Revisiting the angular momentum growth of protostructures evolved from non-Gaussian initial conditions

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by  C. Fedeli

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed23January2013 (MNLATEXstylefilev2.2)
Revisiting the angular momentum growth of protostructures
evolved from non-Gaussian initial conditions
3
C.Fedeli
1
0 DepartmentofAstronomy,UniversityofFlorida,211BryantSpaceScienceCenter,Gainesville,FL32611([email protected])
2

n
a 23January2013
J

2
2 ABSTRACT
I adopta formalismpreviouslydevelopedby Catelan and Theuns(CT) in orderto estimate

] theimpactofprimordialnon-Gaussianityonthequasi-linearspingrowthofcolddarkmatter
O
protostructures.A varietyofbispectrumshapesareconsidered,spanningthe currentlymost
C popular early Universe models for the occurrence of non-Gaussian density fluctuations. In
. their originalwork, CT consideredseveral othershapes, and suggested that only for one of
h
thosedoestheimpactofnon-Gaussianityseemtobeperturbativelytractable.Forthatmodel,
p
and on galactic scales, the next-to-linearnon-Gaussian contribution to the angular momen-
-
o tum variance has an upper limit of ∼ 10% with respect to the linear one. I find that all the
r newmodelsconsideredinthisworkcanalsobeseeminglydescribedviaperturbationtheory.
t
s Consideringcurrentboundson f forinflationarynon-Gaussianityleadstothequasi-linear
NL
a
contributionbeing ∼ 10−20%ofthe linearone.Thisresultmotivatesthe systematicstudy
[
ofhigher-ordernon-Gaussiancorrections,in ordertoattaina comprehensivepictureofhow

1 structuregravitationaldynamicsdescendsfromthephysicsoftheprimordialUniverse.
v
6 Keywords: large-scalestructureoftheUniverse
9
1
5
.
1 1 INTRODUCTION spacetop-hatfilter).Forothertemplatesthenon-Gaussian contri-
0
butioniscomparableto,orlargerthan,thelinearterm,suggesting
3
GravitationalinstabilityandhierarchicalgrowthofColdDarkMat- theimpossibilityofaperturbativeexpansion.
1
ter(CDM)densityperturbationsprovideanelegantdescriptionfor
:
v theacquisitionofangularmomentumbyprotostructures.Accord-
i ingly,patchesofmatterarespunupbytidaltorquesexertedbythe Since then, the issue of angular momentum growth in non-
X Gaussian cosmologies has not been investigated further. On
surroundingLarge-ScaleStructure(LSS,seePeebles1969, 1971;
r Doroshkevich 1970; White 1984; Heavens&Peacock 1988). At the contrary, new and more general models of primordial non-
a
Gaussianityexistnowadays and, mostimportantly, constraints on
thelinearlevelthespingrowthisgivenbythecouplingofthefirst-
thelevel of primordial non-Gaussianity coming fromthe Cosmic
order deformation tensor and the inertia tensor of the patch, and
MicrowaveBackground(CMB)andtheLSShavedramaticallyim-
itagreesquitewellwithmoreaccuratenumerical results(seethe
proved overthelastdecade. Giventhecosmological relevance of
reviewbyScha¨fer2009andthereferencestherein).
primordialnon-Gaussianity(seeBartoloetal.2004;Chen2010for
Acquisition of angular momentum beyond the simple linear
recentreviews)andthesignificanceoftheCDMhaloangularmo-
description has been tackled by means of Lagrangian perturba-
mentumacquisitionfortheformationandevolutionofgalaxies,it
tion theory in a series of papers by Catelan and Theuns (Catelan
is important to update this topic. Specifically, it is interesting to
1995;Catelan&Theuns1996a,b,1997,CThenceforth)almosttwo
exploretheamplitudesandbehaviorsofthequasi-linearcontribu-
decades ago. In particular, they found that the next-to-linear cor-
tionstospingrowthgivenbynon-Gaussianmodelsthatarepopular
rectiontothegrowthofensemble-averaged spinisnon-vanishing
nowadays.Thisisthescopeofthepresentletter.
