THE ASTROPHYSICAL JOURNAL, 562:7È23, 2001 November 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A.

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1 THE ASTROHYSICAL JOURNAL, 56:7È3, 1 November ( 1. The American Astronomical Society. All rights reserved. rinted in U.S.A. A SOLUTION TO THE MISSING LINK IN THE RESS-SCHECHTER FORMALISM MASAHIRO NAGASHIMA Division of Theoretical Astrophysics, National Astronomical Observatory, Mitaka, Tokyo , Japan; masa=th.nao.ac.jp Received 1 March 4; accepted 1 July 5 ABSTRACT The ress-schechter (S) formalism for mass functions of the gravitationally collapsed objects is reanalyzed. It has been suggested by many authors that the S mass function agrees well with the mass function given by N-body simulations, while many too simple assumptions are contained in the formalism. In order to understand why the S formalism works well, we consider the following three e ects on the mass function: the Ðltering e ect of the density Ñuctuation Ðeld, the peak Ansatz in which objects collapse around density maxima, and the spatial correlation of density contrasts within the region of a collapsing halo because objects have nonzero Ðnite volume. According to the method given by Yano, Nagashima, & Gouda (YNG), who used an integral formula proposed by Jedamzik, we resolve the missing link in the S formalism, taking into account the three e ects. While YNG showed that the e ect of the spatial correlation alters the original S mass function, in this paper we show that the Ðltering e ect almost cancels out the e ect of the spatial correlation and that the original S is almost recovered by combining these two e ects, particularly in the case of the top-hat Ðlter. On the other hand, as for the peak Ansatz, the resultant mass functions are changed dramatically and not in agreement, especially at low-mass tails in the cases of Gaussian and sharp k-space Ðlters. We show that these properties of the mass function can be interpreted in terms of the kernel probability (M o M ) in the integral 1 formula qualitatively. Subject headings: cosmology: theory È galaxies: formation È galaxies: luminosity function, mass function È large-scale structure of universe 1. INTRODUCTION It is one of the most important works to determine the number density of collapsed objects from the statistical properties of the density Ñuctuation Ðeld at an early stage of cosmic structure evolution. Such number density, or the mass function of halos, constitutes a main body of analyses of cosmic structures such as galaxies. The pioneering work was done by ress & Schechter (1974, hereafter S). They derived a mass function by relating a one-point distribution function of density Ñuctuation with a multiplicity function of halos. This is very simple and useful and therefore has been applied to many problems related to statistics of cosmic objects by many authors. Moreover, in order to take into account the formation histories of galaxies, this has been extended to estimate the mass function of progenitors of a present-day halo (Bower 1991; Bond et al. 1991, hereafter BCEK; Lacey & Cole 1993, hereafter LC; Kau mann & White 1993; Somerville & Kolatt 1999). Thus, to remove uncertainties in the S formalism and to reðne it are very important. Recent understanding of cosmological structure formation is based on a cold dark matter (CDM) model, which is a kind of hierarchical clustering scenario. In the CDM model, the universe is dominated by cold dark matter gravitationally, and self-gravitating structures are formed via gravitational growth of small initial density Ñuctuations. Since higher density regions collapse earlier and since the power at smaller scales of the Ñuctuations is stronger than that at larger scales, larger objects that are formed by the collapse of large-scale Ñuctuations are formed by the clustering of smaller objects hierarchically. Formation processes of objects such as galaxies and galaxy clusters are qualitatively well understood in the context of such hierarchical clustering scenarios. However, once we would like a quantitative estimate, especially in regard to statistical 7 properties such as the luminosity function of galaxies and the abundance of clusters of galaxies, we face difficulties caused by uncertainties in estimating the mass function of cosmological objects. The S formalism is the most useful even at present. This formula assumes a Gaussian distribution of density contrast and the spherically symmetric collapse of objects (brieñy reviewed in ). In the S formalism, however, there is a problem of the so-called fudge factor of. Because ress & SchechterÏs consideration was only in overdense regions, they simply multiplied by a factor of in order to consider all the matter in the universe. Another approach explaining the distribution of cosmological structures has been developed, that is, the peak formalism. This was Ðrst investigated by Doroshkevich (197) and extensively studied by eacock & Heavens (1985) and Bardeen et al. (1986, hereafter BBKS). In this formula considering conðgurations of local density maxima through the Ðrst and second derivatives of the density Ñuctuation, the number density of the density peaks is derived. However, peaks in a Ðeld with larger smoothing scale often include those with smaller smoothing scale. Thus, the mass function derived from the peak formalism is in agreement with numerical experiments only at very large mass scales (Appel & Jones 199). In order to derive a correct mass function, we must remove such small-scale peaks contained in larger peaks (the so-called cloud-in-cloud problem). By using the conñuent system formalism, Manrique & Salvador-Sole (1995) derived a mass function taking into account the removal of nested peaks in the case of a Gaussian Ðlter. eacock & Heavens (199, hereafter H9) and BCEK attempted to solve the cloud-in-cloud problem by considering the trajectory of a density Ñuctuation against a varying smoothing scale. The method is called the excursion set formalism according to BCEK. Assuming a spherical col-

