Toward the Alleviation of the H₀ Tension in Myrzakulov f(R,T) Gravity

Abstract

In this work, we provide a promising way to alleviate the Hubble tension within the framework of Myrzakulov $f(R,T)$ gravity. The latter incorporates both curvature and torsion under a non-special connection. We consider the $f(R,T)=R+\alpha R^2$ class, which leads to modified Friedmann equations and an effective dark energy sector. Within this class, we make specific connection choices in order to obtain a Hubble function that coincides with that of $\Lambda$CDM at early times while yielding higher $H_0$ values at late times. The reason behind this behavior is that the dark energy equation of state exhibits phantom behavior, which is known to be a sufficient mechanism for alleviating the $H_0$ tension. A full observational comparison with various datasets, including the Cosmic Microwave Background (CMB), is required to test the viability of this scenario. Strictly speaking, the present work does not provide a complete solution to the Hubble tension but rather proposes a promising way to alleviate it.

1. Introduction

The observed accelerated expansion of the Universe, supported by a plethora of cosmological observations such as type Ia supernovae (SNIa), Cosmic Microwave Background (CMB) anisotropies, and baryon acoustic oscillations (BAO), presents a significant challenge to our current understanding of gravity and fundamental physics. While the standard cosmological model, $\Lambda$CDM [1], provides an adequate description of cosmic acceleration through the cosmological constant, it exhibits theoretical and observational issues [2]. These issues have led to various modifications of the underlying theory of gravity [3,4]. One class of theories is obtained within the framework of the standard, curvature-based formulation of gravity, for instance, in $f\left(R\right)$ gravity [5,6,7,8,9,10] or $f\left(G\right)$ gravity [11,12,13,14,15]. Another class of modified gravity is obtained within the torsion formulation of gravity, for example, in $f\left(T\right)$ gravity [16,17,18,19,20,21] or $f(T,T_G)$ gravity [22]. Finally, a third class of modified gravity theories is built within the nonmetricity framework, resulting in $f\left(Q\right)$ gravity [23,24,25,26,27,28,29].

The difference between the above theories lies in the connection that is imposed. In curvature gravity, one uses the standard Levi–Civita connection; in torsion gravity, one uses the Weitzenböck connection; and in nonmetricity gravity, one uses the symmetric teleparallel connection. A more general class of theories is Myrzakulov $f(R,T)$ gravity, which has attracted significant attention due to its approach incorporating both curvature and torsion as dynamical fields [30,31]. In this framework, gravity is formulated in terms of a non-special connection that simultaneously possesses both curvature and torsion, leading to a richer geometric structure with additional degrees of freedom. The theory provides a general function of the curvature scalar R and torsion scalar T in the action, offering a theoretical framework that unifies curvature-based and torsion-based modifications of gravity. Myrzakulov $f(R,T)$ gravity can recover both $f\left(R\right)$ and $f\left(T\right)$ gravity as special cases, and it can be further generalized to allow for various cosmological scenarios, including late-time acceleration and inflationary dynamics [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].

One significant tension in contemporary cosmology is the Hubble tension [47]. This tension relates to the discrepancy between the values of the Hubble constant $H_0$ inferred from early-Universe probes such as Planck CMB observations, and local Universe measurements such as SNIa and Cepheid variable stars. In particular, the Planck collaboration gives $H_0=67.66\pm 0.49$ km/s/Mpc [48], while the SH0ES collaboration finds $H_0=73.17\pm 0.86$ km/s/Mpc [49], a tension at $5.8\sigma$. Hence, the $H_0$ tension indicates a potential problem with the standard $\Lambda$CDM model and has led to the exploration of numerous theoretical solutions. These modifications may include late-time modifications to the expansion history, additional relativistic degrees of freedom, early dark energy models, and new physics beyond the standard model [50,51,52]. Among the proposed solutions, modified gravity theories, in particular, provide a promising way of addressing this issue by altering the effective cosmic expansion rate through new geometrical contributions [53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111]. Note, however, that in [112,113,114], the Hubble tension was solved by a large local void. The local estimates of $H_0$ are measurements of the local gradient, which measures the true background expansion rate only if the Universe is homogeneous on the relevant scales. It is therefore far from certain that a linear regression of redshift with respect to distance in the local Universe is measuring $H_0$ (see also [115]).

