Viscous Cosmology in f(Q,L_m) Gravity: Insights from CC, BAO, and GRB Data

Abstract

In this article, we investigate the influence of viscosity on the evolution of the cosmos within the framework of the newly proposed $f(Q,L_m)$ gravity. We have considered a linear functional form $f(Q,L_m)=\alpha Q+\beta L_m$ with a bulk viscous coefficient $\zeta ={\zeta }_0+{\zeta }_{1}H$ for our analysis and obtained exact solutions to the field equations associated with a flat FLRW metric. In addition, we utilized Cosmic Chronometers (CC), CC + BAO, CC + BAO + GRB, and GRB data samples to determine the constrained values of independent parameters in the derived exact solution. The likelihood function and the Markov Chain Monte Carlo (MCMC) sampling technique are combined to yield the posterior probability using Bayesian statistical methods. Furthermore, by comparing our results with the standard cosmological model, we found that our considered model supports the acceleration of the universe in late time.

1. Introduction

The results of several pieces cosmological evidence, like supernovae search [1,2], WMAP [3,4], and BAO [5,6], show that the cosmos is currently experiencing a phase of accelerated expansion. To explain the accelerated expansion of the universe, the existence of dark energy has been proposed by physicists. The simplest justification for dark energy is the introduction of a cosmological constant, which forms the core for the $\mathsf{\Lambda }$CDM model, highly recommended for its effectiveness in explaining cosmological observations. Although observational data supports the $\mathsf{\Lambda }$CDM model, it also has some drawbacks, such as the difference between the anticipated and observed values of the cosmological constant. Another issue is the cosmic coincidence problem, which raises the question of why dark energy and matter densities are nearly equal in the current universe, a scenario that seems unlikely and unexpected. Due to these challenges, numerous modified theories of gravity have been put forth to provide a more comprehensive explanation for the universe’s accelerated expansion. By changing the Einstein–Hilbert action, Einstein’s general relativity (GR) transforms into modified theories, making it possible to account for cosmic acceleration. In a geometrical context, curvature is not only the core feature; torsion and non-metricity also play significant roles in the connection within a metric space. There are three comparable representations for GR, the first based on curvature (GR itself), where both torsion and non-metricity are absent. The second is torsion $\left(T\right)$-based, and it is the teleparallel equivalent of GR (TEGR). The third is non-metricity $\left(Q\right)$-based, often known as the symmetric teleparallel equivalent of GR (STEGR). The simplest departures from the theories are the functional forms $f\left(R\right)$, $f\left(T\right)$ [7,8,9], and $f\left(Q\right)$ [10,11,12], respectively. Additionally, an extension of $f\left(Q\right)$ theory was introduced by Xu et al. [13], which is known as $f(Q,T)$ gravity. According to this theory, the trace T of the energy–momentum tensor is non-minimally related to the non-metricity scalar Q. Some interesting references on the theory can be found in [14,15,16].

In order to investigate the early inflationary stage of cosmic evolution, viscosity was first incorporated into cosmic fluid models without requiring the assumption of dark energy. Eckart introduced a non-causal viscosity theory by considering first-order deviations from equilibrium. Later, Israel and Stewart developed a causal viscosity theory, which is based on second-order deviations from equilibrium [17,18,19,20]. Late-time acceleration has been analyzed using the causal theory of viscosity. Shear and bulk viscosity are two viscosity coefficients that are obtained from the examination of second-order deviations. The velocity gradient associated with shear viscosity vanishes in a homogeneous universe. Consequently, within an isotropic and homogeneous FLRW framework, only the bulk viscosity coefficient remains relevant. Bulk viscosity refers to the pressure required to maintain thermal stability during the expansion of the cosmic fluid along with the expanding universe. The pressure component responsible for driving cosmic acceleration is influenced by the viscosity coefficients. Recently, extensive research has focused on the influence of bulk viscosity in cosmic evolution, as highlighted by numerous studies [21,22,23,24,25,26,27,28,29,30].