2 8 NAGASHIMA Vol. 56 lapse, the threshold for the collapse is determined by only the value of the density Ñuctuation, independent of mass scale. Thus, we adopt as a mass scale of the density Ñuctuation of a virialized object the maximum mass scale at which the density contrast is Ðrst upcrossing the threshold for decreasing mass scale. When the sharp k-space Ðlter is adopted, they obtained the fudge factor of because of the Brownian motion of the trajectories due to uncorrelativity of phases of Fourier modes. However, in the case of general Ðlters, since the correlation between di erent Fourier modes originates the non-markov behavior, a Monte Carlo method or complicated approximations are required, as done by H9 and BCEK. It should be noted that this problem had already been considered by Epstein (1983, 1984) in the case of a oisson Ñuctuation of point masses. In contrast to the excursion set formalism, which is based on a di erential formula such as the di usion equation, Jedamzik (1995) proposed another approach based on an integral equation. Yano, Nagashima, & Gouda (1996, hereafter YNG) Ðxed an inconsistency of estimation of probability in the original analysis by Jedamzik (1995) and also obtained the fudge factor of in the case of the sharp k-space Ðlter, and also in this case, they extended the Jedamzik integral formula to include the e ect of the spatial correlation, which should be considered within a collapsing region in order to solve the cloud-in-cloud problem. They found that the factor of is lost by the spatial correlation and predicted a larger number density of halos compared to the S mass function at large scale, at which the variance of the density Ñuctuation becomes less than unity. They also connected the peak formalism with the Jedamzik formula and found that the low-mass tail of the mass function is changed. Nagashima & Gouda (1997) conðrmed that the mass function given by YNG is in agreement with that given by the merging cell model proposed by Rodrigues & Thomas (1996), in which the density Ñuctuation Ðeld is realized by the Monte Carlo method and halos are identiðed according to linear theory and the spherical collapse model. On the other hand, Monaco (1995, 1997a, 1997b, 1998) investigated the dynamical e ect of the collapse by using nonlinear dynamics such as the Zeldovich approximation (Zeldovich 197). As is well known, the critical density contrast is 1.69 independent of mass scale for the spherical collapse in an EinsteinÈde Sitter universe. However, the probability of Ðnding objects collapsing spherically is very small, and most halos collapse nonspherically. Taking into account such direction-dependent dynamics, if we interpret a Ðrst shell-crossing epoch as a collapsing epoch, it is found that the resultant mass function is similar to the S mass function with e ectively decreased critical density below The purpose of this paper is to clarify why the S mass function works well and reproduces well mass functions obtained by N-body simulations (e.g., Sheth & Tormen 1999; Jenkins et al. 1). Hereafter, we call this problem the missing link in the S formalism. In this paper we consider the following three e ects based on the integral formula developed by Jedamzik (1995) and YNG. First, the Ðltering e ect is investigated. We use sharp k-space, Gaussian, and top-hat Ðlters. Second, the peak e ect is considered, based on the peak Ansatz in which objects collapse around density maxima. Third, the spatial correlation is introduced in the same way as in YNG, but for the above three Ðlters. Finally, all the e ects are considered simultaneously. Since it is very complicated that the dynamical e ect as considered by Monaco is introduced simultaneously, we will discuss this e ect only in 6. The structure of this paper is as follows: First, in we deðne the smoothed density Ðeld introducing the Ðlters. In 3, we review the S formalism. In 4, we show the formulation by using an integral equation, based on Jedamzik (1995) and YNG. In 5, we show the resulting mass functions and consider the e ects of the Ðlters, peaks, and spatial correlation. In 6, we discuss some uncertainties that we do not investigate in detail. We devote 7 to our summary and conclusions. Many detailed deðnitions and equations are summarized in Appendices A and B.. SMOOTHED GAUSSIAN RANDOM FIELD First, we deðne the density contrast as d(x) 4 [o(x) [ o6 ]/o6, where o(x) is the density at the point x and o6 is the cosmic mean density. The Fourier mode of the density contrast, d, is obtained by k d k \ d(x)eik Õ x dx. (1) This density contrast is smoothed out by a window function W (r) with a smoothing mass scale M, M d (x) 4 W ( o x@ [ x o )d(x@) dx@ () M M \ 1 W3 (kr)dk e~ik Õ x dk, (3) (n)3 where W3 (kr) is the Fourier transform of the window function W (r), R is a scale related to the mass M, and k 4oko. M The following three Ðlters are often used as the window function by using the smoothing length R: 1. top-hat Ðlter, W M (r) \ 3 4nR3 ha 1 [ r RB, (4) W3 (kr) \ 3 (sin kr [ kr cos kr) ; (5) (kr)3. Gaussian Ðlter, A W (r) \ 1 M (nr)3@ exp [ RB r, (6) W3 (kr) \ exp A [ kr 3. sharp k-space Ðlter, B ; (7) W (r) \ sin k c r [ k c r cos k c r, M nr3 (8) W3 (kr) \ h(k [ k) ; c (9) where k is the cuto wave number, k ^ R~1, and h(x) is the Heaviside c step function. c Next we deðne a variance of the density Ðeld with a smoothing mass scale M, p(m). The variance is obtained

3 No. 1, 1 MISSING LINK IN RESS-SCHECHTER FORMALISM 9 by using the power spectrum (k) 4 S o d ot as follows: k p(m) 4 Sd T \ 1 W3 (kr)(k)4nk dk. (1) M (n)3 The one-point distribution function in the Gaussian random Ðeld is characterized by the above variance. The normalization of the power spectrum is given by a characteristic mass M at which p (M ) \ 1. In connection * with the deðnition * of k, here we deðne the mass M with a scale length R or k. c In the case of the top-hat Ðlter, the mass M is clearly related c to 4nR3o6 /3. On the other hand, for the other Ðlters, the relation is not trivial. For example, LC proposed that the mass should be deðned as the enclosed mass by the window function W (r) renormalized to W (r \ ) \ 1. It should be noted that M this uncertainty does not M emerge in the case of the Gaussian Ðlter because it can be renormalized into M in the case of the scale-free power spectrum. As for the * sharp k-space Ðlter, in this paper we adopt a simple relation, k R \ n, while LC adopted k R \ (9n/)1@3. The k [ R relation c will be discussed in 6. c c By using the spherically symmetric collapse model (Tomita 1969; Gunn & Gott 197), the critical density contrast d is equal to 1.69 in an EinsteinÈde Sitter universe. So c the region with d º d is already included in a collapsed c region. We adopt this value throughout this paper. Note that Monaco (1995) found that the e ective critical density contrast, taking into account a nonspherical collapse, is approximated as d ^ 1.5 by dealing with d as a free c c parameter and Ðtting the S mass function to that given by himself. In the following, we restrict ourselves to the EinsteinÈde Sitter cosmological model and scale-free power spectra, (k) kn with n \ and n \[, except for 5.4, in which we will show mass functions in a CDM universe. 