Density of States, Black Holes and the Emergent String Conjecture

We study universal features of the density of one-particle states ρ(E) in weakly coupled theories of gravity at energies above the quantum gravity cutoff Λ, 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 ρ(E) above the black hole threshold M_ 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 Λ. Combining this with thermodynamic arguments we demonstrate that ρ(E) has to grow exponentially for energies Λ≪ E ≪ M_ min. Furthermore we show that the tension of any weakly coupled p-brane with p≥ 1 is bounded from below by Λ^p-1. We use this to argue that any tower of weakly coupled states with mass below Λ 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.