Reproducing Kernel Hilbert Space of A Set Indexed Brownian MotionIJMERJOURNAL
ABSTRACT: This study researches a representation of set indexed Brownian motion { : } X X A A A via orthonormal basis, based on reproducing kernel Hilbert space (RKHS). The RKHS associated with the set indexed Brownian motion X is a Hilbert space of real-valued functions on T that is naturally isometric to 2 L ( ) A . The isometry between these Hilbert spaces leads to useful spectral representations of the set indexed Brownian motion, notably the Karhunen-Loève (KL) representation: [ ] X e E X e A n A n where { }n e is an orthonormal sequence of centered Gaussian variables. In addition, we present two special cases of a representation of a set indexed Brownian motion, when ([0,1] ) d A A and A = A( ) Ls .
Reproducing Kernel Hilbert Space of A Set Indexed Brownian MotionIJMERJOURNAL
ABSTRACT: This study researches a representation of set indexed Brownian motion { : } X X A A A via orthonormal basis, based on reproducing kernel Hilbert space (RKHS). The RKHS associated with the set indexed Brownian motion X is a Hilbert space of real-valued functions on T that is naturally isometric to 2 L ( ) A . The isometry between these Hilbert spaces leads to useful spectral representations of the set indexed Brownian motion, notably the Karhunen-Loève (KL) representation: [ ] X e E X e A n A n where { }n e is an orthonormal sequence of centered Gaussian variables. In addition, we present two special cases of a representation of a set indexed Brownian motion, when ([0,1] ) d A A and A = A( ) Ls .
Holographic Dark Energy in LRS Bianchi Type-II Space Timeinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
Fractales bartolo luque - curso de introduccion sistemas complejosFundacion Sicomoro
¿Qué tienen en común los brócolis, las nubes y los cráteres meteoríticos? Todos exhiben fractalidad. Una nueva ciencia como la de los Sistemas Complejos, requería una nueva manera de caracterizar las formas: la geometría fractal. En esta charla aprenderemos qué es un fractal, dónde aparecen, dónde se usan y qué nos desvelan. Veremos que, en el fondo, la invarianza de escala, que va más allá de la geometría, es el concepto crucial.
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Obtaining three-dimensional velocity information directly from reflection sei...Arthur Weglein
This paper present a formalism for obtaining the subsurface
velocity configuration directly from reflection seismic data.
Our approach is to apply the results obtained for inverse
problems in quantum scattering theory to the reflection
seismic problem. In particular, we extend the results of
Moses (1956) for inverse quantum scattering and Razavy
(1975) for the one-dimensional (1-D) identification of the
acoustic wave equation to the problem of identifying the
velocity in the three-dimensional (3-D) acoustic wave equation
from boundary value measurements. No a priori knowledge
of the subsurface velocity is assumed and all refraction,
diffraction, and multiple reflection phenomena are
taken into account. In addition, we explain how the idea of
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Gauss–Bonnet Boson Stars in AdS, Bielefeld, Germany, 2014Jurgen Riedel
Strong coupling to gravity: self-interacting rotating boson
stars are destabilized.
Sufficiently small AdS radius: self-interacting rotating boson
stars are destabilized.
Sufficiently strong rotation stabilizes self-interacting rotating
boson stars.
Onset of ergoregions can occur on the main branch of boson
star solutions, supposed to be classically stable.
Radially excited self-interacting rotating boson stars can be
classically stable in aAdS for sufficiently large AdS radius and
sufficiently small backreaction.
In this paper we have defined Dk
integral and proved the integration by parts formula.
Key Words and phrases: Absolutely Continuous function, Generalised absolutely continuous function,
Denjoy integration. 2000 Mathematics subject Classification: Primary 26A24 Secondary 26A21, 26A48,
44A10.
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Spectral methods are used in computer graphics, machine learning, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding its eigenvalues and eigenfunctions. We show how to generalize spectral geometry to multiple data spaces. Our construction is based on the idea of simultaneous diagonalization of Laplacian operators. We describe this problem and discuss numerical methods for its solution. We provide several synthetic and real examples of manifold learning, object classification, and clustering, showing that the joint spectral geometry better captures the inherent structure of multi-modal data.
