Single-single gravitational-wave captures in globular clusters: Eccentric deci-Hertz sources observable by DECIGO and Tian-Qin
Johan Samsing Daniel J. D'Orazio Kyle Kremer Carl L. Rodriguez Abbas Askar
astro-ph.HE










^1Niels Bohr International Academy, The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark.

^2Department of Astronomy, Harvard University, 60 Garden Street Cambridge, MA 01238, USA. 

^3Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA. 

^4Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), 2145 Sheridan Road,
Evanston, IL 60208, USA. 

^5Lund Observatory, Department of Astronomy, and Theoretical Physics, Lund University, Box 43, SE-221 00 Lund, Sweden.





We study the formation rate of binary black hole mergers formed through gravitational-wave emission between unbound, single black holes in globular clusters. While the formation of these binaries in very dense systems such as galactic nuclei has been well studied, we show here that this process can operate in lower-density stellar systems as well, forming binaries at a rate similar to other proposed pathways for creating eccentric mergers. Recent advances in post-Newtonian cluster dynamics indicate that a large fraction of dynamically-assembled binary black holes merge inside their host clusters during weak and strong binary-single and binary-binary interactions, and that these systems may retain measurable eccentricities as they travel through the LIGO and LISA sensitivity bands.  Using an analytic approach to modeling binary black holes from globular clusters, we show that the formation of merging binaries from previously unbound black holes can operate at a similar rate to mergers forming during strong binary encounters, and that these binaries inhabit a unique region of the gravitational-wave frequency space which can be identified by proposed
deci-Hertz space-based detectors.




Single-single gravitational-wave captures in globular clusters:

Eccentric deci-Hertz sources observable by DECIGO and Tian-Qin
    Johan Samsing^1, Daniel J. D'Orazio^2, Kyle Kremer^3,4, Carl L. Rodriguez^2, Abbas Askar^5 
    Received 15 May 2019 / Accepted 18 July 2019
===============================================================================================================================





§ INTRODUCTION


Several binary black hole (BBH) mergers have now been observed  by
LIGO (the Laser Interferometer Gravitational Wave Observatory) and VIRGO, through their emission of gravitational waves (GWs)
<cit.>.
However, their astrophysical origin is still unknown, and the observed variety in both BH masses and
spins <cit.>, in addition to the observed merger of binary neutron stars (NSs) <cit.>,
indicate that several formation mechanisms might be operating. Some of the recently proposed include:
field binaries <cit.>,
stellar clusters <cit.>,
active galactic nuclei (AGN) discs <cit.>,
galactic nuclei (GN) <cit.>,
very massive stellar mergers <cit.>,
and single-single GW captures of primordial black holes <cit.>.

From the GW signal of individual BBH merger events, one can measure the (redshifted) mass of the BHs, their
spins <cit.>,
the BBH orbital eccentricity <cit.>, and even
Doppler effects related to a possible movement of the BBH's center of mass (COM) <cit.>.
The question is; how can this information be used to distinguish the proposed astrophysical merger channels?
Regarding BH spins, these are expected to be
isotropically distributed for dynamically assembled BBH mergers, such as those forming in globular clusters (GCs), in contrast to those
forming in isolation in the field <cit.>.
In terms of eccentricity, a significant fraction of BBHs formed dynamically in clusters are expected to lead to a
unique population of GW mergers that have measurable orbital eccentricities in bands from LISA (the Laser Interferometer Space Antenna) <cit.> to LIGO <cit.>, whereas field binaries are most likely to have fully circularized once observable.
Although eccentric sources might form in other ways, e.g., through Lidov-Kozai oscillations <cit.>, quadruple systems <cit.>, and single-single GW
captures <cit.>, eccentric BBH mergers
forming in clusters are not only a natural outcome when BHs are present, but can also accurately be modeled using both simple
analytical <cit.> and numerical techniques <cit.>.
This makes BBH orbital eccentricity a very promising parameter to use for constraining the cluster channel and its astrophysical properties.
In addition to this, it has also been suggested that BBH populations in dense clusters can be probed through their interaction
with nearby stars, e.g., through tidal interactions, which could result in electromagnetic
observables <cit.>.

In this paper we continue our studies on how BBH mergers form in stellar clusters, and in particular GCs; systems which
recently have gained significant attention.
One reason is that these systems are relatively easy to model as they stay more or less isolated for almost their entire life,
during which they evolve through clean physical processes involving Newtonian and Post-Newtonian <cit.>
N-body dynamics. Recent developments in modeling this formation channel include
how BBHs can form through strong binary-single interactions <cit.>, weak
binary-single interactions <cit.>, strong binary-binary interactions <cit.>, and
secular interactions <cit.>. This recent work especially indicates that
∼ 50% of all BBHs assembled in GCs are likely to merge inside their cluster with a variety
of GW peak frequencies and eccentricities <cit.>, which are tightly connected to the properties of their host cluster, whereas
earlier prescriptions only resolved the dynamically ejected population <cit.>.
These recent advances have major observational implications when considering the future of GW astrophysics
where planned observatories such as the Einstein Telescope (ET) <cit.>, and the Cosmic Explorer (CE) <cit.>,
will be able to map out every possible BBH merger in the entire visible Universe.

As one of the last pieces on how BBH mergers might form in GCs through few-body interactions (without the inclusion of a central massive BH),
we here describe the formation of single-single GW capture mergers <cit.>.
Single-single GW captures have in the literature mainly been associated with GN hosting massive central BHs <cit.>; however,
recent work does indicate that single-single GW captures could operate at a
non-negligible rate in GCs partly due to their relative low velocity dispersion <cit.>. In our present paper we build upon these earlier studies,
and by the use of analytical arguments we not only prove that single-single GW captures in GCs lead to significant rates, but we also derive how single-single GW captures
form compared to BBH mergers forming through the dominating three-body interaction channel inside the cluster. Especially, we find that
the rate of single-single GW captures relative to the rate of GW mergers forming during binary-single interactions <cit.>, a
population we loosely will refer to as binary-single GW mergers, does not strongly depend on neither the central velocity dispersion nor the density of the cluster,
but mainly on the binary fraction and the shape of the BH population density profile. In most GCs the binary fraction is at the percent level <cit.>,
for which we find that the rate of single-single GW captures leading to BBH mergers, should be similar to the rate of binary-single GW mergers.

The single-single GW capture mergers distribute differently across GW frequency and orbital eccentricity space
compared to the other before mentioned few-body merger channels. This has been noticed before <cit.>; however, in our
presented paper we derive for the first time the correct normalization of the single-single GW capture mergers
compared to the mergers from binary-single interactions. This allow us to put forward a picture that unifies how BBHs distribute
as a function of the binary fraction and the density profile of the single BH population. We show how the observable GW frequency and orbital
eccentricity distributions change with these properties, which make us propose that single-single GW captures offer unique possibilities to
probe the inner properties of BH subsystems <cit.>.

