Non-Gaussianities in primordial black hole formation and induced gravitational waves

Author(s)

Pi, Shi

Abstract

The most promising mechanism of generating primordial black holes (PBHs) is by the enhancement of power spectrum of the primordial curvature perturbation, which is usually accompanied by the the enhancement of non-Gaussianity that crucially changes the abundance of PBHs. In this review I will discuss how non-Gaussianity is generated in single field inflation as well as in the curvaton scenario, and then discuss how to calculate PBH mass function with such non-Gaussianities. I also show non-Gaussianity only has mild effects on the induced gravitational waves (GWs), which gives robust predictions in the mHz and nHz GW experiments.

Figures

The schematic phase portrait of the ultra-slow-roll inflation on a flat plateau. The blue and green lines with arrows are the inflaton trajectory in the background universe and a perturbed universe, respectively. The red lines are equal-$N$ lines, determined by the conservation law \eqref{eqn8:conservation-usr}, while ($\tilde\pi$, $\tilde\varphi$) at the perturbed trajectory should have the same $N$ as ($\pi$, $\varphi$), i.e. the perturbation is along an equal-$N$ line. Such a perturbed point ($\tilde\pi$, $\tilde\varphi$) satisfies \eqref{eqn8:conservation2-usr}, as is clearly shown geometrically. For a detailed discussion of such a phase portrait of ultra-slow-roll inflation, see \cite{Passaglia:2018ixg}.

The schematic phase portrait of the ultra-slow-roll inflation on a flat plateau. The blue and green lines with arrows are the inflaton trajectory in the background universe and a perturbed universe, respectively. The red lines are equal-$N$ lines, determined by the conservation law \eqref{eqn8:conservation-usr}, while ($\tilde\pi$, $\tilde\varphi$) at the perturbed trajectory should have the same $N$ as ($\pi$, $\varphi$), i.e. the perturbation is along an equal-$N$ line. Such a perturbed point ($\tilde\pi$, $\tilde\varphi$) satisfies \eqref{eqn8:conservation2-usr}, as is clearly shown geometrically. For a detailed discussion of such a phase portrait of ultra-slow-roll inflation, see \cite{Passaglia:2018ixg}.


Inflation potential \eqref{eqn8:bumpV}, and the phase portrait solution to \eqref{eqn8:eom} with $\eta=-4/3$. The initial conditions for the blue, orange, cyan, and gray curves are $\pi_i=14,\, 13,\, 12,\, 11$ at $\varphi_i=-3$ with $\varphi_{t}=100$ (in the unit of $H$). These solutions are not on the attractor initially, which is displayed by the diagonal dotted line, \textit{i.e.} $\pi+\lambda_-\varphi=0$. The equal-$N$ lines, based on \eqref{eqn8:dN4}, are shown in thin red lines with equal difference of $\Delta N=0.5$. It is clearly shown that for the same $\delta\varphi$, the largest $\delta N$ comes from around the critical trajectory (cyan) in the constant-roll stage.

Inflation potential \eqref{eqn8:bumpV}, and the phase portrait solution to \eqref{eqn8:eom} with $\eta=-4/3$. The initial conditions for the blue, orange, cyan, and gray curves are $\pi_i=14,\, 13,\, 12,\, 11$ at $\varphi_i=-3$ with $\varphi_{t}=100$ (in the unit of $H$). These solutions are not on the attractor initially, which is displayed by the diagonal dotted line, \textit{i.e.} $\pi+\lambda_-\varphi=0$. The equal-$N$ lines, based on \eqref{eqn8:dN4}, are shown in thin red lines with equal difference of $\Delta N=0.5$. It is clearly shown that for the same $\delta\varphi$, the largest $\delta N$ comes from around the critical trajectory (cyan) in the constant-roll stage.


Inflation potential \eqref{eqn8:bumpV}, and the phase portrait solution to \eqref{eqn8:eom} with $\eta=-4/3$. The initial conditions for the blue, orange, cyan, and gray curves are $\pi_i=14,\, 13,\, 12,\, 11$ at $\varphi_i=-3$ with $\varphi_{t}=100$ (in the unit of $H$). These solutions are not on the attractor initially, which is displayed by the diagonal dotted line, \textit{i.e.} $\pi+\lambda_-\varphi=0$. The equal-$N$ lines, based on \eqref{eqn8:dN4}, are shown in thin red lines with equal difference of $\Delta N=0.5$. It is clearly shown that for the same $\delta\varphi$, the largest $\delta N$ comes from around the critical trajectory (cyan) in the constant-roll stage.

Inflation potential \eqref{eqn8:bumpV}, and the phase portrait solution to \eqref{eqn8:eom} with $\eta=-4/3$. The initial conditions for the blue, orange, cyan, and gray curves are $\pi_i=14,\, 13,\, 12,\, 11$ at $\varphi_i=-3$ with $\varphi_{t}=100$ (in the unit of $H$). These solutions are not on the attractor initially, which is displayed by the diagonal dotted line, \textit{i.e.} $\pi+\lambda_-\varphi=0$. The equal-$N$ lines, based on \eqref{eqn8:dN4}, are shown in thin red lines with equal difference of $\Delta N=0.5$. It is clearly shown that for the same $\delta\varphi$, the largest $\delta N$ comes from around the critical trajectory (cyan) in the constant-roll stage.


Schematic pictures of the piecewise quadratic potential $V(\varphi)$ in \eqref{eqn8:piesewiseV}, with $m_1^2<0$ (left) and $m_2^2>0$ (right). The potential is continuous at the transition point is $\varphi_{t}$, where $V(\varphi_{t})=V_0$, but $V'(\varphi_{t})$ has a jump. The origin is chosen at the local maximum (upper panel) or minimum (lower panel) of $V(\varphi<\varphi_t)$, for $m_1^2<0$ or $m_1^2>0$, respectively. At $\varphi_{t}$, the potential is continuous, while its slope may not be. The $e$-folding number is also labeled at $\varphi$, $\varphi_{t}$, and $\varphi_f$.

Schematic pictures of the piecewise quadratic potential $V(\varphi)$ in \eqref{eqn8:piesewiseV}, with $m_1^2<0$ (left) and $m_2^2>0$ (right). The potential is continuous at the transition point is $\varphi_{t}$, where $V(\varphi_{t})=V_0$, but $V'(\varphi_{t})$ has a jump. The origin is chosen at the local maximum (upper panel) or minimum (lower panel) of $V(\varphi<\varphi_t)$, for $m_1^2<0$ or $m_1^2>0$, respectively. At $\varphi_{t}$, the potential is continuous, while its slope may not be. The $e$-folding number is also labeled at $\varphi$, $\varphi_{t}$, and $\varphi_f$.


