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# Studies of dark energy with x-ray observatories

Edited by Neta A. Bahcall, Princeton University, Princeton, NJ, and approved March 9, 2010 (received for review January 12, 2010)

## Abstract

I review the contribution of *Chandra* X-ray Observatory to studies of dark energy. There are two broad classes of observable effects of dark energy: evolution of the expansion rate of the Universe, and slow down in the rate of growth of cosmic structures. *Chandra* has detected and measured both of these effects through observations of galaxy clusters. A combination of the *Chandra* results with other cosmological datasets leads to 5% constraints on the dark energy equation-of-state parameter, and limits possible deviations of gravity on large scales from general relativity.

Accelerated expansion of the Universe discovered in 1998 (1 and 2) and the associated problem of dark energy are widely considered as two of the greatest unsolved problems in science (see ref. 3 for a recent review). In this short review, I will summarize the contribution of x-ray astronomy (primarily, *Chandra* and *XMM-Newton*) to the currently emerging picture of empirical properties of dark energy.

There are two main observable manifestations of dark energy: The first is its effect on the expansion rate of the Universe as a whole, which can be probed through the distance-redshift relation using “standard candles” such as type Ia supernovae, or standard rulers such as baryonic acoustic oscillations in the large-scale distribution of galaxies (4). This broad class of cosmological observations is often referred to as “geometric” methods. The second effect is the impact of dark energy on the rate of growth of large-scale structures. As the Universe enters the accelerated expansion phase around *z* ≈ 0.8, it is expected that the rate of structure growth slows down. If this effect is observed accurately—e.g., through the integrated Sachs-Wolfe effect (5), weak lensing on the large-scale structures, redshift-space distortions in the distribution of galaxies (6), or through evolution of galaxy clusters as described below—it should significantly improve constraints on dark energy properties in combination with the geometric methods (4). In addition, the growth of large-scale structures can be used to test, or put limits on, any departures from general relativity on the 10–100 megaparsec scales (7).

X-ray astronomy’s contribution to observational cosmology is primarily through studies of galaxy clusters. Cluster observations provide both the geometric and growth of structure cosmological tests. The distance-redshift relation can be measured either through the Sunyaev-Zel’dovich (8) effect, or using the expected universality of the intracluster gas mass fraction, *f*_{gas} = *M*_{gas}/*M*_{tot} (9 and 10). Both methods can be also used to determine the absolute value of the Hubble constant through observations of low-*z* clusters *. The mass function of galaxy clusters is exponentially sensitive to the underlying amplitude of linear density perturbations and therefore can be used to implement the growth of structure test (13).

In the *Chandra* and *XMM-Newton* era, x-ray observations of galaxy clusters have reached sufficient maturity for a successful implementation of both types of cosmological tests. This success is based on significant advances in our ability to select and statistically characterize large cluster samples, and to get detailed x-ray data at both low and high redshifts. At the same time, quick progress in theoretical modeling of clusters (see ref. 14 for a recent review) resulted in better understanding of their physics and improved ability to obtain reliable mass estimates from the data. These advances are reviewed below.

## Progress in Understanding of Clusters

### Samples.

The *ROSAT* mission which operated in the 1990’s proved to be a great resource for selecting large, complete samples of massive galaxy clusters reaching redshifts beyond *z* = 1 (15). *ROSAT* carried out surveys in a wide range of sensitivity and solid angles. Its sensitivity and angular resolution in the all-sky survey mode are well suited for detection of clusters at low redshifts (e.g., the BCS and REFLEX surveys, (16 and 17)). With substantial effort on the optical identification side, the all-sky survey data can be used to select exceptionally massive clusters out to *z* ∼ 0.5 (MACS survey, (18)). In the pointed mode, *ROSAT PSPC* covered just over 2% of the extragalactic sky. However, the sensitivity and angular resolution in the pointed mode are significantly better than those in the all-sky scans. As a result, the *ROSAT* pointed observations can be used to search for *z* ≈ 0.6 clusters with masses matching those of the low-*z* objects detected in the all-sky survey. Just such a sample of clusters is provided by the 400 degree survey (19). This survey uses inner regions of 1,610 individual *ROSAT* pointings covering 400 deg^{2} to detect clusters above a limiting flux of 1.4 × 10^{-13} erg s^{-1} cm^{-2} in the 0.5–2 keV energy band. The catalog includes 266 galaxy clusters and groups, of which 108 are at *z* > 0.35. Because of a fairly high imposed flux limit—which is above the real sensitivity in many pointings,—the 400 degree catalog does not have any clusters at *z* > 1, but *ROSAT*-selected samples of such clusters are available from other surveys (20). The REFLEX, MACS, and 400 degree surveys, with several hundred clusters each, are the main sources for cosmological observations with *Chandra*.

