The highly redshifted 21 cm line of neutral hydrogen has become recognized as a unique probe of cosmology from relatively low redshifts (z ~ 1) up through the Epoch of Reionization (z ~ 8) and even beyond. To date, most work has focused on recovering the spherically averaged power spectrum of the 21 cm signal, since this approach maximizes the signal-to-noise in the initial measurement. However, like galaxy surveys, the 21 cm signal is affected by redshift space distortions, and is inherently anisotropic between the line-of-sight and transverse directions. A measurement of this anisotropy can yield unique cosmological information, potentially even isolating the matter power spectrum from astrophysical effects. However, in interferometric measurements, foregrounds also have an anisotropic footprint between the line-of-sight and transverse directions: the so-called foreground wedge. Although foreground subtraction techniques are actively being developed, a foreground avoidance approach of simply ignoring contaminated modes has arguably proven most successful to date. In this work, we analyze the effect of this foreground anisotropy in recovering the redshift space distortion signature in 21 cm measurements at both high and intermediate redshifts. We find the foreground wedge corrupts nearly all of the redshift space signal for even the largest proposed EoR experiments (HERA and the SKA), making cosmological information unrecoverable without foreground subtraction. The situation is somewhat improved at lower redshifts, where the redshift-dependent mapping from observed coordinates to cosmological coordinates significantly reduces the size of the wedge. Using only foreground avoidance, we find that a large experiment like CHIME can place non-trivial constraints on cosmological parameters.
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