Galactic angular momenta and angular momentum couplings in the large-scale structure

0.2 MB·English

by  Bjoern Malte Schaefer

Mon.Not.R.Astron.Soc.000,1–10(2008) Printed25January2011 (MNLATEXstylefilev2.2)
Galactic angular momenta and angular momentum couplings in the
large-scale structure
1
Bjo¨rn Malte Scha¨fer1,2,3⋆ and Philipp M. Merkel4
1
0 1InstituteofCosmologyandGravitation,UniversityofPortsmouth,DennisSciamabuilding,BurnabyRoad,PortsmouthPO13FX,UnitedKingdom
2 2Institutd’AstrophysiqueSpatiale,Universite´deParisXI,baˆtiment120-121,Centreuniversitaired’Orsay,91400OrsayCEDEX,France
3AstronomischesRecheninstitut,Zentrumfu¨rAstronomie,Universita¨tHeidelberg,Mo¨nchhofstraße12,69120Heidelberg,Germany
n
4Institutfu¨rTheoretischeAstrophysik,Zentrumfu¨rAstronomie,Universita¨tHeidelberg,Albert-Ueberle-Straße2,69120Heidelberg,Germany
a
J

4
2 25January2011

]
O
ABSTRACT
C Inthispaper,werevisittheaquisitionofangularmomentumofgalaxiesbytidalshearingand
compute the angular momentum variance σ2 as well as the angular momentum correlation
h. functionC (r) froma peak-restrictedGaussiLan randomprocess.Thisstochastic processde-
L
p
scribing the initial conditions treats both the tidal shear as well as the inertia as dynamical
-
o fieldsandexplicitlyaccountsforthediscretenessoftheinertiafield.Wedescribethewayin
r which the correlationsin angularmomentumresult froman interplayof long-rangedcorre-
t
s lationsinthetidalshear,andshortrangedcorrelationsin theinertiafield andwhichreflects
a thecorrelationbetweentheeigensystemsofthesetwosymmetrictensors.Weproposeanew
[ formoftheangularmomentumcorrelationfunctionwhichisabletodistinguishbetweenpar-

1 allelandantiparallelalignmentofangularmomentumvectors,andcommentonimplications
v ofintrinsicalignmentsforweaklensingmeasurements.WeconfirmthescalingL/M M2/3
∝
4 and find the angularmomentumdistribution of Milky Way-sized haloesto be correlatedon
8 scalesof 1Mpc/h.Thecorrelationfunctioncanbewellfitted byanempiricalrelationof
5 theformC∼ (r) exp( [r/r ]β).
L 0
4 ∝ −
. Keywords: cosmology:large-scalestructure,gravitationallensing,methods:analytical
1
0
1
1
v: 1 INTRODUCTION Blazeketal.2011)inparticularwithSDSS-data,andconfirmations
i ofthesealignmentsinnumericalsimulations(Arago´n-Calvoetal.
X Inthecurrentparadigm,haloesacquireangularmomentumbytidal
2007;Betancort-Rijo&Trujillo2009;Schneider&Bridle2010).
shearingfromtheambientmatterdistribution(Heavens&Peacock
r Inthispaper,werevisittheacquisitionofangularmomentum
a 1988; Catelan&Theuns 1996a,b; Lee&Pen 2000; Lee 2006;
ofcosmological objectsinlineartheoryandrecomputethecorre-
Lee&Park 2006; Scha¨fer 2009, for a review), which was first
lationfunctionofangularmomenta.Werestrictourselvestolinear
proposed by Hoyle (1949) and Sciama (1955). Tidal shear-
structureformation,usingtheZel’dovichmappingforthedescrip-
ing is well supported by numerical simulations (White 1984;
tionofthetidalshearingmechanism.Inthispaper,wehopetoim-
Sugermanetal.2000;Catelanetal.2001;Hahnetal.2007,2010)
provepreviousworksonthistopicintheseaspects:
and leads to alignments of theangular momentum direction with
the local tidal shear field. An important observational conse- (i) Weemployanimprovedfunctionalformforthecorrelation
quenceofangularmomentumalignmentsinthelarge-scalestruc- functionwhichisabletodistinguishbetweenparallelandantipar-
ture, are induced intrinsic ellipticty alignments between neigh- allelalignmentsofangularmomenta,andwhichmayassumeneg-
bouring galaxies (Catelan&Porciani 2001; Jing 2002), which ativevalues for antiparallelorientation oftheangular momentum
can be expected to be a significant source of systematics vectors.Thisrequiresthattheangularmomentumcorrelationfunc-
in weak lensing surveys (Croft&Metzler 2000; Heavensetal. tioncannotbeamerequadraticforminthetidalshearandinertia
2000; Crittendenetal. 2001; Hirata&Seljak 2004; King 2005; tensorfields,whicharetherelevantquantitiesforangularmomen-
Sembolonietal.2008)andevengalaxysurveysastheyintroduce tumbuild-up,butneedstobeantisymmetric.
selectioneffectsduetocorrelatedanglesofinclinationofthegalac- (ii) Both the inertia and tidal shear fields will be consistently
ticdisks(Krause&Hirata2011).Bynow, thereisreliableobser- computed from a correlated Gaussian random process, such that
vationalevidence oftidal-shearinginducedellipticitycorrelations thefieldshaveconsistentphaserelations.Theangularmomentum
(Mandelbaumetal. 2006; Hirataetal. 2007; Jonesetal. 2010; correlationwillreflectthedifferentcorrelationlengthsoftheinertia
andtidalshearfields.
(iii) Treating both fields as dynamical quantities improves
⋆ e-mail:[email protected] on the parameterisation introduced by Lee&Pen (2000) and
2 B.M. Scha¨ferandPh.M.Merkel
Crittendenetal.(2001)fortheaveragemisalignmentbetweenthe L(t)= d3r (r ¯r) υ(r,t)ρ(r,t), (1)
eigensystemsofbothtensorsandreflectschangesinaveragemis- ZV − ×
alignmentwithincreasingdistance. whereυ(r,t) isthe(rotational) velocityof thefluidelement with
(iv) Weexplicitlytakeaccountofthediscretenatureoftheiner- density ρ(r,t) = ρ (1 + δ(r,t)) at position r around the cen-
tiafieldastherandomprocessrestrictedtogalaxyformationsitesin tre of gravity r¯. Ihniperturbation theory, δ 1 and the den-
thelarge-scalestructurehasadifferentweightingofcertaininertia- sity field can be approximated by assuming≪a constant density
shearcombinations comparedtothatofcontinuous fields,i.e.the ρ = Ω ρ inside the protogalactic region. Following White
angularmomentumdistributionisbiased. h(19i84),Camteclraitn&Theuns(1996a)andCrittendenetal.(2001),we
describethegrowthofperturbationsonanexpandingbackground
ThetheoryisdevelopedinSect.2,whereweoutlinetheGaus-
in Lagrangian perturbation theory: The trajectory of dark matter
sian model used for determining the angular momentum correla-
particles in comoving coordinates is given by the Zel’dovich ap-
tions in the large-scale structure and where we propose an im-
proximation(Zel’dovich1970):
provedformoftheangularmomentumcorrelationfunction.Inad-
dition, wecomment onthe influenceof dark energy cosmologies x(q,t)=q D (t) Ψ(q) x˙ = D˙ Ψ, (2)
+ +
on the angular momentum acquisition. The results are presented − ∇ → − ∇
which relates the initial particle positions q to the positions x at
inSect.3,where wecompute theangular momentum correlation
timet.Theparticlevelocityx˙ followsfromtheZel’dovich-relation
functionalongwiththeangular momentumvariance, andinvesti-
bydifferentiationbythetime-variable.ThegrowthfunctionD (t)
gate their mass-dependence, followed by a discussion in Sect. 4 +
describesthehomogeneoustimeevolutionofthedisplacementfield
where we summarise our main results and comment on the con-
Ψandcontainstheinfluenceoftheparticulardarkenergymodel.
