Prospects for Determining the Equation of State of the Dark Energy: What Can Be Learned from Multiple Observables?

The dark energy that appears to produce the accelerating expansion of theuniverse can be characterized by an equation of state p=w\rho with w<-1/3. Anumber of observational tests have been proposed to study the value or redshiftdependence of w, including SN Ia distances, the Sunyaev-Zel'dovich effect,cluster abundances, strong and weak gravitational lensing, galaxy and quasarclustering, galaxy ages, the \lya forest, and CMB anisotropies. The proposedobservational tests based on these phenomena measure either thedistance-redshift relation d(z), the Hubble parameter H(z), the age of theuniverse t(z), the linear growth factor D_1(z), or some combination of thesequantities. We compute the evolution of these four observables, and of thecombination H(z)d(z) that enters the Alcock-Paczyznski anisotropy test, inmodels with constant w, in quintessence models with some simple forms of thepotential V(\phi), and in toy models that allow more radical time variations ofw. Measurement of any of these quantities to precision of a few percent isgenerally sufficient to discriminate between w=-1 and w=-2/3. However, thetime-dependence predicted in quintessence models is extremely difficult todiscern because the quintessence component is dynamically unimportant at theredshifts where w departs substantially from its low-z value. Even for the toymodels that allow substantial changes in w at low redshift, there is always aconstant-w model that produces very similar evolution of all of the observablessimultaneously. We conclude that measurement of the effective equation of stateof the dark energy may be achieved by several independent routes in the nextfew years, but that detecting time-variation in this equation of state willprove very difficult except in specialized cases.Comment: 29 pages, 7 figures, many minor corrections, additions, and clarifications, to appear in Ap