The causal structure of the CMB or: how to fit the CMB into a box

Transcription

1 The causal structure of the CMB or: how to fit the CMB into a box L. Raul Abramo University of São Paulo, Physics Dept. & Princeton University, Dept. of Astrophysical Sciences & University of Pennsylvannia, Dept. of Physics and Astronomy R. A., P. Reimberg & H. Xavier, arxiv: (PRD 21) R. A. & H. Xavier, astro-ph/ (PRD 27)

2 Causal view of our local Universe: past light cone

3 Causal view of our local Universe: past light cone

4 Causal view of our local Universe: past light cone

5 Causal view of our local Universe: past light cone Initial (z>>1) conditions of density/temperature fields

6 Causal view of our local Universe: past light cone COSMOS survey Initial (z>>1) conditions of density/temperature fields

7

8 Komatsu

9 Komatsu

10 CMB Temperature: H, Ωcdm, Ωb, ΩΛ, ΩK, ns, σ8 and Polarization: very early universe (inflation or else?) gravity waves adiabatic/isocurvature topological defects reionization magnetic fields etc.

11 CMB Temperature: H, Ωcdm, Ωb, ΩΛ, ΩK, ns, σ8 and WMAP 3y, Page et al. TE TT Polarization: very early universe (inflation or else?) gravity waves adiabatic/isocurvature topological defects reionization magnetic fields etc. BB GW/Inflation EE B from weak lensing of E synchroton+dust, E synchroton+dust, B

12 Why care about the CMB in such exquisite detail? window into gravity at highest energy scales

13 Why care about the CMB in such exquisite detail? window into gravity at highest energy scales

14 Sources of CMB anisotropies: INHOMOGENEITIES Sachs-Wolfe effect ISW Sunyaev-Zel dovich Gravitational lensing Gravity waves

15 Fact #1 about CMB polarization: it s generated by temperature anisotropies Bond & Efstathiou 1984 Kosowski 1996 Seljak & Zaldarriaga 1997 Hu & White 1997

16 Fact #1 about CMB polarization: it s generated by temperature anisotropies Bond & Efstathiou 1984 Kosowski 1996 Seljak & Zaldarriaga 1997 Hu & White 1997 Thompson scattering of CMB photons off free electrons generates polarization:!!!

17 Fact #1 about CMB polarization: it s generated by temperature anisotropies Bond & Efstathiou 1984 Kosowski 1996 Seljak & Zaldarriaga 1997 Hu & White 1997 Thompson scattering of CMB photons off free electrons generates polarization:!!! Integrating over incident radiation field Ii gives the polarization of final state: Polarization quadrupole of incident radiation

18 Fact # 2 about polarization: it s a spin-2 field! Q>, U= Q<, U= Q=, U> Q=, U<

19 Fact # 2 about polarization: it s a spin-2 field! Q>, U= Q<, U= Q=, U> Q=, U< Under rotations: ϕ

20 Fact # 2 about polarization: it s a spin-2 field! Q>, U= Q<, U= Q=, U> Q=, U< Under rotations: ϕ Linear polarization: Polarization has spin 2!

21 ˆn

22 ˆn Temperature: spin- field Polarization: spin-2 field Θ(ˆn) = T (ˆn) T = m Θ m Y m (ˆn) P (ˆn) = (Q iu)(ˆn) 4I = m P m 2 Y m (ˆn) C T T = m Θ m 2 C P P/EE = P m 2 m

23 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster)

24 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster)

25 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster)

26 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster) zez= z=189

27 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster) zez= z=189

28 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster) zez= z=189

29 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster) zez= z=189

30 E.g.: CMB polarization from a cloud of free electrons (e.g., a galaxy cluster) zez= z=189

31 Many scatterings distribution of free electrons visibility µ = η2 η 1 γ = e µ g = d dη 1 γ dη a(η) σ T n e µ: optical depth to Thompson scattering γ: probability that photons do not scatter g (visibility function): scattering probability Recombination Reionization Big bang today

32 Polarization from scatterings at many epochs

33 Polarization from scatterings at many epochs η η η

34 Polarization from scatterings at many epochs x = x + (η η )ˆl η η η x, η P (x, η; ˆl) = Q iu 4I = η Temperature anisotropy (l=2) dη g(η, η )θ 2 (x, η ) +...

