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11th International LISA Symposium IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 840 (2017) 012021 doi :10.1088/1742-6596/840/1/012021

Prospects for observing extreme-mass-ratio inspirals

with LISA

Jonathan R Gair1,†, Stanislav Babak2, Alberto Sesana3, Pau

Amaro-Seoane4,5,6,7, Enrico Barausse8,9, Christopher P L Berry3,

Emanuele Berti10,11 and Carlos Sopuerta4

1School of Mathematics, University of Edinburgh, The King’s Buildings, Peter Guthrie Tait

Road, Edinburgh, EH9 3FD, UK

2Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am M¨uhlenberg 1,

14476 Golm, Germany

3School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15

2TT, UK

4Institut de Ci`encies de l’Espai (CSIC-IEEC), Campus UAB, Carrer de Can Magrans s/n,

08193 Cerdanyola del Vall`es, Spain

5Kavli Institute for Astronomy and Astrophysics, Beijing 100871, China

6Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS,

Beijing 100190, China

7Zentrum f¨ur Astronomie und Astrophysik, TU Berlin, Hardenbergstraße 36, 10623 Berlin,

Germany

8Sorbonne Universit´es, UPMC Univesit´e Paris 6, UMR 7095, Institut d’Astrophysique de

Paris, 98 bis Bd Arago, 75014 Paris, France

9CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis Bd Arago, 75014 Paris, France

10Department of Physics and Astronomy, The University of Mississippi, University, MS 38677,

USA

11CENTRA, Departamento de F´ısica, Instituto SuperiorT´ecnico, Universidade de Lisboa,

Avenida Rovisco Pais 1, 1049 Lisboa, Portugal

E-mail: †[email protected]

Abstract. One of the key astrophysical sources for the Laser Interferometer Space Antenna

(LISA) are the inspirals of stellar-origin compact objects into massive black holes in the centres

of galaxies. These extreme-mass-ratio inspirals (EMRIs) have great potential for astrophysics,

cosmology and fundamental physics. In this paper we describe the likely numbers and properties

of EMRI events that LISA will observe. We present the first results computed for the 2.5 Gm

interferometer that was the new baseline mission submitted in January 2017 in response to

the ESA L3 mission call. In addition, we attempt to quantify the astrophysical uncertainties

in EMRI event rate estimates by considering a range of different models for the astrophysical

population. We present both likely event rates and estimates for the precision with which the

parameters of the observed sources could be measured. We finish by discussing the implications

of these results for science using EMRIs.

1. Introduction

The first gravitational wave (GW) detections by LIGO [1, 2] in 2015 began the new field of GW

astronomy. These events indicated that the ground-based interferometers will see many more

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11th International LISA Symposium IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 840 (2017) 012021 doi :10.1088/1742-6596/840/1/012021

systems over the coming decades [3]. However, seismic noise limits the sensitivity of ground-

based detectors to frequencies above ∼1 Hz and hence to systems with total mass no more than

a few hundred solar masses. Observational evidence indicates that the centres of most galaxies

contain massive black holes (MBHs), with masses between a few tens of thousands and a few

billion solar masses [4]. Mergers involving MBHs are powerful sources of GWs, but these GWs

can only be observed from space. ESA selected GW detection from space as the theme to be

addressed by the L3 mission in the Cosmic Vision programme, scheduled for launch in 2034 [5].

The technology for a space-based interferometer was successfully demonstrated in 2016 by the

LISA Pathfinder satellite [6], which paved the way for an ESA call for L3 mission proposals that

closed in early 2017. Given the success of LISA Pathfinder, it is highly likely that the successful

proposal will be the Laser Interferometer Space Antenna (LISA) [7].

