The measurement of the characteristics of the anisotropies in the temperature
of the CMBR can teach us a lot not only about the processes that gave
rise to the density
perturbations responsible for the appearance of the large-scale
structures we observe today,
but also about cosmological parameters, like the Hubble constant and the
matter and energy
content in the Universe, and physical events that happened between
decoupling and the present,
like re-ionization and the formation of structure (e.g. cluster
formation through the
Sunyaev-Zel'dovich effect). However, there are so many possible ways in
which the properties
of the CMBR anisotropies can be altered, the most troublesome being by
changing the nature of the
initial perturbations and/or of the dark matter (energy), that in order
to extract a tight
interval of confidence for some parameter, e.g. the value of the Hubble
constant, it is
necessary to make *a priori* some restrictions to the nature of the
phenomena
that might affect the characteristics of the CMBR temperature anisotropies.

Therefore, once again the strategy has to be to initialy work within a
simplified framework,
and only if the observational data demands it, expand the initial
assumptions so as to widen the
allowed parameter space. At present, the simplified framework within
which most analysis of the
present CMBR anisotropy data are made has 10 associated parameters,
which may or may not be
allowed to vary freely, and assumes primordial, adiabatic perturbations
with a Gaussian
distribution. The dark matter is taken to be of only two possible types,
cold or massive
standard neutrinos, and any dark energy is assumed to behave dynamically
as a classical
cosmological constant. We will follow the choice of parameters of
(73),
as slightly different, in practice equivalent,
sets of parameters can be worked with. We then have 5 cosmological parameters:
_{k}, the spatial
curvature, equal to
- *k*/*a*^{2}*H*^{2};
_{}, the energy density
associated with a classical cosmological constant,
, which is equal to
/ 3*H*^{2};
*w*_{cdm}
_{cdm}*h*^{2},
the matter density in the form of cold dark matter;
*w*_{}
_{}*h*^{2}, the matter density in the form of
standard neutrinos;
*w*_{b}
_{b}*h*^{2},
the matter density in the form of
baryons. There are also 2 parameters that characterise the primordial
power spectrum of the
scalar adiabatic perturbations, which is assumed to have a power-law
dependence with scale,
_{H}^{2}(*k*)
*k*^{ns -
1}: *n*_{s}, the spectral index of the scalar perturbations;
*A*_{s}, the contribution to the CMBR quadrupole anisotropy
by scalar perturbations, which is in practice equivalent to using
_{H},
the amplitude of the scalar perturbation power spectrum. And 2
parameters that characterise the
primordial power spectrum of tensor perturbations, i.e. gravity waves,
which is also assumed to
have a power-law dependence with scale, *P*_{t}(*k*)
*k*^{nt}:
*n*_{t}, the spectral index of the tensor perturbations;
*A*_{t}, the contribution to the CMBR quadrupole anisotropy
by tensor perturbations. Finally, there is 1 physical parameter:
, the optical depth to the surface
of last scattering (equal to zero if there is no re-ionization after
recombination occurred). From these quantities
other well known parameters can be obtained, like the total matter
density in the Universe,
_{m} = 1 -
_{k} -
_{}, and the value of the
Hubble constant.

Again, it should be emphasised that any conclusions on the value of these parameters, obtained by comparing the theoretical expectation with the CMBR anisotropy data, is restricted to the tight framework imposed. The conclusions could be relaxed substantially if one was to consider introducing the added possibilities of part of the perturbations responsible for the CMBR anisotropies being active, incoherent, isocurvature, or with a non-Gaussian distribution, or the existence of more exotic types of dark matter (warm, or decaying, or self-interacting) or dark energy which does not behave dynamically as a cosmological constant (e.g. an evolving scalar field).

Through the study of the distribution of CMBR temperature anisotropies
in the sky at
different angular scales, it is possible to confirm or disprove the
assumption of a Gaussian
distribution for the perturbations. If these have a Gaussian
distribution, them the anisotropies
should also have one. At present only the *COBE* data has been
searched for evidence of
deviations from Gaussianity. Early results [see e.g.
(47,
30,
34)]
were negative. However, more recently, there have been
reports of strong evidence for deviations from a Gaussian distribution
in the *COBE* data
(29,
63,
65,
28,
60).
However, some analysis continue to dispute the presence
of a clear non-Gaussian signal
(10,
4,
8).

