Revealing the formation of stellar-mass black hole binaries: The need for deci-Hertz gravitational wave observatories

TL;DR

This paper emphasizes the importance of deci-Hertz gravitational wave observatories like DECIGO or Tian Qin for understanding the formation of stellar-mass black hole binaries, especially in dense stellar environments, by complementing LIGO and LISA observations.

Contribution

It demonstrates that deci-Hertz observatories can improve detection prospects and help distinguish formation channels of black hole binaries, especially when LISA cannot detect high-eccentricity mergers.

Findings

High eccentricities may prevent LISA detection of some mergers.

Non-detection by LISA sets a lower eccentricity limit of 0.005.

Deci-Hertz observatories enhance joint detection capabilities.

Abstract

The formation of compact stellar-mass binaries is a difficult, but interesting problem in astrophysics. There are two main formation channels: In the field via binary star evolution, or in dense stellar systems via dynamical interactions. The Laser Interferometer Gravitational-Wave Observatory (LIGO) has detected black hole binaries (BHBs) via their gravitational radiation. These detections provide us with information about the physical parameters of the system. It has been claimed that when the Laser Interferometer Space Antenna (LISA) is operating, the joint observation of these binaries with LIGO will allow us to derive the channels that lead to their formation. However, we show that for BHBs in dense stellar systems dynamical interactions could lead to high eccentricities such that a fraction of the relativistic mergers are not audible to LISA. A non-detection by LISA puts a lower limit of about $0.005$ on the eccentricity of a BHB entering the LIGO band. On the other hand, a deci-Hertz observatory, like DECIGO or Tian Qin, would significantly enhance the chances of a joint detection, and shed light on the formation channels of these binaries.

Equations

$${\dot{a}}_{\mathrm{gw}}=-\frac{8c{R}_S^{3}q(1+q)}{5a^3(1-e^2{)}^{7/2}}\left(1+\frac{73}{24}{e}^2+\frac{37}{96}{e}^4\right),$$

Full text

Introduction.–The first LIGO events, GW150914 and GW151226 (Abbott et al., 2016a, b), are consistent with mergers of General-Relativity black holes (BHs). Data analysis reveal that the orbits started at a semi-major axis of $a\sim 10$ Schwarzschild radii ($R_{S}$) with an eccentricity of $e<0.1$. The BH masses are about $M_{1}\simeq 36$ and $M_{2}\simeq 29~{}M_{\odot}$ for GW150914 and $M_{1}\simeq 14$ and $M_{2}\simeq 7.5~{}M_{\odot}$ for GW151226. The detections can be used to infer new, more realistic event rates, of about $9-240~{}{\rm Gpc^{-3}~{}yr^{-1}}$ (Abbott et al., 2016c). This rate agrees with two formation channels: (i) evolution of a binary of two stars in the field of the host galaxy, where stellar densities are very low (e.g Belczynski et al. (2016)) or (ii) via exchange of energy and angular momentum in dense stellar systems, where the densities are high enough for stellar close encounters to be common (e.g. Rodriguez et al. (2016)).

LIGO and other ground-based gravitational wave (GW) observatories, such as Virgo, are, however, blind with regarding the formation channels of BH binaries (BHBs). Both channels predict populations in the $10-10^{3}~{}{\rm Hz}$ detector band with similar features, i.e. masses larger than the nominal $10\,M_{\odot}$, a mass ratio ($q\equiv M_{2}/M_{1}$) of about $1$, low spin, and nearly circular orbits Abbott et al. (2016d); Amaro-Seoane and Chen (2016).

It has been suggested that a joint detection with a space-borne observatory such as LISA Amaro-Seoane et al. (2012, 2013, 2017) could allow us to study different moments in the evolution of BHBs on their way to coalescence: LISA can detect BHBs when the BHs are still $10^{2}-10^{3}~{}R_{S}$ apart, years to weeks before they enter the LIGO/Virgo band Miller (2002); Amaro-Seoane and Santamaría (2010); Kocsis and Levin (2012); Kocsis (2013); Sesana (2016); Seto (2016); Vitale (2016). At such a separation, the orbital eccentricity bears the imprint of the formation channel because (i) BHBs in dense stellar systems form on systematically more eccentric orbits and (ii) the GW radiation at this stage is too weak to circularize the orbits Miller (2002); Wen (2003); Gültekin et al. (2004, 2006); O’Leary et al. (2006). Therefore, circular binaries typically form in the field, while eccentric ones through the dynamical channel. Recent studies further predict that those BHBs with an eccentricity of $e>0.01$ in the LISA band preferentially originate from the dynamical channel Kyutoku and Seto (2016); Nishizawa et al. (2016, 2017); Breivik et al. (2016); Seto (2016).

