1 Introduction

Contributors: R. Caldwell, G. Nardini.

The Laser Interferometer Space Antenna (LISA) (Amaro-Seoane et al. 2017) is a planned space-borne gravitational-wave (GW) detector that will open a new frontier on astrophysics and cosmology in the mHz frequency band. This European Space Agency-led mission includes participation by ESA member countries and significant contributions from NASA and space agencies of other countries. Phase A work is on track for mission adoption in mid 2020s, and is compatible with a launch in the mid 2030s.

LISA will consist of a trio of satellites, located at the vertices of an equilateral triangle, in an Earth-trailing heliocentric orbit. The 2.5-million km distances between the satellites will be monitored using precision laser interferometry to detect passing GWs. Here we consider a nominal mission of six years with a duty cycle of around 75%, although we understand that the instruments will be engineered to a specification that will enable a possible extension.

LISA will be sensitive to GWs from a wide array of sources (Amaro-Seoane et al. 2017). A primary target will be the inspiral and merger of massive binary black holes (MBBHs), ranging in masses \(10^4\)\(10^7\,M_\odot \), at redshifts out to \(z\sim 10\). A significant foreground signal will be the many galactic white dwarf binaries, each effectively a monotone source.

By the time LISA flies, the state of GW observation will have evolved. The extended Advanced Laser Interferometer Gravitational Wave Observatory (LIGO), Advanced Virgo, and Kamioka Gravitational Wave Detector (KAGRA) family of ground-based GW detectors will have begun implementing third-generation technology demonstration upgrades. The network of pulsar-timing radio telescopes will have grown to include the Square Kilometer Array (SKA). Yet, LISA will be different from its predecessors. The size of the detector will enable access to a completely fresh part of the GW spectrum, leading to observations of new astrophysical sources as well as a new window on primordial stochastic gravitational-wave backgrounds (SGWBs). Many sources will produce overlapping signals, owing to the improved sensitivity. Extracting individual sources and events, and discriminating from an unresolved hum, will be part of the challenge.

According to the mission proposal (Amaro-Seoane et al. 2017), LISA has two main scientific objectives of purely cosmological bearing. The first is to probe the expansion rate of the universe, with specific requirements to measure the dimensionless Hubble parameter by means of GW observations alone, and further to constrain cosmological parameters through joint GW and electromagnetic (EM) observations. The second such objective is to understand SGWBs and their implications for early universe and particle physics. This will entail the characterisation of the astrophysical SGWB, and subsequently a measurement or bound on the amplitude and spectral shape of a cosmological SGWB. There are further scientific imperatives to use LISA to explore the fundamental nature of gravity and to search for unforeseen sources with relevance for cosmology. There is a wealth of cosmological information that may be extracted from LISA observations.

We start with Sects. 2 and 3 on standard sirens and weak gravitational lensing; these are “sure bets” for LISA, based on our current understanding of source populations. These sections are directly related to LISA science objective SO6 “probe the rate of the expansion universe” (LISA Science Study Team 2018). They also identify new opportunities to derive cosmological information from GW astrophysical sources, in connection with LISA science objectives SO1, SO2, SO3 and SO4, which are devoted to understanding the galactic and extragalactic astrophysical source populations (LISA Science Study Team 2018). We follow with sections on more exploratory topics, which are potentially profound and revolutionary. Sect. 4 discusses the constraints on modified gravity theories that may be achieved through measurement of GW sources at cosmological distances. Results on this research subject are aligned with LISA science objectives SO5 “explore the fundamental nature of gravity and black holes” as well as the aforementioned SO6. Section 5 introduces the theoretical foundations, observables, and conventions relevant for subsequent sections. Sections 6, 7 and 8 describe predictions of SGWBs sourced by first-order phase transitions, cosmic strings, and inflationary processes. Section 9 explores how these diverse SGWB signals convey unique information on the expansion rate of the universe at redshift \(\sim \) 1000 or higher. These latter four sections touch on topics that are crucial for LISA science objective SO7 “understand SGWB and their implications”. Inflation not only leads to a SGWB but also to density perturbations which may give rise to the formation of primordial black holes (PBHs), which is the subject of Sect. 10. Finally, Sects. 11 and 12 present existing or planned tools and methods to analyse the GW signals discussed in the previous sections. Such tools and methods potentially constitute key deliverables for SO1-SO7 as well as LISA science objective SO8 “search for GW bursts and unforeseen sources”.

We emphasize that the analyses presented in these final sections need to take into account the superposition between cosmological and astrophysical sources, so that the identification of a potential cosmological contribution necessarily involves modeling of the astrophysical signal. While we briefly discuss the current status of these analyses, we acknowledge that the further development of the astrophysical modeling is by itself a crucial line of reasearch that, while impacting the cosmological searches, goes well beyond what is described in the present document. More in general, certain topics of cosmological interest are intentionally omitted from this document: dark matter particles, some tests of general relativity (GR), waveform uncertainties, astrophysical backgrounds, and the astrophysics of discrete sources such as MBBHs. These and related topics are covered by the Living Reviews maintained by the LISA Astrophysics (Amaro-Seoane et al. 2023), Data Challenge (LISA Data Challenge Working Group n.d.), Fundamental Physics (Arun et al. 2022) and Waveform Working Groups (LISA Waveform Working Group n.d.). Such reviews complement the picture presented herein, by the Cosmology Working Group. The goal of all these documents is to both identify LISA science objectives and corresponding work packages, and to alert the scientific community about novel research opportunities, or potential gaps. The tests of general relativity of Sect. 4 and PBH science of Sect. 10 are exemplary cases of why these Living Reviews are needed. The original LISA proposal (Amaro-Seoane et al. 2017) makes no mention of these science investigations. But in recent years, as the subjects have evolved, a set of new science objectives have been proposed to cover them. The possibility that similar situations arise again justifies the effort and interest for living reviews that report on the thrilling, blooming, and fast-evolving LISA science.

2 Tests of cosmic expansion and acceleration with standard sirens

Section coordinators: J.M. Ezquiaga, A. Raccanelli, N. Tamanini. Contributors: D. Bacon, T. Baker, T. Barreiro, E. Belgacem, N. Bellomo, D. Bertacca, C. Caprini, C. Carbone, R. Caldwell, H-Y. Chen, G. Congedo, M. Crisostomi, G. Cusin, C. Dalang, W. Del Pozzo, J.M. Ezquiaga, N. Frusciante, J. García-Bellido, D. Holz, D. Laghi, L. Lombriser, M. Maggiore, M. Mancarella, A. Mangiagli, S. Mukherjee, A. Raccanelli, A. Ricciardone, O. Sergijenko, L. Speri, N. Tamanini, G. Tasinato, M. Volonteri, M. Zumalacarregui.

2.1 Introduction

Broadly speaking, to learn about the universe and its cosmic expansion we need to measure distances and times. GW astronomy offers a unique perspective in this matter, since the signal emitted by a compact binary coalescence is well predicted by GR. Namely, the amplitude of the GW is inversely proportional to its luminosity distance and it only depends on the masses and orbital inclination of the binary system source. Since cosmological propagation at the background level (namely, excluding the effect of perturbations over the Friedmann–Lemaître–Robertson–Walker (FLRW) geometry) only changes the overall strain amplitude, one can use the frequency evolution of the GW to unveil the masses of the compact binary and the relative amplitude of the two GW polarisations to estimate the orbital inclination, obtaining thus a direct and absolute measurement of the luminosity distance. However, GW signals alone do not provide a way to relate the time (of merger, for example) in the observer frame to the one in the source frame. To access this information, one needs an independent determination of the redshift of the source. In such a case the GW signal from compact binary coalescence can be considered a standard siren (Schutz 1986), namely a cosmological event for which a distance measurement and complementary redshift information are both available. For example, the binary neutron star (BNS) merger GW170817, observed by the Advanced LIGO and Advanced Virgo detectors jointly with several EM facilities which spotted associated EM emissions, has already been used as a proof-of-principles, low-redshift measurement of \(H_0\) (Abbott et al. 2017a). On the other hand, massive black holes (MBHs) seen by LISA with an EM counterpart could be used to map the cosmic expansion up to high redshift (Holz and Hughes 2005; Tamanini et al. 2016; Tamanini 2017; Belgacem et al. 2019c).

