Can We Distinguish Low-mass Black Holes in Neutron Star Binaries?

Merging binary neutron stars (NS) have just resoundingly been shown to produce both strong gravitational-wave (GW) signals and copious electromagnetic (EM) emission covering a large frequency range by the recent event GW170817 (Abbott et al. 2017b, 2017c, 2017d; Alexander et al. 2017; Coulter et al. 2017; Troja et al. 2017; see also, e.g., Kochanek & Piran 1993; Li & Paczyński 1998; Rosswog 2005; Metzger et al. 2010; Metzger & Berger 2012; Ando et al. 2013; Murguia-Berthier et al. 2014). The joint observation of GWs with, for instance, gamma-ray bursts, X-rays, ultraviolet/optical/infrared transients, or radio afterglows is now beginning to provide unprecedented information about the violent dynamics of hot, dense nuclear matter under extreme gravity. With three GW detectors now online, the ability to localize the source of the GW events (albeit within a still rather large window) facilitates identifying EM counterparts (Abbott et al. 2017c, 2017e).

It is natural to assume a compact object is an NS instead of a black hole (BH) if its mass is below the upper bound of a non-rotating NS. Such an assumption has also been supported by the observed mass distribution of NS and BHs in binaries, and applied to distinguish NS–NS and BH–NS binaries through the use of component mass measurements to identify a possible "mass gap" in BHs (Hannam et al. 2013; Littenberg et al. 2015; Mandel et al. 2015). However, the advent of GW astronomy allows us to re-examine preconceptions that might be biased by previously available observations.³ In this paper, we consider the possible existence of low-mass BHs (LMBHs) with a mass range that overlaps that of normal NS. We briefly review possible formation channels of such BHs, determine the prospects for identifying them through GW and/or multi-messenger detections, and discuss the implications upon detecting such objects.

Here we list several possible formation channels to generate LMBHs with masses <3 M_⊙. First, stellar-mass BHs could come from primordial density fluctuations. In the range we are considering (∼1–3 M_⊙), existing constraints stem from microlensing measurements (Carr et al. 2016), indicating that their mass fraction compared to dark matter is f ≤ 5%. We expect that the ratio between such LMBHs and normal NS in a galaxy to be

$$\begin{eqnarray}&&f\displaystyle \frac{{M}_{\mathrm{total}}{{\rm{\Omega }}}_{\mathrm{DM}}}{{N}_{\mathrm{NS}}{M}_{\mathrm{NS}}}\sim 330\displaystyle \frac{{M}_{\mathrm{total}}}{{10}^{12}{M}_{\odot }}\displaystyle \frac{f}{5 \% }{\left(\displaystyle \frac{{N}_{\mathrm{NS}}}{{10}^{8}}\displaystyle \frac{{M}_{\mathrm{NS}}}{1.3{M}_{\odot }}\right)}^{-1}\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{DM}}}{0.845},\end{eqnarray} \tag{ 1 }$$

for a Milky Way-like galaxy (the Milky Way values for the total mass, M_total, and number of NS, N_NS, are estimated in Dehnen & Binney 1998 and Camenzind 2007), where Ω_DM is the mass faction of the dark matter in the total matter density. With this upper bound saturated, if the cross-section for dynamically capturing a NS is approximately the same as the one for a BH with similar mass—which should be the case since such capture is dominated by GW emission (East et al. 2012)—it is possible that the merger rate of an LMBH–NS binary is actually greater than the merger rate of dynamically formed NS–NS binaries. Similarly, motivated by the discussion in Capela et al. (2013) and Fuller et al. (2017), NS, white dwarfs, or even main-sequence stars could capture mini-primordial BHs (PBHs)⁴ causing most of the star's material to be accreted to produce a final BH with stellar mass. It is, however, not clear what fraction of NS could become LMBHs through this process. In fact, a bound on the mini-PBH population was obtained in Capela et al. (2013), assuming that not all NS are destroyed by PBH captures. It has also been proposed that asymmetric dark matter could accumulate in centers of NS through nucleon scattering, and eventually form a seed BH (Goldman & Nussinov 1989; Bramante & Linden 2014; Bramante et al. 2018), providing another scenario for converting an NS to a BH of similar mass.

