Resonant Non-Gaussianity in Inflationary Cosmology

• Resonant non-Gaussianity is a framework where periodic modulations during inflation generate log-periodic oscillations in the power spectrum and higher-point functions.
• It employs both canonical modulated potentials and effective field theory methods to capture oscillatory features that resonate with subhorizon perturbations.
• Its discrete-scale invariance provides key insights into early universe dynamics and influences observables like rare fluctuations and primordial black holes.

Resonant non-Gaussianity is a class of primordial correlation signals generated when inflationary dynamics contain a small, coherent periodic modulation, such as a sinusoidal feature in the inflaton potential, sound speed, or EFT coefficients. The oscillatory time dependence resonates with the subhorizon oscillations of scalar perturbations, imprinting nearly log-periodic oscillations in the power spectrum and higher-point functions, typically as functions of $\ln k$ or $\ln K$ with $K \equiv k_1+k_2+k_3$. A canonical realization is a modulated potential of the form $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$, while modern treatments formulate the phenomenon both in the EFT of inflation and in wavefunction/bootstrap language, with the frequency parameter $\alpha \equiv \omega/H$ controlling the oscillatory running and the resonance regime (Flauger et al., 2010, Creminelli et al., 2024, Pueyo et al., 2023).

1. Physical origin and symmetry structure

The defining ingredient is a small periodic time dependence during inflation. In potential-based realizations, one introduces a modulation such as

$$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$

or, more generally,

$$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$

with $\Lambda^4$ controlling the modulation amplitude and $f$ the period in field space. In EFT-based descriptions, the same physics may be encoded in an oscillatory Hubble rate or oscillatory sound speed, for example through $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$ with $\ln K$0, or through $\ln K$1 (Creminelli et al., 2024, Byrnes, 2014).

The resonance mechanism follows from the approximate de Sitter relation $\ln K$2. A periodic factor such as $\ln K$3 therefore becomes $\ln K$4, so the in-in time integral contains an oscillatory phase from both the coupling and the mode functions. Stationary phase occurs when the physical frequency of the mode matches the background oscillation frequency, yielding a saddle near $\ln K$5 and generating oscillations periodic in $\ln K$6 or $\ln K$7 rather than in $\ln K$8 or $\ln K$9 themselves (Byrnes, 2014).

This structure is naturally associated with discrete scale invariance. In the resonant regime, correlation functions are invariant under a discrete dilation $K \equiv k_1+k_2+k_3$0 up to the homogeneous scaling dictated by the non-oscillatory prefactor. This distinguishes resonant non-Gaussianity from sharp-feature non-Gaussianity: sharp steps generate oscillations periodic in $K \equiv k_1+k_2+k_3$1, tied to a fixed physical scale, whereas resonant signals are periodic in $K \equiv k_1+k_2+k_3$2 and reflect an approximately constant frequency per e-fold (Pueyo et al., 2023, Byrnes, 2014).

2. EFT description and model realizations

In the decoupling limit, scalar fluctuations are described by the Goldstone mode $K \equiv k_1+k_2+k_3$3 of broken time translations, with $K \equiv k_1+k_2+k_3$4 outside the horizon to leading order. For resonant features, a compact nonlinear action is

$K \equiv k_1+k_2+k_3$5

which makes the time dependence of the interaction coefficients explicit and, in the exact de Sitter decoupling limit, makes conservation of $K \equiv k_1+k_2+k_3$6 and hence $K \equiv k_1+k_2+k_3$7 outside the horizon manifest (Creminelli et al., 2024). In the broader EFT of inflation, one instead begins in unitary gauge with time-dependent coefficients such as $K \equiv k_1+k_2+k_3$8, $K \equiv k_1+k_2+k_3$9, and $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$0, and then restores $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$1 through $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$2; soft breaking of the continuous shift symmetry to a discrete subgroup produces the oscillatory vertices responsible for the resonant signal (Behbahani et al., 2011).

Beyond canonical single-field models, general single-field $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$3 theories introduce additional operator weights controlled by

$V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$4

The combination $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$5 multiplies the $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$6 operator and can enhance equilateral-like interactions. Periodic modulations in these coefficients generate resonant bispectra with the same logarithmic running but with shape weights sensitive to $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$7 and $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$8 (Chen, 2010).

A separate refinement is collective symmetry breaking. In that construction, the inflaton shift symmetry is protected by more than one symmetry, so scale invariance in $V(\phi)=V_0(\phi)+\Lambda^4\cos(\phi/f+\psi)$9 correlators is broken only when multiple couplings are turned on simultaneously. The purpose is to suppress radiative transmission of oscillatory symmetry breaking into the quadratic action, allowing the bispectrum to carry a larger resonant signal while keeping the power spectrum nearly scale invariant. Explicit examples include quasi-single-field and strong-mixing models with an additional scalar $\alpha \equiv \omega/H$0 and oscillatory $\alpha \equiv \omega/H$1 interactions (Behbahani et al., 2012).

