Survey of gravitational wave memory in intermediate mass ratio binaries

Tousif Islam,1, 2, 3, 4, ∗ Scott E. Field,2, 3 Gaurav Khanna,1, 3, 5 and Niels Warburton6

1Department of Physics, University of Massachusetts, Dartmouth, MA 02747, USA2Department of Mathematics, University of Massachusetts, Dartmouth, MA 02747, USA

3Center for Scientific Computing and Visualization Research, University of Massachusetts, Dartmouth, MA 02747, USA4Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

5Department of Physics, University of Rhode Island, Kingston, RI 02881, USA6School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

(Dated: September 3, 2021)

The non-linear gravitational wave (GW) memory effect is a distinct prediction in general relativity.While the effect has been well studied for comparable mass binaries, it has mostly been overlooked forintermediate mass ratio inspirals (IMRIs). We offer a comprehensive analysis of the phenomenologyand detectability of memory effects, including contributions from subdominant harmonic modes, inheavy IMRIs consisting of a stellar mass black hole and an intermediate mass black hole. Whenformed through hierarchical mergers, for example when a GW190521-like remnant captures a stellarmass black hole, IMRI systems have a large total mass, large spin on the primary, and possiblyresidual eccentricity; features that potentially raise the prospect for memory detection. We computeboth the displacement and spin non-linear GW memory from the m 6= 0 gravitational waveformscomputed within a black hole perturbation theory framework that is partially calibrated to numericalrelativity waveforms. We probe the dependence of memory effects on mass ratio, spin, and eccentricityand consider the detectability of a memory signal from IMRIs using current and future GW detectors.We find that (i) while eccentricity introduces additional features in both displacement and spinmemory, it does not appreciatively change the prospects of detectability, (ii) including higher modesinto the memory computation can increase singal-to-noise (SNR) values by about 7% in some cases,(iii) the SNR from displacement memory dramatically increases as the spin approaches large, positivevalues, (iv) spin memory from heavy IMRIs would, however, be difficult to detect with futuregeneration detectors even from highly spinning systems. Our results suggest that hierarchical binaryblack hole mergers may be a promising source for detecting memory and could favorably impactmemory forecasts.

I. INTRODUCTION

Detection of gravitational waves (GWs) from the coa-lescence of binary compact objects [1, 2], mostly binaryblack-holes (BBH), not only helps to shape our under-standing about compact objects in the universe, they alsoprovide an unique opportunity to test the predictions ofthe general relativity (GR) [3]. One of the interestingpredictions of GR is the existence of gravitational wavememory, a permanent distortion of an idealized GW de-tector after the wave has passed by [4–7]. Detection ofGW memory would further bolster the validity of GR,while signatures of GW memory in alternative theoriesof GR are expected to deviate from GR predictions [8].Memory effects, therefore, offer a unique way to probethe nature of gravity [9].

GW memory effects could be of different types havingdistinct origins. Earlier studies have primarily focusedon displacement memory which is a lasting change on thegravitational wave strain. Spin memory [10] is sourcedfrom the fluxes of the intrinsic angular momentum ofthe binary while center-of-mass (CM) memory [11] effectis related to changes in the center-of-mass part of theangular momentum. For typical astrophysical sources,

∗ tislam@umassd.edu

displacement memory is expected to be the dominanteffect followed by spin memory and CM memory [11].

The displacement memory effect is the most well studiedflavor of memory. Calculation of the displacement memoryrequires the computation of an angular integral of thegravitational-wave energy flux (see, for example, Eq. (1)of Ref. [12]). Various approximations and models forthis integral have appeared in the literature. Earlierworks [13, 14] have used a post-Newtonian quadrupolarapproximation to compute displacement memory effectsfor BBH inspirals. Subsequent studies [15–17] extendedthe memory calculation based on quadrupole modes tothe full inspiral-merger-ringdown signal.

For purely numerical computations of the gravitationalwave strain, Refs. [18, 19] used detector-adapted coor-dinates to simplify angular integrals, and approximatethe Isaacson stress-energy tensor. The resulting kludgemodel [18, 19] is very simple and applicable to higher har-monic modes, but due to the approximations it employsit is mainly used to study memory phenomenology [20].For example, this approach has been used to computethe memory effects for zoom-whirl extreme mass ratioinspirals (EMRI) orbits around fast spinning Kerr blackholes [19]. Recently Talbot et al. [12] performed a di-rect computation of the memory up to ` = 4 using fullinspiral-merger-ringdown waveforms from numerical rel-ativity simulations including many subdominant modes.This direct computation takes into account coupling be-

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tween modes in the energy flux expression, and it isexpected to be the most accurate because it introducesthe fewest approximations. We note that NR techniqueshave largely been unable to extract memory modes [21],although recent advances [22] have made this possibleby using the SpECTRE code’s [23] Cauchy-characteristicextraction module [24].Detectability of memory effects in current and future

generation GW detectors have recently attracted a lotof interest [18, 20, 21, 25–30]. In particular, Ref. [18]reports that third generation detectors will be able todetect memory effects from optimally oriented GW150914-like [31] events. Ref. [27] has looked into the possibilityof detecting memory effects without being able to detectthe “parent" oscillatory waveform while Ref. [26] looks forthe evidence of memory in BBH signals at the populationscale. Ref. [26] shows that ∼ 100 GW150914-like BBHevents are required to find evidence for the memory effectsin a population of BBHs. Ref. [28] finds that ∼ 2000 GWevents need to be combined in order to recover strongevidence for memory effects in a binary population withcurrent generation detectors while Ref. [20] estimates thatit would take ∼ 5 years of data for the memory modes toreach an SNR threshold of 3 in current detectors. RecentlyRef. [29] takes a different approach and rather attemptsto infer a total memory observable in real GW events.

Previous studies have mostly focused on BBH systemsthat are representative of the events found in the first andsecond observing runs [1]. Similar to the first detection,GW150914, these are comparable mass, moderately (ornon-) spinning systems in quasi-circular orbit. Here weconsider heavy intermediate mass ratio inspirals (IMRIs)with the possibility of high spin and eccentricity. Thesebinaries consist of an intermediate mass black hole (IMBH)with mass ∼ 102 − 104M [32–34] paired with a first-generation black hole produced by stellar collapse of mass∼ 4−40M. The resulting binaries will have a mass ratioin the range q := m1/m2 = 2 − 104 : 1 where m1 (m2)is the mass of the more (less) massive black hole. Theexistence of IMBHs was previously known through indirectelectromagnetic observations [35] and has recently beenconfirmed by GW190521 [36]. The GW190521 remnantblack hole has an estimated mass and dimensionless spinof 142+28

−16M and 0.72+0.09−0.12, respectively.

When formed through hierarchical mergers [37, 38], forexample when a GW190521-like remnant captures a stellarmass black hole, IMRI systems typically have a large totalmass, large spin on the primary, and possibly residualeccentricity; features that potentially raise the prospectfor memory detection especially when subdominant modesare included into the analysis [12, 39]. On the other hand,a competing effect is that higher-mass-ratio sources emitweaker signals. A key goal of this paper is to explorethe dependence of the memory’s SNR as the total mass,spin, mass ratio, and eccentricity parameters are variedfor different IMRI configurations similar to what might beexpected if a GW190521-like remnant captured a stellar-mass BH. IMRIs are likely to exist in dense globular

clusters and galactic nuclei [40, 41] and are one of theprime sources for future-generation detectors [42, 43]. Inparticular, IMRIs with total mass < 2000M may bedetectable with the current generation of detectors [44]with higher mass binaries detectable by future space-basedmissions such as LISA [45] and beyond [46, 47].

