Cosmology with the Laser Interferometer Space Antenna

Author(s)

Auclair, Pierre, Bacon, David, Baker, Tessa, Barreiro, Tiago, Bartolo, Nicola, Belgacem, Enis, Bellomo, Nicola, Ben-Dayan, Ido, Bertacca, Daniele, Besancon, Marc, Blanco-Pillado, Jose J., Blas, Diego, Boileau, Guillaume, Calcagni, Gianluca, Caldwell, Robert, Caprini, Chiara, Carbone, Carmelita, Chang, Chia-Feng, Chen, Hsin-Yu, Christensen, Nelson, Clesse, Sebastien, Comelli, Denis, Congedo, Giuseppe, Contaldi, Carlo, Crisostomi, Marco, Croon, Djuna, Cui, Yanou, Cusin, Giulia, Cutting, Daniel, Dalang, Charles, De Luca, Valerio, Del Pozzo, Walter, Desjacques, Vincent, Dimastrogiovanni, Emanuela, Dorsch, Glauber C., Ezquiaga, Jose Maria, Fasiello, Matteo, Figueroa, Daniel G., Flauger, Raphael, Franciolini, Gabriele, Frusciante, Noemi, Fumagalli, Jacopo, García-Bellido, Juan, Gould, Oliver, Holz, Daniel, Iacconi, Laura, Jain, Rajeev Kumar, Jenkins, Alexander C., Jinno, Ryusuke, Joana, Cristian, Karnesis, Nikolaos, Konstandin, Thomas, Koyama, Kazuya, Kozaczuk, Jonathan, Kuroyanagi, Sachiko, Laghi, Danny, Lewicki, Marek, Lombriser, Lucas, Madge, Eric, Maggiore, Michele, Malhotra, Ameek, Mancarella, Michele, Mandic, Vuk, Mangiagli, Alberto, Matarrese, Sabino, Mazumdar, Anupam, Mukherjee, Suvodip, Musco, Ilia, Nardini, Germano, No, Jose Miguel, Papanikolaou, Theodoros, Peloso, Marco, Pieroni, Mauro, Pilo, Luigi, Raccanelli, Alvise, Renaux-Petel, Sébastien, Renzini, Arianna I., Ricciardone, Angelo, Riotto, Antonio, Romano, Joseph D., Rollo, Rocco, Pol, Alberto Roper, Morales, Ester Ruiz, Sakellariadou, Mairi, Saltas, Ippocratis D., Scalisi, Marco, Schmitz, Kai, Schwaller, Pedro, Sergijenko, Olga, Servant, Geraldine, Simakachorn, Peera, Sorbo, Lorenzo, Sousa, Lara, Speri, Lorenzo, Steer, Danièle A., Tamanini, Nicola, Tasinato, Gianmassimo, Torrado, Jesús, Unal, Caner, Vennin, Vincent, Vernieri, Daniele, Vernizzi, Filippo, Volonteri, Marta, Wachter, Jeremy M., Wands, David, Witkowski, Lukas T., Zumalacárregui, Miguel, Annis, James, Ares, Fëanor Reuben, Avelino, Pedro P., Avgoustidis, Anastasios, Barausse, Enrico, Bonilla, Alexander, Bonvin, Camille, Bosso, Pasquale, Calabrese, Matteo, Çalışkan, Mesut, Cembranos, Jose A.R., Chala, Mikael, Chernoff, David, Clough, Katy, Criswell, Alexander, Das, Saurya, da Silva, Antonio, Dayal, Pratika, Domcke, Valerie, Durrer, Ruth, Easther, Richard, Escoffier, Stephanie, Ferrans, Sandrine, Fryer, Chris, Gair, Jonathan, Gordon, Chris, Hendry, Martin, Hindmarsh, Mark, Hooper, Deanna C., Kajfasz, Eric, Kopp, Joachim, Koushiappas, Savvas M., Kumar, Utkarsh, Kunz, Martin, Lagos, Macarena, Lilley, Marc, Lizarraga, Joanes, Lobo, Francisco S.N., Maleknejad, Azadeh, Martins, C.J. A.P., Meerburg, P. Daniel, Meyer, Renate, Mimoso, José Pedro, Nesseris, Savvas, Nunes, Nelson, Oikonomou, Vasilis, Orlando, Giorgio, Özsoy, Ogan, Pacucci, Fabio, Palmese, Antonella, Petiteau, Antoine, Pinol, Lucas, Portegies Zwart, Simon, Pratten, Geraint, Prokopec, Tomislav, Quenby, John, Rastgoo, Saeed, Roest, Diederik, Rummukainen, Kari, Schimd, Carlo, Secroun, Aurélia, Sesana, Alberto, Sopuerta, Carlos F., Tereno, Ismael, Tolley, Andrew, Urrestilla, Jon, Vagenas, Elias C., van de Vis, Jorinde, van de Weygaert, Rien, Wardell, Barry, Weir, David J., White, Graham, Świeżewska, Bogumiła, Zhdanov, Valery I.

Abstract

The Laser Interferometer Space Antenna (LISA) has two scientific objectives of cosmological focus: to probe the expansion rate of the universe, and to understand stochastic gravitational-wave backgrounds and their implications for early universe and particle physics, from the MeV to the Planck scale. However, the range of potential cosmological applications of gravitational-wave observations extends well beyond these two objectives. This publication presents a summary of the state of the art in LISA cosmology, theory and methods, and identifies new opportunities to use gravitational-wave observations by LISA to probe the universe.

Figures

\small Left panel: $90\%$ (black) and $68\%$ (red) credible intervals for $H_0$/km s$^{-1}$ Mpc$^{-1}$ for each of the LISA configurations considered in Ref.~\cite{DelPozzo:2017kme}. The credible regions are averaged over the galaxy hosts realisations. Right panel: 90\% (black) and 68\% (red) percentiles, together with the median (red dot) of $h$ for the pessimistic, fiducial, and optimistic EMRI models (M5, M1, M6, respectively) and for two different LISA observational scenario (4 and 10 years). Here $h = H_0 / (100\, {\rm km\, s}^{-1} {\rm Mpc}^{-1})$ and the blue dashed horizontal line denotes the true cosmology (set at $h=0.73$ in Refs.~\cite{DelPozzo:2017kme,Laghi:2021pqk}). For each data point, we also report the average number $N$ of EMRIs considered in the analysis. Plots taken from  Refs.~\cite{DelPozzo:2017kme, Laghi:2021pqk}.

\small Left panel: $90\%$ (black) and $68\%$ (red) credible intervals for $H_0$/km s$^{-1}$ Mpc$^{-1}$ for each of the LISA configurations considered in Ref.~\cite{DelPozzo:2017kme}. The credible regions are averaged over the galaxy hosts realisations. Right panel: 90\% (black) and 68\% (red) percentiles, together with the median (red dot) of $h$ for the pessimistic, fiducial, and optimistic EMRI models (M5, M1, M6, respectively) and for two different LISA observational scenario (4 and 10 years). Here $h = H_0 / (100\, {\rm km\, s}^{-1} {\rm Mpc}^{-1})$ and the blue dashed horizontal line denotes the true cosmology (set at $h=0.73$ in Refs.~\cite{DelPozzo:2017kme,Laghi:2021pqk}). For each data point, we also report the average number $N$ of EMRIs considered in the analysis. Plots taken from Refs.~\cite{DelPozzo:2017kme, Laghi:2021pqk}.


