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

Abstract: 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 quadratic kinetic term in the perturbed Lagrangian. Type~3 of NGR has two free parameters and preserves diffeomorphism invariance and spatial rotations, while breaking Lorentz-boost invariance. 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.

• The paper introduces a gauge-invariant formalism in Type 3 NGR to accurately identify tensor, vector, and scalar modes.
• It derives explicit kinetic matrices and imposes ghost-free constraints, outlining a viable parameter space with conditions like 2 < c3/c2 < 4.
• The study establishes background-hierarchy bounds to ensure that perturbative modes dominate over non-linear background dynamics.

Gauge-Invariant Cosmological Perturbations in Type 3 New General Relativity and Background-Hierarchy Bounds

Overview of Type 3 New General Relativity and Motivations

Type 3 New General Relativity (NGR) is a specific two-parameter sector of the general three-parameter NGR framework, an extension of the Teleparallel Equivalent to General Relativity (TEGR). In NGR, gravity is described through spacetime torsion, preserving parity, diffeomorphism invariance, and spatial rotations while breaking Lorentz-boost invariance. The theoretical landscape is motivated by unresolved cosmological phenomena such as dark energy, dark matter, and tensions related to cosmological parameters. Type 3 NGR, distinguished by its absence of ghost instability and compatibility of linear perturbative DOF counting with the full theory, emerges as a particularly promising candidate for cosmological modeling (2605.16869).

Symmetry Structure and Gauge Choices

A major focus is the precise characterization of propagating degrees of freedom (DOFs) arising from symmetry breaking, notably the violation of boost invariance in the local Lorentz group. The Dirac-Bergmann (DB) analysis is utilized to track constraint structures, revealing that Type 3 NGR retains so(3) internal symmetries but breaks full Lorentz invariance except for rotations. This breaking leads to the emergence of new propagating modes inaccessible in TEGR.

The paper conducts a systematic identification of "preferable" gauge choices, ensuring consistency between the symmetries revealed in the Hamiltonian analysis and those retained in linearized perturbative treatments. Notably, Gauge Choice I fixes $\nu'=0$, $B'=0$, and $C_i'=0$, and is shown to be preferable for NGR where boost invariance is broken. In this framework, cosmological perturbation analysis is only reliable when performed in gauge-invariant variables; otherwise, spurious or missing modes may contaminate physical results.

Linear Perturbation Theory: Propagating Modes and Ghost-Free Conditions

Using the preferable gauge (Gauge Choice I), the perturbative analysis around a flat FLRW background correctly identifies the propagating tensor ($h_{ij}$), vector ($a_i$), and scalar ($a$) modes. These modes are attributable directly to symmetry-breaking patterns. The analysis demonstrates that writing the perturbed Lagrangian in gauge-invariant variables is crucial not only for DOF counting but also for avoiding Ostrogradsky instabilities (ghosts) and for correctly estimating energy hierarchy bounds.

All kinetic matrices for scalar, vector, and tensor modes are computed explicitly, with ghost-free conditions emerging as constraints on the two free parameters. For instance, the tensor sector requires $2c_1 - c_2 < 0$ and $c_2 > 0$, while scalar and vector sectors place further bounds, such as $2c_1 - c_2 + c_3 > 0$ and $c_3 > 0$. There is a nontrivial parameter region where all constraints overlap, ensuring perturbative viability.

Background-Hierarchy Bounds and Strong Coupling in Cosmological Contexts

A central technical contribution is the distinction between two forms of strong coupling:

• Conventional EFT strong coupling: Dominance of higher-order interaction terms in perturbative expansion.
• Background-hierarchy bound: Occurs in cosmological setups, where the background dynamics (e.g., expansion rate $B'=0$0) dominates over quadratic kinetic terms in the perturbed Lagrangian. It is shown that cosmological perturbation theory is only valid when the hierarchy bound is absent or sufficiently suppressed.

Background-hierarchy bounds are derived for scalar, vector, and tensor modes, and the parameter space $B'=0$1 is shown to require $B'=0$2 for theoretical health. Three distinct regions in parameter space are identified: (I) no valid modes, (II) only tensor mode survives, (III) all modes propagate and are free from background dominance.

Of particular note is the observation that failure to express the perturbed Lagrangian in gauge-invariant terms will yield qualitatively incorrect strong coupling results, potentially invalidating a model's cosmological predictions.

Practical and Theoretical Implications

The findings establish rigorous criteria for constructing cosmological models within NGR, with explicit bounds on parameter space for phenomenological viability. The absence of ghost instabilities and the existence of a healthy parameter domain underscore the utility of Type 3 NGR for addressing contemporaneous cosmological tensions. Observable constraints on parameters (e.g., $B'=0$3) can be imposed through comparison with cosmological measurements, further refining the model space.

The results also motivate the application of the developed gauge-invariant perturbation formalism to other teleparallel or metric-affine gravity theories, including $B'=0$4 gravity, and suggest directions for exploring hybrid models (e.g., Type 9 scenarios with Higgs inflation). The approach outlined forms a robust basis for future effective field theory construction, structure formation modeling, and non-linear perturbative analysis, with attention to higher-order interactions and potential strong coupling regimes.

Conclusion

This work provides a comprehensive Hamiltonian and perturbative analysis of Type 3 NGR, elucidating the interplay between symmetry breaking, propagating degrees of freedom, ghost-free conditions, and cosmological viability. By enforcing gauge-invariant formulations, stringent energy hierarchy bounds are derived, and the parameter space supporting meaningful cosmological perturbations is explicitly mapped. The results both resolve previous ambiguities in DOF counting and perturbation validity and lay groundwork for systematic phenomenological application and further theoretical development in modified gravity cosmology.