# eLISA eccentricity measurements as tracers of binary black hole formation

###### Abstract

Up to hundreds of black hole binaries individually resolvable by eLISA will coalesce in the Advanced LIGO/Virgo band within ten years, allowing for multi-band gravitational wave observations. Binaries formed via dynamical interactions in dense star clusters are expected to have eccentricities – at the frequencies Hz where eLISA is most sensitive, while binaries formed in the field should have negligible eccentricity in both frequency bands. We estimate that eLISA should always be able to detect a nonzero whenever ; if , eLISA should detect nonzero eccentricity for a fraction () of binaries when the observation time is () years, respectively. Therefore eLISA observations of black hole binaries have the potential to distinguish between field and cluster formation scenarios.

###### pacs:

04.30.Tv,04.25.Nx,97.60.Lf## I Introduction

With the detection of gravitational waves (GWs) by the LIGO/Virgo
scientific collaboration Abbott *et al.* (2016a), black hole (BH)
binaries have entered the realm of observational astronomy. The first
detected binary system (GW150914) has source-frame component masses
, resulting in a
merger remnant of mass .
Its estimated luminosity distance is
Mpc, corresponding to a redshift
Abbott *et al.* (2016b).
The trigger LVT151012 is also likely to be a binary BH system with
masses and
luminosity distance Gpc.
These early GW observations set lower bounds on binary BH merger
rates Abbott *et al.* (2016c), raising interesting questions on the
formation mechanism of compact binary systems. As summarized in the
LIGO/Virgo collaboration paper discussing the astrophysical
implications of the discovery Abbott *et al.* (2016d), BH
binary mergers similar to GW150914 can either result from the
evolution of isolated binaries in galactic fields or from dynamical
interactions in young and old dense star clusters (see
Postnov and Yungelson (2014); Benacquista and Downing (2013) for reviews of these
formation scenarios).

Sesana Sesana (2016) showed that up to hundreds of
GW150914-like BH binaries individually resolvable by a space-based
detector such as eLISA Amaro-Seoane *et al.* (2013) will coalesce in the
LIGO band within ten years. eLISA observations can identify the time
and location of the merger with uncertainties in the merger time
smaller than s, and sky localization accuracies that in many
cases are better than deg. This will allow
multi-wavelength electromagnetic telescopes to point the GW event in
advance and to constrain models of electromagnetic emission
associated with BH binary mergers. Furthermore, BH binaries that
span both the eLISA and Advanced LIGO frequency bands can yield
stringent tests of modified theories of gravity that predict
propagation properties different from general
relativity Abbott *et al.* (2016e); Yunes *et al.* (2016), and in
particular of theories allowing for dipolar radiation in BH
binaries Barausse *et al.* (2016).

The GW150914 signal does not set strong bounds on the eccentricity
of the binary. Ref. Abbott *et al.* (2016b) quotes a
preliminary constraint of at Hz. It is unlikely that
Advanced LIGO observations may use eccentricity measurements to
differentiate between the field and cluster scenarios: as shown
e.g. in Fig. 3 of Ref. Abbott *et al.* (2016d), binaries in
the LIGO band will almost always be circular. Earth-based GW
observations could only differentiate between field and cluster
formation by looking at spin dynamics (see
e.g. Gerosa *et al.* (2013)), redshift distribution and possibly kicks.

However binaries formed in clusters – unlike binaries formed in the field – should have non-negligible eccentricity in the eLISA band. Here we show that eLISA could measure the eccentricity of BH binaries in the last few years or months of their inspiral, constraining their formation mechanism. As a byproduct, we also show how eccentricity affects the estimation of other binary parameters (masses, merger time, distance and sky location).

The possibility of multi-band detections of eccentric
intermediate-mass BH binaries by Earth- and space-based detectors was
pointed out in a series of papers by Amaro-Seoane et
al. Amaro-Seoane *et al.* (2009, 2010); Amaro-Seoane and Santamaria (2010),
but those papers focused on BH binaries with much larger total mass.
Seto Seto (2016) recently studied eccentric BH binaries of the
GW150914 type in the eLISA band, but the focus of his work was
considerably different from ours. He considered monochromatic sources
at frequencies mHz, which have negligible frequency
evolution, and for which the merger will not be visible in the
Advanced LIGO band. On the contrary we focus on binaries that evolve
rapidly in the high-frequency band of the eLISA sensitivity window,
possibly merging in the Advanced LIGO band. Seto used the quadrupole
formula to estimate the signal (which for is dominated
by the second harmonic, i.e. by GWs emitted at twice the orbital
frequency) and estimated the binary eccentricity from the
characteristic amplitude of the third harmonic of the signal. We use
general relativistic waveform models and a Fisher matrix analysis to
estimate errors in the measurement of the eccentricity and of other
parameters characterizing the source (masses, merger time, distance
and sky location). We work within the small-eccentricity waveform
generation formalism proposed in Yunes *et al.* (2009) and further
developed in Tanay *et al.* (2016), which is adequate to address the
present problem, but we note that various groups have recently made
progress in the development of models for the generation, detection
and parameter estimation of GWs from eccentric binaries (see
e.g. Martel and Poisson (1999); Mikoczi *et al.* (2012); Huerta *et al.* (2014); Sun *et al.* (2015); Forseth *et al.* (2016); Hopper *et al.* (2016); Moore *et al.* (2016); Loutrel and Yunes (2016)).

