Primordial Black Hole from Tensor-induced Density Fluctuation: First-order Phase Transitions and Domain Walls

Author(s)

Kumar, Utkarsh, Ghoshal, Anish

Abstract

We present a novel \textit{gauge-invariant and minimal} formation mechanism of primordial black holes (PBHs) in first-order phase transition (FOPT) and domain walls (DW) separately. This is based on the first-order tensor perturbations, generated during FOPT from bubble collisions & sound waves, and from DW annihilation, sourcing curvature, at second-order in perturbation theory. We show that the PBH formation implies \textit{model-independent constraints} on FOPT parameters ($β/H, α, T_{\star}$ ) and on DW parameters, ($α_{\rm ann}, V_{\rm bias}, σ$), from existing PBH constraints. We find that asteroid mass PBHs can become the entire dark matter (DM) of the Universe, for $T_{\star} \in (4 \times 10^{2}, 10^{4})$ GeV, for $β/H \simeq 6$, involving $α>\mathcal{O}(1)$ values. The corresponding FOPT Gravitational Waves (GW) amplitude will have its characteristic peak at $Ω_{\rm GW}^{\rm p} h^2$$\sim \mathcal{O}(10^{-8})$ between frequencies $f_{\rm p} \in ({10^{-5},10^{-2}})$ Hz which is within the reach in LISA and SKA detectors. PBH as entire DM is possible for $σ^{1/3} \in [10^{6}, 10^{8}]$ TeV, for $V_{\rm bias}^{1/4} \in [10^7, 10^{10}]$ MeV with the corresponding GW amplitude peak from DW annihilation $Ω_{\rm GW}^{\rm p} h^2$$\sim \mathcal{O}(10^{-9})$ (for $α_{\rm ann} \sim 10^{-2}$) and peak frequencies between $f_{\rm p} \in (4 \times {10^{-4},10^{-1}})$ Hz with ($T_{\rm ann} \in 4.5 \times [10^3, 10^6] $) GeV within the reach in LISA and ET detectors. We also provide semi-analytical formulae for the tensor-induced density spectrum, $P_{δ^{(2)}}$, $M_{\rm PBH}$ and $f_{\rm PBH}$, relating them in terms of FOPT and DW parameters which in turn, are related to viable particle physics origin of such FOPT and DW, and therefore, constrain such microphysics, either in the visible, or in dark sector models.

Figures

\it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from FOPT (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various FOPT parameters values $\alpha$, $\beta / H$ and $T_{\star}$ on their respective induced curvature spectra. We have used $a=b=2.4$, and $c=4$ (see \cref{eq:ind_PT_ana}) to obtain these results. \textbf{Upper left panel} shows the variation of $\alpha$ with fixed $\beta / H = 5$ and $T_{\star} = 10^4$ GeV, \textbf{upper right panel} shows the variation of $\beta / H$ with fixed $\alpha = 10^2$ and $T_{\star} = 10^4$ GeV, \textbf{lower panel} shows the variation of $T_{\star}$ with fixed $\alpha = 10^2$ and $\beta / H = 5$.
Caption \it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from FOPT (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various FOPT parameters values $\alpha$, $\beta / H$ and $T_{\star}$ on their respective induced curvature spectra. We have used $a=b=2.4$, and $c=4$ (see \cref{eq:ind_PT_ana}) to obtain these results. \textbf{Upper left panel} shows the variation of $\alpha$ with fixed $\beta / H = 5$ and $T_{\star} = 10^4$ GeV, \textbf{upper right panel} shows the variation of $\beta / H$ with fixed $\alpha = 10^2$ and $T_{\star} = 10^4$ GeV, \textbf{lower panel} shows the variation of $T_{\star}$ with fixed $\alpha = 10^2$ and $\beta / H = 5$.
\it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from FOPT (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various FOPT parameters values $\alpha$, $\beta / H$ and $T_{\star}$ on their respective induced curvature spectra. We have used $a=b=2.4$, and $c=4$ (see \cref{eq:ind_PT_ana}) to obtain these results. \textbf{Upper left panel} shows the variation of $\alpha$ with fixed $\beta / H = 5$ and $T_{\star} = 10^4$ GeV, \textbf{upper right panel} shows the variation of $\beta / H$ with fixed $\alpha = 10^2$ and $T_{\star} = 10^4$ GeV, \textbf{lower panel} shows the variation of $T_{\star}$ with fixed $\alpha = 10^2$ and $\beta / H = 5$.
