Primordial Black Holes, Eternal Inflation, and the Inflationary Parameter Space after WMAP5

TL;DR

This work constrains single-field inflation using Slow Roll Reconstruction with WMAP5, ACBAR, and SNLS data, emphasizing how priors on the total number of e-folds and reheating shape the viable parameter space. By employing the Hamilton-Jacobi flow hierarchy and truncating the slow-roll expansion, the authors derive explicit potentials and propagate constraints to the end of inflation through $N$. They find that excluding scenarios with primordial black hole overproduction or eternal inflation imposes meaningful limits on slow-roll parameters, particularly $oldsymbol{\xi}$, with the e-fold priors playing a crucial role. The framework allows translating results to the standard $n_s$, $r$, and $d n_s/d\,ln k$ parametrization, but highlights that current data are still priors-limited for higher-order slow-roll terms; Planck and future polarization data are expected to substantially tighten these constraints.

Abstract

We consider constraints on inflation driven by a single, minimally coupled scalar field in the light of the WMAP5 dataset, as well as ACBAR and the SuperNova Legacy Survey. We use the Slow Roll Reconstruction algorithm to derive optimal constraints on the inflationary parameter space. The scale dependence in the slope of the scalar spectrum permitted by WMAP5 is large enough to lead to viable models where the small scale perturbations have a substantial amplitude when extrapolated to the end of inflation. We find that excluding parameter values which would cause the overproduction of primordial black holes or even the onset of eternal inflation leads to potentially significant constraints on the slow roll parameters. Finally, we present a more sophisticated approach to including priors based on the total duration of inflation, and discuss the resulting restrictions on the inflationary parameter space.

Figures (8)

• Figure 1: The regions of the $\{ \epsilon,\eta\}$ plane excluded by requiring $N(k_\star)>15$ (black), $T_{\rm reh} > 10$ TeV (dark) and instantaneous thermalization (light). The other prior used in this work, $T_{\rm reh} > 10$ MeV, looks very similar to the $T_{\rm reh} > 10$ TeV prior. For this figure only, the amplitude for the scalar spectrum at the fiducial scale has been set to the WMAP5 best fit.
• Figure 2: The joint 68% (dark) and 95% (light) confidence levels obtained on the first two HSR parameters $\epsilon$ and $\eta$ at the fiducial scale $k_\star=0.02$ Mpc$^{-1}$ from the WMAP 5 year data, assuming $\xi$ and all higher order slow roll parameters are zero. The grey (dotted) contours correspond to a "minimal" e-fold prior, $N(k_\star)>15$. The blue (solid) contours are obtained by assuming instantaneous reheating.
• Figure 3: The joint 68% (inner) and 95% (outer) bounds on the slow roll variables (top) and power law spectral parameters (bottom), with $k_\star = 0.02\ \mathrm{ Mpc}^{-1}$, for a High-$\epsilon$ 2-parameter fit. WMAP5 constraints are blue, and WMAP5+SNLS constraints are red. Solid contours come from Slow Roll Reconstruction with prior $T_\mathrm{reh} > 10$ TeV. The running of the scalar index in these models is second order in slow roll and hence very small. The dashed contours show results from WMAP5 chains where the spectrum was specified via $n_s$ and $r$. We superimpose the "trajectories" for three generic slow roll models, $\lambda \phi^4$ (dotted), $m^2 \phi^2$ (solid), and a representative natural inflation model (dashed), along with the position at different values of $N=30$ (triangle), $N=40$ (square), $N=50$ (star), $N=60$ (circle).
• Figure 4: The joint 68% (dark) and 95% (light) confidence levels obtained on the first three HSR parameters $\{\epsilon, \eta, \xi\}$ at the fiducial scale $k_\star=0.02$ Mpc$^{-1}$ from the WMAP 5 year data, assuming all higher order slow roll parameters are zero. The grey (dotted) contours correspond to a "minimal" e-fold prior, $N(k_\star)>15$. The blue (solid) contours are obtained by assuming instantaneous reheating.
• Figure 5: The joint 68% (inner) and 95% (outer) bounds on the first three HSR parameters $\{\epsilon, \eta, \xi\}$ at the fiducial scale $k_\star = 0.02\ \mathrm{ Mpc}^{-1}$. The blue constraints are derived from WMAP 5 year data alone, and the red constraints from the WMAP5+SNLS data combination, with $T_\mathrm{reh} > 10$ TeV.