Boltzmann entropy of a Newtonian Universe

Abstract

A dynamical estimate is given for the Boltzmann entropy of the Universe, under the simplifying assumptions provided by Newtonian cosmology. We first model the cosmological fluid as the probability fluid of a quantum-mechanical system. Next, following current ideas about the emergence of spacetime, we regard gravitational equipotentials as isoentropic surfaces. Therefore gravitational entropy is proportional to the vacuum expectation value of the gravitational potential in a certain quantum state describing the matter contents of the Universe. The entropy of the matter sector can also be computed. While providing values of the entropy that turn out to be somewhat higher than existing estimates, our results are in perfect compliance with the upper bound set by the holographic principle.

Spherical space in the Newtonian limit: The cosmological constant

“…In this section we verify that the results obtained in S 3 correctly reduce to their counterparts in R 3 [5,7,9] as one lets R 0 → ∞.…”

“…a result which certainly saturates, but does not violate, the upper bound 10 123 k B set by the holographic principle [5,9].…”

“…Using this property, in refs. [5,9] we have computed the Boltzmann entropy of a flat, Newtonian Universe R × R 3 when the matter contents of the Universe is represented by a quantum state |ψ ; the entropy operator considered reads S = N k B M H 0 ρ 2 / . Here k B is Boltzmann's constant, M is the total mass (baryonic and dark) of the observable Universe, H 0 is Hubble's constant, and ρ is a radial coordinate with the dimensions of length.…”

“…Newtonian cosmology succeeds in capturing some essential physics of the Universe (for a very incomplete sample see, e.g., refs. [1,2,4,5,7]). It also has the advantage that its mathematics is notably simpler than that of its general-relativistic parent theory.…”

“…(The value of the spatial curvature of our Universe remains the subject of ongoing debate [24], though). Still, this model is worth revisiting from the viewpoint of current trends such as, e.g., the modern thermodynamic approach to gravity and spacetime [5,9,18,25], which is the standpoint adopted here.…”