The Inflationary %Universe: \\ Paradigm: Predictions for CMB
Renaud Parentani
\vskip 1truecm %\centerline{{25-04 }} %\todaydate}}\vskip 1truecm \centerline{{Abstract }} \vskip 0.4truecm {%In this article, we shall briefly %We recall the main reason why t We review why the search for a causal explanation of the large scale %homogeneity and isotropy properties of the universe %and the primordial density fluctuations %seen as temperature fluctuations in the CMB supports the idea that %before nucleosynthesis %present the reasons which lead to the idea that %{the universe experienced} %%there was %, before nucleosynthesis, an extended % long period %(in the units of the local %Hubble time $H^{-1}$) of accelerated expansion, called inflation, preceded primordial nucleosynthesis. %The reason is two-fold: it concerns the possibility that both %the isotropy of the CMB at large angle %and the spectrum of primordial density fluctuations result from some causal process. %of havinga causal origin for the isotropy of the CMB at large angle, %and secondly of identifying a mechanism which could generate a spectrum %of primordial fluctuations whose properties are in agreement with the observations. %We then explain why inflation generically implies that %The first consequence of inflation is that As a consequence of inflation, all pre-existing classical structures are washed out, %.In fact, %in the simplest inflationary models, and the primordial density fluctuations (the seeds of the large scale structures) %which are seen as temperature fluctuations in the CMB %and which are the seeds of the large scale structures %density fluctuations seen in the CMB %is minimal as it only result from the amplification of vacuum quantum fluctuations. %We present t The properties of the spectrum %so obtained are derived and compared to %observational data. % concerning those of the spectrum of CMB anisotropies. %extracted from %%concerning the CMB fluctuation spectrum. The agreement is striking. %The first reason follows from the fact that in the %standard hot big bang scenario the particle horizon prevents %causal processes to account for the %to allow for a causal explanation of the isotropy %of the CMB at large angle. This impossibiity is removed if %the duration of the inflationary phase is large enough for all visible %(comoving) scales to have been inside the Hubble radius during this phase. %homogeneity at large scale, %extreme isotropyhof the CMB at large angle, all visible %(comoving) scales should have been inside the Hubble radius %during the inflationary phase. % %the duration of the %inflationary phase should be large enough for all visible (comoving) scales to %We shall see that this hypothesis leads in turn to the necessity of considering %quantum mechanical fluctuations and to what extend several problems %can be considered as solved.} %
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\begin{document}
%\centerline{{CURRICULUM VITAE}}
%
$\quad$
\vskip .5 truecm
\title{The Inflationary 
%Universe: \\
Paradigm:
Predictions for CMB}
\vskip 1.5 truecm
\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}
%\author{Parentani Renaud~$^{\text{a}}$}
%,\ \ Second author~$^{\text{b}}$,\ \ Third author~$^{\text{c}}$}
%\address{
%
\maketitle
\thispagestyle{empty}
%%%%%%%%%%%%%%%%%%%%%%%%Abstract  
%\begin{Abstract}
\vskip 1truecm
%\centerline{{\bf 25-04 }}
%\todaydate}}\vskip 1truecm
\centerline{{\bf Abstract }}
\vskip 0.4truecm

{%In this article, we shall briefly 
%We recall the main reason why t
We review why the search for a causal explanation of the large scale
%homogeneity and isotropy 
properties of the universe 
%and the primordial density fluctuations
%seen as temperature fluctuations in the CMB
supports the idea that 
%before nucleosynthesis
%present the reasons which lead to the idea that 
%{\it the universe experienced}
%%there was
%, before nucleosynthesis,  
an extended
% long 
period 
%(in the units of the local
%Hubble time $H^{-1}$)
of accelerated expansion, called inflation,
preceded primordial nucleosynthesis. 
%The reason is two-fold: it concerns the possibility that both
%the isotropy of the CMB at large angle
%and the spectrum of primordial density fluctuations result from some causal process.
%of havinga causal origin for the isotropy of the CMB at large angle, 
%and secondly of identifying a mechanism which could generate a spectrum 
%of primordial fluctuations whose properties are in agreement with the observations. 
%We then explain why inflation generically implies that 
%The first consequence of inflation is that
As a consequence of inflation,
all pre-existing classical structures are washed out,
%.In fact, 
%in the simplest inflationary models, 
and the primordial density fluctuations
(the seeds of the large scale structures)
%which are seen as temperature fluctuations in the CMB
%and which are the seeds of the  large scale structures
%density fluctuations seen in the CMB 
%is minimal as it 
only result from the amplification of vacuum quantum fluctuations. 
%We present t
The properties of the spectrum 
%so obtained 
are derived and compared to 
%observational data.
% concerning 
those of the spectrum of CMB anisotropies.
%extracted from 
%%concerning the CMB fluctuation spectrum.
The agreement is striking.

%The first reason follows from the fact that in the 
%standard hot big bang scenario the particle horizon prevents 
%causal processes to account for the 
%to allow for a causal explanation of the isotropy 
%of the CMB at large angle. This impossibiity is removed if 
%the duration of the inflationary phase is large enough for all visible 
%(comoving) scales to have been inside the Hubble radius during this phase.



%homogeneity at large scale, 
%extreme isotropyhof the CMB at large angle, all visible 
%(comoving) scales should have been inside the Hubble radius 
%during the inflationary phase.
%
%the duration of the 
%inflationary phase should be large enough for all visible (comoving) scales to 

%We shall see that this hypothesis leads in turn to the necessity of considering
%quantum mechanical fluctuations and to what extend several problems 
%can be considered as solved.}
%\end{Abstract}
\par\medskip
\centerline
{\rule{2cm}{0.2mm}}\medskip\medskip\medskip
%\setcounter{section}{0}
%\selectlanguage{english}

%\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%The main text (in English)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

%  . Given that the focus of this note concerns the CMB,
%we shall clearly separate the issues and the difficulties which
%concern the dynamical realisation of inflation 
%from the properties inflation implies quite 
%irrespectively of its dynamical realisation. 
This paper is conceived as a pedagogical introduction to
the motivations for inflation, to its mathematical settings, 
and to its implications for 
%particularly those concerning 
% with particular emphasis on
%with inflation. We shall put a particular emphasis on the implications for 
%the properties 
%of the spectrum 
%of temperature fluctuations of 
the Cosmic Microwave Background (CMB) anisotropies.\footnote{
For the other aspects of CMB physics,
we refer to the other contributions of the special
issue of {\it Comptes Rendus Physique} (2003) {\bf 4}, 8.
The full text is available on line at {\bf http://www.sciencedirect.com}.
Notice also that the present text is a slightly expanded
version of that published in the above review.}
%http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=
%23TOC%237237%232003%23999959991%23472304%23FLP
%23Volume_4,_Issue_8,_Pages_819-966_(October_2003)
%2BMDossier%3A_The_Cosmic_Microwave_Background
%2BMEdited_by_F.-X._Desert&_auth=y&view=c&_acct=C000044719&_version=1&_urlVersion=0
%&_userid=829662&md5=5f3ac2b348c15b751111454b016854cf
%http://www.elsevier.fr/html/index.cfm?act=somone&code=HY&date=20031130.}
We shall put a special emphasis on justifying the surprising result that
%, in inflation, 
quantum mechanical rules must be used to compute 
the primordial spectrum.

For the non-specialist, 
two remarks should be made from the outset.
%%we
%it should perhaps first be mentioned 
%first mention that 
First, 
%without recourse to fine-tuning,
the simplest inflationary model 
%(and only inflation)  
has successfully passed the multiple checks
%onstraints recently provided by the new 
based on recent 
%%%and accurate 
observational data. 
%can be presented as (a ?)
%can be conceived as 
Secondly, inflation 
is a rather conservative hypothesis since it rests, 
%on two pilliards:
%the most conservative hypothesis when taking seriously into account, 
on one hand, on the 
%whole 
set of cosmological observations conventionally interpreted,
and on the other hand, on Einstein's equations, i.e., on the hypothesis 
that the action of General Relativity (GR), or a slight generalization thereof, 
governs the evolution of the 
space-time properties from cosmological scales, 
of the order of a thousand megaparsecs ($1$Mpc $ \simeq  
%3
10^{22}$ meters), 
down to some microscopic scale which is not far from the Planck length
($l_P \simeq 
%1,6 \, 
10^{-35}$ meters), 
%``just''
 before the threshold of Quantum Gravity.

Following these preliminary remarks,
%To motivate inflation, 
one should explain why the search for a causal explanation of 
the large scale properties of the universe calls for a long period
of accelerated expansion (where long is defined with respect to
%(in the units of 
the expansion rate during inflation).
%local Hubble time)
% $H^{-1}$).
%of accelerated expansion. 
%In one word,t
This need stems from the fact that the `standard model' of 
cosmology\cite{Weinberg,Peebles},
the Hot Big Bang scenario, is {\it incomplete}: 
%in a 
%precise 
%and strong very contraining sense: 
the causal structure from the big bang
% till today (or till the last scattering surface
%when the universe became transparant)
%obtained with GR 
%to propagates backwards in time the gravitational configurations 
is such that no processes
could have taken place to explain the homogeneity and the isotropy
at large scales. 
(What ``large scales'' means depends on the quantity
under consideration. For the density fluctuations on the 
last scattering surface, at the moment of decoupling, 
%at the time of decoupling $t_{rec}$, 
large scales concern distances larger than hundred 
kiloparsecs. These scales
are seen today as solid angles larger than 1 degree).
%Let us 
%first
%carefully situate this search 
%%explain 
%%To appreciate the 
%physical relevance 
The importance of these considerations
can only be appreciated in the 
%%we shall present how they take place with respect to 
%%%%%%%%%%%%%%%%%%%%%%%the 
%%%show how they emerge from
light of today's understanding of cosmology.
%the cosmological history of the universe.
%and explain how the abovementioned conclusion is obtained.
%%To this end, one should first recall what are the corner stones 
%of standard model.
%Therefore, we shall recall what are the corner stones 
%of standard model of cosmology.

\section{The standard model}
%The standard model: 
%Background quantities and Fluctuations}

The successful predictions of 
%richness of today's understanding of cosmology
the standard model of cosmology concern {\it both} 
%relies in the interplay between 
the evolution of spatial averaged 
quantities (temperature, densities...) 
as well as the evolution of local structures 
from density fluctuations to galaxies and clusters.
%(of temperature, metric...). 
Let's see how these two aspects are woven together.
% emerges.
 
