Myrzakulov Gravity

• Myrzakulov gravity is a family of modified gravitational theories that models curvature, torsion, and non-metricity as dynamic fields to mimic dark sectors.
• It unifies diverse approaches by classifying theories like F(R,T), F(T,Q), and their metric-affine extensions, thereby broadening the modified gravity landscape.
• Cosmological applications demonstrate viable FLRW solutions with effective dark energy behavior, inflation, and bouncing cosmologies comparable to ΛCDM.

Myrzakulov gravity is a family of modified gravitational theories associated with generalized actions built from non-Riemannian geometric invariants. In its central and most widely developed form, it denotes an $F(R,T)$ theory in which $R$ is a curvature scalar and $T$ is a torsion scalar, both constructed from a connection that is neither Levi-Civita nor Weitzenböck, so that curvature and torsion are simultaneously dynamical. In later literature, the label broadens into a taxonomy that also includes theories based on non-metricity, matter-trace couplings, Gauss–Bonnet terms, and metric-affine generalizations. The resulting program treats Myrzakulov gravity less as a single model than as a structured class of non-Riemannian gravities whose cosmological role is to generate effective dark-energy, inflationary, bouncing, or dark-matter-like behavior from geometry itself (Myrzakulov, 2012, Saridakis et al., 2019).

1. Origins and taxonomy

The earliest systematic construction presents Myrzakulov gravity as a hierarchy of modified gravities whose Lagrangians are arbitrary functions of geometric scalars such as $R$, $T$, and $Q$, together with later extensions involving the Gauss–Bonnet scalar $G$, boundary terms $B$, and matter-trace quantities. In that taxonomy, MG-I is $F(R,T)$, MG-II is $F(R,Q)$, MG-III is $R$0, MG-VIII is $R$1, and the classification extends up to MG-XXXVIII, including mixed actions such as $R$2 (Myrzakulov, 2012). A later metric-affine reformulation extends the MG-VIII sector further to $R$3, where $R$4 is the divergence of the dilation current (Iosifidis et al., 2021).

This broad usage has two consequences. First, the phrase “Myrzakulov gravity” can refer narrowly to the original curvature–torsion $R$5 branch or more broadly to an umbrella of metric-affine modified gravities. Second, the literature often distinguishes between the original MG-I curvature–torsion theory and later descendants that unify torsion with non-metricity, matter couplings, or higher invariants. A representative subset is summarized below.

Variant	Defining action or function	Characteristic sector
MG-I	$R$6	Curvature + torsion
MG-III	$R$7	Torsion + non-metricity
Metric-affine extension	$R$8	Curvature + torsion + non-metricity + matter couplings
ECM realization	$R$9	Einstein–Cartan–Myrzakulov hybrid
EGBMG	$T$0	Curvature + torsion + Gauss–Bonnet

A central misconception addressed repeatedly in the literature is terminological. In the core Myrzakulov $T$1 branch, $T$2 denotes a torsion scalar, not the trace of the matter energy-momentum tensor. This differs from the better-known Harko-type $T$3 notation and is essential for reading the Myrzakulov literature correctly (Saridakis et al., 2019, Momeni et al., 28 Jul 2025).

2. Geometric structure of the core $T$4 theory

In the cosmologically most developed branch, the dynamical variables are the tetrad $T$5 and a connection 1-form $T$6, with

$T$7

and zero non-metricity imposed. The curvature and torsion tensors of the general connection are

$T$8

$T$9

The torsion scalar and curvature scalar are then

$R$0

$R$1

This is a Riemann–Cartan-type construction: unlike GR, torsion does not vanish; unlike TEGR, curvature does not vanish; and unlike simply juxtaposing $R$2 and $R$3, both scalars arise from the same chosen non-special connection (Saridakis et al., 2019).