\small Left panel: $90\%$ (black) and $68\%$ (red) credible intervals for $H_0$/km s$^{-1}$ Mpc$^{-1}$ for each of the LISA configurations considered in Ref.~\cite{DelPozzo:2017kme}. The credible regions are averaged over the galaxy hosts realisations. Right panel: 90\% (black) and 68\% (red) percentiles, together with the median (red dot) of $h$ for the pessimistic, fiducial, and optimistic EMRI models (M5, M1, M6, respectively) and for two different LISA observational scenario (4 and 10 years). Here $h = H_0 / (100\, {\rm km\, s}^{-1} {\rm Mpc}^{-1})$ and the blue dashed horizontal line denotes the true cosmology (set at $h=0.73$ in Refs.~\cite{DelPozzo:2017kme,Laghi:2021pqk}). For each data point, we also report the average number $N$ of EMRIs considered in the analysis. Plots taken from  Refs.~\cite{DelPozzo:2017kme, Laghi:2021pqk}.

\small Left panel: $90\%$ (black) and $68\%$ (red) credible intervals for $H_0$/km s$^{-1}$ Mpc$^{-1}$ for each of the LISA configurations considered in Ref.~\cite{DelPozzo:2017kme}. The credible regions are averaged over the galaxy hosts realisations. Right panel: 90\% (black) and 68\% (red) percentiles, together with the median (red dot) of $h$ for the pessimistic, fiducial, and optimistic EMRI models (M5, M1, M6, respectively) and for two different LISA observational scenario (4 and 10 years). Here $h = H_0 / (100\, {\rm km\, s}^{-1} {\rm Mpc}^{-1})$ and the blue dashed horizontal line denotes the true cosmology (set at $h=0.73$ in Refs.~\cite{DelPozzo:2017kme,Laghi:2021pqk}). For each data point, we also report the average number $N$ of EMRIs considered in the analysis. Plots taken from Refs.~\cite{DelPozzo:2017kme, Laghi:2021pqk}.


\small GW lensing and wave effects. Left panel: Amplification $|F(w)|$ caused by a point lens in the wave optics regime. Low frequency signals $w\ll 1$ undergo no magnification. An oscillatory pattern emerges at higher frequencies. Right panel: Imprint of strong lensing on a MBBH source, as observed by LISA. The frequency dependence (onset of magnification, diffraction pattern) carries information about the gravitational lens.

\small GW lensing and wave effects. Left panel: Amplification $|F(w)|$ caused by a point lens in the wave optics regime. Low frequency signals $w\ll 1$ undergo no magnification. An oscillatory pattern emerges at higher frequencies. Right panel: Imprint of strong lensing on a MBBH source, as observed by LISA. The frequency dependence (onset of magnification, diffraction pattern) carries information about the gravitational lens.


\small GW lensing and wave effects. Left panel: Amplification $|F(w)|$ caused by a point lens in the wave optics regime. Low frequency signals $w\ll 1$ undergo no magnification. An oscillatory pattern emerges at higher frequencies. Right panel: Imprint of strong lensing on a MBBH source, as observed by LISA. The frequency dependence (onset of magnification, diffraction pattern) carries information about the gravitational lens.

\small GW lensing and wave effects. Left panel: Amplification $|F(w)|$ caused by a point lens in the wave optics regime. Low frequency signals $w\ll 1$ undergo no magnification. An oscillatory pattern emerges at higher frequencies. Right panel: Imprint of strong lensing on a MBBH source, as observed by LISA. The frequency dependence (onset of magnification, diffraction pattern) carries information about the gravitational lens.


\small SGWB energy density $h^2\Omega_{\rm GW}$ for different cosmological sources compared to the sensitivity of different GW detectors. As cosmological signals we have the vacuum GW contribution coming from inflation (grey dashed line) with $ r=0.044$ and $n_T=-r/8$, the signal expected in axion inflation models (cyan), the signal generated by cosmic string networks with $G\mu=10^{-10}$ (brown), the signal generated by a FOPT with $v_w=0.9$, $\alpha=0.1$, $\beta/H_*=50$, $g_*=100$, $T_*=200 \rm {GeV}$ (pink) and the signal generated at second-order by the formation mechanism of PBHs with $f_{PBH}=1$, $\sigma = 0.5$, $k_* = k_{\rm LISA}$ (orange). For GW detectors we report the sensitivity of Planck (darker green), LITEBird (green), EPTA (blue), SKA (darker blue), LISA (red), DECIGO (purple), LIGO Design (black) and ET (darker black).

\small SGWB energy density $h^2\Omega_{\rm GW}$ for different cosmological sources compared to the sensitivity of different GW detectors. As cosmological signals we have the vacuum GW contribution coming from inflation (grey dashed line) with $ r=0.044$ and $n_T=-r/8$, the signal expected in axion inflation models (cyan), the signal generated by cosmic string networks with $G\mu=10^{-10}$ (brown), the signal generated by a FOPT with $v_w=0.9$, $\alpha=0.1$, $\beta/H_*=50$, $g_*=100$, $T_*=200 \rm {GeV}$ (pink) and the signal generated at second-order by the formation mechanism of PBHs with $f_{PBH}=1$, $\sigma = 0.5$, $k_* = k_{\rm LISA}$ (orange). For GW detectors we report the sensitivity of Planck (darker green), LITEBird (green), EPTA (blue), SKA (darker blue), LISA (red), DECIGO (purple), LIGO Design (black) and ET (darker black).


\small An example of the GW power spectrum from a FOPT, along with the LISA power-law-sensitivity (PLS) curve of SNR\,$=10$. The model parameters used in this example are $v_w = 0.9$, $\alpha = 0.1$, $\beta/H_* = 50$, $T_* = 200$ GeV, $g_* = 100$.

\small An example of the GW power spectrum from a FOPT, along with the LISA power-law-sensitivity (PLS) curve of SNR\,$=10$. The model parameters used in this example are $v_w = 0.9$, $\alpha = 0.1$, $\beta/H_* = 50$, $T_* = 200$ GeV, $g_* = 100$.


\small LISA sensitivity to the SGWB frequency peak of  FOPTs in the early universe. Here the fiducial values $v_w = 1$ and $g=106$ are used to relate the frequency peak of the acoustic spectrum \cite{Caprini:2019egz} to a time and temperature in the early universe, as a function of the transition rate parameter $\beta/H$.

\small LISA sensitivity to the SGWB frequency peak of FOPTs in the early universe. Here the fiducial values $v_w = 1$ and $g=106$ are used to relate the frequency peak of the acoustic spectrum \cite{Caprini:2019egz} to a time and temperature in the early universe, as a function of the transition rate parameter $\beta/H$.


\small The parameter reach of present and future GW observational network for phase transitions in a runaway-like regime with $v_w \simeq 1$ and $\alpha \gg 1$. In each shaded area the corresponding experiment (see labels) detects the FOPT signal with SNR\,$>\!10$. The region labelled BBN, PTA and aLIGO O3 are ruled out. Figure adapted from Refs.~\cite{Figueroa:2018xtu, Megias:2020vek}.

\small The parameter reach of present and future GW observational network for phase transitions in a runaway-like regime with $v_w \simeq 1$ and $\alpha \gg 1$. In each shaded area the corresponding experiment (see labels) detects the FOPT signal with SNR\,$>\!10$. The region labelled BBN, PTA and aLIGO O3 are ruled out. Figure adapted from Refs.~\cite{Figueroa:2018xtu, Megias:2020vek}.