In this work, we propose a promising way to alleviate the $H_0$ tension within the framework of Myrzakulov $f(R,T)$ gravity. By considering specific functional forms of $f(R,T)$ that introduce new geometrical effects, we investigate whether the resulting modifications to the Friedmann equations can accommodate a higher value of the Hubble constant, in agreement with local Universe observations. Our analysis aims to provide a viable theoretical foundation for addressing one of the most pressing challenges in modern cosmology. The organization of this work is as follows. In Section 2, we briefly review Myrzakulov $f(R,T)$ gravity. Then, in Section 3, we use it to provide an alleviation of the tension. Finally, Section 4 contains a summary of our results.

2. Myrzakulov $\mathit{f}(\mathit{R},\mathit{T})$ Gravity

Myrzakulov gravity is an extension of general relativity (GR) that incorporates both curvature and torsion as dynamical entities by employing a non-special connection [30]. The fundamental fields of the theory are the tetrad $e_{\mu }^a$ and the connection 1-forms ${\omega }_{b\mu }^a$, which describe parallel transport. The metric tensor is constructed from the tetrad via the relation

$$g_{\mu \nu }={\eta }_{ab}{e}_{\mu }^a{e}_{\nu }^b,$$

(1)

where ${\eta }_{ab}$ is the Minkowski metric. Imposing zero nonmetricity ensures compatibility with the connection, leading to the condition ${\omega }_{abc}=-{\omega }_{bac}$.

The curvature and torsion tensors in Myrzakulov gravity are defined in terms of the connection as follows:

$$R_{b\mu \nu }^a={\partial }_{\mu }{\omega }_{b\nu }^a-{\partial }_{\nu }{\omega }_{b\mu }^a+{\omega }_{c\mu }^a{\omega }_{b\nu }^c-{\omega }_{c\nu }^a{\omega }_{b\mu }^c,$$

(2)

$$T_{\mu \nu }^a={\partial }_{\mu }{e}_{\nu }^a-{\partial }_{\nu }{e}_{\mu }^a+{\omega }_{b\mu }^a{e}_{\nu }^b-{\omega }_{b\nu }^a{e}_{\mu }^b.$$

(3)

In standard Einstein gravity, one uses the Levi–Civita connection ${\Gamma }_{b\nu }^a$, which leads to zero torsion. The Lagrangian of the Einstein–Hilbert action is the Ricci scalar $R^{\left(LC\right)}$ calculated with the Levi–Civita connection, i.e.,

$$\begin{array}{ccc}& & R^{\left(LC\right)}={\eta }^{ab}{e}_a^{\mu }{e}_b^{\nu }\left[{\Gamma }_{\mu \nu ,\lambda }^{\lambda }-{\Gamma }_{\mu \lambda ,\nu }^{\lambda }+{\Gamma }_{\mu \nu }^{\rho }{\Gamma }_{\lambda \rho }^{\lambda }-{\Gamma }_{\mu \lambda }^{\rho }{\Gamma }_{\nu \rho }^{\lambda }\right].\hfill \end{array}$$

(4)

Similarly, in the teleparallel equivalent of general relativity (TEGR), one uses the Weitzenböck connection $W_{\mu \nu }^{\lambda }=e_a^{\lambda }{e}_{\mu ,\nu }^a$, and the Lagrangian of the theory is the torsion scalar $T^{\left(W\right)}$ [22]

$$\begin{array}{ccc}& & T^{\left(W\right)}={\displaystyle \frac{1}{4}}\left(W^{\mu \lambda \nu }-W^{\mu \nu \lambda }\right)\left(W_{\mu \lambda \nu }-W_{\mu \nu \lambda }\right)+{\displaystyle \frac{1}{2}}\left(W^{\mu \lambda \nu }-W^{\mu \nu \lambda }\right)\left(W_{\lambda \mu \nu }-W_{\lambda \nu \mu }\right)\hfill \\ & & -\left(W_{\nu }^{\mu \nu }-W_{\nu }^{\nu \mu }\right)\left(W_{\mu \lambda }^{\lambda }-W_{\lambda \mu }^{\lambda }\right).\hfill \end{array}$$

(5)

Note that in curvature-based modified theories of gravity, such as $f\left(R\right)$ gravity, one uses the Levi–Civita connection and an arbitrary function of the Ricci scalar $R^{\left(LC\right)}$ in the Lagrangian, while in torsion-based modified gravity, one uses the Weitzenböck connection and an arbitrary function of the torsion scalar $T^{\left(W\right)}$ in the Lagrangian [17].

