Model-independent test of the FLRW metric, the flatness of the Universe, and non-local measurement of $H_0r_\mathrm{d}$

Using measurements of $H(z)$ and $d_\mathrm{A}(z)$ from the Baryon Oscillation Spectroscopic Survey (BOSS) DR12 and luminosity distances from the Joint Lightcurve Analysis (JLA) compilation of supernovae (SN), we measure $H_0 r_\mathrm{d}$ without any model assumption. Our measurement of $H_0 r_\mathrm{d} = $($10033.20^{+333.10}_{-371.81}$ (SN) $\pm$ 128.19 (BAO)) km s$^{-1}$ is consistent with Planck constrains for the flat {\Lambda}CDM model. We also report that higher expansion history rates $h(z)$ (among the possibilities) as well as lower-bound values of $H_0 r\mathrm{d}$ result in better internal consistency among the independent data (H(z)rd and $d_\mathrm{A}(z)/r_\mathrm{d}$ from BAO at $z=0.32$ and z=0.57 and $d_\mathrm{L}$ from JLA) we used in this work. This can be interpreted as an interesting and independent support of Planck cosmology without using any cosmic microwave background data. We then combine these observables to test the Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric and the flatness of the Universe in a model-independent way at two redshifts, namely 0.32 and 0.57, by introducing a new diagnostic for flat-FLRW, $\Theta(z)$, which only depends on observables of BAO and SN data. Our results are consistent with a flat-FLRW Universe within $2{\sigma}$.