Probing Planckian corrections at the horizon scale with LISA binaries

TL;DR

This paper investigates how future LISA gravitational wave observations could detect Planck-scale quantum-gravity effects at the horizons of supermassive compact objects by analyzing tidal heating and deformability signatures.

Contribution

It introduces a model-independent approach to probe quantum-gravity corrections at the horizon scale using gravitational wave data from LISA, focusing on tidal effects.

Findings

Tidal heating effects dominate and can constrain Planck-scale corrections.

Measurement of tidal Love numbers is challenging but can constrain compactness.

Highly-spinning supermassive binaries offer the best tests of quantum-gravity effects.

Abstract

Several quantum-gravity models of compact objects predict microscopic or even Planckian corrections at the horizon scale. We explore the possibility of measuring two model-independent, smoking-gun effects of these corrections in the gravitational waveform of a compact binary, namely the absence of tidal heating and the presence of tidal deformability. For events detectable by the future space-based interferometer LISA, we show that the effect of tidal heating dominates and allows one to constrain putative corrections down to the Planck scale. The measurement of the tidal Love numbers with LISA is more challenging but, in optimistic scenarios, it allows to constrain the compactness of a supermassive exotic compact object down to the Planck scale. Our analysis suggests that highly-spinning, supermassive binaries at 1-20 Gpc provide unparalleled tests of quantum-gravity effects at the horizon scale.

Equations

$$\tilde{h}(f)=\mathcal{A}(f)e^{i({\psi }_{\text{PP}}+{\psi }_{\text{TH}}+{\psi }_{\text{TD}})},$$

$${\dot{E}}_{\mathrm{BH}}^{\mathrm{heating}}$$

$${\psi }_{\text{TH}}^{\mathrm{BH}}={\psi }_{\text{N}}\left(F({\chi }_i,q)v^5\log v+G(q)v^8[1-3\log v]\right),$$

$${\psi }_{\text{TD}}(f)=-{\psi }_{\text{N}}\frac{\Lambda }{6m^5}{v}^{10}\frac{(1+q{)}^2}{q},$$

$$\delta :=r_0-2M\sim 2M{e}^{-1/k}.$$

$$(g|h)=4\mathrm{Re}{\int }_{f_{\min }}^{f_{\max }}df\frac{\tilde{h}(f){\tilde{g}}^{\star }(f)}{S_h(f)},$$

$$f_{\mathrm{crit}}\sim \frac{2^{1/13}}{2\pi m}{\left[\frac{5mQ{\Delta }_i}{m_i\left({\chi }_i+3{\chi }_i^3\right)\left(1+{\Delta }_i\right)\log (r_{+}^{(i)}/\delta )}\right]}^{3/13},$$

Full text

Introduction. Gravitational waves (GWs) are the most direct probes of compact objects down to the horizon scale and can shed light on one of the outstanding open issues in gravitational astronomy: the nature of compact, dark and massive objects Cardoso and Pani (2017a). It has been tacitly assumed that the latter must be black holes (BHs) for a number of compelling reasons: BHs form from classical gravitational collapse of stars, while there are no known, equally-well motivated alternatives which are sufficiently compact to explain observations, especially the recent GW detections Abbott et al. (2016a, b, 2017). Nonetheless, over the years several arguments have been put forward, suggesting that new physics at the horizon scale might set in during gravitational collapse, possibly halting or altering the formation of BHs Mazur and Mottola (2004); Mathur (2005); Skenderis and Taylor (2008); Gimon and Horava (2009); Almheiri et al. (2013). While the end product of the collapse in these scenarios is essentially unknown or model dependent Holdom and Ren (2016); Brustein and Medved (2017); Barceló et al. (2017), the exotic compact objects (ECOs) that might form share two common features: they are extremely compact and horizonless. Regardless of the viability of these objects and the mechanisms behind them, we have, for the first time, the means for testing these scenarios with GWs.

