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Using this type of simulation enables one to examine the velocity field of galaxies and DM independently without an ad hoc assumption of bias between galaxies and DM. Accepting the fact that the existing peculiar velocity data do not allow us to compute the ideally defined as in OS90 , they defined a modified Mach number that incorporates the observational errors in measured distances due to the scatter in the Tully-Fisher relation. They constructed a mock catalog of the observations using SCDM hydrodynamical simulations similar to those that were used by SCO92 and calculated their modified from them. S93 obtained smaller than did OS90 because they included all velocity components on scales less than the bulk flow into the velocity dispersion, whereas OS90 erased the small-scale dispersion by smoothing. As a consequence, the estimates of S93 on are larger, resulting in a smaller. Since on a certain scale R, a larger implies a smaller if the variation of V is weaker than that of.

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Using this type of simulation enables one to examine the velocity field of galaxies and DM independently without an ad hoc assumption of bias between galaxies and DM. Accepting the fact that the existing peculiar velocity data do not allow us to compute the ideally defined as in OS90 , they defined a modified Mach number that incorporates the observational errors in measured distances due to the scatter in the Tully-Fisher relation.

They constructed a mock catalog of the observations using SCDM hydrodynamical simulations similar to those that were used by SCO92 and calculated their modified from them. S93 obtained smaller than did OS90 because they included all velocity components on scales less than the bulk flow into the velocity dispersion, whereas OS90 erased the small-scale dispersion by smoothing.

As a consequence, the estimates of S93 on are larger, resulting in a smaller. Since on a certain scale R, a larger implies a smaller if the variation of V is weaker than that of. We ask ourselves in this paper how typical such a cold region of space would be in the entire distribution of the velocity field. A closely related quantity is the pairwise velocity dispersion 12, which has been much studied due to its cosmological importance in relation to the "cosmic virial theorem.

This is because 12 is a pair-weighted statistic and is heavily weighted by the objects in the densest regions.

Inclusion or exclusion of even 10 galaxies from the Virgo Cluster can change 12 by km s-1, and the correction for the cluster infall also affects the result significantly. To overcome this problem, alternative statistics have been suggested. They analytically showed that 12 is heavily weighted by the densest regions of the sample. In the same spirit, Strauss et al. As we will show in this paper, the above statement for 12 holds for the velocity dispersion as well: it is heavily weighted by the densest regions.

Their results can also be used to estimate m. Since the original work of OS90 , many things have changed. The resolution and the accuracy of simulations have increased significantly due to increased computer power and more realistic modeling of galaxy formation.

The favored cosmology shifted from SCDM to LCDM -dominated flat cold dark matter model , as more and more modern observational data suggest a flat low- m universe with a cosmological constant e. Thus, we are motivated to study this statistic again using a state-of-the-art LCDM hydrodynamical simulation, which allows us to treat baryons and dark matter separately without invoking an ad hoc bias parameter.

All velocities in this paper are presented in the cosmic microwave background frame, and the velocity dispersion is three-dimensional i. The present age of the universe with these parameters is It contains dark matter particles, each weighing 5. The code is implemented with a star formation recipe summarized in Appendix A.

It turns a fraction of the baryonic gas in a cell into a collisionless particle hereafter "galaxy particle" in a given time step once the following criteria are met simultaneously see Appendix A : 1 the cell is overdense, 2 the gas is cooling fast, 3 the gas is Jeans unstable, and 4 the gas flow is converging into the cell.

Each galaxy particle has a number of attributes at birth, including position, velocity, formation time, mass, and initial gas metallicity. The galaxy particle is placed at the center of the cell after its formation with a velocity equal to the mean velocity of the gas and followed by the particle mesh code thereafter as a collisionless particle in gravitational coupling with DM and gas.

Galaxy particles are baryonic galactic subunits with masses ranging from to M ; therefore, a collection of these particles is regarded as a galaxy.

Feedback processes such as ionizing UV, supernova energy, and metal ejection are also included self-consistently. The effect of the cosmological constant on the term 1. We can then later take the ensemble average by This method allows us to observe the distribution of the Mach number before we take the ensemble average.

In the next section, we show the differences between the different definitions. This lack of long-wavelength perturbations results in an underestimate of the bulk flow, as it is determined by the perturbations on scales larger than the patch size R. In particular, for the currently popular low m models, the peak of the power spectrum lies at scales larger than h-1 Mpc. However, the rms bulk flow can be calculated correctly by the above equation at large enough scales.

