ICGC, Goa
19th December 2011
Primordial non-Gaussianity
from 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,
Gianmassimo Tasinato, Jussi Valiviita, Filippo Vernizzi…
review: Classical & Quantum Gravity 27, 124002 (2010)
arXiv:1004.0818
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, , ...)
 k  k  2  P k   3 k1  k 2  , P k   k n  4
3
1
2
– but, many different models, can produce similar power spectra
• Higher-order correlations can distinguish different models
 k  k  k  2  B k1 , k 2 , k3   3 k1  k 2  k3 
3
1
2
3
– non-Gaussianity  non-linearity  interactions = physics+gravity
David Wands
3
Wikipedia: AllenMcC
inflation
Many sources of non-Gaussianity
Initial vacuum
Excited state
S. Das
Sub-Hubble evolution
Higher-derivative interactions
e.g. k-inflation, DBI, Galileons
M. Musso
Hubble-exit
Features in potential
F Arroja
J-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
primordial non-Gaussianity
Primary anisotropies
Last-scattering
Secondary anisotropies
18/2/2008
ISW/lensing + foregrounds
David Wands
F. Lacasa
4
Many shapes for primordial bispectra
• local type (Komatsu&Spergel 2001)
– local in real space (fNL=constant)
– max for squeezed triangles: k<<k’,k’’
 1
1
1 
B k1 , k2 , k3    3 3  3 3  3 3 
 k1 k2 k2 k3 k3 k1 
• equilateral type (Creminelli et al 2005)
– peaks for k1~k2~k3
 3k  k  k k  k  k k  k  k  
B k1 , k2 , k3    1 2 3 2 3 33 3 1 3 1 2 
k1 k2 k3


• orthogonal type (Senatore et al 2009)
– independent of local + equilateral shapes


81


B k1 , k 2 , k3   
3 
 k1k 2 k3 k1  k 2  k3  
18/2/2008
David Wands
5
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
N 
final
initial
x
H dt
on large scales, neglect spatial gradients, solve as “separate universes”
  N  ( x, ti )   N 

I
N
I ( x)  ...
I
Starobinsky 85; Salopek & Bond 90; Sasaki & Stewart 96; Lyth & Rodriguez 05
order by order at Hubble exit
1
I  1I   2I  ...
2
1
   1   2  ...
2


 1
N
N
2 N
 
1I   
 2I  
1I 1 J   ...
I , J  I  I
 I I
 2  I I

e.g., <3>
N’
N’
N’
N’’
N’
sub-Hubble field interactions
N’
super-Hubble classical evolution
Byrnes, Koyama, Sasaki & DW (arXiv:0705.4096)
simplest local form of non-Gaussianity
applies to many inflation models including curvaton, modulated reheating, etc
  ( ) is local function of single Gaussian random field, (x)
1
 ( x)  N ( x)  N ( x) 2  ...
2
  ( x1 ) ( x2 )  N 2 ( x1 )( x2 )  ...
 ( x1 ) ( x2 ) ( x3 ) 
1 2
N  N  ( x1 )( x2 ) 2 ( x3 )  ...
2
3
 f NL  ( x2 ) ( x3 )  ( x1 ) ( x3 )  ...
5
where
• odd factors of 3/5 because (Komatsu & Spergel, 2001, used) 1 (3/5)1
local trispectrum has 2 terms at leading order
NL = (fNL)2
gNL
N’
N’
N’
N’’
N’’
N’’’
N’
N’
• can distinguish by different momentum dependence
• multi-source consistency relation: NL  (fNL)2
18/2/2008
David Wands
9
non-Gaussianity from inflation?
• single slow-roll inflaton field
 N  2  O   1
– during conventional slow-roll inflation f NL
N
– adiabatic perturbations
=>  constant on large scales => more generally: f NL local  n  1
local
L        ...
– e.g. DBI inflation, Galileon fields...
equil
 f NL
 1 2
cs
• sub-Hubble interactions
2
4
• super-Hubble evolution
– non-adiabatic perturbations during inflation =>   constant
– usually suppressed during slow-roll inflation
– at/after end of inflation (modulated reheating, etc)
equil
• e.g., curvaton
f NL
 1
  ,decay
curvaton scenario:
V()

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
1  1  2    2 

 
 
2
3  3 


-
transferred to radiation when curvaton decays with some
efficiency, 0<r<1, where r  ,decay
r    2 
3 2


  r    2  2    G   G 
3 
 
4r
f NL  5
4r
Newtonian potential a Gaussian random field
(x) = G(x)
Liguori, Matarrese and Moscardini (2003)
Newtonian potential a 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 a 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
• 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) - <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? Byrnes, 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 but gNL 106
peak – background split for galaxy bias BBKS’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
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
Constraints 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 collapse
o well-described by Newtonian gravity
 large-scale background needs GR (R≈H-1)
o density perturbation is gauge dependent
~
~
t  t  t ,  m   m   m  3Ht ,  g   g   g  3Ht
 bias is a gauge-dependent quantity
~
~
 g  b m   g  b m  3H (b  1)t
What is correct gauge to define bias?
peak-background split works in GR with right variables
(Wands & Slosar, 2009)
  (N )

Newtonian potential = GR longitudinal gauge metric:

GR Poisson equation:
relates Newtonian potential to density perturbation in comoving2
synchronous gauge:
2  aH 
(c)
m  
 

