuates and vanishes only in the mean.)
%distribution is homogeneous in a strong sense :3-momentum 
%unlike for thermal distributions where the  3-momentum vanishes only
%in the mean, it is here annihilated by the state Eq. \Eqref{inoutvac}. 
The present distribution is also isotropic
since the Bogoliubov coefficients are functions of the norm $k$ only. 
Finally, it is a Gaussian distribution, as can be seen from 
Eq. \Eqref{inoutvac}. 
%This follows from the fact that 
%the hamiltonian is quadratic \cite{Grishchuk90,Albrecht94,PolStar96}.
%since the state Eq. \Eqref{inoutvac} can be shown to 
%results from the unitaty evolution
%of the in-vacuum \cite{albrecht,PolStar}.

To appreciate the peculiar properties of the distribution 
of Eq. \Eqref{inoutvac} it is interesting to consider the most 
general
%When dealing with a 
homogeneous, isotropic, and Gaussian distribution of out quanta.
Its properties are completely specified by three real functions of the norm
$k$ (one real and one complex) through the following expectation values 
%of creation and annihilation operators in the in vacuum : 
\sba \label{momenta}  
 && \langle \hat a_{\bold k}^{out} \rangle
 %_{\rm{in}} 
 =  0 \, , \\
   % \langle \hat a_{\bold k,\, L}^{out} \rangle_{\rm{in}} = 0 \, , \\
 &&\langle \hat a_{\bold k}^{out\, \dagger} \,  
 \hat  a_{\bold k'}^{out} \rangle
% =   \langle \hat a_{\bold k,\, L}^{out\, \dagger}  \, 
% \hat  a_{\bold k',\, L}^{out} \rangle_{\rm{in}}  
 = n_k   \, \delta^3(\bold k - \bold k' ) \, ,
% = \abs{\beta_k}^2 \, \delta^3(\bold k - \bold k' ) 
\\
 &&\langle \hat a_{\bold k}^{out}  \,  
 \hat a_{\bold k'}^{out} \rangle
 = c_k \, \delta^3(\bold k + \bold k' )\, .
% = \alpha_k \beta_k \,
 %\abs{\alpha_k}^2 z_k \,  \delta^3(\bold k + \bold k' )  \, .
\sea 
In the second line, $n_k$ is the mean occupation number.
% of the modes in the in vacuum. 
In the third one, the complex number $c_k$ 
characterizes the quantum coherence of the distribution.
%It is to be noticed that the functions $n_k$ and $c_k$ can be used to characterize 
%the most general homogeneous, isotropic, and Gaussian distribution.
The degree of two-mode coherence is given by $|c_k|/(n_k+1/2)$, see Appendix C.
It is bounded by 1.
%In fact $c_k$ fixes the amplitude of the non-diagonal 
%elements of the density matrix in the occupation number basis.
For a thermal (incoherent) distribution, one has $c_k \equiv 0$: no coherence.

In the  case of pair production from vacuum, 
%described in Eq. \Eqref{inoutvac},
one has 
%$n_k = \abs{\beta_k}^2$
%= 1/4 (k \eta_r)^4 \simeq 10^{100}$ 
%e high occupation limit, one has
\ba
\label{cterm}
 n_k &=& \abs{\beta_k}^2 = \frac{\abs{z_k}^2}{1- \abs{z_k}^2}\, ,
 \nonumber\\
 c_k 
 &=& \alpha_k \beta_k = 
 %\abs{\alpha_k}^2 \, z_k 
 \frac{z_k}{1- \abs{z_k}^2}
 \, .
 %= - n_k (1 + O(k \eta_r)^3) \, .
\ea
Therefore, when considering relevant modes in our inflationary model 
Eq. \Eqref{bogolincosmo},
%for relevant modes,
%in the high occupation number limit, 
one has 
\ba
\label{cterm2}
 n_k &=& \frac{1}{4 (k \eta_r)^4 } \simeq 10^{100}\, ,
 \nonumber\\
% c_k  &=&
%n_k = 1/4 (k \eta_r)^4 \simeq 10^{100}$  and $
\frac{\vert c_k \vert}{ n_k+1/2} &=& 1 + O(k \eta_r)^3\, . 
\ea
That is, the distribution 
which results from inflation is highly populated and, more importantly,
maximally coherent. 
When computing the Green function,
the two-mode coherence of the distribution
will manifest itself in the {\it temporal coherence} of the modes.

\subsection{Two-point function and the neglect of the decaying mode}
%Gaussian distribution of standing waves with fixed phase}


%Second, the power spectrum of $\hat \xi$ for modes  which contribute to
%visible anisotropies in the CMB, i.e. $k\eta_r \ll 1$, is scale invariant. 
%The two-point function at equal times is
%\ba
% \bra{0 in} \hat  \xi(\eta, {\bf x}) \hat  \xi(\eta, {\bf x}') \ket{0 in} &=& 
% \int_0^{\infty} \frac{dk}{k} 
% \frac{\sin k\abs{{\bf x} - {\bf x}'}}{k\abs{{\bf x} - {\bf x}'}} 
% {\cal P}_{\xi}(k,\eta) \, , \nonumber \\
% {\cal P}_{\xi}(k,\eta) &=& \frac{k^3}{2\pi^2} \abs{\xi_k^{in}(\eta)}^2 \, .
%\ea
%Taking into acount the result of Eq. \Eqref{}, one has
%\ba
% {\cal P}_{\xi}(k,\eta) = \left(\frac{H}{2\pi}\right) 
% \left(\frac{\sin k(\eta-2\eta_r)}{k(\eta-2\eta_r)} \right)^2 
%  = \left(\frac{H}{2\pi}\right) \left( 1 +  O(k(\eta-2\eta_r))^2 \right)  \, .
%\ea
When expressed in terms of out modes,
the two-point function associated with the general 
%Gaussian isotropic 
distribution specified by Eqs. \Eqref{momenta} is
% distribution resulting from inflation 
%it can also be characterized by the two-point function:
%of the field $\hat \phi$ :
\ba \label{Ginx}
 G(\eta,{\bf x}, \eta', {\bf x'}) &=& G_{out}(\eta,{\bf x}, \eta', {\bf x'})  
 \nonumber \\
 &&
% \bra{0, \, in} \hat \phi(\eta,{\bf x})  \hat \phi(\eta',{\bf x}')  \ket{0, \, in}
+ \int_0^{\infty} \frac{dk k^2}{\pi^2} 
 \frac{\sin(k\vert {\bf x} - {\bf x}'\vert)}{k\vert {\bf x} - {\bf x}'\vert} 
 2 Re \left[  n_k \, \phi_k(\eta) \phi_k^*(\eta')  
+  c_k \, \phi_k(\eta) \phi_k(\eta') 
\right]  . \quad
\ea
In the first line, we have isolated $G_{out}$,
the contribution of the out vacuum. In the high occupation 
number limit, this contribution is negligible.
%In the second line the expression involves a single integral over the norm 
%of the wave vector 
%since the different two-modes are incoherent and beacuse the distribution is isotropic.
% in the in vacuum. 

It is important to notice that, for a general distribution, 
the sum in bracket in the
above integrand 
%of the second line 
cannot be factorized. 
However it does factorize 
%when the distribution is maximally coherent.
in two cases which are relevant for us: first,
 for coherent states, see Appendix A, and second 
%  unlessthe distribution is maximally coherent, as in the case of 
for distributions resulting 
from pair production from vacuum.
% or as for coherent states, see Appendix A.
%In this case only 
%Moreover when considering the double limit of high occupation number 
%$n_k \gg 1$ and maximal coherence $|c_k|/(n_k+1/2) -1 \ll 1$, this expression simplifies :
%the integrand factorizes. 
Indeed, when considering our cosmological model, 
taking into account the minus sign of Eq. \Eqref{z},
and neglecting correction terms in $O(k\eta_r)^3$
(which amounts to neglect the decaying mode, see Eq. \Eqref{neglectdecay}),
one obtains
%inflation gives 
\ba \label{Ginx2}
%\nonumber \\ 
G_{in}(\eta,{\bf x}, \eta', {\bf x'})
% = &=& 
%G_{out}(\eta,{\bf x}, \eta', {\bf x'})  \nonumber \\&& + 
= \int_0^{\infty} \frac{dk k^2}{\pi^2} 
 \frac{\sin(k\vert {\bf x} - {\bf x}'\vert)}{k\vert {\bf x} - {\bf x}'\vert} 
 \,   n_k \,  \frac{\sin k\eta}{\sqrt{k}}\, \frac{\sin k\eta'}{\sqrt{k}}
 \, .
\ea
%Therefore ieglecting the decaying mode (see Eq. \Eqref{neglectdecay}) 
%and removing the out v.e.v., 
As announced, the integrand in Eq. \Eqref{Ginx} 
factorizes. In inflation, the function which appears is $ \sin k\eta$
where $\eta$ is related to the scale factor by Eq. \Eqref{aeta}.
%$a$ with temporal arguments $\eta$ and $\eta'$ respectively. 
%This property is a manifestation of the coherence of the in vacuum.
This is how the temporal coherence of modes obtains
from the two-mode coherence of the distribution.