In this article, we are working within the framework of $f(Q,L_m)$ gravity, which was recently proposed by Hazarika et al. [31]. The $f(Q,L_m)$ theory is an extension of $f\left(Q\right)$ symmetric teleparallel gravity, obtained by incorporating an arbitrary interaction among the non-metricity Q and the matter Lagrangian $L_m$. Both minimal and non-minimal interactions among matter and geometry are permitted in this framework. The authors were able to obtain the general system of field equations by varying the action with respect to the metric. They also studied the matter energy–momentum tensor non-conservation within the context of this theory.

This manuscript is arranged as follows: Section 2 provides a brief overview of the $f(Q,L_m)$ theory. A flat FLRW universe within $f(Q,L_m)$ gravity incorporating bulk viscous matter is explored in Section 3. In Section 4, using the CC, CC + BAO, and GRB datasets, we obtain constrained values of free parameters in the exact solution. In Section 5, we conclude our findings.

2. Field Equations of $\mathit{f}(\mathit{Q},{\mathit{L}}_{\mathit{m}})$ Gravity

The action principle is used to investigate the dynamics of the physical system. The action for $f(Q,L_m)$ gravity is represented as follows:

$$S=\int f(Q,L_m)\sqrt{-g}{d}^{4}x,$$

(1)

where g denotes the determinant of the metric, and $f(Q,L_m)$ is an arbitrary function of non-metricity scalar Q and the matter Lagrangian $L_m$. One can get the gravitational field equation by varying the action term with respect to the metric tensor; it explains how matter and energy affect the geometry of spacetime:

$$\delta S=\int [(f_Q\delta Q+f_{L_m}\delta L_m)\sqrt{-g}+f\delta \sqrt{-g}]d^{4}x.$$

(2)

Here, $f_Q=\partial f(Q,L_m)/\partial Q$, and $f_{L_m}=\partial f(Q,L_m)/\partial L_m$.

The variation of Q is given by

$$\delta Q=2P_{\alpha \nu \rho }{\nabla }^{\alpha }\delta g^{\nu \rho }-(P_{\mu \alpha \beta }{Q}_{\nu }^{\alpha \beta }-2Q_{\mu }^{\alpha \beta }{P}_{\alpha \beta \nu })\delta g^{\mu \nu }.$$

(3)

$T_{\mu \nu }$ represents the energy–momentum tensor of the matter and is defined as

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

(4)

The variation of the determinant of the metric tensor is

$$\delta \sqrt{-g}=-{\displaystyle \frac{1}{2}}\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }.$$

(5)

From Equations (3)–(5), it follows that Equation (2) can be written as

$$\begin{array}{cc}\hfill \delta S=\int \left[\right(f_Q(2P_{\alpha \nu \rho }{\nabla }^{\alpha }\delta g^{\nu \rho }-(P_{\mu \alpha \beta }{Q}_{\nu }^{\alpha \beta }-2Q_{\mu }^{\alpha \beta }{P}_{\alpha \beta \nu })\delta g^{\mu \nu })& \\ \hfill +{\displaystyle \frac{1}{2}}{f}_{L_m}(g_{\mu \nu }{L}_m-T_{\mu \nu })\delta g^{\mu \nu })-{\displaystyle \frac{1}{2}}f{g}_{\mu \nu }\delta g^{\mu \nu }]\sqrt{-g}{d}^{4}x.\end{array}$$

(6)

After implementing the boundary conditions and performing integration on the first term in Equation (6) becomes $-2{\nabla }^{\alpha }\left(f_Q\sqrt{-g}{P}_{\alpha \mu \nu }\right)\delta g^{\mu \nu }$. The field equation of $f(Q,L_m)$ gravity is obtained by setting the metric variation of the action to zero:

$$\begin{array}{cc}{\displaystyle \frac{2}{\sqrt{-g}}}{\nabla }_{\alpha }\left(f_Q\sqrt{-g}{P}_{\mu \nu }^{\alpha }\right)+f_Q(P_{\mu \alpha \beta }{Q}_{\nu }^{\alpha \beta }-2Q_{\mu }^{\alpha \beta }{P}_{\alpha \beta \nu })\hfill & \\ & \hfill +{\displaystyle \frac{1}{2}}f{g}_{\mu \nu }={\displaystyle \frac{1}{2}}{f}_{L_m}(g_{\mu \nu }{L}_m-T_{\mu \nu }).\end{array}$$

