Monte Carlo reconstruction of the inflationary potential

TL;DR

This work introduces Monte Carlo reconstruction, a stochastic method based on the Hubble slow-roll flow equations to constrain the inflationary potential $V(\phi)$ from observed primordial spectra under the single-field assumption. By sampling random initial slow-roll parameters up to order $M$ and evolving the flow equations, it builds ensembles of potentials consistent with a defined window in observables such as the tensor-to-scalar ratio $r$, the spectral index $n$, and its running $dn/d\ln k$, without imposing a tight parametric form on $V(\phi)$. Planck-like data alone yield qualitative shape information but do not uniquely determine the potential, whereas a fivefold improvement in parameter precision can begin to recover a power-law form $V(\phi) \propto \phi^m$, with $m$ near 4, demonstrating the method’s sensitivity to data quality. The approach highlights fundamental limits of reconstruction and provides a practical, robust framework that can be extended to exact perturbation calculations and alternative slow-roll expansions.

Abstract

We present Monte Carlo reconstruction, a new method for ``inverting'' observational data to constrain the form of the scalar field potential responsible for inflation. This stochastic technique is based on the flow equation formalism and has distinct advantages over reconstruction methods based on a Taylor expansion of the potential. The primary ansatz required for Monte Carlo reconstruction is simply that inflation is driven by a single scalar field. We also require a very mild slow roll constraint, which can be made arbitrarily weak since Monte Carlo reconstruction is implemented at arbitrary order in the slow roll expansion. While our method cannot evade fundamental limits on the accuracy of reconstruction, it can be simply and consistently applied to poor data sets, and it takes advantage of the attractor properties of single-field inflation models to constrain the potential outside the small region directly probed by observations. We show examples of Monte Carlo reconstruction for data sets similar to that expected from the Planck satellite, and for a hypothetical measurement with a factor of five better parameter discrimination than Planck.

Figures (5)

• Figure 1: Paths in the $(\sigma,\epsilon)$ plane (left column) and the corresponding potentials $V(\phi)$ (right column) for $r \simeq 0.0$, $n \simeq 0.93$, $d n / d \log k \simeq 0.0$.
• Figure 2: Paths in the $(\sigma,\epsilon)$ plane (left column) and the corresponding potentials $V(\phi)$ (right column) for a blue spectrum $r \simeq 0.0$, $n \simeq 1.05$, $d n / d \log k \simeq 0.0$.
• Figure 3: Paths in the $(\sigma,\epsilon)$ plane (left column) and the corresponding potentials $V(\phi)$ (right column) for $r \simeq 0.18$, $n \simeq 0.6$, $d n d / \log k \simeq -0.02$. This parameter region is observationally disfavored, but shows the complicated behavior possible for solutions to the flow equations at higher order.
• Figure 4: The upper panel shows 100 reconstructed potentials, assuming $r = 0.02 \pm 0.01$, $n = 0.95 \pm 0.01$, $d n / d \log{k} = 0.0 \pm 0.01$ (the errors bars expected from the Planck mission). This choice implies that the tensor modes are unambiguously detected, and leads to a tight constraint on the normalization of the potential. The field $\phi$ is defined such that the observational parameters are calculated at $\phi = 0$. The lower plot shows 100 reconstructed potentials, for $r = 0.0 + 0.01$, $n = 0.93 \pm 0.01$, $d n / d \log k = 0.0 \pm 0.01$. In this case, tensor modes are not resolved, and the normalization of the potential is poorly constrained. The one anomalous potential in the top figure corresponds to a potential that, by chance, has comparatively large slow roll parameters, but which cancel in just the right way to produce the specified cosmological spectrum.
• Figure 5: These two histograms show the values of the power $m$ (horizontal axes) obtained by fitting Eq. (\ref{['fit']}) to 100 potentials generated by the Monte Carlo reconstruction algorithm, where the measured spectra are assumed to have the central values predicted by $\lambda \phi^4$ inflation. The vertical axes indicate the number of models in a particular bin in $m$. In the top panel we assume that the error bars on the spectral parameters are equal to those expected from Planck, $\delta r \sim 0.01$, $\delta n \sim 0.01$ and $\delta d n / d \log k \sim 0.01$, whereas the bottom panel corresponds to $\delta r \sim 0.002$, $\delta n \sim 0.002$ and $\delta d n / d \log k \sim 0.005$. In the former case, the functional form of the potential cannot be meaningfully recovered, but in the lower case the results are consistent with $V \propto \phi^4$. In both plots we have dropped a few cases for which the least squares solver did not converge.