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:
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%).
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 105 years After 105 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:
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~103 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.
For further, more technical, information see these Lectures given at the 2007 Santa Fe cosmology workshop or the 2008 IUCCA school. or these Lectures from the 2010 Santa Fe cosmology workshop.