Tracing cosmic evolution with clusters of galaxies

The most successful cosmological models to date envision structure formation as a hierarchical process in which gravity is constantly drawing lumps of matter together to form increasingly larger structures. Clusters of galaxies currently sit atop this hierarchy as the largest objects that have had time to collapse under the influence of their own gravity. Thus their appearance on the cosmic scene is also relatively recent. Two features of clusters make them uniquely useful tracers of cosmic evolution. First, clusters are the biggest things whose masses we can reliably measure because they are the largest objects to have undergone gravitational relaxation and entered into virial equilibrium. Mass measurements of nearby clusters can therefore be used to determine the amount of structure in the universe on scales of ${10}^{14}–{10}^{15}{M}_{☉}$, and comparisons of the present-day cluster mass distribution with the mass distribution at earlier times can be used to measure the rate of structure formation, placing important constraints on cosmological models. Second, clusters are essentially “closed boxes” that retain all their gaseous matter, despite the enormous energy input associated with supernovae and active galactic nuclei, because the gravitational potential wells of clusters are so deep. The baryonic component of clusters therefore contains a wealth of information about the processes associated with galaxy formation, including the efficiency with which baryons are converted into stars and the effects of the resulting feedback processes on galaxy formation. This article reviews our theoretical understanding of both the dark-matter component and the baryonic component of clusters, providing a context for interpreting the flood of new cluster observations that are now arriving from the latest generation of x-ray observatories, large optical surveys, and measurements of cluster-induced distortions in the spectrum of the cosmic microwave background.

Figure 1

Relation between velocity dispersion and temperature for a heterogeneous sample drawn from the literature: ∎, data on galaxy groups; 엯, cluster data; dotted lines, best power-law fits for groups; solid lines, best fits for clusters. The best-fitting relation to the combined sample is ${\sigma }_{1\mathrm{D}}={10}^{2.51\pm 0.01}\mathrm{km}{\mathrm{s}}^{-1}{(T∕1\mathrm{keV})}^{0.61\pm 0.01}$, corresponding to ${\beta }_{\mathrm{sp}}=0.97$ at $6\mathrm{keV}$. From 391.

Figure 2

Evolution of energy densities with redshift for different cosmological models: solid lines, the concordance model with ${\Omega }_{\mathrm{M}}=0.3,$ ${\Omega }_{\Lambda }=0.7,$ and $w=-1$; dotted lines, a dark-energy model with ${\Omega }_{\mathrm{M}}=0.3,$ ${\Omega }_{\Lambda }=0.7,$ and $w=-0.8$; long-dashed lines, an open-universe model with ${\Omega }_{\mathrm{M}}=0.3$ and ${\Omega }_{\Lambda }=0.0;$ short-dashed lines, a critical-uiniverse model with ${\Omega }_{\mathrm{M}}=1.0$ and ${\Omega }_{\Lambda }=0.7.$ The upper group of lines illustrates ${\Omega }_M\left(z\right)$, the left-hand group of lines illustrates ${\Omega }_{\Lambda }\left(z\right)$, and the right-hand group of lines illustrates ${\Omega }_R\left(z\right)$. Structure in the universe grows most rapidly while ${\Omega }_{\mathrm{M}}\left(z\right)\approx 1$, because positive density perturbations then exceed the critical density. This period of time occurs between the redshift $z_{\mathrm{eq}}$ when ${\Omega }_{\mathrm{M}}\left(z_{\mathrm{eq}}\right)={\Omega }_{\mathrm{R}}\left(z_{\mathrm{eq}}\right)$ and the redshift at which ${\Omega }_{\mathrm{M}}$ begins to drop. Notice that the redshift $z_{\mathrm{eq}}$ is earlier for larger present-day values of ${\Omega }_{\mathrm{M}}$ and that the redshift at which ${\Omega }_{\mathrm{M}}\left(z\right)$ begins to decline depends on the characteristics of dark energy. Observations of clusters and their evolution provide opportunities to constrain the values of ${\Omega }_{\mathrm{M}}$, ${\Omega }_{\Lambda }$, and $w$ because the timing of both of these epochs influences the properties of the cluster population.

