Dimensional Reduction and (Anti) de Sitter Bounds

Abstract: Dimensional reduction has proven to be a surprisingly powerful tool for delineating the boundary between the string landscape and the swampland. Bounds from the Weak Gravity Conjecture and the Repulsive Force Conjecture, for instance, are exactly preserved under dimensional reduction. Motivated by its success in these cases, we apply a similar dimensional reduction analysis to bounds on the gradient of the scalar field potential $V$ and the mass scale $m$ of a tower of light particles in terms of the cosmological constant $\Lambda$, which ideally may pin down ambiguous $O(1)$ constants appearing in the de Sitter Conjecture and the (Anti) de Sitter Distance Conjecture, respectively. We find that this analysis distinguishes the bounds $|\nabla V|/V \geq \sqrt{4/(d-2)}$, $m \lesssim |\Lambda|^{1/2}$, and $m \lesssim |\Lambda|^{1/d}$ in $d$-dimensional Planck units. The first of these bounds precludes accelerated expansion of the universe in Einstein-dilaton gravity and is almost certainly violated in our universe, though it may apply in asymptotic limits of scalar field space. The second bound cannot be satisfied in our universe, though it is saturated in supersymmetric AdS vacua with well-understood uplifts to 10d/11d supergravity. The third bound likely has a limited range of validity in quantum gravity as well, so it may or may not apply to our universe. However, if it does apply, it suggests a possible relation between the cosmological constant and the neutrino mass, which (by the see-saw mechanism) may further provide a relation between the cosmological constant problem and the hierarchy problem. We also work out the conditions for eternal inflation in general spacetime dimensions, and we comment on the behavior of these conditions under dimensional reduction.