Unimodular Gravity: Theory & Implications

• Unimodular gravity is a theory that fixes the metric determinant, reducing symmetry to volume-preserving diffeomorphisms and resulting in traceless field equations.
• It decouples vacuum energy from spacetime curvature, allowing the cosmological constant to emerge as an integration constant and addressing the old cosmological constant problem.
• The Hamiltonian and quantum analyses reveal distinct constraint structures and renormalization behaviors, offering potential observational signatures compared to general relativity.

Unimodular gravity is a gravitational theory defined by constraining the determinant of the metric tensor, typically fixing it to a constant (commonly set to unity). This restriction reduces the symmetry group of the theory from the full diffeomorphisms group, as in general relativity (GR), to the subgroup of transverse diffeomorphisms (TDiff), often supplemented with local Weyl (conformal) invariance. The resulting field equations correspond to the traceless part of Einstein’s equations, with the cosmological constant emerging not as a coupling but as an integration constant. This structural difference has significant implications at both the classical and quantum levels, particularly regarding the cosmological constant problem, the quantization of gravity, and certain aspects of cosmology and phenomenology.

1. Gauge Symmetries and Foundational Formulation

Unimodular gravity is classically defined by imposing the unimodular constraint on the metric determinant: $\sqrt{|g|} = \text{const}$ or, equivalently in four dimensions,

$\det g_{\mu\nu} = -1$

This requirement restricts the symmetry group of the theory to transverse diffeomorphisms (coordinate transformations with unit Jacobian, i.e., volume-preserving) and, in many formulations, an additional local Weyl invariance. The action can be written in several equivalent forms, most notably in the Henneaux–Teitelboim formalism: $$S = \frac{1}{2\kappa} \int d^4x \sqrt{-g} (R - 2\Lambda + \mathcal{L}_m) + 2\int d^4x\, A\,\partial_\mu \tau^\mu$$ where $A$ is a Lagrange multiplier imposing the unimodular constraint and $\tau^\mu$ is a vector density (Chiou et al., 2010).

Varying the action leads to equations of motion equivalent to the traceless Einstein equations: $$R_{\mu\nu} - \frac14 R\,g_{\mu\nu} = 8\pi G\left(T_{\mu\nu} - \frac14 T g_{\mu\nu}\right)$$ Commensurately, only the traceless part of the energy–momentum tensor couples to gravity. The cosmological constant appears naturally as an integration constant, rather than being prescribed in the action.

UG can also be formulated in a WTDiff-invariant language, with gauge group

$$\text{WTDiff} = \text{(Weyl rescalings)} \ltimes \text{(Volume-preserving diffeomorphisms)}$$

sometimes requiring introduction of an auxiliary background volume form $\omega(x)$ and an auxiliary metric

$$\tilde{g}_{\mu\nu} = g_{\mu\nu}(\omega^2/|g|)^{1/(D+1)}$$

in $D+1$ dimensions (Carballo-Rubio et al., 2022, Garay et al., 2023).

2. Emergence of the Cosmological Constant and the Cosmological Constant Problems

The principal motivation for unimodular gravity is its treatment of the cosmological constant. In GR, $\Lambda$ is a parameter in the action and is subject to radiative corrections, leading to the "old" cosmological constant problem: vacuum energy contributions from quantum fields are gravitationally active and therefore should generate a large spacetime curvature, which is not observed.

In unimodular gravity, the trace of the field equations does not arise from the dynamical variation, instead being fixed only from the contracted Bianchi identities. This enforces the constancy of the Ricci scalar $R$, establishing the cosmological constant as a dynamically generated integration constant determined by initial data (Padilla et al., 2014, Herrera et al., 31 Jan 2024): $R = 4\Lambda$ This decouples vacuum energy contributions from the source of spacetime curvature, leading to "technical naturalness" of the cosmological constant at both the classical and quantum levels (Carballo-Rubio et al., 2022).

However, unimodular gravity does not automatically explain why the measured value of $\Lambda$ is small (the "new" CC problem). Recent work proposes anthropic selection in a multiverse constructed from different cosmological eras within a single big bang, interpreting $\Lambda$ as a random constant determined by the universe's history rather than the action (Salvio, 18 Jun 2024).

3. Constraint Structure and Dynamics

The canonical (Hamiltonian) analysis of unimodular gravity reveals a modified constraint structure distinct from GR (Kluson, 2014, Herrera et al., 31 Jan 2024):

• The theory possesses a primary constraint from the unimodular condition, leading to a chain of secondary constraints.
• There is a reduction in the number of first-class constraints (those generating gauge symmetries), since the restriction to volume-preserving diffeomorphisms removes one generator.
• The cosmological constant arises from the global (zero mode) part of the Hamiltonian constraint.

In the 3+1 decomposition, the evolution remains strongly hyperbolic and well-posed, provided the initial data satisfy the constraints (Herrera et al., 31 Jan 2024). The unimodular restriction is absorbed by fixing the lapse function appropriately: $N = f e^{-6\phi}$ where $f$ is the prescribed density and $\phi$ arises in the BSSN decomposition. This formulation guarantees unique evolution for suitable matter content.

4. Quantum Aspects, Renormalization, and One-Loop Structure

At the quantum level, unimodular gravity is inequivalent to GR except under additional conditions. In the path integral, the conformal (trace) mode of the metric is non-dynamical since the measure is constrained by the unimodular condition. The gauge symmetry is reduced to TDiff, and the Faddeev–Popov ghost structure is accordingly modified; only three independent ghost fields are required, with an additional constraint $\nabla_\mu c^\mu = 0$ for the ghost $c^\mu$ (Eichhorn, 2013, Kugo et al., 2021, Álvarez et al., 2023).

