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PreprinttypesetusingLATEXstyleemulateapjv.5/14/03 MASSFUNCTIONOFLOWMASSDARKHALOS HIDEKIYAHAGI1 DivisionofTheoreticalAstrophysics,NationalAstronomicalObservatoryofJapan,Tokyo181-8588,Japan MASAHIRONAGASHIMA DepartmentofPhysics,UniversityofDurham,SouthRoad,DurhamDH13LE,U.K. AND YUZURUYOSHII2 InstituteofAstronomy,UniversityofTokyo,2-21-1Osawa,Mitaka,Tokyo181-0015,Japan 4 ABSTRACT 0 0 The mass function of dark halos in a Λ-dominated cold dark matter (ΛCDM) universe is investigated. 529 2 output files from five runs of N-body simulations are analyzed using the friends-of-friends cluster finding algorithm. Alltherunsuse5123 particlesintheboxsizeof35h- 1Mpcto140h- 1Mpc. Massofparticlesfor n a 35h- 1Mpcrunsis2.67 107h- 1M⊙. Becauseofthehighmassresolutionofoursimulations,themultiplicity × J functionin the low-massrange, where the mass is wellbelow the characteristicmass and ν =δc/σ.1.0, is 8 evaluated in the present work, and is well fitted by the functionalform proposed by Sheth & Tormen (ST). However,themaximumvalueofthemultiplicityfunctionfromoursimulationsatν 1issmaller,anditslow ∼ 1 masstailisshallowerwhencomparedwiththeSTmultiplicityfunction. v Subjectheadings:cosmology:theory—darkmatter—galaxies:luminosityfunction,massfunction 7 9 0 1. INTRODUCTION of the spherical collapse model (Audit,Teyssier,&Alimi 1 1997; Epstein 1984; Lee&Shandarin 1998; Monaco 1995; The mass function is one of the most important statisti- 0 Sheth,Mo,&Tormen 2001; Sheth&Tormen 2002), an cal quantities of dark halos. Press&Schechter (1974) in- 4 agreementwas improved between the numerical mass func- vented an ansatz to predict the mass function of dark ha- 0 tion and the analytic mass function, especially the ST mass / los formed through hierarchical clustering, assuming that h eachsmoothedmasselementwitharbitrarysmoothinglength function proposed by Sheth&Tormen (1999). In addition, p themassfunctionofclusterprogenitorshasalsobeenstudied evolves independently, in accordance with the spherical - (Okamoto&Habe1999). o top-hat model. The PS formalism was extended so that Ontheotherhand,thenumericalmassfunctiondependson r the merging history of each halo is traceable (Bondetal. t 1991; Bower 1991; Lacey&Cole 1993). This extended the cluster finding algorithm adopted (Gelb&Bertschinger s 1994; Governatoetal. 1999; Lacey&Cole 1993). a PS formalism has been used in a wide range of applica- : tions. Especially the so-called semi-analytic galaxy mod- Jenkinsetal. (2001, hereafter J01) demonstrated that v the friends-of-friends (FoF) algorithm (Davisetal. 1985) els(Coleetal.2000;Kauffmann,White,&Guiderdoni1993; i X Nagashimaetal.2001;Somerville&Primack1999)useitto withafixedlinkinglengthof0.2timesthemeanparticlesep- arationresultsinthenumericalmassfunctionthatshowsthe r successfullyreproducemanypropertiesofobservedgalaxies a The PS mass function and the mass function from N- best universality for various compositions of cosmological parametersandboxsizes. Thus,itisbettertosetthethreshold body simulations agree with each other only qualitatively density proportional to the background mass density rather (Brainerd&Villumsen 1992; Efstathiouetal. 1988), and than to the critical density. White (2002, hereafter W02) modification to threshold linear overdensity leads to a bet- supported the above argument and demonstrated that an teragreement(Efstathiou&Rees1988;Gelb&Bertschinger other definition of mass of halos, M , gives the universal 1994; Lacey&Cole 1994). However, the PS mass 180b numerical mass function. Here, M is the mass within a function gives the smaller number of high-mass halos 180b sphere whose average density is 180 times the background (Grossetal. 