New Parametrization of the Dark-Energy Equation of State with a Single Parameter

Abstract

We propose a novel dark-energy equation-of-state parametrization, with a single parameter $\eta$ that quantifies the deviation from $\Lambda$CDM cosmology. We first confront the scenario with various datasets, from the Hubble function (OHD), Pantheon, baryon acoustic oscillations (BAO), and their joint observations, and we show that $\eta$ has a preference for a non-zero value, namely, a deviation from $\Lambda$CDM cosmology is favored, although the zero value is marginally inside the 1$\sigma$ confidence level. However, we find that the present Hubble function value acquires a higher value, namely, $H_0=66.{624}_{-0.013}^{+0.011}$ Km s⁻¹ Mpc⁻¹, which implies that the $H_0$ tension can be partially alleviated. Additionally, we perform a cosmographic analysis, showing that the universe transits from deceleration to acceleration in the recent cosmological past; nevertheless, in the future, it will not result in a de Sitter phase since it exhibits a second transition from acceleration to deceleration. Finally, we perform the statefinder analysis. The scenario behaves similarly to the $\Lambda$CDM paradigm at high redshifts, while the deviation becomes significant at late and recent times and especially in the future.

1. Introduction

The understanding of fundamental physics is challenged by the universe’s rapid expansion, which is the most exciting subject in modern cosmology. This phenomenon indicates that, contrary to popular belief, the cosmos is expanding faster with time. It was initially discovered in the late 20th century through observations of distant supernovae. Under the general relativistic framework, the first general class of explanation involves introducing new and exotic sectors in the universe content, under the umbrella term dark energy [1,2]. Extending the fundamental principles of gravity constitutes the second general class, in which case the cause of acceleration is of gravitational origin [3,4,5,6,7].

Nevertheless, at the phenomenological level, both approaches can be quantified through the (effective) dark-energy equation-of-state parameter $w_x$. Thus, introducing various parametrizations of $w_x$ allows us to describe the universe’s evolution and confront with observational datasets to reveal the required dark-energy features to obtain agreement. Several researchers have embraced a phenomenological strategy for modeling the equation of state (EoS) parameter, representing it as a function of redshift, denoted as $\omega =\omega \left(z\right)$ [8], or equivalently as a function of the cosmic scale factor $\omega =\omega \left(a\right)$ [9,10]. This methodology streamlines the examination of dark energy (DE) dynamics and enables the exploration of physically significant functions. However, during data fitting, it becomes necessary to simplify the parameterization of the equation of state $\omega \left(z\right)$ and subsequently restrict the evolution of $\omega \left(z\right)$ based on the parameters we introduced in our parametrization. In particular, starting from the simple cosmological constant, a large number of parametrizations have been introduced in the literature, involving one parameter [11,12] or two parameters, such as the Chevallier–Polarski–Linder (CPL) parametrization [13,14], the linear parametrization [15,16,17], the logarithmic parametrization [18], the Jassal–Bagla–Padmanabhan parametrization (JBP) [19], the Barboza–Alcaniz (BA) parametrization [20], $H_0$ problem at low reshift [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50], etc. Additionally, note that one can impose the parametrization at the deceleration parameter level [51,52,53], at the equation-of-sate (EoS) parameter level [54], or even at the Hubble parameter level [55,56,57,58].

