Title: Primordial non-Gaussianity from inflation
1Primordial 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
2Motivation
- 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
3The 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
4Diagrams 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
5Explicit 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
6Bispectrum and trispectrum
CB, Sasaki Wands, 2006 Seery
Lidsey, 2006
7Observable 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
8Single 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
9Loop 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
10Importance 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
11Conclusions
- 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
12Non-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
13Renormalisation
- 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
14defining the primordial density perturbation
gauge-dependent density perturbation, ?? , and
spatial curvature, ?
B
A
15Curvaton 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
16Curvaton 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
17Observational 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
18primordial 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
19The 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
20Inflation
- 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