The connection between the mass power spectrum and the density profile of dark matter halos

Dark matter halos provide the potential wells inside which galaxies form. As a result, understanding their basic properties, including their formation history and internal structure, is an important step for understanding galaxy evolution. In this work (Correa C. A., Wyithe J. S. B., Schaye J., Duffy A. R., 2015b, MNRAS, 450, 1521) we explore the relation between the dark matter halos internal structure (described by their density profile) and their mass accretion history.

In the currently accepted cosmological paradigm, structures are initially in the form of density perturbations, that were seeded by the amplification of quantum fluctuations during inflation. Inflation generates a spectrum of perturbations in different $latex k$-modes, where each defines the typical value of $latex \delta$ (real space density field) on the spatial scale $latex \lambda= 2\pi/k$. The statistical properties of the fluctuations are determined by the variance, $latex \sigma$ of the different $latex k$-modes given by the linear power spectrum, $latex P(k) = (2\pi)-3\langle|\delta k|\rangle 2$, where the angular brackets denote an average over the entire statistical ensemble of modes. The power spectrum is very important because it describes how much power exists at each mode $latex k$ with respect to the background density. The larger the power for a given $latex k$, the more fluctuations in regions of corresponding spatial scale $latex \lambda$.

The collapse of a dark matter halo occurs when fluctuations become nonlinear and the overdensity reach a critical value of the order of unity. The halo then grows in mass, either by accreting material from their neighbourhood or by merging with other halos. In Correa et al. (2015a) we used the extended Press-Schechter formalism to derive the halo mass accretion history from the growth rate of initial density perturbations. We showed that the halo mass history is well described by an exponential function of redshift in the high-redshift regime, and by a power-law in the low-redshift regime.

In this work, we combine the analytic model of Correa et al. (2015a) with the halo mass accretion histories determined from cosmological simulations, to explain the physical connection between the linear rms fluctuation of the density field, $latex \sigma$, and the halo density profile. It is generally believed that the connection raises due to the following. During hierarchical growth, halos form through mergers with smaller structures and accretion from the intergalactic medium. Most mergers are minor (with smaller satellite halos) and do not alter the structure of the inner halo. Major mergers (mergers between halos of comparable mass) can bring material to the centre, but they are found not to play a pivotal role in modifying the internal mass distribution (Wang & White 2009). Halo formation can therefore be described as an ‘inside out’ process, where a strongly bound core collapses, followed by the gradual addition of material at the cosmological accretion rate. Through this process, halos acquire a nearly universal density profile that can be described by a simple formula known as the ‘NFW profile’ (Navarro, Frenk & White 1996).

Fig.1. Median relation between halo formation time and halo mass
Fig. 2. Median relation between halo formation time and halo concentration.

Fig. 2 shows the relation between formation redshift, $latex z_{-2}$ (defined as the time in redshift when the halo was formed), and halo concentration, c (a parameter that characterised the halo density profile), and Fig. 1. shows the relation between formation redshift and halo mass, $latex M_{0}$. The different symbols correspond to the median values of the relaxed halo sample taken from OWLS simulations and the error bars to 1$latex \sigma$ confidence limits. The solid line in Fig. 2 is not a fit but a prediction of the $latex z_{-2}-c$ relation for relaxed halos given by eq. (9) of Correa et al. (2015b). For more details see Correa et al. (2015b).