SlideShare a Scribd company logo
OBSERVATIONAL PARAMETERS OF
INFLATION IN HOLOGRAPHIC COSMOLOGY
MILAN MILOŠEVIĆ
Faculty of Sciences And Mathematics
University of Niš, Serbia
& SEENET-MTP Centre
Workshop on Quantum Fields and Nonlinear Phenomena
27 September 2020
in collaboration with: N. Bilić (Zagreb), G.S. Djordjević, D.D. Dimitrijević and M. Stojanović (Niš)
This work has been supported by the Serbian Ministry for Education, Science and Technological Development under the project No. 176021
and contract No. 451-03-68/2020-14/200124, as well as the ICTP - SEENET-MTP project NT-03 Cosmology-Classical and Quantum Challenges
INTRODUCTION
• The inflation theory proposes a period
of extremely rapid (exponential)
expansion of the universe during the an
early stage of evolution of the universe.
• The inflation theory predicts that during inflation (it takes about 10−34 𝑠) radius of the
universe increased, at least 𝑒60 ≈ 1026 times.
• Although inflationary cosmology has successfully complemented the Standard Model, the
process of inflation, in particular its origin, is still largely unknown.
• Recent years brought us a lot of evidence from WMAP and Planck observations of the CMB
• The most important way to test inflationary cosmological models is to compare the
computed and measured values of the observational parameters.
Figure: Baumann, D. TASI Lectures on Inflation. (2009), arXiv:0907.5424 [hep-th]
BRANEWORLD COSMOLOGY
• Braneworld universe is based on the scenario
in which matter is confined on a brane moving in the higher dimensional bulk with only gravity allowed
to propagate in the bulk.
• One of the simplest models - Randall-Sundrum (RS)
• RS model was originally proposed to solve the hierarchy problem (1999)
• Later it was realized that this model, as well as any similar braneworld model, may have interesting
cosmological implications
• Two branes with opposite tensions are placed at some distance in 5 dimensional space
• RS model – observer reside on the brane with negative tension, distance to the 2nd brane corresponds to the
Netwonian gravitational constant
• RSII model – observer is placed on the positive tension brane, the 2nd brane is pushed to infinity
N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic, M. Milosevic, Tachyon inflation in an AdS braneworld with back-reaction, International Journal of Modern Physics A. 32 (2017) 1750039.
N. Bilic, G.B. Tupper, AdS braneworld with backreaction, Cent. Eur. J. Phys. 12 (2014) 147–159.
HOLOGRAPHIC BRANEWORLD
• Holographic braneworld - a cosmology
based on the effective four-dimensional
Einstein equations on the holographic
boundary in the framework of anti de
Sitter/conformal field theory (AdS/CFT)
correspondence.
• The model is based on a holographic braneworld scenario with an effective
tachyon field on a D3-brane located at the holographic boundary of an
asymptotic AdS5 bulk.
• The cosmology is governed by matter on the brane in addition to the
boundary CFT
time
Conformal
boundary at
z=0
space
z
xRSII brane
at z=zbr
N. Bilić, Randall-Sundrum vs Holographic Cosmology, IRB (2015)
HOLOGRAPHIC TACHYON COSMOLOGY
• The holographic braneworld is a spatially flat FRW universe with line element
൯𝑑𝑠2 = 𝑔 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 = 𝑑𝑡2 − 𝑎2(𝑡)(𝑑𝑟2 + 𝑟2 𝑑𝛺2
• The holographic Friedmann equations
•
Where the scale 𝓁 can be identified with the AdS curvature radius and we introduced a
dimensionless expansion rate ℎ ≡ 𝓁𝐻 and the fundamental dimensionless coupling
ℎ2
−
𝓁2
4
ℎ4
=
𝜅2
3
𝓁4
𝜌
ሶℎ 1 −
𝓁2
2
ℎ2 = −
𝜅2
3
𝓁3(𝑝 + 𝜌)
Standard cosmology:
ℎ2
=
𝜅2
3
𝜌
ሶℎ = −
𝜅2
2
(𝑝 + 𝜌)
𝜅2 =
8𝜋𝐺 𝑁
𝓁2
Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the
holographic braneworld. Journal of Cosmology and Astroparticle Physics 2019, 034–034 (2019).
HOLOGRAPHIC TACHYON COSMOLOGY
• Interesting property - solving the first Friedmann equation as a quadratic
equation
ℎ2
= 2 1 ± 1 −
𝜅2
3
𝓁4 𝜌
• We do not want our modified cosmology to depart too much from the standard
cosmology after the inflation era and demand that this equation reduces to the
standard Friedmann equation in the low density limit 𝜅2 𝓁4 𝜌 ≪ 1
• This demand will be met only by the (−) sign solution. We discard the (+) sign
solution as unphysical.
HOLOGRAPHIC TACHYON COSMOLOGY
• The physical range of the Hubble expansion rate is between ℎmin = 0 and the maximal value
ℎmax = 2
• It corresponds to the maximal energy density 𝜌max = Τ3 𝜅2
𝓁4
• Assuming no violation of the weak energy condition 𝑝 + 𝜌 ≥ 0, the expansion rate will be a
monotonously decreasing function of time.
• The universe starts from 𝑡 = 0 with an initial ℎ𝑖 ≤ ℎ 𝑚𝑎𝑥 with energy density and
cosmological scale both finite.
• The Big Bang singularity is avoided.
Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the holographic braneworld. JCAP, 034–034 (2019).
