General introduction

Gravitational wave (GW) observations have opened a new way to observe and characterize compact objects throughout the Universe and at all cosmic epochs. The Laser Interferometer Space Antenna (LISA; Amaro-Seoane et al. 2017), with its low-frequency band coverage spanning nearly three decades, will allow the detection and study of signals from a strikingly large variety of sources, ranging from stellar-mass binaries in our own galaxy to mergers between nascent massive black holes (MBH), called black hole (BH) seeds, at high redshift. LISA is expected to revolutionize our understanding of these astrophysical sources by allowing reconstruction of their demographics and dynamical evolution, as well as discovery of new types of sources, including some that have been theorized but not yet detected by conventional means. Since the first detection of GWs by the Laser Interferometer GW Observatory (LIGO)/Virgo collaboration in 2015 (Abbott et al. 2016c), ground-based GW observations already had a remarkable impact on astrophysics. For instance, the gravitational-wave facilities LIGO and Virgo have observed the mergers of stellar BHs in the range \(\sim 6\)\(95\,M_{\odot }\) (Abbott et al. 2021), greatly expanding our knowledge of the mass spectrum of BHs. Recently, the first intermediate mass black holes (IMBH), with masses of \(\sim 142\,M_{\odot }\), has been discovered (GW190521, see Abbott et al. 2020c). The existence of stellar-mass BHs with masses higher than observed before, as well as the discovery of an IMBH, have fostered new exciting developments in theoretical models for the formation and evolution of stellar-origin black holes. The discovery of the double neutron star (NS+NS) merger GW170817 with accompanying electromagnetic observations (Abbott et al. 2017b) has had a great impact on our understanding of dense matter and the origin of heavy elements. These discoveries showcase the huge potential that gravitational wave astronomy has to revolutionize our understanding of astrophysical objects and processes.

At the lower frequencies in LISA’s observing band, the stellar-mass systems, in binaries or multiples, provide a very rich source population. The population in the Milky Way is expected to consist of millions of double white dwarf (WD+WD) binaries, with a smaller population of neutron star (NS)/BH binaries, and possibly some of the heavy BHs that LIGO/Virgo have already detected. LISA observations of the BH populations will capture a snapshot of BH systems when their orbital periods are tens of minutes, a few years before their coalescence at the high frequencies observed by LIGO-Virgo. Overall, LISA observations of Galactic binaries will address many open questions in stellar astrophysics, such as the evolution of binary star systems, the origin of different transient phenomena, the origin of the elements and even the structure of the Galaxy. It should be noted that among the stellar-mass binaries in the Milky Way, a few are already known from electromagnetic observing campaigns, and can be used as LISA verification sources. While the vast majority of the stellar-mass binaries are expected to be too dim to be detected by electromagnetic instruments, there will be a substantive number that will be excellent targets for electromagnetic follow-up after LISA discovers them.

The observed BH mass spectrum spans ten orders of magnitude, ranging from a few \(M_{\odot }\) for stellar-mass BHs to up to \(10^{11}\,M_{\odot }\) for the most extreme MBHs. Many of the most massive MBHs, with \(M_{\text{BH}} \gtrsim 10^{8}\,M_{\odot }\), have been discovered in the high-redshift Universe, at \(z > 6\), powering some of the brightest quasars (Fan et al. 2003; Mortlock et al. 2011; Yang et al. 2020a). LISA will open up a wide discovery space for BHs. BH systems that merge at the millihertz frequencies, where LISA is most sensitive, are typically hosted in the most common type of galaxies, namely dwarf and massive spiral galaxies. The fact that LISA observations straddle the frequency bands of merging IMBHs and MBHs suggests that the potential impact on many fields of extragalactic astrophysics is huge. Such foreseen impact, however, relies heavily on our understanding of the astrophysical processes preceding and accompanying the evolution of the binaries during inspiral and into merger (De Rosa et al. 2019b). For MBHs, this knowledge is tightly entangled with the knowledge of the environments in which they evolve, namely their host galaxies and galactic nuclei. It follows then that LISA sources associated with MBH binaries cannot be understood without a robust knowledge of the landscape of galaxy formation and evolution, and in particular without a detailed knowledge of stellar dynamical processes and the interstellar medium inside galactic nuclei. There is thus an inherent multi-disciplinarity in the approach needed to understand these sources, which will naturally bring together various fields of galactic and extra-galactic astrophysics. Furthermore, since LISA will be able to detect MBH binary sources up to very high redshift (\(z \sim 10{-}15\)), one also needs ancillary knowledge of cosmic structure formation, as galaxies, and thus their relevant environments, evolve significantly from high to low redshift (Woods et al. 2019). The endeavour then extends into cosmology, and hints at great possibilities for derivative knowledge, some already expected and others not, coming from the future discovery and characterization of LISA MBH binaries.

The stellar dynamics of the central cluster of stars at the galactic centre (the S-stars, or S0-stars), provides compelling evidence for the existence of a MBH of mass \(\sim 4\times 10^6\,M_{\odot }\), Sgr A* (see for a review Genzel et al. 2010, and references therein). The stars in the centres of galaxies have the potential to interact with MBHs, but only if their pericentres are small enough. LISA will be able to observe the inspiral of a compact object such as a stellar-mass BH, a NS or a WD onto a (light) MBH, i.e., one with a mass between \(\sim 10^4\,M_{\odot }\) and \(\sim 10^7\,M_{\odot }\). Because of the difference in mass between the MBH and the \(\lesssim {\text{few}}\)\({\text{tens}}\) of solar masses of the compact object, we call these extreme mass-ratio inspirals (EMRIs)—where the mass ratio is \(10^{-8}\lesssim q \lesssim 10^{-5}\) (Amaro-Seoane et al. 2007). There is also a potential population of IMBHs with masses between \(10^2\,M_{\odot }\) and \(10^4\,M_{\odot }\), which, through inspiral onto the central MBH, would generate GWs detectable by LISA, this being a class of sources dubbed intermediate mass-ratio inspirals (IMRIs) (Amaro-Seoane et al. 2007). In principle, EMRIs and IMRIs could occur in the nuclei of any galaxy hosting a central MBH. They should be ubiquitous, since most galaxies host a central MBH and undergo a variety of merger events with other galaxies throughout their lives. IMRIs might also occur outside galactic nuclei, for example in a star cluster cannibalizing its own population of compact objects. For EMRIs and IMRIs, astrophysical modelling of their origin are in their earliest theoretical stages; in recent years a number of new astrophysical scenarios have been proposed in which they could form even outside the conventional stellar-dynamical scenarios in the galactic centre or in star clusters. These scenarios have been, for the most part, detached from the notion that their host galaxies are highly dynamical systems with a diverse range of properties, at large scales as well as at the level of galactic nuclei and star clusters. From the astrophysical perspective, this is thus the least explored, albeit potentially most exciting, class of sources in the LISA band. An assessment of the current knowledge and upcoming developments in this area is of paramount importance, to propel new research on the astrophysical impact that the discovery of EMRIs and IMRIs by LISA can have.

The joint exploitation of LISA data with data from terrestrial GW detectors and electromagnetic observations across essentially all possible wavelengths, from infrared and radio to X-ray and gamma-rays, will further enhance its astrophysical impact (Mangiagli et al. 2020). Indeed, essentially all of LISA’s individual sources have potential electromagnetic counterparts. Achieving a quantitative characterization of such counterparts, determining the feasibility of detecting them in one or more wavebands, and assessing the stage at which they would be detectable, relative to the inspiral and/or merger stage of the corresponding GW signal, are the main objectives ahead for current and upcoming research. An assessment of the current knowledge in this area is another important task.

The challenge to bring all these different pieces of knowledge into a coherent, robust picture within the next decade is huge, perhaps the most ambitious that the astrophysical community has ever faced. This review attempts to aid this ambitious, community-wide effort by assessing the status of knowledge in the modelling of LISA sources, and it summarizes our understanding of the astrophysical processes and environments relevant for the interpretation of the LISA data. Furthermore, it discusses the most important challenges ahead of us in the research of galactic binaries/multiples, massive and intermediate-mass black hole binaries, and EMRIs/IMRIs. Among these are the quest for identifying the different astrophysical formation channels for these various sources, including how these might be encoded in the LISA data stream, and the daunting multi-scale modelling needed to reconstruct the full dynamical history of such sources, from their emergence to the final inspiral phase and merger driven by GW radiation. The review material presented will help foster a critical discussion of the major gaps in our knowledge that need to be filled in the next decade, highlighting where disagreement exists between results, and what should be done next to reach beyond the current state of the art. This brings the discussion to important methodological tasks for the immediate future, from exploiting electromagnetic (EM) observations in the next decade, to improving simulation and semi-analytical techniques employed to build astrophysical models for the sources, and to refurbishing analysis and interpretation techniques for the models, for example by employing machine learning, neural networks and other modern inference strategies.

1 Stellar compact binaries and multiples

Coordinators: Silvia Toonen, Tassos Fragos, Thomas Kupfer, Thomas Tauris

1.1 Introduction and summary

Contributors: Silvia Toonen, Tassos Fragos, Thomas Kupfer, Thomas Tauris

Detection of GW emission from binary compact stars is one of the key drivers for the LISA mission. There are already at the time of writing (July 2022) about two dozen known Galactic sources, most of which are guaranteed to be detectable with LISA within a few years of its operation (Sect. 1.2). These are tight binaries (typically with orbital periods of \(P_{\text{orb}}\simeq 5{-}30\,{\text{min}}\)) of WD+WDs which give rise to continuous emission of GWs. Unlike binaries consisting of NSs and BHs, WD binaries (with their larger radii and thus lower orbital frequencies at merger) are not readily detectable by ground-based high-frequency (Hz–kHz) GW observatories, such as LIGO/Virgo/KAGRA, nor by the planned third generation of such detectors. These high-frequency detectors can observe the final a few—a few thousands orbits of inspiral (lasting a fraction of a second–minutes) and the merger event itself for NSs and BHs. Such merger events, however, are rare (of order a dozen events \({\text{Myr}}^{-1}\) for a Milky Way equivalent galaxy) and therefore they are only anticipated to be detected as extra-galactic sources, across volumes that encompass large numbers of galaxies. A major advantage of LISA is that the inspiral phase (due to orbital GW damping in the compact binaries) of the vast population of tight Galactic double WDs, NSs and BHs is in the low-frequency (\(\sim \) mHz) GW window for up to \(\sim 10^6\;{\text{year}}\) prior to their merger event. Thus a significant number of such local sources are anticipated to be detected by LISA, even though their emitted GW luminosity is relatively small compared to that of the final merger process. The possibility that LISA can measure sky locations of its sources will allow for EM follow-up observations which may result in much more precise compact object component masses, e.g., compared to high-frequency GW mergers.

Binary population synthesis studies and early data-analysis work predicts of order \(10^4\) resolved Galactic WD+WD may be detected with LISA. This population includes both detached WD+WD and those undergoing mass transfer (the so-called AM Canum Venaticorum binaries or AM CVns, see Sect. NS+NS systems are also expected to be detected by LISA. Based on the known Galactic population of tight-orbit radio pulsar binaries in combination with population synthesis predictions, an estimated number of \(10^1\)\(10^2\) NS+NS systems with a significant signal-to-noise ratio (SNR) may be detected by LISA within a 4-year mission (Sect. An even larger number of NS+WD systems is expected to be detected too, including ultra-compact X-ray binaries (UCXBs, see Sect., a sub-class of low-mass X-ray binaries, LMXBs). Binary BHs (BH+BH) detectable by LISA are strong candidates to become the first discoveries of such systems in the Milky Way. Given that LISA’s volume sensitivity for a constant SNR scales with chirp mass to the fifth power, \(\mathcal {M}^5_{\text{chirp}}\), BH+BH sources may be detected in distant galaxies, located several hundreds of megaparsecs away (see examples in Fig. 1). Interestingly enough, this fortuitous condition will therefore allow LISA to discover extra-galactic BH+BHs several years before the final merger events that LIGO/Virgo/KAGRA or Einstein Telescope/Cosmic Explorer will detect. Finally, LISA is also expected to detect rare Galactic systems such as (see Sect. 1.2.4): triple stellar systems, tight systems of WDs with exoplanets, or helium star binaries.

