Prepared for submission to JCAP

Strange stars inenergy-momentum-conservedf(R, T ) gravity

G.A. Carvalho,a,b,1 S.I. dos Santos, Jr.,c P.H.R.S. Moraesa,2 andM. Malheiroa,2

aInstituto de Pesquisa e Desenvolvimento (IP&D), Universidade do Vale do Paraíba,12244-000, São José dos Campos, SP, BrazilbDepartamento de Física, Instituto Tecnológico de Aeronáutica,São José dos Campos, SP, 12228-900, BrazilcDipartamento di Fisica, Università degli Studi di Napoli Federico II,Napoli 80126, Italy

E-mail: geanderson.araujo.carvalho@gmail.com, samuel.isidoro.santos@gmail.com,moraes.phrs@gmail.com, malheiro@ita.br

Abstract. For the accurate understanding of compact objects such as neutron stars andstrange stars, the Tolmann-Openheimer-Volkof (TOV) equation has proved to be of greatuse. Hence, in this work, we obtain the TOV equation for the energy-momentum-conservedf(R, T ) theory of gravity to study strange quark stars. The f(R, T ) theory is important,especially in cosmology, because it solves certain incompleteness of the standard model. Ingeneral, there is no intrinsic conservation of the energy-momentum tensor in the f(R, T )gravity. Since this conservation is important in the astrophysical context, we impose thecondition ∇Tµν = 0, so that we obtain a function f(R, T ) that implies conservation. Thischoice of a function f(R, T ) that conserves the momentum-energy tensor gives rise to a stronglink between gravity and the microphysics of the compact object. We obtain the TOV bytaking into account a linear equation of state to describe the matter inside strange stars, suchas p = ωρ and the MIT bag model p = ω(ρ − 4B). With these assumptions it was possibleto derive macroscopic properties of these objects.

1Corresponding author.

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Contents

1 Introduction 1

2 The f(R, T ) gravity 22.1 The energy-momentum conserved formalisms proposed for the f(R, T ) gravity 3

3 Energy-momentum-conserved f(R, T ) gravity for strange quark matter 4

4 The Tolman-Oppenheimer-Volkoff equations and their solutions 54.1 Case: p = ωρ 64.2 Case: MIT Bag Model p = ω(ρ− 4B) 7

5 Discussion 9

1 Introduction

Alternative theories of gravity have the purpose of solving some issues that, in principle,General Relativity cannot, such as the dark energy [1, 2] and dark matter problems [3] andeven the theoretical prediction of observed massive pulsars [4, 5] and super-Chrandrasekharwhite dwarfs [6–8] that hardly can be explained assuming standard structure and equationof state (EoS) for these objects.

Today, probably the most popular of the alternative gravity theories is the f(R) theory[9–11], which takes a general function of the Ricci scalar R in the gravitational action as itsstarting point. Indeed, the presence of general terms in R in the action yields extra terms inthe field equations of the theory, and those, in a cosmological aspect, can explain the presentcosmic acceleration [12, 13] with no need for dark energy [14, 15]. Such extra terms can alsoelevate the maximum mass theoretically expected for neutron stars [16–18] and white dwarfs[19]. Anyhow, some f(R) gravity flaws in the solar system scale were reported, for instance,in [20–22] and should discard most of the f(R) models proposed so far. For the galacticscales, the f(R) theory also does not seem to be suitable [23–25].

In [26], it was proposed a generalization of the f(R) theory, by including in the gravi-tational action, besides the general term in R, a general term in T , the trace of the energy-momentum tensor, yielding the f(R, T ) gravity. The T -dependence on such a theory maybe due to the existence of imperfect fluids in the universe and could generate a theory thatinvolves gravity and quantum mechanics [27].

The f(R, T ) theory describes pretty well the solar system regime [28]. New terms comingfrom this theory attend to describe dark matter galactic effects [29]. It was also shown thatf(R, T ) gravity can give a considerable contribution to the gravitational lensing [30] and adeviation to the usual geodesic equation [31].

