# $$N_l$$ spectra for the CMB-S4 data challenge, and updated $$\sigma_r$$ checkpoints

## 2016-12-20  (Victor Buza) Updated 2017-01-31 to include the new $$\sigma_r$$ checkpoints that switch to the $$f_{sky}=3\%$$ baseline, and take into account the slight updates to the bandpasses and NET's as well as turn dust decorrelation on.

This posting presents the $$N_l$$ spectra to be used for Phase 1 of the CMB-S4 data challenge. These spectra are based on the exact same noise specifications that went into the calculations and results presented in the CMB S4 Science Book Chapter 2. 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:

• 2016 March 31: Fisher projections for σ(r) based on achieved performance
This posting describes the development of a Fisher forecasting framework (developed by Victor Buza, Colin Bischoff, and John Kovac), based on achieved performances (and scalings thereof), specifically targeted towards optimizing tensor-to-scalar parameter constraints in the presence of Galactic foregrounds and gravitational lensing of the CMB. I thoroughly describe the methodology, and then present an example forecast for CMB-S4.

• 2016 May 13: σ(r) forecasting checkpoints
In this posting I describe six checkpoint cases that have been agreed upon by S4 forecasters, for an effective effort of $$1,000,000$$ det-yrs (150 equivalent) on the small-patch part of the S4 survey, for $$f_{sky}=[0.01, 0.05, 0.10]$$ and $$r=[0, 0.01]$$; to understand what the (150 equivalent) stands for, please see bullet point three in section two of the first posting. The case definitions were guided by the full optimization and its caveats, described in the first posting, and were grounded in achieved performances (and scalings thereof).

• 2016 June 3: σ(r) forecasting checkpoints, V2
An update on the above posting in which I introduce: a "no delensing" case, an entry stating the level of residual power for each of the six checkpoints, a detailed description of the delensing assumptions, a Knox implimentation of the forecasts, minor clarifications to the table entries, as well as a direct comparison to forecasts from other groups.

• 2016 June 16: S4 Inflation Chapter Plot Suggestions)
In this posting I put up for discussion the key plots for Chapter 2 of the S4 Science Book. There are five types of plots: $$r-n_s$$, $$r-n_t$$, "$$\sigma_r$$ vs Effort," "$$\sigma_r$$ vs $$f_{sky}$$," and the bin-by-bin $$C_l^{BB}$$ tensor contraints. All have been updated to include decorrelation, and the $$\sigma_r$$ plots have an extra panel depicting the "fraction rms of the lensing residual" and "the fraction of total effort going towards delensing."

• 2016 July 8: S4 Inflation Chapter Plot Suggestions, V2
An update on the above posting in which we propose the switch to $$f_{sky}=0.03$$ for the default scenario, float a proposition for $$r=0.003$$ as the non-zero $$r$$ model, and introduce a new and improved version of the bin-by-bin $$C_l^{BB}$$ plot. Also here, a vast number of stylistic and content suggestions from the S4 forecasting community have been implemented. The consensus is to switch to $$f_{sky}=0.03$$ but keep $$r=0.01$$ as the non-zero $$r$$ model.

• 2016 November 4: Tophat Bands for Data Challenge
In this posting Colin offers a more detailed outlook at the chosen bandpasses for the CMB S4 Inflation Chapter forecasts. It describes a slight update for the 215 and 270 bands, offers foreground scaling calculations, relative foreground brightness, and code to produce relevant bandpass files. These definitions have been chosen for the Phase 1 S4 data challenge for the $$r$$ forecasting, meant as the first step towards testing, developing, and comparing various forecasting methods.

• 2016 November 30: First steps to sim input maps
In this posting Clem offers the first steps towards making signal and noise maps to be used for forecasting sims. The initial goal is to make maps which ought to reproduce the results in the Science Book when run through power spectrum estimation and likelihood analysis.

## 1. Experiment Specification and $$N_l$$ fitting

In this posting I do the following things:

• First, in order to match the Science Book noise prescription, I provide the equivalent of Table 2 of the June 3rd posting with the following updates:

• Switch to the baseline $$f_{sky}=3\%$$ choice. This choice was made in a subsequent posting (on July 18th), and thus the distribution of effort is not present in any of the previous tables.

• Turn dust decorrelation on. In the previous tables, in the May 13th and June 3rd postings, I opted for simplicity and turned dust decorrelation off for direct comparisons with other forecasting frameworks. In this posting I turn it back on, since all the CMB-S4 Sciencebook (Chapter 2) plots have decorrelation turned on. In comparing to previous tables, one can note that one consequence of turning that knob is the increased effort towards degree scale separation.

