2.1 The cosmological
principle
Our current Universe exhibits a wealth of nonlinear structures, but
the zero-th order description of our Universe is based on the
assumption that the Universe is homogeneous and isotropic smoothed
over sufficiently large scales. This statement is usually referred
to as the cosmological principle. In
fact, the cosmological principle was first adopted when
observational cosmology was in its infancy; it was then little more
than a conjecture, embodying ’Occam’s razor’ for the simplest
possible model.
Rudnicki [73] summarized
various forms of cosmological principles in modern-day language,
which were stated over different periods in human history based on
philosophical and aesthetic considerations rather than on
fundamental physical laws:
- The ancient Indian
cosmological principle:
-
The Universe is infinite in space and time and is infinitely
heterogeneous.
- The ancient Greek
cosmological principle:
-
Our Earth is the natural center of the Universe.
- The Copernican
cosmological principle:
-
The Universe as observed from any planet looks much the same.
- The (generalized)
cosmological principle:
-
The Universe is (roughly) homogeneous and isotropic.
- The perfect
cosmological principle:
-
The Universe is (roughly) homogeneous in space and time, and is
isotropic in space.
- The anthropic
principle:
-
A human being, as he/she is, can exist only in the Universe as it
is.
We note that the ancient Indian principle may be
viewed as a sort of ‘fractal model’. The perfect cosmological
principle led to the steady state model, which although more
symmetric than the (generalized) cosmological principle, was
rejected on observational grounds. The anthropic principle is
becoming popular again, e.g., in ‘explaining’ the non-zero value of
the cosmological constant.
Like with any other idea about the physical
world, we cannot prove a model, but only falsify it. Proving the
homogeneity of the Universe is particularly difficult as we observe
the Universe from one point in space, and we can only deduce
isotropy indirectly. The practical methodology we adopt is to
assume homogeneity and to assess the level of fluctuations relative
to the mean, and hence to test for consistency with the underlying
hypothesis. If the assumption of homogeneity turns out to be wrong,
then there are numerous possibilities for inhomogeneous models, and
each of them must be tested against the observations.
For that purpose, one needs observational data
with good quality and quantity extending up to high redshifts. Let
us mention some of those:
- CMB
fluctuations
-
Ehlers, Garen, and Sachs [18] showed that by
combining the CMB isotropy with the Copernican principle one can
deduce homogeneity. More formally their theorem (based on the
Liouville theorem) states that “If the fundamental observers in a
dust space-time see an isotropic radiation field, then the
space-time is locally given by the Friedman-Robertson-Walker (FRW)
metric”. The COBE (COsmic Background Explorer) measurements of
temperature fluctuations (
on
scales of
) give via the Sachs-Wolfe effect (
) and the Poisson equation r.m.s. density
fluctuations of
on
(see,
e.g., [99]), which implies that
the deviations from a smooth Universe are tiny.
- Galaxy redshift
surveys
-
The distribution of galaxies in local redshift surveys is highly
clumpy, with the Supergalactic Plane seen in full glory. However,
deeper surveys like 2dF and SDSS (see Section 6) show that the fluctuations
decline as the length-scales increase. Peebles [69
] has shown that the
angular correlation functions for the Lick and APM (Automatic Plate
Measuring) surveys scale with magnitude as expected in a Universe
which approaches homogeneity on large scales. While redshift
surveys can provide interesting estimates of the fluctuations on
intermediate scales (see, e.g., [72
]), the problems of
biasing, evolution, and
-correction would limit the
ability of those redshift surveys to ‘prove’ the cosmological
principle. Despite these worries the measurement of the power
spectrum of galaxies derived on the assumption of an underlying FRW
metric shows good agreement with the
-CDM (cold dark
matter) model.
- Radio
sources
-
Radio sources in surveys have a typical median redshift of
, and hence are useful probes of clustering at high
redshift. Unfortunately, it is difficult to obtain distance
information from these surveys: The radio luminosity function is
very broad, and it is difficult to measure optical redshifts of
distant radio sources. Earlier studies claimed that the
distribution of radio sources supports the cosmological principle.
However, the wide range in intrinsic luminosities of radio sources
would dilute any clustering when projected on the sky. Recent
analyses of new deep radio surveys suggest that radio sources are
actually clustered at least as strongly as local optical galaxies.
Nevertheless, on very large scales the distribution of radio
sources seems nearly isotropic.
- X-ray
background
-
The X-ray background (XRB) is likely to be due to sources at high
redshift. The XRB sources are probably located at redshift
, making them convenient tracers of the mass
distribution on scales intermediate between those in the CMB as
probed by COBE, and those probed by optical and IRAS redshift
surveys. The interpretation of the results depends somewhat on the
nature of the X-ray sources and their evolution. By comparing the
predicted multipoles to those observed by HEAO1, Scharf et
al. [75] estimate the
amplitude of fluctuations for an assumed shape of the density
fluctuations. The observed fluctuations in the XRB are roughly as
expected from interpolating between the local galaxy surveys and
the COBE and other CMB experiments. The r.m.s. fluctuations
on a scale of
are
less than 0.2%.
Since the (generalized) cosmological principle is
now well supported by the above observations, we shall assume below
that it holds over scales
.
The rest of the current section is devoted to a
brief review of the homogeneous and isotropic cosmological model.
Further details may be easily found in standard cosmology
textbooks [96, 62, 69, 64, 10, 63].
The cosmological principle is mathematically
paraphrased as that the metric of the Universe (in its zero-th
order approximation) is given by
where
is the comoving coordinate, and where we use units in
which the light velocity
. The above Robertson-Walker
metric is specified by a constant
, the spatial curvature, and
a function of time
, the scale factor.
The homogeneous and isotropic assumption also
implies that
, the energy-momentum tensor of the
matter field, should take the form of the ideal fluid:
where
is the 4-velocity of the matter,
is the mean energy density, and
is the mean pressure.