Modified Massless Gravity Theory

• Modified massless gravity theory is a class of covariant models that preserve the graviton’s masslessness while altering the dynamical or symmetry structure of general relativity.
• These frameworks incorporate multi-gravity interactions, enhanced gauge symmetries, and higher-derivative corrections to eliminate ghost instabilities and maintain energy positivity.
• They offer practical implications for cosmology and gravitational wave physics by modifying Friedmann equations and ensuring radiative stability of the cosmological constant.

Modified massless gravity theory refers to a variety of covariant gravitational frameworks in which the graviton remains strictly massless (propagating only helicity-2 modes), but the dynamical and/or symmetry structure of general relativity (GR) is altered. These modifications range from actions with multiple interacting metrics, to non-Einstein backgrounds with enhanced gauge symmetries, to alternative gauge principles that lead to distinct physical or phenomenological consequences, all while maintaining the absence of a graviton mass.

1. Foundations and General Structure

Modified massless gravity models are constructed to preserve the key property that the graviton remains strictly massless, i.e., it propagates only two local tensorial degrees of freedom. However, the underlying Lagrangian, constraint algebra, or symmetries may depart from those of standard Einstein–Hilbert theory. Approaches include (a) models with several interacting massless spin-2 fields (multi-gravity), (b) theories with Weyl, unimodular, or restricted diffeomorphism invariance, (c) ghost-free higher-derivative completions, and (d) constructions embedded in effective field theory or non-metric backgrounds.

The archetypal model is general relativity, defined by the Einstein–Hilbert action,

$$S_{\text{EH}} = \frac{1}{16\pi G} \int d^4x \sqrt{-g}\, R,$$

which is invariant under the full diffeomorphism group $\mathrm{Diff}(M)$ and supports exactly two tensor polarizations. Modified massless gravity generalizes this structure while preventing additional (ghost or physical) degrees of freedom from appearing.

2. Massless Multi-Gravity and Energetic Consistency

Talshir's massless multi-gravity theory (Talshir, 2014) demonstrates the consistent covariant coupling of $N$ massless spin-2 fields (metrics) $g^{(a)}_{\mu\nu}$, $a=1,\ldots,N$, via two-derivative, ghost-free interactions: $$S = \sum_{a=1}^N \alpha_a \int d^4x\, \mathcal{L}_{\rm EH}[g^{(a)}] + \int d^4x\, \mathcal{L}_{\rm int}(g^{(1)},\ldots,g^{(N)}; A^b).$$ The interaction sector $\mathcal{L}_{\rm int}$ consists of non-derivative symmetric functions of the metric determinants and kinetic terms for a set of auxiliary vector fields $A^b_\mu$, ensuring that no higher (third or more) derivatives of any field occur. This structure eliminates Ostrogradsky-type instabilities.

Around the multi-Minkowski vacuum, each metric can be linearized,

$$g^{(a)}_{\mu\nu} = \eta_{\mu\nu} + h^{(a)}_{\mu\nu},$$

with the free action becoming a sum of Pauli–Fierz Lagrangians for massless spin-2 fields, thereby consistently propagating $2N$ massless graviton degrees of freedom. The positivity of energy is demonstrated via canonical (Noether) methods, and the dominant energy condition is strictly enforced for all metric sectors. Importantly, the theory recovers $N$ decoupled copies of Einstein gravity in the zero-interaction limit. The constraints on the interaction terms ensure continuity in the number and nature of propagating degrees of freedom across parameter space, a longstanding issue in prior attempts at multi-graviton models (Talshir, 2014).

3. Alternative Gauge Principles and Symmetry Reductions

Several models realize modifications by restricting, extending, or deforming the gauge symmetry structure of the gravitational sector. Examples include:

• Weyl–Transverse Gravity. This theory, also termed "WT gravity," postulates invariance under transverse diffeomorphisms (TDiff, i.e., volume-preserving diffeomorphisms) and Weyl rescalings, rather than the full $\mathrm{Diff}(M)$. The action is constructed on the unimodular sector, with the dynamical variable $\hat{g}_{ab}$ derived from the metric $g_{ab}$ by enforcing $\det \hat{g} = \det \eta$. The resulting field equations are traceless Einstein equations, and the cosmological constant emerges as an integration constant, inherently protected from radiative corrections (Barceló et al., 2014). Notably, this structure avoids the standard quantum field theory shift-symmetry breaking present in general relativity and thereby addresses the cosmological constant problem at the level of principle.
• Cuscuton/VCDM Theories. The cuscuton or "vacuum constrained dust model" (VCDM) introduces a non-dynamical or infinitely stiff scalar sector that eliminates scalar gravitational degrees of freedom at the level of constraints, while leaving gravitational wave (tensor) propagation unmodified (luminal, massless). The resulting vacuum sector cannot be mapped to Einstein–Hilbert through any local field redefinition, and the cosmological dynamics are modified via a nonlocal ("cuscuton force") term while retaining GR-like propagation for high-frequency gravitational waves (Aoki et al., 2021).
• Type-II Minimally Modified Gravity. The theory defined by De Felice, Doll, and Mukohyama (Felice et al., 2020) employs an auxiliary non-dynamical field $\phi$, together with a deformation potential $V(\phi)$, so that the gravitational action deviates from GR for $V'(\phi) \neq 0$ but reduces to it for $V'(\phi) = 0$. This class propagates strictly two tensor degrees of freedom (no extra scalars), lacks an Einstein frame, and encompasses a scale-dependent effective Newton's constant and modified cosmological background evolution.