Figure 3

Redshift dependence of comoving volume in various cosmologies. The quantity $d{V}_{\mathrm{co}}∕dz$ is the comoving volume of the entire sky between redshift $z$ and $z+dz$, divided by the redshift interval $dz$. If clusters were a nonevolving population of objects, one could distinguish between these cosmologies simply by counting the number of clusters on the sky in each redshift interval.

Figure 4

Cosmological constraints from cluster evolution (358), supernovae (304; 279), and WMAP observations of the cosmic microwave background (332). The horizontal axis labeled $\Omega$ gives the value of ${\Omega }_{\mathrm{M}}$; the vertical axis labeled $\Lambda$ gives the value of ${\Omega }_{\Lambda }$. These particular contraints from cluster evolution are based on the baryonic mass function of clusters (Sec. 3D3), but other measures of cluster evolution give similar results. The complementarity of these constraints is evident from the figure, and the common region of overlap near ${\Omega }_{\mathrm{M}}=0.3$ and ${\Omega }_{\Lambda }=0.7$ is reassuring evidence of consistency in the overall picture. Figure courtesy of Alexey Vikhlinin.

Figure 5

Mass-function evolution in five different cosmologies. The fiducial model in all cases is the $\Lambda \mathrm{CDM}$ model with ${\Omega }_{\mathrm{M}}=0.3$, ${\Omega }_{\Lambda }=0.7$, $w=-1$, and ${\sigma }_8=0.9$. Upper left: panel compares cluster evolution in the $\Lambda \mathrm{CDM}$ case with a standard cold-dark-matter model (SCDM) having ${\Omega }_{\mathrm{M}}=1.0$, ${\Omega }_{\Lambda }=0.0$, and ${\sigma }_8=0.5$. Evolution in the SCDM case is much more dramatic, and the steeper slope of the mass function strongly disagrees with observations of local clusters (300). Lower left: retaining ${\Omega }_{\mathrm{M}}=1.0$ and ${\Omega }_{\Lambda }=0.0$ while adjusting the power spectrum so that $\Gamma =0.21$ gives a $\tau \mathrm{CDM}$ model in which the slope of the low-redshift mass function is more acceptable, but the evolution is still very strong. Upper right: dispensing with dark energy while keeping the matter density low gives an OCDM model (${\Omega }_{\mathrm{M}}=0.3$, ${\Omega }_{\Lambda }=0$, ${\sigma }_8=0.9$) with less evolution than the $\Lambda \mathrm{CDM}$ case because structure formation starts to ramp down earlier in time (see Fig. 2). Lower right: dark energy in a $w\mathrm{CDM}$ model identical to the $\Lambda \mathrm{CDM}$ model except with $w=-0.8$ also slows cluster evolution relative to the $\Lambda \mathrm{CDM}$ case.

Figure 6

Predicted number of clusters on the sky as a function of redshift in different cosmologies. Upper panel shows the number of clusters per unit redshift with $M_{200}>3\times {10}^{14}{h}_{70}^{-1}{M}_{☉}$ over the entire sky. Notice that there are a few tens of thousands of such clusters on the sky in models with ${\Omega }_{\mathrm{M}}=0.3$, most of them at $z<1$. There are many fewer massive clusters at $z>0.5$ in the $\tau \mathrm{CDM}$ model with ${\Omega }_{\mathrm{M}}=1$ because cluster evolution is so rapid in that case. The lower panel shows the numbers of clusters with $M_{200}>1\times {10}^{15}{h}_{70}^{-1}{M}_{☉}$. Differences between models with ${\Omega }_{\mathrm{M}}\approx 0.3$ but differing values of ${\Omega }_{\Lambda }$ and $w$ should be detectable in large cluster surveys containing $\sim {10}^4$ clusters and extending to $z\sim 1$.

