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Window functions corresponding to the spectral localisation of the {\tt NILC} and {\tt MILCA} algorithms. For {\tt NILC} (black) there are10 Gaussian window functions defining 10 needlet scales. {\tt MILCA} (red) uses 11 Gaussian overlapping windows.

Reconstructed \Planck\ all-sky Compton parameter maps for {\tt NILC} (\textit{top}) and {\tt MILCA} (\textit{bottom}) in orthographic projections. The apparent difference in contrast observed between the {\tt NILC} and {\tt MILCA} maps comes from differences in the residual foreground contamination and from the differences in the filtering applied for display purposes to the original Compton parameter maps. For the MILCA method filtering out low multipoles reduces significantly the level of foreground emission in the final $y$\/-map. The wavelet basis used in the NILC method was tailored for tSZ extraction. For details see \citet{planck2014-a28}.

Reconstructed \Planck\ all-sky Compton parameter maps for {\tt NILC} (\textit{top}) and {\tt MILCA} (\textit{bottom}) in orthographic projections. The apparent difference in contrast observed between the {\tt NILC} and {\tt MILCA} maps comes from differences in the residual foreground contamination and from the differences in the filtering applied for display purposes to the original Compton parameter maps. For the MILCA method filtering out low multipoles reduces significantly the level of foreground emission in the final $y$\/-map. The wavelet basis used in the NILC method was tailored for tSZ extraction. For details see \citet{planck2014-a28}.

A small region of the reconstructed \Planck\ all-sky Compton parameter maps for {\tt NILC} (left) and {\tt MILCA} (right) at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

A small region of the reconstructed \Planck\ all-sky Compton parameter maps for {\tt NILC} (left) and {\tt MILCA} (right) at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

In and cross scan contributions in the {\tt NILC} (top line) and {\tt MILCA} (bottom line) $y$\/-maps. From left to right we present the original $y$\/-maps, and their in and cross scan contributions for a small region at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

In and cross scan contributions in the {\tt NILC} (top line) and {\tt MILCA} (bottom line) $y$\/-maps. From left to right we present the original $y$\/-maps, and their in and cross scan contributions for a small region at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

In and cross scan contributions in the {\tt NILC} (top line) and {\tt MILCA} (bottom line) $y$\/-maps. From left to right we present the original $y$\/-maps, and their in and cross scan contributions for a small region at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

In and cross scan contributions in the {\tt NILC} (top line) and {\tt MILCA} (bottom line) $y$\/-maps. From left to right we present the original $y$\/-maps, and their in and cross scan contributions for a small region at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

In and cross scan contributions in the {\tt NILC} (top line) and {\tt MILCA} (bottom line) $y$\/-maps. From left to right we present the original $y$\/-maps, and their in and cross scan contributions for a small region at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

In and cross scan contributions in the {\tt NILC} (top line) and {\tt MILCA} (bottom line) $y$\/-maps. From left to right we present the original $y$\/-maps, and their in and cross scan contributions for a small region at intermediate Galactic latitudes in the southern sky centered at $(0,-45)$ degrees in Galactic coordinates.

{\it Top}: Standard deviation maps for the {\tt NILC} (top) and {\tt MILCA} (middle) $y$\/-maps corresponding to the inhomogenous noise contribution computed from the half difference of the half-rings maps. {\it Bottom}: Angular power spectrum of the homogenous noise contribution for the {\tt NILC} (orange) and {\tt MILCA} (blue) $y$\/-maps (see main text for details).

{\it Top}: Standard deviation maps for the {\tt NILC} (top) and {\tt MILCA} (middle) $y$\/-maps corresponding to the inhomogenous noise contribution computed from the half difference of the half-rings maps. {\it Bottom}: Angular power spectrum of the homogenous noise contribution for the {\tt NILC} (orange) and {\tt MILCA} (blue) $y$\/-maps (see main text for details).

{\it Top}: Standard deviation maps for the {\tt NILC} (top) and {\tt MILCA} (middle) $y$\/-maps corresponding to the inhomogenous noise contribution computed from the half difference of the half-rings maps. {\it Bottom}: Angular power spectrum of the homogenous noise contribution for the {\tt NILC} (orange) and {\tt MILCA} (blue) $y$\/-maps (see main text for details).

{\it Top}: Comparison between the measured tSZ flux reported in the Planck cluster sample and that estimated directly on the {\tt NILC} $y$\/-map for the blindly detected common sources. {\it Bottom}: As above but for the {\tt NILC} and {\tt MILCA} measured fluxes.