The rest of the paper is organized as follows. In Sec. II,we provide an overview of models used to compute thememory effects. Sec. III describes our method for com-puting IMRI waveforms. In Sec. IV, we assess potentialsources of systematic error in our calculation of memoryfor IMRIs. We then present results for the memory signaland its dependence on spins, eccentricity and mass ratio(Sec. V). We further compute the signal-to-noise ratio forthe memory in different detectors. Finally, we discuss theimplication of our results and caveats in Sec. VI.

II. NONLINEAR GRAVITATIONAL WAVEMEMORY MODELS

We model both the displacement and spin memorycomponents. While displacement memory contributespredominantly to hmem

20 and hmem40 spherical harmonics

modes (and therefore to plus polarization of the memorywaveform), spin memory contributions show up in thehmem

30 mode (and therefore in the cross polarization) [25].Below we describe the computation of both memory effectsfrom a given oscillatory gravitational waveform.

A. Computing Displacement Memory

The non-linear displacement memory sourced by grav-itational waves can be computed using the expres-sion [13, 17, 48]

hT T,disjk (Tr, r,Ω) = 4

r

∫ Tr

−∞dt

∫S2dΩ′ dE

dtdΩ′

[n′jn

′k

1− n′lnl

]T T

,

(1)where r is the distance between the source and the ob-server, Tr is the retarded time, Ω = (ι, φ) are the angles(ι is the inclination angle between the angular momentumvector of the binary and the line of sight of the observer,φ is the reference phase at coalescence), and n(Ω) is theunit vector pointing from the source to the observer lo-cated at Ω. Here, Ω′ coordinates describe the sphere overwhich the integral is taken and n′(Ω′) is the associatedunit vector.The gravitational wave energy flux, dE

dtdΩ , can be writ-ten in terms of the time-derivatives of the GW strainexpanded,

dE

dtdΩ = r2

16π∑

`′,`′′,m′,m′′

⟨h`′m′ h∗`′′m′′

⟩(−2)Y`′m′

(−2)Y ∗`′′m′′ ,

(2)

3

in spin-weighted spherical harmonics (−2)Y`m, where 〈.〉denotes an average over a few waveform cycles1 and h∗`mindicates the complex conjugate of h`m. The memoryexpression in Eq. (1) is then projected onto the two or-thogonal polarizations of the GWs by contracting withthe complex polarization tensors eij

+ , eij×:

hdis = hdis+ − ihdis

× = hT T,disjk (ejk

+ − iejk× ) . (3)

It is then convenient to expand the displacement memory

hdis =∑`,m

hdis`m

(−2)Y`m , (4)

in terms of spin-weighted spherical harmonics. Belowwe summarize three approximate methods used in theliterature to compute Eq. (4).

1. Quadrupole approximation

This model uses the dominant (oscillatory) quadrupolemode h22 to compute the displacement memory contribu-tions [17],

h(dis)+ 22(t) = 1

192πrΦ1(ι)∫ t

−∞dt′∣∣h22(t′)

∣∣2 , (5)

where Φ1(ι) := sin2 ι (17 + cos2 ι), an overdot denotesdifferentiation with respect to t′ and |·| denotes the com-plex modulus. For displacement memory sourced by theoscillatory (2,2) mode, h(dis)

× 22 = 0. The quadrupole modelhas often been used to study the phenomenology of dis-placement memory [15, 18, 19].

2. Minimal waveform model

The minimal waveform model (mwm) [15] employs ananalytical expression combining a PN approximation forthe inspiral and a superposition of quasinormal modesduring the merger and ringdown to compute the displace-ment memory both in time and frequency domain. Themodel has been calibrated to an effective-one-body (EOB)model tuned to NR. It is known that themwm model overestimates the memory contribution [12, 15, 20]. Nonethe-less, for the sake of completeness, in Sec. IVB we includethe model in our comparison study.

3. Higher multipole model

The higher multipole model, derived by Talbot et al. [12],evaluates the expression in Eq. (4) by numerically integrat-ing Eq. (1). The model uses both the quadrupole mode

1 In practice, such waveform averaging has not been done in anyof the models considered in this paper as the memory is knownto be relatively insensitive to the waveform averaging [49].

as well as available higher order modes up to ` = 4 andaccounts for the coupling between modes in the energyflux expression. The python package gwmemory [50] usedto carry out the computation is publicly available. Unlessotherwise mentioned, we use this higher multiple modelfrom gwmemory for calculating displacement memory.

B. Computing Spin Memory

Following [25], the spin memory

h(spin)× 22 (t) = 3

64πrΦ2(ι)∫ t

−∞dt′I(U∗22U22). (6)

is computed directly from the oscillatory (2,2) mode ofthe GW signal. Here, Φ2(ι) := sin2 ι cos ι, I denotes theimaginary part, and U22 is given by:

U22(t) = 1√2

[h22 + h∗2,−2] . (7)

III. IMRI WAVEFORM MODEL

Modeling the late inspiral and merger regimes fromIMRI systems is challenging. One reason is that thesesystems are essentially inaccessible to exploration by nu-merical relativity codes due to the small length scaleintroduced by the lighter black hole. This is becausethe inspiral time scales linearly with q and finer grid res-olution is required to resolve a smaller secondary. Forthese reasons the majority of NR simulations have hadq ≤ 15 [51, 52] with a small handful of short-durationsimulations performed at higher mass ratios [53]. As mod-ern gravitational-waveform modeling efforts require NRdata for calibration, a lack of NR data in this regimehas prevented the construction of extensive and accuratemodels. One potential path forward was recently devel-oped and applied to nonspinning, quasi-circular BBHsystems [54]. We now summarize our simple applicationof this technique to spinning and eccentric IMRI systems.We generate our gravitational waveforms using black

hole perturbation theory (BHPT). In this approach, thesmaller black hole with a mass m2 is modeled as a pointparticle, with no internal structure, moving in the space-time of the heavier Kerr black hole with mass m1 andspin angular momentum per unit mass a. The inspirallingtrajectory of the particle is computed using standardenergy and angular momentum balance equations [55–58]. To compute flux radiated to future null infinity andthrough the event horizon for the quasi-circular inspiralswe use the Gremlin code [59–61] from the Black HolePerturbation Toolkit [62]. For eccentric orbits we useSchwarzschild flux data which is available at Ref. [62], andwe arrange the inspirals such that they fully circularizebefore the onset of the plunge. The inspiral trajectory isthen extended to include the plunging trajectory [63–66].With the complete trajectory in hand, the gravitational

4

radiation is computed by numerically solving the time-domain Teukolsky equation in compactified hyperboloidalcoordinates [67–70]. The resulting waveforms include theinspiral, merger, and ringdown of the binary. For anexecutive summary of the methods, see Sec. III of Ref.[19].Waveforms computed as sketched above are suitable

models for extreme mass-ratio inspirals where q 1.Over the past decade, however, there has been mountingevidence that domain of validity of BHPT can be extendedto include moderate mass-ratio binaries [71–73]. Recentlyit was shown waveforms from non-spinning, quasi-circularbinaries generated via BHPT can be made to agree re-markably well with NR waveforms with q ∼ 3 – 15 via asimple rescaling of the binaries total mass [54]. In orderto model IMRIs we follow Ref. [54] and rescale the BHPTwaveforms such that

h`m(t; q, eref , χ) = αh`mBHPT(tα; q, eref , χ), (8)

where h`mBHP T are the spin-weight −2 spherical harmonic

modes of the waveform computed from the Teukolskysolver. Here, χ = a/m1 is the dimensionless spin pa-rameter of the heavier black hole, and eref is the ec-centricity described in Sec. V 4. We use (`,m) =(2, 2), (2, 1), (3, 3), (3, 2), (3, 1), (4, 2), (4, 3) modes in ourcomputation. Negative m modes are computed usingorbital plane symmetry, h`,−m = (−1)l(h`,m)∗. The con-tribution of omitted higher order modes are negligibleand these modes are often dominated by numerical error,and so we exclude them in this analysis.