The CMB is extremely isotropic: 
%at the decoupling time $t_{dec}$, 
the 
%observed 
temperature fluctuations  at the decoupling time $t_{dec}$
have a relative amplitude of the order of $10^{-5}$.
When adopting GR, 
%we learn that 
%at that time 
this implies that 
at that time the 
(today visible)
universe 
%we can see today
%(i.e., the entire patch visible today) universe 
can be accurately described by a linearly perturbed metric. 
%%wherein the 
%whose perturbations can be treated linearly.
%the time of decoupling.
% when the radiation was emitted. 
%This follows from the Sach-Wolfe effect which tells us that 
Indeed the primordial metric fluctuations,
defined on the top of a Robertson-Walker (RW) metric, 
are also of the order of $10^{-5}$
since they are linearly related to the 
%relative 
temperature fluctuations.
We recall that the RW
%Roberston-Walker 
metric reads
\begin{equation}
\bar{d s^2 } = - dt^2 + a^2(t) \,  d\Sigma_3^2  \, ,
%{R_H(t) \over a(t)} = {1 \over \partial_ta(t)} \, .
\label{RW}
\end{equation}
where bar quantities describe smoothed out, spatial averaged,
quantities. The spatial part of the metric $d\Sigma_3^2$ 
describes isotropic and homogeneous 3-surfaces.
% which are iotrpoc  
The actual metric, ignoring gravitons (the two spin 2 fluctuation
modes), can be written as
\begin{equation}
{d s^2 } = - (1+ 2 \Psi )\, dt^2 + (1 - 2 \Psi)\, a^2(t) \, d\Sigma_3^2  \, ,
%{R_H(t) \over a(t)} = {1 \over \partial_ta(t)} \, ,
\label{pRW}
\end{equation}
when the matter stress-tensor 
is isotropic, a simplifying assumption which nevertheless
captures the essential but which limits the fluctuations
% fields to a single field formed with 
to the so-called `adiabatic' modes\cite{MukhB,Langlois}.
%a mild hypothesis\cite{MukhB}.
The local field $\Psi(t, {x})$ acts as a Newtonian potential.
%For simplicity, we have considered that the matter stress-tensor 
%is isotropic, a rather mild hypothesis\cite{MukhB}.

Since the fluctuations are $10^{-5}$, 
%The lesson of these facts is that 
one can first adopt a mean field approximation
and 
%{\it start} the analysis of cosmology by neglecting them and by 
consider only spatially averaged quantities.
In a second step, one can analyze the evolution 
of the fluctuations which ride on the top
of the background quantities formerly obtained.
%and shall lalter seewhich give rise to the structures such as galaxies and clusters. 

\subsection{Mean quantities and primordial nucleosynthesis}

The evolution of the averaged metric Eq. (\ref{RW}) is 
%thus 
governed  
by a single function of time, the scale-factor $a(t)$. 
The derivative of its logarithm defines $H= \partial_t{a} /{a}$
which 
%The latter 
enters in Hubble's law $v = H R$. 
%This law relates the 
$v$ is the relative velocity 
of comoving galaxies separated by a proper distance $R(t)$.
This distance is given by $R(t)= {a}(t) \, d$, 
where $d$ is the fixed comoving distance
between the two galaxies. This distance is defined by the
static line element $d\Sigma_3^2$ of Eq. (\ref{RW}).
%metric
%measured in arbitrary units.
%,conventionaly taken to correspond to the proper distance measured today. 
%$d\Sigma_3^2$ describes the spatial part of the metric. 
%When considering physical processes which took place at or before
%the decoupling (for redshift $z + 1 = a_0/a_{dec} \simeq 1100$,
%where the subscript $_0$ means evaluated today),
%the spatial part of the metric can be taken to be Euclidean 
%since the 3-curvature contribution to these processes is completely negligible.
%This conclusion is obtained as follows.
%follows from the facts that its today value 
%is already small and decreases like $1/z$.
% of its (relative) importance obtained from the
%estimated to be now smaller than 
%was, at that time and before, completely negligeable.
In GR, $a$ obeys the
%is governed by 
%is decrease follows from 
Friedmann equation, 
%the 00 component of Einstein equation,
\begin{equation}
    H^2 =  {8 \pi G \over 3} \bar \rho + { \kappa \over   a^2 }\, .
%{ \Omega^{curv.}_0   H^2_0 \over   a^2 } \, .
%- (1+ \Psi ) dt^2 +   a^2(t) (1+ \Psi) d\Sigma_3^2  \, ,
%{R_H(t) \over a(t)} = {1 \over \partial_ta(t)} \, ,
\label{FE}
\end{equation}
The averaged matter density is denoted $ \bar   \rho$ 
%is the mean matter density (including 
and includes a possible cosmological constant\cite{PeebCC}. 
The last term in Eq. (\ref{FE}) arises from the curvature
of the homogeneous 3-surfaces.
%spatial curvature
%. Its today value is conveniently expressed by whose 
Its today relative contribution is given by
% now estimated as 
$\Omega^{curv.}_0 = { \kappa /( H^2_0   a_0^2)}$
%\vert < 0.1$\cite{WMAP},
(where the subscript $_0$ means evaluated today).
Its value is now observationally tightly constrained:
%the WMAP estimation is  
$ \Omega^{curv.}_0  = 0.02 \pm 0.02$\cite{WMAP}.
% which is smaller than .1.  
% in terms of $\Omega^{curv.}_0$, its relative contribution in 
%the units of $H^2_0$ measured today (recall that $a_0= 1$).
This term is the integration constant of the 
`dynamical' Einstein equation, 
%the 01  Einstein equation,
%When imposing homogeneity and isotropy, 
%matter is, on the average, described by the density and the 
%pressure $   P$ which enters the dynamical equation
\begin{equation}
{  
%\ddot
\partial_t^2
 {   a }\over   a}
 = - {4 \pi G \over 3} ( \bar \rho + 3  \bar P)\, ,
% { k \over   a^2 } \, ,
%- (1+ \Psi ) dt^2 +   a^2(t) (1+ \Psi) d\Sigma_3^2  \, ,
%{R_H(t) \over a(t)} = {1 \over \partial_ta(t)} \, ,
\label{DE}
\end{equation}
and energy conservation 
$d \bar \rho = - 3 ( \bar \rho + \bar P) d \ln a $.
Together with the density $\bar \rho$, the pressure $ \bar P$ 
fully characterizes the matter stress tensor when imposing 
homogeneity and isotropy. 
Furthermore, when considering non-interacting components, 
$\bar \rho$ and $\bar P$ split into terms which obey 
equations of state of perfect fluids:
\begin{equation}
 \bar P_i = w_i \  \bar \rho_i \, .
% =  {4 \pi G \over 3} (  \rho + 3   P)
% { k \over   a^2 } \, ,
%- (1+ \Psi ) dt^2 +   a^2(t) (1+ \Psi) d\Sigma_3^2  \, ,
%{R_H(t) \over a(t)} = {1 \over \partial_ta(t)} \, ,
\label{ESi}
\end{equation}
Dust, (cold) baryons and Cold Dark Matter (CDM) have $w =0 $, photons
and (massless) neutrinos  $w= 1/3 $, whereas the cosmological 
constant has  $w=-1$.

Energy conservation then fixes the scaling 
%in $z$ 
of each component.
For baryons, one gets $ \bar \rho_b 
%=  \bar \rho_{b 0}\,  (  a_0/  a)^{3}$, 
\propto a^{-3}$, 
and for radiation, 
$ \bar \rho_{rad} \propto
%=  \bar \rho_{rad 0} \,  (  a_0/  a)^{
a^{-4}$.
The 3-curvature term scales only as $  a^{-2}$ 
whereas the cosmological constant does not vary. From 
these scalings and the present values of the densities,
one immediately obtains the different stages of our
cosmological history.
In particular, the CMB temperature, $T_0 = 0.23\, {\rm meV}$, 
the effective number of massless fields\cite{Peebles},
%three families of neutrinos and their shifted temperature,
and the relative contribution of cold matter (baryons + CDM), 
$\Omega^{cm}_0 = 0.27 \pm 0.04$ \cite{WMAP}, 
%%(where the subscrip $_0$ means evaluated today),
%one finds that 
fix the redshift of the transition from radiation to matter domination:
$z_{eq} = a_0/a_{eq} -1  \simeq 3200$. 

Notice that this equilibrium occurred 
before the moment of decoupling which occurred 
%the contribution of the 3-curvature 
%obeys at the decoupling, 
when $z_{dec} \simeq a_0/a_{dec} \simeq 1100$.
%%the relative 3-curvature contribution obeys 
Notice also that at that time, the 3-curvature contribution obeyed
$\Omega^{curv.}_{dec} <  \Omega^{curv.}_{0}/(\Omega^{cm}_0 
z_{dec}) < 10^{-4}$. 
Thus, for all processes which took place at or before decoupling,
the spatial 
%part of the 
metric can be taken 
%to be 
Euclidean
and $\kappa/a^2$ 
%the last term 
in Eq. (\ref{FE}) can be dropped.
%see however Uzan's contribution.
%for all processes which took place at or before
%the decoupling, 
%the spatial part of the metric can be taken Euclidean.
%Moreover, for $z> z_{eq}
By a similar reasoning, one also concludes that 
a cosmological term or a term with $w < -1/3$ (inducing
an acceleration: $\partial^2_t a >0$) play no role
before equilibrium. 
So, unless one introduces an additional contribution
to $\bar \rho$ which dominated for a while the r.h.s of
Eq. (\ref{FE}) and then decayed, 
%characterized by a varying $w$, 
one arrives at the conclusion 
that the early cosmology was dominated by the radiation contribution,
$\bar \rho_{rad}$. 
%Moreover since the temperature increases 
%following $T(a)= T_0 a_0/a$, all massive particles behave like radiation, 
%hence HBB 
We shall see in Section II.C
what are the reasons to reject this simple conclusion.
%radiation domination.)


The next step consists in using 
%the standard model of 
{\it particle physics}
to inquire about the processes which took place in the early universe,
before $z_{eq}$. 
% Using these settings and making some choice for the
%matter content of the universe at a given redshift, 
%one can propagate this matter backward (or forward) in time 
%backwards (or forward) in time matter and gravity 
%and compare the outcome with observations.
The greatest success of this standard framework
concerns the prediction of relative abundances of 
light elements by Primordial Nucleo-Synthesis\cite{Weinberg,Peebles,Nucleos}
which occurred for a redshift $z_{\rm PNS} \simeq 10^9$.
%agreement between the theoretical predictions 
%for relative abundances of light elements and their observed values.
This confirms the validity of extrapolating backwards in time 
Einstein's equations in a radiation dominated universe,
at least up to redshifts of the order of 
$10^{9}$ and  temperatures of the 
order of the ${\rm MeV}$.
% with local interactions described  
%by the standard model of particle physics. 
%which governs nuclear reactions. 

