Cosmology

Liverpool John Moores University
School of Engineering

Module Examination, May 1999

ENGAS3055 Cosmology

Duration 2 hours

Instructions to candidates

Do not open this question paper until you have been told to do so by the invigilator.

A figure in [ ] denotes the number of marks available for that question or part of question.

There are 6 questions. Answer 4 questions only.

Questions carry 25 marks each. The total number of marks available is 100.

SI units are assumed unless otherwise specified.

You may take $\rm {H}_0=100\,\rm {km}\,\rm {s}^{-1}\,\rm {Mpc}^{-1}$ unless otherwise instructed.

Examiner: CAC Moderator: PAJ Subject leader: CAC

1.

(a)

State (i) The Copernican Principle and (ii) The Cosmological Principle, and explain how (i) leads to (ii). [4]

Give three observational results which support the Cosmological Principle. [3]

For a universe obeying the Cosmological Principle the Proper Distance $d_{\rm p}(t)$ between two galaxies 1 & 2 can be written in the form

$\begin{displaymath}d_{\rm p}(t) = R(t) f(r_1,\theta_1,\phi_1,r_2,\theta_2,\phi_2).\end{displaymath}$

Explain carefully what R and f represent in this equation. [2]

By considering the rate of change in the Proper Distance for comoving galaxies derive Hubble's law. [3]

(b)

The metric of space-time (ds), which accounts for the expansion of the universe with time can be written

$\begin{displaymath}ds^2=c^2dt^2 - R^2(t)\left [ \frac{dr^2}{(1-kr^2)} +r^2 d\theta^2 + r^2{\rm sin}^2\theta d \phi^2 \right ],\end{displaymath}$

where c is the speed of light. Show that the frequency of light radially leaving a source at time $t_{\rm e}$ is reduced by the factor $R_{\rm e}/R_0$ on arrival at a detector at time t₀, where $R_{\rm e}$ and R₀ are the scale factors at time $t_{\rm e}$ and t₀ repectively. [8]

How is this frequency shift related to Hubble's law? [2]

Show that the power (energy/unit time) of the same light signal leaving the source $P_{\rm e}$ and arriving at the detector P₀ are related by

$\begin{displaymath}\frac{P_{\rm e}}{P_0}=(1+z)^2,\end{displaymath}$

where z is the galaxy redshift. [3]

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2.

(a)

The Friedmann equation for the dynamical evolution of the universe can be written in the form

$\begin{displaymath}\dot{R}^2 = \frac{8 \pi G \rho_0 R^3_0}{3 R} - k, \end{displaymath}$

where k is the curvature of the metric, R the scale factor, $\rho$ the mass density and the suffix 0 refers to present day values. By evaluating this equation at the present time show that

$\begin{displaymath}\frac{k}{R^2_0}={\rm H}^2_0(\Omega_0-1),\end{displaymath}$

where

$\begin{displaymath}\Omega_0=\frac{8 \pi G \rho_0}{3{\rm H}^2_0}.\end{displaymath}$

[3]

By explicit substitution of the present-day equation into the general Friedmann equation show that the look-back time (T) to a redshift z is given by

$\begin{displaymath}T=\int_{t_{\rm z}}^{t_0} dt = \frac{1}{{\rm H}_0} \int_{0}^{z} \frac{dz}{(1+z)^2(z\Omega_0+1)^{1/2}}.\end{displaymath}$

[Note: You may use the following relation:

R₀=R(1+z).]

[12]

Hence show that the age of the universe $T_{\rm u}$ is given by: $T_{\rm u}={\rm H}^{-1}_0$ if $\Omega_0=0$ and $T_{\rm u}=\frac{2}{3}{\rm H}^{-1}_0$ if $\Omega_0=1$ . [4]

(b)

Some of the furthest objects known are quasars at a redshift $z\simeq5$ . How long ago was light from these objects emitted in an Einstein-de Sitter cosmology? Give your answer as a fraction of the age of the universe. [3]

In an Einstein-de Sitter universe, what value of the Hubble constant is just consistent with the age of globular clusters, which are believed to be about 12 Gyr old? [3]

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3.

