  Observations of redshift-space distortions in spectroscopic galaxy surveys
offer an attractive method for measuring the build-up of cosmological
structure, which depends both on the expansion rate of the Universe and our
theory of gravity. Galaxies occupy dark matter halos, whose redshift space
clustering has a complex dependence on bias that cannot be inferred from the
behavior of matter. We identify two distinct corrections on quasilinear scales
(~ 30-80 Mpc/h): the non-linear mapping between real and redshift space
positions, and the non-linear suppression of power in the velocity divergence
field. We model the first non-perturbatively using the scale-dependent Gaussian
streaming model, which we show is accurate at the <0.5 (2) per cent level in
transforming real space clustering and velocity statistics into redshift space
on scales s>10 (s>25) Mpc/h for the monopole (quadrupole) halo correlation
functions. We use perturbation theory to predict the real space pairwise halo
velocity statistics. Our fully analytic model is accurate at the 2 per cent
level only on scales s > 40 Mpc/h. Recent models that neglect the corrections
from the bispectrum and higher order terms from the non-linear real-to-redshift
space mapping will not have the accuracy required for current and future
observational analyses. Finally, we note that our simulation results confirm
the essential but non-trivial assumption that on large scales, the bias
inferred from real space clustering of halos is the same one that determines
their pairwise infall velocity amplitude at the per cent level.
