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Mon.Not.R.Astron.Soc.000,1–??(XXXX) Printed12January2015 (MNLATEXstylefilev2.2) Clues on void evolution III: Structure and dynamics in void shells Andre´s N. Ruiz1,2(cid:63), Dante J. Paz1,2, Marcelo Lares1,2, Heliana E. Luparello1, Laura Ceccarelli1,2, Diego Garc´ıa Lambas1,2 1InstitutodeAstronom´ıaTeo´ricayExperimental,CONICET–UNC,Laprida854,X5000BGR,Co´rdoba,Argentina. 2ObservatorioAstrono´micodeCo´rdoba,UniversidadNacionaldeCo´rdoba,Laprida854,X5000BGR,Co´rdoba,Argentina. 5 1 0 AcceptedXXX.ReceivedXXX;inoriginalformXXX 2 n a ABSTRACT J 9 Inspiredonthewellknowndynamicaldichotomypredictedinvoids,wheresomeunder- denseregionsexpandwhereasotherscollapseduetooverdensesurroundingregions,weex- ] ploredtheinterplaybetweenthevoidinnerdynamicsanditslargescaleenvironment.Theen- O vironmentisclassifieddependingonitsdensityasinpreviousworks.Weanalysethedynam- C icalpropertiesofvoid-centeredsphericalshellsatdifferentvoid-centricdistancesdepending . onthisclassification.Theabovedynamicalpropertiesaregivenbytheangulardistributionof h p theradialvelocityfield,itssmoothness,thefielddependenceonthetracerdensityandshape, - and the field departures from linear theory. We found that the velocity field in expanding o voidsfollowsmorecloselythelinearprediction,withamoresmoothvelocityfield.However r t when using velocity tracers with large densities such deviations increase. Voids with sizes s around 18h−1Mpc are in a transition regime between regions with expansion overpredicted a [ and underpredicted from linear theory. We also found that velocity smoothness increases as thevoidradius,indicatingthelaminarflowdominatestheexpansionoflargervoids(morethan 1 18h−1Mpc).Thecorrelationsobservedsuggestthatnonlineardynamicsoftheinnerregionsof v voidscouldbedependentontheevolutionofthesurroundingstructures.Thesealsoindicate 0 possiblescalecouplingsbetweenthevoidinnerexpansionandthelargescaleregionswhere 2 voids are embedded. These results shed some light to the origin of nonlinearities in voids, 1 goingbeyondthefactthatvoidsjustquicklybecomesnonlinearastheybecomeemptier. 2 0 Keywords: large–scalestructureoftheuniverse-methods:numerical-methods:statistical . 1 0 5 1 : 1 INTRODUCTION studiedandcharacterizedbyseveralauthorsinavarietyofwave- v lengths(Pellegrinietal.1989;Slezaketal.1993;El-Ad&Piran i Voidsareprominentfeaturesofthelarge-scalestructureoftheUni- X 1997;El-Adetal.1997;Aikio&Ma¨ho¨nen1998;El-Ad&Piran verse since they are surrounded by elongated filaments, sheetlike 2000;Mu¨lleretal.2000;Arbabi-Bidgoli&Mu¨ller2002;Plionis& r wallsanddensecompactclusters,settingthepatternofthecosmic a Basilakos2002;Hoyle&Vogeley2002,2004;Hoyleetal.2005; web(Collessetal.2001;Collessetal.2003;Tegmarketal.2004; Colbergetal.2005b;Ceccarellietal.2006;Patirietal.2006;Shan- Huchraetal.2005).Also,voidsarelikelytodeterminelarge-scale darin et al. 2006; Platen et al. 2007; Brunino et al. 2007; Hahn velocityfieldsbytheirradialflowsofmassandgalaxiesinducedby etal.2007;Neyrinck2008;Foster&Nelson2009;Lavaux&Wan- the local mass underdensity (Bertschinger 1985; Melott & Shan- delt2010;Tavasolietal.2013;Elyivetal.2014)andshowsimilar darin1990;Mathis&White2002;Colbergetal.2005a;Shandarin properties,eventhoughavarietyofidentificationmethodsanddata etal.2006;Platenetal.2008;Aragon-Calvo&Szalay2013;Patiri samplesareused(Colbergetal.2008).Also,thevoidphenomenon etal.2012;Pazetal.2013;Michelettietal.2014).Thespatialand hasbeenatargetofmanytheoreticalanalysisinnumericalsimu- dynamicalstatisticsofcosmicvoidsprovidecriticaltestsofstruc- lations(Hoffman&Shaham1982;Hausmanetal.1983;Fillmore tureformationandcosmologicalmodels(e.g.Peebles2001;Ben- & Goldreich 1984; Icke 1984; Bertschinger 1985; Kauffmann & sonetal.2003;Parketal.2012;Kolokotronisetal.2002;Colberg Fairall1991;Padillaetal.2005;Aragon-Calvoetal.2010;Aragon- etal.2005b;Lavaux&Wandelt2010;Biswasetal.2010;Bosetal. Calvo&Szalay2013). 2012a;Parketal.2012;Bosetal.2012b;Herna´ndez-Monteagudo &Smith2013;Clampittetal.2013). Asvoidsexpand,matterissqueezedinbetweenthemandthe The voids in the galaxy distribution have been extensively walls,filamentsandclusters,generatingtheevolutionvoidbound- aries(Martel&Wasserman1990;Regos&Geller1991;Dubinski etal.1993;vandeWeygaert&vanKampen1993;Bondetal.1996; (cid:63) e-mail:[email protected] Goldbergetal.2005;Padillaetal.2005;Cautunetal.2014). (cid:13)c XXXXRAS 2 Andre´sN.Ruizetal. Moreover, at early times the large-scale velocity field out- odsandpresenttheresultsoftheanalysisofthedynamicsofvoid flowingfromunderdenseregionsisassociatedwiththelinearcol- shells in Section 3. In Subsection 3.1 we present the analysis of lapseofoverdenseregionsandshapedbythetidalinfluenceofthe voidshellsonthebasisofanequalareapixelizationscheme,dis- surroundingmassdistribution(TidalTorqueTheoryDoroshkevich tinguishingpixelsaccordingtotheirlocaldensity.Thekinematics 1970;Peebles1969;White1984;Porcianietal.2002b,a).Thispro- ofregionsdefinedbythedifferentsetsofpixelsareareanalyzed cess imprints correlations between the inner dynamics of haloes inSection3.2,whereweusethesesetstocharacterizethenetwork andtheirhostfilamentsandvoiddistributionaswell(Patirietal. structureandthestratumoflowdensitiyregions.Acomparisonof 2006;Pazetal.2008;Pazetal.2011;Schneideretal.2012;Smar- thestructuresdefinedbythedifferentsetsismadeonthebasisof gonetal.2012;vanDaalenetal.2012;Zhangetal.2013;Forero- thepropertiesofminimalspanningtreesonSections3.3and3.4. Romeroetal.2014).However,thisinterplayisnotonlyoriginated Finally,inSection4wepresenttheconclusionsofourresults. bythelineargrowthofinitialfluctuations,butalsoaffectedbythe vorticityofthecosmicvelocityfieldwhicharisesinthenonlinear phase(Pichon&Bernardeau1999;Laigleetal.2013;Wangetal. 