M. Demianski ^{1,2} - A. G. Doroshkevich ^{3,4}
1 - Institute of Theoretical Physics, University of Warsaw,
00-681 Warsaw, Poland
2 - Department of Astronomy, Williams College,
Williamstown, MA 01267, USA
3 - Theoretical Astrophysics Center,
Juliane Maries Vej 30,
2100 Copenhagen Ø, Denmark
4 - Keldysh Institute of Applied Mathematics,
Russian Academy of Sciences, 125047 Moscow, Russia
Received 2 May 2003 / Accepted 19 April 2004
Abstract
We generalize and extend the statistical description of formation
and evolution of Large Scale matter distribution in the Universe
proposed in our previous papers. We investigate the impact of
transverse motions - expansion and/or compression - on properties
of the Large Scale Structure elements - walls, pancakes, filaments
and clouds - and, generalizing the Press-Schechter formalism, derive
their mass functions. Using the Zel'dovich theory of gravitational
instability we show that these mass functions are approximately the
same and the mass of each type of elements is found to be
concentrated near
the corresponding mean mass. At high redshifts, both the mass function
and the mean mass of formed elements depend upon the small scale part
of the initial power spectrum and, in particular, upon the mass of
dominant fraction of dark matter (DM) particles. Using these results
we obtain independent estimates of probable redshifts of the
reionization and reheating periods of the Universe. We show that the
transverse motions do not significantly change the redshift evolution
of the observed mass function and the mean linear number density of
low mass pancakes related to absorption lines in the spectra of the
farthest quasars. Application of this approach to the observed
Lyman-
forest allows one to directly estimate the shape
of initial power spectrum on small scales.
Key words: cosmology: large-scale structure of Universe - cosmology: dark matter - galaxies: formation
In recent years numerical simulations are being more often used to investigate the process of formation and evolution of the Large Scale Structure (LSS) of the Universe. This trend is strongly stimulated by the rapid growth of computer facilities and it provides better and better high quality simulations (see e.g., Benson et al. 2001; Smith et al. 2003; Frenk 2002). Modern simulations performed in large boxes and with high resolution can take into account simultaneous action of many important physical factors and they substantially increase available information about the process of LSS formation and evolution.
At small redshifts the LSS is observed in the large galaxy surveys as a system of filaments and walls (see, e.g., Zel'dovich et al. 1982; Einasto et al. 1984; de Lapparent et al. 1986). At high redshifts the LSS is observed mainly through the distribution of the Lyman-clouds identified through absorption lines in spectra of the farthest quasars. All these elements of the LSS are formed by mildly nonlinear processes and, in contrast with galaxies and clusters of galaxies, they are relaxed along the shorter axes only. They are quite well reproduced in numerical simulations, what indicates their close relation with the initial power spectrum. However, this relation is not yet well understood and a quantitative description of the LSS elements only recently got under way. Now the analysis of both simulated and observed catalogs of galaxies is mainly focused on the discussion of the correlation function, , and the power spectrum, p(k), while other characteristics of the LSS are not determined. This limited description demonstrates that numerical simulations cannot substitute theoretical models of structure formation that provide the basis for much more detailed quantitative description of both the observed and simulated large scale matter distribution.
Theoretical models reveal the main factors that influence the process of structure formation and evolution and clarify links between them and the measured characteristics of the LSS. This opened up a possibility of formulating new and promising approaches of statistical description of both simulated and observed characteristics of the LSS such as, for example, a set of mass functions of structure elements, their separations and so on. These characteristics can be measured with such powerful methods as the Minimal Spanning Tree (MST) technique (Barrow et al. 1985; van de Weygaert 1991; Demianski et al. 2000, hereafter DDMT; Doroshkevich et al. 2001, 2004a,b for the SDSS and 2dF surveys) and the Minkowski Functional method (see, e.g., Mecke et al. 1994; Kerscher 2000; Shandarin et al. 2003). Some others methods had been discussed in Shandarin (1983), Babul & Starkman (1992), Luo et al. (1996), Martinez & Saar (2002). The high efficiency of such approaches is illustrated by the Press-Schechter relation (Press & Schechter 1974) for the mass function of high density clouds which has been widely used and discussed during the last thirty years (see, e.g., Mo & White 1996; Loeb & Barkana 2001; Sheth & Tormen 2002; Scannapieco & Barkana 2002).
Some statistical characteristics of the LSS have been derived in our previous papers (Demianski & Doroshkevich 1999, hereafter DD99; DDMT). They are based on the nonlinear theory of gravitational instability (Zel'dovich 1970; Shandarin & Zel'dovich 1989; for review, Sahni & Coles 1995) applied to a CDM-like broad band initial power spectrum of perturbations. This approach allows to outline the general tendencies of the LSS evolution and demonstrates the leading role of the initial velocity field and of the successive merging of structure elements in the process of formation of large scale matter distribution. Its efficiency and limitations were tested on simulated DM distributions and mock catalogs (see, e.g., DDMT, Doroshkevich et al. 2004a,b).
Indeed, the well known large difference between the coherent lengths of initial density and velocity fields - at least two orders of magnitude (Sect. 2.1) - indicates that the formation of the LSS elements is driven mainly by the velocity field. Using the velocity perturbations the Zel'dovich theory quite successfully describes the process of structure formation in the DM component up to the moment of caustic creation. Both the Zel'dovich theory and our approach do not describe the evolution of compressed matter such as the processes of violent relaxation, the disruption of the LSS elements to the system of high density clouds and so on. However, these processes do not distort the integral characteristics of the LSS such as the fraction of matter accumulated within filaments and pancakes or walls, their number density, the surface density of walls and linear density of filaments, their mass functions, velocities of each element as a whole and velocity dispersion within relaxed elements and some others. These stable statistical characteristics of the LSS are mainly defined by the process of formation of LSS elements and can be derived from the initial power spectrum and cosmological model at all evolutionary stages. Here we consider only such characteristics.
The Zel'dovich theory and our approach do not consider the evolution of baryonic component and accompanied thermal and gasdynamic processes important for the galaxy formation (see, e.g., Rees 1985; Rees & Dekel 1986; Dekel & Silk 1987) and for the appearance of LSS at high redshifts. In all these cases, the observed characteristics of baryonic and DM components are different and to link them special analysis is called for. We do not discuss these problems here. Even so, the application of our statistical approach to the observed galaxy surveys (Doroshkevich et al. 2004a,b) and Ly- forest (Demianski et al. 2003a,b) demonstrate high efficiency of the proposed method for interpretation of the observed characteristics of the LSS.
In contrast with many previously discussed models of the LSS formation such as, for example, the peak - patch picture of Bond & Myers (1996) and the model of Bond et al. (1996), our approach allows one to find the statistical characteristics of the LSS elements without any artificial smoothing of the initial fields. It demonstrates that the possible feedback of formation of galaxies and even cluster of galaxies on the evolution of the LSS is quite moderate because of the small fraction of matter accumulated within such objects.
