Non-Gaussianity from Lifshitz Scalar

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Σωφρονία Ἰουλία Κόρακας
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1 COSMO/CosPA 200 September 27, 200 Non-Gaussianity from Lifshitz Scalar Takeshi Kobayashi (Tokyo U.) based on: arxiv: with Keisuke Izumi, Shinji Mukohyama

2 Lifshitz scalars with z=3 obtain super-horizon, scale-invariant field fluctuations without the need of an inflationary era Mukohyama 09 can source the primordial curvature perturbations this mechanism works regardless of the Lifshitz scalar s selfcoupling strength ( field fluctuations in the inflationary era) large non-gaussianity expected We study the non-gaussian nature of the Lifshitz scalar s intrinsic field fluctuations.

3 Lifshitz scalars action : S Φ = 2 dtd 3 x ] [( t Φ) 2 + ( )z+ M 2(z ) Φ z Φ anisotropic scaling with dynamical critical exponent : z x b x t b z t Φ b (z 3)/2 Φ especially for z =3, Φ b 0 Φ (cf. Hořava 09, Quantum gravity at a Lifshitz point )

4 super-horizon field flucuations of S φ = 2 Lifshitz scalars with z=3 dtd 3 xa(t) 3 [ ( t φ) 2 + φoφ + O(φ 3 ) ] O = M 4 a(t) 6 3 s M 2 a(t) c2 s a(t) 2 m2 a t p /3 < p < curvature perturbations via curvaton mechanism/ modulated reheating

5 scale-invariant field flucuations of S φ = 2 Lifshitz scalars with z=3 dtd 3 xa(t) 3 [ ( t φ) 2 + φoφ + O(φ 3 ) ] O = M 4 a(t) 6 3 φφ M 2, s M 2 a(t) c2 s a(t) 2 m2 Since [φ] = 0, field fluctuations are independent of H: regardless of its self-coupling strength.

6 scale-invariant field flucuations of S φ = 2 Lifshitz scalars with z=3 dtd 3 xa(t) 3 [ ( t φ) 2 + φoφ + O(φ 3 ) ] O = M 4 a(t) 6 3 φφ M 2, s M 2 a(t) c2 s a(t) 2 m2 Since [φ] = 0, field fluctuations are independent of H: regardless of its self-coupling strength. Fluctuations can be scale-invariant & non-gaussian.

7 order estimate of f S φ = dtd 3 xa 3 [ 2 ( tφ) φ 3 φ M 4 a φ αφ2 M 5 a 6 + ] φφ M 2 φφφ αm 3 for linear conversion to curvature pert. ζ = φ µ, ζζ M 2 µ 2 ζζζ α M 3 µ 3 f ζζζ ζζ 2 α µ M

8 order estimate of f S φ = dtd 3 xa 3 [ 2 ( tφ) φ 3 φ M 4 a φ αφ2 M 5 a 6 + ] φφ M 2 φφφ αm 3 for linear conversion to curvature pert. ζ = φ µ, 0 0 ζζ M 2 µ 2 ζζζ α M 3 µ 3 f ζζζ ζζ 2 α µ M 000 α

9 order estimate of f S φ = dtd 3 xa 3 [ 2 ( tφ) φ 3 φ M 4 a φ αφ2 M 5 a 6 + ] φφ M 2 φφφ αm 3 Large f produced as α approaches unity. for linear conversion to curvature pert. ζ = φ µ, 0 0 ζζ M 2 µ 2 ζζζ α M 3 µ 3 f ζζζ ζζ 2 α µ M 000 α

10 shapes of bispectra three independent marginal cubic operators: S3 =! " # dtd x 5 α φ φ + α2 ( φ)( i φ) + α3 ( φ) 3 M a(t) 3 (k k2 k3 )2 F ! x2 0.5 x3 x3 k3 /k x2 k2 /k x2 0.5 x x2 0.5 x !φk (t)φk2 (t)φk3 (t)" = (2π)3 M 3 δ (3) (k + k2 + k3 )F (k, k2, k3 ) Only the α term breaks the shift symmetry φ φ + const., and blows up in the squeezed traiangle limit.

11 S 3 = observational constraints dtd 3 x WMAP7 constraints: for ζ φ, M 5 a(t) 3 { α φ 2 3 φ + α 2 ( 2 φ)( i φ) 2 + α 3 ( φ) 3} 0 <f local < <f orthog. < 6 24 <f equil. < 266 (95% CL) 0.03 < α < < α 2 < < α 3 < 0.3

12 S 3 = observational constraints dtd 3 x WMAP7 constraints: for ζ φ, M 5 a(t) 3 { α φ 2 3 φ + α 2 ( 2 φ)( i φ) 2 + α 3 ( φ) 3} 0 <f local < <f orthog. < 6 24 <f equil. < 266 (95% CL) 0.03 < α < < α 2 < < α 3 < 0.3 NOTE: α i required for validity of perturbative expansions significant orthogonal/equilateral-type non-gaussianity produced within perturbatively controllable regime

