REVIEW
The Measure Problem in Cosmology
High Energy Physics - Theory · Submitted 13 Sep 2006 · 27 references
Pith's verdict: A clean, honest derivation of exp(-3N) for inflation in the canonical measure, with the load-bearing caveat about where the measure surface sits stated plainly in Section 7 rather than buried.
T0 · AI review
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Referee report
The paper
The paper claims
The Hamiltonian structure of general relativity supplies a natural way to count classical universes, and when that count is made properly, inflation turns out to be exponentially rare rather than generic.
The reading
What carries the argument
The load-bearing object is the reduced symplectic form $�omega_C$, the Hamiltonian two-form restricted to the constraint surface $H=0$ and pushed down to the space of classical trajectories, whose $(n-1)$-th power gives the measure. Its dual, a divergence-free 'magnetic field' $B_a$ parallel to the Hamiltonian flow, converts the measure into a flux through a surface each trajectory crosses once; Stokes' theorem then reduces it to a line integral $�oint p_��d��$ over the boundary, which is the standard adiabatic invariant of a scalar field in a time-dependent background. Two further ingredients do the work: the choice of surface $H = H_S$ at a low Hubble rate where the expansion is adiabatic, and the identification of universes with $|�Omega_k| < ��Omega_k$, which cuts off the divergence in the scale factor and makes the answer independent of the cutoff for the inflationary probability.
Extended reading notes
Core claim
The central result is that the canonical measure assigns to $N$ e-folds of slow-roll inflation a probability of order $C(N)e^{-3N}/N$, with $C(N)$ a weak, model-dependent prefactor. The mechanism is a linearised instability: writing $H = H_{SR} + ��H$ around the slow-roll solution gives the exact relation $d(��H)/dN = 3��H$, so deviations from the inflationary trajectory grow as $e^{3N}$ as one runs the clock backwards. Only a band of width $��H_S$ in the phase variable on the late-time measure surface lands on the inflationary solution, and that band shrinks exponentially with the number of e-folds. Because the measure is time-reversal invariant, the same argument makes deflation equally rare, confirming an expectation the authors attribute to Hollands and Wald. The authors also note that the measure for spatial curvature itself is dominated by the flatness cutoff, so while flat universes are favoured, no sharp prediction for $�Omega_k$ follows.
Load-bearing premise
The load-bearing premise is that the measure should be evaluated on a late-time surface where the universe's expansion is adiabatic, with universes flatter than a threshold identified as one; the authors concede that placing the same surface at a high Hubble rate reverses the conclusion.
Editorial extensions
If this is right
- Inflation is not generic in the canonical measure: obtaining $N$ e-folds costs a factor of roughly $e^{-3N}$, so a fundamental theory must supply a sharper prior than indifference to explain why inflation occurred.
- Deflation is predicted to be just as rare as inflation, since the measure treats universes independently of their time orientation.
- In the $m^2��^2/2$ model the probability of inflating all the way back to Planck density is about $e^{-m^{-2}} \approx e^{-10^{12}}$, parametrically the same as the no-boundary instanton estimate with the usual sign of the Euclidean action.
- Flat universes are favoured by the measure, so the observed near-flatness is unsurprising, but the distribution over $�Omega_k$ is dominated by the cutoff and yields no sharp prediction.
- The result is claimed to be insensitive to the value of the flatness cutoff and to the choice of measure surface, provided the surface sits in the adiabatic regime.
Reading between the lines
- Because the answer flips to 'most trajectories inflate' when the same prescription is applied at a large Hubble rate, the late-time choice functions as an initial-condition postulate in disguise; the paper's real content may be that a measure alone cannot explain inflation without a separate statement about which epoch defines distinct universes.
- The $e^{-3N}$ law is a clean, model-independent target: any proposed fundamental prior must overcome an exponentially small phase-space volume at $N \approx 60$, which gives a quantitative benchmark for judging how predictive a landscape or quantum-cosmology proposal actually is.
- The reduction to $�oint p_��\,d��$ assumes the matter phase space is uniformly populated in its angle variable; in multi-field or chaotic minisuperspaces that assumption is exactly what would fail, so extending the flux calculation there is the natural stress test.
- Since the curvature distribution is cutoff-dominated, a future detection of $|�Omega_k| \sim 10^{-3}$ would not discriminate between this measure and its competitors, whereas the predicted rarity of deflation-like histories is in principle a sharper discriminant.
