Intermediate redshift calibration of Gamma-ray Bursts and cosmic constraints in non-flat cosmology

We propose how to calibrate long gamma-ray burst (GRB) correlations employing intermediate redshift data sets, instead of limiting to $z\simeq0$ catalogs. To do so, we examine the most updated observational Hubble data (OHD) and baryonic acoustic oscillations (BAO). We exploit the model-independent technique of B\'ezier polynomial interpolation, alleviating de facto the well-known circularity problem affecting GRB correlations. To get constraints on cosmic parameters, using Markov chain Monte Carlo Metropolis algorithm, we distinguish the influence on BAO scale, $r_{\rm s}$, Hubble constant $H_0$, luminosity distance $D_{\rm L}(z)$ and spatial curvature $\Omega_k$. Inspired by the fact that a few 0.4$\%$ error on $r_{\rm s}$ is got from Planck results, utterly small compared with current BAO measurement errors, we discern two main cases, namely $(r_{\rm s}/r_{\rm s}^{\rm fid})=1$ and $(r_{\rm s}/r_{\rm s}^{\rm fid})\neq1$. For each occurrence, we first fix and then leave free the Universe's spatial curvature. In all our treatments, we make use of the well-consolidated \textit{Amati} correlation, furnishing tighter constraints on the mass density than previous literature. In particular, our findings turn out to be highly more compatible with those got, adopting the $\Lambda$CDM paradigm, with standard candle indicators. Finally, we critically re-examine the recent $H_0$ tension in view of our outcomes.