On using visibility correlations to probe the HI distribution from the dark ages to the present epoch I: Formalism and the expected signal

Redshifted 21 cm radiation originating from the cosmological distribution of neutral hydrogen (HI) appears as a background radiation in low frequency radio observations. The angular and frequency domain fluctuations in this radiation carry information about cosmological structure formation. We propose that correlations between visibilities measured at different baselines and frequencies in radio-interferometric observations be used to quantify the statistical properties of these fluctuations. This has an inherent advantage over other statistical estimators in that it deals directly with the visibilities which are the primary quantities measured in radio-interferometric observations. Also, the visibility correlation has a very simple relation with power spectrum. We present estimates of the expected signal for nearly the entire post-recombination era, from the dark ages to the present epoch. The epoch of reionization, where the HI has a patchy distribution, has a distinct signature where the signal is determined by the size of the discrete ionized regions. The signal at other epochs, where the HI follows the dark matter, is determined largely by the power spectrum of dark matter fluctuations. The signal is strongest for baselines where the antenna separations are within a few hundred times the wavelength of observation, and an optimal strategy would preferentially sample these baselines. In the frequency domain, for most baselines the visibilities at two different frequencies are uncorrelated beyond \Delta \nu ~ 1 MHz, a signature which in principle would allow the HI signal to be easily distinguished from the continuum sources of contamination.