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.