# $$N_l$$ spectra for CMB-S4 DC2.0

## 2017-05-15  (Victor Buza)

This posting presents the $$N_l$$ spectra to be used for Phase 2 of the CMB-S4 data challenge and is the analogue of this posting for Phase 1. These noise spectra are based on the most recent optimization, which includes extra low frequency bands. The documentation and evolution of these specifications have been layed out through multiple postings on the CMB S4 wiki. Below is a useful historical recap that builds on the posting linked above (for Phase1), and includes work from many people:

## Experiment Specification and $$N_l$$ fitting

In this posting I do the following things:

• As before, I present the actual $$N_l$$'s that went into the forecasts (interpolated on $$l=[30,320]$$), and I'm providing noise fit parameters so that they directly match the $$N_l$$'s rather than following some map-depth prescription (that, for instance, conserves survey weight; as was done in Table 2 of the June 3rd posting). In order to offer Clem simple to use analytic $$N_l$$'s for {TT,EE,BB} for making noise maps, I parametrize the noise as follows: $N_{l,fit} = \frac{l(l+1)}{2\pi}\frac{\Omega_{pix}}{B_l^2}\left(1+\left(\frac{l}{l_{knee}}\right)^\gamma\right)\sigma_{map}^2$ and fit for {$$\sigma_{map}$$, $$\gamma$$, $$l_{knee}$$}. Above, $$B^2_{l} = \exp(-\frac{l(l+1)\theta^2}{8log(2)})$$, and $$\theta$$ is the FWHM (in radians) of the Gaussian beam, $$\Omega_{pix}=4\pi$$/(# square arminutes on the sky), and $$\sigma_{map}$$ is the map-noise in $$\mu K$$-arcmin.
A few things to keep in mind:
• As mentioned in the March 31st posting, all the scalings are done from achieved BK performances. For frequencies at which we currently do not have data, we scale from the closest frequency that we have an achieved survey weight for, as that is the performance that should guide us. Prior to the CMB-S4 science book, we had a reasonably accurate measurement of our 220 GHz survey weight, but not of the 220 GHz $$N_l$$'s; hence, for the 215 and 270 channels, we assume noise properties similar to BK_150, while scaling from the BK_220 survey weight. This effectively means that we need to parametrize two noise spectra: BK_95 and BK_150. You will see that the table below reflects that.

• Given that TT and EE were not used in the forecasts for Chapter 2 of the S4 Sciencebook, nor in the most recent optimization, there are no direct calculations of the scalings that take us from BK to S4. Here, for TT and EE, I apply the same noise scalings that were applied to BK BB spectra.

As mentioned in the posting above, the effort distributions in the tables below were calculated given an optimized solution for a minimal $$\sigma_r$$, taking into account contributions from foregrounds and CMB lensing. The assumed unit of effort is equivalent to 500 det-yrs at 150 GHz. For other channels, the number of detectors is calculated as $$n_{det,150}\times \left(\frac{\nu}{150}\right)^2$$, i.e. assuming comparable focal plane area. A conversion between the (150 equivalent) number of det-yrs and (actual) number of det-yrs is given for each band. This is just one way to implement a detector cost-function, and other suggestions are welcomed.

Table 2:
Case: $$r=0$$, Total effort: $$10^6$$ det-yrs (150 equiv)

Note: Given the assumed level of foreground complexity, the fully optimal solution presented in this posting does not necessarily divide effort among all of the eleven bands. To combat this, an equal force split among bands in each atmospheric window has been implemented (aside from Window0, which due to Faraday rotation has all of its effort at 20GHz rather than a lower frequency). You will notice an equal amount of 150 equiv det-yrs being assigned to each of the two bands in each of the atmospheric windows. This effect introduces deviations from the optimal solution which are discussed in the posting linked above. All map-depths are quoted for (Q or U, E or B) polarization.

In addition to the numbers in the tables, I also provide the full $$N_{l,\{BB,EE,TT\}}$$, $$\mu K_{CMB}^2$$ to which I fit -- {Nl(BB), r=0, fsky=3%}, {Nl(EE), r=0, fsky=3%}, {Nl(TT), r=0, fsky=3%}. And the actual fits, linked in the headers below.
$$f_{sky}=0.03$$Analytic Fitting parameters (BB)Analytic Fitting parameters (EE)Analytic Fitting parameters (TT)
$$\nu$$,GHz# det-yrs
(150 equiv)
# det-yrs
(actual)
FWHM, arcmin $$\sigma_{map}$$,
$$\mu K$$-arcmin
$$l_{knee}$$ $$\gamma$$ $$\sigma_{map}$$,
$$\mu K$$-arcmin
$$l_{knee}$$ $$\gamma$$ $$\sigma_{map}$$,
$$\mu K$$-arcmin
$$l_{knee}$$ $$\gamma$$
20 30,000 533 76.6 14.69 50 -2.0 15.06 50 -2.0 17.99 175 -4.1
30 22,500 900 76.6 9.36 50 -2.0 9.59 50 -2.0 11.47 175 -4.1
40 22,500 1,600 57.5 8.88 50 -2.0 9.10 50 -2.0 10.88 175 -4.1
85 182,500 58,600 27.0 1.77 50 -2.0 1.81 50 -2.0 2.17 175 -4.1
95 182,500 73,200 24.2 1.40 50 -2.0 1.43 50 -2.0 1.72 175 -4.1
145 67,500 63,075 15.9 2.19 60 -3.0 2.29 65 -3.0 4.89 230 -3.8
155 67,500 72,075 14.8 2.19 60 -3.0 2.29 65 -3.0 4.89 230 -3.8
220 57,500 118,130 10.7 5.61 60 -3.0 5.87 65 -3.0 12.54 230 -3.8
270 57,500 186,300 8.5 7.65 60 -3.0 8.01 65 -3.0 17.11 230 -3.8
Total Degree Scale Effort 690,000 574,420
Total Arcmin Scale Effort 310,000 289,680
Total Effort 1,000,000 864,100