Initial-condition dependence and rigorous eternality criteria for hilltop eternal inflation

Barenboim, Park and Kinney (JCAP 05 (2016) 030) showed that hilltop inflation - the class of single-field models favored by cosmic microwave background data, in which the inflaton rolls off a local maximum of its potential - supports eternal inflation at any energy scale, under two restrictive assumptions: the field evolves on the slow-roll attractor, and eternality is assessed with leading-order random-walk lifetime estimates. Whether eternal inflation is actually triggered remains open. Non-attractor initial data, for which classical rolling dominates near the hilltop, were excluded; the curvature-perturbation condition $P(k)>1$ is known to be necessary but not sufficient, and no necessary-and-sufficient eternality criterion for generic hilltop potentials has been derived from the full stochastic (Fokker-Planck) dynamics; and the model-class classification (odd-power saddle points, potentials with negative running of the spectral index) is incomplete.

Background

Cosmic inflation is a hypothesized epoch of accelerated expansion of the very early universe, driven by the potential energy of a scalar field $\phi$ (the inflaton), that explains the observed flatness and homogeneity of the cosmos and the origin of the primordial density perturbations now measured precisely in the cosmic microwave background (CMB). In single-field models the expansion rate is the Hubble parameter $H=\dot{a}/a$, with $H\simeq\sqrt{V(\phi)/(3M_P^2)}$ when the potential dominates the kinetic energy; $M_P\simeq2\times10^{18}$ GeV is the reduced Planck mass. CMB data disfavor simple power-law ("chaotic") potentials $V\propto\phi^p$ and favor hilltop (small-field) models, in which inflation occurs as the field rolls away from a local maximum, $V(\phi)\simeq V_0[1-(\phi/\mu)^p]$, with sub-Planckian field excursion.

Eternal inflation is the phenomenon, first described by Vilenkin (1983), that in many models inflation never ends globally: quantum fluctuations of the inflaton, of amplitude $\langle\delta\phi\rangle_Q=H/2\pi$ per Hubble time, compete with the classical field drift $\delta\phi_{Cl}=\dot{\phi}/H$, and when the ratio $H^2/(2\pi\dot{\phi})$ exceeds unity the field random-walks as often up the potential as down, so that ever-inflating regions proliferate and the total inflating volume grows without bound. This ratio equals the amplitude $P(k)$ of the curvature (density) perturbation when a mode crosses the horizon, so the textbook condition for eternal inflation is $P(k)>1$; Vennin and Starobinsky (2015) showed this condition is necessary but not sufficient. The natural framework is the stochastic approach of Starobinsky (1986): the long-wavelength part of the field obeys a Langevin equation $\ddot{\phi}+3H\dot{\phi}+V'(\phi)=N(t,\mathbf{x})$ with Gaussian noise, or equivalently its probability distribution obeys a Fokker-Planck equation.

Barenboim, Park and Kinney (2016) distinguished two classes of eternal inflation. In chaotic-type models, eternality sets in only at expansion rates far above the slow-roll value, $H_{EI}\gg H_*$ (for a pure power-law spectrum one finds $H_{EI}/M_P\simeq1.28$, at the edge of semiclassical control). In hilltop models, by contrast, $H_{EI}\simeq H_*$: eternality arises because the classical drift $\dot{\phi}\propto V'(\phi)$ vanishes near the maximum. For hilltops the field region supporting eternal inflation has finite width $\Delta$, so each Hubble volume leaves the quantum-dominated regime after a finite expected lifetime $\langle t\rangle\simeq4\pi^2\Delta^2/H^3$; comparing this with the volume growth $e^{3Ht}$ of an inflating patch, inflation is eternal in the volume-weighted sense (the inflating volume dominates at any finite time) when $\Delta/\langle\delta\phi\rangle_Q>1/\sqrt{3}$. Using leading-order random-walk estimates on the slow-roll attractor, they showed this criterion is satisfied for all hilltop models consistent with CMB data, at any inflationary energy scale: for $p=2$ it reduces to $|\eta|<\sqrt{3}$, where $\eta=M_P^2V''/V$ is the second slow-roll parameter, and for $p>2$ it reduces to an inequality in $N_*$ (the number of e-folds of expansion remaining before the end of inflation when the reference pivot scale $k_*$ exits the horizon), $p$ and $P(k_*)$ that is automatically satisfied for observed values.

