### A.4 Computation of SIGWs including primordial NGs

In this appendix, we provide additional technical details regarding the MCMC analyses that resulted in figures 25 and 24, as well as the parameter scan leading to figure 23. As reported in section 4.5, the trispectrum arising from the local expansion eq. (4.37) leads to additional contributions to the SIGW spectrum. In particular, from the connected part, one obtains the following two terms

$$ \begin{aligned} \Omega_{\text{GW}}(k, \eta)|_{t} = & \frac{1}{12\pi} \left(\frac{k}{aH}\right)^2 \left(\frac{3}{5} f_{\text{NL}}\right)^2 \int_{0}^{\infty} dt_1 \int_{-1}^{1} ds_1 \int_{0}^{\infty} dt_2 \int_{-1}^{1} ds_2 \\ & \times \int_{0}^{2\pi} d\varphi_{12} \cos 2\varphi_{12} \frac{u_1 v_1}{(u_2 v_2)^2} \frac{1}{w_{a,12}^3} \overline{\tilde{J}(u_1, v_1, x) \tilde{J}(u_2, v_2, x)} \quad (\text{A.3}) \\ & \times \mathcal{P}_{\zeta_g}(v_2 k) \mathcal{P}_{\zeta_g}(u_2 k) \mathcal{P}_{\zeta_g}(w_{a,12} k), \end{aligned} $$

and

$$ \begin{aligned} \Omega_{\text{GW}}(k, \eta)|_{u} = & \frac{1}{12\pi} \left(\frac{k}{aH}\right)^2 \left(\frac{3}{5} f_{\text{NL}}\right)^2 \int_{0}^{\infty} dt_1 \int_{-1}^{1} ds_1 \int_{0}^{\infty} dt_2 \int_{-1}^{1} ds_2 \\ & \times \int_{0}^{2\pi} d\varphi_{12} \cos 2\varphi_{12} \frac{u_1 u_2}{(v_1 v_2)^2} \frac{1}{w_{b,12}^3} \overline{\tilde{J}(u_1, v_1, x) \tilde{J}(u_2, v_2, x)} \quad (\text{A.4}) \\ & \times \mathcal{P}_{\zeta_g}(v_1 k) \mathcal{P}_{\zeta_g}(v_2 k) \mathcal{P}_{\zeta_g}(w_{b,12} k), \end{aligned} $$

were the integration variables $t_i$ and $s_i$ are defined as in eq. (4.19). To keep the equation concise, some terms have been left expressed as functions of $u_i$ and $v_i$, but they have to be intended as depending on the integration variables $t_i$ and $s_i$. Moreover, we introduced $\tilde{J}(u_i, v_i, x) = v_i^2 k^2 \sin^2 \theta I(u_i, v_i, x)$, with $I(u, v, x)$ the integration kernel defined in the main text and $w_{a,12}$ and $w_{b,12}$, defined as

$$ w_{a,12} = [v_1^2 + v_2^2 - 2v_1 v_2 (\cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 \cos \varphi_{12})]^{1/2}, \quad (\text{A.5}) $$

and

$$ w_{b,12} = [1 + v_1^2 + v_2^2 + 2v_1 v_2 (\cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 \cos \varphi_{12}) - 2v_1 \cos \theta_1 - 2v_2 \cos \theta_2]^{1/2}. \quad (\text{A.6}) $$

The sine and cosine functions are related to the integration variables by

$$ \cos \theta_i = \frac{1 - s_i(1 + t_i)}{t_i - s_i + 1}, \quad \sin^2 \theta_i = \frac{(1 - s_i^2)t_i(2 + t_i)}{(t_i - s_i + 1)^2}. \quad (\text{A.7}) $$

As shown in eqs. (A.3) and (A.4), the evaluation of the NG corrections requires a 5 dimensional integration for each of the frequencies at which the final GW spectrum is evaluated. However, $f_{\text{NL}}$ and $A_s$ are multiplicative parameters and the effect of $k_*$ just results in a shift of the spectrum along the k-axis. Hence, once the spectrum is evaluated for a fixed width $\Delta$, it can be used for different values of the parameters reported above, without requiring further evaluation. When $\Delta$ is varied, instead, a new evaluation of the spectrum is required each time. Hence, just a single evaluation of the spectrum would require relatively little time, but the evaluation of the whole spectrum for each point of the MCMC would notably slow down
the run, making it difficult to get the final posterior in a reasonable time, also considering the presence of other parameters in the MCMC evaluation.

