Phys 133 Introduction to Cosmology FINAL EXAM Spring 2021

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Phys 133

Introduction to Cosmology

Spring 2021

FINAL EXAM

Problem 1 (6 pts): Dark Matter

A redshift survey nds a galaxy cluster, a gravitationally bound system of many galaxies. The total luminosity of all the galaxies is L = 5×1012 Lo , and half the total mass is contained within a radius rh = 1 Mpc. The line-of-sight velocity dispersion of the cluster galaxies is computed from the redshifts of the individual galaxies and found to be σLOS = 750士8 km s-1 , where LOS means line of sight. The gravitational potential energy of the cluster is W = 一0.4GM2 /rh , where M is the total mass of the galaxy cluster.

(a, 3 pts) What is the mass of the galaxy cluster Please provide an answer in units of solar mass, Mo .

(b, 1 pt) To appreciate the large amount of dark matter in the cluster, estimate the stellar mass of the galaxy cluster. Assume a stellar mass-to-light ratio of 4 Mo / Lo . Give your answer in units of solar masses.

(c, 2 pts) Find the minimum timescale for this galaxy cluster to form.

Problem 2 (8 pts): Surface of Last Scattering

Just prior to the time of last scattering, redshift zLS ≈ 1090, the sound horizon in the photon-baryon uid was 0.145 Mpc, and the temperature of the gas was 2970 K. Assume the Benchmark Model, where the horizon distance is 14,000 Mpc, H0 = 68 km s-1 Mpc-1 , r,0 = 9 × 10-5 , m,0 = 0.31, and Λ ,0 = 0.69.

(a, 3 pts) Estimate the average mass within the sound horizon at last scattering, i.e., within a sphere of radius 0.145 Mpc.

(b, 2 pts) How large would this structure be today Or, equivalently, what is the comoving size of the sound horizon at last scattering

(c, 3 pts) Calculate the angular size of the sound horizon on the sky Give your answer in degrees.

Problem 3 (3 pts): Cosmic Nucleosynthesis and the Early Universe

At neutron freeze-out, the ratio of neutrons to protons is 1:5. The time delay until the start of nucleosynthesis is not negligible compared to the decay time of the neutron. By the time nucleosynthesis gets underway, neutron decay has decreased the neutron to proton ratio to 3:20. Find the maximum possible 4 He mass fraction, Ymax = ρHe /ρb , that nucleosynthesis could produce.

Problem 4 (5 pts): In ation and the Very Early Universe

Consider a model for in ation where the exponential expansion starts around the GUT scale, ti ≈ 10-36 s and EGUT ≈ 1012 TeV.

(a, 1 pt) What is the temperature of the Universe when the typical energy of a particle is EGUT

(b, 2 pts) We showed in class that in ation requires a minimum number of e-foldings N ≈ 60 to match the observational limits on spatial curvature. What is the minimum scale factor at the end of in ation a(tf ) Write your answer in terms of the scale factor a(ti ) at the beginning of in ation. What is the temperature of the Universe at the end of in ation relative to its temperature at the beginning of in ation In other words, write T (tf ) in terms of T (ti ).

[Hint: Note that e60 ≈ 1026 . And consider whether the Universe should be hotter or colder following rapid expansion.]

(c, 2 pts) The Hubble parameter at the GUT scale is Hi ≈ ti(-)1 ≈ 1036 s-1 . The cosmological constant during in ation would therefore have an energy scale

∈Λ,i = Hi(2) .

Suppose this vacuum energy were instantly converted to radiation at the end of in ation. What temeprature would the Universe be reheated to

Problem 5 (8 pts): Formation of Structure

Consider an empty, negatively curved, expanding universe. Suppose a dynamically in- signi cant amount of matter ( m << 1) is present in such a universe. Answer the following questions about how the density uctuations δ = (ρ(t) 一 ρˉ(t))/ρˉ(t)) grow with time. (a, 3 pts) The dynamics of this nearly-empty universe are dominated by curvature. Solve Freidmann’s equation to nd an expression for the evolution of the scale factor, a(t). (b, 1 pt) Show that the expansion rate of this universe evolves with time as H(t) = t-1 . (c, 2 pts) Write down a di erential equation describing the growth of the matter uctuations. You should assume the linear regime where δ << 1. (d, 1 pt) Solve the di erential equation for δ(t). (e, 1 pt) Describe the behavior of the overdensity at late times. Will it grow, shrink, or stay the same Problem 6 (3 pts): Galaxy Formation Consider an overdense sphere of gas at turn around, the time when it transitions from expanding to contracting. The gas cloud has an intial energy E = PE0 = 一αGM2 /R0 , where M is the cloud mass and R0 is its initial radius. As the cloud collapses, its potential energy becomes more negative, and its kinetic energy must increase. What is the maximum radius of the gas cloud in virial equilibrium [Note: The maximum radius corresponds to the simpli ed case where no energy is lost from the system. A radiating cloud could collapse to smaller radii.] Problem 7 (3 pts): Acceleration of the Universe In the Benchmark Model, at what scale factor a did = 0 This represents the moment when expansion switched from slowing down to speeding up. What was the scale factor amΛ at which the energy density of matter equaled the energy density of the cosmological constant Did the universe start to accelerate before or after the energy densities in matter and radiation became equal Problem 8 (2 pts): Einstein’s Static Universe Prior to Hubble’s discovery of the expanding universe, Einstein favored a static model where the attractive force of the matter density is exactly balanced by the repulsive force of the cosmological constant. Suppose that the matter in Einstein’s universe could be sponta- neously converted into radiation (by stars, for instance). How would the pressure change Would this universe start to expand or contract Explain your answer. Hint: For Einstein’s universe to remain static, both the velocity a˙ and the acceleration must be equal to zero.