Non-Gaussianity and its Evolution in Multi-field Inflation - PowerPoint PPT Presentation

Non-Gaussianity and its Evolution in Multi-field Inflation Gerasimos Rigopoulos (Utrecht) Work in collaboration with E.P.S. Shellard (Cambridge) & B.J.W. van Tent (Orsay) Florence 24 October 2006 Non-Gaussianity and its Evolution in Multi-field Inflation – p. 1/20

Recent Interest Primordial non-Gaussianity has attracted a lot of interest over the past few years. • Precision Cosmology: Non-Linearities may be observable • Gravity is non-linear. Some non-Gaussianity will always be present • Potentially useful for further testing inflation • Consistency check - Identification of new physics beyond inflation (which may produce stronger NG signals) • Discriminant among models One more handle on the physics of Inflation . Primordial NG has been approached via various angles in an effort to compute it and relate it to observables. Non-Gaussianity and its Evolution in Multi-field Inflation – p. 2/20

Observations • Various detections of non-Gaussian signals have been reported in the CMB. However, none has been linked to a primordial source. • Extracting any primordial NG from LSS will probably be very difficult since it will be masked by later non-linear evolution. However, it has been suggested that finding early objects forming at high redshifts is probably a good strategy since they will be tracing the tail of the distribution (you just have to find one). • Thus observations focus on the CMB measuring f NL ∼ � TTT � � TT � 2 or even τ NL ∼ � TTTT � � TT � 2 . Current limits set − 54 < f NL < 114 (95%CL, WMAP). Planck is expected to reach f NL < 5 at best, while an ideal experiment is limited to f NL < 3 . However, ... Non-Gaussianity and its Evolution in Multi-field Inflation – p. 3/20

... a recent paper (astro-ph/0610257) has claimed that even f NL < 0 . 01 is accessible via observations of the 21 cm radio background. If this analysis goes through it will set the whole discussion on inflationary NG under a totally different light. Non-Gaussianity and its Evolution in Multi-field Inflation – p. 4/20

Calculating non-Gaussianity Focusing on Inflation, there are four regimes relevant for NG generation and evolution • Effects before or during horizon crossing. Calculating non-linear corrections from the inflationary mechanism for the generation of perturbations. • Long wavelength evolution during inflation. The subject of this talk... • Long wavelength evolution after inflation. E.g. Reheating, Preheating & The Curvaton scenario. • The relation of this primordial NG to the observed CMB sky. Solving the full system of Boltzman equations at second order. Non-Gaussianity and its Evolution in Multi-field Inflation – p. 5/20

Calculational Approaches • Straightforward second-order perturbation theory: Follow the route of linear theory by extending the perturbative analysis of the Einstein equations to second order. This seems essential for studying scales smaller or close to the horizon → proliferation of terms in the equations. • Long wavelength approximations: First focus on long wavelengths, particularly relevant during inflation, where the dynamics simplifies. This is the approach of the δN formalism as well as the one taken in this talk. Non-Gaussianity and its Evolution in Multi-field Inflation – p. 6/20

Long wavelength approximation ds 2 = − N 2 ( t, x ) dt 2 + e 2 α ( t, x ) h ij ( x ) dx i dx j On long wavelengths ( ∆ x > ( aH ) − 1 , ∇ 2 dropped): K ij ¯ 2 ¯ Ndt H = − 1 1 d K ij − p ( E + S/ 3) • 2 M 2 Ndt ¯ j = − 3 H ¯ ¯ d j ( x ) e − 3 α K i K i K i j = C i ⇒ • j Ndt Π A + 3 H Π A + ∂ A V = 0 D • • H 2 = K ij ¯ 6 ¯ 2 ∇ j ¯ K j 1 p E + 1 1 p Π A ∂ i φ A − 1 K ij , ∂ i H = − 3 M 2 2 M 2 i Π A = ¯ d d K ij = − e 2 α d � Ndt φ A , � H = Ndt α, 2 Ndt h ij ⇒ ∆ N formalism. ‘Separate universe’ evolution Non-Gaussianity and its Evolution in Multi-field Inflation – p. 7/20