Caption \it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from FOPT (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various FOPT parameters values $\alpha$, $\beta / H$ and $T_{\star}$ on their respective induced curvature spectra. We have used $a=b=2.4$, and $c=4$ (see \cref{eq:ind_PT_ana}) to obtain these results. \textbf{Upper left panel} shows the variation of $\alpha$ with fixed $\beta / H = 5$ and $T_{\star} = 10^4$ GeV, \textbf{upper right panel} shows the variation of $\beta / H$ with fixed $\alpha = 10^2$ and $T_{\star} = 10^4$ GeV, \textbf{lower panel} shows the variation of $T_{\star}$ with fixed $\alpha = 10^2$ and $\beta / H = 5$.
\it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from FOPT (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various FOPT parameters values $\alpha$, $\beta / H$ and $T_{\star}$ on their respective induced curvature spectra. We have used $a=b=2.4$, and $c=4$ (see \cref{eq:ind_PT_ana}) to obtain these results. \textbf{Upper left panel} shows the variation of $\alpha$ with fixed $\beta / H = 5$ and $T_{\star} = 10^4$ GeV, \textbf{upper right panel} shows the variation of $\beta / H$ with fixed $\alpha = 10^2$ and $T_{\star} = 10^4$ GeV, \textbf{lower panel} shows the variation of $T_{\star}$ with fixed $\alpha = 10^2$ and $\beta / H = 5$.
Caption \it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from FOPT (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various FOPT parameters values $\alpha$, $\beta / H$ and $T_{\star}$ on their respective induced curvature spectra. We have used $a=b=2.4$, and $c=4$ (see \cref{eq:ind_PT_ana}) to obtain these results. \textbf{Upper left panel} shows the variation of $\alpha$ with fixed $\beta / H = 5$ and $T_{\star} = 10^4$ GeV, \textbf{upper right panel} shows the variation of $\beta / H$ with fixed $\alpha = 10^2$ and $T_{\star} = 10^4$ GeV, \textbf{lower panel} shows the variation of $T_{\star}$ with fixed $\alpha = 10^2$ and $\beta / H = 5$.
\it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from DW (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various DW parameters values $\alpha_{\rm ann}$ and $T_{\rm ann}$ ($ \propto V_{\rm bias} ^{1/4} / \sigma^{1/3}$) on their respective induced curvature spectra. In the \textbf{left panel}, we show the variation of $\alpha_{\rm ann}$ with fixed $T_{\rm ann} = 10^5$ GeV; in the \textbf{right panel}, we show the variation of $T_{\rm ann}$ with fixed $\alpha_{\rm ann} = 10^{-3}$.
Caption \it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from DW (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various DW parameters values $\alpha_{\rm ann}$ and $T_{\rm ann}$ ($ \propto V_{\rm bias} ^{1/4} / \sigma^{1/3}$) on their respective induced curvature spectra. In the \textbf{left panel}, we show the variation of $\alpha_{\rm ann}$ with fixed $T_{\rm ann} = 10^5$ GeV; in the \textbf{right panel}, we show the variation of $T_{\rm ann}$ with fixed $\alpha_{\rm ann} = 10^{-3}$.
\it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from DW (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various DW parameters values $\alpha_{\rm ann}$ and $T_{\rm ann}$ ($ \propto V_{\rm bias} ^{1/4} / \sigma^{1/3}$) on their respective induced curvature spectra. In the \textbf{left panel}, we show the variation of $\alpha_{\rm ann}$ with fixed $T_{\rm ann} = 10^5$ GeV; in the \textbf{right panel}, we show the variation of $T_{\rm ann}$ with fixed $\alpha_{\rm ann} = 10^{-3}$.
Caption \it Plot of the gravitational wave energy density spectrum $\Omega_{\rm GW}$ from DW (dashed) and its total curvature power spectrum including the induced curvature perturbations ($\mathcal{P}_{\zeta} (k)$) with respect to wavenumber $k$ (related to frequency f via k $= 2 \pi$ f) of the GWs. We present the impact of various DW parameters values $\alpha_{\rm ann}$ and $T_{\rm ann}$ ($ \propto V_{\rm bias} ^{1/4} / \sigma^{1/3}$) on their respective induced curvature spectra. In the \textbf{left panel}, we show the variation of $\alpha_{\rm ann}$ with fixed $T_{\rm ann} = 10^5$ GeV; in the \textbf{right panel}, we show the variation of $T_{\rm ann}$ with fixed $\alpha_{\rm ann} = 10^{-3}$.
