Non-Linear f(Q,T) Gravity and the Late-Time Acceleration of the Universe

Abstract

This study examines cosmic acceleration in the framework of $f(Q,T)$ gravity and compares it to the standard $\Lambda$CDM model. It considers a generalized nonlinear form of the nonmetricity, expressed as $f(Q,T)=-Q+{\alpha }_0{Q}^2/H_0^2+{\beta }_{0}T+{\eta }_0$, where ${\alpha }_0,{\beta }_0,$ and ${\eta }_0$ are constants, and $H_0$ is the current value of the Hubble constant. In the solution process, we did not rely on any additional conditions to solve the field equations; instead, the field equations were reduced to a time-dependent closed system of nonlinear first-order coupled differential equations for H and $\rho$. Subsequently, these differential equations were converted to the redshift space for numerical integration alongside the Runge–Kutta method. Furthermore, the study demonstrates that the deceleration parameter q changes sign from being positive in an early period of time at high redshift values to a negative value, passing through a transitional redshift $z_t\approx [0.766,0.769,0.771]$ and $z_t\approx [0.521,0.770,1.010]$, reaching their current values at $q_0=[-0.61,-0.60,-0.59]$ and $[-0.455,-0.595,-0.694]$ for different values of ${\beta }_0$ and ${\alpha }_0$, respectively. Similarly, the effective equation of state $w_{eff}$ shifted from the matter-dominated phase $w_{eff}=0$ at high redshift to a quintessence-like behavior at low redshift. Moreover, a super-accelerated or phantom-like regime with $q_0\simeq -1.59$ and $w_{\mathrm{eff},0}\simeq -1.40$ was obtained when ${\alpha }_0=0.55$ and ${\beta }_0=0.60$ were employed. The model analysis reveals that the universe is presently experiencing an accelerating expansion phase, propelled by a quintessence-type and phantom-like dark energy component, as corroborated by the Om(z) diagnostic test. The results obtained were strongly consistent with the concordance $\Lambda$CDM model.

