# Astrophysics with the Laser Interferometer Space
Antenna

Pau Amaro-Seoane, Jeff Andrews, Manuel Arca Sedda, Abbas Askar, Quentin Baghi, Razvan Balasov, Imre Bartos, Simone Bavera, Jillian Bellovary, Christopher Berry, et al.

► To cite this version:

Pau Amaro-Seoane, Jeff Andrews, Manuel Arca Sedda, Abbas Askar, Quentin Baghi, et al. Astrophysics with the Laser Interferometer Space Antenna. Living Reviews in Relativity, 2023, 26 (1), pp.2. 10.1007/s41114-022-00041-y. hal-03759542

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# Astrophysics with the Laser Interferometer Space Antenna

Pau Amaro-Seoane · Jeff Andrews · Manuel Arca Sedda · Abbas Askar · Quentin Baghi · Razvan Balasov et al. [full author details at the end of the article]

Received: 28 February 2022 / Accepted: 21 October 2022 / Published online: 14 March 2023
© The Author(s) 2023, corrected publication 2023

## Abstract

The Laser Interferometer Space Antenna (LISA) will be a transformative experiment for gravitational wave astronomy, and, as such, it will offer unique opportunities to address many key astrophysical questions in a completely novel way. The synergy with ground-based and space-born instruments in the electromagnetic domain, by enabling multi-messenger observations, will add further to the discovery potential of LISA. The next decade is crucial to prepare the astrophysical community for LISA’s first observations. This review outlines the extensive landscape of astrophysical theory, numerical simulations, and astronomical observations that are instrumental for modeling and interpreting the upcoming LISA datastream. To this aim, the current knowledge in three main source classes for LISA is reviewed; ultra-compact stellar-mass binaries, massive black hole binaries, and extreme or interme-diate mass ratio inspirals. The relevant astrophysical processes and the established modeling techniques are summarized. Likewise, open issues and gaps in our understanding of these sources are highlighted, along with an indication of how LISA could help making progress in the different areas. New research avenues that LISA itself, or its joint exploitation with upcoming studies in the electromagnetic domain, will enable, are also illustrated. Improvements in modeling and analysis approaches, such as the combination of numerical simulations and modern data science techniques, are discussed. This review is intended to be a starting point for using LISA as a new discovery tool for understanding our Universe.

**Keywords** Black holes · Gravitational waves · Stellar remnants · Multi-messenger · Extreme mass ratio in-spirals

## Abbreviations

AGN Active galactic nucleus/nuclei

AM CVn AM Canum Venaticorum

<footnote>Extended author information available on the last page of the article</footnote>
<table><tr><td>AU</td><td>Astronomical Unit</td></tr><tr><td>BD</td><td>Brown dwarf (plural: BDs)</td></tr><tr><td>BH</td><td>Black hole (plural: BHs)</td></tr><tr><td>BH+BH</td><td>(Stellar-mass) binary black hole (plural: BH+BHs)</td></tr><tr><td>CDM</td><td>Cold dark matter</td></tr><tr><td>CE</td><td>Common envelope</td></tr><tr><td>CO</td><td>Compact object</td></tr><tr><td>COM</td><td>Centre-of-mass</td></tr><tr><td>CV</td><td>Cataclysmic variable</td></tr><tr><td>DM</td><td>Dark matter</td></tr><tr><td>ELM</td><td>Extremely low-mass</td></tr><tr><td>EM</td><td>Electromagnetic</td></tr><tr><td>EMRI</td><td>Extreme mass ratio inspiral</td></tr><tr><td>EOS</td><td>Equation of state</td></tr><tr><td>GR</td><td>General relativity/relativistic</td></tr><tr><td>GW</td><td>Gravitational wave (plural: GWs)</td></tr><tr><td>HMXB</td><td>High-mass X-ray binary</td></tr><tr><td>IMBH</td><td>Intermediate-mass black hole</td></tr><tr><td>IMF</td><td>Initial mass function</td></tr><tr><td>IMRI</td><td>Intermediate mass-ratio inspiral</td></tr><tr><td>IR</td><td>Infra-red</td></tr><tr><td>ISCO</td><td>Innermost stable circular orbit</td></tr><tr><td>LMXB</td><td>Low-mass X-ray binary</td></tr><tr><td>MBH</td><td>Massive black hole</td></tr><tr><td>MBHB</td><td>Massive black hole binary</td></tr><tr><td>MHD</td><td>Magnetohydrodynamics/magnetohydrodynamic</td></tr><tr><td>MSP</td><td>Millisecond radio pulsar</td></tr><tr><td>MW</td><td>Milky Way</td></tr><tr><td>MS</td><td>Main sequence</td></tr><tr><td>NFW</td><td>Navarro–Frenk–White</td></tr><tr><td>NS</td><td>Neutron star (plural: NSs)</td></tr><tr><td>NS+NS</td><td>Double neutron star (plural: NS+NSs)</td></tr><tr><td>NSC</td><td>Nuclear star cluster</td></tr><tr><td>PN</td><td>Post-Newtonian</td></tr><tr><td>Pop III</td><td>Population III</td></tr><tr><td>PTA</td><td>Pulsar timing array</td></tr><tr><td>RF</td><td>Radiative feedback</td></tr><tr><td>RLO</td><td>Roche-lobe overflow</td></tr><tr><td>SFH</td><td>Star-formation history</td></tr><tr><td>SGWB</td><td>Stochastic gravitational wave background</td></tr><tr><td>SMBH</td><td>Supermassive black hole</td></tr><tr><td>SMS</td><td>Supermassive star</td></tr><tr><td>SN</td><td>Supernova (plural: SNe)</td></tr><tr><td>SNR</td><td>Signal-to-noise ratio</td></tr><tr><td>SPH</td><td>Smoothed-particle hydrodynamics</td></tr></table>
<table><tr><td>SSO</td><td>Substellar object</td></tr><tr><td>TDE</td><td>Tidal disruption event</td></tr><tr><td>UCB</td><td>Ultra-compact binary</td></tr><tr><td>UCXB</td><td>Ultra-compact X-ray binary</td></tr><tr><td>UV</td><td>Ultra-violet</td></tr><tr><td>WD</td><td>White dwarf (plural: WDs)</td></tr><tr><td>WD+WD</td><td>Double white dwarf (plural: WD+WDs)</td></tr><tr><td>XMRB</td><td>Extremely large mass-ratio burst</td></tr><tr><td>XMRI</td><td>Extremely large mass-ratio inspiral</td></tr><tr><td>ZKL</td><td>Von Zeipel-Kozai-Lidov</td></tr><tr><td>b-EMRI</td><td>Binary-extreme mass ratio inspiral</td></tr></table>

Contents

<table><tr><td>General introduction</td><td>4</td></tr><tr><td>1 Stellar compact binaries and multiples</td><td>7</td></tr><tr><td>1.1 Introduction and summary</td><td>7</td></tr><tr><td>1.2 Classes of LISA binaries</td><td>10</td></tr><tr><td>1.2.1 Known binaries—LISA verification sources</td><td>10</td></tr><tr><td>1.2.2 Detached binaries</td><td>14</td></tr><tr><td>1.2.3 Interacting binaries</td><td>22</td></tr><tr><td>1.2.4 Other potential sources</td><td>25</td></tr><tr><td>1.3 Formation of LISA binaries</td><td>29</td></tr><tr><td>1.3.1 Isolated binaries</td><td>29</td></tr><tr><td>1.3.2 Sources in clusters</td><td>36</td></tr><tr><td>1.3.3 Triple stellar systems</td><td>38</td></tr><tr><td>1.4 Expected LISA observations: numbers and rates</td><td>40</td></tr><tr><td>1.4.1 Binary's detectability</td><td>40</td></tr><tr><td>1.4.2 Detection and parameter estimation expectations</td><td>42</td></tr><tr><td>1.5 Synergies</td><td>44</td></tr><tr><td>1.5.1 Synergies with EM observations</td><td>44</td></tr><tr><td>1.5.2 Synergies with other GW detectors</td><td>52</td></tr><tr><td>1.6 Technical aspects</td><td>55</td></tr><tr><td>1.6.1 How to distinguish between different compact binaries?</td><td>55</td></tr><tr><td>1.6.2 Foreground sources</td><td>57</td></tr><tr><td>1.6.3 Tools</td><td>59</td></tr><tr><td>1.7 Scientific objectives</td><td>60</td></tr><tr><td>1.7.1 Constraining stellar and binary interaction physics</td><td>60</td></tr><tr><td>1.7.2 LISA sources as galactic probes</td><td>76</td></tr><tr><td>2 Massive black hole binaries</td><td>81</td></tr><tr><td>2.1 Introduction</td><td>81</td></tr><tr><td>2.2 MBHs and their path to coalescence</td><td>84</td></tr><tr><td>2.2.1 The galaxy merger and the large-scale orbital decay at kpc scales</td><td>85</td></tr><tr><td>2.2.2 Orbital decay after binary formation at pc scales</td><td>94</td></tr><tr><td>2.2.3 The GW-emission phase at mpc scale</td><td>106</td></tr><tr><td>2.3 MBH origin and growth across the cosmic time</td><td>111</td></tr><tr><td>2.3.1 MBH seeds: formation mechanisms</td><td>112</td></tr></table>
<table><tbody><tr><td>2.3.2</td><td>MBH growth across time and space</td><td>117</td></tr><tr><td>2.4</td><td>Statistics on MBH mergers</td><td>123</td></tr><tr><td>2.4.1</td><td>Modelling MBH evolution in a cosmological context</td><td>126</td></tr><tr><td>2.4.2</td><td>State of the art on MBH merger rates from cosmological simulations</td><td>132</td></tr><tr><td>2.4.3</td><td>How to advance and optimize the scientific return of LISA</td><td>138</td></tr><tr><td>2.5</td><td>Multimessenger on single events: what do we learn about BH physics from the multimesenger view of the coalescence of MBHs?</td><td>142</td></tr><tr><td>2.5.1</td><td>The expected multimessenger signatures</td><td>143</td></tr><tr><td>2.5.2</td><td>Multimessenger observation strategy for MBHB mergers with LISA</td><td>148</td></tr><tr><td>2.5.3</td><td>The path towards LISA</td><td>152</td></tr><tr><td>2.6</td><td>Multimessenger view of MBH populations</td><td>155</td></tr><tr><td>2.6.1</td><td>A landscape of new missions to understand MBH formation, growth, and environment</td><td>156</td></tr><tr><td>2.6.2</td><td>Preparing LISA using prior knowledge on MBHBs from current and upcoming missions</td><td>164</td></tr><tr><td>2.6.3</td><td>The path towards LISA</td><td>168</td></tr><tr><td>3</td><td>Extreme and intermediate mass-ratio inspirals</td><td>169</td></tr><tr><td>3.1</td><td>Introduction</td><td>169</td></tr><tr><td>3.1.1</td><td>Guaranteed science with the detection of EMRIs</td><td>171</td></tr><tr><td>3.1.2</td><td>Plausible science with the detection of EMRIs</td><td>173</td></tr><tr><td>3.1.3</td><td>Speculative science with the detection of EMRIs</td><td>174</td></tr><tr><td>3.1.4</td><td>Data analysis &amp; waveform modelling</td><td>175</td></tr><tr><td>3.2</td><td>Formation channels</td><td>177</td></tr><tr><td>3.2.1</td><td>Gas-poor dynamics: Galactic nuclei including dwarfs, and globular clusters</td><td>177</td></tr><tr><td>3.2.2</td><td>Formation of EMRIs in gas-poor galactic nuclei</td><td>178</td></tr><tr><td>3.2.3</td><td>Physics of IMRI formation: Dwarf galaxies, galactic nuclei and globular clusters</td><td>183</td></tr><tr><td>3.2.4</td><td>Formation of EMRIs and IMRIs in gas-rich galactic nuclei: AGN discs</td><td>188</td></tr><tr><td>3.2.5</td><td>Alternative formation scenarios</td><td>193</td></tr><tr><td>3.3</td><td>Multimessenger prospects</td><td>196</td></tr><tr><td>3.3.1</td><td>Tidal disruption events</td><td>196</td></tr><tr><td>3.3.2</td><td>Electromagnetic counterparts of light IMRIs in AGN discs</td><td>198</td></tr><tr><td>3.3.3</td><td>FeKα lines (or other EM signatures) as probes of small separation MBH-IMBH binaries</td><td>199</td></tr><tr><td>3.3.4</td><td>EMRIs containing a pulsar</td><td>199</td></tr><tr><td>3.4</td><td>Environmental effects on waveforms</td><td>201</td></tr><tr><td>3.4.1</td><td>Gas torques</td><td>202</td></tr><tr><td>3.4.2</td><td>Many-body interactions</td><td>203</td></tr><tr><td>3.4.3</td><td>Moving sources</td><td>204</td></tr><tr><td>3.4.4</td><td>Dark matter as an environmental effect</td><td>205</td></tr><tr><td>3.4.5</td><td>Astrophysical chaos</td><td>205</td></tr><tr><td>3.4.6</td><td>Environment versus PN/self-force degeneracies</td><td>206</td></tr><tr><td>3.5</td><td>EMRI background</td><td>212</td></tr><tr><td>3.5.1</td><td>TDE background</td><td>213</td></tr><tr><td>3.6</td><td>Conclusions</td><td>214</td></tr><tr><td colspan="2">General summary</td><td>218</td></tr><tr><td colspan="2">References</td><td>219</td></tr></tbody></table>

## 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 ~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 ~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 $M_{BH} \gtrsim 10^8 M_{\odot}$, 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 $\lesssim 10^{-5}$ few–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 $P_{\text{orb}} \approx 5-30 \text{ min})$ 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 $Myr^{-1}$ 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 $\sim 10^6$ 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. 1.2.3.1). 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. 1.2.2.3). An even larger number of NS+WD systems is expected to be detected too, including ultra-compact X-ray binaries (UCXBs, see Sect. 1.2.3.2, 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, $M_{\text{chirp}}^5$, 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.

The LISA mission will provide opportunities to learn new physics and answer key scientific questions related to formation and evolutionary processes of tight binary
<figure><img src="image_3.png" /><figcaption>**Fig. 1** 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 $\gg 4$ year the binary will be seen by LISA as a monochromatic GW source, whereas for merger times $< 4$ 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. Image credit: Antoine Klein & Valeriya Korol</figcaption></figure>

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 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
<table><caption>**Table 1** Physical properties (orbital periods, component masses, inclination angles) of the known verification binaries which reach a SNR > 5 after a 10 year LISA mission with 80% duty cycle</caption><thead><tr><th>Source</th><th>P<sub>orb</sub>(s)</th><th>M<sub>1</sub> (M<sub>☉</sub>)</th><th>M<sub>2</sub> (M<sub>☉</sub>)</th><th>t (deg)</th><th>References</th></tr></thead><tbody><tr><td colspan="6"><strong>AM CVn type</strong></td></tr><tr><td>HM Cnc</td><td>321.5</td><td>0.55</td><td>0.27</td><td>≈38</td><td>1, 2</td></tr><tr><td>V407 Vul</td><td>569.4</td><td>[0.8 ± 0.1]</td><td>[0.177 ± 0.071]</td><td>[60]</td><td>3</td></tr><tr><td>ES Cet</td><td>620.2</td><td>[0.8 ± 0.1]</td><td>[0.161 ± 0.064]</td><td>[60]</td><td>4</td></tr><tr><td>SDSS J135154.46-064309.0</td><td>943.8</td><td>[0.8 ± 0.1]</td><td>[0.100 ± 0.040]</td><td>[60]</td><td>5</td></tr><tr><td>AM CVn</td><td>1028.7</td><td>0.68 ± 0.06</td><td>0.125 ± 0.012</td><td>43 ± 2</td><td>6, 7</td></tr><tr><td>SDSS J190817.07+394036.4</td><td>1085.7</td><td>[0.8 ± 0.1]</td><td>[0.085 ± 0.034]</td><td>10-20</td><td>8, 9</td></tr><tr><td>HP Lib</td><td>1102.7</td><td>0.49-0.80</td><td>0.048-0.088</td><td>26-34</td><td>10,11</td></tr><tr><td>PTF1 J191905.19+481506.2</td><td>1347.3</td><td>[0.8 ± 0.1]</td><td>[0.066 ± 0.026]</td><td>[60]</td><td>12</td></tr><tr><td>CXOGBS J175107.6-294037</td><td>1375.0</td><td>[0.8 ± 0.1]</td><td>[0.064 ± 0.026]</td><td>[60]</td><td>13</td></tr><tr><td>CR Boo</td><td>1471.3</td><td>0.67-1.10</td><td>0.044-0.088</td><td>30</td><td>10,14</td></tr><tr><td>V803 Cen</td><td>1596.4</td><td>0.78-1.17</td><td>0.059-0.109</td><td>12-15</td><td>11</td></tr><tr><td colspan="6"><strong>Detached double WDs</strong></td></tr><tr><td>ZTF J1539+5027</td><td>414.8</td><td>0.610<sup>+0.017</sup><sub>-0.022</sub></td><td>0.210 ± 0.015</td><td>84.15<sup>+0.64</sup><sub>-0.57</sub></td><td>15</td></tr><tr><td>ZTF J2243+5242</td><td>527.9</td><td>0.349<sup>+0.093</sup><sub>-0.074</sub></td><td>0.384<sup>+0.114</sup><sub>-0.074</sub></td><td>81.88<sup>+1.31</sup><sub>-0.69</sub></td><td>16</td></tr><tr><td>SDSS J065133.34+284423.4</td><td>765.5</td><td>0.247 ± 0.015</td><td>0.49 ± 0.02</td><td>86.9<sup>+1.6</sup><sub>-1.0</sub></td><td>17, 18</td></tr><tr><td>ZTF J0538+1953</td><td>866.6</td><td>0.45 ± 0.05</td><td>0.32 ± 0.03</td><td>85.43<sup>+0.07</sup><sub>-0.09</sub></td><td>19</td></tr><tr><td>SDSS J093506.92+441107.0</td><td>1188.0</td><td>0.312 ± 0.019</td><td>0.75 ± 0.24</td><td>[60]</td><td>20, 21</td></tr><tr><td>SDSS J2322+0509</td><td>1201.4</td><td>0.34 ± 0.02</td><td>> 0.17</td><td>[60]</td><td>22</td></tr><tr><td>PTF J0533+0209</td><td>1234.0</td><td>0.652<sup>+0.037</sup><sub>-0.040</sub></td><td>0.167 ± 0.030</td><td>72.8<sup>+0.8</sup><sub>-1.4</sub></td><td>19, 23</td></tr><tr><td>ZTF J2029+1534</td><td>1252.0</td><td>0.32 ± 0.04</td><td>0.3 ± 0.04</td><td>86.64<sup>+0.70</sup><sub>-0.40</sub></td><td>19</td></tr><tr><td>ZTF J0722-1839</td><td>1422.5</td><td>0.38 ± 0.04</td><td>0.33 ± 0.03</td><td>89.66 ± 0.22</td><td>19</td></tr><tr><td>ZTF J1749+0924</td><td>1586.0</td><td>0.40<sup>+0.07</sup><sub>-0.05</sub></td><td>0.28<sup>+0.05</sup><sub>-0.04</sub></td><td>85.45<sup>+1.40</sup><sub>-1.15</sub></td><td>19</td></tr><tr><td>SDSS J163030.58+423305.7</td><td>2389.8</td><td>0.298 ± 0.019</td><td>0.76 ± 0.24</td><td>[60]</td><td>20, 24</td></tr><tr><td>SDSS J1235+1543</td><td>3172.6</td><td>0.35 ± 0.01</td><td>0.27<sup>+0.06</sup><sub>-0.02</sub></td><td>27 ± 3.8</td><td>25, 26</td></tr><tr><td>SDSS J092345.59+302805.0</td><td>3883.7</td><td>0.275 ± 0.015</td><td>0.76 ± 0.23</td><td>[60]</td><td>20, 27</td></tr><tr><td colspan="6"><strong>Hot subdwarf binaries</strong></td></tr><tr><td>CD-30°11223</td><td>4231.8</td><td>0.54 ± 0.02</td><td>0.79 ± 0.01</td><td>82.9 ± 0.4</td><td>28</td></tr><tr><td colspan="6"><strong>Ultracompact X-ray binaries</strong></td></tr><tr><td>4U1820-30</td><td>685.0</td><td>[1.4]</td><td>[0.069]</td><td>[60]</td><td>29, 30</td></tr></tbody></table>

Masses and inclination angles in brackets are assumed and based on evolutionary stage and mass ratio estimations

