An excursion set model for the distribution of dark matter and dark matter haloes

A model of the gravitationally evolved dark matter distribution, in the Eulerian space, is developed. It is a simple extension of the excursion set model that is commonly used to estimate the mass function of collapsed dark matter haloes. In addition to describing the evolution of the dark matter itself, the model allows one to describe the evolution of the Eulerian space distribution of the haloes. It can also be used to describe density profiles, on scales larger than the virial radius, of these haloes, and to quantify the way in which matter flows in and out of Eulerian cells. When the initial Lagrangian space distribution is white noise Gaussian, the model suggests that the Inverse Gaussian distribution should provide a reasonably good approximation to the evolved Eulerian density field, in agreement with numerical simulations. Application of this model to clustering from more general Gaussian initial conditions is discussed at the end.Comment: 15 pages, 5 figures, submitted to MNRAS Sept. 199

The Excursion Set Theory of Halo Mass Functions, Halo Clustering, and Halo Growth

I review the excursion set theory (EST) of dark matter halo formation and clustering. I recount the Press-Schechter argument for the mass function of bound objects and review the derivation of the Press-Schechter mass function in EST. The EST formalism is powerful and can be applied to numerous problems. I review the EST of halo bias and the properties of void regions. I spend considerable time reviewing halo growth in the EST. This section culminates with descriptions of two Monte Carlo methods for generating halo mass accretion histories. In the final section, I emphasize that the standard EST approach is the result of several simplifying assumptions. Dropping these assumptions can lead to more faithful predictions and a more versatile formalism. One such assumption is the constant height of the barrier for nonlinear collapse. I review implementations of the excursion set approach with arbitrary barrier shapes. An application of this is the now well-known improvement to standard EST that follows from the ellipsoidal-collapse barrier. Additionally, I emphasize that the statement that halo accretion histories are independent of halo environments is a simplifying assumption, rather than a prediction of the theory. I review the method for constructing correlated random walks of the density field in more general cases. I construct a simple toy model with correlated walks and I show that excursion set theory makes a qualitatively simple and general prediction for the relation between halo accretion histories and halo environments: regions of high density preferentially contain late-forming halos and conversely for regions of low density. I conclude with a brief discussion of this prediction in the context of recent numerical studies of the environmental dependence of halo properties. (Abridged)Comment: 62 pages, 19 figures. Review article based on lectures given at the Sixth Summer School of the Helmholtz Institute for Supercomputational Physics. Accepted for Publication in IJMPD. Comments Welcom

An analytical model of the large neutral regions during the late stage of reionization

In this paper we investigate the nature and distribution of large neutral regions during the late epoch of reionization. In the "bubble model" of reionization, the mass distribution of large ionized regions ("bubbles") during the early stage of reionization is obtained by using the excursion set model, where the ionization of a region corresponds to the first up-crossing of a barrier by random trajectories. We generalize this idea, and develop a method to predict the distribution of large scale neutral regions during the late stage of reionization, taking into account the ionizing background after the percolation of HII regions. The large scale neutral regions which we call "neutral islands" are not individual galaxies or minihalos, but larger regions where fewer galaxies formed and hence ionized later, and they are identified in the excursion set model with the first down-crossings of the island barrier. Assuming that the consumption rate of ionizing background photons is proportional to the surface area of the neutral islands, we obtained the size distribution of the neutral islands. We also take the "bubbles-in-island" effect into account by considering the conditional probability of up-crossing a bubble barrier after down-crossing the island barrier. We find that this effect is very important. An additional barrier is set to avoid islands being percolated through. We find that there is a characteristic scale for the neutral islands, while the small islands are rapidly swallowed up by the ionizing background, this characteristic scale does not change much as the reionization proceeds.Comment: 33 pages, 11 figures, accepted by The Astrophysical Journa

A hierarchy of voids: More ado about nothing

We extend earlier work on the problem of estimating the void-volume function -- the abundance and evolution of large voids which grow gravitationally in an expanding universe -- in two ways. The first removes an ambiguity about how the void-in-cloud process, which erases small voids, should be incorporated into the excursion set approach. The main technical change here is to think of voids within a fully Eulerian, rather than purely Lagrangian, framework. The second accounts for correlations between different spatial scales in the initial conditions. We provide numerical and analytical arguments showing how and why both changes modify the predicted abundances substantially. In particular, we show that the predicted importance of the void-in-cloud process depends strongly on whether or not one accounts for correlations between scales. With our new formulation, the void-in-cloud process dramatically reduces the predicted abundances of voids if such correlations are ignored, but only matters for the smallest voids in the more realistic case in which the spatial correlations are included.Comment: 9 pages, 3 figures; v2 -- improved Eulerian void-assignment algorithm, new figures (including LCDM walks) and clarified discussion. Conclusions regarding walks with correlated steps unchanged. Accepted in MNRA