The Zel'dovich theory describes the process of LSS formation as a successive creation of pancakes, their further compression into filaments and filaments into clouds with a progressive growth of the masses and sizes of structure elements due to their merging. It correctly describes properties of pancakes and walls formed owing to the 1D collapse, but underestimates the rate of collapse and overestimates the expansion rate after caustic formation. This means that our approach decelerates the process of formation of filaments and clouds, and our estimates of their masses and the matter fractions assigned to them correspond to earlier stages of their evolution as compared with observations and simulations. The properties of clouds are also changing in the course of their evolution within the filaments and walls. The same problems appear in the Press-Schechter formalism and its extensions. These relations can be improved after detailed comparison with numerical simulations.
Bearing in mind these comments, we focus our attention on the discussion of pancakes and walls while only some basic characteristics of filaments and clouds are considered. Physical aspects of our approach and main results are also discussed in Sect. 8.
Comparison of the theoretical expectations with measured statistical parameters of the most conspicuous wall-like component of structure was performed in DDMT and in Doroshkevich et al. (2004a,b) for the mock 2dF survey (Cole et al. 1998) and the SDSS Data Release 1 (Stoughton et al. 2002; Abazajian et al. 2003) and 2dF (Colless et al. 2003). The Las Campanas Redshift Survey (Shectman et al. 1996) was considered in Doroshkevich et al. (2001). These investigations confirm that the walls are gravitationally confined and partly relaxed Zel'dovich pancakes formed presumably due to the 1D collapse of matter. Such interpretation was already proposed in Thompson & Gregory (1978) and Oort (1983) just after the first walls have been observed.
From this comparison it follows that the main measured characteristics of both simulated and observed walls are consistent with the theoretical expectations. They are quite well expressed in terms of the time scale, the coherent length and correlation functions of the initial velocity field set by the power spectrum of initial perturbations. The time scale and the coherent length are expressed through the spectral moments and amplitude of the initial perturbations, and the basic parameters of the cosmological model. This analysis provides independent estimates of the amplitude of initial perturbations with a scatter 20% and verifies their Gaussianity.
Results obtained in DD99 and DDMT provide a reliable base for further investigations but impact of some important factors was not considered in these papers. Here we extend these models by taking into account the action of some factors responsible for the evolution of structure elements after their formation. Among the most important are: the expansion and compression of LSS elements in the transverse directions, the small scale damping of initial perturbations caused by the random motions of DM particles and Jeans damping, and the acceleration of cloud formation within larger structure elements, what creates a large scale bias between the spatial distribution of DM and luminous matter.
First of all, considering the expansion and compression of LSS elements in the transverse directions, we approximate the rates of formation of LSS elements and their mass functions in a wide range of redshifts. For all LSS elements - pancakes, filaments and high density clouds - the mass functions are found to be quite similar to each other. They are also similar to those predicted by the Press-Schechter formalism for a suitable initial power spectrum (Loeb & Barkana 2002) and its extension for elliptical clouds (Sheth & Tormen 2002). This similarity demonstrates the generic dependence of the characteristics of the LSS on the initial power spectrum and indicates that the shape of collapsed clouds influences the rate of collapse but does not change significantly the mass functions. These results complement the Press-Schechter approach and allow one to obtain independent estimates of the mass functions for a wider class of objects.
The same factors change the basic properties of long-lived pancakes observed as the Ly- forest in a wide range of redshifts (see, e.g., Cen et al. 1994; Dave et al. 1999). In this paper we discuss the evolution of two most important characteristics of such pancakes: their surface density and their mean number density along the line of sight. Both characteristics are evidently changing with redshift due to the formation of new and merging of old pancakes, and their transverse compression and expansion. Both characteristics play a key role in the discussion of observed evolution of the Ly- forest and we used them (Demianski et al. 2003) to show that the observed properties of absorbers are consistent with theoretical expectations. This approach also allows one to measure the shape of initial power spectrum down to scales of 30h^{-1} kpc (Demianski & Doroshkevich 2003). It is important to notice that the surface density of pancakes is sensitive to the model of absorbers used in the analysis while the mean number density along the line of sight depends only on the position of absorbers and is independent from the physical state of the intergalactic matter and the model of absorbers.
The properties of LSS are also strongly influenced by the interaction of small and large scale perturbations what accelerates or decelerates formation of less massive clouds within compressed or expanded LSS elements. In simulations this interaction manifests itself by a strong concentration of halos within filaments and walls at all redshifts (see, e.g., Mo & White 1996; Lemson & Kauffmann 1999; Sheth & Tormen 1999, 2002; Gottlöeber et al. 2001, 2002, 2003; Sect. 5.7). In observations it is seen as a strong concentration of galaxies within the richer filaments and walls and as an essential difference in richness of galaxy groups situated within walls and filaments (see, e.g., Doroshkevich et al. 2004a,b). In contrast, a significant fraction of both DM and baryons remains within low mass pancakes that are more homogeneously distributed than the luminous matter and are observed as the Ly- forest. We show that the formation of high density halos and galaxies is modulated by the large scale initial velocity field what explains qualitatively the large scale bias. The approach used in this paper allows one to demonstrate and roughly quantify this dependence.
However, the fragmentation of the LSS elements into the system of high density clouds and their further evolution cannot be described with our approach. The analysis of simulations (see, e.g., Gottlöber et al. 2001, 2002, 2003) indicates that the quantitative description of these processes requires special models.
This paper is organized as follows: in Sect. 2 basic relations are introduced. In Sect. 3 the redshift dependence of the expected matter fraction assigned to various types of structure elements is found. In Sect. 4 we discuss the interaction of large and small scale perturbations. In Sect. 5 the joint mass functions of DM structure elements are considered. In Sects. 6 and 7 we discuss the redshift evolution of statistical characteristics of filaments and pancakes. Short conclusions can be found in Sect. 8. Some technical details are given in Appendices A and B.
In this section we present the basic statistical characteristics of Zel'dovich approximate nonlinear theory of gravitational instability used later to describe the process of structure formation and evolution. Main ideas and characteristics were already introduced in DD99 and are repeated here without discussion. Here we generalize and extend this description that allows us to extend its application. Some definitions are improved and corrected what makes the approach more transparent.