\small Typical values of $\alpha$ and $\beta/H_*$ for the general singlet model, superimposed with LISA SNR curves for $T_*=50$ GeV (solid lines). The models in blue (orange) are unlikely (likely) to be probed by the high-luminosity LHC. The dotted straight lines are the contours of the fluid turnover time quantifying the effect of turbulence. In the grey shaded region the decay of sound waves into turbulence is less important than the Hubble damping and the SNR curve reflects this effect. See Ref.~\cite{Caprini:2019egz} for details.

\small Typical values of $\alpha$ and $\beta/H_*$ for the general singlet model, superimposed with LISA SNR curves for $T_*=50$ GeV (solid lines). The models in blue (orange) are unlikely (likely) to be probed by the high-luminosity LHC. The dotted straight lines are the contours of the fluid turnover time quantifying the effect of turbulence. In the grey shaded region the decay of sound waves into turbulence is less important than the Hubble damping and the SNR curve reflects this effect. See Ref.~\cite{Caprini:2019egz} for details.


\small Analytical approximation to the SGWB generated by cosmic string networks with $G\mu=10^{-10}$ and different values of $\alpha$ for Model 1. The solid lines represent the approximation to the SGWB in Eq.~(\ref{approx}), while the dashed lines correspond to the SGWB obtained numerically. Figure taken from Ref.~\cite{Sousa:2020sxs}.

\small Analytical approximation to the SGWB generated by cosmic string networks with $G\mu=10^{-10}$ and different values of $\alpha$ for Model 1. The solid lines represent the approximation to the SGWB in Eq.~(\ref{approx}), while the dashed lines correspond to the SGWB obtained numerically. Figure taken from Ref.~\cite{Sousa:2020sxs}.


\small Figure to demonstrate the validity of the templates for Model III. The dark dashed line is the result from numerical integration with only the fundamental mode and $G(z) = 1$.

\small Figure to demonstrate the validity of the templates for Model III. The dark dashed line is the result from numerical integration with only the fundamental mode and $G(z) = 1$.


\small Solid red curves show cosmic string SGWB curves for a range of $G\mu$ values. From the darkest most high up line to the lightest lowest one these read: $G\mu=10^{-10}$, $G\mu=10^{-13}$, $G\mu=10^{-15}$ and $G\mu=10^{-17}$. The $P_n$ used in computation of these spectra was inferred from simulations~\cite{Blanco-Pillado:2017oxo}, and the loop number density is from Model 2. The dashed orange curve shows the sensitivity of EPTA. The dark orange dash-dotted line shows the projected SKA sensitivity. The dotted black line shows the LISA PLS of SNR\,$=10$.

\small Solid red curves show cosmic string SGWB curves for a range of $G\mu$ values. From the darkest most high up line to the lightest lowest one these read: $G\mu=10^{-10}$, $G\mu=10^{-13}$, $G\mu=10^{-15}$ and $G\mu=10^{-17}$. The $P_n$ used in computation of these spectra was inferred from simulations~\cite{Blanco-Pillado:2017oxo}, and the loop number density is from Model 2. The dashed orange curve shows the sensitivity of EPTA. The dark orange dash-dotted line shows the projected SKA sensitivity. The dotted black line shows the LISA PLS of SNR\,$=10$.


\small Identical to Fig.~\ref{fig:fiducialSGWB}, however, $P_n\propto n^{-4/3}$ and using the loop number density from Model 3~\cite{Lorenz:2010sm}.

\small Identical to Fig.~\ref{fig:fiducialSGWB}, however, $P_n\propto n^{-4/3}$ and using the loop number density from Model 3~\cite{Lorenz:2010sm}.


\small  Amplitude of the SGWB anisotropies for different cosmic string network models, as a function of the string tension. We use a representative LISA-band GW frequency of $1~\mathrm{mHz}$. Note that the spectra here are not normalised with respect to the monopole, so $\sqrt{C_\ell}$ is proportional to $\Omega_\mathrm{GW}$. We note that the angular spectrum is constant with respect to $\ell$, since the anisotropies are driven by local Poisson fluctuations in the number of loops.

\small Amplitude of the SGWB anisotropies for different cosmic string network models, as a function of the string tension. We use a representative LISA-band GW frequency of $1~\mathrm{mHz}$. Note that the spectra here are not normalised with respect to the monopole, so $\sqrt{C_\ell}$ is proportional to $\Omega_\mathrm{GW}$. We note that the angular spectrum is constant with respect to $\ell$, since the anisotropies are driven by local Poisson fluctuations in the number of loops.


\small Projected constraints on $G\mu$ of the LISA mission for cosmic string scenarios characterised by different loop-size parameter $\alpha$ for $n_*=1$ (dashed line) and $n_*=10^5$, with $q=4/3$ (dash-dotted line). The shaded area corresponds to the region of the $(\alpha,G\mu)$ parameter space that will be fully available for exploration with LISA. The dotted line corresponds to scenarios for which $\alpha=\Gamma G\mu$, so that the region above this line corresponds to cosmic string models in which loops are small, while the region bellow corresponds to the large loop regime. Figure taken from  Ref.~\cite{Auclair:2019wcv}.

\small Projected constraints on $G\mu$ of the LISA mission for cosmic string scenarios characterised by different loop-size parameter $\alpha$ for $n_*=1$ (dashed line) and $n_*=10^5$, with $q=4/3$ (dash-dotted line). The shaded area corresponds to the region of the $(\alpha,G\mu)$ parameter space that will be fully available for exploration with LISA. The dotted line corresponds to scenarios for which $\alpha=\Gamma G\mu$, so that the region above this line corresponds to cosmic string models in which loops are small, while the region bellow corresponds to the large loop regime. Figure taken from Ref.~\cite{Auclair:2019wcv}.


\small  SGWB spectrum from a global (solid) and gauge (dashed) string network in standard cosmology with loop length parameter $\alpha = 0.1$, and symmetry breaking scale $\eta =  10^{15}, 5 \times 10^{14}, 10^{14}\,$GeV for red, green, blue, respectively. The grey region is the LISA sensitivity.

\small SGWB spectrum from a global (solid) and gauge (dashed) string network in standard cosmology with loop length parameter $\alpha = 0.1$, and symmetry breaking scale $\eta = 10^{15}, 5 \times 10^{14}, 10^{14}\,$GeV for red, green, blue, respectively. The grey region is the LISA sensitivity.


\small Left panel: SGWB signals from a metastable string network for $G\mu = 2 \times 10^{-7}$ and $\alpha = 0.1$, for different values of $\sqrt{\kappa}$; see Eq.~\eqref{eq:Gammad}. The solid lines represent the numerical result, the dashed lines show the analytical approximation in terms of a flat plateau at high frequencies and an $f^{3/2}$ rise at low frequencies. Right panel: Existing and future constraints on the metastable string parameter space. The hashed vertical band marks the prediction of the $U(1)_{B-L}$ model in Ref.~\cite{Buchmuller:2019gfy}, where both plots appeared for the first time.

\small Left panel: SGWB signals from a metastable string network for $G\mu = 2 \times 10^{-7}$ and $\alpha = 0.1$, for different values of $\sqrt{\kappa}$; see Eq.~\eqref{eq:Gammad}. The solid lines represent the numerical result, the dashed lines show the analytical approximation in terms of a flat plateau at high frequencies and an $f^{3/2}$ rise at low frequencies. Right panel: Existing and future constraints on the metastable string parameter space. The hashed vertical band marks the prediction of the $U(1)_{B-L}$ model in Ref.~\cite{Buchmuller:2019gfy}, where both plots appeared for the first time.


