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BRANEWORLD COSMOLOGY AND TACHYON INFLATION – RSII NUMERICAL MODELS
MILAN MILOŠEVIĆDepartment of Physics
Faculty of Sciences and MathematicsUniversity of Niš, Serbia
XVIII Serbian Astronomical Conference, Belgrade, 17-21 October 2017
In collaboration with N. Bilić (Zagreb), G. Đorđević,D. Dimitrijević, D. Delibašić, M. Stojanović (Niš)
Based on: N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic, M. Milosevic, Tachyon inflation in anAdS braneworld with back-reaction, International Journal of Modern Physics A. 32 (2017) 1750039.
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OUTLINE
• Introduction
• Tachyon Inflation
• Randall - Sundrum Models (RS)
• Numerical results
• Conclusion
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INTRODUCTION AND MOTIVATION
• The inflationary universe scenario in which the early universe undergoes a rapid expansion has been generally accepted as a solution to the horizon problem and some other related problems of the standard big-bang cosmology
• Quantum cosmology: probably the best way to describe the evolution of the early universe.
• Recent years - a lot of evidence from WMAP and Planck observations of the CMB
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OBSERVATIONAL PARAMETERS
• Hubble hierarchy (slow-roll) parameters
• Length of inflation
• The end of inflation• Three independent observational parameters: amplitude
of scalar perturbation , tensor-to-scalar ratio i scalar spectral index
*1 0
ln | |, 0,i
i
d Hi
dN H
Hubble rate at an arbitrarily chosen time
2Pl
1( ) ln ln
end end end
end
t t
end
t t
a H VN d a Hdt d d
a M V
1i
( ) 1i end
1
1 2
16
1 2s
r
n
At the lowest order in parameters 𝜀 and 𝜀
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OBSERVATIONAL PARAMETERS
• Satelite Planck(May 2009 – October 2013)
• Latest results are publishedin year 2016.
Planck 2015 results: XIII. Cosmological parameters, Astronomy & Astrophysics. 594 (2016) A13Planck 2015 results. XX. Constraints on inflation, Astronomy & Astrophysics. 594 (2016) A20
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LAGRANGIAN OF A SCALAR FIELD -
• In general case – any function of a scalar field andkinetic energy
• Canonical field, potential
,
• Non-canonical models
• Dirac-Born-Infeld (DBI) Lagrangian
• Special case – tachyonic
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TACHYONS
• Traditionally, the word tachyon was used to describe a hypothetical particle which propagates faster than light.
• In modern physics this meaning has been changed• The effective tachyonic field theory was proposed by A. Sen
• String theory: states of quantum fields with imaginary mass (i.e. negative mass squared)
• It was believed: such fields permitted propagation faster than light
• However it was realized that the imaginary mass creates an instability and tachyons spontaneously decay through the process known as tachyon condensation
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TACHYION FIELDS
• No classical interpretation of the”imaginary mass”• The instability: The potential of the
tachyonic field is initially at a localmaximum rather than a localminimum (like a ball at the top ofa hill)
• A small perturbation - forces thefield to roll down towards thelocal minimum.
• Quanta are not tachyon any more, but rather an ”ordinary” particle with a positive mass.
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TACHYON INFLATION
• Consider the tachyonic field T minimally coupled to Einstein's gravity with action
• Where R is Ricci scalar, and Lagrangian and Hamiltionian for tachyon potential are
• Homogenous and isotropic space, FRW metrics
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TACHYON INFLATION
• As well as for a standard scalar field i ,however:
.
• Friedman equation:
⁄ .
• Energy momentum conservation equation, takes a form
.
1
8PMG
Reduced Planck mass
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TACHYON INFLATION
• Nondimensional equations
⁄
• Dimensionless constant , a choice of a constant
(brane tension) was motivated by string theory
𝜏 = 𝑡 𝑇⁄
𝑥 =𝑇
𝑇, 𝑈(𝑥) =
𝑉(𝑥)
𝜎, 𝐻 =
𝐻
𝑇.
Energy-momentum conservation eq.
