Precision cosmology with voids: definition, methods, dynamics
aa r X i v : . [ a s t r o - ph . C O ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 22 October 2018 (MN L A TEX style file v2.2)
Precision cosmology with voids: definition, methods,dynamics
Guilhem Lavaux , , Benjamin D. Wandelt , Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801-3080, USA California Institute of Technology, Pasadena, CA 91125, USA
22 October 2018
ABSTRACT
We propose a new definition of cosmic voids based on methods of Lagrangian orbitreconstruction as well as an algorithm to find them in actual data called DIVA. Ourtechnique is intended to yield results which can be modeled sufficiently accurately tocreate a new probe of precision cosmology. We then develop an analytical model ofthe ellipticity of voids found by our method based on Zel’dovich approximation. Wemeasure in N -body simulation that this model is precise at the ∼ ∼ h − Mpc. We estimate that at thisscale, we are able to predict the ellipticity with an accuracy of σ ε ∼ .
02. Finally,we compare the distribution of void shapes in N -body simulation for two differentequations of state w of the dark energy. We conclude that our method is far moreaccurate than Eulerian methods and is therefore promising as a precision probe ofdark energy phenomenology. Large empty regions of space, called voids, represent themajority of the volume of the present Universe. They werefirst discovered in observations by Gregory & Thompson(1978), Joeveer et al. (1978) and Tully & Fisher (1978),followed later by Kirshner et al. (1981) and more largelyin the CfA redshift catalogue (de Lapparent et al. 1986).This discovery was followed by a large amount of theo-retical work. The first gravitational instability model wasgiven by Hoffman & Shaham (1982), quickly followed byHoffman et al. (1983) for an infinite size regular mesh ofvoid and by Hausman et al. (1983) for the impact of cos-mology on their evolution. Other work studied the generalself-similar evolution of voids in Einstein-de-Sitter universes(Bertschinger 1983, 1985).However, as the voids are intrinsically large and thesurveys at that time were small, we only detected a smallnumber of them. This has hindered their use as a cosmo-logical probe for a long time [except some constraints ontheir maximal size compatible with CMB observations bye.g. Blumenthal et al. (1992)]. This situation has changedwith the advent of deep and wide galaxy surveys such asthe Sloan Digital sky survey (SDSS, York et al. 2000), 2dF-GRS (Colless et al. 2001), 2MRS (Huchra 2000) and now the6dFGS (Jones et al. 2009). Still we miss a clear and simpledefinition of voids that would allow us to use them as a pre-cision cosmological probe. In this paper, we investigate, an-alytically and numerically using N -body simulations, a newalgorithm for finding voids in Large Scale structure surveysand an analytical model that accurately predicts the proper-ties of voids found by this method as a function of cosmology. During the last decade, several algorithms to find voidshave been built. They are separated in three broad classes.In the first class, the void finders try to find regions emptyof galaxies (Kauffmann & Fairall 1991; El-Ad et al. 1997;Hoyle & Vogeley 2002; Patiri et al. 2006; Foster & Nelson2009). The second class of void finders try to identify voidsas geometrical structures in the dark matter distributiontraced by galaxies (Plionis & Basilakos 2002; Colberg et al.2005; Shandarin et al. 2006; Platen et al. 2007; Neyrinck2008). The third class identifies structures dynamically bychecking gravitationally unstable points in the distributionof dark matter (Hahn et al. 2007; Forero-Romero et al.2009). At the same time, N -body simulations focused onthe studies of voids in a cosmological context were flour-ishing (Martel & Wasserman 1990; Regos & Geller 1991;van de Weygaert & van Kampen 1993; Gottl¨ober et al.2003; Benson et al. 2003; Colberg et al. 2005). Recently,Colberg et al. (2008) made a comparison which shows that,even if these currently available void finder techniques findapproximately the same voids, the details of the shapes andsizes found by each of the void finders may be significativelydifferent. This problem is further enhanced by the existenceof ad-hoc parameters in most of the existing void finders,which changes the exact definition of voids and does notallow reliable cosmological predictions. One aspect of voidsthat is also often left aside is the hierarchical structuresof voids. So far, apart from ZOBOV (Neyrinck 2008)and the related Watershed Void Finder (WVF) method(Platen et al. 2007), which are parameter free, no voidfinder tries to identify correctly the hierarchy of voids-in-voids and clouds-in-voids (Sheth & van de Weygaert2004). Another problem of these void finders comes from c (cid:13) G. Lavaux & B. D. Wandelt their Eulerian nature: they try to find structures that arenot necessarily in the same dynamical regime (linear ornon-linear), which complicates the building of an analyticalmodel.We propose studying a new void finder that belongsto the third class of these void finders. It is based on thesuccess of both the Monge-Amp`ere-Kantorovitch (MAK)reconstruction of the orbits of galaxies (Brenier et al.2003; Mohayaee et al. 2006; Lavaux et al. 2008) and theZel’dovich approximation (Zel’dovich 1970). This methodis based on finding a way to compute the Lagrangian coor-dinates of the objects at their present position. The study ofvoids in Lagrangian coordinates is not new. The evolutionof voids in the adhesion approximation has been studied bySahni et al. (1994) to understand the formation and evolu-tion of voids and their inner substructure in a cosmolog-ical context. Later, Sahni & Shandarin (1996) emphasizedthe precision of the Zel’dovich approximation for studyingvoid dynamics compared to higher order perturbation the-ory, either Lagrangian or Eulerian. However, no void findermethod has yet tried to take advantage of the Zel’dovich ap-proximation for detecting and studying voids in real data.The voids detected with this method are going to be intrin-sically different than the one found using standard Eulerianvoid finder. This hardens the possibility of making a void-by-void comparison of the different methods.The use of Lagrangian coordinates gives one imme-diate advantages compared to standard void finding: theLagrangian displacement field is still largely in the linearregime even at z = 0 and especially for voids. This allows usfor the first time to make nearly exact analytical computa-tion on the dynamical and geometrical properties of voids inLarge scale structures. The MAK reconstruction is thus par-ticularly adapted to study the dynamics of voids. However,there is an apparent cost to pay: we lose the intuitive way ofdefining voids as “holes” in a distribution of galaxy, that isthe place where matter is not anymore. On the other hand,we gain the physical understanding that voids correspond toregions from which matter is escaping.The dynamics of voids may provide a wealth of infor-mation on dark energy without the need for any new survey.The first obvious probe of dark energy properties comes fromthe study of the linear growth factor. Its evolution with red-shift depends, among the other cosmological parameters, onthe equation of state w of the Dark Energy. In this work,we assume that w is independent of the redshift. We notethat in galaxy surveys, our method is going to be sensitiveto bias but not more than the direct approach to void find-ing. Indeed, void finders of the first class are sensitive to theselection function of galaxies. Generally this is done by limit-ing the survey to galaxies with an apparent magnitude belowsome designated threshold. Changing this selection functionof the galaxies acts on the boundaries of the detected voids,which thus changes the geometry of these voids. From thepoint of view of void finders, this will also act as a “bias”.The method that we propose has a more conventional de-pendence on the bias by using the dark matter distributioninferred from the galaxy distribution. The advantage is thatthis bias could be calibrated. One exact calibration consistsin comparing peculiar velocities reconstructed using MAK toobserved velocities (Lavaux et al. 2008). Additionally, thereare a number of other complementary ways of determining bias from galaxy redshift surveys (e.g. Benoist et al. 1996;Norberg et al. 2001; Tegmark et al. 2004; Erdogdu et al.2005; Tegmark et al. 2006; Percival et al. 2007).This paper is a first of a series studying the proper-ties of voids found by our void finder. It is organised asfollows. First, we recall the theory of the Monge-Amp`ere-Kantorovitch reconstruction in Section 2. Then, we explainhow we can use reconstructed orbits as an alternative way todetect and characterise voids. This corresponds to the coreof DIVA, our void finder through Dynamical Void Analysis,and is explained in Section 3. In Section 4, we model an-alytically the voids found by DIVA. In Section 5, we testour void finder on N -body simulations. We also check ouranalytical model against the results of the simulations fortwo cosmologies. In Section 6, we compare DIVA to earlierexisting void finders. In Section 7, we conclude. The Monge-Amp`ere-Kantorovitch reconstruction (MAK) isa method capable of tracing the trajectories of galax-ies back in time using an approximation of the completenon-linear dynamics. It is a Lagrangian method, as PIZA(Croft & Gaztanaga 1997) or the Least-Action method(Peebles 1989). The MAK reconstruction is discussed ingreat detail in Brenier et al. (2003), Mohayaee et al. (2006)and Lavaux et al. (2008). It is based on the hypothesis that,expressed in comoving Lagrangian coordinates, the displace-ment field of the dark matter particles is convex and poten-tial. Since then, this hypothesis has been justified by thesuccess of the method on N -body simulations. With the lo-cal mass conservation, this hypothesis leads to the equationof Monge-Amp`ere:det i,j ∂ Φ ∂q i ∂q j = ρ ( x ( q )) ρ , (1)with q the comoving Lagrangian coordinates, x ( q ) thechange of variable between Eulerian ( x ) and Lagrangian co-ordinates ( q ), ρ ( x ) the Eulerian dark matter density and ρ the initial comoving density of the Universe, assumed homo-geneous. Brenier et al. (2003) showed that solving this equa-tion is equivalent to solving a Monge-Kantorovitch equation,where we seek to minimise I [ q ( x )] = Z x d x ρ ( x ) ( x − q ( x )) , (2)according to the change of variable q ( x ). Discretising thisintegral, we obtain S σ = X i (cid:0) x i − q σ ( i ) (cid:1) , (3)with σ a permutation of the particles, q j distributed ho-mogeneously, x i distributed according to the distribution ofdark matter. Doing so, we obtain a discretised version of themapping q → x ( q ) on a grid.To solve for the problem of minimising Eq. (3) withrespect to σ , we wrote a high-performance algorithm thathas been parallelised using MPI. This algorithm is based onthe Auction algorithm developed by Bertsekas (1979). It