In Myrzakulov gravity [30], one uses a connection with both non-zero curvature and torsion. Therefore, the resulting theory generally introduces additional degrees of freedom, even for a simple Lagrangian. In particular, one uses a non-special connection that incorporates both curvature and torsion, while allowing for a general functional dependence on the corresponding scalars in the Lagrangian. The action of the theory is given by

$$S=\int d^{4}xe\left[{\displaystyle \frac{f(R,T)}{2{\kappa }^2}}+L_m\right],$$

(6)

where $e=\det \left(e_{\mu }^a\right)=\sqrt{-g}$ and ${\kappa }^2=8\pi G$. The function $f(R,T)$ introduces a dependence on both the curvature scalar R and the torsion scalar T, leading to a richer gravitational dynamics. These scalars are given by [31]

$$\begin{array}{ccc}& & T={\displaystyle \frac{1}{4}}{T}^{\mu \nu \lambda }{T}_{\mu \nu \lambda }+{\displaystyle \frac{1}{2}}{T}^{\mu \nu \lambda }{T}_{\lambda \nu \mu }-T_{\nu }^{\nu \mu }{T}_{\lambda \mu }^{\lambda },\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & R=R^{\left(LC\right)}+T-2T_{\nu ;\mu }^{\nu \mu }.\hfill \end{array}$$

(8)

Thus, they can be expressed as

$$\begin{array}{ccc}& & T=T^{\left(W\right)}+v,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & R=R^{\left(LC\right)}+u.\hfill \end{array}$$

(10)

The quantities v and u embed the properties of the chosen connection, with v being a scalar function determined by the tetrad, its first derivative, and the connection itself, while u depends on the tetrad, its first and second derivatives, as well as the first derivative of the connection.

By choosing specific forms for the connection, Myrzakulov gravity can recover well-known theories such as $f\left(R\right)$ gravity when the torsion vanishes and $f\left(T\right)$ gravity when curvature is set to zero. The presence of both torsion and curvature allows for additional degrees of freedom that can influence cosmological dynamics.

In order to derive the field equations, the action is varied with respect to the tetrad, assuming a specific connection. In the cosmological context, the theory is studied under the homogeneous and isotropic Friedmann–Robertson–Walker (FRW) metric,

$$\begin{array}{c}\hfill d{s}^2=d{t}^2-a^2\left(t\right){\delta }_{ij}d{x}^{i}d{x}^j,\end{array}$$

(11)

where $a\left(t\right)$ is the scale factor, which arises from the tetrad $e_{\mu }^a=\mathrm{diag}[1,a\left(t\right),a\left(t\right),a\left(t\right)]$. In this case, we have

$$\begin{array}{ccc}& & R^{\left(LC\right)}=6\left({\displaystyle \frac{\ddot{a}}{a}}+{\displaystyle \frac{{\dot{a}}^2}{a^2}}\right),\hfill \\ & & T^{\left(W\right)}=-6\left({\displaystyle \frac{{\dot{a}}^2}{a^2}}\right).\hfill \end{array}$$

(12)

This minisuperspace approach simplifies the analysis and allows for an effective description of the cosmic evolution within the Myrzakulov framework.

Let us derive the field equations by following the minisuperspace procedure. Following [116,117,118], we introduce the scalars ${\varphi }_1$ and ${\varphi }_2$; thus, action (6) is rewritten as

$$S={\displaystyle \frac{1}{2{\kappa }^2}}\int d^{4}xe\left[F_{{\varphi }_1}({\varphi }_1,{\varphi }_2)(R-{\varphi }_1)+F_{{\varphi }_2}({\varphi }_1,{\varphi }_2)(T-{\varphi }_2)+F({\varphi }_1,{\varphi }_2)+2{\kappa }^2{L}_m\right],$$

(13)

where $F_{{\varphi }_1}({\varphi }_1,{\varphi }_2)=\partial F({\varphi }_1,{\varphi }_2)/\partial {\varphi }_1$ and $F_{{\varphi }_2}({\varphi }_1,{\varphi }_2)=\partial F({\varphi }_1,{\varphi }_2)/\partial {\varphi }_2$.

