A Broadband Room-Temperature Design for the Einstein Telescope

Introduction

The Einstein Telescope (ET) is a next-generation gravitational-wave observatory designed to achieve an order-of-magnitude improvement in broadband strain sensitivity over current detectors [1, 2, 3, 4, 5]. In particular, extending sensitivity to lower frequencies is crucial for detecting more massive black-hole binaries at high redshifts [6] and for observing low-mass compact binaries earlier in their inspiral. Access to the early inspiral would enable earlier warnings of imminent mergers [7] and provide longer signal baselines for measuring subtle binary dynamics, including spin precession and orbital eccentricity [8, 9]. Improved low-frequency sensitivity is therefore one of the central scientific drivers of the Einstein Telescope.

The baseline ET xylophone architecture employs separate high-power, room-temperature and low-power, cryogenic interferometers optimized for high- and low-frequency detection, respectively [2, 10]. This approach comes at the cost of substantial technical complexity associated with cryogenic operation, including heat extraction [11], molecular adsorption and icing of cold optics [12], and additional noise mechanisms associated with silicon test masses [13, 14]. Moreover, the ET-LF interferometer employs a detuned signal-recycling configuration [15], introducing additional optomechanical couplings and sensing and control challenges [16]. Experience with LIGO and Virgo [17, 18] shows that the measured detector noise can exceed the modelled fundamental-noise budget because of technical noise sources that are difficult to fully characterise. This gap between modelled and realised sensitivity may pose an even greater challenge for ET-LF, where the extreme low-frequency sensitivity requirements demand simultaneous control of numerous mechanical, optical, thermal, seismic, and control-noise mechanisms.

These considerations motivate the exploration of a detector architecture with fewer coupled subsystems and less stringent technical requirements, such as a unified, room-temperature interferometer that can provide broadband sensitivity while avoiding both cryogenic operation and signal-recycling detuning. We refer to this concept as the unified interferometer (UI) for ET. Figure 1 and Figure 2 show the resulting broadband sensitivity and its noise budget, respectively. The parameter set used in the calculation is summarized in Table I. The low-frequency performance is enabled by 440 kg test masses, which reduce the displacement response to radiation pressure, and 4 m-long, high-stress final-stage fused-silica fibres, which suppress suspension thermal noise through enhanced dissipation dilution. A 5 km filter cavity enables low-loss frequency-dependent squeezing to be maintained over the detection band.

In the following sections, we first discuss the design of the fused-silica suspension and the technological solutions adopted to mitigate vertical suspension thermal noise [19]. We then examine the control challenges of the UI and demonstrate how 440 kg test masses mitigate high-power effects such as optical rigidity. Next, we present the broadband and kHz-optimized sensitivity of the detector and investigate its scaling with arm length. Finally, we compare the resulting performance with that of Cosmic Explorer.

Test mass suspension

A key component of the UI for ET is its room-temperature test-mass suspension. Our aim is to achieve the suspension thermal noise [20] at or below the level of the ET facility limit given by the gravity-gradient noises at low frequencies. For the seismic input, we use the underground seismic spectrum measured at Sos Enattos as representative of a seismically quiet underground environment [21]. The gravity-gradient noise arises from the direct gravitational coupling of environmental density fluctuations to the test masses. In this study, we retain the ET model for the Rayleigh-wave, body-wave, atmospheric, and cavern-acoustic contributions [22, 23, 24, 25]. We assume that local seismic and acoustic sensor arrays reduce the amplitude spectral densities of the corresponding Newtonian-noise contributions by a factor of three.

The noise level of the gravity gradients, shown in Figure 2, is at the level of $\approx2.5 \times10^{-25}$ strain/$\sqrt{\mathrm{Hz}}$ and also sets the required limit on the suspension thermal noise of $3.75 \times10^{-21}$ m/$\sqrt{\mathrm{Hz}}$. In this section, we propose a solution to the ET suspension thermal noise problem, covering both the bottom suspension stage and a well-known problem of the vertical suspension thermal noise coming from the metal blades of the final filter stage [26]. In this section, we show that 4 m-long fused-silica fibres can achieve the required suspension thermal-noise performance.

Bottom stage suspension thermal noise

Gains in suspension thermal-noise performance from the final suspension stage arise directly from the deployment of three basic concepts, as previously outlined in [27]. First, a larger test mass reduces the thermal noise, with the relevant scaling inversely proportional to the square root of the test-mass mass, according to the fluctuation-dissipation theorem [20, 28, 29, 30]. Second, longer suspension fibres reduce the resonant frequency of the fundamental pendulum mode, thereby reducing the off-resonance thermal noise in the detection band of the instrument. For the 4 m-long fibres proposed here, the final-stage pendulum frequency is reduced to 0.246 Hz, compared with 0.642 Hz for the 0.6 m-long fibres used in the LIGO suspensions. Longer suspensions also provide additional dissipation-dilution benefits by reducing the ratio of elastic bending energy to gravitational energy.

Parameter	Proposed value
Interferometer configuration
Arm length	15 km
Laser wavelength	1064 nm
Input laser power	500 W
Arm circulating power	3 MW
ITM transmissivity	1.4 %
ETM transmissivity	5 ppm
ITM/ETM radius of curvature	7651 m
Signal-extraction mirror (SEM) transmissivity, broadband/high frequency	8 % / 1 %
Signal-extraction-cavity (SEC) length	100 m
Squeezing and optical losses
Injected squeezing	18 dB
Observed squeezing	$\sim10$ dB
Injected squeezing angle	0 rad
Filter-cavity length	5 km
Arm loss per mirror	37.5 ppm
Signal-extraction-cavity loss	500 ppm
OPA-cavity / injection Faraday-isolator loss	1 % / 2 %
Photodetector / Faraday-isolator / output-mode-cleaner loss	1 % / 1 % / 1 %
Filter-cavity end-mirror transmissivity / round-trip loss	5 ppm / 60 ppm
Squeezer phase noise	10 mrad
Filter-cavity / signal-extraction-cavity length noise (rms)	1 pm / 1 pm
Test masses, beam, and suspension
Test-mass mass	440 kg
Test-mass temperature	290 K
Test-mass radius / thickness	37 cm / 46.5 cm
Test-mass substrate	Fused silica
Beam radius on the test masses	13.5 cm
Coating loss angle	$5 \times10^{-5}/1.3 \times10^{-5}$
Final suspension-fibre material	Fused silica
Number of final-stage fibres	4
Final suspension-fibre length	4 m
Final suspension-fibre neck length	0.03 m
Final suspension-fibre stress	2 GPa
Fibre surface loss	$2.5 \times10^{-12}$ m

