Magnetized strange quark matter solutions in f(R, T ...

Chinese Journal of Physics 55 (2017) 71–78

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Magnetized strange quark matter solutions in f(R, T) gravity with cosmological constant Can Aktas¸ a,∗, Sezgin Aygün b a b

Department of Mathematics, Çanakkale Onsekiz Mart University, Arts and Sciences Faculty, Terziog˘ lu Campus, 17020, Turkey Department of Physics, Çanakkale Onsekiz Mart University, Arts and Sciences Faculty, Terziog˘ lu Campus, 17020, Turkey

a r t i c l e

i n f o

Article history: Received 10 October 2016 Revised 23 November 2016 Accepted 7 December 2016 Available online 24 December 2016 MSC: 04.20.-q 04.20.Jb 04.50.Kd Keywords: Magnetized strange quark matter f(R, T) Cosmology Deceleration parameter FRW universe Static Einstein universe

a b s t r a c t In this research, we have studied magnetized strange quark matter (SQM) solutions for Friedmann-Robertson-Walker (FRW) universe in f(R, T) gravity. To obtain exact solutions of modified field equations we have used f (R, T ) = R + 2 f (T ) and f (R, T ) = f1 (R ) + f2 (T ) models given by Harko et al. (Harko et al. in Phys. Rev. D 84:024020, 2011) and f (R, T ) = R + f3 (T ) model (here f3 is an arbitrary function) with cosmological constant . For t → ∞ we obtain p = −ρ dark energy situation with small constant values of cosmological constant in three different f(R, T) gravitation models. In our solutions magnetic field does not observe also we have transformed our solutions from FRW universe to Static Einstein Universe (SEU) and we get f(R, T) gravity results for SEU universe. Finally we discussed our physical solutions. © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

1. Introduction The interest in alternative gravitational theories is increasing day by day. Because these theories could help to explain the expansion of the universe and dark energy. Some of these theories are called f(R, T) gravity [1], f(R) theory [2], Brans-Dicke Cosmology [3] and Saez-Ballester theory [4] etc. After Harko et al. [1], many scientists have researched f(R, T) gravitation theory for various universe models. A. Alhamzawi and R. Alhamzawi have studied gravitational lensing by f(R, T) [5]. Zubair et al. have researched f(R, T) gravity using conformal killing vectors [6]. Momeni et al. have obtained second law of thermodynamics in f(R, T) theory [7]. Singh and Kumar have studied holographic dark energy models in modified f(R,T) gravity [8]. Singh et al. have researched f(R, T) theory with cosmological constant in various universe models [9–11]. Zubair and his co-authors have contributed significantly to f(R, T) theory. They addressed this issue from different perspectives with various f(R, T) models [12–15]. Reddy et al. have studied bulk viscous cosmological model in f(R, T) gravity [16]. Momeni et al. have discussed cosmological viable mimetic f(R) and f(R, T) theories via Noether symmetry [17]. Also FRW universe model has been studied by Ramesh and Umadevi [18], Amani and Dehneshin [19], Chattopadhyay [20] in f(R, T) gravity. Hovewer, it is important to investigate magnetic fields and quark gluon plasma to understand the early universe. Recent observations indicate that neutron stars, pulsars also galaxy clusters have magnetic fields [21]. The magnetic fields have important role

∗

Corresponding author. E-mail addresses: [email protected] (C. Aktas¸ ), [email protected] (S. Aygün).

http://dx.doi.org/10.1016/j.cjph.2016.12.003 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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C. Aktas¸ , S. Aygün / Chinese Journal of Physics 55 (2017) 71–78