Theory	Gauge Symmetry	Extra Modes	Cosmological Constant Mechanism
GR	Full Diff.	None	Fixed; radiative corrections
Massless Multi-Gravity	$\times_a$ Diff.	None	As in GR (per sector)
Weyl-Transverse Gravity	TDiff $\times$ Weyl	None	Integration constant; radiatively stable
Cuscuton/VCDM	Full Diff.	None	Dynamical via constraint equation
Type-II MMG	Gauge-fixed Diff.	None	Encoded in $V(\phi)$; background-dependent

4. Extension to Non-Einstein and Higher-Derivative Theories

Modified massless gravity theories may also be constructed on backgrounds or with actions that differ substantially from Einstein geometric precepts:

• Partially Massless Gravity. On constant curvature backgrounds (not necessarily Einstein), it is possible to tune the mass term for a massive spin-2 fluctuation so as to enhance the scalar gauge symmetry and eliminate the scalar ("helicity-0") polarization, yielding "partially massless" gravitons with four local degrees of freedom (helicities $\pm2$ and $\pm1$). Generalizations show this condition (analogous to the Higuchi bound) can be solved for Ricci-symmetric but non-Einstein spaces (Bernard et al., 2017). Further, bimetric formulations provide unique nonlinear completions exhibiting these enhanced symmetries for specific values of the interaction couplings (Hassan et al., 2012).
• Born–Infeld and Higher-Derivative Ghost-Free Extensions. Born-Infeld gravity models resum infinite towers of higher-curvature invariants within a determinantal structure. In four dimensions, unitary BI-type actions can be constructed such that all higher-derivative ghost and tachyonic modes are absent in perturbation theory about maximally symmetric backgrounds; crucially, the Gauss–Bonnet density, though topological in four dimensions, must appear in specific linear combinations to avoid the propagation of extra modes (Gullu et al., 2014). More generally, effective field theory approaches allow for the removal of spurious Ostrogradsky ghost modes introduced by higher-derivative operators via order-reduction and constraint analysis. The reduced action only propagates massless helicity-2 gravitons with deformed time-diffeomorphism algebra at short distances (Glavan et al., 24 Sep 2024).

5. Scattering Amplitudes, Positivity Bounds, and Phenomenological Constraints

The low-energy effective field theory of modified massless gravity is further constrained by S-matrix–based positivity bounds on the coefficients of higher-derivative operators that preserve the massless graviton pole. Forward-limit twice-subtracted dispersion relations under unitarity, analyticity, and crossing constrain the allowed sign and magnitude of quadratic curvature couplings (e.g., $R^2$ and $R_{\mu\nu}R^{\mu\nu}$ terms), ensuring that ghost or tachyonic instabilities do not arise, and that semi-classical black hole solutions satisfy the mild Weak Gravity Conjecture (Bellazzini et al., 2019). These positivity bounds restrict the parameter space of admissible modified massless gravity theories, paralleling similar restrictions in generalized scalar-tensor models and effective photon–graviton couplings.

6. Relation to Standard General Relativity and Massive Gravity

A recurring theme is the identification of which modifications are structurally trivial (i.e., equivalent to GR via field redefinitions or gauge choices) and which represent genuine new physics with observable consequences. Several classic theorems assert the uniqueness of GR as the only ghost-free interacting massless spin-2 theory under certain assumptions (notably, the requirement of full diffeomorphism invariance and Fierz–Pauli–like gauge symmetry). However, both Weyl–transverse gravity (Barceló et al., 2014) and the massless multi-gravity construction (Talshir, 2014) show that relaxing or extending these assumptions yields consistent, ghost-free, massless gravity models with physically distinct predictions, particularly at the quantum level (e.g., stability of the cosmological constant). The massless limits of massive gravity, when taken carefully (e.g., with appropriate gauge fixing), can also yield modified couplings and phenomenology not reducible to Einstein gravity (0912.1112, Gambuti et al., 2021).

7. Observational and Theoretical Implications

Despite preserving the graviton's masslessness, modified massless gravity theories often entail distinctive phenomenology in cosmology, black hole physics, and gravitational-wave signals:

• Cosmology: Modified constraint equations or auxiliary fields can alter Friedmann equations, effective dark energy behavior, or the rate of structure formation, subject to stringent observational tests (Felice et al., 2020).
• Quantum Stability: Models with Weyl invariance or modified gauge symmetry offer mechanisms for radiative stability of the cosmological constant, in contrast to the radiative instability in GR (Barceló et al., 2014).
• Tensor Modes: All consistent models ensure that gravitational waves propagate at luminal speed and in the high-energy limit display the same dispersion relation and two physical polarizations as in GR (Aoki et al., 2021).
• Constraints from Positivity: Only those higher-derivative deformations that satisfy rigorous S-matrix positivity constraints are physically viable, bounding possible phenomenological deviations (Bellazzini et al., 2019).

These frameworks form theoretical laboratories for evaluating the uniqueness and foundational assumptions underlying general relativity and exploring the infrared and ultraviolet structure of the gravitational interaction at both the classical and quantum levels.