Minimally Coupled Scalar Field Cosmology in Higher-Order Theories of Gravity

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Candidate: Ciprian SPOREA

Scientific Coordinator:

Prof. Univ. Dr. Dumitru VULCANOV

Minimally Coupled Scalar Field Cosmology in Higher-Order Theories of Gravity

Master Thesis

Timisoara,July 2013

2Master Thesis: Ciprian Sporea

Outline

Theory of a scalar field minimally coupled with Gravity

Scalar field cosmology non-minimally coupled with Gravity

REM for a scalar field minimally coupled with f(R) Gravity

Computational and numerical analysis of the models

3 Appendices


THEORY OF A SCALAR FIELD MINIMALLY

COUPLED WITH GRAVITY

Chapter 1


Theoretical Framework

Principle of least action

Chapter 1

Variation of S

Field equations

The stress-energy tensor

with respect to


Friedman EquationsChapter 1

Einstein equations

FRW metric

The scalar field

The matter field


Friedman EquationsChapter 1

The total stress-energy tensor

Friedman Equations


The "Reverse Engineering Method"Chapter 1

the standard approach for solving Friedman Equations:

• take a known potential (from some background physical suggestions)

• substitute this potential into the Friedman equations

• solve the equations to find : -- the scale factor -- the Hubble parameter


Chapter 1 The "Reverse Engineering Method"

a different approach

• consider "a priori " a certain type of scale factor a(t) (as close as possible to the astronomical observations)

• solve the eq. for V and the scalar field

Ellis and Madsen are among the first authors to use the above described method • this method is more appropriate for modeling the cosmic acceleration

Recently this method was further developed and extended for more general casesin the work of Vulcanov under the name of "reverse engineering method", or REM for short.


Chapter 1 The scalar field dynamics

Following REM, the Friedman eq. can be rewritten as

Hubble parameter Gaussian curvature


Chapter 1 Graphical Results

exponential expansion

V(j) in terms of different wV(j) for w=0.1 (blue curve) and w=0.2 (red curve)


Chapter 1 Graphical Results

linear expansion

V(j) for n=2 (blue curve) and n=5 (red curve)

V(φ) for different values of n


Graphical ResultsChapter 1

Morse type expansion

V(j) in terms of different a at k=0 and w=0.1

V(j) for k=0 and w=0.1 andα = 1 (blue), =1.2 (red), =1.3 (green)


SCALAR FIELD COSMOLOGY NON-MINIMALLY

COUPLED WITH GRAVITY

Chapter 2


Preliminaries Chapter 2

The total action

minimal coupling

conformal coupling


Field EquationsChapter 2

Generalized Einstein Eq.

Stress-energy tensor

stress-energy tensor for minimally coupling

Klein-Gordon eq. for the scalar field


The new Friedman EquationsChapter 2

Ansatz FRW

Friedman type equations


REMChapter 2

the equation for the scalar field

the scalar field potential


Graphical Results Chapter 2

The exponential case

V(j) for k=1, w=1 and ξ = 1/6 (red line) ξ = 1/2 (blue line)ξ = 1/8 (green line)

V(φ) for different initial values of φ ξ = 1/6 (top surface) ξ = 1/2 (bottom surface)




V(j) in terms of different x at k= -1 (left) and k=+1 (right)


w = 1, k=1, x = 0 green line x=-0.1 (left) and x = 0.1 (right) blue line

V(j)

ekpyrotic universe




k = 1 and x = 0.05

ekpyrotic universe

V(j)


REM FOR A SCALAR FIELD MINIMALLY

COUPLED WITH f(R) GRAVITY

Chapter 3


f(R) Theories of GravityChapter 3

The Action S

3 different formulation of f(R) gravity:

Metric formalism

Palatini formalism

Metric-affine formalism


Metric FormalismChapter 3

Variation of S

Einstein form of the equation

Field equation


Metric FormalismChapter 3

Trace of the metric field eq.

Ansatz

Minkovski spacetime

de Sitter ( C>0 )

Anti-de Sitter ( C<0 )

vacuumelectro-vacuum


Palatini FormalismChapter 3

The field equations

Action S


Palatini FormalismChapter 3

a conformal transformation

General Relativity

GR with a cosmological constant & a modifying coupling constant


Generalized Friedman EquationsChapter 3

for the metric formalism

Hubble function


REMChapter 3

the equation for the scalar field

the scalar field potential


An Example Chapter 3

a particular choice for f(R)

time-time component of Friedman eq.

space-space components of Friedman eq.


Graphical ResultsChapter 3


The potential as a function of time for different values of k

The scalar field as a function of time for different values of k


Chapter 3




Graphical Results


Chapter 3

The linear case

V(j) in terms of different a at k=1Time behavior of V in terms of different a at k=0,1 and -1

Graphical Results


Conclusions, remarks …

Applying for the first time the REM method to a scalar field minimally coupled with f(R) gravity

Analyzing Gauss-Bonnet gravity in the REM formalism (in the future? )

Applying REM to non-minimally coupled f(R) gravity (in the future? )

The results where presented ad different national and international meetings and where partially published (or will be published)

Creation of a package programs in Mapple+GRTensor II for doing cosmology with the ‘reverse engineering method’

Finding a new analytical example in the case of standard gravity minimally coupled with a scalar field


References

[1] M.S. Madsen, Class. Quantum Grav., 5, (1988), 627-639

[2] G.F.R. Ellis, M.S. Madsen, Class. Quantum Grav. 8, (1991), 667-676

[3] D.N. Vulcanov, Central European Journal of Physics, 6, 1, (2008), 84-96

[4] G. S. Djordjevic, C.A. Sporea, D.N. Vulcanov, Proc. of the TIM10 Conference, Timisoara, Romania, nov. 2010, in AIP proceedings series.

[5] Cardenas VH , del Campo S, astro - ph /0401031

[6] Tsujikawa S., Phys.Rev.D, 62, 043512, 2000 and references there• • •


“Thank you for your attention”

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Minimally Coupled Scalar Field Cosmology in Higher-Order Theories of Gravity

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