Schematic pictures of the piecewise quadratic potential $V(\varphi)$ in \eqref{eqn8:piesewiseV}, with $m_1^2<0$ (left) and $m_2^2>0$ (right). The potential is continuous at the transition point is $\varphi_{t}$, where $V(\varphi_{t})=V_0$, but $V'(\varphi_{t})$ has a jump. The origin is chosen at the local maximum (upper panel) or minimum (lower panel) of $V(\varphi<\varphi_t)$, for $m_1^2<0$ or $m_1^2>0$, respectively. At $\varphi_{t}$, the potential is continuous, while its slope may not be. The $e$-folding number is also labeled at $\varphi$, $\varphi_{t}$, and $\varphi_f$.

Schematic pictures of the piecewise quadratic potential $V(\varphi)$ in \eqref{eqn8:piesewiseV}, with $m_1^2<0$ (left) and $m_2^2>0$ (right). The potential is continuous at the transition point is $\varphi_{t}$, where $V(\varphi_{t})=V_0$, but $V'(\varphi_{t})$ has a jump. The origin is chosen at the local maximum (upper panel) or minimum (lower panel) of $V(\varphi<\varphi_t)$, for $m_1^2<0$ or $m_1^2>0$, respectively. At $\varphi_{t}$, the potential is continuous, while its slope may not be. The $e$-folding number is also labeled at $\varphi$, $\varphi_{t}$, and $\varphi_f$.


The schematic potential (left panel) and the phase portrait (right panel) of the ``smooth transition'' from ultra-slow-roll to slow-roll, with $\eta'=-0.01$, $\varphi_m=0$. For blue and orange trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$ (in the unit of $H$), respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Equal-$N$ lines are parallel to $\pi\propto-3\varphi$ with almost equal intervals, thus Gaussianity is well preserved.

The schematic potential (left panel) and the phase portrait (right panel) of the ``smooth transition'' from ultra-slow-roll to slow-roll, with $\eta'=-0.01$, $\varphi_m=0$. For blue and orange trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$ (in the unit of $H$), respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Equal-$N$ lines are parallel to $\pi\propto-3\varphi$ with almost equal intervals, thus Gaussianity is well preserved.


The schematic potential (left panel) and the phase portrait (right panel) of the ``smooth transition'' from ultra-slow-roll to slow-roll, with $\eta'=-0.01$, $\varphi_m=0$. For blue and orange trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$ (in the unit of $H$), respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Equal-$N$ lines are parallel to $\pi\propto-3\varphi$ with almost equal intervals, thus Gaussianity is well preserved.

The schematic potential (left panel) and the phase portrait (right panel) of the ``smooth transition'' from ultra-slow-roll to slow-roll, with $\eta'=-0.01$, $\varphi_m=0$. For blue and orange trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$ (in the unit of $H$), respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Equal-$N$ lines are parallel to $\pi\propto-3\varphi$ with almost equal intervals, thus Gaussianity is well preserved.


The schematic potential (left panel) and the phase portrait (right panel) of the ``sharp transition'' from ultra-slow-roll to slow-roll, with $\eta'=0.01$, $\varphi_m=100$ (in the unit of $H$). For the fiducial (blue) and perturbed (orange) trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$, respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Near the origin, the trajectories merge to a unique fast-roll trajectory with different $\pi$, which contribute negligibly to $\delta N$.

The schematic potential (left panel) and the phase portrait (right panel) of the ``sharp transition'' from ultra-slow-roll to slow-roll, with $\eta'=0.01$, $\varphi_m=100$ (in the unit of $H$). For the fiducial (blue) and perturbed (orange) trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$, respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Near the origin, the trajectories merge to a unique fast-roll trajectory with different $\pi$, which contribute negligibly to $\delta N$.


The schematic potential (left panel) and the phase portrait (right panel) of the ``sharp transition'' from ultra-slow-roll to slow-roll, with $\eta'=0.01$, $\varphi_m=100$ (in the unit of $H$). For the fiducial (blue) and perturbed (orange) trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$, respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Near the origin, the trajectories merge to a unique fast-roll trajectory with different $\pi$, which contribute negligibly to $\delta N$.

The schematic potential (left panel) and the phase portrait (right panel) of the ``sharp transition'' from ultra-slow-roll to slow-roll, with $\eta'=0.01$, $\varphi_m=100$ (in the unit of $H$). For the fiducial (blue) and perturbed (orange) trajectories we set $\pi=9.15,~9.05$ at $\varphi=-3$, respectively. To show the difference of the trajectories clearly, $\delta\pi$ is exaggerated. Near the origin, the trajectories merge to a unique fast-roll trajectory with different $\pi$, which contribute negligibly to $\delta N$.


Left: The PBH mass function $f_\mathrm{PBH}(M)$ given by \eqref{eqn8:fPBH(M)} with a monochromatic power spectrum \eqref{eqn8:monoPR} in a ultra-slow-roll inflation with a sharp end, together with the observational constraints from \cite{Carr:2020gox}. The total power and central mass are $\mathcal{A_R}=4.151\times10^{-3}$ and $M_{k_*}=2\times10^{21}~\text{g}$, such that $\int f_\mathrm{PBH}(M)\mathrm{d}\ln M=1$. Right: The PBH abundance $f_\mathrm{PBH}$ as a function of the total power $\mathcal{A_R}$, for a monochromatic power spectrum \eqref{eqn8:monoPR}. From right to left, the black, purple dotted, blue dashed, cyan, brown dot-dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll inflation, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$, respectively.

Left: The PBH mass function $f_\mathrm{PBH}(M)$ given by \eqref{eqn8:fPBH(M)} with a monochromatic power spectrum \eqref{eqn8:monoPR} in a ultra-slow-roll inflation with a sharp end, together with the observational constraints from \cite{Carr:2020gox}. The total power and central mass are $\mathcal{A_R}=4.151\times10^{-3}$ and $M_{k_*}=2\times10^{21}~\text{g}$, such that $\int f_\mathrm{PBH}(M)\mathrm{d}\ln M=1$. Right: The PBH abundance $f_\mathrm{PBH}$ as a function of the total power $\mathcal{A_R}$, for a monochromatic power spectrum \eqref{eqn8:monoPR}. From right to left, the black, purple dotted, blue dashed, cyan, brown dot-dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll inflation, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$, respectively.


Left: The PBH mass function $f_\mathrm{PBH}(M)$ given by \eqref{eqn8:fPBH(M)} with a monochromatic power spectrum \eqref{eqn8:monoPR} in a ultra-slow-roll inflation with a sharp end, together with the observational constraints from \cite{Carr:2020gox}. The total power and central mass are $\mathcal{A_R}=4.151\times10^{-3}$ and $M_{k_*}=2\times10^{21}~\text{g}$, such that $\int f_\mathrm{PBH}(M)\mathrm{d}\ln M=1$. Right: The PBH abundance $f_\mathrm{PBH}$ as a function of the total power $\mathcal{A_R}$, for a monochromatic power spectrum \eqref{eqn8:monoPR}. From right to left, the black, purple dotted, blue dashed, cyan, brown dot-dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll inflation, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$, respectively.

