Using $H(z)$ data as a probe of the concordance model

Direct observations of the Hubble rate, from cosmic chronometers and the radial baryon acoustic oscillation scale, can outperform supernovae observations in understanding the expansion history, because supernovae observations need to be differentiated to extract $H(z)$. We use existing $H(z)$ data and smooth the data using a new Gaussian processes package, GaPP, from which we can also estimate derivatives. The obtained Hubble rate and its derivatives are used to reconstruct the equation of state of dark energy and to perform consistency tests of the $\Lambda \mathrm{CDM}$ model, some of which are newly devised here. Current data are consistent with the concordance model, but are rather sparse. Future observations will provide a dramatic improvement in our ability to constrain or refute the concordance model of cosmology. We produce simulated data to illustrate how effective $H(z)$ data will be in combination with Gaussian processes.

Figure 1

$h(z)=H(z)/H_0$ (top) and $h^{\prime }(z)$ (bottom) reconstructed from cosmic chronometer data (left), BAO data (middle) and $\mathrm{CC}+\mathrm{BAO}$ data (right), using GP. Shaded areas represent 68% and 95% CL. The dashed (red) curve is flat $\Lambda \mathrm{CDM}$ with ${\Omega }_m=0.27$; the solid (blue) curve is the GP mean. Note that while the BAO data appear to give an inconsistent $h^{\prime }(z)$, this is driven by the two highest redshift points, both of which happen to lie below the flat $\Lambda \mathrm{CDM}$ curve.

Figure 2

${\mathcal{O}}_m^{(1)}(z)$ (top), ${\mathcal{O}}_m^{(2)}(z)$ (middle) and ${\mathcal{O}}_K(z)$ (bottom) reconstructed from cosmic chronometers (left), BAO (middle) and $\mathrm{CC}+\mathrm{BAO}$ (right). For ${\mathcal{O}}_m^{(1)}(z)$, the dashed (red) curve is flat $\Lambda \mathrm{CDM}$. For ${\mathcal{O}}_m^{(2)}(z)$ and ${\mathcal{O}}_K(z)$ it is a curved $\Lambda \mathrm{CDM}$ model.

Figure 3

${\mathcal{L}}_m^{(1)}={\mathcal{L}}^{(1)}/(1+z{)}^6$ (top) and ${\mathcal{L}}_m^{(2)}={\mathcal{L}}^{(2)}/(1+z{)}^6$ (bottom) reconstructed from cosmic chronometers (left), BAO (middle) and $\mathrm{CC}+\mathrm{BAO}$ (right). The dashed (red) curve is a $\Lambda \mathrm{CDM}$ model.

Figure 4

$w(z)$ reconstructed from cosmic chronometers (left), BAO (middle—note the different $z$ range) and $\mathrm{CC}+\mathrm{BAO}$ (right) by marginalizing over ${\Omega }_m=0.275\pm 0.016$. The dashed (red) curve is a $\Lambda \mathrm{CDM}$ model.

Figure 5

$h(z)$ (top), $h^{\prime }(z)$ (middle) and $h^{\prime \prime }(z)$ (bottom) reconstructed from simulated data, assuming a concordance model (left) and model (10) with slowly evolving $w(z)$ (right).

Figure 6

${\mathcal{O}}_m^{(1)}(z)$ (top), ${\mathcal{O}}_m^{(2)}(z)$ (middle) and ${\mathcal{O}}_K(z)$ (bottom) reconstructed from simulated data, assuming a concordance model (left) and model (10) (right).

Figure 7

${\mathcal{L}}_m^{(1)}={\mathcal{L}}^{(1)}/(1+z{)}^6$ (top) and ${\mathcal{L}}_m^{(2)}={\mathcal{L}}^{(2)}/(1+z{)}^6$ (bottom) reconstructed from simulated data, assuming a concordance model (left) and model (10) (right).

Figure 8

$w(z)$ reconstructed from simulated data, assuming a concordance model (left) and model (10) (right), by marginalizing over ${\Omega }_m=0.275\pm 0.016$.