In this letter we prove that eccentric BHBs originating in dense stellar environments have a large chance to elude the LISA band.

Inaudible black hole binaries–Non-circular BHBs have two distinct properties. (i) Eccentricity damps the characteristic amplitude ($h_{c}$) of each GW harmonic, as compared to a circular BHB. In Figure 1 we depict two sources similar to GW150914 but originating from two distinct channels, i.e. with two different initial eccentricities. In the low-eccentricity case, the $n=2$ harmonic predominates and it is strong enough to be jointly detected by LISA and LIGO/Virgo. In the (very) eccentric case, however, the amplitudes of the harmonics are orders of magnitude below the noise level of LISA, so that a joint detection is ruled out. When the eccentricity has been significantly damped, about one hour before the merger, the dominant harmonic starts to converge to the $n=2$ one, and later, upon entering the LIGO band, becomes indistinguishable from that in the circular case. Therefore, the imprint about the formation channel is lost.

(ii) Increasing the eccentricity shifts the peak of the relative power of the GW harmonics towards higher frequencies (see Fig. 3 of Peters and Mathews (1963)). Hence, more eccentric orbits emit their maximum power at frequencies farther away from LISA. More precisely, when $e=0$, all the GW power is radiated through the $n=2$ harmonic, so that the GWs have a single frequency of $2/P$, where $P=2\pi(GM_{12}/a^{3})^{-1/2}$ is the orbital period and $M_{12}=M_{1}+M_{2}$. On the other hand, when $e\simeq 1$, the $n=2.16(1-e)^{-3/2}$ harmonic becomes predominant Farmer and Phinney (2003), so most GW power is radiated at a frequency of $f_{\rm peak}=2.16(1-e)^{-3/2}P^{-1}$.

In Figure 2 we display the $a-(1-e)$ plane for a BHB. The boundaries of the stripes have been estimated by looking at the minimum and maximum frequencies audible by the detectors, $f_{1}$ and $f_{2}$, and letting $f_{1}<\,f_{\rm peak}<\,f_{2}$, with $f_{\rm peak}$ defined before. If a BHB is evolving only due to GW emission, it will evolve parallel to the green lines. These track are parallel to the stripes because as long as $e\simeq 1$, the pericenter distance, $r_{p}=a\,(1-e)$, is almost constant during the evolution Peters (1964), and a constant $r_{p}$ corresponds to a constant $f_{\rm peak}$. Because of this parallelism, a BHB cannot evolve into the band of a GW detector if it initially lies below the detector stripe.

Hence, we can see that some binaries will fully miss the LISA/TJ range. A good example is the eccentric BHB we chose for Figure 1. A detector operating at higher frequencies, such as TQ or DECIGO, can however cover the relevant part of the phase-space, so that a joint search is possible. These detectors could alert LIGO/Virgo decades to hours before an event is triggered, as one can read from the isochrones of Figure 2.

*Dense stellar environments.–*BHBs such as the one we have used for our last example completely miss the LISA/TJ band. Eccentric binaries typically originate from dense stellar systems such as globular clusters (GCs) and nuclear star clusters (NSC), as shown by a number of authors in a number of publications (Miller, 2002; Wen, 2003; Gültekin et al., 2004, 2006; O’Leary et al., 2006; Nishizawa et al., 2016, 2017; Breivik et al., 2016). In these systems, BHs diffuse towards the center via a process called mass segregation (see e.g. Peebles, 1972; Bahcall and Wolf, 1976; Amaro-Seoane et al., 2004; Freitag et al., 2006; Alexander and Hopman, 2009; Preto and Amaro-Seoane, 2010). To model it, we adopt a Plummer model Plummer (1911), and we assume that the mean stellar density is $\rho_{*}=5\times 10^{5}~{}M_{\odot}~{}{\rm pc^{-3}}$ and the one-dimensional velocity dispersion is $\sigma_{*}=15~{}{\rm km~{}s^{-1}}$. These values correspond to a typical GC with a final mass of $M_{\rm GC}\approx 10^{5}~{}M_{\odot}$ and a half-mass radius of $R_{h}\approx 0.5$ pc. We note, however, that the main conclusions derived in this work do not significantly change for a NSC.

