Breaking scale invariance from a singular inflaton potential

TL;DR

This paper examines how non-smooth inflaton potentials, motivated by spontaneous symmetry breaking of auxiliary fields, can break the usual scale invariance of the primordial curvature power spectrum $\mathcal{P}(k)$. It develops a general slow-roll/Green's function framework to obtain fully analytic expressions for $\mathcal{P}(k)$ for several sharp and softened features, including slope changes and downward steps. Translating $\mathcal{P}(k)$ into the CMB and matter power spectra, the authors show scale-dependent oscillations and small-scale modulations controlled by feature parameters, and they analyze degeneracies with a scale-invariant spectrum. The results constrain how inflationary microphysics could mimic or modify the standard $\Lambda$CDM signatures and inform model-building and data interpretation under current observational constraints.

Abstract

In this paper we break the scale invariance of the primordial power spectrum of curvature perturbations of inflation. Introducing a singular behaviour due to spontaneous symmetry breaking in the inflaton potential, we obtain fully analytic expressions of scale dependent oscillation and a modulation in power on small scale in the primordial spectrum. And we give the associated cosmic microwave background and matter power spectra which we can observe now and discuss the signature of the scale dependence. We also address the possibility of whether some inflationary model with featured potential might mimic the predictions of the scale invariant power spectrum. We present some examples which illustrate such degeneracies.

Figures (9)

• Figure 1: (Left) plot of $\ln\mathcal{P}$ versus $\ln x_0$, Eq. (\ref{['slopechangelnP']}), and (right) $d\phi/dN$ versus $N$, Eq. (\ref{['slopechangedphidN']}), where $A$ is normalised to 1. Solid lines correspond to $\Delta A/A > 0$ and dashed lines to $\Delta A/A < 0$. $\left| \Delta A/A \right|$ are set to 0.025, 0.05 and 0.1 from the innermost line. As expected, the modulation in power becomes stronger as the slope change is steeper.
• Figure 2: (Dashed) plot of $\ln\mathcal{P}$ versus $\ln x_s$ for a sharp step, Eq. (\ref{['lpsspec']}), and (solid) an arctangent step, Eq. (\ref{['arctanlogp']}). The parameters for the arctangent step are set to $a = 0.001$, $b = 0.02 \times (0.8)^n$$A = (0.8)^{3n + 1}$ where $n = 2, 3$ and 7/2 from the innermost line. These numbers are chosen to roughly match the first few peaks of the sharp step cases with $B = 0.1, 0.5$ and 1.0.
• Figure 3: Plot of $\ln\mathcal{P}$ versus $\ln x_1$ for 2 slope changes, Eq. (\ref{['2slopelogP']}). Here $\Delta A/A = 0.1$, and $\alpha = 0.01, 0.1$ and 1.0 from the innermost line.
• Figure 4: (Left column) plots of the CMB and (right column) matter power spectra associated with a slope change at $k_0$ in $V(\phi)$, Eq. (\ref{['slopechangelnP']}) with (upper row) $\Delta A/A > 0$ and (lower row) $\Delta A/A < 0$. The data and error bars for the CMB power spectrum are taken from WMAP (squares), CBI cbi (circles) and ACBAR acbar (triangles), and those for the matter one from SDSS sdss respectively. $k_0$ is fixed to be $0.05\mathrm{Mpc}^{-1}$.
• Figure 5: (Left column) plots of the CMB and (right column) matter power spectra with (upper row) positive and (lower row) negative slope change, depending on the position of the slope change $k_0$. The size of the slope change is set to $|\Delta A/A| = 0.5$.