Exploring the Geometry, Topology and Morphology of Large Scale Structure using Minkowski Functionals
Abstract
Modern redshift surveys such as the 2 degree field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS) reveal the fully 3 dimensional distribution of a million or so galaxies over a large cosmological volume. Visually galaxies appear to be distributed along sheetlike and/or filamentary superclusters. The CfA Great Wall, Southern Great Wall and the recently discovered SDSS Great Wall are very spectacular superclusters. Clearly theoretical predictions for galaxy clustering must be tested against these rich datasets. This can be achieved by means of the Minkowski Functionals (MFs). A MFbased approach provides an excellent description of superclusters and voids and allows one to quantify the properties of the cosmic web. In this review we give a summary of the progress made in this direction. After reviewing the status of observations and of numerical simulations, we comment upon the nature of bias which serves as a link between theoretical predictions and observations. We also summarise the methods developed for efficient numerical estimation of MFs for cosmological datasets and list several important results obtained using these methods. Specifically, we stress the discriminatory power of MFs and of the derived morphological statistics, the Shapefinders. Shapefinders are an excellent tool with which to study the shapes and sizes of superclusters and voids. We also discuss some of the important effects of scaledependent bias which are brought out by a MFbased study of the mock catalogues of galaxies. Such effects, we note, should be accounted for before comparing theoretical models with observations.
I Introduction
During the past decade, CMB experiments such as WMAP wmap , typeIa supernova searches and observations of galaxy clustering, have helped in turning cosmology into a precision science. The combined set of measurements point to a flat, low density (), accelerating Universe dominated by dark energy (). Modern redshift surveys aim to complement the above information about background cosmological parameters by providing a detailed map of the distribution of baryonic matter on large scales. Redshift surveys have so far been primarily used for two purposes: (1) to get independent constraints on the matter density and several other cosmological parameters (see below) and (2) to measure the abundance and clustering properties of a variety of galaxies. The former exercise provides important consistency tests for cosmic parameters estimated using other methods, whereas the latter gives precious feedback to the physics of galaxy formation. In addition, it is also important to test whether the observed largescale clustering properties of galaxies are consistent with the predictions of the preferred cosmological model(s). In order to carry out such a test, one needs to quantify the information content hidden in redshift space distribution of galaxies. In this article we review the methods developed to study the geometry, connectedness and morphology of large scale structure (LSS) in redshift surveys, which can eventually help us meet such a goal. Such an approach gives valuable insights into the largescale distribution of matter, and provides a framework within which to test the paradigms of gravitational instability and of the Gaussianity of the primordial density field.
Impressive redshift surveys have been undertaken recently to map out the 3dimensional distribution of galaxies. In the next section, we shall outline the history of redshift surveys, and focus on two major recent surveys, 2dFGRS and SDSS. LSS in redshift surveys exhibits phenomenally rich and complex texture, wherein sheetlike and/or filamentary superclusters intersperse with large, empty regions called voids. Several salient aspects of the LSS of the Universe will be dwelt upon and we shall stress why it is important to study the morphology of LSS fn1 . In subsequent sections we shall show that the complex morphological features of LSS are objectively quantifiable and that morphological information supplements traditional approach of understanding LSS using its point correlation functions and can be employed to confront theories with observations in an integrated manner. Statistical methods which probe the geometry and topology of LSS include percolation analysis sh83 ; shandzed89 , genus curve gott86 ; melotopo90 , minimal spanning trees barbhav85 , void probability function white79 , Voronoi tesselations rien94 , as well as more recently introduced Minkowski Functionals (MFs) meckwag94 ; Krofton97 . For excellent reviews of these different approaches, the reader may refer to sc95 ; rien ; martinez02 ; jones04 . In this article, our focus will be on MFs.
Ii Redshift Surveys
Ever since sky maps of galaxy counts, e.g., Lick Catalogue shane54 revealed a rich pattern in LSS of galaxies in projection, it was realised that complementing this information with the redshifts of individual galaxies was essential to get a fully 3dimensional view of the Universe. In such an exercise, one turns the redshift of a galaxy into its distance with the help of Hubble’s law hubble29 , and is thus able to study the redshift space distribution of galaxies.
Various small surveys were carried out in the late 1980’s and mid 1990’s in a controlled fashion; their positive scientific outcome providing impetus for the initiation of bigger ones. Of these perhaps the most influential was the Center for Astrophysics Surveylapp86 . This survey had an angular coverage of and a depth of Mpc. The survey measured redshifts of about 2400 galaxies. The famous Great Wall centered around the Coma cluster was the most striking feature of this survey. The slice showed very clearly the “bubbly” nature of LSS with voids having sharp boundaries; the largest void having a diameter of Mpc. Subsequent surveys including followup CfA slices and the ESO Southern Sky Surveydacosta91 amply confirmed the impression given by the first CfA slice. Indeed, the Southern Sky Redshift Survey (SSRS) discovered a second Great Wall in the southern sky. The emerging picture of filamentary and sheetlike superclustersfairall encircling frothy voids prompted cosmologists to seek an explanation from theory, and this interplay between theory and observations is only now maturing into a truly quantifiable discipline. Another important survey, the Optical Redshift Survey (ORS)sant95 had a depth of 80Mpc, but attempted a complete sky coverage (except for the zone of avoidance). This survey measured 8500 redshifts in total, and was heavily used to estimate luminosity functions, galaxy correlations, velocity dispersions etc. In the mid 1990’s, the StromloAPM redshift surveylove95 and the Durham/UKST redshift surveyrat98 were focussed towards the southern sky and led to fruitful results on correlation functions in real and redshift space, power spectra, redshift distortions, cosmological parameters, bias etc. However, the galaxy redshifts in all these surveys never exceeded .
