Now we have described various methods to determine the cluster mass, it is possible to use clusters for cosmological studies. The cosmological constraining power of clusters arises from two measurements – the baryon fraction fb, and the cluster mass function.
1.5.1 Baryon fraction
Assuming that galaxy clusters are large enough to enclose a representative volume of the universe, the ratio of baryonic mass to total cluster mass is expected to match the cosmic baryon fraction. fb = Mb Mtot = ⌦b ⌦M (1.58) Knowing that the baryon content of clusters can be described by a gas and stellar component, Equation 1.58can be written as
fb= Mb Mtot = Mgas+ Mstars Mtot (1.59) First measurements of the baryon fraction found a value of fb ⇠ 0.1 (White et al., 1993), which provided early evidence that ⌦M was considerably less than 1 and therefore
an Einstein-de Sitter cosmology was incorrect. Another assumption about the baryon fraction is that it is expected to be the same at all redshifts, since after cluster formation, all clusters can be treated as representative “standard buckets” for the universal baryon
fraction. Therefore, fb depends only on the distance to the cluster, which is sensitive to
the underlying cosmology via E(z). To determine the distance to the cluster, we consider a cluster with an angular size on the sky defined by ✓. We can find the proper radius of the cluster via
R = ✓dA (1.60)
The proper volume of the cluster is then, V = 4
3⇡(✓dA)
3 (1.61)
We can then rewrite Mgas as a product of density and volume, using Equation1.61,
Mgas/ ⇢gas(✓dA)3 (1.62)
Recalling Equation1.23, we note that the gas density is related to the X-ray luminosity, which, in turn, is related to the observed flux via Equation 1.12. Therefore we can write the gas density in the following terms,
⇢gas/ (L/V )1/2 (1.63)
which, using Equations 1.61and 1.12, we obtain ⇢gas/ F d 2 L ✓3d3 A !1/2 (1.64) which, rewritten in terms of Mgas from Equation 1.62, becomes,
Mgas/ dLd3/2A (1.65)
When evaluating the total cluster mass Mtot, we can use the fact that Mtot / dA via
the cluster radius, so the estimated value of fb scales with distance according to
fb =
Mgas
Mtot / dL
d1/2A (1.66) Hence, cosmology via the baryon fraction measurement requires knowledge of dA, dL
to the cluster redshift z. However, if we know the predicted value of fb from theory, we
can infer values of dA, dL and see how well it fits the observed data for a given underlying
cosmology (Allen et al., 2002, 2008). The resulting constraints for the fb test are com-
parable to those from independent probes such as the CMB and type-Ia supernovae (see Figure 1.9).
Some caveats to note for this cosmological application is that the value of Mtot might
Figure 1.9: Constraints on parameters ⌦M and ⌦⇤ from the cluster baryon fraction
measurement from Allen et al. (2008). The best fit cosmological values are ⌦M ' 0.27,
⌦⇤ ' 0.86 with 1 and 2 confidence regions shown. The CMB and type-Ia supernovae
constraints are shown in blue and green, respectively.
therefore most reliable when used on a sample of large, relaxed clusters. The value of the baryon fraction might also not be constant if evaluated at smaller radii (e.g. R2500) due
to the ‘missing baryon problem’ thereby deviating from the assumption that the baryonic mass in clusters is representative of the universal baryon fraction (e.g. Ettori,2003).
1.5.2 Cluster mass function
Clusters arise from the gravitational collapse of rare high peaks of primordial density perturbations in the early universe (Peebles,1993). Because they probe the high end of the cosmic density field, the observed abundance of galaxy clusters is sensitive to particular cosmological scenarios (Press and Schechter, 1974). The evolution of galaxy clusters is driven by the growth rate of density fluctuations, which essentially depends on the value of the matter density parameter ⌦M and 8 (seeOukbir and Blanchard, 1992,1997;Eke et al.,1998). Depending on a whether the universe has a high or low density, the cluster population evolves di↵erently, with each case producing a distinct spatial distribution of clusters as a function of mass and redshift in a comoving volume. This is known as the cluster mass function. Various cosmological models are able to predict the number density, which allow for the determination of the best-fit cosmology based on the agreement of the
Figure 1.10: N-body simulation fromBorgani and Guzzo(2001) illustrating cluster evolu- tion in two cosmological models. The top panel shows a dark matter distribution in a flat, low density model with ⌦M = 0.3. The bottom panel describes an Einstein-de-Sitter case
with ⌦M = 1. Each panel consists of three redshift snapshots. The yellow circles mark
the positions of galaxy clusters selected based on their predicted X-ray properties. model with the data.