1. Introduction

Since the discovery of the universe’s expansion by Edwin Hubble, scientists have attributed this expansion to an unknown type of force, named dark energy (DE). DE is characterized by its negative pressure $-p$, expressed by Einstein’s cosmological constant $\Lambda$ in the general theory of relativity (GR). Although multiple investigations have validated its existence [1,2,3,4,5,6,7,8], the essence of DE continues to be enigmatic. This enigma persists as one of the most significant unresolved issues in contemporary cosmology. The problem related to the cosmological constant is beyond the scope of this article. For this, the reader is advised to consult [9]. Recently, the issue of DE has been approached in two ways: through dynamical DE models, such as Chaplygin gas, the Holographic DE model, agegraphic DE models, k-essence, quintessence, tachyon [10,11,12,13,14,15,16], etc., or employing alternative theories of gravity, which in turn involve modifying the Einstein–Hilbert action of general relativity (GR). The modifications include $f\left(R\right)$ [17]. Here, the Ricci scalar R in Einstein’s Hilbert action of GR is replaced by the function $f\left(R\right)$, resulting in a fourth-order gravitational theory. f($\mathcal{T}$) theories [18] have been proposed as an alternative to $f\left(R\right)$ gravity models. In these models, the universe’s late-time acceleration expansion is induced by spacetime torsion $\mathcal{T}$, and the field equations are consistently second order. These theories, grounded in modified teleparallel gravity, present a promising framework for tackling the complexity inherent in $f\left(R\right)$ theories. Numerous authors [19,20,21,22] have investigated the cosmological implications of f($\mathcal{T}$) gravity utilizing current observational data. In $f(R,T)$ theories of gravity [23,24], the gravitational Lagrangian is defined as an arbitrary function of the Ricci scalar R and the trace of the stress–energy tensor T. $f(R,L_m)$ gravity [25] modifies the Einstein–Hilbert action of GR by substituting it with a function of the Ricci scalar R and the matter Lagrangian density $L_m$, rather than using R alone. Additional revisions also encompass $f\left(G\right)$ [26] and $f(R,G)$ gravity [27,28]. In these theories, the gravitational action of GR is integrated with the effects of the Gauss–Bonnet G and both the Ricci scalars R and G, correspondingly. The modifications involve the generalization of symmetrical teleparallel gravity $f\left(Q\right)$, where Q represents the nonmetricity scalar [29]. This theory is also extended to $f(Q,T)$, wherein the Einstein–Hilbert gravitational action is substituted with an arbitrary function f of the nonmetricity Q and the trace of the matter–energy–momentum tensor T [30]. The cosmological aspects of $f(Q,T)$ gravity have been studied by several authors. For example, the work in [31] established the free mode parameter for a nonlinear nonmetricity model within the framework of $f(Q,T)$ gravity utilizing observational data from the Hubble, Pantheon, and BAO datasets. The authors of [32] examined the cosmological implications and constraints of Weyl-type $f(Q,T)$ gravity. This study employed a parameterization of the deceleration parameter to solve the field equations [33]. The solutions to the Friedmann equations in $f(Q,T)$ gravity were derived by introducing a constant jerk parameter, which was subsequently employed to analyze the evolution of other kinematic variables, including the deceleration parameter, energy density, equation of state parameter, and various energy conditions. The work in [34] considered two different forms of a linear function of $f(Q,T)=m{Q}^n+bT$ and $f(Q,T)=Q^{n+1}+bT$, where n, m, and b are constants to constrain the model parameters using different energy conditions. The optimal values for the parameters b, m, and n are constrained using the recently published 1048 Pantheon sample. Additionally, the work in [35] employed a power-law scaling factor $a\left(t\right)$ to evaluate the viability of the model, utilizing 57 data points from Hubble and 580 data points from the Union $2.1$ compilation of supernovae to constrain the model’s free parameters. Furthermore, the work in [36] established the correct equation for the balance of energy in the covariant version of the $f(Q,T)$ theory of gravity. A set of distinct forms of $f(Q,T)$ have been examined to assess the physical efficacy of these models in characterizing diverse cosmic epochs through a dynamical system analysis within a spatially flat Friedmann–Lemaitre–Robertson–Walker spacetime. Using the most recent Hubble and Pantheon+ data and MCMC analysis, the authors of [37] limited the free parameters of the cosmological models in the framework of a generalized form of $f(Q,T)=-{\lambda }_1{Q}^m-{\lambda }_2{T}^2$ theory of gravity, where ${\lambda }_1,{\lambda }_2$, and m are the model parameters. Furthermore, the model parameters were additionally validated using the $BAO$ dataset. In their proposed model, the cosmos exhibits a transitional period of acceleration, shifting from early deceleration to a later accelerating phase. In the same vein, the Om$\left(z\right)$ diagnostics test indicates that the model is in a phase dominated by a phantom field. Again, in [38], cosmic acceleration was examined within the framework of $f(Q,T)=\alpha Q^m+\beta T$ for an isotropic and homogeneous spacetime. Two models and their evolutionary behaviors were examined based on the selection of the free parameters $\alpha$ and m. The geometrical parameters and equation of state (EoS) were determined to be within the optimal range of cosmological measurements. Ultimately, the stability of the models was examined using dynamical system analysis. Similarly, in [39], an accelerating cosmological model of the universe was described within the framework of extended symmetric teleparallel gravity, characterized by the logarithmic form $f(Q,T)=-Q+\beta log{\displaystyle \frac{Q}{Q_0}}+\gamma T$, where $\beta$ and $\gamma$ are free parameters. The field equations were resolved utilizing the Hubble characterization: $H\left(z\right)=H_0{[\alpha +(1-\alpha ){(1+z)}^n]}^{1/32}$, where $H_0$ and n are constants. Subsequently, the Hubble, Baryon acoustic oscillations (BAO), and Type Ia supernovae (SNe Ia) datasets were employed to constrain the free parameters of the models, from which the cosmographic and dynamical parameters were derived. The most intriguing discovery is that the model exhibits the quintessence behavior of the universe in its latter stages, violating the strong energy condition, and its stability was examined using dynamical system analysis.

Inspired by the preceding discussion, this work examines a generalized nonlinear nonmetricity model, expressed as $f(Q,T)=-Q+{\alpha }_0{Q}^2/H_0^2+{\beta }_{0}T+{\eta }_0$, to investigate cosmic acceleration within the framework of $f(Q,T)$ gravity. The adopted approach involves reducing the field equations to a closed system of nonlinear first-order coupled differential equations that can be integrated numerically. This method does not rely on any additional conditions or parameterization to resolve the system of equations.

The paper is organized as follows: Section 2 presents the basic formulations of $f(Q,T)$ gravity and their modified field equations. In Section 3, we investigate the implications in the context of nonlinear model $f(Q,T)$ gravity. Section 4 presents a comparison of our nonlinear $f(Q,T)$ cosmological model with several alternative cosmological models examined in recent years. Section 5 provides a brief introduction to the numerical method employed. The discussion, findings, and results are presented in the conclusion in Section 6.

2. f(Q,T) Gravity Theory

This section outlines the derivations of the modified theory of gravity $f(Q,T)$ introduced by [30]. The generalized Einstein–Herbert action of $f(Q,T)$ gravity is expressed as

$$S=\int \sqrt{-g}{d}^{4}x\left({\displaystyle \frac{1}{16\pi }}f(Q,T)+L_m\right),$$

(1)

where $f(Q,T)$ is a function that associates the trace of the energy–momentum tensor T with its nonmetricity Q, $L_m$ is the the matter Lagrangian density, and $g=det\left(g_{\mu \nu }\right)$ is the metric tensor determinant. The nonmetricity term Q is defined in terms of the disformation tensor $L_{\alpha \gamma }^{\beta }$ as [29]

$$Q\equiv -g^{\mu \nu }(L_{\alpha \mu }^{\beta }{L}_{\nu \beta }^{\alpha }-L_{\alpha \beta }^{\beta }{L}_{\mu \nu }^{\alpha }),$$

