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Balkan Workshop - 2013Vrnjacka Banja - SerbiaREM -- the Shape of Potentials for f(R) Theoriesin Cosmology and TachyonsG.S....
Plan of the presentationReview of the “reverse engineering” methodComputer programs for dealing with REM and cosmologyProc...
Review of the“reverse engineering method”We are dealing with cosmologies based on Friedman-Robertson-Walker ( FRW ) metric...
Thus Einstein equations arewhere the Hubble function and the Gaussiancurvature areReview of “REM”
Thus Einstein equations areIt is easy to see that these eqs . are not independent. Forexample, a solution of the first two...
Thus Einstein equations areThe current method is to solve these eqs . by considering acertain potential (from some backgro...
Thus Einstein equations areEllis and Madsen proposed another method, todayconsidered (Ellis et . al , Padmanabhan ...) mor...
Following this way, the above equations can be rewritten asSolving these equations, for some initially prescribedscale fac...
Review of “REM”
Computer programs for dealing with REM and cosmologyWe used Maple platform with GrTensor IIGrTensorII – a free package (se...
We used Maple platform with GrTensor IIThree steps we done for processing REM, namely :- a library for algebraic computing...
where we denoted with an "0" index all values at the initialactual time. These are the Ellis-Madsen potentials.Examples : ...
Tachyonic potentialsRecently it has been suggested that the evolution of a tachyonic condensatein a class of string theori...
Tachyonic potentialsandNow following the same steps as explained before we have the newFriedmann equations as :With matter...
Tachyonic potentialsWe also have a new Klein-Gordon equation, namely :All these results are then saved in a new library, c...
Tachyonic potentialsTachionic potentials. Here we denoted with R0 the scale factorat the actual time t0 and with a the qua...
Cosmology with non-minimally coupledscalar fieldWe shall now introduce the most general scalar fieldas a source for the co...
Cosmology with non-minimally coupledscalar fieldFor sake of completeness we can compute the Einsteinequations for the FRW ...
Cosmology with non-minimally coupledscalar field••+−=+ )])(()(3)()(21[)(3)(3 2222ttHtVttRktH φξφ•••−+−=+ )])(()(23)()([)(3...
Einstein frameIt is more convenient to transform to the Einsteinframe by performing a conformal transformationµνµν gg 2^Ω=...
where variables with a caret denote those in the Einsteinframe, and2222)81(8)61(1πξϕπξϕξ−−−=Fand22^)81()()(πξϕϕϕ−=VVEinste...
Introducing a new scalar field Φ as∫=Φ ϕϕ dF )(the Lagrangian in the new frame is reduced to thecanonical form:Φ−...
Φ−Φ∇−−= )(21161 ^2^^^VRgLπMain conclusion: we can process a REM in theEinstein frame (using the results from...
Before going forward with some concrete results,let’s investigate some important equations forprocessing the transfer from...
[ ])61(22(sin)61(428)61(1)sgn(34tanh)sgn(23121ςπςϕξξξππξϕξξϕξπξπ−−+−−=Φ−−where sgn(x) represents the sign of x – n...
ExamplesΦ→ϕ^VV →^tt →
Examples : ekpyrotic universeThis is example nr. 6 from [3] - see also (6) - having :)sin()(^^0 tRtR ω=andπωω43cosh2)(22−...
Examples : ekpyrotic universew = 1, k=1, x = 0 green linex=-0.1 (left) and x = 0.1 (right) blue line)(ϕV
Examples : ekpyrotic universe),( ωϕV k = 1 and x = 0.05
Examples : ekpyrotic universe),( ωϕVk=1, x = 0 green surfacex = 0.1 (left) and x = - 0.3 (right) blue
Cosmology with f( R ) gravity andminimally coupled scalar fieldWe shall now move to gravity theories with higherorder lagr...
Cosmology with f( R ) gravity …Now we restrict ourselves to the case when2)( RRRf α+=where a is a real constant. Varying t...
