Holographic origin of TCC and the Distance Conjecture

One of the unique features of quantum gravity is the lack of local observables and the completeness of boundary observables. We show that the existence of boundary observables for particles with mass $\underset{t\to \mathrm{\infty }}{\lim }\frac{m}{H}=\mathrm{\infty }$ in scalar field cosmologies where $a(t)\sim t^p$ is equivalent to TCC, which implies $p\le 1$. Moreover, the mass of weakly-coupled particles must decay like $m\lesssim t^{1-2p}$ to ensure that they yield non-trivial boundary observables. This condition can be expressed in terms of the scalar field that drives the cosmology as $m\lesssim \exp (-c\phi )$ where $c$ depends on the scalar potential. The strongest bound we find is achieved for $V\sim \exp (-2\phi \mathrm{/}\sqrt{d-2})$ where $c=1\mathrm{/}\sqrt{d-2}$. These results connect some of the most phenomenologically interesting Swampland conjectures to the most basic version of holography.