Genus Statistics of the Large-Scale Structure. with Non-Gaussian Density Fields. Takahiko Matsubara

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1 Last Number from YITP Uji Research Center YITP/U UTAP-3/95 September, 995 Genus Statistics of the Large-Scale Structure with Non-Gaussian Density Fields Takahiko Matsubara Department of Physics, The University of Tokyo, Tokyo, 3, Japan and Jun'ichi Yokoyama Uji Research Center, Yukawa Institute for Theoretical Physics Kyoto University, Uji 6, Japan Abstract As a statistical measure to quantify the topological structure of the large-scale structure in the universe, the genus number is calculated for a number of non-gaussian distributions in which the density eld is characterized by a nontrivial function of some Gaussian-distributed random numbers. As a specic example, the formulae for the lognormal and the chi-square distributions are derived and compared with the results of N-body simulations together with the previously known formulae for the Gaussian distribution and second-order perturbation theory. It is shown that the lognormal formula ts most of the simulation data the best. Key words: cosmology: theory galaxies: clustering gravitation large-scale structure of universe methods: statistics and numerical Eective September, 995, Uji Research Center will merge: Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 66-, Japan.

2 Introduction One of the most important purposes of observational cosmology is to probe the power spectrum and the statistical distribution of primordial uctuations out of the redshift survey of large-scale structure, in order to single out the correct model of the evolution of our universe. A number of measures have been used to characterize statistical properties of the large-scale structures. The most commonly used quantity is the two-point correlation function (Totsuji & Kihara 969), which is nothing but the Fourier transform of the power spectrum and does not contain much information on its statistical distribution. On the other hand, the counts-in-cells statistics, including the void probability (White 979), directly measure the statistical distribution of galaxies and various theoretical models based on physical or mathematical arguments have been proposed to t the observational data (Fry 986). Although this measure contains mathematically full information of the statistics in principle, it is dicult to relate the count analysis with the visual image or connectivity of galaxy clustering such as lamentary networks, sheet-like or bubble-like structures, etc.. As a statistical measure to characterize such a topological structure of galaxy distribution, the genus number has been widely used in the analysis of recent redshift surveys (Gott, Melott & Dickinson 986; Gott, Weinberg & Melott 987; Weinberg, Gott & Melott 987; Melott, Weinberg & Gott 988; Gott et al. 989; Park & Gott 99; Park, Gott & da Costa 99; Weinberg & Cole 99; Moore et al. 99; Vogeley et al. 994; Rhoads, Gott & Postman 994). Theoretically, however, the value of the genus had been calculated only for the random Gaussian eld (Adler 98; Doroshkevich 97; Bardeen et al. 986; Hamilton et al. 986) for a long time except for the restricted cases such as Rayleigh-Levy random-walk fractal (Hamilton 988) and union of overlapping balls (Okun 99). Hence what one could have done at best was to estimate the genus number on large scales, which are still in the linear regime, to test the validity of random Gaussian initial conditions. The situation was somewhat improved recently because lowest-order correction to the Gaussian genus number was analytically obtained by one of us using the multidimensional Edgeworth expansion around the Gaussian distribution (Matsubara 994). Detailed comparison has also been done with the results of N-body simulations and it has been shown that the new formula ts the numerical data well in the semi-linear regime but not in the nonlinear regime (Matsubara & Suto 995). This is in accord with the fact that the one-point probability distribution function (PDF) based on the Edgeworth series ceases to t the counts-in-cells once the root-mean-square (rms) value of the density contrast becomes as large as ' =4 (Juszkiewicz et al. 995; Ueda &