in the structure formations in the early universe but the source of the magnetic field is still not fully understood [22]. In this study we have attached magnetic field to strange quark matter in f(R, T) gravitation models for FRW universe due to following reasons: The phase transition which Quark Gluon Plasma (QGP)→ hadron gas has occurred (when T ∼200 MeV) in the early universe. According to quantum chromodynamics in quark fluid electric field vanishes because of Ohm’s Law [21,23,24]. Also quark-gluon matter should fulfill charge neutrality from the reason β -equilibrium [21,25,26]. In the literature there are many studies including quark and strange quark matter solutions in general relativity theory ´ (GRT) also various alternative theories. Çaglar and Aygün have investigated (n + 2 ) dimensional FRW universe model solutions with quark and strange quark matters in Brans-Dicke [27], Self Creation [28] and Lyra [29] cosmologies. Yılmaz and co-authors have investigated quark and strange quark matters in GRT and some modified theories [30–33]. Gholizade et al. have studied thermodynamics and geometry of SQM [34]. Adhav et al. [35] have investigated quark and strange quark matter in Kantowski-Sachs universe model for f(R) gravity. Mahanta et al. have obtained bulk viscous string cloud solutions with SQM in Brans-Dicke theory [36]. Rao et al. have investigated various cosmological models with SQM in GRT and Brans-Dicke gravitation theory [37,38]. Namrata et al. [39] have studied Kaluza-Klein universe model with SQM and . Khadekar and Rajani [40] have researched higher dimensional cosmological model with quark and strange quark matter. SQM attached to string cloud in FRW universe has been researched by Katore [41]. Sahoo and Bivudutta have studied string cloud and domain walls with quark matter in kink cosmological model [42] and in bimetric theory for plane symmetric metric [43]. Also there is a few study about magnetized quark and strange quark matter solutions [21,24] in the literature. Then in this study we will research magnetized strange quark solutions in FRW universe for f(R, T) gravity also we will transform our solutions from FRW universe to Static Einstein Universe with f (R, T ) = R + 2 f (T ), f (R, T ) = f 1 (R ) + f2 (T ) and f (R, T ) = R + f1 (T ) models. 2. Gravitational field equations of f(R, T) modified gravity theory According to Harko et al. [1] the action of new modified f(R, T) gravity is given by [1]

S=

√ f (R, T ) + Lm −gd4 x 16π G

(1)

here R is Ricci scalar, T is the trace of Tαβ , g is the determinant of gαβ also f(R, T) is the arbitrary function of R and T. Also, Lm indicates Lagrangian [1]. Tαβ is defined as [1]

√ 2 δ ( −gLm ) Tαβ = − √ −g δ gαβ

(2)

The Lagrangian Lm is depends only on gαβ and not on its derivatives and shows that

2∂ Lm ∂ gαβ

Tαβ = gαβ Lm −

(3)

By varying Eq. (1), we get

fR (R, T )Rαβ −

1 f (R, T )gαβ + (gαβ − α β ) fR (R, T ) = 8π Tαβ − fT (R, T )Tαβ − fT (R, T ) αβ + gαβ 2

here fR (R, T) and fT (R, T) show derivatives of f(R, T) with respect to R and T respectively and = α α [1,44]. Where αβ is given by

αβ = −2Tαβ + gαβ Lm − 2gik

(4)

α is the covariant derivative;

∂ 2 Lm

(5)

∂ gαβ gik

If we contract Eq. (4), we get the connection between R and T as follows

fR (R, T )R + 3 fR (R, T ) − 2 f (R, T ) = 8π T − fT (R, T )T − fT (R, T ) + gαβ

(6)

where = gαβ αβ [1,44]. From Eqs. (4) and (6), we get gravitational field equations as follows [1]

1 1 1 1 Rg + f (R, T )gαβ = 8π Tαβ − T gαβ − fT (R, T ) Tαβ − T gαβ 3 αβ 6 3 3 1 − fT (R, T ) αβ − gαβ + α β fR (R, T ) + gαβ 3

fR (R, T ) Rαβ −

(7)

and the energy-momentum tensor of perfect fluid is given by [1]

Tαβ = (ρ + p)uα uβ − pgαβ here p is pressure, ρ we get

(8)

is energy density, and uα is the four velocity vector and satisfies the relation uα uα = 1. From Eq. (5),

= −2Tαβ − pgαβ .

(9)


73

The solutions of f(R, T) gravity Harko et al. [1] represented three classes of f(R,T) models as follows

⎧ R + 2 f (T ) ⎪ ⎨ f (R, T ) = f1 (R ) + f2 (T ) ⎪ ⎩ f 1 (R ) + f 2 (R ) f 3 (T )

(10)

In this study we discuss three types solutions i.e. f (R, T ) = R + 2 f (T ), f (R, T ) = f 1 (R ) + f2 (T ) also arbitrary f(T) function model i.e. f (R, T ) = R + f3 (T ) for magnetized strange quark matters with homogeneous isotropic FRW universe model with cosmological constant. FRW space-time describes as follows;

d s2 = dt 2 −

A2 dr 2 − A2 r 2 (dθ 2 + sin(θ 2 )dφ 2 ) 1 − kr 2

(11)

where A is function of t and k = ±1, 0. In this study, we consider the source of gravitational as magnetized strange quark matter. Then our new energy momentum tensor is given by [23,45].

