Universal IMF versus dark halo response in early-type galaxies: Breaking the degeneracy with the Fundamental Plane

We use the relations between aperture stellar velocity dispersion (σ_ap), stellar mass (M_SPS) and galaxy size (R_e) for a sample of ~150 000 early-type galaxies from Sloan Digital Sky Survey/DR7 to place constraints on the stellar initial mass function (IMF) and dark halo response to galaxy formation.We build λ cold dark matter-based mass models that reproduce, by construction, the relations between galaxy size, light concentration and stellar mass, and use the spherical Jeans equations to predict σ_ap. Given our model assumptions (including those in the stellar population synthesis models), we find that reproducing the median σ_ap versus M_SPS relation is not possible with both a universal IMF and a universal dark halo response. Significant departures from a universal IMF and/or dark halo response are required, but there is a degeneracy between these two solutions. We show that this degeneracy can be broken using the strength of the correlation between residuals of the velocity-mass (δlog σ_ap) and size-mass (δlog R_e) relations. The slope of this correlation, ∂_VR = δlog σ_ap/δ logR_e, varies systematically with galaxy mass from ∂_VR ≃ -0.45 at M_SPS ~ 10¹⁰M⊙ to ∂_VR ≃ -0.15 at M_SPS ~ 10^11.6M⊙. The virial Fundamental Plane (FP) has ∂_VR = -1/2, and thus we find that the tilt of the observed FP is mass dependent. Reproducing this tilt requires both a nonuniversal IMF and a non-universal halo response. Our best model has mass-follows-light at low masses (M_SPS ≲ 10^11.2M⊙) and unmodified Navarro, Frenk and White haloes at M_SPS ~ 10^11.5M⊙. The stellar masses imply a mass-dependent IMF which is 'lighter' than Salpeter at low masses and 'heavier' than Salpeter at high masses.