(a)

How does the matter density ( $\rho^{\rm m}$ ) in the universe depend on the expansion scale factor R? What is the dependancy of the radiation energy density ( $\rho^{\rm r}$ ) on R? Explain why it is different to the dependancy of the matter density? [3]

How does the scale factor R vary with time in an Einstein-de Sitter universe? [1]

If the densities of matter and radiation at the present time are taken to be $\rho_0^{\rm m}=10^{-26}$ kg m^-3 and $\rho_0^{\rm r}=4.5\times10^{-31}$ kg m^-3 respectively, calculate the approximate age of the universe when the radiation and energy densities were equal, giving your answer in years. You may assume that the universe is Einstein-de Sitter and that the current age of the universe is $6.7\times10^9$ yrs. [7]

(b)

During the radiation-dominated era the Friedmann equation for the dynamical evolution of the universe can be expressed as

$\begin{displaymath}\dot{R}^2=\frac{8 \pi G \rho R^2}{3}.\end{displaymath}$

Show that under these conditions the age of the universe in seconds is given by the expression

$\begin{displaymath}t=\left ( \frac{3}{32 \pi G \rho} \right ) ^{1/2}.\end{displaymath}$

[10]

Hence estimate the age of the universe when the density was equal to that at the centre of the sun ( $\rho=2\times10^3$ kg m^-3). Aside from expanding, what physical process was going on immediately prior to this epoch? [4]

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4.

(a)

What is the major observational evidence in favour of large quantities of dark matter in (i) galaxies (ii) clusters of galaxies? What are the typical mass-to-optical light ratios of these systems? [5]

What is the largest contribution to the mass of rich galaxy clusters so far directly detected? Explain carefully why this discovery could pose a problem for the standard Einstein-de Sitter cosmological model if clusters formed by gravitational collapse from matter distributed over a large volume of space. [6]

(b)

Why is it inappropriate to probe the dark matter on supercluster-size scales by applying the virial theorem to the velocity of field galaxies? [2]

Suppose the Milky Way has a non-Hubble peculiar velocity ( $\delta v$ ) caused by a small amplitude large-scale gravitational overdensity $\delta \rho/\rho$ (where $\rho$ is the background mass density) located at a characteristic distance R from us. Assuming constant acceleration over the Hubble time, show that

$\begin{displaymath}\delta v\simeq \frac{1}{2}{\rm H}_0\Omega_0 R \frac{\delta \rho}{\rho},\end{displaymath}$

where H₀ and $\Omega_0$ are the present-day Hubble constant and mass density respectively.

[Note: You may use the following relation for the critical density ( $\rho^c$ )

$\begin{displaymath}\rho^c=\frac{3 {\rm H}^2_0}{8 \pi G}.]\end{displaymath}$

[6]

In fact the Milky Way has a non-Hubble peculiar motion of 600 km s^-1 with respect to the cosmic microwave background resulting from the combined gravitational pull of the surrounding galaxies at a depth of 30 Mpc. If the fluctuation in the number density of galaxies at this depth is 2.5 what is the implied value of $\Omega_0$ ? [2]

Explain carefully why this estimate of the mass density is likely to be too small under the assumption that galaxies form at the peaks of the underlying density field. [4]

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5.

(a)

Sketch a logarithmic graph showing the relative abundances by mass of protons, neutrons, deuterium and helium-4 as a function of time after the Big Bang from 10 to 10⁴ secs. Label your curves clearly. [4]

Write down the chain reaction for the production of helium-4. Explain why the abundance of helium-4 is virtually independent of the density while that of deuterium is very sensitive to the density of the universe at nucleosynthesis. [5]

(b)

Describe the terms `bottom-up' and `top-down' in the context of galaxy formation models. What type of fluctuations give rise to these two scenarios in baryonic-dominated cosmologies? [4]

Describe how adiabatic fluctuations are supressed during the radiation dominated era, relative to an initial Harrison-Zeldovich fluctuation spectrum, in a universe dominated by (i) hot non-baryonic dark matter and (ii) cold non-baryonic dark matter. Give an example of a possible dark matter candidate of each of these types. [6]

What major problem of galaxy formation does non-baryonic dark matter help to solve? State the major successes and failures of hot and cold dark matter as theories to explain the origin of large-scale structure. [6]

6.

(a)

Describe the important cosmological observations which the standard Hot Big Bang model explains well. What are the problems with the model which the inflationary picture was designed to solve? [7]

The general equation describing the dynamical evolution of the universe can be written

$\begin{displaymath}\dot{R}^2=\frac{8 \pi G \rho R^2}{3} - k + \frac{\Lambda R^2}{3}.\end{displaymath}$

Describe the physical significance of the term $\Lambda$ in this equation and state the effect it has on the dynamics. Show that if certain conditions prevail the universe will undergo an exponential expansion. [6]

Show that after the end of an inflationary period it is inevitable that $\Omega\simeq 1$ .

[Note: You may use the relation

$\begin{displaymath}\Omega=\frac{8 \pi G \rho}{3 {\rm H}^2}.]\end{displaymath}$

[5]

(b)

Describe the physical process in the early universe which is thought to give rise to a period of inflation. Where does the energy arise from to drive inflation? Illustrate your answer with a sketch showing the vacuum potential of the universe immediately before and after inflation. [7]

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