sequencesoftheimprovedangularmomentummodelonintrinsic
IntheLagrangianframe,theexpressionfortheangularmomentum
ellipticitycorrelations.
becomes
Throughout, the cosmological model assumed is a spatially
flat ΛCDM cosmology with Gaussian adiabatic initial perturba-
L=ρ a5 d3q (x x¯) x˙ ρ a5 d3q(q q¯) x˙, (3)
tionsinthecolddarkmatterdistribution.Choicesfortherelevant 0 − × ≃ 0 − ×
ZVL ZVL
parameter values are: Ω = 0.25 with Ω = 0.04, Ω = 0.75,
m b Λ where theintegration volume is defined in comoving coordinates
H0 =100hkms−1Mpc−1withh=0.72,ns =1andσ8=0.9. as well. Assuming that the gradient Ψ(q) of the displacement
∇
fieldΨ(q) doesnot vary muchacross theLagrangianvolume V ,
L
a second-order Taylor expansion in the vicinity of the centre of
gravityq¯ isapplicable:
2 FORMALISM
∂ Ψ(q) ∂ Ψ(q¯)+ (q q¯) Ψ , (4)
ThissectiondescribestheGaussianmodelusedforderivingthean- α ≃ α − β αβ
β
gularmomentumcorrelationsinthelarge-scalestructure.Sect.2.1 X
explainshow haloesacquirerotationbytidalshearingandrelates TheexpansioncoefficientisthetidalshearΨσγatthepointq¯:
theangularmomentum LtotheinertiaI andgravitationalshear
αβ Ψ (q¯)=∂ ∂ Ψ(q¯), (5)
σγ σ γ
Ψ in the Zel’dovich-approximation. In Sect. 2.2, we outline a
αβ
model for deriving the correlations of shear and inertia from the becausetheZel’dovichdisplacement fieldΨisrelatedtogravita-
fluctuation statistics of the density field, based on a joint multi- tionalpotentialΦandcanbecomputedasthesolutiontoPoisson’s
variate Gaussian probability density. The covariances take a par- equation∆Ψ=δfromthecosmologicaldensityfieldδ.Thegradi-
ticularlysimpleformifexpressed inspherical coordinates,asex- ent∂αΨ(q¯)oftheZel’dovichpotential displacestheprotogalactic
plainedinSect.2.3andweelaborateontheshapeofthecorrela- object,whichisneglectedinthefurtherderivation,asweonlytrace
tionmatricesinSect.2.4.Thecorrelationfunction L (x)L (x) differential advection velocities responsible for inducing rotation.
h α α′ ′ i
of theangular momenta is determined in Sect. 2.5 by integrating Identifyingthetensorofsecondmomentsofthemassdistribution
outtheGaussianprobabilitydensityrestrictedtopeaksintheden- oftheprotogalacticobjectastheinertiaIβσ,
sityfield. Wediscuss a technical issue, namely the misalignment
intheshearandinertiaeigensystemsinSects.2.6and2.7.Dueto Iβσ=ρ0a3 d3q(q−q¯)β(q−q¯)σ (6)
thehighdimensionalityoftheintegration,weemployanumerical ZVL
Monte-Carlointegrationscheme,asexplainedinSect.2.8. oneobtainsthefinalexpressionoftheangularmomentumLα:
L =a2D˙ ǫ I Ψ . (7)
α + αβγ βσ σγ
σ
X
2.1 Acquisitionofangularmomentumbytidalshearing
Itisconvenient torewritethetimedependence of D intermsof
+
Doroshkevich(1970)andWhite(1984)suggestedthattheangular thescalefactorabydD+/dt=aH(a)dD+/da,yielding:
momentumofgalaxiesoriginatesfromtidaltorquingbetweenthe dD
protogalacticregion and thesurrounding matterdistributionprior Lα=a3H(a) da+ ǫαβγ IβσΨσγ. (8)
to collapse. Assuming a non-spherical shape of the protogalactic Xσ
region,theangularmomentumgrowsatfirstorderandlinearlyin Thetheoryofangularmomentumacquisitionbytidalshearinghas
timeinEinstein-deSitteruniverses, whereasinspherical regions, been extended to nonlinear stages by using second order pertur-
theacquisitionofangularmomentumisonlyasecondordereffect bationtheory(Catelan&Theuns1996b)andtoincludeeffectsof
duetoconvectivematterstreamsontheboundarysurface,asshown non-Gaussian initial perturbations (Catelan&Theuns 1997), but
byPeebles(1969). for reasons of analytical computability, we restrict our model of
Quitegenerally,theangularmomentum Lofarotatingmass angular momenta tothe linear regimeof structureformation of a
distributionρ(r,t)containedinthephysicalvolumeVisgivenby: Gaussianrandomfield.
Galacticangularmomentum couplings 3
0.95 densityfieldρ(x),δ(x)=(ρ(x) ρ )/ ρ ,withtheaveragedensity
−h i h i
ρ =Ω ρ .Theseperturbationsareconvenientlydecomposedin
0.9 h i m crit
Fouriermodesδ(k):
0.85
δ(k)= d3xδ(x)exp( ikx). (13)
a) 0.8 Z −
Q( SpecifyingthepowerspectrumP(k)sufficestodescribethestatis-
0.75
o ticalpropertiesofahomogeneous andisotropicGaussianrandom
rati 0.7 field:
h
owt0.65 hδ(k)δ(k′)∗i=(2π)3δ3D(k−k′)P(k). (14)
r
g 0.6 Thepowerspectrum,forwhichwechoosetheansatzP(k) kns
∝ ·
T2(k),isnormalisedtoexhibitavarianceofσ =0.9onscalesof
0.55 8
R=8Mpc/hbytherelation:
0.5
1
σ2 = dkk2W2(kR)P(k), (15)
0.45 R 2π2
0 0.1 0.2 0.3 s0.c4alef0a.5ctor0a.6 0.7 0.8 0.9 1 Z
withaFourier-transformedsphericaltop-hatforthefilterfunction
W(y),i.e.W(y) = 3 sin(y) ycos(y) /y3.Acommonparameteri-
Figure1.Theinfluenceofthedarkenergymodelonthetimeevolutionof −
aanndguwlaar=mo0m,seonltiad:lQin(ea)),afsoarqfuuninctteiosnseonfcesc(awle0f=acto2r/a3foanrdΛwCaDM=0(w,d0a=sh−ed1 wsaatisopnrofoprotsheedsbhyapBearo(cid:2)dfetehneettraanl.sf(e1r98fu6(cid:3)n)c:tionT(q)forCDMmodels
−
line)andadarkenergymodelwithvariableequationofstate(w0 = −2/3 ln(1+2.34q)