1998; Jain&Bertschinger 1994), while giv- density. ing the larger number of low-mass halos when com- However, different authors support different mass func- pared with the N-body mass functions (Governatoetal. tions. Forexample,J01proposedanewfittingformulatothe 1999; Somervilleetal. 2000). This tendency was con- universal mass function, while W02 supported the ST mass firmed over a mass range by connecting the numerical function(seealsoReedetal.2003). Sincethemassfunction mass functions from simulations with different box sizes ofhigh-masshaloshasextensivelybeenstudiedbythem,this (Grossetal. 1998; Jenkinsetal. 2001). Taking into account discrepancyispossiblyduetothelackofinformationonthe the spatial correlation of density fluctuations (Nagashima behaviorofthenumericalmassfunctionforlow-masshalos. 2001; Yano,Nagashima,&Gouda 1996), or incorporating Thus,inordertoinvestigatethefunctionalformoftheuniver- the ellipsoidal collapse model into the PS ansatz instead salmassfunction,weperformedfiverunsofN-bodysimula- Electronicaddress:[email protected] tionswithhighmassresolution. Electronicaddress:[email protected] In this paper, we briefly describe our simulation code and 1NAOJScienceResearchFellow cosmologicaland simulation parameters adopted in §2. Re- 2 Research Center for the Early Universe, University of Tokyo, 7-3-1 sults of the simulations as well as the parameter values for Hongo,Bunkyo-ku,Tokyo113-0033,Japan 2 Yahagi,Nagashima,&Yoshii thebest-fitmassfunctionaregivenin§3,anddiscussionsare 3. SIMULATIONSANDRESULTS givenin§4. We used the Adaptive Mesh Refinement N-bodycode de- veloped by Yahagi (2002), which is a vectorized and par- 2. MULTIPLICITYFUNCTION allelized version of the code described in Yahagi&Yoshii The multiplicity function is the differential distribution (2001). All five runs of simulations we performed adopt functionof the normalizedfluctuationamplitudeof darkha- the ΛCDM cosmologicalparameters of Ωm =0.3, Ωλ =0.7, los for each mass element. According to the definition by h=0.7, and σ =1.0, using 5123 particles in common. The 8 Sheth&Tormen (1999), the multiplicity function is defined sizeofthefinestmeshis1/64ofthebasemesh,andtheforce as dynamicrangeis 215=32768. Othersimulationparameters, suchastheboxsizeandtheparticlemassaregiveninTable1. n(M,z)dlogM νf(ν)=M2 , (1) InitialconditionsweregeneratedbytheGRAFIC2codepro- ρ¯ dlogν videdbyBertschinger(2001)usingthepowerspectrumgiven by Bardeenetal. (1986). Five runs produced 529 files, and wheren(M,z)isthenumberdensityofdarkhalos,ν=δ /σ c M eachofthemwasanalyzedbytheFoFalgorithmwithacon- is the peak height of a halo, δ is the linear overdensity at c stantlinkinglengthof0.2timesthemeanparticleseparation. the collapse epoch of halos given by the spherical collapse Details of the simulations will be given in Yahagi et al., in model, and σ is the standard deviation of the density fluc- M preparation. tuationfieldsmoothedbythetop-hatwindowfunction. Note The mass functions of dark halos are shown in Figure 1. thatthetimeevolutionofthenumberdensityandthedepen- Theupperandlowerpanelsshowthemassfunctionsatz=3 dence of the number density on the initial power spectrum andz=0, respectively. The PSmass functionisrepresented are absorbedin ν. Since this universalityof the multiplicity bysolidlines, andtheSTmassfunctionbydashedlines. At functionis guaranteedunderthe PS