Moreover, the f(R, T ) cosmology evades the dark energy problem, by describing thecosmic acceleration as due to the extra terms in T in the field equations of the model [32, 33],instead of being due to the presence of the cosmological constant.

In opposition to other alternative gravity theories, the modifications here in this the-ory are associated to material terms instead of geometric ones. This new terms yield thenon-vanishing of the covariant derivative of the matter energy-momentum tensor, that is,

– 1 –

∇µTµν 6= 0 [26, 34]. The fact that the energy-momentum tensor is not conserved in thistheory can be related, in a cosmological perspective, to creation (or destruction) of matterduring the universe evolution. This subject was investigated from a thermodynamical per-spective [35]. The same kind of physical property can be noticed in other non-conservativeenergy-momentum theories, such as those presented in [36, 37].

In astrophysics, particularly in the study of hydrostatic equilibrium configurations ofstellar objects, the association of the non-vanishing of the energy-momentum tensor covariantderivative with matter creation is not correct since the Tolman-Oppenheimer-Volkoff (TOV)equation [38, 39] is worked out in a static regime and we do not know exactly what would bethe right mechanism to create matter inside a star.

Therefore, instead of searching for a physical interpretation for this issue, one couldattempt to construct a TOV equation from a “conservative” version of the f(R, T ) gravity. Infact, the energy-momentum conserved (EMC) version of f(R, T ) gravity has been worked outin the literature within different approaches, such as neutron stars hydrostatic equilibriumand even cosmology, and we are going to visit those later (check Section 2.1).

Our purpose here is to take one step further within the EMC f(R, T ) gravity, by con-structing the hydrostatic configurations of strange stars (SSs) [40–42] in the formalism. Aswe are going to revisit, the function h(T ) that conserves the energy-momentum tensor withinf(R, T ) = R + h(T ) depends on the EoS of matter, which in the present case we assume tobe the EoS of strange quark matter.

We naturally wish to confront our results with some observational data of SSs [43–46].We will also compare them with other SS models constructed in alternative gravity, such asthose obtained from f(T ) gravity [47], with T being the torsion scalar, f(R) gravity [48] andeven the non-conservative case of f(R, T ) gravity [49–51].

This article is organized as follows: in Section 2, we describe some important mathe-matical and physical properties of the f(R, T ) gravity. In Section 2.1 we discuss about theEMC f(R, T ) gravity already present in the literature. We derive the EMC f(R, T ) gravityfor SSs in Section 3. We present and solve the referred TOV-like equations in Section 4. Wehighlight and discuss our results in Section 5.

2 The f(R, T ) gravity

In order to obtain the field equations of the f(R, T ) gravity theory, one starts from thefollowing action [26]

S =

∫ [f(R, T )

16π+ Lm

]√−gd4x, (2.1)

with f(R, T ) being a general function of R and T , Lm being the matter lagrangian density,g the determinant of the metric gµν and natural units are assumed throughout the paper.

When varying such an action with respect to gµν we obtain:

Gµν = 8πTµν +1

2h(T )gµν + hT (T )(Tµν − Lmgµν), (2.2)

in which Gµν is the Einstein tensor, Tµν is the energy-momentum tensor and we have consid-ered f(R, T ) = R+ h(T ), with h(T ) being a function of T only, so that one recovers GeneralRelativity in the regime h(T ) = 0. Moreover, hT (T ) ≡ dh(T )/dT .

The covariant derivative of Tµν in (2.2) is

– 2 –

∇µTµν =hT (T )

8π + hT (T )

[(Lmgµν − Tµν)∇µ lnhT (T ) +∇µ

(Lm −

1

2T

)gµν

]. (2.3)

From (2.3) we can see the previously mentioned non-conservation of the energy-momentumtensor in f(R, T ) gravity. In the next section we are going to briefly review some applicationsof EMC f(R, T ) gravity.

