Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 55 / 2

The comoving observer's four-velocity has $u_0=-a$, so its measured photon energy is $E=-u_{\mu }{p}^{\mu }=a{p}^0$, and the comoving momentum is $q=aE=a^2{p}^0$. Let $P^i=p^i/p^0=d{x}^i/d\tau$. The null geodesic condition isDivide the temporal geodesic equation by $p^0=d\tau /d\lambda$, and use the supplied Christoffel symbols:Differentiating $q=a^2{p}^0$ cancels the homogeneous redshift terms. In the remaining first-order term, replace $P^i$ by the unperturbed unit propagation direction $n^i$:This is the synchronous photon momentum redshift. Background physical momentum decreases as $a^{-1}$, while the background comoving momentum stays constant.

The spatial geodesic equation gives $d{p}^i/d\tau =-2\mathcal{H}{p}^i$ in the unperturbed universe. Since the temporal equation gives the identical factor for $p^0$, their ratio $P^i$ is constant at zeroth order. More explicitly, at first order,Every displayed term contains a metric perturbation or its derivative. The orthonormal-frame unit direction differs from $P^i$ by a first-order metric correction, $n^i=P^i+h^i{}_j{P}^j/2+O(h^2)$, whose derivative is also first order. Hence $d{n}^i/d\tau =O(h)$, with no background change of direction. Direction corrections multiplied by an already first-order source contribute only at second order.

Express the Collisionless Boltzmann equation in variables $(\mathbf{x},q,\mathbf{n},\tau )$ and divide the derivative along the photon ray by $d\tau /d\lambda$. Since $f_0(q)$ has no explicit position, direction or conformal-time dependence,This is the chain rule in momentum magnitude and direction. At linear order, $P^i=n^i+O(h)$, while the last two terms are products of first-order perturbations and can be dropped. The synchronous photon momentum redshift from part i then gives, for spatial Fourier convention $e^{i\mathbf{k}\cdot \mathbf{x}}$,The factor $q$ is essential. It is missing in the PDF's displayed intermediate equation, but follows directly from part i and is needed to obtain its final brightness equation.

Let $I_0={\int }_0^{\infty }{q}^3{f}_0(q)dq$. Absorbing the common phase-space normalization into $f$, the homogeneous photon energy density is ${\rho }_{\gamma }=4\pi I_0/a^4$, so the photon brightness perturbation isA small directional temperature change $\Theta =\Delta T/T$ in a Planck distribution gives $f_1=-q{f}_0^{\prime }(q)\Theta$. Integration by parts yieldsbecause the Planck distribution behaves as $q^{-1}$ at low momentum and decays exponentially at high momentum. Thus $\Delta =4\Theta$. Equivalently, frequency-integrated blackbody radiation energy scales as $T^4$. Identifying brightness with a single temperature perturbation assumes this blackbody spectral form; the brightness integral itself is defined more generally.

Finally, multiply the corrected transport equation by $q^3$, integrate and divide by $I_0$. The spatial streaming factor is independent of $q$, while its gravitational source integrates to $(1/2)(-4)h_{ij}^{\prime }{n}^i{n}^j$. ThereforeThis derives the synchronous photon brightness equation with the normalization and sign fixed, rather than inferring it from the dimensionally inconsistent intermediate formula.

There is first an angular-scale typo in the PDF: smaller angles correspond to larger multipoles, approximately $\theta \sim \pi /\ell$. Small scales relative to the first acoustic peak therefore have $\ell \gtrsim 200$, not the printed inequality in the other direction.

The photon-baryon acoustic oscillator produces rapidly varying density and velocity sources around and below the sound horizon. Treating the emission density as a simple long-wavelength gravitational source misses these acoustic peaks and their phases. Silk damping suppresses short-scale perturbations through diffusion before decoupling. The cosmological visibility function has finite width, so different emission times and acoustic phases are averaged rather than sampled on one sharp surface. As tight coupling breaks down, the photon quadrupole and higher moments are generated; Thomson scattering of the quadrupole produces polarization. These require the full photon Boltzmann hierarchy, rather than exactly vanishing emission moments above the dipole. They are the angular-scale limits of instantaneous photon decoupling.

The distinction is important: acoustic evolution and diffusion can already be included in correctly calculated emission ${\delta }_{\gamma }$ and $\mathbf{v}$. They do not themselves invalidate collisionless propagation after decoupling. The two-moment, single-time initial condition and omission of scattering after that surface are the approximations used in the requested line-of-sight formula.

For the stated assumptions, multiply the synchronous photon brightness equation by $e^{ik\mu \tau }$ and integrate:Here ${\tau }_d$ is the decoupling time. Inverse Fourier transformation shifts each source to the unperturbed rayInsert the initial brightness monopole and Doppler dipole, and divide by four:This is the synchronous Sachs-Wolfe line-of-sight formula. The first term is intrinsic emission temperature, the second is the Doppler effect, and the last is the accumulated gravitational redshift in this gauge. The derivative $h_{ij}^{\prime }$ is the partial conformal-time derivative evaluated on the ray, not its total derivative along the ray. Its integral contains endpoint gravitational effects as well as evolving-potential effects when rewritten in Newtonian gauge; it should not be identified solely with the Integrated Sachs-Wolfe effect.

The direction $\mathbf{n}$ is photon propagation toward the observer. The observer-to-source sky direction is its negative, so changing that convention also changes the displayed Doppler sign and the ray parametrization. Correct emitter positions and this direction convention are needed to interpret the formula.