Non-Attractor Inflation Dynamics

Updated 27 August 2026

Non-attractor inflation retains initial inflaton momentum, allowing superhorizon amplification of curvature perturbations and violations of standard single-field consistency relations.
Curvature perturbations during non-attractor inflation can lead to superhorizon growth with observable implications such as enhanced small-scale power and potential primordial black holes (PBH) signals.
Transitions from non-attractor to attractor phases are crucial, affecting the bispectrum and local observables. Local non-Gaussianity is influenced by the nature of these transitions, with sharp non-adiabatic transitions potentially producing significant signals.
Find recent papers about the application of non-attractor inflation in generating primordial black holes.
understand the methodologies involved in detecting and quantifying superhorizon amplification in non-attractor inflation scenarios.
Explore how different sound-speed values impact non-local and local structure in compressed phase space.
What are the primary observable differences between an infinitely extended non-attractor QCS range compared to one of limited extent?
How can primordial black hole signatures from non-attractor inflation be distinguished from those generated by other mechanisms?
Find recent papers about the stochastic dynamics of non-Attractor Inflation and their impact on Emerging Opportunities.

Non-attractor inflation is a class of single-field inflationary dynamics in which the homogeneous trajectory is not uniquely determined by the inflaton value: the field velocity remains an independent initial datum. Consequently, the curvature perturbation need not become conserved outside the horizon. The mode conventionally identified as decaying in attractor inflation can instead grow, producing superhorizon amplification of $\mathcal R$ or $\zeta$ , violations of the usual single-field squeezed-limit consistency relation, enhanced small-scale scalar power, and potentially large primordial black-hole (PBH) and scalar-induced gravitational-wave (SIGW) signals. The canonical ultra-slow-roll (USR) phase, characterized by $\epsilon\propto a^{-6}$ , $\eta=-6$ , and $\zeta\propto a^3$ , is the principal example. Non-attractor inflation is necessarily sensitive to the transition into a subsequent attractor phase, and its observable predictions depend on the evolution of perturbations throughout that transition.

1. Dynamical definition and background evolution

In an attractor phase, the background trajectory in phase space is effectively one-dimensional. For a canonical scalar field, the velocity is determined by the field position,

$\dot\phi=\dot\phi_{\rm att}(\phi),$

and different initial conditions converge toward the same trajectory. In slow roll,

$3H\dot\phi\simeq -V_{,\phi},$

so the influence of the initial velocity is rapidly erased.

A non-attractor phase instead retains sensitivity to the initial inflaton momentum. The phase space is described by $(\phi,\dot\phi)$ , and equal values of $\phi$ can correspond to different future evolutions. The number of e-folds must consequently be regarded as a function of phase-space data,

$N=N(\phi,\dot\phi;\phi_*),$

rather than solely as $\zeta$ 0. This is the background origin of the failure of the usual single-clock argument.

For canonical Einstein gravity,

$\zeta$ 1

and

$\zeta$ 2

The second Hubble-flow parameter is

$\zeta$ 3

The curvature perturbation has an anti-damped nonconstant superhorizon mode whenever

$\zeta$ 4

Indeed, its long-wavelength equation is

$\zeta$ 5

so that

$\zeta$ 6

If $\zeta$ 7 decreases, the nonconstant mode grows.

USR corresponds to an approximately flat potential,

$\zeta$ 8

for which

$\zeta$ 9

Thus

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

and the superhorizon solution is

$\epsilon\propto a^{-6}$ 1

The rolling USR solution is a transient rather than a stable endpoint: the system evolves toward the constant-field de Sitter branch. Large- $\epsilon\propto a^{-6}$ 2 constant-roll solutions likewise do not generally define new attractors. Retaining the full second-order background dynamics reveals a parameter duality,

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

which exchanges the slow-roll and USR branches. The apparent large- $\epsilon\propto a^{-6}$ 4 attractor is mapped to the ordinary slow-roll attractor, while the genuinely large- $\epsilon\propto a^{-6}$ 5 rolling solution is transient (1804.01927).

