the relative roles of the HF and LF instruments, concur to produce the scientific output of ET, and what could be reached in intermediate stages of the commissioning of the detector. Furthermore, this study can suggest directions where changes of the baseline design could results in improvements of the science output. The original structure of the ET design was first laid down 10-15 years ago [10–12], well before the first detections of GWs. At that time, one was not even sure that there was, out there, a population of BBHs that coalesced within a Hubble time. Now we have a statistically significant catalog of detections, and building on the understanding that has accumulated thanks to these discoveries and the studies that they have stimulated, we are now able to present in this paper a study based on astrophysically-motivated populations.<fnref n="46" />

## 8.1 Comparison of different geometries

In this paper we have examined several different geometries, comparing two L-shaped detectors on widely separated sites to a single-site triangle, and with different choices of arm-length (and relative orientations for the two L-shaped detectors), to provide a broad set of options. We will begin by summarizing the main results from the comparison of the 15 km 2L configuration with the 10 km triangle, in section 8.1.1. In section 8.1.2 we will instead summarize the results of the comparison between the 15 km 2L and the 15 km triangle. In section 8.1.3 we also summarize the results for a single third-generation L-shaped detector. Further elements of the comparison between geometries, including correlated noise, the null stream, etc., are summarized in section 8.1.5.

### 8.1.1 Comparison between 15 km 2L and 10 km triangle

Focusing first on the comparison of the 15 km 2L with the 10 km triangle, the main results can be summarized as follows.

* **Binary Black Holes.** The results of section 3.1 show that, for BBHs, the 2L configuration with 15 km arms, oriented at 45°, is superior to a 10 km triangle for the estimation of all parameters, and especially luminosity distance, see figure 5 in the main text and tables 39–42 in appendix C. To put things into perspective, for all parameters (except luminosity distance) the differences are at the level of factors of 2-3, while both configurations outperform LVKI-O5 by orders of magnitudes on all parameters (with less large, but still significant differences for angular localization), see figures 8 and 9. For instance, while all ET configurations detect basically the whole BBH population with SNR ≥ 12, if we restrict to ‘golden events’ with SNR ≥ 100 the 2L-15km-45° configuration detects (in our sample realization) 4933 BBH/yr, while the 10 km triangle 2298, and LVKI-O5 only 4. For SNR ≥ 200 these figures becomes 644, 282 and 2, respectively; see table 39 in appendix C. For most parameters, as mentioned above, the differences between the 15 km 2L at 45° and the 10 km triangle are at the level of factors 2-3, in favor of the 2L; however, they are significantly larger for the accuracy on luminosity distance: in the 2L-15km-45° configuration there are 202 BBH/yr with error on $d_L$ smaller than 1%, to be compared with 28 for the 10 km triangle (and, in our sample realization, just 1 for LVKI-O5). For angular localization, focusing on the events with $\Delta\Omega_{90\%} \le 50$ deg$^2$, we find 10304 BBH/yr for 2L-15km-45°, 6064 for

<footnote n="46">⁴⁶Last but not least, the setting up and the development of the ET Observational Science Board allowed the formation of a coherent scientific community focused on ET, with the necessary expertise across the rather broad range of subjects needed for such a study.</footnote>
the 10 km triangle and 1607 for LVKI-O5, while for $\Delta\Omega_{90\%} \le 10\deg^2$ these numbers become 2124, 914 and 599, respectively; see table 40.<fnref n="47" /> The corresponding difference in the joint accuracies on $d_L$ and angular resolution shows that, also for this metric, the 2L-15km-45° is clearly superior to the 10 km triangle, see figure 6.

In contrast, the 2L configurations with parallel arms are disfavored, because of their worse angular resolution; e.g., for 2L-15km-0° there are 3030 BBH/yr with $\Delta\Omega_{90\%} \le 50\deg^2$), to be compared with 10304 BBH/yr for 2L-15km-45°; for $\Delta\Omega_{90\%} \le 10\deg^2$ the difference is even larger, with 374 BBH/yr for 2L-15km-0° (so, almost a factor of two less than LVKI-O5), compared to 2124 for 2L-15km-45°.

* **Binary Neutron Stars.** Similar conclusions follow from the study of parameter estimation for BNSs in section 3.2, see figure 11 and tables 42–44. Again the 2L with 15 km arms at 45° improves on the already remarkable performances of the 10 km triangle, typically by factors of order 2-3, while both outperform LVKI-O5. For instance, if we restrict to ‘golden events’ with SNR $\ge 50$, the 2L-15km-45° configuration detects 1052 events per year, while the 10 km triangle 458 and LVKI-O5 only 3 (for SNR $\ge 100$ these figures becomes 134, 57 and 0, respectively). If we consider, instead, events with error on luminosity distance smaller than 10%, there is almost a factor of 10 between 2L-15km-45° and the 10 km triangle, with 479 events/yr for 2L-15km-45° and 52 for the 10 km triangle (and only 1 for LVKI-O5; as always, in particular for small numbers, these are the numbers for our specific sample realization). For angular localization, focusing on the events with $\Delta\Omega_{90\%} \le 100\deg^2$, we find 559 events/yr for 2L-15km-45°, 184 for the 10 km triangle and 51 for LVKI-O5, see table 43 (however, for a handful of very close events, localizable to better than 10 $\deg^2$, the five-detector network LVKI-O5 does better, and we get 25, 8 and 31 events, respectively). The left panel of figure 12 shows that, also for BNS, the 2L-15km-45° is clearly superior to the 10 km triangle for the joint accuracy on $d_L$ and angular resolution, that determine the localization volume.

The 2L configurations with parallel arms has again an angular localization capability worse than that of the 2L configuration at 45°, but the effect is less strong compared to BBHs (see the panel for $\Delta\Omega_{90\%}$ in figure 11).<fnref n="48" /> On the other hand, the accuracy on the inclination angle $i$ for the parallel configurations is, comparatively, quite poor. Since $i$ is degenerate with the luminosity distance $d_L$, also for $d_L$ the parallel 2L configuration has significantly lower performance compared to the configuration at 45°. For instance, for the 2L-15km-0° we get 48 BNS/yr with error on $d_L$ better than 10%, a factor of 10 lower than the 479 BNS/yr for 2L-15km-45°. As for angular localization, for 2L-15km-0° we get 293 BNS/yr localized better than $100\deg^2$, so slightly better than the 10 km triangle, but about a factor of 2 worse than the 559 BNS/yr for 2L-15km-45°; see again table 43.

For the parameter $\tilde{\Lambda}$ that encodes the tidal deformability of the neutron stars, we see from figure 11 and table 44 that, again, the 2L with 15 km arms improves by a factor $\sim 2$ the already remarkable performances of the 10 km triangle, and both outperform LVKI-O5 by orders of magnitudes. For instance, considering the events with an error

<footnote n="47">⁴⁷Here the difference with LVKI-O5 becomes smaller: the events with very accurate angular localization are in general the closest, so they are accessible also to LVKI-O5, which can have good localization thanks to its network of five widely separated detectors.</footnote>

<footnote n="48">⁴⁸This is due to the long duration of BNS signals, which allow using the rotation of the Earth to localize the signal, also for the parallel setting.</footnote>
on $\tilde{\Lambda}$ smaller than 10%, we get 2463 BNS/yr for 2L-15km-45°, 1040 for the 10-km triangle, and 2 for LVKI-O5. For events with an error on $\tilde{\Lambda}$ smaller than 5% these numbers become 200, 96 and 0, respectively.

* **ET in a network with CE.** When ET is inserted in a network with Cosmic Explorer (either a single CE 40 km detector or two 2 CE detectors, one with 40 km and one with 20 km arms) the performance of the whole ET+CE network is, naturally, somewhat less sensitive to the choice of geometry for ET. However, the differences are still significant, particularly for the ET+1CE network. For instance, for ET+1CE, the number of BBHs with SNR > 200 is $1.8 \times 10^3$ when ET is in the 10 km triangle configuration, and raises to $2.4 \times 10^3$ for 2L-15km-45°, while the number of BBHs localized better than $10 \deg^2$ raises from $3.0 \times 10^4$ to $3.6 \times 10^4$, and the number of BBHs with luminosity distance measured better than 1% raises from $2.9 \times 10^3$ to $4.3 \times 10^3$ (or from $5.3 \times 10^3$ to $7.0 \times 10^3$, for ET+2CE); see figure 18 and tables 39, 40.

For BNSs, at ET+1CE, the number of detections/yr with SNR > 100 is 312 when ET is in the 10 km triangle configuration, and raises to 418 for 2L-15km-45°; the number of BNS/yr localized to better than $10 \deg^2$ raises from $2.4 \times 10^3$ to $3.8 \times 10^3$; the number of BNS/yr with luminosity distance measured better than 1% raises from $4.1 \times 10^3$ to $7.9 \times 10^3$ (or from $1.4 \times 10^4$ to $1.9 \times 10^4$, for ET+2CE); and the number of BNS/yr with tidal deformability measured better than 5% raises from 243 to 400 (or from 337 to 535 for ET+2CE); see figure 19 and tables 42–44.

In general, we see that the choice of geometry still has significant consequences also when ET is in a 3G network with one or two CE.

* **Multi-messenger astrophysics.** For multi-messenger astrophysics, the results of section 4 show that, again, the 2L-15km-45° allows us to obtain a further improvement on the already remarkable performances of the 10 km triangle (and is comparable to the 15 km triangle), enabling the observation of a larger number of well-localized events, up to a larger redshift. The number of short GRBs with an associated GW signal increases by about 30% making possible a joint detection at larger redshifts, and the number of expected kilonovae counterparts increases by a factor of 2. The better volumetric localization of 2L-15km-45° with respect to 10 km triangle, as shown in figure 6, facilitates the search of the EM counterpart giving the possibility to use galaxy-targeting strategy and enables a more efficient removal of contaminants. For pre-merger alerts with sky-localization smaller than $100 \deg^2$, the 15 km triangle is performing better than the 10 km triangle and the 15 km 2L configuration, reaching almost the capability of the 2L configuration with 20 km arms, see table 3.

* **Stochastic backgrounds.** For stochastic backgrounds, the best configurations are either 2L with parallel arms or the triangles, with the 2L parallel being the best below about 100 Hz and the triangle above 100 Hz. The 2L with arms at 45° is worse than the 2L with parallel arms$^{49}$ below about 200–300 Hz, while the two become comparable above 200–300 Hz (where, however, both are less good than the triangle); see figure 27. However, to exploit the better sensitivity (below 200-300 Hz) of 2L-parallel with respect to 2L-45° for studies of cosmology, the subtraction of

<footnote>$^{49}$See section 2 for the definition that we use for the relative angle between the L-shaped detectors, which differs by about 2.51° from the more standard one using the great circle connecting the two detectors.</footnote>
the astrophysical background is necessary, see section 6.5.3. An imperfect subtraction of CBC sources would pose a serious threat to reaching the nominal sensitivity to a cosmological background (see [427, 428, 523] for recent discussions), possibly even wiping out any advantage of the 2L-parallel with respect to the 2L-45° configuration.