The scaling of the horizontal suspension thermal noise with fibre length, $L_{\mathrm{fibre}}$, and test-mass mass, $M_{\mathrm{tm}}$, relative to the LIGO-estimated value [31], is given by

Since the LIGO horizontal suspension thermal noise is approximately a factor of 19 above the estimated gravity-gradient noise level at the Sos Enattos facility, we need to achieve a comparable reduction in suspension thermal noise by increasing the fibre length and test-mass mass. Choosing $L_{\mathrm{fibre}} = 4\,\mathrm{m}$ and $M_{\mathrm{tm}} = 440\,\mathrm{kg}$, we obtain a reduction in the thermal noise of approximately a factor of 22 relative to the LIGO horizontal suspension noise. This estimate does not require any reduction in the critical fibre surface loss.

The fibre design proposed here follows the standard, well-established approach in which the otherwise dominant thermoelastic noise contribution is minimised by exploiting a particular property of fused silica: its Young’s modulus increases with temperature, i.e., the material stiffens as the temperature rises [32]. This effect can be balanced against the thermal expansion coefficient, allowing the thermoelastic dissipation to be completely cancelled for a specific tensile stress in the fibre’s bending regions, namely 185 MPa. For a 440 kg test mass, this corresponds to a fibre diameter of 2728 $\mu$m in the bending regions. A length of 30 mm at each end of the fibre is specified at this diameter, which is sufficient to accommodate the bending of the fibre.

In addition to horizontal suspension thermal noise, vertical suspension thermal noise can couple into the GW readout channel because, for a long-baseline detector, the local gravity direction cannot be exactly perpendicular to the arm direction due to the curvature of the Earth. The corresponding minimum geometrical coupling of vertical test-mass motion into the GW channel is

where $L$ is the detector arm length and $R_E$ is the Earth’s radius.

The central section of the fibre is operated at a higher stress, $\sigma$, allowing the suspension’s vertical bounce and roll frequencies to be reduced and ultimately shifted out of the detection band for $\sigma\gtrsim1$ GPa, and also providing passive filtering of the vertical suspension thermal noise from the final filter. The higher stress also increases the frequencies of the individual fibre violin modes. The scaling of the vertical suspension thermal noise coupled to the horizontal readout for a 15 km-baseline detector, relative to the LIGO-estimated value [31], is given by

and is significantly below the required suspension thermal noise level for the fibre stress values of $\sigma= 1 - 2$ GPa. Suspensions supporting larger masses at higher stresses have already been demonstrated and provide a viable basis for further development [27, 33]. Ongoing research into stress corrosion in fused-silica fibres [34, 35] suggests that a practical current limit on the ultimate fibre stress is approximately 2 GPa, corresponding to a strength safety factor of approximately 2.5. Previous studies have also found no clear correlation between fibre length and tensile failure strength, suggesting that the increased fibre length does not in itself represent a

significant additional risk. For a 440 kg test mass and four suspension fibres, the stress requires a reduction in the fibre cross-sectional area, giving a central fibre diameter of 828 $\mu$m.

Manufacturing longer fibres is relatively straightforward, requiring only an extension of the pulling apparatus [36]. Suspension fibres with lengths of up to 1.2 m have already been successfully produced [27], and no fundamental difficulties are anticipated in extending the process to 4 m.

Fibres with the required profile have already been successfully demonstrated using 7 mm-diameter fused-silica stock as the starting material, as shown in Figure 3. This enabled the full fibre geometry to be realised, including accurate modelling of the neck profiles at the transitions between different diameters. The experimentally observed variation in diameter achievable during the fibre-pulling process was also incorporated through the thermoelastic-nulling region, as shown in Figure 3. This level of realism is included from the outset in both the design methodology and the thermal-noise modelling.

Suspension thermal noise is modelled using finite element analysis (FEA). The FEA model accurately models the elastic bending energy in the system, allowing accurate calculation of the contributing noise components, together with the dissipation dilution of the pendulum. This permits an advanced level of fidelity of the noise model to be included in the proposed suspension design. Suspension thermal noise is calculated using the FEA-based methodologies extensively detailed in [37, 38], with the FEA model calculating the individual loss components surface, bulk, and thermoelastic- that contribute mechanical dissipation to each low-frequency resonant mode of the pendulum system. The loss components are shown in the inset of Figure 4. The projected single test mass thermal noise is shown in Figure 4, with the projected displacement noise being $1.85 \times10^{-21}\ \mathrm{m}/\sqrt{\mathrm{Hz}}$ at 10 Hz, which translates to a strain sensitivity in a 15 km detector of $2.46 \times10^{-25}\ \mathrm{strain}/\sqrt{\mathrm{Hz}}$ at 10 Hz.

Vertical suspension thermal noise from the final filter

The proposed long fused-silica fibres also provide passive filtering of suspension thermal noise from the upper suspension stages, which have relatively high metal loss angles of $\sim10^{-3}$ [39]. In particular, the vertical springs in the multi-stage suspension attenuate the coupling of ground vibrations to the test mass. However, the final vertical filter is itself a significant source of suspension thermal noise [40]. This is a particular limitation for the low-frequency sensitivity of ET-LF: thermal noise generated by the final vertical filter is transmitted to the test mass and can couple into horizontal motion through the curvature of the Earth [26]. Silicon flexures at the two bottom stages of ET-LF suspensions can inject negligible vertical thermal noise, but do not provide the required filtering of the blades’ thermal noise below 15–25 Hz.

Long fused-silica fibres between the test mass and the marionette in the UI concept address this challenge. Fused silica combines very low mechanical loss with a sufficiently low vertical stiffness and supreme mechanical strength to provide a passive filtering stage. In our design, the bounce mode is at $\approx2$ Hz, providing a $1/f^{2}$ attenuation of vertical motion above this frequency, as shown in Figure 6 for the fibre stress of $\sigma= 1.2 - 2$ GPa.

