# Difference between revisions of "Noise models and sky fractions for WAFTT"

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+ | ===Observing schedules[from Reijo on 1-30-2019]=== | ||

+ | |||

+ | We built the observing schedules using the opportunistic scheduler. First we tiled the sky in celestial coordinates with 10x20 degree (RAxDEC) tiles that overlap by half a tile in each direction. Then we ran the scheduler with three choices of minimum observing elevation: 30, 40 and 50 degrees. We required a 30-degree avoidance region around the Sun and the Moon. The three schedules were run with and without an additional "elevation penalty" which tells the scheduler to favor high elevation observations over low elevation observations. | ||

+ | |||

+ | Originally we tried adjusting the tile priority based on how much of the tile was embedded inside the galactic mask. This turned out to cause undesirable boundary effects, making the final hit distribution around the masked area very uneven. Instead, we ran the scheduler on the full sky and only masked pixels from the resulting hit maps. This approach lead to slightly lower overall observing time but higher effective sky fraction. | ||

+ | |||

+ | Here are the (masked) hit maps for all observing elevations when no elevation penalty is considered: | ||

+ | |||

+ | [[File:Hitmaps2_no_elpenalty.png|1200px]] | ||

+ | |||

+ | Here are the hit maps when the scheduler is told to penalize low elevation observations: | ||

+ | |||

+ | [[File:hitmaps2_with_elpenalty.png|1200px]] | ||

+ | |||

+ | Here is the difference in observing elevations with and without the elevation penalty: | ||

+ | |||

+ | [[File:elevations_el_min30_fsky100.png|520px]] | ||

+ | |||

+ | |||

+ | ===Depth Maps [from Matthew Hasselfield]=== | ||

+ | |||

+ | For this result, I used the hit maps "without elevation penalty" described by Reijo above. The noise model is the one described on [[Survey_Performance_Expectations]], in Large-area Survey Performance Expectation 01, Lat-noise-181121. | ||

+ | |||

+ | The hit maps produced by Reijo were converted into depth maps as follows: | ||

+ | * Define the "survey time", T, as the number of seconds for which the LAT operates. This is specified in the noise model so its value doesn't matter here, but it's a useful parameter for describing the computations. | ||

+ | * The calibration between hit count and observing time was determined, by summing up the hits in all pixels and all elevation bins for the "cut=0" case. It was found that "one uncut survey" is equivalent to (350 * 86400 * 254) = 7.7e9 hits in these maps. So each hit corresponds to (T/7.7e9) of LAT time. Note this computation is done for the cut=0 case but the same conversion is applied to other cases (where galactic cut or other restrictions might yield a smaller total number of hits than in the cut=0 case... we are allowed to suffer inefficiency due to survey strategy). | ||

+ | * Consider a single frequency band, and a single value of "el_min" (the minimum allowed observing elevation): | ||

+ | ** The maps of hits per pixel are combined with the pixel area to give maps of LAT time per unit area (in multiples of T). | ||

+ | ** For each elevation bin, the el-dependent noise model is used to convert LAT time per unit area to depth (the sort of depth that has units of [uK arcmin]^-2). | ||

+ | ** The per-elevation bin depth maps are summed. This produces a single depth map, units [uK arcmin]^-2, for this frequency and el_min. | ||

+ | * The above procedure is performed for each el_min and frequency band. | ||

+ | ** Intra-tube correlations are captured by binning into cross-frequency depth maps (e.g. 90x150). (The mapping speed in off-diagonal elements comes from the off-diagonal elements of the inverse covariance matrix, which is provided by the noise model...) | ||

+ | * The above procedure is performed for the "white noise" limit, and also at ell=[1000,2000,3000]. | ||

+ | |||

+ | The resulting noise maps can be found on NERSC, /global/cscratch1/sd/mhasse/cmb-s4/fskytaskforce/190103/: | ||

+ | * depth_el_min30.zip | ||

+ | * depth_el_min40.zip | ||

+ | * depth_el_min50.zip | ||

+ | * gal_masks_eq.zip (this archive contains Reijo's galactic masks, converted to same pixelization as the depth maps) | ||