The rescaling coefficient

(9)α(ν) = 1− 1.352854ν − 1.2230006ν2

+ 8.601968ν3 − 46.74562ν4 ,

used in Eq. (8) was obtained by fitting the (2, 2)-modeBHPT waveforms against the (2, 2)-mode NR data withnon-spinning, quasi-circular binaries from mass ratio q = 3to q = 10, where ν = q/(1 + q)2 is the symmetric massratio of the binary. The rescaled waveform was also shownto agree with a q = 15, non-spinning NR waveform notused in the fit with a mismatch value of 0.01 [54]. As themass-ratio increases, the scaling factor α approaches unitythereby recovering the fiducial BHPT waveforms. Thecalibrated-BHPT waveform approach provides a methodfor computing IMRI waveforms in a regime currentlyinaccessible to NR simulations.Thus far the α rescaling has only been determined

for non-spinning, quasi-circular binaries. Nonetheless wewill use it to rescale the low eccentricity and spinningBHPT waveform data we use in this work. While we donot expect high-accuracy waveforms to be produced bythis simple method, for our purposes it is sufficient forsurveying gravitational wave memory from IMRIs as wedemonstrate in the Sec. IV.

−2.0 −1.5 −1.0 −0.5 0.0 0.5

time [sec]

0.0

0.5

1.0

1.5

hd

is+

[10−

22]

Higher multipole model

Minimal Waveform Model

Quadrupole Approximation

Figure 1. Displacement memory hdis(t) for a non-spinningBBH with total mass M = 200 M, mass ratio q = 10, a lu-minosity distance D = 250Mpc, and at an inclination ι = π/4computed using the three models described in Sec. IIA. Ashas been previously noted [20], the mwm (dash-dot green line)overestimates the displacement memory effect. The memorysignal computed from the dominant (2,2) mode (dashed redline) slightly underestimates the effect as compared to a com-putation using all available modes (solid blue line). The resultsof our paper use the higher multipole model [12, 50] as it isexpected to be the most accurate.

IV. ROBUSTNESS STUDY

In this section we explore the robustness of our modelfor memory from IMRIs. In doing so, and for later sec-tions, it will be useful to define the signal to noise ratio(SNR), ρ, via

ρ2 = 4∫ fmax

fmin

|h(f)|2

Sn(f) df, (10)

where Sn(f) is the one-sided noise power-spectral den-sity of the detector, h(f) is the Fourier transform of thedetector response given by

h(t) = F+h+ + F×h× , (11)

and where F+ and F× are the antenna response functionsof the detector. The minimum and maximum frequencies,fmin and fmax, in the limits of Eq. (10) are chosen toreflect the sensitivity bandwidth of the detector. Forthe detector configurations considered in this paper, weintegrate over the frequencies 20 Hz to 1 kHz (for aLIGOand KAGRA [74]), 5 Hz to 1 kHz (for Einstein Telescope[75]), and 10 Hz to 1 kHz (for Cosmic Explorer [76])).Before transforming the time domain waveform to thefrequency domain, we taper the time domain oscillatorywaveform using a Planck window [77] while no taperingis used for the memory signal as it introduces additionalnon-physical features.2

2 Tapering the time-domain memory waveform introduces Diracdelta function like structure in the merger-ringdown part. This

5

A. Effect of truncating the weak field inspiral

Memory effects accumulate over the entire evolutionof the binary and formally the lower limit of integrationin Eq. (1) is negative infinity. However, we start theintegration at the start of the waveform at t = −10, 000M ,where t = 0 denotes the time at peak waveform amplitudeand M = m1 + m2 is the total mass of the binary. Weset the memory signal to zero at the start time. Toobtain a physical strain, the dimensionless waveform isthen appropriately scaled using total mass and distance.Unless otherwise specified, this prescription is appliedthroughout the paper.In principle, one can use PN correction to set a non-

zero value for hdis at t = −10, 000M . We investigatewhether such corrections are important to understandthe detectability of memory signals. We generate dis-placement memory waveforms using the mwm model fora binary with q = 10, M = 100 M, D = 250 Mpc andι = π/4. In one case, we set the initial value of hdis

22 (t) tozero. In the other case, we allow the mwm waveforms toretain their non-zero values informed by PN terms. Thewaveforms yield an SNR of 0.94 and 1.1 respectively inadvanced LIGO detector indicating marginal differencesin SNR. We therefore do not use any PN correction inour memory model.

B. Comparison between different displacementmemory approximations

As a first look at our memory calculation, in Fig. 1we plot the displacement memory computed using thethree models described in Sec. II. For a fair comparison,we set hdis

22 (t) to be zero at the start of the waveform asdescribed in Sec IVA. We find that the time evolutionof the memory waveforms are similar for all the models.However, memory computed from the quadrupolar modegenerated through point-particle black hole perturbationtheory exhibits slightly smaller values compared the highermultipole model. The mwm model, on the other hand,overestimates the memory effects. Similar results werefound in, e.g., Ref. [20] – see their Fig. 1.

C. Comparison between displacement memoryeffects computed using ppBHPT and NR waveforms

in the comparable mass ratio regime

As a second check on our memory calculation, we com-pute the memory modes for different binaries in the smallmass ratio regime (1 ≤ q ≤ 10) while fixing M = 200M, χ = 0.0, eref = 0.0 and D = 250 Mpc. We per-form this analysis with two different waveform families: a

results in a flat plateau region in the frequency domain, which isnot a physical feature.

1 2 3 4 5 6 7 8 9 10Mass ratio q

10−23

10−22

10−21

10−20

10−19

max|h

dis

(`,m

)|

(2,0) NRHybSur

(2,0) EMRISur

(2,1) NRHybSur

(2,1) EMRISur

(3,1) NRHybSur

(3,1) EMRISur

Figure 2. Maximum amplitude of three different memorymodes as a function of mass ratio. We compute the displace-ment memory using two different waveform models: EOB-NR hybridized aligned spin surrogate waveform NRHybSur3dq8(solid lines; labeled as NRHybSur) and an α-calibrated ppBHPTwaveform from the surrogate model EMRISur1dq1e4 (dashedlines; labeled as EMRISur). We observe a reasonable matchbetween the memory effects computed with these two differentmodels (details in text; Sec. IVC).

hybridized EOB-NR based aligned-spin surrogate modelNRHybSur3dq8 [78] and EMRISur1dq1e4 [54], a surrogateversion of the ppBHPT waveforms calibrated to NR.

Fig. 2 shows the dominant (2, 0) memory modealong with two important subdominant modes. ForNRHybSur3dq8, we compute the memory modes for 1 ≤q ≤ 9 and, for EMRISur1dq1e4, we show the results forq ≥ 3 to reflect their respective domains of validity. Thefigure shows visual consistency in the computation of (2, 0)memory mode, but noticeable discrepancies for these sub-dominant modes. Differences in the higher order memorymodes arise as the higher order oscillatory modes in the α-calibrated ppBHPT waveforms are not individually tunedto NR. Similarly, small (but noticeable) differences in the(2,0) mode are due to mode coupling between the sub-dominant oscillatory modes that arise in the evaluationof Eq. (2); Sec. IVD considers a memory computationusing only the quadrupole oscillatory mode where we nolonger find any discrepancies.

