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Mon. Not. R. Astron. Soc. 000, 1­10 (0000)

Printed 4 May 2011

A (MN L TEX style file v2.2)

Dark halo mass function in a prescribed spherical host perturbation ­ Press­Schechter theory with statistical constraints
E. P. Kurbatov
1

1

Institute of Astronomy, 48 Pyatnitskaya St., Moscow, 119017, Russia

4 May 2011

ABSTRACT

Here proposed a modification of the Press­Schechter theory allowing for the presence of a host density perturbation ­ host halo or void. The perturbation is accounted as statistical constraints in a form of linear functionals of the random overdensity field. Deviation of the background density within perturbation is interpreted in a pseudo-cosmological sense. Resulting mass function of sub-haloes depends on the perturbation parameters: its mean overdensity, spatial scale, and spatial momenta of higher orders. Applications of the theory to superclusters, voids and bias problem are briefly observed. In its present form, the theory can describe the clustering properties of sub-haloes inside a non-virialized host only. Possible fix of this drawback is also discussed. Key words: large-scale structure of Universe ­ galaxies: luminosity function, mass function

1 INTRODUCTION The theory of Press and Schechter originated as a semi-analytical approach to describing the evolution of the mass function of dark haloes. Today this theory, together with its modifications, perhaps the only one that, using some reservations, most closely matches observations and numerical modeling for the widest mass range of dark haloes, down to the resolution limit of numerical runs. The reservations of the theory concerned to its main provisions: ­ reinterpretation of halo merging as a random walk process allowed to complete the construction of the theory with a mass function of Press and Schechter (Bond et al. 1991); ­ Benson et al. (2005) showed, in the original theory the merger rate, or the merger kernel, in terms of Smoluchowski equation, is asymmetric function of mass which is flaw; the procedure to find a symmetric kernel proposed by the authors can solve this issue; ­ spherical collapse model used to get an overdensity threshold seems too rough; instead were adopted the model of ellipsoidal collapse (Monaco 1997a,b) and more general non-spherical collapse model (Lee & Shandarin 1998). Other reservations related to the environmental effects. Namely, to account to the prescripted large-scale distribution of haloes or the super-/sub-halo relation, were proposed models using considerations about merger process (Mo & White 1996, on alternatives see references therein), and also heuristic models (Peacock & Smith 2000). 1 Modifications to the Press­Schechter theory proposed in

this paper allow for the presence of a host density perturbation ­ host halo or void, and give the mass function of sub-haloes depending on the perturbation parameters. Below in 2nd Section we will describe a modification of the Press­Schechter theory. In 3rd Section we consider applications of the theory to superclusters and voids. Benefits and issues will be discussed in 4th Section.

2 PRESS­SCHECHTER THEORY WITH STATISTICAL CONSTRAINTS 2.1 Outline We assume that the basic model of the theory is canonical, i.e. cumulative distribution function for variance S is set by the random walk in overdensity space with variance acting as a 'time', see Bond et al. (1991) and Lacey & Cole (1993): F (< S ) = erfc where barrier object. ( P DF ) fS = 2S , (1)

is the threshold overdensity for the random process, a for trajectory to pierce to be associated with a collapsed Corresponding differential probability distribution function is
3

F (< S ) = S 2 S

exp -

2 2S

.

(2)

E-mail:kurbatov@inasan.ru 1 Please excuse the author for the reference list is far from completeness. There are over 2000 citations of Press & Schechter (1974) in ADS, it's hard to analyse all of them. c 0000 RAS

Sheth & Tormen (1999) suggested a correction for this formula which gives better fit to their N -body simulations data but we will use the canonical variant for clarity. Mass PDF, fm , is obtained by differentiation F by mass through the mass-dependent variance. Both, threshold overdensity and variance depend on time via the