The Las Campanas Redshift Survey (LCRS)shect96 consisted of six 1.5 slices (3 each in the northern and southern galactic hemisphere), went to the depth of 750 Mpc () and recorded redshifts of about 25000 galaxies. LCRS presented an orderofmagnitude improvement over earlier surveys. This was the first deep survey of sufficient volume which could test ideas such as the statistical homogeneity of the Universe on large scales. Other important results from analysing LCRS included the luminosity function, second and thirdorder correlation functions, power spectra etc. LCRS is perhaps the only large survey wherein both the topology and morphology of LSS have been extensively investigated colley97 ; bharad ; bharad04a and used to confront observations with theoretical models bharad04b .
ii.1 The 2 degree Field Galaxy Redshift Survey (2dFGRS)
The 2dF multifiber spectrograph on the 3.9m AngloAustralian Telescope is a very impressive facility which can obtain spectra for up to 400 objects simultaneously over a field of view with 2 diameter. The spectrograph was put to use to sample two contiguous constantdeclination strips of the sky, one in the northern galactic hemisphere [with angular coverage of in ()] and the other in the southern galactic hemisphere [with angular coverage in ()]. Working in the band, the survey reached a depth of with median redshift of 0.11, and has compiled redshifts of about 2.5 galaxies which are brighter than an extinction corrected magnitude of = 19.45colless03 .
The survey is already complete and the resulting galaxy correlation functions, redshift distortions and pairwise velocity dispersions hawkins03 demonstrate the superb quality of the data set. The science output of this survey and its overall cosmological relevance have been excellentlly reviewed in pea03 . We shall discuss here some of the results which are most relevant to our present discussion.
One of the main science goals of 2dFGRS was to get improved estimates of the 3dimensional power spectrum and the twopoint galaxygalaxy correlation functionfn2 . Since 2dFGRS sampled a cosmological volume as large as [ Gpc], it became possible to measure these quantities over a large range of scales 0.01 Mpc with an unprecedented accuracyperci01 ; tegmark02 . Assuming a constant bias between dark matter and baryons on scales Mpc, Percival et al.(2001) compared the shape of the power spectrum with the primordial power spectrum predicted by linear perturbation theory of structure formationperci01 . This allowed them to constrain the values of , after priors on the power spectrum index and the Hubble constant had been applied (). Apart from this, 2dFGRS was employed to obtain estimates for the luminosity function of early and late type galaxiesmadg02 , and to study the luminosity dependent clustering of galaxies as measured using the correlation function statisticnor02 . The latter study provided clues about the large scale bias between baryonic and dark matter and revealed that the former might follow the latter through a linear relationship (also see lahav02 ).
Cosmological parameter estimation is clearly one of the key results gleaned from large redshift surveys such as 2dFGRS. However, in our view several assumptions made in the process remain to be adequately justified, for instance, the assumption of linear, constant bias used in the exercise mentioned above is clearly inadequate. In fact, the study of biasing of galaxies with respect to the underlying mass distribution has its own history, and the issue has not yet settled completely. Below we shall briefly discuss the currently established ideas about bias.
Figure 1 shows the distribution of galaxies in the fluxlimited 2dFGRS slices. To the eye, galaxies are distributed anisotropically, with sharp qualitative similarities with the distribution of dark matter in Nbody simulations (see below). The 3dimensional powerspectrum, in a more appropriate form , measures the amount of power per unit interval in the logrithmic bin around wavenumber . In simple terms, this is a statistical measure of the density fluctuations within spheres of radius . Clearly, in a measure such as this, the information on lengthscales is averaged out. Hence, one learns relatively little about the nature of the actual distribution of galaxies. Describing (and explaining) the visually rich filamentary pattern of superclusters of galaxies and quantifying the complex nature of the LSS is another, equally important application of redshift surveys. This problem has not been adequately addressed despite considerable recent progress hoyle04 ; hoyle02 ; hoyleetal02 ; hikage03a ; basil03 ; dorosh04 ; hikage02 . Clearly, the contribution of higher order correlation functions is an important ingredient in such a program. Since higher order correlations are difficult to measure in practice, it would be useful if we could quantify the connectedness and nonGaussianity of LSS as well as the shapes and sizes of individual superclusters and voids using geometrical diagnostics. The methods reviewed in the present article are intended to fill this gap in our knowledge and to complement more traditional, correlation function based approaches in developing a quantifiable picture of the LSS and its properties.
ii.2 Sloan Digital Sky Survey (SDSS)
The Sloan Digital Sky Survey (SDSS) is derived from a dedicated 2.5m telescope. The initial photometric program has been to measure the positions and brightness of about 10 objects in steradian (almost 1/4) of the sky. The survey is centered on the northern galactic pole and has an elliptical angular coverage of , where the semimajor axis runs along a line of constant rightascension.
Followup spectroscopy is planned to give redshifts of about 10 galaxies and 10 quasars. At the time of writing, the SDSS team has made its 2 data release, available for free download abaz04 . It contains redshifts of galaxies, with the farthest galaxy at a redshift of 0.3. (See http://cas.sdss.org/astro/en/tools/search/SQS.asp.)
The limiting magnitude of the SDSS survey is in the red band (), and its depth is similar to 2dFGRS. The survey will cover a cosmological volume as large as 1[ Gpc]. Taking the length scale of 100 Mpc as a tentative scale of homogeneity, this is approximately thousand times a representative volume of the Universe. Hence when fully complete, SDSS can be fruitfully used to evaluate the cosmic variance for many statistics associated with LSS. The data are already large enough to permit accurate estimation of power spectrum tegmark04a , 3point correlation function kayo04 , and of cosmological parameters tegmark04b . Complementary studies of LSS in SDSS have been undertaken using MFs by hikage03a and using the minimal spanning treeformalism by dorosh04 . We shall mention later some results from an MFbased study of LSS in mock catalogues of SDSS jsh04 .
In order to make full use of the information encoded in the observed LSS, it is essential to understand the relationship between the number density of galaxies and the underlying mass density field. Since 1980s, it has been realized that the visible density field due to galaxies may be biased relative to the massdensity. In order to appreciate the implications that bias may have in the program of confronting theory with observations, below we briefly review the state of our knowledge in this regard. First we shall discuss the theoretical modelling of LSS using numerical simulations.
