Massachusetts Institute of Technology Department of Physics Physics 8.942 Cosmic Microwave Background Anisotropy c�2001-2003 Edmund Bertschinger. All rights reserved. 1 Introduction These notes present a simplified computation of cosmic microwave background anisotropy induced by primeval gravitational potential or entropy fluctuations. The basic treatment is equivalent to the calculation of anisotropy on large angular scales presented first by Sachs & Wolfe (1967). The discussion here goes beyond the Sachs-Wolfe treatment to include a discussion of the dominant contributions to anisotropy on small angular scales. The presentation given here is based in part on the PhD thesis of Sergei Bashinsky (Bashinsky 2001) and Bashinsky & Bertschinger (2001, 2002). More elementary treat-ments of CMB anisotropy are given by Chapter 18 of Peacock and online by Wayne Hu at http://background.uchicago.edu/). We adopt several assumptions in order to simplify the algebra without losing the main physical effects: 1. Instantaneous recombination. We assume that, prior to hydrogen recombination at a = ar , photons scatter so frequently with electrons that the photon gas is a perfect gas with spatially-varying energy density ρr and fluid three-velocity vi (orthonormal components of peculiar velocity). This gas is tightly coupled to the “baryons” (electrons plus all ionization states of atomic matter). The gradual decoupling of photons and baryons is approximated by instantaneous decoupling at a = ar . We will indicate how to improve on this treatment by treating the radiation field prior to recombination as an imperfect gas coupled to baryons by Thomson scattering. An accurate treatment of hydrogen and helium recombination is needed in this case. With non-instantaneous recombination, photon polarization must also be considered. We ignore polarization here. The resulting errors are a few percent in rms temperature anisotropy at small angular scales. 2. No large-scale spatial curvature. We suppose that the universe is a perturbed Robertson-Walker spacetime. Although the curvature terms (proportional to 1−Ω) 1will be included in the basic equations, our numerical calculations will not fully include them. This simplifies the harmonic decomposition and is consistent with inflation as the origin of fluctuations. The only important effect we miss is the anisotropy produced by the time-changing gravitational potential at redshift z < 10, the integrated Sachs-Wolfe effect. 3. Adopt a simplified description of matter. Aside from baryons prior to recombina-tion, we will treat the matter as cold. This is appropriate for CDM and for other types of matter on scales larger than the Jeans length (an excellent approximation for CMB anisotropy). The complex gravitational interaction between photons, neutrinos, and dark matter prior to recombination will be treated in a simplified manner. However, we include correctly, without approximation, a possible nonzero cosmological constant (vacuum energy). 4. Neglect gravitational radiation. In the linearized treatment of small-amplitude fluc-tuations, gravitational radiation is a distinct mode and so its effect on the CMB may be treated as a separate contribution computed independently of the density and entropy fluctuations studied here. 5. Assume that all perturbations have small (linear) amplitude. This is valid for “pri-mary” anisotropies but not for “secondary” anisotropies caused by nonlinear struc-tures forming at low redshift. Secondary anisotropies are negligible on angular scales larger than about 10 arcminutes. The justification for these simplifications is mainly pedagogical, although in general they do not introduce serious errors on angular scales larger than a few degrees of arc. (Gravitational radiation, if present, can contribute significantly to the CMB anisotropy on large angular scales. Curvature also has a significant effect on large scales.) Once the student understands CMB anisotropy in this simplified model, a more realistic treatment can be undertaken (Ma & Bertschinger 1995; Seljak & Zaldarriaga 1996; Zaldarriaga, Seljak, & Bertschinger 1998; Hu et al 1998, and references therein). Aside from the simplifications given above and some computational approximations stated later, the treatment given herein is rigorous and complete. We will discuss both the “adiabatic” (i.e. isentropic) and “isocurvature” (entropy) modes, the integrated Sachs-Wolfe effect, and both intrinsic and Doppler anisotropies. The computation of microwave background anisotropy has several ingredients. The rest of these notes give a systematic presentation of these ingredients. In Section 2 we present the Einstein equations for a perturbed Robertson-Walker spacetime. In Section 3 we derive and formally integrate the radiative transfer equation for CMB anisotropy. Section 4 gives a physical interpretation of the primary contributions to CMB anisotropy. These contributions cannot be calculated until the evolution of metric, matter, and radia-tion perturbations is given, as they provide the source terms for CMB anisotropy. Section 2� � 5 presents a simplified set of evolution equations. Section 6 solves these equations on large scales to derive the famous Sachs-Wolfe formula. Section 7 presents a real-space Green’s function approach to solving and understanding the small-scale behavior, espe-cially the acoustic peaks. Section 8 shows how to compute the angular power spectrum from the solution of the radiative transfer equation. Numerical results are presented in Section 9. 2 Perturbed Robertson-Walker Spacetimes This section summarizes the elementary treatment of a weakly perturbed Robertson-Walker spacetime. We write the spacetime line element as 2ds2 = a (τ ) −(1 + 2φ)dτ 2 + (1 − 2ψ)dl2 (1) where dl2 is the usual spatial line element in comoving coordinates (e.g. dl2 = dχ2 + r2dΩ2, or dl2 = dx2 + dy2 + dz2 in Cartesian coordinates if the background space is flat). The metric perturbations are characterized by two functions φ(xi, τ ) and ψ(xi, τ ) which we assume are small (we neglect all terms quadratic in these fields). Equation (1) is a cosmological version of the standard weak-field metric used in linearized general