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D. Vulcanov: Symbolic Computation Methods in Cosmology and General Relativity [3]

Lecture form Spring School on Strings, Cosmology and Particles (SSSCP2009), March 31-4 2009, Belgrade/Nis, Serbia

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Symbolic computation methods
     in cosmology and
      general relativity
Part III - Reverse engineering method in cosmology with scalar field



            Dumitru N. Vulcanov
          West University of Timişoara
           Theoretical and Applied Physics Dept.-“Mircea Zăgănescu”




                                                                       1
Cosmology - remember

The Cosmo library is processing Einstein equations :




 for the Friedmann-Robertson-Walker metric




  where k=1,-1,0 and R(t) is the scale factor of the Universe




                                                                2
Cosmology remember

The matter content of the Universe is described by the stress-energy tensor :




 as a scalar field coupled minimally with the gravity and other matter fields
 separately. Thus we have :



  where, as for a perfect fluid :




                                                                                3
Cosmology - remember
For other matter content than the scalar field, the Cosmo library is providing
again the stress-energy tensor as a perfect fluid one :




 with corresponding pressure and density variables.

The main cosmological parameters and functions, namely the Hubble
quot;constantquot; and the deceleration function are :




                                                                                 4
The Cosmo library remember
Cosmo library is providing the Klein-Gordon and the conservation law
for the scalar field, namely :




  and the Friedmann equations :




                                                                       5
Reverse engineerng remember




In the standard cosmology the Friedmann eqs. are solved for a specific potential
of the scalar field, initially prescribed from certain physical arguments, and then
the time function R(t) is obtained and compared with the astrophysical
measurements.

In the quot;reverse-engineeringquot; method, the function R(t) is initially prescribed, as
much as possible close to the measurements, and then the potential V(t) is
obtained from Friedmann eqs, if it is possible !




                                                                               6

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D. Vulcanov: Symbolic Computation Methods in Cosmology and General Relativity [3]