Tik = ( p + ρ + h2 )ui uk +

h2 − p gik − hi hk 2

(12)

here ρ is the energy density, p is the pressure, h2 represents the magnetic field and ui is the four velocity. We can choose magnetic flux in the direction of radial due to hi ui = 0 [24]. Also conservation equation is given by

T;kik =

3A˙ A

2 h2 + hh˙ + ρ˙ = 0 ρ + p+

(13)

3

3. Field equations for f (R, T ) = R + 2 f (T ) model with Modified field equations in f(R, T) theory with cosmological constant are given as follows [1]

Gαβ = [8π + 2 f (T )]Tαβ + [2 p f (T ) + f (T ) + ]gαβ

(14)

where prime denotes differentiation with respect to the argument. For the choice f (T ) = μT (here μ is a constant) the Eq. (14) takes the form

Gαβ = [8π + 2μ]Tαβ + [μρ − pμ + ]gαβ

(15)

Also modified field equations are given by

2A¨ A˙ 2 k + 2 + 2 = (4π − μ )h2 + (8π + 3μ ) p − μρ − A A A 2A¨ A˙ 2 k + 2 + 2 = −(4π + 3μ )h2 + (8π + 3μ ) p − μρ − A A A 3A˙ 2 3k + 2 = −(12π + 5μ )h2 − (8π + 3μ )ρ + μ p − A2 A

(16) (17) (18)

In this study we have five unknowns A, ρ , p, h2 , and three modified field equations. In order to solve the system completely we use the conditions equation of state (EoS) of magnetized strange quark matter

p=

ρ − 4Bc

(19)

3

here Bc is bag constant [46,47] and deceleration parameter is given as follows

q=−

AA¨ = const. A˙ 2

(20)

if we integrate Eq. (20), we get 1

A = (at + b) 1+q

(21)

where a = 0 and b are integration constants. Eq. (21) mentions that the condition for expansion of the universe is 1 + q > 0. From Eqs. (16)–(19) and (21), we get the values of cosmic pressure, density, magnetic field and cosmological constant as follows

p = −Bc −

ρ = Bc −

k 4(4π + μ )(at + b)

2 1+q

3k 4(4π + μ )(at + b)

2 1+q

−

−

a2 4(4π + μ )(1 + q )(at + b)2

4 ( 4π

3a2 + μ )(1 + q )(at + b)2

(22)

(23)

74


h2 = 0

=

(24)

( 6π + 2μ )k [6π (q − 1 ) + (2q − 1 )μ]a2 − 4 ( 2π + μ )Bc − 2 (4π + μ )(1 + q )2 (at + b)2 (4π + μ )(at + b) 1+q

(25)

4. Field equations for f (R, T ) = f1 (R ) + f2 (T ) model with If we choose f (R, T ) = f1 (R ) + f2 (T ) in Eq. (4), modified field equations in f(R, T) theory with cosmological constant [1], we get

f1 (R )Rαβ −

1 f1 (R )gαβ + (gαβ − α β ) f1 (R ) = (8π + f2 (T ))Tαβ + 2

f2 (T ) p +

1 f2 (T ) gαβ + gαβ 2

(26)

If we choose f1 (R ) = μR and f2 (T ) = μT in Eq. (26) with , we get [1,44]

Gαβ =

8π + μ μ

Tαβ +

ρ − p + 2

2

gαβ

(27)

Also the modified field equations with magnetized strange quark matter for f (R, T ) = f 1 (R ) + f2 (T ) model with are given by

4π 1 8π 3 2A¨ A˙ 2 k ρ + 2 + 2 = + h2 + + p− − A μ 2 μ 2 2 A A

(28)

4π 1 8π 3 2A¨ A˙ 2 k ρ + 2 + 2 =− + h2 + + p− − A μ 2 μ 2 2 A A

(29)

3A˙ 2 3k 4π 1 2 + 2 = −3 + h − μ 2 A2 A

8π μ

+

3 2

p ρ + − 2

(30)

here we have five unknowns A, ρ , p, h2 , and three modified field equations. In order to solve the system completely for f (R, T ) = f1 (R ) + f2 (T ) model, we use Eqs. (19), (21) and (28)–(30). In this case, we have the values of cosmic pressure, density, magnetic field and cosmological constant as follows

p = −Bc −

ρ = Bc −

kμ 2(8π + μ )(at + b)

2 1+q

3kμ 2(8π + μ )(at + b)

2 1+q

−

−

2 ( 8π

2 ( 8π

a2 μ + μ )(1 + q )(at + b)2

3a2 μ + μ )(1 + q )(at + b)2

h2 = 0

=

(31)

