Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 57 / 2

Preserving synchronous gauge means keeping both $\delta g_{00}$ and $\delta g_{0i}$ zero. Applying the metric transformation to the background givesThe first equation requires $(a{\xi }^0{)}^{\prime }=0$, so ${\xi }^0=C(\mathbf{x})/a$. For the scalar displacement ${\xi }^i={\partial }^i\lambda$, the second requires ${\lambda }^{\prime }={\xi }^0$, up to a spatially homogeneous coordinate displacement that does not affect scalar perturbations. Integrating givesThis is the residual gauge freedom in synchronous gauge. The lapse and shift conditions do not uniquely choose a slicing and spatial labeling.

A background density is a scalar, so under this remaining time displacementFor a fluid with constant equation of state, background conservation gives ${\bar{\rho }}^{\prime }/\bar{\rho }=-3(1+w)\mathcal{H}$, henceThe synchronous density gauge mode scales as ${\tau }^{-2}$ in radiation domination and as ${\tau }^{-3}$ in matter domination. It can mimic a changing density perturbation without changing physical observables. A density time dependence alone therefore does not identify a physical growing or decaying mode. Choosing coordinates comoving with cold matter fixes the residual time freedom, and a remaining spatial relabeling fixes the initial metric-trace integration convention. Gauge-invariant curvature perturbations provide another way to specify physical initial conditions.

For consistency, linear metric theory requires all background-orthonormal components of $h_{ij}$ to be small. The PDF's small-determinant condition by itself is insufficient: $h_{ij}=\mathrm{diag}(L,0,0)$ has zero determinant even for large $L$. This is small determinant does not imply a small metric perturbation.

For a fixed coordinate cell, the spatial volume element isThus its perturbation relative to the background expansion, or its perturbed comoving volume, is $\Delta V_c/V_c=h/2$. In the cold-matter rest frame each such cell contains a fixed amount of matter. Since density is mass divided by physical volume,Equivalently, stress-energy conservation gives ${\delta }_c^{\prime }=-h^{\prime }/2$; the stated relation fixes the spatially dependent integration constant by the initial mass-coordinate convention. A contraction of comoving volume gives a positive density contrast.

In the matter era $a\propto {\tau }^2$, so $\mathcal{H}=2/\tau$ and the matter density fraction is approximately one. The evolution equation reduces toFor ${\delta }_c\propto {\tau }^p$, the indicial equation is $p(p-1)+2p-6=(p-2)(p+3)=0$. ThereforeThe growing mode is proportional to $a$, and the decaying mode to $a^{-3/2}$. These are the density solutions after the comoving coordinate choice; the residual time mode found in part (i) explains why a similar power in an unfixed density variable cannot be interpreted in isolation.

There is a normalization omission in the printed density parameter if ${\bar{\rho }}_c$ denotes physical density. Its usual definition in conformal time isThe printed expression lacks $a^2$, unless its density is already rescaled by that factor. The matter-era statement ${\Omega }_c\simeq 1$ and the growth equation use the physical density fraction above.

On superhorizon scales during radiation domination, the regular adiabatic cosmological perturbation has approximately constant primordial curvature. The comoving density perturbation is gradient-suppressed, of order $(k/\mathcal{H}{)}^2$ times that curvature. Since $\mathcal{H}\propto {\tau }^{-1}$, it grows as ${\delta }_c\propto k^2{\tau }^2$ for fixed mode amplitude. This is the regular superhorizon solution in the specified slicing, not causal collapse driven by communication across the entire wavelength.

After radiation-era horizon entry, the dominant radiation develops pressure-supported acoustic oscillations. Its potential no longer provides a steady growing gravitational source, while cold matter is too small a fraction of the total density to produce rapid self-gravitating growth. The result is the Mészáros effect: growth is strongly suppressed until equality. The supplied stagnation approximation replaces that slow evolution by a constant. More accurately, ignoring the rapidly oscillating source at leading order gives ${\delta }_c^{\prime \prime }+{\delta }_c^{\prime }/\tau \simeq 0$, with a constant and a logarithmic solution. Thus radiation-era subhorizon growth is at most slow logarithmic growth, rather than the matter-era power law; treating it as constant is the explicit approximation used for the simple transfer estimate below.

Take the regular growing matter mode ${\delta }_c=C{\tau }^2$ and $h=-2C{\tau }^2$. With $\mathcal{H}=2/\tau$, the constraint givesConsequentlyHere ${\zeta }_{\ast }=10C/k^2$ is its early, superhorizon constant value. The fractional time variation is of second gradient order, andThus the primordial curvature is conserved to leading order on superhorizon scales. This is the matter-era density transfer from primordial curvature.

A mode entering after equality evolves in the growing matter solution, givingThis assumes matter-dominated growth to ${\tau }_0$, as in the model. A subsequent dark-energy era would multiply by the appropriate late-time growth suppression. The initial radiation-era adiabatic curvature passes through equality unchanged at leading superhorizon order, so the same ${\zeta }_{\ast }$ is the primordial input.

For $k\gg k_{\mathrm{eq}}$, entry occurs in radiation domination at ${\tau }_h\sim k^{-1}$. The entry contrast is of order ${\zeta }_{\ast }$ because its superhorizon value scales as $k^2{\tau }_h^2{\zeta }_{\ast }$. In the question's stagnation approximation it stays of this order until equality, then grows by $({\tau }_0/{\tau }_{\mathrm{eq}}{)}^2$. Thusindependent of $k$ to that approximation. The precise coefficient needs matching through the equality transition; the horizon convention, including the use of wavelength $2\pi /k$, also changes the matching definition of $k_{\mathrm{eq}}\sim {\tau }_{\mathrm{eq}}^{-1}$ by a fixed factor.

One continuous sharp-equality model, normalizing the plateau by matching the low-$k$ branch at $k_{\mathrm{eq}}$, is the sharp-equality cold-dark-matter transfer approximationThis is a matching approximation, not an exact solution through equality. The function requested here transfers curvature directly to density. It differs from the conventional normalized matter transfer $\mathcal{T}(k)$, where one factors out $k^2{\tau }_0^2/10$: that normalized function tends to one at low $k$ and to $(k_{\mathrm{eq}}/k{)}^2$ in the stagnation model at high $k$.

For the dimensional primordial spectrum $P_{\zeta }=A/k^3$, linear deterministic transfer gives $P_c=|T_{\delta }{|}^2{P}_{\zeta }$. Hence this sharp model yieldsThe spectrum turns over near equality. Retaining the slow radiation-era logarithmic growth refines the high-$k$ transfer to $T_{\delta }\propto ({\tau }_0/{\tau }_{\mathrm{eq}}{)}^2\ln (k/k_{\mathrm{eq}})$, up to constants in the logarithm and matching factors, and therefore refines the small-scale spectrum to $P_c\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$. This is the logarithmic small-scale matter transfer correction to the question's constant-growth idealization. Multiplication by $k^3/(2{\pi }^2)$ would give dimensionless power, whose large-scale slope is $k^4$ rather than the dimensional spectrum's $k$.