In the Zel'dovich theory (Zel'dovich 1970; Shandarin & Zel'dovich
1989) the Eulerian, r_{i}, and the Lagrangian,
,
coordinates of particles (fluid elements) are related by
In this paper we consider only power spectra with the Harrison-Zel'dovich asymptotic,
,
at
,
and CDM-like or WDM-like transfer functions, T^{2}(k), introduced
in Bardeen et al. (1986, hereafter BBKS):
For the spectra (3), the coherent lengths of initial density
and velocity (or displacement) fields,
and l_{v}, are
expressed through the spectral moments, namely, m_{-2} and m_{0},
(DD99):
For spectra under consideration, the normalized correlation
functions of displacement and density fields,
As was shown in DD99, to describe the formation of structure
elements the basic Eq. (1) has to be rewritten using
the differences of particle coordinates and displacements.
The separation of two particles with Lagrangian and Euler
coordinates
and
,
and
and ,
respectively, is described by the
equations (i=1, 2, 3):
According to the relations (11), when two particles with different Lagrangian coordinates and meet at the same Eulerian point a caustic - Zel'dovich pancake - with the surface mass density forms. Following DD99 we assume that all particles situated between these two boundary particles are also incorporated into the same pancake. This assumption is also used in the adhesion approach (see, e.g., Shandarin & Zel'dovich 1989). Comparison of statistical characteristics of pancakes with simulations (DD99; DDMT) verifies this approach and shows that long lived and partly relaxed richer walls accumulate a significant fraction of compressed matter. The same condition can be used to approximate formation of filaments and clouds in the course of successive collapse along two and three axes.
For the general description of structure formation it is
convenient to combine the three Eqs. (11) into
one scalar equation, namely,
In DD99 we assumed that the deformation field is dominated by
two lowest harmonics what allows one to characterize the formation
of structure by three weakly correlated components of the
displacements. Analysis performed in Appendix B shows that,
for the spectra (3), this assumption is valid with
a precision better than 10%. For the most interesting cases
and
,
the normalized
amplitudes of several lowest spherical harmonics of the deformation
field, b_{l}^{2}, are given by:
These results justify the assumption made in DD99 to neglect higher order harmonics of perturbations with and they confirm that the formation of structure can be approximately described by the spherical and quadrupole components of the deformation field. The influence of higher harmonics, even with small amplitude, leads to a small scale disruption of the compressed clouds because of the strong instability of thin pancake-like condensations (Doroshkevich 1980; Vishniac 1983) and, so, to the formation of complex internal structure of clouds.
These results indicate also that when the process of structure formation is described by the function the ellipsoidal or, in the case of two dimensional problem, elliptical volumes/areas are preferable.
When only two spherical harmonics are taken into account, the general deformation of any cloud can be described by its deformations along the three orthogonal principal axes, namely, x_{1}, x_{2}, and x_{3}. In this case, we can use a simpler approach and consider again the three Eqs. (11) instead of (13). With this approach it is possible to obtain some approximate characteristics of the LSS. However, its abilities are restricted and some important problems can be solved only with the general approach (13).
The numbering of principal axes is arbitrary but, further
on, we will usually assume that
The correlations of differences of displacements along the
principal axes are relatively small (DD99),
For quantitative estimates it is convenient to rewrite Eq. (11) in a dimensionless form
(17) |
As is seen from Eqs. (9) and (19), for two limiting
cases
and
,
we have
The parameter q_{0} (4, 10), characterizes the
damping scale in the initial power spectrum and discriminates
regions of strong and small correlations of the initial density
and velocity fields. As is seen from Eq. (22), the condition
introduces also the typical redshift, z=z_{r},
The fractions of matter accumulated by pancakes and filaments were discussed in DD99. Here we extend this approach and introduce three cumulative distribution functions for more detailed description of fractions of matter accumulated by pancakes, filaments and walls with different sizes and rates of expansion and/or compression. In turn, this extension allows one to obtain the approximate mass functions of all three kinds of LSS elements and to take partly into account the further evolution of pancakes after their formation. These problems are discussed in Sects. 5 and 7.
We note in Introduction that, in comparison with numerical simulations, our approach decreases the matter fraction assigned to filaments and clouds. Moreover, it does not describe the evolution of clouds within the filaments and pancakes. This means that these estimates correctly approximate matter concentration within walls and pancakes at all redshifts but shift the moment of formation of filaments and clouds to smaller redshifts. Estimates of clouds characteristics are distorted more strongly than those for filaments.
As is seen from Eq. (19), the conditions
and
are equivalent
and both define collapse of a cloud along the axis of
the most rapid compression. To discriminate between pancakes,
filaments and walls, we have to use also conditions which
restrict the motion along two other directions. For more
detailed description of pancakes we have also to restrict
the rate of pancake expansion. These conditions are consistent with
restrictions (16) and they can be summarized as follows:
The first condition in Eq. (24) assumes that for a given the clouds with , collapse along the first axis. For , the next two conditions exclude from the consideration collapsed filaments with and clouds with . For they restrict the expansion rate of expanding pancakes.
Similar conditions, namely,
Integration of the PDF (20) under conditions (24-26) results in the
cumulative PDFs
that characterize the evolution of the LSS elements:
The functions and describe the fractions of matter assigned to pancakes, filaments and clouds, respectively, under conditions (24-26). For , expressions (27) and (28) become identical to those obtained in DD99.
From the relations (27)-(29) it follows that
in the Zel'dovich theory the maximal fractions of matter that can
be accumulated by clouds, filaments and pancakes, for q_{i}=0,
,
are
The cross correlations of orthogonal displacements as given
by Eq. (17), though small, increase the maximal fraction of
matter accumulated by clouds and filaments up to
and
,
respectively, and decrease the
fraction of matter accumulated by pancakes to
.
These maximal fractions are plotted in Fig. 1 versus
for all three types of structure elements.
As it is apparent from Fig. 1, already at
,
,
25%, 8% and 2% of matter is accumulated
by pancakes, filaments and clouds, respectively. At
,
these mass fractions increase
,
what coincides with Eq. (30).
Figure 1: Fraction of the DM component accumulated by pancakes and walls (solid line), filaments (dashed line) and clouds (long dashed line) plotted versus . These fractions at the moment , , are shown by thin straight lines. | |
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These estimates of the mass fractions are quite reasonable for large redshifts, when the formation of structure elements from the dispersed matter dominates. At small redshifts the merging of the LSS elements distorts these estimates and, for example, some fraction of filaments and clouds is incorporated into richer walls. These estimates can be improved by a comparison with high resolution simulations and observations of Ly- forest at high redshifts.
The relations (27)-(29) are monotonic
functions of
and, for
,
they represent the cumulative three parameters distribution
functions of structure elements with respect of their sizes
measured by differences of Lagrangian coordinates,
q_{1},
q_{2}, q_{3}, for particles bounding these LSS elements along
the principal directions.
This joint PDF for all structure elements is
The expression (31) does not discriminate filaments and
pancakes with respect to their expansion rates described
in Eqs. (27) and (28) by negative
and .
