First-order cosmological perturbations produced by point-like masses: all sca...Maxim Eingorn
This presentation based on the paper http://arxiv.org/abs/1509.03835 was made at Institute of Cosmology, Tufts University, on November 12, 2015. The abstract follows:
In the framework of the concordance cosmological model the first-order scalar and vector perturbations of the homogeneous background are derived without any supplementary approximations in addition to the weak gravitational field limit. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The obtained expressions for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge in all points except the locations of the sources, and their average values are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant it represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggesting itself connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
ALL-SCALE cosmological perturbations and SCREENING OF GRAVITY in inhomogeneou...Maxim Eingorn
M. Eingorn, First-order cosmological perturbations engendered by point-like masses, ApJ 825 (2016) 84: http://iopscience.iop.org/article/10.3847/0004-637X/825/2/84
In the framework of the concordance cosmological model, the first-order scalar and vector perturbations of the homogeneous background are derived in the weak gravitational field limit without any supplementary approximations. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The expressions found for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge at all points except at the locations of the sources. The average values of these metric corrections are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant, this part represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggested connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
First-order cosmological perturbations produced by point-like masses: all sca...Maxim Eingorn
This presentation based on the paper http://arxiv.org/abs/1509.03835 was made at Institute of Cosmology, Tufts University, on November 12, 2015. The abstract follows:
In the framework of the concordance cosmological model the first-order scalar and vector perturbations of the homogeneous background are derived without any supplementary approximations in addition to the weak gravitational field limit. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The obtained expressions for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge in all points except the locations of the sources, and their average values are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant it represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggesting itself connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
ALL-SCALE cosmological perturbations and SCREENING OF GRAVITY in inhomogeneou...Maxim Eingorn
M. Eingorn, First-order cosmological perturbations engendered by point-like masses, ApJ 825 (2016) 84: http://iopscience.iop.org/article/10.3847/0004-637X/825/2/84
In the framework of the concordance cosmological model, the first-order scalar and vector perturbations of the homogeneous background are derived in the weak gravitational field limit without any supplementary approximations. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The expressions found for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge at all points except at the locations of the sources. The average values of these metric corrections are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant, this part represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggested connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
An Approach to the Concept of Energy for Primary School: Disciplinary Framewo...SEENET-MTP
The SEENET-MTP Seminar: Trends in Modern Physics
19–21 August 2011, Niš, Serbia
Talk by Frederico Corni, Faculty of Education, University of Modena аnd Reggio Emilia, Italy
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Classical and Quasi-Classical Consideration of Charged Particles in Coulomb F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb centres is carried out. It is shown that the proton and electron can to create a stable connection with the dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom coincide with the Born orbits.
CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF CHARGED PARTICLES IN COULOMB F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the
Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity
theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb
centres is carried out. It is shown that the proton and electron can to create a stable connection with the
dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the
proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of
the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with
the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy
for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom
coincide with the Born orbits.
Quantum gravitational corrections to particle creation by black holesSérgio Sacani
We calculate quantum gravitational corrections to the amplitude for the emission of a Hawking particle
by a black hole. We show explicitly how the amplitudes depend on quantum corrections to the exterior
metric (quantum hair). This reveals the mechanism by which information escapes the black hole. The
quantum state of the black hole is reflected in the quantum state of the exterior metric, which in turn
influences the emission of Hawking quanta.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
D. Vulcanov - On Cosmologies with non-Minimally Coupled Scalar Field and the "Reverse Engineering Method"
1. ON COSMOLOGIES WITH NON-MINIMALLY
COUPLED SCALAR FIELD AND
THE "REVERSE ENGINEERING METHOD"
G.S. Djordjevic1 , D..N. Vulcanov2
(1) Department of Physics, Faculty of Science and Mathematics, University of Nis,
Visegradska 33, 18001Nis, Serbia
(2) Department of Theoretical and Applied Physics –“ Mircea Zǎgǎnescu”
West University of Timişoara, B-dul. V. Pârvan no. 4, 300223,
Timişoara, Romania
The SEENET-MTP Workshop
BW2011
2. Abstract of the presentation
We studied further the use of the so called “reverse engineering”
method (REM) in reconstructing the shape of the potential in cosmologies
based on a scalar field non-minimally coupled with gravity. We use
the known result that after a conformal transformation to the so
called Einstein frame, where the theory is exactly as we have a
Minimally coupled scalar field. Processing the REM in Einstein frame
And then transforming back to the original frame, we investigated
graphically some examples where the behaviour of the scale factor
is modelling the cosmic acceleration (ethernal inflation)
3. Plan of the presentation
● Introduction – why scalar fields in cosmology
● Review of the “reverse engineering” method
● Cosmology with non-minimally coupled scalar field
● Einstein frame
● Some examples
● Conclusions
4. Plan of the presentation
● Introduction – why scalar fields in cosmology
● Review of the “reverse engineering” method
● Cosmology with non-minimally coupled scalar field
● Einstein frame
● Some examples
● Conclusions
5. Introduction :
Why scalar fields ?
