Weakly parametric approach to stochastic background inference in LISA

Detecting stochastic gravitational wave backgrounds (SGWBs) with the Laser Interferometer Space Antenna (LISA) is one of the mission’s scientific objectives. Disentangling SGWBs of astrophysical and cosmological origin is a challenging task, further complicated by the noise level uncertainties. In this study, we present a Bayesian methodology for inferring SGWBs, drawing inspiration from Gaussian stochastic processes. We assess the effectiveness of this approach for signals with unknown spectral shapes by systematically exploring the model hyperparameters—a preliminary step toward a more efficient transdimensional exploration. To validate our method, we apply it to a representative astrophysical scenario: the inference of the astrophysical background of extreme mass ratio inspirals, as recently estimated [F. Pozzoli et al., Phys. Rev. D 108, 103039 (2023)]. Our findings indicate that the algorithm is capable of recovering the injected signal even with uninformative priors, simultaneously providing an estimate of the noise level.

Figure 1

Realizations from two EGP models, with fixed baseline and knots amplitudes sampled from their respective parameter space. Dashed gray lines represent the knot locations. EGP realizations are shown with solid lines, while dashed red (blue) lines correspond to the fixed power-law (bump) baseline.

Figure 2

Noise realizations from GP and EGP model. Top panel: realizations from GP. The gray shaded region represents the 90% credible interval. Bottom panel: realizations from EGP. The gray area represents the prior 90% credible interval. While GP allows for straightforward inclusion of observed data, EGP smoothly explores a large set of spectral shapes, with the data (i.e. knot amplitudes at chosen locations shown as gray dashed vertical lines) effectively proposed as inference parameters.

Figure 3

Spectral reconstruction of two toy model SGWBs. Injected SGWBs are denoted by black solid lines. They have SNR of 209 and 5, respectively. This places them above and below the nominal power-law sensitivity curve (solid blue line) to an SGWB with SNR of 10 after 4 years of observation. Noise injection and prior 90% confidence intervals are shown as gray solid and purple dashed lines, respectively. Posterior median and 90% confidence intervals on each signal—analyzed independently—are denoted by red dashed lines and light-red shared areas, respectively. The EGP flexibility captures features in the PSD shape.

Figure 4

Top panel: grid of logarithmic Bayes factor for different pairs of number of signal knots ($n$) and integers of kernel length scale $\sigma$. The numbers are computed in relation to the model that has the highest evidence, specifically $(n,m_{\sigma })=(8,32)$. We observe a diagonal pattern: as $m_{\sigma }$ decreases and $n$ increases, the model becomes progressively less preferred. Bottom panel: computational cost (i.e. sampling time as reported by nessai) of parameter estimations as function of integer multiple of $\sigma$, $m_{\sigma }$ (with $\sigma =0.6$) and the number of knots of the signal model, $n$. The numbers are in unit of kilohours. Each inference is run in multithreading over 40 cores.

Figure 5

Posterior distribution for the inference on EMRI background, with a noise-only model. The noise estimation employs three knots, with frequencies corresponding to vertical dashed gray lines in the main panel. Bottom subpanels show posteriors on individual knot amplitudes. The injected spectral noise model corresponds to ${\mathit{\delta }}^n=\mathbf{0}$ and is denoted by solid blue lines. The noise model can compensate with significant biases the presence of an SGWB in the data. Nonetheless, its evidence is significantly lower when compared to the noise and signal models, as illustrated in Fig. 4.

Figure 6

Signal reconstruction of the M1 EMRI background carried out using the EGP parametrization. In both figures, the noise estimation employs three knots. Top panel: the model features eight knots for the signal and a kernel length scale that is 32 times larger than $\sigma$. This configuration aligns with the model exhibiting the highest evidence. Bottom panel: the model features eight knots for the signal, and the kernel length scale is set equal to $\sigma$. This configuration corresponds to the model with the lowest evidence. Blue numbers denote the cumulative SNR integrated from ${10}^{-4}\mathrm{Hz}$ up to the frequency they are located at.

Figure 7

Analog illustration of Fig. 6 where the inference has been conducted without incorporating the T channel in the data. Consequently, the resulting posterior regions exhibit greater dispersion, signifying reduced informativeness. Blue numbers denote the cumulative SGWB SNR integrated from ${10}^{-4}\mathrm{Hz}$ up to the frequency they are located at.

Figure 8

Posterior parameter distribution for the toy-model SGWB with $\mathrm{SNR}=209$ described in Sec. 5a, obtained with an EGP model inference. The signal injected parameters are $\{{\log }_{10}{\mathit{\delta }}^h,{\log }_{10}A,\gamma \}=\{[0.6,-0.2,-0.3,0.4],-43,-2\}$. The injected noise level matches the nominal LISA sensitivity curve [51]. Darker (lighter) shaded areas denote 90% (50%) credible regions, and solid orange lines indicate the injected values. Posteriors are consistent with the injected values at the 90% credible level, and the noise parameters are consistent with zero with accuracies between ${10}^{-3}$ and ${10}^{-2}$.

Figure 9

Posterior parameter distribution for M1 EMRI SGWB for model $(n,m_{\sigma })=(8,32)$ using a larger and narrower prior on ${\mathit{\delta }}^h$. Posteriors shown in this plot correspond to the spectral reconstructions in top panels of Figs. 6 and 7, respectively. They support the result interpretation provided in Sec. 5b2.

Figure 10

Posterior parameter distribution for M1 EMRI SGWB for model $(n,m_{\sigma })=(8,1)$ using a larger and narrower prior on ${\mathit{\delta }}^h$. Posteriors shown in this plot correspond to the spectral reconstructions in bottom panels of Figs. 6 and 7, respectively. They support the result interpretation provided in Sec. 5b2.