The non-linear evolution of CMB: non-Gaussianity and spectral - PowerPoint PPT Presentation

The non-linear evolution of CMB: non-Gaussianity and spectral distortions Cyril Pitrou Institute of Cosmology and Gravitation, Portsmouth 18 Novembre 2010

Outline Motivations for non-Gaussianity search 1 Theory of perturbations 2 Spectral distortions 3 The flat-sky approximation 4 Numerical resolution and analytic insight 5 Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 2 / 50

Motivations for non-Gaussianity Motivations for non-Gaussianity search 1 Theory of perturbations 2 Spectral distortions 3 The flat-sky approximation 4 Numerical resolution and analytic insight 5 Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 3 / 50

Motivations for non-Gaussianity From initial conditions to observations k 3 P � k � of radiation density k 3 P � k � of the potential 40 1 30 0.1 20 0.01 10 0.001 10 � 4 0 � 1.0 � 0.5 0.0 0.5 1.0 1.5 � 1.0 � 0.5 0.0 0.5 1.0 1.5 Log � k � Log � k � Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 4 / 50

Motivations for non-Gaussianity Harmonic analysis Analysis in the space of Y ℓ m � a ℓ m a ℓ ′ m ′ � = δ ℓℓ ′ δ mm ′ C ℓ Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 5 / 50

Motivations for non-Gaussianity Standard lore of perturbation theory Initial conditions : quantization of the free theory implies Gaussian initial conditions: P ( k ) � Φ( k )Φ( k ′ ) � = δ ( k + k ′ ) P ( k ) Evolution : linearisation of GR. Transfer scheme of perturbations Linear equations, modes k are independent, = ⇒ Gaussianity conserved. P ( k ) → Θ( k , η ) → a ℓ m → C ℓ k 3 P � k � of the potential k 3 P � k � of radiation density 40 40 30 30 20 20 10 10 0 0 → → � 1.0 � 0.5 0.0 0.5 1.0 1.5 � 1.0 � 0.5 0.0 0.5 1.0 1.5 Log � k � Log � k � Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 6 / 50

Motivations for non-Gaussianity Motivations for non-Gaussianity search 1 Theory of perturbations 2 Spectral distortions 3 The flat-sky approximation 4 Numerical resolution and analytic insight 5 Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 7 / 50

Motivations for non-Gaussianity non-Gaussianity (NG) Initial conditions non-Gaussian? We want to test the models of inflation with other moments of the statistics. Non-linear dynamics is intrinsic to GR, Statistics of the primordial gravitational potential Φ = Φ ( 1 ) + 1 2 Φ ( 2 ) Gaussian part Φ ( 1 ) and non-Gaussian part Φ ( 2 ) : � Φ( k )Φ( k ′ ) � = δ ( k + k ′ ) P ( k ) � Φ( k 1 )Φ( k 2 )Φ( k 3 ) � = δ ( k 1 + k 2 + k 3 ) f NL F ( k 1 , k 2 , k 3 ) F ( . . . ) = type of non-Gaussianity f NL = its amplitude. Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 8 / 50

Motivations for non-Gaussianity The transfer to temperature fluctuations Θ ℓ m In general Θ ≡ T (Φ) Order 1 Θ ( 1 ) ℓ m ≡ T ℓ m (Φ ( 1 ) ) L Order 2 Θ ( 2 ) ℓ m ≡ T ℓ m (Φ ( 2 ) ) + T ℓ m NL (Φ ( 1 ) Φ ( 1 ) ) L In Fourier space Θ ( 1 ) ( k )Φ ( 1 ) ℓ m ( k ) = T ℓ m L k Θ ( 2 ) ( k )Φ ( 2 ) ℓ m ( k ) = T ℓ m L k NL ( k 1 , k 2 , k )Φ ( 1 ) k 1 Φ ( 1 ) � d 3 k 1 d 3 k 2 δ 3 ( k − k 1 − k 2 ) T ℓ m + k 2 f NL ou T NL ? � Θ ℓ 1 m 1 Θ ℓ 2 m 2 Θ ℓ 3 m 3 � � = 0 because of � Θ ( 1 ) Θ ( 1 ) Θ ( 2 ) � . Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 9 / 50

Theory of perturbations Motivations for non-Gaussianity search 1 Theory of perturbations 2 Spectral distortions 3 The flat-sky approximation 4 Numerical resolution and analytic insight 5 Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 10 / 50

Theory of perturbations Description of perturbations General idea We need to give a precise meaning to δ T ( P ) = T ( P ) − ¯ T ( P ) T ( P ) ¨lives¨ in a perturbed space-time ¯ T ( P ) ¨lives¨ in a background space-time, homogeneous and isotropic Example: metric perturbations d s 2 = a ( η ) 2 � − d η 2 + δ IJ d x I d x J � , d s 2 = a ( η ) 2 � − e 2 Φ d η 2 + 2 B I d x I d η + [ e − 2 Ψ δ IJ + 2 H IJ ] d x I d x J � , Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 11 / 50

Theory of perturbations Correspondence between space-times embedded into 4 + 1 dimensions Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 12 / 50

Theory of perturbations Characteristics of perturbations theory: Get rid of the gauge dependence Gauge-invariant variables A tensor equation is always expressed with such variables Structure of equations in orders of perturbations E [ δ ( 1 ) g , δ ( 1 ) T ] = 0 E [ δ ( 2 ) g , δ ( 2 ) T ] = S [ δ ( 1 ) g , δ ( 1 ) T ] = ⇒ Iterative resolution with gauge-invariant variables Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 13 / 50

