Relativity and Cosmology

$\fbox{ \parbox{1.5in}{\small PAPER CODE NO.\\ {\bf PHYS374} }}$

THE UNIVERSITY
of LIVERPOOL

MAY 2001 EXAMINATIONS

Degree of Master of Physics : Year 3
Degree of Bachelor of Science : Year 3

RELATIVITY & COSMOLOGY

TIME ALLOWED : Three Hours

INSTRUCTION TO CANDIDATES Answer all questions. Question 1 carries 40% of the total marks.

Questions 2 and 3 each carry 30% of the total marks.

The marks allotted to each part of a question are indicated in square brackets.

In the event of a student answering both parts of an either/or question and not clearly crossing out one answer, only the answer to part (a) of the question will be marked.

You may use the following definition for $\Omega$ in terms of the mass density $\rho$ and Hubble constant H:

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

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Section A

(a) Give 3 reasons why Newton's concept of absolute space provides an unsatisfactory framework for the theory of gravity. In what way is Mach's Principle an attempt to overcome these difficulties? [8]

(b) What is the Equivalence Principle? By careful application of the Equivalence Principle show that light starting out perpendicular to a uniform gravitational field of strength g undergoes a fractional deflection by an amount

$\begin{displaymath} \frac{1}{2} \frac{g d}{c^2},\end{displaymath}$

where d is the perpendicular distance travelled.
[6]

(c) By considering the relative velocity vectors of 3 comoving observers, show that the Hubble law is the only velocity field consistent with the cosmological principle. [5]

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

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

where R is the expansion scale factor and $\rho$ the density. Show that under these conditions the age of the universe in seconds is given by

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

State any assumptions you need to make. [10]

(e) Describe carefully what is meant by the term ``baryon catastrophe'' for clusters of galaxies? [5]

(f) Sketch the general form for the predicted curves of density fluctuation against mass for: (i) A fluctuation spectrum generated by inflation. (ii) A universe with mass dominated by hot dark matter. (iii) A universe with mass dominated by a 30 GeV weakly interacting massive particle. Discuss the origin of the differences in the shapes of curves (ii) and (iii). [6]

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Section B

Answer either (a) or (b)

(a)

(i) List 3 observational results which support an isotropic and homogeneous universe. How is this evidence used in conjunction with the Copernican Principle to arrive at the Cosmological Principle? [7]

(ii) The Robertson-Walker 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, R the scale factor and r the comoving coordinate. If a galaxy is located at a coordinate distance r₁ and the geometry of the universe is Einstein-de Sitter, use this equation to derive expressions for the proper distance d_p and the angular diameter distance d_A of the source in terms of the present scale factor R₀, r₁ and the redshift z. [6]

(iii) The ratio of the angular diameter distance d_A to the luminosity distance d_L of a source at redshift z is given by

$\begin{displaymath} \frac{d_L}{d_A} = (1+z)^2. \end{displaymath}$

Explain the origin of the (1+z)² term in this relation. How can this ratio be used to design an observational test for the cosmological origin of the redshift? [7]

(iv) Use the Robertson-Walker metric given above to show that the frequency of light radially leaving a source at redshift z is reduced by a factor (1+z)^-1 on arrival at a detector located at z=0. How is this frequency shift related to Hubble's law? [10]

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(b)

(i) List 6 pieces of evidence that the universe is of finite age. [6]

(ii) 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. List 3 ways in which the application of General Relativity to obtain this equation provides greater physical insight than a Newtonian treatment. [6]

(iii) By evaluating the equation given in part (ii) at the present time show that

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

[4]
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

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

[9]

(iv) Use the look-back time formula above to show that the age of the universe is $\frac{2}{3}{\rm H}_0^{-1}$ , assuming the universe is Einstein-de Sitter. The surface of last scattering for photons making up the Cosmic Background Radiation occured at $z\simeq1000$ . Estimate how long ago this scattering occurred assuming an Einstein-de Sitter cosmology. Give your answer as a percentage of the age of the universe. [5]

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Answer either (a) or (b)

(a)

(i) State 6 pieces of observational evidence for the existence of large quantities of `dark' or `missing' mass in the universe. [6]

(ii) What major problem of galaxy formation does non-baryonic dark matter help to solve? State the major successes and failures of cold and hot dark matter as theories to explain the origin of large-scale structure. [7]

(iii) Explain carefully what is meant by the term `biasing' as applied to theories of galaxy formation and describe the evidence in its favour. How does changing the bias affect the prediction for the amplitude of galaxy clustering in dark matter models? [10]

(iv) One candidate for the dark matter is massive neutrinos. By first calculating the present photon number density, and then assuming that neutrinos and photons were in thermal equilibrium in the past and therefore have the same number density today, estimate the mass of the neutrino necessary to close the universe. Give your answer to the nearest eV. 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. [7]

PAPER CODE PHYS374page 5 of 6 Continued

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(b)

(i) State any 2 major cosmological observations which the standard Big Bang model explains well. Describe carefully the horizon and flatness problems of the standard model. [6]

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

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

What is the usual physical interpretation of the term $\Lambda$ in this equation? Describe the effect it has on the dynamics and timescale of the universe. [3]

(iii) Show that if certain conditions prevail the universe will undergo a rapid inflationary expansion. Hence show that after the end of an inflationary period it is inevitable that $\Omega\simeq1$ . [7]

(iv) What physical conditions are required in order that the early universe went through a period of inflation? Where does the energy arise to drive inflation? Illustrate your answer with a sketch showing the vacuum potential of the universe immediately before and after inflation. [8]

(v) Describe 3 problems facing standard inflation which currently prevents its universal acceptance. [6]

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