Bulk Viscous Fluid in Symmetric Teleparallel Cosmology: Theory versus Experiment

Abstract

The standard formulation of General Relativity Theory, in the absence of a cosmological constant, is unable to explain the responsible mechanism for the observed late-time cosmic acceleration. On the other hand, by inserting the cosmological constant in Einstein’s field equations, it is possible to describe the cosmic acceleration, but the cosmological constant suffers from an unprecedented fine-tuning problem. This motivates one to modify Einstein’s spacetime geometry of General Relativity. The $f\left(Q\right)$ modified theory of gravity is an alternative theory to General Relativity, where the non-metricity scalar Q is the responsible candidate for gravitational interactions. In the present work, we consider a Friedmann–Lemâitre–Robertson–Walker cosmological model dominated by bulk viscous cosmic fluid in $f\left(Q\right)$ gravity with the functional form $f\left(Q\right)=\alpha Q^n$, where $\alpha$ and n are free parameters of the model. We constrain our model with the Pantheon supernovae dataset of 1048 data points, the Hubble dataset of 31 data points, and the baryon acoustic oscillations dataset consisting of 6 data points. We find that our $f\left(Q\right)$ cosmological model efficiently describes the observational data. We present the evolution of our deceleration parameter with redshift, and it properly predicts a transition from decelerated to accelerated phases of the universe’s expansion. Furthermore, we present the evolution of density, bulk viscous pressure, and the effective equation of state parameter with redshift. Those show that bulk viscosity in a cosmic fluid is a valid candidate to acquire the negative pressure to drive the cosmic expansion efficiently. We also examine the behavior of different energy conditions to test the viability of our cosmological $f\left(Q\right)$ model. Furthermore, the statefinder diagnostics are also investigated in order to distinguish among different dark energy models.

1. Introduction

The acceleration of the universe’s expansion is one of the most active discoveries of modern cosmology. Observational studies include Type Ia supernovae [1,2], large-scale structure [3,4], baryon acoustic oscillations [5,6], and cosmic microwave background radiation [7,8]. The reason behind the late-time acceleration is a mystery. Several models hypothesize the existence of a component called dark energy, which makes up around 70% of the entire universe and could possess the feature of speeding up the universe’s expansion.

The cosmological constant $\Lambda$ in the General Relativity (GR) field equations plays the role of dark energy, i.e., a fluid with constant energy density and high negative pressure. There are some issues with the cosmological constant model, such as the cosmic coincidence problem [9], which is the fact that the density of non-relativistic matter and dark energy is the same order today. A more delicate issue surrounding the cosmological constant is the so-called cosmological constant problem, which is the high discrepancy between the astronomically observed value of $\Lambda$ [1,2] and the particle physics theoretically predicted value of the quantum vacuum energy [10].

With the main purpose of solving the above cosmological issues, dynamical (time-varying) dark energy models such as the Chaplygin gas model [11,12], k-essence [13,14], quintessence [15,16], and decaying vacuum models [17,18,19,20] have been proposed for some time in the literature.

Modified theories of gravity have also been intensively investigated to understand the origin of the cosmic acceleration, as well as to address the cosmological constant model problems. It is possible to predict late-time cosmic acceleration by modifying GR action. Some possibilities can be seen within the $f\left(R\right)$ [21,22,23], $f\left(G\right)$ [24,25], $f(R,\mathcal{T})$ [26], and $f\left(T\right)$ [27,28,29,30] theories of gravitation, with R, G, $\mathcal{T}$, and T being, respectively, the Ricci, Gauss–Bonnet, energy–momentum, and torsion scalars.

In the present article, we will work with the recently introduced $f\left(Q\right)$ theory of gravity [31], for which Q is the non-metricity scalar, to be presented below. The $f\left(R\right)$ gravity is an extension of GR that describes the spacetime geometry by the non-vanishing curvature in the absence of torsion and non-metricity, whereas the $f\left(\mathcal{T}\right)$ gravity generalizes the teleparallel equivalent of GR (TEGR), in which the gravitational interactions are attributed to the non-zero torsion with vanishing curvature and non-metricity. Finally, the $f\left(Q\right)$ theory of gravity, which will be described in Section 2, is an extension of the symmetric teleparallel equivalent of GR (STEGR), where the complete charge of gravitational interactions attributed to the non-metricity term with zero curvature and torsion. The non-metricity formulation was discussed earlier by Hehl and Ne’eman (see References [32,33,34,35]). The symmetric teleparallel gravity is proven in the so-called coincidence gauge by imposing that the connection is symmetric [36]. The Weyl geometry is also observed to be a particular example of the Weyl–Cartan geometry, in which torsion disappears. The non-metricity is interpreted as a massless spin three-field in the case of symmetric connections [37,38]. Furthermore, it is noted in the literature that, due to the appearance of non-metricity, the light cone structure is not preserved during parallel transport [39]. Further, fermions are an issue in TEGR because they couple to the axial contorsion of the Weitzenbock connection. This difficulty is eliminated in STEGR since Dirac fermions only couple to the completely antisymmetric component of the affine connection and are unaffected by any disformation piece.

