Cosmological aspects of a hyperbolic solution in gravity
Abstract
This article deals with a cosmological scenario in gravity for a flat FLRW model of the universe. We consider the function as which starts with a quadratic correction of the geometric term having structure , and a linear matter term . To achieve the solution of the gravitational field equations in the formalism, we take the form of a geometrical parameter, i.e. scale factor cha (), where and are model parameters. An eternal acceleration can be predicted by the model for , while the cosmic transition from the early decelerated phase to the present accelerated epoch can be anticipated for . The obtained model facilitate the formation of structure in the Universe according to the Jeans instability condition as our model transits from radiation dominated era to matter dominated era. We study the varying role of the equation of state parameter . We analyze our model by studying the behavior of the scalar field and discuss the energy conditions on our achieved solution. We examine the validity of our model via Jerk parameter, Om diagnostic, Velocity of sound and Statefinder diagnostic tools. We investigate the constraints on the model parameter and (Hubble constant) using some observational datasets: dataset, (Hubble parameter) dataset, (Baryon Acoustic Oscillation data) and their combinations as joint observational datasets + and + + . It is testified that the present study is well consistent with these observations. We also perform some cosmological tests and a detailed discussion of the model.
PACS number: 98.80 cq
Keywords: theory, FLRW metric, Parametrization, Observational constraints.
I Introduction
Einstein field equations (EFE) are
(1) 
In the above equation, is the Ricci tensor, the Ricci scalar, the covariant metric tensor of order 2, the cosmological constant, the gravitational constant, indicates the speed of the light, and the energymomentumtensor (EMT). Despite the fact that general relativity (GR) is extremely well tested, alternatives are always present. According to observations, of the matter content of the Universe is unexplored. GR has several problems, as the problem of initial bigbang spacetime singularity car (); ein (); yil () and it is not yet quantised. GR has to be reconciled with quantum physics to discuss quantum effects. GR together with quantum physics forms the backbone of modern physics. The CDM model of the cosmology is quite successful, but there remain several unresolved issues such as the fine tuning problem bull (). Hence it is worthwhile to examine alternative theories of gravity.
Amongst large range of modified theories of gravity, gravity tho () is considered as an interesting alternative. A more general function of i.e. is considered in the EinsteinHilbert action . An intensively study on gravity seems to indicate that it is an improvement over GR suj (); noji (); cap1 (). It can also explain both phases of cosmic acceleration even in the absence of (early and late times) paul (). gravity behaves extremely well on large scales, but the theory does not hold good on all observational tests, such as on rotation of curved spiral galaxies chi1 (); olmo1 () and the solar system regime eri (); olmo2 (). The generalisation of gravity to , where the matter Lagrangian is considered as a general function of trace of the EMT is termed gravity. Some solar system tests myr1 (); mor2 () have been favorably applied to modified theory of gravity to resolve the abovementioned issues. To introduce exotic imperfect fluids and quantum effects, a trace dependent term is determined. Generally, the source term is a function of the matter Lagrangian , which yields an explicit set of field equations. A lot of remarkable work in cosmology and astrophysics has already been done in gravity by several authors shab2 (); alv (); sin1 (); shab3 (); shab5 (); sin2 (); sin3 (); sin7 () and also this theory can resolve the dark matter issue sin4 (); zar (); sin5 ().
The above study has prompted us to compose a cosmological scenario within gravity. The paper is arranged as follows. In Sect. II, we give a concise discussion on theory. We obtain highly nonlinear field equations by considering the function as the combination of a quadratic Rdependent term and a linear Tdependent term. To determine the solution of the field equations, we use an ansatz for the scale factor , and find the behavior of the other geometrical parameters , . We also present the graphical behavior of , and for the obtained model in Sect. III. Next Sect. IV is devoted to the analysis and interpretation of the obtained solution by examining the potential of the scalar field and energy conditions. In Sect. V, we perform some observational tests using Jerk parameter, Om diagnostic, Velocity of sound and Statefinder diagnostic tools to explore the validity of our model. In Sect. VI, we observe that present study is well behaved with some observational datasets. In Sect. VII, some cosmological tests are discussed to calculate distances in cosmology for the accepted parametrization. Finally, we summarize our results by providing a brief conclusion about the work in Sect. VIII.
