Gravitational waves from inflation in LISA: reconstruction pipeline and physics interpretation
- Braglia, Matteo et al
- arXiv:2407.04356LISA-COSWG-24-03CERN-TH-2024-072
 | \includegraphics[width = 0.2 \textwidth]{preprintlogo_infl_parameter_reconstruction.png} |
 | \small Functions $\mathcal{C}_{0,1,2}(\omega_\textrm{log})$ and $\theta_{\textrm{log},1,2}(\omega_\textrm{log})$ appearing in the template \eqref{eq:resonant-template-const} and as computed in ref.~\cite{Fumagalli:2021cel} for GWs induced during a period of radiation domination. |
 | \small Fisher forecast for the PL template. The color map shows the 68 \% CL errors on the amplitude (left) and spectral index (right) as a function of the value of the injected parameters. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $\sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The black and gray crosses display the benchmarks PL-BNK\_1 and PL-BNK\_2, respectively. We also highlight the line $n_T=2/3$, which corresponds to the spectral index of extra-galactic foregrounds, with a black dashed line. |
 | \small Fisher forecast for the PL template. The color map shows the 68 \% CL errors on the amplitude (left) and spectral index (right) as a function of the value of the injected parameters. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $\sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The black and gray crosses display the benchmarks PL-BNK\_1 and PL-BNK\_2, respectively. We also highlight the line $n_T=2/3$, which corresponds to the spectral index of extra-galactic foregrounds, with a black dashed line. |
 | \small \emph{Left panel}: 1D and 2D posterior distributions derived from the PL template reconstruction of the benchmark PL-BNK\_1 ($\{\log_{10}(h^2\,\Omega_*), n_T\}=\{-12.5,\,2.085\}$) in the presence of galactic and extragalactic foregrounds. The posteriors of foregrounds and noise parameters are omitted for clarity. In the corner plot, the gold and red dots, and corresponding vertical lines, show the injected parameters and their reconstructed mean values. The 68\% C.L.~(95\% C.L.) reconstruction region is displayed in dark (light) gray. The equivalent regions and 1D posteriors obtained with the Fisher approximation are in blue. The top-right inset visualises the injected and reconstructed signals, with 68 and 95 \% CL error bands. The error bands on the galactic foreground and instrumental noise are too small to be visible. The LISA PLS is plotted in solid black. \emph{Right panel}: Like the left panel but for PL-BNK\_2 ($\{\log_{10}(h^2\,\Omega_*), n_T\}=\{-11,\,0.77\}$). In the inset plots of both panels, the error bands on the galactic foreground and instrumental noise are too small to be visible. |
 | \small \emph{Left panel}: 1D and 2D posterior distributions derived from the PL template reconstruction of the benchmark PL-BNK\_1 ($\{\log_{10}(h^2\,\Omega_*), n_T\}=\{-12.5,\,2.085\}$) in the presence of galactic and extragalactic foregrounds. The posteriors of foregrounds and noise parameters are omitted for clarity. In the corner plot, the gold and red dots, and corresponding vertical lines, show the injected parameters and their reconstructed mean values. The 68\% C.L.~(95\% C.L.) reconstruction region is displayed in dark (light) gray. The equivalent regions and 1D posteriors obtained with the Fisher approximation are in blue. The top-right inset visualises the injected and reconstructed signals, with 68 and 95 \% CL error bands. The error bands on the galactic foreground and instrumental noise are too small to be visible. The LISA PLS is plotted in solid black. \emph{Right panel}: Like the left panel but for PL-BNK\_2 ($\{\log_{10}(h^2\,\Omega_*), n_T\}=\{-11,\,0.77\}$). In the inset plots of both panels, the error bands on the galactic foreground and instrumental noise are too small to be visible. |
 | \small \emph{Left panel}: 1D and 2D posterior distributions derived from the PL template reconstruction of the benchmark PL-BNK\_1 ($\{\log_{10}(h^2\,\Omega_*), n_T\}=\{-12.5,\,2.085\}$) in the presence of galactic and extragalactic foregrounds. The posteriors of foregrounds and noise parameters are omitted for clarity. In the corner plot, the gold and red dots, and corresponding vertical lines, show the injected parameters and their reconstructed mean values. The 68\% C.L.~(95\% C.L.) reconstruction region is displayed in dark (light) gray. The equivalent regions and 1D posteriors obtained with the Fisher approximation are in blue. The top-right inset visualises the injected and reconstructed signals, with 68 and 95 \% CL error bands. The error bands on the galactic foreground and instrumental noise are too small to be visible. The LISA PLS is plotted in solid black. \emph{Right panel}: Like the left panel but for PL-BNK\_2 ($\{\log_{10}(h^2\,\Omega_*), n_T\}=\{-11,\,0.77\}$). In the inset plots of both panels, the error bands on the galactic foreground and instrumental noise are too small to be visible.