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Evolution of Inhomogeneous LTB Geometry with Tilted

Congruence and Modified Gravity

Journal: Canadian Journal of Physics

Manuscript ID cjp-2017-0214.R1

Manuscript Type: Article

Date Submitted by the Author: 10-Apr-2017

Complete List of Authors: Yousaf, Z.; Punjab University, Department of Mathematics Bhatti, Muhammad; Punjab University, Mathematics Rafaqat, A; Punjab University, Department of Mathematics

Keyword: Dissipative fluids, Relativistic interiors, f(R) gravity, Self-gravitatring systems, Modified gravity

Is the invited manuscript for

consideration in a Special Issue? :

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Canadian Journal of Physics

For Review O

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and Modified Gravity

Z. Yousaf ∗, M. Zaeem-ul-Haq Bhatti †and Aamna Rafaqat ‡

Department of Mathematics, University of the Punjab,Quaid-i-Azam Campus, Lahore-54590, Pakistan.

Abstract

The goal of this paper is to shed some light on the significanceof congruence of observers which seems to affect the dynamics of theuniverse under Palatini f(R) formalism. Starting by setting up the for-malism needed, we have explored the field equations using Lemaitre-Tolman-Bondi geometry as an interior metric. We have formulatedthe relationship between the matter variables as seen by the observersin both comoving and non-comoving frames. The dynamical equationsare evaluated to study the dynamics of inhomogeneous universe by ex-ploring conservation equations along with the Ellis equations. We havealso explored a collapsing factor describing the bouncing phenomenavia transport equation and conclude the stability region.

Keywords: f(R) gravity; Structure scalars; Relativistic dissipative fluids.PACS: 04.20.Cv; 04.40.Nr; 04.50.Kd.

1 Introduction

During last decades, theoretical as well as observational indications revealedthat Einstein’s gravity theory (GR) cannot demonstrate the flat rotation

∗[email protected]†[email protected]‡[email protected]

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curves of galaxies in the absence of dark matter (DM) and the present ex-pansion rate of the universe without both DM and dark energy (DE), whichis summed up to 96% of the universe’s total energy content. Also, the recentcosmological observations for late-time universe indicate the dominance ofcosmological constant due to mysterious form of DE. However, inclusion ofsuch constant mimics the behavior of standard model of GR and its micro-physical origin remains a mystery. Different puzzling properties have beencame up with recent released information of 2.7 full sky survey throughPlanck data [1] which changed the concept of universe modeling.

Since the universe evolution is mostly governed by the gravitational fieldon large scales, so there exist many alternatives addressing DM and DEduring the modification of theory of gravity. The simplest modification inGR is the inclusion of non-linear interaction of Ricci scalar R in its actionand such theories are named as f(R) theories. It is well established thatGR can be derived from Einstein-Hilbert (EH) action by varying it withrespect to connection and metric independently as well as by standard metricvariation (for reviews on the late-time cosmic acceleration, i.e., DE problem,and modified gravity theories, see, e.g., [2–6]). During the modification of GRto f(R), one can vary the action in the same manner, which leads to differentversions of f(R) theory. The version in which the action is varied by keepingthe connection and metric as independent, is known as Palatini variationalapproach while the other one is named as metric approach (which experiencegravitational instabilities [7]). Moreover, the field equations are second orderin Palatini formulism which are fourth order with metric variation.

Einstein [8] himself introduces this Palatini approach to examine the na-ture of dynamical universe in gravitational theories, however, due to a his-torical misconception, it is named as Palatini approach [9]. In this paper, wefocussed our attention on this particular modification of GR, i.e., Palatiniformulism. Different f(R) models have already been proposed with Palatiniapproach in the literature with the focus on the late-time universe due toinverse powers of the curvature scalar. Sotiriou [10] discussed that the pos-itive powers of the curvature scalar in gravitational action can also be usedto study the late-time universe evolution. An extensive literature is avail-able describing the cosmologically viable f(R) models consistent with thesolar system constraints even with very light scalar field [11]. Amarzguiouiet al. [12] investigated a class of DE models due to the modification in EHLagrangian by using the current cosmological data.

