1. Cross covariance limit cases
For cross qml, the covariance read {$ Cov(C_\ell,C_{\ell'}) = \frac12 (F^{-1} G F^{-1} + F^{-1}) $}. We compare the diagonal of those two terms depending on the noise level :
| 1 muK | 5 muK | 50 muK | 100 muK | 500 muK |
 |  |  |  |  |
2. MC vs TH mode-mode covariance matrix :
- nisde = 4
- nsimu = 1000
- For each figure :
- First case (0,0) is the diagonal of the covariance matrix
- Each following line correspond to a mode
- First column correspond to EE , the second to BB
| muK | mode | log scaled matrices theoretical (left) VS Monte-Carlo (right) |
| 0.1 |  |   |
| 1.0 |  |   |
| 10 |  |   |
| 50 |  |   |
3. Xpol Variance tests :
3.1 Nisde=32, plotting ratio of variance for EE then BB.
- For BB : we recover the xmql results between the ratio : ratio(at low noise) = fsky*ratio(at high noise). This means that the variance goes as 1/fsky for low noise, and 1/fsky^2 for high noise.
3.2 Nside = 128 and 256
| fsky | 0.7 | 0.5 |
| |  |  |
- We observe oscillations for low noise level in BB. Those are actually linked to the signal of EE (see next figure for comparison). When the EE signal is low, leakage is low, and the variance ratio of BB between full and cut sky is
4. New tests :
Comparaison effet de fsky et de masque :
- fsky = 1.0 VS fsky=0.5
- 3 types de masques : polar cap, litebird (=planck dust map), et random.
- nside=4
- nsimu = 1000
- beam =0.5 deg
- delta ell = 1 (non binning)
- r=0.1
- Cross and auto results are similar, thus only cross results are shown.
- Correlation matrices are computed as {$ \displaystyle Corr = \frac{F^{-1}_{ij}}{\sqrt{F^{-1}_{ii} F^{-1}_{jj}}} $} where {$ F $} is the fisher matrix.
- Global variance is computed as follow : {$ \Delta_\alpha = C_\ell^{th} C^{-1}_{\ell \ell'} C_{\ell'}^{th} $} with {$ C^{-1}_{\ell \ell'} = F_{\ell \ell'} $} the inverse of the mode covariance, namely the fisher matrix.
- Global variance computed certain mode range : {$\Delta_\alpha^{fsky=0.5} / \Delta_\alpha^{fsky=1.0} $}, nside=4 for BB
| noise level | Cl type | {$\ell=$} 7 | 8 | 9 | 10 | 11 |
| Analytic muK = 0.1 | classic | 0.4636 | 0.4442 | 0.4242 | 0.4141 | 0.4253 |
| MC muK = 0.1 | classic | 0.4774 | 0.4899 | 0.4552 | 0.4529 | 0.4582 |
| Analytic muK = 10 | classic | 0.3824 | 0.3688 | 0.3598 | 0.3568 | 0.3577 |
| MC muK = 10 | classic | 0.4011 | 0.4065 | 0.3728 | 0.364 | 0.3625 |
| Analytic muK = 1 | classic | 0.4549 | 0.4345 | 0.4144 | 0.4014 | 0.4093 |
| MC muK = 1 | classic | 0.4678 | 0.4829 | 0.4418 | 0.4381 | 0.4444 |
| Analytic muK = 50 | classic | 0.2752 | 0.2812 | 0.2844 | 0.286 | 0.2871 |
| MC muK = 50 | classic | 0.3069 | 0.3151 | 0.3161 | 0.3184 | 0.3184 |
4.1 Polar (BB)
- First plot : spectrum estimation
- Left matrice : Global variance from {$ \ell_{min} $} to {$ \ell_{max} $}.
