# Wormhole Models and Energy Conditions infℛ Gravity with the Hu–Sawicki Model

Authors: Ouziala Ikram; G. Mustafa
Journal: Advances in Astronomy
 (2020)

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## Abstract

In this recent study, we shall investigate the wormhole models with a Hu–Sawicki model in the framework offℛ gravity. Spherically static symmetric space-time is considered to construct wormhole models with the anisotropic fluid distribution. The traceless matter is discussed by imposing a particular equation of state. To address the important conditions of the shape function of the wormhole geometry, we have used the particular values of the involved parameters. Furthermore, different energy conditions are discussed to check the nature of matter against two specific models. The null energy condition is observed to be violated for both of the models. It is mentioned that our inquired results are acceptable.


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## Body

## 1. Introduction

In modern cosmology, considerable independent high-precision observational data have confirmed with startling evidence that our universe is in the accelerating phase [1–5]; this is due to several astronomical probes, such as Ia supernova [6, 7], cosmic microwave background radiation (CMBR) [8, 9], and large structure [10]. This phenomenon led to the breakdown of Einstein’s theory of relativity; it is regarded as an outstanding critical riddle of contemporary physics and becomes the focus of interest for researchers. In this regard, to incorporate this issue, different proposals have appeared which can be grouped into two classes. First, the cosmological constant has been introduced in the field equation [11–14]; second, the approach is to modify the gravitational part of the action by including some extra degree of freedom there. To deal with this problem, several efforts from researchers provided a group of extended theories of gravity [15–25].After the formulation of general relativity, thefℛ theory is regarded as one of the most promising candidates for mysterious dark energy to exploring the accelerated expansion aspect of the cosmos; for cosmological importance, the fℛ model is used by replacing the Ricci scalar ℛ with an arbitrary function of Ricci scalar, i.e., fℛ, in the Einstein–Hilbert Lagrangian function. This theory has been widely used in the literature. Nojiri and Odintsov [26, 27] have explored the accelerated expansion of the universe by introducing the term fℛ=1/ℛ, which is essential at small curvature. Feng [28] reconstructed the fℛ theory from Ricci dark energy, and Felice and Tsujikawa [15] presented several ways to distinguish the several forms of fℛ theory from general relativity. Harko et al. [29] provided a more generic form of fℛ extended gravity by plugging an extra term of matter. Rahaman and his co-authors [30] have tried to calculate the exact wormhole solutions in fℛ gravity in the framework of noncommutative geometry source. In another paper, Harko et al. [31] have calculated some solutions for wormholes in extended fℛ. Pavlovic and Sossich [32] have carried out a wormhole study by employing different models of fℛ without considering the exotic matter. Bronnikov [33] has explored the wormhole models in two different theories (scalar tensor theory and fℛ gravity). Saiedi and Nasr Esfahani [34] have discussed the null and weak energy bounds in the background of wormhole fℛ gravity. Hochberg et al. [35] have presented wormhole and calculated the appropriate field equations. Furthermore, Kuhfittig [36] discussed noncommutative geometry to investigate the distinct forms of fℛ function by taking different shape functions in fℛ theory.A wormhole is a topological passage, which links two discrete chunks of the same or diverse universes through a shortcut way, which is called a channel or bridge. The discussion on wormhole geometry is a hot topic among researchers in the different modified theories of gravity. In 1916, Flamm [37] expressed the notion of wormholes for the first time. After that, in 1935, Einstein and Rosen [38] explored wormhole mathematics. They acquired wormhole solutions, which are known as Lorentzian wormhole. In classical general relativity, the existence of wormholes involves the exotic matter, which requires a stress-energy tensor that violated the null energy condition NEC. Much interest has been provoked in the existence and construction of wormhole solutions in modified gravity theory. Capozziello et al. [39–41] examined the energy condition for modified theories of gravity and developed a generic scheme for such conditions. In the literature, many researchers studied the actuality of wormhole solutions by using different kinds of exotic matter in the context of different modified theories and also tested their stability [42–45]. Recently, Mustafa et al. [46] found the stable wormhole solutions for the exponential gravity model in the fℛ gravity. Samanta and Godani [47] analyze the traversable wormhole solutions with exponential shape function in modified gravity. Most recently, Farasat Shammir and Ahmad [48] have calculated the new equation of motion offering test particle in the equatorial plane around a wormhole geometry in the framework of general relativistic Poynting–Robertson effect. Falco et al. [49] have calculated some feasible regions for the existence of traversable wormhole in modified gravity. Numerous authors [50, 51] have constructed wormholes by including different types of exotic matter such as quantum scalar field models, noncommutative geometry, and electromagnetic field.The plan of this recent study is as follows: In Section2, we shall calculate the basics of fℛ gravity with the Hu–Sawicki model. The next section is dedicated to discussing the energy conditions and traceless matter for wormhole solutions. In the same section, we shall present our acquired solution graphical analysis against the different physical properties of shape function. In Section 4, we shall choose two different kinds of specific shape functions; both are based on special functions to discuss the physical nature of matter through energy conditions. An outlook of this work performed is presented in the last section.


