Statistical Physics And Cosmology

• A1.16

  Statistical Physics and Cosmology

  Part II, 2001

  (i) Introducing the concept of a co-moving distance co-ordinate, explain briefly how the velocity of a galaxy in an isotropic and homogeneous universe is determined by the scale factor $a(t)$. How is the scale factor related to the Hubble constant $H_{0}$ ?

  A homogeneous and isotropic universe has an energy density $\rho(t) c^{2}$ and a pressure $P(t)$. Use the relation $d E=-P d V$ to derive the "fluid equation"

  $$\dot{\rho}=-3\left(\rho+\frac{P}{c^{2}}\right)\left(\frac{\dot{a}}{a}\right)$$

  where the overdot indicates differentiation with respect to time, $t$. Given that $a(t)$ satisfies the "acceleration equation"

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

  show that the quantity

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

  is time-independent.

  The pressure $P$ is related to $\rho$ by the "equation of state"

  $$P=\sigma \rho c^{2}, \quad|\sigma|<1 .$$

  Given that $a\left(t_{0}\right)=1$, find $a(t)$ for $k=0$, and hence show that $a(0)=0$.

  (ii) What is meant by the expression "the Hubble time"?

  Assuming that $a(0)=0$ and $a\left(t_{0}\right)=1$, where $t_{0}$ is the time now (of the current cosmological era), obtain a formula for the radius $R_{0}$ of the observable universe.

  Given that

  $$a(t)=\left(\frac{t}{t_{0}}\right)^{\alpha}$$

  for constant $\alpha$, find the values of $\alpha$ for which $R_{0}$ is finite. Given that $R_{0}$ is finite, show that the age of the universe is less than the Hubble time. Explain briefly, and qualitatively, why this result is to be expected as long as

  $$\rho+3 \frac{P}{c^{2}}>0 .$$

  Part II

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• A3.14

  Statistical Physics and Cosmology

  Part II, 2001

  (i) A spherically symmetric star has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is distance from the star's centre. Stating without proof any theorems you may need, show that mechanical equilibrium implies the Newtonian pressure support equation

  $$P^{\prime}=-\frac{G m \rho}{r^{2}},$$

  where $m(r)$ is the mass within radius $r$ and $P^{\prime}=d P / d r$.

  Write down an integral expression for the total gravitational potential energy, $E_{g r}$. Use this to derive the "virial theorem"

  $$E_{g r}=-3\langle P\rangle V$$

  when $\langle P\rangle$ is the average pressure.

  (ii) Given that the total kinetic energy, $E_{k i n}$, of a spherically symmetric star is related to its average pressure by the formula

  $$E_{k i n}=\alpha\langle P\rangle V$$

  for constant $\alpha$, use the virial theorem (stated in part (i)) to determine the condition on $\alpha$ needed for gravitational binding. State the relation between pressure $P$ and "internal energy" $U$ for an ideal gas of non-relativistic particles. What is the corresponding relation for ultra-relativistic particles? Hence show that the formula $(*)$ applies in these cases, and determine the values of $\alpha$.

  Why does your result imply a maximum mass for any star, whatever the source of its pressure? What is the maximum mass, approximately, for stars supported by

  (a) thermal pressure,

  (b) electron degeneracy pressure (White Dwarf),

  (c) neutron degeneracy pressure (Neutron Star).

  A White Dwarf can accrete matter from a companion star until its mass exceeds the Chandrasekar limit. Explain briefly the process by which it then evolves into a neutron star.

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• A4.18

  Statistical Physics and Cosmology

  Part II, 2001

  (i) Given that $g(p) d p$ is the number of eigenstates of a gas particle with momentum between $p$ and $p+d p$, write down the Bose-Einstein distribution $\bar{n}(p)$ for the average number of particles with momentum between $p$ and $p+d p$, as a function of temperature $T$ and chemical potential $\mu$.

  Given that $\mu=0$ and $g(p)=8 \pi \frac{V p^{2}}{h^{3}}$ for a gas of photons, obtain a formula for the energy density $\rho_{T}$ at temperature $T$ in the form

  $$\rho_{T}=\int_{0}^{\infty} \epsilon_{T}(\nu) d \nu,$$

  where $\epsilon_{T}(\nu)$ is a function of the photon frequency $\nu$ that you should determine. Hence show that the value $\nu_{\text {peak }}$ of $\nu$ at the maximum of $\epsilon_{T}(\nu)$ is proportional to $T$.

