A Search for H I Lyman $\alpha$ Counterparts to Ultra-Fast X-ray
  Outflows
  Prompted by the H I Ly$\alpha$ absorption associated with the X-ray
ultra-fast outflow at -17,300 $\rm km~s^{-1}$ in the quasar PG~1211+143, we
have searched archival UV spectra at the expected locations of H I Ly$\alpha$
absorption for a large sample of ultra-fast outflows identified in XMM-Newton
and Suzaku observations. Sixteen of the X-ray outflows have predicted H I
Ly$\alpha$ wavelengths falling within the bandpass of spectra from either the
Far Ultraviolet Spectroscopic Explorer or the Hubble Space Telescope, although
none of the archival observations were simultaneous with the X-ray observations
in which UFOs were detected. In our spectra broad features with full-width at
half-maximum of 1000 $\rm km~s^{-1}$ have 2-$\sigma$ upper limits on the H I
column density of generally <$2\times10^{13}~\rm cm^{-2}$. Using grids of
photoionization models covering a broad range of spectral energy distributions,
we find that producing Fe XXVI Ly$\alpha$ X-ray absorption with equivalent
widths $>30$ eV and associated H I Ly$\alpha$ absorption with $\rm
N_{HI}<2\times10^{13}~cm^{-2}$ requires total absorbing column densities $\rm
N_{H}>5\times10^{22}~cm^{-2}$ and ionization parameters log $\xi$ > 3.7.
Nevertheless, a wide range of SEDs would predict observable H I Ly$\alpha$
absorption if ionization parameters are only slightly below peak ionization
fractions for Fe XXV and Fe XXVI. The lack of Ly$\alpha$ features in the
archival UV spectra indicates that either the UFOs have very high ionization
parameters, very hard UV-ionizing spectra, or that they were not present at the
time of the UV spectral observations due to variability.
