CMB contraints on primordial non-Gaussianity from the bispectrum (f{sub NL}) and trispectrum (g{sub NL} and Ï{sub NL}) and a new consistency test of single-field inflation

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CMB contraints on primordial non-Gaussianity from the bispectrum (f{sub NL}) and trispectrum (g{sub NL} and Ï{sub NL}) and a new consistency test of single-field inflation

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  CMB Constraints on Primordial non-Gaussianity from the Bispectrum ( f  NL ) andTrispectrum ( g NL  and  τ  NL ) and a New Consistency Test of Single-Field Inflation Joseph Smidt ∗ , 1 Alexandre Amblard † , 1 Christian T. Byrnes ‡ , 2 Asantha Cooray § , 1 Alan Heavens ¶ , 3 and Dipak Munshi ∗∗ 3,4 1 Center for Cosmology, Department of Physics and Astronomy,University of California, Irvine, CA 92697, USA 2  Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany  3  Scottish Universities Physics Alliance (SUPA), Institute for Astronomy,University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK  4 School of Physics and Astronomy, Cardiff University, CF24 3AA (Dated: July 27, 2010)We outline the expected constraints on non-Gaussianity from the cosmic microwave background(CMB) with current and future experiments, focusing on both the third ( f  NL ) and fourth-order( g NL  and  τ  NL ) amplitudes of the local configuration or non-Gaussianity. The experimental focus isthe skewness (two-to-one) and kurtosis (two-to-two and three-to-one) power spectra from weightedmaps. In adition to a measurement of   τ  NL  and  g NL  with WMAP 5-year data, our study providesthe first forecasts for future constraints on  g NL . We describe how these statistics can be correctedfor the mask and cut-sky through a window function, bypassing the need to compute linear termsthat were introduced for the previous-generation non-Gaussianity statistics, such as the skewnessestimator. We discus the ratio  A NL  =  τ  NL / (6 f  NL / 5) 2 as an additional test of single-field inflationarymodels and discuss the physical significance of each statistic. Using these estimators with WMAP5-Year V+W-band data out to  l max  = 600 we constrain the cubic order non-Gaussianity parameters τ  NL , and  g NL  and find  − 7 . 4  < g NL / 10 5 <  8 . 2 and  − 0 . 6  < τ  NL / 10 4 <  3 . 3 improving the previousCOBE-based limit on  τ  NL  <  10 8 nearly four orders of magnitude with WMAP. I. INTRODUCTION We have now entered an exciting time in cosmologicalstudies where we are now beginning to constrain simpleslow-roll inflationary models with high precision obser-vations of the cosmic microwave background (CMB) andlarge-scale structure. In addition to constraining infla-tionary model parameter space with traditional parame-ters such as the spectral index  n s  and the tensor-to-scalarratio  r , we may soon be able to use parameters associ-ated with primordial non-Gaussianity to improve modelselection.In the simplest realistic inflationary models, the field(s)responsible for inflation have minimal interactions. Suchan interaction-less situation should have led to Gaussianprimordial curvature perturbations, assuming that pertu-bations in the inflaton field generates the curvature per-turbation. In this case, the two point correlation functioncontains all the informations on these perturbations. If the early inflation field(s) have non-trivial interactions,higher-order correlation functions of the curvature per-turbations will contain  connected   pieces encoding in-formation about the primordial inflationary interactions.This