\it We compare the scale dependence of the induced density power spectra obtained numerical integration of \cref{eq:PSDf} with the analytical approximations given in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for FOPT and DW scenarios in \textbf{left} and \textbf{right} panels respectively. The parameters used for the FOPT scenario are $\alpha = 10^{3}$, $\beta/H = 5.33$, and $T_{\star} = 10^{5} \,{\rm GeV}$ with $a=b=2.4$, and $c=4$. For the DW scenario, we have used $\alpha_{\rm ann} = 10^{-2}$, $\epsilon = 0.7$, and $T_{\rm ann} = 10^{5} \, {\rm GeV}$ with $\epsilon_{\rm sim} = 3$. To understand the scale dependence of the induced spectrum \textbf{(solid blue)}, we compare it with square of the initial tensor spectrum, $\mathcal{P}_{\chi_{\rm ini}}^2$ for both scenarios \textbf{(solid orange)}. Furthermore, we have explicitly shown the scale dependence given by the analytical approximations in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for both scenarios in \textbf{green dashed lines}.
Caption \it We compare the scale dependence of the induced density power spectra obtained numerical integration of \cref{eq:PSDf} with the analytical approximations given in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for FOPT and DW scenarios in \textbf{left} and \textbf{right} panels respectively. The parameters used for the FOPT scenario are $\alpha = 10^{3}$, $\beta/H = 5.33$, and $T_{\star} = 10^{5} \,{\rm GeV}$ with $a=b=2.4$, and $c=4$. For the DW scenario, we have used $\alpha_{\rm ann} = 10^{-2}$, $\epsilon = 0.7$, and $T_{\rm ann} = 10^{5} \, {\rm GeV}$ with $\epsilon_{\rm sim} = 3$. To understand the scale dependence of the induced spectrum \textbf{(solid blue)}, we compare it with square of the initial tensor spectrum, $\mathcal{P}_{\chi_{\rm ini}}^2$ for both scenarios \textbf{(solid orange)}. Furthermore, we have explicitly shown the scale dependence given by the analytical approximations in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for both scenarios in \textbf{green dashed lines}.
\it We compare the scale dependence of the induced density power spectra obtained numerical integration of \cref{eq:PSDf} with the analytical approximations given in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for FOPT and DW scenarios in \textbf{left} and \textbf{right} panels respectively. The parameters used for the FOPT scenario are $\alpha = 10^{3}$, $\beta/H = 5.33$, and $T_{\star} = 10^{5} \,{\rm GeV}$ with $a=b=2.4$, and $c=4$. For the DW scenario, we have used $\alpha_{\rm ann} = 10^{-2}$, $\epsilon = 0.7$, and $T_{\rm ann} = 10^{5} \, {\rm GeV}$ with $\epsilon_{\rm sim} = 3$. To understand the scale dependence of the induced spectrum \textbf{(solid blue)}, we compare it with square of the initial tensor spectrum, $\mathcal{P}_{\chi_{\rm ini}}^2$ for both scenarios \textbf{(solid orange)}. Furthermore, we have explicitly shown the scale dependence given by the analytical approximations in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for both scenarios in \textbf{green dashed lines}.
Caption \it We compare the scale dependence of the induced density power spectra obtained numerical integration of \cref{eq:PSDf} with the analytical approximations given in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for FOPT and DW scenarios in \textbf{left} and \textbf{right} panels respectively. The parameters used for the FOPT scenario are $\alpha = 10^{3}$, $\beta/H = 5.33$, and $T_{\star} = 10^{5} \,{\rm GeV}$ with $a=b=2.4$, and $c=4$. For the DW scenario, we have used $\alpha_{\rm ann} = 10^{-2}$, $\epsilon = 0.7$, and $T_{\rm ann} = 10^{5} \, {\rm GeV}$ with $\epsilon_{\rm sim} = 3$. To understand the scale dependence of the induced spectrum \textbf{(solid blue)}, we compare it with square of the initial tensor spectrum, $\mathcal{P}_{\chi_{\rm ini}}^2$ for both scenarios \textbf{(solid orange)}. Furthermore, we have explicitly shown the scale dependence given by the analytical approximations in \cref{eq:ind_PT_ana,eq:ind_DW_ana} for both scenarios in \textbf{green dashed lines}.