3 Yokoyama 995). Thus it is also desirable to nd an analytic expression of the genus number for realistic non-gaussian distributions just as various non-gaussian models have been proposed for counts-in-cells statistics (Fry 986). In the present paper we present analytic formulae of genus number for some non- Gaussian distributions in which the density contrast is given by a nontrivial function of either a single Gaussian-distributed random number or its combinations. The former includes the lognormal distribution which has been used as a model of galaxy distribution ever since Hubble (934). The lognormal model is found to t the three-dimensional PDF of observed galaxies (Hamilton 985; Bouchet et al. 993; Kofman et al. 994), and of cell-counts in CDM-type N-body simulations (Kofman et al. 994; Ueda & Yokoyama 995). Coles & Jones (99) argued for the lognormal mapping of the linear density eld to describe its nonlinear evolution in the universe. The latter includes chi-square distribution which is closely related with the negative binomial distribution, another widely used distribution with a hierarchical property of higher-order cumulants (Fry 986; Carruthers 99; Gazta~naga & Yokoyama 993; Bouchet et al. 993). We then compare the new formulae with N-body simulation data with various initial spectra as well as the previously known formulae for the Gaussian distribution and second-order perturbation theory. The rest of the paper is organized as follows. In x we review derivation of the genus number in Gaussian distribution and its lowest-order correction arising from the three-point correlation function. In x3 the genus number is given in the case the density contrast is given by a nontrivial but monotonic function of a Gaussian-distributed quantity. Then in x4 it is extended to the case the density uctuation is a function of several independent Gaussian variables. In x5 various analytic formulae of the genus and the PDF are compared with numerical results obtained from N-body simulations. Finally x6 is devoted to discussion and conclusions. Genus Curve in Gaussian Distribution and its Nonlinear Correction through the Edgeworth Series The genus curve G() is dened by = times the total Euler number per unit volume of isodensity contours of a continuous density eld. It corresponds to [#(holes) #(isolated regions)] =volume: Here the density threshold of the contour is specied by = p h i with being the density contrast which is smoothed appropriately. Mathematically, it is given by the 3

4 following expectation value (Doroshkevich 97; Adler 98; Bardeen et al. 986). G() = E D D ((x) ) D ( ) D ( )j 3 j( ) ; () where i (x) ;i (x), j (x) ;ij (x) and D () is Dirac's delta function. Thus it can in principle be calculated once a seven-point probability distribution function of (x) is known. In the case (x) obeys Gaussian distribution, we nd (x); i (x); and jk (x) are also Gaussian-distributed with a vanishing mean and two-body correlations given by where, and The nal result is h (x)i ; h(x) i (x)i = ; h(x) ij (x)i = 3 ij h i (x) j (x)i = 3 ij; h i (x) jk (x)i = ; () h ij (x) kl (x)i = 5 ( ij kl + ik jl + il jk ); are dened, respectively, by h i; h(r) i; and h(r ) i: (3) G() = e () 3! 3= ( ) G RG (): (4) If the initial uctuation is a Gaussian random eld, linear theory predicts that the genus is described by this Gaussian formula (4) and the PDF by = P RG () = e p : (5) Next we consider correction due to the presence of higher-order correlation functions, which arises as a result of nonlinear gravitational evolution, in terms of the multidimensional Edgeworth expansion following Matsubara (994) and Matsubara & Suto (995). The nal result is! G nd () = e 3 () 3 H () + S 6 H 5() + 3T H 3() + 3UH () In the above expression, H n () () n e = (d=d) n e = polynomial, and S, T, and U are dened as S = 4 h3 i; + O( ) : (6) is the n-th order Hermite T = h r i; (7) U = 3 hr rr i; 4 4 4

5 respectively, which we call generalized skewness. They come from the three-point correlation function that can be evaluated also by second-order perturbation theory (Matsubara 994). One should note that generalized skewness should be evaluated for smoothed uctuations that are evolved nonlinearly. Only after taking into account such smoothing eect, one can compare the second-order perturbation theory with observations. If one uses the Gaussian window with the smoothing length R, and assumes the Gaussian initial uctuations, they are explicitly computed as S = 4 [( + K)L 4 + 3L 3 + ( K)L ] ; T = 6 [5(5 + K)L (9 + K)L L 5 +( K)L 4 + 3( K)L 333 ]; (8) U = 4 [7(3 + K)L L 35 5(3 + 4K)L 44 L 353 6( K)L 444 ] : Here L n (R) stands for the following integral Z L n (R) 4 8 dx Z = () np 4 R 8 Z dx dy Z Z de R (x +y +xy) x y P n ()P (x)p (y) (9) dye R (x +y ) x = y = I n+= (xyr )P (x)p (y); where and are dened by (3) in which is the Gaussian smoothed density uctuation of linear theory over the scale R, P n is the n-th order Legendre polynomial, and I is a modied Bessel function. The above results (8) to () hold for arbitrary values of density parameter and cosmological constant 3. The latter eect manifests only through the function K = K(; ) which very weakly depends on and (Bouchet et al. 99; Bernardeau 994) where 3=(3H ), H is the Hubble parameter. Its explicit form for K has been derived by Matsubara (995) as where K(; ) = 4 Z dxx 3= + 3 Z Z dxx 3= () dxx 5= ; () X(x) =x + x + : () In the two specic models we adopt below, we nd K(; ) = 3=7 = :486 and K(:; :8) = :4335. For the power-law uctuation spectra P (k) / k n, S, T and U can be written down explicitly in terms of the hypergeometric function as n + 3 S = 3F ; n + 3 ; 3 ; n + 3 (n + K)F ; n + 3 ; 5 4 ; ; 4 5

