The cosmological constant and the time of its dominance

TL;DR

The paper develops an anthropic framework in which the cosmological constant $\Lambda$ and the recombination-density contrast $\sigma_{rec}$ are random variables with physics-determined priors. Under the mediocrity principle, typical observers reside in regions where galaxy formation occurs near the epoch of $\Lambda$-domination, explaining why $t_G\sim t_{\Lambda}$ and, together with stellar and biosphere timescales, why $t_0\sim t_G\sim t_{\Lambda}$. Including a cooling boundary $t_{\rm cb}$ and a power-law prior on $\sigma_{rec}$, the most probable $\Lambda$ and $\sigma_{rec}$ lie near the observed values, e.g., $\sigma_{rec}\sim 10^{-3}$, consistent with observational inferences; this supports an anthropic explanation for cosmic coincidences. The approach generalizes to richer models (e.g., multi-field quintessence) and highlights inflationary physics as a source of the priors that shape the observed cosmology.

Abstract

We explore a model in which the cosmological constant $Λ$ and the density contrast at the time of recombination $σ_{rec}$ are random variables, whose range and {\it a priori} probabilities are determined by the laws of physics. (Such models arise naturally in the framework of inflationary cosmology.) Based on the assumption that we are typical observers, we show that the order of magnitude coincidence among the three timescales: the time of galaxy formation, the time when the cosmological constant starts to dominate the cosmic energy density, and the present age of the universe, finds a natural explanation. We also discuss the probability distribution for $σ_{rec}$, and find that it is peaked near the observationally suggested values, for a wide class of {\it a priori} distributions.

Figures (5)

• Figure 1: The probability density per unit logarithmic interval of $t_G/t_{\Lambda}$, Eq.(\ref{['prob3']}), is shown (curve $a$). The maximum is at $t_G/t_{\Lambda}\approx 1.7$ whereas the median value is at $t_G/t_\Lambda\approx 1.5$. The same distribution taking into account the cooling boundary $t_{cb}$ discussed in Section IV is also plotted (curve $b$). The parameters have been chosen so that $t_{cb}=.5\ t_\sigma$ [see Eq. (\ref{["nu'"]})].
• Figure 2: The joint probability density (\ref{['long']}) per unit area in the plane $\log\ (t_\Lambda/t_\sigma)$ (horizontal axis) $\log (t_G/t_\sigma)$ (vertical axis), where $t_{\sigma}$ is defined in Eq. (\ref{['tsigma']}).
• Figure 3: The probability density per unit logarithmic interval of $t_{\Lambda}/t_{cb}$, Eq. (\ref{['tlambda']}), for different values of the parameter $\alpha$ ($\alpha=4,5,10,$ and $15$).
• Figure 4: The probability density per unit logarithmic interval of $\beta\propto \sigma_{rec}^{-3/2}$, Eq. (\ref{['short']}), for $\alpha=4$ and $5$ (thind solid lines). The function $G(\beta)$ is represented by a thick solid line.
• Figure 5: The probability density per unit logarithmic interval of $x_{rec} \sigma_{rec}^{-3}$.