## 2.fℛ Gravity

A modified action forfℛ gravity is(1)SA=12κ∫d4xfℛ−g+Smgμν,ξ,where fℛ and Smgμν,ξ represents the function of the Ricci scalar and the matter term of the action, respectively. In this work, we assumed that κ=1. On variation of the above action, we have the following actual system of equations:(2)Tμνm=fℛℛℛμν−12fℛgμν+gμν□−∇μ∇νfℛℛ.The trace of energy momentum tensor is calculated as(3)T=fℛℛℛ+3□fℛℛ−2fℛ,where □, ∇, and fℛℛ are defined as the d’Alembert operator, covariant derivative operator, and the derivative with respect to ℛ, respectively in the above two lines. By using equation (3) in equation (2), the following expression can be obtained:(4)Tμνeff=ℛμν−12ℛgμν=Gμν,where Tμνeff=Tμνc+T˜μνm represents the effective stress-energy tensor and T˜μνm is calculated as(5)Tμνc=1fℛℛ∇μ∇νfℛℛ−14gμνℛfℛℛ+□fℛℛ+T,Tμνm=T˜μνmfℛℛ.The anisotropic fluid distribution is given as(6)Tμν=ρ+Ptυμυν−Ptgμν+Pr−PtΞμΞν,where υμ mentions the 4-velocity vector with(7)υμ=e−aδ0μ,Ξμ=e−bδ1μ,υμυμ=−ΞμΞμ=1.The space-time for a wormhole geometry is defined as(8)ds2=−e2φfrdt2+r+bsrr−1dr2+r2dΩ2,where φf and bsr represent the red-shift function and shape function, respectively. The expression dΩ2=dθ2+sin2θdϕ2 provides the remaining coordinate quantities. The shape function bsr must satisfy the following conditions: (i) bsr−bsr′r/bsr2>0, (ii) bsrr0=r0, and (iii) bsr′r0<1. The last one condition is known as flaring-out condition.By using (6) and (8) in (4), the following expressions are calculated for the energy density and pressure components:(9)bs′rr2=ρfℛℛ+ψfℛℛ,(10)−bsrr3=Prfℛℛ+1fℛℛ1−bsrrfℛ″ℛ−fℛ′ℛbs′rr−bsr2r21−bsr/r−ψfℛℛ,(11)−bsr′r−bsr2r3=Ptfℛℛ+1fℛℛ1−bsrrfℛ′ℛr−ψfℛℛ,where(12)ψ=ψr=14fℛℛℛ+□fℛℛ+T,with(13)ℛ=2bs′rr2,(14)□fℛℛ=1−bsrrfℛ″ℛ−fℛ′ℛbsr′r−bsr2r21−bsr/r+2fℛ′ℛr.The Hu–Sawicki model [52], which fulfills the cosmological and local gravity conditions, was proposed by Hu and Sawicki and is given as(15)fℛ=R−αγR/γ2mR/γ2m+1,where α,γ, and m are the dimensionless positive parameter. This model has passed all the local tests, and it is a viable fℛ gravity model.By utilizing equations (12)–(15) in equations (9)–(11), we have the following modified field equations:(16)ρ=bs′rr2−αγ4mmX12m4mX12m+12,(17)Pr=12r34mX12m+14bs′r3bsr−28m+1X18mbs′r3−2bs′r3+4mX6X12m+24m+1X8X14m+64mX7X16m+αγ4mmr322m+1X5X12m+16m2m+1X4X14m+2m+1X3X12m,(18)Pt=1−X2bsr2r3−1−X2bs′r2r2+X9+X10,where(19)X1=bs′rγr2,X2=α4mmX12m−14mX12m+12,X3=4m−1r2bs″r2+bs′r2−rbs″r+16m−4+2rbs′r4−8mbs″r+rbs3r+2bs′r3,X4=4m+1r2bs″r2+bs′r2rbs″r+16m+4−2rbs′r8m+4bs″r+rbs3r−2bs′r3,X5=41−4m2r2bs″r2+bs′r2rbs″r−64m2+4−2rbs′r41−8m2bs″r+rbs3r−2bs′r3,X6=4αγm25−8mr2bs′r2−4αγm−1m2m−1r4bs″r2+αγm2m−1r3bs′r×16m−7bs″r−2rbs3r−8bs′r3,X7=−4αγm28m+5r2bs′r2−4αγmm+12m+1r4bs″r2+αγm2m+1r3bs′r×16m+7bs″r+2rbs3r−8bs′r3,X8=64αγm3r2bs′r2+4αγm4m2−1r4bs″r2+αγmr3bs′r7−64m2bs″r+2rbs3r−6bs′r3,X9=αγ4mmX12m4m2m+1X12m−2m+12bs′r−rbs″r4mX12m+13bs′r2,X10=αγ4mmX12m4m2m+1X12m−2m+1bsrrbs″r−2bs′rr4mX12m+13bs′r2.