(32) (33)

k(12π + μ ) 2 ( π + μ )Bc a2 (12π q + 2μq − 12π − μ ) − − 2 μ (8π + μ )(1 + q )2 (at + b)2 1+ q (8π + μ )(at + b)

(34)

5. Field equations for f (R, T ) = R + f3 (T ) model with Sharif and Zubair [12], Moraes et al. [48] also Zubair and Noureen [49] have suggested new cosmological models in f(R, T) gravity. In this section we will solve linear R and arbitrary f3 (T) model in f(R, T) gravity. Here f3 is an arbitrary function of T. For this case modified field equations in f(R, T) theory with cosmological constant are given by

Rαβ −

1 Rg = 8π Tαβ − f3 (T )Tαβ − Tαβ αβ + gαβ 2 αβ

(35)

where prime denotes the derivative w.r.t. T. Also the modified field equations with magnetized strange quark matter for f (R, T ) = R + f3 (T ) model with are given by

2A¨ A˙ 2 k + 2 + 2 = A A A

2A¨ A˙ 2 k + 2 + 2 =− A A A

f3 f3 + 4 π h2 + 8 π p − − 2 2

(36)

f3 f3 + 4 π h2 + 8 π p − − 2 2

(37)


f3 3A˙ 2 3k + = −3 4 π + 2 A2 A2

75

f3 h2 − ( f3 + 8π )ρ − f3 p − − 2

(38)

In order to solve the system completely for f (R, T ) = R + f 1 (T ) model, we use Eqs. (19), (21) and (36)–(38). In this case, we have the values of cosmic pressure, density, magnetic field and cosmological constant as follows

p = −Bc −

ρ = Bc −

k 2(8π + f3 )(at + b)

2 1+q

3k 2 ( 8π

2 + f3 )(at + b) 1+q

−

−

a2 2(1 + q )(8π + f3 )(at + b)2

(39)

3a2 2(8π + f3 )(1 + q )(at + b)2

(40)

h2 = 0

=

(41)

k(12π + f3 ) 16π Bc − f3 a2 (12π q + 2 f3 q − 12π − f3 ) − − 2 2 2 2 (8π + f3 )(1 + q ) (at + b) (8π + f3 )(at + b) 1+q

(42)

6. Discussions As known that, FRW space-time describes today’s universe model. Also modified gravitation theories like f(R), f(R, T), Brans-Dicke etc. use to explain expanding universe with cosmological constant, k-essence, quintessence, dark energy etc. These alternative gravitation theories appears to provide a current explanation for dark energy. For these reasons in this study, we have researched FRW universe model with magnetized strange quark matter in f(R, T) gravity for R + 2 f (T ) and f1 (R ) + f2 (T ) given by Harko et al. [1] and linear R and arbitrary f(T) function models given by [49] with . To obtain exact solutions of modified field equations we used EoS of magnetized strange quark matter and deceleration parameter (q). Because q is an important parameter also shows whether the model accelerates or not [50,51]. From obtained results, we see that a is the important constant and a = 0 for all types f(R, T) models. From Eqs. (24), (33) and (41)easily seen that h2 = 0 i.e. we do not observe the effect of magnetic field in f(R, T) models for FRW universe. Therefore, our results transform strange quark matter solutions in f(R, T) gravity for FRW universe. From Eqs. (13), (21)–(24) and (31)–(33) we obtain the conservation equation as follows 2q

T;kik = k(1 + q )(at + b) 1+q − a2 (q − 1 ) = 0

(43)

for R + 2 f (T ) and f1 (R ) + f2 (T ) models. There are two possibility to providing Eq. (43) for two models. The first possibility is k = 0 and q = 1 the other is k = −a2 and q = 0. From Eqs. (21)–(25) and two cases (k = 0 and q = 1 also k = −a2 and q = 0) to providing conservation equation, we get following exact results for R + 2 f (T ) model as follows in Table. 1. From Eqs. (21), (31)–(34) and two cases (k = 0 and q = 1 also k = −a2 and q = 0) to providing conservation equation, we obtain following exact results for f1 (R ) + f2 (T ) model in f(R, T) gravity as follows in Table. 2. For k = −a2 and q = 0 condition we obtain dark energy ( p = −ρ = −Bc ) situation with small constant values of cosmological constant in Table.1 and Table.2 also in the third model for t → ∞ in Eqs. (39) and (40). From these results we could say that strange quark matter may be source of dark energy in FRW universe model due to Casimir effect [52]. However, we could discuss that the obtained models are beneficial for strange quark stars owing to their constant pressure and density. However for t → ∞ in Eqs. (22), (23), (25) also Eqs. (31), (32) and (34), we get same energy density ( p = −ρ = −Bc ), pressure and values in k = 0 and q = 1. In this study, we have researched linear R and arbitrary f(T) model i.e. f (R, T ) = R + f3 (T ). It is important to choose an arbitrary function f(T). If we take f 3 (T ) = 0 in f(R, T) model, we get general relativity results. Our results in Eqs. (39)–(42) contain arbitrary function f3 (T). We have found that dark energy distribution ( p = −ρ = −Bc ) in the particular choice of f (R, T ) = R + f3 (T ) for increasing value of t. Also, these solutions are Table 1 ρ , p, h2 , and A values for R + 2 f (T ) model in f(R, T) gravity. k = 0 and q = 1