Talk at SIAM-IS 2014 (http://www.math.hkbu.edu.hk/SIAM-IS14/). A big thanks to Michael Bronstein for providing a great set of slides this presentation is a mere extension of.
As a generalization of the concept SLH space, we introduce the concept of slightly strongly locally homogeneous (SSLH) spaces. Also, we introduce the concepts of slightly dense set as well as slightly separable space, and use them to introduce two new types of slightly countable dense homogeneous spaces. Several results, relationships, examples and counter-examples concerning these concepts are obtained.
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.
Holographic Dark Energy in LRS Bianchi Type-II Space Timeinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
Fractales bartolo luque - curso de introduccion sistemas complejosFundacion Sicomoro
¿Qué tienen en común los brócolis, las nubes y los cráteres meteoríticos? Todos exhiben fractalidad. Una nueva ciencia como la de los Sistemas Complejos, requería una nueva manera de caracterizar las formas: la geometría fractal. En esta charla aprenderemos qué es un fractal, dónde aparecen, dónde se usan y qué nos desvelan. Veremos que, en el fondo, la invarianza de escala, que va más allá de la geometría, es el concepto crucial.
D. Mladenov - On Integrable Systems in CosmologySEENET-MTP
Lecture by Prof. Dr. Dimitar Mladenov (Theoretical Physics Department, Faculty of Physics, Sofia University, Bulgaria) on December 7, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
Obtaining three-dimensional velocity information directly from reflection sei...Arthur Weglein
This paper present a formalism for obtaining the subsurface
velocity configuration directly from reflection seismic data.
Our approach is to apply the results obtained for inverse
problems in quantum scattering theory to the reflection
seismic problem. In particular, we extend the results of
Moses (1956) for inverse quantum scattering and Razavy
(1975) for the one-dimensional (1-D) identification of the
acoustic wave equation to the problem of identifying the
velocity in the three-dimensional (3-D) acoustic wave equation
from boundary value measurements. No a priori knowledge
of the subsurface velocity is assumed and all refraction,
diffraction, and multiple reflection phenomena are
taken into account. In addition, we explain how the idea of
slant stack in processing seismic data is an important part
of the proposed 3-D inverse scattering formalism.
Gauss–Bonnet Boson Stars in AdS, Bielefeld, Germany, 2014Jurgen Riedel
Strong coupling to gravity: self-interacting rotating boson
stars are destabilized.
Sufficiently small AdS radius: self-interacting rotating boson
stars are destabilized.
Sufficiently strong rotation stabilizes self-interacting rotating
boson stars.
Onset of ergoregions can occur on the main branch of boson
star solutions, supposed to be classically stable.
Radially excited self-interacting rotating boson stars can be
classically stable in aAdS for sufficiently large AdS radius and
sufficiently small backreaction.
In this paper we have defined Dk
integral and proved the integration by parts formula.
Key Words and phrases: Absolutely Continuous function, Generalised absolutely continuous function,
Denjoy integration. 2000 Mathematics subject Classification: Primary 26A24 Secondary 26A21, 26A48,
44A10.
Building Compatible Bases on Graphs, Images, and ManifoldsDavide Eynard
Spectral methods are used in computer graphics, machine learning, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding its eigenvalues and eigenfunctions. We show how to generalize spectral geometry to multiple data spaces. Our construction is based on the idea of simultaneous diagonalization of Laplacian operators. We describe this problem and discuss numerical methods for its solution. We provide several synthetic and real examples of manifold learning, object classification, and clustering, showing that the joint spectral geometry better captures the inherent structure of multi-modal data.
Talk at SIAM-IS 2014 (http://www.math.hkbu.edu.hk/SIAM-IS14/). A big thanks to Michael Bronstein for providing a great set of slides this presentation is a mere extension of.
As a generalization of the concept SLH space, we introduce the concept of slightly strongly locally homogeneous (SSLH) spaces. Also, we introduce the concepts of slightly dense set as well as slightly separable space, and use them to introduce two new types of slightly countable dense homogeneous spaces. Several results, relationships, examples and counter-examples concerning these concepts are obtained.