To allow such discrimination it is necessary to link
and
with the size of
structure elements. The symmetry of the basic characteristics of
structure elements with respect to
positive and negative
discussed in Sect. 2.2
and the relations (12) indicate that such an extension can
be achieved by imposing the following limits in Eq. (31):
High resolution simulations show that at high redshifts a substantial fraction high density halos is accumulated by filaments. In observed redshift surveys galaxies are mainly found within filaments and walls and the population of isolated galaxies is quite small. This strong galaxy concentration within the LSS elements as compared with more homogeneous distribution of baryons and dark matter - large scale bias - can be naturally explained by the interaction of small and large scale perturbations responsible for the formation of galaxies, filaments and walls, respectively. Of course, such bias is enhanced by the acceleration of evolution after formation of filaments and walls.
The spatial modulation of the rate of matter collapse is not described with the approach discussed above because this problem requires the use of at least two point distribution function. This function was introduced in DD99. Here we return to this problem and propose a simple quantitative measure that illustrates effects of this interaction.
Let us consider the formation of a small pancake within
a large scale perturbation with small amplitude. Following
Sect. 2.1, we will describe the pancake formation by
Eq. (11) with the displacement
Owing to the linearity of Eq. (33), this result is immediately generalized for a set of large scale perturbations. Thus, for the set of successive merging of earlier formed object we need to substitute by in Eq. (35) and z_{3} by in Eq. (36). The expression (36) is valid for while the more general expression (35) can be used for all redshifts.
The analysis of numerical simulations indicates that the early formed halos are later on accumulated by filaments and richer walls and, so, are subjected to two, three and more successive compressions. As the structure formation is a deterministic process (with random initial perturbations), all these future mergings accelerate the formation of the first halos already at high redshifts and determine their location within richer walls or filaments at small redshifts. In the opposite case of halo formation within an expanded region we will have deceleration of the collapse and decrease in richness of the halo.
This interaction can also be described using the two point
correlation function of pancakes introduced in DD99. To do this,
we compare the process of formation of two identical
pancakes with a surface density
,
and
.
We assume that the first - a reference -
pancake is formed at the "time''
and is
characterized by the function
In this case, the interaction of pancakes can be described by
replacing
in Eqs. (27) and (31) with the effective
parameter
defined by the relation (DD99):
The suppression of pancake formation within expanding regions
can be considered in the same manner. Thus, for example, for
negative
and
we have instead of Eq. (38)
The influence of large scale perturbations on the process of formation of small scale objects can be considered as the manifestation of large scale bias. Fraction of matter accumulated within high density clouds increases rapidly with time (see Fig. 1) and halos formed at larger z contain only a small fraction of mass of halos formed at smaller z. But this bias increases the redshift of subclouds formation and their densities, promotes the transformation of DM clouds into observed galaxies, and makes the internal structure of clouds more complex. Regular distribution of young galaxies within high density filaments suppresses their feedback and fosters further formation of galaxies.
We compare these expectations with the high resolution simulation (Klypin et al. 1999; Schmalzing et al. 1999) performed with adaptive code in a box of (60h^{-1} Mpc)^{3} with particles for the Harrison-Zel'dovich primordial power spectrum and the BBKS transfer function. This box size and small scale cutoff of the power spectrum approximately correspond to , keV.
To test the impact of environment on the properties of high density clouds we use the Minimal Spanning Tree technique described in Doroshkevich et al. (2001, 2004). For a given threshold linking length or for the threshold overdensity above the mean density, this technique allows one to select all clouds from the sample without any restrictions of their shape. For each selected cloud its richness is also easily defined.
The matter distribution at redshifts z=4, z=1 and z=0 were analyzed. As the first step, the original samples were decomposed into subsamples of high and low density regions (HDRs and LDRs). The subsamples of high density regions accumulate 40-45% of all DM particles within richer clouds selected for the threshold density equal to the mean density. Subsamples of LDRs were prepared by removing the HDRs from the full samples.
At the second step, the high density structure elements were selected within HDRs and LDRs with a set of the threshold linking lengths. The matter fraction accumulated by the elements is plotted in Fig. 2 versus the threshold overdensity used for the selection. As is seen from this Figure, at all redshifts the majority of highly compressed matter is situated within the HDRs. This excess is moderate at z=0 and progressively increases with redshift. Special test shows that the high density clouds selected at z=4 are equally distributed between HDRs and LDRs selected at z=0. This fact is consistent with almost equal concentration of galaxies within the HDRs and LDRs at z=0 in the observed surveys. It indicates also that the efficiency of the interaction of small and large scale perturbations increases with redshift.
Impact of environment on the properties of galaxy groups
selected within HDRs and LDRs in the observed SDSS DR1 survey are
discussed in Doroshkevich et al. (2004). As before, properties of
such groups selected within HDRs and LDRs are found to be strongly
different. Similar correlation of properties of simulated high
density halos with their environment and their evolutionary
histories was also found in Gottlöber et al. (2001, 2002, 2003)
where different statistical method - the mark correlation
function - was used.
Figure 2: Fractions of matter, , accumulated by structure elements situated within HDRs (solid lines) and LDRs (dashed lines) are plotted vs. the threshold overdensity for simulated DM distribution at three redshifts. | |
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Figure 3: Functions (thick solid line), (thick dashed line) and (thick long dashed line) for two values of q_{0} plotted vs. for . For and , functions are plotted by thin solid and dashed lines, respectively. | |
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Figure 4: Top panel: functions (thick solid line) and (thin solid line) plotted vs. for . Fit (48) is drawn by dashed line. Bottom panel: functions for (thick solid line) and (thin solid line) plotted vs. . Fits (46) and (47) are drawn by thin dashed and long dashed lines, respectively. | |
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The popular Press-Schechter formalism focuses main attention on the spherical collapse and the achieved final critical overdensity (see, e.g., Press & Schechter 1974; Gunn & Gott 1974; Epstein 1983; for review in Loeb & Barkana 2001). It can be extended for elliptical clouds (see, e.g., Sheth & Tormen 2002). In spite of so strong restrictions on the process of collapse it successfully describes the simulated mass distribution of high density halos. In Zel'dovich theory of gravitational instability similar approach allows one to find the mass function of all high density structure elements - clouds, filaments, and walls or pancakes - without any assumptions about their initial and final shapes, stability, relaxation and achieved overdensity. In this regard, our approach is much more general then the Press-Schechter formalism.
The close connection of the observed objects with the initial power spectrum and the generic origin of galaxies and other observed elements of the LSS is clearly seen from the expression (31) which applies to all structure elements. Due to symmetry of the relation (31) with respect to variables q_{1}, q_{2} and q_{3} and because the mass functions are normalized, the restrictions (24)-(26), used for the determination of the mass fractions of different structure elements, are now of no importance and they lead only to renumeration of coordinates. Thus, in the Zel'dovich approach, the mass functions of the various LSS elements differ only by their corresponding survival probability (see, e.g., Peacock & Heavens 1990; Bond et al. 1991).
