Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 57 (Advanced Cosmology)

Use $\mathcal{H}=a^{\prime }/a$ for the conformal expansion rate and $H=a^{\prime }/a^2$ for the physical Hubble parameter. A future unit normal to a constant-$\tau$ slice isIt has norm minus one, and the spatial projection tensor is $P^{\mu }{}_{\nu }=\mathrm{diag}(0,1,1,1)$. The nonzero connections needed here are ${\Gamma }^0{}_{00}=\mathcal{H}$, ${\Gamma }^0{}_{ij}=\mathcal{H}{\delta }_{ij}$ and ${\Gamma }^i{}_{0j}=\mathcal{H}{\delta }^i{}_j$. Thus ${\nabla }_i{n}_j=-{\Gamma }^0{}_{ij}{n}_0=a^{\prime }{\delta }_{ij}$. Projection removes every normal component, yieldingThis uses the positive-expansion convention for extrinsic curvature in the question. With the alternative convention that inserts an overall minus sign, all these extrinsic-curvature quantities change sign.

For a comoving cell, the physical volume is proportional to $a^3$. Its logarithmic derivative in proper time is $3\dot{a}/a=3H=K$. Thus the trace is the expansion rate of the normal congruence: positive for expansion and negative for contraction. The isotropic tensor $K^i{}_j=H{\delta }^i{}_j$ has vanishing trace-free shear. It measures homogeneous expansion, not the intrinsic spatial curvature, which is zero in this flat model.

Write $q_{\alpha \beta }=g_{\alpha \beta }+n_{\alpha }{n}_{\beta }$ for the induced metric. The spatial covariant derivative projects its derivative index and both tensor indices:Four-dimensional metric compatibility removes the derivative of $g$. Each remaining term contains an undifferentiated normal on a projected index and hence vanishes. ThereforeThis establishes metric compatibility of the spatial covariant derivative directly from its projected definition.

Differentiating the normal's normalization givesso $n^{\mu }{\nabla }_{\nu }{n}_{\mu }=0$. Consequently the second projection in the extrinsic curvature definition is redundant:Finally differentiate $P^{\lambda }{}_{\nu }={\delta }^{\lambda }{}_{\nu }+n^{\lambda }{n}_{\nu }$ and project:The first term vanishes by $P^{\nu }{}_{\beta }{n}_{\nu }=0$, and the second is precisely the defining curvature. The positive sign follows from the specified positive-expansion convention, rather than from an independent projector convention.

Let $a_{\mu }=n^{\nu }{\nabla }_{\nu }{n}_{\mu }$ be the normal acceleration. It is tangential by normalization. Resolving the derivative's first index into normal and tangential parts givesThe extrinsic curvature is symmetric because the normal is hypersurface orthogonal: locally $n_{\mu }=-N{\partial }_{\mu }t$, and the antisymmetric derivative of $n$ vanishes after projecting both indices tangentially.

Apply the supplied Ricci commutator to $n^{\nu }$ and project its free index. With the positive-expansion convention it givesTo identify the first two terms, expand the fully projected divergence:Tangency gives $n_{\lambda }{K}^{\lambda }{}_{\mu }=0$. Differentiating it shows that the last term is $-P^{\mu }{}_{\alpha }{K}^{\lambda }{}_{\mu }{n}^{\rho }{\nabla }_{\rho }{n}_{\lambda }=-K_{\alpha \lambda }{a}^{\lambda }$. Substituting into the commutator proves the contracted Codazzi equation with positive expansion:No assumption of geodesic normals was needed; the acceleration terms cancel rather than being discarded.

The metric-trace term in the Einstein field equations drops out of a tangential-normal projection, since $P^{\mu }{}_{\alpha }{g}_{\mu \nu }{n}^{\nu }=0$. Define the normal-frame momentum density $j_{\alpha }=-P^{\mu }{}_{\alpha }{T}_{\mu \nu }{n}^{\nu }$. The equation becomesThus Codazzi supplies the momentum constraint on each spatial slice. In flat homogeneous FRW, $K^i{}_j=H{\delta }^i{}_j$ and $H$ has no spatial gradient, so its left side vanishes. A comoving homogeneous perfect fluid likewise has $j_i=0$. The background satisfies the constraint automatically; the Friedmann expansion law is instead obtained from the normal-normal constraint.

Suitable scalar variables are lapse and shift potentials $A,B$, spatial curvature and scalar shear potentials $\psi ,E$, and a matter momentum potential $J$ with $j_i={\partial }_{i}J$. For example, chooseA density contrast and pressure perturbation can be added for the remaining scalar matter equations. With this positive-shift convention, the linear extrinsic-curvature perturbation isIts scalar shear cancels out of the momentum-constraint combination, leavingThese are the scalar momentum constraint with positive extrinsic curvature variables; a specified gauge can then remove two of the four scalar metric potentials.

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 $k$.