2 VOIDCATALOGUE 2014).Theantisymmetriccomponentofthevelocitydeformation tensoralsoplaysamajorroleinshapingthecosmicweb,producing 2.1 TheN-bodysimulation preferreddirectionsfordarkmatterhaloesandgalaxies(Libeskind In this work we use a dark matter simulation of 5123 particles etal.2014). evolved from an initial redshift of z ∼ 50 to the present time in In previous works we examined the distribution of galaxies acomovingboxofsidelengthL=500h−1Mpc.Thecosmological around voids in the Sloan Digital Sky Survey (SDSS) and per- parameterscorrespondtoaflatΛCDMmodelconsistentwiththe formed a statistical study of the void phenomenon focussing on WMAP7estimations(Jarosiketal.2011),withamatterdensitypa- vvooiidd ednyvniarmonicmse(nPtsaz(Ceetcacla.r2e0ll1i3e,thael.re2a0ft1e3r,PhaepreerafItIe)r.PTaopethraIt)eanndd, rham=e0t.e7r1Ω9man=da1.n0o−rmΩaΛliz=at0io.2n5p8a,raamdiemteernσsio=nle0s.7s9H6u.bTbhleerceosnuslttianngt wSDeShSavbeyecxoammpinuetdintghethdeiisrtrriebdusthioifntsopfagcaeladxeinessitayropurnodfilveo(iPdsapinerthI)e, massparticleismp=6.67×1010h−1M(cid:12).Th8einitialconditionswere generated using GRAFIC2 (Bertschinger 2001) and the simula- and recovering their underlying real space density and velocity tionwasevolvedusingthepublicversionofGADGET-2(Springel profiles (Paper II). The real space profiles were inferred through 2005).TheidentificationofhaloeswasperformedusingaFriends- modeling void-galaxy redshift space distortions on the two-point of-Friends algorithm (Huchra & Geller 1982; Davis et al. 1985, correlation function (Paper II). Sheth & van de Weygaert (2004) hereafterFoF)withapercolationlengthgivenbyl=0.2n¯−1/3,be- presentedtheoreticalfoundationsthatestablishedadynamicaldi- ingn¯ themeannumberdensityofdarkmatterparticles.Thefinal chotomy of voids. According to the authors, there are two dis- cataloguecontains505126haloeswithatleast10particles. tinctvoidbehavioursdependingontheirenvironment:thesocalled “void-in-void” and “void-in-cloud” process. The void-in-void re- gions are embedded in larger-scale underdensities and show ex- 2.2 Voididentification pandingvelocityprofiles.Ontheotherhand,void-in-cloudregions are surrounded by larger overdense environments which undergo Weperformtheidentificationofvoidsusingamodifiedversionof in a former collapse, shrinking at later times the embedded void thealgorithmpresentedbyPadillaetal.(2005)andCeccarellietal. region.Thislastcaseseemstoaffectmorelikelysmallratherthan (2006).Theidentificationisdoneaccordingtothefollowingsteps: largevoids.Themaingoalofourpreviousstudieswastoconfirm (i) Using the halo catalogue as structure tracer, we perform a bythefirsttimeonobservationsthisdynamicalclassification. Voronoi tessellation using the public library VORO++1 (Rycroft WedefinedinPaperIaseparationcriterionbasedonthered- 2009). This allow us to estimate the density field as the inverse shiftspaceintegrateddensityprofiletocharacterizevoidsaccord- oftheVoronoivolumeforeachcell. ingtotheirsurroundingenvironment.Suchcriteriahaveshownsuc- (ii) Thecandidatestounderdenseregionsareselectedatthecen- cessinseparatingexpandingandcontractingvoids(PaperII)and troid position for each cell with estimated overdensity satisfying allow us to define two characteristic void types: (i) voids with a δ<−0.8. density profile indicating an underdense region surrounded by an (iii) Centredoneachcandidateposition,wecomputeiteratively overdenseshell,dubbedS-typevoids,whichbehaveslikea“void- theintegrateddensitycontrast(∆)insidespheresofincreasingra- in-cloud”process;and(ii)voidsshowingacontinuouslyrisingin- dius,denotedbyr.Whentheintegratedoverdensitysatisfy∆(r)> tegrateddensityprofiles,definedasR-typevoids,whichexpandin −0.9, the iteration ceases and the current sphere radius r is de- asimilarwayas“void-in-void”regions.Inthispaperweanalysein finedastheradiusofthevoidcandidate.Ifthethresholdon∆(r) detailthestructureanddynamicsofthevelocityfieldintheregions isneverachievedthecandidateisnotconsideredanymoreasapos- surroundingvoids.Theaimofthisworkistoshedsomelightonthe siblevoid. originoftheobserveddeviationsofvoidvelocityprofilesfromlin- (iv) Onceunderdenseregionsareidentified,thestep(iii)isre- eartheory(PaperII),andaddresswhethersuchdeviationsarisein peatedstartingthistimefromarandomlydisplacedcenterinstead theinnershellstructure(i.e.thefeaturesofthedensityfieldinside of the candidate centroid. Such displacement takes the form of a shells) or they are more related with the surrounding large scale randomjumpproportionaltothevoidradius.Theserandomjumps characteristics (i.e. void of interest classified as R or S-type). To areperformedseveraltimes,obtainingarandomwalkaroundeach thisendwefollowourpreviouslydefinedclassificationschemefor candidate.Eachjumpisonlyacceptedifthenewobtainedradiusis voidenvironments(RandS-types),whereweanalysethevelocity largerthanthelastacceptedvalue.Ifthecurrentstepisaccepted, anddensityfieldinsidevoidcentricshells. thecandidatecenterisupdatedtothenewposition.Thepurposeof Theorganizationofthispaperisasfollows.InSection2we describethenumerical N-bodysimulationusedandtheconstruc- tion of the corresponding void catalogue. We describe the meth- 1 http://math.lbl.gov/voro++/ (cid:13)c XXXXRAS,MNRAS000,1–?? Structureanddynamicsinvoidshells 3 Figure1.Mollweideprojectionofmassdensitycontrast(toppanels)andradialvelocity(bottompanels)pixelmapsoftwovoidswiththeutmostS(“void–in– cloud”,leftpanels)orR-type(“void–in–void”,rightpanels)behaviour.Allsimulationparticlesintherange0.9<r/Rvoid<1.1areusedtocomputeaveraged valueswithineachpixel.ThetwovoidsshowninthisFigurehavebeenchosensothattheyshowthegreatestseparationinthevoidclassificationscheme, basedontheintegrateddensityprofiles(detailedinPaperI).Thistwocasesexemplifyhowthestructureofthevelocityfieldintheinnerpartsofavoidat 1voidradiusexhibitsverydistinctbehaviourdependingontheirenvironment(definedaround3voidradius,i.e.R-orS-typevoid).Suchdifferencesarenot expectedinlineartheory,giventhatbothregionsbydefinitionenclosethesameamountofintegratedoverdensity∆(Rvoid)=−0.9. suchrandomwalkistorecentertheidentifiedvoidsinthelargest thevoidhasnegativeorpositivevelocitydivergenceatthosescales. scalelocalminimumofthedensityfield.Thisprocedurederivea This imply that this region is either in contraction or expansion, well defined center which allows the computing of several stack- respectively. Voids surrounded by an overdense shell are dubbed ingstatisticslikethecorrelationfunctionorradialaveragesprofiles S-type voids, and satisfy the criterion ∆(r ) > 0. On the other max (Ceccarellietal.2008). hand R-type voids are defined as those that satisfy the condition (v) The last step consists on rejecting all the overlapping ∆(rmax) < 0,whichcorrespondstovoidswithcontinuouslyrising spheres,startingforthelargestcandidate. integrateddensityprofiles.Moredetailsaboutthisvoidclassifica- tionschemecanbefoundinPaperIandII.Thesimulationboxis As any void definition, the one presented in the above para- foundtocontain1622voidswithsizesrangingfrom∼9.5h−1Mpc graph have some features which can be related by analogy to to∼29h−1Mpc,where810areS-typeand812areR-type. other identification algorithms. However in order to compare re- sults among the literature corpus some caveats should be taken into account depending on the details of the void definition. For 3 ANALYSISOFTHESTRUCTUREINSPHERICAL instance, several algorithms involve the construction of a hierar- VOID-CENTRICSHELLS chyofvoids,definedasregionsenclosedbymatterridges(seefor instanceColbergetal.2005b;Platenetal.2007;Neyrinck2008; InthisSectionweaimtoinvestigatewhetherthereisarelationbe- Elyivetal.2014,andreferencestherein).Inthosecases,thevalue tween