(7)

For $f(Q,L_m)=f\left(Q\right)+2L_m$, one can get the field equations of $f\left(Q\right)$ gravity as in [31]. Furthermore, the energy momentum tensor is expressed by

$$\begin{array}{cc}\hfill {\displaystyle \frac{2}{\sqrt{-g}}}{\nabla }_{\alpha }\left(f_Q\sqrt{-g}{P}_{\nu }^{\alpha \mu }\right)+f_Q{P}_{\alpha \beta }^{\mu }{Q}_{\nu }^{\alpha \beta }+{\displaystyle \frac{1}{2}}{\delta }_{\nu }^{\mu }f& \\ \hfill ={\displaystyle \frac{1}{2}}{f}_{L_m}({\delta }_{\nu }^{\mu }{L}_m-T_{\nu }^{\mu }).\end{array}$$

(8)

3. Cosmology in $\mathit{f}(\mathit{Q},{\mathit{L}}_{\mathit{m}})$

For our analysis, we adopt the flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric, considering the spatial isotropy and homogeneity of the universe:

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

(9)

For the line element (9), the non-metricity scalar is $Q=6H^2$, where H stands for the Hubble parameter. The energy–momentum tensor for the line element (9), representing a universe filled with perfect fluid matter, is given by the following:

$$T_{\mu \nu }=(\rho +\overline{p})u_{\mu }{u}_{\nu }+\overline{p}{g}_{\mu \nu }.$$

(10)

Here, $\rho$ stands for energy density; $\overline{p}$ is the effective pressure, which is defined as $\overline{p}=p-3\zeta H$, where $\zeta$ is the bulk viscosity coefficient and p is isotropic pressure (without viscosity); and $u^{\mu }$ is the 4-velocity of the fluid, with components $u^{\mu }=(1,0,0,0)$.

Using the FLRW metric, the field equations Equation (7) give the two generalized Friedmann equations:

$$\begin{array}{ccc}\hfill & & 3H^2={\displaystyle \frac{1}{4f_Q}}[f-f_{L_m}(\rho +L_m)],\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill & & \dot{H}+3H^2+{\displaystyle \frac{\dot{f_Q}}{f_Q}}H={\displaystyle \frac{1}{4f_Q}}[f+f_{L_m}(\overline{p}-L_m)].\hfill \end{array}$$

(12)

In this article, we are using the linear $f(Q,L_m)$ model $f(Q,L_m)=\alpha Q+\beta L_m$ (where the dimension of $\alpha$ is $\left[L^{-2}\right]$, which is the same as the dimension of ${\displaystyle \frac{1}{k^2}}$ or ${\displaystyle \frac{1}{G}}$, and $\beta$ is dimensionless, which simply weights the coupling to $L_m$), with the bulk viscosity coefficient $\zeta ={\zeta }_0+{\zeta }_{1}H$ [32] (where the dimensions of ${\zeta }_0$ and ${\zeta }_1$ are $\left[M{L}^{-1}{T}^{-1}\right]$ and $\left[M{L}^{-1}\right]$, respectively). Also, we have considered the matter-dominated universe for which $p=0$.