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.
We apply the recently derived constraintless Clairaut-type formalism (by S. Duplij) to the Cho-Duan-Ge decom- position in SU(2) QCD. We find nontrivial corrections to the physical equations of motion and that the contribution of the topological degrees of freedom is qualitatively different from that found by treating the monopole potential as though it were dynamic. We also find alterations to the field commutation relations that undermine the particle interpretation in the presence of the chromomonopole condensate.
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”.
Five-Dimensional Cosmological Model with Time-Dependent G and Lamda for Const...IOSR Journals
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source with time dependent G and .The Einstein field equations are solvable with the help of constant
deceleration parameter. Physical and kinematical properties of this model are investigated. It has been shown
that the solutions are comparable with recent observations. The behavior of gravitational constant,
cosmological constant, density, critical density and pressure is discussed for dust, radiation dominated and stiff matter of the Universe. It is also examined the behavior of gravitational constant and cosmological constant for expansion law and exponential law for stiff matter
Exact Sum Rules for Vector Channel at Finite Temperature and its Applications...Daisuke Satow
Slides used in presentation at:
“International School of Nuclear Physics 38th Course Nuclear matter under extreme conditions -Relativistic heavy-ion collisions”, in September, 2016 @ Erice, Italy
Similar to Nevenko Bilić "Tachyon inflation on the holographic braneworld" (20)
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
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at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
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z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
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The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Nevenko Bilić "Tachyon inflation on the holographic braneworld"
1. Tachyon inflation in a holographic
braneworld
Neven Bilić
Ruđer Bošković Institute
Zagreb
BW2018 Niš, June 2018
2. Holographic braneworld is a 3-brane located at the
boundary of the asymptotic AdS5. The cosmology is
governed by matter on the brane in addition to
the boundary CFT
time
Conformal
boundary at z=0
space
z
xRSII brane
at z≠0
Basic idea
3. P.S. Apostolopoulos, G. Siopsis and N. Tetradis, Phys. Rev. Lett. 102
(2009) arXiv:0809.3505
P. Brax and R. Peschanski, Acta Phys. Polon. B 41 (2010) arXiv:1006.3054
N.B., Phys. Rev. D 93 (2016) arXiv:1511.07323
Based on:
N.B. , S. Domazet and G. Djordjevic, Class. Quant. Grav. 34, (2017)
arXiv:1704.01072
N.B. , D. Dimitrijevic, G. Djordjevic and M. Milosevic, Int. J. Mod. Phys. A
32, (2017) arXiv:1607.04524
N.B. et al., Tachyon Inflation in a holographic braneworld, in preparation
and related earlier works
5. AdS/CFT correspondence is a holographic duality between
gravity in d+1-dim space-time and quantum CFT on the d-dim
boundary. Original formulation stems from string theory:
Conformal
Boundary
at z=0
AdS bulk
time
Equivalence of 3+1-dim
N =4 Supersymmetric YM Theory
and string theory in AdS5S5
J. Maldacena, Adv. Theor. Math. Phys. 2 (1998)
AdS/CFT
6. Using the solution we can define a functional
Given induced metric on the boundary the geometry is
completely determined by the field equations obtained from the
variation principle
(5)
0
S
G
[ ]G G h
on
shell
[ ] [ ]S h S G h
where is the on shell bulk action
on
shell
[ ]S G h
h
Consider a bulk action with only gravity in the bulk
(5)
5
5
5
1
8 2
R
S d x G
G
7. 4 CFT
[ ] ln exp )(S h d d x h L
AdS/CFT conjecture: S[h] can be identified with the generating
functional of a conformal field theory (CFT)on the boundary
CFT
L – CFT Lagrangian
The induced metric hμν serves as the source for the stress tensor of
the dual CFT so that its vacuum expectation value is obtained from the
classical action
CFT 1
2
S
T
hh
8. The on-shell action is IR divergent and must be regularized and
renormalized. The asymptotically AdS metric in the Fefferman-Graham
form is
(0) 2 (2) 4 (4)
g g z g z g
S. de Haro, S.N. Solodukhin, K. Skenderis, Comm. Math. Phys. 217 (2001)
2
2 2
(5) 2
( )a b
abds G dx dx g dx dx dz
z
Explicit expressions for , in terms of arbitrary can be found in
(2 )n
g
(0)
g
where the length scale ℓ is the AdS curvature radius.