\small Left panel: SGWB signals from a metastable string network for $G\mu = 2 \times 10^{-7}$ and $\alpha = 0.1$, for different values of $\sqrt{\kappa}$; see Eq.~\eqref{eq:Gammad}. The solid lines represent the numerical result, the dashed lines show the analytical approximation in terms of a flat plateau at high frequencies and an $f^{3/2}$ rise at low frequencies. Right panel: Existing and future constraints on the metastable string parameter space. The hashed vertical band marks the prediction of the $U(1)_{B-L}$ model in Ref.~\cite{Buchmuller:2019gfy}, where both plots appeared for the first time.

\small Left panel: SGWB signals from a metastable string network for $G\mu = 2 \times 10^{-7}$ and $\alpha = 0.1$, for different values of $\sqrt{\kappa}$; see Eq.~\eqref{eq:Gammad}. The solid lines represent the numerical result, the dashed lines show the analytical approximation in terms of a flat plateau at high frequencies and an $f^{3/2}$ rise at low frequencies. Right panel: Existing and future constraints on the metastable string parameter space. The hashed vertical band marks the prediction of the $U(1)_{B-L}$ model in Ref.~\cite{Buchmuller:2019gfy}, where both plots appeared for the first time.


\small The GW energy density $\Omega_{\rm GW}$ as a function of frequency sourced by SU(2) axion-gauge field setups (solid blue line and yellow dashed line). A signal outside the reach of certain probes may still be accessed via LISA at small scales (and e.g.~LiteBIRD at very large scales). The vertical bars are the expected 1$\sigma$ error bars estimated with and without the presence of astrophysical foregrounds (light and dark shared areas, respectively). Different colours are used for LiteBIRD (green), SKA (orange), LISA (blue), and ET (purple). Logarithmic binning in wavenumber is employed with $\Delta \ln k=1.2$. For the sake of comparison, the tensor spectrum predicted in some illustrative single-field slow-roll scenario (dashed lines) are displayed, including their BICEP2/Keck/Planck upper bound $r = 0.06$. The quantity $k_p$ is the pivot scale and the quantity $\sigma$ in the upper left box is a parameter of the SU(2) model, typically of order $1-10$. For details see Ref.~\cite{Campeti:2020xwn} from which the figure is taken.

\small The GW energy density $\Omega_{\rm GW}$ as a function of frequency sourced by SU(2) axion-gauge field setups (solid blue line and yellow dashed line). A signal outside the reach of certain probes may still be accessed via LISA at small scales (and e.g.~LiteBIRD at very large scales). The vertical bars are the expected 1$\sigma$ error bars estimated with and without the presence of astrophysical foregrounds (light and dark shared areas, respectively). Different colours are used for LiteBIRD (green), SKA (orange), LISA (blue), and ET (purple). Logarithmic binning in wavenumber is employed with $\Delta \ln k=1.2$. For the sake of comparison, the tensor spectrum predicted in some illustrative single-field slow-roll scenario (dashed lines) are displayed, including their BICEP2/Keck/Planck upper bound $r = 0.06$. The quantity $k_p$ is the pivot scale and the quantity $\sigma$ in the upper left box is a parameter of the SU(2) model, typically of order $1-10$. For details see Ref.~\cite{Campeti:2020xwn} from which the figure is taken.


\small Feynman diagrams for the GW power spectrum.  Left panel: The gauge fields $A_{\mu}$ source GW non-linearly in the Abelian case. Right panel: The $SU(2)$ scenario, where gauge tensor degrees of freedom $t_{ij}$  source GWs already at tree level.

\small Feynman diagrams for the GW power spectrum. Left panel: The gauge fields $A_{\mu}$ source GW non-linearly in the Abelian case. Right panel: The $SU(2)$ scenario, where gauge tensor degrees of freedom $t_{ij}$ source GWs already at tree level.


\small The GW energy density of the EFT when including the contribution of an extra spin-2 field whose helicity-2 mode has a time-dependent sound speed. The two colours correspond to different values of $\rho/H$, respectively $(3 \times 10^{-3}, 4 \times 10^{-4})$, supporting a signal within reach for both SKA and LISA, or for LISA only. In generating both lines, the time dependence of the sound speed has been kept constant to $s_2\equiv\dot{c}_2/(H c_2)\simeq -0.2$. Figure taken from  Ref.~\cite{Iacconi:2020yxn}.

\small The GW energy density of the EFT when including the contribution of an extra spin-2 field whose helicity-2 mode has a time-dependent sound speed. The two colours correspond to different values of $\rho/H$, respectively $(3 \times 10^{-3}, 4 \times 10^{-4})$, supporting a signal within reach for both SKA and LISA, or for LISA only. In generating both lines, the time dependence of the sound speed has been kept constant to $s_2\equiv\dot{c}_2/(H c_2)\simeq -0.2$. Figure taken from Ref.~\cite{Iacconi:2020yxn}.


\small Scalar power spectrum $\mathcal{P}_{\zeta}$ \textbf{(a)} and the scalar-induced GW energy density $\Omega_{\textrm{GW}}$ \textbf{(b)} for two inflation models (see text for details) with a sharp feature and enhanced fluctuations from a strong turn in the inflationary trajectory, together with the PLS for SNR threshold $\textrm{SNR}_\textrm{th}=1$ and total observation time $T=3$ years assumed in the analysis of ~\cite{Fumagalli:2020nvq}. The black dashed line in \textbf{(a)} shows the envelope of $\mathcal{P}_\zeta$ and in  \textbf{(b)} the corresponding GW energy density $\overline{\Omega}_{\textrm{GW}}$. The $\mathcal{O}(1)$-oscillations in $\mathcal{P}_\zeta$ are processed into $\mathcal{O}(10 \%)$ modulations on the principal peak of $\Omega_{\textrm{GW}}$. Over the frequency range of the principal peak, the modulations in $\Omega_{\textrm{GW}}$ can be modelled as cosine-oscillations about $\overline{\Omega}_{\textrm{GW}}$, as can be seen in \textbf{(c)}. To translate $k/k_\star$ into $f$ / Hz, we considered $N_\star=31.5$, i.e.~the peak in $\mathcal{P}_\zeta$ occurs at wavenumber $k=k_\star$, which is 31.5 $e$-folds larger than the CMB value. Figures adapted from Ref.~\cite{Fumagalli:2020nvq}.

\small Scalar power spectrum $\mathcal{P}_{\zeta}$ \textbf{(a)} and the scalar-induced GW energy density $\Omega_{\textrm{GW}}$ \textbf{(b)} for two inflation models (see text for details) with a sharp feature and enhanced fluctuations from a strong turn in the inflationary trajectory, together with the PLS for SNR threshold $\textrm{SNR}_\textrm{th}=1$ and total observation time $T=3$ years assumed in the analysis of ~\cite{Fumagalli:2020nvq}. The black dashed line in \textbf{(a)} shows the envelope of $\mathcal{P}_\zeta$ and in \textbf{(b)} the corresponding GW energy density $\overline{\Omega}_{\textrm{GW}}$. The $\mathcal{O}(1)$-oscillations in $\mathcal{P}_\zeta$ are processed into $\mathcal{O}(10 \%)$ modulations on the principal peak of $\Omega_{\textrm{GW}}$. Over the frequency range of the principal peak, the modulations in $\Omega_{\textrm{GW}}$ can be modelled as cosine-oscillations about $\overline{\Omega}_{\textrm{GW}}$, as can be seen in \textbf{(c)}. To translate $k/k_\star$ into $f$ / Hz, we considered $N_\star=31.5$, i.e.~the peak in $\mathcal{P}_\zeta$ occurs at wavenumber $k=k_\star$, which is 31.5 $e$-folds larger than the CMB value. Figures adapted from Ref.~\cite{Fumagalli:2020nvq}.