3. Correlators, templates, and kinematic limits

At the level of the two-point function, the universal imprint is a logarithmically oscillating correction,

$\alpha \equiv \omega/H$2

with $\alpha \equiv \omega/H$3. In the non-perturbative wavefunction treatment, $\alpha \equiv \omega/H$4, while in the canonical modulated-potential treatment the oscillatory tilt amplitude is

$\alpha \equiv \omega/H$5

These are two parametrizations of the same basic phenomenon: the power spectrum acquires a small oscillatory modulation in $\alpha \equiv \omega/H$6, with amplitude proportional to the feature strength and enhanced by a positive power of the oscillation frequency (Creminelli et al., 2024, Flauger et al., 2010).

The bispectrum is the canonical resonant observable. A standard form is

$\alpha \equiv \omega/H$7

with $\alpha \equiv \omega/H$8, supplemented at next order by a cosine term carrying mild polynomial dependence on the momenta. In the original canonical derivation,

$\alpha \equiv \omega/H$9

with

$$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$0

In the EFT/wavefunction treatment, the bispectrum likewise exhibits $$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$1 oscillations with $$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$2 and a next-to-leading cosine correction (Flauger et al., 2010, Creminelli et al., 2024).

Single-field consistency relations remain intact. In particular,

$$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$3

so the squeezed bispectrum inherits an oscillatory factor from the modulated power spectrum but does not violate the single-field relation itself (Creminelli et al., 2024). This is one of the key points of contrast with more general “clock” signals.

A distinct but related structure is folded resonant non-Gaussianity. Periodic features can generate a small negative-frequency, non-Bunch–Davies component in the mode function, $$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$4. Although $$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$5 is tiny, it becomes important in folded configurations, where one leg uses the negative-frequency branch and the phase becomes approximately stationary. The resulting bispectrum is sharply peaked near folded triangles, has a finite width set by $$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$6, and retains the resonant logarithmic running inherited from the periodic feature (Chen, 2010). By contrast, the pure resonant mechanism by itself does not produce a special flattened enhancement (Byrnes, 2014).

4. Wavefunction and bootstrap formulations

Recent work recasts resonant non-Gaussianity in wavefunction and bootstrap language. In the boostless cosmological bootstrap with discrete scale invariance, the bispectrum wavefunction coefficient takes the form

$$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$7

where $$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$8, $$V(\phi)=V_{\rm sr}(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}\right)$$9 and $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$0 are elementary symmetric polynomials, and $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$1 is the integer derivative count of the corresponding contact interaction. This realizes resonant non-Gaussianity as a deformation of the scale-invariant case to a complex order of the total-energy pole, or equivalently as interactions with a complex number of derivatives. The coefficients are fixed by homogeneity, isotropy, a modified cosmological optical theorem, and the manifestly local test, and the leading large-$$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$2 signal is tied directly to the corresponding flat-space amplitude (Pueyo et al., 2023).

A complementary development is the explicit non-perturbative wavefunction of the universe for inflation with resonant features. In semiclassical form,

$$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$3

where $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$4 is the Gaussian part and $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$5 is the first-order feature correction written in a finite Euclidean form. Expanding $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$6 in powers of $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$7 reproduces the familiar perturbative resonant hierarchy: the $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$8th term scales schematically as $$V(\phi)=V_0(\phi)+\Lambda^4\cos\!\left(\frac{\phi}{f}+\psi\right),$$9, so the perturbative expansion parameter is $\Lambda^4$0 and perturbation theory breaks down for $\Lambda^4$1 (Creminelli et al., 2024).

The non-perturbative wavefunction reveals a qualitative asymmetry between positive and negative fluctuations. After Euclidean rotation, the feature correction contains the characteristic shift $\Lambda^4$2 inside the cosine. A saddle-point analysis then shows that for positive $\Lambda^4$3 the dominant branch picks up an exponential enhancement $\Lambda^4$4, whereas for negative $\Lambda^4$5 the saddle moves onto the imaginary axis and the Euclidean enhancement is canceled. The result is a sharply enhanced positive-$\Lambda^4$6 tail but only order-feature corrections for troughs. The same peaks-versus-troughs asymmetry persists for localized spherical profiles: local maxima acquire the exponential enhancement, local minima do not (Creminelli et al., 2024).

The bootstrap analysis also clarifies several structural points. It identifies an infinite class of resonant shapes beyond the standard templates, fixes subleading $\Lambda^4$7 terms by locality, and isolates the IR-divergent resonant contribution as a local, time-oscillating term that is removed for the observable curvature perturbation in single-field inflation by the appropriate second-order field redefinition (Pueyo et al., 2023).

