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In textbooks on cosmology it is often stated (e.g., Galaxy Formation and Evolution by Mo, van den Bosch & White, p. 108) that the entropy of any comoving volume is constant (as the universe is homogeneous and isotropic, there is no net flow of heat/matter through that volume). Yet, in other sources it is stated that the initial state of the universe must have been extremely low-entropy (e.g., Sean Carroll in his recent The Big Picture: On the Origins of Life, Meaning, and the Universe Itself), and that the total entropy of the universe is increasing, as it should according to the 2nd law of thermodynamics, allowing along the way for such complex phenomena as life. But nothing precludes me from regarding the entire (observable) universe as a single comoving volume (hence, with constant entropy). Does this contradiction follows from my misunderstanding of the issue, or from the fact that there is no general consensus on this question?

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The contradiction primarily boils down to a lack of consensus. The textbook is correct in stating that the entropy of the comoving volume is constant if you consider the dominant source of entropy in the universe to be the Cosmic Microwave Background (CMB). The classical calculation is to consider an expanding photon gas and check to see if its entropy increases if its volume increases. What you should find is that entropy is proportional to the number of quanta which is fixed if you're in a comoving volume. Hence, the expansion of the universe doesn't increase the entropy.

Carroll is in the same camp as Roger Penrose in arguing for a different view on the universe's entropic content. They argue for something called 'gravitational entropy', which (without any conclusive analytical description) basically says that when things gravitationally collapse, the entropy goes up. The maximum entropy state is that of a black hole described by the Bekenstein formula for this. Hence, in a comoving volume while the CMB doesn't increase its entropy, gravitational collapse does. What's crucial isn't just the existence of gravitational entropy but its dominance over other degrees of freedom.

A more subtle point has to do with considering the entire universe as a comoving volume. It depends on what kind of horizon you want to consider (causal, comoving, particle, event). Each horizon will have a different associated entropy because they vary differently with time and hence the ones that aren't comoving let out or let in particles.

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