Early universe thermostatistics in curved momentum spaces
  The theories known as doubly special relativity are introduced in order to
take into account an observer-independent length scale and the speed of light
in the framework of special relativity. These theories can be generally
formulated on the de Sitter and also recently proposed anti-de Sitter momentum
spaces. In the context of these theories, we study the statistical mechanics
and to do this, we consider the natural measure on the corresponding extended
phase space. The invariant measure on the space of distinct microstates is
obtained by restriction of the natural measure of the extended phase space to
the physical phase space through the disintegration theorem. Having the
invariant measure, one can study the statistical mechanics in an arbitrary
ensemble for any doubly special relativity theory. We use the constructed setup
to study the statistical properties of four doubly special relativity models.
Applying the results to the case of early universe thermodynamics, we show that
one of these models that is defined by the cosmological coordinatization of
anti-de Sitter momentum space, implies a finite total number of microstates.
Therefore, without attribution to any ensemble density and quite generally, we
obtain entropy and internal energy bounds for the early radiation dominated
universe. We find that while these results cannot be supported by the standard
Friedmann equations, they indeed are in complete agreement with the nonsingular
effective Friedmann equations that arise in the context of loop quantum
cosmology.
