Inflation in models with conformally coupled scalar fields: An application to the noncommutative spectral action

Abstract

Slow-roll inflation is studied in theories where the inflaton field is conformally coupled to the Ricci scalar. In particular, the case of Higgs field inflation in the context of the noncommutative spectral action is analyzed. It is shown that while the Higgs potential can lead to the slow-roll conditions being satisfied once the running of the self-coupling at two-loops is included, the constraints imposed from the CMB data make the predictions of such a scenario incompatible with the measured value of the top quark mass. We also analyze the rôle of an additional conformally coupled massless scalar field, which arises naturally in the context of noncommutative geometry, for inflationary scenarios.

Coupling of gravity to matter, spectral action and cosmic topology

“…These are indeed the cases considered in [21], [22]. It is worth pointing out that, if one assumes that the field φ is related to the Higgs field, then there are very strong constraints coming from the CMB data, as recently analyzed in [3], which make a Higgs-based inflation scenario, as predicted by this kind of NCG model, incompatible with the measured value of the top quark mass. However, these constraints do not directly apply to other scalar perturbations φ, not related to the Higgs field.…”

Cosmology within Noncommutative Spectral Geometry

“…Normalisation of the kinetic terms implies a relation which was also found for SU(5) and SO (10). Assuming the big desert hypothesis, the running of the couplings α i = g 2 i /(4π) with i = 1, 2, 3 can then be obtained via the RGE.…”

“…(4.14) is sub-dominant and can be neglected. As an explicit analysis has shown [10], for each value of the top quark mass there is a value of the Higgs mass where the effective potential is about to develop a metastable minimum at large values of the Higgs field and the Higgs potential is locally flattened. Calculating [10] the renormalisation of the Higgs self-coupling up to two-loops, we have constructed an effective potential which fits the renormalisation group improved potential around the flat region.…”

“…(4.2), so that the equations of motion read [7] 12) implying that |H| plays the rôle of an effective gravitational constant [7]. The nonminimal coupling between the Higgs field and the Ricci curvature may turn out to be crucial in early universe cosmology [8,10]. Such a coupling has been introduced ad hoc in the literature, in an attempt to drive inflation through the Higgs field, and thus cure one of the main pathologies of the inflationary paradigm, namely the origin of the inflaton field.…”

“…Since running towards lower energies implies that nonperturbative effects in the spectral action cannot be any longer neglected, any results based on the asymptotic expansion and on renormalisation group analysis can only hold for early universe cosmology. Hence, the spectral action at the unification scale Λ offers a framework to investigate early universe cosmological models [7,8,9,10,11,12]. For later times, one should consider the full spectral action, a direction which requires the development of nontrivial mathematical tools.…”