Scalar-Tensor Non-Metricity Gravity

• Scalar-tensor non-metricity gravity is a framework where a scalar field directly couples to the non-metricity tensor defined on a metric-affine spacetime.
• The formulation preserves second-order field equations by using only first derivatives of the metric, and it distinguishes between native scalar–Q models and boundary-coupled Q–B variants.
• These models yield novel cosmological solutions and perturbative dynamics, offering practical insights from dark energy behavior to gravitational wave phenomenology.

Searching arXiv for the cited papers and closely related work on scalar–tensor non-metricity gravity. Scalar–tensor non-metricity gravity denotes a class of theories in which the gravitational sector is formulated in a non-Riemannian geometry with non-vanishing non-metricity, and a scalar field couples either directly to the non-metricity scalar $Q$, to the boundary combination accompanying $Q$, or to an effective Weyl-type non-metricity structure. In the modern symmetric teleparallel formulation, curvature and torsion are set to zero while non-metricity carries the gravitational information; in related Palatini and Weyl-integrable formulations, the scalar can instead determine the non-metricity of an independent connection. The subject therefore comprises both “native” scalar–$Q$ models and non-metricity reformulations of more familiar scalar–tensor or $f(R)$-type theories, with the distinction between coupling to $Q$ alone and coupling to $Q$ plus an accompanying boundary term being structurally decisive (Järv et al., 2018, Capozziello et al., 11 Mar 2025).

1. Geometric setting and basic definitions

Scalar–tensor non-metricity gravity is usually formulated on a metric-affine spacetime with metric $g_{\mu\nu}$ and affine connection $\Gamma^\lambda{}_{\mu\nu}$ treated as independent structures. In the symmetric teleparallel sector one imposes vanishing curvature and vanishing torsion,

$$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$

so that non-metricity remains the only nontrivial field strength. The non-metricity tensor is

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$

or equivalently, in the notation used in some works,

$Q$0

Its two standard traces are

$Q$1

and the disformation tensor is

$Q$2

The non-metricity conjugate, or superpotential, is introduced so that the non-metricity scalar can be written as $Q$3 (Järv et al., 2018, Capozziello et al., 11 Mar 2025).

For the symmetric teleparallel equivalent of general relativity (STEGR), the non-metricity scalar is

$Q$4

A central identity relates it to the Ricci scalar of the Levi-Civita connection: $Q$5 with boundary term

$Q$6

Some papers instead write the same content as $Q$7, reflecting only convention. The identity implies that the Einstein–Hilbert action and the STEGR action differ only by a total divergence at the linear level of the action, but it also anticipates why nonlinear and scalar-coupled extensions separate into distinct classes once $Q$8 and $Q$9 are treated independently (Järv et al., 2018, Capozziello et al., 11 Mar 2025).

A distinct but historically important non-metricity setting is Weyl integrable geometry, where the compatibility condition is

$Q$0

or, in integrable form,

$Q$1

Then the non-metricity is pure trace,

$Q$2

and the scalar $Q$3 acts as the potential for the Weyl one-form. This restricted integrable structure is not generic symmetric teleparallel non-metricity, but it provides a precise scalar-generated non-metricity mechanism and a geometric interpretation of conformal frame changes in Brans–Dicke gravity (Lobo, 2016).

2. Foundational scalar–$Q$4 theories in symmetric teleparallel gravity

The foundational modern scalar–tensor extension of symmetric teleparallel gravity is the theory in which a scalar field couples nonminimally to the non-metricity scalar $Q$5. A standard form is

$Q$6

with a Lagrange-multiplier sector enforcing vanishing curvature and torsion in the original formulation. In the notation of later work, the same structure appears as

$Q$7

Here $Q$8 or $Q$9 is the non-minimal coupling, $f(R)$0 or $f(R)$1 the kinetic coupling, and $f(R)$2 or $f(R)$3 the potential (Järv et al., 2018, Subramaniam et al., 8 Dec 2025).

Variation with respect to the metric yields field equations of Einstein-tensor form but with additional non-metricity braiding terms. One representative form is

$f(R)$4

while variation with respect to the scalar gives

$f(R)$5

Variation with respect to the affine connection gives

$f(R)$6

in the absence of hypermomentum, or equivalently

$f(R)$7

in the earlier notation. Matter is minimally coupled to the metric, and the Levi-Civita covariant conservation law follows on shell (Järv et al., 2018, Subramaniam et al., 8 Dec 2025).