the spatial distribution, shapes and dynamics of structures ofthedensityfieldinsidevoidregionscouldvaryinawiderange. surroundingvoids,dependingonitsclassification(i.e.R-orS-type Inourcaseweselectvoidsbyensuringanencloseddensitybelow void).Aswesummarizedintheprevioussectionandwehaveseen agiventhreshold.Forthisreason,itisnotstraightforwardtorelate onPaperIandPaperII,thisclassificationschemesuccessfullyre- theunderdenseregionsidentifiedonbothmethodologies. flectsthedichotomyonthedynamicsofunderdenseregions.The FollowingtheprocedurepresentedinPaperI,wehavedefined lineartheorypredictsthefuturecollapseandexpansionofS-andR- two types of voids depending on their integrated density contrast typevoids,respectively.Thetheoryalsopredictsthatthebehaviour profiles.Themeanoftheseprofileshavebeencomputedfordiffer- intheinnerpartsofanunderdenseregionisinsensitivetowhether ent void size (Rvoid) intervals. As shown in Paper I, such average it is surrounded by any kind of environment (i.e., S- or R-type): curves have a well defined maximum at a distance rmax from the thevelocityprofileonlydependsontheamountofenclosedmass. voidcentre, exceptforthelargest voidswhichexhibitan asymp- HoweverinPaperII,wehaveseensignificantdeviationsfromthe toticallyincreasingprofile.Thisradiustypicallyreachvoidcentric linearpredictionsintheinnerpartsofvoidvelocitiesprofiles,both distancesaroundrmax≈3Rvoid.Thereforethedensityatsuchscales in R- and S-type voids (see Figure 3, in Paper II). Similar quali- canbethoughtasameasureofthelargescaleenvironmentwhere tativeresultscanbefoundintheworkofHamausetal.(2014a), thevoidisembedded. whofounddeviationsfromthepredictedradialvelocityprofilesof Weclassifyvoidsintotwosubsamplesaccordingtopositive stackedvoidsintheinnerregions(around0.5to1voidradii),being ornegativevaluesoftheintegrateddensitycontrastatr .Dueto thesedifferencesmorerelevantforsmallvoids.Nonetheless,only max thelineartheory,suchdefinitionimpliesthattheregionsurrounding aqualitativecomparisonbetweenourresultsandthoseinHamaus (cid:13)c XXXXRAS,MNRAS000,1–?? 4 Andre´sN.Ruizetal. etal.(2014a)ispossible,duetodifferencesonthevoiddefinition turesathighdensityregions.Bydefinitionofthevoididentification andidentificationalgorithms.Voidsusedintheirworkareidenti- method,bothvoidshellsatr = R ,enclosethesameamountof void fiedaccordinglytoahierarchy(Neyrinck2008),wheresmallun- integrateddensitycontrast∆(R )=−0.9.Evenmore,bothvoids void derdenseregionscanbefoundinsideoflargervoidswithahigher have similar sizes, 12.6 and 15.1h−1Mpc for S- and R-type void cumulativemassoverdensity.Inconsequence,therearedifferences respectively. between definitions of void radius and center. At the same time, Themotionsofshellsenclosinganoverdenseorunderdense theirresultsarenotclassifiedintoS-orR-typesandthereforetheir spherearisenaturallyasaconsequenceofthegravitationalpulland stackedprofilesreflectthetwodynamicalbehaviours(speciallyfor ispredictedinitssimplestformbythelineartheory(Peebles1976). smallsizedvoids). Accordingtothis,inastandardΛCDMmodel,theradialvelocity Atfirstglance,itcouldbeconjecturedthatinnerpartsofun- ofashellatradiusrenclosinganintegratedmassoverdensity∆ m derdenseregionsquicklyreachesthelowerboundsofthevalidity isgivenby: domainofthelineartheory(i.e.nonlineardynamicsbecomesin- H creasinglysignificantasδapproachesto−1).However,thepurpose V (r)=− r∆ (r)Ω0.6, (1) lin 3 m m ofthisSectionistogetadeeperinsightontheissue,byattempting toderivethespecificsourceofsuchdiscrepanciesfromlinearthe- whereHistheHubbleparameterandΩ isthecosmologicaldark m ory.Tothisend,weanalyzethepropertiesofdifferenttracersused matterdensityparameter.Atfirstglance,byEq.(1)weroughlyex- tomapthevoidvelocityfield.Thepropertiestakenintoaccountin- pect the same behavior of the mean radial velocities for the two volvelocaldensity,shapes,andspatialdistributionofmatteraround cases shown at Figure 1 (i.e. by definition both enclose almost voids. the same amount of matter density). However, as it can be seen Weconsidersphericalshellsofvarioussizescenteredoneach inthelowerpanelsofFigure1,thefluctuationsofthevelocityfield voidinthesample.Inallcasestheradialwidthofagivenshellis aroundthismeanvelocitybehavesnotablydifferent.IntheS-type a 20 per cent of the void radius. For each void, the smaller shell shellthereareregionswithnetinfall(bluepixels)oroutflowveloc- is centered at r = Rvoid and the larger one at a distance of r = ities(redpixels).Incontrast,fortheR-typesphericalshellweonly 3Rvoid.Theshellsaresphericallytessellatedwithequalareapixels observepositiveradialvelocities(i.e.,outflow).Also,bycompar- computedusingthesoftware HEALPIX (Go´rskietal.2005).We ingbottomtotoppanelsitcanbeseenthattheregionswithade- needapixelizationschemewherepixelsarenottoolarge,inorder finedsigninradialvelocity(i.e.outfloworinfall)seemtocorrelate toresolvetheshapesofstructuresconveniently.However,theshot withprojectedstructuresinthedensitymap.Wehavefoundsimilar noiseincreaseswhenthesizeofthepixelsdropsdown,sincemost featuresinallofthe6voidsizesubsamplesmentionedabove.Even of the pixels contain no particles. We use a pixelization scheme though,weshouldemphasizethatthemapsshowninFigure1are with 5 levels of refinement (Nside = 32) giving a total of Npix = notthetypicalbehaviourofS-andR-typevoids.Asstatedabove, 12 N2 = 12288 pixels for each map. The results shown in this they are just representative of voids with the largest environment side workdonotdependontheresolutionofthepixelizationscheme. differences.Sinceourvoidclassificationschemeisbasedonlyon twocategories,thereareplentyofvoidswithdensityprofileswhich fall into a grey area between both cases. However, guided by the 3.1 Velocityanddensityangulardistribution insightsobtainedhere,inthefollowingSectionswewillperforma Weappliedthedescribedshelltessellationschemetoallvoidsin statisticalanalysisintendedtocharacterizethefeaturesofthean- thesimulationbox(Sec.2.2)andcomputedthedensityandmean gularvelocity/densityfieldinvoidcentricsphericalshells. radial velocity of particles on each pixel of a given shell (ρ y InordertocomplementthediscussionbasedonFigure1,we pix V ).Sincethispixelizationisperformedonanequalareabasis,the alsoanalysetheutmostR-andS-typevoids(asdefinedpreviously) pix densityissimplyproportionaltothenumbercountofparticlesin insixvoidsizeranges.Forthispurpose,inFigure2weplotthehis- apixel,atadistancewithinthelimitsdefinedforeachshell.Inor- togramsofthepixelaveragedradialvelocityforthesevoids,show- dertostudythedynamicsofstructuresinanexpanding/collapsing ing S-type in black solid lines and R-type in black dashed lines. voidscenario(asdiscussedinPaperII)wecomputedtheaveraged Theresultsfordifferentsizesareshownoneachpanelasindicated