is analogous to the situation encountered in particle ∗  jsmidt@uci.ed † amblard@uci.edu ‡ Byrnes@physik.uni-bielefeld.de § acooray@uci.edu ¶ afh@roe.ac.uk ∗∗ dipak.munshi@astro.cf.ac.uk physics where correlation functions can be separated intounconnected and connected Feynman diagrams, the latercontaining information about the underlying interactions(see Fig. 1 for an example involving the four-point func-tion). A detection of non-Gaussianity therefore gives animportant window into the nature of the inflation field(s)and their interactions.To parameterize the non-Gaussianity of a nearly Gaus-sian field, such as the primordial curvature perturbations ζ  ( x ), we can expand them perturbatively [29] to secondorder as: ζ  ( x ) =  ζ  g ( x ) + 35 f  NL  ζ  2 g ( x ) − ζ  2 g ( x )   + 925 g NL ζ  3 g ( x ) , (1)where  ζ  g ( x ) is the purely Gaussian part with  f  NL  and g NL  parametrizing the first and second order deviationsfrom Gaussianity. This parameterization of the curva-ture perturbations is known as the local model as thisdefinition is local in space.Much effort has already gone into measuring non-Gaussianity at first-order in curvature perturbations us-ing the bispectrum of the CMB anisotropies or large-scalestructure galaxy distribution parametrerized by  f  NL  (seeEq. 1). These studies have found  f  NL  to be consistentwith zero [1–4]. However, there is hope that a significant detection may be possible by future surveys that will leadto improved errors [5].In the trispectrum, two parameters of second-ordernon-Gaussianity at fourth-order in curvature perturba-tions,  τ  NL  and  g NL , can be measured. In this paperwe also introduce a third parameter,  A NL  is an addi-tional parameter that compares  τ  NL  of the trispectrum   a  r   X   i  v  :   1   0   0   4 .   1   4   0   9  v   2   [  a  s   t  r  o  -  p   h .   C   O   ]   2   6   J  u   l   2   0   1   0  2 + MoreUnconnecteContri utionsConnectedMoreContributions + + =  <φ 1 φ 2 φ 3 φ 4  > G  + <φ 1 φ 2 φ 3 φ 4  > c <φ 1 φ 2 φ 3 φ 4  >  = φ 1 φ 1  2  2  3  3 φ 4 φ 4 FIG. 1: Four point correlation function for the  φ 3 theory.The correlation functions breaks up into interaction-less un-connected diagrams and connected diagrams containing in-formation about the interactions. to (6 f  NL / 5) 2 from the bispectrum as a ratio: A NL  =  τ  NL (6 f  NL / 5) 2 .  (2)This ratio can be quite different for many inflationarymodels [6, 7] and, as will be shown below,  A NL   = 1 rulesout single-field inflationary models altogether, includingthe standard curvaton scenario (which neglects pertur-bations from the inflaton field).In this paper we discuss the skewness and kurto-sis power spectra method for probing primordial non-Gaussianity and give constraints for the first ( f  NL ) andsecond-order ( g NL  and  τ  NL ) amplitudes of the local modelin addition to their ratio  A NL . Using the bispectrum of CMB anisotropies as seen by WMAP 5-year data, Smidtet al. (2009) found  − 36 . 4  < f  NL  <  58 . 4 at 95% confi-dence [4]. This is to be compared with the most recentWMAP 7 measurement of   − 10  < f  NL  <  74 [3], wherepart of the discrepancy is due to a difference in optimiza-tion [8]. As outlined in Section VI, using the trispectrumof the same data we find that  − 0 . 6  < τ  NL / 10 4 <  3 . 3and − 7 . 4  < g NL / 10 5 <  8 . 