  A thermally isolated photon gas undergoes a slow change of its volume $V$. Why is $\bar{n}(p)$ unaffected by this change? Use this fact to show that $V T^{3}$ remains constant.

  (ii) According to the "Hot Big Bang" theory, the Universe evolved by expansion from an earlier state in which it was filled with a gas of electrons, protons and photons (with $n_{e}=n_{p}$ ) at thermal equilibrium at a temperature $T$ such that

  $$2 m_{e} c^{2} \gg k T \gg B$$

  where $m_{e}$ is the electron mass and $B$ is the binding energy of a hydrogen atom. Why must the composition have been different when $k T \gg 2 m_{e} c^{2}$ ? Why must it change as the temperature falls to $k T \ll B$ ? Why does this lead to a thermal decoupling of radiation from matter?

  The baryon number of the Universe can be taken to be the number of protons, either as free particles or as hydrogen atom nuclei. Let $n_{b}$ be the baryon number density and $n_{\gamma}$ the photon number density. Why is the ratio $\eta=n_{b} / n_{\gamma}$ unchanged by the expansion of the universe? Given that $\eta \ll 1$, obtain an estimate of the temperature $T_{D}$ at which decoupling occurs, as a function of $\eta$ and $B$. How does this decoupling lead to the concept of a "surface of last scattering" and a prediction of a Cosmic Microwave Background Radiation (CMBR)?

  Part II

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• A1.16

  Statistical Physics and Cosmology

  Part II, 2002

  (i) Consider a one-dimensional model universe with "stars" distributed at random on the $x$-axis, and choose the origin to coincide with one of the stars; call this star the "homestar." Home-star astronomers have discovered that all other stars are receding from them with a velocity $v(x)$, that depends on the position $x$ of the star. Assuming non-relativistic addition of velocities, show how the assumption of homogeneity implies that $v(x)=H_{0} x$ for some constant $H_{0}$.

  In attempting to understand the history of their one-dimensional universe, homestar astronomers seek to determine the velocity $v(t)$ at time $t$ of a star at position $x(t)$. Assuming homogeneity, show how $x(t)$ is determined in terms of a scale factor $a(t)$ and hence deduce that $v(t)=H(t) x(t)$ for some function $H(t)$. What is the relation between $H(t)$ and $H_{0}$ ?

  (ii) Consider a three-dimensional homogeneous and isotropic universe with mass density $\rho(t)$, pressure $p(t)$ and scale factor $a(t)$. Given that $E(t)$ is the energy in volume $V(t)$, show how the relation $d E=-p d V$ yields the "fluid" equation

  $$\dot{\rho}=-3\left(\rho+\frac{p}{c^{2}}\right) H$$

  where $H=\dot{a} / a$.

  Show how conservation of energy applied to a test particle at the boundary of a spherical fluid element yields the Friedmann equation

  $$\dot{a}^{2}-\frac{8 \pi G}{3} \rho a^{2}=-k c^{2}$$

  for constant $k$. Hence obtain an equation for the acceleration $\ddot{a}$ in terms of $\rho, p$ and $a$.

  A model universe has mass density and pressure

  $$\rho=\frac{\rho_{0}}{a^{3}}+\rho_{1}, \quad p=-\rho_{1} c^{2},$$

  where $\rho_{0}$ is constant. What does the fluid equation imply about $\rho_{1}$ ? Show that the acceleration $\ddot{a}$ vanishes if

  $$a=\left(\frac{\rho_{0}}{2 \rho_{1}}\right)^{\frac{1}{3}}$$

  Hence show that this universe is static and determine the sign of the constant $k$.

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• A3.14

  Statistical Physics and Cosmology

  Part II, 2002

  (i) Write down the first law of thermodynamics for the change $d U$ in the internal energy $U(N, V, S)$ of a gas of $N$ particles in a volume $V$ with entropy $S$.

  Given that

  $$P V=(\gamma-1) U,$$

  where $P$ is the pressure, use the first law to show that $P V^{\gamma}$ is constant at constant $N$ and

  Write down the Boyle-Charles law for a non-relativistic ideal gas and hence deduce that the temperature $T$ is proportional to $V^{1-\gamma}$ at constant $N$ and $S$.

