AST5220 – lecture 3 How to compute the CMB power spectrum Hans Kristian Eriksen January 20th, 2010 Overview Question: How do we go from cosmological parameters to a CMB power spectrum? Four main steps: 1. 2. 3. 4. Compute the background cosmology How does the average properties of the universe evolve with time? Compute the ionization history of the universe How many free electrons are there as a function of time? Given some initial structure, compute its evolution until today How do structures grow or decay in time? Given the 3D structures today, project the photon distribution onto a 2D sphere What do the mean CMB fluctuations look like today? Background cosmology Recall the definition of the Friedmann-Robertson-Walker metric for flat space (k=0): ds2 = −c2 dt2 + a2 (t)(dx2 + dy 2 + dz 2 ) This tells us how to measure distances in 4D spacetime (If you’ve never seen this before, don’t worry: More details will be provided next week ☺) a(t) is called the scale factor, and measures the ”size of the universe” as a function of time Today we define this to be 1; a(today) = a0 = 1 We need to know a(t) Cosmological parameters First, we will only consider flat universes in this course Important consequence: Light travels in straight lines The following cosmological parameters will be considered: The Hubble parameter, H0 How fast does the universe expand today? Measured in km/s / Mpc Typical value is 70 km/s / Mpc The dark matter density, Ωm How much dark matter is there, relative to the critical density? Typical value is 0.24 The baryon density, Ωb How much baryonic matter is there, relative to the critical density? Typical value is 0.04 Also A and ns (from inflation), and Ωr and ΩΛ Ωr is fixed by the CMB temperature today (T = 2.73 K) ΩΛ is simply adjusted in order to make the universe flat; Ωm+ Ωb+ Ωr+ ΩΛ = 1 Background evolution Einstein’s equations for a uniform and homogeneous universe lead to the Friedmann equation: H = H0 (Ωm + Ωb )a−3 + Ωr a−4 + ΩΛ where we have defined the Hubble parameter as H= a˙ a = 1 da a dt This is a simple differential equation in a(t) We also define H = aH since this enters into most computations Assignment 1a: Compute H and H as functions of time (or a) Useful time variables We will use four different ”time variables” t = physical time, measured in seconds a(t) = scale factor, relative to today x(t) = log(a(t)) η(t) = ”conformal time” = size of horizon at time t Recall: The horizon is the distance that light may have travelled since the Big Bang. From GR, we know that light travels along null geodesics (ds2 = 0), such that 0 = −c2 dt2 + a2 (t)dx2 To find the total distance light has travelled, we must integrate this from equation from t’=0 to t’=t η= 0 t c dt = a a 0 Assignment 1b: Compute x(t) and η(t) da a2 H Milestone 1: Background cosmology Task: Compute H, H and η as functions of x How? Write a module (”time_mod.f90” in F90) that integrates the Friedmann equation, and stores the resulting functions in lookup tables Write subroutines that can easily get any of these functions at arbitrary times Write a main program driver that initializes the time module, and then outputs each of these functions to file Plot the resulting functions (for instance using xmgrace), and some other interesting functions, and write a short report One of the main purposes of this step is to get the infrastructure (Makefile, libraries, compilers etc.) working, before starting with the harder stuff. Next step: The ionization history Recall: Photons scatter on free electrons by Thompson scattering We need to know the probability for a photon to scatter off an electron as a function of time Quantified by ”optical depth”, τ τ (η) = Here η0 ne σT adη η ne = electron density σT = Thompson cross section (how likely is a single scattering?) The integral goes from a given time until today The optical depth indicates the absorption of a medium Imagine you send a light ray with intensity I0 through a medium −τ The amount of light that survives at optical depth τ is I = I0 e If τ = 0, no absorption occurs If τ >> 1, one is guaranteed absorption How to compute ne? The difficult part is to compute ne as a function of time This is done by solving the Saha equation (high densities) or Peebles equation (intermediate and low densities) Don’t worry: Much more on this when we get there ☺ Saha equation: Xe2 1 ≈ 1 − Xe nb me Tb 2π 3/2 e−ǫ0 /Tb Peebles equation: Cr (Tb ) dXe = [β(Tb )(1 − Xe ) − nH α(2) (Tb )Xe2 ] dx H ne where Xe ≡ nH Auxiliary functions We will not only need τ, but also a few functions derived from this First and second derivatives: dτ ′ τ = dx The visibility function g(x) = −aHτ ′ e−τ and its derivatives g = probability for scattering at time x 2 d τ ′′ τ = 2 dx Milestone 2: Recombination Your task will be to compute τ, τ’, τ’’, g, g’ and g’’ as functions of x How? Write a new module (e.g., rec_mod.f90) that solves the Saha and Peebles’ equations, given a and H from the first project, and Ωb Each function must be interpolated (e.g., splined), to be able to evaluate at arbitrary times There are some numerical issues here and there – these will be explained in detail in the project description After this milestone, we know how the average universe behaves, and how photons scatter in this universe. Next step is to figure out how perturbations behave Gravitational structure formation Gravitational structure formation A quantitative description of the primordial fluid Suppose we simply start out with some random 3D distributions of dark matter (determines the overall gravitational potential) baryons (determines where light comes from) photons (determines what we can see) The fluctuations in each field are quantified in terms a spatially 3D function: Dark matter = δ(x) = (ρm(x) – ρm,0) / ρm,0 Baryons = δb(x) = (ρb(x) – ρb,0) / ρb,0 Photons = Θ(x) = (T(x) – T0 ) / T0 In addition, the dark matter and baryons have velocities, v and vb, and the photon distribution has a momentum, p Our job is to find δ(x), δb(x), Θ(x), v (x), vb(x) as functions of time The perturbed spacetime It is not enough to know how the plasma perturbations behave – we must also know how the perturbed spacetime behave Recall the metric for uniform space: ds2 = −c2 dt2 + a2 (t)δij dxi dxj If small matter perturbations are present, the metric is also perturbed We choose one particular way of perturbing the metric: ds2 = −c2 (1 + 2Ψ)dt2 + a2 (t)δij (1 + 2Φ)dxi dxj Ψ and Φ are scalar functions of both space and time This is called the ”conformal Newtonian gauge” Ψ corresponds to the Newtonian potential at a given time and place Φ corresponds to the local curvature at a given time and place Main point: Spacetime is now no longer uniform – its properties vary in space and time Summary of dynamical quantities Baryon density: Baryon velocity: Dark matter density: Dark matter velocity: Photon distribution: Newtonian potential: Curvature: δb(t, x) vb(t, x) δ(t, x) v(t, x) Θ(t, x, p) Ψ(t, x) Φ(t, x) Need to find the equations that governs the joint evolution of these functions The Boltzmann and Einstein equations Unfortunately, it is impossible to figure out the full evolution of the fluid Instead, we will derive the linearized Boltzmann and Einstein equations: First-order differential equations Note: Don’t worry that this doesn’t make any sense now – we’ll come back to all of this in turn, slow and ”easy” ☺ OK as long as fluctuations are small – and for CMB, they are Good thing: Linear differential equations – straightforward to solve Milestone 3: Evolution Your task is to compute the evolution of all the dynamical variables from a = 10-8 until today How? Write a routine that computes the derivatives listed on the previous slide Insert this into a standard ODE solver (F90 code provided) Solve the equations for a grid in x and k (=Fourier wave number) A few technical issues will arise Very early, the density is very high Equations become unstable Need to make approximations, and solve in two different regimes Tight coupling equations vs. full equations From perturbations to CMB anisotropies Finally, we need to figure out what the CMB anisotropies look like today, from our point of view sc C at M te B rin la g st su rfa ce We use a ”line-of-sight” integration approach + + + + Wave going through the universe Result: We know what the CMB signalfor that singlemode looks like on the sky! Overview of spectrum computation Choose only one single Fourier mode going through the universe Wave number = k Amplitude = 1 (arbitrary normalization) Phase = 0 (arbitrary normalization) Evolve that mode from before recombination until today Linearized Boltzmann and Einstein equations Compute the contribution of that mode to the power spectrum, by the line-of-sight integration method Average over all possible phases Trivial: Simply replace < Φ Φ* > with the Pk assumed set up by inflation Sum up contributions from all k-modes And we’re done! Milestone 4: The CMB spectrum Your task is to compute the expected CMB power spectrum using the results computed for milestone 3 How? Compute a so-called ”source function”, S(k, x) Simply amounts to multiplying previously computed quantities together, but some interpolation is needed Integrate this over x to produce a ”transfer function”, Θl(k) Describes the net contribution from a single Fourier mode to the CMB power spectrum Integrate this over k to produce the final power spectrum, Cl 2 Power spectrum, Cl l(l + 1)/2π (µK ) Finally! 7000 6000 Teoretisk spektrum Observert spektrum 5000 4000 3000 2000 1000 0 10 100 Multipol moment, l 1000 Summary 1. Compute the background evolution of the universe 2. Compute the recombination history of the universe 3. Compute the evolution of perturbations in the universe 4. Compute the power spectrum by projecting these perturbation onto a sphere, and average over Fourier modes