It was recently argued that ECOs can be detected or ruled out through GW measurements in two different regimes: the postmerger ringdown phase of a coalescence – where putative corrections at the horizon scale will produce GW echoes Cardoso et al. (2016a, b); Maselli et al. (2017) (see also Ref. Ferrari and Kokkotas (2000) for an earlier study, and Refs. Abedi et al. (2017a); Ashton et al. (2016); Abedi et al. (2017b) for a debate on the evidence of this effect in aLIGO data) – and the late-time inspiral of the coalescence – through the measurement of the tidal deformability of the two objects Wade et al. (2013); Cardoso et al. (2017); Sennett et al. (2017) and of their spin-induced quadrupole moment Krishnendu et al. (2017).

Here, we discuss another effect that can be used to distinguish ECOs from BHs, namely the absence of tidal heating if the binary components do not possess a horizon. For supermassive binaries detectable by the future Laser Interferometer Space Antenna (LISA) Audley et al. (2017), we show that this effect can be used to constrain putative corrections near the horizon down to the Planck scale, even for sources at cosmological distance111We assume a standard $\Lambda$CDM flat universe with $H_{0}\sim 67.7$, $\Omega_{M}\sim 0.26$ and $\Omega_{\Lambda}\sim 0.69$ Ade et al. (2016). All masses quoted in the text are assumed to be the redshifted masses. With this choice, the waveform (1) is independent of the redshift $z$. if the binary components are highly spinning, as predicted by several models of spin evolution of supermassive objects Sesana et al. (2014).

GW tests of the nature of compact objects. Consider a compact binary, of masses $m_{i}$ ($i=1,2$), total mass $m=m_{1}+m_{2}$, mass ratio $q=m_{1}/m_{2}\geq 1$, and dimensionless spins $\chi_{i}$. In a post-Newtonian (PN) approximation (i.e. a weak-field/slow-velocity expansion of Einstein’s equations), dynamics is driven by energy and angular momentum loss, and particles are endowed with a series of multipole moments and with finite-size tidal corrections Blanchet (2006). Loosely speaking, the nature of the inspiralling objects is encoded in (i) the way they respond to their own field – i.e., on their own multipolar structure, (ii) the way they respond when acted upon by the external gravitational field of their companion – through their tidal Love numbers (TLNs) Poisson and Will (2014), and (iii) on the amount of radiation that they possibly absorb, i.e. on tidal heating Hartle (1973); Hughes (2001). These effects are all included in the waveform produced during the inspiral, and can be incorporated in the (Fourier-transformed) GW signal as (we use $G=c=1$ units)

[TABLE]

where $f$ and ${\cal A}(f)$ are the GW frequency and amplitude, $\psi_{\textnormal{PP}}(f)$ is the “pointlike” phase, whereas $\psi_{\textnormal{TH}}(f),\,\psi_{\textnormal{TD}}(f)$ are the contribution of the tidal heating and the tidal deformability, respectively.

Spin-orbit and spin-spin interactions are included in $\psi_{\textnormal{PP}}$, the latter also depending on the spin-induced quadrupole moment. This property has been recently used to constrain ${\cal O}(\chi_{i}^{2})$ deviations from the Kerr geometry Krishnendu et al. (2017). However, in known models of rotating ECOs – e.g., gravastars Pani (2015); Uchikata et al. (2016) and strongly-anisotropic, incompressible neutron stars Yagi and Yunes (2015) – the multipole moments approach those of a Kerr BH in the high-compactness limit. This suggests that the distinction between ultracompact objects and BHs can only be done, realistically, using finite-size corrections, $\psi_{\textnormal{TH}}(f)$ and $\psi_{\textnormal{TD}}(f)$.

