Ultra-Slow-Roll Inflation Explained

• Ultra-slow-roll (USR) inflation is a non-attractor regime with a very flat potential where the inflaton is governed primarily by Hubble friction and inertia rather than the potential gradient.
• USR inflation allows for scale-invariant or strongly enhanced scalar spectra, order-unity local non-Gaussianity, and primordial black-hole production with applications in cosmology.
• Key phenomenological implications include enhanced curvature perturbations, stochastic effects, and potential for primordial black-hole formation and gravitational-wave detection

Ultra-slow-roll inflation (USR) is a non-attractor inflationary regime in which the inflaton traverses an exceptionally flat region of its potential and its motion is governed primarily by Hubble friction and inertia rather than by the potential gradient. In the canonical limit, $V_{,\phi}\simeq0$ gives $\ddot\phi+3H\dot\phi\simeq0$, with $\dot\phi\propto a^{-3}$, $\epsilon_1\propto a^{-6}$, and $\epsilon_2\simeq-6$. The defining perturbative feature is that the curvature perturbation is not conserved outside the Hubble radius: its would-be decaying mode becomes a growing mode, typically $\zeta\propto a^3$. This behavior permits scale-invariant or strongly enhanced scalar spectra, order-unity local non-Gaussianity, primordial-black-hole production, and departures from standard single-field consistency relations. USR is nevertheless not a unique model but a class of dynamical regimes, encompassing canonical, generalized constant-rate, noncanonical, stochastic, warm-inflation, holographic, and composite-sector realizations.

1. Background dynamics and defining properties

For a canonical scalar field in a spatially flat Friedmann–Lemaître–Robertson–Walker background,

$ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$

the homogeneous equations are

$3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$

and

$\ddot\phi+3H\dot\phi+V_{,\phi}=0.$

The first Hubble-flow parameter is

$$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$

and inflation requires $\ddot\phi+3H\dot\phi\simeq0$0. The higher horizon-flow parameters satisfy

$\ddot\phi+3H\dot\phi\simeq0$1

In ordinary slow roll, the acceleration term is neglected:

$\ddot\phi+3H\dot\phi\simeq0$2

so the inflaton velocity is approximately a local function of the field value. The slow-roll trajectory is consequently an attractor in the $\ddot\phi+3H\dot\phi\simeq0$3 phase space, and initial velocity information is rapidly erased. Conventional slow roll requires $\ddot\phi+3H\dot\phi\simeq0$4 and $\ddot\phi+3H\dot\phi\simeq0$5.

USR arises when the potential slope becomes sufficiently small that the inherited inflaton velocity cannot immediately adjust to the formal slow-roll value. Neglecting $\ddot\phi+3H\dot\phi\simeq0$6 gives

$\ddot\phi+3H\dot\phi\simeq0$7

whose solution is

$\ddot\phi+3H\dot\phi\simeq0$8

The kinetic density and first Hubble-flow parameter therefore evolve as

$\ddot\phi+3H\dot\phi\simeq0$9

while

$\dot\phi\propto a^{-3}$0

The expansion remains approximately de Sitter when the potential dominates the energy density, $\dot\phi\propto a^{-3}$1.

The term “ultra-slow” refers primarily to the rapid decrease of the velocity, not necessarily to an initially small instantaneous velocity. If the inflaton reaches a flat region with substantial residual kinetic energy, it can cross the region more rapidly than the formal slow-roll solution, which would require an extremely small velocity. USR ends when the potential slope becomes sufficiently large to balance Hubble friction again, restoring ordinary slow roll, or when inflation terminates through another mechanism. The physical origin and duration of this transient behavior are analyzed in “Ultra-slow-roll inflation demystified” (Dimopoulos, 2017).

A useful generalized class imposes

$\dot\phi\propto a^{-3}$2

with constant $\dot\phi\propto a^{-3}$3. In the regime $\dot\phi\propto a^{-3}$4,

$\dot\phi\propto a^{-3}$5

Ordinary slow roll corresponds approximately to $\dot\phi\propto a^{-3}$6, exact canonical USR to $\dot\phi\propto a^{-3}$7, and generalized USR to values near $\dot\phi\propto a^{-3}$8. Potentials supporting this family can be obtained by imposing the corresponding background relation on the scalar equations (Martin et al., 2012).

