3.3 Significance of galaxy cluster studies for cosmology
Having established that galaxy clusters form from the seeds of rare primordial density fluctuations, it is therefore expected that their abundance and mass/redshift distribution should give insight into the matter power spectrum, the amount of dark and baryonic matter in the universe as well as the dark energy equation of state. There are several different ways in which galaxy clusters can be used for cosmological studies. Below, I illustrate how different cosmological parameters affect our current picture of structure distribution across a wide redshift range in our universe.
First, it is necessary to find an expression that describes how the cluster number density within a given physical scale changes as a function of redshift and cosmological parameters. In a ’perfect survey’, survey-specific factors such as observable/mass survey limits, the survey spectral and spatial response and coverage do not need to be taken into account. Since, however, I will be discussing the neccessity of low-scatter scaling relations in section 3.4, this discussion will follow the approach and notation by Planck Collaboration XX (2013), which includes a discussion on survey limitations.
The galaxy cluster property which best describes its evolutionary state in the process of hierarchical structure formation is the massM∆within a given radiusr∆, characterized by the radius that encompasses a mean density, which is ∆times bigger than the critical density, ρc(z), at the cluster redshift. Suppose that a survey with a survey completeness fs detects galaxy clusters via the observable O∆ through its signal amplitude Of,∆ and characteristic angular extent Oθ,∆. The number of clusters observed within a redshift element dz and solid angle dΩ is then given by
dN
dzdΩ =, fs(M∆, z,/,η) dN
dzdM∆dΩdM∆, (3.17)
the survey completeness,
fs =, dO(f,∆) ,
dO(θ,∆)P(M∆, z|O(f,∆), O(θ,∆))finstr(O(f,∆), O(θ,∆),/,η), (3.18) being dependent on the intrinsic properties of the measurement characteristics finstr as a function of sky position (/,η) and also on the probability that the measured quantity reflects the true mass, a topic to which I will return in the discussion on scaling relations.
The instrument characteristics may include factors such as scanning strategies, beam char- acteristics as a function of position on the sky and sensitivity limitations of the survey.
As was discussed in the last chapter, in order to predict the evolution of linear density perturbations into the non-linear regime, N-body simulations are needed. Tinker et al.
(2008) propose a mass function, that describes the number density of halos of mass M, based on the assumption that a halo can be described by a spherical overdensity, whose radial extent is defined by a given overdensity within this sphere measured with respect to the mean density at the redshift of the halo1
1 I assume that, following Planck collaboration XX (2013) a conversion from the mean to the critical overdensity convention has been included, allowing the aperture radius to be labelled with∆
dN
dM∆(M∆, z) =fmf(σ)ρm,0 M∆
dlnσ−1
dM∆ , (3.19)
where
fmf(σ) =A /
1 +)σ b
*−a0
exp)− c σ2
*
, (3.20)
describes the evolution of the mass function with redshift since the parameters A, b,a,c depend on redshift. This non-universality is most likely caused by the evolution of the halo concentrations, which themselves are dependent on Ωm. This mass function should therefore only be used up to a redshift of z = 2.5 and for cosmologies for which Ωm is close to the flat ΛCDM cosmology (Tinker et al. 2008).
There are also analytical ways to approach the halo mass function, most notably the Press-Schechter mass function and the Press-Sheth-Torman mass function. The former relies on the spherical collapse model where the condition of collapse is given by the critical overdensity δc, whereas the latter apply an excursion set formalism to ellipsoidal collapse.
Since M∆ defines not only a certain mass but also a volume within which the mass is contained, dzdMdN∆dΩ is obtained from the multiplication of the cosmic volume element
dzdΩdV with the cluster mass function above.
From these considerations, and a given cosmological model, one can subsequently compute:
• The expected number of galaxy clusters above a certain mass threshold at the current epoch as a function of the matter density and σ8 (Rosati et al. 2002, Fig. 3.11a).
