Synergies between radio, optical and microwave observations at high redshift

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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 fσ8 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 mUV<24.5 over 1000 square degrees will make possible fσ8 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 γ parameter to similar precision at these redshifts, enabling us to test the predictions of general relativity at large scales.