Métricas para la calidad interna, externa y en uso ISO/IEC 25022 e

In the last two sections some important aspects of the close interplay between the ob- servable properties of galaxy clusters and the underlying global geometry and structure formation process in the Universe were introduced. This section will shortly summarize a number of cosmological tests that use galaxy clusters as probes and have been successfully applied in the past or will become feasible within the next decade.

For the dawning era of precision cosmology it will be crucial to measure cosmological parameters with tests sensitive to the global geometry of the homogeneous Universe and methods probing the structure evolution of theinhomogeneous Universe. The very different physical nature of the methods and their underlying theoretical frameworks will allow im- portant consistency cross-checks of the emerging cosmological concordance model. Galaxy clusters in this respect enable several independent cosmological tests for both fundamental aspects which qualifies the cluster population as one of the most versatile cosmological probes.

The majority of cosmological tests can be reduced to the measurement of effectively four cosmic evolution functions for which the dependence on the underlying cosmological parameters have been discussed: (i) the cosmic expansion history H(z) (Equ. 3.16), (ii) cosmic distancesD(z) (Equs. 3.19, 3.20, 3.21), (iii) cosmic aget(z) (Equ. 3.28), and (iv) the linear growth factor D+(z) (Equ. 3.32). The first three methods are linked to the global

geometry and probe the homogeneous Universe, the last one traces the structure growth of the inhomogeneous Universe.

(1) Cluster Mass Function: The cluster abundance of the local Universe mainly de- pends on the matter density Ωm and the amplitude of the density fluctuation field

σ8. The mass function is related to the statistics of rare high amplitude peaks in

the primordial matter distribution. For a direct comparison to the observed cluster population, the predicted mass function n(M) is converted to the measured X-ray luminosity function (XLF) via φ(LX) =n[M(LX)]·dM/dLX. The X-ray luminosity

function at z∼0 has been well established (B¨ohringer et al., 2002a) and serves as local reference for evolution measurements. The local XLF is degenerate concerning the parameter combination σ8·Ω∼m0.5, i.e. the effect of a larger fluctuation amplitude

can be compensated by a lower matter density. The degeneracy can be broken in combination with the power spectrum or by measuring the redshift evolution in the next test.

(2) Number Density Evolution: The redshift evolution of the cluster abundance out to z∼1.5 will be the main cosmological test of the final XDCP sample. The evolu- tion of the cluster mass function n(M, z) combines two tests simultaneously: (i) the comoving volume elementsdVcom∝D2/H(z), governed by the expansion history, and

(ii) the growth of the mass fluctuation spectrum driven by D+(z). The cosmological

test also becomes increasingly sensitive to variations of the Dark Energy parameters ΩDE and w at redshifts above z >∼0.8. Up to now, the changing demographics of

the entire cluster population has been investigated out to intermediate redshifts of z∼0.8 by Mulliset al. (2004) and Borgani et al.(2000) (see Fig. 4.3), but only weak evolutionary effects have been found. The redshift at which the cluster number den- sity decreases significantly is yet to be determined and will be one of the goals for XDCP.

(3) Amplitude and Shape of Power Spectrum: The clustering properties and the general shape of the cluster power spectrum depend on the matter density Ωm and

the amplitude of the dark matter fluctuation spectrum σ8 as discussed in Sect. 3.2.2.

Reliable measurements require large area contiguous surveys such as the ROSAT All- Sky Survey. The currently best measured cluster power spectrum has been obtained with the REFLEX survey (see Fig. 4.2) which will soon be improved upon with the extended NORAS 2 and REFLEX 2 samples (see Sect. 4.2). Future X-ray surveys such as eROSITA (see Sect. 12.2) will be able to measure the power spectrum as a function of redshift and will thus directly probe the evolution of the growth function D+(z).

