In (2+1)-dimensional cosmology with a negative cosmological constant, the Fischler-Susskind entropy bound is incompatible with the generalized second law in contracting phases, while the Hubble entropy bound can be reconciled when quantum corrections are added.
Pre-bangian origin of our entropy and time arrow
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I argue that, in the chaotic version of string cosmology proposed recently, classical and quantum effects generate, at the time of exit to radiation, the correct amount of entropy to saturate a Hubble (or holography) entropy bound (HEB) and to identify, within our own Universe, the arrow of time. Demanding that the HEB be fulfilled at all times forces a crucial "branch change" to occur, and the so-called string phase to end at a critical value of the effective Planck mass, in agreement with previous conjectures.
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Testing the Generalized Second Law in $(2+1)$-Dimensional Cosmology: Holographic Entropy Bounds and Observational Constraints
In (2+1)-dimensional cosmology with a negative cosmological constant, the Fischler-Susskind entropy bound is incompatible with the generalized second law in contracting phases, while the Hubble entropy bound can be reconciled when quantum corrections are added.