Galaxy clusters tighten constraints on the cosmic accelerator

Galaxy clusters are the largest gravitationally bound structures in the universe. They began to form about 10 billion years ago when the first galaxies and their massive mantles of dark matter clumped together. Most of a cluster’s matter, however, lies between galaxies. Shocked by the assembly process and trapped by the cluster’s gravitational potential, the baryonic component of the intracluster medium (ICM) reached, and remains at, temperatures in the x-ray-emitting range of 10⁷–10⁸ K.

Their ages, assembly, and x-ray emission make clusters valuable probes of one of the most momentous phenomena ever discovered: The universe’s current rate of expansion is not slowing down, as one expects of a matter-dominated universe, but is speeding up. (See Physics Today, June 1998, page 17 .)

The x-ray glow from a cluster’s ICM is detectable out to a redshift z of about 2. Type Ia supernovae, which featured in the discovery of the accelerated expansion 11 years ago, are detectable out to a z of 1. Because clusters, like supernovae, have certain standard properties and are spread over a range of redshift, they can reveal the expansion history of the universe. Steven Allen’s group at Stanford University carried out such a study last year. ¹ The results corroborated the pioneering supernovae surveys and their more sensitive follow-ups.

Unlike supernovae, clusters can also reveal how the expansion has influenced the growth of objects that inhabit the universe. That extra information could, in turn, distinguish among various explanations of what drives the accelerated expansion. In particular, it could say whether the expansion requires either a new field—dark energy—within general relativity or a new theory of gravity.

Alexey Vikhlinin of the Harvard–Smithsonian Center for Astrophysics and his collaborators have just published a study of how clusters grow as they age. ² The study can’t rule out new gravity, but like other studies it’s consistent with general relativity and with the simplest kind of dark energy: Einstein’s cosmological constant. When combined with other, independent methods, the study yields the tightest constraints so far on the accelerating expansion.

From mass to structure

As the universe ages, matter becomes more diffuse and its energy density falls. Whatever is accelerating the expansion appears to have an energy density that has either remained constant since the Big Bang or barely varied. In the dense early universe, matter and radiation controlled the expansion’s progress. But eventually, when the universe reached about half its current age (z = 0.6), matter’s ability to retard the expansion was overcome by dark energy’s ability to accelerate it.

Galaxies began to assemble into clusters before dark energy attained its dynamical predominance. Observing clusters in x rays, their most luminous band, offers a potentially sensitive probe of dark energy for z up to about 1.

Clusters are so large that each one can be presumed to have scooped up the same universal mix of baryonic and dark matter when it formed. The fraction, by mass, of x-ray-emitting plasma should therefore be the same from cluster to cluster, regardless of size, mass, or redshift.

In their 2008 study, Allen and his group exploited that property to derive redshift-independent distances to the sample’s 42 clusters. Having also determined the redshifts from optical or x-ray spectra, Allen had a set of 42 distance–redshift relations. Only a model with dark energy, or something like it, was consistent with a constant, universal fraction of x-ray-emitting plasma.

Allen used clusters as standard candles and assumed that clusters contain a fair sample of the universe’s matter. Vikhlinin, by contrast, looked at how the expansion has influenced cluster growth. His key starting point was a paradigm called ΛCDM (Λ, Einstein’s cosmological constant, is shorthand for the simplest kind of dark energy; CDM stands for cold dark matter). According to ΛCDM, the density fluctuations that begat the first structures had a simple power-law spectrum. Evidence for that spectrum appears in the cosmic microwave background; further support comes from computer simulations of large-scale structure.

Clusters didn’t form directly from those first fluctuations but were assembled from what those fluctuations produced: protogalaxies. That distinction is important because the nonlinear assembly of clusters goes beyond the linear, perturbative physics of ΛCDM density fluctuations. Observations and computer simulations provide some confidence that the density-fluctuation spectrum relates to the initial cluster mass spectrum in a conceptually straightforward way.

How the cluster mass spectrum evolved over time depends in part on the expansion history of the universe. Expansion weakens the tendency of clusters both to gather together in superclusters and to merge with each other. The more vigorous the expansion, the less mass put on by clusters as they age.

Cluster masses

Determining a cluster’s mass might seem straightforward. The baryons’ x-ray emission depends on the baryons’ number density and temperature. The temperature, in turn, depends on the depth of the gravitational potential and therefore on the total amount of matter, both baryonic and dark.

Relating temperature to total mass requires hydrostatic equilibrium, which doesn’t always prevail. Some clusters harbor active galactic nuclei that blow out vast regions of lower-density plasma. Some clusters, like the Bullet cluster, are undergoing violent mergers. (See Physics Today,November 2006, page 21 .) And some clusters, like Abell 1689, are digesting past mergers (see figure 1).