The small differences shown in Fig. 2 are not a concernfor our SNR computations as the (2, 0) memory mode isexpected to be dominant over other higher order memorymodes (see Figs. 9 and 10). We also observe that thedifferences in the higher order modes decrease as massratio increases as the ppBHPT framework is expected toperform better in the high mass ratio regime that we areinterested in. Interestingly, we observe that the higherorder memory modes have a maxima around q ∼ 2. Thisis consistent with the findings of Talbot et al. [12] who

6

observe a growth in the maximum of the (2,1) and (3,1)memory modes as q increases from 1 to 2 (cf. Fig. 3of Ref. [12]). While we have not explored the origin ofthis behavior, we note that for nonspinning BBH systemsin quasicircular orbit, the odd-m oscillatory modes’ am-plitude are zero at q=1, turn on quickly as q becomesnon-unity, and then transition to ∝ 1/q behavior as qbecomes large.

Further calibration of the higher order radiative modesin the EMRISur1dq1e4 model [79] will improve the agree-ment between the two models for the higher order memorymodes. The comparison in this section for q ≤ 10 showsa good agreement for the dominant (2, 0) memory modeand reasonable qualitative agreement for the subdomi-nant memory modes. As both models are calibrated toNR simulations for q ≤ 10, we next consider mass ratiosq ≥ 10.

D. Comparison between displacement memoryeffects computed using α-calibrated ppBHPT andEOB waveforms in intermediate mass ratio regime

As discussed in Sec. III there are essentially no NRsimulations of IMRIs. In lieu of direct comparison toNR, to explore the robustness of our memory calculationfor IMRIs we now compute displacement memory formass ratios 3 ≤ q ≤ 100 using the α-calibrated ppBHPTwaveforms and an aligned-spin EOB model SEOBNRv2 [80].By construction both models give the correct result inthe geodesic, q →∞, limit, both include some informa-tion from linear-in-the-mass-ratio BHPT, and both arecalibrated against NR simulations for q ≤ 10. Neithermodel is calibrated in the IMRI regime but we find thetwo models give very similar results for the memory frombinaries with intermediate mass ratios with q ≤ 100. Todemonstrate this we use the dominant quadrupolar modeto calculate the memory effects. In Fig. 3, we show themaximum amplitude of the displacement memory as afunction of mass ratio q where we fix M = 200 M,χ = 0.0, eref = 0.0, and D = 250 Mpc. Up to q = 100 wefind the relative difference for the maximum displacementmemory computed using the two models is always lessthan 2.4% for the dominant (2, 0) memory mode.

We now consider the consistency between the two mod-els when the larger black hole is spinning. In this portionof the parameter space the SEOBNRv2 model has beencalibrated using spinning NR simulations. On the otherhand the α-calibrated ppBHPT model computes the GWsfrom an inspiral into a Kerr black hole in the extrememass ratio limit and then rescales to the waveform for IM-RIs using the parameter α which is fitted to NR data fornon-spinning binaries. Nonetheless we find that the mem-ory computed for spinning binaries using the α-calibratedppBHPT and SEOBNRv2 models continue to agree wellfor spinning binaries. In Fig. 4 we show the total displace-ment memory computed using the two models for threedifferent spin configurations χ = −0.8, 0.0,+0.8 for

20 40 60 80 100Mass ratio q

10−26

10−25

10−24

10−23

10−22

10−21

10−20

max|h

dis

(`,m

)|

(2,0) SEOB

(2,0) EMRISur

(4,0) SEOB

(4,0) EMRISur

Figure 3. Maximum amplitude of two different memory modesas a function of mass ratio. We compute the displacementmemory using two different waveform models: an aligned-spin EOB model SEOBNRv2 (solid lines; labeled as SEOB) and appBHPT-based surrogate waveform EMRISur1dq1e4 (dashedlines; labeled as EMRISur). We fix M = 200 M, χ = 0.0,eref = 0.0 and D = 250 Mpc. The observed agreement betweenthese two different models in both small and intermediatemass ratio regime gives us confidence that the memory effectscomputed for IMRIs using either EMRISur1dq1e4, SEOBNRv2,or α-calibrated ppBHPT waveforms accurately capture the truedisplacement memory in this regime.

mass ratio q = 100. All other parameters have remainedthe same as those reported in the previous paragraph.The observed agreement between these two models

gives us confidence in the memory effects computed fromboth in the IMRI regime. It is worth noting that while wefind that the memory computed using the α-calibratedppBHPT and SEOBNRv2 models agree across a widerange of mass ratios, this does not imply the oscillatorywaveforms themselves will necessarily agree. This can beseen directly from Eq. (2), which is less sensitive to smalldephasing than the overlap integral commonly used tocompare waveform models.

E. Comparison between spin memory effectscomputed using different waveform models

As a final sanity check, we compute the spin memory fora binary with mass ratio q = 10, M = 200 M, eref = 0.0,and D = 250 Mpc. We restrict ourselves to a non-spinning system so that we can generate a high-accuracyNRHybSur3dq8 waveform, which can be extrapolated toq = 10 with higher accuracy provided χ = 0. In Fig. 5, weshow the spin memory effect computed using SEOBNRv2,NRHybSur3dq8, and the α-calibrated ppBHPT waveformbased surrogate model EMRISur1dq1e4. Maximum rela-tive differences between the spin memory computed usingEMRISur1dq1e4 and NRHybSur3dq8 (or SEOBNRv2) is al-

7

−8 −7 −6 −5 −4 −3 −2 −1 0

time [sec]

0

2

4

6

8

10

hd

is+

[10−

24]

(q, χ) = (100,−0.8) (SEOB)

(q, χ) = (100,−0.8) (α-calibrated ppBHPT)

(q, χ) = (100, 0.0) (SEOB)

(q, χ) = (100, 0.0) (α-calibrated ppBHPT)

(q, χ) = (100,+0.8) (SEOB)

(q, χ) = (100,+0.8) (α-calibrated ppBHPT)

Figure 4. Displacement memory for three different spin con-figurations χ = −0.8, 0.0,+0.8 with mass ratio q = 100,M = 200 M, χ = 0.0, eref = 0.0 and D = 250 Mpc. Wecompute the displacement memory using two different wave-form models: an aligned-spin EOB model SEOBNRv2 (solidlines; labeled as SEOB) and a α-calibrated ppBHPT waveforms(dashed lines; labeled as α-calibrated ppBHPT). The observedagreement between the memory effects computed with thesetwo different models gives us confidence that the memory ef-fects computed for IMRIs using SEOBNRv2 or the α-calibratedppBHPT waveforms accurately capture the true displacementmemory in this regime.

ways ≤ 10%.

V. PHENOMENOLOGY & DETECTABILITY

In this section, we explore the memory phenomenologyand detectability as the mass ratio, spin, and eccentricityis varied. We report SNRs computed using the designsensitivity of detectors including advanced LIGO, CosmicExplorer (CE), and Einstein Telescope (ET). AssumingGaussian detector noise, an SNR of ≈ 5 is typically con-sidered sufficient for detection. For multiple “stacked"detections some authors have considered a total memorySNR value as low as 3 to be sufficient for claiming hintsof memory [18, 20, 26].

For most of our SNR results, we fix the intrinsic BBHparameters and distance, D, and report the maximumand angle-averaged SNR values. To compute the angle-averaged SNR, we simulate a total of 1125 signal realiza-tions where we sample right accession α and polarizationψ uniformly in [0, 2π], declination δ and inclination ιuniformly in cos δ and cos ι from [−1, 1]. The maximumSNR is taken to be the largest value over the 1125 signalrealizations. As the SNR is proportional to 1/D, our SNRresults can easily be scaled to different luminosity distancevalues. Our default choice of D = 250Mpc is motivated bythe inferred distance for the event GW190814 [81], highestmass ratio event (q ∼ 10) detected by LIGO/Virgo so far.

