Avoiding PBH overproduction in inflation model with modified dispersion relation

TL;DR

The paper addresses the PBH overproduction problem associated with scalar-induced SGWBs suggested by PTA data, by introducing a modified dispersion relation (MDR) for inflationary perturbations with a late-time $k^4$ correction. This MDR yields a broken-power-law curvature spectrum, which, when constrained by NANOGrav 15-year data, can reduce PBH formation to ~2σ; achieving ~1σ compatibility requires a small negative non-Gaussianity, $f_{ m NL} obreak \simeq -1$. The PBH mass distribution is found to peak at sub-solar masses (~$10^{-2} M_$) with the abundance $f_{ m PBH}$ highly sensitive to the amplitude $A$ and pivot scale $k_*$, and to the level of non-Gaussianity. Overall, the work demonstrates that MDR in combination with mild non-Gaussianity offers a viable path to reconciling PTA observations with PBH constraints, guiding future inflation model-building and PBH/SIGW phenomenology.

Abstract

The Pulsar Timing Array (PTA) data of nano-Hertz gravitational waves released in 2023 implies that if such gravitational waves comes from the scalar perturbation induction at the end of inflation, the accompanied primordial black holes (PBHs) will be over-produced, with the fraction exceed the upper bound of unity. This is recognized as the ``overproduction problem", which calls for nontrivial features in the early universe. In this paper, we try to check out whether a modified dispersion relation (MDR) of the primordial perturbations can be helpful for solving the problem. From the constraint on PTA data, we obtain a posterior distribution of the parameters of primordial perturbation, and find that the MDR model, where the $k^4$ term becomes important at later time, can give rise to a broken-power-law (BPL) power spectrum which can alleviate the overproduction problem to nearly $2σ$ level. However, to improve furtherly into $1σ$ still needs small negative non-Gaussianity, e.g. $f_{\rm nl}\simeq -1$. The mass distribution of the PBHs generated is also discussed.

Figures (5)

• Figure 1: The power spectrum of curvature perturbation obtained from Eq. \ref{['zetabar']} with respect to $k$ (in logarithm). Three straight lines mimicking the $P_\zeta-k$ relations are also shown for comparison: red line is for $P_\zeta\sim k^0$, green is for $P_\zeta\sim k^4$, while blue is for $P_\zeta\sim k^{-1}$. The dashed line denotes the pivot scale $k_\ast$ corresponding to the peak of the power spectrum. With the power spectrum in large scales around $2.0\times10^{-9}$, ${\lg|\tau _*|}$ and $\lg \alpha$ are set to be $-4.95$ and $-3.86$ to make the BPL parameters $\lg(k_*/\mathrm{Mpc^{-1}})$ and $\lg A$ to be $7.0$ and $-1.4$.
• Figure 2: Posterior of the parameters of the BPL model, where NANOGrav 15-yr data was used. The contours of the 2D posterior plot from dark to light corresponds to the 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence levels, respectively. Priors of the parameters $\lg (k_*/\mathrm{Mpc^{-1}})$ and $\lg A$ are $U\sim(5.8,12.8)$ and $U\sim(-3,2)$.
• Figure 3: This figure shows the contour lines of PDF of $(\zeta_G, {\cal C}_G)$ with its integral regions under different $r_H$ and $f_\mathrm{nl}$, where the model parameter $\lg A$ is set to ${-1.4}$. Contours from inside to outside represent $P(\zeta_G, {\cal C}_G)$ from $0$ to $-40$. Regions between the solid and the dashed lines with the same color represent the integral regions under the corresponding value $f_\mathrm{nl}$.
• Figure 4: Posterior of Fig. \ref{['fig:SIGW']} v.s. $f_{\mathrm{PBH}}=1$ line for cases of different values of $f_{\rm nl}$. Region above each line represent $f_{\mathrm{PBH}}>1$. The contours of the 2D posterior plot from dark to light correspond to the 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence levels, respectively.
• Figure 5: Mass fractions $f_{\mathrm{PBH}}(M_{\rm PBH})$ in logarithmic form, where $M_{\rm PBH}$ is normalized by $M_\odot$. The parameters of the power spectrum are set as $\lg (k_*/\mathrm{Mpc^{-1}})=7.0$ and $\lg A=-1.4$ (blue curve), which makes $f_{\mathrm{PBH}}\simeq 1$. We also plot $f_{\mathrm{PBH}}(M_{\rm PBH})$ under $\lg (k_*/\mathrm{Mpc^{-1}})=7.0$ and $\lg A=-1.5$ (orange curve) to show that slight decrease of $\lg A$ can obviously suppress the formation of PBHs.