Ii Review of f(R,T)=f(R)+2f(T) cosmology
The general action for gravity har () coupled with the action of a matter field with matter Lagrangian reads
(2) 
where being an arbitrary function of and . Here, we consider which is the first model for inflation and it was proposed by Starobinsky star1 (). This form of function takes its origin in the quantum correction to Friedmann equations. The term appears in the functional form of is the natural correction to GR and it naturally provides an inflationary scenario in early Universe. Also Starobinsky model shows the best compatibility according to the latest observations of the Universe pla () and this model serves as a possible substitute to the scalar field models describing inflation jdb (). Consequently, if one extend the above assumed form with negative exponents of curvature term, then this model is able to express the recent accelerating expansion. Therefore, with the most general model , where and are arbitrary constants, both the acceleration in the Universe (early and late time acceleration) can be explained by the theories beyond GR paul (). To introduce exotic imperfect fluids and taking quantum effects in to account with the abovedefined model, a trace dependent term is much needed. This source term is a function of matter Lagrangian which yields an explicit set of field equations. Here, in this study, we assume as a linear function of defined as . So the complete form of function is .
On defining EMT of matter har ()
(3) 
where its trace is given by . Also if is dependent only on , in that case one can write
(4) 
Taking a variation of action (2) w.r.t. , we have
(5) 
where and represent the derivative of w.r.t. and respectively, is the d’ Alembert operator defined by and indicates the covariant derivative w.r.t. associated with the symmetric LeviCivita connection. is of the following form
(6) 
We consider perfect fluid in the thermodynamic equilibrium, so in this way, in present study, one can simply set the matter Lagrangian and we take EMT of matter as
(7) 
where is the energy density and is the pressure of the fluid present in the Universe. Using (6), the expression for the variation of EMT of perfect fluid is given by
(8) 
Using Eq. (8) in Eq. (5), we get the gravitational equation of motion as
(9) 
The connection between Ricci Scalar and can be seen by contracting the Eq. (9) w.r.t ,
(10) 
On rearranging the terms in Eq. (8), the Ricci tensor takes the form
(11) 
Let us define a new operator as,
(12) 
So the Eq. (11) becomes
(13) 
The expression for Ricci scalar can be written by arranging the terms in Eq. (10)
(14) 
Iii Cosmological dynamics of the Universe
We study the dynamics of the Universe by considering a homogeneous and isotropic Universe in the form of spatially flat FLRW line element given by
(17) 
where being the scale factor. The trace of the EMT (7) and scalar curvature are
(18) 
(19) 
where is the Hubble parameter defined as and overhead dot indicates the differentiation w.r.t. to . Taking and using Eqs. (7), (18), (19) in Eq. (II), we get the following field equations
(20) 
(21) 
where and are the functions of scale factor and its derivatives up to fourth order respectively. Also, we have set the units so that .
Substituting the mentioned choice for function in Eq. (16) leads to
(22) 
Eq. (20) can be rewritten as
(23) 
where we have defined
(24) 
Solving Eqs. (22) and (23) gives
(25)  
(26) 
Solutions (25) and (26) show that to obtain the exact solutions for and and to study dark energy model, we need to adopt a parametrization of either or . This technique is called model independent way to explore dark energy models. This work deals with an ad hoc choice of , which is the outcome of a timedependent deceleration parameter (DP) cha () as
(27) 
where and , are arbitrary constants.
In the present study, we are curious to examine the different regimes of the Universe i.e. the phase transition from decelerated to accelerated expansion by constraining a model parameter . From Eq. (29), DP depends on and inflation in the Universe depends on the sign of . A positive refers the decelerating expansion while a negative corresponds to accelerating phase of the model. For the above parametrization of , our model entirely accelerates and decelerates according as and respectively, and it predicts phase transitions i.e. when . As it is well acknowledged that the Universe experiences an accelerating phase in late time, so it must had a slow expansion in the past rie (); per (), in such case the parametrization of the scale factor is rational.
