We show that the tension introduced by the detection of large amplitude gravitational wave power by the BICEP2 experiment with temperature anisotropy measurements by the Planck mission is alleviated in models where extra light species contribute to the effective number of relativistic degrees of freedom. We also show that inflationary models based on S-dual potentials are in agreement with Planck and BICEP2 data.
Fitting ΛCDM + r to Planck and BICEP2 dataMeasurements of the cosmic microwave background (CMB) and large scale structure (LSS) indicate that we live in a spatially-flat, accelerating, infinite universe composed of 4% of baryons (b), 26% of (cold) dark matter (CDM), and 70% of dark energy (Λ). These observations also reveal that the universe has tiny ripples of adiabatic, scale-invariant, Gaussian density perturbations. The favored ΛCDM model implicitly includes the hypothesis of a very early period in which the scale factor of the universe expands exponentially: a ∝ e Ht , where H = ȧ/a is the Hubble parameter (see e.g. Baumann 2009). If the interval of exponential expansion satisfies ∆t > N/H, with N above about 50 to 60, a small casually connected region can grow sufficiently to accommodate the observed homogeneity and isotropy, to dilute any overdensity of magnetic monopoles, and to flatten the spatial hyper-surfaces (i.e., Ω ≡ 8πρ 3M Pl H 2 → 1, where M PL = G −1/2 is the Planck mass and ρ the energy density; throughout we use natural units, c = = 1). Quantum fluctuations during this inflationary period can explain the observed cosmological perturbations.Fluctuations are created quantum mechanically on subhorizon scales with a spectrum of wavenumbers k. (A mode k is called superhorizon when k < aH and subhorizon when k > aH.) While comoving scales, k −1 , remain constant the comoving Hubble radius, (aH) −1 , shrinks quasi-exponentially during inflation (driving the universe toward flatness) and the perturbations exit the horizon. Causal physics cannot act on superhorizon perturbations and they freeze until horizon re-entry at late times. A mode exiting the horizon can then be described 1