Fast-Roll Inflation

Andrei Linde

REVIEW

Fast-Roll Inflation

Andrei Linde

High Energy Physics - Theory · Submitted 21 Oct 2001 · 34 references

Pith's verdict: A short, honest summary paper: the growth-rate formula and the two side observations are the durable content, and the e-fold numbers are illustrative rather than predictive — which the author mostly says himself.

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arXiv IDhep-th/0110195 (version 2)

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CitationAndrei Linde. "Fast-Roll Inflation." arXiv:hep-th/0110195 (2001). https://pith.science/paper/ZF3NUF4I

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Editor's note

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Worth your time, but read it as a compact note rather than a full paper. Linde says up front that the fast-roll mechanism itself was presented in the companion N=8 supergravity paper [9]; this one is the systematic unpacking. What is genuinely his here: the growth function F(m²/H²) = sqrt(9/4 + m²/H²) − 3/2 and the resulting e^{Ht*} = (φ*/φ0)^{1/F}, which cleanly interpolates between the slow-roll and steep-roll limits and is the formula everyone now quotes. The two side observations are also real. The anti-de Sitter point (Sec. 3.1.1) is almost trivial once stated — H² = ρ/3 has no solution with ρ < 0, so a flat universe after inflation cannot land in AdS — but it is the kind of trivial thing worth having on record. The negative-minimum instability argument (Sec. 3.1.2) is better: the kinetic-domination scaling |φ| ~ sqrt(2/3) ln τ, a(τ) = τ^{1/3} shows the singularity is essentially independent of V(φ), and the two numerical figures back it up. That one has held up as a constraint on quintessence model building with AdS minima. The perturbation estimate in Sec. 3.4, where δρ/ρ = O(1) at the onset of fast roll, is a caution the author raises against his own scenario, which is to his credit. On the soft spots: the stress-test worry about φ0 ~ m/10 is fair in principle but lands softly. That amplitude comes from Minkowski tachyonic preheating [10], and in the regime of interest H ~ m, so the de Sitter variance (H/2π)² is the same order as m²/8π². The import is not a mismatch, just an estimate. Linde flags it as such — 'we will simply replace φ0 by m/10 to find the maximal possible value' — so the 10^200 and 10^56 numbers should be read as illustrative, and they are. The figures in Sec. 3.1.2 come with no code or stated parameters, a minor reproducibility gap. Who should read it: anyone building quintessence or early-universe models with |V''| ~ H², and anyone who wants the AdS-after-inflation argument in citable form. Yes to referees — it is short, correct, and the side observations earn their place.

Referee report

Major objections (4)

[§2, Eqs. (2.9)–(2.11) and (2.20)–(2.31)] The whole e-fold estimate is derived with H held strictly constant, while the field by construction grows until V has dropped by a factor of two at φ*. The paper justifies this in the paragraph before Eq. (2.5) by noting that H falls only by 1/√2, which is fine as an order-of-magnitude argument, but the quoted results in Eqs. (2.22)–(2.31) are then given to two and three significant figures (10⁵⁶·¹, 10⁶·⁶, 10⁴⁴, 10²⁶). For m ~ H, where the headline numbers live, the field spends a comparable number of e-folds on either side of φ*, so the constant-H approximation is least controlled exactly where it is most used. The author should either integrate the coupled Friedmann-plus-field system numerically for m²/H² = 1 and 6 and report the discrepancy, or restate Eqs. (2.22)–(2.31) explicitly as upper bounds with one-significant-figure precision. As written, a reader could reasonably take 10²⁰⁰ as a prediction rather than an estimate.
[§2, Eq. (2.20) and the φ₀ ≈ m/10 input] The initial amplitude φ₀ ≈ m/10 is imported from Ref. [10], which computes the growth of tachyonic fluctuations in Minkowski space (Eq. 2.19, ⟨δφ²⟩ = m²/8π²), into a de Sitter background with H ~ m. The paper is candid that this is an estimate (
[§3.4, Eq. (3.5)] The estimate δρ/ρ ~ H δφ/φ̇ with δφ ~ H/2π and φ̇ ~ m²/2π gives O(1) at the onset of fast roll, and the paper then asserts that perturbations from later stages have
[§§3.2–3.3] The early-universe application requires that the fast-roll stage end and that the energy stored in φ be disposed of, but the paper never specifies how this happens. In §3.2 the mechanism proposed is that fast roll generates large fluctuations of a separate light field χ, which then provides initial conditions for a subsequent slow-roll phase driven by χ; for that to work, the residual energy density in φ after fast roll must be subdominant to V(χ) or must decay efficiently, and neither is shown. Similarly, §3.3 proposes using fast roll at the electroweak scale to dilute moduli and gravitinos by a factor 10⁴–10⁵, which presupposes that φ decays into radiation with a calculable reheat temperature; no decay channel or reheating temperature is given. A short paragraph specifying the exit dynamics, or an explicit citation to standard reheating treatments applied to this potential, would close the gap. Without it the applications in §§3.2–3.3 are suggestive rather than established.