[1] Strohmayer (2005), [2] Roelofs et al. (2010), [3] Marsh and Steeghs (2002), [4] Espaillat et al. (2005), [5] Green et al. (2018a), [6] Skillman et al. (1999), [7] Roelofs et al. (2006), [8] Fontaine et al. (2011), [9] Kupfer et al. (2015), [10] Roelofs et al. (2007b), [11] Solanki et al. (2021), [12] Levitan et al. (2014), [13] Wevers et al. (2016), [14] Provençal et al. (1997), [15] Burdge et al. (2019a), [16] Burge et al. (2020a), [17] Brown et al. (2011), [18] Hermes et al. (2012), [19] Burge et al. (2020b), [20] Brown et al. (2016), [21] Kilic et al. (2014), [22] Brown et al. (2020a), [23] Burge et al. (2019b), [24] Kilic et al. (2011), [25] Breedt et al. (2017), [26] Kilic et al. (2017), [27] Brown et al. (2010), [28] Geier et al. (2013), [29] Stella (1987), [30] Chen et al. (2020a)
<table><caption>**Table 2** Measured EM properties (Galactic coordinates, GW frequency, magnitudes and parallaxes from Gaia early data release 3 (Gaia Collaboration et al. 2021) of the known verification binaries which reach a SNR > 5 after a 10 year LISA mission with 80% duty cycle</caption><thead><tr><th>Source</th><th>f<sub>GW</sub><br>(mHz)</th><th>l<sub>Gal</sub> (deg)</th><th>b<sub>Gal</sub> (deg)</th><th>Gaia G<br>(mag)</th><th>ω (mas)</th></tr></thead><tbody><tr><td colspan="6"><em>AM CVn type</em></td></tr><tr><td>HM Cnc</td><td>6.22</td><td>206.9246</td><td>23.3952</td><td>20.92</td><td>—</td></tr><tr><td>V407 Vul</td><td>3.51</td><td>57.7281</td><td>6.4006</td><td>19.36</td><td>0.0978 ± 0.2384</td></tr><tr><td>ES Cet</td><td>3.22</td><td>168.9684</td><td>— 65.8632</td><td>16.80</td><td>0.5606 ± 0.0677</td></tr><tr><td>SDSS J135154.46–064309.0</td><td>2.12</td><td>328.5021</td><td>53.1240</td><td>18.72</td><td>0.6584 ± 0.2197</td></tr><tr><td>AM CVn</td><td>1.94</td><td>140.2343</td><td>78.9382</td><td>14.06</td><td>3.3106 ± 0.0303</td></tr><tr><td>SDSS J190817.07+<br>394036.4</td><td>1.84</td><td>70.6664</td><td>13.9349</td><td>16.22</td><td>1.0232 ± 0.0335</td></tr><tr><td>HP Lib</td><td>1.81</td><td>352.0561</td><td>32.5467</td><td>13.60</td><td>3.5674 ± 0.0313</td></tr><tr><td>PTF1 J191905.19+<br>481506.2.</td><td>1.48</td><td>79.5945</td><td>15.5977</td><td>19.75</td><td>0.6229 ± 0.2385</td></tr><tr><td>CXOGBS J175107.6<br>–294037</td><td>1.45</td><td>359.9849</td><td>— 1.4108</td><td>16.27</td><td>0.8591 ± 0.1733</td></tr><tr><td>CR Boo</td><td>1.34</td><td>340.9671</td><td>66.4884</td><td>15.47</td><td>2.8438 ± 0.0367</td></tr><tr><td>V803 Cen</td><td>1.25</td><td>309.3671</td><td>20.7262</td><td>15.73</td><td>3.4885 ± 0.0599</td></tr><tr><td colspan="6"><em>Detached double WDs</em></td></tr><tr><td>ZTF J1539+5027</td><td>4.82</td><td>80.7746</td><td>50.5819</td><td>20.40</td><td>— 0.4926 ± 0.5726</td></tr><tr><td>ZTF J2243+5242</td><td>3.79</td><td>104.1514</td><td>— 5.4496</td><td>20.55</td><td>— 1.2372 ± 0.6578</td></tr><tr><td>SDSS J065133.34+<br>284423.4</td><td>2.61</td><td>186.9277</td><td>12.6886</td><td>19.28</td><td>1.0071 ± 0.3091</td></tr><tr><td>ZTF J0538+1953</td><td>2.31</td><td>186.8104</td><td>— 6.2213</td><td>18.80</td><td>0.9617 ± 0.2866</td></tr><tr><td>SDSS J093506.92+<br>441107.0</td><td>1.68</td><td>176.0796</td><td>47.3776</td><td>17.80</td><td>2.7034 ± 0.6648</td></tr><tr><td>SDSS J2322+0509</td><td>1.66</td><td>85.9507</td><td>— 51.2104</td><td>18.75</td><td>1.1558 ± 0.2244</td></tr><tr><td>PTF J0533+0209</td><td>1.62</td><td>201.8012</td><td>— 16.2238</td><td>19.05</td><td>0.7902 ± 0.2396</td></tr><tr><td>ZTF J2029+1534</td><td>1.60</td><td>58.5836</td><td>— 13.4655</td><td>20.47</td><td>0.1240 ± 0.9893</td></tr><tr><td>ZTF J0722–1839</td><td>1.40</td><td>232.9930</td><td>— 1.8604</td><td>19.05</td><td>0.6996 ± 0.2457</td></tr><tr><td>ZTF J1749+0924</td><td>1.26</td><td>34.5093</td><td>17.9025</td><td>20.47</td><td>— 0.2961 ± 0.8222</td></tr><tr><td>SDSS J163030.58+<br>423305.7</td><td>0.84</td><td>67.0760</td><td>43.3604</td><td>19.18</td><td>1.1748 ± 0.1952</td></tr><tr><td>SDSS J1235+1543</td><td>0.63</td><td>284.5186</td><td>78.0320</td><td>17.52</td><td>2.2504 ± 0.1389</td></tr><tr><td>SDSS J092345.59+<br>302805.0</td><td>0.51</td><td>195.8199</td><td>44.7754</td><td>15.92</td><td>3.4795 ± 0.0648</td></tr><tr><td colspan="6"><em>Hot subdwarf binaries</em></td></tr><tr><td>CD–30°11223</td><td>0.47</td><td>322.4875</td><td>28.9379</td><td>12.30</td><td>2.8198 ± 0.0516</td></tr><tr><td colspan="6"><em>Ultracompact X-ray binaries</em></td></tr><tr><td>4U1820–30</td><td>2.92</td><td>2.7896</td><td>— 7.9144</td><td>15.41</td><td>— 0.7676 ± 0.2164</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>**Fig. 2** Sensitivity plot for LISA assuming 10 year of observation with an 80% duty cycle showing the known binaries which reach a 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). Image credit: Thomas Kupfer</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3** Sky position of the verification binaries. The sky positions show a clear bias towards the northern hemisphere and to higher Galactic latitudes. The black line indicates the Galactic equator and $|b| = 10$ deg, with the Galactic Centre located at the black cross. See caption in Fig. 2 for explanation of acronyms. Image credit: Thomas Kupfer</figcaption></figure>

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.
In 2018, Gaia data release 2 (Gaia Collaboration et al. 2018a) announced parallaxes for ≈ 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 orbital period eclipsing WD+WD known to date, ZTF J1539+5027, with an orbital period of just 6.91 min (Burdge et al. 2019a).

## 1.2.2 Detached binaries

Coordinators: Ashley Ruiter, Ross Church

Contributors: Ashley Ruiter (1.2.2.1), Thomas Tauris (1.2.2.2–4), Jeff Andrews (1.2.2.3), Simone Bavera (1.2.2.4), Ross Church (1.2.2.2), Tassos Fragos (1.2.2.4), Gijs Nelemans (1.2.2.1), Milton Ruiz (1.2.2.3), Alberto Sesana (1.2.2.4), Antonios Tsokaros (1.2.2.3), Shenghua Yu (1.2.2.3)
### 1.2.2.1 WD+WD systems
For over three decades it has been known that WD+WD binaries will be the dominant contributor to signals detected by a space-based GW observatory (Hils et al. 1990). While most extra-galactic sources (Farmer and Phinney 2003) as well as a significant fraction of those in the Galactic halo (Rosswog et al. 2009a) are likely too distant to be individually detected by LISA, a large portion of the frequency band observed by LISA will be swamped with GWs from millions of WD+WDs existing in the Galactic disc and bulge. At low frequencies, the combined signal of these millions of WD+WDs will populate just a few frequency bins and merge to form an unresolved confusion foreground (often referred to as the galactic foreground or the galactic confusion noise), with louder resolvable sources standing out above the confusion (see also Sect. 1.6.2). Together with high-frequency sources a large number these form ~$10^4$, (e.g., Nelemans et al. 2001c; Farmer and Phinney 2003; Ruiter et al. 2010; Korol et al. 2017) of resolved WD+WDs and we now discuss these key sources in more detail.

WD+WDs were discovered in the late 1980s and initially were dominated by low-mass ($\lesssim 0.4 M_\odot$) helium-core (He-core) WDs that cannot be formed in single-star evolution within a Hubble time, and, thus, were the targets for radial velocity searches for binarity amongst known WDs (Marsh et al. 1995). Later, (more) unbiased surveys were done, e.g. the Supernova Ia Progenitor surveyY (SPY,

<figure><img src="image_6.png" /><figcaption>**Fig. 4** Characteristic strain amplitude vs GW frequency for LISA. Evolutionary tracks are for an UCXB (blue) and an AM CVn system (magenta) at a distance of $d_L = 1$ kpc. Their slope on the inspiral leg is $\propto f_{gw}^{7/6}$. The stars along the tracks represent (with increasing GW frequency) onset LMXB/CV stage, termination LMXB/CV stage, and onset UCXB/AM CVn stage. The evolutionary timescales along these tracks are shown in Fig. 5. The LISA sensitivity curve (red line, SNR = 1) is based on 4 years of observations. The grey curves are for the UCXB at $d_L = 15$ kpc and 780 kpc (M31), respectively. Comparison tracks are shown for an MBH merger (green) and GW150914 (orange). Their inspiral slopes are $\propto f_{gw}^{-1/6}$. Data from LISA verification sources (Kupfer et al. 2018) include detached double WD binaries (solid squares), AM CVn systems (open circles), and a hot subdwarf binary (solid triangle). Image reproduced with permission from Tauris (2018), copyright by APS</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 5** GW frequency vs dynamical chirp mass for an UCXB and two AM CVn systems, based on detailed mass transfer calculations (including finite-temperature effects of the WD donor star) using the MESA code (Tauris 2018). The end points of the first mass-transfer phases (LMXB and CV) are indicated by red triangles; the starting points of the second mass-transfer phases (UCXB and AM CVn) are indicated by green circles. The time marks along the AM CVn tracks are for the same values as indicated for the UCXB system, unless stated otherwise (in Myr). Time zero is defined at the onset of the second mass-transfer phase. The maximum GW frequencies (strongest LISA signal) in these three examples are 5.45 mHz (UCXB), 5.64 mHz (AM CVn1), and 5.72 mHz (AM CVn2) corresponding orbital periods of 6.12 min, 5.91 min and 5.83 min, respectively. The frequency at the onset of the RLO (green circles) depends on the temperature of the low-mass He WD donor ($T_{\text{eff}} = 10\ 850\ \text{K},\ 9\ 965\ \text{K}\ \text{and}\ 8\ 999\ \text{K}$, respectively)</figcaption></figure>

Napiwotzki et al. 2020, and references therein) and studies using SDSS, discovering also more massive WDs. Over the last decade, it has become more clear that previously-undetected WD+WD systems (and their progenitors, e.g. double-core planetary nebulae) are more easily detectable with today's sophisticated instrumentation (Wesson et al. 2018). A dramatic increase in the number of WD+WDs has come from the ELM WD survey (Kilic et al. 2012; Bell et al. 2017; Brown et al. 2020b) that targets a part of the parameter space in colour–colour diagrams that is occupied by (subdwarf) B-stars, but also by very low mass (below ~0.3 $M_\odot$) WDs that are still approaching the cooling track and are thus relatively large and bright (Istrate et al. 2014b, 2016). In total, the ELM survey alone has discovered 98 WD+WDs so far (Brown et al. 2020b).

Over the past couple of years, ZTF (Bellm et al. 2019) has facilitated a rapid growth in the population of known WD+WDs with orbital periods under an hour. Three of the sources discovered by ZTF so far (Burdge et al. 2020a), the eclipsing WD+WDs: ZTF J1539+5027 ($P_b = 6.91$ min), ZTF J2243+5242 ($P_b = 8.80$ min) and ZTF J0538+1953 ($P_b = 14.4$ min), should all be detected by LISA with a high SNR, enabling precise parameter estimation using GWs (Littenberg and Cornish 2019).
The detached WD+WDs may consist of a pair of He-core WDs, carbon/oxygen-core (C/O-core) WDs, oxygen/neon/magnesium-core (O/Ne/Mg-core) WDs, or any mixed combination thereof. For some systems, LISA measurements of the orbital-decay rate will yield the chirp mass for a given system, which can be combined with EM observations to reveal individual WD component masses. The distribution of WD masses (and their mass ratios), along with the number of detectable sources in a local volume, will provide important information to help understand their formation history (see Sect. 1.3). With enough detached WD+WDs in a sample, it may even be possible to set limits on mass-transfer efficiencies and CE physics (Sect. 1.7) through characterisation of chirp mass distributions (Ruiter et al. 2019). Furthermore, the detected WD+WDs will provide unique information on the formation of progenitors of R Coronae Borealis stars, thought to be formed by the merger of two WDs (e.g., Tisserand et al. 2020), massive carbon-enhanced WDs (Kawka et al. 2020), Type Ia SNe (see Sect. 1.7.1.6), and other transients.

Over the last three decades, several works have made predictions about the scientific impact of LISA detections of Galactic WD+WDs. Different binary evolution population synthesis studies have uncovered how the WD+WD population will look to LISA in terms of characteristic strain amplitude (Nelemans et al. 2001b, 2004b; Yu and Jeffery 2010; Lamberts et al. 2019; Korol et al. 2020), spectral density (Breivik et al. 2020b), as well as how different populations of WD+WDs contribute to the spectral amplitude signal (Rosswog et al. 2009a; Ruiter et al. 2010).

Detached WD+WD binaries that are resolvable with LISA are expected to be on par with or slightly outnumber the resolvable interacting WD+WD binaries (Nelemans et al. 2001a, 2004b; Ruiter et al. 2010), and will be the sole WD+WD contributors to the LISA signal at GW frequencies below ~2 × 10⁻⁴ Hz. Kremer et al. (2017) found that a number of *mass-transferring* WD+WDs (Sect. 1.2.3) will be resolvable with LISA (~200–3000 for SNRs between 10 and 5, respectively), many of which are likely to exhibit a negative chirp (caused by orbital widening)—a diagnostic not applicable for detached WD+WDs. Finally, we expect that the number of detached WD+WDs, composed of a light He-core WD and a more massive C/O-core WD (or possibly an O/Ne/Mg-core WD) detected by LISA must be in accordance with the number of similar interacting AM CVn systems that LISA will detect, given their evolutionary connection (the detached systems being the precursors of the interacting WD+WDs, see Fig. 6). The transitional GW frequency between these two populations (detached and interacting) depends on the mass and temperature of the lighter (last-formed) WD, see examples in Figs. 4 and 5.

### 1.2.2.2 NS+WD and BH+WD systems

The known population of Galactic NS+WD systems can be divided into two classes. Systems with: (i) massive WDs (O/Ne/Mg-core or C/O-core WDs, typically more massive than 0.7 $M_\odot$), and (ii) low-mass He-core WDs (typically less massive than 0.3 $M_\odot$). The massive NS+WD systems can again be subdivided into two populations, depending on the formation order of

<footnote n="1">¹ The empirical rate and its uncertainty measured by the LIGO network of detectors is currently based on only two events (GW170817 and GW190425) but is anticipated to improve substantially in the coming decade.</footnote>
the WD and the NS. The NS+WD systems are observed as binary radio pulsars and the formation order can be clearly distinguished from the properties of the pulsar: if the pulsar has a strong B-field and an eccentric orbit (e.g. Tauris and Sennels 2000; Church et al. 2006), it is the last-formed compact object, whereas if the pulsar is (mildly) recycled with a low-B-field and a fairly rapid spin, and in a near-circular orbit, it is the first-formed compact object (Tauris et al. 2012). For LISA detections, the formation order is irrelevant and among both types of systems examples are known to merge within a Hubble time, thus producing a bright LISA source well before their final merger.

Among the detached low-mass He-core WDs with NS companions, the systems in relatively tight orbits are completely dominated by millisecond radio pulsars. According to the ATNF Pulsar Catalogue (Manchester et al. 2005), there are about 120 such systems known in the MW disc, a handful of which will merge within a Hubble time, producing a bright detached LISA source (depending on their distance) for approximately the last several tens of Myr of the inspiral, before an UCXB is formed (Sect. 1.2.3.2). Based on the observed population of radio pulsars and their selection effects, Tauris (2018) argue that LISA could detect about 50 of these systems while still detached, before they become UCXBs and widen their orbits again, resulting in a negative chirp of the GW signal (see Figs. 4 and 5).

At present, we do not know of any detached BH+WD binaries. However, this is probably due to observational selection effects since the only EM radiation we would expect from such detached systems would be from the cooling of the WD companion —unlike the situation for semi-detached systems or systems containing NSs, which can be detected in X-rays and radio waves, respectively. Nevertheless, several Galactic LMXBs are known with low-mass donor stars and BH accretors (McClintock and Remillard 2006) and thus we expect many of these systems to leave detached BH+WD systems, possibly (although still to be proven) in tight orbits that LISA will detect. A more viable formation channel for more massive WDs in tight orbits with BHs is formation via a CE. Optical follow-up observations of the WD companion, in combination with the measured chirp mass, will constrain the BH mass in these systems. Early simulations (Nelemans et al. 2001b) predict a Galactic merger rate of BH+WD binaries of order ~100 Myr⁻¹ and thus roughly ~100 such systems detectable by LISA.

### 1.2.2.3 NS+NS systems

The known population NS+NS systems so far only manifest themselves as radio pulsars. The first one of these (PSR B1913+16, the *Hulse–Taylor Pulsar*) was discovered in 1974 (Hulse and Taylor 1975). According to the ATNF Pulsar Catalogue (Manchester et al. 2005), there are currently about 20 NS+NS systems detected in our Galaxy. Except for one case, the *double pulsar* PSR J0737−3039 (Lyne et al. 2004), only one of the two NSs is detected—usually the recycled pulsar (Tauris et al. 2017). The other NS, is either not an active radio pulsar anymore or it is not beaming in our direction.

Given the small merger rate of NS+NS systems in our Galaxy¹ (most likely somewhere in the range from a few events up to a hundred events per Myr), it is statistically highly improbable that ground-based high-frequency detectors (LIGO–
Virgo–KAGRA) will detect a NS+NS merger in the Local Group earlier than the LISA era. The advantage of LISA is that it can follow the inspiral of Galactic NS+NS systems up to ~$10^6$ year prior to their merger event, and thus a significant number of NS+NS sources are anticipated to be detected in GWs by LISA.

About half of the 20 known NS+NS systems have orbital periods small enough (or eccentricities sufficiently large) to merge within a Hubble time. As an example, a “standard” NS+NS system with NS masses of $1.35 M_\odot$ and e.g. an orbital period of 16 h will merge in: 11.8 Gyr, 4.4 Gyr or 0.35 Gyr for an initial eccentricity, $e_0$ of 0.1, 0.5 or 0.8, respectively. The number of NS+NS sources that LISA will detect can be evaluated, approximately to first order, from a combination of the Galactic NS+NS merger rate and the distribution of these sources within the Milky Way. The above three standard NS+NS systems will have a remaining lifetime of between ~247 kyr (~243 kyr for $e_0$ = 0.8) and 1.57 Myr (1.48 Myr for $e_0$ = 0.8) by the time they enter the LISA band, if this occurs at a GW frequency of about 2 mHz and 1 mHz, respectively. Thus, if the Galactic merger rate is, say, $10 \text{ Myr}^{-1}$, we can roughly expect to detect between a few and a dozen LISA sources. Of course, the details depend on the Galactic distribution of these sources, the SNR required for a detection, and the duration of the LISA mission. The merger rate can be estimated from population synthesis, but its value is uncertain by, at least, one or two orders of magnitude (Abadie et al. 2010). The merger rate derived from an extrapolation of the LIGO/Virgo empirical merger rate of NS+NSs still has very large error bars due to small number statistics.

Recent works by Lau et al. (2020), Andrews et al. (2020) suggest that LISA may even detect up to ~50–200 Galactic NS+NS sources with a SNR greater than 7 within a 4 year mission. Given that LISA’s volume sensitivity for a constant SNR scales with $M_{\text{chip}}^5$, unlike double BH sources, very few NS+NS sources are anticipated to be detected outside the Milky Way, although a few such binaries may be found in both the LMC and M31 (Seto 2019). Applying a more conservative number, however, for the merger rate of Galactic NS+NS system of about 3–14 $\text{Myr}^{-1}$ (Kruckow et al. 2018) would lead to a substantial reduction in the predicted number of LISA detections. The Galactic merger rate is expected to be significantly better constrained in the coming decade such that we will have a clear idea about the expected number of NS+NS sources detected by LISA prior to its operation. Finally, an expected reduction in LISA SNR for detecting eccentric NS+NS systems, compared to circular NS+NS systems with similar orbital period and NS masses, should be noticed (Randall et al. 2021).