ansatz, we compareour both redshifts, the numericalmass functionsfrom our simu- simulationresultswithanalyticmultiplicityfunctions. lations agree with the ST mass function in a mass range of Along with the PS ansatz, Press&Schechter (1974) pro- 1010M .M.1013M . posedthefollowinganalyticmultiplicityfunction: ⊙ ⊙ Thenumericalmultiplicityfunctionsareshownbycrosses infivepanelsofFigure2. Allthedatafromtheinitialredshift 2 PS: νf(ν)= νexp(- ν2/2). (2) to the present z=0 is compiled to draw the average curves rπ (crosses)witherrorbarsindicatingtheepochtoepochvaria- This PS multiplicity function has successfully predicted the tion. Inthepanel(f),allthenumericalmultiplicityfunctions numericalmultiplicityfunctioninaqualitativemanner.How- are shownby thin lines. Darkhalosthat consistof less than ever,Sheth&Tormen(1999)proposedanalternativeanalytic 600particlesarenotusedincalculatingthemultiplicityfunc- multiplicityfunctionthatcouldbetterreproducethe numeri- tion, and 1/64 dex-sized bins containingless than 100 halos calmultiplicityfunction: are excluded to avoid the contamination of the rare objects. Three analytic multiplicity functions described in the previ- 2 ous section are also shown in this figure, that is PS (solid ST: νf(ν)=A(1+ν′- 2p)rπν′exp(- ν′2/2), (3) lines), ST (dashed lines), and J01 (dotted lines). The best- fitfunctionsbasedontheSTfunctionalformarealsoshown where ν′2 =aν2, and A is a normalization factor defined so bydot-dashedlines. Thebest-fitparametersaregiveninTa- ∞ that f(ν)dν=1. Becauseofthisunityconstraint,Aisnot ble2 andare veryclose tothoseof W02. Since thedataare 0 aninRdependentparameter,butisexpressedinformsof p: availableonlyintheregionatν.3,thesefunctionscouldbe erroneousatν&3. A= 1+2- pπ- 1/2Γ(1/2- p) - 1. (4) Inmostcases,thenumericalmultiplicityfunctionsandthe best-fitfunctionsto themare consistentwith theST and J01 h i multiplicityfunctionsatν&3. However,eachofthenumeri- Sheth&Tormen(1999)gavethebest-fitparametervaluesof calmultiplicityfunctionsresidebetweentheSTandJ01func- a=0.707,p=0.3,andA=0.322.ThePSmultiplicityfunction tions at 1.5.ν .3, and is below the ST function at ν .1 isincludedin thisSTmultiplicityfunctionwitha=1,p=0, exceptforthe35brun. Thenumericalmultiplicityfunctions andA=1/2. haveanapparentpeakatν 1,insteadofaplateauasseenin TheST multiplicityfunctionhasa maximumat ν =νmax ∼ theJ01function. Wehereproposedthefollowingfunctionto thatsatisfiesthefollowingequation: fittothenumericalmultiplicityfunction: (νm′ ax2p+1)(νm′ ax2- 1)+2p=0, (5) νf(ν)=A[1+(Bν/√2)C]νDexp[- (Bν/√2)2], (7) where νm′ ax2=aνmax2. It is trivialto see νmax=1 for the where, A is a normalization factor to satisfy the unity con- caseofthePSmultiplicityfunction. straint, ∞ f(ν)dν=1,therefore 0 J01 also proposed an analytic multiplicity function which R givesafittotheirnumericalmultiplicityfunction: A=2(B/√2)D Γ[D/2]+Γ[(C+D)/2] - 1. (8) { } J01: νf(ν)=0.315exp(- 0.61+logν- logδ 3.8). (6) The best-fit parameters are given as B=0.893, C=1.39, and c | | D=0.408,andfromtheseparameters,Aisconstrainedsothat Thevalidrangeofthisfittingformulais- 1.2<logν- logδ < A=0.298. Thisbest-fitfunctionfromequation7isshownin c 1.05. Hereafter we assume that δ in the above equation is Figure3andisonlyvalidat0.3 ν 3. c ≤ ≤ constantandtakesthevalueintheEinstein-deSitteruniverse, Since the unity constraint is only an assumption, we can i.e. δ =(3/20)(12π)2/3 1.686inordertocomparetheJ01 relax this constraint and treat A as a free parameter. Actu- c ∼ multiplicityfunctionwithotherfunctions. ally, almost a half of the