Figure 7

Evolution of the cluster mass function and its manifestations in temperature and luminosity: solid lines, $\Lambda \mathrm{CDM}$ (${\Omega }_{\mathrm{M}}=0.3$, ${\Omega }_{\Lambda }=0.7$, $w=-1$, ${\sigma }_8=0.9$); dashed lines, $\tau \mathrm{CDM}$ (${\Omega }_{\mathrm{M}}=1.0$, $\Gamma =0.21$, ${\sigma }_8=0.5$). Upper left: evolution of the mass function is far more pronounced in $\tau \mathrm{CDM}$ than in $\Lambda \mathrm{CDM}$ because it is so sensitive to the current matter density. Each set of three lines shows the differential mass function $d{n}_M∕d\ln M$ at $z=0$, 0.5, and 1.0, from top to bottom; black squares show the value of the mass function at a fiducial mass of ${10}^{15}{h}_{70}^{-1}{M}_{☉}$. Upper right: the same mass functions plotted against temperature, assuming $T=T_{200}(M_{200},z)$. Notice that the higher-redshift curves have shifted to the right, weakening the evolution in temperature space, because clusters of a given mass have higher temperatures at higher redshifts. In order to convert these curves to temperature functions, one would need to convolve them with the scatter in the mass-temperature relation and multiply by $d\ln M∕d\ln T\approx 1.5$. The lower two panels show these same curves as a function of luminosity, assuming $L_{\mathrm{X}}=(6\times {10}^{44}{h}_{70}^{-2}\text{erg}{\mathrm{s}}^{-1}){(T_{200}∕6\mathrm{keV})}^3$ at $z=0$ and two different redshift dependences of the $L_{\mathrm{X}}\text{−}T$ relation. Lower left: without $L_{\mathrm{X}}\text{−}T$ evolution, the curves are just relabeled versions of the ones in the upper-right panel. Lower right: the strong $L_{\mathrm{X}}\text{−}T$ evolution $[L_{\mathrm{X}}\propto T^3{(1+z)}^{1.5}]$, however, shifts the three curves in the $\Lambda \mathrm{CDM}$ case nearly on top of one another at $L_{\mathrm{X}}\approx {10}^{44}{h}_{70}^{-1}\text{erg}{\mathrm{s}}^{-1}$. Convolving these curves with the dispersion in the mass-luminosity relation and multiplying by $d\ln M∕d\ln L\approx 0.5$ converts them to luminosity functions.

Figure 8

Observed evolution in the integrated cluster temperature function $n(>kT)$ giving the comoving number density of clusters with temperatures exceeding $kT$: 엯, low-redshift temperature function from a sample of clusters with mean redshift $z=0.051$; ●, observed temperature function of clusters with a mean redshift of $z=0.429$. The data points in each case are correlated because $n(>kT)$ at a given temperature is a cumulative function depending on all data points at higher temperatures. Dotted line, predicted temperature functions at $z=0.051$; solid line, $z=0.429$ for the best-fitting model: ${\Omega }_{\mathrm{M}}=0.28$, ${\Omega }_{\Lambda }=0.98$, ${\sigma }_8=0.68$. Figure from 398.

Figure 9

Observed evolution in the cluster luminosity function, from many different cluster surveys spanning the range $0\lesssim z\lesssim 1$. The vertical axis gives the luminosity functions derived from these surveys in terms of $\varphi \equiv dn∕d{L}_{\mathrm{X}}$, and the shaded region shows the luminosity function at $z\approx 0$. Significant evolution is seen only at $L_{\mathrm{X}}\gtrsim {10}^{45}\text{erg}{\mathrm{s}}^{-1}$, consistent with $\Lambda \mathrm{CDM}$ models with a moderate amount of evolution in the $L_{\mathrm{X}}\text{−}T$ relation (see Fig. 7). From 249.

Figure 10

(Color in online edition) Expected constraints on cosmological parameters from self-calibrated surveys. The SPT and Planck surveys will find 20 000–30 000 clusters through the Sunyaev-Zeldovich effect. DUET is a proposed x-ray survey designed to find $\sim 20000$ clusters. Dotted lines, the expected constraints on $w$ and ${\Omega }_M$ from self-calibration if redshift evolution of the cluster observables behaves exactly according to the standard scaling relations, allowing the redshift dependences of those scaling relations to be fixed; long-dashed lines, loosening of constraints when redshift evolution is determined as part of the self-calibration; dot-dashed lines, tightening of constraints when information about cluster bias is included in the calibration. Solid lines, best-case scenario in which the self-calibration includes both information about cluster bias and supplementary followup calibrations of a small subset of the survey. From 222.

