[Paper Review] Geometric origin of coincidences and hierarchies in the landscape

Raphael Bousso, Freivogel, Ben+2 more|University of Minnesota Digital Conservancy (University of Minnesota)|Dec 13, 2010

Historical Geography and Cartography3 citations

TL;DR

This paper proposes a geometric measure-based explanation for cosmological coincidences and hierarchies in the string theory landscape, using causal patch, fat geodesic, and a new apparent horizon cutoff to regulate eternal inflation. It shows that for vacua with positive cosmological constant, all three measures predict observers emerge just before vacuum domination and curvature domination, naturally explaining the hierarchy between the Planck scale and the cosmological constant without anthropic assumptions.

ABSTRACT

We show that the geometry of cutoffs on eternal inflation strongly constrains predictions for the timescales of vacuum domination, curvature domination, and observation. We consider three measure proposals: the causal patch, the fat geodesic, and the apparent horizon cutoff, which is introduced here for the first time. We impose neither anthropic requirements nor restrictions on landscape vacua. For vacua with positive cosmological constant, all three measures predict the double coincidence that most observers live at the onset of vacuum domination and just before the onset of curvature domination. The hierarchy between the Planck scale and the cosmological constant is related to the number of vacua in the landscape. These results require only mild assumptions about the distribution of vacua (somewhat stronger assumptions are required by the fat geodesic measure). At this level of generality, none of the three measures are successful for vacua with negative cosmological constant. Their applicability in this regime is ruled out unless much stronger anthropic requirements are imposed.

Motivation & Objective

• To explain the observed cosmological coincidences—such as the timing of vacuum and curvature domination—without relying on anthropic selection principles.
• To investigate whether geometric cutoffs in eternal inflation can naturally produce the observed hierarchy between the Planck scale and the cosmological constant.
• To test the viability of three measure proposals—causal patch, fat geodesic, and a new apparent horizon cutoff—across vacua with positive and negative cosmological constants.
• To determine whether the observed small value of the cosmological constant can be explained by geometric selection effects rather than anthropic fine-tuning.
• To assess the robustness of these measures under weak assumptions about the prior distribution of vacua and observer formation.

Proposed method

• Uses three geometric cutoffs—causal patch, fat geodesic, and a novel apparent horizon cutoff—to regulate the infinite multiverse of eternal inflation.
• Computes the mass inside each cutoff region at observation time, which determines the geometric measure factor M in the probability distribution.
• Applies a factor α representing observations per unit mass per logarithmic time, which is marginalized over due to uncertainty.
• Derives the probability distribution for observation time t_obs, vacuum domination t_Λ, and curvature domination t_c using the product of geometric, prior, and observer factors.
• Analyzes the behavior of the probability distribution in different regions of parameter space, especially near the time of recollapse for Λ < 0.
• Compares results across measures, noting differences in time-reversal behavior and applicability in crunching universes.

Experimental results

Research questions

• RQ1 Can geometric cutoffs in eternal inflation naturally explain why observers appear just before vacuum energy domination and just before curvature domination?
• RQ2 To what extent can the hierarchy between the Planck scale and the cosmological constant be explained by the number of vacua and geometric selection effects?
• RQ3 Do the causal patch, fat geodesic, and apparent horizon measures predict similar observer distributions in vacua with positive Λ?
• RQ4 Why do these measures fail for vacua with negative Λ unless strong anthropic assumptions are imposed?
• RQ5 How does the behavior of the observer distribution differ in recollapsing universes, and what does this imply for the viability of the fat geodesic measure?

Key findings

• For vacua with positive cosmological constant, all three measures predict that most observers live just before vacuum energy domination and just before curvature domination, explaining the cosmological coincidence problem geometrically.
• The hierarchy between the Planck scale and the observed value of the cosmological constant is related to the number of vacua in the landscape, as the probability distribution peaks at a specific scale determined by the cutoff geometry.
• The apparent horizon cutoff, introduced here for the first time, yields results consistent with the causal patch and fat geodesic measures in the Λ > 0 regime.
• For Λ < 0, all three measures fail to predict a small cosmological constant without imposing strong anthropic constraints, indicating a fundamental incompatibility in that regime.
• The fat geodesic measure predicts a preference for negative Λ over positive Λ when α grows as t_obs^4, yielding p_−/p_+ ∼ t_Λ^max, suggesting a tension with observation unless α is strongly constrained.
• The time-reversal symmetry of the fat geodesic measure leads to different behavior in recollapsing universes compared to prior analyses, highlighting a discrepancy with earlier scale factor time cutoff studies.