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 $\exp \left(\pi \sqrt{\frac{2}{3}(c-1)J}\right)\mathrm{/}\sqrt{2J}$ in any twist interval at or above $(c-1)\mathrm{/}12$, with a known twist-dependent prefactor. This proves that the large $J$ spectrum becomes dense even without averaging over spins. For twists below $(c-1)\mathrm{/}12$ 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.