Squared-Trace Extended Gravity

• Squared-Trace Extended Gravity is defined as a family of modified gravity theories that incorporate quadratic trace terms to extend Einstein–Hilbert dynamics.
• It encompasses f(R,T), symmetric-teleparallel, EMSG, and Palatini frameworks where quadratic invariants modify effective field equations and strong-field structures.
• Applications include restructured compact objects, viable wormholes without exotic matter, and modified cosmologies, all while balancing conservation laws and symmetry principles.

Squared-Trace Extended Gravity denotes a family of modified-gravity constructions in which the Einstein–Hilbert dynamics are supplemented by terms quadratic in trace-related quantities or by quadratic contractions that play an analogous structural role. In the literature surveyed here, this includes $f(R,T)$ models with $f_2(T)=\lambda_1 T+\lambda_2 T^2$, symmetric-teleparallel models $f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$, energy–momentum squared gravity (EMSG) with $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$, Palatini quadratic-curvature theories such as $f(R,Q)=R+l_P^2(R^2+bQ)$, and foliation-preserving gravitational theories whose holographic trace anomaly acquires an irreducible $R^2$ term. The unifying theme is a nonlinear coupling between geometry and either the trace of the matter sector, the square of that trace, or a quadratic contraction of a tensor whose trace content controls effective gravitational dynamics; the terminology is therefore broader than any single Lagrangian ansatz (Tripathy et al., 2023, Swain et al., 18 Jun 2026, Martinez-Asencio et al., 2013, Nakayama, 2012).

1. Definitions, scope, and model classes

The most direct use of the phrase appears in $f(R,T)$ constructions where the gravitational Lagrangian depends linearly and quadratically on the trace $T\equiv g^{\mu\nu}T_{\mu\nu}$. A representative model is

$f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$

with $\lambda_1$ the linear trace coupling and $f_2(T)=\lambda_1 T+\lambda_2 T^2$0 the genuinely squared-trace coupling. Closely related is the symmetric-teleparallel model

$f_2(T)=\lambda_1 T+\lambda_2 T^2$1

where $f_2(T)=\lambda_1 T+\lambda_2 T^2$2 is the non-metricity scalar and $f_2(T)=\lambda_1 T+\lambda_2 T^2$3 is the quadratic trace contribution. In a broader usage, EMSG is described as a prototypical “squared‑trace” extension of GR because the action contains the quadratic matter scalar $f_2(T)=\lambda_1 T+\lambda_2 T^2$4, even though this is distinct from $f_2(T)=\lambda_1 T+\lambda_2 T^2$5. Palatini theories with $f_2(T)=\lambda_1 T+\lambda_2 T^2$6 and $f_2(T)=\lambda_1 T+\lambda_2 T^2$7 belong to the same quadratic-invariant landscape, while the extended geometric trinity identifies $f_2(T)=\lambda_1 T+\lambda_2 T^2$8, $f_2(T)=\lambda_1 T+\lambda_2 T^2$9, and $f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$0 as the dynamically equivalent nonlinear completions of the curvature, torsion, and non-metricity formulations (Capozziello et al., 11 Mar 2025).

Class	Characteristic term	Representative paper
$f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$1 squared-trace gravity	$f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$2	(Tripathy et al., 2023)
Symmetric-teleparallel squared-trace gravity	$f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$3 in $f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$4	(Agrawal et al., 2022)
EMSG	$f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$5	(Swain et al., 18 Jun 2026)
Palatini quadratic gravity	$f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$6	(Martinez-Asencio et al., 2013)
FPDiff holographic gravity	$f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$7 with induced $f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$8 anomaly	(Nakayama, 2012)

A common misconception is that all such theories are simply variants of $f(Q,T)=-\lambda_1 Q^m-\lambda_2 T^2$9. The cited literature does not support that reduction. The $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$0 models, the $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$1 models, the Palatini $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$2 and Ricci-squared models, and the FPDiff $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$3 deformation are distinct at the level of invariant content, variational principle, and effective field equations. A plausible implication is that “squared-trace” is best treated as an umbrella descriptor for quadratic trace-sector extensions rather than as a single theory.

2. Matter-trace couplings and effective field equations

In the $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$4 realizations, the starting action is

$f=\alpha\,T_{\mu\nu}T^{\mu\nu}$5

with $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$6, and the specialization $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$7, $f=\alpha\,T_{\mu\nu}T^{\mu\nu}$8 yields

$f=\alpha\,T_{\mu\nu}T^{\mu\nu}$9

where

$f(R,Q)=R+l_P^2(R^2+bQ)$0

This is routinely rewritten as

$f(R,Q)=R+l_P^2(R^2+bQ)$1

with an interaction tensor proportional to $f(R,Q)=R+l_P^2(R^2+bQ)$2. The physical interpretation used across the wormhole papers is that the trace-dependent sector acts as an effective fluid sourced by the real matter, while the physical matter stress tensor need not itself provide the exoticity required by the geometry (Tripathy et al., 2023, Moraes et al., 2017).

