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Leading and Next-To-Leading Gravitational Effects on Cosmic Université de Genève Travail de fin de master Leading and next-to-leading gravitational effects on Cosmic Microwave Background polarisation Superviseur : Mme Pr. Ruth Durrer Miguel Vanvlasselaer Superviseur : M. Dc. Pierre Fleury Promo 2018 Juré : Mme Pr. Ruth Durrer Juré : M. Pr. Antonio Riotto 1st October 2017 — 25th Mai 2018 Contents Before to start .......................................... i 1. Fundamentals ........................................ 1 1.1 Stokes parameters, E and B-modes . .1 1.2 Temperature power versus Polarisation power . .6 1.3 Anisotropies and polarisation from Thomson scattering . .8 1.4 Scalar matter perturbations . 11 1.5 Gravitational waves and inflation . 13 1.6 Approximations . 15 2. Geodesic equation and deflection angle ........................... 17 2.1 Preliminaries . 18 2.1.1 Situation . 18 2.1.2 Metric, canonical observers and Christoffel symbols . 20 2.1.3 Redshift along the path . 21 2.2 First order deviation . 22 2.2.1 First order calculation of the geodesic deviation . 22 2.2.2 Time component . 23 2.2.3 Spatial components . 24 2.2.4 Normalisation of the ni vector along the photon geodesic . 26 2.2.5 Different integration parameters: coordinate time and coordinate distance . 27 2.2.6 Work at fixed Recombination time or fixed affine parameter ? Time delay . 28 2.2.7 The remapping function at first order . 30 2.2.8 Choice of the boundary condition . 32 2.3 Second-order deviation . 34 2.3.1 Second-order calculation . 34 2.3.2 Normalisation at second order . 37 Contents 2 2.3.3 Second-order correction to the unit direction vector . 38 2.3.4 Change of variables . 39 2.3.5 Lensing potential and deflection angle . 40 3. Transport of the photon polarisation ............................. 44 3.1 The parallel transport of the polarisation . 45 3.1.1 Situation . 45 3.1.2 Transport of the potential vector and electric vector along a geodesic . 46 3.1.3 A more handy writing for the parallel transport equation . 49 3.2 Parallel transport of the polarisation at first order . 50 3.2.1 Time deviation of the polarisation . 50 3.2.2 Spatial deviation of the polarisation . 51 3.2.3 Normalisation and orthogonality of the Sachs basis vectors . 51 3.3 Parallel transport of the polarisation at second order . 53 3.3.1 Time component . 53 3.3.2 Spatial components . 53 3.3.3 Normalisation and orthogonality of the basis . 54 3.3.4 Rotation in Poisson gauge and constraints . 56 3.4 Rotation: GLC result versus Poisson gauge result . 58 3.4.1 In the Poisson gauge . 59 3.4.2 In the geodesic light-cone gauge . 60 4. Reconciling Poisson and Geodesic Light-Cone gauge .................... 63 4.1 Introduction to the Lie transport . 63 4.2 First-order calculation of the Lie transport . 65 4.3 Second-order calculation of the Lie transport . 65 4.4 Lie transport in GLC gauge . 69 4.5 B-modes power spectrum in each point of view . 70 4.6 Discussion and conclusion . 72 5. The search for a curl in CMB polarisation .......................... 74 5.1 Introduction of the polarisation . 75 5.2 The apparent rotation induced by the change of position . 78 5.2.1 First-order deviation . 78 5.2.2 Second-order deviation . 79 5.3 The apparent rotation induced by parallel transport . 80 Contents 3 5.3.1 Procedure . 80 5.3.2 Motivation for this procedure . 81 5.3.3 First-order correction from projection and transport . 85 5.3.4 Cross terms between remapping and transport . 85 5.3.5 Second-order correction from projection and transport . 87 5.3.6 The curl coming from the rotation in the magnification matrix . 87 5.4 Summary . 88 5.4.1 First order . 88 5.4.2 Second order . 88 5.4.3 Comments . 89 6. Extraction of the power spectrum of B-modes ........................ 91 6.1 Introduction . 92 6.2 E and B-modes . 92 6.3 Lowest order corrections to B-modes power spectrum . 94 6.3.1 Correction from the remapping . 94 6.3.2 Correction from transport . 96 6.3.3 Cross terms . 98 6.4 Higher order corrections to B-modes power spectrum . 98 6.4.1 Corrections from remapping . 100 6.4.2 Correction from rotation . 103 Conclusions ........................................... 107 Appendix 113 A. Post-Born terms ....................................... 1 B. Lie transport and magnification matrix ........................... 3 C. Geodesics light-cone gauge: a rapid exposition ....................... 5 D. Computation of the curl due to parallel transport and projection .............. 8 D.1 First order . .8 D.2 Cross terms . .9 D.3 Second order . 10 Contents 4 E. Power spectrum and correlation function .......................... 