LISA will have sensitivity to GWs in the millihertz band, which are generated by merging

systems having total mass in the range ∼104–107M. LISA is expected to observe a variety of

sources, including hour-period compact binaries (primarily white dwarf–white dwarf binaries)

in the Milky Way, the early inspiral phase of some of the heavier stellar-origin black hole

binaries that LIGO has shown to exist [8], and possibly cosmological sources such as GWs

from cosmic string cusps or a stochastic background generated by phase transitions in the early

Universe. However, the primary source for LISA are the MBHs in the centres of galaxies. These

MBHs will generate GWs either when they merge with other MBHs, which is expected to occur

following mergers between their host galaxies, or when they merge with much smaller compact

objects formed from stars. These latter systems are called extreme-mass-ratio inspirals (EMRIs),

because of the large difference in mass between the two objects involved. Galactic MBHs are

typically surrounded by a stellar cluster, and EMRIs occur when compact objects formed as

the end point of the evolution of stars in that cluster are captured and gradually fall into the

central MBH. EMRIs occur over long timescales, meaning that ∼104–105cycles of gravitational

radiation are generated while the compact object is in the strong gravitational field region close

to the central MBH. In addition, the process by which EMRIs begin, through a scattering capture

of the compact object by the MBH, tends to lead to high initial eccentricities, with the result

that EMRI orbits are expected to retain significant eccentricity and non-equatorial inclinations

when they are radiating in the LISA band. These properties lead to a great richness in EMRI

gravitational waveforms, created by a superposition of the three fundamental frequencies — the

orbital frequency, the perihelion precession frequency and the frequency of precession of the

orbital plane. This complexity contains a great deal of information that can be used for science.

EMRI GWs can be used both to measure the parameters of the system to high precision and

to detect small deviations in the waveforms from the predictions of general relativity that are

indicative of a breakdown of the theory. EMRIs thus have tremendous potential for astrophysics,

cosmology and fundamental physics; this will be discussed again at the end of this article, but

we refer the reader to [9, 10] for a more comprehensive discussion.

In this article we will compute the likely numbers of EMRI events and the precision with which

LISA will be able to measure their parameters. There have been previous studies computing

expected numbers of EMRI events [11]. However, the astrophysical model employed in those

calculations was a combination of simple power laws and no attempt was made to quantify

the uncertainties in that model. EMRI parameter-estimation studies have also been carried

out [12, 13], but only for a small sample of representative cases and not for a full astrophysical

population. We address both of these shortcomings in the current article. We compute event

rates for several different astrophysical models that cover the range that is plausible given current

observational constraints and compute estimates of the parameter-estimation accuracies for all

the events in each population. In addition, we compute these results for the first time for a

2.5 Gm LISA detector with six laser links, which was proposed as the new mission baseline in

the response to the ESA call in January 2017 [7].

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This article is a summary of a paper that will be submitted shortly and we refer the interested

reader to that paper for more details of the calculations and final results. This paper is organised

as follows. In Section 2, we describe the models we employ for the detector sensitivity, the

EMRI waveforms and the astrophysical population of EMRI sources. In Section 3 we present a

comparison of the likely numbers of EMRI events that will be observed under each model and

the parameter-estimation precision. In Section 4 we describe the implications of these results

for science with EMRIs, before finishing in Section 5 with a short summary of conclusions.

2. Model

To estimate the detectability of EMRIs by LISA, we must model three different things — the

intrinsic astrophysical population of EMRI events; the GWs generated by EMRI events; and

the sensitivity of our detector to those GWs. We describe the models we use for each of these

in this section.

2.1. Intrinsic astrophysical population

Our astrophysical population model includes models for the MBH mass and spin distribution,

the intrinsic rate of EMRIs occurring in systems containing black holes with particular properties

and the properties of the compact object in the EMRI.

MBH mass distribution Previous estimates [11] of EMRI rates have used a simple power

law MBH mass function of the form

d log M=AM

3×106Mα

Mpc−3(1)

which is known to be a good fit to the observed MBH population in the relevant range [14].

Values of A= 0.002 and −0.3≤α≤0.3 are usually assumed. More sophisticated models that

self-consistently evolve the MBH population are now available [15], which have been used to

predict the expected number of MBH binary mergers that will be observed by LISA [16]. We

will denote this self-consistent MBH model as “B12”. These models generate mass functions

that are roughly consistent with the above power-law form, with A= 0.005 and α=−0.3. To

obtain a lower bound on the EMRI rate we will also consider a strict power-law MBH mass

function with A= 0.002 and α= 0.3. We will denote this model as “G10”.

