144 Cosmological models
with spatial homogeneity; but once we take evolution into account, we can fit
almost any observational relations by almost any spherically symmetric model
(see [98] for exact theorems making this statement precise). Recent statistical
observations of distant sources support spatial homogeneity on intermediate
scales (between 30 and 400 Mpc [102]), but do not extend to larger scales because
of sample limits.
3.8.5.2 Uniform thermal histories
A strong indication of spatial homogeneity is the fact that we see the same
kinds of object, more or less, at highzas nearby. This suggests that they
must have experienced more or less the same thermal history as nearby objects,
as otherwise their structure would have come out different; and this, in turn,
suggests that the spacetime geometry must have been rather similar near those
objects as near to us, else (through the field equations) the thermal history
would have come out different. This idea can be formulated in thePostulate
of Uniform Thermal Histories(PUTH), stating that uniform thermal histories can
occur only if the geometry is spatially homogeneous. Unfortunately, counter-
examples to this conjecture have been found [7]. These are, however, probably
exceptional cases and this remains a strong observationally-based argument for
spatial homogeneity, indeed probably the most compelling at an intuitive level.
However, relating the idea to observations also involves untangling the effects of
time evolution, and it cannot be considered a formal proof of homogeneity.
3.8.5.3 Almost-EGS theorem
The most compelling precisely formulated argument is a based on our
observations of the high degree of CBR anisotropy around us. If we assume
we are not special observers, others will see the same high degree of anisotropy;
and then that shows spatial homogeneity: exactly, in the case of exact isotropy
(the Ehlers–Geren–Sachs (EGS) theorem [22]) and approximately in the case of
almost-isotropy:
Almost-EGS-theorem.[119]. If the Einstein–Liouville equations are satisfied in
an expanding universe, where there is pressure-free matter with 4-velocity vector
field ua(uaua =− 1 ) such that (freely-propagating) background radiation is
everywhere almost-isotropic relative to uain some domain U , then spacetime is
almost-FLRW in U.
This description is intended to represent the situation since decoupling to
the present day. The pressure-free matter represents the galaxies on which
fundamental observers live, who measure the radiation to be almost isotropic.
This deduction is very plausible, particularly because of the argument just
mentioned in the last subsection: conditions therelookmore or less the same,