\small Left panel: Auto-correlation of GW anisotropies induced by a scalar-tensor-tensor squeezed bispectrum with $F_{\text{NL}}^{\text{stt}}\sim \mathcal{O}(10^{3})$ (red line), and of GW anisotropies arising from propagation through the perturbed background (blue line). Right panel: Cross-correlation of GW anisotropies and CMB temperature anisotropies in the same two cases.

\small Left panel: Auto-correlation of GW anisotropies induced by a scalar-tensor-tensor squeezed bispectrum with $F_{\text{NL}}^{\text{stt}}\sim \mathcal{O}(10^{3})$ (red line), and of GW anisotropies arising from propagation through the perturbed background (blue line). Right panel: Cross-correlation of GW anisotropies and CMB temperature anisotropies in the same two cases.


\small Left panel: Auto-correlation of GW anisotropies induced by a scalar-tensor-tensor squeezed bispectrum with $F_{\text{NL}}^{\text{stt}}\sim \mathcal{O}(10^{3})$ (red line), and of GW anisotropies arising from propagation through the perturbed background (blue line). Right panel: Cross-correlation of GW anisotropies and CMB temperature anisotropies in the same two cases.

\small Left panel: Auto-correlation of GW anisotropies induced by a scalar-tensor-tensor squeezed bispectrum with $F_{\text{NL}}^{\text{stt}}\sim \mathcal{O}(10^{3})$ (red line), and of GW anisotropies arising from propagation through the perturbed background (blue line). Right panel: Cross-correlation of GW anisotropies and CMB temperature anisotropies in the same two cases.


\small The expected 1$\sigma$ error in determing $F_{\rm NL}^{\text{stt}}$ with cross-correlations of CMB temperature anisotropies and GW anisotropies, as a function of $\ell_{\rm max}$.

\small The expected 1$\sigma$ error in determing $F_{\rm NL}^{\text{stt}}$ with cross-correlations of CMB temperature anisotropies and GW anisotropies, as a function of $\ell_{\rm max}$.


\small Left panel: GW energy spectra including changes in the number of relativistic degrees of freedom, depending on whether the  transition from stiff domination  to RD  takes place before (red dashed line) or after (blue solid line) than the QCD phase transition. For comparison we show the corresponding spectra (grey solid line) without correcting for changes in the number of degrees of freedom. Figure taken from Ref.~\cite{Figueroa:2019paj}. Right panel: Parameter space region that LISA can probe after removing the region (above the dashed horizontal line) incompatible with the BBN bound on GW backgrounds. Here $w_s$ refers to the EoS for a {\it stiff} background.

\small Left panel: GW energy spectra including changes in the number of relativistic degrees of freedom, depending on whether the transition from stiff domination to RD takes place before (red dashed line) or after (blue solid line) than the QCD phase transition. For comparison we show the corresponding spectra (grey solid line) without correcting for changes in the number of degrees of freedom. Figure taken from Ref.~\cite{Figueroa:2019paj}. Right panel: Parameter space region that LISA can probe after removing the region (above the dashed horizontal line) incompatible with the BBN bound on GW backgrounds. Here $w_s$ refers to the EoS for a {\it stiff} background.


\small Left panel: GW energy spectra including changes in the number of relativistic degrees of freedom, depending on whether the  transition from stiff domination  to RD  takes place before (red dashed line) or after (blue solid line) than the QCD phase transition. For comparison we show the corresponding spectra (grey solid line) without correcting for changes in the number of degrees of freedom. Figure taken from Ref.~\cite{Figueroa:2019paj}. Right panel: Parameter space region that LISA can probe after removing the region (above the dashed horizontal line) incompatible with the BBN bound on GW backgrounds. Here $w_s$ refers to the EoS for a {\it stiff} background.

\small Left panel: GW energy spectra including changes in the number of relativistic degrees of freedom, depending on whether the transition from stiff domination to RD takes place before (red dashed line) or after (blue solid line) than the QCD phase transition. For comparison we show the corresponding spectra (grey solid line) without correcting for changes in the number of degrees of freedom. Figure taken from Ref.~\cite{Figueroa:2019paj}. Right panel: Parameter space region that LISA can probe after removing the region (above the dashed horizontal line) incompatible with the BBN bound on GW backgrounds. Here $w_s$ refers to the EoS for a {\it stiff} background.


\small The lines show the GW spectrum produced in FOPT by bubble collisions for the indicated transition parameters. From top to bottom, the decay width of the scalar field decreases leading to a lengthening period of effectively matter-dominated expansion as the field oscillates around its minimum before finally decaying. The resulting reheating temperatures read $T_{\rm RD }=10^4$ GeV, $T_{\rm RD}=10^3$ GeV and $T_{\rm RD}=10^2$ GeV, respectively.

\small The lines show the GW spectrum produced in FOPT by bubble collisions for the indicated transition parameters. From top to bottom, the decay width of the scalar field decreases leading to a lengthening period of effectively matter-dominated expansion as the field oscillates around its minimum before finally decaying. The resulting reheating temperatures read $T_{\rm RD }=10^4$ GeV, $T_{\rm RD}=10^3$ GeV and $T_{\rm RD}=10^2$ GeV, respectively.


\small  Range of temperature $T_{\rm RD}$ to which LISA can probe the cosmological expansion rate using a spectrum from a cosmic string network with the indicated string tension $G\mu$. The grey region indicates temperatures where modifications of the expansion rate would already be in tension with BBN.

\small Range of temperature $T_{\rm RD}$ to which LISA can probe the cosmological expansion rate using a spectrum from a cosmic string network with the indicated string tension $G\mu$. The grey region indicates temperatures where modifications of the expansion rate would already be in tension with BBN.


\small  GW spectrum produced by a cosmic string network with $G\mu=10^{-10}$ together with spectra produced if that network evolved in an early matter domination period (with $w=0$) or kination (with $w=1$) lasting up until $T_{\rm RD}=5$ MeV and $T_{\rm RD}=5$ GeV. We note that the blue tilted branches in the figure are only shown for representative purposes, as they violate BBN and CMB constraints [recall discussion at the end of Sec.~\ref{subsec:GWprobeIntro}], so for a realistic effect compatible with those bounds, a much smaller stiff EoS is required, somehow larger than (but very close to) $w \approx 1/3$.

\small GW spectrum produced by a cosmic string network with $G\mu=10^{-10}$ together with spectra produced if that network evolved in an early matter domination period (with $w=0$) or kination (with $w=1$) lasting up until $T_{\rm RD}=5$ MeV and $T_{\rm RD}=5$ GeV. We note that the blue tilted branches in the figure are only shown for representative purposes, as they violate BBN and CMB constraints [recall discussion at the end of Sec.~\ref{subsec:GWprobeIntro}], so for a realistic effect compatible with those bounds, a much smaller stiff EoS is required, somehow larger than (but very close to) $w \approx 1/3$.


