et al., 2012). It can be seen that the use of lensing masses for absolute mass calibration of cluster observables has been possible only in recent years. In this thesis, lensing measurements by Klein et al. (in preparation) of an X-ray selected sample of 27 clusters (eDXL sample with median z ≈ 0.3, introduced in Chapter 3) will be used for measuring the relationship of cluster masses to cluster observables. A summary of the lensing analysis for this sample is presented in Chapter 3.
2.3 Linking cluster observables to mass: scaling relations
Detailed multi-wavelength studies of galaxy clusters are ideal for measuring individual cluster masses. These are especially important for cluster systems that deviate from relaxed dynamical state. Several authors have pursued performing detailed multi-wavelength analyses to trace the baryon and dark matter distribution in few clusters (e.g., Umetsu, Sereno et al., 2015; Jee et al., 2012). Only a fraction of galaxy clusters known today has been studied in galaxy weak-lensing as already described in Section 2.2.3.2. But it stands that for precision cosmology, we need mass measurements for a large sample of clusters and the size of the sample will grow to ∼ 100000 within next few years. Individual and detailed studies are expensive and not feasible for such large number.
In that, the self-similarity of structures becomes handy in providing a baseline model that link the cluster global properties to the cluster mass (Giodini et al., 2013). Below, is a description of the theoretical formalism for self-similar galaxy cluster observable relations to cluster mass.
Self-similar model
Kaiser (1986) showed that structures seeded by scale free initial conditions, and growing hierarchically under the sole action of gravity, follow self-similar relations between cluster mass and their global thermodynamic properties. According to the virial theorem, the total kinetic energy K and the total potential energy U of a virialised system are related as
2K= −U . (2.35)
For a monatomic gas with isothermal temperature T , the average kinetic energy per particle is hKi= 32kBT. Then, the total kinetic energy of the monatomic gas is 32NkBT. It is possible to express the relationship of the kinetic energy to the total gas mass Mgas,∆, i.e., K ∝ NkBT ∝ Mgas,∆kBT. The gravitational potential energy for a self-gravitating sphere of radius R∆and constant density ρ is given by U ∝R0R∆ 4π3 ρr3rρr2dr ∝ M
2 ∆
R∆.
Using the virial relation, the mass of a cluster and temperature of the gas medium are related as T ∝ M∆/R∆, where Mgas,∆ B fgasM∆and fgasis the gas mass fraction. Expressing R∆in terms of mean density of the cluster such that M∆ = 4π3∆ρc(z)R3∆, and substituting for R∆ in the mass and temperature proportionality gives the relation
T ∝ M2/3∆ ρc(z)1/3. (2.36)
Given that ρc(z)= H02E(z)2, the above relation can be written out as
Chapter 2 Clusters of galaxies
The X-ray bolometric luminosities from the bremsstrahlung emitting gas medium is ∝R nenpT1/2dV. The densities of electrons and ions are proportional to the cluster density ρ due to self-similarity. Then
Lbol∝ρ2R3∆T1/2∝ρM∆E(z)1/3M∆1/3 ∝ M∆4/3E(z)7/3, (2.38) where the M–T relation and the substitution of R∆with M∆is used.
The integrated Compton parameter YSZ, which is a product of density and temperature of electron in gas medium, is expected to have tight correlation with mass. Y ∝ R neTdV, which can be written as Y ∝ ρR3∆T ∝ M∆T and combining with Equation 2.37, we get
Y ∝ M 5 3
∆E(z)2/3. (2.39)
YXis the X-ray analogue of the integrated Compton parameter proposed by A. V. Kravtsov, Vikhlinin and Nagai (2006) as a robust X-ray cluster observable that is least affected by the dynamical state of the ICM gas. It is constructed from the product YX ≡ MgasT.
Empirical measurements of scaling relations
A review on scaling relations is given by Giodini et al. (2013). The above self-similar relations are theoretical expectations for gravity only cluster formation process. Deviations from the above relations are expected due to the diversity in the dynamical state of galaxy clusters and non-gravitational astrophysical process affecting the formation. The cluster observables such as Lx, YSZare expected to scatter from the power-law relationship with mass due to the cluster-to-cluster differences.
