The Bispectrum Beyond Slow-Roll in the Unifjed EFT of Infmation - PowerPoint PPT Presentation

The Bispectrum Beyond Slow-Roll in the Unifjed EFT of Infmation Passaglia & Hu, In Prep. Samuel Passaglia University of Chicago

Bispectrum is a test of the Physics of Infmation Size and shape of bispectrum probes infmaton interactions 16)(Beyond Slow-Roll Signals, e.g., Hannestad et al. 2010) (Future Bispectrum Prospects: CMB: Abazajian et al. 16, LSS: Gleyzes et al. 17, Bauldofg et al. Slow-roll violating models produce new discovery modes. Constraints will improve NL NL NL 1 ⟨ ˆ R k 1 ˆ R k 2 ˆ R k 3 ⟩ = (2 π ) 3 δ 3 ( k 1 + k 2 + k 3 ) B R ( k 1 , k 2 , k 3 ) f NL ( k 1 , k 2 , k 3 ) ≡ 5 B R ( k 1 , k 2 , k 3 ) P R ( k 1 ) P R ( k 2 ) + perm. . 6 Planck: f squeeze. ∼ 1 ± 5 , f equil ∼ 0 ± 40 , f ortho. ∼ − 30 ± 20 .

In This Talk Simple expressions for the bispectrum for any* single-fjeld model even when slow-roll is violated This enables precision tests of individual models and of the single-fjeld paradigm 2

Deriving the interactions

The Unifjed EFT of Infmation • EFT of infmation: Explicitly break time difgs in action. • Extended to gravitational operators in Dark Energy context . • Motohashi & Hu 2017 studied these operators in infmationary power spectra . We use this framework to compute bispectrum for general models. (EFT Infmation: Creminelli et al. 06, Cheung et al. 07, Baumann & Green 11)(Operator Extensions: Gleyzes et al. 13, Gleyzes et al. 14, Kase and Tsujikawa 14, Gleyzes et al. 15) 3

Work directly in 3+1 Split • Easily connects to model space and to observables Passaglia & Hu In Prep, Motohashi & Hu 2017 relation • Perturb around FLRW up to cubic order . 4 √ ∫ d 4 xN h L ( N, K i j , R i S = j , t ) , • Only R dynamical. Enforce standard dispersion 1 d dt ( a 3 Q ˙ ∂ 2 R = R ) aQc 2 s

Third-Order Action for Perturbations After integration by parts and use of the equation of Passaglia & Hu, In Prep 5 motion ∫ [ R 2 + aF 2 R ( ∂ R ) 2 a 3 F 1 R ˙ d 3 x d t S 3 = ) 2 R 3 + a 3 F 4 ˙ R ∂ a R ∂ a ∂ − 2 ˙ ( ∂∂ − 2 ˙ + a 3 F 3 ˙ R + a 3 F 5 ∂ 2 R R + F 6 R ∂ 2 R ∂ 2 R + F 7 a 3 ( ∂ a ∂ b R ) 2 ∂ 2 R ˙ a + F 8 a 3 ∂ 2 R ∂ 2 R ∂ 2 R + F 9 ∂ a ∂ b ∂ − 2 ˙ ( )] a ∂ 2 R ( ∂ a ∂ b R ) R , • k-Infmation, Horndeski+GLPV, EFT

Consistency Relation Problem? • Trick (extended from Creminelli et al. 2011, Adshead et Passaglia & Hu, In Prep al. 2013): 6 ∫ [ R 2 + aF 2 R ( ∂ R ) 2 ] a 3 F 1 R ˙ d 3 x d t S squeeze = 3 • F 1 , F 2 ⊃ EFT coeffjcients not in power spectrum. d ( F RH 2 ) = Terms that do not contribute to squeeze dt H + Consistency Relation Terms + Terms that cancel

Cubic Action SR suppressed Passaglia & Hu, In Prep • Manifestly preserves consistency relation in slow-roll. no squeeze no squeeze SR suppressed gives consistency 7 ∫ [ a 3 Q d ( ϵ H + 3 2 σ + q ) R 2 ˙ d 3 x d t S 3 ⇒ R dt 2 [ a 3 Q ] − d 2 ( 1 − n s ) | SR R 2 ˙ R dt + ( σ + ϵ H ) R ( H 2 + 2 L 2 ) ˙ RL 2 + (1 − F ) H + ( F 3 through F 9 terms)

