A Stringy Mechanism for A Small Cosmological Constant

TL;DR

The authors address the puzzling smallness of the observed cosmological constant by proposing a string-theoretic mechanism in which the vacuum energy distribution is dominated by products of random parameters in flux compactifications. Using Mellin-transform techniques, they analyze the distributions of products, ratios, and sums of random variables and show that such products generically peak (or diverge) at zero, a feature that can carry over to the cosmological constant when the vacuum energy is set by moduli stabilization. In Type IIB large-volume scenarios, particularly in a single-modulus Kähler uplifting model, they demonstrate how extremum and stability constraints modulate the naive peaking but do not erase it, allowing $P(\Lambda)$ to favor small $\Lambda$ values under plausible randomness in $W_0$, $A_1$, and related parameters. They further provide simple scenarios and quantitative estimates indicating that moduli stabilization dynamics can suppress $\Lambda$ by several orders of magnitude, though achieving the observed value remains contingent on multi-moduli couplings and zeros in key parameters. Overall, the work provides a probabilistic, string-dynamics-based mechanism for a naturally small cosmological constant, with clear predictions on how the distribution of $\Lambda$ behaves under moduli stabilization and parameter randomness.

Abstract

Based on the probability distributions of products of random variables, we propose a simple stringy mechanism that prefers the meta-stable vacua with a small cosmological constant. We state some relevant properties of the probability distributions of functions of random variables. We then illustrate the mechanism within the flux compactification models in Type IIB string theory. As a result of the stringy dynamics, we argue that the generic probability distribution for the meta-stable vacua typically peaks with a divergent behavior at the zero value of the cosmological constant. However, its suppression in the single modulus model studied here is modest.

Figures (5)

• Figure 1: Some examples of the product distribution of $z$ where each $x_j$ has an uniform ([-1,1]) distribution (on the left hand side) or a normal (Gaussian) distribution with variance $\sigma_j = 1$ (on the right hand side). The product distribution $P(z)$ is for $n =$ 1 ($z=x_1$, solid brown curve), $n=2$ ($z=x_1x_2$, red dashed curve), and $n=3$ ($z=x_1x_2x_3$, blue dotted curve), respectively. The curves on the left are given in (\ref{['s11']}). The curves on the right are given by the Meijer-G function (\ref{['MeijerGn']}), which reduces to the modified Bessel function of the second kind for $n=2$.
• Figure 2: A comparison of the peakiness (or peaking behavior). The left hand side shows the product distributions for $z$ while the right hand side shows cumulative probability distribution beginning from $z=0$. The curves are for $z = x_1$ (solid horizontal line), $z = x_1^2$ (red dashed curve), and $z=x_1^2 x_2$ (blue dotted curve) in each plot, assuming uniform distributions for $x_1$ and $x_2$. .
• Figure 3: The probability distribution of $\Lambda$ in the Bousso-Polchinski type (\ref{['BPsum']}). The probability distribution in the $J=1$ ($J=10, 20$) case peaks at zero ($\sim 50, 200$). Population moves toward large $\Lambda$ as the number of types of fluxes $J$ increases. As a result, tiny cosmological constant is allowed but not preferred in the presence of more than one cycle.
• Figure 4: The plot shows behaviors of the bracket in (\ref{["Westphal's potential"]}) (see Rummel:2011cd) at $C\sim 3.55, 3.65, 3.75, 3.85, 3.95$ from bottom to top respectively, for an illustration of the stability constraint.
• Figure 5: The regions that contribute to $P(z,c)$ in the the $c$-$z$ plane. The left plot shows the valid region in the cases of (\ref{['simple random model from westphal']}), (\ref{['constrained product distribution with xi']}), and (\ref{['constrained product distribution without xi']}), where the region is surrounded by $c=c_1$ and $c = c_0/(1-z)$. The right plot is for (\ref{['including peaking in a_1']}), where the middle curve $c=1/(1-y_m^{9/2} z)$ which separates two regions corresponding each function in (\ref{['cz distribution for the most complicated one']}). Here we assume $\hat{\xi}=1, a_1=1, \gamma_1 = 1$ in the figures.