## 3. Wormhole Solutions

In this current section, we shall discuss the energy conditions and develop a new technique to explore the wormhole construction.


### 3.1. Energy Conditions

The null, weak, strong, and dominant energy conditions are considered main energy conditions in the background offℛ gravity. These energy conditions are defined as(20)NEC⟺Tγξυμυν≥0,WEC⟺TγξΞμΞν≥0,SEC⟺Tγξ−T2gγξΞμΞν≥0,DEC⟺TγξΞμΞν≥0.In the above relation,υμ mentions the null vector and Ξμ represents the timelike vector. The following energy conditions with respect to principal pressure are calculated:(21)NEC⟺∀j,ρ+Pj≥0,WEC⟺ρ≥0and∀j,ρ+Pj≥0,SEC⟺∀j,ρ+Pj≥0,ρ+∑jpj≥0,DEC⟺ρ≥0and∀j,Pjε−ρ,+ρ.The above conditions can be written as(22)NEC:ρ+Pr≥0,ρ+Pt≥0,WEC:ρ≥0,ρ+Pr≥0,ρ+Pt≥0,SEC:ρ+Pr≥0,ρ+Pt≥0,ρ+Pr+2Pt≥0,DEC:ρ≥0,ρ−Pr≥0,ρ−Pt≥0.Theses energy conditions are satisfied by the normal matter because of its positive density and positive pressure. Einstein’s field theory shows that wormholes have required exotic matter. To explore the wormhole construction, we shall check the behavior of the energy conditions, speciallyNEC and WEC. The NEC violation is the important requirement for the existence of wormhole construction.