13 S 3 = observational constraints dtd 3 x WMAP7 constraints: for ζ φ, M 5 a(t) 3 { α φ 2 3 φ + α 2 ( 2 φ)( i φ) 2 + α 3 ( φ) 3} 0 <f local < <f orthog. < 6 24 <f equil. < 266 (95% CL) 0.03 < α < < α 2 < < α 3 < 0.3 NOTE: α i required for validity of perturbative expansions significant orthogonal/equilateral-type non-gaussianity produced within perturbatively controllable regime α required to be further suppressed, e.g. by shift symmetry

k3 (t)φ k4 (t) = (2π) 3 M 4 δ (4) (k + k 2 + k 3 + k 4 )T (k, k 2, k 3, k 4 ) plots of (k k 2 k 3 k 4 ) 9/4 T in the

14 S 3 = S 4 = trispectra dtd 3 { x α M 5 a(t) 3 φ 2 3 φ + α 2 ( 2 φ)( i φ) 2 + α 3 ( φ) 3} dtd 3 { x β M 6 a(t) 3 φ 3 3 φ + β 2 φ 2 ( φ)( 2 φ)+β 3 φ( φ) 3 +β 4 φ 2 ( i φ) 2 + β 5 φ 2 ( i j k φ) 2 + β 6 ( i j k φ)( i φ)( j φ)( k φ) } φ k (t)φ k2 (t)φ k3 (t)φ k4 (t) = (2π) 3 M 4 δ (4) (k + k 2 + k 3 + k 4 )T (k, k 2, k 3, k 4 ) plots of (k k 2 k 3 k 4 ) 9/4 T in the equilateral limit: k = k 2 = k 3 = k 4 in terms of C 2 k k 2 /k k 2, C 3 k k 3 /k k 3 α 2 α 2 2 α 2 3 α α 2 α α 3 α 2 α 3 β β 2 β 3

15 summary a Lifshitz scalar with z = 3 obtains super-horizon, scaleinvariant field fluctuations, regardless of its self-coupling strength the resulting curvature perturbations necessarily leave large non-gaussianity in the sky, unless the field s self-couplings are somehow suppressed significant non-gaussianity of order f = O(00) is generated, within the regime of validity of perturbative expansions self-coupling terms containing spatial derivatives produce non- Gaussianities with various shapes, including the local, equilateral, and orthogonal shapes

16 marginal cubic terms in the UV cubic terms with six spatial derivatives: A ( 3 φ ) φ 2, A 2 ( 2 i φ ) ( i φ) φ, A 3 ( 2 φ ) ( φ), A 4 ( i j φ)( i j φ) φ A 5 ( i φ)( i φ) φ, A 6 ( i j k φ)( i j k φ) φ, A 7 ( 2 φ ) ( i φ)( i φ), A 8 ( i j φ)( i φ)( j A 9 ( i φ)( φ)( i φ), A 0 ( i φ)( i j φ)( j φ), A ( i j k φ)( i j φ)( k φ), A 2 ( φ)( φ)( φ), A 3 ( φ)( i j φ)( i j φ), A 4 ( i j φ)( j k φ)( k i φ). total derivatives with six spatial derivatives: [( i 2 i φ ) φ 2] = A +2A 2, [( i 2 φ ) ( i φ) φ ] = A 2 + A 3 + A 7, i [( i j φ)( j φ) φ] =A 2 + A 4 + A 8, i [( i φ)( φ) φ] =A 3 + A 5 + A 9, i [( j φ)( i j φ) φ] =A 4 + A 5 + A 0, i [( i j k φ)( j k φ) φ] =A 4 + A 6 + A, i [( i φ)( j φ)( j φ)] = A 7 +2A 0, i [( j φ)( i φ)( j φ)] = A 8 + A 9 + A 0, i [( i j k φ)( j φ)( k φ)] = A 8 +2A, i [( φ)( φ)( i φ)] = 2A 9 + A 2, i [( φ)( i j φ)( j φ)] = A 9 + A 0 + A 3, i [( i j φ)( j k φ)( k φ)] = A 0 + A + A 4, i [( j k φ)( j k φ)( i φ)] = 2A + A 3. three independent terms: S 3 = dtd 3 x M 5 a(t) 3 { α φ 2 3 φ + α 2 ( 2 φ)( i φ) 2 + α 3 ( φ) 3}

17 observational constraints on α i φ k (t)φ k2 (t)φ k3 (t) = (2π) 3 M 3 δ (3) (k + k 2 + k 3 )F (k,k 2,k 3 ) scalar product: F template = c local F F 2 k i F (k,k 2,k 3 )F 2 (k,k 2,k 3 ) P k P k2 P k3 F local + c orthog. F orthog. + c equil. F equil. c local = 0.25α c orthog. =0.226α α α 3 c equil. = 0.223α α α 3 for ζ = φ µ, f i = 20 3 µ M ci c i necessary conditions for 0 <f local < <f orthog. < 6 24 <f equil. < 266 are 0.03 < α < < α 2 < < α 3 < 0.3

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