particles are not bound to any ha- Massfunctionoflowmassdarkhalos 3 los as shown in Table 3. On the other hand, the best-fit pa- takevaluesbetweentheSTandJ01functions.Althoughthese rameters without the unity constraint using equation 3 are differencesarewithin1σerrorbars,theyarepossiblydueto A = 0.320, B = 0.664, C = 1.99, and D = 0.36, for which thedifferentboxsizesadopted. Sheth&Tormen(1999)used ∞ f(ν)dν = 1.2713. Since the fraction of bound particle thedatafromtheGIFsimulations(Kauffmanetal.1999)with Rsh0ouldnotexceedunity,weassigntheunityconstrainttoall theboxsizeof144h- 1Mpcorless,andW02mainlyusedthe thebest-fitfunctionsbelow. data from the box size of 200h- 1Mpc. On the other hand, We also investigate the peak position of the multiplicity J01usedthe3000h- 1Mpcsimulationatmaximum. Wehave functions. Since deriving νmax directly from the numer- takenintoaccountthebox-sizeeffectusingtheconditionalPS ical multiplicity function is difficult due to the numerical multiplicity function (equation 9) but this effect is found to fluctuation around the maximum value, we derived νmax betoosmalltoresolvethisproblem. Introducingtheestima- from the best fit ST function by solving equation 5. These tion using the conditionalmultiplicity function based on the νmax are given in Table 2, and are very close to unity. For unconditional multiplicity function which fits the numerical reference, νmax = 0.916 for the J01 multiplicity function, multiplicity function well, such as the ST multiplicity func- and νmax =0.881 for the multiplicity function proposed by tion,wouldresolvethisproblem.However,suchanimproved Lee&Shandarin(1998). estimationofthebox-sizeeffectmightbeweakerthanthees- We also checked the time dependence of the multiplicity timationbasedonthePSfunction,becausetheunconditional function. Figure 4 shows the multiplicity function from the STfunctionatν 1hasabroadpeakthatisbelowthatofthe ∼ 35a run, for four redshift ranges of 0 z<1 (circles), 1 PS function. The fact that our numerical multiplicity func- ≤ ≤ z<3 (squares), 3 z<6 (triangles), and z 6, (crosses). tionskeeptheuniversalitysupportsthislineofargument. ≤ ≥ Athighredshifts, high-ν halosintheexponentialpartofthe Thus, there are two discrepancies remained. One is the best-fit ST function and equation 7 are probed. As redshift discrepancybetween the numericaland analyticalmultiplic- decreases,theprobewindowmovestothelower-νregion. ity functions. Althoughournewlyproposedfunctionalform Since there are no modes of density fluctuations whose (equation7)providesabetterfitwhencomparedwiththeST wavelengthislargerthantheboxsizeonwhichtheperiodic functional form (equation 3), we need an analytic function boundary is placed. The numerical multiplicity function is based on a theoretical background which fits the numerical conditionalsuch that f(ν δ =0), where δ is the density multiplicity function even better. The other one is the dis- box box | contrast smoothed over a mass scale comparable to the box crepancyinthenumericalmultiplicityfunctionsfromvarious size. Sinceourboxsizeissmaller thanthatofothergroups, simulationruns. Therearethreestrategiestoresolvethisdis- wehaveestimatedthisbox-sizeeffectcomparingtheuncon- crepancy. The first is to run simulations having still higher ditional PS multiplicity function using the sharp k-space fil- massdynamicrangefreefromtheboxsizeeffect.Thesecond ter with the conditionalone (Bondetal. 