\small 1D and 2D posterior distributions derived from the LN template reconstruction of the benchmarks LN-BNK\_1 (left panel) and LN-BNK\_2 (right panel). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. LN-BNK\_1 and LN-BNK\_2 are defined as in \cref{eq:master_modelIII_bum} with $\{\log_{10}(h^2\,\Omega_*),\,\log_{10}\rho,\,f_*/{\rm mHz}\}$ set to $\{-9.9,\,-0.13,\, 11\}$ and $\{-11.8,\,-0.37,\,1\}$, respectively.\small 1D and 2D posterior distributions derived from the BPL template reconstruction of the benchmark BPL-BNK\_1. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. In addition, the purple contours display the 1D and 2D posteriors recovered by assuming the injected parameter $\delta=1$ to be known a priori. BPL-BNK\_1 is defined as in eq.~\eqref{eq:master_modelIV_stBBN} with $\log_{10}(h^2\,\Omega_*), f_* /{\rm mHz},\, n_{t,1},\, n_{t,2},\,\delta$ equal to $-10.5, 1,\, 4, 0,\,1$, respectively.\small 1D and 2D posterior distributions derived from the BPL template reconstruction of the benchmark BPL-BNK\_2. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. BPL-BNK\_2 is defined as in eq.~\eqref{eq:master_modelIV_stBBN} with $\log_{10}(h^2\,\Omega_*), f_* /{\rm mHz},\, n_{t,1},\, n_{t,2},\,\delta$ equal to $-9.3, 1,\, 2.65, -2.1,\,5.3$, respectively.\small 1D and 2D posterior distributions derived from the DP template reconstruction of the benchmarks DP-BNK\_1 (top left), DP-BNK\_2 (top right), DP-BNK\_3 (bottom left) and DP-BNK\_4 (bottom right) defined as in \cref{eq:templateDP} with $\{\log_{10}h^2\Omega_* ,\,f_*/\textrm{Hz}\}$ equal to $\{-9.35, 5\}$, $\{-11.35, 5\}$, $\{-9.35, 1\}$ and $\{-11.35, 1\}$, respectively, and $\{\beta , \,\kappa_1,\, \kappa_2,\, \rho, \,\gamma \} = \{0.242, \,0.456,\, 1.234,\, 0.08, \,6.91 \}$ fixed for all benchmarks. Lines' styles and color codes are as in \cref{fig:PL_MCMC}.\small 1D and 2D posterior distributions derived from the ES template reconstruction of the benchmarks ES-BNK\_1 (top left), ES-BNK\_2 (top right) and ES-BNK\_3 (bottom). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. ES-BNK\_1, ES-BNK\_2 and ES-BNK\_3 are defined as in \cref{templateexcited} with $\{\log_{10}h^2\Omega_*,\gamma_{\mathrm{ES}},\omega_{\mathrm{ES}}\mathrm{Hz}\}$ set at $\{-11,10,6 \cdot 10^{3} \}$, $\{-11.5,10,6 \cdot 10^{3} \}$ and $\{-11,10,1.2 \cdot 10^{3}\}$, respectively.\small 1D and 2D posterior distributions derived from the LO template reconstruction of the benchmarks LO-BNK\_1 (left) and LO-BNK\_2 (right). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. LO-BNK\_1 and LO-BNK\_2 are defined as in \cref{eq:sharp-template} with $\{f_*/ \mathrm{mHz},\,\log_{10}(\omega_{\mathrm{lin}} \,\mathrm{Hz}) \}= \{5,\,4\} $ and $\{1,\,4.7\}$ respectively, and fixed $\{\rho, {\cal A}_{\mathrm {lin}}\} = \{ 0.08, 0.3\}$. |
 | \small Fisher forecasts for the LN template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top panels), $f_*$ (central panels) and $\rho$ (bottom panels) as a function of the injected values of $f_*$, $h^2\Omega_*$ and $\rho$ set as specified in the axes and the title of each panel. SNR countour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The gray and black cross display the benchmarks LN-BNK\_1 and LN-BNK\_2 respectively. |
 | \small Fisher forecasts for the LN template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top panels), $f_*$ (central panels) and $\rho$ (bottom panels) as a function of the injected values of $f_*$, $h^2\Omega_*$ and $\rho$ set as specified in the axes and the title of each panel. SNR countour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The gray and black cross display the benchmarks LN-BNK\_1 and LN-BNK\_2 respectively. |
 | \small Fisher forecasts for the LN template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top panels), $f_*$ (central panels) and $\rho$ (bottom panels) as a function of the injected values of $f_*$, $h^2\Omega_*$ and $\rho$ set as specified in the axes and the title of each panel. SNR countour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The gray and black cross display the benchmarks LN-BNK\_1 and LN-BNK\_2 respectively. |
 | \small Fisher forecasts for the LN template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top panels), $f_*$ (central panels) and $\rho$ (bottom panels) as a function of the injected values of $f_*$, $h^2\Omega_*$ and $\rho$ set as specified in the axes and the title of each panel. SNR countour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The gray and black cross display the benchmarks LN-BNK\_1 and LN-BNK\_2 respectively. |