In the present manuscript, we propose a novel dark-energy equation-of-state parametrization, with a single parameter $\eta$ that quantifies the deviation from $\Lambda$CDM cosmology. Additionally, under this scenario, dark energy behaves like a cosmological constant at high redshifts, while the deviation becomes significant at low and recent redshifts, especially in the future. Finally, for $\eta =0$, we recover the $\Lambda$CDM cosmology completely. As we will see, apart from being capable of fitting the data, the new parametrization can partially alleviate the $H_0$ tension too since it leads to a $H_0$ value in between the Planck one and the one from direct measurements [59,60]. Recently, several authors have conducted so many remarkable works regarding the measurements of $H_0$ obtained from cosmological probes for different redshifts, such as a bias-free cosmological analysis with quasars alleviating $H_0$ tension [61]; new statistical insights and cosmological constraints consisting of gamma-ray bursts, quasars, baryonic acoustic oscillations, and supernovae Ia [62]; reduced uncertainties up to $43\%$ on the Hubble constant and the matter density with the SNIa with a new statistical analysis [63]; on the Hubble constant tension in the SNIa Pantheon sample [64]; on the evolution of the Hubble constant with the SNIa Pantheon sample and baryon acoustic oscillations: a feasibility study for GRB-cosmology in 2030 [65]; and $f\left(R\right)$ gravity in the Jordan frame as a paradigm for the Hubble tension, in which the authors provides a subsequent interpretation of the results through an effective Hubble constant that evolves with the redshift in a $f\left(R\right)$ modified gravity theory in the Jordan frame [66].

The article is organized as follows. In Section 2, we present the novel dark-energy parametrization. Then, in Section 3, we perform a detailed confrontation with observations, namely, with Hubble function (OHD), Pantheon, and baryon acoustic oscillations (BAO) data. In Section 4, we perform a cosmographic analysis, and we apply the statefinder diagnostic. Finally, Section 5 is devoted to the conclusions. Lastly, the details of the various datasets and the corresponding fitting procedure are given in the Appendix A.

2. New Single-Parameter Equation-of-State Parametrization

In this section, we first briefly review the basic equations of any cosmological scenario, and then we introduce the new parametrization for the dark-energy equation of state, with just a single parameter. We consider the usual homogeneous and isotropic Friedmann–Robertson–Walker (FRW) metric

$$\begin{array}{c}\hfill \mathrm{d}{s}^2=-\mathrm{d}{t}^2+a^2\left(t\right)\left[\frac{\mathrm{d}{r}^2}{1-K{r}^2}+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)\right],\end{array}$$

(1)

with $a\left(t\right)$ the scale factor and K the spatial curvature parameter ($K=0,-1,+1$ for spatially flat, open, and closed universe, respectively). Furthermore, we consider that the universe is filled with baryonic and dark matter, radiation, and effective dark-energy fluid. Hence, the Friedmann equations that determine the background evolution of the universe are

$$\begin{array}{ccc}\hfill H^2+\frac{K}{a^2}& =& \frac{8\pi G}{3}{\rho }_{tot},\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill 2\dot{H}+3H^2+\frac{K}{a^2}& =& -8\pi G{p}_{tot},\hfill \end{array}$$

(3)

where G is the gravitational constant, and $H=\dot{a}/a$ is the Hubble function, with dots marking time derivatives. The total energy density and pressure are thus given as ${\rho }_{tot}={\rho }_r+{\rho }_b+{\rho }_c+{\rho }_x$ and $p_{tot}=p_r+p_b+p_c+p_x$, where the subscripts $r,b,c,x$ stand respectively for radiation, baryon, cold dark matter, and dark energy. As usual, and without loss of generality, we focus on the spatially flat case, and therefore in the following we impose $K=0$. Finally, assuming that the various sectors do not interact mutually, we deduce that they are separately conserved, following the conservation equations

$$\begin{array}{c}\hfill {\dot{\rho }}_i+3H(1+w_i){\rho }_i=0,\end{array}$$

(4)

where $i\in \{r,b,c,x\}$. In the above expression, we have introduced the equation-of-state parameter of each fluid as $p_i\equiv w_i{\rho }_i$, which yields

$$w_i=-\left(1+\frac{a}{3{\rho }_i}\frac{d{\rho }_i}{da}\right).$$

(5)

We proceed by providing, for completeness, the evolution equations of the universe at the perturbation level. In the synchronous gauge, the perturbed metric reads as