3. THE RESS-SCHECHTER FORMALISM In this section, we review the ress-schechter formalism brieñy. The probability of Ðnding a region whose linear density contrast smoothed on the mass scale M, d, is M greater than or equal to d is assumed to be expressed by c the Gaussian distribution given by 1 = f (ºd ; M) \ exp C d [ c Jnp (M) p dc (M)D dd. (11) This probability corresponds to the volume fraction of the region with more than or equal to d in the density Ðeld with smoothing scale M compared to the c total volume in a fair sample of the universe. Therefore, the di erence between f ( º d ; M) and f ( º d ; M ] dm) represents the volume fraction c of the region for c which d \ d precisely. The density contrast of an isolated collapsed M object c must be precisely equal to d because an object with d[d would eventually be counted c as an object of larger mass scale. c The volume of each object with mass scale M is M/o6. Then we obtain the following relation, Mn(M) o6 dm \[ Lf ( º d c ; M) dm, (1) LM where n(m) means the number density of objects with mass M, that is, the mass function. However, the underdense regions are not considered in the above equation. Hence, ress & Schechter simply multiplied the number density by a factor of, Mn(M) o6 dm \[ Lf ( [d c ; M) LM dm. (13) This factor of has long been noted as a weak point in the S formula (the so-called cloud-in-cloud problem). H9 and BCEK proposed a solution to this problem by taking account of the probability that a subsequent Ðltering of larger scales results in having up d[d at some point, even when at smaller Ðlters d\d at the same c point (the excur- sion set formalism). In the sharp c k-space Ðlter, because of the uncorrelation between di erent Fourier modes d, the trajectory of d behaves as a random walk for adding a k new Fourier mode. M In this case, they found that the factor of in the S formula is reproduced. 4. INTEGRAL EQUATION AROACH Jedamzik (1995) proposed another approach to the cloud-in-cloud problem based on an integral equation. We consider that the regions whose smoothed linear density contrasts with smoothing mass scale M are above 1 d. Each region must be included in an isolated collapsed c object with mass M larger than or equal to M. Therefore, 1 we obtain the following equation: = f (ºd ; M ) \ (M1 o M ) M n(m ) dm, (14) c 1 o6 M1 where (M o M ) means the conditional probability of 1 Ðnding a region of mass scale M in which d is greater than or equal to d, provided it is 1 included in M1 an isolated c overdense region of mass scale M.1 Hereafter, we call the procedure in which the mass functions are estimated by solving equation (14) the Jedamzik formalism. If(M o M ) 1 is simply given by the conditional probability p(d º d o d \ d ), (M o M ) is written as follows by using M1 c M c 1 BayesÏs theorem: (M o M ) \ p(d º d o d \ d ) 1 M1 c M c \ 1 = exp C 1 (d [ d ) [ M1 c D ddm1 Jnp p sub dc sub \ 1, (15) where p \p(m ) [ p(m ) by using the sharp k-space Ðlter. Thus sub we can 1 obtain the S formula, naturally including the factor of as can be seen from equations (14) and (15). The reason originating the factor of is the behavior of the random walk and is essentially the same as the excursion set formalism. However, it is insufficient to use equation (15) for a more realistic estimation of the mass function because it is necessary to consider the spatial correlation of the density Ñuctuations due to the Ðnite size of objects. Besides, the sharp k-space Ðlter is used, which leads p to p(m ) [ p(m ), and then we should evaluate how sub the mass 1 function is changed by other Ðlters. Therefore, we must consider the 1 In YNG, we used the notation (M, M ) according to Jedamzik (1995). In this paper, we adopt the form (M 1 o M ) in order to stress that this shows a conditional probability for d above 1 d in a Gaussian Ðeld M within a collapsing region with M. 1 c 1

4 1 NAGASHIMA Vol. 56 probability (r, M o M ) of Ðnding d º d at a distance r from the center of 1 an isolated object M1 of mass c scale M with general Ðlters. Then we can get the probability (M o M ) by spatially averaging (r, M o M ). 1 Because we expect that the 1 isolated collapsed objects are formed around density peaks, the constraints to obtain the above probability (r, M o M ) are given as follows: 1 1. The linear density contrast, d, of the larger mass scale M should be equal to d \ 1.69 M at the center of the object. c. Each object of the mass scale M must contain a maximum peak of the density Ðeld; i.e., the Ðrst derivative of the density contrast +d must be equal to, and each diagonal component of the M diagonalized Hessian matrix f of the second derivatives must be less than at the center of the object. 3. The density contrast of the smaller mass scale M (¹ M ), which has been already collapsed and which 1 is included in an object of mass scale M, must satisfy the condition d º d at distance r from the center of the larger object. M1 c In order to evaluate the above conditions, in this paper we investigate the following e ects: (1) Ðlters, () density peaks, and (3) spatial correlation. Some other e ects will be discussed in RESULTS 5.1. Filtering E ect In general Ðlters except for the sharp k-space Ðlter, the description of the Markovian random walk in the excursion set formalism cannot be adopted. H9 and BCEK calculated the mass function by the Monte Carlo method realizing Fourier modes d in such a non-markov case. On the k other hand, we can easily calculate the mass function by substituting the conditional probability in equation (15) to general cases as a simple approximation. It is an adequate approximation for the purpose of this paper in order to understand the various e ects mentioned above, at least qualitatively. For this purpose, what we do is estimate the probability, (M o M ) \ p(d º d o d \ d ). (16) 1 M1 c M c Note that in the case of general Ðlters, we cannot reduce this probability to 1 because of the correlation between the Fourier modes. Here we deðne the normalized density contrasts, where the correlation coefficient v is equal to Sl l T \ m ()/p p, which is deðned in Appendix A. The conditional 1 probability r distribution function is obtained by BayesÏs theorem, p(l o l \ l ) dl \ p(l 1, l c ) dl 