Cosmic equation of state from combined angular diameter distances: Does the tension with luminosity distances exist?

Using relatively complete observational data concerning four angular diameter distance (ADD) measurements and combined $\mathrm{SN}+\mathrm{GRB}$ observations representing current luminosity distance (LD) data, this paper investigates the compatibility of these two cosmological distances considering three classes of dark energy equation of state (EoS) reconstruction. In particular, we use strongly gravitationally lensed systems from various large systematic gravitational lens surveys and galaxy clusters, which yield the Hubble constant independent ratio between two angular diameter distances $D_{ls}/D_s$ data. Our results demonstrate that, with more general categories of standard ruler data, ADD and LD data are compatible at $1\sigma$ level. Second, we note that consistency between ADD and LD data is maintained irrespective of the EoS parametrizations: there is a good match between the universally explored Chevalier-Polarski-Linder model and other formulations of cosmic equation of state. Especially for the truncated generalized equation of state (GEoS) model with $\beta =-2$, the conclusions obtained with ADD and LD are almost the same. Finally, statistical analysis of generalized dark energy equation of state performed on four classes of ADD data provides stringent constraints on the EoS parameters $w_0$, $w_{\beta }$, and $\beta$, which suggest that dark energy was a subdominant component at early times. Moreover, the GEoS parametrization with $\beta \simeq 1$ seems to be a more favorable two-parameter model to characterize the cosmic equation of state, because the combined angular diameter distance data ($\mathrm{SGL}+\mathrm{CBF}+\mathrm{BAO}+\mathrm{WMAP}9$) provide the best-fit value $\beta =0.751_{-0.480}^{+0.465}$.

Figure 1

The two-dimensional regions and one-dimensional marginalized distribution with the $1\sigma$ and $2\sigma$ contours for parameters of the XCDM model from $\mathrm{SGL}+\mathrm{CBF}+\mathrm{BAO}+\mathrm{WMAP}9$ (black line), $\mathrm{SN}+\mathrm{GRB}$ (red line), $\mathrm{SGL}+\mathrm{CBF}$(green line), and $\mathrm{BAO}+\mathrm{WMAP}9$ (purple line), respectively.

Figure 2

The two-dimensional regions and one-dimensional marginalized distribution with the $1\sigma$ and $2\sigma$ contours of parameters for CPL parametrization from the combined angular diameter distance data (upper). The lower panel illustrates $1\sigma$ and $2\sigma$ contours in the $w_0-w_{P1}$ parameter space and their one-dimensional marginalized distributions obtained from the combined angular diameter distance data and luminosity distance data, respectively (the matter density parameters ${\mathrm{\Omega }}_b{h}^2$ and ${\mathrm{\Omega }}_c{h}^2$ are fixed at the WMAP9 best-fit values).

Figure 3

The same as Fig. 2, but for the EoS parametrization $w=w_0+w_{P2}z$.

Figure 4

The two-dimensional regions and one-dimensional marginalized distribution with the $1\sigma$ and $2\sigma$ contours of parameters for the truncated GEoS model with $\beta =+2$ from the combined angular diameter distance data (upper). Comparisons between the combined angular diameter distance data and luminosity distance data are also shown (lower).

Figure 5

The same as Fig. 4, but for the truncated GEoS model with $\beta =-2$.

Figure 6

The constraint results from luminosity distance data: SN (dashed line) and $\mathrm{SN}+\mathrm{GRB}$ (solid line) on three parametrizations of cosmic equation of state.

Figure 7

The constraint results from the combined angular diameter distance data ($\mathrm{SGL}+\mathrm{CBF}+\mathrm{BAO}+\text{Planck}$) (dashed line) and luminosity distance data ($\mathrm{SN}+\mathrm{GRB}$) (solid line). The matter density parameters ${\mathrm{\Omega }}_b{h}^2$ and ${\mathrm{\Omega }}_c{h}^2$ are fixed at the Planck best-fit values.

Figure 8

$1\sigma$ and $2\sigma$ confidence level contours for the GEoS parametrization from the combined angular diameter distance data ($\mathrm{SGL}+\mathrm{CBF}+\mathrm{BAO}+\mathrm{WMAP}9$). One-dimensional marginalized parameter likelihood distributions are also added.