Background-Hierarchy Bounds in Type 3 NGR

• Background-hierarchy bounds are conditions ensuring that genuine kinetic terms dominate over background-induced terms in the quadratic action of cosmological perturbations.
• In Type 3 New General Relativity, broken Lorentz boosts lead to distinct kinetic structures and impose parameter-space constraints to maintain perturbative reliability.
• The analysis identifies a viable FLRW region (2 < c3/c2 < 4) where tensor, vector, and scalar modes remain under control, ensuring healthy cosmological evolution.

Searching arXiv for the cited paper and closely related work on Type 3 NGR and background-hierarchy bounds. Background-hierarchy bounds are conditions in linear cosmological perturbation theory that compare the genuine quadratic kinetic terms of perturbative modes with terms induced by the evolution of the background spacetime. In Type 3 of New General Relativity (NGR), these bounds arise because Lorentz-boost invariance is broken while diffeomorphism invariance and spatial rotations are preserved, so the new scalar and vector modes acquire kinetic structure from torsion but also receive background-dependent quadratic contributions proportional to quantities such as the Hubble parameter $H$. When the background contribution becomes comparable to or larger than the kinetic term, the linearized description ceases to be reliable even before conventional EFT strong coupling sets in (Tomonari et al., 16 May 2026). Within the flat FLRW setting, the notion therefore defines amplitude thresholds for tensor, transverse-vector, and scalar perturbations, and organizes a viable region of the Type 3 parameter space for cosmological applications (Tomonari et al., 16 May 2026).

1. Definition and conceptual role

In the terminology of Type 3 NGR, background-hierarchy bounds are conditions ensuring that the quadratic kinetic terms of perturbations dominate over terms induced by the background cosmological evolution in the quadratic action (Tomonari et al., 16 May 2026). The relevant comparison is between a genuine kinetic term of the schematic form

$\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$

and a background-dominated term of the schematic form

$\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$

The associated “background-hierarchy” is the ratio

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$

estimated with $\dot X \sim X/L$ at physical scale $L\sim k^{-1}$ (Tomonari et al., 16 May 2026).

A background-hierarchy bound is then the condition that this ratio remain small in the regime where linear cosmological perturbation theory is supposed to apply. When it becomes $\gtrsim 1$, the background-driven term dominates and the perturbative description of the mode is no longer reliable, even though the EFT may still be weakly coupled in the usual sense (Tomonari et al., 16 May 2026). This distinguishes the mechanism from conventional strong coupling: EFT strong coupling is associated with higher-order interactions overwhelming the quadratic kinetic term, whereas the background-hierarchy bound is already a breakdown internal to the quadratic action itself (Tomonari et al., 16 May 2026).

This framing suggests a cosmology-specific obstruction to perturbation theory. A plausible implication is that the admissible amplitude of a perturbation is controlled not only by couplings and cutoff scales, but also by the background evolution encoded by $H$, $\dot H$, and the theory parameters.

2. Type 3 NGR and the perturbative setting

New General Relativity is a three-parameter teleparallel theory with Lagrangian

$$\mathcal{L}_{\rm NGR} = e\,\Big[ c_1 T^{\rho}{}_{\mu\nu}T_{\rho}{}^{\mu\nu} + c_2 T^{\rho}{}_{\mu\nu}T^{\nu\mu}{}_{\rho} + c_3 T_\rho T^\rho \Big],$$

where $\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$0 is the determinant of the vierbein, $\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$1 is torsion, and $\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$2 is the torsion vector (Tomonari et al., 16 May 2026). TEGR/GR corresponds to $\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$3 (Tomonari et al., 16 May 2026). Type 3 is the subclass satisfying

$\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$4

Dirac–Bergmann analysis implies that Type 3 preserves diffeomorphism invariance and spatial rotations $\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$5, but breaks local Lorentz boosts (Tomonari et al., 16 May 2026). The broken boost sector contributes physical degrees of freedom, and around a flat FLRW background the propagating sector consists of 2 tensor modes $\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$6, 1 transverse vector mode $\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$7, 1 scalar mode encoded in the boost sector, plus the usual matter scalar (Tomonari et al., 16 May 2026). The flat FLRW background is

$\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$8

with a minimally coupled scalar field

$\mathcal{L}_{\text{kin} \sim M_{\rm pl}^2\,a^3\,K_X\,\dot X^2,$9

The background equations are

$\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$0

$\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$1

and

$\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$2

(Tomonari et al., 16 May 2026). The background Lagrangian is

$\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$3

Because the background contribution is proportional to $\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$4, while the kinetic coefficients of the perturbations involve either $\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$5 or $\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$6, the theory naturally exhibits parameter-dependent competition between kinetic and background pieces (Tomonari et al., 16 May 2026).

