# Signal-to-noise on foregrounds from split band difference maps

(C. Bischoff)
UPDATE : Fixed low-ℓ noise parameters

In this posting, I estimate the signal-to-noise on foregrounds that we would achieve with difference maps constructed from adjacent frequency bands. This doesn't feed directly into any of our current CMB-S4 Data Challenge analyses, but you could imagine using it for foreground template construction. Perhaps more useful for our current purposes, this signal-to-noise quantifies how well a difference between adjacent bands can differentiate between CMB, which should difference away cleanly, vs foregrounds, which don't. Using adjacent frequency bands minimizes risks from decorrelation but also decreases signal-to-noise, because of the short frequency lever arm. Fortunately, CMB-S4 will have lots of signal-to-noise…

Estimates are simply calculated from power spectrum signal and noise expectation values.

My dust model is defined by the following parameters:

• $$A_d$$: Dust power, in $$\mu K^2$$, at 353 GHz and ℓ=80
• $$\beta_d$$: Spectral index for modified blackbody law
• $$T_d$$: Dust blackbody temperature. Fixed to 19.6 K
• $$\alpha_d$$: Power law slope for dust power (in $$\mathcal{D}_\ell$$). Fixed to -0.4.

My synchrotron model is defined by the following parameters:

• $$A_s$$: Sync power, in $$\mu K^2$$, at 23 GHz and ℓ=80
• $$\beta_s$$: Spectral index for synchrotron power law SED
• $$\alpha_s$$: Power law slope for sync power (in $$\mathcal{D}_\ell$$). Fixed to -0.4.

The foreground expectation value for a difference map power spectrum is calculated by evaluating the foreground model at the pivot frequency (353 GHz for dust, 23 GHz for sync), then multiply by $$(f_a - f_b)^2$$, where $$f_a$$ and $$f_b$$ are the factors to scale a foreground map from the pivot frequency to frequency maps $$a$$ and $$b$$. These scale factors are calculated using the appropriate foreground SED and the tophat bandpasses listed in Table 1.

For the noise expectation value, I simply add the $$\mathcal{N}_\ell$$ for the two maps under the assumption that their noise is independent. The $$\mathcal{N}_\ell$$ of the maps are calculated for a white + 1/ℓ model using parameters from Table 1. To compare with the foreground signal calculations, I convert the noise spectra from $$\mathcal{C}_\ell$$ to $$\mathcal{D}_\ell$$, i.e. multiply by $$\ell (\ell+1) / 2\pi$$, and divide by $$B_\ell^2$$.

Results of the calculation are shown in the Figure 1 and Figure 2 pagers. In addition to plots showing the foreground and noise spectra, you can select “S/N ratio” to show the signal-to-noise ratio per mode, calculated as the square root of the $$\mathcal{D}_\ell$$ ratio.

I ran for a range of $$A_d$$, $$\beta_d$$, $$A_s$$, and $$\beta_s$$ values. For reference the best-fit model from the most recent BICEP/Keck analysis has $$A_d = 4.7 \, \mu K^2$$, $$\beta_d = 1.6$$, $$A_s = 1.5 \, \mu K^2$$, and $$\beta_s = -3.0$$. For the default pager clicks, which roughly correspond to the BICEP/Keck best-fit model, we see that the 145–155 difference map achieves S/N ∼ 1 around ℓ = 50, and this improves if the dust is brighter or has a flatter SED. The 220–270 difference map will have higher signal-to-noise on dust, but using a longer frequency lever arm and at frequencies further from our main CMB frequencies. The 85–95 difference map approaches signal-to-noise of 1 on dust at low ℓ. Of course these difference maps will contain strong statistical detections of foregrounds even if they are too noisy to detect individual modes.

The 85–95 difference map has low signal-to-noise on synchrotron for all of the models considered here. The 30–40 difference map will measure synchrotron with S/N > 1 for $$\ell \lesssim 130$$, but requires a large extrapolation up to the CMB frequencies.

Code used for this posting: cmbs4_splitbands.py