S-dual Inflation: BICEP2 data without unlikeliness

We show that S-dual inflationary potentials solve the unlikeliness problem manifested in Planck data and explain the excess B-mode power observed by the BICEP2 experiment as arising from primordial tensor fluctuations.

The discovery of primordial gravitational waves in the B-mode power spectrum is fertile testing ground for inflationary models [1]. The B-mode power spectrum recently reported by the BICEP2 Collaboration is well-fit by a lensed-ΛCDM + tensor model, with a tensor-to-scalar ratio r = 0.20 +0.07 −0.05 and is inconsistent with the null hypothesis, r = 0, at a significance of 6 − 7σ depending on the modeling of foreground dust. Furthermore, the new BICEP2 data when combined with observations from the Wilkinson Microwave Anisotropy Probe (WMAP) [2], the Atacama Cosmology Telescope (ACT) [3], the South Pole Telescope (STP) [4], and the Planck mission [5] significantly shrink the space of allowed inflationary cosmologies.
Of particular interest here is that Planck data favor standard slow-roll single field inflationary models with plateau-like potentials V (φ) for which V < 0, over power-law potentials. However, most of these plateaulike inflaton potentials experience the so-called "unlikeliness problem" [6]. As an illustration of this problem consider the simplest example of a plateau-like potential, where V 0 and M are free parameters. It is seen by inspection that the plateau terminates at a local minimum, and then for large φ, the potential grows as a power-law 4 . A problem arises because the minimum of the potential can be reached in two different ways: by slow-roll along the plateau or by slow-roll from the powerlaw side of the minimum. The path from the power-law side requires less fine tuning of parameters, has inflation occuring over a much wider range of φ, and produces exponentially more inflation. Yet the data prefer the unlikely path along the plateau.
The requirement that V < 0 in the de Sitter region, and the avoidance of the unlikeliness problem, must now also accommodate the tensor-to-scalar ratio detected by BICEP2 data. Also, we would like the inflaton potential to possess some connection to particle physics. To this end, we hypothesize that the potential be invariant under the S-duality constraint g → 1/g, or φ → −φ, where φ is the dilaton/inflaton, and g ∼ e φ/M . Here M is expected to be within a few orders of magnitude of the Planck mass M Pl = G −1/2 . (Throughout we use natural units, which solves the unlikeliness problem because it has no power-law wall.
S duality had its origins in the Dirac quantization condition on the electric and magnetic charges, suggesting an equivalence in the description of quantum electrodynamics as either a weakly coupled theory of electric charges or a strongly coupled theory of magnetic monopoles. This was developed in [7] and extended into the S-dualities of Type IIB string theories [8]. We don't attempt a full association with a particular string vacuum, but simply regard the self-dual constraint as a relic of string physics in big bang cosmology.
For the potential in (2), the usual slow-roll parameters are [9] and The density fluctuation at scale k is where H =ȧ/a is the Hubble parameter and the field φ is evaluated at freeze out (f.o.), the time when the scale k leaves the horizon.
A straightforward calculation shows that (1) and (2) share similar behavior, namely that and η are of the scale M 2 Pl /M 2 . For (1) and η quickly grow near the end of inflation (φ ∼ M ), whereas for (2) and η remain small. Thus for (2), as for power-law inflation (with an exponential potential), inflation does not end. We assume that the dynamics of a second field leads to exit from the inflationary phase into the reheating phase. The requirement that there be roughly 60 e-folds of observable inflation is found to be The normalizations of the spectra are given by and the ratio of the amplitudes of the spectra at the pivot k = k * , is Under the approximations of Ref. [10], the spectral indices and their running are The parameters M and φ * (the field value when the k * scale crosses the horizon, k * = aH), are determined from the slow-roll parameters, which in turn are determined by r and n s . V 0 is fixed by the density contrast δρ/ρ ∼ 10 −5 . The parameter space that admits positive solutions for φ * and M is shown in Fig. 1. With and η known, we also compute n t , and the running of the spectral indices. In Table I, we display the parameters of the model, n t and dn/d ln k for values of n s and r within their 1σ uncertainties. The values of M agree with (6) modulo logarithmic corrections from the high field contributions to N .