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THE NON-GAUSSIAN SKY Alex Kehagias NTU Athens Heraklion, March - PowerPoint PPT Presentation

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THE NON-GAUSSIAN SKY Alex Kehagias NTU Athens Heraklion, March 2013 . . . . . . Work in collaboration with A. Riotto: A. Kehagias and A. Riotto, Operator Product Expansion of Inflationary Correlators and Conformal Symmetry of de

  1. THE NON-GAUSSIAN SKY Alex Kehagias NTU Athens Heraklion, March 2013 . . . . . .

  2. Work in collaboration with A. Riotto: A. Kehagias and A. Riotto, “Operator Product Expansion of Inflationary Correlators and Conformal Symmetry of de Sitter,” Nucl. Phys. B 864 , 492 (2012) [arXiv:1205.1523 [hep-th]]. A. Kehagias and A. Riotto, “The Four-point Correlator in Multifield Inflation, the Operator Product Expansion and the Symmetries of de Sitter,” Nucl. Phys. B 868 , 577 (2013) [arXiv:1210.1918 [hep-th]]. A. Kehagias and A. Riotto, “Symmetries and Consistency Relations in the Large Scale Structure of the Universe,” arXiv:1302.0130 [astro-ph.CO]. . . . . . .

  3. J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 0305 , 013 (2003) [astro-ph/0210603]. J. M. Maldacena and G. L. Pimentel, “On graviton non-Gaussianities during inflation,” JHEP 1109 , 045 (2011). I. Antoniadis, P. O. Mazur and E. Mottola, “Conformal Invariance, Dark Energy, and CMB Non-Gaussianity,” arXiv:1103.4164 [gr-qc]. P. Creminelli, “Conformal invariance of scalar perturbations in inflation,” Phys. Rev. D 85 , 041302 (2012) . . . . . .

  4. N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Phys. Rept. 402 , 103 (2004). T. Suyama and M. Yamaguchi, Phys. Rev. D 77 , 023505 (2008). S. Hollands and R. M. Wald, Gen. Rel. Grav. 40 , 2051 (2008). S. Ferrara, R. Gatto and A. F. Grillo, Annals Phys. 76 , 161 (1973). H. Osborn and A. C. Petkou, Annals Phys. 231 , 311 (1994) [hep-th/9307010]. A. Strominger, JHEP 0110 , 034 (2001) [hep-th/0106113]. . . . . . .

  5. Outline 1 Introduction . . . . . .

  6. Outline 1 Introduction 2 Single-Field Slow-Roll Inflation . . . . . .

  7. Outline 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations . . . . . .

  8. Outline 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations 4 Perturbations Cosmological Perturbations NG in multifield inflation . . . . . .

  9. Outline 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations 4 Perturbations Cosmological Perturbations NG in multifield inflation 5 Symmetry Constraints . . . . . .

  10. Outline 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations 4 Perturbations Cosmological Perturbations NG in multifield inflation 5 Symmetry Constraints 6 OPEs 3pt Function in the Squeezed Limit The four-point Correlator . . . . . .

  11. Outline 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations 4 Perturbations Cosmological Perturbations NG in multifield inflation 5 Symmetry Constraints 6 OPEs 3pt Function in the Squeezed Limit The four-point Correlator 7 Suyama-Yamaguchi inequality . . . . . .

  12. Current Section 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations 4 Perturbations Cosmological Perturbations NG in multifield inflation 5 Symmetry Constraints 6 OPEs 3pt Function in the Squeezed Limit The four-point Correlator 7 Suyama-Yamaguchi inequality . . . . . .

  13. The latest cosmological data from Planck The CMB sky as seen from Planck mission agree impressively well with a Universe which at large scales is : . . . . . .

  14. homogeneous, isotropic spatially flat, (well described by a FRW spatially flat geometry). . . . . . .

  15. The Planck CMB pattern as compared to the corresponding pattern of COBE and WMAP . . . . . .

  16. A theoretical puzzle: A flat FRW Universe is extremely fine tuned solution in GR. Many attempts have been put forward to solve this puzzle. However, the most developed and yet simple idea still remains Inflation . Inflation solves homogeneity, isotropy and flatness problems in one go just by postulating a rapid expansion of the early time Universe post Big Bang. A phenomenological implementation of Inflation: “slow rolling” scalar field the Inflaton . . . . . .

  17. Current Section 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations 4 Perturbations Cosmological Perturbations NG in multifield inflation 5 Symmetry Constraints 6 OPEs 3pt Function in the Squeezed Limit The four-point Correlator 7 Suyama-Yamaguchi inequality . . . . . .

  18. Homogeneous and isotropic Universe is described by the FRW metric ds 2 = − dt 2 + a ( t ) 2 d ⃗ x 2 (1) whereas the gravitational dynamics is governed by Einstein equation R µν − 1 2 g µν R = 8 π GT µν (2) . . . . . .

  19. Einstein equations are written for the FRW cosmology a = − 4 π G ¨ a ( ρ + 3 p ) 3 ( ˙ ) 2 = 8 π G a H 2 = ρ (3) a ˙ 3 from where the conservation equation ρ + 3 ˙ a ˙ a ( ρ + p ) = 0 (4) follows. . . . . . .

