On the initial shear field of the cosmic web
aa r X i v : . [ a s t r o - ph . C O ] J a n Mon. Not. R. Astron. Soc. , 1–12 (2011) Printed 10 November 2018 (MN L A TEX style file v2.2)
On the initial shear field of the cosmic web
Graziano Rossi ⋆ Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-Gu, Seoul − , Korea Accepted 2011 December 01. Received 2011 November 30; in original form 2011 July 29
ABSTRACT
The initial shear field, characterized by a primordial perturbation potential, playsa crucial role in the formation of large scale structures. Hence, considerable analyticwork has been based on the joint distribution of its eigenvalues, associated with Gaus-sian statistics. In addition, directly related morphological quantities such as ellipticityor prolateness are essential tools in understanding the formation and structural proper-ties of halos, voids, sheets and filaments, their relation with the local environment, andthe geometrical and dynamical classification of the cosmic web. To date, most analyticwork has been focused on Doroshkevich’s unconditional formulae for the eigenvaluesof the linear tidal field, which neglect the fact that halos (voids) may correspond tomaxima (minima) of the density field. I present here new formulae for the constrainedeigenvalues of the initial shear field associated with Gaussian statistics, which includethe fact that those eigenvalues are related to regions where the source of the displace-ment is positive (negative): this is achieved by requiring the Hessian matrix of thedisplacement field to be positive (negative) definite. The new conditional formulaenaturally reduce to Doroshkevich’s unconditional relations, in the limit of no correla-tion between the potential and the density fields. As a direct application, I derive theindividual conditional distributions of eigenvalues and point out the connection withprevious literature. Finally, I outline other possible theoretically- or observationally-oriented uses, ranging from studies of halo and void triaxial formation, developmentof structure-finding algorithms for the morphology and topology of the cosmic web,till an accurate mapping of the gravitational potential environment of galaxies fromcurrent and future generation galaxy redshift surveys.
Key words: methods: analytical, statistical — cosmology: theory, large-scale struc-ture of Universe — galaxies: formation
The large-scale spatial distribution of dark matter, as revealed from numerical simulations, shows a characteristic anisotropicweb-like structure. This cosmic web , which arises through the gravitational clustering of matter, is mainly due to the effects ofthe tidal field: in fact, the competition between cosmic expansion, the trace, and the traceless part of the tidal field imprintsanisotropies in the large-scale matter distribution in much the same way that gravity and radiation pressure imprints baryonicacoustic oscillations (BAO) on the Cosmic Microwave Background (CMB) sky (Hu & Sugiyama 1995; Lee & Springel 2010).Hence, the initial shear field plays a crucial role in the formation of large scale structures, and a number of studies in theliterature have been devoted to this subject – among the plethora of papers, see for example the classic works by Zeldovich(1970), Icke (1973), Peebles (1980), White (1984), Bardeen et al. (1986), Kaiser (1986), Bertschinger (1987), Bond & Myers(1996), Bond, Kofman & Pogosyan (1996) and van de Weygaert & Bertschinger (1996). In addition, if the cosmic web originatesfrom primordial tidal effects and its degree of anisotropy increases with the evolution of the Universe, then studying this initialfield is crucial in understanding the subsequent nonlinear evolution of cosmic structures (Springel et al. 2005; Shandarin etal. 2006; Desjacques 2008; Desjacques & Smith 2008; Pogosyan et al. 2009), the alignment of shape and angular momentumof halos (West 1989; Catelan et al. 2001; Lee & Springel 2010; Rossi, Sheth & Tormen 2011), the statistical properties ofvoids (Lee & Park 2006; Platen, van de Weygaert & Jones 2008), and more generally for characterizing the geometry and ⋆ Email: [email protected] (cid:13)
G. Rossi morphology of the cosmic web (Shen et al. 2006; van de Weygaert & Bond 2008; Forero-Romero et al. 2009; Aragon-Calvo etal. 2010a,b; Shandarin et al. 2010).The basic theory for the formation and evolution of structures is now well-understood, thanks to the pioneering work ofDoroshkevich & Zeldovich (1964), Doroshkevich (1970), Zeldovich (1970), and Sunyaev & Zeldovich (1972) – the latter in thecontext of galaxy formation. In particular, Doroshkevich (1970) derived the joint probability distribution of an ordered set ofeigenvalues in the tidal field matrix – at random positions – given the variance of the density field, corresponding to a Gaussianpotential; we will refer to it as the unconditional probability distribution of eigenvalues. In addition, Zeldovich (1970) providedthe fundamental understanding of anisotropic collapse on cosmological scales, and recognized the key role of the large scaletidal force in shaping the cosmic web. Subsequently, Doroshkevich and Shandarin (1978) calculated some statistical propertiesof the maxima of the largest eigenvalue of the shear tensor. Their study proved that the most probable formation processstarts first with a one-dimensional collapse (cosmic pancake formation). The directions (orientations) for the one-dimensionalcollapsed sheets are determined by the largest eigenvalue of the deformation tensor, which can be attributed to the initiallinear density perturbations; moreover, the probability that two or even three of the initial eigenvalues are identical or nearlyequal is extremely small, indicating that the collapse is triaxial. Later on, following a constrained field approach pioneeredby Bertschinger (1987), van de Weygaert & Bertschinger (1996) developed an algorithm for setting up tailor-made initialconditions for cosmological simulations, which