Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure

9.4 MB·English

by  M. Liguori

Primordial non-Gaussianity and Bispectrum Measurements in the
Cosmic Microwave Background and Large-Scale Structure
MicheleLiguori∗
Centre for Theoretical Cosmology
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
WilberforceRoad,Cambridge,CB30WA,UK
EmilianoSefusatti†
Institut de Physique The´orique
Commissariat a` l’E´nergie Atomique
0 F-91191Gif-sur-Yvette,France
1
0 JamesR.Fergusson‡ andE.P.S.Shellard§
2
Centre for Theoretical Cosmology

n Department of Applied Mathematics and Theoretical Physics
a University of Cambridge
J WilberforceRoad,Cambridge,CB30WA,UK

5
2 Themostdirectprobeofnon-Gaussianinitialconditionshascomefrombispectrummeasurementsoftemperature
fluctuationsintheCosmicMicrowaveBackgroundandofthematterandgalaxydistributionatlargescales.Such

] bispectrumestimatorsareexpectedtocontinuetoprovidethebestconstraintsonthenon-Gaussianparametersin
O
futureobservations.Wereviewandcomparethetheoreticalandobservationalproblems,currentresultsandfuture
C prospectsforthedetectionofanon-vanishingprimordialcomponentinthebispectrumoftheCosmicMicrowave
Backgroundandlarge-scalestructure,andtherelationtospecificpredictionsfromdifferentinflationarymodels.
.
h
p
-
o
r
t
s Contents
a
[ I. Introduction 3

1
II. Initialconditionsandtheprimordialbispectrum 5
v
A. Theprimordialbispectrumandshapefunction 5
7 B. Generalprimordialbispectraandseparablemodeexpansions 8
0 C. Familiesofprimordialmodelsandtheircorrelations 11
7 1. Theconstantmodel 11
4 2. Equilateraltriangles–centre-weightedmodels 11
. 3. Squeezedtriangles–corner-weightedmodels 12
1
4. Flattenedtriangles–edge-weightedmodels 13
0
5. Features–scale-dependentmodels 14
0
1 III. CosmicMicrowaveBackground 14
: A. TheCMBbispectrum 14
v
B. SeparableprimordialshapesandCMBbispectrumsolutions 16
i
X C. Non-separablebispectrarevisited 18
D. CMBbispectrumcalculationsandcorrelations 19
r
a
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected]
‡Electronicaddress:[email protected]
§Electronicaddress:[email protected]
2
1. Nearlyscale-invariantmodels 19
2. Scale-dependentmodels,cosmicstringsandotherlate-timephenomena 20
E. Theestimationof fNLfromCMBbispectra 22
1. Bispectrumestimatorof fNL 23
2. Optimalityofthecubicestimator 24
3. Breakingrotationalinvariance 27
4. Large fNLregime 28
F. Numericalimplementationofthebispectrumestimator 30
1. Primarycubictermfor fNL 31
2. Linearcorrectiontermfor fNL 33
G. Experimentalconstraintson fNL 35
H. Fishermatrixforecasts 36
1. Ageneralderivation 36
2. Polarization 38
I. Non-Gaussiancontaminants 39
1. Diffuseforegroundemission 43
2. Unresolvedpointsources 44
3. Secondaryanisotropies 45
4. Non-Gaussiannoise 46
5. Othereffects 46
J. Generationofsimulatednon-GaussianCMBmaps 46
1. Algorithmsforlocalnon-Gaussianity 48
2. Algorithmsforarbitrarybispectra 49
IV. Large-ScaleStructure 52
A. Theskewness 53
B. Thematterbispectrum 54
1. Leading-orderresultsinPerturbationTheory 55
2. Second-ordercorrections 59
C. TheGalaxyBispectrum 60
1. Thegalaxybispectrumandlocalbias 61
2. Abispectrumestimator 62
3. Fishermatrixforecasts 63
4. Effectsofcovarianceandcurrentresults 65
5. Primordialnon-Gaussianityandnon-localGalaxyBias 67
6. TheGalaxyBispectrumafterDalaletal.(2008) 70
D. Runningnon-Gaussianity 72
1. Thecaseofascale-dependent fNL 72
2. Runningnon-Gaussianityandbispectrummeasurements 72
V. Conclusions 74
Acknowledgments 75
A. Basicsofestimationtheory 75
References 78
3
I. INTRODUCTION
The standard inflationary paradigm predicts a flat Universe perturbed by nearly Gaussian and scale invariant pri-