In the rest of this introduction we review some literature on BH
formation channels and merger rates, including recent papers that were
not included in the LIGO review on this topic Abadie *et al.* (2010), to
justify our statement that field binaries should typically be
circular, while binaries formed in clusters may have residual
eccentricities. A more realistic study would require astrophysical
models of the mass, spin and eccentricity distribution of BH binaries
in both formation channels and Bayesian model
selection Stevenson *et al.* (2015); such an analysis is beyond the
scope of this paper, where we focus mostly on the preliminary issue of
parameter estimation accuracy. Then we present an executive summary of
our main results on eLISA measurements of eccentricity. Finally we
outline the plan of the paper for the reader’s convenience.

### i.1 Black hole formation channels

Field binaries. Tutukov and Yungelson studied the
evolution of isolated massive binaries before the discovery of the
binary pulsar and predicted the formation of merging binary compact
objects composed of neutron stars (NSs) and/or
BHs Tutukov and Yungelson (1973, 1993). Some early population studies
even predicted that binary BH mergers could dominate detection rates
for ground-based GW detectors Lipunov *et al.* (1997). Several groups
made predictions on the relative rates of BH-BH, BH-NS and NS-NS
binaries over the
years Bethe and Brown (1998); Grishchuk *et al.* (2001); Nelemans *et al.* (2001); Belczynski *et al.* (2001); Voss and Tauris (2003); De Donder and Vanbeveren (2004); Belczynski *et al.* (2008); O’Shaughnessy *et al.* (2008); Kalogera *et al.* (2007). All
of these predictions were largely uncertain, but as late as 2014 some
studies concluded that BH-BH binary detection rates would be
negligible for Advanced LIGO Mennekens and Vanbeveren (2014).

Belczynski et al. Belczynski *et al.* (2010) pointed out that BH-BH
binaries could dominate Advanced LIGO detection rates if a significant
fraction of stars form in low-metallicity environments. This claim
was refined in subsequent work using the Startrack code with
various prescriptions for common envelope evolution, BH kicks and
gravitational
waveforms Dominik *et al.* (2012, 2013); Belczynski *et al.* (2014); Dominik *et al.* (2015); Belczynski *et al.* (2016a, b),
as well as various prescriptions for metallicity evolution as a
function of redshift. These works consistently predicted that BH
mergers should dominate the rates, and that large-mass BH binaries
(including total masses and above) should be
detectable in large numbers out to . Notably, before the
detection of GW150914 Belczynski et al. Belczynski *et al.* (2016a)
found that “the most likely sources to be detected with the advanced
detectors are massive BH-BH mergers with total redshifted mass
.”

Similar conclusions were reached using other population synthesis
codes Spera *et al.* (2015); Eldridge and Stanway (2016). Eldridge and
Stanway Eldridge and Stanway (2016) found that GW150914 has a low
probability of arising from a stellar population with initial
metallicity (or ); when
, a large fraction () of binary BH mergers is
expected to have masses compatible with the GW measurement.
Other groups suggested that common envelope evolution may not be the
only way to form massive BHs. Another channel involves massive, tight
binaries where mixing induced by rotation and tides transports the
products of hydrogen burning throughout the stellar envelopes,
enriching the entire star with helium and preventing the build-up of
an internal chemical
gradient Mandel and de Mink (2015); de Mink and Mandel (2016); Marchant *et al.* (2016). In
these scenarios there would never be a giant phase: both stars would
stay within their Roche lobes and eventually form massive BHs, because
the cores that collapse would be large.
Yet another scenario invokes a Population III origin for massive BH
binaries Kinugawa *et al.* (2014, 2016); Inayoshi *et al.* (2016),
but semi-analytical models suggest that the probability of GW150914
having formed in the early Universe is
Hartwig *et al.* (2016).