Near z=0 the metric can be expanded as
Holographic renormalization
9. We regularize the action by placing a brane (RSII brane) near the AdS
boundary, i.e., at z = εℓ, ε<<1, so that the induced metric on the brane is
(5)
reg 5
5 GH br
5
1
[ ] [ ] [ ]
8 2z
R
S h d x G S h S h
G
(0) 2 2 (2)
2
1
( )h g g
The bulk splits in two regions: 0≤ z ≤εℓ, and εℓ ≤ z ≤∞. We can either
discard the region 0≤ z ≤εℓ (one-sided regularization) or invoke the Z2
symmetry and identify two regions (two-sided regularization). For
simplicity we shall use the one-sided regularization. The regularized
bulk action is
4 4
br matt[ ]S h d x h d x h Lwhere
10. The renormalized boundary action is obtained by adding
counter-terms and taking the limit ε→0
reg ren
1 2 3
0
[ ] lim( [ ] [ ] [ ] [ ])S h S h S h S h S h
The necessary counter-terms are
S.W. Hawking, T. Hertog, and H.S. Reall, Phys. Rev. D 62 (2000)
de Haro et al. Commun. Math. Phys. 217 (2001)
11. Now we demand that the variation with respect to the induced metric hμν of
the regularized on shell bulk action (RSII action) vanishes, i.e.,
reg
[ ] 0S h
Which may be expressed as matter on
the brane
cosmological
constant
Einstein Hilbert term
AdS/CFT prescription
0.=
216
4
5
R
hxd
G
ren 4 4
3 matt
5
3
8
S S d x h d x h
G
L
ren 4 CFT
3
1
( )
2
S S d x h T h
12. The variation of the action yields Einstein’s equations on the
boundary
(0) CFT matt
N
1
8
2
R Rg G T T
This is an explicit realization of the AdS/CFT correspondence:
the vacuum expectation value of a boundary CFT operator is obtained in
terms of geometrical quantities of the bulk.
de Haro et al, Comm. Math. Phys. 217 (2001)
where
13. 2 2
2 2 2 2 2 2 2
(5) 2 2
( ) ( , ) ( , ) kds g dx dx dz t z dt t z d dz
z z
N A
Starting from AdS-Schwarzschild static coordinates
and making the coordinate transformation
the line element will take a general form
Imposing the boundary conditions at z=0:
( ,0) 1, ( ,0) ( )t t a t N A
2 (0) 2 2 2
(0) ( ) kds g dx dx dt a t d
( , ), ( , )t z r r t z
we obtain the induced metric at the boundary in the FRW
form
( , , , , )r
)sin(
)(sin
= 222
2
22
dddd
Holographic cosmology
14. where is the Hubble rate at the boundary and μ is the
dimensionless parameter related to the bulk BH mass
Solving Einstein’s equations in the bulk one finds
22 4
2 2 2
2 4
1
1 ,
4 4
z z
a H
a a
A ,
a
A
N
/H a a
Comparing the exact solution with the expansion
(0) 2 (2) 4 (4)
g g z g z g
we can extract and . Then, using the de Haro et al. expression
for TCFT we obtain
(2)
g
(4)
g
P.S. Apostolopoulos, G. Siopsis, and N. Tetradis, Phys. Rev. Lett. 102, (2009)
P. Brax and R. Peschanski, Acta Phys. Polon. B 41 (2010)
15. CFT CFT (0)1
4
T t T g
The second term is the conformal anomaly
The first term is a traceless tensor with non-zero components
23
2 2
00 2 4 2
5
3 4
3
64
i
i
a
t t H H
G a a a a
3
CFT 2
2
5
3
16
a
T H
G a a
Hence, apart from the conformal anomaly, the CFT dual to the time
dependent asymptotically AdS5 metric is a conformal fluid with the
equation of state
where
CFT CFT 3p
CFT 00 CFT
i
it p t
16. From the boundary Einstein equations we obtain the holographic
Friedmann equation (from now on we assume spatial flatness, i.e., we
put )
2
2
1 4 ( )
2
H H G p
where
matt matt
00 , i
iT p T
quadratic deviation
The second Friedmann equation can be derived from the energy-
momentum conservation
2 2
2 4
4
8 4
4 3
G
H H
