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The Relation between Dynamical Mass-to-Light Ratio and Color for Massive Quiescent Galaxies out to z~2 and Comparison with Stellar Population Synthesis Models

We explore the relation between the dynamical mass-to-light ratio ($M/L$) and rest-frame color of massive quiescent galaxies out to z~2. We use a galaxy sample with measured stellar velocity dispersions in combination with Hubble Space Telescope and ground-based multi-band photometry. Our sample spans a large range in $\log M_{dyn}/L_{g}$ (of 1.6~dex) and $\log M_{dyn}/L_{K}$ (of 1.3~dex). There is a strong, approximately linear correlation between the $M/L$ for different wavebands and rest-frame color. The root-mean-scatter scatter in $\log~M_{dyn}/L$ residuals implies that it is possible to estimate the $M/L$ with an accuracy of ~0.25 dex from a single rest-frame optical color. Stellar population synthesis (SPS) models with a Salpeter stellar initial mass function (IMF) can not simultaneously match $M_{dyn}/L_{g}$ vs. $(g-z)_{rest-frame}$ and $M_{dyn}/L_{K}$ vs. $(g-K)_{rest-frame}$. By changing the slope of the IMF we are still unable to explain the M/L of the bluest and reddest galaxies. We find that an IMF with a slope between $α=2.35$ and $α=1.35$ provides the best match. We also explore a broken IMF with a Salpeter slope at $M<1M_{\odot}$ and $M>4M_{\odot}$ and a slope $α$ in the intermediate region. The data favor a slope of $α=1.35$ over $α=2.35$. Nonetheless, our results show that variations between different SPS models are comparable to the IMF variations. In our analysis we assume that the variation in $M/L$ and color is driven by differences in age, and that other contributions (e.g., metallicity evolution, dark matter) are small. These assumptions may be an important source of uncertainty as galaxies evolve in more complex ways.