Establishing a relationship between the cosmological 21 cm power spectrum and interferometric closure phases

Context. Measurements of the cosmic 21 cm background need to achieve a high dynamic range to isolate it from bright foreground emissions. Instrumental calibration inaccuracies can compromise the spectral fidelity of the smooth foreground continuum, thereby limiting the dynamic range of the measurement and potentially precluding the detection of the cosmic line signal. In light of this calibration challenge, recent work has proposed using the calibration-independent closure phase to search for the spectral fluctuations of the cosmic 21 cm background signal. However, so far there has been only a heuristic understanding of the mapping between closure phases and the cosmological power spectrum of the background line signal. Aims. This work aims to establish a more accurate mathematical relationship between closure phase measurements and the cosmological power spectrum of the background line signal. Methods. Building on previous work, we treat the cosmic signal component as a perturbation to the closure phase and use a delay spectrum approach to estimate the power of the perturbing signal. We establish the relationship between this estimate and the cosmological power spectrum using standard Fourier transform techniques, and validate it using simulated observations from the Hydrogen Epoch of Reionization Array (HERA). Results. We find that, statistically, the power spectrum estimate from closure phases is approximately equal to the true cosmological power spectrum convolved with a foreground-dependent window function, provided that the signal-to-foreground ratio is small. Compared with standard approaches, the foreground dependence of the window function results in an increased amount of mode-mixing and a more pronounced proliferation of foreground power along the line-of-sight dimension of the cylindrical power spectrum. These effects can be mitigated by flagging instances where the window function is broad. Crucial to gaining the necessary sensitivity, this mapping will allow us to average the measurements of closure triads of different shapes based on their imprint in cylindrical Fourier space.