Minimally Coupled Scalar Field Cosmology in Higher-Order Theories of Gravity
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Transcript of Minimally Coupled Scalar Field Cosmology in Higher-Order Theories of Gravity
1
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
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THEORY OF A SCALAR FIELD MINIMALLY
COUPLED WITH GRAVITY
Chapter 1
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Theoretical Framework
Principle of least action
Chapter 1
Variation of S
Field equations
The stress-energy tensor
with respect to
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Friedman EquationsChapter 1
Einstein equations
FRW metric
The scalar field
The matter field
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Friedman EquationsChapter 1
The total stress-energy tensor
Friedman Equations
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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
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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.
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Chapter 1 The scalar field dynamics
Following REM, the Friedman eq. can be rewritten as
Hubble parameter Gaussian curvature
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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)
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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
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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)
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SCALAR FIELD COSMOLOGY NON-MINIMALLY
COUPLED WITH GRAVITY
Chapter 2
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Preliminaries Chapter 2
The total action
minimal coupling
conformal coupling
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Field EquationsChapter 2
Generalized Einstein Eq.
Stress-energy tensor
stress-energy tensor for minimally coupling
Klein-Gordon eq. for the scalar field
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The new Friedman EquationsChapter 2
Ansatz FRW
Friedman type equations
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REMChapter 2
the equation for the scalar field
the scalar field potential
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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)
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Graphical Results Chapter 2
The exponential case
V(j) in terms of different x at k= -1 (left) and k=+1 (right)
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w = 1, k=1, x = 0 green line x=-0.1 (left) and x = 0.1 (right) blue line
V(j)
ekpyrotic universe
Graphical Results Chapter 2
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Graphical Results Chapter 2
k = 1 and x = 0.05
ekpyrotic universe
V(j)
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REM FOR A SCALAR FIELD MINIMALLY
COUPLED WITH f(R) GRAVITY
Chapter 3
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f(R) Theories of GravityChapter 3
The Action S
3 different formulation of f(R) gravity:
Metric formalism
Palatini formalism
Metric-affine formalism
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Metric FormalismChapter 3
Variation of S
Einstein form of the equation
Field equation
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Metric FormalismChapter 3
Trace of the metric field eq.
Ansatz
Minkovski spacetime
de Sitter ( C>0 )
Anti-de Sitter ( C<0 )
vacuumelectro-vacuum
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Palatini FormalismChapter 3
The field equations
Action S
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Palatini FormalismChapter 3
a conformal transformation
General Relativity
GR with a cosmological constant & a modifying coupling constant
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Generalized Friedman EquationsChapter 3
for the metric formalism
Hubble function
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REMChapter 3
the equation for the scalar field
the scalar field potential
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An Example Chapter 3
a particular choice for f(R)
time-time component of Friedman eq.
space-space components of Friedman eq.
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Graphical ResultsChapter 3
The exponential case
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
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Chapter 3
The exponential case
V(j) in terms of different a at k=1 and w=0.1
V(j) in terms of different a at k=0 and w=0.1
Graphical Results
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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
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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
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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• • •
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“Thank you for your attention”
The End