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

Use Newtonian gauge with scalar metric perturbationsLet $\hat{n}$ point from the observer toward the emitter, $\Theta =\delta T/T$, and evaluate the unperturbed ray at $\mathbf{x}(\tau )=({\tau }_0-\tau )\hat{n}$. In the sudden-decoupling approximation the Sachs-Wolfe effect and its linear companions giveHere ${\Theta }_0={\delta }_{\gamma }/4$ is the intrinsic photon monopole; ${\Psi }_{\ast }$ is the emission gravitational redshift; the velocity term is the Doppler CMB anisotropy of the baryon-photon plasma; and the integral is the Integrated Sachs-Wolfe effect from evolving potentials during travel. The last two local terms are an observer monopole and dipole, removed when discussing the higher measured multipoles. Changing the direction convention reverses both velocity signs. A finite cosmological visibility function replaces the sharp emission surface by a weighted source integral.

For scale-invariant primordial potentials, ${\mathcal{P}}_{\Psi }=k^3{P}_{\Psi }/(2{\pi }^2)$ is constant before transfer effects. In matter domination, the adiabatic superhorizon relation is ${\Theta }_0=-2\Psi /3$, so the observable Sachs-Wolfe combination is ${\Theta }_0+\Psi =\Psi /3$. Its amplitude per logarithmic scale is therefore approximately independent of angle. In a flat geometry $\ell \sim k{D}_{\ast }$ and $\theta \sim {\ell }^{-1}$, and projection gives the Sachs-Wolfe plateauBoth the intrinsic and gravitational emission terms have this leading large-angle scaling; they must be combined, not treated as unrelated random sources.

The Euler equation gives $v_b\sim k{\tau }_{\ast }\Psi$ for the growing large-scale matter mode. Thus the Doppler amplitude grows roughly as ${\theta }^{-1}$ relative to the large-scale potential amplitude, while its logarithmic power grows as ${\theta }^{-2}$, until acoustic dynamics and damping invalidate that simple estimate. On subhorizon pressureless scales the Poisson equation gives ${\delta }_m\sim (k\tau {)}^2\Psi$: the corresponding density amplitude has the often-used ${\theta }^{-2}$ scaling. This is a matter-density estimate, not a universal small-angle photon-temperature law; the actual intrinsic photon source undergoes the oscillations in part(b). On superhorizon scales it instead has the constant-potential behavior just described.

The integrated source vanishes for constant linear potentials in pure matter domination. Potential decay during radiation domination or late accelerated expansion makes it nonzero. Its broad, slowly evolving sources favor large angular scales and give a potential-sized contribution there; crossings of many structures cancel progressively on small scales. Scale invariance alone fixes no universal power of angle for the integrated term: its time evolution and transfer functions are also required. Consequently the large-angle plateau, velocity-gradient enhancement and subhorizon density-gradient estimate describe distinct physical regimes, rather than one extrapolated power law across the whole spectrum.

For adiabatic initial conditions, the coupled photon-baryon fluid begins with a coherent displacement and approximately zero initial acoustic velocity. Neglecting slowly varying gravitational driving, its effective temperature satisfies an oscillator equation with phase $k{r}_s$, whereThus the Sachs-Wolfe combination has a dominant cosine phase, while the velocity has a sine phase. At last scattering, the first three temperature Cosmic microwave background acoustic peaks correspond approximately to $k{r}_s=\pi ,2\pi ,3\pi$: first compression, first rarefaction, and second compression of the initially displaced plasma in a potential well. Baryon inertia shifts the oscillator's equilibrium and enhances compressions relative to rarefactions, producing the familiar odd/even peak contrast. Potential decay and radiation driving alter the amplitudes, especially of the earliest peaks.

Adjacent displacement extrema differ by $\pi$, soHere $D_M$ is the transverse comoving distance to last scattering; in a flat model it is ${\tau }_0-{\tau }_{\ast }$. If $c_s$ is treated as constant, this reduces to $\Delta \ell \simeq \pi ({\tau }_0-{\tau }_{\ast })/(c_s{\tau }_{\ast })$. No numerical spacing is determined without a cosmological distance and sound horizon. Gravitational phase shifts, baryon loading and projection displace the exact peak locations from integer multiples of this spacing. These are primarily coherent density-plus-potential acoustic peaks; the label “Doppler peaks” does not mean the velocity term alone creates their sequence.

