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Cosmological Perturbation Theory, Part 1 Hannu Kurki-Suonio 17.3.2020 About These Lecture Notes

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Cosmological Perturbation Theory, Part 1 Hannu Kurki-Suonio 17.3.2020 About These Lecture Notes Cosmological Perturbation Theory, part 1 Hannu Kurki-Suonio 17.3.2020 About these lecture notes: I lectured a course on cosmological perturbation theory at the University of Helsinki in the spring of 2003, in the fall of 2010, in the fall of 2015; and now I am lecturing it in spring 2020. These notes are modified from the 2015 notes for 2020; work in progress. This course cover similar topics as Cosmology II: structure formation, inflation, and a bit of CMB; but on a deeper level and more widely. In Cosmology II we accepted many results as given; now they are derived rigorously from general relativity. Topics not covered: Quantum field theory is not used, except for a brief discussion of gen- eration of primordial perturbations during inflation in Section 6.6. Non-linear effects or higher- order perturbation theory are not covered. Open and closed cosmologies are not covered, i.e., the background (unperturbed) spacetime is assumed flat. Of CMB we discuss only the Sachs–Wolfe effect, in Section 26. I have lectured a couple of times a separate course on CMB Physics, but it is not currently in the teaching program. There is an unfortunate variety in the notation employed by various authors. My notation is the result of first learning perturbation theory from Mukhanov, Feldman, and Brandenberger (Phys. Rep. 215, 213 (1992)) and then from Liddle and Lyth (Cosmological Inflation and Large-Scale Structure, Cambridge University Press 2000, Chapters 14 and 15), and represents a compromise between their notations. 1 Contents 1 Perturbative General Relativity 1 2 The Background Universe 2 3 The Perturbed Universe 4 4 Gauge Transformations 5 4.1 Gauge Transformation of the Metric Perturbations . ............ 8 4.2 Perturbation theory as gauge theory . ........ 10 5 Separation into Scalar, Vector, and Tensor Perturbations 10 6 Perturbations in Fourier Space 13 6.1 Gauge Transformation in Fourier Space . ........ 15 6.2 About the Fourier Convention . ...... 16 7 Scalar Perturbations 17 7.1 BardeenPotentials ............................... 18 8 Conformal–Newtonian Gauge 19 8.1 Perturbation in the Curvature Tensors . ......... 19 9 Perturbation in the Energy Tensor 21 9.1 Separation into Scalar, Vector, and Tensor Parts . ............ 23 9.2 Gauge Transformation of the Energy Tensor Perturbations............. 24 9.2.1 GeneralRule .................................. 24 9.2.2 ScalarPerturbations . 24 9.2.3 Conformal-Newtonian Gauge . 24 9.3 Scalar Perturbations in the Conformal-Newtonian Gauge .............. 24 10 Field Equations for Scalar Perturbations in the Newtonian Gauge 25 11 Energy-Momentum Continuity Equations 26 12 Perfect Fluid Scalar Perturbations in the Newtonian Gauge 27 12.1 FieldEquations................................. 27 12.2 Adiabaticperturbations . ....... 28 12.3 FluidEquations................................. 29 12.4 Adiabatic perturbations at superhorizon scales . .............. 29 13 Scalar Perturbations for a Barotropic Perfect Fluid 29 14 Scalar Perturbations in the Matter-Dominated Universe 30 15 Scalar Perturbations in the Radiation-Dominated Universe 33 16 Other Gauges 35 16.1 SlicingandThreading . ..... 35 16.2ComovingGauge .................................. 36 16.3MixingGauges ................................... 38 16.4 ComovingCurvaturePerturbation . ........ 38 2 16.5 Perfect Fluid Scalar Perturbations, Again . ............ 40 16.5.1 Adiabatic Perfect Fluid Perturbations at Superhorizon Scales . 41 16.6 UniformEnergyDensityGauge . ...... 41 16.7 SpatiallyFlatGauge . ..... 42 17 Synchronous Gauge 43 18 Fluid Components 47 18.1 DivisionintoComponents . ...... 47 18.2 Gauge Transformations and Entropy Perturbations . ............. 48 18.3Equations ...................................... 49 19 Simplified Matter+Radiation Universe 51 19.1 Background solution for radiation+matter . ............ 51 19.2Perturbations .................................. 52 19.3InitialEpoch ................................... 55 19.3.1 Adiabaticmodes ............................... 55 19.3.2 IsocurvatureModes . 56 19.3.3 OtherPerturbations . 57 19.4 Full evolution for large scales . ......... 58 19.5 Initial Conditions in Terms of Conformal Time . ........... 59 20 Effect of a Homogeneous Component 62 20.1 GeneralConsiderations. ....... 62 20.2 Λ+PerturbedComponent . 64 20.3Λ+PerfectFluid................................. 65 20.4Λ+Matter ....................................... 66 21 The Real Universe 67 22 Primordial Era 69 22.1 NeutrinoAdiabaticity . ...... 71 22.2Matter......................................... 72 22.2.1 The Completely Adiabatic Solution . ...... 73 22.2.2 Baryon and CDM Entropy Perturbations . ..... 