In the case of FRW geometry, and inserting the minisuperspace expressions, we finally obtain $S=\int Ldt$ [31], with

$$\begin{array}{ccc}& & L=-{\displaystyle \frac{3}{{\kappa }^2}}\left[f({\varphi }_1,{\varphi }_2)+1\right]a{\dot{a}}^2+a^3\left[{\displaystyle \frac{{\dot{\varphi }}_1^2}{2}}+{\displaystyle \frac{{\dot{\varphi }}_2^2}{2}}-V({\varphi }_1,{\varphi }_2)\right]\hfill \\ & & +a^3\left[{\displaystyle \frac{u(a,\dot{a},\ddot{a})+fv(a,\dot{a})}{2{\kappa }^2}}\right]-a^3{\rho }_m\left(a\right)F_{{\varphi }_1}^{-2}({\varphi }_1,{\varphi }_2),\hfill \end{array}$$

(14)

and where

$$\begin{array}{c}\hfill f({\varphi }_1,{\varphi }_2)={\displaystyle \frac{F_{{\varphi }_2}({\varphi }_1,{\varphi }_2)}{F_{{\varphi }_1}^2({\varphi }_1,{\varphi }_2)}},\end{array}$$

(15)

and

$$V({\varphi }_1,{\varphi }_2)={\varphi }_1{F}_{{\varphi }_1}({\varphi }_1,{\varphi }_2)+{\varphi }_2{F}_{{\varphi }_2}({\varphi }_1,{\varphi }_2)-F({\varphi }_1,{\varphi }_2).$$

(16)

Variation with respect to a, ${\varphi }_1$, and ${\varphi }_2$, under the Hamiltonian constraint $\mathcal{H}=\dot{a}\left[{\displaystyle \frac{\partial L}{\partial \dot{a}}}-{\displaystyle \frac{\partial }{\partial t}}{\displaystyle \frac{\partial L}{\partial \ddot{a}}}\right]+\ddot{a}\left({\displaystyle \frac{\partial L}{\partial \ddot{a}}}\right)+{\dot{\varphi }}_1{\displaystyle \frac{\partial L}{\partial {\dot{\varphi }}_1}}+{\dot{\varphi }}_2{\displaystyle \frac{\partial L}{\partial {\dot{\varphi }}_2}}-L=0$, yields [31]

$$\begin{array}{c}3H^2={\displaystyle \frac{{\kappa }^2}{\left(1+f\right)}}\left[{\displaystyle \frac{1}{2}}\left({\dot{\varphi }}_1^2+{\dot{\varphi }}_2^2\right)+V+F_{{\varphi }_1}^{-2}{\rho }_m\right]\hfill \\ +{\displaystyle \frac{1}{\left(1+f\right)}}\left[{\displaystyle \frac{Ha}{2}}\left(u_{\dot{a}}+v_{\dot{a}}f\right)-{\displaystyle \frac{1}{2}}(u+fv)+{\displaystyle \frac{a{u}_{\ddot{a}}}{2}}\left(\dot{H}-2H^2\right)-{\displaystyle \frac{1}{2}}{\dot{u}}_{\ddot{a}}\right],\hfill \end{array}$$

(17)