The simulation includes a conventional final filter with maraging-steel blades, a vertical eigenfrequency of 0.5 Hz, and a Q-factor of $\approx64$, based on KAGRA data [26] and measurements of the mechanical loss of maraging steel [39]. The resulting vertical thermal noise provides the reference against which the filtering performance of the long fused-silica fibre is evaluated. The vertical suspension thermal-noise coupling to the GW channel estimated in Figure 6 should be regarded as a conservative estimate, assuming no further advances in the design of the vertical springs or sensing system. Over the ET development timescale, this contribution could potentially be reduced further through optimisation of the vertical suspension. Below, we consider three options of reducing the vertical suspension thermal noise below the baseline UI sensitivity.

Fused silica vertical blades

Possible improvements of vertical suspension thermal noise include increasing the length of the maraging-steel blades, thereby reducing their mechanical stiffness, or developing vertical blades from lower-loss materials such as fused silica [41], sapphire [42, 43], or silicon [44].

Fused silica is a natural choice for the vertical blades of the room-temperature UI suspension due to its exceptionally low mechanical dissipation. We estimate that the thermoelastic, surface, and bulk losses of a fused-silica blade are each $< 10^{-8}$, such that the mechanical dissipation is expected to be dominated by clamping losses. In this section, we estimate the vertical suspension thermal noise associated with the fused-silica blades required to support the marionette, test-mass, and reaction masses, which have a combined mass of $M_{\mathrm{tot}} \approx1760$ kg. Assuming 12 fused-silica blades, the load carried by each blade is $F_b = M_{\mathrm{tot}}g/12 \approx1.44$ kN.

For a LIGO-type triangular blade with length $L_b$, clamped-end width $w_b$, and thickness $t_b$, the vertical resonance frequency can be estimated from the static deflection $\delta$ under the applied load,

where $\sigma_b$ is the bending stress and is given by

We choose the blade dimensions to keep $\sigma_b \le80$ MPa and to achieve a margin against mechanical failure.

Realistic geometrical parameters for the fused-silica blades result in a relatively stiff vertical mode of the final filter, with $\omega_v \approx2\pi\times(3.5 - 6.5)$ Hz. This mode can subsequently be softened to approximately 0.3 Hz using magnetic anti-springs, following a concept similar to that employed in the current Superattenuator design. While this softening does not reduce the intrinsic thermal noise of the blades, it can improve the isolation of the test mass from vertical suspension thermal noise generated in the higher suspension stages. Total vertical suspension thermal noise coupling to the beam direction is shown in Figure 7 for $w_b = 10$ cm.

Local sensing of the test mass

Another approach to subtracting vertical suspension thermal noise is the use of local sensing [19]. If asymmetries in the fibre stress prove challenging to control, the long fused-silica fibres in the UI concept provide an additional opportunity to reduce this coupling for symmetric suspensions. Since the bounce mode is at approximately 2 Hz, the test mass is effectively inertial with respect to the marionette at frequencies above the bounce resonance. This enables a direct measurement of the relative vertical motion between the marionette and the test mass, analogous to the measurement principles employed in inertial sensors and gravimeters. The measured relative motion can therefore be used to estimate the vertical suspension thermal noise and construct a feedforward signal for subtraction from the GW channel, either in real time or during post-processing.

The required sensitivity of the interferometric sensor measuring the relative motion between the marionette and the test mass is shown in Figure 8. State-of-the-art compact Michelson interferometers currently achieve sensitivities of approximately 0.03–0.2 pm/$\sqrt{\mathrm{Hz}}$ in the 1–10 Hz frequency band [45, 46, 47]. Fabry–Pérot interferometers with stabilised lasers offer the potential to improve the sensitivity further, reaching the required level of approximately 0.01–0.1 fm/$\sqrt{\mathrm{Hz}}$ over the 2–7 Hz frequency band.

Vertical to pitch coupling

In addition to passive filtering, long fused-silica fibres may further suppress the coupling of vertical suspension thermal noise into the horizontal direction. We propose to achieve this by introducing a 20% asymmetry in the stress between the front and back test-mass fibres. This asymmetric introduces an additional coupling between the vertical motion of the marionette and the pitch motion of the test mass. By appropriately positioning the arm-cavity beam on the test mass relative to the pitch rotation axis, this coupling can be used to cancel the vertical suspension thermal noise, as shown in Figure 9. For our chosen set of parameters, the required beam offset is 7.5 mm. We note that the cancellation of suspension thermal noise through coupling between orthogonal degrees of freedom has been extensively studied in the literature [48, 49].

The vertical offset of the beam position relative to the pitch axis also introduces coupling of the test-mass pitch thermal noise into the horizontal degree of freedom sensed by the beam. However, this additional contribution, shown as P-coupling in Figure 9, remains below the baseline sensitivity because fused silica is a low-loss material. We also note that the fibre asymmetry introduces coupling between the thermal pitch motion of the marionette and the vertical motion of the test mass. We consider this contribution to be negligible, however, because the marionette can be suspended from the final filter stage using a short fused-silica fibre and we leave this contribution for further investigations.

The proposed fibre asymmetry is compatible with current fused-silica fibre manufacturing capabilities [36]. The required difference in fibre length is approximately 22 mm, while the present pulling process is expected to achieve a length accuracy of approximately 0.2 mm, with substantially higher-resolution length encoders available for further improvement. Similarly, the required difference in the central fibre radius is approximately 40 µm. Although current production tolerances are based on a 5% acceptance criterion, improved control of the fibre-pulling process and stabilisation of the heating region make reproducible production of fibres with the required diameters achievable [50]. Some batch production and selection would likely still be required, as is already standard practice, but we do not anticipate this to present a

Suspension inside the vacuum tank

The preliminary ET Superattenuator design [51] demonstrates the feasibility of accommodating a multi-stage suspension within an overall length of approximately 12 m. A comparable suspension height is adopted in the ET baseline detector layout [52]. This height provides a useful geometrical constraint for the UI design considered here: the complete suspension system, including the intermediate attenuation stages and the final suspension of the test mass, must be accommodated within the available vacuum-tank height. In particular, adopting a 4 m-long final fibre requires the remaining suspension stages to be arranged within the remaining envelope. In this paper, we analyse the seismic noise coupling to the GW channel by considering a vibration isolation system that consists of an active platform [53, 54] and a six-stage suspension.