1 c 1 p(l ) 1 c 1 \ Jn(1 C [ v) D ] exp [ (l 1 [ vl c ) dl1, () (1 [ v) where l is the normalized critical density contrast with the smoothing c scale M, d /p, and p(l) is a one-point Gaussian normal distribution, c p(l) dl \ 1 Jn exp A [ l B dl. (1) In the case of the sharp k-space Ðlter, the covariance m () is reduced to p, and then we obtain v \ p /p. Thus, we r obtain equation (15). However, in general cases we must calculate m (r) for speciðed Ðlters. In Figures 1 and, we show (M o M ) \ / = p(l o l \ 1 l1c 1 l ) dl in the cases of the Gaussian Ðlter with the spectral c 1 index n \ and the top-hat Ðlter with n \[ because v accidentally becomes p /p in n \ and then (M o M ) \ 1 r 1 It is evident that the Fourier-mode correlation increases. the value of (M o M ) above 1 when M Z M, in con- 1 1 * trast to the case of the sharp k-space Ðlter, in which the probability is 1 for all regions with M º M. 1 Solving equation (14) by substituting (M o M ), we 1 obtain the mass functions. In Figure 3, we show the mass functions for the Gaussian and the top-hat Ðlters and the S mass function in the form of a di erential multiplicity function, F(M) 4 Mn(M)/o6. Note that the function with the sharp k-space Ðlter is the same as the S mass function. Because (M o M ) increases above 1 at larger M scales 1 1 while f ( º d ; M ) is not changed by Ðlters, the number c 1 density decreases at larger scales. Then, so as to cancel out the decrease, the number density increases at smaller M scales. As a whole, the mass function is seemed to be moved to lower mass scales. This result is similar to that given by the Monte Carlo calculation by H9 and BCEK. Thus, we l 1 4 d M1 p r, (17) l 4 d M, (18) p where the standard deviations p and p are the root mean squares of the variances Sd T1@ r and Sd T1@, which are deðned in Appendix A. Hereafter, M1 we refer M1 to values concerning d with the subscript r. Generally, M1 bivariate Gaussian distribution is given by 1 p(l, l ) dl dl \ 1 1 nj1 C [ v ] exp [ (l 1 [ vl ) [ l (1 [ v) D dl1 dl, (19) FIG. 1.Èrobability (M o M ) for the Gaussian Ðlter in the case of the spectral index n \. The surface 1 of the probability is plotted only against the region M ¹ M. 1

5 No. 1, 1 MISSING LINK IN RESS-SCHECHTER FORMALISM 11 FIG..ÈSame as Fig. 1, but for the top-hat Ðlter and for n \[ conclude that the Ðltering e ect can be interpreted as the correlation e ect between di erent Fourier modes d. It should be noted that in the case of the top-hat Ðlter k and n \, the correlation m () becomes p, so the resultant mass function is exactly the same as the S mass function. FIG. 3.ÈDi erential multiplicity functions F(M) with the Ðltering e ect. In panel a, n \ ; in panel b, n \[. The solid, dashed, and dash-dotted lines denote F(M) for S, the Gaussian Ðlter, and the top-hat Ðlter, respectively. Note that F(M) for the sharp k-space Ðlter and for the top-hat Ðlter with n \ are identical with S. 5.. eak E ect Next we derive the mass function under the assumption that objects collapse around density maxima. Because 1 variables are required to describe one peak, we need to calculate an 11-variate Gaussian distribution function (one is d and others are concerning an object M ). Here we deðne M1 the Ðrst and the second derivatives of the density contrast d, g 4 Ld(x), () i Lx i f 4 Ld(x). (3) ij Lx Lx i j The peak condition is described by using g and f as follows: g \ and the eigenvalues of the Hesse i matrix f ij are less than. The covariances concerning g and f are esti- mated as follows: i ij Sg g T \ p 1 i j 3 d ij, (4) Sx@l T 4 k(r \ ) \ m (r \ ) 1, (5) 1 p p r Sx@l T 4 c \ p 1, (6) p p where x@ is a normalized trace of the diagonalized Hesse matrix, x@ 4 j 1 ] j ] j 3, (7) p and the variances and correlation are deðned in Appendix A. Note that ([j, [ j, [ j ) are the eigenvalues of f and 1 3 the other three components are transformed to variables given by the Euler angles for the diagonalization. By using the above quantities, we can calculate the probability (M o M ) as follows: 1 (M o M 1 S ) \ p(l º l o l \ l, g \, j [ ) 1 c c i/1,,3 \ 1 [ c n(1 [ v[ k[ c] vkc) ] / = dx@ f (x@) / = dl exp [[(Q ] Q )/] l1c 1 a b, (8) / = dx@ f (x@) exp ([Q /) b where Q, Q, and f (x) are deðned in Appendix A. Substituting the above a b probability to equation (15), we obtain the mass functions. It should be noted that in the case of the top-hat Ðlter, the second derivative cannot be done because the top-hat Ðlter does not smooth out the density Ðeld adequately. Then the variance for the derivatives of the density contrast diverges to inðnity, as shown in Appendix B. This fact leads to the resultant mass function, taking into account the peak condition, becoming the same as not taking it into account, if we simply interpret this divergence as c, k ] in equations (5) and (6). The peak formalism should be modiðed to be able to treat the top-hat Ðlter, but this is beyond the scope of this paper. Apparently, a similar thing to the above occurs for the sharp k-space Ðlter. If we do not consider the spatial correlation of the density Ðeld, the peak condition does not add any new properties because the behavior of the Markovian

6 1 NAGASHIMA Vol. 56 random walk owing to adding a new Fourier mode conserves. Mathematically, this is proved by the fact that k \ vc in the case of the sharp k-space Ðlter. Thus, we obtain the same mass function as S, even taking into account the peak condition. Of course, when we consider the spatial correlation, the peak condition will a ect the resultant mass function because k does not coincide with vc at r D (see the next subsection). In Figure 4, we show (M o M ) in the case of the Gaussian Ðlter with the spectral 1 index n \. Note that the viewing angle is changed. We see that (M o M ) shows a behavior such as a function of (M /M ). Thus, 1 the peak condition with the Fourier-mode correlation 1 increases the value of (M o M ) above 1 at M D M even at M > M, in contrast 1 to the results in the previous 1 subsection. 