Exploring the Expanding Universe and Dark Energy using the Statefinder Diagnostic

TL;DR

This paper advocates a geometry-based diagnostic for dark energy using the statefinder pair $\{r,s\}$ (with LCDM fixed point $\{1,0\}$) and the complementary $\{s,q\}$ pair, to distinguish between cosmological constant, quintessence, Chaplygin gas, and braneworld models. It develops a model-independent reconstruction framework from SNAP-like Type Ia supernova data, employing a 3-parameter polynomial for $\rho_{DE}$ and a maximum likelihood approach to infer $H(z)$, $\rho_{DE}(z)$, and $w_{\rm eff}(z)$, from which the statefinders are derived. The study demonstrates that, with realistic priors on $\Omega_m$, the averaged statefinders $\{\bar r,\bar s\}$ and $\{\bar s,\bar q\}$ can distinguish LCDM from many dynamical models at $3\sigma$, and that high-redshift data enhances discrimination especially for Chaplygin gas and tracker models. It also shows the relative success of different fitting strategies, finding that fits based on $H(z)$ or $w(z)$ outperform those based on $D_L(z)$ in recovering statefinders, and emphasizes the synergy between statefinders and SN data for breaking degeneracies in dark energy explanations.

Abstract

The coming few years are likely to witness a dramatic increase in high quality Sn data as current surveys add more high redshift supernovae to their inventory and as newer and deeper supernova experiments become operational. Given the current variety in dark energy models and the expected improvement in observational data, an accurate and versatile diagnostic of dark energy is the need of the hour. This paper examines the Statefinder diagnostic in the light of the proposed SNAP satellite which is expected to observe about 2000 supernovae per year. We show that the Statefinder is versatile enough to differentiate between dark energy models as varied as the cosmological constant on the one hand, and quintessence, the Chaplygin gas and braneworld models, on the other. Using SNAP data, the Statefinder can distinguish a cosmological constant ($w=-1$) from quintessence models with $w \geq -0.9$ and Chaplygin gas models with $κ\leq 15$ at the $3σ$ level if the value of $\om$ is known exactly. The Statefinder gives reasonable results even when the value of $\om$ is known to only $\sim 20%$ accuracy. In this case, marginalizing over $\om$ and assuming a fiducial LCDM model allows us to rule out quintessence with $w \geq -0.85$ and the Chaplygin gas with $κ\leq 7$ (both at $3σ$). These constraints can be made even tighter if we use the Statefinders in conjunction with the deceleration parameter. The Statefinder is very sensitive to the total pressure exerted by all forms of matter and radiation in the universe. It can therefore differentiate between dark energy models at moderately high redshifts of $z \lleq 10$.

Figures (16)