+ | |||

+ | |||

+ | ===Noise characteristics for minimum elevation of 50 degrees=== | ||

+ | |||

+ | |||

+ | {| class="wikitable" style="text-align: right; margin-left: 51px;" | ||

+ | ! Galactic cut (in %) !! 0 !! 10 !! 20 !! 30 | ||

+ | |- | ||

+ | |Sky fraction (in %)||57||52||45||39|| | ||

+ | |- | ||

+ | |Effective sky fraction for noise (in %)||51||47||40||34|| | ||

+ | |- | ||

+ | |Effective sky fraction for signal (in %)||47||43||37||31|| | ||

+ | |- | ||

+ | |} | ||

+ | |||

+ | |||

+ | {| class="wikitable" style="text-align: right; margin-left: 51px;" | ||

+ | ! Frequency (GHz) !! 20 !! 27 !! 39 !! 93 !! 145 !! 225 !! 280 | ||

+ | |- | ||

+ | |white noise level TT (uK-arcmin)||52.6||17.7||9.9||1.7||1.7||5.3||12.7|| | ||

+ | |- | ||

+ | |ell knee TT||449||409||378||1245||4440||4358||4397|| | ||

+ | |- | ||

+ | |1/f exponent TT (uK-arcmin)||-3.5||-3.5||-3.5||-3.0||-2.3||-3.4||-3.4|| | ||

+ | |- | ||

+ | |white noise level E/B (uK-arcmin)||74.4||25.0||14.0||2.4||2.4||7.6||18.0|| | ||

+ | |- | ||

+ | |ell knee E/B||700||467||467||467||467||467||467|| | ||

+ | |- | ||

+ | |1/f exponent E/B (uK-arcmin)||-1.4||-1.1||-1.1||-1.1||-1.1||-1.1||-1.1|| | ||

+ | |- | ||

+ | |Penalty (relative to f_sky scaling)||1.00||1.00||1.00||1.00||1.00||1.00||1.00|| | ||

+ | |- | ||

+ | |} | ||

+ | |||

+ | ===Noise characteristics for minimum elevation of 40 degrees=== | ||

{| class="wikitable" style="text-align: right; margin-left: 51px;" | {| class="wikitable" style="text-align: right; margin-left: 51px;" | ||

− | ! Frequency (GHz) !! 20 !! 27 !! 39 !! 93 !! 145 !! 225 !! 280 | + | ! Galactic cut (in %) !! 0 !! 10 !! 20 !! 30 |

+ | |- | ||

+ | |Sky fraction (in %)||68||63||55||48|| | ||

+ | |- | ||

+ | |Effective sky fraction for noise (in %)||63||58||51||44|| | ||

+ | |- | ||

+ | |Effective sky fraction for signal (in %)||59||54||47||41|| | ||

+ | |- | ||

+ | |} | ||

+ | |||

+ | {| class="wikitable" style="text-align: right; margin-left: 51px;" | ||

+ | ! Frequency (GHz) !! 20 !! 27 !! 39 !! 93 !! 145 !! 225 !! 280 | ||

+ | |- | ||

+ | |white noise level TT (uK-arcmin)||58.0||19.6||10.9||1.8||1.9||6.1||14.6|| | ||

+ | |- | ||

+ | |ell knee TT||480||435||401||1327||4666||4519||4531|| | ||

+ | |- | ||

+ | |1/f exponent TT (uK-arcmin)||-3.5||-3.5||-3.4||-3.1||-2.3||-3.4||-3.4|| | ||

+ | |- | ||

+ | |white noise level E/B (uK-arcmin)||82.0||27.7||15.5||2.6||2.7||8.6||20.6|| | ||

+ | |- | ||

+ | |ell knee E/B||700||467||467||467||467||467||467|| | ||

+ | |- | ||

+ | |1/f exponent E/B (uK-arcmin)||-1.4||-1.1||-1.1||-1.1||-1.1||-1.1||-1.1|| | ||

+ | |- | ||

+ | |Penalty (relative to f_sky scaling)||0.99||1.00||0.99||1.00||1.00||1.02||1.03|| | ||

+ | |- | ||

+ | |} | ||

+ | |||

+ | ===Noise characteristics for minimum elevation of 30 degrees=== | ||

+ | |||

+ | |||

+ | {| class="wikitable" style="text-align: right; margin-left: 51px;" | ||

+ | ! Galactic cut (in %) !! 0 !! 10 !! 20 !! 30 | ||

+ | |- | ||

+ | |Sky fraction (in %)||76||69||62||54|| | ||

+ | |- | ||

+ | |Effective sky fraction for noise (in %)||71||65||57||50|| | ||

+ | |- | ||

+ | |Effective sky fraction for signal (in %)||67||61||54||47|| | ||

+ | |- | ||

+ | |} | ||

+ | |||

+ | |||

+ | {| class="wikitable" style="text-align: right; margin-left: 51px;" | ||

+ | ! Frequency (GHz) !! 20 !! 27 !! 39 !! 93 !! 145 !! 225 !! 280 | ||

|- | |- | ||

|white noise level TT (uK-arcmin)||61.4||20.8||11.6||2.0||2.0||6.7||16.3|| | |white noise level TT (uK-arcmin)||61.4||20.8||11.6||2.0||2.0||6.7||16.3|| | ||