Another possible way to produce LMBHs is through a supernovae explosion, a standard mechanism for creating compact objects. If the explosion is driven by rapidly growing instabilities (10–20 ms), BH with masses >5 M_⊙ are expected but slow ones can produce lower masses (Belczynski et al. 2012). To date, observations point to the former option but the existence of a "mass gap" is by no means a definitive fact (Kreidberg et al. 2012).

Additionally, it is also possible that the final BH produced by an NS–NS merger could become subsequently captured in a new LMBH–NS binary. Such hierarchical mergers were discussed in Fishbach et al. (2017) and Gerosa & Berti (2017) from the detection perspective, and in Antonini & Rasio (2016) as a way to estimate the rate of BH–BH mergers. Similarly, NS could gain mass through accretion and collapse to a BH falling in the mass range considered here (Nakamura 1983; Vietri & Stella 1999; MacFadyen et al. 2005; Dermer & Atoyan 2006). Because of the uncertainty in the upper threshold mass of a normal NS, LMBHs formed through NS accretion-induced collapse or collisions can not be easily distinguished from candidates in other channels through mass measurements.

We also note that the range of BH masses allowed by the above channels could also be modified by possible departures from general relativity, the existence of additional fields in nature, and/or exotic compact objects (e.g., Kaup 1968; Liebling & Palenzuela 2012; Cardoso et al. 2016; Mendes & Yang 2017) whose dynamics can yield LMBHs through collapses or mergers. Consequently, through dynamical captures, LMBH–NS systems could be produced with the masses achievable in each possible scenario. Naturally, with such a range of possible formation channels together with current uncertainties as to their likelihood, the rate of LMBH–NS binaries is unknown.⁵ Thus, future GW observations of such systems (possibly requiring EM counterparts) will be key to understanding this theoretically possible population.

We argue that the leading order tidal effects on the GW signal of an inspiraling compact object binary are, in fact, degenerate between an NS–NS and a BH–NS binary, when considering different equations of state (EOSs) of the star (and hence, setting its radius, etc.). We begin by noting that the phase of the inspiral waveform as a function of frequency can be written as (Vines et al. 2011; Sennett et al. 2017)

$$\begin{eqnarray}{\rm{\Psi }}(f) & = & \displaystyle \frac{3}{128{(\pi { \mathcal M }f)}^{5/2}}[1+{\alpha }_{1\mathrm{PN}}x\,+...\\ & & +({\alpha }_{5\mathrm{PN}}+{\alpha }_{\mathrm{tide}}){x}^{5}+...],\end{eqnarray} \tag{ 2 }$$

where x = (πMf)^2/3, ${ \mathcal M }={({m}_{1}{m}_{2})}^{3/5}/{M}^{1/5}$, and M = m₁ + m₂ is the total mass. The α_1PN, ..., α_5PN terms encode the various order post-Newtonian (PN) effects, while the leading order tidal correction is given by

$$\begin{eqnarray}&&{\alpha }_{\mathrm{tide}}=-24\left[\left(1+12\displaystyle \frac{{m}_{2}}{{m}_{1}}\right)\displaystyle \frac{{m}_{1}^{5}}{{M}^{5}}{{\rm{\Lambda }}}_{1}+(1\leftrightarrow 2)\right],\end{eqnarray} \tag{ 3 }$$

where Λ₁ and m₁ are the dimensionless tidal deformability parameter (normalized by mass to the fifth power) and mass of the first compact object (which here, we will always assume is an NS), respectively, and the second term exchanges these quantities for that of the second compact object (which we take to either be an NS or a BH). It follows from the above that as long as the NS in a BH–NS binary satisfies an EOS that has the tidal deformability of

$$\begin{eqnarray}&&{\rm{\Lambda }}{{\prime} }_{1}={{\rm{\Lambda }}}_{1}+\displaystyle \frac{{m}_{2}+12{m}_{1}}{{m}_{1}+12{m}_{2}}\displaystyle \frac{{m}_{2}^{5}}{{m}_{1}^{5}}{{\rm{\Lambda }}}_{2},\end{eqnarray} \tag{ 4 }$$

we can not distinguish its inspiral waveform from that of an NS–NS system with (Λ₁, Λ₂) for the respective stars, up to the leading PN order in tidal corrections.