The platform performance adopted in the noise budget is shown in Figure 10. The platform utilises state-of-the-art optical seismometers and commercial geophones for inertial stabilisation starting from 10 mHz and provides at least two orders of magnitude of suppression of the support motion across the principal suspension resonances in the 0.1 Hz to 1 Hz band [45, 55, 56, 57, 58].

We propose a six-stage pendulum suspension designed to provide sufficient passive seismic isolation to meet the UI seismic-noise requirements above 3 Hz. The suspension comprises four suspended filter stages followed by the marionette and the test mass, with the upper four significant additional manufacturing challenge. pendulum stages and the marionette stage having lengths of 1.6 m, and a 4 m-long final suspension fibre between the marionette and the test mass. The total length of the wires and fibres at all stages is 12 m. The resulting coupled mechanical system is modelled as a simplified six-degree-of-freedom pendulum. We leave other degrees of freedom for future studies.

Simulations of the resulting seismic transfer function indicate that the proposed configuration can reduce the coupling of ground motion to the test mass by a factor of $\approx5 \times10^{7}$. This approach is therefore particularly attractive for a design constrained by the available 12 m vacuum-tank height, while retaining a 4 m final suspension fibre to provide enhanced low-frequency isolation. The result of the simulation is shown in Figure 10.

Quantum and classical control

The proposed interferometer operates on resonance and avoids detunings in the signal recycling cavity because they cause auxiliary degrees of freedom and technical noises to couple into the GW readout. The detuning also complicates the interferometer’s sensing and control because it modifies the optical response and introduces optical-spring dynamics, resulting in a more strongly coupled length-control system and making lock acquisition and stable operation more challenging [16]. Furthermore, the resulting coupled control system is more sensitive to parameter variations and makes real-time antenna calibration challenging. Finally, detuning rotates the squeezing quadrature in a frequency-dependent manner, complicating the injection of frequency-dependent squeezed vacuum [59].

The choice of an on-resonance broadband interferometer requires $\sim$MW of circulating power in the arms to achieve the baseline sensitivity of UI, both at low and high frequencies. In this section, we discuss the requirements for the quantum states, test-mass geometry, and inertial isolation required to achieve the baseline sensitivity.

Squeezed states of light

Quantum noise has two limiting contributions. At high frequencies, photon-counting statistics give rise to shot noise in the phase quadrature, while at low frequencies, amplitude fluctuations of the optical field exert fluctuating radiation-pressure forces on the test masses. Increasing the circulating arm power reduces shot noise but increases radiation-pressure noise. Injecting squeezed states of light with a properly tuned filter cavity suppresses both noise terms [60, 61, 62, 63, 64].

The proposed design relies on low optical losses that preserve the frequency-dependent squeezed vacuum injected at the detector output. The optical losses of the filter cavity are particularly important at low frequencies.

For a given bandwidth required to achieve the desired quadrature rotation, a longer filter cavity can operate at lower finesse [65, 66] and reduce the total effective optical loss. We therefore adopt a single 5 km-long filter cavity for the proposed UI design. By comparison, the baseline ET-LF model considered here employs two detuned 1 km-long filter cavities. Although the UI filter cavity is longer, the use of a single cavity avoids the need to accommodate two parallel optical paths and their associated vacuum and control systems. This reduces the required tunnel width and simplifies both the underground infrastructure and the optical configuration. We show the detailed quantum noise budget in Figure 11, with relevant parameters listed in Table I.

Optical rigidity

At megawatt circulating power, angular instabilities become an important limitation of the interferometer. In particular, radiation pressure couples the angular motion of the test masses to the optical mode structure, giving rise to the well-known Sidles–Sigg angular instabilities [67]. The radiation-pressure torque can either soften or stiffen the mechanical angular modes, depending on the optical configuration, and can overcome the mechanical restoring torque at sufficiently high power.

Angular stability is determined by the balance between the mechanical restoring torque and the radiation-pressure torque. Importantly, the mechanical restoring torque scales as $\sim M_{\rm tm}^{5/3}$, while the radiation-pressure torque scales as $\sim P_{\rm arm}L$. Compared with the LIGO and Virgo detectors, the mechanical restoring torque increases by a factor of $\approx54$, assuming the same angular-mode eigenfrequencies for the test masses, whereas the radiation-pressure torque increases by a factor of $\approx14$. Therefore, optical rigidity is expected to be less severe in the proposed UI configuration than in existing GW detectors.

To maintain a stable cavity at 3 MW of circulating power, we exploit both the large inertia of the heavy test masses and an “ear” geometry that increases the separation between the suspension fibres. The increased fibre separation provides a longer lever arm and therefore enhances the mechanical restoring stiffness in both pitch and yaw. This design also results in higher angular eigenfrequencies of the suspension and increased coupling of ground vibrations compared with a softer suspension. However, the angular noise level is suppressed by the active platform, as discussed in Sec. III C. The complete test-mass geometry is shown in Figure 5.

The resulting suspension restoring torque coefficient, $k_{\mathrm{res}}$, together with the radiation-pressure torque coefficients for the soft and hard angular modes, $k_{\mathrm{S}}$ and $k_{\mathrm{H}}$, are given in Table II. The resonant frequencies of the soft and hard modes in pitch and yaw as a function of arm-cavity power are shown in Figure 12. The cavity remains stable up to 3.5 MW.

DOF	$k_{\mathrm{res}}$ (N m rad$^{-1}$)	$I$ (kg m$^2$)	$k_{\mathrm{soft}}$ (N m rad$^{-1}$)	$k_{\mathrm{hard}}$ (N m rad$^{-1}$)
Pitch	236.3	23.0	−153.1	7605.6
Yaw	178.9	23.0	−153.1	7605.6

Angular controls

In addition to the longitudinal degrees of freedom, seismic-induced angular motion must also be controlled.

Previous studies have identified angular sensing and control as a potential limitation to ET sensitivity below 20 Hz [68] and proposed inertial sensors to tackle the problem of inertial isolation [69, 70, 71, 72, 73, 74].

