Enhancement of superhorizon scale inflationary curvature perturbations

Andrew R Liddle; David Wands; Misao Sasaki; Samuel M Leach

REVIEW

Enhancement of superhorizon scale inflationary curvature perturbations

Samuel M Leach, Misao Sasaki, David Wands, Andrew R Liddle

Astrophysics · Submitted 23 Jan 2001 · 11 references

Pith's verdict: A clean, physically transparent explanation of superhorizon enhancement in single-field inflation; the analytic criterion is a good diagnostic, but its derivation is truncated exactly where it is applied, and the authors know it.

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arXiv IDastro-ph/0101406 (version 2)

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CitationAndrew R Liddle, David Wands, Misao Sasaki, Samuel M Leach. "Enhancement of superhorizon scale inflationary curvature perturbations." arXiv:astro-ph/0101406 (2001). https://pith.science/paper/ITUJKX7N

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Editor's note

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The thing to know: this paper explains analytically a numerical result that had been sitting unexplained — that the curvature perturbation can grow well after horizon crossing in single-field inflation. The mechanism is mode cancellation. If z = aφ̇/H drops well below its horizon-crossing value, the decaying mode lingers instead of dying, the two modes nearly cancel at η_k, and when the decaying mode finally goes away you are left with a much larger growing-mode amplitude. That is the physics, it is correct, and it is stated in a page and a half. The abstract's claim in [abstract:s0] and [abstract:s1] is exactly this, no more. What is genuinely new is the pair of background integrals D_k and F_k (Eqs. 7, 14, 15, 16) and the α of Eq. 19. The Wronskian step (Eqs. 4–5) is textbook, but using it to express the decaying mode in terms of the growing one and then reading off α from the initial conditions is a neat, self-contained construction. The two worked examples — Starobinsky's step and the Leach–Liddle false-vacuum quartic — are independent numerical checks, and the analytic estimate tracks the exact spectra across the whole range, including the spike at k ~ (A−/A+)^{1/2} H_0 that comes from F_k rather than D_k. That separation of which integral dominates on which scales is the useful practical bit. The soft spot is the one the stress-test flags, and it is real: D_k and F_k are computed with u replaced by u_0, i.e. to O(k^2), and then the criterion is 'enhancement when F_k >> 1' — precisely where the neglected terms are not suppressed. The authors see this themselves; they note that k^2 << |z''/z| fails for k < H_0 in the Leach–Liddle model and call the continued agreement 'rather surprising.' They do not derive why. So Eq. 19 is best read as an order-of-magnitude diagnostic, not a precision formula, and the abstract's 'in general single-field inflation models' is supported by two examples rather than a controlled expansion. That is a bounded claim, not a broken one — the qualitative mechanism is exact and the numerics back it. Minor: the (k/H_k)^2 = 0.1 choice in Eq. 19 is arbitrary and unremarked; a sentence on sensitivity would help. Who it is for: anyone working on primordial features, CMB spectral distortions from inflation, or the general question of when superhorizon evolution matters. Short enough to read in an afternoon. Recommendation: send it out. It is a real result, correctly scoped, and the acknowledged limitation is the kind a referee should push on but not reject over.

Referee report

Significance

If correct, the result is a useful and general piece of inflationary perturbation theory: it identifies a mechanism by which superhorizon enhancement can occur without entropy perturbations, and it supplies a closed-form, parameter-free criterion built only from the background quantity z = aφ̇/H. The derivation chain is internally consistent — I checked Eqs. (9), (10), (17), (18) and (19) against each other and against the definitions of D and F, and the algebra closes — and the two worked examples are compared against exact numerical spectra rather than against the approximation itself, which is the right test. The paper also makes a falsifiable statement: the enhancement is confined to a definite range of scales set by (A_−/A_+)^{1/2}H_0 and (A_+/A_−)^{1/3}H_0, which is the kind of prediction that can be checked against a computed spectrum. The appendix treatment of the case where φ̇ crosses zero, with the regulated integral of Eq. (A7), is a genuine completeness addition rather than an afterthought, and the observation in Section V that the transient-era modes share the slow-roll spectral shape is a nice consistency check on the numerical results. The paper is short, the claims