The key point for us is that BH binaries produced in the field
are expected to be circular in both the Advanced LIGO and eLISA
bands. Typical eccentricity distributions for BH binaries at
frequencies Hz are shown in Fig. 5
of Kowalska *et al.* (2011); predicted values are in the range
. Massive BH binaries of interest
for multi-band astronomy are at the heavy end of the mass spectrum, so
they should receive small kicks (see e.g. Sec. 6
of Belczynski *et al.* (2016a)) and be on the small-eccentricity side of
the distributions predicted in Kowalska *et al.* (2011). For all
practical purposes, massive BH binaries formed in the field can be
assumed to be circular in the eLISA band.

Dense star clusters.
A different scenario for binary BH formation involves dense star
clusters Kulkarni *et al.* (1993); Sigurdsson and Hernquist (1993). In these
environments BHs quickly become the most massive objects. They sink
towards the cluster core, form pairs through dynamical interactions,
and they are most commonly ejected in binary configurations with
inspiral times shorter than the age of the Universe. This basic
scenario was refined by various
authors Portegies Zwart and McMillan (2000); Gurkan *et al.* (2006); Gültekin *et al.* (2006); Fregeau *et al.* (2006); Miller and Lauburg (2009); O’Leary *et al.* (2009); Moody and Sigurdsson (2009); Downing *et al.* (2010, 2011); Morscher *et al.* (2013); Goswami *et al.* (2014); Ziosi *et al.* (2014); Rodriguez *et al.* (2015, 2016); O’Leary *et al.* (2016); Bartos *et al.* (2016); Chatterjee *et al.* (2016).

A dynamical effect that can produce large eccentricities in the LIGO
band is the Kozai mechanism Wen (2003). Recent studies of
Kozai-Lidov resonances showed that binary BH mergers may be more
likely inside the radius of influence of supermassive BHs in galactic
centers Antonini and Perets (2012); VanLandingham *et al.* (2016) or in
hierarchical triples Antonini *et al.* (2016); Samsing *et al.* (2014),
More work is required to understand whether these events can lead to
rates comparable to the other formation channels, and also to
establish the conditions (masses, inclinations, semi-major axes and
eccentricities of both the inner and outer binary) that could result
in non-negligible eccentricities in the eLISA band.

Some predictions for the eccentricity distribution of dynamically
formed binaries can be found in Fig. 10
of Rodriguez *et al.* (2016). The eccentricity at Hz of BH
binaries merging at in the capture scenario peaks at
, with most of the sources having . The classic
results by Peters and Mathews Peters and Mathews (1963) imply that, so long
as , (see e.g. Fig. 1 of
Enoki and Nagashima (2007)). Here we focus on sources emitting at
Hz in the eLISA band. Their typical eccentricity at
frequency is thus , with most sources
having . Almost all relevant eLISA sources (both
resolvable and unresolvable) are at Hz, and their expected
eccentricity is . These numbers are large enough to
require eccentric templates for matched filtering, but the amplitude
and phasing of the signal for binaries with can be
treated in a small-eccentricity approximation.
To summarize: extrapolating the results in
Ref. Rodriguez *et al.* (2016) to lower frequencies, we expect
dynamically formed BH binaries to have small but non-negligible
eccentricities in the eLISA band, and therefore a
small-eccentricity approximation is adequate to study this problem.

### i.2 Executive summary

Consider a binary system with component masses (in the source frame) and , total mass , symmetric mass ratio and chirp mass . Assume that the binary is located at redshift – or equivalently, for a given cosmological model, at luminosity distance – so that the redshifted chirp mass , the redshifted total mass , and similarly for the other mass parameters. Two angles specify the direction of the source in the solar barycenter frame, and for convenience we introduce . Let be the coalescence time, the coalescence phase, the binary’s orbital angular momentum vector (with the corresponding unit vector), and a unit vector pointing in the source direction as measured in the solar barycenter frame. Furthermore, let be the frequency normalized to a reference frequency – here chosen to be – where the eccentricity is , and introduce the standard post-Newtonian (PN) parameter .

We model eLISA as two independent interferometers with non-orthogonal
arms. The sky-averaged noise power spectral density for each of the
two interferometers is denoted by NA, as
in Klein *et al.* (2016); here refers to different
acceleration noise baselines, and denotes different
armlengths (1 or 5 Gm). The observation time is chosen
to be either 5 or 2 years. This choice significantly affects the
signal-to-noise ratio (SNR): if, following Sesana (2016), we
adopt a fiducial 5-year observation time and assume that the binary
merges at the end of the observation, the initial frequency of the
binary will be

(1) |

where we scaled the result by the estimated redshifted chirp mass of GW150914. Our SNR and Fisher matrix calculations are truncated at a maximum frequency , beyond which the eLISA noise is not expected to be under control.

