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We consider gapped fractional quantum Hall states on the lowest Landau levelwhen the Coulomb energy is much smaller than the cyclotron energy. We introducetwo spectral densities, \rho_T(\omega) and \bar \rho_T(\omega), which areproportional to the probabilities of absorption of circularly polarizedgravitons by the quantum Hall system. We prove three sum rules relating thesespectral densities with the shift S, the q^4 coefficient of the staticstructure factor S_4, and the high-frequency shear modulus of the ground state\mu_\infty, which is precisely defined. We confirm an inequality, firstsuggested by Haldane, that S_4 is bounded from below by |S-1|/8. The Laughlinwavefunction saturates this bound, which we argue to imply that systems withground state wavefunctions close to Laughlin's absorb gravitons ofpredominantly one circular polarization. We consider a nonlinear model wherethe sum rules are saturated by a single magneto-roton mode. In this model, themagneto-roton arises from the mixing between oscillations of an internal metricand the hydrodynamic motion. Implications for experiments are brieflydiscussed.

Spectral Sum Rules and Magneto-Roton as Emergent Graviton in Fractional Quantum Hall Effect