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
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33
32
32
31
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111,,
kkkkkkkkkB
33
32
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213132321321
3,,
kkk
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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
...)(),(
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
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JI
JIII
II I
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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
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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