{\it Top}: Comparison between the measured tSZ flux reported in the Planck cluster sample and that estimated directly on the {\tt NILC} $y$\/-map for the blindly detected common sources. {\it Bottom}: As above but for the {\tt NILC} and {\tt MILCA} measured fluxes.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter {\tt MILCA} (top row) and {\tt NILC} (middle row) maps for a selection of PSZ2 cluster candidates with signal to noise ratio of 9.3, 6.2 and 4.6 from left to right. The maps are centred at the positions of the clusters in galactic coordinates, which are given at the bottom of the plot. Color scale is in units of $y \times 10^{-6}$. The bottom row presents the cluster radial profiles for {\tt MILCA} (black) and {\tt NILC} (blue). The beam profile is shown as a blue dashed line.

Compton parameter maps of well known merging systems for MILCA (left column) and for NILC (right column).

Compton parameter maps of well known merging systems for MILCA (left column) and for NILC (right column).

Compton parameter maps of well known merging systems for MILCA (left column) and for NILC (right column).

Compton parameter maps of well known merging systems for MILCA (left column) and for NILC (right column).

Compton parameter maps of well known merging systems for MILCA (left column) and for NILC (right column).

Compton parameter maps of well known merging systems for MILCA (left column) and for NILC (right column).

4$^{\circ}$ x 4$^{\circ}$ average maps for different ranges in $N_{200}$ from 8 to 100. The color scale is in unit of 10$^{-6}$ $y$.

4$^{\circ}$ x 4$^{\circ}$ average maps for different ranges in $N_{200}$ from 8 to 100. The color scale is in unit of 10$^{-6}$ $y$.

4$^{\circ}$ x 4$^{\circ}$ average maps for different ranges in $N_{200}$ from 8 to 100. The color scale is in unit of 10$^{-6}$ $y$.

4$^{\circ}$ x 4$^{\circ}$ average maps for different ranges in $N_{200}$ from 8 to 100. The color scale is in unit of 10$^{-6}$ $y$.

4$^{\circ}$ x 4$^{\circ}$ average maps for different ranges in $N_{200}$ from 8 to 100. The color scale is in unit of 10$^{-6}$ $y$.

4$^{\circ}$ x 4$^{\circ}$ average maps for different ranges in $N_{200}$ from 8 to 100. The color scale is in unit of 10$^{-6}$ $y$.

Integrated tSZ signal as a function of cluster richness (left) and total mass (right). The black points correspond to the average signal obtained for the richness bins considered in Figure \ref{SDSS_stack}. The red line represents the corresponding best-fit power law (Eqs. \ref{sz_rich_rel} and \ref{sz_mass_rel} respectively). Considering $z~\le~$0.42, there are 13814 objects for 8 $< N_{200} \leq$ 10, 37250 for 10 $< N_{200} \leq$ 20, 7458 for 20 $< N_{200} \leq$ 30, 2069 for 30 $< N_{200} \leq$ 40, and 1133 for 40 $< N_{200} \leq$ 60.

Angular cross-power spectra of the \Planck\ {\tt NILC} F/L (left) and {\tt MILCA} F/L (right) reconstructed Compton parameter maps for different Galactic masks corresponding to 30 \% (cyan), 40 \% (green), 50 \% (blue) and 60 \% (pink) of the sky. For comparison we also show {\tt MILCA}-{\tt NILC} F/L (red) and {\tt NILC70} F/L (black) on 50 \% of the sky. See text for details.

Angular cross-power spectra of the \Planck\ {\tt NILC} F/L (left) and {\tt MILCA} F/L (right) reconstructed Compton parameter maps for different Galactic masks corresponding to 30 \% (cyan), 40 \% (green), 50 \% (blue) and 60 \% (pink) of the sky. For comparison we also show {\tt MILCA}-{\tt NILC} F/L (red) and {\tt NILC70} F/L (black) on 50 \% of the sky. See text for details.

Comparison of the tSZ angular power spectrum estimated from the cross-power-spectrum of the {\tt NILC} (left) and {\tt MILCA} (right) F/L maps (black) with the expected angular power spectrum of the confirmed clusters in the \Planck\ Cluster Sample (green line). The angular cross-power spectrum between the {\tt NILCA} and {\tt MILCA} Compton parameter maps and the simulated detected cluster map is shown in red. Tthe correlation between the reconstructed $y$-map and the simulated detected cluster map, to which an arbitrary rotation has been applied, is plotted in grey.

Comparison of the tSZ angular power spectrum estimated from the cross-power-spectrum of the {\tt NILC} (left) and {\tt MILCA} (right) F/L maps (black) with the expected angular power spectrum of the confirmed clusters in the \Planck\ Cluster Sample (green line). The angular cross-power spectrum between the {\tt NILCA} and {\tt MILCA} Compton parameter maps and the simulated detected cluster map is shown in red. Tthe correlation between the reconstructed $y$-map and the simulated detected cluster map, to which an arbitrary rotation has been applied, is plotted in grey.

