SOME COSMOLOGICAL ASPECTS OF CELESTIAL ...

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SOME COSMOLOGICAL ASPECTS OF CELESTIAL

OBJECTS IN MODIFIED GRAVITY

Muhammad Ilyas

A THESIS

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

HIGH ENERGY PHYSICS

Supervised By

Dr. Bilal Masud

Dr. Zeeshan Yousaf

CENTRE FOR HIGH ENERGY PHYSICS

UNIVERSITY OF THE PUNJAB

LAHORE-PAKISTAN

FEBRUARY, 2018

CERTIFICATE

We certify that the research work presented in this thesis is the original work of

Mr. Muhammad Ilyas S/O Nisar Muhammad and is carried out under our

supervision. We endorse its evaluation for the award of Ph.D. degree through the

official procedure of University of the Punjab.

Dr. Bilal Masud and Dr. Zeeshan Yousaf(Supervisors)

Author’s Declaration

I, Mr. Muhammad Ilyas, hereby state that my PhD thesis

titled ”Some Cosmological Aspects of Celestial Objects in

Modified Gravity” is my own work and has not been submitted

previously by me for taking any degree from University of the Punjab,

or anywhere else in the country/word.

At any time if my statement is found to be incorrect even after my

graduation, the university has the right to withdraw my PhD degree.

Name: Muhammad Ilyas

Date: October 03, 2018

DEDICATED

My Loving Parents

Siblings

Table of Contents

Table of Contents v

List of Figures viii

List of Tables xii

List of Publications xiii

Acknowledgements xv

Abstract xvi

Abbreviations xviii

Introduction 1

1 Cosmological Aspects of Celestial Objects and Modified Gravity

Theories 12

1.1 Compact Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.1 White Dwarf Stars . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.3 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Mathematical Formalism Behind Compact Stars . . . . . . . . . . . . 15

1.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.2 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.3 Chandrasekhar Mass Limit . . . . . . . . . . . . . . . . . . . . 17

1.2.4 GR Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Wormholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Some Cosmological Aspects . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.1 Matter Distributions . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.2 Electro-Magnetic distribution . . . . . . . . . . . . . . . . . . 19

1.4.3 Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.4 Anisotropic Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.1 Null Energy Conditions . . . . . . . . . . . . . . . . . . . . . 22

1.5.2 Weak Energy Conditions . . . . . . . . . . . . . . . . . . . . . 22

1.5.3 Dominant Energy Conditions . . . . . . . . . . . . . . . . . . 23

1.5.4 Strong Energy Conditions . . . . . . . . . . . . . . . . . . . . 23

1.6 Equation of State Parameter . . . . . . . . . . . . . . . . . . . . . . . 23

1.7 Equilibrium Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.8 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.9 Different Modified Gravity Theories . . . . . . . . . . . . . . . . . . . 27

1.9.1 f(R) Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.9.2 f(R, T ) Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.9.3 f(G) Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.9.4 f(G, T ) Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.9.5 f(R, ¤R, T ) Gravity . . . . . . . . . . . . . . . . . . . . . . . 31

2 Compact Stars 33

2.1 Anisotropic Relativistic Spheres in f(R) Gravity . . . . . . . . . . . . 34

2.2 Physical Aspects of f(R) Gravity Models . . . . . . . . . . . . . . . . 36

2.2.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.4 Energy Density and Pressure Evolutions . . . . . . . . . . . . 38

2.2.5 TOV Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.6 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.7 Equation of State Parameter . . . . . . . . . . . . . . . . . . . 42

2.2.8 The Measurement of Anisotropy . . . . . . . . . . . . . . . . . 43

2.3 Spherical Anisotropic Fluids and f(R, T ) Gravity . . . . . . . . . . . 45

2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5 Different Models in f(R, T ) gravity . . . . . . . . . . . . . . . . . . . 48

2.5.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6 Physical Aspects of f(R, T ) Gravity Models . . . . . . . . . . . . . . 50

2.6.1 Energy Density and Pressure Evolutions in f(R, T ) gravity . . 51

2.6.2 TOV Equation in f(R, T ) gravity . . . . . . . . . . . . . . . . 53

2.6.3 Stability Analysis in f(R, T ) gravity . . . . . . . . . . . . . . 55

2.6.4 The Measurement of Anisotropy in f(R, T ) gravity . . . . . . 58

2.7 Anisotropic geometry in f(G, T ) gravity . . . . . . . . . . . . . . . . 59

2.8 f(G) Gravity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.8.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.8.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.8.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.9 Physical Aspects of f(G, T ) Gravity Models . . . . . . . . . . . . . . 62

2.9.1 Energy Density and Pressure Evolutions . . . . . . . . . . . . 62

2.9.2 TOV Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.9.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.9.4 The Measurement of Anisotropy . . . . . . . . . . . . . . . . . 68

3 Wormhole Solutions and Energy Conditions 69

3.1 Wormhole Geometry and f(R, T ) Gravity . . . . . . . . . . . . . . . 70

3.1.1 Quadratic Ricci Corrections and Anisotropic Matter Content . 71

3.1.2 Perfect Matter Content . . . . . . . . . . . . . . . . . . . . . . 79

3.1.3 Barotropic State Equation . . . . . . . . . . . . . . . . . . . . 80

3.2 Wormhole Solutions with Cubic Ricci Scalar Model . . . . . . . . . . 84

3.2.1 Equilibrium Condition . . . . . . . . . . . . . . . . . . . . . . 89

3.2.2 Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2.3 Specific Equation of State . . . . . . . . . . . . . . . . . . . . 94

4 Compact Stars and Dark Dynamical Variables 97

4.1 Radiating Sphere and f(R, T ) Gravity . . . . . . . . . . . . . . . . . 98

4.2 Modified Scalar Variables and f(R, T ) Gravity . . . . . . . . . . . . . 102

4.3 Evolution Equations with Constant R and T . . . . . . . . . . . . . . 104

5 Concluding Remarks 110

Appendix A 114

Appendix B 120

Bibliography 121

List of Figures

2.1 Density evolution of the strange star candidates with three different

f(R) models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Radial pressure evolution of the strange stars. . . . . . . . . . . . . . 40

2.3 Transverse pressure evolution of the strange stars. . . . . . . . . . . . 40

2.4 Behavior of dρ/dr with respect to r with three different f(R) models. 41

2.5 Behavior of dpr/dr with respect to r. . . . . . . . . . . . . . . . . . . 41

2.6 Behavior of dpt/dr with increasing r. . . . . . . . . . . . . . . . . . . 41

2.7 The plot of Fg, Fh and Fa with respect to the radial coordinate r(km)

and three different f(R) models. . . . . . . . . . . . . . . . . . . . . . 42

2.8 Variations of v2st − v2

sr with respect radius, r (km) . . . . . . . . . . . 43

2.9 Variations of the radial EoS parameter, ωr with respect to the radial

coordinate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.10 Variations of anisotropicity, ∆, with respect to r. . . . . . . . . . . . 44

2.11 Plot of the density (km−2) evolution of the strange star candidate Her

X-1, SAX J 1808.4-3658, and 4U 1820-30; for three different models. . 52

2.12 Plot of the radial pressure (km−2) evolution of the strange star candi-

date Her X-1, SAX J 1808.4-3658, and 4U 1820-30; for three different

models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.13 Plot of the transverse pressure (km−2) evolution of the strange star

candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; for three

different models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.14 Plot of the dρ/dr with increasing r of the strange star candidate Her

2.15 Plot of the dpr/dr with increasing r of the strange star candidate Her

2.16 Plot of the dpt/dr with increasing r of the strange star candidate Her

2.17 The plot of Fg, Fh and Fa with respect to the radial coordinate r (km)

for f(R, T ) gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.18 Variations of v2sr with respect radius r (km) of the strange star in

f(R, T ) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.19 Variations of v2st with respect radius r (km) of the strange star in

f(R, T ) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

sr with respect radius r (km) of the strange star in

f(R, T ) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.21 Variations of anisotropic measure ∆ with respect to the radial in f(R, T )

gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.22 Plots of the energy density (km−2) for strange star candidates with

three different models. . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.23 Plots of the radial pressure (km−2). . . . . . . . . . . . . . . . . . . . 64

2.24 Plots of the transverse pressure (km−2). . . . . . . . . . . . . . . . . 64

2.25 Plots of dρ/dr with increasing r for compact stars with three different

models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.26 Plots of dpr/dr versus r. . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.27 Plots of dpt/dr versus r. . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.28 Plot of Fg, Fh and Fa with respect to the radial coordinate r (km) for

2.29 Variations of v2sr with radius r (km) for different models. . . . . . . . 66

2.30 Variations of v2st with radius r (km). . . . . . . . . . . . . . . . . . . 67

sr with radius r (km). . . . . . . . . . . . . . . . 67

2.32 Variations of anisotropic (km−2) measure ∆ with radius r (km) for

3.1 Evaluation of ρ with respect to r, α and λ for m = 0.5 and r0 = 1. . . 74

3.2 Evaluation of ρ + Pr and ρ + Pt with respect to r, α and λ for m =

0.5, r0 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3 Evaluation of ρ + Pr and ρ + Pt with respect to r, α and λ= negative

for m = 0.5, r0 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.4 Evaluation of ρ, Pr and Pt for small region. . . . . . . . . . . . . . . 77

3.5 Evolution of Faf and Fhf versus r. . . . . . . . . . . . . . . . . . . . . 79

3.6 Behavior of β(r) and β(r)/r with respect to r for perfect fluid. . . . . 80

3.7 Isotropic case: Evaluation of β′(r) and β(r) − r. . . . . . . . . . . . . 81

3.8 Behavior of ρ(r) and ρ + P . . . . . . . . . . . . . . . . . . . . . . . . 81

3.9 Evaluation of β(r) and β′(r) through EoS having α = 9 and k = 0.001. 83

3.10 Evaluation of β(r)/r and β(r)−r through EoS with α = 9 and k = 0.001 83

3.11 Evaluation of ρ(r) and ρ + Pt through EoS having α = 9 and k = 0.001 83

3.12 Evaluation of ρ with respect to r, α, γ for m = 0.5, r0 = 1 with small

λ values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.13 Evaluation of ρ + Pr and ρ + Pt with respect to r, α, γ for m = 0.5,

r0 = 1 and small λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.14 Evaluation of ρ + Pr and ρ + Pt with respect to r, α, γ for m = 0.5,

r0 = 1 and very small λ . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.15 Evaluation of ρ , Pr and Pt for small region . . . . . . . . . . . . . . . 89

3.16 Equilibrium conditions with different small λ values. . . . . . . . . . . 91

3.17 Isotropic case for α = −0.18, γ = 0.5, Evaluation of β(r) and β(r)/r . 92

3.18 Isotropic case, α = −0.18, γ = 0.5, evaluation of β′(r) and β(r) − r. . 93

3.19 Evaluation of ρ(r) and ρ + P for α = −0.18, γ = 0.5. . . . . . . . . . 93

3.20 Evaluation of β(r) and β′(r) through EoS having α = −20, γ = 0.8, k =

0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.21 Evaluation of β(r)/r and β(r) − r through EoS with α = −20, γ =

0.8, k = 0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.22 Evaluation of ρ(r) and ρ+Pt through EoS having α = −20, γ = 0.8, k =

0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1 Plot of the dynamical variable YT for the strange star candidate 4U

1820-30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 Behavior of the dynamical variable XT for the strange star candidate

4U 1820-30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.3 Role of the dark dynamical variable XTF on the evolution of the strange

star candidate 4U 1820-30. . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4 Plot for the dark dynamical variable YTF on the evolution of the strange

star candidate 4U 1820-30. . . . . . . . . . . . . . . . . . . . . . . . . 108

List of Tables

2.1 The approximate values of the masses M , radii R, compactness µ, and

the constants A and B for the compact stars, Her X-1, SAXJ 1808.4-

3658, and 4U 1820-30. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 The values of parameters involved in f(R, T ) models for the compact

stars Her X-1, SAXJ 1808.4-3658, and 4U 1820-30. . . . . . . . . . . 50

3.1 Different shape function for different choices of m . . . . . . . . . . . 72

List of Publications

The contents of this thesis are based on the following research papers published in

journals of International repute. These papers are also attached herewith.

1. Yousaf, Z., Sharif, M., Ilyas, M., and Bhatti, M. Z. (2017). Influence of f(R)

models on the existence of anisotropic self-gravitating systems.

European Physical Journal C, 77(10), 691.

2. Ilyas, M., Yousaf, Z., Bhatti, M. Z., and Masud, B. (2017). Existence of

relativistic structures in f(R, T ) gravity.

Astrophysics and Space Science, 362(12), 237.

3. Bhatti, M. Z., Sharif, M., Yousaf, Z., and Ilyas, M. (2018). Role of f(G, T )

gravity on the evolution of relativistic stars.

International Journal of Modern Physics D, 27(04), 1850044.

4. Yousaf, Z., Ilyas, M., and Bhatti, M. Z. (2017). Static spherical wormhole

models in f(R, T ) gravity.

European Physical Journal Plus, 132(6), 268.

5. Yousaf, Z., Ilyas, M., and Bhatti, M. Z. (2017). Influence of modification of

gravity on spherical wormhole models.

Modern Physics Letters A, 32(30), 1750163.

6. Bhatti, M. Z., Yousaf, Z., and Ilyas, M. (2017). Evolution of compact stars

and dark dynamical variables.

Also, the following papers related to this thesis have been accepted or submitted

for publication.

1. Bamba, K., Ilyas, M., Bhatti, M. Z., and Yousaf, Z. (2017). Energy conditions

in modified f(G) gravity.

General Relativity and Gravitation, 49(8), 112.

2. Yousaf, Z., Bhatti, M. Z., and Ilyas, M. (2018). Existence of compact structures

in f(R, T ) gravity.

3. Yousaf, Z., Sharif, M., Ilyas, M., and Bhatti, M. Z. (2018). Energy conditions

in higher derivative f(R, ¤R, T ) gravity.

International Journal of Geometric Methods in Modern Physics, 15(9), 1850146.

4. Ilyas, M. (2018). Charged compact stars in f(G) gravity.

5. Bhatti, M. Z., Ilyas, M. and Yousaf, Z.: Existence of wormhole geometries

through curvature matter coupling,

Submitted for Publication.

Acknowledgements

All praise is due to ALLAH Almighty, who begets no offspring, and has no partner

in His dominion, and has no weakness, and therefore no need of any aid” - and

[thus] extol His limitless greatness. I courteously bow before Him for granting and

bestowing the light of knowledge. He has given me power, energy, strength and vigor,

which I felt in every step, in the way of achieving this goal. All esteem and respect

to Hazrat Muhammad (PBUH) for His tender guidance towards the right path

with the indomitable faith in Allah.

I wish to express my heartiest gratitude and deep sense of obligation to my re-

spected supervisors, Dr. Bilal Masud and Dr. Zeeshan Yousaf for his consistent

encouragement, stimulating suggestions and kind guidance which enabled me to com-

plete this thesis. I could not have imagined having better mentors and advisors for my

educational career. Besides my Supervisors, I would like to thank to Dr. M. Zaeem

ul Haq Bhatti, for their encouragement, insightful comments and hard questions.

This acknowledgement will be incomplete without mentioning my feelings with

tearful eyes for my loving parents who taught me to take the first step, to speak the

first word and inspired me throughout of my life. Dear Ammi g and Abbu g! you

sacrificed your own happiness, just so that I could be happy. It is impossible for me

to achieve such a hard task without your love and priceless prayers for me. I express

my gratitude to all my brothers especially Sher Muhammad (late) and my sisters for

their understanding, support and boundless love for me.

Lahore Muhammad Ilyas

February, 2018

Abstract

This thesis is devoted to analyze the existence as well as the dynamics of some celestial

objects in various well-known modified gravity theories. In this framework, we explore

the evolution of spherical self-gravitating structures in the realm of f(R), f(R, T ) and

f(G, T ) theories, where R, T and G are the Ricci scalar, trace of stress energy tensor

and the Gauss-Bonnet term, respectively. It has been a fascinating challenge to find

the realistic configurations of stellar models in different modified theories gravity.

In order to achieve this goal, we have considered the observational values of three

notable strange compact stars, namely Her X-1, SAX J1808.4-3658, and 4U 1820-

30. The solutions obtained by Krori and Barua are used to examine the nature of

particular compact stars with three different modified gravity models. The behavior

of material variables is analyzed through plots. We also discuss the behavior of

different forces, equation of state parameter, measure of anisotropy and Tolmann-

Oppenheimer-Volkoff equation in the modeling of stellar structures. The comparison

from our graphical representations may provide evidences for the realistic and viable

f(R), f(R, T ) and f(G, T ) gravity models at both theoretical and astrophysical scale.

We also investigate the wormhole solutions with spherically symmetric geome-

try in f(R, T ) gravity. We discuss three different cases for matter contents namely,

anisotropic, barotropic and isotropic fluid configurations. We consider few notable

mathematical formulations of f(R, T ) model to analyze the behavior of energy condi-

tions and to explore the general conditions for wormholes in the framework of modified

theories. It is observed that the usual matter in the throat may satisfy the energy

conditions but the gravitational field emerging from higher order terms of modified

gravity favor the existence of the non-standard geometries of wormholes. We represent

this investigation via plots and examined the equilibrium picture in the background

of anisotropic fluid. The stability and existence of these wormholes is also analyzed

in this theory.

Finally, the role of modified versions of structure scalars are analyzed in the mod-

eling of relativistic spheres in f(R, T ) gravity. We assume that non-static diagonally

symmetric geometry is coupled with dissipative anisotropic viscous fluid distributions

in the presence of f(R, T ) dark source terms. A specific distributions of f(R, T ) cos-

mic model has been assumed and the spherical mass function through generic formula

introduced by Misner-Sharp has been formulated. Some very important relations re-

garding Weyl scalar, matter variables and mass functions are being computed. After

decomposing orthogonally the Riemann tensor, some scalar variables in the pres-

ence of f(R, T ) extra degrees of freedom are calculated. The effects of the modified

structure scalars in the modeling of stars through Weyl, shear and expansion scalar

differential equations are investigated. The energy density irregularity factor has been

calculated for anisotropic radiating spherical stars with varying Ricci scalar correc-

tions.

Abbreviations

In this thesis, we shall use the following list of abbreviations.

BH: Black Hole

CS: Compact Star

DE: Dark Energy

DM: Dark Matter

EoS: Equation of State

GC: Gravitational Collapse

GR: General Relativity

MTGs: Modified Theories of Gravity

NS: Neutron Star

TOV: Tolman-Oppenheimer-Volkoff

WD: White Dwarf

WH: Wormhole

Introduction

The phenomenon of self-gravitation is among the important features of astrophysics

that assures the occurrence of stellar bodies in our cosmos. It is note worthy that

the components of these structures are held together under the action of their own

gravitational pull, thereby making such celestial bodies are self-gravitating in na-

ture. Failure to present enough self-gravitation, all celestial bodies will expand and

completely dissipate. The existence of celestial structures has close connection with

the cosmic structure formation. Therefore the investigation of various cosmological

aspects of celestial bodies is among the core problems of the relativistic astrophysics.

The study of the evolution of celestial bodies requires to consider an observa-

tionally consistent gravitational theory. General theory of relativity (GR) is one of

the greatest achievements of 20th century physicists, proposed by Albert Einstein in

1915. This theory states that gravitational field is due to the warping of space-time,

i.e., the more space-time is warped, the effects of gravity becomes stronger. The ob-

servational ingredients of Λ-cold dark matter model is found to be compatible with

all cosmological outcomes but suffers some discrepancies like cosmic coincidence and

fine-tuning (Weinberg, 1989; Peebles and Ratra, 2003; Husain and Qureshi, 2016).

The accelerated expansion of the universe is strongly manifested after the discovery

of unexpected reduction in the detected energy fluxes coming from cosmic microwave

background radiations, large scale structures, redshift and Supernovae Type Ia sur-

veys (Pietrobon et al., 2006; Giannantonio et al., 2006; Riess et al., 2007). These

observations have referred to dark energy (DE) (an enigmatic force) as reason behind

this interesting and puzzling phenomenon. Various techniques have been proposed

in order to modify Einstein gravity in this directions. Qadir et al. (2017) discussed

various aspects of modified relativistic dynamics and proposed that GR may need to

be modified to resolve various cosmological issues like quantum gravity and the dark

matter problem.

The modified theories of gravity (MTGs) are the generalized models that came

into being by modifying only the gravitational portion of the GR action (for further

reviews on DE and modified gravity, see, for instance, (Capozziello and Faraoni, 2010;

Capozziello and De Laurentis, 2011; Bamba et al., 2012; Koyama, 2016; de la Cruz-

Dombriz and Saez-Gomez, 2012; Bamba et al., 2013; Bamba and Odintsov, 2014;

Yousaf et al., 2016a,b; Nojiri and Odintsov, 2007, 2008a,b; Sotiriou and Faraoni,

2010)). The first theoretical and observationally viable possibility of accelerating

cosmos from f(R) gravity (R is the Ricci scalar) was proposed by Nojiri and Odintsov

(2003). There has been interesting discussion of dark cosmic contents on the structure

formation and the dynamics of various celestial bodies in Einstein-Λ (Yousaf, 2017),

f(R) (Sharif and Yousaf, 2015d; Bhatti and Yousaf, 2017a), f(R, T ) (Harko et al.,

2011; Yousaf and Bhatti, 2016a) (T is the trace of energy momentum tensor) and

f(R, T,RµνTµν) gravity (Odintsov and Saez-Gomez, 2013; Haghani et al., 2013; Ayuso

et al., 2015; Yousaf and Farwa, 2017). Recently, Nojiri et al. (2017) have studied

a variety of cosmic issues, like early-time, late-time cosmic acceleration, bouncing

cosmology. They emphasized that some extended gravity theories such as f(R), f(G)

(where G is the Gauss-Bonnet term) and f(T ) (where T is the torsion scalar) can be

modeled to unveil various interesting cosmic scenarios.

Stellar evolution explains how the celestial bodies change with the passage of

time. The characterization of relativistic matter content, in the formation of celestial

interiors, is based on some physical quantities like, energy density, an/isotropic pres-

sure, dissipation and Weyl tensor. The search for the effects of anisotropicity in the

matter configurations of compact objects is the key to various captivating phenom-

ena, like transitions of phase of different types (Sokolov, 1980), condensation of pions

(Sawyer, 1972), existence of a solid as well as Minkowskian core (Herrera et al., 2008,

2009a) etc. One can write all possible exact solutions of static isotropic relativistic

collapsing cylinder in terms of scalar expressions in GR (Herrera et al., 2012) as well

as in f(R) gravity (Sharif and Yousaf, 2015c; Yousaf and Bhatti, 2016b). Sussman

and Jaime (2017) analyzed a class of irregular spherical solutions in the presence of

a specific traceless anisotropic pressure tensor for the choice of f(R) ∝√

R model.

Shabani and Ziaie (2017) used dynamical and numerical techniques to analyze the

effects of a particular f(R, T ) gravity model on the stability of emergent Einstein

universe. Garattini and Mandanici (2017) examined some stable configurations of

various anisotropic relativistic compact objects and concluded that extra curvature

gravitational terms coming from rainbow’s gravity are likely to support various pat-

terns of compact stars. Sahoo and his collaborators (Sahoo et al., 2017; Sahu et al.,

2017) explored various cosmological aspects in the context of anisotropic relativistic

backgrounds.

