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Primoridial perturbations from inflation

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					Cosmo-12, Beijing                                   13th 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,
            work with Chris Byrnes, Jon Emery, Christian Fidler,
       Gianmassimo Tasinato, Kazuya Koyama, David Langlois, David
       Gianmassimo Tasinato, Kazuya Koyama, David Langlois, David
            Lyth, Misao Sasaki, Jussi Valiviita, Filippo Vernizzi…
             Lyth, Misao Sasaki, Jussi Valiviita, Filippo Vernizzi…
          review: Classical & Quantum Gravity 27, 124002 (2010)
          review: Classical & Quantum Gravity 27, 124002 (2010)
                              arXiv:1004.0818
                              arXiv:1004.0818
WMAP7 standard model of primordial cosmology   Komatsu et al 2011
Gaussian random field, z(x)
• normal distribution of values in real space, Prob[z(x)]




• defined entirely by power spectrum in Fourier space



• bispectrum and (connected) higher-order correlations vanish


                             David Wands                        3
non-Gaussian random field, z(x)



        anything else


               David Wands        4
Rocky Kolb



non-Rocky Kolb



                 5
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, e, ...)


   – 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                                       6
                                                                     Wikipedia: AllenMcC
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
inflation




            Super-Hubble evolution       Self-interactions + gravity

            End of inflation             Tachyonic instability
            (p)Reheating                 Modulated (p)reheating
            After inflation              Curvaton decay
                                         Magnetic fields
                                 primordial non-Gaussianity
            Primary anisotropies           Last-scattering
            Secondary anisotropies       ISW/lensing + foregrounds


                                                                            7
                                      David Wands
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                                   8
Primordial density perturbations from quantum field fluctuations
                                t
z = curvature perturbation on
  uniform-density hypersurface
  in radiation-dominated era



 f(x,ti ) during inflation
    field perturbations on initial                                                     x
    spatially-flat hypersurface


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




 Starobinsky 85; Salopek & Bond 90; Sasaki & Stewart 96; Lyth & Rodriguez 05; Langlois &
                                                                               Vernizzi...
order by order at Hubble exit




e.g.,   <z3>
                                                     N’
          N’         N’
                                             N’’
                                                       N’
                     N’


 sub-Hubble field interactions         super-Hubble classical evolution

                                 Byrnes, Koyama, Sasaki & DW (arXiv:0705.4096)
non-Gaussianity from inflation?
• single-field slow-roll inflaton
   – during conventional slow-roll inflation
   – adiabatic perturbations                                          Maldacena 2002

     => z constant on large scales => more generally:
                                                         Creminelli & Zaldarriaga 2004


• sub-Hubble interactions
   – e.g. DBI inflation, Galileon fields...
                                                                     Cheung et al 2008

• super-Hubble evolution
   – non-adiabatic perturbations during multi-field inflation
      => z ¹ constant
       • see talks this afternoon by Emery & Kidani
   – at/after end of inflation (curvaton, modulated reheating, etc)
       • e.g., curvaton
multi-field inflation revisited
• light inflaton field + massive isocurvature fields
   –   Chen & Wang (2010+12)
   –   Tolley & Wyman (2010)
   –   Cremonini, Lalak & Turzynski (2011)
   –   Baumann & Green (2011)
   –   Pi & Shi (2012)
   –   Achucarro et al (2010-12); 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 Þ fNLlocal
simplest local form of non-Gaussianity
applies to many inflation models including curvaton, modulated reheating, etc

  z = z(df ) is local function of single Gaussian random field, df(x)




                                                                         N’
                     where
                                                                   N’’
                                                                          N’


  •   odd factors of 3/5 because (Komatsu & Spergel, 2001, used) F1   =(3/5)z1
Local trispectrum has 2 terms at tree-level



                 tNL                                                 gNL
          N’             N’                                   N’           N’
          N’’            N’’                                  N’’’           N’



 • can distinguish by different momentum dependence
 • Suyama-Yamaguchi consistency relation: tNL = (6fNL/5)2
    – generalised to include loops: < T P > = < B2 >
      Tasinato, Byrnes, Nurmi & DW (2012)            see talk by Tasinato this afternoon
                                       David Wands                                   14
Newtonian potential a Gaussian random field
      F(x) = fG(x)