a
dark radiation
quadratic
deviation
E. Kiritsis, JCAP 0510 (2005); Apostolopoulos et al, Phys. Rev. Lett. 102, (2009)
0
17. The holographic cosmology has interesting properties. Solving the first
Friedmann equation as a quadratic equation for H2 we find
2 2
2
2
(1 1 8 / 3),H G
Demanding that this equation reduces to the standard Friedmann
equation in the low energy limit, i.e., in the limit when
2
1G
it follows that we must discard the + sign solution. Then, it follows
that the physical range of the Hubble rate is between 0 and
starting from its maximal value at an arbitrary initial
time t0. At that time, which may be chosen to be zero, the density
and cosmological scale are both finite so the Big-Bang singularity is
avoided!
2 /
max 2 /H
18. Holographic cosmology appears also in other contexts. In particular it
can be derived in
Modified Gauss-Bonnet gravity
This model was shown to be ghost free.
If in addition one requires that the second Friedmann equation is linear
in , then f must be a function of only one variable f = f(J) where
1/2
21
6
12
J R R G
C. Gao, Phys. Rev. D 86 (2012)
H
4 matt1
( , )
16
S d x g R f R
G
G L
where is the Gauss-Bonnet invariant
2
4R R R R R
G
see, e.g., I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Effective actions in quantum
gravity ( IOP publishing, Bristol, 1992)
The second requirement cannot be fulfilled in a simple f(R) modified
gravity including the Starobinski model
19. In a cosmological context with spatially flat metric one finds ,
the function f becomes a function of H, and the first Friedmann equation
takes the form
/J a a H
2 1 1 8
6 6 3
df G
H f H
dH
The left-hand side is a function of H only and takes the holographic
form if 2 41
( )
2
f H H
Hence, the holographic cosmology is reproduced in the modified
Gauss-Bonnet gravity of the form
2 2
4 2 matt1
6
16 288
S d x g R R R
G
G L
20. One postulates a field, dubbed the inflaton, usually a self-interacting
scalar that evolves towards the minimum of a slow roll potential. In
conjunction with Friedman equation one solves the field equations
from the beginning to the end of inflation. During inflation a slow roll
regime is assumed, i.e., a very slow change of the Hubble rate so the
Universe expands almost as a de Sitter spacetime with a large
cosmological constant.
Quantum fluctuations of the inflaton field generate initial density
perturbations of order at the time of decoupling5
10
300000years (z 1000)t
V
φ
Inflation
21. One of the popular models of inflaton is the tachyon field θ of dimension of
length with the Lagrangian
, ,
4
1( / ) gV
L
Tachyon inflation
Where ℓ is some length scale introduced to make the potential V
dimensionless .
Our aim is to study tachyon inflation in the framework of holographic
cosmology . The model is based on a holographic braneworld scenario
with an effective tachyon field on a D3-brane located at the holographic
boundary of ADS5. In our model we identify ℓ as the curvature radius of
AdS5
22. The covariant Hamiltonian corresponding to L is
224
/1= VV
H
The conjugate momentum is, as usual, related to θ,μ via
Now we assume that the holographic braneworld is a spatially flat
FRW universe with line element
2 2 2 2 2 2
( )ds g dx dx dt a t dr r d
4
= g
,
=
L
where
23. We also demand that these equations reduce to the standard
Friedmann equation in the low energy limit.
