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A Horizon Ratio Bound for Inflationary Fluctuations

Scott Dodelson, Lam Hui

TL;DR

The paper addresses how to constrain inflationary models by bounds derived from the gravity-wave amplitude, introducing a model-independent bound on the horizon parameter $\tilde N \equiv \ln(a_e H_e/k)$ and a model-dependent bound on the number of e-folds $N$ via slow-roll flow analysis. It derives $e^{\tilde N} < e^{60}$ (i.e., $\tilde N<60$) and, for typical slow-roll models, $N<67$, with a general bound $N<105$; a chaotic $\phi^4$ inflation example is shown to be disfavored when combined with CMB data. The methodology combines backward extrapolation from today, post-inflation redshift assumptions, and Monte Carlo exploration of slow-roll trajectories to obtain robust constraints. The results offer a practical framework for incorporating horizon-based limits into observational confrontations and motivate using $\tilde N$ as a primary variable in inflationary dynamics.

Abstract

We demonstrate that the gravity wave background amplitude implies a robust upper bound on the ratio: λ/ H^{-1} < e^60, where λis the proper wavelength of fluctuations of interest and H^{-1} is the horizon at the end of inflation. The bound holds as long as the energy density of the universe does not drop faster than radiation subsequent to inflation. This limit implies that the amount of expansion between the time the scales of interest leave the horizon and the end of inflation, denoted by e^N, is also bounded from above, by about e^60 times a factor that involves an integral over the first slow-roll parameter. In other words, the bound on N is model dependent -- we show that for vast classes of slow-roll models, N < 67. The quantities, λ/ H^{-1} or N, play an important role in determining the nature of inflationary scalar and tensor fluctuations. We suggest ways to incorporate the above bounds when confronting inflation models with observations. As an example, this bound solidifies the tension between observations of cosmic microwave background (CMB) anisotropies and chaotic inflation with a φ^4 potential by closing the escape hatch of large N (< 62).

A Horizon Ratio Bound for Inflationary Fluctuations

TL;DR

The paper addresses how to constrain inflationary models by bounds derived from the gravity-wave amplitude, introducing a model-independent bound on the horizon parameter and a model-dependent bound on the number of e-folds via slow-roll flow analysis. It derives (i.e., ) and, for typical slow-roll models, , with a general bound ; a chaotic inflation example is shown to be disfavored when combined with CMB data. The methodology combines backward extrapolation from today, post-inflation redshift assumptions, and Monte Carlo exploration of slow-roll trajectories to obtain robust constraints. The results offer a practical framework for incorporating horizon-based limits into observational confrontations and motivate using as a primary variable in inflationary dynamics.

Abstract

We demonstrate that the gravity wave background amplitude implies a robust upper bound on the ratio: λ/ H^{-1} < e^60, where λis the proper wavelength of fluctuations of interest and H^{-1} is the horizon at the end of inflation. The bound holds as long as the energy density of the universe does not drop faster than radiation subsequent to inflation. This limit implies that the amount of expansion between the time the scales of interest leave the horizon and the end of inflation, denoted by e^N, is also bounded from above, by about e^60 times a factor that involves an integral over the first slow-roll parameter. In other words, the bound on N is model dependent -- we show that for vast classes of slow-roll models, N < 67. The quantities, λ/ H^{-1} or N, play an important role in determining the nature of inflationary scalar and tensor fluctuations. We suggest ways to incorporate the above bounds when confronting inflation models with observations. As an example, this bound solidifies the tension between observations of cosmic microwave background (CMB) anisotropies and chaotic inflation with a φ^4 potential by closing the escape hatch of large N (< 62).

Paper Structure

This paper contains 4 sections, 12 equations, 1 figure.

Table of Contents

  1. A Brief Review
  2. A Model-independent Bound on $\tilde{N}$
  3. Upper Bound(s) on $N$
  4. Discussion

Figures (1)

  • Figure 1: The probability distribution of $N$-bound (eq. \ref{['Nepsilon']}) among a host of Monte Carlo realizations of inflation models. The spike around $60$ is largely due to fixed points, models where inflation does not terminate at $\epsilon = 1$, but rather $\epsilon \sim 0$. The inset shows two examples of how $\epsilon$ flows with $N$ (i.e. not fixed points).