Theory of perturbations Describing the matter content The fluid approximation T µν = ( P + ρ ) u µ u ν + Pg µν + Π µν Conservation Eq. ∇ µ T µ 0 = 0 ⇒ ρ ′ + · · · = 0 = Euler Eq. ∇ µ T µ i = 0 ⇒ u ′ i + · · · + ∂ j Π ji = 0 = Problems Equation of state P = w ρ ? Expression and evolution of the anisotropic stress tensor? Multifluid: ∇ µ T µν = F ν � = 0. Expression of forces ? Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 14 / 50

Theory of perturbations Statistical description Distribution function f ( x , p a ) Tetrad in order to define locally a free-fall frame e a . e b ≡ e µ a e ν b g µν = η ab e a . e b ≡ e a ν g µν = η ab µ e b Momentum p Decomposed in energy and direction p = E ( e o + n ) Link to the fluid description D ( p . p ) f ( x , p c ) p a p b d p o d 3 p i � T ab ( x ) ≡ δ 1 ( 2 π ) 3 Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 15 / 50

Theory of perturbations Evolution of the distribution function Boltzmann equation L [ f ] = C [ f ] Liouville operator: Free-fall d s = p c ∇ c f ( x , p a ) + ∂ f ( x , p a ) d p c L [ f ] = d f ∂ p c d s Geodesic equation p b ∇ b p a = d p a d s + ω bac p c p b = 0 Collision operator: Compton scattering on free electrons. Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 16 / 50

Theory of perturbations Why do we also need to describe polarization? Because if radiation has a quadrupole, Compton scattering generates polarisation. Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 17 / 50

Theory of perturbations Description of polarisation by the Stokes parameters Tensorial distribution function  0 0 0 0  If n i = ( 0 , 0 , 1 ) : f ab = 1 0 I + Q U + i V 0     0 U − i V I − Q 0 2   0 0 0 0 Covariant expression f µν ( x , p a ) ≡ 1 2 I ( x , p a ) S µν + P µν ( x , p a ) + i 2 V ( x , p a ) e ρ o ǫ ρµνσ n σ Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 18 / 50

Theory of perturbations Covariant description of polarized radiation Tensor valued distribution function A photon is characterized by p µ and ε µ ( p µ ε µ = 0) F µν ( x , p a ) ≡ 1 2 f ( x , p a ) ε µ ε ν Screen projection Screen projector S µν = g µν + e o µ e o ν − n µ n ν S ν µ ε ν is independent of the electromagnetic gauge choice for the polarization. We thus work with f µν ( x , p a ) = S ρ µ S σ ν F ρσ ( x , p a ) Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 19 / 50

Theory of perturbations Boltzmann equation with polarization � x , p h � L [ f ab ( x , p h )] = C ab L [ f ab ( x , p a )] = 1 2 L [ I ( x , p d )] S ab + L [ P ab ( x , p a )]+ i 2 L [ V ( x , p d )] n c ǫ ocab � d 2 Ω ′ � 3 p h �� � p h � � p ′ h � � = n e σ T p o 4 π S c a S d C ab b f cd − f ab 2 We recover the case with no polarization f ab = 1 I = S ab f ab 2 IS ab or ab S ab = 1 + ( n . n ′ ) 2 = 1 + cos 2 θ S ab C ab ∝ S ′ Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 20 / 50

Theory of perturbations Multipolar expansion Multipoles for scalar functions ( I and V ) ∞ I ( x , p o , n a ) = � I a ℓ ( x , p o ) n a ℓ ℓ = 0 � I a ℓ ( x , p o ) = ∆ − 1 I ( x , p o , n a ) n � a ℓ � d 2 Ω ℓ And for polarisation, E and B modes... ∞ ( a B b ) dc ℓ − 2 ( x , p o ) n c ℓ − 2 � TT � E abc ℓ − 2 ( x , p o ) n c ℓ − 2 − n c ǫ cd � P ab ( x , p a ) = ℓ = 2 � E a ℓ ( x , p o ) M 2 ℓ ∆ − 1 n � a ℓ − 2 P a ℓ − 1 a ℓ � ( x , p o , n a ) d 2 Ω , = ℓ � B a ℓ ( x , p o ) M 2 ℓ ∆ − 1 n b ǫ bd � a ℓ n a ℓ − 2 P a ℓ − 1 � d ( x , p o , n a ) d 2 Ω , = ℓ Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 21 / 50

Theory of perturbations Steps to follow g µν + g ( 1 ) 2 g ( 2 ) µν + 1 Perturb the metric g µν = ¯ 1 µν Perturb the tetrad e µ e µ a + e ( 1 ) µ 2 e ( 2 ) µ + 1 a = ¯ 2 a a ω abc + ω ( 1 ) 2 ω ( 2 ) abc + 1 Perturb the connections ω abc = ¯ 3 abc Find the perturbed geodesic equations 4 Compute the perturbed Liouville operator 5 Compute the Thomson scattering for each electron 6 Sum over the electrons distribution to obtain the Collision tensor in 7 full generalities Expand it in perturbations 8 Take the multipoles I a ℓ E a ℓ and B a ℓ of the Boltzmann equation 9 10 Solve it or integrate it numerically Cyril Pitrou (ICG, Portsmouth) 18 Novembre 2010 22 / 50