Detectability of tensor modes in the presence of foregrounds

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In inflationary models gravitational waves are produced in the early universe and generate $B$-type polarization in the cosmic microwave background (CMB). Since $B$ polarization is only generated by gravity waves it does not suffer from the usual cosmic variance. A perfect decomposition of the CMB into $B$-modes and $E$-modes would require data from the entire sky, which in practice is not possible because of the foreground contaminants. This leads to mixing of $E$ polarization into $B$, which introduces cosmic variance contamination of $B$ polarization and reduces sensitivity to gravity wave amplitude even in absence of detector noise. We present numerical results for the uncertainty in the tensor-to-scalar ratio using the Fisher matrix formalism for various resolutions and considering several cuts of the sky, using the foreground model based on dust maps and assuming 90 GHz operating frequency. We find that the usual scaling $△(\frac{T}{S})\ensuremath{\propto}{f}_{\mathrm{sky}}^{\ensuremath{-}1/2}$ is significantly degraded and becomes $△(\frac{T}{S})\ensuremath{\propto}{f}_{\mathrm{sky}}^{\ensuremath{-}2}$ for ${f}_{\mathrm{sky}}>0.7$. This dependence is affected only weakly by the choice of sky cuts. To put this into a context of what is required level of foreground cleaning, to achieve a $T/S={10}^{\ensuremath{-}3}$ detection at $3\ensuremath{\sigma}$ one needs to observe 15% of the sky as opposed to naive expectation of 0.3%. To prevent contamination over this large sky area at required level one must be able to remove polarized dust emission at or better than 0.1% of unpolarized intensity, assuming the cleanest part of the sky has been chosen. To achieve $T/S={10}^{\ensuremath{-}4}$ detection at $3\ensuremath{\sigma}$ one needs to observe 70% of the sky, which is only possible if dust emission is removed everywhere over this region at 0.01% level. Reaching $T/S={10}^{\ensuremath{-}2}$ should be easier: 1% of the sky is needed over which polarized emission needs to be removed at 1% of unpolarized intensity if the cleanest region is chosen. These results suggest that foreground contamination may make it difficult to achieve levels below $T/S={10}^{\ensuremath{-}3}$.