EQUATIONS OF MOTION
• Tachyon matter in the holographic braneworld is described by the DBI Lagrangian
and the Hamiltonian
ℒ = −𝓁−4
𝑉( Τ𝜃 𝓁) 1 − 𝑔 𝜇𝜈 𝜃,𝜇 𝜃,𝜈
ℋ = 𝓁−4
𝑉 1 + 𝜂2
where 𝜂 =
𝑔 𝜇𝜈 𝜋 𝜇 𝜋 𝜈
𝓁4 𝑉
.
• As usual, the conjugate momentum is 𝜋 𝜇
=
𝜕ℒ
𝜕𝜃,𝜇
• The Hamilton equations are
𝜃,𝜇 =
𝜕ℋ
𝜕𝜋 𝜇 𝜋;𝜇
𝜇
= −
𝜕ℋ
𝜕𝜃
EQUATIONS OF MOTION
• The equations of motions
ሶ𝜃 =
𝜂
1 + 𝜂2
ሶ𝜂 = −
3ℎ𝜂
𝓁
−
𝑉,𝜃
𝑉
1 + 𝜂2 +
𝜂2
1 + 𝜂2
• As usual, the pressure and energy density are equal to Lagrangian and Hamiltonian
𝑝 ≡ ℒ = −𝓁−4
𝑉 1 − ሶ𝜃2 = −
𝓁−4 𝑉
1 − 𝜂2
𝜌 ≡ ℋ =
𝓁−4 𝑉
1 − ሶ𝜃2
= 𝓁−4
𝑉 1 − 𝜂2
INITIAL CONDITIONS
• Two natural initial conditions
a) 𝜂𝑖 = 0
b) ሶ𝜂𝑖 = 0
• The condition:
• (a) assures a finite initial ሶℎ, and
• (b) provides the solution consistent with the slow-roll regime
A) 𝜂𝑖 = 0
• 0 < ℎ𝑖 < 2
• From 𝑝 = −
𝓁−4 𝑉
1−𝜂2
and 𝜌 = 𝓁−4
𝑉 1 − 𝜂2 it follows 𝑝𝑖 = −𝜌𝑖
• From ℎ2 −
𝓁2
4
ℎ4 =
𝜅2
3
𝓁4 𝜌 and 𝜌 =
𝓁−4 𝑉
1− ሶ𝜃2
the initial 𝜃𝑖 can be fixed
𝑉 𝜃𝑖 =
3
𝜅2
ℎ𝑖
2
−
ℎ𝑖
4
4
B) ሶ𝜂𝑖 = 0
• From EqM we have
𝜂𝑖 = −
2 𝓁 Τ𝑉,𝜃 𝑉 𝑖
9ℎ 𝑖
2
−4 𝓁 Τ𝑉,𝜃 𝑉 𝑖
2
+3 9ℎ 𝑖
4
−4ℎ 𝑖
2
𝓁 Τ𝑉,𝜃 𝑉 𝑖
2
• From Friedmann equations we obtain
1 −
ℎ𝑖
2
2
2
= 1 −
𝜅2
3
𝑉 𝜃𝑖 1 + 𝜂𝑖
2
• Random numbers: ℎ𝑖, 𝜅 and parameters in the potential 𝑉 𝜃
• Numerical solutions: 𝜂𝑖 and 𝜃𝑖
• Not easy to solve, sometimes the real solution doesn’t exist, etc
𝑉 = 𝑉 𝜃
𝑉′𝜃 =
𝑑𝑉(𝜃)
𝑑𝜃
B) ሶ𝜂𝑖 = 0
• A different way
• Random numbers ℎ𝑖, 𝜃𝑖 and parameters in the potential 𝑉 𝜃
𝜅2
=
)3𝑉(𝜃𝑖
1 + 𝜂𝑖
2
1 − 1 −
ℎ𝑖
2
2
2
and calculate 𝜂𝑖 in the same way as in the previous way.
THE SLOW-ROLL PARAMETERS
• Number of e-folds
𝑁 𝑡 = න
𝑡 𝐶𝑀𝐵
𝑡 𝑒𝑛𝑑
𝐻 𝑡 𝑑𝑡
• Hubble hierarchy (slow-roll) parameters
𝜀𝑖+1 ≡
𝑑ln|𝜀𝑖|
𝑑𝑁
, 𝑖 ≥ 0, 𝜀0 ≡
𝐻∗
𝐻
where 𝐻∗ is the Hubble parameter at some chosen time
• The first two 𝜀 parameters 𝜀1 = −
ሶ𝐻
𝐻2, 𝜀2 = 2𝜀1 +
ሷ𝐻
𝐻 ሶ𝐻
, etc.
• The end of inflation 𝜀𝑖(𝜙 𝑒𝑛𝑑) ≈ 1
𝜙 𝑒𝑛𝑑 = 𝜙(𝑡 𝑒𝑛𝑑)
OBSERVATIONAL PARAMETERS
• Three independent observational parameters: amplitude of
scalar perturbation 𝐴 𝑠, tensor-to-scalar ratio 𝑟 and scalar
spectral index 𝑛 𝑠
𝑟 = 16𝜀1(𝜙𝑖)
𝑛 𝑠 = 1 − 2𝜀1(𝜙𝑖) − 𝜀2(𝜙𝑖)
• Satellite Planck (May 2009 – October 2013)
• Planck Collaboration
• The latest results were published in 2018.
At the lowest order in parameters 𝜀1 and 𝜀2
Planck 2018 results. X Constraints on inflation, arXiv:1807.06211 [astro-ph.CO]
OBSERVATIONAL PARAMETERS 𝑛 𝑠, 𝑟
𝑟 = 16𝜀1 1 + 𝐶𝜀2 +
)2(2 − ℎ2
)3(4 − ℎ2
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 𝜀1
𝑛s = 1 − 2𝜀1 − 𝜀2 − 2 +
8ℎ2
3 4 − ℎ2 2
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 𝜀1
2
− 3 + 2𝐶 +
)2(2 − ℎ2
)3(4 − ℎ2
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 𝜀1 𝜀2 − 𝐶𝜀2 𝜀3
• For 𝑋 = ሶ𝜃2 and 𝑝 = −𝑉 1 − 𝑋 we have
𝑝𝑝,𝑋𝑋
𝑝,𝑋
2 = −1
Bertini, N. R., Bilic, N. & Rodrigues, D. C. Primordial perturbations
and inflation in holographic cosmology. arXiv:2007.02332 [gr-qc]
POTENTIALS
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
60 < 𝑁 < 90
0 < 𝜔 < 0.25
0 < 𝜃𝑖 < 20
Colour represents the number of 𝑛 𝑠, 𝑟 points in a hexagon
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
- THE BEST FITTING RESULTS
𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
60 < 𝑁 < 90
0 < 𝜔 < 0.25
0 < 𝜃𝑖 < 20
𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
- THE BEST FITTING RESULTS
CONCLUSIONS
• We discussed a model of tachyon inflation based on a holographic braneworld
scenario with a brane located at the boundary of the AdS5 bulk.
• We simulated observational parameters of inflation for two potentials
𝑽 𝜽 =
𝟏
𝒄𝒐𝒔𝒉(𝝎𝜽)
, 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
• The agreement of our model with the Planck observational data is good,
especially for a higher number of e-folds.
• Preliminary results are promising and open good opportunity for further
analytical research of these potentials.
This work has been supported by the Serbian Ministry for Education, Science and Technological Development under the project No. 176021
and contract No. 451-03-68/2020-14/200124, as well as the ICTP - SEENET-MTP project NT-03 Cosmology-Classical and Quantum Challenges
REFERENCES
1. Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the
holographic braneworld. Journal of Cosmology and Astroparticle Physics 2019, 034–034 (2019).
2. N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic & M. Milosevic, Tachyon inflation in an AdS braneworld with
back-reaction, International Journal of Modern Physics A. 32 (2017) 1750039.
3. N.R. Bertini, N. Bilic & D.C. Rodrigues, Primordial perturbations and inflation in holographic
cosmology. arXiv:2007.02332 [gr-qc].
4. N. Bilić, Holographic cosmology and tachyon inflation. International Journal of Modern Physics A 33,
1845004 (2018).
5. N. Bilić, Randall-Sundrum versus holographic cosmology, Phys. Rev. D 93 (2016) 066010
[arXiv:1511.07323]

