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How long before the end of inflation were observable perturbations produced?

Andrew R Liddle, Samuel M Leach

TL;DR

The paper investigates how many inflationary e-foldings elapsed before the end of inflation at which observable perturbations were generated, emphasizing that this depends on the entire post-inflation history. It derives a plausible upper bound of about $N_{ m hor}\approx 62$ for the standard cosmology, with the special case $V=\lambda\phi^4$ predicting $N_{ m hor}\approx 64$. It further explores non-standard evolutions (stiff-fluid epochs, early matter domination, thermal inflation, braneworld scenarios) that can raise or lower $N_{ m hor}$, and introduces a refined definition using $\tilde{N}=\ln(aH)$ to address cases with changing $H$. The work highlights the sensitivity of inflationary constraints to reheating and cosmic history, and provides guidance for model-specific analyses, particularly when the energy scale or late-time dynamics deviate from the standard picture.

Abstract

We reconsider the issue of the number of e-foldings before the end of inflation at which observable perturbations were generated. We determine a plausible upper limit on that number for the standard cosmology which is around 60, with the expectation that the actual value will be up to 10 below this. We also note a special property of the $λφ^4$ model which reduces the uncertainties in that case and favours a higher value, giving a fairly definite prediction of 64 e-foldings for that model. We note an extreme (and highly implausible) situation where the number of e-foldings can be even higher, possibly up to 100, and discuss the shortcomings of quantifying inflation by e-foldings rather than by the change in $aH$. Finally, we discuss the impact of non-standard evolution between the end of inflation and the present, showing that again the expected number of e-foldings can be modified, and in some cases significantly increased.

How long before the end of inflation were observable perturbations produced?

TL;DR

The paper investigates how many inflationary e-foldings elapsed before the end of inflation at which observable perturbations were generated, emphasizing that this depends on the entire post-inflation history. It derives a plausible upper bound of about for the standard cosmology, with the special case predicting . It further explores non-standard evolutions (stiff-fluid epochs, early matter domination, thermal inflation, braneworld scenarios) that can raise or lower , and introduces a refined definition using to address cases with changing . The work highlights the sensitivity of inflationary constraints to reheating and cosmic history, and provides guidance for model-specific analyses, particularly when the energy scale or late-time dynamics deviate from the standard picture.

Abstract

We reconsider the issue of the number of e-foldings before the end of inflation at which observable perturbations were generated. We determine a plausible upper limit on that number for the standard cosmology which is around 60, with the expectation that the actual value will be up to 10 below this. We also note a special property of the model which reduces the uncertainties in that case and favours a higher value, giving a fairly definite prediction of 64 e-foldings for that model. We note an extreme (and highly implausible) situation where the number of e-foldings can be even higher, possibly up to 100, and discuss the shortcomings of quantifying inflation by e-foldings rather than by the change in . Finally, we discuss the impact of non-standard evolution between the end of inflation and the present, showing that again the expected number of e-foldings can be modified, and in some cases significantly increased.

Paper Structure

This paper contains 13 sections, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The simplest cosmology
  3. A plausible upper limit
  4. A standard hypothesis
  5. The special case of $\lambda \phi^4$
  6. Extreme cases, and a better definition of inflation
  7. Non-standard cosmologies: upper and lower limits
  8. An upper limit
  9. Early matter domination
  10. Thermal inflation
  11. Braneworld cosmology
  12. An absolute minimum for $N_{{\rm hor}}$
  13. Conclusions

Figures (1)

  • Figure 1: A plot of $\ln(H^{-1}/a)$ versus $\ln a$ shows the different epochs in the $e$-foldings calculation. The solid curve shows the evolution from the initial horizon crossing to the present, with the dashed lines showing likely extrapolations into the past and future. The condition for inflation is that $\ln(H^{-1}/a)$ be decreasing. Lines of constant Hubble parameter (not shown) lie at 45 degrees (running top left to bottom right). The limit of exponential inflation gives a line at this angle, otherwise the inflation line is shallower. During reheating and matter domination $H^{-1}/a \propto a^{1/2}$, while during radiation domination $H^{-1}/a \propto a$. The recent domination by dark energy has initiated a new era of inflation. The horizontal dotted line indicates the present horizon scale. The number of $e$-foldings of inflation is the horizontal distance between the time when $H^{-1}/a$ first crosses that value and the end of inflation.