  State the principle of equipartition of energy and use it to deduce that

  $$U=\frac{3}{2} N k T$$

  Hence deduce the value of $\gamma$. Show that this value of $\gamma$ is such that the ratio $E_{i} / k T$ is unchanged by a change of volume at constant $N$ and $S$, where $E_{i}$ is the energy of the $i$-th one particle eigenstate of a non-relativistic ideal gas.

  (ii) A classical gas of non-relativistic particles of mass $m$ at absolute temperature $T$ and number density $n$ has a chemical potential

  $$\mu=m c^{2}-k T \ln \left(\frac{g_{s}}{n}\left(\frac{m k T}{2 \pi \hbar^{2}}\right)^{\frac{3}{2}}\right)$$

  where $g_{s}$ is the particle's spin degeneracy factor. What condition on $n$ is needed for the validity of this formula and why?

  Thermal and chemical equilibrium between two species of non-relativistic particles $a$ and $b$ is maintained by the reaction

  $$a+\alpha \leftrightarrow b+\beta$$

  where $\alpha$ and $\beta$ are massless particles with zero chemical potential. Given that particles $a$ and $b$ have masses $m_{a}$ and $m_{b}$ respectively, but equal spin degeneracy factors, find the number density ratio $n_{a} / n_{b}$ as a function of $m_{a}, m_{b}$ and $T$. Given that $m_{a}>m_{b}$ but $m_{a}-m_{b} \ll m_{b}$ show that

  $$\frac{n_{a}}{n_{b}} \approx f\left(\frac{\left(m_{a}-m_{b}\right) c^{2}}{k T}\right)$$

  for some function $f$ which you should determine.

  Explain how a reaction of the above type is relevant to a determination of the neutron to proton ratio in the early universe and why this ratio does not fall rapidly to zero as the universe cools. Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Let

  $$Y_{H e}=\frac{\rho_{H e}}{\rho}$$

  be the fraction of the universe that ends up in helium. Compute $Y_{H e}$ as a function of the ratio $r=n_{a} / n_{b}$ at the time of nucleosynthesis.

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• A4.18

  Statistical Physics and Cosmology

  Part II, 2002

  What is an ideal gas? Explain how the microstates of an ideal gas of indistinguishable particles can be labelled by a set of integers. What range of values do these integers take for (a) a boson gas and (b) a Fermi gas?

  Let $E_{i}$ be the energy of the $i$-th one-particle energy eigenstate of an ideal gas in thermal equilibrium at temperature $T$ and let $p_{i}\left(n_{i}\right)$ be the probability that there are $n_{i}$ particles of the gas in this state. Given that

  $$p_{i}\left(n_{i}\right)=e^{-\beta E_{i} n_{i}} / Z_{i} \quad\left(\beta=\frac{1}{k T}\right)$$

  determine the normalization factor $Z_{i}$ for (a) a boson gas and (b) a Fermi gas. Hence obtain an expression for $\bar{n}_{i}$, the average number of particles in the $i$-th one-particle energy eigenstate for both cases (a) and (b).

  In the case of a Fermi gas, write down (without proof) the generalization of your formula for $\bar{n}_{i}$ to a gas at non-zero chemical potential $\mu$. Show how it leads to the concept of a Fermi energy $\epsilon_{F}$ for a gas at zero temperature. How is $\epsilon_{F}$ related to the Fermi momentum $p_{F}$ for (a) a non-relativistic gas and (b) an ultra-relativistic gas?

  In an approximation in which the discrete set of energies $E_{i}$ is replaced with a continuous set with momentum $p$, the density of one-particle states with momentum in the range $p$ to $p+d p$ is $g(p) d p$. Explain briefly why

  $$g(p) \propto p^{2} V$$

  where $V$ is the volume of the gas. Using this formula, obtain an expression for the total energy $E$ of an ultra-relativistic gas at zero chemical potential as an integral over $p$. Hence show that

  $$\frac{E}{V} \propto T^{\alpha},$$

  where $\alpha$ is a number that you should compute. Why does this result apply to a photon gas?

  Using the formula $(*)$ for a non-relativistic Fermi gas at zero temperature, obtain an expression for the particle number density $n$ in terms of the Fermi momentum and provide a physical interpretation of this formula in terms of the typical de Broglie wavelength. Obtain an analogous formula for the (internal) energy density and hence show that the pressure $P$ behaves as

  $$P \propto n^{\gamma}$$

  where $\gamma$ is a number that you should compute. [You need not prove any relation between the pressure and the energy density you use.] What is the origin of this pressure given that $T=0$ by assumption? Explain briefly and qualitatively how it is relevant to the stability of white dwarf stars.

