The Cosmic Microwave Background: How It Works Albert Stebbins Academic Lecture Series Fermilab 2014-03-11 Monday, March 17, 14
General Relativity Metric geometry 10 free functions reduced by 4 to 6 by coordinate freedom Can decompose according to helicity ( 2 scalar+ 2 vector+ 2 tensor ) Dynamics: Einstein’ s Eq’ s: G μν = 8 π G T μν Monday, March 17, 14
Cosmic Relics: Photons: The 2.725K CMBR Neutrinos: ( difficult to see directly ) expect T ν =1.955K Baryons: ( origin of baryon anti-baryon asymmetry unknown ) Dark Matter: (origin unknown) Scalar Perturbation: inhomogeneities ?Tensor Perturbations: gravitational radiation Dark Energy ( origin unknown - only important recently? ) Monday, March 17, 14
Λ CDM Model Thermal: Inhomogeneities: Parameters: T Υ 0 ,H 0 , Λ , Ω m0 , Ω b0 , Ω 0 ,N eff ,A S ,A T ,n S ,n T , τ Monday, March 17, 14
How to Describe the CMBR? Microscopic Description E a [ x , t ] ∝ ∫ d ν e i2 πν t ∫ d 2 ĉ e i2 π ν ĉ · x Ẽ a [ ĉ , ν ] in a small frequency bin: ⟨ ⟩ ∝ ( ) Ẽ x Ẽ x* Ẽ x Ẽ z* I+Q U+i V Light are a collection of electromagnetic Ẽ z Ẽ x* Ẽ z Ẽ z* waves. U-iV I-Q I intensity • There could in principle be a lot of Q , U linear polarization • information in all the detailed V circular polarization • correlations of the EM field. By definition: I , Q , U , V real The interesting information is Schwartz Inequality: I 2 ≥ Q 2 + U 2 + V 2 usually only in the time averaged Elliptically Polarized: I 2 = Q 2 + U 2 + V 2 2nd moments of the E fields. Linearly Polarized: I 2 = Q 2 + U 2 V =0 Circularly Polarized: I =| V | Q = U =0 Expectation: Unpolarized: Q = U = V= 0 CMBR slightly linearly polarized I 2 » Q 2 + U 2 » V 2 Rees (1968) 5 Monday, March 17, 14
How to Describe the CMBR? Macroscopic Description On cosmological length and times-scales (millions to billions of light-years): I [ ĉ , ν , x ,t ] , Q [ ĉ , ν , x ,t ] , U [ ĉ , ν , x ,t ] , V [ ĉ , ν , x ,t ] as we shall see V=0 is a good approximation. Spatial Fourier transform, e.g. I [ ĉ , ν , x ,t ] = ∑ k e i k · x Ĩ [ ĉ , ν , k ,t ] Angular decomposition: spherical harmonics Ĩ [ ĉ , ν , k ,t ] = ∑ ℓ ∑ m Y ( ℓ , ℎ ) [ ĉ ] Ĩ ( ℓ , ℎ ) [ ν , k ,t ] For each k , align “North Pole” of Y (l, ℎ ) to k direction then ℎ gives helicity as we shall see I (l, ℎ ) =0 for | ℎ |>2 is a good approximation. A simple Y (l, ℎ ) decomposition of Q,U is not the best! 6 Monday, March 17, 14
Graphical Representation of Linear Polarization 2d Symmetric Traceless Tensors Q > 0 Q < 0 U < 0 U > 0 7 Monday, March 17, 14
Linear Polarization Patterns Q patterns U patterns 8 Monday, March 17, 14
Linear Polarization Patterns k 0 o -90 o pattern 90 O scalar pattern Stebbins 1996 0 O gradient pattern Kaminokowski, Kosowsky, Stebbins 1997 E-mode Seljak, Zaldarriaga 1997 90 O k +45 O Stebbins 1996 pseudo-scalar pattern Kaminokowski, Kosowsky, Stebbins 1997 curl pattern -45 O Seljak, Zaldarriaga 1997 B-mode +45 O ±45 o pattern 9 Monday, March 17, 14