During a major merger, we expect the stellar cusp around the MBH to be disrupted. The

cusp will regrow, but this takes a quarter of the relaxation time for a Milky Way-like galaxy [17],

during which the standard EMRI picture does not apply. We include this effect in our model

by assuming an MBH is unavailable as a host for EMRIs for a time

tcusp =T0(M/106M)1.19q0.35 Gyr (2)

following a merger with total binary mass Mand mass ratio q. The pre-factor T0depends on

the assumed M–σrelation, which is discussed below. We cannot apply this correction for the

“G10” MBH population, as we do not track mergers in that case.

MBH spin distribution The “B12” model also tracks MBH spins. These are driven

partially by mergers but primarily by accretion. We denote the MBH spin distribution computed

in “B12” by “a98” as it predicts [18] that most MBHs will be spinning close to the maximal

imposed limit, i.e., with a≈0.98. To understand the significance of this assumption we

also consider a “flat” model with spins uniformly distributed in the range a∈[0,0.98], and

a conservative “a0” model in which all spins are set to a= 0.

EMRI rate per black hole We base our EMRI rate per MBH on [19]

R0= 300 M

106M−0.19

Gyr−1.(3)

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IOP Conf. Series: Journal of Physics: Conf. Series 840 (2017) 012021 doi :10.1088/1742-6596/840/1/012021

ID MF MBH spin Erosion M–σrelation NpCO mass [M] EMRI rate [yr−1]

M1 B12 a98 yes Gultekin09 10 10 1600

M2 B12 a98 yes KormendyHo13 10 10 1400

M3 B12 a98 yes GrahamScott13 10 10 2770

M4 B12 a98 yes Gultekin09 10 30 520

M5 G10 a98 no Gultekin09 10 10 140

M6 B12 a98 no Gultekin09 10 10 2080

M7 B12 a98 yes Gultekin09 10 10 15800

M8 B12 a98 yes Gultekin09 100 10 180

M9 B12 flat yes Gultekin09 10 10 1530

M10 B12 a0 yes Gultekin09 10 10 1520

M11 G10 a0 no Gultekin09 100 10 13

M12 B12 a98 no Gultekin09 10 10 20000

Table 1. Summary of the twelve models used in this study. The columns indicate, respectively,

the model identifier, the mass function prescription, the model for MBH spins, whether cusp

erosion following mergers is included or not, the model used for the M-σrelation, the number

of direct plunges per EMRI, the assumed mass of the compact object in all EMRIs, and the

resulting astrophysical rate of EMRI events in the Universe at redshift z < 4.5.

If this rate was maintained over the age of the Universe, it would overgrow lower mass black

holes, particularly when we account for the fact that for every EMRI there will be Np∼1–

100 “failed EMRIs” that directly plunge [20]. Therefore, we add two corrections. First, we

compute Γ = min{td/trelax,1}, where trelax is the relaxation time in the MBH stellar cusp and

td= 0.06M/[mR0(1+Np)] is the time it would take to deplete the stellar cusp of compact objects

through EMRIs and direct plunges. Second, we compute κ= min{exp(−1)M/(Γ∆M),1}, where

∆Mis the total mass that would be accreted by the MBH over its evolution history according

to (3). The final EMRI rate is taken to be κΓR0. The Γ factor corrects for the time taken for

the stellar loss cone to be repopulated by diffusion, while the κfactor imposes a limit (of one

e-fold) on the total mass that an MBH can gain by compact object accretion.

Black hole M–σrelation The factor T0in (2) and the stellar relaxation time depend on

the velocity dispersion σin the stellar cluster. This can be determined from the black hole mass

via the M–σrelation. We use the model of [21] as our reference model “Gultekin09”, which

predicts T0∼6, plus an optimistic case “GrahamScott13” [22], which predicts T0∼2, and a

pessimistic case “KormendyHo13” [23], which predicts T0∼10.

Compact object properties EMRI rates are expected to be dominated by inspirals of

black holes, as these are visible to much greater distances and are intrinsically enhanced by

mass segregation [9, 11]. Typically the mass, m, of the inspiralling object has been taken to be

fixed at m= 10M. Given recent LIGO results [1, 3] we will consider both this standard case

and one in which all EMRIs are assumed to have m= 30M. We take the eccentricity of the

EMRI orbit at plunge to be uniformly distributed in the range [0,0.2]. EMRIs on prograde orbits

into more rapidly spinning MBHs can be more easily captured as inspirals rather than direct

plunges. We account for this by enhancing the EMRI rate by a factor W(a) that is a function

of the MBH spin a, and by using the inclination distribution p(ι)∝sin(ι)[W(a, ι)]−0.83 [24].