void–centricradialvelocitiesoftheparticlesperpixelandpervoid. inthetop-rightlabels.Greycurvesdisplaythecorrespondinghis- Examplesofthepixelizedmapsofdensitycontrastandradialve- togramsforallvoidsinagivensizerange.Asitcanbeseen,inall locity corresponding to a shell located at r = R , are shown in casestheutmostR-typevoidspresentanarrowvelocitydistribution void MollweideprojectioninFigure1.Leftpanelsshowthemapsfora withawelldefinedpeak,andwithalmostallthepixelswithapos- S-typevoid,andrightpanelscorrespondtoacaseofaR-typevoid. itiveradialvelocity(i.e.outflowbehaviour).Incontrast,theutmost We selected the S-type (R-type) void with the maximum (mini- S-type voids show wider distributions and in a variety of shapes, mum) value of integrated density profile at r of the void sam- coveringfromflattomultipeakdistributions,andwithalargecom- max ple (as defined in section 2.2), so that they exhibit the utmost R- ponentofinfallingpixels(i.e.negativeradialvelocity).Ascanbe andS-typebehaviours.ThisselectionoftheutmostR-andS-type seen,thevelocityhistogramsforallvoidsseemtobeenclosedby voidbehaviourshasbeencarriedoutin6voidsubsamplesinvoids thevelocitydistributionoftheextremeR-andS-typevoids.This size intervals of 3h−1Mpc (see Figure 2). For simplicity, in Fig- indicatesthattheselectionofutmostbehavioursbasedonitsden- ure1weonlyshowtheresultsforoneofthesevoidsubsamples. sityprofilesisalsomanifestedinthefluctuationofthevelocityfield Theupperpanelsshowthelogarithmicdensitycontrastofthemass intheinnerpartsofvoids(theshellislocatedatr=R ).Forthe void log (δ +1) = log (ρ /ρ¯),whereρ¯ isthemeandensityofthe last size interval (largest voids, lower-right panel), S- and R-type 10 pix 10 pix simulation.Inthebottomaretheradialvelocitymaps,showingthe voids show similar distributions. This is expected because larger infallingandexpandingregions.Thegridpatternsseeninlowden- voidstendstohaveanR-typebehaviourandthefractionofS-type sityregionsareduetotherelativelysmallsizeofthepixelization voidsdecrease(seePaperIfordetails). schemechosen,andariseasaresultofthesimulationgrid.How- Theresultspresentedinthissubsectionsuggestasystematic ever,theselectedmapresolutionresultsadequatetoanalysestruc- correlationbetweenthefluctuationsofthevelocityfieldattheinner (cid:13)c XXXXRAS,MNRAS000,1–?? Structureanddynamicsinvoidshells 5 Figure2.NormalizedhistogramsofpixelradialvelocitiesfortheS-type(blacksolidlines)andR-typevoids(blackdashedlines)withthelargestdifferences intheirradialprofiles(i.e.theS-typevoidwiththelargestdensityaround3RvoidandtheR-typevoidwiththelowerdensityattheequivalentscale),selected insixvoidsizeintervalsofwidth3h−1Mpc,exceptthefirstandthelastones.Greycurvescorrespondtoallthevoidsinthatsizerange.Thedot-dashedline marksthepositionforVpix=0,thetransitionfromoutflowtoinfallbehaviour.Thevoidsizeincreasesfromtoptobottomandfromlefttoright,asindicated inthetop-rightlabelofeachpanel.ThenumberofvoidswithRorStypeenvironments(NRandNS)areindicatedattheleftbottomlegends. partsofvoids(i.e.r=R ,wheretheintegrateddensityisaround InFigure3weshowthedistributionsoftheseaverageveloci- void ∆=−0.9)andthevoidenvironmentatlargescales(r∼3R ).As tiesasafunctionoftheshellradiusnormalizedtothevoidradius. void discussed at the beginning of this Section, these correlations can Thesedistributionsrepresentthejointprobabilitydensityofshells shedsomelightontheoriginofnonlinearitiesintheinnerpartsof andwereestimatedbysimplycountinghighorlowdensityshells the underdense regions. This kind of systematic deviations in the in bins of r and (cid:104)V(cid:105) . Since by definition we have two veloc- r shell velocityfieldmayhaveinprinciplesomeimpactonstudiesusing ityaverages(highandlowdensitypixelsubsets)atallshellradii void-galaxyredshiftspacestatistics(Lavaux&Wandelt2010;Paz foreachvoid,thesamplinginshellsizesishomogeneous.Thisal- etal.2013;Hamausetal.2014b).Thefactthattheregionsofinfall lowstodirectlycomparebetweentheresultscorrespondingtoeach oroutflowontheutmostS-typevoidsseemtobeclusteredinangu- subsetcenteredratheronR-orS-typevoids,andalsotoperform larpositions(seeleft-lowerpanelinFigure1),indicatesthatitmay comparisonsbetweenbothvoidcategories.SolidlinesinFigure3 beacorrelationbetweenthevelocitydeviationsaroundthemean showisopercentilecurvesofthedistributioncorrespondingtothe flow and the properties of the large scale structures. Inspired on subsetoflowdensitypixels,whereasshadedregionsindicatethe thesefeatures,inthefollowingsections(3.2and3.4)wewillanal- same percentiles for the subset of high density pixels. The inner ysethedependenceofthevelocityfieldonthedensityandshapeof curves/regionsrepresentthe40percentofthesampleandtheouter thesurroundingstructures. onesthe90percent.Thedottedcentrallinecorrespondstotheme- dianofthedistributionofthelinearmodelprediction.Thetopand bottompanelsshowtheresultsforshellscenteredonR-andS-type 3.2 Dynamicsoflowandhighdensityregionsinvoidshells voids, respectively. On the right-hand side of Figure we display Asdescribedintheprevioussection,foreachvoidwecomputean- histogramsofradialvelocitiesfortwointervalsofradialdistance: gularmapsofradialvelocityanddensityinsphericalvoidcentric ashellcenteredatonevoidradiusandashellat2.5voidradius.Fi- shells of varying radii. Shell radii span a range between 1 and 3 nally,theverticaldottedlineindicatesthepositionof(cid:104)Vr(cid:105)shell = 0 timesthevoidradius.Wedividedthefullsetofpixelsineachshell inthehistograms. intothreecategoriesaccordingtoitsdensitycontrastwithrespect ItisworthnoticingthatinFigure3themeanradialvelocity tothemeandensityofthesimulation(δ ):lowdensityregionsare distributionsforallcases(R-orS-typevoidswithhighorlowden- pix definedbypixelswithδ <1,intermediatedensityregionssatisfy sitypixels)qualitativelyfollowthebehaviourofthelineartheory pix 1 < δ < 5andhighdensityregionsareselectedwithδ > 5. prediction. In all cases this model is enclosed within the central pix pix Suchselectionisintendedtoseparatehighandlowdensitystruc- 40percentofthedistributions.Thisisconsistentwiththeprevi- turesfromthestratumofmasswithanaveragebehaviour.Foreach ousresultsreportedonPaperII,whereweshowedthatlinearthe- shell we obtain the average radial velocity (denoted by (cid:104)V(cid:105) ) orycanbeusedtomodelredshiftdistortionsaroundvoids.Inthe r shell by computing the average of pixel radial velocities in one of the same direction, S-type voids exhibit typical negative velocities at three density subsets defined above. This allows to roughly esti- large shell radius (around 1.5 to 3 void sizes). This is the typical matewhetherstructuresofagivendensityaremovingawayfrom behaviourof“void-in-cloud”regionsdescribedbySheth&vande orfallingintothevoidcenter(i.e.theshellpixelsubsetofagiven Weygaert(2004)andobservedontheSDSS(PaperII).Ontheother densityhaveanetexpansionorcontraction).Finally,followingEq. hand,bylookingatthetoppanelofFigure3(S-typevoids),itcan (1),wecalculatethelinearpredictionofradialvelocities,V ,for beseenthatthevelocitydistributionoflowdensitypixelsisnearly lin allvoidsatthedifferentshells. symmetricaroundthelineartheorymedianprediction.Incontrast, (cid:13)c XXXXRAS,MNRAS000,1–?? 