Gravitational-wave memory:
an overview
Marc Favata
Montclair State
University (NJ)
LIGO DCC: xxxx
What is the gravitational-wave memory?
example: nonlinear memory from binary black-hole mergers
Gravita+onal-wave signal vs. +me
The wave no longer returns to the zero-point of its oscilla>on.
This growing-offset is called the memory.
before wave
passage
wave passing through detector
a;er wave passage
( gravita>onal-waves propaga>ng into the screen )
with memory
+me
without memory
Why is this called “memory”?
The linear and nonlinear memory:
Linear memory: (Braginskii, Grishchuck, Thorne, Zeldovich, Polnarev)
§ Arises from the non-oscillatory mo>on of a
source, especially due to unbound masses.
§ Ex: hyperbolic orbits,
mass/neutrino ejec>on
in supernovas/GRBs
[Murphy, O[, &
Burrows ’09]
Nonlinear memory: (Christodoulou, Blanchet, Damour)
§ Arises from the GWs produced by GWs:
⇤h̄jk =
jk
TGW
=
1 dEGW
nj nk
2
R dtd⌦
jk
16⇡( g)(T jk + TGW
[h̄, h̄]) + O(h̄2 )
§ “Unbound par>cles” are the individual “radiated gravitons”. [Thorne ‘92]
§ Produced by all sources of GWs.
§ Allows us to probe one of the most nonlinear features of GR.
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]
Hyperbolic orbit/two-body scaAering
[Turner ‘77, Turner & Will ‘78, Kovacs &
Thorne ‘78 ]
vin
vout
Understanding the memory:
the linear memory effect
Supernova simula+ons and GRBs also show a linear memory due to the asymmetric
ejec>on of masses or neutrinos [Epstein ‘78, Burrows & Hayes ‘96, Murphy, O[, & Burrows ’09,
Segalis & Ori ’01, Sago et al ‘04]:
large memory
Understanding the memory:
the linear memory effect
General formula for the memory jump in a system w/ N components
Thorne ‘87, Thorne ‘92]
[Braginsky &
Understanding the memory:
the nonlinear memory
[Christodoulou ‘91; Blanchet & Damour ‘92]
Mathema>cally, the nonlinear memory arises from the contribu>on of
the gravita+onal-wave stress-energy to Einstein’s equa>ons:
harmonic gauge
EFE…
…has a nonlinear source from
the GW stress-energy tensor…
solve EFE
[Wiseman & Will ‘91]
Understanding the memory:
the nonlinear memory
[Christodoulou ‘91; Blanchet & Damour ‘92]
Mathema>cally, the nonlinear memory arises from the contribu>on of
the gravita+onal-wave stress-energy to Einstein’s equa>ons:
Nonlinear memory can be related to the “linear” memory if we interpret the
component masses as the individual radiated gravitons (Thorne’92):
Understanding the memory:
the nonlinear memory
[Christodoulou ‘91; Blanchet & Damour ‘92]
Can also think of it as a nonlinear correc>on to the mul>poles:
§ Memory piece scales like the radiated energy.
§ So the nonlinear memory is present in all GW sources.
§ The effect is hereditary (depends on en>re past evolu>on).
Understanding the memory:
the nonlinear memory: inspiralling binaries
Although it arises from a 2.5PN correc>on to the mul>pole moments, for
inspiralling binaries the nonlinear affects the waveform at leading
(Newtonian) order:
[Wiseman & Will ‘91]
Why?
Simple analy+c memory model:
h
(mem)
h̃
(t) =
(mem)
h(+1)
h(
1
)
2
✓ ◆
t
h(+1) + h(
tanh
+
⌧
2
1
)
h(mem)
(f ) =
i⇡⌧ csch(⇡ 2 ⌧ f )
2

h(mem)
⇡2
⇡i
1
(⌧ f )2
2⇡f
6
h(mem) ⌘ h(+1)
h(
1)
h̃(mem) =
h(mem) ⇥(t)
(mem)
i h
2⇡f
[ MF, PRD ’11 ]
, 0 < f < fc , f c ⇠
1
⌧
= 0.01sec
= 0.1sec
Step func>on approxima>on:
h(mem) (t) =
⌧ = 0.001sec
hc (f ) = 2f |h̃(f )|
p
hn (f ) = 5f Sn (f )
Simple analy+c memory model:
Using step func>on model and range of rise
>mes, cri>cal characteris>c memory strain for
<SNR>=1 is:
⌧ = 0.001sec
= 0.01sec
= 0.1sec
h(mem)
⇡
c
[MF, PRD ’11 ]
8
>
<
>
:
6 ⇥ 10
5 ⇥ 10
8 ⇥ 10
23
- 10 22 , aLIGO
24
- 10 23 , ET
22
- 2 ⇥ 10 21 , LISA
Memory sources: gravita+onal scaAering
(lin.mem)
hc,et 1
MM
/⌘
R rp
(nonlin.mem)
hc,et =1
/⌘
2M
R
✓
M
rp
◆7/2
Note that the nonlinear is suppressed by
several orders of magnitude in hyperbolic/
parabolic binaries.
⇣ ⌘ ⌘ ✓ M/10M ◆ ✓ 20M ◆
(lin. mem)
hc,et 1
⇡ 10 21
0.25
R/10 Mpc
rp
⇣ ⌘ ⌘2 ✓ M/10M ◆ ✓ 20M ◆
(nonlin. mem)
hc,et =1
⇡ 2 ⇥ 10 22
0.25
R/10 kpc
rp
[ MF, PRD ’11 ]
Memory sources: supernovae
Simula>ons from mul>ple groups
show a memory effect due to
anisotropic ma[er or neutrino
emission:
[Burrows & Hayes ‘94, Murphy, O[,
Burrows ’09, Kotake et al ‘09, Muller
& Janka ’97, Yakunin et al ‘10]
Size of memory varies among
simula>ons depending on input
physics.
[reviews by O[’09 & Kotake ‘11]
[Yakunin et al ’10]
Nonlinear memory for inspiralling binaries:
Survey of previous and recent work
Part I: Inspiral memory in PN approxima+on
• needed to fully describe waveform amplitude correc>ons
(including at 0PN order)
• provide input to merger/ringdown calcula>on
Part II: memory from merger/ringdown
• provides full memory signal; grows rapidly during merger
• semi-analy>c descrip>ons or full NR
Part III: detectability es+mate
• apply above models to evaluate detec>on prospects
Nonlinear memory for inspiralling binaries:
Survey of previous and recent work
ü 0PN inspiral, circular, nonspinning: Wiseman & Will ’91
ü 3PN inspiral, circular, nonspinning: MF ’09a
ü 0PN inspiral, eccentric, nonspinning: MF ’11
ü merger/ringdown, nonspinning, equal-mass: MF ’09b, ‘10
ü merger/ringdown, aligned-spins, equal-masses: Pollney & Reisswig ‘11
ü crude detectability es>mates for LISA & LIGO: MF ‘09, ‘11, Pollney & Reisswig
ü es>mates of recoil-induced QNM Doppler shiu and memory: MF ’09c
ü pulsar >ming studies/searches: Seto ‘09, van Haasteren & Levin ‘10, Pshrikov et
al’10, Cordes & Jenet ‘12, Madison, Cordes, Cha[erjee ‘14, Wang et al ’15,
Arzoumanian et al ’15
üLasky et. al ’16: aLIGO detectability via combina>ons of mul>ple events.
[See also mathema>cal aspects of memory: Bieri, Garfinkle, Tolish, Wald.]
nonlinear memory from circular binaries:
3PN hlm modes and polarization
[MF, Phys. Rev. D ‘09]
nonlinear memory from eccentric binaries
linear memory case:
Hyperbolic orbits:
Parabolic orbits: no linear memory
Ellip>cal orbits: no linear memory
Nonlinear memory case:
Hyperbolic/parabolic orbits:
Ellip>cal orbits:
2.5PN
0PN
[MF PRD’11]
Spin-orbit corrections to nonlinear memory (inspiral):
Aligned-spin case: [ w/ Xinyi Guo]
(mem)
h+
=
i
2⌘M 2 2 h 0PN,nonspin
2 1PN,nonspin
3 1.5PN,spin
6 3PN,nonspin
v sin ⇥ H+
+ v H+
+ v H+
+ · · · v H+
R
0PN,nonspin
H+
1PN,nonspin
H+
1.5PN,spin
H+
17 + cos2 ⇥
=
96
= F (cos2 ⇥, ⌘)
1 X
=
768 i=1,2
i i
S2 =

✓
◆
2
m2i
m
369 2 + 351⌘ + cos2 ⇥ 23 i2 + 57⌘
M
M
§ Spin correc>on maximized for maximally
spinning, aligned binaries.
