Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 4

The emission term comes from the local photon temperature. A blackbody radiation density is proportional to $T^4$, so linearization gives $\delta T/T={\delta }_{\gamma }/4$. It is evaluated at last scattering, with the source position on the observed ray.

The velocity term is the Doppler CMB anisotropy: motion of the emitting plasma changes the measured photon frequency by the component of its peculiar velocity along the line of sight. Its sign here follows the paper's choice of the sky direction $\hat{\mathbf{n}}$.

The metric integral follows the null geodesic energy equation in synchronous gauge. Time dependence of the spatial metric perturbation changes the photon energy by $-h_{ij}^{\prime }{n}_i{n}_j/2$ along the ray. It includes gravitational redshift endpoint terms as well as the part due to evolving gravitational potentials. In particular, the entire synchronous metric integral is not just the integrated Sachs-Wolfe effect. Integrating by parts to express it in conformal Newtonian gauge separates the ordinary Sachs-Wolfe effect from the evolving-potential contribution.

Use one Fourier mode, with $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{n}}$ and unperturbed ray $\mathbf{x}(\tau )=\hat{\mathbf{n}}({\tau }_0-\tau )$. Put $E(\tau )=e^{ik\mu ({\tau }_0-\tau )}$, and let ${\tau }_{\ast }$ denote last scattering. In the supplied branch $h^{\prime }=h_s^{\prime }$, the scalar decomposition givesSince $E^{\prime \prime }=-k^2{\mu }^{2}E$, two integrations by parts give the exact endpoint decomposition of the metric contribution:For the matter-era growing mode, $h_s$ is quadratic in $\tau$, so the last integral vanishes. With $E^{\prime }=-ik\mu E$, the emission endpoint isand the observer endpoint is $-h_s^{\prime \prime }({\tau }_0)/(2k^2)-i\mu h_s^{\prime }({\tau }_0)/(2k)$. The observer terms are only a angular monopole and angular dipole; they are removed when studying the anisotropy multipoles $\ell \ge 2$.

For an adiabatic growing mode outside the horizon at last scattering, ${\delta }_{\gamma }=4{\delta }_m/3$, ${\delta }_m=-h/2$ after fixing the constant trace convention, and $h\propto {\tau }^2$. Hence the intrinsic emission term is smaller than $h_s^{\prime \prime }/(2k^2)$ by order $(k{\tau }_{\ast }{)}^2$. The derivative emission endpoint is smaller by order $k{\tau }_{\ast }$. Stress-energy conservation also makes the peculiar emission velocity subleading; its leading synchronous contribution cancels in ${\delta }_{\gamma }^{\prime }+2h^{\prime }/3$, with gradients supplying the remaining small term. Keeping the Fourier phase, which need not be small over the observer's distance, givesHere $\Theta =\delta T/T$, and the sum over modes is understood. The derivative is the second derivative of $h_s$, as printed in the original PDF. This approximation assumes matter-era growing adiabatic modes and neglects late evolving potentials; large wavelength alone does not remove the last integral in a general cosmology. The equality $h^{\prime }=h_s^{\prime }$ is used as the supplied synchronous condition, rather than as a general identity for every fluid solely from vanishing anisotropic stress.

Write the growing density contrast as ${\delta }_{m,\mathbf{k}}=A_{\mathbf{k}}{\tau }^2$, with $A_{\mathbf{k}}$ having the indicated $k^{n/2}$ spectral amplitude. In the dust-comoving synchronous gauge, $h^{\prime }=-2{\delta }_m^{\prime }$, and the supplied $h_s^{\prime }=h^{\prime }$ therefore implies $h_s^{\prime \prime }=-4A_{\mathbf{k}}$. On the same comoving density slicing, the cosmological Poisson equation isbecause $\mathcal{H}=2/\tau$. This is the comoving-density form of the relativistic scalar constraint, so it can be used on these large scales; replacing its density by the Newtonian-gauge density would not be justified there. Combining the two results givesThus the large-angle Sachs-Wolfe effect has source $\Phi /3$. The potential is constant for this matter-era growing mode, and its power spectrum obeys $P_{\Phi }(k)\propto k^{n-4}$.

Restore the Fourier transform integral and the last-scattering distance:Insert the Rayleigh plane-wave expansion and use orthogonality of spherical harmonics with the complex conjugate on the projection harmonic. The angular integral selects the matching indices, givingThe complex conjugation is present in the original PDF. If $A_{\mathbf{k}}=A{k}^{n/2}$ denotes a mode coefficient with its random amplitude and phase included in $A$, substitution of ${\Phi }_{\mathbf{k}}=-6A_{\mathbf{k}}/k^2$ gives the equivalent factor $-8\pi i^{\ell }{A}_{\mathbf{k}}/k^2$ inside the Fourier integral. The monopole and local observer dipole removed in part (b) do not change this formula for $\ell \ge 2$.

Let $⟨{\Phi }_{\mathbf{k}}{\Phi }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }^3(\mathbf{k}-{\mathbf{k}}^{\prime })P_{\Phi }(k)$. Statistical isotropy and $\int |Y_{\ell m}{|}^{2}d\Omega =1$ applied to the preceding expression give the CMB angular power spectrumFor $n=1$, define the constant dimensionless potential power ${\mathcal{A}}_{\Phi }=k^3{P}_{\Phi }(k)/(2{\pi }^2)$. ThenThe needed logarithmic spherical Bessel square integral can be proved directly. Put $I_{\ell }={\int }_0^{\infty }{j}_{\ell }^{2}dx/x$ and $J_{\ell }={\int }_0^{\infty }{j}_{\ell }{j}_{\ell -1}dx$. From the Spherical Bessel function recurrences,integration of the squares' derivatives gives $J_{\ell }=(\ell +1)I_{\ell }$, and for $\ell >1$ also $J_{\ell }=(\ell -1)I_{\ell -1}$. At $\ell =1$, $j_0(0)=1$ and $j_0(\infty )=0$ give $J_1=1/2$, hence $I_1=1/4$. Induction now proves $I_{\ell }=1/[2\ell (\ell +1)]$. ConsequentlyThis is the flat large-angle Sachs-Wolfe plateau. The useful integral printed in the PDF is missing the square on $j_{\ell }$: without that square the formula is false, since at $\ell =1$ its left-hand side is $\pi /4$, whereas its asserted value would be $1/4$. The squared identity above is the one required by the variance calculation. The plateau describes the matter-era large-angle approximation, not acoustic peaks or late integrated-potential corrections.