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it is now known that our universe is hierarchical . the largest virialized objects in the universe are clusters that are made of hundreds of galaxies . clusters have extensively been used to test structure formation models in the universe , through their abundances and correlations@xcite . superclusters are defined as an ensemble of clusters . they are still assembling , and are even larger structure than clusters ; superclusters have extents of @xmath0 or even larger . thus superclusters offer us information on late evolution of universe , and also on the transition from linear to non - linear regime . examples of the superclusters include a901/902@xcite , ms0302 + 17@xcite , cl1604 + 43@xcite , the sculptor supercluster@xcite , and the shapley supercluster@xcite . detailed investigations have shown that these superclusters are real physical systems where clusters are gravitationally assembling . detailed studies of these superclusters from galaxy distributions and weak lensing also showed that the mass distribution of superclusters is in good agreement with the distribution of early - type galaxies in superclusters . the physical state of gas in superclusters is still unknown , but it is worth mentioning that soft x - ray has been detected in the sculptor supercluster@xcite . although most the studies have been limited to those of individual superclusters so far , the emergence of wide - field surveys / telescopes , such as the sloan digital sky survey ( sdss ) and the subaru suprime - cam , begins to make it possible to do _ statistical _ studies of superclusters . nevertheless , there is no analytic model of superclusters that predicts , e.g. , the mass distribution . in this paper , we construct such an analytic model of superclusters . basically , our model is a modification of the press - schechter theory@xcite , and predicts not only mass distributions of superclusters but also number distribution of clusters in each supercluster . we also compare our model with superclusters in the sdss extracted by einasto _ et al._@xcite . throughout the paper we assume a flat universe @xmath1 , and the hubble constant @xmath2 . in this section , we construct the mass function of superclusters and clusters in each supercluster using the spherical model@xcite . we regard superclusters as overdense regions that are on the course of the spherical collapse . then superclusters can be defined as the regions that have the overdensity @xmath3 , where @xmath4 is the average density within superclusters and @xmath5 is the mean density in the universe . although superclusters are usually characterized by filamentary structure that is quite different from simple spherical overdensity , we consider this model so as to be analytically tractable . the assumption , however , may not be so crude given the similar situation that the press - schechter theory has achieved reasonable success despite its simplicity .      ) and corresponding nonlinear overdensities ( @xmath6 ) as a function of dimensionless cosmic time @xmath7 . different lines mean different values of @xmath8 : from @xmath9 ( _ dark _ ) to @xmath10 ( _ light _ ) . the vertical dashed line indicates the value of @xmath7 corresponding to @xmath11 . in this plot , we adopt @xmath12 . [ fig : sc],scaledwidth=50.0% ]    consider a spherical overdensity with mass @xmath13 and radius @xmath14 . equation of motion of the overdensity is simply given by @xmath15 by integrating this equation , we obtain @xmath16 here @xmath17 for a growing perturbation . we define @xmath18 then equation ( [ ene ] ) reduces to @xmath19 on the other hand , the cosmological time is related with the scale factor as @xmath20 from equation ( [ col ] ) , we can derive the motion of spherical overdensity as a function of @xmath7 . figure [ fig : sc ] shows the evolution of the size of overdense region for various values of @xmath8 . it is shown that the corresponding overdensity @xmath6 increases monotonically as time evolves . since the dimensionless time @xmath7 is related with the scale factor @xmath21 as equation ( [ age ] ) , at fixed time @xmath21 and for fixed @xmath6 we can determine the value of @xmath8 uniquely by solving equations ( [ col ] ) and ( [ age ] ) . then the linear and nonlinear overdensity is easily derived : @xmath22 where @xmath23 is the linear growth rate normalized so that @xmath24 at @xmath25 . from this , arbitrary @xmath6 is converted to the corresponding linear overdensity @xmath26 that is an essential quantity in computing the mass function . in figure [ fig : ov ] , we plot an example of this relationship . it shows one - to - one correspondence between @xmath6 and @xmath27 .    ] ) and nonlinear ( eq . [ [ dnl ] ] ) overdensities at @xmath11 . here we adopt @xmath12 . [ fig : ov],scaledwidth=50.0% ]      to compute the mass function of superclusters , we adopt the press - schechter formalism@xcite . from the linear overdensity @xmath26 ( eq . [ [ dlin ] ] ) , the mass function can be computed as @xmath28 where @xmath29 , and @xmath30 is the ( linear ) mass variance at @xmath31 .    