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### Primordial non-Gaussianityfrom inflation

David Wands

Institute of Cosmology and Gravitation

University of Portsmouth

work with Chris Byrnes, Jose Fonseca, Kazuya Koyama, David Langlois, David Lyth, Shuntaro Mizuno, Misao Sasaki, GianmassimoTasinato, JussiValiviita, FilippoVernizzi…

review: Classical & Quantum Gravity 27, 124002 (2010) arXiv:1004.0818

Primordial Gaussianity from inflation

- Quantum fluctuations from inflation
- ground state of simple harmonic oscillator
- almost free field in almost de Sitter space
- almost scale-invariant and almost Gaussian
- Power spectra probe background dynamics (H, , ...)
- but, many different models, can produce similar power spectra
- Higher-order correlations can distinguish different models
- non-Gaussianity non-linearity interactions = physics+gravity

David Wands

Wikipedia: AllenMcC

Many shapes for primordial bispectra

- local type (Komatsu&Spergel 2001)
- local in real space (fNL=constant)
- max for squeezed triangles: k<<k’,k’’
- equilateral type (Creminelli et al 2005)
- peaks for k1~k2~k3
- orthogonal type (Senatore et al 2009)
- independent of local + equilateral shapes

David Wands

Primordial density perturbations from quantum field fluctuations

t

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

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

x

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

Starobinsky 85; Salopek & Bond 90; Sasaki & Stewart 96; Lyth & Rodriguez 05

order by order at Hubble exit

e.g., <3>

N’

N’

N’

N’’

N’

N’

sub-Hubble field interactions

super-Hubble classical evolution

Byrnes, Koyama, Sasaki & DW (arXiv:0705.4096)

simplest local form of non-Gaussianityapplies to many inflation models including curvaton, modulated reheating, etc

( ) is local function of singleGaussian random field, (x)

where

- odd factors of 3/5 because (Komatsu & Spergel, 2001, used)1 (3/5)1

local trispectrum has 2 terms at leading order

- can distinguish by different momentum dependence
- multi-source consistency relation: NL (fNL)2

NL = (fNL)2

gNL

N’

N’

N’

N’

N’’

N’’

N’’’

N’

David Wands

non-Gaussianity from inflation?

- single slow-roll inflaton field
- during conventional slow-roll inflation
- adiabatic perturbations

=> constant on large scales => more generally:

- sub-Hubble interactions
- e.g. DBI inflation, Galileon fields...
- super-Hubble evolution
- non-adiabatic perturbations during inflation => constant
- usually suppressed during slow-roll inflation
- at/after end of inflation (modulated reheating, etc)
- e.g., curvaton

curvaton scenario:Linde & Mukhanov 1997; Enqvist & Sloth, Lyth & Wands, Moroi & Takahashi 2001

curvaton = a weakly-coupled, late-decaying scalar field

- light field during inflation acquires an almost scale-invariant, Gaussian distribution of field fluctuations on large scales
- energy density for massive field, =m22/2
- spectrum of initially isocurvature density perturbations
- transferred to radiation when curvaton decays with some efficiency, 0<r<1, where r ,decay

Newtonian potential a local function of Gaussian random field

(x) = G(x) + fNL ( G2(x) - <G2> )

fNL=+3000

T/T -/3, so positive fNL more cold spots in CMB

Liguori, Matarrese and Moscardini (2003)

Newtonian potential a local function of Gaussian random field

(x) = G(x) + fNL ( G2(x) - <G2> )

fNL=-3000

T/T -/3, so negative fNL more hot spots in CMB

Liguori, Matarrese and Moscardini (2003)

Constraints on local non-Gaussianity

- WMAP CMB constraints using estimators based on optimal templates:
- -10 < fNL < 74 (95% CL) Komatsu et al WMAP7
- |gNL| < 106 Smidt et al 2010

Newtonian potential a local function of Gaussian random field

- (x) = G(x) + fNL ( G2(x) - <G2> )
- Large-scale modulation of small-scale power
- split Gaussian field into long (L) and short (s) wavelengths
- G (X+x) = L(X) + s(x)
- two-point function on small scales for given L
- < (x1) (x2) >L = (1+4 fNL L ) < s (x1) s (x2) > +...
- X1 X2
- i.e., inhomogeneous modulation of small-scale power
- P ( k , X ) -> [ 1 + 4 fNL L(X) ] Ps(k)
- but fNL<100 so any effect must be small

Inhomogeneous non-Gaussianity? Byrnes, Nurmi, Tasinato & DW

(x) = G(x) + fNL ( G2(x) - <G2> ) + gNL G3(x) + ...

split Gaussian field into long (L) and short (s) wavelengths

G (X+x) = L(X) + s(x)

three-point function on small scales for given L

< (x1) (x2) (x3) >X = [ fNL +3gNL L (X)] < s (x1) s (x2) s2(x3) > + ...

