Synergies between radio, optical and microwave observations at high
  redshift
  We study synergies between three promising methods to measure $2<z<5$
large-scale structure in the next decade. Optical spectroscopic surveys are the
most mature, but become increasingly difficult at $z>2$ and suffer from
interloper problems even for spectroscopic surveys. Intensity mapping of the
21-cm signal can cover large volumes with exquisite fidelity, but is limited
both by loss of information to foreground cleaning and by lack of knowledge of
the mean signal. Cosmic microwave background (CMB) lensing is theoretically
very clean, but ultimately measures just the projected variations in density.
We find that cross-correlation between optical and radio can significantly
improve the measurement of growth rate. Combining these with the CMB provides a
promising avenue to detecting modified gravity at high redshifts, in particular
by independently probing the Weyl and Newtonian potentials and by strengthening
control of systematics. We find that cross-correlating a Stage {\sc ii} 21-cm
survey with DESI quasars with a reasonable brightness temperature prior could
enable measurements of the growth rate $f\sigma_8$ at sub 3\% and sub 8\%
levels at $z = 3, 4$, representing a factor of 4 and 8 improvement over
constraints obtainable from DESI quasars alone. Similarly, cross-correlating
21-cm data with a futuristic LBG survey to $m_{UV}<24.5$ over 1000 square
degrees will make possible $f\sigma_8$ measurements at close to 1\% at $z = 3$
and 3\% at $z = 4$, and improve similar constraints at $z = 5$ by close to a
factor of 3 to sub-10\% precision. Combining the above with CMB lensing from a
Stage 4 CMB survey and LSST data can additionally constrain the gravitational
slip $\gamma$ parameter to similar precision at these redshifts, enabling us to
test the predictions of general relativity at large scales.
