Primordial black hole formation in k-inflation models

TL;DR

This work analyzes primordial black hole (PBH) formation in two k-inflation models with inflection-point potentials: PLLS and a Tachyon-DBI-type model. By numerically solving the background and curvature perturbation equations and evolving the Mukhanov–Sasaki mode functions, the authors obtain enhanced curvature spectra and compute the resulting PBH abundance today using both the Press-Schechter and critical-collapse-peak formalisms, ultimately favoring the CCP approach for realistic spectra. They find that, with careful parameter tuning, significant PBH dark matter fractions are possible, with the Tachyon model yielding broad allowed mass ranges and the PLLS model producing sharp mass windows; normalization to the CMB is maintained via a rescaling invariance. The analysis also shows that non-Gaussianities expected in these models are within observational bounds and have a subdominant but non-negligible impact on PBH predictions, while the PBH abundance remains highly sensitive to the collapse threshold $\delta_c$.

Abstract

The local primordial density fluctuations caused by quantum vacuum fluctuations during inflation grow into stars and galaxies in the late universe and, if they are large enough, also produce primordial black holes. We study the formation of the primordial black holes in $k$-essence inflation models with a potential characterized by an inflection point. The background and perturbation equations are integrated numerically for two specific models. Using the critical collapse and peaks formalism, we calculate the abundance of primordial black holes today.

Figures (8)

• Figure 1: The solution to the Hamilton equations (\ref{['eq7036']}) and (\ref{['eq7034']}) for the PLLS model. The physical field $\phi$ in units of $M_{\rm Pl}$ is ploted for $\alpha=1.5$, fixed initial $\phi_{\rm in}=5 M_{\rm Pl}$, $\phi_{\rm in}'= -8\cdot 10^{-7} M_{\rm Pl}$ and various $\phi_0$ in units of $M_{\rm Pl}$. The remaining parameters $V_0=10^{-16} M_{\rm Pl}^4$ and $\lambda=7.54\cdot 10^{-6}$ are as in Ref papanikolaou.
• Figure 2: The solution to the Hamilton equations (\ref{['eq3236']}) and (\ref{['eq3234']}) for the Tachyon model. The physical field $\phi$ (in units of $M_{\rm Pl}$) is plotted as a function of $N$ for $\phi_{\rm in}=1 M_{\rm Pl}$, $\eta_{\rm in}= -3\cdot 10^3$, $V_0=10^{-16} M_{\rm Pl}^4$, $\lambda =7.502 \cdot 10^{-6}$, and various field shifts $\phi_0$ in units of $M_{\rm Pl}$, as indicated on the plot.
• Figure 3: The slow-roll parameters $\varepsilon_1$ (left panel) and $\varepsilon_2$ (right panel) versus $N$ in the Tachyon model for $\phi_{\rm in}=1.1 M_{\rm Pl}$ (full line) and $1.0520 M_{\rm Pl}$ (dashed line). The remaining corresponding parameters are listed in Table \ref{['table2']}.
• Figure 4: The curvature power spectrum obtained by numerically solving Eq. (\ref{['eq7002']}) (full line) and the spectrum approximated by Eq. (\ref{['eq5021']}) (dashed line) for the PLLS inflation model with $n=3$, $\phi_0=0.8354 M_{\rm Pl}$, and $\alpha=1.5$. The values of $V_0$ and $\lambda$ are as in Fig. \ref{['fig2']}. The initial values are $\phi_{\rm in}=5.20473 M_{\rm Pl}$, $\eta_{\rm in}=-2.81744\cdot 10^{-7}$. The initial perturbation $v_{q\rm in}$ is determined by the Bunch-Davies vacuum and $\mathcal{P}_{\rm S}(q_{\rm CMB})$ as discussed in the text.
• Figure 5: The curvature power spectrum obtained by numerically solving equation (\ref{['eq7002']}) (full blue and long-dashed black lines) combined with the spectrum approximated by Eq. (\ref{['eq5021']}) (short dashed blue and dotted black lines) for the Tachyon model inflation with the potential $U$ as in the PLLS model. The parameters and initial values of the corresponding background solutions are $\phi_{0}=0.2595 M_{\rm Pl}$, $V_0=10^{-16} M_{\rm Pl}^4$, $\lambda=7.502 \cdot 10^{-6}$, $\phi_{\rm in}=1.1 M_{\rm Pl}$, and $\eta _{\rm in}=-5.3\cdot 10^4$ (full and short-dashed lines), and $\phi_{0}=0.259831 M_{\rm Pl}$, $V_0=10^{-16} M_{\rm Pl}^4$, $\lambda =7.600 \cdot 10^{-6}$, $\phi_{\rm in}=1.052 M_{\rm Pl}$ and $\eta _{\rm in}=-5.3625\cdot 10^4$ (long-dashed and dotted lines).