The ability to resolve the angular distribution of stochastic backgrounds, through their lowest multipoles, is significantly better for the 2L configurations (both aligned and misaligned) compared to the triangle, see figure 30.

The analysis in section 6 shows that (as could be expected) the performance of the various geometries, with respect to the specific science cases studied, broadly follows the indications obtained from the more general metrics studied in sections 3–5. The 10 km triangle is a game-changer with respect to LVKI-O5, allowing a jump of orders of magnitudes, accessing physics that it is well beyond reach for 2G detectors. The 15 km 2L with arms at 45° provides a further gain with respect to the 10 km triangle, typically by factors of order 2-3, depending on the specific scientific question. Some examples, from the various studies presented in this paper, are as follows.

* Concerning tests of GR based on BH quasinormal modes, we see from table 13 that the ringdown SNR of GW150914 would have had a remarkable value of 141 in the 10 km triangle, which would further raise to 192 in 2L-15km-45°, inducing a corresponding difference in the reconstruction of ringdown frequencies and damping times, according to eq. (6.1). Furthermore, we see from table 14 that the 10 km triangle would detect 41 events/yr with a ringdown SNR higher than 50, that raise to 110 events/yr for the 2L-15km-45° (for a ringdown SNR higher than 100, these numbers become 4 and 10, respectively), resulting in a further difference in performance when stacking different events.

* For nuclear physics studies, we see from table 17 that the 10 km triangle can reach a remarkable accuracy of 10.0 m on the radius of neutron stars (taken the same for all NS, just as a proxy for the actual hyper-parameters determining the NS equation of state); the 2L-15km-45° further brings it down to 6.4 m, thanks to a more accurate reconstruction of tidal deformability for individual events, and the stacking of a larger number of events. For the postmerger SNR, instead, we see from figure 45 that the differences between different geometries do not exceed the (10 – 20)% level, with the hierarchy fixed by the arm length.

* For the reconstruction of the merger rate, figure 46 and table 24 show that both the 2L-15km-45° configurations and the 10-km triangle lead to a correct reconstruction of the merger rate, although the 2L-15km-45° configuration leads to a more accurate reconstruction, by an average factor ~3 in the central values.

* For primordial BHs, assuming the PBH population model mentioned in the text, the 10 km triangle would detect 77 events/yr at $z > 30$ (a smoking-gun signature for a primordial origin), which is already quite remarkable. The 2L-15km-45° configuration would further rise this by a factor ~3, with 238 events/yr, resulting in a limit on the PBH abundance stronger by a factor about 1.8; see table 25.

* In cosmology, we see from tables 28–34 that, for the tests that we have performed, the results for the accuracy on $H_0$, on the dark energy equation of state and on modified GW propagation from the various geometries analyzed are quite similar.<fnref n="50" /> In general,

<footnote n="50">⁵⁰Here we restricted to the 10 and 15 km triangles and the 15 and 20 km 2L at 45°.</footnote>
the 2L-15km-45° can improve the results of the 10 km triangle on the accuracy of $H_0$, $w_0$ or $\Xi_0$ by factors ranging between ~ (1.3 - 2.5), depending on the test.<fnref n="51" />

To sum up, the first overall conclusion that emerges is that:

1. All the triangular and 2L geometries that we have investigated can be the baseline for a superb 3G detector, that will allow us to improve by orders of magnitude compared to 2G detectors, and allow us to penetrate deeply into unknown territories.

How far we will penetrate into such territories, however, depends to some extent on the configuration chosen. From the above discussion, it follows that

2a. The 2L-15km-45° configuration in general offers better scientific return with respect to the 10 km triangle, improving on most figures of merits and scientific cases, by factors typically of order 2-3 on the errors of the relevant parameters.

### 8.1.2 Comparison between 15 km 2L and 15 km triangle

We next summarize the main results of the comparison between the 15 km triangle and the 2L-15km-45° configuration, so in this case we change the geometry while keeping the arm-length fixed. The main results of this comparison are as follows:

* For parameter estimation of BBHs, the 2L-15km-45° is clearly superior to the 15 km triangle for the estimate of luminosity distance (with 202 events/yr with $d_L$ measured better than 1%, against 77 for the 15 km triangle), and quite similar for all other parameters, see figure 5 and the first row of figure 10, and tables 40 and 41 in appendix C. Correspondingly, also the number of sources with good overall volume localization (say, error less than 1% on $d_L$ and angular resolution better than 50 deg$^2$), is clearly better for the 2L-15km-45°, see the right panel in figure 6.

* The same holds for parameter estimation of BNSs, see figure 11 and the first row of figure 16, and tables 43 and 44 in appendix C. In particular, the 2L-15km-45° has 479 events with $d_L$ measured better than 10%, to be compared with 153 for the 15 km triangle (which become 4328 to be compared with 1756, requiring an error smaller than 30%); see table 43. Instead, for angular resolution, the 15 km 2L with arms at 45° and the 15 km triangle are quite comparable (e.g., for the BNS localized better than 100 deg$^2$, with a threshold SNR > 12 we find 559 events/yr for 2L-15km-45° and 479 for the 15km triangle, see table 43, which become 644 and 764, respectively, with SNR > 8, see table 1. At this level, the comparison is of course affected also by sample variance). The respective joint distribution of events with respect to distance and angular localization is shown in the right panel of figure 12, which again shows an overall preference for 2L-15km-45°. For all other parameters the performances are quite similar.

* This reflects also on several aspects of the science case, where we see that the 2L-15km-45° and the 15 km triangle have similar performances; this can be seen, e.g.,

<footnote n="51">⁵¹Larger differences, in favor of the 15km-2L-45° with respect to the 10km triangle (and, to some extent, also with respect to the 15km triangle), are expected when extracting cosmological information by correlating dark sirens with galaxy catalogs, because of the much better volume localization of the 2L-15km-45° configuration, see figure 6 for BBHs, and figure 12 for BNSs. A quantitative assessment, however, also depends on the completeness of the galaxy catalogs that will be available when ET will be in operation. Work on this is in progress.</footnote>
in tables 1, 6 and 10 for multi-messenger astrophysics (except for the pre-merger alerts for which the 15 km triangle results are better, as shown in table 3); tables 13 and 14 for physics near the BH horizon; figure 40 for tidal deformability of exotic compact objects; table 17 for the measurement of the NS radii; figure 45 for the post-merger signal of BNS coalescences; table 24 for the merger rate reconstruction (where the 2L-15km-45° is a factor of 2-3 better than the 15 km triangle); table 25 and figure 48 for primordial BHs; or tables 28–36 for cosmology.

The overall conclusion, in this comparison, is that:

2b. The 2L-15km-45° configuration and the 15 km triangle have very similar performances on all parameters both for BBHs and BNSs, except for luminosity distance, where the 2L-15km-45° configuration is better by a factor ~3 in the number events with accurately measured distance.

### 8.1.3 A single L-shaped detector

Finally, we have considered the performance of a single L-shaped detector with 20 km arms, still with the ET characteristics in terms of ASD. The conclusion is that such a detector, taken as a single detector not inserted into a network with other 3G detectors, is not capable of delivering the science expected from the next generation of detectors. Its capability of angular localization of the source, and of reconstruction of the luminosity distance, would even be very much inferior to what the LVK network is expected to reach by the end of the O5 run, see the corresponding panels in figures 7 and 13 (actually, the number of well-localized sources detected per year would not even really improve on the results already obtained with the current GWTC-3 catalog of detections). This would result in a complete loss of all the science case aspects of 3G detectors related to multi-messenger astrophysics and to cosmology. Stochastic backgrounds of GWs, that in the frequency band of ground-based detectors can only be detected by correlating the output of two or more detectors, would also not be accessible to a single 3G detector, giving up again the corresponding aspects of the science case related to early Universe cosmology and to astrophysics. Furthermore, with a single detector, unmodeled burst signals and short signals due to massive BBH coalescences would be hard to distinguish from instrumental glitches, rendering problematic any confident detection of transient sources.

Therefore, a single L-shaped detector does not have a valid science case, even with 20 km arms (or longer). If inserted in an international network with 2CE, such as detector could be a useful addition to the network. However, given the scale of the investment, ET must have a solid science case, independently of the decisions that will be taken by other funding agencies, unrelated to a European project. Therefore:

3. A single L-shaped detector is not a viable alternative, independently of arm length. If a single-site solution should be preferred for ET, the detector must necessarily have the triangular geometry.

### 8.1.4 The null stream

The triangle has the advantage of having a null stream, where the GW signal cancels. This will certainly be beneficial. However, while the mathematics of the null stream is elegant, when one is confronted to its use in the concrete setting of the experiment several issues arise, that make it difficult to quantify reliably its impact.
First of all, the triangle null stream is only operative when the interferometers in all three arms are on, so it is quite sensitive to the duty cycle. In this work we have assumed an uncorrelated 85% duty cycle in each L-shaped detector, and in each of the three detectors composing the triangle, see section 3; in this case the null stream will be available for $(0.85)^3 \approx 61\%$ of the run time. Assuming instead an independent duty cycle of 80% for each arm (the best duty cycle for the O3b LVK run is 79% [524]), implies that the null stream will only be available for 51% of the run time. However, the triangle design is considerably more complex than that of 2G detectors, with two interferometers in each detector (an HF interferometer, and a LF interferometer working at cryogenic temperatures), and it is difficult to reliably estimate what the duty cycle will be. In a (hopefully very pessimistic) scenario where each of the six interferometers has an independent duty cycle of 80%, the null stream would only be available for 26% of the run time.

An advantage of the null stream is that it provides a direct estimate of the instrument-noise PSD, see eq. (7.3). Here, however, enters the assumption that noise in different arms are uncorrelated. Noise produced identically in two interferometers (as could be the case for the natural magnetic background or seismic gravity fluctuations) would not appear in the null stream, and would therefore bias the PSD estimates that use the null stream (unless these noise can be identified and characterized, e.g. with witness sensors).

