D. Vulcanov - On Cosmologies with non-Minimally Coupled Scalar Field and the "Reverse Engineering Method"

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The SEENET-MTP Workshop BW2011 …

The SEENET-MTP Workshop BW2011
Particle Physics from TeV to Plank Scale
28 August – 1 September 2011, Donji Milanovac, Serbia

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  • 1. ON COSMOLOGIES WITH NON-MINIMALLY COUPLED SCALAR FIELD AND THE "REVERSE ENGINEERING METHOD" G.S. Djordjevic1 , D..N. Vulcanov2(1) Department of Physics, Faculty of Science and Mathematics, University of Nis, Visegradska 33, 18001Nis, Serbia (2) Department of Theoretical and Applied Physics –“ Mircea Zǎgǎnescu” West University of Timişoara, B-dul. V. Pârvan no. 4, 300223, Timişoara, Romania The SEENET-MTP Workshop BW2011
  • 2. Abstract of the presentationWe studied further the use of the so called “reverse engineering”method (REM) in reconstructing the shape of the potential in cosmologiesbased on a scalar field non-minimally coupled with gravity. We use the known result that after a conformal transformation to the socalled Einstein frame, where the theory is exactly as we have aMinimally coupled scalar field. Processing the REM in Einstein frameAnd then transforming back to the original frame, we investigatedgraphically some examples where the behaviour of the scale factoris modelling the cosmic acceleration (ethernal inflation)
  • 3. Plan of the presentation● Introduction – why scalar fields in cosmology● Review of the “reverse engineering” method● Cosmology with non-minimally coupled scalar field● Einstein frame● Some examples● Conclusions
  • 4. Plan of the presentation● Introduction – why scalar fields in cosmology● Review of the “reverse engineering” method● Cosmology with non-minimally coupled scalar field● Einstein frame● Some examples● Conclusions
  • 5. Introduction : Why scalar fields ?●Recent astrophysical observations ( Perlmutter et . al .) showsthat the universe is expanding faster than the standard modelsays. These observations are based on measurements of theredshift for several distant galaxies, using Supernova type Ia asstandard candles.●As a result the theory for the standard model must be rewritenin order to have a mechanism explaining this !●Several solutions are proposed, the most promising ones arebased on reconsideration of the role of the cosmologicalconstant or/and taking a certain scalar field into account totrigger the acceleration of the universe expansion.●Next figure (from astro - ph /9812473) contains, sintetically theresults of several years of measurements ...
  • 6. Introduction :Cosmic acceleration
  • 7. Review of the “reverse engineering method”We are dealing with cosmologies based on Friedman-Robertson-Walker ( FRW ) metricWhere R(t) is the scale factor and k=-1,0,1 for open, flator closed cosmologies. The dynamics of the systemwith a scalar field minimally coupled with gravity isdescribed by a lagrangian as ⎡ 1 ⎤ R − (∇ ϕ ) − V (ϕ ) ⎥ 1 L= −g⎢ 2 ⎣ 16 π 2 ⎦Where R is the Ricci scalar and V is the potential of thescalar field and G=c=1 (geometrical units)
  • 8. Review of “REM”Thus Einstein equations are where the Hubble function and the Gaussian curvature are
  • 9. Review of “REM”Thus Einstein equations areIt is easy to see that these eqs . are not independent.For example, a solution of the first two ones (calledFriedman equations) satisfy the third one - which isthe Klein-Gordon equation for the scalar field.
  • 10. Review of “REM” Thus Einstein equations areThe current method is to solve these eqs . by consideringa certain potential (from some background physicalsuggestions) and then find the time behaviour of thescale factor R(t) and Hubble function H(t).
  • 11. Review of “REM” Thus Einstein equations areEllis and Madsen proposed another method, todayconsidered (Ellis et . al , Padmanabhan ...) moreappropriate for modelling the cosmic acceleration :consider "a priori " a certain type of scale factor R(t), aspossible as close to the astrophysical observations,then solve the above eqs . for V and the scalar field.
  • 12. Review of “REM”Following this way, the above equations can berewritten asSolving these equations, for some initially prescribedscale factor functions, Ellis and Madsen proposed thenext potentials - we shall call from now one Ellis-Madsen potentials :
  • 13. Review of “REM”
