Past exam of the astrophysics course of the University of Oxford / 2015 / 3

Use $c=1$ and a spatially flat background dominated by the stated perfect fluid, so $4\pi G{a}^2{\rho }_0=3{\mathcal{H}}^2/2$. In Newtonian gauge, the scalar energy and momentum constraints from the linearized Einstein field equations combine to eliminate the potential's time derivative and give the stated relativistic Poisson constraint. Its source contains the velocity correction to the coordinate density perturbation. On sub-horizon scales that correction is small for the ordinary growing modes, recovering the Poisson equation for Newtonian gravity; near or outside the horizon the gravitational field cannot be sourced by $\delta$ alone. Equality of the two metric potentials also assumes negligible scalar anisotropic stress.

The fluid equations are the linear energy-continuity and momentum-Euler parts of ${\nabla }_{\mu }{T}^{\mu \nu }=0$. Here $c_s^2=\delta p/\delta \rho$ is the relevant dimensionless sound-speed squared in the adopted fluid closure; for a constant-$w$ adiabatic barotropic fluid it equals $w$. A fluid with entropy perturbations needs additional pressure information. The equations actually use $\theta =\nabla \cdot \mathbf{v}$ in Fourier space, the peculiar-velocity divergence. Calling it simply momentum density is imprecise: it is the divergence of the velocity, with momentum weighted by ${\rho }_0+p_0$ before normalization. In particular $\theta$ must have inverse-conformal-time units for the displayed continuity equation to be dimensionally consistent.

For the shifted density perturbation in Newtonian gauge, set ${\Delta }_R=\delta -4\Psi$. The radiation continuity equation gives ${\Delta }_R^{\prime }=-4\theta /3$, and its Euler equation givesWriting $D_R=k^2+6{\mathcal{H}}^2$, the relativistic constraint becomesEliminating the potential therefore givesThis has the requested $\mathcal{F}=12{\mathcal{H}}^2/(k^2+6{\mathcal{H}}^2)$. Well inside the sound horizon, $k\gg \mathcal{H}$, the potential contribution is suppressed relative to the density term and $\delta \simeq {\Delta }_R$. ThusThese are sound waves: radiation-pressure gradients restore a compression, while inertia carries the fluid through equilibrium. The perfect-fluid approximation neglects diffusion damping and freely streaming species' anisotropic stress. It gives the high-frequency limiting oscillations, not a claim that every radiation-era perturbation is an undriven sinusoid at horizon entry.

For pressureless matter, ${\Delta }_M=\delta -3\Psi$ obeys ${\Delta }_M^{\prime }=-\theta$. The Euler equation givesThe constraint is nowConsequentlyFor a matter-dominated flat background, $a\propto {\eta }^2$ and $\mathcal{H}=2/\eta$. On small scales $\mathcal{G}\ll 1$ but $\mathcal{G}{k}^2\to 9{\mathcal{H}}^2/2$: the last term must not be dropped with $\mathcal{G}$. The equation becomesSince $\Psi$ is small relative to the growing density term on these scales, this recovers $\boxed{{\delta }_{\mathrm{grow}}\propto {\eta }^2\propto a}$.

There is a printed error in the requested super-horizon decaying mode. At $k\ll \mathcal{H}$, $\mathcal{G}\to 1$ and the leading equation is ${\Delta }_M^{\prime \prime }+4{\Delta }_M^{\prime }/\eta =0$, henceThe printed $\alpha +\beta /\eta$ fails its own equation: for ${\Delta }_M=1/\eta$, the left side is $-2/{\eta }^3\ne 0$ at leading order. The $1/\eta$ decaying solution would instead result from using the radiation-era background $\mathcal{H}=1/\eta$ in this limiting damping term. Finite-$k$ corrections restore terms of relative order $(k\eta {)}^2$ and do not repair the wrong leading power.

A power spectrum describes the variance of Fourier amplitudes as a function of wavenumber. In a statistically homogeneous continuum convention,The paper's $⟨|{\Delta }_M(k){|}^2⟩$ is the corresponding finite-volume or normalization-suppressed statement. For statistical isotropy only $k=|\mathbf{k}|$ remains. The dimensionless variance per logarithmic interval is $k^3{P}_M(k)/(2{\pi }^2)$, a different quantity.

The response changes around $2k^2\sim 9{\mathcal{H}}^2$, or $k\sim 2.1\mathcal{H}$: the velocity contribution to the relativistic constraint and the effective growth equation change relative importance across the horizon-scale transition. Large-scale ${\Delta }_M$ is approximately constant for its regular mode, while short-scale density perturbations grow gravitationally. A smooth initial potential spectrum can therefore acquire a different $k$ dependence in the density spectrum. For illustration, a constant matter-era potential gives ${\Delta }_M=-(5+k^2{\eta }^2/6)\Psi$, so sufficiently small-scale $P_M$ has four additional powers of $k$ relative to $P_{\Psi }$, while the very large-scale limits have the same power dependence.

This identifies a dynamical transition, not a universal prediction of the primordial spectrum or an exact spectral break at $2\mathcal{H}$ for arbitrary initial data. The familiar late-time matter-spectrum turnover also retains the history of horizon entry around radiation–matter equality; it is not generically tied to the current Hubble scale. Initial conditions and the intervening expansion history are needed for an actual spectrum.