%Hence, Since 
Once the cosine is neglected,
%can be neglected in the adiabatic era, 
the quantum distribution 
%resulting from inflation can be 
is effectively replaced by 
a stochastic Gaussian distribution of classical fluctuations 
\ba 
\label{sineeq}
 \phi_{\bf k}(\eta) = S_{\bf k}\frac{\sin k\eta}{\sqrt{k}}  \, ,
\ea
with locked temporal
argument, and random amplitudes with variances given by
\ba \label{Aeff}
 \av{S_{\bf k}S_{\bf k'}^{*}}_{e\!f\!f} =  \av{S_{\bf k}S_{-\bf k'}}_{e\!f\!f} = 
 2n_k \, \delta^{(3)}({\bf k}-{\bf k}') \, .
\ea
Being Gaussian,
the effective probability distribution is simply
\ba \label{Peff}
 {\cal P}_{e\!f\!f} = \widetilde{\prod_{\bold k}} \,
 \frac{1}{2n_k} \exp\left(-\frac{\abs{S_{\bf k}}^2}{2n_k} \right) \, .
\ea
To avoid double counting, one must again
%we have
 use the tilde product which takes into account half the modes
only, as was done in the quantum distribution of Eq. \Eqref{inoutvac}. 
This counting becomes crucial when computing the entropy, see Section III.C.
%requires to consider a
%discrete set of modes normalized with Kronekers (that is, to normalize the 
%modes in a finite conformal 3-volume).

Notice finally that the first equality in Eq. \Eqref{Aeff}
is simply the expression of the reality of the field 
$\phi(\eta,{\bf x})$. This was not the case when noticing that inflation
gives
$\langle \hat a_{\bf k}^{out\, \dagger}  
\, \hat a_{\bf k'}^{out} \rangle = - 
\langle \hat a_{\bf k}^{out}  \, \hat a_{-\bf k'}^{out} \rangle
(1 + O(k \eta_r)^3)$ for the quantum field operators. 
In that case the equality is the expression of the coherence
of the in vacuum, i.e. the absence of 3-momentum fluctuations
and the possibility of factorizing the
%When considering the 
2-point function by discarding the cosines.
%, this translates in the absence of the cosine.

\subsection{Additional remarks}

First
%or completeness,
we %At this point it is appropriate to 
remind the reader why a random 
distribution of both the sine and the cosine 
does not give rise to any temporal 
coherence \cite{Grishchuk90}. 
In fact such a distribution  corresponds to
an incoherent (thermal) distribution.
%....Naively, one would therefore asign a amplitude $v_{\bf k}$ to the positive
%frequency part of $\phi_{\bf k}$. One would get
%Let us verify this.

Writing the field 
%as writing the field 
modes as 
the sum of a sine and a cosine:
\ba
 \hat \phi_{\bf k} &=& \hat a_{\bf k}^{out} \phi_k^{out} + 
 \hat a_{-\bf k}^{\dagger \, out} \phi_k^{out \, *} \, \nonumber \\
 &=& \hat C_{\bf k} \frac{\cos(k\eta)}{\sqrt{k}} + 
 \hat S_{\bf k} \frac{\sin(k\eta)}{\sqrt{k}} \, .
\ea
%In the second line, t
It is to be noticed that 
the operators $\hat C_{\bf k}$ and $\hat S_{\bf k}$ 
are 
%linear combinations of the annihilation and creation operators.They are 
proportional to the field mode operator and its conjugate momentum evaluated
at $\eta = 0$, as if there were no inflation:  
\ba
 \hat C_{\bf k} &=&  
 \frac{1}{\sqrt{2}}  \, 
 \left(\hat a_{\bf k}^{out} + \hat a_{-\bf k}^{\dagger \, out} \right)
 = \sqrt{k} \,  \hat \phi_{\bf k}(0) 
 \, ,  \nonumber \\
 \hat S_{\bf k} &=& \frac{-i}{\sqrt 2}  \, 
 \left( \hat a_{\bf k}^{out} - \hat a_{-\bf k}^{\dagger \, out}\right) 
 = \frac{1}{\sqrt k} \, \partial_{\eta} \hat \phi_{\bf k}(0) 
 \, .
\ea
%Notice however that these are not creation and annihilation operators since 
They
satisfy the canonical equal time commutation relations.
%of the field operator and its conjugate momentum.

%where $v_{\bf k}$ is a Gaussian stochastic variable with 
%following the Gaussian distribution
Consider an incoherent ($c_k =0$) distribution:
\ba
\label{incoh}
\langle \hat a_{\bf k}^{\dagger} \hat a_{{\bf k}'} \rangle_{i\!n\!c}
= n_k \, \delta^{3}({\bf k}-{\bf k}') \, ,\qquad
\langle \hat a_{\bf k} \hat a_{{\bf k}'} \rangle_{i\!n\!c} = 0 \, .
% \av{v_{\bf k} v_{{\bf k}'}}_{i\!n\!c} = 0 \, .
\ea
%The prefactors of the cosine and sine
%define the combinations 
%$C_{\bf k} = \left( v_{\bf k} + v_{- \bf k}^{*} \right) / \sqrt{2}$ and 
%$S_{\bf k} = -i\, \left( v_{\bf k} - v_{- \bf k}^{*} \right) / \sqrt{2}$.
%With the distribution Eq. \Eqref{incoh}, 
%the temporal coherence of the modes is
%completely lost because 
Then $\hat C_{\bf k}$ and $\hat S_{\bf k}$ are
%the real and imaginary parts of $v_{\bf k}$ are
uncorrelated Gaussian operators with equal variance:
%ICI je recoince car je ne comprend pas comment 19 est inevitable et 
%en meme temps le RR expe value peut diferer de 19 ??
\ba
 \langle {\hat  C_{\bf k}\hat C_{{\bf k}'}^\dagger}\rangle_{i\!n\!c} &=& 
\langle  {\hat S_{\bf k}\hat S_{{\bf k}'}^\dagger}\rangle_{i\!n\!c} 
= \left( n_k + \frac{1}{2} \right) \, \delta^{3}({\bf k}-{\bf k}') 
% n_k \, \delta^{3}({\bf k}-{\bf k}')  
 %\, , \qquad 
 %\langle  {\hat S_{\bf k}\hat S_{{\bf k}'}^\dagger}\rangle_{i\!n\!c} = n_k \, \delta^{3}({\bf k}-{\bf k}') 
 \, , \nonumber \\
\langle  {\hat C_{\bf k}\hat S_{{\bf k}'}}\rangle_{i\!n\!c} &=& 0 \, .
\ea
This implies the absence of temporal coherence of the modes,
as can be seen from the temporal behaviour of the bracket in the 
integrand of Eq. \Eqref{Ginx}: when $\eta = \eta'$  the bracket does not 
exhibit any oscillation in $k$ as it did for the inflationary distribution. 
(This absence can also be understood by
considering the $S_{\bf k}$ and $C_{\bf k}$ 
as stochastic variables rather than quantum ones.
%, this absence of oscillations can be understood as follows. One can then write 
%When w
Writing 
the mode in terms of its norm and its phase \cite{Grishchuk90,Allen00}
%completely lost temporal coherence of the modes is
%completely lost  equivalent to say that the temporal phase of the modes 
%is ramdom, as can
%be seen by writing \cite{Allen00, Grishchuk90}
\ba 
 \phi_{\bf k} &=& \Phi_{\bf k} \, \sin(k\eta + \theta_{\bf k}) \, .
\ea
%and compute the distribution for the new stochastic variables 
one can treat $\Phi_{\bf k}$ and $\theta_{\bf k}$ as stochastic variables.
%\ba \label{stupideq2}
% \Phi_{\bf k} &=& \sqrt{C_{\bf k}^2 + S_{\bf k}^2}\, , \nonumber \\
% \tan \theta_k &=& \frac{S_{\bf k}}{C_{\bf k}} \, .\ea 
One then verifies that the distribution for the phase $\theta_{\bf k}$ is uniform
over the interval $\left[0,\, 2\pi \right]$. Hence no temporal coherence 
could obtain.)
%this random distribution cannot therefore lead to acoustic peaks.
%Compare Eq. \Eqref{ABvariances}.


%\subsection{Quantum distribution of standing waves}

%The aim of this subsection is to relate 
%at an operator level, 
As a last remark, to estimate what has been neglected when 
discarding the cosine in Eq. \Eqref{sineeq}, 
we compute the residual fluctuations of 
%the operator 
$\hat C_{\bf k}$ and the cross correlation $\hat C_{\bf k} \hat S_{\bf k}$.
%the quantum mode operators $\hat a_{\bf k}$
%to some new operators $\hat A_{\bf k}$, $\hat B_{\bf k}$ 
%Moreover they are not hermitian, and therefore are not observables in the
%quantum mechanical sense.
To leading order in $n_k$, one has
%the expectation values of these operators are
\ba \label{ABvariances}
 \langle \hat S_{\bf k} \hat S_{{\bf k}'}^{\dagger} \rangle_{in} &=& 
 \left( n_k + \frac{1}{2} - Re(c_k) \right) \, \delta^{3}({\bf k} - {\bf k}') 
 = 2n_k  \, \delta^{3}({\bf k} - {\bf k}')  
 \, , \nonumber \\  
 \langle \hat C_{\bf k} \hat C_{{\bf k}'}^{\dagger} \rangle_{in} &=& 
 \left( n_k + \frac{1}{2} + Re(c_k) \right) \, \delta^{3}({\bf k} - {\bf k}')  
% =  n_k \, O(k \eta_r)^3  \, \delta^{3}({\bf k} - {\bf k}') 
  = O(n_k^{1/4}) \, \delta^{3}({\bf k} - {\bf k}')  
 %= n_k O(k \eta_r)^3 \, \delta^{3}({\bf k} + {\bf k}') 
 \, , \nonumber \\
 \langle \hat C_{\bf k} \hat S_{{\bf k}'}^\dagger \rangle_{in} &=& 
 - \left(Im(c_k) + \frac{i}{2}\right) \, \delta^{3}({\bf k} - {\bf k}') 
 = O(n_k^{1/4}) \, \delta^{3}({\bf k} - {\bf k}') 
%  = n_k \, O(k \eta_r)^3 \, \delta^{3}({\bf k} - {\bf k}')
  \, . 
\ea
When divided by the variance
of $\hat S_{\bf k}$, the variance of $\hat C_{\bf k}$ and 
the cross-correlation are of order $n_k^{-3/4} \sim (k\eta_r)^3
\sim 10^{-75}$. 
One can therefore safely use the distribution Eq. \Eqref{Peff} 
in replacement of the quantum distribution 
Eqs. \Eqref{inoutvac} or \Eqref{ABvariances} 
when calculating the power spectrum.
However, 
in general, this is {\it not} the case for the entropy.



