Analysis with observational constraints in \( \Lambda \)cosmology in f(R, T) gravity
Abstract
An exact cosmological solution of Einstein’s field equations (EFEs) is derived for a dynamical vacuum energy in f(R, T) gravity for Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime. A parametrization of the Hubble parameter is used to find a deterministic solution of the EFE. The cosmological dynamics of our model is discussed in detail. We have analyzed the time evolution of the physical parameters and obtained their bounds analytically. Moreover, the behavior of these parameters are shown graphically in terms of the redshift ‘z’. Our model is consistent with the formation of structure in the Universe. The role of the f(R, T) coupling constant \(\lambda \) is discussed in the evolution of the equation of state parameter. The statefinder and Om diagnostic analysis is used to distinguish our model with other dark energy models. The maximum likelihood analysis has been reviewed to obtain the constraints on the Hubble parameter \(H_0\) and the model parameter n by taking into account the observational Hubble data set H(z), the Union 2.1 compilation data set SNeIa, the Baryon Acoustic Oscillation data BAO, and the joint data set \(H(z) + \mathrm{SNeIa}\) and \(H(z) + \mathrm{SNeIa} + \mathrm{BAO} \). It is demonstrated that the model is in good agreement with various observations.
1 Introduction
The latetime elusive behavior of the Universe is one of the major challenges in modern cosmology. The current observational data of SNeIa confirms the latetime cosmic speed up of the Universe. At present, much observational data is in support of the current acceleration of the Universe [1, 2, 3, 4]. In order to explain this faster rate, a new form of energy is needed in the Universe which has some antigravitational effect that drives the acceleration. This distinct type of energy with negative pressure is termed dark energy (DE) [5, 6, 7]. According to the Planck mission team, it is estimated that the Universe is composed of three main components, \( 4.9\% \) ordinary matter, \( 26.8\% \) dark matter (DM) and \( 68.3\% \) DE. The DM and DE are really different in nature. DM is attractive and responsible for formation of structure and clustering of the galaxies, whereas DE seems to be some kind of energy intrinsic to empty space which keeps getting stronger with time. There are multiple ideas on DE: one idea is that DE is a property of space itself, or some kind of dynamic energy fluid which has some opposite effects on the Universe to ordinary energy and matter. Although DE is a popular explanation for the expansion mystery supported by many observational experiments, there remain many unanswered questions.
The two main models proposed in the literature to explain the nature of DE are the model of the cosmological constant \( \Lambda \), i.e. assuming a constant energy density filling in space homogeneously, and a scalar field model, which considers a dynamical variable energy density in spacetime. The simplest and most favorable candidate of DE is the Einstein cosmological constant \( \Lambda \) [8, 9], which works as a force that counteracts the force of gravity. Adding the cosmological constant \( \Lambda \) to EFE of the FLRW metric leads to the \(\Lambda \mathrm{{CDM}}\) model, which serves as the agent of an accelerating Universe. In spite of its theoretical and phenomenological problems [10], the \( \Lambda \mathrm{{CDM}} \) model has been referred to as the most efficient answer to the question of cosmic acceleration in many aspects because of its consistency with observations. According to the GR, the equation of state (EoS) in cosmology specifies the expansion rate of the Universe. Nowadays, the great challenge in observational cosmology is the attempt to analyze the EoS \(\omega =\frac{p}{\rho }\) of various DE models, where \( \rho \) and p are the energy density and isotropic pressure of the fluid. The quintessence and phantom models, which are dynamic scalar fields, are two specific cases of dark energy models having EoS parameter \( \omega >1 \) and \( \omega <1 \), respectively. The first scenario of the quintessence model was proposed by Ratra and Peebles [11]. The quintessence model differs from \( \Lambda \mathrm{{CDM}} \) in the explanation of DE as a quintessence model is dynamic, changing with respect to time, unlike \( \Lambda \), which always stays constant [12, 13]. The phantom model [14, 15, 16] could cause a big rip in the Universe due to the growing energy density of DE [17, 18, 19]. Also a number of other scalar fields DE models have been proposed: the spintessence [20], kessence [21, 22], quintom [23], tachyon [24, 25], Chameleon [26] having EoS parameter \( \omega \in (1,0) \) model etc. Another class of alternative ideas concerns the theory of dark fluid that unifies both DM and DE as a single phenomenon [27]. In order to understand the behavior of the Chaplygin gas in detail, one can review some excellent work published by Singh et al. [28, 29, 30]. In addition, holographic dark energy (HDE) is also a suitable choice for DE (among other alternatives) that might be originated from the quantum fluctuations of spacetime [31, 32, 33, 34].