### 3.2. Traceless Matter for the Hu–Sawicki Model

Now, we shall discuss the traceless matter [53–55], for the Hu–Sawicki model for fℛ gravity. A special form of equation of state (EoS) for traceless matter is defined as(23)T=0,⟹ρ−Pr−2Pt=0.By using equations (16)–(18) in equation (23), we obtain a nonlinear differential equation, which is calculated as(24)12r24mX12m+14bs′r3αγ4m+1mr322m+14m2−1X12m−2m2−16mm+12m+1X14m+3m−1×X12mr−bsrbs″r2+αγ4mmr2X12mbs′r22m+1X12m64m2−11bsr+43−16m2rbs″r+2r×r−bsrbs3r+16m2m−1X14m44m+3r−16m+11bsrbs″r+2rr−bsrbs3r+2m−111−16mbsr+44m−3rbs″r+2rbsr−rbs3+αγ4mmrX12mbs′r2×−22m+1X12mrrbs″r−64m2+12+232m2−5bsr−16m2m+1X14mrrbs″r+16m+12−28m+5bsr+2m−1rrbs″r−16m+12+28m−5bsr+α22m+1m4mX12m+1×4m2m−1X12m−2m−1X12m−1bs′r4+44mX12m+14bs′r4=0.Equation (24) is a nonlinear differential equation, and we solve it numerically with the following initial conditions: bs0.01=0.001,bs′0.01=0.0002,andbs′′0.01=0.07.


### 3.3. Discussion

In this section, we shall discuss the physical analysis of our acquired results. To improve the real impact of the current study, we have tried to provide a graphical analysis of the intrinsic properties of shape function and energy conditions under particular values of involved parameters. In the Hu–Sawicki model, the parameterα has an important role in our study. The different values of parameter α between zero and one provide different results, which are seen in Figures 1–5. However, the other parameter γ does not change the results. Therefore, we have chosen only one value for γ, i.e., γ=0.07 km for the current study. In this study, there is another parameter m involved. We have taken a very small value of parameter m, i.e., m=0.05. The left part of Figure 1 shows the graphical behavior of the shape function, i.e., bsr. It is observed to be positive and increasing, which describes the realistic nature of wormhole geometry under the Hu–Sawicki model. The flaring-out condition, i.e., the derivative of the shape function concerning radial coordinate r at r0, can be revealed as less than one, which can be confirmed from Figure 1. It is checked from Figure 2 that the expression bsr−r gives the values of r0, which shows the wormhole throat location for the distinct values of parameter α, which are approximately observed at r0≡0.0015,r0≡0.0016,r0≡0.0017,r0≡0.0018, and r0≡0.0019, for α=0.20, α=0.40, α=0.60, α=0.80, and α=1.00, respectively. Figure 2 provides the graphical analysis of bsr/r⟶0 as r⟶∞, which is not observed clearly. It shows that the wormhole geometry is not asymptotically flat, and this conduct agrees with the already prevailing cases described in the literature (see, for example, [43–46]). In the reference of energy conditions, it is verified from Figure 3 that the condition ρ+Pr is seen negative, which shows the presence of exotic matter. From the same figure, it is confirmed that the expression ρ+Pr remains positive. From Figures 4-5, it is confirmed that the expressions ρ−Pr, ρ−Pt, and ρ remain positive.Figure 1
Behavior ofbsr and bs′r. Here, α=0.20 (black), α=0.40 (red), α=0.60 (blue), α=0.80 (green), and α=0.80 (orange).
(a)(b)Figure 2
Development ofbsr−r and bsr/r. Here, α=0.20 (black), α=0.40 (red), α=0.60 (blue), α=0.80 (green), and α=0.80 (orange).
(a)(b)Figure 3
Graphical analysis ofρ+Pr and ρ+Pt.
(a)(b)Figure 4
Graphical analysis ofρ−Pr and ρ−Pt.
(a)(b)Figure 5
Behavior ofρ.


