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 Λ, 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 left|nabla Vright|/Vge sqrt{4/left(d-2right)} , m ≲ |Λ|1/2, and m ≲ |Λ|1/d in d-dimensional Planck units. The first of these bounds is equivalent to the strong energy condition in Einstein-dilaton gravity and precludes accelerated expansion of the universe. It 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.
Highlights
This does not prove that any bound that is preserved under dimensional reduction is necessarily a universal constraint on quantum gravity
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 Λ, which ideally may pin down ambiguous O(1) constants appearing in the de Sitter Conjecture and the (Anti) de Sitter Distance Conjecture, respectively
The lower bound on the gradient of a scalar field potential proposed in the de Sitter Conjecture [4] is likely true in asymptotic regions of scalar field space, and relatedly eternal inflation is likely forbidden in such regions [5]
Summary
We review how the Weak Gravity Conjecture (WGC) [9] bound and the Repulsive Force Conjecture (RFC) [10] bounds are preserved under dimensional reduction. We do not include a Kaluza-Klein photon, but we do include a massless radion λ(x), which controls the radius of the circle Under such a dimensional reduction, the Einstein-Hilbert term reduces as. This result may be generalized to multiple scalar fields, multiple gauge fields, and general scalar field couplings It holds for the case of P reduced to p = P − 1 and for Kaluza-Klein modes when P = 1 [12]: in all cases, the RFC bound is exactly preserved under dimensional reduction. The bound (2.5) is distinguished by the fact that it dictates the self-repulsiveness of a brane at long distances, and simultaneously it is distinguished in that it is preserved under dimensional reduction Both of these bounds are further distinguished by the fact that they seem to be satisfied in all quantum gravity theories. At least in these two cases, dimensional reduction picks out universal, physically meaningful constraints on quantum gravities
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