35 CMB anisotropies: momenta obey non-local equations! Θ m (η) = 4π i P m (η) = 4π i d 3 k (2π) Y m(ˆk) θ 3/2 ( k, η) d 3 k (2π) Y m(ˆk) p 3/2 ( k, η) Seljak & Zaldarriaga 1996 Hu & White 1997

36 CMB anisotropies: momenta obey non-local equations! Θ m (η) = 4π i P m (η) = 4π i d 3 k (2π) Y m(ˆk) θ 3/2 ( k, η) d 3 k (2π) Y m(ˆk) p 3/2 ( k, η) Seljak & Zaldarriaga 1996 Hu & White 1997

37 CMB anisotropies: momenta obey non-local equations! Θ m (η) = 4π i P m (η) = 4π i d 3 k (2π) Y m(ˆk) θ 3/2 ( k, η) d 3 k (2π) Y m(ˆk) p 3/2 ( k, η) Sachs-Wolfe equation (inhomogeneities temperature anisotropies) Seljak & Zaldarriaga 1996 Hu & White 1997 θ () ( k, η) = η dη g(η, η ) θ ( k, η ) + Φ( k, η ) + v b ( k, η ) η + γ(η, η )(Φ + Φ )( k, η ) j (k η)

38 CMB anisotropies: momenta obey non-local equations! Θ m (η) = 4π i P m (η) = 4π i d 3 k (2π) Y m(ˆk) θ 3/2 ( k, η) d 3 k (2π) Y m(ˆk) p 3/2 ( k, η) Sachs-Wolfe equation (inhomogeneities temperature anisotropies) Seljak & Zaldarriaga 1996 Hu & White 1997 θ () ( k, η) = η dη g(η, η ) θ ( k, η ) + Φ( k, η ) + v b ( k, η ) η + γ(η, η )(Φ + Φ )( k, η ) j (k η) θ ( k, η) = θ () ( k, η) η dη g(η, η ) θ 2 ( k, η) 6p 2 ( k, η) (k η) 2 j (k η)

39 CMB anisotropies: momenta obey non-local equations! Θ m (η) = 4π i P m (η) = 4π i d 3 k (2π) Y m(ˆk) θ 3/2 ( k, η) d 3 k (2π) Y m(ˆk) p 3/2 ( k, η) Sachs-Wolfe equation (inhomogeneities temperature anisotropies) Seljak & Zaldarriaga 1996 Hu & White 1997 θ () ( k, η) = η dη g(η, η ) θ ( k, η ) + Φ( k, η ) + v b ( k, η ) η + γ(η, η )(Φ + Φ )( k, η ) j (k η) θ ( k, η) = θ () ( k, η) η dη g(η, η ) θ 2 ( k, η) 6p 2 ( k, η) (k η) 2 j (k η) p ( k, η) = 3 4 ( + 2)! ( 2)! η dη g(η, η ) θ 2 ( k, η) 6p 2 ( j (k η) k, η) (k η) 2

40 CMB anisotropies: momenta obey non-local equations! Θ m (η) = 4π i P m (η) = 4π i d 3 k (2π) Y m(ˆk) θ 3/2 ( k, η) d 3 k (2π) Y m(ˆk) p 3/2 ( k, η) Sachs-Wolfe equation (inhomogeneities temperature anisotropies) Seljak & Zaldarriaga 1996 Hu & White 1997 θ () ( k, η) = η dη g(η, η ) θ ( k, η ) + Φ( k, η ) + v b ( k, η ) η + γ(η, η )(Φ + Φ )( k, η ) Sources: inhomogeneities - S(k,η) j (k η) θ ( k, η) = θ () ( k, η) η dη g(η, η ) θ 2 ( k, η) 6p 2 ( k, η) (k η) 2 j (k η) p ( k, η) = 3 4 ( + 2)! ( 2)! η dη g(η, η ) θ 2 ( k, η) 6p 2 ( j (k η) k, η) (k η) 2 1. Inhomogeneities generate temperature anisotropies 2. Temperature anisotropies generate polarization 3. They feed back into each other

41

42 Now: how about causality? Initial conditions (t=const hypersurface) v. Observations on light-cone

43 Now: how about causality? Initial conditions (t=const hypersurface) v. Observations on light-cone z 11

44 Now: how about causality? Initial conditions (t=const hypersurface) v. Observations on light-cone z 11 z= RLSS we will appear here......in about 14 Gy! RLSS

45 Now: how about causality? Initial conditions (t=const hypersurface) v. Observations on light-cone z 11 z= RLSS S(x) we will appear here......in about 14 Gy! RLSS S( k) or S(x)?