hasan overall time complexity for solving cosmological problems c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics empirically between O ( n ) and O ( n ) (at worst) with n thenumber density of mesh elements { q j } . In this section, we describe our void finder DIVA (for Dy-namIcal Void Analysis). First, we define in Section 3.1 whatwe call a void in this work. Second, in Section 3.2, we makeuse of the displacement field in the immediate neighbour-hood of a void to define the ellipticity arising from tidal fieldeffects, which we also call tidal ellipticity . In Section 3.3, wedefine the Eulerian ellipticity of our voids. In Section 3.4, wediscuss the impact of smoothing in Lagrangian coordinatesto compute void properties.In later sections, we use pure dark matter N -body simu-lations to check the adequacy of the voids found using MAKreconstructed displacement field and the one detected in thesimulated displacement field. The results given by the an-alytical models are then compared to the one given by thesimulated field for two equations of state of the Dark Energy. So far, voids have only been described using a purely geo-metrical Eulerian approach. Typically, as mentioned in theintroduction, a void is an empty region delimited by eithersphere or ellipsoid fitting or by using isodensity contours.We propose here to use a Lagrangian approach and use themapping between Lagrangian, q , and Eulerian coordinates, x as a better probe for voids. In the rest of this article,we will consider these two coordinates to be linked by thedisplacement field Ψ : x ( q ) = q + Ψ ( q ) . (4)We now define the source S Ψ of the displacement field by S Ψ ( q ) = X i =1 ∂ Ψ i ∂q i . (5)As the displacement field is taken to be potential it is strictlysufficient to only look at S Ψ to study Ψ .We now define the position of a candidate void centre bylooking at maxima of S Ψ in Lagrangian coordinates. Thiswill effectively catch the source of displacement and fromwhere the void is expanding. The other, practical, advan-tage is that S Ψ is quite close to the opposite of the linearlyextrapolated initial density perturbations of the consideredpatch of universe (Mohayaee et al. 2006). So we can use theusual power spectrum to study most of the statistics of thisfield. So the main approximation we use in the rest of thisstudy is that the primordial density field power spectrumis a good proxy for the power spectrum of the seed of dis-placement and that this displacement is a Gaussian randomfield. This maxima corresponds in terms of the primordial densityfield to what is sometimes called a protovoid (Blumenthal et al.1992; Piran et al. 1993; Goldwirth et al. 1995).
From Ψ , we define the matrix T l,m of the shear of thedisplacement, which is linked to the Jacobian matrix: J l,m ( q ) = ∂x l ∂q m = δ l,m + ∂ Ψ l ∂q m ( q ) = δ l,m + T l,m , (6) J ( q ) = | J l,m | , (7)with T l,m ( q ) = ∂ Ψ l ∂q m . (8) J is the Jacobian of the coordinate transformation q → x .Geometrically, J specifies how an infinitely small patch ofthe Universe expanded, in comoving coordinates, from highredshift to z = 0. We put λ i ( q ) the three eigenvalues of T l,m ( q ) and sort them such that λ ( q ) > λ ( q ) > λ ( q ).Among the candidate voids, we select only voids that havestrictly expanded, which equivalently means that J > true voids for which λ > , λ > , λ >
0. Theseshould be the most evident and easily detectable voids asthey consist in regions which are expanding in the threedirections of space.- pancake voids for which λ > , λ > , λ <
0. Thepancake voids are closing along one direction of space butexpanding along the two other directions. With a geomet-rical analysis, this case cannot be distinguished from thetrue void case. However the dynamical analysis is capable ofthat, and this will cause a crucial difference to the analysisas we will see later. In practice they represent a substantialfraction of the voids.- filament voids for which λ > , λ < , λ < S Ψ > After having defined the position and the dynamical prop-erties of the void, we may define an interesting property ofthose structures: the ellipticity. Icke (1984) first emphasizedthat isolated voids should evolve to a spherical geometry.But, in the real case, voids are subject to tidal effects. As-suming the present matter distribution evolved from a to- c (cid:13) , 000–000 G. Lavaux & B. D. Wandelt
Figure 1.
Picture of a void and the formation of intrinsic ellipticity – We represent in this figure the central idea of the definitionof a void and its ellipticity. We take voids as maxima in S Ψ . They correspond to first order to minima of the primordial density fieldrepresented here by painted surface. These minima undergo an overall expansion from initial conditions to present time. The shape ofthe Void is defined locally at the minimum. The ellipticity is equal to the square root of the ratio of the axes of the ellipsoid which locallyfits the surface. tally isotropic and homogeneous distribution, Park & Lee(2007) and Lee & Park (2009) have shown that the distri-bution of the ellipticity which is produced by tidal effects isa promising probe for cosmology. More generally, previouswork have shown that a lot of potentially observable statis-tical properties of voids are directly to the primordial tidalfield (e.g. Lee & Park 2006; Platen et al. 2008; Park & Lee2009b,a). However, questions may be raised by the directuse of the formula Doroshkevich (1970), as they applied itin Millennium Simulation. Using the orbit reconstructionprocedure, our approach should be able to treat the prob-lem right from the beginning, even in redshift space (seeLavaux et al. 2008 for a long discussion), though some careshould be taken for the distortions along the line-of-sight.The other advantage of Lagrangian orbit reconstructionis that it offers for free a way of evaluating the ellipticity locally , potentially at any space point. From the mass con-servation equation and the definition of eigenvalues of J l,m we may write the local Eulerian mass density as: ρ E ( q ) = ¯ ρ | (1 + λ ( q ))(1 + λ ( q ))(1 + λ ( q )) | = ¯ ρV L V E ( q ) , (9)with V L the Lagrangian volume of the cell at q and V E ( q )the Eulerian volume of this same cell, ¯ ρ the homogeneousLagrangian mass density. This equation is valid at all times.Now we may also explicitly write the change of volume ofsome infinitely small patch of some universe V E ( q ) = V L | (1 + λ ( q ))(1 + λ ( q ))(1 + λ ( q )) | . (10)Provided the eigenvalues λ i are greater than -1, which isalways the case for voids, we may drop the absolute valuefunction. Now, with the analogy of the volume of an ellip- An eigenvalue less than -1 would mean that the void wouldhave suffered shell crossing at the position of its centre, which isdynamically impossible as we are at the farthest distance possibleof any high density structure. soid, we may write the ratio ν between the minor axis andthe major axis ν ( q ) = s λ ( q )1 + λ ( q ) , (11)and the ratio µ between the second major axis and the majoraxis µ ( q ) = s λ ( q )1 + λ ( q ) . (12)This allows us to define the ellipticity ε ( q ) = 1 − ν ( q ) = 1 − s λ ( q )1 + λ ( q ) . (13)We will define the ellipticity of a void as the value taken by ε at the Lagrangian position of the void.A picture of the concept of voids and ellipticities in thiswork is given by Fig. 1. The painted paraboloid representsa small piece of a larger 2D density field whose value isencoded in the height and the colour. According to our def-inition, the void is at the centre of the paraboloid. At thiscentre, the surface of the volume element is mostly circular.The tidal forces are locally transforming the shape of thissurface, which produces the new elliptic shape on the rightside of the figure. The surface has been here extended alongone direction and slightly compressed along the other.Though we are not strictly limited to study elliptic-ity only at the position of the void it may be promisingin terms of robustness to non-linear effects. Indeed, due tothe absence of shell-crossings inside voids, the MAK recon-struction should give the exact solution (Brenier et al. 2003;Mohayaee et al. 2006; Lavaux et al. 2008) to the orbit re-construction problem. It means that the ellipticity that we We used here the convention of Park & Lee (2007) who takethe square root of the ratio to define µ and ν .c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics will compute will be exact, to the extent that we have takencare of the other potential systematics due to observationaleffects (Lavaux et al. 2008). As any other method relyingon dark matter distribution, we will be sensitive to the factthat the large-scale galaxy distribution is potentially biased.However, if the bias does not depends wildly on redshift, weshould be able to compute statistics on ellipticities and de-rive the evolution of the growth factor of Large Scale Struc-tures.We note that, using MAK reconstruction, we have ac-cess to the joint distribution of the three eigenvalues. Ourcomputation of the ellipticity consists in a projection of thewhole joint 3d joint distribution on a 1d variable. For cos-mological analysis, it is not entirely clear which estimator isthe more robust. On one hand, our intrinsic variables are theeigenvalues and we could include them in the analysis justas well as the ellipticities On the other hand, using ellipticitymay be helpful to average on a lot of different voids. It maybe a more robust estimator with respect to badly modelledtails of the distribution of eigenvalues. In this work, we focuson the use of the ellipticity, as defined in Eq.(13). We define the volume ellipticity ε vol using the eigenvalues ofthe inertial mass tensor (Shandarin et al. 2006): M xx = N p X i =1 m i ( y i + z i ) , M xy = − N p X i =1 m i x i y i , (14)where m i and x i , y i , z i are the mass and the coordinatesof the i -th particle of the void with respect to its centreof mass. The other matrix elements are obtained by cyclicpermutation of x , y and z symbols. We put I j the j -th eigen-values of the tensor M , with I I I . We may nowdefine the volume ellipticity as in ε vol = 1 − (cid:18) I + I − I I + I − I (cid:19) / (15)Even though our work is focused on the tidal ellipticity (Sec-tion 4.2), there is some interest to compare how the Eulerianvolume ellipticity compares to the local tidal ellipticity, asmost of the existing void finders use ε vol as a probe for thedynamics.To have a fair comparison with DIVA results, we arecomputing the inertial mass tensor from the displacementfield Ψ ( q ) smoothed on the scale as for the rest of the anal-ysis. The void domain is defined as specified in Sec. 3.1. Theinertial mass tensor is thus: M xx = Z d q (cid:0) ( q y + Ψ y ( q )) + ( q z + Ψ z ( q )) (cid:1) , (16) M xy = − Z d q ( q x + Ψ x ( q ))( q y + Ψ y ( q )) . (17)with the other elements obtained by cyclic permutations.The volume ellipticity ε vol is compared to the tidal elliptic-ity ε DIVA in Section 5.3.3. Except in that section, we onlyconsider ε DIVA in this paper.