 | \small Fisher forecasts for the LN template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top panels), $f_*$ (central panels) and $\rho$ (bottom panels) as a function of the injected values of $f_*$, $h^2\Omega_*$ and $\rho$ set as specified in the axes and the title of each panel. SNR countour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The gray and black cross display the benchmarks LN-BNK\_1 and LN-BNK\_2 respectively. |
 | \small Fisher forecasts for the LN template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top panels), $f_*$ (central panels) and $\rho$ (bottom panels) as a function of the injected values of $f_*$, $h^2\Omega_*$ and $\rho$ set as specified in the axes and the title of each panel. SNR countour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (black) and in the presence (purple) of foregrounds. The gray and black cross display the benchmarks LN-BNK\_1 and LN-BNK\_2 respectively. |
 | \small 1D and 2D posterior distributions derived from the LN template reconstruction of the benchmarks LN-BNK\_1 (left panel) and LN-BNK\_2 (right panel). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. LN-BNK\_1 and LN-BNK\_2 are defined as in \cref{eq:master_modelIII_bum} with $\{\log_{10}(h^2\,\Omega_*),\,\log_{10}\rho,\,f_*/{\rm mHz}\}$ set to $\{-9.9,\,-0.13,\, 11\}$ and $\{-11.8,\,-0.37,\,1\}$, respectively. |
 | \small 1D and 2D posterior distributions derived from the LN template reconstruction of the benchmarks LN-BNK\_1 (left panel) and LN-BNK\_2 (right panel). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. LN-BNK\_1 and LN-BNK\_2 are defined as in \cref{eq:master_modelIII_bum} with $\{\log_{10}(h^2\,\Omega_*),\,\log_{10}\rho,\,f_*/{\rm mHz}\}$ set to $\{-9.9,\,-0.13,\, 11\}$ and $\{-11.8,\,-0.37,\,1\}$, respectively. |
 | \small Fisher forecasts for the BPL template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $n_{t,1}$ (central left), $n_{t,2}$ (central right) and $\delta$ (bottom) as a function of the injected values of $f_*$, $h^2\Omega_*$ specified in the axes and $n_{t,1}$, $n_{t,2}$ and $\delta$ specified in the title of each panel. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (magenta) and in the presence (red) of foregrounds. The green crosses display the benchmarks BPL-BNK\_1. |
 | \small Fisher forecasts for the BPL template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $n_{t,1}$ (central left), $n_{t,2}$ (central right) and $\delta$ (bottom) as a function of the injected values of $f_*$, $h^2\Omega_*$ specified in the axes and $n_{t,1}$, $n_{t,2}$ and $\delta$ specified in the title of each panel. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (magenta) and in the presence (red) of foregrounds. The green crosses display the benchmarks BPL-BNK\_1. |
 | \small Fisher forecasts for the BPL template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $n_{t,1}$ (central left), $n_{t,2}$ (central right) and $\delta$ (bottom) as a function of the injected values of $f_*$, $h^2\Omega_*$ specified in the axes and $n_{t,1}$, $n_{t,2}$ and $\delta$ specified in the title of each panel. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (magenta) and in the presence (red) of foregrounds. The green crosses display the benchmarks BPL-BNK\_1. |
 | \small Fisher forecasts for the BPL template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $n_{t,1}$ (central left), $n_{t,2}$ (central right) and $\delta$ (bottom) as a function of the injected values of $f_*$, $h^2\Omega_*$ specified in the axes and $n_{t,1}$, $n_{t,2}$ and $\delta$ specified in the title of each panel. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (magenta) and in the presence (red) of foregrounds. The green crosses display the benchmarks BPL-BNK\_1. |
 | \small 1D and 2D posterior distributions derived from the BPL template reconstruction of the benchmark BPL-BNK\_1. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. In addition, the purple contours display the 1D and 2D posteriors recovered by assuming the injected parameter $\delta=1$ to be known a priori. BPL-BNK\_1 is defined as in eq.~\eqref{eq:master_modelIV_stBBN} with $\log_{10}(h^2\,\Omega_*), f_* /{\rm mHz},\, n_{t,1},\, n_{t,2},\,\delta$ equal to $-10.5, 1,\, 4, 0,\,1$, respectively. |
 | \small 1D and 2D posterior distributions derived from the BPL template reconstruction of the benchmark BPL-BNK\_2. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. BPL-BNK\_2 is defined as in eq.~\eqref{eq:master_modelIV_stBBN} with $\log_{10}(h^2\,\Omega_*), f_* /{\rm mHz},\, n_{t,1},\, n_{t,2},\,\delta$ equal to $-9.3, 1,\, 2.65, -2.1,\,5.3$, respectively. |