mordialperturbations. ThesepredictionshavebeenverifiedtoahighdegreeofaccuracybyCosmicMicrowaveBack-
ground(CMB)andLarge-ScaleStructure(LSS)measurements,suchasthoseprovidedbytheWilkinsonMicrowave
AnisotropyProbe(WMAP;Komatsuetal.,2009), the2dFGalaxyRedshiftSurvey(2dFGRS;Percivaletal.,2002)
andtheSloanDigitalSkySurvey(SDSS;Tegmarketal.,2004). Despitethissuccess,ithasprovedtobedifficultto
discriminatebetweenthevastarrayofinflationaryscenariosthathavebeenproposedbyhigh-energytheoreticalinves-
tigations,oreventorule-outalternativestoinflation.. SincemostofthepresentconstraintsontheLagrangianofthe
inflatonfieldhavebeenobtainedfrommeasurementsofthetwo-pointfunction,orpowerspectrum,oftheprimordial
fluctuations,anaturalstepistoextendtheavailableinformationistolookatnon-Gaussiansignaturesinhigherorder
correlators.
Thelowestorderadditionalcorrelatortotakeintoaccountisthethree-pointfunctionoritscounterpartinFourier
space, the bispectrum. Every model of inflation is characterized by specific predictions for the bispectrum of the
primordial perturbations in the gravitational potential Φ(k). The bispectrum BΦ(k1,k2,k3) of these perturbations is
definedas
(cid:104)Φ(k1)Φ(k2)Φ(k3)(cid:105)≡(2π)3δD(k123)BΦ(k1,k2,k3), (I.1)
wherewehaveintroducedthenotationk ≡k +k sothattheDiracdeltafunctionhereisδ (k )≡δ (k +k +k ).
ij 1 2 D 123 D 1 2 3
Togetherwiththeassumptionofstatisticalhomogeneityandisotropyfortheprimordialperturbations,thisimpliesthat
thebispectrumisafunctionofthetripletdefinedbythemagnitudeofthewavenumbersk ,k andk formingaclosed
1 2 3
triangular configuration. The current constraints that we are able to derive on the bispectrum BΦ(k1,k2,k3) provide
additional information about the early Universe; the possible detection of a non-vanishing primordial bispectrum
in future observations would represent a major discovery, especially as it is predicted to be negligible by standard
inflation.
The cosmological observable most directly related to the initial curvature bispectrum is given by the bispectrum
of the CMB temperature fluctuations, which provide a map of the density perturbations at the time of decoupling,
theearliestinformationwehaveabouttheUniverse. Currentmeasurementsofindividualtriangularconfigurationsof
theCMBbispectrumare,however,consistentwithzero. Studiesoftheprimordialbispectrum,therefore,areusually
characterizedbyconstraintsonasingleamplitudeparameter,denotedby fNL,onceaspecificmodelforBΦisassumed.
Since most models predict a curvature bispectrum obeying the hierarchical scaling BΦ(k,k,k) ∼ P2Φ(k), with PΦ(k)
beingthecurvaturepowerspectrum,thenon-Gaussianparameterroughlyquantifiestheratio fNL ∼ BΦ(k,k,k)/P2Φ(k),
definingthe“strength”oftheprimordialnon-Gaussiansignal. Inaddition,wecanwrite
BΦ(k1,k2,k3)≡ fNLF(k1,k2,k3), (I.2)
where F(k ,k ,k ) encodes the functional dependence of the primordial bispectrum on the specific triangle config-
1 2 3
urations. For brevity, the characteristic shape-dependence of a given bispectrum is often referred to simply as the
bispectrumshape(aprecisedefinitionofthebispectrumshapefunctionwillbegiveninsectionII.A).Inflationarypre-
dictionsforboththeamplitude fNL andtheshapeof BΦ thatarestronglymodel-dependent. Noticethatthesubscript
“NL”standsfor“nonlinear”, sinceacommonphenomenologicalmodelforthenon-Gaussianityoftheinitialcondi-
tionscanbewrittenasasimplenonlineartransformationofaGaussianfield. Generically,ofcourse,non-Gaussianity
isassociatedwithnonlinearities,suchasnontrivialdynamicsduringinflation,resonantbehaviourattheendofinfla-
tion(‘preheating’),ornonlinearpost-inflationaryevolution. Attheveryleast,futureCMBandLSSobservationsare
expectedtobeabletoeventuallydetectthesmalllastcontribution.