Cosmology with f( R ) gravity …Working again in FRW metricwe obtained the new Friedmann equationsmuch more complicated, wi...
Cosmology with f( R ) gravity …Here we need to process all three steps …Here are some examples, we plotted for twotypes of...
Cosmology with f( R ) gravity …Expponential case :V(j) in terms of different w at k=0
Cosmology with f( R ) gravity …Expponential case :Time behaviour of V(j) in terms of different a at k=0,1 and -1
Cosmology with f( R ) gravity …Expponential case :V(j) in terms of different a at k=1 and w=0.1
Cosmology with f( R ) gravity …Linear case :Time behaviour of V(j) in terms of different a at k=0,1 and -1
Cosmology with f( R ) gravity …Linear case :V(j) in terms of different a at k=1
Conclusions….
Conclusions….
References[1] M.S. Madsen, Class. Quantum Grav., 5, (1988),627-639[2] G.F.R. Ellis, M.S. Madsen, Class. Quantum Grav.8, (1...
The end !!!Thank you for your attention !
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D. Vulcanov, REM — the Shape of Potentials for f(R) Theories in Cosmology and Tachyons

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D. Vulcanov, REM — the Shape of Potentials for f(R) Theories in Cosmology and Tachyons

  1. 1. Balkan Workshop - 2013Vrnjacka Banja - SerbiaREM -- the Shape of Potentials for f(R) Theoriesin Cosmology and TachyonsG.S. Djordjevic1, D.N. Vulcanov2and C. Sporea2(1)Department of Physics, Faculty of Science and Mathematics, University of Nis,Visegradska 33, 18001Nis, Serbia(2)Department of PhysicsWest University of Timişoara, B-dul. V. Pârvan no. 4, 300223,Timişoara, Romania
  2. 2. Plan of the presentationReview of the “reverse engineering” methodComputer programs for dealing with REM and cosmologyProcessed examples :“Regular” potentials and tachyonic onesCosmology with non-minimally coupled scalar fieldCosmology with f( R ) gravity and scalar fieldConclusions
  3. 3. 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, flat orclosed cosmologies. The dynamics of the system with ascalar field minimally coupled with gravity is described by alagrangian asWhere R is the Ricci scalar and V is the potential of thescalar field and G=c=1 (geometrical units)( ) −∇−−= )(21161 2ϕϕπVRgL
  4. 4. Thus Einstein equations arewhere the Hubble function and the Gaussiancurvature areReview of “REM”
  5. 5. Thus Einstein equations areIt is easy to see that these eqs . are not independent. Forexample, a solution of the first two ones (calledFriedman equations) satisfy the third one - which is theKlein-Gordon equation for the scalar field.Review of “REM”
  6. 6. Thus Einstein equations areThe current method is to solve these eqs . by considering acertain potential (from some background physicalsuggestions) and then find the time behaviour of the scalefactor R(t) and Hubble function H(t).Review of “REM”
  7. 7. Thus Einstein equations areEllis and Madsen proposed another method, todayconsidered (Ellis et . al , Padmanabhan ...) more appropriatefor modelling the cosmic acceleration : consider "a priori " acertain type of scale factor R(t), as possible as close to theastrophysical observations, then solve the above eqs . for Vand the scalar field.Review of “REM”
  8. 8. Following this way, the above equations can be rewritten asSolving these equations, for some initially prescribedscale factor functions, Ellis and Madsen proposed thenext potentials - we shall call from now one Ellis- Madsenpotentials :Review of “REM”