For this reason, to speed up the evaluation we proceed as follows: we numerically pre-compute a grid of NG contributions to the GW spectra as a function of frequency for different widths, in order to explore the range $log_{10} \Delta \in [-1, 1]$. This grid is then used to obtain the NG corrections to the spectrum corresponding to any value of $\Delta$ in the range considered, by interpolating them from the pre-computed ones. In detail, to get the spectrum corresponding to those values of $\bar{\Delta}$ not present in the grid, we first search for $\Delta_{max}$ and $\Delta_{min}$, respectively immediately above and below $\bar{\Delta}$. Then we compute the interpolated spectrum by Taylor expanding around these values, obtaining

$$ \Omega_{GW}(\bar{\Delta}) = \Omega_{GW}(\Delta_{max})w_{min} + \Omega_{GW}(\Delta_{min})(1 - w_{min}), \quad (A.8) $$

with

$$ w_{min} = \frac{(\bar{\Delta} - \Delta_{min})}{(\Delta_{max} - \Delta_{min})}. \quad (A.9) $$

For the evaluation of the scans that require a Fisher forecast and hence the derivatives with respect to the parameters, we proceed in a similar way. We pre-compute a grid of derivatives<fnref n="21" /> in the range $log_{10} \Delta \in [-1, 1]$ and then we interpolate as explained above.

## A.5 Inference

Once $\Omega_{GW}$ has been computed by the SIGWAY, the resulting spectrum is interpolated in log-space and passed to the SGWBinner which computes the posterior distribution according to eq. (5.21). Cobaya [384, 427] is used as an inference-framework. We use different samplers for Monte Carlo sampling depending on the dimensionality, requirements, and structure of the posterior surface:

* The inferences in figures 4, 5, 9, 10, 12, 1322 have been run using the nested sampler nessai [428-430].

* Figures 6, 7, 20, 2425 have been obtained using Cobaya's CosmoMC [431, 432] MCMC, where we started the chains at the injected values and injected FIM estimates of the covariance matrix to speed up convergence.

* The evidences in section 7 have been computed with the nested sampler PolyChord [416, 417].

* The inference for figures 15 and 16 was performed using the active learning algorithm GPry [433, 434] due to the prohibitively slow speed of computing $P_{\zeta}$ stemming from the Bessel functions in its equation.

<footnote n="21">²¹Note that when taking the derivative with respect to $A_s$ and $f_{NL}$ the integrals remain unchanged, hence we consider the same pre-computed NG contributions used in the MCMC runs. When taking the derivative with respect to $k_*$ or $\Delta$, instead, since the integrand is varied, we pre-compute a grid for each of these two derivatives.</footnote>
<figure><img src="image_30.png" /><figcaption>**Figure 28.** Same as figure 6 but showing the marginalised contours for all sampled parameters including the noise parameters $A_{\text{noise}}$, $P_{\text{noise}}$. There is a relatively mild degeneracy between $A_{\text{noise}}$ and $\alpha_{\text{Gal}}$ but no correlations between the signal parameters $A_s$, $\Delta$, $k_*$ and the noise parameters. We found similar correlations (or the lack thereof) for all other injections.</figcaption></figure>

All corner plots have been created with `GetDist` [435]. In the corner plots, we omitted showing the marginalised constraints on $A_{\text{noise}}$ and $P_{\text{noise}}$, as in all cases they were well constrained and showed weak degeneracies with the signal parameters. To give an idea of how tightly these parameters tend to be constrained, figure 28 shows a corner plot including the constraints on $A_{\text{noise}}$ and $P_{\text{noise}}$ for the injected lognormal $P_{\zeta}$ (see eq. (3.5)).

## B Challenges with binned analyses and a large number of bins

The binned approach to performing the double integration going from $P_{\zeta}$ to $\Omega_{\text{GW}}$ introduced in section 4.4 — while in principle extremely powerful at reconstructing any SIGW spectrum without model-dependence — unfortunately suffers from some crucial shortcomings as will be explained in this section.
<figure><img src="image_31.png" /></figure>

**Figure 29.** Three different spectra in $\Omega_{\text{GW}}$ generated with the binned approach with 100 bins where for each spectrum only one of the bins is non-zero. The black dashed line shows the approximate power law integrated sensitivity of LISA assuming a 4-year mission. It is clear from this picture that the three adjacent bins shown are entirely degenerate when trying to resolve them with LISA as the peaks are well outside the sensitivity and the causality tails generated are exactly the same.

For the sake of illustration, we will only consider the case where all modes reenter during radiation domination (see section 4.2) where the kernel is $k$-independent. The situation changes a bit if the kernel has a $k$-dependence such as is the case during an early matter domination era (see section 4.3), however, our main arguments remain unchanged.

It is clear from the structure of the integral in eq. (4.20), that a single wavenumber $k$ in $P_\zeta$ affects multiple frequencies in $\Omega_{\text{GW}}$. An easy way to understand this is to consider a $P_\zeta$ that is sufficiently close to a monochromatic source $P_\zeta(k) = A_s\delta(k - k_*)$. In our binned approach this translates to one single bin $A_*$ being non-zero. Figure 29 shows three such spectra with 100 bins, where each one contains a single non-zero bin $A_*$ (bin nr. 85, 86, 87 in this case). It is evident from this figure that if the peak towards $k_*$ is not resolved, and only the causality tail in the IR regime enters the LISA sensitivity, these spectra become entirely degenerate. This means that there is not necessarily a unique mapping $\Omega_{\text{GW}} \mapsto P_\zeta$. In other words, the power from the bins is “leaking” into adjacent bins.