Several structural properties distinguish the scalar–$f(R)$8 class from scalar–curvature gravity. First, the theory is built from $f(R)$9, which contains only first derivatives of the metric and connection, so the field equations remain second order. Second, the scalar is sourced directly by $Q$0, not by the Ricci scalar. Third, the connection does not play the same role as in torsion-based teleparallel gravity: in symmetric teleparallel geometry it is flat and torsionless, and in the coincident gauge one may locally set

$Q$1

The connection equation must nevertheless be satisfied in the general non-minimally coupled theory; it cannot simply be ignored (Järv et al., 2018).

The Brans–Dicke-type specialization of scalar–$Q$2 gravity is obtained by choosing

$Q$3

which gives

$Q$4

This model is a symmetric teleparallel scalar–tensor theory in which the $Q$5 limit reproduces the scalar representation of $Q$6, in direct analogy with the relation between ordinary Brans–Dicke theory and $Q$7 gravity (Paliathanasis, 2023).

3. $Q$8, $Q$9, and the relation to curvature-based scalar–tensor gravity

A central classification result is that scalar–tensor non-metricity gravity actually contains two conceptually different branches. One branch is “native” scalar–$Q$0 gravity, in which the scalar couples directly to the non-metricity scalar $Q$1. The other is the non-metricity rewriting of curvature scalar–tensor gravity, in which the scalar must couple to the combination $Q$2, not to $Q$3 alone. This follows directly from

$Q$4

Therefore, although STEGR and GR are equivalent at the level of linear actions, it does not follow that nonlinear or scalar-coupled extensions built from $Q$5 alone are equivalent to curvature-based ones. In particular,

$Q$6

in general, whereas

$Q$7

is the exact non-metricity counterpart of $Q$8 gravity (Capozziello et al., 11 Mar 2025).

The same distinction appears in scalar-tensor language. A general scalar–nonmetricity theory may be written schematically as

$Q$9

but this is not the same as the non-metricity representation of curvature scalar–tensor gravity. The latter is instead suggested by the Legendre transform of $g_{\mu\nu}$0,

$g_{\mu\nu}$1

or, up to a boundary term,

$g_{\mu\nu}$2

This shows that the scalar does not naturally couple to $g_{\mu\nu}$3 alone if one wants equivalence to scalar–tensor $g_{\mu\nu}$4; rather, it couples to $g_{\mu\nu}$5, or after integration by parts, to $g_{\mu\nu}$6 plus derivative couplings to the non-metricity traces $g_{\mu\nu}$7 (Capozziello et al., 11 Mar 2025).

This distinction also clarifies the status of $g_{\mu\nu}$8 gravity. Pure $g_{\mu\nu}$9 is not introduced as a fundamental scalar-tensor theory, but it admits an auxiliary-field representation: $\Gamma^\lambda{}_{\mu\nu}$0 with

$\Gamma^\lambda{}_{\mu\nu}$1

It is therefore a restricted scalar–nonmetricity model with linear coupling to $\Gamma^\lambda{}_{\mu\nu}$2, no explicit scalar kinetic term, and potential fixed by the chosen $\Gamma^\lambda{}_{\mu\nu}$3. The scalar is an auxiliary/Legendre mode associated with the nonlinearity $\Gamma^\lambda{}_{\mu\nu}$4, not a fundamental scalar matter field introduced at the outset (D'Ambrosio et al., 2020).

The boundary-extended theory $\Gamma^\lambda{}_{\mu\nu}$5 occupies a different position. Because

$\Gamma^\lambda{}_{\mu\nu}$6

reproduces curvature $\Gamma^\lambda{}_{\mu\nu}$7, it carries the same scalar degree of freedom as $\Gamma^\lambda{}_{\mu\nu}$8. Linearized analysis shows that the boundary term $\Gamma^\lambda{}_{\mu\nu}$9 generates a massive scalar gravitational-wave mode with effective mass

$$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$0

whereas pure $$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$1 in the same regime propagates only the two standard massless tensor modes. The boundary sector is therefore exactly what activates the scalaron-like content on the non-metricity side (Capozziello et al., 2024).