Kainulainen et al. [13] explored the modified Tolman-Oppenheimer-Volkoff

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equation for interior spherical geometry under hydrostatic equilibrium whichdoes not affect the form of vacuum solution. This is one of the main rea-son of the viability of Solar system’s constraints in Palatini formulism. Also,most of the f(R) theories in metric formulism are ruled out due to the failurein finding any interior geometry which can be matched with Schwarzschildde-Sitter metric [14]. Olmo [15] studied the possibility of curvature singular-ities in locality of low curvature spherically symmetric matter configurationswithin the Palatini formulism and polytropic equation of state. Barragan etal. [16] found that Palatini version of f(R) gravity have unusual properties tostudy the collapse process and early universe. They also explored the condi-tions which avoid the big bang singularity and responsible for the emergenceof isotropic and homogeneous models.

Barrow and Ottewill [17] examined the nature of isotropic and homoge-neous solutions within the f(R) gravity theory and explored the constraintsfor which these solutions correspond to those in GR for late and early cosmicepochs. PopÃlawski [18] applied the field equations obtained under Palatiniformalism to a homogeneous and flat isotropic dust universe and check theviability of this Palatini formalism to reveal present cosmic acceleration. Alle-mandi et al. [19] investigated the equivalence between GR and non-minimallycoupled higher order theories in Palatini approach and studied conformal in-variance due to the conjecture of isotropy and homogeneity. Koivisto andMota [20] explored the anisotropic and homogeneous solutions using covari-ant approach on the cosmological equations. Bohmer et al. [21] exploredthe effects of in/homogeneoud perturbations on the Einstein static universeto examine its stability in the background of Palatini gravitational theory.They show that the respective stable regimes exist with certain parametricspace of equation of state.

The inhomogeneities in energy density distribution are described as spher-ical epochs where the geometry takes the form of Lemaitre-Tolman-Bondi(LTB) spacetime. It has been observed that inhomogeneous matter distri-bution is necessary for the applicability of supernova data in the context ofLTB metric. There exist a common scale factor which can demonstrate theexpanding regions of the universe. The role of inhomogeneous matter den-sity has broadly been studied in the discussion of self-gravitating collapsingstars [22]. Herrera et al. [23] explored the role of density irregularities on thestructure and evolution of anisotropic spherical stars. Herrera [24] discussedinhomogeneity factors for adiabatic and non-adiabatic relativistic interiorsand claimed that the system must satisfy these constraints to achieve stable

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configurations. There have been interesting results about the inhomogene-ity parameters in comoving, non-comoving charged and conformally flat [25]relativistic systems. Yousaf and his collaborators found [26] modified grav-ity could be considered as a viable platform to describe new solutions withsome extra degrees of freedom that were not possible in GR. They havealso investigated those factors that are responsible to disturb the stability ofhomogeneous universe [27].

Here, our aim is to compute the relationship of material variables betweencomoving and non-comoving reference frames and discuss its dynamics in thebackground of Palatini f(R) gravity. In the following section, we exploredthe basic formulism including the field equations with a particular form off(R) model. Some particular relations have been explored with and withoutheat flux as well as in the absence of radial pressure. In order the study thecongruence of observers, we have investigated the kinematical quantities insection 3 which have enough physical significance in the study of relativisticastrophysics. The zero and non-zero values of these kinematical quantitiesdemonstrate different realistic regimes. Section 4 is dedicated to the explo-ration of some dynamical variables which have significance to explore theevolution of the universe. These dynamical quantities are related with ma-terial variables by using the contracted form of Bianchi identities. We havealso formulated a collapsing factor in the light of comoving observer. Thelast section summarizes our main findings.

