Christian ByrnesChristian Byrnes

Institute of theoretical physicsInstitute of theoretical physicsUniversity of HeidelbergUniversity of Heidelberg

(ICG, University of Portsmouth)(ICG, University of Portsmouth)

David Wands, Kazuya Koyama and Misao David Wands, Kazuya Koyama and Misao SasakiSasaki

Diagrams: arXiv:0705.4096, Diagrams: arXiv:0705.4096, JCAP 0711:027, 2007;JCAP 0711:027, 2007;

Trispectrum: Trispectrum: astro-ph/0611075,astro-ph/0611075, Phys.Rev.D74:123519, 2006Phys.Rev.D74:123519, 2006

work in progresswork in progress

Primordial non-Gaussianity Primordial non-Gaussianity from inflationfrom inflation

Kosmologietag, Bielefeld, 8th May 2008

Motivation

• Lots of models of inflation, need to predict many observables• Non-Gaussianity, observations improving rapidly • Not just which parameterises bispectrum• ACT, Planck, can observe/constrain trispectrum • 2 observable parameters• What about higher order statistics?• Or loop corrections?• Do they modify the predictions?

• Diagrammatic method • Calculates the n-point function of the primordial curvature perturbation, at tree or loop level • Separate universe approach • Valid for multiple fields and to all orders in slow-roll parameters

Calculate using the formalism (valid on super horizon scales)

Separate universe approach

Efoldings

where and is evaluated at Hubble-exit

Field perturbations are nearly Gaussian at Hubble exit

Curvature perturbation is not Gaussian

The primordial curvature perturbation

Starobinsky `85; Sasaki & Stewart `96; Lyth & Rodriguez ’05

Maldacena ‘01; Seery & Lidsey ‘05; Seery, Lidsey & Sloth ‘06

Diagrams from Gaussian initial fieldsHere for Fourier space, can also give for real spaceRule for n-point function, at r-th order, r=n-1 is tree level

1. Draw all distinct connected diagrams with n-external lines (solid) and r propagators (dashed)

2. Assign momenta to all lines3. Assign the appropriate factor to each vertex and propagator

4. Integrate over undetermined loop momenta5. Divide by numerical factor (1 for all tree level terms)6. Add all distinct permutations of the diagrams

Explicit example of the rulesFor 3-point function at tree level

After integrating the internal momentum and adding distinct permutations of the external momenta we find

Bispectrum and trispectrum

CB, Sasaki & Wands, 2006; Seery & Lidsey, 2006

Observable parameters, bispectrum and trispectrum

We define 3 k independent non-linearity parameters

Note that and both appear at leading order in the trispectrumThe coefficients have a different k dependence,

The non-linearity parameters are

“Single” field inflation

Specialise to the case where one field generates the primordial curvature perturbationIncludes many of the cases considered in the literature:• Standard single field inflation• Curvaton scenario• Modulated reheating

Only 2 independent parametersConsistency condition between bispectrum and 1 term of the trispectrum

Loop corrections

The integrals need a cut off

)log(~1

))(()log(2

~||

11

max3

3

2333

3

Lkq

qd

kLOkLkqkq

qd

k – observed scalek max – smoothing scaleL – IR cut off, large scales, L > 1/HSize of the loop contribution appears to depend on the cut off

)log()log(10)(

)( max101

LkgkLkP

kPNLNLtree

loop

What is L? For L~Horizon scale, loop correction to power spectrum is tinyFor L~eternal inflation, loop correction dominates!Is it just a question of renormalisation?Little agreement about the IR cut off in the literature

The loop correction has k dependence similar to the tree term, hard to observationally distinguish

The bispectrum can have an observable contribution from the loop correction, even with L=1/H

See recent papers by Lyth, Sloth, Seery, Enqvist et al, etc

Importance of the loop correction?