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• A1.16

  Statistical Physics and Cosmology

  Part II, 2003

  (i) Explain briefly how the relative motion of galaxies in a homogeneous and isotropic universe is described in terms of the scale factor $a(t)$ (where $t$ is time). In particular, show that the relative velocity $\mathbf{v}(t)$ of two galaxies is given in terms of their relative displacement $\mathbf{r}(t)$ by the formula $\mathbf{v}(t)=H(t) \mathbf{r}(t)$, where $H(t)$ is a function that you should determine in terms of $a(t)$. Given that $a(0)=0$, obtain a formula for the distance $R(t)$ to the cosmological horizon at time $t$. Given further that $a(t)=\left(t / t_{0}\right)^{\alpha}$, for $0<\alpha<1$ and constant $t_{0}$, compute $R(t)$. Hence show that $R(t) / a(t) \rightarrow 0$ as $t \rightarrow 0$.

  (ii) A homogeneous and isotropic model universe has energy density $\rho(t) c^{2}$ and pressure $P(t)$, where $c$ is the speed of light. The evolution of its scale factor $a(t)$ is governed by the Friedmann equation

  $$\dot{a}^{2}=\frac{8 \pi G}{3} \rho a^{2}-k c^{2}$$

  where the overdot indicates differentiation with respect to $t$. Use the "Fluid" equation

  $$\dot{\rho}=-3\left(\rho+\frac{P}{c^{2}}\right)\left(\frac{\dot{a}}{a}\right)$$

  to obtain an equation for the acceleration $\ddot{a}(t)$. Assuming $\rho>0$ and $P \geqslant 0$, show that $\rho a^{3}$ cannot increase with time as long as $\dot{a}>0$, nor decrease if $\dot{a}<0$. Hence determine the late time behaviour of $a(t)$ for $k<0$. For $k>0$ show that an initially expanding universe must collapse to a "big crunch" at which $a \rightarrow 0$. How does $\dot{a}$ behave as $a \rightarrow 0$ ? Given that $P=0$, determine the form of $a(t)$ near the big crunch. Discuss the qualitative late time behaviour for $k=0$.

  Cosmological models are often assumed to have an equation of state of the form $P=\sigma \rho c^{2}$ for constant $\sigma$. What physical principle requires $\sigma \leqslant 1$ ? Matter with $P=\rho c^{2}$ $(\sigma=1)$ is called "stiff matter" by cosmologists. Given that $k=0$, determine $a(t)$ for a universe that contains only stiff matter. In our Universe, why would you expect stiff matter to be negligible now even if it were significant in the early Universe?

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• A3.14

  Statistical Physics and Cosmology

  Part II, 2003

  (i) The pressure $P(r)$ and mass density $\rho(r)$, at distance $r$ from the centre of a spherically-symmetric star, obey the pressure-support equation

  $$P^{\prime}=-\frac{G m \rho}{r^{2}}$$

  where $m^{\prime}=4 \pi r^{2} \rho(r)$, and the prime indicates differentiation with respect to $r$. Let $V$ be the total volume of the star, and $\langle P\rangle$ its average pressure. Use the pressure-support equation to derive the "virial theorem"

  $$\langle P\rangle V=-\frac{1}{3} E_{g r a v}$$

  where $E_{g r a v}$ is the total gravitational potential energy [Hint: multiply by $4 \pi r^{3}$ ]. If a star is assumed to be a self-gravitating ball of a non-relativistic ideal gas then it can be shown that

  $$\langle P\rangle V=\frac{2}{3} E_{k i n}$$

  where $E_{k i n}$ is the total kinetic energy. Use this result to show that the total energy $U=E_{\text {grav }}+E_{k i n}$ is negative. When nuclear reactions have converted the hydrogen in a star's core to helium the core contracts until the helium is converted to heavier elements, thereby increasing the total energy $U$ of the star. Explain briefly why this converts the star into a "Red Giant". (ii) Write down the first law of thermodynamics for the change in energy $E$ of a system at temperature $T$, pressure $P$ and chemical potential $\mu$ as a result of small changes in the entropy $S$, volume $V$ and particle number $N$. Use this to show that

  $$P=-\left(\frac{\partial E}{\partial V}\right)_{N, S}$$

  The microcanonical ensemble is the set of all accessible microstates of a system at fixed $E, V, N$. Define the canonical and grand-canonical ensembles. Why are the properties of a macroscopic system independent of the choice of thermodynamic ensemble?

