Baryon acoustic oscillations: A cosmological ruler

In a perfectly uniform universe, stars and galaxies would never come to be. But even as a newborn less than 1 million years old, the universe was liberally seeded with small fluctuations in its density. Under the inexorable pull of gravity, those fluctuations formed ever-larger collections of celestial objects as material accreted onto growing structures. Galaxies mark strong concentrations of matter, locations where the seed fluctuations have grown to form so-called halos of dark matter whose mass is 10¹⁰–10¹⁴ times that of the Sun; the baryons (protons and neutrons) that form stars congregate within those halos. On large scales, the pattern of galaxies matches that of the seed perturbations. On small scales, local dynamics have a significant effect on the distribution of galaxies.

The pattern of seed fluctuations is not random, but instead depends on physical processes that occurred in the early universe. One of those processes is the propagation of acoustic waves. Driven by radiation pressure from primordial photons, acoustic waves expand in the form of spherical shells moving outward from primordial fluctuations.

When the universe was less than 400 000 years old, a photon could not travel far before scattering off an electron. Electrons and nuclei were also constantly colliding, so photons, electrons, and nuclei were all coupled together, and the acoustic waves dragged baryons with them. When the universe became 400 000 years old, protons and electrons cooled enough that they could combine in what is called the recombination epoch. After recombination, photons were decoupled from matter, which was no longer dragged along by the radiation. The endpoint was a distribution of material with noninteracting dark matter at the origins of the waves and baryons at the peripheries of the expanding shells. The image on page 32 is an artist’s conception of that endpoint, overlaid with a fanciful consequent galaxy distribution.

The post-recombination perturbations felt the combined gravity of both dark matter and baryons. Eventually the concentrations of matter at the locations of the original perturbations and in the spherical peripheries both had a mix of dark and baryonic matter, as galaxy formation requires. The relic patterns that were left are known as baryon acoustic oscillations (BAOs). The first theoretical calculations of BAO physics, ^{1 , 2} made in 1970, have now led to exquisite predictions of the distribution of fluctuations. (For a review, see the article by Daniel Eisenstein and Charles Bennett, Physics Today, April 2008, page 44 .)

A cosmic ruler

Baryon acoustic oscillations show up as a prominent feature in correlation functions that relate the mass densities $\rho (\mathbf{x})$ at two separate points ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$. Usually the correlations are expressed in terms of the cosmological matter overdensity $\delta (\mathbf{x})\equiv [\rho (\mathbf{x})-\bar{\rho }(\mathbf{x})]/\bar{\rho }(\mathbf{x})$, with $\bar{\rho }(\mathbf{x})$ the mean value of the density. On large scales, $\delta$ should have properties close to those of a Gaussian random field—that is, at any location, its value is drawn from a Gaussian distribution.

Overdensities are not completely random because at different locations they are correlated with each other, as described by the correlation function $\xi ({\mathbf{x}}_1,{\mathbf{x}}_2)\equiv <\delta ({\mathbf{x}}_1)·\delta ({\mathbf{x}}_2)>$. Here the angle brackets denote an expectation value. Due to the homogeneity and isotropy of the cosmos on large scales, the correlation function is a function only of the distance $r$ separating points ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$.

In brief, the matter correlation function quantifies the excess probability of finding a pair of mass concentrations at separation $r$ compared with the case in which concentrations are placed completely at random. It is a key quantity in cosmology, from which can be deduced everything about the large-scale overdensity field except the random locations of individual overdensities.

Baryon acoustic oscillations give rise to an excess in cosmic mass concentrations separated by an amount close to the final size that the acoustic-wave spheres can reach before their propagation is halted at recombination. That BAO feature is on a large enough scale that it participates in the expansion of the universe. It remains approximately fixed in units of comoving length, a scale that grows with cosmological expansion: At present, the BAO scale is approximately 150 megaparsecs (480 million light-years; a cluster of galaxies is 2–10 Mpc across). The width of the cover image corresponds to 2.5 BAO scale lengths.

The near invariance of the BAO scale is manifest in figure 1, which shows the results of a theoretical model of the correlation function for a range of redshifts. At all redshifts, the correlation function has a bump at a similar comoving separation. Because the matter and galaxy distributions are related, the bump implies an excess in the number of pairs of galaxies corresponding to the BAO scale.

Figure 1.

As the redshift drops, the clarity with which the BAO bump can be seen decreases. Note, though, that the growth of structure in the expanding universe acts mainly to smear out the scale rather than to shift its location. Nevertheless, the smearing is pernicious, in that it degrades the fidelity with which the BAOs can be measured. Removing it from observations is an area of research that I will discuss later.

The radiation freed from the baryons at recombination is measured today as the cosmic microwave background (CMB). The signature of the BAOs is imprinted on both the CMB and the matter distribution. In fact, the signal-to-noise ratio is significantly higher for the CMB, as recently observed by the Planck satellite mission. ³ Two factors contribute to the relative degradation of the BAO signal in matter. First, after recombination, the dark matter at the center of the acoustic wave and the baryonic matter at the periphery had to come together under the action of gravity. Then subsequent growth of structure further compromised the signal. The key strength of galaxy surveys is not to tell us about the pre-recombination universe but rather to realize the BAO scale as a standard ruler to measure the geometry of the universe at lower redshifts. And that information, in combination with CMB features, can be used to test cosmological models, including the nature of dark energy.

