Past exam of the astrophysics course of the University of Oxford / 2022 / 6

Treat the long-wavelength nearly uniform spherical perturbation as a separate locally homogeneous expanding region, neglecting pressure gradients and tidal corrections. Its local Friedmann equation isThe curvature term represents the spatial curvature of an FLRW universe in the local patch; for the convention $a_s$ dimensionless, $K=k{c}^2$ has units of inverse time squared. Positive $K$ corresponds to a closed, more tightly bound patch. Here and below the printed $\rho$ is interpreted as mass density or energy density divided by $c^2$; literal SI energy density requires $G\rho /c^2$ on the right.

The region generally has its own scale factor and Hubble parameter. The two printed equations with an identical $H(t)$ can be compared as a leading-order equal-expansion construction: on a slicing where $H_s=H$, use $a_s=a$ in the already first-order curvature term to obtain $K/a^2=(8\pi G/3)\delta \rho$. This equal-expansion curvature overdensity construction does not make them both exact evolution equations with a permanently identical expansion history at common proper time. For example, a growing dust density contrast obeys the continuity equation $H_s-H=-\dot{\delta }/3$, which is nonzero when $\delta \propto a$. This explicit example prevents an inconsistent identification of the two local histories.

Within the equal-expansion construction of a, subtract the flat-background Friedmann equation to obtainThe background density scales as ${\rho }_b\propto a^{-4}$ during radiation domination and ${\rho }_b\propto a^{-3}$ during matter domination. ThereforeThese are the regular adiabatic growing-mode scalings on scales outside the Hubble radius, with a specified time slicing. In a consistent comoving description the numerical relation between curvature and density contrast differs from the naive subtraction, but the powers agree. The result does not imply that every gauge's density contrast, every decaying mode, or subhorizon dark matter during radiation domination has these powers.

For $r_s\gg d_H$, a regular adiabatic perturbation follows the large-scale density contrast growth in b; this does not require causal communication across the perturbation. Since $d_H=c/H$, the Friedmann equation gives $d_H\propto a^2$ during radiation domination and $d_H\propto a^{3/2}$ during matter domination. A fixed comoving perturbation has physical size $r_s=aR$, so its ratio to $d_H$ decreases as $a^{-1}$ or $a^{-1/2}$ until cosmological horizon entry.

The numerical “exactly at equality” assertion is inconsistent with the supplied parameters and $d_H=c/H$. Normalize $a_0=1$ and use the supplied ${\Omega }_{\gamma }$ as the whole radiation component. ThenBoth terms must be retained at matter-radiation equality. The present comoving size corresponding to its Hubble radius isIts physical size then is $0.0355\mathrm{Mpc}$. A mode with $R=79\mathrm{Mpc}$ has physical size only $0.0263\mathrm{Mpc}$ at equality, already inside the Hubble radius. More explicitly, putting $q=R{H}_0/c$ in $aR=c/H(a)$ givesThis is a direct numerical counterexample using the stated data. Neglecting one component exactly at equality instead gives $150.5\mathrm{Mpc}$, which also does not produce $79\mathrm{Mpc}$. Late-time dark energy is negligible at these early times.

For conserved nonrelativistic dark matter and radiation, ${\rho }_{\mathrm{DM}}\propto a^{-3}$ and ${\rho }_{\gamma }\propto a^{-4}$, so $y={\rho }_{\mathrm{DM}}/{\rho }_{\gamma }=a/a_{\mathrm{eq}}$ obeys $\dot{y}=yH$. Denote differentiation with respect to $y$ by a prime. ThenThe supplied Friedmann equations implySubstitution in the linear growth equation and division by $y^2{H}^2$ give the Mészáros equationFor $D_1=1+3y/2$, $D_1^{\prime }=3/2$ and $D_1^{\prime \prime }=0$, making the two remaining terms cancel exactly. For reduction of order, put $\delta =f{D}_1$ and $P=(2+3y)/[2y(1+y)]=1/y+1/[2(1+y)]$. The equation reduces to $f^{\prime \prime }+(2D_1^{\prime }/D_1+P)f^{\prime }=0$, whenceUsing the supplied antiderivative and reversing an irrelevant overall sign gives the independent solutionIts Wronskian is $D_1{D}_2^{\prime }-D_1^{\prime }{D}_2=-1/[y\sqrt{1+y}]\ne 0$. The asymptotic expansions areThus subhorizon dark matter during radiation domination has constant and logarithmic modes, rather than the superhorizon $a^2$ growth of b. A combination with negative $A_2$ supplies positive logarithmic growth. During matter domination, the growing mode is proportional to $a$, agreeing with b, while the second mode decays as $a^{-3/2}$. These results neglect subhorizon radiation clustering; the Mészáros equation is the approximation stated in the question, not a full photon–matter perturbation system through horizon entry.

For a $1\mathrm{Mpc}$ mode, evolve the regular superhorizon density contrast as $\delta \propto a^2$ until cosmological horizon entry. In the radiation domination approximation its entry scale is $a_{\mathrm{ent}}\simeq (H_{0}R/c)\sqrt{{\Omega }_{\gamma }}=2.2\times 10^{-6}$, well before matter-radiation equality. At that point use the Mészáros equation basis of d, fixing both constants by continuity of $\delta$ and $\dot{\delta }$, equivalently $\delta$ and $d\delta /d\ln a$.

Explicitly, for entry data ${\delta }_e$ and ${\delta }_e^{\prime }=2{\delta }_e/y_e$, solveThe nonzero Wronskian guarantees a unique match. This produces slow logarithmic growth before equality, then $\delta \propto a$ with a decaying transient after equality. For example, at $y_e\ll 1$, $A_2\simeq -2{\delta }_e$ and $A_1\simeq {\delta }_e+2{\delta }_e[\ln (4/y_e)-3]$. An abrupt match estimates the growth history; a precise amplitude requires a common gauge and the coupled relativistic perturbation equations through entry. Linear evolution also ceases to apply once $|\delta |$ approaches unity, and late dark energy further suppresses growth.