Non-Gaussianity from explicit $U(1)$-breaking interactions

TL;DR

This work analyzes primordial NG arising from explicit $U(1)$-breaking interactions of a nearly massless axial component during inflation, considering the axial field as either a curvaton or CDM. It develops an explicit cosine-type $U(1)$-breaking potential, derives the local bispectrum from axial self-interactions, and studies how background radial oscillations can generate clock signals, including their impact on the trispectrum. It extends the analysis to couplings with a light scalar $\phi$ (inflaton or curvaton) and to kinetic mixing, showing that NG in curvature and isocurvature sectors can be sizable or suppressed depending on the scenario, with distinctive oscillatory or mixed NG templates that future surveys could test. Overall, the paper identifies parameter regions where $|f_{\mathrm{NL}}^{\rm loc}|$ is reduced to $\mathcal{O}(0.1)$ while the trispectrum remains appreciable, and highlights oscillatory clock signals as a potential observational discriminator of the underlying $U(1)$-breaking self-interactions during inflation.

Abstract

We investigate primordial non-Gaussianity (NG) arising from the explicit $U(1)$ symmetry-breaking interactions during inflation involving a nearly massless axial component of a complex scalar field $P$. We analyze the induced NG parameter $f_{\mathrm{NL}}$ under scenarios where the axial field functions as either a curvaton or cold dark matter (CDM). In the curvaton framework, there is a conventional contribution to the local NG of $f_{\rm NL} \simeq -O(1)$. Additional positive local NG can result from either the self-interactions of axial field fluctuations, their interactions with a light radial partner, or kinetic mixing with the inflaton via $U(1)$ symmetry-breaking terms. We identify parameter regions where the interactions lead to cancellations, suppressing the overall local NG to $|f^{\rm loc}_{\mathrm{NL}}| \lesssim O(0.1)$, while leaving the trispectrum largely unaffected. In the CDM scenario, these interactions enhance the NG in the isocurvature fluctuations. Moreover, interactions between the axial field and another light scalar, such as a curvaton, can generate $O(1)$ curvature NG signals and significant mixed curvature-isocurvature NGs that are within the reach of future experiments with $σ(f^{\rm loc}_{\rm NL})\sim1$. We also explore the role of a heavy radial field in generating oscillating correlation signals, noting that such signals can dominate the shape of the mixed adiabatic-isocurvature bispectrum. In certain cases, an oscillatory isocurvature bispectrum signal may be observable in the future, aiding in distinguishing between certain types of the $U(1)$-breaking self-interactions of the axial field.

Figures (6)

• Figure 1: Feynman diagram representation for the cubic-interaction contribution to the axial field bispectrum, where $\delta A$ is the axial field fluctuation.
• Figure 2: Contours showing the mean mass-squared $m_{A}^{2}/H^{2}$ as a function of $m_{A,I}^{2}/H^{2}$ and initial angular displacement $n\theta_{i}$. The mass of the axial field as it rolls down potential is time-dependent and a function of $\cos(n\theta(t))$. Using the solution for $\theta(t)$ given in Eq. (\ref{['eq:theta-soln']}), we show the time-average mass-squared of the axial field for $N_{\rm inf}=50$. The region where $0.01\leq m_{A}^{2}/H^{2}\leq0.03$ is shaded in light blue.
• Figure 3: From top left to bottom right, figures of the upper bound of NG parameters, $r\Delta f_{{\rm NL}}^{\zeta\zeta \mathcal{S}}$, $-r\Delta f_{{\rm NL}}^{ \mathcal{S}\mathcal{S}\zeta}$, $-r\Delta f_{{\rm NL}}^{\zeta\zeta\zeta}$ and $r\Delta f_{{\rm NL}}^{\mathcal{S}\mathcal{S}\mathcal{S}}$ (normalized with $r$, the fraction of curvaton's energy density to the total energy at the time of the decay) as functions of $\omega_A/r$ with $\omega_A$ as the axial-field dark matter fraction, for various choices of curvaton mass $m_{\phi}$ and isocurvature fraction $\alpha_{\rm iso}$ while $N_k$ is fixed at $50$, $n\sim10$ and $n\theta_0 \sim 1$. These bounds are obtained from Eqs. (\ref{['eq:zzs']}), (\ref{['eq:sszeta']}), (\ref{['zzz']}) and (\ref{['sss']}) respectively by saturating the magnitude of $q_{V,K}$ as defined in Eqs. (\ref{['qV-bound']}) and (\ref{['qK-bound']}). We take $q_K>0$ and $q_V<0$ for positivity of the axial mass. The curves have been truncated as marked by dot(cross) for $r=1(0.1)$ to ensure that $S_I/H \geq 10$. The peak magnitude of the pure and mixed curvature bispectra are primarily dominated by the $q_K$-dependent terms and exhibit an inverse dependence on curvaton mass-squared. The curve for isocurvature NG parameter $f_{\rm NL}^{\mathcal{S}\mathcal{S}\mathcal{S}}$ displays a typical $1/\omega_A$ dependence. The kinks in the curves are due to the piecewise functions given in Eqs. (\ref{['qV-bound']}) and (\ref{['qK-bound']}).
• Figure 4: The figure shows $k^{6}H^{-3}\langle \delta A \delta A\delta A\rangle'$ as a function of $m_{A}^{2}/H^{2}$ for a light scalar field $A$ with the coefficient $c_3$ set to $H\times m_{A}^{2}/H^{2}$. The solid blue and red curves represent numerical results for $N_{k}=50$ and $60$, respectively. The blue dashed and red dotted curves correspond to the analytical approximations as given in Eq. (\ref{['eq:bispec-light-scalar']}), by setting $\epsilon=0.4$ for empirical agreement.
• Figure 5: Clock signal on the “ normalized" axial power spectrum arising from an oscillating background radial field. The blue (circular marker) and the black (box marker) curves are obtained by numerically solving Eqs. (\ref{['inin-eqn']}) and (\ref{['Clock-eom']}) respectively. The solid red curve is the analytical result taken from Eq. (\ref{['clock-inin']}) and is restricted to scales $k/k_{r} \gtrsim 1/2$. In this figure, the amplitude of the normalized scale-invariant power spectrum (without any oscillations) is $1/2$. We fix $m_{S}^{2}=100H^{2}$ and $\beta_{S}=0.1$. The suppression of power on long wavelengths from solving the EoM is explained by the expression in Eq. (\ref{['pwr_suppr']}).