1. Introduction
There is a very important relation in cosmology, and it is the distance-duality relation (DDR). It was founded in 1933 by Etherington [1], which is the relation between the angular diameter distance DA and the luminosity distance DL, and has the following form
$\begin{eqnarray}\displaystyle \frac{{D}_{L}}{{D}_{A}{\left(1+z\right)}^{2}}-1=0,\end{eqnarray}$
where z is the red-shift. There are three general hypotheses about the DDR: photons’ number is constant, the Universe’s space-time is denoted by Riemannian geometry and light travel along zero geodesics. Violation of the first hypothesis can study the cosmic opacity [2–9]. And violation of the next two hypotheses can study the exotic physics [10–14], Tiwari has constrained axion-like particles with the DDR in the paper [14]. Because the DDR does not relate with the matter and the field equations of Einstein, and influences the relationship of the observational data DA and DL, so it plays a very important role in modern cosmology and in cosmic observations [15–20]. Therefore, it is very requisite and worthy to check for the DDR.Uzan et al studied the DDR [21] and got that this relation coincides with the Galaxy cluster observational data in the 68.27% error range and value of η is slightly smaller than 1. Rana et al [22] found that the DDR is consistent with observational data for the entire redshift range (0,2.418) in the 68.27% error range. Santos et al [23] discovered that the DDR is consistent with galaxy clusters observations and Hubble parameter observations in the 68.27% error range. Liao et al [24] got that the DDR is consistent with type Ia supernovae (SNe Ia) data and strong gravitational lensing observations in the 68.27% error range. Using type-Ia supernovae and ultra-compact radio data, Li and Lin [25] discussed the DDR and found that the DDR coincides with observations in the 95.45% error range. Lin, Li and Li [26] constrained the DDR and got that the DDR is true with the newer baryon acoustic oscillation (BAO) data and the type Ia supernovae (SNe Ia) data in the 68.27% error range. Holanda et al [27] have studied the DDR with supernovae Ia and strong gravitational lensing data and find that this relation is consistent the observations in the 68.27% error range. In the preceding time, the DDR was discussed too [28–44].
Lately, Wu et al [45] constrained the DDR with the Union2.1 data and five BAO data, and they found that the DDR is consistent with observational data in the 95.45% error range before h was marginalized. Now that more and new observations have been provided, we will check for the DDR with the observational data in this paper.
2. Method and samples
In order to check for the DDR with a model-independent way, one must get DA and DL at the same redshift from the different cosmic observations, and the DDR is true if the equation (1 ) is zero. The DA data is chosen from the BAO [46–50] observational data. The BAO produces in the early universe and causes an over density of baryons at about the 150Mpc scale now. Because the measurement of the BAO is not related to the number of photons, it is regarded as the standard ruler. The nine data points have been listed in table 1. The DL data are chosen from the standard candles, SNe Ia, include the Union2.1 SNe Ia [51] and the joint light-curve analysis SNe Ia (JLA) [52–54]. The JLA has 740 samples and the Union2.1 has 580 data points. Because one can only get the distance modulus μ directly from the type Ia supernovae data, so the DL can be calculated through equation (2 )
