Robertson-Walker metric In 1930 Robertson and Walker independently showed that the most general metric possible for describing an expanding universe is dr 2 � d θ 2 + sin 2 θ d φ 2 �� ds 2 = ( c dt ) 2 − a 2 ( t ) 1 − kr 2 + r 2 � where ( r , θ, φ ) are spherical comoving coordinates and a ( t ) is the scale factor Spatial geometry is that of a constant curvature: k = 0 flat geometry universe k = − 1 open universe k = + 1 closed universe J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 18 / 131
Horizon Distance travelled by a photon in the whole lifetime of the universe defines the horizon For photons ds =0, so we have that � t dt ′ D horizon ( t ) = a ( t ) = η ( t ) 0 η ( t ) is also called the conformal time Two points in the universe are in casual contact if their distance is smaller than the horizon Horizon problem: why is the universe isotropic and homogeneous on large scales ? The observable universe is today larger than the horizon J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 19 / 131
Redshift Wavelength of light stretches with the scale factor Given a physical rest wavelength at emission λ 0 , the observed wavelength today λ is 1 λ = a ( t ) λ 0 ≡ ( 1 + z ) λ 0 Interpreting the redshift as a Doppler shift, objects recede in an expanding universe Today z = 0 and it increases back on time J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 20 / 131
Deceleration parameter and elapsed time The deceleration parameter q 0 is defined by the series � � 1 + H 0 ( t − t 0 ) − 1 0 q 0 ( t − t 0 ) 2 + . . . 2 H 2 a ( t ) = a ( t 0 ) Taylor expanding a(t) we obtain q 0 = − ¨ a ( t 0 ) a ( t 0 ) ˙ a ( t 0 ) From above we deduce 1 + q 0 0 ( t 0 − t ) 2 � � 1 + z = 1 + H 0 ( t − t 0 ) + H 2 + . . . 2 and inverting t 0 − t = 1 1 + q 0 � z − z 2 � � � + . . . H 0 2 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 21 / 131
Cosmological distances Proper distance, time for a photon to go from z to z + dz d pr = − cdt = − cda ˙ a Comobile distance between observer at z and emitter at z + dz d com = − cdt a = − cda ˙ aa Luminosity distance, d L such that the observed flux, ℓ , of a source of L absolute luminosity L is ℓ = L , 4 π d 2 � � d L = c z + 1 2 ( 1 − q 0 ) z 2 + . . . H 0 Diameter angular distance, relates angular size ∆ θ and physical size, D of a source d A = D d L ∆ θ = ( 1 + z ) 2 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 22 / 131
Cosmological distances J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 23 / 131
Cosmic Distance Ladder Cepheids Parallax: Hipparcos 0-300 pc (GAIA 5 kpc) Cepheids: 100 pc - 20 Mpc (HST) Type Ia SNe: 20 - 400 Mpc J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 24 / 131
Friedmann-Lemaitre equations Apply the Einstein field equations to the R-W metric G µν = − 8 π G T µν From the LHS we obtain �� ˙ � � 2 0 = − 3 a + 1 G 0 a 2 R 2 a � ˙ � � � 2 j = − 1 2 ¨ a a + 1 G i a − a 2 R 2 a for the RHS isotropy demands that T 0 0 = ρ T i j = − p δ i j where ρ is the energy density and p the pressure J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 25 / 131
Dynamics of the universe Finally the FL equations stand � ˙ � 2 a + 1 R 2 = 8 π G 3 a 2 ρ a � ˙ � 2 2 ¨ a a + 1 R 2 = − 8 π Ga 2 p a − a and can be combined into a single one � ˙ � 2 3 a 2 ( ρ + 3 p ) = ad 2 a ¨ = − 4 π G a a a − dt 2 a J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 26 / 131
Curvature and critical density The first FL equation can be written as � ˙ � 2 = 8 π G ( ρ + ρ k ) ≡ 8 π G a H 2 ( a ) ≡ 3 ρ c a 3 ρ c is the critical system and its value today is ρ c ( z = 0 ) = 3 H 2 8 π G = 1 . 8788 × 10 − 29 h 2 g cm − 3 0 Curvature as an effective energy density component 3 ρ K = − 8 π Ga 2 R 2 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 27 / 131