Our main results on eccentricity measurements are summarized in Figs. 1 and 2. Their behavior can be understood, at least qualitatively, using simple scaling arguments. Neglecting correlations between parameters, in a Fisher matrix approximation the error on is

(2) |

where denotes the Fourier transform of the GW amplitude and is the noise power spectral density of the detector. To leading order in a small-eccentricity expansion (what we call the “restricted eccentric waveform” in Section III.1 below) and in the stationary phase approximation, corrections due to the eccentricity enter only in the GW phase through the term proportional to in Eq. (III.1) below, and therefore . Let us approximate the frequency dependence of the noise power spectral density by a power law, . Since the dominant contribution to the Fisher matrix comes from the lowest frequencies, from Eq. (2) we have

(3) |

where on the second line we estimated for a given observation time using the quadrupole formula (1) for a circular binary. In summary, to leading order we expect a rough scaling law of the form

(4) |

with for (N2A5 and N2A1, 2yrs), for (N2A1, 5yrs) and . Note that depends not only on the noise curve, but also on , that is lower for longer : the frequency dependence of the eLISA noise curve is flatter when we consider N2A1 and a 5-year observation time.

This rough approximation will break down when the SNR is small (so the Fisher matrix approximation is invalid), correlations cannot be neglected (as is the case for the “restricted” eccentric waveform), or eccentricities are too small and therefore not measurable. In practice we carry out numerical calculations using the “full” eccentric waveform described in Section III.2 below. Obtaining analytical estimates in this case is more complicated due to the existence of frequency sidebands, but by fitting our numerical data we found that the scaling law with holds well also for these full eccentric waveforms. Because of the breaking of some parameter degeneracies, the scaling with is modified from the previous simple prediction: for . A more accurate scaling law obtained by fitting our numerical data is

(5) |

where the fitting parameters () are listed in Table 1. This scaling is further illustrated in Fig. 2.

The simple scalings of Eqs. (4) and (5) are helpful to understand the numerical results shown in Fig. 1. The error gets larger with decreasing eccentricity: when the typical error is , but when when the typical error . For a given noise curve (N2A5 or N2A1), as expected, longer observation times lead to smaller errors. The effect of changing the armlength is sensibily milder, but (everything else being equal) 5 Gm configurations (A5) yield slightly smaller errors than 1 Gm configurations (A1).

Recall from our previous discussion that binaries formed in dense star clusters are expected to have eccentricities at the frequencies Hz where eLISA is most sensitive, while binaries formed in the field should have negligible eccentricity at these frequencies. eLISA should always be able to detect a nonzero whenever ; if , we find that eLISA will detect nonzero eccentricity for a fraction () of binaries when () years, respectively. Therefore eLISA observations of GW150914-like BH binaries have the potential to distinguish between field and cluster formation scenarios. This is the main result of our paper.

noise | ||||
---|---|---|---|---|

N2A1 | 2yr | 2.57 | 1.5 | |

N2A1 | 5yr | 2.19 | 1.5 | |

N2A5 | 2yr | 2.57 | 1.5 | |

N2A5 | 5yr | 2.57 | 1.5 |

### i.3 Plan of the paper

The rest of the paper provides details on the source catalogs used for our Monte Carlo simulations, on our waveform models, and on the parameter estimation errors for other source parameters (including masses, distance and sky location). In Section II we describe how we generate the source catalogs used in our Monte Carlo analysis. In Section III we describe our “restricted” and “full” eccentric waveform models. In Section IV we show how eccentricity affects errors on the other parameters (time of merger, masses, distance and sky location). We conclude with possible directions for future work. FInally, in Appendix A we show that confusion noise is unlikely to affect our parameter estimation calculations. In the whole paper we use geometrical units ().

## Ii Source catalogs

Following the LIGO/Virgo paper on rate
estimates Abbott *et al.* (2016c), we randomly draw the masses of the
two BHs and from a log-flat mass distribution in the range
, with the additional requirement
that . The binary’s sky location and the orientation
of the angular momentum are distributed uniformly over the sky. The
source redshift is randomly selected assuming a constant binary BH
merger rate and the Lambda-Cold-Dark-Matter
(CDM) flat cosmological model with ,
and
Hinshaw *et al.* (2013).
For each binary we can compute the SNR , defined as

(6) |

where we use analytical approximations to the N2A1 and N2A5 noise
power spectral densities Klein *et al.* (2016). When
computing SNRs we fix the reference eccentricity to zero:
corrections due to nonzero are of order , and they are
less than for the fiducial values considered in
this paper.

