• # Paper 1, Section I, 9B

The continuity, Euler and Poisson equations governing how non-relativistic fluids with energy density $\rho$, pressure $P$ and velocity $\mathbf{v}$ propagate in an expanding universe take the form

$\begin{gathered} \frac{\partial \rho}{\partial t}+3 H \rho+\frac{1}{a} \boldsymbol{\nabla} \cdot(\rho \mathbf{v})=0 \\ \rho a\left(\frac{\partial}{\partial t}+\frac{\mathbf{v}}{a} \cdot \nabla\right) \mathbf{u}=-\frac{1}{c^{2}} \nabla P-\rho \boldsymbol{\nabla} \Phi \\ \nabla^{2} \Phi=\frac{4 \pi G}{c^{2}} \rho a^{2} \end{gathered}$

where $\mathbf{u}=\mathbf{v}+a H \mathbf{x}, H=\dot{a} / a$ and $a(t)$ is the scale factor.

(a) Show that, for a homogeneous and isotropic flow with $P=\bar{P}(t), \rho=\bar{\rho}(t), \mathbf{v}=\mathbf{0}$ and $\Phi=\bar{\Phi}(t, \mathbf{x})$, consistency of the Euler equation with the Poisson equation implies Raychaudhuri's equation.

(b) Explain why this derivation of Raychaudhuri's equation is an improvement over the derivation of the Friedmann equation using only Newtonian gravity.

(c) Consider small perturbations about a homogeneous and isotropic flow,

$\rho=\bar{\rho}(t)+\epsilon \delta \rho, \quad \mathbf{v}=\epsilon \delta \mathbf{v}, \quad P=\bar{P}(t)+\epsilon \delta P \quad \text { and } \quad \Phi=\bar{\Phi}(t, \mathbf{x})+\epsilon \delta \Phi,$

with $\epsilon \ll 1$. Show that, to first order in $\epsilon$, the continuity equation can be written as

$\frac{\partial}{\partial t}\left(\frac{\delta \rho}{\bar{\rho}}\right)=-\frac{1}{a} \boldsymbol{\nabla} \cdot \delta \mathbf{v}$

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• # Paper 1, Section II, 15B

(a) Consider the following action for the inflaton field $\phi$

$S=\int \mathrm{d}^{3} x \mathrm{~d} t a(t)^{3}\left[\frac{1}{2} \dot{\phi}^{2}-\frac{c^{2}}{2 a(t)^{2}} \nabla \phi \cdot \nabla \phi-V(\phi)\right]$

Use the principle of least action to derive the equation of motion for the inflaton $\phi$,

$\ddot{\phi}+3 H \dot{\phi}-\frac{c^{2}}{a(t)^{2}} \nabla^{2} \phi+\frac{\mathrm{d} V(\phi)}{\mathrm{d} \phi}=0$

where $H=\dot{a} / a$. [In the derivation you may discard boundary terms.]

(b) Consider a regime where $V(\phi)$ is approximately constant so that the universe undergoes a period of exponential expansion during which $a=a_{0} e^{H_{\text {inf }} t}$. Show that $(*)$ can be written in terms of the spatial Fourier transform $\widehat{\phi}_{\mathbf{k}}(t)$ of $\phi(\mathbf{x}, t)$ as

$\ddot{\widehat{\phi}}_{\mathbf{k}}+3 H_{\mathrm{inf}} \dot{\hat{\phi}}_{\mathbf{k}}+\frac{c^{2} k^{2}}{a^{2}} \widehat{\phi}_{\mathbf{k}}=0 .$

(c) Define conformal time $\tau$ and determine the range of $\tau$ when $a=a_{0} e^{H_{\text {inf }} t}$. Show that $(* *)$ can be written in terms of the conformal time as

$\frac{\mathrm{d}^{2} \tilde{\phi}_{\mathbf{k}}}{\mathrm{d} \tau^{2}}+\left(c^{2} k^{2}-\frac{2}{\tau^{2}}\right) \widetilde{\phi}_{\mathbf{k}}=0, \quad \text { where } \quad \tilde{\phi}_{\mathbf{k}}=-\frac{1}{H_{\mathrm{inf}} \tau} \widehat{\phi}_{\mathbf{k}}$

(d) Let $|\mathrm{BD}\rangle$ denote the state that in the far past was in the ground state of the standard harmonic oscillator with frequency $\omega=c k$. Assuming that the quantum variance of $\widehat{\phi}_{\mathbf{k}}$ is given by

$P_{\mathbf{k}} \equiv\left\langle\mathrm{BD}\left|\widehat{\phi}_{\mathbf{k}} \widehat{\phi}_{\mathbf{k}}^{\dagger}\right| \mathrm{BD}\right\rangle=\frac{\hbar H_{\mathrm{inf}}^{2}}{2 c^{3} k^{3}}\left(1+\tau^{2} c^{2} k^{2}\right)$

explain in which sense inflation naturally generates a scale-invariant power spectrum. [You may use that $P_{\mathbf{k}}$ has dimensions of [length $\left.]^{3} .\right]$

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• # Paper 2, Section I, 9B

(a) The generalised Boltzmann distribution $P(\mathbf{p})$ is given by

$P(\mathbf{p})=\frac{e^{-\beta\left(E_{\mathbf{p}} n_{\mathbf{p}}-\mu n_{\mathbf{p}}\right)}}{\mathcal{Z}_{\mathbf{p}}}$

where $\beta=\left(k_{B} T\right)^{-1}, \mu$ is the chemical potential,

$\mathcal{Z}_{\mathbf{p}}=\sum_{n_{\mathbf{p}}} e^{-\beta\left(E_{\mathbf{p}} n_{\mathbf{p}}-\mu n_{\mathbf{p}}\right)}, \quad E_{\mathbf{p}}=\sqrt{m^{2} c^{4}+p^{2} c^{2}} \quad \text { and } \quad p=|\mathbf{p}|$

Find the average particle number $\langle N(\mathbf{p})\rangle$ with momentum $\mathbf{p}$, assuming that all particles have rest mass $m$ and are either

(i) bosons, or

(ii) fermions .