3. Gauge structure and gauge-invariant formulation

The perturbative analysis is performed in the pure-vierbein (Weitzenböck gauge) formulation. The tetrad perturbation contains scalar, vector, and tensor pieces in both spacetime and internal indices, with up to 16 perturbation fields appearing in the full decomposition (Tomonari et al., 16 May 2026). Under infinitesimal coordinate transformations $\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$7, the tetrad perturbations transform by the Lie derivative,

$\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$8

which leads to transformation laws such as

$\mathcal{L}_{\text{bkg} \sim M_{\rm pl}^2\,a^3\,B_X(H,\dot H,\dots)\,X^2.$9

(Tomonari et al., 16 May 2026).

The analysis identifies two preferable gauges. Gauge I, the spatially flat gauge,

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$0

is preferred when local Lorentz boosts are broken, because it does not eliminate fields associated with internal symmetry breaking (Tomonari et al., 16 May 2026). Gauge II is

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$1

Gauge I is the one used for the main background-hierarchy analysis (Tomonari et al., 16 May 2026).

Gauge-invariant combinations include

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$2

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$3

(Tomonari et al., 16 May 2026). In Gauge I, these reduce to simple identifications such as

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$4

A central point is that propagating modes can be correctly counted even when the quadratic Lagrangian is not written solely in gauge-invariant variables, but background-hierarchy bounds must be computed from a manifestly gauge-invariant quadratic action (Tomonari et al., 16 May 2026). The paper exhibits a qualitative failure of a non-gauge-invariant estimate: using the non-invariant scalar variable $\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$5 would lead to

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$6

suggesting no bound, whereas the gauge-invariant analysis yields a finite scalar bound (Tomonari et al., 16 May 2026). This indicates that the hierarchy is not merely a feature of variable choice, but of the physical quadratic action.

4. Quadratic sectors and mode content

In gauge-invariant form, the tensor quadratic Lagrangian around FLRW is

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$7

so the tensor kinetic coefficient is $\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$8 (Tomonari et al., 16 May 2026). Ghost freedom requires

$\frac{\mathcal{L}_{\text{bkg}}{\mathcal{L}_{\text{kin}} \sim \frac{B_X(H,\dots)\,X^2}{K_X\,\dot X^2},$9

which, together with $\dot X \sim X/L$0, implies $\dot X \sim X/L$1 and $\dot X \sim X/L$2 (Tomonari et al., 16 May 2026).

In the vector sector, Gauge I leaves the transverse boost mode $\dot X \sim X/L$3, with quadratic Lagrangian

$\dot X \sim X/L$4

Ghost freedom requires

$\dot X \sim X/L$5

(Tomonari et al., 16 May 2026).

In the scalar sector, the new scalar mode is the gauge-invariant variable $\dot X \sim X/L$6, related to the boost scalar by

$\dot X \sim X/L$7

The gauge-invariant scalar quadratic action has kinetic term

$\dot X \sim X/L$8

so the scalar ghost-free condition again is

$\dot X \sim X/L$9

(Tomonari et al., 16 May 2026).

The physical propagating modes in Type 3 around FLRW are therefore: 2 tensor modes $L\sim k^{-1}$0, 1 transverse vector mode $L\sim k^{-1}$1, 1 scalar mode $L\sim k^{-1}$2, plus the matter scalar (Tomonari et al., 16 May 2026). The background-hierarchy analysis applies to the tensor, vector, and gravitational scalar sectors.

5. Derivation of the bounds

The derivation compares, in each sector, the background part of the quadratic action with the corresponding kinetic term. Using

$L\sim k^{-1}$3

and $L\sim k^{-1}$4, one extracts a threshold amplitude $L\sim k^{-1}$5: above this amplitude, background terms dominate (Tomonari et al., 16 May 2026).

For tensor modes, comparing the tensor kinetic term with the background piece proportional to $L\sim k^{-1}$6 yields

$L\sim k^{-1}$7

using $L\sim k^{-1}$8 (Tomonari et al., 16 May 2026). The paper also rewrites the ratio as

$L\sim k^{-1}$9

The schematic form

$\gtrsim 1$0

is used to emphasize the dependence on $\gtrsim 1$1 (Tomonari et al., 16 May 2026).

For the vector mode,

$\gtrsim 1$2

again using $\gtrsim 1$3 (Tomonari et al., 16 May 2026). This diverges as $\gtrsim 1$4, so near the lower ghost-free boundary the vector background-hierarchy threshold is pushed to large amplitudes (Tomonari et al., 16 May 2026).

For the scalar mode, the gauge-invariant action gives

$\gtrsim 1$5

and the same comparison yields

$\gtrsim 1$6

with the same parameter dependence as the vector sector up to factors (Tomonari et al., 16 May 2026).

These thresholds are interpreted as upper bounds on the “healthy” amplitude of each perturbation mode at a given scale. Below the bound, the kinetic term dominates and the mode propagates as expected. Above it, background curvature/torsion effects dominate the quadratic dynamics and the linearized description loses predictivity (Tomonari et al., 16 May 2026).