The same logic appears in the Finsler–Randers/Barthel formulation, where the action is

$f(R,Q)=R+l_P^2(R^2+bQ)$3

and the field equations become

$f(R,Q)=R+l_P^2(R^2+bQ)$4

The same paper introduces

$f(R,Q)=R+l_P^2(R^2+bQ)$5

so the effective coupling itself depends on the trace (Nekouee et al., 1 Nov 2025).

In symmetric teleparallel gravity the corresponding cosmological model is

$f(R,Q)=R+l_P^2(R^2+bQ)$6

with $f(R,Q)=R+l_P^2(R^2+bQ)$7 for spatially flat FLRW and

$f(R,Q)=R+l_P^2(R^2+bQ)$8

The modified Friedmann sector is then expressed in terms of an effective fluid and a nonconserved matter sector. The paper explicitly states that “in the $f(R,Q)=R+l_P^2(R^2+bQ)$9 gravity, there is a violation of the energy-momentum conservation,” and the trace-squared term enters the background dynamics through a square-root relation between $R^2$0 and $R^2$1 (Agrawal et al., 2022).

Across these constructions, the matter Lagrangian is model-dependent: $R^2$2, $R^2$3, and $R^2$4 are all used in the cited papers. This indicates that the precise algebraic form of the effective trace-sector source is not universal even within squared-trace $R^2$5-type gravity.

3. Geometric realizations beyond $R^2$6

EMSG is presented as a prototypical “squared‑trace” extension of GR in which the action depends on the quadratic matter scalar $R^2$7,

$R^2$8

with $R^2$9 recovering GR. For a perfect fluid,

$f(R,T)$0

so geometry is sourced by effective thermodynamic variables rather than by the physical $f(R,T)$1. The cited neutron-star study emphasizes a “clear matter–geometry separation”: the trace anomaly is computed from the fluid sector alone, while curvature scalars are built from the effective quantities that actually source the modified Tolman–Oppenheimer–Volkoff equations (Swain et al., 18 Jun 2026).

Palatini quadratic gravity provides a different realization of trace-sector extension. The basic theory uses

$f(R,T)$2

and the specific black-hole model

$f(R,T)$3

Because metric and connection are varied independently, the connection can be solved algebraically in terms of an auxiliary metric $f(R,T)$4, and the resulting equations remain second order. The paper emphasizes that Palatini $f(R,T)$5 and $f(R,T)$6 theories are free from the perturbative instabilities and ghosts usually present in metric quadratic gravity, and that the higher-curvature corrections appear via algebraic relations such as $f(R,T)$7 and the deformation matrix $f(R,T)$8 relating $f(R,T)$9 and $T\equiv g^{\mu\nu}T_{\mu\nu}$0 (Martinez-Asencio et al., 2013).

In the holographic setting of foliation-preserving diffeomorphic gravity, the bulk action is

$T\equiv g^{\mu\nu}T_{\mu\nu}$1

and the crucial result is not a bulk $T\equiv g^{\mu\nu}T_{\mu\nu}$2 term but an induced boundary anomaly

$T\equiv g^{\mu\nu}T_{\mu\nu}$3

For $T\equiv g^{\mu\nu}T_{\mu\nu}$4, the holographic trace anomaly contains a genuine $T\equiv g^{\mu\nu}T_{\mu\nu}$5 term with nonzero coefficient $T\equiv g^{\mu\nu}T_{\mu\nu}$6, which the paper identifies as compatible with scale invariance but not with conformal invariance. This places quadratic trace-sector structures in direct correspondence with the loss of special conformal symmetry (Nakayama, 2012).

Adler’s trace-dynamics construction is not an explicit $T\equiv g^{\mu\nu}T_{\mu\nu}$7 or $T\equiv g^{\mu\nu}T_{\mu\nu}$8 theory, but it is a trace-based extension in which the induced gravitational action is obtained from the trace-average of the matter action. The leading non-derivative form is

$T\equiv g^{\mu\nu}T_{\mu\nu}$9

and it exactly reduces to a cosmological constant for Robertson–Walker spacetime while diverging as $f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$0 near the Schwarzschild radius (Adler, 2013). This suggests a broader trace-based pathway to extended gravity in which the modification is induced rather than postulated.