Moreover a detailed comparison 
%between 
of theoretical predictions 
and observations leads to a precise 
%(but no longer the best\cite{Nucleos2}) 
determination of the baryonic content of our universe
%\cite{Nucleos}
%. Indeed primordial
%nucleosynthesis fixes the baryon density
%$\Omega_{b \, 0} = 0.04 \pm .01$ with the highest precision, 
(through the determination of $n_b/n_\gamma$, the number 
of baryon per photon, 
%entropy per baryon 
and the absolute normalization of the thermal photons of the CMB).
%bath of photons forming the CMB). 
Using the recent estimation of $H_0$, primordial nucleosynthesis
gives $\Omega_{b 0} = 0.042 \pm 0.005$,
a value in agreement with that obtained from analyzing the CMB\cite{Nucleos2}. 
So, not only the predictions of the extrapolation 
%backwards in time
can be observationally tested, but 
%it increases our knowledge of the universe and the fact that 
independent observations agree. This
puts the standard model on extremely firm foundations.

\subsection{Fluctuations and Structures}

So far, we have ignored
%completly neglected 
the fluctuations around spatial averaged quantities. 
It is remarkable that  
%The second success of 
the standard model of cosmology, as it stands,
%framework  concerns the evolution 
also correctly predicts the evolution 
of {local structures}\cite{Weinberg,Peebles}.
%of the universe\cite{Weinberg,Peebles}.
%,LiLy}.
% and not only the mean evolution.
These structures are described by a perturbed metric, 
as in Eq. (\ref{pRW}), and a local matter density 
$\rho = \bar \rho(t) + \delta\rho(t, {x})$. 
Their combined evolution
is governed by the {\it local} Einstein 
equations (and no longer by their restriction to RW metrics
used so far) and by matter interactions. 

%When starting (at some initial time which can taken
%after primordial nucleosythesis and before $z \simeq 1100$) with t
Since the {primordial} fluctuations $\delta \rho/\bar \rho \simeq 
\Psi $ have an amplitude of the order of $10^{-5}$,
they evolve linearly for a long range of redshifts 
until
%infered from the CMB fluctuation spectrum, 
%the forward evolution GR 
gravitational instabilities and radiation damping
lead to non-linear 
%large scale 
structures. The properties of the latter are in agreement 
with what is now observed. 
%one obtains some thing which correctly correspond to  
%the presently observed large scale structure. 
However, the best agreement is reached 
%for this growth to agree, one should some 
{if} one adds some Cold Dark Matter
whose density is about 10 times larger than that of 
baryons\cite{bert}.
%times largLiLy}. 
%$\Omega_{{\rm CDM}, 0} =.23 \pm $, see \cite{}. 
At first sight this could be considered
as an {\it ad hoc} hypothesis to save the standard model, but 
%But, since 
this is no longer the case since there are now independent observations which 
are consistently interpreted as gravitational effects induced 
by CDM\cite{PeebCC}. 
%indicating (albeit indirectly) since one can also estimate its
%value from the other gravitational effects CDM induces.
%have been observed, albeit indirectly, 
%through different means. 
The most precise estimation of
$\Omega_{{\rm CDM} 0}$ ($= \Omega^{cm}_0 - \Omega_{b0} 
= 0.23 \pm 0.04 $)
is presently extracted from 
%deduced from the 
%gravitational effects on 
%%analysing 
the spectrum of CMB anisotropies. 
%CORRECT ?
% such as gravitational lensing. 
%Eventhough, 
%further theoretical and observational analysis is still needed
%to reach a level of confidence comparable to that of nucleosynthesis,
%%Hence 
%one can already view $\Omega_{CDM\, 0} \simeq .24$ as a 
%success of the standard model.
%prediction
%from structure formation which is now observationally confirmed.
%To reach the level of confidence concerning the primordial nucleosynthesis, 
%   this should be account as a success, and not as an ad hoc hypothesis, 
%see Peebles Cosm cosnt.Eventhough, 
%further theoretical and observational analysis is still needed.
% to confirm the details.
\subsection{The incompleteness of the Hot Big Bang scenario}
%standard model}

Given these two successes, it is tempting to further extrapolate 
%backwards in time 
and to inquire what happened before nucleosynthesis,
for redshifts larger than $10^{10}$ and energies larger 
than the MeV. Was there a 
%period of
baryo-genesis\cite{Riotto} which could explain 
the 
%observed 
baryon-antibaryon asymmetry,
% between baryon and anti-baryon, 
or a lepto-genesis\cite{Buch} which preceded baryo-genesis?
%, a string-genesis from M-theory~? 

In this inquiry one 
%always 
searches 
%%needs to identify
processes 
%in order which allow 
by which
to explain the observed values of quantities which,
for lack of something better,
 have been hitherto treated as {initial conditions.} 
Inflation might constitute a crucial step in this endeavor.
%It is this hope that inflation finds its place. 
The reason is the following.
Irrespectively of the 
%local 
high energy local 
processes which took place,
% have happened, 
%at high temperature, 
one inevitably encounters a problem
%, see p. 525 in \cite{Weinberg},
%\cite{Weinberg,Peebles}
%"severe" problem (p 525 in \cite{Weinberg}, p 362? in \cite{Peebles})
which is {\it non-local} in character:
%which concerns the compatibility of 
Given the {causal} structure obtained by extrapolating 
%backwards  
$a(t)$, the solution of 
Eq. (\ref{DE}) driven by radiation, the
{large scale isotropy} cannot be 
explained\cite{Weinberg},\footnote{In fact
it was the reading of Weinberg's careful explanations which lead
R. Brout, F. Englert, and E. Gunzig to propose one of the first inflationary
models in 
{\it The creation of the universe as a quantum phenomenon},
%HE CREATION OF THE UNIVERSE AS A QUANTUM PHENOMENON, 
Annals Phys. 115:78,1978; and {\it The causal universe},
%HE CAUSAL UNIVERSE
 Gen. Rel. Grav. 10:1-6,1979.}
%and the causal structure obtained when further extrapolating 
%backwards in time Einstein's eqs. driven by radiation, 
as can be seen from Figure 1.

\begin{figure}[ht]
\epsfxsize=7cm
%\epsfysize=7cm
\centerline{{\epsfbox{Fig1new.eps}}}
\caption{ {\it The causal structure from the Hot Big Bang.}
In a radiation dominated universe, there is a {\it space-like}
singularity at $a=0$ which is situated
at a {\it finite} conformal time $\eta = \int_{t_{B\!B}}^{t} dt/a$.
Hence forward light cones from the big bang 
%re exists a {\it particle horizon} whose 
have, at time $t$, a finite (proper) radius
%size 
equal to $a(t) \eta(t)$ which defines the 
so-called {\it particle horizon}.
% (we have put $c=1$). 
%On the last scattering surface ({\it lss}), 
The proper distance between the points $A$ and $B$
is equal to $a\Delta\eta$ evaluated on the last scattering surface ({\it lss}).
%$a\Delta\eta$ is equal to the distance between $A$ and $B$.
%This implies that
%at that time, 
Hence the matter systems at $A$ and $B$ 
have never been in contact. Thus no 
(causal) process could possibly explain why the temperature
fluctuation between 
%seen in the directions towards 
$A$ and $B$ 
%coincide up to relative fluctuations 
is only of the order of $10^{-5}$. 
Moreover 
%To strengthen the problem it should also be said 
%that we presently see 
the today visible universe (in dotted lines)
encompasses 
about $5.\, 10^{4}$ disconnected patches on the {\it lss}
with fluctuations within that range.}
\end{figure}


There is an interesting and complementary way to consider this
incompleteness. It concerns the {origin} 
and the evolution of {\it large scale anisotropies}. 
%(and the metric fluctuations) 
%on super-horizon scales. 
%evolution of modes describing density fluctuations at fixed comoving scales. 
Since the fluctuation modes propagate 
in RW metric,
% is 
%conformally flat, homogeneous,
their comoving wave vectors $ k$ are conserved.
Hence, when treated linearly, the Fourier modes
%  with different $\bb k$, such as 
$\Psi_k = \int\!d^3x \, e^{ikx} \Psi$ and
$\delta \rho_k = \int\!d^3x \, e^{ikx} \delta \rho$
%can be decomposed in Fourrier and , furthermore , the linear equtaions 
%split into uncoupled k equations.
evolve independently of modes
%%$\Psi_{k'}, \delta \rho_{k'}$ 
with $k' \neq k$.
% modes characterized by a different $\bb k$.
%each other when treated linearly.
%modes characterized by a different $\bb k$.
%wave vecteur. 
%Moreover
%, in cosmology, 
%On the other hand 
The important point is that this evolution crucially depends on the
ratio $\lambda/R_H$ where $\lambda=a(t)/k$ is the physical wave length 
and where $R_H= 1/H(t)$ is the Hubble radius ($c=1$).
When $\lambda/R < 1$,
%$1/k < d_H$, 
the modes are `inside' the Hubble radius, and they oscillate.
% as in Minkowski space time.
On the contrary, on super-horizon scales,\footnote{In the Hot Big Bang
cosmology, the Hubble horizon $R_H(t)$, which is locally defined, 
coincides with the globally defined particle horizon $a(t) \eta(t)$
where $\eta$ is the conformal time evaluated from the singularity.
% at $a=0$.
This equality no longer applies in inflation,
thereby offering a solution to the causality problem 
(sometimes also called the {\it horizon problem}) 
we are discussing.}
radiation dominated universe i.e.
when their wave length extends beyond the Hubble radius,
%for  $1/k >   d_H$, 
modes are frozen (or decay).
% ($c=1$)and the physical wave length $a(t)/k$.
%provides 
%a natural a critical length scale when considering 
%which coincides, in a radiation dominated universe,
%with the causal {\it particle horizon} $a \Delta \eta$ defined in the caption of Figure 1.
%in a radiated dominated universe).
%there is a length scale which is  provided by the expansion rate: 
% a length scale,
%the Hubble radius $R_H = 1/  H$ (with $c=1$).
%%and b
Because $k$ is conserved 
%comoving wave vectors are conserved,
 it is appropriate to follow the evolution of $\lambda/R_H$
 by writing it as $k^{-1}/d_H$ where
 %the comoving Hubble radius
%valueevolution of its comoving value:
\begin{equation}
  d_H(t) = {  R_H(t) \over   a(t)} = {1 \over \partial_t{   a}} \, ,
\label{dH}
\end{equation}
gives the comoving size of the Hubble radius at time $t$.
% in comoving coordinate 
%The physical relevance of $d_H$
%follows from the fact that modes behave very 
%differently according to the relative value of 
%whether their wave length $1/k$ and $  d_H$:
%are larger or smaller than $  d_H$:
%the Hubble radius: When $1/k < d_H$, modes oscillate.
% as in Minkowski space time.
%On the contrary, on super-horizon scales, i.e.
%when their wave length extends beyond the Hubble radius,
%for  $1/k >   d_H$, modes are frozen (or decay).
%These considerations become very relevant 

These considerations become crucial when questioning
the {\it origin} of primordial fluctuations on super-horizon 
scales (we know they exist since they determine the CMB anisotropies
at large scales, for solid angles larger than a degree). 
%for angles larger than $1$ degree). 
Indeed since $\partial_t^2 a < 0$ in a radiation dominated universe,
as seen from Eq. (\ref{dH})
%its definition 
and Eq. (\ref{DE}),
% {\it since 
%the Hubble radius (expressed in conformal distances) 
$ d_H$ always increased.
% in a radiation dominated universe}, as seen from Eq. (\ref{dH})
%its definition and Eq. (\ref{DE}),
%(This follows from its very
%definition.) 
%It is of value to represent graphically the situation. From Figure 2,
%one clearly sees that 
Therefore the modes which were still outside 
$d_H$ on the last scattering surface 
have been frozen since the big bang. 
Hence
%, in this model,
%only two options are possible. Either 
%since no causal process could have taken place,
their amplitudes 
%of the various frozen modes 
cannot result from processes and 
can thus only be fixed from the outset
%determined 
%result 
by {\it initial conditions}.
%, that is, the do not result from a process. 
%arbitrarily chosen.

