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  1. 1. Full Fll correlations :
    1. 1.1 LiB :
    2. 1.2 Comparions 1- [F^{-1}]_{\ell\ell} / [F_{\ell\ell}]^{-1}
    3. 1.3 Bicep :
    4. 1.4 Comparions [F^{-1}]_{\ell\ell} / [F_{\ell\ell}]^{-1}
  2. 2. Binning effect, bias, and Full Fll Surveys :
    1. 2.1 Test on LiB
  3. 3. Tests :
    1. 3.1 Cutting modes (V2, surveys) :
    2. 3.2 Adding modes (lmax)
    3. 3.3 Cutting modes

1.  Full Fll correlations :

1.1  LiB :

  • Full Fll plotted in log scale
  • Antidiagonals plotted for paires mode). Colorbar shows the \ell corresponding to the anti-diagonal number (from 2 to 3*lmax+1)

No binning \Delta_b = 1

muKarcm0.1
muKarcm1.0
muKarcm5.0
muKarcm10.0
muKarcm20.0
muKarcm50.0

binning \Delta_b = 2

muKarcm0.1
muKarcm1.0
muKarcm5.0
muKarcm10.0
muKarcm20.0
muKarcm50.0

binning \Delta_b = 3

muKarcm0.1
muKarcm1.0
muKarcm5.0
muKarcm20.0
muKarcm10.0
muKarcm50.0

Max correlations :

    
deltal= 1.0max corr EE=0.037BB=0.1720.10 muK
deltal= 1.0max corr EE=0.046BB=0.1881.00 muK
deltal= 1.0max corr EE=0.057BB=0.2025.00 muK
deltal= 1.0max corr EE=0.064BB=0.20010.00 muK
deltal= 1.0max corr EE=0.072BB=0.17520.00 muK
deltal= 1.0max corr EE=0.078BB=0.13550.00 muK
    
deltal= 2.0max corr EE=0.018BB=0.0370.10 muK
deltal= 2.0max corr EE=0.021BB=0.0401.00 muK
deltal= 2.0max corr EE=0.024BB=0.0635.00 muK
deltal= 2.0max corr EE=0.024BB=0.05810.00 muK
deltal= 2.0max corr EE=0.028BB=0.05120.00 muK
deltal= 2.0max corr EE=0.037BB=0.04450.00 muK
    
deltal= 3.0max corr EE=0.015BB=0.0280.10 muK
deltal= 3.0max corr EE=0.018BB=0.0291.00 muK
deltal= 3.0max corr EE=0.020BB=0.0435.00 muK
deltal= 3.0max corr EE=0.020BB=0.04210.00 muK
deltal= 3.0max corr EE=0.023BB=0.04020.00 muK
deltal= 3.0max corr EE=0.032BB=0.03850.00 muK
  • Observations :
    • Correlation at 10 % for no binning !

1.2  Comparions 1- [F^{-1}]_{\ell\ell} / [F_{\ell\ell}]^{-1}

We compare compare the inverse of the Fisher matrix cuts with the cut of the full Fisher matrix inverse. More precisely, we cute the EE and BB blocks of the matrix, inverse them, then take the diagonals, and compare them with the diagonal of the EE and BB block of the full matrix inverse : 1- [F^{-1}]_{\ell\ell} / [F_{\ell\ell}]^{-1}

Full EE and Full BB

\delta_\ell = 1
\delta_\ell = 2
\delta_\ell = 3

First half EE and BB

\delta_\ell = 1
\delta_\ell = 2
\delta_\ell = 3

Second half EE and BB

\delta_\ell = 1
\delta_\ell = 2
\delta_\ell = 3

1.3  Bicep :

  • Full Fll plotted in log scale
  • Antidiagonals plotted for paires mode). Colorbar shows the \ell corresponding to the anti-diagonal number (from 2 to 3*lmax+1)

No binning :

noiseCorrelation EECorrelation BBAnti diagonalsAnti diagonals (log scale)
muKarcm0.1
muKarcm1.0
muKarcm5.0
muKarcm10.0
muKarcm20.0
muKarcm50.0
  • Observations :
    • first modes are more correlated than the rest (as expected).
    • modes correlations do not exceed the 10th entry.
    • minimal binning advised bandwidth in order to avoid correlation is then \Delta_b = 2*10 = 20 (Current Paper bandwidth is \Delta_b = 14)

Binning , \Delta_b = 10

noiseCorrelation EECorrelation BBAnti diagonalsAnti diagonals (log scale)
muKarcm0.1
muKarcm1.0
muKarcm5.0
muKarcm10.0
muKarcm20.0
muKarcm50.0

Binning , \Delta_b = 14

noiseCorrelation EECorrelation BBAnti diagonalsAnti diagonals (log scale)
muKarcm0.1
muKarcm1.0
muKarcm5.0
muKarcm10.0
muKarcm20.0
muKarcm50.0