Left: The PBH mass function $f_\mathrm{PBH}(M)$ given by \eqref{eqn8:fPBH(M)} with a monochromatic power spectrum \eqref{eqn8:monoPR} in a ultra-slow-roll inflation with a sharp end, together with the observational constraints from \cite{Carr:2020gox}. The total power and central mass are $\mathcal{A_R}=4.151\times10^{-3}$ and $M_{k_*}=2\times10^{21}~\text{g}$, such that $\int f_\mathrm{PBH}(M)\mathrm{d}\ln M=1$. Right: The PBH abundance $f_\mathrm{PBH}$ as a function of the total power $\mathcal{A_R}$, for a monochromatic power spectrum \eqref{eqn8:monoPR}. From right to left, the black, purple dotted, blue dashed, cyan, brown dot-dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll inflation, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$, respectively.


Left: The spectrum of GWs induced by the enhanced narrow-peak power spectrum \eqref{eqn8:monoPR}, with PBH as all the dark matter. The amplitude $\mathcal{A_R}$ for different parameters follows Table \ref{tab8:AR}. From left to right, the black, purple dotted, blue dashed, cyan thick, brown dot-dashed, magenta dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll model, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$ respectively. The designed power-law integrated sensitivity curve of space-borne interferometers LISA (gray thick) \cite{Bartolo:2016ami}, Taiji (gray dashed) \cite{Wang:2021njt}, TianQin (gray dotted)~\cite{Liang:2021bde}, and DECIGO (gray thin) \cite{Schmitz:2020syl} are also shown. Right: Two more realistic signals, the spectrum of GWs induced by ultra-slow-roll inflation with a sharp transition to slow-roll (blue), and a smooth transition to slow-roll, \textit{i.e.} Starobinsky's linear potential model (red). The right panel was drawn with the help of the SIGWfast \cite{Witkowski:2022mtg} package.

Left: The spectrum of GWs induced by the enhanced narrow-peak power spectrum \eqref{eqn8:monoPR}, with PBH as all the dark matter. The amplitude $\mathcal{A_R}$ for different parameters follows Table \ref{tab8:AR}. From left to right, the black, purple dotted, blue dashed, cyan thick, brown dot-dashed, magenta dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll model, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$ respectively. The designed power-law integrated sensitivity curve of space-borne interferometers LISA (gray thick) \cite{Bartolo:2016ami}, Taiji (gray dashed) \cite{Wang:2021njt}, TianQin (gray dotted)~\cite{Liang:2021bde}, and DECIGO (gray thin) \cite{Schmitz:2020syl} are also shown. Right: Two more realistic signals, the spectrum of GWs induced by ultra-slow-roll inflation with a sharp transition to slow-roll (blue), and a smooth transition to slow-roll, \textit{i.e.} Starobinsky's linear potential model (red). The right panel was drawn with the help of the SIGWfast \cite{Witkowski:2022mtg} package.


Left: The spectrum of GWs induced by the enhanced narrow-peak power spectrum \eqref{eqn8:monoPR}, with PBH as all the dark matter. The amplitude $\mathcal{A_R}$ for different parameters follows Table \ref{tab8:AR}. From left to right, the black, purple dotted, blue dashed, cyan thick, brown dot-dashed, magenta dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll model, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$ respectively. The designed power-law integrated sensitivity curve of space-borne interferometers LISA (gray thick) \cite{Bartolo:2016ami}, Taiji (gray dashed) \cite{Wang:2021njt}, TianQin (gray dotted)~\cite{Liang:2021bde}, and DECIGO (gray thin) \cite{Schmitz:2020syl} are also shown. Right: Two more realistic signals, the spectrum of GWs induced by ultra-slow-roll inflation with a sharp transition to slow-roll (blue), and a smooth transition to slow-roll, \textit{i.e.} Starobinsky's linear potential model (red). The right panel was drawn with the help of the SIGWfast \cite{Witkowski:2022mtg} package.

Left: The spectrum of GWs induced by the enhanced narrow-peak power spectrum \eqref{eqn8:monoPR}, with PBH as all the dark matter. The amplitude $\mathcal{A_R}$ for different parameters follows Table \ref{tab8:AR}. From left to right, the black, purple dotted, blue dashed, cyan thick, brown dot-dashed, magenta dashed, and gray curves are for Gaussian case, $f_\mathrm{NL}=1$, $f_\mathrm{NL}=5/2$, ultra-slow-roll model, $f_\mathrm{NL}=10$, $f_\mathrm{NL}=10^2$, and $f_\mathrm{NL}=10^3$ respectively. The designed power-law integrated sensitivity curve of space-borne interferometers LISA (gray thick) \cite{Bartolo:2016ami}, Taiji (gray dashed) \cite{Wang:2021njt}, TianQin (gray dotted)~\cite{Liang:2021bde}, and DECIGO (gray thin) \cite{Schmitz:2020syl} are also shown. Right: Two more realistic signals, the spectrum of GWs induced by ultra-slow-roll inflation with a sharp transition to slow-roll (blue), and a smooth transition to slow-roll, \textit{i.e.} Starobinsky's linear potential model (red). The right panel was drawn with the help of the SIGWfast \cite{Witkowski:2022mtg} package.


