  The dark energy that appears to produce the accelerating expansion of the
universe can be characterized by an equation of state p=w\rho with w<-1/3. A
number of observational tests have been proposed to study the value or redshift
dependence of w, including SN Ia distances, the Sunyaev-Zel'dovich effect,
cluster abundances, strong and weak gravitational lensing, galaxy and quasar
clustering, galaxy ages, the \lya forest, and CMB anisotropies. The proposed
observational tests based on these phenomena measure either the
distance-redshift relation d(z), the Hubble parameter H(z), the age of the
universe t(z), the linear growth factor D_1(z), or some combination of these
quantities. We compute the evolution of these four observables, and of the
combination H(z)d(z) that enters the Alcock-Paczyznski anisotropy test, in
models with constant w, in quintessence models with some simple forms of the
potential V(\phi), and in toy models that allow more radical time variations of
w. Measurement of any of these quantities to precision of a few percent is
generally sufficient to discriminate between w=-1 and w=-2/3. However, the
time-dependence predicted in quintessence models is extremely difficult to
discern because the quintessence component is dynamically unimportant at the
redshifts where w departs substantially from its low-z value. Even for the toy
models that allow substantial changes in w at low redshift, there is always a
constant-w model that produces very similar evolution of all of the observables
simultaneously. We conclude that measurement of the effective equation of state
of the dark energy may be achieved by several independent routes in the next
few years, but that detecting time-variation in this equation of state will
prove very difficult except in specialized cases.
