The Knotted Sky II: Does BICEP2 require a nontrivial primordial power spectrum?

An inflationary gravitational wave background consistent with BICEP2 is difficult to reconcile with a simple power-law spectrum of primordial scalar perturbations. Tensor modes contribute to the temperature anisotropies at multipoles with $l\lesssim 100$, and this effect --- together with a prior on the form of the scalar perturbations --- was the source of previous bounds on the tensor-to-scalar ratio. We compute Bayesian evidence for combined fits to BICEP2 and Planck for three nontrivial primordial spectra: a) a running spectral index, b) a cutoff at fixed wavenumber, and c) a spectrum described by a linear spline with a single internal knot. We find no evidence for a cutoff, weak evidence for a running index, and significant evidence for a "broken" spectrum. Taken at face-value, the BICEP2 results require two new inflationary parameters in order to describe both the broken scale invariance in the perturbation spectrum and the observed tensor-to-scalar ratio. Alternatively, this tension may be resolved by additional data and more detailed analyses.Comment: 14 pages, 5 figures, 4 tables; v2: references added, discussion updated, matches published versio

The Knotted Sky I: Planck constraints on the primordial power spectrum

Using the temperature data from Planck we search for departures from a power-law primordial power spectrum, employing Bayesian model-selection and posterior probabilities. We parametrize the spectrum with $n$ knots located at arbitrary values of $\log{k}$, with both linear and cubic splines. This formulation recovers both slow modulations and sharp transitions in the primordial spectrum. The power spectrum is well-fit by a featureless, power-law at wavenumbers $k>10^{-3} \, \mathrm{Mpc}^{-1}$. A modulated primordial spectrum yields a better fit relative to $\Lambda$CDM at large scales, but there is no strong evidence for a departure from a power-law spectrum. Moreover, using simulated maps we show that a local feature at $k \sim 10^{-3} \, \mathrm{Mpc}^{-1}$ can mimic the suppression of large-scale power. With multi-knot spectra we see only small changes in the posterior distributions for the other free parameters in the standard $\Lambda$CDM universe. Lastly, we investigate whether the hemispherical power asymmetry is explained by independent features in the primordial power spectrum in each ecliptic hemisphere, but find no significant differences between them.Comment: 24 pages, 9 figures, 4 tables, 1 appendix; v2: references added, discussion updated, matches published versio

Designing and testing inflationary models with Bayesian networks

Even simple inflationary scenarios have many free parameters. Beyond the variables appearing in the inflationary action, these include dynamical initial conditions, the number of fields, and couplings to other sectors. These quantities are often ignored but cosmological observables can depend on the unknown parameters. We use Bayesian networks to account for a large set of inflationary parameters, deriving generative models for the primordial spectra that are conditioned on a hierarchical set of prior probabilities describing the initial conditions, reheating physics, and other free parameters. We use $N_f$--quadratic inflation as an illustrative example, finding that the number of $e$-folds $N_*$ between horizon exit for the pivot scale and the end of inflation is typically the most important parameter, even when the number of fields, their masses and initial conditions are unknown, along with possible conditional dependencies between these parameters.Comment: 24 pages, 9 figures, 1 table; discussion update