Jerk, snap and the cosmological equation of state

Taylor expanding the cosmological equation of state around the current epoch p = p(0) + kappa(0)(rho - rho(0)) + 1/2 d(2)p/drho(2)\(0) (rho - rho(0))(2) + O[(rho - rho(0))(3)], is the simplest model one can consider that does not make any a priori restrictions on the nature of the cosmological fluid. Most popular cosmological models attempt to be 'predictive', in the sense that once some a priori equation of state is chosen the Friedmann equations are used to determine the evolution of the FRW scale factor a(t). In contrast, a 'retrodictive' approach might usefully take observational data concerning the scale factor, and use the Friedmann equations to infer an observed cosmological equation of state. In particular, the value and derivatives of the scale factor determined at the current epoch place constraints on the value and derivatives of the cosmological equation of state at the current epoch. Determining the first three Taylor coefficients of the equation of state at the current epoch requires a measurement of the deceleration, jerk and snap-the second, third and fourth derivatives of the scale factor with respect to time. Higher-order Taylor coefficients in the equation of state are related to higher-order time derivatives of the scale factor. Since the jerk and snap are rather difficult to measure, being related to the third and fourth terms in the Taylor series expansion of the Hubble law, it becomes clear why direct observational constraints on the cosmological equation of state are so relatively weak, and are likely to remain weak for the foreseeable future.