Cosmological Perturbation in New General Relativity: Propagating mode from the violation of local Lorentz invariance

This paper establishes a consistent vierbein perturbation framework for New General Relativity to analyze cosmological modes, revealing that Type 3 of the theory possesses five stable propagating modes (tensor, scalar, and vector) that make it particularly suitable for cosmological applications despite its violation of local Lorentz invariance.

The Gist

Imagine the universe as a giant, invisible fabric. For nearly a century, our best map of this fabric was drawn by Albert Einstein's General Relativity (GR). In Einstein's view, gravity isn't a force pulling things down; it's the fabric itself curving and bending under the weight of stars and planets.

But what if that map is missing a few details? What if the fabric has a hidden "twist" or "kink" that Einstein didn't account for?

This is the story of New General Relativity (NGR), a theory proposed by physicists Kyosuke Tomonari, Taishi Katsuragawa, and Shin'ichi Nojiri. They are trying to fix a specific problem in a modern version of gravity called "Teleparallel Gravity," which describes the universe using twists (torsion) instead of curves.

Here is the breakdown of their research, explained with everyday analogies.

1. The Broken Compass (The Problem)

In standard physics, there is a rule called Local Lorentz Invariance. Think of this as a universal "compass" that works the same way no matter how you spin, tilt, or rotate your laboratory. If you measure gravity in New York, it should behave exactly the same as if you measured it in Tokyo, even if you are spinning around.

However, in this "New General Relativity" theory, that compass is broken. The rules change depending on how you rotate your frame of reference.

• The Analogy: Imagine you are playing a video game where the physics engine is glitchy. If you run forward, you move normally. But if you turn your character 90 degrees, suddenly gravity pulls you sideways. The game is "locally broken."

The authors discovered that because this "compass" is broken, the universe might have extra wiggles or new ways to vibrate that don't exist in Einstein's original theory.

2. The 16-Piece Puzzle (The Method)

To find these new wiggles, the authors had to look at the universe not just as a smooth sheet, but as a complex puzzle made of 16 different pieces (mathematical fields).

• The Old Way: Previous scientists looked at the puzzle and only studied the 10 pieces that form the "shape" of the universe (the metric). They ignored the other 6 pieces, assuming they were just static background noise.
• The New Way: The authors realized that because the "compass" is broken, those other 6 pieces are actually active players. They are like the hidden gears in a clock that, when the clock is broken, start ticking loudly and creating new sounds.

They decided to study the universe by shaking this 16-piece puzzle gently (mathematically called "perturbations") to see which pieces start vibrating and sending out waves.

3. The Nine Flavors of Gravity (The Types)

The theory has three adjustable "knobs" (parameters). Depending on how you turn these knobs, you get nine different versions of gravity, which the authors call Type 1 through Type 9.

Think of these like nine different flavors of ice cream:

• Type 6 is Vanilla. It's the standard flavor (Einstein's General Relativity). It's stable and well-known.
• The Others are exotic flavors like "Neutron Star Swirl" or "Dark Matter Mint." Some are stable, some are chaotic, and some might be delicious (useful for explaining the universe) while others are inedible (mathematically broken).

4. The Discovery: Which Flavors Work?

The team shook the puzzle for all nine flavors to see which ones produced stable, healthy waves (propagating modes).

• The "Ghost" Problem: In physics, a "ghost" is a particle that has negative energy. If a theory has ghosts, it's like a car that drives itself backward into a wall—it's unstable and the universe would collapse. The authors checked which flavors were "ghost-free."
• The Winner (Type 3): They found that Type 3 is the star of the show.
  • It has 5 stable ways to vibrate (modes).
  • It includes the usual "gravitational waves" (ripples in space).
  • It includes new "scalar" waves (like a breathing motion of space).
  • It includes new "vector" waves (like a twisting motion).
  • Crucially, it is stable and doesn't have ghosts.

5. Why This Matters for Cosmology

Why do we care about these extra wiggles? Because our current map of the universe has some holes. We don't fully understand Dark Energy (what's pushing the universe apart) or Dark Matter (the invisible glue holding galaxies together).

• The Analogy: Imagine trying to fix a leaky boat. Einstein's theory is a great boat, but it has a slow leak. This new theory (Type 3) suggests that the boat might have a hidden compartment that, if opened correctly, could plug the leak.
• The authors suggest that Type 3 could explain why the universe is expanding faster than expected, without needing to invent new, mysterious substances. It might just be that gravity itself has a "twist" we didn't know about.

6. The "Spatially Flat" Rule

One of the paper's technical breakthroughs was figuring out the right "camera angle" to take a picture of these vibrations.