1D PDF of the \Planck\ $y$-map before (red) and after (orange) masking the PSZ2 clusters, and of the half-difference map (black) for the {\tt NILC} (left) and {\tt MILCA} (right) methods. We also show for comparison the 1D PDF of the simulated PSZ2 cluster map (dark blue).

1D PDF of the \Planck\ $y$-map before (red) and after (orange) masking the PSZ2 clusters, and of the half-difference map (black) for the {\tt NILC} (left) and {\tt MILCA} (right) methods. We also show for comparison the 1D PDF of the simulated PSZ2 cluster map (dark blue).

Bispectra of the {\tt NILC} (green) and {\tt MILCA} (red) $y$\/-maps for four different configurations (equilateral, orthogonal, flat and squeezed), compared with the bispectrum of the projected map of the PSZ2 clusters (blue). $\pm 1\sigma$ uncertainties are indicated as black dotted lines.

Bispectra of the {\tt NILC} (green) and {\tt MILCA} (red) $y$\/-maps for four different configurations (equilateral, orthogonal, flat and squeezed), compared with the bispectrum of the projected map of the PSZ2 clusters (blue). $\pm 1\sigma$ uncertainties are indicated as black dotted lines.

Bispectra of the {\tt NILC} (green) and {\tt MILCA} (red) $y$\/-maps for four different configurations (equilateral, orthogonal, flat and squeezed), compared with the bispectrum of the projected map of the PSZ2 clusters (blue). $\pm 1\sigma$ uncertainties are indicated as black dotted lines.

Bispectra of the {\tt NILC} (green) and {\tt MILCA} (red) $y$\/-maps for four different configurations (equilateral, orthogonal, flat and squeezed), compared with the bispectrum of the projected map of the PSZ2 clusters (blue). $\pm 1\sigma$ uncertainties are indicated as black dotted lines.

{\tt NILC} - {\tt MILCA} F/L cross-power spectrum (black) compared to the power spectra of the physically motivated foreground models. The considered foregrounds are: clustered CIB (green line); infrared sources (cyan line); and radio sources (blue line). Additionally, the best-fit tSZ power spectrum model presented in Sect.~\ref{powerspectcosmo} is also plotted as a solid red line.

2D and 1D likelihood distributions for the combination of cosmological parameters $\sigma_{8} (\Omega_{\mathrm{m}}/0.28)^{3/8}$, and for the foreground parameters $A_{\mathrm{Rad. PS}}$, $A_{\mathrm{CIB}}$ and $A_{\mathrm{IR. PS}}$. We show the 68.3\% and 95.4\% C.L. contours. The red and black contours correspond to a fixed mass bias of 0.2 and 0.4 respectively.

{\tt NILC} - {\tt MILCA} F/L cross-power spectrum after foreground subtraction (red points), compared to the Atacama Cosmology Telescope (ACT; cyan dot) and the South Pole Telescope \citep[SPT; orange, ][]{George_2014} power spectrum estimates. The black line shows the tSZ power spectrum template \citep[EM12][]{2012MNRAS.423.2492E} used in the \Planck\ CMB cosmological analysis \citep{planck2013-p11,planck2014-a13} with its best fit amplitude $A_{tSZ}$ \citep{planck2014-a13}, the grey region allows comparison with the 2$\sigma$ interval.

tSZ power spectrum for existing models in the literature. {\tt NILC}-{\tt MILCA} F/L cross-power spectrum after foreground correction (black dots) compared to the Atacama Cosmology Telescope (ACT; cyan dot) and the South Pole Telescope \citep[SPT; orange, ][]{George_2014} power spectrum estimates. We also show the tSZ power spectrum models from hydrodynamic simulations \citep[][blue]{Battaglia:2012p1841}, from $N$-body simulations plus semi-analytical dust gas models \citep[][cyan; TBO2]{Trac:2011p1795}, and from analytical calculations (\citealt[][green]{Shaw2010}).

Marginalised likelihood distribution for $\sigma_{8} (\Omega_{\mathrm{m}}/0.28)^{3/8}$ for tSZ and CMB based analyses. We represent the tSZ power spectrum analysis results assuming a mass bias, $b$, of 0.2 (red) and 0.4 (orange), the cluster number count analysis results \citep[green;][]{planck2014-a30}, and the combined \Planck\ CMB and BAO analysis \citep{planck2014-a15} with (cyan) and without (blue) extra lensing constraints.