2.1 The energy-momentum conserved formalisms proposed for the f(R, T ) grav-ity

Some different approaches have already been made searching for EMC cases of the f(R, T )theory. Looking for Eq.(2.3), we see that there are at least two possibilities to turn f(R, T )gravity into an EMC theory. Let us briefly present these possibilities and their applicationsbelow.

In [52] it was proposed a form to conserve the energy-momentum tensor in f(R, T ) =R + 2λT cosmology, with λ a constant. In order to illustrate that, the f(R, T ) = R + 2λTfield equations, obtained from the substitution of h(T ) = 2λT in (2.2), were rewritten as

Gµν = 8πT effµν , (2.4)

with T effµν = Tµν + T̃µν and

T̃µν ≡λ

8π[2(Tµν − Lmgµν) + Tgµν ]. (2.5)

Within such a formalism, the application of the Bianchi identities in (2.4) yields∇µ[8π(Tµν+T̃µν)] = 0 or simultaneously ∇µTµν = 0 and ∇µT̃µν = 0. The first of these two equationsyields the usual conservation law of standard cosmology while the second yields the EoSof stiff matter [53]. In this way, the f(R, T ) gravity indicated the existence of a two-fluidcosmological model, in which each of the fluids is conserved during the universe evolution.

S. Chakraborty, on the other hand, has shown that a part of the arbitrary functionf(R, T ) can be determined if one imposes ∇µTµν = 0 [54]. A cosmological model was derivedfrom such a principle [55]. In [55], in order to obtain the EMC cosmological model, theauthors have assumed a Friedmann-Lemaître-Robertson-Walker metric as well as the EoSp = ωρ, with p being the pressure, ω the constant EoS parameter and ρ the density of theuniverse. By solving (2.3), then, they have found h(T ) ∼ T

1+3ω2(1+ω) .

A similar approach was recently applied to the TOV equation for neutron stars [56], thatis, a conservative function h(T ) was found for the polytropic EoS [57] case and the TOV-likeequation was constructed and solved from such a formalism. It is important to remark thatthe conservative case has better results in comparison with the non-conservative version ofthe TOV equation within f(R, T ) gravity [58]. While the contribution of the f(R, T ) gravityfor neutron stars in the latter case resulted in slightly greater masses and greater radii, in theformer conservative case, the maximum masses of neutron stars were substantially increased(> 2M�) while their radii did not vary significantly, getting in touch with massive pulsarsobservations [4, 5].

Another advantage of the EMC f(R, T ) gravity can be seen in the realm of cosmology, inwhich such a model is clearly in advantage when compared to the non-conservative cases forwhat concerns the confrontation of theoretical predictions with supernovae Ia observationaldata [59].

– 3 –

3 Energy-momentum-conserved f(R, T ) gravity for strange quark matter

Now we wish to construct an EMC model for f(R, T ) gravity to be used to obtain thehydrostatic equilibrium configurations of SSs. SSs are stars that contain superdense matteron its fundamental level [40–42], that is, strange matter. We still do not know if this is,indeed, the fundamental level of matter at high densities and that is exactly what makes thestudy of SSs so important. Anyhow, some SSs candidates are well known [43–46] as wellas some methods to prove SSs and, consequently, strange matter existence via gravitationalwave astronomy [60–64].

In order to start the construction of the EMC f(R, T ) gravity for SSs, let us work withEq.(2.3) by forcing ∇µTµν = 0 on it. By assuming Lm = ρ and µ = 1 yields the followingdifferential equation:

(ρ+ p)(lnhT )′ +

1

2(ρ+ 3p)′ = 0, (3.1)

where the comma stands for radial derivative.By assuming the EoS to be p = ωρ to describe the matter inside such objects and solving

Eq.(3.1) yields

hT =1

2

(1− ω1 + ω

)λT−

12(

1+3ω1+ω ), (3.2)

with λ being an arbitrary constant.By integrating (3.3), one has

h(T ) = λT12(

1−ω1+ω ). (3.3)

We observe from (3.3) that for ω = 0, h(T ) ∼√T , which is the same result obtained for

an EMC f(R, T ) gravity cosmological model in the case of pressureless (ω = 0) matter [55].Moreover, since 0 < ω < 1, the exponent of T in (3.3) is restricted to values between 0 and1/2.