6 n + 3 T = 3F ; n + 5 ; 3 ; n + 3 (n + 3 K)F ; n (n )( K) n F ; n + 5 ; 7 5 ; 4 n + 5 U = F ; n + 5 ; 5 ; 4 n + 4 4K F 5 n + 5 ; 5 ; 4 ; n + 5 ; 7 ; 4 ; (3) : The expressions for S in equations (8) and (3) are derived by Lokas et al. (994) which are equivalent to the other form independently derived by Matsubara (994). Similarly we transform the expressions for T and U presented in Matsubara (994; eqs. [6] and [8]) using the function L n (R), which are given in equations (8) and (3). This result of equation (6) is the analog of the second-order Edgeworth series of PDF (Juszkiewicz et al. 995; Bernardeau & Kofman 995): = P nd () = e p + S 6 H 3() + O( ) (4) 3 The Case Density Field is a Monotonic Function of a Random Gaussian Variable Next we turn to evaluation of G() for a non-gaussian distribution whose statistical properties are characterized by a monotonic function F as (x) = F [(x)]; (5) with (x) being a Gaussian-distributed random eld with vanishing mean and unit variance, so that one-point PDF of reads In this case we nd P []d = p jf [F ()]j exp h i F () d: (6) i (x) = F [(x)] ;i (x); (7) ij (x) = F [(x)] ;i (x) ;j (x) + F [(x)] ;ij (x): 6

7 Therefore G() is calculated as = = G() = * D (F () ) D F () ; D F () ; jf () ;3 j i i nhf () ; + F () ; hf 3 () ; + F () ; F () ; ; + F o + () ; () jf ()j D F () h(r) i 3! 3= exp( D ( ; ) D ( ; ) jf ()j jf ()j jf ()jj ;3 jf () ( ; ; ;) 3 F () h i F () ; (8) ) where we have made use of the assumption that F () is monotonic with nonvanishing F (). Thus the shape of the genus curve is obtained from that of the Gaussian distribution simply replacing by F (). This is as expected because mapping in terms of a monotonic function does not change the shape of the density prole. Using the relations = hf () i and = hf () i h(r) i for isotropic spaces, the overall factor can be rewritten to yield the nal result: G() =! hf 3= () i hf G () RG F () : (9) i In the particular case of the lognormal distribution, the function F is dened by F [(x)] = p + exp qln( + )(x) We therefore nd the PDF in the lognormal distribution as n P LN () = p ( + ) ln( + ) exp B ln( + ) and the genus curve in the lognormal distribution as 3 G LN () = () [3( + ) ln( + )] n 3= exp B ln( + ) h( + ) p io + C A : () h( + ) p io + n ln C A ; () h( + ) p io + ln( + ) C A () : One can easily check that the above expression reduces to the Gaussian formula in the limit!. 7

8 4 The Case Density Fields Depends on Multiple Gaussian Fields 4. General consideration We now extend the above analysis to the case statistical properties of the density eld is characterized by a number of independent Gaussian elds a (x) through a function f as (x) = f[ (x); (x); :::; n (x)]: (3) Here we assume that a 's are mutually independent and all have vanishing mean and unit variance. Then introducing new variables a i a a ;i f a a i ; ij = a ;i and a ij a ;ij we b a i b j a ;ij f ab a i b j + f a ij: b (4) i Here and below summation over repeated Latin indices a,b,c,...,h is implicitly assumed. Then the genus number is formally written as D G() = D (f ) D (f e e ) D (f g g )jf h h 3 j ie h(f ab a b + f a a )(f cd c d + f c c ) (f ab a b + f a a ) : (5) Here the two-body correlations are given by where ^ and ^ are dened by h a b i = ab ; h a b i i = ; h a a iji = ^ 3 ab ij ; h a i b j i = ^ 3 ab ij ; h a i b jki = ; h a ij b kl i = ^ 5 ab ( ij kl + ik jl + il jk ); (6) ^ h(r p ) i; and ^ h(r p ) i: (7) Introducing new variables! ij a a + ij ^ ij a =3 we nd a, i b, and!c jk are random- Gaussian variables whose correlations are totally decoupled from each other: h a! b jki = h a i! b jki = ; " ^ h! ij! a kli b = ab 5 ^4 9! ij kl + ^ 5 ( ik jl + il jk ) # : (8) The above fact greatly simplies the subsequent averaging procedures, because we can average over! a ij, b 3, c, and d in turn independently as follows. 8

