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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 $}
muKarcm0.1 | | | | |
muKarcm1.0 | | | | |
muKarcm5.0 | | | | |
muKarcm20.0 | | | | |
muKarcm10.0 | | | | |
muKarcm50.0 | | | | |
Max correlations :
| | | |
deltal= 1.0 | max corr EE=0.037 | BB=0.172 | 0.10 muK |
deltal= 1.0 | max corr EE=0.046 | BB=0.188 | 1.00 muK |
deltal= 1.0 | max corr EE=0.057 | BB=0.202 | 5.00 muK |
deltal= 1.0 | max corr EE=0.064 | BB=0.200 | 10.00 muK |
deltal= 1.0 | max corr EE=0.072 | BB=0.175 | 20.00 muK |
deltal= 1.0 | max corr EE=0.078 | BB=0.135 | 50.00 muK |
| | | |
deltal= 2.0 | max corr EE=0.018 | BB=0.037 | 0.10 muK |
deltal= 2.0 | max corr EE=0.021 | BB=0.040 | 1.00 muK |
deltal= 2.0 | max corr EE=0.024 | BB=0.063 | 5.00 muK |
deltal= 2.0 | max corr EE=0.024 | BB=0.058 | 10.00 muK |
deltal= 2.0 | max corr EE=0.028 | BB=0.051 | 20.00 muK |
deltal= 2.0 | max corr EE=0.037 | BB=0.044 | 50.00 muK |
| | | |
deltal= 3.0 | max corr EE=0.015 | BB=0.028 | 0.10 muK |
deltal= 3.0 | max corr EE=0.018 | BB=0.029 | 1.00 muK |
deltal= 3.0 | max corr EE=0.020 | BB=0.043 | 5.00 muK |
deltal= 3.0 | max corr EE=0.020 | BB=0.042 | 10.00 muK |
deltal= 3.0 | max corr EE=0.023 | BB=0.040 | 20.00 muK |
deltal= 3.0 | max corr EE=0.032 | BB=0.038 | 50.00 muK |
- Observations :
- Correlation at 10 % for no binning !
1.2 Comparions {$ 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} $}
Full EE and Full BB
First half EE and BB
Second half EE and BB
1.3 Bicep :
- Full Fll plotted in log scale
- Antidiagonals plotted for paires mode). Colorbar shows the {$ \ell $} corresponding to the anti-diagonal number (from 2 to 3*lmax+1)
No binning :
noise | Correlation EE | Correlation BB | Anti diagonals | Anti diagonals (log scale) |
muKarcm0.1 | | | | |
muKarcm1.0 | | | | |
muKarcm5.0 | | | | |
muKarcm10.0 | | | | |
muKarcm20.0 | | | | |
muKarcm50.0 | | | | |
- Observations :
- first modes are more correlated than the rest (as expected).
- modes correlations do not exceed the 10th entry.
- minimal binning advised bandwidth in order to avoid correlation is then {$ \Delta_b = 2*10 = 20$} (Current Paper bandwidth is {$ \Delta_b = 14$})
Binning , {$ \Delta_b = 10 $}
noise | Correlation EE | Correlation BB | Anti diagonals | Anti diagonals (log scale) |
muKarcm0.1 | | | | |
muKarcm1.0 | | | | |
muKarcm5.0 | | | | |
muKarcm10.0 | | | | |
muKarcm20.0 | | | | |
muKarcm50.0 | | | | |
Binning , {$ \Delta_b = 14 $}
noise | Correlation EE | Correlation BB | Anti diagonals | Anti diagonals (log scale) |
muKarcm0.1 | | | | |
muKarcm1.0 | | | | |
muKarcm5.0 | | | | |
muKarcm10.0 | | | | |
muKarcm20.0 | | | | |
muKarcm50.0 | | | | |
Binning , {$ \Delta_b = 20 $}
noise | Correlation EE | Correlation BB | Anti diagonals | Anti diagonals (log scale) |
muKarcm0.1 | | | | |
muKarcm1.0 | | | | |
muKarcm5.0 | | | | |
muKarcm10.0 | | | | |
muKarcm20.0 | | | | |
muKarcm50.0 | | | | |
maxmum correlation with binning :
deltal= 10.0 | max corr EE=0.055 | BB=0.057 | 0.10 muK |
deltal= 10.0 | max corr EE=0.057 | BB=0.060 | 1.00 muK |
deltal= 10.0 | max corr EE=0.054 | BB=0.069 | 5.00 muK |
deltal= 10.0 | max corr EE=0.051 | BB=0.070 | 10.00 muK |