For a general $\epsilon\propto a^{-6}$ 6 theory,

$\epsilon\propto a^{-6}$ 7

the sound speed is

$\epsilon\propto a^{-6}$ 8

An example with an $\epsilon\propto a^{-6}$ 9-dominated kinetic sector gives

$\eta=-6$ 0

The non-attractor background can then be realized with approximately constant $\eta=-6$ 1, $\eta=-6$ 2, and $\eta=-6$ 3, followed by a conventional attractor phase in which $\eta=-6$ 4 becomes conserved (Chen et al., 2013).

2. Perturbations, scale invariance, and superhorizon growth

The quadratic action for a single scalar curvature perturbation is

$\eta=-6$ 5

with

$\eta=-6$ 6

For the canonical case, $\eta=-6$ 7 and $\eta=-6$ 8. Introducing the Mukhanov variable

$\eta=-6$ 9

gives

$\zeta\propto a^3$ 0

On scales satisfying $\zeta\propto a^3$ 1, the gradient term is negligible: $\zeta\propto a^3$ 2 Therefore,

$\zeta\propto a^3$ 3

In an attractor phase the integral decays, whereas in USR,

$\zeta\propto a^3$ 4

and the integral grows as $\zeta\propto a^3$ 5. Consequently,

$\zeta\propto a^3$ 6

The power spectrum evolves outside the horizon as

$\zeta\propto a^3$ 7

Its final value therefore cannot be evaluated at horizon exit; it must be evaluated after the non-attractor phase and its transition into an attractor.

For approximately constant $\zeta\propto a^3$ 8, $\zeta\propto a^3$ 9, and $\dot\phi=\dot\phi_{\rm att}(\phi),$ 0, the mode functions are Hankel functions. In the constant-sound-speed realization,

$\dot\phi=\dot\phi_{\rm att}(\phi),$ 1

and the spectral index is

$\dot\phi=\dot\phi_{\rm att}(\phi),$ 2

Scale invariance requires

$\dot\phi=\dot\phi_{\rm att}(\phi),$ 3

The Bunch–Davies vacuum is compatible with this spectrum even though the curvature perturbation subsequently grows outside the horizon (Chen et al., 2013).

Allowing a varying sound speed produces two nearly scale-invariant EFT branches. Defining

$\dot\phi=\dot\phi_{\rm att}(\phi),$ 4

the branches are

$\dot\phi=\dot\phi_{\rm att}(\phi),$ 5

For the first branch, $\dot\phi=\dot\phi_{\rm att}(\phi),$ 6 is conserved: $\dot\phi=\dot\phi_{\rm att}(\phi),$ 7 This is a non-attractor background because $\dot\phi=\dot\phi_{\rm att}(\phi),$ 8 evolves rapidly, but its perturbations behave as an attractor mode. The standard squeezed-limit consistency relation therefore holds.

For the second branch,

$\dot\phi=\dot\phi_{\rm att}(\phi),$ 9

so, for $3H\dot\phi\simeq -V_{,\phi},$ 0,

$3H\dot\phi\simeq -V_{,\phi},$ 1

The $3H\dot\phi\simeq -V_{,\phi},$ 2 limit is USR. This branch violates the standard squeezed-limit relation and generates local non-Gaussianity. The distinction demonstrates that non-attractor background evolution alone does not guarantee consistency-relation violation; the decisive issue is whether the curvature perturbation evolves on superhorizon scales (Akhshik et al., 2015).

3. Squeezed-limit non-Gaussianity

For attractor single-field inflation, a constant long-wavelength mode is locally equivalent to a spatial dilation. The squeezed bispectrum obeys Maldacena’s relation,

$3H\dot\phi\simeq -V_{,\phi},$ 3

$3H\dot\phi\simeq -V_{,\phi},$ 4

For a nearly scale-invariant spectrum this is small.

In USR, the long mode evolves,

$3H\dot\phi\simeq -V_{,\phi},$ 5

and cannot be removed by a time-independent spatial rescaling. A comoving-coordinate calculation gives

$3H\dot\phi\simeq -V_{,\phi},$ 6

corresponding to

$3H\dot\phi\simeq -V_{,\phi},$ 7

For the noncanonical $3H\dot\phi\simeq -V_{,\phi},$ 8 model with constant $3H\dot\phi\simeq -V_{,\phi},$ 9, the full comoving-gauge result is

$(\phi,\dot\phi)$ 0

It reduces to $(\phi,\dot\phi)$ 1 for $(\phi,\dot\phi)$ 2 and becomes parametrically large for $(\phi,\dot\phi)$ 3 (Chen et al., 2013).

