  We examine the signature of dynamic (redshift-space) distortions and
geometric distortions (including the Alcock-Paczynski effect) in the context of
the galaxy power spectrum measured in upcoming galaxy redshift surveys.
Information comes from both the baryon acoustic oscillation (BAO) feature and
the broadband power spectrum shape. Accurate modeling is required to extract
this information without introducing systematic biases in the result. We
consider an analytic model for the power spectrum of dark matter halos in
redshift space, based on the distribution function expansion, and compare with
halo clustering measured in N-body simulations. We forecast that the
distribution function model is sufficiently accurate to allow the inclusion of
broadband information on scales down to k~0.2 h/Mpc, with somewhat better
accuracy for higher bias halos. Compared with a BAO-only analysis with
reconstruction, including broadband shape information can improve unbiased
constraints on distance measures H(z) and D_A(z) by ~30% and 20%, respectively,
for a galaxy sample similar to the DESI luminous red galaxies. The gains in
precision are larger in the absence of BAO reconstruction. Furthermore,
including broadband shape information allows the measurement of structure
growth, through redshift-space distortions. For the same galaxy sample, the
distribution function model is able to constrain f*sigma_8 to ~2%, when
simultaneously fitting for H(z) and D_A(z). We discuss techniques to optimize
the analysis of the power spectrum, including removing modes near the
line-of-sight that are particularly challenging to model, and whether these
approaches can improve parameter constraints. We find that such techniques are
unlikely to significantly improve constraints on geometry, although they may
allow higher precision measurements of redshift-space distortions.
