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Non-Gaussianities in N-flation

Diana Battefeld, Thorsten Battefeld

TL;DR

This work assesses non-Gaussianities in a string-motivated \\mathcal{N}-flation model with a separable quadratic potential and a Marčenko-Pastur mass spectrum. Using slow-roll and the \\delta N formalism, it first derives HC results for the non-linearity parameter \\(f_{NL}\\) and then extends the analysis beyond HC to include post-HC evolution, finding only small (order a few percent) corrections for both narrow and broad spectra. The study shows \\(f_{NL}\\) is effectively indistinguishable from single-field predictions in the slow-roll regime, with larger NG potentially arising from fast-roll episodes or during (p)re-heating, for which an effective single-field with steps is proposed as a modeling tool. The results underscore the robustness of slow-roll predictions in \\mathcal{N}-flation and point to (p)re-heating as the primary avenue for sizeable NG signals in this framework.

Abstract

We compute non-Gaussianities in N-flation, a string motivated model of assisted inflation with quadratic, separable potentials and masses given by the Marcenko-Pastur distribution. After estimating parameters characterizing the bi- and trispectrum in the horizon crossing approximation, we focus on the non-linearity parameter $f_{NL}$, a measure of the bispectrum; we compute its magnitude for narrow and broad spreads of masses, including the evolution of modes after horizon crossing. We identify additional contributions due to said evolution and show that they are suppressed as long as the fields are evolving slowly. This renders $\mathcal{N}$-flation indistinguishable from simple single-field models in this regime. Larger non-Gaussianities are expected to arise for fields that start to evolve faster, and we suggest an analytic technique to estimate their contribution. However, such fast roll during inflation is not expected in N-flation, leaving (p)re-heating as the main additional candidate for generating non-Gaussianities.

Non-Gaussianities in N-flation

TL;DR

This work assesses non-Gaussianities in a string-motivated \\mathcal{N}-flation model with a separable quadratic potential and a Marčenko-Pastur mass spectrum. Using slow-roll and the \\delta N formalism, it first derives HC results for the non-linearity parameter \ and then extends the analysis beyond HC to include post-HC evolution, finding only small (order a few percent) corrections for both narrow and broad spectra. The study shows \ is effectively indistinguishable from single-field predictions in the slow-roll regime, with larger NG potentially arising from fast-roll episodes or during (p)re-heating, for which an effective single-field with steps is proposed as a modeling tool. The results underscore the robustness of slow-roll predictions in \\mathcal{N}-flation and point to (p)re-heating as the primary avenue for sizeable NG signals in this framework.

Abstract

We compute non-Gaussianities in N-flation, a string motivated model of assisted inflation with quadratic, separable potentials and masses given by the Marcenko-Pastur distribution. After estimating parameters characterizing the bi- and trispectrum in the horizon crossing approximation, we focus on the non-linearity parameter , a measure of the bispectrum; we compute its magnitude for narrow and broad spreads of masses, including the evolution of modes after horizon crossing. We identify additional contributions due to said evolution and show that they are suppressed as long as the fields are evolving slowly. This renders -flation indistinguishable from simple single-field models in this regime. Larger non-Gaussianities are expected to arise for fields that start to evolve faster, and we suggest an analytic technique to estimate their contribution. However, such fast roll during inflation is not expected in N-flation, leaving (p)re-heating as the main additional candidate for generating non-Gaussianities.

Paper Structure

This paper contains 11 sections, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $\mathcal{N}$-flation and Slow Roll
  3. Non-Gaussianities
  4. Using Slow Roll and the Horizon Crossing Approximation
  5. Beyond the Horizon Crossing Approximation
  6. Narrow Mass Spectra
  7. Broad Mass Spectra
  8. Discussion
  9. Beyond the Slow Roll Approximation
  10. Conclusions
  11. The $\mathcal{F}_\alpha$ Functions

Figures (4)

  • Figure 1: Probability of a given mass according to the Marčenko-Pastur distribution from (\ref{['mpdistr']}), depending on $\beta$ and the dimensionless square mass $x=m^2/m_1^2$, rescaled with the expectation value $\left<x\right>$ (also dependent on $\beta$): (a) 3D-plot for $0.1<\beta<1$, (b) slices for $\beta_1=1/4$, $\beta_2=1/2$ and $\beta_3=3/4$; the closer $\beta$ is to one, the broader the mass spectrum becomes.
  • Figure 2: Solving (\ref{['eqnfory2']}) numerically leads to $\varphi_1^{2}/\varphi_1^{*2}\equiv\bar{y}(\beta)$ for (a) $-9\leq \log_{10}(\beta)\leq -1$, (b) $0.1 \leq \beta \leq 0.9$. We took $N=60$ in all plots.
  • Figure 3: $-f_{NL}^{(4)}(2N+1)6/5$ over $\log_{(10)}(\beta)$ computed using: a. the horizon crossing approximation $-f_{NL}^{(4)}(2N+1)6/5 =1$, b. the $\delta$-expansion from (\ref{['fnlnhc']}), c. the "exact" expression from (\ref{['fnlexact']}) and d. the approximation from (\ref{['fapprox3']}). We took $N=60$ in all plots. Note that b. and d. are both good approximations up until $\beta\sim 0.1$.
  • Figure 4: $-f_{NL}^{(4)}6/5$ over $\beta$ computed using: a. the horizon crossing approximation $-f_{NL}^{(4)}(2N+1)6/5 =1$, b. the $\delta$-expansion from (\ref{['fnlnhc']}), c. the exact expression from (\ref{['fnlexact']}) and d. the approximation from (\ref{['fapprox3']}). We took $N=60$ in all plots. Both approximations fail to recover the turn of $f_{NL}^{(4)}$ observable in Figure (b).