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The Sunyaev-Zeldovich (SZ) effect provides a powerful cosmological probe,which traditionally is approached independently as cluster number count (CNC)or power spectrum (PS) analysis. Here, we devise a new method for analysing the$y$-map by introducing the survey completeness function, conventionally onlyused in the CNC analysis, in the $yy$-PS modeling. This provides a systematicmethod, based mainly on SZ observables, for obtaining two complementary$y$-maps, one incorporating detected/resolved clusters and the other relyingonly on diffuse/unresolved SZ contributions. We use the catalogue of clustersobtained in the \Planck CNC analysis to define the completeness functionlinking these two $y$-maps. The split depends on the chosen signal-to-noisedetection threshold, which we vary in our discussion. We carefully propagatethe effect of completeness cuts on the non-Gaussian error contributions in the$yy$-PS analysis, highlighting the benefits of masking massive clusters. Ouranalysis of the \Planck $yy$-PS for the unresolved component yields a mass biasof $b=0.15\pm0.04$, consistent with the standard value ($b\approx0.2$), incomparison to $b=0.4\pm 0.05$ for the total $yy$-PS. We find indications forthis drift being driven by the CIB-tSZ cross correlation, which dominantlyoriginates from clusters in the resolved component of the $y$-map. Anotherpossible explanation is the presence of a mass-dependent bias, which has beentheoretically motivated and can be quantified with our novel method. Wefurthermore find first hints for the presence of the 2-halo terms in the$yy$-PS. Finally, the proposed method provides a new framework for combiningthe complementary information of the CNC and PS analyses in upcoming SZsurveys.

Removing the giants and learning from the crowd: a new SZ power spectrum method and revised Compton $y$-map analysis