|- | |- | ||

− | | | + | |ell knee TT||517||465||429||1416||4853||4660||4639|| |

|- | |- | ||

− | |white noise level | + | |1/f exponent TT (uK-arcmin)||-3.5||-3.4||-3.4||-3.1||-2.4||-3.4||-3.4|| |

+ | |- | ||

+ | |white noise level E/B (uK-arcmin)||86.8||29.4||16.4||2.8||2.9||9.5||23.0|| | ||

+ | |- | ||

+ | |ell knee E/B||700||467||467||467||467||467||467|| | ||

+ | |- | ||

+ | |1/f exponent E/B (uK-arcmin)||-1.4||-1.1||-1.1||-1.1||-1.1||-1.1||-1.1|| | ||

+ | |- | ||

+ | |Penalty (relative to f_sky scaling)||0.99||1.00||1.00||1.00||1.02||1.07||1.09|| | ||

|- | |- | ||

|} | |} | ||

+ | |||

+ | ===Forecasting Details [from Joel]=== | ||

+ | Here we will give the details of converting the numbers in the tables above to forecasts for light relic density N<sub>eff</sub>. The forecasts were done for LCDM+N<sub>eff</sub>+M<sub>ν</sub>, assuming BBN consistency to fix Y<sub>p</sub>. TT, TE, EE, and φφ spectra were included, with the lensing reconstruction calculated using the minimum variance combination of quadratic estimators, including the improvement from iterative EB reconstruction. The small improvements which come from delensing T and E spectra (see https://arxiv.org/abs/1609.08143) were also included. | ||

+ | |||

+ | Two sets of forecasts were conducted, one using ILC noise curves provided by Raphael, and another which used an inverse variance weighting of all frequency channels. The results of these forecasts agree to the level of 2 × 10<sup>-3</sup>. | ||

+ | |||

+ | The inverse variance weighted noise was computed by taking white noise for each frequency channel listed above, applying a beam given by 1.4 arcmin × (150 GHz / frequency), and then adding atmospheric noise according to: | ||

+ | |||

+ | N<sub>l</sub><sup>TT</sup> = N<sub>0</sub><sup>TT</sup>(1+ ( l / l<sub>knee</sub><sup>T</sup> )<sup>Texp</sup>) | ||

+ | |||

+ | N<sub>l</sub><sup>EE</sup> = N<sub>0</sub><sup>EE</sup>(1+ ( l / l<sub>knee</sub><sup>E</sup> )<sup>Eexp</sup>) | ||

+ | |||

+ | These noise curves were then combined in an inverse variance sum: | ||

+ | |||

+ | 1 / N<sub>l</sub><sup>S4</sup> = 1 / ∑<sub>freq</sub> N<sub>l</sub><sup>freq</sup> | ||

+ | |||

+ | The procedure from here on is equivalent for the ILC and inverse-variance sum forecasts. The ILC noise curves provided by Raphael are equivalent to N<sub>l</sub><sup>S4</sub> used here. | ||

+ | |||

+ | The S4-only noise was then combined with Planck, in an inverse variance sum, which is important to cover the low l temperature modes contaminated by the atmosphere for S4: | ||

+ | |||

+ | 1 / N<sub>l</sub><sup>Planck+S4</sup> = 1 / (N<sub>l</sub><sup>S4</sup> + N<sub>l</sub><sup>Planck</sup>) | ||

+ | |||

+ | Contamination from point sources was then added. The point source prescription was the same as from https://cmb-s4.org/wiki/index.php/Update_on_Neff_Forecasts. We reproduce the details here: | ||

+ | |||

+ | Point sources act much like an additional source of noise (modulo the correction from the beam). Point sources in TT therefore affect the signal to noise of both TT and TE at high-l. We will estimate the contribute to TT/EE from unresolved point source as | ||

