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Cosmological Constant Problem

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Classical and Quantum Cosmology

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Abstract

A suspicion shared by some, including the author, is that a satisfactory solution of the cosmological constant problem would shed much light on the puzzle of quantum gravity. Quantum field theory, vacuum fluctuations, the microscopic degrees of freedom of gravity and their coarse graining all converge to this Pandora’s box in ways that still fascinate even the most consummate expert. About three quarters of the content of the universe is something whose intimate nature is utterly unknown. Hundreds of explanations, theories, models, conjectures have been put forward without successfully convincing the scientist. This chapter is an account, with neither sad nor happy ending, of the problem and of some of the efforts dedicated to its solution.

What shall we use to fill the empty spaces?

      — Roger Waters (Pink Floyd) , The Wall

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Notes

  1. 1.

    An early attempt to explain the cosmological constant in perturbative quantum gravity is [60], where the bare Λ is compensated by one-loop terms. This case is still of deterministic type, the cosmological constant being driven to zero by quantum effects.

  2. 2.

    When the form factors are left unspecified, the two formulations can be mapped one onto the other by a redefinition of ω [306].

  3. 3.

    First-order formalism is mandatory in order to couple fermions with gravity consistently.

  4. 4.

    For vanishing chemical potential, the enthalpy is \(E - F = \mathcal{T} S\) and measures the difference between the energy and the free energy of a finite-temperature system.

  5. 5.

    An ultra-simplified instance of this mechanism is the recovery of the Friedmann equations with K = 0 = Λ from Newtonian and thermodynamics considerations. Let D = 4 and consider an expanding ball of volume \(\mathcal{V} = 4\pi a^{3}/3\) filled with energy (mass) \(E = M =\rho \mathcal{V}\). Assuming the hypothesis of adiabatic expansion (no change in entropy, dS = 0), the first law of thermodynamics \(\text{d}E + P\,\text{d}\mathcal{V} = 0\) is equivalent to the continuity equation \(\dot{\rho }+3H(\rho +P) = 0\). On the other hand, the first Friedmann equation can be interpreted as an energy conservation equation \(m\dot{a}^{2}/2 - GmM/a = 0\), where the first term is a kinetic energy of a small mass m at distance a from the observer in the uniform medium and the second term is Newton’s potential.

  6. 6.

    In Chap. 2 we used the symbol n μ for a generic null vector but here it will denote a congruence.

  7. 7.

    Without invoking thermodynamical arguments, the equations of motion (7.118) can be obtained in metric formalism by splitting the Einstein–Hilbert Lagrangian \(\mathcal{L}_{\mathrm{EH}} = \mathcal{L}_{\mathrm{bulk}} + \mathcal{L}_{\mathrm{sur}}\) into a bulk and a surface term and, then, varying only the surface term with respect to special variations of the metric encoding a normal displacement to a null surface [511–513]. This is possible thanks to the holographic relation \(\sqrt{-g}\mathcal{L}_{\mathrm{sur}} = -(D/2 - 1)^{-1}\partial _{\sigma }[g_{\mu \nu }\partial (\sqrt{-g}\mathcal{L}_{\mathrm{bulk}})/\partial (\partial _{\sigma }g_{\mu \nu })]\). The same procedure is generalizable to an arbitrary gravitational Lagrangian [514]. The lack of a thermodynamical interpretation in metric formalism, however, does not explain why gravity is holographic.

  8. 8.

    The presence of a minimal length in Padmanabhan’s theory can be inferred also by other arguments independent of (7.137) [525, 526].

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    Gianluca Calcagni

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Calcagni, G. (2017). Cosmological Constant Problem. In: Classical and Quantum Cosmology. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-41127-9_7

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