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ICGC, Goa 19 th December 2011. Primordial non- Gaussianity from inflation. David Wands Institute of Cosmology and Gravitation University of Portsmouth

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Primordial non gaussianity from inflation

ICGC, Goa 19th December 2011

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
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 (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
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
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)


Primordial non gaussianity from inflation
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


Local trispectrum has 2 terms at leading order
local trispectrum has 2 terms at leading order fluctuations

  • 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
non-Gaussianity from inflation? fluctuations

  • 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

V() fluctuations

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, =m22/2

  • spectrum of initially isocurvature density perturbations

  • transferred to radiation when curvaton decays with some efficiency, 0<r<1, where r  ,decay


Primordial non gaussianity from inflation

Newtonian potential a fluctuationsGaussian random field

(x) = G(x)

Liguori, Matarrese and Moscardini (2003)


Primordial non gaussianity from inflation

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)


Primordial non gaussianity from inflation

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 optimal templates:

    • -10 < fNL < 74 (95% CL) Komatsu et al WMAP7

    • |gNL| < 106 Smidt et al 2010


Primordial non gaussianity from inflation

  • 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


Primordial non gaussianity from inflation

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


Primordial non gaussianity from inflation

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

    • |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
Galaxy bias fluctuationsin 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
What is correct gauge to define bias? fluctuations

  • 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
Galaxy power spectrum at z=1 fluctuations

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

bG=2


Angular galaxy power spectrum at z 1
Angular galaxy power spectrum at z=1 fluctuations

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 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)


Outlook
outlook fluctuations

ESA Planck satellite

next all-sky survey

data early 2013…

fNL < 10

gNL < ?

+ future LSS constraints...

fNL < 1??


Non gaussian outlook
Non-Gaussian outlook: fluctuations

  • 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


Scale dependence of f nl

Byrnes, fluctuationsChoi & Hall 2009

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 f nl
    quasi-local model for scale-dependent fluctuationsfNL

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

    • Fourier space:

    • quasi-local non-Gaussianity in real space:

    x’

    x


    Scale dependent f nl from a local two field
    scale-dependent fluctuationsfNLfrom a local two-field

    Byrnes, Nurmi, Tasinato & Wands (2009)

    • power spectrum

    • bispectrum


    Local two field scale dependent f nl
    local two-field scale-dependent fluctuationsfNL

    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 f nl
    scale-dependent fluctuationsfNL

    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
    simplest local form of non-Gaussianity to third order fluctuations

    trispectrum

    where we have two independent parameters from N calculation

    and

    • multi-source consistency relation: NL  (fNL)2


    3 rd order non linearity for curvaton

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

    3rd order non-linearity for curvaton

    for large fNL >>1 find gNL << NL for quadratic curvaton


    Full pdf for from n

    Sasaki, Valiviita & Wands (2006) fluctuations

    full pdf for  from N




    Templates for primordial bispectra
    templates for primordial bispectra fluctuations

    • 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


    Primordial non gaussianity from inflation

    remember: f fluctuationsNL < 100 implies Gaussian to better than 0.1%


    Primordial non gaussianity from inflation

    ekpyrotic non-Gaussianity fluctuationsKoyama, 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 vs ekpyrotic non-Gaussianity? fluctuations

    • 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 gaussianity1
    curvaton vs ekpyrotic non-Gaussianity? fluctuations

    • 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
    outline: fluctuations

    • why Gaussian and why not?

    • local non-Gaussianity and fNL from inflation

    • beyond fNL

      • higher-order statistics

      • scale-dependence

    • conclusions


    Primordial non gaussianity from inflation

    Simple local form for primordial non-Gaussianity fluctuations

    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
    evidence for local non- fluctuationsGaussianity?

    • T/T  -/3, so positive fNL more cold spots in CMB


    Wilkinson microwave anisotropy probe 7 year data february 2010
    Wilkinson Microwave Anisotropy Probe fluctuations7-year data, February 2010