withAs = 0.3222, as = 0.707, andps = 0.3correlates well with mass functions arising from different numerical simulations.
While such models are useful, they require calibration against cosmological simulations.
Furthermore, since they do not encompass the complete complexity of halo formation, their accuracy is likely to be inadequate for precision cosmological constraints (Tinker et al., 2008).
The foundation for precision determination of the mass function from simulations was initi- ated by Jenkins et al. (2001) and Evrard et al. (2002), whose fitting function for the halo abun- dance was accurate to∼ 10%−20%. Extension and improvement to this work has come from many authors, including Tinker et al. (2008) who not only showed their halo mass function to have an accuracy of . 5%, but also demonstrated halo mass function ‘non-universality’ – the same functional form and parameters for the mass function cannot be used for different cos- mologies and redshifts. The universality and evolution of the halo mass function are currently the source of much interest and research.
2.3 Cluster Probes
The complex mechanisms existing inside clusters allow one to probe their dynamics through various observational techniques. In the following sections we outline several methods employed to detect and study clusters.
2.3.1 Clusters in Optical Light
Optical identification of clusters has been ongoing for many decades. Moreover, Charles Messier and William Herschel in the late eighteenth century had already recognised clustering of galaxies in the constellations of Virgo and Coma Berencies. With the improvement of telescope observ- ing power and software analysis techniques, optical cluster catalogues continued to grow over the next two centuries, resulting in the extensive cluster catalogues described in Abell (1958); Abell et al. (1989). The forementioned contain most nearby clusters and have provided a foundation for our understanding of clusters and their physical processes. More recently, large area surveys
2.3 Cluster Probes 28
using the latest detector technology as well as automated cluster detection techniques have pro- duced large catalogues of galaxy clusters (see for example: Koester et al., 2007; Postman et al., 2001, 1996; Gal et al., 2009, 2003, 2000; Gladders and Yee, 2005).
Each cluster observation method has advantages and disadvantages. Clusters are inherently rare objects and therefore we require large survey areas to locate a significant sample. Optical observations offer the opportunity to cover wide areas of the sky with large CCD frames coupled to telescopes with large fields of view. In addition, studies of the colour magnitude relation in red galaxies have allowed accurate distance measurements of clusters out toz ∼1(Gladders and Yee, 2005).
It is well known that cluster studies at this wavelength band have disadvantages. Since optical observations measure flux, which decreases with distance from the source, this technique is distance or redshift limited. To make matters worse, projection effects caused by background or foreground galaxies along the line of sight can contaminate cluster detection results. Moreover, in order for clusters to be cosmologically useful (in terms of parameter estimation) their masses are required. This parameter is difficult to measure straightforwardly from optical means. These issues served to motivate studies of clusters at other frequencies, particularly in X-rays.
2.3.2 Clusters in X-rays
A census of baryons in the local universe indicate that only about a tenth of the universe’s baryons lie within stars in galaxies, leaving the remainder in intergalactic space. These baryons are difficult to detect, however the large potential wells inherent to clusters compress the associated baryonic gas toT ∼ 107 K causing clusters to be conspicuous at X-ray wavelengths. If the gas shares the same properties as the cluster member galaxies, then the temperature is given by
kBT 'µmpσ2v '6
³ σv
103kms−1
´2
keV, (2.3.34)
wherempis the proton mass andµis the mean molecular weight - typically0.6for a primordial composition with76%hydrogen.
2.3 Cluster Probes 29
At the high temperatures described by the above relation, the intra-cluster medium (ICM) is analogous to a fully ionized plasma, whose major emission mechanism is thermal bremsstrahlung.
The emissivity of such a process occurring at frequencyνscales as
²ν ∼nenig(ν, T)T−1/2exp µ−hν
kBT
¶
, (2.3.35)
whereneandniare the electron and ion number densities respectively, andg(ν, T)∝log(kBT /hν) is the Gaunt factor. By integrating Eq. (2.3.35) over the gas distribution and X-ray emission en- ergy range, one obtains typical luminosities of Lx ∼ 1043 −1045erg s−1. These luminosities make it possible to identify clusters at high redshift at X-ray wavelengths.
The connection between local gas pressure, p, and density ρg, is easily understood if one assumes spherical symmetry and hydrostatic equilibrium:
dp
dR =−GM(< R)ρg(R)
R2 . (2.3.36)
Furthermore, by substituting the equation of state for a perfect gas into Eq. (2.3.36) one obtains an expression for the total mass with radiusR
M(< R) =−kBT R Gµmp
µdlogρg
dlogR + dlogT dlogR
¶
. (2.3.37)
IfRdenotes the virial radius, then at redshiftz the mass enclosed isM ∝ R3ρ¯0(1 +z)3∆v(z), where ρ¯0 is the cosmic mean density at the present time, and ∆v is the mean density at the virial radius (as defined in §2.2.6). Furthermore, if one makes the assumption that the universe is of the Einstein-de-Sitter form, then ∆v is constant, and the temperature of an isothermal gas distribution is related to the mass byT ∝M2/3(1+z). Thus, in addition to providing an efficient technique for cluster detection, X-ray observations of the ICM also provide a means to determine cluster masses, which is the parameter predicted by cosmological models of the universe. To complement this, X-ray emission in clusters depends on the square of the gas density, hence clusters standout strongly from regions of lower density. This property, in combination with the relatively low surface density of X-ray sources, mitigates projection effects which tend to plague cluster detection studies at other wavelengths
2.3 Cluster Probes 30
Since the first attempts in the 1970s to map the X-ray sky (Giacconi et al., 1979), obser- vations in this wavelength band have been prolific. Numerous surveys in the 1990s, such as those using the ROSAT satellite, have helped to constrain cosmological parameters (see Rosati et al., 2002, for a review of several cosmologically significant X-ray surveys). Today we are in the era of XMM-Newton and Chandra satellites, which together with the large survey area of XMM-Newton and the high angular resolution of Chandra, have started to shed light on the interplay between the complex dynamics of the intra-cluster medium and the detailed physics of star formation.
Unfortunately, X-ray and optical studies suffer a common drawback. Both of these methods depend on cluster luminosity and thus suffer from redshift dimming. In the next section we introduce the microwave regime as a novel method for cluster detection and outline its advantages over clusters studies at other wavelengths.
2.3.3 Clusters in the Microwave
Two decades after the prediction of the existence of the SZE (Sunyaev and Zeldovich, 1970, 1972) there were still only a few cluster detections, but in the following decade many new clusters were located at high significance using this phenomenon (Birkinshaw, 1999; Carlstrom et al., 2002). As we now enter the forth decade of SZE observations, improved detector sensitivity and large scale surveys, such as the Atacama Cosmology Telescope (ACT; Kosowsky, 2006), South Pole Telescope (SPT; Ruhl et al., 2004) and PLANCK (The Planck Collaboration, 2006), allow one to fully exploit the power of the SZE. Experiments, such as the aforementioned, will provide not only detailed images of clusters, enabling one to study the intra-cluster medium (ICM), but also large catalogues of SZE selected clusters over a wide range of redshift, permitting accurate measurements of cosmological parameters.
The SZE as an observational tool is particularly useful for deep surveys since the detection limit of a particular survey is fixed by the mass of the cluster. Furthermore, SZE surveys will be able to detect all clusters above a particular mass threshold independent of their redshifts. This remarkable property arises due to the fact that although the CMB suffers redshift dimming, the