Gravitational collapse (GC) is an interesting process due to which the stellar

bodies could gravitate continuously to move towards their central points. In this

regard, the singularity theorem (Hawking and Ellis, 1973) states that during this

implosion process of massive relativistic structures, the spacetime singularities may

appear in the realm of Einstein’s gravity. The investigation of the final stellar phase

has been a source of great interest for many relativistic astrophysicists and gravita-

tional theorists. Oppenheimer and Snyder (1939) did the ground breaking work in

the examination of the GC of non-interacting particles. They did this by assuming

the static forms of Friedmann like and Schwarzschild metrics for interior and exterior

regions, respectively. Afterwards, the analysis of GC was put forward in the context

of Einstein-Λ gravity (Markovic and Shapiro, 2000). Capozziello et al. (2011, 2012)

studied GC of non-interacting particles by evaluating dispersion expressions through

perturbation approach and found some unstable regime of the collapsing object under

certain limits. Cembranos et al. (2012) examined GC of non-static inhomogeneous

gravitational sources and studied the large-scale structure formation at early-time

cosmic in different f(R) gravity theories.

Borisov et al. (2012) used a specific relaxation technique to study various features

of GC in f(R) gravity and found some strange density increment in the matter of

self-gravitating relativistic objects. They also claimed that such a behavior could

be considered as a viable platform for testing f(R) gravity. In the context of f(R)

gravity, various results have been found in literature about collapsing stellar interiors

and black holes (BH) (Olmo, 2007; Briscese and Elizalde, 2008; de La Cruz-Dombriz

et al., 2009; Clifton et al., 2012). Alavirad and Weller (2013) examined the influ-

ences of a logarithmic f(R) model on the evolution and structure formation of the

compact objects and concluded that extra degrees of from due to modified gravity

potentially alter some physical properties of the stellar objects, like red shift. Roshan

and Abbassi (2014) explored various instability modes for the self-gravitating celes-

tial systems by deriving Jeans limits on matter content. Modified gravity theories are

likely to host massive celestial objects with smaller radii as compared to GR (Sharif

and Yousaf, 2016; Yousaf and Farwa, 2017). Guo and Joshi (2016) discussed scalar

GC of spherically symmetric spacetime and inferred that relativistic sphere could give

rise to BH configurations, if the source field is strong.

Resco et al. (2016) investigated the influences of f(R) models on the formations of

static spherically self-gravitating compact objects and found much bigger configura-

tions of compact objects as compared to GR. Zhang et al. (2016) studied dynamical

evolution of the relativistic four-dimensional asymptotically flat spherical geometry

by using procedure initially formulated by Choptuik. They presented some inter-

esting features required to understand many physical aspects of BH formation in

f(R) gravity. Panotopoulos (2017) investigated existence of various compact rela-

tivistic bodies by analyzing mass-radius plots in f(R) gravity and found relatively

more stable distributions of stellar interior. Moustakidis (2017) examined the stabil-

ity of static isotropic spherical compact stars and claimed that by selecting proper

parametric model ranges, one can have more compact configurations of stellar struc-

tures. Sharif and Yousaf (2014a,b, 2015a,b,d) performed a detail analysis to analyze

the role of modified gravitational theories for the structure formation of relativistic

interiors. In this respect, various authors (Herrera et al., 1998; Herrera and Santos,

1997; Bhatti and Yousaf, 2017b) explored the problem of GC by taking some realistic

configurations of matter and geometry.

A wormhole (WH) is a topological feature of spacetime that can fundamentally be

considered as a shortcut for a connection of two separate points in spacetime. Barcelo

and Visser (2000) concluded that positive curvature in the presence of non-minimally

coupled scalar field could cause the system to disobey average null energy condi-

tions (NEC), thereby pointing the possible formation of traversable WHs in nature.

Furey and DeBenedictis (2004) studied the modeling of static spherical WH geome-

tries in MTG supplemented with inverse and squared Ricci curvature corrections.

Balakin et al. (2007) explored some exact analytical model that correspond to new

configurations of WHs by taking into account three-parametric Einstein-Yang-Mills

background.

Yue and Gao (2011) introduced a new class of spherical WHs in Brans-Dicke

gravity and claimed that their WH models are supported by matter that obeyed weak

(WEC). Bertolami and Ferreira (2012) presented WH models that are traversable in

nature in MGT and claimed that their solutions are supported by an unusual form

of redshift function. Azizi (2012) discussed the behaviors of ECs in the context of

WH solutions by considering separable formulation of f(R, T ) gravity, i.e., f(R, T ) =

f1(R) + f2(T ). Bronnikov et al. (2012) performed numerical simulations to check

stability of static spherically symmetric WH solutions supplemented by a minimally

coupled scalar field.

Cataldo and Meza (2013) analyzed spherically symmetric WH solutions threaded

regularly with isotropic and irregularly with anisotropic matter gradient in the phan-

ton evolving background. They have also developed some constraints in terms of mat-

ter variables under which stable distributions of WHs could exist. Kuhfittig (2013)

discussed the existence of WH solutions in Einstein-Maxwell gravity and claimed that

construction of traversable WH model (appropriate for a humanoid traveler) could be

possible through coupling of the geometry with both ordinary and quintessential fluid

distributions. Sharif and Yousaf (2014a) tested the existence of stable WH models

supported by isotropic fluid matter in the presence of EoS and R2 corrections through

numerical technique. Bahamonde et al. (2016) investigated the possible formation of

stable WH models coupled with ideal matter content along with equation of state

(EoS) in MGT. They performed their analysis by matching the interior geometry

with Friedman-Lemaıtre-Robertson-Walker (FLRW) universe during the matter and

radiation dominant regimes. Recently, Moraes et al. (2017) investigated some theoret-

ical predictions of f(R, T ) gravity by considering static wormholes models and have

presented a very clear systematic picture of viable WH models in f(R, T ) gravity.

Many attempts have been made to explore the realistic models of the hypothetical

structure of WHs. The occurrence of WH solutions by including scalar fields (Bar-

celo and Visser, 2000), nonsingular spacetimes (Bambi et al., 2016), quantum effects

(Nojiri et al., 1999a,b), in a semi-classical gravity theory (Sushkov, 1992; Garattini

and Lobo, 2007), in the platform of brane-world (Sushkov, 1992; Garattini and Lobo,

2007), supported by Chaplygin gas with its modified and generalized forms (Anchor-

doqui and Bergliaffa, 2000; Bronnikov and Kim, 2003; Sharif and Yousaf, 2014a; Lobo,

2006), in Gauss-Bonnet theory (Richarte and Simeone, 2007; Kanti et al., 2011), f(T )

gravity (Boehmer et al., 2012), etc. have been discussed in literature with great inter-

est. Kar (1994) analyzed several interesting features of WHs in cosmos and evaluated

some relationship between static WH geometries and non-static Lorentzian WH mod-

els. Popov (2001) analyzed the existence of spherical WH geometries with S2 × R2

topological and gradual varying gravitational environment for both massless and mas-

sive scalar fields and found that existence of WH geometries depends on estimation of

curvature coupling parameter. Armendariz-Picon (2002) characterized exotic matter

with the help a generic formulations of microscopic scalar field Lagrangians. Maeda

and Nozawa (2008) analyzed the effects of cosmological constant on the existence of

static n-dimensional WH solutions in Einstein-Gauss-Bonnet gravity.

Lobo and Oliveira (2009) investigated the role of threaded exotic matter on the

geometric form of WHs in modified gravity and found that the extra curvature f(R)

quantities present in the effective energy-momentum tensor are responsible for sus-

taining non-standard WH geometries in nature. Garcia and Lobo (2011) developed

some exact WH models in the realm of non-minimal matter-curvature coupling and

concluded that non-minimal coupling could help to minimize the violation of NEC

of the usual WH throat matter content. Daouda et al. (2011) explored spherically

symmetric WH solutions coupled with anisotropic exotic matter content in the realm

of f(T ) gravity and claimed that theoretical occurrence of WH geometries could be

possible when the radial component of pressure is proportional to a real constant

value of torsion scalar. Boehmer et al. (2012) concluded that WH models obeying

energy conditions (ECs) at its throat are possible with particular choices of shape,

redshift and f(T ) functions in modified gravity. Jamil et al. (2013) evaluated new

class of WH models in f(T ) gravity and claimed that exotic matter within WH throat

obey NEC, when its geometry is supported by isotropic pressure and barotropic state

equation.

The concept of ECs could be considered as viable approach for the better un-

derstanding of the well-known singularity theorem. Santos et al. (2007) developed

viability bounds coming from ECs on generic f(R) formalism. Their approach could

be considered to constrain various possible f(R) gravity models with proper physical

backgrounds. Wang et al. (2010) evaluated some generic expressions for ECs in the

f(R) gravity and employed them on a class of cosmological model to obtain some

viability constraints.

Shiravand et al. (2018) evaluated ECs for f(R) gravity and obtained some stability

constraints against Dolgov-Kawasaki instability. They found special ranges of some

f(R) model parameters under which the theory would satisfy WECs. The investiga-

tion of ECs in modified theories has been carried out under a variety of cosmological

issues like, f(R) gravity (Santos et al., 2010), f(R,Lm) gravity (Wang and Liao,

2012), f(R, T ) gravity (Yousaf et al., 2017b), f(R,G) gravity (Atazadeh and Darabi,

2014), f(G) gravity (Garcıa et al., 2011; Nojiri et al., 2008). The stability of compact

objects along with their ECs have been analyzed in detail by various researchers (Shee

et al., 2017; Maurya and Govender, 2017).

The inhomogeneous state is found to be the predecessor in the process of GC for

the initially homogenous stellar structures. It is pertinent to mention that one can

understand some dynamical properties of self-gravitating systems through investigat-

ing the behavior of pressure anisotropy, tidal forces, inhomogeneous energy density

(IED), etc. There has been extensive work related to check the cause of IED over

the surface of regular compact objects. The work of Penrose and Hawking (1979)

is among pioneers works in this direction. They found Weyl tensor as a key figure

in the emergence of IED in the evolution of spherically symmetric objects. Herrera

et al. (1998) calculated some factors responsible for creating IED over the anisotropic

stellar spheres and inferred that pressure anisotropy may lead the system to develop

naked singularity. Virbhadra et al. (1998); Virbhadra (2009) provided a mathemati-

cal platform under which one can differentiate between the formation of neutron stars

(NS) and BHs.

Herrera et al. (2004) described gravitational arrow of time for the dissipative com-

pact systems by making a relation among Weyl invariant, pressure anisotropy and

IED. Herrera et al. (2011a) examined the influences of IED on the expressions of

shear and expansion evolutions in the presence of electromagnetic field. Yousaf et al.

(2016a) covered this problem for spherical radiating geometries in modified gravita-

tional theory and concluded that a special combination of f(R, T ) gravity model could

significantly interfere in the appearance of IED. Bhatti and his colleagues (Bhatti

and Yousaf, 2016, 2017a) looked into the reasons behind the maintenance of IED

against gravitational collapse of relativistic interiors in modified gravity. Herrera

et al. (2011b) and Herrera (2017a) considered the case of non-comoving coordinate

system and checked the reasons for the start up of the spherical collapse by evalu-

ating transport equations. Yousaf et al. (2017a) modified these results by invoking

Palatini f(R) corrections. Recently, Herrera (2017b) illustrated the answer to the

question that why observations of tilted congruences notice dissipative process in

stellar interiors which seem to be isentropic for non-tilted observers.

This thesis is devoted to study the some dynamical aspects of relativistic structures

(such as WH, compact stars (CS)) in various MTGs. In this context, we formulate

a set of governing equations by considering some-well known MTG models. The

existence of self-gravitating CS and WHs is explored. Spherically symmetric viscous

dissipative stars are investigated through the set of modified structure scalars. The

thesis is outlined as follows.

• Chapter One presents an overview of basic definitions and concepts related to

this thesis.

• Chapter Two deals the existence of spherically symmetric compact stars in

some MTGs models. The dynamics of celestial bodies is discussed by using a

set of governing equations. The corresponding behavior of stellar interiors by

drawing plots are also being checked.

• Chapter Three investigates static forms of spherically symmetric an/isotropic

WH geometries in MTGs. By considering few notable modified gravity models,

the viable regions of spacetime for the stability of such hypothetical structures

are explored through energy conditions. The corresponding graphs are also

being drawn.

• In chapter Four, we analyze the non-static spherical anisotropic sources in

MTGs. We evaluate evolution equations, structure scalars and IED factors.

We also calculate Raychaudhuri and shear evolution expressions in terms of

structure scalars in the presence of modified gravity corrections.

Chapter 1

Cosmological Aspects of CelestialObjects and Modified GravityTheories

In 1916, some of the astronomers estimated the energy density of different binary stars

and found that some of the stars having a very high density of almost 25000 times

greater than the sun density. He thought something was wrong with his assumptions

but today we know that the star like our sun is balanced by the inward gravitational

forces and the pushing outwards fusion process. In the fusion process, the massive

amount of gases ejected out away while the center (core) contracts and leave a stellar

remnant, by which these stars get higher densities and more compact.

In this chapter, we provide some basic ingredients required to understand this thesis.

In particular, we shall describe physical aspects as well as the mathematical formalism

of various compact geometries. We shall also discuss some cosmological aspects of

wormhole through energy conditions. In this respects, the contribution of the equation

of state, energy conditions will be described to perform the stability analysis. At last,

various candidates for modified theories will be discussed.

1.1 Compact Objects

A normal stellar evolution leaves off a compact object. When a star dies, most of its

nuclear fuel at core becomes used up. Then it becomes a compact object or a CS,

thereby producing WD, NS or supermassive BH. All these three types of compact

object are different (from a normal stars) in two distinguished ways.

The first reason is as these objects do not burn their nuclear fuel in a core and

cannot support themselves against the gravitational collapse (by generating thermal

pressure). Due to this, these structures become more self-gravitating in nature. Apart

from this, WD is supported by the pressure of some high-density electrons (degenerate

electrons), while NS is corroborated highly by the pressure of degenerate neutrons.

Further, BHs are the utterly collapsed stars, which could not find any means from

holding the inward pull of gravity (becomes more self-gravitating), therefore these

could collapse to produce singularities.

The second reason which makes these objects distinguished from normal stars

is that their small size with respect to their quantity of matter. These objects are

considered to have relatively smaller radii, thus producing strong gravitational field.

The density range spanned by these objects is enormous. So, their study requires

a very intense physical understanding of the composition of matter with the key role

of all four fundamental interactions which are strong, weak, electromagnetic, and

gravitation.

1.1.1 White Dwarf Stars

These stellar structures are have mass of about one solar mass (M⊙) with radii of

about 5 × 103km. The mean densities of these stars are roughly around 106gcm−3.

The white dwarf stars are no longer to burn their nuclear fuel but they could start

to cool slowly as they emit their residual thermal energy. The name “white” comes

because of its white color (few discover) and characterized by a very low luminosity.

These are believed to be specified from the light stars which have the masses ranging

M < 4M⊙ while there is bound a on the mass of WDs which should be less than

1.4M⊙ .

1.1.2 Neutron Stars

Neutron stars are the remnants of the supernova which are the most interesting as

well as energetic events in the history of the universe after the Big Bang. These are

extremely dense stars with typically the masses of M = 1.4M⊙ and radius merely

10km. The NS is also known as magnetars which can harbor the strong magnetic fields

(about 1013 times stronger than that on the earth). Moreover, some other NS, known

as the pulsar, can spin as fast as almost a dozen millisecond as a period of revolution.

In the interior regions of NS, the gravity is so high and the matter is so dense (at the

core) which we cannot produce in our laboratories. In fact, one of the main reasons

behind this is supposed to be the existence of the equation of states. We do not

know the exact description of matter at the core which affects the bulk properties

of NS, like mass and radius. In order to study various aspects of gravitational fields

produced by such structures, one may need to use an observationally consistent field

theory.

1.1.3 Black Holes

One of the most interesting outcomes of the relativistic astrophysicists is the predic-

tion of compact objects whose gravitational field is so high that even light cannot

escape from its field. Such compact objects deform their corresponding spacetimes.

These objects are named as BHs which are characterized by the presence of an event

horizon (no return boundary in spacetime). Michell and Laplace (in the 18th century)

suggested the existence of such objects whose gravitational fields are very high. Later

on, in 1926, Schwarzschild found the first mathematical solution that has character-

ized a BH. Finkelstein interpreted such stellar bodies as a region of spacetime from

which nothing can escape. Chandrasekhar (1964) termed BH as the most suitable

toy model in our cosmos to understand the geometric properties of a spacetime. The

uniqueness theorems of relativity states that the most of BH solutions with vacuum

background could be described by their masses and angular momentum. Kerr found

the solution which is expecting to describe the astrophysical BHs like Sagittarius A*,

at the center of our own galaxy (for details please see Blandford and Znajek (1977)).

1.2 Mathematical Formalism Behind Compact Stars

In this section, we will summarize some mathematical backgrounds for the above

mentioned compact stars.

1.2.1 Structure

In the modeling of stellar structures, it could be possible to say that there are two

forces acting on a star. The first one is the gravitational force, while the second one

is the thermal or pressure force. The pressure P and force F can be related as

dP =dF

4πr2, (1.2.1)

where 4πr2 is the area. The gravitational force for a spherical symmetric system is

dFg = −Gm(r)dm

r2, (1.2.2)

where r is the radial distance and for varying density, we can write

dm = 4πρ(r)r2dr. (1.2.3)

Using these two equations with F ≡ Fg, we get the coupled equations as

dr= −Gm(r)ρ(r)

dr= 4πρ(r)r2.

(1.2.4)

These are the coupled differential equations for which one should take the initial

conditions as m(0) = 0 with some initial central pressure, P (0) = Pc. As dmdr

positive while dpdr

is negative, so we have some maximum radius R for which P (R) = 0

and m(R) = M .

1.2.2 Equation of State

In the above subsection, we reach at the two coupled mass and pressure stellar dif-

ferential equations with the condition that both of these equations are depending on

the variable ρ. So, one needs to establish the relation between pressure and density.

This relation is called the equation of state.

P = f(ρ) (1.2.5)

The EoS can play an important key role in the compactness of a star. There are

different types of EoS, e.g., the matter in the core of a WD star can be treated as an

ideal Fermi gas of degenerate electrons, while for electrically neutral matter, one need

to add protons to these degenerate electrons. After some straightforward calculations,

the EoS (1.2.5) becomes

P = Kργ (1.2.6)

This is called polytropic EoS, which for non-relativistic case provides γ = 5/3 and

K = Knon−relativistic, while for relativistic case γ = 4/3 and K = Krelativistic.

1.2.3 Chandrasekhar Mass Limit

The maximum mass for WDs for polytropic EoS was found by Chandrasekhar in

1931, while in 1932, Landau found the limit mass for WDs and NS with some simple

arguments. The two coupled equations (1.2.4) can be written as

)= −4πGρ, (1.2.7)

which after some calculations, provides

M = 1.4312

M¯, (1.2.8)

where η is the ratio of atomic mass to its atomic number, i.e., η = A/Z. The WDs

mostly contain carbon 12C, so it could be better to take its atomic mass that gives

the maximum mass of M = 1.4559M¯. This is the famous Chandrasekhar mass limit

for WDs.

1.2.4 GR Correction

If a star is more compact, one may need to consider the effects of GR. These effects

are important, if 2GMc2R

→ 1. For a compact star in the regime of GR, the two coupled

equations become

dr= −Gm(r)ρ(r)

c2ρ(r)

4πr3P (r)

c2m(r)

)(1 − Gm(r)

(1.2.9)

This is the famous Tolman-Oppenheimer-Volkoff (TOV) equation. One can easily

derive this equation by solving the corresponding field equations in the background

of the static, isotropic and ideal fluid spheres at hydrostatic equilibrium.

1.3 Wormholes

Wormholes are hypothetical tunnels or bridges that connect two different regions of

spacetime. These regions of spacetime can be either between two universes or may

be between two different regions of the same universe. A special type of WH through

which an observer can travel easily (theoretically) in either direction is known as

traversable WH. For the first time, Flamm (1916) studied the two-dimensional em-

bedding diagram of Schwarzschild WH, while Einstein and Rosen (1937) introduced

the non-traversable Schwarzschild WH solutions. The interest in the study of WHs

is recently developed by the influential work of Morris and Thorne who investigated

the traversable WH and found that for traversability, a negative energy which is also

called an exotic matter (that violates the NEC) is required. Another major problem

is related to the stability analysis of WHs.

1.4 Some Cosmological Aspects

1.4.1 Matter Distributions

A substance primarily consists of three states, i.e., solid, liquid and gas. The liquid

and gas states of a substance are referred to as fluids. Fluid is a substance which

can move easily and deforms under the effect of shear stress or, in other words, it

is the substance which can flow or float. The two states of fluid, i.e., liquid and gas

are different in the sense that it is not easy to compress the liquids and they contain

fixed volume, while gases can be easily compressed and do not have the fixed volume.

There are two different parts of the Einstein field equation; one is the geometrical

part while the other is the matter part that contains stress-energy momentum tensor.

It is given by

Tαβ =−2√−g

δ (Lmatter

√−g)

δgαβ= −2

δLmatter

δgαβ+ gαβLmatter. (1.4.1)

Now, we discuss some particular types of fluids with the help of stress-energy mo-

mentum tensor.

1.4.2 Electro-Magnetic distribution

The lagrangian density for the electromagnetic field can be written as

Lem = −1

4FαβFαβ, (1.4.2)

where Fαβ is the field strength tensor or Maxwell field tensor. It is a covariant tensor

of rank two and can be expressed in terms of four potential φα as

Fαβ = φβ,α − φα,β, (1.4.3)

which is an anti-symmetric tensor. The presence of charged medium in the fluids can

be described through its stress-energy tensor as

T (em)

αβ = F µα Fµβ − 1

4gαβFµνF

µν . (1.4.4)

1.4.3 Perfect Fluid

As the fluid (or real fluid) are so sticky and conduct heat, so the idealized models

in which these factors are neglected is called a perfect fluid. The perfect fluid is

completely characterized by its rest mass density ρm and isotropic pressure P . The

perfect fluids have no shear stresses, heat conduction or any kind of viscosity. The

stress energy-momentum tensor for perfect fluid can be written as

Tαβ = (ρ + P ) uαuβ + Pgαβ, (1.4.5)

where ρ is the fluid energy density, while uα is the fluid’s four velocity.

1.4.4 Anisotropic Fluid

The fluid which has the property of being directionally depended pressure is called

anisotropic fluid and the relativistic source to be anisotropic matter distribution with

the following mathematical formulation

Tµν = (ρ + Pt)VµVν − Ptgµν + ΠXµXν , (1.4.6)

where Xµ id the four vector, Pt, Pr, and Π are tangential, radial pressure components

and Pr − Pt, respectively.

1.5 Energy Conditions

Any spacetime satisfying the field equations is a solution of these equations even

if no restriction is imposed in the usual energy-momentum tensor, Tαβ. However, in

general, such an energy-momentum tensor may not represent any known matter field.

If this energy-momentum tensor also represents realistic fluid, the spacetime is said

to be an exact solution of the field equations, which is a vacuum solution, if Tαβ = 0.

Now, for the energy-momentum tensor to represent some known matter fields, it

should satisfy certain conditions called the ECs. These conditions are coordinate

invariants (independent of symmetry) restrictions on the energy-momentum tensor.

These conditions limit the arbitrariness of Tαβ and if they are satisfied, then Tαβ

represents realistic sources of energy and momentum. The Raychaudhuri equations

have the key role in finding the mathematical expression of these conditions.

dτ= −1

3θ2 + ωλγω

λγ − σλγσλγ − RλγU

λUγ, (1.5.1)

dτ= −1

2θ2 + ωλγω

λγ − σλγσλγ − Rλγk

λkγ, (1.5.2)

where θ is the trace of expansion tensor θαβ and is called the scalar expansion and

ωλγ, σλγ, kλ, Uλ denote the rotation, shear tensor, null and timelike tangent vectors

in the congruences, respectively. By neglecting the higher terms of rotations and

distortions, Eqs.(1.5.1) and (1.5.2) yield

θ = −τRλγUλUγ, θ = −τRλγk

λkγ.