Although recently proposed, the $f\left(Q\right)$ gravity theory already presents some interesting and valuable applications in the literature [40,41,42,43,44,45,46,47,48,49]. The first cosmological solutions in the $f\left(Q\right)$ gravity appear in References [50,51], while the $f\left(Q\right)$ cosmography and energy conditions can, respectively, be seen in [52,53].

Here, we are going to consider the $f\left(Q\right)$ cosmology in the presence of a viscous fluid. When a cosmic fluid expands too fast, the recovering of thermodynamic equilibrium generates an effective pressure. The high viscosity in a cosmic fluid is the manifestation of such an effective pressure [54,55].

Basically, there are two viscosity coefficients, namely shear viscosity and bulk viscosity. Shear viscosity is related to velocity gradients in the fluid, and by considering the universe as described by the homogeneous and isotropic Friedmann–Lemâitre–Robertson–Walker (FLRW) background, it can be omitted. Anyhow, by dropping the assumption of the homogeneity and isotropy of the universe, several cosmological models with shear viscosity fluid have been constructed, as one can check, for instance, in [56,57,58,59]. On the other hand, bulk viscosity, which we are going to consider here, introduces damping associated with volumetric straining. To become familiar with bulk viscous fluid cosmological models, one can check References [60,61,62,63,64,65,66,67]. Moreover, some interesting applications of bulk viscous cosmology in black holes are presented in [68,69].

Researchers examine dark energy reconstruction with numerous observations as the data increase. The majority of studies have been concentrated on observable evidence from Type Ia supernovae, cosmic microwave background, and baryon acoustic oscillations (BAOs), which are known to be helpful in constraining cosmological models. The Hubble parameter dataset shows the intricate structure of the expansion of the universe. The ages of the most massive and slowly evolving galaxies offer direct measurements of the Hubble parameter $H\left(z\right)$ at various redshifts z, resulting in the development of a new form of standard cosmological probe [70].

In our present work, we include 31 measurements of Hubble expansion spanned using the differential age method [71] and BAO data consisting of six points [72]. Scolnic et al. published a large Type Ia supernovae sample named Pantheon, with 1048 points and covering the redshift range $0.01<z<2.3$ [73]. Our analysis uses the $H\left(z\right)$, BAO, and Pantheon samples to constrain the cosmological model.

This work aims to describe the recently observed late-time acceleration with the help of the bulk viscosity of the cosmic fluid (without including any dark energy component) in the framework of the $f\left(Q\right)$ theory of gravity. The manuscript is organized as follows: in Section 2, we discuss the $f\left(Q\right)$ gravity formalism. In Section 3, we describe the FLRW universe dominated by bulk viscous non-relativistic matter and derive the expression for the Hubble parameter and the deceleration parameter. Further, in Section 4, we analyze the observational data to find the best-fit ranges for the parameters using the Hubble dataset containing 31 points, the BAO sample, and the Pantheon dataset of 1048 samples. Moreover, we analyze the behavior of different cosmological parameters such as Hubble, density, effective pressure, deceleration parameter, and effective equation of state (EoS) parameter. In Section 5, we investigate the consistency of our bulk viscous fluid model by analyzing the different energy conditions. In Section 6, we analyze the behavior of statefinder parameters on the values constrained by the observational data to differentiate between dark energy models. Lastly, we discuss our results in Section 7.

2. Fundamental Formulations in $\mathit{f}\left(\mathit{Q}\right)$ Gravity

In $f\left(Q\right)$ gravity theory, the spacetime is established with the help of the non-metricity and symmetric teleparallelism condition, i.e., ${\nabla }_{\alpha }{g}_{\mu \nu }\ne 0$ and $R_{\sigma \mu \nu }^{\rho }=0$ in an torsionless environment. Hence, the associated affine connection is given by

$${{\rm Y}}_{\mu \nu }^{\alpha }={\Gamma }_{\mu \nu }^{\alpha }+L_{\mu \nu }^{\alpha },$$

(1)

with

$$\begin{array}{c}\hfill {\Gamma }_{\mu \nu }^{\alpha }\equiv \frac{1}{2}{g}^{\alpha \lambda }(g_{\mu \lambda ,\nu }+g_{\lambda \nu ,\mu }-g_{\mu \nu ,\lambda }),\end{array}$$

(2)

$$\begin{array}{c}\hfill L_{\mu \nu }^{\alpha }\equiv \frac{1}{2}(Q_{\mu \nu }^{\alpha }-Q_{\mu \nu }^{\alpha }-Q_{\nu \mu }^{\alpha }),\end{array}$$

(3)

being the Christoffel symbols and the distortion tensor, respectively, where

$$\begin{array}{c}\hfill Q_{\alpha \mu \nu }\equiv {\nabla }_{\alpha }{g}_{\mu \nu },\end{array}$$

(4)

is the non-metricity tensor.