There is an apparent price to pay to go to Lagrangian co-ordinates. One has to set a smoothing scale in Lagrangian coordinates and study the dynamics at corresponding massscale and let go of the evident notion of whether we see ahole in the distribution of galaxies or not. It actually couldbe an advantage. Smoothing at different Lagrangian scalesallows to probe the structures at different dynamical epochof the void formation. Each Lagrangian smoothing scale cor-responds to a different collapse time: the smallest scales be-ing the fastest to evolve. DIVA in this respect allows us tostudy the dynamical properties of a the voids which havethe same collapse time.This approach is related to the peak patch pictureof structure formation (Bond & Myers 1996), which is asimplified but quite accurate model of the dynamic ofpeaks in the density field. This model is even more pre-cise for the void patches, which is the name of the equiv-alent model for studying voids (see e.g. Sahni et al. 1994;Sheth & van de Weygaert 2004; Novikov et al. 2006). Ofcourse, the number of voids depends on the filtering scale(see Section 4.3 and Section 5.3.2). If we smooth on largescales we should erase the smaller voids and leave only thevoids whose size is large enough.Smoothing also affects the ellipticity distribution. Aswe smooth to larger and larger scales the density distri-bution probed by the filter should become more and moreisotropic. This leads voids to become more spherical andthus the ellipticity distribution should be pushed towards aperfect sphere. In this paper, we consider a few scales sep-arately and try to understand what were the properties ofthe minima at each of these scales (see Section 5.1).
In this section, we describe an analytical model of the dis-placement field. This model is based on Zel’dovich approx-imation (Zel’dovich 1970). In a first step (Section 4.1), werecall the statistics of the shear of the displacement field.Then, in Section 4.2, we express the ellipticity defined byEq. (13) in terms of this statistic. Finally, we explicitly writethe required statistical quantity in the model of Gaussianrandom fields and give some expected general properties ofthe voids in this model in Section 4.3.
Park & Lee (2007) described an analytical model of voidellipticities based on the Zel’dovich approximation. Thismodel should be particularly suitable on making predictionsof the result given by DIVA, knowing the previous successesof MAK in this domain (Mohayaee et al. 2006; Lavaux et al.2008). The model that Park & Lee (2007) have proposed isbased on the unconditional joint distribution of the eigenval-ues of the tidal field matrix J l,m (Doroshkevich 1970), giventhe variance of the density field σ (Appendix A): P ( λ , λ , λ | σ ) =33758 √ σ π exp " (cid:0) K − K (cid:1) σ | ( λ − λ )( λ − λ )( λ − λ ) | . (18) c (cid:13)000
Park & Lee (2007) described an analytical model of voidellipticities based on the Zel’dovich approximation. Thismodel should be particularly suitable on making predictionsof the result given by DIVA, knowing the previous successesof MAK in this domain (Mohayaee et al. 2006; Lavaux et al.2008). The model that Park & Lee (2007) have proposed isbased on the unconditional joint distribution of the eigenval-ues of the tidal field matrix J l,m (Doroshkevich 1970), giventhe variance of the density field σ (Appendix A): P ( λ , λ , λ | σ ) =33758 √ σ π exp " (cid:0) K − K (cid:1) σ | ( λ − λ )( λ − λ )( λ − λ ) | . (18) c (cid:13)000 , 000–000 G. Lavaux & B. D. Wandelt with K = λ + λ + λ , (19) K = λ λ + λ λ + λ λ . (20)as defined in Appendix A.This expression however neglects the fact that voids cor-respond to maxima of the source of displacement. As thecurvature of S Ψ = λ + λ + λ is correlated with J l,m , weneed to enforce that we are actually observing the eigen-values in regions where the curvature of S Ψ is negative. Abetter expression would be derived if we could constrainthat the Hessian H (the matrix of the second derivatives)of S Ψ is negative, which is the case in the vicinity of max-ima of S Ψ , the source of the displacement field. We derivein Appendix B a general formalism that allows us to com-pute numerically the probability P ( λ , λ , λ | σ T , r, H < { λ , λ , λ } given that we look inthese regions. This formalism is a natural extension of theformula of Doroshkevich (1970) (for which a simple deriva-tion is given in Appendix A).Of course, “true voids” have the additional constraintthat λ i > i = 1 , ,
3. As we assumed in previoussections that eigenvalues are ordered according to λ > λ >λ , the constraint λ > Whether we use the conditional probability P ( λ , λ , λ | σ T , r, H <
0) or the unconditional one P ( λ , λ , λ | σ T ), both under notation P ( λ , λ , λ | σ T , r ),we may now express the probability to observe δ , ν , µ [defined in Equations (5), (11) and (12)] in terms of P : P ( µ, ν, δ | r, σ T ) = P ( λ , λ , λ | r, σ T ) × δ − µν (1 + µ + ν ) , (21)with λ = − µ − ν + δν µ + ν , (22) λ = − − µ + δµ + ν µ + ν , (23) λ = − − δ + µ + ν µ + ν . (24)The ellipticity distribution of voids is thus P ( ε | σ T , r ) = 1 N Z + ∞ δ = −∞ Z µ =1 − ε P ( µ, ν, δ | σ T , r ) d µ d δ, (25)with N = Z + ∞ δ = −∞ Z ν =0 Z µ = ν d µ d ν d δ P ( µ, ν, δ | σ T , r ) . (26)The alternative distribution for “true voids” is given by en-forcing that λ > λ ) in Eq. (21) and renormalising.We note that the ellipticity that we are consideringhere is of dynamical nature (as emphasized by Park & Lee In terms of primordial density fluctuations, voids correspond tominima of the density field. As MAK is providing a good approx-imation of this field, we may safely jump from one concept to theother.
Shapes of voids –
Now we may compute the ellipticity distri-bution of voids, P ( ε | S Ψ ) for a given cosmology. The varianceof the density field σ T , assuming the power spectrum of thedensity fluctuations P ( k ), is given by σ T = 12 π Z + ∞ P ( k ) W R f ( k ) k d k , (27)with W R f ( k ) = exp − k R f ! (28)the Fourier transform of the filter function used to smooththe density field on the scale R f . In practice, we smooththe displacement field in Lagrangian coordinates, to reducenoise and non-linear effects appearing at small scales in theMAK reconstruction ( . h − Mpc). Thus, we will computethe ellipticity distribution of voids, given that we smoothedon the scale R f in Lagrangian coordinates, and that thelocal source of displacement of the void is S Ψ ( q ).With the model P ( λ , λ , λ | σ T , r, H <
0) of Ap-pendix B, we may also estimate the number of voids in eachclass we defined in Section 3.1. As an illustration, we pickeda usual ΛCDM cosmology, with Ω m = 0 .