andwa=1/3,dash-dottedline). T(q)= 2.34q
1
Fig.1comparesthetimeevolutionoftheangularmomentum 1+3.89q+(16.1q)2+(5.46q)3+(6.71q)4 −4 , (16)
×
indarkenergycosmologieswithSCDM.Wedefinetheratio
wherehthewave-vectorkisgiveninunitsofthesihapeparameterΓ,
Q(a) qDE(a) (9) firstintroduced byEfstathiouetal.(1992).Aconvenient parame-
≡ q (a) terisationofthevalueofΓasafunctionofthematterdensityΩ
SCDM m
andthebaryonicdensityΩ isgivenbySugiyama(1995):
with q(a) = a3H(a)dD /da (with D (a) normalised to unity to- b
+ +
day and we parameterise the dark energy equation of state with q=q(k)= k/Mpc−1h withΓ=Ω hexp Ω 1+ √2h . (17)
(Chevallier&Polarski2001;Linder&Jenkins2003) Γ m − b Ω
  m 
w(a)=w0+(1−a)wa. (10) The mass scale of the objects of interest isset by imposing a
smoothingonhighspatialfrequencies,wherethediameterRofthe
InSCDMtheseformulaesimplifyto H(a) = H a 3/2, D (a) = a
0 − +
isotropicfilterfunctionS (k)correspondstothesizeoftheobjects
andconsequentlyq = H a3/2.Fig.1suggeststhatthespin-up R
SCDM 0
attheonsetofcollapse.Fornumericalreasons,weuseaspherically
ofhaloesindarkenergymodelsissignificantlyslowercomparedto
symmetricGaussianforS (k):
SCDM,andthechoiceoftheequationofstateaffectsthetimeevo- R
dluDtio/ndasigfonlilfiocwasntnluy.mTehriecaglrlyowatshafsuonlcuttiioonntDo+t(hae)garnodwtihtsedqeuraivtiaotniv,e P(k)−→P(k)SR2(k)withSR(k)=exp(−k2R2/2). (18)
+
In order to predict the correlation of the angular momenta L of
d2 1 dlnH d 3 objectswhichformatpeaksintheGaussiandensityfield,oneneeds
D + 3+ D = Ω (a)D (a), (11)
da2 + a dlna da + 2a2 m + to relate the density gradient δ (x), the second derivatives δ (x)
! α αβ
inwhichthedarkenergy model affectsthescalingof theHubble andthetidalfieldΨαβ(x)tothedensityfieldδ(x):
function H(a) andofthematterdensityparameter Ωm(a).Inspa- d3k
tially flat dark energy cosmologies, the Hubble function H(a) = δ(x) = δ(k)exp(ikx), (19)
(2π)3
dlna/dtisgivenby Z
∂δ(x) d3k
δ (x) = =i k δ(k)exp(ikx), (20)
HH2(02a) = Ωa3m +(1−Ωm)exp 3Za1dlna′(1+w(a′))!, (12) δ α(x) = ∂∂2xδα(x) =Z (2π)d33kαk k δ(k)exp(ikx), (21)
with the dark energy equation of state w(a). The value w ≡ −1 αβ ∂xα∂xβ −Z (2π)3 α β
correspondstothecosmologicalconstantΛ. Thetidalshear followsfromthesolutionof thePoissonequation
∆Ψ(x) = δ(x)linkingtheZel’dovichpotentialΨ(x)tothedensity
fieldδ(x):
2.2 Gaussianmodeloftheangularmomentumcorrelations
∂2Ψ(x) d3k k k
Thegoalofthissectionistoderivethe2-pointcorrelationfunction Ψ (x)= = α βδ(k)exp(ikx). (22)
αβ ∂x ∂x (2π)3 k2
oftheangularmomentaofobjectsthatformatpeaksinthecosmic α β Z
densityfield.ThequantitiesneededarethethetidalshearΨ (x)as Animportant consequence ofeqns. (21)and(22)willbethefact
αβ
wellastheinertiaI (x)ofapeakregion,whichbothcanberelated thattheangularmomentumcorrelationisdeterminedbytwomech-
αβ
tothedensityfielditselfanditssecondderivatives.Incosmology, anismswithdifferingcorrelationlength: ashort rangecorrelation
fluctuationsinthedistributionofmatteraredescribedbytheover- ofthepeakshapesandhencetheinertia,andalongrangecorrela-
densityδ(x),whichisdefinedasthefractionalperturbationinthe tionmediatedbythetidalshear.
4 B.M. Scha¨ferandPh.M.Merkel
Thejointdistributionoftheamplitudesofthedensityfield,its agonalinthemasstensoreigenframe.Carryingouttheintegration
derivativesandthetidalshearfollowsfromaGaussianprobability yields:
density(Bardeenetal.1986): η
I = 0 Γdiag A2+A2,A2+A2,A2+A2 . (30)
1 1 αβ 5 y z x z x y
p(υ)dυ= exp υtV−1υ dυ, (23) (cid:16) (cid:17)
(2π)N/2√detV −2 Theevolutionofthedensityfieldisassumedtobehomogeneousto
!
firstorder,η ρ a3 = ρ a3,with ρ =Ω ρ .
wherethequantitiesofinterestatthepoint xhavebeenarranged 0≡ 0 0 h i h i m crit
ina15-dimensional vector υ,i.e.,3values forthedensity gradi-
entδ (x),6valuesforthesecondderivativesδ (x)ofthedensity
α αβ
2.3 Describingthecorrelationsinsphericalcoordinates
field(duetotheinterchangabilityofthesecondderivatives)and6
valuesforthetidalshearΨαβ(x),whichissymmetricunderindex Following the example of Regos&Szalay (1995) and
exchangeaswell.ThecovariancematrixVfollowsfromtheouter Heavens&Sheth (1999), we express the correlations between
pbreoedxutcetndVeijd≡tohiυnicυl∗ujid,e(it,hje)fi=el1d.v.a.l1u5e.sTυh′isatparosbeacboinlditypodienntsxit′y, can cthoeorddeinnasitteys.fiTehled,twitospdeearkivsautinvdeseracnodnstihdeertaitdiaolnsahreeaarssinumsepdhetroicbael
positionedonthez-axis,symmetricabouttheorigin,andseparated
1 1
p(w)dw= exp wtW−1w dw, (24) byadistancer,i.e.theyhavethecoordinates x = (0,0,+z/2)and
(2π)N√detW −2
! x =(0,0, z/2).Thecorrelationstakeaparticularlysimpleshape
′
wherethe30-dimensionalvectorw=(υ,υ)combinesthevectors in the basi−s given by the set of dimensionless complex variables
′
υ and υ at the two points x and x under consideration and the yn (x):
′ ′ ℓm
covariancematrixW,W ww ,(i,j) = 1...30,isdefinedin
ij ≡ h i ∗ji iℓ+2n d3k
completeanalogy. yn (x)= √4π kℓ+2nδ(k)Y (kˆ)exp(ikx), (31)