Original abstract
The Hamiltonian structure of general relativity provides a natural canonical measure on the space of all classical universes, i.e., the multiverse. We review this construction and show how one can visualize the measure in terms of a "magnetic flux" of solutions through phase space. Previous studies identified a divergence in the measure, which we observe to be due to the dilatation invariance of flat FRW universes. We show that the divergence is removed if we identify universes which are so flat they cannot be observationally distinguished. The resulting measure is independent of time and of the choice of coordinates on the space of fields. We further show that, for some quantities of interest, the measure is very insensitive to the details of how the identification is made. One such quantity is the probability of inflation in simple scalar field models. We find that, according to our implementation of the canonical measure, the probability for N e-folds of inflation in single-field, slow-roll models is suppressed by of order exp(-3N) and we discuss the implications of this result.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The Hamiltonian structure of general relativity provides a natural canonical measure on the space of all classical universes, i.e., the multiverse. We review this construction and show how one can visualize the measure in terms of a "magnetic flux" of solutions through phase space. Previous studies identified a divergence in the measure, which we observe to be due to the dilatation invariance of flat FRW universes. We show that the divergence is removed if we identify universes which are so flat they cannot be observationally distinguished. The resulting measure is independent of time and of the choice of coordinates on the space of fields. We further show that, for some quantities of interest, the measure is very insensitive to the details of how the identification is made. One such quantity is the probability of inflation in simple scalar field models. We find that, according to our implementation of the canonical measure, the probability for N e-folds of inflation in single-field, slow-roll models is suppressed by of order exp(-3N) and we discuss the implications of this result.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Hamiltonian structure of general relativity provides a natural canonical measure on the space of all classical universes, i.e., the multiverse. | background 0.91 | 0.16 | 0.08 | asserted 1.00 |
| s1 | We review this construction and show how one can visualize the measure in terms of a "magnetic flux" of solutions through phase space. | contribution 0.63 | 0.83 | 0.04 | asserted 1.00 |
| s2 | Previous studies identified a divergence in the measure, which we observe to be due to the dilatation invariance of flat FRW universes. | result 0.62 | 0.49 | 0.06 | asserted 1.00 |
| s3 | We show that the divergence is removed if we identify universes which are so flat they cannot be observationally distinguished. | result 0.59 | 0.93 | 0.24 | asserted 1.00 |
| s4 | The resulting measure is independent of time and of the choice of coordinates on the space of fields. | result 0.98 | 0.84 | 0.13 | asserted 1.00 |
| s5 | We further show that, for some quantities of interest, the measure is very insensitive to the details of how the identification is made. | result 0.91 | 0.93 | 0.08 | asserted 0.99 |
| s6 | One such quantity is the probability of inflation in simple scalar field models. | contribution 0.60 | 0.53 | 0.06 | asserted 0.88 |
| s7 | We find that, according to our implementation of the canonical measure, the probability for N e-folds of inflation in single-field, slow-roll models is suppressed by of order exp(-3N) and we discuss the implications of this result. | result 1.00 | 0.96 | 0.75 | asserted 1.00 |
What was examined, and what was not
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- relations to other papers
- P abstract
- S sections
- T body sentences
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Figures
Figures reproduced from arXiv:hep-th/0609095 by the authors.
References
How the literature uses it
50 arXiv papers cite this in 111 sentences
Papers citing it · 62 sentences without a confident label
Contests it 3 papers
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The measure of [ref] , however, diverges for flat FLRW universes [as indeed noted by: [ref] ], and the work of [ref] finds a way around this issue by explicitly constructing measures over trajectories for flat FLRW universes in the context of specific SSF potentials.
Effective field theories as a novel probe of fine-tuning of cosmic inflation · The road ahead · contest 0.84
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This contradicts the claims of [ref] that initially $ϕ̇^2 ≫V(ϕ)$ [ref] .
Inflationary Cosmology after Planck 2013 · Initial conditions in the simplest models of chaotic inflation · contest 0.91
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The main problem with [ref] is that the methods developed there are valid for the classical evolution of the universe, but the initial conditions for the classical evolution are determined by the processes at the quantum epoch near the singularity, where the methods of [ref] are inapplicable.
Inflationary Cosmology · Initial conditions · contest 0.89
Builds on it 2 papers
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This construction can be extended to the effective Hamiltonian $⟨ℋ_LQ⟩$ : [equation] From the symplectic form, we can define a divergence-less field [ref] [equation] The field $𝐁$ defines the flow of trajectories across surfaces in the phase space.
Non-minimally coupled loop quantum inflation with inverse-volume corrections · Liouville Measures and Magnetic Flux · 0.85
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As we will show in a separate publication [ref] , if one further develops the methods of [ref] , but imposes the initial conditions at the beginning of inflation, rather than at its end, one finds that inflation is most probable, in agreement with the arguments given in the first part of this section.
Inflationary Cosmology · Initial conditions · 0.80
Uses its method or tool 1 paper
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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
Compares with it 5 papers
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Next, we give an argument based on global properties of the dynamics and the Liouville measure to understand the results of both [ref] and [ref] .
On the measure problem in slow roll inflation and loop quantum cosmology · Introduction · 0.82
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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] ).
Measure and Probability in Cosmology · Difficulties Due to Infinite Total Measure; The Probability of Inflation · 0.97
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Besides that, there is a huge discrepancy between the probability estimated in Gibbons and Turok [ref] and the results obtained, for example, by Linde [ref] .
Warm bounce in loop quantum cosmology and the prediction for the duration of inflation · Introduction · 0.93
Cites it as support 2 papers
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By standard classical statistical mechanical reasoning, even for simple inflaton potentials, there exist more homogeneous and flat cosmic solutions without a long period of inflation than with inflation [ref] .
Inflationary schism after Planck2013 · Introduction · 0.98
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Such `just-enough' inflationary scenarios can be advocated from the fact that a large amount of inflation requires some fine-tunning [ref] and that string landscape models suffer from the $η-$problem [ref] which does not allow them to sustain longer inflation.
Revisiting a pre-inflationary radiation era and its effect on the CMB power spectrum · Introduction · 0.96
Cites it as background 23 papers
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The measure or the probability distribution in eternal inflation have become one of the central problems in cosmology [ref] .
Typicality, Freak Observers and the Anthropic Principle of Existence · Introduction · 0.96
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Recently, Gibbons and Turok overcame some early difficulties in the total normalization and concluded that, for the simplest inflationary potentials in a FRW universe, the probability of inflation was greatly suppressed [ref] .
On the measure problem in slow roll inflation and loop quantum cosmology · Introduction · 0.85
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This is also often interpreted as a problem for inflation (for a range of viewpoints, see [ref] ).
Power-law Inflation Satisfies Penrose's Weyl Curvature Hypothesis · Introduction · 0.84
Discussion
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