Assuming to have obtained an unbiased estimate of the PSD, the most direct impact would be on the measurement of the stochastic gravitational-wave background. An unbiased estimate of the PSD would also be beneficial for compact binary coalescences, to avoid a reduction of the detection horizon from the confusion noise of the unresolved BNS (that, being long-lasting, can produce a continuous background, contrary to BBHs that rather produce a 'popcorn noise, see section 5). However, here the advantage of the null stream should be weighted against the possibility of making longer arms. Figure 69 shows that, assuming a value of the local rate $R_0$ of BNS mergers equal to the median of the currently allowed range, the reduction in the horizon range for the 2L detectors, due to this confusion noise, would be between 2% and 7.5% for CBCs with masses below $20M_{\odot}$, and negligible above. With extreme assumptions on the rate $R_0$, taken at the upper limit of the currently allowed range, the reduction would be between 5% and 20% for CBCs with masses below $20M_{\odot}$, and again negligible above. The triangle null stream would be able to mitigate this reduction, possibly eliminating it completely, which is certainly very beneficial. On the other hand, we see from figure 4, left panel, that, in the same range of masses, the 2L 15km at $45^\circ$ configuration has an horizon distance larger than the 10 km triangle by a factor $\sim (1.4-1.8)$, i.e. $(140-180)\%$.

The confusion noise also impacts the calculation of the false alarm rate. The null stream would allow estimating the noise background distribution directly (e.g. time shifting the null stream with respect to itself). In this way one could obtain more accurate estimates of the false alarm rate.

The null stream can be very effective at subtracting glitches, that would appear as non-Gaussian outliers in the null stream. The most prominent benefit in this case is expected for high-mass distant BBHs, that mimic glitches, as well as for signals that are only described by phenomenological models. In a 2L configuration, one would have to rely only on coincidences between detectors, and environmental sensing.

Another potential virtue of the null stream is that it reduces the uncertainty in the calibration (although it only provides relative calibration errors). However, the level of systematic errors in detector calibration in ET is not known. As discussed in section 7, further studies are needed to demonstrate the performance of the ET null stream calibration, com-
pared to other calibration techniques. 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.

In summary, while in the triangle configuration the null stream would certainly be beneficial, its concrete application is subject to several uncertainties that are difficult/impossible to model before building and commissioning the detector, such as the duty cycle, the level of correlated or identical noise in different arms, the level of non-gaussianity in the noise or, as for instance in the case of detector calibration, the state of advancement of other alternative techniques.

### 8.1.5 Further aspects of the triangle-2L comparison

In this section we summarize our discussions of further aspects that are relevant to the comparison between the triangle and the 2L configurations.

* **Correlated noise.** For the triangle vs. 2L comparison an important aspect, that needs further studies, is that the triangle configurations suffer from a potential threat from correlated Newtonian, seismic and magnetic noise, or lightning strikes; while significant uncertainties exist in their estimate, these correlated noise might spoil the sensitivity to stochastic backgrounds below some frequency and, possibly, even to unmodeled bursts. A network of two widely spaced L-shaped detectors is significantly less sensitive to these problems.

More in detail, an important potential danger, for all triangle configurations, is due to the fact that some mirrors of different nested interferometers are apart by a few hundred meters and, on this scale, correlated seismic and Newtonian noise can be important. This has been studied in detail only recently [141]. According to that analysis, such correlated noise would swamp the sensitivity to stochastic backgrounds of ET, in its 10 km triangle configuration, up to frequencies of order (50 – 100) Hz (blue curve in figure 34), to the extent that the actual sensitivity of ET to stochastic backgrounds, up to these frequencies, would not improve on LIGO’s A+ and Virgo’s AdV+ design. Different assumptions could result in a lower level of noise, also shown in figure 34. With the most optimistic assumptions, the search for a stochastic background would still be impacted up to ~ 10 Hz (pink curve in figure 34). Such problems arise from correlations on the scale of hundreds of meters, and therefore do not concern a 2L configuration with widely spaced detectors.

Magnetic noise will also pose difficulties to the triangle configuration: in the case of a very pessimistic assumption, where the magnetic noise in ET is the same as in the Virgo central building and it is fully correlated between two ET interferometers, it will affect the ASD sensitivity over the entire frequency band, by several orders of magnitudes at low frequencies, $f \lesssim O(30)$ Hz, and by a factor 1-10 at higher frequencies, see the lower panel in figure 35. Again, this would only apply for the co-located interferometers in the triangular configuration. However, further research is needed to understand to which level these infrastructural noise sources would be correlated.

Compared to the triangular design, the 2L configuration is also less prone to effects coming from coherent lightning strikes, which could affect not only stochastic background searches, but, potentially, also the search for unmodeled bursts.
* Site optimization for 2L. Finally, it should be observed that, in this work, no effort was made to optimize the distance between the two L-shaped detectors, even just within Europe, and we simply used the locations of the two current candidate sites. A longer baseline would improve in particular the localization accuracy, as well as the estimation of the parameters more strongly correlated with angular localization.<fnref n="52" />

## 8.2 The role of the low-frequency sensitivity

We now draw our conclusions from the comparison between the two different choices of the amplitude spectral density (ASD) that we have studied, namely the full xylophone design, consisting of a high-frequency instrument together with a low-frequency instrument working at cryogenic temperature ('HFLF-cryo'), and the situation in which there is only the high-frequency instrument ('HF-only'). The main messages that emerge from our analysis are the following:

4. The low-frequency sensitivity is crucial for exploiting the full scientific potential of ET. In the HF-only configuration, independently of the geometry chosen, several crucial scientific targets of the science case would be lost or significantly diminished.

For instance:

* For all geometries, in the HF-only configuration the number of BNS sources localized to better than $10^2 \text{ deg}^2$ (which are the sources relevant for multi-messenger studies) degrades catastrophically; with the threshold SNR $\ge$ 8 that we use for multi-messenger studies, we see from tables 1 and 2 that, for the 10 km triangle, the number of detections with $\Delta\Omega_{90\%} < 10^2 \text{ deg}^2$ (and all orbit inclinations) decreases from 280 to 14 BNS/yr, i.e. by a factor of 20 (and from 26 to 7 for events with viewing angle less than 15°), while for the 2L 15km-45° it decreases from 644 to 76 (for all orbit inclinations), or from 68 to 35 (for viewing angle less than 15°). Note that the reduction is more severe for the 10 km triangle than for the 2L 15km-45° configuration. Increasing the SNR threshold the effect is even stronger: for SNR $\ge$ 12, the number of events localized to better than $10^2 \text{ deg}^2$ decreases from 184 to 4 BNS/yr for the 10 km triangle while, for the 2L 15km-45°, it decreases from 559 to 11; see table 43.

Indeed, the decrease is so severe that the number of BNS/yr localized to better than $10^2 \text{ deg}^2$ would even become much lower than what will be obtained already by LVKI during the O5 run, see the panel on $\Delta\Omega_{90\%}$ in figures 14 and 15, and table 43: as we mentioned, the number of BNS/yr with SNR $\ge$ 12 and $\Delta\Omega_{90\%} < 10^2 \text{ deg}^2$ would be 11 for the 2L 15km-45° HF-only, and 4 for the 10 km triangle HF-only; by comparison, for LVKI-O5 we find 51 BNS/yr with SNR $\ge$ 12 and $\Delta\Omega_{90\%} < 10^2 \text{ deg}^2$.

The accuracy on the measurement of the luminosity distance also degrades dramatically. For instance, for the 2L with 15 km arms at 45°, the number of BNS/yr with $d_L$ measured to better than 10% degrades from 479 in the HFLF-cryo configuration, to 12 in the HF configuration, while for the 10 km triangle it degrades from 52 to 1; for

<footnote n="52">⁵²A third possible candidate site, depending on the outcome of the geological studies, could emerge near Kamenz, in the Lusatia region, Saxony (Germany). The great circle chord distance between the Sardinia site and the site in the Meuse-Rhine is 1165.0 km, while that between the Sardinia site and Kamenz is 1247.3 km, so the two distances are quite comparable, with the Sardinia-Kamenz distance larger by about 7%. In contrast, the great circle chord distance between Kamenz and the site in Meuse-Rhine is 575.4 km, significantly smaller.</footnote>
2L-15km-45°, the number of events/yr with $d_L$ measured to better than 5% degrades from $\mathcal{O}(100)$ in the HFLF-cryo configuration, to about 1. The same trend is observed for BNS 'golden events', see figure 16. The accuracy on many other parameters would also degrade significantly, see again figures 14 and 15. These results are due to the fact that, with the low-frequency sensitivity ensured by the LF instrument, BNSs stay in the bandwidth for a long time, with a corresponding crucial benefit for parameter estimation and, particularly, localization and the partial disentangling of the polarizations, that allows us to alleviate the degeneracy between $d_L$ and $\cos i$.

* For the same reason, for BNS, pre-merger alerts for localized events would become impossible without the low-frequency instrument, compare table 3 with table 4. For instance, for the 10 km triangle the total number of BNS/yr detected 30 min before merger would go from 905 (of which 10 localized, 30 min before merger, to better than $10^2 \text{ deg}^2$ and 85 better than $10^3 \text{ deg}^2$) to zero; for the 2L 15km-45° it would go from 2172 (of which 20 localized, 30 min before merger, to better than $10^2 \text{ deg}^2$ and 194 better than $10^3 \text{ deg}^2$) again to zero. As an example, an event such as GW170817 would enter the ET bandwidth about 1 day before the merger with the full HFLF-cryo sensitivity, but only $\mathcal{O}(10)$ minutes before the merger with an ASD such that it entered the detector bandwidth only at 10 Hz.

This would have a dramatic impact on the possibility to detect precursors, and to probe the prompt/early counterparts, which provide rich information on the physics acting in the GRB engine and jet launch and kilonova ejecta. Of particular interest are prompt/early detection in the very high-energy gamma rays and ultraviolet.

* For the multi-messenger studies, losing the low frequencies reduces the detection of short GRBs with associated gravitational-wave signals; the number of joint detection decreases by about 40% for the 10 km triangle, by about 30% for the 15 km triangle and 2L with 15 km arms at 45°, and 20% for the 2L with 20 km arms at 45°, see tables 6 and 7. The reduction is more severe for kilonova counterparts: the triangle HF configurations detect less than 6% of the kilonovae detected by triangle HFLF-cryo, while the 2L HF-only configurations detect less than 15% of the KNe detected by the 2L HFLF-cryo configuration, see tables 10 and 11. Note that, once again, the triangle geometry is more heavily affected than the 2L by the loss of the low frequency instrument. The reduction of short GRBs with an associated GW signal might impact our understanding of the physics of the GRB engine and relativistic jets, by requiring more years of joint observations to achieve the same science goals. However for cosmology, nuclear physics, and studies of the heavy-elements nucleosynthesis which rely on kilonova detections, losing the LF has a major impact on achieving the goals themselves.

