Sheth, R.

Weygaert, R. van de

English

Abstract: Within the context of hierarchical scenarios of gravitational structure formation we describe how an evolving hierarchy of voids evolves on the basis of {em two} processes, the {em void-in-void} process and the {it void-in-cloud} process. The related analytical formulation in terms of a {it two-barrier} excursion problem leads to a self-similarly evolving peaked void size distribution

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   University of GroningenVoid Hierarchy and Cosmic StructureWeygaert, R. van de; Sheth, R.Published in:ArXivIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2003Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Weygaert, R. V. D., & Sheth, R. (2003). Void Hierarchy and Cosmic Structure. ArXiv.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 11-10-2022arXiv:astro-ph/0310914 v1   31 Oct 2003VOID HIERARCHY AND COSMIC STRUCTURERien van de Weygaert1 & Ravi Sheth2,31Kapteyn Institute, Univ. Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands2Dept. of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara St., PA 15260, U.S.A.3 NASA/Fermilab Astrophysics Group, Batavia, IL 60510-0500, U.S.A.rks12@pitt.edu, weygaert@astro.rug.nlAbstract Within the context of hierarchical scenarios of gravitational structure formationwe describe how an evolving hierarchy of voids evolves on thebasis oftwoprocesses, thevoid-in-voidprocess and thevoid-in-cloudprocess. The relatedanalytical formulation in terms of atwo-barrier excursion problem leads to aself-similarly evolving peaked void size distribution.1. Introduction: ExcursionsHierarchical scenarios of structure formation have been very succesfull in un-derstanding the formation histories of gravitationally bound virialized haloes.Particularly compelling has been the formulation of a formalism in which thecollapse and virialization of overdense dark matter halos within the context ofhierarchical clustering can be treated on a fully analytical basis. This approachwas originally proposed by Press & Schechter (1974), which found a particu-larly useful and versatile formulation and modification in the theexcursion setformalism(Bond et al. 1991).It is based on the assumption that for a structure to reach a particul r nonlin-ear evolutionary stage, such as complete gravitational collapse, the sole condi-tion is that itslinearly extrapolated primordial densityshould attain a certainvalue. The canonic example is that of a spherical tophat overdensity collapsingonce it reaches the collapse barrierδc ≈ 1.69. The successive contributionsto the local density by perturbations on a (mass) resolutionscaleSm may berepresented in terms of a density perturbation random walk,the cumulative ofall density fluctuations at a resolution scale smaller thanSm. By identifyingthe largest scale at which the density passes through the barri r δc it is possible(1) to infer at any cosmic epoch the mass spectrum of collapsed halos and (2)to reconstruct the merging history of each halo (see Fig. 2, top right).In this study we demonstrate that also the formation and evolution of foam-like patterns as a result of the gravitational growth of primordial density pertur-2Figure 1. Illustration of the two essential “void hierarchy modes”: (top) thevoid-in-voidprocess (top), with a void growing through the merging of twoor more subvoids; (bottom)thevoid-in-cloudprocess: a void demolished through the gravitational collapse of embeddingregion.bations is liable to a succesfull description by the excursion et analysis. Thisis accomplished by resorting to a complementary view of clustering evolutionin which we focus on the evolution of thevoidsin the Megaparsec galaxy andmatter distribution, spatiallyth«e dominant component. An extensive descrip-tion of this work can be found in Sheth & Van de Weygaert (2003,MNRAS,submitted).2. Void Evolution: the two processesPrimordial underdensities are the progenitors of voids. Because underdensi-ties are regions of suppressed gravitational attraction, representing an effectiverepulsive gravity, matter flows out of their interior and moves outward to theedges of these expandingvoids. Voidsexpand, become increasinglyemptyanddevelop an increasinglysphericalshape (Icke 1984). Matter from the void’sinterior piles up near the edge: usually a ridge forms aroundthe void’s rim andat a characteristic moment the void’s interior shells take ov r the outer ones.At this shell-crossingepoch the void reaches maturity and becomes a nonlinearobject expanding self-similarly, the implication being that the majority of ob-served voids is at or near this stage (Blumenthal et al. 1992). As voids developfrom underdensities in the primordial cosmic density field,the interaction withinternal substructure and external surrounding structures translates into a con-Void Hierarchy 3tinuing process of hierarchical