# Noise levels in BK15 for S4 sims.

B. Racine

In this short posting, we show how the noise fit changes when choosing different bin center definitions.

## 1. Introduction.

In this posting, our goal is to create a parameter file describing the noise properties of the BK15 dataset. This will be used to construct noise spectra and synthesize noise maps for the S4 map based analysis, in order to check if we recover a $$\sigma(r)$$ similar to the one of BK15. Example of such noise parameters for the different data challenges are shown here.

## 2. Plots.

In figure 3 of the BK15 paper we plot the noise bandpowers normalized by $$\ell_c (\ell_c+1)/(2\pi)$$, where $$\ell_c$$ are the fiducial bandpowers, but we plot them at the position of the bin center computed by weighted_ellbins.m, integrating $$\ell$$ in the bandpower window function (bpwf).

In this posting, we start from the noise bandpowers from the BK15 analysis and fit a 1/f noise model, as described in this posting.

Here we study 3 different cases, where the same $$\ell_c$$ is used for the normalization and for the position:

• We use the fiducial $$\ell_c$$, which are the same for any field or frequency, as is used in the likelihood definition. This is the choice that was made for the previous DCs.
• We use the usual bpwf weighted $$\ell_c$$, as used for the bin position in the BK15 paper, though it is more adapted for a signal flat in $$D_\ell$$.
• We use the flat-$$C_\ell$$ weighted $$\ell_c$$, where we add an extra $$\ell^2$$ factor in front of the bpwf. This is probably the most adapted for the noise bandpowers, which are flat in C_ell.

Note that we fit the noise parameters to the mean of the BK15 noise bandpowers, whereas the previous DCs used the bias removed from the bandpowers which also include a correction of the E-to-B leakage. Since the E-to-B leakage is powerspectrum-method dependent, it might be more "stable" to use the mean of the bandpowers. Here we check that the effect is anyway negligible.

## 3. Tables.

Note that here we report the white noise level in terms of map depth, i.e. the square root of the usual noise level in [$$\mu K^2$$] divided by $$\Omega_{\rm pix}$$, as defined in this posting.

Noise fit parameters using the fiducial $$\ell_c$$
Fields\Parameters $$\ell_{\rm knee}$$ $$\alpha$$ $$\sigma_{\rm map} [\mu \rm K-arcmin]$$
95 GHz TT 149.381 -4.341 10.939
95 GHz EE 64.882 -1.950 7.415
95 GHz BB 72.919 -1.531 6.819
150 GHz TT 228.617 -3.797 9.436
150 GHz EE 65.845 -2.998 4.363
150 GHz BB 61.656 -2.814 4.268
220 GHz TT 220.837 -4.027 83.897
220 GHz EE 59.815 -3.060 39.944
220 GHz BB 58.696 -2.874 38.794
Noise fit parameters using the flat-$$C_\ell$$ weighted $$\ell_c$$.
Fields\Parameters $$\ell_{\rm knee}$$ $$\alpha$$ $$\sigma_{\rm map} [\mu \rm K-arcmin]$$
95 GHz TT 142.640 -5.006 11.223
95 GHz EE 48.396 -2.971 7.991
95 GHz BB 37.384 -2.464 7.793
150 GHz TT 216.217 -4.179 9.733
150 GHz EE 63.744 -3.796 4.387
150 GHz BB 58.033 -4.051 4.325
220 GHz TT 211.406 -4.447 85.796
220 GHz EE 59.083 -3.962 39.740
220 GHz BB 56.702 -4.458 38.767
Noise fit parameters using the flat-$$D_\ell$$ weighted $$\ell_c$$.
Fields\Parameters $$\ell_{\rm knee}$$ $$\alpha$$ $$\sigma_{\rm map} [\mu \rm K-arcmin]$$
95 GHz TT 139.671 -4.816 11.622
95 GHz EE 44.806 -3.929 8.762
95 GHz BB 45.084 -33.934 9.308
150 GHz TT 215.231 -4.051 9.903
150 GHz EE 61.662 -3.538 4.559
150 GHz BB 53.991 -3.978 4.622
220 GHz TT 210.760 -4.309 86.592
220 GHz EE 57.868 -3.504 40.924
220 GHz BB 53.629 -3.773 40.932

## 4. Table for sims.

Here we propose a table to be used for the S4 sims, with some rounded up values, in the spirit of what is reported here.
We use the fiducial $$\ell_c$$ here, as was used for the DC4 numbers.
Note that we obtain map depth at 150 GHz that are only roughly twice higher than, say, DC4 155 GHz. Note again that this is for a $$f_{\rm sky}$$ of roughly 1% whereas the DC4 numbers are for 3%.
Note that we used the same fractional bandwidth as in DC4 or 5: 0.22 for 150 and 220GHz, and 0.24 for 95GHz.

Fields\Parameters 95 150 220
Bandwidth (GHz) 22.8 33 48.4
Beam FWHM (arcmin) 43 30 22
$$\sigma_{\rm map}$$ [$$\mu$$ K-arcmin] TT 10.94 9.44 83.9
$$\ell_{\rm knee}$$ TT 150 230 220
$$\alpha$$ TT -4.3 -3.8 -4
$$\sigma_{\rm map}$$ [$$\mu$$ K-arcmin] EE 7.42 4.36 39.94
$$\ell_{\rm knee}$$ EE 65 65 60
$$\alpha$$ EE -1.9 -3 -3.1
$$\sigma_{\rm map}$$ [$$\mu$$ K-arcmin] BB 6.82 4.27 38.79
$$\ell_{\rm knee}$$ BB 75 60 60
$$\alpha$$ BB -1.5 -2.8 -2.9
$$\ell_{\rm min}$$ 30 30 30
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