\small Most stringent limits on the DM fraction made of PBHs, $f_{\rm PBH}$, coming from the Hawking evaporation producing extragalactic gamma-ray (EG$\gamma$), e$^\pm$ observations by Voyager 1 (Ve$^\pm$), positron annihilations in the Galactic Center (GCe$^+$), gamma-ray observations by INTEGRAL (INT), microlensing searches by Subaru HSC (HSC), MACHO/EROS (E), OGLE (O) and Icarus (I), from CMB limits (CMB), GW observations by LIGO/Virgo (LVC), wide binaires in the galactic halo (WB), the ultra-faint dwarf galaxies Eridanus II (EII) and Segue 1 (S1), X-rays towards the galactic center (XrB) and Lyman-$\alpha$ limits (L$\alpha$).  For microlensing and CMB limits, the different lines indicate some degree of uncertainties, respectively due to PBH clustering and disk vs. spherical accretion.  Microlensing limits only apply to the fraction of PBHs uniformly distributed in galactic halos and are less stringent if a  non-negligible fraction $f_{\rm \rm clust}$ of PBHs are in clusters.  We show the limits for $f_{\rm \rm clust}=0, 0.4$ and $0.8$.  For LVC, the rate suppression of early binaries still allows $f_{\rm PBH} \gtrsim 0.1$.  \emph{All these limits apply to monochromatic models} and can be model dependent.  Recasting them to realistic PBH models with arbitrary mass functions requires a careful analysis. Figure adapted from Ref.~\cite{DeLuca:2020agl}.

\small Most stringent limits on the DM fraction made of PBHs, $f_{\rm PBH}$, coming from the Hawking evaporation producing extragalactic gamma-ray (EG$\gamma$), e$^\pm$ observations by Voyager 1 (Ve$^\pm$), positron annihilations in the Galactic Center (GCe$^+$), gamma-ray observations by INTEGRAL (INT), microlensing searches by Subaru HSC (HSC), MACHO/EROS (E), OGLE (O) and Icarus (I), from CMB limits (CMB), GW observations by LIGO/Virgo (LVC), wide binaires in the galactic halo (WB), the ultra-faint dwarf galaxies Eridanus II (EII) and Segue 1 (S1), X-rays towards the galactic center (XrB) and Lyman-$\alpha$ limits (L$\alpha$). For microlensing and CMB limits, the different lines indicate some degree of uncertainties, respectively due to PBH clustering and disk vs. spherical accretion. Microlensing limits only apply to the fraction of PBHs uniformly distributed in galactic halos and are less stringent if a non-negligible fraction $f_{\rm \rm clust}$ of PBHs are in clusters. We show the limits for $f_{\rm \rm clust}=0, 0.4$ and $0.8$. For LVC, the rate suppression of early binaries still allows $f_{\rm PBH} \gtrsim 0.1$. \emph{All these limits apply to monochromatic models} and can be model dependent. Recasting them to realistic PBH models with arbitrary mass functions requires a careful analysis. Figure adapted from Ref.~\cite{DeLuca:2020agl}.


\small Left panel:  Evolution of the relativistic degrees of freedom $g_{*}$ as a function of the temperature. The grey vertical lines correspond to the masses of the electron, 	pion, proton/neutron, $W$, $Z$ bosons and top quark, respectively.  Right panel:  Effect of the evolution of $g_*$ on the critical overdensity $\delta_{\rm c}$ leading to PBH formation, as a function of the Hubble horizon mass (related to the PBH mass by $M = \gamma m_{\rm H}$).

\small Left panel: Evolution of the relativistic degrees of freedom $g_{*}$ as a function of the temperature. The grey vertical lines correspond to the masses of the electron, pion, proton/neutron, $W$, $Z$ bosons and top quark, respectively. Right panel: Effect of the evolution of $g_*$ on the critical overdensity $\delta_{\rm c}$ leading to PBH formation, as a function of the Hubble horizon mass (related to the PBH mass by $M = \gamma m_{\rm H}$).


\small Left panel:  Evolution of the relativistic degrees of freedom $g_{*}$ as a function of the temperature. The grey vertical lines correspond to the masses of the electron, 	pion, proton/neutron, $W$, $Z$ bosons and top quark, respectively.  Right panel:  Effect of the evolution of $g_*$ on the critical overdensity $\delta_{\rm c}$ leading to PBH formation, as a function of the Hubble horizon mass (related to the PBH mass by $M = \gamma m_{\rm H}$).

\small Left panel: Evolution of the relativistic degrees of freedom $g_{*}$ as a function of the temperature. The grey vertical lines correspond to the masses of the electron, pion, proton/neutron, $W$, $Z$ bosons and top quark, respectively. Right panel: Effect of the evolution of $g_*$ on the critical overdensity $\delta_{\rm c}$ leading to PBH formation, as a function of the Hubble horizon mass (related to the PBH mass by $M = \gamma m_{\rm H}$).


\small PBH density fraction at formation $\beta_{\rm form}$ (left panel) and the corresponding PBH mass function $f_{\rm PBH}$ today (right panel), neglecting the effect of PBH growth by accretion and hierarchical mergers, for two models with a power-law power spectrum and including the effects of thermal history:  Model 1 from Refs.~\cite{Carr:2019kxo,Clesse:2020ghq} with spectral index $n_{\rm s} = 0.97$; Model 2 from Refs.~\cite{DeLuca:2020agl,Byrnes:2018clq} with $n_{\rm s} = 1.$ and a cut-off mass of $10^{-14} M_\odot$.  The transition between the large-scale and small-scale power spectrum is fixed at $k=10^3 {\rm Mpc}^{-1}$.  The power spectrum amplitude is normalized such that both models produce an integrated PBH fraction $f_{\rm PBH} =1$, i.e.~PBH constitute the totality of DM.  A value of $\gamma = 0.8$ was assumed.

\small PBH density fraction at formation $\beta_{\rm form}$ (left panel) and the corresponding PBH mass function $f_{\rm PBH}$ today (right panel), neglecting the effect of PBH growth by accretion and hierarchical mergers, for two models with a power-law power spectrum and including the effects of thermal history: Model 1 from Refs.~\cite{Carr:2019kxo,Clesse:2020ghq} with spectral index $n_{\rm s} = 0.97$; Model 2 from Refs.~\cite{DeLuca:2020agl,Byrnes:2018clq} with $n_{\rm s} = 1.$ and a cut-off mass of $10^{-14} M_\odot$. The transition between the large-scale and small-scale power spectrum is fixed at $k=10^3 {\rm Mpc}^{-1}$. The power spectrum amplitude is normalized such that both models produce an integrated PBH fraction $f_{\rm PBH} =1$, i.e.~PBH constitute the totality of DM. A value of $\gamma = 0.8$ was assumed.


\small PBH density fraction at formation $\beta_{\rm form}$ (left panel) and the corresponding PBH mass function $f_{\rm PBH}$ today (right panel), neglecting the effect of PBH growth by accretion and hierarchical mergers, for two models with a power-law power spectrum and including the effects of thermal history:  Model 1 from Refs.~\cite{Carr:2019kxo,Clesse:2020ghq} with spectral index $n_{\rm s} = 0.97$; Model 2 from Refs.~\cite{DeLuca:2020agl,Byrnes:2018clq} with $n_{\rm s} = 1.$ and a cut-off mass of $10^{-14} M_\odot$.  The transition between the large-scale and small-scale power spectrum is fixed at $k=10^3 {\rm Mpc}^{-1}$.  The power spectrum amplitude is normalized such that both models produce an integrated PBH fraction $f_{\rm PBH} =1$, i.e.~PBH constitute the totality of DM.  A value of $\gamma = 0.8$ was assumed.

