tokens: sky positions of voids, if any, is at the sub-K level, which is negligible. We present results using the SEVEM map in practice. A common mask UT78 is applied to both the temperature and lensing maps (Planck Collaboration X 2016 ). The lensing situation offers less choice, as only a single convergence map is available: the 2015 lensing data are released directly in the form of the spherical harmonic transform of the (masked) field (Planck Collaboration XV 2016 ). 3.1 The optimal radius of the filter Corresponding to each void centre, the CMB signal is taken to be the average temperature T (or ) within a circular aperture r R filter minus the same quantities averaged over an annular aperture , where R filter is the size of the compensated top-hat filter. We will call the filtered temperature and lensing convergence T and , i.e. To maximize the ISW signal, Cai etal. ( 2014 ) showed that the optimal choice was R filter 0.6 r v , using mock void catalogues defined via haloes from N -body simulations. Using our HOD mocks, we re-investigate this scale factor for a possible dependence on void radius. We find that R filter 0.7 r v gives slightly higher amplitudes for the stacked filtered T signal as well as for the lensing signal for voids with 100 r v 150 h 1 Mpc. The corresponding outer radius of the filter is r v . For simplicity, we will use this size of the filter throughout out analysis, even though it may not be the optimal choice for all ranges of voids. 3.2 Stacking with all voids We now look at the results of stacking the CMB sky at the DR12 void locations. Because the predicted signal varies with void radius, as does the fidelity of the void catalogue, we divided the results into different bins of void radius. We sorted the voids in decreasing order of radius, and measured the average filtered T and imprints for several logarithmically spaced bins of r v . The results are shown in the top row of Fig. 2 . The filtered temperature T is negative at large void radii. The deepest temperature dip is approximately 6K between r v 100 and 150 h 1 Mpc, with a significance of 2.4. T crosses zero at r v 90 h 1 Mpc and remains slightly positive at smaller void radii. We can understand the presence of positive filtered temperature as an indication of voids-in-clouds, i.e. voids living in overdense environments. The gravitational potential at the scale of the void for a void-in-cloud is negative; i.e. it is a potential well rather than a potential hill as intuitively expected for a void. The dominant linear ISW effect thus yields a positive temperature perturbation (Cai etal. 2014 ). We also find that the simulated ISW signal crosses zero, though at a similar void radius of 30 h 1 Mpc. This indicates that the stacked signal for the CMB temperature qualitatively resembles an ISW signal in a CDM universe. For the largest voids, the observed T shows consistency with zero at r v 150 h 1 Mpc, which confirms our speculation from simulations that these objects
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