Among the contradicting reports, it has become clear that the one-point
distribution of the
*COBE* anisotropies is indeed Gaussian, but the anisotropies seem
not to be as randomly
distributed across the sky as they should be under the Gaussian
(random-phase) hypothesis,
(10).
The later is supported by the fact
that the statistical techniques that find evidence for non-Gaussianity
seem to be those which
are most sensitive to phase-correlations. The source for these could be
instrument noise, the
method of extraction of the monopole, dipole or the Galactic
contribution, unaccounted for
systematic effects associated with the way *COBE* collected data
(3),
or of course real
features in the microwave sky, which could be at the surface of last
scattering or due to some unknown source of foreground contamination.

Observational evidence for a primordial and adiabatic origin for the
perturbations that produced
the CMBR temperature anisotropies is scarce at present, even after the
release of the Antarctica
long duration flight data from the *BOOMERanG* experiment
(20,
51)
and data gathered by the first flight of the *MAXIMA* balloon
(32).

The evidence for the primordial nature of the perturbations will come
from the signature that the
acoustic oscillations induced in the photon-baryon fluid leave in the
angular power spectrum of the
CMBR anisotropies: a sequence of peaks in
space. Their adiabatic nature is revealed by
the spacing between the peaks:
_{1}
:
_{2}
:
_{3}
... corresponds to ratios of
1 : 2 : 3 : .... Isocurvature primordial perturbations also generate a
sequence of peaks in the
CMBR temperature anisotropy angular power spectrum, but at different
locations. The first
peak due to the acoustic oscillations, generated by the potential wells
that result from the
isocurvature primordial perturbations after they enter the horizon, is
moved to an higher
value, and the ratios between the first three (real) peak locations are
now 3 : 5 : 7 : .... The ratios
between the peaks in both cases only depend weakly on the assumed
cosmological parameters, and
therefore provide an excellent test of the adiabatic or isocurvature
nature of the fluctuations
(37,
38).
Unfortunately, at present, though the *BOOMERanG LDF* and the
*MAXIMA-1*
temperature anisotropy measurements go to sufficiently high
to probe
the expected location of a
second peak in the adiabatic case, the evidence is
inconclusive. Nevertheless, purely isocurvature
perturbations seem to be highly disfavoured just by the shape of the
first peak
(49). So,
presently one cannot use
solely CMBR anisotropy data to distinguish between an adiabatic or an
isocurvature origin for the
density perturbations. In any case, the present known location of the
first acoustic peak
(~ 200) is sufficient to say that if the Universe is spatially flat then
most probably
the initial perturbations were adiabatic, as one then expects
_{1}
220.
Isocurvature perturbations necessarily need the Universe to be closed,
which moves the
horizon scale at decoupling to larger angular scales, given that if the
Universe was flat
then the first acoustic peak for primordial isocurvature perturbations
would be expected at
_{1}
340. If one admits the
possibility that both adiabatic and isocurvature modes are
present, then even with *MAP* it will be difficult to estimate the
relative amplitude of both.
One will have to wait for *Planck* to measure such a quantity with
less than 10 per cent error
(7).

As discussed before, the appearance of acoustic peaks in the angular power spectrum of the CMBR temperature anisotropies is not proof of an inflationary (or primordial) origin for the initial perturbations, though their absence could only be (realistically) explained by early and substantial (i.e. close to unity) re-ionization of the intergalactic medium. However, the fact that at least one acoustic peak has been detected argues against the later scenario, and its presence is evidence for some degree of coherence of the initial perturbations (which does not mean their are primordial, as previously mentioned).

In summary, the primordial, adiabatic and Gaussian nature of the initial
perturbations
seems to be still a viable first assumption, providing the simplest
coherent framework within
which it is possible to explain the present CMBR temperature anisotropy
data. Unfortunately,
even under this framework the CMBR data presently does not tell us much
about the parameters
*n*_{s}, *A*_{s},*n*_{t}, and
*A*_{t}, which characterise the power spectrum of the
scalar and tensor perturbations in the *simplest* inflationary models.

Gravitational waves can only contribute significantly to the CMBR temperature
anisotropies on scales above 1°, or
< 100, given that at the
surface of last scattering any gravitational waves inside the horizon
had their amplitude severely
diminished by redshifting. This means that only the *COBE* and
*Tenerife* data
(73)
might in principle be sensitive to the tensor spectral index
*n*_{t}. However, the
effect of varying *n*_{t} can be easily masked by
simultaneously varying the scalar
spectral index *n*_{s}, or by reducing the tensor
contribution to the large-angle
CMBR anisotropy to levels small enough that (almost) any variation of
*n*_{t}
could be accommodated within the measured errors. Therefore, at present
there are
no useful observational limits on the value of *n*_{t}.

