Gauge-invariant cosmological perturbations in Type 3 New General Relativity and background-hierarchy bounds

In this paper, we investigate background-hierarchy bounds in Type~3 of New General Relativity (NGR). These bounds arise when the contribution associated with the evolution of the background spacetime exceeds that of the kinetic term in the perturbed Lagrangian. Type~3 of NGR has two free parameters and is described in a pure-tetrad formulation while preserving diffeomorphism invariance and spatial rotations. We first review Type~3 and identify preferable gauge choices for metric-affine gauge theories of gravity with Weitzenböck connection, including NGR, from the viewpoint of symmetry in both Dirac–Bergmann analysis and linear perturbation theory. We then revisit the perturbative analysis of Type~3 and show that the propagating modes are correctly identified even when the perturbed Lagrangian is not written solely in terms of gauge-invariant variables. Finally, we derive the background-hierarchy bounds for the scalar, transverse-vector, and tensor modes around a flat FLRW background, and identify the region of parameter space in which the linear perturbation theory of Type~3 remains viable for cosmological applications.