The mass function for di↵erent cosmologies can be derived analytically, however they are most commonly measured from large volume, N-body simulations (see Figure 1.10). Note that the late time structure is a poor discriminator for the two di↵erent cosmolo- gies (right panels in Figure 1.10), and thus requires the leverage of redshift evolution to illustrate the growth of clusters over time.
To measure the mass function observationally, three steps are required. Firstly, cluster surveys are required to detect and count galaxy clusters. Next, the survey selection func- tion determines the volume in which clusters have been detected and counted. Finally, the masses of clusters in a given volume must be calculated, typically through use of scaling relations discussed in Section1.4.
As discussed in Section 1.2, various multi-wavelength signatures for clusters have fa- cilitated their detection. Among these are red sequence based cluster surveys, SZ surveys and X-ray surveys. Each of these detection methods have their benefits and caveats, which are folded into the selection function of the chosen survey. A key property of each survey is the limit at which it can no longer detect clusters, known as the detection limit. To determine the survey volume, we start with the survey area, defined by the solid angle ⌦ on the sky, in which we expect to detect n clusters in some mass bin (M ± M).
The comoving volume element can be written as a product of the comoving area element multiplied by a redshift range according to
dV (z) = c a0H0
(1 + z)2d2 A
E(z)d⌦dz (1.67) This volume be integrated over the solid angle and redshift range of the survey to determine the full survey volume. The redshift range is defined between z = 0 and zmax, at which the
detectability of a cluster drops below the survey limit. Non-trivially, the value of zmax also
depends on the mass of the cluster considered since more massive (e.g. X-ray bright or optically rich) clusters are detectable to higher redshifts. Therefore, the redshift limit can be seen as a function of the cluster mass, zmax(M ). The survey area ⌦ is also dependent
on the cluster mass, due to its dependence on source flux. To estimate the total survey volume or the ‘selection function,’ the volume element is integrated
V (M )⇠
Z zmax(M )
0 ⌦(M, z)dz (1.68)
From Equation 1.68, it is clear that the survey volume scales with the mass of detected clusters. In other words, large survey volumes correspond to massive clusters, which are brighter and easier to detect. For example, in X-ray surveys, the computed value of V (M ) arises from the luminosity-mass relation (see Equation 1.54), since the flux limit of the survey corresponds to some mass on that relation.
Examples of measurements of the cluster mass function can be found in Reiprich and B¨ohringer (2002) and Vikhlinin et al. (2009). Figure 1.11 shows two illustrations of a measured cluster mass function, fitted to two underlying cosmologies. The cluster sample is also split into two redshift ranges (0.025 < z < 0.25 and 0.55 < z < 0.9). It is clear that the cosmological model on the left-hand side is a better fit to the data, thereby ruling out an Einstein-de Sitter model. Another key point is that the both the theoretical and observed cluster mass function are redshift-sensitive, as shown by the fact that the blue and black lines (high and low redshift, respectively) scale di↵erently as a function of cluster mass. Although this is a powerful measurement, it is subject to numerous systematics. The dominant source of error in mass function estimates arises from the choice of mass- observable relation used by a given survey, which has some quantifiable scatter.
Using the earlier example of an X-ray survey, the selection of clusters based on the luminosity-mass scaling relation may not reflect the ‘true’ population of clusters, as clusters with an anomalously high flux for their chosen mass may be over-represented. This is known as Eddington bias, described in Figure 1.12). This bias can be minimised with improved knowledge of the scatter in the luminosity-mass (or indeed any mass-observable)
Figure 1.11: Illustration of sensitivity of the cluster mass function to the cosmological model from Vikhlinin et al. (2009). The left panel shows the measured number density and in a low (black line) and high (blue line) redshift ranges computed for a low density cosmological model. The right panel shows both the data and the model for a cosmology with no dark energy.
relation, and has been implemented in various studies. The motivation for Chapter4is to produce precise stacked weak lensing mass estimates for a sample of clusters, which can be used to measure a cluster mass function and relevant cosmological parameters.