As the gravity is attractive by nature so θ < 0 which implies RλγUλUγ ≥ 0 and

Rλγkλkγ ≥ 0. Generalizing these constraint in the framework of field equation, we

get (Tλγ −

2gλγT

)uλuγ ≥ 0,

(Tλγ −

2gλγT

)kλkγ ≥ 0.

Using these inequalities, some of the ECs are

1.5.1 Null Energy Conditions

We consider an energy-momentum tensor having the form

Tαβ =

ρ 0 0 0

0 p1 0 0

0 0 p2 0

0 0 0 p3

where ρ is the energy density of the fluid and p1, p2, p3 are pressure components

along the three spatial directions. The NEC states that for any null vector kα

Tαβkαkβ ≥ 0

for which the above energy-momentum tensor takes

ρ + pj ≥ 0, ∀j.

The NEC says that the pressure should not be too large as compared to the density.

1.5.2 Weak Energy Conditions

For the WEC, the following condition must be satisfied for all time-like vectors Uα

TαβUαUβ ≥ 0

which implies that

ρ ≥ 0, ρ + pj ≥ 0, ∀j.

The demonstrates that the energy density should be non-negative in addition to NEC.

1.5.3 Dominant Energy Conditions

The dominant energy condition states that the WEC holds and for all time-like vec-

tors, TαβUα is a non-space-like vector, i.e.,

TαβUαUβ ≥ 0, TαβT βλU

αUλ ≥ 0,

which yields

ρ ≥ 0, ρ + pj ≥ 0, ∀j.

This condition implies that no signal can move faster than light.

1.5.4 Strong Energy Conditions

The strong energy condition requires that for all time-like vectors,

TαβUαUβ − 1

2T ≥ 0,

which translates into the conditions

ρ + pj ≥ 0, ρ +∑

pj ≥ 0, ∀j.

The strong energy condition implies that the gravity has an attractive nature. One

can observe that the WEC is the weakest of all these conditions and its violation

signals the violation of all other ECs.

1.6 Equation of State Parameter

The EoS parameter is the dimensionless term that provides the matter state under

some specific physical grounds. The value of this parameter belongs to an open

interval (0, 1) for P = f(ρ) or P = ωρ. In that case, it represents radiation dominated

cosmic era. This EoS for the anisotropic relativistic interior can be defined as

Pr = ωrρ, Pt = ωtρ. (1.6.1)

The values of EoS parameter could describe different stellar as well as cosmic scenar-

ios. In the following, we shall discuss few cases.

• Cold Dust

The zero choice of the EoS parameter, i.e., ω = 0, describes the background of

the cold dust.

• Ultra-Relativistic Matter

The choice ω = 1/3 indicates the ultra-relativistic state of the matter (e.g.

radiations).

• Acceleration of cosmic inflation

The cosmic inflation and the accelerating nature of the universe can be charac-

terized through DE induced by the specific choice of the EoS parameter. For

instance, the EoS of the cosmological constant is ω = −1. More generally, for

any EoS ω < −1/3, the expansion of universe is accelerating and this acceler-

ated expansion was indeed observed. Hypothetical phantom energy can also be

observed by assuming ω < −1 and this would cause a Big Rip. Using the ex-

isting observed data, it is still difficult to distinguish between phantom ω < −1

and non-phantom ω ≥ −1 eras.

• Fluids

It has been analyzed in literature that fluids having enough large EoS parameter

disappear more quickly than those having relatively smaller ω. The origin

of the flatness and monopole problems of the big bang is mediated by the

curvatures with ω = −1/3 and ω = 0, respectively. If they were located at

the time of the early big bang, then they should be still visible at the current

time. These problems are solved by introducing the cosmic inflation which

has ω ≈ −1. Thus, the measuring the DE EoS is one of the greatest efforts

of observational cosmology. By accurately measuring ω, it is hoped that the

nature of cosmological constant could be different than that of quintessence

which has ω 6= −1.

1.7 Equilibrium Condition

We consider a general form of the static spherically symmetric line element as

ds2 = eadt2 − ebdr2 − r2(dθ2 + r2sin2θdφ2), (1.7.1)

where a and b are radial dependent metric coefficients and the TOV equation for the

spherical anisotropic stellar interior is given by

a′(ρ + pr)

2(pr − pt)

r= 0. (1.7.2)

The quantity a′ is the radial derivative of the function appearing in the first metric

coefficient of the line element (1.7.1). However, the quantity a, in general, directly

corresponds to the scalar associated with the four acceleration (aβ = aVβ) of the

anisotropic fluid and is defined as

a2 = aβaβ

Equation (1.7.2) can be relabeled in terms of gravitational (Fg), hydrostatic (Fh) and

anisotropic (Fa) forces as

Fg + Fh + Fa = 0. (1.7.3)

The values of these forces for our anisotropic spherical matter distribution have been

found as follows

Fg = −a′

2(ρ + pr), Fh = −dpr

dr, Fa =

2(pr − pt)

r, (1.7.4)

By making use of these definitions, the behavior of these interactions can be studied

for any CS objects.

1.8 Stability Analysis

Here, we check the stability of our stars by adopting the scheme presented by Herrera

(1992) that was based on the concept of cracking (or overturning). This approach

states that v2sr as well as v2

st must belong to the closed interval [0,1], where vsr indicates

radial sound speed, while and vst denotes transverse sound speed. These are defined

dρ= v2

sr,dpt

dρ= v2

st. (1.8.1)

The system will be dynamically stable, if v2sr > v2

st. The evolution of the radial

and transversal sound speeds for any compact stars should be within the bounds of

stability. This constraint can also be written as

0 < |v2st − v2

sr| < 1. (1.8.2)

1.9 Different Modified Gravity Theories

Here, we present overview of some MTGs that can be used to discuss stellar/cosmological

dynamics such as inflation as well as late-time cosmic expansion.

1.9.1 f(R) Gravity

The standard Einstein-Hilbert action in f(R) gravity can be modified as follows

Sf(R) =1

∫d4x

√−gf(R) + SM , (1.9.1)

where g, κ, SM stand for the determinant of the tensor, the coupling constant and

matter field action. The basic motivation of this theory is to introduce generic al-

gebraic expression of the Ricci scalar rather than cosmological constant in the GR

action. By varying the above equation with respect to gµν , the field equations for

f(R) gravity can be found as

RαβfR − 1

2f(R)gαβ + (gαβ¤ −∇α∇β) fR = κTαβ, (1.9.2)

where Tαβ is the standard energy-momentum tensors, while ∇β is an operator of

covariant derivative, ¤ ≡ ∇β∇β and fR ≡ df/dR. The quantity fR comprises of sec-

ond corresponding derivatives of the metric variables, often termed as scalaron which

propagates new scalar freedom degrees. The trace of Eq.(1.9.2) specifies scalaron

equation of motion as under

¤fR +R

3(2f + R + κT ) . (1.9.3)

The Ricci scalar in terms of cosmological constant can be found from the above

equation by considering limits, fR → 0 along with f(R) → Λ. Equation (1.9.2) can

be remanipulated as

Gαβ =κ

Tαβ + Tαβ) ≡ T effαβ, (1.9.4)

where Gαβ is an Einstein tensor and(D)

Tαβ is termed as effective form of the energy-

momentum tensor in metric f(R) gravity. Its expression is given by

Tαβ =1

{∇α∇βfR − ¤fRgαβ + (f − RfR)

1.9.2 f(R, T ) Gravity

The standard Einstein-Hilbert action for f(R, T ) gravity can be written as follows

(Harko et al., 2011)

∫dx4

√−g [f(R, T ) + Lm], (1.9.5)

where Lm indicates Lagrangian for the usual matter content. Upon varying above

action with respect to gµν , one can get the following equation of motion (Harko et al.,

RµνfR(R, T ) − 1

2gµνf(R, T ) + (gµν∇α∇α −∇µ∇ν) fR(R, T )

= Tµν − fT (R, T )Θµν − fT (R, T )Tµν , (1.9.6)

where fT (R, T ) stands ∂f(R,T )∂T

operator. Here, we have used relativistic units, i.e.,

c = 1 and 8πG=1, where c and G are the light speed and the Newton’s gravitational

constant. The quantity Θµν can be defined though usual energy momentum tensor

Θµν =gαβδTαβ

δgµν= −2Tµν + gµνLm − 2gαβ ∂2Lm

∂gµν∂gαβ, (1.9.7)

Our aim is to analyze the role of anisotropic pressure on the existence and modeling

of WH structures in this gravity. Due to this reason, we assume locally anisotropic

gravitational source whose mathematical form is given by

Tµν = (ρ + Pr)VµVν − Ptgµν + ΠXµXν , (1.9.8)

where Pt, ρ, Pr and Π indicate tangential component of the fluid pressure, fluid

energy density, radial pressure component and Pr − Pt, respectively. The vectors Vµ

and Xµ are four velocity and four vector of the fluid. Under comoving coordinate

system, these satisfy, V µVµ = 1 and XµXµ = −1, relations. Now, we take Lm = ρ

and then Eq.(1.9.7) provides

Θµν = −2Tµν + ρgµν .

The f(R, T ) field equation (1.9.6) can be manipulated as

Rµν −1

2Rgµν = T eff

µν , (1.9.9)

where T effµν include gravitational contribution due to f(R, T ) gravity and is called

effective energy momentum tensor in f(R, T ) theory. Its form is

T effµν =

fR(R, T )[(1 + fT (R, T )) Tµν − ρgµνfR(R, T ) +

2(f(R, T )

− RfR(R, T ))gµν + (∇µ∇ν − gµν∇α∇α) fR(R, T )] . (1.9.10)

1.9.3 f(G) Gravity

The standard Einstein-Hilbert action for f(G) gravity is generalized as follows

∫d4x

√−g

2+ f(G)

(gλγ, ψ

), (1.9.11)

where f, SM(gλγ, ψ) are the arbitrary function of GB invariant and the matter action,

respectively. The GB scalar (G) can be written as follows

G = R − 4RλγRλγ + RλγαβRλγαβ, (1.9.12)

where Rλγ is the Ricci tensor and Rλγαβ is the Riemannian tensor. Upon varying the

above action with respect to gλγ, we get the modified field equations for f(G) gravity

Rλγ −1

2Rgλγ = T eff

λγ , (1.9.13)

where T effλγ is dubbed as effective energy momentum tensor for f(G) gravity. Its

expression can be given as

T effλγ = κ2Tλγ − 8 [Rλργσ + Rλγgλγ − Rλγgλγ − Rλγgλγ + Rλγgλγ

2(gλγgλγ − gλγ gλγ)]∇ρ∇σfG + (GfG − f) gλγ, (1.9.14)

where subscript G defines the derivation of the corresponding term with the GB term.

1.9.4 f(G, T ) Gravity

The action for f(G, T ) gravity can be written as

∫d4x

√−g[R + f(G, T )] +

∫d4x

√−gLm, (1.9.15)

where f depends upon the functions G and T . Here T is the trace of the usual

energy-momentum tensor which is defined through Lm as

Tλγ = − 2√−g

δ(√−gLm)

δgλγ. (1.9.16)

Equation (1.9.16) can be written alternatively as

Tλγ = gλγLm − 2∂Lm

∂gλγ. (1.9.17)

The variation in the the above action (1.9.15) yields

0 = δS =1

∫d4x[(R + f(G, T ))δ

√−g +

√−g(δR + fG(G, T )δG

+ fT (G, T )δT )] +

∫d4xδ(

√−gLm). (1.9.18)

We wish to calculate the variations in the quantities√−g, Rν

αβη, Rαη and R. These

are found respectively as follows

δ√−g = −1

√−ggγβδgγβ, (1.9.19)

δRναβη = ∇β(δΓν

ηα) −∇η(δΓνβα),

= (gαλ∇[η∇β] + gλ[β∇η]∇α)δgνλ + ∇[η∇νδgβ]α, (1.9.20)

δRαη = δRνανη, δR = (Rαβ + gαβ∇2 −∇α∇β)δgαβ, (1.9.21)

where Γναβ describes the Christoffel symbol. Furthermore, the variation of G and T

δG = 2RδR − 4δ(RλγRλγ) + δ(RλγνηR

λγνη), (1.9.22)

δT = (Tλγ + Θλγ)δgλγ, Θλγ = gνη δTνη

δgλγ

. (1.9.23)

By using these variational relations (1.9.19)-(1.9.23) in Eq.(1.9.18), we get the field

equations for f(G, T ) gravity as follows

Rλγ −1

2gλγR = κ2Tλγ − (Tλγ + Θλγ)fT (G, T ) +

2gλγf(G, T ) − (2RRλγ − 4Rξ

λRξγ

− 4RλξγηRξη + 2Rξηδ

λ Rγξηδ)fG(G, T ) − (2Rgλγ∇2 − 2R∇λ∇γ

− 4gλγRξη∇ξ∇η − 4Rλγ∇2 + 4Rξ

λ∇γ∇ξ + 4Rξγ∇λ∇ξ

+ 4Rλξγη∇ξ∇η)fG(G, T ). (1.9.24)

1.9.5 f(R, ¤R, T ) Gravity

The Einstein-Hilbert action for f(R, ¤R, T ) gravity can be casted as

∫d4x

√−gf(R, ¤R, T ) + SM (gµν , ψ), (1.9.25)

By giving variations in the above equation with respect to metric tensor, we have

δS =1

∫d4x[fδ

√−g+

√−g(fRδR+f¤Rδ¤R+fT δT )+2κ2δ(

√−gLM)], (1.9.26)

Using the values of δR, δ¤R and δT and δ√−g in the above equation, we obtain an

equation, which after some manipulations, provides

δS =1

∫d4x

√−ggαβδgαβf +

√−g(Tαβ + Θαβ)fT δgαβ + fR

√−g(Rαβ

+gαβ¤ −∇α∇β)δgαβ +√−gf¤R(∇α∇βR + ¤Rαβ + gαβ¤2 + Rαβ¤ − ¤

×∇α∇β −∇αR∇β + 2gµν∇µRαβ∇ν)δgαβ + 2κ2 δ(

√−gLM)

δgαβδgαβ

], (1.9.27)

where subscript ¤R indicates the derivative of the corresponding quantities with

respect to ¤R. Equation (1.9.27) after simplifications gives rise to

fRRαβ + (gαβ¤ −∇α∇β) fR − 1

2gαβf +

(2f¤R(∇(α∇β)) R −¤Rαβ)

−{Rαβ¤ − ¤∇α∇β + gαβ¤2 −∇αR∇β + 2gµν∇µRαβ∇ν

= κ2Tαβ − fT (Tαβ + Θαβ).

(1.9.28)

This is the required equation of motion for the f(R, ¤R, T ) gravity.

Chapter 2

Compact Stars

This chapter aims to explore some realistic configurations of anisotropic spherical

structures in the background of MTG. The physical characteristics of some anisotropic

compact stars are explored in different MTG and a comprehensive study is performed

with a set of solutions describing the interior of the compact object. The obtained

solutions can be referred to the modeling of compact celestial geometries. The so-

lutions obtained by Krori and Barua are used to examine the nature of particular

compact stars with three different modified gravity models. The behavior of the ma-

terial variables are analyzed through plots. We also discuss the behavior of different

forces, EoS parameter, measure of anisotropy and TOV equation in the modeling of

stellar structures. The comparison from our graphical representations may provide

evidence for the realistic and viable modified gravity models at both theoretical and

astrophysical scale. The results of this chapter have been published in the form of

three research papers (Yousaf et al., 2017c; Ilyas et al., 2017; Bhatti et al., 2018).

2.1 Anisotropic Relativistic Spheres in f (R) Grav-

In this section, we will take the f(R) modified gravity and consider a general form of

a static spherically symmetric line element as mention in Eq.(1.7.1). We assume that

our spherical self-gravitating system is filled with locally anisotropic relativistic fluid

distributions. The energy-momentum tensor of this matter is given in Eq.(1.4.6). The

four-vectors Vµ and Xµ, under non-tilted coordinate frame, obey relations VµVµ = 1

and XµXµ = −1. The f(R) field equations (1.9.4) for the metric (1.7.1) and fluid

(1.4.6) can be given as

ρ =e−b

2r2(r2b′fR

′ + 2fRrb′ + f(−r2eb) + fRr2ebR + 2fReb − 2r2fR′′

− 4rfR′ − 2fR), (2.1.1)

pr = −e−b

2r2(−2fR + 2ebfR − ebfr2 + ebfRr2R − 2fRra′ − 4rfR

′ − r2a′fR′), (2.1.2)

pt =e−b

4r(2fRra′′ − fRra′b′ + 2ra′fR

′ + fRra′2 + 2fRa′ − 2rb′fR′ − 2fRb′

+ 2rfeb − 2rfRebR + 4rfR′′ + 4fR

′), (2.1.3)

where prime stands for radial partial derivations. The corresponding Ricci scalar is

given by

R =e−b

[4 − 4eb + r2a′2 − 4rb′ + ra′ (4 − rb′) + 2r2a′′] . (2.1.4)

In order to achieve some realistic study for the modeling of anisotropic compact stellar

structure, we use a specific combination of metric variables, i.e., a(r) = Br2 + C and

b(r) = Ar2 suggested by Krori and Barua (1975). Here, A, B and C are constants

and can be found by imposing some viable physical grounds. Making use of these

expressions, the f(R) field equations (2.1.1)-(2.1.3)-can be recasted as

2r2e−r2A(−2fR + 2er2AfR − er2Afr2 + 4fRr2A + er2AfRr2R + 2fRr3A′ − 4rfR

+ 2r3AfR′ + r4A′fR

′ − 2r2fR′′), (2.1.5)

pr = − 1

2r2e−r2A(−2fR + 2er2AfR − er2Afr2 − 4fRr2B + er2AfRr2R − 2fRr3B′

− 4rfR′ − 2r3BfR

′ − r4B′fR′), (2.1.6)

4re−r2A(2er2Afr − 4fRrA + 8fRrB − 4fRr3AB + 4fRr3B2 − 2er2AfRrR − 2fRr2

× A′ − 2fRr4BA′ + 10fRr2B′ − 2fRr4AB′ + 4fRr4BB′ − fRr5A′B′ + fRr5B′2 + 4fR′

− 4r2AfR′ + 4r2BfR

′ − 2r3A′fR′ + 2r3B′fR

′ + 2fRr3B′′ + 4rfR′′). (2.1.7)

Now, we consider a three-dimensional hypersurface, Σ that has differentiated our

system into interior and exterior regions. The spacetime for the description of exterior

geometry is given by the following vacuum solution

(1 − 2M

)dt2 −

(1 − 2M

dr2 − r2(dθ2 + sinθ2dϕ2

), (2.1.8)

where M is the gravitating mass of the black hole. The continuity of the structural

variables, i.e., gii, i = 1, 2 and the derivative ∂gtt

∂rover the hypersurface, i.e., r = R,

provide some equations. On solving these equations simultaneously, one can obtain

A =−1

(1 − 2M

), B =

(1 − 2M

, (2.1.9)

C = ln

(1 − 2M

)− M

(1 − 2M

. (2.1.10)

After selecting some particular values of M and R, the corresponding values of metric

coefficients A and B can be found. Some possibilities of such types are mentioned in

Table 2.1.

Table 2.1: The approximate values of the masses M , radii R, compactness µ, andthe constants A and B for the compact stars, Her X-1, SAXJ 1808.4-3658, and 4U1820-30.

Compact Stars M R(km) µ = MR

A(km−2) B(km−2)

Her X-1 0.88M¯ 7.7 0.168 0.006906276428 0.004267364618

SAXJ1808.4-3658 1.435M¯ 7.07 0.299 0.01823156974 0.01488011569

4U1820-30 2.25M¯ 10 0.332 0.01090644119 0.009880952381

2.2 Physical Aspects of f(R) Gravity Models

In this subsection, we consider some well-known viable f(R) models of gravity for

the description of some physical environment of compact stellar interiors. We shall

check evolution of energy density, pressure, EoS parameter, TOV equation and ECs

for some particular stars with three above mentioned f(R) models. We use three

configurations of stellar bodies, i.e., Her X-1, SAX J 1808.4-3658, and 4U 1820-30 of

masses 0.88M¯, 1.435M¯ and 2.25M¯, respectively. We use three f(R) models in

Eqs.(2.1.5)-(2.1.7) to obtain the values of matter variables. These equations would

assist us to investigate various stability features of compact stellar structures (shown

in Table 2.1). We shall also check the corresponding behavior of stellar interiors by

drawing plots. In the diagrams, the stellar structures Her X-1, SAX J 1808.4-3658,

and 4U 1820-30 are labeled with CS1, CS2 and CS3 abbreviations, respectively.

The strange stars are widely known as those quark structures that are filled with

strange quark matter contents. There has been an interesting theoretical evidences

that indicate that quark stars could came into their existence from the remnants of

neutron stars and forceful supernovas (Weissenborn et al., 2011). Surveys suggested

the possible existence of such structures in the early epochs of cosmic history fol-

lowed by the Big Bang (Weber, 2005). On the other hand, the evolution of stellar

structure could end up with white dwarfs, neutron stars or black holes, depending

upon their initial mass configurations. Such structures are collectively dubbed with

the terminology, compact stars.

A maximum permitted mass radius ratio for the case of static background of

spherical relativistic structures coupled with ideal matter distributions should be

2M/R < 89. This result has gained certain attraction among relativistic astrophysi-

cists in order to design the existence of compact structures and is widely known as

Buchdahl-Bondi bound (Bondi, 1964; Buchdahl, 1966; Mak et al., 2000). In this

chapter, we consider the radial and transverse sound speed in order to perform sta-

bility analysis. However, in order to investigate the equilibrium conditions, we shall

study the impact of hydrostatic, gravitational and anisotropic forces, in the possible

modeling of compact stars.

2.2.1 Model 1

Firstly, we assume model in power-law form of the Ricci scalar given by (Starobinsky,

f(R) = R + αR2, (2.2.1)

where α is a constant number. Starobinsky presented this model for highlighting the

exponential growth of early-time cosmic expansion. This Ricci scalar formulation, in

most manuscripts, is introduced for the possible candidate of DE. Einstein’s gravity

introduced can be retrieved, under the limit f(R) → R.

2.2.2 Model 2

Next, we consider Ricci scalar exponential gravity given by (Cognola et al., 2008)

f = R + βR(e(−R/R) − 1

), (2.2.2)

where β and R are constants. The models of such configurations have been studied

in the field of cosmology by Bamba et al. (2010a). The consideration of this model

could provide an active platform for the investigation of late-time accelerating universe

complying with matter-dominated eras.

2.2.3 Model 3

It would be interesting to consider f(R) corrections of the form

f = R + αR2 (1 + γR) , (2.2.3)

in which α and γ are the arbitrary constant. The constraint γR ∼ O(1) specifies this

model relatively more interesting as one can compare their analysis of cubic Ricci

scalar corrections with that of quadratic Ricci term.