Total energy density Energy density today can be given as a fraction of critical density ρ Ω tot ≡ ρ c ( z = 0 ) Note that physical energy density is ∝ Ω h 2 (g cm − 3 ) Likewise the radius of curvature is given by 1 Ω tot − 1 ) − 1 � Ω K = ( 1 − Ω tot ) = 0 R 2 → R = ( H 0 H 2 Ω value defines universe geometry Ω tot = 1, flat universe Ω tot > 1, positively curved Ω tot < 1, negatively curved J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 28 / 131
Experimental astroparticle physics & cosmology L. 2, Section 2: Λ -CDM model J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 29 / 131
The multi-component universe We define the equation of state as p = w ρ Universe consists of multiple components: NR matter ρ m = mn m ∝ a − 3 , w m = 0 1 R radiation ρ r = En r ∝ ν n r ∝ a − 4 , w r = 1 / 3 2 curvature ρ k ∝ a − 2 , w r = − 1 / 3 3 (cosmological) constant energy density ρ Λ ∝ a 0 , w Λ = − 1 4 total energy density summed over all components � � i Ω i a − 3 ( 1 + w i ) ρ ( a ) = i ρ i ( a ) = ρ c ( a = 1 ; z = 0 ) density evolves as � i Ω i exp − � d log a 3 ( 1 + w i ) ρ ( a ) = ρ c ( a = 1 ) and the Hubble constant as 0 exp − � d log a 3 ( 1 + w i ) H 2 ( a ) = H 2 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 30 / 131
General solutions of FL equations Radiation domination H 2 ∝ a − 4 , a ( t ) ∝ t 1 / 2 , H ( t ) = 1 2 t , R H = 2 ct Matter domination H 2 ∝ a − 3 , a ( t ) ∝ t 2 / 3 , H ( t ) = 2 3 t , R H = 3 2 ct Curvature domination k < 0 H 2 ∝ a − 2 , a ( t ) ∝ t , H ( t ) = 1 t , R H = ct Dark energy domination H 2 → constant , a ( t ) ∝ exp (Λ t / 3 ) , H ( t ) = c / R H = � Λ / 3 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 31 / 131
Hubble constant evolution J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 32 / 131
A first set of cosmological parameters and relations H 0 Hubble constant Ω k Curvature energy density Ω m Matter density ( 1 − Ω k ) = Ω tot = Ω m + Ω Λ Ω Λ Dark energy density Ω m = Ω CDM + Ω b + Ω γ + Ω ν Ω CDM Cold Dark matter density Deceleration parameter Ω b Baryonic matter density q 0 = 1 2 Ω NR m − Ω Λ Ω γ Photon density Ω ν Neutrino density m ( 1 + z ) 4 + Ω NR m ( 1 + z ) 3 − Ω k ( 1 + z ) 2 + Ω Λ ) = H 2 H 2 ( z ) = H 2 0 (Ω R 0 E ( z ) 2 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 33 / 131
Experimental astroparticle physics & cosmology L. 2, Section 3: Inflation J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 34 / 131
Motivations for inflation Inflation was motivated by a set of problems encountered by Big Bang theory Flatness problem The universe is observed to be flat today to a great accuracy however the flat solution of the FL equations is unstable Relic abundances Phase transitions in the early universe will lead to relic particles like for example monopoles that are not observed today Horizon problem CMB temperature is uniform and isotropic all over the sky however regions of the sky separated by more than one degree were not in casual contact at the time of CMB formation Origin of cosmological fluctuations All observed structures in the universe were formed by the growth up of primordial fluctuations for which we have no explanation J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 35 / 131
Accelerated expansion To solve the horizon, flatness and relics problem we need � 1 � d < 0 ⇒ ¨ a > 0 ⇒ ρ + 3 p < 0 dt aH So acceleration implies negative pressure p < − 1 / 3 ρ We define the number of e-folds as N = ln a i a f where a i and a f correspond to the scale factors at beginning and end of the accelerated expansion period Notice that N represents some how the amount expansion To solve the horizon, flatness and relics problems we need N ≥ 60 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 36 / 131
