PHYSICS 432/532: Cosmology Midterm Exam Solution Key (2018) 1. [40 points] Short answer (8 points each)

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1 PHYSICS 432/532: Cosmology Midterm Exam Solution Key (2018) 1. [40 points] Short answer (8 points eah) (a) A galaxy is observed with a redshift of How far away is the galaxy, and what is its lookbak time? This is meant to simply be a low redshift alulation, so we an ignore whih kind of distane we re talking about here. The lookbak time is simply: where I used the relation 1p=3.26 ly. d = z = (3 105 km/s)(0.02) = 60h 1 Mp 100h km/s/mp t lookbak = d = 195.6h 1 My (b) Consider a galaxy halo with a density of the form ρ r n. i. Compute the oeffiient, s, in the rotation urve v r s for suh a galaxy. Note that: M R = 4πr 2 ρ(r)dr R n+3 and so v 2 = GM R R s = n Rn+2 ii. (Required for Grad, +2 EC for Undergrad) Now suppose the galaxy has a onstant density out to some fixed radius, R, and has zero density outside. Sketh the rotation urve. Inside R, n = 0, so, s = 1 (linear inrease). Outside, we have a Keplerian potential, so v r 1/2. And, of ourse, the two urves must onnet. Thus: () Beause I don t want you to spend too muh time on unit onversion, I ll let you know that the Einstein radius for a galaxy of mass h 1 M at a redshift of z = 0.1 is about 5.2. Knowing that, what is the approximate Einstein radius (to within a fator of 3) of a massive galaxy luster at a redshift of z = 0.3? Note: To answer this, you need to have some reasonable idea of what the mass of a luster is.

2 First note that the Einstein radius is proportional to: M θ E A very massive galaxy luster might be a few h 1 M. So, taking 3=few (whih should inlude all possible reasonable answers on your part), and noting that the distane inreases by a fator of 10 (roughly) ompared to an individual galaxy, we find that: 300 θ E,luster = 3 θ E,gal = 52. You should find something in the range of a few tens of ar seonds to a few minutes. (d) Please trae the following blank figure into your blue books. D d Please mark the following: i. Draw a large dot,, to indiate the onordane model. ii. Draw a dashed line to indiate all flat universes. On either side, indiate whether the universes are open or losed. iii. Draw a solid line separating those universes whih are aelerating from those whih are deelerating. iv. (Required for Grad, +1 EC for undergrad) I remind you that the age of the onordane model is (to within a few perent) 1/. Knowing that, draw a line representing (approximately) all of the models with age t 0 = 1. Indiate whih side of the line orrespond to older universe, and whih orrespond to the younger ones. Hint: You may need to think of a universe whih also has an age equal to a Hubble time.

3 Flat models have Ω M + Ω Λ = 1. The line separating aelerating from deelerating universes is given by: Ω Λ = 2Ω M The onordane model orresponds to Ω M 0.3, Ω Λ 0.7, and onneting that and the empty universe t 0 = 1 gives the lous of onstant ages. To the left, we have older universes, and to the right, with have younger. E.g. the desitter Universe (Ω Λ = 1) is infinite, while E-dS is 2/3. (e) To within a fator of 2, please give the urrent estimates of Ω B and desribe (in a sentene or two) why observationally at least some of the matter in the universe needs to be non-baryoni. Any of the following are aeptable: Ω B 0.03 The rotation urves of galaxies ontain matter well beyond the luminous part. Cluster X-ray measurements suggest that the gas mass (from luminosity) is insuffiient to aount for the high temperature (whih probes the potential). Cluster dynamis suggest missing mass (via the virial theorem). Lensing systems like the bullet luster indiate that total mass doesn t follow the gas. 2. [30 points] Consider a strange universe filled with Pressureonium, a substane with an equation of state of w P = 1, suh that Ω P = 1 (and with nothing else). The Hubble onstant may be simply expressed as. (a) If the universe doubles in sale fator, by what fator will the density of Pressureonium inrease or derease? For w = 1, ρ a 3(1+w) = a 6 = 1 64 (b) In units of the Hubble time, how old is this universe?

4 We ve done this many times: dt = 1 da a () How does the expansion fator sale with time? Following the previous derivation, t a 3 or = 1 da a = 1 a 2 da Ω X a n 1 a 6 t = 1 1 a 2 da = a t 1/3 (d) What is the horizon sale of this universe? We an do almost the exat same thing but with an extra fator of a downstairs: da χ = = = a 2 2 χ hor = 2 a 2 Ω X a n 3. [30 points] Consider an extremely overdense universe omposed entirely of matter with Ω M = 5. (a) What is the shape and (if not flat) the radius of urvature of this universe? (Express your answer in terms of the Hubble sale.) First, note that: Ω K = 4 a da Thus, the universe is losed. So the radius of urvature is: R 0 = ΩK = 2 (b) At what expansion fator will it ollapse? If never, please justify. The Friedman equation inludes: ( ΩM a 3 + Ω ) K a 2

5 so we reah a turnaround point when: a max = Ω M Ω K = 1.25 () Now onsider an observation of a galaxy with a radius of R = 1h 1 Mp at a redshift of 1. What is the angular size of the galaxy? To assist you, I will note that the omoving distane is First, note that: so D A = S k(χ) 1 + z = R 0 sin(χ/r 0 ) (1 + z) = θ = R D A = rad = 368 (d) What is the ratio of surfae brightness (flux/angular area) of the galaxy as seen at z = 1, ompared to how it would look loally? Hint: I promise you that this is a 1-line alulation. You may get the right answer another way, but you re making life diffiult for yourself. We re looking for: b = f θ 2 D2 A D 2 L The surfae brightness dereases by a fator of 16. = (1 + z) 4 = 1 16