To mitigate the angular controls problem, we introduce active platform stabilization as a pre-isolation stage upstream of the 6-stage suspension chains. The Compact6D seismometer [55, 56, 57] is well suited to this application, having demonstrated six-degree-of-freedom platform stabilization [58] and longitudinal isolation from 10 mHz [45].

The resulting low-frequency isolation provides at least two orders of magnitude of suppression of the tilt motion in the critical 0.1 Hz to 1 Hz band as shown in the inset of Figure 13. This reduction in support motion and the use of wavefront-sensing and control mitigates seismic-induced angular motion below the sensitivity requirement of UI. In addition, the angular control can further improve alignment stability and enhance detector robustness [75, 76]. We implement a 0.7 Hz angular control bandwidth, sufficiently low to prevent wavefront-sensing noise from contaminating the detection band, where the wavefront-sensing noise is assumed at the level of $10^{-15}\,\mathrm{rad}/\sqrt{\mathrm{Hz}}$.

Optical clipping

Having targeted the circulating arm power at 3 MW and the beam radius at 13.5 cm, we now consider the corresponding test-mass geometry. The mirror aperture must be sufficiently large to keep the effective loss of the fundamental arm-cavity mode negligible. For a Gaussian beam incident on a circular mirror, the fraction of power that lies outside the mirror radius is

where $r$ is the mirror radius and $w$ is the beam radius.

Considering the diffraction effect, the amplitude coefficient obtained by projecting the remaining field back onto the original fundamental mode is

The fraction of power retained in the resonant fundamental mode is therefore $\lvert c_{00}\rvert^{2}$, and the corresponding effective single-mirror loss is

This effective loss contains both the power directly removed by the finite aperture and the power scattered from the fundamental mode into higher-order spatial modes (HOMs). The corresponding round-trip effective loss becomes approximately $2l_{\mathrm{eff}}\simeq4l_{\mathrm{c}}$. We select the test mass radius of 37 cm and thickness of 46.5 cm. With the adopted beam radius of 13.5 cm, the direct clipping loss is approximately 0.3 ppm per mirror, while the corresponding effective loss of the resonant fundamental mode in the cavity is approximately 1.2 ppm. The selected mirror therefore provides sufficient aperture for the fundamental arm-cavity mode while retaining margin for alignment offsets.

For amorphous coating systems, large-area ion-beam-sputtering (IBS) coating is already an established technology within the gravitational-wave community. The LMA coating facility, developed for the Virgo and LIGO programmes, has demonstrated low-loss coatings on large interferometer optics [77], while the LMA Grand Coater was designed to accommodate substrates up to 800 mm in diameter [78]. Earlier development of the Virgo coating facility also demonstrated the feasibility of IBS coating systems for optics up to 700 mm in diameter [79]. The existing coating system at the University of Strathclyde is designed to coat optics up to 620 mm in diameter and 200 kg in mass, with the capability to coat two optics simultaneously [80]. More recently, larger-area IBS coating capability has been developed at the Laser Zentrum Hannover (LZH), including a system designed for optics up to 2 m in diameter [81, 82]. The challenge is also not unique to the proposed ET design: Cosmic Explorer plans to employ test masses of comparable mass and size [83], making the development of suitable large-scale coating infrastructure a common requirement for next-generation gravitational-wave observatories. This shared requirement provides an opportunity for coordinated development of coating technologies and facilities across future observatories.

Parametric instabilities

The heavy test mass is beneficial at low frequencies; however, it also brings a challenge associated with its internal acoustic modes. At high circulating power, an acoustic mode can scatter carrier light from the fundamental optical mode into an HOM. If the acoustic-mode frequency is sufficiently close to the corresponding optical-mode separation, the resulting radiation-pressure beat pattern can drive the same mechanical mode and produce parametric instability [84, 85]. If the mechanical mode frequency is low, it can also contaminate the sensitivity region as a narrow thermal-noise feature in the kilohertz band [86]. Preliminary finite-element calculations for the selected test-mass geometry give the lowest internal acoustic mode at approximately 3.36 kHz.

To suppress the parametric instabilities and acoustic-mode thermal noise, we could actively damp the selected modes using a resonant transducer attached to the test mass. The concept is adapted from the mechanical displacement transformers used in resonant-mass gravitational-wave detectors. In these systems, a lightweight secondary resonator, often implemented with a mushroom-like geometry, is tuned to a mode of the primary mass [87, 88, 89]. Near the coupled resonance, the motion of the secondary resonator provides an enhanced local signal that can be sensed with an interferometric readout. For the proposed detector, a mushroom-shaped transducer could be tuned to the approximately 3.36 kHz lowest internal acoustic mode. A compact local interferometer [45, 46, 47] would measure the transducer motion and provide an error signal for feedback actuation, thereby reducing the effective mechanical quality factor of the unstable mode. Active damping of a parametrically unstable test-mass mode has already been demonstrated in Advanced LIGO using electrostatic actuation [91] [90]. The resonant-transducer approach considered here combines this active-damping principle with mechanical displacement enhancement, potentially reducing the required sensing and actuation strength while avoiding the broadband mechanical dissipation associated with a conventional passive damper. Optical feedback provides a complementary approach and has been demonstrated first in an 80 m cavity and, more recently, in Advanced LIGO, suppressing the instability through destructive interference with the responsible higher-order optical mode [91, 92].

Sensitivity analysis

In this section, we discuss the detector parameters optimised for broadband, low-frequency and kHz operation, with the goal of achieving optimal sensitivity to the inspiral phase of compact-binary mergers and exploring the physics of neutron stars. The achievable sensitivity is ultimately constrained not only by the interferometer itself but also by the observatory infrastructure [21, 93].