1 * In Figure 5, we show the resultant mass functions for the Gaussian Ðlter. For reference, we also show the mass functions of S and the top-hat Ðlter derived in the previous subsection. In the case of n \, the Gaussian Ðlter a ects the mass function especially at a lower mass scale M [ M, and the slope at that scale becomes shallower than that of * S. This change reñects the shape of (M o M ) near M D 1 1 M. In the case of n \[, similar to the case of n \, the slope at a lower scale is changed. However, the di erence from S decreases compared with the n \ case. This will lead to the Ðltering and the peak e ects becoming small in the realistic CDM power spectrum with n [ [ at galactic scale. Note that H9 also obtained a similar mass function taking into account the peak condition with another approximation scheme Spatial Correlation Finally, we evaluate the Ðnite-size e ect, that is, the e ect of the spatial correlation on the mass function. As already shown in YNG, this e ect is similar to changing the critical density d in S to a smaller value than 1.69, in the case of c the sharp k-space Ðlter. Here we generalize the YNG result to general Ðlters. First, we consider only the spatial correlation without the peak e ect. In this case, the correlation coefficient v must be generalized to that between two distant points with di erent mass scales, that is, v ] v(r), and the probability equation (16) is interpreted as (r, M o M ). Then, by spatial averag- 1 ing over the region of the object M, we obtain the prob- FIG. 4.Èrobability (M o M ) with the peak e ect for the Gaussian Ðlter in the case of n \. The 1 viewing angle is changed in order to easily see the shape of the probability from Figs. 1 and. FIG. 5.ÈDi erential multiplicity functions F(M) with the peak e ect. In panel a, n \ ; in panel b, n \[. The solid and the dashed lines denote F(M) for S and the Gaussian Ðlter. For reference, the function for the top-hat Ðlter with only the Ðltering e ect is shown by the dash-dotted line because the peak e ect cannot be included in this case. As mentioned in Fig. 3, F(M) for the sharp k-space Ðlter and for the top-hat Ðlter with n \ are identical with S. ability (M o M ), 1 (M o M ) \ / R (r, M 1 o M )4nr dr. (9) 1 / R 4nr dr It should be noted that it is not trivial that the window function W (r) is not needed in this averaging. However, since the averaging M operation has already been executed in the calculations of d and d, we estimate (M o M ) by equation (9). When M1 it is included, M the e ect of the 1 spatial correlation will be enhanced because a more distant point from the center of the object M is considered. In Figures 6, 7, and 8, we show the spatially averaged probabilities (M o M ) for the sharp k-space, top-hat, and Gaussian Ðlters with 1 the spectral index n \. In the cases of the former two Ðlters, in contrast to the Ðltering e ect, the probability becomes less than 1 when M Z M. The reason why the value becomes less than 1 is that 1 the * corre- lation between d and d becomes weaker at more distant points from M1 the center M of the object M. Thus, while the interval of the integration of d [ d is [, O] and then the probability is 1 when the sharp M1 k-space c Ðlter is used and the spatial correlation is neglected (eq. [15]), we must integrate from larger than to O under the consideration of the

7 No. 1, 1 MISSING LINK IN RESS-SCHECHTER FORMALISM 13 FIG. 6.Èrobability (M o M ) with the e ect of the spatial correlation 1 for the sharp k-space Ðlter in the case of n \. Ðnite volume. Then the probability becomes less than 1 On. the other hand, in the Gaussian Ðlter, the decrease of (M o M ) at M Z M emerges only at M ^ M. Except for such 1 region M 1 ^ M * Z M, the shape 1 of (M o M ) is 1 * 1 similar to Figure 1, in which the spatial correlation is not taken into account. This suggests that the Ðltering e ect is stronger than the e ect of the spatial correlation in the Gaussian Ðlter. We show the mass functions for the three Ðlters in Figure 9. While the mass functions with the top-hat and the sharp k-space Ðlters horizontally move to larger mass scales compared with the S mass function, the mass function with the FIG. 9.ÈDi erential multiplicity functions F(M) with the e ect of the spatial correlation. In panel a, n \ ; in panel b, n \[. The solid, dotted, dashed, and dash-dotted lines denote F(M) for S, the sharp k-space Ðlter, the Gaussian Ðlter, and the top-hat Ðlter, respectively. FIG. 7.ÈSame as Fig. 6, but for the top-hat Ðlter FIG. 8.ÈSame as Fig. 6, but for the Gaussian Ðlter Gaussian Ðlter moves to smaller mass scales. These properties can be explained from the shape of (M o M ). In the 1 sharp k-space and the top-hat Ðlters, (M o M ) is less than 1 1 at M Z M. This leads to the multiplicity function F(M) 1 * becoming larger than S at that scale because f (ºd ; M ) in equation (14) is unchanged. Then F(M) decreases c at 1 smaller scale M [ M so as to cancel the increase of F(M) at the larger scale. 1 On * the contrary, in the Gaussian Ðlter, (M o M ) increases above 1 in almost the region of M Z M. 1 This leads to F(M) becoming smaller at that scale 1 and larger * at a smaller scale. Finally, we include the peak condition in the above formalism. In this case, not only v(r) but also k must be generalized to k(r) because k(r) describes the correlation between the second derivative of the density maxima with smoothing scale M and the density height with scale M at distance r from the center of the object M. In order 1 to obtain the probability (M o M ), we substitute v(r) and k(r) into equation (8) and average 1 over the region of the object M according to equation (9). In Figure 1, we show (M o M ) for the Gaussian Ðlter with the peak and spatial correlation 1 e ects. This Ðgure shows the following three characteristics: increase at M? M Z M, increase up to M D M, and steep decrease at 1 * 1 M ^ M. It can be easily interpreted from Figures 1 and 4 1

8 14 NAGASHIMA Vol. 56 FIG. 1.Èrobability (M o M ) with the e ects of the peak and the 1 spatial correlation for the Gaussian Ðlter in the case of n \. that the Ðrst characteristic reñects the Ðltering e ect and the second the peak e ect. On the other hand, the third is a new property generated by the mixture of the Ðltering, peak, and spatial correlation e ects, because the spatial correlation itself a ects (M o M ) only at M Z M, as shown in 1 1 * Figure 8. Figure 11 shows (M o M ) for the sharp k-space 1 Ðlter. Similar to the third e ect in the Gaussian Ðlter case, (M o M ) decreases at M ^ M. This is also a new char- 1 1 acteristic. As seen in the peak e ect (Fig. 4), the change of (M o M ) from 1 at M ^ M a ects the slope of the low- 1 1 mass tail of the multiplicity function. In Figure 1, we show the mass functions taking into account the e ects of Ðltering, peak, and spatial correlation. For reference, we also show the mass functions for the top-hat Ðlter with the e ect of the spatial correlation (Fig. 9), because we cannot derive those with the peak e ect, as mentioned in 5.. At larger scale M Z M, the e ect of 1 * the spatial correlation is similar to that without the peak e ect shown in Figure 9. At smaller scale M [ M, as 1 * shown by YNG, the slope becomes steeper than that of S in the case of the sharp k-space Ðlter. In contrast, in the case of the Gaussian Ðlter, the slope becomes much shallower than S. The e ect of the spatial correlation is too weak to cancel out the peak e ect that makes such a shallow slope, while at a larger scale the mass function is close to S. These results show that the slope at smaller scale M > M is determined by the behavior of (M o M ) from 1 the region * M > M at which (M o M ) is 1 up 1 to M D M. If (M o 1 M ) has a region 1 (M o M ) [ 1, the 1 slope 1 1 FIG. 11.