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E. P. Kurbatov
represented usually in a form of the primeordial power spectrum of the given spectral index n modified with transfer function T : P (k ) k n T 2 (k ) . (7)

cosmological model, power spectrum of fluctuations, and model of collapsing lumps. Defining these dependencies we get an evolution of the mass spectrum of haloes. The value for threshold overdensity is commonly assumed to be 1.69; it is obtained as a critical overdensity for collapse by considering the growth, turnaround and collapse of a uniform spherical overdense region (Bardeen et al. 1986; Lacey & Cole 1993). In general it depends on cosmological density parameters (Eke et al. 1996) but remains close to the conventional. Parameter can be interpreted not only as collapse condition but also as a marker for mass of the 'nonlinear' structures. Since the variance is the monotonic function on mass, PDF reaches its maximum value at the unique mass which sense is the characteristic or nonlinear mass, m . The extremum point for PDFs fS and fm satisfies equation 2 3S (m ) = . On the other hand, in evolved systems the characteristic mass is the mass of a most frequent occurrence. As Zel'dovich et al. (1983) noted, the variance of the density grows rapidly at this mass and crosses unity, so we can reinterpet as an overdensity in objects just evolved to the nonlinear amplitude. For this we assign threshold overdensity to a value marking the characteristic mass (qualitatively the same was mentioned by Jain & Bertschinger (1994)), 3 1.73 . (3) This value is quite close to commonly used 1.69. Later we will assume Eq. 3 against the conventional. Connection of the variance to the mass of the collapsed structures is implemented by filtering the local variance of density fluctuations with some kernel, which is considered isotropic usially. A space scale of the kernel, by-turn, is bound to the mass. Since fluctuations' amplitudes were small at early times, the binding is 3 simply m = (4 /3) m 0 Rf , where Rf is the filtering scale, and m 0 is the mean matter density in the Universe. Denote f the random overdensity field. The filtered one in the real space is f (t, r, RW ) = d3 r W (r - r , RW ) f (t, r ) , (4)

As an example of the transfer function can be mentioned the approximation of Bardeen et al. (1986): T (k ) = ln(1 + 2.34q ) 2.34q
4 -0.25

â 1 + 3.89q + (16.1q )2 + (5.46q )3 + (6.74q )

,

(8)

where q = k/, and is the shape parameter. Transfer function can also be computed numerically as the result of the evolution of perturbations in an early times, e.g. by C M B FA S T code (Seljak & Zaldarriaga 1996). The unconstrained variance calculated for this power spectrum then must be normalized to the given value of 2 8 S (Rf = 8 h-1 Mpc). Calculations for evolution of the power spectrum is a difficult task even in the lowest orders of the perturbation series approach. For Gaussian field these calculations were completed just up to 1loop corrections, i.e. to 2-nd order of accuracy in power spectrum (Jain & Bertschinger 1994), or up to 3-nd order in expressions for filtered statistical momenta (Scoccimarro 1998). In a non-Gaussian case the task became more complicated since the field is not zeromean, and the same precision order requires much more integrals to get (Crocce & Scoccimarro 2006b,a). In our theory we will restrict ourselves to a purely linear evolution, so the overdensity will be proportional to the linear growth factor D(t): f (t ) = D (t ) f L , K (t) = D2 (t) KL , P (t) = D2 (t) PL , S (t ) = D 2 (t ) SL , (9)

where W is the filter; the filtering scale Rf is bound to RW with some filter-dependent relation, see Sect. 2.3. For the given random field the variance is S= f- f
2

~ ~ d3 kd3 k W (k, RW ) W (k , RW ) K (t, k, k ) ,

(5)

~ where W is the Fourier image of the filtering kernel; K (t, k, k ) = ~ ~ f (t, k) f (t, k ) is the correlation function for Fourier modes of unsmoothed field; the proportionality sign means free choose of a normalization constant of the Fourier transformation. The field is often assumed statistically uniform, isotropic, having zero mean and independing Fourier modes, i.e. Gaussian. This reduces the variance to depending only on isotropic power spectrum P in momentum space: S ~ d3 k W 2 (k, RW ) P (t, k) , (6)

where K (t, k, k ) = D (k - k ) P (t, k), i.e. modes are deltacorrelated. With regard to general models, Eq. 5, if non-Gaussianity has form of the statistical constraints applied to the Gaussian field, then the correlation function can be obtained completely from power spectrum. Derivation of the correlation function in this case will be done in next Subsection. The power spectrum itself depends on the cosmology and is