Apart from the gravity-gradient noise considered in Sec. II, residual gas noise is another facility-level contribution to the overall sensitivity. Residual gas adds noise through both gas damping of the test masses at low frequencies and broadband phase noise caused by molecules traversing the laser beam [94]. We include these effects using the residual-gas model adopted for the ET design, with the vacuum pressure and gas composition chosen to represent the expected operating conditions of the proposed facility. The assumed partial pressures of residual H$_2$, N$_2$, H$_2$O, and O$_2$ in the beam tubes are $1 \times10^{-8}$ Pa, $2.5 \times10^{-10}$ Pa, $5 \times10^{-9}$ Pa, and $2.5 \times10^{-10}$ Pa, respectively. The corresponding partial pressures in the vacuum chambers are $2 \times10^{-8}$ Pa, $5 \times10^{-10}$ Pa, $1 \times10^{-8}$ Pa, and $5 \times10^{-10}$ Pa.

Quantum noise

The quantum-noise power spectral density (PSD) of a tuned interferometer without injected squeezed light can be expressed as [15]

where

is the squared strain standard quantum limit [95]. The first and second terms in brackets represent shot noise and quantum radiation-pressure noise, respectively. The optomechanical coupling factor, $\kappa(\omega)$, is given by

where $\omega_{\mathrm{c}}$ is the interferometer bandwidth, $P_{\mathrm{arm}}$ is the circulating arm power, and $\omega_0$ is the laser angular frequency. The total quantum noise reaches the SQL when $\kappa(\omega)=1$.

The interferometer response is characterised by two poles: the optomechanical pole, $\omega_{\mathrm{q}}$, and the coupled-cavity optical pole, $\omega_{\mathrm{c}}$. When these poles are not well separated in frequency, two filter cavities are required to achieve the frequency-dependent rotation of the squeezing angle. In this work, we restrict the implementation of frequency-dependent squeezing to a single filter cavity to simplify the optical configuration and reduce the degradation of squeezing due to optical losses in a second filter cavity. This approach is valid when

In this section, we analyse three interferometer configurations: broadband, high-bandwidth, and low-bandwidth. These are optimised for mid-frequency sensitivity, kilohertz detection, and low-frequency sensitivity at low optical power, respectively, as shown in Figure 14. For each configuration, we set $\omega_{\mathrm{q}} \sim2\pi\times10$ Hz and ensure that the condition in Eq. (11) is satisfied.

Broadband configuration

The broadband configuration of the detector optimises its sensitivity to black hole and neutron star mergers and chooses $\omega_{\mathrm{c}} \approx2\pi\times540$ Hz to balance this quantum noise with the sum of classical noises in the observatory’s most sensitive frequency band from 20 Hz up to a few hundred Hz. Similar to the LIGO and Virgo detectors, the largest classical noise source that determines the choice of the bandwidth, $\omega_{\mathrm{c}}$, is the coating Brownian noise [96]. The scaling of the coating thermal noise for a 15-km baseline detector relative to the LIGO-estimated value [17, 97, 98, 99] is given by

where $w=13.5$ cm is the beam size in the UI and $\phi_{\mathrm{H}}=5 \times10^{-5}$ is the required loss angle of the high-index-of-refraction coating material. The last two factors in the above equation suppress the coating noise by a factor of 5 compared to the Advanced LIGO level.

In the UI noise budget, we adopt target mechanical loss angles of $1.3 \times10^{-5}$ and $5 \times10^{-5}$ for the low- and high-index coating materials, respectively. The latter represents an approximately factor-of-two reduction in the high-index loss angle relative to the current A# baseline value [100, 101] but is more conservative than the A# optimistic coating noise [102, 103, 104]. For the adopted parameters, coating thermo-optic and substrate thermal noises remain below coating Brownian noise as shown in Figure 2.

Several candidate materials and coating architectures may offer routes toward this goal, albeit introducing additional technical challenges relating to material, growth method, and scalability. As an example, improvements of TiO$_2$:GeO$_2$ will likely involve more extensive heat treatment campaigns, which can exacerbate substrate surface figure changes and defect formation in the coating, leading to higher optical loss and scattering. The mixture TiO$_2$–SiO$_2$ is another promising high-index material for which post-deposition annealing is a critical step, with blistering and cracking having been observed after heat treatment [105]. Further pathways are amorphous semiconductor and multi-material coatings [106, 107, 108, 109, 110], whose thermal noise performance must be carefully balanced against their optical absorption. Crystalline coatings made from GaAs/AlGaAs can go lower than the requirement of UI [111, 112], although substantial technological scale-up would be required for the large optics considered here. More generally, coating loss may depend not only on material composition but also on structure across multiple length scales. Structural characterization paired with atomistic modelling has identified specific nm-sized motifs that contribute to the mechanical dissipation, providing key insights to design future coatings [113], while recent measurements of ion-beam-sputtered silica films suggest that elastic and structural anisotropy may represent another relevant factor in the mechanical dissipation of amorphous coatings [114].

kHz detection

In this section, we consider the detector configuration optimised for high-frequency operation, with particular emphasis on the sensitivity required to resolve the post-merger oscillations of neutron stars. These signals, which lie predominantly in the kilohertz frequency range, carry direct information about the dynamics and thermodynamics of nuclear matter and therefore provide a unique probe of the neutron-star equation of state. Achieving high sensitivity at these frequencies is consequently an important scientific goal for the next generation GW detectors.

We exploit the coupled-cavity resonance formed by the arm and signal-extraction cavities (SEC) to enhance sensitivity in the kHz range [115]. The mode-splitting frequency and the bandwidth of each mode can be approximately written as

A resolvable coupled-cavity resonance emerges when $\gamma_c < \omega_c$. The 8% signal-extraction-mirror transmissivity defines the broadband operating point. Reducing the transmissivity to 1% narrows the coupled-cavity response and enhances its peak sensitivity in the kilohertz range, providing improved sensitivity to neutron-star post-merger signals as shown in Figure 16.

We find that transitioning between the broadband and kilohertz-optimised configurations requires only the change in the signal-extraction-mirror transmissivity and the filter-cavity input-mirror transmissivity for incorporating frequency dependent squeezing. Note that the high-frequency sensitivity is unaffected by changing the filter-cavity input-mirror transmissivity, which influences only the low-frequency squeezing. This opens the possibility of making the interferometer dynamically tunable [116, 117] between different observing modes rather than requiring a dedicated optical configuration for each science target. One possible implementation is a signal-extraction-mirror with coatings on both sides, designed such that its effective transmissivity can be varied through controlled changes in the mirror-substrate temperature. Similar to the design of the input test masses in Virgo [118, 119], thermal tuning would provide an actuator for adjusting the mirror transmissivity without replacing the optic, allowing the coupled-cavity response to be moved between a broadband configuration and a narrower, higher-sensitivity response in the kilohertz range.

