  Our current understanding of the Universe depends on the interplay of several
distinct "matter" components, which interact mainly through gravity, and
electromagnetic radiation. The nature of the different components, and possible
interactions, tends to be based on the notion of coupled perfect fluids (or
scalar fields). This approach is somewhat naive, especially if one wants to be
able to consider issues involving heat flow, dissipative mechanisms, or
Bose-Einstein condensation of dark matter. We argue that a more natural
starting point would be the multi-purpose variational relativistic multi-fluid
system that has so far mainly been applied to neutron star astrophysics. As an
illustration of the fundamental principles involved, we develop the formalism
for determining the non-linear cosmological solutions to the Einstein equations
for a general relativistic two-fluid model for a coupled system of matter
(non-zero rest mass) and "radiation" (zero rest mass). The two fluids are
allowed to interpenetrate and exhibit a relative flow with respect to each
other, implying, in general, an anisotropic Universe. We use initial conditions
such that the massless fluid flux dominates early on so that the situation is
effectively that of a single fluid and one has the usual
Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. We find that there is a
Bianchi I transition epoch out of which the matter flux dominates. The
situation is then effectively that of a single fluid and the spacetime evolves
towards the FLRW form. Such a transition opens up the possibility of imprinting
observable consequences at the specific scale corresponding to the transition
time.