1991; Bower 1991; istoincreasethenumberofrealizationsasW02did,because Lacey&Cole1993), thereisascatterfromtherunsusingthesameboxsize. The thirdistorunsimulationswhoseboxsizeissmallerthanthat νf = 2 ν- ǫνbox exp - (ν- ǫνbox)2 , (9) ofthepresentwork, althoughitmightsoundcontradictorily. c rπ(1- ǫ2)3/2 (cid:20) 2(1- ǫ2) (cid:21) Fromsimulationswithsmallerboxsize,wewillobtainthein- formationon the conditionalmultiplicityfunctionwhichco- whereǫ=σ /σ.Settingν =0,thiseffectisfoundtobenot box box incideswiththeunconditionalmultiplicityfunctionatν 1. solargeatν.3fortheboxsizeof35h- 1Mpc,anditmakes Comparingtheunconditionalmultiplicityfunctionfrom≪sim- theuniversalityofthemultiplicityfunctionevenworse. ulationswithalargeboxsizeandtheconditionalmultiplicity function from those of a small box size will offer not only 4. DISCUSSION the clues to resolve the above mentioned discrepancies, but We have performed five runs of N-body simulations with alsoinsightsintothemechanismhowthePSansatzworksto highmassresolutioninordertostudythebehaviorofthenu- reproducethenumericalmultiplicityfunction. merical multiplicity function in the low-mass range, or the low-νregion.Throughoutthepeakrangeof,0.3 ν 3,the ≤ ≤ STfunctionalformprovidesagoodfittothemwithparameter Simulations described in this paper were carried out us- valuesof a=0.664,p=0.321, and A=0.301. These values ing Fujitsu-made vector parallel processors VPP5000 in- are very close to those of W02. Our numerical multiplicity stalled at the Astronomical Data Analysis Center, National functionshaveapeakatν 1asintheSTfunction,instead Astronomical Observatory of Japan (ADAC/NAOJ), under ∼ ofaplateauintheJ01function. the ADAC/NAOJ large scale simulation projects (group-ID: However,somedetaileddiscrepanciesareseenbetweenour myy26a, yhy35b). HY would like to thank Joseph Silk and numericalmultiplicityfunctionsandtheSTandJ01analytic Masahiro Takada for their useful comments. MN acknowl- functions. First, in the low-ν region of ν .1, our numeri- edges support from a PPARC rolling grant for extragalactic calmultiplicityfunctionssystematicallyfallbelowtheSTand astronomy and cosmology. This work has been supported theJ01functions,whiletheyareconsistentwiththatofW02. partly by the Center of Excellence Research (07CE2002)of On the other hand, in the high-ν region, where ν is signifi- the Ministry of Education, Science, Culture, and Sports of cantlylargerthanunity,ournumericalmultiplicityfunctions Japan. REFERENCES Audit,E.,Teyssier,R.&Alimi,J.-M.1997,A&A,325,439 Cole,S.,Lacey,C.G.,Baugh,C.M.,&Frenk,C.S.2000,MNRAS,319, Bardeen,J.M.,Bond,J.R.,Kaiser,N.,&Szalay,A.S.1986,ApJ,304,15 168 Bertschinger,E.2001,ApJS,137,1 Davis,M.,Efstathiou,G.,Frenk,C.S.,&White,S.D.M.1985,ApJ,292, Bond,J.R.,Cole,S.,Efstathiou,G.,&Kaiser,N.1991,ApJ,379,440 371 Bower,R.G.1991MNRAS,248,332 Efstathiou,G.,Frenk,C.S.,White,S.D.M.,&Davis,M.1988,MNRAS, Brainerd,T.G.,&Villumsen,J.V.1992,ApJ,394,409 235,715 4 Yahagi,Nagashima,&Yoshii Efstathiou,G.,&Rees,M.J.1988,MNRAS,230,5p Nagashima,M.,Totani,T.,Gouda,N.,&Yoshii,Y.2001,ApJ,557,505 Epstein,R.I.1984,ApJ,281,545 Okamoto,T.,&Habe,A.1999,ApJ,516,591 Gelb,J.M.,&Bertschinger,E.1994,ApJ,436,467 Press,W.H.,&Schechter,P.1974,ApJ,187,425 Governato, F.,Babul, A.,Quinn, T.,Tozzi, P.,Baugh, C. 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M. 1999, Somerville,R.S.,&PrimackJ.R.1999,MNRAS,310,1087 MNRAS,303,188 White,M.2002,ApJS,143,241 Kauffmann,G.,White,S.D.M.,&Guiderdoni,B.1993,MNRAS,264,201 Yahagi,H.2002,DoctoralThesis,UniversityofTokyo Lacey,C.,&Cole,S.1993,MNRAS,262,627 Yahagi,H.,&Yoshii,Y.2001,ApJ,558,463 Lacey,C.,&Cole,S.1994,MNRAS,271,676 Yano,T.,Nagashima,M.,&Gouda,N.1996,ApJ,466,1 Lee,J.,&Shandarin,S.F.1998,ApJ,500,14 Monaco,P.1995,ApJ,447,23 Nagashima,M.2001,ApJ,562,7 Massfunctionoflowmassdarkhalos 5 101 PS 100 ST 35a 35b -3Mpc] 10-1 771004ab0 3M [h 10-2 g10 dlo 10-3 M)/ < dn( 10-4 10-5 (a) z=3 PS 100 ST 35a 35b -3pc] 10-1 771004ab0 M 3M [h 10-2 g10 dlo 10-3 M)/ n( d 10-4 10-5 (b) z=0 10-6 109 1010 1011 1012 1013 1014 1015 1016 M [h-1M8 ] FIG.1.