Primordial black holes from polynomial potentials in single field inflation

Within canonical single field inflation models, we provide a method to reverse engineer and reconstruct the inflaton potential from a given power spectrum. This is not only a useful tool to find a potential from observational constraints, but also gives insight into how to generate a large amplitude spike in density perturbations, especially those that may lead to primordial black holes (PBHs). In accord with other works, we find that the usual slow-roll conditions need to be violated in order to generate a significant spike in the spectrum. We find that a way to achieve a very large amplitude spike in single field models is for the classical roll of the inflaton to overshoot a local minimum during inflation. We provide an example of a quintic polynomial potential that implements this idea and leads to the observed spectral index, observed amplitude of fluctuations on large scales, significant PBH formation on small scales, and is compatible with other observational constraints. We quantify how much fine-tuning is required to achieve this in a family of random polynomial potentials, which may be useful to estimate the probability of PBH formation in the string landscape.

Figure 1

Example of a power spectrum (blue line) that is put in by hand so that it produces a significant number of PBHs and is consistent with present constraints (shaded region is allowed): CMB and LSS survey (purple) [21, 22, 23, 24], $\mu$ (yellow) and $y$ distortions (green) [25], pulsar timing (orange) [59, 60, 61], and PBH constraints (black dashed). We take ${\delta }_c=0.3$ as an example. In Sec. 4b2, we approximately reconstruct a potential from this blue input power spectrum. To check the consistency of our method, we take the approximate reconstructed potential and then numerically compute the power spectrum (red line) from it exactly.

Figure 2

Example of an input hyperbolic tangent power spectrum (blue curve). We reconstruct a potential from this power spectrum making use of the approximation Eq. (24). To check the consistency of our method, we numerically calculate a power spectrum (red curve) from the reconstructed potential without any approximation.

Figure 3

Reconstructed potential from the power spectrum of Eq. (8), which is representative of a spectrum with a large spike in it on small scales and is shown as the blue curve in Fig. 1. Here we assume $U_{*}=24{\pi }^2{\mathcal{P}}_R(k_{*})/{10}^3$. We plot the case with the spike (blue) and without the spike (green dashed).

Figure 4

Upper panel: Quintic polynomial potential that gives rise to PBHs. The light-green colored region represents the domain of $\varphi$ where the standard slow-roll approximation is accurate. The blue (red) dot represents a local minimum (maximum). Inflation ends when the inflaton reaches the black-dashed line. We take $N_{\max ,0}=65$, $c_3=-0.52$, $c_5=-0.64072504$, and $\mathrm{\Lambda }=0.3$. Lower panel: Corresponding power spectrum. The red curve is the exact result from solving the MS equation and the blue curve is an approximate result using Eq. (24).

Figure 5

Upper panel: Ratio of the amplitude of power spectrum at a maximum ($k=k_p$) and at a pivot scale $(k=k_{*})$ as a function of the amount of $c_5$, with other parameters fixed to be $c_3=-0.52$ and $\mathrm{\Lambda }=0.3$. Lower panel: The number of $e$-foldings after the start of the nonslow-roll phase $N_{\mathrm{NSR}}$ ($=N_{\mathrm{tot}}-N_{\max ,0}$) as a function of $-c_3$, with other parameters fixed to be $|c_5-c_{5,\mathrm{cr}}|/|c_{5,\mathrm{cr}}|\simeq {10}^{-8.5}$ and $\mathrm{\Lambda }=0.3$. Note that we solved the MS equation exactly in this figure.