Iii NBody Simulations
Strong evidence exists to suggest that dark matter is 10 times more abundant than visible baryonic matter. Dark matter is usually assumed to be collisionless and to interact only through gravitation. Due to its relative abundance dark matter is clearly more important than baryons from a dynamical point of view. The best way to understand the evolution of a gravitating system consisting of N dark matter particles would be to actually solve for the force acting on a given particle due to the remaining particles. In a rather different form, by solving for the Poisson’s equation at every stage of evolution of the matterfield, this is precisely the goal which modern versions of Nbody simulation codes achieve. (For an excellent introduction to Nbody simulations, see bert98 ; kly00a ; kly00b ; kly00c ).
Simulations allow us to probe the strictly nonlinear regime of the dark matter density field, and can provide us with detailed predictions of the dark side of the Universe at today’s epoch. (The resulting dark matter distribution can be probed through observations of gravitational lensing – an area of great future promise.) The large scale structure distribution in dark matter could very well
resemble that shown in Figure 2. We note from this figure, how the matter condenses along filamentary/sheetlike ridges between clusters and how voids get progressively more evacuated with the passage of time. Due to weblike and frothy appearance, LSS has been described at various times as being a Cosmic Web bond96 or as Cosmic Froth rien .
During the past two decades the advent of parallel supercomputers have ensured a phenomenal improvement in the computing power available to tackle complex astrophysical processes. Hence, it has become possible to include complex gasdynamical processes into simulations. Various recipes for cooling, star formation and feedback due to supernovae and galactic winds have been incorporated to mimic the formation and evolution of galaxies in a cosmological setup with simulated volumes ranging from 50 to 100 Mpc wein02 ; wein04 . This has made it possible to develop mock catalogues of galaxies inspired by a given cosmological model. Relatively less timeconsuming semianalytic methods have also been proposed which hope to achieve similar goals while bypassing the complex baryonic physics benson02 ; cole00 ; kauf97 ; sc95 ; scocci02 . Such mock catalogues can eventually be compared with clustering properties of galaxies derived from redshift surveys with the help of methods reviewed here.
Iv The Nature of Bias
Above we summarised the theoretical efforts being made to understand the clustering of dark matter. From the predictive point of view, given the fully evolved nonlinear massdensity field (both in the real universe and in Nbody simulations), we would like to know the exact locations where galaxies are formed. An important question related to this is whether the clustering properties of the resulting galaxy distribution (quantified in terms of point correlation functions or some other alternate statistics) are the same as the underlying matterfield, or whether the two are different. The answer to this, clearly depends upon the recipe which we utilise in grafting galaxies on the density field.
During the mid1990’s it was shown that AGNs, IRASselected galaxies and galaxies selected in the optical bands cluster differently pea94 . Earlier (during the 1980’s) it was established that clusters of galaxies are more strongly clustered than galaxies themselves kai84 . Recently it was proven that the more luminous galaxies are more strongly clustered nor01 ; nor02 . Since these different visible tracers of LSS appear to cluster differently, their distributions are clearly biased relative to each other. There is hence, a reason to believe that the visible matter as a whole will be biased relative to the underlying dark matter. This phenomenon is termed as bias. Knowledge of the nature of bias will clearly enable us to connect the two distributions visible and dark; observed and simulated.
In cosmology, it is a conventional practice to quantify the clustering of LSS using the twopoint correlation function or, equivalently, the power spectrum. Both these statistics are known to have a powerlaw form for galaxies groth77 ; baugh96 . However, the correlation function of dark matter was shown to differ from in a complicated, scaledependent fashion for various CDMbased cosmogonieskly96 ; pea97 ; jenkcdm . This suggests that the bias cannot be scaleindependent, i.e., light may not follow the mass in a linear fashion. Recent investigation involving the threepoint correlation evaluated for the SDSS dataset implies that galaxy biasing could be quite a complex and nonlinear process; see kayo04 .
Kaiser(1984) explained the strong clustering of clusters using the high peak model kai84 . This model states that a rare, highdensity fluctuation corresponding to a massive object of a given size, collapses sooner if it lies in a region of a largerscale overdensity. This is because, the smallscale overdensity is aided by the surrounding largescale overdensity so that the collapsecriteria is fulfilled relatively easily for such a halo. This “helping hand” from the longwavelength mode means that overdense regions contain an enhanced abundance of massive objects with respect to the mean. Later this model was generalised to incorporate objects of any mass cole89 ; mowhite96 .
The highpeak model is based on kinematic premises. It traces back the observed clustering of various classes of objects to the inbuilt structure in the primordial density field. It could fruitfully be applied to study the clustering of galaxies only at high redshift () kai84 ; cole89 ; steidel97 , for the galaxyscale fluctuations were prone to collapse at such early times. At such high redshifts, galaxies were shown to be highly biased relative to background steidel97 . Starting from , until the present, galaxies have very likely collapsed in all environments. The underlying massfluctuations have also grown. Thus, one predicts that the large bias (defined as ) which existed earlier, should asymptotically approach unity on large scales. Lahav et al.(2002) showed for 2dFGRS data that the bias averaged over all scales is statistically consistent with a scaleindependent, largescale bias of order unity. However, we should see whether this knowledge helps us achieve our original goal of connecting the simulations with the observed Universe, i.e., in populating the massdensity field with galaxies in the correct manner.
A recent, Halo Model based approach is a fruitful step in this direction. Here all the complications of galaxy formation are encoded via the halo occupation number: the number of galaxies found above some luminosity threshold in a virialised halo of a given mass . In a nutshell, the halo model describes nonlinear structures as virialized halos of different mass, placing them in space according to the linear, largescale density field cooray02 . This model is analytically tractable, and correctly incorporates the dependence of bias on the 2 order correlations in the density field. Consequently, it correctly reproduces the observed 2point correlation function and the power spectrum.