only if primordial density fluctuations are non-Gaussian. CT ex-
plored several non-Gaussian models, and concluded that only for
one of these does the angular momentum acquisition appear per- The rest of the manuscript is organized as follows. In Sec-
turbativelytractable:thelog-normalmodelforthegravitationalpo- tion 2 I review the linear and next-to-linear contributions to the
tentialofMoscardinietal.(1991).Forthistemplate,andadopting ensemble-averaged spin growth of matter patches. In Section 3 I
arepresentativemassscaleof M ∼ 1012h−1M (withh = 0.5and summarize the non-Gaussian cosmologies explored here. In Sec-
⊙
Gaussianfiltering),CTfoundanupperlimitof∼24%forthequasi- tion4resultsaredisplayedandinSection5conclusionsaredrawn.
linearnon-Gaussian contribution tothespinvariance. Thisfigure Whereneeded, I adopted the followingcosmological parameters:
translatestoa ∼ 10%valuewhenrescaledtoasmallerandmore Ωm,0 = 0.272,ΩΛ,0 = 1−Ωm,0,Ωb,0 = 0.046, H0 = 100hkms−1
typicalgalacticmassof M =1010h−1M (withh= 0.7andareal- Mpc−1withh=0.704,σ =0.809,andn =1.
⊙ 8 s
2 C.Fedeli
2 ANGULARMOMENTUMACQUISITION By expanding the Zel’dovich potential around the center of
mass of the patch (assumed to be, without loss of generality, the
2.1 Lagrangiandisplacement
originofthereferenceframe)uptothesecondorder,theprevious
InLagrangian theorythecomoving position xof amasselement equationtakesthecompactform
attimeτcanbewrittenintermsofitsinitialposition qandadis- dD(τ)
placementvectorfieldS,as J1,α(τ)= dτ εαβγD1,βσIσγ(τ). (8)
x(q,τ)=q+S(q,τ). (1) InthepreviousequationεαβγisthefullyantisymmetricLevi-Civita
tensor,D istheZel’dovichdeformationtensor,
1,βσ
Following CT here I used a time variable τ that is related to the
standardcosmictimetbydτ=dt/a2(Shandarin1980),whereais D ≡ ∂2ψ1(0) =− dp p p ψˆ (p), (9)
thescalefactor. 1,βσ ∂qβ∂qσ ZR3 (2π)3 β σ 1
Perturbative approximations to this exact expression can be whileI istheinertiatensorofthepatch,
σγ
foundbyexpandingthedisplacementfieldinaseries,
I (τ)≡a3(τ)ρ dqq q . (10)
∞ σγ m,0 σ γ
S= S , (2) ZΓ
n
Xn=1 Summationoverrepeatedindicesisimplicit.
Likewise,thesecond-ordertermintheseriesexpansionofthe
whereS corresponds totheZel’dovichapproximation S (q,τ) =
1 1 angularmomentumreads
D(τ)∇ψ (q).Here D(τ)isthegrowthfactoroflineardensityper-
1
turbations,whichinaEinstein-deSitteruniversereadsD(τ)=τ−2. J (τ)=a3(τ)ρ dE(τ) dqq×∇ψ (q), (11)
The function ψ1 is the first order (Zel’dovich) displacement po- 2 m,0 dτ ZΓ 2
tential(Zel’dovich1970),relatedtothelineardensityperturbation which,underasecond-orderTaylorexpansionofthedisplacement
field by the Poisson equation ∆ψ1(q) = δ(q), so that in Fourier potentialtakestheform
spaceψˆ (p)=δˆ(p)/p2.