$\begin{eqnarray}{D}_{L}={10}^{0.2(\mu -25)}.\end{eqnarray}$
In order to get the DL(z) from μ(z), all SNe Ia data in the region $[z-0.005,z+0.005]$ have been binned, and the computational formula is $\begin{eqnarray}{\mu }_{{bin}}(z)=\displaystyle \frac{{\sum }_{i}{\mu }_{i}{\sigma }_{i}^{-2}}{{\sum }_{i}{\sigma }_{i}^{-2}},\end{eqnarray}$
and ${\sigma }_{{\mu }_{{bin}}}^{2}$ is $\begin{eqnarray}{\sigma }_{{\mu }_{{bin}}}^{2}=\displaystyle \frac{1}{{\sum }_{i}{\sigma }_{i}^{-2}}.\end{eqnarray}$
Here, ${\sigma }_{{\mu }_{{bin}}}$ is the error of the μi.To check for the DDR with the observational data, three parametrizations of equation (1 ) have been chosen,
$\begin{eqnarray}\displaystyle \frac{{D}_{L}}{{D}_{A}{\left(1+z\right)}^{2}}-1={\eta }_{1},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{D}_{L}}{{D}_{A}{\left(1+z\right)}^{2}}-1={\eta }_{2}z,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{D}_{L}}{{D}_{A}{\left(1+z\right)}^{2}}-1={\eta }_{3}\displaystyle \frac{z}{1+z}.\end{eqnarray}$
Here η1, η2 and η3 are constants. The results of η1, η2 and η3 will be null if the DDR is valid with the observational data.To obtain the best fit results of η1, η2 and η3, we calculate them by using $L\propto {{\rm{e}}}^{{}^{-{\chi }^{2}/2}}$, with
$\begin{eqnarray}{\chi }^{2}({\eta }_{j})=\sum _{i}\displaystyle \frac{{\left({\eta }_{j}-{\eta }_{j,{obs}}({z}_{i}\right)}^{2}}{{\sigma }_{{\eta }_{j,{obs}}}^{2}({z}_{i})}.\end{eqnarray}$
Here j = 1, 2, or 3, ${\eta }_{j,{obs}}({z}_{i})$ is the result gained from the observational data, and ${\sigma }_{{\eta }_{j,{obs}}}({z}_{i})$ is the error of ${\eta }_{j,{obs}}({z}_{i})$. The likelihood of ${\eta }_{j}$ is $P={{\rm{e}}}^{-{\chi }^{2}({\eta }_{j})/2}$.3. Results
Figure 1. The likelihood distributions of η1, η2 and η3 without (st) and with (sy) systematic errors from the Union2.1 plus BAO. |
Figure 2. The likelihood distributions of η1, η2 and η3 without and with systematic errors from the JLA plus BAO. |
Table 2. The best fit values of η and the 68.27% error range without systematic errors and with systematic errors. |
| η | Union2.1+BAO | JLA+BAO |
|---|---|---|
| ${\eta }_{1,{st}}$ | −0.0227 ± 0.0177 | −0.0146 ± 0.0147 |
| ${\eta }_{2,{st}}$ | −0.0485 ± 0.0364 | −0.0256 ± 0.0304 |
| ${\eta }_{3,{st}}$ | −0.0730 ± 0.0550 | −0.0407 ± 0.0464 |
| ${\eta }_{1,{sy}}$ | 0.0066 ± 0.0354 | −0.0106 ± 0.0155 |
| ${\eta }_{2,{sy}}$ | 0.0193 ± 0.0694 | −0.0176 ± 0.0309 |
| ${\eta }_{3,{sy}}$ | 0.0270 ± 0.1069 | −0.0283 ± 0.0473 |
The DDR is consistent with the BAO plus Union2.1 data in the 95.45% error range if the systematic errors are not considered, and in the 68.27% error range if the systematic errors are considered. And the DDR is consistent with the BAO plus JLA data in the 68.27% error range whether the systematic errors are considered or not.
To obtain the distance modulus μ from the SNe Ia [51, 52], the reductive Hubble constant h $({H}_{0}\,=100\,{\rm{h}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1},{H}_{0}$ is Hubble constant) has been taken to be h = 0.7. Because there are minute differences in different observational h, for example, (0.678 ± 0.009 [55], 0.700 ± 0.022 [56, 57], 0.738 ± 0.024 [58], 0.6850 ± 0.0127 [52], ${0.7612}_{-0.0344}^{+0.0347}$ [59] and ${0.672}_{-0.010}^{+0.012}$ [60]), so we must consider the impact of h = 0.7. For the sake of discussing the influence of the uncertainty of h, we assume it is a free parameter. Because ${D}_{L}\propto \tfrac{1}{h}$, one can obtain ${D}_{L}\,=\tfrac{0.7}{h}{10}^{0.2(\mu -25)}$ [45].
