What is the entropy of the universe?

S. D. H. Hsu and D. Reeb, Phys. Rev. D (to appear) [arXiv:0903.2258 [gr-qc]]

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Since the Hawking temperature is T ∼ M − 1 T\sim M^{-1} , the Hawking radiation has entropy S ∼ ∫ 𝑑 Q / T ∼ ∫ d ​ M ​ M ∼ M 2 ∼ A S\sim\int dQ/T\sim\int dM\,M\sim M^{2}\sim A . This is then, at least for large A A , also true for the microcanonical definition of entropy which we employ throughout. See also D. N. Page, Phys. Rev. D 13

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In a black hole spacetime, consider two Cauchy slices Σ 1 \Sigma_{1} , preceding black hole formation, and Σ 2 \Sigma_{2} , after complete black hole evaporation, containing all of the Hawking radiation. Unitarity, or one-to-one mapping, requires the dimensionality of the Hilbert spaces of quantum states on Σ 1 \Sigma_{1} and on Σ 2 \Sigma_{2} to be equal. Thus, unitarity requires equal microcanonical entropies S 1 S_{1} and S 2 S_{2} , which is a puzzle since S 1 < M 3 / 2 S_{1}<M^{3/2} for ordinary matter but S 2 ∼ M 2 S_{2}\sim M^{2} for the general evaporated state DP ; this means that, under unitary evolution, only an exponentially small subset of the possible quantum states on Σ 2 \Sigma_{2} can actually be realized if Σ 1 \Sigma_{1} only contained ordinary matter states

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The number of degrees of freedom of all other systems is tiny compared to that of the CMB (including neutrinos; see Table I without black holes). For this reason only a negligible part of the entropy of the thermal photons and neutrinos could, since the era of decoupling, have been dumped into other forms or into entropy of entanglement with other systems

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