Forecasts for the Complete Survey Sample (z, η)

These final results are based on the assumption of available cluster redshifts and photon counts only, such that we applied the (z, η)-mock catalogue for the four years of eROSITA survey observations, eRASS:8, where in total eight all-sky surveys will be performed. For all tested cosmological mod-els, both the optimistic as well as the pessimistic scenarios were investigated. Tab. 6.3 presents the forecasted constraints on the cosmological as well as on the scaling parameters for the different cos-mological models and simulation approaches, whereas Fig. 6.9 displays the corresponding credibility contours for awCDM-cosmology. These 2-dimensional contours are centred around the input parameter values, while marginalising over all remaining parameters except the two presented ones. The complete triangle diagram, which also includes the contours of the scaling parameters is placed in the appendix (Fig. D.3). Of most interest in this analysis were the constraints on the parameter set{σ₈,Ωm,ns, w₀, w_a} and their dependencies as completely free variables.

Expectedly, the uncertainty on the parameters increased with an extension of the variable set, especially when moving from aw₀CDM-model to awCDM-model. WhereasΔnsremained almost constant,Δσ₈ andΔΩmshowed an increase by∼33% andΔw0was degraded by even a factor of∼3.6. Accordingly, this confirmed the strong degeneracies between the parameters{σ₈,Ωm, w₀, w_a}observed in Fig. 6.9.

For the cosmological parameters, the strong dependence on the knowledge of the scaling relations be-came visible with a significant decrease in the uncertainties for improved scaling information (comp.

Fig. 6.9). The factor of this decrease differed for the individual parameters, such that forσ₈andΩmthe precision improved in general by a factor of∼ 2, whereas for ns the progress was even higher with a factor of∼2.5. On the other hand, the parameters of the dark energy equation of state showed a weaker dependence on the scaling information. For aw₀CDM-cosmology, the constraints onw₀ improved by

6.6 FinalMCMCResults

Figure 6.11: Forecasted joint credibilities for the standard deviation in the matter power spectrumσ₈ and the matter energy densityΩmassuming awCDM-cosmology. Again, the pessimistic as well as the optimistic scenario are presented in blue and red, respectively. The solid lines are computed based on the true parameter distribution obtained in the MCMCsimulation, whereas the dashed lines display the estimated covariance matrices of the chains.

Figure 6.12: Credibility regions for the dark energy parametersw₀ andw_a. As before, we display the results for the optimistic as well as for the pessimistic scenario and additionally approximate theMCMCresults by the corresponding covariance matrices.

6 Cosmological Constraints fromeROSITAGalaxy Clusters

only∼40% when moving from the pessimistic to the optimistic scenario and this development declined to only 10% for awCDM-cosmology. A similar progress was also observed inΔw_a. These trends of the credibility regions were graphically summarised in Fig. 6.9 and emphasised upon in Figs. 6.10, 6.11 &

6.12 for different parameter combinations. When assuming even further knowledge on the scaling rela-tions and comparing the optimistic scenario to the simulation with frozen scaling relarela-tions, the change in the cosmological constraints varied between a factor of∼ 3 to an improvement of only a couple of percent depending on the considered parameter and cosmology. This approach of fixing the scaling relations was of course idealistic, but it supported our aim of quantifying the impact of the knowledge on these relations on the cosmological constraints. The least significant improvement was recorded for thewCDM-cosmology, where the progress read only 30% for ns and only a couple of percent forw₀ andw_a. Since we were mainly interested in studying dark energy and thewCDM-model, we concluded, that a better knowledge on the scaling relations was inevitable to obtain tight cosmological constraints as they were computed for the optimistic scenario. However, a further reduction of the uncertainties in the scaling relations resulted in only small improvements.

Fig. 6.9 also expresses the strong degeneracies betweenh andΩb, which was defined by the applied priors. However, if these priors were not considered, we would allow for a strong degeneracy especially betweenhandnswith several local maxima in the likelihood. This would prevent theMCMC chains from converging. Defining priors onhandΩbthus allowed to localise the chains in the parameter space around the input cosmological values and excluded the other local maxima, depending on the width of the priors. In general, when runningMCMCsimulations for these forecasts, the applied priors did not only improve the constraints, but were required to allow for converging chains.

