A crucial test of the phantom closed cosmological model

We suggest a crucial direct-observational test for measuring distinction between the standard $\Lambda$CDM model and recently proposed phantom dark energy positive curvature cosmological model. The test is based on the fundamental distance--flux--redshift relation for general Friedmann models. It does not depend on the CMBR data, on the large-scale structure growth models, and also on the value of the Hubble constant $H_0$. Our crucial test can be performed by future gamma-ray burst observations with THESEUS space mission and by using gravitational-wave standard siren observations with modern advanced LIGO--Virgo and also forthcoming LISA detectors.
Comment: 4 pages, 1 figure, MNRAS accepted 18.09.2020