Two new diagnostics of dark energy

TL;DR

This work introduces two diagnostics, Om and the acceleration probe $\bar{q}$, to discriminate between a cosmological constant and dynamical dark energy with minimal priors on the matter density $\Omega_{0m}$. Om, constructed from the Hubble parameter, yields a null test for LCDM when comparing $Om(x)$ at different redshifts, and remains robust against $\Omega_{0m}$ uncertainties; $\bar{q}$ provides a model-independent estimate of the onset of cosmic acceleration via the look-back=time-based relation $1+\bar{q}=\frac{1}{\Delta t}(\frac{1}{H_1}-\frac{1}{H_2})$. Applying these diagnostics to Union SN data, BAO, and WMAP5 CMB data with a CPL parametrization shows LCDM remains in excellent agreement, though evolving DE scenarios are also consistent within uncertainties. The results demonstrate the value of combining $H(z)$-based diagnostics with diverse cosmological probes to robustly test the nature of dark energy and potentially infer low-redshift constraints on the equation-of-state parameter $w_0$ when $\Omega_{0m}$ is known.

Abstract

We introduce two new diagnostics of dark energy (DE). The first, Om, is a combination of the Hubble parameter and the cosmological redshift and provides a "null test" of dark energy being a cosmological constant. Namely, if the value of Om(z) is the same at different redshifts, then DE is exactly cosmological constant. The slope of Om(z) can differentiate between different models of dark energy even if the value of the matter density is not accurately known. For DE with an unevolving equation of state, a positive slope of Om(z) is suggestive of Phantom (w < -1) while a negative slope indicates Quintessence (w > -1). The second diagnostic, "acceleration probe"(q-probe), is the mean value of the deceleration parameter over a small redshift range. It can be used to determine the cosmological redshift at which the universe began to accelerate, again without reference to the current value of the matter density. We apply the "Om" and "q-probe" diagnostics to the Union data set of type Ia supernovae combined with recent data from the cosmic microwave background (WMAP5) and baryon acoustic oscillations.

Figures (9)

• Figure 1: The equation of state of a fiducial LCDM model ($w=-1, \Omega_{0m}^{true}=0.27$) is reconstructed using an incorrect value of the matter density. For $\Omega_{0m}^{erroneous} = 0.22$ the resulting EOS shows quintessence-like behavior and its $1-\sigma$ contour is shown in green. In the opposite case, when $\Omega_{0m}^{erroneous} = 0.32$, the EOS is phantom-like and its $1-\sigma$ contour is shown in blue. Note that in both cases the true fiducial model (red) is excluded in the reconstruction. (The parametric reconstruction scheme suggested in statefinder was applied to SNAP-quality data to construct this figure.)
• Figure 2: The Hubble parameter squared is plotted against the cube of $1+z$ for Quintessence ($w=-0.7$, dashed), LCDM ($w=-1$, solid) and Phantom ($w=-1.3$, dot-dash). The universe is assumed to be spatially flat and $\Omega_{\rm DE} = 2/3$ in all models. For LCDM the plot $h^2$ vs $(1+z)^3$ is a straight line whereas for P and Q this line is curved in the interval $-1<z\hbox{$\buildrel < \over \sim$} ~ 1$. This forms the basis for the observation that $Om(x_1,x_2) \equiv Om(x_1) - Om(x_2) = 0$ in LCDM, while $Om(x_1,x_2) > 0$ in Quintessence and $Om(x_1,x_2) < 0$ in Phantom, if $x_1 < x_2$. Thus $Om(x_1,x_2)$ furnishes us with a null test for the cosmological constant. (At $z<0$ the Hubble parameter for Phantom diverges at the 'Big Rip' future singularity, while for Quintessence $h(z) \to 0$ as $z \to -1$. LCDM approaches the de Sitter space-time at late times.)
• Figure 3: The left panel shows the $Om(z)$ diagnostic reconstructed for a fiducial quintessence model with $w=-0.9$ and $\Omega_{0m} = 0.27$ (black line, green shaded region shows $1\sigma$ CL, the red line is the exact analytical result for $Om$). The horizontal blue line shows the value of $Om$ for a $\Lambda$CDM model with the same value of $\Omega_{0m}$ as quintessence. Note that any horizontal line in this figure represents $\Lambda$CDM with a different value of $\Omega_{0m}$. For instance $\Lambda$CDM with $\Omega_{0m} = 0.32$ is shown by the horizontal magenta line. As this figure shows, the negative curvature of quintessence allows us to distinguish this model from (zero-curvature) $\Lambda$CDM independently of the current value of the matter density. The right panel shows the $Om(z)$ diagnostic reconstructed for a fiducial phantom model with $w=-1.1$ and $\Omega_{0m} = 0.27$ (black line, green shaded region shows $1\sigma$ CL). The positive curvature of phantom allows us to distinguish this model from (zero-curvature) $\Lambda$CDM independently of the current value of the matter density. For instance, phantom can easily be distinguished from $\Lambda$CDM both with the correct $\Omega_{0m} = 0.27$ (horizontal blue) as well as incorrect $\Omega_{0m} = 0.22$ (horizontal magenta). (The non-parametric reconstruction scheme suggested in arman has been employed on SNAP quality data for this reconstruction.)
• Figure 4: Reconstructed $Om(z)$ and $w(z)$ from SNLS supernovae data using the CPL ansatz (\ref{['eq:cpl']}) and assuming three different values $\Omega_m = 0.22, 0.27, 0.32$ for the matter density. Notice that while the best fit value of $Om(z)$ is virtually independent of the redshift (top panel, red curve) and is therefore consistent with LCDM (with $\Omega_m = 0.27$: green line), the reconstructed EOS strongly depends upon the value of the matter density. Thus, for the same data set, the best fit value of $w(z)$ is suggestive of quintessence for $\Omega_m = 0.22$, LCDM for $\Omega_m = 0.27$ and phantom for $\Omega_m = 0.32$, while $Om(z)$ favours LCDM throughout. Note that the small variations in $Om(z)$ in the three upper panels are a consequence of the CPL ansatz which requires, as input, the value of the matter density $\Omega_m$. A non-parametric ansatz such as arman, or the parametric ansatz saini00, would have led to a uniquely reconstructed $Om(z)$ with no dependence on $\Omega_m$. Blue lines show $1\sigma$ error bars.
• Figure 5: Reconstructed $Om(z)$ and $w(z)$ from recent Union supernovae data using the CPL ansatz (\ref{['eq:cpl']}) and assuming three different values $\Omega_m = 0.22, 0.27, 0.32$ for the matter density. $Om(z)$ appears to be much more robust against variation in $\Omega_m$ in comparison with $w(z)$. The horizontal green line in the top panel indicates value of $Om (\equiv \Omega_m)= 0.32$ for LCDM model. The blue lines show $1\sigma$ error bars. Though LCDM is still consistent with the Union data, this consistency is not quite as strong as it was for the SNLS data shown in the previous figure. The top panel clearly indicates that evolving DE is also perfectly consistent with Union data.