  9. 9. Review of “REM”
  10. 10. Computer programs for dealing with REM and cosmologyWe used Maple platform with GrTensor IIGrTensorII – a free package (see at http://grtensor.org)embedded in Maple.- the geometry in |GrTensorII is a spacetime withRiemannian structure – adapted for Einstein GR-- It can be easily adapted/exended to alternative theories-- It provides facilities for building dedicated libraries-- simple acces to all Maple facilities – symbolic and-algebraic computation, numerical and graphical facilities
  11. 11. We used Maple platform with GrTensor IIThree steps we done for processing REM, namely :- a library for algebraic computing of Einstein eqs tillFriedmann eqs and calculating the potential and scalarfield time derivative |(as two slides before)- composing algbraic computations routines for analyticprocessing of REM (if possible).If not- composing of numerical and graphical routines forprocessing REM graphicallyComputer programs for dealing with REM and cosmology
  12. 12. where we denoted with an "0" index all values at the initialactual time. These are the Ellis-Madsen potentials.Examples : “regular” potentials
  13. 13. Tachyonic potentialsRecently it has been suggested that the evolution of a tachyonic condensatein a class of string theories can have a cosmological significance.This theory can be described by an effective scalar field with a lagrangian ofof the formwhere the tachyonic potential has a positivemaximum at the origin and has a vanishing minimum where thepotential vanishes Since the lagrangian has a potential, it seems to be reasonable to expectto apply successfully the method of ``reverse engineering for this typeof potentials. As it was shown when we deal with spatially homogeneousgeometry cosmology described with the FRW metric above we can useagain a density and a negative pressure for the scalar field as
  14. 14. Tachyonic potentialsandNow following the same steps as explained before we have the newFriedmann equations as :With matter included also. Here as usual we have
  15. 15. Tachyonic potentialsWe also have a new Klein-Gordon equation, namely :All these results are then saved in a new library, cosmotachi.m which willreplace the cosmo.m library we described in the prevos lecture.Now following the REM method we have finally :which we used to process different types of scale factor, same as inThe Ellis-Madsen potentials above
  16. 16. Tachyonic potentialsTachionic potentials. Here we denoted with R0 the scale factorat the actual time t0 and with a the quantity f(t) – f0
  17. 17. Cosmology with non-minimally coupledscalar fieldWe shall now introduce the most general scalar fieldas a source for the cosmological gravitational field,using a lagrangian as :where x is the numerical factor that describes thetype of coupling between the scalar field and thegravity.( ) −−∇−−= 2221)(21161ϕξϕϕπRVRgL
  18. 18. Cosmology with non-minimally coupledscalar fieldFor sake of completeness we can compute the Einsteinequations for the FRW metric.After some manipulations we have :Although 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. Weappealed to the numerical and graphical facilites of aMaple platform.
  19. 19. Cosmology with non-minimally coupledscalar field••+−=+ )])(()(3)()(21[)(3)(3 2222ttHtVttRktH φξφ•••−+−=+ )])(()(23)()([)(3)(3 222ttHtVttHtH φξφ•••−−−−∂∂=)()(3)()(12)()(6)(6)(22ttHttHttHtRkVtφφξφξξφφwhere 8pG=1, c=1These are the new Friedman equations !!!