1 dE(τ)
The second-order termof thedisplacement fieldcan alsobe J (τ)= ε D I (τ). (12)
separatedintimeandspace, according to S (q,τ) = E(τ)∇ψ (q). 2,α dτ αβγ 2,βσ σγ
2 2
The growth factor E(τ) reads E(τ) = −3τ−4/7 in an Einstein-de Itcanbeshownthatthesecond-orderdeformationtensorinFourier
Sitteruniverse,whileforitsmoregeneralexpressionIrefertoCT. spacereads
Thesecond-orderdisplacementpotentialcanberelatedtoitsfirst-
ordercounterpartinFourierspaceby dp dp (p +p ) (p +p )
D = 1 2 1 2 β 1 2 σK(p ,p )×
2,βσ ZR6 (2π)6 kp1+p2k2 1 2
1 dp dp × ψˆ (p )ψˆ (p ), (13)
ψˆ (p) = − 1 2 (2π)3δ (p +p −p) K(p ,p )× 1 1 1 2
2 p2 ZR6 (2π)6 h D 1 2 i 1 2 intermsoftheZel’dovichpotential.
× ψˆ (p )ψˆ (p ). (3)
1 1 1 2
InthepreviousEquationK(p1,p2)isasymmetricintegrationkernel 2.3 Ensembleaverages
definedas
Inordertosimplifythepreviousresults,itismeaningfultoconsider
1 1
K(p ,p )≡ p2p2− p ·p 2 = p2p2 1−µ2 , (4) the ensemble average of the square of the angular momentum. It
1 2 2 1 2 1 2 2 1 2
h (cid:0) (cid:1) i (cid:16) (cid:17) thenfollowsthat,uptothenext-to-linearorder,
whereµisthecosineoftheanglebetweenthetwowavevectors p
1 kJ(τ)k2 ≃ kJ (τ)k2 +2hJ (τ)·J (τ)i , (14)
and p . 1 1 2
2 D E D E
where
2 dD(τ) 2
2.2 Spingrowth kJ1(τ)k2 = 15" dτ # ν21−3ν2 σ2M . (15)
D E (cid:16) (cid:17)
The angular momentum of the matter initiallycontained in a co- In the previous Equation σ is the mean deviation of the mat-
M
movingLagrangianpatchΓoftheUniverseattimeτcanbewritten ter density field smoothed on a scale corresponding to mass M,
asanintegraloverΓ, while ν and ν are the first and second invariant of the inertia
1 2
tensor, respectively. To be more precise, if λ , λ , and λ are the
∂S(q,τ) 1 2 3
J(τ)=a3(τ)ρm,0ZΓdq(cid:2)q+S(q,τ)(cid:3)× ∂τ . (5) νth2re≡eλe1iλg2en+vaλl1uλe3s+ofλt2hλe3.inertiatensor,thenν1 ≡ λ1+λ2+λ3 and
By considering the series expansion of the Lagrangian displace- Thenexttermisnon-vanishingonlyifdensityfluctuationsare
ment fieldS introduced inEq. (2),the angular momentum of the non-Gaussian,andreads
patchcanbesimilarlywrittenas
2 dD(τ)dE(τ)
hJ (τ)·J (τ)i= ν2−3ν ω , (16)
∞ 1 2 15 dτ dτ 1 2 M
J = J . (6) (cid:16) (cid:17)
m where
Xm=1
Thefirst-ordertermoftheangularmomentumseriestakestheform dp dp
ω = −15 1 2kp +p k2K(p ,p )Wˆ2(kp +p k)×
J (τ)=a3(τ)ρ dD(τ) dqq×∇ψ (q). (7) M ZR6 (2π)6 1 2 1 2 R 1 2
1 m,0 dτ ZΓ 1 × Bψ1(p1,p2,−p1−p2). (17)
Angularmomentumgrowth 3
In the previous Equation B represents the bispectrum of the
ψ1
Zel’dovichpotential,whichisnowexplicitlysmoothedonascale
R = (2GM/Ω H2)1/3.Iassumedthestandardreal-spacetop-hat
m,0 0
smoothing.TheZel’dovichpotentialcanberelatedtothestandard
gravitationalpotentialϕbymakinguseofthePoissonequation,
2 T(p)
ψˆ (p)= ϕˆ(p)≡F(p)ϕˆ(p), (18)
1 3H2Ω
0 m,0
whereT(p)isthecolddarkmattertransferfunction(Bardeenetal.