Finally, from a combination of analytical and numerical techniques we illustrate how single-single GW capture mergers form right where the future planned
GW observatories DECIGO <cit.> and Tian Qin <cit.> are sensitive.
As we clearly demonstrate in this paper, and also previously argued in <cit.>,
deci-Hertz observatories fill out a unique gap in GW frequency space that covers the range where the majority of dynamically assembled BBH mergers form, including those from
single-single, binary-single and binary-binary interactions. Such observations will undoubtedly provide unique information about the distribution of
BHs in dense stellar systems. This greatly adds
to the astrophysical motivation to why such detectors should be built.

The paper is organized as follows. We start in Section <ref>
by calculating the cross sections for close two-body encounters originating from single-single and binary-single interactions, respectively.
In Section <ref> we convert our cross section expressions to formation rates, and estimate the rate of single-single GW capture mergers
compared to binary-single GW mergers for two different cluster profiles.
We then use simple Monte Carlo (MC) techniques in Section <ref> to derive GW peak frequency and
eccentricity distributions for our considered dynamical BBH merger channels, and comment on the observational prospects.
Our study is concluded in Section <ref>.





§ CROSS SECTIONS


In this section we derive close encounter and GW inspiral merger
cross sections for single-single and binary-single interactions, respectively.
These expressions are then used in Section <ref> to derive
absolute and relative merger rates relevant for a system like a GC.
In all our calculations we assume the interacting BHs have the same mass
m, which is a reasonable approximation for interactions happening in GCs due to mass segregation
and frequent exchange interactions <cit.>. An illustration showing the interaction- and
GW merger channels we consider in this paper is presented in Fig. <ref>. 

In terms of notations, throughout the paper `G' denotes Newton's constant, `c'
is the speed of light, and `log' denotes the logarithm to the base 10.











 §.§ Single-Single Interactions


We start by considering the cross section for two single BHs to undergo an encounter with peri-center distance 
less than some distance R_ ss. We denote this cross section by σ_ ss^<R, where `ss' is short for `single-single'.
In the gravitational focusing limit this cross section is given by <cit.>,

    σ_ ss^<R≈ 2 π G 2mR_ ss/v^2,

where m is the BH mass, and v is the velocity dispersion of the BH subsystem (we do not distinguish between relative velocity and velocity dispersion in this paper).

For a single-single interaction to also result in a GW capture merger
the energy radiated at the first peri-center passage from GW emission,
ΔE_ GW≈ (85π/12)G^7/2c^-5m^9/2^-7/2 (see <cit.>),
must be greater than the initial energy between the two singles, E_ ss≈ (1/2) μ v^2, where μ is here the
reduced mass <cit.>.
The maximum peri-center distance from which a single-single GW capture can happen, denoted in this paper by ℛ_ ss,
is the  that satisfies ΔE_ GW = E_ ss, from which one finds,

    ℛ_ ss = (85 π/24√(2))^2/7×ℛ_ m(c^2/v^2)^2/7,

where ℛ_ m is the Schwarzschild radius of a BH with mass m. If one substitutes
R_ ss in Eq. (<ref>) with ℛ_ ss from the above Eq. (<ref>)
one gets the classical single-single GW capture cross section <cit.>, that we here denote by σ_ ss^<ℛ.
The value of ℛ_ ss given above represents the theoretical upper limit for GW capture; however, the
astrophysical upper limit is slightly smaller, and depends on local properties such as the BH density profile and
velocity dispersion. The primary reason is that the definition of
ℛ_ ss given by Eq. (<ref>) only ensures that the two single objects become bound after their first passage,
but not that this newly formed binary actually undergoes a GW inspiral merger before it is interrupted by a later incoming object.
We will take this correction into account later in our numerical experiments (see Eq. (<ref>)), from which we find that this correction for a typical GC is not important.
We therefore keep ℛ_ ss as shown in Eq. (<ref>) throughout our analytical sections. 





 §.§ Binary-Single Interactions


We now describe the derivation of the cross section for a binary-single interaction to 
result in two of the three objects to undergo an encounter with peri-center distance 
less than some characteristic distance R_ bs.
We denote this cross section σ_ bs^<R,
where `bs' is short for `binary-single'. For this, we start by expressing σ_ bs^<R as the following product,

    σ_ bs^<R = σ_ bs× P_ bs^<R,

where σ_ bs is the cross section for a binary to undergo a strong interaction with a single, and P_ bs^<R denotes the
probability for a strong binary-single interaction to result in two of the three objects to undergo an encounter with < R_ bs.
The cross section σ_ bs is in the gravitational focusing limit given by <cit.>,

    σ_ bs≈ 2 π G 3m a/v^2,

where a is the semi-major axis (SMA) of the target binary.
To derive P_ bs^<R we make use of the formalism described in <cit.>.
In short, the approach is to split up the chaotic binary-single interaction into a
series of temporary metastable binary-single states, referred to here as intermediate states (IMSs). Each of these 𝒩 states can be described
by a binary with a bound single. The eccentricity distribution of the IMS binaries follows approximately that of a so-called
thermal distribution, P(e) = 2e <cit.>, where the SMA is approximately equal to the SMA of the initial target binary, a. From this follows that
the probability for a single IMS binary to have a peri-center distance <R_ bs is ≈ 2R_ bs/a <cit.>.
As a binary-single interaction on average assembles 𝒩 such IMS binaries during its chaotic evolution, the total
probability P_ bs^<R is simply given by,

    P_ bs^<R≈2R_ bs/a𝒩.

Using these relations, one now finds that the
cross section σ_ bs^<R can be written as,

    σ_ bs^<R≈  2 π G 6mR_ bs/v^2𝒩.

From this it is clear that the cross section for a binary-single interaction
to result in a close encounter between two of the three objects
with < R_ bs is ∝ R_ bs and independent of the binary SMA a.

As in the single-single case, not all distances R_ bs in the binary-single problem will lead to a GW inspiral merger.
However, in contrast to the single-single case, defining a unique distance for GW inspiral merger in the binary-single problem is not possible,
as the distance from which a GW inspiral is possible changes for each IMS binary depending on the orbital period of the
remaining bound single <cit.>. Therefore, one should in principle work with a distribution of GW inspiral distances; however, this has its own problems, and is therefore out of scope of this paper.
Instead, to simplify our analysis and make it analytically tractable,
we will in this paper work with a single characteristic value for the binary-single GW merger distance, that we
denote by ℛ_ bs. Note here that this assumption still conserves all the right scaling properties of the problem <cit.>.
Following <cit.>, a reasonable value for ℛ_ bs
is the one for which the GW energy loss integrated over one peri-center passage ΔE_ GW (see Section <ref>)
for two of the three objects equals the total initial energy of the three-body system E_ bs, that here
is ≈ Gm^2/(2a). Note here that this requirement is exactly the same as in the single-single case, but with the energy of the initial
binary-single system instead of the energy of the initial single-single system. Now solving for the
peri-center distance  for which ΔE_ GW = E_ bs, one finds,

    ℛ_ bs≈(85 π/24√(2))^2/7×ℛ_ m(a/ℛ_ m)^2/7.

