Inflationary flavor oscillations and the cosmic spectroscopy

Inflationary scenarios motivated by high-energy physics generically contain a plethora of degrees of freedom beyond the primordial curvature perturbation. The latter interacts in a simple way with what we name “inflationary flavor eigenstates,” which differ, in general, from freely propagating “mass eigenstates.” We show that the mixing between these misaligned states results in new striking behaviors in the squeezed limit of the curvature perturbation three-point function, depending not only on the mass spectrum but also on the “mixing angles” of the theory. These results bring about a new perspective on the cosmological collider program: contrary to a widespread belief, the primordial signal needs not be dominated by the lightest extra degree of freedom. Instead, we show that it may display either modulated oscillations, a broken power law, or a transition from oscillations to a power law, thus offering a detailed cosmic spectroscopy of the particle content of inflation.

Figure 1

Diagrammatic expression of the tree-level bispectrum of $\zeta$ (propagators in red), including contributions from the extra scalars ${\sigma }^i$ (propagators in blue). From left to right: usual single-field slow-roll-suppressed contribution, one-particle ${\sigma }^i$ exchange with cubic interaction of strength ${\omega }_i$ (sum over $i$), two-particles ${\sigma }^i$, ${\sigma }^j$ exchange with cubic interaction of strength $R_{ij}$ (sum over $i$, $j$), three-particles ${\sigma }^i$, ${\sigma }^j$, ${\sigma }^k$ exchange with cubic interaction of strength $V_{ijk}$ (sum over $i$, $j$, $k$).

Figure 2

Amplitudes and phase for the different contributions to the squeezed bispectrum. Light fields with parameter $0<{\nu }_i<3/2$ contribute as a power law proportional to $I({\nu }_i)$ (blue line). Heavy fields with parameters ${\mu }_i>0$ contribute with $\ln (\kappa )$ oscillations with an amplitude $\mathcal{A}({\mu }_i)$ (orange line) and phase $\phi ({\mu }_i)$ (green line). We have checked that the apparent divergences of the signal in the ${\nu }_i,{\mu }_i\to 0$ limit are regulated, either by taking into account the other mode $\propto \mathrm{Im}[J_{+}({\nu }_i)]$ in the light case, or by considering the vanishing phase in the heavy case, leading in both situations to the same finite limit, ${\kappa }^{-1/2}{\mathcal{S}}_i\to -2-15/8[\pi -4\ln (2)+\ln (\kappa )]\simeq -2.69-1.88\ln (\kappa )$, and demonstrating the continuity of the signal across the $m_i=3H/2$ threshold. The divergence in the ${\nu }_i\to 3/2$ limit is a manifestation of the usual burden of IR divergences for massless fields.

Figure 3

Rescaled signal in the squeezed limit of the bispectrum, ${\kappa }^{-1/2}{\sum }_i({O^1}_i{)}^2{\mathcal{S}}_i$, for two extra heavy fields. The left panel corresponds to $({\mu }_1,{\mu }_2)=(1.3,1.0)$ and the right one to $({\mu }_1,{\mu }_2)=(1.5,0.5)$, for different values of the mixing angle ${\theta }_{12}\in [0,\pi /2]$. The modulation of the oscillations from the heavier field is clearly visible, even if only for a small nonzero mixing angle ${\theta }_{12}$ when the mass hierarchy is too strong.

Figure 4

Overall signal ${\sum }_i({O^1}_i{)}^2{\mathcal{S}}_i$ in the squeezed limit of the bispectrum, for two extra light fields with $({\nu }_1,{\nu }_2)=(0.5,1.0)$ (left panel), as well as one heavy field and one light one with $({\mu }_1,{\nu }_2)=(0.8,0.4)$ (right panel), for different values of the mixing angle ${\theta }_{12}\in [0,\pi /2]$. The effect of the heavier field is a transition at intermediate squeezing values, between either two different power laws or an oscillatory signal and a power law, but is only visible for a small nonzero mixing angle ${\theta }_{12}$.