References
  • [1] I. D. Zel’dovich, Ya.B.; Novikov, Soviet Astron. AJ (Engl. Transl. ), 10, 602 (1967).
  • [1] I. D. Zel’dovich, Ya.B.; Novikov, Soviet Astron. AJ (Engl. Transl. ), 10, 602 (1967).
  • [2] S. Hawking, Mon. Not. Roy. Astron. Soc. 152, 75 (1971).
  • [3] B. J. Carr and S. Hawking, Mon. Not. Roy. Astron. Soc. 168, 399 (1974).
  • [4] P. Meszaros, Astron. Astrophys. 37, 225 (1974).
  • [5] B. J. Carr, Astrophys. J. 201, 1 (1975).
  • [6] M. Khlopov, B. Malomed, and I. Zeldovich, Mon. Not. Roy. Astron. Soc. 215, 575 (1985).
  • [7] W. H. Press and P. Schechter, Astrophys. J. 187, 425 (1974).
  • [8] M. Shibata and M. Sasaki, Phys. Rev. D 60, 084002 (1999), arXiv:gr-qc/9905064.
  • [9] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).
  • [10] Y. Akrami et al. (Planck), Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211 [astro-ph.CO].
  • [11] S. Matarrese, O. Pantano, and D. Saez, Phys. Rev. D 47, 1311 (1993).
  • [12] S. Matarrese, O. Pantano, and D. Saez, Phys. Rev. Lett. 72, 320 (1994), arXiv:astro-ph/9310036.
  • [13] S. Matarrese, S. Mollerach, and M. Bruni, Phys. Rev. D 58, 043504 (1998), arXiv:astro-ph/9707278.
  • [14] H. Noh and J.-c. Hwang, Phys. Rev. D 69, 104011 (2004).
  • [15] C. Carbone and S. Matarrese, Phys. Rev. D 71, 043508 (2005), arXiv:astro-ph/0407611.
  • [16] K. Nakamura, Prog. Theor. Phys. 117, 17 (2007), arXiv:gr-qc/0605108.
  • [17] K. N. Ananda, C. Clarkson, and D. Wands, Phys. Rev. D 75, 123518 (2007), arXiv:gr-qc/0612013.
  • [18] B. Osano, C. Pitrou, P. Dunsby, J.-P. Uzan, and C. Clarkson, JCAP 04, 003 (2007), arXiv:grqc/0612108.
  • [19] D. Baumann, P. J. Steinhardt, K. Takahashi, and K. Ichiki, Phys. Rev. D 76, 084019 (2007), arXiv:hep-th/0703290.
  • [20] R. Saito and J. Yokoyama, Phys. Rev. Lett. 102, 161101 (2009), [Erratum: Phys.Rev.Lett. 107, 069901 (2011)], arXiv:0812.4339 [astro-ph].
  • [20] R. Saito and J. Yokoyama, Phys. Rev. Lett. 102, 161101 (2009), [Erratum: Phys.Rev.Lett. 107, 069901 (2011)], arXiv:0812.4339 [astro-ph].
  • [21] R. Saito and J. Yokoyama, Prog. Theor. Phys. 123, 867 (2010), [Erratum: Prog.Theor.Phys. 126, 351–352 (2011)], arXiv:0912.5317 [astro-ph.CO].
  • [21] R. Saito and J. Yokoyama, Prog. Theor. Phys. 123, 867 (2010), [Erratum: Prog.Theor.Phys. 126, 351–352 (2011)], arXiv:0912.5317 [astro-ph.CO].
  • [22] E. Bugaev and P. Klimai, Phys. Rev. D 81, 023517 (2010), arXiv:0908.0664 [astro-ph.CO].
  • [23] H. Assadullahi and D. Wands, Phys. Rev. D 81, 023527 (2010), arXiv:0907.4073 [astro-ph.CO].
  • [24] E. Bugaev and P. Klimai, Phys. Rev. D 83, 083521 (2011), arXiv:1012.4697 [astro-ph.CO].
  • [25] G. Domènech, (2024), arXiv:2402.17388 [gr-qc].
  • [26] J. S. Bullock and J. R. Primack, Phys. Rev. D 55, 7423 (1997), arXiv:astro-ph/9611106.
  • [27] P. Ivanov, Phys. Rev. D 57, 7145 (1998), arXiv:astro-ph/9708224.
  • [28] J. Yokoyama, Phys. Rev. D 58, 107502 (1998), arXiv:gr-qc/9804041.
  • [29] P. Pina Avelino, Phys. Rev. D 72, 124004 (2005), arXiv:astro-ph/0510052.
  • [30] D. Seery and J. C. Hidalgo, JCAP 07, 008 (2006), arXiv:astro-ph/0604579.
  • [31] J. C. Hidalgo, (2007), arXiv:0708.3875 [astro-ph].
  • [32] C. T. Byrnes, E. J. Copeland, and A. M. Green, Phys. Rev. D 86, 043512 (2012), arXiv:1206.4188 [astro-ph.CO].
  • [33] S. Young and C. T. Byrnes, JCAP 08, 052 (2013), arXiv:1307.4995 [astro-ph.CO].
  • [34] S. Young, D. Regan, and C. T. Byrnes, JCAP 02, 029 (2016), arXiv:1512.07224 [astro-ph.CO].
  • [35] V. Atal and C. Germani, Phys. Dark Univ. 24, 100275 (2019), arXiv:1811.07857 [astro-ph.CO].
  • [36] C.-M. Yoo, J.-O. Gong, and S. Yokoyama, JCAP 09, 033 (2019), arXiv:1906.06790 [astro-ph.CO].
  • [37] A. Kehagias, I. Musco, and A. Riotto, JCAP 12, 029 (2019), arXiv:1906.07135 [astro-ph.CO].
  • [38] R. Mahbub, Phys. Rev. D 102, 023538 (2020), arXiv:2005.03618 [astro-ph.CO].
  • [39] F. Riccardi, M. Taoso, and A. Urbano, (2021), arXiv:2102.04084 [astro-ph.CO].
  • [40] M. Taoso and A. Urbano, (2021), arXiv:2102.03610 [astro-ph.CO].
  • [41] M. Biagetti, V. De Luca, G. Franciolini, A. Kehagias, and A. Riotto, Phys. Lett. B 820, 136602 (2021), arXiv:2105.07810 [astro-ph.CO].
  • [42] N. Kitajima, Y. Tada, S. Yokoyama, and C.-M. Yoo, JCAP 10, 053 (2021), arXiv:2109.00791 [astro-ph.CO].
  • [43] S. Young, JCAP 05, 037 (2022), arXiv:2201.13345 [astro-ph.CO].
  • [44] A. Escrivà, Y. Tada, S. Yokoyama, and C.-M. Yoo, JCAP 05, 012 (2022), arXiv:2202.01028 [astroph.CO].
  • [45] T. Matsubara and M. Sasaki, JCAP 10, 094 (2022), arXiv:2208.02941 [astro-ph.CO].
  • [46] A. D. Gow, H. Assadullahi, J. H. P. Jackson, K. Koyama, V. Vennin, and D. Wands, (2022), arXiv:2211.08348 [astro-ph.CO].
  • [47] T. Nakama, J. Silk, and M. Kamionkowski, Phys. Rev. D 95, 043511 (2017), arXiv:1612.06264 [astro-ph.CO].
  • [48] J. Garcia-Bellido, M. Peloso, and C. Unal, JCAP 09, 013 (2017), arXiv:1707.02441 [astro-ph.CO].
  • [49] R.-g. Cai, S. Pi, and M. Sasaki, Phys. Rev. Lett. 122, 201101 (2019), arXiv:1810.11000 [astro-ph.CO].