2 Palatini f (R) Formalism

In the universe, different stellar systems are bound together with the helpof gravitational interactions. These celestial bodies provide a powerful toolto study galaxy and inter galactic structure formation. A well-consistentgravitational theory is required to understand these populations. In recenttimes, astrophysicists found MGTs as a powerful tool to trace the cosmicstructures evolution. For f(R) gravity, the usual Einstein-Hilbert action ismodified as [13]

Sf(R) =1

2κ

∫d4x

√−gf(R) + SM , (1)

where SM is action for matter fields, κ is a constant number with appropriatedimensions, for instance κ = 8πG for GR action and f(R) is any viable Ricciinvariant function. In order to attain Palatini f(R) field equation, we shall

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ponder that Γµγδ 6= Γµ

δγ . Making variation on Eq.(1) with gγδ and Γµγδ, we

have

fR(R)Rγδ − [gγδf(R)]/2 = κTγδ, (2)

∇µ(gγδ√−gfR(R)) = 0, (3)

where Tγδ is the standard energy-momentum tensor. Solving Eq.(2) for con-nections and then using this value in Eq.(3), one can achieve the followingPalatini field equation

1

fR

(∇γ∇δ − gγδ¤

)fR +

1

2gγδR +

κ

fR

Tγδ +1

2gγδ

(f

fR

− R

)+

3

2f2R

[1

2gγδ(∇fR)2 − ∇γfR∇δfR

]− Rγδ = 0. (4)

It is worthwhile to note that above equation does not contain connectionterms rather it comprises of matter and metric tensors as variable quantities.The trace of Eq.(4) yields

RfR(R) − 2f(R) = κT. (5)

This indicates that connections Ricci scalar can be manipulated through traceof standard stress-energy tensor, i.e., T . This consequently gives R = R(T )and fR = fR(T ), i.e., R and fR are the function of T that eventually makethem dependent on metric variables not on connections.

The vacuum case, i.e., Tγδ = 0 would necessarily leads the differentialequation to has a constant solution that would secure connections to be well-known Levi-Civita. Further, this would also assign constant value to fR. Inthis scenario, Eq.(5) boils down to

Gγδ = −Λgγδ,

where Λ is a constant term that could be regarded as an affective cosmologicalconstant, this suggests the validity of Birkhoff’s theorem in this modified

gravity. Here, the value of Λ is −(

f−RfR

2fR

), where breve shows that the

corresponding values are evaluated with vacuum environment. Equation (4)can be rewritten through Einstein tensor as

Gγδ =κ

fR

(Tγδ + Tγδ), (6)

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where

Tγδ =1

κ

(∇γ∇δ − gγδ¤

)fR − fR

2κgγδ

(R − f

fR

)+

3

2κfR

[1

2gγδ(∇fR)2 − ∇γfR∇δfR

],

while Gγδ ≡ Rγδ − 12gγδR and ¤ is a de Alembert operator, expressed with

the help of independent connection dependent covariant derivative, ∇γ, as

¤ = ∇γ∇δgγδ.

We wish to consider LTB spherical spacetime with its generic form as [28]

ds2− = dt2 − A′2

(h + ε)dr2 − C2(dθ2 + sin θ2dφ2), (7)

where ε can be have 0 or ±1 constant numerical value, h = h(r) obeying thecondition h + ε ≥ 0 and prime means radial partial derivative of the corre-sponding quantity. This model could be considered as a power and effectivetool to understand many inhomogeneous aspect of this accelerating cosmos.Besides many other well-known cosmological benefits [29], the analysis of ex-act inhomogeneous models such as LTB may provide us better platform forcomprehending the influence of irregularities on the existence of many stellarobjects [30]. By considering B = A′ and h+ε = 1, one can find the followingform of diagonal non-static irrotational LTB line element

ds2− = −dt2 + B2dr2 + C2(dθ2 + sin θ2dφ2). (8)

The geometry of any stellar objects exist due to the gravitational interac-tion of relativistic matter sources. These sources are closely linked withtheir 4-velocities, thus giving importance to the choice of fluid 4-velocities.The general dynamical description of the gravitational source along withits congruence kinematics could be dissimilar, if the two feasible relativis-tic explanations of a given spacetime are related to the boost of one of theobserver congruences regarding to the other one. For example, Friedmann-Robertson-Walker is an ideal fluid solution for the rest observer with referenceto time-like congruence. In the same time, this metric can be interpreted asthe solution of viscous radiating gravitating source, if the observer is movingwith relative velocity respecting the first previous frame. Motivated from