Figure 3 compares the exact nonlinear massdensity field with the halo model representation. The halo model is remarkably successful in obtaining correct correlation function, power spectrum etc. However, as shown in Figure 3 it is manifestly different from the fully evolved density field. This brings out an important aspect relevant to further study of LSS: the loworder correlations cannot be relied upon to provide us with a complete picture of clustering. In addition to the large halos of clusters and groups, today the rich filamentary/sheet like patterns in matter density field are also populated by galaxiesfn4 , and one must theoretically understand the emergence of halos along filaments/sheets in order to complete the picture. In fact, the following experiment can be carried out: the galaxies can be selected in massdensity fields from two rival cosmogonies, so that the observed galaxygalaxy correlation function and the power spectrum are reproduced. In comparing our predictions with the LSSdata, if we confine ourselves merely to twopoint correlation function, such an algorithm would validate both models to be good representatives of our Universe. Evidently this is not true. Clustering properties of galaxies in the two models would of course be different from each other (and perhaps even from LSS in the Universe). This serves to establish that the largescale distribution of galaxies cannot solely be described by the twopoint correlation function alone fn6 . The resulting galaxy distributions can be shown to differ from each other by employing statistics which are sensitive to higher order correlation functions. Following this, we arrive at an important conclusion: the two distributions of galaxies must be compared with the help of higher order correlation functions or statistics derived therefrom.
To summarise, although the high peak model kai84 gives us promising results for galaxies assembling at high redshift and for Abell clusters at present, the predictive power of halo model in populating mass density field with galaxies is clearly inadequate (in fact, due to a variety of reasons, the bias could even be stochastic; see dekel99 ). In particular, for a combination of a cosmological model and a biasing scheme to be proven correct, it is not sufficient that the two explain the twopoint correlation function for galaxies alone. In order to effectively compare the theoretical predictions for galaxydistribution with the data from redshift surveys, one should include information from higher order correlation functions, and/or alternate statistics which can effectively quantify the complex morphology of the galaxy distribution. One such class of diagnostics the Minkowski Functionals – are introduced in the next section.
V Minkowski Functionals
v.1 Minkowski Functionals in Cosmology
It is well known that, in contrast to a Gaussian random field (hereafter, GRF), a fully evolved nonlinear density field cannot be fully quantified in terms of its twopoint correlation function; the latter being simply the lowest and first of an infinite hierarchy of correlation functions describing the galaxy distribution. Furthermore, the bias in galaxies viz a viz dark matter can be nontrivial in nature, and can give rise to different clustering properties of galaxies and the underlying mass density field. Consequently the correlation function determined either for galaxies or for dark matter does not on its own validate a given cosmological model unless support is provided by other, statistically indepdend measures of clustering. It therefore becomes a challenge to compare the theoretical predictions for LSS with real observational data.
If we knew all point correlation functions, we would have a complete description of the galaxy clustering process. However, estimating for a sample of galaxies requires knowing all pairs of galaxies in this sample, whereas calculating the 3point function implies taking all triplets. The amount of computation escalates rapidly with – the order in the point correlation function – especially for a large value of . Besides, it is difficult to extract intuitively useful information from these statistics. The present article focuses on another class of statistics which complement the correlation function approach and which have the advantage of providing a physically appealing interpretation for the evolving density field. These are the Minkowski Functionals.
For an excursion set involving particles embedded in dimensions, the Minkowski Functionals (hereafter, MFs) are defined on an ()dimensional hypersurface. There are MFs in dimensions. In this article, we shall be concerned with the 3dimensional distribution of dark matter and/or galaxies. Hence, MFs will be defined on a 2dimensional surface and they will reflect the physical properties of this surface (in the given instance, an isodensity contour referring to a supercluster of galaxies or a void). For a given surface the four MFs blaschke36 ; min1903 ; meckwag94 are, respectively

Volume V,

Surface area S,

Integrated Mean Curvature C,
(1) and are the two principal radii of curvature of the surface in a given local neighbourhood and the integral is taken over the entire closed surface.

Integrated Gaussian Curvature (or Euler characteristic) ,
(2) A related quantity which is more popular in Cosmology is the genus . The genus is related to by
(3) The 3dimensional genus of an object is a topological invariant. It can be interpreted in terms of the connectivity of the surface. In simple terms it can be viewed as the number of independent cuts which one can make to the surface without breaking it into two separate pieces. A torus can be cut once and yet remain in one piece. Hence, its genus is 1. However, a sphere would separate into two pieces if a cut were made. Hence, its genus is 0. Two objects with the same value of genus are topologically similar: one can be obtained by continuously deforming the other. Thus, a sphere and a cube are topologically equivalent. More concretely, genus of an object is the number of handles that the object has, in excess of the number of holes which it encloses (e.g., see matsub03 ). Thus,
According to this definition, a sphere has no handle, a torus has a single handle (equivalent to a sphere with one handle), and a pretzel has two handles. Introducing a hole or a bubble inside the surface reduces its genus by 1, whereas adding a handle to the surface increases its genus by 1.
An attractive feature of MFs is that they depend upon the entire hierarchy of correlation functions schmalthesis . Thus, MFs can indirectly help us capture the effect of point functions. Higher order point functions gradually become important as a primordial, featureless GRF evolves to develop nonlinear structures. This reflects in the behaviour of MFs for the corresponding density field. MFs are additive in nature, i.e., they can be studied for individual objects (say, clusters or superclusters defined using some prescription) as well as for the entire ensemble of such objects.
Of the 4 MFs listed above, the genus G was already known to the cosmology community dorosh70 ; gott86 ; melotopo90 . Tools like percolation analysis which can also be related to MFs, had earlier been introduced in cosmology by zes82 . There were strong reasons why this was the case.
