Running-Mass Inflation Model and Primordial Black Holes

TL;DR

This work assesses whether PBHs can form during the slow-roll phase of the running-mass inflation model in a way that yields long-lived black holes capable of composing dark matter. It extends the model to include the running of the running of the spectral index and the inflaton mass, and computes $n_S(k)$ exactly rather than relying on a truncated expansion. Using a Press-Schechter PBH-formation formalism with a Gaussian window, it finds that PBH formation into a CDM candidate is possible only in a very narrow region of parameter space where the effective spectral index at PBH scales, $n(k_{PBH})$, becomes sufficiently large while keeping slow-roll and observational constraints satisfied. However, even in favorable cases, the required positive running at PBH scales is difficult to reconcile with current CMB bounds and raises concerns about the overproduction of lighter PBHs, making PBHs as CDM unlikely to be a robust prediction of this model. A sharp end of inflation via a waterfall mechanism could potentially mitigate some issues, but would depart from standard slow-roll assumptions.

Abstract

We revisit the question whether the running-mass inflation model allows the formation of Primordial Black Holes (PBHs) that are sufficiently long-lived to serve as candidates for Dark Matter. We incorporate recent cosmological data, including the WMAP 7-year results. Moreover, we include "the running of the running" of the spectral index of the power spectrum, as well as the renormalization group "running of the running" of the inflaton mass term. Our analysis indicates that formation of sufficiently heavy, and hence long-lived, PBHs still remains possible in this scenario. As a by-product, we show that the additional term in the inflaton potential still does not allow significant negative running of the spectral index.

Figures (6)

• Figure 1: Fraction of the energy density of the universe collapsing into PBHs as a function of the PBH mass, for three different values of $n(R)$ and two different choices of the threshold $\delta_{\rm th} = 0.3 \ (0.7)$ for the solid (dashed) curves. On the upper [lower] of two curves with equal pattern and color we have assumed $n_S(R) = 2 n(R)-1$ [$n_S(R) = 3 n(R) - 2$].
• Figure 2: Scatter plot of $\beta_S(k_0)$ vs. $\alpha_S(k_0)$. Here the model parameters $\tilde{c} \equiv 2 c M_P^2 / V_0, \ g/c$ and $L_0$ are scanned randomly, with flat probability distribution functions.
• Figure 3: The potential parameters $c$ (in units of $V_0/(2 M_{\rm P}^2)$; double--dotted [green] curve), $g/(2c)$ (dotted, blue), the effective spectral index $n_S-1$ at the PBH scale (dot--dashed, red) and the spectral index $n-1$ at the PBH scale (solid, black) are shown as functions of $L_0 = \ln ( \phi_0 / \phi_*)$, for $n_S(k_0) = 0.964$ and $\alpha_S(k_0) = 0.012$. If both solutions in eq.(\ref{['sol3']}) for $g/c$ are acceptable, we have taken the one giving larger $n$ at the PBH scale.
• Figure 4: Evolution of the rescaled inflaton potential, $4 M_{\rm P}^2 (V-V_0) / (V_0 \phi_0^2)$ (solid, black), and of the effective spectral index $n_S$ (dashed, red), as a function of the inflaton field $\phi/\phi_0$ (left frame) or of the ratio of scales $k/k_0$ (right frame). In the left frame the dot--dashed (blue) curve shows $k/k_0$, whereas in the right frame it depicts $\phi/\phi_0$; this curve in both cases refers to the scale to the right. We took $\tilde{c} = -0.1711, \ g/c = 0.09648, \ L_0 = -0.756$; these parameters maximize the spectral index at the PBH scale for $n_S(k_0) = 0.964, \ \alpha_S(k_0) = 0.012$ (see figure \ref{['fig:Fig3']}).
• Figure 5: The solid (black) curve shows the maximal spectral index at the PBH scale $k_{\rm PBH} = 1.6\times10^{19} k_0$ that is consistent with the constraints we impose, as function of $\alpha_S(k_0)$. The other curves show the corresponding model parameters: $\tilde{c}$ (double-dotted, green), multiplied with $-5$ for ease of presentation; $-g/(2c)$ (dotted, blue); and $-L_0$ (dot--dashed, red). $n_S(k_0)$ is fixed at 0.964, but the bound on $n_{\rm PBH}$ is almost independent of this choice.