# LAT Frequency Flowdown

### Summary

(Colin Hill writing 4-22-19)

In this posting I present the results of "flowdown" (i.e., optimization) calculations for the frequency channel distribution of the CMB-S4 LATs.

### Setup and Assumptions

I use the noise calculator for the CMB-S4 LATs posted at https://cmb-s4.org/wiki/images/Lat-noise-190311-py.txt

This noise calculator allows one to vary the sky fraction surveyed (fsky) and the number of optics tubes of each type (LF, MF, UHF). For reference, LF = 27 and 39 GHz, MF = 93 and 145 GHz, UHF = 225 and 280 GHz. I do not consider the 20 GHz channel (ULF) in this work, as its resolution is low enough as to likely be irrelevant for LAT science. I consider fsky = 0.1, 0.2, and 0.4, and all possible optics tube configurations with 18 total tubes (assuming there is 19th tube used for the ULF). I also impose a constraint that there is at least one MF tube, since these are clearly the main 'science frequencies'. This yields a total of 3420 possible configurations. Note that I assume two identical copies of the LAT, in order to reduce the computational complexity. If desired, one could run a full optimization of all 36 optics tubes.

I also run calculations using a modified version of the S4 noise calculator (via Matthew Hasselfield) that allows for the possibility of "XHF" optics tubes, where XHF = 281 and 350 GHz. The atmosphere for these tubes is assumed to match CCAT site specifications.

I include Planck data (30 - 353 GHz) in all calculations, as these channels are useful on large angular scales where the S4 atmospheric noise is large.

### Methodology

I focus on temperature-based observables in the following: thermal SZ, kinematic SZ, CMB lensing via the TT quadratic estimator, and the CMB TT power spectrum. I model the temperature sky at all frequencies from 27 to 353 GHz (13 channels, or 15 when including XHF tubes) using the simulated sky maps described in Sec. 2 of https://arxiv.org/abs/1808.07445 . These maps are also available at https://lambda.gsfc.nasa.gov/toolbox/tb_cmbsim_ov.cfm

I use simple Galactic-emission-thresholded sky masks that leave the cleanest 10%, 20%, or 40% of the sky that is visible from Chile. These are identical to the sky masks used in https://arxiv.org/abs/1808.07445

For each fsky and optics tube configuration option, I use a harmonic-space ILC code to obtain "post-component-separation" noise power spectra for the blackbody CMB temperature and Compton-y fields, using the modeled sky power spectra and the per-frequency noise power spectra computed using the S4 calculator (as well as Planck noise, assumed to be white). I consider the option of explicitly "deprojecting" some contaminants using a constrained ILC, which is a robust way to conservatively assess the frequency coverage that may be needed to sufficiently remove these foregrounds. I consider three deprojection options: no deprojection, deprojection of tSZ (for CMB reconstruction) or CMB (for tSZ reconstruction), and deprojection of a fiducial CIB spectrum (for CMB and tSZ reconstruction). The total number of fsky/configuration/deprojection options is thus 10260.

For each fsky and optics tube configuration option (and deprojection option), I then compute the S/N on four observables: tSZ power spectrum, kSZ power spectrum (conservatively considering only ell>3000), CMB lensing power spectrum via the TT estimator (effectively also a proxy for CMB halo lensing, which is dominated by TT), and the CMB TT power spectrum. These S/N values are used as the 'optimization metric' for 'flowdown', i.e., I seek configurations that maximize these S/N values.

### Results

General result, valid for all observables and all deprojection options: going wide (fsky=0.4) is always optimal for the S/N of all these observables. Even for the kSZ power spectrum, the S4 noise is sufficiently low (i.e., we are close enough to hitting sample variance on a range of modes) that going wide is preferred over going deep. I do not list fsky in the optimal configurations below because it is always 0.4.

I also explicitly highlight below the performance of the fiducial configuration used in all previous calculations (fsky=0.4, LF=2, MF=12, UHF=4).

**No deprojection**, aka 'standard ILC':

There are many joint-optimal configurations that yield S/N values within 93% of maximum for all observables. Reasonable-looking ones include: LF=1, MF=10, UHF=7 LF=1, MF=11, UHF=6 LF=1, MF=12, UHF=5 LF=1, MF=13, UHF=4 LF=1, MF=14, UHF=3 LF=2, MF=10, UHF=6 LF=2, MF=11, UHF=5 LF=2, MF=12, UHF=4 LF=2, MF=13, UHF=3 LF=2, MF=14, UHF=2

If XHF tubes are considered, the joint-optimal configurations are LF=1, MF=9, UHF=5, XHF=3 LF=1, MF=10, UHF=3, XHF=4 LF=1, MF=10, UHF=4, XHF=3 LF=1, MF=11, UHF=2, XHF=4 LF=1, MF=11, UHF=3, XHF=3 LF=2, MF=10, UHF=3, XHF=3 which all yield S/N values within 99% of maximum for all observables.

*The performance of the fiducial configuration is good for everything here, yielding S/N > 93% of maximum for kSZ, lensing, and CMB, and S/N ~ 88% of maximum for tSZ. I think this would serve as a reasonable baseline 'flowdown' justification for the fiducial choice of bands. Note that the S/N maxima are computed when allowing XHF tubes, so the upshot is that allowing XHF tubes would yield ~10% gains in S/N (for the no-deprojection case; gains are larger in other cases below).*

**Deprojection of tSZ (for CMB) or of CMB (for tSZ)**:

The joint-optimal configurations are LF=1, MF=4, UHF=13 LF=1, MF=5, UHF=12 LF=1, MF=6, UHF=11 LF=1, MF=7, UHF=10 yielding S/N values within 92% of maximum for all observables.

If XHF tubes are considered, there are many configurations with S/N values > 95% of maximum for all observables. Reasonable looking ones include: LF=1, MF=8, UHF=4, XHF=5 LF=1, MF=8, UHF=5, XHF=4 LF=1, MF=8, UHF=6, XHF=3 LF=2, MF=5, UHF=6, XHF=5

*The performance of the fiducial configuration is good for everything except kSZ, yielding S/N > 94% of maximum for tSZ, lensing, and CMB, but S/N ~ 84% of maximum for kSZ (still not too bad).*

**Deprojection of CIB (for both CMB and tSZ)**:

The joint-optimal configuration is LF=1, MF=6, UHF=11 yielding S/N values within 93% of maximum for all observables.

If XHF tubes are considered, there are many configurations with S/N values > 95% of maximum for all observables. Reasonable looking ones include: LF=1, MF=10, UHF=0, XHF=7 LF=1, MF=10, UHF=1, XHF=6 LF=1, MF=10, UHF=2, XHF=5 LF=1, MF=9, UHF=0, XHF=8 LF=1, MF=9, UHF=1, XHF=7 LF=1, MF=9, UHF=2, XHF=6 LF=1, MF=9, UHF=3, XHF=5 LF=1, MF=9, UHF=4, XHF=4

*The performance of the fiducial configuration is good for everything except kSZ, yielding S/N > 96% of maximum for tSZ, lensing, and CMB, but S/N ~ 82% of maximum for kSZ (still not too bad).*