No Asymptotic Acceleration without Higher-Dimensional de Sitter Vacua

There has recently been considerable interest in the question whether and under which conditions accelerated cosmological expansion can arise in the asymptotic regions of field space of a $d$-dimensional EFT. We conjecture that such acceleration is impossible unless there exist metastable de Sitter vacua in more than $d$ dimensions. That is, we conjecture that `Asymptotic Acceleration Implies de Sitter' (AA$\Rightarrow$DS). Phrased negatively, we argue that the $d$-dimensional `No Asymptotic Acceleration' conjecture (a.k.a.~the `strong asymptotic dS conjecture') follows from the de Sitter conjecture in more than $d$ dimensions. The key idea is that the relevant field-space asymptotics almost always correspond to decompactification and that the only positive energy contribution which decays sufficiently slowly in this regime is the vacuum energy of a higher-dimensional metastable vacuum. This result is in agreement with recent Swampland bounds on the potential in the asymptotics in field space from e.g. the species bound, but is significantly more constraining. We further note that for our asymptotic decompactification limits based on higher-dimensional de Sitter, the Kaluza-Klein scale always falls below the Hubble scale asymptotically. In fact, this occurs whenever $|V'|/V \leq 2 \sqrt{(d+k-2)/k(d-2)}$ asymptotically, with $k$ the number of decompactifying internal directions. This is steeper than what is needed for accelerated expansion.