Lunar Gravitational-Wave Detection

Abstract

A new era of lunar exploration has begun bringing immense opportunities for science as well. It has been proposed to deploy a new generation of observatories on the lunar surface for deep studies of our Universe. This includes radio antennas, which would be protected on the far side of the Moon from terrestrial radio interference, and gravitational-wave (GW) detectors, which would profit from the extremely low level of seismic disturbances on the Moon. In recent years, novel concepts have been proposed for lunar GW detectors based on long-baseline laser interferometry or on compact sensors measuring the lunar surface vibrations caused by GWs. In this article, we review the concepts and science opportunities for such instruments on the Moon. In addition to promising breakthrough discoveries in astrophysics and cosmology, lunar GW detectors would also be formidable probes of the lunar internal structure and improve our understanding of the lunar geophysical environment.

1 Introduction

The Moon offers a unique geophysical environment for fundamental physics experiments as well as astrophysical observatories (Silk et al. 2023). The permanently shadowed regions (PSRs) at the lunar poles offer the coldest areas in our solar system (Williams et al. 2019), the far side especially during lunar nights might be among the most quiet radio locations in our solar system (Alexander et al. 1975), and seismic vibrations are expected to be several orders of magnitude smaller in a range of frequencies compared to Earth (Lognonné et al. 2009). This opens exciting opportunities for breakthrough observations with electromagnetic observatories and gravitational-wave (GW) detectors (Koopmans et al. 2021; Silk et al. 2021; Harms et al. 2021).

Lunar GW detection has been under consideration since Apollo 17 when the Lunar Surface Gravimeter developed under the coordination of Joseph Weber was deployed on the Moon to observe lunar surface vibrations generated by passing GWs (Giganti et al. 1977). The experiment did not reach its targeted performance, but even if it had, we know today that it would not have been sensitive enough to detect GW signals. Lunar vibrations caused by GWs are expected to be several orders of magnitude weaker than those that the instrument was designed to observe. The main motivation to take such an experiment to the Moon was that with the extremely low level of seismicity observed with previously deployed Apollo seismometers (Nakamura et al. 1981), Weber who had done similar measurements on Earth already (Forward et al. 1961) had reason to suspect that GW signals could be detected on the Moon.

A lunar GW detector would complement a future network of terrestrial and space-based GW detectors. Next-generation terrestrial detectors have been proposed including the Einstein Telescope in Europe (Steering 2020) and the Cosmic Explorer in the US (Evans et al. 2021). Construction of these research infrastructures is expected around 2035 with a targeted lifetime of at least 50 years. The space-borne detector LISA is an approved mission orbiting the Sun expected to be launched in the second half of the 2030s (Bayle et al. 2022). It has a nominal lifetime of four years, but with potential to be extended to a mission of 10 years. Such a network would open opportunities for multiband observations between GW detectors, where especially a decihertz detector on the Moon would make an enormous impact (Sesana 2016; Sedda et al. 2020; Grimm and Harms 2020). If technological hurdles concerning deployment and operation of GW detectors on the lunar surface can be overcome, the Moon might take a key role for the exploration of our Universe through GW observations.

2 Current and Future Facilities for GW Science

2.1 Current Terrestrial Laser-Interferometric GW Detectors

A global network of currently four long-baseline, laser-interferometric gravitational-wave (GW) detectors is operative consisting of Virgo (Acernese et al. 2014), the two LIGO detectors (Aasi et al. 2015), and KAGRA (Akutsu et al. 2019). A fifth detector is under construction in India (Souradeep 2016). Breakthrough observations have been made including the first detection of GWs by LIGO, GW150914 (van den Brand et al. 2016), and the multi-messenger campaign following the merger event GW170817 of two neutron stars (Abbott et al. 2017; LIGO Scientific Collaboration et al. 2017a). Our understanding of compact-binary populations has greatly improved over the past years (Abbott et al. 2021), and the astrophysical processes governing the emissions of electromagnetic signals of neutron-star mergers are much better understood (Abbott et al. 2017b; McCann et al. 2018).

The current detectors are undergoing regular technological upgrades. Recent changes include the implementation of frequency-dependent squeezing, increase of laser power, and the installation of seismic and acoustic monitoring systems for noise cancellation (Cahillane and Mansell 2022; Muciaccia et al. 2023). The laser-interferometric system of a GW detector is very complex and requires many months of commissioning time to benefit from the latest technological upgrades. Much of the effort goes into the isolation of a detector from its environment. The vacuum system, seismic isolation, stray-light control, and environmental monitoring are the most important measures to mitigate environmental disturbances (Zucker and Whitcomb 1996; Acernese et al. 2010; Fiori et al. 2020; Soni et al. 2020; Romero-Rodríguez et al. 2022). An increasingly important technology to mitigate the impact of environmental disturbances is active noise mitigation or noise cancellation (Matichard et al. 2015; Driggers et al. 2019; Harms 2019), where sensor data are being used to correct for the disturbance either via feedback or feedforward control, or by subtracting a noise model from the GW data in post-processing. Through technology upgrades and continuous commissioning of a detector, the facility performance approaches its infrastructural limitations, which makes it harder to improve its sensitivity. At some point, new detector facilities are required to enable new breakthrough science. At low frequencies, performance limitations are linked to environmental disturbances, which are orders of magnitude weaker on the Moon.

2.2 The Laser Interferometer Space Antenna - LISA

LISA is a space-borne gravitational wave detector and is a mission of the European Space Agency (ESA) with NASA as an international partner (Bayle et al. 2022; Thorpe et al. 2019; Amaro-Seoane et al. 2017).

LISA is a constellation of three satellites, 2.5 million km apart, that orbits the Sun in an Earth-like orbit. The constellation trails the Earth by about 20^∘ or 55 million km and is inclined by 60^∘ with respect to the ecliptic (Fig. 1). The shape and orientation of the constellation ensures that no station-keeping manoeuvres have to be undertaken during the lifetime of the mission, which is projected to be approximately 10 years (including a possible extension of the mission lifetime).

Fig. 1

Schematics of LISA orbits. The constellation is trailing the Earth by about 20^∘ and inclined against the ecliptic by 20^∘. This orbital arrangement allows the constellation to be stable without further station keeping manoeuvres

The absence of anthropogenic and other low-frequency noise sources present on Earth, such as seismic, tides, or weather systems in the chosen orbits enables LISA to be sensitive to a band of frequencies between 0.1 mHz and 1 Hz; see figure 2.

Fig. 2

Sensitivity of LISA as characteristic strain (figure from Amaro-Seoane et al. 2017)

The relatively long arm length (or inter-spacecraft distance) allows to employ a comparatively low displacement sensitivity of about 15 pm/$\sqrt{\text{Hz}}$ but comes at a price. The usual interferometric schemes, such as a Michelson interferometer employed by most terrestrial detectors, have to be modified for such long arm lengths. Firstly, the direct reflection of the laser at an end-mirror is not feasible. Sending the laser through a 30 cm telescope (the largest feasible size for this mission) will cause diffraction to widen the laser beam to many kilometers and only a fraction, about $10^{-9}$ of the emitted laser light, will be received by the other spacecraft. Direct reflection being impractical, LISA employs a transponder scheme, where a laser in the receiving spacecraft is phase-locked (with a suitably chosen frequency offset) to the incoming beam before sent out again (see Fig. 3).

Fig. 3

Measurement principle of LISA (figure from Bogenstahl et al. 2017)

Secondly, LISA uses free-falling test masses as fiducial end-points of the arms, instead of optical mirrors. Those test masses are cubes with a side of 46 mm and a mass of 1.96 kg, made out of a Au-Pt metallic mixture that provides reduced magnetic susceptibility to the test mass. The position of those test masses inside the spacecraft is measured interferometrically with respect to an optical bench that also supports the interferometric measurement of the inter-spacecraft distance (see Fig. 4).

Fig. 4

Tri-partition of the measurement in a LISA arm (figure from Bogenstahl et al. 2017)

The demonstration of free-fall for the test masses (by limiting stray forces on the test-masses) was one of the key contributions of the LISA Pathfinder technology demonstrator, in orbit from December 2014 to June 2017. The distance (≈ 38 cm) between the two test masses on LISA Pathfinder was measured interferometrically and the performance of the differential acceleration between the test masses was demonstrated to be smaller than 2 fm/s$^{2}/\sqrt{\text{Hz}}$ (Armano et al. 2016).

The science objectives of LISA are aligned with the low-frequency performance of the constellation. The science of the LISA mission is extremely broad and diverse and has been studied and described in a series of white papers (Littenberg et al. 2019; Caldwell et al. 2019; Colpi et al. 2019; Cutler et al. 2019; Baker et al. 2019; Natarajan et al. 2019; Berry et al. 2019; Cornish et al. 2019; McWilliams et al. 2019; Berti et al. 2019; Bellovary et al. 2019). The sources range from galactic binaries nearby in our own Galaxy (Littenberg et al. 2019) to sources all across the Universe: massive black hole mergers in the range from $10^{4}$ $M_{\odot}$to $10^{7}$ $M_{\odot}$(Colpi et al. 2019; Bellovary et al. 2019), and extreme-mass ratio inspirals (Berry et al. 2019), where neutron stars or stellar-size black holes fall into massive black holes at the center of galaxies, as well as possible observations of a stochastic cosmological background. The science ranges from the astrophysics of binary evolution (Sesana et al. 2020), galaxy formation and black holes to fundamental physics (Arun et al. 2022; Berti et al. 2019) and cosmology (Boileau et al. 2022, 2021; Corman et al. 2021; Caprini et al. 2020; Caldwell et al. 2019).

2.3 Einstein Telescope and Cosmic Explorer

Infrastructural limitations like site conditions and detector size set important constraints on the sensitivities that can be reached with a detector. With increasing lifetime of a detector facility, technological upgrades enabling new breakthrough observations become more complex and more difficult to implement. At some point, new infrastructure is needed to overcome these constraints. A new generation of terrestrial GW detectors has been proposed with the Einstein Telescope in Europe (Steering 2020) and the Cosmic Explorer in the USA (Evans et al. 2021). Figure 5 shows conceptual drawings of the two facilities.

Fig. 5

Artistic conceptions of next-generation terrestrial detector facilities. The Einstein Telescope (left) is a proposed underground facility in Europe to host future-generations of GW detectors with 10 km arm lengths in an equilateral triangle (image credit: Marco Kraan, Nikhef). The Cosmic Explorer (right) is the US proposal for two next-generation detectors located at the surface with 20 km and 40 km arm lengths (image credit: Angela Nguyen, Virginia Kitchen, Eddie Anaya, California State University Fullerton)

The Einstein Telescope is proposed as an underground detector in the shape of an equilateral triangle with 10 km arm length. Each vertex of the triangle is the center station of a pair of interferometers; one optimized for low-frequency observations (ET-LF) between about 3 Hz and 50 Hz, and one optimized for high-frequency observations (ET-HF) from about 50 Hz to a few kHz. The ET-LF interferometers will be operated at cryogenic temperatures and with lower power laser beams. The high-frequency interferometers will be operated with up to a few megawatts of power in the interferometer arms. While the underground environment will help to reduce the ambient seismic and atmospheric noise, ET-LF interferometers require additional mitigation of environmental noise to reach the sensitivity target (Badaracco and Harms 2019; Amann et al. 2020).