Scalar fields in cosmology For a FRWL universe the dynamics of a scalar field is given by φ − ∇ 2 φ φ + 3 H ˙ ¨ + V ′ ( φ ) = 0 a 2 For FRWL universe and assuming φ = φ 0 + δφ we obtain for the homogeneous field φ 2 + ( ∇ φ ) 2 ρ φ = 1 ˙ + V ( φ ) 2 a 2 2 φ 2 − ( ∇ φ ) 2 p φ = 1 ˙ − V ( φ ) 6 a 2 2 So we can write FL equation 3 ρ φ − k 2 H 2 = 8 π G 2 ∼ 8 π G 3 ρ φ J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 37 / 131
Slow roll dynamics We can obtain accelerated expansion of the universe from the scalar field dynamics We neglect the term ∇ 2 φ (somehow diluted by expansion) 1 a 2 ˙ φ We assume 2 ≪ V ( φ ) we have p φ ∼ − ρ φ and thus 2 H 2 ∼ 8 π G 3 V ( φ ) We assume ¨ φ ≪ 3 H ˙ φ 3 Thus : H 2 ≃ 8 π G 3 V φ + V ′ ≃ 0 3 H ˙ J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 38 / 131
Slow roll parameters Net energy is dominated by potential energy and thus acts like a cosmological constant w → − 1 First slow roll parameter � 2 � V ′ ǫ ≡ 3 1 2 ( 1 + w ) = 16 π G V Second slow roll parameter � ˙ ¨ V ′′ φ � a 1 δ ≡ − 1 = ǫ − V = ǫ − η ˙ 8 π G φ a Slow roll conditions imply ǫ, δ, | η | ≪ 1, corresponding to a very flat potential 1 We normally define the reduced Planck mass as M P = 8 π G J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 39 / 131
Potential slowly rolling down J.F. Mac´ ıas-P´ erez (LPSC) Lecture 2: Expanding universe January 22, 2014 40 / 131
Experimental astroparticle physics & cosmology Lecture 3: CMB J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 41 / 131
Cosmic Microwave Background Penzias & Wilson discovered in 1965 an isotropic and homogeneous radiation with a temperature of about 3 K as predicted by Gamow in 1940 the COBE satellite in 1992 showed that the CMB has a black-body spectrum and fluctuations of about 10 − 5 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 41 / 131
Experimental astroparticle physics & cosmology L. 3, Section 1: Thermal history of the Universe J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 42 / 131
Cartoon thermal history of the universe J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 43 / 131
Detailed thermal history of the universe Event T (K) kT (eV) g eff z t Now 2.76 0.0002 3.43 0 13.6 Gyr ∼ 1 Gyr First Galaxies 16 0.001 3.43 6 (?) Recombination 3000 0.3 3.43 1100 38000 yr M-R equality 9500 0.8 3.43 3500 50000 yr e + -e − pairs 10 9 . 7 0.5 10 6 10 9 . 5 11 3 s 10 10 1 10 6 10 10 Nucleosynthesis 11 1 s 10 − 7 s 10 13 1 10 9 10 13 Nucleon pairs 70 10 − 12 s 10 15 . 5 25 10 10 10 15 E-W unification 100 10 − 38 s 10 28 10 24 10 28 GUT 100 (?) 10 − 43 s 10 32 10 28 10 32 Quantum Gravitiy 100 (?) J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 44 / 131
3 Eras: radiation, matter and dark energy The energy density of radiation, matter and dark energy (DE) evolves differently radiation : ρ R ∝ a − 4 matter : ρ M ∝ a − 3 DE : ρ Λ = constant So, the total density of the universe can be written as Ω R x 4 + Ω M x 3 + Ω Λ � � ρ = ρ c ; x = 1 + z Matter-radiation equality is obtained when ρ M = ρ R at z = Ω M − 1 ∼ 3402 Ω R Matter-DE equality when ρ M = ρ Λ at � Ω Λ � 1 / 3 z = − 1 ∼ 0 . 29 Ω M J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 45 / 131
3 Eras: radiation, matter and dark energy J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 46 / 131
Experimental astroparticle physics & cosmology L. 3, Section 2: Physics at recombination J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 47 / 131
Thomson Scattering Thomson scattering of photons off of free electrons is the most important CMB process with a cross section (averaged over polarization states) of σ T = 8 πα 2 = 6 . 65 × 10 − 25 cm 2 3 m 2 e Density of free electrons in a fully ionized x e = 1 universe is given by n e = ( 1 − Y p / 2 ) x e n b ≈ 10 − 5 Ω b h 2 ( 1 + z ) cm − 3 , In general we can write the Thomson scattering rate as Γ = τ ′ = σ t an e x e where τ is the medium optical depth The visibility function g ( η ) = − τ ′ e − τ indicates the probability that a CMB photon last scattered at conformal time η J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 48 / 131
