Spatially covariant gravity with two degrees of freedom: A perturbative analysis up to cubic order

This paper employs a perturbative analysis of polynomial-type spatially covariant gravity Lagrangians up to cubic order around a cosmological background to derive specific coefficient conditions that eliminate the scalar mode, thereby identifying five explicit models that propagate only the two tensorial degrees of freedom characteristic of general relativity.

The Gist

Imagine the universe as a giant, flexible trampoline. In our current best understanding of physics (General Relativity), this trampoline can only wiggle in two specific ways: it can ripple like a wave moving across the surface (these are gravitational waves). These ripples are the only things that carry energy across the universe in this theory.

However, for decades, physicists have been asking: "What if the trampoline could wiggle in a third way?" Maybe there's a hidden "scalar" wiggle—a kind of stretching or breathing motion—that we haven't noticed yet. Some theories suggest this extra wiggle exists to explain dark energy or the expansion of the universe. But here's the problem: we haven't seen it. The gravitational waves detected by LIGO look exactly like the two-dimensional ripples predicted by Einstein, with no sign of that extra "breathing" mode.

So, a group of physicists (Yu, Hu, and Gao) decided to play a game of "What if?" They wanted to build new theories of gravity that, just like Einstein's, only allow those two ripples and strictly forbid the extra "breathing" wiggle.

The Challenge: The "Ghost" in the Machine

The problem is that when you try to write down the math for these new gravity theories, the equations are incredibly messy. It's like trying to bake a cake where you want to ensure there are absolutely no raisins, but the recipe is written in a language where the ingredients are mixed together in a giant, non-linear smoothie.

Usually, to check if a theory has that unwanted "extra wiggle" (which physicists call a "ghost" or a "scalar mode"), they have to do a very difficult, high-level math check called a Hamiltonian analysis. It's like trying to reverse-engineer a complex machine by taking it apart piece by piece in a dark room. It works, but it's hard to see the final picture.

The New Approach: The "Perturbation" Strategy

Instead of taking the machine apart, these authors decided to shake it gently and see what happens. They used a method called perturbation theory.

Think of it like this:

1. The Background: Imagine the trampoline is perfectly flat and still (this is the "cosmological background" of our universe).
2. The Shake: They imagine adding a tiny, tiny wiggle to the trampoline.
3. The Test: They asked, "If I wiggle the trampoline slightly, does that extra 'breathing' mode show up?"
   • If the math says "Yes, a breathing mode appears," the theory is bad.
   • If the math says "No, only the two ripples remain," the theory is good.

They didn't just stop at a tiny wiggle. They shook it harder (up to "cubic order," which is a fancy way of saying they looked at the interaction of three wiggles at once). This is crucial because sometimes a theory looks safe when you wiggle it gently, but when you wiggle it harder, the hidden "breathing" mode suddenly pops out.

The Discovery: Five New Recipes

By carefully tuning the "ingredients" (the coefficients in their math equations) so that the extra wiggle disappears at every level of shaking, they found five specific mathematical recipes (Lagrangians) for gravity.

These five recipes are special because:

• They are built on a framework called Spatially Covariant Gravity (which breaks the rule that time and space must be treated exactly the same, allowing for these new theories).
• They successfully kill the extra wiggle up to the third level of complexity.
• They are concrete, written-down formulas that other scientists can now use to test against real-world data.

The Catch: Not All Recipes Are Tasty

However, the authors found that while all five recipes stop the "breathing" mode, only two of them (called SA1 and SA2) actually taste like General Relativity when you look at them closely.

• The Good Two: These two can smoothly turn into Einstein's theory if you dial down the new effects. They are the most promising candidates for describing our actual universe.
• The Other Three: These stop the breathing mode, but they break the connection to Einstein's theory in a way that makes them unlikely to be the real description of our universe (they don't have a "General Relativity limit").

Why This Matters

This paper is like a chef finding five new ways to bake a cake that has no raisins.

1. It proves that such cakes can exist mathematically.
2. It gives us the specific recipes to make them.
3. It tells us which two recipes are most likely to be the "real" cake we are eating (our universe).

By using this "shaking" method instead of the "taking apart" method, the authors made it much easier to find these hidden, healthy theories of gravity. This helps us narrow down the search for the true laws of the universe, ensuring that whatever theory we pick next, it won't have that pesky, invisible "breathing" mode that we don't see in the sky.

Technical Summary

1. Problem Statement

General Relativity (GR) propagates exactly two tensorial degrees of freedom (DOFs). While many modified gravity theories introduce additional scalar or vector modes to explain cosmological phenomena, recent gravitational wave observations (LIGO-Virgo) and the lack of scalar dark matter evidence suggest that viable theories should ideally preserve the 2-DOF structure of GR.

Constructing such theories within the Spatially Covariant Gravity (SCG) framework—which breaks full spacetime diffeomorphism invariance but preserves spatial diffeomorphism—is challenging.

• The Challenge: Standard Hamiltonian constraint analysis can derive the conditions for a theory to have only 2 DOFs. However, translating these abstract Hamiltonian conditions into explicit Lagrangian forms is notoriously difficult, especially when the Lagrangian depends non-linearly on the extrinsic curvature ($K_{ij}$).
• The Gap: Previous perturbative analyses (e.g., in Ref. [78]) successfully eliminated the scalar mode at the linear (quadratic action) level for SCG theories up to derivative order $d=2$. However, eliminating the scalar mode at the quadratic level does not guarantee its absence at higher (non-linear) orders. The conditions required to remove the scalar mode at the cubic order (cubic in perturbations) were not fully explored for general polynomial-type SCG Lagrangians, particularly those including terms with $d=3$ (three total derivatives).