Our approach describes correctly the formation of pancakes and
walls but, as compared with simulations, it decelerates the
formation of filaments and clouds and
decreases their mean masses for a given redshift. Moreover, the
evolution of high density clouds embedded within filaments and
walls is accelerated due to the impact of environment and,
perhaps, is better described by the coagulation approach. Indeed,
as is noted in Sect. 5.7, at z=4 the mass functions of filaments
and clouds are found to be quite similar to the expected ones.
However, at z=0 these functions can also be fitted by power laws
what demonstrates a possible impact of coagulation processes. This
problem requires further extended investigation similar to that
performed in Gottlöber et al. (2001, 2002, 2003) for high
density halos.
Figure 5: Functions (triangles, q_{0}=10^{-2}) and (up down triangles, q_{0}=10^{-3}) plotted vs. . Fits (51) and (52) are drawn by solid lines, respectively. | |
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Figure 6: Mass functions of galaxy clouds, , selected at redshifts z=0 ( left panels) and z=4 ( right panels) in HDRs and LDRs for two threshold linking lengths. Fits (48) and (57) are plotted by solid and dashed lines, respectively. | |
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To find the mass function of structure elements we
rewrite the general relation (31) in spherical coordinates
This result demonstrates that in the Zel'dovich approximation the distribution of shapes of initial clouds described by the parameters and in Eq. (40) is continuous and the final properties of structure elements, such as their shape, energy and overdensity, depend mainly upon the velocity or deformation field within clouds. In particular, even high density compact relaxed halos can be formed through the collapse of initially asymmetrical clouds. Some information about the initial shape of collapsed clouds is retained in the angular momentum of observed clouds discussed below.
On the other hand, for all clouds, the first step of collapse is the formation of a pancake-like objects (Zel'dovich 1970; Shandarin et al. 1995) that are unstable and will rapidly break up into a system of low mass subclouds (Doroshkevich 1980; Vishniac 1983). This instability stimulates formation of numerous satellites of the largest central object and makes it difficult to observationally distinguish between isolated galaxies and these satellites.
For the more massive clouds with
,
,
we have from Eq. (31):
For low mass clouds with q_{i}< q_{0}, we get for the mass function
As it was shown above formation of low mass structure elements is usually suppressed
because they could be absorbed by larger objects formed at the same
time and in the same region. This process is described by the survival
probability of structure elements and it is different for clouds, filaments and walls.
As was shown in DD99, for walls formed through the 1D compression
along the q_{1} axis the survival probability can be taken as
Integration of the PDF Eq. (41) corrected for the survival probability (43)-(45), , over angular variables for given R, q_{0} and allows one to find the mass functions of structure elements: clouds, filaments, walls and pancakes.
The mass function N_{m} and the function are plotted in Figs. 3 and 4 for q_{0}=10^{-2}, and q_{0}=10^{-3} and for the three more interesting values of and . If corresponds approximately to the present epoch (see Sect. 7.1), then (see Eq. (23)) and describe the early period of structure formation, when the influence of small scale correlations of initial density and velocity fields is more important.
As is seen from Fig. 3 at the mass functions for all three kinds of structure elements are identical, what is a direct consequence of strong correlations of small scale perturbations. However, the difference between these mass functions increases with time and at it becomes significant, especially for q_{0}=10^{-3}.
As is seen from comparison of Figs. 3 and 4, in all the cases the numerous low mass structure elements contain only negligible fraction of mass and for all the main mass is concentrated within structure elements with . For the impact of the parameter q_{0} on the shape of mass functions is negligible and, for both chosen values of q_{0}, the functions are well fitted by the Gauss function (46). This result reflects again a strong small scale correlation of the initial density field and, as was discussed in Sect. 5.2, a small survival probability of low mass structure elements.
At the mass function becomes wider because the continued formation of low mass structure elements is accompanied by progressive mass concentration within massive elements with (6). As is seen in Fig. 3, during this period the small scale cutoff of initial power spectrum and the parameter q_{0} provide the low mass cutoff of the mass function. For such redshifts, the mass functions for objects with and are described respectively by power and exponential laws.
Disregarding the low mass "tails'' of mass functions, we
fit the mass function for both values of q_{0} by:
By definition, the mass functions must satisfy two
normalization conditions
The mean mass of objects is quite sensitive to the coherent
length of initial density field. For the same q_{0} as above,
the mean mass of clouds is plotted in Fig. 5 versus
together with fits
For , the mean masses of pancakes and filaments are smaller than those for clouds by a factor of 1.5-2.
The mass function N_{m} describes all structure elements -
clouds, filaments, and walls or pancakes - without assumptions
about their shapes and achieved overdensity. However, the relation (49) is quite similar to the Press-Schechter mass function
for scale-free power spectra and k corresponding to typical
objects,
:
All mass functions discussed above are related to the process of formation of structure elements and they do not take into account the later nonlinear evolution described by the coagulation equation (Smoluchowski 1916; Silk & White 1978; Sheth & Pitman 1997). This is not so important for walls and filaments for which the merging and coagulation are controlled mainly by the initial velocity field. But the non linear evolution can essentially distort the mass function of high density clouds accumulated by richer walls and filaments.
In contrast with expressions (46)-(49) and
(55) the coagulation process leads to the mass function
The theoretical fits (48) and (56) can be compared with the mass functions found for matter distributions at redshifts z=4, and z=0 in the high resolution simulation (Klypin et al. 1999; Schmalzing et al. 1999) discussed in Sect. 4.3. As was described in Sect. 4.3, at both redshifts the full samples were divided into subsamples of high and low density regions (HDRs and LDRs) dominated by walls and filaments. The selection of structure elements was also performed with the Minimal Spanning Tree technique described in Doroshkevich et al. (2001, 2004).
The mass functions of high density clouds found for the HDRs and LDRs separately are plotted in Fig. 6. The basic parameters of the same samples of selected clouds are listed in Table 1, where and are threshold and mean overdensities of clouds above the mean density of the sample, is the fraction of points accumulated by clouds, and are the number and the mean richness of clouds. At both redshifts, very massive structure elements are formed through the percolation process and they cannot be described by the expressions (48), (55) or (56). For this reason, they were excluded from the analysis of the mass distribution. However, these clouds are included in estimates of the matter fraction accumulated by structure elements as they represent actual filaments and walls. The difference between both and for clouds selected with the same within HDRs and LDRs confirms a significant impact of environment on the properties of clouds (see Sect. 4).
