We use an array of high-resolution N-body simulations to determine the mass function of dark matter haloes at redshifts 10-30. We develop a new method for compensating for the effects of finite simulation volume that allows us to find an approximation to the true ``global'' mass function. By simulating a wide range of volumes at different mass resolution, we calculate the abundance of haloes of mass 10^{5-12} Msun/h. This enables us to predict accurately the abundance of the haloes that host the sources that reionize the universe. In particular, we focus on the small mass haloes (>~10^{5.5-6} Msun/h) likely to harbour population III stars where gas cools by molecular hydrogen emission, early galaxies in which baryons cool by atomic hydrogen emission at a virial temperature of ~10^4K (10^{7.5-8} Msun/h), and massive galaxies that may be observable at redshift ~10. When we combine our data with simulations that include high mass halos at low redshift, we find that the best fit to the halo mass function depends not only on linear overdensity, as is commonly assumed in analytic models, but also upon the slope of the linear power spectrum at the scale of the halo mass. The Press-Schechter model gives a poor fit to the halo mass function in the simulations at all epochs; the Sheth-Tormen model gives a better match, but still overpredicts the abundance of rare objects at all times by up to 50%. Finally, we consider the consequences of the recently released WMAP 3-year cosmological parameters. These lead to much less structure at high redshift, reducing the number of z=10 ``mini-haloes'' by more than a factor of two and the number of z=30 galaxy hosts by more than four orders of magnitude. Code to generate our best-fit halo mass function may be downloaded from http://icc.dur.ac.uk/Research/PublicDownloads/genmf_readme.htmlComment: MNRAS accepted. Changes in response to referee comments, including discussion of uncertainties with 2 additional plots. Corrected version of 2 plots showing mass function dependence on cosmological parameters (WMAP3 vs WMAP1). Conclusions unchanged. Code to reproduce this mass function can be downloaded at http://icc.dur.ac.uk/Research/PublicDownloads/genmf_readme.htm

We use a high‐resolution ΛCDM numerical simulation to calculate the mass function of dark matter haloes down to the scale of dwarf galaxies, back to a redshift of 15, in a 50 h−1 Mpc volume containing 80 million particles. Our low‐redshift results allow us to probe low‐σ density fluctuations significantly beyond the range of previous cosmological simulations. The Sheth & Tormen mass function provides an excellent match to all of our data except for redshifts of 10 and higher, where it overpredicts halo numbers increasingly with redshift, reaching roughly 50 per cent for the 1010–1011 M⊙ haloes sampled at redshift 15. Our results confirm previous findings that the simulated halo mass function can be described solely by the variance of the mass distribution, and thus has no explicit redshift dependence. We provide an empirical fit to our data that corrects for the overprediction of extremely rare objects by the Sheth & Tormen mass function. This overprediction has implications for studies that use the number densities of similarly rare objects as cosmological probes. For example, the number density of high‐redshift (z≃ 6) QSOs, which are thought to be hosted by haloes at 5σ peaks in the fluctuation field, are likely to be overpredicted by at least a factor of 50 per cent. We test the sensitivity of our results to force accuracy, starting redshift and halo‐finding algorithm.

The nature of structure formation around the particle free streaming scale is still far from understood. Many attempts to simulate hot, warm, and cold dark matter cosmologies with a free streaming cutoff have been performed with cosmological particlebased simulations, but they all suffer from spurious structure formation at scales below their respective free streaming scales -i.e. where the physics of halo formation is most affected by free streaming. We perform a series of high resolution numerical simulations of different warm dark matter (WDM) models, and develop an approximate method to subtract artificial structures in the measured halo mass function. The corrected measurements are then used to construct and calibrate an extended Press-Schechter (EPS) model with sharp-k window function and adequate mass assignment. The EPS model gives accurate predictions for the low redshift halo mass function of CDM and WDM models, but it significantly under-predicts the halo abundance at high redshifts. By taking into account the ellipticity of the initial patches and connecting the characteristic filter scale to the smallest ellipsoidal axis, we are able to eliminate this inconsistency and obtain an accurate mass function over all redshifts and all dark matter particle masses covered by the simulations. As an additional application we use our model to predict the microhalo abundance of the standard neutralino-CDM scenario and we give the first quantitative prediction of the mass function over the full range of scales of CDM structure formation.

Abstract. Future galaxy surveys require one percent precision in the theoretical knowledge of the power spectrum over a large range including very nonlinear scales. While this level of accuracy is easily obtained in the linear regime with perturbation theory, it represents a serious challenge for small scales where numerical simulations are required. In this paper we quantify the precision of present-day N -body methods, identifying main potential error sources from the set-up of initial conditions to the measurement of the final power spectrum. We directly compare three widely used N -body codes, Ramses, Pkdgrav3, and Gadget3 which represent three main discretisation techniques: the particle-mesh method, the tree method, and a hybrid combination of the two. For standard run parameters, the codes agree to within one percent at k ≤ 1 h Mpc −1 and to within three percent at k ≤ 10 h Mpc −1 . We also consider the bispectrum and show that the reduced bispectra agree at the sub-percent level for k ≤ 2 h Mpc −1 . In a second step, we quantify potential errors due to initial conditions, box size, and resolution using an extended suite of simulations performed with our fastest code Pkdgrav3. We demonstrate that the simulation box size should not be smaller than L = 0.5 h −1 Gpc to avoid systematic finite-volume effects (while much larger boxes are required to beat down the statistical sample variance). Furthermore, a maximum particle mass of M p = 10 9 h −1 M is required to conservatively obtain one percent precision of the matter power spectrum. As a consequence, numerical simulations covering large survey volumes of upcoming missions such as DES, LSST, and Euclid will need more than a trillion particles to reproduce clustering properties at the targeted accuracy.arXiv:1503.05920v3 [astro-ph.CO]

The dark matter halo mass function is a key repository of cosmological information over a wide range of mass scales, from individual galaxies to galaxy clusters. N-body simulations have established that the friends-of-friends (FOF) mass function has a universal form to a surprising level of accuracy ( 10%). The high-mass tail of the mass function is exponentially sensitive to the amplitude of the initial density perturbations, the mean matter density parameter, Ω m , and to the dark energy controlled late-time evolution of the density field. Observed group and cluster masses, however, are usually stated in terms of a spherical overdensity (SO) mass which does not map simply to the FOF mass. Additionally, the widely used halo models of structure formation -and halo occupancy distribution descriptions of galaxies within halos -are often constructed exploiting the universal form of the FOF mass function. This again raises the question of whether FOF halos can be simply related to the notion of a spherical overdensity mass. By employing results from Monte Carlo realizations of ideal Navarro-Frenk-White (NFW) halos and N-body simulations, we study the relationship between the two definitions of halo mass. We find that the vast majority of halos (80 − 85%) in the mass-range 10 12.5 − 10 15.5 h −1 M ⊙ indeed allow for an accurate mapping between the two definitions (∼ 5%), but only if the halo concentrations are known. Nonisolated halos fall into two broad classes: those with complex substructure that are poor fits to NFW profiles and those "bridged" by the (isodensity-based) FOF algorithm. A closer investigation of the bridged halos reveals that the fraction of these halos and their satellite mass distribution is cosmology dependent. We provide a preliminary discussion of the theoretical and observational ramifications of these results.