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this concept, we first assume the comoving coordinate frame under whichthe 4-velocity for dust energy momentum tensor

Tγδ = ρuγuδ (9)

isuγ = (1, 0, 0, 0), (10)

where ρ is the fluid’s energy density configuring in comoving congruences.Apart from that, we now suppose that matter content has some radial ve-

locity ω with respect to a new frame of reference. Let our system is subjectedto Lorentz boost from locally Minkowskian frame hosting perfect fluid to thisnew frame. This makes us possible to have a tilted congruences, supportedby the following 4-vector field

Uγ =

(1√

1 − ω2,

ω

B√

1 − ω2, 0, 0

). (11)

We consider that observer, in tilted frame, has noted locally anisotropicradiating matter source for LTB geometry with the above mentioned 4-velocity. Its stress-energy tensor is

Tγδ = (ρ + P⊥)UγUδ + εlγlδ − P⊥gγδ + qγUδ + (Pr − P⊥)SγSδ + qδUγ, (12)

where ρ, ε, qγ, P⊥ and Pr are energy density, radiation density, heat fluxvector, tangential and radial pressure, respectively. The quantity lγ is a null4-vector defined as follows

lγ =

(1 + ω√1 − ω2

,1 + ω

B√

1 − ω2, 0, 0

), (13)

while qγ can be defined through another 4-vector, Sγ as

qγ = qSγ. (14)

with

Sγ =

(ω√

1 − ω2,

1

B√

1 − ω2, 0, 0

). (15)

All tilted 4-vectors are obeying

UγUγ = −1 = lγUγ, SγSγ = 1 = lγS

γ, lγlγ = 0 = SγUγ = Uγqγ.

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2.1 Tilted and Non-tilted Relations with Positive andNegative Ricci Scalar Corrections

The Palatini f(R) field equations for tilted environment are obtained asfollows

G00 =κ

fR

[ρ + Prω

2 + 2qω

1 − ω2+

1

κ

{R

2

(fR − f

R

)− fR

(B

B+

9fR

4fR

+2C

C

)

−(

B′

B− 2C ′

C+

f ′R

4fR

)f ′

R

B2+

f ′′R

B2

}], (16)

G11

B2=

κ

fR

[ρω2 + Pr + 2qω

1 − ω2+

1

κ

{−R

2

(fR − f

R

)− f ′

R

B2

(9

4

f ′R

fR

+2C ′

C

)+ fR

−

(fR

4fR

− 2C

C

)fR

}], (17)

G22

C2=

κ

fR

[P⊥ +

1

κ

{−R

2

(fR − f

R

)+ fR +

(B

B− fR

4fR

+C

C

)fR − f ′′

R

C2

+

(B′

B+

f ′R

4fR

− C ′

C

)f ′

R

B2

}], (18)

G01 =κ

fR

[−B

{(ρ + Pr)ω + q(1 + ω2)

(1 − ω2)

}+

1

κ

{f ′

R − 5fRf ′R

2fR

− f ′RB

B

}],

(19)

where x ≡ x + ε and the components of Gγδ are mentioned in [31].In order to present f(R) gravity as an acceptable model, one should con-

sider observationally viable f(R) model. A viable model not only helps toshed light over current cosmic acceleration but also obeys the requirementsimposed by terrestrial and solar system experiments with relativistic back-ground. In this paper, we wish to find some relations that could highlightinflationary and current acceleration of the universe. It is well-known thatRicci scalar function, in f(R) action, incorporating negative and positivepowers endorse recent cosmic acceleration and its inflationary eras, respec-tively [32]. We now consider two models, one is associated with the source(9) and other with the tilted congruence (12). This would eventually givesus relation between parameters depicting inflationary and recent accelerating

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universe. For non-tilted congruence we consider the following f(R) model [33]

f(R) = R + αR2, (20)

where α is any positive constant number. This model could be taken for thedescription of both DM (with α = 1

6M2 [34]) and of DE. For former purpose,M is constrained to be 2.7 × 10−12GeV with α ≤ 2.3 × 1022Ge/V 2 [3].