• Figure 1: The left panel (a) shows the time evolution of the statefinder pair $\lbrace r,s \rbrace$ for quintessence models and the Chaplygin gas. Quintessence models lie to the right of the LCDM fixed point ($r=1,s=0$) (solid lines represent tracker potentials $V=V_0/\phi^{\alpha}$, dot-dashed lines representing quiessence with constant equation of state $w$). For quiessence models, $s$ remains constant at $1+w$ while $r$ declines asymptotically to $1+\frac{9}{2}w(1+w)$. For tracker models, $s$ monotonically decreases to zero whereas $r$ first decreases from unity to a minimum value, then rises to unity. These models tend to approach the LCDM fixed point ($r=1, s=0$) from the right at $t \to \infty$. Chaplygin gas models (solid lines) lie to the left of the LCDM fixed point. For these models, $\kappa$ is the ratio between matter density and the density of the Chaplygin gas at early times. For all Chaplygin gas models, $s$ monotonically increases to zero from -1, whereas $r$ first increases from unity to a maximum value, then decreases (to unity). The dashed curve in the lower right is the envelope of all quintessence models, while the dashed curve in the upper left is the envelope of Chaplygin gas models (the latter is described by $\kappa = \Omega_m/1-\Omega_m$). The region outside the dashed curves is forbidden for both classes of dark energy models. The right panel (b) shows the time evolution of the pair $\lbrace r,q\rbrace$, where $q$ is the deceleration parameter. It is important to note that the the solid line, which corresponds to the time evolution of the LCDM model, divides the $r-q$ plane into two halves. The upper half is occupied by Chaplygin gas models, while the lower half contains quintessence models. All models diverge at the same point in the past ($r=1,q=0.5$) which corresponds to a matter dominated universe (SCDM), and converge to the same point in the future ($r=1,q=-1$) which corresponds to the steady state model (SS) -- the de Sitter expansion (LCDM $\to$ SS as $t \to \infty$ and $\Omega_m \to 0$). The dark dots on the curves show current values $\lbrace r_0, s_0\rbrace$ (left) and $\lbrace r_0, q_0\rbrace$ (right) for different dark energy models. In all models, $\Omega_m = 0.3$ at the current epoch. In both panels quiessence is shown as dot-dashed while dashed lines mark envelopes for Chaplygin gas (upper) and quintessence (lower).
• Figure 2: Trajectories in the statefinder plane $\lbrace r,q \rbrace$ for the braneworld models discussed in (\ref{['eq:hubble_brane']}). BRANE1 models have $w \leq -1$ generically, whereas BRANE2 models have $w \geq -1$. The closed loop represents DDE in which the acceleration of the universe is a transient phenomenon. For braneworld models, parameter values are as follows. BRANE1: solid curves above LCDM; top to bottom: $\Omega_m=0.6, \Omega_l=6.0$, $\Omega_m =0.5, \Omega_l=2.0$, $\Omega_m=0.4, \Omega_l=0.5$. BRANE2: solid curves below LCDM; top to bottom: $\Omega_m=0.3, \Omega_l=0.05$, $\Omega_m=0.2, \Omega_l=0.25$, $\Omega_m=0.1, \Omega_l=0.45$. The thick solid curve in BRANE2 corresponds to the DDG model discussed in (\ref{['eq:ddg']}) with $\Omega_m = 0.24$. For BRANE1 and BRANE2 models $\Omega_{\Lambda_b} = 0$ ( i.e. there is no cosmological constant in the bulk.) Dark dots indicate the current value of $\lbrace r,q \rbrace$ for the models. All models are in reasonable agreement with current supernova data. For DDE, from outer to inner loops, $\Omega_m=0.05, \Omega_{\Lambda_b}=4.9$, $\Omega_m=0.15, \Omega_{\Lambda_b}=1.4$, $\Omega_m=0.20, \Omega_{\Lambda_b}=1.1$.
