Statefinder -- a new geometrical diagnostic of dark energy

TL;DR

It is demonstrated that the Statefinder diagnostic can effectively differentiate between different forms of dark energy and can be determined to very high accuracy from a SNAP-type experiment.

Abstract

We introduce a new cosmological diagnostic pair $\lbrace r,s\rbrace$ called Statefinder. The Statefinder is dimensionless and, like the Hubble and deceleration parameters $H(z)$ and $q(z)$, is constructed from the scale factor of the Universe and its derivatives only. The parameter $r(z)$ forms the next step in the hierarchy of geometrical cosmological parameters used to study the Universe after $H$ and $q$, while the parameter $s(z)$ is a linear combination of $q$ and $r$ chosen in such a way that it does not depend upon the dark energy density $Ω_X(z)$. The Statefinder pair $\lbrace r,s\rbrace$ is algebraically related to the the dark energy pressure-to-energy ratio $w=p/ε$ and its time derivative, and sheds light on the nature of dark energy/quintessence. Its properties allow to usefully differentiate between different forms of dark energy with constant and variable $w$, including a cosmological constant ($w = -1$). The Statefinder pair can be determined to very good accuracy from a SNAP type experiment.

Figures (3)

• Figure 1: The Statefinder pair ($r,s$) is shown for different forms of dark energy. In quiessence (Q) models ($w =$constant $\neq -1$) the value of $s$ remains fixed at $s = 1+w$ while the value of $r$ asymptotically declines to $r(t \gg t_0) \simeq 1 + \frac{9w}{2}(1+w)$. Two models of quiessence corresponding to $w_Q = -0.25, -0.5$ are shown. Kinessence (K) models are presented by a scalar field (quintessence) rolling down the potential $V(\phi) \propto \phi^{-\alpha}$ with $\alpha = 2,4$. These models commence their evolution on a tracker trajectory described by (\ref{['eq:kin_state']}) and asymptotically approach $\Lambda$CDM at late times. $\Lambda$CDM ($r=1,s=0$) and SCDM in the absence of $\Lambda$ ($r=1,s=1$) are the fixed points of the system. The hatched region is disallowed in quiessence models and in the kinessence model which we consider. The filled circles show the current values of the Statefinder pair ($r,s$) for the Q and K models ($\Omega_{m0} = 0.3$).
• Figure 2: The Statefinder pair $\lbrace r,s\rbrace$ is shown for dark energy consisting of a cosmological constant $\Lambda$, quiessence 'Q' with an unevolving equation of state $w = -0.8$ and the inverse power law tracker model $V= V_0/\phi^2$, referred here as kinessence 'K'. The lower left panel shows $r(z)$ while the lower right panel shows $s(z)$. Kinessence has a time-dependent equation of state which is shown in the top right panel. The fractional density in matter and kinessence is shown in the top left panel.
• Figure 3: Confidence levels at $1\sigma$, $2\sigma$, $3\sigma$ of $\bar{r}$ and $\bar{s}$ computed from $1000$ random realizations of a SNAP-type experiment probing a $\Lambda$CDM fiducial model with $\Omega_{m0} = 0.3,~ \Omega_{\Lambda 0} = 0.7$. The filled circles represent the values of $\bar{r}$ and $\bar{s}$ for the quintessence potential $V(\phi) \propto \phi^{-\alpha}$ with $\alpha=1,2,3,4,5,6$ (bottom to top). The filled triangles represent quiessence with $w = -2/3, -1/2, -1/3, 0$ (bottom to top). The cross shows the mean Statefinder value $\bar{r}=0.70,~ \bar{s}=0.27$ for the DGP brane model with $H_0r_c=1.43~(\Omega_{m0}=0.3)$. Note that all inverse power-law models, as well as the DGP model, lie well outside of the three sigma contour centered around the $\Lambda$CDM model.