Apeculiarityworthwhilementioningisthefactthattheden- ℓm σℓ+2n Z (2π)3 ℓm
sityfieldδ(x)isdegeneratewiththetracetrΨαβ(x)ofthetidalshear with kˆ = k/k as the direction of the wave-vector k. σ2 are the
becauseofPoisson’sequation∆Ψ=trΨ =δ.Forthatreason,the j
αβ weightedmomentsofthe(smoothed)matterspectrumP(k):
densityfieldwillappearintheaboveoutlinedrandomprocessasa
derivedquantity,whereastheentriesofthetidalshearmatrixwill σ2= 1 dkk2j+2P(k). (32)
bedrawnfromtheGaussiandistribution,withthepeakrestriction j 2π2
Z
inplace. Thetransformationbetweenthephysicalframeandtheyn -frame
The inertia tensor I of an object forming at a peak in the ℓm
αβ forthescalardensityfieldδ(x)isgivenby
densityfieldatpositionx isrelatedtothesecondderivativesδ of
p αβ
thedensityfieldatthatparticularpoint(Catelan&Theuns1996a): σ0y000(x)=δ(x). (33)
Intheeigenframeofthemasstensor δ ∂ ∂ δatthepeak,the
densityfieldcanbeapproximatedby−aαpβar≡ab−oliαcdβensityprofile, Forthevectorialdensitygradientδα(x,theyread:
σ y0 (x) = √3δ(x), (34)
1 3 1 10 z
δ(x)=δ(xp)− 2 λα(x−xp)2α, (25) σ1y011(x) = − 3/2 δx(x)+iδy(x) . (35)
α=1
X
Thetensorδ (x)pcanbe(cid:16)determinedfro(cid:17)mtheyn -coefficientsby:
whereλ ,α =1,2,3aretheeigenvaluesofthemasstensor.Ifthe αβ ℓm
α
boundary∂ΓofthepeakregionΓisdefinedbytheisodensitysur- σ y0 (x) = 5/4 δ (x)+δ (x) 2δ (x) , (36)
faceδ =0andifthepeakheightisexpressedinunitsofthevari- 2 20 − xx yy − zz
Γ
anceσ ,δ(x)=νσ ,theboundarysurfaceisgivenintheparabolic σ y0 (x) = p15/2(cid:16) δ (x)+iδ (x) , (cid:17) (37)
0 0 2 21 − xz yz
approximationbyanellipsoidequation:
σ y0 (x) = +p15/8 (cid:16)δ (x) δ (x)(cid:17)+2iδ (x) , (38)
2 22 xx − yy xy
∂Γ: 3 (x−xp)α 2 =1, (26) σ2y100(x) = +pδxx(x)+(cid:16) δyy(x)+δzz(x) . (cid:17) (39)
A
Xα=1 α ! Therelationlinkin(cid:16)gthetidalshearΨ to(cid:17)theyn -coefficientscan
αβ ℓm
wherethesemi-axesAαoftheellipsoidarerelatedtotheeigenval- bederivedincompleteanalogytoeqns.(36)through(39).Theynℓm-
uesλαby coefficientsofthetidalsheartensorfielddiffermainlybyafactor
ofσ /σ fromthoseofthemasstensorfield,apartfromthetrace
2νσ 2 0
A = 0. (27) ofthetidalshear:
α λ
r α
The volume Γ of ellipsoidal peak region bounded by isodensity σ0y−201(x) = − 5/4 Ψxx(x)+Ψyy(x)−2Ψzz(x) , (40)
contourδΓ=0intheparabolicapproximationisthengivenby: σ0y−211(x) = −p15/2(cid:16) Ψxz(x)+iΨyz(x) , (cid:17) (41)
Γ= 43π AxAyAz, (28) σ0y−221(x) = +p15/8 (cid:16)Ψxx(x)−Ψyy(x)(cid:17)+2iΨxy(x) , (42)
σ y0 (x) = +pΨ (x)(cid:16)+Ψ (x)+Ψ (x) , (cid:17) (43)
whichwouldimmediatelyyieldanestimateforthemassoftheob- 0 00 xx yy zz
ject: emphasisingthed(cid:16)ifferenceincorrelationle(cid:17)ngthbetweentheden-
4π sityfieldandthepotential.
M=η0Γ=η0 3 AxAyAz. (29) Theynℓm(x)-basisinheritsitssymmetryundercomplexconju-
gationfromthesphericalharmonicsY :
ℓm
TheinertiatensorI followsfromthesecondmomentsofthemass
αβ
distribution,restrictedtothevolumeΓofthepeakregion,andisdi- ynℓm(x)∗=(−1)mynl,−m(x), (44)
Galacticangularmomentum couplings 5
whichwillbecomeimportantatthestageofinvertingtherelations 2.4.2 Cross-correlationmatrix
givenabove.Similarlytothevectorυcontainingthephysicalvari-
Thecross-correlationmatrixC(r)isdefinedanalogously,
ables,theyn (x)-coefficientscanbearrangedinavectorybymap-
ℓm
panindgtthheef3raimndeicoefstnh,eℓya-nvdalmuetsoaarenerwelainteddexbyi.aThlienepahrysuinciatlarυy-ftrraanmse- C(r)=Cℓnℓn′′mm′(r)=hynℓm(x)ynℓ′′m′(x′)∗i. (51)
The steps in simplifying this expression consist in inserting the
formation.Forclarity,weabbreviatey y(x)andy y(x).
≡ ′ ≡ ′ definition of the yn (x)-coefficients, in replacing the variance
ℓm
δ(k)δ(k′)∗ withthematter power spectrum P(k)and inexpand-
h i
ingtheFourierwaveexp(ikr), r x x,byvirtueofRayleigh’s
′
≡ −
2.4 Shapeofthecorrelationmatrices formula,
A(1s99d5e)m, othnestcraotrerdelaitniotnhemfaotrrimceaslisnmeedperdopionsethdebGyauRsesgiaons&proSbzaablaily- exp(ikr)=4π ∞ iLjL(kr) +L YLM(rˆ)∗YLM(kˆ). (52)
L=0 M= L
itydensities(eqns.23and24)assumeaparticularlysimpleshape X X−
intheframegivenbytheyn -coefficientsandcanbeexpressedan- The integration over the three spherical harmonics Y (kˆ) can be
ℓm ℓm
alyticallyintermsofmomentsofthedarkmatterpowerspectrum. simplified by inserting the definition of the Wigner-3j symbols
ThecorrelationmatrixYinthisframeisdefinedastheexpectation (Messiah1962;Abramowitzetal.1988),
value
dΩYℓ1m1(kˆ)Yℓ2m2(kˆ)∗Yℓ3m3(kˆ)=
Yij ≡h(y,y′)i(y,y′)∗ji (45) Z
oftheproductsoftheelementsinthevector(y,y′)andcanbesplit ( 1)m2 Π3i=1(2ℓi+1) ℓ1 ℓ2 ℓ3 ℓ1 ℓ2 ℓ3 . (53)
intotwo15×15submatrices:theauto-correlationmatrixA,defined − s 4π 0 0 0 ! m1 −m2 m3 !
asA = yy andthecross-correlationmatrixC=C(r),givenby
ij h i ∗ji Further reduction is reached by taking advantage of the fact that
C = yy , which depends on the distance r between the two
ij h i ′j∗i bothpeaksareassumedtolieonthez-axis,
pointsxandx,
′
2L+1
A C Y (rˆ)=δ , (54)
Y= C+ A , (46) LM M0r 4π
!