Bispectrum Beyond Slow-Roll

In-In Formalism (Condensed Matter: Schwinger 61, Keldysh 64)(Cosmology: Jordan 86, Calzetta+Hu 87)(Infmation: Maldacena 02, Weinberg 05, Lim et al. 08, Senatore+Zaldarriaga 09, ++++) 8 ⟨ ˆ R k 1 ( t ∗ ) ˆ R k 2 ( t ∗ ) ˆ R k 3 ( t ∗ ) ⟩ = ∫ t ∗ [ ⟨ ⟩] ˆ k 1 ( t ∗ ) ˆ k 2 ( t ∗ ) ˆ R I R I R I d tH I ( t ) Re − i k 3 ( t ∗ ) −∞ (1+ iϵ ) • H I ≃ − ∫ d 3 x L I 3 at this order • R I satisfy quadratic-order equation of motion.

Generalized Slow-Roll GSR is an iterative solution for the modefunctions Stewart 02, Choe et al. 04, Kadota et al. 05, Dvorkin & Hu 09 We compute to fjrst-order in GSR. No general analytic solution to the equation of motion 9 1 d dt ( a 3 Q ˙ ∂ 2 R = R ) aQc 2 s ∫ ∞ d w s ) y ( w ) Im [ y ∗ y ( x ) = y 0 ( x ) − w 2 g (ln ˜ 0 ( w ) y 0 ( x )] , x ( 1 + i ) e ix . y 0 ( x ) = x Where y ∝ R , and orders are suppressed by g = ( f ′′ − 3 f ′ ) /f 2 , where 1 f 2 ∼ ∆ 2 .

GSR Bispectrum Results Combine with a few shape-dependent terms and get Passaglia & Hu In Prep, Adshead et al. 2013 10 Each operator i gives j ( ∼ 5 ) shape-independent integrals ∫ ∞ ds s S ′ I ij ( K ) = S ij (ln s ∗ ) W ij ( Ks ∗ ) + ij (ln s ) W ij ( Ks ) . s ∗ • S ij are sources ∝ F i • W ij are windows , e.g. cos( x ) B R ( k 1 , k 2 , k 3 ) = (2 π ) 4 ∆ R ( k 1 )∆ R ( k 2 )∆ R ( k 3 ) k 2 1 k 2 2 k 2 4 3 9 { ∑ } ∑ × T ij I ij ( K ) + [ T iB I iB (2 k 3 ) + perm . ] . i =2 ij k 1 k 2 k 3 • T ij are k -weights for triangle shapes, e.g. ( k 1 + k 2 + k 3 ) 3

Consistency Relation Revisited: Beyond Slow-Roll Beyond SR, no new interactions contribute to squeeze Analytically show GSR consistency relation enforced when: freeze-out . We return to these conditions shortly Passaglia & Hu, In Prep 11 1. the expansion at fjrst-order is valid (i.e. g ≫ g 2 ) 2. No large power spectrum evolution between k S and k L

Example: transient G-infmation

Slow-Roll Violation in transient G-infmation • Transition from Horndeski G-Infmation (Kobayashi et al. 11, Ohashi & Tsujikawa 12) to Chaotic Infmation. SR Violation: good test of our approach. Ramírez, Passaglia , Motohashi, Hu, Mena, 1802.04290 12 L ⊃ f 3 ( ϕ ) X 2 □ ϕ • f 3 ( ϕ ) tanh step.

Equilateral bispectrum GSR properly handles the transition. Passaglia & Hu, In Prep. SR Formula from De Felice & Tsujikawa 2013 13 0 . 30 GSR Horndeski SR 0 . 25 0 . 20 f Equil . NL 0 . 15 0 . 10 0 . 05 0 . 00 10 − 8 10 − 6 10 − 4 10 − 2 10 0 10 2 10 4 10 6 k Eq

Squeeze-Limit Consistency Relation • Correction: Modefunction evolution between freezeout Passaglia & Hu, In Prep epochs. (see Miranda et al. 2015 for ways to avoid) 14 0 . 30 Consistency GSR Corrected 0 . 25 GSR Original 0 . 20 f squeeze NL 0 . 15 0 . 10 0 . 05 0 . 00 10 − 4 10 − 2 10 0 10 2 10 4 k L • Residual error: Next-order GSR needed, g 2 ∼ g .

Conclusions

Take-Home Messages Our computation of bispectrum beyond slow-roll enables precision model tests. Expressions are easy to use : a few 1-D integrals. Consistency relation explicitly holds beyond slow-roll! Passaglia & Hu, In Prep 15