Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 (Cosmology)

For a component with the barotropic equation of state $P=w\rho$, the cosmological perfect-fluid continuity equation isUsing $H=\dot{a}/a$ makes this a separable differential equation:Taking $a_0=1$ gives the constant-equation-of-state density scalingThe spatially flat Friedmann equation and the cosmological density parameters therefore giveAlong a radial light ray, $d\chi =dt/a$. Since the redshift-time relation is $dt=-dz/[(1+z)H]$ and $a=(1+z{)}^{-1}$,Hence the comoving radial distance is

In the flat Lambda-CDM model,At low redshift, $H(z)=H_0+O(z)$, so the angular diameter distance satisfies $d_A(z)=z/H_0+O(z^2)$ and initially increases from zero. At high redshift, pressureless matter dominates and the integral converges:Thus the positive continuous function $d_A$ rises from zero and returns toward zero, so it attains an interior maximum. This is the angular diameter distance turnover; beyond it, a fixed physical size appears larger at higher redshift.

If the variable dark-energy equation of state differs slightly from $-1$, matter still controls the large-$z$ expansion, while the same low-$z$ limit holds. The turnover therefore persists under such a small change. More generally, it persists whenever the high-redshift comoving distance grows more slowly than $1+z$.

For the transverse baryon acoustic oscillation ruler, $d_A=a\chi$ and the physical ruler length is $a{\chi }_d$. Its observed angle is thereforewhereThe unknown cosmological standard ruler length and $H_0$ occur only through the common amplitude $H_0{\chi }_d$. Measurements at two distinct redshifts givewhich generically determines the one shape parameter ${\Omega }_{m,0}$. Thus two redshifts suffice; additional redshifts overdetermine the model and improve precision.

To first order, the radial and transverse comoving separations arewhereThe local spatial geometry is that of Euclidean space, so the Pythagorean theorem givesConsequently the requested function is

Yes. A radial baryon acoustic oscillation measurement and a transverse baryon acoustic oscillation measurement of the same ruler at one redshift obeyTheir ratio cancels both the unknown ruler length and the overall Hubble scale:This is the Alcock-Paczyński parameter. At a specified redshift its dependence on ${\Omega }_{m,0}$ permits a one-redshift determination within the assumed flat Lambda-CDM model.

Before electron-positron annihilation in cosmology, photons and electron-positron pairs share a temperature. Their effective entropy degrees of freedom areAfter annihilation the electromagnetic plasma contains only the two photon polarizations, so $g_{\ast s}^{\mathrm{after}}=2$. The already decoupled neutrinos receive none of this entropy and simply cool as $T_{\nu }\propto a^{-1}$. Applying cosmological entropy conservation separately to the coupled electromagnetic plasma giveswhereas $T_{\nu }^3{a}^3=T_{\mathrm{before}}^3{a}_{mbefore}^3$. Division yields the Cosmic neutrino background temperature

The energy density from a phase-space distribution function with $g=2$ internal states isIn the ultrarelativistic limit $T_{\nu }\gg m_{\nu }$, set $x=p/T_{\nu }$. The Fermi-Dirac distribution then givesSince $\zeta (4)={\pi }^4/90$,

When ${\mu }_{\nu }/T_{\nu }\gg 1$, the Fermi-Dirac distribution approaches the step function $f(p)=1$ for $p<{\mu }_{\nu }$ and zero for $p>{\mu }_{\nu }$. For one internal state and negligible mass, the degenerate relic neutrino energy density is thereforeso $\boxed{A=1/(8{\pi }^2)}$.

At thermal equilibrium, the antineutrino chemical potential is $-{\mu }_{\nu }$. Its occupation is approximately $e^{-(p+{\mu }_{\nu })/T_{\nu }}$, givingIt is exponentially smaller than the neutrino density; this particle-antiparticle imbalance is a cosmological lepton asymmetry.

Write the neutrino degeneracy parameter as ${\xi }_{\nu }={\mu }_{\nu }/T_{\nu }$. Requiring one degenerate species not to exceed today's critical density givesUsing $T/T_{\nu }=(11/4{)}^{1/3}$,Several equally degenerate species would strengthen the bound by the fourth root of their number.

A large neutrino degeneracy parameter raises the relativistic energy density and therefore the expansion rate. Big Bang nucleosynthesis tests this through primordial light-element abundances; an electron-neutrino chemical potential also shifts neutron--proton chemical equilibrium directly. The Cosmic microwave background power spectrum tests the changed expansion rate, matter-radiation equality, sound horizon, damping scale, and neutrino anisotropic stress. Finally, large-scale structure of the universe and the matter power spectrum test the altered equality scale and neutrino free streaming. These observables distinguish a degenerate-neutrino cosmology from the standard thermal relic scenario in complementary epochs.