Friedman eq.
Friedman acceleration eq.
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CONDITION FOR TACHYON INFLATION
• General condition for inflation
• Slow-roll conditions
• Equations for slow-roll inflation
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INITIAL CONDITION FOR TACHYON INFLATION
• Slow-roll parameters
• Number of e-folds
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RANDAL-SUNDRUM MODELS
• 1999 – one of the simplest models
• Two branes with opposite tensions are placed at some distance in 5 dimensional space
• RSI model – observer reside on the brane with negative tension, distance to the 2nd brane corresponds to the Netwonian gravitational constant
• RSII – observer is placed on the positive tension brane, 2nd brane is pushed to infinity
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RSI MODEL
x
5x y
0y y l y
N. Bilic, “Braneworld Universe”, 2nd CERN – SEENET-MTP PhD School, Timisoara, December 2016
0 0
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RSII MODEL
0y y
0 0
N. Bilic, “Braneworld Universe”, 2nd CERN – SEENET-MTP PhD School, Timisoara, December 2016
x
5x y
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RSII MODEL
• The space is described by Anti de Siter metric
• Extended RSII model include radion backreaction
• Total action
• After integrating out 5th dimension…
2 2 2 2(5) 2 2 22 2
1 11 ( ) ,1 ( )
a babds G dX dX k z x g dx dx dz
k z k z x
1k
AdS curvature radius Radion field
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RSII MODEL
• Action for a 3-brane moving in bulk
• Action for the brane
• Without radion
• Total Lagrangian
4, , br
1 ,16 2RS d x g g SG
Canonicali normalized radion field
2 4sinh3G
4 indbr
, ,4 2 2 24 4 2 2 3
det
(1 ) 1(1 )
S d x g
gd x g k
k k
Brane tension
Tachyon field /kye k
(0) 4br , ,4 41 ,S d x g g
k
2, , 2 2
, , 4 3
1 1 , 1 .2
gg k
N. Bilic, G.B. Tupper, CEJP 12 (2014) 147–159.
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RSII MODEL
• In flat space, FRW metrics
• Hamiltonian equations
, ,
• The Hamiltonian
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RSII MODEL
• The Hamiltonian equations
• The modified Friedman equation
• Combining with a continuity equation it leads to the second Friedman equation
3
3
H
H
2
8 21 .3 3
a G GHa k
2
44 ( ) 13GH Gk
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NONDIMENSIONAL EQUATIONS
• Substitutions:
4
8 2
8 2
2 8 2
10 2
5 8 2
1 /
4 3 /3
2 1 /
4 3 /3
1 /
h
h
2 2
2 2
8
13 12
Gk
ah
a
2
22
2 2
22 2 3
4
22
4 2 3
1 ,
sinh ,6
2sinh ,
6 3
11 / ,
2
1 1
2 1 /
d
d
p
Nondimensional constant
Hubble parameter
Preassure
Energydensity
2 2
( ) 12 6
h p
N h
Additional equations,solved in parallel
2
4
/ ,/ ( ), / ( )),
, / ( )
h H kk kk k
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INITIAL CONDITIONS FOR RSII MODEL
• Initial conditions – from a model without radionfield
• “Pure” tachyon potential
• Hamiltonian
• Nondimensional equation
4
8 2
5 8 2
143 .
1h
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ESTIMATION OF INITIAL CONDITIONS
• The end of inflation , tj. RSII modification can be neglected
• Number of e-folds
• Number of e-folds (standard tachyon inflation)
• Huge difference in number of e-folds RSII extends the period of inflation!!!
2f
1 f 2 f f2
8 8( ) ( ) 1, ( ) .3
h
2 2
2 40 0
1 .8 36
N
2
st.tach 20
1.8
N
st.tach20
95, 0,25 330
NN
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OBSERVATIONAL PARAMETERS
• Scalar spectral index and tensor-to-scalar ratio (the first order of parameters )
• The second order of parameters different
• Always constant , however constant for tachyon inflation in standard cosmology, and
for Randall-Sundrum cosmology
1 2 12
1 2 1 1 2 2 3
16 1 2 ,1 2 2 2 3 2 .s
r Cn C C
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NUMERICAL RESULTS
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OBSERVATIONAL PARAMETERS ( , ),
• Diagram with observational constraints from Planck 2015.
