We briefly summarize the observational probes that play the greatest role in current constraints on dark energy. Further discussion and references can be found in other articles of this Review, in particular Secs. 21 (Cosmological Parameters) and 23 (The Cosmic Microwave Background), and in Ref. .
Cosmic Microwave Background Anisotropies: Although CMB anisotropies provide limited information about dark energy on their own, CMB constraints on the geometry, matter content, and radiation content of the Universe play a critical role in dark energy studies when combined with low redshift probes. In particular, CMB data supply measurements of θs = rs / DA,c(zrec), the angular size of the sound horizon at recombination, from the angular location of the acoustic peaks, measurements of Ωm h2 and Ωb h2 from the heights of the peaks, and normalization of the amplitude of matter fluctuations at zrec from the amplitude of the CMB fluctuations themselves. Planck data yield a 0.4% determination of rs, which scales as (Ωm h2)-0.25 for cosmologies with standard matter and radiation content. The uncertainty in the matter fluctuation amplitude is 3%, dominated by uncertainty in the electron scattering optical depth τ, and it should drop substantially with future analyses of Planck polarization maps. Secondary anisotropies, including the Integrated Sachs-Wolfe effect, the Sunyaev-Zel'dovich (SZ, ) effect, and gravitational lensing of primary anisotropies, provide additional information about dark energy by constraining low-redshift structure growth.
Type Ia Supernovae: Type Ia supernovae, produced by the thermonuclear explosions of white dwarfs, exhibit 10-15% scatter in peak luminosity after correction for light curve duration (the time to rise and fall) and color (which is a diagnostic of dust extinction). Since the peak luminosity is not known a priori, supernova surveys constrain ratios of luminosity distances at different redshifts. If one is comparing a high redshift sample to a local calibrator sample measured with much higher precision (and distances inferred from Hubble's law), then one essentially measures the luminosity distance in h-1 Mpc, constraining the combination h DL(z). With distance uncertainties of 5-8% precision per well observed supernova, a sample of ~ 100 SNe is sufficient to achieve sub-percent statistical precision. The 1-2% systematic uncertainties in current samples are dominated by uncertainties associated with photometric calibration and dust extinction corrections. Another potential systematic is redshift evolution of the supernova population itself, which can be tested by analyzing subsamples grouped by spectral properties or host galaxy properties to confirm that they yield consistent results.
Baryon Acoustic Oscillations (BAO): Pressure waves that propagate in the pre-recombination photo-baryon fluid imprint a characteristic scale in the clustering of matter and galaxies, which appears in the galaxy correlation function as a localized peak at the sound horizon scale rs, or in the power spectrum as a series of oscillations. Since observed galaxy coordinates consist of angles and redshifts, measuring this "standard ruler" scale in a galaxy redshift survey determines the angular diameter distance DA(z) and the expansion rate H(z), which convert coordinate separations to comoving distances. Errors on the two quantities are correlated, and in existing galaxy surveys the best determined combination is approximately Dv(z) = [z DA,c2(z) / H(z)]1/3. As an approximate rule of thumb, a survey that fully samples structures at redshift z over a comoving volume V, and is therefore limited by cosmic variance rather than shot noise, measures DA,c(z) with a fractional error of 0.005(V / 10 Gpc3)-1/2 and H(z) with a fractional error 1.6-1.8 times higher. BAO can also be measured in the Lyman-α forest of intergalactic hydrogen absorption towards background quasars, where the best measured parameter combination is more heavily weighted towards H(z) because of strong redshift-space distortions that enhance clustering along the line of sight. BAO distance measurements complement SN distance measurements by providing absolute rather than relative distances (with precise calibration of rs from the CMB) and by achieving greater precision at high redshift thanks to the increasing comoving volume available. Theoretical modeling suggests that BAO measurements from even the largest feasible redshift surveys will be limited by statistical rather than systematic uncertainties.
Weak Gravitational Lensing: Gravitational light bending by a clustered distribution of matter shears the shapes of higher redshift background galaxies in a spatially coherent manner, producing a correlated pattern of apparent ellipticities. By studying the weak lensing signal for source galaxies binned by photometric redshift (estimated from broad-band colors), one can probe the history of structure growth. For a specified expansion history, the predicted signal scales approximately as σ8 Ωmα, with α ≈ 0.3-0.5. The predicted signal also depends on the distance-redshift relation, so weak lensing becomes more powerful in concert with SN or BAO measurements that can pin this relation down independently. The most challenging systematics are shape measurement biases, biases in the distribution of photometric redshifts, and intrinsic alignments of galaxy orientations that could contaminate the lensing-induced signal. Predicting the large-scale weak lensing signal is straightforward in principle, but exploiting small-scale measurements also requires modeling the effects of complex physical processes such as star formation and feedback on the matter power spectrum.
Clusters of Galaxies: Like weak lensing, the abundance of massive dark matter halos probes structure growth by constraining σ8 Ωmα, where α ≈ 0.3-0.5. These halos can be identified as dense concentrations of galaxies or through the signatures of hot (107-108 K) gas in X-ray emission or SZ distortion of the CMB. The critical challenge in cluster cosmology is calibrating the relation P(Mhalo|O) between the halo mass as predicted from theory and the observable O used for cluster identification. Measuring the stacked weak lensing signal from clusters has emerged as a promising approach to achieve percent-level accuracy in calibration of the mean relation, which is required for clusters to remain competitive with other growth probes.
Redshift-Space Distortions (RSD) and the Alcock-Paczynksi (AP) Effect: Redshift-space distortions of galaxy clustering, induced by peculiar motions, probe structure growth by constraining the parameter combination f(z)σ8(z), where f(z) is the growth rate defined by Equation 3 [25, 26]. Uncertainties in theoretical modeling of non-linear gravitational evolution and the non-linear bias between the galaxy and matter distributions currently limit application of the method to large scales (comoving separations r 10 h-1 Mpc or wavenumbers k 4 0.2 h Mpc-1). A second source of anisotropy arises if one adopts the wrong cosmological metric to convert angles and redshifts into comoving separations, a phenomenon known as the Alcock-Paczynksi effect . Demanding isotropy of clustering at redshift z constrains the parameter combination H(z)DA(z). The main challenge for the AP method is correcting for the anisotropy induced by peculiar velocity RSD.
Direct Determination of H0: The value of H0 sets the current value of the critical density c = 3H02 / 8π GN, and combination with CMB measurements provides a long lever arm for constraining the evolution of dark energy. The challenge in direct H0 measurements is establishing distances to galaxies that are far enough away that their peculiar velocities are small compared to the expansion velocity v = H0 d. This can be done by building a ladder of distance indicators tied to stellar parallax on its lowest rung, or by using gravitational lens time delays or geometrical measurements of maser data to circumvent this ladder.