Compactifying de Sitter space naturally selects a small cosmological constant

We study compactifications of $D$-dimensional de Sitter space with a $q$-form flux down to $D-Nq$ dimensions. We show that for $(N-1)(q-1)\ge 2$ there are double-exponentially or even infinitely many compact de Sitter vacua, and that their effective cosmological constants accumulate at zero. This population explosion of $\mathrm{\Lambda }\ll 1$ de Sitters arises by a mechanism analogous to natural selection.

Figure 1

The Bousso-Polchinski model. Left: the phase diagram of vacua. The lines of isopotential are circles/spheres. Spheres of smaller radius are AdS; spheres of larger radius are dS. If the fluxes get too large, perturbative control is lost. Right: the number density of vacua with a given value of the cosmological constant. The total number of de Sitter minima is finite, and the number density is smooth through ${\mathrm{\Lambda }}_4=0$.

Figure 2

The $N=1$ FR compactification of $D$-dimensional de Sitter. Left: the effective potential as a function of $R$, plotted for several values of the number of conserved flux units $n$. Right: the number density of ${\mathrm{dS}}_{D-q}$ vacua as a function of the lower-dimensional ${\mathrm{\Lambda }}_{D-q}/M_{D-q}^{D-q}$. To make this histogram, we have treated $gn$ as continuous.

Figure 3

The $N=1$ FR compactification of $D$-dimensional Minkowski. Left: the effective potential as a function of $R$, plotted for several values of the number of conserved flux units $n$; the minimum is always AdS. Right: the number density of ${\mathrm{AdS}}_{D-q}$ vacua as a function of ${\mathrm{\Lambda }}_{D-q}/M_{D-q}^{D-q}$.

Figure 4

The number density of granddaughter vacua with a given value of ${\mathrm{\Lambda }}_{D-2q}$ is the sum of the distributions from each of the $(D-q)$-dimensional daughter vacua. Each daughter vacuum contributes a number of granddaughters inversely proportional to its cosmological constant, with a flat distribution that cuts off at its Hubble scale. These contributions pile up at ${\mathrm{\Lambda }}_{D-2q}=0$. (There is also a divergent number density of AdS vacua for every negative ${\mathrm{\Lambda }}_{D-2q}$.)

Figure 5

The phase diagram of $N=2$ compactifications of de Sitter. The de Sitter minima lie in the infinitely long crescent above the Minkowski line and below the decompactification line. The area of this strip is a proxy for the number of dS vacua.