  • [50] C. Unal, Phys. Rev. D 99, 041301 (2019), arXiv:1811.09151 [astro-ph.CO].
  • [51] C. Unal, E. D. Kovetz, and S. P. Patil, (2020), arXiv:2008.11184 [astro-ph.CO].
  • [52] P. Adshead, K. D. Lozanov, and Z. J. Weiner, JCAP 10, 080 (2021), arXiv:2105.01659 [astro-ph.CO].
  • [53] S. Garcia-Saenz, L. Pinol, S. Renaux-Petel, and D. Werth, (2022), arXiv:2207.14267 [astro-ph.CO].
  • [54] K. T. Abe, R. Inui, Y. Tada, and S. Yokoyama, (2022), arXiv:2209.13891 [astro-ph.CO].
  • [55] E. Komatsu and D. N. Spergel, Phys. Rev. D 63, 063002 (2001), arXiv:astro-ph/0005036.
  • [56] Y. Tada and S. Yokoyama, Phys. Rev. D 91, 123534 (2015), arXiv:1502.01124 [astro-ph.CO].
  • [57] S. Young and C. T. Byrnes, JCAP 04, 034 (2015), arXiv:1503.01505 [astro-ph.CO].
  • [58] G. Franciolini, A. Kehagias, S. Matarrese, and A. Riotto, JCAP 03, 016 (2018), arXiv:1801.09415 [astro-ph.CO].
  • [59] K. Ando, M. Kawasaki, and H. Nakatsuka, Phys. Rev. D 98, 083508 (2018), arXiv:1805.07757 [astro-ph.CO].
  • [60] S. Passaglia, W. Hu, and H. Motohashi, Phys. Rev. D 99, 043536 (2019), arXiv:1812.08243 [astroph.CO].
  • [61] M. H. Namjoo, H. Firouzjahi, and M. Sasaki, EPL 101, 39001 (2013), arXiv:1210.3692 [astroph.CO].
  • [62] J. Martin, H. Motohashi, and T. Suyama, Phys. Rev. D 87, 023514 (2013), arXiv:1211.0083 [astroph.CO].
  • [63] X. Chen, H. Firouzjahi, M. H. Namjoo, and M. Sasaki, EPL 102, 59001 (2013), arXiv:1301.5699 [hep-th].
  • [64] H. Motohashi, A. A. Starobinsky, and J. Yokoyama, JCAP 09, 018 (2015), arXiv:1411.5021 [astroph.CO].
  • [65] M. W. Davies, P. Carrilho, and D. J. Mulryne, JCAP 06, 019 (2022), arXiv:2110.08189 [astro-ph.CO].
  • [66] M. H. Namjoo, (2023), arXiv:2311.12777 [astro-ph.CO].
  • [67] M. H. Namjoo and B. Nikbakht, (2024), arXiv:2401.12958 [astro-ph.CO].
  • [68] G. Ferrante, G. Franciolini, A. Iovino, Junior., and A. Urbano, (2022), arXiv:2211.01728 [astroph.CO].
  • [69] A. M. Green, A. R. Liddle, K. A. Malik, and M. Sasaki, Phys. Rev. D 70, 041502 (2004), arXiv:astroph/0403181.
  • [70] C.-M. Yoo, T. Harada, J. Garriga, and K. Kohri, PTEP 2018, 123E01 (2018), arXiv:1805.03946 [astro-ph.CO].
  • [71] C. Germani and I. Musco, Phys. Rev. Lett. 122, 141302 (2019), arXiv:1805.04087 [astro-ph.CO].
  • [72] V. Atal, J. Garriga, and A. Marcos-Caballero, JCAP 09, 073 (2019), arXiv:1905.13202 [astro-ph.CO].
  • [73] V. Atal, J. Cid, A. Escrivà, and J. Garriga, JCAP 05, 022 (2020), arXiv:1908.11357 [astro-ph.CO].
  • [74] S. Young and M. Musso, JCAP 11, 022 (2020), arXiv:2001.06469 [astro-ph.CO].
  • [75] C.-M. Yoo, T. Harada, S. Hirano, and K. Kohri, PTEP 2021, 013E02 (2021), arXiv:2008.02425 [astro-ph.CO].
  • [76] R. Bean and J. Magueijo, Phys. Rev. D 66, 063505 (2002), arXiv:astro-ph/0204486.
  • [77] M. Kawasaki, A. Kusenko, and T. T. Yanagida, Phys. Lett. B 711, 1 (2012), arXiv:1202.3848 [astro-ph.CO].
  • [78] B. Carr and J. Silk, Mon. Not. Roy. Astron. Soc. 478, 3756 (2018), arXiv:1801.00672 [astro-ph.CO].
  • [79] B. Carr, F. Kuhnel, and L. Visinelli, Mon. Not. Roy. Astron. Soc. 501, 2 (2021), arXiv:2008.08077 [astro-ph.CO].
  • [80] T. Nakama, T. Suyama, and J. Yokoyama, Phys. Rev. D 94, 103522 (2016), arXiv:1609.02245 [gr-qc].
  • [81] T. Nakama, B. Carr, and J. Silk, Phys. Rev. D 97, 043525 (2018), arXiv:1710.06945 [astro-ph.CO].
  • [82] T. Nakama, K. Kohri, and J. Silk, Phys. Rev. D 99, 123530 (2019), arXiv:1905.04477 [astro-ph.CO].
  • [83] V. Atal, A. Sanglas, and N. Triantafyllou, (2020), arXiv:2012.14721 [astro-ph.CO].
  • [84] B. Liu and V. Bromm, Astrophys. J. Lett. 937, L30 (2022), arXiv:2208.13178 [astro-ph.CO].
  • [85] M. Biagetti, G. Franciolini, and A. Riotto, Astrophys. J. 944, 113 (2023), arXiv:2210.04812 [astroph.CO].
  • [86] Y. Gouttenoire, S. Trifinopoulos, G. Valogiannis, and M. Vanvlasselaer, (2023), arXiv:2307.01457 [astro-ph.CO].
  • [87] D. Hooper, A. Ireland, G. Krnjaic, and A. Stebbins, (2023), arXiv:2308.00756 [astro-ph.CO].
  • [88] E. Barausse et al., Gen. Rel. Grav. 52, 81 (2020), arXiv:2001.09793 [gr-qc].
  • [89] K. G. Arun et al. (LISA), Living Rev. Rel. 25, 4 (2022), arXiv:2205.01597 [gr-qc].
  • [90] P. Auclair et al. (LISA Cosmology Working Group), Living Rev. Rel. 26, 5 (2023), arXiv:2204.05434 [astro-ph.CO].
  • [91] E. Bagui et al. (LISA Cosmology Working Group), (2023), arXiv:2310.19857 [astro-ph.CO].
  • [92] Z. Ren, T. Zhao, Z. Cao, Z.-K. Guo, W.-B. Han, H.-B. Jin, and Y.-L. Wu, Front. Phys. (Beijing) 18, 64302 (2023), arXiv:2301.02967 [gr-qc].
  • [93] Z.-C. Liang, Y.-M. Hu, Y. Jiang, J. Cheng, J.-d. Zhang, and J. Mei, (2021), arXiv:2107.08643 [astro-ph.CO].
  • [94] N. Bartolo, V. De Luca, G. Franciolini, A. Lewis, M. Peloso, and A. Riotto, Phys. Rev. Lett. 122, 211301 (2019), arXiv:1810.12218 [astro-ph.CO].
  • [95] N. Bartolo, S. Matarrese, and A. Riotto, Phys. Rev. D 69, 043503 (2004), arXiv:hep-ph/0309033.
