CMB Bispectrum and non-Gaussian Inflation James Fergusson and Paul - PowerPoint PPT Presentation

CMB Bispectrum and non-Gaussian Inflation James Fergusson and Paul Shellard (DAMTP, Cambridge) [astro-ph/0612713] (also Michele Liguori (DAMTP)) with Gerasimos Rigopoulos (Utrecht/Helsinki) and Bartjan van Tent (Paris-Orsay) [astro-ph/0511041 etc.]

Dvali & Tye, 2000 B RANE I NFLATION Burgess, Quevedo et al 01 Jones, Stoica & Tye, 2002 KKLMMT, 2003 • Interbrane interaction creates inflationary potential • Brane collision = hybrid inflation reheating ‘Generic’ formation of cosmic strings Sarangi & Tye, 2002 See Majumdar review Extra fields and nonGaussianity hep-th/0512062 Observable signatures of extra dimensions?

Multifield inflation Gravity is inherently nonlinear NonGaussianity! Interacting inflationary potentials CMB observations discriminating inflation models • Gaussian a † + Φ ∗ ˆ Gaussian with � ˆ Φ ˆ Φ ˆ ⇒ Φ � = 0 Φ lin = Φ lin ˆ a lin ˆ • Non-Gaussian Φ = ˆ ˆ Φ lin + ˆ Φ NL where ˆ Φ NL = f NL ˆ Φ 2 lin ⇒ nonGaussian with � ˆ Φ ˆ Φ ˆ Φ � ∼ f NL � ˆ Φ 2 lin � 2 • Observational prospects ( Komatsu astro-ph/0206039) WMAP will observe | f NL | ≥ 20, Planck | f NL | > 5 Current WMAP bound: - 58 < f NL < 134 (95%)

Superhorizon non-Gaussianity • ‘Evolution’ equations (multifield inflation) − κ 2 dH 2 N Π B Π B , (1) = dt D t Π A − 3 NH Π A − NG AB V B , (2) = where V B ≡ ∂ B V ≡ ∂ V/ ∂φ B and κ 2 ≡ 8 π G = 8 π /m 2 pl • ‘Constraint’ equations κ 2 2 Π B Π B + V H 2 � 1 � , (3) = 3 − κ 2 2 Π B ∂ i φ B , ∂ i H (4) = • Separate Universe approach Salopek & Bond, 1990 initial data must respect energy and momentum constraints evolving collection of indpt universes preserve constraints • But how to self-consistently generate fluctuations?

General semi-analytic solution • Recast master equation and perturbatively expand i , . . . ) T , Defining v i a ≡ ( ζ 1 i , θ 1 i , ζ 2 i , θ 2 implies v i a ( t, x ) + A ab ( t, x ) v i b ( t, x ) = 0 , ˙   0 − 1 0 0  with χ ( t, x ) ≡ e A 2 V ; AB e B η ⊥ 0 3 − 6˜ 0 2 η �   A = + ˜ ǫ + ˜   0 0 0 − 1 3 H 2  two-field case slow-roll example 0 0 3 χ 3 (exact case used) Perturbative v (1) i a + A (0) ab ( t ) v (1) b (1) ˙ = i a ( t, x ) , i b expansion: v (2) i a + A (0) ab ( t ) v (2) − A (1) ab ( t, x ) v (1) ˙ = i b , i b where v i a = v (1) i a + v (2) A (0) ab + A (1) ab = A (0) ab + ∂ − 2 ∂ i ( ∂ i A ab ) (1) i a and A ab ( t, x ) = A (0) A (0) ab ( t ) + ¯ abc ( t ) v (1) c ( t, x ) . ≡ � t First order solution: d t ′ G ab ( t, t ′ ) ˙ W ( k, t ′ ) X (1) v (1) bm ( k, t ′ ) . a m ( k, t ) ≡ −∞ Green’s function horizon-crossing linear soln

