The redshift evolution of Λ cold dark matter halo parameters: concentration, spin and shape

J. C. Mu noz-Cuartas 3 A. V. Macci o` 2 S. Gottl ober 3 A. A. Dutton 0 1 0 CITA National Fellow 1 Department of Physics and Astronomy, University of Victoria , Victoria, BC V8P 5C2, Canada 2 Max-Planck-Institut Fur Astronomie , Koningstuhl 17, 69117 Heidelberg, Germany 3 Astrophysikalisches Institut Potsdam , An der Sternwarte 16, 14482 Potsdam, Germany A B S T R A C T We present a detailed study of the redshift evolution of dark matter halo structural parameters in a cold dark matter cosmology. We study the mass and redshift dependence of the concentration, shape and spin parameter in N-body simulations spanning masses from 1010 to 1015 h1 M and redshifts from 0 to 2. We present a series of fitting formulae that accurately describe the time evolution of the concentration-mass (cvir-Mvir) relation since z = 2. Using arguments based on the spherical collapse model, we study the behaviour of the scalelength of the density profile during the assembly history of haloes, obtaining physical insights into the origin of the observed time evolution of the cvir-Mvir relation. We also investigate the evolution with redshift of dark matter halo shape and its dependence on mass. Within the studied redshift range, the relation between the halo shape and mass can be well fitted by a redshift-dependent power law. Finally we show that although for z = 0 the spin parameter is practically mass independent, at increasing redshift it shows an increasing correlation with mass. 1 I N T R O D U C T I O N Observational evidence (e.g. Spergel et al. 2007; Komatsu et al. 2009) favours the hierarchical growth of structures in a universe dominated by cold dark matter (CDM) and dark energy (), the so-called CDM universe. Within this paradigm, dark matter collapses first into small haloes, which accrete matter, and merge to form progressively larger haloes over time. Galaxies are thought to form out of gas which cools and collapses to the centres of these dark matter haloes (e.g. White & Rees 1978). In this picture, the properties of galaxies are expected to be strongly related to the properties of the dark matter haloes in which they are embedded (e.g. Mo, Mao & White 1998; Dutton et al. 2007). It has been shown by several studies that the structural properties of dark matter (DM) haloes are dependent on halo mass; for example, higher mass haloes are less concentrated (Navarro, Frenk & White 1997, hereafter NFW; Eke, Navarro & Steinmetz 2001; Bullock et al. 2001a; Kuhlen et al. 2005; Macci o` et al. 2007; Neto et al. 2007; Duffy et al. 2008; Gao et al. 2008; Macci o`, Dutton & van den Bosch 2008, hereafter M08; Klypin, Trujillo-Gomez & Primack 2010) and are more prolate (Jing & Suto 2002; Allgood et al. 2006; Bett et al. 2007; Gottlober & Yepes 2007; Maccio` et al. 2007; M08) on average. The situation is less clear for the spin parameter; at z = 0 there seems to be no mass dependence (Maccio` et al. 2007; M08) or at least a very weak one (Bett et al. 2007), while for increasing values of the redshift a possible mild correlation between spin and mass seems to arise (Knebe & Power 2008). In M08, the properties of DM haloes were studied in CDM universes whose parameters were fixed by the 1-, 3- and 5-yr release of the Wilkinson Microwave Anisotropy Probe (WMAP) mission (WMAP5; Komatsu et al. 2009). In that study, attention was paid to the structural parameters of virialized haloes and their correlations at the present epoch, z = 0. In this work, we extend this previous analysis to higher redshifts and study how the scaling relations of DM haloes change with time. As in M08 we use a large suite of N-body simulations in a WMAP5 cosmology with different box sizes to cover the entire halo mass range relevant for galaxy formation: from 1010 h1 M (haloes that host dwarf galaxies) to 1015 h1 M (massive clusters). We use these simulations to investigate the evolution of concentrations, spin parameters and shapes of DM haloes through cosmic time. Similar studies have already been conducted in the past, mainly using lower numerical resolution and/or a smaller mass range (but with few recent exceptions). NFW proposed that the characteristic density of DM haloes was directly proportional to the density of the universe at the time of formation, making it possible to connect today the properties of the DM density profile to the halo formation history and to the evolution of the expanding universe. This idea was then expanded by Wechsler et al. (2002), who found a clear connection between the mass growth of DM haloes and the definition of the formation time, connecting directly the growth history of DM haloes to the evolution of their concentration parameter. In a series of papers, Zhao et al. (2003a,b, 2009) have re-addressed the problem of the evolution of DM halo density profile and the mass accretion history. Zhaos main result was the finding of a correlation between rs and the characteristic mass of DM haloes, Ms, defined as the mass inside rs. Thanks to this correlation, they were able to model the time evolution of the concentration parameter in a cosmology-free fashion. A comparison of the different approaches to predict DM halo concentrations was performed by Neto et al. (2007). They made a detailed comparison of the Wechsler et al. (2002) and Zhao et al. (2003a) models for the time evolution of DM halo masses and their resulting predictions for halo concentration. Neto et al. (2007) found that although these models could match the average concentration reasonably well, they performed very poorly in many cases, because their models for halo mass evolution were not able to satisfactorily reproduce real mass growth histories from N-body simulations. However, the evolution of DM halo properties does not reduce to the concentration parameter only; halo shape and spin parameter are also important quantities that could influence the properties of the hosted galaxy. Allgood et al. (2006) studied the mass, radius, redshift and cosmology dependence (via variations of 8) of the shape of DM haloes while the environment dependence of the shape has been addressed in Hahn et al. (2007a,b). The distribution of the spin parameter of DM haloes has been studied in several works (e.g. Bullock et al. 2001b; Bett et al. 2007; Maccio` et al. 2007; Knebe & Power 2008; M08; Davis & Natarajan 2009; AntonuccioDelogu et al. 2010) as well as the correlation between the halo angular momentum and large-scale structure (e.g. Bailin & Steinmetz 2005; Bett et al. 2010). One of the conclusions was that the correlation between spin and mass (almost absent at z = 0) seems to increase with increasing redshift. Such a behaviour could have important influences in the modelling of properties of galaxies at high redshift. Although it has been shown that the inclusion of baryonic physics may affect the properties of the DM distribution (e.g. Gnedin et al. 2004; Kazantzidis et al. 2004; Bett et al. 2010; Knebe et al. 2010; Libeskind et (...truncated)