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

Abstract

Measurements of the cosmic 21 cm signal need to achieve a high dynamic range to isolate it from bright foreground emissions. Calibration inaccuracies can compromise the spectral fidelity of the smooth foreground continuum, thereby limiting the dynamic range and potentially precluding the detection of the cosmic line signal. In light of this challenge, recent work has proposed using the calibration-independent closure phase to search for the spectral fluctuations of the cosmic 21 cm signal. However, so far there has been only a heuristic understanding of how closure phases map to the cosmological 21 cm power spectrum. This work aims to establish a more accurate mathematical relationship between closure phases and the cosmological power spectrum of the background line signal. 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 its power. We establish the relationship between this estimate and the cosmological power spectrum using standard Fourier transform techniques and validate it using simulated HERA observations. We find that, statistically, the power spectrum estimate from closure phases is approximately equal to the 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.

Figures (9)

• Figure 1: Illustration of the closure phase concept. The true visibility phases, $\phi_{pq}$, are corrupted by direction-independent antenna-based gains, $\phi_{p}$. Summing the corrupted phases in a loop eliminates the gain phases. The resulting quantity, $\phi_{123}$, is the closure phase.
• Figure 2: Exaggerated illustration of the visibility phase perturbations due to the weak cosmic signal $\delta V_j$ (green arrow). The horizontal and vertical axes represent the real and complex axes of the complex plane, respectively. The red arrow represents the foreground visibility, $V_j^\mathrm{F}$, and the blue arrow represents the sum of the foreground and cosmic H i 21 cm visibilities, $V_j = V_j^\mathrm{F} + \delta V_j$. The resulting phase perturbation is $\delta \phi$.
• Figure 3: Model visibility amplitudes for a HERA equilateral 14.6 m triad (red, blue and green) as a function of local sidereal time (LST). The thick black line represents the effective visibility, $V_\mathrm{eff}$, which provides the scaling of the closure phase delay power spectrum. This example neglects the frequency dependence of the visibility amplitudes.
• Figure 4: HERA antennas used for validation (red). We only used triads that include antenna 0 to compute a complete and independent set of closure phases.
• Figure 5: Normalised closure phase delay window functions for an equilateral 14.6 m triad before (top) and after (bottom) flagging instances with low dynamic range ($<10^{10}$). The colour indicates the corresponding value of the effective visibility, showing that low dynamic range windows tend to be associated with low visibility amplitudes and hence low values of $V_\mathrm{eff}$. The shaded region indicates delays which lay within the horizon limit of the closure phase delay window function plus buffer.