\it Induced scalar power spectrum obtained for certain choices of phase transition transition parameters $T_{\star}, \beta/H$ and $\alpha$ and DW parameters $\alpha_{\rm ann}, \epsilon$. The shaded area represents constraints from current (solid line) and future (dashed line) experiments.
Caption \it Induced scalar power spectrum obtained for certain choices of phase transition transition parameters $T_{\star}, \beta/H$ and $\alpha$ and DW parameters $\alpha_{\rm ann}, \epsilon$. The shaded area represents constraints from current (solid line) and future (dashed line) experiments.
\it Induced scalar power spectrum obtained for certain choices of phase transition transition parameters $T_{\star}, \beta/H$ and $\alpha$ and DW parameters $\alpha_{\rm ann}, \epsilon$. The shaded area represents constraints from current (solid line) and future (dashed line) experiments.
Caption \it Induced scalar power spectrum obtained for certain choices of phase transition transition parameters $T_{\star}, \beta/H$ and $\alpha$ and DW parameters $\alpha_{\rm ann}, \epsilon$. The shaded area represents constraints from current (solid line) and future (dashed line) experiments.
\it Induced scalar power spectrum obtained for certain choices of phase transition transition parameters $T_{\star}, \beta/H$ and $\alpha$ and DW parameters $\alpha_{\rm ann}, \epsilon$. The shaded area represents constraints from current (solid line) and future (dashed line) experiments.
Caption \it Induced scalar power spectrum obtained for certain choices of phase transition transition parameters $T_{\star}, \beta/H$ and $\alpha$ and DW parameters $\alpha_{\rm ann}, \epsilon$. The shaded area represents constraints from current (solid line) and future (dashed line) experiments.
\it Summary of the PBH constraints. The colored shaded areas are excluded by BBN, CMB, cosmic rays, microlensing and GW observations, as discussed in the text. The future sensitivities of NGRST, LISA and ET are shown by the dotted curves. The FOPT parameters corresponding to each PBH mass and spin is shown in Table \ref{tab:PT} as per the benchmark points A, B, C and D. The green shaded region on the right edge with $T_{\star} \lesssim 10$ MeV is also excluded by BBN. To note that mass of PBH formed M is directly correlated to $T_{\star}$, as shown in the top of X-axis. Larger the M, smaller the $T_{\star}$ as expected since larger $T_{\star}$ denotes the smaller size of the Universe when the PBH is formed. A and B can be the entire DM candidate of the Universe while C and D will be tested in future experiments.
Caption \it Summary of the PBH constraints. The colored shaded areas are excluded by BBN, CMB, cosmic rays, microlensing and GW observations, as discussed in the text. The future sensitivities of NGRST, LISA and ET are shown by the dotted curves. The FOPT parameters corresponding to each PBH mass and spin is shown in Table \ref{tab:PT} as per the benchmark points A, B, C and D. The green shaded region on the right edge with $T_{\star} \lesssim 10$ MeV is also excluded by BBN. To note that mass of PBH formed M is directly correlated to $T_{\star}$, as shown in the top of X-axis. Larger the M, smaller the $T_{\star}$ as expected since larger $T_{\star}$ denotes the smaller size of the Universe when the PBH is formed. A and B can be the entire DM candidate of the Universe while C and D will be tested in future experiments.
\it Summary of the PBH constraints. The colored shaded areas are excluded by BBN, CMB, cosmic rays, microlensing and GW observations, as discussed in the text. The future sensitivities of NGRST, LISA and ET are shown by the dotted curves. The domain wall parameters corresponding to each PBH mass and spin is shown in Table \ref{tab:DW} as per the benchmark points E, F, G and H. The green shaded region on the right edge with $T_{\rm ann} \lesssim 10$ MeV is also excluded by BBN. To note that mass of PBH formed M is inversely proportional to $V_{\rm bias} / \sigma$, as shown in the top of X-axis. Larger the M, smaller the $V_{\rm bias} / \sigma$ as expected since larger bias and surface tension make the collapse happen earlier in cosmic history when the smaller size of the Universe leads to smaller PBH is formed. E and F can be the entire DM candidate of the Universe while G and H will be tested in future experiments.
