Lensing reconstruction using redshifted 21 centimeter fluctuations

We investigate the potential of second generation measurements of redshifted 21 cm radiation from before and during the epoch of reionization (EOR) to reconstruct the matter density fluctuations along the line of sight. To do so we generalize the quadratic methods developed for the cosmic microwave background (CMB) to 21 cm fluctuations. We show that the three- dimensional signal can be decomposed into a finite number of line- of- sight Fourier modes that contribute to the lensing reconstruction. Our formalism properly takes into account correlations along the line of sight and uses all the information contained in quadratic combinations of the signal. In comparison with the CMB, 21 cm fluctuations have the disadvantage of a relatively scale invariant unlensed power spectrum, which suppresses the lensing effect. The smallness of the lensing effect is compensated by using information from a range of observed redshifts. We estimate the size of experiments that are needed to measure this effect. With a square kilometer of collecting area and a maximal baseline of 3 km, one can achieve lensing reconstruction noise levels an order of magnitude below CMB quadratic estimator constraints at L = 1000, and map the deflection field out to less then a tenth of a degree (L > 2000) within a season of observations on one field. Statistical lensing power spectrum detections will be possible to subarcminute scales, even with the limited sky coverage that currently conceived experiments have. One should be able to improve constraints on cosmological parameters by using this method. With larger collecting areas or longer observing times, one could probe arcminute scales of the lensing potential and thus individual clusters. We address the effect that foregrounds might have on lensing reconstruction with 21 cm fluctuations.