(b) The photon total number density $n_{\gamma}$ is given by

$n_{\gamma}=\frac{2 \zeta(3)}{\pi^{2} \hbar^{3} c^{3}}\left(k_{B} T\right)^{3}$

where $\zeta(3) \approx 1.2$. Consider now the fractional ionisation of hydrogen

$X_{e}=\frac{n_{e}}{n_{e}+n_{H}}$

In our universe $n_{e}+n_{H}=n_{p}+n_{H} \approx \eta n_{\gamma}$, where $\eta \sim 10^{-9}$ is the baryon-to-photon number density. Find an expression for the ratio

$\frac{1-X_{e}}{X_{e}^{2}}$

in terms of $\eta,\left(k_{B} T\right)$, the electron mass $m_{e}$, the speed of light $c$ and the ionisation energy of hydrogen $I \approx 13.6 \mathrm{eV}$.

One might expect neutral hydrogen to form at a temperature $k_{B} T \sim I$, but instead in our universe it happens at the much lower temperature $k_{B} T \approx 0.3 \mathrm{eV}$. Briefly explain why this happens.

[You may use without proof the Saha equation

$\frac{n_{H}}{n_{e}^{2}}=\left(\frac{2 \pi \hbar^{2}}{m_{e} k_{B} T}\right)^{3 / 2} e^{\beta I}$

for chemical equilibrium in the reaction $\left.e^{-}+p^{+} \leftrightarrow H+\gamma .\right]$

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• # Paper 3, Section I, 9B

The expansion of the universe during inflation is governed by the Friedmann equation

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3}\left[\frac{1}{2} \dot{\phi}^{2}+V(\phi)\right]$

and the equation of motion for the inflaton field $\phi$,

$\ddot{\phi}+3 \frac{\dot{a}}{a} \dot{\phi}+\frac{\mathrm{d} V}{\mathrm{~d} \phi}=0 .$

Consider the potential

$V=V_{0} e^{-\lambda \phi}$

with $V_{0}>0$ and $\lambda>0$.

(a) Show that the inflationary equations have the exact solution

$a(t)=\left(\frac{t}{t_{0}}\right)^{\gamma} \quad \text { and } \quad \phi=\phi_{0}+\alpha \log t$

for arbitrary $t_{0}$ and appropriate choices of $\alpha, \gamma$ and $\phi_{0}$. Determine the range of $\lambda$ for which the solution exists. For what values of $\lambda$ does inflation occur?

(b) Using the inflaton equation of motion and

$\rho=\frac{1}{2} \dot{\phi}^{2}+V$

together with the continuity equation

$\dot{\rho}+3 \frac{\dot{a}}{a}(\rho+P)=0,$

determine $P$.

(c) What is the range of the pressure energy density ratio $\omega \equiv P / \rho$ for which inflation occurs?

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• # Paper 3, Section II, B

(a) Consider a closed universe endowed with cosmological constant $\Lambda>0$ and filled with radiation with pressure $P$ and energy density $\rho$. Using the equation of state $P=\frac{1}{3} \rho$ and the continuity equation

$\dot{\rho}+\frac{3 \dot{a}}{a}(\rho+P)=0,$

determine how $\rho$ depends on $a$. Give the physical interpretation of the scaling of $\rho$ with

(b) For such a universe the Friedmann equation reads

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3 c^{2}} \rho-\frac{c^{2}}{R^{2} a^{2}}+\frac{\Lambda}{3}$

What is the physical meaning of $R ?$

(c) Making the substitution $a(t)=\alpha \tilde{a}(t)$, determine $\alpha$ and $\Gamma>0$ such that the Friedmann equation takes the form

$\left(\frac{\dot{\tilde{a}}}{\tilde{a}}\right)^{2}=\frac{\Gamma}{\tilde{a}^{4}}-\frac{1}{\tilde{a}^{2}}+\frac{\Lambda}{3} .$

Using the substitution $y(t)=\tilde{a}(t)^{2}$ and the boundary condition $y(0)=0$, deduce the boundary condition for $\dot{y}(0)$.

Show that

$\ddot{y}=\frac{4 \Lambda}{3} y-2$

and hence that

$\tilde{a}^{2}(t)=\frac{3}{2 \Lambda}\left[1-\cosh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)+\lambda \sinh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)\right]$

Express the constant $\lambda$ in terms of $\Lambda$ and $\Gamma$.

Sketch the graphs of $\tilde{a}(t)$ for the cases $\lambda>1, \lambda<1$ and $\lambda=1$.

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• # Paper 4, Section I, B

A collection of $N$ particles, with masses $m_{i}$ and positions $\mathbf{x}_{i}$, interact through a gravitational potential

$V=\sum_{i

Assume that the system is gravitationally bound, and that the positions $\mathbf{x}_{i}$ and velocities $\dot{\mathbf{x}}_{i}$ are bounded for all time. Further, define the time average of a quantity $X$ by

$\bar{X}=\lim _{t \rightarrow \infty} \frac{1}{t} \int_{0}^{t} X\left(t^{\prime}\right) \mathrm{d} t^{\prime}$

(a) Assuming that the time average of the kinetic energy $T$ and potential energy $V$ are well defined, show that

$\bar{T}=-\frac{1}{2} \bar{V}$

[You should consider the quantity $I=\frac{1}{2} \sum_{i=1}^{N} m_{i} \mathbf{x}_{i} \cdot \mathbf{x}_{i}$, with all $\mathbf{x}_{i}$ measured relative to the centre of mass.]