9 First, in G() = (" * D (f ) D (f e e ) D (f g g )jf h h 3 j (9) f a! a ^ 3 a! + f ab a b #" f c! c ^ 3 c! + f cd c d # )+ f a! a + f ab a b ; averaging over! ij a can be readily done using (8). The expression in the curly bracket should be replaced by Dn oe = ^4! 9 f af b ( ab a b ) ^ 3 f ab( a + b a )f b c c +f ab f cd ( a b c d a b c ): d Next a 3average is performed noting that the linear combination f h h 3 is also a Gaussian with a vanishing mean and the variance (3) to yield h(f h h 3 ) i 3 = ^ 3 ~ f ; with ~ f ab f a f b ; (3) hjf h h 3 ji 3 = r 3 ~ f ^ : (3) We assume only nonvanishing ~ f contributes to the nal result. The averaging over a or a is more involved but it can be done utilizing the fact that u i f a i a; p; and i q i (p 6= q) constitute a trivariate Gaussian distribution with the correlation matrix M = ^ ~f f p f q f p f q After some straight forward calculations we nd r 3 h D (u i )i i = r 3 h D (u i ) p i qi i i = : (33) ~f ^ ; (34) ~f ^ ~ f pq f p f q 3 ~ f ^ ; (35) including the case with p = q. We thus obtain the expression for G() leaving only average as *! G() = ^3 (6) 3= D (f ) "(f f ~ a a ) f ~ f aa f abf a f b f c ~f c + (f aa ) #+ f ab f ab ~ f (f abf a f b f cc f ac f bc f a f b ) This average cannot be calculated until we specify the function f( a ). : (36) We therefore move on to a specic example of the chi-square distribution in the next subsection. 9

10 4. Chi-square distribution In chi-square matter distribution the density eld, (x), is given by (x) = n nx p= with being the mean density. The density contrast reads Hence we nd (x) = (x) = n [ p (x)] ; (37) nx p= [ p (x)] : (38) = h(x) i = n : (39) Although n is a positive integer by denition, we can perform analytic continuation to arbitrary positive number and replace n by = using (39) in what follows. Thus in the chi-square distribution the function f() is given by f() = a a ~ ; (4) with f a = a and f ab = ab. We therefore nd from (36), G() = ^3 (6) 3= D (f(~) ) ~ = ^ 3 p (3) 3= + h 4 ~ ~ 4 ~ ( + ) i P CH () G CH (): (4) Here P CH () is the one-point PDF of (x)= as a function of the threshold, which is calculated as Using the relation P CH () h D (f(~)= )i Z d n a ~ = e () n= ( + )n= = n (n=) exp + = ( + ) ( ) exp = D ~ + : (4) = 3h i = 3hf a a i = 4^ n = ^ ; (43) we nally obtain G CH () = () 3= 3! 3= ( + ) ( + ) 3= ( ) exp + : (44)