the conditional mass function of clusters within superclusters , that are already virialized , can be estimated in analog with the extended press - schechter formalism@xcite . the number of clusters with mass @xmath13 that are in a supercluster with mass @xmath32 is given by @xmath33 , \label{cmf}\ ] ] where @xmath34 is the critical overdensity in the standard press - schechter formalism . although the distribution of clusters in each supercluster might be affected by physical processes such as global tides and dynamical frictions@xcite , we adopt this form for simplicity . -band luminosity density @xmath35@xcite to convert into @xmath6 . [ fig : zdel],scaledwidth=50.0% ]    in this section , we compare our analytic model with the results of einasto _ et al._@xcite who made a catalog of superclusters in the sdss early data release ( edr)@xcite . they extracted superclusters from the density field of galaxies with a smoothing length 10@xmath36 , and also clusters from the density field with a smoothing length 0.8@xmath36 . the superclusters and clusters are defined as the regions divided by the threshold density ( @xmath37 ) . since the sdss edr data consists of two slices , the density filed is calculated in 2 dimensions only . thus the total ( @xmath38-band ) luminosity of a supercluster is derived as @xmath39 where @xmath40 is the thickness of the slice , @xmath41 is the average diameter of the supercluster , and @xmath42 is the observed luminosity with the selection function of galaxies corrected . with this way 39 superclusters were identified ( see table 3 of @xcite ) .    since the above definition of superclusters is slightly different from theoretical one , we need to check what the observed superclusters correspond to in our model . we have defined superclusters as the regions that have the overdensity @xmath6 , so we compute @xmath43 for each observed supercluster . figure [ fig : zdel ] shows @xmath6 for the sdss edr superclusters . here we calculated @xmath6 as @xmath44 where @xmath35 is the average @xmath38-band luminosity density@xcite . this coincides with equation ( [ dnl ] ) if the mass - to - light ratio of superclusters is the same as that of the universe . as seen in the figure , the derived value of @xmath6 is scattered at around @xmath45 ( @xmath46 ) . thus we adopt @xmath47 as a fiducial value , though we also examine the cases with @xmath48 to see how the uncertainty of @xmath6 would affect our results .      , @xmath49 , @xmath12 , and @xmath50 . for each panel , we changed one of these parameters while the other parameters are fixed . in all panels , the darker lines mean the larger values . [ fig : lf],scaledwidth=70.0% ]    first , we compare the abundance of superclusters in the sdss edr field . to do so , the volume of the field where superclusters were identified is needed . we simply adopt the sum of volumes of north and south slices of the edr data@xcite . however , this may underestimate the volume because of the correction outside the slices ( eq . [ [ corr ] ] ) . to convert mass functions of superclusters to luminosity functions , the mass - to - light ratio of superclusters . again , we assume the fiducial model @xmath49 , and estimate the sensitivity by changing @xmath51 from @xmath52 to @xmath53 . figure [ fig : lf ] plots luminosity functions of superclusters in theory and observations . first , the abundance of massive superclusters is sensitive to @xmath6 , thus the selection of superclusters is quite important in comparing the abundance . next , the abundance of massive superclusters is also very sensitive to @xmath54 . therefore , surveys of massive superclusters can be a good probe of @xmath54 . finally , the observed number of superclusters shows reasonable agreement with the theory , particularly when we take account of several uncertainties including that of the survey volume . at smaller luminosities the difference seems large , but this may be ascribed to the incompleteness of observations for such small objects . einasto _ et al . _ also estimated the number of clusters in each supercluster . they noted that there exists a well - defined lower limit of cluster luminosities that is a function of distance to clusters ( denoted by @xmath55 ) , particularly for those at larger distances ( see figure 6 of @xcite ) . thus we divide the supercluster sample by the distance @xmath55 . for each subsample , we assume the limiting luminosity shown in figure 6 of @xcite and calculate the number of clusters in a given supercluster by integrating equation ( [ cmf ] ) . we also convert the observed number of clusters in the edr field to the _ total _ number in the similar way as equation ( [ corr ] ) . ( @xmath56 ; _ upper left _ ) , @xmath57 ( @xmath58 ; _ upper right _ ) , @xmath59 ( @xmath60 ; _ lower left _ ) , and @xmath61 ( @xmath62 ; _ lower right _ ) . as limiting masses of clusters , we adopt `` set 1 '' in figure 6 of @xcite . different lines mean different values of @xmath6 . the other parameters are fixed to the fiducial values ( see the caption