• Second, as before, I present the actual $$N_l$$'s that went into the forecasts (interpolated on $$l=[30,320]$$), but I'm now 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, 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 1:
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 eight bands. To combat this, an equal force split among bands in each atmospheric window has been implemented. 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
$$l_{knee}$$ $$\gamma$$
30 27,500 1,100 76.6 10.59 50 -2.0 10.85 50 -2.0 12.97 175 -4.1
40 27,500 1,960 57.5 10.79 50 -2.0 11.06 50 -2.0 13.22 175 -4.1
85 201,250 64,620 27.0 1.88 50 -2.0 1.93 50 -2.0 2.30 175 -4.1
95 201,250 80,720 24.2 1.54 50 -2.0 1.58 50 -2.0 1.89 175 -4.1
145 68,750 64,420 15.9 2.38 60 -3.0 2.49 65 -3.0 5.31 230 -3.8
155 68,750 73,410 14.8 2.45 60 -3.0 2.56 65 -3.0 5.48 230 -3.8
215 56,250 115,560 10.7 5.30 60 -3.0 5.55 65 -3.0 11.86 230 -3.8
270 56,250 182,250 8.5 7.93 60 -3.0 8.30 65 -3.0 17.72 230 -3.8
Total Degree Scale Effort 707,500 583,870
Total Arcmin Scale Effort 292,500 273,325
Total Effort 1,000,000 857,190

## 2. Using our full framework to arrive at updated $$\sigma_r$$ checkpoints

This section is the updated equivalent of Sections 2 of the 2016 May 13: σ(r) forecasting checkpoints and 2016 June 3: σ(r) forecasting checkpoints, V2 postings. The updates consist in switching to the $$f_{sky}=3\%$$ baseline, turning dust decorrelation on, and using the slightly updated bandpasses and NET's described in Colin's November 4th posting.

As before, in this section I use fully descriptive BPCM's, and the assumptions below, as inputs to the Fisher Forecasting framework, to arrive at $$\sigma_r$$ constraints. However, the $$N_l$$ files above should be compatible with the used BPCM's.

• The Fisher matrix I'm considering is 9-dimensional. The 9 parameters we are constraining are: {$$r, A_{dust}, \beta_{dust}, \alpha_{dust}, A_{sync}, \beta_{sync}, \alpha_{sync}, \epsilon, d_{dust}$$}. Where $$\beta_{dust}$$ and $$\beta_{sync}$$ have Gaussian priors of $$0.11, 0.30$$, the rest have flat priors, and $$d_{dust}$$ stands for dust decorrelation.

• The Fiducial Model for the Fisher forecasting is centered at $$r=0$$, with $$A_{dust,l=80}^{\nu=353} = 4.25$$ (best-fit value from BK14) and $$A_{sync, l=80}^{\nu=23}=3.8$$ (95% upper limit from BK14). The spatial and frequency spectral indeces are centered at $$\beta_{dust}=1.59, \beta_{sync}=-3.10, \alpha_{dust}=-0.42, \alpha_{sync}=-0.6$$, the dust/sync correlation is centered at $$\epsilon=0$$, and the dust decorrelation is 3% at $$l=80$$ for the $$217 \times 353$$ cross (and scales with $$l$$ and $$\nu$$ as described in the March 31st posting).

• As one scales $$f_{sky}$$ up, there are of course a number of caveats to keep in mind for this implimentation, all of which are described in a bullet-point list in the preamble to Figure 3 of this posting.

Table 2:
I marginalize over the fully dimensional Fisher Matrix to arrive at the following $$\sigma_r$$ results. Using the total arcminute scale effort specified in the table above, and our assumptions about delensing (found in Section 1 of the June 3rd posting), one can calculate that it corresponds to $$A_L=0.10$$ , i.e. 90% of the lensing power is removed; $$A_L=1$$ stands for no-delensing included, and $$A_L=0$$ stands for perfect delensing.

$$f_{sky}=0.03$$CMB-S4 BookUpdated
$$\sigma_r(r=0, A_L=0), \times 10^{-3}$$0.580.61
$$\sigma_r(r=0, A_L=0.1), \times 10^{-3}$$0.870.91
$$\sigma_r(r=0, A_L=1), \times 10^{-3}$$3.783.82