The ratio between *A*_{t} and *A*_{s}, also
sometimes called *T*/*S*,
should in principle be a more readily measurable quantity, as it affects
the height of
the first acoustic peak relatively to the large-angle Sachs-Wolfe
plateau, and also
the CMBR anisotropy normalisation of the density perturbation power spectrum,
thus affecting the formation of large-scale structure. By analysing
these two effects,
(84) concluded that
values of *T*/*S* as large 4 are still viable, as long as
*n*_{s} is simultaneously
allowed to go up to 1.2. Adding the initial data coming from the
*BOOMERanG* and
*MAXIMA* experiments seems to decrease the maximum possible value
for *T*/*S* to 2
(46).
Restricting *n*_{s} to be lower or equal to unity, given that
this is what is predicted in the simplest inflationary models that produce a
cosmologically significant background of gravitational waves, then the
ratio *T*/*S* would need
to be lower than about 1. In particular, in the case of power-law
inflation models the
observational data seems to require *T*/*S* < 0.5.

Finally, *n*_{s} is also not very well constrained
at present by CMBR anisotropy data alone or in conjunction with
structure formation data.
The CMBR data by itself badly constrains *n*_{s} if all the
10 parameters
associated with the simplest inflationary motivated structure formation
models are allowed
to vary (73).
By fixing the matter density in the form of baryons to be
_{b}*h*^{2}
= 0.02 (as implied by observations if homogeneous standard nucleosynthesis is
assumed), and a reasonable range for the Hubble constant,
0.55 < *h* < 0.75, a interval for
*n*_{s} can be obtained, going from 0.8 to about 1.5
(73). However,
values for *n*_{s} higher than 1.3 can only be obtained at
the cost of having a very large
value for *T*/*S*
(73,
52).
And in any case the the initial data from the *BOOMERanG* and
*MAXIMA* experiments seem to disfavour values for
*n*_{s} above 1.2
(40,
43,
46).
We have seen that structure formation observations do not constrain very
well *n*_{s} either, with values
between 0.7 and 1.4 being possible. Joining all the data together would
thus indicate that a value for *n*_{s}
close to 1 is preferable, with the possibility of going as low as 0.8 or
as high as 1.3. The upper
limit is also confirmed by bounds imposed on the production of
primordial black holes
(42).

Looking into the future, the prospects of measuring the inflationary
parameters *n*_{s}, *A*_{s},
*n*_{t}, and *A*_{t} from CMBR anisotropies
are mixed. Data from balloon experiments like
*BOOMERanG* and *MAXIMA* will not help much in improving
these constraints, except for a better handle on the value of
*n*_{s}. As mentioned, initial data from the
two experiments has already further restricted the value of
*n*_{s} to being within about 10 per cent of
unity (40,
43).
The results from the *MAP*
and *Planck Surveyor* satellites will however change the
situation. All studies that have
been done to date of the precision with which both *MAP* and
*Planck* will be able to
estimate different cosmological parameters use the so-called *Fisher
information matrix*. This
gives the manner in which experimental data depends upon a set of
underlying theoretical parameters
that one wishes to measure. Within this set of parameters, the Fisher
matrix yields a lower limit
to error bars and hence an upper limit on the information that can be
extracted from such a data
set, i.e. the Fisher matrix reveals the best possible statistical error
bars achievable from
an experiment. With the further assumption of Gaussian-distributed
signal and noise, Fisher
matrices can be constructed from the specifications of CMBR
experiments. However, given that
the Fisher matrix formalism in practice asks how well an experiment can
distinguish the true
model of the Universe from other possible models, the results for the
error bars on the parameters
of these models will depend on which *true* model is chosen as
input. Several groups have
tried to estimate how well *MAP* and *Planck* will be able to
extract information
on the cosmological parameters from CMBR anisotropies using the Fisher
matrix formalism
[e.g. (6,
83,
80)].
I will quote the results from
the work by (25),
who were
probably those who more thoroughly explored, within the *simplest*
model context described
before, the parameter degeneracies that may appear when the information
is extracted from the
CMBR data. They even included the possibility of a dependence with scale
of the spectral index of
density perturbations [see also (17)]. Regarding
*n*_{s},
(25)
conclude that among
the four inflationary quantities it is the most readily measurable: just
by using the CMBR temperature
anisotropy angular power spectrum it will be possible to estimate it
with an error of about 0.1
(1) through *MAP* and
around 0.05 for *Planck*. Also using the CMBR polarisation
information the error tends to be halved in the case of *MAP* and
reduced by a factor of 5
for *Planck*. The other quantities, *A*_{s},
*n*_{t}, and *A*_{t}, will not be much better
known
after *MAP* and *Planck*, if just the CMBR temperature
anisotropy spectrum is considered. Only
by including polarisation information it will be possible to vastly improve the
knowledge we have about the values taken by *T*/*S* and
*n*_{t}.

