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Register a new userA Stochastic Theory of the Hierarchical Clustering II. Halo progenitor mass function and large-scale bias
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We generalize the stochastic theory of hierarchical clustering presented in paper I by Lapi & Danese (2020) to derive the (conditional) halo progenitor mass function and the related large-scale bias. Specifically, we present a stochastic differential equation that describes fluctuations in the mass growth of progenitor halos of given descendant mass and redshift, as driven by a multiplicative Gaussian white noise involving the power spectrum and the spherical collapse threshold of density perturbations. We demonstrate that, as cosmic time passes, the noise yields an average drift of the progenitors toward larger masses, that quantitatively renders the expectation from the standard extended Press & Schechter (EPS) theory. We solve the Fokker-Planck equation associated to the stochastic dynamics, and obtain as an exact, stationary solution the EPS progenitor mass function. Then we introduce a modification of the stochastic equation in terms of a mass-dependent collapse threshold modulating the noise, and solve analytically the associated Fokker-Planck equation for the progenitor mass function. The latter is found to be in excellent agreement with the outcomes of $N-$body simulations; even more remarkably, this is achieved with the same shape of the collapse threshold used in paper I to reproduce the halo mass function. Finally, we exploit the above results to compute the large-scale halo bias, and find it in pleasing agreement with the $N-$body outcomes. All in all, the present paper illustrates that the stochastic theory of hierarchical clustering introduced in paper I can describe effectively not only halos abundance, but also their progenitor distribution and their correlation with the large-scale environment across cosmic times.
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We present a new theory for the hierarchical clustering of dark matter (DM) halos based on stochastic differential equations, that constitutes a change of perspective with respect to existing frameworks (e.g., the excursion set approach); this work is specifically focused on the halo mass function. First, we present a stochastic differential equation that describes fluctuations in the mass growth of DM halos, as driven by a multiplicative white (Gaussian) noise dependent on the spherical collapse threshold and on the power spectrum of DM perturbations. We demonstrate that such a noise yields an average drift of the halo population toward larger masses, that quantitatively renders the standard hierarchical clustering. Then, we solve the Fokker-Planck equation associated to the stochastic dynamics, and obtain the Press & Schechter mass function as a (stationary) solution. Moreover, generalizing our treatment to a mass-dependent collapse threshold, we obtain an exact analytic solution capable of fitting remarkably well the N-body mass function over a wide range in mass and redshift. All in all, the new perspective offered by the theory presented here can contribute to better understand the gravitational dynamics leading to the formation, evolution and statistics of DM halos across cosmic times.
The description of the abundance and clustering of halos for non-Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys, which can potentially yield constraints of order unity on the non-Gaussianity parameter f_{NL}. We present tests on N-body simulations of analytical formulae describing the halo abundance and clustering for non-Gaussian initial conditions. We calibrate the analytic non-Gaussian mass function of Matarrese et al.(2000) and LoVerde et al.(2008) and the analytic description of clustering of halos for non-Gaussian initial conditions on N-body simulations. We find excellent agreement between the simulations and the analytic predictions if we make the corrections delta_c --> delta_c X sqrt{q} and delta_c --> delta_c X q where q ~ 0.75, in the density threshold for gravitational collapse and in the non-Gaussian fractional correction to the halo bias, respectively. We discuss the implications of this correction on present and forecasted primordial non-Gaussianity constraints. We confirm that the non-Gaussian halo bias offers a robust and highly competitive test of primordial non-Gaussianity.
We use the Excursion Set formalism to compute the properties of the halo mass distribution for a stochastic barrier model which encapsulates the main features of the ellipsoidal collapse of dark matter halos. Non-markovian corrections due to the sharp filtering of the linear density field in real space are computed with the path-integral technique introduced by Maggiore & Riotto (2010). Here, we provide a detailed derivation of the results presented in Corasaniti & Achitouv (2011) and extend the mass function analysis to higher redshift. We also derive an analytical expression for the linear halo bias. We find the analytically derived mass function to be in remarkable agreement with N-body simulation data from Tinker et al. (2008) with differences smaller than ~5% over the range of mass probed by the simulations. The excursion set solution from Monte Carlo generated random walks shows the same level of agreement, thus confirming the validity of the path-integral approach for the barrier model considered here. Similarly the analysis of the linear halo bias shows deviations no greater than 20%. Overall these results indicate that the Excursion Set formalism in combination with a realistic modeling of the conditions of halo collapse can provide an accurate description of the halo mass distribution.
We use a large suite of N-body simulations to study departures from universality in halo abundances and clustering in cosmologies with non-vanishing neutrino masses. To this end, we study how the halo mass function and halo bias factors depend on the scaling variable $sigma^2(M,z)$, the variance of the initial matter fluctuation field, rather than on halo mass $M$ and redshift $z$ themselves. We show that using the variance of the cold dark matter rather than the total mass field, i.e., $sigma^2_{cdm}(M,z)$ rather than $sigma^2_{m}(M,z)$, yields more universal results. Analysis of halo bias yields similar conclusions: When large-scale halo bias is defined with respect to the cold dark matter power spectrum, the result is both more universal, and less scale- or $k$-dependent. These results are used extensively in Papers I and III of this series.
Using dark matter simulations we show how halo bias is determined by local density and not by halo mass. This is not totally surprising, as according to the peak-background split model, local density is the property that constraints bias at large scales. Massive haloes have a high clustering because they reside in high density regions. Small haloes can be found in a wide range of environments which determine their clustering amplitudes differently. This contradicts the assumption of standard Halo Occupation Distribution (HOD) models that the bias and occupation of haloes is determined solely by their mass. We show that the bias of central galaxies from semi-analytic models of galaxy formation as a function of luminosity and colour is not correctly predicted by the standard HOD model. Using local density instead of halo mass the HOD model correctly predicts galaxy bias. These results indicate the need to include information about local density and not only mass in order to correctly apply HOD analysis in these galaxy samples. This new model can be readily applied to observations and has the advantage that the galaxy density can be directly observed, in contrast with the dark matter halo mass.
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