2. Attractor structure, stability, and phase space

USR differs from slow roll because the field value alone does not specify the state. The trajectory depends on both $\dot\phi\propto a^{-3}$9, and distinct initial velocities can yield distinct velocities at the same field value. In this sense, USR retains initial-condition dependence and is not a one-dimensional slow-roll attractor.

The statement that USR is always transient or always repulsive is not universally valid. A phase-space analysis using the field-acceleration variable

$\epsilon_1\propto a^{-6}$0

shows that USR corresponds to $\epsilon_1\propto a^{-6}$1. Defining $\epsilon_1\propto a^{-6}$2, the USR-inflationary regime requires

$\epsilon_1\propto a^{-6}$3

where

$\epsilon_1\propto a^{-6}$4

For a field rolling down a convex potential, USR is stable when

$\epsilon_1\propto a^{-6}$5

whereas for a field rolling up a concave potential the corresponding condition is

$\epsilon_1\propto a^{-6}$6

Near a sufficiently flat inflection point, $\epsilon_1\propto a^{-6}$7 can decrease along a rolling-down trajectory while $\epsilon_1\propto a^{-6}$8, allowing trajectories to approach USR. This establishes that USR can be an attractor region even though $\epsilon_1\propto a^{-6}$9 is not uniquely determined by $\epsilon_2\simeq-6$0. Attraction in this context means convergence of a finite-width set of trajectories toward the USR regime; it does not imply complete erasure of initial velocity information (Pattison et al., 2018).

The duration of USR is controlled by the relation between the incoming kinetic density and the local slow-roll kinetic density. The latter is

$\epsilon_2\simeq-6$1

If

$\epsilon_2\simeq-6$2

ordinary slow roll can remain valid immediately. If

$\epsilon_2\simeq-6$3

the potential slope cannot balance Hubble friction and USR occurs until the kinetic energy redshifts to the slow-roll value. Under approximately constant $\epsilon_2\simeq-6$4,

$\epsilon_2\simeq-6$5

For a linear potential $\epsilon_2\simeq-6$6, the velocity contains a transient term and a terminal slow-roll term,

$\epsilon_2\simeq-6$7

A sizeable USR phase requires

$\epsilon_2\simeq-6$8

and its approximate duration is

$\epsilon_2\simeq-6$9

Exactly flat potentials do not end USR through the potential slope; an external exit mechanism is then required.

Inflection-point inflation illustrates the distinction between potential flatness and dynamical USR. A very small $\zeta\propto a^3$0 does not guarantee USR: sufficiently small incoming kinetic energy can yield immediate slow roll. Conversely, an inflaton arriving at the inflection region with substantial velocity can overshoot the plateau in a finite number of e-folds, invalidating a slow-roll estimate based on the divergent integral $\zeta\propto a^3$1 (Dimopoulos, 2017).

3. Curvature perturbations and non-attractor evolution

The Mukhanov–Sasaki variable $\zeta\propto a^3$2 satisfies

$\zeta\propto a^3$3

with

$\zeta\propto a^3$4

Equivalently, the long-wavelength curvature perturbation obeys

$\zeta\propto a^3$5

For approximately constant $\zeta\propto a^3$6, the two super-Hubble solutions are

$\zeta\propto a^3$7

In ordinary slow roll, $\zeta\propto a^3$8, so the second mode decays as $\zeta\propto a^3$9 and $ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$0 freezes. In USR, $ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$1, giving

$ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$2

The mode conventionally called “decaying” by continuity with slow roll therefore grows rapidly. More generally, super-Hubble growth occurs for

$ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$3

The growth is a consequence of the rapid decrease of $ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$4:

$ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$5

in canonical USR, so

$ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$6

Because $ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$7 continues evolving after horizon exit, the final power spectrum must be evaluated after the USR growth has ended, not at horizon crossing. For canonical USR,

$ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$8

during the super-Hubble growth phase. This invalidates the ordinary slow-roll horizon-crossing estimate as a final prediction.

For the generalized constant-$ds^2=-dt^2+a^2(t)d\mathbf{x}^2,$9 class, the scalar spectral index is

$3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$0

On the USR branch,

$3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$1

so exact scale invariance occurs at $3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$2. Inserting $3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$3 into the usual slow-roll expression would incorrectly give $3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$4; the failure arises because the curvature perturbation is not conserved outside the horizon (Martin et al., 2012).

