Letters in Mathematical Physics

, Volume 107, Issue 9, pp 1741–1768 | Cite as

Integrable cosmological potentials

  • V. V. Sokolov
  • A. S. Sorin


The problem of classification of the Einstein–Friedman cosmological Hamiltonians H with a single scalar inflaton field \(\varphi \), which possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint \(H=0\), is considered. Necessary and sufficient conditions for the existence of the first-, second- and third-degree integrals are derived. These conditions have the form of ODEs for the cosmological potential \(V(\varphi )\). In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described, and sporadic superintegrable cases are discussed.


Liouville integrability Integrals of motion Nonlinear ODEs 

Mathematics Subject Classification

34A34 37J35 37K10 70H06 



We thank P. Fré, S. O. Krivonos, D. Sternheimer and A. I. Zobnin for useful discussions. We are grateful to an anonymous referee for useful comments and references. V. S. thanks JINR (Dubna) for the hospitality extended during his visits. V. S. was supported by RFBI Grant No. 16-01-00289. The work of A. S. was partially supported by RFBR Grants No. 15-52-05022 Arm-a and No. 16-52-12012-NNIO-a, DFG Grant Le-838/12-2 and by the Heisenberg-Landau program.


  1. 1.
    Linde, A.: Particle Physics and Inflationary Cosmology. Harwood, Chur, Switzerland (1990), Contemp. Concepts Phys. 5 (2005). arXiv:hep-th/0503203
  2. 2.
    Mukhanov, V.: Physical Foundations of Cosmology. University Press, Cambridge (2005)CrossRefMATHGoogle Scholar
  3. 3.
    Gorbunov, D.S., Rubakov, V.A.: Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory. World Scientific, Singapore (2011)CrossRefMATHGoogle Scholar
  4. 4.
    Starobinsky, A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99–102 (1980)ADSCrossRefGoogle Scholar
  5. 5.
    Guth, A.H.: The inflationary universe: a possible solution to the horizon and flatness problems. Phys. Rev. D 23, 347–356 (1981)ADSCrossRefGoogle Scholar
  6. 6.
    Linde, A.D.: A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Phys. Lett. B 108, 389–393 (1982)ADSCrossRefGoogle Scholar
  7. 7.
    Albrecht, A., Steinhardt, P.J.: Cosmology for grand unified theories with radiatively induced symmetry breaking. Phys. Rev. Lett. 48, 1220–1223 (1982)ADSCrossRefGoogle Scholar
  8. 8.
    Martin, J., Ringeval, C., Vennin, V.: Encyclopaedia inflationaris. Phys. Dark Univ. 5–6, 75–235 (2014). doi: 10.1016/j.dark.2014.01.003. arXiv:1303.3787 [astro-ph.CO]CrossRefGoogle Scholar
  9. 9.
    Matveev, V.S., Shevchishin, V.: Two-dimensional superintegrable metrics with one linear and one cubic integral. J. Geom. Phys. 61, 1353–1377 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Valent, G., Duval, G., Shevchishin, V.: Explicit metrics for a class of two-dimensional superintegrable systems. J. Geom. Phys. 87, 461–481 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hietarinta, J.: Direct methods for the search of the second invariant. Phys. Rep. 147, 87–154 (1987)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras. Birkhauser, Verlag, Basel (1990)CrossRefMATHGoogle Scholar
  13. 13.
    Fre, P., Sagnotti, A., Sorin, A.S.: Integrable scalar cosmologies I. Foundations and links with string theory. Nucl. Phys. B877, 1028–1106 (2013). arXiv:1307.1910 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fre, P., Sorin, A.S.: Axial symmetric Kahler manifolds, the \(D\)-map of inflaton potentials and the Picard-Fuchs equation. Fortsch. Phys. 62, 26–77 (2014). doi: 10.1002/prop.201300031. arXiv:1310.5278 [hep-th]ADSCrossRefMATHGoogle Scholar
  15. 15.
    Lie, S.: Untersuchungen uber geodatische Kurven. Math. Ann. 20, 357–454 (1882)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bryant, R., Manno, G., Matveev, V.S.: A solution of S. Lie Problem: normal forms of 2-dim metrics admitting two projective vector fields. Math. Ann. 340, 437–463 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Matveev, V.S.: Two-dimensional metrics admitting precisely one projective vector field. Math. Ann. 352, 865–909 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ferrara, S., Kallosh, R., Linde, A., Porrati, M.: Minimal supergravity models of inflation. Phys. Rev. D 88, 085038 (2013). doi: 10.1103/PhysRevD.88.085038. arXiv:1307.7696 [hep-th]ADSCrossRefGoogle Scholar