More Related Content

What's hot

Parabolic Restricted Three Body Problem
Parabolic Restricted Three Body ProblemParabolic Restricted Three Body Problem
Parabolic Restricted Three Body Problem
Esther Barrabés Vera
 
The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...
The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...
The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...
Eesti Pank
 
Tachyon inflation in DBI and RSII context
Tachyon inflation in DBI and RSII contextTachyon inflation in DBI and RSII context
Tachyon inflation in DBI and RSII context
Milan Milošević
 
Ground Response Analysis - Geotechnical Earthquake engineering
Ground Response Analysis - Geotechnical Earthquake engineeringGround Response Analysis - Geotechnical Earthquake engineering
Ground Response Analysis - Geotechnical Earthquake engineering
Cristian Soriano-Camelo
 
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
SEENET-MTP
 
Introduction to second gradient theory of elasticity - Arjun Narayanan
Introduction to second gradient theory of elasticity - Arjun NarayananIntroduction to second gradient theory of elasticity - Arjun Narayanan
Introduction to second gradient theory of elasticity - Arjun Narayanan
Arjun Narayanan
 
Kgeppt spvm 0_try1
Kgeppt spvm 0_try1Kgeppt spvm 0_try1
Kgeppt spvm 0_try1
foxtrot jp R
 
CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...
CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...
CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...
The Statistical and Applied Mathematical Sciences Institute
 
Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...
Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...
Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...
Cristian Soriano-Camelo
 
Comments_on_Paper_Shuey_1985
Comments_on_Paper_Shuey_1985Comments_on_Paper_Shuey_1985
Comments_on_Paper_Shuey_1985
Benjamin Seive
 
CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...
CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...
CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...
The Statistical and Applied Mathematical Sciences Institute
 
Outgoing ingoingkleingordon spvmforminit1 - copy - copy
Outgoing ingoingkleingordon spvmforminit1 - copy - copyOutgoing ingoingkleingordon spvmforminit1 - copy - copy
Outgoing ingoingkleingordon spvmforminit1 - copy - copy
foxtrot jp R
 
GSA-BBO HYBRIDIZATION ALGORITHM
GSA-BBO HYBRIDIZATION ALGORITHMGSA-BBO HYBRIDIZATION ALGORITHM
GSA-BBO HYBRIDIZATION ALGORITHM
Sajad Ahmad Rather
 
Prediction of the daily global solar irradiance received on a horizontal surf...
Prediction of the daily global solar irradiance received on a horizontal surf...Prediction of the daily global solar irradiance received on a horizontal surf...
Prediction of the daily global solar irradiance received on a horizontal surf...
irjes
 
Application of vector integration
Application of vector integration Application of vector integration
Application of vector integration
Varuna Kapuge
 
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...
IOSR Journals
 
Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...
Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...
Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...
Jiří Šmída
 
Wd equation
Wd equationWd equation
Wd equation
jmguerrac
 
ABOUT MODELLING OF THE GRAVITATIONAL FIELDS
ABOUT MODELLING OF THE GRAVITATIONAL FIELDSABOUT MODELLING OF THE GRAVITATIONAL FIELDS
ABOUT MODELLING OF THE GRAVITATIONAL FIELDS
ijrap
 
Chapter 1: Physical Quantities Unit, And Vectors
Chapter 1: Physical Quantities Unit, And VectorsChapter 1: Physical Quantities Unit, And Vectors
Chapter 1: Physical Quantities Unit, And Vectors
Reema
 

What's hot (20)

Parabolic Restricted Three Body Problem
Parabolic Restricted Three Body ProblemParabolic Restricted Three Body Problem
Parabolic Restricted Three Body Problem
 
The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...
The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...
The Shifting Seasonal Mean Autoregressive Model and Seasonality in the Centra...
 
Tachyon inflation in DBI and RSII context
Tachyon inflation in DBI and RSII contextTachyon inflation in DBI and RSII context
Tachyon inflation in DBI and RSII context
 
Ground Response Analysis - Geotechnical Earthquake engineering
Ground Response Analysis - Geotechnical Earthquake engineeringGround Response Analysis - Geotechnical Earthquake engineering
Ground Response Analysis - Geotechnical Earthquake engineering
 
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
Dragoljub Dimitrijević "Tachyon Inflation in the RSII Framework"
 
Introduction to second gradient theory of elasticity - Arjun Narayanan
Introduction to second gradient theory of elasticity - Arjun NarayananIntroduction to second gradient theory of elasticity - Arjun Narayanan
Introduction to second gradient theory of elasticity - Arjun Narayanan
 
Kgeppt spvm 0_try1
Kgeppt spvm 0_try1Kgeppt spvm 0_try1
Kgeppt spvm 0_try1
 
CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...
CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...
CLIM: Transition Workshop - Overview and New Modeling Directions - Dorit Hamm...
 
Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...
Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...
Soil Dynamics - Local Site Effects - Ground response - Geotechnical Earthquak...
 
Comments_on_Paper_Shuey_1985
Comments_on_Paper_Shuey_1985Comments_on_Paper_Shuey_1985
Comments_on_Paper_Shuey_1985
 
CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...
CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...
CLIM Fall 2017 Course: Statistics for Climate Research, Analysis for Climate ...
 
Outgoing ingoingkleingordon spvmforminit1 - copy - copy
Outgoing ingoingkleingordon spvmforminit1 - copy - copyOutgoing ingoingkleingordon spvmforminit1 - copy - copy
Outgoing ingoingkleingordon spvmforminit1 - copy - copy
 
GSA-BBO HYBRIDIZATION ALGORITHM
GSA-BBO HYBRIDIZATION ALGORITHMGSA-BBO HYBRIDIZATION ALGORITHM
GSA-BBO HYBRIDIZATION ALGORITHM
 
Prediction of the daily global solar irradiance received on a horizontal surf...
Prediction of the daily global solar irradiance received on a horizontal surf...Prediction of the daily global solar irradiance received on a horizontal surf...
Prediction of the daily global solar irradiance received on a horizontal surf...
 
Application of vector integration
Application of vector integration Application of vector integration
Application of vector integration
 
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...
Anisotropic Bianchi Type-III Dark Energy Model with Time-Dependent Decelerati...
 
Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...
Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...
Trends in daily rainfall erosivity in relation with NAO, MO and WeMO for Spai...
 
