Abstract
We consider gauge-dependent dynamical equations describing homogeneous isotropic cosmologies coupled to a scalar field ψ (scalaron). For flat cosmologies (k = 0), we analyze the gauge-independent equation describing the differential χ(α) ≡ ψ (a) of the map of the metric a to the scalaron field ψ, which is the main mathematical characteristic of a cosmology and locally defines its portrait in the so-called a version. In the more customary ψ version, the similar equation for the differential of the inverse map \(\bar \chi (\psi ) \equiv \chi ^{ - 1} (\alpha )\) is solved in an asymptotic approximation for arbitrary potentials v(ψ). In the flat case, \(\bar \chi (\psi )\) and χ−1(α) satisfy first-order differential equations depending only on the logarithmic derivative of the potential, v(ψ)/v(ψ). If an analytic solution for one of the χ functions is known, then we can find all characteristics of the cosmological model. In the α version, the full dynamical system is explicitly integrable for k ≠ 0 with any potential v(α) ≡ v[ψ(α)] replacing v(ψ). Until one of the maps, which themselves depend on the potentials, is calculated, no sort of analytic relation between these potentials can be found. Nevertheless, such relations can be found in asymptotic regions or by perturbation theory. If instead of a potential we specify a cosmological portrait, then we can reconstruct the corresponding potential. The main subject here is the mathematical structure of isotropic cosmologies. We also briefly present basic applications to a more rigorous treatment of inflation models in the framework of the α version of the isotropic scalaron cosmology. In particular, we construct an inflationary perturbation expansion for χ. If the conditions for inflation to arise are satisfied, i.e., if v > 0, k = 0, χ2 < 6, and χ(α) satisfies a certain boundary condition as α→-∞, then the expansion is invariant under scaling the potential, and its first terms give the standard inflationary parameters with higher-order corrections.
Similar content being viewed by others
References
V. Mukhanov, Physical Foundations of Cosmology, Cambridge Univ. Press, New York (2005).
S. Weinberg, Cosmology, Oxford Univ. Press, Oxford (2008).
D. S. Gorbunov and V. A. Rubakov, Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory, World Scientific, Singapore (2010).
A. A. Starobinsky, Phys. Lett. B, 91, 99–102 (1980).
V. F. Mukhanov and G. V. Chibisov, JETP Lett., 33, 532–535 (1981).
A. H. Guth, Phys. Rev. D, 23, 347–356 (1981).
A. D. Linde, Phys. Lett. B, 129, 177–181 (1983).
A. A. Starobinsky, Soviet Astron. Lett., 5, 302–304 (1983).
L. A. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Lett. B, 157, 361–367 (1985).
A. D. Linde, Particle Physics and Inflationary Cosmology, Harwood Academic, Chur (1990); arXiv:hep-th/0503203v1 (2005).
A. Linde, Prog. Theor. Phys. Suppl., 163, 295–322 (2006); arXiv:hep-th/0503195v1 (2005).
R. Kallosh, “Inflation in string theory,” in: Inflationary Cosmology (Lect. Notes Phys., Vol. 738, M. Lemoine, J. Martin, and P. Peter, eds.), Springer, Berlin (2008), pp. 119–156; arXiv:hep-th/0702059v2 (2007).
V. Mukhanov, Eur. Phys. J. C, 73, 1–6 (2013); arXiv:1303.3925v1 [astro-ph.CO] (2013).
J. Martin, K. Ringeval, and V. Vennin, Phys. Dark Univ., 5–6, 75–235 (2014); arXiv:1303.3787v3 [astro-ph.CO] (2013).
A. Linde, “Inflationary cosmology after Planck 2013,” arXiv:1402.0526v2 [hep-th] (2014).
J. Martin, “The observational status of cosmic inflation after Planck,” arXiv:1502.05733v1 [astro-ph.CO] (2015).
Y. Motohashi, A. A. Starobinsky, and J. Yokoyama, J. Cosm. Astropart. Phys., 09, 018 (2015); arXiv: 1411.5021v2 [astro-ph.CO] (2014).
M. Libanov and V. Rubakov, Phys. Rev. D, 91, 103515 (2015); arXiv:1502.05897v1 [hep-th] (2015).
B. Boisseau, H. Giacomini, D. Polarski, and A. A. Starobinsky, J. Cosmol. Astropart. Phys., 7, 002 (2015); arXiv:1504.07927v2 [gr-qc] (2015).
A. T. Filippov, Modern Phys. Lett. A, 11, 1691–1704 (1996).
A. T. Filippov and D. Maison, Class. Q. Grav., 20, 1779–1786 (2003).
A. T. Filippov, Theor. Math. Phys., 146, 95–107 (2006).
A. T. Filippov, “Some unusual dimensional reductions of gravity: Geometric potentials, separation of variables, and static–cosmological duality,” arXiv:hep-th/0605276v2 (2006); “Many faces of dimensional reduction,” in: Gribov Memorial Volume: Quarks, Hadrons, and Strong Interactions (Proc. Memorial Workshop Devoted to the 75th Birthday of V. N. Gribov, Budapest, Hungary, 22–24 May 2005, Yu. L. Dokshitzer, P. Levai, and J. Nyiri, eds.), World. Scientific, Singapore (2006), pp. 510–521.