\section{Inflation}

Inflation
%by definition,
is 
%the hypothesis, or 
the price to pay to reject this vexing outcome.\footnote{
Topological defects resulting
from a phase transition could have been another
%Another 
possibility.
%to explain the primordial spectrum would be 
%could have been to invoke topological defects resulting
%from a phase transition. 
However, 
%the 
recently 
obtained properties of 
%temporal coherence the 
CMB anisotropies (i.e. the temporal coherence of the modes
revealed by the multiple 
``acoustic'' peaks,
% reveal the temporal coherence of the modes,
see later in the text)
%recently obtained 
rule out 
%exclude 
this possibility\cite{PPeter,FirstY}.}
%%%% as a unique source of primordial fluctuations\cite{PPeter}.}
% which allows this second possibility to be realized.
%Indeed, f
From the above considerations, a 
{necessary} condition
for allowing physical processes to have taken place is 
%to assume 
that 
%and the Big Bang, 
there 
%%has been 
was a time when $d_H$ was {\it larger} than today's Hubble
scale $d_{H0}$. Therefore, between then and the 
beginning of the radiation era,
%there must have been a large decrease, i.e., 
$d_H$ must have tremendously {\it decreased}. Thus from 
Eq. (\ref{dH}) there must have been a ``long'' period of accelerated expansion,
see
%%as represented in 
Figure 2. 
%As we shall see below,
%, i.e., 
%%%a period 
%(before $z_{nucleos}$) 
%%%in which $d_H$ sufficiently grew (going backwards in time) 
%%%so as to encompass {\it all} today visible scales. 
%to have once evolved inside the Hubble radius.
%%%%%%%%%%%%%%%%%%That is, 
%As we shall see below, this implies that
%there must have been a ``long'' period of accelerated expansion,
%where long should be unterstood when using the redshift $z$ 
%as a temporal parameter.
%: inflation. 
%When adopting inflation, one thus obtained a situation represented 
%as represented in Figure 2.
% is replaced by Figure 3. 

\begin{figure}[ht]
\epsfxsize=7cm
%\epsfysize=7cm
\centerline{{\epsfbox{Fig2new.eps}}}
\caption{
{\it The evolution of the comoving value of the Hubble radius $d_H$.}
%We have presented the ``trajectory'' of $d_H(t)$ 
%as a function of time.}
A vertical line is a fixed comoving scale. The symbols
$d_{H0}$ and $d_{Hdec}$ designate the Hubble scale 
%of the Hubble radius 
evaluated today 
%($z=0$) 
and at the decoupling respectively. 
The evolution of $d_H$ with or without inflation
splits near some high redshift $z_{reh}$. 
The Hot Big Bang branch is indicated by the dashed line
% which hits $d=0$ at some finite time,
whereas the inflationary branch is marked by the continuous line. 
$z_{in}$, $z_{H0}$,  and  
$z_{reh}$
%%, $z_{H0}$, and  $z_{in}$ 
give the respective values of the redshift when inflation starts,
when the scale $d_{H0}$ exits $d_H(t)$, and at {\it reheating} when inflation stops.
%when the scale $d_{H0}$ exits $d_H(t)$, and when inflation starts.
For inflation to be successful (i.e., to remove
the need of fine tuning the initial conditions of fluctuation modes) 
one must have $d_H(t_{in}) \gg d_{H0}$, as we shall explain in Section III.A.} 
%inflation should last enough so as the largest visible scale $d_{H0}$
%The scales larger than $d_H(t)$ at $t_{dec}$ 
%must have been inside the Hubble radius.  
%always %been outside since the big bang.
%as indicating by the dotted line which hits $d=0$ at some finite time.
%What happened before $z_{in}$ is left as an open 
%question for imaginative physicists. QG?}
\end{figure}

At this point, inflation is simply a kinematic hypothesis 
which allows processes to have taken place.
% explanation for the 
(Notice however that the dilution of scales associated with the 
%tremendous 
{decrease} of $d_H$ during inflation
solves other problems of cosmology\cite{Guth,Linde}).
%,Lindebook}.)
In order to proceed, several issues should be confronted. 
One should first dynamically realize inflation, i.e., 
find an ``engine'' which could be responsible for it. 
%This is not an easy task since Eq. (\ref{DE})
%requires that $ \bar \rho +  3  \bar P$ be negative during a long period. 
Secondly one should 
% issue concerns the identification of 
identify the mechanism giving rise to the primordial spectrum\cite{Mukh}. 
The third issue concerns the {\it reheating}\cite{Lindebook,reheat}: 
%i.e., the fact that at the end of inflation there must be a 
the collective process which damps the expansion
and liberates heat
%at the end of inflation
%in order to produce a  
%resulting in the  
%give rise to 
%the hot 
thereby ending the period of inflation 
and starting
%leading to 
the radiation dominated period
%universe 
which is necessary for 
%required for 
primordial nucleosynthesis to take place.
%Before proceeding, it should be stressed that 

It should be noticed that to a large extend these three issues 
could be addressed separately. In particular, it turns out that the
mechanism giving rise to the spectrum operates quite irrespectively
of the particular dynamical realization of inflation one adopts.
%\cite{Mukh,LiLy}.
Hence, 
%it should cause no suprise that there is a large
%significant degeneracy of 
different dynamical models could end up with very similar spectra.
%same spectrum. 
Therefore, in view of our interest in the CMB,
we shall focus on this mechanism and the properties of 
the spectrum it produces.
%for reasons of simplicity, 
We shall also restrict ourselves to the simplest inflationary model,
that of a single massive scalar field. For a large
panorama of inflationary models we refer to \cite{LiLy}.
%and focus on the mechanism and 
%the properties of the spectrum so produced.
%We shall nevertheless mention in passing
%several difficult questions inflation raises.
%At the end of the paper we shall say a few words about the 
%difficulties to answer the other questions.
%, i.e. to identify the inflaton.

\subsection{The inflationary background}

%This is not an easy task since Eq. (\ref{DE})
%requires that $ \bar \rho +  3  \bar P$ be negative during a long period. 
To have {\it successful} inflation requires two things.
On one hand, the accelerated expansion should last
enough that
% for a sufficiently long period that 
%the Hubble radius% have encompassed 
%largest visible scale 
today's Hubble scale
$d_{H0}$ be  inside the Hubble radius 
at the beginning of inflation, see Figure 2.
To characterize the ``duration'' of inflation it is convenient to
introduce the parameters 
$N_{min}=\ln(a_{reh}/a_{H0})$
%d_{H0}/d_{Hreh})$ 
and $N_{tot} = \ln(a_{reh}/a_{in})$
which respectively give the minimum number of {\it e-folds}
for inflation to be successful and the total number 
of {e-folds} from the beginning of inflation 
to its end at the reheating. 
On the other hand, even though inflation needs not to be homogeneous,
%could 
%should only concern a patch of finite size, 
the patch which inflates
should be sufficiently large and homogeneous that the gradients
%of the metric 
be negligible up to a scale larger than 
%the 
today's Hubble scale $d_{H0}$. If one of these conditions is not met, 
inflation would fail to explain the isotropy of the CMB
up to that scale.
%%that scale.
%which is observed up to  $d_{H0}$.

%in a RW universe
%inflation 
To have accelerated expansion
in GR requires $\bar \rho + 3 \bar P < 0$,
% for a long time,
see Eq. (\ref{DE}).
To have inflation thus requires that this condition
%$\rho(t,{\bb x}) + 3 P (t,{\bb x})< 0$  
be satisfied for a long time but only in the above mentioned patch.
%, as discussed above. 
%nevertheless should be large enough so to encompass
%the largest visible scale $d_{H0}$, cf. Figure 2.
%(The precise meaning of ``long'' and ``large'' shall be given below
%or be found here.)
This can be fulfilled by {introducing} a scalar field,
named {\it inflaton}, 
which 
%%%obeys three conditions it must 
possesses
%in the patch, 
an expectation value $\bar \phi$ which 
obeys three conditions: 
$\bar \phi$ must 
dominate {all} other contributions to $\rho$ and $P$,
%which is large and homogeneous, 
% and large expectation value $\bar \phi$,
it must slowly decay, and it must be sufficiently homogeneous.
%which dominates {\it all} other contributions to $\rho$ and $P$.
%in a sufficiently large patch 
%(The precise meaning of `large` and `slow` shall be given below
%or be found here.)
%In the patch, the inflaton can thus be written as
When these are met, one can write the inflaton as
\begin{equation}
\phi(t,x) = \bar \phi(t) + \delta\phi(t, {x}) \,  ,
\label{inflon}
\end{equation}
and neglect non-linear terms in the fluctuations $\delta\phi, \Psi$
since $ \Psi \propto \delta\phi/\bar \phi \ll 1$.
%as we shall see $\Psi \propto \delta\phi/\phi$. 
%Moreover, when neglecting second order effects and tin the patch, 
Then,  in the patch, $\phi$ 
%where the departure from homogeneity $ \delta\phi$
%satisfies $\bar \phi \gg \delta\phi$,  
%This last condition shall be further discussed below.
%%%Since the patch is
%In the homogeneous patch,
%and isotropic, $\phi$ obeys
%he equation of motion of the inflaton is that of scalar field in a RW metric:
%and it 
obeys the scalar field equation 
%of motion of scalar field 
in a RW metric:
\begin{equation}
%\ddot 
\partial_t^2 \phi + 3 H \partial_t \phi - 
{ \Delta  \phi \over a^2} + m^2  \phi = 0 \, ,
\label{eqinflon}
\end{equation}
%where t The local fluctuation 
%$ \delta\phi$ satisfies $ \delta\phi \ll \bar \phi$and 
where $\Delta $ is the Laplacian associated with the static line
element $d\Sigma_3^2$ of Eq. (\ref{pRW}).
From Eq. (\ref{eqinflon})
%this equation,
one finds that the slow decay of 
%the expectation value 
$\bar \phi$ follows
from $3 H \partial_t{\bar \phi} \simeq  - m^2 \bar \phi$, 
%if it slowly decays. In this case 
i.e., $\bar \phi$ is dragged to zero at a rate given by $m^2/3H$.

