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Register a new userEvolution of the Mass Function of Dark Matter Haloes
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Darren Reed
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We use a high resolution $Lambda$CDM numerical simulation to calculate the mass function of dark matter haloes down to the scale of dwarf galaxies, back to a redshift of fifteen, in a 50 $h^{-1}$Mpc volume containing 80 million particles. Our low redshift results allow us to probe low $sigma$ density fluctuations significantly beyond the range of previous cosmological simulations. The Sheth and Tormen mass function provides an excellent match to all of our data except for redshifts of ten and higher, where it overpredicts halo numbers increasingly with redshift, reaching roughly 50 percent for the $10^{10}-10^{11} msun$ 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 and 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 $simeq$ 6) QSOs, which are thought to be hosted by haloes at 5$sigma$ peaks in the fluctuation field, are likely to be overpredicted by at least a factor of 50%. We test the sensitivity of our results to force accuracy, starting redshift, and halo finding algorithm.
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[ABRIDGED] The unconditional mass function (UMF) of dark matter haloes has been determined accurately in the literature, showing excellent agreement with high resolution numerical simulations. However, this is not the case for the conditional mass function (CMF). We propose a simple analytical procedure to derive the CMF by rescaling the UMF to the constrained environment using the appropriate mean and variance of the density field at the constrained point. This method introduces two major modifications with respect to the standard re-scaling procedure. First of all, rather than using in the scaling procedure the properties of the environment averaged over all the conditioning region, we implement the re-scaling locally. We show that for high masses this modification may lead to substantially different results. Secondly, we modify the (local) standard re-scaling procedure in such a manner as to force normalisation, in the sense that when one integrates the CMF over all possible values of the constraint multiplied by their corresponding probability distribution, the UMF is recovered. In practise, we do this by replacing in the standard procedure the value delta_c (the linear density contrast for collapse) by certain adjustable effective parameter delta_eff. In order to test the method, we compare our prescription with the results obtained from numerical simulations in voids (Gottlober et al. 2003), finding a very good agreement. Based on these results, we finally present a very accurate analytical fit to the (accumulated) conditional mass function obtained with our procedure, which may be useful for any theoretical treatment of the large scale structure.
Galaxy clusters are luminous tracers of the most massive dark matter haloes in the Universe. To use them as a cosmological probe, a detailed description of the properties of dark matter haloes is required. We characterize how the dynamical state of haloes impacts the halo mass function at the high-mass end. We used the dark matter-only MultiDark suite of simulations and the high-mass objects M > 2.7e13 M/h therein. We measured mean relations of concentration, offset, and spin as a function of halo mass and redshift. We investigated the distributions around the mean relations. We measured the halo mass function as a function of offset, spin, and redshift. We formulated a generalized mass function framework that accounts for the dynamical state of the dark matter haloes. We confirm the discovery of the concentration upturn at high masses and provide a model that predicts the concentration for different values of mass and redshift with one single equation. We model the distributions around the mean concentration, offset, and spin with modified Schechter functions. The concentration of low-mass haloes shows a faster redshift evolution compared to high-mass haloes, especially in the high-concentration regime. The offset parameter is smaller at low redshift, in agreement with the relaxation of structures at recent times. The peak of its distribution shifts by a factor of 1.5 from z = 1.4 to z = 0. The individual models are combined into a comprehensive mass function model, as a function of spin and offset. Our model recovers the fiducial mass function with 3% accuracy at redshift 0 and accounts for redshift evolution up to z = 1.5. This approach accounts for the dynamical state of the halo when measuring the halo mass function. It offers a connection with dynamical selection effects in galaxy cluster observations. This is key toward precision cosmology using cluster counts as a probe.
Virial mass is used as an estimator for the mass of a dark matter halo. However, the commonly used constant overdensity criterion does not reflect the dynamical structure of haloes. Here we analyze dark matter cosmological simulations in order to obtain properties of haloes of different masses focusing on the size of the region with zero mean radial velocity. Dark matter inside this region is stationary, and thus the mass of this region is a much better approximation for the virial mass.
We use numerical simulations in a Lambda CDM cosmology to model density profiles in a set of 16 dark matter haloes with resolutions of up to 7 million particles within the virial radius. These simulations allow us to follow robustly the formation and evolution of the central cusp over a large mass range of 10^11 to 10^14 M_sun, down to approximately 0.5% of the virial radius, and from redshift 5 to the present. The cusp of the density profile is set at redshifts of 2 or greater and remains remarkably stable to the present time, when considered in non-comoving coordinates. We fit our haloes to a 2 parameter profile where the steepness of the asymptotic cusp is given by gamma, and its radial extent is described by the concentration, c_gamma. In our simulations, we find gamma = 1.4 - 0.08Log(M/M_*) for haloes of 0.01M_* to 1000M_*, with a large scatter of gamma ~ +/-0.3$; and c_gamma = 8*M/M_*^{-0.15}, with a large M/M_* dependent scatter roughly equal to +/- c_gamma. Our redshift zero haloes have inner slope parameters ranging approximately from r^{-1} to r^{-1.5}, with a median of roughly r^{-1.3}. This 2 parameter profile fit works well for all our halo types, whether or not they show evidence of a steep asymptotic cusp. We also model a cluster in power law cosmologies of P ~ k^n (n=0,-1,-2,-2.7). We find larger concentration radii and shallower cusps for steeper n. The minimum resolved radius is well described by the mean interparticle separation. The trend of steeper and more concentrated cusps for smaller $M/M_*$ haloes clearly shows that dwarf sized Lambda CDM haloes have, on average, significantly steeper density profiles within the inner few percent of the virial radius than inferred from recent observations. Code to reproduce this profile can be downloaded from http://www.icc.dur.ac.uk/~reed/profile.html
We use the halo occupation distribution (HOD) framework to characterise the predictions from two independent galaxy formation models for the galactic content of dark matter haloes and its evolution with redshift. Our galaxy samples correspond to a range of fixed number densities defined by stellar mass and span $0 le z le 3$. We find remarkable similarities between the model predictions. Differences arise at low galaxy number densities which are sensitive to the treatment of heating of the hot halo by active galactic nuclei. The evolution of the form of the HOD can be described in a relatively simple way, and we model each HOD parameter using its value at $z=0$ and an additional evolutionary parameter. In particular, we find that the ratio between the characteristic halo masses for hosting central and satellite galaxies can serve as a sensitive diagnostic for galaxy evolution models. Our results can be used to test and develop empirical studies of galaxy evolution and can facilitate the construction of mock galaxy catalogues for future surveys.
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