In this section we will present the different standard sirens that LISA will detect and the information about the cosmological model that they will provide. The section is organized as follows. We begin by describing the concept of standard siren in Sect. 2.2, detailing the expected LISA bright and dark sirens. We then consider the constraints that could be placed in the standard cosmological constant plus cold dark matter (\(\Lambda \)CDM) cosmological model in Sect. 2.3. Subsequently, we explore LISA capabilities to probe different dark energy (DE) models in Sect. 2.4. Next, we show the synergies of LISA with other EM and GW observatories in Sect. 2.5. Finally, we describe the benefit of cross-correlating LISA data with large-scale structure surveys in Sect. 2.6.

2.2 Standard sirens

GW signals from compact binary coalescences are natural cosmic rulers because of the inverse dependence of the strain with the GW luminosity distance, \(h \propto 1/D_{L}^{{{\rm GW}}}\). In GR and over a Friedman–Lemaître–Robertson–Walker (FLRW) background, the GW luminosity distance is equal to the one of EM radiation and given by

$$\begin{aligned} D_{L}^{{{\rm GW}}}=\frac{c}{H_0}\frac{(1+z)}{\sqrt{\vert \Omega _k\vert }}{\text{sinn}}\left[ H_0 \int _0^z \frac{\sqrt{\vert \Omega _k\vert }}{H(z')}dz'\right] , \end{aligned}$$

where \({\text{sinn}}(x)=\sin (x),\,x,\sinh (x)\) for a positive, zero and negative spatial curvature respectively. Assuming a \(\Lambda \)CDM cosmology, the Hubble parameter is a function of the matter content \(\Omega _m\), the curvature \(\Omega _k\) and the amount of DE \(\Omega _\Lambda \) (radiation at present time is negligible)

$$\begin{aligned} H(z)=H_0\sqrt{\Omega _m(1+z)^3+\Omega _k(1+z)^2+\Omega _\Lambda }. \end{aligned}$$

LISA will attempt to measure \(H_0\), \(\Omega _m\), \(\Omega _k\) and \(\Omega _\Lambda \) using sirens.

GW observations themselves, however, do not provide direct information about the redshift. Therefore, in order to be able to probe the cosmological evolution we need additional input. In the case in which the redshift of the GW source is directly obtained from an EM counterpart, we will refer to the source as a bright siren. A beautiful example of this kind of multi-messenger event was the LIGO/Virgo event GW170817 (Abbott et al. 2017d), which provided the first standard siren measurement of \(H_0\) (Abbott et al. 2017a) (see Abbott et al. 2021c for an update on the measurement of \(H_0\) from standard sirens). As we present in Sect. 2.2.1, LISA will be sensitive to very different bright sirens, but the concept remains the same. On the other hand, when the redshift information is obtained from an analysis that does not include EM counterparts, we will refer to the GW sources as dark sirens. In Sect. 2.2.2 we will present different dark siren classes that LISA will detect and which can be used to obtain cosmological information by cross-matching the sources with galaxy catalogues and looking for correlated features in the mass distribution.

Modern analyses of standard sirens are based on Bayesian inference. We refer the reader to Sect. 11.1 for a glimpse at the actual statistical tools LISA will use and a detailed discussion of their associated systematic uncertainties. In what follows we focus mostly on the different GW sources and their potential as standard sirens. LISA will detect three types of potential standard siren populations: MBBHs at \(1 \lesssim z \lesssim 8\), extreme mass ratio inspirals (EMRIs) at \(0.1 \lesssim z \lesssim 1\) and SOBBHs at \(z \lesssim 0.1\). An example of the expected Hubble diagrams from these three different standard sirens populations can be found in Fig. 1. The details regarding each of these populations will be presented in the following.

2.2.1 Bright sirens: MBBHs with electromagnetic counterpart

LISA will detect the coalescence of MBBHs up to redshift \(z \simeq 15\)–20. However, the mass and redshift distributions of the events are still uncertain. Currently, our knowledge of MBHs is limited to cases where either an active galactic nucleus is present or to quiescent MBHs in nearby galaxies (Kormendy and Ho 2013; Graham 2016; Greene et al. 2020). The population of MBBHs accessible by LISA might be considerably different from our current expectations. Several groups have attempted to address this question with hydrodynamics simulations (Salcido et al. 2016; Katz et al. 2020; Volonteri et al. 2020) or semianalytic formation models (Volonteri et al. 2003; Barausse et al. 2020; Ricarte and Natarajan 2018; Trinca et al. 2022; Valiante et al. 2018). While the former are able to handle more naturally hydrodynamical, thermodynamical, and dynamical processes, the latter are computationally efficient and can be used to explore a larger parameter space.

Two main sources of uncertainties affect the expected redshift and mass distribution of merging MBBHs: black hole seeding and delay time prescription (Klein et al. 2016). If MBHs grow from the remnants of metal-poor population-III stars, the population of MBBHs accessible by LISA is expected to peak at the total mass \({{{\mathrm{M}}}_{{{{\rm tot}}}}}\simeq 10^3 \,M_\odot \). However if MBHs arise from the monolithic collapse of gas in protogalaxies, the mass distribution is expected to range from \(10^4 \,M_\odot \) up to few \(10^7 \,M_\odot \). We note that additional formation mechanisms have been proposed and that they would further modulate the distribution of merging MBHs. Moreover, delay times, between the merger of two galaxies and the merger of their central MBHs, shape the redshift distribution, with short delays leading to more mergers at higher redshift. Further uncertainties arise from the gas inflow to the halo centre, its efficiency and the geometry of accretion. Even if LISA will be able to distinguish different formation scenarios (Sesana et al. 2011), the aforementioned uncertainties reflect in a broad interval for the number of events detected and their distributions (e.g. mass, mass ratio, spins, redshift).

Combining these uncertainties, LISA should be able to detect between a few and several tens of MBBH events per year (Klein et al. 2016). Multiple-body interactions among a MBH binary and one or more intruder MBHs, arising naturally from the hierarchical galaxy formation process, might still produce \(\simeq 10{-}20\) events per year (Bonetti et al. 2019).

LISA will also provide exquisite accuracy on MBBH parameters. For the search of a possible EM counterpart, the sky position accuracy is of paramount importance. Taking into account the full inspiral-merger-ringdown GW signal, MBBHs from few \(10^5 \,M_\odot \) to few \(10^6 \,M_\odot \) can be localised within \(0.4\, {{\mathrm{deg}}}^2\) up to \( z \simeq 3\) (Mangiagli et al. 2020), but with high accuracy obtained only at merger. For these sources the posterior on the sky position is expected to be Gaussian; however, for more massive and distant sources, the recovered sky position is expected to present multimodalities (Marsat et al. 2021). For heavy systems with total mass \({{{\mathrm{M}}}_{{{{\rm tot}}}}}> 10^7 \,M_\odot \) the ringdown portion of the signal might carry most of the information for the source localisation (Baibhav et al. 2020). For cosmology applications, also the estimate on the luminosity distance plays a fundamental role: due to the typical large signal-to-noise ratio (SNR) value for these sources, LISA should be able to constrain the luminosity distance to better than \(10\%\) for most of the events at \( z < 3\) (Tamanini et al. 2016).

If MBBHs evolve in gas-rich environment, EM radiation might be produced by the accretion of gas onto the MBHs close to or after merger. The orbital motion of the binary is expected to open a cavity in the circumbinary disk. Hydrodynamical simulations show that minidisks generally form around each BH from the stream of gas from the circumbinary disk (Farris et al. 2015; Tang et al. 2018).

