# Performance-based Fisher optimization for CMB-S4, v3

## 2018-02-18  (Victor Buza)

This posting is in the style of the v2 optimization posting, which lead to the definition of the noise spectra that in turn were used to define Data Challenge 2.0. In this posting I try to update the previous optimization with the most up to date baseline that most recently appeared in the CDT report. The difference between v2 and v3 is the lack of 10/15 GHz channels, and the improved optimization resolution.

## 1. Worked-out Example; Experiment Specification

Below, similar to Section 2 and 3 of the posting linked above, I present an application of this framework to an optimization grounded in achieved performance.

• For this particular example I assume nine S4 channels: {20, 30, 40, 85, 95, 145, 155, 215, 270} GHz, two WMAP channels: {23, 33} GHz and seven Planck channels: {30, 44, 70,100, 143, 217, 353} GHz.

• This example assumes 0.5m apertures, and scales the beams accordingly. Note: This is not true for the 20 GHz channel, which assumes a beam equivalent to the beam at 30 GHz (i.e we assume that the aperture scales by the required amount to keep the beam fixed to the one at 30).

• 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. The projections run out to a total of 3,000,000 det-yrs (3,000,000 det-yrs, if all at 150 GHz, would be equivalent to 500,000 detectors operating for 6 yrs -- this seems like a comfortable upper bound for what might be conceivable for S4. S4 scale surveys seem likely to be in the range of $$10^6$$ to $$2.5\times10^6$$ det-yrs).

• I first want to emphasize that the NET numbers that follow are only used to determine the scalings between different channels, and NOT to calculate sensitivities. All sensitivities are based on achieved performance. The ideal NET's per detector are assumed to be {214, 177, 224, 270, 238, 309, 331, 747, 1281} $$\mu\mathrm{K}_\mathrm{CMB} \sqrt{s}$$. This is the last column of the table in the Band Definition posting. Note: These updated NET's are calculated for a 100mK bath, as opposed to 250mK before, and are therefore lower than before.

• The BPWF's, ell-binning, and ell-range are assumed to be l=[30,330]; yielding 9 bins with nominal centers at ell of {37.5, 72.5, 107.5, 142.5, 177.5, 212.5, 247.5, 282.5, 317.5}.

• The Fiducial Model for the Fisher forecasting is centered at $$r$$ of 0, with $$A_{dust} = 4.25$$ (best-fit value from BK14) and $$A_{sync}=3.8$$ (95% upper limit from BK14). The spatial and frequency spectral indeces are centered at $$\beta_{dust}=1.59, \beta_{sync}=-3.10, \alpha_{dust}=-0.42, \alpha_{sync}=-0.6$$, and the dust/sync correlation is centered at $$\epsilon=0$$. I also introduce $$\delta_{dust}$$ -- a dust decorrelation parameter (that is always ON), and $$\delta_{sync}$$ -- a sync decorrelation parameter (that is always ON). The dust decorrelation parametrization is exactly as described in Section 5 of the initial optimization posting. The synchrotron decorrelation parameter is being introduced here for the first time, and has the same frequency and spatial form as the dust decorrelation parameter, but is normalized at (23GHz, 33GHz, l=80). While this parameter is let to freely vary in the Fisher optimization, I do center it at zero, given that we have no good information on this value.

• The Fisher matrix is 10-dimensional. The 10 parameters we are constraining are: {$$r, A_{dust}, \beta_{dust}, \alpha_{dust}, A_{sync}, \beta_{sync}, \alpha_{sync}, \epsilon, \delta_{dust}, \delta_{sync}$$}. Where $$\beta_{dust}$$ and $$\beta_{sync}$$ have Gaussian priors of $$0.11, 0.30$$, and the rest have flat priors.

• As before, I implement delensing as an extra band in the optimization. See the description underneath Table 1 in this posting for a more in depth description on how this is done.

## 2. Parameter Constraints; $$\sigma_r$$ performance

Table 0:
Numbers corresponding to the v2 Optimization for DC2.0

$$f_{sky}$$Type of det-yrsEffort in the following channelTotal Effort
$$10$$$$15$$$$20$$$$30$$$$40$$ $$85$$ $$95$$ $$145$$ $$155$$ $$220$$ $$270$$ $$DL$$
0.03 150 GHz Equivalent 1.7k 1.7k1.7k 25.0k 25.0k 187.5k 187.5k 67.5k 67.5k 57.5k 57.7k320k 1000k
Actual 8 16 30 1.0k 1.8k 60.2k 75.2k 63.0k 72.0k 118.1k 186.3k 299.0k 882k
Actual (1/4) 2 4 8 250 450 15.0k 18.8k 15.8k 18.0k 29.5k 46.6k 74.8k 220.5k

Table 1:
Numbers corresponding to this $$N_l$$ posting for DC2.0.

Going from the previous table to this one we had decided to drop the 10 and 15 GHz channels, and up the percentage of effort in this lower window from 0.5% to 3.0%. This of course deviated from the optimal path, and necessitated that some channels get slightly less effort (in particular 30/40, 85/95, and DL).

$$f_{sky}$$Type of det-yrsEffort in the following channelTotal Effort
$$20$$$$30$$$$40$$ $$85$$ $$95$$ $$145$$ $$155$$ $$220$$ $$270$$ $$DL$$
0.03 150 GHz Equivalent 30.0k 22.5k 22.5k 182.5k 182.5k 67.5k 67.5k 57.5k 57.7k310k 1000k
Actual 530 900 1.6k 58.6k 73.2k 63.0k 72.0k 118.1k 186.3k 289.7k 864k
Actual (1/4) 130 225 400 14.7k 18.3k 31.5k 18.0k 29.5k 46.6k 72.4k 216k

Table 2:
For the CDT report, it was decided that slightly more effort was necessary in order to achieve the desired science goal. Instead of re-optimizing, it was decided to simply scale the Table 1 numbers by $$\sqrt{7/6}$$ (except the 20 GHz channel). The numbers in this table correspond to the numbers in the CDT report.
$$f_{sky}$$Type of det-yrsEffort in the following channelTotal Effort
$$20$$$$30$$$$40$$ $$85$$ $$95$$ $$145$$ $$155$$ $$220$$ $$270$$ $$DL$$
0.03 150 GHz Equivalent 30.0k 26.1k 26.1k 211.7k 211.7k 78.3k 78.3k 66.7k 66.7k359.6k 1160k
Actual 530 1040 1.9k 68.0k 84.0k 72.0k 84.0k 136.0k 216.0k 336.0k 1000k
Actual (1/4) 130 260 470 17.0k 21.0k 18.0k 21.0k 34.0k 54.0k 84.0k 250k

Table 3:
Updated Optimization numbers. Comparing to Table 2, we can see that the full optimization agrees perfectly with the scaled version in the 85/95 and 220/DL channels, while preferring slightly more effort in 30/40, 145/155, and slightly less effort in 270.
$$f_{sky}$$Type of det-yrsEffort in the following channelTotal Effort
$$20$$$$30$$$$40$$ $$85$$ $$95$$ $$145$$ $$155$$ $$220$$ $$270$$ $$DL$$
0.03 150 GHz Equivalent 32.5k 37.5k 37.5k 208.7k 208.7k 85.0k 85.0k 62.5k 62.5k360.0k 1160k
Actual 580 1500 2.7k 67.0k 84.0k 79.0k 90.0k 134.0k 203.0k 336.0k 1000k
Actual (1/4) 145 375 670 17.0k 21.0k 20.0k 22.5k 34.0k 51.0k 84.0k 250k