(2)

where $L_{\alpha \gamma }^{\beta }$ satisfies

$$\begin{array}{ccc}\hfill L_{\alpha \gamma }^{\beta }& =& -{\displaystyle \frac{1}{2}}{g}^{\beta \eta }({\nabla }_{\gamma }{g}_{\alpha \eta }+{\nabla }_{\alpha }{g}_{\eta \gamma }-{\nabla }_{\eta }{g}_{\alpha \gamma }),\hfill \\ & =& {\displaystyle \frac{1}{2}}{g}^{\beta \eta }\left(Q_{\gamma \alpha \eta }+Q_{\alpha \eta \gamma }-Q_{\eta \alpha \gamma }\right),\hfill \\ & =& {L^{\beta }}_{\gamma \alpha }.\hfill \end{array}$$

(3)

It is deduced that $L_{\alpha \gamma }^{\beta }$ is symmetric, swapping the index $\alpha$ and $\gamma$. Here, $Q_{\eta \alpha \gamma }$ represents the tensorial form of nonmetricity Q, expressed by the following equation:

$$Q_{\gamma \mu \nu }=-{\nabla }_{\gamma }{g}_{\mu \nu }=-{\partial }_{\gamma }{g}_{\mu \nu }+g_{\nu \sigma }{{\tilde{\Gamma }}^{\sigma }}_{\mu \gamma }+g_{\sigma \mu }{{\tilde{\Gamma }}^{\sigma }}_{\nu \gamma },$$

(4)

and the scalar of nonmetricity is defined as [29]

$$\begin{array}{ccc}\hfill Q& =& -Q_{\beta \mu \nu }{P}^{\beta \mu \nu }=-{\displaystyle \frac{1}{4}}(-Q^{\beta \nu \rho }{Q}_{\beta \nu \rho }+2Q^{\beta \nu \rho }{Q}_{\rho \beta \nu }\hfill \\ & & -2Q^{\rho }{\tilde{Q}}_{\rho }+Q^{\rho }{Q}_{\rho }),\hfill \end{array}$$

(5)

where ${{\tilde{\Gamma }}^{\sigma }}_{\mu \gamma }$ is the Weyl–Cartan connection [23], and $Q_{\beta }=g^{\mu \nu }{Q}_{\beta \mu \nu }$ and ${\tilde{Q}}_{\beta }=g^{\mu \nu }{Q}_{\mu \beta \nu }$ are the vector contractions/or trace vectors of the nonmetricity tensor $Q_{\beta \mu \nu }$. Moreover, the conjugate of the nonmetricity can also be defined as

$$\begin{array}{ccc}\hfill & & P_{\mu \nu }^{\beta }\equiv {\displaystyle \frac{1}{4}}[-Q_{\mu \nu }^{\beta }+2Q_{\left(\mu \nu \right)}^{\beta }+Q^{\beta }{g}_{\mu \nu }-{\tilde{Q}}^{\beta }{g}_{\mu \nu }\hfill \\ \hfill & & -{\delta }_{(\mu }^{\beta }{Q}_{\nu )}]=-{\displaystyle \frac{1}{2}}{L}_{\mu \nu }^{\beta }+{\displaystyle \frac{1}{4}}\left(Q^{\beta }-{\tilde{Q}}^{\beta }\right)g_{\mu \nu }-{\displaystyle \frac{1}{4}}{\delta }_{(\mu }^{\beta }{Q}_{\nu )}.\hfill \end{array}$$

(6)

After presenting the fundamental components of this theory, the field equations are derived by varying the action in Equation (1) with respect to the metric tensor $g_{\mu \nu }$ as follows:

$$\begin{array}{c}\hfill -{\displaystyle \frac{2}{\sqrt{-g}}}{\nabla }_{\beta }(f_Q\sqrt{-g}{P^{\beta }}_{\mu \nu }-{\displaystyle \frac{1}{2}}f{g}_{\mu \nu }+f_T(T_{\mu \nu }+{\mathrm{\Theta }}_{\mu \nu })\\ \hfill -f_Q(P_{\mu \beta \alpha }{Q_{\nu }}^{\beta \alpha }-2{Q_{\mu }}^{\beta \alpha }{P}_{\beta \alpha \nu }))=8\pi T_{\mu \nu }.\end{array}$$

(7)

Here, $f_Q$ and $f_T$ denote the derivatives of the function $f=f(Q,T)$ with respect to nonmetricity Q and the trace of the energy–momentum tensor T, respectively. $T_{\mu \nu }$ represents the energy–momentum tensor that delineates the dispersion of matter within the cosmos, expressed as

$$T_{\mu \nu }=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{L}_m\right)}{\delta g^{\mu \nu }}}.$$

(8)

Furthermore, the variation in the energy–momentum tensor $T_{\mu \nu }$ regarding the metric tensor $g_{\alpha \beta }$ satisfies

$${\displaystyle \frac{\delta g^{\mu \nu }{T}_{\mu \nu }}{\delta g^{\alpha \beta }}}=T_{\alpha \beta }+{\mathrm{\Theta }}_{\alpha \beta },$$

(9)

with

$${\mathrm{\Theta }}_{\alpha \beta }=g^{\mu \nu }{\displaystyle \frac{\delta T_{\mu \nu }}{\delta g^{\alpha \beta }}}$$

(10)

being the effective energy–momentum density of matter and energy of the cosmic fluid.

