We study the statistics of scalar perturbations in models of inflation with

small and rapid oscillations in the inflaton potential (resonant

non-Gaussianity). We do so by deriving the wavefunction

$Psi[zeta(boldsymbol{x})]$ non-perturbatively in $zeta$, but at first order

in the amplitude of the oscillations. The expression of the wavefunction of the

universe (WFU) is explicit and does not require solving partial differential

equations. One finds qualitative deviations from perturbation theory for $

|zeta| gtrsim alpha^{-2}$, where $alpha gg 1$ is the number of

oscillations per Hubble time. Notably, the WFU exhibits distinct behaviours for

negative and positive values of $zeta$ (troughs and peaks respectively). While

corrections for $zeta <0$ remain relatively small, of the order of the

oscillation amplitude, positive $zeta$ yields substantial effects, growing

exponentially as $e^{pialpha/2}$ in the limit of large $zeta$. This

indicates that even minute oscillations give large effects on the tail of the

distribution.

**Text:**

We study the statistics of scalar perturbations in models of inflation with small and rapid oscillations in the inflaton potential (resonant non-Gaussianity). We do so by deriving the wavefunction Ψ[ζ(x)] non-perturbatively in ζ, but at first order in the amplitude of the oscillations. The expression of the wavefunction of the universe (WFU) is explicit and does not require solving partial differential equations. One finds qualitative deviations from perturbation theory for |ζ| ≳ α⁻², where α ≫ 1 is the number of oscillations per Hubble time. Notably, the WFU exhibits distinct behaviors for negative and positive values of ζ (troughs and peaks respectively). While corrections for ζ < 0 remain relatively small, of the order of the oscillation amplitude, positive ζ yields substantial effects, growing exponentially as e^(πα/2) in the limit of large ζ. This indicates that even minute oscillations give large effects on the tail of the distribution.

## Conclusions:

- Scalar perturbations in models of inflation with small and rapid oscillations in the inflaton potential exhibit qualitative deviations from perturbation theory for |ζ| ≳ α⁻².
- The wavefunction of the universe (WFU) has explicit expression and does not require solving partial differential equations.
- The WFU exhibits distinct behaviors for negative and positive values of ζ, with troughs and peaks respectively.
- Corrections for negative ζ are relatively small compared to positive ζ.
- Positive ζ yields substantial effects, growing exponentially as e^(πα/2) in the limit of large ζ.
- Even minute oscillations have large effects on the tail of the distribution.

## Future Roadmap:

The study of scalar perturbations in models of inflation with small and rapid oscillations in the inflaton potential opens up new directions for research. Here are some potential challenges and opportunities on the horizon:

### 1. Further Investigation of Non-Perturbative Analysis:

The non-perturbative analysis of the wavefunction Ψ[ζ(x)] provides valuable insights into the statistics of scalar perturbations. Future research should focus on refining and expanding this analysis. By studying higher orders in the amplitude of the oscillations, a more comprehensive understanding of the effects of rapid oscillations can be gained.

### 2. Exploring the Physical Implications:

The distinct behaviors of the wavefunction of the universe for negative and positive values of ζ have important physical implications. Further investigation is needed to understand the underlying mechanisms that give rise to these behaviors and their consequences for the evolution of the universe. This could lead to new insights into the nature of inflation and its implications for cosmology.

### 3. Experimental Validation:

Experimental validation of the theoretical predictions is crucial to confirm the findings of this study. Future experiments focusing on measuring and characterizing scalar perturbations in models with small and rapid oscillations can provide valuable data to compare with the theoretical predictions. This could involve observations from cosmological probes or laboratory experiments that simulate inflationary scenarios.

### 4. Implications for Cosmological Observations:

The substantial effects of positive ζ on the tail of the distribution have implications for cosmological observations. Future research should investigate the impact of these effects on observables such as the cosmic microwave background radiation and large-scale structure formation. This could lead to new ways of interpreting observational data and refining our understanding of the early universe.

### 5. Exploring Generalizations:

The study focused on models of inflation with small and rapid oscillations in the inflaton potential. It would be interesting to explore the applicability of the non-perturbative analysis to other inflationary models or even beyond inflationary cosmology. Investigating the effects of different potential shapes or additional fields could provide valuable insights into the broader implications of the findings.

### 6. Theoretical Developments:

Building upon the explicit expression of the wavefunction of the universe obtained in this study, further theoretical developments can be pursued. This could involve exploring connections with other areas of physics, such as quantum gravity or string theory. Additionally, investigations into possible connections between resonant non-Gaussianity and other cosmological phenomena, such as primordial black holes or gravitational waves, could yield interesting results.

Overall, the study of scalar perturbations in models of inflation with small and rapid oscillations in the inflaton potential has opened up exciting avenues for future research. By further investigating the non-perturbative analysis, exploring the physical implications, validating the theoretical predictions through experiments, studying the impact on cosmological observations, exploring generalizations, and pursuing theoretical developments, we can deepen our understanding of inflationary cosmology and its implications for the early universe.