Universality of the Microcanonical Entropy at Large Spin

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with c>1. Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin-J operators must grow like expπ23(c-1)J/2J in any twist interval at or above (c-1)/12, with a known twist-dependent prefactor. This proves that the large J spectrum becomes dense even without averaging over spins. For twists below (c-1)/12 we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin-J operators must go to zero as J becomes large.