  The Gibbs "grand potential" $\mathcal{G}(T, V, \mu)$ can be defined as

  $$\mathcal{G}=E-T S-\mu N$$

  Use the first law to find expressions for $S, P, N$ as partial derivatives of $\mathcal{G}$. A system with variable particle number $n$ has non-degenerate energy eigenstates labeled by $r^{(n)}$, for each $n$, with energy eigenvalues $E_{r}^{(n)}$. If the system is in equilibrium at temperature $T$ and chemical potential $\mu$ then the probability $p\left(r^{(n)}\right)$ that it will be found in a particular $n$-particle state $r^{(n)}$ is given by the Gibbs probability distribution

  $$p\left(r^{(n)}\right)=\mathcal{Z}^{-1} e^{\left(\mu n-E_{r}^{(n)}\right) / k T}$$

  where $k$ is Boltzmann's constant. Deduce an expression for the normalization factor $\mathcal{Z}$ as a function of $\mu$ and $\beta=1 / k T$, and hence find expressions for the partial derivatives

  $$\frac{\partial \log \mathcal{Z}}{\partial \mu}, \quad \frac{\partial \log \mathcal{Z}}{\partial \beta}$$

  in terms of $N, E, \mu, \beta$.

  Why does $\mathcal{Z}$ also depend on the volume $V$ ? Given that a change in $V$ at fixed $N, S$ leaves unchanged the Gibbs probability distribution, deduce that

  $$\left(\frac{\partial \log \mathcal{Z}}{\partial V}\right)_{\mu, \beta}=\beta P$$

  Use your results to show that

  $$\mathcal{G}=-k T \log \left(\mathcal{Z} / \mathcal{Z}_{0}\right)$$

  for some constant $\mathcal{Z}_{0}$.

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• A4.18

  Statistical Physics and Cosmology

  Part II, 2003

  Let $g(p)$ be the density of states of a particle in volume $V$ as a function of the magnitude $p$ of the particle's momentum. Explain why $g(p) \propto V p^{2} / h^{3}$, where $h$ is Planck's constant. Write down the Bose-Einstein and Fermi-Dirac distributions for the (average) number $\bar{n}(p)$ of particles of an ideal gas with momentum $p$. Hence write down integrals for the (average) total number $N$ of particles and the (average) total energy $E$ as functions of temperature $T$ and chemical potential $\mu$. Why do $N$ and $E$ also depend on the volume $V ?$

  Electromagnetic radiation in thermal equilibrium can be regarded as a gas of photons. Why are photons "ultra-relativistic" and how is photon momentum $p$ related to the frequency $\nu$ of the radiation? Why does a photon gas have zero chemical potential? Use your formula for $\bar{n}(p)$ to express the energy density $\varepsilon_{\gamma}$ of electromagnetic radiation in the form

  $$\varepsilon_{\gamma}=\int_{0}^{\infty} \epsilon(\nu) d \nu$$

  where $\epsilon(\nu)$ is a function of $\nu$ that you should determine up to a dimensionless multiplicative constant. Show that $\epsilon(\nu)$ is independent of $h$ when $k T \gg h \nu$, where $k$ is Boltzmann's constant. Let $\nu_{\text {peak }}$ be the value of $\nu$ at the maximum of the function $\epsilon(\nu)$; how does $\nu_{\text {peak }}$ depend on $T$ ?

  Let $n_{\gamma}$ be the photon number density at temperature $T$. Show that $n_{\gamma} \propto T^{q}$ for some power $q$, which you should determine. Why is $n_{\gamma}$ unchanged as the volume $V$ is increased quasi-statically? How does $T$ depend on $V$ under these circumstances? Applying your result to the Cosmic Microwave Background Radiation (CMBR), deduce how the temperature $T_{\gamma}$ of the CMBR depends on the scale factor $a$ of the Universe. At a time when $T_{\gamma} \sim 3000 K$, the Universe underwent a transition from an earlier time at which it was opaque to a later time at which it was transparent. Explain briefly the reason for this transition and its relevance to the CMBR.