Substituting this expression for ℛ_ bs into Eq. (<ref>) results in the cross section for the formation of a GW inspiral that merges
during the binary-single interaction; a cross section we denote σ_ bs^<ℛ.
As first noted by <cit.>, the cross section σ_ bs^<ℛ∝ a^2/7, which means that it increases with the initial
SMA a. For a thorough discussion on the subject of binary-single interactions with
dissipative terms (GW emission, tidal dissipation, etc.) see <cit.>.
Finally, as for the single-single GW captures, binary-single interactions and GW inspirals forming during such interactions can in principle be broken up before completion
by incoming objects <cit.>. However, this rarely happens for the systems we consider, and this `infrared' correction will therefore not be
discussed further in this paper.





 §.§ Comparing Binary-Single and Single-Single


One of the key questions we explore in this paper is how important
single-single GW capture mergers are compared to GW inspiral mergers forming during binary-single interactions.
To gain insight into this, we start by considering the following cross section ratio,

    σ_ bs^<R/σ_ ss^<R = 3𝒩 (R_ bs/R_ ss) ≈ 60 (R_ bs/R_ ss),

which follows from the use of Eq. (<ref>) and Eq. (<ref>). Note here that for
the last equality we have used a value of 𝒩 = 20, which follows from numerical experiments <cit.>.
From this ratio we conclude that a BBH is about 3𝒩≈ 60 times more `effective' in forming two-body encounters
with r_ p<R_ ss = R_ bs compared to a single BH. Note here that this efficiency does not depend on any properties
of the cluster or the BBH orbital parameters.

Two values of R_ bs and R_ ss that are particular interesting to compare are the distances from which a
GW merger can form.
By the use of Eq. (<ref>) and Eq. (<ref>) we find that,

    ℛ_ bs/ℛ_ ss≈( a/ℛ_ mv^2/c^2)^2/7 = (v/v_ orb)^4/7,

where v_ orb is the internal orbital velocity of the BBH defined here as the relative velocity between the two BHs assuming a circular orbit. As we only
consider the hard binary (HB) limit in this paper (v_ orb≫ v, see e.g. <cit.>), the above
relations imply that ℛ_ bs will always be  < ℛ_ ss,
which leads to the following inequality for the corresponding GW capture cross
sections: σ_ bs^<ℛ/σ_ ss^<ℛ < 3𝒩.

Finally, as will be shown later in Section <ref>, a highly relevant ratio
to consider is ℛ_ bs, ej/ℛ_ ss,
where the subscript `ej' states that ℛ_ bs is here evaluated at the SMA a = a_ ej, where a_ ej is the (maximum) SMA from which the BBH will
get ejected from the cluster through a binary-single interaction <cit.>.
Assuming that each binary-single interaction decreases the SMA of the BBH by a fixed fraction
δ, such that a →δ a, then it follows from classical mechanics that a_ ej = (1/6)(ϕ - 1)Gm/v_ esc^2,
where v_ esc is the escape velocity of the cluster, and ϕ≡ 1/δ <cit.>.
By substituting this relation for a_ ej into Eq. (<ref>)
one finds,

    ℛ_ bs, ej/ℛ_ ss = ( ϕ - 1/12 f_ ed^2)^2/7,

where f_ ed≡ v_ esc/v (the subscript `ed' refers to that f is the fraction between the ejection (e) and the dispersion (d) velocities, respectively).
Assuming f_ ed = 5, and δ = 7/9 <cit.>, one finds that ℛ_ bs, ej/ℛ_ ss = 1050^-2/7≈ 7.3^-1.
For comparison, if we evaluate this ratio at the HB limit value of the SMA, a_ HB = (3/2)Gm/v^2, which 
is the SMA at which the total energy of the binary, -Gm^2/(2a), equals the energy of incoming single encounters, (1/2)μv^2, we find
ℛ_ bs, HB/ℛ_ ss≈ 1. Note that this also follows directly from Eq. (<ref>) when v ≈ v_ orb.
For our chosen values it therefore follows that
8 ≲σ_ bs^<ℛ/σ_ ss^<ℛ≲ 60.






§ RATES


In this section we convert the cross sections derived in the above
Section <ref> to
formation rates. We assume steady state in all our calculations, which of course is a
simplification of how real clusters evolve; however, this assumption
allows us to explore the rates and observables in closed form expressions, which provides crucial guidance to
what to focus on and include in more sophisticated numerical simulations. Also, this allows for
an easier comparison to the recent analytical literature on both binary-single and single-single GW mergers
forming in both GN and GCs.
We proceed below by first deriving a few general relations,
after which we apply these to study the binary-single and single-single rates of GW capture mergers
from two different cluster models; a uniform density model and the Plummer's sphere model.
Corresponding observables, such as the GW peak frequency and BBH orbital
eccentricity will be discussed in Section <ref>.





 §.§ General Relations


In the following we present general relations for
deriving absolute and relative rates of GW mergers
resulting from single-single and binary-single interactions, respectively.




  §.§.§ Single-Single Rates


We consider a spherical cluster with mass density profile ρ (r) and corresponding number density profile n(r) = ρ (r)/m,
consisting of identical compact objects with mass m.
In a radial shell with width dr at radial position r, the number of single-single encounters with < R_ ss(r) per time interval dt is given by,

    dΓ_ ss(r) = n(r) σ_ ss^<R(r) v(r) × n(r) 4 π r^2 dr ×1/2,

where the first term is the rate of encounters with < R_ ss(r) a single object experiences,
the second term is the number of single objects within the considered shell, and the third term corrects for that
the singles are both targets and encounters (two singles result in one encounter). From here
we do not explicitly show if a quantity x is dependent on, e.g., r by writing x(r). This is done to limit the length of our expressions.
Using the above relation from Eq. (<ref>), the total single-single rate of encounters with < R_ ss from the entire cluster is now given by,

    Γ_ ss = 8 π^2 G/m∫_0^∞R_ ssρ^2/v r^2 dr.

Note here that we have allowed for the characteristic encounter distance R_ ss to be a function of r.
A variable R_ ss will be important to include when deriving the total rate of single-single GW capture mergers.
The above equation can also be written in the following form,

    Γ_ ss     = n_0σ_ ss,0^<R v_01/2 N_ s×4/3π r_ s^3ρ_0/M∫_0^∞R̃_ ssρ̃^2/ṽ 3r̃^2 dr̃
         = n_0σ_ ss,0^<R v_01/2 N_ s×ξ_ss,

where N_ s is the total number of singles, M = N_ s× m is the total mass,
R̃_ ss = R_ ss/R_ ss, 0, ρ̃ = ρ/ρ_0, ṽ = v/v_0, r̃ = r/r_ s, and

    ξ_ss≡4/3π r_ s^3ρ_0/M∫_0^∞R̃_ ssρ̃^2/ṽ 3r̃^2 dr̃.

Here the subscript `0' denotes the corresponding quantity has to be evaluated at r=0 (central cluster values), and r_ s is a characteristic scale.
It is convenient to consider this notation of the rate, as ξ_ss =1 for a simple uniform sphere with constant R_ ss. In other words,
the last term, ξ _ss, represents essentially an `efficiency factor' that depends on how the N_ s objects distribute within the cluster.  




  §.§.§ Binary-Single Rates


We now turn to a derivation of the rate of close two-body encounters forming during binary-single interactions.
To analytically calculate this we here make use of the simple dynamical model described in <cit.>. In short, 
we assume that all binaries form with a SMA equal to their HB value, a_ HB (see Section <ref>),
after which a given binary undergoes interactions with incoming singles that each decreases the binary SMA from a →δ a.
This in-cluster hardening process continues until the binary SMA falls below the critical ejection value
a_ ej (see Section <ref>), at which the binding energy released in a single
binary-single interaction leads to ejection of the binary from its cluster.
The average rate of encounters with r_ p < R_ bs forming during binary-single interactions can in this model be approximated by
Γ_ bs≈ N_ bs/T_ ej, where N_ bs is the total number of encounters with r_ p < R_ bs forming
during the hardening binary-single interactions of the binary from a_ HB to a_ ej, and T_ ej is the time it takes for the binary to transition from
a_ HB to a_ ej.

For estimating this rate, we start by deriving N_ bs.
We do this by first considering the differential version of Eq. (<ref>), dN_ bs = (2R_ bs/a)𝒩 dk,
where k here refers to hardening step k, i.e. a(k) = a_ HBδ^k.
By changing variable from k to SMA a
using the relation da = -a(1-δ) dk now follows,

    N_ bs = 2𝒩/1-δ∫_a_ ej^a_ HBR_ bs/a^2 da,

where we here, and in the rest of the paper, assume that the probability for a BBH to merge inside the cluster is < 1,
which is a reasonable assumption for standard GCs.
As a cross check, if we here substitute R_ bs with the GW inspiral merger distance ℛ_ bs given by Eq. (<ref>),
then one finds that the number of such mergers evaluates to N_ bs^<ℛ≈ 7P_ bs, ej^<ℛ/(5(1-δ)),
where P_ bs, ej^<ℛ is given by Eq. (<ref>) evaluated at a = a_ ej.
This relation was found in <cit.>, which serves as an excellent confirmation of our relations so far. Note here that the `enhancement factor' from including
the whole binary-single hardening sequence, and not only the final SMA (a=a_ ej), evaluates to 7/(5(1-δ)) = 63/10.

We now turn to deriving the time interval T_ ej, for which we assume the binaries
distribute near the cluster center, such that they only relate to the cluster properties
through the central values, n_0, v_0. 
In this approximation we continue by first use that the time between binary-single interactions
is given by the inverse binary-single encounter rate, (n_0 σ_ bs v_0)^-1, which can be converted to
the differential form dt = (n_0σ_ bs v_0)^-1 dk, where t here denotes time.
Using da = -a(1-δ) dk, we can now write the total time it takes for a given binary
to transition from a_ HB to a_ ej
through binary-single scatterings in the following way,

    T_ ej    = ∫_a_ ej^a_ HB1/n_0σ_ bsv_0da/a(1-δ), 
       ≈(6 π G)^-1/(1-δ)v_0/n_0m^-1/a_ ej,

where we have used Eq. (<ref>), and for the last term assumed that a_ HB≫ a_ ej.
We note here that this is also approximately the inverse rate of binary ejections, i.e., Γ_ ej≈ 1/T_ ej.

By the use of Eq. (<ref>) and Eq. (<ref>), and that Γ_ bs≈ N_ bs/T_ ej,
we now finally find,

    Γ_ bs     = n_0σ_ bs,ej^<R v_0 N_ b×∫_1^a'_ HBR_ bs'/a'^2 da'
         = n_0σ_ bs,ej^<Rv_0 N_ b×ξ_bs,

where N_ b is the number of binaries, a' = a/a_ ej, R_ bs' = R_ bs/R_ bs,ej, σ_ bs,ej^<R denotes
the cross section evaluated at a=a_ ej, and

    ξ_bs≡∫_1^a'_ HBR_ bs'/a'^2 da'.

As seen, for a constant value of R_ bs' the scale-factor ξ_bs≈ 1 for a^'_ HB≫ 1. The factor ξ_bs therefore
represents the integrated effect from including a scale dependent R_ bs, as e.g. the GW capture distance ℛ_ bs given by
Eq. (<ref>).
As described in <cit.>, the characteristic distance for dissipative captures, including tidal and GW emission captures,
can often be approximated by R ∝ a^β, where β relates to the energy loss in question, ΔE, as ΔE ∝ r_ p^-β. As seen, for this class of `β-models' the efficiency factor
is given by ξ_bs = (1-β)^-1 for a^'_ HB≫ 1.





  §.§.§ Binary-Single vs. Single-Single


Having derived general forms for the close encounter rate from both single-single, Γ_ ss (Section <ref>),
and binary-single, Γ_ bs (Section <ref>), we are now in a position to compare the two. Using Eq. (<ref>) and
Eq. (<ref>) we find,

    Γ_ bs/Γ_ ss≈ 2N_ b/N_ sσ_ bs,ej^<R/σ_ ss,0^<Rξ_ bs/ξ_ ss
    = 6F_ bs𝒩×R_ bs, ej/R_ ss,0ξ_ bs/ξ_ ss,

where the factor F_ bs denotes the binary
fraction, F_ bs = N_ b/N_ s.
This simple expression constitutes one of our main results from this paper. In the following sections we will evaluate this ratio for different characteristic
distances (constant R and GW capture ℛ) and cluster profiles (`MODEL I' and `MODEL II').
The main purpose of this is to illustrate that single-single interactions could contribute with a non-negligible and unique
population of BBH mergers.





 §.§ MODEL I: Uniform Density Sphere


We here consider a BH subsystem described by a uniform density ρ and velocity dispersion v.
This example therefore represents a classical `n σ v' estimate with a finite number of objects, N_ s.
Considering first the single-single population, it is here natural to set r_ s equal to the size of the cluster, which implies that
ρ̃ = 1 and ṽ = 1 for r < r_ s, and that _ ss = 1 and ξ_ss = 1 for a constant R_ ss and
the GW capture ℛ_ ss.
For binary-single interactions, the efficiency factor ξ_bs from Eq. (<ref>) will in contrast only =1 for constant R_ bs encounters,
but not for GW captures with varying ℛ_ bs.
Below we study the rate of close encounters and GW capture mergers for this model.





  §.§.§ Close Encounters


We start by writing out the rate of single-single encounters with r_ p < R_ ss, where R_ ss is assumed constant.
Since ξ_ ss = 1, as argued above, then the rate is given by,

    Γ_ ss≈ n σ_ ss^<R v 1/2 N_ s,

which follows from Eq. (<ref>).
This is not surprisingly the usual rate of encounters per single, `n σ v', weighted by the total number of singles divided by two, N_ s/2 (see also <cit.>).