2.2.4 Energy Density and Pressure Evolutions

Here, we analyze the domination of star matter as well as anisotropic pressure at the

center with f(R) models. The corresponding changes in the profiles of energy density,

radial and transverse pressures are shown in Figures (2.4)-(2.6), respectively. We see

that dρdr

< 0, dpr

dr< 0 and dpt

dr< 0 for all three models and strange stars. For r = 0,

we obtain

dr= 0,

which is expected because these are monotonically decreasing functions. One can

observe maximum impact of density (star core density ρ(0) = ρc) for small r. The

plot of the density for the strange star candidate Her X-1, SAX J 1808.4-3658, and

4U 1820-30 are drawn. Figure (2.1) shows that as r → 0, the density ρ profile keeps

on increasing its value, thereby indicating ρ as the monotonically decreasing function

of r. This suggests that ρ would decrease its effects on increasing r which indicates

high compactness at the stellar core. This proposes that our chosen f(R) models may

provide viable results at the outer region of the core. The other two plots, shown in

Figures (2.2) and (2.3), indicate the variations of the anisotropic radial and transverse

pressure, pr and pt.

2.2.5 TOV Equation

The TOV equation for the spherical anisotropic stellar interior is given in Eq.(1.7.2)

in which the quantity a′ is the radial derivative of the function appearing in the first

metric coefficient of the line element. Equation (1.7.3) states the contribution of

various interactions, i.e., gravitational (Fg), hydrostatic (Fh) and anisotropic (Fa),

in the modeling as well as the existence of CS. The values of these forces for our

anisotropic spherical matter distribution are mentioned in Eq.(1.7.4). By making

use of these definitions, the behavior of these forces for the onset of hydrostatic

equilibrium are shown for three strange compact stars in Figure (2.7). In the graphs,

we have continued our analysis with different f(R) theories. In Figure (2.7), the left

plot is showing variations due to model 1, middle is for model 2 and the right plot is

describing corresponding changes in principal forces due to model 3.

2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.1: Density evolution of the strange star candidates with three different f(R)models.

2 4 6 8 10r

-0.002

Model 1

2 4 6 8 10r

-0.002

Model 2

2 4 6 8 10r

Model 3

Figure 2.2: Radial pressure evolution of the strange stars.

2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.3: Transverse pressure evolution of the strange stars.

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 1

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 2

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 3

Figure 2.4: Behavior of dρ/dr with respect to r with three different f(R) models.

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 1

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 2

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 3

Figure 2.5: Behavior of dpr/dr with respect to r.

2 4 6 8 10r

-0.0014

-0.0012

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

dpt�drCS1

Model 1

2 4 6 8 10r

-0.0010

-0.0005

dpt�drCS1

Model 2

2 4 6 8 10r

-0.0015

-0.0010

-0.0005

dpt�drCS1

Model 3

Figure 2.6: Behavior of dpt/dr with increasing r.

2 4 6 8 10

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

0.0002 Fg

Model 1

2 4 6 8 10

-0.0008

-0.0006

-0.0004

-0.0002

0.0002 Fg

Model 2

2 4 6 8 10

-0.0020

-0.0015

-0.0010

-0.0005

Model 3

Figure 2.7: The plot of Fg, Fh and Fa with respect to the radial coordinate r(km)and three different f(R) models.

2.2.6 Stability Analysis

Here, we perform the stability analysis of the star candidates with all the three under

observed f(R) models by applying the technique developed by Herrera (1992). After

observing the stability relations mentioned in Eq.(1.8.1), we found that our stellar

systems are dynamically stable under few regions. It has been seen that the evolution

of the radial and transversal sound speeds for all three types of strange stars are within

the bounds of stability for some regions. Figure (2.8) states that all of our stellar

structures (within the background of all f(R) models) obey the following constraint

(1.8.2). Therefore, we infer that all of our proposed model are stable in this theory.

Such kind of results have been proposed by Sharif and Yousaf (2014c,b, 2015a) by

employing different mathematical strategy on compact stellar objects.

2.2.7 Equation of State Parameter

Now, we check the viability range of the EoS parameter mentioned in Eq.(1.6.1) for

the anisotropic relativistic star candidates. We shall analyze their ranges through

2 4 6 8 10r

vst2-vsr

Model 1

2 4 6 8 10r

vst2-vsr

Model 2

2 4 6 8 10r

vst2-vsr

Model 3

Figure 2.8: Variations of v2st − v2

sr with respect radius, r (km)

various plots. The behaviors of ωr for our compact structures is shown graphically

in Figure (2.9). However, the similar behavior of ωt can be observed for all of our

observed compact structures very easily. It has been observed that maximum radius

of compact objects to achieve the limit 0 < ωr < 1 is r ∼ (≤ 7), while the constraint

0 < ωt < 1 is valid for any large value of r. This means that ωi > 1 near its central

point. The spherically symmetric self-gravitating system would be in a radiation

window at the corresponding hypersurfaces. From here, we conclude that that our

relativistic bodies have compact interiors.

2.2.8 The Measurement of Anisotropy

Here, we measure the extent of anisotropy in the modeling of relativistic interiors. It

is well-known that anisotropicity in the stellar system can be measured with the help

of the following formula

∆ =2

r(pt − pr), (2.2.4)

The quantity ∆ is directly related to the difference pt − pr. The positivity of ∆

indicates the positivity of pt − pr. Such a background suggests the outwardly drawn

2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.9: Variations of the radial EoS parameter, ωr with respect to the radialcoordinate.

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 1

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 2

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 3

Figure 2.10: Variations of anisotropicity, ∆, with respect to r.

behavior of anisotropic pressure. However, the resultant pressure will be directly

inward, once ∆ is less than zero. We have drawn the anisotropic factor for our

systems and obtain ∆ > 0, thereby giving pt > pr. All these results are mentioned

through plots as shown in Figure (2.10).

2.3 Spherical Anisotropic Fluids and f (R, T ) Grav-

The f(R, T ) theory of gravity is introduced as the possible modification in f(R)

gravity to unveil many issues of expanding cosmic. This gravity could be regarded as

the viable toy model as it has invoked quantum effects. In this section, we perform the

whole whole analysis mentioned in the previous section for f(R, T ) gravity. Therefore,

we consider the metric (1.7.1) with the relativistic geometry filled with the locally

anisotropic matter content whose energy momentum tensor is mentioned in Eq.(1.4.6).

The corresponding f(R, T ) field equations (1.9.9) becomes

ρ = e−b(r)

[(a′′(r)

2− 1

4a′(r)b′(r) +

4a′(r)2 +

a′(r)

)fR (R, T ) +

(b′(r)

2− 2

′ (R, T ) − 1

2eb(r)fR (R, T ) − fR

′′ (R, T )

], (2.3.1)

Pr =e−b(r)

(1 + fT (R, T ))

[(−a′′(r)

2− 1

4a′(r)b′(r) − 1

4a′(r)2 +

b′(r)

)fR (R, T )

(a′(r)

′ (R, T ) +1

2eb(r)fR (R, T )

]− ρfT

(1 + fT ), (2.3.2)

Pt =e−b(r)

(1 + fT (R, T ))

2r (b′(r) − a′(r)) + eb(r) − 1

)fR (R, T )

2(a′(r) − b′(r))

′ (R, T ) +1

2eb(r)fR (R, T ) + fR

′′ (R, T )

]− ρfT

(1 + fT ). (2.3.3)

It is pertinent to mention that various celestial objects such as galaxies and their

groups are developing in a non-linear nature of stellar medium. To have delve deeply

into their structure formation and evolution, one requires to linearize their analysis.

In this perspective, theoretical physicists have used numerical simulations or some

other mathematical ways. We now wish to take separable forms of the Ricci and

trace of energy momentum tensor in the formulations of f(R, T ) model

f(R, T ) = f1(R) + g(T ). (2.3.4)

The above choice could be considered as possible corrections in the gravitational

dynamics of f(R) gravity. Any f(R, T ) model of the above choice is viable, if one

picks f1(R) formulations from Nojiri and Odintsov (2003) associated with any linear

form of g(T ). We shall consider g to be g(T ) = λT , where λ is a constant number.

Its value should be very small. Keeping in view above equation, the field equations

(2.3.1)-(2.3.3) can be written as

2(1 + 2λ)

[(2 + 5λ)

(1 + λ)Φ1 + λΦ2 + 2λΦ3

], (2.3.5)

Pr =−1

2(2λ + 1)

(λ + 1)Φ1 − (2 + 3λ)Φ2 + 2λΦ3

], (2.3.6)

Pt =−1

2(1 + 2λ)

(1 + λ)Φ1 + λΦ2 − 2(1 + λ)Φ3

], (2.3.7)

Φ1 = e−b(r)

[(a′′(r)

2− 1

4a′(r)b′(r) +

4a′(r)2 +

a′(r)

(b′(r)

2− 2

′ − 1

2eb(r)fR − fR

′′]

Φ2 =e−b(r)

(1 + λ)

[(−a′′(r)

2− 1

4a′(r)b′(r) − 1

4a′(r)2 +

b′(r)

(a′(r)

′ +1

2eb(r)fR

Φ3 =e−b(r)

(1 + λ)

2r (b′(r) − a′(r)) + eb(r) − 1

2(a′(r) − b′(r)) +

)+ × fR

′ +1

2eb(r)fR + fR

′′]

It is seen from the above equations that a and b are the functions of radial coordinate.

These functions can be expressed as the terminated series combinations of their ar-

guments. Krori and Barua (1975) suggested the mathematical formulations of these

variables in terms of three constants A, B and C as a(r) = Br2 + C and b(r) = Ar2.

The values of these constant terms can be evaluated by taking observational values of

stellar structures. Using these expansions, the f(R, T ) field equations (2.3.5)-(2.3.7)

can be rewritten as

ρ =e−Ar2

2r2 (1 + λ) (1 + 2λ)(−eAr2

r2 (1 + λ) f1(R) + (2(−1 − 2λ + 3Br2λ + B2r4λ

+ eAr2

(1 + 2λ) + Ar2(2 + 4λ − Br2λ

)) + eAr2

r2(1 + λ)R)f1′(R) + r((−4 − 6

× λ + 3Br2λ + Ar2(2 + 3λ))R′f1′′(R) − r (2 + 3λ) f1′′(R)R′′ − r (2 + 3λ)

× R′2f1(3)(R))), (2.3.8)

Pr =e−Ar2

2r2 (1 + λ) (1 + 2λ)[eAr2

r2 (1 + λ) f1(R) − (2(−1 − 2λ + B2r4λ + eAr2

(1 + 2λ)

− Br2(2 + λ + Ar2λ

)) + eAr2

r2 (1 + λ) R)f1′(R) + r[{4 + 6λ + Ar2λ + Br2(2 + λ)}

× R′f1′′(R) − rλf1′′(R)R′′ − rλR′2f1(3)(R)]], (2.3.9)

Pt =e−Ar2

2r2 (1 + λ) (1 + 2λ)[eAr2

r (1 + λ) f1(R) − r[2{−B(2 + λ + Br2(1 + λ)) + A(1 + 2λ

+ Br2(1 + λ))} + eAr2

(1 + λ) R]f1′(R) + 2R′f1′′(R) − 2Ar2R′f1′′(R) + 2Br2R′f1′′(R)

+ 2λR′f1′′(R) − 3Ar2λR′f1′′(R) + Br2λR′f1′′(R) + 2rf1′′(R)R′′ + 3rλf1′′(R)R′′

+ 2rR′2f1(3)(R) + 3rλR′2f1(3)(R)]. (2.3.10)

2.4 Boundary Conditions

The boundary conditions have been found to be same as found in f(R) gravity. The

values of A,B and C for the stellar structure shown in Table: 2.1 are Eqs.(2.1.9) and

(2.1.10). In addition, we found the following constraint

ρ(r) = ρc

dr(0) = 0,

b(0) = 0.

2.5 Different Models in f (R, T ) gravity

This section is devoted to study cosmological aspects of some familiar f(R, T ) grav-

ity models. We want to study the impact of these models for understanding the

mathematical modeling of stellar interiors with some physical grounds. We shall ex-

plore various physical characteristics like, compactness, evolution of energy density,

measurement of anisotropic pressure, ECs of the stars. The outcomes from our re-

sults may help to unveil many hidden cosmological results at both theoretical and

astrophysical scales. Depending upon the selection of f(R) models, we shall choose

f(R, T ) models of the type

f(R, T ) = fi(R) + λT, (2.5.1)

where i = i, 2, 3.

2.5.1 Model 1

Firstly, we consider quadratic Ricci scalar corrections in the f(R) model which was

initially proposed by Starobinsky as in Eq.(2.2.1). Then f(R, T ) model (2.5.1) will

become Starobinsky (1980)

f(R, T ) = R + αR2 + λT, (2.5.2)

where α is a constant number. The aim of this model to explore exponential early

time growth of the expanding cosmos as suggested by Starobinsky. This model can

treated as the possible alternative testee for DE. On taking f(R) → R limit, Einstein’s

gravity can be recovered.

2.5.2 Model 2

Secondly, we take the Ricci scalar exponential corrections (as mention in Eq.(2.2.2))

in the action along with g(T ) to have a f(R, T ) model of the form Cognola et al.

(2008)

f(R, T ) = R + βR(e(−R/γ) − 1

)+ λT, (2.5.3)

where β and γ are constants. In order to avoid the appearance of ghost state at

quantum gravity scale, we require that β < exp(R/γ), thereby giving fR = 1 −

β exp(R/γ) > 0. Furthermore, we assume that in the above model β as well as γ are

positive definite to make fRR = βγ

exp(R/β) > 0. This constraint is imposed for the

stable backgrounds of cosmological perturbation and dissolution of tachyonic fields.

This model is also viable for discussing dynamical properties of matter dominated

epoch, under the limit RγÀ 1. The mathematical formulations f2 reduce to Λ-cold

dark matter model as f(R) − R = −βγ =constant with RγÀ 1.

2.5.3 Model 3

It would be interesting to consider some higher terms of Ricci scalar to the quadratic

model of gravity as supposed in Eq.(2.2.3). Then f(R, T ) model (2.5.1) becomes

f(R, T ) = R + αR2 (1 + γR) + λT. (2.5.4)

The constraint γR ∼ O(1) specifies this model relatively more interesting as one can

compare their analysis of cubic Ricci scalar corrections with that of quadratic Ricci

In the above models, we shall take some specific values of parameters (shown in

Table 2.2) to draw diagrams.

f(R, T ) Model Parameter 1 Parameter 2 Parameter 3

Model 1 α ∈ [−1, 1] λ ∈ (−1, 1) —

Model 2 β ∈ [−1, 1] λ ∈ (−1, 1) for very little value of γ

Model 3 α ∈ [−1, 1] λ ∈ (−1, 1) for very little value of γ

Table 2.2: The values of parameters involved in f(R, T ) models for the compact starsHer X-1, SAXJ 1808.4-3658, and 4U 1820-30.

2.6 Physical Aspects of f(R, T ) Gravity Models

Here, This section is devoted to analyze various features of compact stellar struc-

tures. We take three different configurations of compact stars, i.e., Her X-1, SAX J

1808.4-3658, and 4U 1820-30. We use quadratic, exponential and cubic based f(R, T )

models in the gravitational Lagrangian to evaluate Eqs.(2.3.1)-(2.3.3) that give the

values of matter variables in terms of model parameters, α, β, γ and λ. We discuss

various physical properties in order to search for the realistic configurations of stellar

structures (shown in Table 2.1). The comparison between outcomes from these obser-

vations may provide evidences for viability of f(R, T ) gravity models on theoretical

and astrophysical grounds.

2.6.1 Energy Density and Pressure Evolutions in f(R, T ) grav-

In this subsection, we investigate how structural variables of all the three stars are

varying with the evolution of radial coordinate. We check variations not only in the

profiles of energy density and pressure components but also in their radial deriva-

tives. Taking into account Eqs.(2.3.1)-(2.3.3), we draw diagrams (2.11) for all the

three strange star candidate, Her X-1, SAX J 1808.4-3658 and 4U 1820-30 with three

different f(R, T ) models and infer that their energy densities keep on increasing un-

der the limit r → 0. This states ρ monotonically increasing function on decreasing r

values. This demonstrates high compactness of the stellar cores, thereby validating

that our all f(R, T ) models. The other two diagrams (shown in Figures (2.12) and

(2.13)) show the variation in the radial as well as traverser stellar pressure, pr and pt.

The corresponding change in the variations of radial derivative of density and pres-

sure components are shown in Figures (2.14), (2.15) and (2.16). One can observe that

< 0, dpr

dr< 0 and dpt

dr< 0 for all three models and strange stars. For r = 0, we note

that the radial derivatives of fluid variables disappears, thus giving

dr= 0,

dr= 0.

2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.11: Plot of the density (km−2) evolution of the strange star candidate HerX-1, SAX J 1808.4-3658, and 4U 1820-30; for three different models.

2 4 6 8 10r

Model 1

2 4 6 8 10r

-0.002

Model 2

2 4 6 8 10r

Model 3

Figure 2.12: Plot of the radial pressure (km−2) evolution of the strange star candidateHer X-1, SAX J 1808.4-3658, and 4U 1820-30; for three different models.

2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.13: Plot of the transverse pressure (km−2) evolution of the strange starcandidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; for three different models.

Further, the second derivatives of these variables have been found to be less than

zero. These outcomes suggest the heavy profiles of stellar matter variables at their

corresponding central points, thereby indicating compact environments of the testee

stars.

The mathematica code for the plots are given in Appendix A.

2.6.2 TOV Equation in f(R, T ) gravity

It has been noticed that TOV equation contains contributions of three well-known

interactions, that are gravitational (Fg), hydrostatic (Fh) and anisotropic (Fa) forces

as shown in Eq.(1.7.4). The values of these forces by making use of Eqs.(2.3.1)-(2.3.3)

can be expressed as

Fg ≡ −Br(ρ + Pr) =−1

1 + λ

[2B exp(Ar2)r{2(A + B)f ′ + (A + B)rR′f ′′ − f ′′R′′ − R′2f ′′′}

Fh ≡ −dPr

exp(−Ar2)

2r3(1 + λ)(1 + 2λ)[4{1 + 2λ + B2r4λ + A2Br6λ − exp(Ar2)(1 + 2λ) + Ar2

× (1 + 2Br2 + 2λ − B2r4λ)}f ′ + r{−r(4 + 3(2 + Ar2)λ + Br2(2 + λ))R′2f ′′′ + R′{(2

+ 8Ar2 − 6Br2 + 4ABr4 + 2λ + 11Ar2λ − 3Br2λ + 2A2r4λ + 2B2r4λ + exp(Ar2)(2

+ 4λ) + exp(Ar2)r2(1 + λ)R)f ′′ + 3r2λR′′f ′′′} − rf ′′((4 + 6λ + 3Ar2λ + Br2(2 + λ))R′′

− rλR′′′) + r2λR′3f ′′′′}],

Fa ≡ 2(Pr − Pt)

2 exp(−Ar2)

r3(1 + λ)[{exp(Ar2) − (1 + Ar2 − Br2)(1 + Br2)}f ′ + r{−(1

+ Ar2)R′f ′′ + rf ′′R′′ + rR′2f ′′′}].

Using these expressions and observational values from Table 2.1, we plot graphs for

three strange compact stars as shown in Figure (2.17). One can analyze the behavior

of these forces with respect to the radial coordinate r (km) in the modeling of stellar

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 1

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 2

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 3

Figure 2.14: Plot of the dρ/dr with increasing r of the strange star candidate HerX-1, SAX J 1808.4-3658, and 4U 1820-30; for three different models.

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 1

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 2

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 3

Figure 2.15: Plot of the dpr/dr with increasing r of the strange star candidate HerX-1, SAX J 1808.4-3658, and 4U 1820-30; for three different models.

2 4 6 8 10r

-0.0015

-0.0010

-0.0005

dpt�drCS1

Model 1

2 4 6 8 10r

-0.0015

-0.0010

-0.0005

dpt�drCS1

Model 2

2 4 6 8 10r

-0.0015

-0.0010

-0.0005

dpt�drCS1

Model 3

Figure 2.16: Plot of the dpt/dr with increasing r of the strange star candidate HerX-1, SAX J 1808.4-3658, and 4U 1820-30; for three different models.

2 4 6 8 10

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

0.0002 Fg

Model 1

2 4 6 8 10

-0.0008

-0.0006

-0.0004

-0.0002

0.0002 Fg

Model 2

2 4 6 8 10

-0.0020

-0.0015

-0.0010

-0.0005

Model 3

Figure 2.17: The plot of Fg, Fh and Fa with respect to the radial coordinate r (km)for f(R, T ) gravity.

structures. In Figure (2.17), the left, middle and right plots are for model 1, 2 and 3,

respectively.

The mathematica code for the plots are given in Appendix A.

2.6.3 Stability Analysis in f(R, T ) gravity

It is worthy to mention that for relativistic observers, only those stellar structures are

important which are stable against fluctuations. Therefore, the problem of stability is

very important to discuss in the search of realistic model of stellar structure. In this

direction, we check the stability of our stellar structures by employing the technique

demonstrated by Herrera (1992) that was established on the notions of cracking (or

overturning) as mention in Eq.(1.8.1). After using Eqs.(2.3.1)-(2.3.3), their values are

found as follows

Y, vst =

X = r{4r3(B2(1 + λ) + A2(1 + 2λ + Br2(1 + λ)) − AB(3 + 2λ + Br2(1 + λ)))f ′ + r{2(1

+ λ) + Br2(2 + λ) − 3Ar2(2 + 3λ)}R′2f ′′′ + R′{(6Br2 − 2 + 2B2r4 − 2λ + 3Br2λ + 2B2

× r4λ + 2A2r4(2 + 3λ) − Ar2(8 + 11λ + 2Br2(3 + 2λ)) − exp(Ar2)r2(1 + λ)R}f ′′ + 3r2

× (2 + 3λ)R′′f ′′′} + rf ′′{(2(1 + λ) + Br2(2 + λ) − 3Ar2(2 + 3λ))R′′ + r(2 + 3λ)R′′′}

+ r2(2 + 3λ)R′3f ′′′′}, (2.6.1)

Y = −4[exp(Ar2)(1 + λ) − 1 − 2λ − B2r4λ + A2r4{2 + (4 − Br2)λ} + Ar2{−1(4Br2 − 2

+ B2r4)λ}]f ′ + r{r(3(Br2 − 2)λ − 4 + 3Ar2(2 + 3λ))R′2f ′′′ + R′((2 + 14Ar2 − 4A2r4 + 2λ

+ 23Ar2(1 + λ)f ′′ = 3r2(2 + 3λ)R′′f ′′′) + rf ′′((3Br2λ − 4 − 6λ + 3Ar2(2 + 3λ))R′′ − r(2

+ 3λ)R′′′) − r2(2 + 3λ)R′2f ′′′′)}, (2.6.2)

Z = 4[4{1 + 2λ + B2r4λ + A2Br6λ − exp(Ar2)(1 + 2λ) + Ar2(1 + 2Br2 + 2λ − B2r4λ)}f ′

+ r{−r(4 + 3(2 + Ar2)λ) + Br2(2 + λ)}R′2f ′′′ + R′{(2 + 8Ar2 − 6Br2 + 4ABr4 + 2λ

+ 11Ar2λ − 3Br2λ + 2A2r4λ + 2B2r4λ + exp(Ar2)(2 + 4λ) + exp(Ar2)r2(1 + λ)R)f ′′

+ 3r2λR′′f ′′′} − rf ′′{(4 + 6λ + 3Ar2λ + Br2(2 + λ))R′′ − rλR′′′} + r2λR′3f ′′′′].

(2.6.3)

In order to enter in the stable window, the radial and transverse speeds of spherical

relativistic system must satisfy 0 ≤ v2sr ≤ 1 and 0 ≤ v2

st ≤ 1 constraints. We plotted

some graphs, after using Eqs.(2.5.2)-(2.5.4) and (2.6.1)-(2.6.3) along with Table 2.1.

