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Astron. Astrophys. 333, 399-410 (1998)

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Appendix

A.1. Initial density fields

In hierarchical clustering models, structure forms by the gravitational growth of small inhomogeneities in the density field. The fluctuations in the density are conveniently characterised by the density contrast

[EQUATION]

where [FORMULA] is the mean density of the universe. In an Einstein-de Sitter universe, a region with a mean density contrast of zero is marginally bound by gravity. Regions of space where the density contrast is positive are bound and will eventually collapse. At early times [FORMULA] and gravitational dynamics can be handled by linear theory. The mathematical formulation for this is laid out in detail by Bardeen et al. (1986), BBKS henceforth.

To single out sites where objects of mass M are likely to form, the density contrast is first smoothed on the corresponding length scale

[EQUATION]

where

[EQUATION]

s is the smoothing length scale and is given by s [FORMULA] [FORMULA]. If [FORMULA] is a Gaussian random field then [FORMULA] is also a Gaussian random field. The power spectrum of [FORMULA] is given by [FORMULA], where

[EQUATION]

and [FORMULA] is the Fourier transform of the density contrast, and [FORMULA] is the mean value. A peak in the field [FORMULA] is a point where the mass inside a spherical region of size s has a maximum. Such points are plausible sites for the formation of objects of mass M.

Hoffman et al. (1992) extended the formalism of BBKS to investigate the influence of the background density field on a given smaller scale peak. By proceeding in a similar manner we consider two Gaussian random fields, a peak field [FORMULA], and a background field [FORMULA], which are defined by their power spectra

[EQUATION]

and

[EQUATION]

The combined field [FORMULA] is statistically identical to [FORMULA], the density contrast smoothed on a scale s. By requiring that [FORMULA], extremum points in [FORMULA] will also be extrema of [FORMULA]. When smoothing on galactic scales, maxima in the density field are assumed to be progenitors of galaxies.

Dimensionless field values, [FORMULA], are conveniently expressed in units of [FORMULA], i.e., [FORMULA], where

[EQUATION]

is the mean square density contrast (BBKS). Galaxies are believed to form around peaks of height [FORMULA], assuming the density field has been smoothed on an appropriate galactic scale. Specifying an equivalent value [FORMULA] for the background field, at the same point, determines the density field in a larger surrounding of the peak. [FORMULA], [FORMULA] and [FORMULA], are related by (Hoffman et al. 1992)

[EQUATION]

Inside radius s, the statistical properties of the density field are primarily determined by the small scale peak constraint. But, further away from the peak, the statistical properties depends on the value of [FORMULA]. The main effect of the background field is to change the overall density in the vicinity of the smaller scale peak.

The approach then, is to set up a density field [FORMULA], with the constraint that the corresponding field [FORMULA] should have a maximum at the center of the simulated region. The small scale field [FORMULA] is constrained by

[EQUATION]

To ensure that the field has an extremum at the center, it is also required to have vanishing first derivatives there. These constraints specifies the statistical properties of the field. The second derivative of the field could also have been used as a constraint, to ensure that the central extremum is in fact a maximum. This is neglected since it would be a largely redundant constraint. It is very unlikely that a [FORMULA] extremum, as is the case here, is a minimum, and therefore the statistical properties of the field are only marginally affected by adding the contrary as a constraint.

Hoffmann & Ribak (1992) showed how to create a random realization of a Gaussian field, that is subject to constraints that are linear in the field. The method is computationally efficient, and it is easy to handle several constraints. Any quantities, or combinations of, that can be represented within linear theory, can be used as constraints for the field. In addition to the "seeding' peak itself, we have found it useful to also specify [FORMULA], i.e. requiring that the "peak" is stationary. This increases the likelihood that the forming object stays in the center of the simulation volume.

The field is set up in a cubic region of size [FORMULA], where [FORMULA]. The largest wavelengths that can be represented is therefore [FORMULA]. The values used for s are shown in Table A1. The smoothed density field is then split into [FORMULA] and [FORMULA], in such a way that [FORMULA] contains all wavelengths that can be represented in the cube, and [FORMULA] contains all longer wavelengths.


[TABLE]

Table A1. Simulation parameters.


The background field cannot be accurately represented inside the cubic region, where the density field is constructed, since it consists of modes with wavelengths exceeding the size of the region. These long wavelength modes can still have important dynamical effects, because, by changing the overall density inside the cube, these modes change the collapse times for structures in the region. A constant density contrast of [FORMULA] was added to the region to approximate the lowest order effects of the background field. All wavelengths longer than the expansion volume will be assumed to give a constant shift of the density.

There is, however, by no means a one-to-one correspondence between such density peaks and the galaxies that later form (Katz et al. 1993). Objects that form when peaks collapse at a high redshift can at later times merge with other collapsed objects, and form new, more massive, objects. Peaks in the density field at a high redshift will then not be in a one to one correspondence with the galaxies that later form. This is a nonlinear dynamical process that cannot be handled by linear theory. N-body calculations have shown that a significant number of the galactic halos that form, do in fact correspond to peaks in the early density field. It may be argued that the "peaks formalism" would generate atypical initial conditions. This should not be the case however, as long as the parameters are within statistically reasonable limits, i.e. [FORMULA]. The generated field would still be a statistically typical one.

The gravitational particle smoothing used in the simulations are shown in Table A1.

A.2. Two-body heating

There have been recent results showing that there can be undesired, artificial heating of the gas component, due to two-body relaxation like effects, caused by the dark matter halo (Steinmetz & White 1997). This could be the case for our simulations, where the dark matter particles are of higher mass. (i.e. the simulations of [FORMULA] and [FORMULA] total mass.) The effect would be that, e.g., the cooling flow might be reduced. In the hot gas halo, the cooling times are on the other hand very long, as can be seen from Fig. A1, and it cannot cool within a Hubble time. Corresponding temperatures are shown in Fig. A2. Only in the absolute center, (i.e. the disk), the cooling times are short. A few particles just within 10 kpc seems to be possibly prevented from cooling, so we cannot rule out that there are some two-body heating effects. We doubt, however, that a large fraction hot gas halo, is being held at virial temperatures, (or was formed), solely due to two-body effects. However, we believe these two-body heating effects should be taken seriously, and avoided in future work.

[FIGURE] Fig. A1. Cooling times at [FORMULA], for the [FORMULA] run with metallicity. This simulation has the shortest cooling times, making potential two-body heating effects easier to detect. (There's equally long or longer cooling times in the [FORMULA] run. Some particles in the center, have longer cooling times. This is because they have temperatures below the "drop", [FORMULA] K, in the cooling function, and therefore have essentially no cooling.)

[FIGURE] Fig. A2. Temperature at [FORMULA], for the [FORMULA] run with metallicity.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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