X1 X2

local modulation of bispectrum could be significant

< fNL2 (X) > fNL2 +10-8 gNL2

e.g., fNL 10 butgNL 106

peak – background split for galaxy biasBBKS’87

Local density of galaxies determined by number of peaks in density field above threshold

=> leads to galaxy bias: b = g/ m

Poisson equation relates primordial density to Newtonian potential

2 = 4 G => L = (3/2) ( aH / k L ) 2 L

solocal(x) non-local form for primordial density field (x) from

+ inhomogeneous modulation of small-scale power

( X ) = [ 1 + 6 fNL ( aH / k ) 2 L ( X ) ] s

strongly scale-dependent bias on large scales

Dalal et al, arXiv:0710.4560

Constraints on local non-Gaussianity

- WMAP CMB constraints using estimators based on optimal templates:
- -10 < fNL < 74 (95% CL) Komatsu et al WMAP7
- |gNL| < 106 Smidt et al 2010
- LSS constraints from galaxy power spectrum on large scales:
- -29 < fNL < 70 (95% CL) Slosar et al 2008
- 27 < fNL < 117 (95% CL) Xia et al 2010 [NVSS survey of AGNs]

Galaxy bias in General Relativity?

- peak-background split in GR
- small-scale (R<<H-1) peak collapse
- well-described by Newtonian gravity
- large-scale background needs GR (R≈H-1)
- density perturbation is gauge dependent
- bias is a gauge-dependent quantity

What is correct gauge to define bias?

- peak-background split works in GR with right variables
- (Wands & Slosar, 2009)
- Newtonian potential = GR longitudinal gauge metric:
- GR Poisson equation:
- relates Newtonian potential to density perturbation in comoving- synchronous gauge:
- GR spherical collapse:
- local collapse criterion applies to density perturbation in comoving-synchronous gauge: m(c) > * ≈1.6
- GR bias defined in the comoving-synchronous gauge
- see also Baldauf, Seljak, Senatore & Zaldarriaga, arXiv:1106.5507

Galaxy power spectrum at z=1

Bruni, Crittenden, Koyama, Maartens, Pitrou & Wands, arXiv:1106.3999

bG=2

Angular galaxy power spectrum at z=1

using full GR treatment of gauge and line-of-sight effects

Challinor & Lewis, arXiv:1105.5292; Bonvin & Durrer, arXiv:1105.5280

see also Yoo, arXiv:1009.3021

Bruni, Crittenden, Koyama, Maartens, Pitrou & Wands, arXiv:1106

bG=2

observables are independent of gauge used

Beyond fNL?

- Higher-order statistics
- trispectrumgNL(Seery & Lidsey; Byrnes, Sasaki & Wands 2006...)
- -7.4 < gNL/ 105 < 8.2 (Smidt et al 2010)
- N() gives full probability distribution function (Sasaki, Valiviita & Wands 2007)
- abundance of most massive clusters (e.g., Hoyle et al 2010; LoVerde & Smith 2011)
- Scale-dependent fNL(Byrnes, Nurmi, Tasinato & Wands 2009)
- local function of more than one independent Gaussian field
- non-linear evolution of field during inflation
- -2.5 < nfNL < 2.3 (Smidt et al 2010)
- Planck: |nfNL |< 0.1 for ffNL=50 (Sefusatti et al 2009)
- Non-Gaussian primordial isocurvature perturbations
- extend N to isocurvature modes (Kawasaki et al; Langlois, Vernizzi & Wands 2008)
- limits on isocurvature density perturbations (Hikage et al 2008)

outlook

ESA Planck satellite

next all-sky survey

data early 2013…

fNL < 10

gNL < ?

+ future LSS constraints...

fNL < 1??