30, Ω b = 0 . σ = 0 . h = 0 .
65 and estimated the fraction of voids ineach class. The results are, when we smooth at 4 h − Mpc:- true voids : We estimate that these voids represent ∼
40% of the primordial voids, which correspond to under-densities in the primordial density fluctuations.- pancake voids : Doing the same estimation as for “truevoids”, we get that ∼
50% of the primordial voids should bein that case.- filament voids : They correspond to ∼
10% of the pri-mordial voids.We show in Figure 2, the analytical distributionsof ellipticity for the two cases when one constrains (ornot) the curvature of S Ψ to be negative. The solid curvecorresponds to the ellipticity distribution obtained using P ( λ , λ , λ | H < λ > −
1. In the right panel, we restricted our-selves to “true voids”. The difference is striking in the leftpanel between the two models, whereas in the right panelthey essentially give the same prediction. This can be un-derstood by looking at the value of the correlation coeffi-cient between the curvature of S Ψ and T l,m (also defined inEq. (B10)) r = − S √S S , (29)with S n = 12 π Z + ∞ k =0 k n P ( k ) d k (30)This coefficient is equal to ∼ -0.67 for the aforementioned c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics Figure 2.
Comparison of MAK reconstructed and analytical ellipticity distribution –
We represent here the distribution of ellipticity ofthe voids, marginalised over all possible S Ψ . We used black squares for the ellipticity distribution obtained using the MAK reconstructeddisplacement field applied on the simulation. The dashed blue curve is computed using the unconditional Doroshkevich (1970) formula.The red curve is our new formula obtained by conditioning that voids are regions where the density field has a positive curvature. Theleft panel gives the result for all voids (true, pancake and filament). The right panel gives the result for only true voids. The blue dashedcurve and red solid curve overlap. All fields were smoothed with a Gaussian kernel of radius 4 h − Mpc. cosmology. This indicates that the two curvatures are quitestrongly linked. Thus, enforcing that T l,m is positive causesthe curvature of S Ψ to be preferentially negative. So, the twodistributions of the right panel of Fig. 2 should be similar.On the other hand, for the left panel no such implicationexists, which leads to the largely evident discrepancies ofellipticities. We note the distributions of the right panel areonly measurable using our void finder as the other ones can-not distinguish truly expanding voids and pancake voids justby looking at their shape.Number of voids –
Now that we know the shape thevoids should have in the context of linear theory, we wouldlike to know how many of them should be present in the Uni-verse. With our definition of voids, we may conveniently usethe results obtained by Bardeen et al. (1986) for Gaussianrandom fields. In particular in the equation (4.11b), theyshow that the average number density of maxima is equalto n max = 29 − √ / π ) R ∗ ≃ . R − ∗ . (31)with R ∗ = r S S (32)in our notation. We note that this is a mean number, sowe expect some fluctuation according to the mean whichare difficult to compute analytically. Tests on Gaussian ran-dom field seems to point out that the number of voids aredistributed like a Poisson distribution. We also expect thisnumber to slightly overestimate the actual density of voidsthat we will find in simulations (Section 5.2). In the next section, we are now going to confront theanalytical model with the results given by DIVA applied on N -body simulations. N -BODY SIMULATIONS In this section, we are going to identify voids in the N -bodysamples described in Section 5.1. We then give a sketch (Sec-tion 5.2) of the procedure we followed to perform the MAKreconstruction, which corresponds to the one described inLavaux et al. (2008). In Section 5.3, we focus on the resultsobtained at z = 0. First, we give an illustration of a void ofeach class in Section 5.3.1. We then compare the results ob-tained using the displacement field given by the simulationand the one reconstructed by MAK (Section 5.3.2). There,we also detail the number of voids detected and their ellip-ticities for both fields. We compare quantitatively the dis-tribution of Eulerian volume ellipticity to the Lagrangiantidal ellipticity in Section 5.3.3. Finally, we check the valid-ity of the analytical model on the simulated displacementfield (Section 5.3.4). In Section 5.4, we look at the evolutionof the mean ellipticity for a simulation with w = − w is either − − . N -body simulations To test our void finder, we use three large volume N -bodysimulations but with medium resolution of N = 512 , L = c (cid:13) , 000–000 G. Lavaux & B. D. Wandelt h − Mpc, Ω m = 0 .
30, Ω Λ = 0 . H = 65 km s − Mpc − , n s = 1, σ = 0 .
77, Ω b = 0 .
04. The N -body simulations con-tains only dark matter and we include the effect of baryonsonly through power spectrum features in the initial condi-tions. This essentially reduces power on scales smaller thanthe sound horizon and introduces Baryonic Acoustic Oscilla-tions. The two first N -body simulation (ΛSIM and ΛSIM2)corresponds to a standard ΛCDM cosmology for which theequation of state is given by w = −
1. ΛSIM and ΛSIM2have exactly the same cosmology but have different initialconditions. They will allow us to assess the impact of look-ing at two different realisations of the initial conditions. Thethird, wSIM, is assuming an equation of state w = − . GRAFIC (Bertschinger 2001) package to use the powerspectrum generated by the
CAMB package (Lewis et al. 2000).We reach a sub-Mpc resolution scale which is sufficient forproper description of most voids (1-20 h − Mpc). The in-termediately large volume allows accounting for large-scaletidal field effects and cosmic variance effects. We used theparallelised version of the
RAMSES N -body code (Teyssier2002) and run it both on the Cobalt NCSA superclusterand the Teragrid NCSA supercluster through Teragrid fa-cilities (Catlett et al. 2008). We modified RAMSES to simulatecosmologies with a dark energy equation of state differentthan w = − l = 0 . N p = 8. This prescription in practice should mostlyerase the information contained in halos while retaining thedynamics outside of them. Though the use of such a lownumber for the particles of halos are questionable for thestudy of the properties of those halos, we are not here in-terested in them. We are only interested in checking thatwe keep most of the information useful to study voids andtheir overall dynamics, even though we destroy the smallscale information. The above prescription has already beensuccessfully applied for the study of peculiar velocities withMAK (Lavaux et al. 2008). We note that we will only usethe halo catalogue to do the MAK reconstruction. All thetests of the displacement field of the simulation are run onthe particles of the simulation.We note that that the initial conditions of the simu-lation present two particularities that must be taken intoaccount. The largest mode of the simulation box is k low =1 . × − h Mpc − thus introducing a sharp low-k cut.Additionally GRAFIC applies a Hanning filter on the initialconditions to avoid aliasing. In practice,
GRAFIC multipliesthe cosmological power spectrum by the following filter: W hanning ( k ) = (cid:26) (cid:2) cos (cid:0) k ∆ x (cid:1)(cid:3) if k ∆ x π , (33)with ∆ x = 0 . h − Mpc the Lagrangian grid step size ofour simulations. These two features must be introduced inthe power spectrum of Eq. (27) and (30) to make correctpredictions for the simulation. In real observational data,no such features should be present.
Among the different tests that we are going to present inthe following, we have run only one MAK reconstruction forthe full halo catalogue. We chose a resolution of 256 meshelements generated following the algorithm of Lavaux et al.(2008). This algorithm consists in splitting a mass in an in-teger number of MAK mesh elements, with the constraintsthat the splitting must be fair and the number of mesh el-ements is fixed and equal to 256 . This method works welland has, so far, not been prone to biases. Choosing this num-ber of elements gives us a resolution of ∼ h − Mpc on theLagrangian coordinates of the displacement field. We can-not run it at the full simulation resolution due to the highCPU-time cost which hinders doing several reconstruction.One reconstruction takes ∼ O ( N . ), increasing the resolution to 512 wouldhave consumed ∼ CPU-hours. So we limited ourself onrunning the reconstruction at the 256 resolution, the ex-pected worst case for the performance of MAK. At higherredshift, the MAK reconstruction converges faster and givesa more and more precise displacement field compared to theone given by simulation. These two features are both causedby the decrease of small scale non-linearities at earlier times.We took the halo catalogue built on ΛSIM at z = 0 and rana reconstruction on it. The other results presented in thispaper use the displacement field given directly by the simu-lation after having checked that the reconstruction is indeedsufficient for our purpose. This is the case as we will notlook at voids smaller than ∼ h − Mpc scale in Lagrangiancoordinates.We chose two different smoothing scales on which westudy the displacement field for voids: 2.5 h − Mpc and4 h − Mpc. Once the displacement field has been smoothed,we compute the eigenvalues and the divergences in Fourierspace. We then locate the maxima in the divergence of thedisplacement field. At each maxima, we compute the ellip-ticity ε with the help of Eq. (13), where the λ i are taken asthe eigenvalues of T l,m ( q ). The displacement shear tensor,is computed numerically from Ψ ( q ) in Fourier space: T l,m ( q ) = 1(2 π ) Z k d k ik m ˆΨ l ( k )e i k · q , (34)where ˆΨ l ( k ) is the Fourier transform of the displacementfield in Lagrangian coordinates. All these quantities wereevaluated using Fast Fourier Transform on the Lagrangiangrid.In summary, the steps are the following:- First, we prepare a catalogue for a MAK reconstruction.This involves doing fair equal mass-splitting of the objects.- Next, we run the MAK reconstruction.- After the reconstruction is finished, we put the com-puted displacement field given by MAK on the homogeneousLagrangian grid.- We smooth this displacement field and compute thetidal field T i,j in Lagrangian coordinates in Fourier spaceusing Eq.(34).- Now we compute S Ψ ( q ) on the grid using Eq. (5), whichcorresponds to summing the three eigenvalues of T i,j .- We look for local maxima in S Ψ ( q ) using an iterative c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics steepest descent algorithm. This gives us the list of the voidsin Lagrangian coordinates.- Using these coordinates, we now compute the ellipticityusing the eigenvalues of T i,j at the location of the minima.- For computing the void boundary, we execute the mod-ified Watershed transform of Section 3.1. This identifies theLagrangian domain of the voids. The boundary of this do-main is then transported using the displacement field to re-cover the Eulerian void volume.We now look at the results obtained with MAK, the simu-lation and the analytical model at z = 0 in the next section. z = 0 Before going to the statistical study of the local shape ε DIVA properties of void found by DIVA, we look at a visual ex-ample of each of the void type. We chose a filtering scale of4 h − Mpc in Lagrangian coordinates. We selected one void ofeach class. These three voids have the following properties:- The first void is a true void. The eigenvalues of thetidal field T l,m ( q ) (8) at the centre are (1 . , . , . ε =0 .