Recombination When temperature drops to ∼ 1000 K it is thermodynamically favorable for the plasma to form atoms via p + e − ↔ H + γ This is called recombination. If thermal equilibrium hods then the number density of each species is � 3 / 2 � m i T � µ i − m i � n i = g i exp 2 π T and chemical equilibrium impose µ e + µ p = µ H As m H ∼ m p and defining B H = m p + m e − m H = 13 . 6 eV we have � 3 / 2 � m i T f H n H = exp ( B H / T ) n e n p g p g e 2 π J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 49 / 131
Ionization fraction evolution J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 50 / 131
Recombination in a nutshell The Thomson scattering rate evolves as Γ ∝ a − 2 x e The free electron fraction x e starts from 1 at high redshift. Thus, before recombination Γ ≫ a ′ a and the universe is opaque At recombination, about z ∼ 1080, x e decreases sharply and freezes at a very small value Then, after recombination Γ ≪ a ′ a and the universe is transparent At reionization all electrons are free again, however because dilution n e is small and Γ remains much smaller than a ′ a and so most photons do not interact any more J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 51 / 131
Last Scattering Surface Interaction between electrons and photons via Thomson scattering before recombination and after reionization Angular distribution of radiation is the 3D temperature field projected onto a shell - surface of last scattering Integrate along the line of sight in an expanding universe Describe radiation as an statistically isotropic temperature field with fluctuations J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 52 / 131
Experimental astroparticle physics & cosmology L. 3, Section 3: Observing the CMB J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 53 / 131
Brief history of CMB observations Penzias & Wilson discovered in 1965 an isotropic and homogeneous radiation with a temperature of about 3 K In 1992 the COBE satellite demonstrated that the CMB has a black-body spectrum and fluctuations of about 10 − 5 In 1998 Boomerang and Maxima measured the so-called acoustic peaks in the CMB power spectrum The WMAP satellite, launched in 2001, provided first CMB polarization precise measurements The Planck satellite 2013 results has provided best possible CMB temperature anisotropies measurements and much more (polarization analysis expected in 2014) Late 2013 the South Pole telescope and the PolarBear experiment reported first observation of B-lensing modes J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 54 / 131
Observing the sky J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 55 / 131
Foregrounds J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 56 / 131
CMB instruments Radio mm Telescopes dish and horns dish and horns Detectors HEMT + square law detectors bolometer and/or KIDs Cooling 18-50 K 100-300 mK Observing mode Ground, satellite ground, balloon, satellite J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 57 / 131
CMB black-body spectrum Compton scattering of photons with electrons is very efficient to thermalize photons In 1994 the FIRAS spectrograph in the COBE satellite measured the CMB temperature: T CMB = 2 . 726 ± 0 . 001 K J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 58 / 131
Measured CMB anisotropies I Dipole anisotropy induced by Doppler effect (relative motion of the observer with respect to the CMB rest frame) First measured by the COBE satellite in 1992 with an amplitude of 3 . 358 ± 0 . 001 ± 0 . 023 mK in the direction of (l,b)=(264 . 31 ± 0 . 04 ± 0 . 16, + 48 . 05 ± 0 . 02 ± 0 . 09) degrees J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 59 / 131