It is worth noting that, for short GRBs, the HF-only 2L-15km-45° and the HF-only 15km triangle enable the detection of the GW signal for a number of GRBs comparable to the full HFLF-cryo triangle with 10 km arms. On the other hand, for the kilonovae counterparts, no HF-only configuration (even HF-only 2L with 20 km arms at 45°) is comparable to 10 km triangle HFLF-cryo. This is due to a different type of search: while to observe short GRBs, wide FoV satellites in survey mode are used and the temporal coincidence of the GW/GRB signals are used to identify a joint detection, to search for kilonovae the telescopes FoV is typically much smaller than the GW localization uncertainty and thus requires selecting better-localized events to be
followed up. The much-improved localization obtained by accessing low-frequencies
then becomes a key parameter for the search.

* The HF-only configuration has a significantly smaller reach in distance, compared to the HFLF-cryo configuration, in the whole mass range relevant to 3G detectors, as can be seen from the right panels in figures 3 and 4.

This has several significant consequences. In the range of masses relevant to BNS, for the 10 km triangle, the redshift to which a BNS with optimal orientation and sky location can be detected would decrease from $z \simeq 4$ down to $z \simeq 2$ (or, for the 2L-15km-45°, from $z \simeq 6$ to $z \simeq 3$). This is particularly significant since the peak of the star formation rate is around $z \simeq 2-3$, so studies of demography and population of BNS would be strongly impacted.

The lowest reach in redshift would also make it impossible to identify primordial BHs on the basis of the criterion that BBHs at $z \gtrsim 30$ cannot be of astrophysical origin. For instance, for the 10 km triangle, the maximum redshift at which a compact binary coalescence could be seen, for optimal values of the total mass and optimal orientation and sky location, reduces from $z \simeq 100$ in the HFLF-cryo configuration to $z \simeq 20$ in the HF configuration, see figure 3.

Intermediate mass BHs, with masses in the range $(10^3 - 10^4)M_{\odot}$, would also be visible to significantly smaller distances, see again figure 3, reducing significantly the chances of discovery. As an example, a BBH with total mass $5 \times 10^3 M_{\odot}$ would be visible up to $z \simeq 1.1$ in the HFLF-cryo configurations (almost independently of geometry), but only up to $z \simeq 0.6$ in the HF only configurations, corresponding to a reduction by a factor $\sim 5$ in the comoving volume explored and in the corresponding chances of detection.

* The measurement of the eccentricity of subsolar mass compact objects, an important criterion for assessing or excluding the primordial origin of a BBH, would degrade by more than one order of magnitude, see figure 51.

* The degradation of the accuracy in luminosity distance and angular localization would basically render impossible all studies of cosmology, such as accurate measurements of $H_0$ and especially studies of the dark energy equation of state and of modified GW propagation, which crucially rely on accuracy in sky localization and luminosity distance for events at large $z$. Similarly, the exquisite localization accuracy on some high-mass ratio events, needed to extract the Hubble parameter from them, see section 6.4.3, would be impossible without the LF instrument.

* For a number of other aspects of the science case the loss of the LF instrument would not be as disruptive, but would still results in a loss of accuracy on the relevant parameters, by factors of order 2-3. This is the case, for instance, for the possibility of distinguishing BHs from Exotic Compact Objects on the basis of the spin-induced quadrupole moment or of a non-vanishing tidal deformability (figures 39 and 40), or for the accuracy on the reconstruction of NS radii, see table 17.

On the other hand, not all aspects of the Science Case depend on the LF instrument, and another important conclusion that follows from our study is:

5. There are some important targets of the Science Case that depend only on the HF sensitivity, and that could be fully reached with an HF-only instrument.
In particular:

* For multi-messenger astronomy, the number of joint detections of GWs from a BNS merger and the associated X-ray afterglow, as could be detected by an instrument such as THESEUS in survey mode, is basically independent of the LF instrument; see table 8.

* Tests of physics near the BH horizon based on the ringdown signal of the final BH are completely independent of the LF instrument, see table 13 and the discussion in section 6.1.1. The same holds for the search of echoes and near-horizon structures, see section 6.1.2.

* The post-merger signal of BNS coalescences, that contains the information of the NS equation of state, is concentrated in the high-frequency region, and is completely insensitive to the presence of the LF instrument, see the left panel in figure 45.

* The detectability of sub-solar mass primordial BHs is another example of a metric that is not affected by the loss of the LF instrument; see figure 50.

* Stochastic backgrounds of cosmological origin are in general expected to have a smooth power-like behavior, $\Omega_{gw}(f) \propto f^{\alpha}$, over the bandwidth of ground based or space-borne detectors (with the exception of backgrounds from cosmological phase transitions, that will be broadly peaked around a characteristic frequency). At the very low frequencies, say $f \sim 10^{-18}$ Hz, tested by CMB, there are extremely stringent limits $\Omega_{gw} \lesssim 10^{-15}$ (see, e.g. chapter 22 of [85]). Therefore, at the frequencies of ground based detectors, a stochastic background, relic of the early Universe, could only be detectable if it grows with frequency, $\Omega_{gw}(f) \propto f^{\alpha}$ with $\alpha > 0$, at least until a frequency where it saturates, followed by a cutoff. For this reason, the stochastic background predicted by standard single-field slow-roll inflation is not detectable, since it is basically flat (in fact, slightly red, i.e. $\alpha < 0$). However, there are alternative cosmological models that predict spectra with $\alpha > 0$, such as the pre-big-bang model [525, 526] or axion inflation (see [527], or figure 12 of [15]). In these cases, a significant part of the SNR could accumulate in the high-frequency part of the detector bandwidth, where the LF instrument does not contribute to the sensitivity to stochastic backgrounds, see figure 29. A similar behavior takes place, in the ET bandwidth, in the model for cosmic strings whose predictions are shown in figure 64.

A final important conclusion that emerges from our study is that:

6. For some important aspects of the Science Case, the 2L with 15 km arms at 45°, already in the HF-only configuration, is comparable to the 10 km triangle in a full HFLF-cryo configuration.

In particular:

* For parameter estimation of BBHs, the 2L with 15 km arms at 45° in the HF-only configuration is comparable to the 10 km triangle at full HFLF-cryo sensitivity, with better performance of luminosity distance, less good performance on mass reconstruction, and equivalent performances on all other parameters and in SNR distribution, see figure 9. The performance of the 2L-15km-45° configuration is also equivalent to that of the 10 km triangle for what concerns ‘golden BBH events’, see the lower row of figure 10.
* From the right panel of figure 41 we see that, for the measurement of the neutron star radii and the consequences for nuclear physics that we can derive from it, the 2L with 15 km arms at 45° already in the HF-only configuration has performances very similar to that of the full 10 km triangle HFLF-cryo.

* For all the items discussed above, where the LF instrument does not contribute (joint GW+X-ray afterglow and, to some extent, GRB detections, tests of physics near the BH horizon, post-merger signal of BNS coalescences, sub-solar mass BHs, stochastic backgrounds growing as $f^\alpha$ with $\alpha > 0$), the 2L-15km-45° in the HF only configuration will be superior to the 10 km triangle with the full HFLF-cryo sensitivity.

## 8.3 Conclusions

The decisions on the ideal design of a detector such as the Einstein Telescope, which is meant to have a leading role in the field of gravitational waves for decades, are part of a complex process. In this study we have considered alternatives to the geometry (a single-site triangle against two widely separated L-shaped interferometers, with different options for the arm-lengths) and to other aspects of the design (considering the baseline design given by a full xylophone configuration with a high frequency instrument together with a cryogenic low-frequency instrument, and comparing it to a high-frequency only instrument).

The first conclusion of this work, which extends the study presented in [15], is that, whatever geometry is decided between the triangle and 2L configurations considered, ET will clearly be a game-changer. It will over-perform the network of 2G observatories at their best expected O5 sensitivities by several orders of magnitudes in all relevant metrics, accessing physics that it is well beyond reach for 2G detectors. Note, however, that this is only true for the triangle and for the 2L configurations. A single L-shaped detector, not inserted in a global 3G network, even with a very long arm length, does not have a viable Science Case, at the level expected from a 3G detector.

The comparison between the 10 km triangle and a 15 km 2L shows that, from the scientific point of view, the 15 km 2L with arms at 45° is superior on basically all the metrics that we have considered, with the exception of the nominal sensitivity to stochastic backgrounds. However, in this case we have seen that the triangle suffers from a potentially very significant treat from correlated noise, particularly at low frequencies.53 For compact binary coalescences, the better performances of the 15 km 2L with arms at 45° allow us to gain typically further factors of order 2-3 on the number of events which pass given cuts on SNR, or on the accuracy of reconstruction of various parameters and physical quantities. In fact, we have found that the 15 km 2L with arms at 45° is quite comparable to the 15 km triangle: the two have very similar performances for the reconstruction of all parameters of compact binary coalescences, except for luminosity distance, for which the 15 km 2L with arms at 45° is significantly better.

Another interesting element that emerged from our study is that the configuration 2L 15km at 45° gives a better possibility of proceeding in steps, compared to the 10 km triangle. Both for the triangle and 2L configurations, the full scientific outcome expected from a 3G detector will only be reached in the full configuration with the HF and the cryogenic LF instrument. However, the 2L 15km at 45°, already in the HF-only configuration,

<footnote>53Furthermore, more work is needed to assess the effect of an imperfect subtraction of the astrophysical backgrounds, which could limit the sensitivity to cosmological backgrounds to a level fixed in all cases by the accuracy of the subtraction.</footnote>
would be sufficient to obtain some important results (which is true only to a lesser extent for the 10 km triangle). It is therefore natural to investigate whether a staging procedure will be advantageous, where the HF interferometer, which is the simpler to commission, is put into operation first, until the full HFLF-cryo configuration is reached, possibly going through intermediate steps such as, for instance, an LF instrument working, at first, at room temperatures. These issues, however, are beyond the scope of this study, and should be examined within the context of the ET Instrument Science Board.

Finally, the triangle configuration is prone to the effect of correlated Newtonian, seismic and magnetic noise on the scale of hundreds of meters. While more work is needed to fully understand them, these effects might pose a risk/challenge to obtaining the target sensitivity of the triangle, particularly in the low-frequency region.

## Acknowledgments

The research leading to these results has been conceived and developed within the ET Observational Science Board (OSB). We are particularly grateful to Giancarlo Cella, Gianluca Guidi and Frank Ohme for their extremely careful and very useful internal refereeing of the manuscript. We thank Marie-Anne Bizouard, Nelson Christensen, Fernando Ferroni, Harald Lück, Mario Martinez, Ed Porter, Michele Punturo, Tania Regimbau, David Shoemaker, Achim Stahl and Patrice Verdier for very useful discussions and feedback. We express special thanks to Jameson G. Rollins for his consultations about the use of PyGWINC.