At both redshifts, the samples selected with high represent properties of small fraction of high density clouds, while samples selected with small are formed mainly by unrelaxed filaments and walls. The cutoff of the simulated mass functions at low masses caused by a finite resolution increases the measured mean richness of selected clouds as compared with theoretical expectations (51). Because of this, the mean richness was used as a fit parameter in the relations (48) plotted in Fig. 6. However, the shape of the mass function (48) was not altered. As is seen from Fig. 6, this expression fits well the simulated mass functions for both high density clouds and only partly relaxed filaments and walls.
Table 1: Parameters of structure elements selected in HDRs and LDRs at z=0 and 4.
However, at the redshift z=0 the simulated mass functions are well
fitted also by a power law
It is especially important that at both redshifts the relations (48) and (55) successfully reproduce the mass functions of unrelaxed filaments and walls that are far from the spherical shape. This fact demonstrates a moderate influence of the shape of collapsed clouds on their mass and the validity of Zel'dovich approach which considers the dynamical characteristics of collapsed clouds rather than their shape.
Similar analysis of the observed SDSS DR1 sample of galaxies can be found in Doroshkevich et al. (2004).
The tidal interaction of perturbed matter creates both the anisotropy of the collapse and the angular momentum of forming clouds (Hoyle 1949; Peebles 1969; Doroshkevich 1970; White 1984; Bullock et al. 2001; Vivitska et al. 2002). Usually the angular momentum is defined for the high density relaxed clouds and is expressed through the so called -parameter. Here we extend the approach developed in Peebles (1969), Doroshkevich (1970) and White (1984) and express the created angular momentum for structure elements in the framework of statistical approach used in this paper.
For this purpose we use the general Eq. (1) together
with the corresponding expression for the velocity of fluid element
The angular momentum of a cloud is defined by the integral over
the corresponding collapsed volume, V,
For larger clouds with ,
,
we get
For the high density halos formed by the successive mergings of subhalos a statistical model was proposed in Vivitska et al. (2002). Comparison with simulations and the current status of the problem can be found in Bullock et al. (2001) and Vivitska et al. (2002).
In addition to the fraction of matter accumulated by filaments (28) and the mass function of filaments given by Eqs. (46)-(49), we will consider here also the linear density of galaxies along a filament, , defined as a mass per unit length of filament, and the mean surface density of filaments, , defined as the mean number of filaments intersecting a unit area of arbitrary orientation. These characteristics are weakly dependent upon the nonlinear evolution of matter compressed within filaments and are mainly defined by properties of the initial velocity field and the process of filaments formation and merging.
The measured characteristics of filaments depend also upon the threshold linking length, , used for the filament selection, which determines the threshold overdensity bounding the filaments. Now only richer filaments can be selected in both observed and simulated catalogs what restricts their quantitative characteristics. Because of this, here we will only discuss characteristics of richer filaments.
The distribution function of filaments linear density describes
their frequency distribution with respect to the amount of matter
per unit length of filaments. As was discussed in DD99, this
function can be obtained by integrating Eq. (31) over all q_{3}and over the ratio q_{1}/q_{2}. For
and
using the survival probability for low mass filaments given by (44) we have for the PDF of richer filaments with
Figure 7: Distribution function, , for the linear density of DM particles in filaments selected at three linking lengths, . Fits (65) are plotted by solid lines. | |
Open with DEXTER |
The PDF,
,
is well fitted by
For the surface density of filaments,
,
and for their mean separation,
we get, respectively:
From the PDF of the differences of displacements (20) it is possible to extract important approximate characteristics of walls and less massive pancakes which can be directly compared with available observations. Some of them were introduced in DD99 and successfully compared with the simulated and observed characteristics of walls. Here we derive the theoretical expression for the mean linear number density of pancakes for different redshifts and show how the transverse compression and expansion of pancakes after their formation change their observed characteristics. These results are successfully used for description of low mass pancakes observed as Ly- forest at large redshifts (Demianski et al. 2003) and for the determination of the initial power spectrum at small scale (Demianski & Doroshkevich 2003).
The distribution function of sizes of walls
describes their frequency distribution with respect to their
Lagrangian size what is identical to their surface density,
,
defined as the mass per unit surface area of the wall
at the moment of its formation. This distribution function is
obtained by integrating Eq. (31) over all q_{2} and q_{3}for ,
and it is given by:
For some applications we have to estimate the transverse characteristics of pancakes such as the distribution function of Lagrangian size and the mean real size of pancakes. These characteristics can be found with the method used above.
For the frequency distribution of the walls with Lagrangian
transverse sizes, q_{2} and q_{3}, we get from Eqs. (21), (22) and (31)
The mean transverse size of expanded,
,
and compressed,
,
pancakes can be found from relations (11), (20) and (68), we have
After pancake formation, the transverse compression and/or expansion of matter changes its surface density and other characteristics. However, the direct analysis indicates that the PDF of pancakes surface density given by Eq. (67) only weakly depends upon these deformations.
The evolution of the pancake surface area,
,
is described
by the relation (11) as follows:
These results indicate that at high redshifts, when the expansion and compression of pancakes approximately compensate each other and the PDF Eqs. (62) or (66), the mean surface density, , and other average characteristics of pancakes only weakly depend upon the redshift. They indicate also that, in spite of the strong evolution of each individual pancake, the statistical description (67) remains valid also when we consider each wall and each pancake as formed at their current redshifts.
Application of these results to absorbers observed in a wide range of redshifts (Demianski et al. 2003) confirms these conclusions.
Using the relations (31) and (67) it is also possible to
obtain an approximate estimate of the mean comoving linear
number density of recently formed walls, that is the mean number
of walls per unit distance along a straight line. For richer walls
with a threshold surface density
the small scale
fluctuations of density are not important and this function can be
written as follows:
To describe the nonlinear evolution of walls observed in deep galaxy
surveys we can also use
the 1D version of the coagulation equation (Smoluchowski 1916; Silk
& White 1978) which can be written in the comoving space as follows:
The simplest reasonable solution of the coagulation equation, similar
to Eq. (67), can be written as follows:
The approach discussed in Sect. 7.4 neglects the influence of small
scale perturbations and approximately characterizes only the mean
linear number density of richer walls. For low mass pancakes of a
size comparable with the coherent scale of initial density field,
,
the mean linear number density of pancakes with a
threshold size
depends upon the spectral moment, m_{0},
and q_{0} (4). It can be found with the standard technique
(see, e.g., BBKS) used to describe the condition that a random
function exceeds a certain value, we get:
The relation (77) characterizes pancakes by their threshold size at the redshift of formation and neglects evolution of pancakes after they are formed. However, the surface density of formed pancakes is changing because of their transversal compression and/or expansion which shifts some of the pancakes under and/or over the observational threshold. This problem is quite similar to that discussed in Sect. 7.3, where it was noticed that the surface density is a more adequate characteristic of DM pancakes because it takes into account these variations.
In the relations (77) the threshold size of pancakes,
,
together with their transverse sizes appears only in the
function
.
