Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Observational parameters of Inflation in Holographic cosmology

Workshop on Quantum Fields and Nonlinear Phenomena, 27 September 2020. Organized by University of Craiova.

  • Be the first to comment

  • Be the first to like this

Observational parameters of Inflation in Holographic cosmology

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 17. POTENTIALS 𝑽 𝜽 = 𝟏 𝒄𝒐𝒔𝒉(𝝎𝜽) 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
  18. 18. 𝑽 𝜽 = 𝟏 𝒄𝒐𝒔𝒉(𝝎𝜽)
  19. 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
  20. 20. 𝑽 𝜽 = 𝟏 𝒄𝒐𝒔𝒉(𝝎𝜽)
  21. 21. 𝑽 𝜽 = 𝟏 𝒄𝒐𝒔𝒉(𝝎𝜽) - THE BEST FITTING RESULTS
  22. 22. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
  23. 23. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽 60 < 𝑁 < 90 0 < 𝜔 < 0.25 0 < 𝜃𝑖 < 20
  24. 24. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽
  25. 25. 𝑽 𝜽 = (𝟏 + 𝜽)𝒆−𝝎𝜽 - THE BEST FITTING RESULTS
  26. 26. 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
  27. 27. 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]

×