MaNGA DynPop -- V. The dark-matter fraction versus stellar velocity dispersion relation and stellar initial mass function variations in galaxies: dynamical models and full spectrum fitting of integral-field spectroscopy

Using the final MaNGA sample of 10K galaxies, we investigate the dark matter fraction $f_{\rm DM}$ within one half-light radius $R_{\rm e}$ for about 6K galaxies with good kinematics spanning a wide range of morphologies and stellar velocity dispersion. We employ two techniques to estimate $f_{\rm DM}$: (i) Jeans Anisotropic Modelling (JAM), which performs dark matter decomposition based on stellar kinematics and (ii) comparing the total dynamical mass-to-light ratios $(M/L)_{\rm JAM}$ and $(M_{\ast}/L)_{\rm SPS}$ from Stellar Population Synthesis (SPS). We find that both methods consistently show a significant trend of increasing $f_{\rm DM}$ with decreasing $\sigma_{\rm e}$ and low $f_{\rm DM}$ at larger $\sigma_{\rm e}$. For 235 early-type galaxies with the best models, we explore the variation of stellar initial mass function (IMF) by comparing the stellar mass-to-light ratios from JAM and SPS. We confirm that the stellar mass excess factor $\alpha_{\rm IMF}$ increases with $\sigma_{\rm e}$, consistent with previous studies that reported a transition from Chabrier-like to Salpeter IMF among galaxies. We show that the $\alpha_{\rm IMF}$ trend cannot be driven by $M_{\ast}/L$ or IMF gradients as it persists when allowing for radial gradients in our model. We find no evidence for the total $M/L$ increasing toward the centre. We detect weak positive correlations between $\alpha_{\rm IMF}$ and age, but no correlations with metallicity. We stack galaxy spectra according to their $\alpha_{\rm IMF}$ to search for differences in IMF-sensitive spectral features (e.g. the $\rm Na_{\rm I}$ doublet). We only find marginal evidence for such differences, which casts doubt on the validity of one or both methods to measure the IMF.