BHs are very special objects in general relativity and it is no surprise that finite-size effects are different for ECOs than for BHs. If the inspiralling objects are BHs, a small contribution to the dynamics is provided by dissipation of energy and angular momentum at the horizon. To leading PN order, the energy flux at the horizon reads Alvi (2001); Hughes (2001); Taylor and Poisson (2008); Poisson (2009); Nagar and Akcay (2012); Bernuzzi et al. (2012); Chatziioannou et al. (2013); Cardoso and Pani (2013) (henceforth we assume circular orbits and spins aligned with the orbital angular momentum)

[TABLE]

where the expansion parameter $v=(\pi mf)^{1/3}$ is the orbital velocity and $\dot{E}_{\rm GW}\sim v^{10}$ is the leading order, quadrupole GW flux Peters and Mathews (1963). Angular-momentum flux is subleading so the spins remain roughly constant during the evolution Alvi (2001). The GW phase $\psi$ is governed by $d^{2}\psi/df^{2}=2\pi(dE/df)/\dot{E}$, where $E\sim v^{2}$ is the binding energy of the binary. To the leading order, this yields

[TABLE]

where $\psi_{\textnormal{N}}\sim v^{-5}$ is the leading-order contribution to the point-particle phase (corresponding to the flux ${\dot{E}}_{\rm GW}$), $F(\chi_{i},q)$ and $G(q)$ are simple functions of their arguments, and $F\sim\chi_{i}+{\cal O}(\chi_{i}^{2})$. Thus, absorption at the horizon introduces a $2.5$PN ($4$PN)$\times\log v$ correction to the GW phase of spinning (nonspinning) binaries, relative to the leading term (conventionally, one PN order corresponds to a term $\sim v^{2}$). For all known matter, GW absorption is negligible: tidal heating is therefore a good discriminator for the existence of horizons. It is therefore convenient to introduce an absorption coefficient $\gamma$ such that, $\psi_{\textnormal{TH}}\equiv\gamma\,\psi_{\textnormal{TH}}^{\rm BH}$, with $\gamma=0$ for ECOs with a perfectly reflecting surface or whose interior does not absorb GWs, $\gamma=1$ for BHs, and $\gamma\in(0,1)$ for partial absorption.

In addition, while the TLNs of BHs are zero Binnington and Poisson (2009); Damour and Nagar (2009); Fang and Lovelace (2005); Gürlebeck (2015); Poisson (2015); Pani et al. (2015a), those of ECOs are small but finite Pani (2015); Uchikata et al. (2016); Porto (2016); Cardoso et al. (2017). In line with neutron star binaries Poisson and Will (2014); Flanagan and Hinderer (2008); Hinderer (2008), the leading tidal deformability term for ECO binaries reads,

[TABLE]

where $\Lambda=\left(1+{12}/{q}\right)m_{1}^{5}k_{1}+(1+12q)m_{2}^{5}k_{2}$ is the weighted tidal deformability, and $k_{i}$ is the (dimensionless) TLN of the $i-$th object. Thus, tidal deformability introduces a $5$PN correction to the GW phase relative to the leading-order GW term, whereas spin-tidal couplings are subleading Poisson (2015); Pani et al. (2015b); Landry and Poisson (2015) and can be neglected.

The TLNs of a nonspinning ultracompact object of mass $M$ and radius $r_{0}=2M(1+\epsilon)$ (with $\epsilon\ll 1$) in Schwarzschild coordinates vanish logarithmically in the BH limit Cardoso et al. (2017), $k\sim 1/|\log\epsilon|$, opening the way to probe horizon scales. As depicted in Fig. 1, any measurement of the TLN translates into an estimate of the distance of the ECO surface from its Schwarzschild radius,

[TABLE]

For a supermassive object with $M\sim 10^{6}M_{\odot}$, $\delta$ is of the order of the Planck length, $\ell_{P}\approx 1.6\times 10^{-33}\,{\rm cm}$, when $k\approx 0.005$ [cf. Fig. 1]. Therefore, future GW observations should aim at reaching the level of accuracy necessary to measure TLNs as small as $k\sim 0.005$. Below, we show that this will be achievable with LISA.

To summarize, finite-size effects in the inspiral waveform provide two different null-hypothesis tests of BHs. While BHs have zero TLNs but introduce a nonzero tidal heating ($\psi_{\textnormal{TD}}=0$, $\gamma=1$), ECOs have (logarithmically small) TLNs but zero tidal heating ($\psi_{\textnormal{TD}}\neq 0$, $\gamma=0$).