Long Wavelength Coordinate Transformations Consider coordinate transformations which preserves g 0 i = 0 : ds 2 = − ˜ N 2 ( T, X ) dT 2 + e 2˜ α ( T, X ) ˜ h ij ( T, X ) dX i dX j X i = X i ( t, x l ) T = T ( t, x l ) , Then, up to O ( ∇ 2 ) the transformation matrix is: T ≡ ∂x µ ∂ µ T • Λ µ ∂T = N x=const ( ∂ t T ) 2 T=const � ( ∂ i ) 2 � j ≡ ∂x i • Λ i ˜ ∂X j = δ i ˜ j + O X=const � ( ∂ i ) 2 � • Λ 0˜ ∂t ∂ i T ∂X j = − δ i ˜ j ≡ ∂ t T + O j x i = X i + dT N∂ i T � Using these we learn: ( ∂ t T ) 2 • ds 2 = − ˜ N 2 ( T, x ) dT 2 + e 2˜ α ( T, x ) h ij ( x ) dx i dx j 1 ∂t = 1 ∂ ∂T + O ( ∇ 2 ) , ∂ ∂ ∂X i + ∂ i T ∂ ∂ ∂x i = • ˜ N ∂T N Non-Gaussianity and its Evolution in Multi-field Inflation – p. 8/20

Inhomogeneity Separate inhomogeneous evolution from the homogeneous background: • Given a spacetime scalars A ( t, x ) = A ( t ) + ∆ A ( t, x ) one can always set ∆ A = 0 by a suitable choice of time slicing - no coordinate invariant meaning for ∆ A • However, given two scalars A ( t, x ) and B ( t, x ) one can construct a fully non-linear variable which encodes the inhomogeneity and is a scalar (invariant) under long wavelength transformations: ∂ t B ∂ i B = ˜ C i ( t, x ) ≡ ∂ i A − ∂ t A C i ( T, x ) For example: � � ˙ ζ i = ∂ i α − H Ndt ζ i = − H d P ρ ∂ i ρ ⇒ ∂ i P − ρ ∂ i ρ ˙ ρ + P ˙ This is a fully non-linear statement formally similar to that of linear theory. Non-Gaussianity and its Evolution in Multi-field Inflation – p. 9/20

For Inflation... Consider a set of scalar fields during inflation T µν = G AB ∂ µ φ A ∂ ν φ B − g µν 2 ∂ λ φ A ∂ λ φ A + V ( φ ) � 1 � The following spatial vectors are scalar invariants: i = e α � � ∂ i φ A − ∂ t φ A H ≡ ∂ t α Q A � � NH ∂ i α , Nα � � ǫ ∂ i φ A + Π A Π A ≡ ∂ t φ A H 2 , κ 2 ≡ M − 2 ǫ ≡ κ 2 Π 2 ζ A i = − κ Π ∂ i α, N , ˜ √ p 2 2˜ i = ∂ i q A + . . . i = ∂ i ζ A + . . . Q A ζ A where q A Note that: and and ζ A well known linear gauge-invariant variables. phi_2 Define isocurvature and adiabatic A = Π A e_1 e 1 e 2 ˆ Π , ˆ , . . . directions e_2 A e 1 A = 0 , . . . e 2 ˆ A ˆ with phi_1 Non-Gaussianity and its Evolution in Multi-field Inflation – p. 10/20

� ... For more fields an iterative procedure will produce an orthonormal basis adapted to the trajectory with N-1 isocurvature directions. n − 1 Π A n − 1 η A η B e A ( n ) = ( 1 N D t ) ˜ ( n ) − B =1 ˜ ( n ) ˆ η A e A B ˜ ˆ n ≡ η n H n − 1 Π ˜ ( n ) where e A 1 e A n − 1 η A n η n ˜ ( n ) ≡ − ǫ A 1 ··· A n ˆ 1 . . . ˆ n − 1 ˜ ( n ) which gives the basis a definite handedness. Non-Gaussianity and its Evolution in Multi-field Inflation – p. 11/20