Overall, we use 12 different models that are constructed using different combinations of the

above ingredients. The models are summarised in Table 1. M1 is our reference model. M10

and M11 will be used only to estimate event rates, as degeneracies present for non-spinning

EMRIs make parameter-estimation calculations based on the Fisher matrix more challenging.

Full parameter-estimation results will appear in the longer journal article based on this work.

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IOP Conf. Series: Journal of Physics: Conf. Series 840 (2017) 012021 doi :10.1088/1742-6596/840/1/012021

2.2. EMRI gravitational waveforms

The extreme-mass-ratio in an EMRI system means that the emitted GWs can be computed

accurately using black hole perturbation theory. This involves calculating the self-force acting

on the inspiralling object. We refer the reader to [25] for details on this approach. Self-

force calculations have not yet been completed at the necessary order for objects on arbitrary

orbits about spinning black holes, and those calculations that do exist for simpler cases are

computationally intensive and therefore not well suited to studies of populations such as this

one. As an alternative, there are two different families of kludge waveforms available. These

are approximate models that have been constructed to include the most important qualitative

features of EMRI waveforms, while not necessarily being quantitatively accurate. The analytic

kludge (AK) model [12] constructs an EMRI waveform starting from the simple GW emission

from a Keplerian orbit, and then imposing precession and evolution of the orbit onto the system.

The precessional and evolution effects are governed by low-order post-Newtonian equations,

which are accurate in the weak-field, low-velocity regime. The AK model is cheap to generate,

but it rapidly goes out of phase with more accurate waveforms. The numerical kludge (NK)

model [26, 27] uses Kerr geodesics as the basis for constructing EMRI orbits, adding inspiral

determined by evolution equations that are based on post-Newtonian expansions, but augmented

by fits to perturbation theory results. The NK waveform is generated from the trajectory using

a flat-space emission formula [27]. NK waveforms are much more faithful to accurate inspirals

than AK waveforms, staying in phase down to the last stages of inspiral [27]; however, they are

more computationally expensive.

For the purposes of this study we need to be able to generate large numbers of gravitational

waveforms to determine the detectability and parameter-estimation precision for astrophysical

populations of EMRI events. We therefore use the computationally cheapest of the three models,

the AK. While these waveforms are not faithful to the true EMRI signals produced in nature, it

is believed that they capture the richness of real EMRI gravitational waveforms and therefore

will provide an accurate guide to signal to noise ratios and parameter-estimation precisions. To

attempt to characterise the uncertainty from the waveform choice, we consider two different

versions of the AK. In the original AK model [12], the waveforms are terminated when the

object reaches the Schwazrschild innermost stable orbit (ISO). This is inaccurate for prograde

inspirals into spinning black holes, which can get much closer to the central black hole. As the

spins of the black holes in our population tend to be quite high, this is likely to significantly

underestimate the detectability of many events. A simple fix is to continue the inspiral until

the EMRI reaches the Kerr ISO. As the post-Newtonian expressions on which the AK model is

built break down in the strong-field regime at the end of the inspiral, this continuation is not

going to be accurate and will most likely lead to an overestimate of detectability. Therefore, we

expect the two sets of results to bracket the true answer.

2.3. LISA sensitivity to EMRIs

An EMRI will be detectable by LISA if it is sufficiently loud. This can be characterised by a

requirement on the matched filtering signal-to-noise ratio (SNR) ρ, which the source has in the

detector data stream

ρ2=hh|hi,where ha|bi= 2 Z∞

˜a(f)˜

b∗(f) + ˜a∗(f)˜

b(f)

Sn(f)df. (4)

This expression depends on the final detector configuration through Sn(f), the power spectral

density of noise in the detector. We use the LISA noise model described in [7]

Sn(f) = 20

3"L2+2f

0.41c2#4Sacc

n(f)+2Sloc

n(f) + Ssn

n(f) + Somn

n(f),(5)