4. Conformal frames, Brans–Dicke theory, and Weyl-integrable interpretation

The conformal-frame problem in scalar–tensor non-metricity gravity differs sharply from the familiar metric scalar–tensor case. For the scalar–$$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$2 action of the form $$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$3, a conformal transformation

$$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$4

induces

$$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$5

The term proportional to $$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$6 is not present in the original action, so the scalar–nonmetricity action does not preserve its form under conformal rescaling. One cannot in general map it into a conventional Einstein frame purely by conformal transformation and scalar redefinition without generating additional couplings (Järv et al., 2018).

This obstruction appears explicitly in the Brans–Dicke-type non-metricity theory. Starting from

$$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$7

the conformal transformation with

$$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$8

does not yield STEGR plus a minimally coupled canonical scalar. Instead, for the Brans–Dicke choice $$R^\alpha{}_{\beta\mu\nu}=0, \qquad T^\alpha{}_{\mu\nu}=0,$$9, one obtains

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$0

The Einstein frame therefore contains an unavoidable scalar coupling to the boundary term,

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$1

and is not simply minimally coupled scalar gravity (Paliathanasis, 2023).

A later development shows that this limitation can be cured by enlarging the theory to include a scalar coupling to the non-metricity boundary term itself. The action

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$2

uses

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$3

Under a conformal transformation with

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$4

the transformed action can be written in Einstein-frame-like form,

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$5

If

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$6

the transformed theory reduces to STEGR plus a scalar field. In this sense, adding the boundary coupling supplies precisely the missing degree of freedom needed to recover an Einstein-frame theory based on STEGR, something that standard scalar–$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$7 theories generally fail to do (Murtaza et al., 13 Feb 2026).

A different, geometrically older frame interpretation is provided by Weyl integrable geometry in Brans–Dicke gravity. There the Jordan–Einstein conformal transformation may be regarded as a Weyl gauge transformation,

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$8

with

$Q_{\rho\mu\nu}=\nabla_\rho g_{\mu\nu},$9

In that reading, Dicke’s “running units” are the physical manifestation of a pure-trace non-metricity tensor, and the Einstein-frame free-fall curves are geodesics of the Weyl connection rather than of the Levi-Civita connection of the conformally related metric (Lobo, 2016).

5. Cosmology, perturbations, and structure formation

At the homogeneous level, scalar–tensor non-metricity gravity admits a spatially flat FLRW reduction with

$Q$00

In the coincident gauge one finds

$Q$01

for the foundational scalar–$Q$02 theory, and the modified Friedmann equations become

$Q$03

$Q$04

with scalar equation

$Q$05

These equations coincide with those of teleparallel dark energy at the background level for spatially flat FLRW cosmology (Järv et al., 2018).

The Brans–Dicke-type non-metricity theory has also been studied in a non-coincidence gauge, where an additional time-dependent inertial connection function $Q$06 enters through

$Q$07

In that setting exact power-law and de Sitter solutions were constructed in both Jordan and Einstein frames, and the broad conclusion was that the asymptotic physical behavior of the examined solutions remains invariant under the conformal transformation. The same study reports an analytic symmetric teleparallel scalar-tensor cosmology with exponential potential whose qualitative evolution proceeds from a Big Rip singularity through an ideal-gas or matter-dominated era toward a de Sitter attractor (Paliathanasis, 2023).

The role of nontrivial FLRW-compatible connections has been further systematized by classifying them into several branches. One branch reproduces scalar–torsion cosmology, but a second branch carries an additional time-dependent connection function $Q$08 that survives into the background equations whenever $Q$09. In Hubble-normalized variables the genuinely new mode appears as

$Q$10

Phase-space analysis in representative scenarios exhibits a saddle matter point with

$Q$11

and a stable de Sitter attractor with

$Q$12

together with stiff-fluid branches and noncompact asymptotic sectors interpreted as Big Rip or Big Crunch behavior. This suggests that the extra connection mode can reproduce the standard matter-to-de Sitter sequence while also generating genuinely new high-curvature regimes (Murtaza et al., 20 Jun 2025).

Linear perturbation theory for scalar–$Q$13 gravity has recently been developed in full generality for scalar cosmological perturbations. With

$Q$14

the perturbed FLRW metric was taken in the general scalar ansatz

$Q$15

and, under the quasi-static approximation, the theory reduces to an effective Poisson equation

$Q$16

which defines

$Q$17

The matter density contrast then obeys

$Q$18

For the benchmark choice

$Q$19

with

$Q$20

the background expansion is close to $Q$21CDM, but the perturbation sector predicts $Q$22 on the scales shown, hence suppressed growth relative to $Q$23CDM (Subramaniam et al., 8 Dec 2025).