11 Since (4) reduces to the Gaussian distribution for! (44) also coincides with (4) in this limit. Note that (4) indicates that one-point PDF of the density eld obeys the negative binomial distribution P NB [(x)]d = exp d ( ) : (45) The negative binomial distribution is often used to t the data of counts-in-cells analysis, namely, as a model of a PDF of smoothed density eld. A number of observational analyses have shown that it ts the counts-in-cells histogram well for some volume-limited redshift samples (Gazta~naga & Yokoyama 993; Bouchet et al. 993). We also note that this distribution has a hierarchical property of higher-order connected moments. That is, Nth order cumulant N is given by N = (N )!( ) N = (N )! N. On the other hand, in spite of the above similarity, the chi-square distribution and the negative binomial distribution are essentially dierent from each other once spatial dependence is taken into account. To illustrate it let us consider two- and three-point correlation functions, and, respectively, in the chi-square distribution. The former is calculated as (x; y) h(x)(y)i = 4 4 ha (x) a (x) b (y) b (y)i = w (x; y); (46) where h a (x) b (y)i ab w(x; y). Similarly, the latter is given by (x; y; z) h(x)(y)(z)i = 4 w(x; y)w(y; z)w(z; x) (47) = [(x; y)(y; z)(z; x)] = ; which is very dierent from a hierarchical form. On the other hand, if the negative binomial distribution would t the counts-in-cells analysis for any shape and size of the sampling cell, we would expect the following relation to hold (x; y; z) = [(y; z)(z; x) + (z; x)(x; y) + (x; y)(y; z)] ; (48) 3 with the observed slope (Peebles & Groth 975; Groth & Peebles 977) of the power-law two-point correlation function (Gazta~naga & Yokoyama 993). Thus the two distribution should be distinguished from each other even if one-point PDF has exactly the same form. 5 Comparison with Numerical Simulations We measure the genus of the four data sets from cosmological N-body simulations with random-gaussian initial conditions, kindly provided by T. Suginohara and Y. Suto.

12 Three models are evolved in the Einstein-de Sitter universe with the scale-free initial uctuation spectra (at expansion factor a = :): P (k) / k n (n = ; ; and ): (49) The last model corresponds to a spatially-at low-density cold dark matter (LCDM) model. In this specic example, we assume = :, = :8, and h = : (Suginohara & Suto 99). The amplitude of the power spectrum in the LCDM model at a = 6 is normalized so that the top-hat smoothed rms mass uctuation is unity at 8h Mpc. In fact this LCDM model can be regarded to represent a specic example of the most successful cosmological scenarios so far (e.g., Suto 993). All models are evolved with a hierarchical tree code implementing the fully periodic boundary condition in a cubic volume of L 3. The physical comoving size of the computational box in the LCDM model is L = h Mpc. The number of particles employed in the simulations is N = 64 3, and the gravitational softening length is g = L=8 in comoving. Further details of the simulation models and other extensive analyses are described in Hernquist, Bouchet & Suto (99), Suginohara et al. (99), Suginohara & Suto (99), Suto (993), Matsubara & Suto (994), and Suto & Matsubara (994). The computation of the genus from the particle data is performed using the code kindly provided by David Weinberg (Weinberg 988; Gott et al. 989). In short the procedure goes as follows; (i) the computational box is divided into N 3 c (= 8 3 ) cubes, and the density g (r) at the center of each cell is computed using Cloud-In-Cell density assignment. (ii) the Fourier-transform: ~ g (k) L 3 Z g (r)exp(ik r)d 3 r; (5) is convolved with the Gaussian lter, and transformed back to dene a smoothed density of each cell (with the ltering length R f ): Z s (r; R f ) = L3 ~ 8 3 g (k) exp k R f! ik r d 3 k: (5) (iii) the rms amplitude of the density uctuations is computed directly from the smoothed density: (R f ) q h( s = ) i; (5) where is the mean density of the particles. (iv) The isodensity surface of the critical density: c [ + (R f )] (53)