Non-Gaussian outlook:

- Great potential for discovery
- any nG close to current bounds would kill 95% of all known inflation models
- requires multiple fields and/or unconventional physics
- Scope for more theoretical ideas
- infinite variety of non-Gaussianity
- new theoretical models require new optimal (and sub-optimal) estimators
- More data coming
- final WMAP, Planck (early 2013) + large-scale structure surveys
- Non-Gaussianity will be detected
- non-linear physics inevitably generates non-Gaussianity
- need to disentangle primordial and generated non-Gaussianity

Khoury & Piazza 2009

Sefusatti, Liguori, Yadav, Jackson & Pajer 2009

scale-dependence of fNL?Byrnes, Nurmi, Tasinato & Wands (2009); Byrnes, Gerstenlauer, Nurmi, Tasinato & Wands (2010)

- power spectrum
- scale-dependence
- bispectrum
- scale-dependence
- e.g., curvaton
- scale-dependence probes self-interaction, not probed by power spectrum
- could be observable for curvaton models where gNL NL(Byrnes et al 2011)

quasi-local model for scale-dependent fNL

Byrnes, Gerstenlauer, Nurmi, Tasinato & Wands (2010)

- Fourier space:
- quasi-local non-Gaussianity in real space:

x’

x

scale-dependent fNLfrom a local two-field

Byrnes, Nurmi, Tasinato & Wands (2009)

- power spectrum
- bispectrum

local two-field scale-dependent fNL

Byrnes, Nurmi, Tasinato & Wands (2009)

- power spectrum
- bispectrum where
- scale-dependence
- e.g., inflaton + non-interacting curvaton
- for CMB+LSS constraints on this model see Tseliakhovich, Hirata & Slosar (2010)

scale-dependent fNL

Byrnes, Choi & Hall 2009

Khoury & Piazza 2009

Sefusatti, Liguori, Yadav, Jackson & Pajer 2009

Byrnes, Nurmi, Tasinato & Wands 2009

- two natural generalisations of local fNL non-Gaussianity lead to scale-dependent reduced bispectrum
- multi-variable local fNL
- quasi-local fNL

simplest local form of non-Gaussianity to third order

trispectrum

where we have two independent parameters from N calculation

and

- multi-source consistency relation: NL (fNL)2

Sasaki, Valiviita & Wands (astro-ph/0607627)

3rd order non-linearity for curvatonfor large fNL >>1 find gNL << NL for quadratic curvaton

templates for primordial bispectra

- local type (Komatsu&Spergel 2001)
- local in real space (fNL=constant)
- max for squeezed triangles: k<<k’,k’’
- equilateral type (Creminelli et al 2005)
- peaks for k1~k2~k3
- orthogonal type (Senatore et al 2009)

David Wands

ekpyrotic non-GaussianityKoyama, Mizuno, Vernizzi & Wands 2007 (but see also Creminelli & Senatore, Buchbinder et al, Lehners & Steinhardt 2007)

- Two-field model – ekpyrotic conversion isocurvature to curvature perturbations
- tachyonic instability towards steepest descent (-> single field)
- converts isocurvature field perturbations to curvature/density perturbations
- Simple model => clear predictions:
- small blue spectral tilt (for c2 >>1):
- n – 1 = 4 / c2 > 0
- large and negative bispectrum:
- fNL= - (5/12) ci2 < - (5/3) / (n-1)
- Other authors consider corrections (e.g., ci (i)) corrections to tilt + and corrections to fNL
- in general, steep potentials and fast roll => large non-Gaussianity

curvaton vs ekpyrotic non-Gaussianity?

- Curvaton
- fNL > -5/4
- energy density is quadratic
- higher order statistics well described by fNL
- even for multiple curvatons (Assadullahi, Valiviita & Wands 2008)
- unless self-interactions significant (e.g., 4) (Enqvist et al 2009)
- Ekpyrotic
- fNLnegative or positive?
- potentials are steep quasi-exponential
- expect large non-linearities at all orders

curvaton vs ekpyrotic non-Gaussianity?

- Curvaton
- non-interacting curvaton: (Sasaki, Valiviita & Wands 2006)
- gNL = - (10/3) fNL & nfNL = 0
- self-interacting curvaton: (Enqvist et al 2009; Byrnes et al 2011)
- gNL ≈ fNL2 & nfNL = (PT 1/2P 1/2fNL ) -1V’’’/M
- Ekpyrotic
- ekpyrotic or kinetic conversion: (Lehners & Renaux-Petel 2009)
- gNL ≈ fNL2
- exponential potential scale-invariance:
- nfNL = 0 (Fonseca, Vernizzi & Wands, in preparation)

outline:

- why Gaussian and why not?
- local non-Gaussianity and fNL from inflation
- beyond fNL
- higher-order statistics
- scale-dependence
- conclusions

Simple local form for primordial non-Gaussianity

Newtonian potential a local function of Gaussian random field at every point in space

(x) = G(x) + fNL ( G2(x) - <G2> )

Komatsu & Spergel (2001)

evidence for local non-Gaussianity?

- T/T -/3, so positive fNL more cold spots in CMB

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