Let us now call the MIT bag model EoS [41, 42, 49, 65–68] to describe matter insideSSs in the EMC f(R, T ) gravity model. Such an EoS describes a fluid composed of up, downand strange quarks only. The relation between pressure and energy density becomes a linearone, given by p = ω(ρ− 4B), with constant ω and B being the bag constant.

By using the MIT bag model EoS, the EMC functional form h(T ) is calculated from theintegration of

hT (T ) = β(1 + ω)

(1− 3ω)

[(T − 12Bω)

(1 + ω)

(1− 3ω)− 4Bω

]− 12

(1+3ω)(1+ω)

(3.4)

and reads

h(T ) = β

[(T − 12Bω)

(1 + ω)

(1− 3ω)− 4Bω

] (1−ω)2(1+ω)

, (3.5)

where β is an arbitrary constant.

– 4 –

4 The Tolman-Oppenheimer-Volkoff equations and their solutions

Let us now use Eq.(3.3) to construct the TOV-like equation in this model. By using thespherical static metric,

ds2 = eφdt2 − eψdr2 − r2dθ2 − r2 sin2 θdφ2, (4.1)

with φ = φ(r) and ψ = ψ(r) being the metric potentials, as well as assuming the energy-momentum tensor of a perfect fluid, we obtain the 00 and 11 components of the field equationsas

e−ψ

r2(eψ + ψ′r − 1) = 8πρ+

1

2h, (4.2a)

e−ψ

r2

(1− eψ + φ′r

)= 8πp− 1

2h− hT (p− ρ). (4.2b)

We introduce now the quantity m = m(r), such that

e−ψ = 1− 2m

r, (4.3)

and replacing it into (4.2a), we get

dm

dr= 4πρr2 +

1

4hr2, (4.4)

so that m(r) represents the enclosed gravitational mass within a sphere of radius r accordingto the EMC f(R, T ) gravity.

Let us recall that from the conservation of the energy-momentum tensor we have:

∇µTµν = −p′ − (ρ+ p)φ′

2= 0. (4.5)

By isolating φ′ in (4.2b) and substituting in the above equation, one is able to derivethe modified TOV equation as follows

p′ = −(ρ+ p)

{mr2

+[4πp− 1

4h−12hT (p− ρ)

]r}(

1− 2mr

) . (4.6)

The Equations (4.4) and (4.6) can be solved numerically by using the fourth-order Runge-Kutta method and considering a specific model for the functional h and hT . In order to doso, the boundary conditions at the center of the star are as follows: p(0) = pc, ρ(0) = ρc andm(0) = 0, with pc and ρc being the central pressure and central energy density. For r = R,where the pressure and energy density of the star vanish, the enclosed mass m(R) = Mrepresents the total mass of the star, with R being its total radius. By using different valuesof central energy density, one is able to construct the mass-radius relation as well as otherrelations that we further derive in this work.

– 5 –

4.1 Case: p = ωρ

Let us consider now the first case derived in Section 3, where an EoS like p = ωρ was used toobtain the functionals h and hT . For this case the TOV-like equation becomes

p′ = −(ρ+ p)

{mr2

+[4πp+ ζ(ω)ρ

1−ω2(1+ω)

]r}

(1− 2m

r

) , (4.7)

where the parameter ζ(ω) is given by

ζ(ω) =λ

2

[(1 + 8π)(1− 3ω)

− 1+3ω2(1+ω) − 1

2(1− 3ω)

1−ω2(1+ω)

], (4.8)

such that λ = 0 yields the usual TOV equation. It is worth to note that ω = 13 also gives

the standard TOV equation, which is expected since this value for ω yields T = 0, hencecancelling out any contribution from the trace of the energy-momentum tensor in the fieldequations.