  20. Inflation is driven by a scalar field ϕ with a generic potential of the form . . . . . .

  21. Dymanics is described by the Lagrangian 16 π G R − 1 1 2 ∂ϕ 2 − V ( ϕ ) L = (5) with corresponding energy density and pressure ρ = 1 ˙ ϕ + V ( ϕ ) (6) 2 p = 1 ˙ ϕ − V ( ϕ ) (7) 2 When potential energy dominates kinetic energy 1 ˙ ϕ << V ( ϕ ) (8) 2 we get an equation of state a ≈ e Ht , p ≈ − ρ , H = const. (9) . . . . . .

  22. This is an almost de Sitter background, specified by the slow-roll parameters ) 2 ϵ = M 2 ( V ′ ( V ′′ ) P η = M 2 , (10) P 2 V V In the quasi-de Sitter phase ϵ << 1 , η << 1 (11) An important quantity is the number of e-folds ∫ t f N = log a f = (12) Hdt a i t i which, in terms of the scalar is written as ∫ ϕ f 8 π G V N = V ′ d ϕ (13) 3 ϕ i . . . . . .

  23. Current Section 1 Introduction 2 Single-Field Slow-Roll Inflation 3 De Sitter Space Representations 4 Perturbations Cosmological Perturbations NG in multifield inflation 5 Symmetry Constraints 6 OPEs 3pt Function in the Squeezed Limit The four-point Correlator 7 Suyama-Yamaguchi inequality . . . . . .

  24. The four-dimensional de Sitter spacetime of radius H − 1 is described by the hyperboloid defined by 5 = 1 η AB X A X B = − X 2 0 + X 2 i + X 2 ( i = 1 , 2 , 3) , (14) H 2 embedded in 5D Minkowski spacetime M 1 , 4 with coordinates X A and flat metric η AB = diag ( − 1 , 1 , 1 , 1 , 1 ). A particular parametrization of the de Sitter hyperboloid is provided by . . . . . .

  25. x 2 X 0 = 1 ( H η − 1 ) − 1 η , 2 H H η 2 X i = x i H η, x 2 X 5 = − 1 ( H η + 1 ) + 1 η , (15) 2 H H η 2 which may easily be checked that satisfies Eq. (14). The de Sitter metric is the induced metric on the hyperboloid from the five-dimensional ambient Minkowski spacetime d s 2 5 = η AB dX A dX B . (16) . . . . . .

  26. For the particular parametrization (15), for example, we find 1 d s 2 = − d η 2 + d ⃗ x 2 ) ( . (17) H 2 η 2 The group SO (1 , 4) acts linearly on M 1 , 4 . Its generators are ∂ ∂ J AB = X A ∂ X B − X B A , B = (0 , 1 , 2 , 3 , 5) (18) ∂ X A and satisfy the SO (1 , 4) algebra [ J AB , J CD ] = η AD J BC − η AC J BD + η BC J AD − η BD J AC . (19) . . . . . .

  27. We may split these generators as Π + Π − J ij , P 0 = J 05 , i = J i 5 + J 0 i , i = J i 5 − J 0 i , (20) which act on the de Sitter hyperboloid as ∂ ∂ J ij = x i − x j , ∂ x j ∂ x i P 0 = η ∂ ∂η + x i ∂ ∂ x i , ) ∂ i = − 2 H η x i ∂ − H η 2 ∂ Π − x 2 δ ij − 2 x i x j ( ∂η + H , ∂ x j ∂ x i i = 1 ∂ Π + (21) ∂ x i H . . . . . .

  28. They satisfy the commutator relations [ J ij , J kl ] = δ il J jk − δ ik J jl + δ jk J il − δ jl J ik , [ J ij , Π ± k ] = δ ik Π ± j − δ jk Π ± i , [Π ± k , P 0 ] = ∓ Π ± k , i , Π + [Π − j ] = 2 J ij + 2 δ ij P 0 . (22) This is the SO (1 , 4) algebra written in a strange base. . . . . . .

  29. More standard generators are P i = − i Π + K i = i Π − L ij = iJ ij , D = − iP 0 , i , i , (23) we get P i = − i H ∂ i , ( η ∂ ) ∂η + x i ∂ i D = − i , ( ) η ∂ − iH ( − η 2 + x 2 ) ∂ i , ∂η + x i ∂ i K i = − 2 iHx i ( ∂ ∂ ) L ij = i − x j . (24) x i ∂ x j ∂ x i These are also the Killing vectors of de Sitter spacetime. . . . . . .

  30. They generate space translations ( P i ), dilitations ( D ), special conformal transformations ( K i ) and space rotations ( L ij ). They satisfy the conformal algebra in its standard form [ D , P i ] = iP i , (25) [ D , K i ] = − iK i , (26) ( ) [ K i , P j ] = 2 i δ ij D − L ij (27) ( ) [ L ij , P k ] = i δ jk P i − δ ik P j , (28) ( ) [ L ij , K k ] = i δ jk K i − δ ik K j , (29) [ L ij , D ] = 0 , (30) ( ) [ L ij , L kl ] = i δ il L jk − δ ik L jl + δ jk L il − δ jl L ik . (31) . . . . . .

  31. The de Sitter algebra SO (1 , 4) has two Casimir invariants C 1 = − 1 2 J AB J AB , (32) C 2 = W A W A , W A = ϵ ABCDE J BC J DE . (33) Using Eqs. (20) and (23), we find that C 1 = D 2 + 1 2 { P i , K i } + 1 2 L ij L ij , (34) which turns out to be, in the explicit representation Eq. (24), H − 2 C 1 = − ∂ 2 ∂η 2 − 2 ∂ ∂η + ∇ 2 . (35) η . . . . . .

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