Caption \it Summary of the PBH constraints. The colored shaded areas are excluded by BBN, CMB, cosmic rays, microlensing and GW observations, as discussed in the text. The future sensitivities of NGRST, LISA and ET are shown by the dotted curves. The domain wall parameters corresponding to each PBH mass and spin is shown in Table \ref{tab:DW} as per the benchmark points E, F, G and H. The green shaded region on the right edge with $T_{\rm ann} \lesssim 10$ MeV is also excluded by BBN. To note that mass of PBH formed M is inversely proportional to $V_{\rm bias} / \sigma$, as shown in the top of X-axis. Larger the M, smaller the $V_{\rm bias} / \sigma$ as expected since larger bias and surface tension make the collapse happen earlier in cosmic history when the smaller size of the Universe leads to smaller PBH is formed. E and F can be the entire DM candidate of the Universe while G and H will be tested in future experiments.
\it Exclusion regions on FOPT parameters involving its rate of FOPT $\beta/H$ versus $T_{\star}$, also showing the impact of $\alpha$. PBHs are produced corresponding to temperatures $T_{\star}$ shown on the bottom x-axis with the PBH mass shown on the top x-axis. Benchmark points corresponding to \cref{tab:PT} are also shown. See text for details.
Caption \it Exclusion regions on FOPT parameters involving its rate of FOPT $\beta/H$ versus $T_{\star}$, also showing the impact of $\alpha$. PBHs are produced corresponding to temperatures $T_{\star}$ shown on the bottom x-axis with the PBH mass shown on the top x-axis. Benchmark points corresponding to \cref{tab:PT} are also shown. See text for details.
\it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of FOPT parameters, following Eqns \eqref{eq:fpbh_fopt} and \eqref{eq:mpbh_fopt}.
Caption \it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of FOPT parameters, following Eqns \eqref{eq:fpbh_fopt} and \eqref{eq:mpbh_fopt}.
\it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of FOPT parameters, following Eqns \eqref{eq:fpbh_fopt} and \eqref{eq:mpbh_fopt}.
Caption \it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of FOPT parameters, following Eqns \eqref{eq:fpbh_fopt} and \eqref{eq:mpbh_fopt}.
\it Exclusion regions involving DW network parameter space occupying an energy fraction $\alpha_{\rm ann}$ at the onset of the annihilation phase driven by a bias energy density difference $V_{\rm bias}$ between distinct vacua with surface tension $\sigma$. PBHs are produced corresponding to temperatures $T_{\rm ann}$ , ($(V_{\rm bias} / \sigma)^{1/2}$) shown on the bottom x-axis with the PBH mass shown on the top x-axis. We present 3 representative values of DW simulation parameter $\epsilon_{\rm sim}$ to show its impact. Benchmark points corresponding to \cref{tab:DW} are also shown. See text for details.
Caption \it Exclusion regions involving DW network parameter space occupying an energy fraction $\alpha_{\rm ann}$ at the onset of the annihilation phase driven by a bias energy density difference $V_{\rm bias}$ between distinct vacua with surface tension $\sigma$. PBHs are produced corresponding to temperatures $T_{\rm ann}$ , ($(V_{\rm bias} / \sigma)^{1/2}$) shown on the bottom x-axis with the PBH mass shown on the top x-axis. We present 3 representative values of DW simulation parameter $\epsilon_{\rm sim}$ to show its impact. Benchmark points corresponding to \cref{tab:DW} are also shown. See text for details.
\it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of DW parameters, following Eqns \eqref{eq:fpbh_dw} and \eqref{eq:mpbh_dw}.
Caption \it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of DW parameters, following Eqns \eqref{eq:fpbh_dw} and \eqref{eq:mpbh_dw}.
\it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of DW parameters, following Eqns \eqref{eq:fpbh_dw} and \eqref{eq:mpbh_dw}.
Caption \it Comparison between numerical and semi-analytical estimation of fractional abundance of PBH estimation in terms of DW parameters, following Eqns \eqref{eq:fpbh_dw} and \eqref{eq:mpbh_dw}.