It is well known that a system evolving under gravitational instability becomes progressively more nonGaussian. (The hypothesis of a primordial spectrum of density perturbations which is distributed in the manner of a Gaussian random field is supported by observations of the cosmic microwave background carried out by WMAP and other experiments.) For CDMlike cosmological models this results in more matter being concentrated in filamentary and pancakelike distributions whose coherence scale evolves with time. As a result the filling fraction at percolation progressively decreases, from for the initial Gaussian random field to for LCDM ks93 ; sss97 ; shyes98 ; shshs04 . This implies that more matter is being transferred to regions which occupy a progressively smaller amount of space and are also spatially anisotropic and therefore percolate easily. Figuratively this corresponds to the prominance acquired by superclusters of galaxies over time – see Figure 2. This theoretical picture which described the emergence of structure in an initially featureless medium agreed well with what observations were telling us about our universe. The emerging picture of LSS taken from redshift surveys shows us that galaxies are distributed preferentially along filaments and sheets which seem to encircle vast voids. The distribution of galaxies does not occupy much volume, and because of the presence of superclusters, percolates easily. Percolation theory can therefore be used to describe an important quality of the observed galaxy distribution – its connectivity. At the percolation threshold one can visit the far corners of the cosmic web by following filamentary/sheetlike overdensities. The cosmic web is therefore a connected structure and this is clearly revealed in surveys such as Figure 2. Connectivity is of course a mathematical notion, and it is possible to precisely quantify connectivity of an object or a system by evaluating its percolation properties and its genus. Doroshkevich(1970) gave analytic formula for the genus of a GRF which showed that at the median density threshold, GRF would exhibit a spongelike topology dorosh70 . Simulations of LSS within CDMcosmogonies provided similar visual impression when smoothed on sufficiently large scales. Like percolation, the genus too has come to be considered as a useful probe of the initial Gaussianity (as well as the final nonGaussianity) of the cosmic density field; see for instance melotopo90 ; colley97 ; sss97 .
An important indicator of nonGaussianity in a distribution is the percolation curve. The percolation curve describes the volume fraction (or filling factor – FF) in the largest structure (cluster/void) as a function of the density contrast threshold fn7 . Salient features of the percolation curve are illustrated in Fig. 4 in which density fields (with an initial power spectrum ) evolve from an epoch when the scale of nonlinearity is , ( is the fundamental mode corresponding to the boxsize of our Nbody simulation). In Fig. 4 percolation curves for clusters (thick solid lines) and voids (thick dashed lines) are shown as functions of the density contrast Starting from a high density threshold (small ) we find several isolated clusters (corresponding to peaks of the density field). Lowering the density threshold further results in the merger of clusters leading to a rapid growth in the percolation curve and to the onset of percolation. A further lowering of the threshold to very small values results in the merger of almost all clusters so that . An identical procedure followed for underdense regions by gradually increasing the density contrast threshold (increasing ) results in a similar functional form for the volume fraction in the largest void. In our samples the largest cluster percolates between opposite faces of the cube when its filling factor is about half the total . In most cases percolation also coincides with the highest jump in the volume of the largest cluster which was used by delgh91 as a working definition of the percolation threshold.
The solid/dashed vertical line in Fig. 4 represents the density contrast threshold below/above which clusters/voids percolate. We note that both clusters and voids percolate over a range of overlapping density contrasts — a feature that is only possible in three or more dimensions — and corresponds to what is commonly called a sponge topology for the density distribution.
As the simulation evolves increases monotonically for , as structures form and align on increasingly larger scales. From Fig. 4 we see that voids find it easier to percolate as the simulation evolves, as a result the range in densities when both phases percolate initially increases, enhancing the extent of spongelike topology in the distribution.
The right panel in Fig. 4 shows the evolution of the genus curve for the same simulation. The top panel corresponding to an early epoch shows the genus curve largely retaining its bellshaped form which it has for the primordial Gaussian random field (hgw86 ; gwm87 ). During later epochs the density distribution grows progressively more nonGaussian and this is reflected in the change in both shape as well as amplitude of the genus curve in Fig. 4.
Although a study of the genus gives us a useful handle on the connectivity of a density field, we still lack information about the morphology of LSS. For example, if the domain of information is solely restricted to knowing genus, a filament with one handle would be considered to be identical to a sphere with one handle or a pancake with one handle. Therefore, to gain objective insights into the nature of the superclustervoid network, we must complement genus of an object with quantities which in some way, characterise geometry of that object. This is precisely the role played by the first three MFs, Volume V, Surface Area S and Integrated Mean Curvature C. These MFs change when local deformations are applied to the surface, and hence, these may be useful to glean information about the typical size of an object. The final goal here would be to use this geometric information and give an objective meaning to generic sizes and shapes of superclusters and voids belonging to LSS. In this article we will show that MFs can be utilised to characterise the geometry and topology of the cosmic density field, and eventually can be employed to get information about the morphology of the superclustervoid network in LSS. Because MFs depend on the full hierarchy of correlation functions, we can use them to compare and distinguish two rival cosmological models. Recently, Matsubara (2003) derived semianalytical expressions of MFs for weakly nonlinear cosmic density field by using a secondorder perturbative formalism matsub03 . The results obtained by Matsubara can be used in conjunction with the methods reviewed here to obtain insight into the dynamics of the clustering process of a density field smoothed on sufficiently large scales. The role of bias on large scales can also be directly probed using such methods.
v.2 Morphology of LSS with Shapefinders
Sahni, Sathyaprakash & Shandarin (1998) showed that the MFs could be used to evaluate both the size as well as the shape of a threedimensional object such as a supercluster or a void sss98 . The size is given in terms of three Shapefinders which have dimensions of length. These are defined as ratios of MFs, and are conviniently termed as Length (), Breadth () and Thickness ().
(4)  
(5)  
(6) 
The three Shapefinders are defined using spherical normalization, so that for a sphere of radius , .
The above measures quantify the size of the object in question. To further quantify the shape of these objects, two dimensionless Shapefinders have been defined as follows. These are Planarity () and Filamentarity ().