 | \small Fisher forecasts for the DP template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $\beta$ (next-to-top left), $\kappa_1$ (next-to-top right), $\rho$ (next-to-bottom left), $\kappa_2$ (next-to-top right) and $\gamma$ (bottom) as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $\Delta$ and $\mathcal{A}_s$ (and $k_*$ adjusted to yield $f_*=5\,$mHz). SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The white crosses display the benchmarks DP-BNK\_1 and DP-BNK\_2. |
 | \small Fisher forecasts for the DP template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $\beta$ (next-to-top left), $\kappa_1$ (next-to-top right), $\rho$ (next-to-bottom left), $\kappa_2$ (next-to-top right) and $\gamma$ (bottom) as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $\Delta$ and $\mathcal{A}_s$ (and $k_*$ adjusted to yield $f_*=5\,$mHz). SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The white crosses display the benchmarks DP-BNK\_1 and DP-BNK\_2. |
 | \small Fisher forecasts for the DP template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $\beta$ (next-to-top left), $\kappa_1$ (next-to-top right), $\rho$ (next-to-bottom left), $\kappa_2$ (next-to-top right) and $\gamma$ (bottom) as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $\Delta$ and $\mathcal{A}_s$ (and $k_*$ adjusted to yield $f_*=5\,$mHz). SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The white crosses display the benchmarks DP-BNK\_1 and DP-BNK\_2. |
 | \small Fisher forecasts for the DP template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $\beta$ (next-to-top left), $\kappa_1$ (next-to-top right), $\rho$ (next-to-bottom left), $\kappa_2$ (next-to-top right) and $\gamma$ (bottom) as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $\Delta$ and $\mathcal{A}_s$ (and $k_*$ adjusted to yield $f_*=5\,$mHz). SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The white crosses display the benchmarks DP-BNK\_1 and DP-BNK\_2. |
 | \small Fisher forecasts for the DP template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $\beta$ (next-to-top left), $\kappa_1$ (next-to-top right), $\rho$ (next-to-bottom left), $\kappa_2$ (next-to-top right) and $\gamma$ (bottom) as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $\Delta$ and $\mathcal{A}_s$ (and $k_*$ adjusted to yield $f_*=5\,$mHz). SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The white crosses display the benchmarks DP-BNK\_1 and DP-BNK\_2. |
 | \small Fisher forecasts for the DP template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $\beta$ (next-to-top left), $\kappa_1$ (next-to-top right), $\rho$ (next-to-bottom left), $\kappa_2$ (next-to-top right) and $\gamma$ (bottom) as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $\Delta$ and $\mathcal{A}_s$ (and $k_*$ adjusted to yield $f_*=5\,$mHz). SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The white crosses display the benchmarks DP-BNK\_1 and DP-BNK\_2. |
 | \small Fisher forecasts for the DP template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $f_*$ (top right), $\beta$ (next-to-top left), $\kappa_1$ (next-to-top right), $\rho$ (next-to-bottom left), $\kappa_2$ (next-to-top right) and $\gamma$ (bottom) as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $\Delta$ and $\mathcal{A}_s$ (and $k_*$ adjusted to yield $f_*=5\,$mHz). SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The white crosses display the benchmarks DP-BNK\_1 and DP-BNK\_2. |
 | \small Like in \cref{Fisher_double} but with errors as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $k_*$ (expressed in terms of $f_*$) and $\mathcal{A}_s$ with fixed $\Delta =0.2$. |
 | \small Like in \cref{Fisher_double} but with errors as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $k_*$ (expressed in terms of $f_*$) and $\mathcal{A}_s$ with fixed $\Delta =0.2$. |
 | \small Like in \cref{Fisher_double} but with errors as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $k_*$ (expressed in terms of $f_*$) and $\mathcal{A}_s$ with fixed $\Delta =0.2$. |
 | \small Like in \cref{Fisher_double} but with errors as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $k_*$ (expressed in terms of $f_*$) and $\mathcal{A}_s$ with fixed $\Delta =0.2$. |
 | \small Like in \cref{Fisher_double} but with errors as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $k_*$ (expressed in terms of $f_*$) and $\mathcal{A}_s$ with fixed $\Delta =0.2$. |
 | \small Like in \cref{Fisher_double} but with errors as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $k_*$ (expressed in terms of $f_*$) and $\mathcal{A}_s$ with fixed $\Delta =0.2$. |
