  We investigate the adiabatic index $\Gamma_{\mathrm{th}}$ of hot and dense
nuclear matter from chiral effective field theory and find that the results are
systematically larger than from typical mean field models. We start by
constructing the finite-temperature equation of state from chiral two- and
three-nucleon forces, which we then use to fit a class of extended Skyrme
energy density functionals. This allows for modeling the thermal index across
the full range of densities and temperatures that may be probed in simulations
of core-collapse supernovae and neutron star mergers, including the low-density
inhomogeneous mixed phase. For uniform matter we compare the results to
analytical expressions for $\Gamma_{\mathrm{th}}$ based on Fermi liquid theory.
The correlation between the thermal index and the effective masses at nuclear
saturation density is studied systematically through Bayesian modeling of the
nuclear equation of state. We then study the behavior of $\Gamma_{\mathrm{th}}$
in both relativistic and non-relativistic mean field models used in the
astrophysical simulation community to complement those based on chiral
effective field theory constraints from our own study. We derive compact
parameterization formulas for $\Gamma_{\mathrm{th}}$ across the range of
densities and temperatures encountered in core collapse supernovae and binary
neutron star mergers, which we suggest may be useful for the numerical
simulation community.