General E- B- Mode Decomposition in any 2-D Riemannian manifold one has 2 covariant tensors: metric g ab and Levi-Civita symbol ε ab = √ Det[g ab ] {{0,1},{-1,0}} contracting a vector with ε ab rotates by 90 o contracting a tensor with ε ab rotates eigenvectors 45 o starting with any (scalar) function f construct corresponding E- and B- mode vectors E-mode: covariant derivative: f ;a B-mode: rotate by 90 o : f ;b ε ba construct corresponding E- and B- mode traceless symmetric tensors E-mode: 2nd derivative - trace: f ;ab - ½ ( ∇ 2 f) δ ab B-mode: symmetrically rotate by 45 o : ½ ( f ;ac ε cb +f ;bc ε ca ) One can construct E-mode and B-mode tensors of any rank this way! 10 Monday, March 17, 14
E- B- Mode Spherical Harmonics E- B- mode decomposition applied to complete scalar basis gives complete tensor basis! on (direction) 2-sphere use spherical harmonic basis: Y (l,m) gives E- B- mode basis for symmetric traceless tensors on sphere Y E ((l,m) ab ∝ ¡ Y ((l,m) ;ab - ½ ( ∇ 2 Y ((l,m) ) δ ab Y B (l,m) ab ∝ ¡ ½ ( Y ((l,m) ;ac ε cb + Y (l,m) ;bc ε ca ) Y E (0,m) ab ¡= Y B (0,m) ab ¡= Y E (1,m) ab ¡= Y B (1,m) ab ¡=0 these can be used to describe linear polarization: ( ) I+Q U+i V P ab = U-iV I-Q = ∑ k e i k · x ∑ ℓ ∑ ℎ ( ) ( ) I ( ℓ , ℎ ) +i V ( ℓ , ℎ ) Y ( ℓ , ℎ ) + E ( ℓ , ℎ ) Y E ( ℓ , ℎ ) + B ( ℓ , ℎ ) Y B ( ℓ , ℎ ) -i V ( ℓ , ℎ ) I ( ℓ , ℎ ) Equivalent formulation uses spin-weighted spherical harmonic functions Y (s,l,m) Q + i U = ∑ k e i k · x ∑ ℓ ∑ ℎ Y (2, ℓ , ℎ ) 11 Monday, March 17, 14
How to Describe the CMBR? Intensity and Units In astronomy I [ ĉ , ν , x ,t ] usually has units: ergs/cm 2 /sec/steradian/Hz recall Poynting energy flux S=ExB/(8 π )=|E| 2 /(8 π ) (Gaussian CGS units) radio astronomy: often convenient to define a Rayleigh Jeans Brightness temperature kT RJ = ½ (c/ ν ) 2 I this gives the thermodynamic temperature if h ν ≪ kT, theoretically it is most convenient to use the quantum mechanical occupation number n T [ ν ] = ½ (c/ ν ) 2 I/(h ν ) = kT RJ /(h ν ) for a blackbody n = n BB [ ν ,T] = 1/(e (h ν )/(kT) -1) N.B. one can multiply E,B,V by ½ (c/ ν ) 2 /(h ν ) to put them in dimensionless occupation number units: n T , n E , n B , n V 12 Monday, March 17, 14
Spectral Decomposition One may also decompose the spectrum of each component X=I,E,B,V: n X( ℓ , ℎ ) [ ν , k ,t] = ∑ p (-1) p /p! n X( ℓ , ℎ ,p) [k,t] ∂ p n BB [ ν ,T]/ ∂ (ln ν ) p this is a (generalized) Fokker Planck expansion about a blackbody. p=0 corresponds to a pure blackbody - only n T( 0 ,0,0) = 1 ≠ 0 p=1 is spectral deviation from temperature shift Doppler, gravitational redshifts, etc. all 1st order anisotropies and polarizations will have this form p=2 arises from a mixture temperatures shifts it only arises to 2nd order in perturbations theory (small) Thermal Sunyaev-Zel’ dovich (SZ) effect: hot plasma (v e,rms = (m p /m e ) ½ v p,rms = 0.1 c) thermal 13 Monday, March 17, 14