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

(6)

with $\tau$ as the conformal time, and where ${\delta }_{ij}$ and $h_{ij}$ denote the unperturbed and the perturbed metric parts (with $h=h_j^j$ the trace). Perturbing additionally the universe fluids and transforming to the Fourier space, we finally extract [67,68,69]:

$$\begin{array}{ccc}& & {\delta }_i^{\prime }=-(1+w_i)\left({\theta }_i+\frac{h^{\prime }}{2}\right)-3\mathcal{H}\left(\frac{\delta p_i}{\delta {\rho }_i}-w_i\right){\delta }_i-9{\mathcal{H}}^2\left(\frac{\delta p_i}{\delta {\rho }_i}-c_{a,i}^2\right)(1+w_i)\frac{{\theta }_i}{k^2},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & {\theta }_i^{\prime }=-\mathcal{H}\left(1-3\frac{\delta p_i}{\delta {\rho }_i}\right){\theta }_i+\frac{\delta p_i/\delta {\rho }_i}{1+w_i}{k}^2{\delta }_i-k^2{\sigma }_i,\hfill \end{array}$$

(8)

with primes denoting the conformal-time derivative and with $\mathcal{H}=a^{\prime }/a$ denoting the conformal Hubble function, and where k is the mode wave number. Moreover, ${\delta }_i=\delta {\rho }_i/{\rho }_i$ stands for the over density of the i-th fluid, ${\theta }_i\equiv i{k}^j{v}_j$ marks the divergence of the i-th fluid velocity, and ${\sigma }_i$ is the corresponding anisotropic stress. Lastly, $c_{a,i}^2={\dot{p}}_i/{\dot{\rho }}_i$ is the adiabatic sound speed given as $c_{a,i}^2=w_i-\frac{w_i^{\prime }}{3\mathcal{H}(1+w_i)}$.

Let us now introduce the new dark-energy parametrization. As usual, knowing the equation of state of a fluid allows us to extract its time evolution by solving Equation (4). For radiation, we have $w_r=1/3$, and thus we obtain ${\rho }_r={\rho }_{r_0}{a}^{-4}$ (setting the scale factor at present to 1), while for the baryonic and dark matter, we have $w_b=w_c=0$, which leads to ${\rho }_b={\rho }_{b_0}{a}^{-3}$ and ${\rho }_c={\rho }_{c_0}{a}^{-3}$, where ${\rho }_{i_0}$ stands for the present density value of the i-th fluid. Concerning the equation-of-state parameter of the dark-energy sector, since it is unknown, as we mentioned in the Introduction, one can consider various parametrizations. Focusing on the barotropic fluid sub-class, we consider that it is a function of time only, or equivalently, of the scale factor a, i.e., $w_x\left(a\right)$. Hence, the solution of the EoS Equation (5) leads to

$$\begin{array}{c}\hfill {\rho }_x\left(a\right)={\rho }_{x_0}{a}^{-3}exp\left[-3{\int }_1^a\frac{w_x\left(a^{\prime }\right)}{a^{\prime }}d{a}^{\prime }\right].\end{array}$$

(9)

In this work, we consider a novel dark-energy equation-of-state parameter (5) as follows:

$$w_x\left(a\right)=-1+\frac{a^{-\eta }{e}^{-2\eta a}\eta arctan{a}^{-\eta }}{3(1+a^{-2\eta })},$$

(10)

where $\eta$ is the single parameter. Hence, introducing for convenience the redshift z as the independent variable (where $a^{-1}=1+z$), the above relation becomes

$$w_x\left(z\right)=-1+\frac{{(1+z)}^{\eta }{e}^{\frac{-2\eta }{(1+z)}}\eta arctan{(1+z)}^{\eta }}{3(1+{(1+z)}^{2\eta })}.$$

(11)

Relation (11) is the parametrization that we propose, and in the case $\eta =0$, we recover the $\Lambda$CDM concordance model, where $w_x=-1$ and ${\rho }_x={\rho }_{x_0}=const$, but in the general case, the parameter $\eta$ quantifies the deviation from the $\Lambda$CDM scenario. However, note that for general $\eta$, for large redshifts, i.e., for $z\to \infty$, we acquire $w_x\to -1$, which implies that the deviation from the $\Lambda$CDM scenario disappears in this regime, and thus, the Big Bang nucleosynthesis bounds are immediately satisfied.