One can observe from the Figures (2.18) and (2.19) that v2sr and v2

st are within the

stability bounds for all the three observed stellar structures. Moreover, Figure (2.20)

indicates that CS1, CS2 and CS3 are satisfying the stability mode mentioned in

Eq.(1.8.2). Thus our observed three relativistic structures are in the complete range

of stability in the presence of f(R, T ) gravity models.

0 2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.18: Variations of v2sr with respect radius r (km) of the strange star in f(R, T )

gravity

0 2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.19: Variations of v2st with respect radius r (km) of the strange star in f(R, T )

gravity

2 4 6 8 10r

vst2-vsr

Model 1

2 4 6 8 10r

vst2-vsr

Model 2

2 4 6 8 10r

vst2-vsr

Model 3

sr with respect radius r (km) of the strange star inf(R, T ) gravity

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 1

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 2

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 3

Figure 2.21: Variations of anisotropic measure ∆ with respect to the radial in f(R, T )gravity.

2.6.4 The Measurement of Anisotropy in f(R, T ) gravity

The magnitude as well as degree of anisotropicity in the locally anisotropic relativistic

fluid distributions is defined in Eq.(2.2.4). In order to calculate anisotropicity with

the help of Eqs.(2.3.1)-(2.3.3), we find ∆ as

∆ =2 exp(−Ar2)

r3(1 + λ)[{exp(Ar2) − (1 + Ar2 − Br2)(1 + Br2)}f ′ + r{−(1 + Ar2)

× R′f ′′ + rf ′′R′′ + rR′2f ′′′}].

The above expression contains f term. We use different values of f from Eqs.(2.5.2)-

(2.5.4) and obtain three different expressions that correspond to each f(R, T ) model.

We then use observational values of stellar structures mentioned in Table 2.1 in all

the three equations. This results total nine set of equations. We plot these equations,

in order to check the extent of anisotropicity in the realistic modeling of compact

stars. It can be seen from Figure (2.21) that the ∆ is remained positive for all the

three cases of stellar structures that eventually suggests that the role of Pt is greater

than that of Pr. It means that the measure of anisotropy is directed outward.

2.7 Anisotropic geometry in f(G, T ) gravity

In this section, we would like to analyze the role of locally anisotropic pressure as

well as f(G, T ) models on the existence and stability of compact stars. The f(G, T )

gravity is purely based on the proper selection of f(G, T ) model. Like f(R, T ), one

can have separate formulations of G and T in this gravity. We choose following type

of f(G, T ) model

f(G, T ) = f(G) + g(T ). (2.7.1)

This type of model can be treated as a possible candidate for dark sources in the

dynamics of f(G) gravity. Here, we take a linear form of g(T ) in order to have

some striking outcomes on the basis of different degrees of freedom coming from f(G)

gravity. In this direction, the above model can rewritten as

f(G, T ) = f(G) + λT,

where λ is a constant. In the following, we shall continue our analysis with different

choices of f(G) gravity models. We assume that the relativistic system under con-

sideration is a static spherically symmetric metric as in Eq.(1.7.1). Krori and Barua

Krori and Barua (1975) suggested that one can take a and b as the combination of

some arbitrary constants A, B and C given by a(r) = Br2 + C and b(r) = Ar2. The

field equations (1.9.24) for the static anisotropic sphere with the above selection of

metric variables turn out to be

2r2 (1 + λ) (1 + 2λ)e−2Ar2

(−2eAr2

+ 2e2Ar2

+ 4AeAr2

r2 − e2Ar2

f(G)r2

− 4eAr2

λ + 4e2Ar2

λ + 8AeAr2

r2λ + 6BeAr2

r2λ − e2Ar2

f(G)r2λ − 2ABeAr2

+ 2B2eAr2

r4λ + e2Ar2

Gr2 (1 + λ) fG − 4r(−B(5 − eAr2

+ 2Br2)

+ A(−6 − 15λ + 6Br2λ + eAr2

(2 + 5λ)))fG′ − 8fG

′′ + 8eAr2

fG′′ − 20λfG

′′

+ 20eAr2

λfG′′ + 8Br2λfG

′′), (2.7.2)

2r2 (1 + λ) (1 + 2λ)e−2Ar2

(2eAr2 − 2e2Ar2

+ 4BeAr2

r2 + e2Ar2

f(G)r2 + 4eAr2

− 4e2Ar2

λ + 2BeAr2

r2λ + e2Ar2

f(G)r2λ + 2ABeAr2

r4λ − 2B2eAr2

− e2Ar2

Gr2 (1 + λ) fG − 4r(−A(−3 + eAr2

)λ + 2B2r2λ + B(−6 − 7λ − 6Ar2λ

+ eAr2

(2 + 3λ)))fG′ + 4λfG

′′ − 4eAr2

λfG′′ − 8Br2λfG

′′), (2.7.3)

2r2 (1 + λ) (1 + 2λ)e−2Ar2

(−2AeAr2

r2 + 4BeAr2

r2 + e2Ar2

f(G)r2 − 2ABeAr2

+ 2B2eAr2

r4 − 4AeAr2

r2λ + 2BeAr2

r2λ + e2Ar2

f(G)r2λ − 2ABeAr2

+ 2B2eAr2

r4λ − e2Ar2

Gr2 (1 + λ) fG + 4r(A(−3 + eAr2

)λ + 2B2r2(1 + λ)

− B(−2 + λ − eAr2

λ + 6Ar2 (1 + λ)))fG′ + 8Br2fG

′′ + 4λfG′′

− 4eAr2

λfG′′ + 8Br2λfG

′′). (2.7.4)

Now, we match our interior geometry with the appropriate exterior over the

boundary surface by which we get the constant A,B and C as in Table: 2.1. The

approximated values of mass and radius for different compact stars can be used to

evaluate these constants. We will use masses, radii and compactness of three partic-

ular stars namely, Her X-1, SAXJ 1808.4-3658 (SS1) and 4U 1820-30 given in Table

2.8 f (G) Gravity Models

In this section, we consider different well-known f(G) models and check their impact

on various physical features of compact stars such as stability analysis, ECs etc. We

take three different f(G) models and based upon these models, we construct f(G, T )

models, since f(G, T ) = fi(G) + λT with i = 1, 2, 3.

2.8.1 Model 1

Firstly, we take power-law model along with logarithmic corrections of Gauss-Bonnet

term of the type (Schmidt, 2011)

f1 = αGn + βG log(G), (2.8.1)

where α, n and β are arbitrary constants. The degrees of freedom allowed in the

dynamics by the above model could provide observationally well-consistent cosmic

results.

2.8.2 Model 2

Next, we consider Gauss-Bonnet corrections of the form (Bamba et al., 2010b)

f2 = αGn (βGm + 1) , (2.8.2)

where α, β and m are any constant number, while n > 0. The model of the above

type could be fruitful for the better understanding of finite time future singularities.

2.8.3 Model 3

Thirdly, we shall make use of another physically acceptable Gauss-Bonnet function

f3 =a1G

n + b1

a2Gn + b2

, (2.8.3)

here ai’s, bi’s and n are arbitrary constants with n > 0.

2.9 Physical Aspects of f(G, T ) Gravity Models

In this section, we shall take three different configurations of compact stars, i.e.,

Her X-1, SAX J 1808.4-3658, and 4U 1820-30. We shall discuss various physical

conditions, like ECs, TOV equation, stability, evolution of energy density and pressure

etc under the degrees of freedom coming from this gravity. For this purpose, we will

put f(G, T ) models one by one in Eqs.(2.7.2)-(2.7.4) and get ρ, pr and pt. The

corresponding values of A, B and C are shown in Table 2.1.

2.9.1 Energy Density and Pressure Evolutions

Here, we check the evolution of density and pressure gradients for the above mentioned

strange star candidates. We plot the energy density for different gravity models

for all the strange stars as shown in Figure (2.22) which indicates that as r → 0

the density (ρ) attains its maximum value. The increase in the radial distance will

decrease the value of ρ suggesting that ρ is a decreasing function of r. This points

the high compactness at the star core under the effects of three f(G) models, thereby

validating that our models under investigation are viable for the outer region of the

core. Similarly, the profiles of radial and traverse pressure, pr and pt are shown in

Figures (2.23)-(2.24). Continuing in this way, the variations of r-derivatives of density

and pressure components for three different models are shown in Figures (2.25)-(2.27).

We see that dρdr

< 0, dpr

dr< 0 and dpt

dr< 0 for all three models and strange stars. For

r = 0, we obtain

dr= 0,

which is expected because these are the decreasing function and for small r we have

a maximum density (star core density ρ(0) = ρc).

2.9.2 TOV Equation

The generalization of TOV equation for the spherical anisotropic matter content can

be written in form of three different forces as mentioned in Eq.(??). After calculating

these forces in the case of f(G, T ) gravity models, we have plotted graphs for three

strange stars as shown in Figure (2.28). The variations of these fundamental forces,

i.e., gravitational force (Fg), hydrostatic force (Fh) and anisotropic force (Fa), via

radial distance for three different models are shown in Figure (2.28).

2.9.3 Stability Analysis

The stability of stellar interiors has been performed by number of researchers against

perturbation scheme (Yousaf and Bhatti, 2016b; Bhatti and Yousaf, 2016; Yousaf and

Bhatti, 2016a). Here, we check the stability of stellar models in f(G, T ) theory with

the help of radial and transverse sound speeds as mention in Eq.(1.8.1). In order to

achieve stability modes, the radial speed sound, vsr and the transverse sound speed,

vst, should satisfy the constraints 0 ≤ v2sr ≤ 1 and 0 ≤ v2

st ≤ 1. It can be seen from

Figures (2.29) and (2.30) that the radial and transversal sound speeds for all three

strange stars are within the bounds of stability except for Her X-1 and 4U 1820-30

with second model. In the similar fashion, we can see from Figure (2.31) that the

sound speed bound (1.8.2) is valid for all the CSs.

2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

2 4 6 8 10r

Model 3

Figure 2.22: Plots of the energy density (km−2) for strange star candidates with threedifferent models.

2 4 6 8 10r

Model 1

2 4 6 8 10r

-0.005

Model 2

2 4 6 8 10r

Model 3

Figure 2.23: Plots of the radial pressure (km−2).

2 4 6 8 10r

Model 1

2 4 6 8 10r

-0.005

Model 2

2 4 6 8 10r

Model 3

Figure 2.24: Plots of the transverse pressure (km−2).

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 1

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 2

2 4 6 8 10r

-0.005

-0.004

-0.003

-0.002

-0.001

dΡ�drCS1

Model 3

Figure 2.25: Plots of dρ/dr with increasing r for compact stars with three differentmodels.

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 1

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

0.0005

0.0010

dpr�drCS1

Model 2

2 4 6 8 10r

-0.0020

-0.0015

-0.0010

-0.0005

dpr�drCS1

Model 3

Figure 2.26: Plots of dpr/dr versus r.

2 4 6 8 10r

-0.0015

-0.0010

-0.0005

dpt�drCS1

Model 1

2 4 6 8 10r

-0.0015

-0.0010

-0.0005

0.0005

0.0010

dpt�drCS1

Model 2

2 4 6 8 10r

-0.0015

-0.0010

-0.0005

dpt�drCS1

Model 3

Figure 2.27: Plots of dpt/dr versus r.

2 4 6 8 10

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

0.0002 Fg

Model 1

2 4 6 8 10

-0.0012

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

0.0002Fg

Model 2

2 4 6 8 10

-0.0020

-0.0015

-0.0010

-0.0005

Model 3

Figure 2.28: Plot of Fg, Fh and Fa with respect to the radial coordinate r (km) fordifferent models.

0 2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

0 2 4 6 8 10r

Model 3

Figure 2.29: Variations of v2sr with radius r (km) for different models.

0 2 4 6 8 10r

Model 1

2 4 6 8 10r

Model 2

0 2 4 6 8 10r

Model 3

Figure 2.30: Variations of v2st with radius r (km).

2 4 6 8 10r

vst2-vsr

Model 1

0 2 4 6 8 10r

vst2-vsr

Model 2

2 4 6 8 10r

vst2-vsr

Model 3

sr with radius r (km).

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 1

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 2

2 4 6 8 10r

0.0002

0.0004

0.0006

0.0008

0.0010

Model 3

Figure 2.32: Variations of anisotropic (km−2) measure ∆ with radius r (km) fordifferent models.

2.9.4 The Measurement of Anisotropy

The effects of anisotropy can be checked with the help of the relation (2.2.4). We

use field equations with different f(G, T ) models and draw three different plots of

∆. By taking different values of stellar structure, we draw ∆ with respect to radial

coordinate as shown in Figure (2.32). We observe that ∆ > 0, thereby giving pt > pr

which means that the measure of anisotropy is directed outward.

Chapter 3

Wormhole Solutions and EnergyConditions

In this chapter, we explore the possibilities for the existence of wormhole geome-

tries coupled with relativistic matter configurations by taking some particular models

of f(R, T ) gravity. For this purpose, we take the static form of spherically sym-

metric spacetime and after assuming a specific form of matter and combinations of

shape function, the validity of ECs is check. We assume the matter content and find

some of the solutions for shape function along with their viability through flaring-out

conditions and check the ECs. Furthermore, we suppose some barotropic EoS and

find shape function’s solutions and then checked the ECs. We have discussed our

results through graphical representation and studied the equilibrium background of

wormhole models by taking anisotropic fluid. The extra curvature quantities coming

from f(R, T ) gravity could be interpreted as a gravitational entity supporting these

non-standard astrophysical wormhole models. The results of this chapter have been

published in the form of two research papers (Yousaf et al., 2017b,c).

3.1 Wormhole Geometry and f (R, T ) Gravity

We consider the line element (1.7.1), in which a is an arbitrary radial function and is

named as redshift function, while e−b can be expressed by means of shape function,

β(r) as e−b(r) =(1 − β(r)

)(Morris and Thorne, 1988). For the geometry (1.7.1) and

the anisotropic fluid (1.4.6), the four velocity can be defined as V µ = e−a/2δµ0 , while

the unit four vector is defined as Xµ = e−b/2δµ1 . In order to have viable configurations

of a surface corresponding to the throat of WH, we need to consider variations of

radial coordinate from r0 to infinity, where r0 = β(r0). It is well-known that flare out

condition must be obeyed by the WH at its throat. This, in our case, mathematically

we can write it as (β − β′r)/β2 > 0, where β′ at r0 should be less than unity. It is

worthy to mention that these constraints give rise to the occurrence of WH models

threaded with an exotic matter which dissatisfied NEC in the context of GR. The

explicit form of matter variables, i.e., ρ, Pr and Pt can be achieved by solving the field

equations in f(R, T ) gravity.

We consider the particular class of f(R, T ) model mentioned in Eq.(2.3.4). This

choice does not imply the direct non-minimal curvature matter coupling nonetheless,

it can be regarded as correction to f(R) theory of gravity. We shall use the linear form

of g, i.e., g(T ) = λT and will obtain some distinct results based on the non-trivial

coupling as compared to f(R) gravity. In this context, the explicit form of matter

variables, i.e., ρ, Pr and Pt can be found and given in Eqs.(2.3.5)-(2.3.7), respectively.

Alvarenga et al. (2013) have discussed various aspects of ECs in f(R, T ) gravity.

The violation of NEC makes us to assume T effµνkµkν < 0, thereby giving ρeff+P eff

i < 0.

Solving for ρeff + P effr from Eq.(1.9.10), we get

ρeff + P effr =

(ρ + Pr) (1 + λ) +1

(1 − β

′′ + fR′ β − rβ′

2r2(1 − β

)), (3.1.1)

which can be rewritten after using field equations as

ρeff + P effr =

r3(rβ′ − β) . (3.1.2)

This equation has turned out to be same as found in f(R) gravity. The flaring out

constraint restricts us ρeff + P effr to be less than zero. In this case, we have assumed

NEC satisfied for usual energy-momentum tensor nevertheless, the extra curvature

terms coming from modified gravity models may have significant influence on NEC

violation.

3.1.1 Quadratic Ricci Corrections and Anisotropic Matter

Content

We consider the model with quadratic Ricci scalar corrections as mentioned in Eq.(2.2.1).

For α = 0, the model (2.2.1) boils down to give dynamics of GR. Solving the field

equations for this model and then simplifying, we get

2r2 (1 + λ) (1 + 2λ)e−b(−ebr2α (1 + λ) R2 − R(4αλ

+ eb(−4αλ + r2 (1 + λ)

)− 4rαλb′) + rb′ (2λ + rα (2 + 3λ) R′)

+ 2((−1 + eb

)λ − 2rα (2 + 3λ) R′ − r2α (2 + 3λ) R′′)), (3.1.3)

2r2 (1 + λ) (1 + 2λ)e−b(ebr2α (1 + λ) R2 + R(4αλ

+ eb(−4αλ + r2 (1 + λ)

)+ 4rα (1 + λ) b′) + rb′(2 + 2λ

+ rαλR′) + 2(2rα (2 + 3λ) R′ − λ(−1 + eb + r2αR′′))), (3.1.4)

2r2 (1 + λ) (1 + 2λ)e−b(ebr2α (1 + λ) R2 + R((−4α

+ eb(r2 + 4α

)) (1 + λ) + 2rαb′) + rb′ (1 − rα (2 + 3λ) R′)

+ 2((−1 + eb) (1 + λ) + 2rα (1 + λ) R′ + r2α (2 + 3λ) R′′)). (3.1.5)

Now, we assume the red shift function to be a constant quantity which gives a′(r) = 0

and consider the shape function, of the form

e−b(r) =

(1 − β(r)

), (3.1.6)

β(r) = r(r0

. (3.1.7)

In this context, Eqs.(3.1.3)-(3.1.5) provide

r6 (1 + λ) (1 + 2λ)

(−r3 (1 + 2λ) − 2(6 + 5m + m2)rα (2 + 3λ)

α(30 + 45λ + m2(6 + 9λ) + m (26 + 36λ)))] , (3.1.8)

r6 (1 + λ) (1 + 2λ)

(−r3 (1 + 2λ) − 2m (3 + m) rα(4 + (10 + m)

× λ) + mr0

α(20 + 55λ + 3m2λ + m(6 + 28λ)))] , (3.1.9)

2r6 (1 + λ) (1 + 2λ)

((1 + m) r3 (1 + 2λ) + 4m (3 + m) rα(6 + 10λ

+ m (2 + 3λ)) − 2mr0

α(40 + 65λ + 16m (2 + 3λ) + m2 (6 + 9λ)))] .

(3.1.10)

m m = 1 m = 0.5 m = 0 m = −0.5 m = −1 m = −3

β(r) r20/r r0

√r0/r r0

√rr0 r r3r2

Table 3.1: Different shape function for different choices of m

Now, we will check the viability of WEC (ρ > 0) and NEC (ρ+Pr > 0, ρ+Pt > 0)

for different shape function. It can be observed from Eq.(3.1.7) that the quantity β

depends directly on the parameter m. For instance, we take m = 1/2 then Eq.(3.1.7)

gives the following from of shape function

β(r) = r0

√r0/r.

Next, we proceed our analysis by assuming r0 to has a unit value and then, we use

Eq.(3.1.8) in order to plot ρ with respect to radial coordinate r and parameters α

and λ. In order to explore validity epochs for WEC, we draw some plots under which

ρ ≥ 0 and these are shown in the left plot of Figure (3.1).

• It is seen that WEC is valid for small r and is found to be independent of the

choices of α and λ (positive), as shown in left plot of Figure (3.1). In particular,

we have used the optimization method, for 0.1 ≤ r ≤ 10, 0.1 ≤ α < 10 and

1 ≤ λ ≤ 10, to find the minimum regions for which ρ ≥ 0 with respect to the

minimum value of r = 0.1 with α = 4.1541 and λ = 8.28369. In the similar

fashion, the minimum regions for ρ ≥ 0 with respect to the minimum value

of α = 0.1 along with r and λ to be 10, has been explored. Furthermore, the

minimum regions for ρ ≥ 0 respecting the least value of λ = 1 with r = 1.12357

and α = 9.9998 are also been observed.

• If we take λ to be negative, then WEC is also valid for large r. Similarly, if

we take 0.1 ≤ r ≤ 10, 0.1 ≤ α ≤ 10 and −10 ≤ λ ≤ −2, then the minimum

regions for ρ ≥ 0 with respect to the minimum value of r = 1.14171 having

α = 7.88052 and λ = −2 has been found. Similarly, the minimum regions for

which ρ ≥ 0 with respecting the least choice of α = 0.1 along with r = 1.75057

and λ = −4.50503. Furthermore, the least regions for the validity of WEC

relative to minimum choice of λ = −10 with r = 9.95495 and α = 0.614449

Figure 3.1: Evaluation of ρ with respect to r, α and λ for m = 0.5 and r0 = 1.

have been shown in the right plot of Figure (3.1).

Now, we would like to check the validity of NEC. For this purpose, we choose the

range of 0.1 ≤ r ≤ 10, 0.1 ≤ α < 10 and 1 ≤ λ ≤ 10 are shown in Figure (3.2).

In this range, ρ is valid only for small range (< 1.16), while ρ + Pt ≥ 0 is valid also

for large r. By using a well-known optimization method, the minimum value of r for

ρ + Pr ≥ 0 is found to be r = 0.1 with α = 0.869111 and λ = 6.59087. The minimum

zones for ρ + Pr ≥ 0 with respect to the least choices of α and λ have been explored

graphically. These regions for α = 0.1 along with r = 10 and λ = 10 and for λ = 1

with r = 0.114274 and α = 9.21985 have been explored graphically and shown in the

left plot of Figure (3.2).

We then found that the least value of radial coordinate for which ρ + Pt ≥ 0 is

r = 0.949566 with α = 9.13909 and λ = 9.43224. Similarly, the least zones for the

validity of NEC against minimum values of α (α = 0.1) and λ (λ = 1) have been

explored. These regions have been found by taking r = 2.43878 and λ = 5.31504 for

least α value and r = 10 and α = 1.51045 for minimum λ value through graphical

representations. The details can be observed from the right plot of Figure (3.2). It

is concluded that NEC with respect to radial component of pressure (ρ + Pr ≥ 0)

Figure 3.2: Evaluation of ρ + Pr and ρ + Pt with respect to r, α and λ for m =0.5, r0 = 1.

is satisfied for r > 1, however, this EC is obeyed by assuming pressure tangential

pressure component ,i.e., (ρ + Pt ≥ 0) for r < 1. This shows that there are no

common regions between them. For the same region, ρ ≥ 0 and ρ+Pr ≥ 0 are satisfied

while ρ + Pr ≥ 0 does not not satisfy. Now, we will take the range 0.1 ≤ r ≤ 10,

0.1 ≤ α < 10 and −10 ≤ λ ≤ −2 and analyze the behavior with the help of plots.

These are shown in Figure (3.3). By same procedure, we get the least r value under

which ρ + Pr ≥ 0 with α and λ to be 7.09893 and −5.07766, respectively. This r is

1.04202. The minimum values of α and λ have also been found for the validity of

NEC, ρ + Pr ≥ 0. These values are α = 0.1 with r = 1.9211, λ = −4.50503 and

λ = −10 having r = 9.95495 and α = 0.648118 as shown in the left plot of Figure

(3.3).

We have also explored the least choice of radial component r in which ρ + Pt ≥ 0

and its is found to be r = 0.1 for particular choices of α = 4.28403 and λ = −5.91339.

