Level correlations of the CMB temperature angular power spectrum

If the principle of statistical isotropy is valid, then the angular power spectrum (APS) of cosmic microwave background (CMB) radiation is uncorrelated between different multipoles. We propose a novel technique to analyse any possible correlations of the foreground cleaned CMB temperature angular power spectrum (APS) measures ($C_{\ell}$ and $\mathcal{D}_\ell=\frac{\ell(\ell+1)}{2\pi}C_\ell$). This is motivated by the behaviour of level spacings between random matrix eigenvalues. The method helps distinguish uncorrelated statistically isotropic CMB APS from correlated APS, where the latter can arise due to breakdown of isotropy or presence of some residual systematics in the foreground cleaned CMB maps. Spacings of statistically isotropic CMB $C_\ell$'s and $\mathcal{D}_\ell$'s are seen to closely obey Poisson statistics and introduction of correlations changes the distribution to appropriate Wigner-Dyson statistics. For foreground cleaned CMB, we employ the average spacing of consecutive multipole APS for multipoles $\in [2,31]$. This estimator is sensitive to departures from the null hypothesis of zero correlations between the APS measures of statistically isotropic CMB. We study full sky WMAP 9 year ILC and 2018 Planck foreground cleaned maps (Commander, NILC and SMICA). Sans parity distinctions, average spacings are in good agreement with theoretical expectation. With parity distinctions, even multipoles indicate unusually low average spacings for both $C_\ell$'s (at $\geq 98.86\%$ C.L.) and $\mathcal{D}_\ell$'s (at $\geq 95.07\%$ C.L.). We use an inpainting method based on constrained Gaussian realisations and show that for the Planck $U73$ and WMAP $KQ75$ masks, all the foreground cleaned inpainted CMB maps robustly confirm the existence of such unusually low average spacings of even multipole APS. In addition, this signal is independent of the non-Gaussian cold spot.