Inserting the above parametrization in the first Friedmann Equation (2), we obtain

$$H=H_0\sqrt{\left[({\Omega }_{b_0}+{\Omega }_{c_0}){(1+z)}^3+{\Omega }_{r_0}{(1+z)}^4+{\Omega }_{x_0}{e}^{\frac{\eta z}{1+z}}\frac{arctan{(1+z)}^{\eta }}{arctan1}\right]},$$

(12)

where $H_0$ is the present value of the Hubble parameter, and where we have introduced the present values of the density parameters ${\Omega }_{i_0}\equiv \frac{8\pi G}{3H^2}{\rho }_{i_0}$ (hence, the present value of the total matter density parameter is ${\Omega }_{m_0}\equiv {\Omega }_{b_0}+{\Omega }_{c_0}$). This expression allows us to investigate the cosmological evolution in detail, and confront it with observational datasets. Actually, expression (12), which is a simple deviation from $\Lambda$CDM cosmology only at small redshifts, while it recovers $\Lambda$CDM scenario at high redshifts, was the motivation behind parametrization (10).

Lastly, from parametrization (11) and the corresponding Hubble function (12), we can straightforwardly calculate various quantities. In particular, the deceleration parameter $q=-1-\dot{H}{H}^{-2}$ is given by

$$q\left(z\right)=-1+\frac{\left[3{(1+z)}^4{\Omega }_{m_0}+\frac{4\eta {\Omega }_{x_0}{(1+z)}^2{e}^{\frac{\eta z}{1+z}}arctan{(1+z)}^{\eta -1}}{\pi (z^2+2z+2)}+\frac{4\eta {\Omega }_{x_0}{e}^{\frac{\eta z}{1+z}}arctan{(1+z)}^{\eta }}{\pi }\right]}{2(1+z)\left[{(1+z)}^3{\Omega }_{m_0}+\frac{4{\Omega }_{x_0}{e}^{\frac{\eta z}{1+z}}arctan{(1+z)}^{\eta }}{\pi }\right]},$$

(13)

while the higher-order cosmographic parameters [70] read as

$$\begin{array}{ccc}& & j=-q+2q(1+q)+(1+z)\frac{dq}{dz},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & s=j-3j(1+q)-(1+z)\frac{dj}{dz},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & l=s-4s(1+q)-(1+z)\frac{ds}{dz},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & m=l-5l(1+q)-(1+z)\frac{dl}{dz}.\hfill \end{array}$$

(17)

Similarly, for the matter and dark-energy density parameters, we obtain

$${\Omega }_m\left(z\right)=\frac{1}{1+\frac{4{\Omega }_{x_0}{e}^{\frac{\eta z}{1+z}}arctan{(1+z)}^{\eta }}{\pi {\Omega }_{b_0}{(1+z)}^3}},$$

(18)

and

$${\Omega }_x\left(z\right)=\frac{1}{1+\frac{\pi {\Omega }_{b_0}{e}^{\frac{-\eta z}{1+z}}[{(1+z)}^{3}arctan{(1+z)}^{-\eta }]}{4{\Omega }_{x_0}}}.$$

(19)

3. Observational Constraints

In the previous section, we proposed a new parametrization for the dark-energy equation of state, given by (11), which has a single parameter, namely, $\eta$. In this section, we perform a detailed confrontation with various datasets [71,72,73,74,75], focusing on the bounds of $\eta$. In particular, we will use data from (i) Hubble function observations (OHD) with 77 data points [76], (ii) Pantheon with 1048 data points [77], and (iii) baryon acoustic oscillations (BAO). The details of the datasets and the corresponding methodology are given in the Appendix A. In our analysis, we use the following priors: $H_0\in [66,70]$, ${\Omega }_{m_0}\in [0.1,0.4]$, ${\Omega }_{x_0}\in [0.6,0.8]$ and $\eta \in [0,1]$.

