[Paper Review] Measure and Probability in Cosmology

Joshua Schiffrin, Robert M. Wald|arXiv (Cornell University)|Feb 8, 2012

Cosmology and Gravitation Theories4 citations

TL;DR

This paper critically examines the use of the Liouville measure from general relativity’s Hamiltonian formulation to compute probabilities in cosmology, particularly regarding the likelihood of inflation. It identifies four major flaws: lack of equilibration on cosmological scales, infinite phase space measure requiring arbitrary regularization, underappreciated contributions from inhomogeneous degrees of freedom, and the impossibility of retrodiction in a thermodynamically asymmetric universe.

ABSTRACT

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.

Motivation & Objective

• To critically assess the validity of using the canonical Liouville measure from general relativity to compute cosmological probabilities.
• To identify and analyze fundamental conceptual and technical issues in applying measure-theoretic probability to cosmology.
• To demonstrate that the Gibbons-Hawking-Stewart (GHS) measure on minisuperspace fails due to infinite measure and gauge dependence.
• To argue that retrodiction based on trajectory counting is invalid in a universe with a low-entropy past.
• To advocate for alternative justification of cosmological hypotheses based on simplicity and predictive power rather than measure-based likelihood.

Proposed method

• Formal analysis of the Liouville measure in the Hamiltonian formulation of general relativity, including its construction via the symplectic form on phase space.
• Application of the measure to minisuperspace (homogeneous and isotropic universes with scalar fields), leading to the Gibbons-Hawking-Stewart (GHS) measure.
• Identification of the degeneracy of the symplectic form on the constraint surface and the necessity of quotienting by gauge orbits, which eliminates time evolution.
• Investigation of the effects of inhomogeneous degrees of freedom by analyzing perturbations (scalar, vector, tensor) around FLRW spacetime and their contribution to the symplectic structure.
• Use of trajectory counting arguments to illustrate the failure of retrodiction in thermodynamically asymmetric systems.
• Argument that only predictive power and simplicity—not measure-based likelihood—can justify hypotheses about the early universe.

Experimental results

Research questions

• RQ1 Can the Liouville measure from general relativity be reliably used to compute probabilities of cosmological events such as inflation?
• RQ2 Why does the infinite total measure of the phase space invalidate probability computations based on the GHS measure?
• RQ3 How do inhomogeneous degrees of freedom affect the measure and the probability of inflation, even in nearly homogeneous universes?
• RQ4 Why is retrodiction based on trajectory counting invalid in a universe governed by the second law of thermodynamics?
• RQ5 What criteria should be used to justify hypotheses about the early universe if measure-theoretic likelihood fails?

Key findings

• The Liouville measure on general relativity’s phase space is ill-defined in the infinite-dimensional setting without regularization, and probabilities depend sensitively on the choice of cutoffs.
• The Gibbons-Hawking-Stewart measure fails due to the infinite measure of minisuperspace, making probability computations ill-posed without arbitrary regularization.
• Even in nearly homogeneous universes, inhomogeneous degrees of freedom significantly alter the measure, and their inclusion cannot be neglected.
• Trajectory counting arguments correctly predict future evolution but fail as retrodictions in a thermodynamically asymmetric universe, where entropy decreases toward the past.
• Retrodiction cannot be used to infer past conditions; instead, hypotheses about the early universe must be justified by simplicity and predictive success, not by measure-theoretic likelihood.
• The only viable way to assess early-universe scenarios is through simple, elegant hypotheses that successfully predict present-day observations, not through probability arguments based on phase space measures.