The reference design of the Cosmic Explorer foresees two surface detectors in the US; one with 40 km arm length, the other with 20 km arm length. The option to have the vertices and ends of the interferometers underground is under consideration. As surface detectors, the Cosmic Explorer will not match the low-frequency sensitivity of the Einstein Telescope, but its longer arms make it superior at higher frequencies. The low-frequency end of its observation band will ultimately be set by gravitational noise of the ambient fields and limitations in the ability to estimate this noise using environmental data. Assuming that the gravitational noise can be reduced to a level comparable to what is predicted for underground sites (Hall et al. 2021), the targeted observation band is approximately 8 Hz – few kHz.

Figure 6 shows the detection horizons of Einstein Telescope and Cosmic Explorer as a function of total source-frame mass of compact binaries. These would vastly exceed the horizons even of the most ambitious upgrades of current detector facilities like the proposed Voyager (Adhikari et al. 2020). Einstein Telescope would detect almost all of the $10^{5}$ binary black-hole mergers expected to occur per year in our Hubble volume and see a fraction of about 10% – 20% of the $10^{6}$ expected binary neutron-star mergers (Maggiore et al. 2020). Such detection statistics will enable breakthrough science and lead to exciting possibilities also for multi-messenger astronomy with electromagnetic and neutrino counterparts of GW sources. High impact is expected especially in cosmology and astrophysics, and paradigm shifting discoveries like the first signs of quantum gravity, detection of a primordial black-hole population, or the observation of a primordial stochastic GW background are conceivable (Agullo et al. 2021; Ng et al. 2022; Sharma and Harms 2020).

Fig. 6

Detection horizons as a function of total source-frame mass of the compact binary. Dashed lines indicate detection horizons when higher-order modes of the GW signal are included. Terrestrial GW detectors have their best detection capabilities in the mass range 10 – 100 $M_{\odot}$. The plot is taken from (Hall 2022)

2.4 Some Examples of Electromagnetic Observatories Possibly Operating in the Coming Decades

Multi-messenger observations are the new frontier to unveil the most energetic processes associated with white dwarfs (WD), neutron stars, black hole collisions, and stellar explosions. Over the next decades more sensitive and complementary observations, ranging from radio to the very high-energy gamma rays and gravitational waves, will aim at 1) investigating the Universe along its cosmic history up to the dark ages to address open questions about the nature of dark matter and dark energy, 2) probing the nature of matter under extreme conditions, 3) unveiling the physics governing energetic transients, and 4) testing new fundamental physics through extreme astrophysical phenomena. This section gives an overview of some of the major electromagnetic observatories which are expected to play a key role in the future exploration of the Universe in synergy with gravitational-wave detectors.

The Vera C. Rubin Observatory, which is expected to start operations in 2024, will revolutionize time-domain astrophysics. It will perform the so called Legacy Survey of Space and Time (LSST), an unprecedentedly sensitive, ten-years long survey of the optical sky. LSST will observe over 10 million transients every night. The 8.4 m Simonyi Survey Telescope at the Rubin Observatory and its 3200-megapixel LSST Camera are designed to provide an exceptionally wide field of view (9.6 sq.degrees) with a better than 0.2 arcsecond sampling. The telescope’s compact configuration will allow it to rapidly slew across the sky, making LSST capable of surveying the entire southern sky every three nights in six optical bands, ranging from 320 to 1050 nm. The camera will feature solid state detectors with over three billion pixels, providing extremely high quality data. The Rubin observatory will offer target-of-opportunity (ToO) capabilities which enable the early discovery and characterization of the optical counterpart of gravitational-wave sources. For binary-neutron star (BNS) and neutron star/black hole (NSBH) mergers it will detect both the kilonovae and Gamma-Ray Burst (GRB) afterglows (Andreoni et al. 2022). Furthermore, it is expected to detect about 1000 well-sampled tidal disruption events (TDEs) per year. The LSST will observe hundreds of thousands of well-measured Type 1a supernovae (SNe 1a), replicating the current generation of SN 1a cosmology experiments several hundred times over different directions and regions across the sky (Hambleton et al. 2022).

A new generation of 30 – 40 m class telescopes, such as the Thirty Meter Telescope (TMT, Fenchuang 2022), the Giant Magellan Telescope (GMT, Fanson et al. 2022), and the Extremely Large Telescope (ELT, Tamai et al. 2022) will provide unprecedented tools to characterize transients, unveiling the physics of stellar explosions and extreme objects such as black holes. ELT will deliver spectroscopic and high-resolution imaging data at sensitivities never before reached. It will allow us to follow up and characterize the optical and near infrared emission of gravitational-wave sources giving detailed information on the spectral properties and temporal evolution also for the fainter and more distant sources. It will unveil properties of the host galaxies and the local environment of the source. Furthermore, characterizing a sufficiently large sample of SNe 1a detected by LSST, ELT will give insights into the nature of the progenitors of thermonuclear explosions. Despite their crucial role in modern cosmology, it is still unclear whether the explosion is triggered by the merger of a binary system of WDs or is rather due to accretion of material from a non-degenerate companion onto the WD. Supernovae 1a are standard candles providing precise measurements of distances and thus measurements of the expansion rate of the Universe. The inspiral and mergers of binary WDs are expected to be detectable through gravitational waves in the decihertz band, providing another complementary measurement of the source distance. Thus, multi-messenger observations of these events are vital to both understand the progenitors of SN 1a and to solve the tension on the Hubble constant estimate (see e.g., Maselli et al. 2020).

Wide field-of-view (FoV) $\gamma$-ray and X-ray satellites are key instruments operating in synergy with GW detectors and making it possible to detect the short GRB prompt and afterglow emissions associated with BNS mergers (see e.g. Ronchini et al. 2022). Instruments such as the Soft X-ray Imager (SXI, 0.3 – 5 keV, FoV of 0.5 sr) on board of the mission concept THESEUS (Amati et al. 2021) are also important to probe the thermonuclear explosions of massive WDs associated with SN Ia. SXI is expected to detect the thermonuclear shock breakout, the interaction between the SN and the matter bound in the system, and the interaction between the SN ejecta and the circum-binary environment. THESEUS will also sample the soft X-ray light curves of more than 50 TDEs per year. The simultaneous observation of the hard X-ray by the X/Gamma-rays Imaging Spectrometer (XGIS, 2 keV – 10 MeV), and the infrared by the InfraRed Telescope (IRT) can provide information on the geometry and jet orientation (Mereghetti et al. 2021; Report 2021). A more sensitive satellite such as Athena plays a key role in the characterization of high-energy transients enabling the observations of BNS mergers and GRBs afterglows also at late times and when they are seen by an observer not aligned to the jet-axis. Athena is expected to constrain the jet geometry and orientation, to help understanding the rate of BNS which are not able to produce a jet (choked jets), and to probe the fundamental physics of particle shock-acceleration. It is able to detect X-ray emission from the sub-relativistic jet giving rise to the kilonova emission. Athena is also expected to detect the possible high-energy emission from massive BH coalescences, to study the nature of stellar disruption and subsequent accretion onto supermassive black holes during TDEs, and to detect the X-ray emission from accreting WD binaries (Piro et al. 2022).

The Cherenkov Telescope Array (CTA) is the next generation ground-based observatory for detecting very-high energy gamma-rays (Acharya et al. 2013). It consists of an array of more than 100 telescopes located both in the northern and southern hemispheres. CTA is expected to bring dramatic advances in our understanding of the origins and production of non-thermal particles in the Universe thanks to its unique capabilities; it will be 10 times more sensitive than any existing instrument, it will feature a wide energy range (from 20 GeV to 300 TeV), an excellent angular resolution approaching one arcminute, a large field of view of several tens of square degrees, and a rapid slewing time suitable to swiftly catch the early emission from transients such as gamma-ray bursts.

Going to the longer wavelengths, the Square Kilometre Array (SKA) aims at exploring transient radio signals with unprecedented sensitivity, giving access to the physics of extreme phenomena such as exploding stars, compact object mergers, and ultra-relativistic jets from accreting black holes. In its first years of operation, it is expected to detect TDEs and Fast Radio Bursts (FRBs) at rates ranging from 1 to 1000 per week (Fender et al. 2015). The SKA will significantly increase our ability to detect GRB afterglows by following-up several hundred of GRBs in the high frequency bands. It will also be able to routinely detect “orphan afterglow” emission, from the population of GRBs whose relativistic jets are not pointed towards the Earth (Burlon et al. 2015).

2.5 European Space Agency Lunar Exploration Roadmap

In this section, we present the lunar exploration opportunities offered by ESA, the research payloads and the technology demonstrators currently planned for a flight to the Moon, and the available means to mature the design and the technology of a future gravitational-wave detector.

2.5.1 The Terrae Novae Programme and the Underlying Process

The Terrae Novae programme secures Europe’s role in space exploration. Managed by the ESA Human and Robotic Exploration Directorate (HRE), it is a comprehensive programme of human and robotic exploration missions and activities for Low Earth Orbit (LEO), the Moon and Mars. The programme cornerstones are

• Sustained presence in LEO: Extend the International Space Station (ISS) operations with new infrastructure allowing enhanced science in LEO, at the same time promoting commercial exploration services, which will allow access to LEO in the post-ISS era.

• Exploring farther: Continue delivering modules for sending ESA and NASA astronauts to the Moon. This includes ESA contributions (I-HAB and ESPRIT modules) to the first research station orbiting around the Moon, the Lunar Gateway.

• Moon robotic exploration: Fly research and technology demonstrators to the Moon surface in missions of opportunity with international partners and start the construction of the European Argonaut (formerly known as EL3). ESA will land multiple scientific payloads to probe the lunar environment, study its interior and prepare for its sustainable exploration.

• Mars robotic exploration: Continue gathering data from the Trace Gas Orbiter; start the construction of a European lander to take the Rosalind Franklin rover to the surface of Mars; cooperate with NASA on Mars Sample Return to return to Earth physical samples from another planet.

The Terrae Novae programme relies on two pillars, the Science in Space Environment (SciSpacE) programme and the Exploration Preparation Research and Technology (ExPeRT) programme. SciSpacE defines the science and the applications to be developed using a variety of ground and space platforms. On the other hand, the ExPeRT programme integrates, coordinates and manages the development of studies and technologies for future exploration missions to LEO, Moon and Mars destinations. The SciSpacE programme has three main research areas:

• Physical sciences;

• Life sciences;

• Moon and Mars sciences.