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where Sacc

n(f) = 9×10−30 + 3.24 ×10−28 (3 ×10−5Hz/f)10 + (10−4Hz/f)2/(2πf)4m2Hz−1,

Sloc

n(f)=2.89 ×10−24 m2Hz−1,Ssn

n(f)=7.92 ×10−23 m2Hz−1and Somn

n(f)=4×

10−24 m2Hz−1. We set the detector arm length to L= 2.5Gm. The complexities of EMRI

data analysis mean that a relatively high SNR will be required for confident detection. The

value ρthresh = 30 was historically assumed [28], but results from the Mock LISA Data Chal-

lenge suggest that EMRIs with SNR as low as ρthresh = 20 could be identified [29], albeit under

somewhat idealised conditions. We will present results for both thresholds.

We assume conservatively that the duration of the LISA mission is two years and that any

EMRIs that are detected must plunge during the mission lifetime. EMRIs accumulate SNR

gradually over the whole inspiral, so LISA will also observe systems that are in the final stages

of inspiral but do not reach plunge over the mission duration. The event rates reported here

are therefore conservative in this regard. To extrapolate to longer mission durations we can

multiply by the ratio of the mission duration to two years. Again, this will be conservative,

as some EMRI signals that do not accumulate enough SNR over two years will pass the SNR

threshold when integrated over a longer observation.

To evaluate parameter-estimation precision we use the Fisher matrix, defined by

Γij =*∂h

∂λi

∂h

∂λj+(6)

where λidenotes the waveform model parameters. An estimate of the precision of measurement

is given by ∆λ2

i= (Γ−1)ii (no sum over i).

3. Results

3.1. Event rates

Table 2 indicates the number of EMRI events that would be detected over the two year

mission lifetime for each model, using SNRs computed for AK waveforms with the standard

Schwarzschild plunge condition. For the reference model we would expect to observe several

hundred events with SNR above 20, of which approximately one third would have SNR above

30. The number of observed EMRIs can be no more than a factor of 10 larger (M12), if we

assume that MBHs are always available as EMRI hosts and MBHs gain no mass from direct

plunges of compact objects. The EMRI rate could be as low as 0, if we assume a mass function

that falls steeply toward lower MBH masses and that MBHs have small spins (M11). That

model is overly pessimistic, and M5 and M8 probably give more realistic lower bounds, which

are a factor of 10 smaller. The majority of the models predict several hundred EMRI events, so

this figure appears to be fairly robust to astrophysical uncertainties.

If the inspiralling compact objects in EMRIs tend to be more massive (M4), we see a

comparable number of events. While such EMRIs can be seen to larger redshift, this is

compensated by a decrease in the intrinsic rate to prevent MBH overgrowth. Table 3 shows

the corresponding results computed using the Kerr plunge criterion to terminate the EMRI.

These rates are somewhat higher, as expected, but only by a factor ∼1–2.

In our model we have prevented MBH overgrowth through compact object accretion by

imposing a constraint on the EMRI rate. An alternative solution is that the number of light

MBHs is significantly reduced, because these MBHs grow rapidly from EMRI consumption.

Results in the table are divided into mass bins and we see that between 20% and 60% of the

events have MBH mass greater than 105.5M, so even if no lighter MBHs exist we should see

approximately one hundred events in the reference model. The rate drops significantly, by a

factor of 40 or more, if we impose a cut off at 106M, when using the Schwarzschild plunge

condition. However, a higher proportion of the events are heavier mass when we use the Kerr

plunge condition. This is because in that model the EMRI can get closer to the MBH, shifting

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IOP Conf. Series: Journal of Physics: Conf. Series 840 (2017) 012021 doi :10.1088/1742-6596/840/1/012021

Number of events in mass range

Model M10 <5 5 < M10 <5.5 5.5< M10 <6 6 < M10 Total

M1 20 (10) 240 (60) 110 (50) 10 (0) 380 (130)

M2 30 (10) 190 (50) 70 (30) 0 (0) 290 (90)

M3 20 (0) 310 (90) 510 (220) 40 (20) 880 (340)

M4 70 (20) 280 (130) 80 (50) 0 (0) 440 (200)

M5 0 (0) 10 (0) 20 (10) 0 (0) 30 (10)