+ | |||

+ | D<sub>TT,ps</sub>(l=3000) = 6 (μK)² | ||

+ | |||

+ | D<sub>EE,ps</sub>(l=3000) = 3 × 10<sup>-3</sup> × 6 (μK)² | ||

+ | |||

+ | where we used the measured few percent polarization fraction of point sources in Planck/SPT to estimate D<sub>EE,ps</sub>(l=3000). In practice, the polarization fraction would have to be order 1 for the EE point sources to have any effect on our forecasts. | ||

+ | |||

+ | Additionally, there is a cut imposed at l<sub>min</sub> = 30, and we took l<sub>max</sub> = 5000. A model for Planck temperature data was included for l < 30 on f_sky = 0.8. | ||

+ | |||

+ | The resulting noise curves were then used in a Fisher forecast to compute the errors. The sky fraction for S4 is shown in the tables above. Planck was assumed to cover an additional part of the sky, not observed by CMB-S4, up to a total sky fraction of 0.6 (i.e. f<sub>sky</sub><sup>Planck-only</sup> + f<sub>sky</sub><sup>S4+Planck</sup> = 0.6). We also discussed a different prescription for the Planck sky coverage, using f<sub>sky</sub><sup>Planck-only</sup> = 1 - (sky fraction + galactic cut) which is slightly more optimistic, but agrees in the final forecasts to the level of 3 × 10<sup>-3</sup>. The results for both the inverse variance weight and ILC forecasts are shown below. | ||

+ | |||

+ | ===Inverse Variance Weighted Noise Results [from Joel, modified 1-30-2019]=== | ||

+ | [[File:Neff_BBN_S4_psatm_WAFTT2_galcut.png|520px]] | ||

+ | |||

+ | ===Forecasts using Raphael's ILC [from Joel, modified 1-30-2019]=== | ||

+ | [[File:Neff_BBN_S4_psatm_WAFTT_ILC2_galcut.png|520px]] |

## Latest revision as of 11:39, 11 February 2019

## Contents

- 1 Observing schedules[from Reijo on 1-30-2019]
- 2 Depth Maps [from Matthew Hasselfield]
- 3 Noise characteristics for minimum elevation of 50 degrees
- 4 Noise characteristics for minimum elevation of 40 degrees
- 5 Noise characteristics for minimum elevation of 30 degrees
- 6 Forecasting Details [from Joel]
- 7 Inverse Variance Weighted Noise Results [from Joel, modified 1-30-2019]
- 8 Forecasts using Raphael's ILC [from Joel, modified 1-30-2019]

### Observing schedules[from Reijo on 1-30-2019]

We built the observing schedules using the opportunistic scheduler. First we tiled the sky in celestial coordinates with 10x20 degree (RAxDEC) tiles that overlap by half a tile in each direction. Then we ran the scheduler with three choices of minimum observing elevation: 30, 40 and 50 degrees. We required a 30-degree avoidance region around the Sun and the Moon. The three schedules were run with and without an additional "elevation penalty" which tells the scheduler to favor high elevation observations over low elevation observations.

Originally we tried adjusting the tile priority based on how much of the tile was embedded inside the galactic mask. This turned out to cause undesirable boundary effects, making the final hit distribution around the masked area very uneven. Instead, we ran the scheduler on the full sky and only masked pixels from the resulting hit maps. This approach lead to slightly lower overall observing time but higher effective sky fraction.

Here are the (masked) hit maps for all observing elevations when no elevation penalty is considered:

Here are the hit maps when the scheduler is told to penalize low elevation observations:

Here is the difference in observing elevations with and without the elevation penalty:

### Depth Maps [from Matthew Hasselfield]

For this result, I used the hit maps "without elevation penalty" described by Reijo above. The noise model is the one described on Survey_Performance_Expectations, in Large-area Survey Performance Expectation 01, Lat-noise-181121.

The hit maps produced by Reijo were converted into depth maps as follows:

- Define the "survey time", T, as the number of seconds for which the LAT operates. This is specified in the noise model so its value doesn't matter here, but it's a useful parameter for describing the computations.
- The calibration between hit count and observing time was determined, by summing up the hits in all pixels and all elevation bins for the "cut=0" case. It was found that "one uncut survey" is equivalent to (350 * 86400 * 254) = 7.7e9 hits in these maps. So each hit corresponds to (T/7.7e9) of LAT time. Note this computation is done for the cut=0 case but the same conversion is applied to other cases (where galactic cut or other restrictions might yield a smaller total number of hits than in the cut=0 case... we are allowed to suffer inefficiency due to survey strategy).
- Consider a single frequency band, and a single value of "el_min" (the minimum allowed observing elevation):
- The maps of hits per pixel are combined with the pixel area to give maps of LAT time per unit area (in multiples of T).
- For each elevation bin, the el-dependent noise model is used to convert LAT time per unit area to depth (the sort of depth that has units of [uK arcmin]^-2).
- The per-elevation bin depth maps are summed. This produces a single depth map, units [uK arcmin]^-2, for this frequency and el_min.