Binning , \Delta_b = 20

noiseCorrelation EECorrelation BBAnti diagonalsAnti diagonals (log scale)
muKarcm0.1
muKarcm1.0
muKarcm5.0
muKarcm10.0
muKarcm20.0
muKarcm50.0

maxmum correlation with binning :

deltal= 10.0max corr EE=0.055BB=0.0570.10 muK
deltal= 10.0max corr EE=0.057BB=0.0601.00 muK
deltal= 10.0max corr EE=0.054BB=0.0695.00 muK
deltal= 10.0max corr EE=0.051BB=0.07010.00 muK
deltal= 10.0max corr EE=0.049BB=0.07020.00 muK
deltal= 10.0max corr EE=0.062BB=0.07150.00 muK
    
deltal= 14.0max corr EE=0.032BB=0.0340.10 muK
deltal= 14.0max corr EE=0.032BB=0.0311.00 muK
deltal= 14.0max corr EE=0.027BB=0.0315.00 muK
deltal= 14.0max corr EE=0.025BB=0.03510.00 muK
deltal= 14.0max corr EE=0.026BB=0.03720.00 muK
deltal= 14.0max corr EE=0.033BB=0.03850.00 muK
    
deltal= 20.0max corr EE=0.021BB=0.0240.10 muK
deltal= 20.0max corr EE=0.021BB=0.0211.00 muK
deltal= 20.0max corr EE=0.017BB=0.0115.00 muK
deltal= 20.0max corr EE=0.015BB=0.01210.00 muK
deltal= 20.0max corr EE=0.013BB=0.01220.00 muK
deltal= 20.0max corr EE=0.013BB=0.01350.00 muK

1.4  Comparions [F^{-1}]_{\ell\ell} / [F_{\ell\ell}]^{-1}

Full EE and Full BB

\delta_\ell = 10
\delta_\ell = 14
\delta_\ell = 20

First half EE and BB

\delta_\ell = 10
\delta_\ell = 14
\delta_\ell = 20

Second half EE and BB

\delta_\ell = 10
\delta_\ell = 14
\delta_\ell = 20

2.  Binning effect, bias, and Full Fll Surveys :

2.1  Test on LiB

  • deltal = 2

new test interpolation :

  • 1000 simulations
zero order \delta_\ell = 2, et C_b = P_{b\ell} C_\ell exact C_b = [F^{-1}]_{bb'} P_{b'\ell} F_{\ell \ell'} C_{\ell'} exact diagonals only : C_b = [F^{-1}]_{bb} P_{b\ell} F_{\ell \ell} C_{\ell} exact diagonal only for Fll : [F^{-1}]_{bb'} P_{b'\ell} F_{\ell \ell} C_{\ell} diag + interp Fll [F^{-1}]_{bb} P_{b\ell} \hat F_{\ell \ell} C_{\ell} diag only for FLL + interp Fll : [F^{-1}]_{bb'} P_{b'\ell} \hat F_{\ell \ell} C_{\ell}
as expectedno bias as expectedno bias on EE, slight deviation on BBworsenot better than zero orderworse

previous test on LiB (100000simu)

  • 100000 simulations
\delta_\ell = 1 \delta_\ell = 2, et C_b = P_{b\ell} C_\ell \delta_\ell = 2, et C_b = [F^{-1}]_{bb'} P_{b\ell} F_{\ell \ell'} C_{\ell'}
  • conclusion : residues bias is cause by the binned spectrum approximation.

3.  Tests :

3.1  Cutting modes (V2, surveys) :

  • Cutting (first or last) half of modes, then joining the spectra estimated
  • The cut is visible in the middle ( change of color red to black)
  • 1000 simulations
noise [muK-arcmin]0.11.05.010.020.050.0
BICEP
LiB
  • observations :
    • Spectra are more and more biased near the cut. This is due to correlations between neighbourg modes that are not taken into account when cutting the Fisher matrix.
    • The bias is less present for noise dominated cases.

3.2  Adding modes (lmax)

  • noise is 1 muK if not specified

Varying S lmax

 S lmax = 3*ns-1S lmax = 4*ns-1S lmax = 10*ns-1
simu lmax =2*ns-1, fsky=0.5, \ell=[2, 2*ns-1] (cad P_\ell lmin=2 and lmax=7
  • observations :
    • since we the estimator compute exactly all the modes that are simulated, there is no bias.
    • Adding modes in the S matrix increases the variance of the estimator.

Varying noise

 1.0 muK10.0 muK
S lmax = simu lmax = 4*ns-1
  • observations :
    • nothing unexpected, the variance is rising for high multipoles.

Changing fwhm and pixwin

 fwhm 0.0, no pixwinfwhm 0.5 + pixwin
simu lmax = S lmax = 4*ns-1
  • observations :
    • no dependence from the beam

Changing fsky

 fsky = 1.0fsky= 0.5
, simu lmax = S lmax = 5*ns-1
  • observations :
    • Less effect on the variance for full fsky. The impact of modes simulated over 3*ns-1 (simu lmax), or form of 'leakage', depends on the fsky.
    • Even for full map the is some kind of variance 'leakge' (remember that it comes from EE and BB correlations )

Changing EE modes

 fsky = 1.0fsky= 0.5
{$ C_\ell^EE} = 0 , simu lmax = S lmax = 4*ns-1
  • observations :
    • Less effect on the variance for full fsky. The impact of modes simulated over 3*ns-1 (simu lmax), or form of 'leakage', depends on the fsky.
    • Even for full map the is some kind of variance 'leakge' (remebder that it comes from EE and BB correlations! )

Going to lmax = 10*ns

  • we bin the last bins from 3*nside-1 to lmax simu (= S lmax also)

3.3  Cutting modes

  • ns=4, then all bins from 2 to 3*ns-1 are \ell=[2,11]
  • \delta_\ell = 1
  \ell=[2,11] \ell=[2,6] \ell=[6,11]
fsky=1.0
fsky=0.5
  • observations :
    • On full fsky, no bias are found whatever the mode range chosen.
    • On cut fsky, bias are found when not all mode range are chosen. Since in the real world the
  • conclusion :
    • if lmax simu = lmax estimateur (lmax binning) => no bias
    • if lmax simu < lmax estimateur (lmax binning) => no bias from l=2 to l=lmax simu. last modes are missing on the cmb simulations, and the estimator can't find them.
    • if lmax simu > lmax estimateur (lmax binning) => bias for cut skies