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Numerical Calculation of the Hubble Hierarchy Parameters and the Observational Parameters of Inflation
NUMERICAL CALCULATION OF
THE HUBBLE HIERARCHY PARAMETERS AND THE
OBSERVATIONAL PARAMETERS OF INFLATION
Department of Physics
Faculty of Sciences and Mathematics
University of Niš, Serbia
10th International Conference of Balkan Pysical Union (BPU10)
Sofia, 26 - 30 August 2018
In collaboration with N. Bilić (Zagreb), G. S. Đorđević,
D. D. Dimitrijević, M. Stojanović (Niš)
• The inflation theory proposes a period of extremely rapid
(exponential) expansion of the universe during the very early
stage of the universe.
• Inflation is a process in which the dimensions of the universe
have increased exponentially at least times.
• Although inflationary cosmology has successfully
complemented the Standard Model, the process of inflation,
in particular its origin, is still largely unknown.
• Over the past 35 years numerous models of inflationary
expansion of the universe have been proposed.
• The simplest model of inflation is based on the existence of a
single scalar field, which is called inflaton.
• The most important ways to test inflationary cosmological
models is to compare the computed and measured values of
the 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 and scalar spectral index
Hubble expansion rate
at an arbitrarily
At the lowest order in parameters 𝜀 and 𝜀
• Satellite Planck
(May 2009 – October 2013)
• Planck Collaboration
• Latest results are published
in year 2018.
Planck 2015 results. XX. Constraints on inflation, Astronomy & Astrophysics. 594 (2016) A20
• The dynamics of a classical real scalar field ϕ minimally
coupled to gravity
• G – the gravitational constant, R - the Ricci scalar, g - the
determinant of the matric tensor, and is the
Lagrangian, with kinetic term
• We will assume the spatially 4-dimensional flat space-time
with the standard FRW metric
LAGRANGIAN OF A SCALAR FIELD -
• In general case – any function of a scalar field and kinetic
• Canonical field with potential
• Non-canonical models
• Dirac-Born-Infeld (DBI) Lagrangian
• Special case – tachyonic
• 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).
• However it was realized that the imaginary mass creates an instability
and tachyons spontaneously decay through the process known as
• Quanta are not tachyon any more, but rather an ”ordinary” particle with
a positive mass.
• Properties of a tachyon potential
• The corresponding Lagrangian and the Hamiltionian are
• The Friedmann equation
DYNAMICS OF INFLATION
1. The energy-momentum conservation equation
2. The Hamilton’s equations
̇ is the conjugate momentum and the Hamiltonian
• Nondimensional equations
• Dimensionless constant , a choice of a constant
(brane tension) was motivated by string theory
CONDITIONS FOR TACHYON INFLATION
• General condition for inflation
• Slow-roll conditions
• Equations for slow-roll inflation
INITIAL CONDITION FOR TACHYON INFLATION
• Slow-roll parameters
• Number of e-folds
• Randall-Sundrum (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
• 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
• RSII model – observer is placed on the positive tension brane, 2nd
brane is pushed to infinity
• The space is described by Anti de Siter metric
• Extended RSII model include radion backreaction
• Total action
where the second term is the action of the brane, and is the inverse of
AdS5 curvature radius, ϭ is the brane tension, the is tachyon field and is the
rescaled radion field .
• The Lagrangian and the Hamiltonian
• Conjugate momenta
• In flat space, FRW metrics
• The Hamilton’s equations
• The modified Friedman equation
• 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-
• Planck results ( )
• The software we developed and used has been applied to a limited set
of models, mainly to pure tachyonic and RSII inflationary cosmological
• In the present form it is written in such a way that its only inputs are
the Hamilton’s equations and the Friedmann equation, as well as the
• The program can readily be used for a much wider set of models.
• To apply the program to a new model one must determine its
corresponding equations and include these equations in the program
• The next steps are: to extend the program to be applicable for new and
different types of inflationary models, to improve the program in such a way
that only the Hamiltonian or the Lagrangian of a model are their inputs.
• The corresponding system of differential equations would be determined by
• After these improvements the program will be published
as a free software followed by appropriate documentation.
• The best fitting result is obtained for . It
opens good opportunity for further research based on this
potential in the contest of the RSII model and the
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.
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)
• 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–
• 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.