46 The way to address causality is to go to position space

47 The way to address causality is to go to position space Describe the Universe in terms of onion peels: Heavens & Taylor, 1995 A. J. Hamilton, 1997 R.A. & H. Xavier PRD 27 R.A., P. Reimberg and H. Xavier PRD 21 S(x) = m S m (x)y m (ˆx) S( k) = m S m (k)y m (ˆk)

48 The way to address causality is to go to position space Describe the Universe in terms of onion peels: Heavens & Taylor, 1995 A. J. Hamilton, 1997 R.A. & H. Xavier PRD 27 R.A., P. Reimberg and H. Xavier PRD 21 S(x) = m S m (x)y m (ˆx) S( k) = m S m (k)y m (ˆk) Turns out there is a nice relation between the harmonics in Fourier space and in position space - the Hankel transform: S m (x) = S m (k) = 2 π i 2 π ( i) dk k 2 j (kx)s m (k) dx x 2 j (kx)s m (x)

49 The way to address causality is to go to position space Describe the Universe in terms of onion peels: Heavens & Taylor, 1995 A. J. Hamilton, 1997 R.A. & H. Xavier PRD 27 R.A., P. Reimberg and H. Xavier PRD 21 S(x) = m S m (x)y m (ˆx) S( k) = m S m (k)y m (ˆk) Turns out there is a nice relation between the harmonics in Fourier space and in position space - the Hankel transform: S m (x) = S m (k) = 2 π i 2 π ( i) dk k 2 j (kx)s m (k) dx x 2 j (kx)s m (x) dk k 2 j (kx)j (kx ) = π 2 x 2 δ(x x )

50 From Fourier space to position space: temperature (lowest order in scatterings) Θ m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) θ ( k, η)

51 From Fourier space to position space: temperature (lowest order in scatterings) Θ m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) θ ( k, η) = 2 π i dk k 2 θ,m (k, η)

52 From Fourier space to position space: temperature (lowest order in scatterings) Θ m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) θ ( k, η) = 2 π i Temperature anisotropies (momenta) are generated from the sources: dk k 2 θ,m (k, η) θ,m (k, η) η dη S m (k, η ) j (k η) = η dη 2 π ( i) dx x 2 j (kx) S m (x, η ) j (k η)

53 From Fourier space to position space: temperature (lowest order in scatterings) Θ m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) θ ( k, η) = 2 π i Temperature anisotropies (momenta) are generated from the sources: dk k 2 θ,m (k, η) θ,m (k, η) η dη S m (k, η ) j (k η) = η dη 2 π ( i) dx x 2 j (kx) S m (x, η ) j (k η) Substituting this back into Θ we obtain: Θ m (η) dx x 2 η dη S m (x, η ) 2 π dk k 2 j (kx) j (k η)

54 From Fourier space to position space: temperature (lowest order in scatterings) Θ m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) θ ( k, η) = 2 π i Temperature anisotropies (momenta) are generated from the sources: dk k 2 θ,m (k, η) θ,m (k, η) η dη S m (k, η ) j (k η) = η dη 2 π ( i) dx x 2 j (kx) S m (x, η ) j (k η) Substituting this back into Θ we obtain: Θ m (η) η dx x 2 η dη S m (x, η ) 2 π δ(x η) PLC! dk k 2 j (kx) j (k η) dη S m (x = η, η ) Sources averaged over the PLC!

