This work is the second in a series of works aimed at understanding the effect of residual gain errors in different power spectrum estimations in the presence of strong foreground and exploring potential mitigation techniques. In this work, we have not investigated the different possibilities for the presence of residual gain errors and chosen the values of σR and σI same as σg for most of the discussions. In general, the standard deviation of the real and imaginary parts can be different. We observe here that it is essential to assess the time dependence of the gain accurately, as its inaccurate estimation leads to the time-correlated residual gain errors. In this work, we do not consider the effect of frequency correlation in gain error, and all our estimations are done for correlating visibilities in the same frequency channel. Furthermore, this work also uses the foreground subtraction technique, where we expect to have accurate knowledge of the foreground emissions (Jelić et al. 2008; Ghosh et al. 2012). An alternative method, more regularly exercised in literature, is foreground avoidance. It has been established that the foregrounds to the redshifted 21-cm emissions remain correlated across relatively larger bandwidth (Platania et al. 1998; Santos et al. 2005; Ali et al. 2008; Jelić et al. 2008; Chakraborty et al. 2019b), whereas the H i signal decorrelates faster (Bharadwaj & Pandey 2003; Bharadwaj & Ali 2005). As a result, when the power spectrum is observed as a function of (k∥, k⊥), the foreground emission remains concentrated near the low k∥, inside the ‘wedge’ (Datta et al. 2010; Morales et al. 2012; Vedantham et al. 2012). Note that the smaller frequency separation in multifrequency angular power spectrum contributes to larger k∥ modes of the power spectrum. Hence, the effect of frequency-independent residual gain we see here at zero frequency separation may contribute to bias in the power spectrum beyond the wedge. Moreover, the antenna-based gains are functions of both time and frequency; the residual gain is expected to have correlated frequency dependence. Such frequency-correlated calibration errors couple the foreground power beyond the foreground wedge into the EoR window region of the 2D power spectrum space (Barry et al. 2016; Ewall-Wice et al. 2017; Byrne et al. 2019; Pal et al. 2021). At present, we are working towards expanding the formalism presented in this paper to estimate bias and variance in power spectrum estimate when visibility correlation in different frequencies is considered. Here, we also consider that the gain errors arising from the different antennae are uncorrelated. Though this is a fairly good assumption for the gain arising from electronics in the antenna system itself, the ionospheric effects may introduce correlated gains across the antenna. Furthermore, as the calibration procedure uses baseline-dependent gains to solve for the antenna dependent gains, calibration errors can lead to correlated residual gain errors across the antenna. Moreover, asymmetry in the telescope aperture, mechanical fatigue of telescope structure, etc., can lead to parts of the gain errors correlated across different antennae and even across different days of observations. We are investigating these effects, and the result will be presented in future work. Though the demonstrations here are done with the uGMRT as a model for the interferometer, a similar analysis can be carried out for any telescope of concern, and a prior assessment of the effect of the gain errors can be made using the analytical expression presented here with minimum computation cost. Furthermore, this work emphasizes the importance of estimating and establishing the gain statistics for a given interferometer. Though we use a simple model for the residual gain error here, the calculations that lead to the analytical expression can be readily expanded for a more complicated gain error model. We believe that this work provides significant direction in understanding and planning observations to detect redshifted 21-cm power spectrum.
Citation: Datta et al. 2010
Citation function: Uses