The first mechanism is Silk damping: finite photon mean free paths allow radiation to diffuse out of compressions, while velocity slip and radiation shear dissipate acoustic motion. A Fourier diffusion equation has $X^{\prime }=-D(\tau )k^{2}X$, hence $X\propto \exp [-k^2\int Dd\tau ]$. For the polarization-inclusive tightly coupled plasma, a useful expression isThe positive scattering rate is $\dot{\kappa }=a{n}_e{\sigma }_T$, and $R$ is the baryon loading parameter. The two terms represent heat conduction from velocity slip and photon shear viscosity. The mode amplitude has an approximately Gaussian cutoff in $k$; power is suppressed by $e^{-2k^2/k_D^2}$. The formula assumes tight coupling and slowly varying coefficients before decoupling.

The second mechanism is finite-width last-scattering damping. Photons emerge from a range of times and depths, so rapidly varying sources are averaged rather than observed at one sharply defined phase. Approximate the normalized cosmological visibility function by a Gaussian of width ${\sigma }_{\tau }$. For a radial plane-wave phase,The associated power factor is $e^{-(k\mu {\sigma }_{\tau }{)}^2}$. An acoustic factor $e^{\pm ik{c}_s\tau }$ is likewise averaged, replacing $k\mu$ by $k(\mu \pm c_s)$ in the constant-sound-speed approximation. Thus fine radial structure and rapid acoustic time oscillations are suppressed once their phase changes substantially across the visibility width. This is an averaging effect after the perturbations exist; Silk damping physically erases them by diffusion. Angular projection and the evolving source determine the final isotropic damping envelope, so the radial $\mu$ dependence must not be silently discarded.

Thomson scattering generates linear Cosmic microwave background polarization when the incident radiation has a photon quadrupole. The cross-section prefers electric fields perpendicular to the scattering direction. Contributions from an isotropic field cancel, and a dipole alone does not supply the required spin-two angular pattern. Schematically the outgoing Stokes combination is proportional towhich selects the incident quadrupolar angular dependence. Tight coupling nearly isotropizes the radiation, making the quadrupole small; it develops when the photon mean free path becomes finite near decoupling. At leading tight-coupling order its size scales as $(k/\dot{\kappa })v_{\gamma }$.

The polarization is therefore weaker than the temperature anisotropy: a characteristic amplitude is of order a few percent to roughly ten percent of the temperature signal, or several microkelvin compared with temperature fluctuations of several tens of microkelvin on acoustic scales. This is an order-of-magnitude comparison, not a scale-independent ratio of every power-spectrum multipole.

The photon monopole follows the acoustic displacement, approximately $\cos (k{r}_s)$; the velocity sourcing the quadrupole follows $\sin (k{r}_s)$. Hence polarization maxima occur near displacement zeros, between the main temperature peaks. In the leading coherent-oscillator approximation the shift is half the peak spacing, $\Delta \ell /2$, although projection and gravitational driving modify exact locations. Scalar perturbations generate E-mode polarization at linear order, not B-mode polarization. Tensor perturbations and lensing can give B modes, and reionization generates an additional large-angle polarization feature.

Three distinct low-redshift nonlinear contributions are:

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.

A low-frequency example is Galactic synchrotron emission: relativistic electrons spiral in the Galactic magnetic field and emit a broad nonthermal spectrum. In antenna-temperature units its approximate spectrum is $T_A\propto {\nu }^{-{\beta }_s}$ with ${\beta }_s$ around3, varying with the electron population and sky direction. It is a major, often dominant contamination at frequencies below roughly20–40GHz; radio point sources and bremsstrahlung can contribute in the same low-frequency region.

A high-frequency example is Galactic thermal dust emission. Grains heated by starlight have approximate modified-blackbody spectrum $I_{\nu }\propto {\nu }^{{\beta }_d}{B}_{\nu }(T_d)$, with grain temperatures of order tens of kelvin. Their emissivity-weighted emission rises rapidly through the microwave/submillimeter range and commonly dominates anisotropy contamination above roughly100–200GHz, especially at higher frequencies. Dusty extragalactic galaxies provide a related high-frequency foreground.

Synchrotron is the low-frequency contaminant and thermal dust the high-frequency contaminant. There is no universal crossover frequency at which either exceeds CMB anisotropy everywhere: sky masking, angular scale, polarization and source amplitude matter. Cleaner cosmological windows lie between these regimes, and multi-frequency observations distinguish the foreground spectra from a thermodynamic CMB temperature perturbation. The separation into low-frequency emission and high-frequency dust is also used in the Planck astrophysical-component products.