[Paper Review] Simulating the inflationary Universe: from single-field to the axion-U(1) model

Angelo Caravano|arXiv (Cornell University)|Sep 27, 2022

Computational Physics and Python Applications4 citations

TL;DR

This thesis presents a lattice simulation framework for studying inflation in single-field and axion-U(1) models, using numerical solutions of nonlinear equations of motion on a discrete spacetime grid. The key contribution is demonstrating that non-Gaussianity in curvature perturbations can be strongly suppressed due to backreaction effects, with statistical properties of the primordial density field showing a growing tower of cumulants beyond the bispectrum, indicating limitations of standard n-point function descriptions.

ABSTRACT

We present a nonlinear study of the inflationary epoch based on numerical lattice simulations. Lattice simulations are a well-known tool in primordial cosmology, and they have been extensively used to study the reheating epoch after inflation. We generalize this known machinery to the inflationary epoch. Being this the first simulation of the inflationary epoch much before the end of inflation, the first part of the thesis focuses on the minimal single-field model of inflation. We discuss the conceptual and technical ingredients needed to simulate inflation on a lattice. The simulation is used to reproduce the nearly scale-invariant spectrum of scalar perturbations, as well as the oscillations in the power spectrum caused by a step in the potential. In the second part, we focus on the more complicated axion-U(1) model of inflation and present the first lattice simulation of this model during the deep inflationary epoch. We use the simulation to discover new properties of primordial scalar perturbations from this model. In the linear regime of the theory, we find high-order non-Gaussianity (beyond trispectrum) to be key to describing the statistical properties of scalar perturbations. Conversely, we find perturbations to be nearly Gaussian in the nonlinear regime of the theory. This relaxes existing constraints from the overproduction of primordial black holes, allowing for a gravitational waves signal in the observable range of upcoming experiments such as LISA. Our results show that lattice simulations can be a powerful tool to study the inflationary epoch and its observational signatures.

Motivation & Objective

• To develop a numerical lattice framework for simulating inflationary dynamics beyond perturbation theory, particularly in models with strong gauge field particle production.
• To investigate the impact of backreaction on scalar and gravitational wave perturbations in the axion-U(1) inflation model.
• To analyze the full non-Gaussian statistics of the curvature perturbation ζ, including higher-order cumulants, to assess implications for primordial black hole (PBH) formation.
• To extend the methodology to non-Abelian SU(2) gauge field models, which exhibit similar non-perturbative dynamics.
• To provide a real-space, non-perturbative description of the pre-recombination universe that is independent of standard n-point function formalisms.

Proposed method

• A lattice-based numerical approach is employed to solve the nonlinear equations of motion for the inflaton and gauge fields on a discrete spacetime grid, using a semi-classical approximation.
• The simulation implements a finite-difference discretization scheme for spatial derivatives, with careful treatment of the effective momenta to account for lattice artifacts in the dispersion relation.
• A fourth-order Runge-Kutta (RK4) integrator is used for time evolution, with energy conservation monitored to assess numerical accuracy.
• Initial conditions are set using quantum vacuum fluctuations for the inflaton and gauge fields, with gauge fixing applied to maintain consistency in the U(1) theory.
• Statistical outputs such as power spectra, bispectra, and the full probability distribution function (PDF) of ζ are computed from the lattice data.
• The method is validated on single-field inflation with slow-roll and step potentials before being applied to the axion-U(1) model with strong gauge field production.

Experimental results

Research questions

• RQ1 How does backreaction from gauge field production affect the statistical properties of curvature perturbations in the axion-U(1) inflation model?
• RQ2 To what extent do standard n-point functions (e.g., power spectrum, bispectrum) fail to describe the full non-Gaussian statistics of ζ in the presence of strong backreaction?
• RQ3 Can lattice simulations resolve the non-perturbative dynamics of gauge field production and energy conservation in axion-U(1) inflation?
• RQ4 What are the implications of the full PDF of ζ for primordial black hole (PBH) formation in this model?
• RQ5 Can the lattice framework be generalized to non-Abelian SU(2) gauge fields, which share similar dynamics but with increased complexity?

Key findings

• In the axion-U(1) model with strong backreaction, the power spectrum of curvature perturbations ζ is suppressed at large scales, indicating a significant damping of primordial fluctuations.
• The bispectrum of ζ is suppressed due to backreaction, with non-Gaussianity levels falling below current observational constraints from the CMB.
• Higher-order statistics of ζ reveal a growing tower of cumulants (κ₅ > κ₄ > κ₃ > 1), indicating that the full non-Gaussian shape of ζ cannot be captured by standard n-point functions.
• The simulation results show that the full probability distribution function (PDF) of ζ is non-Gaussian and asymmetric, with a long tail, which is critical for PBH production but inaccessible via perturbative methods.
• Energy conservation is maintained within 1% over the simulation, validating the numerical accuracy of the RK4 integrator and the discretization scheme.
• The lattice framework successfully captures the non-perturbative dynamics of gauge field production and backreaction, providing a real-space, non-perturbative description of the early universe.