3 k 
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
g
(c)
 b m
(c)
see also Baldauf, Seljak, Senatore & Zaldarriaga, arXiv:1106.5507
Galaxy power spectrum at z=1
Bruni, Crittenden, Koyama, Maartens, Pitrou & Wands, arXiv:1106.3999
bG=2
Angular galaxy power spectrum at z=1
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 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)
outlook
ESA Planck satellite
next all-sky survey
data early 2013…
fNL < 10
gNL < ?
+ future LSS constraints...
fNL < 1??
Non-Gaussian outlook:
•
Great potential for discovery
–
–
•
Scope for more theoretical ideas
–
–
•
infinite variety of non-Gaussianity
new theoretical models require new optimal (and sub-optimal)
estimators
More data coming
–
•
any nG close to current bounds would kill 95% of all known
inflation models
requires multiple fields and/or unconventional physics
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
Byrnes, Choi & 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

P (k )  N 2 P
 scale-dependence
 bispectrum

k  aH
2

d
ln
N
n  1 
 H 1
 2
d ln k
dt
d ln P
5  N  
f NL (k )   2 
6  N   k  aH
 scale-dependence
n fNL
N 
V  


 2 4  3  

d ln k
N  
3H 2 
d ln f NL
 e.g., curvaton
n fNL
N   V  


2 
N   3H 
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 fNL
Byrnes, Gerstenlauer, Nurmi, Tasinato & Wands (2010)
 Fourier space:

 k 
f NL (k )  f NL (k p )1  n fNL ln   
 k 

 p 

 quasi-local non-Gaussianity in real space:
3
3
2
2
3
 ( x)   1 ( x)  f NL 1 ( x)  n fNL f NL  d x'  1 ( x' ) I ( x  x' )
5
5
x’
x
scale-dependent fNL from a local two-field
Byrnes, Nurmi, Tasinato & Wands (2009)
3
2
 ( x)    ( x)    ( x)  f    ( x)
5
 power spectrum
P (k )  P (k )  P (k )
P (k )
P (k )
ln k
bispectrum
B (k )  B (k )
 f NL (k ) 
ln k
B (k )
P (k )
2

local two-field scale-dependent fNL
Byrnes, Nurmi, Tasinato & Wands (2009)
3
2
 ( x)    ( x)    ( x)  f    ( x)
5
 power spectrum
P (k )  P (k )  P (k )
 bispectrum
f NL  w (k ) f 
2
scale-dependence
where
w (k ) 
P (k )
P (k )
n fNL  2n  n 
 e.g., inflaton + non-interacting curvaton
n fNL 
d ln f NL
d ln k
 4(1  w )2      
for CMB+LSS constraints on this model see Tseliakhovich, Hirata & Slosar (2010)
scale-dependent fNL
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 scaledependent reduced bispectrum
 multi-variable local fNL
 ( x)   1 ( x)   2 ( x)  f11 12 ( x)  f 22 2 2 ( x)  2 f12 1 ( x) 2 ( x)  ...
 quasi-local fNL
3
3
2
 ( x)   1 ( x)  f NL 1 ( x)  n fNL f NL  d 3 x'  12 ( x' ) W ( x  x' )
5
5
simplest local form of non-Gaussianity to third order
 trispectrum
where we have two independent parameters from N calculation
and
• multi-source consistency relation: NL  (fNL)2
3rd order non-linearity for curvaton
Sasaki, Valiviita & Wands (astro-ph/0607627)
for large fNL >>1 find gNL << NL for quadratic curvaton
full pdf for  from N
Sasaki, Valiviita & Wands (2006)
probability distribution for 
probability distribution for 
templates for primordial bispectra
P k   P (k ) / k 3 , B k1 , k2 , k3   (6 / 5) f NL k1 , k2 , k3 P(k1 ) P(k2 )  P(k2 ) P(k3 )  P(k3 ) P(k1 )
• local type (Komatsu&Spergel 2001)
– local in real space (fNL=constant)
– max for squeezed triangles: k<<k’,k’’


local
P (k1 )2  31 3  31 3  31 3 
B k1 , k2 , k3   (6 / 5) f NL
 k1 k2 k2 k3 k3 k1 
• equilateral type (Creminelli et al 2005)
– peaks for k1~k2~k3


equil
P (k1 )2  3k1  k2  k3 k2 3 k33 3 k1 k3  k1  k2  
B k1 , k2 , k3   (6 / 5) f NL
k1 k2 k3


• orthogonal type (Senatore et al 2009)


81
orthog

P (k1 ) 2 
B k1 , k 2 , k3   (6 / 5) f NL
3 


k
k
k
k

k

k
 1 2 3 1 2 3 
David Wands
37
remember: fNL < 100 implies Gaussian to better than 0.1%
ekpyrotic non-Gaussianity
Koyama, 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):
-
large and negative bispectrum:
-
n – 1 = 4 / c2 > 0
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)
Ekpyrotic
• fNL negative or positive?
• potentials are steep quasi-exponential
• expect large non-linearities at all orders
(Enqvist et al 2009)
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:
•
why Gaussian and why not?
•
local non-Gaussianity and fNL from inflation
•
beyond fNL
–
–
•
higher-order statistics
scale-dependence
conclusions
Simple local form for primordial non-Gaussianity
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?
• T/T  -/3, so positive fNL  more cold spots in CMB
Wilkinson Microwave Anisotropy Probe
7-year data, February 2010
T 2
T2
10 10 ,
T 3
T3
10 18