M6 20 (0) 270 (70) 210 (90) 20 (10) 520 (180)

M7 230 (50) 2190 (600) 1040 (480) 60 (40) 3530 (1170)

M8 0 (0) 30 (10) 10 (10) 0 (0) 50 (10)

M9 20 (10) 210 (60) 110 (50) 10 (10) 350 (130)

M10 30 (10) 240 (70) 100 (40) 10 (10) 370 (130)

M11 0 (0) 0 (0) 1 (0) 0 (0) 1 (0)

M12 230 (50) 2420 (670) 1730 (730) 180 (110) 4560 (1560)

Table 2. Number of events detected in each mass range, and total number of events, for each

model. Mass ranges are indicated in terms of M10 = log10(M/M). The primary values in each

cell assume an SNR threshold of 20 is required for detection, while the bracketed numbers given

the corresponding results for an SNR threshold of 30. SNRs are computed using the AK model

with the Schwarzschild plunge condition. All numbers are rounded to the nearest 10 apart from

the M11 results which are rounded to the nearest 1.

Number of events in mass range

Model M10 <5 5 < M10 <5.5 5.5< M10 <6 6 < M10 Total

M1 20 (0) 260 (60) 230 (100) 80 (60) 590 (230)

M2 20 (0) 210 (50) 160 (70) 50 (40) 440 (160)

M3 10 (0) 360 (90) 1000 (470) 240 (180) 1620 (750)

M4 50 (10) 300 (150) 140 (100) 30 (30) 520 (280)

M5 0 (0) 10 (0) 40 (20) 40 (30) 90 (50)

M6 20 (0) 300 (80) 430 (200) 200 (150) 960 (440)

M7 190 (40) 2390 (600) 2110 (930) 730 (510) 5420 (2090)

M8 0 (0) 30 (10) 30 (10) 10 (10) 70 (30)

M9 20 (0) 230 (60) 160 (70) 30 (20) 430 (160)

M10 30 (10) 240 (70) 100 (40) 10 (10) 370 (130)

M11 0 (0) 0 (0) 1 (0) 0 (0) 1 (0)

M12 190 (40) 2700 (680) 3710 (1690) 1830 (1380) 8440 (3790)

Table 3. As Table 2, but now with SNRs computed using the AK model with the Kerr plunge

condition.

the GW emission to higher frequencies and hence providing sensitivity to heavier MBHs. The

prospects for significant numbers of EMRI detections are therefore good even if the number of

lower mass MBHs is significantly depleted.

Table 4 shows how the number of events detected depends on the configuration of the detector

for the reference model M1. If the final LISA configuration is similar to the 1 Gm, four-link

NGO model used in [5], the number of events would be about a factor of 10 smaller. If a more

sensitive 5 Gm configuration was launched, then the event rate could be increased by a factor of

about three. The change in the number of detected events is similar for the other astrophysical

models, except for the high mass compact object model (M4), for which the decrease in number

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Armlength

1 Gm 2.5 Gm 5 Gm

4 links 6 links 4 links 6 links 4 links 6 links

0.1 0.2 0.4 1 2 3

Table 4. Variation in the number of observed EMRI events with detector configuration.

Numbers are computed for the reference model, M1, and are expressed relative to the number

detected for the new baseline 2.5 Gm, six-link configuration.

of events going to the 1 Gm configuration is only a factor of ∼5, and the increase going to the

5 Gm configuration is only a factor of ∼1.5. This difference arises because the EMRIs in M4

are intrinsically louder and so all detector configurations can detect most of the EMRI events

out to moderate redshift.

3.2. Parameter-estimation accuracy

Figure 1 shows the distribution of parameter-estimation precisions that we would expect for

the set of EMRIs observed in each population model. We present results for the redshifted

masses of the MBH and compact object, for the spin of the MBH and for the eccentricity at

plunge. We have normalised results to a fixed SNR of 20 (accuracies scale as ∆X∝1/ρ) and

we see that after doing that the precision is essentially independent of the population model.