Non-linear Isocurvature and Adiabatic Variables e 1 A ∂ i φ A = ∂ i ln a − H • ζ i ≡ ˆ e 1 κ A ζ A i = ∂ i ln a − ǫ ˆ ρ ∂ i ρ √ ˙ 2˜ • σ i ≡ ˆ i = − κ e 2 A ζ A e 2 A ∂ i φ A ǫ ˆ √ 2˜ Define Slow Roll parameters: e 1 B V A η A ≡ − 3 H Π A + ∂ A V ξ A ≡ 3˜ e 1 A − 3˜ η A − ˆ ǫ ≡ κ 2 Π 2 ˜ ˜ H 2 , ˜ , ǫ ˆ B H 2 2 H Π Project isocurvature and adiabatic parts: η � ≡ ˆ η ⊥ ≡ ˆ ξ � ≡ ˆ ξ ⊥ ≡ ˆ ˜ A ˜ ˜ A ˜ e 1 e 2 e 1 e 2 η A , η A , ξ A , ξ A ˜ A ˜ ˜ A ˜ The non-linear equations of motion are formally the same as those of linear perturbation theory with δ → ∂ i f ( t ) → f ( t, x ) Non-Gaussianity and its Evolution in Multi-field Inflation – p. 12/20

Long wavelength equations of motion Choose a gauge with NH = 1 ( ∂ i α = 0 - homogeneous expansion) to simplify expressions � ζ i � ζ i � η ⊥ � 0 − 2˜ 0 � � d + = 0 σ i σ i 0 0 − 1 dt ˙ ˙ σ i σ i ˜ 0 κ ˜ λ where � V 22 ǫ 2 + 4˜ η � + 4(˜ η ⊥ ) 2 + ˜ η � � ξ � , κ ( t, x ) = 3 ˜ 3 H 2 + ˜ ǫ + ˜ + 2˜ ǫ ˜ ˜ η � λ ( t, x ) = 3 + ˜ ǫ + 2˜ All local quantities are given by: √ e m A ∂ i φ A = − • ∂ i ln H = ˜ 2˜ κ ζ m ǫ ǫ ζ i , i √ • e A 2˜ 1 ∂ i Π A = − H ǫ η � ζ i + ˜ η ⊥ σ i � � ˜ κ √ η � + ˜ • e A 2˜ 2 ∂ i Π A = − H ǫ η ⊥ ζ i + � � � � σ i + ˜ ˙ ˜ ǫ σ i κ Non-Gaussianity and its Evolution in Multi-field Inflation – p. 13/20

“Initial Conditions” Assuming that non-linearities are not important on short scales, one can include in a straightforward manner perturbations from shorter wavelengths. This amounts to adding “sources" on the rhs: � � � ζ i � ζ i ζ l ( k ) ˆ α k � � η ⊥ 0 − 2˜ 0 � � d 3 k ˙ d W ( k ) e i kx � σ l ( k ) ˆ + = ∂ i σ i σ i β k 0 0 − 1 dt (2 π ) 3 / 2 ˙ ˙ σ i σ i ˜ σ l ( k ) ˆ 0 ˜ κ λ ˙ β k where ˆ α k = a † ( k ) + a ( − k ) , β k = b † ( k ) + b ( − k ) ˆ a ( k ) , a † ( − k ′ ) = (2 π ) 3 δ ( k − k ′ ) , e.t.c. � � with κ H ζ l ( k ) = − √ √ W ( k ) cuts off short wavelength modes. Simplest 2 k 3 2˜ ǫ κ H σ l ( k ) = − √ √ choice: W ( k ) = Θ( caH − k ) . Final results are 2 k 3 2˜ ǫ κ H independent of the form of W ( k ) . σ l ( k ) = ǫ χ √ √ 2 k 3 2˜ When linearized, these equations are exact and valid to all scales, simply being linear perturbation theory. Non-Gaussianity and its Evolution in Multi-field Inflation – p. 14/20