\subsection{Drawbacks of the simplified description}


The simplified description in terms of a statistical ensemble
of sine standing waves has indeed several shortcomings. 
It is of value to describe them with some attention.

First, it should be pointed out that in the early universe, different physical
processes could give rise to different final density matrices, and hence, to
different entropies. 
As we shall see bellow, only a narrow set of these quantum distributions can
be put in correspondence with that of Eq. \Eqref{Peff}.
To describe the 
most general isotropic and Gaussian distribution,
%describe the general case, 
it is necessary to return to two-mode distributions
characterized by 
%three real functions 
$n_k$ and $c_k$ of Eq. \Eqref{momenta}.
Second, 
%there should be a relationship between 
the classical ensemble of sine waves should be considered only
as an effective description of the density matrix.
% of two-modes. 
The appropriate basis to describe this matrix is provided 
by coherent states. The reasons are the following.
%and coherent states because the latters enjoy the following properties. 
% is not clear. 
%The latters play two important roles. 
On the one hand, they constitute the quantum counterparts of 
%points in the classical 
classical configurations in 
phase space. Therefore they provide an adequate basis for studying 
the semi-classical limit. In particular, since the spread of coherent states
have no preferred direction in phase space, it will 
be easy to see whether or not a distribution has kept some squeezing.
%manisfest itself since coherent states
%have a spread with no prefered direction in phase space.
On the other hand, they are the preferred basis in
which squeezed states decohere when weak interactions are taken into 
account\cite{Zurek93}. 
%This important point is detailed in the next section.
%Therefore one expects that the classical ensemble 
%of sine waves should correspond to a particular distribution 
%of coherent states. We shall see that this is indeed the case.

\section{Two-mode coherent states}

When using coherent states in cosmology, one must pay attention to 
%pair creation process which 
the entanglement between $\bf k$ and $-{\bf k}$ modes. 
A naive use of coherent states which would assign 
amplitudes to each mode separately could erase these
correlations and therefore suppress the information about the temporal phase
of the modes. 
%see Eqs. \Eqref{stupideq1}-\Eqref{stupideq2}.
Taking into account the 
%$\bf k, -{\bf k}$
entanglement leads to the notion of 'two-mode
coherent states'.
%, which we present in the next subsection.

\subsection{Representation of the in vacuum with two-mode coherent states}

To understand the usefulness of two-mode coherent states
it is appropriate to first mention the following properties \cite{CP04}.
%quote an exact result. 
Consider a mode ${\bf k}$ in a coherent state 
$\ket{v,\, {\bf k}}$ constructed with out operators, see Appendix A for its
definition. 
Then compute the one-mode reduced state obtained by
projecting it on the two-mode initial vacuum of Eq. \Eqref{in2mode}
%$\kett{0 in,\, {\bf k}}$ 
%onto the subspace
%$\ket{v_{\bf k}}\bra{v_{\bf k}} \otimes {\bf 1}_{-{\bf k}}$ 
%generated by the one-mode coherent state \cite{CP04}:
\ba
\label{remar}
 \langle v,\, {\bf k} \kett{0 in,\, {\bf k}}  = 
 {\cal A}_k(v) \, \ket{z_k v, \, -{\bf k} } \, .
\ea 
It is remarkable that the state of the $-{\bf k}$ mode
which is entangled to $\ket{v,\, {\bf k}}$ 
is also a coherent state. Its amplitude 
%of this coherent state 
is given by 
%$z_kv_{\bf k}^{*}$. Both the
%the $-{\bf k}$ mode is 
%determined by 
the complex conjugate of $v$ %amplitude of the ${\bf k}$ mode 
times $z_k$ characterizing the pair creation process.
These facts follow from the 
%fact that the in vacuum Eq. \Eqref{inoutvac}
EPR correlations in the initial vacuum displayed in Eq. \Eqref{inoutvac}. 
The prefactor ${\cal{A}}_k(v)$ is
%probability amplitudes are
\ba \label{A(v)}
 {\cal{A}}_k(v) &=& \inv{\abs{\alpha_k}} 
 \exp{\left(-\frac{\abs{v}^2}{2\abs{\alpha_k}^2} \right)}  \, .
\ea
It is the probability amplitude to find the mode ${\bf k}$ in 
the coherent state $\ket{v, \, {\bf k}}$ given that we start with vacuum at the
onset of inflation.

Using the representation of the identity with coherent states,
Eq. \Eqref{coh1}, 
the two-mode in vacuum can be thus 
decomposed as a {\it single} sum of {\it two-mode coherent states}
 rather than two independent integrations over one-mode coherent
states. In fact we have
%\footnote{This result differs from the one previously written in  
%\cite{Kim00}. The two writing differ only by a rescalling of the variable of
%integration, hence they represent the same state. 
%%The identi
%The reason that the decomposition of a two-mode state as a coherent simple sum 
%of coherent states is not unique 
%is that the coherent states constitute an overcomplete basis.}, 
\ba \label{insuperpcoh}
 \kett{0 in,\, {\bf k}}  = 
 \int\!\!\frac{d^2v}{\pi} \, 
 {\cal{A}}_k(v) \, \,  \ket{v,\, {\bf k}}  \otimes \ket{z_k v^*, \, -{\bf k}} 
  \, .
\ea
%We would like to insist on the fact that this %superposition 
This 
result is exact\footnote{Notice however that the above decomposition is not unique
since the coherent states are not orthogonal, compare with \cite{Kim00}. 
%\cite{Kim00}. The writing
Eq. \Eqref{insuperpcoh} has the advantage to be directly related to
%is natural in the sense that is associated with
%it is the one selected 
the detection %e.g. measurement in the language of quantum mechanics, 
of a quasi-classical configuration in the ${\bf k}$ sector.}
%,see Appendix D and \cite{CP04} for further details.} 
%involes only one
%summation over the basis of coherent states instead of two as 
%for a general state.
%This and is not valid only in the limiting case of 
%an approximation valid in 
%the high 
and applies even for low occupation numbers.
% limit. 
It is the consequence of the coherence of
the in vacuum and 
%therefore 
holds for every homogeneous pair creation process.


Another important consequence of Eq. \Eqref{remar} is that
%of this first result, 
%and given the fact that coherent states are
%'almost' orthogonal in the sense that 
%$\abs{\langle u \vert v \rangle}^2 = e^{-\abs{u-v}^2} $, 
the probability to find {\it simultaneously} the ${\bf k}$-mode with
coherent amplitude $v$ and its partner with amplitude $w$ is 
\ba \label{vwdistrib}
 {\cal P}_{2, \, k}(v,w) = 
 \abs{\bra{v,\, {\bf k}}\bra{w,\, -{\bf k}} 0 in, \, {\bf k} \rangle_2}^2 = 
 \abs{{\cal{A}}_k(v)}^2 \times e^{-\abs{w-z_k v^{*}}^2} \, .
\ea 
The second factor arises follows from the overlap between 
two different coherent states: 
$\abs{\langle u \vert v \rangle}^2 = \exp({-\abs{u-v}^2}) $.
Equation \Eqref{vwdistrib} implies that 
once the amplitude of the ${\bf k}$-mode has been measured, the
conditional probability to find its partner in a coherent state $\ket{w}$ is
centered
%extremely peaked 
around $w=z_k v^{*}$. In the high occupation number limit we
are dealing with, the spread ($=1$)
around this mean value is negligible when compared
to the spread in $v$ which is given by $\abs{\alpha_k}^2 = n_k + 1$. 
Therefore, when computing
expectation values in leading order in $n_k$, the
conditional probability acts as a delta function on both 
the real and the imaginary part of $w$. 
This is how the EPR correlations in the in-vacuum 
translate in the coherent states basis.
This 
%is the second 
result 
%of this section, and also explains is at the origin of 
determines the properties 
%is the starting point of the analysis 
%
of the local correlations in the primordial spectra\cite{CP04}. 