where L index means values linearly evolved to the present time with unity growth factor. The linear evolution of perturbations settled on a large scale host overdensity (of both signs) can be represented in a linear perturbation theory for the certain cosmology. Parameters of such a pseudo-cosmological model determined by the mean density of the host: an overdensed background corresponds to higher value of the matter density parameter in the pseudo-cosmology than in the Universe, and vise versa. In the overdense host the perturbations will attain higher amplitudes than in global cosmology, in the underdense one the last will be lower. Since D is the only factor providing the time dependency in the class of the Press­Schechter theories, the bigger values of it will be in charge of the greater age of objects population. As the population evolves mainly by merging of the objects, there is a natural connection between the host overdensity and the intensity of a merging process for sub-haloes of different mass. At this step we should obtain the bias factor for the halo number. The result of the theory should be the number density of objects of a given mass in a predefined host halo, and also secondary values like cumulative mass of a population, etc. The number density per unit mass interval is usually defined as nm / m = m 0 fm /m. In this definition is assumed the spatial homogenity of the distribution, i.e. there is no spatial structure inside the host perturbation. It might not be so in a virialized host halo, where the merger process and therefore the mass function can be affected by the sharp features of the distribution. This may lead to a mass segregation of sub-haloes in space and, hence, to redistribution of the mass spectrum of sub-haloes. We assume this feature can be modelled by a mass-dependent factor m , so the general definition for
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Dark halo mass function in a prescribed spherical host perturbation
the number density is m 0 nm = m fm . (10) m m Factor m obviously makes a contribution to the bias factor. Other distribution functions should be represented similarly. In this work the CDM model is used with following parameters: = 0.7, m = 0.3, 8 = 0.9, spectral index for power spectrum is n = 1. The transfer function was computed using the C M B FA S T code of Seljak & Zaldarriaga (1996). The units adopted are h-1 M for mass, h-1 Mpc for length and (hH100 )-1 for time. 2.2 Constrained correlation function for modes We represent the host halo in terms of a functional constraints for a random field of the overdensity perturbations. In our task it's convenient to make calculations using spherical modes decomposition. To obtain the correlation function for amplitudes of the spherical modes we will use the approach of Hoffman & Ribak (1992), which was proposed by them for plane waves. The field f (r) = f (r, , ) can be transformed to spherical modes via the transformation f (r ) = 2
l 0

3

where the averaging performed over the unconstrained ensemble. Here 1 ~(c) ~ (18) flm (k) = l0 m0 Q- C [f ] ( ) (k) is the image of the mean constrained field but with non-fixed func~ tionals C [f ]; () (k) is the image of cross-correlation function between the field and -th constraint: ~ ~ () (r) = f C [f ] , () (k) = H() (k) P (k) ; (19) and Q Q
-1

is the inversion of the constraints correlation matrix: ~ dk H
()

= C [f ] C [f ] =

~ (k ) H

( )

(k ) P (k ) .

(20)

The unconstrained field is delta-correlated, which means (1) ~ ~ flm (k) fl m (k ) = ll mm D (k - k ) P (k) , where is the one-dimensional is to find the variance of the constr tion of a kind Eq. 5 for correlation ~ filter W . It's easy to show that the K (k , k ) =
(1) D (1) D

(21)

Dirac's delta-function. Our goal ained field, i.e. the full convolufunction Klml m (k, k ) with a only part of last meaning is
-1

(k - k ) P (k ) - Q

~ ~ () (k) ( ) (k ) .

(22)

The filtered variance can finally be expressed as S= ~ ~ dkdk W (k, RW ) W (k , RW ) K (k, k ) . (23)

dk kjl (kr )Y
l=0 m=-l

lm

~ (, )flm (k) ,

(11)

where jl is the radial Bessel's functions, Ylm is the spherical functions with normalization 4 d Yl Yl m = ll mm . Amplitude m of spherical decomposition of the field, or image, is ~ flm (k) = 2


dr r 2 kjl (kr )
0 4

d Y

lm

( , )f (r ) .