The origin of the local shape is superhorizon evolution. In the cubic action, interactions such as

$(\phi,\dot\phi)$ 4

would ordinarily be associated with equilateral non-Gaussianity. During non-attractor evolution,

$(\phi,\dot\phi)$ 5

so these operators become effectively local in the gradient expansion. Spatial-gradient terms are suppressed by powers of $(\phi,\dot\phi)$ 6.

The same result can be obtained with the $(\phi,\dot\phi)$ 7 formalism, provided it is formulated in phase space rather than as a function of the field alone. In the canonical case,

$(\phi,\dot\phi)$ 8

The integration constant $(\phi,\dot\phi)$ 9 carries the dependence on the initial trajectory, and expansion of the final number of e-folds gives

$\phi$ 0

For general sound speed, the phase-space calculation gives the same result as the full comoving-gauge in-in computation (Chen et al., 2013).

The $\phi$ 1 EFT branch behaves differently. Its field-redefinition contribution vanishes because $\phi$ 2, and its squeezed bispectrum is suppressed: $\phi$ 3 Its bispectrum interpolates between equilateral and folded configurations. The equilateral amplitude is

$\phi$ 4

and is below unity in the parameter range considered (Akhshik et al., 2015).

4. Transitions, local observables, and consistency-relation controversies

A non-attractor phase cannot generally constitute a complete inflationary history. It must be followed by an attractor phase in which the curvature perturbation freezes. The final non-Gaussianity is therefore controlled not only by the non-attractor phase but also by the transition.

For canonical USR followed by slow roll, smooth transitions can erase the order-one comoving value $\phi$ 5. The dominant transition interaction is

$\phi$ 6

For a smooth transition, contributions cancel to leading order, leaving a result suppressed by slow-roll parameters. The same cancellation follows after integration by parts because the effective coupling is related to $\phi$ 7 and is small for a sufficiently smooth transition (Cai et al., 2017).

Sharp transitions behave differently. Defining a sharpness parameter $\phi$ 8, the canonical result is

$\phi$ 9

For an immediate perturbation freeze-out, $N=N(\phi,\dot\phi;\phi_*),$ 0,

$N=N(\phi,\dot\phi;\phi_*),$ 1

For the instantaneous background transition $N=N(\phi,\dot\phi;\phi_*),$ 2,

$N=N(\phi,\dot\phi;\phi_*),$ 3

Thus an abrupt change in the background does not by itself imply immediate freeze-out of the perturbations. The original USR result is recovered only in the extremal sharp limit in which the perturbations freeze effectively immediately. The usual consistency relation may remain violated even when the final $N=N(\phi,\dot\phi;\phi_*),$ 4 is small, because the curvature perturbation was not conserved during the earlier evolution (Cai et al., 2017).

The interpretation of the squeezed bispectrum also depends on the observable being defined. In comoving coordinates, the evolving long mode produces a large modulation of short modes. Conformal Fermi Coordinates (CFC), constructed around a freely falling observer, remove leading coordinate effects associated with spatial dilation and time reparametrization. For canonical single-field inflation with adiabatic long-mode evolution, the locally observable squeezed bispectrum is argued to satisfy

$N=N(\phi,\dot\phi;\phi_*),$ 5

In USR, the CFC time shift cancels the comoving-coordinate contribution proportional to the time derivative of the short-scale power spectrum (Bravo et al., 2017).

A related soft-theorem analysis gives a unified comoving-coordinate expression containing Maldacena’s dilation term and additional terms involving $N=N(\phi,\dot\phi;\phi_*),$ 6. It concludes that large observable local non-Gaussianity is not generic for adiabatic non-attractor evolution; sharp non-adiabatic transitions are required for a lasting local signal (Bravo et al., 2020).