For radiation in cosmology, ${\bar{P}}_r={\bar{\rho }}_r/3$ and $\delta P_r=\delta {\rho }_r/3$, so both terms proportional to the equation-of-state difference vanish. During matter domination the gravitational potential is constant, ${\Phi }^{\prime }=0$. Writing ${\theta }_r=\nabla \cdot {\mathbf{v}}_r$, the continuity and Euler equations reduce toDifferentiate the first equation and substitute the second:Thus the radiation perturbation obeys the forced acoustic equation

After a spatial Fourier transform, the preceding equation isThe constant particular solution is ${\delta }_r=-4\Phi$. Hence the Sachs-Wolfe combinationcontains only the homogeneous harmonic oscillator modes and satisfiesFor adiabatic initial conditions on large scales, $S=\Phi /3=\mathcal{R}/5$ and $S^{\prime }=0$ at the start of matter domination. Up to an irrelevant choice of the phase origin,Thus the radiation acoustic transfer function is flat at $1/5$ as $k\to 0$ and oscillates on smaller scales. Projection onto the sky produces a large-angle Sachs-Wolfe plateau and, after the full photon-baryon physics is included, the acoustic structure of the Cosmic microwave background power spectrum.

The baryon transfer function should also oscillate soon after cosmological recombination: before decoupling, Thomson scattering forced baryons to share the acoustic motion of the photon-baryon fluid, and recombination does not instantly erase the resulting density pattern.

Let $k_{\mathrm{eq}}$ be the horizon scale at matter-radiation equality. At a fixed time soon after recombination, the cold-dark-matter transfer function defined as $T_c={\Delta }_c/\mathcal{R}$ has the schematic behaviorwith a smooth turnover near $k_{\mathrm{eq}}$. Large-scale modes enter the horizon during matter domination and the Poisson equation converts an almost scale-independent primordial potential into a density contrast proportional to $k^2$. Small-scale modes enter during radiation domination; radiation controls the potential and pressure prevents it from clustering, so cold-dark-matter growth is only logarithmic until equality. Subsequent matter-era growth multiplies all these modes by the same linear growth factor.

Equivalently, if the conventional matter transfer function is normalized to one as $k\to 0$, it is constant for $k\ll k_{\mathrm{eq}}$ and falls approximately as $\log (k/k_{\mathrm{eq}})/k^2$ for $k\gg k_{\mathrm{eq}}$. Multiplication by the Poisson factor $k^2$ gives the behavior of the definition used here.

Subtracting the baryon equation from the cold-dark-matter equation cancels their common gravitational source and gives the baryon--cold-dark-matter relative density modeBecause the background fractions ${\bar{\rho }}_c/{\bar{\rho }}_m$ and ${\bar{\rho }}_b/{\bar{\rho }}_m$ are constant after recombination, their weighted sum givesDuring matter domination, $a\propto t^{2/3}$. The first equation gives $a^2\dot{D}=\text{constant}$ and henceThe cosmic-time matter density modes giveSolving the definitions for the individual contrasts,soThus $\boxed{f=1}$. Both species feel the same gravitational potential after baryon pressure becomes negligible; the growing total mode overtakes the constant relative mode, so baryons catch up with cold dark matter.

Yes. The baryons carry the oscillatory baryon transfer function inherited from the pre-recombination photon-baryon fluid. Their later infall toward cold-dark-matter overdensities suppresses the relative mode and drives ${\delta }_b/{\delta }_c\to 1$, but gravitational evolution does not remove every scale-dependent acoustic phase. The total cosmological density power spectrum therefore contains the small wiggles known as a baryon acoustic oscillation in the matter power spectrum.

The right-hand side of the slow-roll curvature power spectrum is evaluated at cosmological horizon exit, $k=aH$. Usingrewrites it asAt horizon exit, $d\log k=d\log (aH)=(1-ϵ)dN$, so $d/d\log k=d/dN$ to first slow-roll order. With the first Hubble slow-roll parameter and second Hubble slow-roll parameter,The supplied relations $ϵ={ϵ}_V$ and ${\eta }_V=2ϵ-\eta /2$ imply $\eta =4{ϵ}_V-2{\eta }_V$. Therefore

For a monotonic inflaton trajectory, the potential slow-roll parameter satisfiesThe remaining slow-roll e-fold count is the integral from the present field value to its end value along the direction of motion:

Taking the ratio of the supplied primordial tensor power spectrum and scalar spectrum cancels the common factor $(H/2\pi {)}^2$:To first slow-roll order $ϵ={ϵ}_V$, so the tensor-to-scalar ratio is

For the quadratic hilltop inflation potential while $V_0$ dominates,Since ${ϵ}_V$ is of higher order near the hilltop, the scalar spectral index obeys $n_s-1\simeq 2{\eta }_V$. The e-fold integral becomesand henceSubstitution into $r=C{ϵ}_V$ yields the quadratic hilltop slow-roll predictionso $\boxed{f=1/8}$.

For $n_s-1=-0.04$ the expression decreases as $\tilde{N}$ increases. Its largest allowed value therefore uses $\tilde{N}=40$ and ${\varphi }_e/M_{\mathrm{Pl}}<1$: