Gravitational-wave memory: an overview

Marc Favata Montclair State University (NJ)

LIGODCC:xxxx

What is the gravitational-wave memory? example: nonlinear memory from binary black-hole mergers

Thewavenolongerreturnstothezero-pointofitsoscilla>on.Thisgrowing-offsetiscalledthememory.

Gravita+onal-wavesignalvs.+me

Why is this called “memory”?

beforewavepassage wavepassingthroughdetector a;erwavepassage

with

mem

ory

with

outm

emory+me

(gravita>onal-wavespropaga>ngintothescreen)

The linear and nonlinear memory: Linearmemory:(Braginskii,Grishchuck,Thorne,Zeldovich,Polnarev)§  Arisesfromthenon-oscillatorymo>onofa

source,especiallyduetounboundmasses.

§  Ex:hyperbolicorbits,mass/neutrinoejec>oninsupernovas/GRBs

Nonlinearmemory:(Christodoulou,Blanchet,Damour)

§  ArisesfromtheGWsproducedbyGWs:

§  “Unboundpar>cles”aretheindividual“radiatedgravitons”.[Thorne‘92]

§  ProducedbyallsourcesofGWs.

§  AllowsustoprobeoneofthemostnonlinearfeaturesofGR.

[Murphy,O[,&Burrows’09]

⇤hjk = �16⇡(�g)(T jk + T jkGW[h, h]) +O(h2)

T jkGW =

1

R2

dEGW

dtd⌦njnk

Understanding the memory: the linear memory effect [Zel’Dovich&Polnarev’74;Braginsky&Grishchuk‘85;

Braginsky&Thorne‘87]

Understanding the memory: the linear memory effect [Zel’Dovich&Polnarev’74;Braginsky&Grishchuk‘85;

Braginsky&Thorne‘87]

vin

vout

Hyperbolicorbit/two-bodyscaAering[Turner‘77,Turner&Will‘78,Kovacs&Thorne‘78]

Understanding the memory: the linear memory effect

Supernovasimula+onsandGRBsalsoshowalinearmemoryduetotheasymmetricejec>onofmassesorneutrinos[Epstein‘78,Burrows&Hayes‘96,Murphy,O[,&Burrows’09,Segalis&Ori’01,Sagoetal‘04]:

largememory

Understanding the memory: the linear memory effect

Generalformulaforthememoryjumpinasystemw/Ncomponents[Braginsky&Thorne‘87,Thorne‘92]

solveEFE

[Wiseman&Will‘91]

harmonicgaugeEFE…

…hasanonlinearsourcefromtheGWstress-energytensor…

Understanding the memory: the nonlinear memory [Christodoulou‘91;Blanchet&Damour‘92]

Mathema>cally,thenonlinearmemoryarisesfromthecontribu>onofthegravita+onal-wavestress-energytoEinstein’sequa>ons:

Understanding the memory: the nonlinear memory [Christodoulou‘91;Blanchet&Damour‘92]

Mathema>cally,thenonlinearmemoryarisesfromthecontribu>onofthegravita+onal-wavestress-energytoEinstein’sequa>ons:

Nonlinearmemorycanberelatedtothe“linear”memoryifweinterpretthecomponentmassesastheindividualradiatedgravitons(Thorne’92):

§ Memorypiecescalesliketheradiatedenergy.

§ SothenonlinearmemoryispresentinallGWsources.

§ Theeffectishereditary(dependsonen>repastevolu>on).

Understanding the memory: the nonlinear memory [Christodoulou‘91;Blanchet&Damour‘92]

Canalsothinkofitasanonlinearcorrec>ontothemul>poles:

Understanding the memory: the nonlinear memory: inspiralling binaries

Althoughitarisesfroma2.5PNcorrec>ontothemul>polemoments,forinspirallingbinariesthenonlinearaffectsthewaveformatleading(Newtonian)order:Why?

[Wiseman&Will‘91]

Simpleanaly+cmemorymodel:

[MF,PRD’11]

h(mem)(t) =h(+1) � h(�1)

2tanh

✓t

⌧

◆+

h(+1) + h(�1)

2

h(mem)(f) =�h(mem)

2i⇡⌧ csch(⇡2⌧f)

⇡ i�h(mem)

2⇡f

1� ⇡2

6(⌧f)2

�

�h(mem) ⌘ h(+1) � h(�1)

⌧ = 0.001sec

= 0.01sec

= 0.1sec

hc(f) = 2f |h(f)|

hn(f) =p

5fSn(f)

Stepfunc>onapproxima>on:h(mem)(t) = �h(mem)⇥(t)

h(mem) =i�h(mem)

2⇡f, 0 < f < fc, fc ⇠

1

⌧

Simpleanaly+cmemorymodel:

Usingstepfunc>onmodelandrangeofrise>mes,cri>calcharacteris>cmemorystrainfor<SNR>=1is:

[MF,PRD’11]

⌧ = 0.001sec

= 0.01sec

= 0.1sec

h(mem)c ⇡

8><

>:

6⇥ 10�23 - 10�22, aLIGO

5⇥ 10�24 - 10�23, ET

8⇥ 10�22 - 2⇥ 10�21, LISA

Notethatthenonlinearissuppressedbyseveralordersofmagnitudeinhyperbolic/parabolicbinaries.

Memorysources:gravita+onalscaAering

[MF,PRD’11]

h(lin.mem)

c,et�1

/ ⌘M

R

M

rp

h(nonlin.mem)

c,et=1

/ ⌘2M

R

✓M

rp

◆7/2

h(lin. mem)

c,et�1

⇡ 10�21

⇣ ⌘

0.25

⌘✓M/10M�R/10Mpc

◆✓20M

rp

◆

h(nonlin. mem)

c,et=1

⇡ 2⇥ 10�22

⇣ ⌘

0.25

⌘2

✓M/10M�R/10 kpc

◆✓20M

rp

◆

Memorysources:supernovaeSimula>onsfrommul>plegroupsshowamemoryeffectduetoanisotropicma[erorneutrinoemission:[Burrows&Hayes‘94,Murphy,O[,Burrows’09,Kotakeetal‘09,Muller&Janka’97,Yakuninetal‘10]

[Yakuninetal’10]

Sizeofmemoryvariesamongsimula>onsdependingoninputphysics.[reviewsbyO[’09&Kotake‘11]

Nonlinear memory for inspiralling binaries: SurveyofpreviousandrecentworkPartI:InspiralmemoryinPNapproxima+on

•  neededtofullydescribewaveformamplitudecorrec>ons(includingat0PNorder)

•  provideinputtomerger/ringdowncalcula>on

PartII:memoryfrommerger/ringdown•  providesfullmemorysignal;growsrapidlyduringmerger•  semi-analy>cdescrip>onsorfullNR

PartIII:detectabilityes+mate•  applyabovemodelstoevaluatedetec>onprospects

Nonlinear memory for inspiralling binaries: Surveyofpreviousandrecentwork

ü 0PNinspiral,circular,nonspinning:Wiseman&Will’91

ü 3PNinspiral,circular,nonspinning:MF’09a

ü 0PNinspiral,eccentric,nonspinning:MF’11

ü merger/ringdown,nonspinning,equal-mass:MF’09b,‘10

ü merger/ringdown,aligned-spins,equal-masses:Pollney&Reisswig‘11

ü crudedetectabilityes>matesforLISA&LIGO:MF‘09,‘11,Pollney&Reisswig

ü es>matesofrecoil-inducedQNMDopplershiuandmemory:MF’09c

ü pulsar>mingstudies/searches:Seto‘09,vanHaasteren&Levin‘10,Pshrikovetal’10,Cordes&Jenet‘12,Madison,Cordes,Cha[erjee‘14,Wangetal’15,Arzoumanianetal’15

ü Laskyet.al’16:aLIGOdetectabilityviacombina>onsofmul>pleevents.

[Seealsomathema>calaspectsofmemory:Bieri,Garfinkle,Tolish,Wald.]

[MF,Phys.Rev.D‘09]

nonlinear memory from circular binaries: 3PN hlm modes and polarization

linearmemorycase:

Hyperbolicorbits:

Parabolicorbits:nolinearmemory

Ellip>calorbits:nolinearmemory

Nonlinearmemorycase:

Hyperbolic/parabolicorbits:Ellip>calorbits:

0PN

2.5PN

nonlinear memory from eccentric binaries

[MFPRD’11]

Spin-orbit corrections to nonlinear memory (inspiral):

h(mem)

+

= �2⌘M

Rv2 sin2 ⇥

hH0PN,nonspin

+

+ v2H1PN,nonspin+

+ v3H1.5PN,spin+

+ · · · v6H3PN,nonspin+

i

H0PN,nonspin+

=

17 + cos

2

⇥

96

H1PN,nonspin+

= F (cos

2

⇥, ⌘)

H1.5PN,spin+ =

1

768

X

i=1,2

�ii

369

m2i

M2+ 351⌘ + cos

2⇥

✓23

m2i

M2+ 57⌘

◆�

i = LLLN · sssi

SSS1 = �1m21sss1

SSS2 = �2m22sss2

§  Spincorrec>onmaximizedformaximallyspinning,alignedbinaries.

§  Spintermsproduce～20%maximumcorrec>onatSchwarzchildISCO.

§  Small-inclina>onanglecasealsocomputedanaly>cally.(Dependsonperpendicularspincomponents.)

§  Genericprecessingcasecomputednumerically.

Aligned-spincase:[w/XinyiGuo]

−400 −300 −200 −100 0 100−5

0

5

10

15

20 x 10−4

t/M

(R/M

)h(m

em)

40

η = 0.2500η = 0.2222η = 0.1875η = 0.1600η = 0.1389η = 0.1224η = 0.0988

−400 −300 −200 −100 0 1000

0.02

0.04

0.06

0.08

0.10

t/M(R

/M)h(m

em)

20

η = 0.2500η = 0.2222η = 0.1875η = 0.1600η = 0.1389η = 0.1224η = 0.0988

Merger/ringdown memory (nonspinning):

[w/GoranDojcinoski]§  Expressm=0memory

modesintermsofoscillatorymodes.

§  UsehlmfromSXScatalog.

§  Matchtoinspiralmemory.

11400 11500 11600 11700 11800 11900−0.15

−0.10

−0.05

0

0.05

0.10

0.15

0.20

t/M

(R/M

)h+

h(osc)+ + h(mem)

+

h(osc)+

η = 0.2500Θ = π/2

2900 3000 3100 3200 3300 3400−0.10

−0.05

0

0.05

0.10

0.15

t/M

(R/M

)h+

h(osc)+ + h(mem)

+

h(osc)+

η = 0.1600Θ = π/2

Merger/ringdown memory (nonspinning):

Detectability of memory:

§  Useanaly>cmodelfromMFApJL‘09tocomputeSNRforequal-masscase(extensiontoothermassra>osvianewwaveformsinprogress).

§  MFApJL’09focusedondetectabilitybyLISA.(SMBHmemoryeasilyseentoz=2.)

§  Alsoes>matedaLIGOSNRof8for100M☉binaryat20Mpc.

§  Next:extendtheanalysistofutureground-baseddetectors...

MFApJL’09

Detectability: aLIGO (preliminary) [w/EmanueleBer>]

Detectability: future ground-based (preliminary)

Detectability: future ground-based (preliminary)

§  Goodprospectsformostsensi>ve3rdgenera>ondetectors.

§  Formasses～5to4000solarmasses,memorySNR～O(1%)ofinspiralSNR.

Detectability: stacking multiple events 4

d = 410Mpc and fixed component masses (m1 = 36M�,m2 = 29M�), but random values of inclination, polari-sation, and sky position.

For each binary we calculate three (expectation val-ues of) signal-to-noise ratios. Firstly, we calculate theoscillatory signal-to-noise ratio, hS/Ncbci. Secondly, wecalculate the -degeneracy-breaking signal-to-noise ratio,hS/N�hi (see Eqn. (1)). We include ` 3 modes thatare key to resolving . Measurement of hS/N�hi > 0is, in and of itself, interesting as it is evidence of higher-order modes. We calculate that a single detection of aGW150914-like event at design sensitivity will producean hS/N�hi & 5 detection, suggesting this e↵ect can bedetected well before design sensitivity [see also 21–23].Thirdly, we calculate the memory signal-to-noise ratiohS/Ni. We only retain confident oscillatory detections,hS/Ncbci � 12 [24].

The cumulative hS/Ntoti is shown in Fig. 3. In the toppanel, the solid curves represent the expectation valuewhile the shaded region is the one-sigma uncertainty. Theblue curve sums the memory contributions from all bina-ries. This is unrealistic as it includes binaries where wecannot measure the polarisation, and therefore do notknow the memory sign. The red curve adds the mem-ory contribution only from binaries where we are confi-dent the memory sign is correct. That is, we only addthe memory contribution for signals with hS/N�hi > 2,implying we are & 95% confident that is accuratelymeasured, and the sign of the memory is correct. Wehave verified through simulations3 that we recover thecorrect sign of the memory 95% of the time for signalswith hS/N�hi = 2. Binaries that fail this cut are addedwith memory hS/Ni = 0.

The bottom panel of Fig. 3 shows 20 Monte Carlo reali-sations, highlighting the stochasticity of hS/Ntoti growth,and the contribution of the second cut. We highlight onerealisation in red, and show with blue crosses binarieswith hS/N�hi < 2, and therefore have zero memory con-tribution.

Figure 3 shows that one can expect an hS/Ntoti = 3 (5)detection of memory after ⇠ 35 (90) GW150914-like de-tections with aLIGO at design sensitivity, although thiscould happen with as few as ⇠ 20 (75).

In Fig. 4 we plot the cumulative Bayes factor of thememory signal as a function of the number of events.As with Fig. 3, we plot in blue the cumulative memorysignal from all the binaries, and in red we only add thememory contribution from signals where we are confident

3 We perform a Monte Carlo study with GW150914-like binarymergers at a distance such that

⌦S/N

�h

↵= 2 with aLIGO sen-

sitivity. We compute the maximum likelihood using templateswith the correct sign of the memory, and with the opposite sign,finding that the larger likelihood gives the correct sign of thememory for 95% of binaries.

hS/N

toti

0

1

2

3

4

5

6

allhS/N"hi > 2

number of events20 40 60 80 100

hS/N

toti

0

1

2

3

4

5

FIG. 3: Evolution of the cumulative signal-to-noise⌦S/N

tot

↵

as a function of the number of binary black hole mergers. Allbinaries have the same distance and mass as the maximumlikelihood parameters of GW150914, but have random distri-butions of inclination, polarisation and sky position. In thetop panel, the solid curves represent the expectation valueand the shaded region is the one-sigma uncertainties. Theblue curve sums the memory signal-to-noise contribution fromall binaries, and the red curve assigns memory hS/Ni = 0 forthose binaries where the polarisation angle, and hence thesign of the memory cannot be determined. The bottom panelshows 20 individual realisations of the red curve in the toppanel. One particular realisation is highlighted in red; thebinaries assigned hS/Ni = 0 are shown with blue crosses.In both panels, the horizontal dashed and solid lines show⌦S/N

tot

↵= 3 and 5 respectively.

that the sign of the memory signal is correct. Again, thethick curves show the mean value and the shaded region isthe one-sigma uncertainties. As before we highlight thestochastic nature of the growth of this signal by show-ing 10 individual realisations. The results of Fig. 4 areconsistent with that of Fig. 3: one is likely to be confi-dent of a detection of memory after ⇠ 35 events when(lnBF)tot & 8Repeating the simulations presented in Figs. 3 and 4,

but assuming that events are distributed uniformly involume, we find that the time to detection changes byless than a few percent. This is because the growth ofhS/Nitot is dominated by a relatively small number ofloud events.

4

d = 410Mpc and fixed component masses (m1 = 36M�,m2 = 29M�), but random values of inclination, polari-sation, and sky position.

For each binary we calculate three (expectation val-ues of) signal-to-noise ratios. Firstly, we calculate theoscillatory signal-to-noise ratio, hS/Ncbci. Secondly, wecalculate the -degeneracy-breaking signal-to-noise ratio,hS/N�hi (see Eqn. (1)). We include ` 3 modes thatare key to resolving . Measurement of hS/N�hi > 0is, in and of itself, interesting as it is evidence of higher-order modes. We calculate that a single detection of aGW150914-like event at design sensitivity will producean hS/N�hi & 5 detection, suggesting this e↵ect can bedetected well before design sensitivity [see also 21–23].Thirdly, we calculate the memory signal-to-noise ratiohS/Ni. We only retain confident oscillatory detections,hS/Ncbci � 12 [24].

The cumulative hS/Ntoti is shown in Fig. 3. In the toppanel, the solid curves represent the expectation valuewhile the shaded region is the one-sigma uncertainty. Theblue curve sums the memory contributions from all bina-ries. This is unrealistic as it includes binaries where wecannot measure the polarisation, and therefore do notknow the memory sign. The red curve adds the mem-ory contribution only from binaries where we are confi-dent the memory sign is correct. That is, we only addthe memory contribution for signals with hS/N�hi > 2,implying we are & 95% confident that is accuratelymeasured, and the sign of the memory is correct. Wehave verified through simulations3 that we recover thecorrect sign of the memory 95% of the time for signalswith hS/N�hi = 2. Binaries that fail this cut are addedwith memory hS/Ni = 0.

The bottom panel of Fig. 3 shows 20 Monte Carlo reali-sations, highlighting the stochasticity of hS/Ntoti growth,and the contribution of the second cut. We highlight onerealisation in red, and show with blue crosses binarieswith hS/N�hi < 2, and therefore have zero memory con-tribution.

Figure 3 shows that one can expect an hS/Ntoti = 3 (5)detection of memory after ⇠ 35 (90) GW150914-like de-tections with aLIGO at design sensitivity, although thiscould happen with as few as ⇠ 20 (75).

In Fig. 4 we plot the cumulative Bayes factor of thememory signal as a function of the number of events.As with Fig. 3, we plot in blue the cumulative memorysignal from all the binaries, and in red we only add thememory contribution from signals where we are confident

3 We perform a Monte Carlo study with GW150914-like binarymergers at a distance such that

⌦S/N

�h

↵= 2 with aLIGO sen-

sitivity. We compute the maximum likelihood using templateswith the correct sign of the memory, and with the opposite sign,finding that the larger likelihood gives the correct sign of thememory for 95% of binaries.

FIG. 3: Evolution of the cumulative signal-to-noise⌦S/N

tot

↵

as a function of the number of binary black hole mergers. Allbinaries have the same distance and mass as the maximumlikelihood parameters of GW150914, but have random distri-butions of inclination, polarisation and sky position. In thetop panel, the solid curves represent the expectation valueand the shaded region is the one-sigma uncertainties. Theblue curve sums the memory signal-to-noise contribution fromall binaries, and the red curve assigns memory hS/Ni = 0 forthose binaries where the polarisation angle, and hence thesign of the memory cannot be determined. The bottom panelshows 20 individual realisations of the red curve in the toppanel. One particular realisation is highlighted in red; thebinaries assigned hS/Ni = 0 are shown with blue crosses.In both panels, the horizontal dashed and solid lines show⌦S/N

tot

↵= 3 and 5 respectively.

that the sign of the memory signal is correct. Again, thethick curves show the mean value and the shaded region isthe one-sigma uncertainties. As before we highlight thestochastic nature of the growth of this signal by show-ing 10 individual realisations. The results of Fig. 4 areconsistent with that of Fig. 3: one is likely to be confi-dent of a detection of memory after ⇠ 35 events when(lnBF)tot & 8Repeating the simulations presented in Figs. 3 and 4,

but assuming that events are distributed uniformly involume, we find that the time to detection changes byless than a few percent. This is because the growth ofhS/Nitot is dominated by a relatively small number ofloud events.

Lasky,etal.,PRL’16

§  Buildevidencefornonlinearmemoryviastackingofmul>pleevents.

§  Needtoalsomeasurehigher-ordermodestobreakdegeneracyw/polariza>onangle.

Gravitational Waves from Orphan Memory

Lucy O. McNeill,1 Eric Thrane,1, ⇤ and Paul D. Lasky1

1Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, VIC 3800, Australia

Gravitational-wave memory manifests as a permanent distortion of an idealized gravitational-wave detector and arises generically from energetic astrophysical events. For example, binaryblack hole mergers are expected to emit memory bursts a little more than an order of magni-tude smaller in strain than the oscillatory parent waves. We introduce the concept of “orphanmemory”: gravitational-wave memory for which there is no detectable parent signal. In particular,high-frequency gravitational-wave bursts (& kHz) produce orphan memory in the LIGO/Virgo band.We show that Advanced LIGO measurements can place stringent limits on the existence of high-frequency gravitational waves, e↵ectively increasing the LIGO bandwidth by orders of magnitude.We investigate the prospects for and implications of future searches for orphan memory.

The detection of gravitational waves by LIGO andVirgo [1] has opened up new possibilities for observinghighly-energetic phenomena in the Universe. It was re-cently shown that ensembles of binary black hole detec-tions can be used to measure gravitational-wave memory[2]: a general relativistic e↵ect, manifest as a permanentdistortion of an idealized gravitational-wave detector [3–8]. It is not easy to detect memory. The memory strainis significantly smaller than the oscillatory strain; ⇠ 20times smaller for GW150914.

For gravitational-wave bursts, the memory strain in-creases monotonically with a rise time comparable to theburst duration; e.g., ⌧ ⇠ 10 ms for GW150914 [2]. Forsu�ciently short bursts (with timescales that are shortcompared to the inverse frequency of the detector’s sen-sitive band), the memory is well-approximated by a stepfunction, or equivalently an amplitude spectral densityproportional to 1/f , where f is the frequency. It followsthat the memory of a high-frequency burst introducesa significant low-frequency component which extends tofrequencies arbitrarily below 1/⌧ . If the parent burst isabove the detector’s observing band, this can lead to “or-phan memory”: a memory signal for which there is nodetectable parent.

There are a number of mechanisms that can lead toorphan memory. In the example above, a high-frequencyburst outside the observing band creates in-band mem-ory. This is the premise of memory burst searches inpulsar timing arrays [9–13], which look for memory frommerging supermassive black holes for which the oscilla-tory signal is out of band. Orphan memory can also besourced by phenomena other than gravitational waves,e.g., neutrinos [14, 15], although the probability of de-tection from known sources is small. In principal, it ispossible for beamed gravitational-wave sources to pro-duce orphan memory signals when the oscillatory signalis beamed away from Earth. In practice, however, thenumber of orphan detections from beaming will be smallcompared to the number of oscillatory detections. Inthis Letter, we focus on memory where high-frequencygravitational-wave bursts produce orphan memory inLIGO/Virgo.

Scaling relations. As a starting point it is useful to in-vestigate scaling relations for gravitational-wave bursts.For a gravitational-wave source with timescale, ⌧ , fre-quency f

0

⇡ 1/⌧ , and energy Egw

, the strain amplitudescales as

hosc

0

⇠ E1/2gw

f1/20

d, (1)

where d is the distance to the source, and throughout weuse natural units, c = G = 1. A sine-Gaussian wave-form is well described by these assumptions, and so wework with sine-Gaussian waveforms in the analysis thatfollows. In Figure 1 we show two sine-Gaussian bursts(top panel) with their corresponding memory waveformsapproximated by tanh functions (bottom panel).

time (arb. units)-1

-0.5

0

0.5

1

0.5time (arb. units)

0

0.01

0.02

0.03

0.04

0.05

h(t

)(a

rb.u

nits)

FIG. 1: Strain time series for a gravitational-wave burst.The top panel shows both the burst (solid curves) and mem-ory (dashed curves) strains for two bursts of the same ampli-tude. The high-frequency burst (red) has frequency ten timesthe low-frequency burst (blue). The bottom panel shows anenlarged version of the memory time series’. As the frequencyof the burst increases, the rise time approaches zero and thememory is well-approximated by a step function.

arX

iv:1

702.

0175

9v1

[astr

o-ph

.IM]

6 Fe

b 20

17Detectability: “orphan” memory

McNeill,Thrane,Lasky’17

§  Highfrequencydetectorscoulddetectburstsfromexo>csources(DMcollapseinstars,...)

§  Memorycomponentofhigh-frequencyburstcouldbeintheLIGOband.

3

The dashed black curve in Figure 2 shows the sine-Gaussian amplitude h

0

necessary for an average signal-to-noise ratio hS/Ni = 5 detection in Advanced LIGOoperating at design sensitivity as a function of burst fre-quency f

0

. The solid black curve shows the same h0

ver-sus f

0

sensitivity curve except that we include memoryin the matched filter calculation. This has the e↵ect ofextending the LIGO observing band to sources for whichthe dominant oscillatory component has arbitrarily highfrequencies. For burst frequencies higher than a few kHz,the memory becomes more easily detectable than the os-cillatory burst. The colored curves show h

0

verus f0

sen-sitivity for dedicated high-frequency detectors, which wediscuss presently.

We compare the Advanced LIGO sensitivity curve toseveral dedicated high-frequency detectors. Fermilab’s“Holometer” (labeled with a blue curve in Figure 2) isa pair of co-located ⇠ 40m, high-powered Michelson in-terferometers, sensitive to gravitational wave frequencies105�106 Hz. It has reached an amplitude spectral densityof ⇠ 7⇥10�20 Hz�1/2 [18]. The Bulk Acoustic Wave (la-beled with a red curve in Figure 2) cavity is a proposedresonant mass detector, sensitive to 106 � 109 Hz pro-jected amplitude spectral density of ⇠ 10�22 Hz�1/2 [17].The detector is labeled in plots as “Goryachev” using thefirst author from [17]. The final proposed detector thatwe consider here consists of optically levitated sensors,sensitive to frequencies between 50� 300 kHz, with pro-jected sensitivity to ⇠ 3⇥10�22 Hz�1/2 [16]. It is labeledin Figure 2 with a green curve and denoted “Arvanitaki”after the first author of [16].

Comparing the Figure 2 colored sensitivity curves fordedicated high-frequency detectors with the solid blacksensitivity curve for Advanced LIGO, we see that—givenour fiducial value of —Advanced LIGO will detectorphan memory before currently-proposed, dedicated,high-frequency detectors observe an astrophysical burst.

Two e↵ects, not included in Figure 2, will tend to makeit harder to detect high-frequency bursts compared tolow-frequency memory detection. First, high-frequencydetectors produce false positives at a higher rate thanAdvanced LIGO. Second, the memory search templatebank is trivially small. All orphan memory looks thesame: like a step function. In order to span the space ofoscillatory bursts, it is likely that many more templatesmust be used.

This result has an interesting implication. If high-frequency detectors observe a detection candidate, Ad-vanced LIGO should look for a corresponding memoryburst. A coincident memory burst could provide power-ful confirmation that the high-frequency burst is of as-trophysical origin. Similarly, if Advanced LIGO detectsorphan memory, it may be worthwhile looking for coin-cident bursts in dedicated high-frequency detectors.

In Figure 2, we plot sensitivity curves in terms of h0

:the amplitude of a sine Gaussian burst. It is also useful

to frame our results in terms of amplitude spectral den-sity Sh(f)1/2. In Figure 3, we show the noise amplitudespectral densities for the three high-frequency detectorsincluded in Figure 2, denoted with dashed red, blue, andgreen curves. The dashed black curve shows the noiseamplitude spectral density of Advanced LIGO.

We also plot the amplitude spectral density for threesine-Gaussian bursts with memory. The frequency ofeach burst is matched to the observing bands of di↵er-ent high-frequency detectors. The colors are chosen sothat, e.g., the red burst spectrum matches with the redGoryachev detector. The burst amplitude is tuned sothat hS/Ni = 5 in the associated high-frequency detec-tor. The solid curves show the memory + oscillatorycomponent of the signal while the dotted curves showonly the oscillatory component.

While the oscillatory matched filter signal-to-noise ra-tio is 5 in each high-frequency detector, the associatedLIGO memory signal-to-noise ratio is many times louder:300 for Arvanitaki, 1.4⇥105 for Holometer, and 3.3⇥103

for Goryachev. This is consistent with the conclusiondrawn from Figure 2: for our fiducial value of , Ad-vanced LIGO should be able to easily observe orphanmemory from high-frequency bursts observed in dedi-cated high-frequency detectors.

102 103 104 105 106 107 108

frequency (Hz)

10-2810-2710-2610-2510-2410-2310-2210-2110-2010-1910-18

S1=2

h(f

)(H

z!1=2)

aLIGOArvanitakiHolometerGoryachev

FIG. 3: Strain amplitude spectral density. The dashed curvesrepresent the noise in three di↵erent detectors: AdvancedLIGO (black) and three dedicated high-frequency detectors(colored). For each dedicated detector, we plot the amplitudespectral density for a sine-Gaussian burst in the middle ofthe observing band (colored dotted peaks). The peak heightis tuned so that the oscillatory burst can be observed with asignal-to-noise ratio hS/Ni = 5. The solid colored lines showsthe amplitude spectral density when we include the memorycalculated with our fiducial value of . The memory burstsproduce large signals in Advanced LIGO, with hS/Ni rangingfrom 300 to 105.

3

The dashed black curve in Figure 2 shows the sine-Gaussian amplitude h

0

necessary for an average signal-to-noise ratio hS/Ni = 5 detection in Advanced LIGOoperating at design sensitivity as a function of burst fre-quency f

0

. The solid black curve shows the same h0

ver-sus f

0

sensitivity curve except that we include memoryin the matched filter calculation. This has the e↵ect ofextending the LIGO observing band to sources for whichthe dominant oscillatory component has arbitrarily highfrequencies. For burst frequencies higher than a few kHz,the memory becomes more easily detectable than the os-cillatory burst. The colored curves show h

0

verus f0

sen-sitivity for dedicated high-frequency detectors, which wediscuss presently.

We compare the Advanced LIGO sensitivity curve toseveral dedicated high-frequency detectors. Fermilab’s“Holometer” (labeled with a blue curve in Figure 2) isa pair of co-located ⇠ 40m, high-powered Michelson in-terferometers, sensitive to gravitational wave frequencies105�106 Hz. It has reached an amplitude spectral densityof ⇠ 7⇥10�20 Hz�1/2 [18]. The Bulk Acoustic Wave (la-beled with a red curve in Figure 2) cavity is a proposedresonant mass detector, sensitive to 106 � 109 Hz pro-jected amplitude spectral density of ⇠ 10�22 Hz�1/2 [17].The detector is labeled in plots as “Goryachev” using thefirst author from [17]. The final proposed detector thatwe consider here consists of optically levitated sensors,sensitive to frequencies between 50� 300 kHz, with pro-jected sensitivity to ⇠ 3⇥10�22 Hz�1/2 [16]. It is labeledin Figure 2 with a green curve and denoted “Arvanitaki”after the first author of [16].

Comparing the Figure 2 colored sensitivity curves fordedicated high-frequency detectors with the solid blacksensitivity curve for Advanced LIGO, we see that—givenour fiducial value of —Advanced LIGO will detectorphan memory before currently-proposed, dedicated,high-frequency detectors observe an astrophysical burst.

Two e↵ects, not included in Figure 2, will tend to makeit harder to detect high-frequency bursts compared tolow-frequency memory detection. First, high-frequencydetectors produce false positives at a higher rate thanAdvanced LIGO. Second, the memory search templatebank is trivially small. All orphan memory looks thesame: like a step function. In order to span the space ofoscillatory bursts, it is likely that many more templatesmust be used.