The micro-wave and mm sky We observe a mixture of components: CMB, galactic thermal dust, synchrotron and free-free emissions, extragalactic emission from dusty and radio galaxies J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 60 / 131
From sky observations to CMB maps Component separation algorithms are used to recover the CMB emission J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 61 / 131
Measured CMB anisotropies II Temperature fluctuations of the order of 10 − 5 Planck satellite 2013 results: most precise measurements of the CMB temperature anisotropies J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 62 / 131
Experimental astroparticle physics & cosmology L. 3, Section 4: Physics of CMB anisotropies J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 63 / 131
Spherical harmonics and power spectrum Any scalar field on the sphere, A ( θ, φ ) can be decomposed into spherical harmonics + ℓ � � A ( θ, φ ) = a ℓ m Y ℓ m ( θ, φ ) ℓ m = − ℓ We can define the power spectrum as 1 C ℓ = < a ℓ m a ∗ � | a ℓ m | 2 ℓ m > = 2 ℓ + 1 m For a Gaussian random field then < a ℓ m a ∗ ℓ ′ m ′ > = C ℓ δ ℓℓ ′ δ mm ′ J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 64 / 131
The Boltzmann equation Photons decouple from baryons at recombination so we can not describe them with fluid equations Need to solve the Boltzmann equation for the photon space-phase distribution d d η f γ ( η, x , q ) = C [ f γ ( η, x , q ) , f e ( η, x , q )] at first order in perturbation Notice that as discussed above electrons and baryon are so tightly coupled that it makes no difference to think in terms of photon-electron coupling or photon-baryon coupling In thermal equilibrium the space-phase photon distribution function behaves as a Bose-Einstein distribution 1 f γ ( η, x , q ) = q T ( η, x ) − 1 e J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 65 / 131
Perturbations We expand the photon space-phase distribution function as a background part and first order perturbation f γ = ¯ f γ + δ f γ and so 1 ¯ f γ ( η, x , q ) = q T ( η )+ δ T ( η ) − 1 ¯ e and d ¯ δ T ( η, x ) f γ δ f γ ( η, x , q ) = ¯ T ( η ) d log q Therefore, we can replace f γ ( η, x , q ) by the brightness function Θ( η, x ) ≡ δ T ( η, x ) ¯ T ( η ) In an inhomogeneous universe photons travelling on different geodesic (line-of-sights) experience different redshifts so Θ( η, x , n ) ≡ δ T ( η, x , n ) ¯ T ( η ) J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 66 / 131
Spherical harmonic decomposition The brightness function can be decomposed in Fourier modes such that dk 3 � ( 2 π ) 3 Θ( η, k , n ) e i k . x Θ( η, x , n ) = with power spectrum < Θ( η, k , n )Θ ∗ ( η, k ′ , n ) > = ( 2 π ) 3 P Θ( η, n ) ( k ) Finally Fourier modes can be decomposed in spherical harmonics taking into account the fact that the propagation direction of photons is − n � ( − 1 ) ℓ Θ ℓ, m ( η, k ) Y ℓ. m ( n ) Θ( η, k , n ) = ℓ, m or equivalently in Legendre polynomials � ( − 1 ) ℓ ( 2 ℓ + 1 )Θ ℓ ( η, k ) P ℓ ( k . n / k ) Θ( η, k , n ) = ℓ J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 67 / 131
Power spectrum of the CMB anisotropies We want to compute the power spectrum of the temperature field today as observed from our position, x = 0 , today η = η 0 δ T � T ( n ) = Θ( η 0 , 0 , − n ) = a ℓ m Y ℓ m ( n ) ¯ ℓ m Using previous results and Legendre polynomials to spherical harmonic relations we can write 4 π � ( 2 π ) 3 ( i ) ℓ d 3 k Y ℓ m ( k )Θ ℓ ( η 0 , k ) a ℓ m = Using the orthonormality of spherical harmonics we can write � ∞ Θ ℓ ( η 0 , k ) dk ∆ 2 C ℓ = 4 π k 0 and using the transfer function we obtain � ∞ R ( k ) dk T 2 Θ ℓ ( k )∆ 2 C ℓ = 4 π k 0 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 68 / 131
CMB temperature power spectrum J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 69 / 131
