${\rm {Ly{\alpha}}}$ flux profile as a function of distance from the source. The full ${\rm {Ly{\alpha}}}$ radiative transfer simulations are computationally challenging. Hence we have simply assumed that the ${\rm {Ly{\alpha}}}$ photon flux reduces as $1/R^2$ with the radial distance $R$. This assumption is consistent with the detailed radiative transfer simulations of, e.g., at the large scales.

-   It is then straightforward to calculate the coupling coefficients (collisional, ${\rm {Ly{\alpha}}}$ coupling and coupling with the CMBR) which are used to generate the spin temperature (${T_{\rm S}}$) profile.

-   The differential brightness temperature ${\delta T_{\rm b}}(\vec{\theta}, \nu)$ can be expressed as, $$\begin{aligned}
     {\delta T_{\rm b}}(\vec{\theta}, \nu)  \!\!\!\! & = & \!\!\!\! 27 ~ x_{\rm HI} (\mathbf{x}, z) [1+\delta_{\rm B}(\mathbf{x}, z)] \left(\frac{{\Omega_{\rm B}}h^2}{0.023}\right) \nonumber\\
    &\times& \!\!\!\!\left(\frac{0.15}{{\Omega_{\rm m}}h^2}\frac{1+z}{10}\right)^{1/2}\left[1-\frac{{T_{\gamma}}(z)}{T_{\rm S}(\mathbf{x}, z)}\right]\,\rm{mK},
    \nonumber \\
    \label{brightnessT}\end{aligned}$$ where $\vec{\theta}$ is the sky direction and $\nu$ is the frequency corresponding to the observed region. The quantities $\delta_{\rm B}(\mathbf{x}, z)$, $x_{\rm HI}(\mathbf{x}, z)$ and ${T_{\gamma}}(z)$ = 2.73 $\times (1+z)$ K denote the baryonic density contrast, the neutral hydrogen fraction and the CMBR brightness temperature respectively at the comoving coordinate $\mathbf{x}$ at a redshift $z=1420~{\rm MHz}/\nu -1$. Here $\mathbf{x}$ is related to the sky position $\vec{\theta}$ by the relation $\mathbf{x} 