●Recent astrophysical observations ( Perlmutter et . al .) shows
that the universe is expanding faster than the standard model
says. These observations are based on measurements of the
redshift for several distant galaxies, using Supernova type Ia as
standard candles.
●As a result the theory for the standard model must be rewriten
in order to have a mechanism explaining this !
●Several solutions are proposed, the most promising ones are
based on reconsideration of the role of the cosmological
constant or/and taking a certain scalar field into account to
trigger the acceleration of the universe expansion.
●Next figure (from astro - ph /9812473) contains, sintetically the
results of several years of measurements ...
7. Review of the
“reverse engineering method”
We are dealing with cosmologies based on Friedman-
Robertson-Walker ( FRW ) metric
Where R(t) is the scale factor and k=-1,0,1 for open, flat
or closed cosmologies. The dynamics of the system
with a scalar field minimally coupled with gravity is
described by a lagrangian as
⎡ 1 ⎤
R − (∇ ϕ ) − V (ϕ ) ⎥
1
L= −g⎢
2
⎣ 16 π 2 ⎦
Where R is the Ricci scalar and V is the potential of the
scalar field and G=c=1 (geometrical units)
8. Review of “REM”
Thus Einstein equations are
where the Hubble function and the Gaussian
curvature are
9. Review of “REM”
Thus Einstein equations are
It is easy to see that these eqs . are not independent.
For example, a solution of the first two ones (called
Friedman equations) satisfy the third one - which is
the Klein-Gordon equation for the scalar field.
10. Review of “REM”
Thus Einstein equations are
The current method is to solve these eqs . by considering
a certain potential (from some background physical
suggestions) and then find the time behaviour of the
scale factor R(t) and Hubble function H(t).
11. Review of “REM”
Thus Einstein equations are
Ellis and Madsen proposed another method, today
considered (Ellis et . al , Padmanabhan ...) more
appropriate for modelling the cosmic acceleration :
consider "a priori " a certain type of scale factor R(t), as
possible as close to the astrophysical observations,
then solve the above eqs . for V and the scalar field.
12. Review of “REM”
Following this way, the above equations can be
rewritten as
Solving these equations, for some initially prescribed
scale factor functions, Ellis and Madsen proposed the
next potentials - we shall call from now one Ellis-
Madsen potentials :
17. Cosmology with non-minimally coupled
scalar field
We shall now introduce the most general scalar field
as a source for the cosmological gravitational field,
using a lagrangian as :
⎡ 1 ⎤
R − (∇ ϕ ) − V (ϕ ) − ξ R ϕ 2 ⎥
1 1
L= −g⎢
2
⎣ 16 π 2 2 ⎦
where ξ is the numerical factor that describes the
type of coupling between the scalar field and the
gravity.
18. Cosmology with non-minimally coupled
scalar field
Although we can proceed with the reverse method
directly with the Friedmann eqs. obtained from this
Lagrangian (as we did in [3]) it is rather complicated
due to the existence of nonminimal coupling. In [3] we
appealed to the numerical and graphical facilites of a
Maple platform.
For sake of completeness we can compute the Einstein
equations for the FRW metric.