 | \small Like in \cref{Fisher_double} but with errors as a function of the $\mathcal{P}_\zeta^{\rm ln}$ parameters $k_*$ (expressed in terms of $f_*$) and $\mathcal{A}_s$ with fixed $\Delta =0.2$. |
 | \small 1D and 2D posterior distributions derived from the DP template reconstruction of the benchmarks DP-BNK\_1 (top left), DP-BNK\_2 (top right), DP-BNK\_3 (bottom left) and DP-BNK\_4 (bottom right) defined as in \cref{eq:templateDP} with $\{\log_{10}h^2\Omega_* ,\,f_*/\textrm{Hz}\}$ equal to $\{-9.35, 5\}$, $\{-11.35, 5\}$, $\{-9.35, 1\}$ and $\{-11.35, 1\}$, respectively, and $\{\beta , \,\kappa_1,\, \kappa_2,\, \rho, \,\gamma \} = \{0.242, \,0.456,\, 1.234,\, 0.08, \,6.91 \}$ fixed for all benchmarks. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. |
 | \small 1D and 2D posterior distributions derived from the DP template reconstruction of the benchmarks DP-BNK\_1 (top left), DP-BNK\_2 (top right), DP-BNK\_3 (bottom left) and DP-BNK\_4 (bottom right) defined as in \cref{eq:templateDP} with $\{\log_{10}h^2\Omega_* ,\,f_*/\textrm{Hz}\}$ equal to $\{-9.35, 5\}$, $\{-11.35, 5\}$, $\{-9.35, 1\}$ and $\{-11.35, 1\}$, respectively, and $\{\beta , \,\kappa_1,\, \kappa_2,\, \rho, \,\gamma \} = \{0.242, \,0.456,\, 1.234,\, 0.08, \,6.91 \}$ fixed for all benchmarks. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. |
 | \small 1D and 2D posterior distributions derived from the DP template reconstruction of the benchmarks DP-BNK\_1 (top left), DP-BNK\_2 (top right), DP-BNK\_3 (bottom left) and DP-BNK\_4 (bottom right) defined as in \cref{eq:templateDP} with $\{\log_{10}h^2\Omega_* ,\,f_*/\textrm{Hz}\}$ equal to $\{-9.35, 5\}$, $\{-11.35, 5\}$, $\{-9.35, 1\}$ and $\{-11.35, 1\}$, respectively, and $\{\beta , \,\kappa_1,\, \kappa_2,\, \rho, \,\gamma \} = \{0.242, \,0.456,\, 1.234,\, 0.08, \,6.91 \}$ fixed for all benchmarks. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. |
 | \small 1D and 2D posterior distributions derived from the DP template reconstruction of the benchmarks DP-BNK\_1 (top left), DP-BNK\_2 (top right), DP-BNK\_3 (bottom left) and DP-BNK\_4 (bottom right) defined as in \cref{eq:templateDP} with $\{\log_{10}h^2\Omega_* ,\,f_*/\textrm{Hz}\}$ equal to $\{-9.35, 5\}$, $\{-11.35, 5\}$, $\{-9.35, 1\}$ and $\{-11.35, 1\}$, respectively, and $\{\beta , \,\kappa_1,\, \kappa_2,\, \rho, \,\gamma \} = \{0.242, \,0.456,\, 1.234,\, 0.08, \,6.91 \}$ fixed for all benchmarks. Lines' styles and color codes are as in \cref{fig:PL_MCMC}. |
 | \small Fisher forecasts for the ES template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (left), $\omega_{\rm ES}$Hz (middle) and $\gamma_{\rm ES}$ (right) as a function of the injected values of $h^2\Omega_*$, $\omega_{\rm ES}$Hz and $\gamma_{\rm ES}$ set as specified in the axes and the title of each panel. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (white) and in the presence (black) of foregrounds. The crosses display the benchmarks ES-BNK\_1, ES-BNK\_2 and ES-BNK\_3. |
 | \small Fisher forecasts for the ES template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (left), $\omega_{\rm ES}$Hz (middle) and $\gamma_{\rm ES}$ (right) as a function of the injected values of $h^2\Omega_*$, $\omega_{\rm ES}$Hz and $\gamma_{\rm ES}$ set as specified in the axes and the title of each panel. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (white) and in the presence (black) of foregrounds. The crosses display the benchmarks ES-BNK\_1, ES-BNK\_2 and ES-BNK\_3. |
 | \small Fisher forecasts for the ES template. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (left), $\omega_{\rm ES}$Hz (middle) and $\gamma_{\rm ES}$ (right) as a function of the injected values of $h^2\Omega_*$, $\omega_{\rm ES}$Hz and $\gamma_{\rm ES}$ set as specified in the axes and the title of each panel. SNR contour lines are plotted in white. The pairs of dashed (dotted) lines mark the $ \sigma =0.3$ ($\sigma =0.01$) contours, respectively in the absence (white) and in the presence (black) of foregrounds. The crosses display the benchmarks ES-BNK\_1, ES-BNK\_2 and ES-BNK\_3. |
 | \small 1D and 2D posterior distributions derived from the ES template reconstruction of the benchmarks ES-BNK\_1 (top left), ES-BNK\_2 (top right) and ES-BNK\_3 (bottom). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. ES-BNK\_1, ES-BNK\_2 and ES-BNK\_3 are defined as in \cref{templateexcited} with $\{\log_{10}h^2\Omega_*,\gamma_{\mathrm{ES}},\omega_{\mathrm{ES}}\mathrm{Hz}\}$ set at $\{-11,10,6 \cdot 10^{3} \}$, $\{-11.5,10,6 \cdot 10^{3} \}$ and $\{-11,10,1.2 \cdot 10^{3}\}$, respectively. |