(b) Explain how part (a) can be used, together with observations, to provide evidence in favour of dark matter. [You may assume that time averaging may be replaced by an average over particles.]

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• # Paper 1, Section I, D

The Friedmann equation is

$H^{2}=\frac{8 \pi G}{3 c^{2}}\left(\rho-\frac{k c^{2}}{R^{2} a^{2}}\right)$

Briefly explain the meaning of $H, \rho, k$ and $R$.

Derive the Raychaudhuri equation,

$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3 c^{2}}(\rho+3 P),$

where $P$ is the pressure, stating clearly any results that are required.

Assume that the strong energy condition $\rho+3 P \geqslant 0$ holds. Show that there was necessarily a Big Bang singularity at time $t_{B B}$ such that

$t_{0}-t_{B B} \leqslant H_{0}^{-1}$

where $H_{0}=H\left(t_{0}\right)$ and $t_{0}$ is the time today.

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• # Paper 1, Section II, D

A fluid with pressure $P$ sits in a volume $V$. The change in energy due to a change in volume is given by $d E=-P d V$. Use this in a cosmological context to derive the continuity equation,

$\dot{\rho}=-3 H(\rho+P),$

with $\rho$ the energy density, $H=\dot{a} / a$ the Hubble parameter, and $a$ the scale factor.

In a flat universe, the Friedmann equation is given by

$H^{2}=\frac{8 \pi G}{3 c^{2}} \rho .$

Given a universe dominated by a fluid with equation of state $P=w \rho$, where $w$ is a constant, determine how the scale factor $a(t)$ evolves.

Define conformal time $\tau$. Assume that the early universe consists of two fluids: radiation with $w=1 / 3$ and a network of cosmic strings with $w=-1 / 3$. Show that the Friedmann equation can be written as

$\left(\frac{d a}{d \tau}\right)^{2}=B \rho_{\mathrm{eq}}\left(a^{2}+a_{\mathrm{eq}}^{2}\right)$

where $\rho_{\mathrm{eq}}$ is the energy density in radiation, and $a_{\mathrm{eq}}$ is the scale factor, both evaluated at radiation-string equality. Here, $B$ is a constant that you should determine. Find the solution $a(\tau)$.

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• # Paper 2, Section I, D

During inflation, the expansion of the universe is governed by the Friedmann equation,

$H^{2}=\frac{8 \pi G}{3 c^{2}}\left(\frac{1}{2} \dot{\phi}^{2}+V(\phi)\right)$

and the equation of motion for the inflaton field $\phi$,

$\ddot{\phi}+3 H \dot{\phi}+\frac{\partial V}{\partial \phi}=0 \text {. }$

The slow-roll conditions are $\dot{\phi}^{2} \ll V(\phi)$ and $\ddot{\phi} \ll H \dot{\phi}$. Under these assumptions, solve for $\phi(t)$ and $a(t)$ for the potentials:

(i) $V(\phi)=\frac{1}{2} m^{2} \phi^{2}$ and

(ii) $V(\phi)=\frac{1}{4} \lambda \phi^{4}, \quad(\lambda>0)$.

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• # Paper 3, Section I, D

At temperature $T$, with $\beta=1 /\left(k_{B} T\right)$, the distribution of ultra-relativistic particles with momentum $\mathbf{p}$ is given by

$n(\mathbf{p})=\frac{1}{e^{\beta p c} \mp 1},$

where the minus sign is for bosons and the plus $\operatorname{sign}$ for fermions, and with $p=|\mathbf{p}|$.

Show that the total number of fermions, $n_{\mathrm{f}}$, is related to the total number of bosons, $n_{\mathrm{b}}$, by $n_{\mathrm{f}}=\frac{3}{4} n_{\mathrm{b}}$.

Show that the total energy density of fermions, $\rho_{\mathrm{f}}$, is related to the total energy density of bosons, $\rho_{\mathrm{b}}$, by $\rho_{\mathrm{f}}=\frac{7}{8} \rho_{\mathrm{b}}$.

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• # Paper 3, Section II, D

In an expanding spacetime, the density contrast $\delta(\mathbf{x}, t)$ satisfies the linearised equation

$\ddot{\delta}+2 H \dot{\delta}-c_{s}^{2}\left(\frac{1}{a^{2}} \nabla^{2}+k_{J}^{2}\right) \delta=0,$

where $a$ is the scale factor, $H$ is the Hubble parameter, $c_{s}$ is a constant, and $k_{J}$ is the Jeans wavenumber, defined by

$c_{s}^{2} k_{J}^{2}=\frac{4 \pi G}{c^{2}} \bar{\rho}(t)$

with $\bar{\rho}(t)$ the background, homogeneous energy density.

(i) Solve for $\delta(\mathbf{x}, t)$ in a static universe, with $a=1$ and $H=0$ and $\bar{\rho}$ constant. Identify two regimes: one in which sound waves propagate, and one in which there is an instability.

(ii) In a matter-dominated universe with $\bar{\rho} \sim 1 / a^{3}$, use the Friedmann equation $H^{2}=8 \pi G \bar{\rho} / 3 c^{2}$ to find the growing and decaying long-wavelength modes of $\delta$ as a function of $a$.

