Origins and observations of primordial non-gaussianity. Kazuya Koyama

Transcription

1 Origins and observations of primordial non-gaussianity Kazuya Koyama University of Portsmouth

2 Primordial curvature perturbations Komatsu et.al. 008 Proved by CMB anisotropies nearly scale invariant ns nearly adiabatic 0.16, S / nearly Gaussian local 3 ( x) g( x) f ( x) 5 local g 9 f 111 Generation mechanisms inflation curvaton collapsing universe (Ekpyrotic, cyclic)

3 Generation of curvature perturbations Delta N formalism Starobinsky ;85, Stewart&Sasaki 95 curvature perturbations on superhorizon o scales = fluctuations in local e-folding number t i i Ntx (, ) dtht ' (', x) Ntx (, i ) N( t) t i 4 R a (3) B 3( P) in 0

4 How to generate delta N entropy perturbations Sasaki. 008 adiabatic perturbations single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

5 Suppose delta N is caused by some field fluctuations at horizon crossing Lyth&Rodriguez Lyth&Rodriguez 05 i N 1 N ( tx, ) I IJ... I, J I I J Bispectrum B ( k, k, k ) 1 3 N N N ( k ) ( k ) ( k ) ( k k k ) B ( k, k, k ) D , I, J, IJ 1, I, J, K IJK 1 3 N, I N, I P ( k ) P ( k ) perm. N N N B ( k, k, k ) Local type ( classical ) ca (local in real space =non-local in k-space) Equilateral type (q ( quantum ) (local in k-space)

6 Observational constraints local type maximum signal for k 3 ( x) g( x) f g( x) 5 local WMAP7 local 10 f 74 k, k 3 1 k 1 k 3 k Equilateral type k 3 maximum signal for k k k 1 3 local 4 f 80 Smith et.al. 09 WMAP7 equil 14 f 66 k 1 k

7 Theoretical predictions i Standard inflation Non-standard scenario Single field Multi field f local, equil Maldacena 04 f local depending on the trajectory K-inflation, DBI inflation O(, ) 1 equil f 1/ c 1 s Features in potential Ghost inflation DBI inflation O(1) equil f (1 / c ) / (1 T ) 1 s RS curvaton local f Rigopoulos, Shellard, van Tent 06 new Ekpyrotic (simplest model) Wands and Vernizzi 06 local 1 f ( ns 1) many others isocurvature perturbations (axion CDM) (5 / 4) / curvaton decay f local

8 K-inflation Non-canonical kinetic term 4 1 S d x gp( X, ), X Amendariz-Picon et.al 99 Field perturbations (leading order in slow-roll) 1 H PT P, r 16cs c s M pl P c P,X s P, X XP, XX sound speed G i &M kh Garriga&Mukhanov 99

9 Bispectrum k 1 k 3 k k Babich et.al. 07 P B( k, k, k ) F( k, k, k ), kkk 1 3 DBI inflation Aishahiha et.al PX ( ) 1 f( ) X1 V( ) f ( ) cf local-typel Fk (, k, k) 1 c 1 3 s local 3 local F ( k1, k, k3 ) f ( k1 k k3 ) 10 k 1 k k 3

10 Observational constraints (too) large non-gaussianity eff 35 1 f 108 cs Lyth bound r 1 6r r M N p 16 c 10 s 7 1 n s f equil 300 for equilateral configurations D3 anti-d3 inflaton UV Huston et.al 07, Bean et.al. 07 Kobayashi et.al 08

11 Multi-field model Arroja, Mizuno, Koyama 08 Renaux-Petel, Steer, Langlois Tanaka 08 4 IJ IJ 1 I J (, ), S d x gp X X b P( X ) P( X), X X X X X IJ J I I J Adiabatic and entropy decomposition adiabatic sound speed c P, X ad P X, X 0P, XX entropy sound speed c en s X entropy X bx adiabatic 0

12 Multi-field k-inflation IJ P( X ) P( X) en 1 Multi-field DBI inflation c Huang, Shiu, Underwood 07 Easson et.al. 07 Langlois&Renaux-Petel 08 Renaux-Petel, Steer, Langlois Tanaka 08 1 PX ( IJ ) det( g I J f( ) GIJ ) 1 V ( ) f ( ) c c en ad Unlike k-inflation, the entropy perturbations can develop large non-gaussianity! This is because the DBI action is obtained by the Lorentz boosts where the Lorentz factor is Arroja, Mizuno, Koyama, Tanaka 09 v c c 1 ad en