noise | 95% | |||||
---|---|---|---|---|---|---|

N2A1 | 2yr | 2 | 0–8 | 0.0353 | 78.9 | 11.1 |

N2A1 | 5yr | 4 | 0–14 | 0.0494 | 80.7 | 11.0 |

N2A5 | 2yr | 30 | 5-121 | 0.0803 | 82.8 | 10.6 |

N2A5 | 5yr | 106 | 13-348 | 0.170 | 85.0 | 10.9 |

We generate binary BH sources that are observable by eLISA by imposing a detection threshold for each observation period and noise curve. The mass, redshift and SNR distributions of the events generated in this way are shown in Fig. 3, and the medians of these quantities are listed in Table 2. The SNR and mass distributions are very similar in all four cases, due to the chosen detection threshold in SNR and to the relatively limited mass range for the binary components, respectively. With higher detector sensitivity and longer observation times (corresponding to smaller ) it is possible to detect sources at higher redshifts, because the GW amplitude at small redshifts. Note that the tail of the redshift distribution extends below , corresponding to , below which the galaxy distribution is not continuous. The number of sources we simulated () was chosen arbitrarily to study probability distributions in parameter estimation accuracy. The absolute number of observed events depends, of course, on binary BH merger rates. In Table 2 we list the median and 95% confidence interval of expected eLISA detections for each assumed noise curve and mission duration.

## Iii Eccentric binary waveforms

The most accurate Fourier-domain eccentric waveforms
available at present were computed by Yunes et
al. Yunes *et al.* (2009) and Tanay et al. Tanay *et al.* (2016) in the
small-eccentricity approximation, i.e. using a power series
expansion in . The waveforms in Yunes *et al.* (2009) are
accurate up to (Newtonian, ) order in amplitude and
(Newtonian, ) order in phase. The waveforms
in Tanay *et al.* (2016) used here are accurate up to (Newtonian,
) order in amplitude and (2PN, ) order in phase. The
waveform phase calculation has recently been extended up to 3PN by
Moore et al. Moore *et al.* (2016); however their calculation is
limited to order in amplitude and order in
phase. The waveforms in Yunes *et al.* (2009); Tanay *et al.* (2016) are more
accurate for our present purposes, because eLISA observes the
low-frequency early inspiral of a BH binary, where eccentricity is
larger (recall that ) and PN effects are
relatively less important.

As discussed in the introduction, the sources we are interested in are
expected to have eccentricities at frequencies
Hz, roughly corresponding to the “bucket” of eLISA’s
sensitivity window. Therefore we are justified in using the
small-eccentricity waveform generation formalism proposed
in Yunes *et al.* (2009) and developed in Tanay *et al.* (2016).
Nonspinning eccentric waveforms depend on ten
physical parameters
:
redshifted chirp mass, symmetric mass ratio, time and phase at
coalescence, luminosity distance, eccentricity at
, two angles describing the direction of the
orbital angular momentum, and two angles corresponding to the
orientation of the source in the sky. The angular variables are
measured in the solar barycentric frame. This eccentric waveform
is, in general, quite complicated, and for our parameter estimation
calculations we will further expand the frequency-domain waveforms,
first including only phase corrections up to leading order in
eccentricity (what we will refer to as the “restricted eccentric”
case, Section III.1), and then including up to
next-to-leading order phase corrections as well as amplitude
modulations (”full eccentric” case, Section III.2). As we
will see, restricted eccentric waveforms are useful to gain analytical
understanding of the effects due to nonzero eccentricity, but they are
insufficient for parameter estimation. This happens mainly because
restricted waveforms do not include frequency sidebands to the
dominant harmonic at . These sidebands, which are
present in the “full eccentric” waveforms, carry crucial information
that is necessary to break parameter degeneracies.

### iii.1 Restricted eccentric waveforms

The Fourier transform of the 2PN restricted gravitational waveform for
a nonspinning circular binary with an eccentric-orbit phase
correction reads Krolak *et al.* (1995)

(7) |

where the amplitude includes a factor
because eLISA’s arms have an opening angle of ,
as well as a factor needed to use a sky-averaged
sensitivity Berti *et al.* (2005). Denoting the
th detector’s response functions by and
, the unit vector of orbital angular momentum
by , the unit vector directed to the source by
, and the phase of the detector’s orbital motion
by , the phasing is given by

(8) | ||||

(9) | ||||

(10) |

where . The time variable is related to the frequency by

(11) |

### iii.2 Full eccentric waveforms

A better approximation to the Fourier transform of the gravitational
waveform for a nonspinning eccentric binary
is Yunes *et al.* (2009); Tanay *et al.* (2016)