(iii) Assuming $c_{s}^{2} \approx c_{s}^{2} k_{J}^{2} \approx 0$ in equation $(*)$, find the growth of matter perturbations in a radiation-dominated universe and find the growth of matter perturbations in a curvature-dominated universe.

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• # Paper 4 , Section I, D

At temperature $T$ and chemical potential $\mu$, the number density of a non-relativistic particle species with mass $m \gg k_{B} T / c^{2}$ is given by

$n=g\left(\frac{m k_{B} T}{2 \pi \hbar^{2}}\right)^{3 / 2} e^{-\left(m c^{2}-\mu\right) / k_{B} T},$

where $g$ is the number of degrees of freedom of this particle.

At recombination, electrons and protons combine to form hydrogen. Use the result above to derive the Saha equation

$n_{H} \approx n_{e}^{2}\left(\frac{2 \pi \hbar^{2}}{m_{e} k_{B} T}\right)^{3 / 2} e^{E_{\mathrm{bind}} / k_{B} T}$

where $n_{H}$ is the number density of hydrogen atoms, $n_{e}$ the number density of electrons, $m_{e}$ the mass of the electron and $E_{\text {bind }}$ the binding energy of hydrogen. State any assumptions that you use in this derivation.

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• # Paper 1, Section I, B

[You may work in units of the speed of light, so that $c=1$.]

By considering a spherical distribution of matter with total mass $M$ and radius $R$ and an infinitesimal mass $\delta m$ located somewhere on its surface, derive the Friedmann equation describing the evolution of the scale factor $a(t)$ appearing in the relation $R(t)=R_{0} a(t) / a\left(t_{0}\right)$ for a spatially-flat FLRW spacetime.

Consider now a spatially-flat, contracting universe filled by a single component with energy density $\rho$, which evolves with time as $\rho(t)=\rho_{0}\left[a(t) / a\left(t_{0}\right)\right]^{-4}$. Solve the Friedmann equation for $a(t)$ with $a\left(t_{0}\right)=1$.

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• # Paper 1, Section II, 15B

[You may work in units of the speed of light, so that $c=1$.]

Consider a spatially-flat FLRW universe with a single, canonical, homogeneous scalar field $\phi(t)$ with a potential $V(\phi)$. Recall the Friedmann equation and the Raychaudhuri equation (also known as the acceleration equation)

\begin{aligned} \left(\frac{\dot{a}}{a}\right)^{2} &=H^{2}=\frac{8 \pi G}{3}\left[\frac{1}{2} \dot{\phi}^{2}+V\right] \\ \frac{\ddot{a}}{a} &=-\frac{8 \pi G}{3}\left(\dot{\phi}^{2}-V\right) \end{aligned}

(a) Assuming $\dot{\phi} \neq 0$, derive the equations of motion for $\phi$, i.e.

$\ddot{\phi}+3 H \dot{\phi}+\partial_{\phi} V=0 .$

(b) Assuming the special case $V(\phi)=\lambda \phi^{4}$, find $\phi(t)$, for some initial value $\phi\left(t_{0}\right)=\phi_{0}$ in the slow-roll approximation, i.e. assuming that $\dot{\phi}^{2} \ll 2 V$ and $\ddot{\phi} \ll 3 H \dot{\phi}$.

(c) The number $N$ of efoldings is defined by $d N=d \ln a$. Using the chain rule, express $d N$ first in terms of $d t$ and then in terms of $d \phi$. Write the resulting relation between $d N$ and $d \phi$ in terms of $V$ and $\partial_{\phi} V$ only, using the slow-roll approximation.

(d) Compute the number $N$ of efoldings of expansion between some initial value $\phi_{i}<0$ and a final value $\phi_{f}<0$ (so that $\dot{\phi}>0$ throughout).

(e) Discuss qualitatively the horizon and flatness problems in the old hot big bang model (i.e. without inflation) and how inflation addresses them.

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• # Paper 2, Section I, B

[You may work in units of the speed of light, so that $c=1$.]

(a) Combining the Friedmann and continuity equations

$H^{2}=\frac{8 \pi G}{3} \rho, \quad \dot{\rho}+3 H(\rho+P)=0$

derive the Raychaudhuri equation (also known as the acceleration equation) which expresses $\ddot{a} / a$ in terms of the energy density $\rho$ and the pressure $P$.

(b) Assuming an equation of state $P=w \rho$ with constant $w$, for what $w$ is the expansion of the universe accelerated or decelerated?

(c) Consider an expanding, spatially-flat FLRW universe with both a cosmological constant and non-relativistic matter (also known as dust) with energy densities $\rho_{c c}$ and $\rho_{d u s t}$ respectively. At some time corresponding to $a_{e q}$, the energy densities of these two components are equal $\rho_{c c}\left(a_{e q}\right)=\rho_{d u s t}\left(a_{e q}\right)$. Is the expansion of the universe accelerated or decelerated at this time?

(d) For what numerical value of $a / a_{e q}$ does the universe transition from deceleration to acceleration?

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• # Paper 3, Section I, B

Consider a spherically symmetric distribution of mass with density $\rho(r)$ at distance $r$ from the centre. Derive the pressure support equation that the pressure $P(r)$ has to satisfy for the system to be in static equilibrium.

Assume now that the mass density obeys $\rho(r)=A r^{2} P(r)$, for some positive constant A. Determine whether or not the system has a stable solution corresponding to a star of finite radius.

