Composite Gravity: Emergent Graviton Models

• Composite gravity is a framework where the graviton or metric emerges as a collective excitation of underlying matter fields and energy–momentum tensors.
• The approach employs diverse methods, including metric-independent scalar/fermion theories and functional renormalization group techniques, to derive massless spin-2 poles resembling Einstein gravity.
• Key challenges include regulator dependence, tuning of UV parameters, and achieving full non-linear diffeomorphism invariance to reproduce classical gravitational dynamics.

Searching arXiv for recent and relevant work on composite gravity, emergent/composite gravitons, and related dark-matter/bound-state scenarios. Composite gravity denotes a class of frameworks in which the spacetime metric, the graviton, or gravity-like tensor mediators are not fundamental fields inserted at the microscopic level, but composite operators or collective excitations built from more elementary degrees of freedom. In the recent literature represented here, the term spans several nonidentical constructions: metric-independent scalar and fermion theories whose scattering amplitudes develop a massless spin-2 pole; functional-renormalization-group models in which auxiliary tensor fields sourced by the energy–momentum tensor become dynamical in the infrared; background-independent and stochastic formulations in which a composite metric acquires a semiclassical vacuum expectation value satisfying Einstein’s equations; and phenomenological scenarios where composite spin-2 resonances or gravity-bound microscopic nuggets play the operative role (Batz et al., 2020, Maitiniyazi et al., 9 Dec 2025, Erlich, 3 Apr 2025, Lee et al., 2013, Flambaum, 26 Jan 2025).

1. Conceptual scope and defining structures

A common definition runs through the most formal constructions: the would-be graviton or metric fluctuation is not a fundamental field of the microscopic action, but a composite built from matter fields and, in particular, from their energy–momentum tensor. In the FRG formulation of composite gravity, the microscopic theory is a purely matter theory on flat Euclidean space with nonrenormalizable operators of schematic form

$T_{\mu\nu}T^{\mu\nu},\qquad (T^\rho_{\ \rho})^2,$

which are rewritten by a Hubbard–Stratonovich transformation in terms of an auxiliary symmetric rank-2 tensor field interpreted as a composite spin-2 mode (Maitiniyazi et al., 9 Dec 2025).

A second, closely related line identifies the metric itself as a composite operator. A representative 4D toy model defines

$$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$

with no independent dynamical metric in the microscopic description; Einstein’s equations then arise as a self-consistency condition for a semiclassical background $\langle g_{\mu\nu}\rangle$, and graviton exchange appears in matter correlation functions (Erlich, 3 Apr 2025). In the background-independent scalar construction, the same idea is realized in a manifestly diffeomorphism-invariant, metric-independent action whose Polyakov-like rewriting yields a composite metric

$$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$

with no separate path integral over $g_{\mu\nu}$ (Batz et al., 2020).

These models differ sharply from phenomenological usages in which “composite gravity” refers not to an emergent metric but to gravity-like composite resonances. In the warped-extra-dimensional dark-matter literature, massive spin-2 Kaluza–Klein gravitons and the radion are interpreted in the 4D dual as composite spin-2 and dilaton resonances of a strongly coupled near-conformal sector, with couplings proportional to energy–momentum tensors (Lee et al., 2013). A still different usage appears in the ADD-bound-state scenario, where “composite gravity systems” are microscopic bound states of Standard Model particles whose binding is dominated by short-distance gravity enhanced by extra dimensions rather than by QCD or electromagnetism (Flambaum, 26 Jan 2025).

2. Metric-independent matter theories and the composite graviton pole

The most explicit emergent-graviton models start from a reparametrization-invariant, non-polynomial, metric-independent scalar theory. In the scalar construction with $N$ physical fields $\phi^a$ and $D$ clock-and-ruler fields $X^I$, eliminating the auxiliary metric gives

$$S=\int d^D x\left(\frac{\tfrac D2-1}{V(\phi^a)}\right)^{\frac D2-1} \sqrt{\left|\det\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a+\sum_{I,J=0}^{D-1}\partial_\mu X^I\partial_\nu X^J\eta_{IJ}\right)\right|}.$$

After fixing static gauge for $$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$0 and expanding in $$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$1, the theory becomes a flat-space QFT of free massive scalars plus quartic interactions expressible in terms of the flat-space energy–momentum tensor, making the large-$$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$2 bubble resummation tractable (Carone et al., 2017).