Another possibility of DE that might affect the expansion history of the Universe concerns the dissipative phenomena in the form of bulk and shear viscosity. After the discovery of acceleration of the Universe, the concept of viscous cosmology has been reconsidered again. Some recent papers provide an application of viscous cosmology to the accelerating Universe [35, 36]. For an isotropic and homogeneous cosmic expansion, the negative pressure of a bulk viscous fluid might play the role of an exotic fluid and could account for the effects usually attributed to DE. It is observed that the primordial inflation and present cosmic accelerated expansion can be achieved in the Universe by the inclusion of viscosity concepts in which various cosmic aspects have been considered [37]. There are a number of articles in the literature which discuss the DE phenomenon as an effect of dissipative processes such as bulk viscosity, which has been thoroughly studied in a cosmic medium (for a detailed review see [38, 39]).
In the other direction, the accelerating expansion of the Universe can be revealed by modifying the Einstein–Hilbert action. The standard Einstein Lagrangian can be modified by replacing the scalar curvature R with some arbitrary function of R; this is known as f(R) gravity. Moreover, the replacement of the Ricci scalar R with scalar torsion T is known as f(T) gravity, and replacement with the gravitational constant G is known as f(G) gravity. Many other modifications of the underlying geometry can cause a modified theory different from GR. Among the wide range of alternative ideas of modified gravity, f(R) gravity theory is the most viable alternative theory of gravity [40]. f(R) gravity is considered good on large scales, but it fails to show consistency as regards some of the observational tests, e.g. rotation of curved spiral galaxies [41, 42], or the solar system regime [43, 44]. A more generic extension of f(R) gravity could be considered, as f(R, Sm) , where the matter Lagrangian \( S_{m} \) is a function of the trace T of the energy momentum tensor and is taken as f(R, T) gravity [45]. The main reason to introduce the term T is to take the quantum effects and exotic imperfect fluids into account, and f(R, T) gravity is also capable of explaining the latetime cosmic speed up. Some observational tests [46, 47] have been applied to f(R, T) gravity in order to resolve the issues entailed by f(R) gravity. To understand f(R, T) theory in detail, one may refer to some excellent work [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. For recent reviews on modified gravity theories, see, for instance, [60, 61, 62, 63, 64].
Shabani et al. [65] have studied a minimal \( g(R) + h(T ) \) Lagrangian, a pure nonminimal g(R)h(T) Lagrangian and a nonminimal \( g(R) (1 + h(T)) \) Lagrangian in f(R, T) modified gravity with a dynamical systems approach against the background of the FLRW metric. Shabani et al. have discussed the cosmological and solar system consequences of a noninteracting generalized Chaplygin gas (GCG) with baryonic matter, latetime solutions of the \( \Lambda \mathrm{{CDM}} \) subclass of f(R, T) gravity using a dynamical system approach, latetime cosmological evolution of the Universe in f(R, T) gravity with minimal curvature–matter coupling via considering linear perturbations in the neighborhood of equilibrium, and bouncing cosmological models against the background of \( f(R,T)=R + h(T) \) gravity in FLRW metric with a perfect fluid [66, 67, 68, 69, 70, 71]. Singh et al. [72] have studied a bouncing Universe in the framework of f(R, T) gravity using a specific form of the Hubble parameter.
In the Palatini formalism of f(R) gravity called \( \Lambda (T) \) gravity, first proposed by Poplawski [73], one considered as the most general case where a \( \Lambda \)term is present the general gravitational Lagrangian, which is taken as a function of T, where T is the trace of energy momentum tensor. Moreover, the Palatini version of f(R) gravity can be brought back when we ignore the pressure dependent term from \( \Lambda (T) \) gravity. Also, the dynamical cosmological constant \( \Lambda \) is supported by this theory to solve the cosmological constant problem [10] and it is in good agreement with \( \Lambda (T) \) gravity. A detailed review of \( \Lambda (T) \) cosmology in f(R, T) modified gravity can be found in [74, 75, 76]. Bamba et al. [77] have studied various types of dark energy models e.g. \( \Lambda \mathrm{{CDM}} \), pseudorip Universes, little rip Universes, quintessence and phantom cosmological models with TypeI, II, III, and IV, and nonsingular DE models.