Figure 11

(Color in online edition) Dimensionless entropy $K∕K_{200}$ as a function of scale radius $r∕r_{200}$ for 30 clusters simulated without radiative cooling or feedback: ∎, the median profile; dashed line, the power-law relation $K∕K_{200}=1.32{(r∕r_{200})}^{1.1}$. Most of the entropy profiles shown lie close to this relation in the radial range $0.1\lesssim r∕r_{200}\lesssim 1.0$. At smaller radii, the entropy profiles generally flatten, and their dispersion increases. This flattening is likely to be a real effect, as it sets in well outside the shaded box showing the gravitational softening length of the simulation.

Figure 12

Entropy profiles from smooth accretion and hierarchical accretion. Smoothing of the gas accreting onto a cluster boosts entropy production while maintaining the characteristic $K\left(r\right)\propto r^{1.1}$ entropy profile. The two lower lines show approximate entropy profiles produced by (short-dashed line) hierarchical accretion, including the power-law expression from Fig. 11 and (solid line) the $K_{\mathrm{NFW}}$ model described in the text, with $c_{200}=5$. The two upper lines show entropy profiles resulting from smooth accretion models, including (dotted line) a profile computed by 344 for a ${10}^{15}{h}^{-1}{M}_{☉}$ cluster with $300\mathrm{keV}{\mathrm{cm}}^2$ of preheating and radiative cooling implemented and (long-dashed line) a profile computed from Eq. (72) and preheating amounting to $0.1K_{200}$. The two smooth models run roughly parallel to the hierarchical accretion models but their normalizations are $\sim 1.5$ times higher.

Figure 13

Comparison of entropy measured at $0.1r_{200}$ and the cooling threshold in a large sample of clusters: small points, entropy $K_{0.1}$ measured at $0.1r_{200}$ in a sample of 64 clusters; points with error bars, the mean entropy measurement in temperature bins of eight clusters each; dotted line, mean entropy predicted by simulations of clusters without radiative cooling of feedback; solid line, value of the cooling threshold $K_c\left(T\right)$ computed for heavy-element abundances 0.3 times their solar values and $t_0=14\mathrm{Gyr}$; dashed line, entropy predicted at $0.1r_{200}$ by the simple analytical model. From 372.

Figure 14

Luminosity-temperature relation: ▴, data from 10, who avoided clusters with cool cores; ◻, cluster data from 225 with cool cores excised; ×, group data from 155 and solid octagons, from 267, which were not corrected for cool cores; small points, simulated clusters from 47. These simulations implement radiative cooling and supernova feedback in the form of galactic winds. Lines show modified-entropy models from 366 with entropy truncated at the cooling threshold. There is a slight dependence on ${\sigma }_8$ in these models because higher values of ${\sigma }_8$ lead to dark-matter halos with more concentrated cores. Both the analytical and numerical models agree well with the data at $k_B{T}_{\mathrm{lum}}\gtrsim 2\mathrm{keV}$. Agreement is not as good at lower temperatures, but the reasons for the disagreement are unclear. More feedback may be needed in the numerical models to suppress the luminosities, and the large scatter in the observations at $\lesssim 1\mathrm{keV}$ may reflect a wide range in the effectiveness of feedback.

Figure 15

Mass-temperature relation: Large data points show cluster data from 132 and 260, in which cluster masses were inferred from fitting polytropic beta models (see Sec.); dashed lines, $M_{500}\text{−}{T}_{\mathrm{lum}}$ relation measured in clustered simulated without cooling and feedback by 116, which clearly disagree with the data points. The other lines show the $M_{500}\text{−}{T}_{\mathrm{lum}}$ relations predicted by the analytical models of 366, which agree much better with the data. There is a slight difference between models with ${\sigma }_8=0.9$ (dotted lines) and ${\sigma }_8=1.2$ because higher values of ${\sigma }_8$ lead to clusters with higher halo concentrations that produce slightly higher temperatures. Tiny points show data for clusters simulated by 47 with radiative cooling and feedback in the form of supernova-driven galactic winds. The left-hand panel uses the actual values of $M_{500}$, which agree with the analytical models. The right-hand panel uses values of $M_{500}$ inferred from fitting polytropic beta models to the observations, which underestimate true cluster masses, especially at low temperature, suggesting there may be a systematic observational bias in this method of mass measurement.