PerturbationsintheCMBprovideaparticularlyconvenienttestoftheprimordialdensityfieldbecauseCMBtemper-
atureandpolarizationanisotropiesaresmallenoughtobestudiedinthelinearregimeofcosmologicalperturbations.
Once the effects of foregrounds are properly taken into account, a non-vanishing CMB bispectrum at large scales
wouldbea directconsequenceofanon-vanishing primordialbispectrum. As wewillsee, whileotherCMB probes
ofprimordialnon-Gaussianityareavailable,suchastestsofthetopologicalpropertiesofthetemperaturemapbased
on Minkowski Functionals or measurements of the CMB trispectrum, the estimator for the non-Gaussian parameter
4
f has been shown to be optimal. We will focus mostly on this bispectrum estimator in the section of this review
NL
dedicatedtotheCMB.
In the standard cosmological model, the large-scale structure of the Universe, that is, the distribution of matter
andgalaxiesonlargescales,istheresultofthenonlinearevolutionduetogravitationalinstabilityofthesameinitial
density perturbations responsible for the CMB anisotropies. This is, perhaps, the most important prediction of the
inflationary framework which provides a common origin for the CMB and large-scale structure perturbations as the
resultoftinyquantumfluctuationsstretchedovercosmologicalscalesduringaphaseofacceleratedexpansion. The
large-scalestructureweobserveatlowredshift,however,ischaracterizedbylargevoidsandsmallregionswithvery
large matter density, and it is therefore a much less direct probe of the initial conditions. The distribution of matter
becomes a highly non-Gaussian field precisely as a result of the nonlinear growth of structures, even for Gaussian
initial conditions. This non-Gaussianity is expressed, in particular, by a non-vanishing matter bispectrum at any
measurablescale,includingthelargestscalesprobedbycurrentorfutureredshiftsurveys. Inthiscontext,theeffect
of primordial non-Gaussianity, i.e. of an initial component in the curvature bispectrum, will constitute a correction
tothegalaxybispectrum. Itfollowsthatthepossibilityofconstrainingordetectingthisinitialcomponentisstrictly
relatedtoourabilitytodistinguishitfromother,primarysourcesofnon-Gaussianity,thatisthenonlineargravitational
evolution,and,inthecaseofgalaxysurveys,nonlinearbias.
Thestudyofnon-Gaussianinitialconditionsforlarge-scalestructurehasarelativelylonghistory, withimportant
contributions going back to the mid eighties. The standard picture that has been developed over the years, assumed
that, at large scales, the effect of primordial non-Gaussianity on the galaxy distribution is simply given in terms of
an additional component to the galaxy bispectrum. This is obtained, in perturbation theory, as the linearly evolved
and linearly biased initial matter bispectrum, related to the curvature bispectrum BΦ(k1,k2,k3) by the Poisson equa-
tion. Suchcomponentbecomessubdominantasthegravity-inducednon-Gaussiancontributiongrowsintime. Inthis
framework,asonecanexpect,high-redshiftandlarge-volumegalaxysurveyswouldconstitutethebestprobesofthe
initial conditions. It has been shown, in fact, that proposed and planned redshift surveys, such as Euclid (Refregier
et al., 2010), should be able to provide constraints on the primordial non-Gaussian parameters comparable, if not
better, thanthoseexpectedfromCMBmissionssuchasPlanck. Whatismoreimportant, intheeventofadetection
byPlanck,isthatconfirmationbylarge-scalestructureobservationswillberequired.