In this paper, the work is organized as follows: Sect. 1 provides a brief introduction on dark energy and alternative ideas to cosmic acceleration. In Sect. 2, we review the derivation of the field equations with variable cosmological parameter and obtain exact solutions to the EFE using a specific parametrization of the Hubble parameter. In Sect. 3, we discuss the dynamics of the obtained model and briefly analyze the behavior of the geometrical and physical parameters with the help of some graphical representations. In Sect. 4, we study the energy conditions and perform the diagnostic analysis for our model, and in Sect. 5, we observe the consistency of our model with some cosmological observations. Finally, we conclude with our results in Sect. 6.
2 Basic equations and its solutions
2.1 Field equations in f(R, T) gravity
2.2 Parametrization of H and exact solution
2.3 Bounds on the cosmological parameters
Here, we evaluate the cosmological parameters at two extreme values of time \( t\rightarrow 0 \) and \( t\rightarrow \infty \), to examine the behavior of the model at the initial singularity as well as late time (see Table 1). From Table 1, we can have a range of these cosmological parameters, which depend on the parameter n and the f(R, T) coupling constant \( \lambda \), where \( \lambda =\frac{8\pi }{A1} \). By choosing suitable values of n and \( \lambda \), we can explain the history of the expansion in terms of the various cosmological parameters. The role of the f(R, T) coupling constant \( \lambda \) can be seen clearly from Table 1. The Universe starts with infinite velocity and a finite acceleration and expands indefinitely with constant velocity and constant acceleration. The energy density and isotropic pressure start from infinitely large values at the time of the early evolution of the Universe and decrease gradually to constant values in late time. The EoS parameter \( \omega \) varies in the range \( \left[ \frac{3A+n(2A+1)}{3A+n}, 1 \right] \). The role of the f(R, T) coupling constant bounds and the limit for the EoS parameter will be discussed in another subsection.
Behavior of cosmological parameters
Time (t)  a  H  q  \( \rho \)  p  \( \omega \)  \( \Lambda \) 

\( t\rightarrow 0 \)  0  \( \infty \)  \( n1 \)  \( \infty \)  \( \infty \)  \(\frac{3A+n(2A+1)}{3A+n}\)  \( \infty \) 
\( t\rightarrow \infty \)  \( \infty \)  \( \alpha \)  \( 1 \)  \( \frac{3\alpha ^{2}}{(A+1)} \)  \(\frac{3\alpha ^{2}}{(A+1)} \)  \( 1 \)  \( \frac{3\alpha ^{2}}{(A+1)} \) 
3 Dynamics of the model
3.1 Phase transition from deceleration to acceleration
In Fig. 1, it is clearly observed that the deceleration parameter q is positive and negative for high and low redshift z, respectively, in the range \( 1<n<2 \) of the model parameter n. It has also been noticed explicitly that the phase transition from decelerating to accelerating regimes of the Universe depends on the variation of the model parameter n. For \(n\le 1\), the model exhibits eternal acceleration; for \( n=2 \), the Universe shows no acceleration at present, and for \( n>2 \), acceleration of the Universe is possible in the near future. The plot shows the phase transition redshift \( (z_\mathrm{tr}) \) for various values of n in the feasible range, \( n\in (1,2)\). For \(n=1.25,\) \(q=0\) at \(z_\mathrm{tr}=1.988\), for \(n=1.45,\) \(q=0\) at \(z_\mathrm{tr}=0.73\), for \(n=1.65,\) \(q=0\) at \(z_\mathrm{tr}=0.29\) and for \(n=1.85,\) \(q=0\) at \(z_\mathrm{tr}=0.091\). The present values of the deceleration parameter \(q_{0}\) corresponding to \(n=1.25,\) \( n=1.45,\) \(n=1.65\) and \(n=1.85\) are \(\,0.371,\,0.275,\,0.179\) and \(\,0.077\), respectively. The best fit values of the model parameter n from various observations are discussed in Sect. 5.