The minimum zones for the validity of NEC, i.e., ρ + Pt ≥ 0 with respect to the

minimum value of α = 0.1 having r = 10 and λ = −10 have been explored. The

Figure 3.3: Evaluation of ρ + Pr and ρ + Pt with respect to r, α and λ= negative form = 0.5, r0 = 1.

minimum regions for which ρ+Pt ≥ 0 with respect to the minimum value of λ = −10

having r = 0.208506 and α = 9.93504 is shown in the right plot of Figure (3.2). From

these, we conclude that for m = 0.5, there are no common regions between ρ+Pr ≥ 0

and ρ + Pt ≥ 0 but if we take m = −0.5, then there are some common regions, for

instance, minimum value of r which is common in ρ + Pr ≥ 0 and ρ + Pt ≥ 0 are

r = 0.493601 with α = 10 and λ = −2.80162.

In Figure (3.4), we fixed the value of two parameters coming from f(R, T ) =

R + αR2 + λT model, i.e., α = 6 and λ = −2 along with m = 0.5 and plotted ρ, Pr

and Pt with respect to r. We deduced that, for anisotropic case, the normal matter

threading the WH does not satisfy one of ECs, i.e., ρ + Pt > 0. On varying m in

Eq.(3.1.7), we get some interesting results which are summarized as follows:

1. For m = −2, 2, 3, 4,−4,−1/5, .., there exists some common regions between

ρ + Pt > 0 and ρ + Pr > 0.

2. If m = −3, then all ρ, ρ+Pr and ρ+Pt > 0 are independent of radial coordinate.

2 4 6 8 10

Figure 3.4: Evaluation of ρ, Pr and Pt for small region.

3. The choices m = 1, 1/2, 1/5, .., give almost same results as we discussed for

m = 0.5.

4. If one takes m = 0, then the energy density of the anisotropic system, ρ, becomes

5. If m = −1/2, some common regions between ρ + Pt > 0 and ρ + Pr > 0 are

possible.

In Figures (3.1)-(3.3), we find the regions where the ECs are valid within the particular

range of parameters. For this purpose, we considered a specific form of f(R, T ) model,

i.e., f(R, T ) = R + αR2 + λT which has three parameters, i.e., λ, α, r. Then, we

have plotted (for anisotropic case) the validity regions of different ECs with respect

to these parameters for some defined ranges.

The mathematica code for the plots are given in Appendix B.

Equilibrium Conditions

In this subsection, we analyze the equilibrium distribution of WH models. The equi-

librium scenario for WH can be achieved by solving TOV equation given in Eq.(1.7.2),

in which the principal forces has been found to be anisotropic, hydrostatic and grav-

itational forces, denoted by Faf , Fhf and Fgf , respectively. Their mathematical

formulations are

Fgf = −1

2σ′(σ + Pr), Fhf = −dPr

dr, Faf =

where σ = 2a(r). Then, Eq.(1.7.2) can be recasted as

Fhf + Fgf + Faf = 0. (3.1.11)

It has been observed that in our case the effects coming from gravitational forces are

zero. Therefore, Eq.(3.1.11) becomes

Fhf + Fgf = 0. (3.1.12)

The rest of forces turn out to be

Fhf =1

r7 (1 + λ) (1 + 2λ)(3 + m) r0

(−r3 (1 + 2λ) − 2m (5 + m) rα (4 + (10 + m) λ)

+ 2mr0

α(20 + 55λ + 3m2λ + m (6 + 28λ)

)), (3.1.13)

Faf =−1

r7 (1 + λ)(3 + m) r0

(−r3 − 8m (5 + m) rα + 4m (10 + 3m) r0

(3.1.14)

The behavior of these forces have been investigated and are shown in Figure (3.5).

Here, we have taken m = 0.5, r0 = 1 and α = 0.6 for which the WEC is compatible.

In Figure (3.5), we have tested the possible existence of model and observed that the

two forces Faf and Fhf have alike behaviors but in opposite directions. These forces

tend to cancel effects of each other. This suggests the possible existence of static

spherically symmetric WH models. This result has been achieved through plots by

setting the parameters equal to some numerical values that were compatible with ECs

as discussed in Figures (3.1)-(3.3).

20 40 60 80 100r

-6.´ 10-7

-4.´ 10-7

-2.´ 10-7

2.´ 10-7

4.´ 10-7

6.´ 10-7

Faf & Fhf

Equilibrium Picture

20 40 60 80 100r

-1.5´ 10-6

-1.´ 10-6

-5.´ 10-7

5.´ 10-7

1.´ 10-6

1.5´ 10-6

Faf & Fhf

Equilibrium Picture

Figure 3.5: Evolution of Faf and Fhf versus r.

3.1.2 Perfect Matter Content

Here, we assume that the WH geometry is coupled with an ideal matter distribution.

The isotropic matter environment has been discussed by many relativistic astrophysi-

cists in various astronomy problems, like stability analysis of stellar systems (Yousaf

and Bhatti, 2016a), irregularities in the system energy density (Yousaf et al., 2016b),

rate of gravitational collapse (Yousaf and Bhatti, 2016b), maintenance of smooth con-

figurations of universe (Hernandez-Pastora et al., 2016), etc. Now, we let the system

to keep all pressures components be equal, i.e., Pr = Pt = P or Π = 0. In this

context, Eqs.(3.1.4) and (3.1.5) after simplification provide

r (1 + λ)e−b(−12r2αb′

2 − 12r3αb′3 − rb′(−28α + eb(r2 + 12α) − 28r2αb′′)

+ 2((−1 + eb

(r2 − 4α

)− 28α

)+ 8r2αb′′ − 4r3αb(3))) = 0, (3.1.15)

which is a non-linear third order differential equation. Now, by transferring b(r)

into shape function β(r) and simplifying Eq.(3.1.15), we get another non-linear third

order differential equation which cannot be solved analytically. So, we use numerical

techniques to solve this equation for which we shall use some initial and boundary

conditions and plot their numerical solution in order to check their behavior in the

0 1 2 3 4 5 6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ΒHrL�

Figure 3.6: Behavior of β(r) and β(r)/r with respect to r for perfect fluid.

maintenance of WEC and NECs. We would explore conditions on shape function

under which NECs and WECs are valid. For this purpose, we draw graphs relating

shape function β(r) and r as well as β′(r) verses r in Figure (3.6). From left plot of

Figure (3.6), we have observed that shape function β(r) is an increasing function and

hence the condition β(r) < r is obeyed. The right plot shows the asymptotic behavior

as β(r)/r− > 0 for r− > ∞. Here, the throat is located at r0 = 0.0997567 such that

β(r0) = r0. The derivative of shape function is plotted in the left of Figure (3.7) from

which we see that β′(r0) < 1 and hence the condition β′(r0) < 1 is also obeyed. The

the right plot of Figure (3.7) shows the asymptotic behavior i.e., β(r) − r < 0 which

yields 1 − β(r)/r > 0. One can easily notice from the Figure (3.8) that WECs and

NECs are valid throughout in few observed regions.

3.1.3 Barotropic State Equation

Here, we use a well-known cosmological state equation in order to relate pressure with

energy density through dimension-less parameter, k. This realistic barotropic EoS is

0 1 2 3 4

Β¢ H

0 1 2 3 4 5

ΒHrL-

Figure 3.7: Isotropic case: Evaluation of β′(r) and β(r) − r.

0 2 4 6 8

Figure 3.8: Behavior of ρ(r) and ρ + P .

given by Pr = kρ or equivalently, we can write Pr − kρ = 0.

r (1 + λ) (1 + 2λ)e−b(−ebr2 + e2br2 − ebkr2 + e2bkr2 − 10α + 12ebα + 24ebαλ + 14kα

− 2e2bα − 12ebkα − 2e2bkα − 2ebr2λ + 2e2br2λ − 2ebkr2λ + 2e2bkr2λ − 18αλ

− 6e2bαλ + 30kαλ − 24ebkαλ − 6e2bkαλ − 6 (1 + k) r2α (1 + 2λ) b′2+ 3r3α[2 − (k

− 3)λ]b′3+ 2r2α (2 + 3λ + k (4 + 7λ)) b′′ + rb′(2α(−4 + (−3 + 5k)λ) + eb[−6(1 + k)α

× λ + r2(1 + 2λ)] + 7r2α (−2 + (−3 + k) λ) b′′) + 4r3αb(3) + 6r3αλb(3) − 2kr3αλb(3)) = 0.

(3.1.16)

Making use of Eqs.(3.1.3) and (3.1.4) and transformation between b(r) and β(r), we

obtain a non-linear third order differential equation. After using numerical method,

we have solved this equation and then draw some graphs of β(r). We saw that the

distribution of shape function β(r) for barotropic EoS is increasing, as shown in left

plot of Figure (3.9) and we can see that the WH throat is r0 = 0.501187 where

β(0.501187) = 0.501187 and from the right plot of Figure (3.9), we observed that

β′(0.501187) < 1. The flaring condition and asymptotic conditions can be checked

from the plot of Fig.3.10 in which we have plotted β(r)/r and β(r) − r verses r. In

left plot of Figure (3.10), we observe that β(r)/r → 0 is not valid for r → ∞ while

in right plot of Figure (3.10), one can notice the validity of faring condition, i.e.,

β(r)−r < 0. In Figure (3.11), we see that the NEC is valid but WEC is not satisfied,

thereby suggesting that for this region, no realistic configuration of WH exists.

0 1 2 3 4 5 60.5

0 1 2 3 4 5 6

Β¢ H

Figure 3.9: Evaluation of β(r) and β′(r) through EoS having α = 9 and k = 0.001.

0 1 2 3 4 5 6

0 1 2 3 4 5 6-5

Β¢ H

Figure 3.10: Evaluation of β(r)/r and β(r)−r through EoS with α = 9 and k = 0.001

0 1 2 3 4 5 6

-0.00015

-0.00010

-0.00005

0.00000

0.00005

0.00010

0 1 2 3 4 5 60.00

Figure 3.11: Evaluation of ρ(r) and ρ + Pt through EoS having α = 9 and k = 0.001

3.2 Wormhole Solutions with Cubic Ricci Scalar

To study WH solutions in f(R, T ) gravity, we need to solve system of modified equa-

tions in which f1, λ, Pr, Pt and b are unknown quantities. The explicit formulations

of these expression is not possible as their appearances are non-linear in nature. Due

to this reason, we would adopt physically viable procedure in the context of WH

background. We shall choose well-consistent f1 model and then particular forms of

red shift and shape function. We assume some equation of state and isotropic matter

content within the spherically symmetric line element. In order to discuss the effects

of f(R) model, we take f1 from Eq.(2.2.3) in which α and γ are arbitrary constants

and if γ À R then this model coincides with the quadratic model. The case under

which αR2(1+γR) ¿ R could be used to increase the impact of cubic terms than that

of quadratic terms. For not great magnitudes of interior masses, the consequences of

this model may coincide with R2 f(R) model. This model describes the dynamics of

GR under the limit α = 0.

In this scenario, the f(R, T ) field equations, after some simplifications, turn out

4r2 (1 + λ) (1 + 2λ)e−b(−2ebr2αγ (1 + λ) R3 + r2 (1 + 2λ) a′2 + ra′(4

+ 8λ − (r + 3rλ) b′ + 6rαλR′) − αR2(−3r2γ (1 + 2λ) a′2 + 3rγa′(−4 − 8λ

+ (r + 3rλ)b′) + 2(ebr2 + ebr2λ + 6γλ − 6ebγλ − 6rγλb′ − 3r2γ (1 + 2λ)

× a′′)) + 2(−2λ + 2ebλ − 4rα (2 + 3λ) R′ − 6r2αγ (2 + 3λ) R′2 + rb′(2λ

+ rα (2 + 3λ) R′) + r2a′′ + 2r2λa′′ − 4r2αR′′ − 6r2αλR′′) − 2R(ebr2 + eb

× r2λ + 4αλ − 4ebαλ + 24rαγR′ − r2α (1 + 2λ) a′2 + 36rαγλR′ + rαa′

× (−4 − 8λ + (r + 3rλ)b′ − 9rγλR′) − rαb′ (4λ + 3rγ (2 + 3λ) R′) − 2r2αa′′

− 4r2αλa′′ + 12r2αγR′′ + 18r2αγλR′′)), (3.2.1)

4r2 (1 + λ) (1 + 2λ)e−b(2ebr2αγ (1 + λ) R3 − r2 (1 + 2λ) a′2 − r2a′

× ((1 + λ) b′ − 2α (2 + λ) R′) + αR2(−3r2γ(1 + 2λ)a′2 − 3r2γ (1 + λ)

× a′b′ + 2(ebr2 + ebr2λ + 6γλ − 6ebγλ + 6rγ(1 + λ)b′ − 3r2γ (1 + 2λ)

× a′′)) + 2(2λ − 2ebλ + 4rα(2 + 3λ)R′ − 6r2αγλR′2 + rb′(2 + 2λ + rα

× λR′) − r2a′′ − 2r2λa′′ − 2r2αλR′′) + 2R(ebr2 + ebr2λ + 4αλ − 4ebαλ

− r2α(1 + 2λ)a′2 + 24rαγR′ + 36rαγλR′ + rαb′ (4 + 4λ + 3rγλR′)

− r2αa′((1 + λ)b′ − 3γ (2 + λ) R′) − 2r2αa′′ − 4r2αλa′′ − 6r2αγλR′′)), (3.2.2)

4r2 (1 + λ) (1 + 2λ)e−b(2ebr2αγ (1 + λ) R3 + αR2(2((−6γ + eb(r2

+ 6γ)) (1 + λ) + 3rγb′) + 3rγa′(−2 − 4λ + rλb′)) + ra′(−2 − 4λ + rλb′

+ 2rα (2 + λ) R′) + 2R(ebr2 − 4α + 4ebα + ebr2λ − 4αλ + 4ebαλ + 12rα

× γR′ + 12rαγλR′ + rαa′(−2 − 4λ + rλb′ + 3rγ (2 + λ) R′) + rαb′(2

− 3rγ (2 + 3λ) R′) + 12r2αγR′′ + 18r2αγλR′′) + 2(rb′ (1 − rα (2 + 3λ) R′)

+ 2((−1 + eb

)(1 + λ) + 2rα (1 + λ) R′ + 3r2αγ (2 + 3λ) R′2 + r2α

× (2 + 3λ) R′′))). (3.2.3)

By assuming constant form of redshift function as considered in the previous section

foe which the shape function is given in Eqs.(3.1.6) and (3.1.7). Then, Eqs.(3.2.1)-

(3.2.3) give

r9 (1 + λ) (1 + 2λ)mr0

(−r6 (1 + 2λ) − 2(6 + 5m + m2

)r4α (2 + 3λ)

− 12m(15 + 11m + 2m2

αγ (2 + 3λ) + r3r0

α(30 + 45λ

+ m2 (6 + 9λ) + m (26 + 36λ)) + 2mr02(r0

αγ(99 (2 + 3λ) + 15m2

× (2 + 3λ) + 2m (77 + 113λ))), (3.2.4)

r9 (1 + λ) (1 + 2λ)r0

(−r6 (1 + 2λ) + 2m2r02(r0

αγ(66 + 20m

+ 231λ + 122mλ + 15m2λ) − 2m (3 + m) r4α (4 + (10 + m) λ) − 12m2

× (3 + m) rr0

αγ (4 + (13 + 2m) λ) + mr3r0

α(20 + 55λ

+ 3m2λ + m (6 + 28λ))), (3.2.5)

2r9 (1 + λ) (1 + 2λ)r0

((1 + m) r6 (1 + 2λ) + 4m (3 + m) r4α(6

+ 10λ + m (2 + 3λ)) + 24m2 (3 + m) rr0

αγ (12 + 19λ + m (4 + 6λ))

− 2mr3r0

α(40 + 65λ + 16m (2 + 3λ) + m2 (6 + 9λ)) − 4m2r02(r0

× αγ(15m2 (2 + 3λ) + 33 (7 + 11λ) + m (169 + 256λ)

)). (3.2.6)

Now, we will check the viability of WEC and NEC for various particular choices

of shape function. In order to analyze the viability eras for WEC and NEC with

some particular values of λ, we consider λ = ±2 and calculate the corresponding

expressions for energy density (3.2.4). We plotted these equations for ρ versus r, α

and γ. The plotted regimes ρ > 0 are shown in the left plot of Figure (3.12) by taking

both positive and negative values of α and γ. Further, we found various more regions

in which ρ > 0. These are as follows

• For λ > 1, WEC is valid in some regions and depends upon the sign of α and

γ. For small region 0.1 < r < 1, we required same signs of both α and γ. i.e.,

both positive or negative. For 1 < r < 6, we require positive value of α with

γ < 0 and negative value of α with γ > −70. In order to achieve ρ > 0 regions

Figure 3.12: Evaluation of ρ with respect to r, α, γ for m = 0.5, r0 = 1 with small λvalues.

in 6 < r < 10, α must be fixed less than zero with −100 < γ < 100 as shown in

Figure (3.12).

The NEC validity regions are shown in Figures (3.13) and (3.14) with small λ. De-

pending upon the signs of α and γ in λ < −1, ρ + Pr ≥ 0 is valid in some regions.

For instance, for 0.1 < r < 1, validity of NEC require the same sign of α and γ. For

1 < r < 6, α positive values while γ should attain negative value. Further, in this era,

α < 0 with −100 < γ < 100 also validate NEC with respect to radial pressure com-

ponent. By selecting 6 < r < 10, we require α > −85 along with −100 < γ < 100, as

shown in Figure (3.13). Similarly λ > −1, we get almost the same plots, as discussed

• For λ < −1, the validity of EC ρ + Pt ≥ 0 is controlled by the signs of α and

γ in some zone. For small region 0.1 < r < 5, the validity of WEC needs same

signs of α and γ, i.e., both positive or negative. For 5 < r < 10, the WEC

validity requires α > 0 along with −100 < γ < 100 as shown in Figure (3.13).

Similarly, for WEC validity regions supported by λ > −1, we have obtained

Figure 3.13: Evaluation of ρ+Pr and ρ+Pt with respect to r, α, γ for m = 0.5, r0 = 1and small λ

Figure 3.14: Evaluation of ρ+Pr and ρ+Pt with respect to r, α, γ for m = 0.5, r0 = 1and very small λ

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Figure 3.15: Evaluation of ρ , Pr and Pt for small region

almost the same behaviors of graphs, as mentioned for λ < −1.

The selection m = 1/2 in Eq.(3.1.7) gives those results which can also be analyzed by

taking m = 1, 1/3, 1/5 and so on. By choosing m = −3, the shape function becomes

β(r) = r3r−20 .

If we take this function, then Pr and Pt become independent of r. In this context,

the evolution of ρ, Pr and Pt for m = 0.5, r0 = 1, α = 9 and γ = −20 are shown in

fig.3.15. It is seen from these plots that, for small zones (0.1 < r < 5), ρ > 0, Pr > 0

while Pt < 0.

3.2.1 Equilibrium Condition

Here, we perform an analysis to study equilibrium environment for spherically sym-

metric WH solutions in f(R) gravity. In this respect, the TOV equation can be

recasted as in Eq.(1.7.2). By considering a(r) to be constant quantity, the different

forces for our spherically symmetric WH becomes

Fhf + Faf = 0, (3.2.7)

Fhf =e−

2mr0( r0r )

r16γ4 (1 + λ) (1 + 2λ)(3 + m) r0

(8m6r03(r0

)3m (r0

−r) αλ + 2m5r02(r0

α(−24rr0

+ 24r02(r0

+ 14r4γ

− 17r3r0

γ)λ + r12(e2mr0( r0

r3γ (α − 1) − α)γ4(1 + 2λ) + 2mr9α

× γ3{

10r (2 + 5λ) − 3r0

(7 + 19λ)}

+ m4r0

α(−72rr02

λ + 72r03(r0

λ − 20r7γ2λ + 33r6r0

γ2λ + 4r4r0

γ (4 + 51λ) − 4r3r02(r0

γ(4 + 57λ)) + 2m3r3αγ(r7γ2λ

− 3r6r0

γ2λ − 16r4r0

γ (1 + 6λ) + 12rr02(r0

(2 + 15λ)

− 3r03(r0

(8 + 63λ) + r3r02(r0

γ (18 + 127λ)) + m2r6αγ2(−4r3

γ (3 + 14λ) + 2r4γ (4 + 15λ) − 8rr0

(14 + 53λ)

+ r02(r0

(116 + 469λ))), (3.2.8)

Faf =e−

2mr0( r0r )

r13γ3 (1 + λ)(3 + m) r0

(16m4r02(r0

)2m [−r + r0

− 2mr6(20r − 21r0

)m)αγ2 − r9(e

2mr0( r0r )

r3γ (α − 1) − α)γ3 + 4m3r0

(−12rr0

+ 12r02(r0

+ 8r4γ − 9r3r0

)− 4m2r3αγ

(−28rr0

+ 29r02(r0

+ 2r4γ − 3r3r0

)]. (3.2.9)

The behavior of these forces in Eqs.(3.2.8) and (3.2.9) are mentioned in Figure (3.16).

Here, we have taken m = 0.5, r0 = 1, α = 15, γ = 30 for which the WEC is

compatible. The graphical configurations for positive λ are studied in the left plot,

while the right plot is for negative λ. It is noted that the effects of these forces are

equal but opposite in direction. In order to attain stability, WHs must be configured

20 40 60 80 100r

-2.´ 10-6

-1.´ 10-6

1.´ 10-6

2.´ 10-6

Faf & Fhf

Equilibrium Picture

20 40 60 80 100r

-2.´ 10-6

-1.´ 10-6

1.´ 10-6

2.´ 10-6

Faf & Fhf

Equilibrium Picture

Figure 3.16: Equilibrium conditions with different small λ values.

to cancel the effects of each others.

3.2.2 Isotropic Case

Here, we let the system to be supported by isotropic matter distribution, so we get,

Pr = Pt = P , thereby giving null value to the quantity Π. Equations (3.2.2) and

(3.2.3), after some manipulations yield

r (1 + λ)e−b

[−12r3 α(eb(r2 − 6γ) − γ)b′

3+ 120r4αγb′

4 − 12r2αb′2(eb(r2

− 28γ) + 36γ + 22r2γb′′) + rb′(−e2br4 + 28ebr2α − 12e2br2α − 204αγ

+ 120ebαγ + 84e2bαγ + 4r2α(7eb(r2 − 6γ) + 18γ)b′′ + 48r3αγb(3)) + 2(eb

− 1)(−4ebα(7r2 − 66γ) − 276αγ + e2b(r4 − 4r2α + 12αγ)) + 8r2α(eb(r2

− 18γ) + 18γ)b′′ + 24r4αγb′′2 − 4r3α(eb(r2 − 6γ) + 6γ) b(3))

]= 0. (3.2.10)

This is a third order non-linear differential equation. Now, we use an expression

of b(r) in terms of shape function β(r). This, after substituting and simplifying,

provides another form of again non-linear third order differential equation whose

analytic solution is not possible. We use numerical techniques to solve this equation.

This urges us to take some initial as well as boundary conditions given (for instance)

0 1 2 3 4 5

0.0 0.5 1.0 1.5 2.0

ΒHrL�

Figure 3.17: Isotropic case for α = −0.18, γ = 0.5, Evaluation of β(r) and β(r)/r

as β(1) = 0.1, β′(1) = 0.5 and β(6) = 1. We have then plotted numerical solutions

of this equation. We have also explored three viable conditions on shape function.

The evolution of shape function β(r) verses r and β′(r) verses r are shown in Figure

(3.17).