whichyieldsthefinalformofthecross-correlationmatrixC(r):
where C+ is the Hermitean adjoint of C. The transformation be-
btmwyaetterhinextmhReaatyrcnℓimtxi-nSfgr,aocmnoemtahpneudtvitenhcgeto(pryh,yυysia)cnfadrlofrmreasmwulet=iins(ggυii,vnυenth)b,eyvtehcetocromy,palnexd Cℓnℓn′′mm′(r)=δmm′σ(−ℓ+12n)σm+ℓn′+−2nn′′ L=Xℓ|+ℓ−ℓ′ℓ′|(2L+1)iL+ℓ−ℓ′KL,ℓ+ℓ′+2(n+n′+1)(r)
′ ′
ℓ ℓ L ℓ ℓ L
S= R 0 . (47) × (2ℓ+1)(2ℓ′+1) 0 0′ 0 m m′ 0 , (55)
0 R p ! − !
! where the kmj (kr)-weighted spectral moments are abbreviated
ℓ
The matrices R and S can be constructed from the relations be- withK (r),
ℓm
tweentheyn (x)-coefficientsandthephysicalvariablesδ(x),δ (x),
ℓm α 1
δ (x)andΨ (x)compiledinSect.2.3. K (r)= dkkmj (kr)P(k). (56)
αβ αβ ℓm 2π2 ℓ
Z
j (kr) are the spherical Bessel functions of the first kind
ℓ
(Abramowitzetal.1988).TheCnn′ -coefficientsarealwaysreal:
2.4.1 Auto-correlationmatrix TheWigner3j-symbolsareuneqℓℓu′maml′tozeroif L+ℓ ℓ iseven,
′
−
Theauto-correlationmatrixAintheyn -frameisdefinedby: inwhichcaseiL+ℓ−ℓ′ isarealnumber.Furthermore,thesummation
ℓm over L can be restricted to the range ℓ ℓ 6 L 6 ℓ+ℓ due
′ ′
| − |
A=Anℓℓn′′mm′ =hynℓm(x)ynℓ′′m′(x)∗i. (48) ctorotshse-ctorriraenlgalteioncomndaittriioxnCacpapnliebdebtorotuhgehWttoigbnleorc-3kjdsiaygmobnoallss.hTaphee
InsertingtheFourierexpansionofthevariables,replacingthevari- bysuitablearrangementoftheyn -coefficientsinthevectory,more
ℓm
anceofthedensityfield δ(k)δ(k′)∗ withthematterpower spec- specifically,bygroupingcoefficientswithconstantvalueofmand
h i
trum P(K), and using the orthogonality relation of the spherical increasingthemodulusofmwithincreasingindexi.
harmonicsYℓm(kˆ), IncontrasttotheconstantvaluesinthematrixA,theentriesof
thematrixCdependonthedistancer= x x ofthetwopointsx
′
dΩYℓ1m1(kˆ)Yℓ2m2(kˆ)∗=δℓ1ℓ2δm1m2, (49) andx′.ThesymmetryoftheentriesofC| u−nder| interchangeofthe
Z pointsxandx isgivenbytherelation
′
yieldsfortheauto-correlationmatrix:
σ2 hynℓm(x)ynℓ′′m′(x′)∗i=(−1)ℓ−ℓ′hynℓm(x′)∗ynℓ′′m′(x)i. (57)
Anℓℓn′′mm′ =Anℓmn′δℓℓ′δmm′ =(−1)n−n′σℓ+2ℓn+σn+ℓn+′2n′δℓℓ′δmm′, (50) rTyapreicadlepciocrtreedlaitnioFnigc.oe2ffi.TchieentssmCooℓnℓnt′′hmimn′gasscfaulencRtio(cn.sf.oefqsne.p1a8ra)thioans
wherethedefinitionoftheσ -coefficientsineqn.(32)wasusedfor beensettoR = 1Mpc/handthedensitythresholdwaschosenas
j
substitutingthe(2j+2)thmomentsofthepowerspectrumP(k).The ν = 2, inorder torepresent galaxies. Withthe choice of Ω , the
m
structureofthematrixAisremarkablysimple:Itisdiagonalinthe smoothingofthepowerspectrumatscaleRcorrespondstoamass
indicesℓandmandthesignofitsentirelyrealentriesisdetermined scale of M = 4πρ Ω R3 3.1 1011M /h. For illustration
bywhethern−n′isanevenoroddnumber. purposes, tshcaelecova3riancrcite mmatri≃x Cℓnℓn′′m×m′ has b⊙een transformed to
6 B.M. Scha¨ferandPh.M.Merkel
1 densityp(w)dw,constrainttopeakregionsinthefluctuatingdensity
field.Theregionstowhichtheintegrationisrestrictedarerequired
0.8 toexceedathresholdνindensity,δ(x) > νσ ,tobeofvanishing
) 0
r
( densitygradient,δ (x)=0,andofnegativecurvature,δ (x)<0.
′mm0.6 Thenumberdαensityofmaximainthedensityfieldαisβmodeled
′nnC′ℓℓ by a point-process (Bardeenetal. 1986; Regos&Szalay 1995;
nt 0.4 Heavens&Sheth1999):
e
ffici 0.2 n = δ3(x x). (58)
e peak D − i
o i
c X
n 0 Close to the maximum i at x a Taylor expansion of the density
o i
elati−0.2 gradientδα(x)isapplicable:
r
cor−0.4 δα(x)= δαβ(xi)(x−xi)β. (59)
β
X
Withthisexpansiononeobtainsforthepeakdensity:
−0.6
0 1distan2cer≡3 |x−4x′|in5units6ofR7=1M8pc/h9 10
npeak = δ3D δ−αβ1(xi)δα = detδαβ δ3D(δα). (60)
Fttarioiifgxnr.uamorTefhed2,ei.wssCtmhaoenorrcoreeethltahritnei≡goanus|xctcoao−-leceffixoR′rc|riewienlnaattusisonsCnietsℓtnmℓnt′o′omafmtRrt′ih(xe=r)Afi1=nℓlℓnt′Me′hmrympnℓsm′cc/(iasxhle)eyiqnRnℓ′u′,moat′rlrd(axteon′rs)t∗fthooierarmusenpeadirtfeusimnneantco--t tTthehmreeeorefrleatXahtliieoenmigtaea(cid:16)snksvetaselanusesosimrλ−p(cid:17),lδeiαrβ(cid:12)(cid:12)(cid:12)=:shBa1epi.en.g.w(cid:12)(cid:12)(cid:12)3ahewsnyhmcicomhnesaitldrlieocrwitnesgntsotohrer,eeiptilgahecanessytthhsee-
i
galaxies. determinantbytheinvariantquantity λ λ λ .
Theconstraintscanbecombined|in1a2m3a|sk (υ),whichisde-
finedasafunctiononthevectorυ containingthCederivatesofthe
6
Gaussian random field under consideration: Peaks in the density
fieldaredefinedaspointswithamplitudesinunitsofthevariance
5
()r σ20 =hδ2iexceedingacertainthresholdνandexhibitingavanish-
′mm4 inggradientδαaswellasnegativecurvatureδαβ:
′nnC′ℓℓ (υ)=δ3 [δ (x)] λ λ λ Θ(λ)Θ[δ(x) σ ν]. (61)
nt 3 C D α | 1 2 3| i i − 0
e Y
ffici 2 Θ(x)denotestheHeavisidestep-function. Thepeak densitynpeak,
e i.e. the expecation value for the number density of peaks in the
o
c
n 1 fluctuatingdensityfieldδwhichexceedathresholdνσ0 canthen
atio bederivedfromthemultivariateGaussianrandomprocessp(υ)dυ,
el 0
corr npeak = dυ p(υ)C(υ), (62)
−1 Z
which corresponds to the integral of the differential peak density
−2 n (ν)dνdefinedbyBardeenetal.(1986).