V. de Alfaro and A. T. Filippov, Theor. Math. Phys., 153, 1709–1731 (2007).
V. de Alfaro and A. T. Filippov, Theor. Math. Phys., 162, 34–56 (2010).
A. T. Filippov, “General Properties and some solutions of generalized Einstein–Eddington affine gravity I,” arXiv:1112.3023v1 [math-ph] (2011).
A. T. Filippov, Theor. Math. Phys., 177, 1555–1577 (2013); arXiv:1302.6372v2 [hep-th] (2013).
E. A. Davydov and A. T. Filippov, Gravit. Cosmol., 19, 209–218 (2013); arXiv:1302.6969v2 [hep-th] (2013).
A. T. Filippov, Phys. Part. Nucl. Lett., 11, 844–853 (2014).
A. T. Filippov, “Many faces of scalaron coupled to gravity,” Program of the EU-Italy-Russia@Dubna Round Table, Dubna, 3–5 March 2014, transparencies available at http://theor.jinr.ru (unpublished).
J. E. Lidsey, D. Wands, and E. J. Copeland, Phys. Rep., 337, 343–492 (2000); arXiv:hep-th/9909061v2 (1999).
M. Kawasaki, M. Yamaguchi, and T. Yanagida, Phys. Rev. Lett., 85, 3572–3575 (2000); arXiv:hep-ph/0004243v2 (2000).
S. Ferrara, R. Kallosh, A. Linde, and M. Porrati, Phys. Rev. D, 88, 085038 (2013); arXiv:1307.7696v2 [hep-th] (2013).
R. Kallosh, “Planck 2013 and superconformal symmetry,” arXiv:1402.0527v1 [hep-th] (2014).
R. Kallosh and A. Linde, Comptes Rendus Physique, 16, 914–927 (2015); arXiv:1503.06785v2 [hep-th] (2015).
A. T. Filippov, “On Einstein–Weyl unified model of dark energy and dark matter,” arXiv:0812.2616v3 [gr-qc] (2008).
A. T. Filippov, Theor. Math. Phys., 163, 753–767 (2010).
A. T. Filippov, Proc. Steklov Inst. Math., 272, 107–118 (2011).
A. S. Eddington, The Internal Constitution of Stars, Cambridge Univ. Press, Cambridge (1926).
R. H. Fowler, Quart. J. Math., os-2, 259–288 (1931).
R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York (1953).
L. M. Berkovich, “The generalized Emden–Fowler equation,” in: Symmetry in Nonlinear Mathematical Physics (Kiev, Ukraine, 7–13 July 1997, M. Shkil, A. Nikitin, and V. Boyko, eds.), Vol. 1, Inst. Math., Natl. Acad. Sci. Ukraine, Kiev (1997), pp. 155–163.
K. S. Govinder and P. G. L. Leach, J. Nonlinear Math. Phys., 14, 443–461 (2007).
G. H. Hardy, Proc. London Math. Soc. Ser. 2, s2-10, 451–468 (1912).
R. H. Fowler, Proc. London Math. Soc. Ser. 2, s2-13, 341–371 (1914).
B. Whitt, Phys. Lett. B, 145, 176–178 (1984).
J.-L. Lehners, Phys. Rev. D, 91, 083525 (2015); arXiv:1502.00629v2 [hep-th] (2015).
D. Battefeld and P. Peter, Phys. Rep., 571, 1–66 (2015); arXiv:1406.2790v4 [astro-ph.CO] (2014).
A. A. Starobinsky, JETP Lett., 68, 757–763 (1998); arXiv:astro-ph/9810431v1 (1998); Gravit. Cosmol., 4 (Suppl.), 88–99 (1998); arXiv:astro-ph/9811360v1 (1998).
R. Arnowitt, S. Deser, and C.W. Misner, “The dynamics of general relativity,” in: Gravitation: An introduction to Current Research (L. Witten, ed.), Wiley, New York (1962); arXiv:gr-qc/0405109v1 (2004).
M. Cavaglià, V. de Alfaro, and A. T. Filippov, Internat. J. Mod. Phys. D, 4, 661–672 (1995); arXiv:gr-qc/9411070v2 (1994); 10, 611–633 (1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 188, No. 1, pp. 121–157, July, 2016.
Rights and permissions
About this article
Cite this article
Filippov, A.T. Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron. Theor Math Phys 188, 1069–1098 (2016). https://doi.org/10.1134/S0040577916070072
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577916070072