Fig. \[f3\]). We then obtain two special traversable wormhole solutions supported by the current astrophysical observations: the case of a constant redshift function and that of a specific choice for the shape function. In what follows, utilizing the model parameter pair from the SNe Ia+OHD+Planck+GW constraint, we analyze the traversabilities of the first traversable wormhole solution and the amounts of exotic matter of the second solution.

In contrast to our previous studies [@16; @17; @18; @19], we explored the energy conditions from the point of view of observational cosmology. More precisely, we obtained several model-independent conclusions: (i) According to the violations of different energy conditions, the exotic matter threading the traversable wormhole can be divided into four classes during the evolutionary processes of the universe. (ii) We can impose a strong restriction to the local wormhole space–time structure by using the current astrophysical observations. (iii) A physically gravitational resource for the wormholes supported by astrophysical observations has been identified, i.e., the dark energy of the universe or equivalent space–time curvature effects from the framework of MOG.

We investigated the energy conditions on both astrophysical and cosmological scales and derived the underlying relations between energy conditions and wormhole structure formation for 