1. Vary r fiducial input :
- nside= 4, fsky=0.7, LiB. We vary the true r such that (true r) = p*r
- for r = 0 we do (true r) = p*0.001
| | cross | auto |
| r = 0.0 |  |  |
| r = 0.001 |  |  |
| r = 0.01 |  |  |
| r = 0.1 |  |  |
- observation :
- impact on BB variance non-negigable, especially for high r.
1.1 other noise :
| | cross | auto |
| r = 0.0 |  |  |
| r = 0.001 |  |  |
| r = 0.01 |  |  |
| r = 0.1 |  |  |
2. With {$ El = (C^{AA})^{-1} P_\ell (C^{BB})^{-1} $} approximation
2.1 LiB :
- We vary the estimation of the pixel noise covariance {$ \hat C = S + N$} of a few percent around its true value {$ C = S + p^2 \cdot N$}. Actually, what we vary is the noise level input {$ \sigma \rightarrow p\cdot \sigma $} [muKarcmin] :
- We compare the mc spectrum error using the xQML estimator built using {$ \hat C $} with the optimal spectrum error built from {$ C $} .
- observation: the ratio is under 1.0 for p<1.0 . This is due to the approximation made for the construction of {$ E_\ell $}. Indeed, the same tests on the auto-spectrum give a ratio >1.0. This is expected since, by construction, it should have an optimal variance. And ell other construction of {$ E_\ell $} using {$ \hat C \neq $} should give a variance above the optimal one. We verify this assumption by plotting the auto-spectrum variance ratio :
- also, this effect is only visible for small variation around the true pixel noise matrix {$ N $}, of the order of a few %. For larger variations, of the order of 100%, we recover a ratio above 1.0 :
- anyway, the variations of the spectrum errors compared to the variations of the estimation of noise level {$ \sigma $} muKarmin is at most of the order of 1/100, that is to say, negligible.
3. Using exact solution for {$ El $}
using the kronecker exact solution for the same analysis (see El) .
3.1 LiB
- Observation :
- As expect, we find that find that the variance is indeed optimal with this choice of {$ E_\ell $}. The ratio is always above 1 for cross spectra, similarly as for the auto-spectra case !
