Abstract
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 APS measures (\(C_{\ell }\) and \({\mathcal {D}}_\ell =({\ell (\ell +1)})/({2\pi })C_\ell \)). This is motivated by the behaviour of level spacings between random matrix eigenvalues. This method helps to 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 \(\ge \)98.86% C.L.) and \({\mathcal {D}}_\ell \)’s (at \({\ge }\)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.
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Notes
Here, apostrophes (’) are used to denote plural forms.
Each \(C_\ell \) is a \(\chi ^2\) variable but with \(2\ell +1\) degrees of freedom. Thus the closest to nearly identical distributions for any two \(C_\ell \)’s can be considered by taking spacings of consecutive multipoles, as done here. This minimizes the difference between the distributions of the two \(C_\ell \)’s in a spacing to that of \((2(\ell +1)+1)-(2\ell +1)=2\) degrees of freedom.
Under a parity transform, i.e., \( {\hat{n}} \rightarrow - {\hat{n}}, Y_{\ell m}({\hat{n}}) \rightarrow Y_{\ell m}(-{\hat{n}})=(-1)^\ell Y_{\ell m}({\hat{n}}) \implies a_{\ell m} \rightarrow (-1)^\ell a_{\ell m}\).
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Acknowledgements
We acknowledge the use of publicly available HEALPix (Górski et al. 2005) software package (http://healpix.sourceforge.io). Our analyses are based on observations from Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA), part of the High Energy Astrophysics Science Archive Center (HEASARC). HEASARC/LAMBDA is a service of the Astrophysics Science Division at the NASA Goddard Space Flight Center. MIK would like to thank Ujjal Purkayastha for help with the basics of HEALPix relevant to this project. MIK is grateful to Aamna Ahmed for discussions regarding mean gap ratios.
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Ishaque Khan, M., Saha, R. Level correlations of CMB temperature angular power spectrum. J Astrophys Astron 43, 100 (2022). https://doi.org/10.1007/s12036-022-09893-w
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DOI: https://doi.org/10.1007/s12036-022-09893-w