Error analysis. State-of-the-art waveform models for BH binaries are fitted to numerical simulations in order to reproduce precisely the late-inspiral and merger signal Taracchini et al. (2014); Husa et al. (2016); Khan et al. (2016). Since simulations of ECO binaries are not yet available [with the notable exception of boson stars Bezares et al. (2017), which are however limited to $\epsilon\gtrsim{\cal O}(1)$], there is no fully consistent inspiral-merger-ringdown waveform model for an ECO coalescence.

To overcome this problem, here we focus our analysis on the inspiral phase only, adopting a state-of-the-art PN template in the frequency domain, the so-called TaylorF2 approximant Damour et al. (2001, 2002); Arun et al. (2005). We include spin-orbit, spin-spin Khan et al. (2016) and the known cubic spin corrections to $\psi_{\rm PP}$ up to $3.5$PN order Isoyama and Nakano (2017). As discussed, we assume the quadrupole moment to be that of a Kerr BH. We include tidal heating to leading order (i.e., to $2.5$PN ($4$PN) relative order for spinning (nonspinning) binaries) and tidal deformability terms to next-to-leading order (i.e, to $6$PN relative order Vines et al. (2011); Bini et al. (2012)).

We employ a Fisher matrix analysis, which is accurate at large signal-to-noise ratios (SNRs) Vallisneri (2008), as those relevant for LISA supermassive binaries under consideration. For a given set of parameters $\vec{\xi}$, the error associated with the measurement of parameter $\xi^{a}$ is $\sigma_{a}=\sqrt{\Sigma^{aa}}$, where $\Sigma^{ab}$ is the inverse of the Fisher matrix, $\Gamma_{ab}=\left(\partial_{\xi^{a}}h|\partial_{\xi^{b}}h\right)_{\vec{\xi}=\vec{\xi}_{0}}$, $\vec{\xi}_{0}$ are the injected values of the parameters $\vec{\xi}$, and the inner product is defined as

[TABLE]

where $S_{h}(f)$ is the recently proposed LISA’s noise spectral density Audley et al. (2017). We assume an observation time $T_{\textnormal{obs}}=1\,\textnormal{yr}$, sky-averaging the GW signal Berti et al. (2005), and estimate the initial frequency $f_{\rm min}$ of the binary in terms of $f_{\textnormal{max}}$ and $T_{\rm obs}$ by solving the binary motion to leading order. Since higher-PN corrections are more relevant in the last stages of the inspiral, we are interested in those “golden” binaries which remain in the LISA band for $T_{\rm obs}$ up until the merger. Finally, we choose $f_{\rm max}$ to guarantee that our template is a reliable approximation of the signal in the relevant frequency range. Namely, we choose $f_{\rm max}$ such that the overlap ${\cal O}$ between the TaylorF2 BH-BH template and a more accurate inspiral-merger-ringdown template for spinning BH binaries (the PhenomD waveform Husa et al. (2016); Khan et al. (2016)) is at least equal to the fiducial threshold ${\cal O}\geq 0.95$ (more details are given in the Supplement Material). We note that, to the leading order, the statistical errors estimated through the Fisher matrix are independent of the systematic errors arising from approximating the true signal with an imperfect theoretical template Chatziioannou et al. (2014).

The relevant Fisher-matrix parameters are $\vec{\xi}=(\ln{\cal A},\psi_{c},t_{c},\ln{\cal M},\ln\nu,\chi_{1},\chi_{2})$ [where $\psi_{c}$, $t_{c}$, ${\cal M}=(m_{1}m_{2})^{3/5}/m^{1/5}$ and $\nu=m_{1}m_{2}/m^{2}$ are the phase and time at the coalescence, the chirp mass, and the symmetric mass ratio, respectively] plus possibly $\gamma$ and $\log\Lambda$, depending on the system under consideration. (The subleading term of $\psi_{\rm TD}$ depends on an extra parameter Bini et al. (2012) which, however, turns out to be unmeasurable.)