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• # Paper 3, Section II, B

[You may work in units of the speed of light, so that $c=1 .$ ]

Consider the process where protons and electrons combine to form neutral hydrogen atoms;

$p^{+}+e^{-} \leftrightarrow H^{0}+\gamma$

Let $n_{p}, n_{e}$ and $n_{H}$ denote the number densities for protons, electrons and hydrogen atoms respectively. The ionization energy of hydrogen is denoted $I$. State and derive $S a h a$ 's equation for the ratio $n_{e} n_{p} / n_{H}$, clearly describing the steps required.

[You may use without proof the following formula for the equilibrium number density of a non-relativistic species $a$ with $g_{a}$ degenerate states of mass $m$ at temperature $T$ such that $k_{B} T \ll m$,

$n_{a}=g_{a}\left(\frac{2 \pi m k_{B} T}{h^{2}}\right)^{3 / 2} \exp \left([\mu-m] / k_{B} T\right)$

where $\mu$ is the chemical potential and $k_{B}$ and $h$ are the Boltzmann and Planck constants respectively.]

The photon number density $n_{\gamma}$ is given as

$n_{\gamma}=\frac{16 \pi}{h^{3}} \zeta(3)\left(k_{B} T\right)^{3}$

where $\zeta(3) \simeq 1.20$. Consider now the fractional ionization $X_{e}=n_{e} /\left(n_{e}+n_{H}\right)$. In our universe $n_{e}+n_{H}=n_{p}+n_{H} \simeq \eta n_{\gamma}$ where $\eta$ is the baryon-to-photon number ratio. Find an expression for the ratio

$\frac{\left(1-X_{e}\right)}{X_{e}^{2}}$

in terms of $k_{B} T, \eta, I$ and the particle masses. One might expect neutral hydrogen to form at a temperature given by $k_{B} T \sim I \sim 13 \mathrm{eV}$, but instead in our universe it forms at the much lower temperature $k_{B} T \sim 0.3 \mathrm{eV}$. Briefly explain why this happens. Estimate the temperature at which neutral hydrogen would form in a hypothetical universe with $\eta=1$. Briefly explain your answer.

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• # Paper 4, Section I, B

Derive the relation between the neutrino temperature $T_{\nu}$ and the photon temperature $T_{\gamma}$ at a time long after electrons and positrons have become non-relativistic.

[In this question you may work in units of the speed of light, so that $c=1$. You may also use without derivation the following formulae. The energy density $\epsilon_{a}$ and pressure $P_{a}$ for a single relativistic species a with a number $g_{a}$ of degenerate states at temperature $T$ are given by

$\epsilon_{a}=\frac{4 \pi g_{a}}{h^{3}} \int \frac{p^{3} d p}{e^{p /\left(k_{B} T\right)} \mp 1}, \quad \quad P_{a}=\frac{4 \pi g_{a}}{3 h^{3}} \int \frac{p^{3} d p}{e^{p /\left(k_{B} T\right)} \mp 1},$

where $k_{B}$ is Boltzmann's constant, $h$ is Planck's constant, and the minus or plus depends on whether the particle is a boson or a fermion respectively. For each species a, the entropy density $s_{a}$ at temperature $T_{a}$ is given by,

$s_{a}=\frac{\epsilon_{a}+P_{a}}{k_{B} T_{a}} .$

The effective total number $g_{*}$ of relativistic species is defined in terms of the numbers of bosonic and fermionic particles in the theory as,

$g_{*}=\sum_{\text {bosons }} g_{\text {bosons }}+\frac{7}{8} \sum_{\text {fermions }} g_{\text {fermions }}$

with the specific values $g_{\gamma}=g_{e^{+}}=g_{e^{-}}=2$ for photons, positrons and electrons.]

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• # Paper 1, Section I, B

For a homogeneous and isotropic universe filled with pressure-free matter $(P=0)$, the Friedmann and Raychaudhuri equations are, respectively,

$\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho \quad \text { and } \quad \frac{\ddot{a}}{a}=-\frac{4 \pi G}{3} \rho,$

with mass density $\rho$, curvature $k$, and where $\dot{a} \equiv d a / d t$. Using conformal time $\tau$ with $d \tau=d t / a$, show that the relative density parameter can be expressed as

$\Omega(t) \equiv \frac{\rho}{\rho_{\text {crit }}}=\frac{8 \pi G \rho a^{2}}{3 \mathcal{H}^{2}}$

where $\mathcal{H}=\frac{1}{a} \frac{d a}{d \tau}$ and $\rho_{\text {crit }}$ is the critical density of a flat $k=0$ universe (Einstein-de Sitter). Use conformal time $\tau$ again to show that the Friedmann and Raychaudhuri equations can be re-expressed as

$\frac{k c^{2}}{\mathcal{H}^{2}}=\Omega-1 \quad \text { and } \quad 2 \frac{d \mathcal{H}}{d \tau}+\mathcal{H}^{2}+k c^{2}=0$

From these derive the evolution equation for the density parameter $\Omega$ :

$\frac{d \Omega}{d \tau}=\mathcal{H} \Omega(\Omega-1)$

Plot the qualitative behaviour of $\Omega$ as a function of time relative to the expanding Einsteinde Sitter model with $\Omega=1$ (i.e., include curves initially with $\Omega>1$ and $\Omega<1$ ).

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• # Paper 1, Section II, B

A flat $(k=0)$ homogeneous and isotropic universe with scale factor $a(t)$ is filled with a scalar field $\phi(t)$ with potential $V(\phi)$. Its evolution satisfies the Friedmann and scalar field equations,

$H^{2}=\frac{1}{3 M_{\mathrm{Pl}}^{2}}\left(\frac{1}{2} \dot{\phi}^{2}+c^{2} V(\phi)\right), \quad \ddot{\phi}+3 H \dot{\phi}+c^{2} \frac{d V}{d \phi}=0$

where $H(t)=\frac{\dot{a}}{a}$ is the Hubble parameter, $M_{\mathrm{Pl}}$ is the reduced Planck mass, and dots denote derivatives with respect to cosmic time $t$, e.g. $\dot{\phi} \equiv d \phi / d t$.