addressed the role of the tidal fields in shaping large-scale structures. In thesame period, Bond, Kofman & Pogosyan (1996) developed a cosmic web theory which naturally explains the filamentarystructure present in the Cold Dark Matter (CDM) cosmology, due to the coherent nature of the primordial tidal field. Theyrealized that an “embryonic” cosmic web is already present in the primordial density field, and explained why in overdenseregions sheet-like membranes are only marginal features. Since then, because of the correspondence between structures in theevolved density field and local properties of the linear tidal field pointed out by the same authors, the statistics of the shearhas received more attention in the literature. For example, Lee & Shandarin (1998) computed some probability distributionsfor individual shear eigenvalues and obtained an analytic approximation to the halo mass function, and Catelan & Porciani(2001) explored the two-point correlation of the tidal shear components.However, a variety of studies in the physics of halo formation and cosmic web classification are based on Doroshkevich’sunconditional formulae for the ordered eigenvalues of the initial shear field associated with Gaussian statistics (Doroshkevich1970), but those formulas cannot differentiate between random positions and peak/dips as they neglect the fact that halos(voids) may correspond to maxima (minima) of the density field. According to Bardeen et al. (1986), if one assumes thecosmological density fluctuations to be Gaussian random fields, the local maxima of such fields are plausible sites for theformation of nonlinear structures. Hence, the statistical properties of the peaks can be used to predict the abundances andclustering properties of objects of various types, and in studies of the non-spherical formation of large-scale structures. Thestudy of Bertschinger (1987) goes in this direction, by generalizing the treatment of Bardeen et al. (1986) and proposing apath integral method for sampling constrained Gaussian random fields. The method allows one to study the density fieldaround peaks or other constrained regions in the biased galaxy formation scenario, but unfortunately it is too elaborate andinefficient in its implementation. Instead, by applying the prescription of Hoffman & Ribak (1991) to construct constrainedrandom fields, van de Weygaert & Bertschinger (1996) were able to show that it is possible to generate efficiently initialGaussian random density and velocity fields, and specify the presence and characteristics of one or more peaks and dips atarbitrary locations – with the gravity and tidal fields at the site of the peaks having the required strength and orientation.Some other studies on constrained initial conditions were also pursued by van Haarlem & van de Weygaert (1993), and byvan de Weygaert & Babul (1994). The idea has been expanded in Bond & Myers (1996), who presented a peak-patch pictureof structure formation as an accurate model of the dynamics of peaks in the density field. Their approach goes further, asit involves the explicit formalism for identifying objects in a multiscale field (i.e. it does not restrict to a single scale). Themodel is even more precise for void patches, the equivalent framework for studying voids (Sahni et al. 1994; Sheth & vande Weygaert 2004; Novikov, Colombi & Dore 2006; Colberg et al. 2008). In general, density peaks define a well-behavedpoint-process which can account for the discrete nature of dark matter halos and galaxies, and on asymptotically large scalesare linearly biased tracers of the dark matter field (Desjacques & Sheth 2010).Therefore, it would be desirable to incorporate the peak (or dip) constraint in the statistical description of the initialshear field, in order to characterize more realistically the geometry and dynamics of the cosmic web. The main goal of thispaper is to do so, by providing a set of analytic expressions which extend the work of Doroshkevich (1970) and Bardeen etal. (1986), and are akin in philosophy to that of van de Weygaert & Bertschinger (1996). This is achieved by constraining theHessian of the displacement field (the matrix of the second derivatives) to be positive (negative) definite, which is the case inthe vicinity of minima (maxima) of the source of the displacement field. The new conditional probability distributions derivedin this study include the correlation between the potential and density fields through a reduced parameter r , and naturallyrecover Doroshkevich’s (1970) unconditional formulae in the absence of correlation.Hence, the main focus of this work is to derive explicitly the joint probability distribution of the eigenvalues of the shearfield, given the fact that positions are peaks or dips in the corresponding density field – and not random locations. In thissense the field is termed constrained (i.e. it has constrained eigenvalues, which are the result of looking only at peak/dipregions), and it is statistically described by a conditional probability distribution. Note that even though the formalism isessentially restricted to one scale, the formulae derived here are useful in a variety of applications – some of which will bediscussed at the end of this paper.The layout is organized as follows. Section 2 provides the derivation of the new analytic expressions for the conditionaldistributions of eigenvalues of the initial shear field. In particular, Section 2.1 illustrates the basic notation adopted; Section c (cid:13) , 1–12 nitial shear field In this section, I first introduce the basic notation adopted throughout the paper. I then derive the conditional joint distributionof eigenvalues of the tidal field, under the constraint that the Hessian matrix of the source of the displacement is positive (ornegative) definite. Finally, I present the reverse probability function, useful in relating this work with that of Bardeen et al.(1986) and van de Weygaert & Bertschinger (1996).