ρ

3a 8(4π + μ )(at + b)2

−Bc −

p h

Bc −

2

A(t)

k = −a2 and q = 0

2

2

a 8(4π + μ )(at + b)2

Bc −Bc

0

0

a2 μ − 4 ( 2π + μ )Bc 4(4π + μ )(at + b)2 √ at + b

at + b

−4(2π + μ )Bc

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C. Aktas¸ , S. Aygün / Chinese Journal of Physics 55 (2017) 71–78 Table 2

ρ , p, h2 , and A values for f1 (R ) + f2 (T ) model in f(R, T) gravity. k = 0 and q = 1 3a μ 4(8π + μ )(at + b)2

k = −a2 and q = 0

2

ρ

Bc −

p

a2 μ −Bc − 4(8π + μ )(at + b)2

−Bc

h2

0

0

A(t)

a2 μ 2 ( 4π + μ )Bc − μ 4(8π + μ )(at + b)2 √ at + b

Bc

−

2 ( 4π + μ )Bc

μ

at + b

consistent with Wilkinson Microwave Anisotropy Probe (WMAP5) observations [53]. Three different f(R, T) solutions shows accelerating universe models with small anisotropy which are consistent with current observations of SNe Ia and CMB [54]. For − ab the models have singular points and t = − ab , the results are continue. If we take A(t )2 = 1 in Eq. (11) we transform FRW universe to Static Einstein Universe, also we get new line element as follows

ds2 = −dt 2 +

dr2 + r 2 (dθ 2 + sin(θ 2 )dφ 2 ) 1 − kr 2

(44)

Using Eq. (44) with Eqs. (12)–(20), we obtain the values of pressure, density, magnetic field and cosmological constant solutions for Static Einstein Universe in f(R, T) gravity for R + 2 f (T ) model as follows;

p = −Bc −

ρ = Bc −

k 4 ( 4π + μ )

3k 4 ( 4π + μ )

h2 = 0

=−

(45)

(46) (47)

32π 2 Bc + (6π + μ )(4μBc + k ) 4π + μ

(48)

Using Eq. (44) with Eqs. (28)–(30), we get the pressure, density, magnetic field and cosmological constant for Static Einstein Universe in f(R, T) gravity for f1 (R ) + f2 (T ) model as follows;

p=−

ρ=

16π Bc + 2μBc + kμ 2 ( 8π + μ )

( 2Bc − 3k )μ 8π Bc + 2 ( 8π + μ ) ( 8π + μ )

h2 = 0

=−

(49)

(50) (51)

(2Bc + k )(μ2 + 12π μ ) + 64π 2 Bc μ ( 8π + μ )

(52)

Using Eq. (44) with Eqs. (36)–(38), we get the pressure, density, magnetic field and cosmological constant for Static Einstein Universe in f(R, T) gravity for f1 (R ) + f2 (T ) model as follows;

p = −Bc −

ρ = Bc −

k 2(8π + f3 )

3k 2(8π + f3 )

h2 = 0

= −8π Bc −

(53)

(54) (55)

k(12π + f3 ) f − 3 8π + f 3 2

(56)


77

From Eqs. (47), (51) and (55) we see that vanish magnetic field value for three types f(R, T) gravity models. Also, we obtain non magnetized strange quark matter solutions for static Einstein universe and the Bag constant Bc is effective on cosmic density, pressure and cosmological constant for three f(R, T) gravity models. If we take f 3 (T ) = 0 in Eqs. (53)–(56), we get Static Einstein Universe in General Relativity theory as follows

p = −Bc −

ρ = Bc −

k 16π

3k 16π

(58)

h2 = 0

= −8π Bc −

(57)

(59) 3k 2

(60)

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