(a) Use these equations to derive the Raychaudhuri equation, expressed in the form:

$\dot{H}=-\frac{1}{2 M_{\mathrm{Pl}}^{2}} \dot{\phi}^{2}$

(b) Consider the following ansatz for the scalar field evolution,

$\phi(t)=\phi_{0} \ln \tanh (\lambda t)$

where $\lambda, \phi_{0}$ are constants. Find the specific cosmological solution,

\begin{aligned} H(t) &=\lambda \frac{\phi_{0}^{2}}{M_{\mathrm{Pl}}^{2}} \operatorname{coth}(2 \lambda t) \\ a(t) &=a_{0}[\sinh (2 \lambda t)]^{\phi_{0}^{2} / 2 M_{\mathrm{Pl}}^{2}}, \quad a_{0} \text { constant. } \end{aligned}

(c) Hence, show that the Hubble parameter can be expressed in terms of $\phi$ as

$H(\phi)=\lambda \frac{\phi_{0}^{2}}{M_{\mathrm{P} 1}^{2}} \cosh \left(\frac{\phi}{\phi_{0}}\right),$

and that the scalar field ansatz solution ( $\dagger$ ) requires the following form for the potential:

$V(\phi)=\frac{2 \lambda^{2} \phi_{0}^{2}}{c^{2}}\left[\left(\frac{3 \phi_{0}^{2}}{2 M_{\mathrm{Pl}}^{2}}-1\right) \cosh ^{2}\left(\frac{\phi}{\phi_{0}}\right)+1\right]$

(d) Assume that the given parameters in $V(\phi)$ are such that $2 / 3<\phi_{0}^{2} / M_{\mathrm{Pl}}^{2}<2$. Show that the asymptotic limit for the cosmological solution as $t \rightarrow 0$ exhibits decelerating power law evolution and that there is an accelerating solution as $t \rightarrow \infty$, that is,

$\begin{array}{lll} t \rightarrow 0, & \phi \rightarrow-\infty, & a(t) \sim t^{\phi_{0}^{2} / 2 M_{\mathrm{Pl}}^{2}} \\ t \rightarrow \infty, & \phi \rightarrow 0, & a(t) \sim \exp \left(\lambda \phi_{0}^{2} t / M_{\mathrm{Pl}}^{2}\right) . \end{array}$

Find the time $t_{\mathrm{acc}}$ at which the solution transitions from deceleration to acceleration.

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• # Paper 2, Section I, B

(a) Consider a homogeneous and isotropic universe with a uniform distribution of galaxies. For three galaxies at positions $\mathbf{r}_{A}, \mathbf{r}_{B}, \mathbf{r}_{C}$, show that spatial homogeneity implies that their non-relativistic velocities $\mathbf{v}(\mathbf{r})$ must satisfy

$\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{A}\right)=\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{C}\right)-\mathbf{v}\left(\mathbf{r}_{A}-\mathbf{r}_{C}\right)$

and hence that the velocity field coordinates $v_{i}$ are linearly related to the position coordinates $r_{j}$ via

$v_{i}=H_{i j} r_{j}$

where the matrix coefficients $H_{i j}$ are independent of the position. Show why isotropy then implies Hubble's law

$\mathbf{v}=H \mathbf{r}, \quad \text { with } H \text { independent of } \mathbf{r}$

Explain how the velocity of a galaxy is determined by the scale factor $a$ and express the Hubble parameter $H_{0}$ today in terms of $a$.

(b) Define the cosmological horizon $d_{H}(t)$. For an Einstein-de Sitter universe with $a(t) \propto t^{2 / 3}$, calculate $d_{H}\left(t_{0}\right)$ at $t=t_{0}$ today in terms of $H_{0}$. Briefly describe the horizon problem of the standard cosmology.

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• # Paper 3, Section I, B

The energy density of a particle species is defined by

$\epsilon=\int_{0}^{\infty} E(p) n(p) d p$

where $E(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the energy, and $n(p)$ the distribution function, of a particle with momentum $p$. Here $c$ is the speed of light and $m$ is the rest mass of the particle. If the particle species is in thermal equilibrium then the distribution function takes the form

$n(p)=\frac{4 \pi}{h^{3}} g \frac{p^{2}}{\exp ((E(p)-\mu) / k T) \mp 1}$

where $g$ is the number of degrees of freedom of the particle, $T$ is the temperature, $h$ and $k$ are constants and $-$ is for bosons and $+$ is for fermions.

(a) Stating any assumptions you require, show that in the very early universe the energy density of a given particle species $i$ is

$\epsilon_{i}=\frac{4 \pi g_{i}}{(h c)^{3}}(k T)^{4} \int_{0}^{\infty} \frac{y^{3}}{e^{y} \mp 1} d y$

(b) Show that the total energy density in the very early universe is

$\epsilon=\frac{4 \pi^{5}}{15(h c)^{3}} g^{*}(k T)^{4}$

where $g^{*}$ is defined by

$g^{*} \equiv \sum_{\text {Bosons }} g_{i}+\frac{7}{8} \sum_{\text {Fermions }} g_{i}$

[Hint: You may use the fact that $\left.\int_{0}^{\infty} y^{3}\left(e^{y}-1\right)^{-1} d y=\pi^{4} / 15 .\right]$

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• # Paper 3, Section II, B

The pressure support equation for stars is

$\frac{1}{r^{2}} \frac{d}{d r}\left[\frac{r^{2}}{\rho} \frac{d P}{d r}\right]=-4 \pi G \rho$

where $\rho$ is the density, $P$ is the pressure, $r$ is the radial distance, and $G$ is Newton's constant.

