Radio interferometric observations of the redshifted 21-cm signal directly measures the complex visibilities that are the Fourier components of the intensity distribution on the sky. The radio telescope typically has a finite beam that allows us to use the ‘flat-sky’ approximation. Ideally, the fields κ and δT are expanded in the basis of spherical harmonics. For convenience, we use a simplified expression for the angular power spectrum by considering the flat sky approximation whereby we can use the Fourier basis. Using this simplifying assumption, we may approximately write the cross-correlation angular power spectrum as (Dash & Guha Sarkar 2021)
$$\begin{eqnarray}
C^{ T \kappa }_\ell &amp;=&amp; \frac{1 }{\pi (\chi _2- \chi _1)} \sum _{\chi _1}^{\chi _2} \frac{\Delta \chi }{\chi ^2} ~ \mathcal {A}_T (\chi) \mathcal {A}_\kappa (\chi) D_{+}^2 (\chi) \int _0^{\infty } \mathrm{ d}k_{\parallel }\\
&amp;&amp;\times \,\left[ 1 + \beta _T(\chi) \frac{k_{\parallel }^2}{k^2} \right] P (k),
\end{eqnarray}$$where $k = \sqrt{k_{\parallel }^2 + \left(\frac{\ell }{\chi } \right)^2 }$, D+ is the growing mode of density fluctuations, and βT = f/bT is the redshift distortion factor – the ratio of the logarithmic growth rate f and the bias function and bT(k, z). The redshift-dependent function $\mathcal {A}_{T}$ is given by (Bharadwaj & Ali 2005; Datta, Choudhury & Bharadwaj 2007; Guha Sarkar et al. 2012)
(6)$$\begin{eqnarray}
\mathcal {A}_{T} = 4.0 \, {\rm {mK}} \, b_{T} \, {\bar{x}_{\rm H\,{\small I}}}(1 + z)^2\left(\frac{\Omega _{b0} h^2}{0.02} \right) \left(\frac{0.7}{h} \right) \left(\frac{H_0}{H(z)} \right) .
\end{eqnarray}$$The quantity bT(k, z) is the bias function defined as ratio of H i-21 cm power spectrum to dark matter power spectrum $b_T^2 = P_{\mathrm{ H}\,{\small I}}(z)/P(z)$. In the post-reionization epoch z  6, the neutral hydrogen fraction remains with a value ${\bar{x}_{\rm H\,{\small I}}} = 2.45 \times 10^{-2}$ (adopted from Noterdaeme et al. 2009; Zafar et al. 2013). The clustering of the post-reionization H i is quantified using bT. On sub-Jean’s length, the bias is scale dependent (Fang et al. 1993). However, on large-scale the bias is known to be scale independent. The scale above which the bias is linear, is however sensitive to the redshift. Post-reionization H i bias is studied extensively using N-body simulations (Bagla et al. 2010; Guha Sarkar et al. 2012; Sarkar et al. 2016; Carucci, Villaescusa-Navarro & Viel 2017). These simulations demonstrate that the large-scale linear bias increases with redshift for 1  z  4 (Marín et al. 2010). We have adopted the fitting formula for the bias bT(k, z) as a function of both redshift z and scale k (Guha Sarkar et al. 2012; Sarkar et al. 2016) of the post-reionization signal as
(7)$$\begin{eqnarray}
b_{T}(k,z) = \sum _{m=0}^{4} \sum _{n=0}^{2} c(m,n) k^{m}z^{n} .
\end{eqnarray}$$The coefficients c(m, n) in the fit function are adopted from Sarkar et al. (2016).
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