# $$\sigma_r$$ forecasting checkpoints

## 2016-05-13  (Victor Buza)

In this posting I prescribe six reference cases for $$r$$ forecasting, and use the machinery described in detail in this posting to arrive at a $$\sigma_r$$. These six cases that have been agreed upon are 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 posting linked above. The case definitions are guided by the full optimization and its caveats, described in the posting above, and are grounded in achieved performances (and scalings thereof).

## 1. Experiment Specification

The three tables below should contain all the information necessary for any forecasting machinery to be able to arrive at a corresponding $$\sigma_r$$. The cyan colored boxes represent the $$0^{th}$$ order information for a first-pass forecast. In addition, for a more detailed approach, full bandpower window functions, bandpasses, and $$N_l$$'s have also been provided.

Table 1:
This table offers some case-independent experiment specifications. For each of the eight channels considered, there is a center frequency, a $$\Delta\nu/\nu$$ for a simple bandpass prescription, a beamwidth, a fully detailed bandpass (caculated using a Chebyshev poly-filtering), and a BPWF prescription.

$$\nu$$,GHz$$\Delta \nu/\nu$$FWHM, arcminBandpass, [$$\nu$$,$$B_{\nu}$$]BPWF
30 0.3076.6 bandpass30.txt bpwfS4.dat
40 0.3057.5 bandpass40.txt
85 0.2427.0 bandpass85.txt |
95 0.2424.2 bandpass95.txt |
145 0.2215.9 bandpass145.txt |
155 0.2214.8 bandpass155.txt |
215 0.2210.7 bandpass215.txt |
270 0.188.5 bandpass270.txt |

As mentioned in the text 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.

For each case, I list the fraction of effort spent towards solving the foreground separation problem ("degree-scale effort") and reducing the lensing contribution ("arcmin-scale effort"). For the arcminute scale effort, to calculate an effective level of residual lensing, an experiment with $$1$$ $$arcmin$$ resolution, and mapping speed equivalent to the 145 channel was assumed, hence the conversion between (150 equiv) and (actual); however, all that is necessary to take away (on the delensing front) from these tables are the arcmin-scale map-depths.

For the various cases, given a fixed effort, the map-depths and $$N_l$$'s are scaled accordingly with $$f_{sky}$$.

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 the previous 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. Furthermore, as one increases $$f_{sky}$$, the optimization algorithm allocates more resources towards the foreground separation problem, and results in some bands "kicking in" after the total $$10^6$$ det-yrs threshold. In particular the 145/155 bands for the $$f_{sky}=[0.05, 0.10]$$ cases are significantly underused in the optimal solution. To combat this, some effort from the 85/95 channels has been reallocated towards the 145/155 channels. Both of these effects introduce deviations from the optimal solution which are discussed in the posting linked above.
$$f_{sky}=0.01$$$$f_{sky}=0.05$$$$f_{sky}=0.10$$
$$\nu$$,GHz# det-yrs
(150 equiv)
# det-yrs
(actual)
map depth,
$$\mu K$$-arcmin
$$N_l$$, $$\mu K_{CMB}^2$$# det-yrs
(150 equiv)
# det-yrs
(actual)
map depth,
$$\mu K$$-arcmin
$$N_l$$, $$\mu K_{CMB}^2$$# det-yrs
(150 equiv)
# det-yrs
(actual)
map depth,
$$\mu K$$-arcmin
$$N_l$$, $$\mu K_{CMB}^2$$
30 16,250 650 5.62 Nl_r0_fsky1 28,750 1,150 9.46 Nl_r0_fsky5 28,750 1,150 13.37 Nl_r0_fsky10
40 16,250 1,160 5.73 28,750 2,040 9.64 28,750 2,040 13.63
85 127,500 40,940 0.96 170,000 54,590 1.86 186,250 59,810 2.52
95 127,500 51,140 0.79 170,000 68,190 1.53 186,250 74,710 2.06
145 87,500 81,760 0.84 87,500 81,760 1.87 87,500 81,760 2.65
155 87,500 93,430 0.87 87,500 93,430 1.94 87,500 93,430 2.74
215 55,000 112,990 2.14 42,500 87,310 5.43 55,000 112,990 6.76
270 55,000 178,200 3.20 42,500 137,700 8.14 55,000 178,200 10.11
Total Degree Scale Effort 572,500 560,270 ~ ~ 657,500 526,180 ~ ~ 715,000 604,100 ~ ~
Total Arcmin Scale Effort 427,500 399,470 0.38~ 342,500 320,050 0.94 ~ 285,000 266,320 1.47 ~
Total Effort 1,000,000 959,740 ~~ 1,000,000 846,230 ~~ 1,000,000 870,420 ~~