GW constraints on ECOs. Our requirement ${\cal O}\geq 0.95$ is not satisfied for both large spins and large mass ratios. Because the effects of the former are more important, we focus on $q\in(1,2)$ and $\chi_{i}\in(0,0.95)$.

Let us first consider the effects of tidal heating, by setting $\Lambda=0$. The error $\sigma_{\gamma}$ is shown in the left panel of Fig. 2 as a function of the spin for different systems. The dashed horizontal line marks the threshold $\sigma_{\gamma}=1$: measurements of the tidal heating parameter $\gamma$ below the threshold have less than $100\%$ uncertainty and would discriminate between ECOs ($\gamma=0$) and BHs ($\gamma=1$) at least at $1\sigma$ level. While slowly-spinning, equal-mass binaries are not distinguishable, the accuracy improves dramatically at large spin. In some favorable scenarios, BH coalescences up to luminosity distances of $20\,{\rm Gpc}$ (corresponding to redshift $z\approx 2.5$) with individual masses $m_{i}\sim 10^{6}M_{\odot}$ and spins $\chi_{i}\gtrsim 0.9$ can be confidently distinguished from ECO binaries on the basis of the presence of tidal heating. The enhancement with spin is expected, given that tidal heating enters the GW phase at $2.5$PN$\times\log v$ order [cf. Eq. (3)] and that the merger frequency is higher. We also observe a milder improvement as $q$ increases, due to the larger number of orbits in bandwidth. One of our best-case scenarios ($m_{1}=1.1m_{2}=10^{6}M_{\odot}$, $\chi_{i}=0.9$) corresponds to ${\rm SNR}\approx 2\times 10^{4}$, and relative errors $\simeq(2\times 10^{-3},6\times 10^{-2},6,7,14)\%$ on $({\cal M},\nu,\chi_{1},\chi_{2},\gamma)$, respectively.

The second case of interest is orthogonal to the above: with ECO binaries in mind we now set $\gamma=0=\psi_{\rm TH}$ but a nonvanishing TLN. We are interested in estimating whether a measurement of $\Lambda$ is incompatible with zero and therefore whether ECOs can be distinguished from BHs on these grounds. This case was preliminarily studied in Ref. Cardoso et al. (2017) (and more recently in Ref. Sennett et al. (2017)) only for equal-mass binaries and neglecting spin. The right panel of Fig. 2 shows that the constraints on $\Lambda$ are orders of magnitude more stringent for spinning binaries. In particular, for $\chi_{i}\gtrsim 0.9$ and $q\sim 1$, a constraint on the TLN as stringent as $k\lesssim 0.005$ ($k\lesssim 0.05$) can be obtained for a coalescence within $2\,{\rm Gpc}$ ($20\,{\rm Gpc}$). For several ECO models Cardoso et al. (2017), a bound $k\lesssim 0.005$ on the TLN translates into an impressive constraint on the compactness of ECOs down to the Planck scale near the horizon, i.e. $\delta/M\sim 10^{-45}$ for a supermassive object [cf. Fig. 1].

Thus, finite-size effects open up the tantalizing possibility to know, through GW observations, if BHs do actually exist, and are in fact complementary to searches for new physics at the horizon scale through the detection of GW echoes Cardoso and Pani (2017b).

Effective absorption by ECOs. It might be argued that there should be a continuous transition from BHs to ultracompact horizonless objects, whereas we have assumed that tidal heating of ECOs is negligible. In fact, an ultracompact object can trap radiation within its photon sphere efficiently Cardoso et al. (2014, 2016a, 2016b), thus mimicking the effect of a horizon. This mechanism is interesting per se at a theoretical level and deserves (and requires) a separated study. Nonetheless, we can estimate it as follows. In order for the absorption to affect the orbital motion, it is necessary that the (arrival) time radiation takes to reach the companion, $T_{\rm arr}$, be much longer than the radiation-reaction time scale due to heating, $T_{\rm RR}\sim E/\dot{E}_{\rm BH}^{\rm heating}$. From Eq. (2), the latter reads at most $T_{\rm RR}\sim mv^{-13}/(\chi_{i}+3\chi_{i}^{3})$, when $q=1$ and $\chi_{i}\neq 0$.