 | \small 1D and 2D posterior distributions derived from the ES template reconstruction of the benchmarks ES-BNK\_1 (top left), ES-BNK\_2 (top right) and ES-BNK\_3 (bottom). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. ES-BNK\_1, ES-BNK\_2 and ES-BNK\_3 are defined as in \cref{templateexcited} with $\{\log_{10}h^2\Omega_*,\gamma_{\mathrm{ES}},\omega_{\mathrm{ES}}\mathrm{Hz}\}$ set at $\{-11,10,6 \cdot 10^{3} \}$, $\{-11.5,10,6 \cdot 10^{3} \}$ and $\{-11,10,1.2 \cdot 10^{3}\}$, respectively. |
 | \small 1D and 2D posterior distributions derived from the ES template reconstruction of the benchmarks ES-BNK\_1 (top left), ES-BNK\_2 (top right) and ES-BNK\_3 (bottom). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. ES-BNK\_1, ES-BNK\_2 and ES-BNK\_3 are defined as in \cref{templateexcited} with $\{\log_{10}h^2\Omega_*,\gamma_{\mathrm{ES}},\omega_{\mathrm{ES}}\mathrm{Hz}\}$ set at $\{-11,10,6 \cdot 10^{3} \}$, $\{-11.5,10,6 \cdot 10^{3} \}$ and $\{-11,10,1.2 \cdot 10^{3}\}$, respectively. |
 | \small Fisher forecasts for the LO template with the LN envelope. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $\mathcal{A}_{\rm lin}$ (top right), $f_*$ (central left), $\omega_{\rm lin}$ (central right), $\rho$ (bottom left) and $\theta_{\rm lin}$ (bottom right) as a function of the injected values $h^2\Omega_*$ and $\mathcal{A}_{\rm lin}$ and by fixing the other parameters to $\{f_*= 1\,\mathrm{mHz} ,\,\rho = 0.08, \,\omega_{\mathrm{lin}}\,\mathrm{Hz}= 5\cdot10^{4},\, \theta_{\mathrm{lin}}= 0 \}$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross displays the benchmarks LO-BNK\_1. |
 | \small Fisher forecasts for the LO template with the LN envelope. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $\mathcal{A}_{\rm lin}$ (top right), $f_*$ (central left), $\omega_{\rm lin}$ (central right), $\rho$ (bottom left) and $\theta_{\rm lin}$ (bottom right) as a function of the injected values $h^2\Omega_*$ and $\mathcal{A}_{\rm lin}$ and by fixing the other parameters to $\{f_*= 1\,\mathrm{mHz} ,\,\rho = 0.08, \,\omega_{\mathrm{lin}}\,\mathrm{Hz}= 5\cdot10^{4},\, \theta_{\mathrm{lin}}= 0 \}$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross displays the benchmarks LO-BNK\_1. |
 | \small Fisher forecasts for the LO template with the LN envelope. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $\mathcal{A}_{\rm lin}$ (top right), $f_*$ (central left), $\omega_{\rm lin}$ (central right), $\rho$ (bottom left) and $\theta_{\rm lin}$ (bottom right) as a function of the injected values $h^2\Omega_*$ and $\mathcal{A}_{\rm lin}$ and by fixing the other parameters to $\{f_*= 1\,\mathrm{mHz} ,\,\rho = 0.08, \,\omega_{\mathrm{lin}}\,\mathrm{Hz}= 5\cdot10^{4},\, \theta_{\mathrm{lin}}= 0 \}$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross displays the benchmarks LO-BNK\_1. |
 | \small Fisher forecasts for the LO template with the LN envelope. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $\mathcal{A}_{\rm lin}$ (top right), $f_*$ (central left), $\omega_{\rm lin}$ (central right), $\rho$ (bottom left) and $\theta_{\rm lin}$ (bottom right) as a function of the injected values $h^2\Omega_*$ and $\mathcal{A}_{\rm lin}$ and by fixing the other parameters to $\{f_*= 1\,\mathrm{mHz} ,\,\rho = 0.08, \,\omega_{\mathrm{lin}}\,\mathrm{Hz}= 5\cdot10^{4},\, \theta_{\mathrm{lin}}= 0 \}$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross displays the benchmarks LO-BNK\_1. |
 | \small Fisher forecasts for the LO template with the LN envelope. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $\mathcal{A}_{\rm lin}$ (top right), $f_*$ (central left), $\omega_{\rm lin}$ (central right), $\rho$ (bottom left) and $\theta_{\rm lin}$ (bottom right) as a function of the injected values $h^2\Omega_*$ and $\mathcal{A}_{\rm lin}$ and by fixing the other parameters to $\{f_*= 1\,\mathrm{mHz} ,\,\rho = 0.08, \,\omega_{\mathrm{lin}}\,\mathrm{Hz}= 5\cdot10^{4},\, \theta_{\mathrm{lin}}= 0 \}$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross displays the benchmarks LO-BNK\_1. |
 | \small Fisher forecasts for the LO template with the LN envelope. The panels show the \seCL reconstruction error on $h^2\Omega_*$ (top left), $\mathcal{A}_{\rm lin}$ (top right), $f_*$ (central left), $\omega_{\rm lin}$ (central right), $\rho$ (bottom left) and $\theta_{\rm lin}$ (bottom right) as a function of the injected values $h^2\Omega_*$ and $\mathcal{A}_{\rm lin}$ and by fixing the other parameters to $\{f_*= 1\,\mathrm{mHz} ,\,\rho = 0.08, \,\omega_{\mathrm{lin}}\,\mathrm{Hz}= 5\cdot10^{4},\, \theta_{\mathrm{lin}}= 0 \}$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross displays the benchmarks LO-BNK\_1. |