73 22.3 Neutrinoperturbations. ....... 74 22.3.1 Neutrinos in the early universe . ...... 75 22.3.2 Primordial neutrino density isocurvature mode . ........... 75 22.4 Tight-Coupling Approximation . ........ 77 22.5 Real-Universe Initial Conditions as Expansion in ConformalTime . .. .. 77 23 Superhorizon Evolution and Relation to Original Perturbations 82 24 Gaussian Initial Conditions 83 25 Large Scales 87 25.1 Superhorizonevolution. ....... 88 25.1.1 Adiabaticmode................................ 88 25.1.2 Isocurvaturemodes. 89 25.2 Throughthehorizon .............................. 89 3 26 Sachs–Wolfe Effect 91 26.1 Ordinary Sachs-Wolfe Effect and Primordial Correlations ............. 94 27 Approximate Treatment of the Smaller Scales 95 27.1 Small Scales during Radiation-Dominated Epoch . ............ 96 27.1.1 Radiation .................................... 97 27.1.2 Matter...................................... 98 27.2 Small Scales to the Matter-Dominated Epoch . .......... 100 27.3 TransferFunction. .. .. .. .. .. .. .. .. 101 28 Acoustic Oscillations of the Baryon-Photon Fluid 105 28.1 Insidethehorizon ............................... 106 29 Modified Gravity 108 A General Perturbation 111 B Rules to Convert between Different Time Coordinates 114 C Equations 115 4 1 PERTURBATIVE GENERAL RELATIVITY 1 1 Perturbative General Relativity In the perturbation theory of general relativity one considers a spacetime, the perturbed space- time, that is close to a simple, symmetric, spacetime, the background spacetime, that we already know. In the development of perturbation theory we keep referring to these two different space- times (see Fig. 1). Figure 1: The background spacetime and the perturbed spacetime. This means that there exists a coordinate system on the perturbed spacetime, where its metric can be written as gµν =g ¯µν + δgµν , (1.1) whereg ¯µν is the metric of the background spacetime (we shall refer to the background quantities 1 with the overbar ), and δgµν is small. We also require that the first and second partial derivatives, 2 δgµν,ρ and δgµν,ρσ are small . The curvature tensors and the energy tensor of the perturbed spacetime can then be written as µ ¯µ µ Gν = Gν + δGν (1.2) µ ¯µ µ Tν = Tν + δTν , (1.3) µ µ where δGν and δTν are small. Subtracting the Einstein equations of the two spacetimes, µ µ ¯µ ¯µ Gν = 8πGTν and Gν = 8πGTν , (1.4) from each other, we get the field equation for the perturbations µ µ δGν = 8πGδTν . (1.5) The above discussion requires a pointwise correspondence between the two spacetimes, so that we can perform the comparisons and the subtractions. This correspondence is given by the coordinate system (x0,x1,x2,x3): the point P¯ in the background spacetime and the point P in the perturbed spacetime which have the same coordinate values, correspond to each other. Now, given a coordinate system on the background spacetime, there exist many coordinate systems, 1Beware: the overbars will be dropped eventually. 2Actually it is not always necessary to require the second derivatives of the metric perturbation to be small; but then our development would require more care. 2 THE BACKGROUND UNIVERSE 2 all close to each other, for the perturbed spacetime, for which (1.1) holds. The choice among these coordinate systems is called the gauge choice, to be discussed a little later. In first-order (or linear) perturbation theory, we drop all terms from our equations which contain products of the small quantities δgµν , δgµν,ρ and δgµν,ρσ . The field equation (1.5) becomes then a linear differential equation for δgµν , making things much easier than in full GR. In second-order perturbation theory, one keeps also those terms with a product of two (but no more) small quantities. In these lectures we only discuss first-order perturbation theory. The simplest case is the one where the background is the Minkowski space. Theng ¯µν = ηµν , µ µ and G¯ν = T¯ν = 0. In cosmological perturbation theory the background spacetime is the Friedmann–Robertson– Walker universe. Now the background spacetime is curved, and is not empty. While it is homogeneous and isotropic, it is time-dependent. In these lectures we shall only consider the case where the background is the flat FRW universe. This case is much simpler than the open and closed ones3, since now the t = const time slices (“space at time t”) have Euclidean geometry. This will allow us to do 3-dimensional Fourier transformations in space. The separation into the background and perturbation is always done so that the spatial average of the perturbation is zero, i.e., the background value (at time t) is the spatial average of the full quantity over the time slice t = const. 2 The Background Universe Our background spacetime is the flat Friedmann–Robertson–Walker universe FRW(0). The background metric in comoving coordinates (t,x,y,z) is ds2 =g ¯ dxµdxν = dt2 + a2(t)δ dxidxj = dt2 + a2(t)(dx2 +
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