1986; Sugiyama 1995). Theintegral in Eq. (17) has tobe solved
numerically for realistic bispectrum shapes. Fortunately, the bis-
pectrumusuallydepends onlyonthemagnitudeofitsthreeargu-
ments,sothattheabovesix-dimensionalintegralreducestoathree-
dimensionalone.
3 NON-GAUSSIANSHAPES
I considered five different shapes for the primordial bispectrum,
that are briefly described below. The first four are motivated
by inflationary physics, and the amplitude of non-Gaussianity is
given by the parameter f (assumed to be constant). The fifth
NL
is non-inflationary in nature, and hence independent on fNL. See Figure1.TheshapeofthefunctionωM,quantifyingthenon-Gaussiancon-
Fedelietal.(2011)andreferencestherein. tribution totheangular momentumvariance growth. Different linestyles
andcolors refertodifferent non-Gaussian bispectrum shapes,aslabeled.
Thenon-Gaussianityinducedbythematterbounceisindependenton fNL,
3.1 Localshape whileinallothercases fNL=1hasbeenassumed.
This bispectrum shape arises when a light scalar field, addi-
tional to the inflaton, contributes to the curvature perturbations 3.4 Orthogonalshape
(Bernardeau&Uzan 2002; Babich,Creminelli,&Zaldarriaga
Thisshape isdefined asbeing orthogonal (withrespect toasuit-
2004; Sasaki,Va¨liviita,&Wands 2006). It is the same
ably defined scalar product) to both the local and equilateral
shape produced by the standard model of inflation
forms. The resulting bispectrum is maximized for both equilat-
(Falk,Rangarajan,&Srednicki 1993) but in this case the am-
eraland squashed configurations, andatemplatecan befound in
plitudecanbearbitrary.Thepotentialbispectrumtakesthesimple
Senatore,Smith,&Zaldarriaga(2010).
form
B (p ,p ,p )=2A2f (p p )ns−4+(p p )ns−4+(p p )ns−4 ,(19)
ϕ 1 2 3 NL 1 2 1 3 2 3 3.5 Matterbounceshape
h i
anditismaximizedforsqueezedconfigurations.ThequantityAis
Thisconfigurationistheconsequenceofamodeluniversewithout
thespectralamplitudeofthepotential(givenbyσ ),whilen isthe
8 s inflation,butwithascalefactorthatbouncesinanon-singularway
spectralslope.
(Brandenberger2009;Caietal.2009).Thematterbounceleadsto
a scale-invariant spectrum of density fluctuations, and to a bis-
pectrum whose shape is similar to the local shape. Being non-
3.2 Equilateralshape
inflationary in origin, the non-Gaussianity induced by a matter
This shape is a consequence of the inflaton Lagrangian being bouncemodelhasnodependenceon fNL.Itcaninsteadbeshown
non-standard, and containing higher-order derivatives of the field that thematterbounce bispectrum iscomparable tothelocal one
(Alishahiha,Silverstein,&Tong2004;Arkani-Hamedetal.2004; withafixed fNL =−35/8.Iconsideredexplicitlythematterbounce
Li,Wang,&Wang2008).Theresultingbispectrumismaximized becauseitisinprinciplepossiblethataweightedintegralofthebis-
forequilateralconfigurations.Atemplatefortheequilateralbispec- pectrum,suchastheoneinEq.(17),wouldmagnifyitsdifferences
trumcanbefoundinCreminellietal.(2007),howevertheexpres- withrespecttothelocalmodel.AsIshowbelowthisisactuallynot
sion israthercumbersome and I didnot report ithere. Thesame thecase.
appliestothefollowingshapes.