This means that to go
from the threshold size to the threshold surface density we must
link
and
with the expression (70)
and find a new function
.
This procedure is quite similar to that used in Sect. 7.3,
and, for
,
,
we get instead of Eq. (77) that
Both expressions (77) and (78) were used in Demianski et al. (2003) to describe the observed evolution of the mean linear number density of pancakes and to estimate the important parameter q_{0} and the moment m_{0}of the initial power spectrum.
In this paper we continue the statistical description of the process of LSS formation and evolution based on the Zel'dovich theory of nonlinear gravitational instability. First results obtained in DD99, DDMT and Demianski et al. (2003) show a significant potential of this approach. Here we are allowing for deformation of pancakes after their collapse along the axis of the most rapid compression, interaction of large and small scale perturbations and the impact of small scale cutoff in the initial power spectrum. This extension allows one to consider three important problems.
First of all, we are able to approximate the mass functions and fractions of matter accumulated by the LSS elements, namely, pancakes, filaments and halos for a wide range of redshifts. As was shown in Sect. 6, the mass functions describe reasonably well simulated mass distributions at all redshifts what emphasizes the generic character of the processes of formation of all structure elements. These functions provide quantitative description of the LSS evolution what in itself is an important problem.
Secondly, we discuss the interaction of large and small scale perturbations, that manifests itself as a strong concentration of galaxies within filaments and walls observed at small redshifts. In Sect. 4 we demonstrate the importance of this interaction and can roughly quantify it. However, this interaction is complex and it requires more detailed statistical description.
Thirdly, we derive the mass function and the mean linear number density of pancakes at high redshifts. Both functions play an important role in the interpretation of the Ly-forest observed in spectra of the farthest quasars. Results obtained in Sect. 7 are successfully applied in Demianski et al. (2003) and Demianski & Doroshkevich (2003) for detailed description of the observed absorbers and, in particular, lead to estimates of the spectral moment m_{0} and the mass of the dominant fraction of dark matter particles.
The rapid growth of the observed concentration of neutral hydrogen at redshifts (Djorgovski et al. 2001; Becker et al. 2001; Pentericci et al. 2001; Fan et al. 2001) is an evidence in favor of the reionization of the Universe at this redshift. These observations stimulate discussions of the reheating of the universe and, in particular, of the warm dark matter (WDM) models (see, e.g., Barkana et al. 2001; Loeb & Barkana 2001). This means that it is worthwhile to find direct estimates of the small scale initial power spectrum and its influence on the LSS formation. Results obtained in Sects. 4, 5 and 7 are quite important for such investigations.
In the Zel'dovich theory the evolutionary stage achieved in
the sample under investigation is
suitably characterized by an effective dimensionless "time'',
,
introduced by Eq. (14):
The amplitude of initial perturbations, A, is simply linked
with
and the main parameters of the cosmological model, h and
.
Using the latest estimates (Spergel et al. 2003; Tegmark et al.
2004) for the CDM model (2) we get:
The variance of displacement (7),
and ,
can be
directly expressed through the observed two point correlation
function of galaxies,
.
The correlation functions for the APM survey were found in Loveday
et al. (1996), for the Las Campanas Redshift Survey in Jing et al.
(1998) and for the Durham/UKST Redshift Survey in Ratcliffe et al.
(1998). Using these results we get
Applying the relations (67) to the systems of walls
selected in two largest redshift surveys, namely, the 2dF and the
SDSS DR1 (Doroshkevich et al. 2004) allows one to estimate
as
All these estimates are consistent with each other and the differences between , and given by (80)-(82) reflect the precision actually achieved in modern observations.
The Press-Schechter formalism derived for the spherical collapse describes quite well the mass functions of various observed and simulated structure elements. In spite of this, all attempts to extend it to the collapse of asymmetric objects failed with the exception of recently proposed description of collapse of elliptical clouds (see, e.g., Loeb & Barkana 2001; Sheth & Tormen 2002). In this paper we demonstrate that the Zel'dovich theory successfully describes collapse of any object.
Indeed, formation and relaxation of pancakes along the axis of the most rapid compression does not prevent their deformation in transverse directions due to relatively small gradients of density and pressure in these directions. The growth of density within the pancake changes the rate of evolution and accelerates its compression and further transformation into high density filaments and clouds. However, the masses of filaments and clouds formed due to such compression remain almost unchanged. The same factor also decelerates the expansion of the pancake and its dissipation and, so, increases the fraction of surviving pancakes.
This means that, allowing for these deformations, we obtain approximate time dependent mass functions of structure elements formed due to successive compression in one, two and three directions. These results emphasize the generic character of the formation of all structure elements and link the fundamental characteristics of structure with the initial power spectrum. They extend the Press-Schechter formalism for all LSS elements including the filaments and walls which are far from equilibrium. The unexpected similarity of the mass functions for these elements verifies that the shape of collapsed clouds influences the rate of collapse but it does not change significantly their mass functions.
As is seen from the comparison of results presented in Sect. 5 and in Loeb & Barkana (2001) for the Press-Schechter formalism, the mass functions derived in both approaches are similar and, in particular, the relations (49) and (55) resemble each other. Both mass functions are sensitive to the damping of small scale perturbations caused by the random motion of DM particles and the Jeans damping. They predict the existence of numerous low mass objects which can be identified with isolated dwarf galaxies and a rich population of Ly- absorbers. Both approaches predict a strong suppression of formation of isolated low mass objects with .
This comparison indicates that the differences between these approaches are quantitative rather than qualitative. The similarity of the mass functions demonstrates, in fact, the self similar character of the process of structure evolution that is the successive condensation of matter within clouds, filaments and walls with progressively increased sizes and masses. At the same time, the approximate character of both the Zel'dovich theory and the Press-Schechter formalism implies that the proposed mass functions are only approximate. Detailed analysis of high resolution simulations with application of the approach proposed in this paper allows one to improve results obtained in Sect. 5.
Due to their high overdensity above the mean density, both filaments and walls are easily detected in the deep galaxy surveys using the Minkowski Functional approach (Schmalzing et al. 1999; Kerscher 2000) and the well known friend-of-friend method generalized in the Minimal Spanning Tree technique (Barrow et al. 1985; van de Weygaert 1991; Doroshkevich et al. 2001, 2002). The mass functions of the observed structure elements can be compared with the expected ones. For simulations the same comparison can be performed at all redshifts what allows one to trace the expected redshift dependence of these functions.
In Sect. 5.7 for the first time we compare the expected mass functions with simulated ones. In Doroshkevich et al. (2004) the expected mass functions are compared with the observed ones for the LSS elements selected with various threshold overdensity from the SDSS EDR. These results show that both relations (48) and (55) describe quite well even the mass distribution of filaments and walls with a moderate richness. Stronger disagreement appears for the richest walls and filaments formed due to the process of percolation that is not described by the Zel'dovich theory.