The research topics to be developed in LEO, beyond LEO (Lunar Gateway) and in robotic exploration missions around and on the Moon and Mars are defined in a bottom-up approach. Over 300 scientists were involved in a wide consultation process of the scientific community, which produced 16 white papers reflecting the interests and recommendations on focus areas across a broad range of disciplines that could be addressed within the capabilities of the Terrae Novae programme. The White Papers are streamlined to identify priorities that will drive future Announcements of Opportunities (AOs) for missions. This process is currently ongoing with the involvement of the scientific community and ESA advisory bodies (HESAC, PSWG and LSWG).

2.5.2 Science on the Moon

ESA is currently flying research payloads and technology demonstrators to the lunar surface on missions of opportunity with international and commercial partners. Plans until 2025 are defined and summarized below:

• The Exospheric Mass Spectrometer will land with NASA-CLPS (2022) on the Moon surface. The ion trap mass spectrometer will be used to detect lunar volatiles.

• Landcam-X technology will be installed on board a commercial lander (2023). The camera will be tested to improve precision and safety of future lunar landings.

• The MoonLIGHT laser retroreflector with its automatic pointing mechanism will be launched with NASA-CLPS (2024). The corner cube reflector will be used to perform Lunar Laser Ranging (LLR) experiments from the Earth. The MoonLIGHT ranging measurements will contribute to improve general relativity tests and will be used to study the Moon interior.

• An energetic neutral atoms and ions detector will be placed on the lunar surface with CNSA (2024).

• An Exospheric Mass Spectrometer will also be installed on-board the LUPEX rover in collaboration with JAXA (2024/25).

• Lunar Pathfinder will test commercial communication and navigation services around the Moon (2025).

• PROSPECT will reach the lunar surface with NASA-CLPS (2025). From the PROSPECT payload, ProSEED will drill beneath the surface in the South Pole region of the Moon and extract samples that will be then analyzed in the ProSPA chemical laboratory.

In a mission of opportunity, the availability of resources is negotiated among the international partners. Typical figures allocated to the payload for mass, volume and power consumption are 5 kg, 5 dm³, and 10 W, respectively. Single integrated units are typically preferred and easier to accommodate. Meanwhile, the construction of Argonaut has been approved. With a total mass of delivered cargo of 1500 kg, Argonaut will be compatible with different mission types: cargo, science rover, sample return stage, scientific/technology demonstration packages, in-situ resources, production equipment, power generation equipment, etc.

Future missions in lunar orbit or at the Moon surface will be defined based on the scientific priorities identified for the program and will be selected from the ongoing and future AOs. The following investigations have been proposed in the Fundamental Physics and Astrophysics White Papers:¹

• Einstein’s Equivalence Principle tests through lunar laser ranging experiments to the Moon.

• Tests of de-coherence models in quantum mechanics with entangled photons distributed on the Earth-to-Moon baseline.

• Studies of the origins of primordial fluctuations and primordial black holes using radio interferometry and gravitational wave detectors.

• Studies of the cosmic-ray distribution in the solar system through measurements of fluxes and spectra on board space probes.

• Surface magnetic fields and long-wave radio emissions from the corona using radio telescopes.

• Plasma physics studies relying on small plasma packages, UV and ENA imagers, and transponders for radio science experiments.

Gravitational-wave physics and astronomy are topics of interest both for the Fundamental Physics and the Astrophysics White Papers.

2.5.3 A Path to a Gravitational-Wave Project on the Moon

ESA can offer several means to incubate study ideas, elaborate preliminary mission concepts and advance mission-critical technology to an adequate technology readiness level (TRL).

As first, Topical Teams are working groups that include experts from European universities, research entities, and industries focusing on selected research topics in the areas of interest for the HRE programme. Within a Topical Team contract, ESA provides support for meetings and workshops. A Topical Team can be used to incubate ideas and develop them toward a mission concept.

At the next level, a Concurrent Design Facility (CDF) study might help defining mission design aspects. The ESA CDF infrastructure gathers teams of scientists and engineers from different disciplines and with different expertise to work together, in close coordination, with the purpose of completing complex designs.

Both a Topical Team and a CDF study could be considered to advance the mission concept of a Lunar GW antenna and the pathfinder missions that might be required for site survey. This preliminary study phase could help the team to consolidate the mission concept and prepare for future ESA Announcement of Opportunities.

3 Lunar GW Detection

In the following sections, we present the three main concepts proposed for lunar GW detectors, i.e., (1) the inertial measurement of lunar vibrations produced by GWs (Sect. 3.2), (2) the long-baseline, laser-interferometric measurement of seismic strain produced by GWs (Sect. 3.3), and the long-baseline, laser-interferometric measurement of GW strain with suspended test masses (Sect. 3.4). We include discussions about the lunar GW response (Sect. 3.1) and the cleaning of data from the seismic background (Sect. 3.5).

3.1 Lunar GW Response

Weber suggested that GWs could be detected by observing the induced vibrations of the elastic bodies (Weber 1960). Later, formalisms were developed to apply the theory to planetary bodies, e.g., through generation of seismic waves (Dyson 1969) or by excitation of quadrupolar normal modes (Ben-Menahem 1983). A planet’s GW response in terms of normal modes depends on radial inhomogeneities of the shear modulus. In the case of the Preliminary Reference Earth Model (PREM) (Dziewonski and Anderson 1981) there are two major shear-modulus discontinuities: at the surface and at the core-mantle boundary. A similar structure is predicted by the Weber lunar model (Weber et al. 2011). Comparing the two models, the surface value of the Earth’s radial shear modulus (270 GPa) on basalt surface bed-rock of PREM is two orders of magnitude larger than the lunar surface regolith in the first upper km (0.65 GPa). The lunar GW response at lower frequencies is not significantly affected by the surface layer, while at high enough frequencies, the low shear modulus means that the GW response will be reduced compared to Earth. From preliminary models (Garcia et al. 2019), it can be expected that the transition happens somewhere in the decihertz band with possible resonant effects in the lunar GW response.

To calculate the lunar response using the normal mode theory we use a non-rotating 1D lunar model (Weber et al. 2011) and a Green tensor formalism (Ben-Menahem 1983; Majstorović et al. 2019). The induced GW response is written as

$$ s(\mathbf {r},t)=\int _{-\infty}^{t}\int _{V} \mathbf {G}(\mathbf {r}, \mathbf {r}^{\prime }; t-t^{\prime}) \cdot \mathbf {f}(\mathbf {r}^{\prime},t^{ \prime})dV^{\prime }dt^{\prime }$$

(1)

where in transverse-traceless gauge the force term is defined as the gradient of the shear modulus (Dyson 1969)

$$ \mathbf {f}(\mathbf {r},t)=-\frac{\partial \mu}{\partial r}\mathbf {e}_{r} \cdot \mathbf {h}(\mathbf {r},t) $$

(2)

and $\mathbf {h}(\mathbf {r}, t)$ is the metric perturbation of the space-time.

Considering the GW source as an infinite monochromatic wave with pulsation $\omega _{g}$, $\mathbf {h}(\mathbf {r},t)=\Re \left \{h_{0} \boldsymbol{\epsilon} e^{i( \omega _{g}t-\mathbf {k}_{g}\cdot \mathbf {r})} \right \}$ in the long-wavelength regime, we neglect the ($\mathbf {k}_{g}\cdot \mathbf {r}$) term.

The Green tensor depends on the eigenfunctions computed for the lunar model. After some computation, the induced response for a given mode $k$ (degree $l$ and order $m$) to an infinite monochromatic GW is written (Majstorović et al. 2019) as

$$ \mathbf {s}_{k}(\mathbf {r},t)=h_{0}\mathbf {s}_{k}(\mathbf {r})\bar{g}(t) \delta _{l,2}f^{m}(e,\lambda ,\nu )\zeta _{k}, $$

(3)

where the associated displacement eigenfunctions are

$$\begin{aligned} \mathbf {s}_{k}(\mathbf {r}) =\hat{\mathbf {e}}_{r} U_{k}(r) Y_{lm}(\theta , \phi )+\hat{\mathbf {e}}_{\theta}\left (\kappa ^{-1}V_{k}(r)\partial _{ \theta }P_{lm}(\cos \theta )\right ) \\ +\hat{\mathbf {e}}_{\phi}\left (\kappa ^{-1}V_{k}(r) \frac{1}{\sin \theta}P_{lm}(\cos \theta )\partial _{\phi }e^{i m\phi} \right ) \end{aligned}$$

(4)

with $Y_{lm}$ the real spherical harmonic functions and $P_{lm}$ the associate Legendre polynomials of degree $l$ and order $m$. $\theta$ and $\phi$ are respectively the colatitude and longitude of a point at the Moon surface.

The source-time function defines the resonance with the seismic mode of eigenfrequency $\omega _{k}$

$$ \bar{g}(t)=\frac{1}{2\pi}(i\nu )^{-1} \frac{1}{\gamma _{k}+i(\omega _{g}-\omega _{k})}e^{i\omega _{g} t} $$

(5)

and the pattern function $f^{m}(e,\lambda ,\nu )$ depends on the polarization and incidence angles of the GW. Finally, the function $\zeta _{k}$ depends on the radial interior model and is written

$$ \begin{aligned} \zeta _{k}&=\mu (a)a^{2}\left (U_{k}(a)+ \frac{3}{\sqrt{6}}V_{k}(a) \right ) \\ &\quad -\int _{r} \frac{\partial \mu}{\partial r}\left (U_{k}(r)+ \frac{3}{\sqrt{6}}V_{k}(r)\right )r^{2} {\mathrm{d}}r, \end{aligned} $$

(6)

where $a$ is the Moon mean radius, $U_{k}$ and $V_{k}$ are the mode eigenfunctions for the radial and tangential displacements, and $\gamma _{k}$ is the decay rate of the eigenmode.

Similar theoretical computation was done for the Earth by (Coughlin and Harms 2014a,b,c) later extended to a laterally heterogeneous Earth’s model considering normal mode coupling by (Majstorović et al. 2019).

From equation (1) we see that the lunar induced response depends on the metric perturbation, the GW frequency, GW incidence angles and the 1D lunar model (from which we derive normal mode displacement eigenfunctions and eigenfrequencies). By considering the resonance regime ($\omega _{g} = \omega _{k}$) we can calculate the lunar response shown in Fig. 7 for a seismometer placed at a distance of about 170 km from the lunar pole. The radial response is strongly suppressed since the GW propagation direction is parallel to the Moon’s polar axis.

Fig. 7

Lunar GW response calculated by summing 229 quadruple normal modes and for a sensor location close to the lunar north pole ($\theta =0.1$, $\phi = 0.1$). The GW propagates parallel to the lunar rotation axis. The response is calculated from a 12 h long time series sampled at 0.5 s. As expected, the GW response $s_{r}$ along the radial direction is suppressed near the poles

Numerical simulations implementing topography and regional geological models will be necessary for a more accurate GW response model in the decihertz band. For example, such a model must be able to reproduce the observed effects on moonquake waveforms, from which very high Q-factors around a few 1000 of the ground medium were inferred (Garcia et al. 2019).

3.2 The Lunar Gravitational-Wave Antenna (LGWA)

The first upper limits on a GW power spectrum published in 1961 were obtained in the range 0.3 mHz – 4.5 mHz by measuring the deformation of the Earth produced by GWs with a Benioff strain seismograph (Forward et al. 1961). This experiment was carried out by a group of physicists from University of Maryland (Forward, Zipoy, Weber) together with seismologists from Caltech (Smith, Benioff). To overcome the sensitivity limitations set by the terrestrial background of seismic and atmospheric noise, Weber went on to develop high-precision laboratory-scale experiments known as resonant-bar antennas operating in the kHz range. He also proposed the development of a seismometer to observe the normal modes of the Moon excited by GWs, which was deployed in 1972 by the crew of Apollo 17 (Giganti et al. 1977). The Lunar Surface Gravimeter, as it was called, did not reach the expected performance due to a design error, and today we know that even a working instrument would not have reached the sensitivity to observe GWs.

Nonetheless, it was well known from other seismic experiments during the Apollo era that the seismic background on the Moon is several orders of magnitude lower than on Earth in the range 0.1 Hz to a few Hz (Coughlin and Harms 2014c). The instrument noise of the seismometers limited these searches, and models of the seismic background formed by moonquakes and meteoroid impacts predict that several orders of seismometer sensitivity improvements are possible before meeting limitations from a continuous seismic background (Lognonné et al. 2009; Harms 2022).

The concept for an array of high-precision seismic sensors suitable for GW detection on the Moon was proposed under the name LGWA (Harms et al. 2021). It consists of four seismic stations deployed in a PSR. Starting from an ambient temperature of less than 40 K, as can be found in some permanently shadowed regions (PSRs) at the lunar north and south poles, a sorption cooler will bring the payload to 4 K (Wu et al. 2017) so that niobium, a superconducting material with high mechanical quality (Bilenko et al. 2002), can be used for the proof mass and actuators (van Heijningen et al. 2022). The use of superconductors for the proof mass and for the coil actuators will reduce damping and therefore thermal noise in the system.

The LGWA sensitivity model is shown in Fig. 8 (light gray) together with the sensitivity model of LGWA Soundcheck (dark gray). LGWA Soundcheck was proposed as an explorer of the seismic environment inside a PSR and as a technology demonstrator. The spectra of a few example signals are displayed assuming an observation time of 2 months for each signal. The distances of the signals are chosen so that the signal-to-noise ratio (SNR) of observations with LGWA Soundcheck is about 9. The aim is to achieve a displacement sensitivity of $10^{-12}$ m/Hz$^{1/2}$ from 0.1 Hz to 1 Hz with LGWA Soundcheck, and $10^{-13}$ m/Hz$^{1/2}$ at 0.1 Hz and $2\times 10^{-15}$ m/Hz$^{1/2}$ at 1 Hz with the LGWA array. Below 0.1 Hz, the LGWA sensitivity is fundamentally limited by thermal noise of the proof-mass suspension, while the seismic background is expected to limit the sensitivity above 0.1 Hz together with the noise of the laser-interferometric readout of the proof-mass displacement (van Heijningen et al. 2023). In Sect. 3.5, a technique to reduce the noise from the seismic background is described, which makes use of an optimal, frequency-dependent combination of data from the four seismometers. A careful analysis of achievable noise-reduction factors still needs to be done based on a geological model of the deployment site.

Fig. 8

Characteristic strain noise of LGWA (light gray) and its precursor mission LGWA Soundcheck (dark gray) compared with a few examples of signal spectra. Here, an observation time of 2 months is assumed for all signals. The amplitude modulation of the GW signals due to the rotation of the Moon is visible for signals with slow frequency evolution. The distance of each signal is chosen to yield an SNR of about 9 in LGWA Soundcheck. The plot was produced with GWFish (Dupletsa et al. 2022)

3.3 The Lunar Seismic and Gravitational Antenna (LSGA)

The LSGA project aims for a measurement of strain deformations of the Moon produced by GWs and geophysical signals (Katsanevas et al. 2020). A laser interferometer whose optics are mounted on the lunar surface is proposed to observe GW signals. A more detailed analysis of the geophysical signals is proposed to be achieved with a network of optical fibers (Hilweg et al. 2022). The mounts of the laser optics can be adapted from laser retroreflectors, which are being developed independently for a CLPS mission launch (Intuitive Machines n. 3, or IM-3) to the Reiner Gamma site (59W, 7N) in 2024 (dubbed CP11), under a NASA-ESA MoU.

Laser retroreflector system description. It consists of three main subsystems: (1) the bare 10 cm diameter, high-quality optical retroreflector MoonLIGHT (Moon Laser Instrumentation for General relativity/Geophysics High accuracy tests); (2) MPAc (MoonLIGHT Pointing Actuators, a customized dual gimbal) during system deployment; (3) ADC (Actuated Dust Cover) to protect MoonLIGHT’s optical face from the accumulation of regolith dust. This overall retroreflector system will support Lunar Surface Laser Ranging with a laser deployed on the Moon, as opposed to the standard LLR (Lunar Laser Ranging) up until now performed by ground laser stations located on the Earth and targeting the existing Apollo 11, 14, 15 and Lunokhod 1, 2 reflectors. MoonLIGHT-MPAc-ADC will also support standard LLR for the test of General Relativity including measurements of possible time variations of the gravitational constant, test of the Weak and Strong Equivalence Principle, inverse-square-law deviations, measurements of the beta parameter of the parameterized post-Newtonian formalism (through the Nordtvedt parameter), and geophysics (composition of deep lunar interior) on the 2024 CP11 CLPS mission. The accuracy is ten fold increased with respect to the Apollo/Lunokhod reflectors. Delivery of the first MoonLIGHT-MPAc-ADC proto flight model (PFM1) is currently foreseen by the end of 2023. In parallel and following the PFM1 delivery, a second PFM will be built by ESA and INFN for new opportunities and studies for new applications, like GW detection on the Moon (like this proposal), a Geophysical package for ESA’s Argonaut, Lunar Geophysical Network (LGN) activities. The current CP11 reflector system is required to have an accuracy of equal or less than 3 degrees, although its design can be customized to be capable of 1 degree.

Potential sources of uncertainty and mitigation strategies for the reflector. The heritage that is being acquired through the 2024 CP11 mission (delivery of the PFM1 to IM-3 foreseen in August 2023) represents a significant risk reduction in reflector performance. The availability of a PFM2 is a further risk reduction. All three subsystems have margins for improvement beyond the CP11 baseline. The pointing accuracy can be improved if needed (equal or less than 3 degrees is enough for CP11). The current optical accuracy of MoonLIGHT is: wave/10 in total retro-reflected flatness, and 0.2 arcseconds in the dihedral angle offsets. Improvements for GW detection are possible with extra funding (like for example the flatness of the three back reflecting surfaces and the flatness of the total retro-reflected laser wavefront).

Given its direct inheritance from Apollo 11, 14, and 15, fully exposed optics that work on the surface of the Moon for more than 55 years on the Moon, MoonLIGHT and its basic mounting components/design (Suprasil 1 and/or 311, Aluminum and KEL-F plastic) will resist lunar surface radiation for decades. The same applies to the thermal excursion heritage: Apollo reflectors, which are not too far from the equator, are expected to experience a thermal range of about 300 K based on in situ Apollo temperature measurements and detailed orbit thermal measurements (by DIVINER on LRO). The only Apollo optical performance variation observed in 50 years of LLR observations was a slow reduction of their laser return intensity (due to slow regolith dust deposition over the reflectors), which according to LLR station measurements and associated studies is estimated to be about a factor of 10 in 50 years. But this effect has been compensated by laser station improvements over the same time. If coping with dust accumulations will be needed given the specific deployment scheme (for example a rover) the ADC will have to be considered.

3.4 The Gravitational-Wave Lunar Observatory for Cosmology (GLOC)

Current GW detectors like LIGO and Virgo implement sophisticated isolation and active mitigation techniques to reduce the impact of the environment on the GW measurement (Matichard et al. 2015,?; Fiori et al. 2020). Proposed next-generation detectors like Einstein Telescope and Cosmic Explorer will push the observation band to lower frequencies, which will increase the role of environment disturbances (Amann et al. 2020; Hall et al. 2021). The coupling between environment and detector happens through many different mechanisms. For example, it can be mechanical coupling through the suspension systems, stray light entering the main beam path, direct gravitational coupling to acoustic and seismic fields, magnetic fluctuations, etc. Environmental disturbances also affect the operation of the laser interferometer, where a complex control system must align the suspended optics to bring the interferometer to its most sensitive state and keep it there (Allocca et al. 2020). The Gravitational-wave Lunar Observatory for Cosmology (GLOC) was proposed as a triangular, laser-interferometric concept with suspended optics (Jani and Loeb 2021). It will profit from the quiet lunar environment to simplify the interferometer control and isolation systems, and to exploit the natural high-quality vacuum at the lunar surface. In this way, it might be possible to overcome low-frequency sensitivity limitations known from terrestrial detectors and extend the observation band to decihertz frequencies.

The strain sensitivity goal of $10^{-23}$ Hz$^{-1/2}$ at 1 Hz will nonetheless require seismic isolation. Assuming a baseline of 10 km, the differential displacement of the two suspended test masses of an interferometer arm will have to be less than $10^{-19}$ m/Hz$^{1/2}$ at 1 Hz, which requires about 5 orders of magnitude attenuation with respect to ground motion (Harms 2022). This can in principle be achieved with a staged suspension systems coupled with ultra-sensitive seismometers or strainmeters. However, there is no known material and proposed suspension technology that would produce low enough suspension thermal noise to reach the sensitivity goal at 1 Hz (Harms et al. 2013; Harms and Mow-Lowry 2017). Furthermore, while the interferometer control of GLOC would likely be significantly simpler than what is needed for terrestrial detectors, there are intrinsic effects that need to be counteracted like radiation pressure and strong enhancement of vibrations at the suspension resonances. Such controls are known to inject considerable noise into the system at low frequencies (Andric and Harms 2021).

A possible advantage of GLOC over space-based concepts is that the lifetime of space-based detectors is limited by orbital dynamics eventually making it impossible to maintain a satellite configuration with established laser links. There is no known hard limit to the lifetime of GLOC. The expected science return of GLOC would be much more impactful than of LGWA and LSGA and be close to previously proposed space-based detector concepts like Big Bang Observer (Phinney et al. 2003) and DECIGO (Kawamura et al. 2021). GLOC would be able to observe most types of GW sources from a wide range of compact object masses out to high redshift. It would have an immense impact on cosmology and observe astrophysical sources with high signal-to-noise ratio to enable high-precision tests of general relativity and precise measurements of the properties of compact stellar objects.

