  We present a new model for the redshift-space power spectrum of galaxies and
demonstrate its accuracy in modeling the monopole, quadrupole, and hexadecapole
of the galaxy density field down to scales of $k = 0.4 \ h\mathrm{Mpc}^{-1}$.
The model describes the clustering of galaxies in the context of a halo model
and the clustering of the underlying halos in redshift space using a
combination of Eulerian perturbation theory and $N$-body simulations. The
modeling of redshift-space distortions is done using the so-called distribution
function approach. The final model has 13 free parameters, and each parameter
is physically motivated rather than a nuisance parameter, which allows the use
of well-motivated priors. We account for the Finger-of-God effect from centrals
and both isolated and non-isolated satellites rather than using a single
velocity dispersion to describe the combined effect. We test and validate the
accuracy of the model on several sets of high-fidelity $N$-body simulations, as
well as realistic mock catalogs designed to simulate the BOSS DR12 CMASS data
set. The suite of simulations covers a range of cosmologies and galaxy bias
models, providing a rigorous test of the level of theoretical systematics
present in the model. The level of bias in the recovered values of $f \sigma_8$
is found to be small. When including scales to $k = 0.4 \ h\mathrm{Mpc}^{-1}$,
we find 15-30\% gains in the statistical precision of $f \sigma_8$ relative to
$k = 0.2 \ h\mathrm{Mpc}^{-1}$ and a roughly 10-15\% improvement for the
perpendicular Alcock-Paczynski parameter $\alpha_\perp$. Using the BOSS DR12
CMASS mocks as a benchmark for comparison, we estimate an uncertainty on $f
\sigma_8$ that is $\sim$10-20\% larger than other similar Fourier-space RSD
models in the literature that use $k \leq 0.2 \ h\mathrm{Mpc}^{-1}$, suggesting
that these models likely have a too-limited parametrization.