We obtained the following differential equation for the linear $f(Q,L_m)$ model considered:

$$\dot{H}+{\displaystyle \frac{3H^2}{4\alpha }}(2\alpha +{\zeta }_1\beta )={\displaystyle \frac{-3\beta {\zeta }_{0}H}{4\alpha }}.$$

(13)

The analytic solution for the above differential equation is as follows:

$$H={\displaystyle \frac{{\zeta }_0\beta }{2\alpha +{\zeta }_1\beta }}\left({(1+z)}^{{\displaystyle \frac{3}{4\alpha }}(2\alpha +{\zeta }_1\beta )}-1\right)+H_0{(1+z)}^{{\displaystyle \frac{3}{4\alpha }}(2\alpha +{\zeta }_1\beta )}.$$

(14)

4. Parameter Estimation by Using Observational Data

In this section, we employ the Markov Chain Monte Carlo (MCMC) technique to evaluate the performance of the proposed model by comparing its predictions with observational datasets, specifically those from Hubble parameter measurements, Baryon Acoustic Oscillations (BAO), and Gamma-Ray Bursts (GRBs). MCMC is a powerful statistical tool widely used in cosmology for exploring parameter spaces and estimating the posterior distributions of model parameters. It constructs a Markov Chain that samples the parameter space according to a target probability distribution, where each new sample is generated from the previous one using a proposal distribution. The acceptance of proposed samples depends on their posterior probability, which incorporates both the likelihood from observational data and prior assumptions about the parameters. Once the chain reaches convergence, the posterior distributions provide insight into the most probable parameter values and their associated uncertainties, allowing for robust inference of cosmological observables.

The reliability of the MCMC results depends heavily on ensuring proper convergence and sufficient sampling of the parameter space. To achieve this, multiple Markov Chains are initialized with different starting points, and diagnostic tools, such as trace plots and autocorrelation analysis, are used to assess convergence. Once the chains are confirmed to have mixed well and reached equilibrium, the posterior samples are analyzed to extract marginalized distributions, credible intervals, and best-fit values for the parameters. This statistical framework allows for a rigorous comparison between theoretical models and observational data, providing a quantitative measure of model viability. In this study, MCMC is employed to test the consistency of the linear model with bulk viscosity coefficient against key observational datasets, offering insight into its physical plausibility and parameter constraints.

4.1. Cosmic Chronometer Dataset

We make use of Hubble parameter measurements derived from the differential-age approach. Our study uses a compilation of 31 data points [33,34,35] that includes values of the Hubble parameter derived from observations in the redshift range $0.07\le z\le 2.41$, denoted by $H_{obs}\left(z_i\right)$. The advantage of this method is that it does not rely on assumptions about the early universe, making it a valuable tool for investigating deviations from the standard $\mathsf{\Lambda }$CDM model. The ${\chi }^2$ function for the CC dataset is given by

$${\chi }_{CC}^2=\sum _{i=1}^{31}{\displaystyle \frac{{\left[H_{th}\left(z_i\right)-H_{obs}\left(z_i\right)\right]}^2}{\sigma {\left(z_i\right)}^2}},$$

(15)

where $H_{obs}$ and $H_{th}$ refer to the observed and theoretical values of the Hubble parameter, respectively, and $\sigma$ accounts for the associated uncertainty in the measurement of the Hubble value.

4.2. BAO Dataset

One of the most popular observational methods for studying the large-scale structure of the Universe is baryonic acoustic oscillation (BAO). Acoustic waves in the early cosmos are the source of these oscillations, which result in radiation in the photon–baryon fluid and compression in the baryonic matter. This compression results in a unique peak that serves as a reference scale to calculate distances throughout the cosmos. In this article, we utilize 26 independent line-of-sight BAO data points [36], and the ${\chi }^2$ function for the BAO dataset is

$${\chi }_{BAO}^2=\sum _{i=1}^{26}{\displaystyle \frac{{\left[H(z_i,\odot )-H_{obs}\left(z_i\right)\right]}^2}{{\sigma }_H{\left(z_i\right)}^2}}.$$

(16)

The theoretical Hubble parameter is determined through its cosmographic expansion, which is similar to the CC method.