3.2 Physical significance of \( \lambda \) in the evolution of the Universe
In order to explain formation of structure, we know that, along with decelerated expansion, which is responsible for the structure formation, one must require a kind of matter fluid that could produce a Jeans instability [84], and this is possible only when a low pressure fluid occurs in the Universe. To understand the formation of structure in detail, a force of gravity is required which puts the gas molecules together because as gravity pushes gas molecules closer, pressure and heat are produced, which then tends to push the molecules further apart. In 1902 Jeans was the first person who calculated the region of influence mathematically, called the Jeans length^{1} required to cause gravity to push atoms together and merge into structures like stars, galaxies or in fact global clusters.
Here, we fix \( n=1.45 \) where the phase transition redshift is \( z_\mathrm{tr}\approx 0.7 \). Now we observe the role of coupling constant \( \lambda \) played in the evolution of the EoS parameter \( \omega \) by taking different values of \( \lambda \). For \(\lambda =0\), the case of GR and \(\lambda =10\), EoS parameter \(\omega \) remains negative throughout the evolution. If we increase the value of \(\lambda \) from a small positive number to a large positive number, the redshift transition time gets shifted from right to left. Also if we take high negative values of \(\lambda \) from \(50\) to \(150\), the redshift phase transition time gets shifted from left to right and ultimately remains in the interval \(z\in (0,1)\). For all the cases of \(\lambda \) plotted here other than \(\lambda =0, 10\), it is important to notice that the matter content in the Universe behaves like a perfect fluid in the initial phase of the Universe and later on the Universe enters into a quintessence regime and finally approaches \(\omega =1\) as \(z\rightarrow 1\) but it never crosses the phantom divide line. Hence, the coupling constant \( \lambda \) has great importance in the reconstruction of the cosmic evolution of our model.
3.3 Physical parameters and their evolution
Figure 3a depicts the evolution of the energy density \(\rho \) with respect to the redshift z for different values of the model parameter n mentioned in the plot. For a high redshift, the energy density is very high, as expected; then the energy density falls as time unfolds and later on it approaches \(\frac{3\alpha ^{2}}{(A+1)}\) as \( z\rightarrow 1 \).
The profile of the EoS parameter \(\omega \) and the cosmological constant \(\frac{\Lambda }{H_{0}^{2}} \) is investigated in Fig. 4. The behavior of \(\omega \) with respect to the redshift z can be seen in Fig. 4a. For all values of the model parameter n and a fixed value of the coupling constant \(\lambda =130 \), \(\omega \) takes positive values in the early Universe; then \(\omega \) starts changing its sign from positive to negative and \(\omega =0\) at \(z=1.31678, 0.481009, 1.060876\) and \(z=0.001195\) corresponding to \(n=1.25,1.45,1.65\) and \(n=1.85\), respectively. After this, \(\omega \) enters the quintessence region, and \(\omega \rightarrow 1\) in late time as \(z\rightarrow 1\), which suggests that matter in the Universe behaves like a perfect fluid initially. Later on the model is similar to a dark energy model and behaves like a quintessence model and finally approaches −1 without entering the phantom region. Figure 4b depicts the variation of the cosmological constant \(\left( \frac{\Lambda }{H_{0}^{2}}\right) \) with respect to the redshift z. It has been observed that the cosmological constant remains positive throughout the cosmic evolution, decreasing in nature and reaching a small positive value at present epoch \( z\rightarrow 0 \), favoring the observations [1, 2, 85, 86] and \(\Lambda \rightarrow \frac{3\alpha ^{2}}{(A+1)}\) as \(z\rightarrow 1\). The outcome from these observations suggests a very minute positive value having magnitude \( \sim 10^{123} \).
3.4 Physical significance of jerk, snap, lerk parameters
Figure 5 depicts the evolution of jerk j, snap s and lerk parameter l with respect to the redshift z. In Fig. 5a, the evolution of the jerk parameter is represented for all the four values of n, and it can be observed that the j parameter lies in the positive range throughout its course. Also \(j \rightarrow 1\) as \(z \rightarrow 1 \), \( \forall n\), which is consistent with observations of standard \( \Lambda \mathrm{{CDM}} \) but at present \(z=0\), \(j\ne 1\), \(\forall n\). Therefore, our model is similar to the dark energy model, different from \( \Lambda \mathrm{{CDM}} \), \(\forall n\), at the present time. Figure 5b enacts the profile of snap parameter s during its evolution. In the early Universe s assume value in the negative range \( \forall n\) then as Universe evolves, s take values in the positive range, i.e. in the entire evolution of s, there is one transition from negative to positive range. Also it can be directly seen from Fig. 5b that the transition of s depends on the model parameter n, i.e., the transition redshift of s is delayed as n takes values from 1.25 to 1.85. Figure 5c shows the variation of the lerk parameter l over the redshift z. The lerk parameter l assumes only positive values without any redshift transition. In addition to j, both s and l approach 1 as in late time \( z\rightarrow 1\).