Let us now present the constraints we obtain after applying the above formalism and datasets in the Friedmann equations at hand, focusing on the new model parameter $\eta$. In Figure 1, Figure 2, Figure 3 and Figure 4, we present the likelihood contours with $1\sigma$ and $2\sigma$ confidence levels, around the best-fit values. Additionally, in

Table 1. Summary of the observational constraints on the model parameters $H_0$, ${\Omega }_{m0}$, ${\Omega }_{x0}$ and $\eta$ from various datasets.

Dataset	${\mathit{H}}_0$ (km/s/Mpc)	${\mathbf{\Omega }}_{\mathit{m}0}$	${\mathbf{\Omega }}_{\mathit{x}0}$	$\mathit{\eta }$
$H\left(z\right)$ (77 points data)	$66.{619}_{-0.0094}^{+0.012}$	$0.{2922}_{-0.010}^{+0.0092}$	$0.{650}_{-0.0097}^{+0.012}$	$0.{2392}_{-0.0077}^{+0.0077}$
$Pantheon$	$66.{623}_{-0.011}^{+0.015}$	$0.{2911}_{-0.0094}^{+0.0081}$	$0.{6486}_{-0.0087}^{+0.0059}$	$0.{2399}_{0.0059}^{0.0059}$
$BAO$	$66.{6242}_{-0.0089}^{+0.0089}$	$0.{2939}_{-0.011}^{+0.0087}$	$0.{650}_{-0.010}^{+0.013}$	$0.{2370}_{-0.011}^{+0.0094}$
$H\left(z\right)$ + $Pantheon$ + $BAO$	$66.{623}_{-0.0079}^{+0.011}$	$0.{2927}_{-0.011}^{+0.0090}$	$0.{6508}_{-0.0085}^{+0.0085}$	$0.{2385}_{-0.011}^{+0.0065}$

, we summarize the obtained results. Finally, in

Table 2. Summary of the constraints on the deceleration parameter q, the transition redshift $z_{tr}$ and the current value of the dark-energy equation-of-state parameter $w_{x_0}$.

Dataset	${\mathit{q}}_0$	${\mathit{z}}_{\mathbf{tr}}$	${\mathit{w}}_{{\mathit{x}}_0}$
$H\left(z\right)$ (77 points data)	$-0.449196$	≃0.7804	$-0.704911$
$Pantheon$	$-0.449152$	≃0.7811	$-0.704047$
$BAO$	$-0.449141$	≃0.7759	$-0.707625$
$H\left(z\right)$ + $Pantheon$ + $BAO$	$-0.449532$	≃0.7804	$-0.705775$

, we summarize other cosmological parameters, such as the density parameters and the deceleration parameter, the equation-of-state parameters, and the transition redshift.

As we observe, the new model parameter $\eta$ that quantifies the deviation from $\Lambda$CDM cosmology has a preference for a non-zero value, although zero is marginally inside the 1$\sigma$ confidence level. Concerning $H_0$, we observe that we obtain a higher value compared to the $\Lambda$CDM scenario, although a bit lower than the direct measurements $H_0=(73.04\pm 1.04)$ Km s⁻¹ Mpc⁻¹ at $68\%$ CL, based on the supernovae calibrated by Cepheids [59], which implies that the new dark-energy parametrization at hand can partially alleviate the $H_0$ tension. This is an additional result of the present work.

Figure 2 is plotted using the Pantheon dataset comprising 1048 data points, and a full covariance matrix is used. In the case of Figure 2, we notice that the shape of the contour is not oval, i.e., the posterior distribution for $\eta$ behaves as a bimodal distribution, and therefore we need to test the convergence for the Pantheon dataset. The Gelman–Rubin convergence test [78] is one of the statistical tools in Bayesian inference that is widely used to access the convergence of the chains. The test is based on the idea that multiple MCMC chains with different starting points should converge to the same posterior distribution if they have been run for long enough, that is, after several steps. Essentially, this test evaluates, for each parameter of the discussed model, the term called potential scale reduction $\widehat{R}$, which is the ratio between the variance $W$ within a chain and the variance Var($\theta$) among the chains:

$$\widehat{R}=\sqrt{\frac{\mathrm{Var}\left(\theta \right)}{W}}.$$

(20)

Also, in the likelihood contour of the Pantheon dataset, there is a possibility of $\pm 1.2$ for the $H_0$ value and $\pm 0.29$ for the $\alpha$ value within 100 steps of burn-in, which were used to run the MCMC chains.