The work of Michele Maggiore, Stefano Foffa, Francesco Iacovelli, Michele Mancarella and Niccolò Muttoni is supported by the Swiss National Science Foundation, grant 200020_191957, and by the SwissMap National Center for Competence in Research, and made use of the Yggdrasil cluster at the University of Geneva. M. Branchesi acknowledges financial support from the Italian Ministry of University and Research (MUR) for the PRIN grant METE under contract no. 2020KB33TP. B. Banerjee acknowledges financial support from MUR (PRIN 2017 grant 20179ZF5KS). M. Branchesi and G. Oganesyan acknowledge financial support from the AHEAD2020 project (grant agreement n. 871158). The Gran Sasso Science Institute group acknowledges Stefano Bagnasco, Federica Legger, Sara Vallero, and the INFN Computing Center of Turin for providing support and computational resources. V. De Luca is supported by funds provided by the Center for Particle Cosmology at the University of Pennsylvania. Elisa Maggio acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) — project number: 386119226. G.Franciolini acknowledges financial support provided under the European Union's H2020 ERC, Starting Grant agreement no. DarkGRA-757480 and under the MIUR PRIN programme, and support from the Amaldi Research Center funded by the MIUR program "Dipartimento di Eccellenza" (CUP: B81I18001170001). This work was partly enabled by the UCL Cosmoparticle Initiative. Katarina Martinovic is supported by King's College London through a Postgraduate International Scholarship. M. Mapelli, C. Périgois and F. Santoliquido acknowledge financial support from the European Research Council for the ERC Consolidator grant DEMOBLACK, under contract no. 770017. M. Mancarella and C. Pacilio are supported by European Union's H2020 ERC Starting Grant No. 945155-GWmining and Cariplo Foundation Grant No. 2021-0555. Numerical calculations by C. Pacilio have been made possible through a CINECA-INFN agreement, providing access to resources on MARCONI at CINECA. Anuradha Samajdar thanks the Alexander von Humboldt foundation in Germany for a Humboldt fellowship for postdoctoral researchers. Paolo
Pani acknowledges financial support provided under the European Union's H2020 ERC, Starting Grant agreement no. DarkGRA-757480 and under the MIUR PRIN and FARE programmes (GW-NEXT, CUP: B84I20000100001). BG is supported by the Italian Ministry of Education, University and Research within the PRIN 2017 Research Program Framework, n. 2017SYRTCN. Mairi Sakellariadou is supported in part by the Science and Technology Facility Council (STFC), United Kingdom, under the research grant ST/P000258/1. ACJ was supported by the Science and Technology Facilities Council through the UKRI Quantum Technologies for Fundamental Physics Programme [grant number ST/T005904/1]. Swetha Bhagwat is supported by the UKRI Stephen Hawking Fellowship with grant ref. no EP/W005727. Angelo Ricciardone acknowledges financial support from the Supporting TAle nt in ReSearch@University of Padova (STARS@UNIPD) for the project “Constraining Cosmology and Astrophysics with Gravitational Waves, Cosmic Microwave Background and Large-Scale Structure cross-correlations”. Kamiel Janssens is supported by FWO-Vlaanderen via grant number 11C5720N. F.Gulminelli and C.Mondal acknowledge partial support from the In2p3 Master Project NewMAC. D. Alonso is supported by the Science and Technology Facilities Council through an Ernest Rutherford Fellowship, grant reference ST/P004474. Ssohrab Borhanian acknowledges support from the Deutsche Forschungsgemeinschaft, DFG, project MEMI number BE 6301/2-1. Arnab Dhani, Ish Gupta and B.S. Sathyaprakash were supported by NSF grant numbers PHY-2012083, AST-2006384 and PHY-2207638. G. Cusin is supported by CNRS and by Swiss National Science Fundation (Ambizione grant — *Gravitational wave propagation in the clustered universe*).

## A Basic formalism for stochastic backgrounds

In this appendix we review several basic concepts relevant for the characterization of stochastic GW backgrounds. At the frequencies of ground-based detectors, to obtain an interesting sensitivity one needs to cross-correlate the output of at least two detectors in a network, see e.g. [83, 528] for reviews: as a consequence the observable quantity is not a strain, but rather quantities quadratic in the strain. One usually introduces a polarisation tensor defined (in frequency space) as

$$ \tilde{P}_{ab} = \tilde{h}_a^* \tilde{h}_b, \qquad (A.1) $$

with $a, b = +, \times$ and we defined the superposition of signals in a given direction $\hat{n}$ and at a given frequency $f$ as

$$ \tilde{h}_{ij}(\hat{n}, f) = \tilde{h}_+(\hat{n}, f) e_{ij}^+(\hat{n}) + \tilde{h}_-(\hat{n}, f) e_{ij}^-(\hat{n}). \qquad (A.2) $$

In full generality the polarisation tensor, being a complex $2 \times 2$ tensor, can be decomposed in a basis of the identity and Pauli matrices as

$$ \tilde{P}_{ab}(\hat{n}, f) = \frac{1}{2} [I(\hat{n}, f)1_{ab} + U(\hat{n}, f)\sigma_{ab}^{(1)} + V(\hat{n}, f)\sigma_{ab}^{(2)} + Q(\hat{n}, f)\sigma_{ab}^{(3)}], \qquad (A.3) $$

where the coefficients $I, U, V, Q$ are Stokes parameters describing intensity and polarisation, respectively. Polarisation is expected to be very small if we consider a homogeneous and isotropic AGWB and a CGWB: the amount of polarisation generated via classical diffusion is very small for both background components, see [105]. However, there is a non-negligible amount of circular polarization in the AGWB, generated by Poisson fluctuations in the number of unresolved sources, which can be detected by third-generation interferometers
with SNR> 1 (see [529]). Measuring a highly polarised background component would therefore be interesting both for shedding light on some cosmological parity-violating generation mechanisms (see e.g. [530, 531]) or to estimate whether the SGWB comes from a handful of sources or a relatively large population of binaries. Here we focus on the study of intensity and of its spectrum. It is common in the literature to introduce a dimensionless quantity related to intensity via

$$
\begin{align}
\Omega_{\text{gw}}(\hat{\mathbf{n}}, f) &\equiv \frac{4\pi^2 f^3}{G\rho_c} I(\hat{\mathbf{n}}, f) \nonumber \\
&= \frac{1}{\rho_c} \frac{d\rho_{\text{gw}}(\hat{\mathbf{n}}, f)}{d\ln f \, d^2\hat{\mathbf{n}}}, \tag{A.4}
\end{align}
$$

where the second equality clarifies the physical meaning of this observable: it corresponds
to the background energy density, $\rho_{\text{gw}}(\hat{\mathbf{n}}, f)$, per unit of logarithmic frequency and unit
solid angle (normalised to the critical density of the Universe today, $\rho_c$), that reaches the
observed from a direction $\hat{\mathbf{n}}$.

To characterize the detector sensitivity to stochastic backgrounds, we use the Power-Law integrated Sensitivity curve (PLS), introduced for the first time in [145] for a network of L-shaped detectors, and we extend it to the case of triangular detectors, following [117]. When dealing with triangular detectors (three nested detectors), one has to keep in mind that the noise between two detectors with an arm in common is not uncorrelated, hence the correlation has to be accounted for when computing the PLS. Here we assume that the noise in ET is 20% correlated between detectors with an arm in common. As we discuss in section 5.4, see in particular figure 34, this is a value intermediate between the extremes of modeled noise correlations; we chose here a frequency-independent correlation for simplicity.

Let us assume to have a network of *N* detectors. The normalized overlap reduction function (see [21, 145, 403]) for a detector pair {*A*, *B*} in the network is defined as

$$
\gamma_{AB}(f) = \frac{5}{8\pi} \int d^2\hat{\mathbf{n}} \left[ F_A^+(\hat{\mathbf{n}}) F_B^+(\hat{\mathbf{n}}) + F_A^\times(\hat{\mathbf{n}}) F_B^\times(\hat{\mathbf{n}}) \right] e^{i2\pi \frac{f}{c} \hat{\mathbf{n}} \cdot \Delta\mathbf{x}_{AB}}, \quad (A.5)
$$

where $\mathbf{\Delta x}_{AB} = \mathbf{x}_B - \mathbf{x}_A$ is the separation between the two detectors A and B.

The normalization factor 5/(8π) in eq. (A.5) is conventionally introduced so that
γ<sub>AB</sub>(f) = 1 for two co-located and perfectly aligned detectors with orthogonal arms. The
quantities F<sub>A</sub><sup>+</sup>(n̂) and F<sub>A</sub><sup>×</sup>(n̂) are the detector A pattern functions (and similarly for B).
They are determined by the detector A response tensor D<sub>A</sub> and the polarisation tensors
e<sup>+,×</sup> as F<sub>A</sub><sup>+</sup>(n̂) = Tr[D<sub>A</sub> e<sup>+</sup>(n̂)] and F<sub>A</sub><sup>×</sup>(n̂) = Tr[D<sub>A</sub> e<sup>×</sup>(n̂)].

For a Michelson interferometer with arms pointing along the directions **u**_A and **v**_A, the
detector A response tensor has components D<sub>A</sub><sup>ij</sup> given by (see e.g. [21])

$$
D_A^{ij} = (u_A^i u_A^j - v_A^i v_A^j) / 2. \qquad (A.6)
$$

We start with the effective noise power spectral density. In full generality, this is given by

$$
S_{\text{eff}}(f) = \left[ \sum_{A,B,C>B,D>A} (N_f^{-1})^{AB} \gamma_{BC}(f) (N_f^{-1})^{CD} \gamma_{DA}(f) \right]^{-1/2}, \quad (A.7)
$$

where $N_f^{-1}$ is the inverse of the noise covariance matrix of the network at each frequency $f$. The restrictions on the sum in eq. (A.7) ensure that only a cross-correlation search at
unordered pairs of distinct detectors is considered, thus excluding auto-correlations (and without double counting).

Notice that in the case of uncorrelated detector noises, the noise matrix is diagonal

$$ (N_f^{-1})^{AB} = \frac{\delta^{AB}}{N_f^{AA}}, \qquad (A.8) $$

where $N_f^{AA}$ is the noise power spectral density (PSD) of the detector A. In this case eq. (A.7) can be simplified to:

$$ S_{\text{eff}}(f) = \left[ \sum_{A=1}^{N} \sum_{C>A}^{N} \frac{\gamma_{AC}^2(f)}{N_f^{AA} N_f^{CC}} \right]^{-1/2} . \qquad (A.9) $$

However, for triangular detectors, we need to know both the noise variance (diagonal terms) and the cross-detector noise covariance (off-diagonal).

