Probing Inflationary Origins of Primordial Black Holes with LIGO--Virgo--KAGRA O1--O4a data

Author(s)

An, Haipeng, Guo, Huai-Ke, Qiao, Mai, Wang, Lian-Tao, Yang, Chen, Zhao, Yue

Abstract

Large primordial curvature perturbations not only produce primordial black holes (PBHs) but also inevitably source a scalar-induced stochastic gravitational-wave background upon horizon reentry. We analyze the combined LIGO--Virgo--KAGRA O1--O4a data to constrain two representative inflationary mechanisms for generating such perturbations: ultra-slow-roll inflation and an inflationary phase transition. Detecting no evidence for either scenario, we place 95% credible upper limits on the curvature-spectrum amplitude across the frequency range accessible to ground-based interferometers. Translated into the PBH context, these limits already exceed conventional constraints, probing abundance fractions far below unity. Our results remain robust even when the PBHs themselves are too rare to be directly detected or have evaporated. This work demonstrates that stochastic gravitational-wave observations offer a powerful and complementary probe of small-scale inflationary physics and PBH formation, with upcoming interferometers promising to extend sensitivity to a wider range of inflationary epochs and PBH masses.

Figures

\textit{Constraints on ultra-slow-roll inflation.} \textit{Upper:} The $95\%$ credible upper limit from the combined LVK O1--O4a data on the curvature-power-spectrum amplitude $A_{\rm ref}$ as a function of the reference frequency $f_{\rm ref}$. Also shown are the joint CMB+BBN bound~\cite{Yeh:2022heq}, the contour corresponding to $f_{\rm PBH}=1$, and the projected one-year sensitivities (${\rm SNR}=2$) of A+ and the future GW experiments~\cite{LIGO:T1500293,Liang:2026wwz}. The upper axis translates $f_{\rm ref}$ into the number of e-folds $N_e$ between the end of the USR phase and the end of inflation, assuming $H_{\rm inf}=10^{14}\,\GeV$. \textit{Lower:} The corresponding GW bounds translated into the characteristic PBH mass $M_{\rm PBH}$ and dark-matter fraction $f_{\rm PBH}$ using Eqs.~\eqref{eq:usr-MPBH} and \eqref{eq:usr-fPBH}. Existing constraints from BBN~\cite{Carr:2009jm}, the CMB~\cite{Acharya:2020jbv,Chluba:2020oip}, Hawking evaporation~\cite{Carr:2009jm,Carr:2016hva,Boudaud:2018hqb}, and microlensing~\cite{Smyth:2019whb,Griest:2013esa,Griest:2013aaa,EROS-2:2006ryy} are included for comparison.
Caption \textit{Constraints on ultra-slow-roll inflation.} \textit{Upper:} The $95\%$ credible upper limit from the combined LVK O1--O4a data on the curvature-power-spectrum amplitude $A_{\rm ref}$ as a function of the reference frequency $f_{\rm ref}$. Also shown are the joint CMB+BBN bound~\cite{Yeh:2022heq}, the contour corresponding to $f_{\rm PBH}=1$, and the projected one-year sensitivities (${\rm SNR}=2$) of A+ and the future GW experiments~\cite{LIGO:T1500293,Liang:2026wwz}. The upper axis translates $f_{\rm ref}$ into the number of e-folds $N_e$ between the end of the USR phase and the end of inflation, assuming $H_{\rm inf}=10^{14}\,\GeV$. \textit{Lower:} The corresponding GW bounds translated into the characteristic PBH mass $M_{\rm PBH}$ and dark-matter fraction $f_{\rm PBH}$ using Eqs.~\eqref{eq:usr-MPBH} and \eqref{eq:usr-fPBH}. Existing constraints from BBN~\cite{Carr:2009jm}, the CMB~\cite{Acharya:2020jbv,Chluba:2020oip}, Hawking evaporation~\cite{Carr:2009jm,Carr:2016hva,Boudaud:2018hqb}, and microlensing~\cite{Smyth:2019whb,Griest:2013esa,Griest:2013aaa,EROS-2:2006ryy} are included for comparison.
\textit{Constraints on ultra-slow-roll inflation.} \textit{Upper:} The $95\%$ credible upper limit from the combined LVK O1--O4a data on the curvature-power-spectrum amplitude $A_{\rm ref}$ as a function of the reference frequency $f_{\rm ref}$. Also shown are the joint CMB+BBN bound~\cite{Yeh:2022heq}, the contour corresponding to $f_{\rm PBH}=1$, and the projected one-year sensitivities (${\rm SNR}=2$) of A+ and the future GW experiments~\cite{LIGO:T1500293,Liang:2026wwz}. The upper axis translates $f_{\rm ref}$ into the number of e-folds $N_e$ between the end of the USR phase and the end of inflation, assuming $H_{\rm inf}=10^{14}\,\GeV$. \textit{Lower:} The corresponding GW bounds translated into the characteristic PBH mass $M_{\rm PBH}$ and dark-matter fraction $f_{\rm PBH}$ using Eqs.~\eqref{eq:usr-MPBH} and \eqref{eq:usr-fPBH}. Existing constraints from BBN~\cite{Carr:2009jm}, the CMB~\cite{Acharya:2020jbv,Chluba:2020oip}, Hawking evaporation~\cite{Carr:2009jm,Carr:2016hva,Boudaud:2018hqb}, and microlensing~\cite{Smyth:2019whb,Griest:2013esa,Griest:2013aaa,EROS-2:2006ryy} are included for comparison.