$$\begin{array}{c}2\dot{H}+3H^2={\displaystyle \frac{{\kappa }^2}{\left(1+f\right)}}\left[-{\displaystyle \frac{1}{2}}\left({\dot{\varphi }}_1^2+{\dot{\varphi }}_2^2\right)+V+F_{{\varphi }_1}^{-2}\left({\rho }_m+{\displaystyle \frac{{\dot{\rho }}_m}{3H}}\right)\right]\hfill \\ +{\displaystyle \frac{1}{\left(1+f\right)}}[{\displaystyle \frac{Ha}{2}}\left(u_{\dot{a}}+v_{\dot{a}}f\right)-2H\dot{f}-{\displaystyle \frac{1}{2}}(u+fv)\hfill \\ +{\displaystyle \frac{a}{6}}\left(-u_a-f{v}_a+{\dot{u}}_{\dot{a}}+f{\dot{v}}_{\dot{a}}+\dot{f}{v}_{\dot{a}}\right)\hfill \\ +{\displaystyle \frac{a{u}_{\ddot{a}}}{2}}\left(\dot{H}+3H^2\right)+aH{\dot{u}}_{\ddot{a}}+{\displaystyle \frac{a}{6}}{\ddot{u}}_{\ddot{a}}],\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & {\ddot{\varphi }}_1=-3H{\dot{\varphi }}_1-V_{{\varphi }_1}-{\displaystyle \frac{3}{{\kappa }^2}}{H}^2{f}_{{\varphi }_1}+{\displaystyle \frac{v}{2{\kappa }^2}}{f}_{{\varphi }_1}+2{\rho }_m{F}_{{\varphi }_1}^{-3}{F}_{{\varphi }_1{\varphi }_1},\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& & {\ddot{\varphi }}_2=-3H{\dot{\varphi }}_2-V_{{\varphi }_2}-{\displaystyle \frac{3}{{\kappa }^2}}{H}^2{f}_{{\varphi }_2}+{\displaystyle \frac{v}{2{\kappa }^2}}{f}_{{\varphi }_2}+2{\rho }_m{F}_{{\varphi }_1}^{-3}{F}_{{\varphi }_1{\varphi }_2},\hfill \end{array}$$

(20)

where the indices ${\varphi }_1$ and ${\varphi }_2$ mark partial derivatives with respect to them.

3. Toward the Alleviation of the ${\mathit{H}}_{\mathbf{0}}$ Tension

In this section, we provide ways to alleviate the $H_0$ tension within the framework of Myrzakulov $f(R,T)$ gravity. The Hubble tension, which refers to the significant discrepancy between local measurements of the Hubble constant $H_0$ and its inferred value from early-Universe observations within the $\Lambda$CDM framework, has led to extensive efforts to find possible solutions. Several model-independent approaches have been proposed to alleviate this tension, primarily focusing on modifying the cosmic expansion history in a way that reconciles the different observational datasets [50]. It is known that one of the sufficient ways to obtain higher $H_0$ values compared to $\Lambda$CDM cosmology is by achieving an effective phantom behavior [50,119]. Since phantom dark energy is possible in Myrzakulov gravity, in the following, we examine specific models with this property. Note that in the literature, there is a discussion about whether phantom behavior has theoretical issues and leads to instabilities, since the energy density of the phantom scalar field is unbounded from below and thus it may take negative values. Note, however, that at the classical level, this is not a problem; nevertheless, issues might arise concerning the quantum field theoretical description of phantom fields, since unbounded-from-below states may lead to problems. However, in the literature, there have been various attempts to construct a phantom theory consistent with the basic requirements of quantum field theory, for instance, considering that the phantom fields arise as effective descriptions of more fundamental canonical theories [120,121].

As a straightforward example, we consider the model

$$f(R,T)=R+\alpha R^2.$$

(21)

We choose to consider this ansatz since it is the simplest deviation from standard form, quantified by the parameter $\alpha$, and it is expected to hold in every realistic model as a first (Taylor-expansion) approximation. Additionally, it is known that a similar model in $f\left(R\right)$ gravity, namely the Starobinsky one, is already among the most successful modified gravity models; hence, it is interesting to examine it within the framework of $f(R,T)$ gravity too. In this case, we have $F_{{\varphi }_1}=1+2\alpha {\varphi }_1$, $V({\varphi }_1,{\varphi }_2)=\alpha {\varphi }_1^2$, and $f({\varphi }_1,{\varphi }_2)=0$. Thus, Equations (17)–(20) become [31]

$$3H^2={\kappa }^2\left[{\displaystyle \frac{{\rho }_m}{{(1+2\alpha {\varphi }_1)}^2}}+{\displaystyle \frac{1}{2}}{\dot{\varphi }}_1^2+\alpha {\varphi }_1^2\right]+\left({\displaystyle \frac{Ha{u}_{\dot{a}}-u}{2}}\right)+{\displaystyle \frac{a{u}_{\ddot{a}}}{2}}\left(\dot{H}-2H^2\right)-{\displaystyle \frac{1}{2}}{\dot{u}}_{\ddot{a}},$$

