Primordial Black Holes from Inflation with a Spectator Field

TL;DR

This work investigates primordial black hole (PBH) production during inflation in the presence of a spectator field that is subdominant and decoupled from the inflaton. The key finding is that the spectator can prevent a genuine ultra-slow-roll (USR) phase yet still amplify curvature perturbations through a tachyonic growth of isocurvature modes, which transfer power to curvature during field-space turns, yielding a PBH-compatible power spectrum while preserving CMB observables. The mechanism is shown to be robust to variations in model parameters and is argued to be generic across single-field PBH scenarios, reducing the fine-tuning typically required in USR models and enabling asteroid-mass PBHs without conflicting with CMB constraints. Overall, the study demonstrates a viable multifield route to PBH formation that respects current observational bounds and offers a broader, less fine-tuned framework for PBH dark matter.

Abstract

How is the production of primordial black holes (PBHs) in single-field models of inflation impacted by the presence of additional scalar fields? We consider the effect of a spectator field - a free scalar field with sub-Hubble mass, no direct coupling to the inflaton, and which makes a subdominant contribution to the total energy density - in the context of single-field models of inflation featuring a transient phase of ultra-slow roll (USR) evolution. Despite the modest title, a spectator field can have a dramatic impact: the slow-roll evolution of the spectator prevents the combined inflaton-and-spectator system from entering into USR, which naively might be expected to preclude the production of PBHs. However, we demonstrate that the growth of perturbations is maintained or enhanced by the spectator, through the rich interplay of curvature and isocurvature perturbations. We show in a model-independent way that the single-field phase of ultra-slow-roll is replaced by two turns in field space encompassing a phase of tachyonic instability for the isocurvature perturbations and a transfer of power from isocurvature to curvature modes. Furthermore, we highlight a degeneracy between the fine-tuning of the feature in the inflaton potential and the parameters of the spectator, leading to an overall resilience of model predictions to parameter variations. This makes it easier for the underlying PBH model to accommodate both high-precision CMB constraints and production of PBHs in the asteroid-mass range.

Figures (19)

• Figure 1: The inflationary potential $V_{{\rm PBH},A}$ of Eq. \ref{['eq:VPBH-A']}, in units of $M_{\rm Pl}$. The inset highlights the presence of a local inflection point that causes the inflaton field to slow down as it rolls down the potential, yielding a temporary phase of ultra-slow-roll (USR) evolution. The parameters used in this figure are delineated in the "Reference" row of Table \ref{['tab:PBHA-params']}.
• Figure 2: Evolution of the background field $\varphi$ and of the slow-roll parameters $\epsilon$ and $\eta$ in the single-field regime of Eq. \ref{['eq:action-SF']}. This is shown for the $V_{{\rm PBH},A}$ potential of Eq. \ref{['eq:VPBH-A']}, using the "Reference" model parameters of Table \ref{['tab:PBHA-params']}. The time evolution is shown in terms of the number of e-folds $N$ before the end of inflation, see Eq. \ref{['eq:N-efolds']}. Top panel: The inflaton $\varphi$ rolls slowly down its potential at the beginning and at the end of its evolution. When it reaches the feature in $V_{{\rm PBH},A}$, $\varphi$'s evolution is considerably slowed down ($\varphi$ barely evolves during $20\lesssim N\lesssim 10$) until it evolves beyond the feature. Middle panel: As the inflaton is slowed down by the feature in $V_{{\rm PBH},A}$, the slow-roll parameter $\epsilon$ quickly decreases from ${\cal O}(10^{-4})$ to ${\cal O}(10^{-9})$. It increases again as $\varphi$ picks up speed after the local feature, eventually reaching the value $\epsilon=1$ and ending the inflationary epoch. Bottom panel: The significant slowing down of the field causes a temporary phase of ultra-slow-roll: as $\epsilon$ decreases, $\eta$ becomes large, crossing into the USR regime for $1.4$ e-folds. All panels: The shaded region highlights the phase of ultra-slow-roll $\eta\geq 3$.
• Figure 3: Evolution of the background fields and the Hubble parameter (in units of $M_{\rm Pl}$) for the PBHspec model as compared to the single-field case, for fixed $V_{{\rm PBH},A}$ parameters (reported in Table \ref{['tab:PBHA-params']}: "spectator" refers to the PBHspec model, "no spectator" refers to the Variation model). Top panel: The evolution of $\varphi$ is similar to its single-field behaviour, slowing down when it hits the feature in $V_{\rm PBH}$ and slow-rolling elsewhere. The spectator field $\chi$ slow-rolls at near-constant velocity, since $V_\text{S}(\chi)$ does not exhibit any particular feature. Bottom panel: The Hubble parameter is increased with respect to its single-field value by the presence of $\chi$. This causes additional Hubble friction, which further slows down $\varphi$. All panels: The highlighted 'phase II' corresponds to the period of slower evolution of $\varphi$, encapsulating the single-field USR phase. As seen in Fig. \ref{['fig:PBHA-background-eps-eta-omega']}, it is delimited by the turns in field space.
• Figure 4: Inflationary potential $V(\varphi,\chi)=V_{{\rm PBH},A}(\varphi)+V_\text{S}(\chi)$ (in units of $M_{\rm Pl}$), zoomed in on the region around the $V_{{\rm PBH},A}$ feature (corresponding to the inset in Fig. \ref{['fig:PBHA-V']}). The trajectory of the background field system is superimposed (black curve), as well as the direction of $\hat{\sigma}^I$ (blue) and $\hat{s}^I$ (red) at three points of the inflationary evolution corresponding to phases I--III. The "bump" along the $\varphi$ direction causes the trajectory to temporarily turn from the $\varphi$ direction to the $\chi$ direction; during this phase, $\hat{\sigma}^I$ and $\hat{s}^I$ switch roles. The PBHspec parameter values are listed in Table \ref{['tab:PBHA-params']}.
• Figure 5: Evolution of the slow-roll parameters and the turn rate for the PBHspec and single-field models considered in Fig. \ref{['fig:PBHA-background-fields-H']} (parameter values reported in Table \ref{['tab:PBHA-params']}). Top panel: The presence of $\chi$ sets a floor on $\epsilon\geq\epsilon_\chi$: while $\epsilon_\varphi$ becomes small (tracing the single-field case), $\epsilon_\chi \sim 10^{-4}$ and we never see $\epsilon\rightarrow0$. Middle panel: As $\epsilon$ never becomes unusually small, $\eta$ never enters the USR regime (unlike the single-field case). Bottom panel: The trajectory of the background fields exhibits two turns, corresponding to the transitions between $\epsilon\approx\epsilon_\varphi$ and $\epsilon\approx\epsilon_\chi$ (and vice-versa). All panels: The highlighted 'phase II' is delimited by the turns in field space (the dashed vertical lines correspond to the extrema of $\omega$). During this phase, $\epsilon_\chi\gg \epsilon_\varphi$.