The collision rate $\kappa =a{n}_e{\sigma }_T$ is the scattering probability per unit conformal time. Thomson scattering approximately conserves photon number and energy in the electron rest frame, but redistributes propagation directions. Consequently the collision term drives the photon angular distribution toward an isotropic monopole plus the dipole associated with the baryon/electron bulk velocity. It damps the photon-baryon relative velocity and the quadrupole and higher directional anisotropies. Photon and baryon momentum-transfer terms have equal and opposite forces when weighted by their background enthalpies.

The metric terms describe the change in photon energy along a geodesic in a time-dependent spatial metric. The trace perturbation supplies an isotropic redshift, while the scalar trace-free perturbation supplies a direction-dependent redshift proportional to $3{\mu }^2-1$. Brightness is proportional to the fourth power of temperature, accounting for the factor four relative to the temperature shift. These are gravitational sources, not scattering terms.

When $\kappa \gg k,\mathcal{H}$, angular anisotropy is scattered away faster than free streaming or background expansion can regenerate it. In the photon Boltzmann hierarchy, a moment of order $\ell \ge 2$ has damping of order $\kappa {\Delta }_{\ell }$ and is sourced by neighboring moments with coefficients of order $k$. Successive moments are therefore suppressed by powers of $k/\kappa =k{\tau }_c$. This is the tight-coupling approximation. At leading perfect-fluid order the quadrupole can be neglected; its first nonzero correction must be retained to calculate the diffusion damping in part (iii).

The printed equations do not all share a consistent velocity convention. To fix the intended one, take the Fourier velocity to be $v^i=i{k}^i\theta$. Ignoring the metric source, the synchronous perfect-fluid density equation then givesThe PDF's baryon continuity sign is opposite to this convention. Keeping it literally would make the adiabatic ratio and common velocity in part (ii) incompatible for any nonstationary perturbation. Likewise, the momentum-transfer terms for these potentials are $-\kappa ({\theta }_{\gamma }-{\theta }_b)$ and $-R\kappa ({\theta }_b-{\theta }_{\gamma })$, without the printed $k^{-2}$.

For isotropic scattering in this convention the brightness source is $\kappa [{\delta }_{\gamma }+4ik\mu {\theta }_b-\Delta ]$. Its dipole relaxes toward the baryon velocity. The printed source involving ${\theta }_{\gamma }-{\theta }_b$ is not this source, and cannot yield the stated Euler drag by taking its dipole. The following solutions use the consistent continuity and momentum equations. These repairs do not change the requested tight-coupling reduction because the opposite drag forces cancel from its total-momentum equation. The velocity-divergence convention in the standard photon and baryon evolution equations is converted to this potential convention by $\vartheta =-k^2\theta$.

An initially adiabatic cosmological perturbation leaves the local photon-to-baryon number ratio unchanged. Since radiation density scales as the fourth power of temperature and nonrelativistic particle number as the third, its density contrasts obey ${\delta }_b=3{\delta }_{\gamma }/4$. With the corrected common-velocity continuity equations,so the adiabatic relation persists in tight coupling.

Use the paper's ratio $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$, which is the reciprocal of the commonly used baryon loading parameter. Multiply the photon Euler equation by $R$ and add the baryon Euler equation. The collision forces cancel exactly:This cancellation works even with the common erroneous $k^{-2}$ in the printed forces, but a physical collision rate uses the repaired forces in part (i). At leading tight-coupling order put ${\theta }_{\gamma }={\theta }_b=\theta$ and substitute ${\delta }_b=3{\delta }_{\gamma }/4$. ThenDefine the pressure-to-inertia sound speed, neglecting the very small baryon thermal pressure, byThe combined velocity equation consequently becomesThis is the requested photon-baryon acoustic oscillator system. Photon pressure supplies the restoring force, while both photon enthalpy and baryon mass supply inertia. The retained $c_{sb}^2$ term adds the small thermal-baryon correction rather than being part of the quoted leading $c_s^2$.

For photon-dominated inertia, $R\gg 1$ and $c_s^2\simeq 1/3$. The baryon expansion and pressure terms are suppressed by $1/R$. Ignoring metric driving as stipulated, the common-velocity Euler equation is ${\theta }^{\prime }=-{\delta }_{\gamma }/4+{\sigma }_{\gamma }$. Differentiating the photon continuity equation givesThis limit is compatible with a matter-dominated background: cold dark matter can dominate the expansion while photons dominate the inertia of the photon-baryon subsystem.

In the first nonzero quadrupole approximation, neglect ${\sigma }_{\gamma }^{\prime }$ relative to its collision damping. The shear equation givesSubstitution yields the shear-only photon diffusion damping equationThe sign of the first-derivative term is positive, so finite photon mean free time damps acoustic energy.