This leads to pre-merger EM emission across all wavelengths (D’Ascoli et al. 2018), which can be identified if the pre-merger sky localisation is good (Dal Canton et al. 2019). In the optical band, the Vera C. Rubin Observatory (formerly known as Legacy Survey of Space and Time) (Abell et al. 2009) will reach a magnitude limit of 24.5 in \(30 \, {{{\rm s}}}\) of pointing over a field of view of \(\simeq 10\, {{\mathrm{deg}}}^2\). This survey speed enables the Vera C. Rubin Observatory to cover a sky area of \(\simeq 100 {{\,\mathrm{deg}}}^2\) allowing for possible pre-merger EM detection, though the number of detected counterparts is expected to be low (Tamanini et al. 2016). At or after merger, several transients have been proposed, from spectral changes and brightening (Schnittman and Krolik 2008; Rossi et al. 2010) to jets (Palenzuela et al. 2010; Khan et al. 2018; Yuan et al. 2021). In X-ray, Athena (Nandra et al. 2013), with a field of view of \(0.4 {{\,\mathrm{deg}}}^2\) and a limiting flux of \(\approx 3\times 10^{-16}\)erg cm\(^{-2}\) s\(^{-1}\) in 100 ks, will be optimal to search for possible post-merger signatures. Similarly, in radio, SKA (Dewdney et al. 2009) will observe the launch of putative post-merger radio jets with an initial field of view of \( 1 {{\,\mathrm{deg}}}^2\).

A counterpart detection strategy has been proposed (Tamanini et al. 2016) consisting in first localising the source in the radio with the SKA, and subsequently to proceed to a redshift determination of the host galaxy in the optical with the Extremely Large Telescope.Footnote 1 This strategy is more promising than direct optical identification with the Vera C. Rubin Observatory. Depending on the seed and dynamical evolution models, LISA will detect between \(\simeq 10\) and \(\simeq 25\) MBBH events with EM counterpart during a mission assumed to be of five year (Tamanini et al. 2016). These estimates are however affected by several astrophysical uncertainties that have as yet not been properly characterised. It is, however, robust to expect that MBBH bright sirens will all be detected at relatively high redshift: one study finds standard sirens distributions peaking at around \(z\sim 2-3\) and extending up to \(z\sim 8\) (Tamanini et al. 2016; Tamanini 2017). Therefore, MBBH bright sirens will be of great relevance to test the \(\Lambda \)CDM model, and possible deviations from it, in a redshift range so far scarcely probed by EM observations. In Sects. 2.3 and 2.4 we review the cosmological constraints that LISA will be able to impose both at low and high redshift.

Fig. 1
figure 1

Examples of LISA standard siren data sets at different redshift ranges. Data from low-redshift dark sirens, defined by SOBBHs, are reported in the bottom-left plot. Data at intermediate redshifts correspond to EMRIs and are reported in the top plot. High-redshift data are provided by standard sirens from MBBHs and are reported in the bottom-right plot. Note how SOBBHs and EMRIs, being dark sirens, can only provide broad likelihood regions in the Hubble diagram, while MBBHs, being bright sirens, provide precise data points thanks to the unique redshift association coming from the EM counterpart identification. Images reproduced with permission from [top] Laghi et al. (2021), [bottom left] Del Pozzo et al. (2018), and [bottom right] Speri et al. (2021), copyright by the author(s)

2.2.2 Dark sirens: SOBBH, EMRIs, IMBBHs

In addition to observing the EM counterpart of individual events, there are other methods that one can employ to obtain information about the redshift of the GW source. The most common and widely-used among these methods relies on statistical matching the inferred position of the GW source with a catalogue of galaxies with known redshift (see Abbott et al. 2021c for the application of this method to the Advanced LIGO-Advanced Virgo-KAGRA data). GW events for which an EM counterpart cannot be identified, but which can still be used to extract cosmological information statistically, are usually referred to as dark standard sirens. In this section we outline how dark sirens are treated by correlating galaxy catalogues with the localisation of the source, statistical dark sirens, and using properties of their mass distribution, spectral sirens. Generally speaking, dark standard sirens have the advantage to be applicable to all kind of GW sources for which a distance measurement can be retrieved (not exclusively those emitting EM counterpart signals), although a large number of them are necessary to achieve precise measurements through solid statistics.

In the absence of an EM counterpart, redshift information can be extracted by putting a prior on potential hosts from a galaxy catalogue, assuming that galaxies are good tracers of binary black hole (BBH) mergers. In order to do that (see more details on statistical methods in Sect. 11.1.2), one associates the GW event with every galaxy within the 3D localisation error of the event, assigning to each of them a certain probability of being the true host galaxy of the GW event. In this way, by stacking together the information gathered from several dark sirens, one can statistically infer the true values of the cosmological parameters. For this method to be effective, one requires a large number of events to combine statistics or a very small localisation volume. Errors in the luminosity distance scale with the SNR, \(\Delta D_L/D_L\sim 1/\rho \), while the localisation depends also on the duration of the source, since LISA orbital modulation can be used to help disentangle the location in the sky. One complication is given by the fact that galaxy catalogues are in general not complete, especially at high redshift. This requires one to include information on the missing galaxies within the catalogue, and the catalogue prior must be supplemented by a suitable “completion” term (Chen et al. 2018; Fishbach et al. 2019; Gray et al. 2020; Finke et al. 2021) (again see Sect. 11.1.2 for more details). This aspect will remain a limiting factor until catalogues with very large completeness are available. The availability of complete galaxy catalogues, small GW localisation regions and accurate redshift determination—including characterisation of the uncertainty due to peculiar velocities for low-redshift binaries (Howlett and Davis 2020; Mukherjee et al. 2021c; Nicolaou et al. 2020)—will be crucial in order for the statistical method to give competitive constraints on cosmological parameters. At the time when LISA will be taking data, galaxy catalogues will be available from a plethora of current and future experiments, providing observations of different types of galaxies, with varying number density and sky coverage, over different redshift ranges. In particular, the completed observations from Euclid and Dark Energy Spectroscopic Instrument (DESI), with the addition of redshift-deep catalogues from the Subaru Prime Focus Spectrograph project and the Roman Space Telescope, should be available before LISA is launched. Additionally, the Vera C. Rubin Observatory should have observed hundreds of millions of galaxies, and deep and wide catalogues should be available from SPHEREx. Finally, on time-scales comparable with LISA, the full SKA2 and the ATLAS satellite should provide extremely deep and full-sky catalogues of the sky. Moreover, there are plans to build a next-generation billion-galaxies survey as a successor of DESI. These future surveys are expected to provide well-complete galaxy catalogs up to \(z=1\) and possibly beyond.

Stellar-origin black holes (SOBHs) are guaranteed dark sirens for LISA. From the observations with current LIGO/Virgo detectors (Abbott et al. 2021e), we know that there is a population of BBHs with masses between \(\sim 5\,M_\odot \) and \(\sim 50\,M_\odot \) that LISA will see in their early inspiral. Some of those events will be subsequently detected by ground-based detectors becoming in this way “multi-band” events (see Sect. 2.5.2). SOBBHs with good enough localisation can be used as dark sirens (Kyutoku and Seto 2017; Del Pozzo et al. 2018). Because of their low masses, cosmologically useful SOBHs could only be seen by LISA up to \(z\sim 0.1\), probing essentially the local expansion rate \(H_0\). In practice, only those events with better than 20% accuracy in the luminosity distance and with a typical sky-localisation error better than \(\sim 1 {{\rm deg}}^2\) will be useful as dark sirens. According to current forecasts, LISA could detect \(\sim \) 10–100 dark sirens (Del Pozzo et al. 2018), though uncertainties on the LISA noise level at the high-frequency end of its band (Amaro-Seoane et al. 2017), on the detection threshold (Moore et al. 2019) and merger rate of these systems (Abbott et al. 2021e), might well invalidate the most optimistic expectations. In Sects. 2.3 and 2.4 we will review the cosmological constraints that LISA can impose with SOBBHs as dark sirens.

LISA will also be able to use EMRIs as statistical dark sirens. They will in fact be detected at cosmological distances, possibly in high numbers, though the rates are so far extremely uncertain. EMRIs detected by LISA are expected to be broadly peaked around \(0.5< z < 2\), potentially reaching \(z \sim 4\) (Babak et al. 2017). A recent investigation showed that events up to \(z \sim 0.7\) can safely be used to estimate \(H_0\) (Laghi et al. 2021), provided they can be detected with high SNR. These events can reach relative uncertainties on \(\Delta D_L/ D_L\) below 0.05 and typical sky-localisation errors less than 1 deg\(^2\), representing the best well-localised events suited for a statistical approach on the inference of the GW redshift, comparable to, if not better than, what future third-generation ground-based detectors can achieve (Iacovelli et al. 2022). As we will see in Sects. 2.3 and 2.4, according to the analysis of Laghi et al. (2021), LISA will be able to use from a few to several tens of EMRIs to extract useful cosmological information.

