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Non-minimal Higgs Inflation

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Non-minimal Higgs Inflation and Frame Dependence in Cosmology

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Abstract

In most cosmological models, the inflationary mechanism is described in terms of a scalar field – the inflaton. However, despite its obvious success in fitting observational data and in resolving basic cosmological problems, some fundamental questions still remain unresolved. One of these questions is connected with the physical reality of the inflaton. The origin and the nature of the inflaton field remain unexplained so far. The Standard Model of Particle Physics, whose predictions are experimentally tested to an outstanding level of precision, does also rely on the crucial assumption about the existence of a scalar field. In this context, the scalar field is called Higgs boson and serves to explain the origin of the elementary particle masses, while maintaining gauge invariance. By noticing the formal similarities between the inflaton and the Higgs boson, it is natural to wonder whether there could be a deeper and more fundamental connection between these two scalar particles. Ultimately, this leads to the question whether they could be just two manifestations of one and the same particle–the Higgs-inflaton.

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Notes

  1. 1.

    Although it should be mentioned that beside the striking evidence from the CMB, the explainable large scale structure of matter from tiny primordial quantum fluctuations and the coherent phase argument [30] all support the hypothesis of inflation. Moreover, it is expected that further experiments will measure primordial gravitational waves which ultimately are a measure of the energy scale of inflation. In consideration of all this evidence it seems to be very difficult to find a different mechanism which could explain all these features and does not contradict any observation.

  2. 2.

    Ockham’s razor does not apply if two competing theories make different testable predictions.

  3. 3.

    This is suggested e.g. by the singularity theorems [43], the black hole radiation [41] and the general intention for unification.

  4. 4.

    Even in the restricted scenario of scalar-tensor theories this freedom could correspond to different choices of the non-minimal coupling term parametrized by a general function \(U(\varphi )\).

  5. 5.

    For the relativistic idea, it is not mandatory to rely on a differentiable background manifold and the formalism of General Relativity. In fact, this is rather counter intuitive to a truly background-independent viewpoint of a purely relational setup. In General Relativity we indeed do have a background manifold. However “background independence” is established “afterwards” by considering only covariant equations.

  6. 6.

    The scale which is distinguished by the approximate intersection point of the RG trajectories of the Standard Model gauge couplings.

  7. 7.

    The notion of a cut-off depends on the type of perturbation theory used. Typically, it is determined for the case when all dimensional quantities—fields and their derivatives – are considered on equal footing and treated perturbatively. However, as we will see in Sect. 6.6 the effective Planck mass requires a field dependent cut-off, which allows an extension of the model to scales above \(M_\mathrm{{P}}/\xi \).

  8. 8.

    This is analogous to the approach of [15] where the model was approximated by matching the usual low-energy Standard Model phase with the chiral phase of the Standard Model at the inflation energy scale. This picture, however, takes place in the Einstein frame of the theory and is aggravated by the problem of transition between two different parametrizations of one quantum theory—the Cartesian coordinates in the space of the Higgs multiplet in the low-energy phase versus spherical coordinates in the chiral phase of the Standard Model. This transition is physically non-trivial.

  9. 9.

    At the level of the Schwinger–DeWitt algorithm this corresponds to focusing only on the Higgs-Higgs block \(P_{a}^{b}\) of the potential matrix (4.88). On the level of the general results (5.195)–(5.110), this means neglecting terms associated with inverse powers of \(M_\mathrm{{P}}\).

  10. 10.

    This is certainly justified for an inflationary scenario with \(R\sim H^2\sim V/6U\sim \text {const.}\) and \(\varphi \sim \text {const.}\), cf. (6.36). At the level of the Schwinger–DeWitt formalism, the vanishing of gradients \(\nabla \varphi =0\) leads to \(\hat{\mathcal{R}}_{\mu \nu }=0\) and therefore to a simplified expression for the \(\hat{a}_{2}\) coefficient (4.112).

  11. 11.

    The results for (6.21) in the Appendix of [9] and (5.109) of [82] differ by a global sign which is associated with \(R\) due to the use of a different sign convention.

  12. 12.