Minor objections (9)

[Eq. (2.7)] The right-hand side reads
[§1, paragraph 4] The sentence beginning
[§3.2, paragraph on SU(5)] The sentence
[§3.4, first paragraph] The sentence
[§3.1.2, paragraph after Eq. (3.1)] The phrase
[Eqs. (2.3) and (2.29)] The notation m² is used throughout for the magnitude of a negative mass squared, so that V'' = −m², while the slow-roll parameter is written |m²| ≪ H². This is internally consistent but invites confusion with an ordinary positive mass; a single sentence at Eq. (2.3) fixing the convention, and stating that m² > 0 by definition, would help.
[Figures 1 and 2] Neither figure carries axis labels or units, and the horizontal axes span t = 0–100 in Fig. 1 but t = 0–40000 in Fig. 2 with no indication of the parameter values used. The two panels in each figure should be labelled φ(t) and a(t), and the caption should state V₀, m, λ, v and Λ for the runs shown so that the figures can be reproduced.
[§3.1.2, Eq. (3.1)] The derivation of a(τ) = τ^{1/3} and |φ| = √(2/3) ln τ checks out, but the stated condition for its validity (
[§3.1, paragraph on v] The claim that accelerated expansion is achievable with v as small as 3×10⁻²M_p, violating slow roll by four orders of magnitude, follows from m²/H² = 12/v² ≈ 1.3×10⁴ and is correct; it would be worth stating that relation explicitly in the text, since the reader currently has to reconstruct it from Eq. (2.29).

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The paper

The paper claims

Linde argues that inflation does not need slow roll: a scalar field falling fast from a hilltop, with a tachyonic mass of order the Hubble scale, can still stretch the universe by many e-folds, provided its mass is small in Planck units. He

The reading

The paper asks whether inflation really requires the slow-roll condition $|m^2| = |V''|/V$ to be much smaller than $H^2$. Linde's answer is no: in the simplest symmetry-breaking potentials a field sitting near a maximum with a tachyonic mass of order the Hubble scale grows as $V(0)$, $|m^2| = |V''|/V$, and the total expansion $e^{Ht_*}$ depends on the ratio $V(0)$ raised to $1/F$, where $F = V(0)$. Because that exponent is large when $m$ is small in Planck units, a stage that badly violates slow roll can still deliver 60 or more e-folds, enough to account for the present accelerated expansion and enough to prepare initial conditions for a later slow-roll stage. The paper then makes two general points: a universe that inflated long enough to become flat can never come to be dominated by a negative cosmological constant, and potentials with a stable minimum at $V(0)$ are as dangerous cosmologically as potentials unbounded from below, because the field is driven into a kinetic-energy-dominated collapse whose late-time behaviour hardly depends on the shape of the potential.

What carries the argument

The load-bearing object is the growth exponent $F(m^2/H^2) = V(0)$, which interpolates between the slow-roll result $F = V(0)$ and the fast-roll result $F = V(0)$, together with the expansion formula $e^{Ht_*} = (V(0))^{1/F}$ and its numerical inverses $F^{-1}(1) = 3.3$, $F^{-1}(6) = 0.73$, $F^{-1}(12) = 0.44$. This single function converts a mass ratio into an amount of inflation and is what lets the paper compare spontaneous-symmetry-breaking models, $N=8$ supergravity, and quintessence on one footing. The second piece of machinery is the flatness argument: after long inflation the curvature term $k/a^2$ is negligible, so $H^2 = V(0)$ has no solution with $V(0)$, and anti-de Sitter cannot be reached. The third is the collapse solution $a(0)$ with $V(0)$, which shows why the singularity's structure is fixed by kinetic energy rather than by the potential.