  An ideal gas of fermions $f$ of mass $m$ is in equilibrium at temperature $T$ and chemical potential $\mu_{f}$ with a gas of its own anti-particles $\bar{f}$ and photons $(\gamma)$. Assuming that chemical equilibrium is maintained by the reaction

  $$f+\bar{f} \leftrightarrow \gamma$$

  determine the chemical potential $\mu_{\bar{f}}$ of the antiparticles. Let $n_{f}$ and $n_{\bar{f}}$ be the number densities of $f$ and $\bar{f}$, respectively. What will their values be for $k T \ll m c^{2}$ if $\mu_{f}=0$ ? Given that $\mu_{f}>0$, but $\mu_{f} \ll k T$, show that

  $$n_{f} \approx n_{0}(T)\left[1+\frac{\mu_{f}}{k T} F\left(m c^{2} / k T\right)\right]$$

  where $n_{0}(T)$ is the fermion number density at zero chemical potential and $F$ is a positive function of the dimensionless ratio $m c^{2} / k T$. What is $F$ when $k T \ll m c^{2}$ ?

  Given that $\mu_{f} \ll k T$, obtain an expression for the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{0}$ in terms of $\mu, T$ and the function $F$. Supposing that $f$ is either a proton or neutron, why should you expect the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{\gamma}$ to remain constant as the Universe expands?

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• A1.16

  Statistical Physics and Cosmology

  Part II, 2004

  (i) Consider a homogeneous and isotropic universe with mass density $\rho(t)$, pressure $P(t)$ and scale factor $a(t)$. As the universe expands its energy $E$ decreases according to the thermodynamic relation $d E=-P d V$ where $V$ is the volume. Deduce the fluid conservation law

  $$\dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+\frac{P}{c^{2}}\right) .$$

  Apply the conservation of total energy (kinetic plus gravitational potential) to a test particle on the edge of a spherical region in this universe to obtain the Friedmann equation

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

  where $k$ is a constant. State clearly any assumptions you have made.

  (ii) Our universe is believed to be flat $(k=0)$ and filled with two major components: pressure-free matter $\left(P_{\mathrm{M}}=0\right)$ and dark energy with equation of state $P_{\mathrm{Q}}=-\rho_{\mathrm{Q}} c^{2}$ where the mass densities today $\left(t=t_{0}\right)$ are given respectively by $\rho_{\mathrm{M} 0}$ and $\rho_{\mathrm{Q} 0}$. Assume that each component independently satisfies the fluid conservation equation to show that the total mass density can be expressed as

  $$\rho(t)=\frac{\rho_{\mathrm{M} 0}}{a^{3}}+\rho_{\mathrm{Q} 0},$$

  where we have set $a\left(t_{0}\right)=1$.

  Now consider the substitution $b=a^{3 / 2}$ in the Friedmann equation to show that the solution for the scale factor can be written in the form

  $$a(t)=\alpha(\sinh \beta t)^{2 / 3}$$

  where $\alpha$ and $\beta$ are constants. Setting $a\left(t_{0}\right)=1$, specify $\alpha$ and $\beta$ in terms of $\rho_{\mathrm{M} 0}, \rho_{\mathrm{Q} 0}$ and $t_{0}$. Show that the scale factor $a(t)$ has the expected behaviour for an Einstein-de Sitter universe at early times $(t \rightarrow 0)$ and that the universe accelerates at late times $(t \rightarrow \infty)$.

  [Hint: Recall that $\int d x / \sqrt{x^{2}+1}=\sinh ^{-1} x$.]

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• A3.14

  Statistical Physics and Cosmology

  Part II, 2004

  (i) In equilibrium, the number density of a non-relativistic particle species is given by

  $$n=g_{\mathrm{s}}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} e^{\left(\mu-m c^{2}\right) / k T}$$

  where $m$ is the mass, $\mu$ is the chemical potential and $g_{\mathrm{s}}$ is the spin degeneracy. At around $t=100$ seconds, deuterium $D$ forms through the nuclear fusion of nonrelativistic protons $p$ and neutrons $n$ via the interaction:

  $$p+n \leftrightarrow D$$

  What is the relationship between the chemical potentials of the three species when they are in chemical equilibrium? Show that the ratio of their number densities can be expressed as

