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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-75/1/e/solution

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 75 / 1 / e / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

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 $x = h ν / (k_{B} T_{γ})$ , the thermodynamic-temperature equivalent is $Δ T / T = y [x coth (x /2) - 4]$ , with Compton y parameter $y = \int n_{e} σ_{T} k_{B} T_{e} / (m_{e} c^{2}) d l$ . This is a spectral distortion: a low-frequency decrement, a high-frequency increment, and a nonrelativistic null at $x ≃ 3.83$ , 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, $Δ T / T = - \int n_{e} σ_{T} (v \cdot n / c) d l$ . 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 $\partial B_{ν} / \partial T$ 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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