The CMBR is expected to be partially linearly polarised [see
(39)
for a thorough review].
The polarisation signal can be decomposed into two separate and
orthogonal components, the so-called
*E* and *B* modes, with each mode having their own associated
power spectra, which will
depend in a different way on cosmological parameters. A
cross-correlation signal
between the temperature anisotropy and *E*-mode polarisation maps
will also be present
[see e.g. (44,
45)].
The knowledge of the polarisation signal will clearly contribute to
improving the CMBR constraints on cosmological and perturbation
parameters, but not equally
on all of them. Those parameters that most gain from the use of the CMBR
polarisation
information are the ratio *T*/*S*, *n*_{t} and
the optical depth . The reason is
that only vector or tensor perturbations can give rise to *B*-mode
polarisation, which can also arise from
re-ionization of the intergalactic medium. However, vector perturbations
rapidly decay in an expanding Universe, if all perturbations are
primordial (e.g of inflationary
origin). Consequently, while the total polarisation signal can help in
the better estimation of all
parameters, the power spectrum information associated with the *B*-mode
polarisation will help particularly in bringing down the *relative*
errors in the estimation of *T*/*S*, *n*_{t} and
, most noticeably if the first and
last are small (45,
25).
However, from a practical point
of view, it will not be easy to disentangle the CMBR polarisation signal
from foreground
polarisation sources, given the present lack of knowledge about their
nature (72).

Interestingly, the polarisation signal can also provide one of the strongest tests known of the inflationary paradigm. Because polarisation is generated at the last scattering surface, models in which perturbations are causally produced, necessarily on sub-horizon scales, cannot generate a polarisation signal on angular scales larger than about 2°, approximately the horizon size at photon-baryon decoupling (71). Hence, polarisation correlations on such scales would indicate either a seemingly acausal mechanism for the generation of the perturbations or re-ionization at work. However, the two mechanisms can in principle be disentangled due to their different dependence on large-angular scales (82). If what seems like acausality at work is proven, then it has been argued (54) that inflation provides the only mechanism through which it can be generated, by expanding initialy sub-horizon quantum fluctuations to super-horizon sizes. Unless it is postulated the existence of super-horizon perturbations since the begining of the Universe, which would give rise to a new initial conditions problem.

The measurement of the polarisation signal, by improving the estimation
of *T*/*S* and *n*_{t},
will also provide the possibility of checking whether the so-called
inflationary consistency
relation holds,
*T*/*S* -
7*n*_{t}
[e.g. (56)].
If this is shown than
single-field slow-roll models will receive a tremendous boost. Note,
however, that there are
inflationary models which do not predict such a relation.

Finally, we have until know only discussed what future constraints can
be imposed on the
cosmological parameters and the nature of the perturbations solely
through the CMBR temperature
anisotropy and polarisation signals. Clearly it would help if some of
the degeneracies
inherent to CMBR analysis could be broken by using large-scale structure
data and
direct measurements of the geometry and expansion rate of the
Universe. The most
promising large-scale structure data to be expected in the near future
is the SDSS data,
which together with the already well known local rich cluster abundance,
provides
a means of constraining the slope and normalisation of the present-day
density power spectrum.
Supernova type Ia and direct measurements of the Hubble constant will
enable independent
estimates of respectively the geometry and expansion rate of the
Universe. Further, the
evolution with redshift of the number density of rich galaxy clusters is
a powerful method
for determining the present total matter content,
_{0}, while
gravitational lensing
is able to impose interesting constraints on the value of a possible
cosmological constant.