### Detailed Measurements.

*Chandra* and *XMM-Newton* observations of low-redshift objects now provide detailed measurements of the radial profiles of the density, temperature, and metallicity of the intracluster medium (ICM) over a wide range of radii: Several studies (*Chandra* sample of high-mass relaxed clusters (21); *Chandra* studies of low-*M* groups (22); *XMM-Newton* representative cluster sample (23) including both relaxed and unrelaxed objects) provide a consistent picture. The gas density and temperature profiles show a high degree of regularity and follow simple scalings outside the inner cluster region (Fig. 1 and 2). At large radii, the observed scaling of the ICM entropy with cluster mass is close to that predicted for purely gravitational heating (24 and 25); however, deviations from such a scaling are observed at small radii indicating more complex physics in the inner cluster region. Such measurements are important for cosmological applications of the cluster data for several reasons: First, they provide the necessary observational ingredients for estimation of the cluster total masses via the hydrostatic equilibrium equation. Second, the observed ICM profiles can be used to verify numerical models of the cluster formation (24). The main role of numerical models in the cosmological applications of the cluster data is to provide predictions for the scaling relations between total mass and global x-ray properties. These predictions can be used reliably only because we can verify that numerical models reasonably well reproduce even more complex cluster properties. Lastly, self-similarity of the observed ICM profiles directly confirms a key prediction of the theory of cluster formation and the basis for using clusters as cosmological probes—a notion that the cluster properties are predominantly determined by a single parameter, its mass.

### Mass Measurements.

The existence of scaling relations between various cluster parameters and total mass has long been recognized. However, establishing the absolute scale in such relations is a long-standing problem (see, e.g., ref. 26 for a theorist’s perspective of available observations, circa 2003). The situation today is much improved: A good agreement, at a ∼10% level in mass, exists (27) between normalizations of the mass vs. proxie relations determined from the x-ray measurements in relaxed clusters (e.g., (21)), “measured” in numerical simulations (28 and 29), and obtained from weak-lensing observations of representative samples of intermediate redshift clusters (30 and 31). A 10% accuracy in the absolute cluster mass calibration is indicated not only by the agreement of the results from different methods, but also indirectly by agreement of the amplitude of density perturbations derived from x-ray clusters (32), from optically selected clusters with masses calibrated through weak lensing (33), and from the latest weak-lensing sheer studies (34).

The advances in theoretical and observational studies of galaxy clusters outlined above, which were triggered in large part by the *Chandra* and *XMM-Newton* observations, have enabled efficient application of the geometrical and structure-based cosmological tests.

## Geometric Test with *f*_{gas}

Galaxy clusters are expected to have a nearly cosmic mix of baryonic and dark matter, *f*_{b} = *M*_{b}/*M*_{tot} ≈ Ω_{b}/Ω_{M}, because their mass is orders of magnitude higher than the Jeans mass scale and hence baryons and dark matter are not separated as the clusters grow from large-scale structures (35). The universality of the baryon fraction in cluster was originally used as a method for measuring Ω_{M}, but in the mid-1990’s it was realized that it can be also used as an independent distance indicator (9 and 10). The mass of the intracluster gas (contributing 80%–90% to the total baryonic mass in massive clusters (36)) derived from the x-ray image is proportional to *d*^{5/2} where *d* is the distance to the cluster†, while dynamically derived total mass scales as *d*^{1}. Therefore, the apparent baryon mass fraction is proportional to *d*^{3/2} and is constant as a function of *z* only if we use the correct distance-redshift relation.

