A statistical analysis of stacked Compton$-y$ maps of quasar hosts with a median redshift of $1.5$ using Millennium Simulation is performed to address two issues, one on the feedback energy from quasars and the other on testing dark matter halo models for quasar hosts. On the first, we find that, at the resolution of FWHM=$10$ arcmin obtained by Planck data, the observed thermal Sunyaev-Zeldovich (tSZ) effect can be entirely accounted for and explained by the thermal energy of halos sourced by gravitational collapse of halos, without a need to invoke additional, large energy sources, such as quasar or stellar feedback. Allowing for uncertainties of dust temperature in the calibration of observed Comton$-y$ maps, the maximum additional feedback energy is $sim 25%$ of that previously suggested. Second, we show that, with FWHM=$1$ arcmin beam, tSZ measurements will provide a potentially powerful test of quasar-hosting dark matter halo models, limited only by possible observational systematic uncertainties, not by statistical ones, even in the presence of possible quasar feedback.
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