(4) Baryonic Acoustic Oscillations: Measurements of the locations of the acoustic peaks in the cluster power spectrum as a function of redshift will be one of the main aims and challenges of the eROSITA mission (Sect. 12.2) and will require a total of 50 000–100 000 clusters with approximate redshifts. This cosmological test, however, is one of the cleanest methods to probeDark Energy since it is least affected by systematic uncertainties. It is based on the well-understood ‘standard ruler’ im- printed by the sound horizon at the epoch of last scattering DSH

rec. Measurements of

while measurements in the radial direction trace the expansion historyH(z) through Equ. 3.19.

(5) Cluster Baryon Mass Fraction: This conceptually simple and attractive test rests upon the assumption that clusters, as objects that have collapsed from a region of 10h−1_{Mpc, represent a fair accounting of the relative matter content in the}

Universe, i.e. that their baryon mass fraction fb is representative. The cluster gas

mass fraction fgas can be well measured with X-ray observations. With corrections

for the galaxy mass in clusters, one obtains the universal total cluster baryon fraction which is asymptotically approaching values offb∼0.12–0.14h703/2towards the cluster

outskirts. If the baryon content Ωb of the Universe is taken as prior from cosmic

nucleosynthesis calculations, the total matter density can be determined by Ωm=

Ωb/fb (e.g. Allen et al., 2003).

(6) Evolving Gas Mass Fraction: This method is a generalization of test (5) and uses the cluster gas mass fraction fgas as ‘standard ruler’ to probe the global geometry of

the Universe (Sasaki, 1996). The additional assumption is that the gas mass fraction is universal at all redshifts fgas(z) ≈ const. Since measurements of the gas mass

(Mgas∝h−5/2dang5/2) and total mass (Mtot∝h−1dang) obey different scaling relations

with distance, the resulting observed gas mass fraction scales with fgas(z)∝d3ang/2(z).

The measured gas mass fraction hence depends on the assumed angular diameter distance to the cluster, and will deviate from the (assumed) universal value ¯fgas(z=

0), if the reference cosmology is incorrect. This geometric test can constrain the parameters Ωm, ΩDE, and w and has been applied by Allen et al. (2004) based on

detailed X-ray observations of 26 dynamically relaxed clusters at z <_∼0.9.

(7) Absolute Distance Measurements: When X-ray and SZE observations are com- bined, the absolute distance to a cluster can be determined, allowing the measurement of the Hubble constant H0. The SZ effect (see Sect. 2.3.6) yields the observed tem-

perature decrement ∆T ∝R neTedl∝n0T0Rcl, and the observation of the thermal

X-ray emission (see Sect. 2.3.1) results in the flux SX∝

R

n2

eΛ(Te)dl∝n20Rcl, where

Rcl is the clusters radius,n0the gas density, andT0 the gas temperature. Elimination

ofn0 results in an expression for the physical cluster radiusRcl, which can be related

to the observed angular diameterθcl to yield the angular diameter distance via

dang(z) = Rcl θcl = ∆T 2 SX 1 T2 0 θcl ·const≈ c z H0 . (3.44)

The last step is a low redshift approximation for H0, which can be generalized for

more distant clusters to H(z). Achievable accuracies for H0 based on individual

cluster measurements are about ±20 km s−1_Mpc−1_.

(8) Cluster Number Counts: Cumulative cluster number counts provide a model inde- pendent test since only X-ray flux measurements are needed. Generalizing Equ. 3.26

for the cluster population yields the number of clusters per steradian brighter than flux S (e.g. Borgani, 2006) as

N(> S) = Z _∞ 0 dz cD2_(z) H(z) · Z _∞ S dS n[M(S, z);z] dM dS . (3.45)

Here the first integral accounts for the comoving volume per steradian (Equ. 3.25) and the second term yields the observable clusters in this volume at fluxS. Deviation of the Euclidean slope of Equ. 3.26 are hence related to (i) the flux limit, (ii) evolution effects of the cluster population, and (iii) cosmological distances. A compilation of different log N–log S functions is shown in Fig. 4.6. The observed slope on the bright end starts with the expected Euclidian value 1.5 and then flattens to a value of ' 1 towards faint fluxes (Rosatiet al., 2002). Cluster number counts are nowadays mainly used as a consistency check with other surveys and for an estimation of the survey flux limit.