Redshift information on a GW source can also be obtained in a statistical way performing a population analysis when there are distinctive features in the source mass distribution. This is simply because GW observatories are only sensitive to the redshifted or detector frame masses, which directly relate to the source masses via

$$\begin{aligned} m_z=(1+z)m. \end{aligned}$$

Therefore, if the mass distribution presents a known feature, e.g. a peak or a drop, at a reference scale which is invariant under cosmic evolution, by observing this feature in the GW events at different luminosity distance bins one can infer their redshift, becoming spectral sirens. If such a feature exists, this would be a very convenient probe of the cosmic expansion because it only requires GW data. In case of a time evolution of the reference mass scale, using the full mass distribution helps calibrating against astrophysical uncertainty (Ezquiaga and Holz 2022).

A good example of such features occurs in the mass spectrum of SOBBHs. This is because as stars become more massive, a runaway process induced by electron-positron pair production known as pair-instability supernova (PISN) is triggered (Barkat et al. 1967; Fowler and Hoyle 1964; Heger and Woosley 2002; Fryer et al. 2001; Heger et al. 2003; Belczynski et al. 2016). These PISN result in complete disruption of the stars, preventing the formation of remnant BHs and thus inducing a gap in the mass spectrum starting at around \(50\,M_\odot \). Nonetheless, for sufficiently massive stars the PISN process is insufficient to prevent direct collapse, and a population of BBHs with masses larger than \(\sim 120\,M_\odot \) could arise. Therefore, the theory of PISN predicts a gap in the BBH mass spectrum with two edges that act as reference scales.Footnote 2 While the lower edge of the gap lies within the main sensitivity of present LIGO/Virgo detectors and could lead to precise measurements of H(z) (Farr et al. 2019; Ezquiaga and Holz 2022), LISA will be more sensitive to the upper edge if a population of “far side” binaries in fact exists (Ezquiaga and Holz 2021). These alternative methodologies are currently under development and will need to be further investigated in the future, especially in the framework of GW cosmology with LISA.

Finally, we mention another effect that in principle allows one to access the redshift information directly from the GW signal alone. The variation of the background expansion of the universe during the time of observation of the binary induces an effectively \(-4\) post-Newtonian term in the waveform phase, whose amplitude directly depends on the redshift and on the value of the Hubble parameter both at the source and at the observer (Seto et al. 2001; Nishizawa et al. 2012). Unfortunately, redshift perturbations due to the inhomogeneous distribution of matter between the source and the observer also depend on time, and therefore also contribute to the extra terms in the phase (Bonvin et al. 2017). Among these, the time-varying peculiar velocity of the GW source centre of mass may dominate the signal, effectively preventing the extraction of the redshift information from the amplitude of the dephasing—but possibly allowing a measurement of the binary’s peculiar acceleration (Tamanini et al. 2020; Inayoshi et al. 2017).

2.2.3 Systematic uncertainties on standard sirens

Bright and dark sirens will suffer from some common systematic uncertainties. The measurements of the binary luminosity distances are affected by the detector calibration uncertainty (Karki et al. 2016; Chen et al. 2021a) and the accuracy of the waveform models (see Abbott et al. 2019b for discussions in the context of LIGO/Virgo observations). Accurate waveforms will be particularly needed for the high SNR sources that LISA will detect. Moreover, high redshift sources will be affected by weak lensing uncertainties (Holz and Linder 2005; Hirata et al. 2010; Cusin and Tamanini 2021) (see Sect. 3 for more details). In addition, the parameter estimation of the luminosity distance will be subject to degeneracies with the orbital plane inclination and other parameters, though for long duration signals or when higher harmonics are measured (Baibhav et al. 2020), this degeneracy can be broken. In general, these parameter degeneracies will lead to larger statistical errors but not systematic biases. Finally, our understanding of the possible observational selection effect (Chen 2020) as well as the astrophysical rate evolution (see e.g. Fig. 12 of Finke et al. (2021) for an application to LIGO-Virgo data) are critical to the accuracy of standard siren analysis as well. Not many investigations have so far assessed the systematic uncertainties affecting standard siren measurements with LISA. A thorough exploration of all these effects, needed to consolidate our confidence on LISA cosmological observations, will be necessary in the future.

2.3 Constraints on \(\Lambda \)CDM

In this subsection we present how well LISA will be able to constrain the cosmological parameters of the standard \(\Lambda \)CDM model, by using different classes of standard sirens as presented in Sect. 2.2. We first focus on the Hubble constant \(H_0\) and consider constraints on additional parameters afterwards. We conclude the subsection with a discussion on consistency tests of \(\Lambda \)CDM at high-redshift with LISA MBBH standard sirens.

2.3.1 \(H_0\) tension and standard sirens

The standard model of cosmology is extremely successful and allows one to describe the universe from the time of BBN to the present time of cosmic acceleration. Remarkably, it contains only six parameters, one of which, the present-day Hubble constant \(H_0\) describes the expansion rate of the universe. At small redshifts, it relates the luminosity distance and the redshift of a source such that \(D_L(z\ll 1)= cz/H_0\). Consistency of the model requires the inferred value of \(H_0\) to be independent of the probe and any deviations should be seen as a sign of unaccounted for systematics or more excitingly, new physics.

In recent years, two sets of values for \(H_0\) have emerged from so-called early or late measurements of \(H_0\) depending on the origin of the calibration. As the error bars shrink, it becomes increasingly clear that the two values are in tension, reaching 4–5\(\sigma \) disagreements (Verde et al. 2019). The most precise measurement of \(H_0\) from the early universe comes from the cosmic microwave background (CMB) (at recombination redshift \(z \sim 1100\)) with an inferred value of \(H_0 = 67.4 \pm 0.5\) km s\(^{-1}\) Mpc\(^{-1}\) at 68% C.L. assuming a flat \(\Lambda \)CDM cosmology (Aghanim et al. 2020). In this case the \(H_0\) value is inferred from the angle upon which the scale associated to the horizon at the last scattering surface is projected, which is obtained from the measurement of the density fluctuations. Compatible values of \(H_0\) are obtained also from the Atacama Cosmology Telescope and WMAP5 for which \(H_0 = 67.6 \pm 1.1\) km s\(^{-1}\) Mpc\(^{-1}\) at 68% C.L. (Aiola et al. 2020) and from the joint analysis of Dark Energy Survey (DES) clustering and weak lensing data with baryon acoustic oscillations (BAO) and big bang nucleosynthesis (BBN), \(H_0 = 67.2 ^{+ 1.2}_{-1.0}\) km s\(^{-1}\) Mpc\(^{-1}\) at 60% confidence (Abbott et al. 2018c).

In contrast, several teams have measured a significantly higher value of \(H_0\) in the local universe with redshifts \(z\le 1\) in a model independent fashion. For example, the SH0ES team used Cepheid calibrated supernovae type Ia to measure \(H_0 = 74.03 \pm 1.42\) km s\(^{-1}\) Mpc\(^{-1}\) (Riess et al. 2019) (see also recent updates, Riess et al. 2022). The H0LiCOW collaboration used strong lensing time delays of background quasars to infer \(H_0=73.3^{+1.7}_{-1.8}\) km s\(^{-1}\) Mpc\(^{-1}\) (Wong et al. 2020). The Megamaser Cosmology project used very long baseline interferometric observations of water masers in Keplerian orbits around MBHs to measure \(H_0=73.9\pm 3.0\) km s\(^{-1}\) Mpc\(^{-1}\) (Pesce et al. 2020). The Carnegie-Chicago Hubble Program collaboration used tip of the red giant branch measurements in the large Magellanic cloud to calibrate 18 supernovae type Ia, instead of Cepheids, and found a slightly lower \(H_0=69.8\pm 2.6\) km s\(^{-1}\) Mpc\(^{-1}\) (Freedman et al. 2019).