Two mass-observable relations become very important for near future cluster surveys. The Sunyaev- Zel’dovich observables to mass for the SZ surveys and X-ray luminosities to mass for eROSITA. Many empirical measurements of scaling behaviour between the observables and mass has been studied using observations and, as well as, simulations (e.g., Giodini et al., 2013). X-ray luminosity- mass relations have been measured by multiple authors and they find that the scaling relation deviate from the self-similar relations in their slope, and a large intrinsic scatter of ∼ 40% (Reiprich and Böhringer, 2002; Vikhlinin, A. V. Kravtsov et al., 2009; Giles, Maughan, Dahle et al., 2017; Reichert et al., 2011; Pratt et al., 2009; A. Mantz, Allen, Ebeling et al., 2010). X-ray luminosity is sensitive to the ICM density and variation in gas mass fraction within a cluster and with mass of the structures could lead to deviations from self-similar slope. Additionally, the cores of the ICM can be affected by energy injection processes such as AGN feedback and black hole formation. The presence of denser cores in some clusters leads to more efficient cooling and give rise to the cool-core clusters whose temperatures near the core is cooler than away from the core. Such cool-cores are thought to be responsible for the large scatter in X-ray luminosity relations (Pratt et al., 2009). Some authors excise the core for their luminosity measurements and re-produce self-similar scaling (e.g., Maughan et al., 2012).
A good mass proxy for cluster mass is one that is least affected by cluster-to-cluster differences. A. V. Kravtsov, Vikhlinin and Nagai (2006) proposed the proxy of thermal energy YX B MgasT, which shows small scatter. YX–M relation is measured to be following self-similar scaling (Vikhlinin, Burenin et al., 2009). Simulations predict the Y parameter from the SZ effect to scatter ∼ 6–12% (Stanek, Rasia et al., 2010; Yang, Bhattacharya and Ricker, 2010; Angulo et al., 2012; Sembolini
2.3 Linking cluster observables to mass: scaling relations
Figure 2.10: Covariance of cluster global properties at fixed mass from ∼ 4500 halos in the Millenium Gas Simulations. Off-diagonal terms show the pair-wise correlation between two cluster properties at fixed mass. Two physical processes were considered. A gravity only process (blue) and a pre-heating scenario (red), where the gas was considered to have an entropy floor of 200 keVcm2 at z = 4. The diagonals show the natural logarithmic deviations from the mean mass-scaling relation of each property. Credit: Allen, Evrard and A. B. Mantz (2011), originally adapted from Stanek, Rasia et al. (2010).
et al., 2013). Few authors have measured Y–M scaling relation using X-ray masses (e.g., Andersson et al., 2011; Czakon et al., 2015; Bonamente et al., 2008) and others using lensing masses (Hoekstra, Mahdavi et al., 2012; Marrone et al., 2012; Hoekstra, Herbonnet et al., 2015; Sereno and Ettori, 2015; A. Mantz, Allen, Morris et al., 2016). Most of these work observe self-similar scaling slope with a couple of exceptions (A. Mantz, Allen, Morris et al., 2016; Czakon et al., 2015). In general, the intrinsic scatter found in the Y–M relation is in the range 10–40%.
In the most general case, different cluster properties could be correlated for individual systems. It is becoming increasingly clear that for accurate understanding of the measured scaling behaviours, these intrinsic correlations are important for painting the complete picture (Allen, Evrard and A. B. Mantz, 2011; A. Mantz, Allen, Morris et al., 2016). But these correlations are challenging to measure from current data and, therefore, only few authors have attempted to include these correlations in their analysis of scaling behaviours (A. Mantz, Allen, Morris et al., 2016; A. Mantz, Allen, Ebeling et al., 2010). Due to challenges encountered in numerical simulations in producing detailed gas physics during cluster formation and evolution, only few authors have attempted to give predictions for these intrinsic correlations of cluster properties (Stanek, Rasia et al., 2010; Truong et al., 2016). A result from Stanek, Rasia et al. (2010) is shown in Figure 2.10, which measures the intrinsic correlations of four cluster properties, including the Lboland YSZ. Their simulation used two simple physical
Chapter 2 Clusters of galaxies
Figure 2.11: Systematic effects in cosmological constraints from galaxy clusters due to inaccuracies in the mass calibration of cluster observables. Green, blue and violet contours are 68% and 95% confidence regions of the constraints on the parameters obtained by using mass calibration from WtG, CCCP and CMB lensing estimates respectively. The systematic shift in the confidence levels from using the different mass calibrations demonstrate that the cluster abundance constraints on cosmology are currently limited by the uncertainties in the accuracy of mass-observable calibration. Credit: Planck Collaboration, Ade, Aghanim and al. (2016a).
process, a gravity only heating of gas and another scenario in which the gas medium was heated before collapse by an early energy injection at very early phases of cluster formation. Both these processes reveal that the thermodynamic cluster observables are strongly correlated at fixed mass in the range 0.5–0.8.