Constraining primordial non-Gaussianity with CMB-21cm cross-correlations?

We investigate the effect of primordial non-Gaussianity on the cross-correlation between the CMB anisotropies and the 21 cm fluctuations from the epoch of reionization. We assume an analytic reionization model and an ionization fraction with $f_{\rm NL}$ induced scale dependent bias. We estimate the angular power spectrum of the cross-correlation of the CMB and 21 cm. In order to evaluate the detectability, the signal-to-noise (S/N) ratio for only a single redshift slice is also calculated for current and future observations, such as CMB observations by Planck satellite and 21 cm observations by Omniscope. The existence of the $f_{\rm NL}$ increases the signal of the cross-correlation at large scales and the amplification does not depend on the reionization parameters in our reionization model. However the cosmic variance is significant on such scales and the S/N ratio is suppressed. The obtained S/N ratio is 2.8 (2.4) for $f_{\rm NL}=10$ (100) in our fiducial reionization model. Our work suggests in the absence of significant foregrounds and systematics, the auto-correlations of 21 cm is a better probe of $f_{\rm NL}$ than the cross-correlations (as expected since it depends on $b^2$), while the cross-correlations contain only one factor of $b$. Nevertheless, it is interesting to examine the cross-correlations between 21 cm and CMB, as the signal-to-noise ratio is not negligible and it is more likely we can rid ourselves of systematics and foregrounds that are common to both CMB and 21 cm experiments than completely clean 21 cm of all of the possible foregrounds and systematics in large scales.