(12) | ||||

(13) |

where

(14) | ||||

(15) | ||||

(16) | ||||

(17) |

The coefficients , ,
, depend on the eccentricity
and on the inclination angle , and they are given
in Yunes *et al.* (2009) (where the azimuthal angle determining the
position of the detector relative to the source, in the
notation of Martel and Poisson (1999); Yunes *et al.* (2009), is set to zero).
Here we assume and retain terms up to , with
the following result:

(18) | ||||

(19) | ||||

(20) |

where

(21) | ||||

(22) |

The 2PN phase up to is Tanay *et al.* (2016)

(23) |

The relation between time and frequency up to 2PN can be derived from
Eq. (B8a) in Tanay *et al.* (2016). Keeping terms up to , we
can integrate and obtain . Setting with
, we have

(24) |

noise | [s] | |||||||
---|---|---|---|---|---|---|---|---|

N2A1 | 2yr | 0 | 1.52 | 0.438 | — | |||

2.74 | 0.438 | |||||||

2.74 | 0.438 | |||||||

2.75 | 0.437 | |||||||

N2A1 | 5yr | 0 | 1.75 | 0.469 | — | |||

2.25 | 0.469 | |||||||

2.25 | 0.469 | |||||||

2.26 | 0.469 | |||||||

N2A5 | 2yr | 0 | 2.29 | 0.473 | — | |||

5.11 | 0.473 | |||||||

5.11 | 0.473 | |||||||

5.13 | 0.473 | |||||||

N2A5 | 5yr | 0 | 3.17 | 0.529 | — | |||

4.36 | 0.529 | |||||||

4.37 | 0.534 | |||||||

4.37 | 0.530 |

noise | [s] | |||||||
---|---|---|---|---|---|---|---|---|

N2A1 | 2yr | 0 | 1.52 | 0.438 | — | |||

2.71 | 0.436 | |||||||

2.13 | 0.436 | |||||||

1.43 | 0.432 | |||||||

N2A1 | 5yr | 0 | 1.75 | 0.469 | — | |||

2.27 | 0.469 | |||||||

1.94 | 0.464 | |||||||

1.72 | 0.450 | |||||||

N2A5 | 2yr | 0 | 2.29 | 0.473 | — | |||

5.71 | 0.473 | |||||||

3.43 | 0.473 | |||||||

2.23 | 0.463 | |||||||

N2A5 | 5yr | 0 | 3.17 | 0.529 | — | |||

4.45 | 0.529 | |||||||

3.54 | 0.525 | |||||||

3.14 | 0.505 |

## Iv Parameter estimation errors

Median values of the parameter estimation errors for nonspinning binaries under different assumptions on the eLISA detector noise and on the observation time are listed in Table 3 for restricted eccentric waveforms, and in Table 4 for full eccentric waveforms.

Let us focus first on the restricted eccentric parameter estimation results of Table 3. The phasing of the inspiral signal observed by eLISA is predominantly determined by the mass parameters, which are therefore estimated very well in most cases. The signal is also modulated by the detector’s orbital motion in a way that depends on the position of the source. This allows us to determine the sky location of the source and, to some limited level of accuracy, also the luminosity distance (see e.g. Cutler (1998); Hughes (2002)). For restricted eccentric waveforms enters only in the phasing [cf. Eq. (III.1)], and therefore it has large correlations with the mass parameters and . As a consequence the median errors on and are degraded by a factor of 4–6 with respect to the circular case when . The estimation errors on the merger time and sky location also get worse by several tens of per cent, but the degradation in accuracy due to eccentricity is not as large as in the case of the mass parameters. Quite remarkably, this degradation in parameter estimation is independent of : the high correlation between the eccentricity and the mass parameters is not broken by increasing from to .

As shown in Table 4, this is not the case for full eccentric waveforms: the additional structure in the amplitude and phase due to higher-order effects is crucial to break the degeneracies. Once again, a nonzero eccentricity reduces the accuracy in measuring the other parameters, in particular or , whose determination is degraded by a factor of 4–7 with respect to the circular case when . However, in stark contrast with the restricted waveform, as we increase the correlations are partially broken, and the errors on all parameters (including itself: cf. Fig. 1 above) become smaller. In fact, for the accuracy in determining the mass parameters becomes slightly better than in the circular case. A qualitatively similar (but quantitatively smaller) improvement is seen in other parameter errors, such as and .