55 From Fourier space to position space: temperature (lowest order in scatterings) Θ m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) θ ( k, η) = 2 π i Temperature anisotropies (momenta) are generated from the sources: dk k 2 θ,m (k, η) θ,m (k, η) η dη S m (k, η ) j (k η) = η dη 2 π ( i) dx x 2 j (kx) S m (x, η ) j (k η) Substituting this back into Θ we obtain: Θ m (η) η dx x 2 η dη S m (x, η ) 2 π δ(x η) PLC! dk k 2 j (kx) j (k η) dη S m (x = η, η ) Sources averaged over the PLC!

56 From Fourier space to position space: polarization P m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) p ( k, η) R.A., P. Reimberg and H. Xavier 21

57 From Fourier space to position space: polarization P m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) p ( k, η) = 2 π i R.A., P. Reimberg and H. Xavier 21 dk k 2 p,m (k, η)

58 From Fourier space to position space: polarization P m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) p ( k, η) = 2 π i Polarization momenta temp. anisotropies sources in position space: R.A., P. Reimberg and H. Xavier 21 dk k 2 p,m (k, η) p,m (k, η) η dη θ 2,m (k, η ) j (k η) (k η) 2 η η dη dη 2 π ( i) dx x 2 j (kx) S m (x, η ) j 2 (k η ) j (k η) (k η) 2

59 From Fourier space to position space: polarization P m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) p ( k, η) = 2 π i Polarization momenta temp. anisotropies sources in position space: R.A., P. Reimberg and H. Xavier 21 dk k 2 p,m (k, η) p,m (k, η) η dη θ 2,m (k, η ) j (k η) (k η) 2 η η dη dη 2 π ( i) dx x 2 j (kx) S m (x, η ) j 2 (k η ) j (k η) (k η) 2 Substituting this back into P we obtain: P m (η) dx x 2 η η dη dη S m (x, η ) 2 π dk k 2 j (kx) j (k η) (k η) 2 j 2(k η ) Now what???...

60 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2

61 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y y y ψ x Close triangle if: x y + y y y + x y x + y

62 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y Close triangle if: x y + y y y + x y x + y

63 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η Close triangle if: x y + y y y + x y x + y

64 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η Close triangle if: x y + y y y + x y x + y

65 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η η Close triangle if: x y + y y y + x y x + y

66 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η η η Close triangle if: x y + y y y + x y x + y

67 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η η η Close triangle if: x y + y y y + x y x + y

68 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η η η Close triangle if: x y + y y y + x y x + y x η + η = η η η η + x η x + η

69 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η η η Close triangle if: x y + y y y + x y x + y x η + η = η η η η + x η x + η

70 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η η η Close triangle if: x y + y y y + x y x + y x η + η = η η η η + x η x + η

71 In fact, this is just what s necessary to enforce causality! dk j (kx) j (ky) j 2 (ky )= xy (cos ψ)sin 2 ψ π 4 y P ( 2) 2 y y ψ x x : position of the source y =Δη : interval between scattering and observation y =Δη : interval between source and scattering ψ : angle between x & y η η η η η Close triangle if: x y + y y y + x y x + y x η + η = η η η η + x η x + η η η x η + η

72 Causality constraints like these apply to the exact theory, with an arbitrary number of scatterings! dk j (kx) j (ky) j 2 (ky ) y x y

73 Causality constraints like these apply to the exact theory, with an arbitrary number of scatterings! dk j (kx) j (ky) j 2 (ky ) y x y yxx y

74 Causality constraints like these apply to the exact theory, with an arbitrary number of scatterings! dk j (kx) j (ky) j 2 (ky ) dk k 2 j (kx) j (ky) j 2 (ky ) j 2 (ky ) y y x yxx y y x y y

75 Diagrammatic expansion of the kernel: Θ m (x) = dy 1 dy n K (x, y 1,,y n ) S m (y n ) n=1 K + dk k 2 j (kx) j (ky 1 )= π 2 δ(x y 1) dk k j (kx) j (ky 1 ) j 2 (ky 2 ) + + dk k 2 j (kx) j (ky 1 ) j 2 (ky 2 ) j 2 (ky 3 ) dk k 4 j (kx) j (ky 1 ) j 2 (ky 2 ) j 2 (ky 3 ) j 2 (ky 4 ) +