(9) BCG Hubble Diagram: The brightest cluster galaxies exhibit a remarkably low scatter of about±0.3 mag in their average absolute magnitude ¯MBCG, which qualifies

them as decent ‘standard candles’, similar to supernovae type Ia. With the observed apparent BCG magnitudes mBCG of a given filter, we arrive at the mBCG–log(z)

Hubble diagram by combining Equ. 3.23 & 3.24

mBCG(z) = MBCG+KBCG(z) + 25 + 5 log (dlum[Mpc])−5 logh70

=MBCG+KBCG(z)+25+5 log à c h−1 70 H0 [Mpc] ! +5 log à (1+z) Z _z 0 dz0 E(z0₎ !

=MBCG+KBCG(z)−5 logh70+ 43.16 + 5 log(1+z) + 5 log

ÃZ _z 0 dz0 E(z0₎ ! (3.46) Here KBCG(z) is the redshift dependent K-correction and in the second line the

luminosity distance20 _d

lum was substituted with Equ. 3.20. Assuming a concordance

model cosmology and considering redshift dependent terms up to an order of O(z2₎

in Equ 3.46, we obtain the low-redshift approximation

mBCG(z)≈MBCG+KBCG(z) + 43.16 + 5 logz+ 1.68·z , (3.47)

with the expected linear m(z)∝logz behavior for the local approximation z¿1. Variations of the Hubble constant H0 result in a constant vertical offset, whereas

the sensitivity on the cosmological parameters Ωm, ΩDE, and w change the form

of the Hubble relation at higher redshifts (last term in Equ 3.46). Based on this effect, SNe Ia measurements have established the existence of a non-vanishing Dark 20_{Assuming a flat geometry,i.e.k= 0.}

Cosmological Test Effective Test No _Parameters

1 cluster mass functionn(M) D+ tens Ωm, σ8

2 number density evolution n(M, z) D+(z), Vcom(z) tens Ωm,σ8,ΩΛ,w 3 cluster power spectrumP(k) D+ hundreds Ωm, σ8 4 baryonic acoustic oscillations dang(z), H(z) ten thousands Ωm, ΩΛ, w

5 cluster baryon fraction fb Ωb/Ωm few Ωm

6 evolving gas mass fraction fgas(z) dang(z) tens Ωm, ΩΛ, w

7 absolute distances dang(z) few H0

8 cluster number counts N(> S) D+ hundreds Ωm, σ8 9 BCG Hubble diagram mBCG(z) dlum(z) tens Ωm, ΩΛ

Table 3.2: Cosmological tests using X-ray galaxy clusters as probes. The table summarizes the discussed methods, the effectively measured cosmic property, lower limits on the number of objects required, and the cosmological parameters for which the method is most sensitive to. The first six tests have the potential to provide competitive parameter constraints, while the last three are to be seen as consistency checks without strong constraining power.

Energy component (Perlmutter et al., 1999). The BCG Hubble diagram has been applied for cosmological studies by Collins & Mann (1998) and by Arag´on-Salamanca, Baugh & Kauffmann (1998) using clusters at z <1. With the concordance model in place, the BCG Hubble diagram is nowadays more suitable to probe evolutionary effects in the brightest cluster galaxy population as will be shown in Sect. 11.1. The cosmological tests using X-ray luminous galaxy clusters as probes are summarized in Tab. 3.2. The methods can be loosely classified into two categories: (A) tests 1–6 having the potential to yield competitive constraints on cosmological parameters, and (B) tests 7–9which are more to be considered as consistency checks, either for the underlying model (7), survey cross-comparisons (8), or to disentangle evolutionary effects (9).