Put $\omega =k/\sqrt{3}$ and $\gamma (\tau )=4k^2{\tau }_c(\tau )/15$. For constant ${\tau }_c$, the characteristic roots are $-\gamma /2\pm i\sqrt{{\omega }^2-{\gamma }^2/4}$. The frequency shift begins at second order in the mean free time. To the requested order the general solution iswhere the error describes the frequency expansion at fixed $k$ and time interval. Keeping the exponential retains the accumulated damping rather than expanding away a possibly important effect.

For a general time-dependent coefficient, a first-order solution with prescribed initial conditions can also be written aswhere ${\delta }_0=A\cos [\omega (\tau -{\tau }_i)]+B\sin [\omega (\tau -{\tau }_i)]$. Substitution verifies that the integral solves the oscillator forced by $-\gamma {\delta }_0^{\prime }$, and its value and first derivative vanish initially. This gives the general first-order result without assuming arbitrary time dependence may be absorbed exactly into an exponential.

To obtain the slowly varying subhorizon form, set ${\delta }_{\gamma }=e^{-G}y$ with $G^{\prime }=\gamma /2$. The exact transformed equation isDropping the quadratic term alone leaves ${\omega }^2-{\gamma }^{\prime }/2$, so slow variation is also needed. When $k\gg \mathcal{H}$, $k{\tau }_c\ll 1$, and the mean free time changes on an expansion timescale, $|{\gamma }^{\prime }|/{\omega }^2\ll 1$. The WKB method solution has local frequency $\Omega =({\omega }^2-{\gamma }^{\prime }/2{)}^{1/2}$ and prefactor ${\Omega }^{-1/2}$. At leading acoustic order both reduce to constants; the small derivative corrections to phase and prefactor can be neglected. This slowly varying shear-damped photon oscillator therefore givesA change in the phase origin is absorbed into $A,B$. At zero mean free time the damping vanishes; as time passes the diffusion length grows and $k_D$ decreases. The coefficient is that of the question's isotropic-scattering, shear-only truncation, not the polarization-inclusive full Silk damping formula.

If ionization remains approximately constant during matter domination, $n_e\propto a^{-3}$, so ${\tau }_c\propto a^2\propto {\tau }^4$. Away from the lower limit this givesNear recombination the rapidly changing ionization fraction and eventual failure of tight coupling require the full hierarchy; the integral formula specifies the damping accumulated during the regime used here.

A blackbody photon density perturbation gives an intrinsic temperature fluctuation ${\Theta }_0={\delta }_{\gamma }/4$. The baryon/electron bulk velocity produces a Doppler effect at last scattering. In the common-potential convention of part (ii),Ignoring the slowly varying damping derivative at leading acoustic order, differentiating the density cosine produces a velocity sine. Thus density and velocity acoustic sources are approximately a quarter cycle out of phase. For regular adiabatic initial conditions the density oscillation has approximately a common cosine phase across modes; this phase coherence is what makes a sequence of peaks instead of a smooth incoherent superposition.

Let $D_{\ast }={\tau }_0-{\tau }_{\ast }$ be the comoving distance to last scattering and $r_s({\tau }_{\ast })={\int }^{{\tau }_{\ast }}{c}_{s}d\tau$ its sound horizon. In the present limit $r_s\simeq {\tau }_{\ast }/\sqrt{3}$. Density extrema have $k{r}_s\simeq m\pi$, whereas velocity extrema have $k{r}_s\simeq (m+1/2)\pi$. The projection from spatial wavenumber to high angular multipoles is roughly $\ell \simeq k{D}_{\ast }$, giving an acoustic spacingThe Doppler effect fills some density troughs and modifies the detailed peak shapes; its different projection prevents treating it as an identical second series of narrow peaks.

For clarity, choose the sightline to point outward from the observer and let its angle to $\mathbf{k}$ have cosine $\mu$. The intrinsic-plus-emitter-Doppler source is then $[{\delta }_{\gamma }/4-ik\mu {\theta }_b]e^{ik\mu D_{\ast }}$. Expanding its angular dependence gives the angular projection of density and Doppler acoustic sourcesThe second sign follows from the chosen outward sightline and $v_b^i=i{k}^i{\theta }_b$; it changes with the sightline convention together with the velocity projection. The $j_{\ell }$ are Spherical Bessel functions. Both sources are proportional to the primordial curvature. If ${\mathcal{T}}_{\ell }={\Theta }_{\ell }/{\zeta }_{\ast }$, their angular spectrum in the usual continuum Fourier normalization isThis formula includes the density-velocity cross term, not just a sum of independent source powers.

Photon diffusion multiplies both leading source amplitudes by $e^{-k^2/k_D^2}$, so their power carries $e^{-2k^2/k_D^2}$. The approximate angular damping scale isThis produces the damping tail in the Cosmic microwave background power spectrum. Finite last-scattering thickness adds further small-scale averaging. Gravity, baryon loading and later propagation alter detailed peak locations and heights, but the oscillatory peak sequence, phase-shifted velocity contribution and diffusion damping follow directly from the simplified solution obtained here.