After some manipulations we have :
19. Cosmology with non-minimally coupled
scalar field
•
k 1 • 2
3 H (t ) 2 + 3 = [ φ ( t ) − V ( t ) + 3ξ H ( t ) (φ ( t ) 2 )]
R (t ) 2 2
• • •
3
3 H ( t ) 2 + 3 H ( t ) = [ − φ ( t ) 2 + V ( t ) − ξ H ( t ) (φ ( t ) 2 )]
2
•• ∂V k
φ (t ) = − 6ξ − 6 ξ H ( t )φ ( t )
∂φ R (t ) 2
•
− 12 ξ H ( t ) φ ( t ) − 3 H ( t ) φ ( t )
2
where 8πG=1, c=1
These are the new Friedman equations !!!
20. Einstein frame
It is more convenient to transform to the Einstein
frame by performing a conformal transformation
^
g µν = Ω 2 g µν where Ω 2 = 1 − ξ 8πϕ 2
Then we obtain the following equivalent Lagrangian:
⎡ 1 ^ 1 2 ⎛ ^ ⎞2 ^
^ ⎤
L= −g⎢ R − F ⎜ ∇ ϕ ⎟ − V (ϕ ) ⎥
⎢16π
⎣ 2 ⎝ ⎠ ⎥
⎦
21. Einstein frame
where variables with a caret denote those in the Einstein
frame, and
1− (1− 6ξ )8πξϕ 2
F =
2
(1− 8πξϕ )2 2
and ^
V (ϕ )
V (ϕ ) =
(1 − 8 πξϕ )
2 2
22. Einstein frame
Introducing a new scalar field Φ as
Φ = ∫ F (ϕ )dϕ
the Lagrangian in the new frame is reduced to the
canonical form:
⎡ 1 ^ 1 ⎛ ^ ⎞2 ^
^ ⎤
L= −g⎢ R − ⎜ ∇ Φ ⎟ − V (Φ ) ⎥
⎢16π
⎣ 2⎝ ⎠ ⎥
⎦
23. Einstein frame
⎡ 1 ^ 1 ⎛ ^ ⎞2 ^
^ ⎤
L= −g⎢ R − ⎜ ∇ Φ ⎟ − V (Φ ) ⎥
⎢16π
⎣ 2⎝ ⎠ ⎥
⎦
Main conclusion: we can process a REM in the
Einstein frame (using the results from the minimallly
coupling case and then we can convert the results in
the original frame.
24. Einstein frame
Before going forward with some concrete results,
let’s investigate some important equations for
processing the transfer from Einstein frame to the
original one. First the main coordinates are :
^ ^
t= ∫ Ω dt and R = ΩR
and the new scalar field Φ can be obtained by
integrating its above expression, namely
27. Examples : nr. 1 – exponential expansion
V (ϕ )
ω = 1, ξ = 0 green line
ξ=-0.1 (left) and ξ = 0.1 (right) blue line
28. Examples : nr. 1 – exponential expansion
V (ϕ , ω ) ξ=0.1 (left) and ξ = - 0.1 (right)
29. Examples : nr. 1 – exponential expansion
V (ξ , ω )
ξ = 0 green surface
ξ=-0.1 (left) and ξ = 0.1 (right) blue
30. Examples : nr. 4 - tn
V (ϕ )
n = 3, ξ = 0 green line
ξ=-0.1 (left) and ξ = 0.1 (right) blue line
31. Examples : nr. 4 - tn
V (ϕ , n)
n = 3, ξ = 0 green surface
ξ=-0.3 (left) and ξ = 0.3 (right) blue surface
32. Examples : ekpyrotic universe
This is example nr. 6 from [3] having :
^ ^
R (t ) = R0 sin(ω t )
2
⎡ ⎛ ωΦ ⎞⎤ 3ω
2
and V (Φ ) = 2 ⎢ B cosh⎜ ⎟⎥ −
⎣ ⎝ B ⎠⎦ 4π
1 ⎛ k ⎞
B =
2
⎜1 + 2 ⎟
⎜
with
4π ⎝ R0 ⎟
⎠
33. Examples : ekpyrotic universe
V (ϕ )
ω = 1, k=1, ξ = 0 green line
ξ=-0.1 (left) and ξ = 0.1 (right) blue line