Time ()  Redshift ()  

finite quantity 
From Eq. (29), the model parameters and are related as
(30) 
where denotes the present time and indicates the present value of DP. Providing different values to will give rise to different values of on taking present value of and . Here, we consider and abd () and plot , and for various values of .
Using the relation
(31) 
where is present value of scale factor, we evaluate , and in terms of redshift as
(32) 
(33) 
(34) 
The graphs of scale factor , Hubble parameter and DP w.r.t are shown as:
Substance  EoS parameter  Observations 

Pressureless (Cold) matter  32% of the Universe  
Hot matter  Insignificant at present time  
Radiation  Influential in past  
Hard Universe  Excessive high densities  
Stiff matter  
Ekpyrotic matter  Resist Dominant Energy Condition  
Quintessence  68% of the Universe  
Cosmological constant  Inconsistent with observations  
Phantom Universe  Lead to Big Rip, resist Weak Energy Condition 
From Fig. 1(c), we observe that phase transitions occur when and the model shows eternal acceleration when i.e. the phase transitions w.r.t redshift directly depend on the value of . Also Fig. 1(c) clearly specifies that our model is consistent with the recent observational dataset of , and (Cosmic Microwave Background Radiation) with some fine tuning, corresponding to the value of model parameter abd () when , at , and is supportive with the fitting result of Gold SNIa for with 1Ï errors zha () when , at . Thus we can predict that the Universe started from decelerating phase and ended up with accelerating phase in late times in the case when while the model represents total eternal acceleration right from the evolution of the Universe upto late time when .
The EoS parameter is considered as one of the vital parameter in cosmology, which explains the different cosmic regimes. In a more generic way, this parameter can be defined as , where in the case of solutions (25) and (26) one obtains
(35) 
In GR from the Friedmann equations, it can be observed that there is only one approach to achieve accelerated expanding Universe by considering , which can be realised for an exotic matter, which explicitly refers negative pressure as we considered to be positive always. The various substances present in the Universe lead to different eras of the Universe which can be seen by providing particular values to (see Table II).
It is assumed that there are two major stages in the evolution of the Universe after the big bang known as the radiation and matter eras. A radiation dominated era is requisite to anticipate primordial nucleosynthesis. Therefore, deviation of more than in expanding rate of the Universe related to the CDM at the time of nucleosynthesis epoch clashes with the observed Helium abundance. Radiation and matter dominated stages are defined as the key events that help to shape the Universe. The Universe has the ability to create elements in the matter dominated era defined by the presence and predominance of matter in the Universe. It features three epochs namely atomic, galactic and stellar epochs that span billion of years and includes the present day. All the three epochs are required to formulate the large structure in the Universe that we can observe today.
One can inspect the behavior and from solutions (25) and (26) using the definition (24). Fig. 2(a) highlights the behavior of energy density which is very high i.e. in the beginning of the Universe corresponding to , falls off as time unfolds and as . Fig. 2(b) enacts the trait of matter pressure for all the values of mentioned earlier. For , at , remains negative throughout the evolution and approaches to negative constant value in late times which indicates the eternal cosmic accelerated expansion. The isotropic pressure in the early phase of the Universe for particularized values of reaches an extensively high value and tends to as . Negative pressure in the universe is subjected to the acceleration in the cosmos according to the standard cosmology. Therefore, the present study exhibits accelerating phase at current epoch as well as in the near future. From Fig. 2(b), we can realize that structure formation is achievable in our model for the case because decelerated expansion is required for the structure formation that could appear in the presence of a kind of matter fluid which produces Jeans instability james ().