Vikhlinin and his collaborators drew their sample of 86 clusters from observations made by the ROSAT observatory, which flew from 1990 to 1999. ROSAT ’s x-ray telescope scanned the entire sky during the first phase of its mission. The second, longer phase was devoted to observations of specific targets, some of which were clusters. Clusters also appeared as serendipitous background objects.

ROSAT ’s x-ray telescope was sensitive enough to yield accurate masses for the bright, nearby clusters in the all-sky survey. For the fainter, more distant clusters in the targeted observations, Vikhlinin turned to the Chandra observatory. Even with Chandra’s high-resolution data, Vikhlinin and his colleagues had to carry out several different calibrations to ensure that what they measured (brightness and temperature) yielded accurate total masses.

Some clusters, including Abell 1689, have masses determined independently by weak gravitational lensing. (See the article by Leon Koopmans and Roger Blandford, Physics Today, June 2004, page 45 .) A foreground cluster distorts the shapes of more distant galaxies behind it. You can see the effect in figure 1. With a big enough set of clearly measured distortions, the foreground cluster’s mass—even its mass distribution—can be inferred.

Simulated clusters provided another crosscheck. Vikhlinin’s colleague Daisuke Nagai of Yale University created model clusters of known mass, calculated the expected luminosity and temperature distributions, and then “observed” the clusters to determine which quantities correlate reliably with mass. Nagai’s simulations confirmed a suggestion by another of Vikhlinin’s colleagues, Andrey Kravtsov of the University of Chicago: The mass of the luminous gas times its temperature provides a robust measure of total mass—even for clusters still recovering from a merger. Having determined the clusters’ total masses, the team could plot the cluster mass function—that is, the number density of clusters bigger than mass M as a function of M.

The mass functions for two broad redshift bins are shown in figure 2. Thanks to ongoing merging, massive clusters are more numerous at recent times than they are at earlier times, as reflected in amplitudes of the mass functions. But if the universe’s expansion weren’t accelerating, the difference between the amplitudes would be larger.

To convert that difference into a cosmological measurement, Vikhlinin and his colleagues constructed a model and fitted it to their mass functions. The model has six parameters, three of which characterize the mass-energy content of the universe: Ω_M, the normalized mass-energy density of all the matter in the universe; Ω_Λ, the normalized mass-energy density of dark energy; and w, dark energy’s equation-of-state parameter: its pressure over its density. Two other parameters characterize the past and present structure of the universe: n _s, the power law index of the primordial density fluctuations, and σ₈, the total mass enclosed in a typical sphere of radius 8 megaparsecs in the present universe. Hubble’s constant is the sixth parameter.

Computing the model for each set of parameters was demanding. Each fit required 20 days of computer time on several workstations. Various combinations of free and fixed parameters were evaluated. Overall, the cluster mass functions are consistent with the ΛCDM paradigm. The curves in figure 2 correspond to a model in which all the parameters except σ₈, which serves to normalize both mass functions, were fixed at previously established values.

Perhaps the parameter of most interest is w. If the equations of general relativity require a cosmological constant, then empty space will act as if permeated by fluid that exerts negative pressure and speeds up the universe’s expansion. In 1968 Yakov Zel’dovich showed that the vacuum energy of quantum field theory is mathematically equivalent to Einstein’s cosmological constant. In either case w = −1.

Using the cluster data alone, Vikhlinin and his colleagues find that w = −1.14 ± 0.21. Combining cluster data with other, independent methods of determining w, they obtained a value of 0.991 with a statistical error of ± 0.045 and a systematic error of ± 0.039. The combined result represents an increase in accuracy of a factor of 2.

Vikhlinin’s model presumes w varies with neither time nor space. Both possibilities feature in various alternatives to vacuum energy. And if general relativity fails on large scales, the discrepancy would show up as a variable w.

Current observations, including Vikhlinin’s, don’t place strong constraints on the variation of w, but there are clues from the early universe. The density fluctuations that left their stamp on the cosmic microwave background and the nucleosynthesis of light elements both occurred in the universe’s first billion years. Neither phenomenon requires dark energy to account for its observable effects. That’s no problem for constant w. If, at that early epoch, dark energy had the same strength as it has now, matter and radiation predominated.

More accurate probes will determine whether w does vary. One promising tack is to look for the influence of dark energy on scales far larger than galaxy clusters. Those structures can be tied more directly to the primordial density fluctuations than clusters can. However, observing them will require a three-dimensional map of the universe similar to, but more accurate than, those derived by the Sloan Digital Sky Survey and 2dF (Two Degree Field system). So far, dark energy remains mysterious and general relativity remains safe.