GWs offer an independent test of the tension using bright or dark sirens as described in Sect. 2.2.1 and Sect. 2.2.2. The LIGO-Virgo collaboration used the first bright siren, namely GW170817, to infer \(H_0= 70^{+12}_{-8}\)km s\(^{-1}\) Mpc\(^{-1}\) at 1\(\sigma \) (Abbott et al. 2017a), which lies somewhat in between the early and late universe values but with a worse precision if compared to current EM results. Furthermore, current dark siren measurements reached an inferred value of \(H_0 = 75^{+25}_{-22}\) km s\(^{-1}\) Mpc\(^{-1}\) at 1\(\sigma \) (Fishbach et al. 2019; Finke et al. 2021; Abbott et al. 2021a). The rather large error bars are expected to shrink as \(1/\sqrt{N}\), where N indicates the number of well-localized events. Percent-level precision on \(H_0\) is expected in the 2020s with BNS mergers detected by Advanced LIGO-Virgo and their EM counterpart observations (Nissanke et al. 2013; Chen et al. 2018). Bright sirens in the era of the third generation of ground-based GW detectors could further constrain other cosmological parameters, such as \(\Omega _m\) and \(w_0\) (Sathyaprakash et al. 2010; Zhao et al. 2011; Cai and Yang 2017; Jin et al. 2020; Chen et al. 2021a) and usher us into the era of precise GW cosmology.

LISA will offer an alternative and complementary probe of \(\Lambda \)CDM which might as well provide useful information on the Hubble tension. In the following, we explore the potential of LISA to probe \(H_0\) and beyond.

2.3.2 LISA forecast for \(H_0\)

LISA will be able to contribute measurements of \(H_0\) coming from different classes of standard siren sources (see Sect. 2.2). For the time being, the literature has described only measurements coming from individual classes of sources. A complete analysis combining the constraining power of different LISA GW sources is still missing.

By considering SOBBHs, and by cross-matching with simulated galaxy catalogues, Kyutoku and Seto (2016), Del Pozzo et al. (2018) found that constraints on \(H_0\) can reach the few % level. In particular, in the study presented in Del Pozzo et al. (2018), several different instrumental configurations for LISA were investigated, as well as several coalescence rate models within the range allowed by the LIGO-Virgo observations from O1. This analysis also had \(\Omega _m\) as a free parameter, although due to the low-redshift of the sources results are not informative. The SOBBHs entering the analysis were selected to have SNR \(>8\), a cosmological redshift \(< 0.1\) and an uncertainty on the luminosity distance smaller than 20%. No other selection criteria were applied. With the aforementioned selections, the number of SOBBHs considered ranged from a pessimistic case of 7, yielding an accuracy on \(H_0\) of 7% to a most optimistic case of 259, yielding an accuracy of 1%, see left panel in Fig. 2.

A similar analysis can be done also with EMRIs. A first investigation (MacLeod and Hogan 2008) pointed out that \(\sim \!20\) EMRIs detected at \(z\sim 0.5\) could be enough to constrain \(H_0\) at the 1% level. The analysis provided in MacLeod and Hogan (2008) employed however a simplified approach to estimate cosmological forecasts with LISA, and moreover assumed the more optimistic mission design considered at the time. A recent, more detailed analysis has been performed with LISA EMRIs (Laghi et al. 2021). Using only the most informative, high-SNR (\(> 100\)) EMRIs up to redshift \(z \le 0.7\), cross-matching with the galaxy catalogue obtained from the simulated sky of Henriques et al. (2013), it is shown that constraints at the few % can be forecast for \(H_0\). An analysis of three different EMRI population models taken from Babak et al. (2017), representing a pessimistic, a fiducial, and an optimistic scenario, points out that in a 4-year LISA mission lifetime constraints are expected to be at 3.6%, 2.5%, and 1.6% (68% CL), respectively, while in case of a 10-year mission \(H_0\) can be constrained at the 2.6%, 1.5%, and 1.1% accuracy (68% CL). The different accuracy in the various scenarios reflects the different number of useful EMRIs available in each model, which in case of 10 years of observation and after the SNR selection, ranges from \(\sim \!5\) (in the worst scenario), passing to \(\sim \!30\) (in the fiducial scenario), up to \(\sim \!70\) (optimistic scenario), see right panel in Fig. 2.

Fig. 2
figure 2

Left panel: 90% (black) and 68% (red) credible intervals for \(H_0\)/km s\(^{-1}\) Mpc\(^{-1}\) for each of the LISA configurations considered in Del Pozzo et al. (2018). The credible regions are averaged over the galaxy hosts realisations. Right panel: 90% (black) and 68% (red) percentiles, together with the median (red dot) of h for the pessimistic, fiducial, and optimistic EMRI models (M5, M1, M6, respectively) and for two different LISA observational scenario (4 and 10 years). Here \(h = H_0 / (100\, {\mathrm{km\, s}}^{-1} {{{\rm Mpc}}}^{-1})\) and the blue dashed horizontal line denotes the true cosmology (set at \(h=0.73\) in Del Pozzo et al. 2018; Laghi et al. 2021). For each data point, we also report the average number N of EMRIs considered in the analysis. Images reproduced with permission from [left] Del Pozzo et al. (2018) and [right] Laghi et al. (2021), copyright by the author(s)

Finally the last standard siren class that LISA can employ to constrain \(H_0\) are MBBHs. Although these events are expected to be detected at high-redshift (\(z>1\)), by assuming \(\Lambda \)CDM one can set bounds on \(H_0\), i.e. on the cosmic evolution at low-redshift. By simulating different populations of MBBH mergers, performing (simple) parameter estimations over the expected GW signal, and by simulating the emission and detection of possible EM counterparts, recent works showed that constraints at the few % level can be imposed on \(H_0\) (Tamanini et al. 2016; Tamanini 2017; Belgacem et al. 2019c). These studies predict around four useful standard sirens with observed EM counterpart, per year. Their redshift distribution peaks between redshift 2 and 4, with tails up to \(z\simeq 8\). The dominant contribution on the distance uncertainty of these events is not the LISA measurement error, but rather the systematic effect due to week lensing which dominates at high-z providing an estimated average uncertainty of up to 5-10% (Hirata et al. 2010; Cusin and Tamanini 2021). Nevertheless even if LISA will detect only a few MBBH standard sirens, the fact that these will be at high redshift, with relatively precise distance determination, and of course with the single redshift value identified with the EM counterpart, will allow for interesting constraints on \(H_0\) at the few percent level (Tamanini et al. 2016; Tamanini 2017; Belgacem et al. 2019c). This is comparable with what is expected for low-redshift, more numerous LISA dark sirens such as SOBBHs and EMRIs. As we will show below, MBBHs will however be more interesting for cosmological analyses beyond the simple measurement of \(H_0\).

The joint-inference on \(H_0\) resulting from the combination of the analyses described above for SOBBHs, EMRIs and MBBHs, is expected to provide interesting constraints, possibly reaching the 1% level, or better. Such a combined investigation however has not yet been performed and will be the focus of future studies. Note also that a further class of standard sirens for LISA could be provided by intermediate-mass binary black holes (IMBBHs), ranging from \({\mathcal {O}}(100)\) to \({\mathcal {O}}(1000)\) solar masses. Although recent LIGO/Virgo observations may point towards the existence of this class of BBHs (Abbott et al. 2020b), their merger rate and population properties are completely unknown at the moment (Toubiana et al. 2021), making current LISA forecasts too uncertain to be seriously considered. Given their high-mass and redshift range, IMBBHs could nevertheless well represent the best class of LISA dark sirens if they can be detected in high numbers. Moreover the association with possible EM counterparts, which for these systems are being realistically considered (Graham et al. 2020), could well turn the LISA IMBBH detections that merge within the LIGO/Virgo band in a relatively short time, into useful multi-band bright sirens with a great potential to yield precise cosmological measurements (Muttoni et al. 2022).