3.5 Seismic Background-Reduction Techniques

Even though the lunar environment has an extremely low level of seismic noise, a significant seismic background might still be present in lunar GW measurements in particular due to meteorite impacts. The LGWA seismic array will be exploited to estimate and subtract the seismic background to meet the LGWA sensitivity requirements. Similar techniques might also find applications in GW concepts like LSGA and GLOC.

With this goal in mind, we can make use of the experience gained in terrestrial GW interferometers for what concerns the cancellation of a gravitational-noise background produced by density variations in the environment of a detector associated with seismic and atmospheric fields (Harms 2019; Trozzo and Badaracco 2022). It is predicted to affect the sensitivity of terrestrial GW detectors in their low-frequency band (2–30 Hz), and it is predicted to be important even for future underground GW detectors like the Einstein Telescope, where the gravitational noise is anticipated to be significantly reduced (Badaracco and Harms 2019; Amann et al. 2020). Gravitational noise is widely believed to establish a practical lower frequency limit for terrestrial GW detectors (Harms et al. 2013).

Coherent noise cancellation is an effective way to clean GW data from environmental noise (e.g., see Driggers et al. 2012, 2019) and was proposed to overcome sensitivity limits in future GW detectors (Cella 2000; Driggers et al. 2012). Wiener filters are a possible choice for the optimal processing of environmental data for noise cancellation (Coughlin et al. 2014). In the case of NN cancellation, seismic sensors will be placed around the detector test masses to provide the information for a coherent estimate of the NN associated with the seismic field. In the case of Wiener filtering, the information is encoded in the correlations between the seismic sensors and with the detector data. To maximise the noise-cancellation performance, one needs to optimize the array configuration, which means to find the sensor locations and orientations that minimize the residual noise in the GW data. A study of this nature has already been performed for the GW detector Virgo (Badaracco et al. 2020).

To find the optimal array, one can use recorded seismic data and numerical simulations of the seismic field; in particular, we need to evaluate the cross power spectral density (CPSD) between seismometers. For optimizations of surface seismometer arrays, the domain of this function is a four-dimensional space (the coordinates of two points on the surface). This poses a challenge to the optimization algorithm especially for larger numbers of seismometers. With seismic data from $N$ sensors, we will be able to evaluate $N\times N$ two-point spatial correlations. A technique is required to create a surrogate model of the Wiener filter based on the observed CPSDs, e.g., using Gaussian Process Regression. One can use the surrogate model to estimate the residual of the noise cancellation for an arbitrary number of seismometers in an arbitrary array configuration, whose optimum can then be found.

In the case of lunar GW detectors, the available seismic data from a lunar site will be none or very few before the detector is being deployed. In this case, based on a geological and topographic model of the site, one can employ numerical simulations of the CPSD (Andric and Harms 2020). The results of such simulations can either directly enlarge the set of observed CPSDs, or they can be used in a more sophisticated optimization algorithms as priors of a Bayesian optimization (Andric 2022). Such hybrid optimization schemes, i.e., based on data and simulations, will also be necessary for the Einstein Telescope and are currently under development.

Also for LGWA, the optimal array configuration for background reduction needs to be determined. Lunar seismic data from PSRs are not available yet. Numerical simulations of the seismic field combined with seismic measurements in lunar analogues, e.g., volcanic landscapes, will be fundamental to determine effective array configurations.

4 Lunar Science

4.1 Seismic Exploration of the Moon and Mars

The Moon and Mars are the only two terrestrial bodies other than Earth for which seismological studies have been performed. For the Moon, this has been made thanks the deployment of a network of 4 passive seismometers operated as one of the sensors of the Apollo Lunar Surface Experiment (ALSEP). See (Lognonné and Pike 2015; Lognonné and Johnson 2015; Garcia et al. 2019; Nunn et al. 2021) for a general overview of Apollo instruments (Lognonné and Pike 2015; Nunn et al. 2021) and results (Lognonné and Pike 2015; Garcia et al. 2019). After two unsuccessful attempts (Anderson et al. 1977; Lognonné et al. 1998), Mars has been seismically explored by the SEIS instrument (Lognonné et al. 2019) onboard NASA InSight mission (Banerdt et al. 2020).

More than 13000 Moonquakes have been recorded over the 7.7 years of the Lunar Network operation, most of them being Deep Moonquakes or impacts, with the exception of 30 shallow moonquakes (Nakamura et al. 1981); see among other (Frohlich and Pulliam 1999; Kawamura et al. 2017) for the deep moonquakes seismicity; see (Oberst 1987) for shallow moonquakes and (Lognonné et al. 2009; Gudkova and Zharkov 2004; Gudkova et al. 2015) for impacts.

Slightly more than 1300 Marsquakes have been recorded over the 4 years of InSight operation (Marsquake Service 2023); see (Giardini et al. 2020; Clinton et al. 2020; Ceylan et al. 2022) for more on Mars seismicity.

For the Moon, and because of the four-station network configuration, the internal structure has been constrained from mostly arrival differential travel times of direct waves (e.g. (Lognonné et al. 2003; Gagnepain-Beyneix et al. 2006) for the mantle, crustal conversions (Vinnik et al. 2001) and located impacts (Lognonné et al. 2003; Gagnepain-Beyneix et al. 2006; Chenet et al. 2006) for the crust and core reflected phase (Weber et al. 2011; Garcia et al. 2011) for the core.

For Mars and because of the single station configuration, the use of secondary phases, such as PP, PPP, SS, SSS was necessary for mantle inversion (Khan et al. 2021; Drilleau et al. 2022). The crust below InSight landing site has been characterized by receiver function (Lognonné et al. 2020; Knapmeyer-Endrun et al. 2021) but also by surface wave analysis (Kim et al. 2021; Beghein et al. 2022; Kim et al. 2023) following the recording of two very large impacts (Posiolova et al. 2022) and the largest M = 4.7 marsquake (Kawamura et al. 2022). Last but not least, the core has also been detected thanks ScS phases (Stähler et al. 2021).

With respect to GW detection from the Moon, one of the most critical points related to lunar structure is the extremely low rigidity of the lunar subsurface, with shear velocities ranging from 50 m/s to 125 m/s in the first 15 m (Tanimoto et al. 2008) and increasing deeper toward an average of 500 m/s in the first km (Garcia et al. 2011) associated with a globally distributed cover of regolith. On the other side, the low seismic activity of the Moon will be beneficial to all GW detectors: most of the moonquakes have indeed amplitudes smaller than a few $10^{-10}$ m and furthermore, the Apollo records were flat most of the time, despite their sensitivity of $0.5\cdot 10^{-10}$ m in ground displacement at 0.5 Hz.

4.2 Space Qualified Very Broadband and Long-Period Seismometer Performances

To date, only the Apollo Long Period seismometer (Bates et al. 1979) and the InSight very broadband (VBB) seismometer (Lognonné et al. 2019) have successfully flown. A comparison is shown in Fig. 9. The noise floor was about $2\cdot \ 10^{-10}$ m/s²/Hz$^{1/2}$ and $10^{-10}$ m/s²/Hz$^{1/2}$ for the Apollo Z sensor and InSight VBBZ, respectively, at 0.5 Hz, but with a much better resolution for InSight. In 2025, the FarSide seismic suite (Panning et al. 2022) will operate a vertical VBB, which is a spare unit of InSight with an improved displacement transducer but a lower mechanical gain due to its vertical configuration. A new VBB based on optical sensing (de Raucourt et al. 2022) is now in development for future missions, including Argonaut, Artemis or LGN (Weber et al. 2020), with a targeted noise of $5 \cdot 10^{-12}$ m/s²/Hz$^{1/2}$ in the 0.01 Hz–1 Hz bandwidth ($1.5 \cdot 10^{-13}$ m/Hz$^{1/2}$ at 1 Hz). It has first reached TRL4 in the frame of the H2020 PIONEERS project (Mimoun et al. 2019) and TRL 6 is now targeted by mid 2025 through CNES funding. At 1 Hz, the proof mass Brownian noise is the major source of noise, while the other sources of noise, including those of the interferometric reading, are about $5 \cdot 10^{-13}$ m/s²/Hz$^{1/2}$. Larger proof masses with higher Q could therefore in principle be considered, if such a laser-optical VBB was used for active seismic mitigation of lunar GW detectors.

Fig. 9

Comparison in acceleration spectral density of the self noise of the Apollo sensors, of the InSight VBB sensor for both the Velocity and the POS, tidal outputs, and the future Lunar VBB

4.3 Distributed Acoustic Sensing (DAS)

Use of extended fibre optic cables as seismic antennas in various applications onshore and offshore has increased significantly in the last decade (Jousset et al. 2018; Landrø et al. 2022; Bouffaut et al. 2022; Nishimura et al. 2021; Taweesintananon et al. 2021). For Lunar applications (Landrø et al. 2022; Sollberger et al. 2016) seismic fibre optic cables are attractive due to low weight and long range, which opens for the possibility of designing a lunar seismic gravitational antenna (LSGA).

The basic principle for all DAS (distributed acoustic sensing) is illustrated in Fig. 10, where an acoustic source is creating a stretch in an optic fibre. In a fibre (telecommunication fibre or specially designed fibre) there are always impurities or inhomogeneities that act as optical scatterers (illustrated by the orange sphere in Fig. 10) that will move a small distance $\Delta L$ which can be found by measuring the optical phase difference between two signals recorded by the interrogator, that is $\Delta L = c\Delta t$ where $c$ is the speed of light in the fibre and $\Delta t$ is the time delay between the two signals.

Fig. 10

The acoustic wavefield originating from the acoustic source modulates the fiber. The light propagating inside the fiber, which is backscattered due to inhomogeneities of the glass, is thus reflected from positions being shifted in space by the acoustic wavefield. This is indicated by the orange and light blue sphere. A DAS interrogator can reconstruct this acoustic wavefield present at each location along the fiber by detection of the change/modulation to the phase of the backscattered light, illustrated by the two shifted curves in the interrogator box, also color-coded to see the relation to the two spheres that are shifted by the acoustic signal generated by the whale. Figure from Landrø et al. (2022)

Recently there has been a rapid improvement in DAS-interrogation. First, the signal to noise ratio versus distance has improved significantly, which makes it possible to measure signals 100 to 150 km into a fibre optic cable. A practical example is shown in Fig. 11, where a recent earthquake (August 2022) south of Svalbard was recorded by two DAS-interrogators.