  • [96] K. Enqvist and S. Nurmi, JCAP 10, 013 (2005), arXiv:astro-ph/0508573.
  • [97] M. Sasaki, J. Valiviita, and D. Wands, Phys. Rev. D 74, 103003 (2006), arXiv:astro-ph/0607627.
  • [98] S. Pi and M. Sasaki, Phys. Rev. D 108, L101301 (2023), arXiv:2112.12680 [astro-ph.CO].
  • [99] G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951, L8 (2023), arXiv:2306.16213 [astro-ph.HE].
  • [100] G. Agazie et al. (NANOGrav), Astrophys. J. Lett. 951, L9 (2023), arXiv:2306.16217 [astro-ph.HE].
  • [101] J. Antoniadis et al. (EPTA), Astron. Astrophys. 678, A50 (2023), arXiv:2306.16214 [astro-ph.HE].
  • [102] J. Antoniadis et al. (EPTA), (2023), 10.1051/0004-6361/202346841, arXiv:2306.16224 [astroph.HE].
  • [103] J. Antoniadis et al. (EPTA), (2023), arXiv:2306.16227 [astro-ph.CO].
  • [104] A. Zic et al., (2023), arXiv:2306.16230 [astro-ph.HE].
  • [105] D. J. Reardon et al., Astrophys. J. Lett. 951, L6 (2023), arXiv:2306.16215 [astro-ph.HE].
  • [106] D. J. Reardon et al., Astrophys. J. Lett. 951, L7 (2023), arXiv:2306.16229 [astro-ph.HE].
  • [107] H. Xu et al., Res. Astron. Astrophys. 23, 075024 (2023), arXiv:2306.16216 [astro-ph.HE].
  • [108] G. Agazie et al. (International Pulsar Timing Array), (2023), arXiv:2309.00693 [astro-ph.HE].
  • [109] A. Afzal et al. (NANOGrav), Astrophys. J. Lett. 951, L11 (2023), arXiv:2306.16219 [astro-ph.HE].
  • [110] G. Franciolini, A. Iovino, Junior., V. Vaskonen, and H. Veermae, Phys. Rev. Lett. 131, 201401 (2023), arXiv:2306.17149 [astro-ph.CO].
  • [111] L. Liu, Z.-C. Chen, and Q.-G. Huang, (2023), arXiv:2307.01102 [astro-ph.CO].
  • [112] C. Chen, A. Ghoshal, Z. Lalak, Y. Luo, and A. Naskar, JCAP 08, 041 (2023), arXiv:2305.12325 [astro-ph.CO].
  • [113] Y. Ali-Haı̈moud, Phys. Rev. Lett. 121, 081304 (2018), arXiv:1805.05912 [astro-ph.CO].
  • [114] V. Desjacques and A. Riotto, Phys. Rev. D 98, 123533 (2018), arXiv:1806.10414 [astro-ph.CO].
  • [115] S. Young and C. T. Byrnes, JCAP 03, 004 (2020), arXiv:1910.06077 [astro-ph.CO].
  • [116] T. Suyama and S. Yokoyama, PTEP 2019, 103E02 (2019), arXiv:1906.04958 [astro-ph.CO].
  • [117] V. De Luca, G. Franciolini, and A. Riotto, Phys. Rev. D 104, 063526 (2021), arXiv:2103.16369 [astro-ph.CO].
  • [118] S. Clesse and J. Garcı́a-Bellido, Phys. Dark Univ. 15, 142 (2017), arXiv:1603.05234 [astro-ph.CO].
  • [119] M. Raidal, V. Vaskonen, and H. Veermäe, JCAP 09, 037 (2017), arXiv:1707.01480 [astro-ph.CO].
  • [120] J. Garcı́a-Bellido and S. Clesse, Phys. Dark Univ. 19, 144 (2018), arXiv:1710.04694 [astro-ph.CO].
  • [121] J. Kristiano and J. Yokoyama, (2022), arXiv:2211.03395 [hep-th].
  • [122] J. Kristiano and J. Yokoyama, (2023), arXiv:2303.00341 [hep-th].
  • [123] A. Riotto, (2023), arXiv:2303.01727 [astro-ph.CO].
  • [124] H. Firouzjahi, JCAP 10, 006 (2023), arXiv:2303.12025 [astro-ph.CO].
  • [125] H. Firouzjahi and A. Riotto, JCAP 02, 021 (2024), arXiv:2304.07801 [astro-ph.CO].
  • [126] H. Firouzjahi, Phys. Rev. D 109, 043514 (2024), arXiv:2311.04080 [astro-ph.CO].
  • [127] H. Motohashi and Y. Tada, JCAP 08, 069 (2023), arXiv:2303.16035 [astro-ph.CO].
  • [128] J. Fumagalli, (2023), arXiv:2305.19263 [astro-ph.CO].
  • [129] S.-L. Cheng, D.-S. Lee, and K.-W. Ng, (2023), arXiv:2305.16810 [astro-ph.CO].
  • [130] G. Franciolini, A. Iovino, Junior., M. Taoso, and A. Urbano, (2023), arXiv:2305.03491 [astro-ph.CO].
  • [131] J. Fumagalli, S. Bhattacharya, M. Peloso, S. Renaux-Petel, and L. T. Witkowski, (2023), arXiv:2307.08358 [astro-ph.CO].
  • [132] S. Maity, H. V. Ragavendra, S. K. Sethi, and L. Sriramkumar, (2023), arXiv:2307.13636 [astroph.CO].
  • [133] L. Iacconi, D. Mulryne, and D. Seery, (2023), arXiv:2312.12424 [astro-ph.CO].
  • [134] H. Firouzjahi, (2024), arXiv:2403.03841 [astro-ph.CO].
  • [135] B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, (2020), arXiv:2002.12778 [astro-ph.CO].
  • [136] B. Carr and F. Kuhnel, Ann. Rev. Nucl. Part. Sci. 70, 355 (2020), arXiv:2006.02838 [astro-ph.CO].
  • [137] B. Carr, S. Clesse, J. Garcia-Bellido, M. Hawkins, and F. Kuhnel, Phys. Rept. 1054, 1 (2024), arXiv:2306.03903 [astro-ph.CO].
  • [138] T. Harada, C.-M. Yoo, and K. Kohri, Phys. Rev. D 88, 084051 (2013), [Erratum: Phys.Rev.D 89, 029903 (2014)], arXiv:1309.4201 [astro-ph.CO].
  • [138] T. Harada, C.-M. Yoo, and K. Kohri, Phys. Rev. D 88, 084051 (2013), [Erratum: Phys.Rev.D 89, 029903 (2014)], arXiv:1309.4201 [astro-ph.CO].
  • [139] A. Escrivà, F. Kuhnel, and Y. Tada, (2022), arXiv:2211.05767 [astro-ph.CO].
  • [140] C. Germani and R. K. Sheth, Universe 9, 421 (2023), arXiv:2308.02971 [astro-ph.CO].
  • [141] T. Harada, C.-M. Yoo, T. Nakama, and Y. Koga, Phys. Rev. D 91, 084057 (2015), arXiv:1503.03934 [gr-qc].
  • [142] M. Kawasaki and H. Nakatsuka, Phys. Rev. D 99, 123501 (2019), arXiv:1903.02994 [astro-ph.CO].
  • [143] S. Young, I. Musco, and C. T. Byrnes, JCAP 11, 012 (2019), arXiv:1904.00984 [astro-ph.CO].