  $$\frac{n_{D}}{n_{n} n_{p}} \approx\left(\frac{h^{2}}{\pi m_{p} k T}\right)^{3 / 2} e^{B_{D} / k T},$$

  where the deuterium binding energy is $B_{D}=\left(m_{n}+m_{p}-m_{D}\right) c^{2}$ and you may take $g_{D}=4$. Now consider the fractional densities $X_{a}=n_{a} / n_{B}$, where $n_{B}$ is the baryon number of the universe, to re-express the ratio above in the form

  $$\frac{X_{D}}{X_{n} X_{p}}$$

  which incorporates the baryon-to-photon ratio $\eta$ of the universe. [You may assume that the photon density is $n_{\gamma}=\frac{16 \pi \zeta(3)}{(h c)^{3}}(k T)^{3}$.] From this expression, explain why deuterium does not form until well below the temperature $k T \approx B_{D}$.

  (ii) The number density $n=N / V$ for a photon gas in equilibrium is given by the formula

  $$n=\frac{8 \pi}{c^{3}} \int_{0}^{\infty} \frac{\nu^{2} d \nu}{e^{h \nu / k T}-1},$$

  where $\nu$ is the photon frequency. By considering the substitution $x=h \nu / k T$, show that the photon number density can be expressed in the form

  $$n=\alpha T^{3}$$

  where the constant $\alpha$ need not be evaluated explicitly.

  State the equation of state for a photon gas and explain why the chemical potential of the photon vanishes. Assuming that the photon energy density $\epsilon=E / V=(4 \sigma / c) T^{4}$, use the first law $d E=T d S-P d V+\mu d N$ to show that the entropy density is given by

  $$s=S / V=\frac{16 \sigma}{3 c} T^{3}$$

  Hence explain why, when photons are in equilibrium at early times in our universe, their temperature varies inversely with the scale factor: $T \propto a^{-1}$.

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• A4.18

  Statistical Physics and Cosmology

  Part II, 2004

  (a) Consider an ideal gas of Fermi particles obeying the Pauli exclusion principle with a set of one-particle energy eigenstates $E_{i}$. Given the probability $p_{i}\left(n_{i}\right)$ at temperature $T$ that there are $n_{i}$ particles in the eigenstate $E_{i}$ :

  $$p_{i}\left(n_{i}\right)=\frac{e^{\left(\mu-E_{i}\right) n_{i} / k T}}{Z_{i}},$$

  determine the appropriate normalization factor $Z_{i}$. Use this to find the average number $\bar{n}_{i}$ of Fermi particles in the eigenstate $E_{i}$.

  Explain briefly why in generalizing these discrete eigenstates to a continuum in momentum space (in the range $p$ to $p+d p$ ) we must multiply by the density of states

  $$g(p) d p=\frac{4 \pi g_{s} V}{h^{3}} p^{2} d p$$

  where $g_{s}$ is the degeneracy of the eigenstates and $V$ is the volume.

  (b) With the energy expressed as a momentum integral

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

  consider the effect of changing the volume $V$ so slowly that the occupation numbers do not change (i.e. particle number $N$ and entropy $S$ remain fixed). Show that the momentum varies as $d p / d V=-p / 3 V$ and so deduce from the first law expression

  $$\left(\frac{\partial E}{\partial V}\right)_{N, S}=-P$$

  that the pressure is given by

  $$P=\frac{1}{3 V} \int_{0}^{\infty} p E^{\prime}(p) \bar{n}(p) d p .$$

  Show that in the non-relativistic limit $P=\frac{2}{3} U / V$ where $U$ is the internal energy, while for ultrarelativistic particles $P=\frac{1}{3} E / V$.

  (c) Now consider a Fermi gas in the limit $T \rightarrow 0$ with all momentum eigenstates filled up to the Fermi momentum $p_{\mathrm{F}}$. Explain why the number density can be written as

  $$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{p_{\mathrm{F}}} p^{2} d p \propto p_{\mathrm{F}}^{3}$$

  From similar expressions for the energy, deduce in both the non-relativistic and ultrarelativistic limits that the pressure may be written as

  $$P \propto n^{\gamma}$$

  where $\gamma$ should be specified in each case.

  (d) Examine the stability of an object of radius $R$ consisting of such a Fermi degenerate gas by comparing the gravitational binding energy with the total kinetic energy. Briefly point out the relevance of these results to white dwarfs and neutron stars.

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