Fig. 11

a) DAS-recording (x-axis denotes distance along fibre in km from Ny Ålesund and y-axis shows 75 seconds of data) of an earthquake south of Svalbard. The first event is the P-arrival and the second is the S-arrival. In the present example we have used one interrogator in Ny Ålesund and another in Longyearbyen and spliced the two plots together. The signal to noise ratio is gradually decreasing and at the middle of the cable (approximately at 130 km) we observe that the earthquake signal is hardly visible. b) the path for the trenched fibre optic cable and c) the earthquake position (blue circle) south of the fibre optic cable

Second the sensitivity has increased, enabling a strain sensitivity better than $10^{-10}$. It is expected that new interrogators in the future will push this limit further. The strain sensitivity for frequencies above 1 Hz depends on the gauge length (the length over which coherent phase changes are sampled, see Fig. 10) and the distance from the interrogator to the observation point on the fibre. For instance a gauge length of 10 m and a distance of 10 km yields a strain sensitivity of $10^{-11}$ and a gauge length of 40 m for the same distance yields a strain sensitivity of approximately $10^{-12}$. If the fibre distance is increased to, for instance, 50 km, these numbers drop by a factor of 10, approximately. (Landrø et al. 2022) show that good quality DAS-data can be acquired as low as 10 mHz (and probably lower), which is essential for LSGA-applications. There are several experiments that should be carried out on earth in Lunar-like environments prior to installing DAS-fibres on the Moon. Challenges are related to issues like coupling to the ground. In Fig. 11 the variation in signal strength along the fibre is partly caused by weak coupling to the seabed. Various types of fibres should also be tested in order to determine an optimal Lunar DAS-fibre.

4.4 Seismic Background

Despite the Apollo seismic passive experiment, very little is known about the micro-seismic lunar noise. Records from the Apollo Long period seismometer suggest that the recorded noise was likely instrument self noise on the vertical axis (Lognonné and Johnson 2015), meaning that an upper limit was set for the vertical noise at about $2 \cdot 10^{-10}$ m/s²/Hz$^{1/2}$ in the 0.1–0.5 Hz bandwidth. This translates into a ground displacement noise of 20 pm/Hz$^{1/2}$ at 0.5 Hz. Horizontal seismic noise was reported with amplitudes five times larger at 1 Hz ($10^{-9}$ (m/s²)/Hz$^{1/2}$) and rising as $1/f$ towards lower frequencies. This appears most likely to be tilt noise with a non-seismic origin. Last but not least, the lack of atmosphere of the Moon will prevent from surface pressure noise and from associated gravitational fluctuations as are present on Earth (Fiorucci et al. 2018).

The true micro-seismic background noise of the Moon has therefore not yet been recorded by seismometers. The FarSide Seismic Suite, with an expected 20 times better sensitivity than Apollo peaked mode at 1 Hz (Panning et al. 2022) might measure the noise better above 0.5 Hz but likely not below 0.5 Hz, but future seismometers on Artemis might reach that microseismic noise floor (Weber et al. 2020). A background noise associated with impacts has been proposed based on the extrapolation of Apollo peaked-mode seismic data toward smaller amplitudes (Lognonné et al. 2009), and will likely be mostly related to short periods surface waves trapped in the very low velocity subsurface. Simulations predict a root-mean-square (rms) of background noise that lies 50% of the time below 75 fm and 90% of the time below 0.5 pm based on minute-long stretches of time series (Lognonné et al. 2009). These values correspond to the square-root of the integrated spectral densities within the simulated measurement band. Assuming a flat displacement spectrum of the impact signals (Daubar et al. 2018) and for simplicity a bandwidth of 1 Hz of the ground-displacement spectrum, this translates into a background smaller than $5\cdot 10^{-14}$ m/Hz$^{1/2}$ at 1 Hz 50% of the time. Confirming such low noise will be critical for future lunar GW projects.

Seismic waves in the decihertz band are expected to be strongly scattered by subsurface heterogeneities and will have relatively low horizontal phase or group velocities (about 40 m/s at 5 Hz and possibly 100 – 150 m/s at 1 Hz). These phase velocities for a 50 fm/Hz$^{1/2}$ displacement noise amplitude on the horizontal axes (not considering potentially important contributions from ground tilt and assuming equal displacement spectra along the horizontal and vertical directions), will lead to seismic strain of about $2-3\cdot 10^{-15}$ Hz$^{-1/2}$ at 1 Hz. Background mitigation will require seismic measurements very close to the GW sensors with self noise low enough to prevent noise injection in the process.

4.5 Lunar Geology

The naked eye view of the Moon shows two different types of terrains: the darker Maria volcanic plains and the brighter Highlands, which are topographically higher and rugged. The Highlands constitute about 85% of the total surface of the Moon.

The application of the geological principles to study the observations made with orbital and landed robotic and crewed missions to the Moon since the 1960’s has brought to the identification of two main processes which shaped the surface of the Moon since its formation: volcanism and impact cratering. The Maria are the expression of primarily basaltic flows which filled the topographic lows, overlapping the lighter terrains. Over the Maria, physiographic features like ridges, domes, fissures, and caves skylight support the hypotheses of the volcanic origin of these terrains. Craters are everywhere on the Moon, but rough Highlands have a remarkably higher crater density than the Maria. Impact cratering on the airless Moon happens at hypervelocity and the resulting structures, the craters, witness the effect of impactors at all scales. The craters on the Moon range from 175 km in diameter down to the micrometer scale of micro-impact of space debris, resulting in a continuous abrasion of all exposed material.

The continuous meteoritic bombardment generates a layer of crushed and melted rock called the regolith. In the Apollo era, we had direct access to the regolith, and samples of the regolith can be studied on Earth. The mean grain size of lunar regolith ranges from 40 to 800 μm, with most of the material falling between 45–100 μm (Heiken et al. 1991). The thickness of the regolith ranges from about 5 m on Maria to about 10–20 m on the highlands. However, seismic data from geophysical observations from Apollo 15 seismometers suggest that the first layer of the Moon is made by a highly scattering media a few kilometers up to 20 km thick, referred to as the mega-regolith (Latham et al. 1972). This is consistent with the presence of craters tens of kilometers in diameter, contributing to fracturing the first kilometers of the lunar crust. The compositional diversity between the Maria and the Highlands also implies a lateral variability of the regolith. From the gravitational wave detection perspective, knowing the site-specific geology can be critical in deploying instrumentation on the surface. A geologic model can also drive numerical models simulating the acoustic response of the regolith.

5 Gravitational-Wave Science

Considering all proposed concepts, lunar GW detection might cover an observation band from 1 mHz to a few Hz and open the decihertz band to GW observations (Harms 2022). Several studies have highlighted the importance of a decihertz observatory (Jani and Loeb 2021; Sedda et al. 2020; Mandel et al. 2018; Isoyama et al. 2018; Harms et al. 2021). Considering a network of lunar GW detectors, additional scientific studies can be carried out. For example, antipodal pairs of GW detectors at the two lunar poles would form an ideal network for the search of stochastic GWs exploiting correlations between two detectors (Coughlin and Harms 2014a). It was also pointed out that a distribution of vibration sensors over the surface of the Moon would enable detailed measurements of the GW polarization and interesting tests of general relativity (Wagoner and Paik 1976; Bianchi et al. 1996). In this section, we will briefly outline some of the main science goals of lunar GW detection.

5.1 Populations of Compact Binaries in the Local Universe

Compact binary systems composed of white dwarfs (WDs), neutron stars (NSs) and black holes (BHs) populating our local Universe – from our Solar neighbourhood, through the Milky Way out into the Local Group, up to hundreds of Mpc away for the most massive binaries – are among the sources observable by lunar GW detectors. These detectors have the potential to extend observations in the mHz frequency band of missions like LISA (Amaro-Seoane et al. 2017), TianQin (Luo et al. 2016) and Taiji (Ruan et al. 2020) to the mid-frequency band (about 0.05 Hz – 1 Hz) where mergers of WD binaries occur or to create a bridge to the high-frequency observations of terrestrial GW detectors.

Compact stellar remnant binary systems are guaranteed GW sources as some have are already been discovered with electromagnetic (EM) telescopes. The most famous example of such a binary being the first discovered binary pulsar (Hulse and Taylor 1975), which via measuring its orbital decay over three decades led to the first indirect evidence for GWs (Weisberg and Taylor 2005) and paved the way forward for the field of GW astronomy. Discoveries of many more systems consisting of a NS/WD with another compact companion down to orbital periods as short as several minutes followed since then (Kupfer et al. 2018). Some of the shortest period and better characterised of the known binaries have been even proposed to use for testing space-based GW detectors and maximising their scientific output (e.g. Stroeer and Vecchio 2006; Kupfer et al. 2018; Littenberg 2018; Finch et al. 2022). Therefore known binaries are also called verification binaries in the GW literature. Some of these verification binaries can be detected with LGWA (Harms et al. 2021). Furthermore, known binaries offer a new opportunity to study the astrophysics of compact binaries using both their GW and EM radiation (e.g. Marsh 2011; Korol et al. 2017; Breivik et al. 2018; Tauris 2018; Littenberg and Cornish 2019).

From a theoretical perspective, compact binaries are expected to be several orders of magnitude more abundant than currently observed by EM telescopes, with binaries composed of two WDs being the most abundant in the Milky Way and its immediate neighbourhood. Their formation involves up to several mass transfer episodes; at least one of these episodes has to be unstable leading to a common envelope, a phase during which binary can shrink dramatically (Paczynski 1976; Webbink 1984; Livio and Soker 1988; de Kool et al. 1987; Ivanova et al. 2015). Even though this is one of the least understood phases in binary evolution, the fact that we do observe close double compact objects and mergers is a strong indication that something like a common envelope phase takes place. Thus, the demographics of compact binaries observed via GW radiation will provide opportunities to learn new physics and answer key scientific questions related to formation and evolutionary processes of tight binary systems. Besides the common envelope physics, this includes questions related to the stability and efficiency of mass and angular momentum transfer, tides, accretion onto compact objects, as well as details of their destruction in supernovae explosions (for a review see Amaro-Seoane et al. 2023).

At frequencies $<1$ mHz, binary population synthesis simulations predict ${\mathcal{O}} (10^{7})$ double WDs, followed by ${\mathcal{O}} (10^{5})$ NSWDs in the Milky Way alone (Amaro-Seoane et al. 2023). Studying these binary populations with EM facilities has proven to be technically challenging due to the compact size of these binaries and WD/NS stars themselves, and so GW detectors – characterised by different selection effects – are ideally suited for discovering and characterising them in bulk (Rebassa-Mansergas et al. 2019; Korol et al. 2022). Importantly, GW selection effects enable the study of double compact objects with weak EM signatures at unprecedentedly large distances throughout the Milky Way, including its most massive satellites and possibly reaching out as far as the Andromeda galaxy, which is otherwise impossible with EM telescopes (Korol et al. 2018; Roebber et al. 2020; Wilhelm et al. 2021; Keim et al. 2023).