ÈSame as Fig. 1, but for the sharp k-space Ðlter FIG. 1.ÈDi erential multiplicity functions F(M) with the e ects of the peak and the spatial correlation. In panel a, n \ ; in panel b, n \[. The solid, dotted, and dashed lines denote F(M) for S, the sharp k-space Ðlter, and the Gaussian Ðlter, respectively. For reference, the functions for the top-hat Ðlter with only the e ect of the spatial correlation are shown by the dash-dotted lines. becomes shallow, and if (M o M ) \ 1 near M D M, the 1 1 slope becomes steep. In the case of the top-hat Ðlter, in which the peak e ect cannot be considered, (M o M ) ^ 1 at M D M [ M, and then the slope is close to 1 that of S. Note that the 1 di erence * of the slope from S is prominent in a large spectral index, n. We will comment on the power spectrum in a later section CDM ower Spectrum In this subsection we show the mass function in the case of a Ñat "CDM model as an example of a realistic case. We use a power spectrum given by BBKS with ) \.3, ) \.7, and! \.1, where ), ), and! are the cosmological " density parameter, the cosmological " constant, and the shape parameter, respectively. Here we assign the smoothing scale R for the Gaussian Ðlter to.64r according to BBKS and G deðne the mass of objects M \ 6no6 k~3 according to LC for the sharp k-space Ðlter. The power spectrum c is normalized so as to be unity at 8 h~1 Mpc with the top-hat Ðlter. In Figure 13, we show the mass functions for the three Ðlter functions with the e ect of the spatial correlation and the S mass function. We neglect the peak e ect in order to compare the mass functions under the same condition.

9 No. 1, 1 MISSING LINK IN RESS-SCHECHTER FORMALISM 15 FIG. 13.ÈDi erential multiplicity functions F(M) with the e ect of the spatial correlation in a "CDM model. The solid, dotted, dashed, and dash-dotted lines denote F(M) for S, the sharp k-space, the Gaussian, and the top-hat Ðlters, respectively. The crosses indicate the mass function given by Jenkins et al. (1), which was obtained by Ðtting their N-body simulation. The behavior for Ðlter changing is similar to the scale-free power spectra (see Fig. 9), and the di erence among them becomes a little larger. Note that there is an uncertainty in the estimation of the mass of objects in the cases of the Gaussian and the sharp k-space Ðlters and it cannot be canceled in the case of such a realistic power spectrum with a characteristic scale. If we adopt the same deðnition of mass as previously, the di erence becomes still larger. We also plot the mass function given by Jenkins et al. (1), which is well Ðtted by their high-resolution N-body simulation. At larger mass scale M Z 115 M, it is in good _ agreement with the mass function with the top-hat Ðlter. This agreement suggests that the spherical collapse approximation is good at such a large scale (M Z 115 M ) and _ that an overlapping e ect between similar mass halos becomes important at an intermediate scale (M D 114 M ), as shown by YNG. _ 6. DISCUSSION In this section we discuss some uncertainties in the calculation of the mass function Mass-smoothing Scale Relation So far, we have used a simple mass smoothing scale relation, M \ 4nR3o6 /3 and k R \ n in the case of scale-free power spectra. By changing c this relation, the mass function may be a ected as follows. It should be noted that we calculate the mass function against the normalized mass M/M, and M is deðned such that p (M ) \ 1 with scale-free * power spectra. * Thus, the mass function * does not scale in mass by changing the relation, even in the case of the Gaussian Ðlter. This e ect appears via the spatial correlation in the case of the sharp k-space Ðlter. The ratio of the cuto wavenumber for the simple relation (k ) to that given by the LC method (k ) is k /k \ (n/9)1@3 c D 1.3. Thus, the e ect of the spatial c,lc correlation c c,lc is weakened because the correlation at distance r is described by the combination of k r, and the probability (M o M ) is obtained by integrating c (r, M o M ) from to 1 R with 1 respect to r. We checked this uncertainty, but the e ect does not change our results qualitatively, especially at larger mass scales (M Z M ). If we use a realistic * power spectrum, for example, CDM models, the normalization of the mass spectrum p (M) becomes not arbitrary. In observation, the normalization of the power spectrum is given by the amplitude at 8 h~1 Mpc with the top-hat Ðlter with the bias parameter. This leads to the amplitude at that scale not becoming unity in cases of other Ðlters. In Figure 13, we adopt the LC method in the usual manner. We have found that when we use the simple relation the di erences between S and the mass functions with the Gaussian and the sharp k-space Ðlters become large compared with those in the case of the LC method. art of the remaining di erences is due to the di erence of the amplitude of the power spectrum. 