• Figure 4: This figure shows $3\sigma$ confidence levels in the averaged statefinders (a) $\lbrace\bar{s},\bar{r}\rbrace$ and (b) $\lbrace\bar{s},\bar{q}\rbrace$. The polynomial fit to dark energy, Eq (\ref{['eq:taylor']}) has been used to reconstruct the statefinders for an LCDM fiducial model with $\Omega_m = 0.3$. The dashed line above the LCDM fixed point represents the family of quiessence models having $w=$ constant. The dashed line below the LCDM fixed point shows Chaplygin gas models. It should be noted that the best-fit point in both panels (a) & (b) coincides with the LCDM fixed point (solid star). In the upper half of both panels, the solid rhombi correspond to tracker potentials $V=V_0/\phi^{\alpha}$ while triangles show $w$=constant quiessence models. In the lower half of both panels, solid hexagons show Chaplygin gas models with different values of $\kappa$. (The constant $\kappa$ gives the initial ratio between cold dark matter and the Chaplygin gas. Only models with $\kappa \geq \Omega_m/(1-\Omega_m)$ are permitted by theory, see Eq (\ref{['eq:chap_def']}), (\ref{['eq:chap_lim']}).) All models, with the exception of the braneworld model, have $\Omega_m=0.3$ currently. The braneworld model is marked by a cross and corresponds to the DDG model (\ref{['eq:ddg']}) with $\Omega_m=0.24$ which best-fits current supernova data. Comparing the left and right panels we find that $\lbrace\bar{s},\bar{q}\rbrace$ is a slightly better diagnostic than $\lbrace\bar{s},\bar{r}\rbrace$ for tracker and quiessence models and can be used to rule out a constant equation of state $w\geq-0.9$ at the $3\sigma$ level if the value of $\Omega_m$ is known exactly.
• Figure 5: This figure shows $3\sigma$ confidence levels in the statefinders: (a) $\lbrace \bar{r}, \bar{s} \rbrace$, (b) $\lbrace\bar{q},\bar{s}\rbrace$, (c) $\lbrace r_0, s_0\rbrace$ and (d) $\lbrace q_0, s_0\rbrace$. The fiducial model is assumed to be LCDM and, as in the previous figure, the polynomial fit to dark energy, Eq (\ref{['eq:taylor']}) is used to reconstruct the statefinder pairs. All notations are as in the previous figure. The current observational uncertainty in the value of the matter density is incorporated by marginalizing over the value of $\Omega_m$. The dark grey outer contour shows results for the Gaussian prior $\Omega_m=0.3 \pm \sigma_{\Omega_m}$ with $\sigma_{\Omega_m}=0.05$, the grey contour in the middle uses the Gaussian prior $\sigma_{\Omega_m}=0.015$, and the light grey contour is when $\Omega_m=0.3$ exactly. Comparing panels (a) - (d) we find that $\lbrace s_0, q_0\rbrace$ is an excellent diagnostic of quintessence models which can be used to rule out a constant equation of state $w \geq -0.9$ and tracker potentials $V(\phi) \propto \phi^{-\alpha}, ~\alpha \geq 1$, at the $3\sigma$ level even if $\Omega_m$ is known to an accuracy of only $\sim 17\%$. It is important to note that of all statefinder pairs $\lbrace s_0, q_0\rbrace$ is the least sensitive to the uncertainty in the value of $\Omega_m$. This is reflected in the fact that the $3\sigma$ confidence contour for $\lbrace s_0, q_0\rbrace$ with $\Omega_m = 0.3 \pm 0.05$ is not very much larger than the $3\sigma$ confidence level obtained if $\Omega_m$ is known exactly ($\Omega_m = 0.3$). On the other hand the averaged statefinder pair $\lbrace\bar{s},\bar{q}\rbrace$ is a very good diagnostic of Chaplygin gas models and rules out models with $\kappa \leq 7$ at the $3\sigma$ level if $\Omega_m = 0.3 \pm 0.05$. (The braneworld model marked by the cross can be ruled out by $\lbrace s_0, q_0\rbrace$ as well as $\lbrace\bar{s},\bar{q}\rbrace$.)
• Figure 6: Variation of $<r(z)>$ with $z$ for the cosmological constant model. Solid line shows best-fit $<r(z)>$ averaged over all realizations calculated with the polynomial fit to dark energy, Eq (\ref{['eq:taylor']}), for the prior $\Omega_m=0.3$ exactly. The triple-dot-dashed line represents the exact value of $<r>=1$ for the cosmological constant model. Shaded regions represent the $1 \sigma$ confidence levels for $<r(z)>$. The dark grey outer contour is for the Gaussian prior $\Omega_m=0.3 \pm \sigma_{\Omega_m}$ with $\sigma_{\Omega_m}=0.05$, the grey contour in the middle uses the Gaussian prior $\sigma_{\Omega_m}=0.015$, and the light grey contour uses $\Omega_m=0.3$ exactly. The dotted, dashed and dot-dashed lines represent the exact values of $r(z)$ for different constant $w$ quiessence models, for kinessence models with the tracker potential $V(\phi) = V_0/ \phi^{\alpha}$ for different values of $\alpha$, and for Chaplygin gas models with different $\kappa$ respectively. We see that all the model values plotted lie outside the $1 \sigma$ confidence level even for the most conservative prior of $\sigma_{\Omega_m}=0.05$ at redshifts $\hbox{$\buildrel > \over \sim$} ~ 0.3$.