The central result is that the $$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$3 scalar scattering amplitude develops a massless spin-2 pole. In dimensional regularization, this occurs when

$$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$4

and the residue has precisely the tensor structure of linearized Einstein gravity. The induced Planck mass is

$$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$5

The same framework also reproduces the cubic graviton self-interactions of Einstein gravity, up to higher-derivative corrections, and yields universal coupling to multiple matter species through the total energy–momentum tensor (Carone et al., 2017).

A major concern for this program is regulator dependence. That issue is addressed directly in the Pauli–Villars reanalysis, which replaces dimensional regularization by covariant PV fields in the same metric-independent scalar theory. The massless graviton remains present, again conditioned by fine-tuning. For the minimal $$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$6 realization, the exact tuning condition is

$$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$7

and the emergent Planck mass scales as

$$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$8

This shows that the spin-2 pole is not an artefact of dimensional regularization, although the Planck scale depends on regulator masses and multiplicities (Li et al., 5 Dec 2025).

The fermionic extension preserves the same structural lesson. There one starts from a metric-independent, diffeomorphism-invariant determinant built from clock-and-ruler fields and a fermion bilinear

$$g_{\mu\nu}\equiv \frac{\sum_a \partial_\mu\phi^a\,\partial_\nu\phi^a}{V(\phi^a)},$$9

motivated by the supersymmetric D-brane action modulated by a fermion potential. After gauge fixing, the quartic fermion interactions can again be organized in energy–momentum-tensor form, and the spectrum contains a massless composite graviton, with the gravitational coupling tied to cutoff-scale physics (Carone et al., 2018).

3. Dynamical tensor fields from $\langle g_{\mu\nu}\rangle$0 interactions and functional RG flow

A distinct route to composite gravity begins not from a metric-independent action but from ordinary matter theories on flat Euclidean space with explicit $\langle g_{\mu\nu}\rangle$1 interactions. In the fermionic model,

$\langle g_{\mu\nu}\rangle$2

and in the scalar model,

$\langle g_{\mu\nu}\rangle$3

A Hubbard–Stratonovich transformation introduces symmetric rank-2 auxiliary tensors $\langle g_{\mu\nu}\rangle$4 and $\langle g_{\mu\nu}\rangle$5, which are nondynamical in the UV because they carry only algebraic quadratic terms and no kinetic operator (Maitiniyazi et al., 9 Dec 2025).

The FRG question is whether quantum fluctuations can make these tensor fields propagate. Using the Wetterich equation with Litim regulators and a truncation that includes all two-derivative tensor structures,

$\langle g_{\mu\nu}\rangle$6

the induced tensor two-point functions generate nonzero running kinetic coefficients $\langle g_{\mu\nu}\rangle$7 and $\langle g_{\mu\nu}\rangle$8 even when they vanish at the UV scale. The flow equations take the form

$\langle g_{\mu\nu}\rangle$9

so that in the infrared the auxiliary tensors become dynamical composite fields (Maitiniyazi et al., 9 Dec 2025).

Masslessness is not automatic. It requires additive tuning of the UV mass parameters so that loop-generated quartic running cancels at $$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$0. Once canonically normalized, the infrared kinetic terms have the structure of gauge-fixed linearized gravity. The induced ratios of kinetic coefficients do not coincide with the standard two-parameter covariant gauge choice, but they can be matched by a more general non-covariant gauge-fixing functional. On that basis, the emergent tensor fields are interpreted as gauge-fixed composite metric perturbations rather than generic spin-2 modes. The same analysis identifies an effective Newton constant of order

$$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$1

What is established here is a proof of principle for infrared dynamical spin-2 emergence from matter loops; what is not yet established is full diffeomorphism invariance, the ghost sector, or the nonlinear Einstein–Hilbert self-interaction (Maitiniyazi et al., 9 Dec 2025).