We now consider the rate of binary-single encounters with r_ p < R_ bs, where also R_ bs is here assumed constant.
Since ξ_ bs = 1 also in this case, the rate is simply given by,

    Γ_ bs≈ n σ_ bs^<R v N_ b,

where we have used Eq. (<ref>). Remember here that the cross
section σ_ bs^<R is ∝ mR_ bs/v^2, and therefore independent of the SMA a.

If we now consider the special case for which R_ ss = R_ bs, then the ratio between
the two rates, Γ_ bs and Γ_ ss, reduces to the following expression

    Γ_ bs/Γ_ ss≈6F_ bs𝒩,

where we have used Eq. (<ref>).
This explicitly illustrates that Γ_ bs/Γ_ ss is independent
of the cluster density, its velocity dispersion, and the orbital evolution of the interacting binaries. Instead, it
essentially only depends on the binary fraction F_ bs. As seen, the two rates are therefore
comparable when the binary fraction F_ bs≈ (6𝒩)^-1, from which we conclude that for this model
Γ_ ss≈Γ_ bs when F_ bs≈ 1/120 ∼ 1%. 
Therefore, for our considered model, single-single and binary-single encounters with r_ p < R_ ss = R_ bs contribute at the same level
when the binary fraction is at the percent level. Interestingly, this is exactly the level that
is observed in state-of-the-art GC simulations, which explains why single-single interactions might actually contribute with a non-negligible
fraction of BBH mergers. An illustration of this is shown in Fig. <ref>.

As will be discussed later, a constant R, as considered here, is important for understanding
the rate of GW sources with a particular GW peak frequency (the GW frequency where most of the power is
outputted) which to leading order only depends on the BBH peri-center distance r_ p (see Eq. (<ref>)). Below we continue by deriving the rate of GW mergers forming within the scale dependent capture distance ℛ.








  §.§.§ GW Mergers


We now derive the rates of single-single and binary-single mediated GW mergers for our considered uniform cluster model.
One expects a different ratio than the one derived in Eq. (<ref>), as R is now no longer constant
for the binary-single interactions.

We start by deriving the single-single GW capture rate. In this case, the characteristic distance R_ ss = ℛ_ ss, but
since the cluster is assumed uniform then _ ss is still =1. The single-single GW capture rate is therefore simply given by 

    Γ_ ss≈ n σ_ ss^<ℛ v 1/2 N_ s,

where σ_ ss^<ℛ is evaluated using Eq. (<ref>), and Eq. (<ref>).

We now derive the rate of GW inspiral mergers forming during binary-single interactions. From using Eq. (<ref>)
it follows that R'_ bs = (a/a_ ej)^2/7 = ^2/7, from which we now find using Eq. (<ref>) that,

    ξ_bs = ∫_1^a^'_ HB^2/7/^2 d≈7/5.

For this we assumed the limit a_ HB≫ 1. Note here that this result also follows from setting β = 7/2
in the general relation ξ_bs = (1-β)^-1 derived below Eq. (<ref>). With this value of ξ_bs,
the rate of GW inspiral mergers forming during binary-single interactions is given by,

    Γ_ bs = 7/5 n σ_ bs,ej^<ℛ v N_ b.

The scale dependence of ℛ_ ss gives
rise to a small increase of 7/5, compared to just evaluating the rate at the ejection limit a = a_ ej.

Comparing the rates of GW mergers from the binary-single and single-single channels we find,

    Γ_ bs/Γ_ ss≈6F_ bs𝒩×( ϕ - 1/12 f_ ed^2)^2/77/5,

where we have used Eq. (<ref>) and Eq. (<ref>). As seen, when accounting for all the GW mergers
that form through single-single and binary-single interactions, and not only those for which r_ p < R_ ss = R_ bs as we did in the above
Section <ref>, the relative GW merger rate from single-single interactions increases by a factor of
(5/7)1050^2/7≈ 5.2 for f_ ed = 5 and δ  = 7/9. Therefore, in this case
Γ_ ss≈Γ_ bs for F_ bs≈ 5.2/120 ∼ 5%. This indicates that
single-single GW captures very well could play a role in the formation of eccentric in-cluster mergers, 
as F_ bs is likely < 5% for standard GCs, as shown by Fig. <ref>.
We proceed below by exploring how our results from this section change when considering a more realistic density profile for the single BH population.






 §.§ MODEL II: Plummer's Sphere


Real clusters are not described by the simple uniform density sphere that we considered in the above Section <ref>.
Instead, relaxation processes generally drive systems to a state described by a profile having a high density in the center and a low in the outskirts <cit.>.
To study the effect of a more realistic profile, we here derive absolute and relative rates assuming the single BHs distribute according
to the well-known Plummer's sphere <cit.>. We choose this as it allows for a full analytical treatment of the problem in contrast to other families of profiles.

To start, we first introduce the mass density profile of the Plummer's sphere, which is given by,

    ρ = 3M/4 π b^3( 1 + r^2/b^2)^-5/2,

and the corresponding velocity dispersion,

    v^2 = 1/6GM/b( 1 + r^2/b^2)^-1/2,

where M is the total cluster mass, and b is a characteristic scale <cit.>. Inserting these two expressions
into Eq. (<ref>), one finds that the single-single efficiency factor for this profile reduces to the following form

    ξ_ss = 3 ∫_0^∞_ ss (1+x^2)^-19/4 x^2 dx.

The binary-single efficiency factor ξ_bs is unchanged from what was found in
the above Section <ref>, as we assume the binary-single
encounter rate only depends on the central properties of the cluster.
Finally, from using that the potential of the Plummer's sphere is
given by ϕ(r) = -GM/(b√(1+(r/b)^2)) and that v_ esc(r) = √(-2ϕ(r))
it directly follows that v_ esc,0/v_0 = √(12)≈ 3.5. As this is very close to our
`fiducial' chosen value of f_ ed, we will still be using f_ ed = 5 in the following
sections to make comparisons more clear.
Below we consider the absolute and relative rate of close encounters and GW mergers for the Plummer's sphere.






  §.§.§ Close Encounters


We start by considering the rate of single-single encounters with r_ p < R_ ss, where R_ ss is assumed
constant. In this case _ ss = 1, from which we find by the use of Eq. (<ref>) that

    ξ_ss = 3 ∫_0^∞ (1+x^2)^-19/4 x^2 dx = 3√(π)/4𝒢(13/4)/𝒢(19/4),

where 𝒢(z) = ∫_0^∞ x^z-1 e^-x dx denotes the well-known Gamma function. With this factor it is now straight forward
to derive the single-single rate using Eq. (<ref>). The question is now, does the change from a uniform sphere to a more realistic density profile
leads to an increase or a decreases of the single-single close encounter rate? As seen here, for the Plummer's
sphere ξ_ss = (3√(π)/4)𝒢(13/4)/𝒢(19/4)≈ 0.2, which means that the rate from distributing N_ s singles
in a Plummer's sphere is about 5 times smaller than if one distributes them in a uniform sphere.
The resulting ratio between the binary-single and single-single close encounter rates for
R_ ss = R_ bs is given by,

    Γ_ bs/Γ_ ss≈6F_ bs𝒩×4/3√(π)𝒢(19/4)/𝒢(13/4),

where we have used Eq. (<ref>), and the above Eq. (<ref>). From this
we conclude that Γ_ ss≈Γ_ bs for F_ bs≈ 1/600 ∼ 0.15%,
which suggests that BH subsystems with a realistic density profile is unlikely to
have single-single captures significantly contributing to encounters
with r_ p < R_ ss = R_ bs. Below we study the absolute and relative rate of GW mergers.





