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

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Volume 705 (January 2026)

A&A, 705 (2026) A221

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Table 1.

Comparison of terms used in different delay spectrum approaches.

Term	Visibility	Closure phase prev.	This work
$\hat{P} (τ)$ $\widehat{P}(\tau)$	$\propto {\| {\tilde{V}}^{H} (τ) \|}^{2}$ $\propto \left\|\widetilde{V}^{\mathrm{H}}(\tau)\right\|^2$	$\propto \frac{2}{3} V_{eff} {\| {\tilde{Ψ}}_{▿}^{H} (τ) \|}^{2}$ $\propto \frac{2}{3} V_{\mathrm{eff}} \left\|{\widetilde{\Psi}}_{\triangledown}^{\mathrm{H}}(\tau)\right\|^2$	$\propto 2 V_{eff} {\| {\tilde{Ψ}}_{▿}^{H} (τ) \|}^{2}$ $\propto 2 V_{\mathrm{eff}} \left\|{\widetilde{\Psi}}_{\triangledown}^{\mathrm{H}}(\tau)\right\|^2$
${({\hat{V}}_{j}^{F})}^{- 2}$ $\left({\hat{V}}_j^{\mathrm{F}}\right)^{-2}$	–	${(\frac{\int d ν W (ν) V_{j}^{F} (ν)}{\int d ν W (ν)})}^{- 2}$ $\left(\frac{\int d\nu W(\nu) V_j^{\mathrm{F}}(\nu)}{\int d\nu W(\nu)}\right)^{-2}$	$\frac{1}{B_{eff}} \int d ν {\| \frac{W (ν)}{V_{j}^{F} (ν)} \|}^{2}$ $\frac{1}{B_{\mathrm{eff}}} \int d\nu \left\|\frac{W(\nu)}{V_j^{\mathrm{F}}(\nu)}\right\|^2$
W(k_⊥, k_∥; τ)	$〈 {\| \tilde{W} (τ - \frac{X k_{‖}}{2 π}) \tilde{A} (b - \frac{Y k_{⊥}}{2 π}) \|}^{2} 〉$ $\left\langle \left\|\widetilde{W}\left(\tau - \frac{X k_\parallel}{2\pi}\right)\widetilde{A}\left({\mathbf{b}} - \frac{Y {\mathbf{k}}_\perp}{2\pi}\right)\right\|^2\right\rangle$	$\sum_{j = 1}^{3} q_{j} 〈 {\| {\tilde{Ξ}}_{j} (τ - \frac{X k_{‖}}{2 π}) \tilde{A} (b_{j} - \frac{Y k_{⊥}}{2 π}) \|}^{2} 〉$ $\sum_{j = 1}^{3} q_j \left\langle \left\|{\widetilde{\Xi}}_j\left(\tau - \frac{X k_\parallel}{2\pi}\right)\widetilde{A}\left({\mathbf{b}}_{j} - \frac{Y {\mathbf{k}}_\perp}{2\pi}\right)\right\|^2\right\rangle$	$\sum_{j = 1}^{3} q_{j} 〈 {\| {\tilde{Ξ}}_{j} (τ - \frac{X k_{‖}}{2 π}) \tilde{A} (b_{j} - \frac{Y k_{⊥}}{2 π}) \|}^{2} 〉$ $\sum_{j = 1}^{3} q_j \left\langle \left\|{\widetilde{\Xi}}_j\left(\tau - \frac{X k_\parallel}{2\pi}\right)\widetilde{A}\left({\mathbf{b}}_{j} - \frac{Y {\mathbf{k}}_\perp}{2\pi}\right)\right\|^2\right\rangle$

Notes. This table compares terms used in the standard delay spectrum approach (first column), the closure phase approach according to Thyagarajan & Carilli (2020) (second column) and that according to this work (third column). From top to bottom, the rows show the terms used to estimate the power spectrum, the visibility-like term used to compute the effective visibility (cf. Equation (25); relevant only to the closure phase approaches), and the cylindrical window function. For the power spectrum terms, we have omitted parts that are common to all three approaches and use ${\tilde{V}}^{H} (τ)$ $\widetilde{V}^{\mathrm{H}}(\tau)$ to denote the standard visibility delay spectrum of the cosmic H I signal. In the zeroth (‘Term’) column, we have omitted the ▿ subscript to make the terms representative of both the visibility and the closure phase approaches. See the text for the definitions of the other symbols used in this table.

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