To complete this analysis, and in preparation for studying decoherence, 
it is also interesting to explicitly write the non-diagonal matrix
elements of the in vacuum density matrix. One has
\ba \label{vwrhov'w'}
 \bra{v}\bra{w} \hat \rho_{in} \ket{v'} \ket{w'} &=& 
 {\cal A}_{2, \, k}(v,w) \, {\cal A}_{2, \, k}(v',w')^{*} \, , 
\ea
where the two-mode amplitude is
\ba
 {\cal A}_{2, \, k}(v,w) &=& 
 %\frac{1}{\abs{\alpha}} 
 %\exp\left(\right) 
% e^{-\frac{1}{2}{\abs{v}^2}/{\abs{\alpha}^2}}
{\cal A}_k(v) \,
 e^{-\frac{1}{2}\abs{w-z_k v^*}^2 }\,
 e^{i\, {\rm Im}(w^*z_k v^*)} \, .
\ea
%First we recover that the conditional probability
%amplitude is well peaked around $zv^*$. Second, 
Since the initial vacuum is a pure state and since 
$\abs{\alpha_k}^2 \gg 1$,
%the mean occupation number is large, 
the above matrix elements do not
vanish, {\it even for macroscopically different coherent states}.
Therefore this distribution does not describe a classical ensemble of these
quasi-classical states.
Fortunately, 
%it has been show that 
such quantum distributions are 
%extremely 
unstable to any weak perturbation in that they rapidly evolve 
into statistical mixtures.
% either through a coupling to the external world, or by internal
%coupling through non-linearities in the field amplitude. 
Let us now describe this decoherence process.
%e effective distribution which would result from taking into account such
%interactions.


%When dealing with the original density matrix, the quantum 
%origin of the process is still present even for arbitrary high
%occupation number. Indeed, when taking two different
%semi-classical states described by the two-mode coherent states
%$\ket{v}\otimes\ket{w}$ and $\ket{v'}\otimes\ket{w'}$, the non-diagnamal
%matrix element is
%It is non-vanishing, even for macroscopically different states, as can be seen
%reexpressing the terms in the exponantial in \Eqref{vwrhov'w'} with the
%variables $v-v'$ and $v+v'$.
%This distribution is therefore not completely satisfying to describe the
%classical fluctuations of the metric. First the initial vacuum distribution 
%must decohere.



\subsection{Zurek {\it et al.} analysis and minimal decoherence scheme}


In general, it is a difficult question to determine into 
what mixture an initial density matrix will evolve when taking into account
some interactions amongst modes or with other modes.
%with a reservoir of modes. 
%The basis into which the reduced density matrix is diagonal,  
%depends crutially on the total Hamiltonian. 
There exist  however several cases where clear conclusions
can be drawn. First,
%Notice however that 
when one can neglect the free Hamiltonian, 
%the situation is simple and 
the preferred states (that is the basis into which
the reduced density matrix will become diagonal) 
%after having taken the interactions into account) 
are the eigenstates
of the interaction Hamiltonian \cite{gottfried,Zurek81,Zurek82}.
This approach has been applied in \cite{KiefPolStar98a}, 
%As shown in \cite{KiefPolStar00}, 
%%for 
to primordial density fluctuations when the (physical) modes are 
almost constant because their wave length is much larger than the
Hubble radius. The conclusion 
is that the preferred basis is provided by amplitude (position) 
eigenstates.
%a realistic coupling selects the amplitude
%basis. 
%Although this result is valid while the modes are still outside 
%the horizon, it does no longer hold when they start oscillating at horizon
%reentry. \\
%However, self-interaction cannot be neglected \\
%QUESTION : intensity compared to coupling with other field \\
However this conclusion leaves some ambiguity and
might lead to some difficulties.
%it is unclear to us why ignoring the 
%momentum fluctuations
%neglecting the free Hamiltonian is appropriate.
% in the present case.
% for several reasons.
%case is  for the inflationary case precisely because 
First, position eigenstates are not normalized.
Second, and more importantly, the spread in momentum is infinite
for these states. Therefore, the velocity field would not
be well defined when the modes re-enter
the horizon. Moreover, as pointed out in \cite{KiefPolStar00},
some additional decoherence could be obtained as they re-enter
the horizon.
In this case, the momentum should be treated 
in the same footing as the position. To cure these problems, 
 some finite spread in position should be introduced.
One then needs a physical criterion to choose this spread.
%which also fixes the spread in the momenta.
%Second, the decoherence effects could be so efficient
%that they could occur while 
%will probably be most effective when
%the modes are still oscillating, before they exit 
%and/or re-enter the Hubble horizon. In a similar fashion,
%Moreover, it should not be forgotten
%that even if the physical modes are almost constant
%during the relevant period for decoherence, the
%rescalled modes obeying Eq. \Eqref{PropPhi_k} would still 
%depend on time (in fact grow). This would translate as a 
%rotation in the ????????
%Third, position eigenstates 
%are not normalized. Hence some finite spread should be introduced.
%One therefore needs a criterion to choose this spread.
%, as well as that of the momentum.

To remove this ambiguity
%e ambiguity in the position spread 
%and to avoid the problem of an excessive spread in momentum, 
it is appropriate to appeal to coherent states
both for mathematical and physical reasons. 
In this article, we shall only present the basic mathematical 
results.
We reserve for a forthcoming publication the justification of 
the physical relevance of these states in a cosmological context. 
Let us simply notice the following points. 
In inflationary cosmology, modes are
weakly interacting harmonic oscillators\cite{Maldacena03}. 
Indeed, given that the relative density fluctuations
% modes ????
have small amplitude ($\sim 10^{-5}$), the hypothesis
of weak interactions is perfectly legitimate.
Second, coherent states
%and to recall why they
provide the basis in which the density matrix decoheres 
when considering weakly interacting harmonic oscillators.
This  has been shown by 
Zurek \& {\it al.} \cite{Zurek93}. 
%They have shown that the privileged states are coherent states.
The criterion they used to reach this conclusion is
the minimization of the growth of entropy in the course of the evolution. 
With this criterion,
coherent states are more stable than
squeezed states in that the growth of entropy 
one obtains when they are used as initial states is much slower.
Hence, 
%a corrolary of \cite{Zurek93} is that, 
when starting with a squeezed state, 
there is a phase of rapid growth of the entropy 
which sends the system into a mixture of coherent states
and which is followed by a period of slower increase. 
The entropy growth is in fact directly related to
the decay of the squeezing. 
%(By squee there was a direction in phase space in which
%the spread was smaller than that in the vacuum state.) 
Since it is unlikely that the interactions
in the early cosmology could be sufficiently weak so as
to keep some squeezing, we can use the following mathematical result
to infer that the actual entropy of the final distribution
should be higher than (or equal to) a certain bound.
%lower bound.

Coherent states indeed define a minimal decoherence scheme 
in the following sense. 
Consider the set of final distributions which result from the
initial distribution of Eq. \Eqref{vwrhov'w'}
through some decoherence process
%, which maintain Gaussianity and isotropy,
and which no longer possess any squeezed direction.  
The lowest value of the entropy in this set
is given by the entropy of the incoherent 
superposition of coherent states,
with statistical weights given
by the probabilities to find the corresponding coherent states,
as in Eq. \Eqref{vwdistrib}. 
This distribution gives a lowest entropy simply because 
%We wish now to stress that the above result means that 
%Irrespectively of their physical relevance for cosmology,
%coherent states define a minimal decoherence scheme
%in the following sense.
%as we now argue.
%First, 
coherent states have minimal constant spreads (given quantum uncertainties
and when considering free evolution, see Appendix A).
We shall now explicitely write down this distribution 
and compute the entropy it carries.
%a higher quantum coherence than tha t given by 
%a statistical superposition of coherent states.


%enough to 
%This double behaviour defines a minimal decohered density matrix: that 
%given by the statistical superposition of coherent states whose statistical weights
%are provided by the probabilities to find the corresponding coherent state
%given the initialre state.

%does not vanish as in the present case, Zurek has
%proposed \cite{Zurekbook} that the privileged states 
%are those for which the minimal amount of 
%information is lost on the initial distribution. Equivalently, they are the
%states that produce the least entropy in the course of evolution.
%It has been shown \cite{Zurek93} that these particular states are 
%coherent states. 
%Hence, because of small non-linearities, 
%%when inquiering about the
%%statistical properties of metric perturbations at late time, 
%%e.g. during the radiation dominated era, the 
%primordial metric fluctuations
%will be describded the incoherent superposition of coherent states.

%An important result of Zurek's work is that the growth of entropy is much large for
%squeezed states that for coherent states. Hence, one can introduce a 
%minimal decoherence scheme defined by the coherent states.


%One might object that the intensity of these small non-linearities is too small
%to have an effect on the distribution. But one must bear in mind that first, 
%the strength of a decoherence process is far less sensitive 
%on the coupling than the separation of the eigenstates of the Hamiltonians,
%second highly squeezed states are extremely sensitive to perturbations (for the
%very same reason, since they are highly non-classical state).
%Hence, the self-interactions cannot be neglected and 
%the amplitude basis can no longer be considered as the pointer
%basis. To define the classical basis in this situation, Zurek \cite{Zurekbook}
%proposed that the pointer states are those for which the minimal amount of 
%information is lost on the initial distribution. Equivalently, they are the
%states that produce the least entropy in the course of evolution.