76 Diagrammatic expansion of the kernel: Θ m (x) = dy 1 dy n K (x, y 1,,y n ) S m (y n ) n=1 K x + y1 dk k 2 j (kx) j (ky 1 )= π 2 δ(x y 1) causality y4 x x y3 x y2 + + y3 + y1 y1 y2 y1 y2 dk k j (kx) j (ky 1 ) j 2 (ky 2 ) dk k 2 j (kx) j (ky 1 ) j 2 (ky 2 ) j 2 (ky 3 ) causality dk k 4 j (kx) j (ky 1 ) j 2 (ky 2 ) j 2 (ky 3 ) j 2 (ky 4 ) causalit c

77 Application # I: CMB in the Fourier-Bessel expansion R.A., Reimberg & Xavier 21

78 Application # I: CMB in the Fourier-Bessel expansion R.A., Reimberg & Xavier 21 Information that propagates to the CMB comes only from inside the PLC

79 Application # I: CMB in the Fourier-Bessel expansion R.A., Reimberg & Xavier 21 Information that propagates to the CMB comes only from inside the PLC Transfer of power from inhomogeneities (sources) to anisotropies is regulated by the visibility (optical depth) function of time time vanishing visibility

80 Application # I: CMB in the Fourier-Bessel expansion R.A., Reimberg & Xavier 21 Information that propagates to the CMB comes only from inside the PLC Transfer of power from inhomogeneities (sources) to anisotropies is regulated by the visibility (optical depth) function of time time vanishing visibility Initial conditions only need to be specified in this box

81 The Fourier-Bessel expansion f(x) = m f im Y m (ˆx) j (k i x) i().1 k i = q i R, j (q i )= 1 dz j (q i z)j (q j z)= 1 2 δ ij j 2 +1(q i ) i() j (q i z 1 )j (q i z 2 ) j 2 +1 (q i) = 1 2 δ(z 1 z 2 )

82 The Fourier-Bessel expansion Suppose we have some function defined inside a spherical region, with (e.g.) Dirichlet boundary conditions such that f(r=r)=. Then f can be expanded as: f(x) = m f im Y m (ˆx) j (k i x) i().1 The momenta kil that enter into the Fourier-Bessel expansion are given by the roots of the Bessel functions: k i = q i R, j (q i )= Orthogonality and completeness: 1 dz j (q i z)j (q j z)= 1 2 δ ij j 2 +1(q i ) i() j (q i z 1 )j (q i z 2 ) j 2 +1 (q i) = 1 2 δ(z 1 z 2 )

83 Fourier v. Fourier-Bessel q i

84 Fourier v. Fourier-Bessel Θ m =4πi P m =4πi d 3 k (2π) 3/2 Y m(ˆk)θ ( k) d 3 k (2π) 3/2 Y m(ˆk)p ( k) <k q i

85 Fourier v. Fourier-Bessel Θ m =4πi P m =4πi d 3 k (2π) 3/2 Y m(ˆk)θ ( k) d 3 k (2π) 3/2 Y m(ˆk)p ( k) Θ m = i() P m = i() θ im p im <k k i() = q i R, j (q i )= ???????????????????????????????????????????????????????????????? l=2 l=3 l=4 k q i

86 Application #2: reconstruction of the initial conditions of the Universe ( polarization tomography ) Fourier space: Kamionkowski & Loeb 97 Seto & Sasaki Cooray et al. 3-4 Seto & Pierpaoli 5 Liu, Silva & Aghanim 5 Amblard & White 5 Bunn 6 Complete solution in position space: R.A. & Xavier 7

87 Application #2: reconstruction of the initial conditions of the Universe ( polarization tomography ) Fourier space: Kamionkowski & Loeb 97 Seto & Sasaki Cooray et al. 3-4 Seto & Pierpaoli 5 Liu, Silva & Aghanim 5 Amblard & White 5 Bunn 6 Complete solution in position space: R.A. & Xavier 7 Clusters with known optical depths (e.g., SZ)

88 Application #2: reconstruction of the initial conditions of the Universe ( polarization tomography ) Fourier space: Kamionkowski & Loeb 97 Seto & Sasaki Cooray et al. 3-4 Seto & Pierpaoli 5 Liu, Silva & Aghanim 5 Amblard & White 5 Bunn 6 Complete solution in position space: R.A. & Xavier 7 Clusters with known optical depths (e.g., SZ)

