Kaiser (1986) developed a model to describe the observable properties of galaxy groups and clusters. This modelling is based on three key assumptions:
1. Galaxy groups and clusters form via gravitational collapse from initial peaks of the density field. Therefore, the gravitational collapse is scale-free or self-similar.
2. The initial fluctuations do not have a preferred scale. Hence, the amplitude of the density contrast can be described by a power-law,∆(k) ∝ kn.
3. The physical processes that take place during the formation and evolution of galaxy groups and clusters do not introduce new scales in the problem, i.e. the only source of energy input into the ICM is gravitational.
This modelling has a major effect on the description and study of galaxy groups and clusters: when describing such systems as self-similar it means that they are simply scaled versions of each other, i.e. all galaxy groups and clusters are essentially identical. The self-similarity can be split into two regimes:
• Strong self-similarity:all galaxy clusters of different masses are identical scaled versions of each
other.
• Weak self-similarity: the density of the Universe changes as a function of redshift,
ρcr(z)= ρcr(0)E2(z), i.e. the density was higher at early epochs. This change has to be taken into
5Although, the halo mass determination depends also on the precise determination of the galaxy group and cluster masses. 6In this sense, the halo mass function should be calibrated for a wider range of masses and redshifts, as well to include the
account when comparing low and high redshift galaxy clusters, and it is known as self-similar redshift evolution.
Although the assumptions made by Kaiser (1986) oversimplify the problem of galaxy group and cluster formation and evolution, they predict simple power-law relationships between the different properties of such systems. Since such relations are very important for studies of galaxy groups and clusters, they will be explained in more detail in the following.
Scaling relations
As mentioned before, scaling relations are relations that describe the relationship between different galaxy cluster properties. Such relations are very important since they relate easily observable quantities to other properties which are difficult to determine by observations. For example, the galaxy group and cluster mass is one of the most essential properties to be determined, but cannot be measured directly. In this sense, precise measurements of galaxy cluster masses and their evolution with time are important because they provide constraints on cosmological models (see Section2.2.3).
Some galaxy groups and cluster scaling relations have been already presented throughout this chapter. For example, one of the most obvious is the relation between the mass and radius (Eq.2.23): M ∝ r3. However, such relation should also be translated into its equivalent at earlier epochs, i.e. expressed as a function of redshift. Voit (2005) discusses three different forms of redshift parameterisation for scaling relations, in this work the third form will be used since the scaling involves the analytical solution from the spherical collapse model. Therefore, the mass-radius M − r relations states:
M ∝ r3∆(z)E2(z). (2.38)
Such relation indicates that objects at high redshift have smaller sizes. Bryan & Norman (1998) provide an approximation for∆(z) = 18π2+ 82[Ωm(z) − 1] − 39[Ωm(z) − 1]2, whereΩm(z)= (1 + z)3E−2(z). Since most of the work presented in this thesis focuses on the X-ray aspects of galaxy clusters, the following scaling relations will be related with X-ray observables, skipping the optical and SZ properties (for a more complete review on scaling relations see Böhringer et al.2012and Giodini et al.2013). From X-ray observations one can directly derive the following galaxy cluster properties:
• Mass-Temperature relation, M − T. The self-similar model relates the cluster mass and tem-
perature from Eq.2.8
M ∝ T32∆−12(z)E−1(z). (2.39)
Conversely, one can obtain the T − M relation: T ∝ M23∆13E23(z). This relation shows that
objects of the same mass are hotter at higher redshifts. It is one of the most fundamentals scaling laws since it is obtained directly from the energy budget of the cluster gas. As explained in Section2.1.2, the global temperature can be obtained from the observed spectrum. Observations of galaxy clusters are consistent with such scaling relation (e.g. Reichert et al.2011).
• Mass-Luminosity relation, M − LX. The X-ray luminosity is obtained by integrating the bolo-
metric emissivity (Eq.2.3) over the volume: LX = RVdV ∝ ρ2
gasT12r3 ∝ ∆(z)E2(z)T12M ∝ M
4
3,
using Eq.2.39. Then,
2.2 Galaxy groups and clusters as cosmological probes
This relation shows that objects of the same mass are more luminous at higher redshifts. The X- ray luminosity can be measured from the flux and redshift of the source (see Eq.2.6). This scaling relation is very important for high redshift galaxy clusters, which are usually faint in X-rays, and through this relation one can determine their masses. By using Eq.2.39, the L − T relation can be obtained: LX ∝ T2∆12(z)E(z). Such relation is one of the most widely used since both observables
are derived almost independently. However, the power-law has been measured to be steeper than predicted (e.g. Pratt et al.2009; Reichert et al.2011).
• M − Y
X relation. Motivated by the SZ observations, YX is the product of kBT and Mgas (see Eq.2.12), then YX∝ MgasT ∝ fgasM M23 ∝ M53. Then,
M ∝ Y 3 5 X∆− 1 5(z)E25(z). (2.41)
This scaling relation has been proved to be very robust, and it has low scatter. From observations it has been measured to follow the self-similar expectation (e.g. Arnaud et al.2007).
In general, scaling relations have to be calibrated in large samples of well-determined galaxy cluster properties. Moreover, such relations are expected to have an intrinsic scatter, which has to be understood in order to obtain precise scaling relations. Once the calibration has been achieved, scaling relations can serve as mass proxies of low mass and/or high redshift galaxy groups and clusters, especially for the objects with low-quality data. This part will be an important tool for upcoming X-ray galaxy cluster surveys (see Chapter4).
Deviations from the expected self-similar scaling relations are commonly attributed to non-gravitational effects in the ICM. Such processes are complex baryonic physics phenomena like AGN feedback, star formation, supernova explosions, cooling flows, and shocks. Furthermore, merging systems also add scatter to the expected scaling relations.