2 at 95% confidence level show-ing second order non-Gaussianity is consistent with zeroin WMAP. This paper serves as a guide to the analysisprocess behind our derived limits on  τ  NL ,  g NL  and  A NL .Furthermore, in this paper we analyze what to real-istically expect when measuring non-Gaussianity fromCMB temperature data. We believe establishing whatconstraints can be placed upon  f  NL ,  τ  NL ,  g NL  and  A NL by future experiments is important in determining whatmodels may and may not be tested by future data. Wealso highlight several advantages of our work, includingways to correct the cut-sky and mask through a windowfunction without using linear terms which are computa-tionally prohibative [9, 10].This paper is organized as follows: In Section II wereview how non-Gaussianity may be used to distinguishbetween common inflationary models and stress the phys-ical significance of each statistic. In Section III wedescribe the skewness and kurtosis power spectra andexplain how they may be used to extract informationabout primordial non-Gaussianity from the CMB. In Sec-tion IV, we describe the signal-to-noise of each estimator,how to add the experimental beam and noise to these cal-culations and discuss why these power spectra have theadvantage for dealing with a cut sky. In Section V wecalculate the fisher bounds for upcoming experiments foreach statistic. In Section VI we discuss the technical de-tails for measuring non-Gaussianity in the trispectrumand in Section VII we conclude with a discussion. II. NON-GAUSSIANITY FROM COMMONINFLATIONARY MODELS Non-Gaussinity is a powerful tool that may be usedto distinguish between inflationary models. The sim-plest models do not produce a detectable amount of non-Gaussianity. Maldacena [11] has shown that a single-field, experiencing slow roll with canonical kinetic energyand an initial Bunch-Davies vacuum state produces f  NL  = 512( n s  + f  ( k ) n t ) .  (3)Here  n s  and  n t  are the scalar and tensor spectral indicesrespectively. The function  f  ( k ) has a range 0 ≤ f  ( k ) ≤  56 based on the triangle shapes (see below) of the  k i  suchthat  f   = 0 in the squeezed limit and  f   =  56  for an equilat-eral triangle. For this reason,  f  NL  <  1 will remain unde-tectable in the simple slow roll scenario with CMB dataalone. If any of the above assumptions are violated, veryspecific types of non-Gaussianity are produced [5, 12, 13].In the bispectrum  B ζ  ( k 1 ,k 2 ,k 3 ) defined by  ζ  k 1 ζ  k 2 ζ  k 3  = (2 π ) 3 δ  ( k 1  + k 2  + k 3 ) B ζ  ( k 1 ,k 2 ,k 3 ) ,  (4)where  ζ   is the primordial curvature perturbation, non-Gaussianities show up as triangles in Fourier space. Dif-ferent triangle shapes are be produced by different un-derlying physics, for example: •  squeezed triangle   ( k 1  ∼ k 2  ≫ k 3 ) This is the domi-nating shape from multi-field, curvaton, inhomoge-neous reheating and Ekpyrotic models. •  equilateral triangle   ( k 1  =  k 2  =  k 3 ) Thisshape is produced by non-canonical kinetic energywith higher derivative interactions and non-trivialspeeds of sound. •  folded triangle   ( k 1  = 2 k 2  = 2 k 3 ) These triangles areproduced by non-adiabatic-vacuum models.Additionally, linear combinations of the above shapes orintermediate cases such as  elongated triangles   ( k 1  =  k 2 + k 3 ) and  isosceles triangles   ( k 1  > k 2  =  k 3 ) are possible [5,  3 k        2        0.2    0.4    0.6    0.8         k       3   0.2    0.4    0.6    0.8    2    4    6    8    10    12        k       2    0.2    0.3    0.4    0.5    0.6    0.7    0.8    0.9     k       3       0.2    0.3    0.4    0.5    0.6    0.7    0.8    0.9    0.2    0.4    0.6    0.8     FIG. 2: Plot of the shape functions  S  local (1 ,k 2 ,k 3 ) and S  equil (1 ,k 2 ,k 3 ) normalized such that  S  (1 , 1 , 1) = 1. In theseplots only values satisfying the triangle inequality  k 2  +  k 3  ≥ k 1  = 1 as well as the requirement  k 2  ≤  k 3  to prevent show-ing equivalent configurations are non-zero. The plot on topverified  S  local is maximized when  k 1  ∼  k 3  ≫  k 2  whereas thebottom plot verifies  S  equal is maximised when  k 1  ∼ k 2  ∼ k 3 . 