Measuring the 3D Clustering of Undetected Galaxies Through Cross
  Correlation of their Cumulative Flux Fluctuations from Multiple Spectral
  Lines
  We discuss a method for detecting the emission from high redshift galaxies by
cross correlating flux fluctuations from multiple spectral lines. If one can
fit and subtract away the continuum emission with a smooth function of
frequency, the remaining signal contains fluctuations of flux with frequency
and angle from line emitting galaxies. Over a particular small range of
observed frequencies, these fluctuations will originate from sources
corresponding to a series of different redshifts, one for each emission line.
It is possible to statistically isolate the fluctuations at a particular
redshift by cross correlating emission originating from the same redshift, but
in different emission lines. This technique will allow detection of clustering
fluctuations from the faintest galaxies which individually cannot be detected,
but which contribute substantially to the total signal due to their large
numbers. We describe these fluctuations quantitatively through the line cross
power spectrum. As an example of a particular application of this technique, we
calculate the signal-to-noise ratio for a measurement of the cross power
spectrum of the OI(63 micron) and OIII(52 micron) fine structure lines with the
proposed Space Infrared Telescope for Cosmology and Astrophysics. We find that
the cross power spectrum can be measured beyond a redshift of z=8. Such
observations could constrain the evolution of the metallicity, bias, and duty
cycle of faint galaxies at high redshifts and may also be sensitive to the
reionization history through its effect on the minimum mass of galaxies. As
another example, we consider the cross power spectrum of CO line emission
measured with a large ground based telescope like CCAT and 21-cm radiation
originating from hydrogen in galaxies after reionization with an interferometer
similar in scale to MWA, but optimized for post-reionization redshifts.