(12)

Transformation for radially symmetric field H (r ) consists of only the isotropic modes: ~ H
lm

( k ) = l 0

m0

~ H (k ) ,

Note that unfiltered convolution implies using the image for ~(3) three-dimensional Dirac's delta-function as a filter, D (k) = 1/2 k/(2 ). In general, the filter can be written in the form ~(3) ~ W (k, RW ) = D (k) w(kRW ), where w(0) = 1, w() = 0, and 0 |w| 1. The same applies to the constraining kernel. This is the result of the definition for unconstrained correlator, Eq. 21, where the one-dimensional Dirac's delta-function is used. Let's set a single constraint whose kernel is the filtering kernel also, and the constraint's parameter is a space scale RH . The variance is then S= dk (k, RW , RW ) â 1- ~ W (k, RH ) ~ W (k, RW ) dk (k , RW , RH ) dk (k , RH , RH ) , (24)

(13)

where ij is the Kronecker's delta, and ~ H (k ) = while 1 H (r ) = 4 2
0



4

2



dr r 2 kj0 (kr )H (r ) ,
0

(14)

~ dk kj0 (kr )H (k) .

(15)

Let us write constrains in a form of linear functionals fixing up a value of the convolved field at a point. Since the local extremum is the only constraint we use, we adopt the radially symmetric constraining kernel, and place the point at origin. Using this conditions, the functional for -th constraint can be written in form


~ ~ where (k, RW , RH ) = W (k, RW ) W (k, RH ) P (k). It is obvious that the expression in braces tend to the values close to unity when RW RH or RW RH , and vanish if RW = RH . Therefore, we can expect, in general, that the abundance of the haloes vanishes if halo mass tends to the mass of the host perturbation, and came closer to its unconstrained value if halo mass a lot differs from the host mass. It is not surprising. The spatial halo correlation function is (r) = f (0)f (r) {C } ~(3) = 4 dk D (k)jl (kr )Y
lm

(25)
lm

C [f ] =

d3 r H

()

(r ) f (r ) =
0

~ dk H

()

~ (k)f00 (k) .

(16)

( , ) (26) (k , k ) .

Constraints itself are fixed by assigning values to these functionals. Actually kernels can depend on a set of parameters characterizing a host halo, like a space scale or some momenta. We will not write them as arguments for shortness till next Subsection. Following Hoffman & Ribak (1991, 1992) it can be showed that the pair correlation function for the constrained ensemble of modes is Kl
ml m

â

~(3) dk D (k )Kl

m00

For a spherically symmetric consraints it's image is ~ ~(3) ( k ) = D ( k ) P ( k ) - Q
-1

~ () (k)

~(3) ~ dk D (k ) ( ) (k ) . (27)

~ ~ (k, k ) = flm (k)fl ~ flm (k) - ~(c) flm (k )

m

(k ) {C } ~ fl
m

=

(k ) -

~(c) fl m

(k )

,

(17)

Time dependence of the variance and correlation function is determined by assuming linear growth, Eq. 9, i.e. by renormalization of the variance with the D2 factor (see Subsection 2.4).

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E. P. Kurbatov
0.64RW . For this ratio and specified power spectrum normalization, 8 = 0.9, the nonlinear mass for unconstrained power spectrum is m 3 â 1014 h-1 M . This value is used to adjust characteristic scales for k-sharp and Gaussian kernels. As a result, the ratio for k-sharp kernel as well as Gaussian is Rf = 1.4RW . Ratio for the constraining scale is assumed the same, accordingly to the kernel used. The constraining mass is defined similarly to the 3 filtered one: Mh = (4 /3) m 0 Rh . For the single constraint with H = WTH we can see that the value of the constraining functional is bound to the mass associated with the host perturbation and to the averaged overdensity: (Rh ) C = 3Mh -1 3 4 m 0 Rh (39)

