Since the formation and evolution of individual galaxy groups and clusters cannot be observed given the cosmological time scales that such processes take, it is common to study the statistics of the population of such systems as a function of redshift in order to understand their evolution. Given the close relation between cosmological parameters and the formation and evolution of galaxy groups and clusters, con- straints on the cosmological model can be established by observations of these objects. The distribution of dark matter haloes is termed the halo mass function, and it provides the number density of dark matter haloes, n(M, z), for a given mass at a given redshift. Since galaxy groups and clusters form and evolve together with such haloes, the function can also be interpreted as a galaxy cluster mass function (see Fig.2.6).
The first model for the halo mass function was developed by Press & Schechter (1974). Such model relates the mass function of objects resulting from non-linear collapse to the statistical properties of the initial linear density contrast field, by calculating the probability that such overdensity will collapse into a halo of a given mass. Thus, the halo mass function can be derived as a fraction of the total volume collapsing into halos of mass M, divided by the comoving volume occupied by each such halo within the initial density field (M/¯ρm):
dn(M, z)
dM =
¯ρm
Mg(M, z), (2.25)
where g(M, z) is given by the assumptions of the collapse model. Numerical simulations have shown that the shape of g(M, z) predicted by Press & Schechter (1974) deviates by & 50% when compared with the simulations results (e.g. Jenkins et al. 2001). Nowadays the halo abundance is measured and calibrated from large cosmological simulations.
Sheth & Tormen (1999) generalized the expression for the halo mass function in terms of the scaled differential mass function f (σ, z), which is given by
f(σ, z)= M ¯ρm
dn(M, z)
2.2 Galaxy groups and clusters as cosmological probes z = 0.55 − 0.90 z = 0.025 − 0.25 1014 1015 10−9 10−8 10−7 10−6 10−5 M500, h−1M⊙ N (> M ), h −3 M p c −3 ΩM= 0.25, ΩΛ= 0.75, h = 0.72 z = 0.025 − 0.25 1014 1015 10−9 10−8 10−7 10−6 10−5 M500, h−1M⊙ N (> M ), h −3 M p c −3 ΩM= 0.25,ΩΛ= 0,h = 0.72 z = 0.55 − 0.90
Figure 2.6:Illustration of the sensitivity of the cluster mass function to the cosmological model. Left panel: the measured mass function and predicted models are shown for two redshift bins. Right panel: same as the right panel, but now the data and the models are computed for a cosmology withΩΛ = 0. In this case the predicted
number density of z > 0.55 clusters is in disagreement with the data, and therefore this cosmological model (ΩΛ= 0.25 and ΩΛ= 0) can be rejected. Image adapted from Vikhlinin et al. (2009b).
Here ¯ρm = ρcrΩm is the mean matter density at z = 0, and σ(M, z) is the variance of the linear density field. Then, the halo mass function is given by,
dn(M, z) dM = ¯ρm M d ln σ−1(M, z) dM f(σ, z). (2.27)
This definition of the halo mass function does not explicitly depend on redshift, power spectrum, or cosmology, all of these are encapsulated in σ(M, z), which will be explained more in detail, before presenting a functional form of f (σ, z).
It is required that the spatial average of the density contrast, satisfies hδ(x, z)i= 0. By assuming that the initial density field is described by a Gaussian distribution, the mean, together with variance, completely describe the matter density distribution. The density distribution can be also defined in the Fourier space3, and the variance of δ(x, z) is given by
hδ2i= 1 2π2
Z ∞
0
P(k)k2dk, (2.28)
where P(k, z) is the power spectrum of the density fluctuations as a function of redshift. Objects of mass Marose from initial perturbations of size R(M)= (3M/4πρm)13. In this case, the density field is filtered4
with a window function, W, which smoothes out all the fluctuations of scales smaller than R. Therefore,
3In this case, the density distribution is described as a superposition of plane waves, which evolve independently one of each
other during linear evolution, and density contrast is given by ˜δ(k, z)= R d3xeik·xδ(x, z).
