New estimates of the CMB angular power spectra from the WMAP 5 yrs low resolution data

A Quadratic Maximum Likelihood (QML) estimator is applied to the WMAP 5 year low resolution maps to compute the CMB angular power spectra at large scales for both temperature and polarization. Estimates and error bars for the six angular power spectra are provided and compared, when possible, to those obtained by the WMAP team, without finding any inconsistency. The conditional likelihood slices are also computed for the $C_{\ell}$ of all the six power spectra from $\ell =2$ to 10 through a pixel based likelihood code. Both the codes treat the covariance for $(T,Q,U)$ in a single matrix without employing any approximation. The inputs of both the codes (foreground reduced maps and related covariances) are provided by the WMAP team. The peaks of the likelihood slices are always consistent with the QML estimates within the error bars, however an excellent agreement occurs when the QML estimates are used as fiducial power spectrum instead of the best-fit theoretical power spectrum. By the full computation of the conditional likelihood on the estimated spectra, the value of the temperature quadrupole $C_{\ell=2}^{TT}$ is found to be less than $2 \sigma$ away from the WMAP 5 yrs $\Lambda$CDM best-fit value. The $BB$ spectrum is found well consistent with zero and upper limits on the B-modes are provided. The parity odd signals $TB$ and $EB$ are found consistent with zero.