0 1distan2cer≡3 |x−4x′|in5units6ofR7=1M8pc/h9 10 peak
In analogy, the expectation value of the angular momentum
restrictedtopeakregionsinthedensityfieldcanbeobtainedwith:
Ftfiinroiaggnmusoecrf,aelwd3ei.hsRCtearwonercartesehlesraeta≡itoutnto|ox-cR−coo=exrffir′1e|cliMainetinpuotcnns/iCthmsℓnioaℓnn′tf′mroitmxhr′de(Aerfirnℓ)ℓlntt=′′oemrmrheys′pcnℓimarsel(edsxei)Ranyg,nℓt′′otgmrnaa′aln(axls.x′f)ioTe∗rhism.eaessdmaifonuotnothca-- hLi= npeak1(>ν)Z dυ p(υ)C(υ)L(υ), (63)
andthevarianceoftheangularmomentumfieldcanbecomputed
using:
aframe,wheretheauto-correlations Ann′ arediagonalandnor- 1
ℓℓ′mm′ L2 = dυ p(υ) (υ)L2(υ). (64)
malisedtounity. h i n (>ν) C
Fig. 3 shows the distance dependence of the entries of peak Z
Cdiℓnaℓng′′momn′a(rl)ininstaeafdraomfee,qwuhalerteotthheeauuntoit-cmoarrterilxa.tioNnotmabalterixdiAffnℓeℓnr′′memn′ceiss Idniscbroetthenfeosrsmouflathee, tmheeansuorremdaqliusaantitoitny.factor n−pe1ak accounts for the
to Cnn′ in this frame compared to the frame where to auto- Generalisation of the above relations to include a second
cgoartiroeℓnlℓa′mtoimfo′nthseariesoupnriotybainbcilliutydectohnetofaucrtst,hwathCicℓnhℓn′′mams′suinmdeicsaetemsit-haexieslorna-- pgeualakrrmesoumltesnitna,thweitchotrhreelaGtiaounssfiuanncptiroonbahbLiαli(txy)Ldαe′n(sxi′t)yi po(fwt)hdewan=-
tiosof 5andalessoscillatorybehaviour. p(υ,υ′)dυdυ′:
≃
hLα(x)Lα′(x′)i= (65)
2.5 Ansatzfortheangularmomentumcorrelations n2 1(>ν) dυC(υ) dυ′C(υ′)Lα(υ)Lα′(υ′)p(υ,υ′).
Thecorrelationfunctionoftheangularmomentainthelarge-scale peak Z Z
structurefollowsfromintegratingoutthe30dGaussianprobability In general, the thresholds ν, ν imposed on the peaks are equal.
′
Galacticangularmomentum couplings 7
AsderivedinSect.2.1,theangularmomentumL dependsonthe with(X+)t =+X+.InthederivationoftheangularmomentumL,
α
productoftheinertiatensorI andthetidalshearΨ :
βσ σγ L =a2D˙ ǫ I Ψ =a2D˙ ǫ X , (76)
α + αβγ βσ σγ + αβγ βγ
L =a2D˙ ǫ I Ψ =a2D˙ ǫ X , (66) σ
α + αβγ βσ σγ + αβγ βγ X
σ thepermutationsymbolǫ picksouttheantisymmetriccontribu-
X αβγ
if the acquisition of angular momentum of a protogalactic object tionX−,byvirtueof ǫαβγ(Xβ+γ +Xβ−γ) = Xβ−γ,because thecontrac-
isdescribedintheZel’dovichapproximation.Forconvenience,we tionof thesymmetrictensor X+ withtheantisymmetricpermuta-
introducethematrixXwiththecomponents: tionsymbolvanishes,ǫαβγXβ+γ =0.
Hence,theprotogalacticobjectswillonlyacquireangularmo-
Xβγ(x)= Iβσ(x)Ψσγ(x). (67) mentum if the commutator X− between the inertia and the tidal
Xσ shear isnon-zero, which means that the inertia and shear tensors
Then, the correlation of the angular momentum components be- are not supposed to be simultaneously diagonisable, i.e. they are
comes: notallowedtohaveacommoneigensystem.Inordertocapturethis
mechanism, Lee&Pen (2000) and Crittendenetal. (2001) have
hLα(x)Lα′(x′)i=a4D˙2+ǫαβγǫα′β′γ′hXβγ(x)Xβ′γ′(x′)i. (68) used an effective, parameterised description of the average mis-
Inthenextstepwereplacethe1dvarianceofthecomponentsL in alignment oftheshear andinertiaeigensystems, gauged withnu-
α
thecorrelationfunctionbythe3dvarianceofthefullvector Lby mericaln-bodydata.
takingthetraceofeqn.(68), In the correlation function of the angular momenta CL(r) =
tr Lt(x)L(x) (c.f.eqn.71),thedependenceofLonthecommuta-
′
CL(r)≡trhLα(x)Lα′(x′)i=hLα(x)Lα(x′)i, (69) tohrX− translaitesintotheasymmetricquadraticform X(x)X(x′)
which has the advantage of being a coordinate-frame indepen- X(x)Xt(x′) withthematrixtransposeinthesecondthermcarryin−g
i
dent quantity and allows the usage of the relation ǫ ǫ = thesignal:Acommoneigensystemoftheinertiaandsheartensors,
αβγ αβ′γ′ beingboth symmetricmatrices,would havetheconsequence that
3 δ δ δ δ for reducing the product of the two ǫ -
ββ′ γγ′− βγ′ β′γ αβγ Xwouldbeasymmetricmatrix,X = X+ = Xt,andthecorrelation
symbols:
(cid:16) (cid:17) functionC (r)wouldvanish.
L
hLα(x)Lα(x′)i=3a4D˙2+ hXβγ(x′)Xβγ(x)i−hXβγ(x)Xγβ(x′)i , (70) In contrast, the signal is maximised, if the shear and inertia
eigensystemsareunaligned,i.e.ifXispurelyantisymmetric,X =
wheretheorder oftheihndicesinthelasttermisinterchaniged. In
matrixnotation,thecorrelationfunctionCL(r)reads: Xm−omanednXtu+m=au0t.oI-nctohraretlcaatsioenXftu=nc(tXio−n)tC=m−axX(r−)=is−trXulyanqdutahderaatnicg:ular
L
CL(r)=trhL(x)Lt(x′)i=3a4D˙2+tr hX(x)Xt(x′)i−hX(x′)X(x)i ,(71) a4D˙2
whichisnon-vanishingforgeneralhasymmetricmatricesXβγduieto CLmax(r)=6n2 (>+ν) dυC(υ) dυ′C(υ′)tr X(υ)Xt(υ′) .(77)
peak Z Z
thematrix transposition inthelast term. Finally, one canexpress h i
theangularmomentumcorrelationarisingfromtheGaussianprob- Thecorresponding1-pointvariancewouldbegivenby:
abilitydensityinthenaturalvariableX,Xβγ =IβσΨσγ: L2 = 1 dυ p(υ) (υ)L2 (υ), (78)
tr L(x)Lt(x) =3 a4D˙2+ (72) h maxi npeak(>ν)Z C max
h ′ i n2 (>ν) × which illustrates the effect of partial alignment of the shear and
peak
inertia eigensystems, reducing the variance compared to the case
dυ (υ) dυ′ (υ′)tr X(υ)Xt(υ′) X(υ′)X(υ) p(υ,υ′). where the shear and inertia eigensystems are maximally mis-
C C − aligned.
