Tracing the high energy theory of gravity: an introduction to Palatini inflation

Abstract

We present an introduction to cosmic inflation in the context of Palatini gravity, which is an interesting alternative to the usual metric theory of gravity. In the latter case only the metric \(g_{\mu \nu }\) determines the geometry of space-time, whereas in the former case both the metric and the space-time connection \(\varGamma ^\lambda _{\mu \nu }\) are a priori independent variables—a choice which can lead to a theory of gravity different from the metric one. In scenarios where the field(s) responsible for cosmic inflation are coupled non-minimally to gravity or the gravitational sector is otherwise extended, assumptions of the underlying gravitational degrees of freedom can have a big impact on the observational consequences of inflation. We demonstrate this explicitly by reviewing several interesting and well-motivated scenarios including Higgs inflation, \(R^2\) inflation, and \(\xi \)-attractor models. We also discuss some prospects for future research and argue why \(r=10^{-3}\) is a particularly important goal for future missions that search for signatures of primordial gravitational waves.

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Notes

  1. 1.

    The “Palatini formulation” of GR is usually credited to the paper [118] by Attilio Palatini, but actually it was, apparently, first presented in the paper [49] by Albert Einstein [59]. We will discuss the Palatini formulation of GR in more detail in the next section.

  2. 2.

    For clarity, we note that sometimes the notation \(\{^\sigma _{\mu \nu }\}\) is used instead of \({\bar{\varGamma }}^\sigma _{\mu \nu }\), and the name “Christoffel symbol” or “Riemannian connection” instead of the “Levi-Civita connection” we use in this paper.

  3. 3.

    Parallel transporting a tensor T along the path \(x^\mu (\lambda )\) parameterized by \(\lambda \) means that the covariant derivative of T along the path vanishes,

    figurea

    Also this quantity depends, a priori, only on the connection \(\varGamma ^\sigma _{\mu \nu }\). Note, however, that if the connection is not metric-compatible, parallel transport does not necessarily preserve the norm of vectors, a feature often taken as granted.

  4. 4.

    Derivation of the equations of motion for the metric case requires adding a so-called Gibbons–Hawking–York boundary term to the action to cancel a total derivative term that depends on the second derivatives of the metric [71, 153]. For the possibility of adding a (non-covariant) boundary term that only depends on the first derivatives of the metric, see e.g. Ref. [48].

  5. 5.

    While this definition of the Ricci tensor is not unique, the definition of the curvature scalar (8) is [137].

  6. 6.

    Scenarios where this term is absent and the scalar field is responsible for generating the Einstein–Hilbert term are sometimes called “induced gravity” models [2, 91, 92, 138].

  7. 7.

    Unless the majority of metric perturbations is generated in some other way, for example via the curvaton [56, 110, 116] or modulated reheating mechanism [46, 103]. Here we do not consider such possibilities.

  8. 8.

    For the Goldstone bosons, see Refs. [66, 67, 74, 82, 115].

  9. 9.

    In principle, because the SM field content and couplings are known, the details of (p)reheating can be calculated exactly, which then gives the total number of e-folds between the end of inflation and horizon exit of the scale where measurements are made [22, 65, 128, 142]. However, in practice the SM couplings are not exactly known, and neither is the Beyond-the-Standard-Model (BSM) physics which accommodates e.g. dark matter or baryogenesis, and which may affect the renormalization group running of the SM couplings up to the scales where (p)reheating (or inflation) occurs.

  10. 10.

    This is not a coincidence, as we will discuss in Sect. 3.2.3.

  11. 11.

    In principle, the SM Higgs should not be forgotten, and one can ask what is its effect on the inflationary dynamics. It can be shown that the presence of the Higgs alongside the \(R^2\) term leads to multifield inflation in metric gravity, as recently studied in Refs. [36, 37, 50, 54, 69, 77, 80, 102, 131, 151].

  12. 12.

    This is due to the fact that the scalaron couples to matter differently than the Higgs. For details, see Refs. [27, 73].

  13. 13.

    One can still ask how quickly the slow-roll regime is reached if the non-canonical terms are important in the beginning of inflation. This was recently addressed in Ref. [148].

  14. 14.

    For their relation to the famous “\(\alpha \)-attractor” models [58, 97] and terminology, see Ref. [64].

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Acknowledgements

I thank Félix-Louis Julié and Ryan McManus for useful discussions and the Simons foundation for funding.

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Tenkanen, T. Tracing the high energy theory of gravity: an introduction to Palatini inflation. Gen Relativ Gravit 52, 33 (2020). https://doi.org/10.1007/s10714-020-02682-2

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Keywords

  • Inflation
  • Palatini gravity
  • Metric-affine theory
  • Higgs inflation