The universe in the large

To observe BAO, we astrophysicists need surveys of galaxies that cover volumes of many cubic gigaparsecs and include many hundreds of thousands of galaxies. The angular positions of galaxies are relatively straightforward to obtain, but turning angular positions into a three-dimensional map is more difficult. We obtain the radial information from the redshift in the light from the distant galaxies. It is possible to obtain an approximate measurement of the redshift from the broadband color of a galaxy, but to obtain an accurate redshift, we need a spectrum from which emission and absorption features with known rest-frame wavelengths can be accurately measured.

Given a cosmological model, we can translate from redshift to comoving distance and make a 3D map. But the map is only as valid as the model used to create it: How do we know it’s okay? The key is that galaxy surveys observe the BAO feature at multiple redshifts, as shown in figure 2. If the model is wrong, the comoving size of the BAO feature, as determined by the cosmological model, will not match at one or more wavelengths. In particular, the size of the BAO feature will appear to be inconsistent if the nature of dark energy is significantly different from cosmologists’ expectation.

Figure 2.

In many fields in science, improvement in experimental measurement is driven by improvement in apparatus. For spectroscopic galaxy surveys, the key innovation was the multiobject spectrograph (MOS), which enables the simultaneous measurement of spectra, and hence redshifts, for multiple galaxies. The development and operation of two instruments led to the first observations of BAOs in galaxy surveys: the two-degree Field MOS on the 3.9 m Anglo–Australian Telescope, which was used for the two-degree Field Galaxy Redshift Survey (2dFGRS), ⁴ and the MOS on the Sloan Foundation 2.5 m Telescope, which was used for the Sloan Digital Sky Survey (SDSS). ⁵ As with many experiments during which the data build up slowly over time, it is not possible to pinpoint the instant when the evidence for BAOs first arrived. In 2001, as a young postdoc fortunate to be part of the 2dFGRS team, I analyzed the clustering of the galaxies in the survey when it was approximately two-thirds complete. There, I found the first evidence that baryonic effects in the early universe were needed to explain the observed galaxy distribution. ⁶ As the data improved, the BAO signal became clearer in the 2dFGRS data, ⁷ and it was also convincingly seen by the SDSS team. ⁸

More recent surveys have improved significantly on those early detections. The largest of those is the Baryon Oscillation Spectroscopic Survey (BOSS), which determined the BAO scale to within 1% accuracy and published its final measurements last year. ⁹ Box 1 describes planned and in-process surveys.

Figure 3 shows some of the BOSS maps. The research team observed 1.2 million luminous red galaxies over a volume of 18.7 Gpc³. Its observations covered a redshift range from 0.2 to 0.75, which corresponds to an age of the universe ranging from 12 Gy to 7 Gy; by comparison, the current age of the universe is about 14 Gy. BAO features imprinted on quasar spectra provide information about the cosmos at early times that galaxy surveys can’t access; see box 2.

Figure 3.

The clear detection of BAOs in the BOSS data, together with observations of the CMB, represents a stunning success of the hierarchical structure formation scenario in which today’s structure is built up from small density fluctuations in the early universe. We see the BAOs not just in the initial fluctuations as traced by the CMB, but also in the distribution of the galaxies into which those fluctuations evolved billions of years later. And a comparison of the two manifestations of the BAOs perfectly matches the predictions of our current cosmological model of structure formation.

Across and along the line of sight

Positions of galaxies in the angular and radial directions are obtained in different ways. For that reason, testing the consistency of the BAO scale across the line of sight requires different cosmological quantities than testing the scale along the line of sight. Across the line of sight, we determine the BAO scale from the increase in the number of pairs with a given angular separation and use the cosmological model to relate angular separation to distance. In the radial direction, we see the BAOs as an increase in pairs with a particular, small separation in redshift and use the cosmological expansion rate, also known as the Hubble parameter, to relate the redshift of a pair to distance. As a consequence, the requirement that the BAO scale be the same across and along the line of sight—even at a single redshift—constrains cosmological models. The consistency requirement is known as the Alcock–Paczyński test, ¹⁰ and it contributes a key component of the information provided by galaxy surveys. Incidentally, the statistical test must hold for all structures, which should, on average, be randomly oriented.

The BAO measurements from all surveys to date match the predictions of what has become known as the standard cosmological model; indeed, no observations to date have given convincing evidence for discrepancies with it. The model is based on a universe of energy density acted on by gravity as described by Einstein’s general theory of relativity. Today the dominant energy-density component is dark energy, which leads to an accelerating cosmic expansion. In the standard model, dark energy is taken to be mathematically equivalent to Einstein’s cosmological constant, though experiments are currently under way to test that assumption. Dark energy was not always the main contributor to the cosmic energy density. After all, a cosmological constant does not dilute with the universal expansion, whereas matter does. So as one traces cosmic evolution backward in time, at some point—a redshift of about 0.7—matter and dark energy have the same density. At earlier times, matter dominates.