5.2 Mergers of Massive BHs and AGNs

Based on high spatial resolution observations (van der Marel 1994; Kormendy and Richstone 1995), it is now accepted that at the center of every galaxy (perhaps with the exception of irregular galaxies) lies a BH heavier than about $10^{6}$ M_⊙. The best evidence for this was provided by the microarcsecond angular resolution images of SgrA*, at the Milky Way center, and of the nucleus of M87, the giant elliptical at the heart of the Virgo cluster, obtained by the Event Horizon Telescope at millimetric wavelengths (Event Horizon Telescope Collaboration et al. 2019, 2022).

Virtually no galaxy goes undisturbed during its lifetime, and in fact galaxies often experience episodes of strong interaction with their neighbors, or even mergers (Barnes and Hernquist 1992). In the Local Universe, interacting pairs of galaxies abound, as displayed for instance by the spectacular HST imagery archived in the Hubble Heritage repository .² These legacy snapshots witness the onset of dynamical friction, when the distances between the galaxies central SMBHs are still tens to hundreds of kpc. Radio observations can resolve binary SMBHs down to separations of parsecs in radiogalaxies at $z \sim 0.1$ (Rodriguez et al. 2006; Deane et al. 2014).

From the above follows that super-massive nuclear BHs (SMBH) at the centers of interacting galaxies will become dynamically bound and form close orbits in binary (or multiple) systems. SMBH in binary systems represent a large fraction of the totality of SMBHs in the Universe (Volonteri et al. 2002). Loss of energy via gravitational radiation will cause the orbits to shrink and the binary SMBH system to coalesce in a few million years of the binary “hardening” (Begelman et al. 1980; Komossa and Zensus 2016; Valtonen et al. 2021).

Galactic interaction may trigger star formation in the vicinity of the orbiting SMBH, and nuclear activity when the SMBHs are at less than a parsec distance from one another. This is in fact considered to be a major channel of nuclear activity ignition so that it has been speculated that a large fraction of, or virtually all, AGN should contain a binary SMBH at their centers.

Search for evidence of binary central motion in AGN at all wavelengths from radio to gamma-rays in the form of periodic signals in their light curves has yielded appealing and suggestive results, although the significance is generally not robust, e.g. (Ackermann et al. 2015). The most convincing case appears to be that of OJ287, a blazar source at $z = 0.306$, that shows maxima with a rest-frame period of approximately 9 years in its optical light curve. This was however evenly and densely sampled only in the latest 25 years (Valtonen et al. 2021). The model that describes the binary SMBH system in the nucleus of OJ287 suggests a distance between the two BHs of ∼0.1 pc and a remaining lifetime of $10^{4}$ years before merger.

The final inspiral and merger of binary SMBHs should produce a gravitational signal of sub-Hz frequency, that the future LISA mission and – with higher S/N – the perspective lunar GW experiments could detect from any cosmological distance.

5.3 The Case of SN 1a Progenitors

Type 1a Supernovae (SN 1a) play a crucial role in modern cosmology. In addition to providing the first evidence that the Universe is accelerated by some sort of dark energy, they recently led to a tension with the value of the Hubble constant derived from modelling of the Cosmic Microwave Background (CMB) (Cappellaro 2022, for a review). Yet, and despite of their importance, the nature of their progenitors is still unclear.

SNe Ia are believed to originate from white dwarfs (WD) which, due to mass transfer in close binary systems, grow above the Chandrasekhar limit ($\sim 1.4\, M_{\odot}$) and then are incinerated by a thermonuclear explosion in degenerate matter conditions. This idea dates back to the early 1970s (Whelan and Iben 1973) but the actual nature of the secondary stars is still undetermined. Two alternative scenarios have been proposed: the donor can be a main sequence or a red giant star (the single degenerate scenario, or SD), or another WD (the double degenerate scenario, or DD), which would lose orbital momentum via gravitational waves (GW) radiation and eventually merge.

The evidence collected so far in favour of one or the other option is all indirect. As examples of relevant results we can list: i) the presence (Patat et al. 2011) or absence (Sand et al. 2021) of signatures of circumstellar gas surrounding the progenitor that favour SD and DD, respectively; ii) the failed search for a surviving companion of historical SN Ia in our Galaxy, favoring DD (Ruiz-Lapuente et al. 2019); iii) the failed search of a precursor star in pre-explosion archival images (Graur et al. 2014) favoring DD; iv) the search for candidate DD systems that did not find a sufficient number of short period, massive WDs to account for the observe SN Ia rates (Napiwotzki et al. 2020) favoring SD, or otherwise sub-Chandrasekhar explosions; v) the distribution of delay time from star formation to explosion derived from statistics of SN Ia rate that tends to favor the DD scenario (Maoz et al. 2012), although the strength of this argument is debatable (Greggio and Cappellaro 2019).

The bottom line is that, after decades of research, none of the results are conclusive and, in some cases, the findings appear to be even in contradiction. The new era of gravitational wave (GW) astronomy opens a new avenue for obtaining independent and direct insights to nature of the SN Ia progenitor, which remains an open question in modern astrophysics (Livio 2001). This is because the binary systems from which SN Ia originate are GW sources (cf. Sect. 5.1).

In the case of SD systems, mass transfer to the accreting WD occurs when the secondary donor star fills its Roche lobe. This sets a lower limits for the separation between the two stars and the orbital period (of the order of 10 minutes). In this configuration the GW emission is weak and does not affect significantly the binary evolution. On the other end, loss of orbital momentum via GW emission causes the shrinking of DD system orbital radius. Because of the small tidal radius of WDs, this allows a minimum orbital period ($P$) of a few seconds and a stronger GW signal. In fact, the GW emission grows with the GW frequency, and hence is proportional to $P^{-1}$.

We have to consider that at frequency of $10^{-2}$ Hz the merging time for DD systems is $\sim 10^{4}$ yr. Hence, the Galaxy needs to host few tens WDs with mass $>=0.7\,M_{\odot}$ orbiting with this frequency to allow for 1–5 SN per thousand years. If these systems are not found, either SD dominate the census of progenitor systems or SN Ia can originate from system of smaller mass. We notice that whereas the Sub-Chandrasekhar scenario is not the favoured option to explain the majority of SN Ia (Mazzali et al. 2007), it is gaining support also in accounting for the observed SN Ia diversity (Flörs et al. 2020).

Detectors like LGWA would be able to probe the $10^{-2}$ – 1 Hz GW spectral range. The analysis of mass and orbital period distribution of DD systems would allow for accurate prediction of the SN rate that can be compared with observations.

One may envisage that the ultimate proof of the DD-SNIa connection would be the observation of the merging event followed by the detection of its electromagnetic (EM) counterpart. However, one has to consider that the chance of observing a DD merging is of the order of the SN Ia rate, hence one in a few centuries in the Galaxy. In addition, while in special cases one may expect that the SN explosion, and hence the appearance of the EM source following a few seconds after the merger (Pakmor et al. 2022), more conventional scenarios allow for a long delay between the merger and the explosion, ranging up to $10^{4}$ yr (Shen et al. 2012). This implies that, even in the case such a rare event of a WD merger of appropriate mass occurs and its GW emission is detected, it may not be associated to a SN Ia EM event, and hence it would not provide the long sought and much desired smoking gun.

This is why, in the current state of affairs, the comparison between the known rate of SNe 1a and the rate of GW sources with the properties expected for the merging of DD systems is probably one of the most promising tools to solve this key question.

5.4 Core-Collapse Supernovae and Jets

The acceleration of a jet to relativistic velocity is an interesting source of gravitational waves: Jet-GWs (Segalis and Ori 2001; Piran 2002; Sago et al. 2004; Birnholtz and Piran 2013; Leiderschneider and Piran 2021; Piran 2022). This is a memory type signal whose amplitude satisfies:

$$ h \approx \frac{ 4 G \mathcal{{E} }}{ c^{4} r}\approx 10^{-24} \bigg( \frac{\mathcal{{E}}}{10^{51} ~{\mathrm{erg}}}\bigg)~\bigg( \frac{100 {\mathrm{Mpc}}}{r} \bigg) \ . $$

(7)

where $G$ is Newton’s constant, ℰ is the jet’s energy, $c$ is the speed of light and $d$ the distance to the source. The rise time and the corresponding typical frequency are determined by the longer of the acceleration time or the duration of the jet. Such signals arise from any source in which a relativistic jet is accelerated. AGNs and GRBs are, of course, the classical sources that first come to mind. However, the first are quasi steady state sources (on human time scale) whose typical frequencies are far too low. The second are typically too far and their signals are typically to weak to be detected.

Core collapse SNe (CCSNe) jets are within the frequency range and are possibly sufficiently nearby to be within the expected sensitivity of Moon GW detectors. Such jets were inferred recently to take place in several powerful SNe (Piran et al. 2019). These jets are hidden as they are choked while propagating in the exploding star. However, they deposit their energy in the stellar envelope in powerful cocoons. In some directions, that are not too far from the jet axis, a low-luminosity GRB can be observed as the cocoon breaks out from the envelope (Kulkarni et al. 1998; MacFadyen et al. 2001; Tan et al. 2001). This was noted already for the first low luminosity GRBs 980425. However, the existence of such jets can be inferred even when we do not observe the low-luminosity GRB. The cocoon ejecta escapes with large velocities (Pais et al. 2023) reaching up to 100,000 km/sec in SN 2017iuk (Izzo et al. 2019; Nakar 2019). Those velocities are far higher than typical velocities seen in regular GRBs. The high velocity material spreads around the exploding SN (Piran et al. 2019; Pais et al. 2023) absorbing some of the light of the SNe producing very broad absorption lines. Such lines have been observed already in SN 1997ef (Mazzali et al. 2000) but their origin remained mysterious for more than two decades (Piran et al. 2019). The lines disappear after a few days as the fast material spreads and becomes optically thin. Hence, they have been observed only in a few cases when the SN followup was initiated rather quickly. The existance of such hidden jets is supported by other evidence for jets in SNe (Totani 2003; Mazzali et al. 2005; Maeda et al. 2008; Taubenberger et al. 2009; Grichener and Soker 2017).

Piran et al. (Piran et al. 2019) estimated that the typical energies of these CCSNe jets are of order $10^{51}$ ergs or even higher. This energy is typically insufficient to explode the SN. However, it is an important component in the overall energy budget. Jet-GWs are the only way to directly observe these jets. With typical distances of a few tens of Mpc as compared to a few Gpc for long GRBs and a few hundred Mpc for a nearby short GRB, SNe-Jets are prime candidate sources for detectable Jet-GWs. In fact, estimates suggest that the Jet-GWs from a powerful CCSNe may be stronger and easier to detect than the GWs that arise from the SN collapse itself.