  §.§.§ GW Mergers


For single-single GW capture mergers R_ ss = ℛ_ ss, which implies that _ ss = (v/v_0)^-4/7, where we have used
Eq. (<ref>). Plugging this expression into Eq. (<ref>), and by the use of Eq. (<ref>), we find the following
value for the single-single efficiency factor,

    ξ_ss = 3 ∫_0^∞ (1+x^2)^-129/28 x^2 dx = 3√(π)/4𝒢(87/28)/𝒢(129/28),

which (also) evaluates to ≈ 0.2. Therefore, the rate of single-single GW capture mergers is also greatly reduced for the Plummer's sphere,
compared to the uniform model. Finally, we can also here compare to the rate from binary-single GW mergers.
By the use of Eq. (<ref>), the relation shown in Eq. (<ref>), and our derived values
for ξ_ss (Eq. (<ref>)) and ξ_bs (Eq. (<ref>)), we find

    Γ_ bs/Γ_ ss≈6F_ bs𝒩×( ϕ - 1/12 f_ ed,0^2)^2/77/54/3√(π)𝒢(129/28)/𝒢(87/28),

where f_ ed,0≡ v_ esc,0/v_0.
For δ = 7/9 and f_ ed,0 = 5, the term after the `×'-sign evaluates to ≈ 1. We therefore conclude that
even when accounting for all GW mergers that can form through single-single and binary-single interactions, the two rates
are only expected to be similar for F_ bs≈ 1/120 ∼ 1%. However, as previously described, and also shown in Fig. <ref>,
F_ bs is in fact likely to be at the percent level (see Fig. <ref>). 








§ OBSERVABLE IMPLICATIONS




Having shown using analytical arguments that BH single-single GW captures in GCs might give rise to an observable population of mergers,
we now explore what the observable characteristics are of these mergers. In particular, we explore if single-single GW capture mergers
can be distinguished from the other BBH merger types in GCs. These include BBHs dynamically ejected from their host cluster,
BBHs merging in-between their binary-single interactions, and BBHs merging during their binary-single interactions.
In the following sections we will refer to these GW merger types as ejected mergers, 2-body mergers, and 3-body mergers, respectively,
to shorten labels and descriptions (see also <cit.>). As pointed
out by <cit.>, these BBH
merger types have different eccentricity- and GW peak frequency distributions. For example, in <cit.>
it was pointed out that 2-body mergers will naturally form near the LISA band with high eccentricity, where 3-body mergers will form at higher frequencies making
them eccentric LIGO sources <cit.>.

We here explore how the single-single GW capture mergers will distribute, and in particular what their properties are at 1 Hz and 10 Hz in GW peak frequency. The 1 Hz regime is potentially interesting for the planned deci-Hertz
observatories DECIGO and Tian Qin, where the 10 Hz limit is naturally interesting for currently operating observatories such as LIGO/VIRGO,
but has also relevance for third generation observatories including ET/CE, which likely will be able to resolve eccentricities
down to ∼ 0.01 in this range <cit.>.
It is clear that single-single GW captures are not the dominating source of BBH mergers forming in GCs; however, with future observatories such as ET/CE
we are entering an age where we can expect to see every BBH merger in the visible Universe. Therefore, it is important to have a solid understanding for
how BBH mergers might form in different environments, and what their corresponding observable distributions can tell us about their host systems.
Below we present results from a simple analytical MC method.





 §.§ Analytical Monte Carlo Approach


To derive BBH merger distributions with the inclusion of single-single interactions, we start by deriving the distribution of 2-body and 3-body mergers
using the semi-analytical MC approach first described in <cit.>. After this, we super impose the distribution from
single-single interactions given some density profile. As for our analytical results, the 2-body and 3-body mergers
are all derived assuming that the binary-single encounter rate is determined by the central quantities of the cluster only. In the following we describe this procedure in detail.



  §.§.§ Modeling Binary-Single Mergers


For building up the distribution of 2-body and 3-body mergers we follow a large ensemble of uncorrelated BBHs undergoing binary-single interactions
in an environment described by a constant velocity dispersion, v, and number density, n, equal to the central values of the cluster.
Following <cit.>, each of these BBHs are assumed to form with an initial SMA equal to their hard-binary value
a_ HB; however, for GCs the exact upper value of the SMA
a is not important as long as a_ HB≫ a_ ej.
Considering now the evolution of one of these BBHs, this BBH will after formation undergo strong binary-single interactions, each of which is assumed to lead
to a constant decrease in the SMA from a to δ a. At the same time, the interactions will also change the eccentricity of the BBH,
where we here assume this change is sampled from the distribution P(e) = 2e <cit.>.
In the point-particle Newtonian limit, this series of binary-single interactions will always end with
a dynamical ejection of the BBH from its host cluster when its SMA falls below a_ ej.
This classical hardening process therefore leads to an ejection of BBHs with a SMA ∼ a_ ej,
and an eccentricity distribution that is thermally distributed; however, when GR effects are included in this process, a given BBH can also
merge inside its cluster through (at least) 2-body and 3-body mergers. To account for these two in-cluster
merger types we follow the approach outlined in the paragraphs below. 

3-body Mergers: For determining if a BBH merges during a strong 3-body interaction,
we model this often highly chaotic and resonating state using the approach put forward in <cit.> and briefly
described in Section <ref>. In short, the interaction is here divided up into 𝒩 IMSs,
each of which is described by a BBH with a bound single BH.
For each IMS we assign the corresponding BBH an eccentricity sampled from the distribution P(e) =2e, but keep
the SMA fixed to its initial value a. To determine if the BBH undergoes a GW inspiral merger during this IMS, i.e. merge while the
third objects is still bound to it, we compare the energy radiated over one orbit of the BBH (ΔE_ GW),
to the total orbital energy of the bound 3-body state (E_ bs).
If ΔE_ GW > E_ bs we label the IMS assembled BBH as a 3-body merger. This energy threshold is equivalent of saying that
the BBH will undergo a GW inspiral merger during the interaction if its peri-center distance r_ p < ℛ_ bs.
If instead r_ p > ℛ_ bs the BBH does not merge during the considered IMS.
We repeat this process, i.e. first assigning the IMS assembled BBH an eccentricity from P(e) = 2e and then compare its r_ p to ℛ_ bs,
up to 𝒩 =20 times per interaction.

2-body Mergers: If a BBH does not undergo a 3-body merger we determine right after its binary-single interaction if the BBH instead will undergo a 2-body merger. We do this by first assigning an eccentricity to the BBH sampled from P(e) = 2e, after which we calculate its inspiral time
t_ insp∝ a^4(1-e^2)^7/2 and the time it takes for the next strong single encounter to interact with the BBH, t_ int∝ (n_0 σ_ bs v_0)^-1.
If t_ insp < t_ int then we label the BBH as a 2-body merger, if instead t_ insp > t_ int the BBH survives and we move
on to the next binary-single interaction. 