§ Spin terms produce ～20% maximum
correc>on at Schwarzchild ISCO.
§ Small-inclina>on angle case also computed
analy>cally. (Depends on perpendicular
spin components.)
§ Generic precessing case computed
numerically.
LN · ŝsi
i = L̂
2s
2 m2ŝ
2
S1 =
2s
1 m1ŝ
1
Merger/ringdown memory (nonspinning):
0.10
[w/ Goran Dojcinoski]
(mem)
0.08
(R/M ) h 20
§ Express m=0 memory
modes in terms of
oscillatory modes.
η
η
η
η
η
η
η
0.06
=
=
=
=
=
=
=
0.2500
0.2222
0.1875
0.1600
0.1389
0.1224
0.0988
0.04
0.02
§ Use hlm from SXS catalog.
0
§ Match to inspiral memory.
−400
−300
−200
−100
0
100
−100
0
100
t/M
−4
x 10
η
η
η
15
η
η
η
10
η
(R/M ) h 40
(mem)
20
=
=
=
=
=
=
=
0.2500
0.2222
0.1875
0.1600
0.1389
0.1224
0.0988
5
0
−5
−400
−300
−200
t/M
Merger/ringdown memory (nonspinning):
0.20
(R/M ) h +
0.15
(osc)
(mem)
h+ + h+
(osc)
h+
η = 0.2500
Θ = π/2
0.10
0.05
0
−0.05
−0.10
−0.15
0.15
(R/M ) h +
0.10
11400
(osc)
11500
(mem)
h+ + h+
(osc)
h+
11600
t/M
11700
11800
11900
3300
3400
η = 0.1600
Θ = π/2
0.05
0
−0.05
−0.10
2900
3000
3100
t/M
3200
Detectability of memory:
§ Use analy>c model from MF ApJL ‘09 to
compute SNR for equal-mass case
(extension to other mass ra>os via new
waveforms in progress).
MF ApJL’09
§ MF ApJL ’09 focused on detectability by
LISA. (SMBH memory easily seen to z=2.)
§ Also es>mated aLIGO SNR of 8 for 100 M☉
binary at 20 Mpc.
§ Next: extend the analysis to future
ground-based detectors...
Detectability: aLIGO (preliminary)
[w/ Emanuele Ber>]
Detectability: future ground-based (preliminary)
Detectability: future ground-based (preliminary)
§ Good prospects for most sensi>ve 3rd genera>on detectors.
§ For masses ～5 to 4000 solar masses, memory SNR ～O(1%) of inspiral SNR.
Detectability: stacking multiple events
top panel, the solid curves represent the expectation value
and the shaded region is the one-sigma uncertainties. The
S/N
i is sums
shown
in Fig.
3. In thecontribution
top
blue
the memory
signal-to-noise
from
totcurve
all
binaries,
and
the
red
curve
assigns
memory
hS/Ni
=
0 for
ves represent the expectation value
those binaries where the polarisation angle, and hence the
ionsign
is ofthe
The
theone-sigma
memory cannotuncertainty.
be determined. The
bottom panel
shows 20 individual
realisations
of theall
redbinacurve in the top
memory
contributions
from
panel. One particular realisation is highlighted in red; the
stic
as itassigned
includes
where
binaries
hS/Nibinaries
= 0 are shown
withwe
blue crosses.
In
both
panels,
the
horizontal
dashed
and
solid
lines show
⌦
↵
polarisation,
and
therefore
do
not
S/Ntot = 3 and 5 respectively.
6
5
hS/Ntot i
4
3
2
all
hS/N"h i > 2
1
0
5
4
hS/Ntot i
ed component masses (m1 = 36 M ,
random values of inclination, polarition.
Lasky, et al., PRL ’16
we§ calculate
three (expectation
valBuild evidence
for nonlinear
ise ratios. Firstly, we calculate the
memory via stacking of mul>ple
-noise ratio, hS/Ncbc i. Secondly, we
events. signal-to-noise ratio,
neracy-breaking
(1)).
We include
 3 modeshigher-order
that
§ Need
to also` measure
. Measurement of hS/N h i > 0
modes
toisbreak
degeneracy
nteresting
as it
evidence
of higher- w/
polariza>on
angle.
alculate
that a single
detection of a
nt at design sensitivity will produce
ection, suggesting this e↵ect can be ⌦
↵
FIG.