\small PBH density fraction at formation $\beta_{\rm form}$ (left panel) and the corresponding PBH mass function $f_{\rm PBH}$ today (right panel), neglecting the effect of PBH growth by accretion and hierarchical mergers, for two models with a power-law power spectrum and including the effects of thermal history: Model 1 from Refs.~\cite{Carr:2019kxo,Clesse:2020ghq} with spectral index $n_{\rm s} = 0.97$; Model 2 from Refs.~\cite{DeLuca:2020agl,Byrnes:2018clq} with $n_{\rm s} = 1.$ and a cut-off mass of $10^{-14} M_\odot$. The transition between the large-scale and small-scale power spectrum is fixed at $k=10^3 {\rm Mpc}^{-1}$. The power spectrum amplitude is normalized such that both models produce an integrated PBH fraction $f_{\rm PBH} =1$, i.e.~PBH constitute the totality of DM. A value of $\gamma = 0.8$ was assumed.


\small SGWB sourced at second order by the density perturbations at the origin of PBH formation, for Model 2 of Fig.~\ref{fig:fPBH}. On top of the plot, we show the PBH mass associated to a given GW frequency as in Eq.~\eqref{PBHmassfreq}. The LISA sensitivity \cite{Audley:2017drz} and the hint for a detection by NANOGrav 12.5 yr \cite{Arzoumanian:2020vkk} are represented, as well as the constraints coming EPTA \cite{Lentati:2015qwp}, PPTA \cite{Shannon:2015ect}, NANOGrav 11 yrs \cite{Arzoumanian:2018saf, Aggarwal:2018mgp} and future sensitivity curves for planned experiments like SKA \cite{Zhao:2013bba}, DECIGO/BBO \cite{Yagi:2011wg}, CE \cite{Evans:2016mbw}, ET \cite{Hild:2010id}, Advanced LIGO + Virgo collaboration \cite{TheLIGOScientific:2016dpb}, Magis-space (MS) and Magis-100 (M100) \cite{Coleman:2018ozp}, AEDGE \cite{Bertoldi:2019tck} and AION \cite{Badurina:2019hst}. Figure taken from Ref.~\cite{DeLuca:2020agl}.

\small SGWB sourced at second order by the density perturbations at the origin of PBH formation, for Model 2 of Fig.~\ref{fig:fPBH}. On top of the plot, we show the PBH mass associated to a given GW frequency as in Eq.~\eqref{PBHmassfreq}. The LISA sensitivity \cite{Audley:2017drz} and the hint for a detection by NANOGrav 12.5 yr \cite{Arzoumanian:2020vkk} are represented, as well as the constraints coming EPTA \cite{Lentati:2015qwp}, PPTA \cite{Shannon:2015ect}, NANOGrav 11 yrs \cite{Arzoumanian:2018saf, Aggarwal:2018mgp} and future sensitivity curves for planned experiments like SKA \cite{Zhao:2013bba}, DECIGO/BBO \cite{Yagi:2011wg}, CE \cite{Evans:2016mbw}, ET \cite{Hild:2010id}, Advanced LIGO + Virgo collaboration \cite{TheLIGOScientific:2016dpb}, Magis-space (MS) and Magis-100 (M100) \cite{Coleman:2018ozp}, AEDGE \cite{Bertoldi:2019tck} and AION \cite{Badurina:2019hst}. Figure taken from Ref.~\cite{DeLuca:2020agl}.


\small Expected merger rates of PBH of mass $m_1$ and $m_2$ for the  Model 1 (top panels) and Model 2 (bottom panels) mass distributions displayed in Fig.~\ref{fig:fPBH} due to the binary formation channels ``primordial binaries" (left panels) and ``tidal capture in halos" (right panels) coming from  Eq.~(\ref{eq:Rprimbinaries}) and Eq.~(\ref{eq:ratescatpure2}) respectively.

\small Expected merger rates of PBH of mass $m_1$ and $m_2$ for the Model 1 (top panels) and Model 2 (bottom panels) mass distributions displayed in Fig.~\ref{fig:fPBH} due to the binary formation channels ``primordial binaries" (left panels) and ``tidal capture in halos" (right panels) coming from Eq.~(\ref{eq:Rprimbinaries}) and Eq.~(\ref{eq:ratescatpure2}) respectively.


\small Expected merger rates of PBH of mass $m_1$ and $m_2$ for the  Model 1 (top panels) and Model 2 (bottom panels) mass distributions displayed in Fig.~\ref{fig:fPBH} due to the binary formation channels ``primordial binaries" (left panels) and ``tidal capture in halos" (right panels) coming from  Eq.~(\ref{eq:Rprimbinaries}) and Eq.~(\ref{eq:ratescatpure2}) respectively.

\small Expected merger rates of PBH of mass $m_1$ and $m_2$ for the Model 1 (top panels) and Model 2 (bottom panels) mass distributions displayed in Fig.~\ref{fig:fPBH} due to the binary formation channels ``primordial binaries" (left panels) and ``tidal capture in halos" (right panels) coming from Eq.~(\ref{eq:Rprimbinaries}) and Eq.~(\ref{eq:ratescatpure2}) respectively.


\small Example of SGWB expected for PBH binaries formed by tidal capture in clusters, with rates given by Eq.~(\ref{eq:ratescatpure2}) and different virial velocities ($2$km/s in green, $20$km/s in blue and $200$km/s in red) and a peaked PBH mass distribution around $30 M_\odot$.  Figure taken from Ref.~\cite{Clesse:2016ajp}.

\small Example of SGWB expected for PBH binaries formed by tidal capture in clusters, with rates given by Eq.~(\ref{eq:ratescatpure2}) and different virial velocities ($2$km/s in green, $20$km/s in blue and $200$km/s in red) and a peaked PBH mass distribution around $30 M_\odot$. Figure taken from Ref.~\cite{Clesse:2016ajp}.


\small Redshift range of LISA according to the analysis of \cite{Burke-Spolaor:2018bvk} for equal-mass BBH coalescences as a function of the total system mass, and comparison with the range of other detectors and pulsar timing arrays.  The color scale represents the expected SNR emerged from the study. Figure taken from Ref.~\cite{Burke-Spolaor:2018bvk}.

\small Redshift range of LISA according to the analysis of \cite{Burke-Spolaor:2018bvk} for equal-mass BBH coalescences as a function of the total system mass, and comparison with the range of other detectors and pulsar timing arrays. The color scale represents the expected SNR emerged from the study. Figure taken from Ref.~\cite{Burke-Spolaor:2018bvk}.


\small Left panel: IMS and acceleration noise power spectra expressed in the simplified scenario where all test-mass and laser noises are equal for all space-crafts. Right panel: LISA noise spectra in the $XYZ$ and $AET$ basis as given in e.g.~in Ref.~\cite{Flauger:2020qyi}. Plots taken from Ref.~\cite{Flauger:2020qyi}.

\small Left panel: IMS and acceleration noise power spectra expressed in the simplified scenario where all test-mass and laser noises are equal for all space-crafts. Right panel: LISA noise spectra in the $XYZ$ and $AET$ basis as given in e.g.~in Ref.~\cite{Flauger:2020qyi}. Plots taken from Ref.~\cite{Flauger:2020qyi}.


\small Left panel: IMS and acceleration noise power spectra expressed in the simplified scenario where all test-mass and laser noises are equal for all space-crafts. Right panel: LISA noise spectra in the $XYZ$ and $AET$ basis as given in e.g.~in Ref.~\cite{Flauger:2020qyi}. Plots taken from Ref.~\cite{Flauger:2020qyi}.

\small Left panel: IMS and acceleration noise power spectra expressed in the simplified scenario where all test-mass and laser noises are equal for all space-crafts. Right panel: LISA noise spectra in the $XYZ$ and $AET$ basis as given in e.g.~in Ref.~\cite{Flauger:2020qyi}. Plots taken from Ref.~\cite{Flauger:2020qyi}.