By surveying the considerations made above, we can expect LISA to deliver constraints on \(H_0\) at the few percent level or better, similarly to what ground-based detectors may produce on comparable timescales (Chen et al. 2021a). The robustness of these constraints against uncertainties in the overall analysis, including for example calibration and waveform modelling issues, will be something to carefully assess in the future. In any case, a precise and accurate measurement of \(H_0\) with LISA will provide useful insights on the Hubble tension, should the tension still persist. On the other hand, a further independent and complementary measurement of \(H_0\) will help strengthen our confidence in the value of the Hubble constant, especially if hints of physics beyond \(\Lambda \)CDM appear.

2.3.3 \(\Lambda \)CDM beyond \(H_0\)

The relatively high-redshift reach for some of the standard sirens sources detected by LISA, will allow the inference on cosmological parameters beyond \(H_0\). In particular MBBHs will be extremely useful to test the cosmic expansion at high-redshift (\(z\lesssim 8\)) while EMRIs could be useful for cosmological applications up to \(z\sim 1\), or more generally up to the redshift at which we will be able to employ fairly complete galaxy catalogues.

As shown in Tamanini et al. (2016), Tamanini (2017), MBBH mergers with an identified EM counterpart could be used by LISA to infer the values of \(\Omega _m\) and \(\Omega _k\), albeit with large uncertainties. Assuming \(\Lambda \)CDM, constraints on \(\Omega _m\) are forecast to reach the \(\sim 10\%\) level (68% C.L.) only in the most optimistic scenarios (Tamanini et al. 2016; Belgacem et al. 2019c; Spergel et al. 2015), while allowing for spatial curvature degrades these estimates for \(\Omega _m\) to \(\sim 25\%\) and yields a measurement of \(\Omega _k\) at similar precision (Tamanini et al. 2016). Needless to say these results will certainly not be competitive with EM observations, but will at least provide an independent and complementary measurement.

Similarly, EMRIs will be able to provide information on \(\Omega _m\) only in the most optimistic scenarios. In fact recent estimates (Laghi et al. 2021) indicate that using the loudest \({\mathcal {O}}(70)\) events, as predicted in an optimistic population scenario, it is possible to get constraints with \(\sim 20\%\) accuracy (68%CL) on \(\Omega _m\) when jointly inferred with \(H_0\). Even though this result is not expected to compete with current and future EM observations (Amendola et al. 2018b; Scolnic et al. 2018), it is somehow not surprising since the EMRIs used in Laghi et al. (2021) are distributed up to \(z \lesssim 0.7\). Future analysis, possibly including a larger number of events, thus including low-SNR events which are typically coming from high redshifts, are expected to give more informative results on cosmological parameters beyond \(H_0\).

Preliminary investigations of the full \(\Lambda \)CDM cosmological model where the curvature term \(\Omega _k\) is not fixed to 0, seem to indicate that, even for moderate redshift sources such as the loudest EMRIs (SNR \(> 100\)) at \(z\lesssim 0.7\), LISA will provide some simultaneous constraints on all cosmological parameters. In the fiducial scenario of Laghi et al. (2021), \(\Omega _k\) can be constrained with an accuracy of \(\sim 30\%\), while \(\Omega _m\) can be constrained with an accuracy of \(\sim 55\%\), while retaining an accuracy on \(H_0\) of \(\sim 2\%\) (all 68% CL) (Laghi 2021). These results will need to be further investigated in the future, however they are suggestive that a possible simultaneous inference of different cosmological parameters with LISA standard sirens will indeed be possible. In particular the full LISA cosmological analysis with results obtained from the combination of all classes of LISA standard sirens, namely SOBBHs, EMRIs and MBBHs, should substantially increase the accuracy of the estimates on \(\Omega _m\) and \(\Omega _k\) above, thanks especially to the combination of cosmological datasets from different redshift ranges which might break some of the degeneracy between the cosmological parameters. A future combined investigation will thus be needed to thoroughly assess the ability of LISA to constrain parameters beyond \(H_0\).

2.3.4 Tests of \(\Lambda \)CDM at high-redshift

The most interesting class of LISA standard sirens are MBBHs, not only because they are expected to produce detectable EM counterparts (bright sirens) but also because they will be detected at high redshift and thus can be employed to test the cosmic expansion at early epochs largely unexplored by current EM cosmological surveys. As we will show in Sect. 2.4 and in Sect. 4, MBBHs will have the potential to test different cosmological models alternative to \(\Lambda \)CDM. Here we briefly mention how well LISA can test \(\Lambda \)CDM itself at high z, taking into account possible general deviations (to be discussed shortly) and by comparing with EM probes at similar redshifts. Note also that in analogy to EM distance observations, LISA MBBHs can as well be employed to probe the fundamental assumptions of \(\Lambda \)CDM, e.g. the cosmological principle (Cai et al. 2018).

Let us first of all recall how well MBBHs can test \(\Lambda \)CDM. As shown in the previous section, in the most optimistic cases \(\Omega _m\) can be tested at the \(\sim 10\%\) level, which reflects the fact that at \(z\gtrsim 1\) the universe is expected to be matter dominated and thus to provide information mainly on \(\Omega _m\). Any deviation from the standard matter-dominated evolution at redshift \(1\lesssim z \lesssim 4\) however will be constrained by LISA, at a level not attained by current EM observations.

To put in context the potential of LISA we can compare its constraining power with other EM measurements of the cosmic expansion at high z. LISA will in fact provide independent and complementary data which will not only deliver useful and accurate information on deviations from \(\Lambda \)CDM but will also be used to check and cross-validate EM measurements, expected to be sparse and inaccurate at such high-redshift. As a clear example, LISA can successfully compete with quasar cosmological observations at \(z\sim 2\) and above (Speri et al. 2021). Current quasar distance measurements indicate possible issues in the Hubble diagram at \(z\gtrsim 2\), where deviations from \(\Lambda \)CDM have already been claimed (Risaliti and Lusso 2019) (see also Velten and Gomes 2020; Yang et al. 2020b; Banerjee et al. 2021). To understand if such apparent deviations are due to systematic effects or are indeed due to new physics beyond \(\Lambda \)CDM, complementary measurements at the same redshift range will be needed. As shown in Speri et al. (2021), four MBBH standard sirens detected by LISA are on average enough to unequivocally confirm or rule out the apparent deviation claimed in Risaliti and Lusso (2019), and thus to reveal if indeed this is due to systematics in the quasar Hubble diagram or to new physics. Standard siren observations will play a crucial role to test deviations from the \(\Lambda \)CDM.

GW observations provide a more reliable distance luminosity measurement than other EM observations, such as quasars. Standard sirens rely in fact on fundamental predictions of GR and not on phenomenological relations between observed quantities. Although the quasar Hubble diagram will become more precise and accurate in the time between now and when LISA will fly, for example with observations by the eROSITA (Merloni et al. 2012) and Euclid (Amendola et al. 2018b; Barnett et al. 2019) missions, intrinsic systematics on the quasar cosmological measurement might still be unresolved. LISA will thus provide a unique complementary test of the cosmic expansion at \(z\gtrsim 2\), which not only will yield accurate measurement of deviations from \(\Lambda \)CDM but it will also be used to cross-validate any results obtained by other EM observations at the same redshift range, notably with quasars.

2.4 Probing dark energy

This subsection discusses how LISA can probe the fundamental nature of DE through GW standard sirens. Here only simple alternative DE models are considered, and in particular we assume that the underlying gravitational theory remains GR. Models of DE based on modification of GR are discussed in details in Sect. 4.

2.4.1 Equation of state of dark energy: \(w_0\) and \(w_a\)

Deviations from the standard cosmological model include the presence of a DE fluid with effective equation of state (EoS) given by: \(p_{{{{\rm DE}}}}=w_{{{{\rm DE}}}}(a)\rho _{{{{\rm DE}}}}\), where \(p_{{{{\rm DE}}}}\) and \(\rho _{{{{\rm DE}}}}\) are the effective pressure and density of the DE fluid respectively. Although the functional dependence of \(w_{{{{\rm DE}}}}(a)\) may be very non-trivial (see e.g. self-accelerating cosmologies), for practical purposes we consider here only a simple phenomenological parametrization introduced by Chevalier-Polarski-Linder (CPL) (Chevallier and Polarski 2001; Linder 2003):

$$\begin{aligned} w_{{{{\rm DE}}}}(a)=w_0 + w_a(1-a), \end{aligned}$$

where \(w_0\) and \(w_a\) are constants and indicate, respectively, the value and the time derivative of \(w_{{{{\rm DE}}}}\) today. We refer to Sect. 4 about modified gravity for more complicated, model dependent forms.

