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Adiabatic and entropy perturbations from inflation

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Christopher Gordon, David Wands, Bruce A. Bassett, Roy Maartens (Portsmouth)

Astrophysics · Submitted 8 Sep 2000 · 51 references

Pith's verdict: The paper that gave multi-field inflation its working language; the central asymmetry between entropy and curvature sourcing is correct and the soft spots are bounded and mostly self-disclosed.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv IDastro-ph/0009131 (version 2)
Pith IDHJT4OUEV, Pith's permanent record of this paper and its review
CitationBruce A. Bassett, Christopher Gordon, David Wands, Roy Maartens (Portsmouth). "Adiabatic and entropy perturbations from inflation." arXiv:astro-ph/0009131 (2000). https://pith.science/paper/HJT4OUEV
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The paper

The paper claims

In multi-field inflation, entropy perturbations can drive the curvature perturbation on arbitrarily large scales, but only when the background trajectory bends in field space.

The reading

The paper asks a sharp question about inflation driven by more than one scalar field: can a non-adiabatic (isocurvature, or 'entropy') fluctuation change the adiabatic curvature perturbation after the mode has left the Hubble radius? The authors answer yes, and they answer it by rotating the field-space basis at each instant so that one direction, $Q_σ$, points along the classical trajectory and the other, $δs$, points orthogonal to it. In those variables the equations separate cleanly: on large scales the entropy perturbation obeys a homogeneous second-order equation with no source from the adiabatic mode, while the adiabatic mode picks up a source term proportional to $δs$ times the bending rate $θ$-dot of the trajectory. The practical payoff is a criterion: whether entropy fluctuations survive inflation at all is set by an effective mass $μ_s^2 = V_{ss} + 3θ$-dot$^2$, so a heavy entropy direction squeezes them to nothing before preheating begins. Two worked applications follow, a double-inflation cross-correlation and a two-field preheating study, and the second doubles as a demonstration that the rotated variables are numerically far more stable than the original field perturbations.

What carries the argument

The machinery is a position-dependent rotation of the field-space basis. The adiabatic field $σ$ measures path length along the classical trajectory with $̇{σ} = (cos θ)̇{φ} + (sin θ)̇{χ}$, and the entropy field $s$ is defined so that $δs = (cos θ)δχ - (sin θ)δφ$ is the component of the perturbation orthogonal to the trajectory. Because $s$ is constant along the classical path, $δs$ is automatically gauge-invariant. This rotation turns the coupled two-field perturbation problem into a pair of equations with a transparent structure: a homogeneous entropy equation whose mass term $V_{ss} + 3̇{θ}^2$ controls suppression, and an adiabatic equation sourced by $̇{θ}δs$. The curvature perturbation itself takes the single-field form $R = ψ + (H/̇{σ})δσ$, so the familiar intuition carries over once the right direction in field space is chosen.

Extended reading notes

Core claim

The central claim is an asymmetry between the two kinds of perturbation. Combining the perturbed field equations with the metric constraints, the authors show that the comoving curvature perturbation evolves as $̇{R} = (H/̇{H})(k^2/a^2)Ψ + (2H/̇{σ})̇{θ}δs$ (their Eq. 45), so on super-Hubble scales the only unsuppressed source is the entropy perturbation multiplied by the rate at which the background path curves; a straight trajectory gives no effect. The entropy perturbation, by contrast, satisfies $̈{δs} + 3Ḣ{δs} + (k^2/a^2 + V_{ss} + 3̇{θ}^2)δs = (̇{θ}/̇{σ})(k^2/2πGa^2)Ψ$ (Eq. 52), whose source vanishes as $k → 0$, so adiabatic perturbations cannot regenerate entropy at long wavelengths. From this the authors extract a survival condition: large-scale entropy perturbations exist only if $μ_s^2 = V_{ss} + 3̇{θ}^2 < 3H^2/2$. They then apply the formalism twice — computing the adiabatic–entropy cross-correlation in double inflation with two non-interacting massive fields, and following the same quantities through two-field preheating, where they show that the entropy mass during inflation, not the resonance parameter, decides whether preheating can move the curvature perturbation.

Load-bearing premise

The whole construction divides by the rate at which the combined field rolls: the rotation angle, the bending rate, and the source terms all carry factors of $1/̇{σ}$, so if both fields momentarily stop the adiabatic and entropy directions become undefined and the comoving curvature perturbation itself diverges; the authors acknowledge this in the preheating section and note it can cause numerical instability in a narrow valley.

Editorial extensions

If this is right

  • Any model in which the inflationary trajectory is straight in field space — for instance double inflation with equal masses, where the authors show the cross-correlation vanishes identically — produces uncorrelated adiabatic and entropy perturbations.
  • Correlated adiabatic and isocurvature spectra are a generic prediction of curved trajectories, giving a distinctive target for cosmic microwave background analyses rather than a model-specific curiosity.
  • An entropy field heavier than the Hubble scale during inflation leaves no large-scale entropy perturbation for preheating to amplify, so strong resonance alone is not enough to alter the curvature perturbation.
  • The rotated variables remove a numerical cancellation that otherwise demands roughly 32 significant figures when $δs$ is exponentially small, making long simulations of preheating tractable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The survival criterion $μ_s^2 < 3H^2/2$ is a sharp discriminator: a model-builder can compute $V_{ss} + 3̇{θ}^2$ during inflation and know, before running any lattice or mode-by-mode simulation, whether isocurvature effects are worth pursuing.
  • Because the source term scales with $̇{θ}$, the size of any isocurvature imprint is a direct probe of the geometry of the inflationary path, so a future measurement of adiabatic–isocurvature correlation would constrain trajectory curvature rather than just the potential's slope.
  • The same rotation should apply to the post-inflationary era for any multi-component matter sector, suggesting a general bookkeeping in which only the component of the pressure perturbation orthogonal to the total momentum flow can generate curvature at super-horizon scales.
Original abstract

