Density of States, Black Holes and the Emergent String Conjecture

We study universal features of the density of one-particle states $\rho(E)$ in weakly coupled theories of gravity at energies above the quantum gravity cutoff $\Lambda$, defined as the scale suppressing higher-derivative corrections to the Einstein--Hilbert action. Using thermodynamic properties of black holes, we show that in asymptotically flat spacetimes, certain features of $\rho(E)$ above the black hole threshold $M_{\rm min}$ are an indicator for the existence of large extra dimensions, and cannot be reproduced by any lower-dimensional field theory with finitely many fields satisfying the weak energy condition. Based on the properties of gravitational scattering amplitudes, we argue that there needs to exist a (possibly higher-dimensional) effective description of gravity valid up to the cutoff $\Lambda$. Combining this with thermodynamic arguments we demonstrate that $\rho(E)$ has to grow exponentially for energies $\Lambda \ll E \ll M_{\rm min}$. Furthermore we show that the tension of any weakly coupled $p$-brane with $p\geq 1$ is bounded from below by $\Lambda^{p+1}$. We use this to argue that any tower of weakly coupled states with mass below $\Lambda$ has to be a Kaluza--Klein (KK) tower. Altogether these results indicate that in gravitational weak-coupling limits the lightest tower of states is either a KK tower, or has an exponentially growing degeneracy thereby resembling a string tower. This provides evidence for the Emergent String Conjecture without explicitly relying on string theory or supersymmetry.