Nonparametric reconstruction of the dark energy equation of state

A basic aim of ongoing and upcoming cosmological surveys is to unravel the mystery of dark energy. In the absence of a compelling theory to test, a natural approach is to better characterize the properties of dark energy in search of clues that can lead to a more fundamental understanding. One way to view this characterization is the improved determination of the redshift-dependence of the dark energy equation of state parameter, $w(z)$. To do this requires a robust and bias-free method for reconstructing $w(z)$ from data that does not rely on restrictive expansion schemes or assumed functional forms for $w(z)$. We present a new nonparametric reconstruction method that solves for $w(z)$ as a statistical inverse problem, based on a Gaussian process representation. This method reliably captures nontrivial behavior of $w(z)$ and provides controlled error bounds. We demonstrate the power of the method on different sets of simulated supernova data; the approach can be easily extended to include diverse cosmological probes.

Figure 1

Redshift distribution of supernovae from the three simulated data sets investigated. In addition to JDEM measurements of supernovae, we assume a low redshift sample of 300 supernovae for $z\le 0.1$. The bin width is $\Delta z=0.1$.

Figure 2

Three simulated data sets. The upper row shows $\Delta {\tilde{\mu }}_B$ (the data itself with the corresponding value for a $\Lambda \mathrm{CDM}$ model subtracted, red crosses) as a function of redshift, $z$, and the exact $\Delta {\tilde{\mu }}_B$ for the corresponding model (blue line). The lower panels show the behavior of the equation of state $w(z)$ as a function of redshift. The underlying model for the first data set is a cosmological constant. The second and third data sets are based on quintessence models. The third data set has been chosen to test our reconstruction method on a nontrivial equation of state.

Figure 3

Simulated data for Model 1 ($w=-1$) with error bars in red. The green and blue line show the exact distance modulus ${\tilde{\mu }}_B$ for Models 2 and 3. Note the very small size of the difference.

Figure 4

Reconstruction results for $w$ for data sets 1–3 (left to right) assuming $w=\mathrm{const}$ and ${\Omega }_m$ and ${\Delta }_{\mu }$ fixed at their fiducial values. The black dashed curve shows the “truth” and the red curve, the reconstruction results. The dark blue shaded region indicates the 68% confidence level, while the light blue shaded region extends it to 95%. The assumption $w=\mathrm{const}$ makes it impossible to capture the time dependence in data sets 2 and 3.

Figure 5

Results as in Fig. 4, but letting ${\Omega }_m$ and ${\Delta }_{\mu }$ vary. The result for data set 1 is very accurate—$w_0$ is very close to the true values. The predictions for data sets 2 and 3 are poor, not only for $w_0$, but also for the incorrect biasing of ${\Omega }_m$ (see text).

Figure 6

Upper row: same as in Fig. 4, but with the reconstruction based on the parameterization of $w$ represented by Eq. (16). The parameterization captures the variation in data set 2 reasonably well, but is still not flexible enough to reconstruct an equation of state with less smooth changes—as in data set 3. The lower panels shows the 68% and 95% confidence contours for the fitting parameters $w_0$ and $w_a$ in Eq. (16) for the three data sets. Note that the axes of the contour plots have different ranges; the uncertainties for data set 3 are the largest.

Figure 7

Results shown as in Fig. 6 but allowing ${\Omega }_m$ and ${\Delta }_{\mu }$ to vary.

Figure 8

Example for the priors (red lines) and posteriors (black lines) for ${\sigma }^2$, $\rho$ and ${\kappa }^2$ for data set 1. The parameters ${\sigma }^2$ has a noninformative prior, ${\kappa }^2$ has an inverse Gamma prior, and $\rho$ has a $Beta$ prior. The posteriors for the three different data sets for these parameters are very similar.

Figure 9

Results displayed as in Fig. 4, but with GP model-based reconstruction. For all three data sets the GP model succeeds in capturing the true behavior of $w$; the error bars at higher $z$ are slightly larger due to the sparser supernova sampling beyond $z=1.1$.

Figure 10

As in Fig. 9 but with ${\Omega }_m$ and ${\Delta }_{\mu }$ free to vary. The GP reconstruction performs extremely well for all the first two cases and captures the third case reasonably well (within error bands).

Figure 11

Residuals for $\mu$ considering the predictions for $w$ for data set 3 with ${\Omega }_m$ and ${\Delta }_{\mu }$ free. The left plot shows the result for the $w_0-w_a$ parametrization, the right plot shows the results for the GP approach.