Caption \textit{Constraints on ultra-slow-roll inflation.} \textit{Upper:} The $95\%$ credible upper limit from the combined LVK O1--O4a data on the curvature-power-spectrum amplitude $A_{\rm ref}$ as a function of the reference frequency $f_{\rm ref}$. Also shown are the joint CMB+BBN bound~\cite{Yeh:2022heq}, the contour corresponding to $f_{\rm PBH}=1$, and the projected one-year sensitivities (${\rm SNR}=2$) of A+ and the future GW experiments~\cite{LIGO:T1500293,Liang:2026wwz}. The upper axis translates $f_{\rm ref}$ into the number of e-folds $N_e$ between the end of the USR phase and the end of inflation, assuming $H_{\rm inf}=10^{14}\,\GeV$. \textit{Lower:} The corresponding GW bounds translated into the characteristic PBH mass $M_{\rm PBH}$ and dark-matter fraction $f_{\rm PBH}$ using Eqs.~\eqref{eq:usr-MPBH} and \eqref{eq:usr-fPBH}. Existing constraints from BBN~\cite{Carr:2009jm}, the CMB~\cite{Acharya:2020jbv,Chluba:2020oip}, Hawking evaporation~\cite{Carr:2009jm,Carr:2016hva,Boudaud:2018hqb}, and microlensing~\cite{Smyth:2019whb,Griest:2013esa,Griest:2013aaa,EROS-2:2006ryy} are included for comparison.
\textit{Constraints on Inflationary Phase Transition.} Same as Fig.~\ref{fig:sen_usr}, but for the InPT scenario. We show the constraints on $A_{\rm ref}$ and $f_{\rm ref}$ (\textit{upper}) and their translation into the PBH parameter space (\textit{lower}). In the upper panel, $f_{\rm ref}$ is mapped to the number of e-folds $N_e$ between the phase transition and the end of inflation.
Caption \textit{Constraints on Inflationary Phase Transition.} Same as Fig.~\ref{fig:sen_usr}, but for the InPT scenario. We show the constraints on $A_{\rm ref}$ and $f_{\rm ref}$ (\textit{upper}) and their translation into the PBH parameter space (\textit{lower}). In the upper panel, $f_{\rm ref}$ is mapped to the number of e-folds $N_e$ between the phase transition and the end of inflation.
\textit{Constraints on Inflationary Phase Transition.} Same as Fig.~\ref{fig:sen_usr}, but for the InPT scenario. We show the constraints on $A_{\rm ref}$ and $f_{\rm ref}$ (\textit{upper}) and their translation into the PBH parameter space (\textit{lower}). In the upper panel, $f_{\rm ref}$ is mapped to the number of e-folds $N_e$ between the phase transition and the end of inflation.
Caption \textit{Constraints on Inflationary Phase Transition.} Same as Fig.~\ref{fig:sen_usr}, but for the InPT scenario. We show the constraints on $A_{\rm ref}$ and $f_{\rm ref}$ (\textit{upper}) and their translation into the PBH parameter space (\textit{lower}). In the upper panel, $f_{\rm ref}$ is mapped to the number of e-folds $N_e$ between the phase transition and the end of inflation.
Here we show $F^{\rm USR}$ and $F^{\rm InPT}$ as functions of $f/f_{\text{ref}}$. We note that, for InPT, $F^{\rm InPT}$ has a very mild dependence on $\beta/H_{\text{inf}}$.
Caption Here we show $F^{\rm USR}$ and $F^{\rm InPT}$ as functions of $f/f_{\text{ref}}$. We note that, for InPT, $F^{\rm InPT}$ has a very mild dependence on $\beta/H_{\text{inf}}$.
\textit{Left:} Marginalized posterior probability distribution of the combined model of CBCs and USR using the LVK O1--O4a data for parameters $\Omega_\text{ref}$, $f_\text{ref}$ and $A_\text{ref}$ in $\log_{10}$ space. The blue-dashed vertical lines stand for the $16\%$ and $84\%$ percentiles, respectively. The dark-blue (light-blue) contours enclose the 68\% (95\%) credible regions. \textit{Right:} Same as the left panel, but for the combined CBC+InPT model with $\beta/H_{\text{inf}}=5$.
Caption \textit{Left:} Marginalized posterior probability distribution of the combined model of CBCs and USR using the LVK O1--O4a data for parameters $\Omega_\text{ref}$, $f_\text{ref}$ and $A_\text{ref}$ in $\log_{10}$ space. The blue-dashed vertical lines stand for the $16\%$ and $84\%$ percentiles, respectively. The dark-blue (light-blue) contours enclose the 68\% (95\%) credible regions. \textit{Right:} Same as the left panel, but for the combined CBC+InPT model with $\beta/H_{\text{inf}}=5$.
\textit{Left:} Marginalized posterior probability distribution of the combined model of CBCs and USR using the LVK O1--O4a data for parameters $\Omega_\text{ref}$, $f_\text{ref}$ and $A_\text{ref}$ in $\log_{10}$ space. The blue-dashed vertical lines stand for the $16\%$ and $84\%$ percentiles, respectively. The dark-blue (light-blue) contours enclose the 68\% (95\%) credible regions. \textit{Right:} Same as the left panel, but for the combined CBC+InPT model with $\beta/H_{\text{inf}}=5$.
Caption \textit{Left:} Marginalized posterior probability distribution of the combined model of CBCs and USR using the LVK O1--O4a data for parameters $\Omega_\text{ref}$, $f_\text{ref}$ and $A_\text{ref}$ in $\log_{10}$ space. The blue-dashed vertical lines stand for the $16\%$ and $84\%$ percentiles, respectively. The dark-blue (light-blue) contours enclose the 68\% (95\%) credible regions. \textit{Right:} Same as the left panel, but for the combined CBC+InPT model with $\beta/H_{\text{inf}}=5$.
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