Hierarchical Clustering: Structure Formation in the Universe

In order to understand galaxy formation models it is necessary to have a reasonably clear idea of dark matter clustering. This because, in the standard cosmological scenario, galaxies are thought to reside in larger dark matter haloes, extending beyond the galaxy observable radius. Haloes form as consequence of gravitational instability of dark matter density perturbations, and collapse at a density about two hundred times that of the surrounding environment. Clustering happens at allmasses at any time. Until now no direct observations of the existence of these darkmatter haloes have been done; however, their presence may be indirectly tested by their gravitational influence. For example, galaxies in groups have a velocity dispersion much higher than that caused only by visiblematter. Astronomers thus assumed the existence of large amounts of dark matter, an hypothesis later found consistent with other independent observations like gravitational lensing, galaxy clustering on very large scales and anisotropies in the cosmic microwave background radiation. In particle physics, supersymmetry predicts the existence of a particle named neutralino (Jungman et al., 1996; Bertone et al., 2005), today regarded as the most likely candidate for the darkmatter. This particle is heavy and slow-moving (mass ? 100 Gev), so that dark matter density fluctuations can collapse for any mass larger than 10? 6M? (Hofmann et al., 2001; Green et al., 2004, 2005). This places amass cut-off on the smallest darkmatter haloes that can collapse. Neutralino can also annihilatewith its anti-particle, generating ? -ray photons (Bergstrom, 2000; Bertone et al., 2005), with annihilation rates growing as the square of the density. Due to this process, it is expected that future ? -ray telescopes (like GLAST, Morselli (1997)) should be able to detect some excess in the ? -ray background signal from the center of theMilky-Way halo and from its satellites. This would be the first time of an in-”direct” detection of dark matter. In this PhD dissertation we study the evolution of dark matter haloes, using two complementary approaches: numerical simulations and analytical modeling (through the extended Press & Schechter formalism). The work is organized as follows. In the first three chapters we describe and review some properties of the early universe and the theory underlying models of dark matter clustering. We discuss how density perturbations evolve and formdarkmatter haloes inside which baryons can shock and cool, eventually form stars and galaxies. We also show how the number density of haloes can be estimated at any redshift using the excursion set approach, both for the spherical and ellipsoidal collapsemodels. These model mass functions are compared with those from numerical simulations in Chapter 4. We show that the ellipsoidal collapse model (Sheth et al., 2001; Sheth and Tormen, 2002) perfectly reproduces the global mass function in N-Body simulations, while, on the other hand, the spherical collapse model (Press and Schechter, 1974; Lacey and Cole, 1993; Bond et al., 1991) overpredicts the aboundace of smallmasses and underpredicts that of large ones. Dark matter clustering is hierarchical, i.e. small systems collapse first (at higher redshift), and subsequentlymerge together forming larger haloes. In this scenario, if we define a formation time as the earliest redshift when an halo assembles half of its present-daymass, small haloes formfirst and large ones form later. The top of the hierarchical pyramid is occupied by galaxy clusters, which represent the largest virialized structures in the universe. Another important quantity describing dark matter clustering is any conditional mass function. One example is the probability that an halo observed at redshift z1, will be part of a larger halo at z0 < z1. This distribution is also called progenitor mass function; theoretical predictions and N-Body simulations are compared at the end of Chapter 4. There we show that, also in this case, the ellipsoidal collapse prediction well reproduces the distribution found in numerical simulations atmost redshifts. In Chapter 5 we will discuss how it is possible to estimate the formation time distribution from the conditional mass function, and present a new formula, based on the ellipsoidal collapse, that better fits the formation redshift distributionmeasured in N-Body simulations. The progenitors accreted along themerging history tree of a halo can survive today in their host system, and constitute the so-called substructure population (Ghigna et al., 1998; Tormen et al., 2004; Gao et al., 2004; De Lucia et al., 2004; van den Bosch et al., 2005). In Chapter 6 we discuss how it is possible to analytically estimate this population using the conditionalmass function, assuming no tidal stripping andmerging among substructures. By extrapolating the power spectrumof density perturbations down to the typical neutralino Jeansmass, we estimate the substructure population in aMilky-Way size halo, both for a spherical and ellipsoidal collapse model. Modeling the neutralino annihilation rate, we then estimate the ? -ray emission from this population and its detectability with a GLAST-like telescope. In Chapter 7 we study the growth of the main progenitor halo, and the mass it accretes along itsmerging history tree using numerical simulations. Themass function of accreted haloes, called “unevolved subhalomass function”, turns out to be independent of the final host halo mass, both before and after its formation redshift. The accreted haloes, called satellites, are then followed snapshot by snapshot in order to compute their mass loss rate. This allow us to interpret the present-day subhalo population in term of the mass loss from the accreted satellites. Since smaller hosts form earlier than larger ones, the former will accrete satellites at earlier times; these satellites will therefore spend a longer time inside the host halo and lose a larger fraction of their initialmass. This translates the (mass-independent) unevolved subhalo population in a present-day subhalo distribution that depends on the host halo mass: at fixed subhalo-to-host halo mass: msb/M0, more massive hosts contain more subhaloes than smaller hosts do. Subhaloes defined in this way may contain other subhaloes within themselves (Diemand et al., 2007b; Li and Helmi, 2007), which were accreted when they were still isolated systems. In Chapter 8 we show how subhaloes within subhaloes can be identified following all branches of themerging history tree of an host halo. We also compare our definition of substructures with that of other authors (Gao et al., 2004), finding very good agreement. In the last chapter of this dissertation, we show how the merging history tree of a halo can be followed using Monte Carlo realizations. The partition code, on which the tree is based, is very fast, time step independent, and provides results in excellent agreement with the spherical collapse conditional mass function down to any required mass resolution (Sheth and Lemson, 1999). The tree has been run following the main branch and resolving all satellites down to the typical neutralino Jeans mass, in order to study the Milky-Way subhalo population.