$B$-mode forecast of CMB-Bh$\overline{a}$rat
  Exploring Cosmic History and Origins (ECHO), popularly known as
`CMB-Bh$\overline{a}$rat', is a space mission that has been proposed to the
Indian Space Research Organisation (ISRO) for the scientific exploitation of
the Cosmic Microwave Background (CMB) at the next level of precision and
accuracy. The quest for the CMB polarization $B$-mode signals, generated by
inflationary gravitational waves in the very early universe, is one of the key
scientific goals of its experimental design. This work studies the potential of
the proposed ECHO instrumental configuration to detect the target
tensor-to-scalar ratio $r \sim 10^{-3}$ at $3\sigma$ significance level, which
covers the predictions of a large class of inflationary models. We investigate
the performance of two different component separation pipelines, NILC and
Commander, for the measurement of $r$ in presence of different physically
motivated models of astrophysical foregrounds. For a simplistic foreground
model (only polarized dust and synchrotron), both component separation
pipelines can achieve the desired sensitivity of ECHO, i.e. $\sigma (r =0) \sim
(0.4 - 0.7)\times 10^{-3}$. NILC performs better than Commander in terms of
bias on recovered $r$ for complex spectral models (power-law and curved
power-law) of the synchrotron emission and complex dust models (dust
decorrelation). Assuming 84 % delensing, we can achieve an improvement of
$\sigma (r = 0)$ by approximately 50 % as compared to the results obtained for
the same configuration without any lensing correction.
