Nonparametric reconstruction of the dark energy equation of state from diverse data sets

The cause of the accelerated expansion of the Universe poses one of the most fundamental questions in physics today. In the absence of a compelling theory to explain the observations, a first task is to develop a robust phenomenological approach: If the acceleration is driven by some form of dark energy, then the phenomenology is determined by the form of the dark energy equation of state $w(z)$ as a function of redshift. A major aim of ongoing and upcoming cosmological surveys is to measure $w$ and its evolution at high accuracy. Since $w(z)$ is not directly accessible to measurement, powerful reconstruction methods are needed to extract it reliably from observations. We have recently introduced a new reconstruction method for $w(z)$ based on Gaussian process modeling. This method can capture nontrivial $w(z)$ dependences and, most importantly, it yields controlled and unbiased error estimates. In this paper we extend the method to include a diverse set of measurements: baryon acoustic oscillations, cosmic microwave background measurements, and supernova data. We analyze currently available data sets and present the resulting constraints on $w(z)$, finding that current observations are in very good agreement with a cosmological constant. In addition, we explore how well our method captures nontrivial behavior of $w(z)$ by analyzing simulated data assuming high-quality observations from future surveys. We find that the baryon acoustic oscillation measurements by themselves already lead to remarkably good reconstruction results and that the combination of different high-quality probes allows us to reconstruct $w(z)$ very reliably with small error bounds.

Figure 1

Results for reconstructing $w(z)$ from currently available data. The top row shows reconstruction results for $w(z)$ (blue line; the red dashed line shows $w=-1$) for an exponential covariance function, i.e. $\alpha =1$, including different cosmological probes. The bottom panels show the priors (normal distribution in the first panel, flat priors in the rest) and the posterior distributions for ${\Omega }_m$ (red lines: priors; black lines: posteriors). The first column shows the results from supernova data only, the second column includes CMB measurements, the third column uses supernova and BAO data, and the fourth column shows the results for a combined supernova-BAO-CMB analysis. The light blue contours show the 95% confidence level, and the dark blue contours show the 68% confidence level. As is to be expected, the error bars shrink somewhat if more data sources are included, though the effect is small due to the limited number of extra data points. As found previously by others (e.g., Ref. 13), current data are consistent with a cosmological constant.

Figure 2

A few random GP realizations for $w(z)$ for the SNe-only data (left upper panel in Fig. 1).

Figure 3

Dark energy equation of state $w(z)$ for our three simulated models.

Figure 4

Redshift distribution of the small supernova data set. The simulated data have exactly the same distribution as the real data.

Figure 5

Two realizations of 20 simulated BAO points for a BigBOSS-like survey. The left column shows the angular distance diameter, while the right column shows the Hubble parameter, both appropriately scaled by the sound horizon as relevant to BAO measurements. Model 1 ($w=\mathrm{const}$) is shown with error bars with 1 standard deviation. For model 2 (green dashed line) and model 3 (blue dotted line) we show the exact predictions.

Figure 6

Left column: reconstruction result for $w(z)$ for realization I of the BAO data later used in combination with the small supernova sample. Right column: results for realization II later used in combination with the large supernova sample. Top to bottom: results for models 1–3. The dashed line shows the underlying theoretical model, the dark blue region shows the 68% confidence level, the light blue region shows the 95% confidence level, and the dark blue line shows the mean reconstructed history. The error bands represent the single realization reconstruction error. In all cases, the reconstruction results reliably capture the truth within the error bands.

Figure 7

Reconstruction results for the model with $w=-1$. From left to right different probes are considered: SNe, $\mathrm{SNe}+\mathrm{CMB}$, $\mathrm{SNe}+\mathrm{BAO}$, and a combination of all three measurements. The red dashed line shows the truth, the blue solid line the mean result for the reconstruction. The blue shaded region shows the 68% confidence level, the light blue shaded region the 95% confidence level. The upper row shows the results for the small supernova data set (557 supernovae with ${\tau }_i=0.15$ out to $z=1.4$), while the lower row shows potential space-based supernova measurements (2298 supernovae with ${\tau }_i=0.13$ out to $z=1.7$). The CMB data point is the same in all cases where it is included; the BAO data are of the same quality but two different realizations are used out to $z=2$. Note that the redshift range varies in the different panels depending on which probes are included.

Figure 8

Same as in Fig. 7 but for model 2, the quintessence model.

Figure 9

Same as in Fig. 7 but for model 3.

Figure 10

Reconstruction results for $w(z)$ from 2298 simulated SNe measurements for different priors on $\rho$. Left column: $w(z)$ reconstruction with 68% and 95% confidence limits. Right column: prior (red line) and posterior distribution (black line) for $\rho$. Upper row: flat prior on $\rho$, $\pi (\rho )\sim U(0,1)$. Bottom row: informative prior on $\rho$, $\pi (\rho )\sim Be(6,1)$.

Figure 11

Reconstruction results following Fig. 10, but now in addition to using a uniform prior for $\rho$, we loosen the ${\kappa }^2$ prior: ${\kappa }^2\sim G(1,0.5)$, $\rho \sim U(0,1)$. The left upper panel shows the reconstruction result for $w(z)$, and the right upper panel shows some realizations (similar to Fig. 2). The error bands in the prediction for $w(z)$ do increase in this case, as expected. This case shows the most relaxed priors we study for this data set.