Other mass estimation methods

Hydrostatic mass estimates have limited precision due to the required assumptions. Com- plementary observations of the galaxy population of clusters in the optical give rise to mass estimation methods which do not require the same assumptions. Microwave observations probe the ICM through the Sunyaev-Zel’dovich (SZ) effect and yield additional information about the hot cluster gas due to e.g. the different density dependence. Mass estimates from the optical or microwave range can thus be used to calibrate X-ray mass measurements.

Velocity dispersion of cluster galaxies

Optical observations probe the galaxy population and enabled early mass estimates based on the dynamics of the member galaxies (e.g. Zwicky 1937). Using the virial theorem and the knowledge of the galaxy positions and redshifts, the mass can be derived as

M = π 2 3σ2_vRV G with RV = N 2         X i>j r_{i j}−1         −1 , (1.8)

whereσvis the line-of-sight velocity dispersion of the member galaxies, RVthe virialization

radius, G the gravitational constant andπ/2 a geometrical factor.

Apart from the question to which extent the virial theorem holds for clusters, the definition of a galaxy member is most crucial in this method. Galaxy clusters are no isolated systems and

1.2 Mass estimates 15

it is often difficult to distinguish between galaxy members and the spurious inclusion of non- members lying in the line-of-sight. Including non-members would lead to an overestimation of the cluster mass, but different analysis techniques such as clipping in the velocity distribution minimize this bias (e.g. Beers et al. 1990; Biviano et al. 2006).

Regarding the validity of the virial theorem, many authors now solve the Jeans equation instead, which assumes that the cluster is in dynamical equilibrium and uses the radial depend- ence of the projected galaxy velocity dispersion (e.g. Carlberg et al. 1997; Biviano & Girardi 2003).

Weak lensing mass measurements

Gravitational lensing provides a mass estimation method, which is independent of the assump- tion of hydrostatic equilibrium and directly traces the depth and shape of the cluster potential (for a recent review of mass measurements from lensing see e.g. Hoekstra et al. 2013). Struc- tures along the line-of-sight, in our case a galaxy cluster, deflect photons which are emitted from sources more distant than the cluster and act as gravitational lenses (for a review on gravitational lensing see e.g. Bartelmann 2010). The deflection angle depends on the gradient of the cluster potential, decreases with distance from the lens and produces distorted (sheared) and slightly magnified images of background sources, typically high-redshift galaxies. Meas- uring the distortions provides information about the gravitational tidal field, independent of the dynamical state of the cluster. In the case of large deflection angles in the context of the small angle approximation (i.e.. 30′′), multiple images of the background source and arcs are observed. Such cases are called strong gravitational lensing and provide good mass estim- ates for the region of the lens which is enclosed by the distorted images. The most accurate estimates can be derived when the underlying potential is modeled to reproduce the observed signatures such as multiple images and arcs (e.g. Kneib et al. 1996; Broadhurst et al. 2005; Meneghetti et al. 2010). For smaller deflection angles and thus less obvious distortions, so- called weak lensing techniques are applied. The small shear distortion of a large number of background sources is measured and enables the reconstruction of the projected surface mass density.

This method is not based on the assumptions of hydrostatic equilibrium and spherical shape, but it requires a model for the underlying mass distribution and is thus also not an unbiased mass estimator. However, owing to the different assumptions made, weak lensing estimates can be used to calibrate X-ray mass estimates, which are observationally cheaper than weak lensing analyses (see Sect. 1.2.1 and Fig. 1.5).

The integrated Compton parameter YSZ

Information about the ICM can also be obtained from microwave observations through the thermal Sunyaev-Zel’dovich (SZ) effect (Sunyaev & Zeldovich 1970, 1972). Cosmic mi- crowave background (CMB) photons are Compton-scattered by free ICM electrons and shif- ted to slightly higher energies, which results in distortions of the black body CMB spectrum. The shape of the distorted spectrum is characterized by the Compton parameter y, which is proportional to the probability that a photon, which passes through the ICM, will be Compton scattered and the typical energy gain of the scattered photon. Since y gives the integrated thermal pressure of the ICM along the line-of-sight, it is a good proxy for the gas mass Mg

and consequentially the total cluster mass. For cosmological purposes, y is integrated over the solid angle A, which yields the integrated Compton parameter Y

Y =

Z

y dA ∝

Z

neTedV ∝ Mg Te, (1.9)

where A is the projected surface area, nethe electron density of the ICM, V the cluster volume

and Tethe electron temperature.

YSZ is a low-scatter mass proxy which is quite insensitive to the dynamical state of the

cluster (e.g. da Silva et al. 2004; Motl et al. 2005; Andersson et al. 2011; Planck Collabora- tion et al. 2013b), and hence is an ideal tool for precision cosmology. Contrary to X-rays, the SZ signal is redshift-independent and not subject to surface-brightness-dimming since the dimming is exactly compensated by the increase of the CMB intensity. This makes SZ ob- servations ideal for the detection of high-redshift clusters. On the other hand, this leads to the problem of projection effects due to overlapping SZ signals from structures at different redshifts (e.g. White et al. 2002).

Mass estimates derived from several wavelengths and methods have complementary ad- vantages and disadvantages. For example, X-ray data requires the assumption of hydrostatic equilibrium and spherical shape, which is not valid for dynamically young clusters. Projec- tion effects are problematic for the measurement of velocity dispersions, SZ signals or weak- lensing shear, but not for X-ray data. In addition, most SZ and weak lensing analyses are currently limited to larger scales because of their limited spatial resolution. Combining differ- ent measurements therefore yields the best way towards robust, morphology-independent and precise mass estimates.