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Inflaton potential reconstruction without slow-roll

Ian J Grivell, Andrew R Liddle

TL;DR

The paper introduces a direct method to reconstruct the inflaton potential from CMB observations without relying on slow-roll, by numerically solving the Mukhanov mode equations for a parametrized $V(\phi)$ and inferring potential parameters from $C_\ell$ data. It employs a local Taylor expansion of $V(\phi)$ about $\phi_*$ and uses the Fisher information matrix to forecast uncertainties, demonstrated on a $\lambda\phi^4$ potential with Planck-like data, where $V_*$ and $V'_* / V_*$ are detectable but higher derivatives are not. A key strength is the direct extraction of the full covariance of the potential parameters, enabling unbiased, correlated reconstructions of the potential over the region probed by the data. This approach offers a model-consistent, non-slow-roll trajectory to test single-field inflation and potential deviations from slow-roll, complementing traditional reconstruction methods and providing a principled framework for interpreting inflationary dynamics.

Abstract

We describe a method of obtaining the inflationary potential from observations which does not use the slow-roll approximation. Rather, the microwave anisotropy spectrum is obtained directly from a parametrized potential numerically, with no approximation beyond linear perturbation theory. This permits unbiased estimation of the parameters describing the potential, as well as providing the full error covariance matrix. We illustrate the typical uncertainties obtained using the Fisher information matrix technique, studying the $λφ^4$ potential in detail as a concrete example.

Inflaton potential reconstruction without slow-roll

TL;DR

The paper introduces a direct method to reconstruct the inflaton potential from CMB observations without relying on slow-roll, by numerically solving the Mukhanov mode equations for a parametrized and inferring potential parameters from data. It employs a local Taylor expansion of about and uses the Fisher information matrix to forecast uncertainties, demonstrated on a potential with Planck-like data, where and are detectable but higher derivatives are not. A key strength is the direct extraction of the full covariance of the potential parameters, enabling unbiased, correlated reconstructions of the potential over the region probed by the data. This approach offers a model-consistent, non-slow-roll trajectory to test single-field inflation and potential deviations from slow-roll, complementing traditional reconstruction methods and providing a principled framework for interpreting inflationary dynamics.

Abstract

We describe a method of obtaining the inflationary potential from observations which does not use the slow-roll approximation. Rather, the microwave anisotropy spectrum is obtained directly from a parametrized potential numerically, with no approximation beyond linear perturbation theory. This permits unbiased estimation of the parameters describing the potential, as well as providing the full error covariance matrix. We illustrate the typical uncertainties obtained using the Fisher information matrix technique, studying the potential in detail as a concrete example.

Paper Structure

This paper contains 7 sections, 2 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Direct estimation of inflationary parameters
  3. Uncertainty and covariance of inflaton potential parameters
  4. Parametrizing the potential
  5. Parameter uncertainty
  6. Parameter uncertainty covariance
  7. Discussion

Figures (3)

  • Figure 1: The traditional route from model to observables and back is the two-stage process at the top. The procedure outlined in this paper enables a direct route without approximations beyond linear perturbation theory.
  • Figure 2: Twenty Monte Carlo reconstructions of the potential, compared against the true potential which is shown as a dashed line. The upper panel shows the potential itself, and the lower one the combination $V'/V^{3/2}$ which is a combination coming primarily from the density perturbations alone. The dotted vertical lines indicate the region of the potential directly probed by the microwave background, ranging from the current horizon scale to the horizon scale when the $\ell = 1500$ mode was generated (evaluated in the underlying model). The upper panel shows that the gradient is quite well recovered but the overall amplitude much less so, while the lower highlights the obvious fact that the reconstruction is accurate only where there is data available to constrain it.
  • Figure 3: Twenty Monte Carlo reconstructions of the combination $V'/V^{3/2}$, as in the lower panel of Fig. \ref{['f:recons']}, but for the model investigated by Wang et al. Wangetal.