%Explicitely, initial distribution Eq. \Eqref{vwrhov'w'} 
%is replaced by a decohered matrix 
%Noticing that the interactions are to small to effectevely couple different
%modes \cite{Maldacena03}, the density matrix factorizes into the 
%tensorial product over half the momenta of two-mode density matrixes preserving
%the $({\bf k},\, -{\bf k})$ entanglement : 

\subsection{Application to cosmology}
 
When considering the initial distribution 
Eq. \Eqref{vwrhov'w'}, one obtains the following 
%minimal decohered 
density matrix:
%, see Appendix B for a justification :
\ba \label{rhodec}
 \hat \rho_{min} = \int\!\!\frac{d^2v}{\pi} \frac{d^2w}{\pi}\, 
 {\cal{P}}_{2, \, k}(v,w) \, \,
 \ket{v,\, {\bf k}}\bra{v,\, {\bf k}} \otimes 
 \ket{w, \, -{\bf k} }\bra{w, \, -{\bf k}} \, ,
\ea
where the probability distribution is given in Eq. \Eqref{vwdistrib}. 
%\ba
% {\cal{P}}_{z}(v,w) = \abs{{\cal A}_{2,\, k}(v,w)}^2 = 
% %\alpha}^2}
% \inv{\abs{\alpha}^2} \exp{\left(-\frac{\abs{v}^2}{\abs{\alpha}^2} \right)}
% \exp{\left( \abs{w-zv^*} \right)} \, .
%\ea
In Appendix B we show that Eq. \Eqref{rhodec} is indeed 
the resulting normalized distribution. The technical point 
which requires clarification is the extension 
%from one-mode squeezed states
of the above mentionned minimal scheme
%of the results of \cite{Zurek93} 
to two-mode squeezed states.

It should first be noted that when computing expectation
values in leading order in $n_k$, the above distribution can be simplified and
written as a single sum of two-mode coherent states,
as in eq. \Eqref{insuperpcoh}:
\ba \label{rhodec2}
 \hat \rho_{min} \simeq \int\!\!\frac{d^2v}{\pi} \,
 %\frac{d^2w}{\pi}\, {\cal{P}}_{2, \, k}(v,w) 
 \abs{{\cal A}_k(v)}^2  \,\,
 \ket{v,\, {\bf k}} \bra{v,\, {\bf k}} 
 \otimes
 \ket{z v^*, \, -{\bf k} }
 %\bra{v,\, {\bf k}} 
 %\otimes \ket{w, \, -{\bf k} }
 \bra{z v^*, \, -{\bf k}} \, .
\ea
As we shall progressively see, this distribution should be conceived
as the quantum counterpart of the effective distribution of sine
functions discussed in section II.B.

%Using again the relation 
%$\abs{\langle u \vert v \rangle}^2 = \exp({-\abs{u-v}^2})$, one verifies that
%the distribution of Eq. \Eqref{rhodec} is still normalized, 
%${\rm Tr}(\hat \rho_{min}) = 1$, and that 
Secondly by reducing the density matrix, some
entropy has been introduced. One verifies indeed that
 ${\rm Tr}(\hat \rho_{min}^2) < 1$.
%One is drawn to the conclusion that in order to take into account both the
%inevitable decoherence of the initial quantum distribution into a mixture of
%coherent states and the necessity to preserve the temporal coherence of the
%modes, one must write a two-mode distribution of coherent states. 
%In fact, this is both a necessary and a natural refinement of the naive
%distribution considered above, because of the entanglement between 
%${\bf k}$ and $-{\bf k}$ modes in the initial distribution.
The important point is 
that this decohered distribution has retained all the information about 
the temporal coherence of the modes. 
In fact one has
%present in the initial vacuum, as an explicit calculation shows :
\ba \label{RLcoherence}
 {\rm Tr}(\hat \rho_{min} \hat a_{\bf k} \hat a_{-{\bf k}} ) &=&
 \int\!\!\frac{d^2v}{\pi} \frac{d^2w}{\pi}
 \left[ {\cal{P}}_{2, \, k}(v,w) \, v\, w \right] 
 =\int\!\!\frac{d^2v}{\pi} \left[ \abs{{\cal A}_k(v)}^2 v\, (z_kv^*)\right]  = 
 %\abs{v}^2 
 %\int\!\!\frac{d^2u}{\pi}\frac{d^2w}{\pi} \,  \, 
 %\abs{\langle u \vert v \rangle}^2 \, \abs{\langle w \vert zv^*\rangle}^2 = 
 z_k \abs{\alpha_k}^2 \,  ,
 \nonumber\\
 {\rm Tr}(\hat \rho_{min} \hat a_{\bf k}^{\dagger} \hat a_{\bf k} ) &=& 
 {\rm Tr}(\hat \rho_{min} \hat a_{-{\bf k}}^{\dagger} \hat a_{-{\bf k}} ) =
 \abs{\alpha_k}^2 = n_k + 1 \, .
\ea
The first line shows that the cross term is equal 
to that of the original distribution,
see Eqs. (\ref{momenta}, \ref{cterm}). 
In the second line, one sees that
the occupation numbers slightly differ, but to order $1/n_k$ only.
%One has
%\ba \label{occupnumbers}
% {\rm Tr}(\hat \rho_{min} \hat a_{\bf k}^{\dagger} \hat a_{\bf k} ) &=& 
% {\rm Tr}(\hat \rho_{min} \hat a_{-{\bf k}}^{\dagger} \hat a_{-{\bf k}} ) =
% \abs{\alpha_k}^2 = n_k + 1 \, ,\ea
%in the place of $\abs{\beta_k}^2 = n_k$.

The effect  of having increased by $1$ the occupation number while having kept
untouched the cross-correlation means two things. 
First, the (relative) degree of coherence
has been reduced and therefore
%is the manifestation that the new distribution has lost cohe
some entropy has been created.
Secondly, in the high occupation limit, the two-point function of
Eq. \Eqref{Ginx} is 
not affected by this loss of coherence: for relevant modes,
 the relative change being of the order of $1/n_k \sim 10^{-100}$.

%hat the degree of coherence has been reduced and that therefore
%is the manifestation that the new distribution has lost cohe
%some entropy has  as we now discuss. 
%However, since all the information about the relative phase between ${\bf k}$
%and $-{\bf k}$ is preserved, it is at most half the maximal (thermal) entropy,
%see bellow. 
%found by 
%\cite{MukhaBrandyProkopec92,MukhaBrandyProkopec93,Giovannini93a}.


\subsection{Minimal entropy and the neglect of the decaying mode}

The entropy of any Gaussian two-mode distribution can be exactly
calculated \cite{Zeh85,Serafini04} by using the fact that  
%This calculation is instructive since it will enable us to 
%compute the entropy associated with any given decoherence scheme, 
%and in particular for the procedure which consists in neglecting 
%the decaying mode. 
%To compute it, one uses the fact that 
the density matrix of a two-mode squeezed state 
%vacuum 
%CORRECT?
%
is unitarily equivalent to the tensorial product of 
two thermal density matrices of oscillators, see Appendix C.
We shall name $a$ and $b$ these two real oscillators.
%thermal density matrices 
One has
\ba \label{map}
 \hat \rho_{2,\, k} = {\cal M}^{\dagger} \, \hat \rho_{th,\, a} \otimes 
 \hat \rho_{th,\, b} \, {\cal M} \, ,
\ea
where $\cal M$ is a unitary operator acting on the two-mode Hilbert space. 
The expression of the (von Neumann) entropy immediately follows:
\ba
 S\left[ \hat \rho_{2,\, k} \right] &=& S\left[ \hat \rho_{th,\, a} \right] +
 S\left[ \hat \rho_{th,\, b} \right] \, ,
\ea
where the entropy of a thermal bath with mean occupation $\bar n$ is
\ba
 S\left[ \hat \rho_{th} \right] = (\bar n +1 ) \ln(\bar n +1 ) - 
 \bar n \ln(\bar n) \, .
\ea
%This mapping can be constructed for any Gaussian density matrix 
%but we 
When considering only distributions preserving homogeneity and isotropy,
%whose only non-vanishing momenta are
%\ba \label{expectvalues}
% &&\langle \hat a_{\bf k}^{\dagger}\hat a_{\bf k} \rangle = n_{k} \, , \qquad 
% \langle \hat a_{-{\bf k}}^{\dagger}\hat a_{-{\bf k}}\rangle = n_{k} \, , \nonumber \\
% &&\langle \hat a_{\bf k}\hat a_{-{\bf k}} \rangle =  c = c_r + i c_i \, , 
%\ea
%where $\pm$ designate the two modes ${\bf k}, -{\bf k}$ we
%considere, $n_\pm$ the corresponding
%occupation numbers, and $c$ is the complex number which 
%characterizes the coherence. 
%When $n_+=n_-=n_k$, Then 
the occupation numbers of the thermal matrices are equal and
%equal
%to each other and 
given by
%related to $n_{\pm}$ and $c$ by
\ba \label{tilde n}
 \bar n_{k} + \frac{1}{2} = 
 \left( (n_{k} + \frac{1}{2})^2 - \abs{c_{k}}^2 \right)^{1/2} \, ,
\ea
where $n_k$ and $c_k$ are defined in Eq. \Eqref{momenta}.

Let us apply this result to several cases.
%density matrices. 
First, for the two-mode in vacuum of Eq. \Eqref{inoutvac},
%a pure squeezed state
the occupation number and the coherence term are 
related by Eq. \Eqref{cterm}, one has
$\bar n_k =0$ as expected. Hence 
%von Neumann 
the entropy vanishes.
% in thiscase. 

