Data Provided: A formula sheet and table of physical constants is attached to this paper. DARK MATTER AND THE UNIVERSE

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1 Data Provided: A formula sheet and table of physical constants is attached to this paper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn Semester ( ) DARK MATTER AND THE UNIVERSE 2 HOURS Answer question ONE (Compulsory) and TWO other questions. All questions are marked out of twenty. The breakdown on the right-hand side of the paper is meant as a guide to the marks that can be obtained from each part. Please clearly indicate the question numbers on which you would like to be examined on the front cover of your answer book. Cross through any work that you do not wish to be examined. 1 TURN OVER

2 COMPULSORY 1 (a) Define the term critical density ρ c in cosmology. State its importance in the study of dark matter and outline a simple derivation for ρ c in terms of the Hubble constant and the gravitational constant G. (b) Explain the term Hot Dark Matter (HDM) and outline the characteristics that distinguish it from Cold Dark Matter (CDM). Give a possible candidate particle for HDM. (c) Outline how liquid xenon can be used as a detection medium for WIMP dark matter. Include two reasons why detectors built using liquid xenon can be efficient at reducing background events. (d) A region of our Galaxy is found to have a total mass to luminosity ratio of η = 10. If the average luminosity in this region is 3.0 x W pc -3, estimate the mass density present in units of M pc -3. (e) Explain briefly why observations of the cosmic microwave background radiation (CMBR) support both the concept of inflation and the existence of non-baryonic dark matter. 2 END

3 2 (a) Like all galaxies our own Milky Way contains both luminous material, mainly stars, and dark matter. Show on a plot of density from the centre of our Galaxy vs. distance the likely distributions of dark matter and luminous matter in the Milky Way. Indicate on this plot the approximate distance of the Sun from the centre of the Milky Way in kpc and the likely density of dark matter at this position in GeV/cm 3. [6] (b) An object orbiting the Galaxy in a circular motion at radial distance a from the centre does so with orbital period P. Write an expression P in terms of a and M, the mass contained within the orbital path. Write an expression for the orbital velocity v in terms of P for such a circular orbit. Hence write an expression for v in terms of M to show that v " a #1/ 2. Assume that the mass is spherically distributed. (c) Write down a value for the orbital velocity of the Sun around the Milky Way in units of km s -1. Hence, using your value for the distance of the Sun from the centre of the! Milky Way given in part (a), show that the enclosed mass within the orbital path is ~10 11 M. (d) Assume that the Sun is at the edge of the luminous part of the galaxy (roughly this is true) and that the matter within the Sun s orbital path is only luminous matter. Observations show that the orbital velocity of objects at radial distances greater than that of the Sun remain constant and approximately equal to the Sun s orbital velocity. Assuming that this observation is caused by the presence of dark matter, estimate the percentage of the galaxy within a radius of 100 kpc that is composed of luminous material. (e) The theory of Modified Newtonian Dynamics (MOND) has been proposed as an explanation for the observed flat rotation curves of galaxies. What would be the answer to part (d) if this theory is correct? State your reasons. [2] 3 TURN OVER

4 3 (a) Astronomical observations of the Universe provide a powerful tool for understanding the particle physics properties of WIMP dark matter and how such particles might be detected in the laboratory. For instance, we observe that galaxies comprise a visible disk of stars embedded in a much larger, roughly spherical, distribution of dark matter. Assuming WIMPs are the dark matter briefly outline how the observed structure of galaxies likely arose. What does the observed distribution of baryonic and non-baryonic matter in galaxies tell us in particular about the properties of WIMPs? (b) WIMPs on Earth are expected to produce nuclear recoils in detectors built using suitable target materials. Explain what particular property of the two components of matter in galaxies opens the possibility of using the direction of nuclear recoils in such an experiment to provide a strong signal for the existence of WIMPs in our Galaxy. (c) Consider a WIMP search detector on Earth using a Ge semiconductor detector. Suppose a WIMP of mass 1.07 "10 #25 kg and initial speed 230 km s -1 collides with a Ge nucleus of mass 1.19 "10 #25 kg in the detector. The WIMP is deflected by angle φ and the Ge nucleus recoils at angle θ with an energy of 10 kev (see the figure below). By using conservation of energy show that the final speed of the Ge nucleus is m s -1! 1.64 "10 5 and of the WIMP is 1.52 "10 5 m s -1.! [5]!! (d) Now by using conservation of momentum and the values from part (c) calculate the recoil angle θ of the Ge nucleus. [5] (e) In practice, although it may be possible to measure the energy of the nuclear recoil in the Ge detector above, it is likely impossible to measure the direction. Briefly explain why this is the case and suggest an alternative means by which this measurement might be feasible. [2] 4 END