- The above procedure is performed for each el_min and frequency band.
- Intra-tube correlations are captured by binning into cross-frequency depth maps (e.g. 90x150). (The mapping speed in off-diagonal elements comes from the off-diagonal elements of the inverse covariance matrix, which is provided by the noise model...)

- The above procedure is performed for the "white noise" limit, and also at ell=[1000,2000,3000].

The resulting noise maps can be found on NERSC, /global/cscratch1/sd/mhasse/cmb-s4/fskytaskforce/190103/:

- depth_el_min30.zip
- depth_el_min40.zip
- depth_el_min50.zip
- gal_masks_eq.zip (this archive contains Reijo's galactic masks, converted to same pixelization as the depth maps)

### Noise characteristics for minimum elevation of 50 degrees

Galactic cut (in %) | 0 | 10 | 20 | 30 | |
---|---|---|---|---|---|

Sky fraction (in %) | 57 | 52 | 45 | 39 | |

Effective sky fraction for noise (in %) | 51 | 47 | 40 | 34 | |

Effective sky fraction for signal (in %) | 47 | 43 | 37 | 31 |

Frequency (GHz) | 20 | 27 | 39 | 93 | 145 | 225 | 280 | |
---|---|---|---|---|---|---|---|---|

white noise level TT (uK-arcmin) | 52.6 | 17.7 | 9.9 | 1.7 | 1.7 | 5.3 | 12.7 | |

ell knee TT | 449 | 409 | 378 | 1245 | 4440 | 4358 | 4397 | |

1/f exponent TT (uK-arcmin) | -3.5 | -3.5 | -3.5 | -3.0 | -2.3 | -3.4 | -3.4 | |

white noise level E/B (uK-arcmin) | 74.4 | 25.0 | 14.0 | 2.4 | 2.4 | 7.6 | 18.0 | |

ell knee E/B | 700 | 467 | 467 | 467 | 467 | 467 | 467 | |

1/f exponent E/B (uK-arcmin) | -1.4 | -1.1 | -1.1 | -1.1 | -1.1 | -1.1 | -1.1 | |

Penalty (relative to f_sky scaling) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

### Noise characteristics for minimum elevation of 40 degrees

Galactic cut (in %) | 0 | 10 | 20 | 30 | |
---|---|---|---|---|---|

Sky fraction (in %) | 68 | 63 | 55 | 48 | |

Effective sky fraction for noise (in %) | 63 | 58 | 51 | 44 | |

Effective sky fraction for signal (in %) | 59 | 54 | 47 | 41 |

Frequency (GHz) | 20 | 27 | 39 | 93 | 145 | 225 | 280 | |
---|---|---|---|---|---|---|---|---|

white noise level TT (uK-arcmin) | 58.0 | 19.6 | 10.9 | 1.8 | 1.9 | 6.1 | 14.6 | |

ell knee TT | 480 | 435 | 401 | 1327 | 4666 | 4519 | 4531 | |

1/f exponent TT (uK-arcmin) | -3.5 | -3.5 | -3.4 | -3.1 | -2.3 | -3.4 | -3.4 | |

white noise level E/B (uK-arcmin) | 82.0 | 27.7 | 15.5 | 2.6 | 2.7 | 8.6 | 20.6 | |

ell knee E/B | 700 | 467 | 467 | 467 | 467 | 467 | 467 | |

1/f exponent E/B (uK-arcmin) | -1.4 | -1.1 | -1.1 | -1.1 | -1.1 | -1.1 | -1.1 | |

Penalty (relative to f_sky scaling) | 0.99 | 1.00 | 0.99 | 1.00 | 1.00 | 1.02 | 1.03 |