We study adiabatic (curvature) and entropy (isocurvature) perturbations produced during a period of cosmological inflation that is driven by multiple scalar fields with an arbitrary interaction potential. A local rotation in field space is performed to separate out the adiabatic and entropy modes. The resulting field equations show explicitly how on large scales entropy perturbations can source adiabatic perturbations if the background solution follows a curved trajectory in field space, and how adiabatic perturbations cannot source entropy perturbations in the long-wavelength limit. It is the effective mass of the entropy field that determines the amplitude of entropy perturbations during inflation. We present two applications of the equations. First, we show why one in general expects the adiabatic and entropy perturbations to be correlated at the end of inflation, and calculate the cross-correlation in the context of a double inflation model with two non-interacting fields. Second, we consider two-field preheating after inflation, examining conditions under which entropy perturbations can alter the large-scale curvature perturbation and showing how our new formalism has advantages in numerical stability when the background solution follows a non-trivial trajectory in field space.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,709 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.35 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study adiabatic (curvature) and entropy (isocurvature) perturbations produced during a period of cosmological inflation that is driven by multiple scalar fields with an arbitrary interaction potential. A local rotation in field space is performed to separate out the adiabatic and entropy modes. The resulting field equations show explicitly how on large scales entropy perturbations can source adiabatic perturbations if the background solution follows a curved trajectory in field space, and how adiabatic perturbations cannot source entropy perturbations in the long-wavelength limit. It is the effective mass of the entropy field that determines the amplitude of entropy perturbations during inflation. We present two applications of the equations. First, we show why one in general expects the adiabatic and entropy perturbations to be correlated at the end of inflation, and calculate the cross-correlation in the context of a double inflation model with two non-interacting fields. Second, we consider two-field preheating after inflation, examining conditions under which entropy perturbations can alter the large-scale curvature perturbation and showing how our new formalism has advantages in numerical stability when the background solution follows a non-trivial trajectory in field space.

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 study adiabatic (curvature) and entropy (isocurvature) perturbations produced during a period of cosmological inflation that is driven by multiple scalar fields with an arbitrary interaction potential. contribution 0.99 0.90 0.07 asserted 0.81
s1 A local rotation in field space is performed to separate out the adiabatic and entropy modes. method 0.99 0.90 0.03 asserted 0.97
s2 The resulting field equations show explicitly how on large scales entropy perturbations can source adiabatic perturbations if the background solution follows a curved trajectory in field space, and how adiabatic perturbations cannot source entropy perturbations in the long-wavelength limit. result 0.93 0.86 0.55 asserted 0.87
s3 It is the effective mass of the entropy field that determines the amplitude of entropy perturbations during inflation. result 0.94 0.62 0.12 asserted 1.00
s4 We present two applications of the equations. contribution 0.95 0.81 0.02 asserted 0.81
s5 First, we show why one in general expects the adiabatic and entropy perturbations to be correlated at the end of inflation, and calculate the cross-correlation in the context of a double inflation model with two non-interacting fields. contribution 0.73 0.95 0.08 asserted 0.95
s6 Second, we consider two-field preheating after inflation, examining conditions under which entropy perturbations can alter the large-scale curvature perturbation and showing how our new formalism has advantages in numerical stability when the background solution follows a non-trivial trajectory in field space. contribution 0.83 0.94 0.12 asserted 0.93

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • 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

Figures
Figure 1. FIG. 1. An illustration of the decomposition of an arbi
Figure 2. FIG. 2. Numerical simulations of the entropy and comov

Figures reproduced from arXiv:astro-ph/0009131 by the authors.

References
How later papers used it

301 later papers cite it

Evidence: substantial. 285 of the citing papers share no author with it, and 83 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 513 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

Builds on it 3 of 8 papers with this label

  • Generalized this paper's two-field result, summing over 3N^2-1 entropy modes with angles between each mode and the inflaton trajectory in the space of 3N^2 scalar fields.

    M-flation: Inflation From Matrix Valued Scalar Fields (2009) · 1.00 · paraphrased: the licence does not allow quoting

  • Extended the two-field formalism of this paper and another work to three fields to calculate the power spectra for scalar and tensor perturbations.

    Inflation and Scale-invariant $R^2$-Gravity (2021) · 0.99 · paraphrased: the licence does not allow quoting

  • Extended this paper's formalism for the canonical kinetic term case.

    Spinflation (2007) · 0.97 · paraphrased: the licence does not allow quoting

Uses its method or tool 19 of 66 papers with this label
How the literature uses it

285 arXiv papers cite this in 500 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 · 295 sentences without a confident label

Builds on it 8 papers
Uses its method or tool 62 papers
Uses its data or values 1 paper
Compares with it 9 papers
Cites it as support 5 papers
Cites it as background 87 papers

Discussion

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