Primordial non-Gaussianity

from inflation

Primordial non-Gaussianity

from inflationDavid 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, Gianmassimo Tasinato, Jussi Valiviita, Filippo

Vernizzi…review: Classical & Quantum Gravity 27, 124002 (2010)

arXiv:1004.0818

David WandsInstitute of Cosmology and

GravitationUniversity of Portsmouth

work with Chris Byrnes, Jose Fonseca, Kazuya Koyama, David Langlois, David Lyth, Shuntaro Mizuno, Misao Sasaki, Gianmassimo Tasinato, Jussi Valiviita, Filippo

Vernizzi…review: Classical & Quantum Gravity 27, 124002 (2010)

arXiv:1004.0818

ICGC, Goa 19th December 2011

WMAP7 standard model of primordial cosmology Komatsu et al 2011

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 3Wikipedia: AllenMcC

421

33 ,221

nkk kkPkkkP

3213

3213 ,,2

321kkkkkkBkkk

Many sources of non-GaussianityInitial vacuum Excited state S. Das

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

M. Musso

Hubble-exit Features in potential F ArrojaJ-O Gong

Super-Hubble evolution Self-interactions + gravity R. Rangarajan

End of inflation Tachyonic instability

(p)Reheating Modulated (p)reheating

After inflation Curvaton decay

Magnetic fields P. Trivedi

Primary anisotropies Last-scattering

Secondary anisotropies ISW/lensing + foregrounds F. Lacasa

18/2/2008 David Wands4

primordial non-Gaussianity

inflation

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

18/2/2008 David Wands 5

31

33

33

32

32

31

321

111,,

kkkkkkkkkB

33

32

31

213132321321

3,,

kkk

kkkkkkkkkkkkB

3

321321

321

81,,

kkkkkkkkkB

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

...)(),(

xN

NtxN II I

i

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

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

t

x

order by order at Hubble exit

...2

1

...2

1

...2

1

,11

2

21

21

21

JI

JIII

II I

II I

III

NNN

sub-Hubble field interactions super-Hubble classical evolution

N’’

N’

N’

N’ N’

N’

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

e.g., <3>

( ) 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

...)()()()(5

3

...)()()(2

1)()()(

...)()()()(

...)(2

1)()(

3132

32

212

321

212

21

2

xxxxf

xxxNNxxx

xxNxx

xNxNx

NL

gNLNL = (fNL)2

local trispectrum has 2 terms at leading order

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

18/2/2008 David Wands 9

N’’ N’’ N’’’

N’ N’ N’ N’

N’

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

12 ON

Nf localNL

21

s

equilNL c

f

1 nf localNL

...42 L

decay

equilNLf

,

1

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

- 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

2

22

3

1

3

1

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

rfr

rr NLGG 4

54

32

32

2

2

V()

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-GaussianityConstraints 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) - <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-GaussianityConstraints 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 collapseo well-described by Newtonian gravity

large-scale background needs GR (R≈H-1)o density perturbation is gauge dependent

bias is a gauge-dependent quantity

tHtHttt gggmmm 3~

,3~

,

tbHbb mgmg )1(3~~

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

2)(

3

2

k

aHcm

)( N

)()( cm

cg b

Galaxy power spectrum at z=1Bruni, Crittenden, Koyama, Maartens, Pitrou & Wands, arXiv:1106.3999

bG=2

Angular galaxy power spectrum at z=1

observables are independent of gauge used

using full GR treatment of gauge and line-of-sight effectsChallinor & 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

Beyond fNL?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)

outlookESA Planck satellitenext all-sky surveydata early 2013…

fNL < 10gNL < ?

+ future LSS constraints...fNL < 1??

Non-Gaussian outlook: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

scale-dependence of fNL?

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)

2ln

ln

ln1

)(

21

2

dt

NdH

kd

Pdn

PNkPaHk

2

2

3342

ln

ln

6

5)(

H

V

N

N

kd

fdn

N

Nkf

NLfNL

aHkNL

Byrnes, Nurmi, Tasinato & Wands (2009); Byrnes, Gerstenlauer, Nurmi, Tasinato & Wands (2010)

23H

V

N

Nn fNL

Byrnes, Choi & Hall 2009Khoury & Piazza 2009

Sefusatti, Liguori, Yadav, Jackson & Pajer 2009

quasi-local model for scale-dependent fNL

Fourier space:

quasi-local non-Gaussianity in real space:

pfNLpNLNL k

knkfkf ln1)()(

)'()'('5

3)(

5

3)()( 2

132

11 xxIxxdfnxfxx NLfNLNL

x’

x

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

scale-dependent fNL from a local two-field

power spectrum

bispectrum

)(5

3)()()( 2 xfxxx

)()()( kPkPkP

Byrnes, Nurmi, Tasinato & Wands (2009)

)(kP

)(kP

kln

kln

)()( kBkB

2)()(

)(kP

kBkfNL

local two-field scale-dependent fNL

power spectrum

bispectrum where

scale-dependence

e.g., inflaton + non-interacting curvaton

for CMB+LSS constraints on this model see Tseliakhovich, Hirata & Slosar (2010)

)(5

3)()()( 2 xfxxx

)()()( kPkPkP

fkwfNL )(2 )()(

)( kPkP

kw

nnn fNL 2

2)1(4ln

lnw

kd

fdn NL

fNL

Byrnes, Nurmi, Tasinato & Wands (2009)

scale-dependent fNL

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

multi-variable local fNL

quasi-local fNL

...)()(2)()()()()( 21122

2222

11121 xxfxfxfxxx

)'()'('5

3)(

5

3)()( 2

132

11 xxWxxdfnxfxx NLfNLNL

Byrnes, Choi & Hall 2009Khoury & Piazza 2009

Sefusatti, Liguori, Yadav, Jackson & Pajer 2009Byrnes, Nurmi, Tasinato & Wands 2009

trispectrum

where we have two independent parameters from N calculation

and

simplest local form of non-Gaussianity to third ordersimplest local form of non-Gaussianity to third order

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

3rd order non-linearity for curvaton3rd order non-linearity for curvatonSasaki, Valiviita & Wands (astro-ph/0607627)

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

full pdf for from Nfull pdf for from N Sasaki, Valiviita & Wands (2006)

probability distribution for probability distribution for

probability distribution for probability distribution for

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 37

)()()()()()(,,)5/6(,,,/)( 1332213213213 kPkPkPkPkPkPkkkfkkkBkkkP NL P

31

33

33

32

32

31

21321

111)()5/6(,,

kkkkkkkfkkkB local

NL P

33

32

31

21313232121321

3)()5/6(,,

kkk

kkkkkkkkkkfkkkB equil

NL P

3

321321

21321

81)()5/6(,,

kkkkkkkfkkkB orthog

NL P

remember: fNL < 100 implies Gaussian to better than 0.1%

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

• fNL negative 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 ) -1

V’’’/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:

• why Gaussian and why not?

• local non-Gaussianity and fNL from inflation

• beyond fNL

– higher-order statistics– scale-dependence

• conclusions

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

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

2> )

Komatsu & Spergel (2001)

Simple local form for primordial non-Gaussianity

evidence for local non-Gaussianity?evidence for local non-Gaussianity? T/T -/3, so positive fNL more cold spots in CMB

Wilkinson Microwave Anisotropy Probe 7-year data, February 2010

Wilkinson Microwave Anisotropy Probe 7-year data, February 2010

183

310

2

2

10,10 T

T

T

T