Histograms of for full eccentric waveforms were shown in the introduction (Fig. 1), where we presented analytical arguments to justify why decreases as the chirp mass and increase. Since frequency sidebands break the correlation between parameters, parameter estimation errors decrease more rapidly with in the full eccentric case than in the restricted eccentric case. A best fit to our numerical results for yields the scaling relation of Eq. (5), which is compared against the data in Fig. 4. The accuracy of the scaling relation degrades for eLISA designs with shorter armlength and for shorter mission durations. The scattering of the data is also larger for small eccentricities, where correlations between and the other parameters are larger.

In Fig. 5 we compare the error on the merger time for full (left) and restricted (right) eccentric waveforms. This plot shows quite clearly that as we increase (bottom to top in each figure) the determination of gets better in the full eccentric case, where the more complex waveform breaks the correlation between the parameters, but not in the restricted eccentric case. This general trend applies to all measurement errors, so in the following we focus on full eccentric waveforms.

In Fig. 6 we use full eccentric waveforms to compute parameter estimation errors on the chirp mass (top left), symmetric mass ratio (top right), luminosity distance (bottom left) and sky location (bottom right) for full eccentric nonspinning binaries. The most notable feature of this plot is that the errors on the mass parameters decrease with , while the errors on source localization and distance are not sensibly affected by .

Looking at the sky location determination in Fig. 6, a careful reader will notice the seemingly counterintuitive result that binaries observed for 5 years will be located with worse precision than binaries observed for 2 years. This is simply a selection effect. Our catalogs were constructed by imposing an SNR threshold of , therefore catalogs corresponding to shorter observation times include systems with smaller luminosity distance and more optimal orientation. To show that selection effects are indeed responsible for this counterintuitive trend, in Fig. 7 we plot histograms of the angular resolution accuracy rescaled by the luminosity distance . When normalized to , the angular resolution distributions for the 5-year catalogs are indeed almost indistinguishable from those computed for the 2-year catalogs.

## V Discussion

In this section we discuss how our parameter estimation calculations would change if we were to relax some of the approximations involved in our waveform models and parameter estimation techniques. In particular, we focus on the effect of high eccentricity, spins, confusion noise, and the Fisher matrix approximation.

### v.1 Highly eccentric binaries

One important limitation of our approach is the small- expansion
adopted in our waveform models. All BH binaries we consider are
evolving in frequency above , and our results are
accurate at the level of . Expected astrophysical
eccentricities for field binaries and binaries in a dense stellar
cluster are . For these populations our phasing is
accurate to within , so we expect our parameter estimation
results to be representative of the capabilities of eLISA when more
accurate waveforms will be available. For binary populations models
which predict large numbers of binaries with ,
however, our small-eccentricity approximation is not good enough. In
principle one could keep terms up to using
currently available waveforms, but even the detection of highly
eccentric () binaries requires nonperturbative (in )
eccentric waveform. The development of accurate high-eccentricity
waveforms is a very active research area and it is beyond the scope of
this
study Martel and Poisson (1999); Mikoczi *et al.* (2012); Huerta *et al.* (2014); Sun *et al.* (2015); Forseth *et al.* (2016); Hopper *et al.* (2016); Moore *et al.* (2016); Loutrel and Yunes (2016).

### v.2 Spinning binaries

In this paper we considered nonspinning BH binaries, but the introduction of spin parameters in the full eccentric waveforms should not degrade parameter estimation accuracy. For binaries with aligned spins, spin effects enter the waveform at 1.5PN order, while eccentricity enters the waveform at Newtonian level and it is proportional to . This implies that spin effects are more important at higher frequencies and eccentricity dominates at lower frequencies, so that degeneracies between spin and eccentricity effects should be small. In fact we have computed errors on for nonspinning and aligned-spin binaries using the “restricted” eccentric waveforms of Section III.1, and confirmed that relative variations in the errors are below 60% (in the worst cases) for all eLISA configurations considered in this study.

### v.3 Confusion noise

If many binaries emit in a given observational frequency band, their signal will constitute a source of confusion noise that can limit detectability and parameter estimation accuracy. A simple estimate of this confusion noise is given in Appendix A, and it allows us to conclude that our signals are unlikely to be contaminated by confusion noise. To verify this statement we can compare the typical starting frequency of a BH binary for a given eLISA observation time with the “confusion noise frequency” below which more than two GW signals exist simultaneously in a single frequency bin. The former is () for 2-year (5-year) eLISA observations, respectively. Using Eq. (29), the confusion noise frequency can be estimated to be () for a typical BH binary merger rate of and 2-year (5-year) eLISA observations, respectively. Therefore, in general, the signal should be relatively easy to resolve and disentangle in the frequency region of interest for multiband binaries. In principle extreme mass ratio inspirals may overlap in frequency with some multiband binaries, but their waveform is expected to be quite different (because of high eccentricity and spin precession). Note, moreover, that we do not expect significant contributions to confusion noise from other galactic sources (such as WD-WD binaries) at the frequencies of interest.