This conclusion is not universal across all treatments. A separate cosmological-soft-theorem analysis emphasizes that a realistic USR-to-slow-roll transition involves two phase-space variables, $N=N(\phi,\dot\phi;\phi_*),$ 7 and $N=N(\phi,\dot\phi;\phi_*),$ 8. The long mode can perturb the momentum carried into the transition, and the final short-scale power can depend on that momentum. Parameterizing this dependence by

$N=N(\phi,\dot\phi;\phi_*),$ 9

and introducing a transition-response factor $\zeta$ 00, the late-time squeezed signal becomes

$\zeta$ 01

Thus the final squeezed bispectrum need not vanish after the system reaches slow roll (Suyama et al., 2021).

These results distinguish several notions that are sometimes conflated:

Comoving-coordinate bispectrum: can be large during USR and can yield $\zeta$ 02.
Locally observable bispectrum: may be suppressed by CFC coordinate corrections.
Transition-generated physical signal: can survive when the transition is non-adiabatic and depends on the phase-space momentum.
Final bispectrum after a smooth transition: can be negligible in canonical models, while remaining parametrically large in suitable noncanonical models.

5. Transient spectra, PBHs, and induced gravitational waves

Non-attractor phases are frequently used to amplify curvature perturbations over a finite range of scales. A transient USR stage can be embedded between an initial slow-roll phase and a final attractor phase. The pump field $\zeta$ 03 decreases during the non-attractor interval and later increases again, converting the growing superhorizon mode into a final constant perturbation.

A gradient expansion provides a systematic description. The growing solution is expanded as

$\zeta$ 04

with recursive contributions controlled by integrals containing inverse powers of $\zeta$ 05. The final perturbation can be written in terms of a transfer coefficient,

$\zeta$ 06

so that

$\zeta$ 07

The transfer coefficient contains $\zeta$ 08 and $\zeta$ 09 terms. Their interference can produce a dip when

$\zeta$ 10

The imaginary component prevents the spectrum from vanishing exactly.

Immediately above the dip, the $\zeta$ 11 term can dominate: $\zeta$ 12 giving a transient slope

$\zeta$ 13

The slope can then pass through

$\zeta$ 14

as different gradient terms become dominant. Near the peak, realistic intermediate phases generally yield a gentler growth with

$\zeta$ 15

After the peak, the return to an attractor phase produces a mild decay and possibly damped oscillations (Özsoy et al., 2019).

The same transient structure is relevant to PBHs. PBH production requires a small-scale curvature spectrum amplified by many orders of magnitude relative to CMB scales. The detailed peak shape, rather than only its maximum amplitude, controls the mass function. The transient $\zeta$ 16 regime is important for the buildup of power, but the spectrum near the peak is more directly relevant to collapse. Non-Gaussianity can broaden the range of smoothing scales with appreciable PBH formation probability, shift the scale at which the probability is largest, and modify the inferred mass distribution. In one transient-USR example, the second-highest power-spectrum peak produces more PBHs than the highest peak because it has a larger effective non-Gaussian amplitude, despite having slightly smaller power (Namjoo et al., 11 Dec 2025).

Large scalar perturbations also source SIGWs at second order. Schematically,

$\zeta$ 17

The resulting frequency profile depends on the complete scalar spectrum, including its slopes, width, oscillations, and post-peak decay. For realistic spectra generated by transient non-attractor evolution, the infrared SIGW slope can differ from the $\zeta$ 18 behavior of simplified sharply peaked spectra, and the post-peak scalar decay generates an extended ultraviolet tail (Özsoy et al., 2019).

A three-phase model with initial slow roll, intermediate non-attractor evolution, and final slow roll has been studied in connection with PTA-frequency SIGWs. The scalar amplification scales approximately as

$\zeta$ 19

where $\zeta$ 20 is the duration of the non-attractor phase. The transition sharpness $\zeta$ 21 modifies the normalization, dip position, and oscillatory structure. Selected parameter choices, especially lower sound speeds such as $\zeta$ 22 and $\zeta$ 23, give qualitative comparisons with NANOGrav distributions, but these comparisons are not statistical detections or definitive evidence that the PTA signal is primordial (Nassiri-Rad et al., 2023).

The full bispectrum has scale-dependent geometry. Non-attractor evolution tends to produce squeezed correlations, whereas sharp transitions and particle production preferentially generate equilateral correlations around the scales affected by the transition. The bispectrum may therefore be strongly equilateral near a power-spectrum peak while retaining squeezed behavior on scales governed by superhorizon non-attractor evolution. A scale-dependent local-like estimator can approximate the full bispectrum over relevant scales, but a constant $\zeta$ 24 is generally inadequate (Namjoo et al., 11 Dec 2025).