The maximum Gelman–Rubin diagnostic across the model parameters is labeled as Max Gelman–Rubin $\widehat{R}$ in the header and is less than $1.2$. Refs. [78,79] suggest that diagnostic $\widehat{R}$ values greater than $1.2$ for any of the model parameters should indicate non-convergence. The contour plots in the plane $\eta -H_0$ with $1\sigma$ and $2\sigma$ errors are given in Figure 2, and the corresponding best-fit values of $H_0$, and $\eta$ for different datasets are given in Table 1.

4. Cosmographic Analysis and Statefinder Diagnostic

In this section, for completeness, we perform the cosmographic analysis of the cosmological scenario with the new dark-energy parametrization, and we apply the statefinder diagnostic. For simplicity, we neglect the radiation sector.

Let us start with the deceleration parameter given in (13). Using the best-fit values of the model parameters given in Table 1 and Table 2, we plot $q\left(z\right)$ in Figure 5. As we can see, we obtain the transition from deceleration to acceleration at the transition redshift $z_{tr}$ in agreement with observations. However, it is interesting to note that the novel parametrization at hand will lead to a second transition in the future, at redshift $z_{t{r}_2}$, from acceleration to deceleration (at around $z_{t{r}_2}\approx -0.9$).

We proceed to the examination of the other cosmographic parameters given in Equations (14)–(17). In particular, we use the best-fit values of the model parameters given in Table 1 and Table 2, and in Figure 6, we present their evolution. Additionally, in

Table 3. Summary of the constraints on the present values of the cosmographic parameters, namely jerk j, snap s, lerk l and m parameters, as well as on the present values of the statefinder diagnostic parameters r and $s^{*}$.

Dataset	j	s	l	m	r	${\mathit{s}}^{*}$
$H\left(z\right)$ (77 points data)	$0.506092$	$-1.02864$	$2.37997$	$-10.7706$	$0.{506}_{-0.269}^{+0.341}$	$0.{174}_{-0.123}^{+0.124}$
$Pantheon$	$0.504616$	$-1.02915$	$2.37783$	$-10.7529$	$0.{505}_{-0.200}^{+0.238}$	$0.{174}_{-0.076}^{+0.074}$
$BAO$	$0.51083$	$-1.02745$	$2.38876$	$-10.8356$	$0.{511}_{-0.201}^{+0.227}$	$0.{172}_{-0.073}^{+0.074}$
$H\left(z\right)$ + $Pantheon$ + $BAO$	$0.507417$	$-1.02752$	$2.37906$	$-10.7734$	$0.{508}_{-0.198}^{+0.225}$	$0.{1729}_{-0.074}^{+0.073}$

we summarize their values at present. Since in the $\Lambda$CDM paradigm, the value of the jerk parameter is equal to unity ($j=1$), the deviation from $j=1$ quantifies the deviation of a dark-energy scenario from the concordance model. Again, we find that the new proposed dark-energy parametrization behaves similarly to $\Lambda$CDM scenario at high redshifts, while the deviation becomes more significant at late and recent times, and especially in the future. Finally, the same features can be obtained from the evolution of the snap s, lerk l, and m parameters.