  20. 20. Einstein frameIt is more convenient to transform to the Einsteinframe by performing a conformal transformationµνµν gg 2^Ω= where2281 πϕξ−=ΩThen we obtain the following equivalent Lagrangian:−∇−−= )(21161 ^2^2^^ϕϕπVFRgL
  21. 21. where variables with a caret denote those in the Einsteinframe, and2222)81(8)61(1πξϕπξϕξ−−−=Fand22^)81()()(πξϕϕϕ−=VVEinstein frame
  22. 22. Introducing a new scalar field Φ as∫=Φ ϕϕ dF )(the Lagrangian in the new frame is reduced to thecanonical form:Φ−Φ∇−−= )(21161 ^2^^^VRgLπEinstein frame
  23. 23. Φ−Φ∇−−= )(21161 ^2^^^VRgLπMain 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.Einstein frame
  24. 24. Before 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 :∫Ω= dtt^and RR Ω=^and the new scalar field F can be obtained byintegrating its above expression, namelyEinstein frame
  25. 25. [ ])61(22(sin)61(428)61(1)sgn(34tanh)sgn(23121ςπςϕξξξππξϕξξϕξπξπ−−+−−=Φ−−where sgn(x) represents the sign of x – namely +1 or -1Einstein frame
  26. 26. ExamplesΦ→ϕ^VV →^tt →
  27. 27. Examples : ekpyrotic universeThis is example nr. 6 from [3] - see also (6) - having :)sin()(^^0 tRtR ω=andπωω43cosh2)(22− Φ=ΦBBVwith += 202141RkBπ
  28. 28. Examples : ekpyrotic universew = 1, k=1, x = 0 green linex=-0.1 (left) and x = 0.1 (right) blue line)(ϕV
  29. 29. Examples : ekpyrotic universe),( ωϕV k = 1 and x = 0.05
  30. 30. Examples : ekpyrotic universe),( ωϕVk=1, x = 0 green surfacex = 0.1 (left) and x = - 0.3 (right) blue
  31. 31. Cosmology with f( R ) gravity andminimally coupled scalar fieldWe shall now move to gravity theories with higherorder lagrangian, so alled “f( R ) theories” whereWhere we have again a scalar field minimally coupledwith gravity and we have also regular matter fieldsdescribed in LM24 41 1( ) ( ) ( , )2 8 2PM MMS d x g f R V d xL gµµ µνφ φ φπ = − + ∇ ∇ − + Ψ  ∫ ∫
  32. 32. Cosmology with f( R ) gravity …Now we restrict ourselves to the case when2)( RRRf α+=where a is a real constant. Varying the above actionwe get the new field equations as (G=c=1) :2; ; ; ;, , , ,1(1 2 ) ( ) 2 ( )21 1( )2 2R R g R R g R g g g Rg g Vµν µν λσαβ αβ µν α β αβ µν λ σµνα β αβ µ να α αφ φ φ φ φ+ − + − − = = − −  
  33. 33. Cosmology with f( R ) gravity …Working again in FRW metricwe obtained the new Friedmann equationsmuch more complicated, with extra second and higher orderterms …),...)(),(,...()(21)(41)(43)( 222tHtHktRktHtHV  +++=πππϕ),...)(),(,...()(41)(41 222tHtHktRktH  ++−=ππϕ
  34. 34. Cosmology with f( R ) gravity …Here we need to process all three steps …Here are some examples, we plotted for twotypes of unverses :1) The exponential expansion unverse withteRtR ω0)( =2) The linear expansion unverse withnttRtR 00 RR(t)or)( ==
  35. 35. Cosmology with f( R ) gravity …Expponential case :V(j) in terms of different w at k=0
  36. 36. Cosmology with f( R ) gravity …Expponential case :Time behaviour of V(j) in terms of different a at k=0,1 and -1
  37. 37. Cosmology with f( R ) gravity …Expponential case :V(j) in terms of different a at k=1 and w=0.1
  38. 38. Cosmology with f( R ) gravity …Linear case :Time behaviour of V(j) in terms of different a at k=0,1 and -1
  39. 39. Cosmology with f( R ) gravity …Linear case :V(j) in terms of different a at k=1
  40. 40. Conclusions….
  41. 41. Conclusions….
  42. 42. 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 ofPhysics, 6, 1, (2008), 84-96[4] V. Bordea, G. Cheva, D.N. Vulcanov, Rom. Journ.of Physics,55,1-2 (2010), 227-237[5] G. S. Djordjevic, C.A. Sporea, D.N. Vulcanov, Proc.of the TIM10 Conference, Timisoara, Romania, nov.2010, in AIP proceedings series.[6] Cardenas VH , del Campo S, astro - ph /0401031[7] Tsujikawa S., Phys.Rev.D, 62, 043512, 2000 and referencesthere
  43. 43. The end !!!Thank you for your attention !

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