  • 1. Symbolic computation methods in cosmology and general relativity Part III - Reverse engineering method in cosmology with scalar field Dumitru N. Vulcanov West University of Timişoara Theoretical and Applied Physics Dept.-“Mircea Zăgănescu” 1
  • 2. Cosmology - remember The Cosmo library is processing Einstein equations : for the Friedmann-Robertson-Walker metric where k=1,-1,0 and R(t) is the scale factor of the Universe 2
  • 3. Cosmology remember The matter content of the Universe is described by the stress-energy tensor : as a scalar field coupled minimally with the gravity and other matter fields separately. Thus we have : where, as for a perfect fluid : 3
  • 4. Cosmology - remember For other matter content than the scalar field, the Cosmo library is providing again the stress-energy tensor as a perfect fluid one : with corresponding pressure and density variables. The main cosmological parameters and functions, namely the Hubble quot;constantquot; and the deceleration function are : 4
  • 5. The Cosmo library remember Cosmo library is providing the Klein-Gordon and the conservation law for the scalar field, namely : and the Friedmann equations : 5
  • 6. Reverse engineerng remember In the standard cosmology the Friedmann eqs. are solved for a specific potential of the scalar field, initially prescribed from certain physical arguments, and then the time function R(t) is obtained and compared with the astrophysical measurements. In the quot;reverse-engineeringquot; method, the function R(t) is initially prescribed, as much as possible close to the measurements, and then the potential V(t) is obtained from Friedmann eqs, if it is possible ! 6
  • 7. Reverse engineerng remember Ellis-Madsen potentials…. 7
  • 8. More examples A new type of density factor is introduced : suggested in the recent literature and from the experimental measurements on the quantitative proportions between barionic, radiative, dark matter and dark energy. Remark : it is very improbable that this density factor is constant for long time evolution ! Thus we have here an approximation - our results are good for prescribing initial data for numerical simulations ! This was the purpose of our next investigations ilustrated here in what follows 8
  • 9. More examples Thus, for matter as a perfect fluid we have, as usual : where, of course, we have : dust pressureless matter radiative matter Following the same steps as in the previous simple example - I.e. solving the Friedman eqs, Klein-Gordon and the conservation law eqs. (Ecunr1...Ecunr3, EcuKG) we have, finally : 9
  • 10. More examples These were obtained after a series of solve, subs and simplify commands. Only Maple commands manipulation ! Conclusion : Cosmo library can be used even by those nonfamiliar with GrTensorII ! 10
  • 11. More examples From now one the things are depending on how complex are the above equations. Mainly we have troubles with the second one ! Some of the examples we processed are with exact analytical solutions, some need certain approximation assumptions. In searching initial data for numerical simulations, the last approximative solutions can be good, at least a short period after the initial time ! 11
  • 12. Linear expansion An example with simple analytic solution : for linear expansion we obtained Where Now comes some numerical simulations with these results as initial data. We used Cosmo thorn within the Cactus code ! 12
  • 13. Linear expansion Convergence test : linear expansion with dust matter 13
  • 14. Linear expansion Convergence test : linear expansion with radiative matter 14
  • 15. Linear expansion Scale factor time behavior with dust matter 15
  • 16. Linear expansion Scale factor time behavior with radiative matter 16
  • 17. Linear expansion Scalar field time behavior with dust matter 17
  • 18. Linear expansion Hubble function time behavior with radiative matter 18
  • 19. DeSItter exponential expansion Another example : again the DeSitter exponential expansion : Here we obtained, for the closed case (k=1) : Where And 19
  • 20. DeSItter exponential expansion The final step : expressing V(t) in terms of the scalar field to obtain : where C and D have certain complicated expressions in term of the cosmological parameters. We investigated again the convergence of the numerical simulations having these results as initial data : 20
  • 21. DeSItter exponential expansion Convergence test for p=0 and Omega = 0.1 21
  • 22. DeSItter exponential expansion Convergence test for p=0 and Omega = 0.3 22
  • 23. DeSItter exponential expansion Hubble function time evolution 23
  • 24. DeSItter exponential expansion Scalar field time evolution 24
  • 25. Tachionic potentials Recently it has been suggested that the evolution of a tachyonic condensate in a class of string theories can have a cosmological significance (T. Padmanabham, Phys.rev. D, 66, 021301(R) 2002). This theory can be described by an effective scalar field with a lagrangian of of the form where the tachyonic potential has a positive maximum at the origin and has a vanishing minimum where the potential vanishes Since the lagrangian has a potential, it seems to be reasonable to expect to apply successfully the method of ``reverse engineering'' for this type of potentials. As it was shown when we deal with spatially homogeneous geometry cosmology described with the FRW metric above we can use again a density and a negative pressure for the scalar field as 25
  • 26. Tachyonic potentials and Now following the same steps as explained before we have the new Friedmann equations as : With matter included also. Here as usual we have 26
  • 27. Tachionic potentials We also have a new Klein-Gordon equation, namely : All these results are then saved in a new library, cosmotachi.m which will replace the cosmo.m library we described in the previous lecture. Now following the REM method we have finally : which we used to process different types of scale factor, same as in The Ellis-Madsen potentials above 27
  • 28. Tachionic potentials Tachionic potentials. Here we denoted with R0 the scale factor at the actual time t0 and with α the quantity φ(t) – φ0 28
  • 29. 1. J.P. Ostriker, P.J. Steinhardt, Nature, 377, 600 (1995) 2. S. Perlmutter et.al., Astrophys. J, 517, 565, (1999) 3. C.W. Misner, K.S. Thorne, J.A. Wheeler : Gravitation, Freeman, San Francisco, (1973) 4. J.N. Islam, An Introduction to Mathematical Cosmology, Cambridge University Press, Cambridge, (1992) 5. G.F.R. Ellis, M.S. Madsen, Class. Quantum Grav., 8,667 (1991) 6. G.F.R. Ellis, Murugan J., Tsagas C.G., Class. Quantum Grav. 21 233 (2004) 7. D.N. Vulcanov, V.D. Vulcanov, Analele Univ. de Vest din Timisoara, Seria Matematica-Informatica, vol. XLII, Fasc. special 2, p. 181 (2004), electronic version at ar.Xiv:cs.SC/0409006, 8. D.N. Vulcanov, IEEE Computer Society SYNASC06 Conference Post-proceedings, p. 47, (2006), electronic version at IEEE Digital Library and arXiv:gr-qc/0210006 9. D.N. Vulcanov, Central Europ. J. of Physics, vol. 6 (1) 2008, p. 84 29
  • 30. End of part III and The End Thank you ! 30