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We present the velocity dispersion measurements of four massive$\sim10^{11}M_\odot$ quiescent galaxies at $3.2 < z < 3.7$ based on deep H andK$-$band spectra using the Keck/MOSFIRE near-infrared spectrograph. We findhigh velocity dispersions of order $\sigma_e\sim250$ km/s based on strongBalmer absorption lines and combine these with size measurements based onHST/WFC3 F160W imaging to infer dynamical masses. The velocity dispersion arebroadly consistent with the high stellar masses and small sizes. Together withevidence for quiescent stellar populations, the spectra confirm the existenceof a population of massive galaxies that formed rapidly and quenched in theearly universe $z>4$. Investigating the evolution at constant velocitydispersion between $z\sim3.5$ and $z\sim2$, we find a large increase ineffective radius $0.35\pm0.12$ dex and in dynamical-to-stellar mass ratio$<$log(M$_{\textrm{dyn}}$/M*)$>$ of 0.33$\pm0.08$ dex, with low expectedcontribution from dark matter. The dynamical masses for our $z\sim3.5$ sampleare consistent with the stellar masses for a Chabrier initial mass function(IMF), with the ratio $<$log(M$_{\textrm{dyn}}$/M$^*_{\textrm{Ch}})>$ =-0.13$\pm$0.10 dex suggesting an IMF lighter than Salpeter may be common formassive quiescent galaxies at $z>3$. This is surprising in light of theSalpeter or heavier IMFs found for high velocity dispersion galaxies at$z\sim2$ and cores of present-day ellipticals, which these galaxies are thoughtto evolve into. Future imaging and spectroscopic observations with resolvedkinematics using the upcoming James Webb Space Telescope could rule outpotential systematics from rotation, and confirm these results.

the astrophysical journal. letters

Consistent dynamical and stellar masses with potential light IMF in  massive quiescent galaxies at $3 < z < 4$ using velocity dispersions  measurements with MOSFIRE