= Solution
Three distinct low-redshift nonlinear contributions are:
* The <thermal Sunyaev-Zeldovich effect> arises when hot cluster electrons inverse-Compton scatter the <cosmic microwave background>. Photons receive energy from the thermal electron distribution. With $x=h\nu/(k_BT_\gamma)$, the thermodynamic-temperature equivalent is $\Delta T/T=y[x\coth(x/2)-4]$, with <Compton y parameter> $y=\int n_e\sigma_T k_BT_e/(m_ec^2)\,dl$. This is a spectral distortion: a low-frequency decrement, a high-frequency increment, and a nonrelativistic null at $x\simeq3.83$, about217GHz for the present CMB temperature. Relativistic electron corrections shift the detailed spectrum and null.
* The <kinetic Sunyaev-Zeldovich effect> is the Doppler shift caused by scattering off the bulk motion of an electron cloud. At lowest order, $\Delta T/T=-\int n_e\sigma_T(\mathbf v\cdot\widehat n/c)\,dl$. It can have either sign. Unlike the thermal effect, it has the spectrum of a small <blackbody> temperature shift: frequency independent in thermodynamic temperature units, or proportional to $\partial B_\nu/\partial T$ in intensity units.
* The <Rees-Sciama effect> arises from potentials changing during nonlinear collapse, accretion or motion of structures. Energy gained when a photon enters a well need not equal energy lost on departure. Its gravitational line-of-sight source is the nonlinear evolving-potential counterpart of the <Integrated Sachs-Wolfe effect>. It too is achromatic in thermodynamic temperature units, not a Compton spectral distortion. Linear potentials can be constant in a matter era while nonlinear potentials still evolve.
Gravitational lensing supplies another nonlinear modification by remapping pre-existing temperature and polarization patterns. It preserves the local blackbody spectrum; it does not generate temperature differences by lensing a perfectly isotropic background.
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