LISA may give us the opportunity to probe a hidden subpopulation of NS+NS systems with different properties compared to those of the well-known radio pulsar NS+NSs. The nature of GW190425, a presumed NS+NS merger detected by the LIGO/Virgo network with a total mass of $3.4 M_\odot$ (Abbott et al. 2020a), is still a mystery. With such a large total mass, GW190425 stands at five standard deviations away from the total mass distribution of Galactic NS+NSs detected in the Milky Way as radio pulsars (Farrow et al. 2019). If a subpopulation of heavy GW190425-

<footnote n="2">² Notice, the $2.6 M_\odot$ compact object in GW190814 might well have been a massive NS rather than a BH.</footnote>
like NS+NSs exists in our Galaxy, it is not yet clear why it should be radio-quiet (e. g. Safarzadeh et al. 2020b). Thus, LISA may actually be the most suited instrument for detecting the population of GW190425-like binaries (Galaudage et al. 2021; Korol and Safarzadeh 2021). In particular, Korol and Safarzadeh (2021) demon- strated that if GW190425-like binaries constitute a fraction larger than 10% of the total Galactic population, LISA should be able to recover this fraction with better than ~15% accuracy, assuming the merger rate of 42 Myr⁻¹.

Additional recent investigations (Thrane et al. 2020; Kyutoku et al. 2019) have discussed the importance of sky-localization on LISA NS+NS sources for multi-messenger follow-ups that may allow to impose constraints on the equation-of-state of NSs by measuring the Lense–Thirring precession (Thrane et al. 2020) or test general relativity through the detection of radio pulses from Galactic NS+NS binaries in a very tight orbit with the period shorter than 10 min (Kyutoku et al. 2019). Sky-localization may also help disentangle NS+NS systems from others sources by either knowing their position in the Milky Way, or in nearby galaxies, thus enhancing the possibility of EM follow-ups (e.g. Lau et al. 2020). In particular, in addition to differences in chirp masses, it will allow us to distinguish between eccentric NS+NS and eccentric WD+WD systems—the latter only expected to be formed in globular clusters and ejected into the Galactic halo via dynamical interactions, while the former systems have an eccentricity encoded from the last SN explosion.

#### 1.2.2.4 BH+NS and BH+BH systems
For several decades, a number of high-mass X-ray binaries (HMXBs) containing BH accretors have been identified in the Milky Way and nearby galaxies (e.g. Cyg X-1, LMC X-1, LMC X-3, MCW 656, M33 X-7, see van den Heuvel 2019). It has been shown that a fraction of these known wind-accreting HMXBs may eventually form BH+BHs or BH+NS systems (Belczynski et al. 2012, 2013, see also Fig. 9), while others will merge in an upcoming CE phase (Sect. 1.7.1.3), once the companion star evolves to a giant-star size and possibly initiates dynamically unstable Roche-lobe overflow (RLO), depending on its stellar

<figure><img src="image_8.png" /><figcaption>**Fig. 6** Illustration of the formation of an AM CVn system and a detached WD+WD binary. LISA sources are indicated with waves. Acronyms. ZAMS: zero-age main sequence; RLO: Roche-lobe overflow (mass transfer); CE: common envelope; He star: helium star; WD: white dwarf; CV: cataclysmic variable; AM CVn: AM Canum Venaticorum binary. Image reproduced with permission from Tauris and van den Heuvel (2023), copyright by PUP</figcaption></figure>
structure and the mass ratio between the two binary components. The masses of compact objects in X-ray binaries can be estimated with astrometry and the Galactic stellar-mass BHs are found to have masses between roughly $5-21 M_\odot$ (Gandhi et al. 2019; Arnason et al. 2021; van den Heuvel 2019; Miller-Jones et al. 2021). The astrometric satellite Gaia, can also be used to detect optical emission from the HMXB companion stars (Barstow et al. 2014; Kawanaka et al. 2017; Mashian and Loeb 2017; Breivik et al. 2017; Yamaguchi et al. 2018). Finally, non-interacting binaries with a BH component have also been discovered by combining radial velocity measurements with photometric variability data (Breivik et al. 2017; Thompson et al. 2019; Liu et al. 2019a), although their interpretations can, in some cases, be subject to alternative explanations (van den Heuvel and Tauris 2020).

The LIGO–Virgo GW detectors have detected BH+BH mergers in distant galaxies, out to ~5 Gpc. (Abbott et al. 2021). The inferred BH masses² of their inspiralling BH components potentially range all the way from ~2.6 ± 0.1 $M_\odot$ (Abbott et al. 2020b) to ~95 ± 10 $M_\odot$ (Abbott et al. 2021), thus significantly more massive than the known Galactic stellar-mass BHs. This difference is mainly attributed to the relatively high metallicity content of the Galaxy (Belczynski et al. 2016a; Kruckow et al. 2018). The LIGO–Virgo GW detectors have also identified two sources in close proximity or within the mass ranges expected for BH+NS binaries: GW190426 and GW190814. The first event has marginal significance (i.e. a high false-alarm rate, FAR = 1.4 year⁻¹) and the second is likely to be a BH+BH, not a BH+NS. Nevertheless, LISA like LIGO is much more sensitive to the masses (as opposed to matter content) of the binaries, so the presence of similar binaries suggests LISA will copiously find similar sources.

Interestingly enough, although the LIGO–Virgo GW sources are located in distant galaxies at Gpc distances, their low-frequency GWs during the last few years of inspiral prior to the merger event is often so luminous that it allows for detection with LISA (Sesana 2016). For example, the very first GW source (GW150914) would have been observable by LISA several years before its merger (see Fig. 4). Similarly, the extreme event GW190521 (Abbott et al. 2020c; Toubiana et al. 2021), with a total stellar mass of ~160 $M_\odot$ and located at a distance of about 5 Gpc, would also have been detected during its inspiral in the LISA band.

### 1.2.2.5 Stochastic background
As discussed above, stellar-mass compact binaries (BH+BH, BH+NS, NS+NS) are one of the primary targets for LISA, with expected detection rates of between a few and a few thousands per year, as summarized in Sect. 1.4. Nevertheless, many of these sources will not be detected, either because they are too distant and thus have a low SNR, or because the signals from multiple long-lived sources will overlap in time and will be difficult to disentangle. These unresolved signals will combine incoherently and produce a stochastic GW background (SGWB). Current predictions of the amplitude of this background typically rely on the merger rates measured in the local Universe by LIGO-Virgo (Abbott et al. 2016a, 2018b), but since most of the unresolved sources reside at higher redshifts, these predictions depend on the detailed population synthesis and galaxy evolution models.
The expected amplitude of the SGWB from BH+BH binaries in the LISA band (without source subtraction) varies in the range $\Omega_{GW}(f) \sim 10^{-13} - 10^{-11}$ at $f = 3$ mHz (Sesana 2016; Dvorkin et al. 2016a; Cusin et al. 2020; Périgois et al. 2021). Up to a few thousands of these binaries (likely responsible for about 10% of the background) will be individually detected by LISA (Sesana 2016; Périgois et al. 2021). The prediction for the background from NS+NS is significantly lower at $\Omega_{GW}(f) \sim 10^{-14}$ at $f = 3$ mHz (Périgois et al. 2021), and the background from BH+NS in the model of Périgois et al. (2021) is slightly higher but similar to that of NS+NS. The uncertainties on these rates will be significantly reduced in the coming years with more detections of stellar-mass binaries by ground-based detectors, and improved modelling of source formation and evolution. Detection of this background and in particular its shape, will provide important information about the population at periods too short to be directly observed, but before the merger phase probed by ground-based detectors (see Sect. 1.5.2.1).

### 1.2.3 Interacting binaries

**Coordinators:** Shenghua Yu, Thomas Tauris

**Contributors:** Ashley Ruiter (1.2.3.1), Thomas Kupfer (1.2.3.1), Thomas Tauris (1.2.3.1–2), Gijs Nelemans (1.2.3.1), Shenghua Yu (1.2.3.2)

#### 1.2.3.1 AM CVn binaries (AM Canum Venaticorum binaries—accreting WDs)

AM CVn binaries consist of a WD accreting from a hydrogen-deficient star (or WD) companion (Warner 1995; Solheim 2010). In their formation history (Fig. 6 and Sect. 1.3.1.1), AM CVns form after at least one CE phase of their progenitor system. The current RLO is initiated, due to orbital damping caused by GW radiation, at orbital periods of typically 5–20 min (depending on the nature and the temperature of the companion star), and the mass-transfer rate is determined by a competition between orbital angular momentum loss through emission of GWs and orbital widening due to RLO from the less-massive donor star to the more-massive WD accretor.

If the system survives the onset of the semi-detached phase, a stable accreting AM CVn binary is formed in which the orbital separation widens shortly after onset of RLO (Fig. 5), and the system evolves to longer orbital periods (see Fig. 5). When they reach an orbital period of $\sim 60$ min (after a few Gyr), the donor star has been stripped down to about 5 Jupiter masses ($5 M_\oplus$, Tauris 2018). These systems have been hypothesised to be possible progenitors of faint thermonuclear explosions (flashes, or Type “.Ia” SNe, Nelemans et al. 2001a; Bildsten et al. 2007).

Figure 4 shows examples of computed evolutionary tracks of AM CVn (and UCXB) systems in the characteristic strain amplitude vs GW frequency diagram. As can be seen, AM CVn systems are indeed anticipated to be detected by LISA—in some cases even with a SNR $> 100$ (for the sources located within 1 kpc).
Though there are currently ~65 AM CVn binaries known in the Galaxy (Ramsay et al. 2018), their formation pathways are still a puzzle (Green et al. 2018b). Their compact orbits and the lack of hydrogen in their spectra, led to three different proposed formation channels: (i) the donor is a low-mass (likely He-core) WD (Paczyński 1967; (ii) the donor is a semi-degenerate hydrogen-stripped, helium-burning star (e.g. main-sequence helium star, or hot subdwarf); or (iii) the donor is a helium-rich core of a main-sequence star that has not undergone helium-burning since it had a rather low mass to begin with (see Solheim 2010). Further discussions on their formation is given in Sect. 1.3.1.1.

Only for eclipsing AM CVns is it possible to fully determine all binary parameters and put constraints on the donor type. Recent results from eclipsing systems revealed that the donor stars are likely larger and more massive than previously assumed (Copperwheat et al. 2011; Ramsay et al. 2018), implying that a semi-degenerate donor is more likely for such systems, unless the donor star is a low-mass He-core WD which can remain bloated on a Gyr timescale (Istrate et al. 2014b). If that is the norm rather than the exception, it will lead to more AM CVn GW sources than previously predicted.

Based on binary population synthesis, Nelemans et al. (2001a) predicted a space density of AM CVn stars in a range of 0.4–1.7 × 10⁻⁴ pc⁻³ and a number of resolvable AM CVn systems for LISA roughly equal to the number of detached WD+WDs (Nelemans et al. 2004b). More recently, Kremer et al. (2017) predicts that ~2700 systems will be observable by LISA with a negative chirp of 0.1 year⁻² (i.e. $\dot{f}_{gw} < 0$, resulting from orbital expansion due to mass transfer, see Figs. 4 and 5). Until very recently, when ZTF reported a large number of eclipsing WD+WDs (Burdge et al. 2019a, 2020a), the majority of known LISA verification binaries was dominated by AM CVn systems (Roelofs et al. 2007a, 2010; Kupfer et al. 2015; Green et al. 2018a; Kupfer et al. 2018). However, observational space density estimates from SDSS data are in strong disagreement with theoretical predictions from these binary population studies. Roelofs et al. (2007c), Carter et al. (2013) derived an observed space density about an order of magnitude below the prediction by Nelemans et al. (2001a), Kremer et al. (2017) which would result in only ≲1000 resolvable systems in the LISA band. The discrepancy could be real with the population synthesis predicting too many systems, related to assumptions in binary evolution physics (especially the treatment of mass transfer in close binaries), and/or possibly because some of the systems that are predicted to evolve into AM CVn binaries in fact merge in a CE shortly after the less-massive star fills its Roche lobe. On the other hand, it could also be that AM CVn stars are more difficult to find than expected (when not in outburst) or that they are distributed with relative high concentration in the thick disc (Nissanke et al. 2012). Ramsay et al. (2018) argued, based on Gaia data release 2 parallaxes, that a significant fraction of AM CVn systems, even within 100 pc, could still be undiscovered. Future transient sky surveys, such as LSST using the Vera C. Rubin Observatory, could have great success in detecting short-period binary systems with the implementation of appropriate cadence intervals (e.g. very short, ~15 s sub-exposures). Indeed already some AM CVn systems are thought (or known) to be eclipsing (Burdge et al. 2020b).
Nonetheless, AM CVn binaries are expected to be extremely useful for LISA because they simultaneously provide EM information across different wavelengths, as well as being observable in LISA's GW frequency range. For this reason, AM CVn systems have been cited as being important verification sources for LISA (e.g. Kupfer et al. 2018). In other words, AM CVn binaries will be multi-messenger sources once LISA flies. See Sect. 1.5.1 for further discussion on the multi-messenger opportunities for LISA.

### 1.2.3.2 UCXBs (ultra-compact X-ray binaries)

It has been known for many years that tight-orbit post-LMXB systems, leaving behind a NS+WD binary that spirals-in due to GW radiation, may avoid a catastrophic event, once the WD fills its Roche lobe. The outcome is expected to be a long-lived UCXB (Webbink 1979; Nelson et al. 1986; Podsiadlowski et al. 2002; Nelemans et al. 2010; van Haaften et al. 2012; Heinke et al. 2013). These sources are tight X-ray binaries observed with an accreting NS and a typical orbital period of less than 60 min. Because of the compactness of UCXBs, the donor stars are constrained to be either a WD, a semi-degenerate dwarf or a helium star (Rappaport et al. 1982). UCXBs are not only excellent laboratories for testing binary-star evolution, but also important GW sources. Studies of their orbital parameters (mass, orbital period and eccentricity), chemical composition and spatial distribution may provide important information and clues to understand both the accretion processes of compact binaries (including spin-orbit and tidal interactions) and the long-term evolution of double compact object binaries.

Depending on the mass-transfer rate, the UCXBs are classified in two categories: persistent and transient sources (e.g. Heinke et al. 2013). Only about 14 UCXBs have been confirmed so far (9 persistent, 5 transient), and an additional ~14 candidates are known. Thus UCXBs are difficult to detect or represent a rare population. Earlier studies (e.g. Istrate et al. 2014a) have suggested the need for extreme fine tuning of initial parameters (stellar mass and orbital period of the LMXB progenitor systems) in order to produce an UCXB from an LMXB system. UCXBs are detected with different chemical compositions in the spectra of their accretion discs (e.g. H, He, C, O and Ne, see Nelemans et al. 2010). To explain this diversity requires donor stars which have evolved to different levels of nuclear burning and interior degeneracy, and therefore to different scenarios for the formation of UCXBs. Since a large fraction of the UCXBs are found in globular clusters, some of these UCXB systems could also have formed via tidal captures, direct collisions or stellar exchange interactions (Fabian et al. 1975; Mutantyo 1975; Hut and Bahcall 1983).

Figure 5 displays the evolution of AM CVns and UCXBs in the GW frequency vs dynamical chirp mass diagram. These systems undergo stable RLO and will start to widen their orbits again within a few Myr after the onset of the mass transfer. LISA will detect such Galactic systems continuously both during the inspiral phase for a few tens of Myr, while the systems are still detached, and after the onset of RLO on a timescale of up to 100 Myr (depending on their distance).

The long-term stability of UCXBs has been a topic of debate. From an analytical investigation, van Haaften et al. (2012) argued that for a 1.4 $M_{\odot}$ NS accretor, only C/
O-core WDs with a mass of $\lesssim0.4 M_\odot$ lead to stable UCXB configurations. Subsequent hydrodynamical simulations suggested that this critical WD mass limit could be lower (Bobrick et al. 2017). The first successful numerical calculations of RLO from a WD to a NS were presented by Sengar et al. (2017), and they were able to follow the entire evolution until the low-mass He-core WD donor star has become a $\sim 0.005 M_\odot$ planet-like companion. These systems were further evolved (Tauris 2018), including the finite-temperature (entropy) effects of the WD donor stars, and the first evolutionary tracks of such sources across the LISA GW band were produced, see e.g. Figs. 4 and 5. Further independent studies on the detectability of UCXBs as LISA sources have been provided by e.g. Chen et al. (2020a), Yu et al. (2021).

It is also anticipated that LISA may detect interacting BH+WD systems (Bahramian et al. 2017; Sberna et al. 2021). It has been estimated in some studies (Yungelson et al. 2006) that the Galaxy contains some $10^4$ of these systems. However, their formation process (especially those with low-mass WD companions) remains uncertain (Podsiadlowski et al. 2003).

### 1.2.4 Other potential sources

**Contributors:** Camilla Daniëlski (1.2.4.1-2), Silvia Toonen (1.2.4.2), Thomas Kupfer (1.2.4.4), Jan van Roestel (1.2.4.3), Nicola Tamanini (1.2.4.1), Valeriya Korol (1.2.4.1)

#### 1.2.4.1 Helium-star binaries
Subdwarf B stars (sdBs) are stars of spectral type B with luminosities below that of main-sequence stars. The formation mechanism and evolution of sdBs are still debated, although most sdBs are likely He-burning stars with masses $\sim 0.5 M_\odot$, radii as small as $\sim 0.1 R_\odot$ and thin hydrogen envelopes (Heber 2016). A large fraction are found in binary systems and, due to their compact nature, the most compact ones have orbital periods $\lesssim 1$ h (Vennes et al. 2012; Geier et al. 2013; Kupfer et al. 2017, 2020b, a), making them potentially detectable sources for LISA.

The most compact systems have WD companions and as such they are prime progenitor systems for double detonation Type Ia SNe. In this scenario a WD is orbited by a core He-burning sdB star in an ultra-compact orbit ($P_{orb} < 80$ min). Due to the emission of GWs, the binary shrinks until the sdB star fills its Roche lobe and starts mass transfer. He-rich material is then transferred to the C/O-core WD companion which will lead to the accumulation of a He-layer on top of the WD. After accreting about $0.1 M_\odot$, He-burning is predicted to be ignited in this shell. This in turn triggers the ignition of carbon in the core, even if the WD mass is significantly lower than the Chandrasekhar limit (Fink et al. 2010). So far, the only known candidate for this scenario is the ultra-compact sdB+WD binary $CD-30^\circ 11223$ with an orbital period of $P = 71$ min (Vennes et al. 2012; Geier et al. 2013). This system was also found to be detectable for LISA with an expected SNR of $\sim 5$ after 4 years of LISA observations (Kupfer et al. 2018). More recently, the first members of ultra-compact sdB binaries which have started to transfer material to the WD companion
were discovered. The most compact system, ZTF J2130+4420, consisting of a low-
mass sdB star with $M_{\text{sdB}} = 0.337 M_{\odot}$ has an orbital period of 39 min. The system has
well measured properties from EM observations and is expected to have a SNR of
~3 after 4 years of LISA observations, adding to the growing number of LISA
detectable He-burning stars.

Götberg et al. (2020) modelled the Galactic population of stripped stars, which contain the low-mass sdB stars as well as more massive He-core burning stars, in tight orbits with compact companions, focusing on those that will be detectable by LISA. Their analysis predicts up to 100 stripped star + WD binaries and up to 4 stripped star + NS binaries with SNR > 5 after 10 years of observations with LISA. Although the expected numbers are significantly smaller than for WD+WDs or AM CVns, Götberg et al. (2020) finds that all of the LISA detectable sources are within 1 kpc and therefore bright in EM flux which makes them ideal targets for multi-messenger studies (see Sect. 1.5.1 for more details on multi-messenger opportunities).

**1.2.4.2 Period bouncing CVs**
Period bouncing CVs are highly evolved cataclysmic variables where the donor has lost almost all of its mass and has become degenerate. These systems have reached the minimum orbital period for a hydrogen donor (70 min) and are evolving to longer orbital periods (up to 100 min). Model predictions are that 40–70% of all CVs are period bouncers (Kolb 1993; Knigge et al. 2011). However, only a few have been identified so far because of the low accretion rate and low temperature of the WD and donor (e.g. Pala et al. 2018). While the donor-mass is low and the orbital periods are relatively long, nearby period bouncers are detectable with LISA. Given their high space density, a dozen of these systems are close enough to be detected by LISA.

**1.2.4.3 Exoplanets, brown dwarfs and substellar companions**
In the Galaxy, due to the slope of the Salpeter-like initial mass function (IMF), more than 97% of all stars will terminate their lives as a WD, meaning that the vast majority of the known 4000+ planet-hosting stars will end their lives as WDs. In the last couple of decades, most of the attention in exoplanetary searches has been focused on the formation and characterisation of exoplanets orbiting host stars on the main sequence, but very little is known on planetary systems in which the host star evolves off the main sequence, to become a red giant. Theoretical models indicate that, if planets avoid engulfment and evaporation throughout the red-giant or/and the asymptotic-giant branch phases of the host star, they can survive (see e.g. Livio and Soker 1984; Duncan and Lissauer 1998; Nelemans and Tauris 1998). This is expected to be the fate of the planet Mars, and other planets orbiting further out in our Solar System (Schröder and Smith 2008). Observational evidence, in the form of photospheric contamination by the accreted debris (Zuckerman et al. 2010; Koester et al. 2014), dusty (Farihi et al. 2009) and gaseous circumstellar discs (Gänsicke et al. 2006; Manser et al. 2016), supports the existence of dynamically active planetary systems around WDs. Up to very recently, only two planetesimals had been observed orbiting a WD (Vanderburg et al. 2015; Manser et al. 2019). However, within the last years, two giant planets
have been detected orbiting single WDs (Gänsicke et al. 2019; Vanderburg et al. 2020), showing that planets can survive single host-star evolution.

Today, over 1000 brown dwarfs (BDs) have been detected in the Solar neighbourhood (Burningham 2018). Some of them have been discovered also around single WDs, and examples of BDs orbiting at distances beyond the tidal radius of the asymptotic-giant branch progenitor (but also within it, e.g. WD 0137−349 B, Maxted et al. 2006), show that BDs can survive stellar evolution of their host star, whether or not they are engulfed by its expanding envelope. Farihi et al. (2005) predicted that few tenths of percent of Galactic single WDs hosts a BD.

The most straightforward way with which LISA could detect sub-stellar objects, such as planets or BDs, would be the direct detection of GWs emitted by a binary system composed of a sub-stellar object in a tight orbit around a single star. However, the absolute orbital period minimum for a hydrogen-rich body (i.e. a star, BD or a gas giant planet) in a binary system is about $P_{orb} \approx 37$ min (Rappaport et al. 2021). This corresponds to a GW frequency of at most $f_{GW} \approx 0.9$ mHz. Such a system could be detected only at close distances (say, within 1 kpc) and only for relatively high sub-stellar masses ($M \gtrsim 13 M_J$), possibly excluding all exoplanets. Furthermore, the mass of the sub-stellar object cannot be directly inferred from direct detection, and at best only the chirp mass of the binary system can be retrieved. Further investigations and EM observations are necessary to better understand the detectability and the rates of these sub-stellar objects, although at the moment it seems unlikely that a large number of these systems will be observed by LISA (Wong et al. 2019b).

Another option is to search for circumbinary planets around WD+WD through a modulation of the WD+WD signal (Tamanini and Daniëlski 2019), that can probe regions of parameter space not probed by EM observations (far away and not towards the Galactic Centre). The discovery of evolved planetary systems will statistically increase the current sample of post-main-sequence planets, filling an area of the planetary Hertzsprung-Russell diagram that is currently not explored (Tamanini and Daniëlski 2019). LISA will provide observational constraints on both planets that can survive two CE stellar evolution phases and on a possible second-generation planet population produced from CE ejecta material (Schleicher and Dreizler 2014). Even in the case where LISA will prove no detection anywhere in the Milky Way, it will be possible to set strong unbiased constraints on planetary evolution and dynamical theories, and in particular on the fate of exoplanets bound to a binary that undergoes two CE phases.

#### 1.2.4.4 Triples and multiples
LISA's stellar sources will also contain multiple body systems, such as triples and quadruples. Hierarchical systems that consist of nested orbits represent stable configurations that can remain intact for several Gyr and throughout (despite) the evolution of the stellar components, as evidenced by observations. Within 20 pc of the Sun, there are already two such systems that harbor close WD+WDs. These are WD 0326−273 (Luyten 1949; Poveda et al. 1994; Nelemans et al. 2005; Giammichele et al. 2012; Toonen et al. 2017) and WD 0101+048 (Saffer et al. 1998; Maxted et al. 2000a; Caballero 2009; Giammichele et al.
2012; Toonen et al. 2017). The former is a triple that consists of a close WD+WD with a period of ~1.8 d, and an M5 star in a wide orbit. The latter is a quadruple consisting of a close WD+WD (with a period of ~1.2 d, but see (Maxted et al. 2000a) and a MS+MS binary. Two triple systems with three WD components are known as well, J1953−1019 (Perpinyà-Vallès et al. 2019) and WD 1704+481 (Maxted et al. 2000b). The inner binary of the latter system has a period of ~0.15 d, just inside the LISA frequency range. Even millisecond pulsars have been found to be part of triple-architectures; the PSR J0337+1715 system harbors a compact NS+WD (1.6 d orbital period) inner binary which is orbited by another (tertiary) WD every 327 d (Ransom et al. 2014; Tauris and van den Heuvel 2014). The globular cluster (M4) pulsar B1620−26 has a WD companion in a half-year orbit, and a planetary companion in a 100-year orbit (Thorsett et al. 1999; Sigurdsson et al. 2003).

Theory suggests that exoplanets (and BDs) also exist around WD+WDs in the Galactic disc, and that such objects are more likely to survive around evolving close binary stars than around evolving single stars (Kostov et al. 2016). The eclipse timing variation technique allowed the detection of a few post-CE systems (that is WD+low-mass star), and a BD companion(s) (see e.g. Goździewski et al. 2015; Beuermann et al. 2012; Almeida et al. 2019), nevertheless today no BDs or exoplanets orbiting WD+WDs have been observed yet.

LISA will be able to detect outer companions to compact (inner) binaries when they impose eccentricity oscillations in the inner orbit due to three-body dynamics (von Zeipel 1910; Lidov 1962; Kozai 1962; Naoz 2016). In particular Hoang et al. (2019) showed such oscillations would be observable with LISA to distances up to a few Mpc for compact binaries near supermassive BHs, which can also be considered a three-body system. Furthermore, LISA can detect outer companions by exploiting the Doppler frequency modulation on the GW waveform due to their gravitational pull (Robson et al. 2018). The acceleration imparted by the hierarchical companions can be detected in the GW signal for outer periods as large as 100 year (Robson et al. 2018; Tamanini et al. 2020). For systems with orbital periods that are shorter than, or comparable to, the mission lifetime, the perturbation allows for the determination of the orbital period, eccentricity, initial orbital phase and radial velocity parameter of the companion (Robson et al. 2018; Tamanini and Daniëlski 2019). On a general level, the sensitivity of LISA will be able to detect WD+WDs companions with masses down to ~$M_J$ (Daniëlski et al. 2019), and therefore allow not only for the detection of stellar companion and compact objects, but also BDs and exoplanets. This being an indirect detection, i.e. the observation of a periodic Doppler shift modulation of an existing strong binary GW signal, we are able to probe a wider mass range, whose inferior limit also covers the giant planets range. The novelty of using LISA for the detection of planetary/low-mass companions is that GWs provide a much larger spatial coverage than the one provided by EM techniques, enabling us to probe regions of our Galaxy currently not accessible to other methods. More specifically, Daniëlski et al. (2019) showed that during a 4 year nominal mission LISA will detect from 3 to 83 exoplanets, and from 14 to 2218 BDs everywhere in the Milky Way. The sensitivity of LISA is such that in the most optimistic cases exoplanets could be detected orbiting WD+WDs in the Milky Way's satellites, in
particular in the Large Magellanic Cloud (LMC, Daniëlski and Tamanini 2020). Such an observation could represent the first detection of an extra-galactic bound exoplanetary system.

**1.2.4.5 Capturing the inspiral of a CE system** It has been suggested by Renzo et al. (2021) that LISA may be able to detect the inspiral of binaries undergoing a CE phase. Depending on various assumptions, they anticipate that LISA could detect between 0.1 and 100 such GW sources in the Galaxy during the mission duration. Detecting this GW signal would provide direct insight into the gas-driven physics of CE evolution.

## 1.3 Formation of LISA binaries

**Coordinators:** Katelyn Breivik

**Contributors:** Michela Mapelli (1.3.2-3), Simone Bavera, Katelyn Breivik, Martyna Chruslinska, Gijs Nelemans, Pau Amaro Seoane (1.3.2), Manuel Arca Sedda (1.3.2-3), Thomas Tauris, Silvia Toonen (1.3.2-3), Jeff Andrews, Tassos Fragos (1.3.1), Luca Graziani, Daryl Haggard (1.3.2), Melvyn B. Davies (1.3.2-3)

In the following section, we discuss the formation of LISA binaries. For a review and broader description of the many physical aspects of stellar evolution and binary star interactions that are referred to below in the context of the formation of compact object binaries, we refer to the textbooks by, for example, Shore et al. (1994), Hilditch (2001), Eggleton (2006), Chaty (2022), Tauris and van den Heuvel (2023).

### 1.3.1 Isolated binaries

The formation pathways of isolated binaries observable by LISA are marked with several phases of mass loss or exchange. In the following Section, we refer to the initially more massive star as the primary and the initially less massive star as the secondary. Stable mass transfer can occur either through wind mass loss/accretion or RLO. Wind mass loss is generally assumed to be non-conservative across all phases of stellar evolution, with mass accretion efficiency ranging from $\lesssim 10\%$ for the Bondi-Hoyle-Lyttleton mechanism and 20-50% if the accretion is focused (de Val-Borro et al. 2017). In this case, the orbit widens as the mass lost from the system causes an increase of the remaining specific angular momentum. In practice, the dynamics are complicated and dependent on physics related to the geometry and structure of the wind, tidal effects, orbital characteristics and in some cases magnetic fields and radiation transport, thus calling for three-dimensional, multi-physics, hydrodynamical simulations (e.g. Saladino et al. 2018, 2019).

RLO occurs when the donor star expands to the point that its radius exceeds the Roche radius (Eggleton 1983). RLO mass transfer can proceed in a dynamically stable or unstable fashion, depending on the structure of the donor and accretor as well as their mass ratio. The stability of RLO mass transfer is commonly described
using the Webbink radius–mass exponents (Webbink 1985) that determine the timescales on which mass transfer will become unstable. In the case of dynamically stable mass transfer, the orbital evolution depends strongly on the mass ratio of the binary: if $M_{\text{donor}}/M_{\text{accretor}} > 1$, the orbit tightens (for fully conservative mass-transfer), while in the converse case the orbit widens. Mass-transfer efficiency, as well as the assumed specific angular momentum carried away from mass lost from the system, play a crucial role in mass-transfer stability and the orbital evolution of the binary.

Dynamically unstable mass transfer is believed to generate a CE phase where the donor star's core and companion are enshrouded in the donor's envelope (for a review see Ivanova et al. 2013). The precise dynamics of how CE proceeds are still not fully understood. In the context of compact object binary formation and population synthesis studies, energy budget arguments are most often employed to estimate the post-CE properties of a binary. In the “$\alpha_{\text{CE}}$” prescription, it is assumed that a fraction $\alpha_{\text{CE}}$ of the released orbital energy is used to unbind the donor's envelope and eject it from the system (van den Heuvel 1976; Webbink 1984). Several recent studies have suggested that other sources of energy may be needed to successfully eject the envelope, including recombination energy (e.g. Zorotovic et al. 2014; Nandez and Ivanova 2016) or jets launched by the companion (e.g. Shiber et al. 2019). Each of these will change the overall energy budget of the CE evolution and lead to differences in the final orbital separation (e.g. Iaconi et al. 2018). Alternatively, in the “$\gamma_{\text{CE}}$” prescription, angular momentum conservation arguments, which lead to less dramatic inspiral, have been considered to explain the orbital period distribution of WD+WDs (Nelemans et al. 2000).

### 1.3.1.1 WD+WD systems and AM CVn binaries
The progenitors of isolated WD+WD and AM CVn binaries begin with zero age main sequence stars with masses below $8-10 M_{\odot}$. The formation pathways of close WD+WD and AM CVn binaries contain several stages of stable and unstable mass transfer, or CE (see Fig. 6). The uncertain outcomes of these interactions determine whether the progenitor binary

<figure><img src="image_9.png" /><figcaption>**Fig. 7** Illustration of the formation of a detached NS+WD binary and an UCXB system. See Fig. 6 for details. Additional acronyms: SN: supernova; NS: neutron star; LMXB: low-mass X-ray binary; BH: black hole. Image reproduced with permission from Tauris and van den Heuvel (2023), copyright by PUP</figcaption></figure>
continues on in its evolutionary path toward becoming a LISA source or if it merges with its companion. Conversely, LISA observations of the populations of WD+WD and AM CVn sources will constrain these interactions.

Virtually all close WD+WD and AM CVn progenitors experience an interaction as the primary star advances off the main sequence and fills its Roche lobe. This interaction can either proceed stably or unstably on dynamical timescales. In either case, the orbit will shrink because of the donor's higher mass relative to the accretor. For systems with late red giant and asymptotic giant branch donors, initially dynamically stable but thermally unstable mass transfer can produce mass loss from the L2 Lagrange point, which leads to a delayed dynamical instability and a CE phase (Ge et al. 2020; Misra et al. 2020). In the rare case of close WD+WDs where the more massive WD forms second (converse to stellar lifetime expectations), a phase of stable mass transfer, followed by a CE generated by the initially lower mass star could be necessary (Woods et al. 2012).

After each interaction, the star which donates mass becomes stripped leaving behind a He core that can have varying structure depending on the evolutionary phase at which the donor filled its Roche lobe. Such stripped stars orbiting main sequence companions have been widely observed throughout the Galaxy (Rebassa-Mansergas et al. 2007) and been used to constrain CE ejection efficiencies (Zorotovic et al. 2010; Toonen and Nelemans 2013).

<figure><img src="image_10.png" /><figcaption>**Fig. 8** Evolutionary sequence showing how ultra-compact X-ray binaries (prime LISA source candidates) are formed from merging NS+WD binaries, descending from LMXBs. Plotted here is mass-transfer rate of the donor star as a function of stellar age. The initial MS star + NS binary has components of $1.40 M_\odot$ and $1.30 M_\odot$, respectively. The system evolves through two observable stages of mass transfer: an LMXB for 4 Gyr, followed by a detached phase lasting about 3 Gyr where the system is detectable as a radio millisecond pulsar orbiting the helium WD remnant of the donor star, until GW radiation brings the system into contact again, producing a UCXB. The colour bars indicate detectability in different regimes resulting in synergies between LISA and EM detectors. Image reproduced with permission from Tauris (2018), copyright by APS</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 9** Illustration of the formation of a tight BH+NS binary that evolves towards a merger. See Figs. 6 and 7 for explanation of acronyms. Image reproduced with permission from Tauris and van den Heuvel (2023), copyright by PUP</figcaption></figure>

Since the previous interaction brings the two stars together, further interactions are likely. Interactions can occur while the secondary is still on the main sequence. In this case, if the mass transfer is stable, a cataclysmic variable is formed (Zorotovic et al. 2011). Conversely, mass transfer can also occur as the secondary advances up the giant branch. Due to previous envelope stripping, the mass ratio of the secondary donor to the WD accretor can be either greater or less than one. If the secondary is already less massive than the WD, stable mass transfer will occur and widen the orbit, removing the possibility of detection by LISA. However, if the secondary is more massive than the WD companion, the mass exchange will lead to orbital tightening. If the mass transfer is unstable, another CE phase takes place, potentially bringing the stars even closer together and leaving behind a WD with a stripped He core companion. The structure of the He core again depends on the evolutionary phase at which the secondary overflows its Roche lobe. At this point, a close WD+WD binary is assured and the slow evolution due to GW emission brings the WD+WD toward the LISA band.

A key uncertainty in the formation pathways of AM CVn binaries is the nature of the donor star. AM CVn binaries consist of a WD accreting He-rich material originating from a WD, semi-degenerate helium star, or evolved MS donor (Solheim 2010). Indeed, it could be the case that the observed AM CVn population is a combination of all three with different relative contributions (Nelemans et al. 2004b). If the donor star is a non-degenerate evolved star, magnetic braking is required, along with GW emission, to maintain the ultra-short periods of observed AM CVn systems (van der Sluys et al. 2005). Magnetic braking is a process in which orbital angular momentum in a tight synchronized binary is converted into spin angular momentum via a magnetic stellar wind (a process that therefore requires a low-mass stellar component with a convective envelope). The ultra-compact orbital configuration is less problematic for semi-degenerate and fully-degenerate donors which originate from the ejection of a second CE, with tighter orbits allowed by more degenerate donors (Yungelson 2008). In the case of fully-degenerate He-core WD donors, the
orbit can become so small that the mass lost from the donor directly impacts the accretor leading to a rapid decrease in orbital size followed by a long-lived phase of accretion which widens the orbit (Nelemans et al. 2004b; Marsh et al. 2004; Deloye and Taam 2006; Kremer et al. 2017). Regardless of the donor, a significant uncertainty still remains in how much He-rich material the accretor can handle until novae erupt on the WD's surface. While detailed binary evolution calculations (e.g. Tauris 2018) have shown that RLO mass transfer in WD+WD can be stable, it has been suggested that interactions of the donor star with the expanding nova shells will likely lead to a rapid orbit shrinkage and eventually a merger (Shen 2015).

### 1.3.1.2 White-dwarf binaries with neutron-star or black-hole companions
Compared to WD+WDs, the formation of detached binaries with WD and NS or BH companions occur in binaries with stars that are massive enough to explode in a SN (see Fig. 7). Similar to WD+WD formation, the more massive primary evolves first and, because of the relatively large mass ratio, begins RLO mass transfer that is often expected to be unstable and lead to a CE. Soon after, the primary evolves to become a compact object, through either a supernova explosion (NS or BH) or direct collapse (BH only). Since a NS is thought to receive a kick during its formation, there is a significant probability that the binary disrupts at this point.

The subsequent evolution, of a lower-mass non-degenerate star with a NS or BH, will typically go through a phase of stable mass transfer in which the binary becomes observable as X-ray binary, due to the strong heating of the accretion disc in the deep potential of the NS or BH. When the onset of the mass-transfer occurs after the secondary star has evolved past its main sequence, the core of the star has already contracted. Thus after the X-ray binary phase, when the envelope of the expanding star has been completely transferred, the NS or BH is left with a WD companion that was the core of the donor star.

In some cases, the NS/BH+WD binary is tight enough that angular momentum loss due to GW emmission will bring the two objects together as LISA sources (Fig. 8). At periods of ~10–20 min, i.e. within the LISA band, the WD will start to transfer mass to the NS/BH, forming an X-ray binary again, but now of ultra-short period, called an UCXB. Detailed numerical calculations, including finite-temperature (entropy) effects, have shown that UCXBs can indeed form via stable RLO from post-LMXBs systems (Sengar et al. 2017; Tauris 2018).

### 1.3.1.3 Double neutron star/black hole binaries
NS+NS formation has been extensively discussed in the literature (see Tauris et al. 2017, for a review). The standard scenario (see Fig. 9 for a schematic diagram) involves several phases of interaction, starting with a stable RLO, during which the primary loses part of its envelope before it undergoes a SN to form a NS. The newly formed NS HMXB is likely too dim to be detectable in X-rays, as the orbital separation is still large and the NS may only be able to capture an appreciable fraction of the companion's stellar wind when the latter evolves to the giant phase. During the subsequent evolution the orbital separation needs to decrease from ~$10^3 R_⊙$ to a few $R_⊙$ for the final binary to merge within the Hubble time. Significant tightening is typically achieved through a
CE phase that occurs when the secondary this time fills its Roche lobe. The post-CE binary is expected to encounter another phase of mass transfer, initiated by a stripped helium-burning secondary (i.e., so-called Case BB mass transfer, often initiated when $a \sim 2.5-3.5 M_{\odot}$ helium star expands during shell-helium burning, Habets 1986). This leads to further orbital tightening, stripping of the secondary's envelope, and NS spin-up. If this last mass-transfer episode is unstable and leads to a second CE phase, a fast merging NS+NS will be formed; a scenario invoked to explain r-process element enrichment observed in some stellar systems (Safarzadeh et al. 2019; Zevin et al. 2019a). Such NS+NSs would be effectively unobservable with current radio surveys, and if they exist within the Galaxy, their presence will be revealed by LISA (Kyutoku et al. 2019; Andrews et al. 2020). However, recent detailed binary evolution calculations have shown that this last phase of Case BB mass transfer is expected to be stable (Tauris et al. 2015; Vigna-Gómez et al. 2018) and do not support the existence of the aforementioned fast-merging channel (in contrast to earlier works Ivanova et al. 2003; Dewi and Pols 2003).

Besides the pre-HMXB evolution, the most important and uncertain aspects of our current understanding of NS+NS and mixed BH+NS formation are related to: (i) CE evolution and spiral-in of the NS, (ii) momentum kicks (magnitude and direction) imparted onto newborn NSs, and (iii) the mass distribution of NSs.

From an energetics point of view, it has been shown that an inspiralling NS may indeed be able to eject the envelope of its massive star companion (e.g. Xu and Li 2010; Loveridge et al. 2011; Wang et al. 2016a; Kruckow et al. 2016). However, predicting the final post-CE separation is difficult for several reasons, including: estimating the location of the bifurcation point within the massive star (Tauris and Dewi 2001), separating the remaining core from the ejected envelope (Tauris and Dewi 2001; Fragos et al. 2019), additional energy sources such as accretion energy (MacLeod and Ramirez-Ruiz 2015; Murguia-Berthier et al. 2020), energy and radiation transport during the CE inspiral (Fragos et al. 2019) and the effect of an inflated envelope of the exposed naked helium core (Sanyal et al. 2015, see also Sect. 1.7.1.3).

Newly formed NSs gain velocity (natal kicks; e.g. Gunn and Ostriker 1970; Hobbs et al. 2005; Verbunt et al. 2017) due to asymmetries arising during their formation (e. g. Janka 2012). The properties of the modelled NS+NS population (e.g. number of systems formed, orbital parameters, merger locations relative to formation site) are highly dependent on the adopted natal kick prescription (e.g. Portegies Zwart and Yungelson 1998; Bloom et al. 1999; Chruslinska et al. 2018; Giacobbo and Mapelli 2018; Andrews and Zezas 2019). To match the current observational constraints on the NS+NS merger rate and parameters of several of the observed systems (e.g. van den Heuvel 2007), it is necessary to assume that a fraction of NS forms with natal kicks smaller than typically found for young single pulsars. Some scenarios involve low-mass NS progenitors and electron-capture triggered explosions (e.g. Dessart et al. 2006; Jones et al. 2013). Others postulate a link between the natal kick magnitude and the mass of the NS progenitor and SN ejecta (e.g. Beniamini and Piran 2016; Bray and Eldridge 2016; Janka 2017). These claims have been supported by 3D NS simulations of ultra-stripped stars (Müller et al. 2019). In fact, it has been demonstrated that close-orbit, low-eccentricity NS+NS and BH+NS systems most
**Fig. 10** Illustration of the formation of LISA sources via examples of exchange encounters. See Figs. 6 and 7 for explanation of acronyms. Image reproduced with permission from Tauris and van den Heuvel (2023), copyright by PUP

EXCHANGE ENCOUNTERS

<figure><img src="image_12.png" /></figure>

likely form via ultra-stripped SNe when the last star explodes (Tauris et al. 2013, 2015). The reason being that the last Case BB RLO mass-transfer phase causes the NS to significantly strip its evolved helium-star companion, almost to a naked metal core prior to its explosion, and thus there is very little SN ejecta (see also Sect. 1.7.1.7).

Finally, a clear correlation has been predicted between the spin period of the recycled pulsar and the orbital period of the system after the second SN (Tauris et al. 2017). This correlation can be tested in LISA binaries, if the spin period is measured, since only short orbit systems will enter the LISA band within a Hubble time, and these binaries should therefore contain the most rapidly spinning NSs of this population. Another hypothesis that can be tested by LISA, is the resulting mass distribution among NS+NS systems (e.g. Özel and Freire 2016, and references therein).

Merging BH+BHs and NS+BHs in the field are thought to occur under some specific binary interactions which either (i) bring the parent stars closer together during their evolution or (ii) prevent stars in close obits from expanding.

The former one (i) occurs in a similar manner to the formation of NS+NSs described above, and involves many of the main uncertainties. In contrast to NS+NS formation, BH+BHs and to a lesser degree BH+NSs are sensitive to the metallicity of the progenitor stars, and they favor low-metallicity environments. In addition the second mass transfer episode, after the first compact object formation, can be either dynamically stable (e.g. van den Heuvel et al. 2017; Inayoshi et al. 2017a; Neijssel et al. 2019) or unstable (e.g., Smarr and Blandford 1976; van den Heuvel 1976; Tutukov and Yungelson 1993; Kalogera et al. 2007; Postnov and Yungelson 2014; Belczynski et al. 2016a). In the latter case this leads to a CE phase. The resulting tight system composed of a compact object and a Wolf-Rayet star can eventually undergo a tidal spin up of the star (Qin et al. 2018; Bavera et al. 2020). On the other end, if the second mass transfer is stable the binary will result in wider orbits compared to the evolution through CE and avoid a subsequent tidal spin up phase (Bavera et al. 2021). Eventually, following wind-driven mass loss, the secondary will collapse to a compact object. This leave us with either a BH+BH system or a NS+ BH system with either a first- or second-born NS.
The latter possibility (ii) occurs when two massive stars are born in a tight orbit (orbital periods less than 4 days) in low-metallicity environments which due to their tidal interactions can maintain the stars at almost critical rotation. Such rapidly rotating stars develop a temperature gradient between the poles and the equator leading to chemical homogeneous evolution (e.g., de Mink et al. 2009; Mandel and de Mink 2016; Marchant et al. 2016; du Buisson et al. 2020). In these stars meridional circulation transport hydrogen from the surface into the core and helium out into the envelope until nearly all the hydrogen in the star is fused into helium. At the end of their main sequence these stars are essentially Wolf-Rayet stars and do not expand, hence, avoiding any additional mass-transfer phase.

LISA may answer whether or not mixed binaries of BHs and NSs, in which the NS formed first, are produced in the Galaxy. It is possible that the in-spiralling NS is unable to eject the envelope of the relatively massive BH progenitor star (Kruckow et al. 2018). Since we currently do not know of the existence of mixed BH+NS systems in the Galaxy, any LISA detections of such systems, as well as double BH systems, will provide crucial information about their formation process.

### 1.3.2 Sources in clusters

The inner regions of stellar clusters are cosmic factories of compact binaries (i.e., binaries containing two compact objects; BHs, NSs or WDs), owing to the dominant role played by stellar dynamics in such environments. Massive stars in stellar clusters lose kinetic energy to lighter stars and accumulate into the cluster centre. In just a few Myr, these stars evolve into stellar BHs and NSs.

The production of compact binaries can take one of two routes. If only a small fraction of BHs are retained within a cluster, encounters between BHs and binary stars lead to dynamical exchanges, where the BH replaces a less massive star within the binary (Hills and Fullerton 1980). Globular clusters have a considerably-enhanced population of X-ray binaries (Heinke et al. 2003), which might have formed when a NS or a BH exchanges into a binary star (e.g., Hills 1976). After the first exchange, evolution of the stellar companion (which might also become a BH or a NS) or a second dynamical exchange can produce a compact binary. Binary-single interactions represent an efficient mechanism to harden these binaries (Heggie 1975) to the point where they can merge via the emission of gravitational radiation (see Fig. 10 for a schematic representation).

Alternatively, if a large fraction of BHs are retained in the cluster, they can form a BH subsystem (Spitzer 1969; Mackey et al. 2007, 2008; Arca Sedda et al. 2018; Kremer et al. 2020b, but see Breen and Heggie 2013). BHs in the subsystem tend to strongly interact with each other, undergoing frequent pairing, exchanges, and ejections. The most efficient mechanism driving binary formation in globular clusters is via three-body scatterings (e.g., Morscher et al. 2015). After formation, a binary can undergo dozens of interactions with passing stars and binaries, which can lead to the production of very hard binaries, capable of merging via the action of GW emission (Portegies Zwart and McMillan 2000). If the star cluster centre harbours a BH subsystem, the BHs dominate the dynamics, quenching mass segregation and preventing the formation of binaries containing other compact objects (see e.g. Ye
<figure><img src="image_13.png" /><figcaption>**Fig. 11** Illustration of the formation of a LISA source in a triple system. See Figs. 6 and 7 for explanation of acronyms. Image reproduced with permission from Tauris and van den Heuvel (2023), copyright by PUP</figcaption></figure>

et al. 2020). However, dynamically evolved clusters can lose a substantial fraction of the BHs. In these BH-poor clusters, binary-single interactions can allow the formation of binary NS and BH+NS binaries. Furthermore, it has been proposed that a parabolic encounter between two compact objects could potentially lead to the formation of a binary due to an abrupt loss of energy emitted as gravitational radiation (e.g. Hansen 1972; Quinlan and Shapiro 1989; Kocsis et al. 2006; Hong and Lee 2015). However, the event rate of this mechanism, which is often referred to as the “gravitational brake” capture, is very likely to be negligible due to the small cross-section (Kochanek et al. 1990).

Old BH-poor clusters may also be ideal for dynamical formation of WD+WD binaries as well as BH/NS+WD binaries (Kremer et al. 2020b), see Fig. 10. In old globular clusters, WDs are by far the most abundant type of compact object (roughly $10^5$ WDs are expected in a $10^6 M_{\odot}$ cluster). A number of analyses have studied ways WD binaries, both accreting and detached, may be dynamically assembled in stellar clusters (Grindlay et al. 1995; Ivanova et al. 2006; Belloni et al. 2016; Kremer et al. 2018b). Furthermore, a handful of the stellar-mass BH binary candidates observed in Galactic globular clusters are suspected to be ultra-compact accreting BH+WD binaries (Strader et al. 2012; Bahramian et al. 2017; Church et al. 2017). Overall, up to a few dozen dynamically formed WD binaries are expected to be resolved by LISA in the Galactic globular clusters, likely constituting the largest class of dynamically-formed LISA sources in the Galaxy (Willems et al. 2007; Kremer et al. 2019b). Currently two candidates to AM CVns in globular clusters have been identified (Zurek et al. 2016; Rivera Sandoval et al. 2018) and several more are expected to be discovered in upcoming globular cluster surveys.

Comparing nuclear stellar clusters with globular clusters, the former tend to have somewhat larger escape speeds (due in part to the presence of a central massive BH, MBH: Graham and Spitler 2009). This means that a larger fraction of BHs are likely to be retained (e.g. Miller and Lauburg 2009), while the higher dispersion velocity inhibits both exchange encounters and the dynamical formation of binaries (e.g. Heggie and Hut 2003). The presence of a dense nuclear cluster surrounding the MBH can significantly affect the formation process of compact binaries in a number of ways. Dynamical three body encounters can form at least one compact BH+BH if the nuclear cluster-to-MBH mass ratio exceeds 10, whereas at lower values the
reservoir of compact binaries might be replenished via star cluster inspiral (e.g. Arca Sedda 2020a). The presence of an MBH can leave significant imprints on the BH+ BH evolution, owing to the possible development of von Zeipel–Kozai–Lidov cycles (von Zeipel 1910; Kozai 1962; Lidov 1962), which can boost the rate of BH+BH mergers (Blaes et al. 2002; Antonini and Perets 2012; Hoang et al. 2018; Fragione et al. 2019; Arca Sedda 2020a) and significantly affect the BH mass spectrum in these extreme environments (e.g. Arca Sedda 2020a).

Young star clusters and open clusters, because of their relatively low total masses (10²–10⁵ M☉), host a smaller population of BHs with respect to globular and nuclear clusters (e.g., Portegies Zwart and McMillan 2000; Banerjee et al. 2010; Banerjee 2017, 2021). BH+BHs in young/open star clusters mostly originate from dynamical exchanges or even from the evolution and hardening of primordial binaries (Ziosi et al. 2014; Di Carlo et al. 2019, 2020b; Kumamoto et al. 2019, 2020). Furthermore, dynamical exchanges favour the formation of BH+NS binaries in young and open clusters (Rastello et al. 2020).

Finally, hierarchical mergers in globular/nuclear clusters (e.g. Miller and Hamilton 2002a; Rodriguez et al. 2019; Antonini et al. 2019; Arca Sedda 2020a; Arca Sedda et al. 2020b) or runaway collisions of massive stars (e.g. Portegies Zwart et al. 2004; Giersz et al. 2015; Mapelli 2016; Rizzuto et al. 2020) and binary star mergers in young star clusters (Di Carlo et al. 2020a) might even lead to the formation of intermediate-mass BHs (Graham et al. 2019) and BHs with mass in the pair-instability gap (e.g. Arca Sedda et al. 2020b), similar to GW190521 (Abbott et al. 2020c, d).

### 1.3.3 Triple stellar systems

Some of the LISA sources may form as part of triples and higher-order multiples. This includes sources in the Galactic disc (formed through e.g. isolated triple evolution) as well as those in dense environments. Three-body (or more) interactions are important in the formation of compact sources in two ways: during short-lived dynamical interactions and in hierarchical triple systems (Fig. 11).

Hierarchical triples, in which two bodies orbit each other, and a third body orbits the centre of mass of the inner orbit, can remain stable for secular timescales, and therefore stay intact for Hubble times (Kiseleva et al. 1994; Mardling and Aarseth 1999; Georgakarakos 2008; He and Petrovich 2018). They may form in clusters (where they may interact with interloper stars in the densest environments) or exist in the Galactic disc (and evolve in pure isolation). Their evolution differs from that of isolated binaries due to three-body effects. Hence, triples that live their lives in isolation bridge the gap between classically isolated LISA sources and dynamically-evolving sources (often used to mean cluster sources). The importance of three-body interactions in hierarchical systems has been recognised for the evolution of stellar triples (Thompson 2011; Hamers et al. 2013; Silsbee and Tremaine 2017; Antonini et al. 2017; Liu and Lai 2017; Toonen et al. 2018; Fragione and Loeb 2019; Fragione and Kocsis 2019; Toonen et al. 2020), triples that consists of a combinations of stars and planets (Hamers and Portegies Zwart 2016; Hamers 2017; Veras et al. 2018;
Stephan et al. 2018, 2020), as well as stellar binaries in dense environments (Antonini and Perets 2012; Antonini et al. 2016; Petrovich and Antonini 2017; Stephan et al. 2016, 2019; Hamilton and Rafikov 2019b, a; Fragione et al. 2020; Martinez et al. 2020; Fragione et al. 2020).

The general formation of binaries and multiples (compact and wide) in clusters is boosted during the collapse of the dense cluster core, which is halted by frequent stellar interactions (Spitzer 1987; Hut et al. 1992). The formation of the first binaries takes place most likely via three-body scatterings, involving three initially unbound objects (Goodman and Hut 1993; Lee 1995). As soon as binaries start forming, binary-single (Hut and Bahcall 1983; Sigurdsson and Phinney 1993) and binary-binary (Mikkola 1983; Miller and Hamilton 2002a) interactions take over and become the dominant dynamical processes at play. Even for relatively low triple fractions, dynamical interactions involving triples occur roughly as often as encounters involving either single or binary stars alone, particularly in low-mass star clusters (Leigh and Geller 2013). When the objects involved in the interaction cross their mutual sphere of influence, a strong interaction can trigger the formation of a short-lived bound triple system (Goodman and Hut 1993). During these chaotic resonances, a pair of objects has a non-negligible probability of experiencing a very close passage, triggering the formation of a compact binary and subsequent merger (Samsing et al. 2014). Depending on the cluster structure, binary mergers developing through resonant interactions can be highly eccentric at LISA frequencies and even still when entering the frequency range typical of ground-based detectors (Samsing 2018; Samsing and D'Orazio 2018; Arca Sedda et al. 2021b). Binary-binary interactions (Mikkola 1984; McMillan et al. 1991; Miller and Hamilton 2002a) represent another efficient mechanism to form triples, either in the form of short lived, resonant unstable triples (Hut and Bahcall 1983; Zevin et al. 2019b; Arca Sedda et al. 2021b), or in a hierarchical configuration (Antonini et al. 2016; Zevin et al. 2019b; Arca Sedda et al. 2021b; Martinez et al. 2020; Fragione et al. 2020).

The best known manifestation of three-body dynamics are the von Zeipel–Kozai–Lidov cycles (von Zeipel 1910; Lidov 1962; Kozai 1962) regime in which the inner orbit eccentricity and the inclination between the two orbits vary periodically. Strictly speaking this applies to the (inner) test particle regime, an axisymmetric outer potential and the lowest-order expansion of the Hamiltonian (i.e., quadrupole). Relaxing either one of these assumptions leads to qualitative different dynamical evolution, which include extreme eccentricity variations and orbital flips (see Naoz 2016, for a review). The high eccentricities can lead to close passages between the bodies, mass transfer, and enhancement of dissipative processes such as from tides or by GW emission. Over time, the latter can lead to a significant reduction of the inner orbital separation during the nuclear burning stages or during the compact object phase of the stars (Mazeh and Shaham 1979; Kiseleva et al. 1998; Fabrycky and Tremaine 2007; Thompson 2011). Through the internal stellar evolution, a triple may transition from one dynamical regime to another, enhancing (or diminishing) the three-body effects (Shappee and Thompson 2013; Michaely and Perets 2014; Toonen et al. 2016).
## 1.4 Expected LISA observations: numbers and rates

**Coordinators:** Abbas Askar, Simone Bavera

**Contributors:** Abbas Askar, Quentin Baghi, Simone Bavera, Tassos Fragos, Valeriya Korol, Kyle Kremer, Manuel Arca Sedda

### 1.4.1 Binary's detectability

The detectability of Galactic stellar binaries with LISA primarily depends on the parameters involved in the GW amplitude:

$$A = \frac{2(GM)^{5/3}(\pi f)^{2/3}}{c^4 d}, \qquad (1)$$

where *f* is the binary's GW frequency, *M* = (m₁m₂)³⁵/³/(m₁ + m₂)¹⁵/³ is the chirp mass (with m₁ and m₂ being the primary and secondary masses), *d* is the luminosity distance, *G* and *c* are respectively the gravitational constant and the speed of light. This follows from the fact that the signal-to-noise ratio scales linearly with amplitude $\rho \propto A\sqrt{T}$ with *T* being the observation time (Babak et al. 2021). Besides, in contrast

<table><caption>**Table 3** Estimated absolute number of compact binaries from *isolated binary evolution* in the Milky Way</caption><thead><tr><th>Source</th><th><i>N</i></th><th><i>N</i><sup>detected</sup></th></tr></thead><tbody><tr><td>WD+WD</td><td>~ 10<sup>8</sup></td><td>6000–10,000</td></tr><tr><td>NS+WD</td><td>~ 10<sup>7</sup></td><td>100–300</td></tr><tr><td>BH+WD</td><td>~ 10<sup>6</sup></td><td>0–3</td></tr><tr><td>NS+NS</td><td>~ 10<sup>5</sup></td><td>2–100</td></tr><tr><td>BH+NS</td><td>~ 10<sup>4</sup>–10<sup>5</sup></td><td>0–20</td></tr><tr><td>BH+BH</td><td>~ 10<sup>6</sup></td><td>0–70</td></tr></tbody></table>

The columns show the source type, the total number of binaries in the galaxy at any frequency and the total number of estimated sources detected by LISA. We report values from independent studies which assume different LISA mission lifetimes and SNR. WD+WD models assume a frequency range 0.5–10 mHz while models for the other sources assume a frequency range 0.1–10 mHz. At lower frequencies the total number of LISA sources is so high that it might become impossible to distinguish individual sources from the GW foreground. The ranges of the expected intrinsic number of each binary type are extracted from Nissanke et al. (2012), Breivik et al. (2020a), Belczynski et al. (2010a), Kruckow et al. (2018), Nelemans et al. (2001c), van Oirschot et al. (2014), Lamberts et al. (2018) while the ranges of the expected number of LISA sources are extracted from Nelemans et al. (2001b), Korol et al. (2017), Lamberts et al. (2019), Korol et al. (2018), Liu et al. (2010), Ruiter et al. (2010), Tauris (2018), Breivik et al. (2020a), Belczynski et al. (2010a), Kruckow et al. (2018), Lau et al. (2020), Andrews et al. (2020), Sesana et al. (2020)
<table><caption>**Table 4** Estimated absolute number of compact binaries in *globular clusters* in the Milky Way</caption><thead><tr><th>Source</th><th><i>N</i></th><th><i>N</i><sup>detected</sup></th></tr></thead><tbody><tr><td>WD+WD</td><td>~2 × 10<sup>4</sup></td><td>4–20</td></tr><tr><td>NS+WD</td><td>~10<sup>3</sup></td><td>3–6</td></tr><tr><td>BH+WD</td><td>~10<sup>2</sup></td><td>2–4</td></tr><tr><td>NS+NS</td><td>~40</td><td>1</td></tr><tr><td>BH+NS</td><td>~4</td><td>0</td></tr><tr><td>BH+BH</td><td>~2 × 10<sup>2</sup></td><td>4–7</td></tr></tbody></table>

The columns show the source type, the total number of binaries in the galaxy at any frequency and the total number of estimated sources detected by LISA assuming a frequency range of $10^{-5}$–1 Hz and a mission lifetime of 4 years with a SNR ranging between 2 and 7 (Kremer et al. 2018a)

with electromagnetic observations, the observed GW signal scales as $1/d$, rather than $1/d^2$.