Piecewise-constant large-$3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$5 models show that the amplification depends on which modes enter the non-attractor phase. Modes already outside the horizon when USR begins can receive an enhancement proportional to

$3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$6

which becomes $3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$7 for $3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$8. A causal analysis of the equal-time two-point function constrains the growth rate in the idealized toy model to approximately

$3M_{\rm Pl}^2H^2=\frac12\dot\phi^2+V(\phi),$9

in the convention $\ddot\phi+3H\dot\phi+V_{,\phi}=0.$0 (Cheng et al., 2018).

4. Bispectrum and consistency relations

Canonical USR generates a local bispectrum even when the scalar power spectrum is scale invariant. Using the Maldacena cubic-action formalism, the dominant contribution arises from the nonlinear field redefinition involving $\ddot\phi+3H\dot\phi+V_{,\phi}=0.$1. This term is important because $\ddot\phi+3H\dot\phi+V_{,\phi}=0.$2 is not negligible outside the horizon in USR.

For the generalized constant-$\ddot\phi+3H\dot\phi+V_{,\phi}=0.$3 family, the local nonlinearity parameter is

$\ddot\phi+3H\dot\phi+V_{,\phi}=0.$4

At exact scale invariance, $\ddot\phi+3H\dot\phi+V_{,\phi}=0.$5 and $\ddot\phi+3H\dot\phi+V_{,\phi}=0.$6, giving

$\ddot\phi+3H\dot\phi+V_{,\phi}=0.$7

The bispectrum has local structure for arbitrary triangle configurations in the calculation, rather than only in the squeezed limit. The result differs from the conventional attractor single-field relation,

$\ddot\phi+3H\dot\phi+V_{,\phi}=0.$8

The apparent violation does not contradict the standard theorem under its usual assumptions. The consistency relation assumes a single-field attractor in which the long-wavelength curvature perturbation becomes an adiabatic coordinate rescaling and remains constant outside the horizon. USR evades the relation because it is non-attractor and retains a growing super-Hubble mode. The relation is therefore not violated by an ordinary attractor single-field model; rather, one of its essential hypotheses is absent (Martin et al., 2012).

Stochastic treatments reproduce the classical USR value $\ddot\phi+3H\dot\phi+V_{,\phi}=0.$9. In perturbative stochastic $$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$0, the leading correction is

$$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$1

where $$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$2 is proportional to the final curvature power. The fractional corrections are of order the power spectrum in the regime considered (Firouzjahi et al., 2018).

Finite USR phases can restore the usual squeezed consistency relation after the curvature perturbation freezes. Numerical studies of finite slow-roll-to-USR-to-slow-roll models find

$$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$3

to good accuracy over most scales, despite potentially large equilateral non-Gaussianity near sharp spectral features. The distinction between pure, indefinitely extended USR and finite USR followed by an attractor phase is therefore essential (Ragavendra et al., 2020).

5. Stochastic dynamics and quantum diffusion

USR is particularly sensitive to stochastic effects because the classical drift decays as $$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$4 while the field fluctuation per e-fold remains of order $$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$5. A phase-space stochastic description must retain both $$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$6 and the normalized momentum $$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$7:

$$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$8

in the idealized flat-potential limit. The leading noise correlations are

$$\epsilon_1\equiv-\frac{\dot H}{H^2} =\frac{\dot\phi^2}{2M_{\rm Pl}^2H^2},$$9

while momentum noise and cross-correlations are negligible in the approximation used.

A stochastic $\ddot\phi+3H\dot\phi\simeq0$00 formulation identifies

$\ddot\phi+3H\dot\phi\simeq0$01

where $\ddot\phi+3H\dot\phi\simeq0$02 is the random first-passage time to an exit surface. In a finite USR well with absorbing lower and reflecting upper boundaries, the first-passage distribution interpolates between a drift-dominated regime and a diffusion-dominated regime. Classical drift gives a sharply peaked duration, whereas diffusion produces an exponential tail,

$\ddot\phi+3H\dot\phi\simeq0$03

The corresponding full probability distribution is relevant for PBH formation because PBH abundance depends on the far tail rather than only on the variance (Pattison et al., 2021).

