  We show that the distance measures (such as the luminosity and angular
diameter distances) are linear functionals of the equation-of-state function
w(z) of the dark energy to a fair degree of accuracy in the regimes of
interest. That is, the distance measures can be expressed as a sum of (i) a
constant and (ii) an integral of a weighting function multiplied by the
equation of state parameter w(z). The existence of such an accurate linear
response approximation has several important implications: (a) Fitting a
constant w model to the data drawn from an evolving model has a simple
interpretation as a weighted average of w(z). (b) Any polynomial (or other
expansion coefficients can also be expressed as weighted sums of the true w(z).
(c) A replacement for the commonly used heuristic equation for the effective
w(z), as determined by the CMB, can be derived and the result is found to be
quite close to the heuristic expression commonly used. (d) The reconstruction
of w(z) by Huterer et al. (2002) can be expressed as a matrix inversion. In
each case the limitations of the linear response approximation are explored and
found to be surprisingly small.