However, the potential of both approaches is limited as they cannot describe incorporation of filaments and walls into a joint network (percolation process) and the final disruption of collapsed clouds what leads to formation of numerous low mass satellites of the central object. Neither the Press-Schechter formalism nor the Zel'dovich theory can describe the faster evolution of clouds accumulated by richer walls as compared with isolated clouds. As was discussed in Sect. 5, both approaches do not describe the coagulation of high density clouds what distorts their mass function and makes it similar to the power low. These problems remain open for further investigations.
The analysis of redshift distribution of absorbers shows almost homogeneous spatial distribution of baryonic and DM components of the matter on scales 1 h^{-1} Mpc. At the same time, the observed spatial distribution of luminous matter is strongly inhomogeneous. Thus, about half of the observed galaxies are concentrated within large walls while, for example, within the Böotes void the number of galaxies is very small. The observed walls and voids are associated with compressed and expanded regions of the Universe and these observations point out to the possible correlations between the galaxy formation and the large scale deformation field (see, e.g., Rees 1985; Dekel & Silk 1986; Dekel & Rees 1987). Some of such correlations were already noticed in simulations (see, e.g., Sahni et al. 1994).
As was shown in Sect. 4, the Zel'dovich theory allows one to quantify this interaction and demonstrates that the formation rate of high density objects is modulated by large scale perturbations. This modulation cannot significantly change the redshift evolution of the fraction of mass accumulated by structure elements. However, this interaction accelerates the formation of high density halos and galaxies within regions associated with the future LSS elements such as clusters of galaxies, filaments and walls. These results suggest that the poorer sample of isolated galaxies and invisible DM halos situated within low density regions were formed later then similar galaxies and DM halos situated within filaments and walls. The results presented in Sect. 4.3 illustrate this influence for simulated matter distribution. This problem was widely discussed during last years (see, e.g., Hoffman et al. 1992; Little & Weinberg 1994; Mo & White 1996; Kauffmann et al. 1999a,b,c; Benson et al. 2001; Mathis & White 2002). The same effect was also found in Gottlöber et al. (2001, 2002, 2003) for simulated spatial distribution of halos and in Doroshkevich et al. (2004) for properties of observed groups of galaxies.
Numerous simulations performed recently (see, e.g., Zhang et al. 1998; Weinberg et al. 1999) demonstrate that the absorption lines observed in spectra of the farthest quasars - the Ly- forest - are related to the numerous low mass clouds formed at high redshifts. As was discussed in Demianski et al. (2003), some of the statistical characteristics of observed absorbers can be successfully explained on the basis of Zel'dovich theory. At the same time, the analysis shows that the approximate description based on results obtained in DD99 have to be essentially improved. Indeed, the absorbers are observed in a wide range of redshifts and the properties of long-lived absorbers are changing with time at least due to their transverse expansion and compression.
Discussion of this problem in Sect. 7 shows that for statistically homogeneous sample of absorbers we can neglect this influence, at least for their two important characteristics, namely, for the PDF of their surface density (67) and their mean linear number density (77). Results obtained in Sect. 7 link these characteristics with the properties of the initial power spectrum at small scale. This means that the analysis of these characteristics allows one to measure the variance of initial density perturbations and to restrict the mass of dominant fraction of dark matter particles.
Results obtained in Sect. 7 were applied to study the Lyman- forest in Demianski et al. (2003). This analysis confirms that the relations (67) and (77) describe quite well the PDF and the redshift distribution of 5000 observed absorbers. These results also verify the Gaussianity of initial perturbations. Comparison of other characteristics of pancakes derived in Sects. 3 and 7 with observations can be found in Demianski et al. (2003). The shape of small scale initial power spectrum derived from the absorber characteristics was discussed in Demianski & Doroshkevich (2003).
The relations (29) and (47) show that in the Zel'dovich theory of gravitational instability the rate of matter collapse at high redshifts depends upon both the amplitude of perturbations, , as given by Eq. (14), and the coherent length of density field, and q_{0}, as given by Eq. (4), which, in turn, depends on the shape of the initial power spectrum at large k and the mass of the dominant type of DM particles.
As was noticed in Sect. 3, the most effective matter condensation within high density clouds occurs at (see Eq. (23)) when these clouds already accumulate 2% of matter and the main fraction of mass is concentrated within clouds with .
For
we can estimate the redshift of the period
of most efficient condensation and the mean mass of DM clouds as
follows:
According to available estimates (Haiman & Loeb 1999; Loeb & Barkana 2001) the reionization of the universe occurs after 1-3% of matter is concentrated within high density halos. From Eqs. (83) and (84) it follows that this fraction of collapsed matter can be reached already at at redshifts , for . The effective reionization and reheating of the universe at such redshifts is consistent with the observed concentration of neutral hydrogen at redshifts , which characterizes mainly the rate of generation and achieved intensity of UV background. This means that these ranges of q_{0} and z_{r}provide a reasonable estimate of the period of reheating.
The observations of environment of high redshift quasars (Fan et al. 2001) provide an evidence in favor of reheating of the Universe at , what is more consistent with . However, our analysis shows that, for the standard WDM model with the Harrison - Zel'dovich large scale power spectrum and exponential cutoff caused by a mass keV of the dominant fraction of DM particles, some problems can appear with formation of low mass isolated galaxies with . Perhaps, some excess of power at small scales can help to solve this problem. First observational indications of such excess are presented in Demianski & Doroshkevich (2003).
Acknowledgements
A.D. thanks V. Lukash and E. Mikheeva for useful discussions. This paper was supported in part by Denmark's Grundforskningsfond through its support for an establishment of Theoretical Astrophysics Center and a grant 1-P03D-014-26 of the Polish State Committee for Scientific Research. This work was supported in part by EC network HPRN-CT-2000-00124. A.D. also wishes to acknowledge support from the Center for Cosmo-Particle Physics "Cosmion'' in the framework of the project "Cosmoparticle Physics''. Furthermore, we wish to thank the anonymous referee for valuable discussion and many useful comments.
In this appendix we present a few correlation and structure functions which describe the relative spatial distribution of important parameters of the initial perturbations. These functions have been introduced in DD99, where more details can be found.
The structure function of gravitational potential perturbations
characterizes correlation of the gravitational potential at two
points
and
.
As the power
spectrum is a function of only the absolute value of wave number |k|, this structure function depends on
and for the perturbations
of gravitational potential we have
The general deformation of a spherical cloud with a diameter
can be suitably characterized by the
dimensionless random scalar function
The function
is symmetrical with respect to the
replacement
and, therefore, b^{2}_{l}=0for odd l. Using the relations (A.3) we have for the relative
amplitude of even spherical harmonics, for the most interesting cases
1 and
1, respectively: