<table><caption>**Table 13.** Projected ringdown SNR of GW150914 as detected by ET with different detector geometries (left column headers) and PSDs (top row headers). As explained in the main text, we do not observe differences between different PSDs because the signal is dominated by a monochromatic component at a frequency insensitive to the LF part of the PSD.</caption><thead><tr><th>SNR<sub>GW150914</sub></th><th>HFLF-cryo</th><th>HF-only</th></tr></thead><tbody><tr><td>Δ-10 km</td><td>141</td><td>141</td></tr><tr><td>Δ-15 km</td><td>190</td><td>190</td></tr><tr><td>2L-15 km-0°</td><td>196</td><td>196</td></tr><tr><td>2L-15 km-45°</td><td>192</td><td>192</td></tr><tr><td>2L-20 km-0°</td><td>240</td><td>240</td></tr><tr><td>2L-20 km-45°</td><td>235</td><td>235</td></tr></tbody></table>

In the following, we discuss the impact of detector designs on specific tests of the nature of compact objects and the physics at the BH horizon.

### 6.1.1 Testing the GR predictions for space-time dynamics near the horizon

The ringdown waveform originates from the perturbed remnant object, and can be approximated as a superposition of damped sinusoids characterized by the complex quasinormal modes (QNMs) of the remnant. The frequencies, damping times, amplitudes, and phases of QNMs depend on the binary progenitors and on the underlying theory of gravity.

There are countably infinite QNMs indexed by $(l, m, n)$, where $(l, m)$ denote the angular dependence of the mode and $n = 0, 1, 2$ is the overtone index (with $n = 0$ labelling the fundamental tone). The $(l, m, n) = (2, 2, 0)$ is the dominant mode in a binary BH ringdown signal and the excitation of the subdominant modes depends on the initial configuration of the progenitor binary [167–173].

If the remnant is a Kerr BH and the underlying theory is GR, the entire QNM spectrum is fully characterized by its mass $M_f$ and dimensionless spin $\chi_f$. The *BH spectroscopy* program [169, 174–178] aims at detecting more than one QNM, providing us with multiple independent null-hypothesis tests of GR. In addition, even if the remnant is described by the Kerr solution but the underlying theory is not GR or the progenitors are not BHs (for example NSs or more exotic objects), then the QNM amplitudes and phases would be different from the GR prediction [173]. Thus, measuring the ringdown modes in the post-merger signal of a binary coalescence provides a clean and robust way to test GR and the nature of the remnant.

The fundamental QNM frequency and damping time have been measured by the LVK Collaboration only for a few events, providing an independent measurement of the mass and spin of the remnant which is in agreement with what inferred from the inspiral-merger phase [8]. The first GW event, GW150914, stands out among those with the highest accuracy.

If measured with ET, GW150914 would have had a ringdown signal-to-noise ratio between 141 and 240 depending on the detector configuration, see table 13. It would therefore constitute a golden event with exquisite precision in the measurability of the QNM spectrum. We can quantify the measurability of a GW150914-like event with ET using a Fisher matrix formalism [176, 179]. We include the first three subdominant modes (3, 3, 0), (2, 1, 0) and (4, 4, 0), modeling their amplitudes through the numerical fits in [169, 173], and assuming a start time for the ringdown at $t \approx 10M$ after the peak of the waveform. The relative uncertainties on the dominant frequencies $f_{220}$ and damping times $\tau_{220}$ are prospected to scale
with the ringdown SNR as

$$ \frac{\Delta f_{220}}{f_{220}} \sim 0.2\% \left( \frac{100}{\text{SNR}} \right), \quad \frac{\Delta \tau_{220}}{\tau_{220}} \sim 2\% \left( \frac{100}{\text{SNR}} \right), \qquad (6.1) $$

which propagate similar relative errors on the mass and spin of the remnant (assuming the latter to be a Kerr BH) thus giving an estimate of the performance of consistency tests informed by the inspiral part of the signal [8]. On the other hand, the subdominant frequencies scale as

$$ \frac{\Delta f_{330}}{f_{330}} \sim 2\% \left( \frac{100}{\text{SNR}} \right), \quad \frac{\Delta f_{210}}{f_{210}} \sim 4\% \left( \frac{100}{\text{SNR}} \right), \quad \frac{\Delta f_{440}}{f_{440}} \sim 3\% \left( \frac{100}{\text{SNR}} \right). \qquad (6.2) $$

This shows that, by detecting high-SNR events, and in particular golden events with SNR ≥ 100, ET will be able to measure the QNM spectrum at the percent level for individual events. Moreover, by stacking together multiple events (see table 14 below), it will be possible to test deviations from the GR predictions at the sub-percent level.

Next, using the BBH population catalog generated with the FASTCLUSTER code [53, 54] as described in section 3, we estimate the event rates for BH spectroscopy with different ET designs. Figure 36 shows the distributions of source-frame mass $M_f^z = (1+z)M_f$ and $\chi_f$ for the final BH, as predicted from the simulated BBH merger events in the catalog, and computed from the analytical fits in [180, 181].<fnref n="29" /> The spin $\chi_f$ clusters around 0.7 as a consequence of the fact that the majority of the BBH events in the catalog have progenitor mass ratio close to unity (with only a small portion having mass ratio $q \equiv m_1/m_2 \gtrsim 5$) and moderate progenitor spins. Furthermore, we see that the distribution of the QNM fundamental frequency peaks at around 100 Hz, which sets the typical scale of the (almost monochromatic) ringdown signal.

In order to assess the feasibility of BH spectroscopy with different ET configurations, we compute the number of ringdown detections per year. Here a detection corresponds to SNR ≥ 12 in the ringdown-only portion of the signal. Moreover, we compute the number of ringdown detections per year with ringdown signal-to-noise ratio SNR ≥ 50 and with SNR ≥ 100, the latter representing golden events for BH spectroscopy. All detector designs are assumed in the HFLF-cryo configuration. We did not find any significant difference when the LF instrument is absent. This is due to the fact that the ringdown signal is almost monochromatic and, as shown in figure 36, there is a negligible portion of events for frequencies $f_{220} \lesssim 20$ Hz, where the contribution of the LF instrument impacts on the ET sensitivity curve. Results are reported in table 14 for the particular realization of the catalog considered here, where the rates are subjected to the statistical Poisson counting uncertainty $\sigma(N_{det}) \approx \sqrt{N_{det}}$.

In conclusion, for BH spectroscopy, the baseline 10 km triangle configuration allows for ~ 34 high-SNR events and 3 golden events per year. While these numbers are already extremely interesting, both the 15 km triangle design and the 15 km 2L design allow to increase by a factor ~ 3 – 4 the detection rate w.r.t. the baseline, with 90+ high-SNR events and 9+ golden events per year and a slight preference for the 15-km triangle. The 20 km 2L configurations show the best performances in terms of detected events, with ~ 240 high-SNR events<fnref n="30" /> and ~ 20 golden events per year. Table 14 also reports the event rates as measured

<footnote n="29">²⁹We also benchmarked them against surfinBH [182], which is a machine learning predictor trained on numerical simulations, finding excellent agreement within the common regimes of validity.</footnote>

<footnote n="30">³⁰A back-of-the-envelope estimate, assuming that by stacking $N_{det}$ signals the errors shown in eq. (6.2) are reduced by a factor $\approx \sqrt{N_{det}}$, suggests that by stacking multiple ringdown detections with SNR > 50, the 2L-15 km-45° configuration can reach $\approx 0.1\%$ accuracy for the subdominant QNMs.</footnote>
<figure><img src="image_45.png" /><figcaption>**Figure 36.** Top: distribution of the redshifted final masses and dimensionless final spins from the simulated BBH events as detailed in section 3. Bottom: corresponding distribution of the ringdown frequency and damping time of the dominant (2, 2, 0) mode.</figcaption></figure>

<table><caption>**Table 14.** Number of ringdown events per year above the detection threshold (SNR ≥ 12), with high signal-to-noise ratio (SNR ≥ 50), and golden events (SNR ≥ 100) for different proposed detector designs (all taken in the HFLF-cryo configuration) and compared to the most optimistic prospect for LVKI in O5. The last column displays the SNR of the loudest event in the catalogue. Here SNR refers to the ringdown part of the signal only.</caption><thead><tr><th></th><th>N<sub>det</sub>(SNR ≥ 12)</th><th>N<sub>det</sub>(SNR ≥ 50)</th><th>N<sub>det</sub>(SNR ≥ 100)</th><th>max(SNR)</th></tr></thead><tbody><tr><td>LVKI-O5</td><td>15</td><td>0</td><td>0</td><td>38</td></tr><tr><td>ET</td><td></td><td></td><td></td><td></td></tr><tr><td>Δ-10 km</td><td>4665</td><td>34</td><td>3</td><td>262</td></tr><tr><td>Δ-15 km</td><td>11692</td><td>117</td><td>13</td><td>317</td></tr><tr><td>2L-15 km-0°</td><td>11627</td><td>110</td><td>11</td><td>281</td></tr><tr><td>2L-15 km-45°</td><td>10175</td><td>97</td><td>9</td><td>330</td></tr><tr><td>2L-20 km-0°</td><td>18972</td><td>255</td><td>21</td><td>327</td></tr><tr><td>2L-20 km-45°</td><td>17185</td><td>228</td><td>18</td><td>384</td></tr></tbody></table>

by the LVKI network in O5. We see that ET will constitute a game changer for probing near-horizon physics.

Next, we investigate the redshift reach of BH spectroscopy for the two representative cases of a 10-km triangle and a 15km-45°-2L configuration. Figure 37 (left) shows the event rate per year as a function of the redshift: while ET is capable of detecting ringdown events up to redshift $z \lesssim 14$, high-SNR are limited to $z \lesssim 4$ and golden events are limited to
<figure><img src="image_46.png" /><figcaption>**Figure 37.** Ringdown event rates as a function of redshift (left) and mass ratio (right) for the baseline 10km triangle and the 15km-45°-2L configuration. Different colors indicate detectors configurations; continuous, dashed, and dotted contours indicate different ringdown SNR thresholds.</figcaption></figure>

$z \lesssim 2$. Figure 37 (right) displays the complementary information of the event rate as a function of the mass ratio: we see that the loud ringdown events are limited to small mass ratios $q \lesssim 3$ while high mass-ratio events are detected with moderate or small SNR. Since different mass ratios excite different modes in the ringdown spectrum, high mass-ratio events offer valuable complementary information and it is, therefore, desirable to develop efficient stacking techniques [183–185] to take advantage from them.

Finally, in table 15 we also computed the event rates when ET operates in synergy with a single 40 km CE detector and with a network of one 40 km CE and one 20 km CE. The results show that joint detections by ET in its baseline configuration and CE produce ringdown rates comparable with ET alone in its 2L-20 km configurations, and that a 15 km or 20 km ET in synergy with CE will constitute a major improvement for BH spectroscopy as it will result in between 30 to 50 golden events per year. We also observe that the differences among the results, induced by different choices of the ET geometry, remain quite significant even when ET is in a network with 1CE or with 2CE.

### 6.1.2 Searching for echoes and near-horizon structures

Exotic compact objects (ECOs) are horizonless objects predicted in certain quantum-gravity extensions of GR and in the presence of exotic matter fields [186–188]. Some broad motivations for ECOs are (see [187] for a review): i) resolving the singularities inevitably present inside BHs within GR; ii) evading deep conceptual conundrums associated with the presence of a horizon (most notably, the information loss paradox); or iii) simply providing effective models for new species of compact objects that might co-exist in the universe along with BHs and NSs.

Several models of ECOs have been conceived, including BH “fuzzball” microstates in string theories [189, 190] and boson stars as self-gravitating objects made of massive bosonic fields minimally coupled to GR [191, 192]. One can devise a model-independent framework [193] in which the ECO properties are parametrized in terms of their effective radius $r_0 = r_+(1 + \epsilon)$ (where $r_+$ is the would-be location of the Kerr BH horizon and $\epsilon$ is a dimensionless closeness parameter), and the reflectivity $R$ at the effective radius with a generic phase $\phi(R)$. The BH case is described by a null reflectivity at the horizon for any frequency,
<table><caption>**Table 15.** Same as table 14 but with a single 40 km CE detector (top) or with one 40 km CE detector and one 20 km CE detector (bottom) added to the network.</caption><thead><tr><th>ET (+1CE)</th><th>N<sub>det</sub>(SNR ≥ 12)</th><th>N<sub>det</sub>(SNR ≥ 50)</th><th>N<sub>det</sub>(SNR ≥ 100)</th><th>max(SNR)</th></tr></thead><tbody><tr><td>Δ-10 km</td><td>17268</td><td>188</td><td>18</td><td>298</td></tr><tr><td>Δ-15 km</td><td>23634</td><td>311</td><td>29</td><td>350</td></tr><tr><td>2L-15 km-0°</td><td>23342</td><td>311</td><td>29</td><td>317</td></tr><tr><td>2L-15 km-45°</td><td>22262</td><td>290</td><td>27</td><td>362</td></tr><tr><td>2L-20 km-0°</td><td>29307</td><td>504</td><td>41</td><td>359</td></tr><tr><td>2L-20 km-45°</td><td>28126</td><td>439</td><td>38</td><td>412</td></tr><tr><td>ET (+2CE)</td><td></td><td></td><td></td><td></td></tr><tr><td>Δ-10 km</td><td>20990</td><td>268</td><td>25</td><td>308</td></tr><tr><td>Δ-15 km</td><td>27065</td><td>383</td><td>38</td><td>359</td></tr><tr><td>2L-15 km-0°</td><td>26691</td><td>384</td><td>37</td><td>327</td></tr><tr><td>2L-15 km-45°</td><td>25749</td><td>357</td><td>37</td><td>370</td></tr><tr><td>2L-20 km-0°</td><td>32266</td><td>597</td><td>49</td><td>368</td></tr><tr><td>2L-20 km-45°</td><td>31197</td><td>531</td><td>45</td><td>420</td></tr></tbody></table>

whereas an ECO reflects part of the incoming radiation, i.e. $R \neq 0$ (the reflectivity is gener- ically complex and frequency dependent). Because of this reflectivity, ECOs emit a different GW signal relative to the BH case. One of their smoking guns is the emission of a modu- lated train of GW echoes in the late-time ringdown stage of a compact binary coalescence, associated to signals reflected off the object interior [194, 195]. When an ECO remnant with $\epsilon \ll 1$ forms, a ringdown analogous to the BH one is emitted at early times, followed by a new signal in the form of late-time echoes. Several searches for GW echoes in LVK data have been performed, claiming no evidence for their observation [8, 196–199] (although some results are controversial, see [188]).

Here, we assess the detectability of GW echoes with different ET configurations. We focus on a GW150914-like system (final mass $M_f = 70M_\odot$ and final spin $\chi_f = 0.68$) placed at a fiducial distance $d_L = 1$ Gpc. We use the frequency-domain echo template developed in ref. [200] for a representative choice $\epsilon = 10^{-5}$ where we vary the reflectivity in the range $R \in [0.01, 0.99]$ and set $\phi(R) = 0$. Figure 38 shows the fractional percentage errors (obtained with a Fisher matrix analysis) on the reflectivity of an ECO for different ET configurations. The fractional error is a proxy for the detectability of ECOs, since the BH case corresponds to $R = 0$ and therefore any putative measurement of $R$ should exclude $R = 0$ with the highest confidence level. We report the values of $\sigma_R/R$ after averaging over the sky position and the inclination angle. We take the detectors in the cryogenic HFLF configuration but, as in the ringdown analysis reported in section 6.1.1, we do not find significant deviations when the LF instrument is not operating.
<figure><img src="image_47.png" /><figcaption>**Figure 38.** Fractional percentage errors on the ECO reflectivity $\mathcal{R}$ for a GW150914-like event placed at $d_L = 1$ Gpc with final mass $M_f = 70M_\odot$ and final spin $\chi_f = 0.68$. The detector is taken in the cryogenic HFLF configuration and errors are averaged over the sky position and the inclination angle. There are no significant deviations in the figure for the HF-only configuration. The Fisher-matrix parameters are $\{M_f, \chi_f, \mathcal{R}, \phi(\mathcal{R}), \epsilon\}$, where $\mathcal{R}$ is generically a complex number, and the amplitude and phases $\{A_{lmn}, \phi_{lmn}\}$ of the excited modes.</figcaption></figure>

<table><caption>**Table 16.** Fractional percentage errors on the measurability of the ECO reflectivity $\mathcal{R}$ for small ($\mathcal{R} = 0.01$), medium ($\mathcal{R} = 0.5$) and large ($\mathcal{R} = 0.99$) reflectivity for the same system and detector configuration of figure 38.</caption><thead><tr><th rowspan="2"></th><th colspan="3">σ<sub>R</sub>/R [%]</th></tr><tr><th>R = 0.01</th><th>R = 0.5</th><th>R = 0.99</th></tr></thead><tbody><tr><td>Δ-10 km</td><td>422</td><td>7</td><td>0.06</td></tr><tr><td>Δ-15 km</td><td>308</td><td>5</td><td>0.04</td></tr><tr><td>2L-15 km</td><td>326</td><td>5</td><td>0.04</td></tr><tr><td>2L-20 km</td><td>265</td><td>4</td><td>0.03</td></tr></tbody></table>

*The main result is that the detector configuration affects the accuracy on the reflectivity of compact objects by a factor of ~2 between the 10 km designs and the 20 km designs, with the 15 km designs representing the best compromise.*

This is further quantified in table 16 for three values of the reflectivity (small, medium and large, respectively). We can ascribe these differences to the larger SNR of longer arm length configurations. The distribution of the SNR follows the same statistics of the ringdown SNR as reported in table 14 and figure 37 and therefore the same considerations apply about the advantage of a 15-km design relative to the baseline 10-km design.

### 6.1.3 Constraining tidal effects and multipolar structure

The multipolar structure of a Kerr BH can be elegantly written as [201],

$$M_{\ell}^{\text{BH}} + iS_{\ell}^{\text{BH}} = M^{\ell+1} (i\chi)^{\ell}, \qquad (6.3)$$

where $\chi := J/M^2$ is the dimensionless spin, $M = M_{00}$, $J = S_{10}$, $M_{\ell} = M_{\ell 0}$, $S_{\ell} = S_{\ell 0}$, with $M_{lm}$ and $S_{lm}$ being the mass and current multipole-moment tensors, respectively. For a Kerr BH, the only nonvanishing multipole moments have $m = 0$ and the mass (current) multipole
<figure><img src="image_48.png" /><figcaption>Figure 39. 1-σ uncertainty on the parameter controlling deviations on the symmetric combination of the spin-induced ECO quadrupole moment, $κ_s = 1 + δκ_s$, with $κ_s = 1$ corresponding to binaries with Kerr BHs. We assume the Kerr case ($δκ_s = 0$) and optimally-oriented sources with mass ratio $q = 2$, non-precessing spins $χ_1 = 0.9$, $χ_2 = 0.7$ at a luminosity distance of $d_L = 100$ Mpc. The optimal orientation depends on the detector location and configuration. The Fisher-matrix parameters are {$M_c, η, d_L, θ, φ, ι, ψ, t_c, Φ_c, χ_{1,z}, χ_{2,z}, δκ_s$}.</figcaption></figure>

moments vanish when $ℓ$ is odd (even). These properties are a consequence of the axial and equatorial symmetry of the Kerr solution.

In general, objects other than BHs violate these symmetries and the unique relation (6.3) between all multipole moments and the BH mass and spin. Their multipolar structure can be schematically written as

$$M_{lm} = M_{l}^{BH} + \delta M_{lm}, \quad S_{lm} = S_{l}^{BH} + \delta S_{lm}, \qquad (6.4)$$

where $δM_{lm}$ and $δS_{lm}$ are some model-dependent corrections to the mass and current multipole moments.

Generically, the mass quadrupole moment $M_{2m}$ is the dominant multipolar contribution, entering the inspiral GW signal at second PN order [202]. When $m \neq 0$, this term can induce binary precession even in the absence of progenitor spins [203]. Deviations from standard BH predictions can be parameterized by defining $M_{20} = -κM^3χ^2$, where $κ = 1$ corresponds to the Kerr case, while $κ(χ)$ in all other cases. At the 2PN order, contributions from the quadrupole moments of the binary components enter the waveform through symmetric and antisymmetric combinations given by $κ_s = (κ_1 + κ_2)/2$ and $κ_a = (κ_1 - κ_2)/2$, respectively. The latter can be recast in terms of deviations from the Kerr baseline as $κ_s = 1 + δκ_s$, and $κ_a = δκ_a$. Figure 39 shows the absolute errors on the parameter $δκ_s$ as a function of the binary total mass, for representative systems with mass ratio $q = 2$ and spins $(χ_1, χ_2) = (0.9, 0.7)$ aligned to the orbital angular momentum (and therefore nonprecessing). We use a TaylorF2 waveform model, truncating the inspiral phase at the ISCO of the Kerr remnant, also including corrections induced by the self-force and the spin of the less massive object [204].

We find that the differences on the uncertainties obtained assuming the triangle and the L-shaped configurations are typically of a factor two in favor of the triangle, across the range of masses we considered.

Another smoking-gun deviation from the standard Kerr BH in GR is given by the tidal deformability, considered in section 3.2 for BNSs. A remarkable result in GR is that the tidal
<figure><img src="image_49.png" /></figure>

**Figure 40.** Absolute (1-σ) uncertainty on the tidal deformability $\tilde{\Lambda}$ as a function of total mass $M_{\text{tot}}$ in the source frame for the various configurations. We assume negligible deformability of the source, nonspinning equal-mass binaries, and an optimally oriented source at a distance $d_L = 100$ Mpc. The optimal orientation depends on the detector location and configuration. The Fisher-matrix parameters are $\{M_c, \eta, d_L, \theta, \phi, \iota, \psi, t_c, \Phi_c, \chi_{1,z}, \chi_{2,z}, \tilde{\Lambda}, \delta\tilde{\Lambda}\}$. These results complement what presented in section 3.2 for BNSs to the case of ECOs or subsolar BHs (i.e. PBHs) potentially observable with ET, see more details in section 6.3.3.

deformability of BHs is precisely zero [205–207], at variance with any other object in GR or with BHs in modified gravity [208–213].

Figure 40 shows the absolute error on the binary tidal parameter $\tilde{\Lambda}$ as a function of the source frame total mass, for non-spinning, equal-mass, optimally oriented binaries, at luminosity distance $d_L = 100$ Mpc. We assume the TaylorF2 waveform,<fnref n="31" /> cutting the inspiral phase at the ISCO frequency of a remnant Schwarzschild BH with mass $\approx M_{\text{tot}}$. Extending the Fisher analysis to higher frequencies would significantly improve measurement errors but would require specific ECO models.

*We conclude that different ET configurations do not impact significantly on the accuracy in the measurement of the tidal deformability for $M_{\text{tot}} \lesssim O(M_{\odot})$, while differences are slightly larger, within typically a factor of order 2, for heavier binaries (with the 2L configurations performing again better than the triangle ones).*

## 6.2 Nuclear physics

One of the longstanding open challenges in nuclear physics is to determine the properties of matter at supranuclear densities. Given the extremely high central densities in neutron stars and the possibility to form merger remnants that not only probe the highest densities in our Universe before black hole formation but also matter at nonzero temperature, the Einstein Telescope will provide important nuclear-physics insights, see e.g., refs. [15, 214–222]. In particular, it will be possible to probe the cold EoS from the inspiral phase with unprecedented accuracy and furthermore, in contrast to existing gravitational-wave detectors, there is a high chance of detecting gravitational-wave emission from the merger and its

<footnote n="31">³¹We checked that, at the level of the comparison between different configurations, assuming the IMRPhe-nomD_NRTidalv2 waveform approximant gives analogous results. However, in the range of masses of its validity, the latter provides a smaller uncertainty on $\tilde{\Lambda}$ due to the fact that the waveform (calibrated on BNS coalescences) is valid up to higher frequencies, as well as to a dependence of the amplitude on the deformability (which is only included in the phase of TaylorF2).</footnote>
remnant which not only probes the cold EoS but would also allow us to place constraints at the EoS at temperatures up to ~50MeV; cf. e.g. [223] and references therein. For these reasons, it is essential to understand to which extent the different detector configurations impact possible nuclear-physics constraints.

### 6.2.1 Radius estimation from Fisher-matrix computation

We start by assessing the accuracy of constraining the EoS by providing estimates for the uncertainty in the inferred radius based on the Fisher matrix approach. The results used here differ from those in section 3.2 due to the assumption of a common EoS for the NSs when producing the events catalog. In particular, we adopt the DD-LZ1 model<fnref n="32" /> [224–226] and compute the adimensional tidal deformability parameters of the objects from their source-frame masses $m$, using the $\Lambda(m)$ relation predicted by this EoS. The choice of DD-LZ1 is dictated by its high allowed maximum mass (of $2.56 M_{\odot}$); this makes it consistent with the BNS population model used throughout this work which, as mentioned in section 3, assumes a source-frame mass distribution of the component objects uniform between $1.1 M_{\odot}$ and $2.5 M_{\odot}$. Moreover, the behavior of DD-LZ1 complies with all the present constraints from low energy nuclear physics [227]. As in section 3.2, the results are produced using the GWFAST package [17, 32] and adopting the IMRPhenomD_NRTidalv2 waveform model. A detailed study of the effect of the injection EoS and waveform model is ongoing.

We use two ways to transform the error of the tidal deformability $\tilde{\Lambda}$ [defined in eq. (3.1)] into the error on the measurement of the NS radius $R$. The first approach assumes that $\tilde{\Lambda} \propto R^6$, which leads to

$$ \frac{\Delta R}{R} = \frac{1}{6} \frac{\Delta \tilde{\Lambda}}{\tilde{\Lambda}}, \qquad (6.5) $$

and arises from relations derived in ref. [228]. This allows us directly to compute the individual radius uncertainty $\Delta R/R$ for all detections. We summarize these results in figure 41. Comparing different configurations (left panel), we find that the main limiting factor is the arm-length of the gravitational-wave detectors, so the 10 km triangle performs less well than a 15 km 2L. Consistently with a number of other results in section 3, the right panel of figure 41 shows that the 2L with 15 km arms at 45° in the HF-only configuration performs at a level comparable to the 10 km triangle in the full HFLF cryo configuration.

An important consequence of the large number of BNS detections with ET is that we can noticeably improve the EoS constraints by stacking information from multiple detections at different masses, which enables constraining the EoS at different densities. To quantify this, we perform a simple analysis in which we assume a common radius of 12km radius for all neutron stars. While this assumption is a priori unrealistic, it nevertheless captures the feature that different neutron star measurements will probe the same underlying hyper-parameters characterizing the EoS. To obtain a combined ansatz, we compute the average $(\Delta\tilde{\Lambda}/\tilde{\Lambda})$ and use

$$ \Delta R_{\text{all}} = \frac{1}{6\sqrt{N_{\text{det}}}} \left( \frac{\Delta \tilde{\Lambda}}{\tilde{\Lambda}} \right) 12\text{km}. \qquad (6.6) $$

<footnote n="32">³²We point out that the EoSs employed here do not contain phase transitions, however, we expect that the presence of a strong phase transition will likely influence the accuracy to which we can measure the neutron star radius.</footnote>
<figure><img src="image_50.png" /><figcaption>**Figure 41.** Distribution of the relative errors on the NS radius attainable from the adimensional tidal deformability combination $\tilde{\Lambda}$, as measured by different detector configurations and sensitivity curves. In particular, the left panel shows the results for the six geometries considered, all with their best sensitivity, the central panel for the 10 km triangle with the two different ASDs, and the right panel for the 2L with 15 km arms at 45° and the two ASDs considered.</figcaption></figure>

We present results for $R_{\text{all}}$ in table 17 for all ET configurations, from which we see that *loosing the LF instrument, the constraints gets worse by about a factor of 2, irrespectively of the geometry*.

In table 18, complementing the results in section 3.3, we also report the results for some ET geometries, all with their full HFLF cryogenic ASD, working in a network with 1 or 2 CE detectors. To improve the reliability of the Fisher matrix approach, we restrict the analysis to events with a relative error on the adimensional tidal deformability of $\Delta\tilde{\Lambda}/\tilde{\Lambda} \le 50\%$. This threshold is chosen to have a large number of events in each configuration and with each ASD, and also reflects the fact that the loudest events — with smaller errors — will be the ones providing most of the information. We emphasize that this error represents only the statistical uncertainty; however, systematic uncertainties, e.g., due to modeling errors in the gravitational waveform will also be important [229]. Moreover, we stress that the results shown in figure 41 and in tables 17 and 18 depend on the choice of the EoS. From the dependence of the tidal deformability on the star's compactness, it is evident that, with increasing mass, the tidal deformability decreases. Thus EoS models predicting large maximum masses in general also predict very low values of $\Lambda$ close to the maximum mass, which is difficult to constrain. Indeed, DD-LZ1 predicts tidal deformability as small as $\Lambda \sim 5$ for $m \approx 2.56$ M$_{\odot}$. Assuming a different EoS [which might require a lowering of the maximum mass in the mass distribution to accommodate the maximum mass allowed by the EoS] could result in tighter or broader constraints. This will be discussed in further details in section 6.2.2. Furthermore, the uniform distribution adopted in section 3.2 predicts smaller values for the error on the radius, but using an EoS enables including the physical effect of the tidal deformability parameter decreasing with increasing mass. The results presented here should be taken only as indicative, however, they still provide a relevant benchmark for comparing different detector configurations.

Our second approach is based on first generating a prior distribution of EoS by Monte-Carlo sampling of a large parameter set of 10 independent, uniformly distributed empirical parameters. These parameters characterize the density dependence of the energy in the symmetric matter (i.e. equal number of protons and neutrons) and of the symmetry energy (i.e. the variation of binding energy as a function of the neutron-to-proton ratio). They are given by the successive derivatives with respect to the density of the symmetric matter and symmetry energy functionals, calculated at the equilibrium density of symmetric matter.
<table><caption>**Table 17.** Number of detections, $N_{det}$ with $\Delta\tilde{\Lambda}/\tilde{\Lambda} \le 50\%$, and statistical error on the NS radius obtained combining all of them according to (6.6), $\Delta R_{all}$ (in meters), for various geometries and ASDs.</caption><thead><tr><th rowspan="2">Geometry</th><th colspan="2">HFLF cryo</th><th colspan="2">HF only</th></tr><tr><th>N<sup>ΔΛ/Λ≤0.5</sup><sub>det</sub></th><th>ΔR<sub>all</sub> [m]</th><th>N<sup>ΔΛ/Λ≤0.5</sup><sub>det</sub></th><th>ΔR<sub>all</sub> [m]</th></tr></thead><tbody><tr><td>Δ 10 km</td><td>4878</td><td>10.0</td><td>1412</td><td>19.2</td></tr><tr><td>Δ 15 km</td><td>15285</td><td>5.7</td><td>4047</td><td>11.3</td></tr><tr><td>2L 15 km 45°</td><td>12013</td><td>6.4</td><td>3194</td><td>12.4</td></tr><tr><td>2L 20 km 45°</td><td>25489</td><td>4.4</td><td>7387</td><td>8.2</td></tr><tr><td>2L 15 km 0°</td><td>11884</td><td>6.5</td><td>3357</td><td>12.1</td></tr><tr><td>2L 20 km 0°</td><td>23988</td><td>4.5</td><td>7081</td><td>8.3</td></tr><tr><td>1L 20 km</td><td>5464</td><td>9.4</td><td>1493</td><td>19.1</td></tr></tbody></table>

<table><caption>**Table 18.** Same as in table 17 with some selected geometries, all with the full HFLF cryogenic ASD, in combination with 1 CE detector with a length of 40 km and 2 CE detectors, one with a length of 40 km and the other of 20 km.</caption><thead><tr><th rowspan="2">Geometry</th><th colspan="2">HFLF cryo + 1CE</th><th colspan="2">HFLF cryo + 2CE</th></tr><tr><th>N<sup>ΔΛ/Λ≤0.5</sup><sub>det</sub></th><th>ΔR<sub>all</sub> [m]</th><th>N<sup>ΔΛ/Λ≤0.5</sup><sub>det</sub></th><th>ΔR<sub>all</sub> [m]</th></tr></thead><tbody><tr><td>Δ 10 km</td><td>24129</td><td>4.6</td><td>35694</td><td>3.8</td></tr><tr><td>2L 15 km 45°</td><td>35172</td><td>3.8</td><td>47791</td><td>3.2</td></tr><tr><td>2L 15 km 0°</td><td>35217</td><td>3.8</td><td>48581</td><td>3.2</td></tr></tbody></table>

Their prior distribution is consistent with the present empirical knowledge for a large set of nuclear data [227]. The use of the same functional to describe the core and the inhomogeneous crust [230] guarantees a consistent estimation of the crust-core transition inside the neutron star and thus consistent predictions for its radius. This approach enables incorporating priors from nuclear physics on the EoS and including the uncertainties at high densities. The only limiting assumption is that matter is composed of charged leptons, nucleons and nuclei only, and in particular that no first-order phase transition occurs. There might also be some dependence on the different approximations applied to treat inhomogeneous matter, e.g. assuming vanishing temperature, which could slightly modify the crust properties (see e.g. [231]). However, this should not affect the comparison of different configurations. The assumed nuclear prior complies with the chiral EFT energy per particle band for symmetric and pure neutron matter as given by [232] for baryon number densities $0.02 \le n_B \le 0.18\text{fm}^{-3}$, see [233] for details of the current implementation.

In figure 42 we show the mass-radius distribution for the nuclear prior described above along with the posteriors at 68% and 95% confidence intervals assuming 5970 simulated “observations” obtained from a Fisher matrix calculation. We also indicated the DD-LZ1 EoS used as input EoS in the calculation. The full compatibility of the DD-LZ1 model with nuclear physics constraints is shown by the fact that the injected mass-radius relation sits in the middle of the nuclear informed prior. This particular simulation was performed with geometry 2L -20 km with aligned arms (cryogenic), but we have checked that different configurations do not make any significant difference in the outcome, provided that the number of detections exceeds $N_{det} \sim 6000$.
<figure><img src="image_51.png" /><figcaption>**Figure 42.** Posterior distribution for NS mass-radius relation obtained assuming 5970 detections with geometry 2L-20 km with aligned arms. For comparison, the results obtained from the nuclear prior as well the underlying DD-LZ1 EoS model used in the Fisher matrix calculations are displayed.</figcaption></figure>

<table><caption>**Table 19.** Number of detections, $N_{det}$ along with $1\sigma$ uncertainty on radii for 1.4 and 2.0 solar mass NSs obtained for geometry 2L-20km-0° using nuclear prior.</caption><thead><tr><th>N<sub>det</sub></th><th>R<sub>1.4M<sub>☉</sub></sub><sup>+ΔR<sub>+</sub></sup><sub>-ΔR<sub>-</sub></sub> [km]</th><th>R<sub>2.0M<sub>☉</sub></sub><sup>+ΔR<sub>+</sub></sup><sub>-ΔR<sub>-</sub></sub> [km]</th></tr></thead><tbody><tr><td>Prior</td><td>12.983<sup>+0.420</sup><sub>-0.420</sub></td><td>13.156<sup>+0.447</sup><sub>-0.454</sub></td></tr><tr><td>54</td><td>13.163<sup>+0.221</sup><sub>-0.227</sub></td><td>13.358<sup>+0.234</sup><sub>-0.242</sub></td></tr><tr><td>592</td><td>13.146<sup>+0.122</sup><sub>-0.136</sub></td><td>13.355<sup>+0.099</sup><sub>-0.083</sub></td></tr><tr><td>5970</td><td>13.107<sup>+0.148</sup><sub>-0.037</sub></td><td>13.332<sup>+0.050</sup><sub>-0.013</sub></td></tr></tbody></table>

To understand the impact of different number of detections using the same geometry as above, we summarize the $1\sigma$ uncertainty on radii for 1.4 and 2.0 solar mass NSs with 54, 592 and 5970 detections in table 19. We can observe that the uncertainty on the radii only roughly follows the Poissonian law as described in eq. (6.6). The reason is probably that the radius is not a direct observable but, rather, is obtained in a convoluted way from the TOV equation and the underlying hypothesis on the EoS formalism. As a general statement, we can see that the Poisson hypothesis tends to underestimate the radius uncertainty from the Fisher matrix analysis, unless the number of detections is extremely large (more than $N_{det} \sim 6000$). Moreover, we should also stress that the uncertainty estimations of table 19 might be underestimated themselves, because of the restrictive hypothesis of a continuous equation of state at any density. Nevertheless, these results give us an indicative impact of ET measurements on the radii uncertainties using a fairly general EoS, within the hypothesis of purely nucleonic degrees of freedom.

### 6.2.2 Full parameter estimation results

In addition to our estimates based on the Fisher matrix computations, we also perform full parameter estimation studies.

For this analysis, we consider three different physical systems (A,B,C) corresponding to different input equations of state (APR4 and H4), mass ratios (symmetric or asymmetric), and spins (zero or non-zero). The values of those injection parameters can be seen on the
<figure><img src="image_52.png" /><figcaption>**Figure 43.** Left: constraints on the chirp mass $M_c$, mass ratio $q$, and tidal deformability $\tilde{\Lambda}$ for a binary neutron star system with parameters listed at the top of the panel. A decreasing minimum frequency noticeably increases the accuracy of measuring the chirp mass and also has an important effect on determining the mass ratio but has a smaller effect on the measurement of the tidal deformability. Right: constraints on the tidal deformability for three different physical systems and different detector designs using an initial frequency of 6Hz.</figcaption></figure>

<table><caption>**Table 20.** Source properties used for the full PE injections in section 6.2.2.</caption><thead><tr><th>Name</th><th>Injection model</th><th>m<sub>1</sub>, m<sub>2</sub> [M<sub>☉</sub>]</th><th>M<sub>c</sub> [M<sub>☉</sub>]</th><th>Λ<sub>1</sub>, Λ<sub>2</sub></th><th>Λ̃</th><th>χ<sub>1</sub>, χ<sub>2</sub></th></tr></thead><tbody><tr><td>Source A</td><td>APR4</td><td>1.68, 1.13</td><td>1.19479</td><td>77, 973</td><td>303</td><td>0,0</td></tr><tr><td>Source B</td><td>APR4</td><td>1.38, 1.37</td><td>1.19700</td><td>275, 309</td><td>292</td><td>0.02, 0.03</td></tr><tr><td>Source C</td><td>H4</td><td>1.38, 1.37</td><td>1.19700</td><td>1018, 1063</td><td>1040</td><td>0, 0</td></tr></tbody></table>

different panels of figure 43, and on table 20. The choice of the particular systems follows the injection study of [234] done by the LVK.

For these studies, we use the relative binning technique [235–237] to reduce the computational cost of our analysis. Figure 43 shows some of our main findings, and we refer to [238] for more details. We analyze signals injected in zero noise, to avoid the impact of fluctuations from noise when analyzing different starting frequencies. The left panel shows the impact of varying the initial frequency on the recovery of the chirp mass, the mass ratio, and the
<table><caption>**Table 21.** Values for the nuclear-matter parameters up to second order in units of MeV except for $n_{sat}$, which is in units of fm$^{-3}$. For ET, the two configurations leading to the smallest and largest uncertainties have been chosen; all other configurations give uncertainties in-between these.</caption><thead><tr><th>source</th><th>Geometry</th><th>n<sub>sat</sub></th><th>E<sub>sat</sub></th><th>K<sub>sat</sub></th><th>E<sub>sym</sub></th><th>L<sub>sym</sub></th><th>K<sub>sym</sub></th></tr></thead><tbody><tr><td rowspan="2">source-A</td><td>2L 20km 0°</td><td>0.164<sup>+0.005</sup><sub>-0.004</sub></td><td>-16.18<sup>+0.41</sup><sub>-0.40</sub></td><td>228<sup>+34</sup><sub>-24</sub></td><td>31.65<sup>+2.85</sup><sub>-2.15</sub></td><td>40.9<sup>+9.1</sup><sub>-13.9</sub></td><td>-262<sup>+61</sup><sub>-70</sub></td></tr><tr><td>Δ 10km</td><td>0.164<sup>+0.005</sup><sub>-0.007</sub></td><td>-16.14<sup>+0.44</sup><sub>-0.42</sub></td><td>229<sup>+27</sup><sub>-25</sub></td><td>31.59<sup>+3.41</sup><sub>-2.99</sub></td><td>40.1<sup>+14.8</sup><sub>-15.1</sub></td><td>-269<sup>+83</sup><sub>-70</sub></td></tr><tr><td rowspan="2">source-B</td><td>2L 20km 0°</td><td>0.162<sup>+0.007</sup><sub>-0.007</sub></td><td>-16.06<sup>+0.54</sup><sub>-0.47</sub></td><td>227<sup>+29</sup><sub>-27</sub></td><td>30.91<sup>+2.69</sup><sub>-2.81</sub></td><td>42.0<sup>+21.9</sup><sub>-19.0</sub></td><td>-197<sup>+155</sup><sub>-141</sub></td></tr><tr><td>Δ 10km</td><td>0.163<sup>+0.006</sup><sub>-0.007</sub></td><td>-16.05<sup>+0.53</sup><sub>-0.49</sub></td><td>227<sup>+27</sup><sub>-26</sub></td><td>30.86<sup>+3.34</sup><sub>-2.86</sub></td><td>40.9<sup>+17.1</sup><sub>-18.9</sub></td><td>-208<sup>+165</sup><sub>-137</sub></td></tr><tr><td rowspan="2">source-C</td><td>2L 20km 0°</td><td>0.155<sup>+0.005</sup><sub>-0.004</sub></td><td>-15.94<sup>+0.44</sup><sub>-0.56</sub></td><td>236<sup>+21</sup><sub>-17</sub></td><td>30.96<sup>+1.54</sup><sub>-2.06</sub></td><td>70.3<sup>+11.7</sup><sub>-10.3</sub></td><td>74<sup>+78</sup><sub>-87</sub></td></tr><tr><td>Δ 10km</td><td>0.156<sup>+0.004</sup><sub>-0.005</sub></td><td>-15.92<sup>+0.47</sup><sub>-0.49</sub></td><td>236<sup>+23</sup><sub>-19</sub></td><td>30.92<sup>+1.58</sup><sub>-2.02</sub></td><td>70.8<sup>+11.2</sup><sub>-11.8</sub></td><td>74<sup>+78</sup><sub>-87</sub></td></tr></tbody></table>

<table><caption>**Table 22.** Radii (km) for $1.4M_{☉}$ neutron stars as obtained for different geometries and different sources from the full parameter estimation runs under the nucleonic hypothesis. Again, as in table 21, only the geometries leading to the smallest and largest uncertainties, respectively, are shown here.</caption><thead><tr><th>source</th><th>Geometry</th><th>R<sub>1.4M<sub>☉</sub></sub> [km]</th></tr></thead><tbody><tr><td rowspan="2">source-A</td><td>2L 20km 0°</td><td>11.44<sup>+0.13</sup><sub>-0.13</sub></td></tr><tr><td>Δ 10km</td><td>11.34<sup>+0.15</sup><sub>-0.15</sub></td></tr><tr><td rowspan="2">source-B</td><td>2L 20km 0°</td><td>11.30<sup>+0.19</sup><sub>-0.21</sub></td></tr><tr><td>Δ 10km</td><td>11.32<sup>+0.20</sup><sub>-0.20</sub></td></tr><tr><td rowspan="2">source-C</td><td>2L 20km 0°</td><td>13.69<sup>+0.07</sup><sub>-0.06</sub></td></tr><tr><td>Δ 10 km</td><td>13.68<sup>+0.06</sup><sub>-0.07</sub></td></tr></tbody></table>

tidal deformability. Overall the width of the 90% confidence interval for the recovery of the chirp mass reduces by more than an order of magnitude when going from 20 to 6Hz, and also the size of the 90% confidence interval for the mass ratio shrinks by a factor of 2. However, the constraint on the tidal deformability reduces only by about 25%. Comparing the $\Delta 10\text{km}$ and the $2L$ 15km setup, we find that the $2L$ configuration is overall better for measuring tidal deformability by about 25%, where this difference is more or less independent of the initial frequency. As shown in [238], this is due mainly to the different arm lengths. In the right panel of figure 43, we compare the $\tilde{\Lambda}$ posteriors for three different physical systems, and five different detector configurations. In this case, we use an initial frequency $f_{\text{low}} = 10$ Hz for all the runs, and a noise simulated with the HFLF cryogenic sensitivity. We find that in general, the aligned or misaligned $2L$ detector configurations lead to similar results. The $2L$ configurations result in tighter bounds on $\tilde{\Lambda}$ than the $\Delta 10\text{km}$ one; as above, these improvements arise mainly from the different arm lengths. It is also worth pointing out that the error on $\tilde{\Lambda}$ for the three selected sources, as reported in the right panel of figure 43, is consistent with that found by Fisher matrix calculations within a few percent, for the various detector configurations considered.

### 6.2.3 Connected uncertainty of nuclear-physics parameters

An immediate question is to what extent the different proposed configurations of the instrument can improve on our knowledge of nuclear matter and NS properties. In section 6.2.1 we have already discussed estimates for the uncertainties on the NS radius resulting from the different detector configurations. Here we will continue the discussion by analysing the full
<figure><img src="image_53.png" /><figcaption>**Figure 44.** Posterior distribution for NS mass-radius relation (left) and the nuclear symmetry energy as function of baryon number density (right) for source A with geometry 2L-15 km with aligned arms. For comparison, the results obtained from the nuclear prior are also shown.</figcaption></figure>

parameter estimation results for the three sources discussed above. As above, we employ the prior on the EoS from present-day nuclear physics knowledge [233] as a baseline. Because of the nuclear physics constraints, a number of popular equations of state might be only marginally compatible with our nuclear physics informed prior. This is particularly the case of the APR4 model, that was never confronted to nuclear physics data. This can be seen from the NS mass-radius relation shown in figure 44 (left panel) for source A and the 2L geometry with 15 km arm length. In this figure, we can see that the APR model predicts too small radii for the heaviest neutron stars, with respect to our nuclear prior.<fnref n="33" /> This explains why the injection value is not precisely recovered in the posterior distribution. For this reason, the radius uncertainty estimation will not be reliable for the higher masses and is not analyzed in the following. In table 21 we summarise the posterior results for the different nuclear-matter parameters up to second order at saturation density with the three sources for the two configurations of the instrument leading to the smallest and largest uncertainty, respectively. These parameters represent the equilibrium density ($n_{\text{sat}}$) and energy ($E_{\text{sat}}$), the symmetry energy at saturation ($E_{\text{sym}}$) and its slope ($L_{\text{sym}}$), and the nuclear matter compressibility in the isoscalar ($K_{\text{sat}}$) and isovector sector ($K_{\text{sym}}$). For the symmetry energy, the posterior as a function of density is shown in figure 44 (right panel) for source A and the 2L geometry with 15 km arm length. The results indicate first of all that the injected properties are reasonably well recovered, even if the input model is only marginally compatible with our nuclear physics informed prior. On the scale of this figure, the differences between the proposed configurations of the instrument are insignificant and the 2L geometry is only slightly favored concerning the constraints on the nuclear-matter parameters.

In table 22 we summarise the posterior results for the NS radius at the fiducial mass 1.4 $M_{\odot}$, with the three sources for the two configurations of the instrument leading to the smallest and largest uncertainty, respectively. We can see that the uncertainties depend stronger on the injection model than on the configuration of the instrument. The absolute values are different from those obtained with the Fisher matrix approach, and reported in table 17 and 19. This is essentially due to the fact that the results in table 22 are

<footnote n="33">³³In this figure, in order to allow for slightly smaller radii, we have relaxed the 2 solar mass constraint for the nuclear prior which is applied in figure 42 above.</footnote>
obtained on single events with very high SNR, which thus result in tight constraints on the source parameters, while the ones in section 6.2.1 are produced from runs on a full realistic population of sources, the vast majority of which have much higher statistical uncertainties, resulting in looser constraints on the EoS parameters. Also, we again stress that the results presented in table 17 are obtained through the extremely simplified metric in eq. (6.6), and should thus only be considered as indicative for the comparison of the various detector configurations.

To conclude, even if the quantitative uncertainties depend on many not fully controlled factors, from the NS mass-radius posterior shown in figure 44 (left panel), the improvement with respect to current knowledge by Einstein Telescope is clearly seen. In addition, the present analysis assumes at each time one detected source, i.e., an inspiral with two given masses, whereas the Einstein Telescope will detect tens of thousands of different BNS merger events with a distribution of NS mass. This staggering will considerably reduce the uncertainty on the density dependence of the dense matter EoS and hence the nuclear-matter parameters and the information on the NS radius; see also the discussion in section 6.2.1. *The impact of the instrument geometry on these conclusions is only marginal; as above, the results indicate that the 2L geometry is favored, but the difference with the other geometries is not significant.*

### 6.2.4 Postmerger detectability

A BNS system typically merges at a frequency of ~1 kHz. After the merger, there are a number of physical processes, such as thermal effects, turbulences, phase transitions, magnetohydrodynamical instabilities, dissipative processes etc. [215, 239-244], which drive the postmerger physics leading to different remnants, depending on the masses of the neutron stars and EOS of the system. The merger remnant can be either a black hole, a hypermassive NS (HMNS), a supramassive NS (SMNS) or a massive NS. If the mass of the merger remnant is above a threshold mass it will collapse to a BH [245-247]. Otherwise, there is a post-merger gravitational-wave signal that can last over a time scale of about 10ms up to several 100ms [248] emitted at frequencies of ~2-4 kHz. While the postmerger signal from a BNS merger is not expected to be detectable for 2G detectors due to their insufficient sensitivity at high frequencies, it will be accessible to 3G detectors. Due to the complicated physical effects involved, postmerger signals are most accurately represented by numerical relativity (NR) simulations. For this study we will use NR waveforms [222, 249-252].

To determine the detectability of the postmerger signal with different detector configurations, we will compute and compare the SNR from the postmerger signal for a set of NR BNS waveforms as seen by the various configurations considered in this study. The postmerger SNR is defined as the SNR of the signal from the merger frequency [253] of the BNS system, up to 8192 Hz. We use a set of 6 BNS-NR waveforms from the SACRA catalogue [254], the masses and tidal deformabilities of which are given in table 23. For each detector geometry and PSD configuration, and each NR waveform, we use the same random set of 60 (ra, dec, $\psi$) values and quote the averaged postmerger SNR ($\rho_{pm}^{\text{Avg}}$) for each of the NR waveforms.

The left panel of figure 45 shows $\rho_{pm}^{\text{Avg}}$ for all the detector geometry — PSD combinations. for the system 15H_135_135_00155. As the high-frequency sensitivity of the HFLF cryo and HF-only PSDs are quite similar, the postmerger SNRs for these two PSDs are almost equal. We see the majority of variation occurring with varying arm-length for a given detector shape and changing detector shapes (triangular vs. 2L configurations). The maximum $\rho_{pm}^{\text{Avg}}$
<table><caption>**Table 23.** Table of the BNS-NR waveforms and their physical parameters used to compute the postmerger SNR.</caption><thead><tr><th>Configuration name</th><th>mass1</th><th>mass2</th><th>Λ<sub>1</sub></th><th>Λ<sub>2</sub></th></tr></thead><tbody><tr><td>15H_135_135_00155</td><td>1.35</td><td>1.35</td><td>1211</td><td>1211</td></tr><tr><td>125H_107_146_0015</td><td>1.07</td><td>1.46</td><td>3196</td><td>535</td></tr><tr><td>H_117_156_00155</td><td>1.17</td><td>1.56</td><td>1415</td><td>238</td></tr><tr><td>H_135_135_00155</td><td>1.35</td><td>1.35</td><td>607</td><td>607</td></tr><tr><td>125H_121_151_00155</td><td>1.21</td><td>1.51</td><td>1621</td><td>435</td></tr><tr><td>H_118_155_00155</td><td>1.18</td><td>1.55</td><td>1354</td><td>249</td></tr></tbody></table>

<figure><img src="image_54.png" /><figcaption>**Figure 45.** Left: averaged postmerger SNR for the 15H_135_135_00155 system for different detector configuration and PSD combinations. Right: averaged postmerger SNR for all the BNS-NR systems as given in table 23 for different detector configuration and HLFL cryo PSD. Each BNS-NR system is plotted with a different color with the detector configuration given on the x-axis.</figcaption></figure>

for this system is seen by the 2L 45° 20km HLFL configuration which is ~20% larger than the minimum value, which corresponds the Δ 10km HF configuration.

The right panel of figure 45 shows $\rho_{pm}^{\text{Avg}}$ for all the BNS-NR systems for the different detector configurations and only the HFLF cryo PSD. For all the BNS-NR systems, 2L 20km 45° gives the highest postmerger SNR. For $\rho_{pm}^{\text{Avg}}$, there is not much difference between the SNR for 2L 0° and 45° (~0.7%) irrespective of the arm length. Going from 15km 2L to 20km 2L gives a ~6% increase in $\rho_{pm}^{\text{Avg}}$. Between Δ 10 km and Δ 15 km, we see ~11% increase.

These results show that the performances of the different geometries are relatively similar within a 10–20% level, with 2L 45° 20km detector configuration being the best for detecting postmerger signals.

### 6.2.5 Conclusions: nuclear physics with ET

The Einstein Telescope will significantly advance our ability to constrain fundamental nuclear-physics properties, going well beyond the capabilities with any existing observatory
or experiment, and providing complementary constraints to explore large domains of the QCD phase diagram. For example, the upgrade of the FAIR at the GSI Helmholtz Centre for Heavy Ion Research will probe matter which is only slightly neutron-rich and at lower baryon densities and higher temperatures than what is expected for ET, which can reach the most extreme densities. Furthermore, the X-ray observatory NICER and proposed future instruments, such as eXTP, Strobe-X or ATHENA will be able to improve on current knowledge of cold dense matter by observing more pulsars and increasing the length of the observational data period and thus obtain NS radii with a precision of a few percent, similar to the anticipated improvements with the existing advanced LIGO and advanced Virgo detectors (even in combination with KAGRA and LIGO India) or proposed upgrades such as LIGO A# and Virgo_next. ET, in addition to being able to determine NS radii with a statistical precision of sub-percent order due to the immense statistics with $10^5$ mergers expected per year, will also be the only instrument able to add to the zero temperature β-equilibrated NS matter the information on the hot ultra-dense matter from the postmerger phase. With regards to the payoffs for nuclear physics, there is no significant difference between the different detector configurations, with longer arm-lengths leading to slightly better results.

## 6.3 Population studies

### 6.3.1 Merger rate reconstruction

Since ET will have a long distance range, it will see binary coalescences out to cosmological distances. Therefore, it will probe the merger rate up to high redshifts. Knowing how the BBHs are distributed in the Universe would help constrain their various formation channels, leading to a better understanding of how such binaries form [54, 255–258]. Different models exist, often assuming that the merger rate follows the star formation rate with some additional contribution coming from primordial black holes. Models differ in the details but most of them agree on the general shape of the distribution: the merger rate increases up to a redshift between 1 and 2 before decaying for higher redshifts (see [255, 258] for examples). The merger rate as a function of the redshift is written as [259]

$$ \frac{dR}{dz}(z) = \rho(z) \frac{dV}{dz}, \qquad (6.7) $$

where $V(z)$ is the comoving volume up to redshift $z$ and

$$ \rho(z) \propto \int_{t_d^{\min}}^{+\infty} \frac{\rho_*(z_f(z, t_d))}{1 + z_f(z, t_d)} P(t_d) dt_d \qquad (6.8) $$

is the binary formation rate, where $\rho_*$ is the star formation rate, $z$ is the redshift of interest, $z_f$ is the redshift at which the binary is formed, $t_d$ is the time delay between the formation of the binary and its merger, and $P(t_d)$ is the probability to have a given time delay.

To evaluate how well different configurations for ET would be able to identify the merger rate density, we follow a procedure similar to [260], with a theoretical population made of 10 years of BBHs as given by [258], and discussed in section 3. Then, we start by assigning a random sky location to the binaries. We compute their (network) SNRs and keep only the events with an SNR higher than 9 for a given configuration. The others are considered non-detectable. These events are then used to estimate the detector's efficiency as a function of the redshift. For each event, we perform a Fisher matrix analysis to obtain a multivariate Gaussian likelihood and sample from it to have an approximation for the
<figure><img src="image_55.png" /><figcaption>**Figure 46.** *Top:* representation of the reconstructed merger rate with the 90% confidence interval for 10 years of observation of ET in a triangular shape with 10km arms (orange) and ET with 2 L-shaped detectors with 15km arms at 45° (green). The blue line represents the true merger rate distribution. The two configurations enable one to reconstruct the merger rate but the accuracy is better for the 2 L-shaped detectors. *Bottom:* ratio of the width of the 90% confidence interval for the triangular 10km configuration with the other possible configuration. The red line corresponds to the case where the two detectors are equivalent, while the blue line is the average ratio. The hierarchy to be seen in this figure is that the 2-L aligned detectors are better than the triangular ones, and the 2L with arms at 45° are more accurate than the 2 L's with arms at 0°. For a given configuration, the increase in arm length gives a small improvement, mainly due to the increase in SNR.</figcaption></figure>
<table><thead><tr><td>Other configuration</td><td>Ratio width 90% confidence interval</td></tr></thead><tbody><tr><td>Δ 15 km</td><td>1.2</td></tr><tr><td>2 L 0° 15 km</td><td>1.8</td></tr><tr><td>2 L 0° 20km</td><td>2.0</td></tr><tr><td>2 L 45° 15 km</td><td>2.9</td></tr><tr><td>2 L 45° 20 km</td><td>3.3</td></tr></tbody></table>

**Table 24.** Mean values for the ratios of the 90% confidence intervals in the merger rate reconstruction between ET triangle with 10 km arm-lengths and the other possible configuration envisaged in this work. 2 L's are more accurate than one triangle, with L's rotated by 45° the one with respect to the other being better. This happens because of the reduced error on the sky location. For a given configuration, increasing the arm length by 5km does not strongly increase the accuracy since the only source of improvement is the increased SNR for the observed events.

measurements [219]. For each detectable BBH, we make a Gaussian centered on the mean value of the recovered posterior (which corresponds roughly to the injected value since the result of a Fisher matrix is centered on the true value), with a width equal to the standard deviation of the luminosity distance posterior obtained from sampling the Fisher likelihood. We then draw a *measured* luminosity distance. This is done for all the detectable BBHs. Then, we convert the luminosity distances to redshifts and count the number of events that we have in the redshift bins. This gives an observed merger rate, which is then corrected by the efficiency to account for the number of mergers we theoretically expect to miss for each redshift bin. We repeat the drawing of measurements 250 times to get error bars on the observed merger rate.

The top panel of figure 46 shows the reconstructed merger rate for 10 years of detections with ET seen as a single triangle with 10 km arms versus ET as 2 L's with 15km arms and a 45° rotation between the two detectors. The other panels show the ratio in accuracy between the single triangle with 10 km arms and the other configurations considered in this work. Although the two configurations lead to a correct reconstruction of the merger rate, one sees that the configuration made of 2 L-shaped detectors leads to a more accurate reconstruction as it has smaller error bars and closer central values. The increase in the ratio is such that a hierarchy can be established between the different configurations. 2 L's are better than a single triangle and L's rotated by 45° are better than aligned ones. Table 24 shows the mean ratio found between the detectors. The improvement from one detector configuration to the other is driven by a reduction in error on the luminosity distance. On the other hand, for a given configuration, 5 km longer arms are slightly better, which can be understood by the increased SNR of events. Nevertheless, it is still possible to have a good grasp of the merger rate density for all the configurations within 10 years of observation. This should be quite helpful in the quest to reveal the formation channels for BBH mergers.

### 6.3.2 Constraints on PBHs from high-redshift mergers

Primordial black holes (PBHs) may form in the early universe and represent a dark matter (DM) candidate (see e.g. ref. [261] for a review). Many constraints were set on their abundance in a wide range of masses [262] and recent GW detections suggest that, within standard scenarios, PBHs can only account for a subdominant DM component in the stellar mass range [122, 263–286]. Nonetheless, a discovery of a sub-population of PBHs would have fundamental consequences for our understanding of cosmology and particle physics (see e.g. [287–291]).
One “smoking-gun” signature of the primordial scenario is the production of high redshift mergers, which are unique targets of 3G detectors. The PBH model predicts a merger rate density that grows monotonically with redshift [267, 269, 271, 292, 293], with a power-law scaling of the form $R_{\text{PBH}}(z) \propto [t(z)/t(z=0)]^{-34/37}$, where $t(z)$ indicates the age of the universe at redshift $z$. This rate evolution extends up to redshifts $z \gtrsim \mathcal{O}(10^3)$. On the contrary, even though large uncertainties are still present, astrophysical-origin mergers are generically not expected to occur at $z \gtrsim 30$ in standard cosmologies [294] (see also [124–127, 295–308]).

Recently, it was shown that a PBH population which may be compatible with a subset of the recent LVK observations would also lead to a large number of high-redshift detections at 3G experiments [280]. Observations of such distant events may be characterised by large measurement uncertainties on the inferred luminosity distance due to the low SNR [309–312]. However, focusing on constraining the merger rate evolution at redshift larger than $\mathcal{O}(10)$ may allow probing PBH populations up to abundances as low as $f_{\text{PBH}} \approx 10^{-5}$ [313] in the solar mass range (even including the contamination of a population of Pop-III binaries), which would improve current capabilities by orders of magnitude. Therefore, the search for high-redshift primordial mergers is one of the scientific targets of the ET experiment.

In order to test the performance of different configurations to search and constrain a PBH population, we consider the benchmark PBH model resulting from the analysis in ref. [282]. The PBH model hyper-parameters are: $[M_c, \sigma]$ determining the PBH mass distribution, modeled as a lognormal function with mean $M_c$ and width $\sigma$; the abundance $f_{\text{PBH}}$ normalised with respect to the dark matter; the redshift $z_{\text{cut-off}}$ characterising PBH accretion [314]. These are assumed to be given by the population inference result of ref. [282], constraining the potential contribution from a PBH subpopulation to the GWTC-2 catalog and representing an upper bound on the PBH abundance in this mass range. In particular, we adopt $M_c = 34.54M_{\odot}$, $\sigma = 0.41$, $f_{\text{PBH}} = 10^{-3.64}$ and $z_{\text{cut-off}} = 23.90$, such that the PBH channel may be adequate in explaining around (1–21)% of the detections in the O1/O2/O3a run of LVC, and given the associated astrophysical models considered, the mass gap event GW190521.

We compute the number of detectable PBH events, $N_{\text{det}}$, by estimating the detector network selection bias based on an injection of $N = 10^6$ events at redshift $z \ge 10$. We randomise event directions and orientations, and we assume negligible PBH spins (compatible with the prediction of the standard scenario at high redshift [271, 314, 315]). The computation is performed using the IMRPhenomHM waveform model. We set a detection threshold SNR $\ge 9$. The minimum testable abundance is conservatively estimated by requiring at least one detectable event per year with $z \ge 30$ [280, 316]. This simplified approach provides an estimate of the ET capabilities which is consistent with actual population analyses, as performed in ref. [313].

We report the result of the analysis for the various configurations in tables 25, 26 and 27, detailing how the detection prospects change as a function of the configuration geometry, the inclusion of a single 40 km CE detector in the network, and removal of the low-frequency instrument, respectively. In each table, we indicate the number of detectable events above redshift $z > 10$ and $z > 30$, alongside the estimated minimum testable abundance. The comparison of tables 25 and 27 shows the important of the LF instrument for the observation of PBH at large redshifts. In particular, without the LF instrument, there are no detection at $z \ge 30$, for all the geometries considered.

The importance of the LF instrument when searching high redshift mergers can be intuitively understood by looking at figure 47, where we compare the signal amplitude for a
<table><caption>**Table 25.** Capabilities for constraining high-redshift PBH mergers of the various ET geometries (all taken with their HFLFcryo ASD). From left to right, columns report the number of detectable events per year at redshift larger than 10 or 30 assuming the PBH population from ref. [282]. The rightmost column reports the minimum testable PBH abundance, see more details in the main text. In bold we indicate the reference configuration.</caption><thead><tr><th>Configuration</th><th>N<sub>det</sub>(z > 10) [1/yr]</th><th>N<sub>det</sub>(z > 30) [1/yr]</th><th>f<sub>PBH</sub><sup>constrained</sup> [×10<sup>-5</sup>]</th></tr></thead><tbody><tr><td>Δ-10km</td><td>1140</td><td>77</td><td>2.61</td></tr><tr><td>Δ-15km</td><td>1764</td><td>261</td><td>1.42</td></tr><tr><td>2L-15km-0°</td><td>1597</td><td>238</td><td>1.48</td></tr><tr><td>2L-15km-45°</td><td>1651</td><td>221</td><td>1.54</td></tr><tr><td>2L-20km-0°</td><td>1984</td><td>434</td><td>1.10</td></tr><tr><td>2L-20km-45°</td><td>2080</td><td>416</td><td>1.12</td></tr></tbody></table>

<table><caption>**Table 26.** Same as table 25 for four ET geometries in a network configurations including one 40km Cosmic Explorer. ET is only taken here in its HFLFcryo configuration.</caption><thead><tr><th>Configuration</th><th>N<sub>det</sub>(z > 10) [1/yr]</th><th>N<sub>det</sub>(z > 30) [1/yr]</th><th>f<sub>PBH</sub><sup>constrained</sup> [×10<sup>-5</sup>]</th></tr></thead><tbody><tr><td>CE40km</td><td>1373</td><td>47</td><td>3.34</td></tr><tr><td>Δ-10km + CE40km</td><td>1940</td><td>180</td><td>1.71</td></tr><tr><td>Δ-15km + CE40km</td><td>2276</td><td>372</td><td>1.19</td></tr><tr><td>2L-15km-45° + CE40km</td><td>2210</td><td>333</td><td>1.26</td></tr><tr><td>2L-20km-45° + CE40km</td><td>2476</td><td>522</td><td>1.00</td></tr></tbody></table>

<table><caption>**Table 27.** Same as table 25 for configurations with the HF-only ASD.</caption><thead><tr><th>Configuration</th><th>N<sub>det</sub>(z > 10) [1/yr]</th><th>N<sub>det</sub>(z > 30) [1/yr]</th><th>f<sub>PBH</sub><sup>constrained</sup> [×10<sup>-5</sup>]</th></tr></thead><tbody><tr><td>Δ-10km-HF</td><td>15</td><td>0.00</td><td>—</td></tr><tr><td>Δ-15km-HF</td><td>85</td><td>0.00</td><td>—</td></tr><tr><td>2L-15km-0°-HF</td><td>75</td><td>0.00</td><td>—</td></tr><tr><td>2L-15km-45°-HF</td><td>69</td><td>0.00</td><td>—</td></tr><tr><td>2L-20km-0°-HF</td><td>178</td><td>0.00</td><td>—</td></tr><tr><td>2L-20km-45°-HF</td><td>170</td><td>0.00</td><td>—</td></tr></tbody></table>

merger with source frame mass $M_{\text{src}}^{\text{tot}} = 20M_{\odot}$ located close to the horizon (i.e. having SNR = 9 in the Δ-10km-HFLF-Cryo configuration). Due to the large source redshift, the detector frame mass is redshifted by a large factor, and the majority of SNR can only accumulate in the LF portion of the sensitivity curve.

In order to fully exploit the high redshift smoking-gun signature of a PBH population of mergers, sufficient precision on the source distance is also necessary, to distinguish PBHs from potential Pop-III contaminants. The performance of ET on single-event-identification was studied in details ref. [309, 312] adopting a Bayesian parameter estimation. A limited precision on the source redshift may also impact the ability to reconstruct the merger rate
<figure><img src="image_56.png" /></figure>

**Figure 47.** Comparison between the amplitude of a PBH binary signal $2f|\tilde{h}(f)|$ and the ET detector sensitivity curves ($fS_n)^{1/2}$ shown in green (10km-HFLF-Cryo) or yellow (10km-HF). We consider a distant merger with source frame mass $M_{\text{src}}^{\text{tot}} = 20M_{\odot}$ at $z = 52$. Also, we show a subsolar merger with $M_{\text{src}}^{\text{tot}} = 1M_{\odot}$ at $z = 0.88$. In both cases, we assume $q = 1$, optimal orientation of the binary and the redshift is fixed in order to have SNR=9 in the $\Delta$-10km-HFLF-Cryo configuration.

evolution at high redshift within a population analysis, as shown in refs. [311, 313]. Based on a Fisher Matrix analysis, we compare the performance of different configurations in figure 48, which reports the maximum source redshift above which a relative precision on $d_L$ better than 10% and 1% cannot be achieved. The relative error $\sigma_{d_L}/d_L$ is highly impacted by the performance of the detector configurations at low frequencies and follows the general trend observed in the behaviour of the horizon (see figure 3) and number of detectable high redshift PBH events (i.e. tables 25, 26 and 27). We refer to [317] for further details on the analysis.

As far as the comparison of the performance of the various configurations is concerned, the results can be summarised as follows.

i) The 10 km triangle is the one that performs less well. In its full HFLF-cryo configuration, for the assumed PBH population, the 2L-15km-45° configuration would detect about three times more events with $z > 30$ than the 10 km triangle.

ii) A dramatic reduction in the number of high-redshift event is observed in the absence of the LF instrument, to the extent that no events are detected at redshifts $z > 30$, where a detection is identified as a PBH. This is also reflected in the behaviour of the uncertainties on the luminosity distance.

This is due to the reduced capabilities at small frequencies, where most of the high-$z$ events are expected, as they are characterised by a large detector frame total mass.

### 6.3.3 Other PBH signatures

In this section, we analyse some other observables which may be crucial to discover/constrain the population of PBHs (see ref. [310] for a systematic overview and [317] for further details on the present analysis).
<figure><img src="image_57.png" /><figcaption>**Figure 48.** Contour lines of constant relative error on $d_L$ as a function of source frame total mass $M_{\text{tot}}$ and source redshift $z$. Different colors indicate different configurations while each panel (from left to right) assume HFLF-Cryo and HF, respectively. The solid (dashed) lines correspond to $\sigma_{d_L}/d_L = 10\% (1\%)$. We assume equal masses $m_1 = m_2$ and optimal orientation of the binary. As a consequence of setting $\iota = 0$, aligned and misaligned 2L configurations result in the same performance for this kind of estimation.</figcaption></figure>

○ **Subsolar mass range.** The distribution of PBH masses $m_{\text{PBH}}$ is determined by the characteristic size and statistical properties of the density perturbations, corresponding to curvature perturbations generated during the inflationary epoch. Within this standard formation scenario, $m_{\text{PBH}}$ is related to the mass contained in the cosmological horizon at the time of the collapse, and a wider range of masses is accessible compared to astrophysical BHs [318]. PBHs, in particular, can have subsolar masses, which are not expected from standard stellar evolution, while they can also populate the astrophysical mass gaps [291, 319, 320]. For masses $m_i \gtrsim 3M_{\odot}$ PBHs should be distinguished from stellar-origin BHs by mass-spin measurements [310, 314, 321]. The range $M_{\odot} \lesssim m_i \lesssim 3M_{\odot}$ can be distinguished from NS thanks to tidal deformability measurements (see section 6.1.3). Finally, the subsolar range represents a smoking-gun signature of the non-standard, i.e. potentially primordial, origin of a binary.34,35

The Einstein Telescope will dramatically extend the reach of current technology when searching for mergers with at least one component below the Chandrasekhar limit [316, 324]. In figure 49 we show the detection horizon in the subsolar mass range, extending figure 3 in the range of masses where only PBHs, white dwarfs, brown dwarfs, or exotic compact objects [187] (e.g. boson stars [325]) can appear. As a large portion of the SNR is built up in the HF range of the sensitivity curve, the horizons are only slightly dependent on the LF instrument. This feature is shown in figure 47, where we compare the signal amplitude for a subsolar merger with source frame mass $M_{\text{src}}^{\text{tot}} = 1M_{\odot}$ located close to the horizon (i.e. having SNR = 9 in the $\Delta$-10km-HFLF-Cryo configuration).

<footnote>34See, however, ref. [322] for models in which subsolar BHs are born out of dark sector interactions. Also, in this case, the detection of subsolar mergers would indicate new physics.</footnote>

<footnote>35While in the main text we focus on nearly equal mass mergers with $q \approx 1$, other potentially interesting systems for the detection of subsolar objects are intermediate mass ratio inspirals, given the high precision in measuring the mass of the light object at subpercent level [323]. Due to the relevance of the low-frequency range for these observations, the implementation of the LF interferometer could in principle enhance the capability of detecting these systems.</footnote>
<figure><img src="image_58.png" /><figcaption>**Figure 49.** Detection horizons for non-spinning PBH binaries assuming a mass ratio $q = 1$ or $q = 1/4$, for the various detector configurations considered. This plot extends figure 3 in the sub-solar mass range, of interest for PBHs.</figcaption></figure>

<figure><img src="image_59.png" /><figcaption>**Figure 50.** Relative error on the primary source frame mass (top) and source redshift (bottom) as a function of $m_1$ in the subsolar range. We consider a source located at a distance such that the SNR = 30 in the Δ-10km HFLF-Cryo configuration.</figcaption></figure>
<figure><img src="image_60.png" /><figcaption>**Figure 51.** Relative error on the eccentricity $e_0$ at $f_0 = 10$Hz as a function of its injected value for a binary with a total source frame mass of $M_{\text{tot}} = 2M_{\odot}$ and $d_L = 100$ Mpc in the upper panel, while $M_{\text{tot}} = 20M_{\odot}$ and $d_L = 500$ Mpc in the lower panel. For the range of eccentricities in the horizontal axis (that are too large for PBH binaries), one is able to exclude (at $3\sigma$ level) a PBH origin for events that fall below the horizontal black dashed line.</figcaption></figure>

In figure 50 we compare the performance of different ET configurations when measuring the source frame mass and distance of subsolar PBH mergers. In order to perform a proper comparison between different designs, for each mass $m_1$, we arbitrarily assume a source located at a distance fixed in order to have SNR= 30 in the Δ-10km HFLF-cryo configuration. We adopt the IMRPhenomHM waveform model and test both face-on ($i = 0$) and edge-on ($i = \pi/2$) binaries. In this case we conclude that the loss of the LF sensitivity results only in a minor reduction of the uncertainty on $m_1$ and $z$. The length of the arms, which leads to a larger SNR, is the main factor responsible for the different performance of the various designs.

○ **Eccentricity measurements.** Another key prediction of the PBH model involves the eccentricity $e$. While formed with large eccentricity at high redshift [326, 327], PBH binaries then have enough time to circularize before the GW signal can enter the observation band of current and future detectors [310]. Therefore, PBH binary candidates must have small eccentricities. Let us mention here that, on a more general scope, the measurement of eccentricity may also inform us about the possible astrophysical formation pathway of a binary. While isolated formation channels predict small values of $e$ in the observable range of
frequencies of 3G detectors, dynamical channels predicts a fraction $O(10\%)$ with $e > 0.1$ [54, 328–333]. This can be used to distinguish the origin of compact objects (see e.g. [334–337]).

To test the sensitivity of the various configurations to the measurement of a possible orbital eccentricity, we use the TaylorF2 inspiral-only waveform model [338–340] with the extension presented in [341] to account for a small eccentricity in the orbit. Using as reference values for the total source-frame mass $M_{\text{tot}} = 2M_{\odot}$ and $20M_{\odot}$, for each detector configuration, we compute the relative error that can be attained on equal mass, non spinning systems, as a function of the eccentricity $e_0$ defined at $f_{0,\text{ecc}} = 10\text{Hz}$. The distance to the source is fixed to $d_L = 100 \text{ Mpc}$ for the case $M_{\text{tot}} = 2M_{\odot}$ and to $d_L = 500 \text{ Mpc}$ for the other. The results for the relative errors attainable on $e_0$ are reported in figure 51. *We notice, in particular, the great relevance of the LF instrument to perform eccentricity measurements.* This can be traced to the fact that eccentricity gives larger effects during the inspiral (i.e. at low frequencies) since when going closer to a merger, a binary system tends to circularize, with $e_0 \propto f_{\text{GW}}^{-19/18}$. The range of values of $e_0$ shown on the horizontal axis of figure 51 corresponds to eccentricities that are too large for a PBH binary, that are expected to have $e_0 \sim 10^{-6}$ when they reach $f = 10 \text{ Hz}$ [310]. Therefore, in this range, if at the same time the relative error on $e_0$ for a given detection is sufficiently small, e.g. below the horizontal dashed line in the figure (which corresponds to $\Delta e_0/e_0 = 0.33$), one would be able to exclude that this event has a primordial origin.

The information in figures 49, 50 and 51 can be summarised as follows:

i) among the considered geometries, the triangle is the less performing giving e.g., uncertainties on the estimates of redshift about 1.4 times larger as compared to the 2L configurations, with both the considered ASDs.

ii) the LF instrument turns out to be of great relevance for the reconstruction of the eccentricity (with a gain in relative error that can be larger than one order of magnitude) and to a lesser extent also in the reconstruction of the masses and distances of sub-solar objects that, thanks to the LF sensitivity, can stay in the band for more cycles.

## 6.4 Cosmology

### 6.4.1 Hubble parameter and dark energy from joint GW/EM detections

In the context of cosmological studies, a key property of compact binary coalescences is that, from their GW signal, one can reconstruct the luminosity distance to the source [342]. Compact binary coalescences are then typically called ‘standard sirens’, by analogy with the ‘standard candles’ that provide absolute cosmological distance measurements from electromagnetic observations. Since the GW signal does not provide a direct measurement of the redshift, the ideal situation to constrain cosmological parameters is to measure the redshift of the host galaxy of the gravitational-wave source. Due to the poor localization capability of the gravitational-wave detectors, the host galaxy is typically identified only if we detect an electromagnetic counterpart (with some possible exceptions, see section 6.4.3). There are, however, a variety of statistical methods, based either on spatial correlations with galaxies or large-scale structures, or on features in the mass distribution of the sources, that can be used to extract cosmological information even in the absence of an electromagnetic counterpart [9, 343–371]. Standard sirens, either with counterpart or combined to statistical methods, allow us to measure the Hubble parameter $H_0$ at small redshifts while, at the larger redshifts that become accessible to a 3G detector, we can also explore the dark energy (DE) sector, through the study of the DE equation of state and of modified GW propagation. In
this section we explore the relative potential of different ET configurations for cosmology,
limiting ourselves to the study of GW sources with an electromagnetic counterpart, while in
sections 6.4.2 and 6.4.3 we will discuss alternative possibilities. For this purpose, among the
instruments discussed in section 4.2, the most useful are those which enable to obtain precise
(spectroscopic) measurement of the host galaxy redshift. In particular, following the analysis
in [372], we consider joint GW/EM detections between ET and THESEUS. We consider this
instrument, for the detection of the associated short GRB, since the localisation capabili-
ties (~ arcmin/arcsec) of both the XGIS and SXI instruments of THESEUS facilitate the
follow-up by ground-based telescopes, guaranteeing a high chance of obtaining the redshift
determination (see the discussion in section 4.2). We consider only the prompt emission (see
tables 6 and 7 for the number of joint detection per year). We further consider the joint
detections between ET and VRO, which is able to observe the kilonova emission produced
by BNS mergers in the optical band and give a precise localization for accurate redshift mea-
surements. As discussed in section 4, this allows us to study multi-messenger prospects for
ET both at the high redshifts probed by GRB observations and at $z \lesssim (0.3-0.4)$, where it
will be possible to detect kilonova emission using wide field telescopes.

We stress that the absolute estimates for the accuracy that we will obtain below for $H_0$, as well as for the parameters describing the DE equation of state and modified GW propagation introduced in the following, should not be understood as forecasts for the overall capability of ET for measuring these cosmological parameters. We are focusing here only on standard sirens observed from two specific instruments (in this case, THESEUS and VRO). Apart from the fact that the results also rely on assumptions on their performances and specific observational strategies, other EM observatories are expected to operate together with ET, increasing the sample of detections and/or adding further complementary information. On top of this, ‘dark sirens’, i.e. coalescing binaries without an observed electromagnetic counterpart, can provide very significant information through various statistical techniques, as mentioned above. In particular, the correlation with galaxy catalogs already provides valuable cosmological information at the level of current LVC data: the statistical method has been applied to extract $H_0$ from the recent LIGO/Virgo detections in a number of papers [9, 359, 364, 373–376], and to studies of the DE sector in [359, 365, 377, 378]. These studies in the ET era will greatly benefit from Euclid and the Large Synoptic Survey Telescope (LSST) surveys by the VRO which will map billion of galaxies giving more precise photometric redshifts, and spectroscopic surveys by multi-object survey spectrographs such as WAVE or 4MOST. Furthermore, several other ideas have been put forward to extract cosmological information from ET, see e.g. [348–350, 355, 356, 362, 363, 377, 379]. Therefore, an assessment of the full capabilities of ET for estimating cosmological parameters must take into account all these opportunities. Here, we focus on two examples, the joint GW/EM detections by ET+THESEUS and by ET+VRO, with the purpose of evaluating the relative performances of different configurations of ET in these specific situations, rather than the absolute precision to which cosmological parameters can be reconstructed at ET.

For the ET+THESEUS analysis, the population of BNSs and the associated EM signal
at high energies are generated following the approach in ref. [18]. Following the same approach
adopted in section 4, we fix the EM signal for each BNS of the population, for the entire
set of simulations corresponding to the different ET configurations and geometries. As a
reference, we consider 5 years of observations, and we select from our simulation a sample
of events, such that among them there are 75 joint detections with the configuration 2L
20 km HFLF cryo (see table 6). Considering the same BNS sample, we then compute the
number of joint detections for all the other configurations. With this method, we are correctly evaluating the capabilities of the ET configurations considering all of them to have the same total observational time, the same BNS sample and the same EM signal. In this derivation, the duty cycle of each ET configuration is taken into account.

We have performed parameter estimation using both GWFISH [31] and GWFAST [17, 32], finding full agreement. A delicate point of the parameter estimation procedure is that the events that give a detectable GRB are close to face-on, where the degeneracy between the luminosity distance $d_L$ and the inclination angle $\iota$ results in large errors on $d_L$, and often unreliable inversion of the Fisher matrices. This can be cured by imposing a prior on $\iota$, which reflects the information that, if a jet was observed from cosmological distances, the binary was close to face-on. A full analysis of the corresponding selection effects is quite non-trivial, and is in progress. For this study, we observe that all the GRBs of our catalog have $|\iota| < (5 - 10)$ deg.$^{36}$ We also keep fixed the position of the source in the sky, having assumed that the host galaxy has been identified. We then use the simple (and common) prescription of setting $\Delta d_L/d_L = 2/SNR$. This expression, with a factor of two included to be conservative, corresponds to the large SNR limit of the Fisher-matrix analysis when all other parameters are fixed. Since $\iota$ and the position in the sky are the parameters that are most degenerate with $d_L$, once these are kept fixed or strongly constrained by priors, this limiting value can become a reasonable approximation. A more detailed study on this is in progress. We then add in quadrature an error $\Delta d_L/d_L = 0.05z$ due to lensing, as in [380, 381] (see also [382] for more recent justification). From the value of $z$ of a joint GW+EM event of our catalog, we then compute the nominal value of its luminosity distance using a fiducial $\Lambda$CDM model (we use the fiducial values $H_0 = 67.66 \text{ km s}^{-1} \text{ Mpc}^{-1}$ and $\Omega_M = 0.31$, from Planck-2018 [383]), and the error $\Delta d_L$ from the above procedure and then, in order to simulate the realistic setting of an experiment, we scatter the ‘observed’ value of $d_L$ using a Gaussian distribution with the given $\Delta d_L$.

In a similar way, we study the capabilities of the synergy between ET and VRO. The population of sources and their respective KN emission are generated through the procedure described in section 4. In this case, the number of events considered corresponds to one year of observations, rather than 5, due to the larger number of potential joint detections (see section 4 and table 10 for details). We consider the observational strategy consisting on following-up all the events localized with $\Delta\Omega_{90\%} < 40$ deg$^2$, two filter ($g$ and $i$) observations repeated the first and second night after the merger and an exposure time for each pointing of 600 s. In particular, for each configuration we consider a number of sources matching the ones reported in column 3 of table 10. Note that, thanks to its isotropic nature, the KN emission can be observed irrespectively of the binary inclination, resulting in a sample of events with better conditioning for the Fisher matrix analysis, thanks to a milder degeneracy between $\iota$ and $d_L$.

We now present the results obtained for $(H_0, \Omega_{m,0})$, for the DE equation of state, and for modified GW propagation.

**Hubble constant.** In a flat $\Lambda$CDM model, the relation between the luminosity distance $d_L$ and the redshift is

$$d_L(z) = \frac{c}{H_0} (1+z) \int_0^z \frac{d\tilde{z}}{\sqrt{\Omega_M(1+\tilde{z})^3 + \Omega_\Lambda}}, \quad (6.9)$$

<footnote>$^{36}$More precisely, to take into account the periodicity of $\iota$, they have $\min(\iota, 180^\circ - \iota) < (5 - 10)$ deg.</footnote>
<figure><img src="image_61.png" /><figcaption>**Figure 52.** Reconstruction of the parameters $H_0$ and $\Omega_M$ in $\Lambda$CDM, from the joint GW+EM events obtained with ET+THESEUS in 5 yr of observations, for the different geometries of ET shown, all with their HFLF-cryo sensitivity.</figcaption></figure>

<table><caption>**Table 28.** Relative errors on $H_0$ and $\Omega_M$ in $\Lambda$CDM (median and symmetric 68% CI), from the joint GW+EM events obtained with ET+THESEUS, for the different geometries of ET shown, all with their HFLF-cryo sensitivity. We stress that no prior from electromagnetic observations, such as CMB+BAO+SNe, is used here; with such priors, the accuracy on $H_0$ becomes sub-percent.</caption><thead><tr><td>Configuration</td><td>Δ$H_0/H_0$</td><td>Δ$\Omega_M/\Omega_M$</td></tr></thead><tbody><tr><td>Δ-10km</td><td>0.057</td><td>0.546</td></tr><tr><td>Δ-15km</td><td>0.035</td><td>0.290</td></tr><tr><td>2L-15km-45°</td><td>0.040</td><td>0.370</td></tr><tr><td>2L-20km-45°</td><td>0.029</td><td>0.276</td></tr></tbody></table>

where $\Omega_M$ is the present matter fraction and $\Omega_\Lambda$ is the energy density fraction associated with the cosmological constant (we neglect the contribution from radiation, which is completely negligible at the redshifts relevant for standard sirens. In this approximation $\Omega_\Lambda = 1 - \Omega_M$ for a flat $\Lambda$CDM cosmology). At low redshift this reduces to Hubble's law $d_L(z) \simeq (c/H_0)z$, so standard sirens at low redshift can allow us to measure $H_0$.

The first measurement of $H_0$ from a standard siren with counterpart has been obtained from GW170817 [384]. However, the error from this single detection is still too large to discriminate between the value of $H_0$ obtained from late-Universe probes [385, 386], and that inferred from early-Universe probes assuming $\Lambda$CDM [387, 388], which are currently in disagreement at 5.3$\sigma$ level.
<figure><img src="image_62.png" /><figcaption>**Figure 53.** Reconstruction of the parameters $H_0$ and $\Omega_M$ in $\Lambda$CDM, from the joint GW+EM events obtained with ET+VRO in 1 yr of observations, for the different geometries of ET shown, in the left panel with their HFLF-cryo sensitivity and in the right panel with the HF instrument only.</figcaption></figure>

<table><caption>**Table 29.** Relative errors on $H_0$ and $\Omega_M$ in $\Lambda$CDM (median and symmetric 68% CI), from the joint GW+EM events obtained with ET+VRO in 1 yr of observations, for the different geometries of ET shown, in the left table with their HFLF-cryo sensitivity and in the right table with the HF instrument only.</caption><thead><tr><th colspan="3">HFLF cryogenic</th><th colspan="3">HF only</th></tr><tr><th>Configuration</th><th>ΔH<sub>0</sub>/H<sub>0</sub></th><th>ΔΩ<sub>M</sub>/Ω<sub>M</sub></th><th>Configuration</th><th>ΔH<sub>0</sub>/H<sub>0</sub></th><th>ΔΩ<sub>M</sub>/Ω<sub>M</sub></th></tr></thead><tbody><tr><td>Δ-10km</td><td>0.009</td><td>0.832</td><td>Δ-10km</td><td>0.065</td><td>1.23</td></tr><tr><td>Δ-15km</td><td>0.007</td><td>0.303</td><td>Δ-15km</td><td>0.057</td><td>1.86</td></tr><tr><td>2L-15km-45°</td><td>0.006</td><td>0.370</td><td>2L-15km-45°</td><td>0.066</td><td>1.31</td></tr><tr><td>2L-20km-45°</td><td>0.004</td><td>0.243</td><td>2L-20km-45°</td><td>0.031</td><td>1.22</td></tr></tbody></table>

Our results for $H_0$ and $\Omega_M$ are shown in figure 52 and table 28 for ET+THESEUS (that shows the result of a typical realization of the catalog of joint detections, among several that we have generated), and in figure 53 and table 29 for ET+VRO. We should stress that these are the results obtained from standard sirens only, without any prior from cosmological electromagnetic probes. Of course, CMB, BAO, SNe, structure formation, and other electromagnetic probes provide much stronger constraints on $\Omega_M$; assuming $\Lambda$CDM, Planck-2018 fixes $\Omega_M = 0.315 \pm 0.007$ [383], i.e. $\Delta\Omega_M/\Omega_M \simeq 0.022$, much smaller than what can be obtained from standard sirens alone. In principle, one could then combine cosmological and standard sirens data, as in [372], reducing significantly the error on $H_0$.<fnref n="37" /> In any case, for our present

<footnote n="37">³⁷The limits on $H_0$ reported here, obtained from GWs+GRB only, are consistent with the results in [372], after taking into account that we are considering 5 years of data (which is more consistent with updated estimates for the expected duration of the THESEUS mission), while the data in [372] were relative to a 10 yr period, and that we use a model for short GRBs normalized to the observations (which takes into account the fact that not all BNS merger produce a GRB), resulting in a smaller number of joint GW+GRB detections. As shown in [372], if one further combines the constraints from joint GW+EM detection with purely</footnote>
purpose of comparing the different geometries, figures 52 and 53 and tables 28 and 29 contain
the information that we need, on the relative performances of the different ET configurations.

We see that the best results for $H_0$ are obtained by ET+VRO, whose detections are potentially more numerous and at small redshift, where the impact of $H_0$ is bigger. However, we notice that the results obtained for THESEUS are very solid and conservative: they are based on the current observations of short GRBs and consider only the prompt emission (including the afterglow detections by XGIS and SXI will increase the detections per year by a factor of about 3). In contrast, the VRO estimates strongly depend on the BNS merger rate normalization (which, on the basis of the GW observations by the LIGO and Virgo detectors, can be one order of magnitude smaller or larger).

Concerning the comparison of configurations we find that the triangle-15km and the 2L-15km-45° have quite similar accuracy on $H_0$ for this test, and improve by a factor ~ (1.3–1.6) the result of the 10 km triangle, for ET+THESEUS and ET+VRO (with ET taken in both cases in the HFLF-cryo configuration). From the ET+VRO results, we also see the huge impact of the LF instrument, which allows us to observe a larger number of sources with good localization and results in constraints more than a factor of 7 tighter on $H_0$ and more than a factor of 2 tighter on $\Omega_{m,0}$.

Observe that the reduction of joint ET+VRO detections is dramatic if we lose the low-frequency: only 6% of the events detected by the triangle-10 km HFLF-cryo, and only 11% of those detected by the 2L-15km HFLF-cryo, are observed by the corresponding HF-only configurations, see columns 4 of table 11 and table 10. In contrast, for the joint ET+THESEUS detections the reduction is much smaller: 60% of the events detected by the triangle-10 km HFLF-cryo and 70% of those detected by the 2L-15km HFLF-cryo are observed by the corresponding HF-only configurations, see columns 4 of table 11 and table 10. Taking into account that the detections are distributed in the same redshift range (see for example figure 24), the observations of THESEUS together with ET-HF become comparable to those obtained by THESEUS together with ET at full HFLF-cryo sensitivity, in a observation time longer by a factor ~ 1.5 (which could also be partly compensated if the duty cycle of the HF-only instrument will be larger than that of the full HFLF-cryo configuration).

**Dark energy equation of state.** An accurate measurement of the Hubble parameter with GWs would be of course very interesting a priori. However, it is in principle possible (although by no means guaranteed) that, by the time that 3G detectors will be operative, the discrepancy on $H_0$ between late-Universe and early-Universe probes might have already found a resolution. More specific to 3G detectors is the possibility of testing the cosmological model with GWs up to moderate and large redshifts, where signatures of a dynamical dark energy could be found.

In general, on cosmological scales, one performs a separation between the homogeneous Friedmann-Robertson-Walker (FRW) background, and scalar, vector and tensor perturbations over it; tensor perturbations, when they are well inside the cosmological horizon, are just GWs propagating on the FRW background. Whenever gravity is modified on cosmological scales, both the background evolution and the dynamics in the sectors of scalar and tensor perturbations are modified (vector perturbations usually decay and are irrelevant). Here we discuss the effect of dark energy on the background evolution, which, as we will recall, can be described by an effective dark energy (DE) equation of state, while below we will discuss the modification of the tensor sector related to modified GW propagation.

<footnote>electromagnetic observations from CMB, BAO, SNe, etc, the accuracy on $H_0$ reaches the subpercent level.</footnote>
<figure><img src="image_63.png" /><figcaption>**Figure 54.** Reconstruction of the DE equation of state parameters $w_0$ and $w_a$, from the joint GW+EM events obtained with ET+THESEUS in 5 yr of observations, for the different geometries of ET shown, all with their HFLF-cryo sensitivity.</figcaption></figure>

<table><caption>**Table 30.** Absolute errors on the DE equation of state parameters $w_0$ and $w_a$ (symmetric 68% CI), from the joint GW+EM events obtained with ET+THESEUS, for the different geometries of ET shown, all with their HFLF-cryo sensitivity.</caption><thead><tr><th>Configuration</th><th>Δw<sub>0</sub></th><th>Δw<sub>a</sub></th></tr></thead><tbody><tr><td>Δ-10km</td><td>0.49</td><td>3.81</td></tr><tr><td>Δ-15km</td><td>0.40</td><td>2.65</td></tr><tr><td>2L-15km-45°</td><td>0.35</td><td>2.55</td></tr><tr><td>2L-20km-45°</td><td>0.34</td><td>2.40</td></tr></tbody></table>

At the background level, the effect of a dynamical dark energy component is described by the DE density $\rho_{DE}(z)$ and by its pressure $p_{DE}(z)$. Equivalently, one can introduce the DE equation of state $w_{DE}(z)$, defined by $p_{DE}(z) = w_{DE}(z)\rho_{DE}(z)$. The standard $\Lambda$CDM model is recovered for $w_{DE}(z) = -1$. Using the conservation of the DE energy-momentum tensor, one finds that the DE density is given as a function of redshift by

$$ \rho_{DE}(z) = \rho_0 \Omega_{DE} \exp \left\{ 3 \int_0^z \frac{d\tilde{z}}{1+\tilde{z}} [1+w_{DE}(\tilde{z})] \right\}, \quad (6.10) $$

where $\Omega_{DE} = \rho_{DE}(0)/\rho_0$ is the DE density fraction and $\rho_0 = 3H_0^2/(8\pi G)$ is the critical density. At the background level, the properties of DE are therefore encoded just in one function $w_{DE}(z)$. The corresponding expression for the luminosity distance (neglecting again
<figure><img src="image_64.png" /><figcaption>**Figure 55.** Reconstruction of the DE equation of state parameters $w_0$ and $w_a$, from the joint GW+EM events obtained with ET+VRO in 1 yr of observations, for the different geometries of ET shown, in the left panel with their HFLF-cryo sensitivity and in the right panel with the HF instrument only.</figcaption></figure>

<table><caption>**Table 31.** Absolute errors on the DE equation of state parameters $w_0$ and $w_a$ (symmetric 68% CI), from the joint GW+EM events obtained with ET+VRO, for the different geometries of ET shown, in the left table with their HFLF-cryo sensitivity and in the right table with the HF instrument only.</caption><thead><tr><th colspan="3">HFLF cryogenic</th><th colspan="3">HF only</th></tr><tr><th>Configuration</th><th>Δw<sub>0</sub></th><th>Δw<sub>a</sub></th><th>Configuration</th><th>Δw<sub>0</sub></th><th>Δw<sub>a</sub></th></tr></thead><tbody><tr><td>Δ-10km</td><td>0.31</td><td>3.92</td><td>Δ-10km</td><td>0.85</td><td>4.26</td></tr><tr><td>Δ-15km</td><td>0.23</td><td>3.69</td><td>Δ-15km</td><td>0.56</td><td>3.84</td></tr><tr><td>2L-15km-45°</td><td>0.14</td><td>2.49</td><td>2L-15km-45°</td><td>0.37</td><td>4.13</td></tr><tr><td>2L-20km-45°</td><td>0.13</td><td>2.89</td><td>2L-20km-45°</td><td>0.27</td><td>3.52</td></tr></tbody></table>

radiation) is

$$d_L(z) = \frac{c}{H_0} (1+z) \int_0^z \frac{d\tilde{z}}{\sqrt{\Omega_M (1+\tilde{z})^3 + \rho_{\text{DE}}(\tilde{z}) / \rho_0}}, \quad (6.11)$$

and reduces to eq. (6.9) when $\rho_{\text{DE}}(\tilde{z})$ is a constant, i.e. when $w_{\text{DE}}(z) = -1$. Any redshift dependence of $\rho_{\text{DE}}$, i.e. any deviation of the DE equation of state from the $\Lambda$CDM value $w_{\text{DE}} = -1$, would provide evidence for a dynamical dark energy.

In general, it is difficult to extract from the data a full function of redshift, such as $w_{\text{DE}}(z)$, and a parametrization in terms of a small number of parameters is useful. For the DE equation of state a standard choice is the ($w_0, w_a$) parametrization [389, 390],

$$w_{\text{DE}}(z) = w_0 + \frac{z}{1+z} w_a. \qquad (6.12)$$

Combining eq. (6.11) with eqs. (6.10) and (6.12), from the measurements of the luminosity distance and redshift of an ensemble of sources we can perform the inference on the cosmolog-
ical parameters $H_0, \Omega_M, w_0, w_a$. The constraining power of GWs alone is too weak to provide useful information on all these parameters (as we already saw just for $(H_0, \Omega_M)$ in table 28), so one is obliged to combine standard sirens with other cosmological probes. In this context, it is then common to assume that $H_0, \Omega_M$ are simply fixed, or strongly constrained by priors obtained from electromagnetic probes, and keep only $(w_0, w_a)$ in the inference. Several forecasts of $(w_0, w_a)$ for ET already exist in the literature [372, 380, 381, 391]. Here we present updated results using our catalog of joint ET+THESEUS detections, for the various ET geometries considered, as well as ET+VRO for the various geometries and the two ASDs (as we saw above, in the case of ET+THESEUS, the loss of the LF instrument is much less dramatic, and basically can be compensated by increasing by a factor $\sim 1.5$ the observation time, which does not affect the relative performances of the geometries). The results are shown in figure 54 and table 30 for ET+THESEUS and in figure 55 and table 31 for ET+VRO.38

For ET+THESEUS, we see that the differences in performances are not large between the various geometries: the 2L 15km 45° and the 2L 20km 45° are quite comparable and slightly better than the 15-km triangle, while the 10 km triangle performs less well (e.g. by a factor 1.4 on $w_0$, with respect to the 2L 15km 45°). For ET+VRO the situation is similar, with slightly larger differences (the 10 km triangle performs less well, with respect to the 2L 15km 45°, by a factor 2.2 on $w_0$). We also notice again the improvement brought by the LF instrument for the VRO+ET case, with results tighter on $w_0$ by a factor $\sim (2.4 - 2.7)$, depending on the geometry.

**Modified GW propagation.** As we have discussed, the DE equation of state characterizes the impact of dark energy on the background evolution and, as such, can also be studied with purely electromagnetic probes. Current Planck-2018 analysis from CMB, combined with SNe and BAO, including both $w_0$ and $w_a$ in the inference, find $w_0 = -0.961 \pm 0.077$ and $w_a = -0.28^{+0.31}_{-0.27}$ (68% c.l.) [383]. While GW observation can provide an independent probe of these quantities, it is not evident that even 3G detectors will be able to improve quantitatively on these accuracies. As we see from tables 30 and 31, this is certainly not the case for the specific example of joint ET+THESEUS or ET+VRO observations that we are considering, where we are well above the $\sim 7\%$ accuracy on $w_0$ that we already have from CMB+BAO+SNe. For instance (recalling that in $\Lambda$CDM $w_0 = -1$, so the absolute errors on $w_0$ reported in tables 31 and 30 are the same as the relative errors), for ET+VRO the configuration 2L-15km-45° reaches an accuracy on $w_0$ of 14%, while the 10 km triangle of 31%.

As emphasized in [391, 392], in the tensor sector the situation is different, and potentially much more interesting, for two reasons: (1) Modified GW propagation can only be tested from GW observations at large redshift, and therefore current limits (see below) are much broader. (2) There exist phenomenologically viable and theoretically motivated modified gravity models [393] that comply with all observational bounds at the background level and in the scalar perturbation sector (where, therefore, they differ from $\Lambda$CDM by at most a few percent level) and, nevertheless, in the tensor sector predict deviations from GR and $\Lambda$CDM that could be as large as 80% [394, 395].

The effect is due to a modification of the equation that governs the propagation of tensor perturbations over FRW. In GR, this is given by

$$ \tilde{h}''_A + 2\mathcal{H}\tilde{h}'_A + c^2k^2\tilde{h}_A = 0, \qquad (6.13) $$

<footnote>38The results for ET+THESEUS, obtained with GWFAST, have been double-checked using TiDoFM, finding broad agreement.</footnote>
<table><caption>**Table 32.** Errors on the parameters $Ξ_0$ and $n$ that describe modified GW propagation (median and symmetric 68% CI), from the joint GW+EM events obtained with ET+THESEUS, for the different geometries of ET shown, all with their HFLF-cryo sensitivity.</caption><thead><tr><td>Configuration</td><td>ΔΞ<sub>0</sub>/Ξ<sub>0</sub></td><td>Δn</td></tr></thead><tbody><tr><td>Δ-10km</td><td>0.071</td><td>2.87</td></tr><tr><td>Δ-15km</td><td>0.040</td><td>3.01</td></tr><tr><td>2L-15km-45°</td><td>0.049</td><td>2.91</td></tr><tr><td>2L-20km-45°</td><td>0.030</td><td>3.12</td></tr></tbody></table>

where $\tilde{h}_A(\eta, \mathbf{k})$ is the Fourier-transformed GW amplitude, $A = +, \times$ labels the two GW polarizations, the prime denotes the derivative with respect to cosmic time $\eta$ [which is defined by $d\eta = dt/a(t)$], $a(\eta)$ is the FRW scale factor, and $H = a'/a$. Whenever gravity is modified on cosmological scales, unavoidably this propagation equation is also modified, as has been seen on many explicit examples [394–402]. A change in the coefficient of the $k^2\tilde{h}_A$ term induces a speed of GWs, $c_{gw}$, different from that of light. After the observation of GW170817, this is now excluded at a level $|c_{gw} - c|/c < \mathcal{O}(10^{-15})$ [3]. However, the modified gravity models that pass this constraint still, in general, induce a change in the ‘friction term’, so the propagation equation for tensor modes becomes,

$$ \tilde{h}_A'' + 2H[1 - \delta(\eta)]\tilde{h}_A' + c^2 k^2 \tilde{h}_A = 0, \quad (6.14) $$

for some function $\delta(\eta)$ that encodes the modifications from GR. In GR, using eq. (6.13), one finds that the GW amplitude decreases over cosmological distances as the inverse of the FRW scale factor. From this one can show that, after propagation from the source to the observer, the amplitude of the GW from a compact binary coalescence is proportional to the inverse of the luminosity distance to the source (see, e.g. section 4.1.4 of [403]). This is at the origin of the fact that compact binaries are standard sirens, i.e. that the luminosity distance of the source can be extracted from their gravitational signal. However, when the GW propagation is rather governed by eq. (6.14), this result is affected. It can then be shown that the quantity extracted from GW observations is no longer the standard luminosity distance $d_L(z)$ of the source [that, in this context, we will denote by $d_L^{\text{em}}(z)$, since this is the quantity that would be measured, for instance, using the electromagnetic signal from a counterpart]. Rather, the quantity extracted from GW observation is a ‘GW luminosity distance’ $d_L^{\text{gw}}(z)$ [392], related to $d_L^{\text{em}}(z)$ by [391, 392]

$$ d_L^{\text{gw}}(z) = d_L^{\text{em}}(z) \exp \left\{ -\int_0^z \frac{dz'}{1+z'} \delta(z') \right\}, \quad (6.15) $$

where the function $\delta$ that appears in eq. (6.14) has now been written as a function of redshift.

Similarly to what we have discussed for the DE equation of state, inference on a full function of redshift such as $\delta(z)$ is difficult (although it can be performed, to some extent, with the technique of Gaussian process reconstruction [404]), and it is convenient to introduce a parametrization. A very convenient choice, in terms of two parameters $(\Xi_0, n)$, has been proposed in [391]. Rather than parametrizing $\delta(z)$, it is simpler to parametrize directly the ratio $d_L^{\text{gw}}(z)/d_L^{\text{em}}(z)$ (which is also the directly observed quantity), in the form

$$ \frac{d_L^{\text{gw}}(z)}{d_L^{\text{em}}(z)} = \Xi_0 + \frac{1 - \Xi_0}{(1+z)^n}. \quad (6.16) $$
<figure><img src="image_65.png" /><figcaption>**Figure 56.** Reconstruction of the parameters $\Xi_0$ and $n$ in modified gravity, from the joint GW+EM events obtained with ET+THESEUS in 5 yr of observations, for the different geometries of ET shown, all with their HFLF-cryo sensitivity.</figcaption></figure>

<figure><img src="image_66.png" /><figcaption>**Figure 57.** Reconstruction of the parameters $\Xi_0$ and $n$ in modified gravity, from the joint GW+EM events obtained with ET+VRO in 1 yr of observations, for the different geometries of ET shown, in the left panel with their HFLF-cryo sensitivity and in the right panel with the HF instrument only.</figcaption></figure>
<table><thead><tr><th colspan="3">HFLF cryogenic</th></tr><tr><th>Configuration</th><th>ΔΞ<sub>0</sub>/Ξ<sub>0</sub></th><th>Δn</th></tr></thead><tbody><tr><td>Δ-10km</td><td>0.097</td><td>3.04</td></tr><tr><td>Δ-15km</td><td>0.051</td><td>2.95</td></tr><tr><td>2L-15km-45°</td><td>0.038</td><td>2.95</td></tr><tr><td>2L-20km-45°</td><td>0.027</td><td>2.59</td></tr></tbody></table>

<table><caption>**Table 33.** Errors on the parameters $Ξ_0$ and $n$ that describe modified GW propagation (median and symmetric 68% CI), from the joint GW+EM events obtained with ET+VRO, for the different geometries of ET shown, in the left table with their HFLF-cryo sensitivity and in the right table with the HF instrument only.</caption><thead><tr><th colspan="3">HF only</th></tr><tr><th>Configuration</th><th>ΔΞ<sub>0</sub>/Ξ<sub>0</sub></th><th>Δn</th></tr></thead><tbody><tr><td>Δ-10km</td><td>0.617</td><td>2.80</td></tr><tr><td>Δ-15km</td><td>0.207</td><td>2.87</td></tr><tr><td>2L-15km-45°</td><td>0.108</td><td>2.88</td></tr><tr><td>2L-20km-45°</td><td>0.063</td><td>3.00</td></tr></tbody></table>

This parametrization reproduces the fact that, at $z = 0$, $d_L^{gw}/d_L^{em} = 1$ since, as the distance to the source goes to zero, there can be no effect from modified propagation. In the opposite limit of large redshifts, in contrast, eq. (6.16) predicts that $d_L^{gw}/d_L^{em}$ approaches a constant value $Ξ_0$. This is motivated by the fact that, in typical modified gravity models, the deviations from GR only appear in the recent cosmological epoch, so $\delta(z)$ goes to zero at large redshift, and the integral in eq. (6.15) saturates to a constant. The parametrization (6.16), interpolates between these two limiting behaviors, with a power-law determined by $n$, and GR corresponds to $Ξ_0 = 1$ (for any $n$). This simple parametrization has been shown to work remarkably well for practically all best-studied modified gravity models [402]. A particularly interesting example is given by the model proposed in [393] (see [395, 405] for reviews) that, while complying with all current cosmological constraints, predicts modified GW propagation with a value of $Ξ_0$ that (depending on a single parameter related to the initial conditions of the cosmological evolution) can differ from the GR value $Ξ_0 = 1$ by an amount between a few percent up to a value $Ξ_0 \simeq 1.8$ [394, 395], that would correspond to a 80% deviation from GR. Limits on $Ξ_0$ have already been obtained from the current set of LVC detections, either from GW170817 as a standard siren with counterpart [391], or using BBH dark sirens from the O1, O2 and O3a LIGO/Virgo run, and performing a correlation with galaxy catalogs. This results in the value $Ξ_0 = 2.1_{-1.2}^{+3.2}$ [359].$^{39}$ Using instead a joint cosmology-population analysis that exploits the mass scales in the BBH mass function [355, 377] and the GWTC-3 catalog of detections [6] one finds the constraints $Ξ_0 = 1.2 \pm 0.7$ [365].

Here we show the constraint on $Ξ_0$ that can be obtained from the joint set of GW and electromagnetic observations obtained from ET+THESEUS and ET+VRO. As we have stressed at the beginning of this section, the absolute estimates for the accuracy that we will present should not be understood as the forecast for the overall capability of ET for measuring these cosmological parameters, since we are studying the information that could be obtained from two examples, namely the joint detections of ET+THESEUS and ET+VRO, and, as discussed at the beginning of this section, there are other ways of obtaining information on $Ξ_0$ from the ET data. Again, here we are mainly interested in the relative performances of the different ET configurations on some concrete examples.

Figure 56 and table 32 show the results for the parameters $Ξ_0$ and $n$ that characterize modified GW propagation from the ET+THESEUS joint detections, while figure 57, and ta-

<footnote>$^{39}$An even more stringent result is obtained if one accepts the tentative identification of the flare ZTF19abahr as the electromagnetic counterpart of the BBH coalescence GW190521. Then, the analysis in [359] gives $Ξ_0 = 1.8_{-0.6}^{+0.9}$. (68% c.l.). This is consistent with similar results obtained in [406].</footnote>
ble 33 show the results for ET+VRO. The hierarchy of performances is similar to that for $H_0$, with the 2L 20 km providing the best results, followed by the 15 km triangle and the 2L 15 km with similar performances (with 2L-15 km better than triangle-15km for ET+VRO), and the 10 km triangle providing the less good performances. For ET+VRO, the huge improvement as a consequence of the inclusion of the LF instrument is also apparent.

Observe that, in view of the discussion above, the accuracy that can be obtained from $\Xi_0$ even from these two specific examples, which ranges between 3% and 10% depending on the geometry with the full sensitivity curve, is already very interesting in absolute terms, since current $1\sigma$ limits are at the 60% - 100% level, and furthermore there are models that predicts a potentially much higher signal.

### 6.4.2 Hubble parameter and dark energy from BNS tidal deformability

BNS have an intrinsic mass scale and can only exist in a narrow range of masses. This mass scale is imprinted in the tidal interaction between the component NSs. Therefore, if the nuclear EoS is known, one can determine the source-frame masses by a measurement of the tidal deformability. This, in turn, would allow the measurement of the redshift directly from a GW observation because it is the redshifted mass that is inferred from the point-particle approximation of the waveform. Such a method was first proposed in [348] and further explored in [407, 408]. A measurement of the Hubble constant using a known relationship between the tidal parameter and source-frame mass was explored in [351, 361, 369] while [368] showed that one can simultaneously estimate both the nuclear EoS and the Hubble constant using future observatories. A measurement of the dark energy EoS was explored in [357, 366].

In this section, we explore the potential of different ET configurations to constrain the expansion history of the Universe assuming that the nuclear EoS is known. It is found in [361] that up to a 15% uncertainty in the knowledge of the EoS does not affect the measurement of the Hubble constant in a meaningful manner. We use the TaylorF2 waveform model augmented with the 5PN and 6PN tidal terms in the phase, terminating the signal at theISCO frequency corresponding to the total mass of the binary. In section 6.1.3, it was found that the results using this model versus an inspiral-merger-ringdown (IMR) waveform gives similar results. Since the analysis in this section is along similar lines, we expect the same to hold here too. Additionally, we assume the DD-LZ1 EoS for the NS for reasons elucidated in section 6.2.1. We fit the logarithm (base 10) of the tidal deformability as a function of the mass of the NS using a fourth-order polynomial given by

$$ \log_{10} \Lambda(m) = -0.3550 m^4 + 2.162 m^3 - 4.652 m^2 + 2.514 m + 3.892, \quad (6.17) $$

where $m$ is in units of $M_\odot$. We verify that the fit reproduces the slope of the curve accurately with maximum errors at a few percent. This is crucial because it is the slope of the curve that contributes to the Fisher errors on the redshift.

The Fisher errors from the $d_L-z$ space are then propagated to the space of cosmological parameters, $\vec{\phi}$, via another Fisher matrix given by [409]

$$ \mathcal{G}_{ij} = \sum_{k=1}^{N} \frac{1}{\sigma_{d_L, k}^2} \frac{\partial d_L^k(z)}{\partial \phi^i} \frac{\partial d_L^k(z)}{\partial \phi^j}, \quad (6.18) $$

where $N$ is the total number of observations in the catalog and $\sigma_{d_L, k}^2$ is the total luminosity distance error for the k-th event given by

$$ (\sigma_{d_L})^2 = (\sigma_{d_L}^h)^2 + (\sigma_{d_L}^z)^2. \quad (6.19) $$
<figure><img src="image_67.png" /><figcaption>**Figure 58.** Constraints on the parameters $H_0$ and $\Omega_M$ in $\Lambda$CDM model using one year GW observations from BNS alone for different ET geometries. The covariance for a standalone ET is shown in the left panel while ET in a network with 2CE is shown in the right panel. $H_0$ is measured in km s$^{-1}$ Mpc$^{-1}$.</figcaption></figure>

<table><thead><tr><th>Configuration</th><th>ΔH<sub>0</sub>/H<sub>0</sub></th><th>ΔΩ<sub>M</sub>/Ω<sub>M</sub></th></tr></thead><tbody><tr><td>Δ-10km</td><td>9.63 × 10<sup>-3</sup></td><td>1.10 × 10<sup>-1</sup></td></tr><tr><td>Δ-15km</td><td>7.20 × 10<sup>-3</sup></td><td>6.62 × 10<sup>-2</sup></td></tr><tr><td>2L-15km-45°</td><td>7.59 × 10<sup>-3</sup></td><td>7.47 × 10<sup>-2</sup></td></tr><tr><td>2L-20km-45°</td><td>5.90 × 10<sup>-3</sup></td><td>5.04 × 10<sup>-2</sup></td></tr></tbody></table>

<table><caption>**Table 34.** Standard deviation on the parameters $H_0$ and $\Omega_M$ in $\Lambda$CDM using one year of GW observations from BNS alone for different geometries of ET alone (left) and ET in a network with 2CE (right). ET is always taken with the full HFLF-cryo sensitivity.</caption><thead><tr><th>Configuration</th><th>ΔH<sub>0</sub>/H<sub>0</sub></th><th>ΔΩ<sub>M</sub>/Ω<sub>M</sub></th></tr></thead><tbody><tr><td>Δ-10km + 2CE</td><td>7.35 × 10<sup>-4</sup></td><td>4.40 × 10<sup>-3</sup></td></tr><tr><td>Δ-15km + 2CE</td><td>6.35 × 10<sup>-4</sup></td><td>3.71 × 10<sup>-3</sup></td></tr><tr><td>2L-15km-45° + 2CE</td><td>6.54 × 10<sup>-4</sup></td><td>3.84 × 10<sup>-3</sup></td></tr><tr><td>2L-20km-45° + 2CE</td><td>5.79 × 10<sup>-4</sup></td><td>3.30 × 10<sup>-3</sup></td></tr></tbody></table>

Here, $\sigma_{d_L}^h$ is the contribution to the luminosity distance error due to the error in the GW amplitude while $\sigma_{d_L}^z$ is that due to the error in the redshift measurement, given by

$$ \sigma_{D_L}^z = \left| \frac{\partial D_L}{\partial z} \right| \sigma_z. \qquad (6.20) $$

In writing eq. (6.18), we have neglected the correlations in the $d_L-z$ space for simplicity.

**Hubble constant.** The results for $H_0$ and $\Omega_M$ are shown in figure 58 and table 34. A description of $\Lambda$CDM cosmology is given in section 6.4.1. We see that a single year of observing run can achieve sub-percent errors on $H_0$ with ET alone, albeit the errors improve by an order of magnitude when ET operates in a network with 2CE. The ET configuration with two 20km L-shaped detectors at 45° gives the best constraints. Similar trends follow for the dark matter energy density parameter, $\Omega_M$, which can be constrained at percent level precision, except for the case of a standalone 10km triangular ET which can only achieve an accuracy of $\sim 10\%$.

**Dark energy equation of state.** The results for the dark energy equation of state parameters are shown in figure 59 and the $1\sigma$ errors tabulated in table 35. A description of the dark energy model considered here is in section 6.4.1. We see that the 2L 20km at 45° obtains
<figure><img src="image_68.png" /><figcaption>**Figure 59.** Constraints on the $w_0-w_a$ plane in a dynamical dark energy model using one year GW observations from BNS alone for different ET geometries. The covariance for a standalone ET is shown in the left panel while ET in a network with 2CE is shown in the right panel.</figcaption></figure>

<table><thead><tr><th>Configuration</th><th>Δw<sub>0</sub></th><th>Δw<sub>a</sub></th></tr></thead><tbody><tr><td>Δ-10km</td><td>0.348</td><td>5.79</td></tr><tr><td>Δ-15km</td><td>0.217</td><td>2.98</td></tr><tr><td>2L-15km-45°</td><td>0.210</td><td>3.19</td></tr><tr><td>2L-20km-45°</td><td>0.153</td><td>2.04</td></tr></tbody></table>

<table><thead><tr><th>Configuration</th><th>Δw<sub>0</sub></th><th>Δw<sub>a</sub></th></tr></thead><tbody><tr><td>Δ-10km +2CE</td><td>0.0241</td><td>0.231</td></tr><tr><td>Δ-15km +2CE</td><td>0.0207</td><td>0.193</td></tr><tr><td>2L-15km-45° +2CE</td><td>0.0211</td><td>0.198</td></tr><tr><td>2L-20km-45° +2CE</td><td>0.0189</td><td>0.169</td></tr></tbody></table>

<table><thead><tr><th>Configuration</th><th>Δw<sub>0</sub></th><th>Δw<sub>a</sub></th></tr></thead><tbody><tr><td>Δ-10km</td><td>0.129</td><td>1.02</td></tr><tr><td>Δ-15km</td><td>0.0940</td><td>0.644</td></tr><tr><td>2L-15km-45°</td><td>0.111</td><td>0.773</td></tr><tr><td>2L-20km-45°</td><td>0.0823</td><td>0.532</td></tr></tbody></table>

<table><caption>**Table 35.** Errors on the parameters $w_0$ and $w_a$ (68% c.l.) in a dynamical dark energy model using one year of GW observations from BNS alone for different geometries of ET alone (left) and ET in a network with 2CE (right), all with their HFLF-cryo sensitivity. The top panel shows the constraints marginalised over the parameters $H_0$ and $\Omega_M$ whereas, in the bottom panel, the $\Lambda$CDM parameters are assumed to be known a priori.</caption><thead><tr><th>Configuration</th><th>Δw<sub>0</sub></th><th>Δw<sub>a</sub></th></tr></thead><tbody><tr><td>Δ-10km +2CE</td><td>7.07 × 10<sup>-3</sup></td><td>0.0405</td></tr><tr><td>Δ-15km +2CE</td><td>6.08 × 10<sup>-3</sup></td><td>0.0343</td></tr><tr><td>2L-15km-45° +2CE</td><td>6.29 × 10<sup>-3</sup></td><td>0.0356</td></tr><tr><td>2L-20km-45° +2CE</td><td>5.50 × 10<sup>-3</sup></td><td>0.0306</td></tr></tbody></table>

the best constraints on the parameters, albeit the EoS parameter $w_a$ is not constrained by a standalone ET if one marginalises over the $\Lambda$CDM parameters. In such a scenario, it is common to fix the $\Lambda$CDM parameters to that measured with EM probes and measure only the dark energy parameters using GWs. In this case, except for the single 10km triangular ET configuration, the EoS parameters can be constrained.

**Modified GW propagation.** We next consider modified gravitational wave propagation, using the parametrization (6.16). The bounds on the corresponding parameters $\Xi_0$ and $n$ are shown in figure 60 and the corresponding error (at 68% c.l.) are tabulated in table 36. A difference from the analysis of the previous section using EM counterparts is that we assume fiducial values of $\Xi_0 = 1.1$ and $n = 2.5$. As described in section 6.4.1, these values are consis-
<figure><img src="image_69.png" /><figcaption>**Figure 60.** Constraints on the $\Xi_0-n$ plane in a modified gravity model with modified tensor perturbations using one year GW observations from BNS alone for different ET geometries. The covariance for a standalone ET is shown in the left panel while ET in a network with 2CE is shown in the right panel. The $\Lambda$CDM parameters are assumed to be known a priori.</figcaption></figure>

<table><thead><tr><th>Configuration</th><th>ΔΞ<sub>0</sub>/Ξ<sub>0</sub></th><th>Δn</th></tr></thead><tbody><tr><td>Δ-10km</td><td>1.06 × 10<sup>-2</sup></td><td>5.91 × 10<sup>-2</sup></td></tr><tr><td>Δ-15km</td><td>7.01 × 10<sup>-3</sup></td><td>4.45 × 10<sup>-2</sup></td></tr><tr><td>2L-15km-45°</td><td>8.11 × 10<sup>-3</sup></td><td>4.29 × 10<sup>-2</sup></td></tr><tr><td>2L-20km-45°</td><td>5.88 × 10<sup>-3</sup></td><td>4.37 × 10<sup>-2</sup></td></tr></tbody></table>

<table><caption>**Table 36.** Errors (at 68% c.l.) on the parameters $\Xi_0$ and $n$ that describe modified GW propagation, using one year of GW observations from BNS alone for different geometries of ET alone (left) and ET in a network with 2CE (right), all with their HFLF-cryo sensitivity. The $\Lambda$CDM parameters are assumed to be known a priori.</caption><thead><tr><th>Configuration</th><th>ΔΞ<sub>0</sub>/Ξ<sub>0</sub></th><th>Δn</th></tr></thead><tbody><tr><td>Δ-10km +2CE</td><td>3.71 × 10<sup>-4</sup></td><td>5.78 × 10<sup>-3</sup></td></tr><tr><td>Δ-15km +2CE</td><td>3.07 × 10<sup>-4</sup></td><td>4.99 × 10<sup>-3</sup></td></tr><tr><td>2L-15km-45° +2CE</td><td>3.22 × 10<sup>-4</sup></td><td>5.21 × 10<sup>-3</sup></td></tr><tr><td>2L-20km-45° +2CE</td><td>2.71 × 10<sup>-4</sup></td><td>4.99 × 10<sup>-3</sup></td></tr></tbody></table>

tent with all current observations. We do this because the form of the parameterization (6.16) renders the Fisher matrix (6.18) singular for GR value of $\Xi_0 = 1$ and for $n = 0$. Any value of $n$ away from $n = 0$ is sufficient to make the Fisher matrix well-defined. We choose $n = 2.5$ for reasons outlined in [372]. We see that the hierarchy among the configurations is analogous to that found in table 32, with the 2L 20km with arms at 45° providing the best results, followed by 2L 15km with arms at 45° and triangle 15km, that gives quite similar results, while the triangular 10km ET gives the largest errors.

### 6.4.3 Hubble parameter from high-mass ratio events

A main limitation for the reconstruction of the luminosity distance $d_L$ is the fact that it is highly correlated with the angle $\iota$ between the binary's orbital angular momentum and the observer's line-of-sight.<fnref n="40" /> Purely based on the signal-to-noise ratio of an event, for large SNR we should expect that the luminosity distance $d_L$ of a source is measured to an accuracy of $\Delta d_L/d_L = 1/\text{SNR}$. However, this limit is never reached since this correlation (as well as, to lesser extent, the correlation with sky location) degrades the accuracy, typically leading

<footnote n="40">⁴⁰For precessing binaries $\iota$ would be a function of time since the orbital angular momentum precesses about the total angular momentum, which includes the companions' spin angular momenta. For such systems, $\iota$ should be defined at some fixed, but arbitrary, instant of time.</footnote>
<figure><img src="image_70.png" /><figcaption>**Figure 61.** The accuracy with which the Hubble constant can be measured by different detector networks by high-mass ratio binary black holes, with events corresponding to 1 and 2 yrs of observation time picked randomly from the 10-year catalog of BBH events located within $z = 0.1$. In the left plot, $H_0$ is measured with no prior imposed on the dark matter energy density $\Omega_M$, while for the right plot we assume a gaussian prior with a width of 0.017. The markers show the median value of the fractional error in $H_0$ and the error bars denote the 68% confidence region.</figcaption></figure>

to error in the distance measurement an order of magnitude larger [410] (unless the angle $\iota$ and the sky location of the source can be fixed by the observation of an associated GRB, see the discussion on page 96). For instance, when the inclination angle changes from 0 to $\pi/6$ radians, the relative amplitudes of the $h_+$ and $h_-$ polarizations of the dominant quadrupole mode change by no more than 1%, while the overall amplitude reduces by 12%. This change in amplitude can be entirely compensated by moving the binary 12% closer to the observer without an appreciable change in the polarization content of the observed signal.

This scenario changes dramatically for high-mass ratio events from either neutron star-black hole or double black hole binaries. Significant energy in the higher-order modes can break the distance-inclination angle degeneracy since the polarization modes have a different dependence on the inclination angle. In a recent paper [411] it has been shown that high-mass ratio events with large SNR would not only break the $d_L-\iota$ degeneracy, but a small population of such events would also localize the source to a region small enough to contain, on average, just one $L_*$ galaxy. This helps in obtaining the redshift of the source from the galaxy, although the source itself may not have an EM afterglow. As already mentioned, these events are called *dark sirens* and can measure the Hubble constant without the need for an extra distance calibrator. Some of these events, albeit very rare, could measure the Hubble constant to an accuracy of better than 2% with a single event with no known systematics and are called *golden $H_0$ events*. This method could be one of the most *precise* and *accurate* way of measuring the Hubble constant.
Since BBHs are not likely to be associated with an EM counterpart, the only way to identify the true host is to localize the source to a single galaxy. For NSBH events, however, we expect EM afterglows if the mass ratio is not too small (say, greater than about 1:4). While asymmetric masses in a NSBH binary could help break the distance-inclination degeneracy, the host galaxy could be identified from the kilonova. Additionally, for NSBH binaries with large mass asymmetry, it is also possible to identify the host within the sky localization from gravitational wave observations. In this sense, NSBH binaries could be used as both bright and dark sirens or *gray sirens*.

For BBH systems, we restrict our population to events within the redshift $z = 0.1$, or 476 Mpc. This leaves us with 86 BBH mergers in a span of 10 years. Among these, we choose those events for which the sky area is constrained to better than 0.04 deg$^2$, as we expect only one $L_*$ galaxy to lie in this sky patch up to a redshift of 0.1 [412]. For the events that qualify, we obtain the measurement errors in luminosity distance by calculating the Fisher information matrix. Assuming the $\Lambda$CDM cosmology, the errors in $d_L$ can be converted to measurement uncertainty in $H_0$ after accounting for the measurement errors in the matter density, $\Omega_M$. We present the bounds on $H_0$ for two cases: without taking a prior on $\Omega_M$, and with a gaussian prior on $\Omega_M$. The width of the gaussian prior is fixed to the sum of the squares of the bounds placed on $\Omega_M$ by Planck [383] and SH0ES [413], which is equal to 0.017. Using binary black holes as dark sirens, the accuracy with which $H_0$ would be measured by different detector configurations in a network of ET and two CE detectors, one of which is a 40 km arm length detector and the other 20 km, is shown in figure 61.

For NSBH systems, the Hubble constant can be measured with gravitational waves both without (dark sirens) and with (bright sirens) electromagnetic counterpart. From the 10 year population, we choose events that qualify as dark or bright sirens (and make sure to not double-count events that qualify as both). The NSBH events that qualify as dark sirens are chosen in the same way as was done for BBH systems. For bright sirens, we only consider events that lie within a redshift of $z = 0.5$, as we do not expect to detect the kilonovae from events that lie farther. In addition, only those events are selected for which the sky position is measured better than 9.6 deg$^2$, i.e. the field of view of the Rubin observatory. For these events, we generate the kilonova light curves with DD2 as the neutron star equation of state using the numerical recipes from [414, 415]. We claim that a particular kilonova is detected with the Rubin observatory by using the same criteria that were used in section 4.3, i.e. the luminosity of the kilonova should be brighter than the limiting magnitude (corresponding to 600s of exposure) of the g and the i filters on two consecutive nights. We also assume a 50% duty cycle for the Rubin observatory. The events for which the corresponding kilonova is detected qualify as bright sirens and are considered for the measurement of the Hubble constant. The luminosity distance errors for both dark and bright siren events are combined and the measurement errors for $\Omega_M$ are accounted for in the same way as was done for BBH dark siren events. The accuracy in measurement of $H_0$ using this gray siren approach with NSBH systems is shown in figure 62.

Among the events that contribute to the measurement of the Hubble constant, there are certain *golden* events for which the accuracy in the measurement of the luminosity distance is better than 2%. Such events can individually measure $H_0$ with a precision better than 2% and resolve the Hubble tension. The number of such golden binaries is given in table 37, both for the dark siren case with BBHs, and for the gray siren case with NSBH systems, and an observation span of 10 years. *We see that, when in a network with 2CE detectors, the performances of all the ET geometries considered are comparable in the measurement of the Hubble*
<figure><img src="image_71.png" /><figcaption>**Figure 62.** Same as figure 61 except the results are obtained for the NSBH population instead of the BBH population. Moreover, the results were obtained by combining measurement accuracies from systems that produce kilonovae in addition to those that identify a unique host within the localization error box.</figcaption></figure>

<table><caption>**Table 37.** The number of dark siren BBH golden events and gray siren NSBH golden events for different ET detector configurations in an observation time of 10 years.</caption><thead><tr><th rowspan="2">Configuration</th><th colspan="2">BBH</th><th colspan="2">NSBH</th></tr><tr><th>With 2CE</th><th>Without 2CE</th><th>With 2CE</th><th>Without 2CE</th></tr></thead><tbody><tr><td>Δ 10 km</td><td>68</td><td>5</td><td>21</td><td>0</td></tr><tr><td>Δ 15 km</td><td>74</td><td>7</td><td>30</td><td>0</td></tr><tr><td>2L 15 km 45°</td><td>73</td><td>4</td><td>25</td><td>1</td></tr><tr><td>2L 15 km 0°</td><td>69</td><td>0</td><td>21</td><td>0</td></tr><tr><td>2L 20 km 45°</td><td>77</td><td>8</td><td>37</td><td>1</td></tr><tr><td>2L 20 km 0°</td><td>70</td><td>0</td><td>22</td><td>0</td></tr></tbody></table>

constant using the dark and gray siren approaches. In contrast, for ET alone, the L-shaped configurations with parallel arms perform significantly worse compared to the other configurations, as they are unable to detect a single golden event in an observing time of 10 years.

## 6.5 Cosmological stochastic backgrounds

In this section we consider the performances of the various ET geometries for the detection of specific examples of stochastic backgrounds of cosmological origin. In section 6.5.1 we compare the ET sensitivities with the predictions from models of cosmic strings, while in section 6.5.2 we consider the stochastic background produced by a phase transition in
the early Universe. Finally, in section 6.5.3, we study the prospects for separating these cosmological backgrounds from the astrophysical stochastic background.

### 6.5.1 Cosmic strings

Phase transitions followed by spontaneously broken symmetries may lead to the production of topological defects as relics of the previous more symmetric phase of the Universe. Within Grand Unified Theories, it has been shown [416] that cosmic strings (line-like defects) are generically formed. At the energy scales relevant to cosmology, the width of a string is negligible relative to its characteristic size. Such strings (Nambu-Goto strings) are parameterised by a single dimensionless quantity, the string tension $G\mu$ related to the energy scale $\eta$ at which cosmic strings are formed through $G\mu \sim 10^{-6}[\eta/(10^{16}\text{GeV})]^2$ (we set the speed of light $c = 1$).

Cosmic strings emit bursts of beamed gravitational radiation. The main sources of bursts are kinks, cusps or kink-kink collisions. Kinks are discontinuities in the tangent vector of the string that propagate at the speed of light and appear in pairs. Cusps are points on the string that briefly travel at the speed of light. The incoherent superposition of these bursts through the history of the Universe produces a strongly non-Gaussian stochastic gravitational wave background [417]. Occasionally there may also be sharp and high-amplitude bursts of GWs above this stochastic background. The production of gravitational waves by cosmic strings offers a tool to probe particle physics beyond the Standard Model at energy scales much above the ones reached by accelerators. A non-detection of a stochastic background of gravitational waves imposes bounds on the cosmic string tension and therefore on particle physics models beyond the Standard Model. ET will be able to considerably improve on current 2G bounds [418].

In figures 63 and 64 we show the prediction for $\Omega_{\text{GW}}(f)$ for two different models of the loop distribution of cosmic strings, denoted as models A [419] and model B [420], and different values of the string tension $G\mu$. In model A the loop production function can be determined from Nambu-Goto simulations of cosmic string networks. In model B the distribution of non-self-intersecting scaling loops is the extracted quantity from different simulations. Within the latter model, loops are formed at all sizes following a power-law while the scaling loop distribution is cut-off on small scales by the gravitational back-reaction. In model B are produced many more tiny loops than in the model A. These two models for the loop distributions have been used by the LVK collaboration to put constraints on cosmic strings from O1, O2 and O3 data [418].

The various curves define the power-law integrated sensitivity for each ET configuration, assuming SNR=1 and $T_{\text{obs}} = 1$ yr. As already remarked in section 5.1, the triangular configurations perform better at high frequency, because the overlap reduction function remains constant at high frequencies for co-located detectors, while it falls to zero for widely separated detectors. The 2L shape configuration does better around a frequency of 10 Hz, but only for co-aligned detectors (this maximises the coherence for the stochastic search). The case of 45 deg 2L configurations is universally the worst. We conclude that, for the considered loop distributions (models A and B), the best stochastic constraints on the string tension are $\sim 10^{-19}$, corresponding to an energy scale between $10^8$ GeV and $3 \times 10^{10}$ GeV, 6 to 8 orders of magnitude above the electroweak scale.

### 6.5.2 First-order phase transition

As the Universe cooled down, it may have undergone a first-order phase transition (FOPT) with bubbles of true vacuum forming, expanding and colliding [421–423]. This in turn leads
<figure><img src="image_72.png" /><figcaption>**Figure 63.** Energy density of gravitational waves emitted from cosmic string loops with loop distribution given by model A [419]. The various curves define the power-law integrated sensitivity for each ET configuration, assuming SNR$_{th}$ = 1 and T$_{obs}$ = 1 yr.</figcaption></figure>

to gravitational waves generated from the collision of bubbles, and from the sound waves and turbulence in the surrounding plasma (see e.g. [85, 424] for reviews). The frequency of gravitational radiation emitted depends on the temperature. For a temperature of the order (10⁵ – 10¹⁰) GeV, energy scales inaccessible to particle colliders, we find a SGWB that peaks in the ET frequency range. Here we briefly discuss ET detection prospects of supercooled FOPTs, a special class of FOPTs which is expected to create the “loudest” gravitational waves [425, 426].

The gravitational wave spectrum from a supercooled FOPT depends only on two parameters, namely the reheating temperature $T_{RH}$ and inverse duration $\beta/H_{RH}$, where we denote $H_{RH} \equiv H(T_{RH})$.<fnref n="41" /> Note that the strength of transition $\alpha$ that affects the SGWB for the more general FOPTs, does not impact the supercooled spectrum. In figure 65 we simulate a SGWB containing the residual from unresolved CBCs and an FOPT signal for a range of $T_{RH}$ and $\beta/H_{RH}$ values. The left panel includes a FOPT signal that is dominated by bubble collisions, while the right panel is for a FOPT signal dominated by sound waves. Turbulence is expected to be subdominant to both bubble collisions and sound waves, so we exclude it from our study. It is clear that we will be able to exclude a large part of the parameter space with the next generation of GW detectors.