For the decohered matrix Eq. \Eqref{rhodec}, 
using Eq. (\ref{RLcoherence}), 
%\ref{occupnumbers}),
the occupation number of the two thermal baths are
\ba  
 \bar n_{k} = 
 \frac{1}{2} \left( -1+\sqrt{8(n_k +1 ) +1}\right) \sim
 \sqrt{2  n_k} \, , 
% \tilde n_{2} &\sim& \sqrt{2 \bar n} \, ,
\ea
where the last term is the leading order when 
$n_k \gg 1$.
%high occupation numbers.
The two-mode entropy of this mixture is then
\ba
\label{Smin}
 S\left[ \hat \rho_{min} \right] &=& 2 \, S\left[ \hat \rho_{th}
 \right] = 2 \ln \bar n_k + O(1) \, , 
%2 \left[(\bar n_k +1 ) \ln(\bar n_k +1 ) - \bar n_k \ln(\bar n_k) \right] 
\nonumber \\
 &=& \ln n_k + O(1) = 2\, r_k +  O(1) \, , 
 \nonumber \\
 &\simeq& 100 \ln(10)  \, .
\ea
In the second line, we have expressed the occupation number in term of the
squeezing parameter $r_k$ : $n_k = \sh^2 r_k$. 
Hence, a two-mode squeezed vacuum
state which decoheres in the two-mode coherent basis goes along with an entropy
of $S_{{\bf k},\, -{\bf k}} = 2 \, r_k$ {\it per two-mode}. 
This value is large, but not maximal. Indeed, 
had the coherence term $c_k$ vanished while keeping the occupation numbers 
fixed\cite{MukhaBrandyProkopecg}, 
one would have found the maximal value of the entropy 
which is given by twice this above value, i.e.
\ba
S_{{\bf k},\, -{\bf k}}^{max}= S_{{\bf k},\, -{\bf k}}^{inc} = 4 \, r_k\, ,
\ea 
or $S^{inc}_{\bf k} = 2 \, r_k$ per mode ${\bf k}$.
%As previously noticed, a vanishing coherence term can be obtained from averaging
%over the relative phase.  

It is interesting to notice that the entropy
associated with the Gaussian distribution Eq. \Eqref{Peff} 
of sine functions
%simplified description based on the neglection of the decaying mode gives 
%the same entropy as 
equals that of Eq. \Eqref{rhodec}, up to an arbitrary constant
which arises from the usual 
ambiguity of attributing an entropy to a classical distribution. 
(This ambiguity can be lifted when introducing $\hbar$ to normalize
the phase space integral.) Using this trick, the entropy associated
with 
%the Gaussian distribution 
Eq. \Eqref{Peff} is $S_{e\!f\!f}= 2 \, r_k$ for each 
{\it independent} mode, since this entropy is maximal. 
%Hence its entropy is $S_{e\!f\!f}= 2 \, r_k$ for each 
%{\it independent} mode, that is,
However, for each 
{independent} mode here
means for each two-mode since the mode $-{\bf k }$ 
is no longer independent of the ${\bf k }$ mode once
the cosines have been neglected, see Eq. \Eqref{Aeff}.
From this equality of entropies, we conclude that the
quantum density matrix which corresponds to the Gaussian
ensemble of sine functions is precisely given by Eq. \Eqref{rhodec}.

A priory one might think that many quantum distributions 
can be associated with the classical distribution Eq. \Eqref{Peff}.  
This is not the case when imposing that Gaussianity is preserved
and that the entropies coincide. Indeed, 
%Finally, iIt should be also stressed that 
in the high squeezing limit, 
%notice also the extreme sensitivity of the value of 
the entropy is an
extremely sensitive function of the relative coherence. 
To see this dependence,
let us calculate the entropy of a generic distribution \Eqref{momenta},  
%a density matrix with equal occupation numbers
%$\bar n_{+} = \bar n_{+} = n_k$
and let us write the norm of the coherence term as
\ba
\label{paramd}
 \abs{c_k}^2 = n_k(n_k + 1 - \delta_k) \, ,
\ea
where $\delta_k$ is a real number between $0$ and $n_k + 1$.
The equation \Eqref{tilde n} has the solution
$\bar n(\bar n +1 ) =  n_k \delta_k $. 
%Several cases occur for
%large occupation numbers.
They are three characteristic values. 
$\delta_k =0$ obviously corresponds to the pure squeezed state: the in vacuum.
%: $\bar n=0$.
$\delta_k = 1$ corresponds to the minimal decoherence scheme
with entropy given in Eq. \Eqref{Smin}. $\delta_k = n_k + 1$
corresponds to the thermal case with maximum entropy. 
From this analysis we see that 
the uncertainty in the definition of the quantum distributions
which give rise to the entropy $S_{{\bf k},\, -{\bf k}}= 2 r_k$ is 
very limited: $\delta_k$ must be of order $1$ in the following sense.
%Let us c
Consider that the loss of coherence scales as
%has a spectra index $\gamma$: 
$\delta_k \propto n_k^\gamma$. Then the thermal occupation number and the entropy
respectively scale as $\bar n_k \propto n_k^{(1+\gamma)/2}$ and
$S_{{\bf k},\, -{\bf k}} = (1+\gamma) 2 r_k + Const$. 
This linear dependence in $r_k$
implies that the distributions with entropy given
by Eq. \Eqref{Smin} all have $\gamma =0$. 


We re-emphasize that a value of $\delta_k$ smaller than $1$ is 
unlikely in the context of
primordial fluctuations since it would mean that the distribution
has kept some of its quantum squeezeness. The remaining question thus
concerns the computation of $\delta_k$, noticing that it can receive
contributions both from the inflationary period and from the adiabatic 
era\cite{KiefPolStar00}.
The challenge is to determine which one is more important and
what could be a realistic value of $\delta_k$.
% \cite{envrac}.

As a final comment, we provide an upper bound for the 
decoherence entropy which could have resulted from processes
in the inflationary phase. Because increasing the decoherence implies
increasing the power of the growing mode,
% and because
%at that time the decaying mode is not yet as small as at recombination, 
one obtains a bound on the decoherence level 
when requiring that the power of the decaying mode
be equal to that of the growing mode at the onset of the adiabatic era.
(This requirement follows from the fact
that the rms value of the primordial 
fluctuations (of the Bardeen potential) cannot be 
much higher than that obtained from in vacuum
because otherwise this would invalidate the whole
framework of linear metric perturbations.)
% evolving on the top a FRW model.)
Using Eq. \Eqref{Ginx} evaluated at $\eta_r$
and the parameterization of Eq. \Eqref{paramd}, one obtains
\ba
\delta_k = n_k^{1/2}\, , \qquad S^{upper}_{{\bf k},\, -{\bf k}}= 3 r_k + O(1)\ .
\ea
If no further decoherence is added in the adiabatic
area, this should be the maximum amount of entropy stored
in the primordial spectrum. Notice that when
evaluated at recombination, the two-point
function is still unaffected by this modification of the coherence
because at that time the decaying mode has still further decreased.
Indeed the residual modifications are then of the order of $n_k^{-1/2}
\sim 10^{-50}$. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%r%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vskip .5 truecm

{\bf Acknowledgments:} %The acknowledgements.
We are grateful to Dani Arteaga, 
Claus Kiefer, Serge Massar, 
Jihad Mourad and Alexei Starobinsky
for useful remarks.
%D$^{\rm r}$ Obadia and D. Campo for their comments and advice during the

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{appendix}
\section{Coherent states}

This appendix aims to present 
%give a self-contained presentation of coherent states, with special emphasis on
the properties which we shall use
in the body of the manuscript. 
For more details, we refer to \cite{Glauber63a,Glauber63b,Zhang90}.

%They are several equivalent ways to define c
Coherent states (of a real oscillator) can be 
%The definition we adopt \cite{Glauber63b} 
defined as eigenstates of the annihilation operator:
\ba \label{DefCoh1}
 \hat a \ket{v} = v \ket{v} \, ,
\ea
where $v$ is a complex number.
%The development on this state
%The components of this state in the
%occupation number representation $\{ \ket{n} \} $ are on Fock basis is
In Fock basis it is written as
\ba \label{DefCoh2}
 \ket{v} = e^{-\frac{\vert v \vert^2}{2}}  \sum_{n=0}^{\infty}
 \frac{v^n}{\sqrt{n!}} \ket{n} \, ,
\ea  
where the exponential prefactor guarantees that the state is normalized 
to unity $\langle v \vert v \rangle=1$.
They are also obtained by the action of a displacement operator on the vacuum :
\ba \label{displacement}
 \ket{v} = \hat D(v) \ket{0} = e^{v^* \hat a - v \hat a^{\dagger}} \, .
\ea

The first interesting property of coherent states is that they correspond 
to states with a well defined complex amplitude $v$. 
Indeed, by definition \Eqref{DefCoh1}, 
the expectation values of the annihilation and creation operators are
\ba
 \bra{v} \hat a \ket{v} = v \, , \quad \bra{v} \hat a^{\dagger} \ket{v} = v^{*} 
 \, .
\ea
Thus the mean occupation number is
\ba \label{occupnumber}
 \bra{v} \hat a^{\dagger} \hat a \ket{v} = \abs{v}^2 \, .
\ea 
It is to be also stressed that the variances vanish:
\ba \label{nullvariance}
 \Delta \hat a^2 = \bra{v} \hat a^2 \ket{v} - \bra{v} \hat a \ket{v}^2 = 0 \, , 
 \quad \Delta \hat a^{\dagger \, 2} = \bra{v} \hat a^{\dagger \, 2} \ket{v} - 
 \bra{v} \hat a^{\dagger} \ket{v}^2 =  0 \, .
\ea 

