Cosmology

Liverpool John Moores University
School of Engineering

Module Examination, May 2000

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 5 questions. Answer 3 questions only.

Questions carry 25 marks each. The total number of marks available is 75.

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: JMP Subject leader: CAC

1.

Consider the time dependent evolution of a point mass m located on the inside edge of a sphere of uniform density $\rho(t)$ , and by applying Birkhoff's theorem and the Cosmological Principle, show that for an isotropic and homogeneous universe

$\begin{displaymath}\frac{d^2R(t)}{dt^2}=-\frac{4\pi}{3} G \rho(t) R(t),\end{displaymath}$

where G is the gravitational constant, t is time and R a scale factor.[10]

Hence show that if the density of the universe is dominated by matter then the dynamical equation for the evolution of the universe (known as the Friedmann equation) is,

$\begin{displaymath}\left (\frac{\dot{R}}{R} \right )^2=\frac{8 \pi G \rho_0 R^3_0}{3R^3} - \frac{k}{R^2}.\end{displaymath}$

where the suffix `0' refers to present day. [5]

By considering this equation at the present time, derive an expression for the density of an Einstein-de Sitter universe and calculate its value in kg m^-3. [4]

Explain carefully in what 3 ways an application of General Relativity to obtain the dynamical equation for the evolution of the universe provides greater physical insight than the Newtonian treatment above. [6]

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2.

Explain carefully the concept of comoving coordinates as applied to an expanding universe. What do you understand by the terms (i) proper distance d_p and (ii) angular diameter distance d_A. [6]

The Robertson-Walker metric of space-time (ds), which accounts for the expansion of the universe with time (t), can be written in spherical coordinates as

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

where c is the speed of light, R the scale factor and r the comoving coordinate. If a galaxy is located at co-ordinate distance r₁ from the Earth and the geometry of the universe is Einstein-de Sitter, use this equation to derive expressions for d_p and d_A of the source in terms of the present scale factor R₀, r₁ and the redshift z. [6]

Using the expression

$\begin{displaymath}r_1=\frac{c( zq_0 + (q_0-1)(-1 + (1+2q_0z)^{1/2}))}{R_0 {\rm H}_0 q^2_{0} (1+z)},\end{displaymath}$

where q₀ is the present deceleration parameter, c the speed of light and H₀ Hubble's constant, show that

$\begin{displaymath}d_A=\frac{2c}{{\rm H}_0}\frac{(1-(1+z)^{-1/2})}{(1+z)}\end{displaymath}$

and find the particular redshift z_max at which d_A goes through a maximum. What is the implication of this result for observations of galaxies at $z \geq z_{max}$ ? [10]

If a galaxy in this universe has a diameter = 100 kpc find the angular diameter to the nearest arcsec at z=1.50. Assume 1 radian= $2.063\times10^5$ arcsecs. [3]

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3.

How does the observation that the night sky is dark (Olber's paradox) provide evidence for a universe of a finite age? Write down 5 more pieces of evidence that the universe is of a finite age? [6]

The dynamical equation describing the expansion of the universe at the present time, with zero cosmological constant, can be written as

$\begin{displaymath}\frac{k}{R^2_0} = \rm {H}^2_0(\Omega_0 - 1), \mbox{ with } \Omega_0=\frac{8 \pi G \rho_0}{3\rm {H}^2_0},\end{displaymath}$

where k is the curvature of the metric, R₀ the present scale factor, $\rho_0$ the present mass density and $\rm {H}_0$ the present Hubble constant. Use the general Friedmann equation for the dynamical evolution assuming a matter dominated universe, given by

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

to show that the variation of Hubble's constant with redshift $\rm {H}(z)$ is given by,

$\begin{displaymath}{\rm H}(z)={\rm H}_0 (1+z)[1+z \Omega_0]^{1/2}.\end{displaymath}$

[6]

Hence show that $\Omega(z)$ is given by

$\begin{displaymath}\Omega(z)=\frac{\Omega_0 (1+z)}{(1+z\Omega_0)}.\end{displaymath}$

[5]

Comment on the value of $\Omega(z)$ at high redshift and using the general Friedmann equation, given above, show that this behaviour of $\Omega$ at high z is expected if the universe went through a period of inflation and $R\rightarrow \infty$ . [8]

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4.

State the observational evidence in favour of large quantities of dark matter. [6]

Explain carefully what is meant by the term `biasing' as applied to galaxy formation theories and describe the evidence for biasing from observations of large-scale structure. Which cosmological model seems to require a positive biasing parameter? [9]

Suppose the Milky Way has a non-Hubble peculiar velocity ( $\delta v$ ) caused by a small amplitude large-scale gravitational overdensity $\delta \rho/\rho_0$ (where $\rho_0$ is the present day background mass density) located at a characteristic distance R from us. Assuming constant acceleration over the Hubble time, show that

$\begin{displaymath}\delta v\simeq \frac{1}{2}{\rm H}_0\Omega_0 R \frac{\delta \rho}{\rho_0},\end{displaymath}$

where H₀ and $\Omega_0$ refer to the present-day Hubble constant and mass density respectively. [6]

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

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

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 mean depth of 30 Mpc. If the fluctuation in the number density of galaxies at this depth is 3.0, what is the implied value of $\Omega_0$ for (i) a biasing parameter of 1, (ii) a biasing parameter of 3. [4]

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5.

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

Explain why in the hot big bang model of the universe, effectively no nucleosynthesis occurs until the temperature drops below $3\times10^9$ K. What is the ``deuterium bottleneck'' and how does this affect the build up of heavier elements? What factors govern the neutron/proton ration during the nuceosynthesis era? Explain why helium-4 formation does not simply occur via the reaction

$\begin{displaymath}2n + 2p \rightarrow ^4_2 He.\end{displaymath}$

[9]

The constraint on the baryon density $\Omega_B$ obtained from a comparison between nucleosynthesis predictions and observational data gives

$\begin{displaymath}0.01\leq \Omega_Bh^2 \leq 0.015.\end{displaymath}$

What is the largest contribution to the mass of rich galaxy clusters so far detected? Explain carefully why this discovery together with the prediction from nucleosynthesis poses a problem for the standard Einstein-de Sitter cosmological model? [6]

One possible non-baryonic component of the mass density is massive neutrinos. Assuming that neutrinos and photons were in thermal equilibrium in the past and therefore have the same number density now, by calculating the present photon number density estimate the mass of the neutrino necessary to close the universe. Give your answer to the nearest eV. [6]

[Note: You may assume that the critical density $\rho_c=10^{-26}$ kg m^-3, the present radiation energy density $\rho^{r}_0=10^{-31}$ kg m^-3 and the temperature of the Cosmic Background Radiation is 2.7 K.]

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