2.3 Constraining and filtering kernels Let H (r ) be the kernel for the functional constraining peak (or dip) amplitude of a field at origin. Besides amplitude, the shape can be constrained also via derivations of the kernel: H
(1st deriv)

(r ) =

d H (r ) , dr

H

(2nd deriv)

(r ) =

d 2 H (r ) dr 2

(28)

and so on. Using reccurence relations for radial functions jl and their derivations it's easy to show that constraints for odd-order radial derivations turn to zero because of symmetry. Images for the second and fourth derivations are expressed as ~ H
(2nd deriv)

(k ) = -

k2 ~ H (k ) , 3

~ H

(4th deriv)

(k ) =

k4 ~ H (k) . (29) 5

Filtering kernels generally accepted to use are k-sharp, Tophat and Gaussian: ­ k-sharp W ~ W
KS

(r, RW ) =

KS

1 j1 (r /RW ) , 3 2 RW r /RW ~(3) (k, RW ) = D (k) (1 - kRW ) ; 3 (1 - r /RW ) , 3 4 RW j1 (kRW ) ~(3) (k, RW ) = D (k) 3 ; kRW (r, RW ) = 1 (2 )
3/2 3 RW -r 2 /(2R2 ) W

(30) (31)

­ Top-hat W ~ W
TH

(32) (33)

(note the remark on constraining scale made in the previous paragraph). This quantity carries the growth factor of the host perturbation for a given epoch, as a multiplier. Choice of the constraining kernel as well as filtering one is not unique and motivated by convenience usually. Top-hat kernel is a natural choice for a mass constraint as it has no 'tails' in the real space. On the other hand the k-sharp filter suits better for excursion set formalism (Bond et al. 1991). Also k-sharp kernel gives simple expressions in momentum space. Indeed, consider the single constraint with H = W = WKS . Denoting S (0) an unconstrained variance (Eq. 6) we get the trivial relation for subhaloes (RW < RH ): S=S
(0)

TH

(RW ) - S

(0)

(RH ) .

(40)

­ Gaussian WG (r, RW ) = e , (34) (35)

~(3) ~ WG (k, RW ) = D (k) e

2 -k 2 RW / 2

;

~(3) where D (k) = k/(21/2 ) is the image of the three-dimensional Dirac's delta-function. Consider one of these kernels as the constraining kernel which depends on only the single parameter, a space scale RH . Volume integral for any of these filters is normalized to unity, although it is not necessary for constraining kernels since any factor will be reduced in Eq. 22. However, this normalization allows for clear physical treatment of constraints. For example, the mass of the host perturbation with overdensity (c) (r ) inside a sphere of radius r is M (< r ) = = d3 r (r ) (1 - r /r ) 4 m 0 r 3
3

(36)
(c)

1+
0

~ dk

~ (k ) W

TH

(k , r )

, (37)

Something similar should take place for other kernels, although in a case of a larger number of constraints it might not be so. At Fig. 1 is presented the function SL (m) for different kernels and different numbers of constraints. The models were computed for the host halo mass Mh = 1012 h-1 M . As we see the Top-hat kernel lead to the variance stable with respect to adding more constraints. Also the k-sharp has the same property for masses m < Mh but it gives oscillations for greater masses, seemingly due to oscillations in the real space. In the case of the Gaussian filter we see a more logical behaviour: the increase in the number of statistical constraints lead to the decrease in the variance in total. Note, since the function SL (m) is monotonically decreases, the lesser values of the variance cause the lesser amount of sub-haloes of associated mass, shifting mass PDF to the lower masses while the total amount of mass keeps unchanged and equals to Mh . Nevertheless, the adopting of Tophat kernel instead of k-sharp, or Gaussian instead of Top-hat lead to effectively lower host mass and to younger sub-halo population. Applying different kernels for constraint and filter is probably not a good idea since kernels may interfere giving oscillations in real space. 2.4 Linear growing of perturbations A linear evolution law for density perturbations is known for an arbitrary uniform and isotropic cosmology model (Bildhauer et al. 1992). It depends on global density parameters of the Universe and gives the amplitude of decaying or growing modes as a function of scale factor or redshift. The solution for a growing mode is D
(0)

where the second term in braces is the overdensity averaged in the volume of the host. The host overdensity profile can be obtained using Eq. 18 for the fixed values of the constraining functionals C , then the averaged one will be =Q
-1

C



~ dk W

TH

~ (k , r ) H

( )

(k, RH ) P (k) .