Figure 1 below shows the behaviour of the total mass with total radius of the star, wherefive values of λ were used and ω = 0.28 in reference to the MIT bag model EoS with B = 0.For this case of ω, the approach presented in Section 2.1 yields h(T ) = λT 0.28. It is worth toquote that λ = 0 corresponds to the result found within General Relativity framework.

5 10 15 20 25 30R[km]

0

1

2

3

4

M/M

= 0.28

= 2 × 10 4

= 1 × 10 4

= 0= 0.5 × 10 4

= 1 × 10 4

Figure 1. Mass-radius relation for p = ωρ within the conservative model of the f(R, T ) gravity forthe interval 50−800 MeV/fm3 for the central density. Several values of λ were employed and ω = 0.28.

From Figure 1 we can note that for λ > 0, less massive and smaller stars are found. Sucha behaviour can be understood as a strong gravity regime effect, such like the case of GeneralRelativity Theory in comparison with Newtonian gravitation [69]. On the other hand, forthe cases where λ < 0 we observe an increasing in the total mass and total radius of the starwhen |λ| increases. We can see that the increasing in the mass and radius of the star is verysensitive to the value of |λ|.

In the left panel of Fig.2 below we show the mass against central energy density for theconserved model of f(R, T ) gravity, where several values of λ, ω = 0.28 and B = 0 were

– 6 –

0 200 400 600 800

c[MeV/fm3]

0

1

2

3

4

5

6

7

8M

/M= 0.28

400 600 800 1000 1200

c[MeV/fm3]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

M/M

= 0.28

Figure 2. Left panel: total mass versus central energy density for the conservative model of thef(R, T ) gravity using several values of λ, ω = 0.28 and B = 0. The value λ = −1.5 × 10−4, in theinterval of central energy density we have used (50 − 800 Mev/fm3), does not produce any stablestars. Right panel: total mass versus central density for the non-conservative linear model of f(R, T )gravity, i.e., f(R, T ) = R+ 2χT . Some values of χ and ω = 0.28 were used.

employed. For the values of λ larger than −1.5 × 10−4 we observe that the mass initiallyincreases with central density until it attains a maximum value. After that point, the massdecreases with the increasing of central density.

From the regular criterion of stability, ∂M/∂ρc > 0, we conclude that the maximum masspoints mark the onset of instability in each of those curves of Fig.2. However, the value ofλ = −1.5× 10−4 does not produce any stable stars in the considered range of central energydensity (50 − 800 Mev/fm3) since its mass-density relation does not respect the stabilitycriterion, in this way, setting up a lower limit for λ and a maximum stable mass of ∼ 6M�.

In the right panel of Fig.2 we consider the same EoS, with ω = 0.28, in the context of alinear non-conservative f(R, T ) gravity model, namely, f(R, T ) = R+ 2χT gravity, with χ afree parameter, as the one discussed in [70].

4.2 Case: MIT Bag Model p = ω(ρ− 4B)

Now, using the MIT bag model EoS we derive a different TOV-like equilibrium equation as

dp

dr= − (ρ+ p)(

1− 2mr

) {mr2

+ [4πp− βξ(ω, ρ)(p− ρ)] r}, (4.9)

where ξ(ω, ρ) is given by

ξ(ω, ρ) =1

2

{1

2[ρ(1 + ω)− 4ωB]

1−ω2(1+ω) − 1 + ω

1− 3ω[−ρ(1 + ω) + 4ωB]−

12

1+3ω1+ω

}. (4.10)

We will consider ω = 0.28, which gives a quark mass of 250 MeV/fm3, and the bag constantwill be taken as B = 60 MeV/fm3.

The mass-radius and mass-density relations are shown in Fig.3. The considered valuesfor β range from −1× 10−3 to 1× 10−3, and the value β = 0 corresponds to the results foundwithin General Relativity framework. It can be seen that the positive values of β tend to

– 7 –

Table 1. Physical parameters of observed strange star candidates derived using β = −0.5 × 10−3

and B = 60MeV/fm3. Zs represents the surface redshift and it is calculated as: Zs =1√

1−2M/R− 1.