4Mathematically spatial filtering is equivalent to a convolution of the density field with a window function:
the variance of the perturbed field at scale R is given by σ2(M, z)= 1 2π2 Z P(k)| ˜W(k)|k2dk. (2.29) ˜
W(k) is the Fourier transform of the window function W. The usual shape for the window function is the top-hat filtering, which is constant within a sphere of radius R and zero outside. Its Fourier transform is given by
˜
W(k)= 3[sin(kR) − kR cos(kR)]
(kR)3 . (2.30)
The redshift dependence of σ(M, z) enters only through the growth factor D+(z) (see Eq.2.21), σ(M, z) = σ(M, 0)D+(z)
D+(0), (2.31)
i.e. it is assumed that the density perturbations continue to grow according to the linear growth factor, even when they have entered into the non-linear regime.
The power spectrum in Eq.2.29is a statistical description of the large-scale structure of the Universe. In the linear regime of structure formation, each fluctuation evolves independently and hence the evolution density is a linear function of the initial conditions. The growing solution (see Eq.2.21) is the mode that dominates the evolution, therefore the power spectrum is given by
P(k, z)= T2(k) D+(z) D+(0)
!2
Pin(k). (2.32)
T(k) is the transfer function which contains all the non-gravitational effects that modify the original lin- ear evolution of the power spectrum. In general, T(k) is redshift dependent. The initial power spectrum, Pin(k), is obtained by normalizing the linear growth factor, D+(z), to 1 at z= 0 in Eq.2.32: Pin(k)= Akns.
Here, nsis the index of the initial power spectrum, and can be measured through observations (ns∼1), and A is the amplitude at z= 0. A is directly linked to the normalization of the power spectrum, which is defined as the variance computed for at top-hat window having a comoving radius of R= 8 h−1Mpc (see Eq.2.29). This value of R is motivated by early results of galaxy surveys where it was found that δ(R = 8 h−1Mpc) ' δM/M ' 1, i.e. the variance of galaxy number density in spatial bins of 8 h−1Mpc is about unity (Davis & Peebles1983). This variance is better known as σ8 and determines the height of density peaks and consequently the number of haloes in the Universe. σ8and nsare two of the most important cosmological parameters that can be directly measured. The Planck mission has determined σ8 = 0.8159 ± 0.0086 and ns= 0.9667 ± 0.0040 (Planck Collaboration et al.2015a).
Coming back to the halo mass function determination, there have been many attempts to calibrate the halo mass function from cosmological simulations (e.g. Reed et al.2003; Warren et al.2006; Reed et al. 2007). In particular, the halo abundance function obtained by Tinker et al. (2008) is one of the most widely used in the galaxy cluster field. The parameterised Tinker et al. (2008) mass function is given by f(σ)= AT σ bT −aT + 1e−cT/σ2. (2.33)
Here, AT sets the overall amplitude of the mass function, aT and bT are the slope and amplitude of the low-mass power law, respectively. cT determines the cut-off scale at which the abundance of halos exponentially decreases. An advantage of this study is the publication of the fitting parameters as a
2.2 Galaxy groups and clusters as cosmological probes
function of the overdensity∆. Tinker et al. (2008) also found that the overall shape of the halo mass function has a redshift dependence. The redshift evolution of the parameters is given by
AT(z) = AT,0(1+ z)−0.14, (2.34) aT(z) = aT,0(1+ z)−0.06, (2.35) bT(z) = bT,0(1+ z)−αT, (2.36) log10αT(∆) = − 0.75 log10(∆/75) 1.2 . (2.37)
The simulations by Tinker et al. (2008) have a 5% statistical precision in halo number at z = 0 for a ΛCDM cosmology.
A final remark on the importance of the halo mass function is that it depends strongly (through σ(M, z)) on all the cosmological parameters that have been mentioned throughout this entire section (Ωm,ΩΛ, h, w, σ8, and ns, see Fig.2.6). Therefore, the abundance of massive systems, such as galaxy groups and clusters, is a determining probe of the current cosmological model5. Several on-going and future surveys
that aim to detect galaxy groups and clusters via optical, X-ray, and SZ observations (see Chapter4). It is expected that the number of detected galaxy groups and clusters will range from thousands to tens thousands. Then, in order to maximize the extraction of cosmological information from such surveys, the halo mass function should be known to a few per cent accuracy.6.