4.3. Gamma-Ray Bursts (GRBs)

Gamma-Ray bursts (GRBs) are the most powerful explosions in the Universe, releasing immense amounts of energy. These cosmic phenomena can be observed at relatively higher redshifts. The relationship between $\nu$ $F_{\nu }$, $E_{p,i}$, and $E_{iso}$ is expressed as follows:

$${log}_{10}\left({\displaystyle \frac{E_{iso}}{1\mathrm{erg}}}\right)=b+a{log}_{10}\left({\displaystyle \frac{E_{p,i}}{300\mathrm{keV}}}\right).$$

(17)

Here, $\nu$ $F_{\nu }$ represents the peak energy of the rest frame spectrum, $E_{iso}$ represents the equivalent isotropic radiated energy [37], a and b are constants, and $E_{p,i}$ represents the peak spectral energy in the cosmic rest frame of the GRBs. $E_{p,i}$ corresponds with the observer-frame quantity $E_P$ through the expression $E_{p,i}=E_p(1+z)$. Additionally, $E_{iso}$ can be determined with the help of luminosity distance $d_L$ and the bolometric fluence $S_{bolo}$ as

$$E_{iso}=4\pi d_L^2(z,cp)S_{bolo}{(1+z)}^{-1}.$$

(18)

In this study, we examine a dataset consisting of 162 redshift measurements, with values ranging from 0 to 9.3 [38]. This presents an opportunity to extend the research beyond observing supernovae and explore regions at higher redshifts.

Due to their brightness and high-energy nature, GRBs serve as valuable cosmological probes, especially in the redshift range where other standard candles like Type Ia Supernovae become sparse. The calibration of GRBs is often model-dependent; however, various empirical correlations, such as the Amati relation used here, allow for constructing Hubble diagrams from GRB observations.

To utilize GRBs as distance indicators, we compute the theoretical distance modulus ${\mu }_{th}$ from the cosmological model and compare it with the observed one derived from the Amati relation. The observed distance modulus is given by

$${\mu }_{obs}=5{log}_{10}\left({\displaystyle \frac{d_L}{1\mathrm{Mpc}}}\right)+25.$$

(19)

The luminosity distance ($d_L$) is a key observable in cosmology, defined as the distance at which an astronomical source would lie in a static Euclidean universe to produce the observed flux for a given intrinsic luminosity. For a spatially flat FLRW metric, it is expressed as

$$d_L\left(z\right)=(1+z){\int }_0^z{\displaystyle \frac{d{z}^{\prime }}{H\left(z^{\prime }\right)}}.$$

(20)

This expression assumes natural units where the speed of light $c=1$. We calibrate the Amati relation parameters $(a,b)$ either using low-redshift GRBs or through a simultaneous fitting with cosmological parameters, ensuring self-consistency. Here, we obtain the optimal parameter space by maximizing the likelihood function $L\propto e^{-{\displaystyle \frac{{\chi }^2}{2}}}$, where ${\chi }^2$ is expressed as follows:

$${\chi }^2=\Delta V^T\left(C^{-1}\right)\Delta V$$

(21)

where C represents the covariance matrix for uncertainties, and $\Delta V$ is the discrepancy between the theoretical and observational measurements of the distance modulus function.

This statistical framework allows for robust constraints on cosmological models using GRB observations. Furthermore, when combined with other cosmological datasets such as BAO or cosmic chronometers, GRBs enhance the redshift leverage and help test the consistency of dark energy models across cosmic time.

4.4. Results

This subsection is dedicated to employ the formulations defined above for each dataset to constrain our model using the MCMC technique. We began with the CC sample by minimizing Equation (15). The 2D contour results up to $3\sigma$ can be observed in Figure 1, where the innermost region is the confidence level $1\sigma (68\%)$, the middle region represents the confidence level $2\sigma (95\%)$, and the outer faded region is the confidence level $3\sigma (99.7\%)$. In addition, we combined the 31 points of the CC with the 26 points of the BAO sample to constrain the linear viscous model. To maximize the likelihood in the MCMC analysis, we considered the sum of Equations (15) and (16). The combination gives rise to 2D contours up to $3\sigma$ CL, which can be observed in Figure 2. Finally, the 162 points of Gamma-Ray Burst data, which are collected from a wider range of redshift up to 9.3, are combined with the CC and BAO datasets. Following the same methodology, with the Equations (15), (16) and (21), we perform an MCMC simulation to obtain the contours presented in Figure 3. The respective values of the $1-\sigma$ ranges are summarized in

Table 1. The obtained values of parameters using CC, CC + BAO, and GRB dataset.