  • [144] V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto, and C. Ünal, JCAP 07, 048 (2019), arXiv:1904.00970 [astro-ph.CO].
  • [145] I. Musco, Phys. Rev. D 100, 123524 (2019), arXiv:1809.02127 [gr-qc].
  • [146] A. Escrivà, C. Germani, and R. K. Sheth, Phys. Rev. D 101, 044022 (2020), arXiv:1907.13311 [gr-qc].
  • [147] S. Young, Int. J. Mod. Phys. D 29, 2030002 (2019), arXiv:1905.01230 [astro-ph.CO].
  • [148] M. Kopp, S. Hofmann, and J. Weller, Phys. Rev. D 83, 124025 (2011), arXiv:1012.4369 [astroph.CO].
  • [149] R. Brout, F. Englert, and E. Gunzig, Annals Phys. 115, 78 (1978).
  • [150] A. H. Guth, Phys. Rev. D23, 347 (1981).
  • [151] A. A. Starobinsky, Phys. Lett. 91B, 99 (1980), [Adv. Ser. Astrophys. Cosmol.3,130(1987)].
  • [151] A. A. Starobinsky, Phys. Lett. 91B, 99 (1980), [Adv. Ser. Astrophys. Cosmol.3,130(1987)].
  • [152] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532 (1981), [Pisma Zh. Eksp. Teor. Fiz.33,549(1981)].
  • [152] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532 (1981), [Pisma Zh. Eksp. Teor. Fiz.33,549(1981)].
  • [153] A. D. Linde, Phys. Lett. B 108, 389 (1982).
  • [154] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
  • [155] V. F. Mukhanov, JETP Lett. 41, 493 (1985).
  • [156] M. Sasaki, Prog. Theor. Phys. 76, 1036 (1986).
  • [157] J. M. Maldacena, JHEP 05, 013 (2003), arXiv:astro-ph/0210603.
  • [158] P. Creminelli and M. Zaldarriaga, JCAP 10, 006 (2004), arXiv:astro-ph/0407059.
  • [159] Y. Akrami et al. (Planck), Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211 [astro-ph.CO].
  • [160] T. M. C. Abbott et al. (DES), Phys. Rev. D 105, 023520 (2022), arXiv:2105.13549 [astro-ph.CO].
  • [161] S. M. Leach, M. Sasaki, D. Wands, and A. R. Liddle, Phys. Rev. D 64, 023512 (2001), arXiv:astroph/0101406.
  • [162] C. T. Byrnes, P. S. Cole, and S. P. Patil, JCAP 06, 028 (2019), arXiv:1811.11158 [astro-ph.CO].
  • [163] P. S. Cole, A. D. Gow, C. T. Byrnes, and S. P. Patil, (2022), arXiv:2204.07573 [astro-ph.CO].
  • [164] M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95, 71 (1996), arXiv:astro-ph/9507001.
  • [165] D. Wands, K. A. Malik, D. H. Lyth, and A. R. Liddle, Phys. Rev. D 62, 043527 (2000), arXiv:astroph/0003278.
  • [166] D. H. Lyth, K. A. Malik, and M. Sasaki, JCAP 05, 004 (2005), arXiv:astro-ph/0411220.
  • [167] V. Vennin and D. Wands, (2024), arXiv:2402.12672 [astro-ph.CO].
  • [168] S. Pi and M. Sasaki, Phys. Rev. Lett. 131, 011002 (2023), arXiv:2211.13932 [astro-ph.CO].
  • [169] A. A. Starobinsky, JETP Lett. 55, 489 (1992).
  • [170] G. Domènech, G. Vargas, and T. Vargas, (2023), arXiv:2309.05750 [astro-ph.CO].
  • [171] J. H. P. Jackson, H. Assadullahi, A. D. Gow, K. Koyama, V. Vennin, and D. Wands, (2023), arXiv:2311.03281 [astro-ph.CO].
  • [172] Y.-F. Cai, X. Chen, M. H. Namjoo, M. Sasaki, D.-G. Wang, and Z. Wang, JCAP 05, 012 (2018), arXiv:1712.09998 [astro-ph.CO].
  • [173] M. Biagetti, G. Franciolini, A. Kehagias, and A. Riotto, JCAP 07, 032 (2018), arXiv:1804.07124 [astro-ph.CO].
  • [174] G. Tasinato, Phys. Rev. D 103, 023535 (2021), arXiv:2012.02518 [hep-th].
  • [175] S. Pi and J. Wang, JCAP 06, 018 (2023), arXiv:2209.14183 [astro-ph.CO].
  • [176] T. Moroi and T. Takahashi, Phys. Lett. B 522, 215 (2001), [Erratum: Phys.Lett.B 539, 303–303 (2002)], arXiv:hep-ph/0110096.
  • [176] T. Moroi and T. Takahashi, Phys. Lett. B 522, 215 (2001), [Erratum: Phys.Lett.B 539, 303–303 (2002)], arXiv:hep-ph/0110096.
  • [177] K. Enqvist and M. S. Sloth, Nucl. Phys. B 626, 395 (2002), arXiv:hep-ph/0109214.
  • [178] D. H. Lyth, C. Ungarelli, and D. Wands, Phys. Rev. D 67, 023503 (2003), arXiv:astro-ph/0208055.
  • [179] S. Kasuya and M. Kawasaki, Phys. Rev. D 80, 023516 (2009), arXiv:0904.3800 [astro-ph.CO].
  • [180] M. Kawasaki, N. Kitajima, and T. T. Yanagida, Phys. Rev. D 87, 063519 (2013), arXiv:1207.2550 [hep-ph].
  • [181] M. Kawasaki, N. Kitajima, and S. Yokoyama, JCAP 08, 042 (2013), arXiv:1305.4464 [astro-ph.CO].
  • [182] K. Ando, K. Inomata, M. Kawasaki, K. Mukaida, and T. T. Yanagida, Phys. Rev. D 97, 123512 (2018), arXiv:1711.08956 [astro-ph.CO].
  • [183] D.-S. Meng, C. Yuan, and Q.-G. Huang, Sci. China Phys. Mech. Astron. 66, 280411 (2023), arXiv:2212.03577 [astro-ph.CO].
  • [184] T. Suyama and J. Yokoyama, Phys. Rev. D 84, 083511 (2011), arXiv:1106.5983 [astro-ph.CO].
  • [185] G. Ferrante, G. Franciolini, A. Iovino, Junior., and A. Urbano, JCAP 06, 057 (2023), arXiv:2305.13382 [astro-ph.CO].
  • [186] A. D. Gow, T. Miranda, and S. Nurmi, JCAP 11, 006 (2023), arXiv:2307.03078 [astro-ph.CO].
  • [187] M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993).
  • [188] C. R. Evans and J. S. Coleman, Phys. Rev. Lett. 72, 1782 (1994), arXiv:gr-qc/9402041.
  • [189] T. Koike, T. Hara, and S. Adachi, Phys. Rev. Lett. 74, 5170 (1995), arXiv:gr-qc/9503007.
  • [190] J. C. Niemeyer and K. Jedamzik, Phys. Rev. Lett. 80, 5481 (1998), arXiv:astro-ph/9709072.
  • [191] I. Hawke and J. M. Stewart, Class. Quant. Grav. 19, 3687 (2002).