The non-metricity tensor given by Equation (4) has the following two traces,

$$Q_{\alpha }=Q_{\alpha }{}^{\mu }{}_{\mu },$$

(5)

$${\tilde{Q}}_{\alpha }=Q^{\mu }{}_{\alpha \mu }.$$

(6)

In addition, the superpotential tensor or non-metricity conjugate is given by

$$\begin{array}{ccc}\hfill 4P^{\lambda }{}_{\mu \nu }& =& -Q_{\mu \nu }^{\lambda }+Q_{\mu \nu }^{\lambda }+Q_{\nu \mu }^{\lambda }+(Q^{\lambda }-{\tilde{Q}}^{\lambda })g_{\mu \nu }\hfill \\ & -& \frac{1}{2}({\delta }_{\mu }^{\lambda }{Q}_{\nu }+{\delta }_{\nu }^{\lambda }{Q}_{\mu }).\hfill \end{array}$$

(7)

Then, the trace of the non-metricity tensor can be acquired as [74]

$$Q=-Q_{\lambda \mu \nu }{P}^{\lambda \mu \nu }.$$

(8)

The definition of the energy–momentum tensor for matter is  

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

(9)

Furthermore, one can obtain the following relation between the curvature tensor $R_{\sigma \mu \nu }^{\rho }$ and ${\mathring{R}}_{\sigma \mu \nu }^{\rho }$ corresponding to the connection ${\rm Y}$ and $\Gamma$ as

$$R_{\sigma \mu \nu }^{\rho }={\mathring{R}}_{\sigma \mu \nu }^{\rho }+{\mathring{\nabla }}_{\mu }{L}_{\nu \sigma }^{\rho }-{\mathring{\nabla }}_{\nu }{L}_{\mu \sigma }^{\rho }+L_{\mu \lambda }^{\rho }{L}_{\nu \sigma }^{\lambda }-L_{\nu \lambda }^{\rho }{L}_{\mu \sigma }^{\lambda }$$

(10)

and so,

$$R_{\sigma \nu }={\mathring{R}}_{\sigma \nu }+\frac{1}{2}{\mathring{\nabla }}_{\nu }{Q}_{\sigma }+{\mathring{\nabla }}_{\rho }{L}_{\nu \sigma }^{\rho }-\frac{1}{2}{Q}_{\lambda }{L}_{\nu \sigma }^{\lambda }-L_{\nu \lambda }^{\rho }{L}_{\rho \sigma }^{\lambda }$$

(11)

$$R=\mathring{R}+{\mathring{\nabla }}_{\lambda }{Q}^{\lambda }-{\mathring{\nabla }}_{\lambda }{\tilde{Q}}^{\lambda }-\frac{1}{4}{Q}_{\lambda }{Q}^{\lambda }+\frac{1}{2}{Q}_{\lambda }{\tilde{Q}}^{\lambda }-L_{\rho \nu \lambda }{L}^{\lambda \rho \nu }$$

(12)

Now, the connection (1) can be parameterized as [31]

$${{\rm Y}}^{\alpha }{}_{\mu \beta }=\frac{\partial x^{\alpha }}{\partial {\xi }^{\rho }}{\partial }_{\mu }{\partial }_{\beta }{\xi }^{\rho }.$$

(13)

Here, ${\xi }^{\alpha }={\xi }^{\alpha }\left(x^{\mu }\right)$ is an invertible relation. Hence, it is always possible to find a coordinate system so that the connection ${{\rm Y}}_{\mu \nu }^{\alpha }$ vanishes. This situation is called the coincident gauge, and the covariant derivative ${\nabla }_{\alpha }$ reduces to the partial one ${\partial }_{\alpha }$. However, in any other coordinate system in which this affine connection does not vanish, the metric evolution will be affected and result in a completely different theory [75,76]. Thus, in the coincident gauge coordinate, we have

$$Q_{\alpha \mu \nu }={\partial }_{\alpha }{g}_{\mu \nu }$$

(14)

while, in an arbitrary coordinate system,

$$Q_{\alpha \mu \nu }={\partial }_{\alpha }{g}_{\mu \nu }-2{{\rm Y}}_{\alpha (\mu }^{\lambda }{g}_{\nu )\lambda }.$$

(15)

Locally, GR does not distinguish between gravitational and inertial effects; however, by invoking frame fields, it is possible to covariantly define gravitational energy in the teleparallel approach [77]. The canonical frame is identified in the absence of both curvature and torsion and the canonical coordinates in the absence of inertial effects. New physics can emerge from such a formalism. Symmetric teleparallel gravity is broadly derived in three gravity theories based on the coordinate transformations [78]. In this case, we use the spatially flat case, i.e., the $f\left(Q\right)$ gravity, whose field equations are much easier to understand.

The $f\left(Q\right)$ gravity is described by the action [31]:

$$S=\int \frac{1}{2}f\left(Q\right)\sqrt{-g}{d}^{4}x+\int L_m\sqrt{-g}{d}^{4}x,$$

(16)

in which $f\left(Q\right)$ is an arbitrary function of the non-metricity, $g=\det g_{\mu \nu }$, and $L_m$ is the Lagrangian density of matter. One can check in Reference [31] that the functional form $f\left(Q\right)=-Q$ corresponds to the symmetric teleparallel equivalent to General Relativity (STEGR) limit.

