An explanation for the slopes of stellar cusps in galaxy spheroids

The stellar surface mass density profiles at the centres of typical ~ L_* and lower mass spheroids exhibit power-law 'cusps' with Σ α R^-η, where 0.5 ≲ η ≲ 1 for radii ~ 1-100 pc. Observations and theory support models in which these cusps are formed by dissipative gas inflows and nuclear starbursts in gas-rich mergers. At these comparatively large radii, stellar relaxation is unlikely to account for, or strongly modify, the cuspy stellar profiles. We argue that the power-law surface density profiles observed are a natural consequence of the gravitational instabilities that dominate angular momentum transport in the gravitational potential of a central massive black hole. The dominant mode at these radii is an m = 1 lopsided/eccentric disc instability, in which stars torquing the gas can drive rapid inflow and accretion. Such a mode first generically appears at large radii and propagates inwards by exciting eccentricities at smaller and smaller radii, where M_*(< R) « M_BH. When the stellar surface density profile is comparatively shallow with η < 1/2, the modes cannot efficiently propagate to R = 0 and so gas piles up and star formation steepens the profile. But if the profile is steeper than η = 1, the inward propagation of eccentricity is strongly damped, suppressing inflow and bringing η down again. Together these results produce an equilibrium slope of 1/2 ≲ η ≲ 1 in the potential of the central black hole. These physical arguments are supported by non-linear numerical simulations of gas inflow in galactic nuclei. Together, these results naturally explain the observed stellar density profiles of 'cusp' elliptical galaxies.