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Register a new userA derivation of (half) the dark matter distribution function
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All dark matter structures appear to follow a set of universalities, such as phase-space density or velocity anisotropy profiles, however, the origin of these universalities remains a mystery. Any equilibrated dark matter structure can be fully described by two functions, namely the radial and the tangential velocity distribution functions (VDF), and when we will understand these two then we will understand all the observed universalities. Here we demonstrate that if we know the radial VDF, then we can derive and understand the tangential VDF. This is based on simple dynamical arguments about properties of collisionless systems. We use a range of controlled numerical simulations to demonstrate the accuracy of this result. We therefore boil the question of the dark matter structural properties down to understanding the radial VDF.
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The velocity distribution of dark matter near the Earth is important for an accurate analysis of the signals in terrestrial detectors. This distribution is typically extracted from numerical simulations. Here we address the possibility of deriving the velocity distribution function analytically. We derive a differential equation which is a function of radius and the radial component of the velocity. Under various assumptions this can be solved, and we compare the solution with the results from controlled numerical simulations. Our findings complement the previously derived tangential velocity distribution. We hereby demonstrate that the entire distribution function, below 0.7 v_esc, can be derived analytically for spherical and equilibrated dark matter structures.
We find the distribution function f(E) for dark matter (DM) halos in galaxies and the corresponding equation of state from the (empirical) DM density profiles derived from observations. We solve for DM in galaxies the analogous of the Eddington equation originally used for the gas of stars in globular clusters. The observed density profiles are a good realistic starting point and the distribution functions derived from them are realistic. We do not make any assumption about the DM nature, the methods developed here apply to any DM kind, though all results are consistent with Warm DM. With these methods we find: (i) Cored density profiles behaving quadratically for small distances rho(r) r -> 0 = rho(0) - K r^2 produce distribution functions which are finite and positive at the halo center while cusped density profiles always produce divergent distribution functions at the center. (ii) Cored density profiles produce approximate thermal Boltzmann distribution functions for r < 3 r_h where r_h is the halo radius. (iii) Analytic expressions for the dispersion velocity and the pressure are derived yielding an ideal DM gas equation of state with local temperature T(r) = m v^2(r)/3. T(r) turns to be constant in the same region where the distribution function is thermal and exhibits the same temperature within the percent. The self-gravitating DM gas can thermalize despite being collisionless because it is an ergodic system. (iv) The DM halo can be consistently considered at local thermal equilibrium with: (a) a constant temperature T(r) = T_0 for r < 3 ; r_h, (b) a space dependent temperature T(r) for 3 r_h < r < R_{virial}, which slowly decreases with r. That is, the DM halo is realistically a collisionless self-gravitating thermal gas for r < R_{virial}. (v) T(r) outside the halo radius nicely follows the decrease of the circular velocity squared.
(Abridged) We apply a very general statistical theorem introduced by Cramer (1936) to study the origin of the deviations of the halo spin PDF from the reference lognormal shape. We find that these deviations originate from correlations between two quantities entering the definition of spin, namely the ratio $J/M^{5/2}$ (which depends only on mass) and the total gravitational binding energy $E$. To reach this conclusion, we have made usage of the results deduced from two high spatial- and mass resolution simulations. Our simulations cover a relatively small volume and produce a sample of more than 16.000 gravitationally bound halos, each traced by at least 300 particles. We verify that our results are stable to different systematics, by comparing our results with those derived by the GIF2 and by a more recent simulation performed by Maccio et al. We find that the spin probability distribution function shows systematic deviations from a lognormal, at all redshifts z <= 1. These deviations depend on mass and redshift: at small masses they change little with redshift, and also the best lognormal fits are more stable. The J-M relationship is well described by a power law of exponent $alpha$ very near to the linear theory prediction (alpha=5/3), but systematically lower than this at z<= 0.3. We argue that the fact that deviations from a lognormal PDF are present only for high-spin halos could point to a role of large-scale tidal fields in the evolution of the spin PDF.
We introduce the position-dependent probability distribution function (PDF) of the smoothed matter field as a cosmological observable. In comparison to the PDF itself, the spatial variation of the position-dependent PDF is simpler to model and has distinct dependence on cosmological parameters. We demonstrate that the position-dependent PDF is characterized by variations in the local mean density, and we compute the linear response of the PDF to the local density using separate universe N-body simulations. The linear response of the PDF to the local density field can be thought of as the linear bias of regions of the matter field selected based on density. We provide a model for the linear response, which accurately predicts our simulation measurements. We also validate our results and test the separate universe consistency relation for the local PDF using global universe simulations. We find excellent agreement between the two, and we demonstrate that the separate universe method gives a lower variance determination of the linear response.
Studies of flux anomalies statistics and perturbations in stellar streams have the potential to constrain models of warm dark matter (WDM), including sterile neutrinos. Producing these constraints requires a parametrization of the WDM mass function relative to that of the cold dark matter (CDM) equivalent. We use five WDM models with half-mode masses, $M_mathrm{hm}=[1.3,35]times10^{8}$~$M_{odot}$, spread across simulations of the Local Group, lensing ellipticals and the $z=2$ universe, to generate such a parametrization: we fit parameters to a functional form for the WDM-to-CDM halo mass function ratio, $n_mathrm{WDM}(M_{X})/n_mathrm{CDM}(M_{X})$, of ($1+(alpha M_mathrm{hm}/M_{X})^{beta})^{gamma}$. For $M_{X}equiv$ virial mass of central halos we obtain $alpha=2.3$, $beta=0.8$, and $gamma=-1.0$, and this fit is steeper than the extended Press-Schechter formalism predicts. For $M_{X}equiv$ mass of subhalos we instead obtain $alpha=4.2$, $beta=2.5$ and $gamma=-0.2$; in both mass definitions the scatter is $sim20$~per~cent. The second fit typically underestimates the relative abundance of $z=2$ WDM subhaloes at the tens of per cent level. We caution that robust constraints will require bespoke simulations and a careful definition of halo mass, particularly for subhalos of mass $<10^{8}M_{odot}$.
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