\it Plot showing the entire range of FOPT parameter space involving $\beta / H$ and $T_{\rm \star}$. Same as in Fig. \ref{fig:PT_exclusion} with now showing the SGWB regions where the GW detectors will be able to detect a signal with SNR $>$ 10. See text for details. Observability of stochastic GW background (SGWB) produced by FOPT bubbles \& sound waves (shaded regions) compared to PBH abundance. The orange line shows advanced LIGO run $O_3$ from LIGO-Virgo-KAGRA (LVK) \cite{KAGRA:2021kbb}, assuming the Signal-to-Noise Ratio detection thresholds $\rm SNR =2$. The shaded regions indicate the future prospects from SKA (\textbf{magenta}) \cite{Janssen:2014dka}, LISA (\textbf{red}) \cite{Audley:2017drz,Robson:2018ifk,LISACosmologyWorkingGroup:2022jok} and ET/CE (\textbf{blue/maroon})\cite{Punturo:2010zz,Maggiore:2019uih,Reitze:2019iox}. The regions labelled ``BBN'' in rule out from excessive dark radiation production~\ref{fig:DW_exclusion}.
Caption \it Plot showing the entire range of FOPT parameter space involving $\beta / H$ and $T_{\rm \star}$. Same as in Fig. \ref{fig:PT_exclusion} with now showing the SGWB regions where the GW detectors will be able to detect a signal with SNR $>$ 10. See text for details. Observability of stochastic GW background (SGWB) produced by FOPT bubbles \& sound waves (shaded regions) compared to PBH abundance. The orange line shows advanced LIGO run $O_3$ from LIGO-Virgo-KAGRA (LVK) \cite{KAGRA:2021kbb}, assuming the Signal-to-Noise Ratio detection thresholds $\rm SNR =2$. The shaded regions indicate the future prospects from SKA (\textbf{magenta}) \cite{Janssen:2014dka}, LISA (\textbf{red}) \cite{Audley:2017drz,Robson:2018ifk,LISACosmologyWorkingGroup:2022jok} and ET/CE (\textbf{blue/maroon})\cite{Punturo:2010zz,Maggiore:2019uih,Reitze:2019iox}. The regions labelled ``BBN'' in rule out from excessive dark radiation production~\ref{fig:DW_exclusion}.
\it Plot showing the entire range of DW parameter space involving $\alpha_{\rm ann}$ and $(\frac{V_{\rm bias}}{\sigma})^{1/2}$ Same as in Fig. \ref{fig:DW_exclusion} with now showing the SGWB regions whee the GW detectors will be able to detect a signal with SNR $>$ 10. See text for details. Observability of stochastic GW background (SGWB) produced by annihilating DW networks (shaded regions) compared to PBH abundance. The orange line shows advanced LIGO run $O_3$ from LIGO-Virgo-KAGRA (LVK) \cite{KAGRA:2021kbb}, assuming the Signal-to-Noise Ratio detection thresholds $\rm SNR =2$. The shaded regions indicate the future prospects from SKA (\textbf{magenta}) \cite{Janssen:2014dka}, LISA (\textbf{red}) \cite{Audley:2017drz,Robson:2018ifk,LISACosmologyWorkingGroup:2022jok} and ET/CE (\textbf{blue/maroon})\cite{Punturo:2010zz,Maggiore:2019uih,Reitze:2019iox}. The black solid, dashed and dotted lines represent thickness of the brown bands represents the uncertainty on $\epsilon_{\rm sim}$ of the DW network. The regions labelled ``No Domain'' and ``DW domination'' are the same as in Fig.~\ref{fig:DW_exclusion}.
Caption \it Plot showing the entire range of DW parameter space involving $\alpha_{\rm ann}$ and $(\frac{V_{\rm bias}}{\sigma})^{1/2}$ Same as in Fig. \ref{fig:DW_exclusion} with now showing the SGWB regions whee the GW detectors will be able to detect a signal with SNR $>$ 10. See text for details. Observability of stochastic GW background (SGWB) produced by annihilating DW networks (shaded regions) compared to PBH abundance. The orange line shows advanced LIGO run $O_3$ from LIGO-Virgo-KAGRA (LVK) \cite{KAGRA:2021kbb}, assuming the Signal-to-Noise Ratio detection thresholds $\rm SNR =2$. The shaded regions indicate the future prospects from SKA (\textbf{magenta}) \cite{Janssen:2014dka}, LISA (\textbf{red}) \cite{Audley:2017drz,Robson:2018ifk,LISACosmologyWorkingGroup:2022jok} and ET/CE (\textbf{blue/maroon})\cite{Punturo:2010zz,Maggiore:2019uih,Reitze:2019iox}. The black solid, dashed and dotted lines represent thickness of the brown bands represents the uncertainty on $\epsilon_{\rm sim}$ of the DW network. The regions labelled ``No Domain'' and ``DW domination'' are the same as in Fig.~\ref{fig:DW_exclusion}.
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