Early pilot studies based on this test were inconclusive (10 and 37). Comparison of the *Chandra* results (38) with these early works exemplifies just how revolutionary *Chandra* has been for cluster cosmology (Fig. 3). The object-to-object scatter is now low and the trends in the *f*_{gas}(*z*) data arising from assuming a “wrong” cosmological model are clearly detectable. In particular, the expected absence of redshift trends in the *f*_{gas} measurements is observed only for the range of parameters corresponding to the “concordant” cosmological models, while strong trends in *f*_{gas}(*z*) are found if, e.g., one assumes an Ω_{M} = 1 model without a cosmological constant (38).

Unfortunately, the assumption that *f*_{gas} (and even the total baryon fraction including stellar mass) in clusters is constant and universal is only approximately accurate because there are observed trends with radius within individual clusters. The *f*_{gas} values measured at a fixed fraction of the virial radius also show a trend with cluster mass (21, 22, 25). The nature of these trends remains uncertain. Feasible explanations include different star formation efficiencies in high- and low-mass clusters, and some form of nongravitational heating of the gas in the central regions. Existence of *f*_{gas} trends in the low-*z* clusters almost certainly implies that *f*_{gas} should slightly vary with redshift. Allen et al. (38) corrected for some of these effects using results from numerical simulations. Unfortunately, nonnegligible systematic uncertainties must be assigned (e.g., Allen et al. allowed for ± 10% variations of intrinsic *f*_{gas} between *z* = 0 and 1), and they dominate the final error budget when the *f*_{gas} test is used, for example, to constrain the dark energy equation-of-state parameter, *w*. Even with the current level of systematic uncertainties, the *f*_{gas} test provides interesting constraints on the value of *w* (Fig. 4).

## Growth of Structure Test

Evolution of the cluster mass function traces (with exponential magnification) the growth of linear density perturbations. Growth of structure and the distance-redshift relation are similarly sensitive to properties of dark energy, and also are mutually highly complementary sources of cosmological information (e.g., (39)). Pre-*Chandra* works which used the cluster mass function as a cosmological probe were limited by small sample sizes. They also had to use either poor proxies for the total mass (e.g., the x-ray flux) or inaccurate measurements (e.g., temperatures with large uncertainties). Despite these limitations, reasonable constraints could still be derived on Ω_{M} (e.g., (40 and 41)). However, constraints on the dark energy equation-of-state parameter from such studies were weak.

As discussed above, the situation with the cluster mass function data has been dramatically improved in the past three years, and the new measurements allow us to track the growth of density perturbations over the redshift interval *z* = 0 - 0.7. These measurements confirm the slow down of the perturbations growth caused by cosmic acceleration, improve constraints on the equation-of-state parameter, and even put limits on possible departures from general relativity on the ∼10 Mpc scales.

The sensitivity of the cluster mass function to the presence of dark energy is illustrated in Fig. 5. The cluster sample used in (32) provides sufficient statistics to measure the amplitude of density perturbations independently in the redshift intervals *z* = 0.015 - 0.15, 0.35 - 0.45, 0.45 - 0.55, and 0.55 - 0.9. Together with the amplitude of perturbations at *z* ≈ 1,000 derived from the cosmic microwave background fluctuations, these data track the growth of perturbations over a wide redshift interval (Fig. 6). The slowdown of the perturbations growth at low redshifts is clearly seen, and the data indicate that the transition from fast to slow growth was fast and occurred at *z* ∼ 1, as expected for models with dark energy (see, e.g., the *solid red line* in Fig. 5 and compare it with the growth histories for low-density models without dark energy shown by *blue dashed lines*).

The evolution of the cluster mass function measured from the 400 degree survey provides sufficient statistics to constrain the dark energy equation-of-state parameter (Fig. 7). The combination of the structure growth data with other cosmological datasets results, as was long anticipated, in dramatic improvement of the constraints: For example, a nonevolving equation-of-state parameter is constrained to be *w*_{0} = -0.99 ± 0.045 (*inner ellipse* in Fig. 7); without the cluster data, the statistical and systematic uncertainties on *w*_{0} are a factor of 1.5–2 worse (32).