Extended reading notes

Core claim

The central claim is that fast-roll inflation is a genuine regime, not a failed limit. For $V(0)$ near a maximum, the homogeneous mode obeys $V(0)$ and grows as $V(0)$ with $F = V(0)$; the total expansion is $e^{Ht_*} = (V(0))^{1/F}$. Slow roll is the small-$m$ limit $F = V(0)$, fast roll is the large-$m$ limit $F = V(0)$, and the interesting middle case $m^2 = O(H^2)$ is where the paper's estimates live. Taking the initial amplitude to be the level of the exponentially growing quantum fluctuations, $V(0)$, Linde finds expansions of order $10^{200}$ for $m = H = 10^{-60}M_p$, $10^{56}$ for the electroweak scale, and still $10^{44}$ in the gauged $N=8$ supergravity case where $|V''| = 2V$, i.e. $m^2 = 6H^2$. He is explicit that fast-roll inflation at the very start produces adiabatic perturbations of order one with a steep red spectrum, so it is safe only if followed by slow roll or if it occurs at the present epoch; a short fast-roll stage at an intermediate scale risks overproducing black holes.

Load-bearing premise

The size of every quoted expansion factor rests on taking the starting displacement of the field to be about one tenth of its mass, $V(0)$, a value borrowed from simulations of the growth of quantum fluctuations rather than derived within the model; a starting value nearer the hilltop edge would shrink the predicted inflation to almost nothing.

Editorial extensions

If this is right

A present-day accelerated expansion driven by a field near a hilltop with $|V''| = O(H^2)$ is consistent with the data, since $V(0)$ stays nearly constant during the fast roll, so the equation of state is indistinguishable from a cosmological constant over the observed epoch.
A short fast-roll stage at nearly Planckian density can homogenise a region of Planck size and thereby supply the initial conditions that low-scale slow-roll and hybrid inflation otherwise lack.
Because $V(0)$ can be enormous for small $m$, models with potentials unbounded from below, or with minima at $V(0)$, remain viable descriptions of the current acceleration even though they eventually collapse.
Fast-roll inflation generates isocurvature perturbations with a flat spectrum from light fields, which can be converted into adiabatic perturbations if those fields later decay into radiation.
Inflationary cosmology predicts that we cannot end up in an anti-de Sitter universe, which is a constraint on any proposed resolution of the cosmological constant problem.
A fast-roll stage at the electroweak scale or below can add the $10^4$ to $10^5$ e-folds needed to dilute moduli and gravitinos, and can extend thermal inflation beyond the point where the field leaves the top of its potential.

Reading between the lines

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

The paper's expansion estimates are exponentially sensitive to the assumed initial displacement: since $e^{Ht_*} = (V(0))^{1/F}$, any mechanism that places the field closer to the hilltop than the fluctuation level $V(0)$ lengthens inflation dramatically, and any mechanism that places it further shortens it. This makes the initial-condition story, not the growth law, the place where a numerical si
The observation that a flat universe cannot reach anti-de Sitter can be turned around: a detection of negative spatial curvature at the percent level would reopen the anti-de Sitter route, so the claim is in principle a cosmological test of the inflationary prediction $\Omega = 1$.
The statement that minima with $V(0)$ are as dangerous as unbounded potentials suggests a selection criterion for quintessence model building: what matters is not whether the potential is bounded below, but how long the field lingers near a positive plateau before the kinetic energy takes over.
The paper treats the fast-roll and slow-roll stages as sequential; a natural extension is to ask whether the transition itself leaves a feature in the scalar spectrum at the scale that left the horizon when $F$ changed, which would be a distinctive signature of a two-stage early history.

Original abstract

We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation. In this regime the standard slow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can be rather long if |m| is sufficiently small. Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe. It may also be responsible for the present stage of accelerated expansion of the universe. We also make two observations of a more general nature. First of all, the universe after a long stage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitter regime even if the cosmological constant is negative. Secondly, the theories with the potentials with a "stable" minimum at V(\phi)<0 in the cosmological background exhibit the same instability as the theories with potentials unbounded from below. This instability leads to the development of singularity with the properties practically independent of V(\phi). However, the development of the instability in some cases may be so slow that the theories with the potentials unbounded from below can describe the present stage of cosmic acceleration even if this acceleration occurs due to the fast-roll inflation.