We are now in position to verify that $ \bar \phi $ leads to 
%an accelerated expansion and 
inflation when the above  conditions 
%concerning the initial value of $\phi$ and the decay of $\bar \phi$ 
are satisfied.
%the decay is slow and long enough. 
%Accelerated expansion is obtained from the fact that t
The energy density
%$\bar \rho_\phi = (\partial_t\bar \phi)^2/2 + m^2 \bar\phi ^2/2 $ whereas its 
and the pressure 
%are such that $\bar P_\phi / \bar \rho_\phi \simeq -1 $.
%This follows from the fact that they 
are dominated by the contribution of $\bar \phi$ given by 
$\bar \rho_\phi, \bar P_\phi = 
(\partial_t \bar \phi)^2 /2 \pm m^2 \bar\phi ^2/2$. 
When the decay is slow enough,  
%$(m/3H)^2 \ll 1$, 
the kinetic term is negligible. In this case
one gets $\bar P_\phi / \bar \rho_\phi \simeq -1 $
and hence accelerated expansion, see Eq. (\ref{DE}).
% then follows from the fact that they are both dominated by 
%of  $\bar \phi$) are both dominated by 
%the mass term when $(m/3H)^2 \ll 1$, cf. the  
%Hence they are both dominated by the mass term when 
%$ \partial_t \bar \phi \ll m \bar \phi $.
%Given the above mentioned decay rate.
% they are both dominated by the mass term when $(m/3H)^2 \ll 1$.
% must be fulfilled.
%since the kinetic term $\partial_t{\bar \phi}^2/2$ is much smaller
%In this case their ratio obeys
%$\bar \rho_\phi /\bar P_\phi \simeq -1 $.
%since the kinetic term $\partial_t{\bar \phi}^2/2$ is much smaller than
%the potential term. 
As long as the mass term is larger than the kinetic term, 
$ \bar \phi $ thus acts as a (slowly decaying) cosmological constant. 
%synthetize the situation it \
To characterize this decay, it is appropriate to introduce 
%express the relevant quantities in terms of 
the {\it slow-roll} parameter 
\begin{equation}
\epsilon= - \frac{\partial_t H}{H^2} = -
\frac{d\ln H }{d \ln a } \ll 1 \, .
\end{equation}
%and to see how it enters the relevant quantities. 
Algebra then gives $m^2/3H = \epsilon H$ for the decay rate,
and
%This is confirmed by
%a more detailed analysis. $\epsilon$ is
%the {\it slow-roll parameter} defined by $\partial_t H/H^2$.
%From Eq. (\ref{DE}) one gets
$\bar P_\phi /\bar \rho_\phi = -1 + 2 \epsilon/3 $
for the equation of state.  
% where $\epsilon$ is
%the {\it slow-roll parameter} defined by $\partial_t H/H^2$.
One also finds that 
$N_{tot} = 1/2\epsilon = 2 \pi G \, \bar \phi_{in}^2$, thereby relating 
the total number of e-folds to $\epsilon$ and to the initial value
of the inflaton in Planck units.
%to the number of e-folds. 

It remains to make contact with (micro)physics. It is generally 
believed that the reheating process has something to do
%with will shoot the temperature 
with Grand Unification Theories~\cite{Guth,Lindebook}.
If this is correct $T_{reh}$, the reheating
%adiation dominated expansion should start with a 
temperature, should be close to GUT scale, near $10^{14} {\rm GeV}$. 
%close to the Grand Unification Theory scale.
In this case, there must be at least 60 e-folds of inflation.
This condition follows from:
% the sequence:
$N_{tot}
% \simeq \ln(d_H(t_{in})/d_{Hreh})  
> N_{min} 
%= \ln(a_{Hreh}/a_{H0})
\simeq \ln(d_{H0}/d_{Hreh})
\simeq \ln(a_0/a_{reh}) 
\simeq \ln (T_{reh}/T_0)
\simeq 60$.
%where $N_{H0}$ is the minimum number of e-folds for inflation to
%be successful. 
(The first two $\simeq$
%e simplicity of the 
%first $\simeq$ arises equations, 
follow from
%we have used 
scaling laws:
% of $d_H$: 
$d_H \propto 1/a$ during inflation and $d_H
\propto a$ during the radiation era.)
%one should  completes our analysis of the homogeneous quantities.

Before considering the fluctuations of the inflaton, 
%s a final comment 
it should be stressed  that
the initial value of $\bar \phi$ obeys 
$\bar \phi_{in} > N_{min}^{1/2} M_{P}$, 
where $M_P= G^{1/2}$ is the Planck mass when $c=\hbar =1$.
That is, the slow-roll conditions send us inevitably 
{\it above} the Planck 
scale. However there is no particular reason to 
believe that the settings we are using make sense in this regime.
More complicated inflationary models based on two fields do 
not suffer from this disease\cite{LiLy}.
Nevertheless the question of the nature of the inflaton is still open. 
There are basically two attitudes.
The dominant attitude\cite{Guth,Lindebook}
%Either one considers 
is that inflation belongs to particle
physics and occurs sufficiently below the Planck scale so that gravity 
can be safely treated by classical GR.
%\footnote{
This  
%first 
option raises a very difficult question\cite{Brandy} related to the 
cosmological constant problem: If gravity can all the way be 
treated classically, why the mechanism which screens the
vacuum energy during the whole radiation era (where several 
transition should have taken place thereby giving rise to 
non-vanishing vacuum energy) did not screen as
well the potential energy of the inflaton during inflation?
%or, Therefore, o
One should therefore 
%However one might 
not exclude the alternative possibility 
%as e.g. in $R^2$-inflation\cite{Staro,Mukh}, one considers 
that the inflaton be merely a 
%convenient
phenomenology which aptly characterizes,
as e.g. in $R^2$-inflation\cite{Staro},
the background and the 
fluctuations in a domain wherein quantum gravity 
(or stringy effects) could still play an important role.
As far as the mechanism giving rise to
the primordial spectrum is concerned, 
these alternatives are equivalent\cite{Mukh}. 

%CORRECT ?

%background 

\subsection{The primordial spectrum}

To identify this mechanism,
%at the origin of the primordial density perturbations, 
we need to consider the 
%coupled equations governing the 
combined evolution of
the inflaton and the metric perturbation, $\delta \phi$ and $\Psi$.
% and the metric perturbation $\Psi(t,x)$. 
Assuming we can 
work to first order in these fluctuations,
GR gives us a set of equations for their Fourier components,
$\delta \phi_k$ and $\Psi_k$,
see \cite{MukhB} for details. 
Two important points should be mentioned.
First the $0i$ Einstein equation gives 
%the nice equation is  
\begin{equation}
{1 \over a}\partial_t (a \Psi_k) = 
{4 \pi G }  \partial_t \bar \phi  \, \delta\phi_k \, .
\label{0ieq}
\end{equation}
From this constraint equation one learns that $\Psi_k$ decays like $1/a$
in the absence of matter density fluctuations. 
Hence the primordial metric fluctuations
%``seen'' in the CMB must have been driven 
are sustained by matter fluctuations.
From the linearization procedure, one also learns that, 
during inflation, the {\it dominant} matter fluctuations are
those of the inflaton because the background energy
$\bar \rho$ is dominated by $\bar \phi$. (When considering inflationary models
with several scalar fields, 
%the same is true for 
a particular combination of their fluctuations, named {\it adiabatic},
%\cite{Langlois},
drives the metric fluctuations $\Psi_k$. The other fluctuations, 
called iso-curvature, could nevertheless play important roles\cite{Langlois}.)
%which already governed the background evolution.
%if $\delta\phi_k$ is a constant, then $\Psi_k$ is also constant and given by
%$H \Psi_k \simeq 4 \pi G }  \partial_t \bar \phi  \, \delta\phi_k$.
%when $H$ is slowly varying.

The second important fact is that the modes $v_k$, defined by
$v_k /a = \delta \phi_k +   \Psi_k (\partial_t \bar \phi/ H )$, 
%is the mode 
are those of a canonical (and gauge invariant) field. 
Since all fluctuations obey linear equations 
it is indeed crucial to identify the combination
which forms a canonical field  
%(in the sense that the commutator $[\phi, \partial_t phi] = i \hbar$)
because its {\it normalized} quantum mechanical
fluctuations will be taken into account in the sequel.
%we shall ...below. 
The modes $v_k$ behave like
 harmonic oscillators with time dependent frequency:
\begin{equation}
\partial_\eta^2 v_k + (k^2 - \xi) v_k = 0 \, ,
\label{Meq}
\end{equation}
where $\eta$ is the conformal time ($d\eta = dt/a$) 
and where $\xi$ is a function of the background solution
approximatively given, in the slow-roll limit, by
% function of the background solution which, roughly speaking, behaves like 
$2/\eta^2$. Hence, in the early past, for $\eta \to -\infty$, Eq. (\ref{Meq}) 
reduces to a standard harmonic oscillator. However when $k^2 = {\xi}$, 
which corresponds to $k \simeq 1/d_H(t)$, 
%i.e., $k/a = H$, 
the mode exits the Hubble radius (also often called Hubble horizon) 
and stops oscillating.
One gets a growing ($\propto a$) and decaying mode
($\propto 1/a)$. Thus $v_k/a$ becomes constant determined
by the growing mode. To evaluate the norm of this constant 
one needs an initial condition for $v_k$. 
%Straightforward algebra gives the following result.
In preparation of the subsequent analysis,
let us impose that $v_k$ has a unit Wronskian
and contains only positive frequencies
for large values of $kd_H(t_{in})$, i.e., 
when the mode is well inside the Hubble radius.\footnote{In 
the present classical treatment, this choice is purely mathematical. 
It will become physically meaningful when we shall
use second quantized rules in the next subsection.
Let us also point out that we shall consider only the norm 
of $v_k/a$ since we have in mind that the fluctuations should
be described by a statistical ensemble which fixes only the
power of the spectrum.} 
%From Eq. (\ref{Meq}), one gets that t
%for large values of $k$, i.e., for $k \gg 1/d_H(t_{in})$,
%and if $v_k$ is a positive frequency mode of unit Wronskian at 
%the onset of inflation, $t_{in}$, 
After horizon exit, one finds that the {\it frozen} value of $v_k/a $ is  
\begin{equation}
\vert (v_k/a)^{fr} \vert^2 = {H_k^2 \over 2 k^3} = 
\vert \delta \phi_k^{fr} \vert^2 \, \left(1+ O\!\left(\epsilon\right)\right)\, ,
\label{Fsp}
\end{equation}
where $H_k$ is the value of $H(t)$ at horizon exit,
when $k=1/d_H(t)$. 
%The above equation is correct to the leading order in 
%$\epsilon$. 

