Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 53 / 3

In Questions 3–4, use primes for conformal-time derivatives and $\mathcal{H}=a^{\prime }/a$; proper-time derivatives will be stated separately. Let ${\xi }^{\mu }=(T,L^i)$. The metric is a covariant tensor, so under the passive coordinate change ${\tilde{x}}^{\mu }=x^{\mu }+{\xi }^{\mu }$,The inverse Jacobian is ${\delta }_{\mu }^{\alpha }-{\partial }_{\mu }{\xi }^{\alpha }$ to first order. At the same new coordinate argument, the homogeneous background satisfies ${\bar{g}}_{\alpha \beta }(x)={\bar{g}}_{\alpha \beta }(\tilde{x})-T{\bar{g}}_{\alpha \beta }^{\prime }$. Shifting the argument of $\delta g$ or multiplying it by a Jacobian perturbation produces only second-order terms, which we discard. Subtracting the unperturbed metric therefore givesThis is the printed Jacobian expression expanded to linear order, equivalently $\delta \tilde{g}=\delta g-{\mathcal{L}}_{\xi }\bar{g}$. It is a gauge transformation in cosmological perturbation theory, not a change of the physical spacetime.

The perturbations obey $\delta g_{00}=2a^2\psi$, $\delta g_{0i}=-a^2{B}_i$, and $\delta g_{ij}=2a^2(\varphi {\delta }_{ij}-E_{ij})$. Apply part (i). The time-time component changes by $-2a^2(T^{\prime }+\mathcal{H}T)$, and the time-space component changes by $a^2(L_i^{\prime }-{\partial }_{i}T)$. HenceFor the spatial components,Taking the trace and the symmetric trace-free part separately giveswhere ${\partial }_{⟨i}{L}_{j⟩}=\frac{1}{2}({\partial }_i{L}_j+{\partial }_j{L}_i)-{\delta }_{ij}{\partial }_k{L}^k/3$. This explicitly accounts for the trace term in $\varphi$; it must not be absorbed into the trace-free shear.

A matter worldline has coordinate velocity $v^i=d{x}^i/d\eta$. On that same worldline,Thus ${\tilde{v}}^i=v^i+L^{i\prime }$. A time-dependent spatial relabelling makes a physically unchanged worldline move relative to the new coordinate grid. This is the frame dependence of the coordinate peculiar velocity, not an added physical acceleration.

For the scalar gauge transformation with trace-free spatial shear, put $L_i={\partial }_{i}L$. Part (ii) and the velocity rule giveDefine $C=\varphi +{\nabla }^{2}E/3$ and $w=B+v$. Then $\tilde{C}=C+\mathcal{H}T$ and $\tilde{w}=w+T$. Therefore $\mathcal{R}=-C+\mathcal{H}w$ is gauge-invariant, because the two time-shift terms cancel.

To identify the hypersurfaces, lower the index on the matter four-velocity. To first order its spatial components are $u_i=-a(B_i+v_i)=-a{\partial }_{i}w$. The matter worldlines are orthogonal to constant-time slices when $u_i=0$, which for the inhomogeneous scalar modes is $w=0$. Choose $T=-w$ to reach this comoving slicing. Then $C_{\mathrm{co}}=C-\mathcal{H}w=-\mathcal{R}$. Applying the supplied intrinsic-curvature formula on those slices givesSpatially homogeneous additions to the scalar potentials do not affect this curvature.

Work in conformal Newtonian gauge, where $\mathcal{R}=-\varphi +\mathcal{H}v$. SetThe momentum equation gives $v=-({\varphi }^{\prime }+\mathcal{H}\varphi )/A$, henceThis assumes nonzero background enthalpy, as required for the matter comoving slicing. Background energy conservation gives ${\bar{\rho }}^{\prime }=-3\mathcal{H}(\bar{\rho }+\bar{P})$ and thus

The density constraint gives $4\pi G{a}^2\delta \rho ={\nabla }^2\varphi -3\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )$. Insert this into the pressure equation to obtainDifferentiating the expression for $\mathcal{R}$, using the formula for $A^{\prime }/A$ and then $A={\mathcal{H}}^2-{\mathcal{H}}^{\prime }$, groups the terms asThe preceding equation proves the gradient evolution identity for comoving curvature:For a Fourier mode, ${\nabla }^2\varphi =-k^2\varphi$. On super-Hubble scales, $k/\mathcal{H}\ll 1$, its evolution is gradient-suppressed. With the stated regular adiabatic relation between $\mathcal{R}$ and $\varphi$, $\mathcal{R}$ is constant at leading long-wavelength order. At finite wavelength there are corrections; the conclusion is not an exact claim that the Laplacian vanishes.

Inflation produces the primordial comoving curvature perturbation while modes exit the Hubble radius. Its adiabatic super-Hubble conservation allows that initial amplitude to be carried through the end of inflation and subsequent changes of the background equation of state without solving each mode's full evolution at every intervening time. After Hubble re-entry, cosmological transfer functions relate it to the cosmic microwave background temperature and polarization fluctuations and to the matter power spectrum. The conserved curvature perturbation connects inflationary initial conditions to later cosmological observables. This bridge uses the adiabatic conditions stated in part (v); entropy perturbations can supply an additional evolving source.