Major objections (2)

[Abstract; §III, Eqs. (16)–(19); §IV] The abstract states that the authors give 'a criterion for this enhancement in general single-field inflation models', but the criterion Eq. (19) is built from D_k and F_k as approximated in Eqs. (14) and (16), which replace the exact growing-mode solution u by the constant u_0 inside the integrands. Equation (17) is explicitly described as 'accurate to O(k^2)', yet the criterion is then applied in the regime F_k ≫ 1 or D_k ≫ 1, where the neglected higher-order terms are not parametrically suppressed. The authors are aware of the neighbouring problem: in §IV they note that in the Leach–Liddle model the long-wavelength condition k^2 ≪ |z''/z| is violated for k < H_0, and they describe the continued success of the approximation as 'rather surprising' without offering an argument for why it should hold. The two worked examples are persuasive evidence but they are two models with similar transient structure, so the generality asserted in the abstract is supported by example rather than by a controlled derivation. This does not threaten the qualitative mechanism, which is exact and numerically confirmed, but it does bound the quantitative criterion. I would ask the authors either to add a short argument for why the truncated α tracks the exact enhancement when F_k or D_k is large (for instance, by showing that the leading correction to Eq. 19 is suppressed by the same small quantity that makes z small, rather than by k^2), or to soften the abstract and the §VI summary to state that the criterion is derived in the long-wavelength approximation and validated in the models considered.
[§IV, Eqs. (24) and (25)] The branch conditions as printed in Eqs. (24) and (25) cannot be right as they stand. The condition 'k > (k/H_k)H_0' is equivalent to H_k > H_0, which is satisfied for every mode, so the two branches are not distinguished by the stated inequality; the same applies to the second pair. This is inconsistent with the text immediately following, which states that F_k dominates on scales k < H_0 and D_k on scales k > H_0, and with the quoted ranges (A_−/A_+)^{1/2}H_0 ≲ k ≲ (A_+/A_−)^{1/5}H_0 and (A_−/A_+)^{1/3}H_0 ≲ k ≲ (A_+/A_−)^{1/3}H_0, which are stated relative to H_0 alone. The factors '(k/H_k)(H_0/k)' inside the brackets also appear degenerate. Since these equations are the explicit evaluation of the paper's central criterion in its flagship example, a reader cannot reproduce the quoted scale ranges from them. I assume this is a typesetting corruption of expressions such as (H_0/k)^5 and (k/H_0)^3, but it should be corrected and checked against the numerical spectrum in Fig. 1.

Minor objections (7)

[§III, Eq. (15)] In the inner integral of Eq. (15) the integrand is printed as 'z^2(η'') u(η'')u_0 dη'''; from the structure of the iteration and from the comparison with Eq. (16) it should presumably be u(η'')/u_0. Please check.
[§III, below Eq. (19)] The statement 'for definiteness we will take (k/H_k)^2 = 0.1' defines the matching epoch η_k as the time at which the mode is still marginally inside the horizon. Since α is defined relative to R_c(η_k), and both D_k and F_k are evaluated at that epoch, the quoted enhancement factor depends on this convention. The final amplitude R_c(η_*) is of course convention-independent, but it would help to state explicitly how sensitive α is to the choice of η_k, or at least to note that the convention only affects the split between α and R_c(η_k) at the O(1) level.
[§IV, Eq. (23)] The estimate z_min ≈ (A_−/A_+)^{2/3} z_0 drops O(1) numerical factors arising from minimising the two-term expression; the exact minimum of A_−e^x + (A_+−A_−)e^{−2x} sits at e^{3x} = 2(A_+−A_−)/A_−. Since this ratio feeds directly into the quoted ranges of k, a footnote giving the numerical coefficient would make the estimates reproducible.
[Appendix A, Eq. (A7)] The counterterm in Eq. (A7) is printed as '−2ε z_k^2 u_k^2/(z'0^2 u_0^2)', with ε in the numerator. Near η = η_0 each of the two integrals diverges as 1/(z'0^2 ε), so the subtraction required to make the sum finite should scale as 1/ε, not ε. Please verify the printed expression.
[Figs. 1 and 2] The captions do not state the vertical normalisation of the plotted power spectra, nor the pivot or the units of the horizontal axis beyond the implicit H_0 scaling. Since the figures are the main evidence that the analytic α tracks the exact result, labelling both axes explicitly and stating what the normalisation of R_c^2 is would make them self-contained.
[§II, Eq. (10) and following] The estimate that R'_c/(H_k R_c) is 'at most of order unity' at horizon crossing rests on the assumption of an initial vacuum state for the quantum fluctuations. It would be worth stating the vacuum choice (Bunch–Davies at k ≫ aH) explicitly at this point, since the whole enhancement factor is measured relative to that initial amplitude.
[§V] The duality discussion (Eq. 26 and the two-parameter family) is interesting but sits somewhat apart from the main argument, and the connection to the enhancement result is asserted rather than demonstrated. A sentence explaining what the duality buys the reader in the context of the enhancement — for instance, that it explains why the transient-era modes inherit the slow-roll spectral shape seen in Figs. 1–2 — would tie the section to the rest of the paper.