- Right matrix : Correlation matrice
Cl
Cl classic
Cl flat = 1e-17
Cl null = 0
5. ds_dcb tests :
ns=4 :
5.1 Comparison Full Matrix ds_dcb
| | BB {$\ell=2$} | EE {$\ell=2$} | BB {$\ell=3$} | EE {$\ell=3$} | BB {$\ell=4$} | EE {$\ell=4$} | BB {$\ell=5$} | EE {$\ell=5$} | BB {$\ell=6$} | EE {$\ell=6$} | BB {$\ell=7$} | EE {$\ell=7$} |
| my ds_dcb |  |  |  |  |  |  |  |  |  |  | |  |
| mc ds_dcb |  |  |  |  |  |  |  |  |  |  | |  |
5.2 Comparison lines of Matrix ds_dcb
| | {$\ell=0$} | {$\ell=1$} |
| BB |  |  |
| EE |  |  |
5.3 Comparison Matrix Signal S
6. Previous tests :
6.1 Temperature
Cl
| Masks |  |  |
Cl classic
| noise [muK arcmin] | | | |
| 10 |  |  |  |
| 10 |  |  |  |
| 1 |  |  |  |
| 1 |  |  |  |
| 0.1 |  |  |  |
| 0.1 |  |  |  |
Cl flat = 1e-17
Cl null = 0
6.2
Comparaison effet de fsky et de masque :
- fsky = 1.0 VS fsky=0.6
- 3 types de masques : polar cap, litebird (=planck dust map), et random.
- nside=4
- nsimu = 1000
- beam =0.5 deg
- delta ell = 1 (non binning)
- r=0.1
6.3 ClEE ≠ 0
Estimateur (plots partie haut) et variance (plots partie bas), EE et BB.
| Masks |  | | |
| {$\sigma=0.1 \mu Karcm$} |   |   |   |
| {$\sigma=1.0 \mu Karcm$} |   |   |   |
| {$\sigma=5.0 \mu Karcm$} |   |   |   |
Histogramme Cl à ell=2 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} |  |  |  |
| {$\sigma=1.0 \mu Karcm$} |  |  |  |
| {$\sigma=5.0 \mu Karcm$} |  |  |  |
Histogramme Cl à ell=6 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} |  |  |  |
| {$\sigma=1.0 \mu Karcm$} |  |  |  |
| {$\sigma=5.0 \mu Karcm$} |  |  |  |
6.4 ClEE = 0
Estimateur (plots partie haut) et variance (plots partie bas), EE et BB.
| Masks |  |  |  |
| {$\sigma=0.1 \mu Karcm$} |   |   |   |
| {$\sigma=1.0 \mu Karcm$} |   |   |   |
| {$\sigma=5.0 \mu Karcm$} |   |   |   |
Histogramme Cl à ell=2 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} |  |  |  |
| {$\sigma=1.0 \mu Karcm$} |  |  |  |
| {$\sigma=5.0 \mu Karcm$} |  |  |  |
Histogramme Cl à ell=6 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} |  |  |  |
| {$\sigma=1.0 \mu Karcm$} |  |  |  |
| {$\sigma=5.0 \mu Karcm$} |  |  |  |
Comparaison effet de fsky et de masque :
- fsky = 1.0 VS fsky=0.6
- 3 types de masques : polar cap, litebird (=planck dust map), et random.
- nside=4
- nsimu = 1000
- beam =0.5 deg
- delta ell = 1 (non binning)
- r=0.1
6.5 ClEE ≠ 0
Estimateur (plots partie haut) et variance (plots partie bas), EE et BB.
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} | | | |
| {$\sigma=1.0 \mu Karcm$} | | | |
| {$\sigma=5.0 \mu Karcm$} | | | |
Histogramme Cl à ell=2 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} | | | |
| {$\sigma=1.0 \mu Karcm$} | | | |
| {$\sigma=5.0 \mu Karcm$} | | | |
Histogramme Cl à ell=6 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} | | | |
| {$\sigma=1.0 \mu Karcm$} | | | |
| {$\sigma=5.0 \mu Karcm$} | | | |
6.6 ClEE = 0
Estimateur (plots partie haut) et variance (plots partie bas), EE et BB.
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} | | | |
| {$\sigma=1.0 \mu Karcm$} | | | |
| {$\sigma=5.0 \mu Karcm$} | | | |
Histogramme Cl à ell=2 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} | | | |
| {$\sigma=1.0 \mu Karcm$} | | | |
| {$\sigma=5.0 \mu Karcm$} | | | |
Histogramme Cl à ell=6 :
| Masks | | | |
| {$\sigma=0.1 \mu Karcm$} | | | |
| {$\sigma=1.0 \mu Karcm$} | | | |
| {$\sigma=5.0 \mu Karcm$} | | | |