## 3.1. Energy Conditions

The null, weak, strong, and dominant energy conditions are considered main energy conditions in the background offℛ gravity. These energy conditions are defined as(20)NEC⟺Tγξυμυν≥0,WEC⟺TγξΞμΞν≥0,SEC⟺Tγξ−T2gγξΞμΞν≥0,DEC⟺TγξΞμΞν≥0.In the above relation,υμ mentions the null vector and Ξμ represents the timelike vector. The following energy conditions with respect to principal pressure are calculated:(21)NEC⟺∀j,ρ+Pj≥0,WEC⟺ρ≥0and∀j,ρ+Pj≥0,SEC⟺∀j,ρ+Pj≥0,ρ+∑jpj≥0,DEC⟺ρ≥0and∀j,Pjε−ρ,+ρ.The above conditions can be written as(22)NEC:ρ+Pr≥0,ρ+Pt≥0,WEC:ρ≥0,ρ+Pr≥0,ρ+Pt≥0,SEC:ρ+Pr≥0,ρ+Pt≥0,ρ+Pr+2Pt≥0,DEC:ρ≥0,ρ−Pr≥0,ρ−Pt≥0.Theses energy conditions are satisfied by the normal matter because of its positive density and positive pressure. Einstein’s field theory shows that wormholes have required exotic matter. To explore the wormhole construction, we shall check the behavior of the energy conditions, speciallyNEC and WEC. The NEC violation is the important requirement for the existence of wormhole construction.


## 3.2. Traceless Matter for the Hu–Sawicki Model

Now, we shall discuss the traceless matter [53–55], for the Hu–Sawicki model for fℛ gravity. A special form of equation of state (EoS) for traceless matter is defined as(23)T=0,⟹ρ−Pr−2Pt=0.By using equations (16)–(18) in equation (23), we obtain a nonlinear differential equation, which is calculated as(24)12r24mX12m+14bs′r3αγ4m+1mr322m+14m2−1X12m−2m2−16mm+12m+1X14m+3m−1×X12mr−bsrbs″r2+αγ4mmr2X12mbs′r22m+1X12m64m2−11bsr+43−16m2rbs″r+2r×r−bsrbs3r+16m2m−1X14m44m+3r−16m+11bsrbs″r+2rr−bsrbs3r+2m−111−16mbsr+44m−3rbs″r+2rbsr−rbs3+αγ4mmrX12mbs′r2×−22m+1X12mrrbs″r−64m2+12+232m2−5bsr−16m2m+1X14mrrbs″r+16m+12−28m+5bsr+2m−1rrbs″r−16m+12+28m−5bsr+α22m+1m4mX12m+1×4m2m−1X12m−2m−1X12m−1bs′r4+44mX12m+14bs′r4=0.Equation (24) is a nonlinear differential equation, and we solve it numerically with the following initial conditions: bs0.01=0.001,bs′0.01=0.0002,andbs′′0.01=0.07.


## 3.3. Discussion

In this section, we shall discuss the physical analysis of our acquired results. To improve the real impact of the current study, we have tried to provide a graphical analysis of the intrinsic properties of shape function and energy conditions under particular values of involved parameters. In the Hu–Sawicki model, the parameterα has an important role in our study. The different values of parameter α between zero and one provide different results, which are seen in Figures 1–5. However, the other parameter γ does not change the results. Therefore, we have chosen only one value for γ, i.e., γ=0.07 km for the current study. In this study, there is another parameter m involved. We have taken a very small value of parameter m, i.e., m=0.05. The left part of Figure 1 shows the graphical behavior of the shape function, i.e., bsr. It is observed to be positive and increasing, which describes the realistic nature of wormhole geometry under the Hu–Sawicki model. The flaring-out condition, i.e., the derivative of the shape function concerning radial coordinate r at r0, can be revealed as less than one, which can be confirmed from Figure 1. It is checked from Figure 2 that the expression bsr−r gives the values of r0, which shows the wormhole throat location for the distinct values of parameter α, which are approximately observed at r0≡0.0015,r0≡0.0016,r0≡0.0017,r0≡0.0018, and r0≡0.0019, for α=0.20, α=0.40, α=0.60, α=0.80, and α=1.00, respectively. Figure 2 provides the graphical analysis of bsr/r⟶0 as r⟶∞, which is not observed clearly. It shows that the wormhole geometry is not asymptotically flat, and this conduct agrees with the already prevailing cases described in the literature (see, for example, [43–46]). In the reference of energy conditions, it is verified from Figure 3 that the condition ρ+Pr is seen negative, which shows the presence of exotic matter. From the same figure, it is confirmed that the expression ρ+Pr remains positive. From Figures 4-5, it is confirmed that the expressions ρ−Pr, ρ−Pt, and ρ remain positive.Figure 1
Behavior ofbsr and bs′r. Here, α=0.20 (black), α=0.40 (red), α=0.60 (blue), α=0.80 (green), and α=0.80 (orange).
(a)(b)Figure 2
Development ofbsr−r and bsr/r. Here, α=0.20 (black), α=0.40 (red), α=0.60 (blue), α=0.80 (green), and α=0.80 (orange).
(a)(b)Figure 3
Graphical analysis ofρ+Pr and ρ+Pt.
(a)(b)Figure 4
Graphical analysis ofρ−Pr and ρ−Pt.
(a)(b)Figure 5
Behavior ofρ.