RecentresultsfromN-bodysimulationswithnon-Gaussianinitialconditions,however,haverevealedamorecom-
plex picture. The effect of primordial non-Gaussianity at large scales is not limited to an additional contribution to
thegalaxybispectrum,butitquitedramaticallyaffectsthegalaxybiasrelationitself,thatis,therelationbetweenthe
matter and galaxy distributions. A surprising consequence is that it induces a large correction even for the galaxy
power spectrum. Such an effect has attracted considerable recent attention and, remarkably, have placed constraints
onthenon-GaussianparameterfromcurrentLSSdata-setswhichalreadyappeartomarginallyimproveonCMBlim-
its. However, from a theoretical point of view, a proper understanding of the phenomenon remains to be properly
developed and, for example, reliable predictions for the galaxy bispectrum are not yet available. Most importantly,
asforgeneralcosmologicalparameterestimation,acompletelikelihoodanalysisaimedatconstraining,ordetecting,
primordialnon-Gaussianityinlarge-volumeredshiftsurveysshouldinvolvejointmeasurementsofthegalaxypower
spectrumandbispectrum,aswellaspossiblyhigher-ordercorrelationfunctions. Whilewearestillfarfromaproper
assessmentofwhatsuchanalysiswouldbeabletoachieve,currentresultsinthisdirectionareveryencouraging.
This review is divided in four parts. In section II we will first discuss initial conditions as defined in terms of
the primordial curvature bispectrum and its phenomenology. We will then review the observational consequences
of primordial non-Gaussianity on the CMB bispectrum, section III, and on the large-scale structure bispectrum as
measuredinredshiftsurveys,sectionIV.Inbothcaseswewilldiscusstheoreticalmodelsfortheobservedbispectra
and technical problems related to the estimation of the non-Gaussian parameters, with the differences that naturally
characterize such distinct observables. We also give an example of joint analysis using both CMB and large-scale
structurewhenweconsiderthepossibilityofconstrainingastronglyscale-dependentnon-Gaussianparameter f (k),
NL
emerginginsomerecentlyproposedinflationarymodels.
5
k1 k1 k3
k k
2 1
k
3
k2 k3
k
2
FIG.1 Triangletypescontributingtothebispectrumcorrespondingto‘squeezed’orlocalconfigurationswithk (cid:28) k ,k (left),
3 1 2
equilateralconfigurationswithk ≈k ≈k (centre)andflattenedconfigurationswithk ≈k +k (right).
3 1 2 3 1 2
II. INITIALCONDITIONSANDTHEPRIMORDIALBISPECTRUM
A. Theprimordialbispectrumandshapefunction
ThestartingpointforthisdiscussionistheprimordialgravitationalpotentialperturbationΦ(x, t)whichwasseeded
by quantum fluctuations during inflation or by some other mechanism in the very early universe (t (cid:28) t ). When
dec
characterizingthefluctuationsΦweusuallyworkinFourierspacewiththe(flatspace)transformdefinedthrough
(cid:90)
d3k
Φ(x, t)= e−ik·xΦ(k,t). (II.3)
(2π)3
TheprimordialpowerspectrumPΦ(k)ofthesepotentialfluctuationsisfoundusinganensembleaverage,
(cid:104)Φ(k)Φ∗(k(cid:48))(cid:105)=(2π)3δD(k−k(cid:48))PΦ(k), (II.4)
where we have assumed that physical processes creating the fluctuations are statistically isotropic so that only the
dependenceonthewavenumberremains,k = |k|. Recallthatfornearlyscale-invariantperturbations,thefluctuation
varianceonthehorizonscalek≈ Hisalmostconstant∆2k∼H ≈k3PΦ(k)/2π2 ≈const.,implyingPΦ(k)∼k−3.
TheprimordialbispectrumBΦ(k,k2, k3)isfoundfromtheFouriertransformofthethree-pointcorrelatoras
(cid:104)Φ(k1)Φ(k2)Φ(k3)(cid:105)=(2π)3δD(k123)BΦ(k1,k2,k3). (II.5)
Here, the delta function enforces the triangle condition, that is, the constraint that the wavevectors in Fourier space
mustclosetoformatriangle,k +k +k = 0. Examplesofsuchtrianglesareshowninfig.1,illustratingthebasic
1 2 3
squeezed,equilateralandflattenedtrianglestowhichwewillreferlater.Notethataspecifictrianglecanbecompletely
describedbythethreelengthsofitssidesandso, intheisotropiccase, weareabletodescribethebispectrumusing
onlythewavenumbersk ,k ,k . Thetriangleconditionrestrictstheallowedwavenumberconfigurations(k ,k ,k )to
1 2 3 1 2 3
theinteriorofthetetrahedronillustratedinfig.2.
