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Gravitational Lensing of the CMB by Tensor Perturbations

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Gravitational Lensing of the CMB by Tensor Perturbations Thesis Presented to the Faculty of Sciences of the University of Geneva for a degree of Master in Physics Gravitational Lensing of the CMB by Tensor Perturbations SUPERVISORS: Prof. Ruth Durrer Dr. Julian Adamek Vittorio Tansella April 2015 Abstract We investigate the properties of Gravitational Lensing by tensor perturbations and the e↵ect on the CMB temperature field. After a first introductory chapter and a second chapter where the formalism of Lensing by density perturbations is introduced as a reference for comparison, in the third chapter we discuss some aspects of lensing by tensor perturbations. The displacement vector of a null geodesic travelling through a tensor perturbed FRW universe is computed. This displacement vector has the unique feature of a curl- component at the linear level and so it is decomposed into its gradient-mode and curl-mode lensing potentials. We compute the power spectra of the two lensing potentials and we get a qualitative understanding of their properties by giving plots for a matter dominated universe. In the last section the modifications induced by this type of lensing to the temperature fluctuations field of the CMB are investigated and we find them to 7 be of the order of 10− , smaller than the cosmic variance of intrinsic CMB anisotropies for every observable multipole. Contents 1 Introduction 2 1.1 HistoricalRemarks.......................................... 2 1.2 TheUnlensedCMB ......................................... 3 1.3 SimpleExample:TheSchwarzschildLens . 5 1.4 LensMapping............................................. 7 2 Lensing by Scalar Perturbations 9 2.1 Perturbed Photon Path and Lensing Potential . 9 2.2 TheCMBLensedPowerSpectrum ................................. 13 3 Lensing by Tensor Perturbations 15 3.1 PerturbedPhotonPath ....................................... 15 3.2 EulerRotationmethod........................................ 16 3.3 Scalar and Vector Lensing Potential . 18 3.3.1 Angular decomposition of and $ ............................. 21 3.4 Curl-mode&Gradient-modePowerSpectra . 24 3.5 TheCMBLensedPowerSpectrum ................................. 26 3.5.1 Full-Sky - Small angle . 27 3.5.2 Flat-Sky - Arbitrary angle . 28 4 Summary 32 Appendices 34 AppendixA-StatisticsoftheCMB ................................ 34 ↵ Appendix B - Γµ⌫ fortensorperturbations............................. 37 AppendixC-Sphericalharmonics ................................. 39 AppendixD-Besselfunctions ................................... 43 AppendixE-LimberApproximation. 45 Appendix F - Total deflection angle power . 46 1 Chapter 1 Introduction 1.1 Historical Remarks Figure 1.1: Extract from Einstein’s notebook showing a rough sketch of a lens system and a few formulae. Image from The Collected Papers of Albert Einstein, Volume 3, http://www.einstein-online.info/. The idea that massive bodies could act upon light through gravity can be traced back long before Ein- stein’s theory of General Relativity. Isaac Newton was the first to have the suspicion that gravity influences the behavior of light, as he states in the first edition of Opticks in 1704. After that the idea was not carried further for almost a century but, in the meantime, through the work of Peter Simon Laplace, another aspect of this interaction was discovered: by computing the escape velocity 2 of light from a massive body, he introduced what today we call Schwarzschild radius RS =2GM/c and anticipated the existence of black-holes. In the XIX century Johann Georg von Soldner published a paper in which he considered the error induced by the deflection of light in the determination of the angular position of stars and, as the small angle limit of the classical mechanic result, he used a deflection angle ↵ = Rs/r for a light ray with impact parameter r. The same result was computed by Einstein in 1911 without using the full formalism of General Relativity, not yet developed. As the times were right for a breakthrough in the field, history got in the way: first, World War I broke out and spoiled Freundlich’s expedition to verify experimentally the deflection of light and second, even when Einstein used the full equations of General Relativity to derive the correct expression of the deflec- tion angle ↵ =2Rs/r, the growing anti-Semitism in Berlin made it hard for him to be taken seriously [1, 2, 3]. Was Arthur Eddington that than took the reins of the subject, not only with his famous expedition to observe the solar eclipse of 29 May 1919, but also as the first to point out that light deflection in the universe can generate multiple images of the same object, introducing the phenomenon of Gravitational Lenses. The newborn field of Gravitational Lensing (GL) grew through the work of Zwicky, who understood the 2 possibility of reconstructing the