## 4. Analysis of Energy Conditions for Two Specific Shape Functions

In this section, we shall analyze the behavior of energy condition to check the nature of matter for two specific shape functions.


### 4.1. First Model

Here, we consider a specific model of shape function, i.e.,bsr=r0logr+1/logr0+1 [56]. By plugging this special function-based specific model of shape function into equations (16)–(18), we obtain the following expressions for energy density and pressure components:(25)ρ=r0r2r+1logr0+1−αγ4mmψ12m4mψ12m+12,Pr=12r3r+1r04mΨ12m+14logr0+1αγ4mmr3−22m+1ψ5ψ12m+16m2m+1ψ4ψ14m+2m−1×ψ3ψ12mlogr0+1+r0logr+1αγ4mmr2ψ2ψ12mlogr0+1−2r+1r04mψ12m+14,Pt=8αγm23r+2r2rlogr0+1−r0logr+1/4mψ12m+13+r02r+1logr+1−r/r+1logr0+1+ψ6+ψ72r3r0,where(26)ψ1=r0γr2r+1logr0+1,ψ2=−8m23r+22−4m+1ψ12m+16mψ14m+1−7rr+14mψ12m+12−2mr27r+35+10×16mψ14m−1,ψ3=4m3r+22−r+13r+1logr0+1+3r+2r0,ψ4=4m3r+22+r+13r+1logr0+1−3r+2r0.


### 4.2. Second Model

Now, we take another specific model of shape function, i.e.,bsr=r/er−r0 [57]. By using bsr=r/er−r0 in equations (16)–(18), the expressions for energy density and pressure components are calculated as follows:(27)ρ=−αγ4mmΩ12m4mΩ12m+12−er0−rr−1r2,Pr=−12r−13r34mΩ12m+14e3r−3r0αγ4mme3r0−3rr3Ω2Ω12m+αγ22m+1me2r0−2rr3Ω3Ω12m+2e4r0−4rr−13r4mΩ12m+14,Pt=12e−r08αγm2er0−err2−2r−124mΩ12m+13+e2r0−rr+Ω4+Ω5,where(28)Ω1=−er0−rr−1γr2,Ω2=−8m2r2−22−4m+1Ω12m+16mΩ14m+1−rrr2+r−10+10+24mΩ12m+12−2mrr3r2+r−16+6+1216mΩ14m−1,Ω3=4m2r2−22−4m+1Ω12m+16mΩ14m+1+rr3−5r+4+24mΩ12m+12+2m×r2r3−9r+4+616mΩ14m−1,Ω4=αγm26m+1err2−2+er0−34m+1r2+24m+r+4r−124mΩ12m+12,Ω5=αγmer04mr2−2+r+13r−4−22m+1err2−2r−124mΩ12m+1.