(38)

Relation between the characteristic scale RW and filtering radius Rf , as well as between the scale RH and actual constraining radius Rh is determined by the choise of the kernel. As Bardeen et al. (1986) noted, the characteristic scale for the Top-hat filter is not the actual space scale which is filtered out if we interested in the mass enclosed inside, but connected to it via Rf =

=

5 a 2

-3/2

a

(a )
0

db

b3/2 , 3 ( b)
1/2 3

(41)

where (a ) = 1+ k a+ a m m . (42)

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Dark halo mass function in a prescribed spherical host perturbation
103 102 SL 10
1

5

KS TH G

1. 6 1. 4 1. 2 1 D 0. 8 0. 6 = = = = 20 3 0 -0.75

100 10- 10-
1

2

One constraint KS TH G

0. 4 0. 2 0 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 a 1

10

3

102 SL 101 100 10- 10-
1

Figure 2. Linear growth factor as the function of scale factor. Perturbations are settled on a background with different overdensity . The case = 0 corresponds to the unperturbed background. The solid line is the upper limit of the dependence when D (z = 0) = 5 (see text).

2

Two constraints KS TH G

103 102 SL 101 100 10- 10-
1

2

Three constraints 104 106 108 1010 m [h-1 M ] 1012 1014 1016

averaged over the volume of the host perturbation, Eq. 39, namely = (Rh ), understanding the last as evolved to the present time. Substituting these definitions into equations above we get a law of growth on the given background. Properties of solutions are following. If the linear overdensity of the host is positive, then the amplitude will grow faster turning to a greater value at present time. However, an arbitrarily large value of can lead to only finite linear amplitude at present time with limiting value D(0) = 5 (Bildhauer et al. 1992, eq. 21 for m 1). Otherwise, the arbitrarily small value of the linear overdensity lead to smaller values of amplitude at present time, giving D(0) = 0 as a limiting case for = -1. 3 APPLICATIONS Our theory involves several parameters in addition to those in the original Press­Schechter approach. The parameters are the spatial scale of constraints Rh or associated mass Mh , and the mean overdensity . Constraining kernel can also be chosen fairly free. As we seen in Subsec. 2.3 the, different kernels with the same spatial scale effectively correspond to the different masses of the host halo, since they give different rates of decrease of the variance with spatial scale. Such an effective mass or scale can be used to calibrate Rh for various kernels. The effective value of this parameter can be calculated as the spatial momentum of (c) (r ) profile or so. Other way is to bind the effective scale to the scale where constrained variance crosses unity. Seems reasonable to use the model with H = W = WTH as a reference because the relations of the kind Eq. 39 looks naturally in the real space for this case. In the following examples of applications we'll use the Top-hat kernel both for filter and for constraints. We need to check the correctness of our theory for various types of the real objects. As was mentioned in Subsec. 2.1, the structure of the host halo may affect the mass function of the subhalo population. This impact should take place through the mass segregation and to be reflected in the sub-halo merging process. It is reliably enough to assume that such situation occurs in virialized haloes only. Hence, the knowingly non-virialized structures such as voids and friable superclusters, having low overdensities ( 100, see Lacey & Cole (1993)), are free from this effect of the 'structure biasing'. Last means that for these objects the m

Figure 1. Linearly evolved variance calculated for various number of constraints, from top to bottom: one (just kernel), two (the kernel and its 2nd derivation) and three (the kernel, its 2-nd & 4-th derivations). Host halo mass is 1012 h-1 M . Kernels used are: k-sharp (solid line), Top-hat (longdashed) and Gaussian (short-dashed). Upper dotted line is the unconstrained variance, and horizontal dotted line marks the level of nonlinearity, SL = 1. As seen, the Top-hat kernel gives variance which is invariant to the change of the number of constraints. For k-sharp kernel this property is true only for masses lesser than host. In both cases, the nonlinear mass is of order of the host halo mass. Using Gaussian kernel leads to the substantial dependence of the variance on the constraints number.