of figure [ fig : lf ] ) . [ fig : cl_dis],scaledwidth=70.0% ]    . the observational data ( _ filled triangles _ ) are taken from @xcite . for the errorbars , only statistical errors are included . for each panel , we changed one parameters while the other parameters are fixed to the fiducial values ( see the caption of figure [ fig : lf ] ) . in all panels , the darker lines mean the larger values . [ fig : cl_par],scaledwidth=70.0% ]    in figure [ fig : cl_dis ] , we plot the number of clusters within superclusters for different distances @xmath55 . we find that theoretical curves and observations show good agreement with each other . it seems that the model overpredicts the number of clusters , but this is perhaps because the assumption of the existence of lower limit of cluster luminosities becomes inaccurate for lower @xmath55 ( see figure 6 of @xcite ) . next we examine the dependence of several parameters on the number of clusters , which is shown in figure [ fig : cl_par ] . the figure indicates that the number of clusters in superclusters hardly depends on these parameters , though that was already clear from the expression of the conditional mass function shown in equation ( [ cmf ] ) . thus the model predictions are quite robust , and the good agreement means that our model works well . in this paper , we have constructed an analytic model of superclusters . our model is based on the press - schechter theory , and defines superclusters as overdense regions that are on the course of the spherical collapse . our model allows us to compute the mass function of superclusters , and also the number distributions of clusters in each supercluster .    to test the model , we have compared the model with the supercluster catalog in the sdss edr data . we have shown that the model shows in reasonable agreement with the observation . we have also examined parameter dependences of the mass function of superclusters , and found that the mass function of massive superclusters is sensitive to @xmath54 . thus superclusters will be a useful probe of @xmath54 if superclusters are selected more rigorously . on the other hand , the number of clusters in superclusters shows only weak dependences on parameters .    our model would be useful not only in constraining cosmological parameters from the abundance of superclusters but also in many applications . one of such applications is hot gas in superclusters . recently it has been claimed that soft x - ray@xcite and sunyaev - zeldovich effects@xcite in the supercluster regions may be detected , and this might suggest the existence of relatively hot gas in superclusters . if this is true , the statistical significance of the signals that are caused by the hot gas can be computed by using our model , once the we measure the temperature and density structure of hot gas . other application is gravitational lensing ; gravitational lensing measures the total mass along the line of sight , thus a supercluster outside a cluster may affect , e.g. , the mass estimate of the cluster . our model is useful since it allows us to estimate the distribution of the host superclusters for a given cluster . in practice , we found that the effect is negligible when we consider superclusters with @xmath63 ; assuming a homogeneous sphere , convergence of a supercluster with @xmath64 can be estimated as @xmath65 where we assumed a source at @xmath66 and a supercluster at @xmath67 . however , it is of interest to consider the effect more carefully , because the effect is dependent on the density profile of superclusters as well as the definition of superclusters , @xmath6 . in particular , superclusters could have much more significant effect if superclusters have centrally concentrated density distributions . although our model looks successful at this moment , definitely we need more tests . the more direct way to test the model is to compare it with numerical simulations , since we know the initial condition and cosmological parameters in numerical simulations . of course , comparison with observations is also of great importance in order to test our understanding of the universe . now the sdss data have been becoming public@xcite , and the quite large survey region of the sdss would allow us to make three - dimensional catalog of superclusters . the follow - up work is currently in progress , and the results will be presented elsewhere . we thank issha kayo for useful discussions . mo , kt , and ki are supported by jsps through jsps research fellowship for young scientists . n.  a.  bahcall and r.  m.  soneira , astrophys .  j.   * 270 * , 20 ( 1983 )
we construct an analytic model for the mass distributions of superclusters and clusters in each supercluster . 
 our model is a modification of the press - schechter theory , and defines superclusters as the regions that have some overdensity smaller than those for usual virialized objects . 
 we compare the mass functions with a catalog of superclusters in the sloan digital sky survey early data release and found that they are in reasonable agreement with each other .