Large-$N$ Random Matrix Gravity and the Double Hierarchy Problem

Abstract

Why are the cosmological constant, electroweak and Planck scales so different? This ``double hierarchy" problem, where $\Lambda \ll M^2_{EW} \ll M^2_p$, is one of the most pressing in fundamental physics. We show that in a theory of $N$ randomly coupled massive gravitons at the electroweak scale, these scales are linked precisely by such a double hierarchy for large $N$, with intriguing cosmological consequences. Surprisingly, in all the physical scales, only one massless graviton emerges which is also, effectively, the only one that is coupled to matter, giving rise to standard Einstein gravity, with $M_p^2\, G_{\mu\nu}= T_{\mu\nu}$ at large $N$. In addition there is a tower of massive gravitons, the lightest of which can drive late-time acceleration. In this scenario, the observed empirical relation $\Lambda\, M_p^2 \sim M_{EW}^4$ as well as the double hierarchy, arise naturally since $\Lambda \sim M^2_{EW}/\sqrt{N}$ and $M^2_p \sim \sqrt{N}M_{EW}^2$.