Wd equation
Wd equationWd equation
Wd equation
 
ABOUT MODELLING OF THE GRAVITATIONAL FIELDS
ABOUT MODELLING OF THE GRAVITATIONAL FIELDSABOUT MODELLING OF THE GRAVITATIONAL FIELDS
ABOUT MODELLING OF THE GRAVITATIONAL FIELDS
 
Chapter 1: Physical Quantities Unit, And Vectors
Chapter 1: Physical Quantities Unit, And VectorsChapter 1: Physical Quantities Unit, And Vectors
Chapter 1: Physical Quantities Unit, And Vectors
 

Similar to Observational parameters of Inflation in Holographic cosmology

Observational Parameters in a Braneworld Inlationary Scenario
Observational Parameters in a Braneworld Inlationary ScenarioObservational Parameters in a Braneworld Inlationary Scenario
Observational Parameters in a Braneworld Inlationary Scenario
Milan Milošević
 
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
Milan Milošević
 
The Dark Side of the Universe
The Dark Side of the UniverseThe Dark Side of the Universe
The Dark Side of the Universe
RohanSrivastava56
 
Biermann - Quarks and the Cosmos
Biermann - Quarks and the CosmosBiermann - Quarks and the Cosmos
Biermann - Quarks and the Cosmos
Atner Yegorov
 
Ms 1341-p touze-final
Ms 1341-p touze-finalMs 1341-p touze-final
Ms 1341-p touze-final
ThomasTouz
 
Quantum gravity phenomenology: Minimal length
Quantum gravity phenomenology: Minimal lengthQuantum gravity phenomenology: Minimal length
Quantum gravity phenomenology: Minimal length
Milad Hajebrahimi
 
Cosmology
CosmologyCosmology
Cosmology
DanielBaumann11
 
The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...
The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...
The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...
Lake Como School of Advanced Studies
 
My Prize Winning Physics Poster from 2006
My Prize Winning Physics Poster from 2006My Prize Winning Physics Poster from 2006
My Prize Winning Physics Poster from 2006
Dr. Catherine Sinclair She/Her
 
Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...
Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...
Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...
JeremyHeyl
 
The Past, Present and Future of Cosmology
The Past, Present and Future of Cosmology The Past, Present and Future of Cosmology
The Past, Present and Future of Cosmology
DanielBaumann11
 
Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)
Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)
Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)
Christian Ott
 
Day 6 cosmo_inflation_ss09
Day 6 cosmo_inflation_ss09Day 6 cosmo_inflation_ss09
Day 6 cosmo_inflation_ss09
Atner Yegorov
 
Higgs inflation
Higgs inflationHiggs inflation
Higgs inflation
OUSEPHCJ
 
Euler lagrange equations of motion mit-holonomic constraints_lecture7
Euler lagrange equations of motion  mit-holonomic  constraints_lecture7Euler lagrange equations of motion  mit-holonomic  constraints_lecture7
Euler lagrange equations of motion mit-holonomic constraints_lecture7
JOHN OBIDI
 
Neil Lambert - From D-branes to M-branes
Neil Lambert - From D-branes to M-branesNeil Lambert - From D-branes to M-branes
Neil Lambert - From D-branes to M-branes
SEENET-MTP
 
E03503025029
E03503025029E03503025029
E03503025029
theijes
 
Wave Mechanics and Large-scale Structure
Wave Mechanics and Large-scale StructureWave Mechanics and Large-scale Structure
Wave Mechanics and Large-scale Structure
Peter Coles
 
Dark energy by david spergel
Dark energy by david spergelDark energy by david spergel
Dark energy by david spergel
Kumar
 
Adamek_SestoGR18.pdf
Adamek_SestoGR18.pdfAdamek_SestoGR18.pdf
Adamek_SestoGR18.pdf
garfacio30
 

Similar to Observational parameters of Inflation in Holographic cosmology (20)

Observational Parameters in a Braneworld Inlationary Scenario
Observational Parameters in a Braneworld Inlationary ScenarioObservational Parameters in a Braneworld Inlationary Scenario
Observational Parameters in a Braneworld Inlationary Scenario
 
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
Numerical Calculation of the Hubble Hierarchy Parameters and the Observationa...
 
The Dark Side of the Universe
The Dark Side of the UniverseThe Dark Side of the Universe
The Dark Side of the Universe
 
Biermann - Quarks and the Cosmos
Biermann - Quarks and the CosmosBiermann - Quarks and the Cosmos
Biermann - Quarks and the Cosmos
 
Ms 1341-p touze-final
Ms 1341-p touze-finalMs 1341-p touze-final
Ms 1341-p touze-final
 
Quantum gravity phenomenology: Minimal length
Quantum gravity phenomenology: Minimal lengthQuantum gravity phenomenology: Minimal length
Quantum gravity phenomenology: Minimal length
 
Cosmology
CosmologyCosmology
Cosmology
 
The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...
The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...
The Analytical/Numerical Relativity Interface behind Gravitational Waves: Lec...
 
My Prize Winning Physics Poster from 2006
My Prize Winning Physics Poster from 2006My Prize Winning Physics Poster from 2006
My Prize Winning Physics Poster from 2006
 
Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...
Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...
Gwendolyn Eadie: A New Method for Time Series Analysis in Astronomy with an A...
 
The Past, Present and Future of Cosmology
The Past, Present and Future of Cosmology The Past, Present and Future of Cosmology
The Past, Present and Future of Cosmology
 
Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)
Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)
Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)
 
Day 6 cosmo_inflation_ss09
Day 6 cosmo_inflation_ss09Day 6 cosmo_inflation_ss09
Day 6 cosmo_inflation_ss09
 
Higgs inflation
Higgs inflationHiggs inflation
Higgs inflation
 
Euler lagrange equations of motion mit-holonomic constraints_lecture7
Euler lagrange equations of motion  mit-holonomic  constraints_lecture7Euler lagrange equations of motion  mit-holonomic  constraints_lecture7
Euler lagrange equations of motion mit-holonomic constraints_lecture7
 
Neil Lambert - From D-branes to M-branes
Neil Lambert - From D-branes to M-branesNeil Lambert - From D-branes to M-branes
Neil Lambert - From D-branes to M-branes
 
E03503025029
E03503025029E03503025029
E03503025029
 
Wave Mechanics and Large-scale Structure
Wave Mechanics and Large-scale StructureWave Mechanics and Large-scale Structure
Wave Mechanics and Large-scale Structure
 
Dark energy by david spergel
Dark energy by david spergelDark energy by david spergel
Dark energy by david spergel
 
Adamek_SestoGR18.pdf
Adamek_SestoGR18.pdfAdamek_SestoGR18.pdf
Adamek_SestoGR18.pdf
 

More from Milan Milošević

Inflacija, crne rupe i Fizika u Nišu
Inflacija, crne rupe i Fizika u NišuInflacija, crne rupe i Fizika u Nišu
Inflacija, crne rupe i Fizika u Nišu
Milan Milošević
 
Kako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fiziku
Kako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fizikuKako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fiziku
Kako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fiziku
Milan Milošević
 
Sunce - zvezda iz Sunčevog sistema
Sunce - zvezda iz Sunčevog sistemaSunce - zvezda iz Sunčevog sistema
Sunce - zvezda iz Sunčevog sistema
Milan Milošević
 
Od crne rupe do Nobelove nagrade za fiziku
Od crne rupe do Nobelove nagrade za fizikuOd crne rupe do Nobelove nagrade za fiziku
Od crne rupe do Nobelove nagrade za fiziku
Milan Milošević
 