4. Compact objects and strong-field structure

In neutron-star interiors, the EMSG analysis asks whether the QCD trace anomaly still organizes curvature when gravity couples nonlinearly to matter. The normalized trace anomaly is defined as

$f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$1

and is computed only from the physical fluid sector. For five relativistic mean-field equations of state, the radial trace-anomaly profiles increase monotonically from core to surface in all accepted EMSG models, as in GR, but split systematically with the EMSG coupling strength; the splitting grows with stellar compactness. Curvature invariants still fall onto organized bands when plotted against the trace anomaly, extending the GR thermodynamic-geometric correspondence. The Ricci contraction

$f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$2

shows the tightest organization, whereas the Ricci scalar remains the most equation-of-state sensitive. EMSG effects are modest for observationally accessible stars but largest in stiff, ultracompact configurations (Swain et al., 18 Jun 2026).

Palatini quadratic gravity gives explicit black-hole realizations of squared-curvature corrections. In pure Palatini $f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$3 coupled to Born–Infeld nonlinear electrodynamics, the paper reports black holes with up to three horizons and a softened central singularity whose leading divergence behaves as $f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$4, instead of $f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$5 or $f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$6. In the full $f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$7 model with Maxwell electrodynamics, the charged solution takes the form

$f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$8

with $f(R,T)=R+\lambda_1 T+\lambda_2 T^2,$9, $\lambda_1$0, and $\lambda_1$1. For $\lambda_1$2, the curvature invariants are finite and the geometry extends through a wormhole throat; when $\lambda_1$3, the event horizon disappears and the mass spectrum becomes

$\lambda_1$4

The same paper also derives a null-fluid collapse metric,

$\lambda_1$5

interpreted as a Reissner–Nordström solution with a wrong-sign charge term induced purely by the quadratic corrections (Martinez-Asencio et al., 2013).

These results collectively show that squared-trace or quadratic-invariant extensions can regularize or restructure strong-field cores without necessarily introducing higher-derivative dynamics. A plausible implication is that compact objects are the natural testing ground because the nonlinear corrections scale with high density, high compactness, or near-horizon redshift.

5. Wormholes and the problem of exotic matter

The original $\lambda_1$6 wormhole construction within $\lambda_1$7 gravity uses

$\lambda_1$8

in the action

$\lambda_1$9

with anisotropic matter $f_2(T)=\lambda_1 T+\lambda_2 T^2$00, $f_2(T)=\lambda_1 T+\lambda_2 T^2$01, and barotropic equations of state $f_2(T)=\lambda_1 T+\lambda_2 T^2$02, $f_2(T)=\lambda_1 T+\lambda_2 T^2$03. For the illustrative choice

$f_2(T)=\lambda_1 T+\lambda_2 T^2$04

the paper finds that for $f_2(T)=\lambda_1 T+\lambda_2 T^2$05, $f_2(T)=\lambda_1 T+\lambda_2 T^2$06 is positive, NEC and WEC are satisfied, SEC is satisfied, and DEC is satisfied radially but violated tangentially. The exoticity needed for the wormhole is therefore supplied by the modified gravity sector rather than by the physical matter (Moraes et al., 2017).

A later analysis of the specific model

$f_2(T)=\lambda_1 T+\lambda_2 T^2$07

reaches a more restrictive conclusion. The wormhole field equations force

$f_2(T)=\lambda_1 T+\lambda_2 T^2$08

for non-vanishing trace and relate the anisotropic equations-of-state through

$f_2(T)=\lambda_1 T+\lambda_2 T^2$09

For the exponential shape function $f_2(T)=\lambda_1 T+\lambda_2 T^2$10, non-exotic matter wormholes exist only for a narrow range

$f_2(T)=\lambda_1 T+\lambda_2 T^2$11

For the power-law shape $f_2(T)=\lambda_1 T+\lambda_2 T^2$12, the allowed interval is broader,

$f_2(T)=\lambda_1 T+\lambda_2 T^2$13

If one requires both geometries to admit non-exotic matter wormholes, only the overlap interval remains. The paper therefore states that “the existence of non-exotic matter traversable wormholes is not obvious” and that the possibility may depend on the choice of the wormhole geometry (Tripathy et al., 2023).

The Finsler–Randers/Barthel extension modifies this picture by introducing an anisotropy parameter $f_2(T)=\lambda_1 T+\lambda_2 T^2$14 through the osculating metric and by working with

$f_2(T)=\lambda_1 T+\lambda_2 T^2$15

The consistency conditions again fix $f_2(T)=\lambda_1 T+\lambda_2 T^2$16, while $f_2(T)=\lambda_1 T+\lambda_2 T^2$17 is required to avoid a traceless matter sector. For both the exponential and the power-law shape functions, the representative choice

$f_2(T)=\lambda_1 T+\lambda_2 T^2$18

gives

$f_2(T)=\lambda_1 T+\lambda_2 T^2$19

so NEC, WEC, SEC, and DEC are all satisfied. The paper further states that the Barthel connection significantly extends the parameter space for non-exotic, physically viable wormholes compared to purely Riemannian models (Nekouee et al., 1 Nov 2025).