The solid and the dashed lines in Figure 1 show the predicted rms Mach number calculated from equations 1 , 2 , and 3 the latter definition. The solid line is calculated by using the P k obtained from the COSMICS package, 1 which was also used to generate the initial conditions of our simulation.

This nonlinear P k is known to provide a good fit to the observed optical galaxy power spectrum Peacock The nonlinear P k has more power on small scales and, therefore, predicts smaller than the linear P k due to larger velocity dispersion. The scale dependence of the predicted for the case of the full linear P k can be well fitted by a power law R The slope becomes shallower than this on smaller scales R 10 h-1 Mpc for the nonlinear P k case.

The power index steepens to The nonlinear P k was evolved from an empirical double power-law linear spectrum that is known to provide a good fit to the observed optical galaxy power spectrum Peacock The top two lines are calculated using the full P k and the bottom two using the cutoff P k at h-1 Mpc to show the effect of the limited simulation box size.

The three vertical lines indicate the range of simulated raw values of before the correction of the bulk flow for the lack of long-wavelength perturbations.

On all scales, many of the simulated are smaller than the theoretical prediction by a factor of about 1. The source of this slight discrepancy between the simulated and the predicted is probably due to the use of the linear theory equations; i.

We wish to correct our simulated values of V and for the lack of long-wavelength perturbations, but this is not a trivial task Strauss et al. We first followed the method of Strauss et al. In Figure 2 , we show the distributions of the simulated bulk flow before and after this process, calculated with the grouped galaxy velocities.

The raw simulated bulk flow is shown by the short-dashed histogram. The solid histogram is the one after the addition of the random Fourier components. The dotted histogram is obtained by simply multiplying the raw simulated bulk flow by the numerical factors of 1. The smooth curves are the "eyeball" fits to the histograms by a Maxwellian distribution. The raw simulated bulk flow is shown by the dashed histogram.

The smooth curves are the eyeball fits to the histograms by the Maxwellian distribution. See text for discussion. We find that the change in the distribution is fairly well approximated by simply multiplying a numerical factor to the raw simulated bulk flow. We also confirm that the distribution does not change very much on the -V plane when the random Fourier components of the bulk flow are added.

Another thing is that the method of Strauss et al. On the other hand, if we simply take the ratio of the two rms Mach numbers calculated with the full P k and the cutoff P k Fig. For these reasons, we choose to correct for the lack of long-wavelength perturbations in the latter manner, as it is sufficient for our purpose.

For the dashed lines [nonlinear P k case], the ratios are slightly smaller: 1. These factors are larger than the factors obtained by adding the random Fourier components. However, even if these correction factors turned out to be overestimates, our conclusion would be strengthened in that case, because our corrected are still well below the observed.

Hereafter, we adopt the correction factors of 1. We explore various options of calculations to see if they cause any difference in. We are also interested in the difference in of different tracers of the velocity field.

There are many ways one can place the patches in the simulation. One also has to decide whether to use the particle-based ungrouped data set or to apply a grouping algorithm and identify galaxies and dark matter halos.

Here, we consider the following cases: Particle based: a Centered on grouped galaxies: use galaxy particles gal-pt , b Centered on grouped galaxies: use DM particles dm-galctr-pt , c Centered on grouped DM halos: use DM particles dm-dmctr-pt ; 2. Group based: a Centered on grouped galaxies: use grouped galaxy velocity gal-gp , b Centered on grouped DM halos: use grouped DM halo velocity dm-gp.

This cutoff is motivated by looking at Figure 3 , where grouped objects that occupy more than two cells in the simulation are shown. DM halos are not affected by the latter cutoff.

We have confirmed that the results are robust to this pruning. We are left with galaxies and DM halos after this pruning. Changing the grouping parameters certainly affects the number of objects, which in turn affects the estimate of the velocity dispersion.

Without grouping, for example, the velocity dispersion would be overestimated, as it would include the internal motions of particle in each object. But this is an unavoidable numerical uncertainty, and one should keep this in mind upon reading the results below. We pick those that lie above this boundary line as the centers of the spherical top-hat patches.

DM halos are not affected by this cutoff.

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