Ellipsoidal collapse and an improved model for the number and spatial distribution of dark matter haloes

TL;DR

This work addresses shortcomings of the standard spherical Press--Schechter/excursion set framework by incorporating ellipsoidal collapse, which introduces a mass-dependent moving barrier. A simple fitting form δ_ec(σ,z) = δ_sc(z)[1 + β (σ^2/σ_*^2(z))^γ] with β=0.47 and γ=0.615 (σ_* ≡ δ_sc) allows the excursion set approach to capture the halo mass function and large-scale bias more accurately in simulations. The authors show that using this moving barrier yields substantially better object-by-object mass predictions and reproduces the GIF mass function with a comparable barrier form, as well as a realistic bias relation with an upturn at low masses. The results provide a practical analytic tool for predicting halo abundances and clustering, improving cosmological parameter constraints from large-scale structure analyses.

Abstract

The Press--Schechter, excursion set approach allows one to make predictions about the shape and evolution of the mass function of bound objects. It combines the assumption that objects collapse spherically with the assumption that the initial density fluctuations were Gaussian and small. While the predicted mass function is reasonably accurate at the high mass end, it has more low mass objects than are seen in simulations of hierarchical clustering. This discrepancy can be reduced substantially if bound structures are assumed to form from an ellipsoidal, rather than a spherical collapse. In the spherical model, a region collapses if the initial density within it exceeds a threshold value, delta_sc. This value is independent of the initial size of the region, and since the mass of the collapsed object is related to its initial size, delta_sc is independent of final mass. In the ellipsoidal model, the collapse of a region depends on the surrounding shear field, as well as on its initial overdensity. Therefore, there is a relation between the density threshold value required for collapse, and the mass of the final object. We provide a fitting function to this delta_ec(m) relation for initially Gaussian fields which simplifies the inclusion of ellipsoidal dynamics in the excursion set approach. We discuss the relation between the excursion set predictions and the halo distribution in high resolution N-body simulations, and show that our simple parametrization of the ellipsoidal collapse model represents a significant improvement on the spherical model on an object-by-object basis. Finally, we show that the associated statistical predictions, the mass function and the large scale halo-to-mass bias relation, are also more accurate than the standard predictions.

Figures (6)

• Figure 1: The evolution of an ellipsoidal perturbation in an Einstein-de Sitter universe. Symbols show the expansion factor when the longest axis collapses and virializes, as a function of initial $e$ and $p$, in steps of 0.025, if the initial overdensity was $\delta_i$. The solid curve shows our simple formula for the $p=0$ result, and the dashed curves show $|p|=e/2$. The time required to collapse increases mononically as $p$ decreases. The axis on the right shows the associated critical overdensity required for collapse, and the axis on the top shows the result of using our simple formula to translate from $e$ to $\sigma(m)$ when $p=0$.
• Figure 2: The mass of the halo in which a randomly chosen particle is, $M_{\rm halo}$, is plotted versus the mass predicted by the spherical (left panel) and ellipsoidal collapse (right panel) models. A randomly chosen $10^4$ of the $10^6$ particles in a simulation of an Einstein-de Sitter universe with white noise initial conditions were used to make the plot.
• Figure 3: The mass of a halo in a simulation of an Einstein-de Sitter universe with white noise initial conditions versus that predicted by the excursion set approach. The panel on the left shows the prediction associated with the 'standard' spherical collapse approximation to the dynamics; the panel on the right shows the prediction associated with our moving barrier parametrization of the ellipsoidal collapse model.
• Figure 4: The effect of changing $p$ at a given $e$ on the predicted mass of a halo: as $p$ becomes more negative(positive), $\delta_{\rm ec}(e,p)$ increases(decreases), so the predicted mass decreases(increases). The filled circles show the $p=0$ prediction used to produce the previous figure, and the bars show the range $|p|=0.33\,e$. The two panels on the top show the result for white noise initial conditions, and the bottom panels were constructed from simulations in which the slope of the initial power spectrum was $n=-1.5$.
• Figure 5: The large scale bias factor $b(m)$ as a function of halo mass. Dotted curves show a fit to this relation measured in numerical simulations by Jing (1998), though his Figure 3 shows that the bias factor for massive haloes in his simulations is slightly smaller than the one given by his fitting function. Dashed curves show the spherical collapse prediction of Mo & White (1996), and solid curves show the elliposidal collapse prediction of this paper. At the high mass end, our solid curves and the simulation results differ from Jing's fitting function (dotted) in the same qualitative sense.