And then, we constrain the DDR by allowing h to be a free parameter rather than a definite value. The results are shown in figures 3 and 4, and the best fit values are $({\eta }_{2,{st}}=-0.0872,h=0.714)$, $({\eta }_{2,{sy}}=0.0840,h=0.677)$, $({\eta }_{3,{st}}=-0.2203,h=0.736)$ and $({\eta }_{3,{sy}}=0.1432$, h = 0.674) with the Union2.1 plus BAO data; $({\eta }_{2,{st}}=0.0348$, h = 0.679), $({\eta }_{2,{sy}}=0.0369$, h = 0.681), $({\eta }_{3,{st}}=0.0763,h\,=0.673)$ and $({\eta }_{3,{sy}}=0.0806,h=0.675)$ with the JLA plus BAO data. So the DDR is consistent with the observational data in the 68.27% error range whether the systematic errors are considered or not while h is a free parameter.
Figure 3. The contour plots of $h-{\eta }_{2}$ and $h-{\eta }_{3}$ from BAO plus Union2.1 observational data. |
Figure 4. The contour plots of $h-{\eta }_{2}$ and $h-{\eta }_{3}$ from BAO plus JLA observational data. |
To eliminate the influence of h, we marginalize the h. We assume that h is a Gaussian distribution, $\overline{P}(h)\,=\exp \left(-\tfrac{{\left(h-{h}_{{obs}}\right)}^{2}}{2{\sigma }_{{h}_{{obs}}}^{2}}\right)$, and one can get P(ηj) by calculating
$\begin{eqnarray}P({\eta }_{j})={\int }_{-\infty }^{+\infty }\overline{P}(h)\cdot P({\eta }_{j},h){\rm{d}}{h},\end{eqnarray}$
where $P({\eta }_{j},h)={\rm{\exp }}(-{\chi }^{2}({\eta }_{j},h)/2)$. The Hubble constant is chosen ${H}_{0}=70.0\pm 2.2\,\mathrm{km}\,{{\rm{s}}}^{-1}{\mathrm{Mpc}}^{-1}$ [56, 57] in this paper. The likelihood distributions of η2 and η3 are shown in figures 5 and 6. And find the best fit values and 68.27% error range are ${\eta }_{2,{st}}=-{0.0564}_{-0.0760}^{+0.0680}$, ${\eta }_{2,{sy}}={0.0214}_{-0.0909}^{+0.0924}$, ${\eta }_{3,{st}}\,=-{0.0887}_{-0.1035}^{+0.1071}$ and ${\eta }_{3,{sy}}={0.0283}_{-0.1414}^{+0.1439}$ with the Union2.1 plus BAO data; ${\eta }_{2,{st}}=-{0.0167}_{-0.0622}^{+0.0633}$, ${\eta }_{2,{sy}}\,=-{0.0099}_{-0.0627}^{+0.0648}$, ${\eta }_{3,{st}}=-{0.0329}_{-0.0995}^{+0.1033}$ and ${\eta }_{3,{sy}}\,=-{0.0215}_{-0.1002}^{+0.1041}$ with the JLA plus BAO data. Though the best fit values are not zero, but the DDR is accommodated in the 68.27% error range after h is marginalized, whether the systematic errors are considered or not.
Figure 5. The likelihood functions of η2 and η3 with a marginalization over h from BAO plus Union2.1 observational data. |
Figure 6. The likelihood functions of η2 and η3 with a marginalization over h from BAO plus JLA observational data. |
4. Conclusion
The DDR is very important in modern cosmology because it is independent with any cosmic model, and it can reflect a lot of information about the Universe when one tests it.
In this letter, we check for the DDR with the new standard candles (SNe Ia) and the new standard rulers (BAO), the best fit values of ηj support that the DDR is valid to the observational data in the 68.27% error range, except for the instance of Union2.1 and BAO regardless of the systematic errors where the DDR is valid to observations in the 95.45% error range.
When we treat h as a free parameter, the DDR is consistent with observations in the 68.27% error range no matter how the systematic errors are included. So this constraint is dependent on the precise Hubble constant. And then, assuming h is a Gaussian distribution and making a marginalization over it, the likelihood functions of η2 and η3 are obtained, the best fit values of η2 and η3 are nearer zero, the DDR is consistent with observations in the 68.27% error range, whether systematic errors are considered or not. So, the more precise observations are expected to constrain the DDR in future.