Primordial non-Gaussianity from inflation

1
Primordial non-Gaussianity from inflation
Christian Byrnes Institute of theoretical
physics University of Heidelberg (ICG,
University of Portsmouth) David Wands, Kazuya
Koyama and Misao Sasaki Diagrams
arXiv0705.4096, JCAP 0711027,
2007 Trispectrum astro-ph/0611075,
Phys.Rev.D74123519, 2006 work in progress
Kosmologietag, Bielefeld, 8th May 2008
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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

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The primordial curvature perturbation

• 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

Starobinsky 85 Sasaki Stewart 96 Lyth
Rodriguez 05
Maldacena 01 Seery Lidsey 05 Seery, Lidsey
Sloth 06
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Diagrams from Gaussian initial fields

• Here for Fourier space, can also give for real
  space
• Rule for n-point function, at r-th order, rn-1
  is tree level
• Draw all distinct connected diagrams with
  n-external lines (solid) and r propagators
  (dashed)
• Assign momenta to all lines
• Assign the appropriate factor to each vertex and
  propagator
• Integrate over undetermined loop momenta
• Divide by numerical factor (1 for all tree level
  terms)
• Add all distinct permutations of the diagrams

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Explicit example of the rules
For 3-point function at tree level
After integrating the internal momentum and
adding distinct permutations of the external
momenta we find
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Bispectrum and trispectrum
CB, Sasaki Wands, 2006 Seery
Lidsey, 2006
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Observable parameters, bispectrum and trispectrum
We define 3 k independent non-linearity parameters
Note that and both appear at
leading order in the trispectrum The coefficients
have a different k dependence,
The non-linearity parameters are
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Single field inflation

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

Only 2 independent parameters Consistency
condition between bispectrum and 1 term of the
trispectrum
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Loop corrections
The integrals need a cut off
k observed scale k max smoothing scale L IR
cut off, large scales, L gt 1/H Size of the loop
contribution appears to depend on the cut off
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Importance of the loop correction?
What is L? For LHorizon scale, loop correction
to power spectrum is tiny For Leternal
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 L1/H
See recent papers by Lyth, Sloth, Seery, Enqvist
et al, etc
Boubekeur and Lyth, 05
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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

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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

Rigopoulos et al 05,Vernizzi Wands 06
Battefeld Easther 06, Yokoyama et al 07
Easier to generate non-Gaussianity after
inflation E.g. Curvaton, modulated (p)reheating,
inhomogeneous end of inflation
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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 07 and many others
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defining the primordial density perturbation
gauge-dependent density perturbation, ?? , and
spatial curvature, ?
B
A
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Curvaton scenario

• 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 curvatons energy density to the
  total energy density is

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Curvaton scenario cont.

• In the case that r ltlt1
• The non-linearity parameters are given by
• In general this generates a large bispectrum and
  trispectrum.

Sasaki, Valiviita and Wands 2006
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Observational constraints
WMAP3 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
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primordial perturbations from scalar fields
t
in radiation-dominated era curvature perturbation
? on uniform-density hypersurface

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

x
on large scales, neglect spatial gradients, treat
as separate universes
the ? N formalism
Starobinsky 85 Sasaki Stewart 96 Lyth
Rodriguez 05 works to any order
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The n-point function of the curvature perturbation

• 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
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Inflation

• Inflation generates the primordial density
  perturbations from vacuum fluctuations in the
  scalar field
• The simplest models predict
• A 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