The distribution of SNRs is determined by the spatial distribution of sources, which is uniform

in comoving volume and hence also similar in each model. However, the models that predict

larger numbers of events will have some sources at high SNRs which will have correspondingly

more precise parameter determinations. We see that we would expect to measure the redshifted

masses of the components and the orbital characteristics such as eccentricity to precisions of

∼10−6–10−3, while also measuring MBH spins to ∼10−5–10−3. Figure 2 shows corresponding

results for EMRI measurements of source distances and sky positions. Luminosity distance

should typically be determined to a few to a few tens of percent, and sky location should be

measured to about a thousandth of a steradian, which is a few square degrees. Sky position will

be slightly less well measured for EMRIs with larger mass inspiraling objects, since such sources

are observable for a smaller fraction of a LISA orbit. The luminosity distance accuracy is also

approximately the accuracy with which the intrinsic masses of the sources will be determined,

since the redshift will be inferred from the luminosity distance. We would therefore expect to

determine the intrinsic MBH mass to better than 10% in the majority of cases.

These results were computed using the Schwarzschild plunge condition. Results computed

with the Kerr plunge condition are typically a factor of ∼10 better for the intrinsic parameters

(and comparable for the extrinsic parameters), so these results are most likely conservative.

4. Implications for EMRI science

4.1. Astrophysics

In [14] it was shown that the classic 5 Gm LISA configuration would be able to measure the slope

of the mass function of MBHs in the LISA range to a precision of ∼ ±0.3 if 10 EMRI events were

observed. This is the precision with which the mass function is currently known. This conclusion

is robust to assumptions about the configuration of the detector. In the astrophysical models

presented here, we expect more than 10 EMRI events in two years for all but the most pessimistic

model. In most cases, the number of EMRIs expected is several hundreds. The precision scales

as the square root of the number of events, so with a factor of 36 more EMRIs we would measure

the slope of the mass function to a precision of ±0.05. This would provide constraints on the

low mass end of the MBH mass function that can not be determined in any other way. We note,

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11th International LISA Symposium IOP Publishing

IOP Conf. Series: Journal of Physics: Conf. Series 840 (2017) 012021 doi :10.1088/1742-6596/840/1/012021

Figure 1. Expected parameter-estimation accuracies for redshifted MBH mass (top left),

redshifted compact object mass (top right), MBH spin (bottom left) and eccentricity at plunge

(bottom right).

Figure 2. Expected parameter-estimation accuracies for luminosity distance (left) and sky

position (right).

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Schwarzschild plunge condition Kerr plunge condition

Model N(z < 0.5) N(z < 0.5; small error) N(z < 0.5) N(z < 0.5; small error)

M1 30 5 29 7

M2 23 4 22 4

M3 62 15 60 16

M4 11 4 11 4

M5 2 0 3 1

M6 35 6 35 8

M7 298 48 285 52

M8 4 0 4 1

M9 25 3 25 5

M10 24 0 24 0

M11 0 0 0 0

M12 354 60 354 74

Table 5. Number of EMRIs detected at redshift z < 0.5 for each model, computed using each of

the two waveform plunge conditions. The first column in each case gives all EMRIs at z < 0.5,

while the second gives those EMRIs at z < 0.5 that also satisfy the error conditions that were

assumed in [30], i.e., ∆(ln DL)<0.07zand ∆Ω <16z2.

however, that [14] assumed the dependence of the EMRI rate on MBH mass was known, which

is not the case due to the various complications discussed here. LISA will actually determine

the product of the MBH mass function with the rate per MBH to the aforementioned precision.

Combining EMRI observations with those of MBH mergers could break this degeneracy, but

more work is required to properly understand this.

4.2. Cosmology

EMRI observations can be used to probe cosmological parameters. A GW observation cannot

determine the redshift of an EMRI source and electromagnetic counterparts are unlikely to be

observed. However, constraints can also be determined statistically. McLeod and Hogan [30]

showed that by using the electromagnetically determined redshifts of all galaxies within any given