CMB power spectrum and cosmological parameters Ω m , Ω b , Ω Λ (P1) Peak Scale Ω b (P2) Odd/even peak amplitude ratio Ω m (P3) Overall peak amplitude Ω m , Ω b , Ω Λ (P4) Damping enveloppe (P5) Global Amplitude A s (P6) Global tilt n s (P7) Additional SW plateau tilting via ISW Ω Λ Amplitude for l > 40 only τ reio (P8) J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 70 / 131
CMB temperature power spectrum and parameters J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 71 / 131
Planck measured CMB temperature spectrum J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 72 / 131
Measured CMB temperature spectrum at small angular scales J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 73 / 131
Experimental astroparticle physics & cosmology L. 3, Section 5: Secondary CMB temperature anisotropies J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 74 / 131
Main secondary temperature anisotropies We call secondary CMB anisotropies those that are generated after recombination either by gravitational effects of interaction of photons with electrons: Integrated Sachs Wolfe (ISW) effect: Sachs-Wolfe effect originated by changes in the gravitational potentials along the line-of-sight. The non-linear contribution is generally called Vishniac effect. Gravitational Lensing: gravitational lensing induced by mass distribution along the line-of-sight Sunyaev-Zel’dovich effect: Compton inverse between CMB photons and hot free electrons on clusters of galaxies Reionization: Thomson interaction of CMB photons with free electrons at the time global reionization of the universe when first star form. J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 75 / 131
Gravitational lensing in a nutshell Gravitational potentials along the line of sight n to some source at comoving distance D s gravitationally lens the image We can define an effective potential � dDD s − D φ ( n ) = 2 Φ( D n , η ( D )) DD s such that the image is remapped as n I = n S + ∇ n φ ( n ) In the case of CMB lensing we are in the weak lensing regime and we expect small distortions of the image In particular we can observe that the convergence is simply the projected mass J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 76 / 131
CMB lensing cartoon J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 77 / 131
Integrated gravitational potential J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 78 / 131
Lensing power spectrum J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 79 / 131
Lensing power spectrum J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 80 / 131
Sunyaev-Zeldovich (SZ) effect Thermal (t)SZ effect corresponds to a small spectral distortion of the CMB spectrum ∆ T tSZ � k B T e = f ( x ) y = f ( x ) n e m e c 2 σ T d ℓ T CMB where x = h ν k B T and xe x + 1 � � f ( x ) = e x − 1 − 4 Kinetic (k)SZ effect If clusters are moving with respect to the CMB frame there is an additional spectral distortion due to the Doppler effect of the cluster bulk velocity on the scattered CMB photons. In the non-relativistic limit the kSZ is just a thermal distortion ∆ T kSZ � v pec � � v pec � � = − τ e = − n e σ T d ℓ T CMB c c J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 81 / 131
tSZ effect with Planck J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 82 / 131
Examples of cluster of galaxies observed via the tSZ effect J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 83 / 131
The COMA cluster Detailed observations of the Coma cluster including the outskirts Direct observation of compression shocks on the tSZ data J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 84 / 131
The Planck cluster sample 1227 cluster candidates: 861 clusters and 366 candidates being confirmed J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 85 / 131
Compton parameter map All-sky map of cluster of galaxies and maybe filaments Unfortunately foreground contribution is important, more work needed, keep tuned next year. J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 86 / 131
