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Measure and Probability in Cosmology
General Relativity and Quantum Cosmology · Submitted 8 Feb 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | General relativity has a Hamiltonian formulation, which formally provides a canonical (Liouville) measure on the space of solutions. | background 1.00 | 0.05 | 0.02 | asserted 0.92 |
| s1 | 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. | background 1.00 | 0.04 | 0.03 | hedged possibility 0.88 |
| s2 | 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. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s3 | 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. | contribution 0.93 | 0.48 | 0.55 | asserted 0.86 |
| s4 | (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. | limitation 0.32 | 0.29 | 0.09 | asserted 1.00 |
| s5 | (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. | contribution 0.43 | 0.69 | 0.08 | asserted 1.00 |
| s6 | 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. | result 0.39 | 0.26 | 0.24 | asserted 1.00 |
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The paper
Original 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.
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References
How the literature uses it
30 arXiv papers cite this in 49 sentences
Papers citing it · 26 sentences without a confident label
Uses its method or tool 2 papers
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The probability of inflation is defined using the Liouville measure and its induced measure on the constraint surface after gauge fixing [ref] .
Non-minimally coupled loop quantum inflation with inverse-volume corrections · Probability of inflation from the induced Liouville measure · 0.94
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As above, we consider the space of perturbations expanded in Fourier modes, $f_j(x⃗)$: $Φ(x) = ∑_j b_j f_j(x⃗)$ Following [ref] we impose the linearized field equations and ignore terms that vanish on integration over a Cauchy slice, to obtain
$w=-1$ as an Attractor · Perturbations about FRW · 0.99
Compares with it 1 paper
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Our analysis will be similar to that performed in [ref] , however an extra gauge symmetry in reducing from phase-space variables to physical observables will render a different result.
$w=-1$ as an Attractor · GR Action and Symplectic Structure · 0.90
Cites it as support 1 paper
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Such probability measures in cosmology cannot be defined without introducing arbitrary cut-offs to regularize for divergent integrals and furthermore, the resulting verdict is very sensitive to such arbitrary choices [ref] .
Cosmological Inflation and Meta-Empirical Theory Assessment · MIA Argument and Cosmological Inflation · 1.00
Cites it as background 15 papers
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Much effort has been invested in understanding how much (if any) fine-tuning of the initial conditions is required for inflation to begin [ref] .
A Relativistic Gas of Inflatons as an Initial State for Inflation · Introduction · 0.98
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Thus, in this article we take as working hyposthesis that the landscape idea combined with anthropic reasoning is correct, leaving aside the issues raised by the measure problem [ref] or the potentially short lifetime of metastable vacua on higher dimensional field spaces [ref] .
A Smooth Landscape: Ending Saddle Point Inflation Requires Features to be Shallow · Inflation near a Saddle Point · 0.96
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This question has prompted a great deal of effort devoted to understanding the question of initial conditions [ref] .
How Likely are Constituent Quanta to Initiate Inflation? · Introduction · 0.98
What it cites · 19 sentences without a confident label
Uses the method or tool of 2 papers
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Following Carroll and Tam [ref] , we will choose the variables $Φ$ and $δ$ to indicate the “size” of scalar perturbations, since $|Φ|≪1$ means that the metric perturbation is small compared to the background metric, and $|δ|≪1$ means that the density perturbation is small compared to the background density.
Unitary Evolution and Cosmological Fine-Tuning · Modifying the GHS Measure by Including Inhomogeneities, and the Effect on the Probability of Inflation · 0.92
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We follow the prescription of [ref] (see also [ref] ), to construct the symplectic form on the phase space of this Lagrangian field theory.
Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy · The Phase Space of General Relativity · 1.00
Compares with 2 papers
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As we shall now show, if one evaluates $P(N)$ at $H_* ≫m$, one obtains a value very close to $1$ (in agreement with Carroll and Tam [ref] ), whereas if one evaluates $P(N)$ at $H_* ∼m$ one obtains a value very close to $0$ (in agreement with Gibbons and Turok [ref] ).
Unitary Evolution and Cosmological Fine-Tuning · Difficulties Due to Infinite Total Measure; The Probability of Inflation · 0.97
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As we shall now show, if one evaluates $P(N)$ at $H_* ≫m$, one obtains a value very close to $1$ (in agreement with Carroll and Tam [ref] ), whereas if one evaluates $P(N)$ at $H_* ∼m$ one obtains a value very close to $0$ (in agreement with Gibbons and Turok [ref] ).
The Measure Problem in Cosmology · Difficulties Due to Infinite Total Measure; The Probability of Inflation · 0.97
Cites as support 1 paper
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On the other hand, most universes that are nearly homogeneous and isotropic at a Hubble constant corresponding to the present value, $H_0$, in our universe were extremely inhomogeneous at a much earlier phase [ref] .
Unitary Evolution and Cosmological Fine-Tuning · Difficulties Arising in the Treatment of Inhomogeneous Degrees of Freedom · 0.97
Cites as background 6 papers
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Thus, we shall not consider anthropic arguments in this paper, nor shall we consider any issues that involve the likelihood of the existence of observers and/or civilizations with given characteristics [ref] , including the “measure problem” as discussed in, e.g., [ref] .
Boltzmann babies in the proper time measure · Introduction · 0.81
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Thus, we shall not consider anthropic arguments in this paper, nor shall we consider any issues that involve the likelihood of the existence of observers and/or civilizations with given characteristics [ref] , including the “measure problem” as discussed in, e.g., [ref] .
Boltzmann brains and the scale-factor cutoff measure of the multiverse · Introduction · 0.81
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Thus, we shall not consider anthropic arguments in this paper, nor shall we consider any issues that involve the likelihood of the existence of observers and/or civilizations with given characteristics [ref] , including the “measure problem” as discussed in, e.g., [ref] .
Possible Anthropic Support for a Decaying Universe: A Cosmic Doomsday Argument · Introduction · 0.81
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