We repeat this process of checking for 3-body and 2-body mergers while the BBH gradually hardens inside its GC until the BBH either merges
or is ejected. For all 2-body and 3-body mergers we record their orbital parameters, a and e, at formation, i.e. 
before they start their inspiral, which allows us to quickly calculate eccentricity and GW peak frequency distributions. These distributions are then
normalized by assuming that the BBHs are formed at a constant rate given by 1/T_ ej, where T_ ej is given by Eq. (<ref>).

This model clearly represents a simplified picture of how a BH population evolves in a real GC; however, it is very fast, have resulted in very precise
estimates so far, and provides therefore an ideal test-bed for exploring what effects that might be important to included for an accurate modeling
of such systems <cit.>.







  §.§.§ Modeling Single-Single Mergers


Calculating the single-single GW capture merger distribution is a well defined problem for a non-evolving density profile consisting of equal mass objects,
and can be done in several ways. For this paper we made a small MC routine that simply samples single-single encounters up to the
local GW capture distance ℛ_ ss.
For each of these encounters we calculate the corresponding orbital parameters, a = -Gm^2(2ΔE_ GW + 2E_ ss)^-1 and
e = 1- r_ p/a, of the newly formed BBH right after capture, from which we can
determine eccentricity- and GW peak frequency distributions similar to the binary-single case. In this routine, for the local maximum GW capture
distance ℛ we do not only require that the two single BHs are bound after the encounter, but also that their GW inspiral time,
t_ insp(a,e), is shorter than the local binary-single interaction time, t_ int(a).
This requirement results in a local maximum GW capture distance ℛ that fulfills the following equation,

    Am^9/2ℛ^-7/2 - 1/2μv^2 = Bm^2(C/Dn/v1/m^2)^2/3ℛ^7/3,

where A = (85π/12)G^7/2c^-5, B = G/2, C = (768/425)(2^7/25/512)G^-3c^5,
and D = (6 π G)^-1.
This unfortunately does not have a closed form solution, but can easily be solved numerically. However, we did not
find a significant difference between using the classical capture distance given by Eq. (<ref>),
and the one found from the above Eq. (<ref>). Results are discussed in the next section.






 §.§ Results


In this section we present results from numerical experiments, where we follow a population of binary and single
BHs using the routines described in the above Section <ref>. All our results
are based on a system characterized by m = 20M_⊙, v_0 = 10 kms^-1, n_0 = 10^5pc^-3, f_ ed,0 = 5,
and F_ bs = 0.01.



  §.§.§ Gravitational Wave Peak Frequency


We start by considered Fig. <ref>, which shows the GW peak frequency distributions of the merging population of
BBHs right when they form in our MC routine.
For making this, we used the orbital parameters a,e of each of the assembled BBHs to derive their corresponding peri-center distance r_ p = a(1-e), from which
the GW peak frequency, denoted here by f_ GW, can be approximated by <cit.>,

    f_ GW≈1/π√(2Gm/r_ p^3).

Focusing on the single-single (grey/black) and binary-single (red) mergers we first notice that they peak at different
locations: roughly about 10^-1 Hz and 10^0.3 Hz, respectively.
For both of these channels, their peak location is near the lowest f_ GW
at which all of the mergers from the specific channel is able to contribute (see also <cit.>). The two peak locations can therefore be expressed
as, f_ GW,ss^ peak≈π^-1√(2Gm/ℛ_ ss,0^3)∝ v_0^6/7/m, and
f_ GW,bs^ peak≈π^-1√(2Gm/ℛ_ bs,ej^3)∝ v_ esc^6/7/m, respectively.
Now taking the ratio between these two terms we find the following relation,

    f_ GW,bs^ peak/f_ GW,ss^ peak = ( 12f_ ed,0^2/ϕ - 1)^3/7.

This interestingly shows that the single-single and binary-single peak locations are separated by a constant factor, that in our model
only depends on the properties of the system through f_ ed,0.
For δ = 7/9 and f_ ed,0 = 5 follows that f_ GW,bs^ peak/f_ GW,ss^ peak≈ 20,
which agrees with the results shown in Fig. <ref>. As f_ ed,0 is always > 1, our derived ratio further implies that
f_ GW,bs^ peak/f_ GW,ss^ peak >  (12/(ϕ - 1))^3/7≈  5, therefore, the peaks will always be separated
with f_ GW,bs^ peak > f_ GW,ss^ peak.

If we now consider the actual shapes of the single-single and binary-single distributions in the region where their GW peak frequencies are greater
than their corresponding distribution peaks, i.e. for f_ GW > f_ GW^ peak, we see that they follow the same functional form. This is most easily seen
when comparing the `3-body' with the `sin-sin (uniform)' distributions. The reason is simply that the single-single and the binary-single cross sections
for close encounters with r_ p < R are both ∝ R, as seen in Eq. (<ref>) and Eq. (<ref>).
The shape of both of the distributions is therefore given by,

    d Γ (>f_ GW)/dlog f_ GW∝ f_ GWd σ (< R)/df_ GW∝ f_ GW^-2/3,

where we have used that σ^<R∝ R for both single-single and binary-single interactions, and the relation R ∝ f_ GW^-2/3 from Eq. (<ref>). For further discussions on the single-single population and this distribution see <cit.>.

Finally, we now consider the relative normalizations of the single-single and binary-single distributions. As seen, for our assumed value of F_ bs = 0.01,
the `sin-sin (uniform)' follows very closely the `3-body' distribution at high values of f_ GW. This is exactly what we derived in Eq. (<ref>),
in which we argued that the rates should be similar for encounters with r_ p < R, i.e. for encounters with f_ GW > f_ GW(R), when the binary
fraction is at the percent level. This, combined with the ∝ f_ GW^-2/3 dependence, means that they must
overlap at high f_ GW. 
The normalizations of the `sin-sin (Plummer)' and `3-body' distributions, over all f_ GW, derived using our analytical methods in Eq. (<ref>), are also very similar.
To conclude, our analytical and numerical methods agree fully on how and where the different BBH merger populations distribute in f_ GW space.




  §.§.§ Orbital Eccentricity


We now consider Fig. <ref>, which shows the orbital eccentricity distributions of the merging BBHs from Fig. <ref>
at 1 Hz and 10 Hz. The 1 Hz regime is relevant for planned detectors such as DECIGO and Tian Qin, where the 10 Hz
regime is relevant for currently operating LIGO/VIRGO, and future ground-based observatories such as the ET and CE (the ET and CE are likely to be sensitive at even lower GW frequencies).