We can illustrate that this leading order difference in the tidal effects of an NS–NS versus BH–NS system can be readily accommodated into uncertainties in the EOS. To give a concrete example of this, we consider a one parameter family of EOSs given by the SLy equation (Douchin & Haensel 2001) of state at low densities, and a Γ = 3 polytrope at high densities, also roughly consistent with SLy, where the parameter sets the pressure at some reference density (Read et al. 2009; we consider P = 10^34.1 to 10^35.1 dyne cm⁻² at ρ = 5 × 10¹⁴ gm cm⁻³). Then, for a given set of binary parameters, we find the mapping between equations of state in this family such that Equation (4) is satisfied (see Gagnon-Bischoff et al. 2017 for details on computing Love numbers). We show this mapping, in terms of the amount by which the NS radius in the BH–NS binary has to be larger, relative to the radius of the corresponding NS in an NS–NS binary, in Figure 1. Typically this increase is less than 2 km. Furthermore, since the tidal effects in a binary NS are dominated by the star of a larger radius, the required increase can be quite small when the smaller object is taken to be the BH. We note in passing that attributing measured tidal effects to a BH–NS binary generically implies a stiffer EOS, and so could be favored, if the softer EOS implied by an NS–NS binary is in tension with other observations (e.g., of the maximum allowed NS mass).

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In addition, this mapping assumes we know the component masses exactly, and the uncertainty just lies in the EOS. If we also fold in the uncertainty in component masses (Hannam et al. 2013; Chatziioannou et al. 2014), there is a greater degeneracy.

As discussed above, the leading order tidal effect is degenerate between LMBH–NS and NS–NS systems, as long as there is sufficient uncertainty in EOS to allow for the mapping in Equation (4). This degeneracy may be resolved in several ways: through the measurement of the next-to-leading PN order tidal effects, as they contain different mass and frequency dependence; through the difference between two types of waveforms in the late-inspiral stage, where the tidal disruption of the NS strongly influences the waveform; or through the accumulation of many NS–NS events and the consequential reduction in the uncertainty of the star's EOS (notably its radius), to break the degeneracy. Since the effect of PN corrections to tidal effects is ∼10%–20%,⁶ we focus on the latter two possibilities here.

In order to distinguish the two waveforms h_NSNS and h_BHNS, we adopt the measure of

$$\begin{eqnarray}&&{\mathrm{S/N}}_{{\rm{\Delta }}}^{2}=4\int {df}\displaystyle \frac{| {h}_{\mathrm{NSNS}}(f)-{h}_{\mathrm{BHNS}}(f){| }^{2}}{{S}_{n}(f)},\end{eqnarray} \tag{ 5 }$$

with S_n(f) being the spectra density of Advanced LIGO detector noise. If S/N_Δ ≥ 1, we shall say that the two waveforms are marginally distinguishable (Lindblom et al. 2008). This threshold has to be raised if we require higher statistical significance. The inspiral signals of LMBH–NS and NS–NS waveforms terminate at different respective characteristic frequencies. For an LMBH–NS system, the signal terminates at the cut-off frequency f_cut, which is related to the tidal disruption of an NS within a BH–NS binary. For simplicity, in the following estimate we will assume that in LMBH–NS binaries when the star is disrupted, the GW is negligible, and ignore the post-merger part of the waveform for both types of systems. Since Advanced LIGO/VIRGO's sensitivity degrades considerably at the high frequencies where contact (for NS–NS systems) or disruption (for LMBH–NS systems) occur, a rather good approximation to ${\mathrm{S/N}}_{{\rm{\Delta }}}^{2}$ can be readily obtained this way. Based on Shibata et al. (2009), f_cut is actually much higher than the frequency that the NS undergoes during mass shedding, and it depends on the mass ratio of the system and the NS EOS (in particular the NS compactness ${ \mathcal C }\equiv {M}_{\mathrm{NS}}/{R}_{\mathrm{NS}}$). Here, we adopt the fitting formula from Pannarale et al. (2015),⁷ (under a simplified assumption that the BH is nonspinning)

$$\begin{eqnarray}&&{f}_{\mathrm{cut}}=\displaystyle \frac{1}{M}\displaystyle \sum _{i,j}{f}_{{ij}}{{ \mathcal C }}^{i}{Q}^{j},\end{eqnarray} \tag{ 6 }$$

where Q = M_BH/M_NS is the mass ratio and f_ij are numerical coefficients given in Pannarale et al. (2015).