The most studied primordial bispectrum is the local model in which contributions from ‘squeezed’ triangles are
dominant, that is, with e.g. k (cid:28) k , k (as illustrated in the left of fig. 1). This is well-motivated physically as it
3 1 2
encompasses ‘superhorizon’ effects during inflation when a large scale mode k (say) which has exited the Hubble
3
radius exerts a nonlinear influence on the subsequent evolution of smaller scale modes k , k . Although this effect
1 2
is small in single field slow-roll inflation, it can be much larger for multifield models. In a weakly coupled regime,
the potential can be split into two components, the linear term Φ , representing a Gaussian field, giving the usual
L
perturbationresultsplusasmalllocalnon-GaussiantermΦ (Salopek&Bond,1990),
NL
Φ(x) = Φ (x)+Φ (x)
L NL
= Φ (x)+ f (cid:2)Φ2(x)−(cid:104)Φ2(x)(cid:105)(cid:3), (II.6)
L NL L L
6
k
3
(0,K,K)
k
2
(K,0,K)
K
(K,K,0)
0 K k
1
FIG.2 Tetrahedraldomainforallowedwavenumberconfigurationsk ,k ,k contributingtotheprimordialbispectrumB(k ,k ,k )).
1 2 3 1 2 3
Aregulartetrahedronisshownsatisfyingk +k +k ≤2k ≡2K.
1 2 3 max
where f iscalledthenonlinearityparameter. InFourierspace,thenonlineartermisthengivenbytheconvolution
NL
(cid:34)(cid:90) (cid:35)
d3k
Φ (k)= f Φ (k+k(cid:48))Φ (k(cid:48))−(2π)3δ (k)(cid:104)Φ2(cid:105) . (II.7)
NL NL (2π)3 L L D L
Fromthiswecaninfer,using(II.4),thattheonlynon-vanishingcontributionstothebispectrum(II.5)taketheform
(cid:104)Φ(k1)Φ(k2)Φ(k3)(cid:105)=2(2π)3δD(k123)PΦ(k1)PΦ(k2). (II.8)
Accountingforpermutations,thelocalbispectrumbecomes
BΦ(k1,k2,k3) = 2fNL[PΦ(k1)PΦ(k2)+PΦ(k2)PΦ(k3)+PΦ(k3)PΦ(k1)]
(cid:39) 2fNL(k k∆2Φk )2 kkk12 + kkk22 + kkk32  . (II.9)
1 2 3 2 3 1 3 1 2
Although this is a rather pathological function which diverges along the edges of the tetrahedron (i.e. when any
k → 0),wecaninferfromitsomebasicpropertiesofthebispectrumforanymodelwhichisnearlyscale-invariant.
i
Forexample,wecanobservethatthebispectrumatequalk hasthecharacteristicscaling,
i
BΦ(k, k, k)=2fNL∆2Φ/k6. (II.10)
If we remove this overall k−6 scaling by multiplying (II.9) by the factor (k k k )2, we note that on transverse slices
1 2 3
throughthetetrahedrondefinedbyk˜ ≡(k +k +k )/2=const.(seefig.2)thebispectrumonlydependsontheratios
1 2 3
ofthewavenumbers,sayk /k andk /k . Indeed,itcanproveconvenienttocharacterizethebispectrumintermsof
2 1 3 1
thefollowingtransverseparameters(Fergusson&Shellard,2007;Rigopoulosetal.,2006a)
k˜ = 1(k +k +k ), α˜ =(k −k )/k˜, β˜ =(k˜−k )/k˜, (II.11)
2 1 2 3 2 3 1
withthedomainsk˜ ≤ k ,0 ≤ β˜ ≤ 1and−(1−β˜) ≤ α˜ ≤ 1−β˜. Thevolumeelementontheregulartetrahedronof
max
allowedwavenumbersthenbecomesdk dk dk =k2dk˜dα˜dβ˜.
1 2 3
7
FIG.3 Shapefunctionsforthescale-invariantequilateral(left)andlocal(right)models,S(k ,k ,k )=S(α˜,β˜)ontransverseslices
1 2 3
with2k˜ =k +k +k =const.