Starting with the distribution at 1 Hz (top plot), we see that the single-single population (we here only show results for the Plummer's sphere) clearly
dominates the distribution for all eccentricities log(e) > -1.5. That is, in our model, eccentric deci-Hertz sources are likely to originate from single-single
GW captures. This was also noticed in <cit.>; however, we have here been able to derive the correct normalization and how it relates to
the 2-body and 3-body mergers. Note here, that we have in our model not included binary-binary <cit.> and any weak encounter driven mergers
<cit.>, which in principle also could contribute to eccentric deci-Hertz sources.

If we now consider the distribution at 10 Hz (bottom plot), we see that the single-single GW capture mergers don't
contribute much to the population with an eccentricity resolvable by LIGO/VIRGO (e>0.1). Instead, the single-single GW captures seem to fully dominate the
region near -2.5< log (e) - 1.5, which interestingly could possibly be resolved by future detectors such as the ET/CE <cit.>.
Therefore, to get a complete picture of what future detectors might observe and how we can use it to constrain the origin of BBH mergers,
the inclusion of single-single GW capture mergers seems to be very important. 

Finally, because highly eccentric orbits will emit qausi-periodic burst of GWs at pericenter, it is observational relevant to consider the timescale between bursts for the single-single and 3-body mergers. As the time between bursts
is equal to the BBH orbital time, T_ orb, the period of interest is simply given by,

    T_ orb = 2/f_ GW(1-e)^-3/2,

where we have used the gravitational wave peak frequency, f_ GW (Eq. (<ref>)), together with Kepler's law T_ orb = 2 π√(a^3/2Gm). 
In the high eccentricity limit, i.e. for e ∼ 1 where bursts are relevant, it is useful to express the eccentricity as e = 1-10^-x, from which
T_ orb can be written as T_ orb = (2/f_ GW)10^3x/2.
In the low eccentricity limit, i.e. for e ≪ 1, the orbital period T_ orb is to leading order given by T_ orb≈ (2/f_ GW)(1+3e/2).

For representative single-single GW capture mergers that have circularized to the point where they are detectable (near the end of the vertical rise of the grey and red tracks in Figure <ref>), but still on eccentric orbits, e.g., e ∼ 0.9 and f_ GW=0.1 Hz, T_ orb, ss≈ 10 minutes. For the 3-body capture mergers, in a similar regime, e.g., e ∼ 0.9 and f_ GW=10^0.3 Hz, T_ orb, bs≈ 30 seconds. Work is currently being done on how to detect signals from eccentric burst-like sources <cit.>.








  §.§.§ Multi-band GW Observations




We end this section by showing how the 2-body mergers, single-single GW capture mergers, and 3-body mergers distribute and evolve as a function of GW peak frequency
and GW strain to complement recent studies <cit.>.
Results are shown in Fig. <ref>, where we have assumed a uniform source population up to a redshift of z = 0.3, sampled from the data
shown in Fig. <ref>. On the figure is also shown sensitivity curves for LISA, DECIGO, LIGO 3G (ET), and LIGO O2 (current operational mode).
As is clear from this figure, the single-single GW capture mergers form where DECIGO
is most sensitive, and a future joint multi-band GW network, including instruments like LISA, DECIGO/Tian Qin, and ET/CE, will therefore be able to
put tight constraints on exactly how BHs are brought to merger in dense clusters <cit.>.

Note here that the tracks shown in Fig. <ref>
show a representative characteristic strain at any point in the binary lifetime.
They are only corrected for
the observation-time or life-time of the detector in the sense that the characteristic strain is computed differently for binaries that will not significantly evolve in frequency over a chosen four-year observation time. This results in the `knee' in the 2-body tracks (green) that is described more thoroughly in <cit.>. Binaries for which observation begins anywhere on the track to the left of the knee will not merge within an observation time.
These tracks therefore serve more like an illustrative overview along side with Fig. 1
from <cit.>. However, we will provide a more in-depth signal-to-noise (S/N) analysis in an upcoming paper (for a discussion on multi-band S/N calculations see <cit.>), together with how the observational prospects change with cluster properties and BH masses.






§ CONCLUSIONS




Recent work on the formation of BBH mergers in GCs has shown that the inclusion of PN few-body dynamics leads to distinctive merger
populations with measurable orbital eccentricities in both
LISA and LIGO <cit.>.
For these studies, the dynamical few-body channels that so far have been systematically explored include strong- and weak
binary-single interactions <cit.>,
and strong binary-binary interactions <cit.>.

In this paper we have expanded on this emerging picture of how BBHs are driven to merger in GCs,
by studying the formation of single-single GW capture mergers. For this we have focused on
deriving the rate of BBH mergers from this single-single channel relative to that of BBH mergers forming during binary-single interactions.
This approach gives us a much better handle on exactly how frequent single-single GW capture mergers might be, in contrast to other studies
that report (highly uncertain) absolute rates.

Using analytical arguments, we find that the contribution from single-single GW
capture mergers relative to those forming during binary-single interactions, does not strongly depend on either the absolute value of the central velocity dispersion,
or the BH number density (see e.g. Eq. (<ref>)).
Instead, it mainly depends on the number of BBHs relative to single BHs, i.e. the BH binary fraction, and the shape of the density profile of the
single BH population. Assuming the single BHs follow either a uniform- or a Plummer's distribution,
we have shown that the rate of single-single GW capture mergers should be comparable
to that of mergers forming during binary-single interactions when the binary fraction is at the percent level. Interestingly, recent MC simulations
(see Fig. <ref>) do in fact indicate that the fraction of dynamically formed BBHs is at the percent level,
which leads us to conclude that single-single GW capture mergers are expected to form at an observable rate.
Knowing the relative contributions from both single-single and binary-single interactions is extremely important, as their interplay and
resulting GW observables provide the key to probe how BBHs might form in dense stellar clusters.

Finally, using a semi-analytical MC framework we derived GW peak frequency- and eccentricity distributions for
single-single GW capture mergers, and mergers forming during and in-between strong binary-single interactions. As also noticed by <cit.>,
the single-single GW capture mergers form right in the deci-Hertz regime where the proposed DECIGO and Tian Qin detectors are sensitive.
For a binary fraction of F_ bs = 0.01 and assuming a Plummer's sphere for the single BH population, we further illustrated that BBHs with eccentricities log(e) > -1.5
at 1 Hz and -1.5 > log(e) > -2.5 at 10 Hz are dominated by the single-single GW capture channel. This has major implications for mapping out how BBHs
form as a function of redshift in dense stellar clusters with DECIGO/Tian Qin as well as third generation GW detectors such as the ET and CE.







Acknowledgments. —
It is a pleasure to thank the organizers of the
CIERA Mini-workshop on Post-Newtonian Dynamics in Stellar Clusters, December 5-8, 2018,
where part of this work was initiated. The authors also thank Mirek Giersz for useful
comments on the manuscript.
JS acknowledges support from the Lyman Spitzer Fellowship
and the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie
grant agreement No. 844629.
DJD acknowledges financial
support from NASA through Einstein Postdoctoral Fellowship award
number PF6-170151.
KK acknowledges support by the National
Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1324585.
CR acknowledges supported from the Pappalardo Postdoctoral Fellowship at MIT.
AA is supported by the Carl Tryggers Foundation through
the grant CTS 17:113.


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