4 RESULTS
3.3 Enfoldedshape
InFigure1Ishowω asafunctionofthemassscaleforthefive
M
The enfolded shape results from primordial non-Gaussianity be- non-Gaussian cosmologies considered in this letter. In all cases,
ing evaluated without the regular Bunch-Davies vacuum hy- exceptforthematterbounce,Iselected f =1inordertopurely
NL
pothesis (Chenetal. 2007; Holman&Tolley 2008). In this highlighttheeffectofthebispectrumshape.Itishowevereasyto
case the bispectrum is maximized for squashed configura- seethatω issimplyproportionalto f .Allcurvesdecreasewith
M NL
tions. A template for such a bispectrum is reported in increasingmass,implyingthatlargermatterpatchesacquirelower
Meerburg,vanderSchaar,&Corasaniti(2009). amounts of angular momentum than smaller ones. This behavior
4 C.Fedeli
Einstein-deSittercosmological model thisimplies τ2 = σ ,and
∗ M
thus
12ω
Υ (τ )=− M . (22)
M ∗ 7 σ3
M
InFigure2IshowthemassdependenceofthefunctionΥ (τ )
M ∗
forthevarious non-Gaussian cosmologies that havebeen consid-
ered in this work. For models with inflationary non-Gaussianity
I assumed f = 1. As can be seen the mass dependence is in
NL
all cases relatively weak. In the local and matter bounce mod-
els Υ (τ ) is basically unchanged for masses ranging between
M ∗
M = 108h−1M and M = 1015h−1M .Fortheequilateralanden-
⊙ ⊙
foldedmodelsΥ (τ )increasesby∼ 50%overthesameinterval,
M ∗
whilefortheorthogonalcaseitdecreasesbyafactorof∼3.Hence,
despitethefactthatthelinearandnon-Gaussiancontributionstothe
spingrowthbothdecreasewithmass,theirrelativeimportancere-
mainsrelativelyunchanged. Theonlyexceptionisrepresentedby
theorthogonalmodel.
Next, I selected a reference mass scale of M = 1010h−1M
⊙
andcomputedthedependenceofΥ (τ )on f ,showninFigure
M ∗ NL
3.Aspreviouslymentioned,thisdependenceisalwayslinear,how-
evertheFigureisimportantinordertounderstandforwhatvalue
of f acertainnon-Gaussianmodelprovidesagivencontribution
Figure2.Thenon-Gaussiancontributiontothespinvariancegrowthnor- NL
tothetotalangularmomentum variance.Thematterbounce non-
malized by the linear contribution, evaluated at the collapse time for an
Gaussianityisindependentof f ,henceitscontributionisalways
overdensityofagivenmass.LinetypesandcolorsarethesameasinFigure NL
1,and fNL =1hasbeenassumedforallmodelsexceptthematterbounce atthepercent(negative)levelcomparedtothelinearone.Asforthe
(whichis fNL-independent). othermodels,inorderforthenon-Gaussiancontributiontobecom-
parabletothelinearone, primordial non-Gaussianity wouldneed
tobeat theunrealisticlevel of f ∼ 400 forthelocal case, and
NL
issimilartothe massdependence of thelinear termgiven inEq.
substantiallylargerthanthatforotherbispectrumshapes.
(15). Also, curves referring to different models are rather similar
Figure 3 allows one to determine the non-Gaussian contri-
inshape.Besidesthematterbouncemodel,thelocalmodelisthe
bution, in units of the linear contribution, for the current bounds
one havingthelargest effect, whiletheorthogonal model hasthe
on f . Constraints from the CMB (Komatsuetal. 2011) imply
NL
lowest. This isdifferent fromthe behavior of, e.g., thehalo bias,
−13< f <96at95%ConfidenceLevel(CL)forthelocalshape1,
NL
for which the equilateral model displays the smallest effect. The
meaningthatthenon-Gaussiancontributioncanbeatmost∼19%
functionω forthematterbounceisvirtuallyidenticaltothatfor
M ofthelinearone.ThesameCMBdataconstrain−278< f <346
thelocal model when assuming f = −35/8, inagreement with NL
NL for the equilateral shape, implying a ∼ 18% relative importance.
thepreviousdiscussion.