From left plot of Figure (3.17), one can easily say that shape function β(r) is an

increasing function, thereby obeying the condition β(r) < r. The right plot of this

figure shows the asymptotic behavior as β(r)/r− > 0 for r− > ∞. The throat is

located here at r0 = 0.111252 such that β(r0) = r0. The derivative of shape function

is also plotted in the left of Figure (3.18), from which one can see that β′(r0) < 0 and

hence the condition β′(r0) < 1 is obeyed. The right plot of Figure (3.18) represents

the asymptotic behavior, i.e., β(r) − r < 0 which yields 1 − β(r)/r > 0. One can

easily observe the evolution of WEC and NECs through Figure (3.19). This figure has

shown that spherically symmetric WH model filled with isotropic matter at throat

has obeyed WEC and NEC.

0 1 2 3 4

Β¢ H

0 1 2 3 4 5 6

ΒHrL-

Figure 3.18: Isotropic case, α = −0.18, γ = 0.5, evaluation of β′(r) and β(r) − r.

1 2 3 4 5 6 7

-0.010

-0.005

1 2 3 4 5 6

-0.010

-0.005

Figure 3.19: Evaluation of ρ(r) and ρ + P for α = −0.18, γ = 0.5.

3.2.3 Specific Equation of State

Here, we use a relation connecting energy density ρ with the radial pressure Pr through

a relation, known as EoS or more specifically, we shall use barotropic EoS given by

Pr = kρ giving Pr − kρ = 0. We have

r (1 + λ) (1 + 2λ)

[−ebr2 + e2br2 − ebkr2 + e2bkr2 + 10α − 12ebα + 2e2bα

− 14kα + 12ebkα + 2e2bkα − 2ebr2λ + 2e2br2λ − 2ebkr2λ + 2e2bkr2λ + 18αλ

− 24ebαλ + 6e2bαλ − 30kαλ + 24ebkαλ + 6e2bkαλ + 6 (1 + k) r2α(1 + 2λ)

× b′2+ 3r3α (−2 + (−3 + k) λ) b′

3 − 2r2α (2 + 3λ + k (4 + 7λ)) b′′ + rb′

× (2α (4 + (3 − 5k) λ) + eb(6 (1 + k) αλ + r2 (1 + 2λ)) − 7r2α{+(k − 3)λ − 2}

× b′′) − 4r3αb(3) − 6r3αλb(3) + 2kr3αλ b(3)]

In order to solve above equation, we have obtained transformation between b and β

from Eqs.(3.2.1) and (3.2.2). Using this result and numerical method, we get different

types of behavior of β(r) as shown in plots. We have observed that even in the case

of barotropic EoS, shape function β(r) serves as an increasing function with the WH

throat at r0 = 0.12729 where β(0.12729) = 0.12729 (as shown in left plot of Figure

(3.20). It is also seen from the right plot of Figure (3.20) that β′(0.12729) < 1.

The viability of flaring-out as well as asymptotic conditions can be checked from

the plot of Figure (3.21) in which we have plotted β(r)/r and β(r) − r verses radial

coordinate r. In left plot of Figure (3.21), we have analyzed that β(r)/r− > 0 is not

valid for r− > ∞ while in right plot of Figure (3.21), we have seen the validity of

faring-out condition β(r) − r < 0 at the throat of spherically symmetric WH throat.

On can clearly see from Figure (3.22) that NEC and WEC are not satisfied for WH

model supported by barotropic equation of state. Thus, we inferred that no realistic

1 2 3 4 5 60.0

0 1 2 3 4 5 6

Β¢ H

Figure 3.20: Evaluation of β(r) and β′(r) through EoS having α = −20, γ = 0.8, k =0.001

0 1 2 3 40.00

ΒHrL�

0 1 2 3 4 5

Β¢ H

Figure 3.21: Evaluation of β(r)/r and β(r) − r through EoS with α = −20, γ =0.8, k = 0.001

WH solutions exist in nature at that particular configurations.

1 2 3 4 5 6

-0.004

-0.002

0 1 2 3 4 5 60

Figure 3.22: Evaluation of ρ(r) and ρ+Pt through EoS having α = −20, γ = 0.8, k =0.001

Chapter 4

Compact Stars and DarkDynamical Variables

This chapter is devoted to explore the effects of extra curvature ingredients of f(R, T )

gravity theory on the dynamical variables of the compact spherical star. The matter

contents in the stellar interior are taken to be imperfect due to anisotropic stresses,

shear viscosity, and dissipative terms. A particular form of f(R, T ) function i.e.,

f(R, T ) = f1(R) + f2(R)f3(T ), is utilized to explore the modified field equations.

The Misner-Sharp mass function is generalized by including the higher curvature

ingredients of f(R, T ) theory. We have disintegrated the Weyl tensor, which describes

the distortion in the shapes of celestial objects due to tidal forces, into two parts

named as its electric and magnetic parts. The magnetic part vanishes due to the

symmetry of spherical star and all the tidal effects are due to its electric component.

The results presented in this chapter has been published in an international research

paper (Bhatti et al., 2017).

4.1 Radiating Sphere and f (R, T ) Gravity

Let us consider an irrotational diagonal non-static form of spherically symmetric

metric

ds2 = H2(t, r)dr2 − A2(t, r)dt2 + C2(dθ2 + sin2 θdφ2), (4.1.1)

in which A, B and C depend on t and r. It is assumed that above geometry is being

coupled with radiating shear locally anisotropic fluid represented by

Tλν = µVλVν + P⊥hλν + Πχλχν − 2ησλν + εlλlν + q(χνVλ + χλVν), (4.1.2)

where ε is radiation density, qβ is heat flux, Π ≡ Pr − P⊥, hαβ, σαβ are projection

and shear tensor, P⊥, Pr are tangential and radial pressure elements, µ is the energy

density and η is coefficient of shear viscosity. The projection tensor is defined as

hαβ = gαβ + VαVβ, while χβ and lβ are radial and null four-vectors, respectively.

Under co-moving coordinate system, the definitions of these vectors are found as

V ν = 1Aδν0 , χν = 1

Cδν1 , lν = 1

Aδν0 + 1

Bδν1 , qν = q(t, r)χν . In order to maintain comoving

coordinate frame, these obey relations

χνχν = 1, V νVν = −1, χνVν = 0,

lνVν = −1, V νqν = 0, lνlν = 0.

With reference to Eq. (4.1.1), the shear tensor and scalar corresponding to expansion

tensor are

H− C

), ΘA =

where overdot describes ∂∂t

In order to have observationally well-consistent gravitational theory, one need to

cope appropriate f(R, T ) gravity model. In this perspective, we take the following

combinations of f(R, T ) model (Houndjo and Piattella, 2012)

f(R, T ) = f1(R) + f2(R)f3(T ). (4.1.3)

This form of model description states a minimal background of matter and geometry

coupling, thereby indicating higher order corrections in well-known a f(R) theory.

Realistic f(R, T ) models can be achieved by picking any Ricci scalar function from

Nojiri and Odintsov (2003) along with any linear form of T function. In this context,

we shall take f(R, T ) = R+λR2T 2, where λ ¿ 1. The dynamics proposed by Einstein

can be found on setting λ = 0 in the above model. The f(R, T ) field equations for

Eqs.(4.1.1)-(4.1.3) are

G00 =A2

1 + 2RλT 2

[µ + ε + 2TλR2 − λ

2T 2R2 +

], (4.1.4)

G01 =AH

1 + 2RλT 2

[−(1 + 2TλR2)

1 + 2RλT 2(q + ε) +

], (4.1.5)

G11 =H2

1 + 2RλT 2

[µ2TλR2 + (1 + 2TλR2)(Pr + ε − 4

3ησ) +

2T 2R2 +

(4.1.6)

G22 =C2

1 + 2RλT 2

[(1 + 2TλR2)(P⊥ +

3ησ) + µ2TλR2 +

2T 2R2 +

], (4.1.7)

ψ00 = 2∂ttfR + ∂tfR

∂rfR

(−2AA′ + A2H ′

H− 2A2C ′

ψ11 = −H2

A2∂ttfR +

∂tfR

(−2H2 C

C+ H2 A

A− 2HH

)+ ∂rrfR

C− 2

)∂rfR,

ψ01 = −A′

A∂tfR + ∂t∂rfR − H

H∂rfR,

ψ22 = −C2∂ttfR

A− 3

C− H

)∂tfR +

H2∂rfR

C− H ′

while Gγδ are mentioned in (Herrera et al., 2011a). Here, prime indicates ∂∂r

relativistic fluid 4-velocity, U , can be given as

U = DT C =C

A. (4.1.8)

The spherical mass function via Misner-Sharp formulations can be re-casted as (Mis-

ner and Sharp, 1964)

m(t, r) =C

A2− C ′2

). (4.1.9)

The temporal and radial derivatives of the above equation after using Eqs.(4.1.4)-

(4.1.6) and (4.1.8) are found as follows

DT m =−1

2(1 + 2RλT 2)

{(1 + 2TλR2)(Pr −

3ησ) + 2TλR2µ − λ

2R2T 2 (4.1.10)

{(1 + 2TλR2)

1 + 2RλT 2q − ϕ01

DCm =C2

2(1 + 2RλT 2)

[µ + 2TλR2 − λ

2R2T 2 +

A2− U

{ ϕ01

−(1 + 2TλR2)

1 + 2RλT 2q

}], (4.1.11)

where over bar notation describes X = ε + X, while DT = 1A

∂∂t

. The second equation

from the above set of equations provides

1 + 2RλT 2

[µ + 2TλR2 − λ

2R2T 2 +

A2− U

{ ϕ01

−(1 + 2TλR2)

1 + 2RλT 2q

}]dC, (4.1.12)

where E ≡ C′

H, whose value can be written through mass function as

E ≡ C ′

[1 + U2 − 2m(t, r)

. (4.1.13)

Equations (4.1.10)-(4.1.13) yield

[µ + 2TλR2 − λ

2R2T 2 +

{(1 + 2TλR2)

1 + 2RλT 2q

− ϕ01

}C2C ′

]dr, (4.1.14)

that connects various structural variable elements, like energy density, mass function,

etc with f(R, T ) extra curvature terms. It is well known that in the spherical case,

one can decompose the Weyl tensor into two different tensors, i.e., the magnetic Hαβ

part and the electric Eαβ part. These two are defined respectively as

Hαβ =1

2εαγηδC

ηδβρV

γV ρ = CαγβδVγV δ =, Eαβ = CαφβϕV φV ϕ,

where ελµνω ≡√−gηλµνω with ηλµνω as a Levi-Civita symbol. The electric component

of the Weyl tensor can be expressed through fluid’s 4 vectors as

Eλν =

[χλχν −

3− 1

3VλVν

in which E represents scalar corresponding to the Weyl tesnor. The value of E through

spherical geometric variables are found as

E = − 1

B− C

−[−

A− C ′

)(C ′

C ′′

C− A′′

2B2. (4.1.15)

Another way of writing E with the inclusion of f(R, T ) extra curvature terms is

2(1 + 2RλT 2)

[µ + 2R2λT − (1 + 2R2λT )(Π − 2ησ) − λ

2T 2R2 +

−ϕ11

]− 3

1 + 2RλT 2

[µ + 2R2λT − λ

2T 2R2 +

{(1 + 2R2λT )

1 + 2RλT 2q − ϕ01

}C2C ′

]dr, (4.1.16)

where the bar over Π indicates Π = Pr − P⊥.

4.2 Modified Scalar Variables and f (R, T ) Gravity

Here, we shall compute structure scalars corresponding to radiating spherical bodies

in R +λR2T 2 gravity. In this background, we would use two well-known tensors, i.e.,

Xαβ and Yαβ. These tensors were proposed by Bel (1961) and Herrera et al. (2004,

2009b, 2011a) after orthogonal splitting of Riemann curvature tensor. These are

Xαβ = ∗R∗αγβδV

γV δ =1

2ηερ

αγR∗ερβδV

γV δ, Yαβ = RαγβδVγV δ, (4.2.1)

where steric on the right, left and both sides of the tensor describe operation related

to right, left and double dual of that term, respectively. These tensors with the help

of 4-vector Vα and projection tensor, hαβ, can be written as

Xαβ =1

3XT hαβ + XTF

(χαχβ − 1

3hαβ

), (4.2.2)

Yαβ =1

3YT hαβ + YTF

(χαχβ − 1

3hαβ

), (4.2.3)

here XT and YT indicate trace parts of the tensors Xαβ and Yαβ, respectively, while

XTF and YTF stand for the trace-free components of the tensors Xαβ and Yαβ, re-

spectively. Using Eqs.(4.1.4)-(4.1.8), (4.2.2) and (4.2.3), we obtain

1 + 2RλT 2

{µ + 2R2λT +

2R2T 2

}, (4.2.4)

XTF = −E − 1

2(1 + 2RλT 2)

{(2R2λT + 1)(−2ση + Π) − ϕ22

}, (4.2.5)

2(1 + 2RλT 2)

{6µR2λT + µ + 2R2λT + 3(1 + 2R2λT )Pr − 2Π(2R2λT + 1)

C2+ +2λT 2R2

}, (4.2.6)

YTF = E − 1

2(1 + 2RλT 2)

{(Π − 2ησ)(2R2λT + 1) − ϕ22

}. (4.2.7)

The value of YTF can be followed from Eqs.(4.1.16) and (4.2.7) as

YTF =1

2(1 + 2RλT 2)

(µ + 2R2λT − 2(1 + 2R2λT )(Π − 4ησ) +

2T 2R2

A2− 2ϕ11

)− 3

1 + 2RλT 2

[µ + 2R2λT

2T 2R2 +

{(1 + 2R2λT )

1 + 2RλT 2q − ϕ01

}C2C ′

]dr. (4.2.8)

One can define few particular collections of fluid and dark source terms as dagger

variables as

µ† ≡ µ + 2R2λT +ϕ00

A2, P †

r ≡ Pr +ϕ11

H2− 4

3ησ,

P †⊥ ≡ P⊥ +

3ησ,

Π† ≡ P †r − P †

⊥ = Π − 2ησ − ϕ22

In this context, it follows from Eqs.(4.2.4)-(4.2.7) that

XTF =3κ

{1 + 2RλT 2}

{µ† − λ

2T 2R2 +

(q − ϕq

}×C2C ′] dr − 1

2{1 + 2RλT 2}

[µ† − λ

2T 2R2

], (4.2.9)

YTF =1

2(1 + 2RλT 2)

[µ† − λ

2T 2R2 − 2(1 + 2R2λT )Π† + 4R2λT

×(ϕ11

H2− ϕ22

)]− 3

{1 + 2RλT 2}{µ†

2T 2R2 +

(q − ϕq

}C2C ′

]dr, (4.2.10)

2(1 + 2RλT 2)

[(1 + 6R2λT )µ† − 6εR2λT + 3(1 + 2R2λT )P †

r − 2(1 + 2R2λT )Π†

−2R2λT(ϕ11

)+ 2(2 + 2R2λT )

C2− 2λT 2R2

], (4.2.11)

(1 + 2RλT 2)

[µ† − λ

2T 2R2

]. (4.2.12)

The GR structure scalars (Herrera et al., 2009b, 2011a) can be retrieved by taking

f(R, T ) = R in the above equations. These quantities have utmost relevance in the

study of some important dynamical features of self-gravitating objects, for instance

IED, quantity of matter content. In order to understand the the role of f(R, T ) terms

on the shear and expansion evolution of radiating relativistic interiors, we shall like to

compute Raychaudhuri equations. These relations were also evaluated independently

by Landau (Albareti et al., 2014). With the help of f(R, T ) structure scalars, one

can write

−(YT ) =Θ2

3σαβσαβ + V αΘ;α − aα

;α, (4.2.13)

thereby describing the importance of one of the f(R, T ) scalar functions in the mod-

eling of expansion scalar evolution equation. In the similar fashion, we shall calculate

shear evolution equation as

YTF = a2 + χαa;α − aC ′

BC− 2

3Θσ − V ασ;α − 1

3σ2. (4.2.14)

It is pertinent to mention that this equation has been expressed successfully via

f(R, T ) structure scalar, YTF . Using field equations and Eq.(4.2.9), one can write the

differential equation[XTF +

κµ†

2(1 + 2RλT 2)

= −XTF3C ′

κ(Θ − σ)

2(1 + 2RλT 2)

(qB − ϕq

). (4.2.15)

On solving it for XTF , one can identify that it is the XTF which is controlling IED

of the spherical dissipative celestial bodies.

4.3 Evolution Equations with Constant R and T

In this section, we shall investigate the influences of R + λR2T 2 corrections on the

formulations of shear, expansion and Weyl evolution equation for the relativistic dust

cloud with constant curvature quantities. In order to represents constant values of R

and T , we shall use the tilde over the corresponding mathematical quantities. In this

framework, the spherical mass function in the presence of R + λR2T 2 corrections is

found to be

2{1 + 2RλT 2}

(µ + 2TλR2)C2C ′dr − λR2T 2

2{1 + 2RλT 2}

C2C ′dr, (4.3.1)

while the Weyl scalar turns out to be

2C3{1 + 2RλT 2}

µ′C3dr − λR2T 2

4{1 + 2RλT 2}. (4.3.2)

The widely known equation relating spherical mass with radiating structural param-

eters can be recasted as

2{1 + 2RλT 2}

[µ + 2TλR2 − 1

µ′C3dr

λR2T 2

2{1 + 2RλT 2}. (4.3.3)

The f(R, T ) structure scalars with R + λR2T 2 corrections boil down to be

{1 + 2RλT 2}

[µ + 2TλR2 − λ

2R2T 2

], (4.3.4)

YTF = −XTF = E , (4.3.5)

2{1 + 2RλT 2}[µ + 2TλR2 + 6µTλR2 − 2λR2T 2

]. (4.3.6)

These equations indicate that XT , YT and YTF , XTF are controlling effects induced

by fluid energy density and tidal forces caused by Weyl scalar, respectively in an en-

vironment of f(R, T ) extra degrees of freedom. An equation describing the evolution

of inhomogeneity factors in the emergence of IED for dust fluid is[µ

2{1 + 2RλT 2}− λT 2R2

4{1 + 2RλT 2}+ XTF

= − 3

CXTF C ′. (4.3.7)

This equation involves XTF that was pointed out to be inhomogeneity factor in the

context of GR. It is seen from the above equation that µ = µ(t) if and only if

XTF = 0 = λ. This shows that, even in R + λR2T 2 gravity, XTF is a IED factor. In

the famework of non-interacting particles evolving with constant R and T , the shear

as well as expansion evolution equations turn out to be

V αΘ;α +2

3σ2 +

3− aα

;α =1

{1 + 2RλT 2}

[µ + 2TλR2 − λ

2T 2R2

]= −YT , (4.3.8)

V ασ;α +σ2

3σΘ = −E = −YTF . (4.3.9)

These equations have been expressed with the help of YTF and YT . It is worthy to

notice that one can use these set of equations to check the structure formation of

compact objects under expansion-free scenario (Herrera et al., 2008, 2009a; Sharif

and Yousaf, 2013).

The study of compact objects is amongst the most burning issues of our myste-

rious dark universe in which, stars came into being during the dying phenomenon

of relativistic massive stars. Such celestial bodies are having size as a big city and

generally contain mass atleast 40% more mass than solar mass. Due to this fact, their

core density exceeds the density of an atomic nucleus. This specifies that the compact

stars could be treated as test particle to study some physical features beyond nuclear

density.

Rossi X-ray Timing Explorer gathered information based on satellite observations

about the structure of a neutron star, named 4U1820-30. They found mass of this

star to be 2.25M¯ containing high amount of exotic matter. We now apply our results

of dynamical dark variables on the observational values of this compact star. As our

f(R, T ) field equations are non-linear in nature, therefore we suppose that our star

consists of non-interacting particles. We suppose that our geometry is demarcated

with the three-dimensional boundary surface. The interior to that is given by (4.1.1),

Figure 4.1: Plot of the dynamical variable YT for the strange star candidate 4U1820-30.

Figure 4.2: Behavior of the dynamical variable XT for the strange star candidate 4U1820-30.

Figure 4.3: Role of the dark dynamical variable XTF on the evolution of the strangestar candidate 4U 1820-30.

Figure 4.4: Plot for the dark dynamical variable YTF on the evolution of the strangestar candidate 4U 1820-30.

while the exterior vacuum geometry is given by

ds2+ = −Z2dν2 + Z−1dρ2 + ρ2(dθ2 + sin2 θdφ2). (4.3.10)

where Z =(1 − 2M

)with M and ν are total matter content and retarded time,

respectively. We use Darmois junction conditions (Darmois, 1927) to make continuous

connections between Eqs.(4.1.1) and (4.3.10) over hypersurface. These conditions,

after some manipulations, provide

AdtΣ= dν

(1 − 2

Σ= ρ(ν), (4.3.11)

MΣ= m(t, r), (4.3.12)

These constraints should be fulfilled by both manifolds in order to remove jumps over

the boundary.

It is well-known from the literature that the dynamical variable, YT has the same

role as that of the Tolman mass density in the evolutionary phases of those relativistic

systems which are in the state of equilibrium or quasi-equilibrium. Figures (4.1)

and (4.2) state the evolution of YT and XT variables with the increase of r and T ,

respectively. Other very important dark scalar functions are ˜XTF and ˜YTF . These

two variables have opposite behaviors on the dynamical phases of our relativistic 4U

1820-30 star candidate. The modified structure scalar ˜XTF is controlling appearance

of inhomogeneities on the initially regular compact object. It can be observed from

the Figure (4.3) that the inhomogeneity of the compact star keep on decreasing by

increasing the radial coordinate of the spherical self-gravitating object. The totally

reverse behavior of YTF can be observed from Figure (4.4).

Chapter 5

Concluding Remarks

This thesis is devoted to explore some stable configurations of celestial geometries

coupled with relativistic matter contents in various MGTs. The investigation of such

celestial models is of great value in understanding widely different physical gravita-

tional phenomena. The significance of analyzing ECs lies in the fact that the math-

ematical modeling of many stellar bodies could involve a number of various types

of matter contents. Several types of relativistic matter configurations are reported

to obey ECs in order to present realistic stellar models. The aim of this work is to

analyze the behavior of ECs as well as the stability of celestial objects by considering

various spacetimes coupled with some specific forms of matter. The extra curvature

quantities coming from different MGTs could be interpreted as a gravitational fluid,

supporting some non-standard astrophysical models, like WHs. The obtained results

may provide a promising way to understand the accelerating cosmic expansion and

the formation of large structures.

Chapter TWO aims to explore some realistic configurations of anisotropic spher-

ical structures in the background of metric f(R), f(R, T ) and f(G, T ) gravity. We

consider the relativistic compact stellar objects whose interior geometry is based on

anisotropy within the framework of modified gravity. We have used the Krori-Barua

solutions for metric functions of the spherical star whose arbitrary constants are ex-

plored over the boundary surface by matching it with the suitable exterior. The

arbitrary constants of the Krori-Barua solutions can be written in the form of mass

and radius for any compact star. We have used the observational data of three partic-

ular star models to explore the influence of extra degrees of freedom on CSs. For this

purpose, few physically viable gravity models are used. By using the values of these

star and gravity models, we have plotted the material variables like energy density

and anisotropic stresses against radial distance. It is found that as the radius of the

star increases, the density tends to decrease, thereby indicating the maximum dense

configurations of stellar interiors. A similar behavior is analyzed in the evolutionary

phases of the tangential and radial pressures.

It is seen that the first radial derivative of the material variables vanishes at r = 0

for all the CS. It is found that our spherically symmetric anisotropic systems obey

NECs, WECs, SECs and DECs, hence the CSs under the effects of extra degrees

of freedom modified gravity is physically valid. We have seen that the gravitational

forces in the overcome the corresponding repulsive forces, thus indicating the collaps-

ing nature of compact relativistic structures. It is well-known that a stellar system

would be stable against fluctuations if it satisfies the bounds of [0,1] for radial and

tangential sound speeds. Figures (2.8), (2.20) and (2.31) indicate that all of our stel-

lar models are stable (i.e., v2sr > v2

st). Further, we have observed that the equation

of state parameter lies in the interval (0,1) for all the CSs. The anisotropic param-

eter remains positive which is necessary for a realistic stellar configuration. We can

conclude our discussion as follows.

