Second Order CMB Perturbations

Transcription

1 Second Order CMB Perturbations Looking At Times Before Recombination September 2012

2 Evolution of the Universe Second Order CMB Perturbations 1/ 23

3 Observations before recombination Use weakly coupled particles (Neutrinos) Detection is challenging Going back in time, even weakly coupled particles will have strong interactions Primordial gravitational waves Hard to detect Indirect measurements in the CMB B polarisation Nongaussianity Second Order CMB Perturbations 2/ 23

4 Single field inflation [Baumann] Field slowly rolling down flat potential, causing inflation Quantum fluctuations induce primordial perturbations Inflation decays during reheating Second Order CMB Perturbations 3/ 23

5 Single field inflation [Baumann] Scalar perturbations responsible for structure in the Universe Tensor perturbations or gravity waves suppressed by slow roll parameter Generates gaussian perturbations Second Order CMB Perturbations 3/ 23

6 CMB anisotropies [WMAP] At leading order the CMB is homogenous and isotropic Horizon problem: Inflation as possible explanation Second Order CMB Perturbations 4/ 23

7 CMB anisotropies [WMAP] Small anisotropies of order 10 5 induced from quantum fluctuations Define cosmological perturbation theory as deviations from homogeneity Inflation constrained by power spectra of the CMB fluctuations Second Order CMB Perturbations 4/ 23

8 CMB anisotropies [WMAP] Inflation also generates primordial gravity waves Tensor to scalar ratio linked to slow roll parameter Polarisation can be used to measure the primordial tensor perturbations Second Order CMB Perturbations 4/ 23

9 Definition of polarisation modes Define matrix of distribution functions f ab f ++ and f : Distribution function of photons with helicity up / down f + and f +: Correlations between photons with different helicity Multipole decomposition f ab (k, q) = 4π ( i) l 2l + 1 f ab,lm(k, q)ylm(q) s lm ab,lm (k) = Definition of polarisation modes dqq 3 f ab,lm (k, q) dqq3 f (0) I (q) I,lm = 1 2 ( ++,lm +,lm ) E,lm = 1 2 ( +,lm + +,lm ) B,lm = i 2 ( +,lm +,lm ) Second Order CMB Perturbations 5/ 23

10 Parity of E and B polarisation [Hu] B polarisation not induced by scalar sources Ô Can only be generated by gravity waves Second Order CMB Perturbations 6/ 23

11 E and B polarisation [Zaldarriaga] E polarisation: pure gradient field in analogy to electric field Induced by scalar and tensor fluctuations B polarisation: pure rotation field in analogy to magnetic field Only induced by tensor fluctuations Second Order CMB Perturbations 7/ 23

12 E and B polarisation [Zaldarriaga] E polarisation: pure gradient field in analogy to electric field Induced by scalar and tensor fluctuations B polarisation: pure rotation field in analogy to magnetic field Only induced by tensor fluctuations Second Order CMB Perturbations 7/ 23

13 Nongaussianity Statistical information of primordial perturbations imprinted in CMB Gaussian distribution determined by mean value and variance. Nongaussianity parameterised by f NL Second Order CMB Perturbations 8/ 23

14 Non gaussian temperature maps Even distribution of cold and warm spots Second Order CMB Perturbations 9/ 23

15 Non gaussian temperature maps Warm spots: numerous moderate warm regions Cold spots: few very cold spots Second Order CMB Perturbations 9/ 23

16 Higher order effects E polarisation was measured in 2002 by WMAP [Kovac et al., 02] B polarisation is not measured yet Tensor perturbations are small and the tensor-to-scalar ratio might be of the order of cosmological perturbations (10 5 ): Tensor induced B polarisation is formally a second order effect Second order B mode sources Weak lensing and time delay E mode converted to B mode by travel through an inhomogeneous universe [Lewis et al., 06] Higher order metric perturbations [Mollerach et al., 04] Scattering sources [Beneke, Fidler, 10] [Pitrou, 08] Second Order CMB Perturbations 10/ 23

17 Higher order effects E polarisation was measured in 2002 by WMAP [Kovac et al., 02] B polarisation is not measured yet Tensor perturbations are small and the tensor-to-scalar ratio might be of the order of cosmological perturbations (10 5 ): Tensor induced B polarisation is formally a second order effect Second order B mode sources Weak lensing and time delay E mode converted to B mode by travel through an inhomogeneous universe [Lewis et al., 06] Higher order metric perturbations [Mollerach et al., 04] Scattering sources [Beneke, Fidler, 10] [Pitrou, 08] Second Order CMB Perturbations 10/ 23