mass of clusters of galaxies using their lensing properties [4], but it was considered a fairly esoteric field until 1979 when Walsh, Carswell and Weymann discovered and described the first GL candidate: two images of a quasar at z 1.4 lensed by a galaxy at z 0.34 [5] (nowadays hundreds of GL have been detected, see e.g. http://www.cfa.harvard.edu/castles/⇠ ⇠). In the mean time cosmologist were developing cosmological perturbation theory and it was not long until they realized that light propagating through an inhomogeneous universe is lensed and di↵erent from light propagating through an unperturbed Friedmann-Lemaitre universe. This, together with the discovery of the Cosmic Microwave Background in 1964, made GL an important subject for modern cosmology: in fact, as cosmologists realized the importance of CMB anisotropies and as the experimental precision in the measure of these anisotropies grew, it became important to consider the e↵ect of lensing on the CMB. 1.2 The Unlensed CMB Preliminaries The universe is described by a four-dimensional spacetime ( ,g). The Cosmological principle dictates the universe to be isotropic and homogeneous and this means thatM the spacetime admits a slicing into maximally symmetric 3-spaces. There is a preferred coordinate ⌧, the cosmic time1, such that the 3-spaces of constant time ⌃⌧ are indeed maximally symmetric and hence of constant curvature K. The generic form of the metric g is then ds2 = g dxµdx⌫ = d⌧ 2 + a2(⌧)γ dxidxj = a2(t)( dt2 + γ dxidxj) , (1.1) µ⌫ − ij − ij d⌧ where we have introduced the conformal time dt a and γij is the metric of the 3-spaces of constant curvature. Among the many ways to describe γ we⌘ should write it in spherical coordinates since it is the most convenient for this work γ dxidxj = dr2 χ2(r) d✓2 +sin2 ✓dφ2 , (1.2) ij − where r Flat 3-space K = 0, 1 sin( K r) Spherical 3-space K>0 χ(r)=8p K | | (1.3) > | | <> 1 sinh(p K r) Hyperbolic 3-space K<0. p K | | | | > p The symmetry of space time:> allows only an energy-momentum of the form ⇢g 0 (T )= 00 , (1.4) µ⌫ − 0 Pg ✓ ij◆ where ⇢ and P are the energy density and the pressure of the components of the universe. The Einstein’s equations, including the cosmological constant ⇤, are then written a˙ 2 8⇡G a2⇤ + K = a2⇢ + , a 3 3 ✓ ◆ 2 (1.5) a˙ • a˙ 2 + + K = 8⇡Ha2P + a2⇤ , a a − ✓ ◆ ✓ ◆ where = d and H = a˙ is the Hubble parameter. • dt a2 1⌧ is the proper time of an observer who experiences an isotropic and homogeneous universe. 3 The Cosmic Microwave Background A complete treatment of the thermal history of the universe is beyond our purposes and can be found, for example, in [6] or [11]. Here we just give a general overview of recombination and decoupling. At early times in the history of the universe reaction rates for particle interactions were much faster than the expansion rate so that the cosmic plasma was in thermal equilibrium. As the universe expands it also 1 cools adiabatically following T a− . As long as the temperature is above the ionization energy of neutral hydrogen T>1Ry = ∆= 13.6eV/ all hydrogens atoms that forms through e– +p+ H+γ are rapidly dissociated but, at a temperature of about T 4000K 0.4eV, the number density of! photons with energies above ∆drops below the baryon density of the⇠ Universe.⇠ As a consequence e– and p+ begin to (re)combine to neutral hydrogen. This behavior can be described with the Saha equation 2 3/2 xe 45σ me ∆/T = e− (1.6) 1 x 4⇡2 2⇡T − e – ⇣ ⌘ 2 where xe is the e ionization fraction and σ is the entropy per baryon . We can see from Figure 1.2 that for – + T Trec 0.4eV we have xe 1, meaning that far behind recombination epoch e and p are not in form ⇠ ⇠ – of neutral hydrogen, while for T Trec the ionization fraction xe 0 meaning that almost all the e in the universe are bonded in H atoms.⌧ Defining the temperature of recombination⇠ as T T (x =0.5) we find3 rec ⌘ e Trec = 3757K. 1.0 0.8 0.6 2 Wb h =0.01 e x 2 Out[17]= Wb h =0.02 0.4 2 Wb h =0.03 0.2 0.0 0.25 0.30 0.35 0.40 0.45 T eV – 2 Figure 1.2: The e ionization fraction as a function of temperatureH L for di↵erent values of ⌦bh ,togetherwiththelinexe =0.5which is our definition of recombination temperature. Photons and baryons are tightly coupled before recombination by Thompson scattering of electrons. During recombination the free (ionized) e– density drops as in Eq. 1.6 and the mean free path of the photons grows larger than the Hubble scale. Usually one says that when the scattering rate Γof γ on e– falls below the expansion rate of the Universe Γ <H, photons become free to propagate without (almost) any interac- tion. This epoch is called decoupling (zdec 1100). The radiation of free photons with a perfect black-body spectrum (due to thermal equilibrium before⇠ recombination) free streaming after decoupling is the CMB, its 2 sγ 4⇡2 T 3 The entropy per baryon is σ = = ,wheresγ is the photon entropy density and nB is the conserved baryon nB 45 nB number density so that σ is a constant.
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