SS candidate Observed mass M/M� Predicted radius (km) M/R ZsPSR J1614-2230 1.97±0.04 [4] 11.49+0.02

−0.01 0.253 0.423Vela X-1 1.77±0.08 [71] 11.40+0.07

−0.08 0.229 0.3594U 1608-52 1.74±0.14 [72] 11.33+0.06

−0.12 0.227 0.353PSR J1903+327 1.667±0.021 [73] 11.28+0.07

−0.04 0.218 0.3324U 1820-30 1.58±0.06 [74] 11.18+0.08

−0.1 0.209 0.310Cen X-3 1.49±0.08 [71] 11.03+0.12

−0.14 0.199 0.290EXO 1785-248 1.3±0.2 [75] 10.65+0.38

−0.47 0.180 0.250LMC X-4 1.29±0.05 [71] 10.64+0.12

−0.11 0.179 0.248SMC X-1 1.04±0.09 [71] 10.04+0.22

−0.26 0.153 0.200SAX J1808.4-3658 0.9±0.3 [76] 9.62+0.83

−1.13 0.138 0.1754U 1538-52 0.87±0.07 [71] 9.54+0.21

−0.27 0.135 0.169HER X-1 0.85±0.15 [77] 9.49+0.46

−0.57 0.132 0.166

reduce the star mass and shrink the star radius, which can be understood as a gravitationalforce “stronger” than the General Relativity one (when β = 0). The opposite behavior isfound for negative β, where larger and more massive stars are found and this behavior canbe understood as a “weaker” gravitational force. One interesting feature of assuming negativevalues of β is that it allows a larger maximum mass. For instance, the maximum mass forβ = −10−3 is Mmax ≈ 2.6M�. On the other hand, for the case of β = 0 the maximum massis ∼ 2M�, which represents a value ∼ 30% smaller.

6 8 10 12R[km]

0.0

0.5

1.0

1.5

2.0

2.5

M/M

= 10 3

= 0.5 × 10 3

= 0= 0.5 × 10 3

= 10 3

250 500 750 1000 1250 1500 1750 2000

C [MeV/fm3]0.0

0.5

1.0

1.5

2.0

2.5

M/M

= 10 3

= 0.5 × 10 3

= 0= 0.5 × 10 3

= 10 3

Figure 3. Left panel: Mass-radius relation for the MIT bag model of the EMC f(R, T ) gravity.Right panel: Mass versus central energy density for the EMC f(R, T ) gravity using the MIT bagmodel EoS. Different values of β, ω = 0.28 and B = 60 MeV/fm3 were employed on both plots.

Strange quark stars have been studied in the non-conserved f(R, T ) theories in severalrecent works by using the MIT bag model equation of state [49, 50, 58, 70, 78]. In this work,rather than consider a non-conserved f(R, T ) theory of gravity we derived the conservedform of the theory by using also the MIT equation of state. The outcomes of our conserved

– 8 –

Table 2. Physical parameters of the strange star candidate LMC X-4 for different values of β andB = 60MeV/fm3.

β Predicted radius (km) M/R Zs−10−3 10.84+0.11

−0.12 0.176 0.242−0.5× 10−3 10.64+0.12

−0.11 0.179 0.2480 10.38+0.08

−0.09 0.184 0.2570.5× 10−3 10.02+0.03

−0.05 0.19 0.27010−3 9.44+0.28

−0.07 0.212 0.295

model can be compared with observed parameters of strange star candidates. Table 1 showsthe observed mass of strange star candidates and the predicted radius, compactness andgravitational redshift with use of the energy-momentum conserved f(R, T ) gravity for thevalue of β = −0.5 × 10−3, and table 2 shows also the predicted radius, compactness andgravitational redshift for the star LMC X-4 for several values of β. From table 2 one cansee that the increasing of the parameter β yields to smaller radius for the object LMC X-4,which means that the object would be more compact and with a larger surface gravitationalredshift. However, when β is negative the increasing in its magnitude leads to larger radiiand hence the behavior of the compactness and redshift are reversed. Table 1 also allows usto confirm the feasibility of our work concerning the confrontation with observational data ofcompact objects.

