Inflationary perturbations from a potential with a step

TL;DR

The paper shows that a sharp step in the inflaton potential during inflation can imprint scale-dependent oscillations in the primordial perturbation spectrum ${\cal P}_{{\cal R}}(k)$, even for a small step in amplitude. By numerically evolving the full mode equations using a step potential ${V(\phi)=\tfrac{1}{2}m^2\phi^2[1+c\tanh((\phi-\phi_{\rm step})/d)]}$ and the Stewart–Lyth formalism, the authors obtain both scalar and tensor spectra without relying on slow-roll. They demonstrate that the scalar spectrum exhibits oscillations spanning roughly two decades in $k$ with potentially large amplitude, while the tensor spectrum remains largely unaffected; comparison with CMB and LSS data via ${\rm CMBFAST}$ constrains the step size $c$ and gradient $d$. The results imply tight cosmological bounds on such features and provide a mechanism (as discussed by Adams et al.) for injecting scale dependence into inflationary perturbations, linking high-energy physics to observable spectra. The work highlights the power of numerical mode evolution to capture sharp potential features and their observable consequences.

Abstract

We use a numerical code to compute the density perturbations generated during an inflationary epoch which includes a spontaneous symmetry breaking phase transition. A sharp step in the inflaton potential generates $k$ dependent oscillations in the spectrum of primordial density perturbations. The amplitude and extent in wavenumber of these oscillations depends on both the magnitude and gradient of the step in the inflaton potential. We show that observations of the cosmic microwave background anisotropy place strong constraints on the step parameters.

Figures (6)

• Figure 1: The scalar and tensor power spectrum for $c=0.002$ and $d=0.01$. The $z"/z$ term for these parameters is shown in Figure 2.
• Figure 2: Evolution of $z"/z$ and $\ddot{a}$ for $c=0.02$ and $d=0.01$ with the number of $e$-folds of inflation, $N$. We have set $N=0$ at the step in the potential.
• Figure 3: Evolution of the independent modes $u^1_k$ and $u^2_k$ (with initial conditions for $u^1_k$ and $u^2_k$ given in Eq.(\ref{['initcond1']}-\ref{['initcond4']})) and the linear combination of their amplitude, Eq. (\ref{['comb']}) for $k=0.3$.
• Figure 4: The effect of changing $c$ and $d$ on the scalar perturbation spectrum.
• Figure 5: The CMB angular power spectrum for the primordial spectra shown in Figure 4. The normalization in each case is to COBE. The data are from COBE (circles), BOOMERanG (squares) and MAXIMA (triangles).