(7)  
(8) 
Sahni et al.(1998) showed that () quantify the shape of both simple as well as complicated objects. Thus, for an oblate ellipsoid its planarity is relatively large and its filamentarity is small. For a prolate spheroid, the reverse is true. The statistics respond monotonically to deformations of these surfaces.
If we include the genus then, the triplet of numbers (), can define a threedimensional ShapeSpace which may be used to represent the distribution of shapes of superclusters and their topologies in a given cosmological density field. Note that the inclusion of genus is significant, since most superclusters at moderate density thresholds can have a spongy texture with several loops and/or branches emanating from a central body (see Figure 11 for an illustration).
v.3 Methods of Estimation
In cosmology the distribution of galaxies in space can frequently be regarded as being an example of a point process. There have been several attempts of evaluating MFs for a given point process. For example, the Boolean grain model meckwag94 ; schmalthesis would decorate the input set of points with spheres of varying radii, and would study the morphology of the structures which contain overlapping spheres. According to another approach Krofton97 one smooths the distribution of points using a suitable window function, and defines the density field on a grid. MFs can be evaluated for connected structures defined on a grid by using Crofton’s formulae or Koenderink Invariants Krofton97 .
A new approach for determining MFs has been reported in shsssh03 . Rather than working with a gridapproximation for connected structures, these workers go a step forward, and model surfaces for these objects. Minkowski Functionals are evaluated for the isodensity surfaces online, i.e., while these are being modelled using an elaborate surface triangulation technique. Sheth et al.(2003) describe the algorithms for surface construction and for the evaluation of MFs on these surfaces. Sheth (2004) discusses the method of implementation of these algorithms which has culminated into a robust and accurate software SURFGEN (short for “SURFace GENerator”)sheth04 . SURFGEN has been tested for accuracy against Gaussian random fields and against a variety of simply and multiply connected eikonal surfaces. SURFGEN evaluates the geometric MFs to better than 1% accuracy in most cases, while the genus is evaluated exactly shsssh03 . SURFGEN has since been employed to study geometry, topology and morphology of LSS both in dark matter and galaxies within a variety of cosmogonies. (SURFGEN could also be relevant in areas of science where the properties of surfaces becomes important such as in medical imaging (tomography) and in condensed matter physics.)
In the next section we shall discuss a set of new results obtained by applying MFbased techniques to LSSdata.
Vi Results
vi.1 Morphology of Large Scale Structure using Shapefinders
Given a cosmic density field, Minkowski Functionals (MFs) can be evaluated for a supercluster (void) defined as a connected overdense (underdense) region above (below) certain physically motivated threshold of density. SURFGENlike technique would isolate such a structure, and would model its isodensity contour in order to precisely estimate its MFs. MFs referring to a given structure are called partial MFs. By knowing partial MFs, one can further measure shape and size of the concerned structure using Shapefinders. Further, MFs are additive in nature. Hence, at any given threshold of density, global MFs pertaining to the entire density field can be obtained by summing over partial MFs of the constituent overdense/underdense objects. While partial MFs are useful to measure shapes and sizes of individual superclusters and voids, global MFs can be employed (1) to discriminate between models, (2) to probe the effect of nonlinear gravitational evolution on the density field and (3) to capture the nontrivial effects of bias on the structure of the cosmic web. We shall review here some of the promising results obtained in these directions.
vi.2 Discriminating between rival models of LSS
The growthrate of density perturbations differs between various cosmogonies. As a result, the clustering properties of the matterdistribution also turn out to be different. A serius candidate statistic of LSS should be able to capture such differences and discriminate between rival cosmological models.
The three upper panels of Figure 5 show clustered distribution of dark matter at in three rival cosmological models SCDM, CDM and CDM respectively, simulated by Virgo group jenkcdm . The three models are simulated such that the observed abundance of rich clusters of galaxies can be reproduced in all of them. SURFGEN was employed to quantify the visually apparent differences between these models. Global MFs were evaluated at a set of density levels, and were studied with respect to FF, the fractional overdense volume shsssh03 . The lower 3 panels of Figure 5 show variations in 3 global MFs area, curvature and genus as density level is progressively brought down. Notice that MFs for SCDM (dashed lines) show the largest amplitudes, those for CDM (solid lines) show the smallest amplitudes with CDM curves (dotdashed lines) falling between the two. Since the number of rich clusters and their relative spatial distribution in all the models is the same, this difference can be attributed to difference in matterdistribution along sheets and/or filaments connecting the clusters shsssh03 ; sheth04 . Global MFs are thus noted to be remarkable at discriminating between models (also see schmal99 ).
In Section IV we noted that the distribution of galaxies cannot be fully quantified using its 2point correlation function alone. However, the galaxy distribution predicted by a given cosmological model must at least reproduce the observed . Cole et al.(1998) generated mock catalogues of galaxies in a variety of cosmogonies subject to the above constraint. Thus, galaxies were selected from the simulated massdistributions so that the observed is reproduced. The catalogues were further used to construct mock 2dFGRS and SDSS redshift surveys in such models cole2df98 . Since MFs depend on the hierarchy of correlation functions, these can be relied upon in distinguishing the galaxy distributions resulting from different models. Sheth (2004) made such a comparison between mock SDSS catalogues of galaxies from CDM and CDM models by evaluating global MFs of density fields smoothed with a Gaussian window 6 Mpc wide.
Figure 6 shows the global mean curvature and genus for the two models evaluated using SURFGEN. Note that the MFamplitudes in CDM are larger than those for CDM model, and that the two models are successfully distinguished from each other using MFs jsh04 , even in the case when the distributions share the same . (The CDM curves are averaged over 10 realizations made available by cole2df98 .) MFs can thus be reliably adopted to compare mock catalogues with the observed LSS in redshift surveys.
vi.3 MFs and the scaledependent bias
We reviewed the status of our knowledge about bias in Section IV. We raised an important question as to whether the clustering properties of dark matter and the biased galaxy distribution could be different or otherwise. This question was investigated by jsh04 by employing the mock SDSS catalogues of galaxies in CDM and CDM models. The clustering properties of galaxies were measured in terms of MFs.