For an NS–NS system, the inspiral ends at the contact frequency, f_contact, of the two NS (Damour et al. 2012) is

$$\begin{eqnarray}&&{f}_{\mathrm{contact}}=\displaystyle \frac{1}{\pi M}{\left(\displaystyle \frac{{m}_{1}}{M}\displaystyle \frac{1}{{{ \mathcal C }}_{1}}+\displaystyle \frac{{m}_{2}}{M}\displaystyle \frac{1}{{{ \mathcal C }}_{2}}\right)}^{-3/2}.\end{eqnarray} \tag{ 7 }$$

Equation (5) can then be approximated by

$$\begin{eqnarray}&&{\mathrm{S/N}}_{{\rm{\Delta }}}^{2}\approx 4{\int }_{{f}_{\min }}^{{f}_{\max }}\,{df}\displaystyle \frac{| {h}_{3\mathrm{PN}}(f){| }^{2}}{{S}_{n}(f)},\end{eqnarray} \tag{ 8 }$$

where we have used a sky-averaged 3PN inspiral waveform (Kidder 2008) in this range. In Figure 2, we plot S/N_Δ as a function of the mass ratio of the binary and the radius of the lighter NS. We assume that the binary is at a distance of 100 Mpc, and the mass of the lighter NS is assumed to be 1.35 M_⊙. Within most of the parameter regime we consider here, S/N_Δ is bounded below ∼1.2. The regime with very low S/N_Δ corresponds to the cases with f_cut ≈ f_merger. Nevertheless, the distinguishability will be improved if the source is closer, or if we include the post-merger part of the waveform (regarding that there are still significant modeling uncertainties). For example, the S/N of the full post-merger waveform is estimated to be around 1.5 (Clark et al. 2016) for a 1.35 M_⊙ + 1.35 M_⊙ NS binary at a distance of d = 50 Mpc and with the "TM1" EOS, while the S/N contribution from the dominant mode is significantly lower (Yang et al. 2018), depending on the EOS.

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On the other hand, multiple detections of NS–NS mergers will be able to constrain the NS EOS, which could help break the degeneracy indicated by Equation (4). According to Hinderer et al. (2010), a single Advanced LIGO event from a distance, d, typically constraints Λ to an accuracy of

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Lambda }}\sim 2.9\times {10}^{3}{\left(\displaystyle \frac{M}{2.7{M}_{\odot }}\right)}^{-2.5}{\left(\displaystyle \frac{{m}_{2}}{{m}_{1}}\right)}^{0.1}\\ &&\qquad \times {\left(\displaystyle \frac{{f}_{\mathrm{end}}}{500\mathrm{Hz}}\right)}^{-2.2}\left(\displaystyle \frac{d}{100\ \mathrm{Mpc}}\right).\end{eqnarray} \tag{ 9 }$$

Assuming a similar end frequency for integration f_end ∼ 500 Hz as in Hinderer et al. (2010),⁸ and an equal mass binary with M ∼ 2.7 M_⊙, we obtain an estimate on the tidal deformability of a star: ΔΛ₁(1.35M_⊙) ∼ 2.9 × 10³. Assuming a low-spin prior, GW170817 places on upper bound Λ₁(1.4M_⊙) ≤ 800 at the 90% confidence level, and an upper bound of ≤1400 if the prior is relaxed to allow for high spin (Abbott et al. 2017c). Such constraints on Λ₁ also limit the allowed radius of NS, given the family of EOS assumed in this paper. In Figure 3, we compare this uncertainty in the tidal deformability to the amount by the tidal deformability that has to be changed in order for a BH–NS binary to have the same leading order tidal effects as a corresponding NS–NS binary with the same masses.

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With N identical detections, and under the same high-spin prior, such uncertainty scales as Δ_NΛ₁ (1.4M_⊙) ∼ 1.4 × 10³ N^−1/2. In reality, the component masses and source distances are different for different events, and it is possible that the best event of previous N detections dominates the constraint of NS EOS.