### 4.3. Discussion

Here, we shall discuss about the behavior of energy conditions for two different specific models of shape function. For the first model, i.e.,bsr=r0logr+1/logr0+1, we have shown the graphical analysis of energy conditions in Figures 6–8. To explore the wormhole models, we check the behavior of the energy conditions, specially NEC and WEC. The NEC violation is the important requirement for the existence of wormhole solutions. From Figure 6, the violation of NEC can be confirmed, which shows the presence of exotic matter. The presence of exotic matter is responsible to open the wormhole throat. Other energy conditions are seen satisfied. The positive nature of energy density shows the supremacy of this study. The validity of other remaining energy conditions has no effect on the wormhole solutions. In the second case, i.e., bsr=r/er−r0, the violation of NEC is also perceived for small radial coordinate, which can be checked from Figure 9. The graphical behavior of ρ−Pr and ρ−Pt can be checked from Figure 10 for the second model. The behavior of energy density for the second model can be depicted from Figure 11. Furthermore, the detailed summary for both models can be seen from Table 1.Figure 6
Graphical analysis ofρ+Pr and ρ+Pt.
(a)(b)Figure 7
Graphical analysis ofρ−Pr and ρ−Pt.
(a)(b)Figure 8
Behavior ofρ.Figure 9
Graphical analysis ofρ+Pr and ρ+Pt.
(a)(b)Figure 10
Graphical analysis ofρ−Pr and ρ−Pt.
(a)(b)Figure 11
Behavior ofρ.Table 1
Summary for first and second models of shape function.
Energy conditionsModels/conditionsbsr=r0logr+1/logr0+1bsr=r/er−r0ρ+Ptρ+Pt>0ρ+Pt>0ρ+Prρ+Pr<0ρ+Pr<0ρ−Ptρ−Pt>0ρ−Pt<0ρ−Prρ−Pr<0ρ−Pr<0ρρ>0ρ>0


## 4.1. First Model

Here, we consider a specific model of shape function, i.e.,bsr=r0logr+1/logr0+1 [56]. By plugging this special function-based specific model of shape function into equations (16)–(18), we obtain the following expressions for energy density and pressure components:(25)ρ=r0r2r+1logr0+1−αγ4mmψ12m4mψ12m+12,Pr=12r3r+1r04mΨ12m+14logr0+1αγ4mmr3−22m+1ψ5ψ12m+16m2m+1ψ4ψ14m+2m−1×ψ3ψ12mlogr0+1+r0logr+1αγ4mmr2ψ2ψ12mlogr0+1−2r+1r04mψ12m+14,Pt=8αγm23r+2r2rlogr0+1−r0logr+1/4mψ12m+13+r02r+1logr+1−r/r+1logr0+1+ψ6+ψ72r3r0,where(26)ψ1=r0γr2r+1logr0+1,ψ2=−8m23r+22−4m+1ψ12m+16mψ14m+1−7rr+14mψ12m+12−2mr27r+35+10×16mψ14m−1,ψ3=4m3r+22−r+13r+1logr0+1+3r+2r0,ψ4=4m3r+22+r+13r+1logr0+1−3r+2r0.


## 4.2. Second Model

Now, we take another specific model of shape function, i.e.,bsr=r/er−r0 [57]. By using bsr=r/er−r0 in equations (16)–(18), the expressions for energy density and pressure components are calculated as follows:(27)ρ=−αγ4mmΩ12m4mΩ12m+12−er0−rr−1r2,Pr=−12r−13r34mΩ12m+14e3r−3r0αγ4mme3r0−3rr3Ω2Ω12m+αγ22m+1me2r0−2rr3Ω3Ω12m+2e4r0−4rr−13r4mΩ12m+14,Pt=12e−r08αγm2er0−err2−2r−124mΩ12m+13+e2r0−rr+Ω4+Ω5,where(28)Ω1=−er0−rr−1γr2,Ω2=−8m2r2−22−4m+1Ω12m+16mΩ14m+1−rrr2+r−10+10+24mΩ12m+12−2mrr3r2+r−16+6+1216mΩ14m−1,Ω3=4m2r2−22−4m+1Ω12m+16mΩ14m+1+rr3−5r+4+24mΩ12m+12+2m×r2r3−9r+4+616mΩ14m−1,Ω4=αγm26m+1err2−2+er0−34m+1r2+24m+r+4r−124mΩ12m+12,Ω5=αγmer04mr2−2+r+13r−4−22m+1err2−2r−124mΩ12m+1.