Z Z
Inthecalculationoutlinedabohveweaimtoavoidadeciomposition The symmetric contribution X+, which measures the degree
ofthetidalshearandinertiacorrelationsaccordingto of alignment of the inertia and shear eigensystems, causes an
anisotropic deformation of the protogalactic region during the
I(x)Ψ(x)I(x′)Ψ(x′) =P(IΨ)P(IΨ) Ψ(x)Ψ(x′) , (73) courseoflinearstructureformationpriortogravitationalcollapse.
h i | | h i
whichusesthedifferenceincorrelationlengthsoftheΨandI-fields Consequently,thedeterminationofellipticitydistributionsislikely
andisvalidonscalesonwhichtheI-tensorsareuncorrelated. tobeaffectedeven inthestageof linear structureformation, and
predictionsoftriaxialitybasedonpeakshapesinGaussianrandom
fields(Bardeenetal.1986)couldberefinedusinganadaptationof
2.6 Misalignmentoftheshearandinertiaeigensystems theformalismoutlinedabove.
ThetensorX = IΨcanbedecomposedaccordingtoX = X++X−
intoanantisymmetriccontributionX−,definedviathecommutator 2.7 Symmetryofthecancellationmechanism
[I,Ψ],
A possible objection concerning the symmetry of the correlation
1 1
X− ≡ 2[I,Ψ], Xβ−γ = 2 IβσΨσγ−ΨβσIσγ , (74) functionCL(r),
wmiatthritxhetrsaynmspmoseittriyon(X(I−)atn=dXσΨ21(Ia(cid:16)Ψre−syΨmI)mte=tri21c(ΨmIa(cid:17)−triIcΨe)s)=an−dXi−nutondtheer CmLig(rh)t∝beDtthra(cid:16)tXt(hxe)Xmte(cxh′)a−nisXm(xc′o)Xm(pxa)r(cid:17)iEn,gthealignmentsofthesh(7e9a)r
andinertiaeigensystemsoutlinedaboveisonlypresentinthema-
corresponding symmetricmatrixX+ byusing theanticommutator
trix X(x) at position x, which appears transposed in the second
I,Ψ betweeninertiaIandtidalshearΨ: ′ ′
{ } term, but not in the matrix X(x) at position x. The expression,
X+ ≡ 21{I,Ψ}, Xβ+γ = 12 IβσΨσγ+ΨβσIσγ , (75) thro(Xw(exve)rX,(cxa)n be(Xr(exf)oXrm(xul)at)tte)d=ustirn(gX(txr()XX((xx′))X(xX)(−xX)X(x(x)X)t(),x′u)st)in=g
Xσ (cid:16) (cid:17) ′ − ′ ′ − ′
8 B.M. Scha¨ferandPh.M.Merkel
thepropertiesofthetracetr(A+B)=tr(A)+tr(B),tr(AB)=tr(BA) 102
athnedmtre(Acht)an=istmr(Ais),paresswenetllinasth(Aetfi)trs=ttAenasnodra(sAwBe)tll=. BtAtsuchthat 2] σσLL//MM−∝v Mal0u.e64s6 w9±i0th.0 0e1rrors × 100
h) theoretical expectation
AnotherobjectionmightbethatthecorrelationfunctionC (r) /
L c
vanishesifoneofthematricesXissymmetric.Asymmetricshape Mp
ofthematrixX,however, neveroccursintheangular momentum [(
build-up,becauseofthefactthatsymmetricmatricesformagroup M
/L
undermatrixmultiplication:AsymmetricmatrixXcouldonlyhave σ
emergedfromsymmetricmatricesIandΨwithacommoneigen- on101
systemduetoavanishingcommutator[I,Ψ],makingitimpossible ati
vi
forthehalotoacquireangularmomentumasdiscussedearlier,and e
d
consequently,thecorrelationfunctionhastobezero. d
r
a
d
n
a
2.8 NumericsoftheconstraintGaussianintegration st
ThecovariancematrixYintheframegivenbytheyn -coefficients 100
ℓm 1011 1012 1013
is transformed to the physical frame yielding the correlation ma- halomass[M⊙/h]
trixW.Then, wedeterminenumerically thecorrelationfunctions
C (r) = tr Lt(x)L(x) of the angular momentum L as a func- Figure4.Standard deviation σL/M ofthespecificangular momentumas
L h ′ i a function of halo mass, as obtained from numerical sampling (circles)
tion of distance r = x x , by carrying out the integration over
| − ′| including error bars (artificially enlarged byafactor of100), along with
the multivariate Gaussian probability density p(w)dw, subjected
a power-law fit to the data (solid line) and the theoretical expectation
to(wbe)i=ng c(oυn)str(aυin)t.tTohteheinperetaiaksIinotfheandeonbsjietcytfifoerlmdibnygtahteampeaaskk σL/M ∝M2/3(dashedline).
C C C ′ αβ
inthedensityfieldisconsistentlyderivedfromthelocalcurvature
∂ ∂ δ(x)ofthedensityfieldatthepeak. sampleswecouldreproducetheiranalyticalresulttoarbitraryaccu-
α β
Due to the high dimensionality the numerical constraint in- racy.Wecouldalsorecoverthelimitingcaseν .Toprovide
→−∞
tegrationisadifficultandtimeconsuming task.Thistaskismost anerrorestimateforourresultsoftheangularmomentumcorrela-
efficientlyaddressedbyexploitingourpriorknowledgeoftheun- tionfunctionsgiveninthenextsectionwecarriedouteverysam-
derlyingprobabilitydensityfunction:Sincethedistributionisjust plingprocessseveraltimesseedingtherandomnumbergenerator
– tough very high dimensional – Gaussian, it is advantageous to differentlyandcomputedtheresulting1σerror.Forallnumerical
sampletheintegralofeqn.(72)directlyinsteadofrelyingblindly calculationsweassuredthatthenumberofacceptedsamplesisof
onacommonMonte-Carloscheme.Generatingsampleswhichfol- theorderof105foreachstepindistance.
lowthedistributiongivenineqn.(23)andeqn.(24)respectivelyis
straightforward since this can be mapped onto the generation of
unit Gaussian variates, for example via theCholesky decomposi- 3 RESULTS
tionofthecorrespondingcovariancematrix.Gaussian(unit)vari-
ates,however,canbeobtainedfromavarietyofveryfastandeffi- Forvisualisation,wedefinethespecificangularmomentum
cientrandomnumbergenerators,e.g.thezigguratmethod. L
L˜ = , (80)
Following thestrategy of reducing thenumerical integration
H M
0 scale
of eqn. (72)toadirect sampling process whichonly requiresthe
generation of unit Gaussian variates is the most important sim- normalisedbythemassscale Mscale = 43πρcritΩmR3 anddivideout
plification and acceleration in comparison to the use of standard theHubble-constant H0 definingthecosmologicaltimescalesuch
Monte-Carlo techniques. In addition, our method allows to carry thattheresultingquantityhasunitsofasquared lengthscaleand
outthesamplinginthephysicalframewheretheconstraintsCcan depends only on a single fundamental unit, for which we choose
beevaluatedmost easily.Furtheraccelerationcanbeachieved by Mpc/h.
usingsimplelinearalgebrainordertominimizethenumberofdi- AsshowninPeebles(1969)andHeavens&Peacock(1988),
agonalisationsofthemasstensorMαβ=∂α∂βδ.Insteadofchecking thestandarddeviationσL/Mofthisquantityscales∝M2/3,whichis
whetheralleigenvaluesofthemasstensorarenegativewecheckits avaluablecheckforournumericalsamplingcode.Infact,asshown
negativedefinitenessbycombiningtheinvariantstr(Mn),n = 1,2 inFig.4, werecover thisbehaviour toahighdegree of accuracy
andthedeterminantdet(M).Finally,itisimportanttonotethatthe overtwoordersofmagnitudeinhalomass.Afittothedatapoints
actualdimensionalityoftheintegrationisreducedbytherequire- includingtheirsamplingerroryields
mineonutrosfabmepinligngatparopceeaskstihnethdeerdiveantsiviteysfioefldth,ei.ed.enδsDi(t∇yδfi(exl)d).aTrehunso, σL/M = M 0.6469±0.001, (81)
σ M
degree of freedom but rather fixed (namely to be zero). This re- L/M,0 0!