  • [192] I. Musco, J. C. Miller, and A. G. Polnarev, Class. Quant. Grav. 26, 235001 (2009), arXiv:0811.1452 [gr-qc].
  • [193] B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Phys. Rev. D 81, 104019 (2010), arXiv:0912.5297 [astro-ph.CO].
  • [194] L. T. Witkowski, G. Domènech, J. Fumagalli, and S. Renaux-Petel, JCAP 05, 028 (2022), arXiv:2110.09480 [astro-ph.CO].
  • [195] G. Domènech, Universe 7, 398 (2021), arXiv:2109.01398 [gr-qc].
  • [196] N. Bartolo et al., JCAP 12, 026 (2016), arXiv:1610.06481 [astro-ph.CO].
  • [197] G. Wang and W.-B. Han, (2021), arXiv:2108.11151 [gr-qc].
  • [198] K. Schmitz, JHEP 01, 097 (2021), arXiv:2002.04615 [hep-ph].
  • [199] L. T. Witkowski, (2022), arXiv:2209.05296 [astro-ph.CO].
  • [200] P. Amaro-Seoane et al., GW Notes 6, 4 (2013), arXiv:1201.3621 [astro-ph.CO].
  • [201] P. Amaro-Seoane et al., Class. Quant. Grav. 29, 124016 (2012), arXiv:1202.0839 [gr-qc].
  • [202] P. Amaro-Seoane et al. (LISA), (2017), arXiv:1702.00786 [astro-ph.IM].
  • [203] W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, Int. J. Mod. Phys. A 35, 2050075 (2020), arXiv:1807.09495 [gr-qc].
  • [204] J. Luo et al. (TianQin), Class. Quant. Grav. 33, 035010 (2016), arXiv:1512.02076 [astro-ph.IM].
  • [205] K. Kohri and T. Terada, Phys. Rev. D 97, 123532 (2018), arXiv:1804.08577 [gr-qc].
  • [206] J. Crowder and N. J. Cornish, Phys. Rev. D 72, 083005 (2005), arXiv:gr-qc/0506015.
  • [207] V. Corbin and N. J. Cornish, Class. Quant. Grav. 23, 2435 (2006), arXiv:gr-qc/0512039.
  • [208] S. Kawamura et al., Class. Quant. Grav. 23, S125 (2006).
  • [209] S. Kawamura et al., Class. Quant. Grav. 28, 094011 (2011).
  • [210] R.-G. Cai, S. Pi, and M. Sasaki, (2019), arXiv:1909.13728 [astro-ph.CO].
  • [211] V. Atal and G. Domènech, (2021), arXiv:2103.01056 [astro-ph.CO].
  • [212] S. Shandera, A. L. Erickcek, P. Scott, and J. Y. Galarza, Phys. Rev. D 88, 103506 (2013), arXiv:1211.7361 [astro-ph.CO].
  • [213] T. Matsubara, T. Terada, K. Kohri, and S. Yokoyama, Phys. Rev. D 100, 123544 (2019), arXiv:1909.04053 [astro-ph.CO].
  • [214] H. V. Ragavendra, P. Saha, L. Sriramkumar, and J. Silk, (2020), arXiv:2008.12202 [astro-ph.CO].
  • [215] E. Silverstein and D. Tong, Phys. Rev. D 70, 103505 (2004), arXiv:hep-th/0310221.
  • [216] M. Alishahiha, E. Silverstein, and D. Tong, Phys. Rev. D 70, 123505 (2004), arXiv:hep-th/0404084.
  • [217] C. Armendariz-Picon, T. Damour, and V. F. Mukhanov, Phys. Lett. B 458, 209 (1999), arXiv:hepth/9904075.
  • [218] J. Garriga and V. F. Mukhanov, Phys. Lett. B 458, 219 (1999), arXiv:hep-th/9904176.
  • [219] T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Phys. Rev. Lett. 105, 231302 (2010), arXiv:1008.0603 [hep-th].
  • [220] D. Artigas, S. Pi, and T. Tanaka, To appear.
  • [221] K. Uehara, A. Escrivà, T. Harada, D. Saito, and C.-M. Yoo, (2024), arXiv:2401.06329 [gr-qc].
  • [222] Y.-F. Cai, X.-H. Ma, M. Sasaki, D.-G. Wang, and Z. Zhou, Phys. Lett. B 834, 137461 (2022), arXiv:2112.13836 [astro-ph.CO].
  • [223] Y.-F. Cai, X.-H. Ma, M. Sasaki, D.-G. Wang, and Z. Zhou, (2022), arXiv:2207.11910 [astro-ph.CO].
  • [224] R. Kawaguchi, T. Fujita, and M. Sasaki, JCAP 11, 021 (2023), arXiv:2305.18140 [astro-ph.CO].
  • [225] A. Escrivà, V. Atal, and J. Garriga, JCAP 10, 035 (2023), arXiv:2306.09990 [astro-ph.CO].
  • [226] S. Hooshangi, M. H. Namjoo, and M. Noorbala, Phys. Lett. B 834, 137400 (2022), arXiv:2112.04520 [astro-ph.CO].
  • [227] S. Hooshangi, M. H. Namjoo, and M. Noorbala, JCAP 09, 023 (2023), arXiv:2305.19257 [astroph.CO].
  • [228] M. Celoria, P. Creminelli, G. Tambalo, and V. Yingcharoenrat, JCAP 06, 051 (2021), arXiv:2103.09244 [hep-th].
  • [229] A. A. Starobinsky, Lect. Notes Phys. 246, 107 (1986).
  • [230] A. A. Starobinsky, Phys. Lett. B 117, 175 (1982).
  • [231] A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50, 6357 (1994), arXiv:astro-ph/9407016.
  • [232] V. Vennin and A. A. Starobinsky, Eur. Phys. J. C 75, 413 (2015), arXiv:1506.04732 [hep-th].
  • [233] C. Pattison, V. Vennin, H. Assadullahi, and D. Wands, JCAP 10, 046 (2017), arXiv:1707.00537 [hep-th].
  • [234] J. M. Ezquiaga, J. Garcı́a-Bellido, and V. Vennin, JCAP 03, 029 (2020), arXiv:1912.05399 [astroph.CO].
  • [235] V. Vennin, Stochastic inflation and primordial black holes, Other thesis (2020), arXiv:2009.08715 [astro-ph.CO].
  • [236] D. G. Figueroa, S. Raatikainen, S. Rasanen, and E. Tomberg, Phys. Rev. Lett. 127, 101302 (2021), arXiv:2012.06551 [astro-ph.CO].
  • [237] C. Pattison, V. Vennin, D. Wands, and H. Assadullahi, JCAP 04, 080 (2021), arXiv:2101.05741 [astro-ph.CO].
  • [238] D. G. Figueroa, S. Raatikainen, S. Rasanen, and E. Tomberg, JCAP 05, 027 (2022), arXiv:2111.07437 [astro-ph.CO].
  • [239] C. Animali and V. Vennin, (2022), arXiv:2210.03812 [astro-ph.CO].
  • [240] J. M. Ezquiaga, J. Garcı́a-Bellido, and V. Vennin, Phys. Rev. Lett. 130, 121003 (2023), arXiv:2207.06317 [astro-ph.CO].