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7. GALAXY CLUSTER DISTANCE VIA X-RAY AND SUNYAEV-ZEL'DOVICH EFFECT

Galaxy clusters are the largest virialized astronomical structures in the Universe and observations of their physical properties can provide important cosmological information. An important phenomenon occurring in galaxy clusters is the Sunyaev-Zel'dovich effect (Sunyaev and Zeldovich 1972, Birkinshaw 1999, Carlstrom et al, 2002), a small distortion of the cosmic microwave background radiation (CMBR) spectrum provoked by the inverse Compton scattering of the CMBR photons passing through a population of hot electrons. The Sunyaev-Zel'dovich effect (SZE) is proportional to the electron pressure integrated along the line of sight, i.e., to the first power of the plasma density. The measured temperature decrement ∆TSZ of the CMBR is given by De Filippis et al (2005)

Equation 11

(11)

where Te is the temperature of the intra-cluster medium, kB the Boltzmann constant, TCMBR = 2.728° K is the temperature of the CMBR, σT the Thompson cross section, me the electron mass and f(ν, Te) accounts for frequency shift and relativistic corrections (Itoh et al, 1998).

Other important physical phenomena occurring in the intra-galaxy cluster medium are the X-ray emission caused by thermal bremsstrahlung and line radiation resulting from electron-ion collisions. The X-ray surface brightness SX is proportional to the integral along the line of sight of the square of the electron density. This quantity may be written as follows

Equation 12

(12)

where ΛeH is the X-ray cooling function of the intra-cluster medium (measured in the cluster rest frame), DA and DL are the angular diameter and luminosity distances of the galaxy cluster, respectively, and ne is the electron number density. It thus follows that the SZE and X-ray emission both depend on the properties (ne, Te) of the intra cluster medium.

As is well known, it is possible to obtain the angular diameter distance (ADD) of galaxy clusters by their SZE and X-ray surface brightness observations (SZE/X-ray technique). The calculation begins by constructing a model for the cluster gas distribution. Assuming, for instance, the spherical isothermal β-model such that ne is given by (Cavaliere and Fusco-Femiano 1978)

Equation 13

(13)

equations (11) and (12) can be integrated. Here rc is the core radius of the galaxy cluster. This β model is based on the hydrostatic equilibrium equation and constant temperature (Sarazin 1988). In this way, we may write for the SZE

Equation 14

(14)

where θc = rc / DA is the angular core radius and ∆T0 is the central temperature decrement that includes all physical constants and terms resulting from the line-of-sight integration. More precisely:

Equation 15

(15)

with

Equation 16

(16)

where Γ(α) is the gamma function and the others constants are the usual physical quantities. For X-ray surface brightness, we have

Equation 17

(17)

where the central surface brightness SX0 reads

Equation 18

(18)

Here μ is the molecular weight given by μi ≡ ρ/ni mp.

One can solve equations (15) and (18) for the ADD by eliminating ne0 and taking for granted the validity of cosmic distance duality relation (CDDR), DL(1 + z)−2 DA−1 = η = 1. In this case one obtains

Equation 19

(19)

where zc is the galaxy cluster redshift. Recently, such a technique has been applied for a fairly large number of clusters (Reese et al, 2002, De Filippis et al, 2005, Bonamente et al, 2006) with systematic and statistical errors around 20% and 13%, respectively (see table 3 in Bonamente et al, 2006). The crucial point in the SZE/X-ray technique is the choice of morphology used to describe the galaxy cluster. The standard spherical geometry has been severely questioned, since Chandra and XMM-Newton observations have shown that clusters usually exhibit an elliptical surface brightness (Sereno et al, 2006). The assumed cluster shape can affect considerably the SZE/X-ray distances, and, consequently, the H0 estimates and other astrophysical quantities.