Next, we suppose f(R) model widely known as Carroll-Duvvuri-Trodden-Turner (CDTT) model [35], i.e.,

f(R) = R + ρδ4

R, (21)

where δ > 0 and ρ = +1. For the specific value of δ, i.e., δ−1 ∼ (1033eV )−1 ∼1018sec, this model could support current accelerating cosmic regimes. Forvacuum case, this model boils down to present de-Sitter solution.

Since the Einstein tensor for both tilted and non-tilted reference framesare equal, therefore we obtain some important relations

κρ

1 + 2αR+

αR2

2(1 + 2αR)=

κR2(ρ + Prω2 + 2qω)

(R2 − δ2)(1 − ω2)− Rδ2

R2 − δ4, (22)

(ρ + Pr)ω + q(1 + ω2) = 0, (23)

κR2(ρω2 + Pr + 2qω)

(R2 − δ2)(1 − ω2)=

−αR2

2(1 + 2αR)− Rδ2

R2 − δ2, (24)

P⊥ =−α(R2 − δ4)

2(1 + 2αR)κ− δ4

Rκ. (25)

It is noted that due to Palatini f(R) gravity, the tangential pressure is non-zero and purely depends upon the degrees of freedom coming from inflation-ary as well as late time cosmic acceleratory terms.

The relations between tilted congruence with late time accelerating cor-rections and non-tilted congruence with inflationary degrees of freedom canbe achieved through Eqs.(9) and (12). These are

ε =(R2 − δ4)

1 + 2αR

{ρ

R2(1 − ω2)+

α

2κ

}+

δ4

Rκ− ρ, (26)

Pr = ρ − (R2 − δ4)ρ

R2(1 + 2αR)− (R2 − δ4)α

κ(1 + 2αR)− 2δ4

Rκ, (27)

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R2(1 − ω)(1 + 2αR)− α(R2 − δ4)

2κ(1 + 2αR)− δ4

Rκ, (28)

=Pr − ρω

(1 − ω)+

α(R2 − δ4)(1 + ω)

2κ(1 + 2αR(1 − ω))+

δ4(1 + ω)

Rκ(1 − ω), (29)

=−ρ(R2 − δ4)ω

R2(1 + 2αR)(1 − ω2)− ε. (30)

It is worthy to mention that, these relations have been found by taking intoaccount constant Ricci scalar condition. Under the limit f(R) → R, all ourrelations reduce to that of GR [28]. For bounded laboratory distributions,it is found that, like GR, the junction conditions are not satisfied therebyindicating the existence of thin shell over the hypersurface. In the followingsubsections, we shall discuss some particular cases.

2.1.1 ε = 0 with Non-zero Heat Flux

In this scenario, Eqs.(26)-(30) yield

Pr = ρω2 − α(R2 − δ4)(1 + ω2)

2κ(1 + 2αR)− δ4(1 + ω2)

κR, (31)

ρ =(R2 − δ4)

(1 + 2αR)

{ρ

R2(1 − ω2)+

α

2κ

}+

δ4

Rκ, (32)

q =−ρ(R2 − δ4)ω

R2(1 + 2αR)(1 − ω2). (33)

2.1.2 ε 6= 0 with Zero Heat Flux

For this case, it is evident from Eqs.(26)-(30) that

Pr = ρω − α(R2 − δ4)(1 + ω)

2κ(1 + 2αR)− δ4(1 + ω)

κR, (34)

ρ =(R2 − δ4)

(1 + 2αR)

{ρ

R2(1 − ω)+

α

2κ

}+

δ4

Rκ, (35)

ε =−ρ(R2 − δ4)ω

R2(1 + 2αR)(1 − ω2). (36)

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2.1.3 Pr = 0

In this scenario, we have from Eqs.(26)-(30) that

ρ =ρ(R2 − δ4)

R2(1 + 2αR)+

α(R2 − δ4)

