An excursion set model of hierarchical clustering: Ellipsoidal collapse and the moving barrier

TL;DR

This work extends the excursion-set theory by introducing an ellipsoidal-collapse moving barrier and deriving analytic first-crossing distributions valid for a broad class of barriers. It provides practical tools: an analytic expression for the unconditional mass function, a fast Monte Carlo method to generate halo masses, and a robust analytic form for the conditional mass function and its dependence on local density. Comparisons with N-body simulations show that the ellipsoidal moving-barrier model generally outperforms the spherical constant-barrier model, especially at high redshift, and better captures the density- and progenitor-dependent aspects of halo populations. However, discrepancies at small lookback times point to missing correlations between scales, indicating that incorporating scale correlations is essential for fully accurate merger histories; the framework nonetheless offers a flexible, generalizable analytic approach to halo statistics and to non-cold initial conditions such as warm dark matter.

Abstract

The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately. The predictions of this approach depend on how the nonlinear processes of collapse and virialization are modelled. We present simple analytic approximations which allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse. In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today. We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times. Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function which quantifies the tendency for massive haloes to populate denser regions than less massive haloes. This happens because low density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high density regions are collapsed haloes viewed at small lookback times. Although we have only applied our analytic formulae to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers. We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold.

Figures (16)

• Figure 1: First crossing distributions and the universal unconditional halo mass function. Histogram shows the distribution obtained by simulating random walks which are absorbed on the ellipsoidal collapse moving barrier; solid curve shows our analytic approximation to this distribution (equation \ref{['taylors']}). Dashed curve shows the distribution which fits the halo mass function in cosmological simulations (equation \ref{['giffit']}), and dotted curve shows the distribution associated with spherical collapse.
• Figure 2: Examples of random walks used to construct the conditional mass functions associated with the ellipsoidal collapse moving barriers at $z=0$ (dotted curve) and $z=2$ (solid curve).
• Figure 3: Excursion set conditional mass functions at $z$, for parent haloes identified at the present time. The two sets of curves in each panel are for parent haloes in the mass range $1\le M/M_*\le 2$ (upper curves) and $16\le M/M_*\le 32$ (lower curves); the upper curves have been shifted upwards by a factor of ten for clarity. Symbols with error bars show the distributions measured in the $\Lambda$CDM simulations, histograms show the result of generating the first crossing distribution by simulating an ensemble of $10^4$ random walks, smooth solid curves show the analytic approximation discussed in the text, and dotted curves show the distribution associated with barriers of constant height.
• Figure 4: Conditional mass functions in the SCDM simulations (symbols with error bars). The two sets of curves in each panel show parent haloes with masses in the range $1\le M/M_*\le 2$ (upper curves) and $16\le M/M_*\le 32$ (lower curves); the upper curves have been shifted upwards by a factor of ten for clarity. Dotted curves show the spherical collapse prediction (Lacey & Cole 1993), dashed curves show the distribution one gets by rescaling the unconditional mass function (equation \ref{['giffit']}), and solid curves show our analytic approximation to the random walk with moving-barrier simulations (equation \ref{['fsS']}).
• Figure 5: Mass functions as a function of local density in the SCDM simulations (symbols with error bars), plotted as a function of relative mass, so as to resemble the conditional mass functions presented earlier. Dotted curves show the spherical collapse prediction, dashed curves show the distribution one gets by rescaling the unconditional mass function (equation \ref{['giffit']}), and solid curves show our analytic approximation to the random walk with moving-barrier simulations (equation \ref{['fsS']}). The curves have been offset upwards by a factor of ten and a hundred, in the case of the middle and topmost curves, respectively. The upper most curves show the densest cells.