Spacetime and Modified Field Equations (MFEs)

Throughout this subsection, the homogeneous, isotropic, and spatial universe, whose geometry is described as a flat FLRW metric, is assumed:

$$d{s}^2=-N^2\left(t\right)d{t}^2+a^2\left(t\right)\left[d{x}^2+d{y}^2+d{z}^2\right],$$

(11)

where $a\left(t\right)$ is a time-dependent scale factor of the universe that measures the rate of cosmic expansion, and $N\left(t\right)$ is the lapse function; the units $G=c=1$ are also used. In accordance with [40], the use of the coincident gauge, defined by ${\Gamma }_{\mu \nu }^{\alpha }=0$, results in the forfeiture of diffeomorphism gauge symmetry within the general framework. Consequently, the liberty to choose time parameterizations, usually permitted in general relativity, is no longer accessible. Nonetheless, as Q exhibits a residual time reparameterization invariance as articulated in [29], we can set $N\left(t\right)=1$ for the sake of simplicity. We also consider that the universe is filled by a perfect fluid, whose distribution is given by the following energy–momentum tensor:

$$T_{\nu }^{\mu }=diag(-\rho ,p,p,p).$$

Moreover, the nonmetricity scalar Q for this type of metric is derived and given as $Q=6H^2$, where $H=\dot{a}/a$ is the Hubble parameter, and $a=a\left(t\right)$ is the universe scale factor. The generalized Friedmann equations are given below by using the metric (11) and the field Equation (7):

$$\kappa \rho ={\displaystyle \frac{f}{2}}-6F{H}^2-{\displaystyle \frac{2\tilde{G}}{1+\tilde{G}}}(\dot{F}H+F\dot{H}),$$

(12)

$$\kappa p=-{\displaystyle \frac{f}{2}}+6F{H}^2+2(\dot{F}H+F\dot{H}),$$

(13)

where $\kappa =8\pi$, the dot (·) denotes a derivative with respect to time, and the symbols $F=f_Q$ and $\kappa \tilde{G}=f_T$, respectively, signify differentiation with respect to Q and T.

On the other hand, the generalized energy balance equation is given by

$$\dot{\rho }+3H(\rho +p)={\displaystyle \frac{\tilde{G}\left[\dot{S}-\frac{(2+3\tilde{G})}{(1+\tilde{G})}\frac{\dot{\tilde{G}}}{\tilde{G}}S+6HS\right]}{2\kappa (1+\tilde{G})(1+2\tilde{G})}},$$

(14)

where $S=2\dot{F}H+2F(\dot{H}-H\tilde{T})$, and $\tilde{T}=\dot{N}/N$ are the lapse functions. For more detailed derivations, the reader is directed to [23]. The analogous general relativity formulation of (12) and (13) is given by

$$3H^2=8\pi {\rho }_{eff}={\displaystyle \frac{f}{4F}}-{\displaystyle \frac{4\pi }{F}}\left[(1+\tilde{G})\rho +\tilde{G}p\right],$$

(15)

and

$$\begin{array}{c}\hfill 2\dot{H}+3H^2=-8\pi p_{eff}={\displaystyle \frac{f}{4F}}-{\displaystyle \frac{2\dot{F}H}{F}}+\\ \hfill {\displaystyle \frac{4\pi }{F}}\left[(1+\tilde{G})\rho +(2+\tilde{G})p\right],\end{array}$$

(16)

where ${\rho }_{eff}$, and $p_{eff}$ are the effective pressure and density, respectively. Equations (15) and (16) can be re-arranged for $\dot{H}$ as

$$2\dot{H}=-2{\displaystyle \frac{\dot{F}}{F}}H+{\displaystyle \frac{\kappa }{F}}(1+\tilde{G})(\rho +p).$$

(17)

The deceleration parameter (DP) is a significant cosmological quantity that can be used to measure the accelerating/decelerating of the universe’s history. The DP is given by

$$q=-1-{\displaystyle \frac{\dot{H}}{H^2}}={\displaystyle \frac{1}{2}}\left(1+3w_{eff}\right),$$

(18)

where $w_{eff}=p_{eff}/{\rho }_{eff}$ is an effective equation of state. The universe demonstrates acceleration when $q<0$ and deceleration when $q>0$, whereas $q=0$ signifies a linear (or coasting) expansion, corresponding to $\ddot{a}=0$. Consequently, the third-order time derivative of the universe’s scale factor is dimensionless and denoted as the jerk parameter j, described by the equation

$$j={\displaystyle \frac{\stackrel{⃛}{a}}{a{H}^3}}.$$

(19)

The significance of the jerk lies in the fact that it initiated the rate of change of the universe’s acceleration; in other words, this indicates a shift from a decelerating expansion phase to an accelerating cosmic expansion.