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, 7,427 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.47 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.16 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation. In this regime the standard slow-roll condition |m^2| << H^2 is violated. Nevertheless, this stage can be rather long if |m| is sufficiently small. Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe. It may also be responsible for the present stage of accelerated expansion of the universe. We also make two observations of a more general nature. First of all, the universe after a long stage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitter regime even if the cosmological constant is negative. Secondly, the theories with the potentials with a "stable" minimum at V(\phi)<0 in the cosmological background exhibit the same instability as the theories with potentials unbounded from below. This instability leads to the development of singularity with the properties practically independent of V(\phi). However, the development of the instability in some cases may be so slow that the theories with the potentials unbounded from below can describe the present stage of cosmic acceleration even if this acceleration occurs due to the fast-roll inflation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that in the simplest theories of spontaneous symmetry breaking one can have a stage of a fast-roll inflation.	contribution 0.55	0.92	0.64	asserted 0.95
s1	In this regime the standard slow-roll condition \|m^2\| << H^2 is violated.	result 0.88	0.46	0.08	asserted 1.00
s2	Nevertheless, this stage can be rather long if \|m\| is sufficiently small.	result 0.99	0.73	0.10	hedged possibility 0.64
s3	Fast-roll inflation can be useful for generating proper initial conditions for the subsequent stage of slow-roll inflation in the very early universe.	interpretation 0.96	0.42	0.22	hedged possibility 0.87
s4	It may also be responsible for the present stage of accelerated expansion of the universe.	interpretation 0.95	0.30	0.12	hedged possibility 1.00
s5	We also make two observations of a more general nature.	contribution 0.74	0.83	0.03	unclear 0.63
s6	First of all, the universe after a long stage of inflation (either slow-roll or fast-roll) cannot reach anti-de Sitter regime even if the cosmological constant is negative.	result 0.99	0.73	0.13	asserted 1.00
s7	Secondly, the theories with the potentials with a "stable" minimum at V(\phi)<0 in the cosmological background exhibit the same instability as the theories with potentials unbounded from below.	result 0.99	0.70	0.13	asserted 1.00
s8	This instability leads to the development of singularity with the properties practically independent of V(\phi).	result 0.97	0.73	0.14	asserted 1.00
s9	However, the development of the instability in some cases may be so slow that the theories with the potentials unbounded from below can describe the present stage of cosmic acceleration even if this acceleration occurs due to the fast-roll inflation.	result 0.54	0.60	0.13	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.91.

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.** Evolution of the scalar field and the scale factor in the model with V (φ) = V0 − m2φ 2 2 . The second model has the standard potential used for the description of spontaneous symmetry breaking, Eq. (2.29), but we added to it a negative cosmological constant Λ < 0 so that V (φ) = λ 4 (φ 2 − v 2 ) 2 + Λ becomes negative in the minimum of V (φ) at φ = v. As we see in

**Figure 2.** Evolution of the scalar field and the scale factor in the model with V (φ) = λ 4 (φ 2−v 2 ) 2+Λ, with Λ < 0. not depend on the choice of the effective potential, unless the potential is exponentially steep. Indeed, let us assume for a second that in our investigation of the singularity one can neglect V (φ), and then check whether such an approximation is self-consistent. In this approximation the energy density is determined not by V (φ) but by φ˙2/2, and the corresponding equation of state is p = ρ. This leads to the collapse of the universe with a(τ ) = τ 1/3 , where τ is th

Figures reproduced from arXiv:hep-th/0110195 by the authors.

References

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A. D. Linde and V. Mukhanov, “Nongaussian isocurvature pert urbations from inﬂation,” Phys. Rev. D 56, 535 (1997) [arXiv:astro-ph/9610219]. – 20 –

View on Pith Pith review arXiv 1997

Forward citations

Cited by 13 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

An analytic approach to non-slow-roll inflation
hep-th 2020-12 conditional novelty 6.0 of 10

Multiple short non-slow-roll epochs during inflation can steepen the small-scale curvature power spectrum to n-1 = 8, twice the single-epoch bound of n-1 = 4.
The attractive behaviour of ultra-slow-roll inflation
astro-ph.CO 2018-06 conditional novelty 6.0 of 10

Ultra-slow-roll inflation is a dynamical attractor whenever the inflaton potential satisfies M_Pl V'' > |V'| while the field rolls downhill, overturning the common claim that this regime is always transient.
Constraining f(R) Gravity as a Scalar Tensor Theory
astro-ph 2006-12 conditional novelty 6.0 of 10

Viable f(R) gravity models that pass solar system tests cannot produce dark energy distinguishable from a cosmological constant, whether they hide the scalar via large mass or via the chameleon effect.
Hilltop Inflation
hep-ph 2005-02 conditional novelty 6.0 of 10

Hilltop inflation, starting near a potential maximum, is shown to be generic, to predict r < 0.002, and to arise naturally from F- and D-term models once Planck-suppressed terms are included.
Cosmology With Negative Potentials
hep-th 2002-02 accept novelty 6.0 of 10

A scalar field potential that dips negative forces the universe to turn from expansion to collapse, and this rules out the simplest version of the cyclic universe scenario.