Od velikog praska do Nobelove nagrade za fiziku za 2019. godinu
Od velikog praska do Nobelove nagrade za fiziku za 2019. godinuOd velikog praska do Nobelove nagrade za fiziku za 2019. godinu
Od velikog praska do Nobelove nagrade za fiziku za 2019. godinu
Milan Milošević
 
Evolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivo
Evolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivoEvolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivo
Evolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivo
Milan Milošević
 
Kako videti nevidljivo? - prva fotografija crne rupe
Kako videti nevidljivo? - prva fotografija crne rupeKako videti nevidljivo? - prva fotografija crne rupe
Kako videti nevidljivo? - prva fotografija crne rupe
Milan Milošević
 
Kako preživeti internet?
Kako preživeti internet?Kako preživeti internet?
Kako preživeti internet?
Milan Milošević
 
CERN mesto gde je nastao "internet"
CERN mesto gde je nastao "internet"CERN mesto gde je nastao "internet"
CERN mesto gde je nastao "internet"
Milan Milošević
 
Kako je svet postao globalno selo?
Kako je svet postao globalno selo?Kako je svet postao globalno selo?
Kako je svet postao globalno selo?
Milan Milošević
 
NETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacija
NETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacijaNETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacija
NETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacija
Milan Milošević
 
NETCHEM CPD: Video konferencijsko povezivanje
NETCHEM CPD: Video konferencijsko povezivanjeNETCHEM CPD: Video konferencijsko povezivanje
NETCHEM CPD: Video konferencijsko povezivanje
Milan Milošević
 
Fizika mobilnog telefona
Fizika mobilnog telefonaFizika mobilnog telefona
Fizika mobilnog telefona
Milan Milošević
 
30 godina World Wide Web-a
30 godina World Wide Web-a30 godina World Wide Web-a
30 godina World Wide Web-a
Milan Milošević
 
"Svet nauke" o svetu nauke
"Svet nauke" o svetu nauke"Svet nauke" o svetu nauke
"Svet nauke" o svetu nauke
Milan Milošević
 
NETCHEM Forum
NETCHEM ForumNETCHEM Forum
NETCHEM Forum
Milan Milošević
 
Overview of collected WARIAL data from NETCHEM consortium
Overview of collected WARIAL data from NETCHEM consortiumOverview of collected WARIAL data from NETCHEM consortium
Overview of collected WARIAL data from NETCHEM consortium
Milan Milošević
 
Agreement of protection of intellectual property
Agreement of protection of intellectual propertyAgreement of protection of intellectual property
Agreement of protection of intellectual property
Milan Milošević
 
Invision power. Visual design of Forum. Demo
Invision power. Visual design of Forum. DemoInvision power. Visual design of Forum. Demo
Invision power. Visual design of Forum. Demo
Milan Milošević
 
Multilingual approach for video-clips and learning material for courses offer...
Multilingual approach for video-clips and learning material for courses offer...Multilingual approach for video-clips and learning material for courses offer...
Multilingual approach for video-clips and learning material for courses offer...
Milan Milošević
 

More from Milan Milošević (20)

Inflacija, crne rupe i Fizika u Nišu
Inflacija, crne rupe i Fizika u NišuInflacija, crne rupe i Fizika u Nišu
Inflacija, crne rupe i Fizika u Nišu
 
Kako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fiziku
Kako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fizikuKako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fiziku
Kako smo videli nevidljivo - od crne rupe do Nobelove nagrade za fiziku
 
Sunce - zvezda iz Sunčevog sistema
Sunce - zvezda iz Sunčevog sistemaSunce - zvezda iz Sunčevog sistema
Sunce - zvezda iz Sunčevog sistema
 
Od crne rupe do Nobelove nagrade za fiziku
Od crne rupe do Nobelove nagrade za fizikuOd crne rupe do Nobelove nagrade za fiziku
Od crne rupe do Nobelove nagrade za fiziku
 
Od velikog praska do Nobelove nagrade za fiziku za 2019. godinu
Od velikog praska do Nobelove nagrade za fiziku za 2019. godinuOd velikog praska do Nobelove nagrade za fiziku za 2019. godinu
Od velikog praska do Nobelove nagrade za fiziku za 2019. godinu
 
Evolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivo
Evolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivoEvolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivo
Evolucija zvezda i nastanak crnih rupa - kako smo videli nevidljivo
 
Kako videti nevidljivo? - prva fotografija crne rupe
Kako videti nevidljivo? - prva fotografija crne rupeKako videti nevidljivo? - prva fotografija crne rupe
Kako videti nevidljivo? - prva fotografija crne rupe
 
Kako preživeti internet?
Kako preživeti internet?Kako preživeti internet?
Kako preživeti internet?
 
CERN mesto gde je nastao "internet"
CERN mesto gde je nastao "internet"CERN mesto gde je nastao "internet"
CERN mesto gde je nastao "internet"
 
Kako je svet postao globalno selo?
Kako je svet postao globalno selo?Kako je svet postao globalno selo?
Kako je svet postao globalno selo?
 
NETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacija
NETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacijaNETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacija
NETCHEM CPD: Audio prezentovanje jednosmerna i dvosmerna komunikacija
 
NETCHEM CPD: Video konferencijsko povezivanje
NETCHEM CPD: Video konferencijsko povezivanjeNETCHEM CPD: Video konferencijsko povezivanje
NETCHEM CPD: Video konferencijsko povezivanje
 
Fizika mobilnog telefona
Fizika mobilnog telefonaFizika mobilnog telefona
Fizika mobilnog telefona
 
30 godina World Wide Web-a
30 godina World Wide Web-a30 godina World Wide Web-a
30 godina World Wide Web-a
 
"Svet nauke" o svetu nauke
"Svet nauke" o svetu nauke"Svet nauke" o svetu nauke
"Svet nauke" o svetu nauke
 
NETCHEM Forum
NETCHEM ForumNETCHEM Forum
NETCHEM Forum
 
Overview of collected WARIAL data from NETCHEM consortium
Overview of collected WARIAL data from NETCHEM consortiumOverview of collected WARIAL data from NETCHEM consortium
Overview of collected WARIAL data from NETCHEM consortium
 
Agreement of protection of intellectual property
Agreement of protection of intellectual propertyAgreement of protection of intellectual property
Agreement of protection of intellectual property
 
Invision power. Visual design of Forum. Demo
Invision power. Visual design of Forum. DemoInvision power. Visual design of Forum. Demo
Invision power. Visual design of Forum. Demo
 
Multilingual approach for video-clips and learning material for courses offer...
Multilingual approach for video-clips and learning material for courses offer...Multilingual approach for video-clips and learning material for courses offer...
Multilingual approach for video-clips and learning material for courses offer...
 