3. Nonlinear Model of f(Q,T)

In this work, the most general version of $f(Q,T)$ is adopted, as introduced by [41]. This model posits a distinction from [37,38] that enables us to retrieve the $\Lambda$CDM model when ${\alpha }_0=0={\beta }_0$. The expression is provided by following formula:

$$f(Q,T)=-Q+{\displaystyle \frac{{\alpha }_0}{H_0^2}}{Q}^2+{\beta }_{0}T+{\eta }_0,$$

(20)

where ${\alpha }_0$, ${\beta }_0$, ${\eta }_0$, and $H_0$ are the free model’s and Hubble parameters, respectively. For this model, $F=f_Q=-1+2{\alpha }_{0}Q/H_0^2$, $\dot{F}={\dot{f}}_Q=2{\alpha }_0\dot{Q}/H_0^2$, $\kappa \tilde{G}=f_T={\beta }_0$, $Q=6H^2$, and $\kappa =8\pi$; then, the generalized Friedman Equation (17) yields

$$\dot{H}={\displaystyle \frac{(\kappa +{\beta }_0)(p+\rho )H_0^2}{72{\alpha }_0{H}^2-2H_0^2}}.$$

(21)

Equation (21) is a time-dependent first-order coupled differential equation that measures the universe expansion rate in terms of the energy density $\rho$, the pressure p, and the free parameters ${\alpha }_0$ and ${\beta }_0$. If ${\beta }_0>0$ and ${\alpha }_0>0$, $\dot{H}$ is positive $+ve$, and the expansion of the universe is accelerating. If ${\beta }_0>0$ and ${\alpha }_0<0$, $\dot{H}$ is negative $-ve$; in this case, the expansion of the universe is decelerating. If ${\beta }_0=-\kappa$ and $\dot{H}=0$, then the universe undergoes an exponential expansion. However, the general relativity limits are recovered when ${\alpha }_0=0$ and ${\beta }_0=0$, indicating that $\dot{H}=-\kappa (p+\rho )/2$. Equation (21) cannot be solved unless $\rho$ is given or an additional assumption about the density of the energy mass is imposed. In this model, the non-conservation equation gives rise to the density evolution as

$$\begin{array}{c}{\displaystyle \frac{\left(2\kappa +(3-w){\beta }_0\right)}{2\kappa +4{\beta }_0}}\dot{\rho }+{\displaystyle \frac{6H(w+1)(\kappa +{\beta }_0)\rho }{2\kappa +4\beta }}=0,\hfill \end{array}$$

(22)

which can be integrated to give

$$\rho ={\rho }_0{a}^{{\displaystyle \frac{6(w+1)(\kappa +{\beta }_0)}{{\beta }_0(w-3)-2\kappa }}}\mathrm{or}\rho ={\rho }_0{a}^{{\displaystyle \frac{-6(w+1)(\kappa +{\beta }_0)}{2\kappa -{\beta }_0(w-3)}}},$$

(23)

or

$${\Omega }_i=\sum {\Omega }_{i0}{a}^{{\displaystyle \frac{-6(\kappa +{\beta }_0)(1+w)}{2\kappa -(w-3){\beta }_0}}},$$

(24)

where ${\rho }_0$ is the integration constant that corresponds to the initial density. In order to relate the model to the observations, we transform all the physical quantities to the redshift space using the following relations:

$${\displaystyle \frac{d}{dt}}=-(1+z)H\left(z\right){\displaystyle \frac{d}{dz}},a={\displaystyle \frac{1}{1+z}},h=H/H_0,$$

where $a\left(0\right)=1$ is used. Thus, Equation (21) can be expressed as

$${\displaystyle \frac{dh}{dz}}={\displaystyle \frac{-3h}{1+z}}{\displaystyle \frac{(\kappa +{\beta }_0)(1+w)}{72{\alpha }_0{h}^2-2}}{\Omega }_i.$$

(25)

Now, for a matter-dominated universe, where $w=0$, Equation (24) provides the redshift dependence for ${\Omega }_m\left(z\right)$ as

$${\Omega }_m={\Omega }_{m0}{(1+z)}^{{\displaystyle \frac{6(\kappa +{\beta }_0)}{2\kappa +3{\beta }_0}}}.$$

(26)

Then, by substituting Equation (26) into Equation (25), the following result is obtained.