κ(1 + 2αR)+

2δ4

κR, (37)

q =−ρω

1 − ω+

α(R2 − δ4)(1 + ω)

2κ(1 + 2αR)(1 − ω)+

δ4(1 + ω)

Rκ(1 − ω), (38)

ε =ρω2

(1 − ω2)− α(R2 − δ4)(1 + ω2)

2κ(1 + 2αR)(1 − ω2)− δ4(1 + ω2)

Rκ(1 − ω2). (39)

3 Kinematical Quantities

For the complete description of non-rotating relativistic matter distribution,three kinematical quantities, i.e., 4-acceleration, shear tensor and and ex-pansion scalar occupy enticing importance. One can find four accelerationthrough its general formula

aγ = Uγ;δU

δ. (40)

This kinematical variable has been found to be expressed by means of 4-vector field and Palatini dark source terms as

aγ = aSγ − gγγ ∂γfR

2fR

, (41)

where

a =1

3√

1 − ω2

[ω +

ωω′

B+ (1 − ω2)

(ωB

B+

ωfR

fR

+f ′

R

BfR

)]. (42)

It is noted that in tilted Palatini f(R) frame, the congruences always behavenon-geodesically. By making a = 0, one can achieve non-trivial solution fora particular LTB geometry. In other words, a special kind of constraint canbe obtained that would express velocity fields with the help of dark sourceterms coming from Palatini f(R) gravity.

Further, the mathematical expression for expansion scalar is given by

Θ = Uµ;µ, (43)

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which, for our tilted system, turns out to be

Θ =1

3√

1 − ω2

[ωω +

ω′

B+ (1 − ω2)

(B

B+ 2

fR

fR

+2C

C+

2ωC ′

CB+

ωf ′R

BfR

)].

(44)

The shear tensor is defined as

σγδ = U(γ;δ) + a(γUδ) −1

3Θhγδ, (45)

where hγδ is a projection tensor. The non-vanishing components of the aboveequation are

σ00 =2ω2

3(1 − ω2)

[σ +

3fR

2ω2fR

√1 − ω2

+3f ′

R

4BfRω

√1 − ω2

], (46)

σ11 =2B2

3(1 − ω2)

[σ +

√1 − ω2

(3fR

4fR

− 3ωf ′R

2BfR

)], (47)

σ22 =−C2

3

[σ +

1√1 − ω2

(3fR

2fR

− ωf ′R

2BfR

)], (48)

where

σ =1

3√

1 − ω2

[ωω +

ω′

B+ (1 − ω2)

{B

B− fR

fR

− C

C− ωC ′

CB+

ωf ′R

BfR

}]. (49)

By taking ω = 0 along with f(R) = R, the kinematical quantities for non-tilted LTB spacetime can be recovered.

4 Some Basic Auxiliary Equations

In this section, we shall evaluate some basic expressions required to un-derstand the complete general discussion of the the tilted LTB relativisticmatter configurations. The equations of motion describing the dynamics ofthe evolving cosmic system occupy alluring importance as such equationscould efficiently provide the energy variation of the stellar population gradi-ents respecting time and proximate surfaces. These are generally found viacontracted Bianchi identities, i.e.,

Xγδ;γ

= 0, where Xγδ = T γ

δ + T γδ.

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Using above relation along with Eqs.(16)-(19), (42), (44) and (49), we get

ρ∗ + ρΘ + q† + q

{ωΘ +

√1 − ω2

B

(2C ′

C+

f ′R

fR

)+

2ω√1 − ω2

}+

ρf∗R

fR

+qf †

R

fR

+ωP ′

⊥B

+ P⊥

(Θ +

fR

fR

+ωf ′

R

fR

)+ D0 = 0, (50)

Pr†+ a(ρ + Pr) +

2q

3

[2Θ + σ − 3ω(ln C)†

]+ q∗ +

ωf ∗R

fR

(ρ + P⊥) − q√

1 − ω2

×

(B

B+

2C

C

)+

1

fR

√1 − ω2

(qω2fR − ρf ′

R

B− P⊥f ′

R

B

)−

√1 − ω2(P⊥ω)

− ω2P ′⊥

B√

1 − ω2+

ω√1 − ω2

[ ˙µ + (ωq)] + D1 = 0, (51)

where

z† = z,µSµ, z∗ = z,µU

µ,

while D0 and D1 are terms containing Palatini f(R) corrections in generalform and are mentioned in Appendix.