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The paper

The paper claims

A brief slowdown of the inflaton can amplify curvature perturbations long after they cross the horizon, even in single-field inflation, because the decaying mode lingers and cancels the growing mode at horizon crossing before dying away.

The reading

The standard story is that in single-field inflation the curvature perturbation $R_c$ freezes in once a scale crosses the Hubble horizon, so its final amplitude is just $(H^2/2πφ̇)$ evaluated at $k=aH$. This paper argues that this freezing is not automatic: it holds only when the background quantity $z=aφ̇/H$ grows steadily. If $z$ instead drops well below its horizon-crossing value for a stretch of inflation, the mode that is supposed to decay stays nearly constant, so the two independent modes nearly cancel at horizon crossing and the perturbation starts out small. When the decaying mode finally disappears, what remains is the growing mode alone, and the final amplitude is the small initial value multiplied by a large factor $|α|$. The authors give a criterion for this amplification in terms of two background integrals, $D_k$ and $F_k$, and show it is realized in a model where the inflaton potential has a sudden change of slope.

What carries the argument

The load-bearing object is $z=aφ̇/H$, the pump field of the Mukhanov-Sasaki equation, together with the two integrals built from it: $D_k=3H_k∫_{η_k}^{η_*}dη' z^2(η_k)/(z^2(η')u^2(η'))$, which measures how long the decaying mode lingers, and $F_k$, the $O(k^2)$ correction to the growing mode obtained by iterating the mode equation. When either integral exceeds unity, the final curvature perturbation differs substantially from its horizon-crossing value, and the enhancement factor is $|α|≈|1+(D_k/3H_k)(R_c'/R_c)-F_k|$. The mechanism is mode cancellation at horizon crossing followed by release once the decaying mode disappears.

Extended reading notes

Core claim

The central claim is that superhorizon enhancement of the curvature perturbation is possible in single-field inflation, and that the size of the effect is controlled by the behavior of $z=aφ̇/H$. Working from the exact mode equation $R_c''+2(z'/z)R_c'+k^2R_c=0$, the authors express the general solution as a growing mode $u$ plus a decaying mode $v$, related by a Wronskian that scales as $1/z^2$. If $z$ falls far below $z_k=z(η_k)$ after horizon crossing, the decaying mode does not decay promptly; it stays almost constant, forcing the two modes to nearly cancel at $η_k$ so that $R_c(η_k)$ is small. Once the decaying mode dies out, $R_c(η_*)=αR_c(η_k)$ with $|α|$ large. The criterion is that either $D_k$ or $F_k$ exceeds unity, where $D_k$ measures the accumulated $1/z^2$ integral and $F_k$ is the second-order correction to the growing mode. In Starobinsky's model with a step in the potential slope and $A_+/A_-=10^4$, the enhancement covers scales from about $(A_-/A_+)^{1/2}H_0$ to $(A_+/A_-)^{1/3}H_0$, with $F_k$ dominating for $k<H_0$ and $D_k$ for $k>H_0$. The authors also note that the usual slow-roll amplitude formula can remain a good approximation even when the background is far from slow-roll at horizon crossing, an instance of the duality invariance of perturbation spectra under $z̃=C_1z+C_2z∫dη'/z^2$.