— Themassfunctionsofdarkhalosat(a)z=3and(b)z=0.TheSTmassfunction(dashedline)agreeswiththenumericalmassfunctions(symbols) fairlywellatbothredshifts,whilethePSmassfunction(solidline)doesnotagreewiththem. 6 Yahagi,Nagashima,&Yoshii 0.5 PS PS PS ST ST ST (a) J01 (b) J01 (c) J01 0.4 ST-fit ST-fit ST-fit 35a 35b 70a 0.3 0.2 0.1 0 PS PS PS ST ST ST (d) (e) (f) J01 J01 J01 0.4 ST-fit ST-fit ST-fit 70b 140 all 0.3 0.2 0.1 0 0.5 1 2 3 0.5 1 2 3 0.5 1 2 3 FIG. 2.— Thenumericalmultiplicityfunctionsfromfiverunsofoursimulations(crosseswitherrorbars)areshowninthepanels(a-e),exceptforthepanel (f)whichshowstheresultsfromalltheruns(thinlines). Thenumericalmultiplicityfunctionsarederivedbycompilingallthedatafromtheinitialredshiftto thepresentz=0. Crossesanderrorbarsrepresenttheiraverageandrms,respectively. AlsoshownineachpanelarethePSmultiplicityfunction(solidline), theSTmultiplicityfunction(dashedline),theJ01multiplicityfunction(dottedline),andthebest-fitfunctionusingtheSTfunctionalform(dot-dashedline)for whichadoptedparametersaregiveninTable2.TheJ01multiplicityfunction,originallygivenasafunctionofσ(Jenkinsetal.2001),isexpressedhereinterms ofν=δc/σ,assumingthatδcisconstantalthoughitvariesslightlyintheΛCDMuniverse. Inthehigh-massrange(ν>1),thenumericalmultiplicityfunctions residebetweentheSTandJ01functions,anditsmaximumvalueatν∼1isbelowthoseoftheSTandJ01functions. Massfunctionoflowmassdarkhalos 7 0.35 ST J01 0.3 ST-fit Eq. 7 35a 0.25 35b 70a 70b 0.2 140 0.15 0.1 0.05 0 0.5 1 2 3 FIG. 3.— Thebest-fitmultiplicity functionusingequation 7(solidline). Thisfunctionwellrepresents thenumerical multiplicity functions indicated by symbolswitherrorbars.AlsoshownforcomparisonaretheST(dashedline),J01(dottedline),andbest-fitST(dot-dashedline)functions.For1.5.ν.3,all thenumericalfunctionsresidebetweentheSTandtheJ01functions.Eventhebest-fitSTfunctioncannotrepresentthemultiplicityfunctionswellinthisregion. 8 Yahagi,Nagashima,&Yoshii FIG.4.— Timedependenceofthemultiplicityfunctionfromthe35arun;0≤z<1(circles),1≤z<3(squares),3≤z<6(triangles),andz≥6(crosses). AlsoshownarethebestfitfunctionforalltherunsusingtheSTfunctionalform(dot-dashedline)andequation7(solidline).Athighredshifts,high-νhalosin theexponentialpartoftheSTandequation7areprobed.Asredshiftdecreases,theprobewindowmovestothelower-νregion. Massfunctionoflowmassdarkhalos 9 TABLE1. SIMULATIONPARAMETERS Model L[h- 1Mpc] mptcl[h- 1M⊙] zstart 35a.. 35 2.67×107 50 35b.. 35 2.67×107 50 70a.. 70 2.13×108 41 70b.. 70 2.13×108 41 140.. 140 1.70×109 33 NOTE. —AllthesimulationsadoptthecosmologicalparametersofΩm=0.3,Ωλ=0.7,h=0.7,andσ8=1.0. Thenumberofparticlesis5123andtheforcedynamicrangeof simulationsis215=32768incommon.Listheboxsize,mptclistheparticlemass,andzstartistheinitialredshift. 10 Yahagi,Nagashima,&Yoshii TABLE2. BEST-FITPARAMETERS Model a p A νmaxa 35a.. 0.665 0.317 0.305 0.998 35b.. 0.715 0.303 0.320 0.975 70a.. 0.666 0.327 0.293 0.988 70b.. 0.614 0.325 0.296 1.031 140.. 0.658 0.321 0.300 0.999 all... 0.664 0.321 0.301 0.996 PS... 1 0 0.5 1 ST... 0.707 0.3 0.322 0.983 W02. 0.64 0.34 — 0.995 NOTE.—TheSTfunctionalforminequation3isusedtofittothenumericalmultiplicityfunction. aThevalueofνatwhichthebest-fitSTfunctionattainsamaximum.