--- title: Myrzakulov Gravity url: https://www.emergentmind.com/topics/myrzakulov-gravity type: topic --- # Myrzakulov Gravity 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 [1205.5266] [1912.03882]. ## 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 \(F(T,Q)\), **MG-VIII** is \(F(R,T,Q,\mathcal{T})\), and the classification extends up to **MG-XXXVIII**, including mixed actions such as \(F(R,T,Q,G,B,\mathcal{T})\) [1205.5266]. A later metric-affine reformulation extends the MG-VIII sector further to \(F(R,T,Q,\mathcal{T},\mathcal{D})\), where \(\mathcal{D}\) is the divergence of the dilation current [2106.05083]. This broad usage has two consequences. First, the phrase “Myrzakulov gravity” can refer narrowly to the original curvature–torsion \(F(R,T)\) 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 | \(F(R,T)\) | Curvature + torsion | | MG-III | \(F(T,Q)\) | Torsion + non-metricity | | Metric-affine extension | \(F(R,T,Q,\mathcal{T},\mathcal{D})\) | Curvature + torsion + non-metricity + matter couplings | | ECM realization | \(\alpha R+F(T,Q)\) | Einstein–Cartan–Myrzakulov hybrid | | EGBMG | \(R+F(T,G)\) | Curvature + torsion + Gauss–Bonnet | A central misconception addressed repeatedly in the literature is terminological. In the core Myrzakulov \(F(R,T)\) branch, \(T\) denotes a **torsion scalar**, not the trace of the matter energy-momentum tensor. This differs from the better-known Harko-type \(f(R,T)\) notation and is essential for reading the Myrzakulov literature correctly [1912.03882] [2507.21359]. ## 2. Geometric structure of the core \(F(R,T)\) theory In the cosmologically most developed branch, the dynamical variables are the tetrad \(e^a{}_\mu\) and a connection 1-form \(\omega^a{}_{b\mu}\), with \[ g_{\mu\nu}=\eta_{ab}e^a{}_\mu e^b{}_\nu, \] and zero non-metricity imposed. The curvature and torsion tensors of the general connection are \[ R^{a}{}_{b\mu\nu}=\omega^{a}{}_{b\nu,\mu}-\omega^{a}{}_{b\mu,\nu} +\omega^{a}{}_{c\mu}\omega^{c}{}_{b\nu} -\omega^{a}{}_{c\nu}\omega^{c}{}_{b\mu}, \] \[ T^{a}{}_{\mu\nu}=e^{a}{}_{\nu,\mu}-e^{a}{}_{\mu,\nu} +\omega^{a}{}_{b\mu}e^{b}{}_{\nu} -\omega^{a}{}_{b\nu}e^{b}{}_{\mu}. \] The torsion scalar and curvature scalar are then \[ T=\frac{1}{4}T^{\mu\nu\lambda}T_{\mu\nu\lambda} +\frac{1}{2}T^{\mu\nu\lambda}T_{\lambda\nu\mu} -T_{\nu}{}^{\nu\mu}T^{\lambda}{}_{\lambda\mu}, \] \[ R=R^{(LC)}+T-2T_{\nu}{}^{\nu\mu}{}_{;\mu}. \] This is a Riemann–Cartan-type construction: unlike GR, torsion does not vanish; unlike TEGR, curvature does not vanish; and unlike simply juxtaposing \(f(R)\) and \(f(T)\), both scalars arise from the same chosen non-special connection [1912.03882]. The effective parametrization \[ T=T^{(W)}+v,\qquad R=R^{(LC)}+u \] is the standard cosmological device in this framework. Here \(u\) measures the deviation of the curvature scalar from the Levi-Civita Ricci scalar, while \(v\) 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 [1912.03882]. This parametrization reproduces familiar limits. If the connection becomes Levi-Civita, then \(u=0\) and \(v=-T^{(W)}\), reducing the theory to ordinary \(F(R)\), and to GR for \(F(R)=R\). If the connection becomes Weitzenböck, then \(v=0\) and \(u=-R^{(LC)}\), reducing the theory to \(F(T)\), and to TEGR for \(F(T)=T\). 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 [1912.03882] [2202.10871]. ## 3. FLRW reduction and effective geometrical fluid For spatially flat FRW cosmology, \[ ds^2=dt^2-a^2(t)\delta_{ij}dx^i dx^j,\qquad e^a{}_\mu=\mathrm{diag}[1,a(t),a(t),a(t)], \] the benchmark scalars are \[ R^{(LC)}=6\left(\frac{\ddot a}{a}+\frac{\dot a^2}{a^2}\right),\qquad T^{(W)}=-6\left(\frac{\dot a^2}{a^2}\right). \] In minisuperspace one takes \(u=u(a,\dot a,\ddot a)\), \(v=v(a,\dot a)\), and the matter Lagrangian \(L_m=-\rho_m(a)\). The simplest and most studied model is the linear choice \[ F(R,T)=R+\lambda