12, 13]. The most recent WMAP 7 constraints on the amount of non-Gaussinaity from each shape is  − 10  <f  localNL  <  74,  − 214  < f  equilNL  <  266 and  − 410  < f  orthogNL  <  6at 95% confidence [3].A convenient way to distinguish between shapes is tointroduce the shape function defined as S  ( k 1 ,k 2 ,k 3 ) ≡  1 N  ( k 1 k 2 k 3 ) 2 B ζ  ( k 1 ,k 2 ,k 3 ) ,  (5)where N is a normalization factor often taken to be1 /f  NL . Using a notation introduced by Fergusson andShellard [14], we can give the shape function for the morecommon configurations as: S  local ( k 1 ,k 2 ,k 3 )  ∝  K  3 K  111 ,  (6) S  equil ( k 1 ,k 2 ,k 3 )  ∝ ˜ k 1 ˜ k 2 ˜ k 3 K  111 ,  (7) S  folded ( k 1 ,k 2 ,k 3 )  ∝  1 K  111 ( K  12 − K  3 ) + 4  K  2 ˜ k 1 ˜ k 2 ˜ k 3 , (8)where K   p  =  i ( k i )  p with  K   =  K  1 ,  (9) K   pq  = 1∆  pq  i  = j ( k i )  p ( k j ) q ,  (10) K   pqr  = 1∆  pqr  i  = j  = l ( k i )  p ( k j ) q ( k l ) r ,  (11)˜ k ip  =  K   p − 2( k i )  p with ˜ k i  = ˜ k i 1 ,  (12)with ∆  pq  = 1 +  δ   pq  and ∆  pqr  = ∆  pq (∆ qr  +  δ   pr ) (nosummation). Plots for the local and equilateral shapesare given in Figure 2.In addition to  f  NL  being generated by different shapes,it also may vary with scale. Recently, a new parameterhas been introduced to measure this scale dependancedefined as: n f  NL ( k ) =  d ln | f  NL ( k ) | d ln k .  (13)This scale dependance has the ability to test the ansatz 1to test whether the local model should allow for  f  NL to vary with scale [15]. Using the results of Smidt etal.(2009) (Fig. 16 of Ref [4]) and assuming f  NL ( l ) =  f  NL 200   ll 200  n f  NL ( l ) ,  (14)we can constrain  n f  NL ( l ) to roughly − 2 . 5  < n f  NL ( l )  <  2 . 3at 95% confidence. We therefore find  f  NL  is consistentwith having no scale dependance.In this paper we focus on the local model that probesnon-Gaussianty of a squeezed shape. As mentionedabove, simple inflationary models can not produce a de-tectable amount of non-Gaussinity for local models. Wenow review the prediction for local non-Gaussianity forthe most common models. A. Review Of The  δN   formalism. The curvature perturbation can be conveniently de-scribed using the  δN   formalism [16–20]. During infla- tion, spacetime expands by a certain number of e-foldsN. By Heisenberg’s uncertainty principle, expansion foreach point in space ends at slightly different times pro-  4ducing a spatially dependent total e-fold: N  ( x ) =    t f  t i H  ( x,t ) dt,  (15)where  H  ( x,t ) is the Hubble parameter allowing us to de-fine  N  ( x ) = ¯ N   + δN  ( x ). The fluctuations in e-fold aboutthe mean value ¯ N  , which correspond to perturbations inlocal expansion, are the curvature perturbations  ζ   =  δN  .In addition to a spatial parameterization, we may pa-rameterize the number of e-folds by the underlying fields ζ   =  N  ( φ A )  −  ¯ N   where  φ A represents the initial val-ues for the scalar fields . If we write out the fields as φ A = ¯ φ A +  δφ A we can expand the curvature perturba-tions as ζ   =  δN   =  n 1 n ! N  A 1 A 2 ...A n δϕ A 1 δϕ A 1 ...δϕ A n .  (16)The  N  x  means the derivative of N with respect to thefields  x . For example,  N  A 1 A 2  ≡  ∂  2 N ∂ϕ A 1 ∂ϕ A 2  . In this equa-tion there is an implicit sum over the  A i . Einstein sum-mation is implicit in all equations relating to the  δN  formalism.Using this formalism we may compute to first orderfrom  ζ   =  N  A δϕ A :  ζ  k ζ  k ′   =  N  A N  B C  AB ( k )(2 π ) 3 δ  3 ( k + k ′ ) ,  (17)where  C  AB ( k ) in the slow roll limit becomes to leadingorder  δ  AB P  ( k ).Likewise, we can calculate the bispectrum and trispec-trum in this formalism; B ζ  ( k 1 ,k 2 ,k 3 ) =  N  A N  BC  N  D [ C  AB ( k 1 ) C  BD ( k 2 ) (18)+  C  AB ( k 1 ) C  BD ( k 2 ) + C  AB ( k 1 ) C  BD ( k 2 )] ,T  ζ  ( k 1 ,k 2 ,k 3 ,k 4 ) =  N  A 1 A 2 N  B 1 B 2 N  C  N  D  (19) × [ C  A 2 B 2 ( k 13 ) C  A 1 C  ( k 2 ) C  B 1 D ( k 2 ) + (11 perms)]+ N  A 1 A 2 A 3 N  B N  C  N  D × [ C  A 1 B ( k 13 ) C  A 2 C  ( k 2 ) C  A 3 D ( k 2 ) + (3 perms)] , where  k ij  =  | k i  +  k j | . In the slow roll limit to leadingorder these expressions may be rewritten as: B ζ  ( k 1 ,k 2 ,k 3 ) = 65 f  NL [ P  ζ  ( k 1 ) P  ζ  ( k 2 ) + (20) P  ζ  ( k 2 ) P  ζ  ( k 3 ) + P  ζ  ( k 3 ) P  ζ  ( k 1 )] ,T  ( k 1 ,k 2 ,k 3 ,k 4 ) =  τ  NL [ P  ζ  ( k 13 ) P  ζ  ( k 3 ) P  ζ  ( k 4 ) + (11 perms)]+ 5425 g NL [ P  ζ  ( k 2 ) P  ζ  ( k 3 ) P  ζ  ( k 4 ) + (3 perms)] , (21)where  P  ζ  ( k ) =  N  A N  B C  AB ( k ) and therefore in the slowroll limit  P  ζ  ( k ) =  N  A N  A P  ( k ).From the above two expressions we can read off thevalues for each statistic: f  NL  = 56 N  A N  B N  AB ( N  C  N  C  ) 2  ; (22) τ  NL  =  N  AB N  AC  N  B N  C  ( N  D N  D ) 3  ; (23) g NL  = 2554 N  ABC  N  A N  B N  C  ( N  D N  D ) 3  ; (24) A NL  =  τ  NL (6 f  NL / 5) 2 .  (25) B. General Single-Field Models For a single scalar field  ϕ  perturbing  N  ( ϕ ) we mayexpand  ζ  , using the above formalism [18], as: ζ   =  N  ′ δϕ + 12 N  ′′ δϕ 2 + 16 N  ′′′ δϕ 3 + ...,  (26)where N  ′ =  dN/dϕ . Note that we do not require that ϕ isthe inflaton field, it could be the curvaton or a field whichmodulates the efficiency of reheating. From equations 22-24 we may immediately read off  f  NL  = 56 N  ′′ ( N  ′ ) 2 ; (27) τ  NL  = ( N  ′′ ) 2 ( N  ′ ) 4  ; (28) g NL  = 2554 N  ′′′ ( N  ′ ) 3 ; (29) A NL  = 1 .  (30)Equations 27 and 28 yield a very important conse- quence of single-field models namely  τ  NL  = (6 f  NL / 5) 2 .This is a general result and therefore  A NL   = 1 may beused to rule out single-field models all together. C. Multi-Field Inflationary Models Suyama and Yamaguchi showed in general  τ  NL  ≥ (6 f  NL / 5) 2 by the Cauchy-Schwartz inequality and equal-ity only holds if   N  A  is an eigenmode of   N  AB  [20]. Modelswhere equality does not hold can not be those of a single-field. We now examine such models.Unlike the single-field case, using the  δN   formalismto make general statements about multi-field models isnearly impossible. Instead, one is forced to work withspecific models that utilize simplifying assumptions. Wenow present a class of multi-field models that we believeis sufficiently general to uncover many details that arecharacteristic of multi-field models in general.Recently, Byrnes and Choi reviewed two field modelswith scalar fields  ϕ  and  χ  that have a separable potential  5 W  ( ϕ,χ ) =  U  ( ϕ ) V  ( χ ) [6, 21–24]. The slow roll parame- ters for these models are: ǫ ϕ  = M  2  p 2  U  ,ϕ U   2 , ǫ χ  = M  2  p 2  V  ,χ V   2 ,  (31)(32) η ϕϕ  =  M  2  p U  ,ϕϕ U  , η ϕχ  = M  2  p U  ,ϕ V  ,χ W  , η χχ  = M  2  p V  ,χχ V  , from which we can define˜ r  =  ǫ χ ǫ ϕ e 2( η ϕϕ − η χχ ) N  .  (33)For this class of models, in the regions where | f  NL | >  1we have f  NL  = 56 η χχ ˜ r (1 + ˜ r ) 2 e 2( η ϕϕ − η χχ ) N  ; (34) g NL  = 103˜ r ( η ϕϕ − 2 η χχ ) − η χχ 1 + ˜ r f  NL ; (35) τ  NL  = 1 + ˜ r ˜ r  6 f  NL 5  2 ; (36) A NL  = 1 + ˜ r ˜ r .  (37)It is worth noting that both  τ  NL  and  g NL  are related to f  NL  for this class of models. Here we have  | g NL | < | f  NL | which will therefore be much harder to detect. On thecontrary,  τ  NL  >  (6 f  NL / 5) 2 so that non-Gaussinity mayin fact be easier to detect in the trispectrum than thebispectrum for some multi-field models. Here we find A NL  = (1 + ˜ r ) / ˜ r >  1. The scale dependance of   f  NL  hasalso been worked out for this class of models and wasfound to be  n f  NL  = − 4( η ϕϕ − η χχ ) / (1 + ˜ r )  <  0. D. Curvaton Models In the curvaton scenario, a weakly interacting scalarfield  χ  exists in conjunction to the inflaton  ϕ  [6, 18, 25– 28]. During inflation, the curvaton field is subdominant,but after inflation  χ  can dominate the energy density.The decay of the inhomogeneous curvation field in thisscenario produces the curvature perturbations and notthe inflaton.If such a curvaton field is the soul contributor to cur-vature perturbations, we can write out the perturbationsusing the  δN   formalism as we did in the single field case: ζ   =  N  ′ δχ + 12 N  ′′ δχ 2 + 16 N  ′′′ δχ 3 + ...,  (38)where now  N  ′ =  dN/dχ . Immediately we recover therelations 27-29 and find for such curvaton models  A NL  =1 as should be expected from curvature perturbationsgenerated by a single-field.Recently, curvation models with generic potentials of the form V   = 12 m 2 χ 2 + λχ n +4 ,  (39)have been analyzed [27, 28]. Here  m  is the curvaton’smass and  λ  is a coupling constant. For such models  N  in Equation 38 has been worked out giving: f  NL  = 54 r χ (1 + h ) −  53  −  5 r χ 6  ,  (40) g NL  = 2554   94 r 2 χ (˜ h + 3 h ) −  9 r χ (1 + h ) (41)+12(1 − 9 h ) + 10 r χ  + 3 r 2 χ  , where r χ  = 3Ω χ,D 4 − Ω χ,D , h  =  χ 0 χ ′′ 0 χ ′ 20 ,  ˜ h  =  χ 20 χ ′′′ 0 χ ′ 30 .  (42)Here Ω χ,D  is the energy density at time of curvaton decay, χ 0  is the curvation field during oscillations just beforedecay and the primes here denote derivatives with respectto time.Unlike single scalar field inflation, curvaton models canhave large self interactions. Enqvist et al. pointed outthat even if   f  NL  is small,  g NL  can be large for significantlevels of self-interactions [28]. This places a physical sig-nificance on  g NL  that can be thought of as parameterizinglarge self-interactions. E. Brief Summary In this section we have discussed the physical signif-icance of each statistic  f  NL ,  g NL ,  τ  NL  and  A NL . Inthe bispectrum,  f  NL  receives contributions from differ-ent shaped triangles in Fourier space related to differentunderlying physics. By analyzing the amount of non-Gaussianity from these different shapes we can distin-guish between models with multiple fields, non-canonicalkinetic energy and non-adiabatic vacuums.In addition we stressed the physical significance of localnon-Gaussianity in the trispectrum. The relation  A NL  = τ  NL / (6 f  NL / 5) 2 is an important constraint of multi-fieldmodels. A general result for single-field models is  A NL  =1. Lastly,  g NL  will place important constraints on thelevel of self-interactions. III. POWER SPECTRA ESTIMATORS FORFIRST AND SECOND-ORDERNON-GAUSSIANITY. We would like to find a way to measure the non-Gaussianity of these fields from something directly ob-servable. Fortunately, information about the curvature
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