(a) What two boundary conditions should we impose on the above equation for it to describe a star?

(b) By assuming a polytropic equation of state,

$P(r)=K \rho^{1+\frac{1}{n}}(r)$

where $K$ is a constant, derive the Lane-Emden equation

$\frac{1}{\xi^{2}} \frac{d}{d \xi}\left[\xi^{2} \frac{d \theta}{d \xi}\right]=-\theta^{n}$

where $\rho=\rho_{c} \theta^{n}$, with $\rho_{c}$ the density at the centre of the star, and $r=a \xi$, for some $a$ that you should determine.

(c) Show that the mass of a polytropic star is

$M=\frac{1}{2 \sqrt{\pi}}\left(\frac{(n+1) K}{G}\right)^{\frac{3}{2}} \rho_{c}^{\frac{3-n}{2 n}} Y_{n}$

where $Y_{n} \equiv-\left.\xi_{1}^{2} \frac{d \theta}{d \xi}\right|_{\xi=\xi_{1}}$ and $\xi_{1}$ is the value of $\xi$ at the surface of the star.

(d) Derive the following relation between the mass, $M$, and radius, $R$, of a polytropic star

$M=A_{n} K^{\frac{n}{n-1}} R^{\frac{3-n}{1-n}}$

where you should determine the constant $A_{n}$. What type of star does the $n=3$ polytrope represent and what is the significance of the mass being constant for this star?

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• # Paper 4, Section I, B

A constant overdensity is created by taking a spherical region of a flat matterdominated universe with radius $\bar{R}$ and compressing it into a region with radius $R<\bar{R}$. The evolution is governed by the parametric equations

$R=A R_{0}(1-\cos \theta), \quad t=B(\theta-\sin \theta)$

where $R_{0}$ is a constant and

$A=\frac{\Omega_{m, 0}}{2\left(\Omega_{m, 0}-1\right)}, \quad B=\frac{\Omega_{m, 0}}{2 H_{0}\left(\Omega_{m, 0}-1\right)^{3 / 2}}$

where $H_{0}$ is the Hubble constant and $\Omega_{m, 0}$ is the fractional overdensity at time $t_{0}$.

Show that, as $t \rightarrow 0^{+}$,

$R(t)=R_{0} \Omega_{m, 0}^{1 / 3} a(t)\left(1-\frac{1}{20}\left(\frac{6 t}{B}\right)^{2 / 3}+\ldots\right)$

where the scale factor is given by $a(t)=\left(3 H_{0} t / 2\right)^{2 / 3}$.

that, when the spherical overdensity has collapsed to zero radius, the linear perturbation has value $\delta_{\text {linear }}=\frac{3}{20}(12 \pi)^{2 / 3}$.

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• # Paper 1, Section I, C

In a homogeneous and isotropic universe, describe the relative displacement $\mathbf{r}(t)$ of two galaxies in terms of a scale factor $a(t)$. Show how the relative velocity $\mathbf{v}(t)$ of these galaxies is given by the relation $\mathbf{v}(t)=H(t) \mathbf{r}(t)$, where you should specify $H(t)$ in terms of $a(t)$.

From special relativity, the Doppler shift of light emitted by a particle moving away radially with speed $v$ can be approximated by

$\frac{\lambda_{0}}{\lambda_{\mathrm{e}}}=\sqrt{\frac{1+v / c}{1-v / c}}=1+\frac{v}{c}+\mathcal{O}\left(\frac{v^{2}}{c^{2}}\right)$

where $\lambda_{e}$ is the wavelength of emitted light and $\lambda_{0}$ is the observed wavelength. For the observed light from distant galaxies in a homogeneous and isotropic expanding universe, show that the redshift defined by $1+z \equiv \lambda_{0} / \lambda_{\mathrm{e}}$ is given by

$1+z=\frac{a\left(t_{0}\right)}{a\left(t_{\mathrm{e}}\right)}$

where $t_{\mathrm{e}}$ is the time of emission and $t_{0}$ is the observation time.

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• # Paper 1, Section II, C

The evolution of a flat $(k=0)$ homogeneous and isotropic universe with scale factor $a(t)$, mass density $\rho(t)$ and pressure $P(t)$ obeys the Friedmann and energy conservation equations

$\begin{array}{r} H^{2}(t)=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3} \rho+\frac{\Lambda c^{2}}{3} \\ \dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+P / c^{2}\right) \end{array}$

where $H(t)$ is the Hubble parameter (observed today $t=t_{0}$ with value $H_{0}=H\left(t_{0}\right)$ ) and $\Lambda>0$ is the cosmological constant.