Table 3:
Case: $$r=0.01$$, Total effort: $$10^6$$ det-yrs (150 equiv)

Note: See Note from Table 2.
$$f_{sky}=0.01$$$$f_{sky}=0.05$$$$f_{sky}=0.10$$
$$\nu$$,GHz# det-yrs
(150 equiv)
# det-yrs
(actual)
map depth,
$$\mu K$$-arcmin
$$N_l$$, $$\mu K_{CMB}^2$$# det-yrs
(150 equiv)
# det-yrs
(actual)
map depth,
$$\mu K$$-arcmin
$$N_l$$, $$\mu K_{CMB}^2$$# det-yrs
(150 equiv)
# det-yrs
(actual)
map depth,
$$\mu K$$-arcmin
$$N_l$$, $$\mu K_{CMB}^2$$
30 28,750 1,150 4.23 Nl_r01_fsky1 41,250 1,650 7.89 Nl_r01_fsky5 41,250 1,650 11.16 Nl_r01_fsky10
40 28,750 2,040 4.31 41,250 2,930 8.04 41,250 2,930 11.38
85 151,250 48,570 0.88 195,000 62,620 1.74 211,250 67,830 2.36
95 151,250 60,670 0.72 195,000 78,220 1.43 211,250 84,730 1.94
145 50,000 46,720 1.11 50,000 46,720 2.48 50,000 46,720 3.50
155 50,000 53,390 1.45 50,000 53,390 2.56 50,000 53,390 3.62
215 42,500 87,310 2.43 42,500 87,310 5.44 42,500 87,310 7.69
270 42,500 137,700 3.64 42,500 137,700 8.13 42,500 137,700 11.50
Total Degree Scale Effort 545,000 437,550 ~ ~ 657,500 470,540 ~ ~ 690,000 482,280 ~ ~
Total Arcmin Scale Effort 455,000 425,170 0.37 ~ 342,500 320,050 0.95 ~ 310,000 289,680 1.41 ~
Total Effort 1,000,000 862,720 ~~ 1,000,000 790,590 ~~ 1,000,000 771,960 ~~

## 2. Worked-out implementation; parameter constraints and $$\sigma_r$$ performance

In this section, I use fully descriptive BPCM's (more details about the treatment of noise and signal in the formation of the BPCM can be found in Section 1 of this posting) as inputs to the Fisher Forecasting framework (described in Sections 1 and 2 of the same posting linked above). However, the $$N_l$$ files above should be compatible with the used BPCM's.

• The Fisher matrix I'm considering is 8-dimensional. The 8 parameters we are constraining are: {$$r, A_{dust}, \beta_{dust}, \alpha_{dust}, A_{sync}, \beta_{sync}, \alpha_{sync}, \epsilon$$}. Where $$\beta_{dust}$$ and $$\beta_{sync}$$ have Gaussian priors of $$0.11, 0.30$$, and the rest have flat priors. For a more detailed description of the parameters used in the model, see the "Multicomponent Model" subsection in Section 1 of the posting linked above.

• The Fiducial Model for the Fisher forecasting is centered at either $$r$$ of 0 or 0.01, 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 $$\beta_{dust}=1.59, \beta_{sync}=-3.10, \alpha_{dust}=-0.42, \alpha_{sync}=-0.6$$, and the dust/sync correlation is centered at $$\epsilon=0$$.

• For an illustration of the full dimensionality and the degeneracies of the Fisher matrix being constrained in this implementation, see this Fisher ellipse plot for the $$r=0$$, $$f_{sky}=0.01$$ case.

• 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 the above posting.

Table 4:
For each of the six cases decribed above, I marginalize over the fully dimensional Fisher Matrix to arrive at the following $$\sigma_r$$ results.
$$f_{sky}=0.01$$$$f_{sky}=0.05$$$$f_{sky}=0.10$$
$$\sigma_r(r=0), \times 10^{-4}$$5.528.069.51
$$\sigma_r(r=0.01), \times 10^{-3}$$1.851.441.42

It is worth noting that this framework has been validated against simulations at the BKP and BK14 noise-levels, and further development is in progress to perform validations with map-level simulations of skies with various degrees of complexity, provided by groups such as Jo/Ben/David or Ghosh/Aumont/Boulanger.