• In previous attempts, scientists tried to take a photo from a spinning, tilted angle, which made the picture blurry and confusing.
• The authors realized that to see the truth in a theory where the compass is broken, you must take the photo from a flat, non-rotating angle (called the "spatially flat gauge").
• The Metaphor: If you are trying to measure the wind in a room where the fans are broken and blowing randomly, you can't stand on a spinning chair to measure it. You have to stand still on the floor. This paper tells us exactly how to stand still to get the right measurement.

Summary

This paper is a repair manual for a new theory of gravity.

1. The Problem: A new theory of gravity breaks the rules of rotation (Lorentz invariance).
2. The Investigation: The authors looked at the "broken" parts of the theory that others ignored.
3. The Result: They found that one specific version of this theory (Type 3) is stable, healthy, and has five distinct ways to vibrate.
4. The Hope: This version might be the key to solving the biggest mysteries of the universe, like Dark Energy, by treating gravity as a twisting, rather than just a curving, force.

It's like discovering that the universe isn't just a trampoline that bends; it's also a trampoline that can twist, and that twist might be the secret to understanding the cosmos.

Technical Summary

Here is a detailed technical summary of the paper "Cosmological Perturbation in New General Relativity: Propagating mode from the violation of local Lorentz invariance" by Tomonari, Katsuragawa, and Nojiri.

1. Problem Statement

New General Relativity (NGR) is an extension of Teleparallel Equivalent to General Relativity (TEGR) characterized by three free parameters ($c_1, c_2, c_3$). While NGR offers richer degrees of freedom (DOFs) potentially useful for explaining dark energy, dark matter, and cosmological tensions, it suffers from a fundamental theoretical issue: the violation of local Lorentz invariance (LI).

Previous studies on cosmological perturbations in NGR (e.g., Refs. [19, 22, 48]) faced several critical limitations:

1. Incomplete Framework: They often relied on metric-based perturbations or failed to clearly map perturbation variables to the underlying vierbein (tetrad) components, specifically neglecting the anti-symmetric part of the vierbein where LI violation manifests.
2. Inconsistent Gauge Fixing: Standard gauges (like the Conformal Newtonian gauge) were applied without accounting for the broken LI symmetry. Fixing gauges associated with broken symmetries can inadvertently remove genuine propagating modes.
3. Discrepancy with Hamiltonian Analysis: Previous perturbative results often did not align with the Dirac-Bergmann (DB) analysis regarding the number of non-linear DOFs, raising doubts about whether the perturbative analysis captured all physical modes.
4. Order of Approximation: Many analyses were limited to first-order perturbations, potentially missing higher-order contributions that could generate new propagating modes.

The paper aims to resolve these issues by performing a rigorous second-order linear perturbation analysis of NGR around a flat Friedmann-Lemaître-Robertson-Walker (FLRW) background, explicitly incorporating the anti-symmetric vierbein components and respecting the constraints of the DB analysis.

2. Methodology

A. Theoretical Framework

• Lagrangian: The authors utilize the NGR Lagrangian defined by three torsion invariants with coefficients $c_1, c_2, c_3$.
• Vierbein Decomposition: The co-vierbein field $e^I_\mu$ is decomposed into a symmetric part ($\bar{e}^I_\mu$, related to the metric) and an anti-symmetric part ($\tilde{e}^I_\mu$, related to LI violation).
  • The symmetric part contains standard scalar ($\psi, \phi, B, F$), vector ($G_i, C_i$), and tensor ($h_{ij}$) perturbations.
  • The anti-symmetric part introduces new fields: scalar ($\alpha$), pseudo-scalar ($\tilde{\sigma}$), vector ($\alpha_i$), and pseudo-vector ($\tilde{V}_i$).
• Matter Source: A real scalar field $\Phi$ is introduced as matter to generate a dynamical FLRW background, avoiding the triviality of a static Minkowski background.

B. Gauge Fixing Strategy

A crucial methodological contribution is the identification of the Spatially Flat Gauge as the appropriate choice for theories with broken LI.

• Reasoning: In theories where LI is violated, the anti-symmetric components of the vierbein represent physical propagating modes. Fixing gauges that eliminate these components (e.g., setting $\alpha=0$ or $\tilde{V}_i=0$) would artificially suppress physical degrees of freedom.
• Implementation: The authors fix the diffeomorphism-related gauges ($\xi^0, \xi, \xi^{(v)}_i$) to set $\phi=0$, $B=0$, and $C_i=0$, while leaving the LI-violating fields ($\alpha, \tilde{\sigma}, \alpha_i, \tilde{V}_i$) unfixed to ensure they can propagate if the theory allows.

