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

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.

💡 Research Summary

The paper addresses a central challenge in 21 cm cosmology: extracting the faint cosmological signal from foregrounds that are orders of magnitude brighter, while also coping with calibration errors that can corrupt the spectral smoothness of the foregrounds. Recent proposals suggested using interferometric closure phases—quantities that are immune to antenna‑based, direction‑independent gain errors—as a way to bypass calibration altogether. However, prior work only offered heuristic arguments for how closure‑phase fluctuations might relate to the underlying 21 cm power spectrum, leaving a gap in quantitative understanding.

The authors begin by modeling the measured visibility on any baseline as the sum of a dominant foreground component (V_F) and a much smaller cosmological perturbation (\delta V) ((|\delta V/V_F|\ll1)). Expanding the visibility phase to first order yields (\phi \approx \phi_F + \Im(\delta V/V_F)). The closure phase, defined as the sum of three baseline phases around a triangle of antennas, therefore splits into an unperturbed foreground closure phase (\phi_F^\triangle) and a perturbation term proportional to the sum of the imaginary parts of (\delta V_j/V_{Fj}) for the three baselines. Crucially, only the component of the cosmological signal that is orthogonal to the foreground vector in the complex plane contributes to the phase perturbation, making closure phases a direct probe of the “phase‑wise dissimilarity” between signal and foreground.

To access line‑of‑sight fluctuations, the authors adopt a delay‑spectrum approach. Rather than Fourier transforming the raw closure phase (which suffers from (2\pi) phase wrapping), they Fourier transform the complex exponential (e^{i\phi^\triangle(\nu)}) across frequency, applying a spectral taper (W(\nu)). In the small‑perturbation limit this yields a closure‑phase delay spectrum (\Psi^\triangle(\tau)) that consists of a foreground‑only term (\Psi_F^\triangle(\tau)) plus a convolution of three quantities: (i) the foreground visibility weighted by the foreground closure phase (\Xi_j(\tau)), (ii) the Fourier transform of the cosmological perturbation (g\delta V_j(\tau)), and (iii) the complex conjugate of the same product for the opposite triangle orientation. This decomposition mirrors the standard visibility delay‑spectrum formalism but now includes an extra foreground weighting factor.

The power spectrum estimator is constructed by squaring the closure‑phase delay spectrum and taking an ensemble average. Assuming statistical isotropy of the cosmological signal and its independence from foregrounds, the only non‑zero correlations involve products of (\Xi_j) and (g\delta V_j). By invoking the standard mapping between delay (\tau) and line‑of‑sight wavenumber (k_\parallel) ((k_\parallel = 2\pi\tau/X)) and between baseline length (b) and transverse wavenumber (k_\perp) ((k_\perp = 2\pi b\nu/Y)), the authors express the expected value of (|\Psi^\triangle|^2) as an integral over the true 21 cm power spectrum (P(k_\perp,k_\parallel)) convolved with a window function that depends on the foreground beam pattern and on the specific triangle geometry. In other words, \