void evolution (Dubinski et al. 1993, Van deWeygaert & Van Kampen 1993).The evolution of voids resembles that of dark halos in that large voids formfrom mergers of smaller voids that have matured at an earliercosmic time(Fig. 1, top row). However, in contrast to dark halos, the fatof voids is ruledby two processes. Crucial is the realization that the evolving void hierarchydoes not only involve thevoid-in-void process but also an additional aspect,the void-in-cloud process. Small voids may not only merge into larger en-compassing underdensities, they may also disappear througcollapse whenembedded within a larger scale overdensity (Fig. 1, bottom rw). In terms oftheexcursion set approach, it means that theone-barrierproblem for the halopopulation has to be extended to a more complextwo-barrier problem. Voidsnot only should ascertain themselves of having decreased their density belowthevoid barrier δv of theshell-crossingtransition. For their survival and sheerexistence it is crucial that they take into account whether ty are not situatedwithin a collapsing overdensity on a larger scale which crossed the collapsebarrierδc. They should follow a random walk path like type “3” in Fig. 2 (bot-tom right), rather than thevoid-in-cloudpath “4”. The repercussions of thisare far-reaching and it leads to a major modification of the void properties anddistribution.Figure 2. Left: Void Size Probability Function for various values of thevoid-and-cloudparameterδc/δv: 1/2 (short dashed), 1 (dashed), 2 (dot-dashed), and∞ (solid). For reference,the dotted curves are predictions by the “adaptive peaks model” f Appel & Jones (1990).Right: Excursion Set Random Walks, local densityδ(x) as function of mass resolutionSm.Clearly, the fluctuations are larger as resolutionSm increases. (Top) Halo formation. Horizontaldashed bar is the collapse barrierδc. (Bottom) Fate of void V2, illustrated by 2 possible randomwalks: curve “4” is avoid-in-cloudprocess, curve “3” avoid-in-voidprocess.43. Void Distribution: Universal, Peaked and Self-SimilarAnalytically, the resulting expression follows by evaluating the fraction ofwalks which first crossδv atS, and which do not crossδc until after they havecrossedδv (Sheth & Van de Weygaert 2003). An insightful approximationofthis distribution, in terms of the “self-similar” void density ν ≡ δ2v/S isνf(ν) ≈√ν2πexp(−ν2)exp(−|δv|δcD24ν− 2D4ν2), (1)in which theD ≡ (|δv|/(δc − δv) parameterizes the the relative impact of voidand halo evolution on the hierarchically evolving population of voids.The resulting distributions, for various values ofD, are shown in Figure 2(left). The void size distribution is clearlypeaked about a characteristic size:thevoid-in-cloudmechanism is responsible for the demise of a sizeable popu-lation of small voids. The halo mass distribution diverges towards small scalemasses, so that in terms of numbers the halo population is dominated by smallmass objects. The void population, on the other hand, is “void” f small voidsand has a sharp small-scale cut-off.Four additional major observations readily follow from this analysis: (•)At any one cosmic epoch we may identify acharacteristic void sizewhichalso explains why in the present-day foamlike spatial galaxy distribution voidsof ∼ 20 − 30h−1Mpc are the predominant feature; (•) The void distributionevolves self-similarlyand thecharacteristic void size increases with time: thelarger voids present at late times formed from mergers of smaller voids whichconstituted the dominating features at earlier epochs (Fig. 1, top panels); (•)Volume integration shows that at any given timeth population of voids ap-proximately fill space, apparently squeezing the migrating high-density matterin between them; (•) As the size of the major share of voids will be in theorder of that of the characteristic void size this observation implies that thecosmic matter distributionresembles afoamlike packing of spherical voids ofapproximately similar size and excess expansion rate.In all, a slight extension and elaboration on the original extension formulismenables the framing of an analytical theory explaining how the characteris-tic observed weblike Megaparsec scale galaxy distribution, and the equiva-lent frothy spatial matter distribution seen to form in computer simulations ofcosmic structure formation, are natural products of a hierarchical process ofgravitational clustering. A continuously evolving hierarchy of voids producesa dynamical foamlike pattern whose characteristic dimension grows continu-ously along with the evolution of cosmic structure, a Universe which at anyone cosmic epoch is filled with bubbles whose size corresponds to the scalejust reachingmaturity.

Void Hierarchy and Cosmic Structure

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Void Hierarchy and Cosmic Structure

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