4. Background-independent composite metrics, stochastic dynamics, and thermodynamic interpretations

In the background-independent scalar program, the vacuum expectation value of the composite metric is determined self-consistently rather than prescribed. Introducing nondynamical clock-and-rod fields $$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$2 as a coordinate scaffold, the vacuum metric is defined by

$$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$3

and then fixed by the tadpole condition $$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$4. Using the DeWitt–Schwinger expansion of the scalar effective action, this condition becomes Einstein’s equation with cosmological constant,

$$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$5

plus four-derivative corrections from the $$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$6 Seeley–DeWitt coefficient. The Planck scale is induced as

$$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$7

and a graviton propagator emerges from the resummed $$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$8 kernel in vacuum correlation functions (Batz et al., 2020).

A more radical version embeds the same composite-metric idea in a stochastic microphysics. In the stochastic composite gravity framework, the fundamental fields $$g_{\mu\nu}=\frac{D/2-1}{V(\phi^a)}\left(\sum_{a=1}^N\partial_\mu\phi^a\partial_\nu\phi^a\right),$$9 evolve by a discretized Itô process with causal Poisson sprinkling,

$g_{\mu\nu}$0

and the metric is again composite,

$g_{\mu\nu}$1

with stochastic forward and backward derivatives used for the time components. The sprinkling scale $g_{\mu\nu}$2 acts as a physical Lorentz-invariant UV regulator, and the induced Planck mass obeys

$g_{\mu\nu}$3

Within this framework, the Bekenstein–Hawking entropy is recovered as an induced quantity, the species problem is softened because $g_{\mu\nu}$4, and the theory predicts qualitative suppression of low-$g_{\mu\nu}$5 CMB power when the early universe is out of stochastic equilibrium (Erlich, 3 Apr 2025).

The thermodynamic reading of composite metrics is sharpened by the horizon-thermodynamics analysis. Once a diffeomorphism-invariant effective action for a composite metric exists, a boundary-of-a-boundary decomposition of the equations of motion yields integrated horizon relations of Clausius type,

$g_{\mu\nu}$6

The important limitation is conceptual rather than algebraic: Clausius-like relations are generic, but a genuine thermodynamic interpretation of the metric requires equilibration, an H-theorem, and decoherence driven by the pregeometric degrees of freedom (Sindoni, 2012). The first stochastic analysis makes the same point differently: long-distance gravity is expected only once the stochastic system relaxes toward equilibrium, and gravity can switch off at ultra-short distances where the smooth metric description fails (Erlich, 2022).

5. Composite operators within quantum gravity: geons, CDT, and supergravity extensions

Composite gravity can also mean that the physically observable excitations of quantum gravity must themselves be composite, diffeomorphism-invariant operators, even if the microscopic theory is already a theory of the metric. In the CDT analysis of geons, the operative analogy is not to emergent spacetime from matter, but to glueballs in Yang–Mills theory: the graviton is gauge dependent, whereas physical excitations must be created by curvature composites such as $g_{\mu\nu}$7, $g_{\mu\nu}$8, or their lattice avatars (Maas et al., 24 Oct 2025).

The basic observables are connected two-point functions of the Quantum Ricci Curvature Scalar $g_{\mu\nu}$9, the extracted curvature scalar $N$0, and $N$1, at fixed cosmological time $N$2,

$N$3

fitted by

$N$4

In CDT configurations fluctuating around an emergent Euclidean de Sitter universe, these correlators exhibit an intermediate-distance exponential regime consistent with a massive geon-like excitation. At maximal spatial extent, $N$5, the extracted mass is

$N$6

using the lattice-spacing estimate $N$7. The mass rises sharply during periods of rapid expansion, coinciding with enhanced curvature fluctuations, which suggests a nontrivial dependence of composite gravitational excitations on cosmological time (Maas et al., 24 Oct 2025).

The same paper extends the composite-operator logic to supergravity. There the proposal is that physical operators must be supergauge invariant, and that a Fröhlich–Morchio–Strocchi-type expansion of a composite invariant,

$N$8

could explain how local supersymmetry remains fundamentally present while ordinary low-energy observables are dominated by one component of a composite supermultiplet. The claim is explicitly tentative, but it recasts the non-observation of low-energy supersymmetry as a possible compositeness effect rather than solely a high-scale symmetry-breaking effect (Maas et al., 24 Oct 2025).

