Improved constraints on the dark energy equation of state using Gaussian processes

We perform a comprehensive study of the dark energy equation of state (EoS) utilizing the model-independent Gaussian processes (GP). Using a combination of the Union 2.1 data set, the 30 newly added H(z) cosmic chronometer data points and Planck’s shift parameter, we modify the usual GaPP code and provide a tighter constraint on the dark energy EoS than the previous literature about GP reconstructions. Subsequently, we take the “controlling variable method ” to investigate directly the effects of the variable matter density parameter ${\mathrm{\Omega }}_{m0}$, variable cosmic curvature ${\mathrm{\Omega }}_{k0}$, and variable Hubble constant $H_0$ on the dark energy EoS. We find that too small or large ${\mathrm{\Omega }}_{m0}$, ${\mathrm{\Omega }}_{k0}$, and $H_0$ are all disfavored by our GP reconstructions based on current cosmological observations. Subsequently, we find that variables ${\mathrm{\Omega }}_{m0}$ and ${\mathrm{\Omega }}_{k0}$ affect the reconstructions of the dark energy EoS but hardly affect the reconstructions of the normalized comoving distance $D(z)$ and its derivatives $D^{\prime }(z)$ and $D^{\prime \prime }(z)$. However, variable $H_0$ affects the reconstructions of the dark energy EoS by affecting obviously those of $D(z)$, $D^{\prime }(z)$, and $D^{\prime \prime }(z)$. Furthermore, we find that the results of our reconstructions support substantially the recent local measurement of $H_0$ reported by Riess et al.

Figure 1

The GP reconstructions of $f(x)$, $f^{\prime }(x)$, and $f^{\prime \prime }(x)$ by using SNe Ia. The shaded regions are reconstructions with 68% and 95% confidence levels. The blue lines represent the underlying true model (the mean value of reconstructions). Since this figure is aimed at demonstrating the correctness of our GP reconstruction, we take the labels $x$, $f(x)$, $f^{\prime }(x)$, and $f^{\prime \prime }(x)$ as distinctions with those of $z$, $D(z)$, $D^{\prime }(z)$, and $D^{\prime \prime }(z)$ used in Fig. 2.

Figure 2

The GP reconstructions of $D(z)$, $D^{\prime }(z)$, and $D^{\prime \prime }(z)$ using SNe $\mathrm{Ia}+\mathrm{H}(z)+\mathrm{CMB}$. The blue and magenta lines represents the underlying true model (the mean value of reconstructions) and the $\mathrm{\Lambda }\mathrm{CDM}$ model, respectively. The 30 $H(z)$ data points are exhibited in the lower right panel.

Figure 3

The GP reconstructions of $D(z)$, $D^{\prime }(z)$, and $D^{\prime \prime }(z)$ using different observations. The upper left panel, upper right panel, lower left panel, and lower right panel correspond to SNe Ia, SNe $\mathrm{Ia}+\mathrm{CMB}$, SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})$, and SNe $\mathrm{Ia}+\mathrm{H}(z)+\mathrm{CMB}$, respectively. We have assumed ${\mathrm{\Omega }}_{m0}=0.3$, ${\mathrm{\Omega }}_{k0}=0$, and $H_0=70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.

Figure 4

The GP reconstructions of the dark energy EoS $\omega (z)$ using different observations. The different panels from left to right correspond to SNe Ia, SNe $\mathrm{Ia}+\mathrm{CMB}$, SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})$, and SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})+\mathrm{CMB}$, respectively. We have assumed ${\mathrm{\Omega }}_{m0}=0.3$, ${\mathrm{\Omega }}_{k0}=0$, and $H_0=70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.

Figure 5

The GP reconstructions of the dark energy EoS $\omega (z)$ using SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})+\mathrm{CMB}$. The upper panels from left to right correspond to the cases of ${\mathrm{\Omega }}_{m0}=0.308\pm 0.012$, 0.1, and 0.26, respectively. The lower panels from left to right correspond to the cases of ${\mathrm{\Omega }}_{m0}=0.268$, 0.35, and 0.5, respectively. We have assumed ${\mathrm{\Omega }}_{k0}=0$ and $H_0=73.24\pm 1.74\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.

Figure 6

The GP reconstructions of the dark energy EoS $\omega (z)$ using SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})+\mathrm{CMB}$. The upper panels from left to right correspond to the cases of ${\mathrm{\Omega }}_{k0}=0.005$, $-0.005$, 0.05, and $-0.04$, respectively. The lower panels from left to right correspond to the cases of ${\mathrm{\Omega }}_{k0}=0.2$, $-0.2$, and $-0.08$, respectively. We have assumed ${\mathrm{\Omega }}_{m0}=0.308\pm 0.012$ and $H_0=73.24\pm 1.74\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.

Figure 7

The GP reconstructions of $D(z)$, $D^{\prime }(z)$, and $D^{\prime \prime }(z)$ using SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})+\mathrm{CMB}$. The upper panels from left to right correspond to the cases of $H_0=65$ and $68.1\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, respectively. The medium panels from left to right correspond to the cases of $H_0=70$ and $73.8\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, respectively. The lower panels from left to right correspond to the cases of $H_0=80$ and $100\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, respectively. We have assumed ${\mathrm{\Omega }}_{m0}=0.308\pm 0.012$ and ${\mathrm{\Omega }}_{k0}=0$.

Figure 8

The GP reconstructions of the dark energy EoS $\omega (z)$ using SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})+\mathrm{CMB}$. The upper panels from left to right correspond to the cases of $H_0=65$, 68.1, and $70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, respectively. The lower panels from left to right correspond to the cases of $H_0=73.8$, 80, and $100\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, respectively. We have assumed ${\mathrm{\Omega }}_{m0}=0.308\pm 0.012$ and ${\mathrm{\Omega }}_{k0}=0$.

Figure 9

The GP reconstructions of $D(z)$, $D^{\prime }(z)$, $D^{\prime \prime }(z)$, and the dark energy EoS $\omega (z)$ using SNe $\mathrm{Ia}+\mathrm{H}(\mathrm{z})+\mathrm{CMB}$. We have assumed ${\mathrm{\Omega }}_{m0}=0.308\pm 0.012$, ${\mathrm{\Omega }}_{k0}=0$, and $H_0=66.93\pm 0.62\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.