1. Primordial non-Gaussianity from inflation
Cosmo-12, Beijing th September 2012
Primordial non-Gaussianity from inflation
David Wands
Institute of Cosmology and Gravitation
University of Portsmouth
work with Chris Byrnes, Jon Emery, Christian Fidler, Gianmassimo Tasinato, Kazuya Koyama, David Langlois, David Lyth, Misao Sasaki, Jussi Valiviita, Filippo Vernizzi…
review: Classical & Quantum Gravity 27, (2010) arXiv:

2. WMAP7 standard model of primordial cosmology Komatsu et al 2011

3. 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

4. non-Gaussian random field, (x)
anything else
David Wands

5. Rocky Kolb
non-Rocky Kolb

6. 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

7. Many sources of non-Gaussianity
Initial vacuum
Excited state
Sub-Hubble evolution
Higher-derivative interactions
e.g. k-inflation, DBI, Galileons
Hubble-exit
Features in potential
Super-Hubble evolution
Self-interactions + gravity
End of inflation
Tachyonic instability
(p)Reheating
Modulated (p)reheating
After inflation
Curvaton decay
Magnetic fields
Primary anisotropies
Last-scattering
Secondary anisotropies
ISW/lensing + foregrounds
inflation
primordial non-Gaussianity
David Wands

8. 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

9. Primordial density perturbations from quantum field fluctuations
 = 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...

10. order by order at Hubble exit
e.g., <3>
N’’ N’ N’
sub-Hubble field interactions
super-Hubble classical evolution
Byrnes, Koyama, Sasaki & DW (arXiv: )

11. non-Gaussianity from inflation?
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-adiabatic perturbations during 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

12. multi-field inflation revisited
light inflaton field + massive isocurvature fields
Chen & Wang ( )
Tolley & Wyman (2010)
Cremonini, Lalak & Turzynski (2011)
Baumann & Green (2011)
Pi & Shi (2012)
Achucarro et al ( ); 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

13. 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)
where
odd factors of 3/5 because (Komatsu & Spergel, 2001, used) 1 (3/5)1
N’’ N’

14. Local trispectrum has 2 terms at tree-level
NL
gNL N’ N’ N’ N’
N’’
N’’
N’’’ N’
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
David Wands

15. Newtonian potential a Gaussian random field (x) = G(x)
Liguori, Matarrese and Moscardini (2003)

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

17. T/T  -/3, so negative fNL  more hot spots in CMB
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)

18. Constraints on local non-Gaussianity
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

19. (x) = G(x) + fNL ( G2(x) - <G2> )
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) > +...
X X2
i.e., inhomogeneous modulation of small-scale power
P ( k , X ) -> [ fNL L(X) ] Ps(k)
but fNL <100 so any effect must be small

20. 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) > + ...
X X2
local modulation of bispectrum could be significant
< fNL2 (X) >  fNL gNL2
e.g., fNL  10 but gNL 106

21. 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 ) = [ fNL ( aH / k ) 2 L ( X ) ]  s
 strongly scale-dependent bias on large scales
Dalal et al, arXiv:

22. Constraints on local non-Gaussianity
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]

23. Tantalising evidence of local fNLlocal?
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

24. Beyond fNL? Higher-order statistics
trispectrum  gNL (Seery & Lidsey; Byrnes, Sasaki & Wands )
-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)

25. new era of second-order cosmology
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 

26. outlook ESA Planck satellite next all-sky survey data early 2013…
fNL < 5
+ future LSS constraints...
Euclid satellite: fNL < 3?
SKA ??

27. Non-Gaussian outlook:
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