4 Physical analysis and geometrical diagnostic
4.1 Energy conditions

NEC \(\Leftrightarrow \) \(\rho +p_i \ge 0\), \( \forall i \),

WEC \(\Leftrightarrow \) \(\rho \ge 0\), \( \rho +p_i \ge 0 \), \( \forall i \),

SEC \(\Leftrightarrow \) \(\rho +\sum _{i=1}^3 p_i \ge 0\), \(\rho +p_i \ge 0\), \( \forall i \),

DEC \(\Leftrightarrow \) \( \rho \ge 0\) , \(p_i \le \rho \), \( \forall i \), \( i=1,2,3 \).

NEC: \(\forall \, V(\phi )\),

WEC \(\Leftrightarrow \) \(V(\phi ) \ge \frac{\dot{\phi }^2}{2}\),

SEC \(\Leftrightarrow \) \(V(\phi ) \le \dot{\phi }^2\),

DEC \(\Leftrightarrow \) \(V(\phi ) \ge 0\).
Here NEC, WEC, SEC and DEC are defined as the null energy condition, weak energy condition, strong energy condition and dominant energy condition, respectively.
Here, we present the graphs of NEC, SEC and DEC for all four values of the model parameter n and a fixed value of f(R, T) coupling constant \(\lambda \). We observe that, from Fig. 6a–c, NEC and DEC hold for all values of n, while SEC fails for all values of n.
4.2 Statefinder diagnostic
As is well known, the role of geometric parameters has great importance in order to study the dynamics of a cosmological model. In what follows, in Sect. 4.2, we have discussed the different phases of the evolution of the deceleration parameter and concluded that the deceleration parameter alters its sign from positive to negative, corresponding to high redshift to low redshift, respectively. The phase transition of the deceleration parameter provides hope to discover the source of the recent acceleration. Through the requirement of a more general dark energy model other than \( \Lambda \mathrm{{CDM}} \) and the development in the accuracy of current cosmological observational data, there arises the problem of looking into the quantities involving higher derivatives of the scale factor a.
Various dark energy scenarios can be examined by the distinct evolutionary trajectories of the geometric pair \(\{r,s\}\) emerging in the r–s plane in Fig. 7a. A symbolic feature of the SFD is that the standard \( \Lambda \mathrm{{CDM}} \) model of cosmology is represented by the pegged point \(\{r,s\}=\{1,0\}\), whereas the standard matter dominated Universe, SCDM, corresponds to the fixed point \(\{r,s\}=\{1,1\}\). Other than the \( \Lambda \mathrm{{CDM}} \) and SCDM model, the SFD analysis can successfully discriminate among the several dark energy candidates such as quintessence, braneworld dark energy models, Chaplygin gas and some other interacting dark energy models by locating some particular region in the said diagram in the distinctive trajectories [28, 99, 100, 101].
Figure 7a represents the time evolution of four trajectories for different values of n in r–s plane. All the trajectories corresponding to different values of n evolve with time but deviate from the point SCDM, i.e., \( \{r,s\}=\{1,1\} \), which corresponds to a matter dominated Universe. The directions of the r–s trajectories in the plane diagram are represented by arrows. Initially, we have examined that cases corresponding to \( n=1.25 \) and \( n=1.45 \); the trajectories remain in the domain \( r<1, s>0 \), which relate our dark energy model to the quintessence model. Also trajectories corresponding to \(n=1.65\) and \(n=1.85\) start evolving from the region \( r>1,s<0\), which resembles the behavior of dark energy with a Chaplygin gas and this region is highlighted by CG in the top leftmost part of the plot. The downward pattern of trajectories representing the CG behavior and the upward trend of the trajectories representing quintessence behavior are eventually met at the point \(\{r,s\}=\{1,0\}\), i.e., we have the \( \Lambda \mathrm{{CDM}} \) model. This suggests that our model behaves like \( \Lambda \mathrm{{CDM}} \) in the late time of cosmic evolution. In addition we have presented one more horizontal line in the above diagram, which shows the transformation of trajectories from a matter dominated Universe SCDM to \( \Lambda \mathrm{{CDM}} \) as time unfolds. The point having coordinates \(\{r,s\}=\{1,\frac{2}{3}\}\) on the horizontal line represents the holographic dark energy model with future event horizon as IR cutoff labeled as HDE in Fig. 7a, begins the evolution from the point \( \{r,s\}=\{1,\frac{2}{3}\}\) and ultimately ends its evolution at \( \Lambda \mathrm{{CDM}} \) [102, 103, 104]. Therefore, the plot of \(\{r,s\}\) for our model is effectively discriminant among other dark energy model for different n.