13. The volume of the void, in Lagrangian coordinates,is V L = 75240 h − Mpc , which corresponds to an equiva-lent spherical volume given by a sphere of radius R equiv =(3 / (4 π ) ∗ V ) / = 26 h − Mpc.- The second void is a pancake void. The eigenvalues ofthe tidal field are (1 . , . , − . . V L = 1560 h − Mpc , with R equiv = 7 . h − Mpc.- The last void is a filament void. Again, the eigenvaluesof the tidal field are (0 . , − . , − . ε = 0 .
557 and the Lagrangian volume is V L = 260 h − Mpc ,with R equiv = 4 . h − Mpc.Those three voids are represented in three dimensions, alongwith the particles of the simulation in the same region, inFig. 3. We note that the true voids is the largest one. Weexpect to observe that effect as true voids expands in threedirections and thus are more likely to be bigger than pan-cake voids and filament voids. For these three cases, thesurface delimited by DIVA seems to nicely fit to the struc-tures located on the boundaries. In the case of the pancakeand filament voids, we note that the cavity seem to be splitinto two pieces. This splitting is due to the Watershed trans-form criterion which isolated two void cavities separated bya structure.
We now concentrate on the properties of the voids at z = 0.This is the time where the density is the most non-linearand the reconstruction is the most difficult and thereforerepresents a worst case example. We take the displacementfield given by the simulation as the field of reference to studyvoids. Indeed, this field has been obtained by solving com-pletely the equation of motion for each particle. In this sec-tion, we will first compare the properties of the voids foundusing this field and the MAK reconstructed field. Then, wewill focus on how it compares to the analytic model. Filter Predicted Displacement Number ofaverage number field candidates2.5 h − Mpc 42908 Simulated 31002Reconstructed 283974 h − Mpc 11706 Simulated 10643Reconstructed 9412
Table 1.
Unconditioned number of voids in Λ SIM – we give herethe unconditioned number of voids found within the volume ofthe simulation, (500 h − Mpc) . The predictions are obtained us-ing Equation (31) applied on the power spectrum of primordialdensity fluctuations multiplied by the Fourier transform of the in-dicated filter in the first column. The last column gives the actualnumber of void candidates that we found using the displacementfield. We represent on Fig. 4 the distribution of ellipticitiesobtained using both the reconstructed and the simulated dis-placement field. We also give the number of voids found insimulated displacement field, in the reconstructed displace-ment field and the expected number of maxima according toEq. (31) (Table 1). To allow for a void-by-void comparison,we tried to match the voids found using the two displace-ment fields. We considered that two voids are the same if thedistance between the two voids is less than a Lagrangian gridcell ( d √ l cell , with l cell the length of the side of a cell).At 2.5 h − Mpc smoothing, the agreement of the elliptic-ity distribution derived from the simulated displacement andthe MAK reconstructed displacement is very good, thoughthe high ellipticity tail seems a little different in the twocases. This is actually due to a selection effect which, unfor-tunately, is correlated with the ellipticity. If we look at thenumber of voids detected using the two fields (Table 1), wesee that MAK is missing about 10% of the voids detected inthe simulation. The distribution of ellipticity of those voidshappens to be skewed towards higher ellipticities. Thus itseems that we tend to miss the most elliptical voids. Thisbehaviour is expected as these voids tend to be pancakevoids. So they are closing along one direction, and the moreelliptical they are the faster they are closing. If they close,the particles of these voids shell cross and MAK is not ableto reconstruct the displacement field. Looking at this samedistribution, but for a 4 h − Mpc smoothing, we now barelysee a difference between the two curves. We indeed checkedthat the ellipticity distribution of the voids that have notbeen identified using the MAK reconstructed displacementfield is similar to the distribution of the other voids.The number of voids detected in the simulation looksless than the expected average number of minima (Table 1).This is also due to the destruction of minima by the collapseof large scale structures. When we look at the beginning ofthe simulation we find 11,485 minima (for R f = 4 h − Mpc),and this number steadily decreases to the value we put in thetable as the simulation evolves. We estimated the mean andthe variance in the number of minima using 20 realisationsof a Gaussian random field with the same cosmology as thesimulation. We found that the mean should be 11,762 andthe standard deviation is 58, which is in agreement with theresult given by the analytic computation.Using the match between voids coming from the two c (cid:13) , 000–000 G. Lavaux & B. D. Wandelt
Figure 3.
Example of voids – We illustrate here the voids that are found using DIVA. Each of these belong to one of the void class thatwe listed in Section 3.1. The scale of the box is the same for the three cases: the side corresponds to 50 h − Mpc. fields, we can build a statistical error model in the form ofthe joint probability distribution P ( ε MAK , ε
SIM ) of gettingan ellipticity ε MAK using MAK displacement and an ellip-ticity ε SIM ) using simulated displacement. It is importantto check P ( ε MAK , ε
SIM ) if, as we will do in future, we wantto estimate cosmological parameters from ellipticity distri-bution. This function tells us what accuracy we may expecton the ellipticity measurements. We represent this probabil-ity in the left panel of Figure 5. We see a strong correlation,already seen in Fig. 4, indicating a high accurate reconstruc-tion. We also see that the error seems low. We represent atthe left side of the thick red solid line of the right panel theconditional probability P ( ε MAK | ε SIM ) that we computed us- ing: P ( ε MAK | ε SIM ) = P ( ε MAK , ε
SIM ) R ε MAK =0 P ( ε, ε SIM )d ε (35)wherever it was possible to evaluate the denominator. Thisconditional probability is mostly Gaussian, so we tried toestimate the standard error by computing the mean varianceof the error on the ellipticity for the distribution between thetwo dotted red line ε SIM ∈ [ ε S,min ; ε S,max ] with ε S,min = c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics Figure 4.
MAK reconstructed ellipticity distribution vs. ellipticity distribution from simulated displacement field – This figure gives theellipticity distribution computed using either the MAK reconstructed displacement field (solid black line) or the simulated displacementfield (solid thick gray line). We represent the dispersion expected if the error on estimating the distribution come from the number ofvoids in a given bin. We assumed a Poisson distribution for the estimation of the error bar. The displacement fields were both smoothedwith a Gaussian kernel of radius R f = 2 . h − Mpc in the left panel and to R f = 4 h − Mpc in the right panel.
Figure 5.
Ellipticity derived from the simulated displacement field vs. the MAK displacement field – We represent here a scatter plotbetween the ellipticities of the voids that were both detected in the MAK reconstructed displacement field and the simulated displacementfield, both smoothed at 4 h − Mpc. In the left panel, we show the raw joint probability distribution of the two ellipticities. The gray-scaleis linear in the density of probability. In the right panel, we represent the conditional distribution of ε MAK given ε SIM , on the left of thethick vertical line. On the right of this same line, we represent this same distribution if one uses the estimated standard deviation σ ε ofthe error. It is estimated using the distribution between the two vertical dotted lines. The estimated standard deviation is σ ε = 0 . (cid:13) , 000–000 G. Lavaux & B. D. Wandelt ε DIVA ε v o l ε DIVA ε v o l Figure 6.
Tidal ellipticities vs Volume ellipticity – This plotgives a comparison of the ellipticity of the void as determinedeither using the shear of the displacement field [ ε DIVA , Eq. (13)]or using the overall shape of the void [ ε vol , Eq. (15)]. .
