  We demonstrate that the high isotropy of the Cosmic Microwave Background
(CMB), combined with the Copernican principle, is not sufficient to prove
homogeneity of the universe -- in contrast to previous results on this subject.
The crucial additional factor not included in earlier work is the acceleration
of the fundamental observers. We find the complete class of irrotational
perfect fluid spacetimes admitting an exactly isotropic radiation field for
every fundamental observer and show that are FLRW if and only if the
acceleration is zero. While inhomogeneous in general, these spacetimes all
possess three-dimensional symmetry groups, from which it follows that they also
admit a thermodynamic interpretation. In addition to perfect fluids models we
also consider multi-component fluids containing non-interacting radiation, dust
and a quintessential scalar field or cosmological constant in which the
radiation is isotropic for the geodesic (dust) observers. It is shown that the
non-acceleration of the fundamental observers forces these spacetimes to be
FLRW. While it is plausible that fundamental observers (galaxies) in the real
universe follow geodesics, it is strictly necessary to determine this from
local observations for the cosmological principle to be more than an
assumption. We discuss how observations may be used to test this.