Show 8 more citing papers

Purely quadratic non-Gaussianity from tachyonic instability: Primordial black holes and scalar-induced gravitational waves
astro-ph.CO 2026-04 unverdicted novelty 5.0 of 10

Purely quadratic non-Gaussianity from tachyonic instability allows narrow curvature spectra to exponentially suppress primordial black hole overproduction via correlation coefficient ρ approaching -1 while retaining s...
Hybrid $\alpha$-attractors, primordial black holes and gravitational wave backgrounds
astro-ph.CO 2022-11 conditional novelty 5.0 of 10

A tiny linear term in the hybrid alpha-attractor potential lets the model keep CMB-compatible tilt while producing a tunable small-scale power peak that seeds primordial black holes and a testable stochastic gravitati...
Inflation after Planck: Judgment Day
astro-ph.CO 2019-02 conditional novelty 5.0 of 10

Working through nine classes of criticism against cosmic inflation in light of the final Planck data, the authors find most objections defused and the paradigm strengthened, with inflation's ability to homogenize gene...
Primordial Black Holes from the inflating curvaton
hep-ph 2012-11 conditional novelty 5.0 of 10

A two-stage inflation model in which the curvaton field inflates can generate primordial black holes with masses between 10^14 and 10^38 grams, with the mass set by the number of curvaton e-foldings.
Non-Gaussianity from Curvatons Revisited
astro-ph.CO 2011-07 conditional novelty 5.0 of 10

For curvatons with non-quadratic potentials, the non-uniform onset of oscillation can drive large non-Gaussianity of either sign, and hilltop pseudo-Nambu-Goldstone curvatons predict 10 <~ f_NL <~ 30.
Extended Slow-Roll Conditions and Rapid-Roll Conditions
astro-ph 2008-07 accept novelty 5.0 of 10

Generalized slow-roll conditions for a non-minimally coupled inflaton are derived, and the scalar and tensor spectral indices are shown to be conformally invariant to first order in slow-roll parameters.
How long before the end of inflation were observable perturbations produced?
astro-ph 2003-05 accept novelty 5.0 of 10

Observable perturbations were produced about 55 e-foldings before the end of inflation in standard cosmology, with a plausible ceiling near 62 and a firm value of 64 for the quartic potential.
Particle physics models of inflation and curvaton scenarios
hep-ph 2010-01 accept novelty 3.0 of 10

The review argues that inflatons and curvatons built from MSSM flat directions are the class most tightly tied to known particle physics, and therefore the class most likely to be falsified by CMB and collider data.

How the literature uses it

40 arXiv papers cite this in 58 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 · 45 sentences without a confident label

Compares with it 3 papers

The above agrees nicely with Ref. [ref] .

Primordial Black Holes from Thermal Inflation · Perturbations from thermal inflation · 0.98
It is worth noticing some analogies between asymptotic states under considerations and fast-roll (or rapid-roll) inflationary states [ref] .

The non-minimal coupling constant and the primordial de Sitter state · $ξ=3/16$ and instability of the initial de Sitter state · 0.94
We should notice some analogies between asymptotic state under consideration and fast-roll (or rapid-roll) inflationary state [ref] .

A new generic and structurally stable cosmological model without singularity · Instability of the initial de Sitter state · 0.94

Cites it as background 7 papers

We emphasize here that it is far from trivial to build a model in which the universe is initially expanding, then contracts, and finally expands again, although each step was relatively well studied (see Refs. [ref] for the turnaround to contraction by a negative potential and Refs. [ref] for the bounce with positive curvature) There are many attempts to realize bouncing or cyclic cosmology by violating the NEC [ref] , but we do not consider such a possibility throughout the paper.

Bouncing Universe from Nothing · Introduction · 0.92
See, for example [ref] , for different cases of `fast roll' inflation, depending on which (and how many) of the slow roll parameters in ([ref]) are actually large during the inflationary period.

On Non-slow Roll Inflationary Regimes · Other non-slow roll regimes · 0.97
They encompass fast roll [ref] and constant rate of roll [ref] models.

A Gravity Dual of Ultra-slow Roll Inflation · Ultra-slow roll inflation · 0.91

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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