A gravitationally consistent long-wavelength treatment imposes the momentum constraint

$\ddot\phi+3H\dot\phi\simeq0$04

On a nondegenerate Hamilton–Jacobi branch, this constraint prevents the momentum from acquiring an independent stochastic source: all super-Hubble patches remain on one global Hamilton–Jacobi trajectory. When the field reaches the degenerate de Sitter surface

$\ddot\phi+3H\dot\phi\simeq0$05

it can subsequently diffuse freely. This produces a “stochastic conveyor belt”: trajectories first move along a unique nondegenerate USR branch and are then deposited onto a diffusive de Sitter branch.

If the exit surface is reached before the asymptotic zero-momentum point, all trajectories can exit with finite moments of $\ddot\phi+3H\dot\phi\simeq0$06. If diffusion carries trajectories onto an infinite flat half-line, the exit-time distribution has a normalizable $\ddot\phi+3H\dot\phi\simeq0$07 tail, but its positive moments diverge. A finite flat region or another boundary regulates this behavior (Prokopec et al., 2019).

The stochastic results do not imply that quantum diffusion generically enhances the power spectrum dramatically. In perturbative regimes, the fractional correction to the power spectrum is of order the final curvature power, and the absolute correction is of order $\ddot\phi+3H\dot\phi\simeq0$08. Quantum diffusion becomes structurally important when the classical drift is extremely small, but its quantitative effects depend on the field range, exit boundary, and duration of USR.

6. Phenomenology, extensions, and theoretical realizations

The most prominent phenomenological application of USR is the enhancement of small-scale curvature perturbations. A finite USR episode near an approximate inflection point can generate a spectrum that rises sharply, develops a localized peak, and later decreases when slow roll is restored. The enhanced fluctuations can form PBHs after horizon re-entry and source scalar-induced secondary gravitational waves. Canonical models and reconstructed $\ddot\phi+3H\dot\phi\simeq0$09 scenarios can produce PBHs with asteroid-scale masses and induced gravitational-wave signals potentially relevant to PTAs, LISA, BBO, DECIGO, Einstein Telescope, and advanced LIGO/Virgo. The predicted abundance is highly sensitive to the collapse threshold, window function, reheating history, non-Gaussian density tails, and the scalar-spectrum shape (Ragavendra et al., 2020).

Quantum loop corrections impose an additional model-building constraint. If the slow-roll-to-USR transition is sharp, loop-induced effective masses can regenerate a small velocity for long-wavelength CMB modes. USR then amplifies that velocity, potentially overproducing large-scale power. A representative condition is

$\ddot\phi+3H\dot\phi\simeq0$10

where $\ddot\phi+3H\dot\phi\simeq0$11 measures the USR amplification. PBH formation requires large $\ddot\phi+3H\dot\phi\simeq0$12, whereas CMB safety requires the loop-induced source to remain sufficiently small. This tension is model dependent and does not establish a universal one-loop no-go theorem for single-field USR PBH formation (Cheng et al., 2023).

Noncanonical kinetic theories generalize the canonical value $\ddot\phi+3H\dot\phi\simeq0$13. In G-inflation, with

$\ddot\phi+3H\dot\phi\simeq0$14

one can obtain

$\ddot\phi+3H\dot\phi\simeq0$15

with $\ddot\phi+3H\dot\phi\simeq0$16. An early noncanonical phase followed by canonical USR can generate a broken power-law spectrum, blue on the largest scales and nearly scale invariant on smaller scales, with oscillations around the transition. In potential-dominated models, stability enforces $\ddot\phi+3H\dot\phi\simeq0$17 and therefore a red tensor tilt. A stable blue tensor spectrum is possible instead in kinetically driven de Sitter-attractor models with controlled null-energy-condition violation (Hirano et al., 2016).

USR has also been embedded in gauge/gravity duality. A type IIB supergravity construction with de Sitter slicing realizes a glueball-driven USR regime with

$\ddot\phi+3H\dot\phi\simeq0$18

where the inflaton is a scalar glueball mode rather than a fundamental scalar or D-brane position. The analytic solution has

$\ddot\phi+3H\dot\phi\simeq0$19

and provides a holographic realization of the characteristic $\ddot\phi+3H\dot\phi\simeq0$20 behavior. The construction is perturbative and does not compute a complete four-dimensional perturbation spectrum, reheating process, or compactification (Anguelova, 2015).

