Primordial black holes save $R^2$ inflation

Abstract

In light of the latest Planck and Atacama Cosmology Telescope (P-ACT) joint results on the primordial scalar power spectrum, we show that the $R^2$ inflation model extended with a non-minimally coupled scalar field $χ$--namely the $χ$-extended $R^2$ inflation model--can naturally accommodate a larger spectral index $n_s$ and a small positive running $α_s$ at cosmic microwave background (CMB) scales, both of which are consistent with the latest P-ACT constraints. This is because the $χ$ field contributes a blue-tilted component to the primordial power spectrum, which both modifies the large-scale power and, as a result, significantly enhances power on small scales. The deviation of the $n_s$ and $α_s$ from the single field $R^2$ inflation is related to the non-minimal coupling constant $ξ$. The consequent enhancement in the primordial power spectrum can be large enough to lead to the formation of primordial black holes (PBHs) of mass $\lesssim 10^{20}\mathrm{g}$ as dark matter candidates. Furthermore, future observations of the small-scale power spectrum, CMB spectral distortions, and stochastic gravitational waves will provide decisive tests of this model and its predictions for PBHs. We stress its strong connection to the seesaw mechanism for the generation of the observed small masses.

Figures (6)

• Figure 1: The shape of potential and the background solution with different initial conditions. The solid red and blue lines show the background trajectory obtained by numerically solving the background equations of motion with different initial conditions $\chi_{\mathrm{ini}}>0$ (red) and $\chi_{\mathrm{ini}}<-0$ (blue).
• Figure 2: A schematic diagram of the simplified trajectory. We assume that the transition stage occurs instantaneously at $n_*$, therefore the background evolution consists of two separate phases of effective single-field inflation along $\phi$ and $\chi$ directions, respectively. The fiducial background trajectory is indicated by black solid arrows, while the perturbed trajectory is shown with red solid arrows. For the perturbed trajectory, the boundary condition is perturbed by the field perturbations $\delta\phi_k$ (dashed blue arrow) and $\delta\chi_k$ (dashed purple arrows) in the spatially flat gauge at the horizon exiting stage of mode $k$, denoted by $n_k$.
• Figure 3: The $n_s-\alpha_s$ relation shown in eq. \ref{['nsas']} for different $M_{\mathrm{PBH}}$s together with the joint constraints from Planck, P-ACT and P-ACT-LB (The constraint contours are taken from AtacamaCosmologyTelescope:2025nti). To plot the figure, we use $\mathcal{P}_\mathcal{R}^{\mathrm{max}}/\mathcal{P}_\mathcal{R}^{\mathrm{CMB}}=10^{7}$ in order to have $f_{\mathrm{PBH}}=1$. The hollow black dots refer to cases that $a=3$, or equivalently $\xi\geq 3/16$ when the small-scale amplification is well isolated from the large scales.
• Figure 4: The primordial power spectrum evaluated at the end of inflation shown alongside the (future) observational constraints. The black dashed line shows the result for $\xi>3/16$, whose power spectrum converges to the effective single-field $R^2$ inflation. The curves are obtained by numerically solving for the primordial power spectrum by the Transport method using the model parameters from Table \ref{['tab:parameters']}, Case 2 except for a varying $\xi$. From the darkest to the lightest solid curves, we take $\xi=0.0725$, $0.0715$, $0.0705$, $0.0695$, $0.0689$ (Case 2), $0.0685$. The constraint plot comes from Kavanagh:2019, including CMB constraints (red) from Planck Planck:2018jri, Lyman-$\alpha$ forest constraints Zaldarriaga:2000mz (blue), $\mu$-distortion constraints (green) from FIRAS Hu:1994bz, and PIXIE (future) Abitbol:2017vwaKogut:2011xw, and gravitational wave constraints (purple) from PTAs NANOGrav:2023gorEPTA:2023fykXu:2023wog, SKA Janssen:2014dka (future), LISA LISA:2017pwj (future) and BBO Harry:2006fi (future).
• Figure 5: The $n_s-r$ relation in the $R^2$ model and $\chi$-extended $R^2$ model shown alongside the joint constraints from BICEP/Keck observations BICEP:2021xfz (denoted as BK18) with Planck-LB (orange), ACT-LB (blue), and P-ACT-LB (purple). The constraint contours are taken from AtacamaCosmologyTelescope:2025nti. The dark blue dashed line with hollow dots refers to the $n_s-r$ relation in the $R^2$ model, and the light cyan dashed line and markers represent the $\chi$-extended $R^2$ model with PBH production. For the light cyan plot, we use the model parameters from Table \ref{['tab:parameters']}, Case 2 except for a varying $\xi$. From lower $n_s$ (left) to higher $n_s$ (right), the coupling constants used are $\xi=0.3125$ (hollow dot), $0.0725$, $0.0715$, $0.0705$, $0.0695$, $0.0689$ (star-shaped, Case 2), $0.0685$.