� � Massachusetts Institute of Technology Department of Physics Physics 8.942 Fall 2001 Problem Set #2 Due in class Tuesday, September 25, 2001. 1. Cosmological Redshift as a Doppler Shift For nearby objects with recession speed Hr � c, the first-order Doppler shift gives λo/λe = 1 + Hr/c. To get the redshift factor R ≡ λo/λe for a distant object, we can break the light path into many small segments such that HΔr � c for each, where Δr is the proper length of a segment. Multiplying together the redshift factors for all segments gives the net redshift factor R = R1R2 · · · Rn. a) Using the first-order Doppler shift, show that for one segment, lasting in time from t to t + dt, the redshift factor is 1 da Rt = exp dt . (1) a dt b) Multiply the results from all segments to obtain the usual cosmological red- shift formula λo/λe = ao/ae (where usually one sets ao = 1). 2. Cosmic Time-Redshift-Distance Relation Newspaper articles presenting cosmological discoveries of distant objects usually do not quote redshifts, but instead give the time since the big bang. One such account reported that a galaxy has been found to be fully formed a mere 1 billion years after the big bang. Suppose that you are asked by an astronomer friend what redshift was measured for the galaxy. Additionally, a non-astronomer friend asks you the distance to the galaxy in light-years. After explaining that distance is ambiguous, your friend says alright, just tell me the comoving distance in light-years. Estimate the redshift and the comoving distance using the following three cosmolog-ical models. In each case, assume that the Hubble constant is 72 km s−1 Mpc−1 . a) SCDM: flat Ωcdm = 1, ΩΛ = 0; b) OCDM: open Ωcdm = 0.35, ΩΛ = 0; c) ΛCDM: flat Ωcdm = 0.35, ΩΛ = 0.65. 1Note: you may have to perform a numerical integral. The 8.942 webpage has links to information on using Mathematica, Matlab, Maple, as well as a function (C and F77) to perform numerical integration. 3. Angular Size vs. Redshift An object of linear size L (perpendicular to the line of sight) subtends a small angle α. Suppose that the object and the observer are both comoving and that the redshift of the object is z. For a fixed linear size the angle is a function of redshift: α = α(z). For α2 � 1, α ∝ L. Find the constant of proportionality in terms of τe and τo (the conformal times when light is emitted by the object and received by the observer). For an Einstein-de Sitter universe (k = 0 and a ∝ t2/3), α(z)/L has a minimum at a certain redshift. What is that redshift? If galaxies formed long ago and all have about the same typical size, say 5 kpc, what is the minimum angular size of a galaxy in an Einstein-de Sitter universe, in arcseconds? Assume a Hubble constant of 72 km s−1 Mpc−1 . 4. Number Counts Suppose that there exists a population of luminous sources distributed homogeneously throughout space. For simplicity, assume that all sources have identical proper bolomet-ric luminosity L and that they emit isotropically. (If the sources have a distribution of luminosities we could integrate over that later.) The comoving number density of these sources is n(t), i.e., the proper number density at cosmic time t is (a0/a)3 n(t) where a(t) is the expansion scale factor and a = a0 today. A terrestrial astronomer surveys the entire sky and finds N sources with measured bolometric flux greater than S. Assuming that we live in a Robertson-Walker universe, find N (S). Evaluate it fully in the sim-ple case n(t) = constant and an Einstein-de Sitter universe (k = 0 matter-dominated). Show that as S → ∞, N (S) approximates the Euclidean result N = (4πn/3)(L/4πS)3/2 . Why? In the opposite limit, S → 0, what is the behavior of N (S), and why?