Let Ψ denote the displacement field, Φ the potential of the displacement field, S Ψ the source of the displacement field. Indicatewith q the Lagrangian coordinate, with x the Eulerian coordinate, where x ( q ) = q + Ψ( q ) . (1)Use T ij to denote the shear tensor of the displacement field, H ij for the Hessian matrix, J ij for the Jacobian of the displacementfield ( i, j = 1 , , J ij ( q ) = ∂x i ∂q j = δ ij + T ij (2) T ij = ∂ Ψ i ∂q j = ∂ Φ ∂q i ∂q j (3) H ij = ∂ S Ψ ∂q i ∂q j (4) S Ψ ( q ) = X i=1 ∂ Ψ i ∂q i ≡ X i=1 ∂ Φ ∂q . (5)The eigenvalues of T ij are λ , λ , λ , those of H ij are ξ , ξ , ξ . In general, the ordering of the eigenvalues is assumed to besuch that λ > λ > λ . Let the potential Φ be a Gaussian random field determined by the power spectrum of matter densityfluctuations P ( k ), with k denoting the wave number and W ( k ) the smoothing kernel. The density field described by the source S Ψ is also a Gaussian random field. The correlations between these two fields are expressed by: h T ij T kl i = σ
15 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) (6) h H ij H kl i = σ
15 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) (7) h T ij H kl i = Γ
15 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) (8)where σ = S ≡ σ , σ = S ≡ σ , Γ TH = − S ≡ − σ , δ ij is the Kronecker delta, and S n = 12 π Z ∞ k n P ( k ) W ( k )d k (9) σ = 12 π Z ∞ k P ( k ) W ( k ) d k ≡ S . (10) T ij and H ij are real symmetric tensors, so they are specified by 6 components (note however that only 5 are independent oncethe height of the density peak or dip has been set, because the trace of the shear field tensor is entirely constrained by thelocal density field value via the Poisson equation). Label them as A = (1 , B = (2 , C = (3 , D = (1 , E = (1 , F = (2 , α or β indicate the various couples, in a compact notation, where α, β = A, B, C, D, E, F . For example, c (cid:13) , 1–12 G. Rossi if α = D then T α = T D ≡ T , and so forth. In the subsequent derivations, for clarity all the various dependencies on σ ’s aredropped. This is done by introducing the “reduced” variables ˜ T and ˜ H , defined as˜ T α = T α /σ T , ˜H α = H α /σ H (11)and the “reduced” correlation r = Γ TH /σ T σ H = − σ σ σ ≡ = − γ (12)where γ is the same as in Eq. (4.6a) of Bardeen et al. (1986). In this notation, the eigenvalues of ˜ T and of ˜ H are ˜ λ i ’s and˜ ξ i ’s, respectively, with i = 1 , ,
3. Of course, the σ dependence can be restored at any time, if desired, by using the previousEquations (11) and (12). The main goal of this section is to derive the joint distribution of the eigenvalues of J ij when the ordering is λ > λ > λ , inregions where the source of displacement S Ψ is a minimum (or maximum). Being a minimum (maximum) of S Ψ implies thatthe displacement field is such that the gradient of S Ψ is zero, and the Hessian matrix ˜ H α is positive (or negative) definite. Inblock notation, the covariance matrix ˜ V of the 12 reduced components of Y = ( ˜ T , ˜ H ) is:˜ V = (cid:18) h ˜ T α ˜ T α i h ˜ T α ˜ H β ih ˜ H β ˜ T α i h ˜ H β ˜ H β i (cid:19) = 115 (cid:18) A rArA A (cid:19) (13)where A = (cid:18) B ⊘⊘ I (cid:19) , B = (14)with I a (3 ×
3) identity matrix and ⊘ a (3 ×
3) null matrix. The inverse of the reduced covariance matrix is simply:˜ C = ˜ V − = 15(1 − r ) (cid:18) A − − rA − − rA − A − (cid:19) (15)where A − = (cid:18) B − ⊘⊘ I (cid:19) , B − = 110 − − − − − − . (16)The joint probability of observing a tidal field ˜ T for the gravitational potential and a curvature ˜ H for the density field (fromnow on, the understood indices α and β are dropped) is a multivariate Gaussian in Y = ( ˜ T , ˜ H ): p ( Y | r )d Y = e − Y T · ˜ C · Y (2 π ) q | ˜ V | d Y ≡ p ( ˜ T , ˜ H | r )d ˜ T d ˜ H. (17)Therefore p ( ˜ T | r, ˜ H >
0) = R ˜H > p ( ˜ T , ˜ H | r ) d ˜ Hp ( ˜ H >
0) = R ˜H > p ( ˜ H ) · p ( ˜ T | r, ˜ H ) d ˜ H R ˜H > p ( ˜ H ) d ˜ H , (18)where p ( ˜ T | r, ˜ H ) = p ( ˜ T , ˜ H | r ) p ( ˜ H ) = p ( ˜ T , ˜ H | r ) R ˜T p ( ˜ T , ˜ H | r ) d ˜ T = e − (˜T − b) T · A − · (˜T − b) (2 π ) p | A | . (19)The marginal distributions p ( ˜ T ) and p ( ˜ H ) are simply multidimensional Gaussians with covariance matrix A/
15, and thereforethey can be expressed using Doroshkevich’s formulae as p ( ˜ T ) = 15 √ π e − (2k − ) (20)with k = ˜ T + ˜ T + ˜ T (21) k = ˜ T ˜ T + ˜ T ˜ T + ˜ T ˜ T − ˜ T − ˜ T − ˜ T (22)and similarly p ( ˜ H ) = 15 √ π e − (2h − ) (23) c (cid:13)000