15 and ε S,max = 0 .
40. With this notation, we computed σ ε = 1 ε S,max − ε S,min Z ε S,max ε = ε S,min d ε sZ ε mak =0 d ε mak ( ε mak − ε ) P ( ε mak | ε ) . (36)Within the interval limited by the two dotted red line, weestimate σ ε ≃ . P ( ε | ε SIMU , σ ε ) = 1 √ πσ ε e − σ ε ( ε − ε SIMU ) (37)with σ ε as estimated above. We note that this model ofthe conditional probability (right of the vertical solid line)looks much like the actual ellipticity discrepancy (left of thevertical solid line). ε vol vs. Tidal ellipticity ε DIVA
In Fig. 6, we represented a comparison between the elliptic-ity of the volume, ε vol , and the local tidal ellipticity, ε DIVA .The volume ellipticity is computed using the Eq. (15), withthe inertial mass tensor M as computed using the smootheddisplacement field. Visually, the two variables seem looselycorrelated. We observe they do follow each other but thiscorrelation just get worse when the ellipticity increases. Thisis expected as the volume ellipticity is a non-local quan-tity and thus is sensitive to all local ellipticities in the voidvolume. This is what makes ε vol more difficult to use as aprecise cosmology probe. We show in Appendix C that thesetwo quantities are indeed related but only to first order. Thisexplains that the scatter seems smaller for small ellipticitiesthan for high ellipticities. The volume ellipticity, which has been used so far, seems thus to be a poor proxy of the tidal el-lipticity, which we manage to recover with extreme precisionusing our Lagrangian orbit reconstruction technique.
For therest of this paper, we will only use the tidal ellipticity.
In this section, we compare the results obtained on the simu-lated displacement field and the prediction given by the an-alytical model of Section 4. We represented in the left panelof Fig. 2 the ellipticity distribution of all observable voidsas defined in Section 3.1. The black points give the elliptic-ity distribution for voids in the reconstructed displacementfield. We estimated error bars assuming a Poisson distri-bution of the number of voids in each bin. The red line isobtained using the method of Appendix B. The dashed blueline is obtained through the formula of Park & Lee (2007),where no constraints are put on the curvature of the densityfield where the ellipticity of the void is measured.The success of the comparison between the black pointsand the solid red curve is striking. It shows the importanceof our constraint that we only look in the negatively curvedpart of S Ψ . We note that this should be a robust featurefor voids found with any void finder. Any probe of the el-lipticity in voids looks in regions of the density field thatis strongly underdense, and thus should come mainly frominitially underdense regions. In these primordial underdenseregions, the curvature of the density field is likely to be pos-itive, which corresponds to a negatively curved S Ψ in ourcase. Our measured ellipticity distribution in the simulationis very clean because we rely on features of the displacementfield which can be understood in terms of linear theory evenat redshift z = 0 . In the right panel of Fig. 2, we show this same ellipticitydistribution but only for “true voids”. The comparison be-tween simulation and analytic is also here successful. As wenoted in Section 4.3, there is no real difference between thetwo models in this case. However, there is no way a purelygeometrical analysis could yield this curve from the obser-vation of galaxy catalogues, as this requires the knowledgeof the sign of eigenvalues of T l,m (Eq. 6). We note a smallshift of ∼ In this section, we focus on the evolution with redshift ofthe ellipticity of voids. This evolution has been shown to beanalytically a sensitive probe of w by Lee & Park (2009).We took snapshots of the simulation at different redshiftsand computed the ellipticity distribution in each of thesesnapshots. We note that we must at least have two main dif-ferences compared to what would happen with observations.First, we may have a systematic effect in the evolution of themean ellipticity as we are studying only a single realisationof initial conditions. Second, the number of available voidsshould be a non-trivial function of redshift. Second, becauseof both volume and selection effects, the error bars should c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics Figure 7.
Evolution of the mean ellipticity with redshift – We represent the evolution of the mean ellipticity with redshift for the twoΛCDM simulations (left panel) and the wCDM simulation (right panel). The mean ellipticities deduced from ΛSIM are represented usingsquare symbols, and the ones from ΛSIM2 using triangular symbols. The solid curve is obtained using the statistical model of Appendix Band changing σ according to the growth factor as specified in Eq. (38). The bottom panels give the relative difference, in percentage,between the simulation and the analytical model. In the bottom left panel, the points at ∼
2% correspond to the square symbols be large at both small and large redshift, while attaining aminimum at some intermediate redshifts The exact numbersfor this second problem depends on the specific consideredgalaxy survey, in particular the apparent magnitude limitand the incompleteness function.To compute the predicted ellipticity distribution at anygiven redshift z , we simply scale the σ ( z ) parameter usingthe growth factor D ( z ): σ ( z ) = σ ( z = 0) × D ( z ) D ( z = 0) . (38)For clarity we only represent on Fig. 7 the mean ellip-ticity ¯ ε , defined as,¯ ε ( z ) = Z εP ( ε | z )d ε (39)with P ( ε | z ) the probability distribution of the ellipticity ε at redshift z . The red solid line gives the prediction givenby the analytic model of Section 4. The black points areobtained from the simulated field. The error bar on the meanellipticity is estimated using σ ¯ ε ≃ σ ε √ N voids (40)with σ ε = 0 .
02 as estimated in Section 5.3.2 for a smoothingscale of 4 h − Mpc. For N voids ≃ , this gives a typical errorof σ ¯ ε = 2 × − on the mean. This error bar correspondsto the uncertainty of the ellipticity derived from the MAKreconstructed displacement field with respect to the one ob- tained from the simulated displacement field. This gives aninterval on which the mean ellipticity can be trusted.In the left panel of Fig. 7, we see that the agreementbetween the analytical model and the one from the simu-lated displacement field (square symbols, “Simulation 1”)is very good for w = −
1. Looking at the relative error be-tween the model (lower-left panel) and the simulation yieldsa systematic ∼
2% deviation relative to the expectation. Themain reason is the inexactness of the initial conditions tothe finite number of modes available in the simulation box.Even though the power spectrum is normalised to σ = 0 . σ of the displacement field is 0 . σ = 0 . − . c (cid:13) , 000–000 G. Lavaux & B. D. Wandelt σ . Thus the points should be scattered according to ourdashed horizontal line “0%” and not systematically pushedup or down. w = − . vs w = − . w = −
1. However, we firststarted to look at voids to check if they may be good tracersof the properties of the dark energy, and in particular of thegrowth factor. We now focus on the results obtained fromwSIM, a wCDM simulation with w = − . σ = 0 . .
33% above 0.77. We again note that the discrepancy in thelower-right panel in Fig. 8 has the correct systematic shiftat low redshift. Taking into account this shift, as previously,the analytical model seems to follow the results given bysimulation at the 0 .
1% level, taking into account the statis-tical uncertainty. The points obtained from the simulationare in excellent agreement with the simulation.Current redshift galaxy catalogues map the Universe atintermediate redshifts 0 . z .
1. We check if our methodis sufficiently sensitive to distinguish a w = − . w = − σ as measured in the simulations to compute the analyticalpredictions (red and blue solid curves). We note that evenat z ≃ .
2, the behaviour of the two curves is already sig-nificantly different and above statistical uncertainties. If weconsider the whole interval between z = 0 and z = 1, thedifference is very significant compared to the uncertainties,with an ellipticity that changes by ≃ ∼ z = 0 . ∼ h − Mpc), and we take a Lagrangian smoothing scaleof 5 h − Mpc. This smoothing scale is motivated by the av-erage density of galaxies in the SDSS, which in band r isabout 1 − × − h Mpc (Blanton et al. 2003). This givesa mean separation of ∼ − h − Mpc. The equation (31)predicts that we should observe ∼ h − Mpc, taking intoaccount the survey coverage. If we go to z = 0 .
2, this numbershould increase to ∼ z . .
2. The Fishermatrix analysis is done in a companion paper.
In this section, we discuss how our technique is related tothe other existing void finders. We try to make a qualitative
Simulation 2 w=−1Simulation w=−0.5Model w=−1Model w=−0.5
Redshift (z) < ε > ( z ) Figure 8.
Difference between w=-1 and w=-0.5 – We plottedhere the evolution of the mean ellipticity with redshift. We usedthe simulation ΛSIM2 (square) and wSIM (triangle). The red solidline gives the prediction for σ = 0 . w = −
1. The blue solidline gives the prediction σ = 0 .
773 and w = − .