 | \small 1D and 2D posterior distributions derived from the LO template reconstruction of the benchmarks LO-BNK\_1 (left) and LO-BNK\_2 (right). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. LO-BNK\_1 and LO-BNK\_2 are defined as in \cref{eq:sharp-template} with $\{f_*/ \mathrm{mHz},\,\log_{10}(\omega_{\mathrm{lin}} \,\mathrm{Hz}) \}= \{5,\,4\} $ and $\{1,\,4.7\}$ respectively, and fixed $\{\rho, {\cal A}_{\mathrm {lin}}\} = \{ 0.08, 0.3\}$. |
 | \small 1D and 2D posterior distributions derived from the LO template reconstruction of the benchmarks LO-BNK\_1 (left) and LO-BNK\_2 (right). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. LO-BNK\_1 and LO-BNK\_2 are defined as in \cref{eq:sharp-template} with $\{f_*/ \mathrm{mHz},\,\log_{10}(\omega_{\mathrm{lin}} \,\mathrm{Hz}) \}= \{5,\,4\} $ and $\{1,\,4.7\}$ respectively, and fixed $\{\rho, {\cal A}_{\mathrm {lin}}\} = \{ 0.08, 0.3\}$. |
 | \small Fisher forecasts for the RO template with the flat envelope $h^2\Omega_{\rm GW}= h^2\Omega_*$. The panels show the \seCL reconstruction error on $\log_{10}(h^2\Omega_*)$ (top left), $A_{\rm log}$ (top right), $\omega_{\rm log}$ (bottom left) and $\phi_{\rm log}$ (bottom right) as a function of the injected values of $h^2\Omega_*$ and $\omega_{\rm log}$ and by fixing $A_{\rm log}= 0.5$ and $\phi_{\rm log} = 0$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross display the benchmark RO-BNK\_2. |
 | \small Fisher forecasts for the RO template with the flat envelope $h^2\Omega_{\rm GW}= h^2\Omega_*$. The panels show the \seCL reconstruction error on $\log_{10}(h^2\Omega_*)$ (top left), $A_{\rm log}$ (top right), $\omega_{\rm log}$ (bottom left) and $\phi_{\rm log}$ (bottom right) as a function of the injected values of $h^2\Omega_*$ and $\omega_{\rm log}$ and by fixing $A_{\rm log}= 0.5$ and $\phi_{\rm log} = 0$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross display the benchmark RO-BNK\_2. |
 | \small Fisher forecasts for the RO template with the flat envelope $h^2\Omega_{\rm GW}= h^2\Omega_*$. The panels show the \seCL reconstruction error on $\log_{10}(h^2\Omega_*)$ (top left), $A_{\rm log}$ (top right), $\omega_{\rm log}$ (bottom left) and $\phi_{\rm log}$ (bottom right) as a function of the injected values of $h^2\Omega_*$ and $\omega_{\rm log}$ and by fixing $A_{\rm log}= 0.5$ and $\phi_{\rm log} = 0$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross display the benchmark RO-BNK\_2. |
 | \small Fisher forecasts for the RO template with the flat envelope $h^2\Omega_{\rm GW}= h^2\Omega_*$. The panels show the \seCL reconstruction error on $\log_{10}(h^2\Omega_*)$ (top left), $A_{\rm log}$ (top right), $\omega_{\rm log}$ (bottom left) and $\phi_{\rm log}$ (bottom right) as a function of the injected values of $h^2\Omega_*$ and $\omega_{\rm log}$ and by fixing $A_{\rm log}= 0.5$ and $\phi_{\rm log} = 0$. SNR contour lines are plotted in white. Depending on the panel, the pairs of dashed lines mark the absolute error $ \sigma =0.3$ ($\sigma =0.01$) and relative error 30\% ($1\%$) contours, respectively in the absence [white] and in the presence [black] of foregrounds. The cross display the benchmark RO-BNK\_2. |
 | \small 1D and 2D posterior distributions derived from the RO template reconstruction with a flat envelope run on the benchmarks RO-BNK\_1 (left) and RO-BNK\_2 (right). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. Benchmarks RO-BNK\_1 and RO-BNK\_2 are defined as in eq.~(\ref{eq:resonant-template}) with $\{h^2 \Omega_* \, , \ A_\textrm{log} \, , \ \omega_\textrm{log} \, , \ \phi_\textrm{log} \, \}$ set at $\{-11,1,12,0\}$ and $\{-12,0.5,6,0\}$ respectively. |
 | \small 1D and 2D posterior distributions derived from the RO template reconstruction with a flat envelope run on the benchmarks RO-BNK\_1 (left) and RO-BNK\_2 (right). Lines' styles and color codes are as in \cref{fig:PL_MCMC}. Benchmarks RO-BNK\_1 and RO-BNK\_2 are defined as in eq.~(\ref{eq:resonant-template}) with $\{h^2 \Omega_* \, , \ A_\textrm{log} \, , \ \omega_\textrm{log} \, , \ \phi_\textrm{log} \, \}$ set at $\{-11,1,12,0\}$ and $\{-12,0.5,6,0\}$ respectively. |
 | \small Constraints on the parameters of axion inflation parameters (left panels) and massive-graviton inflation (right panels) derived from the PL template-based reconstruction of the benchmarks PL-BNK\_1 (top panels) and PL-BNK\_2 (bottom panels). The hatched region is excluded by the Higuchi bound~\cite{Higuchi:1986py} ($m_h/H<\sqrt{2}$) in models respecting de Sitter isometries. |
 | \small Constraints on the parameters of axion inflation parameters (left panels) and massive-graviton inflation (right panels) derived from the PL template-based reconstruction of the benchmarks PL-BNK\_1 (top panels) and PL-BNK\_2 (bottom panels). The hatched region is excluded by the Higuchi bound~\cite{Higuchi:1986py} ($m_h/H<\sqrt{2}$) in models respecting de Sitter isometries. |