−5 −4 −3 −2 −1 0

time [sec]

0

1

2

3

hsp

in×

[10−

24] NRHybSur

EMRISur

SEOB

Figure 5. Spin memory computed for a non-spinning binarywith mass ratio q = 10 using three different waveform models:an aligned-spin EOB model SEOBNRv2 (dashed dotted lines; la-beled as SEOB), an EOB-NR hybridized aligned spin surrogatewaveform NRHybSur3dq8 (solid lines; labeled as NRHybSur), anda α-calibrated ppBHPT waveform from the surrogate modelEMRISur1dq1e4 (dashed lines; labeled as EMRISur). We fixM = 200 M, eref = 0.0 and D = 250 Mpc. The observedagreement between the spin memory effects computed withthese three different models gives us confidence that the mem-ory effects computed for IMRIs using either NRHybSur3dq8,SEOBNRv2, or the α-calibrated ppBHPT waveforms accuratelycapture the true spin memory in this regime.

As GW190814 is one of the closest BBH detections todate, our choice of D = 250Mpc is represents a plausibly-optimistic default value. While we will broadly considerIMRI systems, one particular focus is on memory fromhierarchical mergers involving second- or third-generationblack holes. These systems present, on average, bothlarger masses and larger spins [38]. Many of our experi-ments consider systems with M = 200 M, q = 10, andlarge-spin systems, which is consistent with a GW190521-like remnant capturing a first-generation, stellar-massblack hole.

1. Structure of the memory signal

To understand the structure of memory waveforms, wepick a non-spinning GW signal with mass ratio q = 10,total massM = 200 M and luminosity distance D = 250Mpc. In Fig. 6, we plot both the displacement memory(middle panel) and spin memory (lower panel) contri-butions from the dominant ` = 2,m = 2 mode. Forcomparison, we also show ` = 2,m = 2 mode waveform(upper panel). We fix the inclination to be ι = π/4 suchthat the memory effect fits in between the maximumand conservative cases (discussed more in Fig. 8). Dis-placement memory effects are found to be two orders ofmagnitude smaller than the oscillatory waveform, and thespin memory contributions are another ∼ two orders ofmagnitude smaller compared to its displacement memorycounterpart. We note that the displacement memory in-creases gradually during the inspiral and reaches a flatmaximum following the merger. Spin memory, on the

8

−6 −5 −4 −3 −2 −1 0

−3.0

0.0

3.0h

+[1

0−21

] ` = 2,m = 2

−6 −5 −4 −3 −2 −1 0

0.0

2.5

5.0

7.5

hd

is+

[10−

23]

−6 −5 −4 −3 −2 −1 0

time [sec]

0.0

1.5

3.0

hsp

in×

[10−

24]

Figure 6. Upper panel: Plus polarization of the h22 spheri-cal harmonic mode for a non-spinning BBH with total massM = 200 M, mass ratio q = 10 at a luminosity distanceD = 250 Mpc and at an inclination ι = π/4. Middle panel:Gravitational waveforms associated with the non-linear dis-placement memory contributions computed using all avail-able modes using the higher multipole model. Lower panel:Spin memory contributions computed from the dominant` = 2,m = 2 mode.

other hand, drops sharply after the merger.In Table I we report the maximum and angle-averaged

SNR for a BBH with q = 10, M = 200 M and D =250 Mpc in four different detectors. We find that whiledisplacement memory modes will have significant SNRsin future detectors and could possibly result in confidentdetection, spin memory modes would still have very lowSNRs. These SNR values are consistent with earlierstudies done in the context of comparable mass binaries[18, 25, 27].

To probe the effects of higher modes, in Fig. 7 we showthe total memory computed using different mode con-tent for a binary with q = 10 and M = 200M. Tocompute the displacement memory we use (i) only thedominant (2,±2) modes (solid red line), (ii) all modeswith ` ≤ 2 (dashed green line), (iii) all modes with ` ≤ 3(dash-dot blue line), (iv) and all modes in our α-calibratedppBHPT waveforms (dashed black line). We notice thatthe memory contribution from the quadrupolar mode al-ready accounts for most of the signal content. To quantifythe importance of the higher modes in the memory compu-tation we compute SNRs of displacement memory signalsobtained using only the quadrupolar mode (Table I; inparenthesis). We find that SNRs increase by about ∼ 7%across detectors when higher order modes are included inmemory computation.

−1.0 −0.8 −0.6 −0.4 −0.2 0.0

time [sec]

1.0

3.0

5.0

7.0

hd

is+

[10−

23]

(`,m) = (2, 2) Mode

` <= 2 Modes

` <= 3 Modes

All Modes

Figure 7. Gravitational waveform associated with the non-linear displacement memory contributions computed usingdifferent combinations of spherical harmonics modes. Oursystem is a non-spinning BBH with total mass M = 200 M,mass ratio q = 10 at a luminosity distance D = 250 Mpc andat an inclination ι = π/4. We zoom in the late inspiral, mergerand ringdown part of the waveform.

We now probe the dependence of both the displacementand spin memory on the inclination angle ι. In Fig. 8, weplot the maximum of the displacement and spin memorycomputed using all available modes as a function of theι. We fix q = 10, M = 200 M, and D = 250 Mpc.We find that the maximum effects for the displacementmode is obtained for ι = π/2 whereas spin memory modesare loudest for ι ∼ π/4 − π/3. This is due to the factthat (2, 0) displacement memory is dominant over othermodes. The angular dependency of the (2, 0) mode canbe approximated as ∼ sin2 ι (17 + cos2 ι) (as done inquadrupole approximation) which clearly shows that thedisplacement memory effect is expected to reach a maximaat ι = π/2.

2. Mode Decomposition of the Memory Waveform

Following the prescription provided in Ref. [12], wedecompose the memory waveform into spin-weighted har-monics modes to explore their dependence on spin andeccentricity (Fig. 10), and mass ratio (Fig. 9). In all cases,as is well known, we find that the (2, 0) mode is dominant.Many of the subdominant modes (except the (2, 1) and(3, 1)) are negligible compared to the (2, 0) mode and couldsafely be ignored for SNR computations, although we willcontinue to include them. Another interesting aspect ofthe memory mode decomposition seen in Figs. 9,10 is theasymmetry observed between (`,m) and (`,−m) memorymodes. Such asymmetry originates due to the non-linearbeating of different oscillatory modes that generates thememory (also seen in Fig. 3 of [12]).

Fig. 9 shows the mode decomposition of q = 10 binarieswhile varying spin and eccentricity configurations. By

9

Table I. Angle-averaged and maximum SNRs a of the dis-placement memory mode and spin memory mode in differentdetectors (aLIGO, KAGRA [74], ET and Cosmic Explorer(CE) [76]). The BBH source parameters are: q = 10, M = 200M, D = 250 Mpc. For a comparison we also show the SNRs(in parenthesis) for memory signals computed using only thedominant l = 2,m = ±2 mode.

Displacement Memory Spin Memoryρavg ρmax ρavg ρmax

aLIGO 0.41 1.76 0.002 0.01(0.38) (1.64)

KAGRA 0.29 1.29 0.001 0.004(0.28) (1.21)

ET 4.63 20.04 0.04 0.26(4.34) (18.59)

CE 15.31 66.07 0.16 1.05(14.32) (61.31)

a Symbols: ρavg: average SNR; ρmax: maximum SNR.