The GW frequency, defined as $f = 2/P$ with $P$ being the binary’s orbital period, has the strongest impact on the signal’s detectability. Binaries emitting at $f > 3$ mHz fall in the most sensitive part of the LISA frequency band (Amaro-Seoane et al. 2017). As a result, these high-frequency binaries—even if consisting of the lowest-mass WD components—can be detected across the Milky Way also reaching satellite galaxies (cf. Figs. 1 and 15). At a set frequency, the maximum distance at which the binary can be detected increases with its chirp mass, as shown in Fig. 1. In addition, the chirp mass defines how fast the GW frequency changes during the in-spiral phase. This so-called chirp is given by

$$ \dot{f} = \frac{96}{5} \pi^{8/3} \left( \frac{GM}{c^3} \right)^{5/3} f^{11/3}. \qquad (2) $$

The chirping rate for stellar binaries in the LISA band is generally very slow ($\dot{f} < 10^{-15}$ Hz$^2$ for typical WD+WD and NS+NS binaries emitting at frequencies lower than a few mHz). Note that the frequency derivative (not the chirp mass) can be measured directly from GW data. The measurements is possible for binaries emitting at sufficiently large frequencies ($f > 3$ mHz; Roebber et al. 2020), such that during the observation time ($T$) with LISA the binary sweeps through at least a few frequency bins ($1/T$). If $\dot{f}$ is measured and assuming that the inspiral is driven by radiation reaction only, the luminosity distance can be recovered by plugging in the measured value of the chirp mass into Eq. (1). At lower frequencies, binaries will be effectively “seen” by LISA as monochromatic. In which case, only measurements of $f$ and $A$ will be possible, while the chirp mass estimate will be degenerate with that of the distance (cf. Eq. 1).

Unlike circular binaries, eccentric ones emit GWs at multiple harmonics. Each signal can be thought as a collection of $n$ signals from circular binaries emitting at $f_n = nf/2$ and the amplitude $A_n = A(2/n)^{5/3}g(n, e)^{1/2}$, where $g(n, e)$ is given in Peters and Mathews (1963). The total signal-to-noise ratio can be estimated as the
quadrature sum of the individual harmonics’ signal-to-noise ratios. Thus, to measure the chirp mass in case of the eccentric binary, in addition to $\dot{f}$, one needs to simultaneously measure the eccentricity. This is possible when at least two harmonics are detected (e.g. Seto 2016), otherwise only an upper limit on the chirp mass can be derived.

Another aspect of Galactic binaries’ detectability is that they will be observed continuously during the course of the mission. However, it is likely that the measurements will undergo occasional interruptions due to communication antenna re-pointing or spacecraft housekeeping. Additionally, spurious disturbances may affect the interferometer signals, such as the transient glitches observed in LISA Pathfinder (Armano et al. 2022; Baghi et al. 2022), leading to corrupted data spans. These operating conditions may impact the duty cycle of the mission, with a minimum requirement of 89% at the time of writing. If the masking events are frequent, they could impact the detection of low-frequency sources (around 0.1 mHz and below). However, mitigation techniques have been developed and show promising results for restoring the optimal detectability (Baghi et al. 2019; Blelly et al. 2022).

### 1.4.2 Detection and parameter estimation expectations

It is expected that the Galactic stellar binaries will dominate the number of individually detected GW sources at mHz frequencies (Table 3). Out of ~$O(10^7)$ stellar binaries emitting in the LISA frequency band, LISA is expected to deliver ~$O(10^4)$ individual detections (e.g. Littenberg et al. 2020; Karnesis et al. 2021). Current estimates suggest that at frequencies >3 mHz, Galactic WD+WD detectable by LISA will be counted in thousands, NS+NSs in few up to hundreds and BH+BHs in a few (see Tables 3 and 4). At frequencies <3 mHz the number of stellar binaries is so large, that only a small fraction—the closest and more massive ones—will be individually detected, while the rest of the population will form an unresolved stochastic foreground (see Sect. 1.6.2).

Amongst the resolvable systems, population synthesis simulations suggest that hundreds will be exceptionally strong LISA sources (e.g. Karnesis et al. 2021). These binaries will be detectable within weeks from the start of mission operations, and over the full mission lifetime can accumulate SNRs up to $10^3$. Based on an SNR evaluation via an iterative scheme for the estimate of the confusion foreground generated by Milky Way’s GW sources (e.g. Littenberg et al. 2020; Karnesis et al. 2021), synthetic population analysis (e.g. Korol et al. 2017; Lamberts et al. 2019) yields about up to $10^4$ binaries detectable with SNR > 20, several $10^3$ with SNR > 100, a few with SNR > 1000. For all binaries, frequencies predicted be measured very accurately with $\sigma_f/f < 10^{-5}$, which corresponds to $\sigma_P < 0.025$ ms for a typical mHz WD+WD binary. Frequency derivative is estimated to be measured to better than 30% for up to $10^4$ binaries, leading to the measurement of the chirp mass via Eq. (2). Consequently, also distances can be derived to better than 30% for several $10^3$ binaries. The sky localisation uncertainty depends strongly on the SNR and GW frequency ($\propto \rho^{-2}f^{-2}$). Additionally, it is also dependent on the ecliptic
latitude: a source on the ecliptic has an order of magnitude more uncertainty than a
source at the poles (Roebber et al. 2020). On average, sky localisation error for
Galactic WD+WD systems is expected to be of several deg², improving with
increasing GW frequency and SNR down to sub-square degree precision and below.
These expectation are well supported by Bayesian-based data analysis exercises
(Littenberg et al. 2020) with mock data simulating observations over 1 year, and
including 30 millions of injected Galactic sources. Consolidation of these results are
expected once more realistic data simulations featuring mixed source types are
analysed (see https://lisa-ldc.lal.in2p3.fr).

Beyond the Milky Way, the Local Group galaxies are expected to harbour from a
few to a few hundreds LISA sources (mainly WD+WDs and some NS+NSs)
depending on the total mass and the distance of the galaxy (Seto 2019; Andrews et al.
2020; Korol et al. 2020; Lau et al. 2020) (see Fig. 1). For instance, the number of
WD+WDs and NS+NS in the largest Milky Way satellites, the Large and Small
Magellanic Clouds, will be high enough to overcome Galactic foreground and to
unambiguously identify these galaxies in the LISA data (Roebber et al. 2020). Even
further away, with the total mass comparable to the Milky Way’s mass, the
Andromeda galaxy could also be visible on the LISA sky as a group of GW sources
(Korol et al. 2018).

The outside limits of the Local Group, LISA can access distances up to ~1 Gpc
through GW signal from stellar-mass BH+BHs, which will ultimately be observable
during merger to ground-based detectors (see Sect. 1.5.1). Studies based on
cosmological simulations of galaxies at z = 0 find that the present-day dwarf galaxies
can accommodate a larger BH+BH merger rate compared to massive galaxies
(O'Shaughnessy et al. 2017). Specifically, for massive BH+BHs similar to
GW150914, about 40% of mergers are expected to be in galaxy progenitors of
Milky Way-like systems and the rest in smaller satellite or isolated dwarf galaxies
(Marassi et al. 2019). This translates into a large number of potential hosts within the
estimated LISA horizon distance. Not only BH+BHs formed from the evolution of
isolated binaries, but also BH+BHs formed in extragalactic globular clusters may be
detectable by LISA, with initial studies predicting the number of such sources to be
in the range of 1–10² (Kremer et al. 2018a).

Given LISA's selection effects, extra-galactic LIGO-like BH+BH are expected to
have quite low signal-to-noise ratios (~10). For this class of stellar binaries,
Buscicchio et al. (2021) report sky localisation errors of a few tens of squared
degrees and constrains on the detector-frame chirp mass down to ±0.01 M☉. They
also show that at 10 mHz the eccentricity for these binaries can be measured down to
10⁻³, while the merger time can be determined within a time window of 1 h.

For sources that form through isolated binary evolution, the rates (such as those
mentioned above and in Table 3) are often estimated with the population synthesis
approach. There are many uncertainties that affect the rate calculations. These
uncertainties can be divided in five broad categories: (i) binary evolution physics (e.
g., CE evolution, mass transfer stability, mass-accretion efficiency, etc.), (ii) stellar
evolution physics (e.g., metallicity dependent stellar winds, core-collapse mecha-
nisms, natal kicks, pair-instability SN and pulsational pair instability, etc.), (iii) initial
stellar and binary properties (e.g., initial mass function, binary fraction, initial distribution of separation, mass ratio, eccentricity, etc.), (iv) different assumptions for the Galactic spacial distribution (thin/thick disc and bulge) and star-formation history and (v) LISA selection effects (e.g., GW foreground, mission length, sensitivity curve and SNR detection threshold).

As discussed in Sect. 1.3.2, LISA sources may also form dynamically in dense stellar environments such as globular clusters (and may have markedly different features compared to sources that form through isolated binary evolution, Sect. 1.3.1). In Table 4, we show the estimated number of sources expected to be found in globular clusters in a Milky Way-like galaxy (Kremer et al. 2018a). The number of detectable LISA BH+BHs originating from young massive clusters and open clusters is expected to be several tens to ~100 (or about 0.5–3 times the density of those clusters in the local volume in units of Mpc⁻³; Banerjee 2020). Merger rate estimates for GW sources produced in dynamical environments from cluster simulations, can also be influenced by many uncertain physical processes, some of which are common to the ones outlined for sources formed via isolated binary evolution. For instance, natal kicks for compact objects have a direct impact on retention of those objects in stellar clusters (Morscher et al. 2013; Arca Sedda et al. 2018; Webb et al. 2018; Pavlík et al. 2018; Banerjee et al. 2020) which influences the number of dynamically formed binary systems. The number of compact objects that form in dense environments also depends on their metallicity and the initial mass function of their stars which may vary with their formation environment (Dabringhausen et al. 2009; Geha et al. 2013; Krumholz 2014; Chruślińska et al. 2020). Additionally, dissolved or tidally disrupted open and globular clusters can also contribute to the number of GW sources that may have dynamically formed (Muratov and Gnedin 2010; Fragione et al. 2018; Giersz et al. 2019).

Apart from population synthesis-like simulations, another approach to predict the expected LISA rates is to derive empirical estimates from the already observed population of sources. For NS+NSs for instance, one can use the inferred merger rate coming from the known Galactic NS+NS population, and accounting for survey selection effects (Phinney 1991; Kim et al. 2003), or the inferred merger rate from LIGO-Virgo (Abbott et al. 2021b), to predict that LISA should be able to detect 50–300 NS+NSs in the Milky Way (Andrews et al. 2020). In a similar manner, based on O1 LIGO-Virgo detections, it was estimated that LISA maybe able to detect up to ~50 BH+BHs (Sesana 2016), but the inferred BH+BH merger rate density decreased in the most recent O3 run by a factor of ~2. Empirical estimates of the Galactic NS+WD population have been derived, based on the observed pulsar population, with 100–150 being predicted to be observable by LISA (Tauris 2018).

# 1.5 Synergies

## 1.5.1 Synergies with EM observations

Coordinators: Thomas Kupfer
**Contributors:** Thomas Kupfer, John Tomsick, Nicole Lloyd-Ronning, John Quenby, Thomas Tauris, Thomas J. Maccarone

Many new verification binaries are expected to be discovered before the launch of LISA, in particular with wide-field optical surveys such as ZTF, BlackGEM and LSST on the Rubin Observatory. Once flying, LISA will be complemented with surveys across different frequency bands (radio to gamma-rays). New sources discovered by LISA can be studied with the next generation of follow-up facilities such as ESO/ELT, CTA, SKA, ngVLA or even Athena as well as smaller space missions which could be approved, built and launched in the early 2030s (eXTP or STROBE-X). This provides a plethora of large-scale follow-up resources for detailed multi-messenger studies but it requires a well planned follow-up strategy to generate the most useful results.

The large number of facilities running in the 2020s and 2030s will provide a bright future for the research on compact Galactic binaries, with hundreds of additional EM discovered systems ready to be studied in detail through EM+GW observations as soon as LISA is operational. A significant sample of binaries, observed with EM+ GW observations, will open up possibilities to explore and study astrophysical phenomena which are crucial to our understanding of the Universe. This includes the long-standing questions of the progenitors of SNe Ia, the formation and evolution of compact objects in binaries and accretion physics under extreme conditions.

**1.5.1.1 UV/Optical/IR observations** Previous studies predict that we will be able to observe several thousand Galactic binaries in both GW and optical emission (Littenberg et al. 2013; Korol et al. 2017). A subset of the known UCBs have orbital periods that lie in the LISA band and these will be individually detected by LISA due to their strong GW signals, some on a timescale of weeks or a few months. Combined GW and EM multi-messenger studies of UCBs will allow us to derive population properties of these systems such as masses, radii, orbital separations, and inclination angles but in many cases EM observations are required to complement GWs and break degeneracies in the GW data. Shah et al. (2012), Shah et al. (2013), Shah and Nelemans (2014b) and Kupfer et al. (2018) present several studies on how EM observations can complement LISA GW data and vice versa. The GW amplitude and inclination is strongly correlated, but the GW amplitude can be improved by a factor of six when including EM constraints on the inclination and the sky position and inclination can reduce the uncertainty in amplitude by up to a factor of 60 (Shah and Nelemans 2014b).

Additionally, knowing the distance from EM, e.g., from parallaxes measured by Gaia, will help to derive a chirp mass from GWs even for non-chirping sources because for many LISA sources only the frequency and amplitude will be measured. This leaves a degeneracy between chirp mass and distance and inclination, since a more massive binary further away can have the same amplitude as a lower mass one that is closer by and inclined systems closer by look like face on systems further away (Shah et al. 2012; Shah and Nelemans 2014a).

If the chirp mass has been measured from GWs using, e.g., Gaia parallaxes and the mass ratio has been measured from EM, through radial velocities or ellipsoidal
<figure><img src="image_14.png" /><figcaption>**Fig. 12** Upper panel: GW frequency derivative, $|\dot{f}_{GW}|$ as a function of GW frequency, $f_{GW}$ for the UCXB shown in Figs. 4 and 5. The black coloured points correspond to the inbound leg (orbital shrinking) and the blue points correspond to the outbound leg (orbital expansion, after reaching the orbital period minimum) including mass transfer/loss from the system and finite-temperature effects of the WD donor. Each point represents a binary stellar MESA model. The green solid circle indicates the onset of the UCXB stage. (Before this point, the system is a detached NS+WD binary). Bottom panel: pure general relativistic (GR, blue) and astrophysical (orange) chirps as a function of GW frequency for WD+WD (AM CVn) systems. The astrophysical contribution from mass transfer and tides leads to a significant deviation from the contribution from GR alone and will cause the systems to widen their orbit upon mass transfer. The decoupling of the astrophysical chirp from the GR chirp will be possible with combined measurements from LISA and Gaia. The error bars show the anticipated $1\sigma$ measurement errors. On average, Breivik et al. (2018) find that about 50 AM CVn systems in the Mikly Way with $P_{\text{orb}} < 800$ s have resolvable GR and astrophysically driven chirps. Images reproduced with permission from [top] Tauris (2018), copyright by APS; and [bottom] Breivik et al. (2018), copyright by AAS</figcaption></figure>
deformation of one component, both measurements can be combined to the measure
the masses of both components with a few percent precision. Comparing the
measured orbital decay with the predicted orbital decay from general relativity will
allow us to measure the effects of tides compared to GWs. Tides are predicted to
contribute up to 10% of the orbital decay (Piro 2011) but has not been measured so
far and is very difficult with GW and EM alone. Some of the known AM CVn and
detached WD+WD binaries have well constrained distances from Gaia (Kupfer et al.
2018; Ramsay et al. 2018). Measuring the distance from EM constrains the
uncertainty in chirp mass to 20%, whereas adding the period derivative $\dot{P}$ reduces it
to 0.1% (see e.g. Fig. 12). A GW chirp mass measurement would provide the first
detection of tidal heating in a merging pair of WDs from the deviations in predicted $\dot{P}$
(Shah and Nelemans 2014b). With a large enough sample of WD+WD binaries
whose chirp masses can be determined, we can plausibly extract constraining
information about CE phase evolution physics. In studying massive WD+WD
binaries that are likely progenitors of merger-induced collapse NSs (Ruiter et al.
2019), chirp mass distributions had different shapes depending on the adopted CE
phase prescription in the binary evolution population synthesis model.

Several studies have shown that spectral and photometric analysis of detached WD+WD can provide precise sky positions, orbital periods and in some cases mass ratios, inclinations, and the rate of orbital period decay (e.g. Maxted et al. 2002; Brown et al. 2011; Hermes et al. 2012; Burdge et al. 2019a). All of the known verification binaries have precise sky positions and orbital periods and five systems have a measured orbital decay from photometric monitoring (Kupfer et al. 2018). SDSSJ0651, a 12 min orbital period detached WD+WD (Brown et al. 2011), and ZTFJ1539, a 7 min orbital period detached WD+WD (Burdge et al. 2019a), are prime example of what can be accomplished. Using only 1 year of eclipse timing measurements, Hermes et al. (2012) found an orbital decay of $\dot{P} = (-9.8 \pm 2.8) \times 10^{-12} \text{ s s}^{-1}$ in SDSSJ0651 which has not been updated since then. Burge et al. (2019a) used photometric data from PTF/iPTF and ZTF covering a total of 10 years and measured a very precise orbital decay of $\dot{P} = (-2.373 \pm 0.005) \times 10^{-11} \text{ s s}^{-1}$. However, neither of the two systems has a precision of WD component masses good enough from optical observations to see if their $\dot{P}$ differs from the predictions of the theory of General Relativity (GR). GW observations can solve that. Tidal theory predicts a 10% deviation from GR if the WDs are tidally heating up. Which means that combining EM+GW observations from LISA will allow a measurement of the amount of tidal heating in these merging pairs of WDs for the first time.

Combined EM+GW observations of the Galactic WD+WD population will help
to solve another major problem in astrophysics: the SNe Ia progenitor problem.
Although only the thermonuclear explosion of a WD following the interaction with a
binary companion can explain the observed features in the SN light, much less is
known about their progenitors. Recent results have shown that SN Iae show a large
range of explosion energies and decay times, photometric and spectroscopic
signatures indicating different progenitor systems (Jha et al. 2019). Several different
explosion scenarios are under discussion, including the merger and subsequent
explosion of an ultra-compact WD+WD system (double-degenerate model) or the
explosion triggered by ignition of an helium shell accreted from a helium star in a
UCB (double-detonation scenario). However, the number of known progenitor
systems is limited. Rebassa-Mansergas et al. (2019) studied the probability of finding
WD+WD progenitors of SNe Ia using a binary population synthesis approach, and
found that the chance of identifying such progenitors purely in EM data is ~10⁻⁵.
These include both double-lined spectroscopic binaries and the eclipsing systems.
Even with the next generation of 30-m class telescopes, the probability for detection
only goes up by a factor of ~10. Korol et al. (2017) predicts that LISA will
individually resolve ~25,000 detached WD+WD systems including the most
massive systems. EM follow-up observations in combination with GW measure-
ments will allow us to measure masses of individual systems and find and
characterize the population of double degenerate SN Ia progenitors.

Kilonovae (see Metzger 2019) are optical/IR emission accompanying the merger of NS+NS, possibly NS+BH, and in special cases, WD+WD mergers (Rueda et al. 2019). Although LISA is not sensitive to the actual merger that can produce a kilonova, it is sensitive to the GW emission from their progenitors. Therefore it is worthwhile to consider kilonova events in the nearby universe because they allow constraints on these degenerate stellar populations.

### 1.5.1.2 X-ray observations
Many of the LISA sources also emit X-rays, thus allowing for a number of joint LISA + X-ray investigations. The donor stars in UCXBs appear to be a mixture of C/O-core and He-core WDs and abundance measurements can help identify their formation scenario (Nelemans et al. 2010). In the oxygen-rich systems, oxygen is the dominant coolant in the accretion discs, and the iron emission lines are suppressed; the strength of the iron lines is broadly in agreement with the thermonuclear burst properties of the sources, strengthening the case that this donor classification process works reasonably well even in its simplest form (Koliopanos et al. 2020). However, in some cases, the abundances of the WD inferred from X-ray data are at odds with those inferred from Type I burst properties. In a few cases, the inference has been made from neon-to-oxygen ratios (Juett et al. 2001), and for this scenario, it has been shown that there is a channel of binary evolution that allows a He-core WD to have such neon-to-oxygen ratios (in't Zand et al. 2005). Alternatively, Bildsten et al. (1992) show that spallation of CNO elements in a NS atmosphere is quite likely. The combination of LISA measurements with X-ray (and optical) abundance measurements thus opens a window to determining which of these scenarios is correct. If the apparent C/O-core WD donors are paired with high mass NSs, then the spallation scenario is strongly preferred.

More detailed X-ray spectroscopy should potentially be able to make detailed
abundance estimates for the donor stars, allowing, e.g., identification of systems in
which the CNO processing may not have gone to completion, and perhaps estimating
the time of formation through estimates of the abundances of non-CNO elements,

<footnote n="3">³ Most imaging X-ray telescopes will not be able to make such measurements because of pile-up or deadtime issues.</footnote>
which could yield the initial metallicity of the star. The X-ray measurements are essential given that a large fraction of the UCXBs are located in globular clusters, or deep in the Galactic Plane where ultraviolet and optical spectroscopy are more challenging. Substantive work has also been done in the optical wavebands (see e.g. Nelemans et al. 2004a, 2006). Furthermore, when combined with LISA data the UCBs then may provide a means of testing how conservative accretion onto NSs is; C/O-core WD donors must start at masses of at least ≈ 0.5$M_⊙$, but are generally observed with masses of 0.1 $M_⊙$ or less. If the early stages of mass transfer in these systems are conservative, the NSs should typically be about 1.8 $M_⊙$, while if they show much lower masses, this implies that the mass transfer was strongly non-conservative.