From the covariant Hamilton’s equations
, ;=
H H
we obtain two first order differential equations in comoving frame
2 2
= ,
V
,
2 2
= 3
VV
H
V
The Lagrangian and Hamiltonian are identified with the pressure and
energy density: , and we employ the holographic
Friedmann equations
,p L H
2
2 4 8
=
4 3
G
H H
2
2
1 = 4 ( )
2
H H G p
24. In the following we will examine a simple exponential potential
where ω is a free dimensionless parameter. We will also consider the
initial value hi
2 as a free parameter ranging between 0 and 2.
Inflation on the holographic brane
Tachyon inflation is based upon the slow evolution of the field θ
with the slow-roll conditions
2
1, | | 3 .H
2 2 2 2
2(1 1 / 3),h H V
2 2
= 8 /G
Then, during inflation we find
where
/
=V e
and the evolution is constraint to the physical range of the Hubble rate
2
0 2h
25. Another important quantity is the so called number of e-folds defined as
where the subscripts i and f denote the beginning and the end of
inflation. Typically is sufficient to solve the flatness and
horizon problems
The most important parameters that characterize inflation are the slow-
roll inflation parameters εj defined recursively
2 2 2
1
1 2 12 2 2 2 2
1
(4 ) 2
= , = 2 1
6 (2 ) (2 )(4 )
H h h
H h h H h h
1 / ( )j j jH
starting from , where H* is the Hubble rate at some
chosen time. The next two are given by
0 * /H H
f
i
t
t
N Hdt
During inflation and inflation ends once either ε1 or ε2 exceeds 1.
Near the end of inflation and .
1,2 1
2 2
/ 3 1h V 2 12
50 60N
26. 2 2 2 2
i i
2
12
= 1 1 ln 2 ln 1 1
3 2 2 3
h h
N
The number of e-folds can also be calculated explicitly yielding an
expression that relates our free parameters hi and ω to N
From the field equations we find an approximate equation
3H
which can be easily integrated yielding the time as a function of H
in the slow roll regime
2
3 ( 2 1)(2 )
= 2( 2 ) ln ,
( 2 1)(2 )
h
t h
h
h H
Hence, for a fixed chosen N we have only one free parameter
27. /ℓ
Slow roll parameters ε1 (dashed red line) and ε2 (blue line)
versus time for fixed N=60 and ω2=0.027 corresponding
to the initial hi
2=0.6
28. The slow-roll parameters εj are related to the observational quantities
such as the tensor-to-scalar ratio r and the scalar spectral index ns
defined by
T S
s
S
ln
= , =
ln
d
r n
d q
P P
P
where and are the power spectra of scalar and tensor
perturbations, respectively, evaluated at the horizon, i.e., for a wave-
number satisfying q=aH. One finds at the lowest order in ε1 and ε2
SP TP
= 2 ln2 0.72C where
2
2
1 2 12
2
s 1 22
4 2
=16(1 ) 1
12 3
2 3
1= 3
2
h
r h C
h
h
n
h
Deviations from the
standard tachyon
inflation
29. r versus ns for fixed N and varying initial value hi
2 ranging from 0 to 2
along the lines. The parameter ω is also varying in view of the functional
dependence N=N(hi, ω). The black lines denote the Planck constraints
contours of the 1σ (dash-dotted) and 2σ (dotted) confidence level
N=70
N=90
2σ
1σ
30. Conclusions and outlook
We have shown that the slow-roll equations of the tachyon
inflation with exponentially attenuating potential on the
holographic brane are quite distinct from those of the
standard tachyon inflation with the same potential
The ns - r relation depends on the initial value of the Hubble
rate and on the assumed value of the number of e-folds N
and show a reasonable agreement with the Planck 2015
data for N >70.
The presented results obtained in the slow roll
approximation are preliminary. What remains to be done is
to solve the exact equations numerically for the same
potential and for various other potentials that have been
exploited in the literature.