6 Andre´sN.Ruizetal. Figure3.Jointprobabilitydensitydistributionofshellmeanradialvelocity(cid:104)Vr(cid:105)shell,andtheshellradiusr(invoidsizeunitsRvoid),forS-type(toppanel) andR-type(bottompanel)voids.Solidlinesshowisopercentilesofthedistributionoflowdensitypixelsandshadedregionsshowthesamepercentiles correspondingtonetworkstructures,formedbyhighdensitypixels.Innercurves/regionsrepresentthe40percentofthesampleandtheouteronesthe90 percent.ThedottedcentrallinecorrespondstothemedianofthelinearmodelpredictiongivenbyEq.(1)andthedot-dashedlinemarksthepositionof (cid:104)Vr(cid:105)shell =0,i.e.theinfall/outflowtransition.Plotsontheright-handsideofFigureshowthenormalizedhistogramsofradialvelocitiesfortwointervalsof radialdistance:ashellatnearlyonevoidradiusandashellatroughly2.5voidradius.Thedottedlinethatcrossesthehistogramsindicatesthepositionof (cid:104)Vr(cid:105)shell=0. highdensitypixelsexhibitashifttowardsgreaterinfallvelocities from∼ 1.5to∼ 3voidradiusarethemainresponsibleforthede- with respect to the median linear theory. Also the outer contours viationsfromlineartheory. (whichenclosethe90percentofthepopulation)indicateabroader Wehaveanalyseddeviationsofthevelocityfieldfromtheme- distributionofvelocities atthesedensities.Thissuggeststhat the dian of linear theory predictions depending on the density of the largerdeviationsfromlineartheoryobservedonS-typevoids(Pa- pixelandthetypeofthevoidcentre.Inordertoseewhethersuch perII)aremainlyreflectedinthedynamicsofthesurroundinghigh deviations relies in the first order approximation of linear theory, densitypeaks.However,weshouldemphasizethatthedeviations we study the distribution of radial velocity in void shells relative describedherearearoundthemedianlineartheoryvalues.These to its corresponding linear prediction. To this end, for each shell deviationsaresourceofdispersionanderrorsinstackingstatistics and pixel density subset, we compute the difference between the asthosepresentedinPaperII,wherethedynamicsofagivenvoid meanradialvelocityandthevalueexpectedfromlineartheory(i.e. sampleismodeledbyapplyinglineartheorytothemeanormedian (cid:104)V(cid:105) −V ).Lookingforamorequantitativeanalysis,foralldis- density profile. Studies using a mixture of S- and R- type voids r shell lin tributionswecomputefourmoments:thefirstandsecondmoments (see for instance Hamaus et al. 2014a), have also report system- (mean and standard deviation) plus third and fourth standardized atics deviations from linear theory, in qualitative agreement with moments(skewnessandkurtosis).TheresultsareshowninFigure ourresults.ForthecaseofR-typevoids(bottompanel),itisworth 4,whereS-typevoidsareindicatedwithsolidlineswhereasR-type noticingthatmorethan90percent(outercontours)oflowdensity voidsaredrawnwithdashedlines.Thethicknessindicateslowand pixelaveragesdistributesonlyoverpositivevelocities.According highdensityregionsforthinandthicklines,respectively.Thefirst tothe“void-in-void”dynamics,thistypeofvoidsshouldonlydis- threepanelsfromuptobottomcorrespondtothemean,standard play expansion velocity curves. However, the high density pixels deviationandskewness,whereasthekurtosisisnotshowbecause show a significant fraction of structures with net infall averaged it is statistically consistent with zero at all radii. In the panel (d) velocities.Hereagainthehighdensitypeaksatdistancesranging ,weshowtheskewnessofthedistributionsof(cid:104)V(cid:105) .Ascanbe r shell (cid:13)c XXXXRAS,MNRAS000,1–?? Structureanddynamicsinvoidshells 7 seen, the results shown in this panel are consistent with distribu- 20 < V r >shell-Vlin (a) tionsdisplayedinFigure3.TheS-typevoidvelocitydistributions 10 show asymmetry toward infall velocities (negative skewness) for ean 0 both high and low density regions. Such asymmetry could arise m -10 density: low - high S-type fromthesystematiccollapseofthesurroundingvoidregion.When -20 R-type thelinearvelocityissubtracted(panel(c)),theskewnessofthedis- ttrhiebudteivoinastivoannsisfrhoemsograduismsiianniisthyeisnctohnesvideeloracbitlyy.dTishtriisbiuntdioicnastecsanthbaet viation 7600 < V r >shell-Vlin (b) explained mostly by means of linear theory. On the other hand, de 50 d R-type voids exhibit marginally positive skewness in its velocity ar 40 d distribution(panel(d)inFigure4),whichisconsistentwiththeob- an 30 servedsuppressionofnegativevelocitiesinFigure3.Respectingto st 1.0 themeanofvelocitydeviations(panel(a)inFigure4),S-typevoids < V r >shell-Vlin (c) 0.5 showlargerdeviationsfromlineartheorythanR-typevoids,along ss e thewholescalerange.AmongS-typevoids,thelargestdeviations wn 0 e arepresentwhenusinghighdensityregionstocomputeradialve- sk -0.5 locities.Howeveratshellradiibelow1.5voidsizes,R-typevoids -1.0 alsopresentimportantdeviationsfromlineartheoryvalues.These 1.0 results suggest that low density tracers for S-type voids and high < V r >shell (d) 0.5 density tracers for R-type voids are more likely to reproduce lin- ss e eartheorypredictions.Thiscanbeimportantinstudiesthatdepend wn 0 e sensitivelyonthemodelingofthevelocityfieldsurroundingvoids. k s -0.5 Asanexample,itcouldbementionedworksintendedtoperform -1.0 the Alcock-Paczynski test (Lavaux & Wandelt 2012; Sutter et al. 1.0 1.5 2.0 2.5 3.0 2012)onvoidsidentifiedonsparsegalaxysamples,whichareex- r [R ] void pectedtobedominatedbyR-typevoids(PaperI)andhighlumi- nositytracers.Thesestudiesaresensibletoanisotropiesinducedby Figure4.Momentsofthedistributionsofradialvelocityrelativetothelin- redshiftspacedistortions(PaperII),andthereforeapropermodel- earprediction((cid:104)Vr(cid:105)shell−Vlin)asafunctionofthevoid–centricdistance (threefirstpanelsfromuptobottom).S-typevoidsareindicatedwithsolid ingofthevelocityfieldisrequired.Inthesamedirection,studies linesandR-typevoidswithdashedlines.Thehighdensityregionsareplot- based on low redshift samples, as the main galaxy sample of the tedwiththethicklinesandthelowdensityoneswiththinlines.Inthefourth SDSS,canbebenefitedbyanappropriatetreatmentoftheveloc- panel(bottom),itisdisplayedthethirdstandardizedmoment(skewness)for itytracerpopulation.Thelastrelyinthefactthatsuchsamplesare thesamedistributionsofradialvelocitiesofFigure3. expectedtobedominatedbyS-typeatsmallsizevoidswhereasat largersizeR-typevoidsbecomemorefrequent(PaperI). Finally,inFigure5weshowthemedianof(cid:104)V(cid:105) normalized notably