The two driving and competing mechanisms in the evolution of any BHB in the center of the cluster are (i) interaction with other stars, “interlopers”, which come in at a rate of $\Gamma\sim 2\pi G\rho_{*}a(M_{12}/M_{*})/\sigma_{*}$, with $M_{*}=10~{}M_{\odot}$ the mean mass of the interlopers because the cluster has gone through mass segregation, and (ii) gravitational radiation, which shrinks the orbital semi-major axis at a rate of

[TABLE]

Peters (1964). We can readily separate the phase-space in two distinct regimes according to these two competing processes by equating their associated timescales: $t_{\rm int}:=1/\Gamma$ and $t_{\rm gw}:=(1/4)\,\left|a/\dot{a}_{\rm gw}\right|$, which defines the threshold shown as the thick, black line in Figure 3. The reason for the $1/4$ factor is given in Peters (1964). Below the curve, BHBs will evolve due to GW emission. Above it, close encounters with interlopers are the main driving mechanism, so that BHBs can be scattered in both directions in angular momentum in a random-walk fashion. The scattering in energy is less significant but also present (see Alexander (2017) and discussion in Amaro-Seoane (2012)).

Possible ways of forming relativistic BHBs.– Different mechanisms have been proposed in the literature to form a BHB which eventually might end up emitting detectable GWs.

(1) Primordial binaries: In stellar dynamics this term refers to binaries already present in the cluster which form via stellar evolution. Population synthesis models predict that these binaries populate the area of phase-space displayed as the grey thick-dashed box of Figure 3 (see e.g. Belczynski et al. (2004)). We note that only a small fraction of them are in the LISA/TJ band.

(2) Dynamics: (2.1) Close encounters of multiple single, i.e. initially not bound, objects also form BHBs (see e.g. Kulkarni et al. (1993); Sigurdsson and Hernquist (1993); Miller and Hamilton (2002); Ivanova et al. (2005)). Their formation follows a thermal distribution in $e$ (e.g. Antognini and Thompson (2016)), like primordial binaries, but the distribution of $a$ is better constrained: When the binding energy of the binary, $E_{b}=GM_{1}M_{2}/(2a)$ becomes smaller than the mean kinetic energy of the interlopers, $E_{*}=3M_{*}\sigma_{*}^{2}/2$, the binary ionizes Binney and Tremaine (2008). The threshold condition $E_{b}=E_{*}$ can be expressed in terms of a “hard radius”, $a_{h}=GM_{1}M_{2}/(3M_{*}\sigma_{*}^{2})$. These “hard” binaries heat up the system, meaning that they deliver energy to the rest of the stars interacting with them: Binaries with $a<a_{h}$ impart on average an energy of $\Delta E\simeq kG\mu M_{*}/a$ to each interloper, where $\mu$ is the reduced mass of the binary and $k$ is about $0.4$ when $M_{1}\simeq M_{2}\simeq M_{*}$ (Heggie, 1975). The interloper hence is re-ejected into the stellar system with a higher velocity because of the extra energy, $v\sim\left(3\sigma_{*}^{2}+2kG\mu/a\right)^{1/2}$, and the center-of-mass of the BHB recoils at a velocity of $v_{b}\sim M_{*}v/(M_{1}+M_{2})$. Occasionally, the BHB will leave the system if this velocity exceeds the escape velocity of the GC, $v_{\rm esc}=\sqrt{2.6GM_{\rm GC}/R_{h}}$ Rodriguez et al. (2016). The threshold for this to happen is defined by the condition $v_{b}=v_{\rm esc}$, i.e. the binary must have a semi-major axis smaller than the “ejection radius”, $a_{\rm ej}$. Therefore, all of these BHBs are confined in $a_{h}<a<a_{\rm ej}$ of Figure 3. Because of their thermal distribution, we have that $95\%$ of them have $e<0.975$. Therefore, they populate an even smaller area than those primordial binaries.