From these properties one sees that the expectation values
of the position and momentum operators (in the Heisenberg picture) 
\ba 
 \hat q(t)=\sqrt{\frac{\hbar}{2\omega}}\left(\hat a e^{-i \omega t} + 
 		\hat a^{\dagger} e^{i \omega t}\right) \, , \qquad 
 \hat p(t)=-i\sqrt\frac{\hbar\omega}{2}(\hat a e^{-i \omega t} - 
 				\hat a^{\dagger}e^{i \omega t}) 
 \, , \nonumber
\ea
are
\ba \label{cltrajincohstate}
 \bar q(t) &=& \bra{v} \hat q(t) \ket{v} = 
 \sqrt{\frac{\hbar}{2\omega}}( v e^{-i \omega t} + v^* e^{i \omega t}) 
            = \sqrt{\frac{2\hbar}{\omega}} \abs{v} \cos(\omega t - \phi_v) 
 \, , \nonumber \\ 
 \bar p(t) &=& \bra{v} \hat p(t) \ket{v} = 
  -i\sqrt{\frac{\hbar\omega}{2}} ( v e^{-i \omega t} - v^* e^{i \omega t}) 
            = -\sqrt{2\hbar\omega} \abs{v} \sin(\omega t - \phi_v) 
	    = \partial_t \bar q(t) \, .
\ea
We have used the polar decomposition $v=\abs{v} e^{i \phi_v}$.
These expectation values have a well defined amplitude and phase and 
follow a classical trajectory of the oscillator. 
This is due to the ``stability'' of coherent states under the
evolution of the free Hamiltonian $2 H_0= p^2 + \omega^2 q^2$:
%in the Sch\"{o}dinger picture. If the 
if the state is $\ket{v}$ at time $t_0$, one immediately gets from 
\Eqref{DefCoh2} that at a later time $t$, the state is 
%a coherent state
given by $\ket{v(t)} = \ket{ve^{-i\omega (t-t_0)}}$.
Notice that the variances of the position and the momentum are 
\ba
 \Delta \hat q^2 = \frac{\hbar}{2\omega} \, , \qquad  
 \Delta \hat p^2  = \frac{\hbar\omega}{2} \, .
\ea
%when $\hbar = 1$.
They minimize the Heisenberg uncertainty relations and are time-independent.
Hence, in the phase space $(q,p)$, a coherent state can be
considered as a unit quantum cell $2\pi \hbar$ in physical units 
(see also \Eqref{measure} for the measure of
integration over phase space) centered on the classical position and momentum of
the harmonic oscillator $(\bar q(t),\bar p(t))$. 
In the large occupation number limit $\abs{v}\gg 1$,
coherent states can therefore be
interpreted as classical states since 
$\Delta \hat q / \sqrt{\bar q^2+ \bar p^2/\omega^2} =
\Delta \hat p / \sqrt{\omega^2\bar q^2+ \bar p^2} = 1/2\abs{v}$.
%This is 
%one reason 
%why they are considered as the quantum counterparts of
%classical points in phase space. 
This is a special application of the fact that coherent states
can in general be used to define the classical limit 
of a quantum theory, see \cite{Zhang90} and references therein.

One advantage of coherent  states \cite{Glauber63a} is that the 
calculations of Green functions resembles closely to those of the 
corresponding classical theory (i.e. treating the fields not as operators 
but as c-numbers) provided either one uses normal ordering,
or one considers only the dominant contribution when $\abs{v}\gg 1$.
%In preparation for the calculations with field
We compute the Wightman function in the coherent state $\ket{v}$
\ba
\widetilde G_v(t,t') &=& \bra{v} \hat q(t) \hat q(t') \ket{v} \nonumber \\
           &=& \langle :\hat q(t)\hat q(t') : \rangle_v + 
	   \frac{\hbar}{2\omega} e^{i\omega(t-t')} \, ,
\ea
where we have isolated the contribution of the vacuum.
The normal ordered correlator is order $\abs{v}^2$: 
\ba \label{cohgreen}
 \langle :\hat q(t)\hat q(t') : \rangle_v &=& 
 \frac{\hbar}{\omega} \, 
 Re\left[\langle \hat a^2 \rangle_v e^{-i\omega(t+t')} + 
    \langle  \hat a^{\dagger} \hat a \rangle_v e^{i\omega(t-t')} \right]
    \nonumber  \\ 
 &=& \frac{2\hbar}{\omega} \vert v \vert^2 \cos(\omega t - \phi_v) 
     \cos(\omega t' -\phi_v)   = \bar q(t) \,  \bar q(t') \, .
\ea
We see that the perfect coherence of the state, namely 
$\abs{\langle \hat a  \hat a  \rangle_v } 
= \langle \hat a^\dagger  \hat a  \rangle_v  $ 
is necessary to 
combine the contributions of the diagonal and the interfering term so as to
bring the time-dependent classical position $ \bar q(t)$ in 
Eq. \Eqref{cohgreen}.


The wave-function of a coherent state in the coordinate
representation is given by 
\ba \label{CohWaveFunct}
  \psi_{v}(q) = \left(\frac{\omega}{\pi\hbar}\right)^{1/4} 
 \exp\left(-\frac{\omega}{2\hbar}(q-\bar q)^2 -i \frac{\bar p q}{\hbar} \right) 
 \, ,
\ea
where $v=(\omega \bar q + i \bar p)/\sqrt{2\omega\hbar}$.
This follows from the definition 
$\bra{q} \hat a \ket{v} = v \langle q \vert v \rangle$. 
From this equation one notes that two coherent states are not orthogonal.
The overlap between two coherent states is
\ba \label{overlap}
 \langle v \vert w \rangle = \exp\left(v^{*} w - \inv{2}\abs{v}^2 - 
 \inv{2}\abs{w}^2 \right)  \, .
\ea
Nevertheless they form an (over)complete
basis of the Hilbert space in that 
the identity operator in the
coherent state representation $\{ \ket{v} \}$ reads
\ba
\label{coh1}
 {\bf 1} = \int\!\!\frac{d^2v}{\pi} \, \ket{v} \bra{v} \, .
\ea 
The measure is 
\ba \label{measure}
 \frac{d^2v}{\pi} = \frac{d({\mathrm Re} v) d({\mathrm Im} v)}{\pi} = 
 \frac{d\bar q d\bar p}{2\pi \hbar} \, .
\ea
The representation of identity can be established by calculating 
the matrix elements of both sides of
the equality in the coordinate representation $\{ \ket{q} \}$, 
with the help of \Eqref{CohWaveFunct}. 
%This writing permits to expand states and 
%operators on the coherent state basis, leading to interesting 
%analytical properties 
%\cite{Glauber63b}, but this is out of the main scope of this paper.
%For us the importance of Eq. \Eqref{coh1} is to use coherent states
%to represent the detection of classical configurations in quantum terms.


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

\section{Application of Zurek \& al. results to the cosmological case}

In this appendix we show that Eq. \Eqref{vwrhov'w'}
is indeed the minimal decohered distribution by
decomposing the 
complex 
%The field 
mode $\hat \phi_{\bf k}$ into
two real oscillators $\hat \phi_{1}$ and $\hat \phi_{2}$ given by
%s a complex harmonic oscillator that can be separated into 
its real and imaginary parts.
% $\hat \phi_{1}$ and $\hat \phi_{2}$.
Since the two-mode Hamiltonian is Hermitian, it splits into the sum 
of two identical one-mode oscillator Hamiltonians 
for $\hat \phi_{1}$ and $\hat \phi_{2}$ separately. 
Notice that the annihilation operators of these two real oscillators,
$\hat a_1 = \left(\hat a_{\bf k} + \hat a_{-\bf k}\right)/\sqrt{2} , \, 
\hat a_2 = -i\, \left(\hat a_{\bf k} - \hat a_{-\bf k}\right)/\sqrt{2}$, 
mix ${\bf k}$ and $-{\bf k}$ annihilation operators. 
Hence they can easily take
into account the entanglement between these two modes.    

A two-mode squeezed state $\kett{0 in, {\bf k}}$ 
can always be written as the tensorial
product of the two one-mode squeezed states\cite{schum}.
In our case, the one-mode squeezed states
are those of the oscillators $1$ and $2$ because the
Hamiltonian separates. Thus we have
\ba
 \kett{0 in, {\bf k}} = \ket{0 in, 1} \otimes \ket{0 in, 2} \, . 
\ea
The one-mode squeezed states are governed by the same parameter $z/2$ :
\ba
 \ket{0 in, 1} = \frac{1}{\sqrt{\abs{\alpha}}} \sum\limits_{n=0}^{\infty} 
 \left( \frac{z}{2} \right)^n \frac{\sqrt{2n !}}{n!} \ket{2 n,\, 1} \, .
\ea
%and a similar 
The same expression holds for the ket  $\ket{0 in, 2}$.
% \kett{0 in, {\bf k}}$.