18 Higher order effects Second order nongaussianity Nongaussianity is naturally induced at second order: At first order lm can be written as: (1) lm = Φ(k)T l(k)y lm (k) < Φ(k 1 )Φ(k 2 )Φ(k 3 ) >= 0 < l1m 1 l2m 2 l3m 3 >= 0 Non vanishing bispectrum linked to primordial nongaussianity Second Order CMB Perturbations 11/ 23

19 Higher order effects Second order nongaussianity Nongaussianity is naturally induced at second order: At second order we obtain: (2) lm = K [Φ(k 1)Φ(k 2 )T lm (k, k 1, k 2 )Y lm (k)] < Φ(k 1 )Φ(k 2 )Φ(k 3 ) >= 0 < (2) l 1m 1 l2m 2 l3m 3 >= 0 Primordial nongaussianity contaminated by second order background Calculation of shape and magnitude of second order nongaussianity needed if primordial nongaussianity is small Second Order CMB Perturbations 11/ 23

20 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) A (2) n : Second order metric sources σ n : Weak lensing and time delay C nm (2) m : Free streaming term κ (2) n : Suppression term κ ς nm (2) m : Counter term κ ρ n : Scattering source term Second Order CMB Perturbations 12/ 23

21 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) A (2) n : Second order metric sources σ n : Weak lensing and time delay Convolutions over first order photon and metric perturbations Sources exist at any multipole moment C nm (2) m : Free streaming term κ (2) n : Suppression term κ ς nm (2) m : Counter term κ ρ n : Scattering source term Second Order CMB Perturbations 12/ 23

22 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) A (2) n : Second order metric sources σ n : Weak lensing and time delay C nm (2) m : Free streaming term Couples neighbouring moments, generating higher moments over time Couples E and B polarisation κ (2) n : Suppression term κ ς nm (2) m : Counter term κ ρ n : Scattering source term Second Order CMB Perturbations 12/ 23

23 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) A (2) n : Second order metric sources σ n : Weak lensing and time delay C nm (2) m : Free streaming term κ (2) n : Suppression term Induces gradient suppressing every moment Only relevant before recombination when κ is large κ ς nm (2) m : Counter term κ ρ n : Scattering source term Second Order CMB Perturbations 12/ 23

24 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) A (2) n : Second order metric sources σ n : Weak lensing and time delay C nm (2) m : Free streaming term κ (2) n : Suppression term κ ς nm (2) m : Counter term Counters suppression term for monopole Couples dipole and electron velocity Couples I mode quadrupole to E mode quadrupole κ ρ n : Scattering source term Second Order CMB Perturbations 12/ 23

25 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) A (2) n : Second order metric sources σ n : Weak lensing and time delay C nm (2) m : Free streaming term κ (2) n : Suppression term κ ς nm (2) m : Counter term κ ρ n : Scattering source term Convolutions over first order photon and electron perturbations Sources exist at any multipole moment, but high multipoles are suppressed Second Order CMB Perturbations 12/ 23

26 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) Tight coupling: Before recombination κ dominates Only monopole and dipole exist Baryon acoustic oscillations No polarisation exists Free streaming: After recombination κ vanishes All scattering sources vanish Higher multipoles are generated Lensing generates B polarisation During recombination: All terms must be calculated Scattering contributions generate B polarisation Second Order CMB Perturbations 13/ 23

27 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) Tight coupling: Before recombination κ dominates Only monopole and dipole exist Baryon acoustic oscillations No polarisation exists Free streaming: After recombination κ vanishes All scattering sources vanish Higher multipoles are generated Lensing generates B polarisation During recombination: All terms must be calculated Scattering contributions generate B polarisation Second Order CMB Perturbations 13/ 23

28 Second order toy equations (2) n + A (2) n + σ n + C nm (2) m = κ ( (2) n + ς nm (2) m + ρ n ) Tight coupling: Before recombination κ dominates Only monopole and dipole exist Baryon acoustic oscillations No polarisation exists Free streaming: After recombination κ vanishes All scattering sources vanish Higher multipoles are generated Lensing generates B polarisation During recombination: All terms must be calculated Scattering contributions generate B polarisation Second Order CMB Perturbations 13/ 23

29 Numerical challenges 1) The equations are statistical differential equations 2) We have to deal with a infinite system of coupled equations Solutions 1) We use Transfer functions to separate the statistical and analytical part 2) The line-of-sight integration can be used to solve the full hierarchy [Seljak, Zaldarriaga 96] n (η 0 ) = η 0 dη e κ(η) j nm (k(η 0 η))σ m (η) 0 Second Order CMB Perturbations 14/ 23