In all the scenarios studied by Belgacem et al. (2019c) (which depend on the BH seeds and on the assumptions about the error on redshift), the GW sources considered are expected not to contribute relevantly to improve the knowledge on \(w_0\), with respect to what is already known from current cosmological observations. More precisely, the 1\(\sigma \) error \(\Delta w_0\) only goes from \(\Delta w_0=0.045\) (using CMB, type Ia supernovae and BAO data) to \(\Delta w_0=0.044\) adding MBBH standard sirens to the datasets, even in the best scenario where LISA alone can only reach a 20% relative 1\(\sigma \) uncertainty on \(w_0\) (Tamanini 2017). It is important to remark, however, that these outcomes are based on a mission duration of 4 years and sources are limited to MBBHs with EM counterparts. Significant improvements are expected by extending the data taking time or by combining with information from other sources, notably EMRIs.

Indeed, recent investigations (Laghi et al. 2021) suggest that EMRIs will deliver constraints on \(w_0\) of the order of \(\sim 10\%\), when inferred simultaneously with \(w_a\). When assuming prior knowledge of \(H_0\) and \(\Omega _m\), constraints on \(w_0\) are estimated at the \(\sim 7\%\) level in a realistic EMRI scenario, reaching \(\sim 5\%\) in the best case (all 90% C.L.). These results are better than what expected with dark sirens analysis for 3 G ground-based detectors (Belgacem et al. 2018b), and compatible with what they can achieve with bright sirens (Belgacem et al. 2019b). While relevant information on \(w_0\) can be obtained with moderately low-redshift events, \(w_a\) is expected to be measurable only with higher-redshift events, which in the joint cosmological inference of \(w_0\) and \(w_a\) of Laghi et al. (2021) are not considered, and thus no relevant measurement of \(w_a\) is obtained.

Although from these estimates it seems that LISA standard sirens will not be competitive with future EM observations, we stress that they will anyway provide independent and complementary measurements which will increase our confidence on any insight on the nature of DE. This is strikingly important for modified gravity models of DE, where GWs can indeed provide orthogonal information with respect to EM observations; see Sect. 4.

2.4.2 Alternative dark energy models

The CPL phenomenological parametrization discussed in the previous subsection constitutes only a simple approach for detecting deviations from the \(\Lambda \)CDM. The theoretical description of DE may require more sophisticated models. Here, we briefly discuss such a possibility focusing on scenarios that change the background cosmological expansion with respect to \(\Lambda \)CDM. We assume the standard evolution equation for GWs. Possible modifications to GW propagation through cosmological spacetimes—motivated by modified graviton dispersion relations, or non-minimal couplings of the dark-energy sector with curvature—are described in Sect. 4.

Models for DE can include quite a large number of parameters, leading to rich dynamics for the DE sector as a function of redshift. Some investigations aim to dynamically explain the puzzling small value for the present-day acceleration rate, leading to a time-dependent evolution of the DE density. These include quintessence scenarios (Wetterich 1988; Zlatev et al. 1999), which can be generalised to kinetically-driven (Armendariz-Picon et al. 2000) and kinetic-braiding models (Deffayet et al. 2010) without modifying the propagation properties of GWs. Other scenarios aim to alleviate the coincidence problem relating DE with DM at intermediate redshifts, for example in the DE-DM interacting models (Wetterich 1995; Amendola 2000), or in Chaplygin gas cosmology (Kamenshchik et al. 2001). More exotic possibilities include holographic DE, associating the present-day acceleration of the universe with the size of the particle horizon (Li 2004). See e.g. Copeland et al. (2006b), Li et al. (2011), Ishak (2019), Huterer and Shafer (2018) for reviews, including analysis of cosmological implications and observational prospects of DE scenarios. A recent resurgence of interest on DE model building, based on the previous approaches, has been motivated by the \(H_0\) tension discussed in Sect. 2.3.1; see Di Valentino et al. (2021) for a review. Among many examples, such tension can be alleviated in scenarios with DM-DE interactions or with features at small or intermediate redshifts (Di Valentino et al. 2017, 2016; Keeley et al. 2019; Raveri 2020), in ranges that might be probed with GW sirens. See e.g. Knox and Millea (2020) for a comprehensive discussion on this topic, including comparison between theoretical ideas and existing cosmological constraints.

These theoretical models suggest that distinctive DE effects can occur at different redshifts, from very small to relatively large values of z. The capability of LISA to probe cosmological expansion in a large range of redshifts, as discussed in the previous sections, provides invaluable opportunities for building independent cosmological tests (see Sect. 2.3.4) and thus to probe different DE scenarios, in a complementary way with respect to EM probes. A clear example is given by the investigations of Cai et al. (2017), Caprini and Tamanini (2016) where LISA forecasts for testing cosmological models allowing for DE-DM interactions or for early DE have been produced using MBBH as bright sirens. Similar analyses using LISA dark sirens are still missing in the literature and will constitute material for future explorations. By considering the results obtained with both SOBBHs and EMRIs for standard cosmological models (see Sect. 2.3), it would be very interesting to further develop these studies by designing an efficient, unified method to reconstruct the redshift dependence of DE with GWs, similar for example to what already done with EM observations; see e.g. Sahni and Starobinsky (2006). Assumptions going beyond the linear CPL parameterisation discussed in the previous section might be better suited to test specific alternative scenarios, or for improving the DE reconstruction in a wider redshift interval, e.g. through polynomial fitting (Alam et al. 2003). Alternatively, methods based on a principal component analysis of a binned parametrization of the signal as a function of redshift, as proposed in Huterer and Starkman (2003), might be adapted and applied to GW observations with LISA.

Besides the alternative DE models already considered in the literature, there are plenty of others that can still be tested by LISA and for which detailed analyses will be needed in the forthcoming years in order to understand how well LISA will constrain them and thus to better assess and expand the science case of the mission. We conclude by mentioning again that the nature of DE can be further investigated if this is connected to an underlying gravitational theory beyond GR. In this case new observational signatures might appear, giving rise to a richer phenomenology and to more promising LISA results. Such models and analyses will be discussed in Sect. 4 in the context of modified theories of gravity (Baker et al. 2021).

2.5 Synergy with other cosmological measurements

GW observations by LISA will provide a wealth of cosmological information as we have seen in the previous sections. But LISA will not be alone in the quest of understanding the cosmic history. It is thus of great importance to assess how LISA could complement with other facilities. In what comes next, we discuss LISA synergies with EM observatories and other GW detectors.

2.5.1 Integration with standard electromagnetic observations

To date, most studies constraining cosmological parameters with GWs have focused on the Hubble parameter, see Sect. 2.3.2. This is because the cosmological information in GW amplitudes is primarily held in the luminosity distance of the source, \(D_L\), which for low redshifts (\(z\ll 1\)) can be approximated to \(D_L\simeq c z /H_0\). The majority of LISA sources will exist at higher redshifts where this approximation breaks down. Instead the full expression for the luminosity distance must be used (see Eq. (1)) and so GW amplitudes in principle have sensitivity to further cosmological parameters traditionally measured electromagnetically, such as the fractional densities \(\Omega _m\) and \(\Omega _\Lambda \), and the DE EoS parameters \(w_0\) and \(w_a\), see Sects. 2.3.3 and 2.4.

A key outstanding question is whether specific combinations of EM and GW probes have the ability to improve constraints on these parameters and break degeneracies between them. Details about the galaxy surveys co-temporal with LISA are not available at present, but as a conservative approach we can consider ‘Stage IV’ experiments planned over the next decade, such as DESI, Euclid, the Vera C. Rubin Observatory, and the Roman Space Telescope (Aghamousa et al. 2016; Laureijs et al. 2011; Schmidt et al. 2020; Spergel et al. 2015). The corresponding CMB data will come from the LiteBIRD mission (Hazumi et al. 2012). Direct cross-correlation of GW sources with galaxy catalogues will be considered in Sect. 2.6. Here we focus instead on probe combination, though we note that there is a lack of comprehensive studies on this topic in the current literature.