An important practical limitation on high-frequency sensitivity is optical loss in the signal-extraction cavity. Scattering, mode mismatch, thermally induced wavefront distortions, and higher-order-mode generation all contribute to the effective loss, reducing the signal response and degrading the benefits of squeezed light. In the sensitivity calculation, we assume a total round-trip signal-extraction-cavity loss of 500 ppm.

Low-power regime

In this section, we show that UI can maintain its low-frequency sensitivity in a low-power operating mode. While reducing the circulating arm power, the GW signal gain can be proportionally increased by adjusting the finesse of the signal-recycling cavity to preserve the quantum-noise level of the broadband configuration below the coupled-cavity pole frequency, $\omega_c$. However, reducing the arm power also lowers $\omega_c$, potentially bringing it closer to the optomechanical pole frequency, $\omega_q$. Therefore, (11) imposes a lower limit on the circulating arm power required to maintain UI sensitivity in the 3–30 Hz band.

In this configuration, we choose $\omega_c \approx2\pi\times50$ Hz, $\omega_q \approx2\pi\times11$ Hz to satisfy the single-filter-cavity condition. The corresponding circulating arm power is

with a required input power of 17.4 W and a signal-recycling mirror transmissivity of 60%.

This low-power configuration allows UI to approach the low-frequency infrastructure noise limit for a representative ET facility at Sos Enattos, while addressing quantum radiation-pressure noise using 440 kg test masses without detuning the signal-recycling cavity. It also mitigates the risk of technical low-frequency noise that may arise during high-power operation in the broadband configuration.

Sensitivity scaling with the arm length

Arm length is a key parameter in the system-level design of a gravitational-wave detector. For displacement noise with a physical amplitude spectral density that is independent of the arm length $L$, the corresponding strain noise scales as $L^{-1}$. For a fixed interferometer bandwidth, with the circulating arm power and test-mass mass held constant, the strain-referred quantum shot noise scales approximately as $L^{-1/2}$, whereas quantum radiation-pressure noise scales as $L^{-3/2}$. Coating thermal noise scales as $L^{-1}$ for a fixed beam size, or approximately as $L^{-3/2}$ if the arm-cavity $g$ factor is held constant.

In this section, we compare designs with arm lengths of 10 km, 15 km, 20 km, and 25 km. The circulating arm power and input test mass transmissivity are held fixed, while the signal extraction mirror transmissivity is adjusted for each arm length to maintain the same effective interferometer bandwidth as the 15 km reference design. We keep the beam radius, rather than the arm-cavity $g$ factor, fixed because further increasing the beam size is constrained by the finite test-mass aperture. The resulting sensitivities for the broadband configuration are

shown in Figure 15.

We find that increasing the arm length produces the most pronounced improvement at low frequencies, where quantum radiation-pressure noise is a dominant contribution in the few-hertz band. At higher frequencies, the scaling depends on the dominant noise source. For shorter arm lengths, where coating thermal noise limits the sensitivity, the strain noise improves approximately as $L^{-1}$ from 15 km up to 20 km. As the arm length increases and quantum shot noise becomes dominant, the improvement becomes weaker, scaling approximately as $L^{-1/2}$ for arm lengths above $\approx20$ km. The precise transition between these two scaling regimes depends on the coating loss angle and circulating arm power.

Comparison with the Cosmic Explorer design

The proposed detector shares several key design features with the Cosmic Explorer 40 km and 20 km concept [124], including room-temperature operation based on fused-silica technology and an optical configuration that retains strong sensitivity to neutron-star merger signals in the kilohertz band [115]. The underground environment of ET, however, provides a substantially lower environmental noise floor. In particular, reduced ground motion and Newtonian noise make it possible to extend the detector sensitivity to lower frequencies.

The enhanced sensitivity of the two crossed 15 km UI detectors for ET compared with a single 40 km Cosmic Explorer detector, as shown in Figure 16, can be understood, to first order, by considering the arm lengths and circulating optical powers. Similar to ET-HF, the proproposed UI design assumes 3 MW of circulating power in each arm, compared with a more conservative baseline of 1.5 MW adopted by Cosmic Explorer. The ultimate circulating optical power is expected to be limited by optical absorption in the mirror coatings, including spatially non-uniform absorption that can lead to thermal lensing and wavefront distortions [125, 126]. We note, however, that achieving higher circulating powers in longer interferometers may present additional technical challenges, for example due to the negative optical rigidity discussed in Sec. III B. The factor of 2.67 in arm length between a 40-km Cosmic Explorer and UI for ET does not lead to a linear scaling in sensitivity, but rather an improvement approximately proportional to $\sqrt{L}$, as discussed in the following section. This factor is also partially compensated, by a factor of approximately 1.3, by the use of two crossed UI detectors for ET. Increasing the arm length of UI for ET from the baseline 15 km to 20 km would further reduce the risk associated with the assumptions made regarding the arm power and coating thermal noise.

To quantify their performance in astrophysics, we compute the cumulative annual yield, $N(> \rho)$, of compact-binary signals with network signal-to-noise ratio (SNR) larger than $\rho$. We retain the population shapes and one-year realizations of the public BBH and BNS catalogs constructed for the ET design-comparison study [4], and calculate the SNRs with GWFAST [127], including the source-dependent antenna response and the rotation of the Earth. We adopt a low-frequency cutoff of 2 Hz and the catalog luminosity distances based on [128]. The ET curve represents the quadrature network SNR of two L-shaped interferometers with 15 km arms and a relative rotation of 45°; the CE 40 km broadband and CE 20 km high-frequency curves each represent a single L-shaped interferometer, using the corresponding CE design sensitivities [124]. The comparison is therefore between the specified observatory configurations, rather than between equal numbers of interferometers. We update the overall population normalizations to the GWTC-5.0 merger-rate constraints [123]; the shaded intervals in Figure 17 propagate only this rate uncertainty, while leaving the catalog mass, spin, and redshift distributions fixed.