different. This result becomes more relevant if we notice r shell tothelinearpredictionV asafunctionofthevoidsizeR for thatthebehaviourofthecumulativeoverdensitywithvoidsizehave lin void negligibledifferenceforbothR-andS-typevoids(thecorrespond- S-typevoids(top-leftpanel,solidlines)andR-typevoids(top-right ingcurvescannotbeseenduetothefactthattheylieinsidethe panel,dashedlines).Thesevaluescorrespondtotheshelllocated greyshadedareainthebottompanelofFigure5).Therefore,even atadistanceofonevoidradiusfromthecenter.Thevaluesforhigh whenthediscrepancieswithlineartheoryaremostlyexplainedby densitypixelsareplottedinthicklinesandthelowdensityregions theemptinessoftheinnerpartsofunderdenseregions,theyexhibit areshownwhitthinlines.ItisclearfromthisFigurethatthereisa anotablydifferentbehaviourdependingonthelargescaleenviron- dependenceofthemeanradialvelocityoftheshellonthevoidsize, ment. inbothtypesanddensities.Thisdependenceismorepronounced forS-typevoids,andthehighdensityregionshavemoredispersion thanthelowones.Thenormalizationwiththelinearmodelensures 3.3 Structuresinthehighdensityregions thatthiscorrelationisafeaturebeyondthelineardependencewith sizepresentinEq.(1).Thevaluesofthemedianradialvelocityfor Once pixels have been classified according to its particle density S-typevoidsareunderpredictedbya∼ 10percentinthemedian containedwithinthesphericalshell,wecanseparatetheminden- withrespecttothelineartheoryforsmallvoids(andmoreevident sityintervalsinordertostudytheirproperties.Theselectedpixels forthehighdensitypixels)andoverpredictedbya∼ 20percent inadensitylayerformclusteredregionsor“chunks”,whichrep- in the median for the larger voids in our sample. For the sample resent the projection of structures surrounding the void within a ofR-typevoidswenoteanoverpredictionfrom∼ 10to∼ 20per spherical slice of a given size. These structures appear in a rich cent in the median in all the sizes range. At the bottom panel of variety of shapes and complexity, which depend on the density Figure5weshowthemedianofthecumulativemassoverdensity oftheirconstituentpixels,thedistancetothevoidcentreandthe insidetheshelllocatedatR asafunctionofthevoidradiusfor dynamics of the corresponding void. We argue that these depen- void thewholesample,irrespectiveofthevoidenvironment.Ascanbe dences encode useful information about the large–scale structure seen, there is a trend with the void radius which can be used to ofthevoid–filamentnetwork,andalsopossiblyabouttheirforma- explaintheincreasingdeviationsfromlineartheorywithvoidsize. tionanddynamics.Weaimtostudythemorphologicalproperties Thatis,largervoidstendtohavemorenon-linearvaluesofcumula- of the pixel structures in order to seek for and quantify such de- tivemassoverdensity,whichasaresultmakethedynamicofthese pendencies. As a first step, we isolated the structures by forming voidslessdescribedbythelineartheory.However,thedifferences groupsofadjacentornearpixelsinthesamedensityrange.This between the deviations when comparing R- and S-type voids are canbeachievedbyapplyingaFoFpercolationalgorithmtotheset (cid:13)c XXXXRAS,MNRAS000,1–?? 8 Andre´sN.Ruizetal. 1.3 caldistancesbetweenthem.AnimportantMSTcharacteristicisits lengthL,equaltothesumoftheweightsofallitsedges.Themax- 1.2 imum distance between any pair of nodes, D , is also a useful max quantity,whichcombinedtothelengthallowstodefineanestima- 1.1 torofstructureelongation,E = D /L.Thislastparameterisdi- max V lin mensionlessandbyconstructionitsvaluerangesbetween0and1, < >V / rshell 10..09 hloigwh d deennssiittyy woarrhetehcroreemagdpro–auclitpksoerwsctiortuhnccateunnrteerslao,tneadgnadsttirgourncotuvuparsleuswe.iothfraoulogwhlyel1onagraetifiolnamvaelnutes Wecomputethemaximumseparation,lengthandelongation 0.8 estimatorsforallstructuresarisingfrompixelsinthehighdensity S-type voids R-type voids layer,onceagainasinprevioussubsections,inashellcenteredat r=R .Aswediscussedinsection3.1,weareinterestedinchar- 10 15 20 10 15 20 25 void R [h- 1 Mpc] R [h- 1 Mpc] acterizinginastatisticallyrobustbasis,thedeviationsfromlinear -0.50 void void theoryintheangularstructurearoundvoids,andwhethertheyare -0.55 relatedornotwiththevoidlargescaleenvironment(i.e.R-orS- median of the cumulative R )void -0.60 matter density typevInoiFdi)g.ure6weshowthemedianvalues(blacklines)andthe (m -0.65 standarddeviation(greylines)ofMSTparametersasafunctionof -0.70 the void radius. S-type voids are drawn in solid lines and R-type -0.75 voids in dashed lines. In the top panel we show the angular size 10 15 20 25 R v o id [h- 1 Mpc] oftheMSTlength(L)andthemaximumseparation(Dmax),andin themiddlepaneltheelongationparameterE = D /L.Asitcan max Figure5.Medianoftheaveragedradialvelocitiesofeachvoidrelativeto beseeninthisFigure,theseparametersdonotdependsignificantly thelinearpredictionasafunctionofthevoidradiusforS-type(upper-left, onvoidsize,indicatingthatthedefinitionofthesequantitiesisnot solid lines) and R-type(upper-right, dashedlines) voids.The thicklines biasedbythespatialsizeofthevoidshell.Inorderhaveavisual correspondtothehighdensitypixelsandthethinlinestothelowdensity insightonthestructureshapesandtheelongationvalues,weshow regions.Errorbarsshowthestandarddeviations.Thelowerpanelshowsthe inthelowerpanelanexampleofthedistributionofstructuresand medianofthecumulativemassoverdensity.Thedot-dashedhorizontalline indicatesthepredictionforradialvelocitiesaccordingtothelineartheory. theirelongationparametersinavoidshell. Allpanelsreferstotheshellslocatedatr=Rvoid. InthenextSectionweusetheseparameterstoexplorethedy- namicsofstructuresdependingonthevoidclassificationandsize. of pixels, with an angular linking length equal to the mean pixel 3.4 Dynamicsofstructures separationforthecompletevoidsampledividedbythevoidradius. The mean separation is around to ∼ 3h−1Mpc for the case of the Sincethemassintheshellsaroundvoidsisarrangedinstructures, densest pixels. This procedure leads to a number of pixel groups speciallywhendensitylevelsareconsidered,thestudyoftheveloc- withadiversityofarrangements,fromcompacttomultibranchfil- itiesofthesestructuresmeritsfurtherinvestigation.InSection3.3 amentaryor”spiderlike”structures.Anumberofstrategiescould wedescribedtheproceduretocharacterizetheshapesofgroupsof beusedinordertoquantifythegeometryofthearrangementsof pixels,asaproxytotheprojectionofstructureswithintheshell,by pixels.Thesimplestmethodtocharacterizetheirformsispossibly meansoftheminimalspanningtree.Thisapproachhasanumber thecomputationofanormalizedinertiatensortoobtaintheelonga- ofadvantages,mainlyitssimplicityandtheabilitytodifferentiate tionastheratiooftheminor–to–majoraxes.Thesemi–axismethod betweenfilamentarystructureswithseveraldegreesofcomplexity. hasbeenusedtocharacterizeboth3Dand2Dstructures(Babul& InthisSectionweaimtocharacterizethesmoothnessofthe