15, and thereforethey can be expressed using Doroshkevich’s formulae as p ( ˜ T ) = 15 √ π e − (2k − ) (20)with k = ˜ T + ˜ T + ˜ T (21) k = ˜ T ˜ T + ˜ T ˜ T + ˜ T ˜ T − ˜ T − ˜ T − ˜ T (22)and similarly p ( ˜ H ) = 15 √ π e − (2h − ) (23) c (cid:13)000 , 1–12 nitial shear field with h = ˜ H + ˜ H + ˜ H (24) h = ˜ H ˜ H + ˜ H ˜ H + ˜ H ˜ H − ˜ H − ˜ H − ˜ H . (25)The conditional distribution p ( ˜ T | r, ˜ H ) is also a multidimensional Gaussian, with mean b and covariance matrix A , where b = r ˜ H, A = 115 (1 − r ) A. (26)Carrying on the calculation indicated in (19) yields: p ( ˜ T | r, ˜ H ) = 15 √ π − r ) exp h − − r ) (2 K − K ) i (27)where K = ( ˜ T − r ˜ H ) + ( ˜ T − r ˜ H ) + ( ˜ T − r ˜ H ) = k − rh K = ( ˜ T − r ˜ H )( ˜ T − r ˜ H ) + ( ˜ T − r ˜ H )( ˜ T − r ˜ H )+ ( ˜ T − r ˜ H )( ˜ T − r ˜ H ) − ( ˜ T − r ˜ H ) − ( ˜ T − r ˜ H ) − ( ˜ T − r ˜ H ) = k + r h − rh k + rη (28)and η = ˜ T ˜ H + ˜ T ˜ H + ˜ T ˜ H + 2 ˜ T ˜ H + 2 ˜ T ˜ H + 2 ˜ T ˜ H (29)accounts for cross-correlation terms.The previous expression (27) generalizes Doroshkevich’s formula (20) to include the fact that halos (voids) may correspondto maxima (minima) of the density field. Note that, if r = 0, Doroshkevich’s formula (20) is indeed recovered – as expected inthe limit of no correlations between the potential and density fields. Equation (27) is written in a Doroshkevich-like format,and it is one of the key results of this paper.It is also possible to express (27) in terms of the constrained eigenvalues ζ ′ s of the ( ˜ T | r, ˜ H ) matrix. The result is: p (˜ ζ , ˜ ζ , ˜ ζ | r ) ≡ p (˜ λ , ˜ λ , ˜ λ | r, ˜ ξ , ˜ ξ , ˜ ξ ) (30)= 15 √ π − r ) exp h − − r ) (2 K − K ) i · | (˜ ζ − ˜ ζ )(˜ ζ − ˜ ζ )(˜ ζ − ˜ ζ ) | where in terms of constrained eigenvalues: K = ˜ ζ + ˜ ζ + ˜ ζ = k − rh (31) K = ˜ ζ ˜ ζ + ˜ ζ ˜ ζ + ˜ ζ ˜ ζ = k + r h − rh k + rη (32) η = ˜ λ ˜ ξ + ˜ λ ˜ ξ + ˜ λ ˜ ξ (33) k = ˜ λ + ˜ λ + ˜ λ (34) k = ˜ λ ˜ λ + ˜ λ ˜ λ + ˜ λ ˜ λ (35) h = ˜ ξ + ˜ ξ + ˜ ξ (36) h = ˜ ξ ˜ ξ + ˜ ξ ˜ ξ + ˜ ξ ˜ ξ (37)with the partial distributions expressed by Doroshkevich’s unconditional formulae as p (˜ λ , ˜ λ , ˜ λ ) = 15 √ π e − (2k − ) (˜ λ − ˜ λ )(˜ λ − ˜ λ )(˜ λ − ˜ λ ) , (38) p (˜ ξ , ˜ ξ , ˜ ξ ) = 15 √ π e − (2h − ) (˜ ξ − ˜ ξ )(˜ ξ − ˜ ξ )(˜ ξ − ˜ ξ ) . (39)and ˜ ζ i ≡ (˜ λ i | r, ˜ ξ i ) = ˜ λ i − r ˜ ξ i (40)which is obvious from (28), with ˜ λ i ’s and ˜ ξ i ’s the corresponding unconstrained eigenvalues of the matrices ˜ T and ˜ H . Equation(30) is another key result of this work. This relation is easily derived from (27) because the volume element in the six-dimensional space of symmetric real matrices is simply given, in terms of constrained eigenvalues, by Y α d( ˜ T α | r, ˜ H α ) = | (˜ ζ − ˜ ζ )(˜ ζ − ˜ ζ )(˜ ζ − ˜ ζ ) | d˜ ζ d˜ ζ d˜ ζ dΩ S (41)where dΩ S is the volume element of the three-dimensional rotation group SO (3), and also of the three-sphere (see AppendixB in Bardeen et al. 1986). In addition, from (28) it is direct to see that one has ˜ T ii ≡ ˜ λ i and ˜ H ii ≡ ˜ ξ i in the system whereboth ˜ T and ˜ H are diagonal; therefore K = P i (˜ λ i − r ˜ ξ i ) ≡ P i ζ i , with ζ i the constrained eigenvalues.The fact that the system in which ˜ T is diagonal is also the system in which ˜ H is diagonal has a simple geometrical c (cid:13) , 1–12 G. Rossi explanation: in order to obtain the conditional distribution ( ˜ T | r, ˜ H ), a standard procedure for multivariate Gaussians (whentransforming from uncorrelated to correlated variates) is to use a Cholesky decomposition. However, this decomposition doesnot sample symmetrically with respect to the principal axes. An alternative method which takes care of the alignment withthe principal axes (and used here) is the following: decompose spectrally the covariance matrix of ( ˜ T | r, ˜ H ), A , i.e. find theeigensystem D and diagonal eigenvalue matrix Λ of the covariance matrix such that A = D · Λ · D T ; then set W = D · Λ / Y ,with Y a set of independent zero mean unit variance Gaussians. It is direct to show that W is indeed the correct covariancematrix, and this scheme takes care of the alignment with the principal axes. Equation (18) expresses the probability of observing a tidal field ˜ T in regions where the curvature ˜ H of the density field ispositive/negative (i.e. peak or dip regions). One may be also interested in the reverse joint conditional distribution, namely p ( ˜ H | r, ˜ T > p ( ˜ H | r, ˜ T >