(22)

$$\begin{array}{ccc}& & 2\dot{H}+3H^2={\kappa }^2\left(-{\displaystyle \frac{1}{2}}{\dot{\varphi }}_1^2+\alpha {\varphi }_1^2\right)+\left[{\displaystyle \frac{Ha{u}_{\dot{a}}-u}{2}}-{\displaystyle \frac{a}{6}}\left(u_a-{\dot{u}}_{\dot{a}}\right)\right]\hfill \\ & & +{\displaystyle \frac{a{u}_{\ddot{a}}}{2}}\left(\dot{H}+3H^2\right)+aH{\dot{u}}_{\ddot{a}}+{\displaystyle \frac{a}{6}}{\ddot{u}}_{\ddot{a}},\hfill \end{array}$$

(23)

$$\begin{array}{c}\hfill {\ddot{\varphi }}_1=-3H{\dot{\varphi }}_1-2\alpha {\varphi }_1+{\displaystyle \frac{4\alpha {\rho }_m}{{(1+2\alpha {\varphi }_1)}^3}}.\end{array}$$

(24)

As we can see, we can rewrite the two Friedmann equations in the standard form

$$\begin{array}{ccc}\hfill 3H^2& =& {\kappa }^2\left({\rho }_m+{\rho }_{de}\right)\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill 2\dot{H}+3H^2& =& -{\kappa }^2\left(p_m+p_{de}\right),\hfill \end{array}$$

(26)

by defining the energy density

$${\rho }_{de}={(1+2\alpha {\varphi }_1)}^2\left[{\displaystyle \frac{1}{2}}{\dot{\varphi }}_1^2+\alpha {\varphi }_1^2+{\displaystyle \frac{Ha{u}_{\dot{a}}-u}{2{\kappa }^2}}+{\displaystyle \frac{a{u}_{\ddot{a}}\left(\dot{H}-2H^2\right)-{\dot{u}}_{\ddot{a}}}{2{\kappa }^2}}\right]-{\displaystyle \frac{12\alpha {\varphi }_1}{{\kappa }^2}}(1+\alpha {\varphi }_1)H^2,$$

(27)

and pressure

$$p_{de}={\displaystyle \frac{1}{2}}{\dot{\varphi }}_1^2-\alpha {\varphi }_1^2-{\displaystyle \frac{1}{{\kappa }^2}}\left[{\displaystyle \frac{Ha{u}_{\dot{a}}-u}{2}}-{\displaystyle \frac{a}{6}}\left(u_a-{\dot{u}}_{\dot{a}}\right)+{\displaystyle \frac{a{u}_{\ddot{a}}}{2}}\left(\dot{H}+3H^2\right)+aH{\dot{u}}_{\ddot{a}}+{\displaystyle \frac{a}{6}}{\ddot{u}}_{\ddot{a}}\right].$$

(28)

Thus, the dark energy equation of state is

$$\begin{array}{ccc}& & w_{de}\equiv {\displaystyle \frac{p_{de}}{{\rho }_{de}}}.\hfill \end{array}$$

(29)

Finally, we make the choice $u=c_1\dot{a}-c_2$, with $c_1$ and $c_2$ the model parameters. Such a choice is made for phenomenological reasons, and as shown in [31], it can describe an accelerating Universe, allowing also for some analytical expressions. In this case, we have

$$\begin{array}{ccc}& & {\rho }_{de}={(1+2\alpha {\varphi }_1)}^2\left[{\displaystyle \frac{1}{2}}{\dot{\varphi }}_1^2+\alpha {\varphi }_1^2+{\displaystyle \frac{c_2}{{\kappa }^2}}\right]-{\displaystyle \frac{12\alpha {\varphi }_1}{{\kappa }^2}}(1+\alpha {\varphi }_1)H^2,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & p_{de}={\displaystyle \frac{1}{2}}{\dot{\varphi }}_1^2-\alpha {\varphi }_1^2-{\displaystyle \frac{c_2}{{\kappa }^2}}.\hfill \end{array}$$

(31)

It proves convenient to use the redshift $z=-1+a_0/a$ instead of the time variable. We fix the scale factor at present as $a_0=1$, so we have $\dot{H}=-(1+z)H\left(z\right)H^{\prime }\left(z\right)$, where the prime denotes derivatives with respect to z. The Hubble function behavior of $\Lambda$CDM cosmology in terms of the redshift is

$$\begin{array}{c}\hfill H_{\Lambda \mathrm{CDM}}\left(z\right)\equiv H_0\sqrt{{\Omega }_{m0}{(1+z)}^3+1-{\Omega }_{m0}},\end{array}$$