6. Stochastic dynamics, tensor modes, and field-range constraints

Stochastic non-attractor evolution

Because the velocity is an independent variable, stochastic inflation in a non-attractor phase must initially be formulated in phase space. For shift-symmetric $\zeta$ 25 theories, the long-wavelength variables are $\zeta$ 26 with $\zeta$ 27. The classical velocity obeys

$\zeta$ 28

$\zeta$ 29

Quantum fluctuations entering at the sound horizon generate a Langevin equation for the coarse-grained field. In the strict superhorizon limit, the velocity noise is suppressed, and the field obeys an effectively one-dimensional stochastic equation with diffusion coefficient

$\zeta$ 30

The corresponding Fokker–Planck equation contains deterministic drift from the decaying classical velocity and diffusion from the field noise.

In the weak-diffusion regime, characterized by a perturbative parameter $\zeta$ 31, the stochastic corrections to the mean number of e-folds and the curvature power spectrum are fractional corrections of order the classical power spectrum: $\zeta$ 32 Thus superhorizon growth does not by itself imply order-one stochastic corrections. In a large-diffusion regime, quantum kicks dominate the classical drift, boundary conditions become essential, and first-passage observables can depend sensitively on absorbing or reflecting boundaries. The resulting power spectrum can scale as $\zeta$ 33 rather than the usual $\zeta$ 34 expression (Firouzjahi et al., 2020).

Tensor non-attractor evolution

Non-attractor behavior can also occur in the tensor sector. In Horndeski gravity, the quadratic tensor action is

$\zeta$ 35

with

$\zeta$ 36

The tensor pump field is

$\zeta$ 37

On superhorizon scales,

$\zeta$ 38

If $\zeta$ 39 decreases, the second tensor mode grows. A realization with

$\zeta$ 40

has tensor power growing as

$\zeta$ 41

The tensor bispectrum is enhanced in the squeezed limit because the long tensor mode is dynamically growing rather than a conserved adiabatic rescaling. A Horndeski interaction containing $\zeta$ 42 is particularly important because

$\zeta$ 43

during the tensor non-attractor phase. The resulting squeezed enhancement can be parametrically larger than the standard tensor consistency-relation contribution and is frame independent under the relevant conformal/disformal transformation (Ozsoy et al., 2019).

A squeezed tensor bispectrum induces a quadrupolar modulation of the stochastic gravitational-wave background. The modulation has zero statistical mean but nonzero variance and can, for suitable parameters, generate percent-level anisotropy. A ground-based Michelson interferometer responds to this anisotropy through an orientation-dependent correction to its detector response, potentially producing a diurnal modulation as Earth rotates. A complete signal-to-noise analysis remains model dependent.

Inverse Lyth bound

Canonical non-attractor evolution obeys an upper field-range bound complementary to the conventional Lyth bound. Since

$\zeta$ 44

and anti-damping requires $\zeta$ 45, one has

$\zeta$ 46

Therefore,

$\zeta$ 47

For constant $\zeta$ 48 with $\zeta$ 49,

$\zeta$ 50

USR corresponds to $\zeta$ 51: $\zeta$ 52

The nonconstant curvature-mode velocity is amplified as

$\zeta$ 53

while the field excursion approaches a finite limit as $\zeta$ 54. In the quasi-de Sitter limit, if the amplified mode dominates the final curvature perturbation, the scalar power amplification is approximately

$\zeta$ 55

and

$\zeta$ 56

This result is an upper bound on the field distance accumulated during a continuous canonical non-attractor interval, not a bound on the total field range of a model containing additional attractor stages. It does not directly apply to noncanonical theories, multifield systems, or modified gravity (Kinney, 26 Aug 2026).

The principal organizing principle is that non-attractor inflation reverses the usual relation between background motion and perturbation evolution. Rapid loss of inflaton kinetic energy can make the field-space excursion finite while amplifying the nonconstant curvature mode, and the final observational consequences are determined by the subsequent transition, the sound-speed and kinetic structure, and the distinction between coordinate-dependent correlators and locally measurable observables.