Figure 7b represents the time evolution of four trajectories for different values of n in r–q plane. Since we have seen the complete description of the phase transition of deceleration parameter in Sect. 3.2, we can again observe the phase transition of our model by looking into the trajectories of r–q diagram as q changes its sign from positive to negative. The evolution of the trajectories for different values of n commences in the vicinity of a matter dominated Universe SCDM but never converges to SCDM. As time evolves, the values of r and q start to decline and they attain their minimum position, after which both r and q start to increase towards SS, which is located in the diagram at \( (1,1) \). The progression of the trajectories to SS suggests that our dark energy model may behave like the steady state model in late time.
4.3 Om diagnostic
The contrasting behavior of dark energy models from \( \Lambda \mathrm{{CDM}} \) depend on the slope of the Om(z) diagnostic. A quintessence \((\omega >1)\) type behavior of dark energy can be identified by its negative curvature with respect to z, and a phantom type behavior \( (\omega <1) \) can be diagnosed by its positive curvature with respect to z, and a zero curvature of Om(z) represents the standard \( \Lambda \mathrm{{CDM}} \).
Figure 8 exhibits the evolution of different trajectories of the function Om(z) with respect to the redshift z, corresponding to different values of the model parameter n. From the plot of Fig. 8, we can observe that all the trajectories show negative slope, i.e., all the trajectories move in an upward direction as time increases or the redshift decreases. The negative curvature pattern suggests that our model is behaving similar to quintessence for all values of n.
5 Observational constraints on the model parameters
According to the current survey, as is well known, a wide variety of observational data including the Cosmic Microwave Background (CMB) data (which is the relic radiations from the baby Universe), Slogan Digital Sky Survey (SDSS) data (the observations of the distribution of galaxies with position redshift which essentially encode the fluctuations in the Universe, spectra of quasars), Type Ia supernovae data (usually known as standard candles and used to measure the expansion of Universe), Baryon Acoustic Oscillations (BAO) data (which measures the structure in the Universe), and largescale structure (LSS) data (having provided a very strong tool to test our cosmological framework for many years [108]) produce various measurements of challenging issues in our Universe like the evolution of the Universe, properties of dark matter and dark energy. More significantly, the growth rate of structure tests are independent and complementary to the constraints, which may be obtained from the analysis of the temperature and polarization fluctuations in the CMB and other observations such as Type Ia supernovae and BAO. The consistency between these observational data sets function as one of the strongest reasons in favor of the current standard model, the \( \Lambda \mathrm{{CDM}} \) model.
5.1 Hubble parameter H(z)
In this subsection, we compare our model with the 29 points of H(z) data set [109] in the redshift range \( 0.1\leqslant z\leqslant 2.5 \) and compare with the \( \Lambda \mathrm{{CDM}} \) model. We choose the value of the current Hubble constant from Planck 2014 results [110] as \( H_{0}=67.8\,\mathrm{{km/s/Mpc}} \) to complete the data set.
5.2 Type Ia Supernova
The left panel of Fig. 9 displays the best fitting curve of our model compared with the \( \Lambda \mathrm{{CDM}}\) model for the H(z) data set and the right panel shows best fitting curve of our model compared with the \( \Lambda \mathrm{{CDM}} \) model for SNeIa data set.
5.3 Baryon acoustic oscillations
In the early Universe, before matter decouples, baryons and photons form a plasma through which sound waves can propagate. Sound waves can leave a very distinct imprint on the statistical properties on matter that we can call BAO. It measures the structures in the Universe from very large scales which allow us to understand dark energy better. In this study we adopt a sample of BAO distances measurements from different surveys, namely SDSS(R) [112], the 6dF Galaxy survey [113], BOSS CMASS [114] and three parallel measurements from the WiggleZ survey [115].