Then Eq. (\ref{0ieq}) implies that 
%if $\delta\phi_k$ is a constant, then 
$\Psi_k$ also tends to a constant after horizon exit.
The frozen value is given by
%$H \Psi_k^{fr} \simeq 4 \pi G }  \partial_t \bar \phi  \, \delta\phi_k^{fr}$, i.e.
$\Psi_k^{fr} \simeq - \delta\phi_k^{fr}/\bar \phi $,
i.e. $\vert \Psi_k^{fr} \vert^2 = 4\pi G\, \epsilon \vert \delta\phi_k^{fr}\vert^2$.
%$H \Psi_k \simeq 4 \pi G }  \partial_t \bar \phi  \, \delta\phi_k$.
%when $H$ is slowly varying.
%Hence to get the metric fluctuation spectrum, 
%it suffices to determine the spectrum of $\delta\phi_k^{fr}$.
However,
%To be complete 
there is still a subtlety:
% which however does not affect this conclusion. 
%It concerns the fact that this constant
$\Psi_k^{fr}$ gets multiplied by 
$2/3\epsilon$ when inflation stops. 
This follows from the existence
of a conserved quantity\cite{MukhB} which is
proportional to $\Psi_k (w+5/3)/(w+ 1)$, where $w=\bar P/\bar \rho$.
Hence when the equation of state changes
from inflation 
%with $w + 1= 2\epsilon/3$ 
to radiation,  
the amplification factor is $2/3 \epsilon$.

In brief, from the combined evolution of 
the background variables $a, \, \bar \phi$
and the modes $\Psi_k, \, \delta \phi_k$, inflation tells us that
in the radiation dominated era, before horizon re-entry, the {\it primordial}
%(adiabatic\cite{Langlois}) 
spectrum of the metric fluctuations $\Psi_k$  is given by
\begin{equation}
\vert \Psi_k^{prim} \vert^2 = {4 \over 9}  {4\pi G \over \epsilon_k } \vert (v_k/a)^{fr} \vert^2  \, .
\label{prS}
\end{equation}
%where $\xi$ is a constant (close to 1)
%which depends of the equation of state upon re-entry.
We have added a subscript $k$ to  $\epsilon$ because, in general,
it (slowly) varies with time and hence on $k$ through $k/a=H$.
%is determined by the frozen spectrum of Eq. (\ref{Fsp})
%$\vert (v_k/a)^{frozen} \vert^2$
%which is fixed at horizon exit, during inflation.
The primordial spectrum 
%(today observable) 
it thus given in terms of the background equation
of state $\bar P/\bar \rho$ and the norm of the frozen value of $v_k/a$
%the r.h.s of Eq. (\ref{prS}) 
evaluated during inflation, just after horizon exit. 
This is the first non-trivial outcome of inflation.
%(Notice that we have added a subscript $k$ to 
%$\epsilon$ because, in general, it is time dependent, 
%and hence $k$ dependent through $k/a=H$.) 
%This is the first remarquable outcome of inflation.
From Eq. (\ref{Meq}), the r.h.s. of Eq. (\ref{prS}) 
can be computed {\it if} we know 
%by the knowledge of 
the amplitude
of $v_k$ at the onset of inflation.
%, as in Eq. (\ref{Fsp}).
Therefore, so far,
%by adopting inflation, 
we have only ``postponed'' 
the problem of the initial conditions, from some high redshift 
before nucleosynthesis, to some still higher redshift before 60 e-folds
of inflation.

\subsection{Initial conditions and QFT}

It thus remains to confront the 
question of the initial conditions of the inflaton fluctuations.
%Before doing so, it should be clear that this question 
This question only concerns 
%concerns 
the values of $k$ which correspond to scales which
are today visible. We shall call them the {\it relevant} scales.
%  it remains to determine what is (or better could be) 
%the (classical) value of $v_k$ at the onset of inflation. 
When inquiring about their initial conditions,
% of the {relevant} scales, 
%conbination In trying to determine how to compute the spectrum of $v_k$,
one encounters a major surprise: {\it one must abandon classical settings.}
%and go over quantum field theory. 
%This spectrum cannot be compute in classical terms. 
The reason is the following. 
When a mode $v_k$ is well inside the Hubble horizon, 
it behaves as 
%are effectively
a massless mode, see Eq. (\ref{Meq}).
%since its momentum $p(t)=k/a(t)$ is much larger than $H$. 
Hence,
its energy density behaves like radiation
and scales like $1/a^4$. 
Thus, {if} $v_k$ were to possess a classical amplitude inside the Hubble horizon,
one would reach an inconsistency because when propagating  
%$v_k$ 
backwards in time, 
% after few e-folds,  
its energy density would, after few e-folds, overtake
the background density $\bar \rho_{\phi}$ 
%since the latter 
which is almost constant. 
But this would violate our assumption that there {\it is} inflation
which implies that $\bar \rho_{\phi}$ {is} the dominant contribution.
So, unless one fine tunes 
%in an artificial manner
the number of {\it extra} e-folds, $N_{extra}= N_{tot} - N_{min}$, 
%the conclusion is that 
the relevant modes cannot have
classical amplitudes at the onset of inflation.\footnote{For the 
same reason, the curvature relative contribution 
should be exponentially small when evaluated today:
$\Omega_0^{curv.} \ll 1$. In other words, in inflation, 
the {\it relevant} part of the spatial sections is flat.}
%,unless 
%in an artificial manner the number of {extra} e-folds is small.} 
%In other words, by consistency, inflation requires that the 

%Therefore 
One must therefore look for a quantum mechanical origin
of the primordial spectrum.\footnote{ 
%It is truely a supprise that the 
%origin of the largest scales of the universe ought to be 
%of quantum origin, because quantum mechanics was, after all, 
%invented to deal with atomic and molecular spectra.  
If it turns out that the {\it large scale structures} of our universe 
are of quantum origin, 
this would constitute the triumph of quantum mechanics 
which was elaborated from 
%to deal with 
atomic and molecular spectra.}
When adopting the settings of quantum field
theory in curved space\cite{BDavies}, 
one should re-address the question of the initial
conditions. It is now formulated  
in terms of the occupation number $n_k$ 
of the quanta of the field operator $\hat v$ in  Fock space. 
%of the quanta of the field operator $\hat v$. 
%As in classical terms, one 
For relevant $k$,
one finds that $n_k$ must vanish at the onset of inflation.
Indeed, the energy density carried by these quanta 
($\propto n_k k^4/a^4$) also scales like $a^{-4}$ and
would therefore violate, as in classical settings, the hypothesis that
the background energy density is dominated by $\bar \rho_{\phi}$.
So one is left with the vacuum as the unique possibility.
 It is then remarkable that the vacuum energy contribution evades
a potential inconsistency which would have otherwise ruined inflation:
Because of the subtraction of the zero-point energy\cite{BDavies},
the vacuum energy does {\it not} grow like $1/a^4$ and stays
much smaller than the background density $\bar \rho_{\phi}$. 
Hence vacuum is the only initial state of relevant modes 
which is consistent with inflation.
%, but perfectly consistent.
Notice that nothing can be said about the (irrelevant) infra-red 
modes 
%(i.e. 
with $k \ll 1/d_{H0}$
%which exit the Hubble horizon after a few e-folds. Indeed
because their energy density does not diverge at the onset of inflation. 
Notice also that  nothing should be said about them
because they are viewed today as part of the homogeneous background
and cannot be identified as modes. 
%initial state is that 
%it is perfectly consistent to assume the the initial
%state of all today visible modes be in their vacuum state.

There is a complementary way to understand how inflation
answers the question of the initial conditions.
%\footnote{It
%should be clear that we are here only concerned by the 
%fluctuation modes, and not the background variables. 
%Whether or not inflation can also satisfactorily solve
%the initial conditions of those variables remains an open question.}
Instead of asking what happens when propagating modes
backwards in time, lets directly consider what could possibly be the initial state
at the onset of inflation. 
In the homogeneous patch, there is an energy scale: $H$.
Hence, the occupation number of quanta of (proper) frequency 
$\omega_{in} =k/a(t_{in})$ 
%much larger than $H$ 
must obey a kind of Wien law 
$n(\omega_{in}) \propto e^{-\omega_{in}/H}$ for $\omega_{in} \gg H$.
On the other hand, at the onset of inflation,
the initial frequencies of all relevant modes
obey $\omega^{\rm relevant}_{in} > H z_{in}/z_{H0}$, where $z_{H0}$ is the redshift when 
the largest visible scale 
$d_{H0}$ exits the Hubble radius, see Figure 2. 
Therefore, when the number of {extra} e-folds, 
$N_{extra}= N_{tot} - N_{min}$, 
is larger than, say $5$, one finds that 
all relevant modes {\it are} in their ground state.
Concomitantly to this vacuum condition, at the onset of inflation, 
%the today Hubble scale 
the proper size of scale $d_{H0}$ 
%had an initial
% at the onset of inflation, a proper size equal to 
was given by $H^{-1} z_{H0}/z_{in }=  H^{-1} e^{-N_{extra}}$. 
So unless one constraints $N_{extra}$ to be smaller than $\ln(M_{Planck}/H)
\simeq 13$, our universe was initially inside a Planck cell.
% at the onset of inflation.
To appreciate the unavoidable character of this
conclusion, we invite the reader to draw the ``trajectory'' of the
(proper) Planck  length $l_P$ in ($a,d$) plane of Figure 2.
%, more on this in the next footnote.} 
% $d_H(t)$.   
%%%Notice again that nothing could be said about  infra-red modes.
% (i.e. with $k< 1/d_{H0}$)which are view today as constant. 