2. Methodology

The authors adopt a fully Lagrangian perturbative approach around a Friedmann-Lemaître-Robertson-Walker (FLRW) background.

• Starting Point: They consider a general polynomial-type SCG action containing monomials up to total derivative order $d=3$. This action includes terms involving the lapse function ($N$), spatial metric ($h_{ij}$), extrinsic curvature ($K_{ij}$), spatial Ricci tensor ($R_{ij}$), and their derivatives.
• Perturbation Expansion: The action is expanded around a cosmological background up to cubic order in scalar perturbations ($A, B, \zeta$), where $\zeta$ is the curvature perturbation.
• Constraint Solving: Since the lapse ($N$) and shift ($N^i$) are non-dynamical (or auxiliary) in this framework, their perturbation variables ($A$ and $B$) are solved in terms of $\zeta$ using their constraint equations.
• Degeneracy Conditions: The core strategy is to substitute the solutions for $A$ and $B$ back into the action to obtain an effective action for $\zeta$. To ensure only 2 DOFs propagate, the kinetic term of $\zeta$ must vanish (degeneracy condition) at every order of perturbation.
  • The authors first ensure the quadratic action is degenerate (removing the scalar mode at linear order).
  • They then impose further constraints on the coefficient functions of the Lagrangian to ensure the cubic action is also degenerate (removing the scalar mode at the non-linear level).
• Classification: The analysis is split into two main branches (Case I and Case II) based on the solutions found for the quadratic degeneracy conditions.

3. Key Contributions

1. Extension to Cubic Order: The paper extends the perturbative analysis of 2-DOF SCG theories from quadratic to cubic order in perturbations. This is crucial because conditions derived at lower orders are often insufficient to guarantee the absence of the scalar mode in the full non-linear theory.
2. Inclusion of $d=3$ Terms: Unlike previous works that focused on $d=2$, this analysis explicitly includes monomials with three total derivatives ($d=3$), such as $K^3$, $K_{ij}K^{jk}K_k^i$, and terms involving spatial derivatives of the acceleration ($a_i = \nabla_i \ln N$).
3. Systematic Derivation of Coefficients: The authors derive a complete set of differential equations and algebraic constraints on the time- and lapse-dependent coefficient functions ($c(t, N)$) required to eliminate the scalar mode up to cubic order.
4. Identification of Viable Models: The analysis filters the vast space of polynomial SCG Lagrangians down to a specific set of candidate theories that satisfy the 2-DOF condition up to cubic order.

4. Key Results

The analysis yields five explicit candidate Lagrangians (labeled $S_{A1}, S_{A2}, S_{B1}, S_{B2}, S_{C}$) that propagate only 2 DOFs up to cubic order.

• Case I (Main Branch): This branch assumes specific relations between coefficients derived from the quadratic analysis (specifically $\tilde{b}_3 = 0$).

  • Solution A1 and A2: These are the most physically significant results. They satisfy the cubic degeneracy conditions and, crucially, admit a General Relativity (GR) limit when the coefficients are tuned appropriately (specifically when $C_4 \neq 0$).
    • $S_{A1}$ and $S_{A2}$ contain terms up to $K^3$ and $K_{ij}K^{jk}K_k^i$.
    • They are shown to be compatible with the Extended Cuscuton theory and the Quadratic 2-DOF theory (which is known to be non-perturbatively 2-DOF).
  • Solution B1, B2, and C: These solutions also eliminate the scalar mode up to cubic order. However, they correspond to cases where $C_4 = 0$, meaning their quadratic part is incompatible with the Einstein-Hilbert action. Consequently, they do not admit a GR limit and are considered less physically viable for describing our universe, though they remain valid mathematical examples of 2-DOF SCG.

• Case II (Ruled Out): This branch corresponds to a different set of quadratic constraints involving the acceleration terms ($f_3, \tilde{f}_3$).

  • The authors perform a detailed analysis (including non-local operators arising from the constraint equations) and find that no consistent solution exists for Case II.
  • The conditions required to eliminate the scalar mode at cubic order are mutually inconsistent in this branch. This implies that SCG theories with this specific structure inevitably reintroduce a scalar degree of freedom at the non-linear level.

5. Significance

• Validation of Perturbative Methods: The paper demonstrates that a perturbative analysis up to a finite order (cubic) is sufficient to identify robust 2-DOF theories. It confirms that the conditions derived perturbatively can reproduce known non-perturbative results (like the quadratic 2-DOF theory) and extend them to higher orders.
• Concrete Model Building: By providing explicit Lagrangians ($S_{A1}, S_{A2}$), the authors offer concrete candidates for modified gravity theories that are consistent with current gravitational wave constraints (2 tensor modes) while allowing for deviations in the cosmological regime.
• Unification: The work clarifies the relationship between SCG, the Extended Cuscuton theory, and Minimally Modified Gravity (MMG). It shows that these previously distinct theories can be viewed as specific limits or subclasses of the broader polynomial-type SCG framework derived here.
• Exclusion of Pathological Branches: The rigorous exclusion of Case II provides a clear boundary for model building, indicating that certain classes of acceleration-dependent SCG theories cannot be made 2-DOF without breaking other consistency conditions.

In conclusion, this work provides a systematic, Lagrangian-based construction of 2-DOF spatially covariant gravity theories, identifying specific models that are free of scalar ghosts up to cubic order and compatible with General Relativity in the low-energy limit.