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Dark energy and fundamental physics

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Abstract

The acceleration of the expansion of the Universe which has been identified in recent years has deep connections with some of the most central issues in fundamental physics. At present, the most plausible explanation is some form of vacuum energy. The puzzle of vacuum energy is a central question which lies at the interface between quantum theory and general relativity. Solving it will presumably require to construct a quantum theory of gravity and a correspondingly consistent picture of spacetime. To account for the acceleration of the expansion, one may also think of more dynamical forms of energy, what is known as dark energy, or modifications of gravity. In what follows, we review the vacuum energy problem as well as the basic models for dark energy or modification of gravity. We emphasize the conceptual aspects rather than the techniques involved. We also discuss the difficulties encountered in each approach. This review is intended for astrophysicists or physicists not specialized in particle physics, who are interested in apprehending the issues at stake in fundamental physics.

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Notes

  1. The metric signature we adopt throughout is Einstein’s choice: (+,−,−,−).

  2. These equations can be obtained from the Einstein–Hilbert action:

    $$ \mathcal{S} = {1 \over8 \pi G_{N}} \int d^4x \sqrt{-g} \biggl[-{1 \over 2} R - \lambda \biggr] + \mathcal{S}_m(\psi, g_{\mu\nu}) , $$
    (1.1)

    where the generic fields ψ contribute to the energy-momentum: \(T_{\mu\nu} = (2/\sqrt{-g})(\delta\mathcal{S}_{m} /\delta g^{\mu\nu})\).

  3. The latter case corresponds to a cosmological constant as can be seen from (1.11)–(1.12) where the cosmological constant can be replaced by a component with ρ Λ =−p Λ =λ/(8πG N ).

  4. In case w X is not constant during the evolution, this term has the form:

    $$ \varOmega_{X} e^{3 \int_a^{a_0} [1+w_{X}(a') ]\,da'/a'} . $$
    (1.23)

  5. In Fig. 3, there is a rotational symmetry around the axis OL; in the general case, the angles θ I , θ S are replaced by 2-dimensional vectors θ I , θ S representing the angular position in the sky, and similarly for α.

  6. Beware that some authors use a different definition (e.g. Copeland et al. 2006).

  7. We assume that the evolution of the scale factor is set by the background: H∼2/(n B t) does not depend on ϕ to a first approximation.

  8. We will return below to the questions raised by the fact that the field has to reach a regime where ϕ becomes larger than the Planck scale (just as in standard chaotic inflation).

  9. For example, the quantum electrodynamics Lagrangian reads, with our conventions, \(\mathcal{L} = -(1/4e^{2}) F^{\mu \nu} F_{\mu\nu}\). The extra term would lead to an effective electromagnetic coupling \((1/4e_{{\rm eff}}^{2}) = (1/4e^{2}) + \beta(\phi/m_{P})\), hence a time-dependent fine structure constant \(\alpha= e_{{\rm eff}}^{2}/(\hbar c)\).

  10. We use the following results: if \(\bar{g}_{\mu\nu} = F g_{\mu\nu}\), then \(\bar{g}^{\mu\nu} = F^{-1} g^{\mu\nu}\), \(\sqrt{-\bar{g}} = F^{2} \sqrt{-g}\) and \(F \bar{R} = R - 3 D^{\mu}(\partial_{\mu}\ln F ) -{3 \over2} ( \partial_{\mu}\ln F ) ( \partial^{\mu}\ln F )\).

  11. Note that, as w ϕ →−1, the dependence of α with time vanishes.

  12. Using \(\operatorname{det} ( \delta^{m}_{n} + A^{m} B_{n} ) = 1 + A^{m} B_{m}\).

  13. In D dimensions, the corresponding behaviour is expected to be r −(D−2) for r≪R.

  14. Branes are higher-dimensional objects, similar to higher-dimensional membranes, which play a dynamical role in string theory; in particular, the ends of open strings are attached to branes.

  15. See Sect. 5.2.

  16. We consider only positive values of ρ Λ .

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Authors and Affiliations

  1. AstroParticule et Cosmologie, Université Paris Diderot, CNRS, CEA, Observatoire de Paris and Sorbonne Paris Cité, Bâtiment Condorcet, 10, rue Alice Domon et Léonie Duquet, 75205, Paris Cedex 13, France

    P. Binétruy

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  1. P. Binétruy
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Correspondence to P. Binétruy.

Additional information

Partially based on lectures given at the First TRR33 Winter School, Paso Tonale, 3–7 December, 2007 and at the UniverseNet School, Barcelona, 28 September–2 October, 2009.

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Binétruy, P. Dark energy and fundamental physics. Astron Astrophys Rev 21, 67 (2013). https://doi.org/10.1007/s00159-013-0067-2

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  • DOI: https://doi.org/10.1007/s00159-013-0067-2

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