Use these two equations to derive the acceleration equation

$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)+\frac{\Lambda c^{2}}{3}$

For pressure-free matter $\left(\rho=\rho_{\mathrm{M}}\right.$ and $\left.P_{\mathrm{M}}=0\right)$, solve the energy conservation equation to show that the Friedmann and acceleration equations can be re-expressed as

$\begin{gathered} H=H_{0} \sqrt{\frac{\Omega_{\mathrm{M}}}{a^{3}}+\Omega_{\Lambda}} \\ \frac{\ddot{a}}{a}=-\frac{H_{0}^{2}}{2}\left[\frac{\Omega_{\mathrm{M}}}{a^{3}}-2 \Omega_{\Lambda}\right] \end{gathered}$

where we have taken $a\left(t_{0}\right)=1$ and we have defined the relative densities today $\left(t=t_{0}\right)$ as

$\Omega_{\mathrm{M}}=\frac{8 \pi G}{3 H_{0}^{2}} \rho_{\mathrm{M}}\left(t_{0}\right) \quad \text { and } \quad \Omega_{\Lambda}=\frac{\Lambda c^{2}}{3 H_{0}^{2}}$

Solve the Friedmann equation and show that the scale factor can be expressed as

$a(t)=\left(\frac{\Omega_{\mathrm{M}}}{\Omega_{\Lambda}}\right)^{1 / 3} \sinh ^{2 / 3}\left(\frac{3}{2} \sqrt{\Omega_{\Lambda}} H_{0} t\right)$

Find an expression for the time $\bar{t}$ at which the matter density $\rho_{\mathrm{M}}$ and the effective density caused by the cosmological constant $\Lambda$ are equal. (You need not evaluate this explicitly.)

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• # Paper 2, Section I, C

In a homogeneous and isotropic universe $(\Lambda=0)$, the acceleration equation for the scale factor $a(t)$ is given by

$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right),$

where $\rho(t)$ is the mass density and $P(t)$ is the pressure.

If the matter content of the universe obeys the strong energy condition $\rho+3 P / c^{2} \geqslant 0$, show that the acceleration equation can be rewritten as $\dot{H}+H^{2} \leqslant 0$, with Hubble parameter $H(t)=\dot{a} / a$. Show that

$H \geqslant \frac{1}{H_{0}^{-1}+t-t_{0}}$

where $H_{0}=H\left(t_{0}\right)$ is the measured value today at $t=t_{0}$. Hence, or otherwise, show that

$a(t) \leqslant 1+H_{0}\left(t-t_{0}\right)$

Use this inequality to find an upper bound on the age of the universe.

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• # Paper 3, Section I, C

(a) In the early universe electrons, protons and neutral hydrogen are in thermal equilibrium and interact via,

$e^{-}+p^{+} \leftrightharpoons H+\gamma$

The non-relativistic number density of particles in thermal equlibrium is

$n_{i}=g_{i}\left(\frac{2 \pi m_{i} k T}{h^{2}}\right)^{\frac{3}{2}} \exp \left(\frac{\mu_{i}-m_{i} c^{2}}{k T}\right)$

where, for each species $i, g_{i}$ is the number of degrees of freedom, $m_{i}$ is its mass, and $\mu_{i}$ is its chemical potential. [You may assume $g_{e}=g_{p}=2$ and $g_{H}=4$.]

Stating any assumptions required, use these expressions to derive the Saha equation which governs the relative abundances of electrons, protons and hydrogen,

$\frac{n_{e} n_{p}}{n_{H}}=\left(\frac{2 \pi m_{e} k T}{h^{2}}\right)^{\frac{3}{2}} \exp \left(-\frac{I}{k T}\right)$

where $I$ is the binding energy of hydrogen, which should be defined.

(b) Naively, we might expect that the majority of electrons and protons combine to form neutral hydrogen once the temperature drops below the binding energy, i.e. $k T \lesssim I$. In fact recombination does not happen until a much lower temperature, when $k T \approx 0.03 I$. Briefly explain why this is.

[Hint: It may help to consider the relative abundances of particles in the early universe.]

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• # Paper 3, Section II, C

(a) The scalar moment of inertia for a system of $N$ particles is given by

$I=\sum_{i=1}^{N} m_{i} \mathbf{r}_{i} \cdot \mathbf{r}_{i}$

where $m_{i}$ is the particle's mass and $\mathbf{r}_{i}$ is a vector giving the particle's position. Show that, for non-relativistic particles,

$\frac{1}{2} \frac{d^{2} I}{d t^{2}}=2 K+\sum_{i=1}^{N} \mathbf{F}_{i} \cdot \mathbf{r}_{i}$

where $K$ is the total kinetic energy of the system and $\mathbf{F}_{i}$ is the total force on particle

Assume that any two particles $i$ and $j$ interact gravitationally with potential energy

$V_{i j}=-\frac{G m_{i} m_{j}}{\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|}$

Show that

$\sum_{i=1}^{N} \mathbf{F}_{i} \cdot \mathbf{r}_{i}=V$

where $V$ is the total potential energy of the system. Use the above to prove the virial theorem.

(b) Consider an approximately spherical overdensity of stationary non-interacting massive particles with initial constant density $\rho_{i}$ and initial radius $R_{i}$. Assuming the system evolves until it reaches a stable virial equilibrium, what will the final $\rho$ and $R$ be in terms of their initial values? Would this virial solution be stable if our particles were baryonic rather than non-interacting? Explain your answer.

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• # Paper 4, Section I, C

(a) By considering a spherically symmetric star in hydrostatic equilibrium derive the pressure support equation

$\frac{d P}{d r}=-\frac{G M(r) \rho}{r^{2}},$

where $r$ is the radial distance from the centre of the star, $M(r)$ is the stellar mass contained inside that radius, and $P(r)$ and $\rho(r)$ are the pressure and density at radius $r$ respectively.

(b) Propose, and briefly justify, boundary conditions for this differential equation, both at the centre of the star $r=0$, and at the stellar surface $r=R$.

Suppose that $P=K \rho^{2}$ for some $K>0$. Show that the density satisfies the linear differential equation

$\frac{1}{x^{2}} \frac{\partial}{\partial x}\left(x^{2} \frac{\partial \rho}{\partial x}\right)=-\rho$

where $x=\alpha r$, for some constant $\alpha$, is a rescaled radial coordinate. Find $\alpha$