As noted by melotopo90 ; springtop , the amplitude of the genuscurve drops as the Nbody system develops phase correlations. Given two density fields, the system with larger genusamplitude shows many more tunnels/voids which are, therefore, smaller in size. As time progresses, the voids are expected to expand and merge, leading to a drop in the genusamplitude, while the phase correlations continue to grow. With this simple model, one could correlate the amount of clustering with the relative smallness of the amplitude of genus, and therefore, of the area and the mean curvature. Going by this reasoning, we may infer from Figure 6 that the CDM galaxydistribution is relatively more porous and would show less coherence on large scales compared to that due to CDM. As illustrated in Figure 5(see shsssh03 ), the dark matter distribution of CDM model due to Virgo group shows considerably larger amplitudes for the MFs compared to the CDM model, whereas, we find the reverse trend in the MFs of the galaxy distributions due to the same two models. Evidently biasing appears to be a source of this effect.
To establish this effect more firmly, the CDM and CDM dark matter Virgo simulations were analysed by adopting the same resolution (=3.5Mpc) and the smoothing scale (=6Mpc) as utilised in the analysis of mock catalogues. So as not to introduce any bias due to redshift space distortions, global MFs of the galaxy catalogues were computed in real space.
The effect of biasing is most dramatically seen in the global genuscurve of the dark matter and galaxies (see Figure 7). We notice that the amplitudes , whereas . Here subscripts and stand for “dark matter” and “galaxies”, respectively. We may conclude from here that the matter distribution in CDM is less porous than that in CDM, but this trend reverses as we investigate biased distributions of galaxies due to the same two models: the galaxy distribution due to CDM appears to be more porous than that due to CDM. This could perhaps mean that a simple scheme of scaledependent biasing leads to different degree of phasecorrelations in the galaxy distribution as compared to the underlying mass distribution. This can be dubbed as a phenomenon of phasemismatch between the two distributions visible and dark jsh04 . This is supported by the fact that cole2df98 require large antibias in high density regions of the CDM model.
To conclude, the study of the global MFs reveals that, local, densitydependent bias could lead to an apparent phasemismatch or different phasecorrelations among the dark matter and the galaxies. It therefore becomes important to include realistic treatment of bias before comparing theoretical predictions about LSS with redshift surveys.
Model  Volume  Area  Curvature  Genus  Shapefinders (Mpc)  

Mpc  
CDM  6  5.63  7.30  70.03  
1  5.63  7.34  92.74  
1  5.47  6.93  70.00  
0  6.27  7.90  102.7  
0  5.71  7.22  69.95  
1  5.25  6.70  40.53  
0  5.3  6.68  72.90  
0  5.57  6.65  65.24  
0  5.44  7.00  56.33  
1  5.72  7.44  22.74 
We noted earlier that morphology of LSS appears weblike with frothy voids separating sheetlike and/or filamentary superclusters. The rich texture of LSS is a culmination of two interconnected factors: (1) strong nonGaussianity induced in cosmic density field due to gravitational dynamics and (2) biased formation of galaxies relative to the underlying mass. Superclusters are thus strongly nonGaussian, baryonic structures. Unlike galaxies and their clusters, superclusters are as yet unrelaxed and dynamically evolving; their morphological properties may be expected to be sensitive to the cosmological parameters of the Universe. It is hence important to quantify their shapes and sizes and relate them with physically relevant other quantities such as the mass enclosed or volume occupied by them. Similarly, voids too can be studied, and the combined properties of the superclustervoid network could provide us with yet another, independent check on the cosmological parameters. Since modern redshift surveys are deep and large enough, this may perhaps be the most effective method to test theoretical predictions.
Recently, a comprehensive morphological study of the superclustervoid network in CDM cosmogony was carried out by shshs04 . Dark matter distribution at was smoothed with 5 Mpc Gaussian window to produce a cosmic density field sampled over a box of size 239.5 Mpc. Superclusters and voids were defined as overdense and underdense connected regions, respectively and their MFs and Shapefinders were measured using SURFGEN.
Superclusters in CDM model were found to percolate through the Universe while occupying fractional overdense volume as small as 7% shshs04 . This points to the connectedness of the overdense regions in the Universe. From their tendency to effectively percolate the volume, it is apparent that superclusters must show strong departure from sphericity. Voids, on the other hand, occupy as large a fractional volume as 22% before the underdense regions percolate shshs04 . Evidently, the morphology of voids would be markedly different from superclusters. In shape, they are likely to be more isotropic.
Shandarin, Sheth and Sahni (2004) further supported above inferences by studying the sizes of superclusters and voids with respect to the enclosed mass and occupied volume, respectively.
Figure 8 shows the results. Notice that all three sizes of the structures show significant correlation with mass enclosed (for superclusters) and volume occupied (for voids): the larger the mass, the larger the size of SC and more the volume, the larger is the size of the void, as expected. The solid lines show the radius of a sphere having the same volume as a given object (). The thickness and breadth approximately double their values and length grows by over an order of magnitude when the mass increases from about to . Both the thickness and breadth are considerably smaller than the radius of a sphere having similar volume for large superclusters (). On the other hand the length is considerably greater than . This is a clear manifestation of the anisotropy of largescale superclusters (see Figure 11). Voids demonstrate similar correlations with their volumes as shown in the right panels of Figure 8. Voids, especially large ones, are also anisotropic, e.g., see shshs04 for illustrations. It was shown for example, that both superclusters and voids show significantly larger filamentarity in proportion to the mass enclosed and volume occupied, respectively. Superclusters show negligible planarity whereas, voids could be relatively more planar shshs04 .