Recently uploaded

RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem studentsRHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
Himanshu Rai
 
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
مصحف القراءات العشر   أعد أحرف الخلاف سمير بسيوني.pdfمصحف القراءات العشر   أعد أحرف الخلاف سمير بسيوني.pdf
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
سمير بسيوني
 
Benner "Expanding Pathways to Publishing Careers"
Benner "Expanding Pathways to Publishing Careers"Benner "Expanding Pathways to Publishing Careers"
Benner "Expanding Pathways to Publishing Careers"
National Information Standards Organization (NISO)
 
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
A Free 200-Page eBook ~ Brain and Mind Exercise.pptxA Free 200-Page eBook ~ Brain and Mind Exercise.pptx
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
OH TEIK BIN
 
Juneteenth Freedom Day 2024 David Douglas School District
Juneteenth Freedom Day 2024 David Douglas School DistrictJuneteenth Freedom Day 2024 David Douglas School District
Juneteenth Freedom Day 2024 David Douglas School District
David Douglas School District
 
Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"
Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"
Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"
National Information Standards Organization (NISO)
 
A Visual Guide to 1 Samuel | A Tale of Two Hearts
A Visual Guide to 1 Samuel | A Tale of Two HeartsA Visual Guide to 1 Samuel | A Tale of Two Hearts
A Visual Guide to 1 Samuel | A Tale of Two Hearts
Steve Thomason
 
Oliver Asks for More by Charles Dickens (9)
Oliver Asks for More by Charles Dickens (9)Oliver Asks for More by Charles Dickens (9)
Oliver Asks for More by Charles Dickens (9)
nitinpv4ai
 
Bonku-Babus-Friend by Sathyajith Ray (9)
Bonku-Babus-Friend by Sathyajith Ray  (9)Bonku-Babus-Friend by Sathyajith Ray  (9)
Bonku-Babus-Friend by Sathyajith Ray (9)
nitinpv4ai
 
How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17
Celine George
 
Bossa N’ Roll Records by Ismael Vazquez.
Bossa N’ Roll Records by Ismael Vazquez.Bossa N’ Roll Records by Ismael Vazquez.
Bossa N’ Roll Records by Ismael Vazquez.
IsmaelVazquez38
 
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptxNEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
iammrhaywood
 
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
ImMuslim
 
Skimbleshanks-The-Railway-Cat by T S Eliot
Skimbleshanks-The-Railway-Cat by T S EliotSkimbleshanks-The-Railway-Cat by T S Eliot
Skimbleshanks-The-Railway-Cat by T S Eliot
nitinpv4ai
 
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdfREASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
giancarloi8888
 
MDP on air pollution of class 8 year 2024-2025
MDP on air pollution of class 8 year 2024-2025MDP on air pollution of class 8 year 2024-2025
MDP on air pollution of class 8 year 2024-2025
khuleseema60
 
How to Predict Vendor Bill Product in Odoo 17
How to Predict Vendor Bill Product in Odoo 17How to Predict Vendor Bill Product in Odoo 17
How to Predict Vendor Bill Product in Odoo 17
Celine George
 
Educational Technology in the Health Sciences
Educational Technology in the Health SciencesEducational Technology in the Health Sciences
Educational Technology in the Health Sciences
Iris Thiele Isip-Tan
 
Wound healing PPT
Wound healing PPTWound healing PPT
Wound healing PPT
Jyoti Chand
 
Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...
PsychoTech Services
 

Recently uploaded (20)

RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem studentsRHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
 
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
مصحف القراءات العشر   أعد أحرف الخلاف سمير بسيوني.pdfمصحف القراءات العشر   أعد أحرف الخلاف سمير بسيوني.pdf
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
 
Benner "Expanding Pathways to Publishing Careers"
Benner "Expanding Pathways to Publishing Careers"Benner "Expanding Pathways to Publishing Careers"
Benner "Expanding Pathways to Publishing Careers"
 
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
A Free 200-Page eBook ~ Brain and Mind Exercise.pptxA Free 200-Page eBook ~ Brain and Mind Exercise.pptx
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
 
Juneteenth Freedom Day 2024 David Douglas School District
Juneteenth Freedom Day 2024 David Douglas School DistrictJuneteenth Freedom Day 2024 David Douglas School District
Juneteenth Freedom Day 2024 David Douglas School District
 
Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"
Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"
Jemison, MacLaughlin, and Majumder "Broadening Pathways for Editors and Authors"
 
A Visual Guide to 1 Samuel | A Tale of Two Hearts
A Visual Guide to 1 Samuel | A Tale of Two HeartsA Visual Guide to 1 Samuel | A Tale of Two Hearts
A Visual Guide to 1 Samuel | A Tale of Two Hearts
 
Oliver Asks for More by Charles Dickens (9)
Oliver Asks for More by Charles Dickens (9)Oliver Asks for More by Charles Dickens (9)
Oliver Asks for More by Charles Dickens (9)
 
Bonku-Babus-Friend by Sathyajith Ray (9)
Bonku-Babus-Friend by Sathyajith Ray  (9)Bonku-Babus-Friend by Sathyajith Ray  (9)
Bonku-Babus-Friend by Sathyajith Ray (9)
 
How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17How to Manage Reception Report in Odoo 17
How to Manage Reception Report in Odoo 17
 
Bossa N’ Roll Records by Ismael Vazquez.
Bossa N’ Roll Records by Ismael Vazquez.Bossa N’ Roll Records by Ismael Vazquez.
Bossa N’ Roll Records by Ismael Vazquez.
 
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptxNEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
 
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
 
Skimbleshanks-The-Railway-Cat by T S Eliot
Skimbleshanks-The-Railway-Cat by T S EliotSkimbleshanks-The-Railway-Cat by T S Eliot
Skimbleshanks-The-Railway-Cat by T S Eliot
 
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdfREASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
REASIGNACION 2024 UGEL CHUPACA 2024 UGEL CHUPACA.pdf
 
MDP on air pollution of class 8 year 2024-2025
MDP on air pollution of class 8 year 2024-2025MDP on air pollution of class 8 year 2024-2025
MDP on air pollution of class 8 year 2024-2025
 
How to Predict Vendor Bill Product in Odoo 17
How to Predict Vendor Bill Product in Odoo 17How to Predict Vendor Bill Product in Odoo 17
How to Predict Vendor Bill Product in Odoo 17
 
Educational Technology in the Health Sciences
Educational Technology in the Health SciencesEducational Technology in the Health Sciences
Educational Technology in the Health Sciences
 
Wound healing PPT
Wound healing PPTWound healing PPT
Wound healing PPT
 
Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...
 