### 6.5.3 Source separation

The unresolved astrophysical mergers are expected to be the dominant contribution to the background observed by ET. It is therefore important to subtract the individual sources, as discussed in the previous section, in the hope to reveal the cosmological background. Note

<footnote n="41">⁴¹For simplicity, we assume wall velocity to be equal to the speed of light.</footnote>
<figure><img src="image_73.png" /><figcaption>**Figure 64.** Energy density of gravitational waves emitted from cosmic string loops with loop distribution given by model B [420]. The various curves define the power-law integrated sensitivity for each ET configuration, assuming SNR$_{th}$ = 1 and T$_{obs}$ = 1 yr.</figcaption></figure>

<figure><img src="image_74.png" /><figcaption>**Figure 65.** Detectability of bubble collisions (left) and sound wave (right) dominated supercooled FOPT with next-generation ET detector. The dashed line is the detection threshold using 10 km triangular ET configuration, the dotted line is 15 km triangular configuration, while the solid and dot-dashed contours are for two L-shaped detectors with 15 km and 20 km arms, respectively.</figcaption></figure>

that the imperfect subtraction of CBC sources may pose a serious threat to digging out a cosmological background, and extensive research must be done to improve the subtraction methods [427, 428]. Here we take as an example two cosmological sources: cosmic strings and an early-Universe first-order phase transition.

By limiting the study to the frequency range $f \in [10, 100]$ Hz, we can assume the CBC background to follow $f^{2/3}$ and the cosmic strings background to be frequency-independent, i.e. $\Omega_{CS} = \text{const}$. The first-order phase transition is modeled as a broken power-law (with
<figure><img src="image_75.png" /><figcaption>**Figure 66.** Upper limits we can place on the amplitude of the background from cosmic strings (left panel) and a first-order phase transition (right panel) with ET's triangular 10 km configuration (dashed line) and 2L 20 km configuration (solid line). We assume a strong preference for a signal that contains an astrophysical and a cosmological contribution, relative to a signal containing only unresolved astrophysical objects for log Bayes factors greater than 8.</figcaption></figure>

$\Omega_*$ amplitude and $f_*$ break frequency), with the assumption that the sound-wave component is the dominant contribution to the cosmological SGWB. We simulate astrophysical and cosmological SGWB and perform a model selection study of the signal. For details on the parameter estimation Bayesian tools, see [429]. We compute Bayes factors for a CBC signal to noise, and for a combination of cosmological and CBC signal to noise. Comparing the two Bayes factors demonstrates if one model is preferred over the other, i.e. if we can detect the presence of a cosmological signal. From figure 66 we conclude that a cosmic strings background of amplitude $2.2 \times 10^{-11}$ and a first-order phase transition background of amplitude up to $4.0 \times 10^{-10}$ (at 25 Hz) will be probed with this 3G detector.

*Note that the various ET configuration makes little quantitative difference to the results, but including next-generation Cosmic Explorer can lead to great improvement in SGWB source separation.*

## 6.6 Continuous waves

Broadly speaking, the class of continuous waves (CW) comprises all deterministic$^{42}$ GW signals that last much longer than typical transient signals, whose durations vary from a fraction of a second to tens or hundreds seconds at the most. No CW has been detected so far, although interesting upper limits have been established, see e.g. [430–432] for recent reviews. The prototypical source of CWs is a spinning neutron star, asymmetric with respect to the rotation axis, that emits a quasi-monochromatic gravitational wave whose frequency changes extremely slowly over time. For such sources, the expected signal duration is much longer than one day, meaning that the frequency modulation due to the Doppler effect, caused by the motion of the Earth, and the phase and amplitude modulations due to the detector radiation pattern functions, play relevant roles.

Actually, other sources of CW-like signals have been investigated. In particular, transient CW signals (tCW) have been considered, such as those expected to be emitted by a newborn magnetar, in the early stages of its evolution [433–436]. In this case, the frequency secular variation is caused by a combination of the electromagnetic emission, due to the

<footnote>$^{42}$In contrast to GW backgrounds, which are described by means of stochastic processes.</footnote>
super-strong magnetic field, and the gravitational-wave emission, due to the strong asymmetry induced by the electromagnetic field itself. As a consequence, the resulting source spin-down is orders of magnitudes faster than for standard neutron stars, producing a signal that stays in the sensitivity band of detectors for hours or days.

Recently, several theoretical works have shown that GW detectors can also be used to probe dark matter, see e.g. [437] for a comprehensive review. In many of the proposed mechanisms, the resulting signal has a CW-like nature. This is, for instance, the case for depleting scalar or vector ultra-light boson clouds that could form around Kerr black holes through superradiance [438], the interaction of dark matter particles with detector components [439],<fnref n="43" /> or the early inspiral of sub-solar mass primordial black holes [440].

The typical duration of CW signals has some important consequences. First, once detected, a CW source can be monitored for long periods of time, becoming a true laboratory for fundamental physics and astrophysics, and potentially allowing observations of tiny deviations from the assumed models (for instance, a deviation from GR, in the form of non-standard polarizations [441]). At the same time, signal persistence allows us to reduce the false alarm probability of any candidate virtually to zero, because if a signal is present in a given dataset — even with low significance — it will also appear in new datasets, and its significance will increase until a detection can be claimed (of course, this is not necessarily true for tCWs). Moreover, CW source parameters can be measured with very high accuracy, as their uncertainties decrease with the observation time. On the other hand, CW signals are predicted to have small amplitudes, compared to e.g. typical CBC signals, and to search for them can require a huge computational cost. For this reason, the development of increasingly sensitive, robust, and computationally efficient algorithms is a very active field that grows in parallel to current detector upgrades and R&D for future detectors, such as ET. In the following, we give more details on different sources of CW-like signals, focusing attention on the impact that different ET configurations have on the chance of detection.

### 6.6.1 CWs from spinning neutron stars

Spinning neutron stars (NS), isolated or in a binary system, asymmetric with respect to the rotation axis, emit a persistent gravitational wave signal with frequency in a given ratio to the rotation frequency of the star.

The typical amplitude of CWs signals is difficult to estimate, as it depends on poorly known quantities, especially the star's ellipticity, which is a measure of its degree of asymmetry. In the case of a CW emitted by a non-accreting, spinning NS, assuming it is asymmetric with respect to its rotation axis, the signal amplitude is given by

$$h_0 = \frac{4\pi^2 G \epsilon I_{zz} f^2}{c^4 d} \approx 1.06 \times 10^{-26} \left(\frac{\epsilon}{10^{-6}}\right) \left(\frac{I_{zz}}{10^{38} \text{ kg m}^2}\right) \left(\frac{f}{100 \text{ Hz}}\right)^2 \left(\frac{1 \text{ kpc}}{d}\right), \quad (6.21)$$

where d is the source distance, f is the GW frequency (equal to two times the star's spin frequency in the prototypical case of a NS rotating around one of its principal axes of inertia), $\epsilon \equiv (I_{xx} - I_{yy})/I_{zz}$ is the ellipticity of the star, and $I_{zz}$ is the moment of inertia of the star with respect to the principal axis aligned with the rotation axis.

The search for CWs from spinning NSs depends on the amount of information we have or we assume to have on the sources. For *targeted* searches, we use accurate measurements

<footnote n="43">⁴³In this case the effect of the interaction is not a true GW signal, but is such to induce a relative motion of the optical components which can be searched as if it was a standard gravitational wave.</footnote>
of the main NS parameters, namely the position, spin frequency and its derivative(s), and Keplerian parameters for NSs in binary systems, obtained from electromagnetic (EM) observations (typically radio, gamma, or X-rays) to perform very sensitive searches based on matched filtering. Data from the detector network can be coherently combined in order to increase the sensitivity; see [442] for the most recent O3 results obtained by the LIGO and Virgo Collaborations for targeted searches. On the other hand, there are all-sky searches for NSs without an EM counterpart, in which a portion of the parameter space as large as possible is explored. Due to computational load constraints, such searches are based on hierarchical semi-coherent methods which sacrifice some sensitivity in order to make them computationally feasible. See [443] for the latest LVK all-sky search results. Searches covering “intermediate” cases are also routinely carried out. For instance, narrow-band searches slightly extend the parameter space around the EM-inferred parameters of known pulsars, in order to account for possible small — fraction of a Hz — mismatches between the EM and GW emission [444]. Directed searches assume the source position is known, while the star’s rotational parameters are unknown, as in the case of the supernova remnants or the galactic center [445, 446]. All-sky and directed searches typically use data from different detectors (or runs of the same detector) in order to make coincidences among outliers, with the aim of suppressing the false alarm probability. Although no detection of CWs has been made so far, interesting upper limits on the star’s ellipticity have been placed, see references above. These can be interpreted in terms of constraints on the residual initial star’s deformation frozen after crust solidification or as due to an inner magnetic field misaligned with the rotation axis, using a given ratio with the external poloidal one.

Although the range of CW searches for spinning NSs is very wide, in the following we provide some figures of merit about detection perspectives by ET focusing on two relevant cases, targeted and all-sky searches, that bracket the span of typical searches in terms of dimension of the explored parameters space, sensitivity and computational cost.

For targeted searches, we provide some summary results in table 38 where, for different ET configurations, the number of potentially detectable sources (among those contained in the ATNF catalogue$^{44}$) is given, under three different assumptions of the source ellipticity. We consider a total observation time $T_{obs} = 1$ year, a duty cycle of 85% and assume to make a fully-coherent search based on matched filtering. In the case of 2L-shaped network, results do not depend on the relative orientation of the two detectors. Columns 2 indicates the number of detectable pulsars under the assumption that they emit at their spin-down limit, i.e. that all their spin-down is due to the emission of GWs. This provides a theoretical upper limit to the emitted strain, given by

$$h_{0,\text{sd}} = \frac{1}{d} \left( \frac{5GI_{zz}}{2c^3} \frac{|\dot{f}|}{f} \right)^{1/2}, \qquad (6.22)$$

where $\dot{f}$ is the signal frequency first time derivative. The corresponding limit on the star ellipticity is

$$\epsilon_{\text{sd}} = \frac{h_{0,\text{sd}}}{I_{zz}} \left( \frac{c^4 d}{4\pi^2 G f^2} \right) \approx 9.46 \times 10^{-6} \left( \frac{h_{0,\text{sd}}}{10^{-25}} \right) \times \left( \frac{10^{38} \text{ kg m}^2}{I_{zz}} \right) \left( \frac{100 \text{ Hz}}{f} \right)^2 \left( \frac{d}{1 \text{ kpc}} \right). \quad (6.23)$$

<footnote>$^{44}$https://www.atnf.csiro.au/research/pulsar/psrcat/.</footnote>
<table><caption>**Table 38.** Expected number of detectable sources, assuming three different conditions for the ellipticity: $\epsilon = \epsilon_{sd}$ ($n_1$), $\epsilon = \min(\epsilon_{sd}, 10^{-6})$ ($n_2$), $\epsilon = \min(\epsilon_{sd}, 10^{-9})$ ($n_3$), assuming a total observation time $T_{\text{obs}} = 1$ year and a duty cycle of 85%. For each case, we give in parentheses the minimum and median value of ellipticity for detectable signals.</caption><thead><tr><th>Configuration</th><th>n<sub>1</sub></th><th>n<sub>2</sub></th><th>n<sub>3</sub></th></tr></thead><tbody><tr><td>Δ 10km</td><td>866 (2.5 × 10<sup>-10</sup>, 1.3 × 10<sup>-4</sup>)</td><td>180 (2.5 × 10<sup>-10</sup>, 4.4 × 10<sup>-9</sup>)</td><td>19 (2.5 × 10<sup>-10</sup>, 7.5 × 10<sup>-10</sup>)</td></tr><tr><td>Δ 10km HF-only</td><td>398 (2.5 × 10<sup>-10</sup>, 6.2 × 10<sup>-6</sup>)</td><td>178 (2.5 × 10<sup>-10</sup>, 4.4 × 10<sup>-9</sup>)</td><td>19 (2.5 × 10<sup>-10</sup>, 7.5 × 10<sup>-10</sup>)</td></tr><tr><td>Δ 15km</td><td>983 (2.1 × 10<sup>-10</sup>, 1.1 × 10<sup>-4</sup>)</td><td>214 (2.1 × 10<sup>-10</sup>, 4.4 × 10<sup>-9</sup>)</td><td>33 (2.1 × 10<sup>-10</sup>, 7.9 × 10<sup>-10</sup>)</td></tr><tr><td>2L 15km</td><td>959 (2.0 × 10<sup>-10</sup>, 1.2 × 10<sup>-4</sup>)</td><td>206 (2.0 × 10<sup>-10</sup>, 4.2 × 10<sup>-9</sup>)</td><td>29 (2.0 × 10<sup>-10</sup>, 8.1 × 10<sup>-10</sup>)</td></tr><tr><td>2L 15km HF-only</td><td>451 (2.0 × 10<sup>-10</sup>, 5.6 × 10<sup>-6</sup>)</td><td>203 (2.0 × 10<sup>-10</sup>, 4.0 × 10<sup>-9</sup>)</td><td>29 (2.0 × 10<sup>-10</sup>, 8.1 × 10<sup>-10</sup>)</td></tr><tr><td>2L 20km</td><td>1035 (1.8 × 10<sup>-10</sup>, 1.1 × 10<sup>-4</sup>)</td><td>227 (1.8 × 10<sup>-10</sup>, 4.3 × 10<sup>-9</sup>)</td><td>33 (1.8 × 10<sup>-10</sup>, 7.3 × 10<sup>-10</sup>)</td></tr></tbody></table>

The number of pulsars given in column 3 is computed by assuming that every star's ellipticity is the minimum of the spin-down limited value $\epsilon_{sd}$ and $10^{-6}$, which is of the order of the maximum theoretical ellipticity a NS with a standard EOS could have [447]. In column 4, the number of detectable pulsars is computed assuming the minimum ellipticity between $\epsilon_{sd}$ and $10^{-9}$, which is considered as a plausible value for millisecond pulsars [448, 449]. For each case, we also report the minimum and median value of the ellipticity for the potentially detectable sources.

From this table, we can draw some considerations. *The main factor affecting the number of detectable sources is the arm length, rather than the network geometry.* Indeed, by coherently combining individual detector data streams, the triangle configuration would provide a sensitivity gain of $\sqrt{3}/2$, with respect to the 2L network, which is nearly fully balanced by the shape factor, so that at the end what mostly contributes to the sensitivity is the arm length. *We also note that, for the more realistic choices on star's ellipticities, corresponding to columns $n_2$ and $n_3$, most of the potentially detectable sources are millisecond or young fast pulsars, spinning at frequencies larger than a few tens of Hertz, for which the low frequency sensitivity of the detectors is not particularly relevant.* The corresponding typical values of the ellipticity are of about $2 \times 10^{-10}$, corresponding to “mountains” extending from the star’s surface not higher than about $1\mu m$. Lastly, focusing on the most conservative condition, given by column $n_3$, we find that *the number of potential detections increases by 50-70% going from 10km-arm detectors to 15 or 20km-arm detectors*.

Wide-parameter searches are based on semi-coherent methods in which data segments of given duration are incoherently combined. In principle, they are less sensitive than targeted ones, but can cover a huge parameter space and are intrinsically more robust against non-predicted deviations from the assumed signal model. Concerning, in particular, all-sky searches, figure 67 shows, for different detector configurations, the minimum detectable ellipticity as a function of the signal frequency, assuming source rotational evolution is dominated by the emission of GWs and a fixed source distance of 8 kpc. These results have been obtained considering an analysis done by means of the FrequencyHough pipeline [450], a semi-coherent procedure routinely used in current CW searches, whose sensitivity can be estimated (under the assumption the noise is Gaussian) as

$$h_{\min,95\%} \approx \frac{4.97}{N^{1/4}} \sqrt{\frac{S_n(f)}{T_{\text{FFT}}}} \sqrt{\text{CR}_{\text{thr}} + 1.6449}, \quad (6.24)$$

where $N = T_{\text{obs}}/T_{\text{FFT}}$ is the ratio between the total observation time and the duration of the data segments to be incoherently combined, $S_n(f)$ is the detector average noise power
<figure><img src="image_76.png" /><figcaption>**Figure 67.** ET sensitivity reach to all-sky searches of CWs (95% C.L.). The various continuous curves are the minimum detectable ellipticity as a function of frequency for different detector configurations, assuming a source distance of 8 kpc, an observation time $T_{\text{obs}}=1$ yr, a duty cycle $D=85\%$ and data segments duration $T_{\text{FFT}}=10$ days. The dashed curve indicates a constant spin-down of $10^{-8}$ Hz/s. Right plot is a zoom of the left plot in the high frequency region. On the scale of the right plot, the curves corresponding to $\Delta$ 10 km HF and $\Delta$ 10 km coincides, and similarly for those corresponding to the 2L 15 km HF and 2L 15 km configurations.</figcaption></figure>

spectrum and $CR_{\text{thr}}$ is the threshold Critical Ratio to select outliers. In the plot, we used $T_{\text{obs}}=1$ year, $T_{\text{FFT}}=10$ days and $CR_{\text{thr}} = 3.4$ for triangular configurations and $CR_{\text{thr}} = 4$ for L-shape configurations. The presence of more that one detector is exploited in such kind of searches to strongly reduce the false alarm probability. For the standard search parameters considered here, the number of points in the parameter space (sky position, frequency and frequency derivatives) is of the order of $10^{23}$. Assuming that for each detector of the order of $10^{12}$ outliers are selected,<fnref n="45" /> it is possible to show that double coincidences are enough to reduce the number of candidates to $O(10)$, and that triple coincidences would allow — in principle — to shrink this number to a negligible level, assuming ideal Gaussian noise [450]. As a matter of fact, in practice we expect a significant number of candidates to survive the coincidence step, due to non-Gaussian instrumental artefacts which, inevitably, pollute the data. Having triple coincidences, as for the triangle configuration, allows to better reduce false candidates with respect to the 2L configuration. This implies that in the former case we can use a slightly lower threshold for outliers selection, with the aim of having a similar number of candidates after the coincidences. A lower threshold, according to eq. (6.24), corresponds to a better sensitivity. Of course the exact value of the threshold would depend on the non-Gaussian, non-stationary features of the actual dataset, which are difficult to predict in advance. As a rule of thumb, we decide to reduce the threshold on the $CR$ from 4 to 3.4 in the case of the triangular configuration. This choice would correspond to a reduction of one order of magnitude in the false alarm probability in presence of Gaussian noise.

As for targeted search, the relative orientation of the two triangular detector in the 2L-shaped configuration does not impact on the analysis outcome. A clear feature of these results is that different configurations do not produce very different results, especially in the high frequency range, which corresponds to smaller ellipticities. In general, however, longer-

<footnote n="45">⁴⁵Currently, the typical number of selected outliers is $O(10^{10})$, but we devise an increase of two orders of magnitude as reasonable in the ET era.</footnote>
arm detectors perform better than shorter ones, with differences up to a few tens percent. For instance, a source located at 8 kpc, and emitting a signal at about 500 Hz, would be detected if its ellipticity is at least ~ (4 − 5) × 10⁻⁸. At a given frequency, the ellipticity scales linearly with the distance. Nearby sources, located within ~ 1 kpc, would produce detectable signals with an ellipticity as small as few times per 10⁻⁹, if spinning at around 500 Hz. Although current searches typically use a shorter data segment duration, values as large as 10 days are reasonable, and likely conservative, for the ET era, given the algorithmic developments and the increase in available computing power. It is interesting to stress, however, that while the sensitivity of semi-coherent searches scales as the fourth root of the segment duration, the dimension of the parameter space scales at least as the duration to the fifth power. This means, for instance, that using segments of 30 days, instead of 10, would provide a theoretical sensitivity gain of about 30%, in front of a computing cost at least 3⁵ times bigger. Moreover, and probably even more importantly, the use of very long segment durations reduces the search robustness with respect to unmodeled signal features, which can translate into a sensitivity loss. As an example, a source may have a significant intrinsic transversal velocity with respect to the line of sight. If not taken into account, it may determine a sub-optimal Doppler correction as a consequence of the change of sky position during the observation time. Another possibility is that a signal may be characterized by some level of frequency wandering due to some unpredicted mechanism, involving the interaction with other objects, or due to intrinsic processes in the source (see e.g. [451, 452] for analysis methods that tackle this issue). It is then important to note that a reasoned balancing among theoretical sensitivity, available computing power and algorithm robustness must be always taken into account when new algorithms for wide-parameter searches are developed.

6.6.2 Transient CWs

Newly-born neutron stars could be extremely deformed and emit very strong GWs imme-
diately after they have formed from a binary neutron star merger or supernova explosion.
See [453, 454] and references therein for recent observational evidences.

Such systems are astrophysically very interesting: detecting a remnant of a neutron star merger would help to improve constraints on the equation of state [455–458] compared to only considering the inspiral of the system alone [234, 459, 460] in the case of binary neutron star mergers or only electromagnetic observations of the supernova. Furthermore, detecting such a system would be a huge step in multi-messenger astronomy, and would allow us to determine the nature of the remnant, something at which electromagnetic observations can only hint. As we saw from GW170817, astronomers were able to see the remnant in multiple ways — UV, IR, visible — and each way seemed to imply a different expected remnant. Some studies report the formation of a hypermassive neutron star lasting for $O(s)$ [461–465], while others saw continued energy emission from a long-lived remnant [466–469]. In the latter case, the signal emitted would last longer than the canonical $O(s)$ expected from mergers, but shorter than the durations $O(\text{years})$ expected from much older, asymmetrically rotating neutron stars. This “intermediate” regime of signal durations would contain very interesting physics, and yet until recently, not much effort had been made to search for these kinds of sources.

Such searches for a remnant of GW170817 were performed on O2 data, analyzing data
in segments of $O(\text{seconds})$ for burst-like signals and $O(\text{hours-days})$ for tCWs, and set the
first-ever upper limits on GW emission from a remnant [470, 471]. In contrast to ordinary
CWs, tCWs last for a much shorter time, have very large spin-downs, and have frequencies
that evolve non-linearly over time. Thus, new analysis methods had to be designed to search
<figure><img src="image_77.png" /><figcaption>**Figure 68.** Left: ET distance reach to transient CW signals emitted by remnants of binary neutron star mergers or supernovae for various arm lengths and shapes, fixing the configuration to allow both high and low-frequency enhancements. The ellipticites used to construct these curves are the maximum allowed by energy conservation, assuming that all rotational energy is emitted in GWs, and imposing $\epsilon \le 5 \times 10^{-3}$, the largest expected degree of deformation. At frequencies for which the ellipticity given by energy conservation exceeds $5 \times 10^3$, we impose $\epsilon = 5 \times 10^{-3}$. Right: ET minimum detectable ellipticity for different configurations for newborn neutron stars 6 Mpc away. Both plots use $\dot{f}_0 \le -1 \times 10^{-4}$ Hz/s, $T_{FFT} \sim 70$ s, $T_{obs} = 1$ day, and $I_{zz} = 10^{38}$ kg·m$^2$.</figcaption></figure>

for these signals [472–477]. The sensitivity of the O2 search allowed us to reach distances $\mathcal{O}(1)$ Mpc; however, with ET, we will be able to reach distances of $\mathcal{O}(10-100)$ Mpc, depending on the degree of deformation of the neutron star. To compute this sensitivity, we use an equation similar to eq. (6.24), that has been generalized for power-law signals and gives the maximum distance reach at a given confidence level:

$$d_{\max} = 4.63 \cdot 10^{-9} \text{ m} \left( \frac{I_{zz}}{10^{38} \text{ kg} \cdot \text{m}^2} \right) \left( \frac{\epsilon}{10^{-3}} \right) \frac{1}{N^{1/4}} \frac{T_{FFT}}{\sqrt{T_{obs}}} \left( \sum_i \frac{\mathcal{F}_i^2}{S_n(f_i)} \right)^{1/2} (\text{CR}_{thr} + 1.6449)^{-1/2}. \quad (6.25)$$

Here, $\mathcal{F}_i = f_i^2$ is short-hand for the $i$th frequency, and the sum goes over all frequencies during $T_{obs}$.