Additionally, for 4U 1820−30, X-ray measurements provide a straightforward way to monitor the source’s period derivative (Tan et al. 1991), as the source is deep in the potential well of a globular cluster whose gravitational acceleration leads to its negative period derivative. It is easy to track the source’s orbital period in the X-ray band (Stella et al. 1987; Tan et al. 1991), and hard in other bands, due to the crowding in the cluster. With more intensive X-ray data, the period derivative of 47 Tuc X-9, the best candidate BH+WD binary in the Milky Way (Bahramian et al. 2017), could be tracked. Imposing these constraints, which are likely to lie outside the range of normal templates, can help improve the quality of the LISA GW detection. Globular clusters are likely good hosts for LISA sources, but the globular cluster sources’ periods have come from ultraviolet photometry (Dieball et al. 2005; Zurek et al. 2009) and the best non-cluster source’s period comes from optical spectroscopy (Madej et al. 2013). Additional intensive monitoring campaigns would be required for the period derivative to be estimated.

For the AM CVn systems, X-ray emission is also valuable. The same abundance issues can be studied in the X-ray band in AM CVn systems, although primarily from the emission lines from the boundary layer of the accretor, rather than disc reflection. Relatively short period AM CVn systems will be detectable to large distances, where reddening may be important, and in these cases, restricting the set of sources to those which are in a reasonable range of fluxes. For the faintest AM CVn sources with periods less than half an hour, the X-ray luminosities are typically about $10^{31}$ erg s⁻¹ (e.g. Nelemans et al. 2004b; Strohmayer 2004; Ramsay et al. 2005, 2006; Zurek et al. 2016), meaning that eROSITA should detect them to distances of about 3 kpc. Combining with radio surveys to remove background Active Galactic Nuclei (AGN), and optical surveys to remove foreground stars and CVs should then yield a much more manageable list of candidates for high time resolution optical follow-up (which usually has limited fields of view) than without the X-ray data and potentially add more LISA verification sources.

X-rays are also likely to provide the best EM distance estimators for many of these UCXBs (see Fig. 8) complementing LISA data. Some are located in globular clusters, where the cluster distance can be used. None of the Galactic field UCXBs is bright enough for Gaia in the optical, and most are also too faint for radio parallaxes with current facilities. Thermonuclear bursts with radius expansion can be used to estimate the Eddington luminosities (Kuulkers et al. 2003), and these can then be
used in conjunction with the GW estimates of the masses to establish self-consistent
properties for the sources. For the persistent sources that burst in an appropriate
manner, these data are already in hand, but obtaining such data for transients would
require instruments with large collecting area and small deadtime (e.g., NICER,
STROBE-X, eXTP Gendreau et al. 2016; Ray et al. 2019; Zhang et al. 2019).³ The
other approach that can be used to estimate distances is that of dust-scattering halos
(Trümper and Schönfelder 1973), something that requires good angular resolution,
good collecting area, and the ability to observe bright sources; while Chandra has
done some work in this area, Athena should be able to help dramatically (Corrales
et al. 2019).

Some UCXBs may contain BHs as well. The first strong globular cluster BH
candidate (Maccarone et al. 2007) in NGC 4472 is an ultracompact system (Zepf
et al. 2008), probably with an orbital period near 5 min, and 47 Tuc X-9 (Miller-
Jones et al. 2015; Bahramian et al. 2017) is also a strong candidate ultracompact BH
X-ray binary. At the shortest orbital periods, BH+WD binaries should be
detectable by LISA to distances of a few megaparsecs. For these cases, imaging
X-ray data would be needed, along with follow-up optical spectroscopy to look for
[O III] nebulae as well as hydrogen emission similar to that in the globular cluster
RZ 2109 in NGC 4472 (Zepf et al. 2008; Steele et al. 2014; Dage et al. 2019). In the
Milky Way, UCXBs with BH accretors at relatively long orbital periods could be
quite faint X-ray sources in quiescence (being considerably fainter than accreting
NSs at the same mass transfer rate due to advection dominated accretion, Narayan
and Yi 1994). They could also exhibit only rare outbursts, meaning that sensitive X-
ray observations would be needed to detect their counterparts. Furthermore, these
objects may be preferentially in globular clusters, meaning that excellent angular
resolution, from Chandra or a Chandra successor mission like Lynx or AXIS would
be needed to find their counterparts. If some new BH UCXBs are discovered with X-
ray outbursts, they may become bright enough to make BH spin estimates using
reflection and/or disc continuum modelling.

For most of the topics related to accretion, there is a need for developing better
spectral models that treat unusual abundances. Development of reflection models that
include both the reflection from the surface of the WD, and discs made from
hydrogen-poor material is thus vital. It has already been found that details of how
atomic physics is incorporated into the disc models can affect inferred spins and
abundances (García et al. 2016; Tomsick et al. 2018).

X-ray observations are also vital for understanding the detached systems with NS
members. In most of these systems, the older NS will have experienced significant
spin-up, and will be a millisecond pulsar. Pulsar beam opening angles are larger at
high energy than at radio wavelengths (a phenomenon exhibited by objects like
Geminga (Halpern and Holt 1992) and many of the Fermi-discovered pulsars),
meaning that some fraction of these objects will be radio-quiet pulsars (Marelli et al.
2015). X-ray observations will then provide the most comprehensive means for
estimating the spins of these systems and determining what fraction of them have
become recycled. Furthermore, in combination with radio searches for pulsations,
having a gravitationally-selected sample will allow a clean determination of the ratio
of pulsars with radio and X-ray emission, allowing an important constraint for
developing a full picture of the pulsar beam geometry. In an ideal case we may
identify an object with thermal cap emission from the NS, such that pulse-profile
measurement and modelling could be done to estimate its radius. If this comes in
conjunction with sufficiently good LISA GW measurements to provide an
independent, precise, estimate of the NS's mass, this would give a constraint on
the equation of state for NSs, even from a single object (Watts 2019).

**1.5.1.3 Radio observations** The synergies from joint, multi-messenger observations of radio pulsar binaries entering the LISA band are very promising and will provide significantly more information than observations in the EM or GW bands alone (see Fig. 8). Such benefits include better measuring the orbital inclination angle (Shah et al. 2012) and sky position (Shah et al. 2013) and potentially even constraining the NS mass-radius relation to within ~0.2% (Thrane et al. 2020). Additionally, radio astrometry can give parallax distances out to a few kpc already, and with ngVLA (Murphy et al. 2018), that should increase dramatically. Most LISA sources would be nearby enough that with ngVLA, one would be able to get 10% or better geometric parallaxes over about 3/4 of the sky.

Binary NSs in NS+NS and NS+WD systems enter the LISA band at a GW
frequcey of order 1 mHz (depending on their distance), corresponding to orbital
periods of about 30 min. Doppler smearing of radio pulsations from pulsars in such
tight binary orbits (Eatough et al. 2013) could cause a selection bias against detection
of e.g., rapidly spinning millisecond radio pulsars in many previous and present day
acceleration searches (at least for dispersion measures, < 100 cm⁻³ pc). However,
using neural networks, Pol et al. (2021) develop accurate modelling of the observed
binary pulsar population and argue for a ~50–80% chance of detecting at least one
of these systems with P_orb ≤ 15 min using data from surveys with the Arecibo radio
telescope, and ~80–95% using optimal integration times of ~50 s in the next
several years. The chances of a radio detection of a binary pulsar in the LISA GW
band is expected to be significantly enhanced by the completion of the Square
Kilometre Array (Keane et al. 2015).

It has been argued (Pol et al. 2021) that unequal mass NS+WD systems are easier to detect compared to the usually near-equal mass NS+NS systems. It should be kept in mind that RLO from these (often bloated) WD companions begins when P<sub>orb</sub> has decreased to 25 – 15 min, depending on their temperature (Tauris 2018) (see also Figs. 4 and 5). This will exclude radio detection of such pulsar binaries once accreted plasma enters the NS magnetosphere.

**1.5.1.4 Particle observations** For high-frequency GW detections, there are prospects for detection of neutrino's or cosmic rays, from jets produced in mergers or from supernovae (Adrián-Martínez et al. 2016). For LISA, the prospects are not so clear, even though associations of LISA GW sources with AGN jets and tidal disruption events could be possible.
## 1.5.2 Synergies with other GW detectors

**Coordinators:** Lijing Shao; Paul Groot

**Contributors:** Ilya Mandel (1.5.2.1), Alberto Sesana, Emanuele Berti, Lijing Shao (1.5.2.4), Davide Gerosa (1.5.2.1), Pau Amaro Seoane, Paul Groot, Thomas Tauris (1.5.2.2), Valeriya Korol (1.5.2.3)

### 1.5.2.1 High-frequency GW merger precursors seen by LISA
LISA has a unique capability of covering the full frequency spectrum for stellar-mass binary BHs and NSs when combined with the higher-frequency ground-based GW detectors advanced LIGO (Aasi et al. 2015) and Virgo (Acernese et al. 2015), and their third generation successors such as the proposed Einstein Telescope (Punturo et al. 2010) and Cosmic Explorer (Abbott et al. 2017a).

Some individual sources can be tracked on human timescales from the LISA band to the ≳ few Hz ground-based detector sensitive frequency band (Sesana 2016). The GW driven merger timescale for a circular binary with components of equal mass *m* from a starting frequency *f* is (Peters 1964a)

$$ \tau_{\text{GW}} \approx 5 \left( \frac{f}{0.01 \text{ Hz}} \right)^{-8/3} \left( \frac{m}{68 M_{\odot}} \right)^{-8/3} \text{ year.} \quad (3) $$

Thus, a signal like GW190521 (Abbott et al. 2020c) could be tracked from 10 mHz to merger across the full range of frequencies with the combination of LISA and ground-based detectors. Rate estimates have been presented by Shannon et al. (2015), Kyutoku and Seto (2016), Sesana (2016, 2017), Gerosa et al. (2019) and Moore et al. (2019), with predictions ranging from 0 to roughly a dozen detections during the LISA mission.

The high mass and correspondingly rapid orbital evolution of IMBHs with masses in the 100–1000 $M_{\odot}$ range makes them particularly appealing targets for tracking across the LISA and ground-based detector frequency bands. IMBHs with these masses are a challenge for EM observations: their dynamical signature is relatively insignificant, while their X-ray emission can be confused with that of super-Eddington accretors (e.g., Miller and Colbert 2004; Feng and Soria 2011; Greene et al. 2020). On the other hand, IMBH mergers have been proposed in the context of both isolated binary evolution of very massive stars (Belczynski et al. 2014) and globular cluster dynamics (Amaro-Seoane and Freitag 2006). The latter can also be responsible for intermediate-mass ratio inspirals of stellar-mass compact objects into IMBHs (Mandel et al. 2008; Haster et al. 2016a). Meanwhile, hierarchical mergers of few-hundred $M_{\odot}$ seeds have been proposed as seeds of today’s massive BHs (Volonteri et al. 2003a, see Sect. 2). Joint observations with LISA and third-generation ground-based detectors (Sesana et al. 2011a; Gair et al. 2011) would provide the perfect tools for studying these elusive IMBHs.

Observations of the same individual source across a broad range of frequencies can improve the accuracy of source parameter measurement. Some parameters are
likely to be best measured at low frequencies. For example, sky localisation accuracy depends on timing precision (Fairhurst 2009; Wen and Chen 2010; Grover et al. 2014). The sky localization accuracy can be estimated as the timing accuracy divided by the light travel time across the detector baseline (Mandel et al. 2018), which is ~ astronomical unit (AU) for LISA, yielding a relative position error of:

$$ \sigma_{\theta} \sim 0.025 \left( \frac{0.01 \text{ Hz}}{f} \right) \left( \frac{8}{\rho} \right), \qquad (4) $$

where $\rho$ is the detection signal-to-noise ratio. For heavier sources that can evolve faster than the LISA observing duration, the LISA frequency bandwidth $f_{bandwidth}$ should be used in place of $f$. The angular resolution scales inversely with baseline. Therefore, despite the lower observing frequency (lower bandwidth), for high SNR sources, LISA sky localisation is likely to be superior to the capabilities of ground-based detectors, whose baseline, even in a network, is limited by the size of the Earth (unless the signal is sufficiently long-lived to allow the effective baseline to be extended by the detector motion over the duration of the observation).

On the other hand, some source parameters will be better measured at higher frequencies, allowing ground-based detectors to provide complementary information to LISA observations. These include measurements of the ringdown of the post-merger BH, which yield the final mass and spin, and the tidal effects for NSs.

Yet other measurements could benefit from the joint constraints placed by low-frequency and high-frequency observations. These include measurements of spin magnitudes and spin-orbit misalignment angles, which could carry information about formation scenarios (e.g., Gerosa et al. 2013; Vitale et al. 2017; Stevenson et al. 2017a; Zevin et al. 2017; Farr et al. 2017; Gerosa et al. 2018a). Spin-orbit and spin-spin coupling enter the waveform at higher post-Newtonian orders in an expansion in the orbital frequency (Poisson and Will 1995), and so may be better measured at higher frequencies by ground-based detectors. On the other hand, massive binaries like GW190521 may spend a million cycles in the LISA band (only 4 cycles were observed in the LIGO band when this signal was detected in 2019; Abbott et al. 2020c). Further analysis is necessary to explore the quantitative benefits of LISA for parameter estimation of such signals (but see e.g. Vitale 2016; Moore et al. 2019; Mangiagli et al. 2019; Cutler et al. 2019)

Lower-mass GW sources such as double NSs (Lau et al. 2020; Andrews et al. 2020) will not be individually trackable on a human timescale from the LISA band to the band of ground-based detectors. However, they may still benefit from tracking the entire population of sources as the sources evolve from the LISA frequency band to the frequency band of ground-based detectors. For example, binaries circularise through GW emission (very roughly, the eccentricity scales inversely with the increase in frequency), making eccentricity challenging to observe with ground-based detectors (e.g., Romero-Shaw et al. 2019; Lenon et al. 2020). Thus, LISA observations at lower frequencies, where eccentricities are still significant, could help to distinguish compact binary formation scenarios (Breivik et al. 2016; Nishizawa et al. 2016, 2017).
This can be further aided by the detection of a stochastic background from a superposition of GWs emitted by multiple individually unresolvable binaries (see Sect. 1.2.2.6). For circular binaries, the stochastic background should be a simple power-law in frequency, and any deviations from that could indicate the emergence of new binaries, particularly eccentric binaries, at high frequencies. Moreover, the combined low-frequency and high-frequency stochastic background observations may make it easier to subtract the astrophysical background and reveal a possible GW background of cosmological origin (e.g. Mandic et al. 2012; Lasky et al. 2016; Callister et al. 2016).

LISA precursors to ground-based detector mergers could also have important repercussions in fundamental physics, allowing stringent tests of the BH no-hair theorems, as well as more stringent bounds on low-Post-Newtonian deviations from GR (Toubiana et al. 2020; Barausse et al. 2016; Tso et al. 2019; Gnocchi et al. 2019; Carson and Yagi 2020; Shao et al. 2017).

### 1.5.2.2 Dual-line GW sources
A possibility in upcoming GW astronomy will be the potential discovery of a dual-line Galactic GW source (Tauris 2018), where ground-based detectors detect the continuous high-frequency GW emission from the rapid spinning (recycled) NS (Andersson 2019) and LISA simultaneously detects the gravitational damping of the system’s orbital motion via continuous low-frequency GW emission. Such a system could very well be a UCXB (e.g. van Haaften et al. 2012; Heinke et al. 2013). Combining the expressions for the strain amplitudes of the ground-based and LISA observations ($h_{\text{spin}}$ and $h_{\text{orb}}$ respectively) yields (Tauris 2018):

$$I_{zz} \varepsilon = \sqrt{\frac{2}{5}} \left(\frac{\sqrt{G}}{2\pi}\right)^{4/3} \left(\frac{f_{\text{orb}}^{1/3}}{f_{\text{spin}}}\right)^2 \mathcal{M}^{5/3} \left(\frac{h_{\text{spin}}}{h_{\text{orb}}}\right). \quad (5)$$

Once the right-hand-side of this equation is determined observationally, and assuming that the NS mass, $M_{\text{NS}}$ can be determined from the chirp mass, $\mathcal{M}$ (see required assumptions on component mass determinations in Tauris 2018), constraints can be made on the NS moment of inertia, $I_{zz}$, and thus the NS radius (Ravenhall and Pethick 1994). Suvorov (2021) recently examined the dual-line detectability of tight Galactic binaries, and found that at least two of the known systems (4U 1820-30 and 4U 1728-34) may be visible to both ground-based and space-based instruments simultaneously. Although only measuring the moment of inertia in combination with the ellipticity, $\varepsilon$, it will still help in pinning down the long sought-after equation of state (EOS) of NS matter. The maximum spin rate and $\varepsilon$ for accreting NSs (Andersson 2019) remain to be constrained firmly.

### 1.5.2.3 TianQin
TianQin is a space-based GW observatory conceived as an equilateral triangle constellation of three drag-free satellites with frequency sensitivity at $10^{-3}-10^{-1}$ Hz (Luo et al. 2016), between LISA and DECIGO. Unlike LISA, TianQin will follow a geocentric orbit with a radius of about $10^5$ km (Hu et al. 2018; Ye et al. 2019). Its constellation plane will be nearly perpendicular to the
ecliptic plane and will have a fix orientation pointing toward RX J0806.3+1527 (Strohmayer 2005), a 5-min orbital period Galactic binary that is expected to be the strongest GW source among currently known systems (Kupfer et al. 2018). Planned for the launch around 2035, TinQin will see the same GW sources as LISA (Mei et al. 2020). Consequently, many synergies can be envisioned between the two missions. For instance, TianQin and LISA will simultaneously detect several thousand Galactic WD+WD binaries, which will improve the parameters estimation including the amplitude, inclination and sky localisation for these binaries (Huang et al. 2020).

### 1.5.2.4 Mid-band observatories, e.g., DECIGO
After the discovery of GW150914 (Abbott et al. 2016c), it was realized that massive stellar-mass BHs will be detectable in both LISA and LIGO/Virgo bands (Sesana 2016; Amaro-Seoane and Santamaría 2010). However, the SNRs are not expected to be large in the mHz band, and because of the use of template bank searching, in order to claim a confident detection, BH+BH signals in LISA require a larger SNR threshold than 15 (Moore et al. 2019). Fortunately, these sources will have large SNRs if they are seen in the decihertz band (Isoyama et al. 2018; Arca Sedda et al. 2020a; Liu et al. 2020). DECihertz laser Interferometer Gravitational wave Observatory (DECIGO) is a representative GW detector in the relevant frequency band (Yagi and Seto 2011; Kawamura et al. 2011, 2020). Studies showed that, not only will observations in the decihertz band provide profound insights to astrophysics (see Arca Sedda et al. 2020a, for a comprehensive discussion), they will also provide unprecedented playgrounds for fundamental physics (e.g., testing the dipolar radiation (Barausse et al. 2016; Liu et al. 2020). The mid-band observations of decihertz frequency are natural means to bridge the gap between LISA and LIGO/Virgo observatories.

## 1.6 Technical aspects

**Coordinators:** Irina Dvorkin

**Contributors:** Emanuele Berti, Sylvain Chaty, Astrid Lamberts, Alberto Sesana, Kinwah Wu (1.6.3), Shenghua Yu, Shane Larson, Irina Dvorkin (1.6.1, 1.6.3, 1.6.5), Kinwah Wu already cited above, Pau Amaro Seoane, Giuseppe Lodato, Xian Chen, Valeriya Korol (1.6.2), Silvia Toonen (1.6.5)

### 1.6.1 How to distinguish between different compact binaries?

One of the outstanding challenges of LISA will be to analyze a datastream that consists of multiple overlapping signals from astrophysical and possibly cosmological sources as well as instrumental noise. Since many LISA sources will remain in band for multiple orbits, from several days or months up to the entire duration of the mission, there will be an overlap between multiple sources in any given data stretch. Data analysis techniques suitable for this unique problem are currently under development, including in the context of the LISA Data Challenge (Babak et al.
2010; Cornish and Shuman 2020; Littenberg et al. 2020). A standard procedure that allows us to extract WD+WD signals from a noisy datastream uses waveform templates that span a large parameter space (Owen 1996). The most studied case (and the only class detected so far by LIGO-Virgo) is that of isolated compact binaries (see Sect. 1.2.2), which are characterized by the component masses, the distance to the binary, its position on the sky and the orbital eccentricity, as well as the orbital frequency. Contrary to the case of ground-based interferometers, these binaries are long-lived LISA sources. In other words their orbital evolution timescale due to the emission of GW is very slow compared to the mission duration. This will allow us to collect data from many cycles of each binary, increasing the SNR, but also puts stringent requirements on the accuracy of the template waveform. The waveform of quasi-monochromatic sources, such as WD+WD binaries, is relatively simple and can be quite accurately described by the leading order terms in the orbital dynamics (Littenberg et al. 2020). On the other hand, binaries that evolve in the LISA band (such as BH+BH) require a more detailed computation to higher Post-Newtonian order (Mangiagli et al. 2019). Accurate waveform templates are crucial for measuring the source parameters and distinguishing between various source classes since any error in the predicted phase of the template waveform will accumulate over the many cycles the binary stays in band.