The effective parametrization

$R$4

is the standard cosmological device in this framework. Here $R$5 measures the deviation of the curvature scalar from the Levi-Civita Ricci scalar, while $R$6 measures the deviation of the torsion scalar from the Weitzenböck torsion scalar. These deformation functions encode the cosmologically relevant imprint of the underlying connection without requiring an explicit tensorial reconstruction of that connection in each model (Saridakis et al., 2019).

This parametrization reproduces familiar limits. If the connection becomes Levi-Civita, then $R$7 and $R$8, reducing the theory to ordinary $R$9, and to GR for $T$0. If the connection becomes Weitzenböck, then $T$1 and $T$2, reducing the theory to $T$3, and to TEGR for $T$4. Myrzakulov gravity is therefore best understood as a deformation of both curvature-based and torsion-based gravity, with the connection itself supplying extra degrees of freedom (Saridakis et al., 2019, Papagiannopoulos et al., 2022).

3. FLRW reduction and effective geometrical fluid

For spatially flat FRW cosmology,

$T$5

the benchmark scalars are

$T$6

In minisuperspace one takes $T$7, $T$8, and the matter Lagrangian $T$9. The simplest and most studied model is the linear choice

$Q$0

Its cosmological significance is that even this linear theory is not a trivial rescaling of GR or TEGR. The Friedmann equations contain additional terms involving $Q$1, $Q$2, and derivatives such as $Q$3, $Q$4, and $Q$5, showing that the non-special connection alone already generates new dynamics (Saridakis et al., 2019).

The modified equations are commonly rewritten as

$Q$6

thereby defining an effective Myrzakulov geometrical fluid. This effective sector obeys its own conservation law,

$Q$7

provided ordinary matter satisfies

$Q$8

The geometric sector is thus treated as an emergent dark-energy component sourced by the connection rather than by an explicit cosmological constant or scalar field (Saridakis et al., 2019).

For the simple ansatz

$Q$9

one obtains

$G$0

In this case, $G$1 gives quintessence-like behavior, $G$2 gives phantom-like behavior, and $G$3 gives exactly $G$4, reproducing $G$5CDM. Moreover, for $G$6 and $G$7,

$G$8

so the cosmological constant arises geometrically from the connection sector (Saridakis et al., 2019).

4. Cosmological dynamics: exact solutions, attractors, inflation, and bounce

The late-time background dynamics of linear Myrzakulov gravity have been studied through exact FLRW solutions. In one such analysis, two exact models based on $G$9 produce the same hyperbolic functional form for $B$0, $B$1, $B$2, $B$3, and $B$4, while differing in how the parameters enter the effective dynamics. Both models yield a Big-Bang-type initial singularity, a matter-like decelerating phase with $B$5 at high redshift, and an asymptotic de Sitter-like future with $B$6. The first model has

$B$7

while the second has

$B$8

Both are transit-phase universes with transition redshift in the range

$B$9

and present age

$F(R,T)$0

with late-time statefinder limit $F(R,T)$1 (Maurya et al., 2024).

A complementary dynamical-systems treatment analyzes two classes of $F(R,T)$2 models. Class 1 possesses $F(R,T)$3CDM as a limit and exhibits the standard sequence of a matter-dominated saddle followed by a dark-energy-dominated de Sitter attractor. In this class, $F(R,T)$4 yields quintessence-like dark energy, $F(R,T)$5 yields phantom-like dark energy, and $F(R,T)$6 gives the exact $F(R,T)$7CDM limit. Class 2 does not contain $F(R,T)$8CDM as a limit, but still has a de Sitter late-time attractor and allows quintessence-like, phantom-like, or phantom-divide-crossing behavior during evolution (Papagiannopoulos et al., 2022).

The early-universe sector is likewise nontrivial. The original cosmological analysis showed that the same linear model can produce a purely geometrical de Sitter phase and admits a reconstruction method for inflation: one specifies a desired $F(R,T)$9 or $F(R,Q)$0 and solves for the deformation functions $F(R,Q)$1 and $F(R,Q)$2 that realize it. In this sense, inflation can be reconstructed from the connection sector analogously to reconstructing an inflaton potential in scalar-field inflation (Saridakis et al., 2019).