1 2 3
Theseconsiderationsleadnaturallytothedefinitionoftheprimordialshapefunction(Babichetal.,2004)
1
S(k1,k2,k3)≡ N(k1k2k3)2BΦ(k1,k2,k3), (II.12)
whereNisanormalizationfactorwhichisoftenchosensuchthatS isunityfortheequalk case,thatis,S(k, k, k)=1
i
(weshalldiscussalternativestothisratherarbitraryconventionlater). Forexample,thecanonical‘local’model(II.9)
hastheshape
 
Slocal(k1,k2,k3)= 31kkk12 + kkk22 + kkk32  . (II.13)
2 3 1 3 1 2
Thus it is usual to describe the primordial bispectrum in terms of an overall amplitude f and a transverse two-
NL
dimensionalshapeS(k ,k ,k ) = S(α˜, β˜), whichincorporatesanydistinctivemomentumdependence. Ofcourse, if
1 2 3
thereisanon-trivialscaledependence, thenthefullthree-dimensionaldependenceofS(k ,k ,k )onthek mustbe
1 2 3 i
retained.
There are other physically well-motivated shapes in the literature which have also been extensively studied. The
simplestshapeistheconstantmodel
Sconst(k ,k ,k )=1, (II.14)
1 2 3
which,likethelocalmodel,hasalarge-angleanalyticsolutionfortheCMBbispectrum(Fergusson&Shellard,2009).
The local model tends to be the benchmark against which all other models are compared and normalized, but for
practical purposes the constant model is much more useful, given its regularity at both late and early times. The
equilateralshapeisanotherimportantcasewith(Babichetal.,2004)
(k +k −k )(k +k −k )(k +k −k )
Sequil(k ,k ,k )= 1 2 3 2 3 1 3 1 2 . (II.15)
1 2 3 k k k
1 2 3
While not derived directly from a physical model, it has been chosen phenomenologically as a separable ansatz for
higher derivative models (Creminelli, 2003) and DBI inflation (Alishahiha et al., 2004). The equilateral shape is
contrastedwiththelocalmodelinfig.3.
Another important early result was the primordial bispectrum shape for single-field slow roll inflation derived by
Acquavivaetal.(2003);Maldacena(2003)
SMald(k1,k2,k3) ∝ (3(cid:15)−2η)kkk12 + kkk22 + kkk32 +(cid:15)(k1k22+5perm.)+4k12k22+kkk22kk32+k32k12
2 3 1 3 1 2 1 2 3
5
(cid:39) (6(cid:15)−2η)Slocal(k ,k ,k )+ (cid:15)Sequil(k ,k ,k ), (II.16)
1 2 3 1 2 3
3
8
where (cid:15), η are the usual slow roll parameters. In the second line, we have noted that this shape can be accurately
representedasthesuperpositionoflocalandequilateralshapes. Thecoefficientsin(II.16), whichincludethescalar
spectral index n−1 = −6(cid:15) +2η (cid:39) −0.05, confirm that f (cid:28) 1 and so standard single slow roll inflation cannot
NL
produceanobservationallysignificantsignal. Nevertheless,itisinterestingtodeterminewhichshapeisdominantin
(II.16)andtowhatextentotherprimordialshapesareindependentfromoneanother.
Whethertwodifferentprimordialshapescanbedistinguishedobservationallycanbedeterminedfromthecorrela-
tion between the corresponding two CMB bispectra weighted for the anticipated signal to noise, as in the estimator
(seenextsection)andtheFishermatrixanalysis(seesectionIII.H).However,directcalculationsoftheCMBbispec-
trumcanbeverycomputationallydemanding. Amuchsimplerapproachistodeterminetheindependenceofthetwo
shapefunctionsS andS(cid:48)fromthecorrelationintegral(Fergusson&Shellard,2009,seealsoBabichetal.(2004))
(cid:90)
F (S,S(cid:48))= S(k ,k ,k )S(cid:48)(k ,k ,k )ω (k ,k ,k )dV , (II.17)
(cid:15) 1 2 3 1 2 3 (cid:15) 1 2 3 k
V
k
wherewechoosetheweightfunctiontobe
1
ω (k ,k ,k )= , (II.18)
(cid:15) 1 2 3 k +k +k
1 2 3
reflectingtheprimaryscalingoftheCMBcorrelator. Theshapecorrelatoristhendefinedby
F(S,S(cid:48))
C¯(S,S(cid:48))= √ . (II.19)
F(S,S)F(S(cid:48),S(cid:48))
Here, the integral is over the tetrahedral region shown in fig. 2 taken out to a maximum wavenumber k (cid:46) k
max
corresponding to the experimental range l ≤ (cid:96) for which forecasts are sought (with (cid:96) ≈ τ k ). The weight
max max 0 max
function ω (k ,k ,k ) appropriate for mimicking the large-scale structure bispectrum estimator (see section IV.C.2),
(cid:15) 1 2 3
wouldbedifferentwithvaryingscalinglawasintroducedbythetransferfunctionsforwavenumberskaboveandbelow
k ,theinversecomovinghorizonatequalmatter-radiation. Nevertheless,the1/k weightgivenin(II.18)providesa
eq
compromisebetweenthesescalings,andsoshapecorrelationresultsshouldofferausefulfirstapproximation.