The main parameter that can help to distinguish the different classes of isolated compact binaries is the chirp mass of the system. In order to establish the class of a quasi-monochromatic source one may use the fact that the chirp mass distribution of WD+WD binaries peaks around $\simeq 0.25 M_\odot$ (Korol et al. 2017) with the tail up to $1 M_\odot$, while the chirp mass of NS+NS systems is expected to lie around $\simeq 1.2 M_\odot$. BH+NS and BH+BH systems will have higher chirp masses. However, high-mass WD+WD systems at the tail of the distribution with chirp masses of up to $\simeq 1.2 M_\odot$ may be confused for a NS+NS or a NS+WD binary. Similarly, BH+NS binaries may be confused with NS+NS if the NS has an extremely high mass, or the BH has an extremely low mass. Indeed, the discovery by LIGO-Virgo of GW190814, a binary consisting of a $23 M_\odot$ BH and a $2.6 M_\odot$ compact object (Abbott et al. 2020b) is very difficult to interpret: the secondary component is either the lightest BH or the heaviest NS discovered to date.

Additional clues as to the identity of the source are somewhat model-dependent, although priors from copious ground-based observations will help with NS+NS, NS+BH, BH+BH scale events. Thus, it may be possible to use eccentricity measurements to distinguish between WD+WDs and NS+NSs (Lau et al. 2020). Since WD+WDs are expected to have formed via isolated binary evolution, their progenitors are expected to have circularised via multiple mass transfer episodes (see Sect. 1.3.1.1). This assumption is supported by the lack of observed eccentric galactic WD+WD binaries. On the other hand, NS+NSs in the LISA band could have measurable eccentricities: e.g. in the fiducial model by Lau et al. (2020) half of LISA NS+NS sources have eccentricities $e > 0.1$. Thus, a detection of an eccentric source with chirp mass of around $\simeq 1.2 M_\odot$ can be interpreted as a likely NS+NS. Nevertheless, some rare eccentric WD+WD can be produced in globular clusters of the Milky Way, or via triple interactions (e.g. Kremer et al. 2018a).
The case of interacting binaries (see Sect. 1.2.3) is potentially even more complex, since their orbital evolution is influenced not only by gravity, but also mass transfer and magnetic braking, which lead to qualitatively different waveforms (such as anti-chirping phases) depending on the evolution stage of the binary (e.g. Kremer et al. 2017; Tauris 2018). For example, as discussed in Sects. 1.2.3.1 and 1.2.3.2, AM CVns and UCXBs can be detectable by LISA either during their inspiral phase (when the binary components are detached and the orbits shrinks) or during mass transfer (when the orbit expands). The upside of this complexity is that anti-chirping signals are easier to distinguish from isolated compact binaries.

Clearly, an EM counterpart to a GW detection will help to identify the source. Indeed, as discussed in Sect. 1.5.1, EM observations can help in distinguishing between interacting and isolated sources, as well as identifying NS+NS or BH+NS binaries. For the technical aspects of EM synergies, see Sect. 1.5.1.

## 1.6.2 Foreground sources

The Milky Way hosts a large variety of stellar binaries (Sect. 1.2), numbering in the millions below mHz frequencies (Table 3). They will appear as nearly monochromatic (constant frequency) sources emitting over the whole duration of the mission (continuous GW sources). Up to tens of thousands—those with frequencies larger than a few mHz and/or located closer than a few kpc—will be individually resolvable. The rest of Galactic binaries will blend together into the confusion-limited foreground that is expected to affect the LISA data stream at frequencies below 3 mHz (e.g. Bender and Hils 1997; Edlund et al. 2005; Ruiter et al. 2010; Cornish and Robson 2017). The optimal detection, characterization, and subtraction

<table><caption>**Table 5** Population synthesis codes used by the community at the time of writing</caption><thead><tr><td>Code name</td><td>References</td><td>Publicly available</td></tr></thead><tbody><tr><td>Binary_c</td><td>Izzard (2004), Izzard et al. (2006, 2009)</td><td>No</td></tr><tr><td>BSE</td><td>Hurley et al. (2002)</td><td>Yes</td></tr><tr><td>BPASS</td><td>Stanway and Eldridge (2018)</td><td>No</td></tr><tr><td>ComBinE</td><td>Kruckow et al. (2018)</td><td>No</td></tr><tr><td>COMPAS</td><td>Stevenson et al. (2017b)</td><td>Yes</td></tr><tr><td>COSMIC</td><td>Breivik et al. (2020a)</td><td>Yes</td></tr><tr><td>MOBSE</td><td>Giacobbo et al. (2018)</td><td>No</td></tr><tr><td>POSYDON</td><td>Fragos et al. (2022)</td><td>Yes</td></tr><tr><td>Scenario Machine</td><td>Lipunov et al. (1996, 2009)</td><td>No</td></tr><tr><td>SEVN</td><td>Spera et al. (2015)</td><td>Yes</td></tr><tr><td>SeBa</td><td>Portegies Zwart and Verbunt (1996), Toonen et al. (2012)</td><td>Yes</td></tr><tr><td>StarTrack</td><td>Belczynski et al. (2008)</td><td>No</td></tr><tr><td>TRES</td><td>Toonen et al. (2016)</td><td>Yes</td></tr></tbody></table>

Binary_c, COSMIC, MOBSE are based on the BSE code (Hurley et al. 2002)
of Galactic binaries from the data stream has been recognized as one of the fundamental tasks for the LISA analysis. Over-fitting the population of Galactic binaries can result in a large contamination fraction in the catalogue of detected sources, while under-fitting it can degrade the analyses of extra-galactic GW sources in the data due to the excess residual.

The waveforms for Galactic binaries are well predicted using only leading order terms for the orbital dynamics of the binary (Peters and Mathews 1963) and can be computed at low computational cost using a fast/slow decomposition of the waveform combined with the instrument response (Cornish and Littenberg 2007). Nevertheless, their identification in the LISA data will be laborious due to the sheer number of sources expected to be in the measurement band (~$10^4$) and the large number of parameters required to model each source (between 5 and 10, depending on if the source is chirping and if spins are important in the modelling of one or both of the components). In addition, the high density of Galactic binaries in the LISA band (to the extent that sources are overlapping) and the modulation effects caused by LISA's orbital motion, which spreads a source's spectral power across multiple frequency bins, makes the true number of signals at a given frequency difficult to identify. Several techniques have been developed to address this challenge.

*A hierarchical/iterative scheme* The detectable binaries can be identified by using an iterative process that utilizes a median smoothing of the power spectrum to estimate the effective noise level at each iteration, regresses binaries from the data with signal-to-noise ratios above the established threshold as detected sources, repeating the process until the convergence (Cornish and Larson 2003; Timpano et al. 2006; Nissanke et al. 2012). However, each iteration can leave behind some residual due to “imperfect subtraction” that can affect further analysis and bias parameter estimation. In practice, the stochastic signal is not actually subtracted but rather included in the covariance matrix during the likelihood calculation.

*Global fit* A number of studies show that a global fit to the resolvable binaries, while simultaneously fitting a model for the residual confusion or instrument noise and using Bayesian model selection to optimize the number of detectable sources can provide an effective solution to the Galactic binaries challenge (Cornish and Crowder 2005; Umstätter et al. 2005). These global fit methods have been demonstrated on the

<table><caption>**Table 6** N-body and few-body dynamics codes used by the community at the time of writing</caption><thead><tr><td>Code name</td><td>References</td><td>Publicly available</td></tr></thead><tbody><tr><td>NBODY6/NBODY6++GPU/NBODY7</td><td>Aarseth (2012), Nitadori and Aarseth (2012), Wang et al. (2015)</td><td>Yes</td></tr><tr><td>phiGRAPE/phiGPU</td><td>Berczik et al. (2013, 2011)</td><td>No</td></tr><tr><td>HiGPU</td><td>Capuzzo-Dolcetta et al. (2013)</td><td>No</td></tr><tr><td>CMC</td><td>Kremer et al. (2020b, 2019b, 2018a)</td><td>Yes</td></tr><tr><td>MOCCA</td><td>Hypki and Giersz (2013), Giersz et al. (2013)</td><td>No</td></tr><tr><td>clusterBH</td><td>Antonini and Gieles (2020)</td><td>No</td></tr></tbody></table>
Galactic binaries using data from the LISA Data Challenges (Littenberg et al. 2020), but work still needs to be done to extend this into a fully-developed analysis pipeline with the full variety of overlapping LISA sources.

It should be noted that there will possibly be SGWBs of unresolved extragalactic sources or of cosmological origin (see Sect. 1.2.2.5). Such backgrounds will similarly be a broadband confusion signal, and similar considerations to identify and characterize SGWB in LISA data may be needed in order to reveal some of the fainter signals from astrophysical and cosmological sources. Techniques to identify and subtract the SGWB in LISA data are currently being developed by several groups (Karnesis et al. 2020; Caprini et al. 2019; Pieroni and Barausse 2020).

### 1.6.3 Tools

#### 1.6.3.1 Modelling isolated binary evolution and populations
The long-term evolution of stars and binaries is typically modelled in either of two methods; by solving the stellar structure equations, i.e. referred to as detailed calculations, or by faster approximate methods typically aimed at the population synthesis approach. The latter either interpolates in a grid of detailed calculations or uses parametrised stellar evolution tracks which are fitted to detailed calculations. The advantage of this method is the highly boosted computational speed (the simulation of the evolution of a single binary takes a fraction of a second in stead of hours or days), at the cost of detail; one only has access to those parameters included in the grid of tracks. Due to the speed, the effect of different assumptions for poorly understood stellar physics (e. g., stellar mass-loss, interaction physics, supernova kick physics) can be tested in a statistical way, which leads to a deeper understanding of the underlying physical processes involved. The population synthesis approach has proven to work well in retrieving the general characteristics of large binary populations (Toonen et al. 2014) and has led to many insights in binary evolution.

Population synthesis codes are crucial for LISA science, both in order to make forecasts for source rates, but also to develop data analysis pipelines. Indeed, ongoing work on building fast and reliable waveforms relies on the knowledge of the expected properties of the sources (masses, spins, eccentricities) and the accuracies required to detect them and measure these properties. The codes currently in use by the LISA community are listed in Table 5. Further development of these codes, in particular the inclusion of additional physical processes as well as cross-checks between the codes will help to prepare for LISA observations.

#### 1.6.3.2 Modelling binary evolution in dense environments
Stellar-origin LISA GW sources can also be formed in dense stellar environments through dynamical interactions. Hence simulation codes that follow the evolution of dense stellar systems, either by direct integration or using Monte-Carlo techniques, are crucial for their study. Many of these codes follow simultaneously the evolution of single and binary stars within the dense stellar system, using one of the tools described in the previous Section. A list of stellar dynamics codes currently used in the community
for the study of the formation of stellar-origin LISA GW sources can be found in Table 6.

#### 1.6.3.3 GW signal tools
In order to calculate the GW signal of Galactic binary systems there are a number of approaches that can be used, ranging from detailed TDI based methods, e.g., via codes from the LISA Data Challenges (LDC, https://lisa-ldc.lal.in2p3.fr) to (more) analytic methods to calculate the signal and SNR of specific objects (e.g. Cornish and Larson 2003; Robson et al. 2019; Korol et al. 2017; Kupfer et al. 2018; Smith and Caldwell 2019). There are also some web-based tools to explore sensitivity of different GW detectors, including LISA and also detectability of sources in LISA, such as https://gwplotter.com (Moore et al. 2015) and the Gravitational Wave Universe Toolbox (https://www.gw-universe.org, Yi et al. 2022).

# 1.7 Scientific objectives

## 1.7.1 Constraining stellar and binary interaction physics

**Coordinators:** Fritz Röpke, Alina Istrate

**Contributors:** Karel Temmink (1.7.1.1), Mike Lau (1.7.1.1, 1.7.1.7-8), Stephan Rosswog (1.7.1.2, 1.7.1.6), Vasileios Paschalidis (1.7.1.2), Alina Istrate (1.7.1.2), Fritz Röpke, (1.7.1.3,1.7.1.6), Kinwah Wu (1.7.1.4-5), Stéphane Mathis (1.7.1.4), Thomas Tauris (1.7.1.5), Stéphane Blondin (1.7.1.6), Ashley Ruiter(1.7.1.6, 1.7.1.8), Chris Fryer (1.7.1.7), Thierry Foglizzo (1.7.1.7) Manuel Arca Sedda (1.7.1.8), Kyle Kremer (1.7.1.8), Simone Bavera (1.7.1.8), Abbas Askar (1.7.1.8), Silvia Toonen (1.7.1.8), Gijs Nelemans (refs for 1.7.1.8), Irina Dvorkin (refs for 1.7.1.8), Valerya Korol (refs for 1.7.1.8), Astrid Lamberts (refs for 1.7.1.8)

Throughout this section, we will highlight science questions related to LISA that can/ should be addressed before the launch of the mission with the label **pre-LISA-launch objective**, while science questions that can only be addressed by using LISA data will be highlighted with the label **post-LISA-launch objective**. It should be emphasized, that in most cases LISA detection of individual binaries is limited to GW frequency and a somewhat poor sky location. For a fraction of these thousands of systems, however, high SNR and/or high frequency will enable the measurement of the GW frequency derivative, a good sky localization, and possibly constraining the eccentricity of the binary system. The GW frequency derivative will reveal the chirp mass of the binary and thus the distance. Although these cases will be exceptional systems in the overall global perspective, they will be the key science drivers that deliver deep insight and new breakthroughs in our understanding of binary compact object systems—often because of the enhanced chances of finding the EM counterpart of these well-localized systems.
<figure><img src="image_15.png" /><figcaption>**Fig. 13** Hydrodynamical investigation of dynamical stability in a UCXB with a $0.15 M_{\odot}$ WD donor star and a $1.4 M_{\odot}$ accreting NS, in an orbit with an initial eccentricity of 0.04. Plotted here is the mass density in the orbital plane after roughly 13 orbits of RLO. The density plot shows eccentric structures in the accretion disc, the complex character of the flow near the circularization radius and a strong density cusp near the NS. The envelope surrounding the binary is sparse but its total mass is significant compared to the disc. Image reproduced with permission from Bobrick et al. (2017), copyright by the authors</figcaption></figure>

### 1.7.1.1 Dynamical stability and efficiency of mass transfer in the formation of LISA sources

The formation of compact binary systems with two compact objects is still relatively poorly constrained. Typically, at least two phases of mass transfer are required to form a general observable stellar LISA system: one for each component star to lose their hydrogen envelope, and additional phases are possible to remove the helium-rich envelope. To explain the compactness of the orbit, typically one or more of these mass-transfer phases are considered to proceed in an unstable fashion in order to get the necessary amount of orbital shrinkage i.e. a CE phase (e.g. Paczyński and Sienkiewicz 1972; Paczynski 1976; Webbink 1984). Hence, it is crucial to understand for which binary configurations mass transfer proceed stably, and for which it will be unstable.

**Pre-LISA-launch objective** The precise value of the stability boundary (i.e. a critical mass ratio, *q* between the two stellar components above which no stable mass transfer is possible) remains under debate. Theoretical work has shown that mass transfer can proceed significantly more stable than classical results have previously implied (e.g. Hjellming and Webbink 1987; Chen and Han 2008; Woods and Ivanova 2011; Passy et al. 2012b; Pavlovskii and Ivanova 2015; van den Heuvel et al. 2017; Misra et al. 2020). Similarly, the mass-retention fraction of the accreting companion remains relatively poorly understood (e.g. Paczyński and Sienkiewicz 1972; Kato and Hachisu 1999; Hachisu et al. 1999; Tauris et al. 2000; Hurley et al. 2002; Nomoto et al. 2007; Vinciguerra et al. 2020). These issues severely affect the predicted formation details of compact binaries, and determine which evolutionary pathways are dominating in the formation rate of LISA sources (Sect. 1.4), and hence would leave characteristic imprints on the numbers and properties (orbital periods, masses) of the resulting LISA population (e.g. Korol et al. 2017; Ruiter et al. 2019).
<figure><img src="image_16.png" /><figcaption>**Fig. 14** Stability regions in the donor mass—accretor mass plane of WD+WDs and UCXBs. Donor star masses are on the vertical axes, accretor star mass on the horizontal axes. Upper panel: analytical results of Marsh et al. (2004) where the solid line marks the transition between disc and direct impact accretion, and the other lines show how the strict stability limit of Nelemans et al. (2001a) is relaxed when dissipative torques feed angular momentum from the accretor back to the orbit. Bottom panel: ballistic calculation approach by Kremer et al. (2017) using zero-temperature mass–radius relations for the WD donor. Red systems are stable throughout their lifetimes, through stages of both direct-impact and disc accretion; black systems are unstable; and blue systems have an accretor that exceeds the Chandrasekhar limit during their evolution. The solid black line marks again the boundary between disc and direct-impact accretion for initially synchronous and circular binaries. The yellow region indicates systems with a total mass in excess of the Chandrasekhar limit, i.e. potential SN Ia progenitors. It is evident from these figures that UCXBs with low-mass (≈0.2 M<sub>☉</sub>) He WD donor stars, and NS accretors, are always dynamically stable. Images reproduced with permission from [top] Marsh et al. (2004), copyright by RAS; and [bottom] from Kremer et al. (2017), copyright by AAS</figcaption></figure>
For instance, whether or not the first phase of mass transfer in the formation of a WD+WD leads to a shrinkage or widening of the orbit (i.e. unstable or stable mass transfer) determines to which size the secondary star can evolve before filling its Roche lobe, which dictates the mass of its core at RLO, i.e. the mass of the resulting WD, and hence the mass ratio of the WD+WD (Nelemans et al. 2000, 2001c; van der Sluys et al. 2006; Toonen et al. 2012).

Stability of mass transfer depends rather sensitively not only on the intricate details of the structure of the donor star, but also on the transfer and potential loss of mass and angular momentum (Soberman et al. 1997). **Pre-LISA-launch objective**
Hydrodynamical simulations can help settle the question of dynamical stability (see Fig. 13). The mass that is transferred to the companion can not always be fully accreted by the companion star: spin-up to critical rotation rates and/or strong optically thick winds or outflows (including jets) can result in significant fractions of transferred material being lost from the accreting star. Since mass that is not accreted leaves the binary system, carrying with it an amount of orbital angular momentum, the efficiency of accretion and the stability of mass transfer are linked. In the case of compact object accretors, it is expected that less conservative mass transfer is typically more stable than mass transfer where all mass is accreted (Soberman et al. 1997).

**Post-LISA-launch objective** Observed samples are required to reverse-engineer the progenitor evolution, and constrain the evolutionary pathways (e.g. Nelemans et al. 2000; van der Sluys et al. 2006; Zorotovic et al. 2010; De Marco et al. 2011; Portegies Zwart 2013). On the EM side, only relatively small samples exist currently, with relatively large biases towards the lower-mass and hotter WDs, since they have longer (observational) lifetimes and are brighter. However, LISA will be sensitive to WD+WDs throughout the whole Galaxy and will be able to provide properties of the entire WD+WD population with relatively few selection biases. This will allow for stronger and more meaningful statistical analyses. Additionally, LISA will be able to almost directly measure the Galactic rate and masses of merging WD+WDs, which is another useful tool in constraining progenitor evolution.

The detection of NS+NSs (and potentially NS+BH systems) with LISA allow for a direct view of their formation pathways through their eccentricity, that is induced by the supernova kick associated with the formation of the second NS (Vigna-Gómez et al. 2018; Lau et al. 2020). The population of eccentric LISA NS+NS binaries originate from NS+NS binaries born right in or near the sensitivity window of LISA, so that there is little time for GWs to circularise the orbit. The tight orbit prior to the second SN, leading to NS+NS formation, is characterized by the last phase of mass transfer (see Sect. 1.3.1.3 and Fig. 9). This is Case BB mass RLO initiated by the expansion of the naked helium-star after core-helium depletion. The Case BB mass transfer episode is believed to be predominantly stable from detailed simulations (Tauris et al. 2015) and in order to match the observed period distribution of Galactic NS+NS systems (Vigna-Gómez et al. 2018). However, unstable Case BB RLO would lead to an additional CE phase that produces NS+NSs with sub-hour periods. Yet, because such NS+NSs that have gone through unstable Case BB RLO prior to the second SN are formed with higher GW frequencies, they also have a more rapid