3:
Evolution
of
the
cumulative
signal-to-noise
S/N
tot
e design sensitivity [see also 21–23].
as a function of the number of binary black hole mergers. All
te binaries
the memory
signal-to-noise
have the same
distance and mass ratio
as the maximum
likelihood
parameters
of
GW150914,
but
have
random distriain
confident
oscillatory
detections,
butions of inclination, polarisation and sky position. In the
3
2
1
0
20
40
60
number of events
80
100
arXiv:1702.01759v
or equivalently
an amplitude
spectral
density
e compare
thefunction,
Advanced
curveorphan
to
vanced LIGO
should be LIGO
able tosensitivity
easily observe
of the signal while the dotted curves show
proportional to 1/f , where f is the frequency. It component
follows
ral memory
dedicated
high-frequency
detectors.
Fermilab’s
from high-frequency bursts observed in dedi- only the oscillatory component.
that the memory of a high-frequency burst introduces
ometer”
(labeled
withdetectors.
alow-frequency
blue curve component
in Figure which
2) is extends to
cated high-frequency
1
a significant
While
the
oscillatory
matched filter signal-to-noise rair of co-located
⇠ 40 m,arbitrarily
high-powered
infrequencies
below Michelson
1/⌧ . If the parent
burst is
0.5
tio is 5 in each high-frequency detector, the associated
rometers, sensitive
to gravitational
waveband,
frequencies
above the
detector’s observing
this can lead to “or0
6
McNeill,
Thrane,
’17spectral
memory”:
a memory
signal for
which thereLIGO
is no memory signal-to-noise ratio is many times louder:
10
Hz.
It hasphan
reached
anLasky
amplitude
density
10-18
1/2
20
300 for Arvanitaki,
1.4 ⇥ 105 for Holometer, and 3.3 ⇥ 103
-0.5
7 ⇥ 10 -19
Hz detectable
[18]. parent.
The Bulk Acoustic aLIGO
Wave (la10
There
a number
of mechanisms
that can lead
to
for Goryachev.
This is consistent with the conclusion
frequency
detectors
Arvanitaki
d§
withHigh
a red curve
in are
Figure
2) cavity
is acould
proposed
-1
-20
(arb. units)
orphan
memory.
In
the
example
above,
a
high-frequency
Holometer
6
9
drawn from Figure
2: for our time
fiducial
value of , Ad10
0.05
nant mass
detector,
sensitive
to
10
10
Hz
prodetect
bursts
from
exo>c
sources
Goryachev
burst
outside
the
observing
band
creates
in-band
memvanced LIGO0.04should be able to easily observe orphan
10-21
d amplitude
spectral
of ⇠ 10 of22memory
Hz 1/2burst
[17]. searches
ory. Thisdensity
is the premise
in
memory
from0.03high-frequency bursts observed in dedi(DM
collapse
in
stars,
...)
-22
detector
is
labeled
in
plots
as
“Goryachev”
using
the
pulsar timing arrays [9–13], which look for memory from
10
0.02
detectors.
supermassive
black holes
for which
oscilla- high-frequency
author -23
from merging
[17]. The
final proposed
detector
that the cated
0.01
10
§ Memory
component
ofOrphan
highsignal
out of band.
memory can also be
onsider
here tory
consists
ofis optically
levitated
sensors,
0
-24
0.5
10
sourced
by
phenomena
other
than
gravitational
waves,
time (arb. units)
tive tofrequency
frequencies between
50
300
kHz,
with
proburst[14,could
be inthethe
e.g., neutrinos
15], although
probability of de--18
10-25
d sensitivity
to ⇠ 3 ⇥ 10 22 Hz 1/2 [16]. It is labeled
tection from known sources is small. In principal, it10is
FIG. 1: Strain time series for a gravitational-wave burs
-26
band.
gure 2LIGO
apossible
green curve
and denoted
“Arvanitaki”
10with
-19 The top panel shows both the burst (solidaLIGO
for beamed
gravitational-wave
sources to procurves) and mem
10
Arvanitaki
the first
of
[16].