0) = R ˜ T> p ( ˜ H, ˜ T | r ) d ˜ Tp ( ˜ T >
0) = R ˜ T > p ( ˜ T ) d ˜ T · p ( ˜ H | r, ˜ T ) R ˜ T> p ( ˜ T ) d ˜ T (42)where now (symmetrically) p ( ˜ H | r, ˜ T ) = p ( ˜ H, ˜ T | r ) p ( ˜ T ) = p ( ˜ H, ˜ T | r ) R ˜H p ( ˜ H, ˜ T | r ) d ˜ H = e − (˜H − b ′ ) T · A − · (˜H − b ′ ) (2 π ) p | A | . (43)The distribution p ( ˜ H | r, ˜ T ) is also a multidimensional Gaussian with mean b ′ and covariance matrix A , where b ′ = r ˜ T , A = 115 (1 − r ) A. (44)Along the lines of the previous calculation, it is direct to obtain p ( ˜ H | r, ˜ T ) = 15 √ π − r ) exp h − − r ) (2 H − H ) i (45)where in analogy with (27) one has now: H = ( ˜ H − r ˜ T ) + ( ˜ H − r ˜ T ) + ( ˜ H − r ˜ T ) = h − rk H = ( ˜ H − r ˜ T )( ˜ H − r ˜ T ) + ( ˜ H − r ˜ T )( ˜ H − r ˜ T )+ ( ˜ H − r ˜ T )( ˜ H − r ˜ T ) − ( ˜ H − r ˜ T ) − ( ˜ H − r ˜ T ) − ( ˜ H − r ˜ T ) = h + r k − rk h + rη. (46)In addition, if ǫ , ǫ , ǫ are the constrained eigenvalues of ( ˜ H | r, ˜ T ), then p (˜ ǫ , ˜ ǫ , ˜ ǫ | r ) ≡ p (˜ ξ , ˜ ξ , ˜ ξ | r, ˜ λ , ˜ λ , ˜ λ ) (47)= 15 √ π − r ) exp h − − r ) (2 H − H ) i · | (˜ ǫ − ˜ ǫ )(˜ ǫ − ˜ ǫ )(˜ ǫ − ˜ ǫ ) | where in terms of constrained eigenvalues H = ˜ ǫ + ˜ ǫ + ˜ ǫ = h − rk (48) H = ˜ ǫ ˜ ǫ + ˜ ǫ ˜ ǫ + ˜ ǫ ˜ ǫ = h + r k − rh k + rη, (49)with the ǫ i ’s simply given by ˜ ǫ i ≡ (˜ ξ i | r, ˜ λ i ) = ˜ ξ i − r ˜ λ i . (50)Note an interesting symmetry: K , H , k , h are always the traces of their corresponding matrices.New analytic formulae derived from (45) and (47), which generalize the work of Bardeen et al. (1986), will be presented ina forthcoming publication. The previous relations are also useful in making the connection with the work of van de Weygaert& Bertschinger (1996). For example, Equation (50) confirms their finding of the correlation between the shifted mean valuesfor density and shear (see for example Equation 108 in van de Weygaert & Bertschinger 1996, which provides the reversemean value of the shear given a density field shape, and their Section 4.4 for more details). A direct application of the main formulae derived in Section 2.2 is to compute the individual distributions of eigenvalues(along with some other related conditional probabilities), given the peak/dip constraint. Starting from Equation (30), withsome extra complications due to the inclusion of correlation terms, it is possible to extend the work of Lee & Shandarin (1998) c (cid:13) , 1–12 nitial shear field Figure 1.
Individual conditional distributions p ( ζ i | r ) of the initial shear field in the peak/dip picture (Equations 53, 54 and 55), fordifferent values of the reduced correlation parameter r – as indicated in the panels. to account for those regions where the source of the displacement is positive (negative) – see the Appendix A for a compactsummary of their results. Introducing the following notation: l = 112564 √ π (51) l ′ = l · (1 − r ) − / n = √ π/ L ′ ( x, y | r ) = p − r ( x − y )(3 x − y ) · exp h − − r ) (cid:16) x − xy + 32 y (cid:17)i N ′ ( x, y | r ) = ( x − y )[8(1 − r ) + 3(3 x − y )(3 y − x )] · exp h − − r ) (cid:16) x − xy + 3 y (cid:17)i · erfc h √
34 1 √ − r ( x − y ) i and dropping the understood ∼ symbol (but note that the eigenvalues are still “reduced variables”), the two-point probabilitydistributions of the constrained eigenvalues of the shear field are given by: p ( ζ , ζ | r ) = l ′ h L ′ ( ζ , ζ | r ) + n · N ′ ( ζ , ζ | r ) i (52) p ( ζ , ζ | r ) = l ′ h L ′ ( ζ , ζ | r ) + n · N ′ ( ζ , ζ | r ) i p ( ζ , ζ | r ) = l ′ n L ′ ( ζ , ζ | r ) + L ′ ( ζ , ζ | r ) + n · h N ′ ( ζ , ζ | r ) + N ′ ( ζ , ζ | r ) io . Performing the various partial integrations, it is straightforward to obtain: p ( ζ | r ) = l ′ n Z ζ −∞ L ′ ( ζ , ζ | r )d ζ + n Z ζ −∞ N ′ ( ζ , ζ | r )d ζ o (53)= √ π h − r ) ζ exp (cid:16) − ζ − r ) (cid:17) − √ π (1 − r ) / exp (cid:16) − ζ − r ) (cid:17) · erfc (cid:16) − √ ζ √ − r (cid:17) [(1 − r ) − ζ ]+ 3 √ π √ − r exp (cid:16) − ζ − r ) (cid:17) erfc (cid:16) − √ ζ √ − r (cid:17)i p ( ζ | r ) = l ′ n Z + ∞ ζ L ′ ( ζ , ζ | r )d ζ + n Z + ∞ ζ N ′ ( ζ , ζ | r )d ζ o (54)= √ √ π √ − r exp h − ζ (1 − r ) i p ( ζ | r ) = l ′ n Z ζ ζ L ′ ( ζ , ζ | r )d ζ + n Z ζ ζ N ′ ( ζ , ζ | r )d ζ o (55)= − √ π h − r ) ζ exp (cid:16) − ζ − r ) (cid:17) + √ π (1 − r ) / exp (cid:16) − ζ − r ) (cid:17) · erfc (cid:16) √ ζ √ − r (cid:17) [(1 − r ) − ζ ] − √ π √ − r exp (cid:16) − ζ − r ) (cid:17) erfc (cid:16) √ ζ √ − r (cid:17)i . c (cid:13) , 1–12 G. Rossi
Figure 2.