quirestoadjusttheoverallnormalisationofourconstrainedGaus- withσ = 11.362σ (Mpc/h)2 and M = 1012M /h,within3%
L/M 0 0
siansamplingprocessbyanappropriatefactor,i.e.wehavetouse of the theoretical value. For convenience, we hav⊙e divided L/M
thenormalisationoftheunconstrainedGaussiandistribution. in Fig. 4 by σ . Physically, the 1σ value of L/M corresponds to
0
Inordertoinvestigatetheperformance andreliabilityofour a homogeneous sphere with the radius 1 Mpc/h and a mass of
samplingstrategyusedtoevaluateintegralslikethoseofeqn.(64) 1012M /hrevolvingonceevery 109years.
and eqn. (72) we computed in a first step the mean peak density T⊙hecorrelationfunctionCL˜∼(r)ofthespecificangularmomen-
givenineqn.(62)andcomparedourresultwiththeanalyticalso- tumisdepictedinFig.5foramassscaleof1012M /hcorrespond-
lutionderived by Bardeenetal.(1986).Increasing thenumber of ingtoaMilkyWay-sizedhalo.Itisdecreasingrap⊙idlyafteradis-
Galacticangularmomentum couplings 9
105 weightingofthetidalshear-configurationscomparedtoacontinu-
correlation C(r)
L ousfield.Theinertiaof apeakregion isdetermined fromalocal
fitting model
parabolicdensityprofile,andintegratedwithintheboundarygiven
4] byδ = 0,whichmightbeatoocoarseapproximationinthecom-
/h)104 putationoftheinertia,becauseoftheweightingofmasselements
c
p withthesquareofthedistancefromthecentreofgravity.
M
( (iv) Thecovariancematricesusedfordescribingtherelationbe-
[
) tweentheindividualGaussianderivates,areexpressedintheyn -
(r103 ℓm
˜L basis(i.e. inspherical coordinates), and transformed tothe phys-
C
n ical variableswithalinear transformation. Animportant detail is
o
ati the degeneracy of the density field δ with the trace of the tidal
rel102 shear α∂α∂αΨ = ∆ΨduetothePoissonequation. Forthatrea-
r
o sonthedensityfielditself,althoughitisusedforderivingtherela-
c
P
tionsbetweenallvariablesintheGaussianrandomprocess,isnot
partoftherandomprocessitself,butderivedfromthetidalshear.
101 Consequentlythesamplingcanapriorinotberestrictedtovalues
1/3 1 3
distancer[Mpc/h] δ > νσ , but δ needs to be recomputed for each sampling point,
0
andeventually berejected, whichresultsinasamplingefficiency
Figure5.AngularmomentumcorrelationfunctionCL˜(r)foraMilkyWay- amountingto 2.5%forthe2-pointfunction.
sizedhaloof1012M /h(circles)includingsamplingerrors,togetherwith (v) Compu≃ting the standard deviation from of the distribu-
anempiricalfittingfo⊙rmulaoftheformCL˜(r)∝exp(−[r/r0]β)(solidline). tion of the specific angular momentum L/M confirms the ex-
pectedscaling M2/3 proposedbyPeebles(1969)andderivedby
∝
Heavens&Peacock(1988)inananalyticalapproach,andprovides
tance of 1 Mpc/h corresponding to the correlation length of avaluablecheckforourcode.
∼
theangularmomentumfieldandassumestheasymptoticvalueof (vi) TheresultingcorrelationfunctionCL˜(r)ofthespecifican-
CL˜(r) → σ2L˜/npeak forr → 0infulfilmentoftheCauchy-Schwarz gularmomentum L˜ = L/(H0Mscale)(wheretherescalingmakesit
inequalitywithourvaluesforthepeakdensity.Anempiricalfitto possibletostatetheangularmomentuminunitsofasinglefunda-
thespectrumisgivenby mentalunit)isdeterminedbytheinterplayoftwofieldswithdif-
CL˜(r)=aexp −[r/r0]β (82) rfeelraetnitoncos,rraenldattihoentliednagltshhse:aTrhΨeinehratsiathIαeβseaxmheibcitosrrsehloartitornanlegnegdtchoars-
αβ
witha=(2501(cid:16)0 165)σ(cid:17)2(Mpc/h)4,r =(0.8628 0.008)Mpc/h thedensityfield.Theresultingcorrelationsinangularmomentum
± 0 0 ±
(both error bounds correspond to 1σ) and β = 3/2. In addition, have a range of about 1 Mpc/h for Milky Way-sized haloes, and
wefindtheangularmomentaofhaloestobepositivelycorrelated, canbefittedwellwithanempiricalformulaCL(r)∝exp(−[r/r0]β).
i.e.angularmomentumvectorstendtobealignedinaparallelway,
Futureinvestigationswillincludetheapplicationoftheangu-
becausetheformofourcorrelationfunctionwouldbeabletodis-
larmomentumcorrelationfunctionforcomputingellipticitycorre-
tinguishbetweenparallelandantiparallelalignment.
lationfunctions,whichplayanimportantroleingravitationallens-
ing:There,acommonassumptionareuncorrelatedgalaxyshapes,
butcoupledangularmomentagiverisetotheeffectthatthegalac-
ticdisksofneighbouringgalaxiesareviewedundersimilarangles
4 SUMMARY
of inclination, such that their ellipticitiesarenaturally correlated.
In this paper, we recompute the angular momentum correlation Theformalismoutlinedabovecanservetocomputetheellipticity
function arising from tidal torquing in an improved statistical correlationfunction,andfurthermore,theintrinsicshapeE-andB-
model,whichisbasedonapeak-restrictedGaussianrandompro- modecorrelationfunctions,CE(ℓ)andCB(ℓ)ofthe(tensor-valued)
ǫ ǫ
cess. ellipticityfieldǫ–apaperaboutthistopicisinpreparation.
(i) Darkenergyinfluencesangularmomentumaquisitionbythe
timederivativeH(a)dD /daofthegrowthfunctionD (a).Angular
+ +
ACKNOWLEDGEMENTS
momentumbuild-upindarkenergymodelsissignificantlyslower
comparedtotheSCDM-cosmology:Atearlytimestheangularmo- We would like to thank in particular Rob Crittenden, Matthias
menta grow 30% slower in ΛCDM and 20% slower in classical BartelmannandAlanHeavensforvaluable comments, andSarah
quintessencewithw0 = −2/3,withqualitativedifferencesindark Bridlefororganisingtheworkshoponintrinsicalignments.BMS’s
energymodelswithepoch-varyingequationofstatew(a). work is supported by the German Research Foundation (DFG)
(ii) Duetothefact thattheangular momentum arisingintidal withintheframeworkoftheexcellenceinitiativethroughtheHei-
shearing is proportional to the commutator X− = [I,Ψ] between delberg Graduate School of Fundamental Physics, and PMM ac-
the inertiaI and the tidal shear Ψ theangular momentum field is knowledges funding from the Graduate Academy Heidelberg as
sensitivetotherelativemisalignmentsoftheprincipalaxissystems wellasfromtheGraduateSchoolofFundamentalPhysics.
ofthetidalshearandinertiatensors.Forthatreasonbothfieldsare
includedintheGaussianrandomprocess.
(iii) Theangularmomentumcorrelationfunctionandvarianceis
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