One advantage leveraged by Stage IV galaxy surveys is the ability to combine multiple probes from the same instrument. Most commonly the main probes are shear power spectra from galaxy weak lensing, BAO measurements from galaxy clustering, supernovae, and, additionally, strong gravitational lenses in some analyses. As examples of the expected constraints, Zhan and Tyson (2018) provides forecasts for standard cosmological parameters using weak lensing, BAO and supernovae data from the Vera C. Rubin Observatory, combined with current Planck CMB data. The resulting 68% confidence intervals on \(\left\{ w_0, w_a\right\} \) and \(\omega _m = \Omega _m h^2\) are \(\left\{ 7.05\times 10^{-2},1.86\times 10^{-1}\right\} \) and \(7.73\times 10^{-4}\) respectively, with h being the normalised Hubble parameter. Similar results are expected from the Euclid space mission, which should yield 68% confidence intervals around \(\left\{ 3\times 10^{-2}, 1.3\times 10^{-1}\right\} \) for \(\left\{ w_0, w_a\right\} \). Comparing these values to the forecasts using LISA EMRI detections presented in Laghi et al. (2021), it seems unlikely that LISA will be able to offer competitive constraints on \(\Omega _m\). However, in mildly optimistic scenarios they may offer moderate constraints on the DE parameters, with (Laghi et al. 2021) forecasting \(95\%\) confidence intervals on \(\left\{ w_0, w_a\right\} \) of approximately \( \left\{ 6.5\times 10^{-2},6.5\times 10^{-1}\right\} \) for a four-year LISA mission and optimistic MBH population models.

An alternative strategy is to find probe combinations where GW data can break degeneracies existing between EM probes. One such example is put forwards in Qi et al. (2021), which determines forecasts for the combination BNS data from the DECihertz Interferometer Gravitational wave Observatory (DECIGO) with measurements of redshift drift (the Sandage–Loeb effect, Loeb 1998) from the SKA and the European-Extremely Large Telescope (E-ELT). Redshift drift measurements use high-resolution spectroscopy of the HI emission line (SKA) or Lyman-\(\alpha \) absorption lines in quasar spectra (E-ELT) over long time frames (10 years+) to directly measure tiny shifts the line frequencies. Although experimentally challenging, this constitutes a direct measurement of H(z), as opposed to the integrated effect of H(z) probed by luminosity distances; see Eq. (1).

GWs can likewise offer a direct measurement of H(z) through the dipole of the luminosity distance. Equation (1) gives the luminosity distance-redshift relation in a perfectly homogeneous and isotropic universe; in fact small corrections to this are induced by gravitational lensing and peculiar velocities. As shown in Bonvin et al. (2006a), Bonvin et al. (2006b) for the EM case (see e.g. Nishizawa et al. 2011 for the GW case), the peculiar motion of our Galaxy with respect to the CMB frame induces a dipole mode in the luminosity distances. This dipole moment is given by

$$\begin{aligned} D_L^{(1)}(z)=\frac{|\mathbf{v_0}|\,(1+z)^2}{H(z)}, \end{aligned}$$

where \(\mathbf{v_0}\) is the dipole anisotropy in the CMB, estimated to be approximately \(369.1\pm 0.9\) km/s. Combining measurements of this dipole from DECIGO BNS sources with the redshift drift measurements above, Qi et al. (2021) finds substantial breaking of degeneracies in the \(H_0-\Omega _m\) plane, leading to 1-\(\sigma \) bounds on \(\{H_0, \Omega _m\}\) of \(\{0.78\, {{\mathrm{km\, s}}}^{-1}{{\rm Mpc}}^{-1}, 0.006\}\), competitive with current bounds. Mild improvements on the constant DE EoS were also obtained (\(\sigma _{w_0}\sim 0.03\)), though no meaningful constraint on \(w_a\) was possible. Further analyses are needed to properly understand if such method can equally be applied to LISA.

On cosmological scales, a bias prescription is used to model how GW events trace the large-scale DM distribution. Analogously to galaxy bias, the GW bias is modelled as scale-independent on large scales. Using the parameterisation \(b_{{\text {GW}}}(z)=b^0_{{\text {GW}}}\,(1+z)^\alpha \), Mukherjee et al. (2021d) finds the parameters \(b^0_{{\text {GW}}}\) and \(\alpha \) to be uncorrelated with \(H_0\) and \(\Omega _m\) for a \(\Lambda \)CDM model (\(w_0=-1\) fixed). This suggests cautious optimism that combined GW and EM constraints on cosmological parameters should not be strongly sensitive to the GW bias prescription.

2.5.2 Complementarity with other gravitational-wave observatories

In addition to the synergies with other cosmological surveys, LISA will also build new synergies with other GW detectors. In particular, LISA will be able to detect the early inspiral phase of compact binary coalescences that later merge within the frequency bands of Earth-based interferometers. Some example signals are displayed in the left panel of Fig. 3. Because the same signal is detected across different frequencies, these events are known as multi-band. Different populations of BBHs could become multi-band sources. Most notably, the SOBBHs that current LIGO/Virgo detectors are observing could have been seen if LISA was online a few years before these detections (Sesana 2016). Nonetheless, LISA high frequency sensitivity limits their number (Moore et al. 2019). If present in nature, IMBBHs would be more promising candidates (Amaro-Seoane and Santamaria 2010; Jani et al. 2019; Sedda et al. 2020), and possibly yield interesting cosmological results (Muttoni et al. 2022). There is however a limit to their masses, because if they are too heavy they will merge before reaching the frequencies of ground-based detectors. (This limitation could be removed with a deci-Hertz observatory (Sedda et al. 2020).)

The right panel of Fig. 3 displays the fraction of multi-band events, defined as the subset of LISA detections merging within 10 years and being detected by a ground-based instrument (Ezquiaga and Holz 2021). For concreteness we consider Advanced LIGO (aLIGO), its possible upgrade (A+), Voyager and the third-generation detectors Cosmic Explorer (CE) and Einstein Telescope (ET). Interestingly, the multi-band fraction peaks where the upper end of the PISN mass gap is expected to be found, implying that far-side binaries could be promising multi-band sources (Ezquiaga and Holz 2021). As noted in Gerosa et al. (2019), for \(M_{{{\rm tot}}}\lesssim 100\,M_\odot \), there is no difference for the multi-band ratio between 2 G and 3 G detectors because the fraction is limited by LISA high-frequency range. On the contrary, for \(M_{{{\rm tot}}}>200\,M_\odot \) the difference among ground-based detectors are sizeable and depends mostly on their low frequency sensitivity.

Besides individual sources, LISA and ground-based detectors could share SGWBs. For example the background of unresolved SOBBHs could be within the reach of LISA detectability (Sesana 2016). Similarly, binaries above the PISN gap could leave an additional background (Mangiagli et al. 2019).

Finally, LISA could also complement other space-based detectors. In particular, there are several proposals such as Taiji (Ruan et al. 2020a) and TianQin Wang et al. (2020a) that could potentially fly at the same time as LISA. These additional detectors would help improve the cosmological inference (Wang et al. 2022a, b; Baral et al. 2020; Zhao et al. 2020). Moreover, any further detector in the deci-Hertz frequency band would be an excellent addition to LISA science (Sedda et al. 2020).

Altogether, LISA and the other GW observatories will all contribute to a standard siren measurement of H(z) across a broad redshift range.

Fig. 3
figure 3

On the left, GW signals as seen by LISA and different ground-based detectors. The times indicate the time to merge which depends on the initial frequency. On the right, fraction of total events observed by LISA and that will merge within 10 years, being detected by a ground-based detector. Image [right panel] reproduced with permission from Ezquiaga and Holz (2021), copyright by AAS

2.6 Cross-correlation and interaction with large scale structure

GW maps of resolved events or SGWB can be cross-correlated with other large-scale-structure (LSS) tracers to perform a variety of astrophysical and cosmological measurements. Here we focus on the use of the SGWB measurements while technicalities about this signal are postponed to Sec. 5.