Belowwewillsurveyprimordialmodelsintheliterature, showinghowclosetheshapecorrelatorcomestoafull
Fisher matrix analysis. However, here we note that the local shape (II.13) and the equilateral shape (III.53) have
only a modest 46% correlation. For the natural values of the slow roll parameters (cid:15) ≈ η we find the somewhat
surprisingresultthatSMald is99.7%correlatedwithSlocal (anditcannotbeeasilytunedotherwisebecause3(cid:15) ≈ ηis
notconsistentwithdeviationsfromscale-invariancefavoredobservationallyn−1<0). Suchstrongcorrespondences
areimportantindefiningfamiliesofrelatedprimordialshapes,thusreducingthenumberofdifferentcasesforwhich
separateobservationalconstraintsmustbesought.
B. Generalprimordialbispectraandseparablemodeexpansions
Thethreeshapefunctions(II.13),(II.14)and(III.53)quotedabovesharetheimportantpropertyofseparability,that
is,theycanbewrittenintheform
S(k ,k ,k )= X(k )Y(k )Z(k )+5perms., (II.20)
1 2 3 1 2 3
or as the sum of just a few such terms. As we shall see, if a shape S is separable, then the computational cost of
evaluatingthecorrespondingCMBbispectrum B isdramaticallyreduced. Infact,withoutthisproperty,thetask
(cid:96)1(cid:96)2(cid:96)3
of estimating whether a non-separable bispectrum is consistent with observation appears to be intractable (for large
(cid:96) ). Ofcourse,thenumberofmodelswhichcanbeexpresseddirectlyintheform(II.20)isverylimited,despitethe
max
usefulnessofapproximateansa¨tzesuchastheequilateralshape(III.53). Indeed,approximatingnon-separableshapes
byeducatedguessesforfortheseparablefunctionsX, Y, Zisneithersystematicnorcomputationallyefficient(because
arbitrarynon-scalingfunctionscreatenumericaldifficulties,asweshallexplainlater).
9
n = 5
n = 4
4 n = 3
n = 2
n = 1
2
n = 0
0.2 0.4 0.6 0.8 1.0
-2
-4
FIG.4 Theone-dimensionaltetrahedralpolynomialsq (k)onthedomain(II.23),rescaledtotheunitintervalforn = 0–5. Also
n
plottedaretheshiftedLegendrepolynomialsP (2x−1)(dashedlines)whichsharequalitativefeaturessuchasnnodalpoints.
n
Instead, we shall present a separable mode expansion approach for efficient calculations with any non-separable
bispectrum,asdescribedindetailinFergussonetal.(2009)(andoriginallyproposedinFergusson&Shellard,2007).
