A practical theorem on using interferometry to measure the global 21-cm
  signal
  The sky-averaged, or global, background of redshifted $21$ cm radiation is
expected to be a rich source of information on cosmological reheating and
reionizaton. However, measuring the signal is technically challenging: one must
extract a small, frequency-dependent signal from under much brighter spectrally
smooth foregrounds. Traditional approaches to study the global signal have used
single antennas, which require one to calibrate out the frequency-dependent
structure in the overall system gain (due to internal reflections, for example)
as well as remove the noise bias from auto-correlating a single amplifier
output. This has motivated proposals to measure the signal using
cross-correlations in interferometric setups, where additional calibration
techniques are available. In this paper we focus on the general principles
driving the sensitivity of the interferometric setups to the global signal. We
prove that this sensitivity is directly related to two characteristics of the
setup: the cross-talk between readout channels (i.e. the signal picked up at
one antenna when the other one is driven) and the correlated noise due to
thermal fluctuations of lossy elements (e.g. absorbers or the ground) radiating
into both channels. Thus in an interferometric setup, one cannot suppress
cross-talk and correlated thermal noise without reducing sensitivity to the
global signal by the same factor -- instead, the challenge is to characterize
these effects and their frequency dependence. We illustrate our general theorem
by explicit calculations within toy setups consisting of two short dipole
antennas in free space and above a perfectly reflecting ground surface, as well
as two well-separated identical lossless antennas arranged to achieve zero
cross-talk.