Load-bearing premise

The criterion assumes that the growing-mode solution can be replaced by a constant inside the integrals $D_k$ and $F_k$, which requires the long-wavelength condition $k^2≪|z''/z|$ to hold through the transient; the authors themselves note this condition fails for $k<H_0$ in the Leach-Liddle model yet report that the approximation still works there without showing why.

Editorial extensions

If this is right

Any single-field model in which $z$ dips well below its horizon-crossing value for a sustained interval should show a boosted curvature perturbation on the scales that cross during that interval.
The enhancement factor can be estimated analytically from the background trajectory alone, without solving the perturbation equations mode by mode, using $D_k$ and $F_k$ in the long-wavelength approximation.
In a potential with a sharp change of slope, the amplified range of scales is bounded, running roughly from $(A_-/A_+)^{1/2}H_0$ to $(A_+/A_-)^{1/3}H_0$, so the effect is a localized feature in the spectrum rather than a global rescaling.
Because the underlying spectrum can be unchanged by the transient, the standard slow-roll amplitude formula may still describe the result even when the background is not slow-rolling at horizon crossing.

Reading between the lines

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

If the mechanism operates in models with a sharp feature, the resulting localized bump or dip in the primordial power spectrum could be sought in the small-scale tail of the cosmic microwave background and in primordial black hole abundance constraints.
The cancellation picture suggests a general diagnostic: any inflationary background where $z$ is non-monotonic is a candidate for superhorizon amplification, so a scan over $z(η)$ histories, rather than over potentials, may be the efficient way to map where the effect is large.
The paper's reliance on the constant-$u_0$ approximation inside the integrals invites a numerical survey of models where $k^2\ll|z''/z|$ fails badly, to see whether the closed-form $α$ tracks the exact enhancement or merely the qualitative trend.

Original abstract

We show that there exists a simple mechanism which can enhance the amplitude of curvature perturbations on superhorizon scales during inflation, relative to their amplitude at horizon crossing. The enhancement may occur even in a single-field inflaton model, and occurs if the quantity $a\dot\phi/H$ becomes sufficiently small, as compared to its value at horizon crossing, for some time interval during inflation. We give a criterion for this enhancement in general single-field inflation models.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.18 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.17 · mentions code or data availability 0.02.

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

We show that there exists a simple mechanism which can enhance the amplitude of curvature perturbations on superhorizon scales during inflation, relative to their amplitude at horizon crossing. The enhancement may occur even in a single-field inflaton model, and occurs if the quantity $a\dot\phi/H$ becomes sufficiently small, as compared to its value at horizon crossing, for some time interval during inflation. We give a criterion for this enhancement in general single-field inflation models.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that there exists a simple mechanism which can enhance the amplitude of curvature perturbations on superhorizon scales during inflation, relative to their amplitude at horizon crossing.	result 0.54	0.92	0.88	asserted 1.00
s1	The enhancement may occur even in a single-field inflaton model, and occurs if the quantity $a\dot\phi/H$ becomes sufficiently small, as compared to its value at horizon crossing, for some time interval during inflation.	result 0.92	0.76	0.27	hedged possibility 0.98
s2	We give a criterion for this enhancement in general single-field inflation models.	contribution 0.98	0.93	0.12	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

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Figures

**Figure 2.** FIG. 2. Power-spectrum for the false-vacuum quartic

Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/0101406 by the authors.

References

Works cited (11)

[1]

V. F. Mukhanov, Sov. Phys. JETP Lett. 41, 493 (1985); 67, 1297 (1988)

View on Pith 1985
[2]

M. Sasaki, Prog. Theor. Phys. 76, 1036 (1986)

View on Pith 1986
[3]

S. M. Leach and A. R. Liddle, [astro-ph/0010082]

View on Pith Pith review arXiv
[4]

H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984)

View on Pith 1984
[5]

V. F. Mukhanov, H. A. Feldman and R. H. Branden- berger, Phys. Rep. 215, 203 (1992)

View on Pith 1992
[6]

