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Problems with Modified Theories of Gravity, as Alternatives to Dark Energy

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Beyond Einstein

Part of the book series: Einstein Studies ((EINSTEIN,volume 14))

Abstract

In this contribution to the conference “Beyond Einstein: Historical Perspectives on Geometry, Gravitation and Cosmology in the Twentieth Century,” we give a critical status report of attempts to explain the late accelerated expansion of the universe by modifications of general relativity. Our brief review of such alternatives to the standard cosmological model addresses mainly readers who have not pursued the vast recent literature on this subject.

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Notes

  1. 1.

    See, e.g., Copeland et al. (2006) and Straumann (2007) and references therein.

  2. 2.

    For an extensive review and literature, we refer to Sotiriou and Faraoni (2010).

  3. 3.

    If this transcendental equation has a solution, any vacuum solution of GR with the corresponding Λ is obviously a vacuum solution of (10.2).

  4. 4.

    Therefore, one expects that the Cauchy problem is well-posed. This is certainly the case for the vacuum theory, but with matter the problem is not completely settled; see Salgado (2006) and Salgado et al. (2008). In Salgado et al. (2008) two first-order strongly hyperbolic formulations of scalar-tensor theories are presented, which however do not include the exceptional case ω = −3∕2.

  5. 5.

    For symmetry reasons T μν has the form of an ideal fluid of T μν.

  6. 6.

    For a dynamical system analysis of this reconstruction, see Fay et al. (2007).

  7. 7.

    It turns out (Faraoni 2005) that the nonnegativity of the expression in (10.19) is the stability condition of the de Sitter spacetime with respect to small inhomogeneous perturbations of the f(R) model (without matter).

  8. 8.

    For a simplified discussion in the Einstein frame, see Capozziello and Tsujikawa (2008).

  9. 9.

    The effective potential is defined by \(\frac {\partial V_{\mathrm {eff}}}{\partial \phi }=-\frac {\kappa ^2}{3}\rho +\frac {1}{3}(2f-Rf')\).

  10. 10.

    Note that V (ϕ) is closely related to U in (10.11). It is easy to see that the de Sitter value for R is mapped to the value of ϕ, where V  takes its minimum.

  11. 11.

    This paper contains a discussion of a generic instability of Lagrangian systems in mechanics with higher derivatives that was discovered by Ostrogradski (1850).

  12. 12.

    This approach was actually first introduced by Einstein (1925). This is correctly stated in Pauli’s classical text on relativity (p. 215).

  13. 13.

    It is shown in Sotiriou (2006) that if the matter action is independent of Γ, the theory is dynamically equivalent to a Brans-Dicke theory with the special Brans-Dicke parameter − 3∕2, plus a potential term.

  14. 14.

    In Lanahan-Tremblay and Faraoni (2007), it is shown that the basic system of equations in vacuum cannot be rewritten as a system of only first order, since \(\square \phi \) cannot be eliminated, except of course if \(\square \phi =0\) (e.g., for the vacuum theory).

  15. 15.

    Especially from the evolution of linear cosmological perturbations for such models (Song et al. 2006; Lanahan-Tremblay and Faraoni 2007; Bean et al. 2007; Pogosian and Silvestri 2007).

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Authors and Affiliations

  1. Physik-Institut, University of Zurich, Winterthurerstrasse 190, CH–8057, Zurich, Switzerland

    Norbert Straumann

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  1. Norbert Straumann
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  1. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

    David E. Rowe

  2. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

    Tilman Sauer

  3. Centre François Viète, Université de Nantes, Nantes Cedex, France

    Scott A. Walter

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Straumann, N. (2018). Problems with Modified Theories of Gravity, as Alternatives to Dark Energy. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein. Einstein Studies, vol 14. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7708-6_10

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