Primordial trispectrum and orthogonal non-Gaussianities

Primordial trispectrum andorthogonal non-GaussianitiesSébastien Renaux-PetelLagrange Institute - LPTHE, Paris.Cosmo 13, Cambridge,03.09.13

Motivations / message• Data analysis of the trispectrum has received much lessattention than the bispectrum: only• Until recently, no counterpart to orthogonal bispectrum• Generation of ‘orthogonal-type’ trispectra in asignificant fraction of parameter space in the simplesttheoretical context• Use as a diagnostic for the appearance of a largeorthogonal bispectrum (pre-Planck)Renaux-Petel,1302.6978and1303.2618

EFT of single-clock inflation• Unitary gauge action:Cheungetal,0709.0293• Lowest order in derivatives, up to 4th order in fluctuations:• Stückelberg time diffeomorphism in thedecoupling regime (neglect the mixing with gravity)with approximate shift-symmetry: constant

EFT of single-clock inflation(leading-order terms in the low sound speed limit)For A and B of order one, all operators introduce thesame strong coupling scale technically naturalWell known bispectrum: one-parameter family ofshapes; equilateral, and orthogonal for 3.1 < A < 4.2

TrispectrumSet-up computationaly equivalent to k-inflationChen et al,0905.3494s=scalarexchangec=contactinteractions

‘Equilateral’ trispectrumAll 4 shapes are similar, and peak near the regulartetrahedron (RT) limitSimple representative ansatz:Simple measure of the amplitude:Only constraint to date on a trispectrum fromquantum origin (non-local type) :AssumingFergusson,Regan,Shellard(10)

A diagnostic of a large orthogonalbispectrum (pre-Planck)WMAP9 (weakly) favoredvalues of c s of order 10 -2For such values, thetrispectrum, scaling likeWMAP9is of order 10 8 and is greaterthan the observationalboundStrong constraints on models that can generate the largeorthogonal bispectrum suggested by WMAP9 … unless theobservational bound is not applicable!

TrispectrumTwo-parameter family of shapes. For any A, T interpolatesbetween highly correlated and anti-correlated with Tc1 as Bgoes from large negative to positive values. A shapequalitatively different from Tc1 necessarily arises in theneighborhood of a particular value of B (function of A).Around which value? How narrow is this region?What does the shape look like? How does it vary with A?

Orthogonal trispectrumIn 1302.6978, we concentrated on the region 3.1< A < 4.2where the orthogonal bispectrum arises, motivated by thehints of a large orthogonal bispectrum in WMAP9 analysis.Visual inspection of shapes invarious representative limits:Tc1 is representative only forA qualitatively different shapearises in the complementaryregion, centered around:The shapes of these various orthogonal trispectra dependvery weakly on A

Orthogonal trispectrumPlots of the dimensionless quantitiesSpecializedplanar limitFolded limitMore in 1302.6978

Family of orthogonal trispectraMore generally, the cancellation of tNL gives a goodestimate of when Tc1 is not representative of the totalshape.One-parameter family of ‘orthogonal-type’ trispectra:This is in full agreement with the visual inspection of shapes

Conclusion• Data analysis of the trispectrum has received much lessattention than the bispectrum. Until recently, nocounterpart to orthogonal bispectrum.• We have shown that the simple ‘equilateral’ ansatz usedto represent trispectra from derivative operators is notenough: qualitatively new ‘orthogonal-type’ trispectra aregenerated in a substantial fraction of parameter space inthe simplest theoretical context. Current constraints aremostly blind to them. Need for dedicated data analysis.• See 1303.2618 for a discussion of the trispectrum in thecase of DBI-Galileon inflation, which can generate abispectrum of orthogonal shape.