In the above discussion, we have not accounted for the effect of spin, which is particularly important for LMBHs formed through hierarchical mergers, as they are expected to have relatively high spin (a ∼ 0.7; Fishbach et al. 2017; Gerosa & Berti 2017).⁹ Such a spin magnitude will generate ∼0.15 mismatch between a nonspinning BH–NS waveform and a generic pretcessing waveform if the mass ratio is around 2 (see Figure 8 of Harry et al. 2014). The relation between distinguishable mismatch and S/N is discussed in Baird et al. (2013).

An important question is whether multi-messenger signals can help us identify an LMBH. In other words, what are the possible features of LMBH–NS systems that distinguish them from NS–NS systems, besides direct GW observation of the merger waveform? (BH–BH systems in stellar-mass ranges are not expected to produce EM signals, so the clear presence of such a signal would favor a system with at least one NS.) We argue that current limitations in our theoretical understanding of the underlying astrophysical process giving rise to EM counterparts make it difficult to clearly distinguish a binary with only one NS versus a binary with two NS. In what follows, we discuss several leading counterpart prospects (see also, e.g., Metzger & Berger 2012), but note that the era of multi-messenger astronomy will bring an increased understanding of them, as well as awareness of further ones.

Several EM counterparts have been proposed that occur within (tens of) milliseconds prior to merger, including possible emissions related to crust-cracking due to tidal effects (with associated luminosities which could reach levels of order L ≈ 10⁴⁸ erg s⁻¹; Tsang et al. 2012) and magnetosphere interactions (with associated luminosities which could reach levels of order L ≈ 10⁴³ (B/10¹⁴ G)² erg s⁻¹; McWilliams & Levin 2011; Piro 2012; Palenzuela et al. 2013; Metzger & Zivancev 2016). However, uncertainties in the EOS and magnetization level of the NS makes distinguishing such signals seem unlikely.

As the merger proceeds, the star will be disrupted by the LMBH and give rise promptly to an accreting BH—the most popular central engine model for a short gamma-ray burst (sGRB). On the other hand, binary NS can themselves power a jet, which, as discussed in Murguia-Berthier et al. (2014), can escape if the jet breaks in a sufficiently short time. Thus, an sGRB expected to take place nearly coincident with the peak in GWs would not provide a clear discerning prospect. It is important to keep in mind that the newly formed massive NS will reach very high magnetizations (B ≈ 10^15–17 G; Anderson et al. 2008; Kiuchi et al. 2017). If it collapses, large amounts of energy (L ≈ 10⁴⁹ (B/10¹⁵ G)² erg s⁻¹) could be released rather isotropically (Lehner et al. 2012) setting the stage for possible less -intense high-energy (gamma, X-ray) emissions, which do not necessarily require the observer to be specially aligned. Interestingly, the (short) GRB170817A associated with the GW event GW170817 is less luminous than typical sGRBs (Abbott et al. 2017b). This fact could be explained by the viewing angle but also through different burst mechanisms/models.

The collapse of a hypermassive NS to a BH, however, can take place in a significantly delayed fashion, and the resulting accreting BH state would fit naturally in the "canonical picture" of an accreting BH launching the jet, especially if the jet is Poynting flux dominated. It is tempting to speculate that, in such a paradigm, a significantly delayed sGRB would favor an NS–NS system (as timescales for launching a jet could be around ≈100 ms; e.g., Paschalidis et al. 2015; Ruiz et al. 2016); however, the time required to set the right topology and strength of the magnetic fields required for launching a jet¹⁰ (e.g., McKinney & Blandford 2009) introduces a delay that can potentially blur the differences between a BH-accretion scenario set up promptly after the merger (through BH–NS or NS–NS mergers) or delayed (via an NS–NS merger that produces a long-lived remnant). Furthermore, current uncertainties in key effects like effective viscosity of the forming disk, magnetization levels of the star, accretion characteristics, as well as the sGRB model itself (e.g., Narayan et al. 2001; Piran 2004) currently stand in the way of clearly distinguishing the progenitors based on such a delay.

