Detectability of tensor modes in the presence of foregrounds

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 $△(\frac{T}{S})\propto f_{\mathrm{sky}}^{-1/2}$ is significantly degraded and becomes $△(\frac{T}{S})\propto f_{\mathrm{sky}}^{-2}$ 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 $T/S={10}^{-3}$ detection at $3\sigma$ 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 $T/S={10}^{-4}$ detection at $3\sigma$ 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 $T/S={10}^{-2}$ 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 $T/S={10}^{-3}$.

Figure 1

The masks used for cutting the sky based on HEALPix resolution 3 galactic dust maps in Aitoff projection. Various levels of gray show the way the masks were built by adding areas with higher dust temperature. The units are of black body temperature.

Figure 2

$\mathrm{w}{C}_l^{EE}$, $\mathrm{w}{C}_l^{BB}$, and the $\mathrm{lensing}+\mathrm{detector}$ noise level after applying the Gaussian window for $T/S\simeq 1/2$. $\mathrm{w}{C}_l^{EE}$ is represented by the dashed line, $\mathrm{w}{C}_l^{BB}$ by the continuous line, and the $\mathrm{lensing}+\mathrm{detector}$ noise level by the thick horizontal line. These are the values used for resolution 6, where $l_{\mathrm{Ny}}=196$.

Figure 3

$\mathrm{w}{C}_l^{BB}$ for two different values of the optical depth to the reionization $\tau$. The dashed line corresponds to a value of $\tau =0.07$ and the continuous line to $\tau =0.17$. The thick horizontal line is the $\mathrm{lensing}+\mathrm{detector}$ noise level.

Figure 4

$△(\frac{T}{S})$ as a function of sky fraction. The dashed line is computed based on the exact method for resolution 4, while the points with error bars are based on the Monte Carlo method for resolution 6. The continuous lines represent the theoretical scaling of $△(\frac{T}{S})$. The lower line corresponds to the idealized scaling $△(\frac{T}{S})\propto f_{\mathrm{sky}}^{-1/2}$, while the upper line is $△(\frac{T}{S})\propto f_{\mathrm{sky}}^{-2}$ and fits better the actual results for $f_{\mathrm{sky}}>0.7$.

Figure 5

The lines represent the fraction of the sky for which the polarized fraction of the dust thermal emission is less than the polarized signal, as defined in Eq. (10), assuming the polarized fraction of the dust thermal emission to be 1% (top), 0.1% (middle), and 0.01% (bottom). Dashed area is excluded if we require a detection of tensor-to-scalar ratio $\frac{T}{S}$ at a $3\sigma$ confidence. For example, to detect $T/S={10}^{-3}$ at $3\sigma$ we need to observe 15% of the sky and if that is chosen to be the cleanest region of the sky then the dust polarization in this region needs to be cleaned at 0.1% level of its intensity.

Figure 6

$△(\frac{T}{S})$ computed based on the exact method for resolution 4. The dashed line corresponds to an optical depth $\tau =0.07$, while the continuous line corresponds to $\tau =0.17$.

Figure 7

$△(\frac{T}{S})$ for HEALPix resolution 7 for a map confined to Earth’s southern hemisphere, declinations $\delta <-30°$. The graph is obtained by ignoring the reionization peak for the $B$ modes and using the Monte Carlo method, with $1\sigma$ error bars shown on the graph. The continuous line is a cubic spline through the considered points.

Figure 8

The difference in $△(\frac{T}{S})$ for two different types of cuts. The continuous line is obtained for the parallel equatorial cut. Both results are obtained through the exact method for HEALPix resolution 4.