It can be useful to study the matter density and pressure in the limit of small and large times. Straightforward calculations show that both quantities tend to infinity whose signature depend upon the model constants , and . In the limit of large times one obtains
(36) 
The limit value (36) includes some interesting information. It is noted that it does not depend on the coupling constant which incorporates the curvature correction term in the Lagrangian. On the contrary, result (36) depends on the coupling constant of the matter part, i.e. . From (36) we see that matter behaves like the DE in the late times. Also, value (36) implies that to have a gravitational model with observationally accepted values for the matter density and pressure in the late times, the constraint must hold. More precisely, by taking the limit value of the matter density in the early times one finds that to guarantee the weak energy condition (WEC), , constraints which are indicated in the Table III must hold.
It is worth discussing the behavior of the EoS parameter (35). In the early times the EoS parameter (35) goes to a constant value and in the late times it mimics the DE, i.e.,
(37) 
where the subscript ’’ represents the values of in the early times. For the late times one obtains
(38) 
which is independent of the model constants. Some simple calculations show that constraining the EoS parameter as (which is plausible in cosmology) forces one to choose for . Also, for and , the EoS parameter can behave like ekpyrotic matter which resists DEC followed by matter dominated era where energy density is extensively high to radiation dominated era in the early Universe. Results (37) and (38) contains an important informations about different epochs of cosmic evolution. From (37) we learn that the underlying gravitational model can describe two different cosmological transitions:

for one obtains . That is independent of other model parameters and only for the model describe a transition from a state in which stiff matter dominates in the early eras to a state which behaves like the DE in the late times.

it is also possible a transition from a pressureless matter dominated era with in the early Universe to a DE like Universe in the late times. In this case one obtains the following relation
(39)
It has been seen that the parameters and do not play any role in determining the initial and final states of cosmological evolution. It means that only modifications in the matter part of the Lagrangian are responsible for different states of the Universe. The evolution of EoS parameter w.r.t. time as well as redshift are represented graphically in Fig. 3(a) and Fig. 3(b) when .
The profound discussion on the behavior of EoS parameter which depends on the range of mattergeometry coupling constant in gravity is worthy and it has been investigated in Fig. 3. For a fix value of , and , quintessence region for high redshift and as time unfolds, in infinite future (i.e. ) when , which is consistent with the observations of temperature fluctuation in cosmic microwave background radiations (CMBR) hin (). Also, the EoS parameter indicates another possibility for evolution of the Universe when . In this case our model evolves from a state of a stiffmatter fluid dominated era to a DE like era in the late times.
The present value of the EoS parameter for the mentioned types of evolution era can be obtained respectively, as follows
(40) 
for models with and
(41) 
for , where stands for the current value of the EoS parameter. From Eqs. (40) and (41), it can be seen, both coupling constants and in the matter Lagrangian affect the present value of the EoS parameter. Therefore, some suitable astronomical data can be used to constrain the values of the coupling constant. The EoS parameters and at present epoch for respectively, which is in good agreement with the observation abd ().
Iv Interpretation of the model
iv.1 Scalar field correspondence
In section 3, we have already discussed that the idea to predict the acceleration in the Universe is to filled with an exotic form of matter which satisfy . According as the observations, the energy which produces the acceleration satisfies . If , there are many models that can explain inflation exactly such as quintessence model, phantom model etc. rat (); sam (); cald (); sah1 (). In section 3, we have also study above the construction of EoS parameter for our model, so it is appropriate to consider a matter field which shows exotic behavior and is able to produce antigravitational effects. Here, we consider dark energy as quintessence to explain the cosmic acceleration whose action is given by
(42) 
with the matter Lagrangian density
(43) 
where is the timedependent scalar field. Therefore, we can consider scalar field as a perfect fluid with energy density and pressure as
(44) 
(45) 
Here is the kinetic energy and is the potential energy of the scalar field. So it can be noticed that i.e. it can no more be treated as a constant. The quintessence or phantom model is consistent with the observations provided . Thus, we need i.e. the of is insignificant in comparison to the . In this study, we consider that is the only source of DE with , so one can consider energy density and pressure of scalar field as and respectively for flat FLRW spacetime under Barrow’s scheme bar () using Eqs. (44) and (45) as
(46) 
(47) 
The and can be obtained by solving the Eqs. (46) and (47). Fig. 4 demonstrates the potential energy plots w.r.t. scalar field for the same considered values of model parameters as we have taken in Fig. 1, 2. From Fig. 4, we notice that the potential is present in the interval and at . Therefore, we can predict that the scalar field is the only source of DE with potential . Thus we conclude that our model is an accelerating dark energy model.