0.0 π/4 π/2 3π/4 π0.0

0.5

1.0

1.5

|hd

ism

ax|[

10−

22]

0.0 π/4 π/2 3π/4 π

Inclination ι

0.0

1.0

2.0

3.0

|hsp

inm

ax|[

10−

24]

Figure 8. Maximum of the total displacement memory andspin memory as a function of the inclination angle ι. Oursystem is a non-spinning BBH with total mass M = 200 M,mass ratio q = 10 at a luminosity distance D = 250 Mpc.

comparing the quasi-circular (blue circle) and eccentric(red triangle) cases, we see that eccentricity brings almostno change in the maximum value of the memory modes.This has been observed for the dominant (2, 0) memorymode in the context of comparable mass ratio binaries[39, 49], and our result extends this finding in the interme-diate mass ratio regime and for the subdominant memorymodes. Spinning systems, however, have noticeably dif-ferent mode content as compared to their non-spinningcounterparts; Sec. V 3 considers the impact of spin onthe memory’s SNR.Fig. 10 shows the different memory modes for non-

spinning, non-eccentric binaries as the mass ratio is in-creased. The effect of mass ratio is clearly observed inFig. 10 where the maximum value in all of the memorymodes decreases with mass ratio. Ref. [12], however, ob-served an increased contribution to higher order modesfrom the q ≤ 2 asymmetric mass binaries they considered,which is also apparent in Fig. 2.

3. Effect of spin

Next, we provide a systematic study of both displace-ment and spin memory’s spin dependence. We computethe memory effects for a set of binaries with mass ratioq = 10 but for different values of the primary black-hole’sdimensionless spin χ. We fix q = 10, M = 200 M,D = 250 Mpc and ι = π/4. In Fig. 11, we show the dis-placement and spin memory effects as a function of timefor three different BBH with spins χ = [−0.8, 0.0, 0.8].We observe that the memory effect increases as the spinχ increases. This is due to the fact that prograde inspi-rals spend a longer in the strong field as the last stableorbit’s radius shrinks. As a consequence, the SNR ofemitted GWs is also expected to be larger than the corre-sponding non-spinning binary system. Our findings areconsistent with results obtained in Ref. [19] that observesan increased memory amplitude for lager values for spins.

In Fig. 12, we report the maximum and angle-averagedSNR for the memory modes in a BBH with q = 10,M = 200 M and D = 250 Mpc. The maximum SNRs forthe displacement memory in ET is sufficient for confidentdetection across the entire range of spins considered here.The angle-averaged SNR for the displacement memory isbetween ∼ 1 (for χ = −.8) and ∼ 20 (for χ = .99). Forhierarchical mergers with second-generation (or higher)component black holes, spins of up to χ ≈ .9 are ex-pected [37, 38], suggesting that memory from some ofthese system may be directly observed with ET even inthe typical case. Indeed, a handful of high-spin blackholes have been identified [82, 83] and may therefore bethe most promising candidates for memory detections.For aLIGO, angle-averaged SNRs are always below thevalue 5 at our fiducial distance D = 250 Mpc, althoughoptimally oriented binaries cross the detection thresholdfor χ >= 0.9. Increased SNR at larger spin values wouldtherefore favorably contribute to forecasts for memorydetection. The spin memory, however, is unlikely to bedetected in aLIGO or ET.

4. Effect of eccentricity

Depending on the formation channel, some IMRIs areexpected to retain significant eccentricity even at thefinal stage of the inspiral [42, 84, 85]. Memory fromeccentric systems has typically been studied using PNapproximations [49, 86] or using a kludge model [19]. Inthis paper we use the higher multipole model and focus on

10

(2,−2)(2,−1) (2, 0) (2, 1) (2, 2) (3,−3)(3,−2)(3,−1) (3, 1) (3, 2) (3, 3) (4,−4)(4,−3)(4,−2)(4,−1) (4, 0) (4, 1) (4, 2) (4, 3) (4, 4)

Modes

10−26

10−25

10−24

10−23

10−22

10−21

max

(|hd

is`,m|)

q, χ, eref = 10, 0.0, 0.0q, χ, eref = 10, 0.8, 0.0

q, χ, eref = 10,−0.8, 0.0q, χ, eref = 10, 0.0, 0.15

Figure 9. The spherical harmonic decomposition of the displacement memory waveform for different spin and eccentricconfigurations. We fix mass ratio q = 10, M = 200 M and D = 250 Mpc. The absolute value of the late-time memory is shownas a function of the (`,m) spherical harmonic decomposition of the memory. This figure extends Fig 3 of Talbot et al. [12] inthe high mass ratio regime - focusing on mass ratio q = 10 and for different configurations of spins and eccentricities. The (3,0)mode’s amplitude is extremely small and omitted from this figure.

(2,−2)(2,−1) (2, 0) (2, 1) (2, 2) (3,−3)(3,−2)(3,−1) (3, 1) (3, 2) (3, 3) (4,−4)(4,−3)(4,−2)(4,−1) (4, 0) (4, 1) (4, 2) (4, 3) (4, 4)

Modes

10−27

10−26

10−25

10−24

10−23

10−22

10−21

max

(|hd

is`,m|)

q, χ, eref = 10, 0.0, 0.0q, χ, eref = 25, 0.0, 0.0

q, χ, eref = 50, 0.0, 0.0q, χ, eref = 75, 0.0, 0.0

Figure 10. The spherical harmonic decomposition of the displacement memory waveform for different mass ratios. We fix spinχ = 0.0, eref = 0.0, M = 200 M and D = 250 Mpc. The absolute value of the late-time memory is shown as a function of the(`,m) spherical harmonic decomposition of the memory. This figure extends Fig 3 of Talbot et al. [12] in the high mass ratioregime - covering mass ratios 10 ≤ q ≤ 75. The (3,0) mode’s amplitude is extremely small and omitted from this figure.

small to moderate eccentricities with e . 0.2. To estimatethe eccentricity at a given time, we use [87]:

e(t) =√ωp(t)−

√ωa(t)√

ωp(t) +√ωa(t)

, (12)

where ωa and ωp are the orbital frequencies at apocenterand pericenter, respectively. We let eref be the value ofe(t) measured three cycles before the merger. A detaileddescription of the method is given in Refs. [87, 88].We explore how the memory effect changes as the bi-

nary becomes increasingly more eccentric. To do this, wesimulate gravitational waveforms for q = 10, M = 200M, D = 250 Mpc, and ι = π/4 with different values

of eccentricity. In Fig. 13, we show the memory contri-butions as a function of time for one particular valueof eccentricity eref = 0.17. Eccentricity introduces addi-tional modulation in both displacement and spin memorycomponents. Such modulations are small in the displace-ment memory but prominent for the spin memory. Themodulations are strongly correlated to the modulationsin the oscillatory gravitational waveform (upper panel;Fig. 13) and these features become more evident as theeccentricity increases (Fig. 14). For the displacementmemory, these modulations roughly resembles the stair-case structure found in the zoom-whirl orbits [19]. Thesemodulations are consistent with results obtained in Refs.

11

−4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0

0.0

0.5

1.0

1.5h

dis

+[1

0−22

]χ = +0.8

χ = 0.0

χ = −0.8

−4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0

time [sec]

0.0

1.5

3.0

hsp

in×

[10−

24]

χ = +0.8

χ = 0.0

χ = −0.8

Figure 11. Time evolution of the displacement memory hdis(t)(upper panel) and spin memory hspin(t)(lower panel) for threedifferent spin values, χ, of the primary black hole. Our systemis a BBH with total mass M = 200 M, mass ratio q = 10 ata luminosity distance D = 250Mpc and inclination ι = π/4.

