LEGISLACIÓN COMPARADA

We have used Nm =50 realizations from the Low resolution Baryon Acoustic Simulation at

ICC (L-BASICCII)N-body simulations kindly provided by Ra´ul Angulo. We have used these simulations to construct a set of mock catalogues for the REFLEX II galaxy cluster survey (see chapter 4). Thus it is useful to consider some properties of this simulation such like the halo abundance, the halo-mass bias and the clustering. The latter has been analyzed

3.6 Halo bias and two-point statistics from N₋body simulations 49

Figure 3.8: Top: L-BASICC II halo mass function (points with error bars) for the three outputs of the L-BASICC II simulation. On top of the measurements is shown the pre- diction based on the fit of Jenkins et al. (2001) for the corresponding redshift, follow- ing Equation (3.44). Bottom: measured vari- ance (points) from the 50 realizations at each redshift compared with the theoretical predic- tion from Equation (3.46) (lines).

and discussed in real configuration space (correlation function) and in Fourier space (power spectrum) (Angulo et al., 2008; S´anchez et al., 2008).

These simulations have outputs at redshiftsz=0,0.5 and z=1.0 and consist of 4483 dark matter particles. Dark matter haloes are identified via a FoF algorithm with a Plummer softening length of 50 kpc h−1 _{and a linking length of b =0.2 times the mean inter-particle}

separation. The halo mass resolution is M=1.75_×1013M_⊙h−1 (corresponding to ten particles,

and each particle with mass m = ρ¯(L/448)3_{) in a box with side L = 1.34 Gpc h}−1_{, yielding} ≈ 4.6_×105 identified dark matter haloes in each realization. The low resolution allow us only to probe the high mass tail of the halo mass function. The cosmological parameters used in the simulation consist of a flatΛCDM model with a matter energy-density parameter

Ωmat = 0.237, baryon energy-density Ωba = 0.046, dimensionless Hubble parameter h = 0.73, dark matter equation of state w=−1, the rms of mass fluctuations σ8 =0.773and the scalar

spectral index ns = 0.997. These values are close to the latest constraints on cosmological parameters (Spergel et al., 2007; S´anchez et al., 2009; Percival et al., 2010).

The L-BASICC II mass function

TheL-BASICCII halo mass function has been determined by counting the number of haloes

∆N in a mass bin∆M in each realization. We compare the result of the mean∆N/V∆M (where

50 3. Dark matter haloes: abundance, bias and clustering

averaged in the corresponding bin,

ˆn(Mi,z)= 1 ∆Mi Z ∆Mi dMn(M,z). (3.41)

This theoretical prediction is evaluated at a mass

Mi= 1 ¯n Z ∆Mi dMMn(M,z), (3.42)

The mean number density of haloes in thei₋th mass bin ¯n(Mi)

¯ni=

Z

∆Mi

dMn(M,z). (3.43)

In Fig. 3.7 we show the ratio of the measured halo mass function of theL-BASICCII simulation

to some models and fits provided in the literature. This figure shows that the Press-Schechter mass function overestimates high mass haloes and understimates the abundance of low-mass haloes. Note that this ratio decreases at low masses because of limited resolution. Therefore we can only describe main features on masses≥1014_M

⊙/h. In that range, deviations of.5%

are found in the Jenkins et al. (2001)(J), Warren et al. (2006)(W) and Tinker et al. (2008)(T) fitting formulae. The mass function of Sheth and Tormen (1999) acquires deviations of_≈40%. Figure 3.8 shows the measurements of the three outputs of the L-BASICC II simulations compared with Jenkins et al. (2001) mass function averaged in the same mass bins of the measurements. The points show the mean value for theNs=50realizations,

¯n(Mi)= 1 Nm Nm X j=1 ˆnj(Mi), (3.44)

where the error bars represent the1σ variance of the ensemble given by:

σ2(Mi) ¯n2 = 1 Nm−1 Nm X j=1 (¯n(Mi)−ˆnj(Mi))2 ¯n(Mi)2 . (3.45)

We find that the Jenkins et al. (2001) fitting formula describes the measured halo mass function fairly well. Small discrepancies are found on both tails of the mass function. In the low mass end, resolution effects leas to an underestimation of the halo abundance. Being the fiting formulae of Jenkins et al. (2001) calibrated withN₋body simulations with higher reso- lution, it has been proven to fairly well describe the halo mass function on larger dynamical ranges as the one explored by the L-BASICC II simulations. On the other hand, on the high mass end small discrepancies (below1σ) can be observed, especially at redshift z=1.

The expected variance in the halo mass function can be determined from the estimation of the bias treatment in Mo and White (1996). This is given as σ2

h(M,z) = b

2_{(M,z) ˜σ}2 m(M)

whereσ˜m is the rms of the matter fluctuation given by Equation (2.11) evaluated at a scale

R=(3V/4π)1/3 where V is the volume of the simulation. Since we measure the mass function

in bins of mass, the comparison must be done with averaged quantities in those bins. In addition, the discrete nature of the sample adds a shot noise contribution. Assuming a Poisson shot-noise, uncorrelated with the matter overdensity, this reads

σ2_h(Mi,z) ¯n2 i =ˆb2(Mi,z) ˜σ2m(z)+ 1 ¯niV , (3.46)

3.6 Halo bias and two-point statistics from N₋body simulations 51

Figure 3.9: Mean halo power spectrum at z=0 in real space for five different bins on halo mass: from lower to higher amplitudes (13.7 _≤ log M _≤ 13.9, 13.9 _≤ log M _≤ 14.1, 14.1_≤log M_≤14.3, 14.3_≤log M_≤ 14.5, 14.5 _≤ log M _≤ 14.7). The dot- ted lines represent the halo model prediction with a scale independent bias.

where the effective bias reads

ˆb(Mi)= 1 ¯ni Z ∆Mi dMn(M,z)b(M,z). (3.47)

The bottom panel of Fig.3.8 shows the comparison between the variance determined from the ensemble average of Equation (3.45) and the prediction from Equation (3.46), where a top-hat window function has been used, with the assumption of a linear halo-mass bias. This plot shows that Equation (3.46) can reproduce the ensemble averaged variance in the simu- lation with a very good agreement, especially at the high-mass end. A systematic increment (compared with the prediction) in the variance measured from the ensemble is detected on masses M.1014M_⊙/h. This deviation is stronger at redshiftz=0.

Note that the Jenkins et al. (2001) mass function has been calibrated with a numerical simulation (Evrard et al., 2002) that have the same characteristics of the L-BASICC II (except for the redshift). Therefore it is not surprising to find good agreement between the their fitting formula and the measurements we have shown here from the L-BASICC II simulation. The Tinker et al. (2008) mass function provides also a good fit for the mass range of the L-BASSIC II. When required, we use the Jenkins et al. (2001) mass function.