• The anisotropic stresses and energy density are positive throughout the star

configurations.

• The r−derivatives of density and anisotropic stresses (i.e., density and pressure

gradients) remain negative.

• All types of ECs are valid.

• The sound speeds remain within the bounds of [0, 1] (i.e, compact stars are

stable).

• The equation of state parameter lies between 0 and 1 for each star radius.

• The measure of anisotropy remains positive at the star core.

Chapter THREE explores the possibility of the existence of wormhole geometries

coupled with relativistic matter configurations by taking a particular model of f(R, T )

gravity. For this purpose, we take the static form of spherically symmetric spacetime

and after assuming a specific form of matter and combinations of shape function,

the validity of energy conditions is checked. We have discussed our results through

graphical representation and studied the equilibrium background of WH models by

taking an anisotropic fluid. The extra curvature quantities coming from f(R, T )

gravity could be interpreted as a gravitational entity supporting these non-standard

astrophysical WH models. We have shown that in the context of anisotropic fluid

with R + αR2 + λT and R + αR2(1 + γR) + λT gravity, WH models could possibly

exist in few zones in the space of parameters without the need of exotic matter.

Chapter FOUR is devoted to explore the dark dynamical effects of f(R, T ) grav-

ity theory on the dynamics of compact celestial stars. We have taken the interior

geometry as a spherical star which is filled with imperfect fluid distribution. The

modified field equations are explored by taking a particular form of f(R, T ) model,

i.e., f(R, T ) = f1(R) + f2(R)f3(T ). These equations are then utilized to formulate

the well-known structure scalars under the dark dynamical effects of this higher order

gravity theory. Also, the evolution equations for expansion and shear are formulated

with the help of these scalar variables. Further, all this analysis have been made

under the condition of constant R and T . We found a crucial significance of the dark

source terms and of dynamical variables on the evolution and density inhomogeneity

of compact objects.

Appendix A

In order to draw Figures (2.11), (2.12), (2.13), (2.14), (2.15), (2.16), (2.18), and (2.19)

for CS1, CS2 and CS3 under model 1, we have used the following mathematical cod-

ρc1 = (1/(r∧4)) E∧(-0.028 r∧2) (E∧(0.028 r∧2) (-0.06 + 1. r∧2) + E∧(0.021 r∧2)

(0.37 - 1. r∧2 + 0.014 r∧4 + (2.4*10∧(-7)) r∧6) + E∧(0.014 r∧2) (-0.31 - 0.0017 r∧2 -

0.000034 r∧4 - (1.4*10∧(-7)) r∧6 + (4.*10∧(-10)) r∧8));

ρc2 = (1/(r∧4)) E∧(-0.073 r∧2) (E∧(0.073 r∧2) (-0.06 + 1. r∧2) + E∧(0.055 r∧2)

(0.37 - 1. r∧2 + 0.036 r∧4 + (1.1*10∧(-6)) r∧6) + E∧(0.036 r∧2) (-0.31 - 0.0045 r∧2 -

0.000057 r∧4 - (4.1*10∧(-6)) r∧6 + (1.2*10∧(-8)) r∧8));

ρc3 = (1/(r∧4)) E∧(-0.044 r∧2) (E∧(0.044 r∧2) (-0.06 + 1. r∧2) + E∧(0.033 r∧2)

(0.37 - 1. r∧2 + 0.022 r∧4 + (2.2*10∧(-7)) r∧6) + E∧(0.022 r∧2) (-0.31 - 0.0027 r∧2

+ (7.6*10∧(-6)) r∧4 - (9.4*10∧(-7)) r∧6 + (8.9*10∧(-10)) r∧8));

prc1 = (1/(r∧4)) E∧(-0.028 r∧2) (E∧(0.028 r∧2) (0.06 - 1. r∧2) + E∧(0.021 r∧2)

(0.37 + 1. r∧2 + 0.009 r∧4 - (2.4*10∧(-7)) r∧6) + E∧(0.014 r∧2) (-0.43 - 0.0033 r∧2 -

0.00001 r∧4 + (2.1*10∧(-8)) r∧6 + (7.2*10∧(-11)) r∧8));

prc2 = (1/(r∧4)) E∧(-0.073 r∧2) (E∧(0.073 r∧2) (0.06 - 1. r∧2) + E∧(0.055 r∧2)

(0.37 + 1. r∧2 + 0.031 r∧4 - (1.1*10∧(-6)) r∧6) + E∧(0.036 r∧2) (-0.43 - 0.0089 r∧2 -

0.00014 r∧4 - (4.*10∧(-7)) r∧6 + (3.1*10∧(-9)) r∧8));

prc3 = (1/(r∧4)) E∧(-0.044 r∧2) (E∧(0.044 r∧2) (0.06 - 1. r∧2) + E∧(0.033 r∧2)

(0.37 + 1. r∧2 + 0.021 r∧4 - (2.2*10∧(-7)) r∧6) + E∧(0.022 r∧2) (-0.43 - 0.0053 r∧2 -

0.000058 r∧4 - (1.9*10∧(-7)) r∧6 + (2.6*10∧(-10)) r∧8));

ptc1 = (1/(r∧4)) E∧(-0.028 r∧2) (E∧(0.028 r∧2) (-0.063 - (1.2*10∧(-16)) r∧2) + E∧(0.021

r∧2) (-0.37 - 0.00099 r∧2 + 0.0019 r∧4 - 0.000012 r∧6) + E∧(0.014 r∧2) (0.43 + 0.0044

r∧2 + 0.000026 r∧4 + (9.2*10∧(-8)) r∧6 - (3.1*10∧(-10)) r∧8));

ptc2 = (1/(r∧4)) E∧(-0.073 r∧2) (E∧(0.073 r∧2) (-0.063 - (1.2*10∧(-16)) r∧2) + E∧(0.055

r∧2) (-0.37 - 0.0013 r∧2 + 0.013 r∧4 - 0.000052 r∧6) + E∧(0.036 r∧2) (0.43 + 0.01 r∧2

+ 0.000088 r∧4 + (2.8*10∧(-6)) r∧6 - (8.7*10∧(-9)) r∧8));

ptc3 = (1/(r∧4)) E∧(-0.044 r∧2) (E∧(0.044 r∧2) (-0.063 - (1.2*10∧(-16)) r∧2) + E∧(0.033

r∧2) (-0.37 - 0.00038 r∧2 + 0.0097 r∧4 - 0.000011 r∧6) + E∧(0.022 r∧2) (0.43 + 0.0058

r∧2 + 0.000019 r∧4 + (6.3*10∧(-7)) r∧6 - (6.1*10∧(-10)) r∧8));

dρbydrc1 = (1/(r∧5)) E∧(-0.028 r∧2) (E∧(0.028 r∧2) (0.24 - 2. r∧2) + E∧(0.021 r∧2)

(-1.5 + 2. r∧2 + 0.014 r∧4 - 0.00019 r∧6 - (3.3*10∧(-9)) r∧8) + E∧(0.014 r∧2) (1.2

+ 0.012 r∧2 + 0.000048 r∧4 + (6.6*10∧(-7)) r∧6 + (5.4*10∧(-9)) r∧8 - (1.1*10∧(-11))

r∧10));

dρbydrc2 = (1/(r∧5)) E∧(-0.073 r∧2) (E∧(0.073 r∧2) (0.24 - 2. r∧2) + E∧(0.055

r∧2) (-1.5 + 2. r∧2 + 0.037 r∧4 - 0.0013 r∧6 - (3.9*10∧(-8)) r∧8) + E∧(0.036 r∧2) (1.2

+ 0.031 r∧2 + 0.00033 r∧4 - (4.*10∧(-6)) r∧6 + (3.5*10∧(-7)) r∧8 - (9.*10∧(-10)) r∧10));

dρbydrc3 = (1/(r∧5)) E∧(-0.044 r∧2) (E∧(0.044 r∧2) (0.24 - 2. r∧2) + E∧(0.033 r∧2)

(-1.5 + 2. r∧2 + 0.022 r∧4 - 0.00047 r∧6 - (4.7*10∧(-9)) r∧8) + E∧(0.022 r∧2) (1.2 +

0.019 r∧2 + 0.00012 r∧4 - (2.2*10∧(-6)) r∧6 + (4.5*10∧(-8)) r∧8 - (3.9*10∧(-11)) r∧10));

dprbydrc1 = (1/(r∧5)) E∧(-0.028 r∧2) (E∧(0.028 r∧2) (-0.24 + 2. r∧2) + E∧(0.021

r∧2) (-1.5 - 2. r∧2 - 0.014 r∧4 - 0.00012 r∧6 + (3.3*10∧(-9)) r∧8) + E∧(0.014 r∧2) (1.7

+ 0.018 r∧2 + 0.000092 r∧4 + (3.3*10∧(-7)) r∧6 - (3.*10∧(-10)) r∧8 - (2.*10∧(-12))

r∧10));

r∧2) (-1.5 - 2.1 r∧2 - 0.037 r∧4 - 0.0011 r∧6 + (3.9*10∧(-8)) r∧8) + E∧(0.036 r∧2) (1.7

+ 0.049 r∧2 + 0.00065 r∧4 + (9.3*10∧(-6)) r∧6 + (4.2*10∧(-8)) r∧8 - (2.3*10∧(-10))

r∧10));

r∧2) (-1.5 - 2. r∧2 - 0.022 r∧4 - 0.00045 r∧6 + (4.7*10∧(-9)) r∧8) + E∧(0.022 r∧2) (1.7

+ 0.029 r∧2 + 0.00023 r∧4 + (2.2*10∧(-6)) r∧6 + (9.4*10∧(-9)) r∧8 - (1.1*10∧(-11))

r∧10));

dptbydrc1 = (1/(r∧5)) E∧(-0.028 r∧2) (E∧(0.028 r∧2) (0.25 + (2.4*10∧(-16)) r∧2)

+ E∧(0.021 r∧2) (1.5 + 0.0071 r∧2 + 0.000014 r∧4 - 0.00005 r∧6 + (1.6*10∧(-7)) r∧8)

+ E∧(0.014 r∧2) (-1.7 - 0.021 r∧2 - 0.00012 r∧4 - (5.2*10∧(-7)) r∧6 - (3.8*10∧(-9)) r∧8

+ (8.5*10∧(-12)) r∧10));

dptbydrc2 = (1/(r∧5)) E∧(-0.073 r∧2) (E∧(0.073 r∧2) (0.25 + (2.4*10∧(-16)) r∧2)

+ E∧(0.055 r∧2) (1.5 + 0.016 r∧2 + 0.000046 r∧4 - 0.00057 r∧6 + (1.9*10∧(-6)) r∧8)

+ E∧(0.036 r∧2) (-1.7 - 0.052 r∧2 - 0.00075 r∧4 - (8.4*10∧(-7)) r∧6 - (2.4*10∧(-7)) r∧8

+ (6.4*10∧(-10)) r∧10));

dptbydrc3 = (1/(r∧5)) E∧(-0.044 r∧2) (E∧(0.044 r∧2) (0.25 + (2.4*10∧(-16)) r∧2)

+ E∧(0.033 r∧2) (1.5 + 0.0088 r∧2 + (8.4*10∧(-6)) r∧4 - 0.00023 r∧6 + (2.3*10∧(-7))

r∧8) + E∧(0.022 r∧2) (-1.7 - 0.03 r∧2 - 0.00025 r∧4 + (4.5*10∧(-7)) r∧6 - (3.*10∧(-8))

r∧8 + (2.7*10∧(-11)) r∧10));

vsrc1 = (E∧(0.027625 r∧2) (2.1497*10∧(10) - (1.8295*10∧(11)) r∧2) + E∧(0.020719

r∧2) (1.3173*10∧(11) + (1.834*10∧(11)) r∧2 + (1.2635*10∧(9)) r∧4 + (1.1164*10∧(7))

r∧6 - 296.43 r∧8) + E∧(0.013813 r∧2) (-1.5322*10∧(11) - (1.6579*10∧(9)) r∧2 - (8.2836

10∧(6)) r∧4 - 29455. r∧6 + 26.561 r∧8 + 0.17857 r∧10))/(E∧(0.027625 r∧2) (-2.1497*10∧(10)

+ (1.8295*10∧(11)) r∧2) + E∧(0.020719 r∧2) (1.3173*10∧(11) - (1.825*10∧(11)) r∧2

- (1.2635*10∧(9)) r∧4 + (1.7073*10∧(7)) r∧6 + 296.43 r∧8) + E∧(0.013813 r∧2) (-

1.1023*10∧(11) - (1.0713*10∧(9)) r∧2 - (4.2823*10∧(6)) r∧4 - 58985. r∧6 - 479.93 r∧8

+ 1. r∧10));

vsrc2 = (E∧(0.0729263 r∧2) (2.66231*10∧(8) - (2.26579*10∧(9)) r∧2) + E∧(0.0546947

r∧2) (1.63137*10∧(9) + (2.28061*10∧(9)) r∧2 + (4.13079*10∧(7)) r∧4 + (1.27014*10∧(6))

r∧6 - 42.9183 r∧8) + E∧(0.0364631 r∧2) (-1.8976*10∧(9) - (5.42644*10∧(7)) r∧2 -

717160. r∧4 - 10315.5 r∧6 - 46.3716 r∧8 + 0.2543 r∧10))/(E∧(0.0729263 r∧2) (-

2.66231*10∧(8) + (2.26579*10∧(9)) r∧2) + E∧(0.0546947 r∧2) (1.63137*10∧(9) - (2.25087

10∧(9)) r∧2 - (4.13079*10∧(7)) r∧4 + (1.46548*10∧(6)) r∧6 + 42.9183 r∧8) + E∧(0.0364631

r∧2) (-1.36514*10∧(9) - (3.4963*10∧(7)) r∧2 - 367344. r∧4 + 4386.21 r∧6 - 383.719

r∧8 + 1. r∧10));

vsrc3 = (E∧(0.043626 r∧2) (6.1584*10∧(9) - (5.2412*10∧(10)) r∧2) + E∧(0.032719 r∧2)

(3.7736*10∧(10) + (5.2617*10∧(10)) r∧2 + (5.7162*10∧(8)) r∧4 + (1.166*10∧(7)) r∧6 -

120.67 r∧8) + E∧(0.021813 r∧2) (-4.3895*10∧(10) - (7.5128*10∧(8)) r∧2 - (5.9451*10∧(6))

r∧4 - 55335. r∧6 - 240.13 r∧8 + 0.29175 r∧10))/(E∧(0.043626 r∧2) (-6.1584*10∧(9)

+ (5.2412*10∧(10)) r∧2) + E∧(0.032719 r∧2) (3.7736*10∧(10) - (5.2205*10∧(10)) r∧2

- (5.7162*10∧(8)) r∧4 + (1.2105*10∧(7)) r∧6 + 120.67 r∧8) + E∧(0.021813 r∧2) (-

3.1578*10∧(10) - (4.8343*10∧(8)) r∧2 - (3.0325*10∧(6)) r∧4 + 56734. r∧6 - 1143.7 r∧8

+ 1. r∧10));

vstc1 = (E∧(0.027625 r∧2) (-2.2412*10∧(10) - 0.00002115 r∧2) + E∧(0.020719 r∧2)

(-1.3173*10∧(11) - (6.323*10∧(8)) r∧2 - (1.2254*10∧(6)) r∧4 + (4.4978*10∧(6)) r∧6 -

14525. r∧8) + E∧(0.013813 r∧2) (1.5414*10∧(11) + (1.8516*10∧(9)) r∧2 + (1.0872*10∧(7))

r∧4 + 46761. r∧6 + 337.69 r∧8 - 0.76026 r∧10))/(E∧(0.027625 r∧2) (-2.1497*10∧(10)

+ (1.8295*10∧(11)) r∧2) + E∧(0.020719 r∧2) (1.3173*10∧(11) - (1.825*10∧(11)) r∧2

- (1.2635*10∧(9)) r∧4 + (1.7073*10∧(7)) r∧6 + 296.43 r∧8) + E∧(0.013813 r∧2) (-

1.1023*10∧(11) - (1.0713*10∧(9)) r∧2 - (4.2823*10∧(6)) r∧4 - 58985. r∧6 - 479.93 r∧8

+ 1. r∧10));

vstc2 = (E∧(0.0729263 r∧2) (-2.7756*10∧(8) - (2.6193*10∧(-7)) r∧2) + E∧(0.0546947

r∧2) (-1.63137*10∧(9) - (1.76619*10∧(7)) r∧2 - 50878.6 r∧4 + 630004. r∧6 - 2103.

r∧8) + E∧(0.0364631 r∧2) (1.90893*10∧(9) + (5.75251*10∧(7)) r∧2 + 828525. r∧4 +

936.12 r∧6 + 263.153 r∧8 - 0.707083 r∧10))/(E∧(0.0729263 r∧2) (-2.66231*10∧(8) +

(2.26579*10∧(9)) r∧2) + E∧(0.0546947 r∧2) (1.63137*10∧(9) - (2.25087*10∧(9)) r∧2

- (4.13079*10∧(7)) r∧4 + (1.46548*10∧(6)) r∧6 + 42.9183 r∧8) + E∧(0.0364631 r∧2)

(-1.36514*10∧(9) - (3.4963*10∧(7)) r∧2 - 367344. r∧4 + 4386.21 r∧6 - 383.719 r∧8 +

1. r∧10));

vstc3 = (E∧(0.0436258 r∧2) (-6.42043*10∧(9) - (6.03662*10∧(-6)) r∧2) + E∧(0.0327193

r∧2) (-3.77364*10∧(10) - (2.25537*10∧(8)) r∧2 - 215427. r∧4 + (5.95716*10∧(6))

r∧6 - 5912.83 r∧8) + E∧(0.0218129 r∧2) (4.41568*10∧(10) + (7.77155*10∧(8)) r∧2

+ (6.44704*10∧(6)) r∧4 - 11430.9 r∧6 + 772.095 r∧8 - 0.686299 r∧10))/(E∧(0.0436258

r∧2) (-6.15837*10∧(9) + (5.24117*10∧(10)) r∧2) + E∧(0.0327193 r∧2) (3.77364*10∧(10)

- (5.22055*10∧(10)) r∧2 - (5.71621*10∧(8)) r∧4 + (1.21047*10∧(7)) r∧6 + 120.67 r∧8)

+ E∧(0.0218129 r∧2) (-3.1578*10∧(10) - (4.83426*10∧(8)) r∧2 - (3.03247*10∧(6)) r∧4

+ 56734.4 r∧6 - 1143.69 r∧8 + 1. r∧10));

Plot[{ρc1, ρc2, ρc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red, Green}, AxesLabel ->

{“r”, “ρ”}, PlotLegends -> Placed[PointLegend[{“CS1”, “CS2”, “CS3”}, Legend-

Function -> “Frame”, LegendLayout -> “Column”], {{1.2, 1}, {0.5, 1}}], PlotLabel

-> “Model 1”]

Plot[{prc1, prc2, prc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red, Green}, AxesLabel ->

{“r”, “pr”}, PlotLegends -> Placed[PointLegend[{“CS1”, “CS2”, “CS3”}, Legend-

-> “Model 1”]

Plot[{ptc1, ptc2, ptc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red, Green}, AxesLabel ->

{“r”, “pt”}, PlotLegends -> Placed[PointLegend[{“CS1”, “CS2”, “CS3”}, Legend-

-> “Model 1”]

Plot[{dρbydrc1, dρbydrc2, dρbydrc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red, Green},AxesLabel -> {“r”, “dρ/dr”}, PlotLegends -> Placed[PointLegend[{“CS1”, “CS2”,

“CS3”}, LegendFunction -> “Frame”, LegendLayout -> “Column”], {{1.2, 1}, {0.5,

1}}], PlotLabel -> “Model 1”]

Plot[{dprbydrc1, dprbydrc2, dprbydrc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red,

Green}, AxesLabel -> {“r”, “dpr/dr”}, PlotLegends -> Placed[PointLegend[{“CS1”,

“CS2”, “CS3”}, LegendFunction -> “Frame”, LegendLayout -> “Column”], {{1.2,

1}, {0.5, 1}}], PlotLabel -> “Model 1”]

Plot[{dptbydrc1, dptbydrc2, dptbydrc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red,

Green}, AxesLabel -> {“r”, “dpt/dr”}, PlotLegends -> Placed[PointLegend[{“CS1”,

“CS2”, “CS3”}, LegendFunction -> “Frame”, LegendLayout -> “Column”], {{1.2,

1}, {0.5, 1}}], PlotLabel -> “Model 1”]

Plot[{vsrc1, vsrc2, vsrc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red, Green}, Axes-

Label -> {“r”, “v2sr”}, PlotLegends -> Placed[PointLegend[{“CS1”, “CS2”, “CS3”},

LegendFunction -> “Frame”, LegendLayout -> “Column”], {{1.2, 1}, {0.5, 1}}],PlotLabel -> “Model 1”]

Plot[{vstc1, vstc2, vstc3}, {r, 0.1, 10}, PlotStyle -> {Blue, Red, Green}, Axes-

Label -> {“r”, “v2st”}, PlotLegends -> Placed[PointLegend[{“CS1”, “CS2”, “CS3”},

LegendFunction -> “Frame”, LegendLayout -> “Column”], {{1.2, 1}, {0.5, 1}}],PlotLabel -> “Model 1”]

Appendix B

In order to draw Figures (3.1), (3.2), and (3.3) with corrections mentioned in Eq.(2.5.2),

we have used the following mathematical coding.

ρ = -((0.5 (1/r)∧6.5 ((1/r)∧0.5 α (-22.25 - 32.625 λ) + r∧3 (0.5 + λ) + r α (17.5

+ 26.25 λ)))/((1. + λ) (0.5 + 1. λ)));

ρppr = ((1/r)∧3.5 (-0.75 - 1.5 λ) + α (-15.75 (1/r)∧5.5 + 16.875 (1/r)∧7. - 31.5

(1/r)∧5.5 λ + 33.75 (1/r)∧7. λ))/((1. + λ) (0.5 + 1. λ));

ρppt = ((1/r)∧3.5 (0.125 + 0.25 λ) + α (3.5 (1/r)∧5.5 - 3.25 (1/r)∧7. + 7. (1/r)∧5.5

λ - 6.5 (1/r)∧7. λ))/((1. + λ) (0.5 + 1. λ));

RegionPlot3D[ρ >= 0, {r, 0.1, 10}, {α, 0.1, 10}, {λ, -10, -2}, PlotRange -> Au-

tomatic, Mesh -> 10, AxesLabel -> Automatic, BoxRatios -> {1, 2, 2}, PlotStyle ->

{Orange, Blue, Red} , PlotLabel -> “ρ > 0”]

RegionPlot3D[ρppr >= 0, {r, 0.1, 10}, {α, 0.1, 10}, {λ, 1, 10}, PlotRange -> Auto-

matic, Mesh -> 10, AxesLabel -> Automatic, BoxRatios -> {1, 2, 2}, PlotStyle ->

RegionPlot3D[ρppt >= 0, {r, 0.1, 10}, {α, 0.1, 10}, {λ, 1, 10}, PlotRange -> Auto-

matic, Mesh -> 10, AxesLabel -> Automatic, BoxRatios -> {1, 2, 2}, PlotStyle ->

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