PHY 16 FINAL EXAM ANSWERS DEC 1 2007 1. Answer the following multiple choice questions and for each one give a brief and clear explanation of your answer. [ 1.0 pt each] 1.1. Lyman-alpha absorbers are clouds of mostly neutral H, believed to be the building blocks of galaxies. The mass fraction of Deuterium in such systems should be: A. Independent of redshift. B. Larger for higher redshift systems. C. Smaller for higher redshift systems. C. Zero, since these are Hydrogen clouds. A. Assuming that these clouds are primordial, the mass fraction of Deuterium is fixed to the mass fraction just after nucleosynthesis was over. 1.2. Consider a closed Universe with no cosmological constant. We observe galaxies after such a Universe has started contracting. What would the Hubble diagram of velocities vs distance look like? Galaxies would appear to have velocities that are: A. Positive (receding) and still proportional to their distance. B. Positive nearby, but negative at larger distances. C. Negative nearby, but positive at larger distances. D. Inversely proportional to their distances. C. Since our case study Universe has started contracting, nearby galaxies will be seen approaching us (having negative velocities), while more distant ones will still appear receding (having positive velocities), since we see them with the velocity that they had in the past, when the Universe was still expanding. 1.. According to the Steady-State cosmology: A. There are galaxies in the Universe that are several times older that our galaxy. B. There was no Big Bang. C. The space density of quasars should look the same at all redshifts. D. All of the above. D. All these things are true for a Steady-State Universe. There is no special time, no special redshift. There are galaxies of all ages.
1.4. Which change in the appearance of a normal star will occur when a massive dark object passes directly between you and the star? A. No change, dark matter cannot affect light. B. Star light dims momentarily because the massive object blocks the light. C. The star brightens temporarily with blue light brightening more that red light. D. The star brightens temporarily, with all colors exhibiting the same behavior. D. Gravitational microlensing will increase the star brightness temporarily. This is an achromatic process, all colors respond the same. A B Some time after recombination C D 1.5 The central panel of the sketch above depicts an area of our Universe some time after recombination. The darker the area the shade of gray, the higher the density. Which of the four graphs A,B,C,D better describes this part of the Universe at a letter time? D. The average (background) density will decrease with time. This eliminates B and C that have the same background density. The thinner ellipse will grow and get less dense, while the darker ellipse will shrink and get denser. So it is D.
2. When you go back home for the break, your uncle Tony, who studied physics some time ago, asks you to describe to him the sequence of important steps that cosmology took during the 20th century. Enumerate here the breakthroughs that, according to you, brought us to our current understanding of the Universe. Do it in a concise and clear manner to keep uncle Tony happy. For each brakthrough, describe briefly and clearly how it shaped our understanding, and what new quantitative results brought in. When applicable, plot the constraints on the Ω m,0 Ω Λ,0 plot. [5 pt] This was an essay type question. You should mention and briefly discuss at least the following points: 1. Olbers paradox: we live in a Universe which is either finite in size or had a beginning. 1. Hubble s law: the Universe is expanding.. CMB detection: Strongly supports the Big Bang theory. 4. Galaxy rotation curves, cluster dispersion velocities and gravitational lensing: most of the matter in the Universe is in the form dark matter. 5. Big Bang Nucleosynthesis shows that the observed light element anbundances require Ω m 0.04. 6. Supernovae distances: the Universe is accelerating, there is a cosmological constant. 7. The power spectrum of the CMB shows that we live in a flat Universe with Ω m = 0., Ω Λ = 0.7. 8. Structure formation models connect the currently observed matter distribution in the Universe to the CMB fluctuations, if the dark matter is mostly cold. 9. Inflation theory solves the horizon, flatness, and lack of magnetic monopole problems.. Jeans length and the growth of density perturbations in the Universe. Consider a sphere of radius R and density ρ. Calculate the acceleration of a test particle at the surface of the sphere and, through this, estimate the time it takes for the sphere to collapse under its own gravity, assuming no other effect takes place. Does the collapse time depend on the radius of the sphere? Under what conditions a pressure built-up can act quickly enough to stop the collapse? Use this to obtain the Jeans length, the minimum size that a disturbance must have in order to collapse. Recalling that the speed of sound in a gas is given by c s = w 1/2 c, where c is the speed of light and w is the dimensionless costant relating pressure to energy density (P = wɛ), compare the Jeans length to the Hubble distance (d H = c/h 0, where
H 0 = (8πG ɛ/c 2 ) 1/2, with ɛ = ρc 2 ). Do you expect density perturbations to grow significantly in a radiation dominated Universe? Explain. Using now the ideal gas law (P = nkt ), and assuming a pure Hydrogen gas, find an expression for w as a function of temperature. What is w for the baryonic matter just after decoupling (kt = 0.26 ev )? Here you will need the proton mass m p c 2 = 940 MeV. Do you expect matter density perturbations to grow significantly after recombination? Explain. [5pt]. The acceleration is: a = GM R 2 = 4πGR ρ, where M = 4πR ρ. (1) Assuming that the acceleration is constant during collapse, the collapse time is ( 2R t dyn = a ) 1/2 =. (2) Apparently, the collapse time does not depend on the radius of the sphere, but only on its density. A pressure built up will propagate with the speed of sound c s, and it will require time t s = R/c s to propagate throughout the sphere. If t s < t dyn, the collapse can be stopped by the pressure built up. This condition is written as The Jeans length can be written as while the Hubble distance as R < λ j = c s. () λ j = cw 1/2, (4) d H = c. (5) 8πG ρ These two lengths are equal for w = 1/4, not very different from w = 1/ that applies to radiation, so we do not expect perturbations to grow in a radiation dominated Universe. For a pure H gas, P = nkt = nm p c 2 (kt/m p c 2 ) = ɛ(kt/m p c 2 ). Therefore, w = kt m p c 2 = 0.26 ev 940 MeV 2.8 10 10. (6)
Just after recombination, the Jeans length will become smaller than the Hubble radius by w 1/2 10 5, and density perturbations will grow rapidly. 4. Imagine that you are blindfolded in the middle of a forest. If the density of trees is n and the average radius of the trunk of the trees is r, how far will you go on the average before you hit a tree? If the fraction of the forest area covered by trees is k 1, write an expression for k as a function of n and r. Express now the average length that you will travel before hitting a tree as a function of k and r. For a given forest coverage k, will you go further in a denser forest with thin trees of in a less dense forest of thicker trees? For k = 0.001, what is the smallest tree radius that, statistically, will permit you to escape from a circular forest of 1 km diameter without hitting a tree? How many trees such a forest has? [5 pt] The average distance you will go is l = 1/(2rn), where 2r is the diameter of a tree. If there are n trees per unit area, and each tree has an area of πr 2, then the coverage of the forest is k = πr 2 n. Solving this for n we have n = k/πr 2. Substituting this in l = 1/2rn, we obtain l = 1 2rn = πr 2k. Thicker (and therefore fewer) trees increase the distance you will travel before you hit a tree. Setting k = 0.001, l = 1 km, r = 2kl/π = 2/π m= 0.64 m, a good size trunk. The density of trees is n = k/πr 2, and in a circular forest of radius l and area πl 2, the number of trees is N = πl 2 n = kl 2 /r 2 = π 2 /4k = 2467 trees.