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The Mass Function of Dark Halos in Sup erclusters and Voids
E. P. Kurbatov1
1 ,*

Institute of Astronomy of the Russian Academy of Sciences

A mo dification of the Press­Schechter theory allowing for presence of a background large-scale structure (LSS) ­ a sup ercluster or a void is prop osed. The LSS is accounted as the statistical constraints in form of linear functionals of the random overdensity field. The deviation of the background density within the LSS is interpreted in a pseudo-cosmological sense. Using the constraints formalism may help us to prob e non-trivial spatial statistics of halo es, e.g. edge and shap e effects on b oundaries of the sup erclusters and voids. Parameters of the constraints are connected to features of the LSS: its mean overdensity, a spatial scale and a shap e, and spatial momenta of higher orders. It is shown that presence of a non-virialized LSS can lead to an observable deviation of the mass function. This effect is exploited to build a pro cedure to recover parameters of the background p erturbation from the observationally estimated mass function. PACS numb ers:

1. INTRODUCTION Mass function of Press & Schechter [18] (hereafter PS) for dark halo es depends on global cosmological parameters so do es not incorporate the presence of the background large-scale disturbances which can form superclusters and voids. Presence of the LSS alter the matter density and can change the growth rate of cosmological fluctuations. This problem was first considered by Bond et al. [2] and Bower [3] as the `two barrier' problem and improved by Mo & White [14], Sheth & Tormen [26], and others. These mo dels provided a connection between the mass function and the background overdensity making possible to solve the inverse problem of recovering the last, as was done by Munoz & Lo eb [15] and Sheth & Diaferio ~ [23]. In these mo dels, however, the background perturbation was considered spherical. Also
*

Electronic address: kurbatov@inasan.ru


2 there was not provided a possibility to examine effects of transition between structures of different densities, e.g. between superclusters and voids. In this paper a mo dification of the PS theory is proposed allowing for a background perturbation (name it `host') which is asso ciated with the supercluster or the void. The host perturbation is defined by a set of statistical constraints for the cosmological perturbation field. Growth rate of the random perturbations depends on the host overdensity and determines dynamical age of the population of halo es. As a demonstration of applicability of the mo del the inverse problem is considered in a simple case for recovering the host overdensity. Below, in 2nd Section described the mo dification of the PS theory. In the 3rd Section considered an application of the theory to superclusters and voids. Benefits and issues are discussed in the 4th Section.

2. PRESS­SCHECHTER THEORY WITH STATISTICAL CONSTRAINTS 2.1. Outline The key element of the excursion set theory is the assumption that the virialized halo es form in a hierarchical sto chastic pro cess of absorbtion of halo es of lower masses. Herewith, only those halo es are counted for mass function which are on a top level of a hierarchical tree, i.e. are not sub-halo es of any other halo. The sto chastic nature of the pro cess is governed by properties of the initial cosmological fluctuations. Conditions for fluctuations to form a virialized ob ject may be implied to linear stage of their evolution, and with a go o d approximation they are conditions for the filtered overdensity only. A halo condensed when its characteristical linearly evolved overdensity exceeded some threshold value given from the spherical collapse mo del. Defining the statistical properties of the fluctuations, their growth rate, and the threshold overdensity allows for completely determining of the function of mass of the halo es for any redshift [2, 11]. The idea is necessary for this research to be done is to describe a large-scale perturbation using formalism of statistical constraints in such a way so that would be possible to define various parameters of the host, e.g. volume averaged density, spatial scale, momenta of inertia etc. The cosmological fluctuations, from which the sub-structures developed, should evolve on top of the background mo dified by the host. Let's define as the residual between a total overdensity field and overdensity of the host (which is the ensemble mean for ,


3 and will be defined in Subsec. 2.2.2): (r) (r) - (r) , (1 )

where r is the spatial position. Also define the field which is filtered over a volume with characteristic scale R: (r, R) d3 r W (|r - r |, R) (r ) , (2 )

where W (r, R) is the filtering function. In a simple approximation all the interesting information on the features of the host contains in variance of the filtered field (r, R) 2 (r, R) . According to excursion set theory, the hierarchical halo formation pro cess can be interpreted as a random pro cess for the overdensity filtered over the scale R acting as a parameter of the random pro cess. The pro cess starts at the parameter R = moving toward R = 0. The mass function can be expressed then via the distribution function for the parameter values when the pro cess first crossed a certain threshold. In presence of the constraints the statistical homogeneity and isotropy of the fluctuations field may be broken and spectral mo des of the fluctuations may have non-trivial correlations. Because of this, the excursion set approach in its standard form may not work. This issue will be touched in Subsection 2.2.5. In theories of PS class the evolution of amplitude of the fluctuations can be accounted either as a time-dependent overdensity threshold or as evolution of the variance. We will stick to the second scheme since it makes possible, in principle, to consider the non-linear corrections to the field evolution. Exact calculation for evolution of the fluctuations' amplitude is difficult even in lowest orders of the perturbation series approach. For Gaussian field these calculations were completed just up to 1-lo op corrections, i.e. to 2-nd order of accuracy in power spectrum [10], or up to 3-rd order in expressions for filter