--- title: Background-Hierarchy Bounds in Type 3 NGR url: https://www.emergentmind.com/topics/background-hierarchy-bounds type: topic --- # Background-Hierarchy Bounds in Type 3 NGR 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 [2605.16869]. 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 [2605.16869]. ## 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 [2605.16869]. 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 \(L\sim k^{-1}\) [2605.16869]. 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 [2605.16869]. 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 [2605.16869]. 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 \(e\) is the determinant of the vierbein, \(T^\rho{}_{\mu\nu}\) is torsion, and \(T_\mu := T^\rho{}_{\rho\mu}\) is the torsion vector [2605.16869]. TEGR/GR corresponds to \(c_1=-\tfrac14,\;c_2=\tfrac12,\;c_3=1\) [2605.16869]. Type 3 is the subclass satisfying \[ 2c_1 + c_2 = 0. \] Dirac–Bergmann analysis implies that Type 3 preserves diffeomorphism invariance and spatial rotations \(SO(3)\), but breaks local Lorentz boosts [2605.16869]. The broken boost sector contributes physical degrees of freedom, and around a flat FLRW background the propagating sector consists of 2 tensor modes \(h_{ij}\), 1 transverse vector mode \(a_i\), 1 scalar mode encoded in the boost sector, plus the usual matter scalar [2605.16869]. The flat FLRW background is \[ ds^2 = -dt^2 + a^2(t)\delta_{ij}dx^i dx^j, \] with a minimally coupled scalar field \[ \phi(t,x)=\phi_0(t)+\delta\phi(t,x). \] The background equations are \[ -3(2c_1 - c_2 + 3c_3)H^2 = -\dot\phi_0^2 + V_0, \] \[ -(2c_1 - c_2 + 3c_3)(3H^2 + 2\dot H) = - 2V_0 + 4\dot\phi_0^2, \] and \[ \ddot\phi_0 + 3H\dot\phi_0 + V'(\phi_0) = 0 \] [2605.16869]. The background Lagrangian is \[ \mathcal{L}_{\text{Background} = - 3M_{\rm pl}^2(2c_1 - c_2 + 3c_3)a^3H^2 + 2a^3\dot\phi_0^2 - a^3V_0. \] Because the background contribution is proportional to \(2c_1-c_2+3c_3\), while the kinetic coefficients of the perturbations involve either \(2c_1-c_2\) or \(2c_1-c_2+c_3\), the theory naturally exhibits parameter-dependent competition between kinetic and background pieces [2605.16869]. ## 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 [2605.16869]. Under infinitesimal coordinate transformations \(x^\mu\to x^\mu+\xi^\mu\), the tetrad perturbations transform by the Lie derivative, \[ \delta e^A{}_\mu = - \mathcal{L}_\xi e^A{}_\mu, \] which leads to transformation laws such as \[ \Phi'=\Phi + H\xi^0,\qquad v'=v-\xi^0 \] [2605.16869]. The analysis identifies two preferable gauges. Gauge I, the spatially flat gauge, \[ v' = 0,\quad B' = 0,\quad C_i' = 0, \] is preferred when local Lorentz boosts are broken, because it does not eliminate fields associated with internal symmetry breaking [2605.16869]. Gauge II is \[ \Psi' = 0,\quad B' = 0,\quad C_i' = 0. \] Gauge I is the one used for the main background-hierarchy analysis [2605.16869]. Gauge-invariant combinations include \[ \mathcal{B} = \Psi + H(B - a\dot F),\quad \mathcal{Y} = \Phi + \dot{(B - a\dot F)},\quad \mathcal{A} = F - a B, \] \[ \mathcal{B}_i = G_i - 2a C_i,\quad \mathcal{W}_i = V_i - 2\epsilon_{ijk}\partial^j C^k \] [2605.16869]. In Gauge I, these reduce to simple identifications such as \[ \mathcal{A}' = F'=\mathcal{A},\quad \mathcal{B}_i' = G_i'=\mathcal{B}_i,\quad \mathcal{W}_i' = V_i'=\mathcal{W}_i. \] 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 [2605.16869]. The paper exhibits a qualitative failure of a non-gauge-invariant estimate: using the non-invariant scalar variable \(\alpha\) would lead to \[ \delta \alpha \sim \sqrt{\frac{2c_1 - c_2 + 3c_3}{2c_1 - c_2 + c_3} \sim \sqrt{\frac{2c_3}{k}\to 0\quad (k\to\infty), \] suggesting no bound, whereas the gauge-invariant analysis yields a finite scalar bound [2605.16869]. 