 | \small Constraints on the parameters of axion inflation parameters (left panels) and massive-graviton inflation (right panels) derived from the PL template-based reconstruction of the benchmarks PL-BNK\_1 (top panels) and PL-BNK\_2 (bottom panels). The hatched region is excluded by the Higuchi bound~\cite{Higuchi:1986py} ($m_h/H<\sqrt{2}$) in models respecting de Sitter isometries. |
 | \small Constraints on the parameters of axion inflation parameters (left panels) and massive-graviton inflation (right panels) derived from the PL template-based reconstruction of the benchmarks PL-BNK\_1 (top panels) and PL-BNK\_2 (bottom panels). The hatched region is excluded by the Higuchi bound~\cite{Higuchi:1986py} ($m_h/H<\sqrt{2}$) in models respecting de Sitter isometries. |
 | \small Effect on the SGWB spectrum of the power law parameters $\Omega_*$ (left) and $n_t$ (right). For reference, we indicate with a gray dashed line the LISA power-law integrated sensitivity curve \cite{Babak:2021mhe,Colpi:2024xhw}. The black solid line corresponds to PL-BNK\_2, whose parameters are $\{ \log_{10}\Omega_*, n_t\} = \{ -11, \,0.77\}$. |
 | \small Effect on the SGWB spectrum of the broken power law template parameters. The black line indicates the spectrum assuming $\{\log_{10}\Omega_*, n_{t,1}, n_{t,2}, \delta \} = \{-10,\,2,\,-2,\,1\}$. |
 | \small Effect on the SGWB spectrum of the log-normal template parameters. The black line indicates the spectrum assuming $\{\log_{10}\Omega_*,\,f_*\,[{\rm Hz}],\, \rho \} = \{-10,\, 10^{-3},\,0.5 \}$. |
 | \small Effect on the SGWB spectrum of the double peak template parameters. The black line indicates the spectrum assuming $\{\log_{10}\Omega_*, \beta, \kappa_1, \kappa_2, \rho, \gamma \} = \{ -10,0.242,0.456,1.234,0.08,6.96\}$ (corresponding to a log-normal primordial power spectrum with $\Delta=0.2$ as the benchmarks in the main text). |
 | \small Effect on the SGWB spectrum of the double peak template parameters. The black line indicates the spectrum assuming $\{\log_{10}\Omega_*, \beta, \kappa_1, \kappa_2, \rho, \gamma \} = \{ -10,0.242,0.456,1.234,0.08,6.96\}$ (corresponding to a log-normal primordial power spectrum with $\Delta=0.2$ as the benchmarks in the main text). |
 | \small Effect on the SGWB spectrum of the excited states template parameters. The black line indicates the spectrum assuming $\{\log_{10}h^2\Omega_*, \log_{10}\gamma_{\rm ES}, \log_{10}\omega_{\rm ES}\} = \{ -9, 1, 3.778\}$. (The latter corresponding to $\omega_{\mathrm{ES}} = 6\cdot 10^3 \,\mathrm{Hz}^{-1}$). |
 | \small Effect on the SGWB spectrum of the linear oscillations template parameters. The black line indicates the spectrum assuming $\{\mathcal{A}_{\mathrm{lin}},\log_{10}(\omega_{\mathrm{lin}}\,\mathrm{Hz}),\theta_{\mathrm{lin}} \} = \{0.3, 4.7 ,0\}$, where the envelope is assumed to be the log-normal with $\vec{\theta}_{\rm env} = \{\log_{10}\Omega_*, \rho \} = \{-9, 0.08\}$. In these plots, the range of frequencies was zoomed in close to the peak of the signal, to highlight spectral oscillations. For this reason only a small portion of the LISA power-law integrated sensitivity is visible (gray dashed line). |
 | \small Effect on the SGWB spectrum of the logarithmic resonant oscillation template parameters. The black lines corresponds to $\{\log_{10}\Omega_*, A_\textrm{log}, \omega_\textrm{log}, \phi_\textrm{log}\} = \{-9, 0.5, 12 , 0\}$ and a flat envelope. |
 | \small Plot of eq.~\eqref{calcnom}, choosing $\Delta=0.2$. The spectral index of the \GWB\ approaches $n_\Omega=3$ for $\kappa\to0$, but at larger $\kappa$ it is more like $2.5$, in agreement with the numerical results. |
 | \small \textit{Left panel:} The red points indicate the scalar-induced GW spectrum from a log-normal curvature power spectrum and the solid curves show the fit by the DP template to these points. \textit{Right panel:} Best-fit parameters characterising the DP template when matched with the GW spectrum induced from a log-normal curvature power spectrum of the form \eqref{eq:Pk_LN} as a function of $\Delta$. |
 | \small \textit{Left panel:} The red points indicate the scalar-induced GW spectrum from a broken power-law curvature power spectrum and the solid curves show the fit by the DP template to these points. \textit{Right panel:} Fit of the DP template to the scalar-induced GW spectrum from a broken power-law curvature power spectrum as a function of $p_2$. |
 | \small Fit of the DP template to the scalar-induced GW spectrum from a broken power-law curvature power spectrum as a function of $p_1$ in the limit $p_2\to \infty$. |