6. Phenomenological realizations: composite resonances, dark matter, and gravity-bound nuggets

In the warped-extra-dimensional dark-matter scenario, gravity-mediated dark matter is dual to a 4D picture of composite dark matter coupled through composite gravitational resonances. The 5D KK graviton and radion become, in the 4D dual, a composite spin-2 resonance and a dilaton of a strongly coupled near-conformal sector. Their interactions are

$N$9

with a characteristic hierarchy of overlap coefficients,

$\phi^a$0

Dark matter freeze-out then proceeds through exchange of these gravity-like composite resonances rather than through gauge or Higgs portals, and indirect-detection analyses constrain the resulting $\phi^a$1, $\phi^a$2, $\phi^a$3, $\phi^a$4, and gamma-box signatures (Lee et al., 2013, Lee et al., 2014).

A separate phenomenological branch considers composite objects bound directly by modified gravity. In the ADD model with $\phi^a$5 compact extra dimensions, the short-distance potential scales as $\phi^a$6, and a self-gravitating many-fermion system with Gaussian density profile has kinetic energy $\phi^a$7 but gravitational energy so singular for $\phi^a$8 that the variational minimum collapses to the UV cutoff scale $\phi^a$9. The bound-state condition for identical fermions becomes

$D$0

and with the assumption that gravity matches gauge strength at $D$1, the paper obtains a characteristic lower bound $D$2, with a representative value $D$3 (Flambaum, 26 Jan 2025).

For quark constituents with $D$4, this gives a typical composite mass

$D$5

while the size is controlled by the cutoff,

$D$6

Because the interaction cross section is effectively nuclear,

$D$7

the ratio

$D$8

is far below the cited astrophysical bound $D$9 for realistic $X^I$0, making such strong-gravity nuggets viable dark-matter candidates within the assumptions of the model (Flambaum, 26 Jan 2025).

7. Limitations, controversies, and open questions

No current composite-gravity program surveyed here is complete. In the metric-independent scalar and fermion models, the massless graviton is contingent on tuning $X^I$1; the same tuning also enforces vanishing effective cosmological constant and removes momentum-independent graviton self-interactions. The Pauli–Villars analysis shows that the graviton pole is regulator-robust, but it also makes explicit that the induced Planck mass depends on regulator masses and multiplicities, so the gravitational coupling is UV-scheme dependent rather than universal in a Wilsonian sense (Li et al., 5 Dec 2025, Carone et al., 2017).

The FRG construction demonstrates dynamical tensor emergence but stops at the quadratic level. Full diffeomorphism invariance is not shown explicitly; the induced kinetic operator is interpreted as Einstein–Hilbert plus gauge fixing, but the symmetry structure, ghosts, Ward identities, and nonlinear graviton vertices are left open. The truncation keeps only two-derivative tensor terms, fixes matter wavefunction renormalization to unity, and does not attempt a UV completion of the nonrenormalizable $X^I$2 operators (Maitiniyazi et al., 9 Dec 2025).

Background-independent and stochastic composite-metric theories have a different set of unresolved issues. They give a mechanism for selecting $X^I$3 and for inducing $X^I$4, but realistic matter content, precise cosmological perturbation theory, and explicit low-energy tests remain incomplete. In the stochastic program, the low-$X^I$5 CMB suppression is qualitative rather than yet fully computed, and the thermodynamic interpretation of the metric requires an H-theorem and equilibration mechanism that are argued for but not derived from first principles (Erlich, 3 Apr 2025, Sindoni, 2012).

The CDT geon results likewise remain preliminary. The reported mass scale $X^I$6 is extracted on relatively coarse lattices and in a specific Euclidean de Sitter phase; a larger operator basis, finite-volume control, and continuum scaling analysis are still needed before one can regard the geon spectrum as established. The supergravity extension is explicitly conjectural (Maas et al., 24 Oct 2025).

Taken together, these programs suggest that composite gravity is better understood as a family of research agendas than as a single theory. What unifies them is the replacement of a fundamental graviton or metric by composite, induced, or gauge-invariant tensorial structures. What remains unsettled is whether any such structure can simultaneously reproduce full nonlinear GR, explain the value of the Planck scale and cosmological constant without ad hoc tuning, and deliver a UV-complete microscopic dynamics with controlled phenomenology.