5. These twovalues of σ have been computed using the initial conditions ofthe two simulations. assessment of its strengths and weaknesses compared to thethree classes of void finders define in Section 1.The void finders of the first class try to find emp-tier regions in a distribution of points, which in ac-tual catalogues correspond to galaxies. The void finderdeveloped by El-Ad et al. (1997), and one of its laterversions by Hoyle & Vogeley (2002), is popularly usedin observations (Hoyle & Vogeley 2004; Hoyle et al. 2005;Tikhonov & Karachentsev 2006; Foster & Nelson 2009). Inthese void finders, the first step consists in classifying galax-ies in two types. Galaxies may lie in a strongly overdenseregion, in this case we consider it as a “wall galaxy”. Theother possibility is that they lie in a mildly underdense re-gion, and they are then called “field galaxies”. The exactseparation between “wall galaxies” and “field galaxies” de-pends on an ad-hoc parameter. This parameter specifies howthe local density of galaxy control the type (field or wall) ofthe galaxy. The voids are then grown from regions emptyof wall galaxies. This classification thus gives a non-trivialdependence of the void sizes and shapes on the galaxy biasand catalogue selection function. Additionally, this defini-tion depends on an ad-hoc empirical factor. These issuesmake the quantitative study of the geometry of voids dif-ficult, while they find voids that correspond to the visualimpression given by large scale structures in redshift cata-logues of galaxies.Void finders belonging to the second class look for par-ticularities in the continuous three dimensional distributionof the dark matter traced by galaxies. Of course, from obser-vational data, one has then to first project and then smooththe distribution of galaxies to obtain this distribution. Dif-ferent techniques are used: c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics - One technique is to adaptively smooth the galaxydistribution either using an SPH technique (see e.g.Colombi et al. 2007) or a Delaunay Tesselation Field Esti-mator (Schaap & van de Weygaert 2000). Voids must thenbe identified from the smooth distribution derived using ei-ther of these techniques. One option is to use a scheme simi-lar to El-Ad et al. (1997) or Hoyle & Vogeley (2002) to iden-tify shapes of underdensity in the vicinity of a minima of thedensity field (Colberg et al. 2005). As with void finders ofthe first class, the galaxy bias does not affect the positionsof the voids but their overall properties. A second option isto use a Watershed Transform (Platen et al. 2007) to iden-tify the cavities of the voids. In this case, the galaxy biasdoes not affect the structure of the cavities. However, devis-ing an efficient way of relating the geometrical properties ofthese cavities to the cosmology, which corresponds to study-ing Morse theory, could well be non-trivial (Jost 2008). Somework to study the skeleton (also called the “cosmic spine”)of Large Scale structures in this theory has been recentlydone by Aragon-Calvo et al. (2008), Pogosyan et al. (2009)and Sousbie et al. (2009).- A second technique consists in using the local densityestimated using the Voronoi diagram of a Delaunay tessel-lation to locate minima (Neyrinck 2008). The particles arefirst grouped in zones. Each particle is assigned to a zone onto where this particle is attracted if it followed the densitygradient as in the watershed technique. Each zone is definedto be a void. But it is also possible to define a hierarchyof voids by checking, for two neighbouring voids, which ofthe two has the lowest density at the minima. The zones arethen grouped and a probability of being a void is assigneddepending on the contrast between the density at the ridgeof the void and its depth.This void finder has the advantage of defining voids in termsof topology of the density field, which is easier to handle froma theoretical point of view and may better define a voidin terms of dark matter. Still, we are faced with the taskof relating the shape of the voids that are found by thesealgorithms, which is non-local by nature, to cosmology. Asmentioned previously for void finders of the first class, thisseems to be non-trivial.Void finders of the third class use the inferred dark mat-ter distribution but they do a dynamical analysis to infer thelocation of these voids. DIVA belongs to these class of voidfinders. There are two advantages of looking at dynamicsfor voids. (i) It gives a much more physical and intuitivedefinition of these structures: voids corresponds to places inthe universe from which the matter is really escaping andnot gravitationally unstable at present time. (ii) Using thiscriterion, one is bound to use either the velocity field or thedisplacement field. These two quantities are highly linear. Ithas been directly shown for velocity fields by Ciecielg et al.(2003) and indirectly shown by Mohayaee et al. (2006) andLavaux et al. (2008) for the displacement field. This linear-ity helps us at constructing an analytical statistical modelof the voids. Hahn et al. (2007) and Forero-Romero et al.(2009) attempted to classify structures according to a cri-terion on the gravitational field. However we may highlighttwo very important differences compared to the approachwe are following here:- we are using a purely Lagrangian method and it takes into account the true evolution of the void and not howvirtual tracers in the void should move now,- we put an exact natural threshold on eigenvalues toclassify voids. This is in contrast with Forero-Romero et al.(2009) where they need to to put a threshold on eigenval-ues depending on an estimated collapse time. In our case,everything is already integrated in the definition of the dis-placement field.Moreover, the Monge-Amp`ere-Kantorovitch reconstructionpresents the two advantages of: (i) never diverging in theneighbourhood of large density fluctuations, compared toa pure gravitational approach, (ii) recovering exactly theZel’dovich approximation in the neighbourhood of centresof voids.As for the other void finders, DIVA depends on galaxybias. We recall that the bias b is defined with δ g ≃ bδ m , (41)with δ g the density fluctuations of galaxies and δ m the den-sity fluctuations of matter. As MAK is essentially recon-structing the Zel’dovich displacement in underdense regions,and that the Zel’dovich potential is proportional to densityfluctuations, the MAK displacement should also be mostlylinear with the bias.We describe in a second paper (Lavaux & Wandelt 2009,in prep.) how to relate the volume of the voids that we findin Lagrangian coordinates to the voids that we observe inEulerian coordinates. We have described a new technique to identify and char-acterise voids in Large Scale structures. Using the MAKreconstruction, we have been able to define void centres inLagrangian coordinates by assimilated them to the maximaof the divergence S Ψ of the displacement field, interpretedas its source term. The scalar field S Ψ has the interestingproperty of being nearly equal to the opposite of the lin-early extrapolated primordial density field (Mohayaee et al.2006). This allowed us to consider that the statistics of thosetwo fields were equal. Using that, we made predictions onthe number of voids in Lagrangian coordinates, along withtheir ellipticities defined using the eigenvalues of the curva-ture of S Ψ .We tested our model using N -body simulations withdifferent cosmologies ( w = − w = − . ∼ c (cid:13) , 000–000 G. Lavaux & B. D. Wandelt the evolution of the number of voids and their size distribu-tion. We will make further robustness tests using mock cat-alogues, including especially redshift distortion effects. Wealso would like to apply our method to the Luminous RedGalaxy sample of the SDSS DR7 (Abazajian et al. 2009).
ACKNOWLEDGEMENTS
This research was supported in part by the National Sci-ence Foundation through TeraGrid resources provided bythe NCSA. TeraGrid systems are hosted by Indiana Uni-versity, LONI, NCAR, NCSA, NICS, ORNL, PSC, PurdueUniversity, SDSC, TACC and UC/ANL. The authors ac-knowledge financial support from NSF grant AST 07-08849.We acknowledge the hospitality of the California Institute ofTechnology, where the authors completed most of this work.We thank the referee, Rien van de Weygaert, for his usefulcomments and suggestions.
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APPENDIX A: EIGENVALUES PROBABILITYDISTRIBUTION
In this appendix, we derive the unconditional distribution ofthe eigenvalues of the Jacobian matrix of the displacementfield in the Zel’dovich regime. Starting from the potentialΦ( q ) of the displacement field at the comoving Lagrangiancoordinate q , we define the Jacobian matrix of the displace-ment field T i,j = ∂ Φ ∂q i ∂q j . (A1)This matrix is real and symmetric. We assume the com-ponents to be normally distributed and to be as given by(Bardeen et al. 1986, Appendix A): h T i,j T k,l i = σ
15 ( δ i,j δ k,l + δ i,k δ j,l + δ i,l δ j,k ) , (A2)with σ the standard deviation of the density field. As itis a 3 × A = 1 . . .
