Clues on void evolution III: Structure and dynamics in void shells

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by  A. N. Ruiz

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
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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
.
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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–??