Response of distance measures to the equation of state

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. (1) Fitting a constant-w model to the data drawn from an evolving model has a simple interpretation as a weighted average of w(z). (2) Any polynomial (or other expansion coefficients) can also be expressed as weighted sums of the true w. (3) A replacement for the commonly used heuristic equation for the effective w, as determined by the cosmic microwave background, can be derived and the result is found to be quite close to the heuristic expression commonly used. (4) The reconstruction of w(z) by Huterer & Starkman 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