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 \[ \frac{\mathcal{L}^{(2)}_{T}{M_{\rm pl}^2} = - a^3(2c_1 - c_2)\,\dot h_{ij}\dot h^{ij} + \dots, \] so the tensor kinetic coefficient is \(- (2c_1-c_2)\) [2605.16869]. Ghost freedom requires \[ 2c_1 - c_2 < 0, \] which, together with \(2c_1+c_2=0\), implies \(c_2>0\) and \(c_1<0\) [2605.16869]. In the vector sector, Gauge I leaves the transverse boost mode \(a_i\), with quadratic Lagrangian \[ \mathcal{L}^{(2)}_{V,\text{Type 3} = M_{\rm pl}^2 \left[ (2c_1 - c_2 + c_3)a^3 \dot a_i \dot a^i + \dots \right]. \] Ghost freedom requires \[ 2c_1 - c_2 + c_3 > 0 \] [2605.16869]. In the scalar sector, the new scalar mode is the gauge-invariant variable \(Y\), related to the boost scalar by \[ \alpha = c\,aH^{-1} \dot Y,\qquad c = -\frac{1}{aH^2(2c_1 - c_2 + 3c_3)}. \] The gauge-invariant scalar quadratic action has kinetic term \[ \mathcal{L}^{(2)}_{S,\text{Type 3} = M_{\rm pl}^2\Big[ (2c_1 - c_2 + c_3)aH^{-2}\,\dot Y\,\dot Y + \dots \Big], \] so the scalar ghost-free condition again is \[ 2c_1 - c_2 + c_3 > 0 \] [2605.16869]. The physical propagating modes in Type 3 around FLRW are therefore: 2 tensor modes \(h_{ij}\), 1 transverse vector mode \(a_i\), 1 scalar mode \(Y\), plus the matter scalar [2605.16869]. 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 \[ M_{\rm pl}^2 a^3 B_X(H,\dots)\,X^2 \sim M_{\rm pl}^2 a^3 K_X\,\dot X^2 \] and \(\dot X \sim X/L\), one extracts a threshold amplitude \(\delta X\): above this amplitude, background terms dominate [2605.16869]. For tensor modes, comparing the tensor kinetic term with the background piece proportional to \(2c_1-c_2+3c_3\) yields \[ \delta h_{ij} \sim \sqrt{\frac{2c_1 - c_2 + 3c_3}{2c_1 - c_2} = \sqrt{\frac{3c_3}{-c_2} = \sqrt{-1 + \frac{3c_3}{2c_2}, \] using \(2c_1+c_2=0\) [2605.16869]. The paper also rewrites the ratio as \[ \frac{2c_1 - c_2 + 3c_3}{2c_1 - c_2} = \frac{-2c_2 + 3c_3}{-2c_2} = 1 - \frac{3c_3}{2c_2}. \] The schematic form \[ \delta h_{ij}\sim \sqrt{-1 + \frac{3c_3}{2c_2} \] is used to emphasize the dependence on \(c_3/c_2\) [2605.16869]. For the vector mode, \[ \delta a_i \sim \sqrt{\frac{2c_1 - c_2 + c_3}{2c_1 - c_2 + 3c_3} = \sqrt{\frac{2c_3}{2c_2 - c_3}, \] again using \(2c_1+c_2=0\) [2605.16869]. This diverges as \(c_3\to 2c_2\), so near the lower ghost-free boundary the vector background-hierarchy threshold is pushed to large amplitudes [2605.16869]. For the scalar mode, the gauge-invariant action gives \[ \frac{\mathcal{L}_{S,\text{Type 3}^{(2)}{M_{\rm pl}^2} = (2c_1 - c_2 + c_3)aH^{-2}\dot Y^2 + \dots, \] and the same comparison yields \[ \delta Y \sim \sqrt{\frac{2c_1 - c_2 + 3c_3}{2c_1 - c_2 + c_3} = \sqrt{1 - \frac{2c_3}{2c_2 - c_3}, \] with the same parameter dependence as the vector sector up to factors [2605.16869]. 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 [2605.16869]. ## 6. Parameter-space structure and cosmological viability A convenient parameterization is \[ x := \frac{c_3}{c_2}. \] Ghost freedom for tensor, vector, and scalar sectors implies \[ c_2>0,\qquad c_3>2c_2, \] equivalently \[ x>2 \] [2605.16869]. The paper then imposes the requirement that scalar and vector hierarchy scales not be worse than the tensor one, \[ \delta h_{ij} < \delta a_i = \delta Y, \] which implies \[ 2c_2 < c_3 < 4c_2 \quad\Leftrightarrow\quad 20\) is introduced to remain close to the lower ghost-free boundary: \[ 0 < c_3 - 2c_2 < \varepsilon. \] This gives \[ 2 < x < 2 + \frac{\varepsilon}{c_2}. \] The tensor bound near this upper limit is written as \[ \delta h_{ij}\sim (L M_{\rm pl})^{-1}\sqrt{\varepsilon/c_2}, \] and one obtains the parameterization \[ c_2 = \frac{\varepsilon}{(LM_{\rm pl})^2},\qquad c_3 = (1 + L^2 M_{\rm pl}^2)\varepsilon \] [2605.16869]. 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 \(\varepsilon/c_2>5\), the background-hierarchy bounds become too large and all modes are affected already at relatively small amplitudes, whereas if \(\varepsilon/c_2\lesssim 5\), there exists a region in the \((x,\delta X)\) plane where perturbations are under control [2605.16869]. The qualitative structure is organized into three regions for \(2