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On matching conditions for cosmological perturbations

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Nathalie Deruelle, V.F. Mukhanov

General Relativity and Quantum Cosmology · Submitted 27 Mar 1995 · 15 references

Pith's verdict: A clean, useful clarification of matching conditions across a sharp transition; the constant-energy choice is a modeling assumption the authors own, and the algebra holds.

T0 · AI review

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arXiv IDgr-qc/9503050 (version 1)
Pith IDTR5SL3K4, Pith's permanent record of this paper and its review
CitationNathalie Deruelle, V.F. Mukhanov. "On matching conditions for cosmological perturbations." arXiv:gr-qc/9503050 (1995). https://pith.science/paper/TR5SL3K4
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The paper

The paper claims

Matching cosmological perturbations across a sharp equation-of-state transition on a constant-energy hypersurface confirms the standard inflationary result that scalar perturbations are amplified by (1+p-)/(1+p+) while gravitational waves are not.

The reading

When the early universe went through a phase transition, such as the end of inflation, the equation of state changed abruptly. Cosmologists model this as a sharp jump and ask how small ripples in the spacetime metric (the seeds of galaxies and of gravitational waves) pass from the earlier phase to the later one. The answer depends on which three-dimensional surface you use to stitch the two phases together. The authors show that the physically correct choice is a surface of constant energy density, because the pressure depends only on the energy density, so a sharp change in the equation of state is naturally located there. With that choice, the matching conditions take a clean gauge-invariant form. The consequences are the standard ones: long-wavelength gravitational waves pass through essentially unchanged, while the gravitational potential that characterizes density perturbations is amplified by a factor (1+p-)/(1+p+), which can be very large when the earlier phase is strongly inflating. The paper uses this to argue that a recent claim of a scale-dependent spectrum from inflation rests on matching on an arbitrary synchronous time slice rather than a constant-energy surface. It also applies the machinery to string cosmology, showing that a growing mode in a contracting (superinflationary) phase converts almost entirely into a decaying mode after the transition to a Friedmann era.
Extended reading notes

Core claim

The matching conditions for cosmological perturbations across a sharp jump in the equation of state must be imposed on a hypersurface of constant energy density, and when imposed there they yield the standard results h+/h- -> O(1) and Phi+/Phi- -> (1+p-)/(1+p+), rebutting the contrary claim of Grishchuk (1994). The paper states this as: 'We on the other hand shall make the matching on hypersurfaces of constant energy for physical reasons and clarify and confirm the standard results (1.1)-(1.2).'

Load-bearing premise

The load-bearing premise is that the transition occurs on a hypersurface of constant energy density (Sigma: epsilon_0 + delta epsilon = const), introduced in Sect. III and used throughout. The paper itself demonstrates in Sect. V.A that in a pure de Sitter limit 'any hypersurface is an hypersurface of constant energy' and 'the physical result is uncertain,' so the constant-energy choice is a physically motivated modeling assumption rather than a derived consequence. If the true transition surface deviates from constant energy, the matching conditions (3.4) and the resulting amplitudes change.

Original abstract

We derive the matching conditions for cosmological perturbations in a Friedmann Universe where the equation of state undergoes a sharp jump, for instance as a result of a phase transition. The physics of the transition which is needed to follow the fate of the perturbations is clarified. We dissipate misleading statements made recently in the literature concerning the predictions of the primordial fluctuations from inflation and confirm standard results. Applications to string cosmology are considered.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,505 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.76 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We derive the matching conditions for cosmological perturbations in a Friedmann Universe where the equation of state undergoes a sharp jump, for instance as a result of a phase transition. The physics of the transition which is needed to follow the fate of the perturbations is clarified. We dissipate misleading statements made recently in the literature concerning the predictions of the primordial fluctuations from inflation and confirm standard results. Applications to string cosmology are considered.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We derive the matching conditions for cosmological perturbations in a Friedmann Universe where the equation of state undergoes a sharp jump, for instance as a result of a phase transition. contribution 0.99 0.96 0.28 asserted 1.00
s1 The physics of the transition which is needed to follow the fate of the perturbations is clarified. contribution 0.66 0.71 0.12 asserted 0.96
s2 We dissipate misleading statements made recently in the literature concerning the predictions of the primordial fluctuations from inflation and confirm standard results. contribution 0.80 0.88 0.28 asserted 1.00
s3 Applications to string cosmology are considered. contribution 0.99 0.71 0.04 asserted 0.83

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How later papers used it

68 later papers cite it

Evidence: substantial. 68 of the citing papers share no author with it, and 39 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 93 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 4 of 35 papers with this label
How the literature uses it

58 arXiv papers cite this in 81 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 38 sentences without a confident label

Contests it 1 paper
  • As initially pointed out in [ref] , the result for the spectrum of fluctuations in the expanding phase depends quite sensitively on the details of the matching, and it is in fact not clear whether the matching prescription of [ref] can be applied at all due to the inconsistency of the matching of the background.

    On the Transfer of Adiabatic Fluctuations through a Nonsingular Cosmological Bounce · Introduction · contest 0.74

Uses its method or tool 30 papers
  • To this surface we apply the matching conditions of [ref] which state that the induced 3-metric on this hypersurface and its extrinsic curvature are continuous.

    Dark Energy Perturbations Revisited · Dark energy perturbation with parameterized EoS and matching conditions · 0.95

  • In the analytical analysis, we apply matching conditions away from the bounce point, in regions where both the background and the perturbations can be matched consistently via the Deruelle-Mukhanov [ref] (initially derived in [ref] ) conditions [For a discussion of the dangers in applying the matching conditions at the transition point between contraction and expansion see [ref] .].

    On Perturbations of Quintom Bounce · Introduction · 0.99

  • To perform the matching of cosmological perturbations, we demand that both the intrinsic and extrinsic curvatures are continuous on the hyper-surface separating the two inflationary phases [ref] .

    Brane Annihilations during Inflation · Matching Conditions at $t_c$ · 0.87

Compares with it 3 papers
  • Thus, the results obtained here by evolving the perturbations through the bounce and in [ref] by using the constant energy density matching conditions given in [ref] are in agreement, at least for predictions of the scalar index $n_s$.

    Ekpyrotic loop quantum cosmology · The Expanding Branch · 0.99

  • This is in contrast to the examples studied in [ref] in which the transition was characterized by a surface of constant energy density and hence the uniform density gauge is more appropriate since $δρ=0$ in that gauge.

    On primordial equation of state transitions · Matching Conditions · 0.98

  • Note that the same result was also derived in [ref] in an inflationary context.

    Resolving the Hubble Tension with New Early Dark Energy · Perturbation Matching · 0.81

Cites it as background 3 papers

Discussion

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