Fig. 1
figure 1

Image credit: Antoine Klein & Valeriya Korol

Distance to which LISA binaries can be detected as a function of GW frequency. The coloured lines represent the SNR threshold of 7 (here computed assuming a mission duration of 4 year with 100% duty cycle) for (quasi-)stationary equal-mass circular binaries of different total masses in the distance–GW frequency parameter space. The shaded range represents angle-averaged curve limits for the optimal and worst binary orientation. The ticks on the curves represent binary merger times: for merger times \(\gg 4\;{\text{year}}\) the binary will be seen by LISA as a monochromatic GW source, whereas for merger times \(< 4\;{\text{year}}\) the binary will be seen as evolving. Note in particular that evolving sources like GW190521 and GW150914 remain within the LISA band for less than the mission lifetime.

The LISA mission will provide opportunities to learn new physics and answer key scientific questions related to formation and evolutionary processes of tight binary and multiple stellar systems containing compact objects. This includes questions related to the stability and efficiency of mass transfer, common envelopes (CEs), tides and stellar angular momentum transport, irradiation effects, as well as details of their formation and destruction in core-collapse supernovae (SNe) and Type Ia SNe (and related transients), respectively. Furthermore, information about the environments of these sources will be available too, and the number and Galactic distribution of LISA sources are excellent probes to gain new knowledge on the star formation history and the structure of the Milky Way. Finally, the sheer numbers of LISA sources will provide crucial knowledge concerning their formation and evolution processes and help to place constraints on key physical parameters related to binary (and triple-star) interactions.

The current catalogue of known LISA “guaranteed sources” consists of detached WD+WDs, accreting AM CVn binaries, a hot subdwarf binary, and an UCXB. Although the sample is still small and inhomogeneous, binary population synthesis predicts a large population of multi-messenger sources that are EM bright and also detectable by LISA. This includes up to a few thousand detectable WD+WDs as well as a few tens of NS or BH binaries, with a population strongly peaking towards the Galactic Plane/Bulge. Many sources will be detected across different EM bands. Detached WD+WDs and NS+WDs are typically seen in optical and UV bands, whereas AM CVn systems and UCXBs are also seen in X-rays. NSs in compact binaries can potentially be detected as pulsars in the radio band. Therefore, in parallel with the LISA mission, we expect an EM bright future of thousands of resolved Galactic LISA binaries.

Systems with orbital periods <20 min will be the strongest Galactic LISA sources and will be detected by LISA within weeks after science operations begin. These verification binaries, as well as other so far unknown loud sources, are crucial in facilitating the functional tests of the instrument and maximize LISAs scientific output. Combined GW and EM multi-messenger studies of UCXBs will allow us to derive population properties of these systems with unprecedented quality including for the first time the effects of tides compared to GW radiation. Tides are predicted to contribute up to 10% of the orbital decay. For accreting WDs as well as NS binaries, multi-messenger observations give us the possibility to study the angular momentum transport due to mass transfer. In particular for monochromatic GW sources, EM observations are required to break degeneracies in the GW data (e.g. between masses and distance).

1.2 Classes of LISA binaries

1.2.1 Known binaries—LISA verification sources

Coordinators: Thomas Kupfer, Thomas Tauris

Contributors: Thomas Kupfer, Thomas Tauris, Silvia Toonen, Tassos Fragos

The most abundant sources in the LISA band will be binary stars with orbital periods <60 min, so-called ultra-compact binaries (UCBs). They are a class of binary stars with ultrashort orbital periods, consisting of a WD or NS primary and a compact helium-star/WD/NS secondary. A subset of the known UCBs have predicted GW strains high enough that they will be individually detected due to their strong GW signals (e.g. Burdge et al. 2020b). These LISA guaranteed sources are termed verification binaries with some being expected to be detected on a timescale of weeks or a few months (Stroeer and Vecchio 2006). Currently, we know of only about two dozen of these systems although hundreds are predicted by theory to be detectable in our Galaxy (e.g. Nelemans et al. 2004b; Timpano et al. 2006; Littenberg et al. 2013; Korol et al. 2017; Kremer et al. 2017; Kupfer et al. 2018; Lamberts et al. 2019).

At present, the catalogue of verification binaries include 13 WD+WDs, 11 semi-detached accreting WDs (AM CVn binaries, a subclass of Cataclysmic Variables, CVs), one hot subdwarf star with a WD companion, and one semi-detached UCXB. Tables 1 and 2 present an overview of the known systems with observed EM properties. Figure 2 shows the characteristic strain of the known verification binaries which reach a predicted \({\text{SNR}}\ge 5\) in LISA assuming an optimistic 10 year mission with an 80% duty cycle. So far large-scale searches for verification binaries have been conducted almost exclusively in the northern hemisphere, because large-scale survey instruments (e.g., the Sloan Digital Sky Survey, SDSS, and the Zwicky Transient Facility, ZTF) are located in the Northern Hemisphere, and mostly at high Galactic latitudes, to avoid stellar crowding. Figure 3 shows the sky location of the known verification systems which presents the strong bias towards sources in the Northern Hemisphere.

Fig. 2
figure 2

Image credit: Thomas Kupfer

Sensitivity plot for LISA assuming 10 year of observation with an 80% duty cycle showing the known binaries which reach a \({\text{SNR}}\ge 5\). Filled symbols represent eclipsing sources and open symbols represent non-eclipsing sources from Kupfer et al. (2018). The black lack solid line represents the LISA sensitivity curve. Acronyms for binaries: AM Canum Venaticorum (AM CVns), WD+WD (DWDs), subdwarf B-star (sdB) and ultracompact X-ray binary (UCXB).

In 2018, Gaia data release 2 (Gaia Collaboration et al. 2018a) announced parallaxes for \(\approx 1.3\) billion sources. The Gaia catalogue contains the distances of many of the known LISA verification binaries, allowing accurate prediction of their GW strains. Using the Gaia distances, Kupfer et al. (2018) found 13 sources will exceed an SNR of 5 after 4 year of LISA observations. This sample consists of 13 verification binaries from the current, known list; it is strongly biased and incomplete. It includes AM CVn, CR Boo, V803 Cen and ES Cet, which were all found as outliers in surveys for blue, high-Galactic latitude stars. HM Cnc and V407 Vul are the most compact known AM CVn systems and were discovered during the course of the ROSAT All-Sky Survey showing an on/off X-ray profile modulated on a period of 321 and 569 s respectively. The known WD+WD verification binaries, such as SDSS J0651 and SDSS J0935, were found as part of the extremely low-mass (ELM) WD survey (Brown et al. 2020b and references therein).

More recently, more systematic searches for UCBs were performed. UCBs show up in lightcurves with variations on timescales of the orbital period (e.g.  due to eclipses or tidal deformation of the components). Therefore, photometric surveys are well suited to identify UCBs in a homogeneous way. A number of fast cadence ground-based surveys, including the Rapid Temporal Survey (RATS; Ramsay and Hakala 2005; Barclay et al. 2011), OmegaWhite (Macfarlane et al. 2015) survey as well as the ZTF high-cadence Galactic plane survey (Masci et al. 2019; Kupfer et al. 2021), have been executed to study the variable sky down to a few minute period aiming to find UCBs and increase the number of known verification binaries. The ELM survey targets a colour-selected sample of B-type hypervelocity candidates from SDSS (Anderson et al. 2005; Roelofs et al. 2007c), which are being followed up systematically (Brown et al. 2020b and references therein). ELM WDs can be separated efficiently from the bulk of WDs with a colour selection (Brown et al. 2010).

Over the last few years the number of known verification binaries has almost doubled thanks to these large scale surveys. The two most significant contributors were the ELM survey (Brown et al. 2020b and references therein) and ZTF (Burdge et al. 2019a, 2020b, a). The ELM survey discovered six WD+WD verification binaries including SDSS J0651: a detached eclipsing system with an orbital period of 12 min. Most recently ZTF released seven new WD+WD verification binaries, five systems found as eclipsing sources. Remarkably, one of the first ZTF discoveries was the shortest orb