Constraints on the redshift dependence of the dark energy potential

We develop a formalism to characterize the shape and the redshift evolution of the dark energy potential. Our formalism makes use of quantities similar to the horizon-flow parameters in inflation and is general enough that can deal with multiscalar quintessence scenarios, exotic matter components, and higher-order curvature corrections to General Relativity. We show how the shape of the dark energy potential can be recovered nonparametrically using this formalism and we present approximations analogous to the ones relevant to slow-roll inflation. Since presently available data do not allow a nonparametric and exact reconstruction of the potential, we consider a general parametric description. This reconstruction can also be used in other approaches followed in the literature (e.g., the reconstruction of the redshift evolution of the dark energy equation of state $w(z)$). Using observations of passively evolving galaxies and supernova data we derive constraints on the dark energy potential shape in the redshift range $0.1<z<1.8$. Our findings show that at the $1\sigma$ level the potential is consistent with being constant, although at the same level of confidence variations cannot be excluded with current data. We forecast constraints achievable with future data from the Atacama Cosmology Telescope.

Figure 1

Left panel: the absolute age for the 32 passively evolving galaxies in our catalogue (see text for more details) determined from fitting stellar population models is plotted as a function of redshift. Note that there is a clear age-redshift relation: the lower the redshift the older the galaxies. Right panel: the value of the Hubble parameter as a function of redshift as derived from the differential ages of galaxies in the left panel. The determination at $z\sim 0.1$ indicated by the ’$+$’ symbol is the Hubble constant determination of $H$ from 41. The dotted line is the value of $H(z)$ for the LCDM model.

Figure 2

Regions in the ${\lambda }_1/{\rho }_c$ vs ${\lambda }_0/{\rho }_c$ (left panel) and ${\lambda }_2/{\rho }_c$ vs ${\lambda }_0/{\rho }_c$ (right panel) excluded at the $1\sigma$ and $2\sigma$ joint confidence level, by the priors and the constraints that the kinetic energy in the quintessence field must be positive and that at all redshifts ${\rho }_m+{\rho }_q$ must be positive.

Figure 3

Constraints in the ${\lambda }_1/{\rho }_c$ vs ${\lambda }_0/{\rho }_c$ (left panel) and ${\lambda }_2/{\rho }_c$ vs ${\lambda }_0/{\rho }_c$ (righ panel) obtained from $H(z)$ measurement based on relative galaxy ages. Contour levels are $1\sigma$ marginalized, $1\sigma$ joint, and $2\sigma$ joint. The diamond shows the location of the maximum of the marginalized likelihood.

Figure 4

One and two sigma joint constraints in the ${\lambda }_1/{\rho }_c$ vs ${\lambda }_0/{\rho }_c$ plane and ${\lambda }_2/{\rho }_c$ vs ${\lambda }_0/{\rho }_c$ obtained from the Riess et al. (2004) supernovae data.

Figure 5

Reconstructed $V(z)$ from relative galaxy ages (left) and from Supernovae (right). The gray regions represent the 1 and $2\sigma$ confidence regions. In the left panel the dotted line shows the constraint imposed by the prior.

Figure 6

Reconstructed $V(q)$ from relative galaxy ages.

Figure 7

Predicted constraints for an experiment with 2000 galaxies for which ages are measured with an accuracy of $\sim 10\%$. The constraint in the Chebyshev coefficients (left panels; circles show the location of the maximum marginalized likelihood while ’$+$’ show the location of the maximum of the joint 5D likelihood) and the reconstructed dark energy potential (right panel) are significantly better than current constraints (see text). We have a LCDM model as fiducial. The Atacama Cosmology Telescope will identify about 500 galaxy clusters in the redshift range $0.1<z<1.5$, for at least 2000 galaxies there will be spectroscopic follow-up and therefore galaxy ages can be derived.

Figure 8

Left panel: Constraints in the ${\omega }_0$ vs ${\omega }_1$ obtained from our galaxy sample with the differential ages method. The contours show the $1\sigma$ marginalized, $1\sigma$, and $2\sigma$ joint confidence levels. The degeneracy constrains $w_0\equiv {\omega }_0-{\omega }_1$. Three points in the parameter space are selected. Right panel: difference between the Hubble constant in a given model and the Hubble constant in the LCDM model. The points with error bars are our data points, the long-dashed line corresponds to the LCDM model ($\genfrac{}{}{0ex}{}{*}{}$-point in the left panel), the dot-dashed line corresponds to the diamond-point, and the dot-dot-dot-dashed one to the ’$+$’-point.

Figure 9

Interval in time where the slow-roll approximation is valid for some choices of $V_0^{\prime }$ and $K_0$, when $V_0/{\rho }_c=0.7$.