1. 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 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 ICGC, Goa19 th December 2011

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

3. 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 Wands3 Wikipedia: AllenMcC

4. Many sources of non-Gaussianity Initial vacuumExcited stateS. Das Sub-Hubble evolutionHigher-derivative interactions e.g. k-inflation, DBI, Galileons M. Musso Hubble-exitFeatures in potentialF Arroja J-O Gong Super-Hubble evolutionSelf-interactions + gravityR. Rangarajan End of inflationTachyonic instability (p)ReheatingModulated (p)reheating After inflationCurvaton decay Magnetic fieldsP. Trivedi Primary anisotropiesLast-scattering Secondary anisotropiesISW/lensing + foregroundsF. Lacasa 18/2/2008David Wands 4 primordial non-Gaussianity inflation

5. 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 18/2/2008David Wands5

6. Primordial density perturbations from quantum field fluctuations  (x,t i ) during inflation field perturbations on initial spatially-flat hypersurface  = curvature perturbation on uniform-density hypersurface in radiation-dominated era on large scales, neglect spatial gradients, solve as “separate universes” Starobinsky 85; Salopek & Bond 90; Sasaki & Stewart 96; Lyth & Rodriguez 05 t x

7. order by order at Hubble exit sub-Hubble field interactionssuper-Hubble classical evolution N’’ N’ Byrnes, Koyama, Sasaki & DW (arXiv:0705.4096) e.g.,

8.    (  ) 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-Gaussianity applies to many inflation models including curvaton, modulated reheating, etc

9. g NL  NL = (f NL ) 2 local trispectrum has 2 terms at leading order can distinguish by different momentum dependence multi-source consistency relation:  NL  (f NL ) 2 18/2/2008David Wands9 N’’ N’’’ N’

10. 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

11. 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,   =m 2  2 /2 -spectrum of initially isocurvature density perturbations -transferred to radiation when curvaton decays with some efficiency, 0<r<1, where r   ,decay curvaton  = a weakly-coupled, late-decaying scalar field  V(  )

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

13. Liguori, Matarrese and Moscardini (2003) f NL =+3000 Newtonian potential a local function of Gaussian random field  (x) =  G (x) + f NL (  G 2 (x) - )  T/T  -  /3, so positive f NL  more cold spots in CMB

14. Liguori, Matarrese and Moscardini (2003) f NL =-3000 Newtonian potential a local function of Gaussian random field  (x) =  G (x) + f NL (  G 2 (x) - )  T/T  -  /3, so negative f NL  more hot spots in CMB

15. Constraints on local non-Gaussianity WMAP CMB constraints using estimators based on optimal templates:  -10 < f NL < 74 (95% CL) Komatsu et al WMAP7  |g NL | < 10 6 Smidt et al 2010

16. Newtonian potential a local function of Gaussian random field  (x) =  G (x) + f NL (  G 2 (x) - )  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 L = (1+4 f NL  L ) +... X 1 X 2 i.e., inhomogeneous modulation of small-scale power P ( k, X ) -> [ 1 + 4 f NL  L (X) ] P s (k) but f NL <100 so any effect must be small

17. Inhomogeneous non-Gaussianity? Byrnes, Nurmi, Tasinato & DW  (x) =  G (x) + f NL (  G 2 (x) - ) + g NL  G 3 (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 X = [ f NL +3g NL  L (X)] +... X 1 X 2 local modulation of bispectrum could be significant  f NL 2 +10 -8 g NL 2 e.g., f NL  10 but g NL  10 6

18. 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 f NL ( 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

19. Constraints on local non-Gaussianity WMAP CMB constraints using estimators based on optimal templates:  -10 < f NL < 74 (95% CL) Komatsu et al WMAP7  |g NL | < 10 6 Smidt et al 2010 LSS constraints from galaxy power spectrum on large scales :  -29 < f NL < 70 (95% CL) Slosar et al 2008  27 < f NL < 117 (95% CL)Xia et al 2010 [NVSS survey of AGNs]

20. 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  bias is a gauge-dependent quantity

21. 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

22. Galaxy power spectrum at z=1 Bruni, Crittenden, Koyama, Maartens, Pitrou & Wands, arXiv:1106.3999 b G =2

23. Angular galaxy power spectrum at z=1 observables are independent of gauge used 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 b G =2

24. Beyond f NL ? Higher-order statistics –trispectrum  g NL (Seery & Lidsey; Byrnes, Sasaki & Wands 2006...) -7.4 < g NL / 10 5 < 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 f NL (Byrnes, Nurmi, Tasinato & Wands 2009) –local function of more than one independent Gaussian field –non-linear evolution of field during inflation -2.5 < n fNL < 2.3 (Smidt et al 2010) Planck: |n fNL | < 0.1 for f fNL =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. outlook ESA Planck satellite next all-sky survey data early 2013… f NL < 10 g NL < ? + future LSS constraints... f NL < 1??