The priors ofΔh=±0.022 (Riess et al. 2011) and ofΔΩbh²=±0.00046 (Cooke et al. 2014), or equiva-lently ofΔΩb=±0.00304, were reproduced in all considered cosmologies with a deviation of only less than 10% from the initial values. The precision on these parameters showed a statistical scatter around the prior values independent of the simulation scenario. Accordingly, the cluster data added only little information to the constraints on these parameters. As there was no trend for this deviation with cosmo-logical model or scaling information, we neglectedΔhandΔΩbfrom our further interpretations (comp.

also Fig. 6.9).

In contrast, the large sample of observed clusters allowed for a self-calibration of the scaling relations, such that the constraints on some of the scaling parameters were reduced significantly when compared to their initial prior values (comp. Fig. D.3, Tabs. 6.1 & 6.3). WhereasΔαLM reproduced the initial value for all considered simulation scenarios, the estimated uncertainties onβ_LMandγ_LM reduced the prior in all set-ups by∼50% and by∼10−65%, respectively. The larger deviations were commonly ob-served for the pessimistic approach, such that in these cases alsoΔσLMwas improved by∼20%. These improvements of the prior knowledge indicated that the extendedeROSITAcluster catalogue contained additional information on the investigated parameters. On the other hand, these deviations between the priors and the computedMCMC uncertainties were also partially explained by the discrepancy of the MCMCcontours from a normal distribution. Investigating Figs. 6.9 & D.3, these differences were espe-cially visible in the shapes of the 1-dimensional, marginalised histograms for the pessimistic simulation scenarios and for the parametersh,Ωb,βLM, when considering parameters with prior constraints. Ac-cordingly, theMCMCapproach allowed for additional information on these prior values by inreasing the allowed freedom on the shape of the credibility regions.

Deviations from a normal distribution were to some extent also observed for the remaining cosmolog-ical parameters. In general, these discrepancies increased with increasing parameter uncertainties and thus with the number of free cosmological parameters as well as with the decreasing knowledge on the scaling relations. When analysing e.g. Figs. 6.10 & 6.12, we presented theMCMC parameter distributions as solid lines and additionally displayed the approximated Gaussian shapes, based on the

6.6FinalMCMCResults Table 6.3:MCMCforecasts of the cosmological and scaling relation parameters for the modelsΛCDM,w₀CDM andwCDM and for both the optimistic and the pessimistic scenarios as well as for the idealistic approach of full knowledge on the scaling relations. The simulations for theeROSITAdata alone were based on the four years of all-sky survey (eRASS:8) including the stated priors onhand onΩbh². ThePlanckdata were combined with external information on BAO, supernovae type Ia andH₀(Planck+BAO+H0+JLA), where for the combination ofeROSITAandPlanckdata, we applied the optimistic scenario.

Data Δσ₈ ΔΩm Δns Δh ΔΩb Δw₀ Δwa ΔαLM ΔβLM ΔγLM ΔσLM

eRASS:8+pes. 0.0187 0.0117 0.0748 0.0192 0.0027 – – 0.0525 0.0334 0.1106 0.0283 eRASS:8+opt. 0.0082 0.0061 0.0288 0.0177 0.0024 – – 0.0141 0.0124 0.0540 0.0095

eRASS:8+fixed 0.0027 0.0031 0.0209 0.0214 0.0029 – – – – – –

Planck 0.0143 0.0133 0.0062 0.0096 0.0011 – – – – – –

eRASS:8+Planck 0.0068 0.0047 0.0039 0.0036 0.0004 – – 0.0083 0.0120 0.0454 0.0093 eRASS:8+pes. 0.0195 0.0126 0.0823 0.0208 0.0029 0.0543 – 0.0517 0.0342 0.1735 0.0279 eRASS:8+opt. 0.0087 0.0064 0.0329 0.0218 0.0030 0.0329 – 0.0143 0.0123 0.0688 0.0096 eRASS:8+fixed 0.0059 0.0048 0.0217 0.0201 0.0028 0.0255 – – – – –