For distant cluster cosmology, the uncertainties in the mass calibration derived from observed properties and the redshift evolution of the scaling relations pose the largest current limitations. Improving these uncertainties and characterizing z > 1 clusters as cosmological probes are important goals for the XDCP survey. The main cosmological application of the XDCP sample will be test 2, which is expected to yield competitive parameter constraints 4–5 years from now. For tests 6 &7, XDCP can provide additional high-z clusters in order to increase the redshift leverage. Methods 8 & 9 are applicable even for an uncompleted survey and are presented in Sect. 9.1.2 & 11.1.

X-ray Cluster Surveys

This last introductory chapter will shortly summarize some of the major achievements of galaxy cluster X-ray surveys. In addition, the state-of-the-art of distant X-ray luminous clusters at the start of this thesis will be discussed.

4.1

X-ray Selection

Cosmological evolution studies of the galaxy cluster population impose three essential requirements on any suitable survey sample (e.g. Rosati, Borgani & Norman, 2002):

1. An efficient method to find clusters over a wide redshift range; 2. An observable estimator of cluster mass;

3. A method to evaluate the survey volume within which clusters are found. Finding distant clusters is the main aim of this thesis and will be discussed in detail in the next four chapters. A suitable observable proxy for the cluster mass has to be calibrated with local surveys and is shown in the next section. The survey volumeVmaxor equivalently

the selection function Ssky[f(L, z)]1,i.e.the effective sky coverage as a function of limiting

flux, are essential to determine the comoving density of clusters and hence to relate the observations to model predictions.

For a strictly flux-limited X-ray survey with limit flim and corresponding sky coverage

S[flim], the comoving maximum search volume Vmax(L) for objects with luminosity L

is straightforward to calculate using Equ. 3.25. The maximum redshift zmax is obtained

from the luminosity distance (Equ. 3.20) by solvingdlum(zmax)≡

q

L/(4πflim) forzmax. The

search volume (see Fig. 3.2) is then given by Vmax(L) = Vcom[zmax(flim, L)] =

Z _zmax 0 Ssky[flim] Ã dlum(z) 1 +z !₂ c dz H(z) . (4.1) 1_{Here the flux is denoted withf and the sky coverage in units of steradian withS}

sky, not to be confused

with the fluxS often used as a notation for number counts.

For an inhomogeneous sensitivity distribution, as typical for serendipitous surveys, the sky coverage2 _{rises gradually over an extended flux range from the most sensitive regions with}

limit flow to the flux fhigh covering the full survey area. In this case, the search volume is

obtained by a second integration over the flux limit and proper accounting of the additional survey area dS/df at higher fluxes

Vmax(L) = Z _fhigh flow dVcom[zmax(f, L)] df df= Z _fhigh flow Z _zmax(f) 0 dSsky[f] df à dlum(z) 1 +z !₂ c dz H(z)df . (4.2) In principle, galaxy clusters can be systematically selected in several independent ways: (i) Optical/NIR selection based on the cluster signature of a spatial galaxy overdensity (and clustering in color or redshift space), (ii) X-ray detection of the extended ICM emission, and for future surveys (iii) selection based on the SZE, or (iv) the weak gravitational lensing signature of clusters. The main important advantages of X-ray galaxy cluster surveys can be summarized as follows:

• Bright thermal X-ray emission is only observed for well evolved clusters with a deep gravitational well (Sect. 2.3);

• The X-ray emission is highly peaked which minimizes projection effects (Sect. 2.3);

• The X-ray luminosity is tightly correlated with the gravitational mass and is provid- ing a reliable mass proxybased on the survey data (Sect. 4.2);

• The X-ray selectioncan be accurately assessed allowing a simple evaluation of the survey volume (this section);

• Cosmological applications with X-ray clusters can build upon several decades of survey experience (Chaps. 1 & 4).