Application of the Abel equation of the 1st kind to inflation analysis of non-exactly solvable cosmological models

Artyom V. Yurov, Anna V. Yaparova, Valerian A. Yurov1


We revisit the relationship between the Einstein-Friedmann and Abel equations to demonstrate how the latter might be applied to the inflationary analysis in a spatially-flat Friedmann universe filled with a real-valued scalar field. The analysis is performed for three distinct cases of polynomial potentials. As a result of numerical integration of the Abel equation, necessary and sufficient conditions for both slow rolling and inflation are estimated with respect to the initial value of the field. In addition, the relationship between the slow-rolling condition and the inflation is ascertained.


  1. A. D. Linde, Phys. Lett. B 129, 177 (1983).
  2. A. D. Linde, Mod. Phys. Lett. A 1, 81 (1986).
  3. A. D. Linde, Phys. Lett. 108, 389 (1982).
  4. A. Albrecht and P. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
  5. A. A. Starobinsky, JETP Lett. 30, 682 (1979).
  6. A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
  7. J. Barrow and A. Ottewill, J. Phys. A:Math. Gen. 16, 2757 (1983).
  8. A. Guth, Phys. Rev. D 23, 347 (1981).
  9. S. Hawking, I. Moss, and J. Stewart, Phys. Rev. D 26, 2681 (1982).
  10. A. Guth and E. Weinberg, Nucl. Phys. B 212, 321 (1983).
  11. A. Linde, Particle Physics and Inflationary Cosmology (Chur, Switzerland: Harwood, 1990), pp. 40??57.
  12. J. D. Barrow, Phys. Rev. D 49, 3055 (1994).
  13. R. Maartens, D. R. Taylor, and N. Roussos, Phys. Rev. D 52, 3358 (1995).
  14. A. V. Yurov, Class. Quantum Grav. 18, 3753 (2001).
  15. A. V. Yurov and S. D. Vereschagin, Theor. Math. Phys. 139, 787 (2004).
  16. A. V. Yurov and V. A. Yurov, J. Math. Phys. 51, 082503 (2010).
  17. S. V. Chervon and V. M. Zhuravlev, Russ. Phys. J. 43, 11 (2000).
  18. V. M. Zhuravlev and S. V. Chervon, J. Exp. Theor. Phys. 91, 227 (2000).
  19. S. V. Chervon and V. M. Zhuravlev, The cosmological model with an analytic exit from inflation, gr-qc/9907051.
  20. G. Felder, A. Frolov, L. Kofman, and A. Linde, Phys. Rev. D 66, 023507 (2002).
  21. K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests, arXiv: 1205.3421.
  22. R. Caldwell and M. Kamionkowski, Ann. Rev. Nucl. Part. Sci. 59, 397 (2009).
  23. J. Frieman and M. Turner, Ann. Rev. Astron. Astrophys. 46, 385 (2008).
  24. A. Silvestri and M. Trodden, Rept. Prog. Phys. 72, 096901 (2009).
  25. M. Li, X. Li, S. Wang, and Y. Wang, Commun. Theor. Phys. 56, 525 (2011).
  26. A. V. Yurov and S. D. Vereshchagin, Theor. Math. Phys. 139, 787 (2004).
  27. E. Elizalde, S. Nojiri, and S. D. Odintsov, Latetime cosmology in (phantom) scalar-tensor theory: dark energy and the cosmic speed-up, hep-th/0405034.
  28. A. A. Andrianov, F. Cannata, and A. Yu. Kamenshchik, Phys. Rev. D 72, 043531 (2005).
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