### v.4 Fisher matrix analysis

The Fisher matrix approximation is well known to break down for
low-SNR systems (see e.g. Vallisneri (2008)). A comparison of
Fisher-matrix results with Markov-Chain Monte Carlo results can be
found in Rodriguez *et al.* (2013). Their study focuses on Advanced
LIGO, but their typical SNRs () are similar to those of
interest in our work. Ref. Rodriguez *et al.* (2013) shows that there
Fisher-matrix parameter estimation results have large scatter, but
median values are relatively robust. In this sense, our Fisher
analysis should be relatively reliable for Monte-Carlo studies of
source populations.

In the LISA context, parameter estimation studies beyond the Fisher
matrix approximation were implemented in some studies of WD binaries
end EMRIs, most notably in the Mock LISA Data Challenges
Babak *et al.* (2010); Blaut *et al.* (2010). Some results of those studies
concern low SNR sources that remain in band for a long time, and they
support the validity of our analysis. The data challenge is to dig out
the signal from the data by matching a sufficient number of cycles,
but once a signal is detected, the precision to which the parameters
are estimated is comparable to Fisher matrix estimates. This has been
demonstrated both for galactic WDs (similar to BH binary signals that
hardly evolve in frequency during the eLISA observation, i.e. those at
frequencies Hz) and for EMRIs (similar to massive BH
signals chirping and “crossing over” to the Advanced LIGO band at
Hz). In both cases, once the signal is above the detection
threshold (usually assumed to be SNR for WD binaries and
SNR for EMRIs), parameters are estimated with very high
precision and usually also with good accuracy. In a few cases, EMRI
parameters are not accurately recovered because of failures in
identifying the global maximum in the likelihood function, but this is
an issue related to the search algorithms: the likelihood function
exploration fails to correctly identify the sources. This issue is
unlikely to be as relevant here, since BH binary eccentricities are
usually small, implying a smoother behavior of the likelihood function
(prominent secondary maxima associated to strong higher harmonics of
the signal should be absent). In any case, both WD binary and EMRI
parameters have been recovered with high accuracy and precision in the
aforementioned numerical experiments, and the errors are not too far
from Fisher Matrix estimates (usually within a factor of five in the
worst cases).

## Vi Conclusions

Binaries formed via dynamical interactions in dense star clusters are
expected to be at least mildly eccentric
(–) at the frequencies Hz
where eLISA is most sensitive Rodriguez *et al.* (2016). On the
contrary, binaries formed in the field are expected to have negligible
eccentricities (–) in the eLISA
band Kowalska *et al.* (2011). In this paper we carried out Monte Carlo
simulations over a catalog of BH binaries that merge in the Advanced
LIGO band to assess eLISA’s potential to measure eccentricity, and
therefore differentiate between competing BH formation scenarios. We
showed that eLISA should always be able to detect a nonzero
whenever . If , eLISA will detect
nonzero eccentricity for a fraction () of
binaries when the observation time is () years,
respectively. Therefore eLISA observations of BH binaries have the
potential to distinguish between field and cluster formation
scenarios.

In the future we plan to refine this analysis using better waveform
models and more realistic astrophysical assumptions. It is
particularly interesting to consider binaries inspiralling at lower
frequencies: these binaries will not necessarily “cross over” to the
band accessible by Earth-based detectors, but they may have higher
eccentricity, e.g. because of the Kozai
mechanism Wen (2003); Antonini and Perets (2012); VanLandingham *et al.* (2016); Antonini *et al.* (2016); Samsing *et al.* (2014). These
highly eccentric systems present a harder challenge in terms of data
analysis, and they motivate further efforts to develop accurate
waveform models and reliable parameter estimation schemes.

## Appendix A Confusion noise

At low frequencies the frequency evolution of a binary is slower, and the number of sources in a given frequency bin is larger. If there are more than two signals simultaneously in a single bin, these signals are indistinguishable and can produce confusion noise. In this Appendix we estimate this effect, and we show that confusion noise is unlikely to affect our conclusions.

The number of inspiral GW signals in a bin of frequency resolution is given by

(25) |

Here is the merger rate per unit time, which can be obtained by integrating over redshift:

(26) |

where is the comoving distance to redshift , and is the merger rate per unit comoving volume and unit proper time at redshift . For a constant merger rate , Eq. (26) reduces to

Substituting the frequency derivative at Newtonian order Cutler and Flanagan (1994)

into Eq. (25), we have

For a power-law mass distribution of the form