Now, we would find factors disturbing the energy density inhomogeneityof the tilted LTB system coupled with anisotropic dissipative relativisticmatter. We found the well known equation that has related the Weyl tensorwith fluid source variables. The detail for the evaluation of such expressions(for various non-tilted frames) has been found in literature with both GR [24]and modified gravity [36] backgrounds. For tilted observer, we found[E − κ

2fR

(ρ − Pr + P⊥ + T00 −

T11

B2+

T22

C2

)]′

= −3C ′

C

[E +

κ

2fR

(Pr − P⊥

+T11

B2− T22

C2

)]+

3κC

2CfR

{(ρ + Pr)ω + q(1 + ω2)

(1 − ω2)− T01

B

}. (52)

This equation is in complex form whose general analytic equation is a difficulttask. But one can find its analytic solution by taking some constraints. Here,we wish to check the influence of pressure gradient on the evolution of tiltedLTB geometry. The dynamics of tilted environment can be examined throughlocally anisotropic source material. In this context, we consider q = ε = 0

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along with Pr = P⊥ = P . Under this framework, the solution of abovedifferential equation is

E ′ +3C ′

CE =

κ

2

[1

fR

(ρ + T00 −

T11

B2+

T22

C2

)]′

+3κC

2CfR

(ρ + P )

(1 − ω2)ω. (53)

In GR, the Weyl scalar is responsible for disturbing homogeneities in theisotropic system, but here the situation is quite different. The two importantparameters are trying to produce hindrances for the system to enter in theinhomogeneous phase. First one is Palatini dark source terms and second oneis the fluid velocity ω, emerging due to tilted congruences. From the aboveexpression, it is clear that the tilted congruence has given birth to pressure forhelping the system to be remain in the regular stage. If one assumes, ω = 0,then the contribution of pressure will be vanished. On taking f(R) = R andω = 0 in the above equation, we get

ρ′ = 0 ⇔ E = 0. (54)

It is pertinent to mention that various interesting aspects about the evolu-tion of the cosmic system, dissipating in the form of diffusion approximation,can be highlighted by exploring transport equation. Such mathematical in-terpretation can be achieved with the help of casual dissipative theory. In thisframework, relaxation time, τ , corresponding to radiating phenomena occupybasic prominence. For example, in the field thermodynamics, this definitepositive term can not be ignored for the discussion of system’s transientand hydrodynamical epochs. For heat conducting vector field, the transportequation is given by

τhγδUµqγ,µ + qγ + Khγδ(∆,δ + ∆aδ) +K

2∆2

(τU δ

K∆2

);δ

qγ = 0, (55)

where ∆ and K stand for temperature and thermal conductivity, respec-tively. Using Eqs.(16)-(19) and (55), we get the following single independentcomponent of the above equation

τ

(q − qfR

2fR

+ω

Bq′

)+ q

√1 − ω2 = −τ

2q(1 − ω2)1/2Θ − K

[∆ω +

∆′

B+ ∆

×

(a√

1 − ω2 − ωfR

2fR

− f ′R

2BfR

)]− K∆2

2q

{( τ

K∆2

),0

+ω

B

( τ

K∆2

),1

}.

(56)

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This equation shows that relaxation time for tilted radiating source is influ-enced by dark source terms induced by Palatini f(R) gravity.