(2.2) Binary-single interactions: Initially we have a hard BHB which interacts with a single object in a chaotic way. During the interaction the interloper might excite the eccentricity of the inner binary to such high values that the binary is on an almost head-on-collision orbit, to soon merge and emit a detectable burst of GWs Gültekin et al. (2006); Samsing et al. (2014); Amaro-Seoane and Chen (2016). This happens only if $t_{\rm gw}$ is shorter than the period of the captured interloper $P_{\rm int}$. The event rate for BHBs has not been calculated for this scenario but earlier calculations for neutron-star binaries find it to be $1~{}{\rm Gpc^{-3}~{}yr^{-1}}$ Samsing et al. (2014). We derive now the eccentricities of these BHBs: Suppose the semi-major axis of a BHB changes from $a$ (with, of course, $a_{\rm ej}<\,a<\,a_{h}$) to $a^{\prime}$, and $e$ to $e^{\prime}$ during the three-body interaction, and the final orbit of the interloper around the center-of-mass of the BHB has a semi-major axis of $a_{\rm int}$. Energy conservation results in the following relations, $a^{\prime}>a$ and $a_{\rm int}\simeq 2a/(1-a/a^{\prime})$ (see Samsing et al. (2014)), where we neglect the initial energy of the interloper because the BHB is assumed to be hard. Then using a conservative criterion for a successful inspiral, $t_{\rm gw}(a^{\prime},\,e^{\prime})=P_{\rm int}(a_{\rm int})$, we derive $e^{\prime}$ for the BHB, which allows us to confine the range of eccentricities as the dashed, blue curve of Figure 3.

(2.3) Hierarchical triple: This is similar to the previous configuration, but now we only consider $1<a^{\prime}/a<1.5$, because this requires that $a_{\rm int}>6\,a$, in which case the configuration is stable Mardling and Aarseth (2001). This leads to a secular evolution of the orbital eccentricity of the inner BHB which is known as the Lidov-Kozai resonance (see Lidov (1962); Kozai (1962) and also Miller (2002); Wen (2003); O’Leary et al. (2006); Naoz et al. (2013); Antognini et al. (2014); Liu et al. (2015)). The inner BHB will decouple via GW emission and merge at a critical eccentricity, and the merger rate has been estimated to be $0.3-6~{}{\rm Gpc^{-3}~{}yr^{-1}}$ Antonini et al. (2014, 2016); Kimpson et al. (2016); Silsbee and Tremaine (2017). We follow the scheme of Antonini et al. (2014) of isolated hierarchical triples but impose four additional requirements which are fundamental for a realistic estimation of the threshold eccentricity in our work: (a) The BHB has $a_{\rm ej}<\,a<\,a_{h}$. (b) The third body orbiting the BHB has a mass of $M_{\rm int}=10~{}M_{\odot}$ because of mass segregation, and an eccentricity of $e_{\rm int}=2/3$, which corresponds to the mean of a thermal distribution Antognini and Thompson (2016). (c) The outer binary, i.e. the third object and the inner BHB, is also hard, so that $a_{\rm int}<GM_{12}/(3\sigma_{*}^{2})$. (d) The pericenter distance of the outer binary, $a_{\rm int}(1-\,e_{\rm int})$ should meet the criterion for a stable triple (Eq. 90 in Ref. Mardling and Aarseth (2001)). These conditions delimit the range of eccentricities as shown by the dashed, orange lines in Figure 3.

(3) Gravitational braking: There is a small probability that two single BHs come to such a close distance that GW radiation dissipates a significant amount of the orbital energy, leaving the two BHs gravitationally bound Turner (1977); Quinlan and Shapiro (1987); Kocsis et al. (2006); O’Leary et al. (2009); Lee et al. (2010); Hong and Lee (2015). For GCs, and using optimistic assumptions, these binaries contribute an event rate of $0.06-20~{}{\rm Gpc^{-3}~{}yr^{-1}}$ in the LIGO band Lee et al. (2010); Antonini et al. (2016), while in NSCs it has been estimated to range between $0.005-0.02~{}{\rm Gpc^{-3}~{}yr^{-1}}$ Tsang (2013). The boundaries in Figure 3 for BHBs formed via this mechanism can be calculated using the formulae of O’Leary et al. (2009). For that, we choose an initial relative velocity $v$ in the range $\sigma_{*}<v<3\sigma_{*}$ and an initial impact parameter $b$ in the range $0.3b_{\rm max}<b<0.99b_{\rm max}$ to account for the majority of the encounters, because the encounter probability is proportional to $b^{2}$, and $b_{\rm max}$ is the maximum impact parameter that leads to a bound binary. The first LIGO detections, had they been originated via this mechanism, should originate from the red area above the green line.