  • 14. Review of “REM”where we denoted with an "0" index all values at theinitial actual time. These are the Ellis-Madsen potentials.
  • 15. Review of “REM”
  • 16. Review of “REM”
  • 17. Cosmology with non-minimally coupled scalar fieldWe shall now introduce the most general scalar fieldas a source for the cosmological gravitational field,using a lagrangian as : ⎡ 1 ⎤ R − (∇ ϕ ) − V (ϕ ) − ξ R ϕ 2 ⎥ 1 1 L= −g⎢ 2 ⎣ 16 π 2 2 ⎦ where ξ is the numerical factor that describes the type of coupling between the scalar field and the gravity.
  • 18. Cosmology with non-minimally coupled scalar fieldAlthough we can proceed with the reverse methoddirectly with the Friedmann eqs. obtained from thisLagrangian (as we did in [3]) it is rather complicateddue to the existence of nonminimal coupling. In [3] weappealed to the numerical and graphical facilites of aMaple platform.For sake of completeness we can compute the Einsteinequations for the FRW metric. After some manipulations we have :
  • 19. Cosmology with non-minimally coupled scalar field • k 1 • 23 H (t ) 2 + 3 = [ φ ( t ) − V ( t ) + 3ξ H ( t ) (φ ( t ) 2 )] R (t ) 2 2 • • • 33 H ( t ) 2 + 3 H ( t ) = [ − φ ( t ) 2 + V ( t ) − ξ H ( t ) (φ ( t ) 2 )] 2 •• ∂V k φ (t ) = − 6ξ − 6 ξ H ( t )φ ( t ) ∂φ R (t ) 2 • − 12 ξ H ( t ) φ ( t ) − 3 H ( t ) φ ( t ) 2 where 8πG=1, c=1 These are the new Friedman equations !!!
  • 20. Einstein frameIt is more convenient to transform to the Einsteinframe by performing a conformal transformation ^g µν = Ω 2 g µν where Ω 2 = 1 − ξ 8πϕ 2Then we obtain the following equivalent Lagrangian: ⎡ 1 ^ 1 2 ⎛ ^ ⎞2 ^ ^ ⎤ L= −g⎢ R − F ⎜ ∇ ϕ ⎟ − V (ϕ ) ⎥ ⎢16π ⎣ 2 ⎝ ⎠ ⎥ ⎦
  • 21. Einstein framewhere variables with a caret denote those in the Einsteinframe, and 1− (1− 6ξ )8πξϕ 2 F = 2 (1− 8πξϕ )2 2 and ^ V (ϕ ) V (ϕ ) = (1 − 8 πξϕ ) 2 2
  • 22. Einstein frameIntroducing a new scalar field Φ as Φ = ∫ F (ϕ )dϕthe Lagrangian in the new frame is reduced to thecanonical form: ⎡ 1 ^ 1 ⎛ ^ ⎞2 ^ ^ ⎤ L= −g⎢ R − ⎜ ∇ Φ ⎟ − V (Φ ) ⎥ ⎢16π ⎣ 2⎝ ⎠ ⎥ ⎦
  • 23. Einstein frame ⎡ 1 ^ 1 ⎛ ^ ⎞2 ^ ^ ⎤L= −g⎢ R − ⎜ ∇ Φ ⎟ − V (Φ ) ⎥ ⎢16π ⎣ 2⎝ ⎠ ⎥ ⎦Main conclusion: we can process a REM in theEinstein frame (using the results from the minimalllycoupling case and then we can convert the results inthe original frame.
  • 24. Einstein frameBefore going forward with some concrete results,let’s investigate some important equations forprocessing the transfer from Einstein frame to theoriginal one. First the main coordinates are :^ ^t= ∫ Ω dt and R = ΩRand the new scalar field Φ can be obtained byintegrating its above expression, namely
  • 25. Einstein frame 3 ⎡ 4 3π ϕξ sgn( ξ ) ⎤Φ= sgn( ξ ) tanh ⎢ −1 ⎥ 2 π ⎢ 1 − (1 − 6ξ )8πξϕ 2 ⎣ ⎥ ⎦ + 4 2 πξ [ ξ (1 − 6ξ ) sin −1 ( 2ϕ 2πς (1 − 6ς ) ]where sgn(ξ) represents the sign of ξ – namely +1 or-1
  • 26. Examplesϕ → Φ ^ ^ t → tV → V
  • 27. Examples : nr. 1 – exponential expansionV (ϕ ) ω = 1, ξ = 0 green line ξ=-0.1 (left) and ξ = 0.1 (right) blue line
  • 28. Examples : nr. 1 – exponential expansionV (ϕ , ω ) ξ=0.1 (left) and ξ = - 0.1 (right)
  • 29. Examples : nr. 1 – exponential expansionV (ξ , ω ) ξ = 0 green surface ξ=-0.1 (left) and ξ = 0.1 (right) blue
  • 30. Examples : nr. 4 - tnV (ϕ ) n = 3, ξ = 0 green line ξ=-0.1 (left) and ξ = 0.1 (right) blue line
  • 31. Examples : nr. 4 - tnV (ϕ , n) n = 3, ξ = 0 green surface ξ=-0.3 (left) and ξ = 0.3 (right) blue surface
  • 32. Examples : ekpyrotic universe This is example nr. 6 from [3] having : ^ ^ R (t ) = R0 sin(ω t ) 2 ⎡ ⎛ ωΦ ⎞⎤ 3ω 2and V (Φ ) = 2 ⎢ B cosh⎜ ⎟⎥ − ⎣ ⎝ B ⎠⎦ 4π 1 ⎛ k ⎞ B = 2 ⎜1 + 2 ⎟ ⎜with 4π ⎝ R0 ⎟ ⎠
  • 33. Examples : ekpyrotic universeV (ϕ ) ω = 1, k=1, ξ = 0 green line ξ=-0.1 (left) and ξ = 0.1 (right) blue line
  • 34. Examples : ekpyrotic universeV (ϕ , ω ) κ = 1 and ξ = 0.05
  • 35. Examples : ekpyrotic universeV (ϕ , ω ) k=1, ξ = 0 green surface ξ = 0.1 (left) and ξ = - 0.3 (right) blue
  • 36. Conclusions….
  • 37. Conclusions….
  • 38. References[1] M.S. Madsen, Class. Quantum Grav., 5, (1988), 627-639[2] G.F.R. Ellis, M.S. Madsen, Class. Quantum Grav. 8, (1991), 667-676[3] D.N. Vulcanov, Central European Journal of Physics, 6, 1, (2008), 84-96[4] V. Bordea, G. Cheva, D.N. Vulcanov, Rom. Journ. Of Physics, 55,1-2 (2010), 227-237[5] Padmanabhan T, PRD 66 (2002), 021301(R)[6] Cardenas VH , del Campo S, astro - ph /0401031
  • 39. The end !!!Thank you for your attention !