The evolution of perturbations inside an overdense or underdense region can be examined in a pseudo-cosmological notation, if parameters of such notation are chosen appropriately (Peebles 1993). Namely, the need to set the density of the matter and dark energy to corresponding values in a host halo or void, then do the integral for growth factor using the new density parameters: m = (1 + ) m = k = 1 - - m Growth factor calculated in thise case we will denote D. Here is the overdensity of a host perturbation linearly evolved to the present day, i.e. using the notation of Eq. 9, it can be written as D(0) (z = 0) L . It is natural to bind this value to the overdensity
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(43)


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E. P. Kurbatov
106 = = = = = 200 20 3 0 -0.75

factor in Eq. 10 equals to unity and number density PDF becomes m 0 nm = fm , m m
Mh

fm

S fS , = m

(44)
104 E 102 100 10-2 106 104 E 102 100 10-2 1010

where fS from Eq. 2. The cumulative distribution function is N (> m ) =
m

dm

Ms f, mm

(45)

where Ms is the total mass of an area of interest, i.e. of the sub-area of a supercluster or void. This value can be bound to the observed scale Rs at the given redshift via relation Ms = 3 (4 /3) m 0 Rs (1 + ). Without resorting to calculations we can find qualitative properties of these distributions. Since the growth factor is limited for all possible values of the host overdensity, and the variance depends on the mass continuously (if constraining kernels are continuous), there are the limiting distribution finction for given constraining scale Rh or Mh , corresponding to the limit of the evolution of sub-haloes population when the high-mass end of the PDF is the heaviest possible. Other limiting property is the lowmass end of the distribution where PDF tends to the unconstrained law which is rescaled accordingly to the growth factor value and independent of Mh . Below we will very briefly touch three possible applications of the theory: the mass function in superclusters and also in the voids, and the bias relation for haloes in non-virialized hosts. The primary goal here to show the dependence of the distribution functions on the host overdensity. 3.1 Superclusters Superclusters exhibit a very broad distribution of sizes from 10h-1 Mpc extending up to 150h-1 Mpc, and mass up to 1016 h-1 M . These objects form mainly in shells and filaments, although lumps are also observed. Clusters of galaxies are the characteristic components of superclusters. The typical dynamical mass of rich clusters is about 1013 - 1015 h-1 M . A number of rich clusters in supercluster can vary from several to over ten, and their mass fraction can be about 50 per cent (Bahcall 1999). Superclusters can be characterized with density enhancement factor which is the number of objects (clusters or galaxies) in the supercluster related to the mean number of objects in the same volume (Bahcall & Soneira 1984). On the theoretical base the factor can be represented as the relation E= N (< m ) ( 1 + ) , N (0) (< m) (46)



= = = = =

200 20 3 0 -0.75

1011

1012

1013 m [h-1 M ]

1014

1015

1016

Figure 3. Density enhancement factor as the function of the lowest sub-halo mass used for counting, and the host overdensity. Top panel: Ms = Mh = 1015 h-1 M . Bottom panel: Ms = Mh = 1016 h-1 M . As seen, there is no dependense of the factor on the host halo mass or on the lowest subhalo mass for small values of last, although the dependense is strong for moderate and high masses. In theory with unconstrained statistics we have E 1 (not shown).

where m is the lowest mass of objects we are counting. Here the Eq. 45 is used with Ms = Mh , and the factor 1 + is applied to transform the Lagrange volume of the evolved host perturbation to the unperturbed Eulerian one. The density enhancement factor keeps information about total mass or radius and depth of the host perturbation. On the Fig. 3 this function is presented for Mh = 1015 and 1016 h-1 M , and for various overdensity values. In the low-mass limit (m 1010 h-1 M , which corresponds to the dwarf galaxies) the density enhancement factor reduces to dependence on only, D(0) E ( 1 + ) , D so measuring E by cou