T. \] 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 \(u\), \(v\), and derivatives such as \(u_{\ddot a}\), \(\dot u_{\ddot a}\), and \(\ddot u_{\ddot a}\), showing that the non-special connection alone already generates new dynamics [1912.03882]. The modified equations are commonly rewritten as \[ 3H^2=\kappa^2(\rho_m+\rho_{MG}),\qquad 2\dot H+3H^2=-\kappa^2(p_m+p_{MG}), \] thereby defining an effective Myrzakulov geometrical fluid. This effective sector obeys its own conservation law, \[ \dot\rho_{MG}+3H(\rho_{MG}+p_{MG})=0, \] provided ordinary matter satisfies \[ \dot\rho_m+3H(\rho_m+p_m)=0. \] 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 [1912.03882]. For the simple ansatz \[ u=c_1\dot a-c_2,\qquad v=c_3\dot a-c_4, \] one obtains \[ w_{DE}=-1+\frac{2\lambda\dot H}{c-3\lambda H^2},\qquad c\equiv c_2+c_4. \] In this case, \(\lambda<0\) gives quintessence-like behavior, \(\lambda>0\) gives phantom-like behavior, and \(\lambda=0\) gives exactly \(w_{DE}=-1\), reproducing \(\Lambda\)CDM. Moreover, for \(\lambda=0\) and \(u=c_1\dot a-c_2\), \[ \rho_{MG}=-p_{MG}=\frac{c_2}{2\kappa^2}\equiv\Lambda, \] so the cosmological constant arises geometrically from the connection sector [1912.03882]. ## 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 \(F(R,T)=R+\lambda T\) produce the same hyperbolic functional form for \(a(t)\), \(H(t)\), \(q(t)\), \(r(t)\), and \(s(t)\), 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 \(q\to 1/2\) at high redshift, and an asymptotic de Sitter-like future with \(q\to -1\). The first model has \[ -1\le \omega_{\rm eff}\le 0, \] while the second has \[ -1.031\le \omega_{\rm eff}\le 0. \] Both are transit-phase universes with transition redshift in the range \[ 0.60\) yields phantom-like dark energy, and \(\lambda=0\) gives the exact \(\Lambda\)CDM limit. **Class 2** does not contain \(\Lambda\)CDM 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 [2202.10871]. 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 \(H(t)\) or \(a(t)\) and solves for the deformation functions \(u\) and \(v\) that realize it. In this sense, inflation can be reconstructed from the connection sector analogously to reconstructing an inflaton potential in scalar-field inflation [1912.03882]. Myrzakulov gravity has also been used to realize nonsingular bouncing cosmologies. For \(F(R,T)=R+\lambda T\), suitable choices of \(u(a,\dot a,\ddot a)\) and \(v(a,\dot a)\) 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 \[ P_\zeta=\frac{\sigma G}{36\pi}. \] The bounce solutions are therefore presented as proof-of-principle realizations of regular background evolution and finite scalar perturbations in the curvature–torsion framework [2501.18524]. ## 5. Observational status The first dedicated background-level observational constraints on the core \(F(R,T)\) 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 \(1000\) walkers and \(3500\) steps. In both models the coupling \(\lambda\) was constrained to an interval around zero, with contours slightly shifted toward positive values. Model 1 was found to be very close to \(\Lambda\)CDM, while Model 2 resembled it at low redshift but allowed earlier-time deviations. AIC and BIC favored \(\Lambda\)CDM because of parameter penalization, but the combined DIC criterion found both Myrzakulov models statistically equivalent to \(\Lambda\)CDM, even though Model 2 has no \(\Lambda\)CDM limit [2012.06524]. Independent late-time fits based on exact FLRW solutions used \(H(z)\) 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 \[ -1\le \omega_{de}\le -0.5176, \] with present ages reported as \[ t_0=13.8486_{-0.0640}^{+0.1005}\ \text{Gyr},\qquad t_0=12.0135_{-0.2743}^{+0.6206}\ \text{Gyr}, \] for the \(H(z)\) and Pantheon fits respectively [2401.00686]. A related exact-solution analysis found transition redshifts in the observationally acceptable range \(0.6