## Testing for Departures from General Relativity

Perhaps a more interesting application of the cluster mass function is to test for possible deviations from general relativity on the ∼10 Mpc scales. NonGR gravity theories modify the distance-redshift relations; however, the changes in *d*(*z*) generally can be mimicked by variations of the equation-of-state parameter for “true” dark energy and therefore nonGR models cannot be tested by geometric methods alone. We can test the nonGR models using a combination of geometric measurements with the growth of structure data. Each of the essential ingredients of the cluster mass function theory—the growth of linear density perturbations, nonlinear collapse of large-amplitude perturbations, and relations between the cluster mass and its observed properties—is potentially modified in nonGR gravity models.

Unfortunately, self-consistent predictions for the properties of the cluster population in nonGR models are still rare. Usually, the published analyses are restricted to predicted modifications of the structure growth rate in the linear regime. It has been suggested (7) that a useful parametrization for such deviations is the linear growth index, γ, defined as [1]where *D* is the perturbations growth factor at redshift *z*. If *D* is measured at a set of redshifts, γ can be constrained by fitting the model curves given by Eq. **1** to the data. The test is useful because it was found that for a wide range of models in which dark energy is represented by some form of a scalar field, *γ* ≃ 0.55 with high precision (7). Therefore, if γ is found to significantly deviate from 0.55, this potentially would imply that gravity does not follow GR on the cluster scales. Unfortunately, implementations of this method using cluster data necessarily ignore potential effects of nonGR gravity on the nonlinear collapse and relations between the cluster mass and observables. However, γ derived from the cluster data still provides a useful null test. The best published results from the x-ray cluster mass function constrain the growth index to be *γ* = 0.44 ± 0.16 (42); no other cosmological test currently provides useful constraints on γ.

As of this writing, the only self-consistent test of a nonGR theory with the cluster data is presented by Schmidt et al. (43). They consider a specific variant of a so-called *f*(*R*) models, in which two terms are added to the GR Lagrangian, one corresponding to Einstein’s cosmological constant and another to a genuine modification of GR, [2]where *R*_{0} is the average present-day curvature in the Universe, and *f*_{R} characterizes the fractional (with respect to *R*) modification of the Lagrangian density of the gravitational field. Schmidt et al. showed that a combination of the 400 degree survey cluster data with other cosmological datasets constraints the nonGR term to be *f*_{R} < 10^{-3}.

## Conclusions

X-ray observations of massive galaxy clusters with *Chandra* and *XMM-Newton* resulted in robust implementations of the geometric and growth of structure cosmological tests. Cluster data independently confirm the accelerated expansion of the universe; show that the empirical properties of dark energy are very close to those of cosmological constant; and start to provide interesting constraints on possible deviations of gravity from general relativity on large scales.

## Footnotes

^{1}E-mail: alexey{at}head.cfa.harvard.edu.Author contributions: A.V. designed research; performed research; analyzed data; and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵

^{*}A combination of the Sunyaev-Zel’dovich effect observations and x-ray data for the same cluster naturally provides the absolute distance to the object (11). In the gas fraction method, it is assumed that the baryon mass fraction within clusters,*f*_{b}, approximates the mean cosmic value, Ω_{b}/Ω_{M}. The absolute value of this ratio is now very well known from the CMB data (12). On the other hand, the mass fraction of the hot intracluster gas, the dominant baryonic component in clusters, derived from the x-ray data is proportional to*h*^{-3/2}(see below in the text), therefore*h*can be extracted from these measurements after correcting*f*_{gas}for the contribution of stellar mass to the total baryon budget.↵

^{†}The origin of these scalings is purely geometrical. For example, the x-ray volume emissivity is proportional to*ρ*^{2}. Therefore, given the observed flux,*f*, we have the following relations for the gas mass within a sphere of angular size*θ*: If the total mass is derived from a method based on a virial theorem, we have where*v*^{2}is the velocity dispersion of particles. Therefore, the derived total mass is proportional to*d*^{1}. The same scaling is true for the total mass derived from the hydrostatic equilibrium equations.

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