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).

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.

\(\nu\),GHz | \(\Delta \nu/\nu\) | FWHM, arcmin | Bandpass, [\(\nu\),\(B_{\nu}\)] | BPWF |
---|---|---|---|---|

30 | 0.30 | 76.6 | bandpass30.txt | bpwfS4.dat |

40 | 0.30 | 57.5 | bandpass40.txt | ↑ |

85 | 0.24 | 27.0 | bandpass85.txt | | |

95 | 0.24 | 24.2 | bandpass95.txt | | |

145 | 0.22 | 15.9 | bandpass145.txt | | |

155 | 0.22 | 14.8 | bandpass155.txt | | |

215 | 0.22 | 10.7 | bandpass215.txt | | |

270 | 0.18 | 8.5 | 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 \(f_{sky}\).

\(f_{sky}=0.01\) | \(f_{sky}=0.05\) | \(f_{sky}=0.10\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

\(\nu\),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 | ~ | ~ |

\(f_{sky}=0.01\) | \(f_{sky}=0.05\) | \(f_{sky}=0.10\) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

\(\nu\),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 | ~ | ~ |

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.

\(f_{sky}=0.01\) | \(f_{sky}=0.05\) | \(f_{sky}=0.10\) | |
---|---|---|---|

\(\sigma_r(r=0), \times 10^{-4}\) | 5.52 | 8.06 | 9.51 |

\(\sigma_r(r=0.01), \times 10^{-3}\) | 1.85 | 1.44 | 1.42 |

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.