The gravitational field equations obtained by varying the action (16) with respect to the metric are

$$\frac{2}{\sqrt{-g}}{\nabla }_{\lambda }\left(\sqrt{-g}{f}_Q{P}^{\lambda }{}_{\mu \nu }\right)+\frac{1}{2}{g}_{\mu \nu }f+f_Q(P_{\mu \lambda \beta }{Q}_{\nu }{}^{\lambda \beta }-2Q_{\lambda \beta \mu }{P}^{\lambda \beta }{}_{\nu })=-{\mathcal{T}}_{\mu \nu },$$

(17)

in which we define $f_Q=df/dQ$.

3. The Cosmological Model

We consider the flat FLRW metric for our analysis [79], such that:

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

(18)

In Equation (18) above, $a\left(t\right)$ is the cosmic scale factor, as usual.

From now on, we will fix the coincident gauge so that connection becomes trivial and the metric is only a fundamental variable.

We are going to assume a bulk viscous fluid, and below, we present some considerations favoring such an assumption.

Firstly, considering bulk viscosity in a fluid can be seen as an attempt to refine its description, minimizing its ideal properties. This can be checked, for instance, in the stellar astrophysics realistic models in References [80,81,82].

Under the conditions of spatial homogeneity and isotropy (which refers to the cosmological principle, as one can check, for instance, in Reference [83]), the bulk viscous pressure is the unique admissible dissipative phenomenon. In a gas dynamical model, the existence of an effective bulk pressure can be traced back to a non-standard self-interacting force on the particles of the gas [63]. The bulk viscosity contributes negatively to the total pressure, as one can check, for instance, in References [84,85,86].

Due to spatial isotropy, the bulk viscous pressure is the same in all spatial directions and, hence, proportional to the volume expansion $\theta =3H$, with $H=\dot{a}/a$ being the Hubble parameter and the dot representing the time derivative.

The effective pressure of the cosmic fluid becomes [87,88,89]

$$\overline{p}=p-\zeta \theta =p-3\zeta H,$$

(19)

in which p is the usual pressure and $\zeta >0$ is the bulk viscosity coefficient, which we will assume as a free parameter of the model.

The corresponding energy–momentum tensor is given by

$${\mathcal{T}}_{\mu \nu }=(\rho +\overline{p})u_{\mu }{u}_{\nu }+\overline{p}{g}_{\mu \nu },$$

(20)

in which $\rho$ is the matter–energy density and the four-velocity $u^{\mu }$ is such that its components are $u^{\mu }=(1,0,0,0)$.

The relation between normal pressure and matter–energy density follows [90] $p=(\gamma -1)\rho$, with $\gamma$ being a constant lying in the range $0\le \gamma \le 2$. Then, the effective equation of state for the bulk viscous fluid is given by the following:

$$\overline{p}=(\gamma -1)\rho -3\zeta H.$$

(21)

The Friedmann-like equations for our $f\left(Q\right)$ gravitational model are obtained from the substitution of Equations (18)–(21) into Equation (11) and read as follows (check, for instance, References [74,91]):

$$3H^2=\frac{1}{2f_Q}\left(-\rho +\frac{f}{2}\right),$$

(22)

$$\dot{H}+\left(3H+\frac{\dot{f_Q}}{f_Q}\right)H=\frac{1}{2f_Q}\left(\overline{p}+\frac{f}{2}\right).$$

(23)

In particular, for $f\left(Q\right)=-Q$, we retrieve the usual Friedmann equations [91], as expected, since, as we have mentioned above, this particular choice for the functional form of the function $f\left(Q\right)$ is the STEGR limit of the theory.

For our investigation of the bulk viscosity fluid cosmological model, we consider the following $f\left(Q\right)$ functional form:

$$f\left(Q\right)=\alpha Q^n,$$

(24)

with $\alpha \ne 0$ and constant n. This particular functional form for $f\left(Q\right)$ was motivated by a polynomial form applied, for instance, in Reference [53]. Note that the parameter n considered in our model is not the spectral index; rather, it is a free model parameter.

For the above choice of the $f\left(Q\right)$ function (Equation (24)), we rewrite Equations (22)–(23) as follows

$$\rho =\alpha 6^n\left(\frac{1}{2}-n\right)H^{2n},$$

(25)

$$\dot{H}+\frac{3}{2n}{H}^2=\frac{6^{1-n}\overline{p}}{2\alpha n(2n-1)}{H}^{2(1-n)},$$

(26)

in which, for the former, we isolated $\rho$.

