Predicting the cosmological constant with the scale-factor cutoff measure

TL;DR

The paper investigates predicting the cosmological constant $\Lambda$ in an eternal-inflation multiverse by applying the scale-factor cutoff measure, a global-time regulator with time variable $t=\ln a$. It first recaps the measure problem and then derives $P(\Lambda)$ under both pocket-based and scale-factor cutoff frameworks, including both $\Lambda>0$ and $\Lambda<0$ scenarios with anthropic prescriptions. A key finding is that the scale-factor cutoff suppresses large positive $\Lambda$ values and, when combined with simple structure-formation weighting, yields a distribution in good agreement with the observed $\Lambda$; it also avoids the youngness bias and the $Q$ catastrophe and is largely insensitive to initial conditions. The work also discusses qualitative implications for the curvature parameter $\Omega$ and density contrast $Q$, suggesting that measurable curvature remains a possibility, and notes robustness of results to choices of $M_G$ and $\Delta\tau$ and potential avoidance of Boltzmann-brain issues. Overall, the scale-factor cutoff emerges as a viable and empirically consistent regulator for multiverse predictions of low-energy observables like $\Lambda$.

Abstract

It is well known that anthropic selection from a landscape with a flat prior distribution of cosmological constant Lambda gives a reasonable fit to observation. However, a realistic model of the multiverse has a physical volume that diverges with time, and the predicted distribution of Lambda depends on how the spacetime volume is regulated. We study a simple model of the multiverse with probabilities regulated by a scale-factor cutoff, and calculate the resulting distribution, considering both positive and negative values of Lambda. The results are in good agreement with observation. In particular, the scale-factor cutoff strongly suppresses the probability for values of Lambda that are more than about ten times the observed value. We also discuss several qualitative features of the scale-factor cutoff, including aspects of the distributions of the curvature parameter Omega and the primordial density contrast Q.

Figures (6)

• Figure 1: The normalized distribution of $\Lambda$ for $\Lambda>0$, with $\Lambda$ in units of the observed value, for the pocket-based measure. The vertical bar highlights the value we measure, while the shaded regions correspond to points more than one and two standard deviations from the mean.
• Figure 2: The normalized distribution of $\Lambda$, with $\Lambda$ in units of the observed value, for the pocket-based measure. The left column corresponds to anthropic hypothesis $A$ while the right column corresponds to anthropic hypothesis $B$. Meanwhile, the top row shows $P(\Lambda)$ while the bottom row shows $P(|\Lambda|)$. The vertical bars highlight the value we measure, while the shaded regions correspond to points more than one and two standard deviations from the mean.
• Figure 3: The normalized distribution of $\Lambda$ for $\Lambda>0$, with $\Lambda$ in units of the observed value, for the scale-factor cutoff. The vertical bar highlights the value we measure, while the shaded regions correspond to points more than one and two standard deviations from the mean.
• Figure 4: The normalized distribution of $\Lambda$, with $\Lambda$ in units of the observed value, for the scale-factor cutoff. The left column corresponds to anthropic hypothesis $A$ while the right column corresponds to anthropic hypothesis $B$. Meanwhile, the top row shows $P(\Lambda)$ while the bottom row shows $P(|\Lambda|)$. The vertical bars highlight the value we measure, while the shaded regions correspond to points more than one and two standard deviations from the mean.
• Figure 5: The normalized distribution of $\Lambda$, with $\Lambda$ in units of the observed value, for anthropic hypothesis $B$ in the scale-factor cutoff. The left panel displays curves for $\Delta\tau=3$ (solid), $5$ (dashed), and $7$ (dotted) $\times 10^9$ years, with $M_G= 10^{12}\,M_\odot$, while the right panel displays curves for $M_G=10^{10}\,M_\odot$ (solid), $10^{11}\,M_\odot$ (dashed), and $10^{12}\,M_\odot$ (dotted), with $\Delta\tau=5\times 10^9$ years. The vertical bars highlight the value of $\Lambda$ that we measure.