Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 66 / 3

For pressureless matter with curvature and a possible constant cosmological constant, write the background Friedmann equation asDifferentiating on the expanding branch givesConsequentlyThis proves that the Hubble parameter as a solution of the dust growth equation is valid, including constant-$\Lambda$ and curvature contributions to the background. It does not extend without change to a radiation background or arbitrary dynamical dark energy.

In an Einstein-de Sitter universe, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G\bar{\rho }=2/(3t^2)$. The linear matter perturbation growth equation becomesSubstitute $D=t^p$. The indicial equation is $p(p-1)+4p/3-2/3=0$, with roots $2/3$ and $-1$. ThereforeThe Hubble-parameter solution is the decaying mode in the Einstein-de Sitter universe.

Choose the Fourier transform conventionStatistical homogeneity makes the Fourier covariance diagonal, and isotropy makes the matter power spectrum depend only on $k=|\mathbf{k}|$. The two-point correlation function $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$ follows by inserting the transform and using this covariance:Integrating the polar angle relative to $\mathbf{r}$ gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceThe inverse isotropic transform is $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$ whenever these transforms exist, with distributions or regularization otherwise.

For a spherical average of comoving radius $R$, the spherical top-hat window function isAt a fixed epoch let $P(k)=A{k}^n$. The scale-free smoothed density variance isThus the RMS fractional mass fluctuation scales as $R^{-(n+3)/2}$; the RMS mass fluctuation itself scales as $\bar{\rho }{R}^3{\sigma }_{\delta }\propto R^{(3-n)/2}$ at fixed epoch.

The comoving Poisson equation gives ${\Phi }_{\mathbf{k}}=-4\pi G{a}^2\bar{\rho }{\delta }_{\mathbf{k}}/k^2$. For the growing irrotational mode the linearized cosmological continuity equation gives ${\mathbf{v}}_{\mathbf{k}}=-iaHf\mathbf{k}{\delta }_{\mathbf{k}}/k^2$, where $f=d\log D_{+}/d\log a$. The averaged-potential and bulk-velocity variances are thereforeFor a single Cartesian component of the isotropic bulk velocity, divide its variance by three. Rescaling $kR$ givesThese are dimensional scaling laws, with an important convergence qualification. Since $W(y)\to 1$ at small $y$ and has large-$y$ envelope $O(y^{-2})$, the density average converges for $-3<n<1$, the absolute potential average for $1<n<5$, and the bulk velocity for $-1<n<3$. No one unbroken power law makes all three absolute averages finite on the entire range $0<k<\infty$. In particular, long-wavelength modes can dominate the potential. Physical infrared/ultraviolet cutoffs, potential differences, or fluctuation amplitudes in a band around $k\sim R^{-1}$ resolve the relevant divergences; with cutoffs the simple powers need not describe the full integrated variance. The scale-free potential and bulk-flow variance laws should be understood with these conditions, not as a claim that divergent RMS integrals are finite.

Use the same negative-exponent Fourier transform as in the preceding part and let $f=d\log D_{+}/d\log a$. In linear theory, $\dot{\delta }=Hf\delta$, and the linearized cosmological continuity equation is $\dot{\delta }+a^{-1}{\nabla }_x\cdot \mathbf{v}=0$. Its irrotational solution isHere $\mathbf{r}$ labels the comoving spatial position for the Fourier expansion. The quantity being differentiated is also $-\Phi /(4\pi G{a}^2\bar{\rho })$, by the Poisson equation; thus this explicitly obtains the velocity from the Fourier components of the peculiar gravitational potential.

Conjugating the Rayleigh plane-wave expansion, and using the symmetric addition formula for spherical harmonics, givesFor the radial component $U=\mathbf{v}\cdot \hat{r}$, differentiate at fixed angular position. Only the Spherical Bessel function depends on $r$, and $d{j}_{\ell }(kr)/dr=k{j}_{\ell }^{\prime }(kr)$. Therefore the radial peculiar velocity in spherical harmonics isThe coefficient is $4\pi /(2\pi {)}^3=1/(2{\pi }^2)$, and $(-i{)}^{\ell }=(i^{\ell }{)}^{\ast }$, so the phase and conjugation agree with the requested expression. At the present epoch, $a=1$ and $H=H_0$. Adopting the traditional approximation $f\simeq {\Omega }_m^{0.6}$ givesThe printed expression is therefore the present-epoch result in units with $H_0=1$, or with its velocity interpreted as $U/H_0$. For distances and velocities in ordinary physical units, it needs the factor $H_0$; at a general epoch it needs $aH$. The approximation $f\simeq {\Omega }_m^{0.6}$ is exact in the Einstein-de Sitter limit but is not an exact growth formula for arbitrary cosmologies.