deltal= 10.0 | max corr EE=0.049 | BB=0.070 | 20.00 muK |
deltal= 10.0 | max corr EE=0.062 | BB=0.071 | 50.00 muK |
| | | |
deltal= 14.0 | max corr EE=0.032 | BB=0.034 | 0.10 muK |
deltal= 14.0 | max corr EE=0.032 | BB=0.031 | 1.00 muK |
deltal= 14.0 | max corr EE=0.027 | BB=0.031 | 5.00 muK |
deltal= 14.0 | max corr EE=0.025 | BB=0.035 | 10.00 muK |
deltal= 14.0 | max corr EE=0.026 | BB=0.037 | 20.00 muK |
deltal= 14.0 | max corr EE=0.033 | BB=0.038 | 50.00 muK |
| | | |
deltal= 20.0 | max corr EE=0.021 | BB=0.024 | 0.10 muK |
deltal= 20.0 | max corr EE=0.021 | BB=0.021 | 1.00 muK |
deltal= 20.0 | max corr EE=0.017 | BB=0.011 | 5.00 muK |
deltal= 20.0 | max corr EE=0.015 | BB=0.012 | 10.00 muK |
deltal= 20.0 | max corr EE=0.013 | BB=0.012 | 20.00 muK |
deltal= 20.0 | max corr EE=0.013 | BB=0.013 | 50.00 muK |
1.4 Comparions {$ [F^{-1}]_{\ell\ell} / [F_{\ell\ell}]^{-1} $}
Full EE and Full BB
First half EE and BB
Second half EE and BB
2. Binning effect, bias, and Full Fll Surveys :
2.1 Test on LiB
new test interpolation :
zero order {$ \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} $} |
| | | | | |
as expected | no bias as expected | no bias on EE, slight deviation on BB | worse | not better than zero order | worse |
previous test on LiB (100000simu)
{$ \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'} $} |
| | |
- conclusion : residues bias is cause by the binned spectrum approximation.
3. Tests :
3.1 Cutting modes (V2, surveys) :
- Cutting (first or last) half of modes, then joining the spectra estimated
- The cut is visible in the middle ( change of color red to black)
- 1000 simulations
noise [muK-arcmin] | 0.1 | 1.0 | 5.0 | 10.0 | 20.0 | 50.0 |
BICEP | | | | | | |
LiB | | | | | | |
- observations :
- Spectra are more and more biased near the cut. This is due to correlations between neighbourg modes that are not taken into account when cutting the Fisher matrix.
- The bias is less present for noise dominated cases.
3.2 Adding modes (lmax)
- noise is 1 muK if not specified
Varying S lmax
| S lmax = 3*ns-1 | S lmax = 4*ns-1 | S lmax = 10*ns-1 |
simu lmax =2*ns-1, fsky=0.5, {$ \ell=[2, 2*ns-1] $} (cad {$ P_\ell $} lmin=2 and lmax=7 | | | |
- observations :
- since we the estimator compute exactly all the modes that are simulated, there is no bias.
- Adding modes in the S matrix increases the variance of the estimator.
Varying noise
| 1.0 muK | 10.0 muK |
S lmax = simu lmax = 4*ns-1 | | |
- observations :
- nothing unexpected, the variance is rising for high multipoles.
Changing fwhm and pixwin
| fwhm 0.0, no pixwin | fwhm 0.5 + pixwin |
simu lmax = S lmax = 4*ns-1 | | |
- observations :
- no dependence from the beam
Changing fsky
| fsky = 1.0 | fsky= 0.5 |
, simu lmax = S lmax = 5*ns-1 | | |
- observations :
- Less effect on the variance for full fsky. The impact of modes simulated over 3*ns-1 (simu lmax), or form of 'leakage', depends on the fsky.
- Even for full map the is some kind of variance 'leakge' (remember that it comes from EE and BB correlations )
Changing EE modes
| fsky = 1.0 | fsky= 0.5 |
{$ 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
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