Let us now come to the statefinder diagnostic, which is based on higher derivatives of the scale factor [80,81,82]. In particular, one introduces a pair of geometrical parameters $\{r,s^{*}\}$ in order to examine the dynamics of different dark-energy models [83,84]. The pair of parameters $\{r,s^{*}\}$ are defined as:

$$r=\frac{\stackrel{⃛}{a}}{a{H}^3},s^{*}=\frac{r-1}{3(q-\frac{1}{2})},$$

(21)

with $q\ne \frac{1}{2}$. For our parametrization (9), the expression for r is found to be

$$r=\frac{r_1+r_2+r_3}{{(1+z)}^2{[2+z(2+z)]}^2{tan}^{-1}{(1+z)}^2\left[\pi {(1+z)}^3{\Omega }_{b_0}+4e^{\frac{\eta z}{1+z}}{\Omega }_{x_0}{tan}^{-1}{(1+z)}^{\eta }\right]},$$

(22)

where

$$\begin{array}{ccc}& & r_1=\pi {(1+z)}^5{[2+z(2+z)]}^2{\Omega }_{m_0}{tan}^{-1}{(1+z)}^2+2e^{\frac{\eta z}{1+z}}{(1+z)}^4(\eta -1)\eta {\Omega }_{x_0}{tan}^{-1}{(1+z)}^{\eta },\hfill \\ & & r_2=4e^{\frac{\eta z}{1+z}}\eta \left\{2\eta -3+z[2\eta -7+z(\eta -2z-6\left)\right]\right\}{\Omega }_{x_0}{tan}^{-1}{(1+z)}^{1+\eta },\hfill \\ & & r_3=2e^{\frac{\eta z}{1+z}}{[2+z(2+z)]}^2[2{(1+z)}^2-4(1+z)\eta +{\eta }^2]{\Omega }_{x_0}{tan}^{-1}{(1+z)}^{2+\eta }.\hfill \end{array}$$

(23)

Finally, the expression for $s^{*}$ is obtained using (13) and (22).

In Figure 7, we present trajectories for different observational datasets in the $q-r$ plane. As we can see, all trajectories start from the decelerating zone, enter into the accelerating zone behaving close to $\Lambda$CDM at present, and in the far future, they converge to the CDM model without a cosmological constant (namely, the $SCDM$ model) without resulting to the de Sitter ($dS$) phase. The present values of parameters $\{r,s^{*}\}$ of the statefinder diagnostic are also given in Table 3. As we can see, the new parametrization at hand at high redshifts behaves like $\Lambda$CDM, while the deviation appears at small redshifts and present time.

5. Conclusions

In this work, we proposed a novel dark-energy equation-of-state parametrization, with a single parameter $\eta$ that quantifies the deviation from $\Lambda$CDM cosmology. Firstly, we confronted the scenario with various datasets, from the Hubble function (OHD), Pantheon dataset, baryon acoustic oscillations (BAO) observations, and their joint dataset, and we presented the corresponding likelihood contours. As we saw, the new model parameter $\eta$ has a preference for a non-zero value, namely, a deviation from $\Lambda$CDM cosmology is favored, although the zero value is marginally inside the 1$\sigma$ confidence level. However, interestingly enough, we found that $H_0$ acquires a higher value compared to the $\Lambda$CDM scenario, which implies that the new dark-energy parametrization at hand can partially alleviate the $H_0$ tension.

Additionally, we performed a cosmographic analysis, examining the cosmographic parameters, namely, deceleration q, jerk j, snap s, lerk l, and m parameters. As we showed, in the scenario at hand, the universe transits from deceleration to acceleration in the recent cosmological past; however, in the future it will not result in a de Sitter phase since a second transition (at $z_{t{r}_2}\approx -0.9$) will lead from acceleration to deceleration. Additionally, we found that the scenario behaves similarly to the $\Lambda$CDM paradigm at high redshifts, while the deviation becomes significant at late and recent times (and thus, the $H_0$ tension is alleviated), and especially in the future.

Finally, we performed a statefinder diagnostic analysis. As we saw, all trajectories start from the decelerating zone, enter into the accelerating zone behaving close to $\Lambda$CDM at the present time, while in the far future, they converge to deceleration without resulting in the de Sitter phase.

In summary, the new parametrization of the dark-energy equation of state with a single parameter is efficient in describing the data, as well as being able to alleviate the $H_0$ tension. Hence, it would be worthy to proceed to more detailed investigations, such as examining it at the perturbation level, and in particular, relating to the ${\sigma }_8$ tension. Such an analysis will be performed in a separate project.