Conditional probability distribution p ( K | r, ζ > ≡ p (∆ | r, ζ >
0) given by Equation (61), for different values of r , asindicated in the panels. Solid vertical lines show the maximum of each distribution ( K max1 ≃ . √ − r ), dotted lines represent theircorresponding average values as given by Equation (62). Plots of these distributions are shown in Figure 1, for different values of the reduced correlation r . Note the symmetrybetween p ( ζ | r ) and p ( ζ | r ), evident from (53) and (55). In particular, p ( ζ | r ) is simply a Gaussian with variance 2(1 − r ) / p ( λ | r, ξ ) is a Gaussian with the same variance and shifted mean rξ (recall that ζ = λ − rξ ). In addition, it is easy to show that, with ∆ = ζ + ζ + ζ ≡ K , one obtains p (∆ , ζ | r ) = 3 √ π − r ) h − ζ + 135 ζ − − r ) i · exp h − − r ) (3∆ − ζ + 15 ζ ) i + 3 √ π − r ) exp h − − r ) (cid:16) ∆ − ζ + 152 ζ (cid:17)i . (56)By integrating the previous equation over ζ , it is direct to verify that p (∆ | r ) ≡ p ( K | r ) is a Gaussian with zero mean andvariance (1 − r ), namely: p (∆ | r ) = 1 p π (1 − r ) · exp h − ∆ − r ) i ≡ p ( K | r ) . (57)Since K = k − rh (i.e. Eq. 31), the previous expression implies that p ( k | r, h ) is therefore a Gaussian with mean rh andvariance (1 − r ). In addition: p ( ζ > | r ) = 2325 , h ζ | r i = 3 √ π p − r , σ ζ | r = 13 π − π (1 − r ) (58) p ( ζ > | r ) = 12 , h ζ | r i = 0 , σ ζ | r = 215 (1 − r ) (59) p ( ζ > | r ) = 225 , h ζ | r i = − √ π p − r , σ ζ | r = 13 π − π (1 − r ) . (60)Moreover, the probability distribution of ∆ confined in the regions with ζ > p (∆ | r, ζ >
0) = p (∆ , ζ > | r ) p ( ζ >
0) = R ∆ / p (∆ , ζ | r )d ζ p ( ζ >
0) (61)= − √ π ∆(1 − r ) exp (cid:16) −
98 ∆ (1 − r ) (cid:17) ++ 254 √ π √ − r exp (cid:16) − ∆ − r ) (cid:17)h erf (cid:16) √ √ − r (cid:17) + erf (cid:16) √ √ − r (cid:17)i . Using the previous formula, it is direct to prove that h ∆ | r, ζ > i = 25 √ √ π (3 √ − p − r ≃ . p − r . (62)Figure 2 shows the conditional probability distribution p ( K | r, ζ > ≡ p (∆ | r, ζ >
0) (i.e. Equation 61), for different valuesof r . The dotted vertical lines display the average values of those functions, as given in (62), while the vertical solid linesindicate their maximum values. In fact, it is easy to see that the maximum of p (∆ | r, ζ >
0) is reached when ∆ ≃ . √ − r .This result is readily obtained by computing the derivative of p (∆ | r, ζ > r = 0 . c (cid:13)000
0) is reached when ∆ ≃ . √ − r .This result is readily obtained by computing the derivative of p (∆ | r, ζ > r = 0 . c (cid:13)000 , 1–12 nitial shear field Figure 3.
Determination of the maxima of (61). The derivative p ′ ( K | ζ > , r ) is shown, for an arbitrary value of the reduced correlation r , i.e. r = 0 .