Bispectrum expression Second-order equation with linear source terms Linear mode functions ζ lin Linear Green’s function Equiv. to linear mode fns Analytic soln at horizon-crossing v (2) i a + A (0) ab ( t ) v (2) A (0) abc ( t ) v (1) − ¯ i b ( t, x ) v (1) ˙ = c ( t, x ) , � t i b d t ′ G 1 a ( t, t ′ ) ¯ A (0) abc ( t ′ ) v (1) Solution for three-point adiabatic correlator bm ( k, t ′ ) v (1) cn ( k ′ , t ′ ) ζ 1 ζ 1 ζ 1 � (2) ( k 1 , k 2 , k 3 , t ) = (2 π ) 3 δ 3 ( � −∞ � � � s k s ) f ( k 1 , k 2 ) + f ( k 1 , k 3 ) + f ( k 2 , k 3 ) Perturbed coefficient in ζ A evolution equation Integrated/cumulative effect over time with � t f ( k , k ′ ) ≡ − k 2 + k · k ′ d t ′ G 1 a ( t, t ′ ) ¯ | k + k ′ | 2 v (1) 1 m ( k, t ) v (1) A (0) abc ( t ′ ) v (1) bm ( k, t ′ ) v (1) 1 n ( k ′ , t ) cn ( k ′ , t ′ ) −∞ + k ↔ k ′ . Slow-roll approximate bispectrum ( power spectrum ) 2 = � ζ 1 ζ 1 ζ 1 � bispectrum η ⊥ ) 2 ∆ t , f NL ≡ ≈ 2(˜ ( � ζ 1 ζ 1 � ) 2

Momentum dependence Approach suited to calculating < ζ ( k 1 ) ζ ( k 2 ) ζ ( k 3 )> ‘shape’ information • Triangular parametrisation appropriate (scale out k = k 1 + k 2 + k 3 ) k 1 = ka = k (1 − β ) k 2 = kb = 1 2 k (1 + α + β ) k 3 = kc = 1 2 k (1 − α + β ) , • General momentum dependent f NL B Ψ ( k 1 , k 2 , k 3 ) 2 f NL ( k 1 , k 2 , k 3 ) = P Ψ ( k 1 ) P Ψ ( k 2 ) + P Ψ ( k 2 ) P Ψ ( k 3 ) + P Ψ ( k 3 ) P Ψ ( k 1 ) .

‘Local’ vs ‘Equilateral’ • In the new parametrisation local and approx. equilateral are: • local ( a, b, c ) = a 3 + b 3 + c 3 equilateral ( a, b, c ) = (1 − a )(1 − b )(1 − c ) B SI B SI . • a b c a b c • (1 )(1 )(1 ) *! ' !"+ %! !"& (! !"* $! !"% !"( )! !"$ #! !") !"# '! !"' ! ' ! ' ' !"& !"& !"( !"( !"% !"% ! ! !"$ !"$ ! !"( ! !"( !"# !"# ! ! ! ' ! '

Primordial and CMB bispectra • The angle-averaged bispectrum Wigner 3j symbol � (2 l 1 + 1)(2 l 2 + 1)(2 l 3 + 1) � � l 1 l 2 l 3 B l 1 l 2 l 3 = (8 π ) 3 0 0 0 4 π Primordial bispectrum � � � � dk 3 ( xk 1 k 2 k 3 ) 2 B Ψ ( k 1 , k 2 , k 3 ) dx dk 1 dk 2 × Transfer × ∆ l 1 ( k 1 ) ∆ l 2 ( k 2 ) ∆ l 3 ( k 3 ) functions × j l 1 ( k 1 x ) j l 2 ( k 2 x ) j l 3 ( k 3 x ) . More problems • If the primordial bispectrum is separable this simplifies N � B Ψ ( k 1 , k 2 , k 3 ) = X i ( k 1 ) Y i ( k 2 ) Z i ( k 3 ) , i • Example: the local approximation B Ψ ( k 1 , k 2 , k 3 ) = 2 P Ψ ( k 1 ) P Ψ ( k 2 ) + P Ψ ( k 2 ) P Ψ ( k 3 ) + P Ψ ( k 3 ) P Ψ ( k 1 ) � � . The integral reduces to products of 1D integrals � b L k 2 dkP Ψ ( k ) ∆ l ( k ) j l ( kx ) l ( x ) = � x 2 dx b L l 1 ( x ) b L l 2 ( x ) b NL � where l 3 ( x ) + perms b NL k 2 dk ∆ l ( k ) j l ( kx ) , l 3 ( x ) = f NL

Adaptive integration �� • Assuming an overall scale-dependence f(k) �� � � dk 1 dk 2 dk 3 ( k 1 k 2 k 3 ) 2 B Ψ ( k 1 , k 2 , k 3 ) ∆ l 1 ( k 1 ) ∆ l 2 ( k 2 ) ∆ l 3 ( k 3 ) x 2 dxj l 1 ( k 1 x ) j l 2 ( k 2 x ) j l 3 ( k 3 x ) . � d α d β B SI ( α , β ) I T ( α , β ) I G ( α , β ) , B SI ( α , β ) ≡ ( abc ) 2 B Ψ ( α , β ) , � I G ( α , β ) ≡ j l 1 ( ax ) j l 2 ( bx ) j l 3 ( cx ) x 2 dx � ∆ l 1 ( ak ) ∆ l 2 ( bk ) ∆ l 3 ( ck ) k n dk I T ( α , β ) ≡ k • Hierarchical adaptive mesh refinement methods