32. Why AdS?
Anti de Sitter space is dual to a conformal field theory at
its boundary (AdS/CFT correspondence)
AdS is a maximally symmetric solution to Einstein’s
equations with negative cosmological constant. In 4+1
dimensions the symmetry group is AdS5≡ SO(4,2)
The 3+1 boundary conformal field theory is invariant under
conformal transformations: Poincare + dilatations + special
conformal transformation = conformal group ≡ SO(4,2)
Basic idea
Braneworld cosmology is based on the scenario in which
matter is confined on a brane in a higher dimensional
bulk with only gravity allowed to propagate in the bulk.
The brane can be placed, e.g., at the boundary of
a 5-dim asymptotically Anti de Sitter space (AdS5)
33. 4 CFT
[ , ] ln exp ( )) ( ( )) ( )(S h d d x h x O x x L
AdS/CFT conjecture: S[φ,h] can be identified with the generating
functional of a conformal field theory on the boundary
CFT
L – conformal field theory Lagrangian
O – operators of dimension Δ
2
( ( )) ( ( )) ( ( )) ( ( ))
( ) ( )
S
O x O y O x O y
x y
where the boundary fields serve as sources for CFT operators
The induced metric hμν serves as the source for the stress tensor of the
dual CFT so that its vacuum expectation value is obtained as
In this way the CFT correlation functions can be calculated as functional
derivatives of the on-shell bulk action, e.g.,
2
CFT1
2
S
T
hh
34. One usually imposes the RS fine tuning condition
0 2
5 N
3 3
8 8G G
The Planck mass scale is determined by the curvature of
the five-dimensional space-time
2 /
N 5 50
1
2
y
e dy
G G G
1 one-sided
2 two-sided
which eliminates the 4-dim cosmological constant.
35. Bound on the AdS5 curvature radius ℓ:
The classical 3+1 dim gravity is altered on the RSII brane
For the Newtonian potential of an isolated source
on the brane is given by
r
2
N
2
2
( ) 1
3
G M
r
r r
J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778 (2000)
Table top tests of Long et al find no deviation of Newton’s
potential and place the limit
1 12
0.1mm or > 10 GeV
36. Holographic type cosmologies appear also in other contexts:
1. The saddle point of the spatially closed mini superspace partition
function dominated by matter fields conformally coupled to gravity
A. O. Barvinsky, C. Deffayet and A. Y. Kamenshchik, JCAP 0805, (2008)
arXiv:0801.2063
2. A modified Friedmann equation with a quartic term ~H4 derived from
the generalized uncertainty principle and the first low of
thermodynamics applied to the apparent horizon entropy.
J. E. Lidsey, Phys. Rev. D 88 (2013) arXiv:0911.3286
3. The quartic term as a quantum correction to the Friedmann equation
using thermodynamic arguments at the apparent horizon [37]
S. Viaggiu, Mod. Phys. Lett. A, 31 (2016) arXiv:1511.06511
4. Modified Gauss-Bonnet gravity
G. Cognola et al Phys. Rev. D 73 (2006), C. Gao, Phys. Rev. D 86 (2012);
37. Inflation
A short period of rapid expansion at the very early
universe (starting at about 10-43 s after the Big Bang)
1025 times in 10-32 s
38. The main problems of the standard
cosmology solved by inflation
• Horizon problem – homogeneity and isotropy of the
CMB radiation
• Flatness problem – fine tuning of the initial conditions
• Large scale structure problem – the origin of
initial density perturbations that serve as seeds of the
observed structure today
• Monopole problem – absence of topological defects:
monopoles, cosmic strings, domain walls
39. One of the popular models of inflaton is the tachyon. Our aim is to study
tachyon inflation in the framework of holographic cosmology . The model
is based on a holographic braneworld scenario with an effective tachyon
field on the D3-brane located at the holographic bound of ADS bulk. A
tachyon Lagrangian of the form
can be derived in the context of a dynamical brane moving in a 4+1
background with a general warp
2 2
(5) 2
1
= ( )
( )
ds g dx dx dz
z
, ,
4
1( / ) gV
L
4 4
( / ) / ( )V
The field θ is identified with the 5-th coordinate z and the potential is related
to the warp
Tachyon inflation
N.B., S. Domazet and G. Djordjevic, Class. Quant. Grav. 34, (2017) arXiv:1704.01072.