    The name rolling force stems from the fact that the Klein–Gordon equation (re-introducing the second derivative term \(\ddot{\varphi }\)) corresponds to the equation for a damped harmonic oscillator with a damping term \(\propto H\dot{\varphi }\) and a driving force \(\propto F\). The inflationary picture suggests that this force drives the field to roll down the potential \(V\).

  13. 13.

    In [7] only a single scalar field has been considered. Therefore, the Goldstone contributions to \({\mathbf {A}}\) and \({\mathbf {C}}\) did not appear.

  14. 14.

    The whole formalism presented here has been derived in [7] for a single scalar field. Thus, there are neither Goldstone boson contributions to \(V_\mathrm{{eff}}\) (no terms \(\propto \,6\,\lambda \) in (6.14)) nor to \(U_\mathrm{{eff}}\) (\({\mathbf {C}}\ll 1\) and thus \({\mathbf {A}}={\mathbf {A_{I}}}\)).

  15. 15.

    In [12], the contributions from Goldstone bosons have not been taken into account and all parameters have been calculated at the electroweak scale \(\nu =246\)  GeV only. However, in view of (6.9) and \(\lambda (0)\ll 1\) this does not make not make a difference at this scale.

  16. 16.

    At a Landau pole, the running coupling constants \(g_{i}(t)\) become infinite for finite values of the RG parameter \(t\).

  17. 17.

    In the inflationary analysis this choice might seem unnatural. A more natural choice would be \(\mu =M_\mathrm{{P}}/\sqrt{\xi }\) in order to keep the one-loop logarithms small.

  18. 18.

    The analytical algorithm of transition between different scales for \(g_{s}^2\) was presented in [64, 75, 76].

  19. 19.

    No modification of the spectral index due to \(Z\) occurs in the one-loop RG running because the difference \(n_\mathrm{{s}}^\varphi -n_\mathrm{{s}}^\phi =2\gamma dt/dN=(\gamma \,{\mathbf {A_I}}/48\pi ^2)e^x/(e^x-1)\) belongs to the two-loop order.

  20. 20.

    It is clear that \(|\xi _{0}|\) should roughly be of order \(10^4\) to satisfy (6.118), but for the algorithm we describe here it does not matter in principle, albeit choosing a very different \(\xi _{0}\) will increase the number of iterations as we will see in a moment.

  21. 21.

    An over-Planckian scale of \(\varphi \) does not signify a breakdown of the semi-classical expansion, because the energy density \(\sim 10^{-10} M_P^4\) stays far below the Planckian value.

  22. 22.

    The existence of this vacuum can perhaps be probed by wavelengths longer than that of a pivotal \(N\simeq 60\), but this requires a deeper analysis.

  23. 23.

    We have shown in Sect. 5.3.1 that such a transition between Jordan frame and Einstein frame is always possible for a \(O(N)\) symmetric multiplet of scalar fields non-minimally coupled to gravity, in contrast to the claim of [46].

  24. 24.

    Like the logarithmic term of (6.47) which dominates over the nearly flat classical part of the inflaton potential and qualitatively modifies tree-level predictions of the theory [9].

  25. 25.

    It is clear that even in the absence of quantum corrections the scale invariance in the Jordan frame is broken by the mass scale \(M_\mathrm{{P}}\), when inflation comes to an end (remember that this scale was determined by the condition \(\hat{\varepsilon }=3/4\)). This is of course necessary since an exact scale invariance would imply a strictly constant (for all times) potential in the Einstein frame and would therefore not permit a phase of reheating.

  26. 26.

    It is well known that despite the perturbative range of coupling constants, the two-loop RG improvement essentially lowers the electroweak instability threshold compared to the one-loop RG running [32].

  27. 27.

    In fact, this is a problem of consistent transition from \(\langle O[V(\varphi )]\rangle \) to \(O[V^\mathrm{eff}(\langle \varphi \rangle )]\) which brings the problem of gauge dependence in the formalism of the mean field \(\langle \varphi \rangle \) and its effective action.

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    Christian Friedrich Steinwachs

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Steinwachs, C.F. (2014). Non-minimal Higgs Inflation. In: Non-minimal Higgs Inflation and Frame Dependence in Cosmology. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-01842-3_6

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