Black hole constraints on the running-mass inflation model

TL;DR

The paper investigates primordial black hole (PBH) constraints on the running-mass inflation model, which predicts a strongly scale-dependent spectral index as slow-roll ends. It combines analytic (extended slow-roll) and numerical (Mukhanov formalism) methods to compute the perturbation spectra and then translates the short-scale amplitude into PBH limits via the dispersion $\sigma_{\rm hor}$, enforcing $\sigma_{\rm hor} \lesssim 0.04$. The results show that PBH production excludes a substantial region of parameter space, especially for $N_{\rm COBE}=45$, with tighter bounds for larger COBE-scale $n$ and for negative $\tilde{\alpha}_0$; the constraints weaken for smaller $N_{\rm COBE}=25$ and positive $\tilde{\alpha}_0$. Overall, the work emphasizes the need to assess perturbations all the way to the end of inflation in models with strong blue tilts, as PBH constraints can decisively test otherwise viable scenarios.

Abstract

The running-mass inflation model, which has strong motivation from particle physics, predicts density perturbations whose spectral index is strongly scale-dependent. For a large part of parameter space the spectrum rises sharply to short scales. In this paper we compute the production of primordial black holes, using both analytic and numerical calculation of the density perturbation spectra. Observational constraints from black hole production are shown to exclude a large region of otherwise permissible parameter space.

Figures (5)

• Figure 1: A sketch of the potential. The inflaton starts near the maximum with $\eta_{{ V}}$ negative. As it rolls towards the origin the mass passes through zero and $\eta_{{ V}}$ grows to 1. For the purposes of this diagram the bump is greatly exaggerated; in reality the potential is extremely flat.
• Figure 2: An example power spectrum, taking $\tilde{\alpha}_0 = 0.01$, $A_0 = 1.0$, $\mu_0^2 = 0.5$ and $N_{{\rm COBE}} = 45$. The numerical calculation of the perturbation amplitude breaks down towards the end of inflation. The end point of the power spectrum ($k_{{\rm end}} = e^{N_{{\rm COBE}}} \, h/3000 \, {\rm Mpc}^{-1}$) is defined to be where $\eta = 1$. The position where $\eta = 0.5$ is shown for comparison.
• Figure 3: Parameter space constraints for $\tilde{\alpha}_0 = 0.01$ for two choices of $N_{{\rm COBE}}$. The solid lines show the spectral index on COBE scales. The dashed lines are $\sigma_{{\rm hor}} = 0.02, 0.04, 0.08$ contours, where $\sigma_{{\rm hor}}$ has been evaluated at the end of slow-roll inflation. Parameter space below this region is excluded, and these models will violate the bound on $\sigma_{{\rm hor}}$before the end of slow-roll inflation. From Eq. (\ref{['eqn:instant_r']}), instant reheating requires $V_0/M^4_{{\rm P}} \gtrsim 10^{-36}$ for $N_{{\rm COBE}}=45$, indicated by the dot-dashed contour; the parameter space above this contour is excluded.
• Figure 4: As Fig.\ref{['fig:alpha_pos']}, but for $\tilde{\alpha}_0 = -0.01$. The thick line to the right is a bound on the allowed values of the parameters ($\left|A_0\right| > \mu_0^2 + 1)$. The dashed lines, reading from left to right, are the $\sigma_{{\rm hor}} = 0.02, 0.04, 0.08$ contours, and the region to the right of these contours is excluded by PBH constraints. From Eq. (\ref{['eqn:instant_r']}), instant reheating requires $V_0/M^4_{{\rm P}} \gtrsim 10^{-36}$ for $N_{{\rm COBE}}=45$ indicated by the dot-dashed contour; the parameter space above this contour is excluded.
• Figure 5: The effect of varying the coupling constant, $\tilde{\alpha}_0$ ($N_{{\rm COBE}}=45$). $n_{{\rm crit}}$ is the spectral index on COBE scales above which the model is ruled out, assuming the constraint is $\sigma_{{\rm hor}} < 0.04$. For $\tilde{\alpha}_0>0$ the constraint is weakened as $\tilde{\alpha}_0$ is increased, while for $\tilde{\alpha}_0<0$ the constraint becomes more restrictive as $\left|\tilde{\alpha}_0\right|$ is increased.