Towards $21$-cm intensity mapping at $z=2.28$ with uGMRT using the tapered gridded estimator III: Foreground removal

Neutral hydrogen (HI) $21$-cm intensity mapping (IM) is a promising probe of the large-scale structures in the Universe. However, a few orders of magnitude brighter foregrounds obscure the IM signal. Here we use the Tapered Gridded Estimator (TGE) to estimate the multi-frequency angular power spectrum (MAPS) $C_{\ell}(\Delta\nu)$ from a $24.4\,\rm{MHz}$ bandwidth uGMRT Band $3$ data at $432.8\,\rm{MHz}$. In $C_{\ell}(\Delta\nu)$ foregrounds remain correlated across the entire $\Delta\nu$ range, whereas the $21$-cm signal is localized within $\Delta\nu\le[\Delta \nu]$ (typically $0.5-1\,\rm{MHz}$). Assuming the range $\Delta\nu>[\Delta \nu]$ to have minimal $21$-cm signal, we use $C_{\ell}(\Delta\nu)$ in this range to model the foregrounds. This foreground model is extrapolated to $\Delta\nu\leq[\Delta \nu]$, and subtracted from the measured $C_{\ell}(\Delta\nu)$. The residual $[C_{\ell}(\Delta\nu)]_{\rm res}$ in the range $\Delta\nu\le[\Delta\nu]$ is used to constrain the $21$-cm signal, compensating for the signal loss from foreground subtraction. $[C_{\ell}(\Delta\nu)]_{\rm{res}}$ is found to be noise-dominated without any trace of foregrounds. Using $[C_{\ell}(\Delta\nu)]_{\rm res}$ we constrain the $21$-cm brightness temperature fluctuations $\Delta^2(k)$, and obtain the $2\sigma$ upper limit $\Delta_{\rm UL}^2(k)\leq(18.07)^2\,\rm{mK^2}$ at $k=0.247\,\rm{Mpc}^{-1}$. We further obtain the $2\sigma$ upper limit $ [\Omega_{{\rm HI}}b_{{\rm HI}}]_{\rm UL}\leq0.022$ where $\Omega_{{\rm HI}}$ and $b_{{\rm HI}}$ are the comoving HI density and bias parameters respectively. Although the upper limit is nearly $10$ times larger than the expected $21$-cm signal, it is $3$ times tighter over previous works using foreground avoidance on the same data.