%It is amusing to notice that 
%%when there are $60$ e-folds of inflation, 
%the today's Hubble scale $1/H_0
%\simeq 10^{26} {\rm meters} \simeq  e^{140}\, l_P$ 
%has, at $z_{H0}$ see Figure 2, a proper length equal to $l_P $.
%To further appreciate the unavoidable character of these
%conclusions, it is also instructive to draw the ``trajectory'' of the
%fixed proper distance $L=l_P$ in Figure 2.}


So, in inflation, the choice of the initial state does not follow from 
a principle (e.g. some symmetry) but it is fixed by the 
kinematics of inflation itself.
This stems from the blue-shift effect encountered in a
backward in time propagation which sends the proper
frequencies $\omega = k/a$ of relevant modes way above $H$. 
%which force to appeal to ultra-high initial frequencies 
%which are in their ground state. 
%it is not a question of symmetry which leads to the choice of 
At this point it is interesting to mention the
%all these aspects are also find , 
%there is a 
precise analogy between these aspects and 
black hole physics. 

%{\bf Link with black hole physics} \noindent
Classically black holes cannot radiate and this is 
deeply connected to the {\it no hair} theorem:
stationary solutions are characterized only
by mass, angular momentum and electric charge because 
all multipoles are radiated away\cite{MTW} after a few {e-folds},
where the unit of time is here given by the inverse of
the ``surface gravity'' of the hole. 
%in terms of the ``surface gravity'' which is proportional to 
(For a neutral spherically symmetric hole, this characteristic
time is the Schwarzschild radius.)
So if after some e-folds some radiation is emitted by a black hole,
it must be of quantum origin, i.e. it must be Hawking radiation\cite{Hawk,GO}.
Similarly, for the relevant scales
%classically, for the relevant modes 
in inflationary cosmology, 
the geometry would classically be bald,  no hair;
because, when $N_{extra}=N_{tot} - N_{min} \gg 1$,
all pre-existing classical structures are so much
%stretched 
diluted that today they would still be part of the homogeneous background. 
%, at least for the relevant modes. 
So if some structure is found at smaller scales, 
it must be of quantum origin. 
In addition, in both cases, the properties of the spectrum is 
%unequivocally
fully determined by the geometry because the initial state 
of relevant modes is the ground state.\footnote{It should also
be noticed that this conclusion 
%Notice however that this 
has recently been 
questioned both in black hole physics
%\cite{bhtp} 
and in inflationary cosmology
under the appellation {\it the trans-Planckian question}, for a review 
see \cite{tp}.
The question arises from the fact that
quantum fields in a curved space-time
%propagating in a given classical background
should be a valid approximation of quantum gravity 
only below the Planck scale.
However, as we saw, the initial state is specified for
%the proper 
frequencies well above that scale. Hence
% $\omega_{in}= k/a_{in} \gg M_P$, 
we should worry about the validity of 
%the settings, and 
the predictions we obtained. 
%since quantum fields propagating in a given classical background
%is suppose to be a valid approximation of quantum gravity 
%only below the Planck scale. 
%Following a suggestion of Unruh\cite{unruh},
To confront this question, a phenomenological approach
proposed by Unruh\cite{unruh} has been thoroughly  investigated. 
The (disputed) conclusion\cite{bhtp,ictp} is that modifications 
of the standard results are unlikely because the properties of the
spectrum are generically {\it robust} even when 
drastically modifying the physics above the Planck scale.}
%It is amusing to notice that 
%%when there are $60$ e-folds of inflation, 
%the today's Hubble scale $1/H_0
%\simeq 10^{26} {\rm meters} \simeq  e^{140}\, l_P$ 
%has, at $z_{H0}$ see Figure 2, a proper length equal to $l_P $.
%To further appreciate the unavoidable character of these
%conclusions, it is also instructive to draw the ``trajectory'' of the
%fixed proper distance $L=l_P$ in Figure 2.}
% So one should also worry 
%about the validity of the predictions obtained by using these settings.}

\subsection{Predictions and observational data}

Having understood how inflation fixes 
%the question of 
the initial condition of fluctuations, we can bring together the 
various results. Since $v(t,x)$ is a canonical field, the positive 
frequency modes $v_k(t)$ should have unit Wronskians in order for 
the creation and destruction operators to obey the canonical commutation
rules. Therefore $<\hat v_k \hat v_{k}^\dagger>$, 
the v.e.v. of the Fourier transform of the field operator,
% and since the state of its relevant modes is vacuum, the expectation value
%$\hat v_k \hat v_{k}^\dagger$ 
is equal to $1/2k$ ($\hbar = 1$)
at the onset of inflation. 
Then Eq. (\ref{Fsp}) gives the v.e.v. after horizon exit.
(In quantum optical terms, one would say that the freezing out of the 
modes at horizon exit induces a
%there has been a  
%the fact that 
parametric amplification of vacuum fluctuations
which leads to extremely squeezed {two}-mode states\cite{Grishuk}. 
When considering the modes near horizon re-entry, 
the high squeezing manifests itself by the 
%translates into properties of 
{\it temporal coherence} of the modes, a 
property specific to inflation, see e.g. \cite{Albrecht} and below.)
%distiguish inflation from other models for primordial sprectra.)
%physical relevance that inflation
%leads to  {\it two}-mode squeezed states 
%(and not one mode) will be discussed below.)
%described modes $v_k$ of unit Wronskian, see Eq. (\ref{}))
%The amplitude of the fluctuations are minimal in that they result 
%from amplification of vacuum fluctuations.
Using Eq. (\ref{prS}), the primordial power spectrum $P^{prim}_k$,
defined by
%using Eq. (\ref{prS}), 
%In the simplest inflationary model,
%After this long digression concerning initial conditions, 
%we conclude that in inflation all relevant modes
%are initially in their ground state.
%Hence the expectation value of $v_k$ oveys Eq. (\ref{}).
%$\langle \hat  v_k \hat v_k^\dagger \rangle= 1/2k$ at the onset 
%of inflation. Then Eq. (\ref{Meq}) tells us that the frozen
%spectrum is $\vert () \vert = H^2/2k^3$. 
%One finds that the primordial spectrum of metric fluctuations is given by 
the two-point function
\begin{equation}
\langle \Psi(t, x+y) \, \Psi(t, y) \rangle 
= {1 \over 2 \pi^2} \int_0^\infty {dk \over k}{\sin(kx) \over kx} \, P^{prim}_k\, ,
%P^{prim}_k= k^3 \vert \Psi_k \vert^2 = {\xi \over \epsilon}  H^2_k \, ,
%\vert (v_k/a)^{fr} \vert^2 = {H_k^2 \over 2 k^3} = 
%\vert \delta \phi_k^{fr} \vert^2 \, \left(1+ o\left(\epsilon\right)\right)\, ,
\label{Psp}
\end{equation}
is given by
%using Eq. (\ref{prS}),
%where  the primordial power spectrum $P^{prim}_k$ is given by 
\begin{equation}
P^{prim}_k= k^3 \vert \Psi_k^{prim} \vert^2 = { 8 \pi G \over 9}
{H^2_k \over \epsilon_k} \, .
%\vert (v_k/a)^{fr} \vert^2 = {H_k^2 \over 2 k^3} = 
%\vert \delta \phi_k^{fr} \vert^2 \, \left(1+ o\left(\epsilon\right)\right)\, ,
\label{Posp}
\end{equation}
%To get this equation, we have used Eq. (\ref{Fsp}) and Eq. (\ref{prS}). 
In Eq. (\ref{Psp}), $t$ should be in the radiation dominated era,
well before horizon re-entry. When getting close to horizon re-entry,
$\Psi_k$ starts re-oscillating.
%to  such that the relevant modes
%are still inside the Hubble horizon, before horizon exit. 

In brief,
%From these results we can say that 
the simple model of inflation on a single scalar field
{\it predicts} that the primordial spectrum 
should enjoy the following properties:
\begin{itemize}
%\item 
%The amplitude of the fluctuations are minimal in that they result 
%from amplification of vacuum fluctuations.
\item
The power spectrum Eq. (\ref{Posp})
%given in Eq. (\ref{Fsp})
is nearly scale invariant.  This 
%simply 
results from the stationarity
of the process: parametric amplification of vacuum fluctuations, $k$ after $k$.
% of vacuum
%fluctuations described modes $v_k$ of unit Wronskian, see Eq. (\ref{}))
%which occurs in slowly evolving accelerated universe, 
%$\epsilon = - \partial_t H/H^2 \ll 1$.
%a succession of near de Sitter space. 
In fact the $k$-dependence of $P_k^{prim}$
only originates from the slow 
%($\epsilon = - \partial_t H/H^2 \ll 1$)
evolution of the background with $t$
evaluated at horizon exit, when $k=1/d_H(t)$. 
% and re-entry.
%: $k/a=H$. 
\item
The fluctuation spectrum forms a Gaussian {\it ensemble} --
%Gaussian, 
because the vacuum is a Gaussian state,
%and an ensemble, 
and because relevant modes were initially in their ground state. 
This means that the {\it probability} to find $\Psi_k^{prim}$ 
with a given amplitude is Gaussian, and that the l.h.s. of
Eq. (\ref{Psp}) should be 
interpreted as
%in terms of 
an {ensemble} average.
(Non-Gaussianities are not expected to develop in the early universe
since the rms fluctuations are $10^{-5}$\cite{malda}.)
% $A_k$.
% is given by $P(A_k) \propto e^{- \vert A_k\vert^2/2 \sigma_k^2}$ where 
%the dispersion $\sigma_k$ is given by Eq. (\ref{prS}).
%equal to the rms value $......$.
%Since we have linearized the dynamical equations for the 
%fluctuations, this result should cause no suprise. 
%Notice that the {\it validity} of the linearization
%rests on the fact that the mean 
%no non-linearities souldn't have altered because rms
%amplitude is of sethe order of $10^{-5}$, see however  \cite{BerUzan}.
%Malda}
\item
The modes $\Psi_k$ are {coherent} in time
at horizon re-entry. By coherent one means the following.
When $k/a\simeq H$, each $\Psi_k$ starts to 
re-oscillate. Hence the general solution is governed by {two} 
independent quantities.
Inflation predicts that
%, for all relevant $k$, 
the combination representing
the decaying mode vanishes, 
%simply 
because it has had all the time to do so.  
%is a necessary condition for acoustic peaks to form\cite{Hu}.  
%\item
%The density perturbation spectrum (mainly?)
%results from the `adiabatic' mode which is always present
%what ever the dynamical model is. More complicate models
%will have more iso-curvature modes.
\end{itemize}
%When confronting these predictions to 
%One can now confront these predictions
Observational data tell us that the spectrum does enjoy the following properties:
%, one finds:
% We find the following:
\begin{itemize}
\item
The power spectrum is nearly scale invariant:
It has a rms amplitude $\simeq 10^{-5}$ (which fixes $H_k 
\simeq 10^{-6} \, M_P$) and it is parameterized
% of the power spectrum is now\cite{WMAP} given 
by a spectral index, defined by $n_k = d\ln P/d \ln k$,
which obeys\cite{WMAP} 
$n_k = 0.93 \pm 0.04$ and $dn_k/d\ln k = -0.03 \pm .02$ at a scale
which represents $0.5${\%} of $d_{H\, 0}$, the conformal
scale of 
%whose today's propecr distance is $a_0/k= 20 \, {\rm Mpc}$.
%Remember that 
today's Hubble radius.
% $\simeq 4000 {\rm Mpc}$.
%which is defined by $n_k = d\ln P/d \ln k$.
%which enters the power spectrum as $P_k= k^{n-1}$. 
\item
Non-Gaussianities have not been found, and
%If they exist, the should be smaller that . On the other hand, 
the spread of the data 
(the {\it cosmic variance}) is compatible
with 
%the {\it cosmic variance} , i.e., the deviations around mean values is
%in agreement with 
standard deviations given the finite number of independent observables. 
\item
Narrow ``acoustic'' peaks are observed in the temperature fluctuations spectrum.
These are by-products of the temporal coherence of the $\Psi_k$ at horizon re-entry. 
Moreover, when considering the {\it polarization}
of the CMB, one finds an anti-correlation peak which is 
``the distinct signature of primordial adiabatic fluctuations''\cite{FirstY}.
This peak excludes that cosmic strings or textures could be a relevant
%the principal 
mechanism for primordial fluctuations.
%%%%\cite{PPeter}. 
%the 
%maximum amount of non-coherent spectrum is about 10% \cite{PPeter}. 
\end{itemize}