(32)

where ${\Omega }_{m0}$ is the current value of the matter density parameter ${\Omega }_m\equiv {\displaystyle \frac{8\pi G{\rho }_m}{3H^2}}$. Our strategy is as follows. We choose model parameters for which $H\left(z\right)$ from (25) coincides with $H_{\Lambda \mathrm{CDM}}\left(z\right)$ at very high redshifts, i.e., at $z=z_{\mathrm{CMB}}\approx 1100$, but today gives $H(z\to 0)>H_{\Lambda \mathrm{CDM}}(z\to 0)$.

In Figure 1, we show the normalized Hubble function $H\left(z\right)/{(1+z)}^{3/2}$ for $\Lambda$CDM cosmology and for our model of Myrzakulov gravity. Indeed, our model coincides with $\Lambda$CDM cosmology at high redshifts, while at small redshifts, we obtain higher $H_0$ values. Specifically, $H_0$ depends on $\alpha$, and it can be around $H_0\approx 72.5$ km/s/Mpc for $\alpha =0.2$.

We now make some additional comments. In the literature, there is a discussion about whether late-time solutions or early-time solutions alone are able to solve the $H_0$ tension [122]. Although one can construct models in which the Hubble function deviates from $\Lambda$CDM at both early and late times, such a condition is sufficient but not necessary. As shown in a recent review [47], there are many theories, models, and scenarios that alleviate the $H_0$ tension just by obtaining higher $H\left(z\right)$ values at small redshifts, without deviating from $\Lambda$CDM at early times. In this context, our proposed scenario also falls within this class of models and can thus serve as a candidate for at least a partial alleviation of the tension.

As we discussed above, the mechanism behind the alleviation of the $H_0$ tension in the scenario at hand is that the dark energy equation of state exhibits phantom behavior. In order to illustrate this more clearly, we present $w_{de}$ for the above example in Figure 2. Indeed, we verify that $w_{de}$ lies in the phantom regime. Note that according to the recently released DESI DR2 dataset [123], $w_{de}$ lay in the phantom regime in the past and crossed the phantom divide from below at small redshifts (less than 0.4), reaching values slightly larger than $-1$ today, which provides strong evidence for dynamical dark energy. In the above analysis, we showed that within our model, it is possible to obtain a phantom $w_{de}$ in the past, which increases and approaches values very close to −1 at small redshifts. Hence, the essential features of the DESI dataset can be satisfied. Naturally, a detailed comparison with the full dataset, including the extraction of contour plots for various model parameters, will offer more information on the capabilities of the model.

We conclude this section with an important comment. Since the scenario under consideration does not alter early-Universe physics and focuses on late-time modifications, mainly at redshifts smaller than 1, the CMB features should remain relatively close to those predicted by Planck. In particular, the angular scale of the sound horizon at recombination is defined as

$${\theta }_{*}={\displaystyle \frac{r_s\left(z_{*}\right)}{D_A\left(z_{*}\right)}}={\displaystyle \frac{r_s\left(z_{*}\right)}{{\chi }_{*}/(1+z_{*})}},$$

(33)

where $r_s\left(z_{*}\right)$ is the comoving sound horizon at the redshift of recombination $z_{*}$ and $D_A\left(z_{*}\right)$ is the angular diameter distance to recombination. Using $z_{*}=1089$ and assuming a flat Universe, the comoving distance to recombination is

$${\chi }_{*}={\int }_0^{z_{*}}{\displaystyle \frac{dz}{H\left(z\right)}},$$

(34)

For models with no new physics at early times, $r_s\left(z_{*}\right)$ remains essentially unchanged, as it depends primarily on the pre-recombination evolution of the baryon–photon fluid. Therefore, ensuring that ${\chi }_{*}$ is close to the $\Lambda$CDM value guarantees that ${\theta }_{*}$ remains within observational constraints.

