Non-Gaussianities of cosmological perturbations

1. Paolo Creminelli (ICTP, Trieste) Non-Gaussianities of cosmological perturbations with N. Arkani-Hamed, D. Babich, L. Boubekeur, S. Mukohyama, A. Nicolis, L. Senatore, M. Tegmark, and M. Zaldarriaga

2. Is there any correlation among modes?

3. Slow-roll = weak coupling V Friction is dominant To have ~ dS space the potential must be very flat: The inflaton is extremely weakly coupled. Leading NG from gravity. Completely model independent as it comes from gravity Unobservable (?). To see any deviation you need > 1012 data. WMAP ~ 2 x 106 Maldacena, JHEP 0305:013,2003, Acquaviva etal Nucl.Phys.B667:119-148,2003

4. Smoking gun for “new physics” Any signal would be a clear signal of something non-minimal • Any modification enhances NG • Modify inflaton Lagrangian. Higher derivative terms, ghost inflation, DBI inflation… • Additional light fields during inflation. Curvaton, variable decay width… • Potential wealth of information Translation invariance: Scale invariance: F contains information about the source of NG Note. We are only considering primordial NGs. Neglect non-linear relation with observables. Good until primordial NG > 10-5. see P.C. + Zaldarriaga, Phys.Rev.D70:083532,2004 Bartolo, Matarrese and Riotto, JCAP 0606:024,2006

5. Higher derivative terms P.C. JCAP 0310:003,2003 Change inflaton dynamics and thus density perturbations Potential terms are strongly constrained by slow-roll. Impose shift symmetry: Most relevant operator: 3 point function: In EFT regime NG < 10-5 Difficult to observe We get large NG only if h. d. terms are important also for the classical dynamics One can explicitly calculate the induced 3pf:

6. DBI inflation Alishahiha, Silverstein and Tong, Phys.Rev.D70:123505,2004 Example where higher derivative corrections are important A probe D3 brane moves towards IR of AdS. Geometrically there is a speed limit AdS The dual description of this limit is encoded in h.d. operators. DBI action: The scalar is moving towards the origin of the moduli space. H.d. operators come integrating out states becoming massless at the origin. Conformal invariance • It helps inflation slowing down the scalar (potential?) • Generic in any warped brane model of inflation (reconstruct the shape of the throat?) • 3pf can be as large as you like • Generic 3pf for any model with: (see e.g. S. Kecskemeti etal, hep-th/0605189) S. Kachru etal. hep-th/0605045

7. General approach X. Chen, M.x.Huang, S.Kachru and G.Shiu, hep-th/0605045 We want to calculate for a generic: Let us rephrase the calculation in a way which emphasis symmetries: EFT around FRW EFT around a given FRW background . Focus on the perturbation Goldstone of broken time-translation • This mode can be reabsorbed in the metric going to unitary gauge • In this gauge, only gravity with time-dependent spatial diff. : • The Goldstone can be reintroduced with Stuckelberg trick • ADM variables are useful: • Most generic action invariant under spatial diff. + reintroduce the Goldstone At 0th order in derivative I have only: Tadpole terms to make the given a(t) a solution: Coefficients can be time dependent:

8. E.g. this is all for a standard scalar field: Fix the coefficients from the background: Reintroducing At 0th derivative level up to 3rd order in pert: Only two operators are allowed by symmetries: 2 contributions (up to slow-roll) The first operator changes the speed of sound M4 > 0 to avoid superluminality See P.C., M.Luty, A.Nicolis, L.Senatore, hep-th/0606090 and Leonardo’s talk Contribution to NGs:

9. Ghost inflation with Arkani-Hamed, Mukoyama and Zaldarriaga, JCAP 0404:001,2004 Ghost condensation: WRONG SIGN • Spontaneous breaking of Lorentz symmetry: • Consistent derivative expansion: • Non Lorentz-invariant action, standard spatial kinetic term NOT allowed • IR modification of gravity. “Higgs phase” for gravity.

10. Large non-Gaussianities • Use the “ghost” field as the inflaton. It triggers the end of dS. • Non Lorentz-invariant scaling • The leading non-linear operator: Reheating has dimension 1/4 Quite large. Close to exp. bound. Derivative interactions are enhanced wrt standard case by NR relativistic scaling

11. Perturbations generated by a second field • Curvaton • Variable inflaton decay • 2 field inflation • Perturbation of parameters • relevant for cosmo evolution Every light scalar is perturbed during inflation. Its perturbations may become relevant in various ways: Example: variable decay of right-handed neutrinos with L.Boubekeur, hep-ph/0602052 • Parallel Universes: • NG is generated by inefficiency: RHN neutrinos will not be completely dominant. To match 10-5 we need larger fluctuations and thus larger NGs • In general: NG > 10-5, but model dependent. Possible isocurvature contributions. The RHN goes out of equilibrium and decay in ≠ way in ≠ regions of the Universe

12. The shape of non-Gaussianities Babich, P.C., Zaldarriaga, JCAP 0408:009,2004 • LOCAL DISTRIBUTION Typical for NG produced outside the horizon. 2 field models, curvaton, variable decay… • EQUILATERAL DISTRIBUTIONS Derivative interactions irrelevant after crossing. Correlation among modes of comparable . F is quite complicated in the various models. But in general Quite similar in different models

13. Shape comparison The NG signal is concentrated on different configurations. • They can be easily distinguished (once NG is detected!) • They need a dedicated analysis

14. Consistency relation for 3-p.f. J. Maldacena, JHEP 0305:013,2003 P.C. + M. Zaldarriaga, JCAP 0410:006, 2004 Underthe usual “adiabatic” assumption (a single field is relevant), INDEPENDENTLY of the inflaton Lagrangian The long wavelength mode is a frozen background for the other two: it redefines spatial coordinates. In the squeezed limit the 3pf is small and probably undetectable • Models with a second field have a large 3pf in this limit. • Violation of this relation is a clear, model independent evidence for a second field • (same implications as detecting isocurvature). • This is experimentally achievable if NG is detected.

15. Analysis of WMAP data with Nicolis, Senatore, Tegmark, Zaldarriaga, astro-ph/0509029 WMAP alone gives almost all we know about NG. Large data sample + simple. Not completely straightforward! It scales like Npixels5/2 ~ 1016for WMAP!!! Too much… But if F is “factorizable” the computation time scales as Npixels3/2 ~ 109. Doable! Use a fact. shape with equilateral properties

16. Real space VS Fourier space CMB signal diagonal in Fourier space (without NG!!). Foreground and noise in real space. Non-diagonal error matrix + linear term in the estimator Minimum variance estimator: Reduces variance wrt WMAP coll. analysis. N_obs varies across the sky. Smaller power in more observed regions. On a given realization it looks like a NG signal. Bigger variance.

17. Results No detection! Limits (1yr): -27 < fNLlocal < 121 at 95% C.L. (WMAP 3yr: -54 < fNLlocal < 114 ) -366 < fNLequil. < 238 at 95% C.L. 3 yr data. 1 sigma error for local 37 ----> 33 only 15% improvement The quite different limits is just a consequence of the definition of fNL

18. Summary • Non-Gaussianities are enhanced in: 1. Single field models with modified Lagrangian (single operator, DBI inflation, ghost inflation…) 2. Models with a second field (curvaton, variable decay width, variable RHN decay…) • Shape of the 3-point function: • Local (more than 1 field) VS Equilateral (higher derivative terms) • Consistency relation for 3pf for any single field model • WMAP data analysis for different shapes