In the left-hand panel of figure 68, we plot the expected distance reach for different ET configurations, for a braking index $n = 5$, i.e. pure GW emission. The ellipticity at each frequency is the maximum allowed by energy conservation, and we restrict to ellipticities less than $5 \times 10^{-3}$, the largest expected for newborn neutron stars [478]. However, at frequencies below $\sim 1200$ Hz, the ellipticity given by energy conservation is much larger than that allowed theoretically; hence, we impose $\epsilon = 5 \times 10^{-3}$ up to 1200 Hz. In this regime, we see that the distance reach steadily increases with frequency at a fixed $\epsilon = 5 \times 10^{-3}$, which is consistent with eq. (6.25). For frequencies greater than 1200 Hz, the distance reach decreases slightly, since at this point, the ellipticity allowed by energy conservation is in fact less than $5 \times 10^{-3}$. Magnetars formed in core collapse supernovae look more promising, in terms of event rate, with respect to those following the merger of NS binary systems. The overall core collapse supernova rate is of the order of one event per year within 10 Mpc (4 events/yr within 20 Mpc) [479]. The fraction of core collapses leading to the formation of a magnetar is uncertain and, while past estimations pointed to the $(1-10)\%$ range [480], more recent results, based on observations and a more proper consideration of selection effects, typically provide higher values, even of the order of 40%, see e.g. [481]. This would correspond to a magnetar
formation rate of ~2 per year within 20 Mpc, which is well within the search distance reach for a wide range of initial spin frequencies, for the optimistic deformation rate discussed above.

In the right-hand panel of figure 68 we plot the minimum detectable ellipticity at a fixed distance of 6 Mpc, at which we expect to detect magnetars forming at a rate of one per 5-10 years. At such a distance, the only low-frequency ($\lesssim 20$ Hz) sources that could produce a detectable gravitational-wave signal are those with unphysical ellipticities, i.e. $\epsilon > 1$, so we omit those on the plot. We only consider frequencies at which the minimum detectable ellipticity is less than $5 \times 10^{-3}$, which is consistent with the left-hand panel. In fact, at such distance, we would be able to detect the tCW from newborn magnetars with ellipticity of the order $10^{-3} - 10^{-4}$, much lower than the maximum predicted value.

We see that the differences among configurations are mostly driven by the arm-length, and there is little difference between the 2L-15km and 15km triangle. As in the CW case, we have applied $CR_{thr} = 4$ for the 2L case, and $CR_{thr} = 3.4$ in the triangle case to capture the anticipated improvements in both computational power in the 3G era and improved coincidences resulting from a triangle-shaped detector. Furthermore, these plots are created with the full HFLF xylophone configuration. If we were to remove the low-frequency instrument in the left-hand panel, the distance at low frequencies $\lesssim 100$ Hz (not shown) decreases from $O(1-10)$ kpc to $O(0.1-1)$ kpc. This could be relevant, since the enhancements due to the LF instrument allow us to reach the galactic center at low frequencies. However, in the right-hand panel, the results do not vary at all, since the ellipticities are already unphysical at frequencies below 30 Hz (not shown).

### 6.6.3 Search for dark matter with CWs

GW detectors could probe the existence of some kinds of dark matter (DM) [482]. In many cases, DM has been predicted to produce CW-like signals, so that data analysis methods used for the search of “standard” CW or tCW have been extended, see e.g. [483], and applied to the search for DM signatures. Ultralight bosons are interesting DM candidates, including dark photons and Quantum ChromoDynamics axions [438, 484–486]. Potential detectable systems might be boson clouds formed around spinning BHs or vector bosons in form of dark photons. Detectable continuous signals can arise also from compact dark objects or primordial black holes.

**Boson clouds around black holes.** Ultra-light bosons fields present in the nearby regions of a Kerr black hole can clump around it through a superradiant mechanism, forming a BH-boson “cloud” system at the expense of the BH angular momentum [438, 486, 487]. This formation channel is maximally efficient when the particles’s Compton wavelength is comparable to the size of the BH, $\hbar/(m_b c) \simeq GM_{BH}/c^2$, where $m_b$ is the boson mass, $M_{BH}$ is the black hole mass, $\hbar$ is the reduced Planck’s constant, and $c$ is the speed of light. Once formed, the cloud starts to dissipate via particle annihilation to gravitons, emitting a quasi-monochromatic and long-duration signal with a frequency dependent mainly on the boson mass,

$$f_{gw} \approx 483 \text{ Hz} \left( \frac{m_b c^2}{10^{-12} \text{ eV}} \right). \qquad (6.26)$$

Earth-bound detectors are mostly sensitive to boson masses in the $10^{-14} - 10^{-11}$ eV/c² range [452]. The BH-boson cloud system is often referred to as a gravitational atom. The emitted signal is characterized by a spin-up due to the cloud mass decrease, which reduces
its binding energy [488, 489], as well as to boson self-interaction [490]. The GW signal amplitude is

$$h(t) = h_0 (1 + t/\tau_{\text{gw}})^{-1}, \qquad (6.27)$$

where $h_0$ is mainly determined by the BH and boson masses, the BH initial spin and the distance, through

$$h_0 \approx 3 \times 10^{-24} \left(\frac{\alpha}{0.1}\right)^7 \left(\frac{\chi_i - \chi_c}{0.5}\right) \left(\frac{M_{\text{BH}}}{10 \text{ M}_\odot}\right) \left(\frac{1 \text{ kpc}}{d}\right), \qquad (6.28)$$

where $\alpha$ is the fine-structure constant of the gravitational atom, given by

$$\alpha = \frac{GM_{\text{BH}} m_b c^2}{c^3 \hbar}. \qquad (6.29)$$

The gravitational-wave timescale $\tau_{\text{gw}}$ is also dependent on the BH mass, the boson mass and the BH initial spin [438, 452, 489]. The detection of CW signals from boson clouds would allow to establish a fascinating connection among particle physics and black holes. Null results, as those obtained so far, see e.g. [452, 491, 492], brings anyway valuable constraints on the permitted boson masses.

**Dark-matter particles interacting with GW detectors.** Another interesting scientific scenario for ET are ultralight particles behaving as a classical field interacting coherently with the atoms of the test masses. For instance, dark photons coupled to the baryon or baryons minus leptons number $U(1)_B/U(1)_{B-L}$ have been considered in literature. These particles can produce an oscillating force on dark charged objects [439, 493, 494]. The same type of behavior is expected for tensor particles [495]. Different production mechanisms have been proposed for the dark photon production, such as misalignment mechanism associated with the inflationary epoch, light scalar decay or cosmic strings [496, 497]. The dark photon DM field oscillations impinge a time-dependent Equivalence Principle-violating force acting on the test masses. This will produce a change in the relative length of the detector's arms, causing a signal strain at the detector output. The time-dependent force acting on the test masses produces a strain oscillating at the same frequency and phase as the dark photon field [439, 498–500]. It should be noted that a similar signal may be produced in the case of scalar dark matter particles directly interacting with the mirrors. In particular, as described by [501], the interaction of the scalar bosons with the detector beam-splitter will induce oscillations in the thickness of the mirror due to the oscillations of the fundamental constants [502–504]. They could also interact with the particles in the reference cavity in the detector, resulting in a similar measurable oscillation of fundamental constants [505]. The signal frequency is determined by the dark photon mass, $f_0 = m_A c^2/(2\pi\hbar)$, corresponding for Earth-based detectors to dark photon masses in the range $10^{-14} - 10^{-11} \text{ eV/c}^2$. In fact, the signal is not exactly monochromatic due to the fact dark photon particles follows a Maxwell-Boltzmann velocity distribution.

Taking into account the physics of dark-matter signals, different CW methods based on cross correlation and excess power have been applied to look for these types of signals [439, 498, 506, 507], and have resulted in competitive constraints on $U(1)_B$ coupling using data from the most recent observing run of advanced LIGO/Virgo/KAGRA [499, 508], and on scalar dilaton dark-matter coupling to electrons and photons using GEO600 data [501]. Furthermore, projected constraints for axions altering the polarization of light shining down the interferometer arms have been produced [509, 510].
When considering the different configurations and geometries, we note that the 2L geometry would experience a larger strain due to dark photons by a factor of $2/\sqrt{3} \sim 1.15$ compared to a triangle-shaped detector, independent of the arm-length [439]. Since any frequency (or mass) for the dark photon or other dark matter particle is possible, both the low- and the high-frequency instruments of the ET xylophone configuration are important. Furthermore, increased arm length is also the primary way to improve sensitivity to these particles, since they behave just as traditional CWs, though $T_{\text{FFT}}$ could not be as long as 10 days because of the frequency modulation induced by the Maxwell-Boltzmann distributed velocities. Finally, the triangle-shaped detector would allow triple coincidences, but since the dark-photon signal would show up looking like “correlated noise” between the detectors, having a single triangle-shaped detector in one place may increase the number of correlated noise disturbances, reducing the sensitivity to dark-matter particles.

**Inspiring compact dark objects and PBHs.** Dark matter can be present in our universe in form of macroscopic objects, we generically refer to as compact dark objects (CDOs). The actual formation channel and the origin of CDOs is still widely debated. CDOs can form pairs and emit an almost monochromatic signal while being far from the coalescence phase. Primordial BHs (PBHs), that we have discussed in detail in sections 6.3.2 and 6.3.3, are an example of CDOs. In the context of CW signals, binary systems made of sub-solar mass PBHs as well as more generic CDOs, represent potential targets. Indeed, the GW signals emitted by these systems, when the two compact objects are far away from the coalescence and their masses are small enough, say $< 10^{-2} M_{\odot}$, i.e below the mass range considered in section 6.3.3, can be modeled as CW or tCW signals with a spin-up described by a power law, see e.g. [440, 511].

If the chirp mass is small enough, the frequency can be modeled exactly as the linear frequency Taylor expansion used for standard CW searches. This means that, for instance, a pair of inspiralling PBHs with chirp masses below $10^{-5} M_{\odot}$ would emit a GW signal indistinguishable from those arising from non-axisymmetric rotating NSs spinning up, and upper limits have already been placed on the fraction of dark matter that PBHs could compose ($f_{\text{PBH}}$) using CW results from the O3a and O3 observing runs [443, 512].

These upper limits on $f_{\text{PBH}}$ derived from CW searches are of particular importance, since they lie within the “asteroid-mass” PBH mass regime. In this mass range, there are almost no existing constraints on $f_{\text{PBH}}$ [261]. ET will allow us to probe realistic $f_{\text{PBH}}$ in the asteroid-mass regime (currently upper limits on $f_{\text{PBH}} > 1$ from these searches), and will result in an improvement of 1-2 orders of magnitude, regardless of the detector configuration or geometry [440]. The same comparison of different detector designs discussed in section 6.6.1 for all-sky searches would be applicable here, since the constraints on $f_{\text{PBH}}$ are derived from all-sky search results.

### 6.6.4 Conclusions

We have considered here the impact of different ET geometries and configurations on the detectability of persistent GW signals. In the case of CWs from slowly spinning down asymmetries rotating neutron stars, we find that the number of detectable sources in targeted searches is mostly sensitive to the arm length while, for realistic degrees of deformation to high-frequency enhancements, the shape of the detector does not have a relevant role. Furthermore, in all-sky searches, we expect to be able to use a value of $T_{\text{FFT}}$ that is $\sim 100$ times longer than used in analyses today, and thus find that we can constrain the ellipticity
of unknown neutron stars to be less than $1 \times 10^{-7}$ at frequencies above 300 Hz from the galactic center. This sensitivity is also mostly a function of the arm-length, and does not vary significantly between design choice.

For transient CWs, the arm length is also the most important factor that determines the relativity sensitivity of the different configurations considered, with the low-frequency sensitivity only helping to reach the galactic center at signal frequencies below $\sim$ 30 Hz. However, we considered sources with initial spin-downs one to three orders of magnitude smaller than those searched for in [471], determined by the maximum allowed ellipticity predicted theoretically. This limited our distance reach by a couple of orders of magnitude compared to the ellipticities $10^{-2} - 10^{-1}$ considered in [471].

We have hinted at the possibility of constraining exotic physics with CW and tCW searches, and note that the results presented here can be recast in terms of constraints on, e.g. the fraction of dark matter that PBHs could compose, the maximum coupling strength to baryons that dark-matter particle could have, or boson/black hole mass pairs that could be excluded from existing.

All these results must be interpreted carefully, since they assume Gaussian noise and a lack of detector artifacts. In practice, glitches and noise lines will affect the sensitivity to all of these sources. Noise lines will inhibit dark-matter and CW searches the most, since they focus on persistent, long-lived GWs, while glitches are more problematic for the transient CW searches. We have tried to capture the impact of improved coincidences by lowering $CR_{thr}$ for the triangle-shaped detector compared to the 2L one; however, these thresholds are subject to change in practice based on available computing power and the characteristics of the noise. We do not consider explicitly how the so-called “null stream” would impact the detectability of these sources, though we expect that our methods, already designed to be robust against noise disturbances and glitches, can achieve comparable sensitivities as those obtained in Gaussian noise, based on current GW searches.

# 7 The role of the null stream in the triangle-2L comparison

The null stream is a signal-free linear combination of the interferometer strain data, and it plays an important role in the comparison of 2L and triangle ($\Delta$) configurations of ET. A null stream can be formed for an arbitrary GW detector network with at least three detectors, but generally only for one GW signal at a time [513]. However, the $\Delta$ configuration allows one to form a null stream that cancels out all gravitational-wave signals simultaneously [120], because of its closed geometry of component arms and almost negligible light travel times between components. The null stream in the $\Delta$ configuration (from now on just “null stream”) has the unique ability to provide access to a signal-free channel, and therefore access to the properties of (incoherent) instrumental noise without contamination from gravitational-wave signals [149].

It should be stressed that inference from data with the null stream noise discarded is identical to coherent inference from the full set of data so that access to the null stream does not directly improve inference [514]. Moreover, a 2L configuration without access to the null stream may still achieve good noise mitigation but requires more complex methods that introduce uncertainty into the process. Access to the null stream provides a more straightforward way (compared to the 2L configuration) to extract the power-spectral density (PSD) of the detector noise and enables unique data-analysis techniques for improved science
extraction. In the following, we will describe some of these techniques and their impact on science extraction.

**Estimation of unbiased noise power spectrum.** Within the $\Delta$ configuration of ET, the null stream $d_{\text{null}}$ can be straightforwardly formed by

$$d_{\text{null}} = d_1 + d_2 + d_3, \qquad (7.1)$$

where $d_i$ are the strain data for the individual detectors in the triangle. If the noise properties are homogeneous and incoherent among these detectors, then the noise PSD of each detector $S_n^i$ can be estimated by

$$S_n^i = \frac{1}{3} S_n^{\text{null}}, \qquad (7.2)$$

where $S_n^{\text{null}}$ is the PSD of the null stream. In the absence of a null stream, it is non-trivial to disentangle the detector noise from the unresolvable GW signals, which is also referred to as “confusion noise”.

Assuming uncorrelated noise, another method to estimate the instrument-noise PSD is by calculating the cross power-spectral density (CSD) of the null stream with the data streams [149]:

$$\langle d_{\text{null}} d_i^* \rangle = S_n^i, \qquad (7.3)$$

where $S_n^i$ is the PSD of detector $i$. This method does not require the PSD of different interferometers to be the same.

Noise can be produced identically in two interferometers, which would therefore not appear in the null stream and bias PSD estimates that use the null stream. Technically, the bias in PSD estimates is produced by the real part of the complex-valued instrument-noise CSDs between interferometers. The natural magnetic background is a possible source of such noise. Also seismic gravity fluctuations are considered a possible source of identical noise. However, over the > 400 m distances between any two test masses of two LF interferometers, this so-called Newtonian noise is most likely weakly correlated [515], which means that one should only expect a mild suppression in the null stream. See section 5.4 for a more in-depth discussion of these noise sources. Ref. [150] shows that, given knowledge of the correlated/non-identical noise sources (e.g. through witness sensors), unbiased instrument-noise PSD estimates can in principle be obtained even in the presence of correlated noise.

**Impact on science extraction.** The most direct impact of an unbiased estimate of the noise power spectrum is on the measurement of the stochastic gravitational-wave background. By subtracting the null stream PSD from the individual detector PSD, one obtains the PSD of the GW signals $S_h$ present in the data

$$S_h \approx S_n^i - \frac{1}{3} S_n^{\text{null}}, \qquad (7.4)$$

where the equality holds when the noise is identical amongst individual detectors (see e.g. refs. [149] and [150] for discussion on deviations from this assumption). $S_h$ can be directly related to the SGWB if the data contains no resolvable signals. Otherwise, one has to first subtract the loud sources from the data (see e.g. refs. [516, 517]).

The inability to disentangle detector noise from confusion noise also has the effect of raising the overall perceived noise level. In a templated GW search, the effect manifests as a loss
<figure><img src="image_78.png" /><figcaption>**Figure 69.** The loss in the ET detection horizon, as estimated in [516], due to the confusion of instrumental noise and compact binary inspirals. Dashed lines show the effect of the median local merger rate and dotted lines represent the upper merger rate. A higher merger rate introduces a stronger confusion noise and hence the higher horizon redshift loss. This effect can be mitigated by the ET null stream available in the *Δ* configuration.</figcaption></figure>

of matched filtering SNR, and it is demonstrated in figure 69. Figure 69 shows the CBC horizon redshift (left) and its percentage loss (right) as a function of the total source-frame mass and for different merger rates. The loss in the redshift reach of ET increases with a higher merger rate of compact binaries, and for BBHs it may appear between ≈ 2.5% and ≈ 20%, depending also on the mass of the CBC. For the *Δ* configuration, the efficiency for mitigating this effect approaches 100% as the sensitivity of all ET components approaches the same level.

Furthermore, the confusion noise impacts a calculation of the false alarm rate (FAR), the probability of incorrectly rejecting the noise hypothesis. The FAR is calculated by comparing the noise-induced (background) distribution of a detection statistic (e.g. SNR for matched filtering) with the one measured from a signal candidate. For example, one could perform matched filtering on time shifted data (between the individual detectors) to estimate the background distribution (see e.g. ref. [518]). FAR calculations typically assume that the number of genuine GW detectable signals is low enough not to impact the estimate, which will not be the case for ET anymore. By estimating the noise background distribution directly from the null stream (e.g. time shifting the null stream with respect to itself), one could mitigate the contamination from confusion noise and thereby improving the detection sensitivity and accuracy.

**Mitigation of transient detector glitches.** A number of known instrumental noise artefacts such as glitches invalidate the standard assumption that the data is described by a superposition of the Gaussian instrumental noise and GW signals. If unmodeled, they will appear as false positives, whereas modeling glitches introduces additional uncertainties. Unlike GWs, these artefacts will appear as non-Gaussian outliers in the null stream, which provides an opportunity to remove them. Ref. [149] outlines two methods for eliminating glitches with the null stream. First, one can identify glitches by match-filtering them in the null stream. Second, one can identify null stream data segments that are inconsistent with Gaussian noise. The authors show that it is possible to end up with a clean Gaussian background in the *Δ* configuration in the limit where ET components approach the same sensitivity.
*Impact on science extraction.* The non-Gaussian part of the background causes extended tails in the noise background and therefore limits the FAR. The exact extent to which this happens will only be known when ET is operational. Current gravitational wave observatories not only discover more GWs while achieving a better sensitivity but also retract more spurious signals [519]. This increasing trend might continue down to the design sensitivity of the ET. The null stream will reduce the uncertainty associated with modeling glitches, and allow us to mitigate the effects of non-Gaussian noise. The most prominent benefit of the null stream is expected for high-mass distant BBHs that mimic glitches as well as for signals that are only described by phenomenological models.

**Control of known and unknown systematic errors.** Any errors in the detector calibration can propagate into the null stream to cause incomplete cancellations of gravitational wave amplitudes [520]. If the signal waveform is a-priori well understood, and if its parameters are well-measured by the network of detectors, then the residual signal in the null stream will be the product (in the frequency domain) of the calibration error and known weighted amounts of signal [521]. One can detect this residual by performing matched filtering on the null stream. The calibration error can be obtained by fitting with a family of specific functions supplemented by the SNR output of the matched filters over a number of detected events. The ET null stream in the triangle configuration reduces the uncertainty in calibration parameters, which may propagate to GW signal parameters. However, the level of systematic errors in detector calibration in ET is not known, and even for current detectors it is only known to be less than 2% [522]. Therefore, it is not clear how many signals in ET will be affected by these systematic errors. Further studies are also necessary to demonstrate the performance of ET null stream calibration compared to other techniques, as well as the impact on the gravitational wave parameter estimation. In the worst case scenario, in 2L detectors calibration uncertainties will have to be estimated simultaneously with gravitational-wave source parameters, which would increase the computational cost of the analyses and introduce additional uncertainty. Note also that ref. [521] estimates that, in a network of at least three L-shaped 3G detectors, the calibration error can be inferred at the percent level if supplemented with $O(100)$ relatively loud (SNR=20) events.

## 8 Summary

In this work we have performed a detailed study of the Science Case of ET, significantly expanding the study in ref. [15]. We have considered both ET in its reference design (a 10 km triangle in a xylophone configuration, with an HF instrument and a cryogenic LF instrument), as well as variations on the reference design, both in the geometry (triangle vs 2L, with different arm lengths) and in the ASD (full sensitivity vs HF instrument only).

First of all, the significantly more detailed study of the Science Case performed here confirms and provides more detail and evidence for the picture summarized in ref. [15]: *ET, in its reference configuration, is a superb detector, with an extraordinary discovery potential in the domains of astrophysics, cosmology and fundamental physics.* With order $10^4-10^5$ BBH, BNS and NSBH detections per year, it will address and provide answers to an extremely rich and varied set of scientific questions; at the same time, it will penetrate deeply into unknown territories, where revolutionary discoveries could await for us.

Starting from the extraordinary potential of the reference design, it is, however, still mandatory to study what happens under variations of this baseline. At the very least, this allows us to understand how the different elements of the design, such as its geometry or