89 Application #2: reconstruction of the initial conditions of the Universe ( polarization tomography ) Fourier space: Kamionkowski & Loeb 97 Seto & Sasaki Cooray et al. 3-4 Seto & Pierpaoli 5 Liu, Silva & Aghanim 5 Amblard & White 5 Bunn 6 Complete solution in position space: R.A. & Xavier 7 Clusters with known optical depths (e.g., SZ) By studying polarization from many clusters, we can learn about the Universe in many different places inside our PLC! But... how?

90 We have shown that, to a good approximation (2 nd order in g): P m (η) dk K (η, k) S i m(k) P m (η) R dx K (η, x) S i m(x)

91 We have shown that, to a good approximation (2 nd order in g): P m (η) P m (η) dk K (η, k) Sm(k) i R dx K (η, x) S i m(x) Can we invert this to get the S s, and reconstruct the I.C. of the local Universe? If we can invert these expressions (Fredholm integral equations), we could then connect initial and final state of our local Universe (observed on the PLC):

92 We have shown that, to a good approximation (2 nd order in g): P m (η) P m (η) dk K (η, k) Sm(k) i R dx K (η, x) S i m(x) Can we invert this to get the S s, and reconstruct the I.C. of the local Universe? If we can invert these expressions (Fredholm integral equations), we could then connect initial and final state of our local Universe (observed on the PLC): I.C., Φ~δρ/ρ~1-5 Structure formation Final state: LSS w/ δρ/ρ >> 1!!!

93 So: can it work -- even if only in theory?

94 So: can it work -- even if only in theory? X Fourier space reconstruction does not work, even in theory, because of the intrinsic statistical uncertainties of the Fourier modes!

95 So: can it work -- even if only in theory? X Fourier space reconstruction does not work, even in theory, because of the intrinsic statistical uncertainties of the Fourier modes! Position space reconstruction works perfectly - in theory. In practice, limited by: Noisy data (signal very small, <~ µk) Unresolved sources (e.g., filaments) with high optical depth Partial sky coverage (fsky) Angular resolution of polarization data (~ number of clusters)

96 For a hypothetical survey of ~1 5 clusters with polarization data, covering 2/3 of the sky, and up to z=5, we get (with 1 4 simulations of the reconstruction scheme): Slm(reconstr.)/Slm(th)

97 For a hypothetical survey of ~1 5 clusters with polarization data, covering 2/3 of the sky, and up to z=5, we get (with 1 4 simulations of the reconstruction scheme): Slm(reconstr.)/Slm(th) (Gaussian errors)

98 For a hypothetical survey of ~1 5 clusters with polarization data, covering 2/3 of the sky, and up to z=5, we get (with 1 4 simulations of the reconstruction scheme): Slm(reconstr.)/Slm(th) (Gaussian errors) Cluster polariz. signal: <~ μk. Very, very difficult! With SPT: > 1 6 hs obs. time! (Would need a > 2m aperture telescope) Observations a bit futuristic, but we can reconstruct our init. cond. s!

99 Conclusions

100 Conclusions Line-of-sight solution in position space reveals causal structure of CMB All the information that determines the physical observables (temperature and polarization maps) are a consequence of initial conditions inside our PLC -- like it should We can consistently put the CMB in a (spherical) box, and ignore what s outside of that box! Cluster polarization surveys may one day be used to reconstruct the initial conditions (@ z~11) of our entire observable Universe (r<rlss.) That would be a formidable test of the structure formation paradigm! More applications: Constrained simulations of temperature/polarization Non-gaussian/inhomogeneous/anisotropic models

101 The End

102 CMB ingredients: inhomogeneities + free electrons + anisotropies Primary: z ~ LSS z ~ 3-6 (?) Reionization z < 3 galaxy clusters (SZ)

103 CMB ingredients: inhomogeneities + free electrons + anisotropies scattering into/out of the l.o.s. Primary: z ~ LSS z ~ 3-6 (?) Reionization z < 3 galaxy clusters (SZ)

104 Penrose, 1959: the least-squares solution to a singular linear problem is given by the Moore-Penrose pseudo-inverse: The pseudo-inverse is built using the Singular Value Decomposition, which states that any matrix can be written in the form: where U and V are square orthogonal matrices, and Λ is the singular values list (eigenvalues list): Quite generically, the more null singular values, the more intrinsically uncertain is the inversion.