5. The maxima correspond to those points where K max1 ≃ . √ − r , indicated with solid vertical lines in the figure. Finally, the probability that a given region with density ∆ will have all positive eigenvalues ζ ’s is: p ( ζ > | ∆ , r ) = p (∆ , ζ > | r ) p (∆ | r ) = − √ √ π ∆ √ − r exp (cid:16) −
58 ∆ (1 − r ) (cid:17) + 12 h erf (cid:16) √ √ − r (cid:17) + erf (cid:16) √ √ − r (cid:17)i . Note that, in absence of correlations between the potential and density fields (i.e. when r = 0), all the previous expressionsreduce consistently to the unconditional limit of Lee & Shandarin (1998) – see again Appendix A. Several other results onprobability distributions along these lines were also already derived by van de Weygaert & Bertschinger (1996): here weconfirm their findings that the inclusion of the condition for a peak or dip in the Gaussian density field involves a 1 / √ − r factor, and that the conditional distributions have shifted mean and reduced variance (see again their Section 4). Since the initial shear field associated with Gaussian statistics plays a major role in the formation of large scale structures,considerable analytic work has been based on the joint distribution of its eigenvalues – i.e. Doroshkevich’s formulae (Equation20). However, Doroshkevich’s equations neglect the fact that halos (voids) may correspond to maxima (minima) of the densityfield. The main goal of this work was to provide new analytic expressions, in the context of the peak/dip picture (Bardeen etal. 1986; Bond et al. 1991), which include the fact that the eigenvalues of the linear shear field are related to regions where thesource of the displacement is positive (negative). These new conditional probabilities, derived in Section 2, are Equations (27),(30), (45) and (47): they represent the main results of this paper. Written in Doroshkevich-like format, they naturally reduceto Doroshkevich’s (1970) unconditional relations in the limit of no correlation between the potential and the density fields(i.e. when r = 0). As a first direct application of (30), the individual conditional distributions of eigenvalues were obtained inSection 3; these relations extend some previous work by Lee & Shandarin (1998) – see Appendix A for a compact summaryof their results. Much more analytic work can be carried out using these new formulae, especially in connection with thestatistics of peaks developed by Bardeen et al. (1986): their calculations can be extended within this framework (see Section2.3), and results of this extension will be presented in a forthcoming publication – along with more insights on the equationsderived in Section 2.2.To obtain the conditional distribution p ( ˜ T | r, ˜ H >
0) (i.e. Equation 18), in principle one needs to sample numericallythe probability (19) as done in Lavaux & Wandelt (2010). However, the new analytic results of this paper suggest a simplergeneralized excursion set algorithm, which will be also presented in a forthcoming study. The algorithm allows for a fastsampling of (19), and permits to test the new relations derived here (i.e. Sections 2 and 3, and Appendix A) against mockdata. In addition, along these lines it is possible to extend the main shape distributions involved in the triaxial formation ofnonlinear structures (i.e. ellipticity e , prolateness p , axis ratios ν and µ , etc.). Halos and voids are in fact triaxial rather thanspherical (i.e. Rossi, Sheth & Tormen 2011), and the initial shear field plays a crucial role in their formation. For example,in the ellipsoidal collapse framework the virialization condition depends on ellipticity and prolateness, which are directly c (cid:13) , 1–12 G. Rossi related to the eigenvalues of the external tidal field. Starting from p ( ζ , ζ , ζ | r ), it is then straightforward to turn this jointconditional distribution into p ( e, p, δ | r ) – to include the peak constrain, – and then to characterize p ( e | r ), p ( p | r ), p ( µ | r ), p ( ν | r )and so forth. Applications to halos and voids will be also presented next, including implications for the skeleton of the cosmicweb. The extension to non-Gaussian fields, along the lines of Lam et al. (2009), is a natural follow-up of this work and will bepresented in a forthcoming publication as well. In the context of cosmic voids, other interesting uses of the new formulae (27),(30), (45) and (47) involve the Monge-Amp`ere-Kantorovitch reconstruction procedure – see for example Lavaux & Wandelt(2010).The analytic framework described here can be also useful in several observationally-oriented applications, and in particularfor developing algorithms to find and classify structures in the cosmic web. For example, Bond, Strauss & Cen (2010) presentedan algorithm that uses the eigenvectors of the Hessian matrix of the smoothed galaxy distribution to identify individualfilamentary structures. They used the distribution of the Hessian eigenvalues of the smoothed density field on a grid to studyclumps, filaments and walls. Other possibilities include a web classification based on the multiscale analysis of the Hessianmatrix of the density field (Aragon-Calvo et al. 2007), the skeleton analysis (Novikov et al. 2006), as well as a morphological(Zeldovich-based) classification (for instance Klypin & Shandarin 1983; Forero-Romero et al. 2009). More generally, the factthat the eigenvalues of the Hessian matrix can be used to discriminate different types of structure in a particle distributionis fundamental to a number of structure-finding algorithms (Forero-Romero et al. 2009), shape-finders algorithms (Sahni etal. 1998), and structure reconstruction on the basis of tessellations (Schaap & van de Weygaert 2000; Romano-Diaz & vande Weygaert 2007), etc. In addition, the classification of different environments should provide a framework for studying theenvironmental dependence of galaxy formation (see for example Blanton et al. 2005). To this end, recently Park, Kim andPark (2010) extended the concept of galaxy environment to the gravitational potential and its functions – as the shear tensor.They studied how to accurately estimate the gravitational potential from an observational sample finite in volume, biaseddue to galaxy biasing, and subject to redshift space distortions, by inspecting the dependence of dark matter halo propertieson environmental parameters (i.e. local density, gravitational potential, ellipticity and prolateness of the shear tensor). Itwould be interesting to interpret their results with the theoretical formalism developed here, and ultimately to study thegravitational potential directly from a real dataset such as the SDSS Main Galaxy sample within this framework.The formalism presented in this paper is restricted to one scale (i.e. peaks and dips in the density field, as in Bardeenet al. 1986), but the extension to a multiscale peak-patch approach along the lines of Bond & Myers (1996) is duable andsubject of ongoing work. This will allow to account for the role of the peculiar gravity field itself, an important aspect notconsidered here but discussed for example in van de Weygaert & Bertschinger (1996). In fact, these authors introduced thepeak constraints to describe the density field in the immediate surroundings of a peak, and then addressed the constraints onthe gravitational potential perturbations; in particular, they constrained the peculiar gravitational acceleration at the positionof the peak itself, in addition to characterizing the tidal field around the peak. Including all these effects in our formalism isongoing effort. Finally, the interesting and more complex question of the local expected density field alignment/orientationdistribution as a function of the local field value (or the other way around – see Bond 1987; Lee & Pen 2002; Porciani et al.2002; Lee 2011) can be addressed within this framework, and is left to future studies. I would like to thank Ravi K. Sheth especially for bringing to my attention the work of Lee & Shandarin (1998), andChangbom Park for his always wise advices – along with numerous discussions. I would also like to thank the referee, Rienvan de Weygaert, for his insightful report and useful suggestions – which have been all included in the paper. Part of this workwas carried out in June 2011 during the “APCTP-IEU Focus Program on Cosmology and Fundamental Physics” at Postechin Pohang (Korea), and completed during the “Cosmic Web Morphology and Topology” meeting at the Nicolaus CopernicusAstronomical Center in Warsaw (Poland), on July 12-17, 2011 – where I had stimulating conversations with Bernard Jonesand Sergei Shandarin. I would like to thank all the organizers of these workshops, and in particular Changrim Ahn for theformer and Wojciech A. Hellwing, Rien van de Weygaert and Changbom Park for the latter.
