  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.