Summary of the numerical results for flat Universe
Data  \( \chi ^2_{\mathrm{min}} \)  Hubble parameter \(H_0\) (km/s/Mpc)  Parameter n 

H(z) (29 points data )  24.5790  66.9762  1.410 
SNeIa (Union 2.1 compilation data)  586.173  66.6213  1.390 
BAO  29.4699  68.8486  1.509 
\( H(z) + \text {SNeIa} \)  611.0960  67.2050  1.390 
\( H(z) + \text {SNeIa} + \text {BAO} \)  673.6844  65.8202  1.509 
The likelihood contours for the parameters n and \( H_{0} \) with \( 1 \sigma \), \( 2 \sigma \) and \( 3 \sigma \) errors in the n–\( H_{0} \) plane are shown in Figs. 10 and 11. The best fit values of n are found to be 1.41, 1.30, 1.509, 1.39 and 1.509 according to the Hubble data set and SNeIa (Union 2.1 compilation data set), BAO , joint data set \( H(z) + \text {SNeIa} \) and the joint data set \( H(z) + \text {SNeIa} + \text {BAO} \) for which the corresponding best fit values of \( H_0 \) are constrained as 66.9762, 68.5583, 68.8486, 67.2050 and 65.8202, respectively (see Table 2).
6 Discussions and Conclusions
 (i)
In order to study a cosmological model capable of explaining the recent astronomical observations of accelerating expansion of the Universe with a decelerating phase of evolution in the past, we have considered a geometrical parametrization of the Hubble parameter H used by Singh [80] and Banerjee et al. [81], which leads to a variable deceleration parameter q. The obtained form of q describes both the scenario of early deceleration and of present acceleration. The behavior of the geometrical parameters a, H and q at two extremities (\(t\rightarrow 0,t\rightarrow \infty \)) have been analyzed in Table 1.
 (ii)
For the considered parametrization of H, the different phases of evolution of the deceleration parameter has been examined. From the expression of q(z), we have found the range of the deceleration parameter, i.e., \( q\in [n1,1] \), which clearly shows the signature flipping behavior because the model parameter \(n>1\). For a close view of q, we can observe the decelerating to accelerating regimes of the Universe depending on the variation of the model parameter n in Fig. 1. As the values of n increase from 1.25 to 1.85, the phase transition redshift \( z_\mathrm{tr} \) could be delayed.
 (iii)
To discuss the role of the f(R, T) coupling constant \( \lambda \) played in the evolution of the EoS parameter \(\omega \), we fix the value of \( n = 1.45 \) and vary \( \lambda \). In Fig. 2, we have examined the special characteristic of the coupling constant \( \lambda \), and observed the variation in \(\omega \) as \(\lambda \) takes both negative and positive values. This shows the contribution of f(R, T) gravity in this model on considering the acceptable range of \( \lambda \).
 (iv)
In Table 1, we have shown the behavior of a, H, q, \( \rho \), p, \( \omega \), and \( \Lambda \) at \( t\rightarrow 0 \) and \( t\rightarrow \infty \) and studied the physical significance of \( \rho \), p, \( \omega \), and \( \Lambda \) with respect to the redshift z in Sect. 3.3. In Fig. 3, the energy density and isotropic pressure reduce from their dense state to a constant value, which depends on \( \lambda \). Figure 3b depicts the isotropic pressure p starting from a very large value at the initial singularity and approaching \( \frac{3\alpha ^{2}}{(A+1)} \) in the future \( z\rightarrow 1 \) for some specific values of n. The negative values of the cosmic pressure are corresponding to the cosmic acceleration according to standard cosmology. Hence our model exhibits accelerated expansion at present as well as in the infinite future. The model is consistent with the formation of structure of the Universe. In Fig. 4a, for all values of n and a fixed value \( \lambda =130 \), the EoS parameter \( \omega \) transits from positive to negative and ultimately approaches the quintessence region, which suggests that matter in the Universe behaves like a perfect fluid initially and as dark energy in late time. In Fig. 4b, the cosmological constant \( \Lambda \) starts decreasing from a very high value at high redshift and approaches a small positive value at present epoch \( (z\rightarrow 0) \), which is in good agreement with the current observations [1, 2, 85, 86].