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APPENDIX A: INDIVIDUAL UNCONDITIONAL DISTRIBUTIONS AND PROBABILITIES
Starting from Doroshkevich’s formulae for the unconditional distribution of the ordered eigenvalues λ > λ > λ (Eq. 38),Lee & Shandarin (1998) derived a set of useful probability functions. Their expressions can be considerably simplified by usingvarious symmetries, and by introducing the following notation: l = 112564 √ π (A1) n = √ π/ L ( x, y ) = ( x − y )(3 x − y ) · exp h − (cid:16) x − xy + 32 y (cid:17)i N ( x, y ) = ( x − y )[8 + 3(3 x − y )(3 y − x )] · exp h − (cid:16) x − xy + 3 y (cid:17)i · erfc h √
34 ( x − y ) i . c (cid:13) , 1–12 G. Rossi
Dropping the understood ∼ symbols for the eigenvalues (although all the quantities are still “reduced”, i.e. the σ dependenceis not shown), the two-point probability distributions are: p ( λ , λ ) = l h L ( λ , λ ) + n · N ( λ , λ ) i (A2) p ( λ , λ ) = l h L ( λ , λ ) + n · N ( λ , λ ) i (A3) p ( λ , λ ) = l n L ( λ , λ ) + L ( λ , λ ) + n · h N ( λ , λ ) + N ( λ , λ ) io . (A4)Performing the various integrations: p ( λ ) = l n Z λ −∞ L ( λ , λ )d λ + n Z λ −∞ N ( λ , λ )d λ o (A5)= √ π h λ exp (cid:16) − λ (cid:17) − √ π exp (cid:16) − λ (cid:17) erfc( −√ λ )(1 − λ ) + 3 √ π exp (cid:16) − λ (cid:17) erfc (cid:16) − √ λ (cid:17)i p ( λ ) = l n Z + ∞ λ L ( λ , λ )d λ + n Z + ∞ λ N ( λ , λ )d λ o (A6)= √ √ π exp h − λ i p ( λ ) = l n Z λ λ L ( λ , λ )d λ + n Z λ λ N ( λ , λ )d λ o (A7)= − √ π h λ exp (cid:16) − λ (cid:17) + √ π exp (cid:16) − λ (cid:17) erfc( √ λ )(1 − λ ) − √ π exp (cid:16) − λ (cid:17) erfc (cid:16) √ λ (cid:17)i . Note the symmetry between p ( λ ) and p ( λ ). In addition, with δ = λ + λ + λ ≡ k , one gets p ( δ, λ ) = 3 √ π h δ − δλ + 135 λ − i · exp h −
38 (3 δ − δλ + 15 λ ) i + 3 √ π exp h − (cid:16) δ − δλ + 152 λ (cid:17)i . (A8)By integrating the previous expression over λ , it is direct to verify that p ( δ ) is a zero-mean unit-variance Gaussian distribution.In addition: p ( λ >
0) = 2325 , h λ i = 3 √ π , σ λ = 13 π − π (A9) p ( λ >
0) = 12 , h λ i = 0 , σ λ = 215 (A10) p ( λ >
0) = 225 , h λ i = − √ π , σ λ = 13 π − π . (A11)Moreover, the probability distribution of δ confined in the regions with λ > p ( δ | λ >
0) = p ( δ, λ > p ( λ >
0) = R δ/ p ( δ, λ )d λ p ( λ >
0) (A12)= − √ π δ exp (cid:16) − δ (cid:17) + 254 √ π exp (cid:16) − δ (cid:17)h erf (cid:16) √ δ (cid:17) + erf (cid:16) √ δ (cid:17)i . Using the previous formula, it is direct to show that h δ | λ > i = 25 √ √ π (3 √ − ≃ . . (A13)It is also easy to see that the maximum of p ( δ | λ >
0) is reached when δ ≃ .
5: this can be readily achieved by computingthe derivative of the previous expression and by finding the corresponding zeroes, as done in the main text (see Figure 3).Finally, the probability that a given region with density δ will have all positive eigenvalues is: p ( λ > | δ ) = p ( δ, λ > p ( δ ) = − √ √ π δ exp (cid:16) − δ (cid:17) + 12 h erf (cid:16) √ δ (cid:17) + erf (cid:16) √ δ (cid:17)i . c (cid:13)000