[39, 49] for comparable mass ratio binaries.Next, we compute the SNRs for memory signals from

eccentric binaries. In Table II we report SNRs for thebinary with highest eccentricity (eref = 0.17) consideredin our study. For comparison, we also show the SNRvalues computed for the non-eccentric binary. The SNRvalues change by at most 4 percent. We find that for agiven mass ratio and detector sensitivity, the computedSNRs are roughly constant for eref ≤ 0.17 despite therich phenomenology that larger eccentricity offers. Thisis perhaps not surprising in light of the fact that most ofthe SNR is accumulated around the merger, and maxi-mum value for the memory signal changes modestly witheccentricity for the values of eref considered here. Wefurther confirm that the differences in SNR computedusing ppBHPT waveforms with and without α scaling[defined in Eq. (8)] are small, suggesting that the rescalingobtained from the non-eccentric binaries can reasonablybe used for eccentric binaries - at least for the purpose ofthis study.

5. Effect of mass ratio

We explore how the memory effect changes as the binarybecomes increasingly asymmetric. To do this, we simu-late gravitational waveforms with M = 200 M, χ = 0.0,eref = 0.0, D = 250 Mpc and ι = π/4 while varying themass ratio. In Fig. 15, we plot both the displacement

−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00

10−1

100

101

102

ρd

is

M = 200M, D = 250 Mpcρmax

aLIGO

ρavgaLIGO

ρmaxET

ρavgET

−0.75−0.50−0.25 0.00 0.25 0.50 0.75 1.00

χ

10−4

10−3

10−2

10−1

100

ρsp

in

Figure 12. Maximum SNR ρmax (solid line) and angle-averageSNR ρavg (dashed line) for the memory modes computed usingthe design sensitivity of the advanced LIGO (blue) and ET(green) as a function of the spin on the primary black-hole.We use q = 10, M = 200 M and D = 250 Mpc. Upper panel(lower panel) shows the SNR for the displacement memory(spin memory). Black dashed line denotes an SNR of 5, typicalthreshold for detection.

Table II. Angle-averaged and maximum SNRs of the displace-ment memory mode and spin memory mode in different detec-tors for a M = 200 M, χ = 0.0, D = 250 Mpc binary withhighest eccentricity (eref = 0.17) considered in our study. Fora comparison, we also show the SNRs of the correspondingnon-eccentric binary in parenthesis.

a

Displacement Memory Spin Memoryρavg ρmax ρavg ρmax

aLIGO 0.42 1.85 0.002 0.011(0.41) (1.76) (0.002) (0.011)

ET 4.86 21.07 0.041 0.266(4.63) (20.04) (0.040) (0.264)

a Symbols: ρavg: average SNR; ρmax: maximum SNR.

memory and spin memory for three different mass ratiosq = 10, 15, 25. The memory signals become weaker asthe mass ratio increases, which follows from the fact thatthe oscillatory mode’s amplitude decreases as 1/q in thelarge-mass-ratio limit. In Fig. 16, we show the maximumand angle-averaged SNR for memory signals with differentmass ratios. We find that the SNRs for the memory modesdecrease as mass ratio increases for q ≥ 10. We comple-ment this analysis by computing SNRs for the memorysignals in the comparable mass ratio regime (1 ≤ q ≤ 9)using the NRHybSur3dq8 model (red shaded region). Weconfirm that the overall trend of SNR scaling (∼ 1/q) con-

12

−10 −8 −6 −4 −2 0

0.0

1.0

2.0|h

+|[1

0−21

]

−10 −8 −6 −4 −2 0

0.0

3.0

6.0

hd

is+

[10−

23]

−10 −8 −6 −4 −2 0

time [sec]

0.0

1.5

3.0

hsp

in×

[10−

24]

Figure 13. Time evolution of the displacement memory hdis(t)(middle panel) and spin memory hspin(t) (lower panel) for theBBH with eccentricity eref = 0.17 measured three cycles beforethe merger. All other details are same as in Fig. 6. For acomparison, we show the time evolution of the ` = 2,m = 2mode amplitude in the upper panel. Eccentricity introducesadditional modulation in both the amplitude of ` = 2,m = 2oscillatory mode and both flavors of memory.

tinues almost up to q = 1. Furthermore, we observe thatthe SNR values have a mostly smooth transition from thecomparable mass regime (obtained using NRHybSur3dq8model) to intermediate mass ratio regime (obtained usingα-scaled ppBHPT waveforms) providing further evidenceon the robustness of the SNR computation in the q ≥ 10regime where waveform modeling is more challenging.

6. Detectability in aLIGO

In the previous subsections, we have focused on thedependence of the memory’s SNR (with and without sub-dominant modes) as the total mass, spin, mass ratio, andeccentricity are varied. We now consider the regions of themass-distance parameter space that are most promisingfor the direct detection of displacement memory. In lightof Fig. 16, we again focus on q = 10 systems as they offerthe best chance for direct detection for the mass ratiosprimarily considered in this paper.

In Fig. 17, we map out regions of plausible detectabilityfor binaries with mass ratio q = 10. The shaded areas be-low each solid line indicate the region where displacementmemory modes have a maximum SNR (optimally orientedwith respect to the detector) of more than 3 (solid line;for “hints" of memory) and 5 (dashed line). We show

−2.0 −1.5 −1.0 −0.5 0.0

0.0

3.0

6.0

hd

is+

[10−

23]

eref = 0.17

eref = 0.08

eref = 0.00

−2.0 −1.5 −1.0 −0.5 0.0

time [sec]

0.0

1.5

3.0

hsp

in×

[10−

24]

eref = 0.17

eref = 0.08

eref = 0.00

Figure 14. Time evolution of the displacement memory hdis(t)(upper panel) and spin memory hspin(t) (lower panel) for threedifferent eccentricity values eref . Our system is a BBH withtotal mass M = 200 M, mass ratio q = 10 at a luminositydistance D = 250 Mpc and inclination ι = π/4.

boundaries for binaries with χ = 0.8 (blue), χ = 0.0 (red),and χ = −0.8 (orange). With larger positive spins, dis-placement memory becomes increasingly easier to detect.These are particularly promising as the primary black holeof an IMRI system is generally expected to have largepositive spins (∼ 0.7) when formed through hierarchicalmergers [89–91].

VI. DISCUSSION & CONCLUSION

In this work, using a recently-developed spin [25] andhigher multipole displacement memory model [12], wesystematically investigate the total memory effects forintermediate mass ratio inspirals (IMRIs) while primar-ily focusing on the potential detectability of these sig-nals. Our work is motivated by binary systems formedthrough hierarchical mergers [37, 38], for example, whena GW190521-like remnant captures a stellar-mass blackhole. Such systems typically have a large total mass, largespin on the primary, and possibly residual eccentricity;features that potentially raise the prospect for memory de-tection especially when subdominant modes are includedinto the analysis.

To generate the oscillatory part of the IMRI waveform(which is used in the computation of memory), we usepoint particle black hole perturbation theory (ppBHPT)waveforms computed by solving the Teukolsky equation.The ppBHPT waveforms are then calibrated to NR simu-

13

−4 −3 −2 −1 0

0.0

2.5

5.0

7.5h

dis

+[1

0−23

]q = 10

q = 15

q = 25

−4 −3 −2 −1 0

time [sec]

0.0

1.5

3.0

hsp

in×

[10−

24]

q = 10

q = 15

q = 25

Figure 15. Time evolution of the displacement memory hdis(t)(upper panel) and spin memory hspin(t) (lower panel) for BBHswith three different mass ratio q. Our system is a non-spinningBBH with total mass M = 200 M, at a luminosity distanceD = 250 Mpc and at an inclination ι = π/4.