We define an effective dimensionless noise energy spectrum for $S_{\text{eff}}(f)$ in eq. (A.7) as

$$ \Omega_{\text{eff}}(f) = \frac{10\pi^2}{3H_0^2} f^3 S_{\text{eff}}(f) \qquad (A.10) $$

and consider an isotropic and unpolarized stochastic GW background with fractional energy density contribution $\Omega_{\text{gw}}(f)$ with respect to the critical energy density $\rho_c$ of the Universe today. With the definition (A.10), the integrated signal-to-noise ratio $\rho$ for a cross-correlation search at the network in a frequency range from $f_{\text{min}}$ to $f_{\text{max}}$, for a coincident observation time $T$, is [403]

$$ \rho = \left[ 2T \int_{f_{\text{min}}}^{f_{\text{max}}} df \frac{\Omega_{\text{gw}}^2(f)}{\Omega_{\text{eff}}^2(f)} \right]^{1/2} . \qquad (A.11) $$

For a set of power-law indices $\beta$, we write

$$ \Omega_{\text{gw}}(f) = \Omega_{\beta}(f/f_{\text{ref}})^{\beta} \qquad (A.12) $$

where $f_{\text{ref}}$ is a reference frequency such that, at $f = f_{\text{ref}}$, $\Omega_{\text{gw}}(f)$ has the value $\Omega_{\beta}$, and we compute the value of the amplitude $\Omega_{\beta}$ such that the integrated signal-to-noise ratio $\rho$ has some fixed value. Using eq. (A.11), this is given by

$$ \Omega_{\beta} = \frac{\rho}{\sqrt{2T}} \left[ \int_{f_{\text{min}}}^{f_{\text{max}}} df \frac{(f/f_{\text{ref}})^{2\beta}}{\Omega_{\text{eff}}^2(f)} \right]^{-1/2} . \qquad (A.13) $$

In the following we will set $\rho = 1$. For each pair of values $(\beta, \Omega_{\beta})$, we compute $\Omega_{\text{GW}} = \Omega_{\beta}(f/f_{\text{ref}})^{\beta}$. The envelope is the power-law integrated sensitivity curve. Formally it is given by

$$ \Omega_{\text{PLS}}(f) = \max_{\beta} \left[ \Omega_{\beta} \left( \frac{f}{f_{\text{ref}}} \right)^{\beta} \right] . \qquad (A.14) $$

Any line (on a log-log plot) that is tangent to the power-law integrated sensitivity curve corresponds to a gravitational-wave background power-law spectrum with an integrated signal-to-noise ratio $\rho = 1$. This implies that, if the curve for a predicted background lies everywhere below the sensitivity curve, then $\rho < 1$ for such a background. Note that, from the definition (A.12) of $\Omega_{\beta}$ and $f_{\text{ref}}$, a change in the choice of $f_{\text{ref}}$ is compensated by a corresponding change of $\Omega_{\beta}$, such that $\Omega_{\beta}f_{\text{ref}}^{-\beta}$ remains constant. Therefore, the value of $\Omega_{\beta}$ obtained from eq. (A.13) depends on the arbitrary choice of $f_{\text{ref}}$, but $\Omega_{\text{PLS}}(f)$, given by eq. (A.14), does not.
<figure><img src="image_79.png" /><figcaption>**Figure 70.** Normalized overlap reduction function (ORF) $\gamma(f)$ for different values of the angle $\alpha$.</figcaption></figure>

## B Sensitivity to stochastic backgrounds of misaligned 2L configurations

Focusing on the stochastic gravitational-wave background, we highlight below the effect of the orientation of the two L-shaped interferometers on the peak sensitivity. As discussed in section 2, in this paper we have fixed the arms of the Sardinia interferometer so that they point locally East and North, and rotate anti-clockwise the arms of the Netherlands interferometer by an angle $\alpha$ with respect to the local East-North directions. The angle $\alpha = 0$ differs from the usual definition of the alignment of two interferometers with respect to the great circle [21, 22] by 2.51 degrees and is clearly not the maximum possible alignment, due to the curvature of the Earth between the two sites.

In figure 70 we show the normalized overlap reduction function (ORF) defined in eq. (A.5) as a function of frequency, for the $\alpha = 0$ deg and $\alpha = 45$ deg cases, which are the examples considered in the main text. However, the best and worst cases for a stochastic search are the values $\alpha = 2.51$ deg and $\alpha = 47.51$ deg, respectively. Clearly, while varying the angle $\alpha$ near the 0 deg case has very little impact, the consequences of varying $\alpha$ just a few degrees around the 45 deg are very important. These results reflect into a significant dependence on $\alpha$ of the peak sensitivity to the stochastic gravitational-wave background.

Figure 71 illustrates the peak sensitivity to the stochastic gravitational-wave background (minimum value of the power-law integrated curve) as a function of the angle $\alpha$. The choices $\alpha = 0$ deg and 45 deg, considered in the main text of this document, are not respectively the best and worst cases for stochastic searches. Varying the angle $\alpha$ near the 0 deg case has very little impact, but varying $\alpha$ just a few degrees around the 45 deg case varies the peak sensitivity to the stochastic gravitational-wave background by orders of magnitude. As one can clearly see, the best and worst cases for stochastic searches, are when $\alpha = 2.51$ deg and 47.51 deg, respectively.
<figure><img src="image_80.png" /><figcaption>**Figure 71.** Peak sensitivity to the stochastic gravitational-wave background as a function of the angle $\alpha$ in the 2L-20km-HFLF-Cryo configuration. The SNR threshold used is $\rho = 1$ and the observation time is $T = 1$ yr.</figcaption></figure>

To be strictly optimal the alignment of two interferometers would have the two detectors such that each had an arm along the great circle connecting them for 0 deg (which is how the two LIGO detectors are aligned), and rotating one by 45 deg for the minimal value [22]. For such a minimal alignment the overlap reduction function would be exactly zero in the limit $f\Delta x \to 0$, where $\Delta x$ is the separation between the two detectors, and very small basically for all frequencies, as we see from figure 70. This would also give the best polarization separation for the CBC searches.

### C Tables of figures of merit for BBHs and BNSs

To make the comparison among different geometries and ASDs easier, we provide here tables with the number of sources detected by each configuration with various cuts on the match-filtered SNR, or on measurement errors on some of the most relevant quantities, corresponding to the results shown in figures 5, 8, 9 and 10 for BBHs, and to figures 11, 14, 15 and 16 for BNS. We further show the same quantities for the networks ET+1CE and ET+2CE, corresponding to the results shown in figures 18 and 19.

In particular, in table 39 we show the number of detected BBH sources with different cuts in SNR, ranging from 8 to 200; in table 40 with cuts on the relative error on the luminosity distance and the 90% sky localisation [notice that the parameter estimation analysis has been performed only for the events detected with SNR $\ge 12$]; and in table 41 with cuts on the relative error on the detector-frame chirp mass and the error on the spin magnitude of the primary object.
<table><caption>**Table 39.** Summary table of the number of BBH detections with various cuts in SNR for the considered ET geometries and ASDs, also in combination with one 40 km CE detector or two CE detectors and for a LVKI network during O5.</caption><thead><tr><th>Configuration</th><th>SNR ≥ 8</th><th>SNR ≥ 12</th><th>SNR ≥ 50</th><th>SNR ≥ 100</th><th>SNR ≥ 200</th></tr></thead><tbody><tr><td>Δ-10km-HFLF-Cryo</td><td>103 528</td><td>87 568</td><td>13 674</td><td>2298</td><td>282</td></tr><tr><td>Δ-15km-HFLF-Cryo</td><td>111 231</td><td>101 308</td><td>26 092</td><td>5730</td><td>759</td></tr><tr><td>2L-15km-45°-HFLF-Cryo</td><td>107 661</td><td>97 205</td><td>23 491</td><td>4933</td><td>644</td></tr><tr><td>2L-20km-45°-HFLF-Cryo</td><td>110 698</td><td>103 773</td><td>34 009</td><td>8828</td><td>1267</td></tr><tr><td>2L-15km-0°-HFLF-Cryo</td><td>104 935</td><td>94 015</td><td>24 088</td><td>5143</td><td>642</td></tr><tr><td>2L-20km-0°-HFLF-Cryo</td><td>106 417</td><td>98 274</td><td>32 915</td><td>8551</td><td>1246</td></tr><tr><td>Δ-10km-HF</td><td>87 125</td><td>65 092</td><td>5595</td><td>773</td><td>98</td></tr><tr><td>Δ-15km-HF</td><td>102 149</td><td>85 698</td><td>13 697</td><td>2360</td><td>292</td></tr><tr><td>2L-15km-45°-HF</td><td>97 881</td><td>81 210</td><td>12 089</td><td>1987</td><td>248</td></tr><tr><td>2L-20km-45°-HF</td><td>105 032</td><td>93 050</td><td>20 551</td><td>4144</td><td>515</td></tr><tr><td>2L-15km-0°-HF</td><td>89 707</td><td>73 696</td><td>10 688</td><td>1732</td><td>201</td></tr><tr><td>2L-20km-0°-HF</td><td>104 558</td><td>92 308</td><td>21 970</td><td>4540</td><td>569</td></tr><tr><td>Δ-10km-HFLF-Cryo+CE-40km</td><td>115 179</td><td>110 118</td><td>44 676</td><td>12 590</td><td>1805</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+CE-40km</td><td>116 328</td><td>112 661</td><td>50 947</td><td>15 545</td><td>2355</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+CE-40km</td><td>114 816</td><td>110 265</td><td>49 034</td><td>14 820</td><td>2243</td></tr><tr><td>Δ-10km-HFLF-Cryo+2CE</td><td>117 045</td><td>113 910</td><td>52 092</td><td>16 109</td><td>2505</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+2CE</td><td>117 436</td><td>115 166</td><td>57 678</td><td>19 028</td><td>3126</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+2CE</td><td>116 639</td><td>113 597</td><td>55 218</td><td>17 849</td><td>2917</td></tr><tr><td>LVKI-O5</td><td>8603</td><td>2861</td><td>47</td><td>4</td><td>2</td></tr></tbody></table>

In table 42 we report the number of BNS sources detected with different cuts in SNR, in table 43 with cuts on the luminosity distance relative error and 90% sky localisation area,<fnref n="54" /> and in table 44 with cuts on the relative errors on the chirp mass and adimensional tidal deformability parameter $\tilde{\Lambda}$ [defined in eq. (3.1)].