iv.2 Energy conditions
Energy conditions (EC) have a great utility in classical GR which discuss the singularity problems of spacetime and explain the behavior of null, spacelike, timelike or lightlike geodesics. It provides some extra freedom to analyse certain ideas about the nature of cosmological geometries and some relations that the stress energy momentum must satisfy to make energy positive. In general, the EC can be classified as (i) NEC (Null energy condition), (ii) WEC (Weak energy condition), (iii) SEC (Strong energy condition), and (iv) DEC (Dominant energy condition). The EC can be formulated in many ways such as geometric way (EC are well expressed in terms of Ricci tensor or Weyl tensor), physical way (EC are expressed purely by the help of stress energy momentum tensor), or effective way (EC are expressed in terms of energy density , which serves as the timelike component and pressures , which represent the spacelike component). The formulation of these four types of EC in GR are pointwise expressed effectively as

NEC , ,

WEC , , ,

SEC , , ,

DEC , , .
The graphical representation of NEC, SEC and DEC for a fix value of and varying range of are shown in Fig. 5
Using the above mentioned relations, we discuss all four energy conditions in theory for all different values of and by providing different range of coupling constant . We observe the evolution of energy density and validation for all the EC for both positive and negative range of . We examine that for positive , NEC, WEC and DEC hold but SEC violates for , which directly implies the accelerated expansion of the Universe. Also, as it is clear that is any arbitrary coupling constant so it can also accept the negative values, so if we extend our domain of upto negative values, then it is worth emphasizing that SEC does not hold good for all the models , which exactly leads to accelerating phase of the Universe. Here as a matter of discussion, we graphically sketch the figures of the energy conditions for only.
V Validation of the model
v.1 Jerk parameter
As we know that the Hubble parameter measures the fractional rate of change of scale factor i.e. the instantaneous expansion and the second derivative of scale factor measures the cosmic acceleration. Similarly higher derivatives of scale factors are allow us to study the cosmic expansion history and they can potentially differentiate the various dark energy models. Jerk parameter is an extensive kinematical quantity which measures the rate of change of third derivative of scale factor w.r.t. time . On expanding the Taylor series for scale factor around , the fourth term of the Taylor series contains the jerk parameter sin6 (). The Taylor’s expansion around containing the jerk parameter is given by
(48) 
where , and being the the current values of the parameters. Therefore the jerk parameter is defined as
(49) 
and in terms of , jerk parameter reads vis (); rap ()
(50) 
(51) 
Also it will always be suitable to express jerk parameter in terms of redshift when is given sah3 (); ala (). The Jerk parameter in terms of is expressed as
(52) 
In Fig. 6(a), the cosmic jerk parameter highlights the dynamics of the Universe. Universe transits from decelerated to accelerated phase in a cosmic jerk with a positive value and negative value in accordance with CDM. It shows the evolution of parameter for different values of and is freely seen that remains positive in all the cases and approaches to in late times. Jerk parameter at present () is positive different from in all the four cases. Therefore, we can expect another dark energy model instead of CDM.
v.2 Om diagnostic
In this section we discuss the most popular diagnostic known as Om diagnostic denoted by , used to distinguish standard model from various dark energy models sah4 (); zun (). This diagnostic is related to Hubble parameter and redshift . It is noted that different trajectories of facilitate significant differences among various DE models without actually mentioning the current value of (density parameter of matter). Om diagnostic is defined as
(53) 
This tool suggest a quintessence type behavior of dark energy corresponding to its negative curvature (i.e. below the line), phantom type behavior corresponding to its positive curvature (i.e. above the line) and corresponding to zero curvature. Fig. 6(b) explains the behavior of different dark energy models corresponding to different values of . For model shows quintessence type behaviour as graph of shows a downward trend as redshift increases and for , model represents phantom behavior as has positive slope.
v.3 Velocity of sound
The velocity of sound is one of the stringent attempt to investigate the validity of a cosmic model. A model is said to be physically acceptable if velocity of sound is less than the speed of light . The stability condition for the model is given by the relation . In this study we have taken the speed of light is 1. Therefore, the model is physically realistic provided the condition is satisfied.