Now, we shall talk about the existence of non-tilted congruence againstperturbations. We assume that our non-tilted distributions is subjected tosmall oscillations at initial time, t = 0. We would analyze oscillating non-tilted relativistic distribution with the time duration smaller than taken byhydrostatic and thermal relaxation processes. Upon employing perturbation,we found that

ω = q = 0, ω ≈ q 6= 0. (57)

In order to lessen down the order of complexity in our system, we haveassumed that radiations stem from streaming out approximations are negli-gible, i.e., ε = 0. The second dynamical equation provides

ρω + q +ρf ′

R

BfR

− ρf ′R

BfR

= 0,⇒ ωρ + q = 0. (58)

Since, initially, the system was in complete geodesically thermal equilibrium,therefore, we can take ∆′ = 0. Then, Eq.(56) reduces to

τ q = −K∆ω − K∆f ′R

2BfR

. (59)

Equations (58) and (59) yield

ω

(ρ − K∆

τ

)− K∆f ′

R

2BfRτ≡ ω(ρ − α1ρ) − K∆f ′

R

2BfRτ, (60)

where α1 ≡ K∆/(ρτ). If α1 = 1, then perturbation will induce instabilityto the system evolving with non-tilted congruences. Infact, this scenarioimplies impart zero value to inertial mass density of the system. The detailanalysis in this context under the variety of cosmological applications hasbeen discussed in literature [37,38].

5 Summary

In relation with recent observations of supernova type Ia, a renewed interestof researchers have been developed in LTB geometry to study the expansion

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history of the accelerating universe. We have defined and revisited the basicequations describing evolution of the universe in the framework of Palatinif(R) theory from the congruence of different observers. A congruence maybe tilted or non-tilted depending upon the motion of fluid and observer.The idea is to expose the role of observers in the depiction of any physicalprocess. The physical interpretation of both tilted and non-tilted models isquite different, thus one can ask that which model is better than the otherone? However, one cannot characterize what is the correct interpretation asboth are physically viable and each analysis is related to a specific congruenceof observers.

We explore the cosmologies from that point where the universe is inho-mogeneously expanding by a field of anisotropic and dissipative fluid. Oursearch will depend upon two different types of f(R) models. For the observercomoving with the fluid, we consider quadratic f(R) model while for non-comoving observer, i.e., in tilted frame, we choose CDTT model to constructour systematic analysis. A better understanding of these models in inflation-ary and late time cosmic epochs could help us deepen into such situations.We have formulated a link between the matter variables of both tilted andnon-tilted congruences in the background of Palatini f(R) gravity with LTBmetric as interior geometry. The matter distribution is dust cloud for non-tilted observer which becomes anisotropic for tilted observer when one applythe Lorentz boost (rotation free transformation) to a locally Minkowskianframe in which the fluid element has radial velocity. The effects of darksource terms are also explored in these relations.

We have explored the kinematical quantities under the influence of ex-tra degrees of freedom of f(R) gravity with the reference of tilted observer.The four acceleration for LTB geometry takes the general form as obtainedin Eq.(41) with f(R) extra curvature terms. We have investigated the non-vanishing values of the shear tensor and expansion scalar which have sig-nificance in the light of relativistic astrophysics. We have formulated theconservation equations in a precise manner to distinguish the role of darksource terms as well as the congruence of observers. The evolution equationhave been established to explore the inhomogeneities appearing due to theexistence of Weyl tensor. The necessary and sufficient condition for a homo-geneous system is found as the system should be conformally flat for non-tilted observer. We have obtained a differential equation describing the heattransfer phenomena for a collapsing matter distribution. We found a similarfactor α1 which describe the stability of non-tilted congruence in the frame-

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work of GR. It is pertinent to mention that one can identify more irregularityfactors except that one which we have formulated here using Eqs.(50)-(52).All results obtained here are well consistent with the literature.

Appendix

The parts of Eqs.(50) and (51) are

D0 = −T00 +

(T10

B2

)′

− T00

(B

B+

3fR

2fR

+2C

C

)+

T01

B2

(B′

B+

2f ′R

fR

+2C ′

C

)

− T11

B2

(B

B+

fR

2fR

)− 2T22

C2

(C

C+

fR

2fR

), (A1)

D1 = −T10 +

(T11

B2

)′

+ T00f ′

R

2fR

− T10

(B

B+

2fR

fR

+2C

C

)+

T11

B2

(2C ′

C+

3f ′R

2fR

)− 2T22

C2

(C ′

C+

f ′R

2fR

).

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