Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 (Cosmology)

The cosmological continuity equation and $P_i=w_i{\rho }_i$ giveWith $a_0=1$, $1+z=a^{-1}$ andthe Friedmann equation becomesSpatial curvature may be included as an effective component with $w=-1/3$ and ${\Omega }_{k,0}=-k/H_0^2$.

Given the component parameters, solve the first-order equationA cosmic string network has $w=-1/3$, so $\rho \propto a^{-2}$ and $H\propto a^{-1}$. Therefore $\dot{a}$ is constant and a string-dominated expanding universe hasafter choosing the Big Bang as $t=0$.

For pressureless matter, ${\rho }_m{a}^3={\rho }_{m,0}$ when $a_0=1$. Setthen ${\dot{a}}^2=D/a-k$. Substitution of $a=A(1-\cos \theta )$ makes this identity hold whenIndeed $dt/d\theta =B(1-\cos \theta )=Ba/A$, and henceafter setting $t=0$ at $\theta =0$. This is the closed matter-dominated Friedmann solution.

The small-$\theta$ expansions areWriting $x=(6t/B{)}^{1/3}$ and reverting the second series gives $\theta =x(1+x^2/60+\cdots )$. Thereforeso $C=(A/2)(6/B{)}^{2/3}$. The leading $a\propto t^{2/3}$ is the flat matter-dominated limit.

During recollapse $H<0$, so nearby comoving galaxies are increasingly blueshifted rather than redshifted. Their physical separations and angular-diameter distances shrink, making them appear larger and generally brighter; in a closed geometry sufficiently old light can also produce repeated or strongly focused images.

The cosmic microwave background temperature obeys $T_{\gamma }\propto a^{-1}$. It therefore rises without bound as $\theta \to 2\pi$, and its photons are blueshifted. Before the formal Big Crunch, the growing temperature reionizes matter, scattering makes the universe opaque, and the idealization that old galaxies and the original last-scattering surface remain directly visible eventually fails.

Chemical equilibrium for $e^{-}+p\leftrightarrow H+\gamma$ gives ${\mu }_e+{\mu }_p={\mu }_H$, since ${\mu }_{\gamma }=0$. Insert the supplied nonrelativistic equilibrium densities. Neglecting $m_e/m_p$ in the translational reduced mass and using the stated degeneracy convention givesCharge neutrality gives $n_p=n_e$, soThis is the hydrogen Saha ionization equation.

Since $n_e=n_p=X_e{n}_b$ and $n_H=(1-X_e)n_b$,Using $n_b=\eta n_{\gamma }$ and $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$ yieldsCosmological recombination occurs far below $B_H$ because the baryon-to-photon ratio is only about $10^{-9}$. There are roughly a billion photons per baryon, so the high-energy tail of the blackbody distribution continues to photoionize hydrogen until the exponential Boltzmann factor overcomes this enormous entropy factor.

Neglecting baryon loading, the Sachs-Wolfe combination obeys a harmonic-oscillator equation with sound speed $1/\sqrt{3}$. Adiabatic initial conditions give a nonzero initial displacement and negligible initial velocity, henceProjection maps $k$ approximately to $\ell /{\chi }_{\ast }$, and the angular power is quadratic in the transfer function, producing the observed approximate ${\cos }^2(k{r}_s)$ sequence of acoustic peaks. A ${\sin }^2$ phase would instead indicate vanishing initial displacement and nonzero initial velocity, characteristic of an isocurvature rather than adiabatic primordial mode.

Multiplying every pre-recombination density by $\lambda$ changes $H$ by $\sqrt{\lambda }$ and therefore shrinks the sound horizon at fixed recombination epoch by ${\lambda }^{-1/2}$. Changing $B_H$ shifts the recombination temperature approximately as $T_{\mathrm{rec}}\propto B_H$, so $a_{\ast }\propto {\mu }^{-1}$. Delaying recombination with $\mu <1$ can compensate the faster expansion; schematically the acoustic scale can be retained by choosing $\mu \sqrt{\lambda }\simeq 1$.

This can preserve the peak positions approximately because post-recombination distances are unchanged. It cannot make the entire spectrum exactly identical: the photon-diffusion scale, visibility-function width, early integrated Sachs-Wolfe effect and relative peak heights scale differently. Thus a tuned pair $(\lambda ,\mu )$ creates an approximate degeneracy, with residual observables breaking it.

During matter domination, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. Trying ${\delta }_m=t^p$ giveswith roots $p=2/3,-1$. The growing linear cosmological density perturbation is therefore

After all species are nonrelativistic, each background density scales as $a^{-3}$, sois constant. Since $\delta {\rho }_i={\bar{\rho }}_i{\delta }_i$,

Neutrino free streaming gives ${\delta }_{\nu }\simeq 0$ on the stated scales, so ${\delta }_m=(1-f_{\nu }){\delta }_{cb}$. With derivatives with respect to $N=\log a$, the growth equation becomesFor ${\delta }_{cb}\propto a^p$,Henceto first order.

Relative to massless-neutrino growth after $z_{\nu }$,The total contrast has the additional factor $1-f_{\nu }$. Squaring its amplitude to obtain the cosmological density power spectrum givesThe two terms respectively encode the unclustered neutrino fraction and the accumulated slowing of cold-matter growth.

Canonical quantization imposesThe two-point correlation function has mode power $P_{\delta \varphi }=|f_k{|}^2/a^2$. After cosmological horizon exit, $|k\tau |\ll 1$, soThereforeThis is the scale-invariant inflationary power spectrum of a light canonical scalar.

Linearizing the curvaton equation and Fourier transforming givesWriting $f_k^{\sigma }=a\delta {\sigma }_k$ yieldsSince $m_{\sigma }\ll H$, the mass term is negligible and this is the equation stated. It has the same Bunch-Davies mode as the inflaton perturbation, so after horizon exitup to a small mass-induced spectral tilt.

When $H\ll m_{\sigma }$ over an oscillation, the background equation reduces toTime averaging gives $⟨{\dot{\bar{\sigma }}}^2⟩=m_{\sigma }^2⟨{\bar{\sigma }}^2⟩$, henceThe oscillating quadratic scalar is therefore pressureless matter. Its continuity equation giveswith the slowly varying amplitude obeying $\Sigma \propto a^{-3/2}$.

After reheating, inflaton decay products are radiation with ${\rho }_{\gamma }\propto a^{-4}$, whereas the oscillating curvaton has ${\rho }_{\sigma }\propto a^{-3}$. Its fractional density therefore grows, so only the curvaton can become dynamically important at late times.

The curvaton mechanism can generate the observed primordial curvature perturbation even though the inflaton drives inflation. A viable pure-curvaton realization requires the field to be light during inflation, to become sufficiently important before decay, to decay into the visible sector before nucleosynthesis, and to avoid excessive residual isocurvature and primordial non-Gaussianity. If it dominates before complete decay, these conditions can be compatible with observations.