### Noise characteristics for minimum elevation of 30 degrees

Galactic cut (in %) | 0 | 10 | 20 | 30 | |
---|---|---|---|---|---|

Sky fraction (in %) | 76 | 69 | 62 | 54 | |

Effective sky fraction for noise (in %) | 71 | 65 | 57 | 50 | |

Effective sky fraction for signal (in %) | 67 | 61 | 54 | 47 |

Frequency (GHz) | 20 | 27 | 39 | 93 | 145 | 225 | 280 | |
---|---|---|---|---|---|---|---|---|

white noise level TT (uK-arcmin) | 61.4 | 20.8 | 11.6 | 2.0 | 2.0 | 6.7 | 16.3 | |

ell knee TT | 517 | 465 | 429 | 1416 | 4853 | 4660 | 4639 | |

1/f exponent TT (uK-arcmin) | -3.5 | -3.4 | -3.4 | -3.1 | -2.4 | -3.4 | -3.4 | |

white noise level E/B (uK-arcmin) | 86.8 | 29.4 | 16.4 | 2.8 | 2.9 | 9.5 | 23.0 | |

ell knee E/B | 700 | 467 | 467 | 467 | 467 | 467 | 467 | |

1/f exponent E/B (uK-arcmin) | -1.4 | -1.1 | -1.1 | -1.1 | -1.1 | -1.1 | -1.1 | |

Penalty (relative to f_sky scaling) | 0.99 | 1.00 | 1.00 | 1.00 | 1.02 | 1.07 | 1.09 |

### Forecasting Details [from Joel]

Here we will give the details of converting the numbers in the tables above to forecasts for light relic density N_{eff}. The forecasts were done for LCDM+N_{eff}+M_{ν}, assuming BBN consistency to fix Y_{p}. TT, TE, EE, and φφ spectra were included, with the lensing reconstruction calculated using the minimum variance combination of quadratic estimators, including the improvement from iterative EB reconstruction. The small improvements which come from delensing T and E spectra (see https://arxiv.org/abs/1609.08143) were also included.

Two sets of forecasts were conducted, one using ILC noise curves provided by Raphael, and another which used an inverse variance weighting of all frequency channels. The results of these forecasts agree to the level of 2 × 10^{-3}.

The inverse variance weighted noise was computed by taking white noise for each frequency channel listed above, applying a beam given by 1.4 arcmin × (150 GHz / frequency), and then adding atmospheric noise according to:

N_{l}^{TT} = N_{0}^{TT}(1+ ( l / l_{knee}^{T} )^{Texp})

N_{l}^{EE} = N_{0}^{EE}(1+ ( l / l_{knee}^{E} )^{Eexp})

These noise curves were then combined in an inverse variance sum:

1 / N_{l}^{S4} = 1 / ∑_{freq} N_{l}^{freq}

The procedure from here on is equivalent for the ILC and inverse-variance sum forecasts. The ILC noise curves provided by Raphael are equivalent to N_{l}^{S4 used here.
}

The S4-only noise was then combined with Planck, in an inverse variance sum, which is important to cover the low l temperature modes contaminated by the atmosphere for S4:

1 / N_{l}^{Planck+S4} = 1 / (N_{l}^{S4} + N_{l}^{Planck})

Contamination from point sources was then added. The point source prescription was the same as from https://cmb-s4.org/wiki/index.php/Update_on_Neff_Forecasts. We reproduce the details here:

Point sources act much like an additional source of noise (modulo the correction from the beam). Point sources in TT therefore affect the signal to noise of both TT and TE at high-l. We will estimate the contribute to TT/EE from unresolved point source as

D_{TT,ps}(l=3000) = 6 (μK)²

D_{EE,ps}(l=3000) = 3 × 10^{-3} × 6 (μK)²

where we used the measured few percent polarization fraction of point sources in Planck/SPT to estimate D_{EE,ps}(l=3000). In practice, the polarization fraction would have to be order 1 for the EE point sources to have any effect on our forecasts.

Additionally, there is a cut imposed at l_{min} = 30, and we took l_{max} = 5000. A model for Planck temperature data was included for l < 30 on f_sky = 0.8.

The resulting noise curves were then used in a Fisher forecast to compute the errors. The sky fraction for S4 is shown in the tables above. Planck was assumed to cover an additional part of the sky, not observed by CMB-S4, up to a total sky fraction of 0.6 (i.e. f_{sky}^{Planck-only} + f_{sky}^{S4+Planck} = 0.6). We also discussed a different prescription for the Planck sky coverage, using f_{sky}^{Planck-only} = 1 - (sky fraction + galactic cut) which is slightly more optimistic, but agrees in the final forecasts to the level of 3 × 10^{-3}. The results for both the inverse variance weight and ILC forecasts are shown below.