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 (−5 ΔT ∕T = 10 on scales of ∘ 10) give via the Sachs–Wolfe effect (1 2 ΔT ∕T = 3Δ ϕ∕c) and the Poisson equation r.m.s. density fluctuations of δρ∕ ρ ∼ 10−4 on 1000h −1 Mpc (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 [69Jump To The Next Citation Point] 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., [72Jump To The Next Citation Point]), the problems of biasing, evolution, and K-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 Λ-CDM (cold dark matter) model.
Radio sources
Radio sources in surveys have a typical median redshift of ¯z ∼ 1, 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 z < 5, 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 δρ∕ρ on a scale of ∼ 600h −1 Mpc 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 −1 l > 100h Mpc.

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 [966269641063].

The cosmological principle is mathematically paraphrased as that the metric of the Universe (in its zero-th order approximation) is given by

( dx2 ) ds2 = − dt2 + a(t)2 -------2 + x2d𝜃2 + x2sin2𝜃d ϕ2 , (1 ) 1 − Kx
where x is the comoving coordinate, and where we use units in which the light velocity c = 1. The above Robertson–Walker metric is specified by a constant K, the spatial curvature, and a function of time a(t), the scale factor.

The homogeneous and isotropic assumption also implies that Tμν, the energy-momentum tensor of the matter field, should take the form of the ideal fluid:

T μν = (ρ + p)u μuν + pgμν, (2 )
where uμ is the 4-velocity of the matter, ρ is the mean energy density, and p is the mean pressure.
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