Another way in which an LMBH–NS may potentially differ from an NS–NS binary is in the amount (and neutron richness) of NS material that is unbound during the merger. This ejecta will undergo r-process nucleosynthesis, building up heavy elements that decay, powering a so-called kilonova/macronova (Li & Paczyński 1998; Kulkarni 2005; Metzger 2017). In a rather spectacular fashion, such observations have been identified as counterparts to GW170817 (e.g., Smartt et al. 2017; Drout et al. 2017). The greater the mass, M_ej, of material that is ejected, the brighter the transient, and the longer the timescale on which it will peak. In terms of the velocity of the ejecta, such an EM transient will peak on timescales of t_peak ∼ 0.3 (M_ej/0.01 M_⊙) (v/0.2c) days, in the ultraviolet/optical to near-infrared frequencies, with peak luminosities of L ∼ 1.6 × 10⁴¹ (M_ej/0.01 M_⊙) (v/0.2c) erg s⁻¹ (Barnes & Kasen 2013). On longer timescales of ∼2.6 (E_ej/10⁵⁰ erg)^1/3 (v/0.2c)^−5/3 years (where E_ej is the kinetic energy of the ejecta), there may also be a radio transient associated with the collision of this material with the interstellar medium (Nakar & Piran 2011; Hallinan et al. 2017).

Simulations of NS–NS mergers typically find ejecta of ≲0.01 M_⊙, with the most ejecta coming from mergers with soft EOSs. With unequal mass ratios (Lehner et al. 2016; Sekiguchi et al. 2016), the ejected material is highly neutron rich, and the amount is on the higher end across EOSs. Higher mass ratio simulations of BH–NS mergers find significant ejecta when the BH has non-negligible spin aligned with the orbital angular momentum and/or the NS has a larger radius (Foucart et al. 2014; Kawaguchi et al. 2016), in which case the amount of ejecta can be up to ∼0.1M_⊙. Hence, an unusually bright ejecta-powered transient would seem to favor an LMBH–NS merger, though a transient consistent with ≲0.01 M_⊙ ejecta could be attributed to either. We note, however, BH–NS mergers with nearly equal masses are not well studied (see Etienne et al. 2008; Shibata et al. 2009 for some early studies), and further scrutiny will be required to delineate their properties across parameter space.

An additional caveat to the above discussion is that non-negligible NS spin, on the order of a ∼ 0.1, has also been shown to enhance the amount of ejecta to the level of a few percent of a solar mass (this falls within the allowed range for the spin along the orbital angular momentum estimated in GW170817, notice, however, that the component orthogonal to it is not constrained; East et al. 2016a, 2016b; Dietrich et al. 2017). Orbital eccentricity at the merger can also significantly increase the amount of ejecta (East & Pretorius 2012; Radice et al. 2016), though presumably this will be well constrained by the GW signal.

It is conceivable that LMBHs may be produced through PBH capture, supernovae, NS–NS mergers, the collapse of exotic compact objects, or other such phenomena. Therefore, their existence is tightly connected to the astrophysical population/distribution of these seeding objects and the underlying fundamental physics that governs them. Because of the uncertainty in the NS EOS and the degeneracy in the tidal effect of LMBH–NS and NS–NS systems in the inspiral stage, it appears challenging for Advanced LIGO to definitively identify such objects. The ability to differentiate between the two can be improved by better understanding their respective post-merger waveforms, as well as achieving better GW detector sensitivity (Miao et al. 2014, 2017) and accumulating statistics from many detections (Bose et al. 2018; Yang et al. 2018, 2017). The similarities in the potential EM counterparts to the two systems, within theoretical uncertainties, also makes distinguishing them with multi-messenger astronomy challenging, and calls for a better understanding of the underlying astrophysical processes. Such a task of refining models and honing in on the relevant parameter space will benefit tremendously from a dialog with observations as they take place.

If such an LMBH were discovered, the problem of identifying its formation channel would naturally arise. One possible indicator could be the spin of the LMBH—one can compute its prior distributions in each formation channel and compare them with the posterior distributions of each detection. The mass and redshift information of these objects may also help distinguish their origins. Excitingly, third-generation GW detectors will be capable of detecting non-vacuum compact binary mergers up to z ∼ 6 (Abbott et al. 2017a). If LMBHs are present, even in a small portion of such mergers, they will guide fruitful discoveries in physics and astronomy.

We thank Jérémie Gagnon-Bischoff and Nestor Ortiz for sharing their code for computing tidal Love numbers. This research was supported in part by NSERC, CIFAR, and by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada, and by the Province of Ontario through the Ministry of Research and Innovation.