<footnote n="54">⁵⁴Some differences with the results shown in tables 1 and 2 are due to the different choices for the cut in SNR for the parameter estimation analysis; in section 3 (to which the result of this appendix refer) was used SNR ≥ 12, while in section 4 the threshold was lowered to SNR ≥ 8 because of the increased statistical significance provided by the coincidence with an electromagnetic counterpart. As can be seen from table 42, passing from one cut to the other, the number of detections changes by more than factor of 2, in particular for the HF-only configurations. Part of the discrepancy is also due to the different inversion procedure adopted in the two sections, as already outlined in footnote 9, with some events being discarded in the results presented here and in section 3, while their Fisher matrices are regularized in the ones reported in section 4, resulting in higher numbers.</footnote>
<table><caption>**Table 40.** Same as table 39, for BBH sources, with cuts on the relative error on the luminosity distance and on the 90% sky localisation area.</caption><thead><tr><th>Configuration</th><th>Δd<sub>L</sub>/d<sub>L</sub> ≤ 0.1</th><th>Δd<sub>L</sub>/d<sub>L</sub> ≤ 0.01</th><th>ΔΩ<sub>90%</sub> ≤ 50 deg<sup>2</sup></th><th>ΔΩ<sub>90%</sub> ≤ 10 deg<sup>2</sup></th></tr></thead><tbody><tr><td>Δ-10km-HFLF-Cryo</td><td>10 969</td><td>28</td><td>6064</td><td>914</td></tr><tr><td>Δ-15km-HFLF-Cryo</td><td>17 321</td><td>77</td><td>10 470</td><td>2273</td></tr><tr><td>2L-15km-45°-HFLF-Cryo</td><td>22 237</td><td>202</td><td>10 304</td><td>2124</td></tr><tr><td>2L-20km-45°-HFLF-Cryo</td><td>28 801</td><td>365</td><td>14 920</td><td>3648</td></tr><tr><td>2L-15km-0°-HFLF-Cryo</td><td>13 865</td><td>79</td><td>3030</td><td>374</td></tr><tr><td>2L-20km-0°-HFLF-Cryo</td><td>17 008</td><td>144</td><td>4706</td><td>608</td></tr><tr><td>Δ-10km-HF</td><td>3919</td><td>6</td><td>2409</td><td>281</td></tr><tr><td>Δ-15km-HF</td><td>8083</td><td>26</td><td>5156</td><td>817</td></tr><tr><td>2L-15km-45°-HF</td><td>11 193</td><td>56</td><td>5263</td><td>835</td></tr><tr><td>2L-20km-45°-HF</td><td>16 155</td><td>113</td><td>8448</td><td>1566</td></tr><tr><td>2L-15km-0°-HF</td><td>4111</td><td>17</td><td>1054</td><td>120</td></tr><tr><td>2L-20km-0°-HF</td><td>9693</td><td>57</td><td>2936</td><td>362</td></tr><tr><td>Δ-10km-HFLF-Cryo+CE-40km</td><td>80 676</td><td>2901</td><td>69 268</td><td>29 924</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+CE-40km</td><td>82 358</td><td>4301</td><td>73 164</td><td>36 457</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+CE-40km</td><td>64 471</td><td>2995</td><td>57 497</td><td>25 782</td></tr><tr><td>Δ-10km-HFLF-Cryo+2CE</td><td>101 912</td><td>5250</td><td>90 889</td><td>46 744</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+2CE</td><td>104 289</td><td>7006</td><td>95 387</td><td>54 640</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+2CE</td><td>93 832</td><td>5824</td><td>85 267</td><td>45 429</td></tr><tr><td>LVKI-O5</td><td>767</td><td>1</td><td>1607</td><td>599</td></tr></tbody></table>

<table><caption>**Table 41.** Same as table 39, for BBH sources, with cuts on the relative error on the detector-frame chirp mass and on the error on the spin magnitude of the primary object.</caption><thead><tr><th>Configuration</th><th>ΔM<sub>c</sub>/M<sub>c</sub> ≤ 10<sup>-3</sup></th><th>ΔM<sub>c</sub>/M<sub>c</sub> ≤ 10<sup>-4</sup></th><th>Δχ<sub>1</sub> ≤ 0.05</th><th>Δχ<sub>1</sub> ≤ 0.01</th></tr></thead><tbody><tr><td>Δ-10km-HFLF-Cryo</td><td>48 922</td><td>4549</td><td>27 877</td><td>2811</td></tr><tr><td>Δ-15km-HFLF-Cryo</td><td>64 469</td><td>7703</td><td>41 612</td><td>4856</td></tr><tr><td>2L-15km-45°-HFLF-Cryo</td><td>58 371</td><td>6456</td><td>35 943</td><td>3958</td></tr><tr><td>2L-20km-45°-HFLF-Cryo</td><td>67 999</td><td>9073</td><td>45 666</td><td>5706</td></tr><tr><td>2L-15km-0°-HFLF-Cryo</td><td>57 330</td><td>6472</td><td>33 236</td><td>3653</td></tr><tr><td>2L-20km-0°-HFLF-Cryo</td><td>63 154</td><td>8279</td><td>40 068</td><td>4935</td></tr><tr><td>Δ-10km-HF</td><td>21 146</td><td>1580</td><td>11 715</td><td>1438</td></tr><tr><td>Δ-15km-HF</td><td>32 643</td><td>2818</td><td>19 956</td><td>2564</td></tr><tr><td>2L-15km-45°-HF</td><td>28 442</td><td>2405</td><td>16 382</td><td>2025</td></tr><tr><td>2L-20km-45°-HF</td><td>36 969</td><td>3547</td><td>23 205</td><td>2940</td></tr><tr><td>2L-15km-0°-HF</td><td>25 863</td><td>2146</td><td>13 669</td><td>1652</td></tr><tr><td>2L-20km-0°-HF</td><td>38 537</td><td>3740</td><td>23 065</td><td>2935</td></tr><tr><td>Δ-10km-HFLF-Cryo+CE-40km</td><td>73 785</td><td>8901</td><td>54 789</td><td>6649</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+CE-40km</td><td>79 425</td><td>11 187</td><td>60 709</td><td>7988</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+CE-40km</td><td>77 772</td><td>10 885</td><td>57 471</td><td>7519</td></tr><tr><td>Δ-10km-HFLF-Cryo+2CE</td><td>79 895</td><td>10 523</td><td>62 826</td><td>8226</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+2CE</td><td>84 507</td><td>12 703</td><td>68 382</td><td>9557</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+2CE</td><td>82 473</td><td>12 307</td><td>65 287</td><td>9012</td></tr><tr><td>LVKI-O5</td><td>78</td><td>1</td><td>155</td><td>20</td></tr></tbody></table>
<table><caption>**Table 42.** Same as table 39 for BNS sources.</caption><thead><tr><th>Configuration</th><th>SNR ≥ 8</th><th>SNR ≥ 12</th><th>SNR ≥ 50</th><th>SNR ≥ 100</th><th>SNR ≥ 150</th></tr></thead><tbody><tr><td>Δ-10km-HFLF-Cryo</td><td>107 902</td><td>36 985</td><td>458</td><td>57</td><td>19</td></tr><tr><td>Δ-15km-HFLF-Cryo</td><td>213 583</td><td>89 910</td><td>1206</td><td>159</td><td>38</td></tr><tr><td>2L-15km-45°-HFLF-Cryo</td><td>190 528</td><td>77 458</td><td>1052</td><td>134</td><td>33</td></tr><tr><td>2L-20km-45°-HFLF-Cryo</td><td>275 595</td><td>129 821</td><td>2018</td><td>243</td><td>64</td></tr><tr><td>2L-15km-0°-HFLF-Cryo</td><td>192 030</td><td>78 675</td><td>1040</td><td>136</td><td>33</td></tr><tr><td>2L-20km-0°-HFLF-Cryo</td><td>274 395</td><td>132 486</td><td>2048</td><td>250</td><td>65</td></tr><tr><td>Δ-10km-HF</td><td>44 713</td><td>13 410</td><td>166</td><td>18</td><td>9</td></tr><tr><td>Δ-15km-HF</td><td>116 349</td><td>41 181</td><td>516</td><td>55</td><td>17</td></tr><tr><td>2L-15km-45°-HF</td><td>101 550</td><td>34 956</td><td>447</td><td>52</td><td>15</td></tr><tr><td>2L-20km-45°-HF</td><td>176 396</td><td>70 441</td><td>961</td><td>115</td><td>32</td></tr><tr><td>2L-15km-0°-HF</td><td>103 539</td><td>35 817</td><td>443</td><td>57</td><td>17</td></tr><tr><td>2L-20km-0°-HF</td><td>184 799</td><td>74 805</td><td>989</td><td>124</td><td>37</td></tr><tr><td>Δ-10km-HFLF-Cryo+CE-40km</td><td>348 434</td><td>177 925</td><td>2836</td><td>312</td><td>87</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+CE-40km</td><td>392 680</td><td>212 260</td><td>3677</td><td>418</td><td>116</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+CE-40km</td><td>402 234</td><td>220 023</td><td>3770</td><td>414</td><td>119</td></tr><tr><td>Δ-10km-HFLF-Cryo+2CE</td><td>406 630</td><td>220 725</td><td>3961</td><td>436</td><td>120</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+2CE</td><td>442 526</td><td>252 136</td><td>4900</td><td>559</td><td>152</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+2CE</td><td>448 798</td><td>258 615</td><td>4974</td><td>531</td><td>162</td></tr><tr><td>LVKI-O5</td><td>250</td><td>71</td><td>3</td><td>0</td><td>0</td></tr></tbody></table>