Let us calculate the comoving distance to recombination. For convenience, and as a first approximation, we neglect the radiation sector, since it has not been considered in this work. In our scenario, using the numerically obtained $H\left(z\right)$ for the case $\alpha =0.2$, $c=2$, which yields the largest $H_0$ value and matches the Planck result at high redshifts while differing only at $z\lesssim 1$, we obtain

$${\chi }_{*}^{\mathrm{model}}\approx 13850\mathrm{Mpc}.$$

(35)

On the other hand, for the Planck 2018 $\Lambda$CDM baseline values ($H_0=67.4$ km/s/Mpc, ${\Omega }_m=0.315$), and neglecting radiation, we obtain

$${\chi }_{*}^{\Lambda \mathrm{CDM}}\approx 13940\mathrm{Mpc}.$$

(36)

Hence, we acquire a relative deviation of approximately

$${\displaystyle \frac{|{\chi }_{*}^{\Lambda \mathrm{CDM}}-{\chi }_{*}^{\mathrm{model}}|}{{\chi }_{*}^{\Lambda \mathrm{CDM}}}}\approx 0.4\%.$$

(37)

This level of deviation can lie within the tight observational constraint on ${\theta }_{*}$ provided by Planck 2018:

$${\theta }_{*}^{\mathrm{Planck}}=(1.04109\pm 0.00030)\times {10}^{-2}\mathrm{rad}.$$

A ∼0.4% deviation in ${\chi }_{*}$ (with $r_s$ fixed) translates to a similar percent-level deviation in ${\theta }_{*}$, which remains within current observational bounds. Therefore, the scenario under consideration does not significantly change the angular scale of the CMB power spectrum and may thus serve as a viable late-time solution to the Hubble tension while remaining consistent with Planck measurements. Naturally, it would be preferable to reduce this percentage difference even further, possibly by incorporating minor early-Universe physics changes as well.

4. Conclusions

In this work, we explored Myrzakulov $f(R,T)$ gravity, an extension of general relativity that incorporates both curvature and torsion as dynamical elements through the use of a non-special connection. This framework allows for a more generalized description of gravitational interactions, offering additional degrees of freedom that can modify the cosmic expansion history. By considering a function of both the curvature scalar R and the torsion scalar T, Myrzakulov gravity unifies curvature-based and torsion-based modifications, recovering well-known theories such as $f\left(R\right)$ and $f\left(T\right)$ gravity as limiting cases.

The Hubble tension, a significant challenge in modern cosmology, refers to the discrepancy between local measurements of the Hubble constant $H_0$ and its inferred value from early-Universe observations. This tension, currently exceeding $5\sigma$, may indicate a possible problem with the standard cosmological model in fully describing cosmic evolution. Various solutions have been proposed, including modifications to the expansion history, the introduction of additional relativistic species, and exotic dark energy models. Modified gravity theories constitute a significant class of such solutions by introducing new geometrical effects that alter the late-time expansion rate, potentially leading to higher $H_0$ values.

Within the framework of Myrzakulov $f(R,T)$ gravity, we examined specific functional forms of $f(R,T)$ that introduce specific contributions capable of alleviating the $H_0$ tension. In particular, we examined the $f(R,T)=R+\alpha R^2$ class, which leads to modified Friedmann equations and an effective dark energy sector. Within this class, we employed specific connection choices in order to obtain a Hubble function that coincides with that of $\Lambda$CDM at early times but yields higher $H_0$ values at late times. This behavior is driven by a phantom-like dark energy equation of state, which is known to be a promising mechanism for alleviating the $H_0$ tension. As reviewed in [47], similar behavior arises in other modified gravity theories, although not in broad classes of them.

Naturally, further investigations are required in order to assess the viability of Myrzakulov gravity in light of current and future observational constraints. In particular, one should perform a full observational comparison with various datasets, such as type Ia supernovae (SNIa), $H\left(z\right)$, baryon acoustic oscillations (BAO), and Cosmic Microwave Background (CMB) observations, which would also allow for the construction of the corresponding likelihood contours for the model parameters, including $\alpha$ and c. In particular, a full comparison with the CMB data would be a crucial test of the scenario’s viability, since, strictly speaking, the present work does not solve the Hubble tension but rather proposes a promising way to alleviate it. However, such a detailed analysis lies beyond the scope of the present work and will be addressed in future projects. Additionally, one could explore other functional forms of $f(R,T)$ and their impact on cosmic evolution, which is also left for future work.