Theoretical foundations of the reduced relativistic gas in the
  cosmological perturbed context
  The Reduced Relativistic Gas (RRG) is a simplified version of the ideal
relativistic gas, which assumes that all particles have the same momentum
magnitude. Although this is a very idealized situation, the resulting model
preserves the phenomenology of Maxwell-Boltzmann distribution and, in some
situations, can be described as a perfect fluid, without introducing large
errors in both cosmological background and first-order perturbations. The
perfect fluid description of RRG model was already used to study the warmness
of dark matter, massive neutrinos and interaction of baryons and photons before
recombination, showing very good agreement with previous works based on the
full Einstein-Boltzmann system of equations. In order to understand these
results and construct a more general and formal framework for RRG, we develop a
theoretical description of first-order cosmological perturbations of RRG, based
on a distribution function which encodes the simplifying assumption that all
particles have the same momentum magnitude. The full set of Einstein-Boltzmann
equations for RRG distribution are derived and quantities beyond the perfect
fluid approximation are studied. Using RRG to describe warm dark matter, we
show that, for particles with $m \sim \text{keV}$, the perfect fluid
approximation is valid on scales $k < 10\, \text{h}/\text{Mpc}$, for most of
the universe evolution. We also determine initial conditions for RRG in the
early universe and study the evolution of potential in a toy model of universe
composed only by RRG.