5 Discussion

Recent theoretical studies show that alternative theories to General Relativity provide impor-tant insights to solve complex issues of present astrophysical and cosmological observations.On this regard, one could also check Refs.[79, 80].

In the paper we have studied one of these alternative theories, the f(R, T ) theory. Thedependency on T in such a theory is motivated by quantum effects [81] and the possibleexistence of imperfect fluids in the universe. A consequence of the T -dependence is thenon-conservation of the energy-momentum tensor, which can be evaded from the approachespresented in Section 2.1.

The present literature contained EMC f(R, T ) models in cosmology and hydrostaticequilibrium configurations of neutron stars. In both cases, the physical features obtained aresignificantly more desirable than the non-conservative cases.

Take, for instance, the EMC f(R, T ) ∼ T 1/2 cosmological model, derived in [55] andobservationally tested in [59]. It has been shown in [59] that the EMC f(R, T ) cosmologicalmodel is the only one that passes cosmological tests such as the confrontation with supernovaeIa observational data.

In parallel, in [56], the macroscopical features of neutron stars obtained for the EMCf(R, T ) gravity are in touch with massive pulsars observations [4, 5], while the non-conservativecase is not [58].

Here we have implemented a method to find the conservative functional form within thef(R, T ) function for SSs. We, then, have derived and solved the referred TOV-like equations.

One may wonder about the dependence of the function h(T ) on the EoS and the re-liability of such a feature. This may be due to the geometry-matter coupling predicted by

– 9 –

the f(R, T ) theory. Note that the inception of material terms in the gravitational action ofa given theory yields the possibility of non-minimally couple geometry to matter. In thisway, in a fundamental level, the geometrical aspect of the function that shall replace R inthe gravitational action is expected to depend on the material features of the system. In thisway, different E’soS yield different functional forms for h(T ). It is important to quote herethat geometry-matter coupling have shown to yield the cosmic acceleration [82] and to mimicthe dark matter effects [29, 83, 84].

Regarding the results obtained for the equilibrium configurations of SSs, in the left panelof Fig.3 we have seen that for negative values of β, larger and more massive stars are obtainedwith the increasing of |β|. On the other hand, for positive values of β, we obtain smaller andless massive stars according to the increasing of the parameter.

In the right panel of Fig.3 we have plotted the star mass against its central energydensity for approximately the interval 250−2000 Mev/fm3 of the latter. Also in Fig.1 the leftpanel refers to our EMC model while the right panel is obtained from the f(R, T ) = R+2χTnon-conservative model, as the one of Ref.[70] (although the referred authors have considereda different value for ω), among others [58, 85]. Our conservative model presents a moresensitive contribution to the increasing of mass for the changes in λ < 0 in comparison withthe non-conservative case. We also see in Fig.2 that the maximum mass points of our EMCmodel are attained for smaller values of ρc when compared to the results of Ref.[70].

Furthermore, the left panel of Fig.2, together with the regular criterion of stability,indicates a lower limit for λ in the present model, which reads λ > −1.5 × 10−4 and is inagreement with the constraint found in Ref.[85], being more stringent than the latter by afactor of 2.

In what concerns Fig.3, which is related to the MIT bag model EoS, with non-null bagconstant, we see that for negative values of β, more massive and greater stars are obtainedwhen the results are confronted to General Relativity. In particular, we have shown the feasi-bility of our model by comparing our theoretical values of mass and radius with observationaldata of strange star star candidates and with the results of the non-conserved f(R, T ) modelsas indicated by tables 1 and 2.

Acknowledgments

GAC thanks Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) grantsPDSE 88881.188302/2018-01 and PNPD 88887.368365/2019-00. PHRSM would like to thankSão Paulo Research Foundation (FAPESP), grants 2015/08476-0 and 2018/20689-7, for finan-cial support.

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