Predictability crisis in inflationary cosmology and its resolution

TL;DR

The paper tackles the predictability crisis in inflationary cosmology arising from gauge-dependent probability distributions in eternally inflating spacetimes. It introduces a gauge-invariant spherical cutoff on the thermalization surface $\Sigma_*$ to define the distribution $P(\chi)$ of slowly varying fields and observable constants, and develops two complementary computational methods: a Fokker-Planck equation approach using the inflaton as a time variable and direct numerical spacetime simulations (both comoving and physical-space). The FP framework yields analytic and numerical expressions for $P_*(\chi)$ and its equilibrium limits, with results that agree with simulation-based distributions to within a few percent for continuous $\chi$; the discrete case, however, remains ill-defined under eternal-inflation sampling. The work provides a principled path to probabilistic predictions of varying constants and density-perturbation spectra in inflationary cosmology, with potential implications for interpreting cosmological data and guiding model-building in the presence of multiple light fields.

Abstract

Models of inflationary cosmology can lead to variation of observable parameters ("constants of Nature") on extremely large scales. The question of making probabilistic predictions for today's observables in such models has been investigated in the literature. Because of the infinite thermalized volume resulting from eternal inflation, it has proven difficult to obtain a meaningful and unambiguous probability distribution for observables, in particular due to the gauge dependence. In the present paper, we further develop the gauge-invariant procedure proposed in a previous work for models with a continuous variation of "constants". The recipe uses an unbiased selection of a connected piece of the thermalized volume as sample for the probability distribution. To implement the procedure numerically, we develop two methods applicable to a reasonably wide class of models: one based on the Fokker-Planck equation of stochastic inflation, and the other based on direct simulation of inflationary spacetime. We present and compare results obtained using these methods.

Figures (13)

• Figure 1: The inflaton potential for the double-well model (\ref{['v']}).
• Figure 2: A two-dimensional simulation for the double-well model at four consecutive moments of proper time: $t = 5 H_0^{-1}$(a), $t = 5.5 H_0^{-1}$(b), $t = 6 H_0^{-1}$(c), $t = 6.5 H_0^{-1}$(d). We evolved a comoving region of initial size $l=H_0^{-1}$ with the initial value of $\varphi=0$ at $t=0$. Inflating regions are shown white, while thermalized regions with $\varphi=+\eta$ and $\varphi=-\eta$ are shown with different shades of grey. Thermalized regions of the same type tend to join in the course of the simulation. For example, regions labeled $A$ and $B$ in snapshot (c) have merged into a single region in snapshot (d).
• Figure 3: Spacetime structure in a one-dimensional simulation for the double-well model. It can be thought of as a spacetime slice through the ($2+1$)-dimensional simulation illustrated in Fig. 2. Inflating regions are white, and thermalized regions of different type are shown with different shades of grey.
• Figure 4: ${\cal S}_1$ is a surface of a constant proper time. It crosses many thermalized regions of different types. ${\cal S}_2$ is a spacelike surface which crosses regions of only one type. ${\cal S}_3$ is a spacelike surface which does not cross any thermalized regions.
• Figure 5: A snapshot of a simulation for the two-field model (\ref{['vphichi']}). Inflating regions are white, while regions that thermalized with different values of $\chi$ are shown with different shades of grey. The shading code is indicated in the bar on the right of the figure.