Starkman1992;Luo&Vishniac1995;Sathyaprakashetal.1998; radialflowofstructures.Todothis,wecomputetheaveragedvoid– Pazetal.2006,andreferencestherein).Thisprocedure,however, centricradialvelocities,(cid:104)V(cid:105) ,andtheaveragedtangentialveloci- r str does not distinguish between different levels of compactness or ties,(cid:104)V(cid:105) ,ofallthestructuresresultingfromtheFoFpercolation t str filamentarity. Other approaches have been proposed to character- ofhighdensitypixelsoneachshell.Thesmoothnessoftheradial izethegeometryofcomplexstructures,mostnotablyMinkowsky flowcanbequantifiedbymeasuringthemedianoftheratioofra- functionals(Bharadwajetal.2000;Basilakos2003;Einastoetal. dialtotangentialvelocities,namely(cid:104)V(cid:105) /(cid:104)V(cid:105) . r str t str 2007;Costa-Duarteetal.2011),surfacemodelingviatriangulated In Figure 7 we show the results of (cid:104)V(cid:105) , (cid:104)V(cid:105) and r str t str networks(Shethetal.2003)andshapefindersofseveralkinds(e.g. (cid:104)V(cid:105) /(cid:104)V(cid:105) asfunctionofvoidradiusconsideringseparatelyS- r str t str Sahnietal.1998;Arago´n-Calvoetal.2007). and R-type voids. It can be seen that the radial flow smoothness We find useful to represent the set of pixels in a group by a of structures in void shells increases with void radius so that the two-dimensionalgraph,whereeachnodeisapixelandanypairof largestthevoids,themoresignificanttheoutwardvoid–centricra- nodesisconnectedbyanedgewithaweightequaltotheangular dialvelocityascomparedtotheirtangentialmotions.Wefindstruc- distancebetweenthecentersofthepixelscorrespondingtothead- tures in R-type voids to have systematically smoother radial out- jacent nodes. The disposition of nodes and the general geometry flowscomparedtostructuresinS-typevoids,andthatradialflows canbeeasilycomputedbymeansoftheminimalspanningtreeof dominatethedynamicsforvoidslargerthan∼18h−1Mpc.Besides, thegraph(Barrowetal.1985,hereafterMST),definedastheshort- itcanbeseeninthelowerpanelsthatthisbehavioriscausedbythe estweightedpaththatconnectsallthenodeswithoutcycles.This combined effect of structures in S-types to have a systematically tree can be constructed for a set of pixels with the Kruskal algo- lowerradialflowandlargertangentialmotions.Theseresultscould rithm(Kruskal1956),usingthepositionsofpixelsandthespheri- beusefulwhenmodelingthevelocityfieldinvoids.Forinstance, (cid:13)c XXXXRAS,MNRAS000,1–?? Structureanddynamicsinvoidshells 9 100 S-type voids elongated structures (E > ~0.6) R-type voids 80 L (MST length) 1.5 round structures (E < ~0.6) g] size [de 6400 < >V / V rlinstr10..05 R-type voids 20 (maximum separation) D max 0.0 0.8 10 12 14 16 18 20 22 24 E0.6 1.5 linear fit to the 0.4 total sample (R+S) 10 12 14 R v1 o i6d [h- 1 Mp1c8] 20 22 24 < >V / V rlinstr10..05 00..3344 00..5588 S-type voids 00..4488 0.0 00..5544 00..2288 10 12 14 16 18 20 22 24 00..2288 R [h- 1 Mpc] 00..4455 void 00..6655 00..5588 00..8800 Figure8.Medianofthemeanradialvelocitiesofstructuresofeachvoid 00..8844 relativetothelinearpredictionofEq.(1)forR-type(upperpanel)andS- typevoids(lowerpanel)asfunctionofvoidsize.Thesampleshasbeendi- videdintoelongated(dot-dashedlines)androundstructures(dottedlines), examples of structures and elongation parameters asdescribedinthetext.Thetrendforthewholesample(withoutseparating Figure6.ShapeparametersofstructureswithintheshellscenteredatRvoid, inelongationorvoidtype)isshowedwiththethinsolidline.Greylines ofS-type(solidlines)andR-type(dashedlines)voids.Medianvaluesare correspondtothestandarddeviation. shown in black and the corresponding errors in grey. In the upper panel we show the minimal spanning tree length (L) and the maximum sepa- ration (Dmax). In the middle panel, the median of the elongation values inthecaseofmeasurementsinredshiftspace(eithercorrelationor E = Dmax/L are given as a function of void radius. In the lower panel stackingstatistics),forcertainvoidsizes,theresultswouldbemore weshowanexampleoftheangulardistributionofstructuresandtheircor- affectedbyuncertaintiesinthepair-wisevelocitydispersionthanin respondingelongationparametervalues. themeanflowvelocity.Thiscouldbeinferredfromthementioned abovedominanceoftangentialvelocitiesonvoidswithradiusbe- low18h−1Mpc.ThisisalsothecaseforS-typevoidsatallsizes: theyseemstobemoreaffectedbytangentialvelocitiesthanR-type regions. We have additionally explored the behavior of the mean ra- dialvelocitiesofstructuresrelativetothelinearpredictionofEq. (1)asfunctionofR .TheresultsaregiveninFigure8wherewe void have considered separately the sample of structures identified in R-type(upperpanel)andS-typevoids(lowerpanel),andeachof thissubsamplesintoelongated(dot-dashedlines)andround(dot- tedlines)structures.Thislastseparationwasperformedaccording totheresultsofthemiddlepanelofFigure6:structureswithelon- gationvalueshigherthanthemean(E >∼ 0.6)areconsideredas elongated,andstructureswithlowervaluesasround.Thesolidthin linerepresentstheglobaltrendofthewholesample(alinearfitto the total void sample), with no shape restrictions. It can be seen thatforbothR-andS-typevoids,thelinearpredictionprovidesan overestimationoftheradialvelocityofabout∼ 20percentinthe median for small voids and an underestimation of ∼ 20 per cent inthemedianforthelargervoids.TheglobalbehaviourforS-type Figure7.Medianofthemeanradial(middlepanel)andtangential(bottom voids in Figure 6 agree with the trend shown in the left panel of panel)velocities,in km/s,ofstructuresofS-type(solidlines)andR-type Figure5.However,forR-typevoidswenoteadiscrepancysince (dasheslines)voids,asfunctionofvoidradius.Ontheupperpanelweshow theradialmotionofshellsisalwaysunderestimatedbythelinear theratioofradial-to-tangentialvelocity,whichgivesanestimateofthemass model(seerightpanelofFigure5),whilestructuresidentifiedin flowsmoothness.Greylinesindicatetheerrorsofthemedians.Thedot- thehighdensitypixelspresentthesamevelocitytrendthanstruc- dashed line mark the transition from more turbulent motions to a radial turesinS-typevoids.Depictthisqualitativesimilarbehavioursin flow. bothFigures(8and5),itshouldbeemphasizedthatthequantities shownoneachcasehaveadifferentmeaning.Forinstance,results onFigure5canbethoughtasthetypicalerrorobtainedwhenthe meanvelocityofagivenshellispredictedbylineartheory.Mean- (cid:13)c XXXXRAS,MNRAS000,1–?? 