For BHs, $T_{\rm arr}\to\infty$ because of time dilation, so that the condition $T_{\rm arr}\gg T_{\rm RR}$ is always satisfied. For ECOs in the $\epsilon\to 0$ limit, $T_{\rm arr}$ turns out to be just the GW echo delay time of the $i$-th object, $T_{\rm arr}\sim m_{i}|\log{\epsilon}|$ Cardoso et al. (2016a, b). For a given compactness, the condition $T_{\rm arr}\gg T_{\rm RR}$ implies that an effective heating can take place only when $f\gg f_{\rm crit}$. Including spins Abedi et al. (2017a) and generic mass ratios in the computation, we find,

[TABLE]

where $\Delta_{i}=\sqrt{1-\chi_{i}^{2}}$, $Q=(1+q)^{4}/[q(1+q(q-1))]$, and $r_{+}^{(i)}=m_{i}(1+\Delta_{i})$. When $\chi_{i}\to 0$, the subleading term in Eq. (2) becomes dominant, but its effect is negligible. As expected, $f_{\rm crit}$ decreases as the compactness increases. However, owing to the logarithmic dependence, even for $\delta\sim\ell_{P}$, $f_{\rm crit}\gtrsim f_{\rm ISCO}$ when $\chi_{i}<0.9$ or when $q\gg 1$ for any spin. In the least favorable case [$0.9<\chi_{i}<0.99$ and $q\sim 1$], $f_{\rm crit}\approx 0.4f_{\rm ISCO}$. Therefore, in the entire region where the PN expansion is valid the “effective” tidal heating of a Planck-scale ECO can be neglected.

Another possible caveat concerns GW dephasing due to mode excitation. The fundamental QNMs of an ECO have small frequency and are extremely long lived. Because $\omega_{\rm QNM}\sim{2\pi}/{T_{\rm arr}}\sim 1/(M|{\log\epsilon}|)$ Cardoso et al. (2016a, b), the QNMs can be possibly excited only when the orbital frequency $\simeq 2\pi/T_{\rm arr}$ which, for Planck-scale ECOs, occurs only near the ISCO and with extremely narrow resonances that should absorb a negligible amount of energy.

In summary, our results seem robust even when relaxing some of the assumptions: mode excitation and tidal heating can be expected to be negligible for Planck-scale ECOs, validating our analysis.

Discussion and Prospects. The future space-based GW detector LISA Audley et al. (2017) will be able to distinguish whether supermassive dark objects in a binary coalescence have a horizon or not by measuring two distinct and complementary finite-size effects on the waveform: tidal heating and tidal deformability. These effects become stronger for highly spinning binaries – as those predicted in several models of BH spin cosmic evolution Sesana et al. (2014) – and allow us to constrain the location of the ECO surface down to Planck scales, even for cosmological sources. This is a truly spectacular potential. GW observations will possibly provide the most impressive tests of near-horizon physics, and will challenge our understanding of quantum gravity backreaction effects.

This work is intended as a proof-of-principle; including eccentricity, spin misalignment, higher-order PN terms, and performing a Bayesian analysis are certainly relevant extensions. In the near future, we expect that a substantial effort will be devoted to perform fully relativistic simulations of exotic binaries. The latter would be extremely precious to extend the domain of validity of current waveforms to the merger and ringdown phases, matching semi-analytical templates to the numerical data, in order to build complete ECO templates (e.g. using the hybrid waveform approach Husa et al. (2016); Khan et al. (2016), or an effective-one-body framework Buonanno and Damour (1999); Bernuzzi et al. (2012); Taracchini et al. (2013, 2014)).

LISA rates of these events at $z\lesssim 2.5$ are uncertain, but they are expected to be $\sim 1-10$ per year Klein et al. (2016). Since even a single detection of such systems during LISA lifetime is enough to impose exquisite constraints, we claim that highly spinning, supermassive binaries may be wonderful probes of putative quantum-gravity effects at the horizon scale.