How to Describe the CMBR? Summary Mode decomposed each Stokes parameter w/ “quantum numbers” k spatial dependence ℎ helicity: =0 scalar, =1 vector, =2 tensor ℓ angular wavenumber p spectral mode 14 Monday, March 17, 14
Statistical Description of CMBR Assume CMBR can be described as a realization of statistical distribution Assume statistical homogeneity and isotropy These assumptions severely restricts form of 2-point statistics translation symmetry requires different k modes uncorrelated rotational symmetry requires different ℎ modes uncorrelated ⟨ n X( ℓ , ℎ ,p) [k,t] n Y( ℓ ’ , ℎ ’,p’) [k’,t’]* ⟩ = C XY( ℓ , ℓ ’; ℎ ;p,p’) [|k|;t,t’] ¡δ k,k’ ¡δ ℎ , ℎ ’ 15 Monday, March 17, 14
Statistical Description of Observed CMBR We only get to measure CMBR from one vantage point at one time ( ) I+Q U+i V = ½ (c/ ν ) 2 /(h ν ) ∑ p (-1) p /p! ∂ p n BB [ ν ,T]/ ∂ (ln ν ) p ∑ ℓ ∑ m U-iV I-Q ( ) ( ) n T( ℓ , m ,p) +i n V( ℓ , m ,p) Y ( ℓ , m ) + n E( ℓ , m ,p) Y E ( ℓ , m ) + n B( ℓ , m ,p) Y B ( ℓ , m ) -i n V( ℓ , m ,p) n T( ℓ , m ,p) where n X( ℓ , m ,p) = ∑ k ∑ ℎ D ℓ m ℎ [ k ] n X( ℓ , ℎ ,p) [k,t] since the k’ s are isotropically distributed our sky is isotropic: ∫ d 2 ĉ D ℓ m ℎ [ ĉ ] D ℓ ’m’ ℎ [ ĉ ] = 4 π δ ℓ , ℓ ’ ¡δ m , m’ ⟨ n X( ℓ , m ,p) n Y( ℓ ’ , m’ ,p’) * ⟩ = ¡ C XY( ℓ ; p,p’) ¡δ ℓ , ℓ ’ ¡δ m , m’ where C XY( ℓ ; p,p’) = ¡ ∑ k ∑ ℎ C XY( ℓ , ℓ ’; ℎ ;p,p’) [|k|;t 0 ,t 0 ] ¡ 16 Monday, March 17, 14
Statistical Description of Observed CMBR To first order we only observe p=1: C XY ℓ = C XY( ℓ ; 1,1) Circular polarization damped possible modes: parity even: C TT ℓ , C EE ℓ , C BB ℓ , C TE ℓ parity odd: C TB ℓ , C EB ℓ 17 Monday, March 17, 14
Boltzmann Equation Dynamics determined by free-streaming and scattering D t n X =C X ∂ t n X [ ĉ , ν , x ,t]+c ĉ · ∇ n X [ ĉ , ν , x ,t]+( ∂ t ĉ ) · ∇ ĉ n X [ ĉ , ν , x ,t]+( ∂ t ln ν ) ∂ ln ν n X [ ĉ , ν , x ,t]=C X [ ĉ , ν , x ,t] only Thompson (non-relativistic Compton) scattering is important! absorption and emission unimportant d ν ’ ) = 3/16 π σ T (1+ ĉ · ĉ ’) δ [ ν - ν ’ ] d σ [ ĉ , ĉ ’; ν , ν ’ ]/(d 2 ĉ ’ S X [ ĉ , ν , x ,t] = 3/16 π c σ T n e [ x ,t] ∑ Y ∫ d 2 ĉ ’(1+ ĉ · ĉ ’) n Y [ ĉ ’ , ν , x ,t] lensing term ( ∂ t ĉ ) · ∇ ĉ n X [ ĉ , ν , x ,t] is 2nd order ∂ t ln ν = - ĉ · ∇Φ + ∂ t Φ + ĉ · ∂ t H ⫠ tr · ĉ independent of ν 18 Monday, March 17, 14
Boltzmann Equation ∂ t τ = c σ T n e 19 Monday, March 17, 14
Thomson Scattering Monday, March 17, 14
Thomson Scattering Monday, March 17, 14
Thomson Scattering Monday, March 17, 14
Thomson Scattering Monday, March 17, 14
Linear Polarization Patterns k 0 o -90 o pattern 90 O 0 O 90 O k +45 O -45 O +45 O ±45 o pattern 24 Monday, March 17, 14
http:/ /background.uchicago.edu/~whu/animbut/anim1.html Baryon Density Monday, March 17, 14
http:/ /background.uchicago.edu/~whu/animbut/anim2.html Dark Matter Density Monday, March 17, 14
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