Problem with a paradox concerning the entropy in the universe The Entropy Paradox :
I am going to explain what I use to call "the entropy paradox" which
is of a cosmological nature. I believe that it is a "paradox" due to my lack
of knowledge and I would be very grateful if someone could, in return, explain
me how the paradox is solved. It relies on two points of view, concerning the
evolution of the universe, which are as far as I can understand contradictory.
(i) The first point of view is as follows :
Let us consider the cosmic
microwave background (CMB). Since COBE (whose observations since then
have been confirmed and refined), we know that the CMB satisfies the black
body radiation law. It is admitted that this CMB is the relic of a radiation
in which, at the moment of the decoupling (say 380 000 years after the BB),
photons were in thermal equilibrium, corresponding to a huge entropy density.
When time goes on, thermal equilibrium remains a thermal equilibrium, with
huge entropy density. Before the decoupling, the universe was opaque and the
thermodynamic equilibrium should be shared by radiation and massive matter
(matter in short) due to the strong level of interactions between all components
of the universe, i.e. both light and matter were in a high entropy density state.
(ii) The second point of view is as follows :
Today, the universe is not
in a state of thermodynamic equilibrium, as we can easily check even in our
everyday life. Now, the second law of thermodynamics tells us that entropy
increases (or stays constant) in time. Clearly, the entropy nowadays does not
remain constant but increases. Therefore, it will be larger in the future, and it
was smaller in the past. Therefore, the universe, in the past, had a low level of
entropy density (otherwise, there would not be any arrow of time).
Hence a contradiction. I am pretty sure that someone knows how to explain
this paradox. For the time being, I believe that there is no paradox at all, but
a lack of skill from my part. I would be grateful to anyone who would provide
me a solution to the issue, I mean, a "simple" clear-cut solution that even the layman could understand.
Thanks in advance for providing me with a solution to the issue.
 A: We know from observing the CMBR that matter and radiation in the universe soon after the Big were distributed almost uniformly. At first glance this would appear to be a highly disordered state with very high entropy. However, once you take gravity into account, it turns out that this near-uniform distribution is actually a very ordered and very unlikely state, and so has extremely low entropy. The universe is therefore evolving from this initial near-uniform low entropy state towards a final high entropy state in which almost all matter and radiation has been absorbed by black holes. This Wikipedia article provides more details.
A: For a complete accounting of the entropy one has to consider the effects of gravitation. Roger Penrose stresses in his books "A road to reality" and "Cycles of Time" that our universe started from very special initial conditions where spacetime was always rather homogeneous and isotropic as one can deduce from the homogenity of the CMB radiation.
According to Penrose the universe could have started from very different initial conditions, for instance with a more complex spacetime structure, even including black holes, a spacetime structure that would have had much more entropy.
But as the universe started from rather homogeneous and isotropic conditions its entropy was actually quite low even if the matter inside it was at thermodynamical equilibrium.
When matter starts to agglomerate due to the effect of gravitation actually the entropy increases, in particular if it comes to the formation of black holes. Black holes contain a huge amount of entropy according to the Hawking-Bekenstein formula which relates the surface $A$ of a black hole to its entropy (where $k_B$ is the Boltzmann constant and $\cal{l}_P$ the Planck length):
$$ S =\frac{k_B A}{4 \cal{l}_P^2}$$
Actually, black holes are the objects of the universe which bear most of the entropy.
Therefore the entropy has increased with respect to the beginning of the universe, and in particular due to the formation of many black holes which make the spacetime structure of the universe more complex.
This also means that the second law of Thermodynamics during the development of the universe has not been violated.
