Accuracy of slow-roll formulae for inflationary perturbations: implications for primordial black hole formation

TL;DR

The paper assesses the reliability of the slow-roll framework for inflationary perturbations by comparing the Stewart-Lyth analytic spectrum to numerically evolved Mukhanov perturbations across diverse inflationary histories generated via the Hamilton-Jacobi flow formalism. It shows that slow-roll can fail during fast-roll or rapidly evolving backgrounds, necessitating numerical solutions to accurately predict the curvature spectrum and, consequently, the primordial black hole (PBH) abundance. Applying this to PBHs, the authors demonstrate that standard slow-roll inflation cannot yield significant PBHs without violating CMB-scale constraints, while a carefully engineered fast-roll episode near the end of inflation can; such models require precise numerical evaluation of perturbations and are highly fine-tuned. The work provides practical methods to design epsilon(N) profiles that produce PBHs while remaining consistent with large-scale observations, highlighting both the potential and the extreme fine-tuning involved in inflationary PBH scenarios.

Abstract

We investigate the accuracy of the slow-roll approximation for calculating perturbation spectra generated during inflation. The Hamilton-Jacobi formalism is used to evolve inflationary models with different histories. Models are identified for which the scalar power spectra computed using the Stewart-Lyth slow-roll approximation differ from exact numerical calculations using the Mukhanov perturbation equation. We then revisit the problem of primordial black holes generated by inflation. Hybrid-type inflationary models, in which the inflaton is trapped in the minimum of a potential, can produce blue power spectra and an observable abundance of primordial black holes. However, this type of model can now be firmly excluded from observational constraints on the scalar spectral index on cosmological scales. We argue that significant primordial black hole formation in simple inflation models requires contrived potentials in which there is a period of fast roll towards the end of inflation. For this type of model, the Stewart-Lyth formalism breaks down. Examples of such inflationary models and numerical computations of their scalar fluctuation spectra are presented.

Figures (5)

• Figure 1: Panel (a) shows the parameter $\epsilon$ as a function of $N$, the number of e-folds of inflation (with $N=0$ at the end of inflation). Rise in $\epsilon$ towards unity at the end of inflation gives rise to the small scale feature of the power spectrum $\mathcal{P_R}(k)$, shown in panel (b). The spectrum in bold line uses the Mukhanov variable (\ref{['pmukh']}) while the thin line shows the Stewart-Lyth approximation (\ref{['bigPR']}).
• Figure 2: Panel (a) shows $\epsilon(N)$ for a model in which $\epsilon(N)$ peaks prominently during inflation as a consequence of a bumpy potential with a pronounced bump or drop. The origin of the dip in the Mukhanov spectrum (panel (b), bold line) and its magnitude in comparison with that of the Stewart-Lyth spectrum are discussed in the text.
• Figure 3: Panel (a) shows $\epsilon(N)$ for a model which temporarily fast-rolls (when $\epsilon>1$), giving rise to a typical 'ringing' in the power spectrum shown in panel (b). The origin of the large discrepancy between the Mukhanov (bold) and the Stewart-Lyth spectra in panel (b) is discussed in the text.
• Figure 4: Plot of the primordial black hole abundance $\Omega_{\hbox{\scriptsize{pbh}}}(M>10^{15}$g) against the spectral index $n_s$ for $10^6$ models where inflation ends suddenly. There is virtually no dispersion around the curve given approximately by equation (\ref{['fit']}). The exponential sensitivity of $\Omega_{\hbox{\scriptsize{pbh}}}$ on $n_s$ at the steep part of the curve highlights the level of fine-tuning needed to produce an interesting density of primordial black hole .
• Figure 5: Left panel shows $\epsilon(N)$ designed using Hermite interpolating polynomials. Fast-roll regime has been modelled so as to produce primordial black holes at an interesting abundance. The reconstructed power spectrum $\mathcal{P_R}(k)$ is shown in the right panel. For this model, $\Omega_{\hbox{\scriptsize{pbh}}}(M>10^{15}\space\hbox{g})\simeq1.6\times10^{-20}$. The observables at CMB scales are $r\simeq0.01$, $n_s\simeq0.99$ and $dn_s/d\ln k\simeq10^{-5}$, broadly compatible with constraints from WMAP and other data.