E. D. Stewart and D. H. Lyth, Phys. Lett. B 302, 171 (1993)

View on Pith 1993
[7]

A. A. Starobinsky, private communication: We are grate- ful to him for pointing this out to us

View on Pith
[8]

A. A. Starobinsky, JETP Lett. 55, 489 (1992)

View on Pith 1992
[9]

D. Roberts, A. R. Liddle and D. H. Lyth, Phys. Rev. D 51, 4122 (1995) [astro-ph/9411104]

View on Pith Pith review arXiv 1995
[10]

D. Wands, Phys. Rev. D 60, 023507 (1999) [gr- qc/9809062]

View on Pith 1999
[11]

O. Seto, J. Yokoyama and H. Kodama, Phys. Rev. D 61, 103504 (2000) [astro-ph/9911119]. 5

View on Pith Pith review arXiv 2000

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Primordial black holes as dark matter and gravitational waves from bumpy axion inflation
astro-ph.CO 2020-08 conditional novelty 5.0 of 10

For a tuned axion-monodromy parameter point, gauge fields amplified during a brief fast roll source a peaked scalar spectrum that collapses into 10^-13 solar-mass primordial black holes accounting for all dark matter,...
Stochastic inflationary dynamics beyond slow-roll and consequences for primordial black hole formation
astro-ph.CO 2020-06 accept novelty 5.0 of 10

At linear order, stochastic inflation reproduces the standard Mukhanov-Sasaki curvature power spectrum even through an ultra-slow-roll phase, so the claimed quantum-diffusion enhancement of the primordial-black-hole p...
Primordial black holes as dark matter and gravitational waves from single-field polynomial inflation
astro-ph.CO 2020-01 conditional novelty 5.0 of 10

Single-field polynomial inflation with a tuned near-inflection point can generate primordial black holes comprising all dark matter, plus an induced gravitational-wave signal within reach of LISA and DECIGO.
Primordial Black Holes in Higgs-$R^2$ Inflation as a Whole Dark Matter
hep-ph 2019-12 conditional novelty 5.0 of 10

Adding an R² term to Higgs inflation creates a second plateau that amplifies curvature perturbations enough to form primordial black holes of 10^-16 to 10^-15 solar masses, potentially the bulk of dark matter.
Primordial Black Holes from $\alpha$-attractors
astro-ph.CO 2018-05 conditional novelty 5.0 of 10

Alpha-attractor inflation with a tuned inflection point generates a curvature power spectrum peak that collapses into 10^-16 to 10^-14 solar-mass primordial black holes at up to 10% of the dark matter density.
Quantum diffusion beyond slow-roll: implications for primordial black-hole production
astro-ph.CO 2018-05 conditional novelty 5.0 of 10

Including quantum diffusion beyond slow-roll in a quasi-inflection-point inflation model raises the predicted primordial black-hole abundance by orders of magnitude, mainly through a diffusion peak in the power spectr...
Mechanisms for Primordial Black Hole Production in String Theory
hep-th 2018-03 conditional novelty 5.0 of 10

A bumpy string axion potential drives a brief ultra slow-roll phase that amplifies curvature perturbations enough to produce light primordial black holes, potentially a large fraction of dark matter.
Obstructions to Bell CMB Experiments
astro-ph.CO 2017-06 conditional novelty 5.0 of 10

Formal Bell-inequality violation in the inflationary CMB state is shown to be unmeasurable, because one sky map fixes at most one pseudo-spin component and a Bell test needs two non-commuting ones.
Inflationary Magnetogenesis without the Strong Coupling Problem
astro-ph.CO 2013-05 conditional novelty 5.0 of 10

Optimizing the coupling slope and reheating history subject to backreaction and strong-coupling constraints raises the ceiling on today's Mpc-scale magnetic field from inflation to 10^-13 G at low inflation scales.
{\delta}N formalism
gr-qc 2012-08 accept novelty 5.0 of 10

The delta N formalism is valid because on superhorizon scales the perturbed Hamiltonian constraint and scalar-field equations take the same form as their unperturbed counterparts, with momentum-constraint violations d...
On Features and Nongaussianity from Inflationary Particle Production
astro-ph.CO 2010-06 conditional novelty 5.0 of 10