Warm inflation introduces further constraints because USR must coexist with a thermalized radiation bath. With dissipative coefficient $\ddot\phi+3H\dot\phi\simeq0$21,

$\ddot\phi+3H\dot\phi\simeq0$22

For temperature-only dissipation, the USR phase drives the temperature downward rapidly and destroys thermal equilibrium. A field-dependent choice such as

$\ddot\phi+3H\dot\phi\simeq0$23

can compensate the temperature evolution for a short interval, allowing a transient warm-USR phase that returns smoothly to warm slow roll. Perturbations during warm USR remain to be analyzed (Biswas et al., 2023).

Other extensions include composite pseudo-Nambu–Goldstone boson models, in which a periodic non-minimal coupling flattens the Einstein-frame potential and produces a near-inflection-point USR phase. Such models have been used to obtain PBH masses around $\ddot\phi+3H\dot\phi\simeq0$24–$\ddot\phi+3H\dot\phi\simeq0$25 and induced gravitational waves in a frequency gap between standard interferometers and resonant-cavity experiments, subject to assumptions about the survival of ultra-light PBHs (Merchand, 17 Oct 2025). USR transitions can also excite Affleck–Dine spectator condensates, linking PBH production and baryogenesis through the same change in the inflaton rolling rate (Wu et al., 2021).

A further modification arises when a seed black hole is present in a de Sitter background. In a Schwarzschild–de Sitter geometry, the transition from slow roll to USR excites a combination of de Sitter and black-hole quasi-normal modes. Small black holes produce predominantly damped de Sitter-like evolution, while sufficiently large black holes produce oscillatory Schwarzschild-like ringing. The resulting modification of the scalar background may alter the curvature spectrum and PBH abundance, but a gauge-invariant perturbation calculation on the black-hole background has not yet been carried out (Croney et al., 2024).

7. Conceptual status and open problems

USR is a mathematically consistent and physically distinctive non-attractor regime, but its interpretation depends strongly on the model and on the duration of the phase. Exact canonical USR gives the simple relations

$\ddot\phi+3H\dot\phi\simeq0$26

These relations should not be conflated with all large-$\ddot\phi+3H\dot\phi\simeq0$27 phases, generalized constant-rate solutions, warm-USR, or noncanonical USR.

Several commonly used statements require qualification. USR is not necessarily always repulsive: it can be stable near suitable convex or concave regions of the potential. It is not necessarily limited to a few e-folds: broad inflection regions can support many classical e-folds, although quantum diffusion can eventually dominate. Extreme potential flatness does not by itself guarantee USR, because the incoming kinetic density is decisive. Conversely, the usual horizon-crossing power-spectrum formula is not reliable during a non-attractor phase because $\ddot\phi+3H\dot\phi\simeq0$28 evolves outside the horizon.

The principal unresolved issues concern the transition into and out of USR, nonlinear stability, quantum diffusion, loop backreaction, reheating, non-Gaussian PBH abundance, and the ultraviolet completion of very flat potentials. Pure USR lasting throughout inflation is generally an idealized laboratory: it yields analytically transparent mode evolution but does not reproduce the observed spectral tilt, naturally terminate inflation, or generically provide realistic amplitudes for magnetic fields or PBHs. Finite USR embedded between slow-roll phases is more relevant phenomenologically but introduces sensitivity to transition profiles and matching conditions.

The fate of primordial quantum signatures is also distinct from the fate of the classical growing perturbation. Although the mode called “decaying” during USR grows as $\ddot\phi+3H\dot\phi\simeq0$29, after the transition to radiation domination it maps predominantly into the late-time growing mode. Under Gaussian, linear, single-field evolution with a Bunch–Davies initial state, the residual late-time decaying-mode amplitude remains suppressed by approximately $\ddot\phi+3H\dot\phi\simeq0$30 after imposing the observed scalar amplitude. Thus USR changes the intermediate squeezing and mode decomposition without generically leaving an observable late-time noncommuting mode (Putter et al., 2019).

USR is therefore best regarded as a broad non-attractor framework rather than a single inflationary model. Its significance lies in exposing the assumptions behind curvature conservation and single-field consistency relations, supplying mechanisms for localized small-scale power enhancement, and providing a testing ground for stochastic inflation, nonlinear perturbation theory, modified kinetic sectors, holographic constructions, warm dynamics, magnetogenesis, PBH formation, secondary gravitational waves, and inflationary model building.