• The dots represent the calculation in the tachyon model for various ,
• 35% of calculated results for pairs of free parameters is represented in the plot.
• Red solid line represents the slow-roll approximation of the standard tachyon model with inverse quartic potential.
45 120
1 25
N
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OBSERVATIONAL PARAMETERS ( , ),RSII MODEL
• Free parameters from the interval:
• Approximate relation:• RS model
• Tachyon model (FRW)
0
60 1201 120 0,5
N
32 17 sr n
16(1 )3 sr n
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( , ) AS A FUNCTION OF , ,
• 65% is plotted,12% in range
0 0
60 120, 0,51 12, 0,5
0 0,5, 0.05
N N
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THE BEST FITTING RESULTS ( , )
0
85 1101 8
0 0,5
N
0
60 1202
0 0,25
N
0
115 1201,250.05
N
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CONCLUSION
• We have investigated a model of inflation based on the dynamics of a D3-brane in the AdS5 bulk of the RSII model. The bulk metric is extended to include the backreaction of the radion excitations.
• The agreement with observations is not ideal, the present model is disfavored but not excluded. However, the model is based on the brane dynamics which results in a definite potential with one free parameter only.
• The simplest tachyon model that stems from the dynamics of a D3-brane in an AdS5 bulk yielding basically an inverse quartic potential.
• The same mechanism could lead to a more general tachyon potential if the AdS5 background metric is deformed by the presence of matter in the bulk, e.g. in the form of a minimally coupled scalar field with an arbitrary self-interaction potential.
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• This work is supported by the SEENET-MTP Network under the ICTP grant NT-03.
• The financial support of the Serbian Ministry for Education and Science, Projects OI 174020 and OI 176021 is also kindly acknowledged.
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THE MOST IMPORTANT REFERENCES• N. Bilic, G.B. Tupper, AdS braneworld with backreaction, Cent. Eur. J. Phys. 12 (2014) 147–159.
• D. Steer, F. Vernizzi, Tachyon inflation: Tests and comparison with single scalar field inflation, Phys. Rev. D. 70 (2004) 43527.
• P.A.R. Ade, N. Aghanim, M. Arnaud, F. Arroja, M. Ashdown, J. Aumont, et al., Planck 2015 results: XX. Constraints on inflation, Astron. Astrophys. 594 (2016) A20.
• L. Randall, R. Sundrum, Large Mass Hierarchy from a Small Extra Dimension, Physical Review Letters. 83 (1999) 3370–3373; L. Randall, R. Sundrum, An Alternative to Compactification, Physical Review Letters. 83 (1999) 4690–4693.
• 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.
• M. Milosevic, D.D. Dimitrijevic, G.S. Djordjevic, M.D. Stojanovic, Dynamics of tachyon fields and inflation -comparison of analytical and numerical results with observation, Serbian Astronomical Journal. 192 (2016) 1–8.
• M. Milosevic, G.S. Djordjevic, Tachyonic Inflation on (non-)Archimedean Spaces, Facta Universitatis (Niš) Series: Physics, Chemistry and Technology. 14 (2016) 257–274.
• N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic, M. Milosevic, M. Stojanovic, Dynamics of tachyon fields and inflation: Analytical vs numerical solutions, AIP Vol 1722 No 1 (2016) 50002.
• G.S. Djordjevic, D.D. Dimitrijevic, M. Milosevic, On Canonical Transformation and Tachyon-Like ”Particles” in Inflationary Cosmology, Romanian Journal of Physics. 61 (2016) 99–109.
• D.D. Dimitrijevic, G.S. Djordjevic, M. Milosevic, Classicalization and quantization of tachyon-like matter on (non)archimedean spaces, Romanian Reports in Physics. 68 (2016) 5–18.
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NUMERICAL (PSEUDO)ALGORITAM