LISA EMRI error box, a statistical constraint on the Hubble constant could be determined. A

determination of H0to 1% precision would be possible provided LISA observes ∼20 EMRIs at

redshift z < 0.5. This calculation was done for the classic 5 Gm LISA configuration and assumed

that LISA could determine luminosity distance and sky position to precisions ∆(ln DL) = 0.07z

and ∆Ω = 16z2. The new 2.5 Gm baseline will not be able to measure parameters to the

same precision. However, these precisions will be reached for some events. In Table 5 we show

how many EMRI events will be observed at redshift z < 0.5 in each model and how many

will be observed with the assumed parameter-estimation accuracy. We see that most models

predict the requisite number of EMRIs, but we would only expect ∼5 events with the assumed

precision. We would expect a factor of 2 worse precision with a factor of 4 fewer events, and

so it is clear that a 2% measurement of H0will be possible in most cases. The LISA mission

is likely to be longer than the two years assumed here, which will increase the number of

accurately determined EMRIs. Moreover, all of the EMRIs will contribute something to the

H0measurement, so assuming only the well localised EMRIs are useful is clearly conservative.

Thus it is likely EMRI observations with LISA will provide interesting cosmological constraints,

unless the total number of observed EMRIs is toward the low end.

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Figure 3. Precision with which LISA could detect a deviation in the quadrupole moment of

the MBH spacetime away from the Kerr value.

4.3. Fundamental physics

EMRIs are excellent probes of fundamental physics, as they generate many cycles in the strong

field regime. In particular, EMRIs can be used to map out the space-time structure outside

the central MBH, and identify any deviations from the Kerr metric structure predicted by

general relativity. There has been a lot of research in this area and a full discussion of tests of

fundamental physics with LISA can be found in [10]. An example of the application of EMRIs

to fundamental physics is to ask with what precision an EMRI could measure a deviation in

the quadrupole moment of a black hole away from the Kerr metric value. Such a study was

performed in [31], using a gravitational waveform model constructed by adding the leading

order effect of a quadruple deviation into the AK model [12]. We have repeated the same

calculation for the current EMRI populations. We assume that EMRIs are occurring in a Kerr

MBH spacetime, i.e., the true ∆Q= 0, but compute a Fisher matrix including this quadrupole

deviation parameter. The corresponding element of the inverse Fisher Matrix is then a measure

of how large a quadrupole deviation could be present in an observed EMRI before we would

be able to identify that it was there. These results are shown in Figure 3, again normalised to

a fixed SNR ρ= 20. We see that every EMRI should provide a constraint on deviations from

the Kerr metric at a level of 0.01–1%, irrespective of our assumptions about the underlying

astrophysical model.

5. Summary

We have described a comprehensive study of the prospects for detection of EMRIs with LISA.

Our study has attempted to quantify, for the first time, the uncertainties in EMRI rate

predictions arising from astrophysical uncertainties, as well as updating predictions for the new

LISA baseline configuration used in [7]. We find that LISA should observe several hundred EMRI

events over two years, with uncertainties of about one order of magnitude in each direction.

These predictions are robust to the distribution of MBH spins and the possible depletion of

MBHs at low masses. For all of these events LISA will determine the intrinsic parameters to high

precision (sub-percent accuracy), determine sky location to a few square degrees and determine

luminosity distance to O(10%). The parameter-estimation results are largely independent of the

astrophysical model assumptions.

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These EMRI detections have tremendous scientific potential. EMRIs will provide new

constraints on the mass function and spin distribution of MBHs in the LISA range. EMRIs

will provide constraints on cosmological parameters that are competitive to existing constraints,

but with completely independent systematics. Finally, EMRIs will provide strong tests of aspects

of the theory of general relativity. We have shown that all of these scientific objectives will be

realised irrespective of the true nature of the astrophysical population. The science will be

enhanced if the number of observed EMRIs is at the high end of the plausible range, but these

goals will still be met even if the rates are nearer to the low end.

Acknowledgments

AS is supported by the Royal Society. E. Barausse and E. Berti acknowledge support from

the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904. This work has made use of the

Horizon Cluster, hosted by the Institut d’Astrophysique de Paris. We thank Stephane Rouberol

for running smoothly this cluster for us. E. Berti was supported by NSF Grant No. PHY-

1607130 and by FCT contract IF/00797/2014/CP1214/CT0012 under the IF2014 Programme.

PAS acknowledges support from the Ram´on y Cajal Programme of the Ministry of Economy,

Industry and Competitiveness of Spain. PAS’s work has been partially supported by the CAS

President’s International Fellowship Initiative.

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