Gravitational waves themselves are modeled as tensor perturbations of the metric. Einstein first derived gravitational waves from a linearly perturbed Minkowski metric. Ringdown analysis of compact binary black hole mergers is done in the context of the perturbed Kerr metric (see section 5 of this). Therefore, mathematically, tensor perturbations are gravitational waves.
So, if you have tensor perturbations of a metric used to model cosmology, you then have a type of gravitational waves that are cosmologically relevant. An example are "primordial" gravitational waves that could appear in the cosmic microwave background (CMB) due time-varying quadrupole moments (and higher order currents) of the early Universe matter-energy distribution. There are also scalar density perturbations that are possible.
Inflationary models of early Universe cosmology offer an explanation for primordial gravitational waves, and if detected as a gravitational wave background would provide observational support for inflation.
The claim that B-modes were observed in the BICEP2 experiment were not confirmed confidently, and later the Planck experiment showed that the BICEP2 results can be explained by cosmic dust. Therefore, it is generally accepted that B-mode oscillations in the CMB have not been observed. For an explanation of what B-modes and E-modes are, please see here.
Joint detections using space based gravitational wave interferometers, pulsar timing arrays, and CMB probes in the future may provide measurements of the primordial gravitational wave background.