from electrons with a bulk flow velocity $\vec v$, the photons are Doppler shifted to higher or lower frequencies depending on the orientation of $\vec v$ with respect to the line of sight. This kinematic SZ (kSZ) effect has a black-body spectrum, and the contribution from each volume element scales as $(v/c) n_e$ where $c$ is the speed of light and $n_e$ is the density of free electrons. In massive galaxy clusters, photons scattering from hot electrons tend to gain energy, distorting the primary CMB black-body spectrum. This is known as the thermal SZ (tSZ) effect and leads to fewer observed photons below 217GHz and more above 217GHz, with a contribution from each volume element scaling as $(k_B T_e/m_ec^2)n_e$, where $m_e$ is the mass of the electron and $T_e$ is the temperature of the electrons. The different frequency scalings of the tSZ and kSZ effects allows for their simultaneous determination in multi-frequency observations.

The tSZ anisotropy power depends strongly on the normalization of the matter power spectrum, commonly parametrized by the RMS of the $z=0$ linear mass distribution on 8 Mpc/$h$ scales, $\sigma_8$ [e.g., @komatsu02]. This steep scaling (see Section \[sec:tsz\] for details) means that a reasonably precise measurement of 