Slow-roll evolution is an attractor: a broad set of initial conditions converges to $\dot{\phi}=\dot{\phi}_{SR}$. The source analysis assumes this attractor. But if the field has not relaxed - for example $\dot{\phi}\gg\dot{\phi}_{SR}$, the transient branch studied by Tzirakis and Kinney (2007) - classical rolling dominates over quantum diffusion even near the maximum, and the eternality criterion does not apply. The paper states its own limitations: whether eternal inflation actually occurred is a question of boundary/initial conditions that cannot be settled without an understanding of the pre-inflationary state; the measure problem (how to define probabilities in a spacetime with infinite future four-volume) is avoided by restricting to a finite initial ensemble but is not solved; power-law models push eternality to $H\sim M_P$ where the semiclassical stochastic treatment fails; saddle-point potentials with odd $p$ can avoid eternal inflation for suitable parameters (Kinney and Freese (2015) likewise showed that negative running of the spectral index - a decrease of the tilt $n_s$ of $P(k)$ with increasing scale - can quench eternality); and the lifetime and volume estimates are leading-order rather than full Fokker-Planck results.

Problem Statement

Barenboim, Park and Kinney established that hilltop potentials $V(\phi)\simeq V_0[1-(\phi/\mu)^p]$ with even $p$ support eternal inflation at any energy scale, under two restrictive assumptions: the inflaton evolves on the slow-roll attractor, and eternality is assessed with leading-order random-walk lifetime estimates. The open problem is to determine when hilltop inflation is genuinely eternal - i.e. to decide whether the eternal phase is actually triggered, not merely possible - through the following connected aspects of that single question, all three of which are mandatory:

(1) Rigorous criterion. Derive necessary and sufficient conditions for volume-weighted eternality in hilltop models from the full stochastic dynamics (Fokker-Planck, or - wherever a classical background trajectory exists - the stochastic-$\delta N$ formulation, which tracks the number of e-folds of expansion accumulated along each stochastic realization), replacing the leading-order estimates. The conditions are to hold across the class of hilltop potentials $V(\phi)\simeq V_0[1-(\phi/\mu)^p]$, with their dependence on the model parameters $(V_0,\mu,p)$ and the observables $(N_*,P(k_*))$ made explicit, rather than being established for one fixed potential only - i.e. settle where the necessary condition $P(k)>1$ fails to be sufficient and where the competition between escape from the finite-width eternal region and exponential volume growth is won by volume. The class here is the same one the source treats, with $p$ even, so the odd-$p$ saddle-point case is settled under (3) rather than by this criterion.

(2) Initial-condition dependence. Determine the basin in phase space of initial data $(\phi_0,\dot{\phi}_0)$ - including non-attractor data with $\dot{\phi}\gg\dot{\phi}_{SR}$ and the transient branch that crosses the maximum faster than slow roll - that flows into the quantum-dominated, self-reproducing regime, as opposed to data for which classical evolution traverses the hilltop region without triggering eternal inflation, with the dependence of the basin on the potential parameters made explicit; i.e. decide whether triggering is generic in the initial data or finely tuned. Since no single convention is canonical, the verdict must be stated with respect to an explicit measure or typicality criterion on the $(\phi_0,\dot{\phi}_0)$ plane (candidate conventions - Liouville measure for the second-order system, uniform measure in field value, or the uniform ensemble of initial Hubble patches on a finite initial manifold used by the source - are not equivalent and can support different verdicts for the same dynamics).

(3) Classification completeness. Complete the model-class classification left open by the source: odd-$p$ saddle-point hilltops and potentials with negative running of the spectral index, delineating the parameter-space boundaries at which eternality is quenched.

Addressing (1) and (2) together settles the verdict on whether the eternal phase is triggered; (3) is completed only when the parameter-space quenching boundaries are given for both the odd-$p$ and the negative-running classes.

The objective is a defensible answer to the question the source leaves open: whether hilltop eternal inflation actually occurs, given a well-specified stochastic dynamics and a well-specified treatment of initial conditions. Per the source, the measure problem - defining probabilities on the full eternally inflating spacetime - is a logically separate question and is not part of this problem.

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