6, where each number refer to the ( i, j ) couples(1 , , (2 , , (3 , , (1 , , (1 ,
3) and (2 , V of the variances V = σ (A3)thus the covariance matrix C is C = V − = 1510 σ − − − − − − (A4)The unconditional probability d P ( { T A } ) of observing thesecomponents at a point in the universe is given byd P ( { T A }| σ ) = 337516 √ σ π exp − X A,B =1 C A,B T A T B ! Y A =1 d T A (A5)We want now to make a change of variables and get the prob-ability distribution of the eigenvalues of T i,j . The appendixB of Bardeen et al. (1986) shows that the infinitesimal vol-ume is transformed according to: Y A =1 d T A = | ( λ − λ )( λ − λ )( λ − λ ) | d λ d λ d λ dΩ S , (A6)with dΩ S the infinitesimal rotation of the hypersphere ofdimension 3. The quadratic form Q in the argument of theexponential of Eq. (A5) may be expanded Q = X A,B =1 C A,B T A T B = 15 σ (cid:0) T + T + T ) − T T + T T + T T ) + T + T + T (cid:1) . (A7)Linear algebra tells us that the scalar quantities K = T + T + T and K = T T + T T + T T − T − T − T (A8)are invariant by change of vector basis. The quadratic formmay then be rewritten: Q = 6 σ (cid:18) K − K (cid:19) (A9)We may now express K and K in terms of the eigenvalues: K = λ + λ + λ , (A10) K = λ λ + λ λ + λ λ . (A11)After integrating the expression (A5) over S , we obtain theunconditional probability that the Jacobian matrix has thethree ordered eigenvalues λ , λ and λ : P ( λ , λ , λ | σ ) =33758 √ σ π exp " (cid:0) K − K (cid:1) σ | ( λ − λ )( λ − λ )( λ − λ ) | . (A12)This equation corresponds to the distribution derived byDoroshkevich (1970). c (cid:13)000
3) and (2 , V of the variances V = σ (A3)thus the covariance matrix C is C = V − = 1510 σ − − − − − − (A4)The unconditional probability d P ( { T A } ) of observing thesecomponents at a point in the universe is given byd P ( { T A }| σ ) = 337516 √ σ π exp − X A,B =1 C A,B T A T B ! Y A =1 d T A (A5)We want now to make a change of variables and get the prob-ability distribution of the eigenvalues of T i,j . The appendixB of Bardeen et al. (1986) shows that the infinitesimal vol-ume is transformed according to: Y A =1 d T A = | ( λ − λ )( λ − λ )( λ − λ ) | d λ d λ d λ dΩ S , (A6)with dΩ S the infinitesimal rotation of the hypersphere ofdimension 3. The quadratic form Q in the argument of theexponential of Eq. (A5) may be expanded Q = X A,B =1 C A,B T A T B = 15 σ (cid:0) T + T + T ) − T T + T T + T T ) + T + T + T (cid:1) . (A7)Linear algebra tells us that the scalar quantities K = T + T + T and K = T T + T T + T T − T − T − T (A8)are invariant by change of vector basis. The quadratic formmay then be rewritten: Q = 6 σ (cid:18) K − K (cid:19) (A9)We may now express K and K in terms of the eigenvalues: K = λ + λ + λ , (A10) K = λ λ + λ λ + λ λ . (A11)After integrating the expression (A5) over S , we obtain theunconditional probability that the Jacobian matrix has thethree ordered eigenvalues λ , λ and λ : P ( λ , λ , λ | σ ) =33758 √ σ π exp " (cid:0) K − K (cid:1) σ | ( λ − λ )( λ − λ )( λ − λ ) | . (A12)This equation corresponds to the distribution derived byDoroshkevich (1970). c (cid:13)000 , 000–000 G. Lavaux & B. D. Wandelt
APPENDIX B: PROBABILITY DISTRIBUTIONOF TIDAL FIELD IN VOIDS
We show in this Appendix how to compute the form of theprobability distribution of the gravitational tidal field in thecase of voids. The technique involved in this derivation looksmuch like the one used in Appendix A but with an extracomplication due to correlations with the density field. Bydefinition, void centres are chosen to be maxima of S Ψ ( q ) = X i =1 ∂ Φ ∂q i . (B1)with Φ defined as in §
2, the scalar potential of the displace-ment field. We will assume that, for voids, we are fully inthe Zel’dovich regime and thus we can equate Φ as givenby MAK to the potential of the Zel’dovich displacement athigh redshift. In this case, S Ψ corresponds the primordialdensity fluctuations scaled linearly to z = 0.Being a maxima of S Ψ means the displacement fieldsatisfies two properties: the gradient of S Ψ is null and theHessian matrix H i,j = ∂S Ψ ∂q i ∂q j (B2)of S Ψ is negative-definite. Thus our aim is to compute theprobability of the Jacobian matrix of the displacement fieldgiven that H i,j is symmetric negative-definite. We write thisprobability P ( T i,j | H l,m < P ( k ), with k a wave number. Wecompute the correlation between those two fields: h T i,j T l,m i = σ T δ i,j δ l,m + δ i,l δ j,m + δ i,m δ j,l ) , (B3) h H i,j H l,m i = σ H δ i,j δ l,m + δ i,l δ j,m + δ i,m δ j,l ) , (B4) h T i,j H l,m i = Γ TH δ i,j δ l,m + δ i,l δ j,m + δ i,m δ j,l ) , (B5)with σ T = S = σ and σ H = S and Γ TH = −S , (B6) σ the variance of the density field and S n = 110 π Z + ∞ k =0 k n P ( k )d k . (B7)To reduce the complexity of the correlation matrix, we nowuse the reduced random variables defined as follow˜ T i,j = T i,j σ T (B8)˜ H i,j = H i,j σ H . (B9)and the reduced correlation r = Γ TH σ T σ H . (B10)As T and H are 3 × A = 1 . . .
6, where each number refer to the ( i, j ) couples(1 , , (2 , , (3 , , (1 , , (1 ,
3) and (2 , V ofthe variance of the 12 reduced components may be formallywritten, using a block-matrix representation: V = (cid:18) A rArA A (cid:19) (B11)and A =
13 13
13 13 . (B12)The inverse may be computed straightforwardly C = M − = 11 − r (cid:18) A − − rA − − rA − A − (cid:19) (B13)Now, we may express the joint probability P ( T, H ) ofobserving a tidal field T for the gravitational potential and acurvature H for the density field using the covariance matrix C = M − . P ( T, H | r ) = p | det C | (2 π ) exp (cid:18) − t Y CY (cid:19) (B14)with Y = ( T, H ). Now the probability of observing somematrix T given that H must be positive could be computedformally: P ( T | r, H <
0) = Z H< P ( T, H | r )d H (B15)However, it is quite involved to find an analytic expressionof this integral in function of the eigenvalues of T . We pro-pose to sample this distribution instead of computing thisintegral.One can prove that the conditional probability P ( T | H )may be written P ( T | H, r ) = P ( T, H | r ) R T P ( T, H | r ) = p | det A | (2 π ) exp (cid:18) − t ( T − rH ) A ( T − rH ) (cid:19) (B16)Thus, Equation (B15) may be re-expressed as: P ( T | r, H <
0) = Z H< P ( T | r, H ) P ( H )d H (B17)with P ( H ) the probability of getting a random symmetricmatrix H , with the covariance matrix A . The method is nowthe following:- we generate a random symmetric matrix H , if it is neg-ative, we accept it, in the other case we try again;- we generate a matrix T , following the probability givenby Equation (B16);- we compute and store the eigenvalue of this matrix T ,after multiplication by σ T .That way the joint probability distribution P ( λ , λ , λ | σ T , r, H <
0) is correctly sampled, eventhough we do not have any explicit expression of it. c (cid:13) , 000–000 recision cosmology with voids: definition, methods, dynamics APPENDIX C: LOCAL TIDAL ELLIPTICITYVS. GLOBAL VOLUME ELLIPTICITY
In this appendix, we try to relate the two ellipticities ε vol and ε DIVA defined in Eq.(13) and (15). To do that, we willmake use of Zel’dovich approximation in voids, which hasbeen shown to be a relatively precise modeling of the voidevolution. The inertial mass tensor writes as: M = a I − K (C1)with I the 3 × a = Z V d q || x ( q ) − ¯ x || (C2) K i,j = Z V d q ( x i ( q ) − ¯ x i )( x j ( q ) − ¯ x j ) , (C3)with q the Lagrangian coordinates, V the Lagrangian do-main of the considered void, ¯ x the centre of mass of the V in Eulerian coordinates. With this parametrization, thevolume ellipticity ε vol simplifies as ε vol = 1 − (cid:18) J J (cid:19) / (C4)with J and J the smallest and largest eigenvalues of K .We may write exactly: x ( q ) = q + Ψ ( q ) . (C5)We now expand Ψ to first order around the position of thecentre of mass in Lagrangian coordinates ¯ q Ψ i ( q ) = Ψ i (¯ q ) + ∂ Ψ i ∂q j ( q j − ¯ q j ) . (C6)Using Zel’dovich approximation we identify ∂ Ψ i /∂q j and T i,j given in Eq. (8). We now reexpress K i,j K i,j = L i,j + T i,k L k,j + T j,k L k,i (C7) L i,j = Z V d q ( q i − ¯ q i )( q j − ¯ q j ) . (C8)As voids, in Lagrangian coordinates, should be mostlyisotropic the Lagrangian inertial tensor L i,j must be diago-nal: L i,j = 13 a δ i,j , (C9)with a = Z V d q || q − ¯ q || . (C10)This assumption is verified in average by linear theory butmay be broken for some specific voids. In the case wherevoids are effectively isotropic, the inertial mass tensor K i,j is extremely simplified: K i,j = a δ i,j + 2 T i,j ) . (C11)The eigenvalues of K are thus J i ∝ λ i ≃ (1 + λ i ) . (C12)The volume ellipticity may thus be related to the tidal el-lipticity as ε vol = 1 − (cid:18) J J (cid:19) / ≃ − (cid:18) λ λ (cid:19) / = ε DIVA (C13) c (cid:13)000