Baryon Acoustic Oscillations

editors: Elisa Ferreira, Bryce Cyr, Erik Madsen

What? Baryons have gravity too? The lumpy CMB leads to lumps in the galaxies

Baryon Acoustic Oscillations are a rich topic of study, with many good references. A solid review paper that we will be drawing from by Bassett and Hlozek can be found here.

Like usual, Wayne Hu's tutorials are a great resource for cosmological information. A couple good illustrations of his can be found here.

Sound Waves in the Cosmic Landscape (Peebles, Eisenstein et al.)

The CMB provides excellent evidence through its anisotropies that there were regions in the early universe that were slightly overdense (ρ>ρ_universe), and slightly underdense (ρ<ρ_universe). Consider these overdense regions, wells, comprised of dark matter, photons, and baryonic matter (before last scattering). An overdense region attracts matter towards it due to gravitational forces, while photon-matter interactions produce a pressure in the opposite direction. Once the overdense region attracts enough matter, the photon-matter interactions become dominant. After a while, after this pressure pushes away enough matter, the gravitational forces kick in again and become dominant. We see that we are in an oscillatory state, in where the counteracting gravity and pressure forces produce sound waves in the early universe. (Image credit: Wayne Hu)


Potential hills appear in under-density regions. Compression into the wells corresponds to rarefaction in the hills.

Consider now a spherical sound wave propagating out from this overdensity. The dark matter stays at the origin of the overdensity since it interacts only through gravitational forces, but the baryons and photons push outwards at some sound speed velocity near half the speed of light (peebles). Once the photons decoupled from matter (at the era of last scattering), the pressure force essentially went to 0 due to the lack of photon-matter interactions. This lack of pressure stopped the baryons from propagating any further outwards, and this distance between the original overdensity site and the effective ring of baryons is called the ‘sound horizon’.

During the maximal compression (or rarefaction) in the potential wells, the modes reach an extrema. After recombination, this picture is frozen and those modes will correspond to the peaks of the CMB power spectrum, since those modes will have enhanced temperature. (Image: Wayne Hu)


Due to the fact that there were many such overdensities in the primordial universe, one would expect the distribution of galaxies throughout the universe to look like the pattern formed by many overlapping ripples in a pond.

Left: Many galaxies on a few rings formed by the primordial overdense instabilities. We can clearly see a typical length scale for the sound horizon. Right: A few galaxies on many rings. This is a more realistic scenario, but it is harder to see the sound horizon here. Image Credit from Bassett & Hlozek.

Animated version of Figure 1.10 from Bassett & Hlozek (originally from Eisenstein et al. 2007). This shows an evolving spherical density perturbation in one dimension. Note the tight coupling of baryon matter (blue) and photons (red) up until z ~ 1100. Also note how dark matter follows baryons with a lag since it is not directly subject to photon interactions.

2D animations of density perturbations evolving. On the left is a single perturbation rippling outward and then freezing in place at decoupling. On the right is essentially the same thing superposed in many locations, which is a better reflection of reality.baryon_acoustic_peak.png
From Eisenstein et al. 2005, the Baryon Acoustic Peak is a bump in the two-point correlation function (in this case, of SDSS luminous red galaxies) on a scale of ~150 Mpc (~105 on this x-axis since h = H0 / (100 km/s/Mpc)).
Standard Rulers (Basset & Hlozek)

Baryon Acoustic Oscillations (BAO) are a great tool in modern cosmology, as they provide a ‘standard ruler’ for measuring objects on a cosmological distance scale. The idea of a standard ruler is quite simple; we judge the distance to an object of known size by how large of an angular size they make in our frame of reference. In cosmology, we need an object of known size at a redshift z, or a number of objects at different redshifts with sizes that change in well-known way. A variety of intergalactic radio sources and galaxy clusters have been used previously as standard rulers. In particular, a study by Allen et al. use X-ray emission flux coming from a galaxy cluster to measure the size of the cluster, which can then be compared to its angular size to produce a distance measure. Physically, BAO are just periodic fluctuations in the matter density of baryons throughout the universe, and can be used to constrain cosmological parameters such as the dark energy content of the universe. A statistical standard ruler utilizes the fact that galaxies may cluster on a preferred scale. If this preferred scale is known, observing galaxy clusters at different redshifts can constrain angular diameter distances.

Oscillations are frozen in at recombination, so we know the scale of the comoving sound horizon fixed during recombination. The wavenumbers (or spatial frequency) of the peaks of the CMB power spectrum are harmonically related to the fundamental scale - the distance sound can travel by recombination – Standard ruler. We can use the measurement of the BAO to probe the expansion of the universe at different redshifts. With that we can probe the acceleration of the expansion history of the Universe and the properties of dark energy.

Using the CMB, we can fix the oscillation scale. Using redshift surveys, we can observe the preferred clustering scale set by the BAO at different redshifts to constrain the Hubble parameter and the angular diameter distance. The BAO scale can be measured along and across the line of sight (see figure bellow).
The BAO peak (or ring) at a redshift z appears at an angular separation ∆θ = rd/[(1 + z)DA(z)] and at a redshift separation ∆z = rd/DH(z), where DA and DH = c/H are the angular and Hubble distances, and rd is the sound horizon at the drag epoch. So, if you measure the separation size transversal to the line of sight, we can measure the angular diameter distance:

\begin{align} d_A(z)=\frac{s_{\perp}(z)}{\Delta \theta (1+z)}. \end{align}

and along the line of sight, where we can determine the H(z):

\begin{align} H(z)=\frac{c\Delta z}{s_{\parallel}(z)} \end{align}

Early universe is permeated by many of the spherical acoustic waves. The final density distribution is a linear superposition of the small-amplitude sound waves. However, the acoustic signature is still detectable statistically through the two-point correlation function of the matter distribution, ξ(r), making the BAO a statistical standard ruler. The BAO will appear as a peak in the correlation function and as oscillations in its power spectrum.


In fact, there is a BAO standard ruler experiment with heavy McGill involvement that is currently far along in construction: the Canadian Hydrogen Intensity Mapping Experiment (CHIME). This is a stationary radio telescope that uses interferometry techniques to observe the whole northern sky once per day in the radio band between 400 MHz and 800 MHz. Since hydrogen is an excellent tracer of baryonic matter, the plan is to map the 21-cm neutral hydrogen line (intrinsically this is at a radio frequency of 1420 MHz) over the range of redshifts (0.8 - 2.5) for which it's shifted to frequencies in CHIME's band. This leads to a 3D map of structure over these redshifts which can be used to track the BAO scale over the history of the Universe and thus to trace the expansion history and test dark energy theories.


CHIME's projected distance-vs-redshift measurements based on BAO and assuming a standard ΛCDM cosmology, along with some existing measurements at lower redshifts.

Still beyond the CMB, galaxy surveys probe the BAO scale. This measurement, it is of particular interest to constraint DE (equation of state) and the curvature of the Universe. Some examples of experiments are SDSS, DES, …

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