In addition to these three points, the properties of the
CMB anisotropy spectrum tell us that 
%$\Omega_0^{curv.}$, 
the contribution of the spatial curvature
obeys $\Omega_0^{curv.}= 0.02 \pm 0.02$ in agreement with the
prediction that it
%the 3-curvature 
should vanish,
%of inflation, 
see footnote 5.
In brief, the 
outcome of the above comparison is that {\it the generic predictions of
the {\rm simplest} inflationary model are in accord
%are in agreement 
%{\it all} found in 
%extremely good agreement with the last 
with observational data.}

\subsection{Perspectives}

%To conclude it should be mentioned 
It should be clear to the reader that 
%that in this introduction
several aspects  have not been discussed
in the present text.
% which is only an introduction. 
These include polarization,
%\cite{JHu},
galaxy spectra (to have access to the power spectrum at 
smaller scales) as well as gravitational waves and iso-curvature modes\cite{JHu}. 
%When all taken into account,
%altogether, 
%both theoretically and observationally, 
By taking all these  into account,
%one can search for 
a finer understanding of the predictions of inflation
will be obtainable and one might envisage 
constraining  inflationary models
%ary models and envisage 
by observational data (already available and also to come).
%to put constraints on the various inflationary models.
%But this is next week news.
To give a flavour of this 
program, let us briefly present
%list what are the main 
some of the points under current
investigation. They concern {\it refinements} of the simple model
we presented.
\begin{itemize}
\item
%Since relevant modes are initially in their grounds states,
The primordial power spectrum can be {\it phenomenologically} expressed in terms of
{several} {slow-roll} parameters which characterize the inflaton action and/or
the evolution of the background\cite{Lid}.
It is hoped that these parameters, and therefore the inflaton action
(or its potential), could be determined from observational data.
In particular, degeneracies amongst inflationary models
could be removed by the detailed properties of the polarization spectra.
\item
Primordial non-Gaussianities can be obtained when
iso-curvature modes are considered\cite{Uz}. 
Theoretical studies might orient and ease observational search.
\item
Inflation also predicts that gravitational waves should be produced.
Their spectrum is given by $P^{grav}_k= 4\pi G  \, H_k^2$,
in the place of Eq. (\ref{Posp}).
Being characterized by 
%Phenomenologically, it could be characterized by 
different spectral indices, the observation of both 
$P^{grav}_k$ and $P^{prim}_k$
%this spectrum
%gravitational  simultaneous knowledge of both spectral 
%primordial gravitational and density 
would allow
%, together with Eq. (\ref{Posp}), 
to perform consistency checks which 
could confirm (or rule out) inflation.
(Notice however that the lower value of $P^{grav}_k$
might delay (or even exclude) the detection of primordial
gravitational waves.)
%titute the final proof that their was inflation.
\end{itemize}

Besides these phenomenological questions, 
deep fundamental questions persist, in 
particular: Who is the inflaton? What is the re-heating temperature?


%Finally, could one still wonder if one can get rid of inflation,
%but at which cost ?

%Isn't this remarquable ? 
%In conclusion, we invite the reader to meditate
%on this question.

%%%  Acknowledgements  %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
%\
\vskip .7 truecm 

{\bf Acknowledgements}
%The acknowledgements.}
I am grateful to N. Obadia and D. Campo for their comments and advice during the
writing of this paper. I am also indebted to P. Peter and J-Ph. Uzan 
for useful remarks.}
\vskip .7 truecm 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Bibliography  
%%%%%%%%%%%%%%%%%%%%%%%
%\newpage

\begin{thebibliography}{99}
%\selectlanguage{english}
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\end{document}



\end{document}

astro-ph0404022.extracted_bib
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%\newpage

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\bibitem{PPeter} Bouchet F.R., {\it et al.,}
{\it Evidence against or for topological defects in the BOOMERanG data} ?
 Phys. Rev. D65 (2002) 021301.
\bibitem{FirstY} Peiris H.V., {\it et al.}, {\it First 
Year {\it WMAP} Observations, Implications for Inflation}, 
 arXiv.org: astro-ph/0302225.
\bibitem{Guth} Guth A., {\it The inflationary universe: A possible solution to the horizon 
and flatness problems},
Phys. Rev. D23 (1981) 347. 
\bibitem{Linde} Linde A.,  {\it A new inflationary universe scenario: 
a possible solution of the horizon, 
flatness, homogeneity, isotropy and primordial monopole problems}, 
Phys. Lett. B108 (1982) 389.
\bibitem{Mukh} Mukhanov V., Chibishov G.,{\it  Quantum fluctuations
and a nonsingular Universe}, JEPT Lett. 33 (1981) 532, 
{\it see also}  arXiv.org: astro-ph/0303077.
\bibitem{Lindebook} Linde A., {\it Particle Physics and Inflationary Cosmology},
Harwood academic, Chur, 1990.
\bibitem{reheat} Greene P., Kofman L., Linde A., Starobinsky A., 
{\it Structure of resonance in preheating after inflation}, 
Phys. Rev. D56 (1997) 6175. 
\bibitem{Staro} Starobinsky A., {\it A new type of isotropic cosmological
models without singularity}, Phys. Lett. B91 (1980) 99.
\bibitem{Brandy} Brandenberger R.,
{\it Principles, Progress and Problems in Inflationary Cosmology},
AAPPS Bull. 11 (2001) 20.
\bibitem{Langlois} Langlois D., {\it Isocurvature cosmological
perturbations}, C. R. Physique. {\bf 8} (2003) 953.
\bibitem{BDavies} Birrell N.D., Davies P.C.W., {\it Quantum Fields in 
Curved Space}, Cambridge University Press,
Cambridge, 1982.
\bibitem{MTW} Misner C.W., Thorne K.S., Wheeler J.A.,{\it  Gravitation}, Freeman,
San Fransisco, 1973.
\bibitem{Hawk} Hawking S., {\it Particle Creation By Black Holes}, Commun. Math. Phys. 43 (1975) 199.
\bibitem{GO} Brout R., {\it et al.}, {\it A primer of black hole quantum physics}, 
Phys. Rept. 260 (1995) 329.  
\bibitem{tp} 
%Jacobson T., Black hole evaporation and ultrashort distances,
%Phys. Rev. D44 (1991) 1731; 
%Black hole radiation in the presence of a short distance cutoff,
%Phys. Rev. D48 (1993) 728; {\it for reviews see}
Jacobson T., 
{\it Trans-Planckian redshifts and the substance of the space-time river},
Prog. Theor. Phys. Suppl. 136 (1999) 1.
%, {\it and} Helfer A., 
%Do black holes radiate?, Rep. Prog. Phys. 66 (2003) 943. 
\bibitem{unruh}
%m{bhtp} 
Unruh W.,
{\it Sonic analogue of black holes and the effects of high frequencies 
on black hole evaporation},
Phys. Rev. D51 (1995) 2827.
\bibitem{bhtp} 
Brout R., {\it et al.}, 
{\it Hawking radiation without trans-Planckian frequencies}, Phys. Rev. D52 (1995) 4559. 
\bibitem{ictp} Martin J., Brandenberger R., 
 {\it Trans-Planckian problem of inflationary cosmology}, Phys. Rev. D63 (2001) 123501; 
Niemeyer J., Parentani R., {\it Trans-Planckian dispersion and scale-invariance 
of inflationary perturbations},
    Phys. Rev. D64 (2001) 101301; {\it for a recent work see}
Martin J., Brandenberger R., 
{\it On the Dependence of the Spectra of Fluctuations in Inflationary Cosmology
 on Trans-Planckian Physics},
arXiv.org: hep-th/0305161. 
\bibitem{Grishuk} Grishuk L.P., 
{\it Density perturbations of quantum mechanical origin and
anisotropy of the microwave background},
Phys. Rev. D50 (1994) 7154.
\bibitem{Albrecht} Albrecht, A., {\it Cosmic Inflation}, NATO Advanced Studies 
Institute ``Structure Formation in the Universe'', Cambridge 1999, astro-ph/0007247;
for recent work, see also 
Campo D. and Parentani R., {\it Inflationary spectra, decoherence,
and two-mode coherent states}, to appear on the ArXives.
\bibitem{malda} J. Maldacena,
 JHEP $\bf 0305$, 013 (2003).
\bibitem{JHu} Hu W.,  
{\it CMB Temperature and Polarization Anisotropy Fundamentals},
Annals Phys. 303 (2003) 203.
\bibitem{Lid} Lidsey J.E., {\it et al.}, 
{\it Reconstructing the Inflaton potential --
an Overview}, Rev. Mod. Phys. 69 (1997) 373.
\bibitem{Uz} Bernardeau F., Uzan J.-P., 
{\it Non-Gaussianity in multi-field inflation}, 
 Phys.Rev. D66 (2002) 103506.
\end{thebibliography}