C. Perturbative Analysis

• The Lagrangian density is expanded up to second order in perturbation fields.
• The analysis is performed for Tensor, Scalar, Pseudo-scalar, and Vector/Pseudo-vector sectors separately.
• Constraint equations derived from the variation of non-dynamical fields (like $\psi$ and $F$) are solved and substituted back into the Lagrangian to decouple the propagating modes.
• Ghost-free conditions are derived by ensuring the kinetic terms of the propagating modes have the correct sign.

3. Key Contributions

1. Clarification of Perturbation Origins: The paper explicitly maps every perturbation variable to its origin in the vierbein field (symmetric vs. anti-symmetric), resolving ambiguities in previous literature regarding the definition of scalar and vector modes in NGR.
2. Resolution of Gauge Issues: It establishes that the Spatially Flat Gauge is the correct framework for analyzing NGR. This prevents the accidental elimination of physical modes arising from LI violation, a flaw in previous conformal gauge analyses.
3. Second-Order Analysis: By expanding to second order, the authors capture kinetic terms and interactions that are absent in first-order analyses, revealing propagating modes that were previously thought to be non-existent or constrained.
4. Comprehensive Classification: The study provides a complete classification of propagating modes for all nine types of NGR (based on the parameter space of $c_1, c_2, c_3$), comparing them directly with the non-linear DOF counts from the Dirac-Bergmann analysis.

4. Key Results

The authors analyzed the propagation of modes for each of the 9 NGR types. The results are summarized as follows:

• Type 3 (The Most Promising Candidate):
  • Non-linear DOFs: 5 (from DB analysis).
  • Propagating Modes: 5 modes found in perturbation theory: Tensor ($h_{ij}$), Scalar ($\alpha$), and Vector ($\alpha_i$).
  • Significance: This type preserves $SO(3)$ rotational invariance (crucial for cosmological isotropy) while allowing for stable propagation of tensor, scalar, and vector modes. It possesses a ghost-free parameter region, making it a viable candidate for cosmological applications.
• Type 6 (TEGR):
  • Reduces to General Relativity. Only the Tensor mode ($h_{ij}$) propagates (2 DOFs).
• Other Types:
  • Type 1: Can support up to 8 modes (Tensor, Scalar, Pseudo-scalar, Vector, Pseudo-vector), but requires specific parameter tuning to decouple modes.
  • Type 2 & 9: Support 5 and 3 modes respectively, including pseudo-vector modes ($\tilde{V}_i$).
  • Type 4, 5, 7, 8: Show varying numbers of propagating modes (2–6). Some are "irregular" systems where the linear perturbation count may differ from the non-linear DB count due to potential strong coupling or symmetry restoration at the linear level.
• Ghost-Free Conditions:
  • Tensor: $2c_1 - c_2 < 0$
  • Scalar ($\alpha$): $2c_1 - c_2 + c_3 > 0$
  • Pseudo-scalar ($\tilde{\sigma}$): $2c_1 + c_2 < 0$
  • Vector/Pseudo-vector: Specific combinations of parameters must be satisfied to avoid ghosts.

Comparison with Previous Work:
The results differ significantly from Ref. [22] (Golovnev et al.) for Types 2, 3, 4, 5, 7, and 8. The authors attribute these discrepancies to the gauge choice in Ref. [22]. By fixing gauges that eliminate the anti-symmetric components, Ref. [22] missed the propagation of modes like $\alpha$ (Type 3) and $\tilde{V}_i$ (Type 2). The current work demonstrates that these modes are physical and propagate when the correct gauge (Spatially Flat) is used.

5. Significance and Conclusion

• Viability of NGR for Cosmology: The study identifies Type 3 NGR as a particularly robust candidate for cosmological applications. It offers 5 propagating degrees of freedom (matching the DB analysis), maintains $SO(3)$ isotropy (consistent with CMB observations), and possesses a stable, ghost-free parameter space.
• Theoretical Consistency: The work bridges the gap between Hamiltonian constraint analysis (Dirac-Bergmann) and linear perturbation theory. It confirms that the violation of local Lorentz invariance introduces new physical degrees of freedom (anti-symmetric vierbein modes) that must be treated carefully in cosmological models.
• Future Directions: The authors suggest that if Type 3 is free of strong coupling issues, it could provide a new framework for understanding dark matter and large-scale structure formation. If strong coupling arises, screening mechanisms may be required. Additionally, the violation of boost invariance in NGR could leave observable imprints on gravitational wave signals, offering a potential observational test.

In summary, this paper provides the most rigorous and consistent perturbative analysis of NGR to date, correcting previous gauge-fixing errors and establishing a solid foundation for using NGR (specifically Type 3) in modern cosmology.