Statefinder — a new geometrical diagnostic of dark energy

We introduce a new cosmological diagnostic pair $\{ r,s\}$ called Statefinder. The Statefinder is a geometrical diagnostic and allows us to characterize the properties of dark energy in a model independent manner. The Statefinder is dimensionless and is constructed from the scale factor of the Universe and its time derivatives only. The parameter $r$ forms the next step in the hierarchy of geometrical cosmological parameters after the Hubble parameter $H$ and the deceleration parameter $q$, while $s$ is a linear combination of $q$ and $r$ chosen in such a way that it does not depend upon the dark energy density. The Statefinder pair $\{ r,s\}$ is algebraically related to the equation of state of dark energy and its first time derivative. The Statefinder pair is calculated for a number of existing models of dark energy having both constant and variable $w$. For the case of a cosmological constant the Statefinder acquires a particularly simple form. We demonstrate that the Statefinder diagnostic can effectively differentiate between different forms of dark energy. We also show that the mean Statefinder pair can be determined to very high accuracy from a SNAP-type experiment.