The wormhole literature therefore contains both a positive and a restrictive message. Squared-trace gravity can shift exoticity from physical matter to an effective interaction sector, but the existence of non-exotic solutions is sensitive to the invariant employed, the geometric background, and the chosen shape function.

6. Cosmology, holography, and conceptual tensions

In cosmology, the symmetric-teleparallel model

$f_2(T)=\lambda_1 T+\lambda_2 T^2$20

was confronted with 32 cosmic-chronometer $f_2(T)=\lambda_1 T+\lambda_2 T^2$21 points, the Pantheon$f_2(T)=\lambda_1 T+\lambda_2 T^2$22 sample of 1701 supernovae, and BAO data. In the small-coupling regime $f_2(T)=\lambda_1 T+\lambda_2 T^2$23, the background solution reduces to

$f_2(T)=\lambda_1 T+\lambda_2 T^2$24

and for a two-component cosmology the paper uses

$f_2(T)=\lambda_1 T+\lambda_2 T^2$25

The best-fit values reported are

$f_2(T)=\lambda_1 T+\lambda_2 T^2$26

from Hubble data, and

$f_2(T)=\lambda_1 T+\lambda_2 T^2$27

from Pantheon$f_2(T)=\lambda_1 T+\lambda_2 T^2$28. The model shows an early deceleration transitioning to an accelerating phase at $f_2(T)=\lambda_1 T+\lambda_2 T^2$29, and the $f_2(T)=\lambda_1 T+\lambda_2 T^2$30 diagnostic has a positive slope, favoring a phantom-dominated phase (Agrawal et al., 2022).

The nonconservative traceless theory provides a different trace-centered cosmology. Its basic field equations are

$f_2(T)=\lambda_1 T+\lambda_2 T^2$31

with divergence relation

$f_2(T)=\lambda_1 T+\lambda_2 T^2$32

For $f_2(T)=\lambda_1 T+\lambda_2 T^2$33, the background behaves as radiation plus de Sitter; for the constraint

$f_2(T)=\lambda_1 T+\lambda_2 T^2$34

the Friedmann equation becomes

$f_2(T)=\lambda_1 T+\lambda_2 T^2$35

and choosing $f_2(T)=\lambda_1 T+\lambda_2 T^2$36 reproduces the $f_2(T)=\lambda_1 T+\lambda_2 T^2$37CDM background. The perturbations, however, are not $f_2(T)=\lambda_1 T+\lambda_2 T^2$38CDM-like, because the effective radiative fluid has perturbations that behave like matter (Daouda et al., 2018).

The holographic FPDiff analysis sharpens a conceptual tension already present in several of these theories: quadratic trace-sector terms often coexist with restricted symmetry. The presence of the $f_2(T)=\lambda_1 T+\lambda_2 T^2$39 term in the trace anomaly,

$f_2(T)=\lambda_1 T+\lambda_2 T^2$40

is identified as the holographic diagnostic of scale invariance without conformal invariance (Nakayama, 2012). In the extended geometric trinity, equivalence survives only for the special combinations $f_2(T)=\lambda_1 T+\lambda_2 T^2$41, $f_2(T)=\lambda_1 T+\lambda_2 T^2$42, and $f_2(T)=\lambda_1 T+\lambda_2 T^2$43; once independent quadratic trace or squared-trace invariants are added, one moves beyond the extended trinity into genuinely new dynamical territory (Capozziello et al., 11 Mar 2025).

A recurrent issue across the literature is therefore the trade-off between phenomenological flexibility and structural economy. The $f_2(T)=\lambda_1 T+\lambda_2 T^2$44 and $f_2(T)=\lambda_1 T+\lambda_2 T^2$45 sectors can regularize compact objects, enlarge the space of wormhole solutions, or generate late-time acceleration, but they also tend to introduce nonconservation, preferred-frame effects, anomaly constraints, or degeneracies between microphysics and modified gravity. The cited neutron-star study explicitly notes that EMSG can mimic or counteract equation-of-state stiffness and that fully self-consistent perturbation theory in EMSG is still needed for refined observational predictions (Swain et al., 18 Jun 2026).

Squared-Trace Extended Gravity is therefore not a single mature theory but a research program spanning several geometric languages. Its central idea is stable: nonlinear trace-sector couplings provide effective sources that can reorganize curvature, interior structure, cosmological expansion, and energy-condition bookkeeping. What remains unsettled is how much of that reorganization can be achieved while retaining conservation laws, symmetry principles, and robust observational discriminants.