Previous studies have highlighted model-dependent scenarios in which spectator fields can enhance primordial curvature perturbations, and hence PBH formation [63, 64, 65, 66, 67, 68].

We define a scalar spectator field as being subdominant in potential energy density, ${|V_{\mathrm{S}}/V_{\mathrm{PBH}}|}_{t_{\mathrm{CMB}}}\ll 1$, with no direct coupling to the inflaton, and with a sub-Hubble mass ${|m_{\chi }/H|}_{t_{\mathrm{CMB}}}\ll 1$, where $H$ is the Hubble parameter. Both conditions are evaluated at the time $t_{\mathrm{CMB}}$ when the comoving CMB pivot scale $k_{\mathrm{CMB}}$ first exits the Hubble radius.

To compare Eqs. (7) and (8) with the corresponding expressions in Ref. [100], note that the term ${\mu }_s^2$ that appears in Eq. (7) of Ref. [100] is equal to the shifted mass ${\tilde{\mu }}_s^2={\mu }_s^2+4{\omega }^2$, as defined below Eq. (9) of the present paper. To demonstrate the equivalence with Eqs. (6) and (7) of Ref. [100], one may use the equation of motion for $\dot{\sigma }$ together with the definition of the slow-roll parameter $\eta =2\epsilon -\dot{\epsilon }/(2H\epsilon )$, which is equivalent to $\ddot{\sigma }/(H\dot{\sigma })=\epsilon -\eta$.

By adding a light spectator field to single-field models, the PBH-forming dynamics more closely resemble the well-studied physics of hybrid inflation [47, 48, 49, 50, 51, 52, 53, 54, 55]: A sharp turn in field space combined with a brief phase of tachyonic growth seeds PBH formation.