10 Andre´sN.Ruizetal. while,inFigure8,thedisplayedtrendindicatethetypicaldiffer- shell, ∆ (R ). As expected, larger voids have stronger non lin- m void enceobservedbetweentheactualvelocityofagivenstructurewith earvaluesofmassdensity,thereforethedynamicsoflargervoids themeanpredictedbythelinearmodel. arelessdescribedbyalinearapproximation(inagreementforin- stancewithHamausetal.2014a,andreferencestherein).However, itisimportanttonotethatalthoughthevaluesof∆ areindistin- m guishablebetweenbothS-andR-typevoids,thetrendofradialve- 4 CONCLUSIONS locitywithvoidsizeobservedpresentsnotablydifferencesdepend- Based on the dynamical dichotomy first predicted by Sheth & ingofthelargescaleenvironment.Theseresultsreinforcetheidea vandeWeygaert(2004),andconfirmedonobservationsinprevious repeated along this work: the nonlinearties observed in the inner works(PaperI,PaperII),wehaveexploredtheinterplaybetween partsofvoidscouldarisefromacouplingofscales.Largescaleen- the inner dynamics of voids and its large scale environment. The vironmentdefinedarround3Rvoidhasasignificantcorrelationwith presentedstudygoesbeyondourpreviousanalysis,thatexamined dynamicsatinnerscales(−0.8<∆m<−0.6). the mean velocity profile of voids, here we have studied the ve- The structures defined in the high density regions of void locityfieldstructure(highermoments,angulardistribution,etc.)in shellsprovidefurtherdynamicalstudies.Wefindthatthesestruc- shellsaroundthem.Theaimofthisanalysiswasnotonlytoinquire turesshownsimilardeviationsfromlineartheorythanpreviousre- theexistenceofcorrelationsbetweentheinnervoiddynamicsand sults, that is larger departures are seen at larger sizes. We found itsenvironment,butalsowastolookforcluesontheoriginofthe againthatvelocitiestracedbystructuresinbothS-andR-typehave nonlinearities(PaperII,Hamausetal.2014a)inthevelocityfield acommontransitionscale,fromlinearmodeloverpredictiontoun- ofvoids. derprediction,atvoidsizesaround∼18h−1Mpc.Besides,elongated We have analysed voids in different dynamical regimes, ac- structuresexhibitsystematicallylarger(cid:104)V(cid:105) /V valueswithre- cording to the R/S type classification presented in Paper I. In or- r str lin specttotherounderstructuresby∼20percentinthemedian.We dertoaccomplishthis,wehaveconsideredsphericalshellsaround have analysed the radial to tangential velocity ratio of the struc- voidsatseveralvoid–centricdistances,analysingthetransitionbe- tures as a suitable measure of the void-centric flow smoothness. tweenthedomainsofvoidsandsurroundingstructures.Anstatis- The results of this study show an increasing trend of radial flow ticalstudyonthespatialandvelocitydistributionsonvoid-centric smoothnessasafunctionofvoidsizeproducedbyasystematically shellsshowsnotablydifferencesbetweenvoidtypesandsizes.At increasingmeanradialvelocityandacontinuouslydecreasingtan- theinnerpartsoftheutmostS-typevoids,thevelocitydistribution gentialvelocity.Theobservedcorrelationsuggeststhatlargevoids takestheformofangularpatcheswithpredominantinfallorout- (withsizesgreaterthanR ∼ 18h−1Mpc)havemorefrequently flowradialvelocities,eventhoughthemeanvelocityinallcasesis void boundaries with structures dominated by radial outflows, in con- outwards.Meanwhile,forR-typevoids,thevelocityangulardistri- trastwithamoreturbulentmotionofthestructuresatboundaries butionisconsistentwithapureoutflowpattern.Asaresult,thedis- ofsmallvoids.WealsofindstructuresinR-typevoidstohavesys- tributionsofdeviationsfromthebulkflowonR-typevoidsaretyp- tematicallysmootherradialoutflowsby∼25percentinthemedian icallynarrow,wellcenteredonitsmeanvalues.Incontrast,S-type comparedtostructuresinS-typevoids. voidstypicalexhibitsaninnerbroaddistributionofradialveloci- ties,withoutaneatbulkflowpeakor,insomecases,incipientbi- As a final remark, is worth to notice that R-type voids with modality.Itshouldbenoticedthatbetweenbothutmostbehaviours larger sizes are the regions best described by linear theory and thereisacontinuousrangeofvelocitypatterns. dominated by radial outflows. In addition, low density structures Intheanalysisfortheradialvelocityfieldatshellsinincreas- seemtotracetheinnervelocityfieldinbetteragreementwithlin- ingradiiwefindthatR-typevoidsthevelocityfieldremainswith eartheory.Thiscouldbeimportantwhendesigningcosmological positiveoutflowatdifferentshellradii,whichcontrastwiththebe- testsbasedonvoidssamplesandtheoncomingnextgenerationof haviour of S-type voids that show a significant infall beyond 1.5 large volume galaxy surveys as HETDEX (Hill et al. 2008), Eu- voidradii,inagreementwithPaperII.Thespreadofvoid–centric clid (Laureijs et al. 2011), SDSS-III (Eisenstein et al. 2011) and radialvelocitiesforS-typevoidsissignificantlylargerthanthatof VIPER(Michelettietal.2014).Inparticularmeasurementswhich R-type voids, particularly at large distances from the center. We dependsonapropermodelingofthevelocityordensityvoidpro- alsofoundthatthedistributionofvelocitiesofthehighdensityre- files(e.g.theAlcock-PaczynskitestortheIntegratedSachs-Wolfe gionsexhibitaninfallingtail,whichisremarkableforS-typevoids effect, Granett et al. 2008; Pa´pai et al. 2011; Lavaux & Wandelt butalsonotablyonR-typevoids.Thelinearmodelprovidesasuit- 2012; Sutter et al. 2012; Herna´ndez-Monteagudo & Smith 2013; able fit to the observed radial infall/outflow; however, in S-type Ilic´ etal.2013;Caietal.2014b,a).Forthesenextgenerationsur- voids, there is a systematic offset towards smaller values by ap- veys, given the large volume sampled, it is expected to identify proximately∼ 30−50km/s.Wehavealsoseenthatlineartheory voidsbyusinghighluminositytracers.Voidscataloguesobtained successfullypredictsGaussiandeviationsonradialvelocityfluctu- fromsuchgalaxysamplesareexpectedtohavepredominantlyR- ations (i.e. skewness and kurtosis moments are negligible on lin- type underdense regions. Even though, these type of regions be- eartheorydeviations).Theresultssummarizedintheabovepara- havesonbetteragreementwithlineartheory,thebrightersample graphs,couldbeimportantwhenmodelingthepair-wisevelocity of galaxies is expected to trace the velocity field of high density function on redshift space measurements and models (see for in- structure,whichhasshowninthisworkhavesystematicdepartures stancePaperII). fromlinearityattheinnerregions.Therefore,itcouldberequired The ability of the linear theory is also tested when compar- a proper model of the velocity expansion profile of this systems ing the ratio of the observed radial velocity to the predicted one whenmeasurementsareperformedinredshiftspace.Inthesame in the shell located the void radius, founding that this ratio is an direction,studiesbasedonlowredshiftsamples,asthemaingalaxy increasing function of the void size for both S- and R-type and sampleoftheSDSS,canbebenefitedbyanappropriatetreatment highandlowdensityregions.Anexplicationofthistrendcanbe ofthevelocitytracerpopulationforbothS-andR-typeunderdense found by analysing the cumulative mass overdensity inside the regions. (cid:13)c XXXXRAS,MNRAS000,1–??

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