In reality, this means that, for many bins and towards low SNR, the binned reconstructed $P_\zeta$ is highly degenerate, and valid reconstructions include “oscillations” between the bins as lower power in one bin can be compensated by higher power in an adjacent one.

The aforementioned degeneracies are not unexpected and are really just a feature of the physical properties of the process of scalar-induced gravitational waves. However, they do induce some practical complications. In an ideal template-agnostic pipeline, we would want to perform inference on the $N_{\text{bins}}$ bins in $P_\zeta$ to reconstruct the signal with the highest evidence parameterization. Due to the large (non-linear) degeneracies between the bins, the likelihood is far from Gaussian and the FIM approximation is invalid, making MC-sampling necessary. Sampling over this space is very computationally challenging due to (a) the very narrow
<figure><img src="image_32.png" /><figcaption>**Figure 30.** Same as figure 4, but for an injected BPL spectrum following the benchmark USR model and using 50 bins.</figcaption></figure>

degeneracies (b) the resulting large number of posterior modes and (c) the high dimensionality of the parameter space. In practice, this leads to overconfidence in the reconstruction, as some posterior modes are inevitably missed or underexplored by the MC sampler. Figure 30 shows the binned reconstruction of the USR model injection from section 2.4 with 50 bins. The oscillation effect is clearly visible in the low SNR region, where $P_ζ$ oscillates between high power and low power, thus overconstraining certain bins. These degeneracies are partially broken by fewer bins, as visible in figure 4.

This leaves us in a dilemma: we would like to bin $P_ζ$ as finely as possible to increase the frequency resolution of the template, but as one increases $N_{\text{bins}}$ the posterior becomes much more difficult to sample. In our tests, we found $N_{\text{bins}} = 15$ to be reliable in terms of convergence for the BPL signal (figure 4) and $N_{\text{bins}} = 40$ for the injected lognormal signals in $\Omega_{\text{GW}}$ and $P_ζ$ (figure 26) that occupy less of the frequency range. However, it is clear that this low number of bins cannot reconstruct the shape of $P_ζ$ with high fidelity.

Luckily, this problem does not appear when no signal is present, as the posterior distribution in $A_i$ becomes a simple upper bound, an unimodal structure that is easy to map by a nested sampler, even in high dimensions. We can therefore conclude that the upper bounds obtained by this method are reliable even with many bins.

Future work on this approach could include studying improved bases for the reconstructed bins (a basis of Gaussians or other wide kernels in $P_ζ$ may be less multi-modal), or improving sampling by manually adding jump proposals to the degenerate modes of the posterior in a given basis (e.g. as is done for LISA black hole binary sampling in *BBHx* [58]).
<figure><img src="image_33.png" /><figcaption>Figure 31. Same as figure 6 for the same injected parameters as in section 6.5.</figcaption></figure>

### C Testing the resolvability of non-Gaussian corrections: additional plots

Figure 31 shows a corner plot that was obtained by injecting a purely Gaussian SIGW signal with a lognormal shape in $P_ζ$ (see eq. (3.5)) that is equivalent to the cases discussed in figures 24 and 25 with $f_{NL} = τ_{NL} = 0$. The remaining injected parameters are the same as in section 6.5: $log_{10} A_s = -2$, $log_{10} Δ = -0.75$, $log_{10}(k_*/s^{-1}) = -2.3$. By comparing these results to the one obtained for $f_{NL} = 1$ shown in figure 24 and $f_{NL} = 10$ shown in figure 25, we see that the NG contribution neither significantly improves, nor worsens the constraints on the signal and foreground parameters.

### D Testing the scalar-induced hypothesis: additional plots

Figure 32 provides further insight into the quality of reconstruction when comparing the SGWBinner and the SIGWAY methods. The noise is accurately reconstructed in all cases, with only a slight underestimation in case 1 (left panel) using the SIGWAY template reconstruction. In contrast, the extragalactic background is reconstructed less accurately with the SGWBinner compared to the other methods.

A particularly notable observation is that in case 1 (left column), the SIGWAY template method significantly underestimates both the extragalactic and galactic foregrounds. This underestimation can be attributed to the model compensating for excess power in the causality tail by reducing the power allocated to the foregrounds, due to the limited flexibility in the shape of $Ω_{GW}$ provided by the template. Interestingly, in case 2 (right column), the SIGWAY method — using the template or not — performs better than the SGWBinner in reconstructing the foregrounds. This improvement arises because the assumption of SIGWs
<figure><img src="image_34.png" /><figcaption>**Figure 32.** Reconstruction of ΩGW for case 1 (left) and case 2 (right) including the noise and foreground. See section 7 for more details. The TT-channel component of the noise is above 10⁻⁷ and we therefore omit it.</figcaption></figure>

enforces a specific shape for ΩGW, which cannot be easily mimicked by the foregrounds. In contrast, the SGWBinner does not impose such restrictions during reconstruction. This result stresses the importance of adopting optimal modeling of the eventual cosmological signal even when reconstructing the astrophysical properties of the foreground sources.

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