From Equations (26) and (21), we have the following:

$$\dot{H}+\frac{3\gamma }{2n}{H}^2=-\frac{6^{2-n}\zeta }{4\alpha n(2n-1)}{H}^{3-2n}.$$

(27)

Now, we replace the term $d/dt$ by $d/dlna$ via the expression $d/dt=Hd/dlna$, such that Equation (27) becomes

$$\frac{dH}{dlna}+\frac{3\gamma }{2n}H=-\frac{6^{2-n}\zeta }{4\alpha n(2n-1)}{H}^{2(1-n)}.$$

(28)

The integration of Equation (28) yields the following solution:

$$H\left(a\right)={\left\{{\left(H_0{a}^{-\frac{3\gamma }{2n}}\right)}^{2n-1}+\frac{6^{1-n}\zeta }{\gamma \alpha {(2n-1)}^2}[a^{-\frac{3\gamma (2n-1)}{2n}}-1]\right\}}^{\frac{1}{2n-1}},$$

(29)

with $H_0$ being a constant of integration to be found below.

We obtain the Hubble parameter in terms of redshift by using relation [83] $a\left(t\right)=1/(1+z)$ in Equation (30). By making $z=0$ in (30), we find that $H\left(0\right)=H_0$. The deceleration parameter is defined as $q=-\ddot{a}a/{\dot{a}}^2=-\ddot{a}/\left(H^{2}a\right)$. Henceforth, from Equation (29), we have

$$\begin{array}{ccc}\hfill H\left(z\right)& =& {\left\{{\left[H_0{(1+z)}^{\frac{3\gamma }{2n}}\right]}^{2n-1}+\frac{6^{1-n}\zeta }{\gamma \alpha {(2n-1)}^2}[{(1+z)}^{\frac{3\gamma (2n-1)}{2n}}-1]\right\}}^{\frac{1}{2n-1}},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill q\left(z\right)& =& \frac{3}{2n}\left\{\frac{\zeta }{\alpha 6^{n-1}(2n-1)\left\{{\left[H_0{(1+z)}^{\frac{3\gamma }{2n}}\right]}^{2n-1}+\frac{6^{1-n}\zeta }{\gamma \alpha {(2n-1)}^2}[{(1+z)}^{\frac{3\gamma (2n-1)}{2n}}-1]\right\}}+\gamma \right\}-1.\hfill \end{array}$$

(31)

4. Observational Constraints

To examine the observational features of our cosmological model, we used the cosmic Hubble and supernovae observations. We used the 31 points of the Hubble datasets, the 6 points of the BAO datasets, and the 1048 points from the Pantheon supernovae samples. We applied the Bayesian analysis and likelihood function along with the Markov chain Monte Carlo (MCMC) method in the emcee python library [92].

4.1. Hubble Datasets

The Hubble parameter can be expressed as $H\left(z\right)=-dz/\left[dt\right(1+z\left)\right]$. As $dz$ is derived from a spectroscopic survey, the model-independent value of the Hubble parameter may be calculated by measuring the quantity $dt$.

We incorporated the set of 31 data points that are measured from the differential age approach [93] to avoid extra correlation with the BAO data. The mean values of the model parameters $\zeta$, $\alpha$, $\gamma$ and n are calculated using the chi-squared function as follows:

$${\chi }_H^2(\zeta ,\alpha ,\gamma ,n)=\sum _{k=1}^{31}\frac{{[H_{th}(\zeta ,\alpha ,\gamma ,n,z_k)-H_{obs}\left(z_k\right)]}^2}{{\sigma }_{H\left(z_k\right)}^2}.$$

(32)

Here, $H_{th}$ is the Hubble parameter value predicted by the model, $H_{obs}$ represents its observed value, and the standard error in the observed value of H is ${\sigma }_{H\left(z_k\right)}$.

Figure 1 shows the error bar plot of the considered model and the $\Lambda$CDM or standard cosmological model, with cosmological constant density parameter ${\Omega }_{{\Lambda }_0}=0.7$, matter density parameter ${\Omega }_{m_0}=0.3$, and $H_0=69$ km/s/Mpc.

4.2. BAO Datasets

The BAO distance dataset, which includes the 6dFGS, SDSS, and WiggleZ surveys, comprises BAO measurements at six different redshifts in

Table 1. Values of $d_A\left(z_{*}\right)/D_V\left(z_{BAO}\right)$ for distinct values of $z_{BAO}$.

${\mathit{z}}_{\mathit{B}\mathit{A}\mathit{O}}$	$0.106$	$0.2$	$0.35$	$0.44$	$0.6$	$0.73$
$\frac{d_A\left(z_{*}\right)}{D_V\left(z_{BAO}\right)}$	$30.95\pm 1.46$	$17.55\pm 0.60$	$10.11\pm 0.37$	$8.44\pm 0.67$	$6.69\pm 0.33$	$5.45\pm 0.31$

. The characteristic scale of BAO is ruled by the sound horizon $r_s$ at the epoch of photon decoupling $z_{*}$, which is given by the following relation:

$$r_s\left(z_{*}\right)=\frac{c}{\sqrt{3}}{\int }_0^{\frac{1}{1+z_{*}}}\frac{da}{a^{2}H\left(a\right)\sqrt{1+(3{\Omega }_{b0}/4{\Omega }_{\gamma 0})a}}.$$

(33)

Here, ${\Omega }_{b0}$ and ${\Omega }_{\gamma 0}$ correspond to the present densities of baryons and photons, respectively.

