This indicates that:

- Our theory of gravity is wrong, or
- The universe is dominated by a material which violates the strong energy condition.

We see the dark energy through its effects on the expansion rate of the Universe. Indeed dark energy was first found by measurements of cosmic expansion, although models containing it had been around for several years. To constrain the nature of dark energy we need to be able to measure the expansion rate of the Universe and there are three main approaches:

- Standard candles: which measure the luminosity distance as a function of redshift.
- Standard rulers: which measure the angular diameter distance and expansion rate as a function of redshift.
- Growth of fluctuations.

Consider (2) further. Suppose we had an object whose length (e.g. in meters) we knew as a function of cosmic epoch. By measuring the angle subtended by this ruler as a function of redshift we map out the angular diameter distance, d(z). By measuring the redshift interval associated with this distance we map out the Hubble parameter, H(z).

To get competitive constraints on dark energy we need to be able to see changes in H(z) at the 1% level -- this would give us statistical errors in the DE equation of state of O(10%).

- We need to be able to calibrate the ruler accurately over most of the age of the universe.
- We need to be able to measure the ruler over much of the volume of the universe.
- We need to be able to make ultra-precise measurements of the ruler.

Cosmological objects can probably never be uniform enough so we use statistics of the large-scale distribution of matter and radiation. If we work on large scales or early times perturbative treatment is valid and calculations under control. Preferred length scales arising from physics of the early universe are imprinted on the distribution of mass and radiation and form time-independent rulers.

Let us consider the early universe, which was composed of a coupled plasma
of energetic photons and ionized hydrogen (protons and electrons) plus other
trace elements and the mysterious dark matter.
Start with a single perturbation.
The plasma is totally uniform except for an excess of matter at the origin.
High pressure drives the gas+photon fluid outward at speeds approaching the
speed of light. In the panels below we show some snapshots from this
process, with the baryon density shown in the left panel, the photon
density in the right panel and the mass profile as a graph in the final
panel.
Initially both the photons and the baryons move outward together, the radius of the shell moving at over half the speed of light
This expansion continues for 10^{5} years
After 10^{5} years the universe has cooled enough the protons
capture the electrons to form neutral Hydrogen.
This decouples the photons from the baryons.
The former quickly stream away, leaving the baryon peak stalled.
The photons continue to stream away while the baryons, having lost their
motive pressure, remain in place.
The photons have become almost completely uniform, but the baryons remain
overdense in a shell 100Mpc in radius. In addition, the large gravitational
potential well which we started with starts to draw material back into it.
As the perturbation grows by O(1000) the baryons and DM reach equilibrium
densities in the ratio Wb/Wm. The final configuration is our original peak
at the center (which we put in by hand) and an echo in a shell roughly
100Mpc in radius. The radius of this shell is known as the sound horizon.

Features of baryon oscillations:

- Firm prediction of models with baryons
- Positions well predicted once (physical) matter and baryon density known - calibrated by the CMB.
- Oscillations are sharp, unlike other features of the power spectrum.
- Internal cross-check: d should be the integral of H(z).
- Since have d(z) for several zs can check spatial flatness: d(z1+z2) = d(z1)+d(z2)+O(curvature)
- Ties low-z distance measures (e.g. SNe) to absolute scale defined by the CMB.

Thus the BAO program is (in principle) straightforward. One finds a
tracer of the mass density field (e.g. galaxies or the
Ly-a forest)
and computes its 2-point function. The features in the 2-point function
correspond to the sound horizon, roughly 100Mpc in size. By knowing the
angle (and redshift interval) this distance subtends one measures d(z).
Comparing to the value at z~10^{3} allows us to constrain the
evolution of the dark energy.

The problem is that the ruler we are using is inconveniently large. It is only with the latest generation of large galaxy redshift surveys that we are able to probe the giga-parsec volumes required to make a precision measurement of the BAO signal.

In order to turn this idea into a workable measurement there are number of higher order effects which need to be taken into account. These involve the details of how the statistics are measured on the galaxy redshift survey and the corrections for non-linearity, galaxy bias and redshift space distortions. You can find more details on these technical aspect by following the links below.

Lectures given
at the 2007 Santa Fe cosmology workshop or the 2008
IUCCA school.

BAO pages
for researchers.