Particle production during inflation generates a localized bispectrum feature whose skewness, for coupling g squared of 0.01, is equivalent to a local f_NL of about minus 53.
Exploring Gravitational Wave Signatures Due to Primordial Non-gaussianity and Large Scale Structure Using SKAO
astro-ph.CO 2026-06 unverdicted novelty 4.0 of 10

Explores SKAO detection of scalar-induced GW backgrounds as probes of primordial non-Gaussianity and parity violation, with LSS cross-correlation to improve SNR.
One-Loop Tensor Power Spectrum from a Non-Canonical Spectator Field during Inflation
astro-ph.CO 2026-01 conditional novelty 4.0 of 10

One-loop scalar-sourced tensor corrections in non-minimal spectator PBH models are O(1) for a Gaussian dip and up to 10^12 times the tree level for an oscillatory coupling.
Note on the bispectrum and one-loop corrections in single-field inflation with primordial black hole formation
hep-th 2023-03 conditional novelty 4.0 of 10

In single-field inflation with a temporary ultraslow-roll period, the bispectrum obeys Maldacena's consistency relation and the one-loop correction to the large-scale power spectrum, computed two independent ways, agr...
Primordial black holes dark matter from inflection point models of inflation and the effects of reheating
astro-ph.CO 2019-07 conditional novelty 4.0 of 10

A symmetric sextic inflection-point inflation model can produce asteroid-mass primordial black holes accounting for all dark matter, and a matter-dominated reheating epoch shifts the mass window upward and increases t...
Axion inflation with gauge field production and primordial black holes
astro-ph.CO 2013-12 conditional novelty 4.0 of 10

Axion inflation with a gauge-field coupling predicts a large small-scale curvature spectrum; matching the resulting primordial black hole abundance to existing limits bounds the coupling at xi < 1.8 (quadratic potenti...
Primordial black holes from a type-IIB/F theory inflaton potential
gr-qc 2026-10 reject novelty 3.0 of 10

Adding a Gaussian bump tuned by three free parameters to a string-derived inflaton potential can produce a primordial black hole population near 10^{-6} solar masses, but the bump's shape and amplitude are not derived...
Inflation and Primordial Black Holes
astro-ph.CO 2023-01 accept novelty 3.0 of 10

The paper is a pedagogical review that classifies inflationary mechanisms for producing primordial black holes into single-field non-slow-roll enhancement and multi-field tachyonic or gauge-field amplification.
String Cosmology: from the Early Universe to Today
hep-th 2023-03 conditional novelty 2.0 of 10

A comprehensive survey that organizes moduli stabilisation, string inflation, post-inflationary moduli domination, dark energy, and alternative early-universe scenarios into a single reference with observational targets.
Stochastic inflation and primordial black holes
astro-ph.CO 2020-09 conditional novelty 2.0 of 10

Stochastic quantum diffusion during inflation can change predicted primordial-black-hole abundances by several orders of magnitude, according to this synthesis of the author's published stochastic-delta-N results.

How later papers used it

103 later papers cite it

Evidence: substantial. 86 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 191 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

2 later papers by other authors are listed below: 2 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 3 of 7 papers with this label · 1 by the same authors

Used the gradient expansion formalism introduced in this paper to compute the evolution of fluctuations on super-horizon scales in models with non-attractor phases.

CMB $\mu T$ cross-correlations as a probe of PBH scenarios (2021) · 1.00 · paraphrased: the licence does not allow quoting
Re-derived the consistency conditions using the rigorous gradient expansion approach first introduced in this paper.

Consistency conditions and primordial black holes in single field inflation (2021) · 0.97 · paraphrased: the licence does not allow quoting

By the same authors

Used the technique developed in this paper to show explicitly that the spectrum can be modified only near k=k_0 and quickly approaches the standard result at k much less than k_0.

Effects of particle production during inflation (2008) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

103 arXiv papers cite this in 191 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 · 142 sentences without a confident label

Builds on it 3 papers

In this work, extending the work in [ref] , we will push our formulas up to order $k^4$ ($n=2$).