The overlap of this one-mode squeezed state with a one-mode coherent state is
\ba
 \langle v ,\, 1 \vert 0 in, 1  \rangle = 
 %\int\!\!\frac{d^2 v_1}{\pi} \, 
 \frac{1}{\sqrt{\abs{\alpha}}} \exp\left(- \abs{v_1}^2/2+ zv_1^{*\, 2}/2 \right) \, .
\ea

According to \cite{Zurek93}, when taking small
interactions into account, the density matrix of a
one-mode squeezed state will preferable decohere into the mixture
\ba
 \hat \rho_{red,\, 1} =  \int\!\!\frac{d^2 v_1}{\pi} \, 
  {\cal P}_1(v_1)  \, 
 \ket{v_1} \bra{v_1} \, .
\ea
where the statistical weight is given by the probability to find a coherent
state starting from the in vacuum :
\ba \label{proba1}
 {\cal P}_1(v_1) = \abs{\langle v ,\, 1 \vert 0 in, 1  \rangle }^2 &=&
 \frac{1}{\abs{\alpha}} \exp\left(- \abs{v_1}^2 + Re(zv_1^{*\, 2}) \right) \, ,
 \nonumber \\
 &=& \frac{1}{\abs{\alpha}}e^{- 2 R_1^2} \,
 e^{-  I_1^2/2 \abs{\alpha}^2 + O(n^{-3/4}R_1 I_1)} \, .
\ea
In the second line we have introduced the real and imaginary parts
of $v_1$ in order to show that one gets an ellipse of great axis
equal to $\abs{\alpha}^2$ which is oriented along the imaginary axis.
The width of the small axis is $1/2$, as in vacuum. 
%%%The surface is thus $1/2

%$R_1 = Re (v_1)$ 

For this decoherence procedure to be valid,
as noticed in \cite{KiefPolStar98a}, it is important that the 
interactions do not break the
coherence between ${\bf k}$ and $-{\bf k}$ modes, or equivalently do not mix 
$\phi_1$ and $\phi_2$. 

%The final step concerns the fact that 
The product of two one-mode coherent states $1$ and $2$ is also the
product of a coherent state for the ${\bf k}$ and $-{\bf k}$ modes:
%for the and a coherent state $w_{-{\bf k}}$ :
\ba
 \ket{v_1}\otimes\ket{v_2} &=&  
 \hat D_1(v_1) \hat D_2(v_2) \ket{0 in, 1} \otimes \ket{0 in, 2} 
 \, , \nonumber \\
 &=& \hat D_{\bf k}(v) \hat D_{-{\bf k}}(w) 
 \ket{0 in, {\bf k}} \otimes \ket{0 in, -{\bf k}} 
 \, , \nonumber \\
 &=&  \ket{v, {\bf k}} \otimes \ket{w, -{\bf k}} \, ,
\ea 
where the amplitudes are related by 
\ba
 v=\frac{v_1+iv_2}{\sqrt{2}} \, , \qquad w=\frac{v_1-iv_2}{\sqrt{2}} \, .
\ea
Finally, the product of the probabilities \Eqref{proba1} give the probability 
\Eqref{vwdistrib}. Performing the change of variables from $(v_1,\, v_2)$ to 
$(v,\, w)$ completes the proof.

%As a final comment, we wish to draw the attention to the following
%point. When using the real oscillators $\phi_1$ and $\phi_2$ 
%the highly squeezed character show up in a certain direction
%whose orientation depends of arbitrary conventions. For instance
%a shift in the spatial origin introduces a relative phase 
%shift of the operators $\hat a_{\bf k} $ and $\hat a_{-\bf k}$
%which enter in $\hat a_1$. ???????????
%Still requires clarification........








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

\section{Diagonalization of the covariance matrix}

%The transformation Eq. \Eqref{map} amounts to diagonalize 
%the covariance matrix of the distribution.

Introducing the position and momentum variables for each mode, i.e.
$\hat a_{{\bf k}} = (\hat q_{{\bf k}} + i\hat p_{{\bf k}})/\sqrt{2}$ and 
$\hat a_{-{\bf k}} = (\hat q_{-{\bf k}} + i\hat p_{-{\bf k}})/\sqrt{2}$, 
and defining the vector 
\ba
 {\bf \zeta}^{\dagger} = \left( \hat q_{{\bf k}} \, \, \hat p_{{\bf k}} \, \, 
 \hat q_{-{\bf k}} \,\,  \hat p_{-{\bf k}} \right) \, ,
\ea
one has the covariance matrix 
%\footnote{This is not the covariance matrix of
%the most general two-mode Gaussian operator. We restric to the matrices of the
%form \Eqref{expectvalues}. For a general study, see \cite{Serafini04}.} 
\ba
 C = \langle \left[ \, {\bf \zeta}_i , \, {\bf \zeta}_j \right]_+ \rangle = 
 \left( 
 	\begin{array}{c c c c}
	n_{k} + \frac{1}{2} & 0 & c_r & c_i \\
	0 & n_{k} + \frac{1}{2} & c_i & -c_r \\
	c_r & c_i & n_{k} + \frac{1}{2} & 0 \\
	c_i & -c_r & 0 & n_{k} + \frac{1}{2} 
	\end{array}
 \right) \, ,
\ea
where $\left[ \, , \, \right]_+$ is the anticommutator. 
Notice that $\left( n_{k} + \frac{1}{2}\right)^2 - \abs{c_k}^2 > 0$ 
is a necessary condition for the matrix to have positive eigenvalues.

The transformation Eq. \Eqref{map} amounts to diagonalize 
this matrix:
% of the distribution.This matrix can be diagonalized
\sba
 C &=& M^{t} T  M \, , \\
 T &=&  \left( \bar n_{k} + \frac{1}{2}\right) {\bf 1}
 %{\rm diag}( \tilde n_{+} + \frac{1}{2} , \, \tilde n_{+} + \frac{1}{2} ,
 % \, \tilde n_{-}+ \frac{1}{2} , \, \tilde n_{-}+ \frac{1}{2} ) \, .
\sea
The matrix $T$ is the covariance matrix of the two thermal density matrices
$\hat \rho_{th,\, a} \otimes \hat \rho_{th,\, b}$ in Eq. \Eqref{map}. 
The matrix $M$ is the
product of two local transformations  
and one global rotation $R$. The latter brings the covariance matrix $C$
under a $2\times 2$ bloc diagonal form. A product of 
local rotations $R_1(\theta_1) \oplus R_2(\theta_2)$
diagonalize each bloc, and the product of 
local squeezing $S_1(r_1) \oplus S_2(r_2)$ brings the
resulting matrix under the form $T$. Explicitly one has
\ba
 R(\phi) &=& 
 \left( 
 	\begin{array}{c c c c}
	\cos \phi & 0 & -\sin \phi &  0 \\
	0 & \cos \phi & 0 & -\sin \phi \\
	\sin \phi &  0 & \cos \phi & 0  \\
	 0 & \sin \phi& 0 & \cos \phi 
	\end{array}
 \right) \, , \nonumber \\
 R(\theta) &=& 
 \left( 
 	\begin{array}{c c}
	\cos \theta & \sin \theta \\
	-\sin \theta & \cos \theta  
	\end{array}
 \right)  \, ,\qquad
 S(r) = 
 \left( 
 	\begin{array}{c c}
	e^{r/2} & 0 \\
	0 &  e^{-r/2} 
	\end{array}
 \right)  \, ,
\ea
with the rotation angles $\phi = \pi/4$, $\theta_1=\theta_2$ given by 
$\tan 2\theta = -c_i/c_r$, and the
squeezing parameter $r_1=-r_2$ defined by $\th r = - \abs{c_k}/(n_k + 1/2)$. 

The eigenvalue $\bar n_k$ of the thermal matrices
is easily obtained by conservation of the determinant: 
%and trace
\ba
 {\rm det} \, C &=& \left( (n_{k} + \frac{1}{2})^2 - \abs{c_k}^2 \right)^2
 = \left(\bar n_{k} + \frac{1}{2}\right)^4 
 %\left( (n_{+} + \frac{1}{2})^2 - \abs{c}^2 \right) 
 %\, \left( (n_{-} + \frac{1}{2})^2 - \abs{c}^2 \right)  = 
 %\left(\tilde n_{+} + \frac{1}{2}\right) ^2 
 %\left(\tilde n_{-} + \frac{1}{2}\right) ^2  
 %\, \nonumber \\
 %{\rm Tr} \, C &=& 4 \left(n_{k} + \frac{1}{2} \right) = 
 %4 \left(\bar n_{k} + \frac{1}{2} \right) 
 %2 (n_{+} + n_{-} + 1) =  2 (\tilde n_{+} + \tilde n_{-} + 1) 
 \, . \nonumber
\ea

\end{appendix}


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\end{thebibliography}

\end{document}

astro-ph0404021.extracted_bib
\begin{thebibliography}{9}

\bibitem{Mukhanov81} V. Mukhanov, C. Chibisov, 
JETP Lett. ${\bf 33}$, No. 10, 532 (1981), astro-ph/0303077.

\bibitem{Staro79} A.A. Starobinsky,
JETP Lett. ${\bf 30}$, 682 (1979), 
Pisma Zh.Eksp.Teor.Fiz. ${\bf 30}$, 719 (1979). 

\bibitem{Grishchuk90} L.P. Grishchuk and Yu.V. Sidorov, 
Phys. Rev. D$\bold{42}$, 3413 (1990).

\bibitem{Zurekbook} W. H. Zurek, in
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Perez-Mercader, and W. H. Zurek, 
(Cambridge University Press, Cambridge, UK, 1993): Theor. Phys.

\bibitem{Zurek93} W. H. Zurek, S. Habib, J. P. Paz,
Phys.Rev.Lett. $\bold{70}$, 1187 (1993). 

\bibitem{Maldacena03} J. Maldacena,
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R. Brandenberger, V. Mukhanov, and T. Prokopec,
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