Probability Distribution for $Ω$ in Open-universe Inflation

TL;DR

The paper develops a time-reparametrization-invariant framework to compute the probability distribution of the density parameter $\Omega$ in open-universe inflation formed by bubble nucleation. It introduces the ε-prescription regularization to regularize infinite bubble volumes and applies it to a continuous spectrum of bubbles in the Linde–Mezhlumian hybrid-inflation model. A diffusion-based treatment yields how bubble-type distributions weight by the local expansion history via $a_{*}(\phi_{0})$ and instanton actions, producing a final $d\mathcal P/d\Omega$ that can peak near $\Omega=1$ or at intermediate values, with anthropic considerations via $\nu_{\text{civ}}(\Omega)$ able to shift peaks. The results quantify how model parameters control the likely range of present-day curvature in open universes, offering a principled way to connect fundamental inflationary dynamics with observable cosmology.

Abstract

The problem of making predictions in eternally inflating universe that thermalizes by bubble nucleation is considered. A recently introduced regularization procedure is applied to find the probability distribution for the ensemble of thermalized bubbles. The resulting probabilities are shown to be independent of the choice of the time parametrization. This formalism is applied to models of open ``hybrid'' inflation with $Ω<1$. Depending on the parameters of the model, the probability distribution for $Ω$ is found to have a peak either very close to $Ω=1$, or at an intermediate value of $Ω$ in the range $0.03\lesssim Ω<1$.

Figures (6)

• Figure 1: A conformal diagram of bubbles nucleating in an inflating background. The shaded regions of spacetime inside the bubbles are thermalized. The thermalization surfaces are the boundaries of these regions. They have an infinite $3$-volume which can be regularized by introducing a cutoff hypersurface $t=t_c$ and keeping only the part of the volume below this hypersurface.
• Figure 2: Geometry of the bubble interior.
• Figure 3: The shape of the potential $V_0(\sigma )$ in Eq. (\ref{['HybridPot']}).
• Figure 4: Probability distribution $d\tilde{{\cal {P}}}(\Omega )/d\Omega$ with $\phi _{\max }^2/\phi _{*}^2=100$, shown logarithmically up to a normalization. (a): $\mu =0.01$. The peak at $\Omega \approx 1$ is extremely sharp; the ratio of the values at $\Omega =.99$ and at $\Omega =0$ is $\sim \exp 20$, while the peak value differs from that at $\Omega =0$ by a factor of $\sim \exp \left( 25000\right)$. (b): $\mu =0.5$, there are two local maxima near $\Omega =0$ and $\Omega =1$. (c): $\mu =2$. The function monotonically decreases. The maximum value at $\Omega =0$ differs from a typical intermediate value ($\Omega \sim 1/2$) by a factor of $\sim \exp \left( 60\right)$.
• Figure 5: Dependence of $\nu _{\text{c}iv}$ on $\Omega$ for $\bar{\delta}=10^{-2}$. The ratio of the maximum and the minimum values is $\sim \exp \bar{\delta} ^{-2}$. The origin on the vertical axis is arbitrary.