A hierarchy of voids: Much ado about nothing

We present a model for the distribution of void sizes and its evolution in the context of hierarchical scenarios of gravitational structure formation. We find that at any cosmic epoch the voids have a size distribution which is well-peaked about a characteristic void size which evolves self-similarly in time. This is in distinct contrast to the distribution of virialized halo masses which does not have a small-scale cut-off. In our model, the fate of voids is ruled by two processes. The first process affects those voids which are embedded in larger underdense regions: the evolution is effectively one in which a larger void is made up by the mergers of smaller voids, and is analogous to how massive clusters form from the mergers of less massive progenitors. The second process is unique to voids, and occurs to voids which happen to be embedded within a larger scale overdensity: these voids get squeezed out of existence as the overdensity collapses around them. It is this second process which produces the cut-off at small scales. In the excursion set formulation of cluster abundance and evolution, solution of the cloud-in-cloud problem, i.e., counting as clusters only those objects which are not embedded in larger clusters, requires study of random walks crossing one barrier. We show that a similar formulation of void evolution requires study of a two-barrier problem: one barrier is required to account for voids-in-voids, and the other for voids-in-clouds. Thus, in our model, the void size distribution is a function of two parameters, one of which reflects the dynamics of void formation, and the other the formation of collapsed objects.Comment: 23 pages, 9 figures, submitted to MNRA

Why Do Stars Form In Clusters? An Analytic Model for Stellar Correlation Functions

Recently, we have shown that if the ISM is governed by super-sonic turbulent flows, the excursion-set formalism can be used to calculate the statistics of self-gravitating objects over a wide range of scales. On the largest self-gravitating scales ('first crossing'), these correspond to GMCs, and on the smallest non-fragmenting self-gravitating scales ('last crossing'), to protostellar cores. Here, we extend this formalism to rigorously calculate the auto and cross-correlation functions of cores (and by extension, young stars) as a function of spatial separation and mass, in analogy to the cosmological calculation of halo clustering. We show that this generically predicts that star formation is very strongly clustered on small scales: stars form in clusters, themselves inside GMCs. Outside the binary-star regime, the projected correlation function declines as a weak power-law, until a characteristic scale which corresponds to the characteristic mass scale of GMCs. On much larger scales the clustering declines such that star formation is not strongly biased on galactic scales, relative to the actual dense gas distribution. The precise correlation function shape depends on properties of the turbulent spectrum, but its qualitative behavior is quite general. The predictions agree well with observations of young star and core autocorrelation functions over ~4 dex in radius. Clustered star formation is a generic consequence of supersonic turbulence if most of the power in the velocity field, hence the contribution to density fluctuations, comes from large scales. The distribution of self-gravitating masses near the sonic length is then imprinted by fluctuations on larger scales. We similarly show that the fraction of stars formed in 'isolated' modes should be small (\lesssim10%).Comment: 8 pages, 3 figures, accepted to MNRAS (minor revisions to match accepted version

An Excursion-Set Model for the Structure of GMCs and the ISM

The ISM is governed by supersonic turbulence on a range of scales. We use this to develop a rigorous excursion-set model for the formation and time evolution of dense gas structures (GMCs, massive clumps, and cores). Supersonic turbulence drives the density distribution to a lognormal with dispersion increasing with Mach number; we generalize this to include scales >h (the disk scale height), and use it to construct the statistical properties of the density field smoothed on a scale R. We then compare conditions for self-gravitating collapse including thermal, turbulent, and rotational support. We show this becomes a well-defined barrier crossing problem. As such, an exact 'bound object mass function' can be derived, from scales of the sonic length to above the disk Jeans mass. This agrees remarkably well with observed GMC mass functions in the MW and other galaxies; the only inputs are the mass and size of the galaxies (to normalize the model). This explains the mass function cutoff and its power-law slope (close to, but shallower than, -2). The model also predicts the linewidth-size and size-mass relations of clouds and the dependence of their residuals on surface density/pressure. We use this to predict the spatial correlation function/clustering of clouds and star clusters; these also agree well with observations. We predict the size/mass function of ISM 'bubbles' or 'holes', and show this can account for observed HI hole distributions without any local feedback. We generalize the model to construct time-dependent 'merger/fragmentation trees' which can be used to follow cloud evolution and construct semi-analytic models for the ISM. We provide explicit recipes to construct the trees. We use a simple example to show that, if clouds are not destroyed in ~1-5 crossing times, then all ISM mass would be trapped in collapsing objects even if the large-scale turbulence were maintained.Comment: 21 pages, 11 figures, accepted to MNRAS (revised to match accepted version; predictions for high-redshift galaxies added