30 Numerical challenges 1) The equations are statistical differential equations 2) We have to deal with a infinite system of coupled equations Solutions 1) We use Transfer functions to separate the statistical and analytical part 2) The line-of-sight integration can be used to solve the full hierarchy [Seljak, Zaldarriaga 96] n (η 0 ) = η 0 dη e κ(η) j nm (k(η 0 η))σ m (η) 0 Second Order CMB Perturbations 14/ 23

31 Transfer functions Problem: First oder solutions needed, but only statistical properties can be calculated. Transfer functions (1) (1) lm (k) = Φ(k)T lm (k)y lm(k) [ ] (2) lm (k) = K Φ(k 1 )Φ(k 2 )T (2) lm (k, k 1, k 2 )Y lm (k) We obtain a system of ordinary differential equations for T (2) lm Statistical properties can be related to properties of primordial perturbations Second Order CMB Perturbations 15/ 23

32 Transfer functions Problem: First oder solutions needed, but only statistical properties can be calculated. Transfer functions (1) (1) lm (k) = Φ(k)T lm (k)y lm(k) [ ] (2) lm (k) = K Φ(k 1 )Φ(k 2 )T (2) lm (k, k 1, k 2 )Y lm (k) We obtain a system of ordinary differential equations for T (2) lm Statistical properties can be related to properties of primordial perturbations Second Order CMB Perturbations 15/ 23

33 Line-of-sight integration The line-of-sight integration is an analytical solution of the differential equation (2) n + C nm (2) m = κ (2) n + ξ n We identify ξ n = A (2) n σ n κ (ς nm (2) m + ρ n ) Which leads to the integral equation (2) n (η 0 ) = η dη κ e κ(η) j nm (k(η 0 η))(ς mp (2) p (η) + ρ m (η)) η 0 0 dη e κ(η) j nm (k(η 0 η))(a (2) m (η) + σ m (η)) Second Order CMB Perturbations 16/ 23

34 Line-of-sight integration η 0 0 dη κ e κ(η) j nm (k(η 0 η))(ς mp (2) p (η) + ρ m (η)) Scattering contributions Visibility function κ e κ(η) is non-vanishing only around recombination ς mp vanishes for l p > 2 ρ m vanishes for large l m at early times Second order metric contributions Lensing and time-delay Second Order CMB Perturbations 17/ 23

35 Line-of-sight integration η 0 0 dη e κ(η) j nm (k(η 0 η))a (2) m (η) Scattering contributions Second order metric contributions Important also at late times During matter domination and later A (2) m photon perturbations Lensing and time-delay evolves almost independent of Second Order CMB Perturbations 17/ 23

36 Line-of-sight integration η 0 0 dη e κ(η) j nm (k(η 0 η))σ m (η) Scattering contributions Second order metric contributions Lensing and time-delay Only depends on first order perturbations Sources at any l m Second Order CMB Perturbations 17/ 23

37 Approximations Solve the early hierarchy using transfer functions Line-of-sight integration done using numerical methods Only use sources for the lowest l 4. Higher sources depend on higher multipoles which are induced later, when the scattering rate is neglectable Second Order CMB Perturbations 18/ 23

38 Results Second order transfer functions Second Order CMB Perturbations 19/ 23

39 Results 4e-20 1e-20 l(l+1) 2π C BB,vector l 3.5e-20 3e e-20 2e e-20 1e-20 nl = 2 5e-21 nl = 3 nl = 4 nl = l l(l+1) 2π C BB,tensor l 9e-21 8e-21 7e-21 6e-21 5e-21 4e-21 3e-21 2e-21 nl = 2 nl = 3 1e-21 nl = 4 nl = l Peaks around l 1000 Double peak structure for vector modes Second Order CMB Perturbations 20/ 23

40 Results Comparable to a tensor-to-scalar ratio of 10 6 at l = 100 and 10 4 at l = 1000 Comparable to effect of second order metric perturbations [Mollerach et al., 04] Second Order CMB Perturbations 21/ 23

41 Results Bispectrum preliminary!!! Second Order CMB Perturbations 22/ 23

42 Outlook We have written a second order CMB code which computes Second order nongaussianity Second order B-polarisation Induced from Scattering sources Metric sources Work in progress Lensing and time delay Second Order CMB Perturbations 23/ 23

43 Outlook We have written a second order CMB code which computes Second order nongaussianity Second order B-polarisation Induced from Scattering sources Metric sources Work in progress Lensing and time delay Second Order CMB Perturbations 23/ 23

44 Outlook We have written a second order CMB code which computes Second order nongaussianity Second order B-polarisation Induced from Scattering sources Metric sources Work in progress Lensing and time delay Thank you for your attention! Second Order CMB Perturbations 23/ 23