The boundary-coupled model $Q$24 admits a similarly rich cosmological phase structure. Using a unified autonomous system across several affine-connection branches, one finds standard matter eras, scaling solutions, and stable de Sitter attractors, especially in the $Q$25 and $Q$26 branches. The key late-time result is that both branches possess a stable point with

$Q$27

showing that the boundary coupling supplies a viable late-time acceleration mechanism while preserving geometric flexibility across coincident and non-coincident gauges (Murtaza et al., 13 Feb 2026).

6. Propagating modes, broader formulations, and open directions

The propagating content of scalar–tensor non-metricity gravity depends sharply on which invariant is used. In pure scalar–$Q$28 theories the scalar is explicit in the action, but the geometric sector remains second order and is structurally distinct from scalaron-based $Q$29 gravity. In $Q$30 itself, the auxiliary-field form contains no explicit scalar kinetic term, so the scalar-like content is tied specifically to the nonlinearity $Q$31 (D'Ambrosio et al., 2020).

By contrast, once the boundary term enters through $Q$32, linearized gravitational-wave analysis shows that the theory propagates three degrees of freedom: $Q$33 The scalar mode is encoded in the linearized boundary term

$Q$34

which obeys

$Q$35

It is massive, helicity zero, and transverse in the detector response at leading order, while the tensor sector reproduces the standard plus and cross modes of general relativity (Capozziello et al., 2024).

Beyond symmetric teleparallel theory, scalar–tensor non-metricity also appears in broader metric-affine and Palatini settings. In Palatini scalar–tensor gravity with an independent torsionless connection, one may include explicit couplings of the scalar to non-metricity traces through the action

$Q$36

where $Q$37. In this framework enlarged frame transformations include both conformal rescalings and almost-geodesic transformations of the independent connection, and the resulting Palatini frames naturally include non-metric descriptions. The connection is algebraically eliminable, but non-metricity becomes part of the frame-covariant structure rather than merely a fixed background artifact (Kozak et al., 2018).

A related metric-affine route starts from $Q$38 gravity supplemented with all parity-even quadratic invariants in torsion and non-metricity. There the non-Riemannian connection no longer remains an auxiliary projective gauge redundancy; after solving the connection equations, the theory becomes equivalent on shell to a Riemannian scalar–tensor theory,

$Q$39

with the extra scalar originating from the trace sectors $Q$40, $Q$41, and $Q$42. This is not a pure symmetric teleparallel model, but it demonstrates that non-metricity traces can act as the geometric progenitors of effective scalar degrees of freedom once projective symmetry is broken (Iosifidis, 2024).

More generally, scalar–nonmetricity theories can also be reorganized in unitary gauge as spatially covariant gravity with nonmetricity. In that language the disformation tensor is decomposed into six spatial building blocks, and the scalar monomials of the resulting theory can be exhausted up to total derivative order $Q$43. For the $Q$44 sector, the disformation acts as an auxiliary variable that can be integrated out, producing an effective spatially covariant metric theory with modified coefficients. This suggests that a broad portion of scalar–nonmetricity model space may be understood as an auxiliary affine extension of the familiar unitary-gauge scalar-tensor framework, although the presence of nonmetricity can also generate additional degrees of freedom once higher-order structures are admitted (Yu et al., 2024).

Taken together, these developments support a precise but nontrivial picture. “Scalar–tensor non-metricity gravity” is not a single theory but a family of geometrically distinct constructions. One part of that family consists of native symmetric teleparallel models with couplings $Q$45. Another consists of boundary-extended theories involving $Q$46 or $Q$47, which provide the exact non-metricity counterpart of curvature scalar–tensor gravity and recover scalaron-like dynamics. A third consists of Weyl-integrable and Palatini formulations in which the scalar generates non-metricity of an independent connection. The main unresolved issues are therefore not definitional but structural: which coupling class is being considered, which connection branch or gauge is adopted, and whether the scalar is fundamental, auxiliary, or emergent from the non-metric geometry itself (Capozziello et al., 11 Mar 2025, Murtaza et al., 13 Feb 2026).