ANALYSIS-OF-SIMULATED-COSMIC-MICROWAVE-BACKGROUND-POLARIZATION-DATA-

Plots of powers spectrum ($C_\ell^{TT}$, $C_\ell^{EE}$, $C_\ell^{BB}$, $C_\ell^{TE}$)

Study of some cosmological dependencies ($r$ and $\tau$) :

map T :

map Q : map U :

data simulated using stokes parameters maps (T,Q,U) :

Estimation of r by gaussian Log-Likelihood minimization

We can show (from J.Errard et al 2019) :

$$-2 \log(L) = fsky * \sum_{l} (2l+1)[C_{\ell}^{-1} D_{\ell} + ln(C_{\ell})] = \chi^{2}$$

$$ L = \exp\Big(-\left(\chi^{2} - \min(\chi^{2})\right)^{2}\Big) $$

Where $C_{\ell}$ is the theoretical spectrum (the model) and $D_{\ell}$ is the reconstructed spectrum with white noise.

For the covariance, we can write :

$$C_{\ell}(r) = \frac{1}{r_{0}} r C_{\ell, tensor}^{BB}(r = r_{0}) + C_{\ell,lensing}^{BB} + N_{\ell}$$

Where $N_{\ell}$ is the white noise.

MCMC (Marlov Chain Monte Carlo) method to estimate $r$ and $Alens$ :

Exemple of map with mask (via PyMaster / NaMaster) :