<table><caption>**Table 43.** Same as table 40 for BNS sources.</caption><thead><tr><th>Configuration</th><th>Δd<sub>L</sub>/d<sub>L</sub> ≤ 0.3</th><th>Δd<sub>L</sub>/d<sub>L</sub> ≤ 0.1</th><th>ΔΩ<sub>90%</sub> ≤ 100 deg<sup>2</sup></th><th>ΔΩ<sub>90%</sub> ≤ 10 deg<sup>2</sup></th></tr></thead><tbody><tr><td>Δ-10km-HFLF-Cryo</td><td>748</td><td>52</td><td>184</td><td>8</td></tr><tr><td>Δ-15km-HFLF-Cryo</td><td>1756</td><td>153</td><td>479</td><td>23</td></tr><tr><td>2L-15km-45°-HFLF-Cryo</td><td>4328</td><td>479</td><td>559</td><td>25</td></tr><tr><td>2L-20km-45°-HFLF-Cryo</td><td>7821</td><td>919</td><td>1028</td><td>43</td></tr><tr><td>2L-15km-0°-HFLF-Cryo</td><td>774</td><td>48</td><td>293</td><td>12</td></tr><tr><td>2L-20km-0°-HFLF-Cryo</td><td>1499</td><td>104</td><td>565</td><td>23</td></tr><tr><td>Δ-10km-HF</td><td>4</td><td>1</td><td>4</td><td>0</td></tr><tr><td>Δ-15km-HF</td><td>7</td><td>1</td><td>11</td><td>1</td></tr><tr><td>2L-15km-45°-HF</td><td>126</td><td>12</td><td>11</td><td>0</td></tr><tr><td>2L-20km-45°-HF</td><td>262</td><td>22</td><td>24</td><td>1</td></tr><tr><td>2L-15km-0°-HF</td><td>20</td><td>1</td><td>11</td><td>1</td></tr><tr><td>2L-20km-0°-HF</td><td>28</td><td>2</td><td>24</td><td>1</td></tr><tr><td>Δ-10km-HFLF-Cryo+CE-40km</td><td>32 053</td><td>4100</td><td>54 994</td><td>2427</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+CE-40km</td><td>45 252</td><td>7949</td><td>75 828</td><td>3838</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+CE-40km</td><td>16 999</td><td>2079</td><td>29 821</td><td>1515</td></tr><tr><td>Δ-10km-HFLF-Cryo+2CE</td><td>72 335</td><td>13 630</td><td>112 705</td><td>6570</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+2CE</td><td>89 877</td><td>19 129</td><td>145 272</td><td>9841</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+2CE</td><td>78 798</td><td>14 909</td><td>125 640</td><td>7592</td></tr><tr><td>LVKI-O5</td><td>12</td><td>1</td><td>51</td><td>31</td></tr></tbody></table>
<table><caption>**Table 44.** Same as table 41, for BNS sources, with cuts on the relative error on the detector-frame chirp mass and on the relative error on the tidal deformability parameter $\tilde{\Lambda}$.</caption><thead><tr><th>Configuration</th><th>ΔM<sub>c</sub>/M<sub>c</sub> ≤ 10<sup>-3</sup></th><th>ΔM<sub>c</sub>/M<sub>c</sub> ≤ 10<sup>-4</sup></th><th>ΔΛ̃/Λ̃ ≤ 0.1</th><th>ΔΛ̃/Λ̃ ≤ 0.05</th></tr></thead><tbody><tr><td>Δ-10km-HFLF-Cryo</td><td>35 127</td><td>18 401</td><td>1040</td><td>96</td></tr><tr><td>Δ-15km-HFLF-Cryo</td><td>84 835</td><td>39 607</td><td>2783</td><td>227</td></tr><tr><td>2L-15km-45°-HFLF-Cryo</td><td>68 391</td><td>36 645</td><td>2463</td><td>200</td></tr><tr><td>2L-20km-45°-HFLF-Cryo</td><td>115 695</td><td>59 051</td><td>5189</td><td>386</td></tr><tr><td>2L-15km-0°-HFLF-Cryo</td><td>67 023</td><td>29 813</td><td>2225</td><td>179</td></tr><tr><td>2L-20km-0°-HFLF-Cryo</td><td>113 726</td><td>49 445</td><td>4703</td><td>374</td></tr><tr><td>Δ-10km-HF</td><td>10 831</td><td>687</td><td>248</td><td>21</td></tr><tr><td>Δ-15km-HF</td><td>31 663</td><td>1548</td><td>667</td><td>45</td></tr><tr><td>2L-15km-45°-HF</td><td>25 058</td><td>2033</td><td>634</td><td>47</td></tr><tr><td>2L-20km-45°-HF</td><td>50 464</td><td>3644</td><td>1494</td><td>90</td></tr><tr><td>2L-15km-0°-HF</td><td>22 839</td><td>1463</td><td>445</td><td>36</td></tr><tr><td>2L-20km-0°-HF</td><td>42 731</td><td>2292</td><td>780</td><td>72</td></tr><tr><td>Δ-10km-HFLF-Cryo+CE-40km</td><td>171 997</td><td>46 008</td><td>2669</td><td>243</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+CE-40km</td><td>197 631</td><td>66 214</td><td>4753</td><td>400</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+CE-40km</td><td>198 171</td><td>63 219</td><td>4488</td><td>371</td></tr><tr><td>Δ-10km-HFLF-Cryo+2CE</td><td>216 875</td><td>58 522</td><td>3847</td><td>337</td></tr><tr><td>2L-15km-45°-HFLF-Cryo+2CE</td><td>244 198</td><td>80 406</td><td>6319</td><td>535</td></tr><tr><td>2L-15km-0°-HFLF-Cryo+2CE</td><td>250 033</td><td>79 746</td><td>6106</td><td>505</td></tr><tr><td>LVKI-O5</td><td>54</td><td>2</td><td>2</td><td>0</td></tr></tbody></table>
D Correlation between parameters for typical events

In this appendix we discuss the correlation between parameters, in the BBH parameter estimation. This is instructive to obtain a physical understanding of some of the results presented in the main text. In particular, we wish to understand why the 15 km triangle and the 15 km 2L at 45° have performances very similar for the reconstruction of all parameters, except for luminosity distance, where the 15 km 2L at 45° is clearly superior, see figure 5 and tables 40, 41. To this purpose, we consider in detail some selected events (the correlation patterns that we will discuss are in fact generic). We select some BBH event with very high SNR, so that the Fisher matrix analysis is expected to be more reliable. As a first example we consider, in our ensemble of detected events, a ‘light’ BBH event (that we denote as “event 1”) with source-frame masses $m_1 \simeq 7.9M_\odot$, $m_2 = 7.6M_\odot$, at a distance $d_L \simeq 0.30$ Gpc ($z \simeq 0.065$). Due to its close distance, its SNR is very high, and is 513 in the 15 km triangle and 478 in the 15 km 2L at 45°.

In figure 72 we show the correlation between some parameters. For readability, within the 15-dimensional parameter space that we use for BBHs, we restrict to the most interesting correlations, and we compare the results for the 15 km 2L at 45° with the 15 km triangle, both in their full HFLF-cryo configuration. The interesting point is that, in the correlation of various parameters with the luminosity distance, the contours for the 15 km triangle (red) are in general more tilted than the blue contour referring to the 15 km 2L at 45°. In figure 72 we see this in the correlation with all the parameters shown. This implies that, when one marginalizes over these parameters, the same error on these parameters induces a larger marginalized error on $d_L$. We see indeed that, for all parameters except $d_L$, for this event the marginalized errors are the same for the 15 km triangle and the 15 km 2L at 45°, while the marginalized distribution for $d_L$ is narrower for the 15 km 2L at 45°. Observe that this happens despite the fact that, for this event, the SNR in the triangle was higher than in the 15 km 2L at 45°.

Figure 73 shows the analogous results for an event (“event 2”) with source-frame masses $m_1 \simeq 29.0M_\odot$, $m_2 = 24.6M_\odot$, at a distance $d_L \simeq 1.19$ Gpc ($z \simeq 0.23$). Its SNR is again very high, 424 in the 15 km triangle and 318 in the 15 km 2L at 45°. As we reminded in section 2, the differences in the SNR between configurations, alone, is not a good indicator of the relative performances of parameter estimation (the most obvious example being that, on an ensemble of events, a 2L parallel configuration has higher SNR but worse angular localization than a 2L at 45°). This is also clearly visible from this plot: in this case, despite the larger SNR in the 15 km triangle, most parameters are significantly better estimated by the 15 km 2L at 45°. We also see that, again, the correlation contours of $d_L$ with various parameters are more tilted in the triangle case, contributing to enlarging the marginalized posterior of $d_L$. This effect is also particularly evident, for this event, in the correlation between $\theta$ and $\phi$.
<figure><img src="image_81.png" /><figcaption>**Figure 72.** A corner plot showing the correlations between various parameters for a BBH event with light masses and very close distance ("event 1", see the text).</figcaption></figure>
<figure><img src="image_82.png" /><figcaption>**Figure 73.** As in figure 72 for a BBH event with heavier masses and greater distance ("event 2", see the text).</figcaption></figure>
## Acronyms

**3G** Third Generation. 2

**AGWB** Astrophysical Gravitational Wave Background. 50

**ASD** Amplitude Spectral Density. 6

**BAO** Barionic Acoustic Oscillation. 98

**BBH** Binary Black Hole. 1

**BNS** Binary Neutron Star. 1

**CBC** Compact Binary Coalescence. 5

**CDO** Compact Dark Object. 126

**CE** Cosmic Explorer. 2

**CGWB** Cosmological Gravitational Wave Background. 50

**CIB** Cosmic Infrared Background. 51

**CMB** Cosmic Microwave Background. 51

**CSD** Cross-power Spectral Density. 57

**CW** Continuous Wave. 117

**DE** Dark Energy. 95

**DM** Dark Matter. 88

**ECO** Exotic Compact Object. 73

**EM** Electromagnetic. 39

**EoS** Equation of State. 39

**FAR** False Alarm Rate. 129

**FOPT** First-Order Phase Transition. 114

**FoV** Field of View. 24

**FRW** Friedmann-Robertson-Walker. 100

**GR** General Relativity. 15

**GRB** Gamma Ray Burst. 3

**GW** Gravitational Wave. 1
**GWTC** Gravitational Wave Transient Catalog. 89

**HF** High-Frequency. 2

**HMNS** Hypermassive Neutron Star. 84

**IMBH** Intermediate Mass Black Hole. 15

**IMR** Inspiral Merger Ringdown. 107

**ISB** Instrument Science Board. 4

**ISCO** Innermost Stable Circular Orbit. 75

**KN** Kilonova. 39

**LF** Low-Frequency. 2

**LSST** Large Synoptic Survey Telescope. 96

**LVC** LIGO-Virgo Collaboration. 1

**LVKI** LIGO Hanford, LIGO Livingston, Virgo, KAGRA, LIGO India. 6

**NN** Newtonian Noise. 56

**NR** Numerical Relativity. 84

**NSBH** Neutron Star-Black Hole. 1

**ORF** Overlap Reduction Function. 148

**OSB** Observational Science Board. 2

**PBH** Primordial Black Hole. 88

**PLS** Power Law Sensitivity. 3

**PN** Post Newtonian. 10

**PSD** Power Spectral Density. 57

**QNM** Quasinormal Mode. 68

**SFH** Star Formation History. 55

**SGWB** Stochastic Gravitational Wave Background. 50

**SMNS** Supramassive Neutron Star. 84

**SN** Supernova. 98

**SNR** Signal-to-Noise Ratio. 7
**SXI** Soft X-ray Imager. 47

**tCW** Transient Continuous Wave. 117

**TOV** Tolman-Oppenheimer-Volkoff. 79

**UV** Ultra Violet. 42

**VRO** Vera Rubin Observatory. 48

**XGIS** X/Gamma-ray Imaging Spectrometer. 44

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