Primordial tensor modes from quantum corrected inflation

Abstract

We analyze quantum corrections on the naive φ 4 -Inflation. These typically lead to an inflaton potential which carries a non-integer power of the field. We consider both minimal and non-minimal couplings to gravity. For the latter case we also study unitarity of inflaton-inflaton scattering. Finally we confront these theories with the Planck and BICEP2 data. We demonstrate that the presence of nonvanishing primordial tensor modes requires sizable quantum departures from the φ 4 -Inflaton model for the non-minimally coupled scenario which we parametrize and quantify. We compare the results with the minimally coupled case and elucidate the main distinctive features. I. NON-MINIMALLY COUPLED THEORIES WITH QUANTUM POTENTIALSThe underlying origin of the inflationary paradigm constitutes a prominent problem in cosmology [1][2][3][4][5][6]. Inflation is traditionally modelled via the introduction of new scalar fields. Many models have been put forward to describe the dynamics of these scalar fields and their interactions with other fields, as it has been recently reviewed in [7].On general grounds any renormalizable field theory will recieve quantum corrections to the potential. One can think of the E. Weinberg and Coleman perturbative quantum corrections to the classical scalar potential of any field theory as a simple example of these type of corrections [8,9]. We phenomenologically characterize these corrections to the φ 4 theory by introducing a real parameter γ as follows:with Λ a given energy scale. Of course, model by model, one can compute the specific potential as in [10]. Nevertheless we will show that it is possible to provide useful information on a large class of models corresponding to different values of γ using this simple approach. For completeness we analyze the cases in which φ couples both minimally and non-minimally to gravity. We find that for the non-minimally coupled case, the recent results by BICEP2 indicating the presence of primordial tensor modes [11] constrains γ to lie in the region 0.08 − 0.12, at the two-sigma confidence level. However, independently on the validity of the BICEP2 results [12,13], it is fundamental to know whether quantum corrected potentials can account for nonzero tensor modes.Interestingly, we also discover that for large primordial tensor modes the results are largely independent on the number of e-foldings. Relevant examples of nonminimally coupled models are the Higgs-Inflation model [14] and the ones in which the inflaton is a composite state [15][16][17][18]. II. COUPLING TO GRAVITY AND SLOW-ROLL INFLATIONWe consider the action of a scalar field non-minimally coupled to gravity:The subscript J refers to the Jordan frame, and indicates that the gravity sector is not of the Einstein Hilbert form. Generally in the Jordan frame, the scalar background contributes to the effective Planck mass: M 2 P = M 2 +ξ φ 2 . However, φ 2 = 0 in the present case and we can safely identify M with M P .

Agravity

“…4 10 18 GeV is the reduced Planck mass. Special assumptions such as V (S) ∝ f (S) 2 make the Einstein-frame potential V E flat at S M Pl , with predictions compatible with present observations [7][8][9][10][11][12][13][14][15][16][17][18][19]. However this flattening is the result of a fine-tuning: in presence of generic Planck-suppressed operators V and f and thereby V E are generic functions of S/M Pl .…”

“…In the agravity context, such field must have a dimensionless logarithmic potential: this is why our predictions for r ≈ 8/N ≈ 0.13 differ from the tentative prediction r ≈ 12/N 2 ≈ 0.003 of a generic ξ-inflation model with mass parameters in the potential [7][8][9][10][11][12][13][14][15][16][17][18][19].…”

“…In the limit where h or χ feel the vacuum expectation value of s as a constant mass term, one obtains Starobinsky inflation and Higgs ξ-inflation [7][8][9][10][11][12][13][14][15][16][17][18][19]: agravity dictates how they can hold above the Planck scale. Leaving a full analysis to a future work, we here want to explore the possibility that the inflaton is the field s that dynamically generates the Planck scale, as discussed in section 3.…”

“…However, this is a quite unusual outcome of quantum field theory: it requires special models with flat potentials, and often field values above the Planck scale. Let us discuss this issue in the context of Starobinsky-like inflation models [7][8][9][10][11][12][13][14][15][16][17][18][19]: a class of inflation models favoured by Planck data [20]. Such models can be described in terms of one scalar S (possibly identified with the Higgs H) with a potential V (S) and a coupling to gravity − 1 2 f (S)R. Going to the Einstein frame (i.e.…”

Is Higgs inflation ruled out?

“…For example, one may hope that λ or γ acquires a significant running at high scales so as to give a workable solution consistent with both the measured scalar and tensor power spectra. (See, e.g., [41]). …”