5. Validity regimes, perturbativity, and observational status

In the standard EFT treatment with softly broken discrete shift symmetry, the oscillatory signal in higher-point functions is generically subdominant to that in the power spectrum. The connected $\Lambda^4$8-point functions obey a hierarchy

$\Lambda^4$9

with perturbativity requiring $f$0. In that regime, the two-point oscillation is expected to be the easiest signal to detect, while higher-order correlators provide a subleading but important consistency check of the discrete symmetry structure (Behbahani et al., 2011).

The non-perturbative wavefunction calculation imposes additional bounds. Its regime of validity is first order in the feature amplitude $f$1, large $f$2, and late-time profiles satisfying the decoupling-limit condition

$f$3

together with further restrictions ensuring that one does not probe widely separated epochs in a single configuration. Loop corrections are suppressed in the wavefunction by $f$4, and for equal-time correlators the bulk and boundary loop contributions cancel at $f$5 in dimensional regularization (Creminelli et al., 2024).

A more recent analysis identifies a qualitatively different regime at very high frequency and very small amplitude. There the actual cutoff can lie above the naive value $f$6, with a logarithmic uplift when the oscillation amplitude is sufficiently small. In that window, $f$7-point functions with $f$8 can dominate the total signal-to-noise, and the signal can exhibit $f$9 oscillations per decade in $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$0 (Creminelli et al., 26 Aug 2025). This suggests that the familiar statement that the power spectrum dominates is regime-dependent: it holds in the standard weakly coupled EFT window, whereas the uplifted-cutoff regime opens a separate high-frequency sector in which higher-$\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$1 correlators can become the primary observables (Behbahani et al., 2011, Creminelli et al., 26 Aug 2025).

Observationally, no compelling evidence for resonant non-Gaussianity has been found. CMB analyses have scanned resonant templates over frequencies and phases and have reported no statistically significant detection, with power-spectrum oscillations consistent with zero within current sensitivity (Byrnes, 2014). A dedicated Planck analysis based on the cosmological bootstrap, scanning $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$2, likewise found no compelling evidence: the raw significance $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$3 was generally below $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$4, with occasional $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$5–$\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$6 peaks consistent with a look-elsewhere effect (Pueyo et al., 2023).

6. Phenomenology, degeneracies, and related oscillatory signals

The most striking phenomenological consequence of the non-perturbative wavefunction is its effect on rare positive fluctuations. Since the probability density is $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$7, the exponentially enhanced positive-$\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$8 tail can produce large fractional changes to Gaussian tail probabilities even when the oscillation amplitude is minute. For primordial black holes, whose abundance depends exponentially on the probability of exceeding a threshold $\dot H(t)=\dot H_\star[1-b\cos(\omega t+\delta)]$9, the feature-induced correction can compete with or even overwhelm the Gaussian suppression at suitable phases and frequencies. The same logic applies to other tail-sensitive observables, including rare transitions and aspects of eternal inflation (Creminelli et al., 2024).

Resonant signals can also be degenerate with more familiar bispectrum shapes. A notable example is the superposition of multiple resonant contributions: by summing $\ln K$00 oscillatory terms with different frequencies and phases, one can “Fourier synthesize” an approximately equilateral bispectrum even in canonical single-field inflation. This leads to a possible degeneracy with the equilateral signal of non-canonical models such as DBI inflation. However, if oscillations are absent in the power spectrum and the usual resonant $\ln K$01-point hierarchy holds, then a detection of equilateral non-Gaussianity at a level greater than the Planck sensitivity of $\ln K$02 would rule out a resonant origin, unless collective symmetry breaking is operative (Gwyn et al., 2012, Behbahani et al., 2012).

Standard resonant non-Gaussianity is also not the only oscillatory mechanism. Periodic production of heavy fields coupled to the inflaton yields a distinct “product-shape” bispectrum,

$\ln K$03

which oscillates in each external momentum individually rather than only through the total momentum. In controlled parameter windows its bispectrum signal-to-noise can be comparable to, or even larger than, that of the corresponding power-spectrum oscillations, unlike standard resonant non-Gaussianity. Interference terms additionally generate resonant-in-$\ln K$04 contributions at effective frequency $\ln K$05, again distinct from the canonical resonant frequency $\ln K$06 (Flauger et al., 2016).

Taken together, these developments establish resonant non-Gaussianity as a broad framework rather than a single template. Its core signature is discrete-scale-invariant oscillatory running, but its detailed realization depends on the interaction basis, initial state structure, perturbative regime, and observable being considered. In the standard weak-feature regime it is most naturally sought through oscillations in the power spectrum and bispectrum; in newer non-perturbative and uplifted-cutoff regimes, it also becomes a theory of rare tails and potentially observable large-$\ln K$07 correlators (Flauger et al., 2010, Creminelli et al., 2024, Creminelli et al., 26 Aug 2025).