The following relations are used in the BAO measurements:

$$\begin{array}{ccc}\hfill ▵\theta & =& \frac{r_s}{d_A\left(z\right)},\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill d_A\left(z\right)& =& {\int }_0^z\frac{d{z}^{\prime }}{H\left(z^{\prime }\right)},\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill ▵z& =& H\left(z\right)r_s,\hfill \end{array}$$

(36)

where $▵\theta$ represents the measured angular separation, $d_A$ is the angular diameter distance, and $▵z$ represents the measured redshift separation of the BAO feature in the two-point correlation function of the galaxy distribution on the sky along the line of sight.

In this work, the BAO datasets of six points for $d_A\left(z_{*}\right)/D_V\left(z_{BAO}\right)$ is taken from References [5,6,72,94,95,96], where the redshift at the epoch of photon decoupling is taken as $z_{*}\approx 1091$ and $d_A\left(z\right)$ is the co-moving angular diameter distance together with the dilation scale $D_V\left(z\right)={\left[d_A{\left(z\right)}^{2}z/H\left(z\right)\right]}^{1/3}$. The chi-squared function for the BAO dataset is taken to be [96]

$${\chi }_{BAO}^2=X^T{C}^{-1}X,$$

(37)

$$X=\left(\begin{array}{c}\frac{d_A\left(z_{\star }\right)}{D_V\left(0.106\right)}-30.95\\ \frac{d_A\left(z_{\star }\right)}{D_V\left(0.2\right)}-17.55\\ \frac{d_A\left(z_{\star }\right)}{D_V\left(0.35\right)}-10.11\\ \frac{d_A\left(z_{\star }\right)}{D_V\left(0.44\right)}-8.44\\ \frac{d_A\left(z_{\star }\right)}{D_V\left(0.6\right)}-6.69\\ \frac{d_A\left(z_{\star }\right)}{D_V\left(0.73\right)}-5.45\end{array}\right),$$

(38)

The inverse covariance matrix $C^{-1}$ is defined in [96]

$$C^{-1}=\left(\begin{array}{cccccc}0.48435& -0.101383& -0.164945& -0.0305703& -0.097874& -0.106738\\ -0.101383& 3.2882& -2.45497& -0.0787898& -0.252254& -0.2751\\ -0.164945& -2.454987& 9.55916& -0.128187& -0.410404& -0.447574\\ -0.0305703& -0.0787898& -0.128187& 2.78728& -2.75632& 1.16437\\ -0.097874& -0.252254& -0.410404& -2.75632& 14.9245& -7.32441\\ -0.106738& -0.2751& -0.447574& 1.16437& -7.32441& 14.5022\end{array}\right).$$

(39)

4.3. Pantheon Datasets

Scolnic et al. [73] put together the Pantheon samples consisting of 1048 Type Ia supernovae in the redshift range $0.01<z<2.3$. The PanSTARSS1 Medium Deep Survey, SDSS, SNLS, and numerous low-z and HST samples contribute to it. The empirical relation used to calculate the distance modulus of SNeIa from the observation of light curves is given by $\mu =m_B^{*}+\alpha X_1-\beta C-M_B+{\mathrm{\Delta }}_M+{\mathrm{\Delta }}_B$, where $X_1$ and C denote the stretch and color correction parameters, respectively [73,97], $m_B^{*}$ represents the observed apparent magnitude, and $M_B$ is the absolute magnitude in the B-band for SNeIa. The parameters $\alpha$ and $\beta$ are the two nuisance parameters describing the luminosity stretch and luminosity color relations, respectively. Further, the distance correction factor is ${\mathrm{\Delta }}_M$, and ${\mathrm{\Delta }}_B$ is a distance correction based on the predicted biases from simulations.

The nuisance parameters in the Tripp formula [98] were reconstructed using a novel technique called BEAMS with bias corrections [99,100], and the observed distance modulus was reduced to the difference between the corrected apparent magnitude $m_B$ and the absolute magnitude $M_B$, which is $\mu =m_B-M_B$. We shall avoid marginalizing over the nuisance parameters $\alpha$ and $\beta$, but marginalize over the Pantheon data for $M_B$. Hence, we ignored the values of $\alpha$ and $\beta$ for the present investigation of the model.

The luminosity distance read as

$$D_L\left(z\right)=(1+z){\int }_0^z\frac{cd{z}^{\prime }}{H\left(z^{\prime }\right)},$$

(40)

with c being the speed of light.

