Anisotropic power spectrum and bispectrum in thef(ϕ)F2mechanism

Abstract

A suitable coupling of the inflaton ϕ to a vector kinetic term F 2 gives frozen and scale invariant vector perturbations. We compute the cosmological perturbations ζ that result from such coupling by taking into account the classical vector field that unavoidably gets generated at large scales during inflation. This generically results in a too anisotropic power spectrum of ζ. Specifically, the anisotropy exceeds the 1% level (10% level) if inflation lasted ∼ 5 e-folds (∼ 50 e-folds) more than the minimal amount required to produce the CMB modes. This conclusion applies, among others, to the application of this mechanism for magnetogenesis, for anisotropic inflation, and for the generation of anisotropic perturbations at the end of inflation through a waterfall field coupled to the vector (in this case, the unavoidable contribution that we obtain is effective all throughout inflation, and it is independent of the waterfall field). For a tuned duration of inflation, a 1% (10%) anisotropy in the power spectrum corresponds to an anisotropic bispectrum which is enhanced like the local one in the squeezed limit, and with an effective local fNL ∼ 3 (∼ 30). More in general, a significant anisotropy of the perturbations may be a natural outcome of all models that sustain higher than 0 spin fields during inflation.

Anisotropy in solid inflation

“…The rederivation is not more complete, since, by admission, it disregards the contribution from the modes in the quantum regime. We argue that it is also not simpler, since also the results of [16,17] are due to the classical super-horizon contribution, as repeatedly stressed in [17]. 2 Notice that in the f (φ) F 2 model with a non-vanishing vev of the vector field, a bispectrum that breaks statistical isotropy is generated, and its angle-average does assume the form (2).…”

“…The modes that leave the horizon in the final 60 e-folds see this sum as a classical homogeneous background quantity. This last statement is commonly accepted in the case of scalar fields (this is the origin of the coherent vev in the Affleck-Dine [65] and in the curvaton [66] mechanisms), but -as remarked in [17] -its validity has nothing to do with the spin of the field, but only with the property of the super-horizon modes. Any field (of any spin) that has a frozen spectrum of perturbations outside the horizon develops a coherent vev, that is locally observed as a homogeneous quantity.…”

“…Possibly, the similarities between the two models will also include the infra-red sensitivity to anisotropic superhorizon modes that characterizes the f (φ) F 2 model [17]. Assume that inflation starts from an isotropic configuration, for instance with a triad of orthogonal vectors of equal magnitude, and choose the function f (φ) to produce a frozen scale invariant spectrum of vector perturbations outside the horizon.…”

“…Quite interestingly, a nontrivial angular dependence in that limit had previously been obtained in [16] and further studied in [13,[17][18][19][20][21] 1 in the model f (φ) F 2 , where φ is the inflaton and F 2 the square of a vector field strength F µν , in the case in which f (φ) is chosen so to produce a scale invariant spectrum for the vector field. In this case, the nontrivial angular dependence of the bispectrum is due to the fact that a homogeneous vector breaks isotropy locally, and so the anisotropic modulation survives also in the k 3 → 0 limit.…”

“…The first possibility has been suggested as a 1 Ref. [22] rederived the results of [17], claiming that their rederivation uses only the classical mode functions, and it is therefore "simpler and more complete" than the computation of [17]. The rederivation is not more complete, since, by admission, it disregards the contribution from the modes in the quantum regime.…”