Myrzakulov gravity has also been used to realize nonsingular bouncing cosmologies. For $F(R,Q)$3, suitable choices of $F(R,Q)$4 and $F(R,Q)$5 generate both a simple bounce and a matter bounce, with the effective modified-gravity sector violating the null energy condition near the bounce. Scalar perturbations were studied with the Mukhanov–Sasaki formalism, and in the matter-bounce case the primordial spectrum amplitude was reported as

$F(R,Q)$6

The bounce solutions are therefore presented as proof-of-principle realizations of regular background evolution and finite scalar perturbations in the curvature–torsion framework (Lymperis et al., 30 Jan 2025).

5. Observational status

The first dedicated background-level observational constraints on the core $F(R,Q)$7 theory used cosmic chronometers, Pantheon supernovae, BAO, and a BBN prior. Two phenomenological models based on the linear theory were fit with an affine-invariant MCMC sampler implemented in emcee, using $F(R,Q)$8 walkers and $F(R,Q)$9 steps. In both models the coupling $R$00 was constrained to an interval around zero, with contours slightly shifted toward positive values. Model 1 was found to be very close to $R$01CDM, while Model 2 resembled it at low redshift but allowed earlier-time deviations. AIC and BIC favored $R$02CDM because of parameter penalization, but the combined DIC criterion found both Myrzakulov models statistically equivalent to $R$03CDM, even though Model 2 has no $R$04CDM limit (Anagnostopoulos et al., 2020).

Independent late-time fits based on exact FLRW solutions used $R$05 and Pantheon SNe Ia data and again found viable deceleration-to-acceleration histories. In one transit model the effective dark-energy equation of state remains in the quintessence-like interval

$R$06

with present ages reported as

$R$07

for the $R$08 and Pantheon fits respectively (Maurya et al., 2024). A related exact-solution analysis found transition redshifts in the observationally acceptable range $R$09 and present ages near $R$10 Gyr (Maurya et al., 2024).

A later and conceptually distinct observational program treats a torsion-based curvature–torsion version of Myrzakulov gravity on Weitzenböck spacetime with

$R$11

Using SPARC galaxy rotation curves, Planck 2018, DES, KiDS-1000, BOSS/SDSS, Pantheon+, and Euclid forecast data, the paper reports MCMC constraints

$R$12

$R$13

all at $R$14. It also reports a growth index

$R$15

compared with

$R$16

and states that the tensor propagation speed remains

$R$17

At the same time, the study provides no explicit $R$18, AIC, or BIC comparison against $R$19CDM, so its claims of improvement are qualitative rather than model-selection results (Momeni et al., 28 Jul 2025).

6. Generalizations and related branches

The metric-affine extension is among the most technically ambitious developments. It promotes the metric and affine connection to independent variables and constructs a full field-equation system for

$R$20

where $R$21 is the non-metricity scalar, $R$22 the trace of the metrical energy-momentum tensor, and $R$23 the divergence of the dilation current. In this formulation, torsion and non-metricity are genuine geometric fields, and matter with hypermomentum can source the connection directly (Iosifidis et al., 2021).

Another major branch is MG-III, the Myrzakulov $R$24 theory, which the literature presents as a unification of $R$25 and $R$26 gravity. In the linear model

$R$27

exact FLRW solutions have been derived, and MCMC fitting to $R$28 observed $R$29 points has been used to constrain three models. The first two models exhibit deceleration-to-acceleration transition and asymptotic $R$30CDM-like behavior, while the third yields a power-law cosmology with nearly constant acceleration (Maurya et al., 2024).