The ${\chi }^2$ function for Type Ia supernovae is obtained by correlating the theoretical distance modulus:

$$\begin{array}{ccc}\hfill \mu \left(z\right)& =& 5lo{g}_{10}{D}_L\left(z\right)+{\mu }_0,\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill {\mu }_0& =& 5log(1/H_{0}Mpc)+25,\hfill \end{array}$$

(42)

such that

$${\chi }_{SN}^2(\zeta ,\alpha ,\gamma ,n)=\sum _{k=1}^{1048}{\displaystyle \frac{{\left[{\mu }_{obs}\left(z_k\right)-{\mu }_{th}(\zeta ,\alpha ,\gamma ,n,z_k)\right]}^2}{{\sigma }^2\left(z_k\right)}},$$

(43)

where ${\mu }_{th}$ is the theoretical value of the distance modulus and ${\mu }_{obs}$ is the observed value, whereas ${\sigma }^2\left(z_k\right)$ is the standard error in the observed value.

Figure 2 shows the error bar plot of $\mu \left(z\right)$ for the considered model and the $\Lambda$CDM or standard cosmological model, with cosmological constant density parameter ${\Omega }_{{\Lambda }_0}=0.7$, matter density parameter ${\Omega }_{m_0}=0.3$, and $H_0=69$ km/s/Mpc.

The $1-\sigma$ and $2-\sigma$ likelihood contours for the model parameters using the Hubble, BAO, and Pantheon datasets is presented in Figure 3. The obtained best-fit values are presented in

Table 2. The table shows the constraints on the model parameters corresponding to the Hubble, BAO, and Pantheon datasets.

Data	$\mathit{\zeta }$	$\mathit{\alpha }$	$\mathit{\gamma }$	n
Hubble	$0.69\pm 0.11$	$-0.{0145}_{-0.0039}^{+0.0077}$	$1.{39}_{-0.11}^{+0.13}$	$0.982\pm 0.024$
BAO	$0.69\pm 0.13$	$-0.{0141}_{-0.0035}^{+0.0055}$	$1.{157}_{-0.071}^{+0.064}$	$0.{997}_{-0.014}^{+0.010}$
Pantheon	$0.69\pm 0.13$	$-0.{0118}_{-0.0028}^{+0.0062}$	$1.36\pm 0.14$	$0.994\pm 0.025$

.

4.4. Cosmological Parameters

The evolution of the Hubble parameter, deceleration parameter, energy density, pressure with the bulk viscosity, and the effective EoS parameter for the redshift range $-1<z<8$ are presented below, in order to test the late-time cosmic expansion history and the future of the expanding universe [101]. In order to do so, we used the set of values constrained by the Hubble, BAO, and Pantheon datasets for the model parameters.

From Figure 4, it is clear that the deceleration parameter shows the transition from a decelerated ($q>0$) to an accelerated ($q<0$) phase of the universe’s expansion for the constrained values of the model parameters. The transition redshift is $z_t\approx 0.236$, $z_t\approx 0.691$, and $z_t\approx 0.428$ corresponding to the Hubble, BAO, and Pantheon datasets, respectively. The present value of the deceleration parameter is, respectively, $q_0=-0.211$, $q_0=-0.384$, and $q_0=-0.346$.

From Figure 5 and Figure 6, it is clear that the Hubble and density parameters show the positive behavior for all the constrained values of the model parameters, which is expected.

Figure 7 indicates that the bulk viscous cosmic fluid exhibits, for lower redshifts, the negative pressure that makes the bulk viscosity a viable candidate to drive the cosmic acceleration. Furthermore, the effective EoS parameter presented in Figure 8 indicates that the cosmic viscous fluid behaves like quintessence dark energy. The present values of the EoS parameter corresponding to the Hubble, BAO, and Pantheon samples are ${\omega }_0=-0.48$, ${\omega }_0=-0.59$, and ${\omega }_0=-0.56$.

5. Energy Conditions

In the present section, we are going to construct the energy conditions for the solutions of the present model. The energy conditions are the relations applied to the matter energy–momentum tensor with the purpose of satisfying positive energy. The energy conditions are derived from the Raychaudhuri equation and are written as [102]:

• Null energy condition (NEC):${\rho }_{eff}+p_{eff}\ge 0$;
• Weak energy condition (WEC):${\rho }_{eff}\ge 0$ and ${\rho }_{eff}+p_{eff}\ge 0$;
• Dominant energy condition (DEC):${\rho }_{eff}\pm p_{eff}\ge 0$;
• Strong energy condition (SEC):${\rho }_{eff}+3p_{eff}\ge 0$.

with ${\rho }_{eff}$ being the effective energy density.

In Figure 9 and Figure 10, it is evident that the NEC and DEC exhibit the positive behavior for all the constrained values of the model parameters. As the WEC is the combination of energy density and the NEC, we conclude that the NEC, DEC, and WEC are all satisfied in the entire domain of redshift. Figure 11 indicates that the SEC exhibits, for lower redshifts, the negative behavior that is related to cosmic acceleration [53,103]. This is also reflected in the deceleration parameter behavior in Figure 4.