5.5 Tidal Disruption of WDs

Main sequence or giant stars nearby stellar black holes (BH) or super-massive black holes (SMBH) have been observed to be disintegrated or disrupted during their passage at the periastron; these are the so-called Tidal Disruption Events (TDEs). The disrupted star undergoes extreme acceleration that stretches the lighter star along the line of gravitational force and squeezes it along the perpendicular directions; hence the star is flattened in an elongated pancake shape, aligned to the plane of its orbit around the compact star (Carter and Luminet 1982). These TDEs are mainly transient phenomena that lasts for days to months and observed at multi-wavelength by EM observatories (see e.g. Rossi et al. 2021, and references therein). A catalog of observed TDE candidates can be found here .³

TDE requires that the passage of the lighter star near the compact object lie within the tidal radius (so called Roche or Hill radius) (Hills 1976) written as

$$ R_{t} \approx R_{*}\left (\frac{M_{c}}{M_{*}}\right )^{1/3}, $$

(8)

where ($R_{*}$, $M_{*}$) are the radius and mass of the lighter star and $M_{c}$ is the mass of the compact star. The strength of the TDE is defined as the penetration parameter

$$ \beta _{p} = \frac{R_{t}}{R_{p}}, $$

(9)

while $R_{p}$ is the periastron radius.

For a TDE, three conditions have to be satisfied: (i) $\beta _{p} \ge 1$, which can be achieved if the star approaches the BH ⁴ in a highly eccentric or parabolic trajectory. For a given $M_{*}$, (ii) $R_{t} > R_{*}$, and $R_{\mathrm{BH}} > R_{*}$; where $R_{\mathrm{BH}}$ is event horizon of BH, otherwise the whole BH enters into the star, and (iii) $R_{t} > R_{\mathrm{BH}}$ otherwise, the BH would engulf the entire star at once (Maguire et al. 2020).

The disrupted star results in debris, in which most of the matter is dispersed away from the BH but a smaller part of the stellar matter form closed orbits (Rees 1988) that results in the EM radiation via various processes. In addition, a burst-like GW signal is also generated because the quadrupole moment of the star due to the tidal deformation varies in time as it makes the closest approach towards the BH. Since the GW strain increases with the $M_{*}$ and decreases with the $R_{*}$ for a given $\beta _{p}$, hence a WD or an NS will have stronger GW signals with higher frequencies as well compared to a main-sequence or giant star (Kobayashi et al. 2004). However, a WD-BH TDE is more likely than a NS-BH TDE because NSs are around $10^{8}$ times denser than WDs, so they often result in mergers without a TDE. Another reason is that WDs are much more prevalent in the universe compared to NSs (Shapiro and Teukolsky 1983).

The most relevant feature of the WD-BH TDEs is that the associated GW frequencies lie in the deci-Hz band which is between the observation bands of by LISA, $10^{-1}$ Hz and lower, and that of the ground based GW detectors above 10 Hz (Casalvieri and Ferrari 2006). Hence, the deci-Hz sensitivity of lunar GW detectors fills this observational gap and opens up future possibilities to detect Intermediate Mass Black holes (IMBHs) and to probe the spin of the responsible BH for the TDE (Gezari 2014). For establishing the detection of a WD-BH TDE, the EM-counterparts for the GW signal will be of great importance, since during the disruption the WD is subjected to extreme pressure causing a sharp increase in the temperature, triggering explosive nuclear burning for $\beta _{p} \ge 3$ (Rosswog et al. 2009), which is seen as a transient increase in the luminosity followed by an slow decay. Fast X-ray transients and optical Type-Ia supernovae-like events are also the possible observational manifestations of a WD-BH TDE (Haas et al. 2012). EM signatures may also include observations of jets due to the Blandford-Znajek process (Lam et al. 2022).

5.6 GW Cosmology

By accessing GW signals in a frequency range from 0.1 – 1 Hz with a sensitivity of $\sim 10^{-21}$ in strain measurement, lunar GW detector concepts like LGWA and GLOC could potentially survey black holes across several orders of magnitude in mass to cosmological distances. The enhanced low-frequency sensitivity permits the survey of primordial and intermediate-mass black holes (IMBHs) practically across the entire universe. Such cosmological reach is crucial for connecting IMBHs with the Pop-III remnants (Madau and Rees 2001) and the seeds of supermassive black holes (Gair et al. 2009; Pacucci and Loeb 2020). Furthermore, the cosmological reach of lunar GW detectors at their peak sensitivities is not expected to have any astrophysical foregrounds from the white dwarf binaries (Robson et al. 2019). Thus, any GW signal with redshifted frequency $f_{\mathrm{det}} = (1+z)f_{\mathrm{src}} \gtrsim 0.2$ Hz could be identified in the data without contamination.

As an illustration, we show in Fig. 12 the cosmological reach of GLOC for various binaries consisting of neutron stars and black holes of several orders of masses. While we do not expect observations of such binaries beyond $z{\sim} 30$ (Loeb and Furlanetto 2013), even one such detection would violate $\Lambda$CDM cosmology (Koushiappas and Loeb 2017; Ng et al. 2022).

Fig. 12

Cosmological reach of a lunar GW detector concept GLOC in comoving coordinates. The concentric circles represent the percentage fraction of the comoving volume of the observable universe $(V_{\mathrm{obs}} = 1.22 \times 10^{4}~\mathrm{Gpc}^{3} )$ out to a given cosmological redshift, with the outermost being the cosmic microwave background. The highlighted slices refer to the reach of GLOC in units of redshift. For reference, the circle in the center represents the maximum reach of Advanced LIGO at the design sensitivity for a 100 solar mass binary. The plot was adapted from Jani and Loeb (2021)

One of the strongest science cases of lunar GW astronomy program is towards studying Type 1a supernovae (SNe) mechanisms. The access to 0.1 – 1 Hz enables direct discrimination between the single (Falta et al. 2011; Seitenzahl et al. 2015) and double degenerate (mergers of two white dwarfs) scenarios of Type 1a SNe. Joint observation of such an event with GWs and electromagnetic signals can be used to constrain the unknown masses and explosion mechanism of the white dwarfs. This can potentially reduce the error budget in using SNe as standard candles. Further, such multi-messenger observations could constrain cosmological parameters to sub-percent precision.

Lunar GW detectors can put the tightest bounds on the speculative population of sub-solar dark matter objects ($0.1-1\, M_{\odot}$) (Shandera et al. 2018). There are no known astrophysical phenomena that can create detectable GWs at such low masses; however, primordial black holes or dark matter within neutron star cores offer possible scenarios (Abbott et al. 2019). The deci-Hz reach of lunar GW detectors allows us to measure the dark matter density of such exotic objects to 30% of the entire observable volume of the universe ($z\sim 10$).

For stellar black hole binaries, lunar GW detectors would start measuring the inspiral phase a day before the merger. A multi-band observation (Jani et al. 2019) of these black holes binaries between the Moon and earth-based GW detector on Earth can reduce the sky-location error to the angular scale of a single galaxy. This opens a new population of high redshift dark sirens to independently measure the evolution of the Hubble parameter as a function of redshift (Schutz 1986). Furthermore, combining these high redshift dark sirens with GW lensing would constraint cosmological parameters to increased precision (Congedo and Taylor 2019).

6 Conclusion

In this article, we provided an overview of the proposed concepts for lunar GW detectors and discussed some of the key technologies to realize the instruments. We highlighted the unique properties of the Moon that enable an exclusive contribution of lunar GW detection to the field including the Moon’s extremely low background of seismic disturbances, and the low-temperature, thermally stable conditions of the Moon’s permanently shadowed regions. The Moon can either be used as an antenna for GWs and a sensor reads out its deformations caused by the fluctuating spacetime, or it can serve as an ultra-quiet environment for a long-baseline laser interferometer similar in design to the current detectors on Earth.

Having access to the frequency band 0.1 Hz – 1 Hz, lunar GW detectors will be able to observe GW sources that LISA or future terrestrial GW detectors cannot see. These sources include intermediate-mass black-hole binaries at the cosmic dawn and beyond, mergers of white-dwarf binaries in our local universe, tidal disruption events, and low-frequency signals from supernovae and jets. Some of these sources will emit electromagnetic (EM) counterparts, and the association of GW and EM signals can teach us more about the environment of these sources and provide ground-breaking insight into source mechanisms, e.g., revealing the nature of SN-1a progenitors.

Lunar GW detectors will also play a crucial role in what is known as multiband GW observations. A neutron-star binary can be detected months before its merger, which can then be observed with terrestrial GW detectors and EM observatories. An early warning of the upcoming merger can be issued together with a precise sky-localization. Similarly, an intermediate-mass black-hole binary can be observed first in the millihertz band with a LISA-type detector and later its merger in the decihertz band with a lunar GW detector for high-precision waveform analyses.

Finally, the deep connection that is established by lunar GW detection between observations of the universe through GWs and of the Moon through its seismic properties is an immense and unique opportunity for an inter-disciplinary science case. In fact, the present article is an early product of this collaboration.

7 Dedication to Stavros Katsanevas

ISSI Forum on Lunar Gravitational-wave Detection, 5–6 Oct, 2022. Stavros Katsanevas was a Greek-French physicist born in Athens in 1953. He was director of the European Gravitational Observatory, former director of the AstroParticle and Cosmology (APC) laboratory, and former chairman of the Astroparticle Physics European Consortium (APPEC). He spearheaded the inter-disciplinary approach to gravitational-wave research and conceived, under the aegis of the NASA Artemis mission, a lunar project for the detection of gravitational radiation that uses the Moon itself as a detector.

The launch of the Artemis program by NASA, aimed at following up the Apollo missions and setting up the conditions for cis-lunar colonization, stirred a lot of interest and enthusiasm in the European community, and sparked ideas and proposals to foster various areas of science from astroparticle to biophysics, from seismology to condensed matter physics.

With characteristic vision and scientific foresight, Stavros caught this opportunity to conceive a gravitational detection experiment that used the Moon itself as a detector and adopted Engineered Fiber Distributed Acoustic Sensors (EFDAS) to offer sensitivity to the signal. With his holistic approach to science, he went a step further and involved the astrophysicists to identify the multi-wavelength characteristics of the potential gravitational wave emitters and the electromagnetic facilities that would optimize their detection and monitoring. Ultimately, this should lead to establish a fully-fledged multi-messenger observational program from the Moon.

This ISSI Forum was part of these efforts and was intended as a kickoff of the possible European collaborations in GW Lunar projects and activities. Very sadly, Stavros left us less than two months after the Workshop, after having dedicated to this, and a multitude of other projects, up to the last minute of his very creative and intellectually rich life.

However, the seeds Stavros has sown bear a solid promise to bring EFDAS to the Moon and all the technology involved in gravity waves detection using lunar experiments. Most importantly, he spearheaded progress in new physical and technological avenues, both experimental and theoretical. He then contributed to establish a network of scientists dedicated to the manifold aspects of research development and exploitation of the lunar environment, and saw into it that this team and its projects could start and grow in a coordinated and choral way and could play a dominant role in the global lunar race.

The baton of this concept has been passed on to us, his colleagues, friends and followers, who will endeavour to further it into completion and implementation.