$${\displaystyle \frac{dh}{dz}}={\displaystyle \frac{c_{0}h}{36{\alpha }_0{h}^2-1}}{(1+z)}^{{\displaystyle \frac{6(\kappa +{\beta }_0)}{2\kappa +3{\beta }_0}}-1},$$

(27)

where $c_0=-3(\kappa +{\beta }_0){\Omega }_{m0}/2$ is constant. In Equation (25), w remained generic; however, in Equation (26), a value of 0 is assigned to derive for the matter-dominated formulation of ${\Omega }_m\left(z\right)$, which is applicable to nonrelativistic matter ($p=0$). Using Equations (18) and (25), the DP and Jerk parameters are given as follows:

$$q\left(z\right)=-1-{\displaystyle \frac{3(\kappa +{\beta }_0)}{72{\alpha }_0{h}^2-2}}{\Omega }_{m0}{(1+z)}^{{\displaystyle \frac{6(\kappa +{\beta }_0)}{2\kappa +3{\beta }_0}}}.$$

(28)

Hence, the DP is an important tool for measuring the cosmic acceleration. For example, the universe demonstrates an acceleration phase of expansion when $q<0$ and under the influence of dark energy (DE), and the universe decelerates when $q>0$ and matter dominates over DE, whereas $q=0$ signifies a linear (or coasting) expansion. On the other hand, the jerk parameter can also be expressed in terms of q and its derivative as

$$j\left(z\right)=q\left(z\right)\left(1+2q\left(z\right)\right)+\left(1+z\right){\displaystyle \frac{dq\left(z\right)}{dz}},$$

(29)

or in its redshift dependence form

$$\begin{array}{cc}\hfill j\left(z\right)& =\left(-1-{\displaystyle \frac{3(\kappa +{\beta }_0){\Omega }_{m0}{(1+z)}^{\frac{6(\kappa +{\beta }_0)}{2\kappa +3{\beta }_0}}}{(72{\alpha }_0{h}^2-2)}}\right)\times \left(-1-{\displaystyle \frac{3(\kappa +{\beta }_0){\Omega }_{m0}{(1+z)}^{\frac{6(\kappa +{\beta }_0)}{2\kappa +3{\beta }_0}}}{36{\alpha }_0{h}^2-1}}\right)\hfill \\ \hfill & -{\displaystyle \frac{9{(\kappa +{\beta }_0)}^2{\Omega }_{m0}{(1+z)}^{\frac{6(\kappa +{\beta }_0)}{2\kappa +3{\beta }_0}}}{(2\kappa +3{\beta }_0)(36{\alpha }_0{h}^2-1)}}.\hfill \end{array}$$

(30)

If ${\beta }_0=-\kappa$, $j=1$, this is exactly the same as in the $\Lambda$CDM models.

Om$\left(z\right)$ Diagnostic Test

The authors of [42] introduced a novel diagnostic test called Om$\left(z\right)$. It is used to distinguish between the $\Lambda$CDM model and alternative dark energy (DE) scenarios based only on its slope. If Om(z) has a $+ve$ slope, then the model indicates a phantom-like behavior $(w<-1)$; if the slope of Om$\left(z\right)$ is $-ve$, then the model indicates a quintessence-like behavior with $(w>-1)$. Finally, if the slope of Om$\left(z\right)$ remains constant at various redshifts z, then the model indicates a $\Lambda$. Mathematically, it is expressed as follows:

$$Om\left(z\right)={\displaystyle \frac{h{\left(z\right)}^2-1}{{(1+z)}^3-1}},$$

(31)

where $h\left(z\right)=H\left(z\right)/H_0$ is the normalized Hubble parameter.

4. Comparison with Other Work

In this section, the nonlinear $f(Q,T)$ cosmological model is compared with a group of alternative cosmological models studied in recent years-see

Table 1. Comparison of the adopted model with different forms of the $f(Q,T)$ formula used by other authors. Note: N.M.—numerical method; D.S.A—dynamical system analysis.

Ref	$\mathit{f}(\mathit{Q},\mathit{T})$	Parameterization	N.M. Used	Compare with $\mathbf{\Lambda }$CDM
[32]	$\alpha Q+{\displaystyle \frac{\beta }{6{\kappa }^2}}T$	$q\left(z\right)=1-{\displaystyle \frac{2}{1+\chi {(1+z)}^4}}$	MCMC	Good fit
	${\displaystyle \frac{\gamma }{6H_0^2{\kappa }^2}}QT$
[33]	$Q+\alpha Q^2+2\gamma T$		MCMC
[34]	$m{Q}^n+bT$		Exact + MCMC
	$Q^{n+1}+bT$		Exact + MCMC
[35]	$Q^{n+1}+bT$			Suitable to describe
[37]	$-{\lambda }_1{Q}^m-{\lambda }_2{T}^2$	$H\left(z\right)=H_0{\left[\sqrt{{(1+z)}^{3(1+w)}}\right]}^{1/m}$	MCMC	Better fit
[38]	$\alpha Q^m+\beta T$	$a\left(t\right)\propto {\left[sinh\left({\displaystyle \frac{3}{2}}\sqrt{{\displaystyle \frac{\Lambda }{3}}t}\right)\right]}^{{\displaystyle \frac{2}{3}}}$	D.S.A	Similar at late times
[39]	$-Q+\beta log{\displaystyle \frac{Q}{Q_0}}+\gamma T$	$H\left(z\right)={\left[\alpha +(1-\alpha ){(1+z)}^n\right]}^{{\displaystyle \frac{3}{2n}}}$	MCMC + D.S.A	Supports

