Cosmic string gravitational wave backgrounds at LISA: II. Reconstruction of conventional signals over astrophysical foregrounds

Author(s)

Dimitriou, Androniki, Figueroa, Daniel G., Simakachorn, Peera, Stomberg, Isak, Zaldivar, Bryan

Abstract

We study the reconstruction of conventional cosmic-string signals with LISA in the presence of all major known astrophysical foregrounds expected in the LISA band. These include stellar-origin black-hole binaries (SOBHBs), galactic (WDs) and extragalactic (ExWDs) white dwarfs, extreme-mass-ratio-inspirals (EMRIs), and massive black-hole binaries (MBHBs). Using the Simulation-based Inference package GWBackFinder, we perform a joint inference on the LISA noise, foregrounds, and signal, across a range of injected string tensions $Gμ$. We find that reconstructing tensions with an error $\lesssim 10\%$ requires values as large as $Gμ\gtrsim 10^{-11}$, i.e. a factor $\sim10^5$ larger than previous estimates with no foregrounds, and $\sim 10^2$ larger compared to estimates accounting only for SOBHB and WD foregrounds. This work is the second in a series initiated in Ref. arXiv:2508.05395, which aims to quantify LISA's ability to measure representative cosmic-string models.

Figures

Caption

An example of marginalized posterior distributions for a single mock LISA observation with astrophysical foregrounds included, comparing the SBI posterior (green) to an independent MCMC analysis (black) at an injected string tension $G\mu_{\rm inj}=10^{-14}$. Red lines indicate the injected parameter values. The interval on top of each column denotes the 95\% highest-density interval of each parameter's posterior. The predictions on the instrumental-noise parameters ($A_{\rm acc}$ and $A_p$) are omitted for clarity, but they have been jointly inferred together with the rest of the parameters.

Caption

Reconstructed spectra for the cosmic string signal of $G\mu_{\rm inj} = 10^{-14}$ (red) and the different foregrounds considered in this work (except the MBHB which is not reconstructed): Galactic binaries (WD, blue), stellar-origin Black-hole binaries (SOBHBs, green), extragalactic white dwarfs (ExWDs, pink), extreme-mass-ratio- inspirals (EMRIs, yellow), and massive black-hole binaries (MBHB, gray), as well as LISA instrumental noise (AA channel). We show the injected contributions (dashed), posterior median (solid), as well as the 95\% CI bands.

Caption

Reconstruction precision $\delta G\mu$, defined in Eq.~\eqref{eq:precision}, as a function of the injected string tension $G\mu_{\rm inj}$. \emph{Upper:} For the fiducial foreground model, blue bars show the foreground-free precision, while the stacked orange bars indicate the degradation due to simultaneously inferring the astrophysical foreground amplitudes. The error bars indicate the standard deviation among 20 data realizations. \emph{Lower:} Consider three foreground-amplitude scenarios: The fiducial values (orange, cf. Table \ref{tab:foreground_fiducial}), a conservative case where these fiducials are shifted by $+1\sigma$ (dark gray) according to their corresponding priors, and an ``optimistic'' case where the fiducials are shifted by $-1\sigma$ (light gray). The red arrows in both panel indicate that the reconstruction precision exceeds the plot range.

Caption

Reconstruction precision $\delta G\mu$, defined in Eq.~\eqref{eq:precision}, as a function of the injected string tension $G\mu_{\rm inj}$. \emph{Upper:} For the fiducial foreground model, blue bars show the foreground-free precision, while the stacked orange bars indicate the degradation due to simultaneously inferring the astrophysical foreground amplitudes. The error bars indicate the standard deviation among 20 data realizations. \emph{Lower:} Consider three foreground-amplitude scenarios: The fiducial values (orange, cf. Table \ref{tab:foreground_fiducial}), a conservative case where these fiducials are shifted by $+1\sigma$ (dark gray) according to their corresponding priors, and an ``optimistic'' case where the fiducials are shifted by $-1\sigma$ (light gray). The red arrows in both panel indicate that the reconstruction precision exceeds the plot range.

Caption

Foreground-aware reconstruction precision from SBI (orange bars) compared with independent MCMC results (red circles) and Fisher-matrix forecasts (green stars). Error bars denote the standard deviation in SBI analysis across 20 independent mock LISA realizations. See also Tab.~\ref{tab:results}.

Caption

Reconstruction precision $\delta G\mu$, defined in Eq.~\eqref{eq:precision}, as a function of the injected string tension $G\mu_{\rm inj}$, assuming three different foreground budgets. Blue bars show the foreground-free precision; the stacked orange bars indicate the degradation due to all five foregrounds; and the green bars assume only two foregrounds (WD+SOBHB) as in \cite{Blanco-Pillado:2024aca}. The error interval on each bar indicates the standard deviation among 20 data realizations. The red arrows in both panels indicate that the reconstruction precision exceeds the plot range.

Caption

Example marginalized posterior distributions for two single mock LISA observations with astrophysical foregrounds included, comparing the SBI posterior (green) to an independent MCMC analysis (black) at injected string tensions $G\mu_{\rm inj}=10^{-13}$ (left) and $10^{-12}$ (right). Red lines indicate the injected parameter values. The interval on top of each column denotes the 95\% highest-density interval of each parameter's posterior. The prediction on the instrumental-noise parameters ($A_{\rm acc}$ and $A_p$) are omitted for clarity, but they have been jointly inferred together with the rest of parameters.

Caption

Example marginalized posterior distributions for two single mock LISA observations with astrophysical foregrounds included, comparing the SBI posterior (green) to an independent MCMC analysis (black) at injected string tensions $G\mu_{\rm inj}=10^{-13}$ (left) and $10^{-12}$ (right). Red lines indicate the injected parameter values. The interval on top of each column denotes the 95\% highest-density interval of each parameter's posterior. The prediction on the instrumental-noise parameters ($A_{\rm acc}$ and $A_p$) are omitted for clarity, but they have been jointly inferred together with the rest of parameters.

Caption

P--P plots for the foreground-free SBI model, showing empirical coverage as a function of confidence level for the three inferred parameters ($G\mu$, $A_p$, and $A_{\rm acc}$). The black curves show the empirical coverage, the green dashed lines correspond to perfect calibration, and the grey bands indicate the uncertainty due to the finite test set. The close agreement with the diagonal confirms well-calibrated posteriors.

Caption

P--P plots for the foreground-aware SBI model, showing empirical coverage for all seven inferred parameters: $G\mu$, $A_p$, $A_{\rm acc}$, $A_{\rm Gal}$, $A_{\rm ExB}$, $A_{\rm ExWD}$, and $A_{\rm EMRI}$. All parameters closely follow the diagonal within the finite-sample uncertainty band, demonstrating that the inclusion of astrophysical foreground nuisance parameters does not degrade posterior calibration.