Dataset	${\mathit{H}}_0$	$\mathsf{\alpha }$	$\mathsf{\beta }$	${\mathsf{\zeta }}_0$	${\mathsf{\zeta }}_1$
CC	$67.3\pm 1.9$	$2.74\pm 0.10$	$-4.{18}_{-0.69}^{+1.0}$	$9.0\pm 2.3$	$0.{99}_{-0.23}^{+0.17}$
CC + BAO	$64.4\pm 1.3$	$2.23\pm 0.20$	$-3.{64}_{-0.51}^{+0.66}$	$9.1\pm 1.7$	$1.{16}_{-0.31}^{+0.19}$
CC + BAO + GRB	$63.7\pm 1.2$	$2.1\pm 0.21$	$-3.35\pm 0.50$	$7.2\pm 1.8$	$1.{01}_{-0.18}^{+0.15}$

. As seen from Table 1, although the model parameters obtained from all dataset combinations are numerically close, subtle discrepancies are present. To examine which combination provides greater consistency, we analyze the behavior of the Hubble parameter and the distance modulus functions across the redshift range.

The concordance model, along with the error bars of the datasets, has been considered for comparison purposes with our model. The ability of the concordance model to describe late-time observations is well known. In this context, we consider the Hubble function plot in Figure 4 and the error bar of the datasets that measure $H\left(z\right)$. The curves for our model, obtained from the mean values of the MCMC results are in excellent agreement with the standard cosmological model and the error bars. In addition, we follow the methodology for the distance modulus function against the redshift. As shown in Figure 5, all model curves exhibit good agreement with the observational data and the $\mathsf{\Lambda }$CDM model in the low-redshift regime. At higher redshifts, however, the CC + BAO + GRB combination demonstrates improved consistency with both the standard cosmological model and the GRB data points.

5. Conclusions

In this article, we explored the influence of bulk viscosity on cosmic evolution within the framework of the recently proposed $f(Q,{\mathcal{L}}_m)$ gravity, where Q denotes the non-metricity scalar and ${\mathcal{L}}_m$ represents the matter Lagrangian. In a recent work [39], the authors successfully reconstructed the $\mathsf{\Lambda }$CDM scenario within the $f(Q,{\mathcal{L}}_m)$ theory.

We commenced with the assumption of the simplest linear form, i.e., $f(Q,{\mathcal{L}}_m)=\alpha Q+\beta {\mathcal{L}}_m$, and incorporated a bulk viscous coefficient of the form $\zeta ={\zeta }_0+{\zeta }_{1}H$ in our analysis. We derived the analytical solution for the Hubble parameter $H\left(z\right)$ based on this linear model. This expression, which encapsulates both the model parameters and the viscosity parameters, was subsequently used for Markov Chain Monte Carlo (MCMC) analysis.

The free parameters of the model were constrained using three different combinations of observational data: CC, CC + BAO, and CC + BAO + GRB. The reliability of the results was assessed by comparing them with the predictions of the standard $\mathsf{\Lambda }$CDM model and the associated observational uncertainties. One of the most noteworthy aspects of the model is its compatibility with both the data and the standard model.

Furthermore, the inclusion of GRB data allows us to extend the analysis into a redshift range that is beyond the reach of conventional probes such as Type Ia supernovae or BAO alone. This highlights the capability of our model to remain viable across a wide range of cosmic epochs, from recent acceleration to early structure formation.

Overall, the agreement of our model with multiple datasets, including high-redshift GRBs, supports its potential as a compelling alternative framework for explaining late-time cosmic acceleration. The model shows promising prospects for future investigations into the cosmic expansion history, structure growth, and possible deviations from the standard paradigm.