\small Theoretical models for the LISA sensitivity in the self-correlation of the $A$ and $T$ TDI data channels ($AA$ and $TT$, respectively) and for the SGWBs due to GBs and SOBBHs as specified in Ref.~\cite{Flauger:2020qyi}. Mock data for the sum of all these theoretical templates are shown for reference.

\small Theoretical models for the LISA sensitivity in the self-correlation of the $A$ and $T$ TDI data channels ($AA$ and $TT$, respectively) and for the SGWBs due to GBs and SOBBHs as specified in Ref.~\cite{Flauger:2020qyi}. Mock data for the sum of all these theoretical templates are shown for reference.


\small Reconstruction of a broken power law signal in presence of GB foreground. Figure taken from Ref.~\cite{Flauger:2020qyi}.

\small Reconstruction of a broken power law signal in presence of GB foreground. Figure taken from Ref.~\cite{Flauger:2020qyi}.


\small LISA Sensitivity (\ref{SNR-sensitvity}) to various multipoles of the SGWB. Figure taken from Ref.~\cite{Bartolo:2022pez}.

\small LISA Sensitivity (\ref{SNR-sensitvity}) to various multipoles of the SGWB. Figure taken from Ref.~\cite{Bartolo:2022pez}.


\small Normalised auto-correlated response of TDI channel X at time $t = 0$ in the Solar System barycentre reference frame, at frequencies $f = 10^{-4}$ Hz, $f = 10^{-1}$ Hz, $f = 5\times10^{-1}$ Hz from left to right respectively. Estimates based on Ref.~\cite{Contaldi:2020rht}.

\small Normalised auto-correlated response of TDI channel X at time $t = 0$ in the Solar System barycentre reference frame, at frequencies $f = 10^{-4}$ Hz, $f = 10^{-1}$ Hz, $f = 5\times10^{-1}$ Hz from left to right respectively. Estimates based on Ref.~\cite{Contaldi:2020rht}.


\small Normalised auto-correlated response of TDI channel X at time $t = 0$ in the Solar System barycentre reference frame, at frequencies $f = 10^{-4}$ Hz, $f = 10^{-1}$ Hz, $f = 5\times10^{-1}$ Hz from left to right respectively. Estimates based on Ref.~\cite{Contaldi:2020rht}.

\small Normalised auto-correlated response of TDI channel X at time $t = 0$ in the Solar System barycentre reference frame, at frequencies $f = 10^{-4}$ Hz, $f = 10^{-1}$ Hz, $f = 5\times10^{-1}$ Hz from left to right respectively. Estimates based on Ref.~\cite{Contaldi:2020rht}.


\small Normalised auto-correlated response of TDI channel X at time $t = 0$ in the Solar System barycentre reference frame, at frequencies $f = 10^{-4}$ Hz, $f = 10^{-1}$ Hz, $f = 5\times10^{-1}$ Hz from left to right respectively. Estimates based on Ref.~\cite{Contaldi:2020rht}.

\small Normalised auto-correlated response of TDI channel X at time $t = 0$ in the Solar System barycentre reference frame, at frequencies $f = 10^{-4}$ Hz, $f = 10^{-1}$ Hz, $f = 5\times10^{-1}$ Hz from left to right respectively. Estimates based on Ref.~\cite{Contaldi:2020rht}.


\small Transfer functions $T_{\ell}$ for the average reconstructed $C_{\ell}$s obtained with different frequency cutoffs $f_{max}$, drawn from Ref.~\cite{Contaldi:2020rht}. Each simulation set consists of 50 maps, each a different realisation of the same $C_{\ell}$ input  (dashed grey baseline). There appears to be a clear one-to-one relation between the resolution $\ell_{\rm max}$ of the instrument and the frequency cutoff.

\small Transfer functions $T_{\ell}$ for the average reconstructed $C_{\ell}$s obtained with different frequency cutoffs $f_{max}$, drawn from Ref.~\cite{Contaldi:2020rht}. Each simulation set consists of 50 maps, each a different realisation of the same $C_{\ell}$ input (dashed grey baseline). There appears to be a clear one-to-one relation between the resolution $\ell_{\rm max}$ of the instrument and the frequency cutoff.


\small Simulation (left) and recovery (right) of the  white dwarf GB foreground energy density in GWs at 1\,mHz, using one year of data in the Solar System barycentric frame. The bright spot in the map corresponds to the galactic central bulge. Figure taken from Ref.~\cite{Banagiri:2021ovv}.

\small Simulation (left) and recovery (right) of the white dwarf GB foreground energy density in GWs at 1\,mHz, using one year of data in the Solar System barycentric frame. The bright spot in the map corresponds to the galactic central bulge. Figure taken from Ref.~\cite{Banagiri:2021ovv}.


\small Simulation (left) and recovery (right) of the  white dwarf GB foreground energy density in GWs at 1\,mHz, using one year of data in the Solar System barycentric frame. The bright spot in the map corresponds to the galactic central bulge. Figure taken from Ref.~\cite{Banagiri:2021ovv}.

\small Simulation (left) and recovery (right) of the white dwarf GB foreground energy density in GWs at 1\,mHz, using one year of data in the Solar System barycentric frame. The bright spot in the map corresponds to the galactic central bulge. Figure taken from Ref.~\cite{Banagiri:2021ovv}.


\small Measurement of the orbital modulation of the white dwarf binary foreground. In grey: 1500 estimates of $\Omega_{\textrm{Mod},i}=  \frac{4\pi^2}{3H_0} \left( \frac{c}{2\pi L}\right)^2 A_i^2$ ($A_i$ amplitude of the characteristic strain). In red: 50 MCMC results with 8 parameters (2 parameters for BBHs, 4 parameters for white dwarf binaries, 2 parameters for the LISA noise). In green, fit to the 50 MCMC run results to estimate the modulation from the LISA antenna pattern amplitude at 3 mHz. Modulation model: $\Omega_{\textrm{Mod},i} =\Omega_{\rm DWD,LF}^u\left(F^2_{+,i} + F^2_{\times,i} \right)$. Figure taken from Ref.~\cite{Boileau:2021sni}.

\small Measurement of the orbital modulation of the white dwarf binary foreground. In grey: 1500 estimates of $\Omega_{\textrm{Mod},i}= \frac{4\pi^2}{3H_0} \left( \frac{c}{2\pi L}\right)^2 A_i^2$ ($A_i$ amplitude of the characteristic strain). In red: 50 MCMC results with 8 parameters (2 parameters for BBHs, 4 parameters for white dwarf binaries, 2 parameters for the LISA noise). In green, fit to the 50 MCMC run results to estimate the modulation from the LISA antenna pattern amplitude at 3 mHz. Modulation model: $\Omega_{\textrm{Mod},i} =\Omega_{\rm DWD,LF}^u\left(F^2_{+,i} + F^2_{\times,i} \right)$. Figure taken from Ref.~\cite{Boileau:2021sni}.


\small Cosmological amplitude uncertainty estimates from the Fisher Information study denoted by solid lines and from the MCMC by crosses. The upper horizontal dashed line represents the error level $50\%$. Figure taken from Ref.~\cite{Boileau:2021sni}.

\small Cosmological amplitude uncertainty estimates from the Fisher Information study denoted by solid lines and from the MCMC by crosses. The upper horizontal dashed line represents the error level $50\%$. Figure taken from Ref.~\cite{Boileau:2021sni}.


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