Equal multipole bispectra • Local vs equilateral bispectra with full radiation transfer fns ()*+,-./0123245678.)- % 2 '9-67 :;0)76.,/67 & " ! & ! % ! $ ! # 2 !" !"" !""" ' • Equilateral errors for the large angle approx. (stringent) ()*+,-,./012+34 !#"!& , 5"" ! 5 !6""75 !#"! • !#""& ! "#$$& "#$$ "#$%& "#$% , !" !"" !""" '

Local vs equilateral bispectra !"# !"# ! ! ' ' ! !"# ! !"# !"& ' ' !"& !"% !"( !"( !"% !"$ ! ! !"$ !"# ! !"( ! !"( !"# ! ! ' ! ! ' !"# !"# ! ! ' ' ! !"# !"& ! !"# ' !"& ' !"% !"( !"% !"( !"$ ! !"$ ! !"# ! !"( !"# ! !"( ! ! ' ! ! ' !"' !"' ! ! ' ' ! !"' ! !"' !"& ' !"& ' !"% !"( !"% !"( !"$ !"$ ! ! !"# ! !"( !"# ! !"( ! ! ' ! ! '

DBI Inflation »»»» ««« Multifield Inflation

Non-separable DBI bispectrum DBI vs equilateral bispectra Difference with equilateral approx.

Likelihood analysis • Minimising ‘least squares’ for general primordial bispectra E = 1 � � B l 1 l 2 l 3 l 1 l 2 l 3 � a l 1 m 1 a l 2 m 2 a l 3 m 3 m 1 m 2 m 3 N C l 1 C l 2 C l 3 l i m i Planck full sky map Theoretical model Wigner 3j symbol ( B l 1 l 2 l 3 ) 2 � N = where C l 1 C l 2 C l 3 l 1 l 2 l 3 • Estimator with bispectrum in separable form N fact b l 1 l 2 l 3 = 1 a lm � � X ( i ) l 1 Y ( i ) l 2 Z ( i ) X ( i ) � � X ( i ) l 3 + 5 perms a ( ˆ n ) = Y lm ( ˆ n ) , , • l 6 C l i =1 lm • N fact S = 1 � � n X ( i ) n ) Y ( i ) n ) Z ( i ) a ( ˆ a ( ˆ a ( ˆ n ) . d ˆ N i =1

Separable expansion • Smooth bispectrum implies accurate sum with basis functions b l 1 l 2 l 3 = 1 � � � X ′ α ( l 1 ) X ′ β ( l 2 ) X γ ( l 3 ) + 2 permutations a αβγ , 3 αβγ • Here the coefficients in the sum are given by X α ( l ) = P α (2 l − l max α ( l ) = X α ( l ) X ′ ) , l ( l + 1) . l max • With expansion coefficients given by ... � � l 1 ( l 1 + 1) l 2 ( l 2 + 1) l 3 ( l 3 + 1) � b l 1 l 2 l 3 = a αβγ X α ( l 1 ) X β ( l 2 ) X γ ( l 3 ) l 1 ( l 1 + 1) + l 2 ( l 2 + 1) + l 3 ( l 3 + 1) αβγ � dl 1 dl 2 dl 3 � l 1 ( l 1 + 1) l 2 ( l 2 + 1) l 3 ( l 3 + 1) � a αβγ = (2 α + 1)(2 β + 1)(2 γ + 1) b l 1 l 2 l 3 X α ( l 1 ) X β ( l 2 ) X γ ( l 3 ) l 3 l 1 ( l 1 + 1) + l 2 ( l 2 + 1) + l 3 ( l 3 + 1) max • So the estimator becomes ... X α ( l ) X α ( l ) a lm a lm ¯ � ¯ � X ′ X α ( ˆ n ) = Y lm ( ˆ n ) , α ( ˆ n ) = Y lm ( ˆ n ) , l ( l + 1) C l C l lm lm � ¯ S = 1 M αβγ = 1 � � n ) ¯ n ) ¯ � X ′ X ′ α ( ˆ β ( ˆ X γ ( ˆ n ) + 2 perms a αβγ M αβγ where d ˆ . n 3 N αβγ