On the slope of curvature power spectrum in non-attractor inflation · Solving the mode equations at super-horizon scales · 0.97
Parallel to the second-order differential equation ([ref]) for $R_c^NL$, which is a natural extension of the well-known linear version ([ref]), the above expression for the final value of $R_c^NL$ turns out to be a natural extension of the result obtained in linear theory in [ref] .

Non-Gaussianity of superhorizon curvature perturbations beyond $\delta$ N formalism · General formalism · 0.81
As discussed in [ref] , the expression [ref], along with the considerations above, generalizes to a time-dependent sound speed and Planck mass the arguments first developed in [ref] .

Inflation and Primordial Black Holes · Enhancement through the resurrection of the decaying mode · 0.98

Uses its method or tool 7 papers

For this purpose, we make use of the gradient expansion formalism introduced in [ref] , which allows one to compute the evolution of fluctuations on super-horizon scales in models with non-attractor phases [ref] .

CMB $\mu T$ cross-correlations as a probe of PBH scenarios · Introduction · 1.00
Making use of duality arguments developed in [ref] , we also get analytic control, in representative scenarios, on the behavior of the scalar power spectrum after the peak in the power occurs: for modes associated with the transition from non-attractor to final attractor phase that we dub as graceful exit epoch.

On the slope of curvature power spectrum in non-attractor inflation · Introduction · 0.98
To express its solution, we implement a gradient expansion approach (see e.g. [ref] ), starting from the solution in the limit of small $k/(a H)$, and including its momentum-dependent corrections which solve [ref] order-by-order in a $k/(a H)$ expansion.

Inflation and Primordial Black Holes · The dynamics of curvature perturbation · 0.99

Uses its data or values 1 paper

where $C$ is an arbitrary constant which is fixed to be 1 in [ref] .

Primordial Black Hole Formation in Starobinsky's Linear Potential Model · Superhorizon enhancement factors · 0.89

Compares with it 2 papers

An alternative, systematic approach more similar to ours was proposed in [ref] , solving the mode functions implementing a gradient expansion in powers of $k$ (although using a slightly different Ansatz with respect to us) and studying the role of the decaying mode after horizon crossing.

An analytic approach to non-slow-roll inflation · Brief comparison with previous analytical analysis of the spectrum · 0.95
[ref] (often dubbed Wands duality, see ref.

Primordial black hole dark matter from inflation: the reverse engineering approach · On the formation of a raised plateau in the power spectrum · 0.84

Cites it as support 4 papers

These modes undergo superhorizon evolution after the USR transition into phase 2 [ref] , and eventually settle in the final boundary measured at the end of inflation as a $P_ζ∼k^4$ growth [ref] .

Induced gravitational waves from the cosmic coincidence · Generalised ultra-slow-roll inflation · 0.96
For example, in the context of the standard linear perturbation theory, Leach et al. pointed out that there can be enhancement of the comoving curvature perturbation on superhorizon scales, where it is usually conserved, even in a single field model if the slow-roll condition is temporarily violated [ref] .

Gradient expansion approach to nonlinear superhorizon perturbations II -- a single scalar field -- · Introduction · 0.96
However, a rough quantatitive analysis of the behaviour of the curvature perturbation modes through the feature as in [ref] , supports the possibility that such a setup could provide a novel realisation of power enhancement in the primordial curvature perturbations and thus an interesting mechanism to produce PBHs.

Mechanisms for Primordial Black Hole Production in String Theory · Conclusions · 0.99

Cites it as background 21 papers

book Main References: Our discussion in this section is based on the papers [ref] .

Inflation and Primordial Black Holes · Enhancement of scalar perturbations during single-field inflation · 0.87
To do so, it is helpful to recall the physical conditions underlying this amplification, first discussed by Leach et al. [ref] [Note however, the first toy model that can amplify the power in scalar fluctuations is proposed by Starobinsky [ref] .].

Mechanisms for Primordial Black Hole Production in String Theory · Introduction · 0.94
In this work we investigate the role of non-gaussianities in the context of inflationary models featuring an ultra slow roll (USR) phase [ref] .

Non-gaussianities for primordial black hole formation · Motivations · 0.84

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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