On likely values of the cosmological constant

TL;DR

The paper interrogates anthropic explanations for the small observed value of the vacuum energy $\rho_\Lambda$ and the coincidence $t_0\sim t_\Lambda$. It shows that the commonly assumed flat a priori distribution ${\cal P}_*(\rho)$ is not universally valid when the vacuum energy arises from slowly varying scalar potentials or from quantum-cosmological ensembles, and that the resulting final distributions depend sensitively on the potential shape. In particular, power-law potentials $V(\phi)\propto\phi^n$ with $n>1$ can shift the peak of $\mathcal{P}(\rho_\phi)$ toward anthropically allowed values, sometimes improving agreement with observations, while quantum-cosmology volume weighting can select a single $\rho_\Lambda$ outside the anthropic range. The results extend to quintessence models, where similar priors lead to broadly consistent but still model-dependent predictions for the timing of domination and the observed vacuum energy. Overall, anthropic reasoning remains viable in a wide class of models but is highly sensitive to the detailed form of the potential and cosmological dynamics, offering potential falsifiability through scalar-field phenomenology and gravitational-wave constraints.

Abstract

We discuss models in which the smallness of the effective vacuum energy density $ρ_Ł$ and the coincidence of the time of its dominance $t_Ł$ with the epoch of galaxy formation $t_G$ are due to anthropic selection effects. In such models, the probability distribution for $ρ_Ł$ is a product of an {\it a priori} distribution ${\cal P}_*(ρ_Ł)$ and of the number density of galaxies at a given $ρ_Ł$ (which is proportional to the number of observers who will detect that value of $ρ_Ł$). To determine ${\cal P}_*$, we consider inflationary models in which the role of the vacuum energy is played by a slowly-varying potential of some scalar field. We show that the resulting distribution depends on the shape of the potential and generally has a non-trivial dependence on $ρ_Ł$, even in the narrow anthropically allowed range. This is contrary to Weinberg's earlier conjecture that the {\it a priori} distribution should be nearly flat in the range of interest. We calculate the (final) probability distributions for $ρ_Ł$ and for $t_G/t_Ł$ in simple models with power-law potentials. For some of these models, the agreement with the observationally suggested values of $ρ_Ł$ is better than with a flat {\it a priori} distribution. We also discuss quantum-cosmological approach in which $ρ_Ł$ takes different values in different disconnected universes and argue that Weinberg's conjecture is not valid in this case as well. Finally, we extend our analysis to models of quintessence, with similar conclusions.

Figures (2)

• Figure 1: The probability distribution (\ref{['Pxrec']}) for the effective cosmological constant $\rho_{\phi}$, for different values of $n$. As explained in the text, an observed value of $\Omega_\phi \approx .7$ corresponds to $x_{rec}/\sigma_{rec}^3 \approx .1$. There is at present some uncertainty in this estimate, because a number of assumptions must be made in order to infer the value of $\sigma_{rec}$ from observations. Notice, however, that this value lies at the tail of the $n=1$ curve, corresponding to Weinberg's conjecture (a flat a priori distribution). On the other hand, for $2\leq n \lesssim 5$ the value $x_{rec}/\sigma_{rec}^3 \approx .1$ is well within the broad peak of the distribution. Thus, it is possible that a departure from Weinberg's conjecture may actually fit the observations better (more so if it turns out that the cosmological constant is smaller than .7). The median of each distribution is indicated by a round bead.
• Figure 2: Probability distribution for $t_G/t_{\phi}$, Eq. (\ref{['prob3']}), for different values of $n$. The round beads indicate the median of each distribution. Note that the time coincidence $t_G\sim t_\phi$