5 4 (a) The virial theorem provides a powerful tool for understanding the dynamics of galaxy clusters and hence the nature of the non-luminous dark matter. An interesting application is in the study of merging clusters such as the Bullet Cluster. Briefly explain the importance of recent observations of the Bullet Cluster and similar cluster mergers to our understanding of the nature of dark matter. Explain why observations of such cluster mergers are hard to reconcile with the theory of Modified Newtonian Dynamics (MOND). (b) Write down a simple expression for the virial theorem in terms of the total gravitational potential energy U of a galaxy cluster and the total kinetic energy T. State any significant assumptions required for this equation to be applied to a galaxy cluster. [6] (c) Consider two identical clusters of galaxies, each of mass M, potential energy U i and radius R i. Initially very far away from each other moving at zero relative velocity, they eventually merge into a single bigger cluster and relax. Show that the gravitational potential energy of each cluster at the start is given by U i = αgm 2 /R i, where α is a constant. Also, by using the virial theorem, show that the total energy of the combined system is given by E T = U i. Explain any assumptions. (d) Now, applying the virial theorem to the final merged cluster and assuming that α = 3/5, write an expression for the potential energy of the merged cluster U f in terms of its radius R f. Hence derive a value for the ratio R f /R i relating the radius of the merged cluster to those of the original individual clusters. (e) The result for the radius ratio of part (d) does not in fact apply to the Bullet Cluster. Explain why this is and state qualitatively how the ratio is different. [2] 5 TURN OVER

6 5 (a) Briefly describe the concept of weak gravitational lensing and contrast this with strong gravitational lensing. Include a description of the astronomical phenomena observed and how such observations can be used to study dark matter. [6] (b) Consider a case of so-called strong gravitational lensing in which a galaxy of mass M (the lensing object) is situated at distance D LE from Earth. At distance D SL directly behind this lens is another galaxy (the source) which has light bent to form an Einstein ring. In the case where D SL = D LE = 0.5D SE, where D SE is the total distance from Earth to the background source galaxy, the radius α (in radians) of the Einstein ring can be given by α 2 = 4 MG. c 2 D SE Draw a sketch to illustrate the above scenario. Write a modification to the equation above using D SL and D LE that is suitable for use when D SL does not equal D LE. (c) Now consider the figure below depicting a rare example of a double Einstein ring that results when there are two source galaxies, which we call A and B, each at different distances behind the lensing object. It is found that the lens is 1 Gpc from Earth and that the source galaxy A that causes the outer ring is 2 Gpc from Earth. Use this information and the equation given in (b) to calculate the mass M of the lens in solar masses assuming the radius of the outer ring is 0.2 arcsec. Note that the assumption here is that the outer arcs shown in the image are all components of the same outer ring (d) The inner ring is most likely caused because galaxy B is nearer to the lens than galaxy A. Estimate the radius of the inner ring and hence, using your value for M in (c) calculate the distance between galaxy B and the lens in Gpc. (e) Explain briefly why observation of double Einstein rings such as shown here might provide a particularly powerful tool for observational cosmology. [2] END OF EXAMINATION PAPER 6 END

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