In their visual appearance, filaments stand out remarkably well in our impression of LSS. However, the occasional occurrence of vast, sheetlike superclusters such as the SDSS Great Wall
(which is almost twice as large as the CfA Great Wall gott03 ; tegmark04a ; see Figure 9.) raises questions as to how significant are sheets with respect to filaments on large scales in the actual Universe. In future, if the sheetlike superclusters in redshift surveys are confirmed and their counterpart structures not found in Nbody simulations, it may present a clear dichotomy between theory and observations.
To investigate this question in the context of simulations, shsssh03 studied the correlation in the Shapefinders of CDM superclusters identified at percolation.
Figure 10 is a scatter plot of Shapefinders , , for clusters in CDM defined at the percolation threshold. The strong correlation between and in the left panel indicates that two (of three) dimensions defining any given cluster assume similar values and are of the same order as the correlation length. (5Mpc for the largest superclusters.) The clustering of objects near () in this panel suggests that the superclusters are either quasispherical or filamentary. The superclusters in simulations thus do not appear to be planar. The scatter plot between & in the right panel of Figure 10 breaks the degeneracy between spheres and filaments. The massdependence of morphology is highlighted in this panel in which larger dots denote more massive objects. This figure clearly reveals that more massive clusters/superclusters are, as a rule, also more filamentary, while smaller, less massive objects, are more nearly spherical shsssh03 .
For Shapefinderlike statistics, it is important that the measured properties of an object also conform with its visual appearance.
Figure 11 shows the largest percolating supercluster in the CDM cosmogony for which shsssh03 . Note that the supercluster is indeed highly filamentary in agreement with the measured, high value of .
Earlier we noted results from MFbased study of mock SDSS catalogues due to CDM and CDM models. Global MFs were shown to successfully discriminate between the two models. Further, we noted the phenomenon of phasemismatch between dark matter and galaxies due to biasing. In this subsection, we noted that superclusters in a typical CDMcosmogony are generically filamentary. Could we use the length of the filamentary superclusters as discriminatory statistic between models? In other words, are the morphological properties of superclusters indeed sensitive to the underlying cosmic parameters? The 10 most voluminous superclusters in dark matter Virgo simulations of CDM, SCDM and CDM models were shown to have considerably different morphological properties. In fact, the percolating CDM supercluster (shown in Figure 11) was found to be more filamentary and topologically more simpler () than its counterpart structures in SCDM and CDM ().
To answer the same question in case of galaxy distributions, sheth04 studied the abundance of superclusters larger than certain length from within a given large ensemble of superclusters selected from mock SDSS catalogues of CDM and CDM models. This quantity is referred to as Cummulative Probability Function (CPF). Figure 12 shows the results, where the CPF() have been evaluated at FF=7.8 per cent (at the onset of percolation) and at FF=11.7 per cent (after the percolation). Note that at the onset of percolation, the length of the longest CDM superclusters could be as large as 90 Mpc, whereas their CDM counterpart structures, which exhibit same degree of statistical significance, are relatively shorter with 55 Mpc. We conclude that the CDM superclusters tend to be statistically longer than their CDM counterpart structures. The largescale coherence in the superclusters is attributed to the phasecorrelations in the density field; the higher the degree of phasecorrelations, the larger the lengthscale of coherence. Based on this, and the results reported earlier, we would anticipate the CDM superclusters to be longer than those in CDM. As we can see, the result reported here agrees well with this anticipation. This is a considerable success for the ansatz of the Shapefinder quantifying length, for as we noted above, it helps us capture the relative effect of phasecorrelations among rival models of structure formation.
vi.4 Applications to Redshift Surveys
We end this section by providing a brief summary of the results which have emerged out of the analysis of the observed LSS. We further outline the scope of these methods in light of the newly available large datasets.
The methods of topological analysis of LSS have been well developed since late 1980s melotopo90 . Hence, most of the investigations complementing the standard correlation function based approach are in the wake of studying the connectivity and topology of LSS. Almost all the topological analyses are carried out to test whether the galaxydistribution is consistent with the underlying Gaussian, randomphase primordial density field. Topology of LSS revealed both in LCRS data colley97 and SDSS data hoyleetal02 has been found to be consistent with Gaussian, randomphase hypothesis for the primordial density fluctuations. SDSS Early Data Release has been further found to agree well with the predictions of CDM concordance cosmology model hikage02 . Springel et al.(1998) reached similar conclusions based on their analysis of the 1.2 Jy redshift survey springtop . Connectivity of LSS of LCRS has been studied by shyes98 . These authors detected a network of filamentary superclusters in LCRS slices, which they found to be consistent with the gravitational instability paradigm of structure formation.
Recently there has been an upthrust of interest in complementing earlier topological studies with geometry of LSS. Probably the first morphological analysis of LSS was done using 1.2 Jy redshift survey data by sathya98 . MFs for SDSS early data release (EDR) have been evaluated by hikage03a who find the data to be consistent with the clustering predictions of the CDM cosmogony fn5 . Basilakos (2003) has measured shapes of superclusters in SDSS(EDR) and finds them to be predominantly filamentary. Due to their large volume, SDSS and 2dFGRS can be fruitfully employed to test predictions for MFs in a weakly nonlinear regime. An interesting application has been illustrated by colley00 , who use mock SDSS catalogues (inspired by CDM cosmology) for the purpose.
Morphology of LSS in LCRS has been studied by bharad who detect significant filamentarity in LCRS slices compared to an equivalent random distribution. The LSSdata from recent redshift surveys such as SDSS and 2dFGRS yet await an integrated morphological analysis and a detailed comparison with theoretical predictions. Meanwhile, encouraging efforts have been recently made to constrain the scale of homogeneity of the LSS by investigating the largest lengthscale of filaments in LCRS slices bharad04a . Robust 2dimensional morphological statistic such as the Shapefinder have been used for the purpose. The scale of homogeneity is found to be 7080 Mpc bharad04a . CDM mock LCRS catalogues have been further confronted with the LCRS slices using the same techniques. A uniform, moderate large scale bias of 1.15 seems to give good agreement between the model and the LCRS slices.
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