105

106 Thickness of the Last Scattering Surface (as a function or redshift)

107 Polarization: brief overview 1 Individual photons possess fixed linear polarization, given by the direction of the electric field:

108 Polarization: brief overview 1 Individual photons possess fixed linear polarization, given by the direction of the electric field: A generic radiation field is a multi-photon state mix of linear polarizations For a beam propagating in the direction z the Stokes parameters are:

109 Polarization: brief overview 1 Individual photons possess fixed linear polarization, given by the direction of the electric field: A generic radiation field is a multi-photon state mix of linear polarizations For a beam propagating in the direction z the Stokes parameters are: Intensity Polarizations - Polarizations / - \ Circular polarization (= f/ monochromatic wave)

110 Polarization: linear modes Q (N-S, E-W) and U (SW-NW, SW-NE): Q>, U= Q<, U= Q=, U< Q=, U> The Stokes parameters depend explicitly on the choice of reference frame Sometimes it s better to use scalars/pseudoscalars under rotation: GRADIENT mode ( electric, E) ROTATIONAL mode ( magnetic, B) E-mode does not change under parity (mirroring) B-mode switches sign under parity

111 Gravity waves are spin-2 fields in 3D! Any combination of x and + GW linear modes are possible, so circular polarizations (L and R) are equally good measures of GW polarization

112 Gravity waves are spin-2 fields in 3D! Any combination of x and + GW linear modes are possible, so circular polarizations (L and R) are equally good measures of GW polarization The temperature pattern of photons as seen by some scattering center acquires also a rotation (curl) after projection

113 Gravity waves are spin-2 fields in 3D! Any combination of x and + GW linear modes are possible, so circular polarizations (L and R) are equally good measures of GW polarization The temperature pattern of photons as seen by some scattering center acquires also a rotation (curl) after projection E and B modes

114 Density (scalar) perturbations only lead to E (even) modes

115 Density (scalar) perturbations only lead to E (even) modes Gravity waves are spin-2 fields They generate E and B modes

116 Density (scalar) perturbations only lead to E (even) modes Gravity waves are spin-2 fields They generate E and B modes However, weak lensing (a spin-2 operator!) can also generate a B mode from a pre-existing E mode!

117 CMB Temperature: Precision cosmology H, Ωcdm, Ωb, ΩΛ, ΩK, ns, σ8, etc. WMAP 3y, Spergel et al.

118 CMB Temperature: Precision cosmology H, Ωcdm, Ωb, ΩΛ, ΩK, ns, σ8, etc. ISW/ Dark Energy Curvature Baryons/CDM WMAP 3y, Spergel et al. CDM: SDSS & clusters

119 CMB anisotropies: the line-of-sight integral solution Seljak & Zaldarriaga 1996 Hu & White 1997

120 CMB anisotropies: the line-of-sight integral solution Seljak & Zaldarriaga 1996 Hu & White 1997 Between scatterings, photons are freely propagating plane waves separate angular modes from radial modes: e i k x =4π m i j (k x) Y m(ˆk) Y m ( ˆx) x η ˆx ˆl ˆl

121 CMB anisotropies: the line-of-sight integral solution Seljak & Zaldarriaga 1996 Hu & White 1997 Between scatterings, photons are freely propagating plane waves separate angular modes from radial modes: e i k x =4π m i j (k x) Y m(ˆk) Y m ( ˆx) x η ˆx ˆl ˆl Anisotropies at x=, in terms of their Fourier modes ( momenta ): Θ m (η) = 4π i P m (η) = 4π i d 3 k (2π) 3/2 Y m(ˆk) θ ( k, η) d 3 k (2π) 3/2 Y m(ˆk) p ( k, η) Momenta of temperature and polarization anisotropies MUST COMPUTE THEM!