## 4.3. Discussion

Here, we shall discuss about the behavior of energy conditions for two different specific models of shape function. For the first model, i.e.,bsr=r0logr+1/logr0+1, we have shown the graphical analysis of energy conditions in Figures 6–8. To explore the wormhole models, we check the behavior of the energy conditions, specially NEC and WEC. The NEC violation is the important requirement for the existence of wormhole solutions. From Figure 6, the violation of NEC can be confirmed, which shows the presence of exotic matter. The presence of exotic matter is responsible to open the wormhole throat. Other energy conditions are seen satisfied. The positive nature of energy density shows the supremacy of this study. The validity of other remaining energy conditions has no effect on the wormhole solutions. In the second case, i.e., bsr=r/er−r0, the violation of NEC is also perceived for small radial coordinate, which can be checked from Figure 9. The graphical behavior of ρ−Pr and ρ−Pt can be checked from Figure 10 for the second model. The behavior of energy density for the second model can be depicted from Figure 11. Furthermore, the detailed summary for both models can be seen from Table 1.Figure 6
Graphical analysis ofρ+Pr and ρ+Pt.
(a)(b)Figure 7
Graphical analysis ofρ−Pr and ρ−Pt.
(a)(b)Figure 8
Behavior ofρ.Figure 9
Graphical analysis ofρ+Pr and ρ+Pt.
(a)(b)Figure 10
Graphical analysis ofρ−Pr and ρ−Pt.
(a)(b)Figure 11
Behavior ofρ.Table 1
Summary for first and second models of shape function.
Energy conditionsModels/conditionsbsr=r0logr+1/logr0+1bsr=r/er−r0ρ+Ptρ+Pt>0ρ+Pt>0ρ+Prρ+Pr<0ρ+Pr<0ρ−Ptρ−Pt>0ρ−Pt<0ρ−Prρ−Pr<0ρ−Pr<0ρρ>0ρ>0


## 5. Outlook

In this recent work, we have employed the Hu–Sawicki model infℛ gravity to explore the wormhole construction. In this regard, the anisotropic source of matter is used with spherically symmetric static space-time. We have tried to calculate the modified equations for energy density and pressure components by using the Hu–Sawicki model. By employing the Hu–Sawicki model, our obtained solutions under the particular values of parameters show the possible way to explore traversable asymptotically hyperbolic and flat wormhole solutions that respect NEC around the throat. In the case of the other models, asymptotically flat wormholes can also admire the WEC through the whole space outside the throat. The Hu–Sawicki model also satisfied the cosmological and local gravity constraints. Furthermore, we have plugged the idea of traceless matter to calculate the shape function. The following important outcomes have been presented(1)
The wormhole throat location for the distinct values of parameterα are calculated at r0≡0.0015,r0≡0.0016,r0≡0.0017,r0≡0.0018, and r0≡0.0019, for α=0.20, α=0.40, α=0.60, α=0.80, and α=1.00, respectively.(2)
The ratio of shape function and radial coordinate approaches a very small number, i.e.,bsr/r⟹0.067,bsr/r⟹0.095,bsr/r⟹0.115,bsr/r⟹0.135,andbsr/r⟹0.150 for α=0.20, α=0.40, α=0.60, α=0.80, and α=1.00, respectively.(3)
The shape function, i.e.,bsr, with positive behavior can be verified from Figure 1.(4)
The flaring condition can be confirmed from Figure1, which is observed at less than 1.(5)
The conditionρ+Pr is noticed negative, which can be verified from Figure 3. The negative nature of ρ+Pr shows the presence of exotic matter. The expression ρ−Pr has been depicted positively.(6)
The expressionsρ+Pt and ρ−Pt have remained positive.(7)
The energy densityρ has remained positive in all cases.(8)
The graphical behavior of energy conditions for both models can be checked from Figures6–11.Furthermore, theNEC has been revealed to be violated for two different specific shape functions. The negative nature of NEC shows the presence of exotic matter which is necessary for wormhole construction. All our calculated results can be seen from the summary, which is presented in Table 1 for two different specific shapefunctions. It is very interesting to comment here that our calculated results in fℛ gravity with the Hu–Sawicki model for anisotropic fluid sources are physically acceptable.



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