13 is approximated by the boundary surface of the high-density ( s > c ) and low-density ( s < c ) cells. (v) Then the genus of the surface is computed by summing up the angle decit D(i; j; k) at the vertex of cell (i; j; k): g s () = 4 XN c i;j;k= D(i; j; k): (54) The way to compute D(i; j; k) is detailed in Gott et al. (986). The genus curve G() is dened to be the number of genus per unit volume as a function of the threshold. (vi) We repeated the above procedure 5 times using the bootstrap resampling method (Ling, Frenk & Barrow 986) in order to estimate the statistical errors of G(). It should be noted that earlier papers (e.g., Gott et al. 989; Rhoads, Gott, & Postman 994; Vogeley et al. 994) dened the density threshold of genus curves so that the volume fraction on the high-density region of the isodensity surface is equal to f = p Z e t = dt: (55) Adopting this method, the lognormal model and random-gaussian model would not be distinguished from each other (Coles & Jones 99). Since we intend to distinguish them so as to see whether or not the lognormal model can t the genus number, we adopt the straightforward denition = of the density threshold throughout this paper. To obtain the normalized genus curve G()=G(), we follow the method developed in Matsubara & Suto (995). In practice, we rst compute G() at 5 bins (in equal interval) for 3 3. Then we estimate the amplitude of G() by -tting the 7 data points around = to the lognormal formula () so that thus computed value of G() is less aected by the statistical uctuation at one data point. The result is almost insensitive to which tting formula we use in estimating G(). The PDFs, P (), which have been similarly computed with 5 bootstrap resampling errors, and normalized genus curves, G()=G(), are plotted in Figures to 3 for powerlaw models with n =,, and, respectively. We select three dierent sets of the expansion factor a (= at the initial epoch) and the ltering length R f for each model so that the resulting (R f ) covers from weakly to fully nonlinear regimes. The upper panels show the PDF and lower panels show the genus curves. The results of random Gaussian distribution, (5) and (4), are plotted in dotted curves, those of second-order perturbation theory, (4) and (6) with (3) in dashed curves, lognormal formulae, () and (), in solid curves, chi-square formulae, (4) and (44), in dot-dashed curves. Symbols indicate the results of N-body simulations. The curves for random Gaussian and second-order perturbation theory using Edgeworth series are not guaranteed that the genus for the 3

14 .5 a =.6 n = - a = 6.9 RG a = 8. PDF().5 σ =.7 σ =.69 LN CH PT σ =.9 G()/G().5.5 a =.6 σ =.7 a = 6.9 σ =.69 a = 8. σ = Figure : The PDFs (upper panels) and normalized genus curves (lower panels) from the N-body simulation data for n = power-law model ( =, = ) are plotted by open circles. Three dierent sets of the expansion factor a (= at the initial epoch) are selected and the Gaussian window function with the ltering length R f = L=5 is used. The values of the expansion factor and the resulting variance are indicated in the gure. The theoretical prediction for the random Gaussian eld is plotted by dotted lines, lognormal model by solid lines, chi-square model by dot-dashed lines, second-order perturbation theory by dashed lines. 4

15 .5 a =.7 n = a = 8.8 RG a = 37.6 PDF().5 σ =.76 σ =.3 LN CH PT σ =. G()/G().5.5 a =.7 σ =.76 a = 8.8 σ =.3 a = 37.6 σ = Figure : Same as Fig. 5 for n = power-law model. 5

16 .5 a =.63 n = a = 8. RG a = 77.7 PDF().5 σ =.44 σ =.5 LN CH PT σ =.78 G()/G().5.5 a =.63 σ =.44 a = 8. σ =.5 a = 77.7 σ = Figure 3: Same as Fig. 5 for n = power-law model. 6

17 negative density vanish. We have forced these theoretical curves to be zero in the relevant regions in the plot. The comparison of genus curves between the N-body results and the theories based on the Edgeworth series are described in Matsubara & Suto (995) in detail, so we do not repeat the detailed argument here. In short, these two curves agree well for :< < :4 where perturbation theory is expected to be valid, but the extrapolation of the second-order formula beyond this regime does not work. As for PDFs, the lognormal model ts the simulation results fairly well from weakly to fully nonlinear regimes. The degree of agreement, however, depends on the initial power spectrum. It works best for n = model. Note that we use the Gaussian window. Usually, the lognormal model is applied to the count-in-cells analysis which corresponds to the top-hat window. For the top-hat window, n = model ts the lognormal model better than n = model (Bernardeau & Kofman 995). Other non-gaussian curves including chi-square distribution t simulation data well on weakly nonlinear regimes, while they deviate considerably from simulation results on fully nonlinear regimes. For the genus, the lognormal model is also the best among the non-gaussian distributions considered here. The degree of agreement between the lognormal model and simulation results also depends on the initial power spectrum. The tting is the best for n = model. For other models, the lognormal model does not t well in positivethreshold regions. For all the simulation results, the agreement of the lognormal model happen to be better than weakly nonlinear formula for low-density regions. This may be partly because the weakly nonlinear formula cannot naturally take into account the positivity of the density, but the lognormal model achieves it by construction. On the other hand, the chi-square model does not agree well with any simulation results except for the initial linear regime, even if it assures the positivity of the density. In Figure 4 are plotted the PDFs and normalized genus curves for LCDM model for an example of a realistic cosmological scenario. The smoothing length R is 4h Mpc. If galaxies trace mass, a = 6 corresponds to the present epoch (z = ). Thus a = 4 and 5 correspond to z = :5 and :, respectively. We nd our simulation results of LCDM model and the lognormal model for Gaussian smoothed PDFs agrees fairly well except for some dierences of the peak height. See also Ueda & Yokoyama (995) for top-hat smoothed PDF. For the normalized genus curves, the lognormal model ts the simulation results well. 7