lations for q ≤ 10 using a rescaling discussed in Sec. III.As IMRI waveform models are still under active develop-ment, we have furnished extensive comparisons (Figures2, 3, 4, and 5) between a hybrid EOB-NR surrogateNRHybSur3dq8, an aligned-spin effective one body modelSEOBNRv2, and our calibrated ppBHPT waveforms. Wefind these models agree surprisingly well for the dominantcontributions to the GW memory for mass ratios q ≤ 100despite being calibrated to NR simulations mostly in thecomparable mass ratio regime.To assess the detectability of the memory signal in

current and future gravitational wave detectors (primar-ily considering Advanced LIGO and ET), we computeboth the optimal and angle-averaged signal to noise ratios(SNRs) for different binary configurations. Specifically, wehave explored the SNR’s dependence on the the total mass,mass ratio, spin of the primary black hole, and eccentricityusing memory signals with and without including sub-dominant harmonic modes. We find that memory signalsbecome stronger when the primary black hole has positivespins, with the SNR growing by as much as a factor of 10as the spin is varied from χ ≤ 0 to χ ≈ .99. Fig. 12 showsthat memory signals from nonspinning BBH systems farfrom the detectability threshold may be detected for spinsnear χ ≈ 0.95. We find that when mild to moderate eccen-tricity is introduced, memory signals show rich structures(see Figures 14 and 13) – with additional modulationsboth in the displacement and spin memory. However,the memory signal’s amplitude hardly changes due to

100 101 102

10−2

10−1

100

101

102

103

ρd

is

M = 200M, D = 250 Mpc ρmaxaLIGO

ρavgaLIGO

ρmaxET

ρavgET

100 101 102

Mass ratio q

10−4

10−3

10−2

10−1

100

ρsp

in

Figure 16. Maximum SNR ρmax (solid line) and angle-averageSNR ρavg (dashed line) for the memory modes computed usingthe design sensitivity of the advanced LIGO (blue) and ET(green) as a function of the mass ratio q. We use M = 200M, χ = 0.0, eref = 0.0 and D = 250 Mpc. Both displacementand spin memory decreases as ∼ 1/q. The shaded red regionshows SNRs computed using the NRHybSur3dq8 model over1 ≤ q ≤ 9 (details are in text).

eccentricity, and consequently the SNRs for the memorymodes in different eccentric configurations remain largelyunchanged (see Table II). We’ve also explored the SNR’sdependence on mass ratio, largely confirming the overallexpectation of that the memory signal (and hence SNR)will become weaker as the mass ratio increases. This fol-lows from the fact that the oscillatory mode’s amplitudedecreases as 1/q in the large-mass-ratio limit. This trendis seen most clearly in Fig. 16 and continues almost upto equal-mass systems.All of our main results have been obtained using

the higher multipole displacement memory model [12]using an oscillatory waveform model with (`,m) =(2, 2), (2, 1), (3, 3), (3, 2), (3, 1), (4, 2), (4, 3) modes in ourcomputation. Unlike most previous works, our displace-ment memory results include contributions from higher-order modes that have been typically omitted in similarstudies. We have therefore provided some comparisonsto displacement memory effects computed with the (2,2)mode only. The inclusion of subdominant modes in thememory computation will “activate" modes such as the(3,1) memory mode, which (for non-spinning systems) wesee from Fig. 2 has maximum power around q ≈ 2.5. Arepresentative sample of the mode hierarchy is shown inFigs. 9 and 10, which shows an asymmetry between (`,m)and (`,−m) memory modes that was also observed inFig. 3 of Ref. [12]. We find that including subdominantmodes has a small but non-negligible impact on the sys-tems considered here. Table I directly compares SNR

14

200 300 400 500 600 700 800Mass [M]

300

400

500

600

700

800

900

1000

Dis

tan

ce[M

pc]

χ = +0.8

χ = 0.0

χ = −0.8

Figure 17. We show the region (shaded in blue/red/orange)in parameter space (for a q = 10 binary) important for the de-tection of displacement memory with a single advanced LIGOdetector operating at design sensitivity. The solid red linedenotes the total mass and distances for which an optimally-oriented nonspinning (χ = 0.0) binary has an SNR=3 (“hintsof memory"). The red shaded area below this line indicates theregion where the memory modes have SNR≥ 3. We also showsimilar boundaries for χ = 0.8 (solid blue line) and χ = −0.8(solid orange line). Dashed lines are used to mark the locationof SNR= 5.

values with and without higher order modes finding adifference by about ∼ 7% across different detectors.

Our results indicate that displacement memory effectsin IMRIs could be detected in future generation detectorssuch as the Einstein Telescope. Detection in current gen-eration detectors would, however, require some amountof luck (e.g. systems merging very close and/or witha large, positive spin on the primary) and/or combin-ing many events to compute the evidence (similar toRef. [27]). Figure 17, for example, identifies regions ofthe total mass/distance parameter space where memoryfrom a q = 10 binary may have SNRs above 5 using asingle advanced LIGO detector operating at design sensi-tivity. On the other hand, the spin memory would stillbe difficult to detect even for highly spinning, optimallyoriented systems. Furthermore, as our SNR computationshave been done assuming only one detector, repeatingthis study using a network of detectors would naivelyincrease the memory SNRs by a factor of ∝

√Ndet where

Ndet is the number of GW detectors. However, a fullstudy would be needed to include each detector’s PSD aswell as the relative orientation factors that may suppressor enhance any particular detector’s sensitivity to theincoming memory signal.Heavy binaries with large positive spins are particu-

larly promising for memory detections. For hierarchical

mergers with second-generation (or higher) componentblack holes, spins of up to χ ≈ .9 are expected [37, 38].The increased SNR at larger spin values would favorablycontribute to forecasts for memory detection. However,to the best of our knowledge, all memory forecasts thathave appeared in the literature (e.g. [20, 26]) use popu-lation models that favor nonspinning systems similar toGW190514. We expect that future work on memory fore-casts that include both subdominant modes and mixedpopulation models (1g+1g, 1g+2g, etc.) may find moreoptimistic forecasts.Finally, we note that in this work we have focused on

IMRIs composed of a stellar origin BH and an IMBH. Itis also possible to form IMRIs with the combination ofan IMBH and a massive BH. Though their event ratesare very uncertain, these are exciting and potentially veryhigh SNR sources for the LISA detector [42]. Their highSNR should also make them good candidates for detectingGW memory.

ACKNOWLEDGMENTS

We thank Everett Gaige Field and Hamish Warburtonfor helpful discussions and interactions throughout thiswork, and Colm Talbot for assistance with the Pythonpackage gwmemory. A portion of this work was carriedout while a subset of the authors were in residence at theInstitute for Computational and Experimental Researchin Mathematics (ICERM) in Providence, RI, during theAdvances in Computational Relativity program. ICERMis supported by the National Science Foundation underGrant No. DMS-1439786. Simulations were performedon CARNiE at the Center for Scientific Computing andVisualization Research (CSCVR) of UMassD, which issupported by the ONR/DURIP Grant No. N00014181255and the MIT Lincoln Labs SuperCloud GPU supercom-puter supported by the Massachusetts Green High Per-formance Computing Center (MGHPCC). This researchwas supported in part by the Heising-Simons Foundation,the Simons Foundation, and National Science FoundationGrant No. NSF PHY-1748958. The authors acknowl-edge support of NSF Grants No. PHY-2106755 (G.K),No. PHY-1806665 (T.I. and S.F), and No. DMS-1912716(T.I., S.F, and G.K). T.I. acknowledges additional sup-port from the Kavli Institute for Theoretical Physics,University of California, Santa Barbara through KavliGraduate Fellowship. This research was supported in partby the Heising-Simons Foundation, the Simons Founda-tion, and National Science Foundation Grant No. NSFPHY-1748958. N.W. acknowledges support from a RoyalSociety–Science Foundation Ireland Research Fellowship.This publication has emanated from research conductedwith the financial support of Science Foundation Irelandunder Grant number 16/RS-URF/3428.

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