3 April 2012 Ay 127 E.S. PhinneyCosmology and Galaxy FormationProblem Set 1 Due in class, Friday the 13th April 2012Reading: See the on-line syllabus for lecture-by-lecture readings. Before answering thefirst two homework questions, also read the following paper http://arxiv.org/pdf/0707.0380v1.Collaboration policy: See the on-line collaboration policy.Homework Problem:1. MIT is located 1.35 × 10−16Mpc from Caltech. Hubble’s law says it should therefore bereceding from Caltech at a speed of 9.5 × 10−15km s−1= 0.3mm y−1. Is it? If not, why not?2. In general relativity, mass-energy curves space-time. So a space empty of matter and energyshould not be curved. An FRW “expanding universe” in which the density of matter andradiation vanishes (Ω0→ 0) describes an empty space, and so this spacetime (“Milne (1932)spacetime”) should be merely the non-expanding Minkowski metric in unusual moving coor-dinates (in the jargon of GR, such changes of coordinates are called “gauge transformations”).Unmask this by finding that coordinate transformation.Specifying the problem more explicitly: the FRW metric of constant negative curvature isds2= −c2dT2+ a2(T )(dχ2+ sinh2χ(dθ2+ sin2θ dϕ2) (1)(this more elegant form can be obtained by substituting r = sinh χ into Serjeant eq. 1.6when k = −1, and flipping the overall sign. Sterl uses the MTW metric sign convention,where space-like intervals have positive ds2as God and Euclid intended. Serjeant, under theinfluence of some satanic particle physicists, uses the opposite sign for ds2)).a) Show that the dynamical equations (e.g. Serjeant eq. 1.14 or 1.7) for an empty Ω0= 0universe require the Hubble parameter H2∝ a−2and thus a(T ) ∝ T .b) Find the coordinate transformation r(T, χ), t(T, χ) that converts this FRW metric tothe familiar flat Minkowski space (in spherical coordinates):ds2= −c2dt2+ dr2+ r2(dθ2+ sin2θ dϕ2) (2)c) Describe the motion of cosmological Milne observers (observers at rest in the χ, θ, φcoordinates, for which a constant T defines a negatively curved spatial slice) with respectto the Minkowski-space observers.d) You do not need to use the full Riemann/Ricci tensor machinery of general relativity toanswer this: given that mass-energy curves space-time, what is the space-time curvatureof the metric 1, when the (curved) space it describes is empty? Notice that curvature ofspace-time is not the same as the curvature of the spatial slice T = const or t = const.e) Conversely, could a universe with flat spatial slices have a curved space-time?13. Explain how Hubble’s law is consistent with homogeneity and isotropy of the universe. Onwhat scales is the universe homogeneous and isotropic?4. Explain the difference between co-moving distance and physical distance, and their connectionto cosmological redshift.5. Inhomogeneity in the present universe. When computing, adopt the presently preferred Ωm=0.27, Ωb= 0.044, H0= 72km s−1Mpc−1.a) The mass within the sun’s 8kpc orbital radius about the Milky Way is ∼ 8 × 1010M.i. By what factor does the average density within this sphere exceed the mean densityof matter in the universe?ii. But ∼ 70% of mass within the sun’s 8kpc orbital radius is accounted for by thebaryons of stars and gas. By what factor does the average baryon density withinthis sphere exceed the mean density of baryons in the universe? Can you think of areason the baryons would concentrate more than the dark matter?b) The mass of the Milky Way’s dark halo is 2×1012M, and that of the nearest comparablylarge galaxy, the Andromeda galaxy M31 is 1.2× 1012M. M31 is 780 kpc distant, fallingtowards the Milky Way at 116km s−1. By what factor does the average density withina sphere of radius 780 kpc centered between the Milky Way and M31 exceed the meandensity of matter in the