*Discussions and conclusions.–*A joint detection of BHBs with LIGO/Virgo and LISA/TJ would be desirable because of the science payback. In this paper we show that the actual number of BHBs to be coincidentally detected is very uncertain. As Figure 2 shows, LISA/TJ is already deaf to mildly eccentric BHBs: For example, a BHB at milli-Hertz orbital frequencies starting at $a\sim 10^{-3}$ AU and $0.7\lesssim e\lesssim 0.9$ will also be missed by LISA/TJ, but later be detectable by LIGO/Virgo.

BHBs can form via the five mechanisms which we discussed in the list of possible formations. This allows us to pinpoint the regions in phase-space which produce BHBs that eventually will merge via gravitational radiation. The total area of these five regions is a small subset of phase-space. It is an error to assume that all binaries born in this subset are jointly detectable by LIGO/Virgo and LISA/TJ.

Only a subset of that subset of phase-space will lead to successful joint detections. This sub-subset depends on the masses of the BHBs. We can see this in Figures 3 and 4. While in the first figure the hierarchical triple gets into the LISA/TJ band, it does not in the second one.

On the other hand, up to 95% of primordial and dynamical binaries (1 and 2.1 in the list of possible formations) are produced in the box delimited by grey dashed lines. In that box, and in principle, the BHBs can lead to sources jointly detectable by LIGO/Virgo and LISA/TJ. However, exceptions might occur if a scatter results in a BHB jumping towards high eccentricities. This probability has not been fully addressed. It requires dedicated numerical scattering experiments with relativistic corrections (e.g. Amaro-Seoane and Santamaría (2010)), as well as a proper star-cluster model to screen out BHBs that can decouple from the stellar dynamics (e.g. our model as presented in Figures 3 and 4).

We have shown that mergers in GCs produced by the mechanisms (2.2), (2.3), and (3) are inaudible to LISA. The event rates corresponding to these mergers have been largely discussed in the literature, but are uncertain, due to questionable parameters, such as the cosmic density of GCs and the number of BHs in them. Nevertheless, it has been estimated that the rate could be as large as $20~{}{\rm Gpc^{-3}~{}yr^{-1}}$ Lee et al. (2010), while the current LIGO detections infer a total event rate of $9-240~{}{\rm Gpc^{-3}~{}yr^{-1}}$. Moreover, these mergers could also originate in NSCs, (Kocsis et al., 2006; Miller and Lauburg, 2009; O’Leary et al., 2009; Tsang, 2013; Hong and Lee, 2015; Antonini and Perets, 2012; Addison et al., 2015; Antonini and Rasio, 2016), and the event rates there are higher, up to $10^{2}~{}{\rm Gpc^{-3}~{}yr^{-1}}$ (VanLandingham et al., 2016).

Therefore, future multi-band GW astronomy should prepare for LIGO/Virgo BHBs that do not have LISA/TJ counterparts. A non-detection by LISA/TJ is also useful in constraining astrophysics: It puts a lower limit on the eccentricities of the LIGO/Virgo sources, which according to Figures 3 and 4 is about $0.005$.

A deci-Hz detector, by covering the gap in frequencies between LISA/TJ and LIGO/Virgo, would drastically enhance the number of jointly detectable binaries.

Acknowledgement.–This work is supported partly by the Strategic Priority Research Program “Multi-wavelength gravitational wave universe” of the Chinese Academy of Sciences (No. XDB23040100) and by the CAS President’s International Fellowship Initiative. PAS acknowledges support from the Ramón y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain. We thank Bence Kocsis and Fukun Liu for many fruitful discussions, and Eric Peng for a thorough reading of our manuscript.