26. 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

27. scale-dependence of f NL ?  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 g NL   NL (Byrnes et al 2011) Byrnes, Nurmi, Tasinato & Wands (2009); Byrnes, Gerstenlauer, Nurmi, Tasinato & Wands (2010) Byrnes, Choi & Hall 2009 Khoury & Piazza 2009 Sefusatti, Liguori, Yadav, Jackson & Pajer 2009

28. quasi-local model for scale-dependent f NL  Fourier space:  quasi-local non-Gaussianity in real space: x’ x Byrnes, Gerstenlauer, Nurmi, Tasinato & Wands (2010)

29. scale-dependent f NL from a local two-field  power spectrum  bispectrum Byrnes, Nurmi, Tasinato & Wands (2009)

30. local two-field scale-dependent f NL  power spectrum  bispectrum where scale-dependence  e.g., inflaton + non-interacting curvaton for CMB+LSS constraints on this model see Tseliakhovich, Hirata & Slosar (2010) Byrnes, Nurmi, Tasinato & Wands (2009)

31. scale-dependent f NL two natural generalisations of local f NL non-Gaussianity lead to scale- dependent reduced bispectrum  multi-variable local f NL  quasi-local f NL Byrnes, Choi & Hall 2009 Khoury & Piazza 2009 Sefusatti, Liguori, Yadav, Jackson & Pajer 2009 Byrnes, Nurmi, Tasinato & Wands 2009

32. trispectrum where we have two independent parameters from  N calculation and simplest local form of non-Gaussianity to third order multi-source consistency relation:  NL  (f NL ) 2

33. 3 rd order non-linearity for curvaton Sasaki, Valiviita & Wands (astro-ph/0607627) for large f NL >>1 find g NL <<  NL for quadratic curvaton

34. full pdf for  from  N Sasaki, Valiviita & Wands (2006)

35. probability distribution for 

37. 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 Wands37

38. remember: f NL < 100 implies Gaussian to better than 0.1%

39. 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 c 2 >>1 ): -n – 1 = 4 / c 2 > 0 -large and negative bispectrum: -f NL = - (5/12) c i 2 < - (5/3) / (n-1) -Other authors consider corrections (e.g., c i (  i ) ) corrections to tilt + and corrections to f NL -in general, steep potentials and fast roll => large non-Gaussianity

40. curvaton vs ekpyrotic non-Gaussianity? Curvaton f NL > -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 f NL negative or positive? potentials are steep quasi-exponential expect large non-linearities at all orders

41. curvaton vs ekpyrotic non-Gaussianity? Curvaton non-interacting curvaton: (Sasaki, Valiviita & Wands 2006) g NL = - (10/3) f NL & n fNL = 0 self-interacting curvaton: (Enqvist et al 2009; Byrnes et al 2011) g NL ≈ f NL 2 & n fNL = (P T 1/2 P  1/2 f NL ) -1 V’’’/M Ekpyrotic ekpyrotic or kinetic conversion: (Lehners & Renaux-Petel 2009) g NL ≈ f NL 2 exponential potential  scale-invariance:  n fNL = 0 (Fonseca, Vernizzi & Wands, in preparation)

42. outline: why Gaussian and why not? local non-Gaussianity and f NL from inflation beyond f NL –higher-order statistics –scale-dependence conclusions

43. Newtonian potential a local function of Gaussian random field at every point in space  (x) =  G (x) + f NL (  G 2 (x) - ) Komatsu & Spergel (2001) Simple local form for primordial non-Gaussianity

44. evidence for local non-Gaussianity?  T/T  -  /3, so positive f NL  more cold spots in CMB

45. Wilkinson Microwave Anisotropy Probe 7-year data, February 2010