Acknowledgments. We are grateful to Leor Barack and Eric Poisson for useful correspondence regarding self-force effects in ultracompact backgrounds, to Ryuichi Fujita for clarifying the convergence properties of the PN series in the extreme-mass ratio limit, to Emanuele Berti for interesting discussions on the Fisher-matrix analysis, and to Michele Vallisneri for useful correspondence and for comparing with us the results of Ref. Vallisneri (2008). We are indebted to Tanja Hinderer and Scott Hughes for a critical reading of the manuscript and for many useful suggestions for improvement. We acknowledge interesting discussions with the participants of the workshops “New Frontiers in Gravitational-Wave Astrophysics” and “Strong Gravity Universe”, especially with Alessandra Buonanno, Kent Yagi, and Nico Yunes. P.P. acknowledges financial support provided under the European Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480 and is grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work. V.C. acknowledges financial support provided under the European Union’s H2020 ERC Consolidator Grant “Matter and strong-field gravity: New frontiers in Einstein’s theory” grant agreement no. MaGRaTh–646597. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development $\&$ Innovation. The authors would like to acknowledge networking support by the COST Action CA16104. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690904, the “NewCompstar” COST action MP1304, and from FCT-Portugal through the projects IF/00293/2013. The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by CENTRA/IST. Computations were performed at the clusters “Baltasar-Sete-Sóis” and Marenostrum, and supported by the MaGRaTh–646597 ERC Consolidator Grant.

Appendix A Reliability of the TaylorF2 template

In order to guarantee that the TaylorF2 template adopted in our analysis provides a reliable approximation of the inspiral waveform in the relevant frequency range, we compare it to the PhenomD waveform Husa et al. (2016); Khan et al. (2016); the latter provides an accurate inspiral-merger-ringdown template for spinning BH binaries. We choose $f_{\rm max}$ such that the overlap ${\cal O}\equiv{\rm max}_{t_{c},\phi_{c}}(h_{\rm TF2}|h_{\rm PD})>0.95$, where $h_{\rm TF2}$ is our waveform in the BH-BH case (i.e., $\gamma=1$, $k_{i}=0$), whereas $h_{\rm PD}$ is the PhenomD template for the same parameters.

For all results presented in this work, the overlap between our model and PhenomD is at least ${\cal O}\gtrsim 0.95$ (and often ${\cal O}\gtrsim 0.99$). We have also checked that using a more stringent threshold (${\cal O}\gtrsim 0.97$) would increase the statistical errors $\sigma_{\gamma}$ and $\sigma_{\Lambda}$ just by a factor of a few in the optimal configurations. A less conservative choice (often adopted in previous exploratory studies) is to set a sharp cut-off, $f_{\rm max}=\min(1\,{\rm Hz},f_{\rm ISCO})$, where $f_{\rm ISCO}$ is the GW frequency at the innermost-stable circular orbit (ISCO) of the Kerr metric222Because the external geometry of an ECO is arbitrarily close to that of a Kerr BH when $\epsilon\to 0$, the Kerr ISCO provides a good estimate of the cutoff., including corrections due to the self-force and the spin of the less massive object Favata (2011). This choice would provide more stringent constraints than those reported in the main text and, to the leading order, would not affect the statistical errors estimated through the Fisher matrix Chatziioannou et al. (2014).

Finally, we have performed a consistency check of our numerical approach, computing the maximum mismatch $|\ln r|$, i..e. the ratio between the Fisher-approximate likelihood and the exact probability distribution Vallisneri (2008). Specifically, the mismatch creterion quantifies whether the strength of the signal is large enough to justify the high SNR approximation, i.e. whether the Fisher matrix can be consistently used to represent a given parameter’s uncertainty. Following Vallisneri (2008), we have computed $|\ln r|$ requiring that $90\%$ of its cumulative distribution is below the threshold $|\ln r|<0.1$. The result of this analysis is shown in Fig. 3, for some of the most representative configurations below the threshold of Fig. 2. This analysis indicates that the statistical errors estimated from the Fisher matrix should be a reliable estimate of the measurement precision achievable by LISA for the sources under consideration.