Inflationary steps in the Planck data

We extend and improve the modeling and analysis of large-amplitude, sharp inflationary steps for second order corrections required by the precision of the Planck cosmic microwave background power spectrum and for arbitrary Dirac-Born-Infeld sound speed. With two parameters, the amplitude and frequency of the resulting oscillations, step models improve the fit by $\mathrm{\Delta }{\chi }^2=-11.4$. Evidence for oscillations damping before the Planck beam scale is weak: damping only improves the fit to $\mathrm{\Delta }{\chi }^2=-14.0$ for one extra parameter, if step and cosmological parameters are jointly fit, in contrast to analyses which fix the latter. Likewise, further including the sound speed as a parameter only marginally improves the fit to $\mathrm{\Delta }{\chi }^2=-15.2$ but has interesting implications for the lowest multipole temperature and polarization anisotropy. Since chance features in the noise can mimic these oscillatory features, we discuss tests from polarization power spectra, lensing reconstruction and squeezed and equilateral bispectra that should soon verify or falsify their primordial origin.

Figure 1

Minimum ${\chi }^2$ relative to the best fit smooth (no step) model as a function of the oscillation damping scale $x_d$ for the $c_s=1$ potential step model (top panel). The minimization is performed jointly over cosmological and step parameters with step position $s_s$ and amplitude of oscillations $C_2$ shown in the middle and bottom panels respectively.

Figure 2

Best fit models for a potential step with $c_{sa}=1.0$ (red) and a warp step at $c_{sa}=0.7$ (blue). Both models have the same few percent oscillations around the best fit smooth model at the first and second peaks (lower panel). The warp step is a marginally better fit of $\mathrm{\Delta }{\chi }^2=-15.2$ versus the potential step with $\mathrm{\Delta }{\chi }^2=-14.0$, due to suppression of the lowest multipoles, but introduces the sound speed as an additional parameter.

Figure 3

${\chi }^2$ improvement as a function of the sound speed after the step $c_{sa}$ for the warp ($\text{blue}+$) and potential ($\text{red}\times$) steps. Other parameters are fixed to their minimum ${\chi }^2$ values in the $c_s=1$ potential step model including the oscillation amplitude $C_2$. Note that warp steps with $c_{sa}\gtrsim 0.7$ cannot generate the required $C_2$ (see Eq. (a44), as marked by the vertical line.

Figure 4

Polarization (left) and temperature polarization cross (right) power spectra for the best fit models of Fig. 2. Step oscillations provide falsifiable predictions for the polarization which would not be mimicked by chance features in the noise.

Figure 5

Temperature power spectrum derivatives for the best fit models of Fig. 2. Low amplitude, high frequency oscillations produce new signals for lensing reconstruction and squeezed bispectra.

Figure 6

Evolution of $c_s$ (upper) and ${\epsilon }_H$ (lower) across a warp step. The step in both parameters and transient behavior right after the step is modeled by Eqs. (a17) and (a39) to excellent approximation. Model parameter choices are given in Appendix pp2.

Figure 7

Maximum oscillation amplitude $C_2$ for a warp step as a function of $c_{sa}$, the sound speed after the step. The blue dashed line corresponds to $C_2=1/3$, where the GSR approximation breaks down in the oscillatory regime. Note that for $c_s\gtrsim 1/4$ the oscillation amplitude is limited by physicality rather than the GSR approximation.

Figure 8

Evolution of $c_s$ (upper) and ${\epsilon }_H$ (lower) across a potential step. The transient change in both parameters is modeled by Eqs. (a17) and (a39) to excellent approximation. Model parameter choices are given in Appendix pp2.

Figure 9

GSR approximations versus exact solutions for the curvature power spectrum (top panel) and the fractional error of the GSR0 and GSR1 analytical solutions (bottom panel). The models are warp step (left) and potential step (right).

Figure 10

GSR approximations versus exact solutions for the temperature power spectrum (top panel) and the fractional error of the GSR0 and GSR1 analytical solutions (bottom panel). The models are warp step (left) and potential step (right).