Ay 127 – Homework 4[version 2: corrected typo in expression fo r v in problem #1a]Due: Friday, May 31, 2013Note: This is a technically complicated homework (much harder than what you’ll find o n the final exam).The point is that after working it you should have a much better understanding of how to s olve the types oftime-dependent radiative transfer problems that occur in cosmology, hig h-energy astrophysics, etc.1. Diffusion damping in the CMB – analytical background. [25 points]In this problem and the following problem, you will es timate the scale at which the C MB power sp e c trumbecomes exponentially damped by diffusion of photons (Silk damping). Such diffusion results in a slowdecay o f sound waves in the photon-bar yon plasma. This problem will also give you some practice with theradiation-hydrodynamical methods that are actually use d in CMB codes (albeit with some simplifications).Let’s consider the following idealized problem. Suppose we have a uniform background of photons andbaryons, and turn off gravity, dark matter, neutrinos ...; and furthermore, let’s assume the energy density isdominated by photons, so that the inertia of the baryons can be neglected. The photons have an integratedintensity I(x,ˆn) =R∞0Iν(x,ˆn) dν with units of erg/cm2/s/sr, that depends on position x and direction ofpropagationˆn. Our job is to determine how sound wave s propagate and damp in this plasma.We’ll consider linear perturbation theory in which there is a sing le Fourier mode in the z-direction. Onlylongitudinal waves will be considered (since these are sourced by density perturbations). ThenI(x,ˆn) = I0[1 + 4eikzΘ(µ)], (1)where µ = nzis the cosine of the angle between the photon’s direction of pr opagation and the z-axis. HereΘ(µ) is a small perturbation.1Only in problem #2 will we worry about putting this in the expandingUniverse; until then we will do everything in physical (as opposed to comoving) units. And for this pro blem,you may completely neglect polar ization and the fa c t that Thomson scattering is not isotropic (these changethe answer by only ∼ 10%). This problem should be worked only to linear order in perturbation theory inΘ(µ) and the baryon velocity v.(a) We will consider I(x,ˆn) to represent the intensity in an inertia l frame. Show that if the baryons aremoving with velocity v in the z-direction, then the intensity seen by the baryons isIbary(x,ˆn) = I0h1 + 4eikzΘ(µ) − 4vcµi. (2)Show that the ba ryons experience no net force from radiation pressure ifv =32ceikzZ1−1µΘ(µ) dµ; (3)this will be our expression for their velocity since we are neglecting their inertia (i.e. if we neglect m inF = ma).(b) Suppose the mean free path of the photo ns is λp(due to Thomson scattering: independent offrequency). Show that the general radiative transfer equation gives˙I(x,ˆn) + cµ∂∂zI(x,ˆn) =cλp[Jbary(x) − Ibary(x,ˆn)], (4)where Jbary(x) denotes the angle-average of Ibary. [This is very much like the radia tive transfer equationyou learned about in stellar atmospheres: the main thing you want to explain is why the right-hand side is1The factor of 4 is conventional: it is because we usually think in terms of fractional temperature perturbations, and theintensity scales as T4.1in the ba ryon frame rather than the inertial frame.] Simplify this to give the result˙Θ(µ) + ikcµΘ(µ) =cλp12Z1−1Θ(µ′) dµ′− Θ(µ) +32µZ1−1µ′Θ(µ′) dµ′≡cλpǫ(µ). (5)Let’s define the object in bra ckets here to be ǫ(µ).Also at this point define the integrals Θ0=12R1−1Θ(µ) dµ and Θ1=32R1−1µΘ(µ) dµ. Let’s take the limitof small λp: then ǫ(µ) has to be small, and we may write (with some algebraic rearrangement)Θ(µ) = Θ0+ Θ1µ − ǫ(µ) (6)andǫ(µ) = λp[c−1˙Θ(µ) + ikµΘ(µ)]. (7)These are the “key equations” that we will use to der ive the behavior of sound waves when λpis small.(c) From Eq. (6), show thatR1−1ǫ(µ) dµ = 0 andR1−1µǫ(µ) dµ = 0.(d) Show tha t to first order in λp:Z1−1µ2−13Θ(µ) dµ = −845ikλpΘ1. (8)[You will need both Eq. (6) and (7) for this.](e) Using the result from parts (c) and (d) and Eq. (7), show that˙Θ0= −13ikcΘ1and˙Θ1= −ikcΘ0−415λpk2cΘ1. (9)Show that the “zeroeth moment” of the intensity Θ0thus obeys a second-order linear differential equation.It corresponds to a damped harmonic oscillator. What are its frequency ω and damping rate γ (i.e. solutionsare proportional to e±iωt−γt) as functions of k – ag ain working only to first order on λp? Give a qualita-tive/intuitive explanation for how ω and γ s c ale with k: in particular why these are what you might expectfor a diffusively damped so und wave.2. Diffusion damping in the CMB – cosmological context. [2 0 points]Now it’s time to actua lly apply the machinery of Problem #1 to the CMB.(a) Express the physical mean free path λpof a photon in ter ms of r edshift z, ionization fraction xe, andcosmologic al parameters.(b) Tr anscribe the results from Problem #1e into a damping rate γ in terms of comoving (insteadof physical) k, but with the correct factors of scale factor a included. [You shouldn’t need to re-deriveeverything, just use scaling arguments.](c) Show that the number of e-folds of damping of the amplitude is given byN = σ2Dk2, (10)where σ2D= CRa−2λpdt. Wha t is C? Ex plain why σDis often called the “damping length” o r “dampingscale.”(b) From your solution to HW#2 Problem 1b (o r from the curve given in the lecture no tes), give a roughestimate (a factor of a few is fine) of σD. What is the value of k and hence of multipole ℓ at which the CMBacoustic wave amplitude s hould be suppresse d by one