2012/02/08 by Joshua S. Schiffrin, Joshua Schiffrin, Robert M. Wald
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Classical mechanics #Computer science #Cosmology #Cosmology and Gravitation Theories #Einstein #General relativity #Inflation (cosmology) #Mathematical analysis #Mathematics #Measure (data warehouse) #Minisuperspace #Noncommutative and Quantum Gravity Theories #Physics #Probability measure #Quantum cosmology #Quantum gravity #Quantum mechanics #Scalar field #Theoretical physics #Universe #astro-ph.CO #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevd.86.023521
published as Phys.Rev.D.86.023521,2012 · 43 pages, 2 figures
arxiv created 2012/02/08 · openalex publication_date 2012/07/12 · arxiv updated 2012/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
General relativity has a Hamiltonian formulation, which formally provides a canonical (Liouville) measure on the space of solutions. In ordinary statistical physics, the Liouville measure is used to compute probabilities of macrostates, and it would seem natural to use the similar measure arising in general relativity to compute probabilities in cosmology, such as the probability that the Universe underwent an era of inflation. Indeed, a number of authors have used the restriction of this measure to the space of homogeneous and isotropic universes with scalar field matter (minisuperspace)---namely, the Gibbons-Hawking-Stewart measure---to make arguments about the likelihood of inflation. We argue here that there are at least four major difficulties with using the measure of general relativity to make probability arguments in cosmology: (1) Equilibration does not occur on cosmological length scales. (2) Even in the minisuperspace case, the measure of phase space is infinite and the computation of probabilities depends very strongly on how the infinity is regulated. (3) The inhomogeneous degrees of freedom must be taken into account (we illustrate how) even if one is interested only in universes that are very nearly homogeneous. The measure depends upon how the infinite number of degrees of freedom are truncated, and how one defines ``nearly homogeneous.'' (4) In a Universe where the second law of thermodynamics holds, one cannot make use of our knowledge of the present state of the Universe to retrodict the likelihood of past conditions.