. Most scholars used polynomial or power-law variables in Q and T. We considered the most generic form of $f(Q,T)$ by adding a quadratic part to the nonmetricity scalar Q and a linear part to the matter trace T to allow our system to replicate the concordance $\Lambda$ CDM model as a limiting case. This choice could yield different dynamics, such as quintessence or ghosts, depending on the values of the free parameters ${\alpha }_0$ and ${\beta }_0$. The easiest way to determine these parameters is to use the MCMC techniques with recent data to constrain them. However, we solved the nonlinear modified field equation through a straightforward process, directly avoiding any additional assumptions and ensuring that the obtained solution would comply with the theoretical framework.

5. Numerical Integration

The solution to the nonlinear differential Equation (27) was obtained using the local Runge–Kutta method (RK4). This method approximates the solution by calculating four weighted slopes at each step, leading to a local truncation error (LTE) of order $O\left(h^{-5}\right)$. The explicit slope averaging approach of RK4 guarantees a high level of accuracy and computational effectiveness. In this study, the LTE was determined using a step-doubling method with step sizes $\Delta z=0.1$ and ${\displaystyle \frac{\Delta z}{2}}=0.05$.

Table 2. Adaptive RK4 results showing step size control for ${\alpha }_0=-0.75$.

z	$\mathbf{\Delta }\mathit{z}$	$\mathit{h}\left(\mathbf{\Delta }\mathit{z}\right)$	$\mathit{h}(\mathbf{\Delta }\mathit{z}/2)$	LTE
0.0	0.1000	1.045256916944	1.045256886319	2.0 × ${10}^{-9}$
0.5	0.1000	1.094333050753	1.094333027138	1.6 × ${10}^{-9}$
1.0	0.1000	1.157917980845	1.157917901190	5.3 × ${10}^{-9}$
1.5	0.1000	1.233634788613	1.233632097590	1.8 × ${10}^{-7}$
2.0	0.0860	1.277992045787	1.277982181618	6.6 × ${10}^{-7}$
2.5	0.0648	1.279593520688	1.279580900700	8.4 × ${10}^{-7}$
3.0	0.0459	1.257626921555	1.257617313515	6.4 × ${10}^{-7}$
3.5	0.0364	1.255923194858	1.255913231958	6.6 × ${10}^{-7}$
4.0	0.0297	1.255675705098	1.255665353770	6.9 × ${10}^{-7}$
4.5	0.0248	1.256293373635	1.256282599088	7.2 × ${10}^{-7}$
5.0	0.0210	1.257469925505	1.257458692019	7.5 × ${10}^{-7}$
5.5	0.0181	1.259032651702	1.259020922640	7.8 × ${10}^{-7}$
6.0	0.0158	1.260877454939	1.260865192660	8.2 × ${10}^{-7}$
6.5	0.0140	1.262938047331	1.262925213082	8.6 × ${10}^{-7}$
7.0	0.0124	1.265170187596	1.265156741395	9.0 × ${10}^{-7}$
7.5	0.0112	1.267543108716	1.267529009238	9.4 × ${10}^{-7}$
8.0	0.0101	1.270034616902	1.270019821378	9.9 × ${10}^{-7}$

illustrates the fluctuation in the LTE throughout the domain $z\in [0,8]$. The LTE demonstrates a steady rise with z, while staying within the limits of ${10}^{-6}$ to ${10}^{-9}$. These outcomes correspond to the expected theoretical error limits of RK4, which confirms the method’s accuracy and reliability.

6. Discussion and Conclusions

This study analyzed the cosmological implications of a nonlinear $f(Q,T)$ gravity theory. Specifically, we employed the most general functional form $f(Q,T)=-Q+{\alpha }_0{\displaystyle \frac{Q^2}{H_0^2}}+{\beta }_{0}T+{\eta }_0$, where ${\alpha }_0$, ${\beta }_0$, and ${\eta }_0$ are independent parameters, and $H_0$ denotes the present value of the Hubble parameter. Comparative studies between the adopted model and similar studies are shown in Table 1. To solve the modified field equations, they were changed to a time-dependent closed system of nonlinear first-order coupled differential equations (DEs) for the Hubble parameter H and mass energy density $\rho$. The auxiliary expression for $\rho$ was derived from the nonconservation of the energy–momentum tensor, as shown in Equation (14). This technique does not rely on any additional conditions or parameterization while solving the system of equations, as indicated in Table 1. Subsequently, the governing DE was converted into a redshift space z and numerically integrated employing the fourth-order Runge–Kutta method for the dust-dominated universe, utilizing various choices of ${\alpha }_0$ and ${\beta }_0$. Having obtained the numerical value of h, the model characterization parameters, the mass energy density parameter, the deceleration parameter, the effective equation of state, the jerk, and the Om(z) diagnostic test parameter, ${\Omega }_m,q,w_{eff}$, and j were presented and compared with the standard $\Lambda$CDM model, as illustrated in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.