18 .5 a = 4. LCDM a = 5. RG a = 6. PDF().5 σ =.57 σ =.7 LN CH PT σ =.9 G()/G().5.5 a = 4. σ =.57 a = 5. σ =.7 a = 6. σ = Figure 4: Same as Fig. 5 for low-density cold dark matter model ( = :, = :8, h = :) The adopted Gaussian ltering length R f corresponds to 4h Mpc (comoving). The top-hat smoothed rms mass uctuation at a = 6 is unity at 8h Mpc. 8

19 6 Conclusions and Discussion In the present paper, we have derived theoretical prediction of the genus statistics for some non-gaussian distributions. We have generally calculated the expectation value of genus number for non-gaussian elds that are given by nontrivial functions of Gaussian random elds. Two specic elds of this category, the lognormal distribution and the chi-square distribution were investigated in detail. As is seen in the gures the one-point PDF of the smoothed density eld is tted by the lognormal model fairly well for all the simulated models adopted here, i.e., powerlow models with n = ; ; and the LCDM model. The lognormal formula for the genus curve also works well to t all the simulation data in the low density regions. But considerable deviation is observed in the positive-threshold regions in the nonlinear regime of the power-law models with n = and. On the other hand, the formula ts the entire regions of n = and the LCDM models fairly well. Thus the genus statistics are more appropriate than one-point PDF to distinguish between various initial power spectra for Gaussian smoothed eld. This is because the former depends on the spatial derivatives of density eld, too. Several arguments exist to explain the validity of the lognormal model as a statistical distribution in nonlinear regimes. Coles & Jones (99) argues that the lognormal distribution is obtained from the continuity equation in the nonlinear regime but with linear or Gaussian velocity uctuations, so that it may be adopted as a model to describe statistics in the weakly nonlinear regime. On the other hand, Bernardeau & Kofman (995) claims its successful t to the PDF of CDM type simulations is just a coincidence due to the particular shape of the CDM power spectrum based on the top-hat smoothing. Since LCDM power spectrum has a similar structure to n = power law in the interested scales, our results of the genus curves are consistent with these arguments. Observationally, the negative binomial distribution, which has the same one-point PDF as the chi-square distribution, has been shown to t the counts-in-cells of various redshift samples well (Gazta~naga & Yokoyama 993; Bouchet et al. 993). However, our results indicate that chi-square formulae reproduce neither one-point PDF nor genus curve of the simulations. Does that imply these N-body simulations have nothing to do with the real universe? Not necessarily, because here we are extracting information on relatively small length scales R 4h Mpc using all the 64 3 particles in the simulations, while the observational counts analysis has been performed on larger scales using volumelimited samples with much smaller number density of galaxies. The eect of sparse sampling in the counts analysis has been analyzed by Ueda & Yokoyama (995) for the LCDM model and it has been shown that the chi-square or the negative binomial PDF 9

20 does not t the data well if we use all the particles in the simulation but it ts well if we use sparsely-sampled data with a similar number density to that of volume-limited samples currently available. Thus our results do not rule out the LCDM model. On the contrary, it remains one of the most promising models of our universe (e.g., Suto 993). To test the LCDM model further, we can use the genus curve by examining if it is tted by the lognormal formula. Although the presently available redshift data are not statistically signicant enough (e.g., Vogeley et al. 994) to extract a specic conclusion, we can reasonably expect the statistical signicance of observed genus curve will improve rapidly in the near future. Now that we have obtained a number of theoretical formulae for the genus curve we can make use of it not only to test the Gaussianity of the primordial uctuations but also to discriminate between various models of structure formation. ACKNOWLEDGMENTS We are grateful to Yasushi Suto for providing us his N-body simulation data and helping us to plot the genus curves from simulation data and to David Weinberg for providing us the routines to compute genus curve from numerical data. T. M. gratefully acknowledges the fellowship from the Japan Society of Promotion of Science. This research was supported in part by the Grants-in-Aid by the Ministry of Education, Science and Culture of Japan (No. 4).

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