7.1. SZE/X-ray technique applications: the Hubble constant

The Hubble constant, H0, sets the scale of the size and age of the Universe and its determination from independent methods is still worthwhile to be investigated. Severeal authors used the SZE/X-ray technique to estimate the Hubble constant (see table in Holanda et al, 2012a), however, since the samples used has galaxy clusters in high redshifts (up to z ≈ 0.90), a cosmological model had to be assumed in analysis, usually a flat ΛCDM model. Moreover, due to degeneracy on the cosmological parameters, the matter density parameter, ΩM, was taken as being ΩM = 0.3. An interesting method was adopted by Holanda et al (2012a). In this case, the authors used a sample of 25 ADD of galaxy clusters in redshift range 0.023 < z < 0.784 described by an elliptical β model (De Filippis et al, 2005) to constrain H0 in dark energy models. In order to avoid the use of priors on the cosmological parameters, a joint analysis involving the ADD, the baryon acoustic oscillations (BAO) and the CMBR Shift Parameter signature was proposed. By taking into account the statistical and systematic errors of the SZE/X-ray technique it was obtained for nonflat ΛCDM model H0 = 74+5.0−4.0 km/s/Mpc (1σ) whereas for a flat universe with constant equation of state parameter it was obtained H0 = 72+5.5−4.0 km/s/Mpc (1σ). These values are in full agreement with the latest local estimate performed by Riess et al (2016): H0 = 73.24 ± 1.74 km/s/Mpc (1σ).

7.2. The search for two numbers: H0 and q0

A kinematic method to access cosmic acceleration and the H0 value based exclusively on the SZE and X-ray surface brightness data from galaxy clusters also was investigated recently. This is a very important task since until very recently the type Ia supernovae observations was the unique direct access to the late time accelerating stage of the Universe. By assuming the current observational error distribution of the samples of 25 ADD from De Filippis et al (2005), Holanda et al (2013) performed Monte Carlo simulations based on a well-behaved parametrization for the deceleration parameter, q0, to generate samples with different characteristics and study the improvement on the determination of the cosmographic parameters: q0 and H0. As a interesting result it was shown that, even keeping the current statistical observational uncertainty, an increase in the number of data points increases considerably the figure-of merit for the cosmographic plane (hq0), where h = H0/100 (see figure 12).

Figure 12

Figure 12. Confidence contours (1, 2 and 3σ) on the plane (hq0). The panel left corresponds to the real data.

7.3. The search for new physics

The SZE/X-ray technique also has been shown to being a powerful tool to investigate fundamental physics. Uzan et al (2004) argued that this technique is strongly dependent on the validity of the CDDR, DL(1 + z)−2 DA−1 = η = 1, valid for all cosmological models based on Riemannian geometry, being dependent neither on Einstein field equations nor on the nature of matter (Etherington 1933, reprinted as Etherington 2007), playing an essential role in modern cosmology. It only requires that source and observer are connected by null geodesics in a Riemannian spacetime and that the number of photons are conserved. However, if η ≠ 1, instead of the real ADD, the measured quantity is DAdata(z) = DA(z) η2(z). By using different expressions for η(z), several authors have tested the CDDR by using ADD of galaxy clusters and DL from type Ia supernovae observations (Holanda et al, 2010, 2012b, Li et al, 2011b, Nair et al, 2011, Yang et al, 2013, Holanda and Barros 2016). No significant deviation was obtained when an elliptical β model was used in analysis (for results from different cosmological observations see table I in Holanda et al, 2016.

As was showed by Holanda et al (2016), this technique depends on the fine structure constant, α. If α is a time-dependent quantity, e.g., α = α0 ϕ(z), where ϕ is a function of redshift, the current ADD data do not provide the real ADD to the cluster but instead DAdata(z) = η2(z) ϕ(z) DA(z). Constraints on a possible variation of α for a class of dilaton runaway models was performed considering the sample of DAdata(z) from De Filippis et al (2005) and estimates of DA(z) from type Ia supernovae observations. It was found no significant indication of variation of α with the present data.

Finally, it is very important to stress that the SZE/X-ray technique is independent of any calibrator usually adopted in the determinations of the distance scale. The above results, therefore, highlight the cosmological interest in ADD measurements of galaxy clusters.

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