  19. 19.
    Whitt, B.: Fourth order gravity as general relativity plus matter. Phys. Lett. B 145, 176–178 (1984)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Barrow, J.D.: The premature recollapse problem in closed inflationary universes. Nucl. Phys. B 296, 697–709 (1988)ADSCrossRefGoogle Scholar
  21. 21.
    Kofman, L.A., Linde, A.D., Starobinsky, A.A.: Inflationary universe generated by the combined action of a scalar field and gravitational vacuum polarization. Phys. Lett. B 157, 361–367 (1985)ADSCrossRefGoogle Scholar
  22. 22.
    Kofman, L.A., Mukhanov, V.F.: Evolution of perturbations in an inflationary universe. JETP Lett. 44, 619–622 (1986). [Pisma Zh. Eksp. Teor. Fiz. 44, 481–483 (1986)]ADSGoogle Scholar
  23. 23.
    Galajinsky, A., Lechtenfeld, O.: On two-dimensional integrable models with a cubic or quartic integral of motion. JHEP 1309, 113 (2013). doi: 10.1007/JHEP09(2013)113. arXiv:1306.5238 [math-ph]ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Polyanin, A.D., Zaitsev, V.F.: Handbook of exact solutions for ordinary differential equations, 2nd edn. CRC Press, Boca Raton (2003). and references thereinMATHGoogle Scholar
  25. 25.
    Covacic, J.: An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comput. 2, 3–43 (1986)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yehia, H.M.: Completely integrable 2D Lagrangian systems and related integrable geodesic flows on various manifolds. J. Phys. A Math. Theor. 46, 325203 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rosquist, K., Uggla, C.: Killing tensors in two-dimensional space-times with applications to cosmology. J. Math. Phys. 32, 3412 (1991). doi: 10.1063/1.529455 ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Karlovini, M., Rosquist, K.: Third rank killing tensors in general relativity: the (1+1)-dimensional case. Gen. Rel. Grav. 31, 1271–1294 (1999). doi: 10.1023/A:1026724824465. arXiv:gr-qc/9807051 ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Valent, G.: On a class of integrable systems with a cubic first integral. Commun. Math. Phys. 299, 631–649 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kruglikov, B.: Invariant characterization of Liouville metrics and polynomial integrals. J. Geom. Phys. 58, 979–995 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kruglikov, B., Matveev, V.S.: The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta. Nonlinearity 29, 1755–1768 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Fre, P., Sorin, A.S., Trigiante, M.: Integrable scalar cosmologies II. Can they fit into gauged extended supergravity or be encoded in N=1 superpotentials? Nucl. Phys. B 881, 91–180 (2014). doi: 10.1016/j.nuclphysb.2014.01.024. arXiv:1310.5340 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ferrara, S., Roest, D.: General sGoldstino Inflation. JCAP 1610, 038 (2016). doi: 10.1088/1475-7516/2016/10/038. arXiv:1608.03709 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Fre, P., Sorin, A.S.: Inflation and integrable one-field cosmologies embedded in rheonomic supergravity. Fortsch. Phys. 62, 4–25 (2014). doi: 10.1002/prop.201300030. arXiv:1308.2332 [hep-th]ADSCrossRefMATHGoogle Scholar
  35. 35.
    Ferrara, S., Fre, P., Sorin, A.S.: On the Gauged Kähler isometry in minimal supergravity models of inflation. Fortsch. Phys. 62, 277–349 (2014). doi: 10.1002/prop.201400003. arXiv:1401.1201 [hep-th]ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.L.D. Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Bogoliubov Laboratory of Theoretical Physics and Veksler and Baldin Laboratory of High Energy PhysicsJoint Institute for Nuclear ResearchDubnaRussia
  3. 3.National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)MoscowRussia
  4. 4.Dubna International UniversityDubnaRussia

Personalised recommendations