6. Parameter-space structure and cosmological viability

A convenient parameterization is

$\gtrsim 1$7

Ghost freedom for tensor, vector, and scalar sectors implies

$\gtrsim 1$8

equivalently

$\gtrsim 1$9

(Tomonari et al., 16 May 2026).

The paper then imposes the requirement that scalar and vector hierarchy scales not be worse than the tensor one,

$H$0

which implies

$H$1

This is identified as the viable region of Type 3 parameter space for cosmological applications (Tomonari et al., 16 May 2026).

A small parameter $H$2 is introduced to remain close to the lower ghost-free boundary: $H$3 This gives

$H$4

The tensor bound near this upper limit is written as

$H$5

and one obtains the parameterization

$H$6

(Tomonari et al., 16 May 2026). The paper interprets this as a narrow region in parameter space where tensor hierarchical breakdown is postponed beyond the regime of interest.

The analysis also states that if $H$7, the background-hierarchy bounds become too large and all modes are affected already at relatively small amplitudes, whereas if $H$8, there exists a region in the $H$9 plane where perturbations are under control (Tomonari et al., 16 May 2026).

The qualitative structure is organized into three regions for $\dot H$0. Region I is one in which all modes are under background-hierarchy control and no mode propagates reliably. Region II is one in which tensor modes are under control, but scalar and vector modes are still background-dominated. Region III is one in which tensor, vector, and scalar modes all have their kinetic term dominating, so perturbation theory is fully viable (Tomonari et al., 16 May 2026). The physically relevant requirement is that observable cosmological amplitudes lie in Region III.

7. Relation to gauge invariance, strong coupling, and broader hierarchy notions

A central conclusion is that background-hierarchy bounds must be computed from a gauge-invariant quadratic action (Tomonari et al., 16 May 2026). The distinction between using $\dot H$1 and $\dot H$2 in the scalar sector shows that variable choice can obscure the hierarchy completely. This suggests that any future EFT strong-coupling analysis in Type 3 NGR must likewise be organized in gauge-invariant variables.

The comparison with GR is instructive. In TEGR/GR, the parameters satisfy

$\dot H$3

local Lorentz symmetry is intact, and there are no extra boost-sector modes. The paper states that no analogous background-hierarchy pathology appears: only tensors and the usual scalar from matter propagate, and standard cosmological perturbation theory remains valid without extra bounds (Tomonari et al., 16 May 2026). In Type 3 NGR, by contrast, broken Lorentz boosts decouple the coefficients of kinetic and background terms for the new modes, allowing background-hierarchy issues to arise (Tomonari et al., 16 May 2026).

Within the broader landscape of “hierarchy bounds” in the supplied literature, the phrase denotes structured constraints organized by an underlying parameter. In the informational setting of $\dot H$4-designs, the hierarchy is ordered by design order $\dot H$5, interpolating between Holevo and subentropy bounds (Dall'Arno, 2015). In multiparameter quantum metrology, the hierarchy is one of commutativity-based saturation conditions $\dot H$6 governing which precision bounds coincide (Imai et al., 12 Feb 2026). In binary search trees, multiple lazy-finger bounds form a proper hierarchy $\dot H$7 (Chalermsook et al., 2016). In the cosmological context, the organizing parameter is instead the relation between kinetic coefficients and background coefficients in the quadratic perturbation action, and the hierarchy is expressed as amplitude thresholds sector by sector (Tomonari et al., 16 May 2026). This suggests a family resemblance rather than a common formalism: in each case, a structured sequence of bounds quantifies how an auxiliary property—uniformity, commutativity, search locality, or background evolution—limits the effective behavior of the system.

The outlook identified for Type 3 NGR includes extension to other backgrounds, comparison with higher-order EFT strong-coupling scales, observational constraints on $\dot H$8, and applications to other teleparallel theories such as $\dot H$9 gravity and other NGR types (Tomonari et al., 16 May 2026). The governing conclusion is that Type 3 NGR can remain cosmologically viable only when the theory sits in the region

$$\mathcal{L}_{\rm NGR} = e\,\Big[ c_1 T^{\rho}{}_{\mu\nu}T_{\rho}{}^{\mu\nu} + c_2 T^{\rho}{}_{\mu\nu}T^{\nu\mu}{}_{\rho} + c_3 T_\rho T^\rho \Big],$$0

and sufficiently close to the lower ghost-free boundary $$\mathcal{L}_{\rm NGR} = e\,\Big[ c_1 T^{\rho}{}_{\mu\nu}T_{\rho}{}^{\mu\nu} + c_2 T^{\rho}{}_{\mu\nu}T^{\nu\mu}{}_{\rho} + c_3 T_\rho T^\rho \Big],$$1, so that tensor, vector, and scalar background-hierarchy bounds lie above amplitudes relevant for cosmological observables (Tomonari et al., 16 May 2026).