[Paper Review] Properties of Cold Dark Matter Halos at z>6

Renyue Cen, Fengqiu Adam Dong+2 more|arXiv (Cornell University)|Mar 15, 2004

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This study uses a high-resolution N-body simulation to analyze cold dark matter halos in the mass range $10^{6.5}-10^9\,M_\odot$ at redshifts $z=6-11$ in the standard LCDM model. It finds significant halo bias ($b \sim 2-6$), a mass function slope of $2.05 \pm 0.15$, and a mass- and redshift-dependent central density profile with $\alpha_0 = 0.75((1+z)/7.0)^{-1.25}(M/10^7\,M_\odot)^{0.11(1+z)/7.0}$, along with a median spin parameter of $0.03-0.04$ and peculiar velocities exceeding $30-40$ km/s.

ABSTRACT

We compute the properties of dark matter halos with mass $10^{6.5}-10^9\msun$ at redshift $z=6-11$ in the standard cold dark matter cosmological model, utilizing a very high resolution N-body simulation. We find that dark matter halos in these mass and redshift ranges are significantly biased over matter with a bias factor in the range 2-6. The dark matter halo mass function displays a slope of $2.05\pm 0.15$ at the small mass end. We do not find a universal dark matter density profile. Instead, we find a significant dependence of the central density profile of dark matter halos on halo mass and epoch with $α_0=0.4-1.0$; the high-mass ($M\ge 10^8\msun$) low-redshift ($z\sim 6$) halos occupy the high end of the range and low-mass ($M\sim 10^{7}\msun$) high-redshift ($z\sim 11$) halos occupy the low end. Additionally, for fixed mass and epoch there is a significant dispersion in $α_0$ due to the stochastic assembly of halos. Our results fit a relationship of the form $α_0=0.75((1+z)/7.0)^{-1.25}(M/10^7\msun)^{0.11(1+z)/7.0}$ with a dispersion about this fit of $\pm 0.5$ and no systematic dependence of variance correlated with environment. The median spin parameter of dark matter halos is $0.03-0.04$ but with a large lognormal dispersion of $\sim 0.4$. Various quantities are tabulated or fitted with empirical formulae.

Motivation & Objective

• To investigate the properties of cold dark matter halos at high redshift ($z > 6$) in the standard LCDM cosmology, where observational constraints are emerging from quasar absorption and CMB data.
• To quantify the mass function, bias, density profiles, spin parameters, angular momentum, and peculiar velocities of halos in the $10^{6.5}-10^9\,M_\odot$ mass range at $z=6-11$.
• To determine whether universal density profiles or spin parameters exist at high redshift, or if they depend on mass and epoch.
• To provide empirical fits and statistical distributions for key halo properties to support theoretical modeling of early galaxy and reionization processes.

Proposed method

• A high-resolution Tree-Particle-Mesh (TPM) N-body simulation with $512^3$ particles in a $4\,h^{-1}$ Mpc box, particle mass $3.57 \times 10^4\,h^{-1}\,M_\odot$, and softening length $0.14\,h^{-1}$ kpc.
• Dark matter halos were identified using the DENMAX scheme with Gaussian smoothing at $300\,h^{-1}$ kpc.
• The halo mass function, bias, and clustering were computed by comparing halo distributions to the underlying matter field.
• Density profiles were fitted using a power-law form $\rho \propto r^{-\alpha_0}$, with $\alpha_0$ measured in the central region and fitted as a function of mass and redshift.
• Spin parameters and angular momentum profiles were computed via $\lambda = J M^{3/2} / \sqrt{G |E|}$, with profiles fitted using a modified lognormal function.
• Peculiar velocities were computed as the bulk motion of halos, with distributions fitted using a lognormal model including a minimum velocity offset $\epsilon$.

Experimental results

Research questions

• RQ1 What is the bias of dark matter halos relative to the underlying matter field at $z > 6$?
• RQ2 What is the slope of the dark matter halo mass function at the low-mass end in the $z=6-11$ redshift range?
• RQ3 Is there a universal dark matter density profile at high redshift, or does it depend on halo mass and redshift?
• RQ4 How do the spin parameters and angular momentum profiles of high-redshift halos vary with mass and epoch?
• RQ5 What are the typical peculiar velocities of dark matter halos at $z=6$, and how do they depend on simulation resolution and missing large-scale modes?

Key findings

• Dark matter halos at $z=6-11$ are significantly biased over the matter distribution, with a bias factor in the range $2-6$.
• The halo mass function has a slope of $2.05 \pm 0.15$ at the low-mass end, indicating a steep rise in number density toward lower masses.
• The central density profile slope $\alpha_0$ varies from $0.4$ to $1.0$, depending on halo mass and redshift, and is well-fitted by the empirical relation $\alpha_0 = 0.75((1+z)/7.0)^{-1.25}(M/10^7\,M_\odot)^{0.11(1+z)/7.0}$ with a dispersion of $\pm 0.5$.
• The median spin parameter of halos is $0.03-0.04$, but with a large lognormal dispersion of $\sim 0.4$, indicating strong stochasticity in angular momentum assembly.
• The fraction of mass in a halo with specific angular momentum less than a given value is approximately $0.5$ times the ratio of that value to the average specific angular momentum.
• The peculiar velocity of halos exceeds $30-40$ km/s, with a lower limit of $38 \pm 2$ km/s in the simulation, suggesting that missing large-scale modes may increase the true value.