More recent variants extend the Myrzakulov idea in several directions. The Einstein–Cartan–Myrzakulov realization adopts

$R$31

and treats curvature, torsion, and non-metricity together in a Weitzenböck-inspired metric-affine setting, with ansatz-based reconstructions for dust, $R$32CDM-like, quintessence-like, and phantom-like histories (Momeni et al., 29 Jul 2025). A metric-affine model

$R$33

has been used to construct hyperbolic transit solutions with quintom-A-type dark-energy behavior and observational fits from cosmic chronometers and Pantheon data (Maurya et al., 3 Aug 2025). Inflationary applications have also been developed comparatively across the metric $R$34, teleparallel $R$35, symmetric teleparallel $R$36, and hybrid $R$37 sectors, including the benchmark polynomial model

$R$38

with predictions discussed in the $R$39–$R$40 plane (Momeni et al., 15 Jul 2025).

The Myrzakulov framework has also been combined with other modified-gravity sectors. One example is Myrzakulov $R$41 quasi-dilaton massive gravity, which merges the linear curvature–torsion sector

$R$42

with ghost-free dRGT massive gravity and a quasi-dilaton field; the model admits self-accelerating branches and modifies the effective graviton mass and gravitational-wave dispersion relation (Kazempour et al., 2023). Another is Einstein–Gauss–Bonnet–Myrzakulov gravity, formulated through

$R$43

which extends the Myrzakulov program toward torsion–Gauss–Bonnet dynamics in Weitzenböck spacetime (Momeni et al., 23 May 2025).

7. Open problems, ambiguities, and controversies

The literature leaves several foundational issues unresolved. The first is terminological heterogeneity. In some papers, “Myrzakulov gravity” means specifically the non-special-connection $R$44 theory with simultaneous curvature and torsion; in others, it denotes a broad program that includes $R$45, $R$46, metric-affine $R$47, Einstein–Cartan–Myrzakulov, or even $R$48. This suggests that the term functions as a family label rather than a uniquely fixed theory (Myrzakulov, 2012, Momeni et al., 29 Jul 2025).

A second issue is notation. The core $R$49 literature is explicit that $R$50 is the torsion scalar, not the trace of the matter energy-momentum tensor. Yet later metric-affine extensions introduce separate matter-trace variables $R$51 or $R$52, and some expository papers become internally inconsistent. One vielbein-based overview explicitly uses $R$53 as torsion scalar through most of the paper but later describes it once as a matter-trace variable; the same work also gives competing definitions of the torsion scalar (Momeni et al., 2024).

A third issue concerns local Lorentz invariance in pure-tetrad teleparallel-inspired formulations. One recent observational paper is explicit that $R$54 is Lorentz invariant, $R$55 is not, and the pure-tetrad setup therefore involves “controlled Lorentz violation,” with future covariant teleparallel formulations using an inertial spin connection proposed as the cure. The same paper later states that its field equations “respect local Lorentz invariance,” a claim that sits uneasily with its earlier caveat; the more cautious reading is therefore the appropriate one (Momeni et al., 28 Jul 2025).

Finally, many cosmological realizations remain ansatz-driven. The deformation functions $R$56 and $R$57 in the core $R$58 branch are often chosen to generate desired FLRW histories, exact transit solutions, or bounce backgrounds, but their deeper derivation from a fully specified connection is left open. Analogously, several extended models are programmatic rather than fully rigorous: the ECM paper gives multiple expressions for $R$59 and $R$60 and no separate explicit connection equation, while a number of observational studies remain background-only, without full perturbative, ghost, or Laplacian-stability analyses (Maurya et al., 2024, Momeni et al., 29 Jul 2025).

Taken together, these limitations do not erase the central content of the subject. They indicate instead that Myrzakulov gravity is best viewed as an evolving non-Riemannian research program: a set of curvature–torsion–non-metricity modified gravities in which the connection itself is a model-building input, geometry can mimic dark sectors, and cosmological viability depends crucially on how the extra geometric degrees of freedom are parameterized, constrained, and ultimately derived.