Ouraimwillbetoexpressanyshapefunctionasanexpansioninmodefunctions
(cid:88)(cid:88)(cid:88) (cid:88)
S(k ,k ,k )= α q (k )q (k )q (k )≡ αQQ (k ,k ,k ), (II.21)
1 2 3 prs {p 1 r 2 s} 3 n n 1 2 3
p r s n
where,here,forconveniencewehaverepresentedthesymmetrizedproductsoftheseparablebasisfunctionsq (k)as
p
(cid:104) (cid:105)
Q (k ,k ,k )= 1 q (x)q (y)q (z)+5perms ≡ q q q , (II.22)
n 1 2 3 6 p r s {p r s}
withaone-to-onemappingorderingtheproductsasn↔{prs}. Theimportantpointisthattheq (k)mustbeaninde-
p
pendentsetofwell-behavedbasisfunctionswhichcanbeusedtoconstructcompleteandorthogonalthree-dimensional
eigenfunctionsonthetetrahedralregionV definedby(seefig.2)
T
k ,k ,k ≤k , k ≤k +k fork ≥k , k , + 2perms. (II.23)
1 2 3 max 1 2 3 1 2 3
Theintroductionofthecut-offatk ismotivatedbothbyseparabilityandthecorrespondencewiththeobservational
max
domain l ≤ (cid:96) . In the shape correlator (II.19), we have already seen what is essentially an inner product between
max
twoshapesonthistetrahedralregion,whichwecandefinefortwofunctions f, gas
(cid:90)
(cid:104)f, g(cid:105)= f(k ,k ,k )g(k ,k ,k )ω(k ,k ,k )dV , (II.24)
1 2 3 1 2 3 1 2 3 T
VT
withweightfunctionw.
Satisfactory convergence for known bispectra can be found by using simple polynomials q (k) in the expansion
p
(II.21), that is, using analogues of Legendre polynomials on the domain (II.23). With unit weight, the polynomials
satisfying(cid:104)q (k ), q (k )(cid:105) = δ canbefoundbygeneratingfunctionswiththefirstthreegivenby(Fergussonetal.,
p 1 r 1 pr
2009)
√ (cid:16) (cid:17)
q (x)= 2, q (x)=5.79(−7 +x), q (x)=23.3 54 − 48x+x2 , ... (II.25)
0 1 12 2 215 43
Thefirstfewpolynomialsq (k)areplottedinfig.4,wheretheyarecontrastedwithLegendrepolynomials.
p
Thethree-dimensionalseparablebasisfunctionsQ in(II.22)reflectthesixsymmetriesofthebispectrumthrough
n
thepermutedsumoftheproductterms. Theycouldhavebeenconstructeddirectlyfromsimplerpolynomials,suchas
10
FIG. 5 Orthonormal eigenmode decomposition coefficients (II.26) for the equilateral and DBI models (upper panel) and shape
correlations (II.19) of the original bispectrum against the partial sum up to a given mode n (lower panel). The correlation plot
includesbothprimordialandlate-timeCMBbispectrafortheequilateralandDBImodels,aswellasthelate-timeCMBbispectrum
from cosmic strings (refer to section III). In all cases, we find that we need at most 15 three-dimensional modes to obtain a
correlationgreaterthan98%(primordialconvergencewithouttheacousticpeaksrequiresonly6modes).
1, k +k +k , k2+k2+k2, ...,however,theq polynomialshavetwodistinctadvantages. First,theq ’sconferpartial
1 2 3 1 2 3 p p
orthogonalityontheQ and,secondly,theseremainwell-behavedwhenconvolvedwithtransferfunctions.
n
InordertorapidlydecomposeanarbitraryshapefunctionS intothecoefficientsαQ ↔ αQ ,itismoreconvenient
n prs
toworkinanon-separableorthonormalbasisR ((cid:104)R , R (cid:105)=δ . ThesecanbederiveddirectlyfromtheQ through
n n m nm n
Gram-Schmidtorthogonalization,sothatR = (cid:80)n λ Q withλ alowertriangularmatrix(seeFergussonetal.,
n p=0 mp p mp
2009). Thuswecanfindtheuniqueshapefunctiondecomposition
(cid:88)N (cid:88)N (cid:88)N
S(k ,k ,k )= αRR (k ,k ,k )= αQQ (k ,k ,k ), with αR =(cid:104)S, R (cid:105) and αQ = (λ(cid:62)) αR. (II.26)
1 2 3 n n 1 2 3 n n 1 2 3 n n n np p
n n p
In the orthonormal R frame, Parseval’s theorem ensures that the autocorrelator is simply (cid:104)S, S(cid:105) = (cid:80) αR2. Hence,
n n n
with a simple and efficient prescription we can construct separable and complete basis functions on the tetrahedral
domain (II.23) providing rapidly convergent expansions for any well-behaved shape function S. These eigenmode
expansionswillprovetobeofgreatutilityinsubsequentsections. Examplesofthisbispectraldecompositionandits
rapidconvergencefortheequilateralandDBImodelsareshowninfig.5.