Cluster number counts and cosmology Clusters of galaxies are the largest gravitational bound structures in the universe and can be assimilated to dark matter halos The number of cluster of galaxies in terms of their mass and redshift is very sensitive to cosmological parameters and non-linear physics J.F. Mac´ ıas-P´ erez (LPSC) Lecture 3: CMB January 22, 2014 87 / 131
Experimental astroparticle physics & cosmology Lecture 4: CMB polarization J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 88 / 131
Experimental astroparticle physics & cosmology L. 4, Section 1: polarization power spectra J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 88 / 131
Stokes parameters Polarised light can be described using Stokes parameters For a light beam propagating on the z direction, the polarization plane is defined by x − y plane The electric field can be decomposed as E ( t , z ) = E x ( t , z ) e x + E y ( t , z ) e y where E x ( t , z ) and E y ( t , z ) are plane waves E x ( t , z ) = A x e φ x e i ( kz − wt ) E y ( t , z ) = A y e φ y e i ( kz − wt ) Stokes parameters are defined are I = < E x E ∗ x + E y E ∗ y > = A 2 x + A 2 y Q = < E x E ∗ x − E y E ∗ y > = A 2 x − A 2 y U = < E x E ∗ y + E y E ∗ x > = 2 A x A y cos ( φ y − φ x ) V = − i < E x E ∗ y − E y E ∗ x > = 2 A x A y sin ( φ y − φ x ) J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 89 / 131
Stokes parameters II J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 90 / 131
Stokes parameters III: some special cases Right-handed (left handed) circularly polarised light, E x = E y and 1 cos ( φ y − φ x ) = ± π 2 I = S Q = 0 U = 0 V = ± S Linearly polarized light cos ( φ y − φ x ) = 0 2 I = S Q = pS cos ( 2 ψ ) U = pS sin ( 2 ψ ) V = 0 √ Q 2 + U 2 where p = and ψ are the degree and polarization angle. I J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 91 / 131
Linear polarization properties In the case of linearly polarised light a change of reference frame modify the Stokes parameters as follows I ′ = I Q ′ = Q cos ( 2 θ ) + U sin ( 2 θ ) U ′ = − Q sin ( 2 θ ) + U cos ( 2 θ ) So we can form a spin ± 2 object Q ± iU that transforms as Q ′ ± iU ′ = e ∓ 2 i θ [ Q ± iU ] Thus, Stokes parameters on the sphere can be decomposed as ℓ m a T T ( n ) = � ℓ m Y ℓ m ( n ) [ Q ± iU ] = � ℓ m [ a E ℓ m ± ia B ℓ m ] ± 2 Y ℓ m ( n ) J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 92 / 131
polarization power spectra We can define three scalar fields T , E , B which are independents of the chosen reference frame Using those we can form 3 auto-power spectra 1 C TT m | a T ℓ m | 2 = � ℓ 2 ℓ + 1 C EE 1 m | a E ℓ m | 2 = � ℓ 2 ℓ + 1 1 ℓ m | 2 C BB � m | a B = ℓ 2 ℓ + 1 and 3 cross-spectra ∗ ) 1 C TE � m ( a T ℓ m a E = ℓ ℓ m 2 ℓ + 1 ∗ ) C TB 1 m ( a T ℓ m a B = 2 ℓ + 1 ( � ℓ ℓ m ∗ ) C EB 1 m ( a E ℓ m a B = � ℓ 2 ℓ + 1 ℓ m C TB and C EB vanish if parity is conserved J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 93 / 131
Experimental astroparticle physics & cosmology L. 4, Section 2: CMB polarization physics J.F. Mac´ ıas-P´ erez LPSC J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 94 / 131
Thomson scattering As discussed before polarization state of radiation along the line-of-sight is described by the components of the electric field E The differential cross section of Thomson scattering is given by d Ω = 3 σ T d σ 8 π | E ′ . E | 2 where E ′ and E are the incoming and outgoing directions of the electric field To get final polarization state along the line-of-sight n we sum over angle and incoming polarization d n ′ d σ � � d Ω i = 1 , 2 J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 95 / 131
Cartoon polarization generation Only quadrupole anisotropies generate polarization J.F. Mac´ ıas-P´ erez (LPSC) Lecture 4: CMB polarization January 22, 2014 96 / 131
Recommend
More recommend