Observational parameters of Inflation in Holographic cosmology

  • 1. OBSERVATIONAL PARAMETERS OF INFLATION IN HOLOGRAPHIC COSMOLOGY MILAN MILOŠEVIĆ Faculty of Sciences And Mathematics University of Niš, Serbia & SEENET-MTP Centre Workshop on Quantum Fields and Nonlinear Phenomena 27 September 2020 in collaboration with: N. Bilić (Zagreb), G.S. Djordjević, D.D. Dimitrijević and M. Stojanović (Niš) This work has been supported by the Serbian Ministry for Education, Science and Technological Development under the project No. 176021 and contract No. 451-03-68/2020-14/200124, as well as the ICTP - SEENET-MTP project NT-03 Cosmology-Classical and Quantum Challenges
  • 2. INTRODUCTION • The inflation theory proposes a period of extremely rapid (exponential) expansion of the universe during the an early stage of evolution of the universe. • The inflation theory predicts that during inflation (it takes about 10−34 𝑠) radius of the universe increased, at least 𝑒60 ≈ 1026 times. • Although inflationary cosmology has successfully complemented the Standard Model, the process of inflation, in particular its origin, is still largely unknown. • Recent years brought us a lot of evidence from WMAP and Planck observations of the CMB • The most important way to test inflationary cosmological models is to compare the computed and measured values of the observational parameters. Figure: Baumann, D. TASI Lectures on Inflation. (2009), arXiv:0907.5424 [hep-th]
  • 3. BRANEWORLD COSMOLOGY • Braneworld universe is based on the scenario in which matter is confined on a brane moving in the higher dimensional bulk with only gravity allowed to propagate in the bulk. • One of the simplest models - Randall-Sundrum (RS) • RS model was originally proposed to solve the hierarchy problem (1999) • Later it was realized that this model, as well as any similar braneworld model, may have interesting cosmological implications • Two branes with opposite tensions are placed at some distance in 5 dimensional space • RS model – observer reside on the brane with negative tension, distance to the 2nd brane corresponds to the Netwonian gravitational constant • RSII model – observer is placed on the positive tension brane, the 2nd brane is pushed to infinity N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic, M. Milosevic, Tachyon inflation in an AdS braneworld with back-reaction, International Journal of Modern Physics A. 32 (2017) 1750039. N. Bilic, G.B. Tupper, AdS braneworld with backreaction, Cent. Eur. J. Phys. 12 (2014) 147–159.
  • 4. HOLOGRAPHIC BRANEWORLD • Holographic braneworld - a cosmology based on the effective four-dimensional Einstein equations on the holographic boundary in the framework of anti de Sitter/conformal field theory (AdS/CFT) correspondence. • The model is based on a holographic braneworld scenario with an effective tachyon field on a D3-brane located at the holographic boundary of an asymptotic AdS5 bulk. • The cosmology is governed by matter on the brane in addition to the boundary CFT time Conformal boundary at z=0 space z xRSII brane at z=zbr N. Bilić, Randall-Sundrum vs Holographic Cosmology, IRB (2015)
  • 5. HOLOGRAPHIC TACHYON COSMOLOGY • The holographic braneworld is a spatially flat FRW universe with line element ൯𝑑𝑠2 = 𝑔 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 = 𝑑𝑡2 − 𝑎2(𝑡)(𝑑𝑟2 + 𝑟2 𝑑𝛺2 • The holographic Friedmann equations • Where the scale 𝓁 can be identified with the AdS curvature radius and we introduced a dimensionless expansion rate ℎ ≡ 𝓁𝐻 and the fundamental dimensionless coupling ℎ2 − 𝓁2 4 ℎ4 = 𝜅2 3 𝓁4 𝜌 ሶℎ 1 − 𝓁2 2 ℎ2 = − 𝜅2 3 𝓁3(𝑝 + 𝜌) Standard cosmology: ℎ2 = 𝜅2 3 𝜌 ሶℎ = − 𝜅2 2 (𝑝 + 𝜌) 𝜅2 = 8𝜋𝐺 𝑁 𝓁2 Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the holographic braneworld. Journal of Cosmology and Astroparticle Physics 2019, 034–034 (2019).
  • 6. HOLOGRAPHIC TACHYON COSMOLOGY • Interesting property - solving the first Friedmann equation as a quadratic equation ℎ2 = 2 1 ± 1 − 𝜅2 3 𝓁4 𝜌 • We do not want our modified cosmology to depart too much from the standard cosmology after the inflation era and demand that this equation reduces to the standard Friedmann equation in the low density limit 𝜅2 𝓁4 𝜌 ≪ 1 • This demand will be met only by the (−) sign solution. We discard the (+) sign solution as unphysical.
  • 7. HOLOGRAPHIC TACHYON COSMOLOGY • The physical range of the Hubble expansion rate is between ℎmin = 0 and the maximal value ℎmax = 2 • It corresponds to the maximal energy density 𝜌max = Τ3 𝜅2 𝓁4 • Assuming no violation of the weak energy condition 𝑝 + 𝜌 ≥ 0, the expansion rate will be a monotonously decreasing function of time. • The universe starts from 𝑡 = 0 with an initial ℎ𝑖 ≤ ℎ 𝑚𝑎𝑥 with energy density and cosmological scale both finite. • The Big Bang singularity is avoided. Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the holographic braneworld. JCAP, 034–034 (2019).