Renormalization group analyses show that unimodular gravity admits a nontrivial ultraviolet (UV)-attractive fixed point, differing in its β-function numerically from that of full quantum Einstein gravity (QEG) due to the absence of volume (conformal) fluctuations (Eichhorn, 2013). The cosmological constant does not get renormalized, and the quantum effective action lacks running of $\Lambda$ under RG flow.

The one-loop divergences in unimodular gravity are absent for the cosmological constant, as no term proportional to $\Lambda$ arises in the on-shell counterterm structure (Álvarez et al., 2023). However, for matter with non-minimal coupling, the running of certain physical couplings can be distinct between GR and unimodular gravity, indicating the possibility, in principle, to probe WTDiff versus Diff-invariant quantum structure in the presence of nonminimal matter couplings (Herrero-Valea et al., 2020).

5. Physical Observables and Phenomenology

On-shell (scattering) amplitudes and linearized perturbation theory reveal that both GR and unimodular gravity yield identical predictions for experiments in the weak-field regime, such as Newtonian gravity and classical gravitational wave propagation (Álvarez et al., 2012, Herrero-Valea, 2018, Álvarez et al., 2023). This is consistent with the fact that the S-matrix is only sensitive to the transverse (TDiff) part of the gauge redundancy and not the full diffeomorphism invariance, and is supported by analysis of graviton three- and four-point amplitudes.

At the cosmological and strong-field level, both background solutions (such as FLRW cosmologies or black holes) match those of GR when energy–momentum conservation is imposed, although the preservation or violation of energy–momentum conservation enables the existence of evolving cosmological "constants" and "decaying vacuum" scenarios (Fabris et al., 2023, Fabris et al., 2021). For spherically symmetric configurations, stability analyses can reveal differences: the rigidity enforced by the unimodular constraint can prevent certain perturbative instabilities that are present in GR (Fabris et al., 2023).

In cosmology, loop quantum cosmology built atop unimodular gravity replaces singularities (e.g., the big bang) with quantum "bounces" under the dynamics generated by the unimodular clock variable, with the bounce occurring when the total energy density reaches a critical value. Semiclassicality of the quantum state requires the cosmological constant expectation value $\langle A \rangle$ be small in Planck units (Chiou et al., 2010).

In modified gravity models (e.g., unimodular $f(T)$, $f(R,T)$, or Brans–Dicke frameworks), implementation of the unimodular constraint alters the field equations' trace structure, sometimes converting the cosmological constant into a dynamical variable or enforcing new relationships among the effective stress–energy contributions (Nassur et al., 2016, Rajabi et al., 2017, Almeida et al., 2022).

6. Embedding in Extended Frameworks: String Theory and Gauge/Gravity Duality

String theory can accommodate both GR-like (Diff-invariant) and unimodular (WTDiff-invariant) low-energy effective actions, leading to identical massless mode scattering amplitudes. The difference resides in how the cosmological constant is introduced: as a radiatively corrected coupling in GR, or as an integration constant in the unimodular setting (Garay et al., 2023).

In the context of gauge/gravity duality, calculation of on-shell gravity actions for Unimodular Gravity and GR in AdS backgrounds reveals that physical (IR-finite) boundary correlators are equivalent after subtracting IR-divergent contact terms, ensuring that the two formulations yield the same dual CFT correlators at the two- and three-point level (Anero et al., 2022).

7. Open Issues, Extensions, and Observational Prospects

Although classical predictions and weak-field tests are universally consistent with general relativity, the different treatment of the cosmological constant in semiclassical and quantum regimes remains a distinguishing feature (Carballo-Rubio et al., 2022). Unimodular gravity introduces new possibilities for addressing the old cosmological constant problem and offers a technically natural setting for a vanishing or small $\Lambda$. The "new" CC problem, the observed coincidence of dark energy and matter density, is likely not solved within unimodular gravity and may require supplemental anthropic or multiverse hypotheses (Salvio, 18 Jun 2024).

Quantum-level divergences, renormalization group flow of non-gravitational couplings, and the structure of perturbations in modified unimodular gravity and string-inspired frameworks provide further directions for discriminating unimodular gravity from GR. In some cases, such as quantum cosmological perturbations in quadratic gravity, unimodular gravity predicts the absence of certain isocurvature modes, offering potential future observational signals (Salvio, 18 Jun 2024).

Table: Distinguishing Features of Unimodular Gravity vs General Relativity

Aspect	GR	Unimodular Gravity (UG)
Gauge Symmetry	Full Diff	TDiff + Weyl (WTDiff)
Metric Determinant	Dynamical	Fixed (e.g., $\sqrt{-g} = 1$)
Cosmological Constant	Parameter in action	Integration constant
Vacuum energy (QFT)	Direct source	Decoupled from field equations
Renormalization of $\Lambda$	Radiative corrections	No renormalization
Weak-field S-matrix	As in classical GR	Identical (on-shell)
Non-minimal scalar coupling	Standard QFT running	Modified β-functions (Herrero-Valea et al., 2020)

Unimodular gravity thus provides a rigorously defined, symmetry-based alternative to general relativity that addresses some key conceptual problems in gravitational physics, notably the cosmological constant. Its close agreement with GR on most observable phenomena is underpinned by significant algebraic and quantum-structural differences, which are the focus of ongoing research in both theoretical formulation and potential empirical discrimination.