N-body simulations calibration

As we just mentioned, the ST semi-analytical mass function, which generalised the PS one, is a cosmology dependent fitting formula, which can be expressed as

fST(ν) =AST r 2αST π ν 1 + (ν2αST)βST exp −ν 2_α ST 2 , (2.21) where the coefficients are obtained by a fit to GIF/Virgo Collaboration simulations of clustering (Kauffmann et al. 1999a). These simulations were performed with 2563

particles, in two boxes of sizesL1 = 85 Mpc/handL2 = 141 Mpc/h, for three cosmo-

logical models. They used a spherical overdensity (SO) group finder (Tormen 1998) to measure the mass function in the simulations. The best-fitting values they ob- tained are αST = 0.707 and βST = 0.3, while AST = 0.3222 is derived assuming that

all mass is collapsed into haloes (i.e. the integral of the mass function to infinity is equal to unity). The PS case follows easily fromαST = 1,βST = 0 andAST = 0.5. ST

improve the analytical fit to N-body simulations results, but nevertheless remaining still poor in the agreement for rare high redshift haloes (Reed et al. 2007).

By means of larger and more elaborated N-body simulations, it has been found empirically that the mass function determined for a wide range of redshifts and cosmological models can be fitted accurately by a universal function (Jenkins et al. 2001; Evrard et al. 2002; Warren et al. 2006; Tinker et al. 2008). Some expressions, based on fits to simulation data, have been calculated and they agree at the 10₋30%, with the largest discrepancy on the high mass tail. Jenkins et al. (2001) showed that the mass function of DM haloes from galaxies to clusters masses is quite well described by the ST function up to redshift z = 5, with some suppression at high masses. He analysed the results of the Hubble Volume simulation, a simulation of DM clustering in a cubic volume of size L = 3 Gpc/h, with 10243 _{DM particles.}

This yields a DM particle mass of 2.2 _×1012_M

⊙, implying that a galaxy cluster

halo typically contains 100-1000 particles. Despite the poor mass resolution, the very large volume allowed to explore the mass function on a broad range of masses, including the very high mass end, where clusters reside. They identified DM halos using the friends-of-friends algorithm (Davis et al. 1985). Jenkins proposed finally the following alternative analytic fitting formula to the simulations:

fJ(σM) =AJexp −|lnσM−1+αJ|βJ

, (2.22)

where AJ = 0.301, αJ = 0.64 and βJ = 3.82. Its accuracy is well tested by the Mil-

2.2 Mass function 47

Figure 2.2: Left panel: Tinker’s f(σ) at z = 0 and for ∆ = 200, from all simulations. The solid line is the best fit function. The lower window shows the residuals with respect to the fitting function. Left panel: f(σ) atz = 1.25 and for ∆ = 200. The lower window shows that the z = 1.25 mass function is offset by ∼20% with respect to the z = 0 one. Credit: Tinker et al. (2008)

at redshift zero measured by Warren et al. (2006). This last work was aimed to test the mass function over a wider range of mass scales than the one obtained from a single simulation. For this purpose, they simulated 16 boxes of different physical size but the same number of DM particles (10243_{), nested in such a way that they defined}

a composite halo mass function covering five orders of magnitude in mass scale. The current state-of-the-art halo mass function has been estimated by Tinker et al. (2008, 2010), who achieved a fit at the 5% precision level, for a ΛCDM cosmology. The simulations used to obtain this result were based on variants of the flat ΛCDM model, where the parameters referred to the first-year or three-year WMAP results (Spergel et al. 2003, 2007). They used fifty realizations of a simulation on a cubic box of L= 1280 Mpc/h size, performed with the GADGET2 _{code (Springel 2005) and} six simulations using the adaptive refinement technique of Kravtsov et al. (1997). They employed the standard SO algorithm Lacey & Cole (1994), but relocating the

centres of haloes at their density peaks, instead that on the centre of mass of the particles within the sphere. The results of all these simulations can be visualised in Fig. 2.2, showing the mass function best fit (solid line) at z = 0, 1.25, for ∆ = 200. The model they obtained, valid over wide redshift and mass ranges, has the following form: fT(ν) =AT h 1 + (βTν)−2φT i ν2ηT+1_e−γTν2/2 _{, (2.23)}

whereAT = 0.368 and the other parameters evolve in redshift as

βT = 0.589 (1 +z)0.20 ,

φT = −0.729 (1 +z)−0.08 ,

ηT = −0.243 (1 +z)0.27 ,

γT = 0.864 (1 +z)−0.01 . (2.24)

The above results are only valid for cluster mass at R200, with an overdensity of

∆ = 200 in units of the mean mass density of the Universe, which is the case we restrict to.

In none of the above calibrated mass functions, the effects of baryon physics is taken into account. An interesting analysis on the effect on the halo mass function caused by the inclusion of baryons has been performed by Cui et al. (2012). They employed two hydrodynamical simulations: one including radiative cooling, star for- mation and kinetic feedback from Supernovae, and a non-radiative simulation. These were based on the TreePM/SPH GADGET-3 code (Springel 2005), having a cubic volume of size L = 410 Mpc/h, with 2×10243 _{DM particles and using a SO halo}

finder algorithm. They obtained that the inclusion of baryons increases the mass of 1-2% at ∆ = 200 and of 4-5% at ∆ = 500.