Although the standard cosmological model considers dark energy to be the cosmological constant, the truth is that the physics of dark energy is unknown, and understanding that physics is one of the great problems facing modern physics. Theorists have devised many alternatives to the cosmological constant! In effect, dark energy is just a name for the unknown mechanism responsible for the accelerated expansion, but naming something does not imply understanding.

The BAO standard ruler has become an important tool in tests of whether dark energy really is equivalent to the cosmological constant (see figure 4). Other techniques—for example, one based on standard-candle supernovae—can also address the properties of dark energy (see the article by Josh Frieman, Physics Today, April 2014, page 28 ). But methods based on BAOs reduce to simple geometrical measurements that are less prone to systematic errors than most other techniques. That advantage is important as the next generation of experiments attempts an order-of-magnitude improvement on measurements of the cosmological acceleration.

Figure 4.

Bulk motions and reconstruction

Galaxies are in motion with respect to each other, and strictly speaking, their locations do not stay fixed in comoving coordinates. On large scales, that “peculiar” motion is driven by the growth of structure: Individual galaxies tend to fall into clusters of galaxies and out of voids, which leads to an overall increase in the amplitude of the clustering, as shown in figure 1.

The galactic motion also means that by the time galaxy pairs are observed, some of them have moved toward each other and some have separated relative to the seed perturbations from which they formed. Consequently, as seen in figure 1, as the cosmos evolves, the peak in the correlation function smooths out, reducing the fidelity with which the BAO feature can be observed and the length of the standard cosmological ruler measured.

The observed pattern of galaxies contains information about their motion. With that information, coupled with the fact that initial distribution of seed perturbations would have been close to uniform, one can in effect run simulations of the universe backward in time so as to undo the smoothing. ¹¹ That process, which the BAO community calls reconstruction, provides a significant boost to BAO measurements. Developing more accurate reconstruction algorithms remains an active area of research.

A gold mine for cosmology

Galaxy surveys furnish information about galactic masses, star-formation histories, and the physics driving galactic evolution. But they also provide a wealth of cosmological information. In this article I have emphasized the BAOs, but before closing I should acknowledge some of the other cosmological nuggets that can be mined from galaxy surveys.

► Cosmological structure growth. A galaxy’s redshift is caused by the Hubble expansion plus the peculiar motion of the galaxy. Maps that ignore the peculiar motion and use only the Hubble expansion to convert from redshift to distance are called redshift-space maps. The differences between those and the true maps that include peculiar motion are called redshift-space distortions (RSDs).

To get a handle on the RSDs, astrophysicists measure clustering across and along the line of sight in redshift-space maps; differences in the two measurements reflect the influence of peculiar velocities. After all, across the line of sight, the maps are not affected by peculiar motion, whereas they are affected along the line of sight. The information revealed about the peculiar motions in turn gives insight into the gravitational growth of cosmological structure and complements the geometric information provided by BAOs. In particular, theorists have suggested that dark energy can be explained in terms of a modification to general relativity, without the need to introduce a new and mysterious energy density. Some of their models give similar expansion histories to the standard cosmological model or to models that include a new energy-density component. However, they predict a different rate of structure growth within the background expansion, and thus the combination of BAO and RSD measurements can potentially distinguish between them.

► Neutrino masses. High-energy-physics experiments have probed the mass differences between neutrino species by observing neutrino oscillations. However, those experiments do not determine the absolute masses of neutrino species. Cosmological measurements of neutrino mass exploit the differences between the behavior of cold (nonrelativistic) dark matter and relativistic neutrinos in the early universe.

Fast-moving neutrinos wash out small-scale structure that would have formed in a universe with just cold dark matter. To first approximation, the suppression depends on Ω_ν, the ratio of the mass density comprising neutrinos to the critical mass density of matter and dark energy that would yield a spatially flat universe. That density, in turn, is related to the sum of the masses of all neutrino species m_ν = 94.1 Ω_ν h² eV, where h is the Hubble constant divided by 100 km/s/Mpc. Thus m_ν can be probed by galaxy survey measurements that compare the small-scale and large-scale clustering of galaxies. ¹²

► Cosmological inflation. The isotropy of the CMB suggests that in the instants after the Big Bang, the universe underwent a period of rapid expansion now known as inflation. Cosmological inflation explains why the universe appears homogeneous and isotropic on large scales and why the total energy density is close to the critical value. Inflation also explains the origin of large-scale structure: During inflation quantum fluctuations in the microscopic inflationary region are magnified to cosmic size and thus provide the initial seed perturbations that lead to BAOs. The physics underlying inflation is currently unknown, but some models predict that at very large separations, the clustering of galaxies and the clustering of matter will be very different—a conjecture that can be tested with galaxy surveys covering large volumes of the universe.