Fig. 6(c) shows the profile of for . The constant decides the stability of the model for a fix value of . The stability of the cosmic model depends on the coupling constant . By considering different values of , we have different stability scenarios of the model. The model satisfies the condition throughout the evolution with time for all the cases when is taken in the range . The condition does not hold for other cosmic ranges. Therefore, we can say that our model is partially stable.
v.4 Statefinder diagnostic
The present cosmic acceleration which is rational with the recent cosmological observations and the signature flipping behaviour of deceleration parameter from to in accordance with high redshift to low redshift enforce us to study beyond and , and find some more cosmological models of DE other than CDM. The behavior of higher derivatives of scale factor other than and are the essential components to explain the dynamics of the Universe. Due to this reasons, we generalize our domain to construct geometrical parameters which involves higher derivatives of . A technique named as Statefinder diagnostic in which a pair of geometrical parameters proposed by sah3 (); ala () is taken in to account to describe the dynamics of various DE models. These parameters are defined as
(54) 
where .
For our parametrization of in Eq. (27), the expressions of and are given as follows:
(55) 
where the parameter is same as the jerk parameter , which is defined in subsection .
(56) 
This technique facilitate us that how one can differentiate various DE models easily by plotting the different trajectories of and (see Fig. 7). For a brief and recent review on statefinder diagnostic, see sin2 (); rit2 (); srt ().
Fig. 7(a) highlight the evolution of four trajectories with time for and in plane. Each trajectory for ; and exhibit the same pattern as all begin in the region and evolving with time, approaches to CDM model i.e. the point . The time evolution of the trajectory corresponding to starts from the region and and eventually approaches to CDM model. From the plot 7(a), we observe that all the trajectories deviate from SCDM which is resemble to matter dominated universe, exhibit different dark energy candidates as Chaplygin gas for , quintessence for ; and , CDM for and SCDM for . Thus the various DE scenarios can be observed by these evolutionary trajectories which are the remarkable features of statefinder diagnostic.
Fig. 7(b) states the evolution of the four trajectories with time for and in plane. Each trajectory begins in the neighbourhood of at the time of evolution of the Universe without passing through CDM and converge to , the steady state model of the Universe. The downward pattern of a trajectory corresponding to , and upward trend of the trajectories corresponding to , ; , and , converge to the point , denoted by SS i.e. the steady state model of the Universe which suggest the steady state behavior of dark energy model in late times.
Vi Observational constraints on the model parameters
An impressive feature of astronomy is associated with its recent progress in observational cosmology. Study of the origin, evolution, structure formation, properties of dark matter and dark energy in the Universe with the help of cosmic instruments and ray detectors is called observational cosmology. There are several types of observational data available today for different measurements in the field of cosmology. Some of them are Sloan Digital Sky Survey () which provide the map of the galaxy distribution and encode the current fluctuations in the Universe, that serves as the evidence of the big bang theory, Quasi Stellar Radio Sources () which are considered as the most metal thing in the Universe and extract the matter between observer and quasars, Baryon Acoustic Oscillations () that measures the large scale structures in the Universe in order to understand dark energy better, observations from type Ia Supernova are the tools for measuring the cosmic distances usually known as standard candles. In the subsequent sections, we have presented a statistical analysis by using some observational datasets of , and to constrain Hubble parameter and model parameter involved in our model. To constraint model parameter , we restrict the inverse hyperbolic of sine series and hyperbolic of cotangent series in Eq. (33) upto first term and then integrate the approximate series to calculate the Chisquare value i.e. using each observational data set.
vi.1 Hubble observation H(z)
The Hubble parameter can be observed in terms of some physical observable quantity such as length, time and redshift . In terms of redshift , reads