We use the relations between aperture stellar velocity dispersion ( \sigma _ { ap } ) , stellar mass ( M _ { SPS } ) , and galaxy size ( R _ { e } ) for a sample of \sim 150 000 early-type galaxies from SDSS/DR7 to place constraints on the stellar initial mass function ( IMF ) and dark halo response to galaxy formation .
We build \Lambda { CDM } 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 \sigma _ { ap } .
Given our model assumptions ( including those in the stellar population synthesis models ) , we find that reproducing the median \sigma _ { ap } vs 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 ( \Delta \log \sigma _ { ap } ) and size-mass ( \Delta \log R _ { e } ) relations .
The slope of this correlation , \partial _ { VR } \equiv \Delta \log \sigma _ { ap } / \Delta \log R _ { e } , varies systematically with galaxy mass from \partial _ { VR } \simeq - 0.45 at M _ { SPS } \sim 10 ^ { 10 } { M } _ { \odot } , to \partial _ { VR } \simeq - 0.15 at M _ { SPS } \sim 10 ^ { 11.6 } { M } _ { \odot } .
The virial fundamental plane ( FP ) has \partial _ { VR } = -1 / 2 , and thus we find the tilt of the observed FP is mass dependent .
Reproducing this tilt requires both a non-universal IMF and a non-universal halo response .
Our best model has mass-follows-light at low masses ( M _ { SPS } \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \sim$ } \hss } \raise 2.0 pt% \hbox { $ < $ } } 10 ^ { 11.2 } { M } _ { \odot } ) and unmodified NFW haloes at M _ { SPS } \sim 10 ^ { 11.5 } { M } _ { \odot } .
The stellar masses imply a mass dependent IMF which is “ lighter ” than Salpeter at low masses and “ heavier ” than Salpeter at high masses .