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Cosmo-12, Beijing 13 th September 2012. Primordial non- Gaussianity from inflation. David Wands Institute of Cosmology and Gravitation University of Portsmouth

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Primordial non gaussianity from inflation

Cosmo-12, Beijing 13th September 2012

Primordial non-Gaussianityfrom inflation

David Wands

Institute of Cosmology and Gravitation

University of Portsmouth

work with Chris Byrnes, Jon Emery, Christian Fidler, GianmassimoTasinato, Kazuya Koyama, David Langlois, David Lyth, Misao Sasaki, JussiValiviita, FilippoVernizzi…

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



Gaussian random field x
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


Non gaussian random field x
non-Gaussian random field, (x)

anything else

David Wands


Rocky kolb
Rocky Kolb

non-Rocky Kolb


Primordial gaussianity from inflation
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 sources of non gaussianity
Many sources of non-Gaussianity

inflation

primordial non-Gaussianity

David Wands


Many shapes for primordial bispectra
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


Primordial density perturbations from quantum field fluctuations
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; Langlois & Vernizzi...


Order by order at hubble exit
order by order at Hubble exit fluctuations

e.g., <3>

N’

N’

N’

N’’

N’

N’

sub-Hubble field interactions

super-Hubble classical evolution

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


Non gaussianity from inflation
non-Gaussianity from inflation? fluctuations

  • 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-adiabaticperturbationsduring multi-field inflation =>  constant

      • see talks this afternoon by Emery & Kidani

    • at/after end of inflation (curvaton, modulated reheating, etc)

      • e.g., curvaton

Maldacena 2002

Creminelli & Zaldarriaga 2004

Cheung et al 2008


Multi field inflation revisited
multi-field inflation revisited fluctuations

  • 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


simplest local form of non-Gaussianity fluctuationsapplies 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

N’

N’’

N’


Local trispectrum has 2 terms at tree level
Local trispectrum has 2 terms at tree-level fluctuations

  • 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

NL

gNL

N’

N’

N’

N’

N’’

N’’

N’’’

N’

David Wands


Newtonian potential a fluctuationsGaussian random field

(x) = G(x)

Liguori, Matarrese and Moscardini (2003)


Newtonian potential fluctuationsa 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 fluctuationsa 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
Constraints on local non- fluctuationsGaussianity

  • 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 fluctuationsGaussian 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? fluctuationsByrnes, 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 bias fluctuationsBBKS’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 gaussianity1
Constraints on local non- fluctuationsGaussianity

  • 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 f nl local
Tantalising evidence of local fluctuationsfNLlocal?

  • 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 f nl
Beyond fluctuationsfNL?

  • 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
new era of second-order cosmology fluctuations

  • 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




Outlook
outlook fluctuations

ESA Planck satellite

next all-sky survey

data early 2013…

fNL < 5

+ future LSS constraints...

Euclid satellite: fNL < 3?

SKA ??


Non gaussian outlook
Non-Gaussian outlook: fluctuations

  • 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


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