                        Liguori, Matarrese and Moscardini (2003)
Newtonian potential a local function of Gaussian random field
      F(x) = fG(x) + fNL ( fG2(x) - <fG2> )
fNL=+3000




DT/T » -F/3, so positive fNL Þ more cold spots in CMB
                         Liguori, Matarrese and Moscardini (2003)
Newtonian potential a local function of Gaussian random field
      F(x) = fG(x) + fNL ( fG2(x) - <fG2> )
fNL=-3000




DT/T » -F/3, so negative fNL Þ more hot spots in CMB
                         Liguori, Matarrese and Moscardini (2003)
Constraints on local non-Gaussianity
• WMAP CMB constraints using estimators based on
  matched templates:

   Ø -10 < fNL < 74 (95% CL) Komatsu et al WMAP7
   Ø -5.6 < gNL / 105 < 8.6 Ferguson et al; Smidt et al 2010
Newtonian potential a local function of Gaussian random field

       F(x) = fG(x) + fNL ( fG2(x) - <fG2> )
Þ Large-scale modulation of small-scale power
 split Gaussian field into long (L) and short (s) wavelengths
       fG (X+x) = fL(X) + fs(x)
 two-point function on small scales for given fL
       < F(x1) F(x2) >L = (1+4 fNL fL ) < fs (x1) fs (x2) > +...


                 X1                           X2
 i.e., inhomogeneous modulation of small-scale power
       P ( k , X ) -> [ 1 + 4 fNL fL(X) ] Ps(k)
       but fNL <100 so any effect must be small
Inhomogeneous non-Gaussianity? Byrnes, Nurmi, Tasinato & DW
        F(x) = fG(x) + fNL ( fG2(x) - <fG2> ) + gNL fG3(x) + ...
split Gaussian field into long (L) and short (s) wavelengths
        fG (X+x) = fL(X) + fs(x)
 three-point function on small scales for given fL
  < F(x1) F(x2) F(x3) >X = [ fNL +3gNL fL (X)] < fs (x1) fs (x2) fs2 (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 = dg/ dm




  Poisson equation relates primordial density to Newtonian potential
        Ñ 2F = 4p Gdr =>          fL = (3/2) ( aH / k L ) 2 dL
  so local F(x) Þ non-local form for primordial density field d(x) from
  + inhomogeneous modulation of small-scale power

        s ( X ) = [ 1 + 6 fNL ( aH / k ) 2 dL ( X ) ] s s
  Þ 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:

   Ø -10 < fNL < 74 (95% CL) Komatsu et al WMAP7
   Ø -5.6 < gNL / 105 < 8.6 Ferguson et al; Smidt et al 2010


• LSS constraints from galaxy power spectrum on large
  scales:

   Ø -29 < fNL < 70 (95% CL) Slosar et al 2008 [SDSS]
   Ø 27 < fNL < 117 (95% CL) Xia et al 2010 [NVSS survey of AGNs]
Tantalising evidence of local fNL                     local
                                                              ?
• Latest SDSS/BOSS data release (Ross et al 2012):

   Prob(fNL>0)=99.5% without any correction for systematics
   Ø 65 < fNL < 405 (at 95% CL) no weighting for stellar density

   Prob(fNL>0)=91%
   Ø -92 < fNL < 398 allowing for known systematics

   Prob(fNL>0)=68%
   Ø -168 < fNL < 364 marginalising over unknown systematics
Beyond fNL?
•       Higher-order statistics
    –       trispectrum Þ gNL (Seery & Lidsey; Byrnes, Sasaki & Wands 2006...)
          •     -7.4 < gNL / 105 < 8.2 (Smidt et al 2010)
    –       dN(j) 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 dN to isocurvature modes (Kawasaki et al; Langlois, Vernizzi & Wands 2008)
    –      limits on isocurvature density perturbations (Hikage et al 2008)
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)




    Ø templates for secondary non-Gaussianity (inc. lensing)
    Ø induced tensor and vector modes from density perturbations
    Ø testing interactions at recombination dr
        Ø e.g., gravitational wave production                 h
                                             dr
            outlook
      ESA Planck satellite
       next all-sky survey
        data early 2013…
                   fNL < 5


+ future LSS constraints...

 Euclid satellite: fNL < 3?
                     SKA ??
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

				
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posted:7/25/2013
language:English
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