  • 8. EQUATIONS OF MOTION • Tachyon matter in the holographic braneworld is described by the DBI Lagrangian and the Hamiltonian ℒ = −𝓁−4 𝑉( Τ𝜃 𝓁) 1 − 𝑔 𝜇𝜈 𝜃,𝜇 𝜃,𝜈 ℋ = 𝓁−4 𝑉 1 + 𝜂2 where 𝜂 = 𝑔 𝜇𝜈 𝜋 𝜇 𝜋 𝜈 𝓁4 𝑉 . • As usual, the conjugate momentum is 𝜋 𝜇 = 𝜕ℒ 𝜕𝜃,𝜇 • The Hamilton equations are 𝜃,𝜇 = 𝜕ℋ 𝜕𝜋 𝜇 𝜋;𝜇 𝜇 = − 𝜕ℋ 𝜕𝜃
  • 9. EQUATIONS OF MOTION • The equations of motions ሶ𝜃 = 𝜂 1 + 𝜂2 ሶ𝜂 = − 3ℎ𝜂 𝓁 − 𝑉,𝜃 𝑉 1 + 𝜂2 + 𝜂2 1 + 𝜂2 • As usual, the pressure and energy density are equal to Lagrangian and Hamiltonian 𝑝 ≡ ℒ = −𝓁−4 𝑉 1 − ሶ𝜃2 = − 𝓁−4 𝑉 1 − 𝜂2 𝜌 ≡ ℋ = 𝓁−4 𝑉 1 − ሶ𝜃2 = 𝓁−4 𝑉 1 − 𝜂2
  • 10. INITIAL CONDITIONS • Two natural initial conditions a) 𝜂𝑖 = 0 b) ሶ𝜂𝑖 = 0 • The condition: • (a) assures a finite initial ሶℎ, and • (b) provides the solution consistent with the slow-roll regime
  • 11. A) 𝜂𝑖 = 0 • 0 < ℎ𝑖 < 2 • From 𝑝 = − 𝓁−4 𝑉 1−𝜂2 and 𝜌 = 𝓁−4 𝑉 1 − 𝜂2 it follows 𝑝𝑖 = −𝜌𝑖 • From ℎ2 − 𝓁2 4 ℎ4 = 𝜅2 3 𝓁4 𝜌 and 𝜌 = 𝓁−4 𝑉 1− ሶ𝜃2 the initial 𝜃𝑖 can be fixed 𝑉 𝜃𝑖 = 3 𝜅2 ℎ𝑖 2 − ℎ𝑖 4 4
  • 12. B) ሶ𝜂𝑖 = 0 • From EqM we have 𝜂𝑖 = − 2 𝓁 Τ𝑉,𝜃 𝑉 𝑖 9ℎ 𝑖 2 −4 𝓁 Τ𝑉,𝜃 𝑉 𝑖 2 +3 9ℎ 𝑖 4 −4ℎ 𝑖 2 𝓁 Τ𝑉,𝜃 𝑉 𝑖 2 • From Friedmann equations we obtain 1 − ℎ𝑖 2 2 2 = 1 − 𝜅2 3 𝑉 𝜃𝑖 1 + 𝜂𝑖 2 • Random numbers: ℎ𝑖, 𝜅 and parameters in the potential 𝑉 𝜃 • Numerical solutions: 𝜂𝑖 and 𝜃𝑖 • Not easy to solve, sometimes the real solution doesn’t exist, etc 𝑉 = 𝑉 𝜃 𝑉′𝜃 = 𝑑𝑉(𝜃) 𝑑𝜃
  • 13. B) ሶ𝜂𝑖 = 0 • A different way • Random numbers ℎ𝑖, 𝜃𝑖 and parameters in the potential 𝑉 𝜃 𝜅2 = )3𝑉(𝜃𝑖 1 + 𝜂𝑖 2 1 − 1 − ℎ𝑖 2 2 2 and calculate 𝜂𝑖 in the same way as in the previous way.
  • 14. THE SLOW-ROLL PARAMETERS • Number of e-folds 𝑁 𝑡 = න 𝑡 𝐶𝑀𝐵 𝑡 𝑒𝑛𝑑 𝐻 𝑡 𝑑𝑡 • Hubble hierarchy (slow-roll) parameters 𝜀𝑖+1 ≡ 𝑑ln|𝜀𝑖| 𝑑𝑁 , 𝑖 ≥ 0, 𝜀0 ≡ 𝐻∗ 𝐻 where 𝐻∗ is the Hubble parameter at some chosen time • The first two 𝜀 parameters 𝜀1 = − ሶ𝐻 𝐻2, 𝜀2 = 2𝜀1 + ሷ𝐻 𝐻 ሶ𝐻 , etc. • The end of inflation 𝜀𝑖(𝜙 𝑒𝑛𝑑) ≈ 1 𝜙 𝑒𝑛𝑑 = 𝜙(𝑡 𝑒𝑛𝑑)
  • 15. OBSERVATIONAL PARAMETERS • Three independent observational parameters: amplitude of scalar perturbation 𝐴 𝑠, tensor-to-scalar ratio 𝑟 and scalar spectral index 𝑛 𝑠 𝑟 = 16𝜀1(𝜙𝑖) 𝑛 𝑠 = 1 − 2𝜀1(𝜙𝑖) − 𝜀2(𝜙𝑖) • Satellite Planck (May 2009 – October 2013) • Planck Collaboration • The latest results were published in 2018. At the lowest order in parameters 𝜀1 and 𝜀2 Planck 2018 results. X Constraints on inflation, arXiv:1807.06211 [astro-ph.CO]
  • 16. OBSERVATIONAL PARAMETERS 𝑛 𝑠, 𝑟 𝑟 = 16𝜀1 1 + 𝐶𝜀2 + )2(2 − ℎ2 )3(4 − ℎ2 𝑝𝑝,𝑋𝑋 𝑝,𝑋 2 𝜀1 𝑛s = 1 − 2𝜀1 − 𝜀2 − 2 + 8ℎ2 3 4 − ℎ2 2 𝑝𝑝,𝑋𝑋 𝑝,𝑋 2 𝜀1 2 − 3 + 2𝐶 + )2(2 − ℎ2 )3(4 − ℎ2 𝑝𝑝,𝑋𝑋 𝑝,𝑋 2 𝜀1 𝜀2 − 𝐶𝜀2 𝜀3 • For 𝑋 = ሶ𝜃2 and 𝑝 = −𝑉 1 − 𝑋 we have 𝑝𝑝,𝑋𝑋 𝑝,𝑋 2 = −1 Bertini, N. R., Bilic, N. & Rodrigues, D. C. Primordial perturbations and inflation in holographic cosmology. arXiv:2007.02332 [gr-qc]
  • 17. POTENTIALS 𝑽 𝜽 = 𝟏 𝒄𝒐𝒔𝒉(𝝎𝜽) 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
  • 19. 𝑽 𝜽 = 𝟏 𝒄𝒐𝒔𝒉(𝝎𝜽) 60 < 𝑁 < 90 0 < 𝜔 < 0.25 0 < 𝜃𝑖 < 20 Colour represents the number of 𝑛 𝑠, 𝑟 points in a hexagon
  • 22.
  • 23. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
  • 24. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽 60 < 𝑁 < 90 0 < 𝜔 < 0.25 0 < 𝜃𝑖 < 20
  • 25. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
  • 26. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽 - THE BEST FITTING RESULTS
  • 27. CONCLUSIONS • We discussed a model of tachyon inflation based on a holographic braneworld scenario with a brane located at the boundary of the AdS5 bulk. • We simulated observational parameters of inflation for two potentials 𝑽 𝜽 = 𝟏 𝒄𝒐𝒔𝒉(𝝎𝜽) , 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽 • The agreement of our model with the Planck observational data is good, especially for a higher number of e-folds. • Preliminary results are promising and open good opportunity for further analytical research of these potentials. This work has been supported by the Serbian Ministry for Education, Science and Technological Development under the project No. 176021 and contract No. 451-03-68/2020-14/200124, as well as the ICTP - SEENET-MTP project NT-03 Cosmology-Classical and Quantum Challenges
  • 28. REFERENCES 1. Bilić, N., Dimitrijević, D. D., Djordjevic, G. S., Milošević, M. & Stojanović, M. Tachyon inflation in the holographic braneworld. Journal of Cosmology and Astroparticle Physics 2019, 034–034 (2019). 2. N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic & M. Milosevic, Tachyon inflation in an AdS braneworld with back-reaction, International Journal of Modern Physics A. 32 (2017) 1750039. 3. N.R. Bertini, N. Bilic & D.C. Rodrigues, Primordial perturbations and inflation in holographic cosmology. arXiv:2007.02332 [gr-qc]. 4. N. Bilić, Holographic cosmology and tachyon inflation. International Journal of Modern Physics A 33, 1845004 (2018). 5. N. Bilić, Randall-Sundrum versus holographic cosmology, Phys. Rev. D 93 (2016) 066010 [arXiv:1511.07323]