Estudio conformacional del mimético de horquilla

The mass function of dark matter haloes can be estimated by using the spherical collapse model and the statistics of the (linearly-extrapolated) initial overdensity

the scale, the smaller the variance of fluctuations (Figure 2.2). Intuitively one can think of this as follows. For the very largest scales, smoothed fluctuations are very small, and the densities of such large regions approach the background density of the universe. For these smoothing scales, the distribution of overdensities approaches a Dirac delta. If the same field is smoothed at smaller scales, perturbations from the mean become more evident. With this higher resolution, a greater diversity in overdensities results, making the width of their distribution (the variance) broader. Figure 2.4 illustrates this, showing the same initial field as a function of ~x, times D(t0)/D(ti), smoothed at two different scales. On average, the peaks and troughs of the smoothed field with the smaller filter tend to be more pronounced than those with the larger filter.

Now pick a point~x at random early in the universe (at timet =ti). Pick a very large sphere and smooth this field with a sharp-k space filter. Gradually shrink the radius of this filter, and smooth the field to increasingly smaller regions. Then linearly extrapolate the result to the present. One such process is shown in Fig- ure 2.7. For this particular point, the smoothed overdensity enclosed in the filter’s radius tends to increase as the region gets smaller. However, for some portions the opposite happens, indicating that a dense region at one scale can itself be embedded in a rather underdense region, a void.

For comparison, Figure 2.3 shows the smoothed field at only two scales and several ~x. Figure 2.7, on the other hand, shows to only one point~x, and it shows

Figure 2.7: The smoothed overdensity field at various scales. The jagged line re- pressents the run of smoothing the initial overdensity field with a sharp-k space filter in spheres of increasing radius R around a randomly chosen point ~x. The dotted horizontal lines are the spherical collapse thresholds at two redshifts.

the value of the smoothed field at all scales. Choosing a different point ~x in the universe would yield a different jagged trajectory. For this particular location, we ended up with an overdensity as R _→ 0. In other words, this point is found in an overdense patch of the initial density field. Had we picked a point in an underdense patch, the jagged line would have ended at a negative height at small R. In both situations, the jagged lines deviate more from zero (in either direction), on average, as the radius of the smoothing filter decreases.

In Figure 2.7, we have included the present-time critical collapse threshold, and that associated with a higher redshift (dotted horizontal lines). Take, for instance, the lower of the two. Notice that the jagged line crosses this barrier at several points. Identifying all the scales as collapsed objects would miscount regions that are embedded in larger collapsed regions. This is the so-called ‘cloud-in cloud’ problem (Epstein, 1983; Bardeen et al., 1986; Peacock & Heavens, 1990; Jedamzik, 1995; Sheth, 1995; Avelino & Viana, 2000). The excursion set formalism of Bond et al. (1991) solves this by selecting only the largest of such regions as the collapsed object. In other words, the largest scale that crosses the barrier carries with it information about the mass of the virialized halo surrounding the point ~x. Notice that the higher the barrier (z > 0), the smaller the value of R is at which this happens. In other words, haloes tend to be less massive at earlier times, in agreement with the hierarchical picture of gravitational clustering.

Figure 2.8: The smoothed overdensity field with increasing variance. This is the same as Figure 2.7, but in terms of the variance S, which increases with decreasing R. The jagged line is a Brownian motion random walk.

filter. A more natural choice is to use the variance S = σ2(R) itself, which truly describes how fluctuations deviate from zero at each scale. This process is described in Figure 2.8. Since large scales correspond to small values ofS, virialized regions are identified with the first place where the jagged line upcrosses the collapse barrier. The excursion set approach maps the fraction of random trajectories that first upcross a barrier of height δc(z) between S and S+dS with the fraction of mass at redshift z in haloes with mass between m and m+dm:

f(Sm|δc(z))dSm =f(m|z)dm. (2.58)

This situation is illustrated in Figure 2.9, where the red solid circle indicates where the random trajectory first crosses the barrier of height δc(z). Very few random walks will upcross this barrier immediately – most of them will wander around before crossing. Since S is a monotonically decreasing function of mass, most of the mass will be in low-mass haloes. At higher redshift, the height of the barrier increases, making it more difficult for walks to cross it. This means that most haloes will, on average, have smaller masses. This is consistent with the idea that smaller haloes formed before massive ones.

Finding this crossing distribution is nontrivial. Nevertheless, having a sharp-k space filter makes matters much easier. With such a filter, the action of replacing the smoothing scale as R → R+dR amounts to changing the size of the top-hat filter in Fourier space from somekTHtokTH+dk. The Fourier modes that contribute to this new scale come only from a thin spherical shell of width dk. This means

Figure 2.9: The first crossing of a barrier by a random walk. The jagged trajectory denotes a Brownian random walk and the horizontal dotted line is a constant barrier. The solid red circle denotes the location where the walk first upcrosses the barrier.

that the new contribution carries no information on the smaller scales, making consecutive scales uncorrelated. In other words, the steps in these trajectories are uncorrelated, and such trajectories are true Brownian random walks. Random walks – also known as excursion sets – have been studied in great detail for many years in pure mathematics (where they are known as Wiener processes, Grimmett & Stirzaker, 2001), and have been applied to subjects as diverse as genetics, gambling, colloids, neurology, stellar dynamics, and finance. Bachelier (1900) was the first to compute the first crossing distribution of a constant barrier. The next section is devoted to the derivation of this result.