Primordial non-Gaussianity

from inflationDavid Wands

Institute of Cosmology and Gravitation

University of Portsmouthwork with Chris Byrnes, Jon Emery, Christian Fidler,

Gianmassimo Tasinato, Kazuya Koyama, David Langlois, David Lyth, Misao Sasaki, Jussi Valiviita, Filippo Vernizzi…review: Classical & Quantum Gravity 27, 124002 (2010)

arXiv:1004.0818

Cosmo-12, Beijing 13th September 2012

WMAP7 standard model of primordial cosmology Komatsu et al 2011

Gaussian random field, (x)• normal distribution of values in real space, Prob[(x)]

• defined entirely by power spectrum in Fourier space

• bispectrum and (connected) higher-order correlations vanish

David Wands 3

'2 33' kkkPkk

0''' kkk

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non-Gaussian random field, (x)

anything else

David Wands 4

Rocky Kolb

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non-Rocky Kolb

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 6Wikipedia: AllenMcC

421

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nkk kkPkkkP

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Many sources of non-GaussianityInitial vacuum Excited state

Sub-Hubble evolution Higher-derivative interactionse.g. k-inflation, DBI, Galileons

Hubble-exit Features in potential

Super-Hubble evolution Self-interactions + gravity

End of inflation Tachyonic instability

(p)Reheating Modulated (p)reheating

After inflation Curvaton decay

Magnetic fields

Primary anisotropies Last-scattering

Secondary anisotropies ISW/lensing + foregrounds

David Wands7

primordial non-Gaussianity

inflation

Many shapes for primordial bispectra• local type (Komatsu&Spergel 2001)

– local in real space– 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

• separable basis (Ferguson et al 2008)

David Wands 8

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Primordial density perturbations from quantum field fluctuations

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

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

final

initialdtHN

...)(),(

xNNtxN II I

i

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

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

t

x

order by order at Hubble exit

...21

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sub-Hubble field interactions super-Hubble classical evolution

N’’

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N’N’ N’

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

e.g., <3>

non-Gaussianity from inflation?• single-field slow-roll inflaton

– 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 multi-field inflation

=> constant• see talks this afternoon by Emery & Kidani

– at/after end of inflation (curvaton, modulated reheating, etc)• e.g., curvaton

12 ON

Nf localNL

21

s

equilNL c

f

1 nf localNL

...42 L

decay

localNLf

,

1

Maldacena 2002

Creminelli & Zaldarriaga 2004

Cheung et al 2008

multi-field inflation revisited• light inflaton field + massive isocurvature fields

– Chen & Wang (2010+12)– Tolley & Wyman (2010)– Cremonini, Lalak & Turzynski (2011)– Baumann & Green (2011)– Pi & Shi (2012)– Achucarro et al (2010-12); Gao, Langlois & Mizuno (2012)

• integrate out heavy modes coupled to inflaton, M>>H• effective single-field model with reduced sound speed

• effectively single-field so long as • c.f. effective field theory of inflation: Cheung et al (2008)

• see talk by Gao this afternoon

• multiple light fields, M<<H fNLlocal

222

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4 s

equilNL

eff

effs c

fM

Mc

effM

( ) is local function of single Gaussian random field, (x)

where

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

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

...)()()()(53

...)()()(21)()()(

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NL

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Local trispectrum has 2 terms at tree-level

• can distinguish by different momentum dependence• Suyama-Yamaguchi consistency relation: NL = (6fNL/5)2

– generalised to include loops: < T P > = < B2 >Tasinato, Byrnes, Nurmi & DW (2012) see talk by Tasinato this afternoon

David Wands 14

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Liguori, Matarrese and Moscardini (2003)

Newtonian potential a Gaussian random field(x) = G(x)

Liguori, Matarrese and Moscardini (2003)

fNL=+3000

Newtonian potential a local function of Gaussian random field(x) = G(x) + fNL ( G

2(x) - <G2> )

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

Liguori, Matarrese and Moscardini (2003)

fNL=-3000

Newtonian potential a local function of Gaussian random field(x) = G(x) + fNL ( G

2(x) - <G2> )

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

Constraints on local non-Gaussianity• WMAP CMB constraints using estimators based on

matched templates:

-10 < fNL < 74 (95% CL) Komatsu et al WMAP7 -5.6 < gNL / 105 < 8.6 Ferguson et al; Smidt et al

2010

Newtonian potential a local function of Gaussian random field

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

2> )

Large-scale modulation of small-scale power

split Gaussian field into long (L) and short (s) wavelengthsG (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 powerP ( 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) - <G

2> ) + gNL G3(x) + ...

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

2 (x3) > + ...

X1 X2

local modulation of bispectrum could be significant

< fNL2 (X) > fNL

2 +10-8 gNL2

e.g., fNL 10 but gNL 106

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

so local (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

peak – background split for galaxy bias BBKS’87

Constraints on local non-Gaussianity• WMAP CMB constraints using estimators based on optimal

templates:

-10 < fNL < 74 (95% CL) Komatsu et al WMAP7 -5.6 < gNL / 105 < 8.6 Ferguson et al; Smidt et al 2010

• LSS constraints from galaxy power spectrum on large scales:

-29 < fNL < 70 (95% CL) Slosar et al 2008 [SDSS] 27 < fNL < 117 (95% CL) Xia et al 2010 [NVSS survey of AGNs]

Tantalising evidence of local fNL

local?• Latest SDSS/BOSS data release (Ross et al 2012):

Prob(fNL>0)=99.5% without any correction for systematics 65 < fNL < 405 (at 95% CL) no weighting for stellar density

Prob(fNL>0)=91% -92 < fNL < 398 allowing for known systematics

Prob(fNL>0)=68% -168 < fNL < 364 marginalising over unknown systematics

Beyond fNL?• Higher-order statistics

– trispectrum gNL (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)

new era of second-order cosmology• Existing non-Gaussianity templates based on non-linear primordial perturbations + linear Boltzmann codes (CMBfast, CAMB, etc)

• Second-order general relativistic Boltzmann codes in preparation

• Pitrou (2010): CMBquick in Mathematica: fNL ~ 5?• Huang & Vernizzi (Paris)• Fidler, Pettinari et al (Portsmouth)• Lim et al (Cambridge & London)

templates for secondary non-Gaussianity (inc. lensing) induced tensor and vector modes from density perturbations testing interactions at recombination

e.g., gravitational wave production

h

outlookESA Planck satellitenext all-sky surveydata early 2013…

fNL < 5

+ future LSS constraints...

Euclid satellite: fNL < 3?SKA ??

Non-Gaussian outlook:• Great potential for discovery

– detection of primordial non-Gaussianity would kill textbook single-field slow-roll 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

– 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