Abstract
We will start with the most technical chapter of this thesis because the results obtained in this part will serve as a basis for the cosmological investigations of the Higgs inflation scenario we will discuss in Chap. 6. We will make use of the generalised Schwinger–DeWitt algorithm discussed in Sect. 4.5 in order to calculate the one-loop effective action for gravity coupled to scalar fields—a setup especially interesting in the cosmological context of inflation.
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Notes
- 1.
A minimal coupling to gravity means no coupling with derivative terms of the metric \(R\sim \partial ^2\,g\).
- 2.
In general, this is important in order to obtain the correct symmetrization structure and—especially in the mixed sectors—to obtain the correct contributions to the structures with derivatives acting on different perturbations. However, in this case it does not matter because \(S_\mathrm{{gb}}\) only contributes to the graviton-graviton sector.
- 3.
An additional factor of \(1/2\) in [24] is due to a different definition of the pole in dimension \(\epsilon \).
- 4.
A more general dependence of the functional couplings on the multiplet \(\Phi ^{a}\) would destroy the \(O(N)\) invariance of the action.
- 5.
We will specify precisely what we mean by Jordan frame and Einstein frame and give a mathematical definition in Sect. 5.3.1.
- 6.
This is similar to the situation of choosing an ordinary space-time coordinate system. For example, we can imagine a rotational invariant problem and try to describe it once in Cartesian coordinates and once in polar coordinates. Of course, the physical content of the problem is the same, described in both coordinate systems. But the polar coordinates will be better suited to interpret the physical results, since they are optimally adapted to the symmetries of the problem. This freedom of the choice of configuration space variables—the “generalized coordinates”—lies at the very heart of the Lagrangian formalism.
- 7.
As in the case of spherical coordinates in a flat space, we can of course have a non-vanishing connection despite the fact that there is no curvature. The difference to a curved space is connected with the fact that we can transform the connection away globally in a flat space.
- 8.
In the following, we will neglect the issue of gauge transformations. In general, we have to separate the gauge transformations from the arbitrary field reparametrizations, which complicates the formalism and is of minor interest for the purpose of this section.
- 9.
The choice of the configuration space metric is by itself a subtle point. One of the criteria to fix \(G_{ij}\) proposed in [26] is that \(G_{ij}\) should be determined by the coefficients of the highest derivative terms, present in the classical action \(S[\phi ]\).
- 10.
A frequent argument in favour of the Jordan frame is that this is the frame in which real physical distances are measured in terms of the physical Jordan frame metric.
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Steinwachs, C.F. (2014). One-Loop Cosmology and Frame Dependence. In: Non-minimal Higgs Inflation and Frame Dependence in Cosmology. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-01842-3_5
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