 (v)
Next, we have compared our dark energy model with the standard \( \Lambda \mathrm{{CDM}} \) model by examining the behavior of the other geometrical parameters, e.g., the jerk j, snap s and lerk l parameters. From Fig. 5a, it can be seen that, for every value of n, our model behaves different from the \( \Lambda \mathrm{{CDM}} \) model at the present time \( z=0 \), but in the late future \( j\rightarrow 1 \), which is in accordance with the \( \Lambda \mathrm{{CDM}} \) model. In addition to j, the behaviors of the snap s and lerk l parameters are graphically demonstrated in Fig. 5. The snap parameter s shows one transition from negative to positive throughout its evolution with respect to the redshift z, while the lerk parameter l is decaying in nature with no transition.
 (vi)
In Sect. 4, some physical analysis is made and geometrical diagnostics of the model has been studied. The physical viability of the model has been analyzed by verifying the energy conditions of our model. In Fig. 6, it can be seen easily that NEC and DEC hold good but SEC fails for all values of the model parameter n and the fixed value \( \lambda = 130 \).
 (vii)
In Sect. 4.2, Fig. 7 represents the time evolution of four trajectories for different values of n in \( \{s,r\} \) and \( \{q,r\} \) plane diagram. The directions of the s–r trajectories in the plane diagram are represented by arrows, showing different dark energy models and they ultimately approach \( \Lambda \mathrm{{CDM}} \) (see Fig. 7a). In the q–r plane diagram, the evolution of the trajectories, for different values of n, commences in the vicinity of SCDM and, as time evolves, the trajectories of the q–r approach the steady state model SS (see Fig. 7b).
 (viii)
Also, one more geometrical diagnostic has been interpreted to gain understanding of the different dark energy models for every value of n. A plot of Om(z) against redshift z has been displayed in Fig. 8. All the trajectories of Om(z) exhibit a negative slope, which suggests that our model is behaving similar to a quintessence model for all n and in late time, i.e., \( z\rightarrow 1 \), \( Om(z)\rightarrow k \), where k is a positive finite quantity. This means that our model may correspond to \( \Lambda \mathrm{{CDM}} \) in the future.
 (ix)
The likelihood contours for the model parameters n and \( H_{0} \) with \( 1 \sigma \), \( 2 \sigma \) and \( 3 \sigma \) errors in the n–\( H_{0} \) plane are shown in Figs. 10 and 11. The model parameter n is constrained using the 29 points of the H(z) and Union 2.1 compilation data. The obtained model is in good agreement with the H(z) and SNeIa (Union 2.1 compilation data) and nearly follows the \( \Lambda \mathrm{{CDM}} \) behavior (see Fig. 9). The constrained best fit values of the model parameters n are 1.41, 1.30, 1.509, 1.39 and 1.509 according to the Hubble data H(z) and SNeIa (Union 2.1 compilation data), BAO, joint data H(z) + SNeIa and \(H(z) + \text {SNeIa} + \text {BAO}\) for which the corresponding best fit values of \( H_0 \) are evaluated to be 66.9762, 68.5583, 68.8486, 67.2050 and 65.8202, respectively (see Table 2).
Footnotes
 1.
The Jeans length in a region can be calculated by the formula \( L_J=\sqrt{\frac{\pi k T}{m G \rho }} \), where k is the Boltzmann constant, T is the temperature of the gas, m is the mass of the atom in the gas, G is the gravitational constant and \(\rho \) is the density of the gas. The Jeans length at the time of decoupling was approx. 108 light years.
Notes
Acknowledgements
The authors express their gratitude to Prof. M. Sami and Prof. S. G. Ghosh, Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi, India, and Prof. J. P. Saini, Director, NSIT, New Delhi, India, for some fruitful discussions, and providing necessary facilities to complete the work. The author JKS expresses his gratitude to the Department of Mathematical Sciences, University of Zululand, KwaDlangezwa 3886, South Africa, and Department of Mathematics, Statistics and Computer Sciences, University of KwaZuluNatal, Westville 4001, South Africa, for some fruitful discussions with Prof. S. D. Maharaj, their financial support and providing necessary facilities as well as hospitality; part of the work was completed there. Moreover, the work of KB was supported by the JSPS KAKENHI Grant Number JP25800136 and Competitive Research Funds for Fukushima University Faculty (17RI017 and 18RI009). The authors also express their gratitude to the referee for valuable comments and suggestions.
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