-27author
duce
orphan
memory
signals
when
the
oscillatory
signal
ory
(dashed
curves)
strains
for
two
bursts
of
the same ampl
10
-20
Holometer
tude. The high-frequency burst (red) has frequency ten time
10
beamed
from
Earth. In curves
practice,for
however, the
omparing
the isFigure
2 away
colored
sensitivity
Goryachev
10-28
panel shows a
number
orphan
detections
from
beaming
will be small-21 the low-frequency burst (blue). The bottom
2
3 ofdetectors
4
5 with6 the
7solid
8black
cated high-frequency
10
10
10
10
10
10
10
10
enlarged
version
of
the
memory
time
series’.
As
the frequenc
compared
to the
number
of oscillatory
detections. In
tivity curve for
Advanced
LIGO,
we
see
that—given
frequency (Hz)
-22 of the burst increases, the rise time approaches zero and th
this Letter, we focus on memory where high-frequency
10
memory is well-approximated by a step function.
fiducial value of —Advanced LIGO will detect
gravitational-wave bursts produce orphan memory in-23
3: Strain
amplitude
spectral density. Thededicated,
dashed curves
10
an FIG.
memory
before
currently-proposed,
LIGO/Virgo.
represent detectors
the noise in
three di↵erent
detectors: burst.
Advanced
-frequency
observe
an astrophysical
10-24
LIGO (black) and three dedicated high-frequency detectors
wo (colored).
e↵ects, not
in Figure
2, will
tendthe
toamplitude
make
Forincluded
each dedicated
detector,
we plot
10-25
arder
to detect
bursts
to of
spectral
densityhigh-frequency
for a sine-Gaussian
burstcompared
in the middle
the observing
band detection.
(colored dotted
peaks).
The peak height
10-26
requency
memory
First,
high-frequency
is tuned
so that
the positives
oscillatory burst
be observed
with a
-27
ctors
produce
false
at a can
higher
rate than
10
signal-to-noise ratio hS/N i = 5. The solid colored lines shows
anced
LIGO. Second,
memory
-28
the amplitude
spectral the
density
when wesearch
includetemplate
the memory
10
iscalculated
triviallywith
small.
All
orphan
memory
looks
the
2
3
4
5
6
7
8
our fiducial value of . The memory bursts
10
10
10
10
10
10
10
produce
large
signals in In
Advanced
LIGO,
with
i ranging
e: like
a step
function.
order to
span
thehS/N
space
of
frequency (Hz)
5
from
300
to
10
.
latory bursts, it is likely that many more templates
h(t) (arb.units)
1=2
Sh (f ) (Hz!1=2 )
1=2
Sh (f ) (Hz!1=2 )
Detectability: “orphan” memory
Spin memory:
• Mo>vated by papers: Strominger, Zhiboedov, Pasterski.
• Related works by Flanagan & Nichols, Madler & Winicour
• Recent paper by Nichols ‘17 explains “spin memory” in PN context:
It is the “nonlinear, nonhereditary memory” discussed in MF PRD ‘09 & originally found in
2.5PN order amplitude correc>on of Arun et al ‘04.
Current mul>pole moments can also source linear and nonlinear memory effects—these are (I
think) what the more recent literature refers to as “spin memory”:
(l)
(tail)
(nonlin. mem)
U L = IL + U L
+ UL
V L = J L + VL
+ VL
(l)
(tail)
+ ···
(nonlin. spin mem)
Leads to polariza>on
correc>on [Arun et. al.’04]
+ ···
(nonlin. spinmem)
h⇥
=
12 2 M 7/2 2
⌘
x sin ⇥ cos ⇥
5
R
Detec>on is difficult (need mul>ple
event and ET) since this is a 2.5PN effect
instead of a 0PN effect (like the
hereditary nonlinear memory). [See
Nichols ‘17]
[ MF PRD ’09 ]
Summary:
§ Linear and nonlinear memories are interes>ng non-oscillatory
components to the gravita>onal-wave signal.
§ Linear memory has the poten>al to tell us about non-periodic
sources (binary sca[ering, supernovae, GRB jets, ...)
§ Nonlinear memory let’s us probe nonlinear wave genera>on in GR
(“waves that produce waves”).
§ Detec>on of linear memory relies on “ge{ng lucky” with a nearby
source.
§ Nonlinear memory from BBH mergers is clearly detectable by 3rd
genera>on detectors or LISA;
poten>ally within reach of LIGO with ~100 detec>ons.
[ This work supported by NSF Grant No. PHY-1308527. ]