Thetighterconstraintsonthelevelofnon-Gaussianityfortheen-
Ascanbeseen fromthestructureofEqs. (15) and(16),the
foldedshapecomefromtheLSS(Xiaetal.2011),correspondingto
relativeimportanceofthenon-Gaussiancontributionwith respect
−16< f <465at2σconfidencelevel.Thismeansthatthenon-
NL
tothelinearoneis
Gaussiancontributionisatmost∼ 16%ofthelinearone.Finally,
Υ (τ)≡2hJ1(τ)·J2(τ)i =2dE(τ)/dτ ωM . (20) for the orthogonal shape the CMB data by Komatsuetal. (2011)
M kJ1(τ)k2 dD(τ)/dτσ2M bear−533< fNL <8,implyingamaximum∼10%(negative)rela-
Theacquisiti(cid:10)onofang(cid:11)ularmomentumbyprotostructureshappens tivestrength.Thesenumberscanbeappreciatedalsobylookingat
thepositionsofthefilledcirclesinFigure3.Forcomparison, the
athighredshiftwhere,undertheassumptionofflatspatialgeom-
black dotted lineshows theupper limit tothenon-Gaussian con-
etry, the Universe is well approximated by an Einstein-de Sitter
tributionfoundbyCT,afterassumingalog-normaldistributionfor
model.Correctionsduetothepresenceofacosmologicalconstant
the primordial gravitational potential and after rescaling it to the
canbeconsideredtobenegligible.Ifthisisthecase,then
scale M = 1010h−1M . I stress the fact that, while CT adopted a
Υ (τ)=−12τ−2ωM =−12(3t)2/3ωM . (21) Gaussianwindowfunc⊙tionandh=0.5,Iusedareal-spacetop-hat
M 7 σ2M 7 σ2M filterandh=0.704.Moreover,whileCTcalibratedthelevelofpri-
Note that the negative sign cancels with the negative sign in the mordialnon-GaussianityusingavalueSR =4fortheskewnessof
definitionofω ,sothatthenon-Gaussiancontributionispositive thematterdensityfieldonascaleR=8h−1Mpc,Iadopted,conser-
M
(increasesthespingrowth)foramodelwithpositiveskewnessand vatively,SR=0.1.Thisvalueresultsfromthelarge-scaleskewness
negativeotherwise.ThishasalreadybeennoticedbyCT.Thepre- per unit fNL for local non-Gaussianity (∼ 10−3, e.g., Figure 1 of
vious equation also shows that the non-Gaussian contribution to Fedelietal.2011)multipliedbythemostrecentupperlimitonthe
spinacquisitiongrowsfasterthanthelinearone. levelofnon-Gaussianityforthesameshape(fNL∼100).
If the matter patch under consideration is an overdense re-
gion, it is reasonable to assume that the spin growth induced by
tidaltorques occurs until theoverdensity detaches fromtheover- 1 IconvertedallconstraintsmentionedheretotheLSSconvention,which
allexpansionoftheUniverseandcollapsesintoaboundstructure. hasbeen adopted throughout. SeeFedeli&Moscardini (2010)andrefer-
Thismomentτ∗ canbenaivelyidentifiedasD(τ∗)σM = 1.Foran encesthereinforadiscussion.
Angularmomentumgrowth 5
becausehigher-ordernon-Gaussiancorrections,thathavenotbeen
consideredhere,dependonthetrispectrumoftheZel’dovichpoten-
tial,andhencealsoreacttoprimordialnon-Gaussianity.Theresults
presentedinthislettermotivatethestudyofthesehigher-ordercon-
tributions,andshowhowitispossibletoconsistentlydescribethe
dynamicsofprotostructuresbasedonlargelygeneralcosmological
initialconditions.
ACKNOWLEDGEMENTS
IthanktheUniversityofFloridaforsupportthroughtheTheoret-
ical Astrophysics Fellowship. I credit L. Moscardini for insight-
ful comments onthe manuscript andI amdeeply indebted tothe
anonymousrefereeforhelpinsubstantiallyimprovingthiswork.
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