6.. E ect of Nonspherical Collapse Throughout this paper, we have analyzed the mass function under the simpliðcation of spherical collapse, in which the critical density contrast for collapse is d \ However, in a more realistic condition, objects c collapse nonspherically. Monaco (1995, 1997a, 1997b, 1998) showed that the Ðrst collapse, which means a one-dimensional or pancake collapse, occurs during a shorter timescale. Fitting the S mass function to the resultant mass function, he obtained an e ective density contrast d ^ 1.5. This predicts c a larger number density of halos at larger scales compared to S, similar to the e ect of the spatial correlation. Apparently, this seems to enhance the di erence of the mass function, including the e ects of the Ðltering, peak, and spatial correlation. It will be correct when we consider the subsequent evolution of baryonic components included in halos because shock heating of gas occurs at the Ðrst collapse. Even so, in analyses of the mass function by using N-body simulations, which are in good agreement with the theoretical S mass function, it is not trivial that the condition of the identiðcation of isolated halos corresponds to the Ðrst collapse. Actually, such identiðed halos are rather spherical. This means that the e ective critical density is larger than 1.5 and that the second- or third-collapse criterion will be better than the Ðrst-collapse one. If so, the nonspherical e ect will hardly a ect our results, unless the mixture of the e ects emerges as shown in Figure SUMMARY AND CONCLUSIONS In this paper, we have analyzed the following three e ects on the mass function: the Ðltering e ect of the density Ñuctuation Ðeld, the peak Ansatz in which objects collapse around density maxima, and the spatial correlation of density contrasts within the region of a collapsing halo because objects have nonzero Ðnite volume. It has been shown by many authors that the S mass function agrees with N-body results well. However, it had been unclear why the S mass function works well. Taking into account the above three e ects and using an integral formulation proposed by Jedamzik (1995), we have tried to solve the missing link in the S formalism. While YNG showed that the e ect of the spatial correlation alters the original S mass function, in this paper we showed that the Ðltering e ect almost cancels out the e ect of the spatial correlation and that the original S is almost recovered by combining these two e ects, particularly in the case of the top-hat Ðlter. As shown in Figure 13, the mass function with this Ðlter is in

10 16 NAGASHIMA Vol. 56 good agreement with that given by an N-body simulation (Jenkins et al. 1). On the other hand, as for the peak e ect, the resultant mass functions are changed dramatically and not in agreement, especially at low-mass tails in the cases of the Gaussian and the sharp k-space Ðlters. In the sharp k-space Ðlters, as shown by H9 and BCEK, the density contrast at a point changes as the smoothing scale decreases because a new Fourier mode with an uncorrelated phase with other modes gives the density contrast an additional value similar to a Markovian random walk, which originates the so-called fudge factor of. On the other hand, when using general Ðlters, we cannot expect that the trajectory of the density contrast against the smoothing scale behaves like a Markovian random walk. Then the factor of is lost. We conðrmed the result of H9 and BCEK by using the other formalism proposed by Jedamzik (1995), against the Ðltering e ect to which we referred in this paper. The number density of halos decreases at a larger mass scale compared to a typical mass scale, which gives unity variance, and increases at a smaller mass scale. Besides, we found that the slope at a smaller scale does not change. Thus, apparently the shape of the multiplicity function F(M) scales horizontally against the mass scale. This e ect can be interpreted as the correlation of the Fourier modes, d, in a statistical sense. k We also investigated the e ect of the peak Ansatz in which objects collapse around density maxima and which is often considered in structure formation theories. In the case of the Gaussian Ðlter, this makes the low-mass tail of the multiplicity function shallower than S, dramatically. We found that this di erence can be interpreted by the behavior of the probability (M o M ) near M D M. (M o M ) increases above 1 at that scale, while it remains 1 in the sharp k-space Ðlter. On the other hand, in the case of the top-hat Ðlter, we cannot deal with this e ect correctly because this Ðlter does not smooth out the density Ðeld adequately so as to be able to di erentiate the density contrasts with respect to the spatial coordinate because of the sharpness on the edge of the Ðlter. While this trouble may be removed by adopting another Ðlter with a rounded edge, this will originate complicated mathematical difficulty. The e ect of the spatial correlation only in the case of the sharp k-space Ðlter was already considered in YNG. They showed that, in contrast to the Ðltering e ect shown here, the mass function predicts a greater number of halos at a larger mass scale, M Z M, and a smaller number at a smaller scale, M [ M. This * apparently corresponds to scaling the mass scale * to a larger one. This property is caused by decreasing the correlation between density contrasts with di erent smoothing scales. By considering the other Ðlters, we showed that such properties are weakened and the resultant mass functions become close to the original S because the Ðltering e ect strengthens the mode correlation, as mentioned above. We also showed, however, that the spatial correlation cannot correct the change of the slope of the low-mass tail in the case of the Gaussian Ðlter, which is caused by the peak e ect and, besides, steepens the tail in the case of the sharp k-space Ðlter. This is because the probability (M o M ) is mainly a ected at M D M almost independently 1 of the absolute values of M and 1 M by the peak e ect, while mainly a ected at M Z M 1 by the Ðltering e ect and the spatial correlation. robably, 1 * we can conðrm which Ðlter is better by using a high-resolution N-body simulation, while it may be difficult to obtain an adequate dynamic mass range to much smaller scale than M. * As a conclusion, the discrepancy with S shown by YNG, in which only the sharp k-space Ðlter was considered, is almost canceled out by the Ðltering e ect. The peak Ansatz, which does not a ect the mass function in the case of the top-hat Ðlter because of the mathematical reason, causes a new discrepancy at low mass scales in the cases of the Gaussian and the sharp k-space Ðlters. A new Ðlter similar to the top-hat Ðlter but di erentiable will be required in order to consider more realistic situations of collapsing halos. It should be noted that the discrepancy is not perfectly canceled by the Ðltering e ect and the spatial correlation, especially at a larger mass scale, M Z M. This scale corresponds to galaxy clusters. Several authors * have tried to determine the cosmological parameters and the normalization factor of the power spectrum by comparing the S mass function to the observational cluster abundance (e.g., Bahcall ). We should pay attention to the fact that the results by such an approach may depend on the model mass function. We discussed