13 Transfer from entropy mode Tensor to scalar ratio r 16 c s 1 1 T RS T RS T S * RS * H H, S s s Bispectrum k-dependence is the same as single field case! f eff c 1 T eff f RS 3 s RS RS 3, 1 T 1 T large transfer from entropy mode eases constraints Renaux-Petel, Steer, Langlois Tanaka 08

14 Trispectrum Mizuno, H I 4 Mizuno, Arroja,Koyama, Tanaka 09 Mizuno, Arroja, Koyama 09 H c s 1 1 I H 3 I 3 I 1 H4 H c s H3 H c s H I 3 r T, f T, T f equi equi equi RS RS RS The amplitude of the trispectrum in multi-field models is enhanced for a given amplitude of the bispectrum

15 Transfer coefficient Conversion during inflation Instability along angular direction? s Detailed analysis with potentials in string theory cf. Baumann et.al depending on the embedding of D7 brane there appears an instability along angular directions Gong, Chen, KK, Tasinato 10

16 Observational constraints -CMB Estimator theory 1 B N lm m i i 1 m m 3C C C 1 3 observation a a a m m m We need a theory template to derive a constraint Theory template local type models give a separable form B( k, k, k ) P( k ) P( k ) perm

17 Bispectrum in DBI inflation is not separable K k k k Equilateral non-gaussianity 1 3

18 Theoretical template - It is essential to find a good template which captures the shapes of the bispectrum in theoretical models and at the same time reduces the computational time in evaluating the estimator. - We could miss an interesting signal if there is a signal which does not match the template c.f. orthogonal type Senatore, Smith, Zaldaliaga 08 cos Forthg equil F cosforthg Flocal

19 Trispectrum Local type T ' T T T T T 3 T T ' WMAP7 Munshi et.al 10 Equilateral type? Galliano, Crittenden, KK work in progress

20 Late time effects Non-linear evolution of perturbations gives additional contributions to bispectrum a noise for the primordial non-gaussianity! Primordial, primary and secondary effects Linear evolution Non-linear evolution Primordial Power spectrum Bispectrum f Primary Sachs-Wolfe Doppler effects nd order effects Secondary Integrated Sachs-Wolfe Lensing-SZ Reionization Lesning-ISW Sunyaev-Zel dovich

21 Secondary effects Temperature anisotropies LSS n ISW (+ Rees-Sciama) 3 ISW 1 lensing

22 Effective local non-gaussian parameter f Local type Equilateral type This secondary effect gives a bias f = + O(10) to the local type non-gaussianity must be subtracted! Hanson, Smith, Chalinor, Liguori 09 Mangrilli, Verde 09

23 Primary effects nd order Boltzman equation coupled to nd order Einstein equations Pitrou, Uzan, Bernardeau (see also Bartolo, Matarrese, Riotto ) Temperature anisotropies multi-pole expansion 1 4 I I ( k, ) Q ( k), Q ( k) i Y, m 4 1 m m m m

24 Boltzmann equation (Liouville operator) first order z k second order vector tensor z k

25 quadratic terms Nitta, Komatsu, Bartolo, Matarrese, Riotto 09

26 Final results Planck sensitivity nd primary nd primary lensing-isw lensing-isw equil local contamination f 6, f 4.5 (cf. lensing-isw equil local f, f 8 ) Two late time effects add up and they should be detected by Planck

27 Full results are now available in the Poisson gauge (flat sky approximation, no ISW) Given cosmological parameters, we can calculate the non-primordial contributions without ambiguity and in principle they can be removed Need to find an effective way to remove the non-primordial contributions use the shape of the bispectrum? Need to speed-up a code -Synchronous gauge - implementation in the CAMB code - Inclusion of TTE, TEE, EEE Pitrou, Pettinari Portsmouth

28 Scalar, vector, tensor decomposition at second order, vectors and tensors are sourced by 1 st order scalars and we cannot separate their contributions from scalars nd order and quadratic of 1 st order contributions again a separation of these contributions does not have any physical meaning as the nd order variables is sourced by quadratic of 1 st order variables in Einstein equations In fact these separations are gauge dependentd This becomes clear if we write down the nd order Boltzmann equation without fixing a gauge Naruko, KK, Sasaki, Pitrou 10

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30 Temporal gauge transformation mixes all the terms but the nd order Boltzmann equation is invariant! provides a very non-trivial i check of the calculation l nd order tensor contribution mixed with the lensing term etc.

31 Conclusions Power spectrum from pre-wmap to post-wmap Bispectrum WMAP 9year Planck Large scale structure (halo power spectrum, bispectrum) f O (1)