This result has an interesting implication. If high-frequency detectors observe a detection candidate, Ad-vanced LIGO should look for a corresponding memoryburst. A coincident memory burst could provide power-ful confirmation that the high-frequency burst is of as-trophysical origin. Similarly, if Advanced LIGO detectsorphan memory, it may be worthwhile looking for coin-cident bursts in dedicated high-frequency detectors.

In Figure 2, we plot sensitivity curves in terms of h0

:the amplitude of a sine Gaussian burst. It is also useful

to frame our results in terms of amplitude spectral den-sity Sh(f)1/2. In Figure 3, we show the noise amplitudespectral densities for the three high-frequency detectorsincluded in Figure 2, denoted with dashed red, blue, andgreen curves. The dashed black curve shows the noiseamplitude spectral density of Advanced LIGO.

We also plot the amplitude spectral density for threesine-Gaussian bursts with memory. The frequency ofeach burst is matched to the observing bands of di↵er-ent high-frequency detectors. The colors are chosen sothat, e.g., the red burst spectrum matches with the redGoryachev detector. The burst amplitude is tuned sothat hS/Ni = 5 in the associated high-frequency detec-tor. The solid curves show the memory + oscillatorycomponent of the signal while the dotted curves showonly the oscillatory component.

While the oscillatory matched filter signal-to-noise ra-tio is 5 in each high-frequency detector, the associatedLIGO memory signal-to-noise ratio is many times louder:300 for Arvanitaki, 1.4⇥105 for Holometer, and 3.3⇥103

for Goryachev. This is consistent with the conclusiondrawn from Figure 2: for our fiducial value of , Ad-vanced LIGO should be able to easily observe orphanmemory from high-frequency bursts observed in dedi-cated high-frequency detectors.

102 103 104 105 106 107 108

frequency (Hz)

10-2810-2710-2610-2510-2410-2310-2210-2110-2010-1910-18

S1=2

h(f

)(H

z!1=2)

aLIGOArvanitakiHolometerGoryachev

FIG. 3: Strain amplitude spectral density. The dashed curvesrepresent the noise in three di↵erent detectors: AdvancedLIGO (black) and three dedicated high-frequency detectors(colored). For each dedicated detector, we plot the amplitudespectral density for a sine-Gaussian burst in the middle ofthe observing band (colored dotted peaks). The peak heightis tuned so that the oscillatory burst can be observed with asignal-to-noise ratio hS/Ni = 5. The solid colored lines showsthe amplitude spectral density when we include the memorycalculated with our fiducial value of . The memory burstsproduce large signals in Advanced LIGO, with hS/Ni rangingfrom 300 to 105.

Spin memory: •  Mo>vatedbypapers:Strominger,Zhiboedov,Pasterski.•  RelatedworksbyFlanagan&Nichols,Madler&Winicour•  RecentpaperbyNichols‘17explains“spinmemory”inPNcontext:Itisthe“nonlinear,nonhereditarymemory”discussedinMFPRD‘09&originallyfoundin2.5PNorderamplitudecorrec>onofArunetal‘04.Currentmul>polemomentscanalsosourcelinearandnonlinearmemoryeffects—theseare(Ithink)whatthemorerecentliteraturereferstoas“spinmemory”:

UL = I(l)L + U (tail)

L + U (nonlin. mem)

L + · · ·

VL = J (l)L + V (tail)

L + V (nonlin. spinmem)

L + · · ·

[MFPRD’09]

h

(nonlin. spinmem)

⇥ = �12

5

⌘

2

M

R

x

7/2sin

2

⇥ cos⇥

Leadstopolariza>oncorrec>on[Arunet.al.’04]

Detec>onisdifficult(needmul>pleeventandET)sincethisisa2.5PNeffectinsteadofa0PNeffect(likethehereditarynonlinearmemory).[SeeNichols‘17]

Summary: § Linearandnonlinearmemoriesareinteres>ngnon-oscillatorycomponentstothegravita>onal-wavesignal.

§ Linearmemoryhasthepoten>altotellusaboutnon-periodicsources(binarysca[ering,supernovae,GRBjets,...)

§ Nonlinearmemorylet’susprobenonlinearwavegenera>oninGR(“wavesthatproducewaves”).

§ Detec>onoflinearmemoryrelieson“ge{nglucky”withanearbysource.

§ NonlinearmemoryfromBBHmergersisclearlydetectableby3rdgenera>ondetectorsorLISA;poten>allywithinreachofLIGOwith~100detec>ons.

[ThisworksupportedbyNSFGrantNo.PHY-1308527.]