• The expected number of galaxy clusters for different matter densities and cosmolog- ical constant values as a function of redshift above a given mass threshold, given a uniform normalization at the current epoch (Rosati et al. 2002, Fig. 3.11b).
• The expected number of galaxy clusters as a function of redshift for a given dark energy content and σ8 but for different dark energy equation of state evolution parameters (Mohr et al. 2003, Fig. 3.11c).
As one can see from Fig. (3.11 a), the cumulative mass function atz= 0 is most sensitive to Ωm in comparison to ΩX since, for a small redshift range, the sensitivity for any evolutionary charateristics is not given. Given a measured value for the cumulative mass function, a higherΩm predicts a lower σ8. This is to be expected because a higher matter density requires lower initial amplitude fluctuations than a lower matter density for a given cumulative mass function value above a certain mass threshold at a given epoch.
A pedagogically intuitive illustration of the mass function dependence on the dark energy equation of state parameter is given in Mohr et al. (2003).
As one see in Fig. 3.11c , for a more positive w parameter than −1, i.e. a dark energy model instead of a cosmological constant model, structure forms more slowly at high redshift and the volume element is smaller than in the w=−1 case.
By investigating the cumulative mass function above a given mass threshold over a
3.3. Significance of galaxy cluster studies for cosmology 37
Figure 3.11: Cosmological predictions with galaxy clusters. a) The expected number of galaxy clusters above a certain mass threshold, M > 5×1014 h−1 M), at the current epoch as a function of the matter density and σ8 by Rosati et al. (2002). The coloured lines correspond to the same cosmological model colouring adopted in b. b) The expected number of galaxy clusters for different matter densities and cosmological constants as a function of redshift above a given mass threshold, given a uniform normalization at the current epoch by Rosati et al. (2002). c) The effect of different dark energy models on the cluster number counts (figure by Mohr et al. 2003).
wide redshift range in conjunction with knowledge of the current normalization, further studies on Ωm and ΩX can be made as is illustrated by Rosati et al. (2002), Fig. 3.11b.
In addition, more recently, Vikhlinin et al. (2009) studied 86 galaxy clusters separated into a low-redshift sample of 49 cluster inz<0.25 and a high-redshift sample of 37 clusters for which 0.35<z<0.9. They examined the mass function as a function of mass and redshift. From their low-redshift sample, they derive constraints on σ8 and Ωm. Using this information, by comparing their high and low-redhsift samples, Vikhlinin et al. (2009) were able to constrain ΩΛ and w0 (see table 2 of Vikhlinin et al. (2009) for the results, including the information from other experiments).
The above discussion, focussed on examining the masses of galaxy clusters over a wide mass and redshift range. Single cluster studies that have higher masses than expected from the currently accepted cosmological model, can however also contribute to the testing of
these models. The high-mass/redshift test is particularly sensitive at high redshift. Recent high-redshift, high-mass clusters have all passed the ΛCDM model test, the most famous being ’El Gordo’ from the ACT sample (Menanteau et al. 2012).
Under the assumption of spherical symmetry and exploiting the different distance dependences of X-ray and Sunyaev-Zel’dovich measurements, one can use galaxy clusters to measure the angular diameter distance as a function of redshift (Fig. 2.2).
Cosmological simulations show that the baryonic gas fraction in galaxy clusters should stay constant as a function of redshift. Hence, any evolution of the gas fraction must be attributed to an evolution due to the acceleration of the universe. This has been studied for a SZA/CARMA observed sample of clusters by LaRoque et al. (2006).
The lately reported tension in Planck number count derived and CMB-derivedσ8−Ωm
parameter space (Planck Collaboration XX 2013) has led to numerous discussions as to its origins. It is currently believed to be caused by a combination of mass function estimates, total mass/mass-proxy scaling relations and neutrino mass considerations - the respective contribution of these possible explanations is still an active area of research. This, once again, stresses the need for accurate and precise mass estimations for galaxy cluster studies.