an important e ect of nonspherical collapse, which has been investigated by Monaco (1995, 1997a, 1997b, 1998). He found that if we adopt the Ðrst shell crossing as the collapse criterion, the resultant mass function predicts a larger number density of halos at a larger scale M Z M and a smaller number density at a smaller scale * M [ M* compared to S, because the timescale of collapse decreases. This e ect will enhance the di erence between S and the mass function given by a full statistics with the e ects taken into account in this paper and with nonspherical collapse. Thus, when we consider the evolution of the baryonic component, we should pay attention to this e ect. However, the agreement of S with the mass function given by many N-body simulations will be obtained by adopting the second or the third shell crossing as the collapse because objects identiðed as halos are rather spherical, not sheetlike or Ðlamentary. We found that the e ects that we considered become more prominent in a larger spectral index n. In a CDM model, the e ective spectral index n is less than [ at galactic scales. Therefore, in a realistic power spectrum the di erence in the low-mass tail of the mass function from S may be negligible. However, at a larger scale M Z M, the di erence remains. This may lead to a systematic error * when the cosmological parameters are estimated by comparing observations such as cluster abundances with S. Thus, we need more detailed study of the mass function. Finally, we should note that the approximation method used here is di erent from the excursion set approach. In the calculation of (M o M ), we considered only two di erent mass scales, M and 1 M. However, more strictly speaking, the probability 1 must be subject to an additional condition, that is, the object with M is an isolated object. In other words, the collapsing condition of objects M must describe the Ðrst upcrossing above d. This is essentially the same as the cloud-in-cloud problem c and the overlapping problem that was pointed out by YNG. Even so, we believe that the approximation used here is adequate to resolve the mystery in the S formalism because the fact that the Ðltering e ect correlates di erent Fourier modes leads to enhancing the probability that the objects M in our formu- lation are isolated. This conjecture can be stressed by the

11 No. 1, 1 MISSING LINK IN RESS-SCHECHTER FORMALISM 17 similarity of the mass functions shown here and the Monte Carlo approach to solve the excursion set formalism with the general Ðlters, as mentioned above. In future, this will be proved by using a high-resolution N-body simulation numerically experimentally. I wish to thank N. Gouda, T. Yano, and F. Takahara for useful suggestions. Numerical computation in this work was partly carried out at the Yukawa Institute Computer Facility. FORMULATION OF AENDIX A (M 1 o M ) FOR GENERAL FILTERS In this paper, we have considered only Gaussian random Ðelds. It is well known that the Gaussian Ðeld is perfectly determined by the two-point correlation function. In the following, we Ðrst show the form of the probability required to analyze the mass function. Then, we show the correlation and the autocorrelation functions, which are introduced as covariances in multivariate Gaussian distribution functions in the case of the power-law power spectrum, (k) 4 S o d ot \ Akn, where A is the amplitude, which is normalized so as to be unity at M \ M, and the angle brackets stand k for the ensemble average. * The kernel in equation (14), (M o M ), is generally expressed as a conditional probability. Since we need at least two Gaussian Ðelds with smoothing scales 1 M and M (M ¹ M ), or R and R (R ¹ R ) in terms of length as mentioned in, multivariate Gaussian distribution functions 1 are required. 1 Here we 1 deðne VN 1 4 (x,..., x ) as a vector of N Gaussian distributed variables and CN as a condition for N variables. The distribution function 1 for a variable N x under the condition CN~1 is written by BayesÏs theorem, N p(x o CN~1) dx \ p(x N, CN~1) dx N, (A1) N N p(cn~1) where p(v) is a multivariate Gaussian distribution function. In this paper, the condition CN means a criterion of collapse for larger objects, M. When we do not consider the peak condition, the vector V is a two-variable one, V\(d, d ), and the 1 condition C constrains one variable d, C1\(d \ d ). Note that the points for d and d are generally not the same if the c 1 spatial correlation is taken into account. When we consider the peak condition, 11-variate Gaussian distribution is required as V11 \ (d, g, f, d ), where i \ 1,, 3 and i ¹ j. Note that g and f are derivatives of d with respect to the coordinate x. i ij 1 i ij i Conditions for g and f are determined such that the density has a local maximum at the point M. i ij By using a covariance matrix M, the N-variate Gaussian distribution function is generally written as exp ([Q/) p(vn) dvn \ J(n)N det (M) dvn, (A) where Q \ VM~1VT, (A3) M \ S(x [ Sx T)(x [ Sx T)T. (A4) ij i i j j This shows that the multivariate Gaussian distribution is perfectly determined by correlations for any two variables. Thus, what we do is calculate the covariance matrix M for given variables. As shown in YNG, the following covariances have nonzero values: SdT \ p, SdT \ p, Sd d T \ m, Sg g T \ p 1 1 r 1 i j 3 d ij, Sf f T \ p ij kl 15 (d ij d kl ] d ik d jl ] d il d jk ), Sd f ij T \[p 1 3 d ij, Sd 1 f ij T \[m 1 d, (A5) 3 ij where d is the Kronecker delta. Variances and correlations p, p, and m are deðned as ij r i i m (r) \ 4n = sin kr W3 (kr1 )W3 (kr )(k) kl` dk \ 4nA 1 l (n)3 kr (n)3 Rn`l`3 I n`l` (r), (A6) p\ 4n = W3 (kr )(k)kl` dk \ 4nA 1 l (n)3 (n)3 Rn`l`3 I n`l`, (A7) p\ 4n = W3 (kr1 )(k)k dk \ 4nA 1 r (n)3 (n)3 Rn`3 I n`. (A8) 1 Clearly, m (r) means the lth moment of the two-point correlation function, p is the lth moment of the variance with the smoothing l scale R, and p is the variance with the scale R. The subscript r l stands for the value with the scale R because generally it is separated by r r from the center of the object with 1 the scale R in this paper. It should be noted thatm (r) 1 ] p for r ] and R ] R. The integrals I (r) and I are calculated for speciðc window function W3 (kr) in the next section. l l 1 m m In the following, the peak condition is taken into account. In the case without the condition, the probabilities are obtained by neglecting quantities concerning g and f. Here we diagonalize the Hesse matrix f by introducing the eigenvalues ([j, i ij ij 1