Figure 1 and Figure 2 illustrate the evolution of the normalized Hubble parameter with redshift, converging towards its anticipated value at $z=0$. Figure 3 and Figure 4 illustrate the progression of the deceleration parameter $q\left(z\right)$ in the $f(Q,T)$ model and its comparison with the $\Lambda$CDM model for various values of ${\alpha }_0$ and ${\beta }_0$. The deceleration parameter q changes from being positive at an early period of time at high redshift values to a negative value passing through a transitional redshift $z_t\approx [0.766,0.769,0.771]$ and $z_t\approx [0.521,0.770,1.01]$, reaching their current values at $q_0=[-0.61,-0.60,-0.59]$ and $[-0.455,-0.595,-0.694]$ for different values of ${\beta }_0$ and ${\alpha }_0$, respectively. This indicates that the cosmos has undergone a transition from a slowdown phase of expansion in the early period to accelerating at the current time, at $z=0$ and $t=t_0$, “late-time acceleration”.

Additionally, Figure 5 and Figure 6 illustrate the redshift progression of the effective equation of state $w_{eff}$ in the $f(Q,T)$ model, as compared with the $\Lambda$CDM model for varying values of ${\alpha }_0$ and ${\beta }_0$. The effective equation of state $w_{eff}$ shifts from the matter-dominated phase $w_{ff}=0$ at high redshift to a quintessence-like behavior in the present epoch at values of $z=0$, $w_{eff0}=[-0.741,-0.735,-0.731]$, and $w_{eff0}=[-0.741,-0.735,-0.731]$, as predicted by $\Lambda$CDM for different values of ${\beta }_0$ and ${\alpha }_0$, respectively. Figure 7 and Figure 8 illustrate the evolution of the parameter of the energy mass density ${\Omega }_m$ as a function of the redshift z in the $f(Q,T)$ and $\Lambda$CDM models. The evolution of ${\Omega }_m$ depends solely on ${\beta }_0$, not on ${\alpha }_0$, and grows positively with redshift z. The redshift evolution of the jerk parameter in the $f(Q,T)$ and $\Lambda$CDM models is shown in Figure 9 and Figure 10. The figures reveal that, for different values of ${\alpha }_0$ and ${\beta }_0$, the $j\left(z\right)$ behaves similarly to the $\Lambda$CDM model and shows a small shift in its amplitude. This result is consistent with the work of [38]. Moreover, Figure 11 show the evolution of the diagnostic Om$\left(z\right)$ in the $f(Q,T)$ model for different values of ${\alpha }_0$ and ${\beta }_0$ and its comparison with the standard cosmological model $\Lambda$CDM. We observe that, for ${\alpha }_0=-0.55$ and $-0.75$, Om(z) has a negative slope, and for ${\alpha }_0=-1$, the slope remains negative and then becomes positive near $z=0$. Then, it again becomes negative, indicating a quintessence-like behavior followed by a phantom-like behavior near $z=0$ and settles to a quintessence-like behavior at $z=0$, as shown in Figure 11b. Figure 11a shows Om(z) with a negative slope for ${\beta }_0=1.01$, indicating a quintessence-like behavior, and for ${\beta }_0=0.511$, the slope remains negative, except near $z=0$, where it becomes positive, indicating the changing behavior of DE from quintessence-like to phantom-like and then to quintessence-like. For ${\beta }_0=0.011$, the Om(z) has a positive slope, indicating the phantom-like behavior of DE. Additionally, for cases with ${\alpha }_0>0$ and ${\beta }_0>0$, Equation (27) is numerically solved with ${\alpha }_0=0.55$ and ${\beta }_0=0.60$, resulting in $q_0\simeq -1.59$ and $w_{\mathrm{eff},0}\simeq -1.40$, as shown in Figure 12. These findings indicate a super-accelerated or phantom-like state ($w_{\mathrm{eff}}<-1$); this adds to this work’s focus on the decelerating cases (${\alpha }_0<0,{\beta }_0>0$). These potential solutions present evidence that the model could address both quintessence-like and phantom-like dynamics at late times. Further research should explore the adopted model’s reliability, elucidating cosmic acceleration in the background, stability, cosmic perturbations, structure formation, and comparison with other existing cosmological models utilizing the recent observations.