Boubekeur and Lyth, ‘05

Conclusions• Non-Gaussianity is a topical and powerful way to

constrain models of inflation• We have presented a diagrammatic approach to

calculating n-point function including loop corrections at any order

• Trispectrum has 2 observable parameters

- only in single field inflation

• Loop correction poorly understood, appears to grow with cut off

Non-Gaussianity from slow-roll inflation?

single inflaton field– can evaluate non-Gaussianity at Hubble exit (zeta is conserved)

– undetectable with the CMB

multiple field inflation– difficult to get large non-Gaussianity during slow-roll inflation

No explicit model has been constructed

Easier to generate non-Gaussianity after inflation E.g. Curvaton, modulated (p)reheating, inhomogeneous end of inflation

Rigopoulos et al 05,Vernizzi & Wands 06 Battefeld & Easther 06, Yokoyama et al 07

Renormalisation• There is a way to absorb all diagrams with dressed vertices, this deals with some of the divergent terms • A physical interpretation is work in progress• We replace derivatives of N evaluated for the background field to the ensemble average at a general point• Renormalised vertex = Sum of dressed vertices

• Remaining loop terms still have a large scale divergence• For chaotic inflation starting at the ‘self reproduction’ scale the loops dominate

Boubekeur & Lyth ’05; Seery ’07and many others…

defining the primordial density perturbationgauge-dependent density perturbation, , and spatial curvature,

gauge-invariant combination: dimensionless density perturbation on spatially flat

hypersurfaces

constant on large scales for adiabatic perturbations

H

Wands, Malik, Lyth & Liddle (2000)

x

t

(3+1) dimensional spacetime

B

x

t

(3+1) dimensional spacetime

B

A

• In the curvaton scenario the primordial curvature perturbation is generated from a scalar field that is light and subdominant during inflation but becomes a significant proportion of the energy density of the universe sometime after inflation.

• The energy density of the curvaton is a function of the field value at

Hubble-exit

• The ratio of the curvaton’s energy density to the total energy density is

Curvaton scenario

• In the case that r <<1• The non-linearity parameters are given by

• In general this generates a large bispectrum and trispectrum.

Curvaton scenario cont.

Sasaki, Valiviita and Wands 2006

• If the bispectrum will be small

In this case the first non-Gaussianity signal might come from the trispectrum through .1NLg

Enqvist and Nurmi, 2005

Observational constraintsWMAP3 bound on the bispectrum (but see Jeong and Smoot 07 and Yadav and Wandelt 07)

Hence CMB is at least 99.9% Gaussian!

Bound on the trispectrum?

Not yet but should come this year Hopefully with WMAP 5 year data

Assuming no detection, Planck is predicted to reach

In the future 21cm data could reach exquisite precision

Kogo and Komatsu ‘06

primordial perturbations from scalar fields

during inflation field perturbations (x,ti) on initial spatially-flat hypersurface

in radiation-dominated era curvature perturbation on uniform-density hypersurface

final

initialdtHN

II I

initial

NNNN

on large scales, neglect spatial gradients, treat as “separate universes”

Starobinsky `85; Sasaki & Stewart `96 Lyth & Rodriguez ’05 – works to any order

t

x

the N formalism

• This depends on the n-point function of the fields

• The first term is unobservably small in slow roll inflation • Often assume fields are Gaussian, only need 2-point function• Not if non-standard kinetic term, break in the potential…

• Work to leading order in slow roll for convenience, in paper extend to all orders in slow roll • Curvature perturbation is non-Gaussian even if the field perturbations are

Maldacena ‘01; Seery & Lidsey ‘05; Seery, Lidsey & Sloth ‘06

The n-point function of the curvature perturbation

Inflation• Inflation generates the primordial density perturbations

from vacuum fluctuations in the scalar field

• The simplest models predictA nearly scale invariant spectrum of adiabatic (curvature)

perturbations with a nearly Gaussian distribution

There are LOTS of models of inflation:single field, multi field, new, chaotic, hybrid, power-law, natural, supernatural, assisted, Nflation, curvaton, eternal, F-term, D-term, brane, DBI, k- ....

With so many models we need as many observables as possible to distinguish between them

Not just the spectral index and tensor-scalar ratio