Planck 0.0201 0.0093 0.0053 0.0105 0.0015 0.0476 – – – – –

eRASS:8+Planck 0.0072 0.0049 0.0042 0.0054 0.0008 0.0243 – 0.079 0.01108 0.0472 0.0094 eRASS:8+pes 0.0265 0.0190 0.0864 0.0198 0.0028 0.1308 0.5259 0.0552 0.0355 0.1980 0.0353 eRASS:8+opt. 0.0129 0.0096 0.0315 0.0200 0.0028 0.1169 0.4316 0.0141 0.0126 0.0678 0.0095 eRASS:8+fixed 0.0107 0.0078 0.0217 0.0200 0.0028 0.1136 0.4222 – – – –

Planck 0.0207 0.0102 0.0057 0.0107 0.0016 0.1121 0.4467 – – – –

eRASS:8+Planck 0.0085 0.0062 0.0043 0.0063 0.0009 0.0771 0.2759 0.0079 0.0114 0.0480 0.0094

131

6 Cosmological Constraints fromeROSITAGalaxy Clusters

covariance matrix of the chains, as dotted curves. The normalisation of the dark energy equation of state w₀showed small indications for a non-Gaussian 1-dimensional uncertainty distribution in thew₀ CDM-model for the pessimistic scenario. This deviation was enhanced in thewCDM-model. The uncertainty on the time evolution of the dark energyw_apresented the strongest non-normal shapes with a tilt of the 1-dimensional histogramm to higher values in the pessimistic approach (Fig. 6.12). And even for the optimistic scenario, slight substructures were observed in the distribution. Accordingly, the joint credi-bility contours ofw₀ andw_adid not resemble a Gaussian ellipse, which was also observed for the joint uncertainty regions of the other parameter combinations including the dark energy characteristics (Fig.

6.9). In contrast to the dark energy parameters, however,σ8andΩmshow mainly Gaussian constraints with only slight deviations forΩmin the pessimistic scenario (Fig. 6.11). These likelihood shapes were expected, as these two parameters are commonly best constrained by cluster experiments (comp. e.g.

Reiprich & Böhringer 2002; Voit 2005; Allen et al. 2011; Mantz et al. 2015).

In summary and following the above considerations, the final constraints on the cosmological param-eters for a dark energy cosmology were computed asΔσ₈ = 0.0129, ΔΩm = 0.0096, ns = 0.0315, w0=0.1169,wa =0.4316 (Tab. 6.3). This related to uncertainties of3% for the first three parameters and of∼12% forw0. These results implied, that theeROSITAcluster samplealonewas able to achieve the same precision in the dark energy characteristics as thePlanckdata, when considering all external knowledge (Planck+BAO+H0+JLA). Though the estimated Δns was downgraded by a factor of∼ 5 when compared to the combinedPlanckresults, the precision onΩmandσ8was improved by 10−60%

by theeROSITAcluster abundance. What is more, current cluster catalogues have not been sensitive for the index of the power spectrumns, yet, such that a tight prior was commonly applied on this parameter when applying clusters for cosmological studies (comp. e.g. Mantz et al. 2015). Though the constraints onnswere not as precise as for other cosmological probes, the extendedeROSITAcluster catalogue was able to break the degeneracy in this parameter (comp. also Pillepich et al. 2012).

As the main science driver foreROSITAis the study of dark energy, we computed the figure of merit (Sect. 6.4.6) for the joint credibility region ofw₀andw_a to test the significance of the forecasted con-straints. We obtained values of FoM^2σ_w

0,wa =12 for the pessimistic scenario and of FoM^2σ_w

0,wa =26 for the optimistic scenario. This investigation characterisedeROSITA’s cosmological constraining power alone as an advanced Stage III study according to the DETF (Albrecht et al. 2006). Their report requested a FoM^2σof [8,43] for a Stage III dark energy mission and of [27,645] for a Stage IV mission. However, the final classification is defined for the combination of the considered probe with thePlanckdata. Ac-cordingly, already precise estimates on the dark energy equation of state will be obtained fromeROSITA data only, where these results will even be improved by the combination withPlanckdata (Sect. 6.6.4).