6. Statefinder Analysis

The cosmological constant $\Lambda$ suffers from two major drawbacks, namely the aforementioned cosmological constant and cosmic coincidence problems. To surmount these problems, dynamic models of dark energy have been introduced in the literature, as we have also mentioned in the Introduction. To discriminate between these time-varying dark energy models, an appropriate tool is required. In this direction, V. Sahni et al. introduced a new pair of geometrical parameters known as statefinder parameters $(r,s)$ [104]. The statefinder parameters are defined as

$$\begin{array}{ccc}\hfill r& =& \frac{\stackrel{⃛}{a}}{a{H}^3},\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill s& =& \frac{(r-1)}{3(q-\frac{1}{2})}.\hfill \end{array}$$

(45)

The parameter r can be rewritten as $r=2q^2+q-\frac{\dot{q}}{H}$.

For different values of the statefinder pair $(r,s)$, it represents the following dark energy models:

$r=1,s=0$ represents the $\Lambda$CDM model;

$r>1,s<0$ represents the Chaplygin gas model;

$r<1,s>0$ represents the quintessence model.

In Figure 12 and Figure 13, we plot the $s-r$ and $q-r$ diagrams for our cosmological model by taking the values of the parameters constrained by the Hubble, BAO, and Pantheon datasets.

Figure 12 and Figure 13 show that our bulk viscous model lies in the quintessence region. Furthermore, the evolutionary trajectories of our model depart from the $\Lambda$CDM point. The present values of the statefinder parameters corresponding to the values of the model parameters constrained by the Hubble, BAO, and Pantheon samples are $r_0=0.828$ and $s_0=0.08$, $r_0=0.592$ and $s_0=0.15$, and $r_0=0.773$ and $s_0=0.089$, respectively.

7. Discussions and Conclusions

In the present section, we will discuss the results obtained in Section 4, Section 5 and Section 6 for the bulk viscous symmetric teleparallel cosmological model here developed and presented.

Cosmology has been on the agenda mainly for two reasons: dark energy and dark matter. While dark energy was deeply discussed throughout the paper, dark matter is predicted within the $\Lambda$CDM model as a sort of matter that does not interact electromagnetically, so that it cannot be seen, but its gravitational effects otherwise can well be detected. Still, we have not yet detected or even associated dark matter with a particle of the standard model or beyond [105,106,107]. Modified (or alternative) theories of gravity have also been used to describe dark matter’s effects [108,109]. In these cases, dark matter is simply an effect of the modification of gravity.

Returning to the dark energy question, it is highly counter-intuitive that the expansion of the universe is actually accelerating. Although the vacuum quantum energy can well explain this dynamical effect via the cosmological constant in Einstein’s field equations of GR, the aforementioned important and persistent problems related to $\Lambda$ supply the search for alternative explanations.

In the present article, as an attempt to describe dark energy, we assumed the symmetric teleparallel gravity as the underlying gravity theory.

The $f\left(Q\right)$ gravity was recently proposed by Jiménez et al. in [31] as a ramification of the geometric trinity, which says that the spacetime manifold can be described by curvature, torsion, or non-metricity. In particular the symmetric teleparallel gravity describes gravitational interactions via the non-metricity scalar, with null curvature and torsion.

Our $f\left(Q\right)$ cosmological model was based on a spatially homogeneous and isotropic flat metric and an energy–momentum tensor describing a bulk viscous fluid. We let the $f\left(Q\right)$ function be a power of n as $f\left(Q\right)\sim Q^n$, with n a free parameter.

In Section 4, we started testing our cosmological solutions. We started confronting the Hubble parameter (30) with 31 data points that are measured from the differential age approach.We then plotted Figure 1 and Figure 2, in which $H\left(z\right)$ and $\mu \left(z\right)$ for our $f\left(Q\right)$ model are confronted with cosmological data and compared with the $\Lambda$CDM prediction. We can see the $f\left(Q\right)$ model describes observations with good agreement, and it is clear that it provides a better fit when compared to the $\Lambda$CDM model. Further, in Figure 3, we obtained the best-fit values for the model free parameters presented in Table 2.

From Figure 5 and Figure 6, we found that the Hubble and density parameters show the expected positive behavior for all the constrained values of the model parameters. Figure 7 indicates that the bulk viscous cosmic fluid exhibits the negative pressure that makes the bulk viscosity a viable candidate to drive the cosmic acceleration. This is also reflected in the deceleration parameter behavior in Figure 4, which shows a transition from decelerated to accelerated phases of the universe’s expansion. Furthermore, the effective EoS parameter presented in Figure 8 indicates that the cosmic viscous fluid behaves like quintessence dark energy.

In Section 5, we investigated the consistency of our model by analyzing the different energy conditions. We found that the NEC, DEC, and WEC are all satisfied in the entire domain of redshift (presented in Figure 9 and Figure 10), while the SEC, presented in Figure 11, is violated for lower redshifts, which implies the cosmic acceleration, and satisfied for higher redshifts, which implies a decelerated phase of the universe.

In Section 6, Figure 12 and Figure 13 show that the evolutionary trajectories of our model depart from the $\Lambda$CDM fixed point $r=1,s=0$. In the present epoch, they lie in the quintessence region $r<1,s>0$. The present model is, therefore, a good alternative to explain the universe’s dynamics, particularly with no necessity of invoking the cosmological constant.