14 F. Correlation functions of the lensing potential in the flat sky approximation ......... 16 Abstract e investigate the weak lensing corrections to the CMB with emphasis on the polarisation. W Starting from fundamentals and working in Poisson gauge, we deduce the technology and the explicit expressions for the lensing and the transport of the polarisation for the leading order and beyond. We then compare the results found in the Poisson gauge with the ones found using the Geodesic Light-Cone gauge and underline a discrepancy involving the vorticity term. The demonstration of the equivalence between the two points of view comes in two steps: we first show that the rotation angle worked out in Poisson gauge has the same mathematical expression than the angle obtained from Geodesic Light-cone gauge means. In the second step, we demonstrate, using only the formalism of E- and B-modes decomposition, that both methods lead to the same B-modes power spectrum. After this satisfactory reconciliation, we turn to the computation of the curl induced by the corrections up to second order, assuming vanishing primordial gravitational waves. Confronted with the problem of the inversion of the Laplacian operator and the lack of observational meaning of those quantities, we then complete the analysis by the computation of the corrections to the B-modes power spectrum at the leading order and beyond. Before to start Notations and preliminary definitions Generally, a dot will denote a derivative with respect to time. For example, dxµ =x ˙ µ. dt The Greek indices, for example µ, will run over the four space-time components: µ = 0, 1, 2, 3, while the Latin letters will run over the three spatial components of space-time slices: i = 1, 2, 3. The first letters in the alphabet will be used from time to time to span 2-dimensional hypersurfaces (for example two angles): a, b, c, d = 1, 2. The vector kµ will denote the components of the four-impulsion of the photon, defined by dxµ kµ = . (0.1) dλ The notation k, without indices, will denote the four-dimensional geometrical object, while a bold letter, n, will denote a three-dimensional vector (with space-like components). We will mainly use ω for the frequency of the photon at the emission.A δ will denote a first order correction induced by perturbative gravitational effects, evaluated at the same value of the parameter λ: for example δki(λ) = ki(λ) − ki(λ) . perturbed up to first order unperturbed Similarly, δ(n) will denote the nth correction. The gravitational (that we will call Weyl) field, in Poisson gauge, will be written φ(λ, n). To simplify a bit our notations in the following, we will also introduce the integrated fields λ Z 0 1 Φ(λ, n) = φ(x(λ0))dλ0, (0.2) λ0 λ and λ Z 1 Ψ(λ, n) = φ(x(λ0))dλ0, (0.3) λ0 0 and use often the fact that the derivatives commute with the integrations along λ, up to the appearance 1 of a factor λ which will be explained soon. This will prove very convenient in the following. Before to start ii The parameter λ will be defined to be zero at the observer and to be λ0 at the emitter, the last scattering surface. Therefore it runs in the opposite direction with respect to the physical photon. It does not really matter, because we can imagine a virtual photon starting at the observer and going back to the point of emission. It will be useful to decompose the four-impulsion vector into its spatial and time parts kµ = ω(uµ + nµ), (0.4) where uµ = (u0, 0) is the four velocity of a static observer and nµ = (0, n) is the “unit direction” four-vector. Notice that the direction of the vector n is from the observer to the source. At this stage, it is also important to fix the dimension of the different quantities we will work with. The k vector has the dimension of the inverse of a length, [k] = L−1. (0.5) ω has the same dimension while u and n are defined to be dimensionless unit vector: [n] = 1, [u] = 1, [ω] = L−1, (0.6) and [xµ] = 1. (0.7) As a consequence, (0.1) forces us to take [λ] = L. (0.8) We will also decompose the spatial derivatives in transverse and parallel parts with respect to n. Explicitly, we will use the notation ∂i = ∇⊥i + ∇ki, (0.9) where the ∇⊥i, the angular derivative, is understood to be written in the spherical basis, (t, r, θ, φ), where it takes the form 1 ∂ 1 ∂ ! ∇i = 0, 0, , . ⊥ r ∂θ r sin θ ∂φ 1 The factor r will give rise to maybe surprising effects when we will exchange integrations and derivatives. R Indeed, if we take as an example the θ component and apply it on the integral R φ(r, θ, φ)dr, we 0 have R R R R Z 1 ∂ Z 1 Z ∂ 1 Z ∇θ φ(r, θ, φ)dr = φ(r, θ, φ)dr = φ(r, θ, φ)dr = r∇θ φ(r, θ, φ)dr.
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