For BBHs, the upper panel of Figure 17 shows that the crossed broadband configuration of UI for ET produces the largest cumulative yield throughout the plotted SNR range. At the reference cut $\rho\ge12$, used in the ET design-comparison study as the entry criterion for parameter-estimation forecasts, the central expectation is $1.06 \times10^{5}$ events per year, compared with $9.8 \times10^{4}$ for CE 40 km broadband and $5.1 \times10^{4}$ for CE 20 km high-frequency. The distinction is stronger in the scientifically valuable loud tail: the expected numbers above $\rho= 100$ are $2.1 \times10^{4}$, $9.1 \times10^{3}$, and $3.4 \times10^{2}$, respectively, while above $\rho= 200$ they are $2.0 \times10^{3}$, $1.3 \times10^{3}$, and approximately 48. The proposed room-temperature configuration therefore lies in the high-SNR regime identified as central to the ET science case [1, 4]. Such a large sample of loud BBH signals can support precise measurements of masses and spins, the resolution of subdominant multipoles, studies of formation channels and their evolution with redshift, and precision tests of the merger and ringdown in the strong-field regime. The cumulative distribution is particularly informative for these applications: once the total number of detections approaches the underlying annual population, further scientific gain is driven mainly by the number of events with very large SNR.

The BNS result in the bottom panel of Figure 17 gives a complementary test of both the low- and high-frequency performance. For the GWTC-5.0 central rate, the 15 km crossed-L configuration yields $4.4 \times10^{4}$ BNS signals per year above $\rho= 12$, compared with $3.4 \times10^{4}$ for CE 40 km broadband and $1.3 \times10^{3}$ for CE 20 km high-frequency. Using $\rho\ge30$, the “golden-event” benchmark adopted for BNSs in Ref. [4], the corresponding yields are $3.9 \times10^{3}$, $2.6 \times10^{3}$, and approximately 76; above $\rho= 100$ they are approximately 93, 63, and 2.4. Thousands of loud inspirals would provide a powerful ensemble for measuring tidal deformabilities and constraining the neutron-star equation of state, while the best-localized events could be combined with electromagnetic counterparts or galaxy information for multi-messenger studies and standard sirens [1, 4]. Sensitivity from a few hertz allows a BNS to be followed for hours, so the time-dependent antenna response generated by the Earth’s rotation can add localization information; the two crossed L-shaped interferometers additionally provide complementary polarization response. At the other end of the band, the proposed detector’s separate kilohertz-tuned operating point targets the tidal and post-merger signals that carry direct information about hot, dense matter [115, 129]. The cumulative-SNR curve shown here uses the broadband configuration and an inspiral–merger waveform without a post-merger signal, and therefore does not itself quantify that kilohertz science case.

For comparison, Figure 17 also shows the cumulative event counts for a configuration in which the two L-shaped interferometers are rotated by $30^\circ$ relative to one another. The predicted rates differ by less than 1% from those of the $45^\circ$ configuration. Previous studies have shown that the relative orientation of the interferometers can significantly affect their sensitivity to a stochastic gravitational-wave background [130, 131]. The $30^\circ$ configuration therefore represents a promising alternative to enhance stochastic-background science while preserving essentially the same compact-binary detection yield.

Conclusion

We have presented a 15 km room-temperature interferometer concept for the Einstein Telescope with the aim of exceeding its broadband sensitivity with lower complexity. The concept exploits the quiet underground environment of ET and proposes heavier test masses, low-loss and strongly dissipation-diluted suspensions, optimised quantum-noise reduction, and enhanced seismic isolation. The proposed configuration employs 440 kg fused-silica test masses, 4 m-long high-stress fused-silica suspension fibres, and a 5 km filter cavity to preserve low-loss frequency-dependent squeezing. The intermediate-frequency sensitivity is maintained through a large beam radius and low-loss coating materials, while the high-frequency performance retains the 3 MW circulating power of the ET-HF design.

The concept can achieve the low-frequency baseline sensitivity even with a 100 kW arm power and can also be optimised for high-frequency GW signals by tuning the transmission of the signal-extraction mirror. The coupled-cavity resonance of the signal-extraction cavity enhances sensitivity in the kilohertz regime and provides a promising route to improved measurements of neutron-star post-merger dynamics and the nuclear equation of state. The signal-extraction mirror can also be made tunable, allowing the detector response to be adjusted between low-frequency, broadband and kHz-optimised operating modes without changing the core interferometer.

We have also investigated the dependence of the achievable sensitivity on arm length. Increasing the arm length provides the largest improvement at low frequencies, where quantum radiation-pressure noise is important, while the gains at higher frequencies depend on the transition between coating thermal noise and quantum shot noise. For the parameters considered here, arm lengths of 15 km to 20 km mark an important regime in which further increases in length provide progressively smaller broadband sensitivity improvements.

Our results indicate that the broadband performance targeted by ET does not necessarily require the separation of low- and high-frequency optimization into two interferometers. A single room-temperature interferometer can instead redistribute the noise budget across the frequency band and exploit the advantages of the underground environment, while retaining the mature high-power technologies developed for current-generation detectors.

Acknowledgements

We acknowledge the members of the Belgian delegation, led by Prof. Giacomo Bruno, who visited the University of Birmingham in June 2026, for the valuable discussions that helped initiate this work. We acknowledge Jan Harms for reviewing this paper. We acknowledge Helios Vocca, Flavio Travasso, Piero Chessa, and Aniello Grado for useful discussions on the suspension thermal noise. We also acknowledge Massimo Granata, Silvia Corezzi, Nelson Christensen and the LIGO Suspension Working Group for useful discussions, and the support of the Institute for Gravitational Wave Astronomy at the University of Birmingham, STFC Consolidated Grant “Astrophysics at the University of Birmingham” (No. ST/S000305/1) and UKRI “The next-generation gravitational-wave observatory network” project (Grant No. ST/Y00423X/1). H.M. would like to acknowledge the support from the National Key R&D Program of China “Gravitational Wave Detection” (Grant No.: 2023YFC2205800), and support from Quantum Information Frontier Science Center. H.Y. is supported by the Natural Science Foundation of China (Grant 12573048). A.V. acknowledges the support of the Royal Society and Wolfson Foundation.

References