Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 (Cosmology)

The cosmological perfect-fluid continuity equation isFor constant equation-of-state parameter $w=p/\rho$, integration giveswhere $a_0=1$. Define the present cosmological density parameterThe Friedmann equation then givesSpatial curvature may be included as an effective $w=-1/3$ component. Once the parameters are specified, the scale factor follows from the first-order equation $\dot{a}=aH(a)$, or equivalently

For a variable equation of state, continuity givesFor the Chevallier-Polarski-Linder parametrizationthe integral isConsequentlyand

For successive wavecrests from the same comoving source, $1+z=a(t_0)/a(t_s)$. ThereforeThe same radial null path implies equality of the elapsed conformal time,Expanding both scale factors to first order gives the redshift drift

Measurements at many source redshifts directly sample the function $H_0[1+z-E(z)]$. In principle, sufficiently many precise and well-spaced measurements can fit $H_0,{\Omega }_{m,0},{\Omega }_{\mathrm{DE},0},w_0,w_a$ simultaneously; a single redshift supplies only one combination and cannot. In a spatially flat two-component model, ${\Omega }_{m,0}+{\Omega }_{\mathrm{DE},0}=1$, reducing the number of independent parameters by one. In practice the signal accumulated over ten years is extremely small and parameter degeneracies require broad redshift coverage and complementary data.

Chemical equilibrium for $e^{-}+p\leftrightarrow H+\gamma$ requires ${\mu }_e+{\mu }_p={\mu }_H$, since the photon chemical potential vanishes. Inserting the nonrelativistic Maxwell-Boltzmann distribution and using charge neutrality $n_p=n_e$ giveswhere the proton-to-hydrogen mass ratio and the stated degeneracy factors have been approximated by one. Thus

With $X_e=n_e/n_b$, charge neutrality and $n_b=n_p+n_H$ giveMultiplying part a by $n_b=\eta n_{\gamma }$ and using $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$ yields the Saha ionization equationAlthough $T\ll B_H$ suppresses ionization of an individual atom, the baryon-to-photon ratio is only about $10^{-9}$. The enormous number of photons per baryon leaves enough photons in the high-energy thermal tail to ionize hydrogen until $T\simeq 0.3\mathrm{eV}$, far below $B_H$.

Photon decoupling occurs when the Thomson interaction rate falls below the expansion rate,At decoupling the universe is approximately matter dominated, soUsing $n_e=X_e\eta (2\zeta (3)/{\pi }^2)T^3$ in ${\Gamma }_{\gamma }=H$ givesFor $X_e\ll 1$, the Saha equation makes $X_e(T)$ exponentially steep through $e^{-B_H/(2T)}$. Power-law changes of $H_0$ or ${\Omega }_{m,0}$ therefore cause only a small shift in $T_{\mathrm{dec}}$, explaining why $T_{\mathrm{dec}}\simeq T_{\mathrm{rec}}$ is a good approximation.

At fixed $\eta$, the recombination temperature is determined by atomic physics and the Saha equation, not by today's CMB temperature. Under the stated approximation,Since $1+z_{\mathrm{dec}}=T_{\mathrm{dec}}/T_0$,and the same ratio holds for the large redshifts themselves to excellent accuracy. The comoving distance ${\chi }_{\mathrm{dec}}={\int }_0^{z_{\mathrm{dec}}}dz/H(z)$ can nevertheless be held fixed by adjusting the late-time expansion history, for example $H_0$, ${\Omega }_{m,0}$, spatial curvature, or dark-energy density and equation-of-state parameters.

During matter domination, $4\pi G{\bar{\rho }}_m=3H^2/2$ and $a\propto t^{2/3}$. Substitution of ${\delta }_m\propto t^p$ gives $p=2/3,-1$, henceDuring radiation domination, neglecting the rapidly oscillating radiation perturbation and the subdominant matter source leavesThereforeThis logarithmic behavior is the Mészáros effect for subhorizon cold dark matter.

For an approximately scale-invariant primordial spectrum, the late cosmological density power spectrum behaves schematically asModes with $k\ll k_{\mathrm{eq}}$ enter the horizon after matter-radiation equality and retain $T(k)\simeq 1$, so $P(k)\sim k^{n_s}$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination and grow only logarithmically until equality. Their transfer function behaves as $T(k)\sim \log (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$, giving a strongly falling small-scale spectrum. The result is a turnover near the matter-radiation equality scale $k_{\mathrm{eq}}$.

The Fourier-space cosmological Poisson equation givesInsert this into the line-of-sight expression for the CMB lensing potential, Fourier transform ${\delta }_m$, and use the Rayleigh plane-wave expansion. Projection onto $Y_{lm}$ and spherical-harmonic orthogonality give

Define the radial transfer integralwhere ${\delta }_m(\mathbf{k},\tau )=D_i(\tau ){\delta }_m(\mathbf{k},{\tau }_i)$ and $D_i({\tau }_i)=1$. Withthe angular and radial integrations giveThis expression makes the lensing signal a weighted projection of the evolving matter power spectrum.

The creation and annihilation operators obey the canonical commutation relationsSince $\delta \hat{\varphi }=\hat{f}/a$, the vacuum two-point function identifiesOn superhorizon scales, $|k\tau |=k/(aH)\ll 1$, so

For each mode, the right side ofis evaluated at cosmological horizon exit, $k=aH$. Sincefirst order in the Hubble slow-roll parameters gives

For the monomial inflation potential $V=\lambda M_{\mathrm{Pl}}^4(\varphi /M_{\mathrm{Pl}}{)}^{\alpha }$,andSlow-roll inflation is possible when both are much smaller than one:Inflation ends once one of these conditions fails, commonly near $|\varphi |\simeq \alpha M_{\mathrm{Pl}}/\sqrt{2}$ when $ϵ\simeq 1$.

The expressions in part c implySince the tensor-to-scalar ratio is $r=16ϵ$,For positive $\alpha$, this class predicts a red scalar tilt andA blue tilt with positive $r$, or $r\ge 8(1-n_s)$, is incompatible. An allowed pair fixes $\alpha =2r/[8(1-n_s)-r]$, subject to slow roll and observational bounds.