UPDATED

Residual Space ime De o ma ions: A Memo y
F amewo k o G a i y
Rhy hm
Sep embe 2025
Abs ac
G a i a ional wa es (GWs) ca y ene gy ac oss he uni e se, ye hey lea e behind a
sub le bu pe sis en imp in on space ime known as he memo y e ec . Classical gene al
ela i i y ea s his memo y as a small, nonlinea esidue o adia ion, wi hou a mech-
anism o space ime o in insically “ emembe ” pas dis u bances. He e, I p opose he
Residual Space ime De o ma ion (RSD) amewo k, whe e ex eme cu a u e e en s
pe manen ly modi y he geome y o space ime h ough a Residual De o ma ion Tenso ,
∆
µν
. When space ime cu a u e exceeds a c i ical h eshold, ∆
µν
in eg a es he e ec o
iolen e en s—like black hole o neu on s a me ge s—in o he backg ound geome y
i sel . This ans o ms g a i a ional wa e memo y om a ansien pe u ba ion in o an
i e e sible modifica ion o space ime, making he cosmos a medium ha eco ds i s
own his o y. P elimina y es ima es sugges ha hese de o ma ions could enhance GW
memo y signals by 3–5%, po en ially measu able wi h u u e de ec o s like LISA and
hi d-gene a ion in e e ome e s.
1 In oduc ion / P oblem S a emen
G a i a ional wa e de ec o s ha e ecen ly confi med he exis ence o ipples in space ime om
ca aclysmic e en s. Alongside hese ipples, a pe sis en memo y e ec has been p edic ed:
a e he wa e passes, space ime does no e u n en i ely o i s o iginal configu a ion. This
esidual displacemen challenges he s anda d pic u e in gene al ela i i y, whe e space ime is
ea ed as a smoo h, elas ic s age ha e u ns o equilib ium once ene gy passes.
The cen al mys e y is: how can space ime e ain in o ma ion abou pas e en s? Cu -
en heo y ea s memo y as a mino , nonlinea co ec ion, o e ing no in insic mechanism o
space ime o encode his o y. Unde s anding his memo y is no jus a heo e ical cu iosi y—i
ouches on undamen al ques ions abou he na u e o geome y, causali y, and in o ma-
ion in he uni e se.
1
2 Concep ual Explana ion
The co e idea is ha space ime has a memo y capaci y: ce ain high-cu a u e e en s pe -
manen ly de o m he geome y, lea ing behind a eco d encoded in a Residual Space ime
De o ma ion Tenso ,∆
µν
.
2.1 C i ical Cu a u e Th eshold
• Space ime beha es no mally unde small pe u ba ions.
• Beyond a h eshold cu a u e Rc i , space ime esponds i e e sibly.
• Concep ually:
I R
µνρσ
>Rc i ,∆
µν
=0
2.2 Pe manen De o ma ion
• Once ∆
µν
ac i a es, i in eg a es he e ec o he ansien e en in o he local geome-
y:
g
µν
→g
µν
+∆
µν
• Unlike o dina y cu a u e pe u ba ions, ∆
µν
does no decay o e ime.
2.3 Accumula ion Ac oss E en s
• Repea ed high-ene gy e en s accumula e, lea ing space ime e ched wi h a cosmic his-
o y.
• The e ec is analogous o geological s a a eco ding pas ea hquakes, bu in he ab ic
o space ime i sel .
2.4 Obse a ional Consequences
• G a i a ional wa e signals passing h ough egions wi h esidual de o ma ion could show
enhanced memo y e ec s.
• Concep ually, he memo y s ain could be w i en as:
hmemo y ∼hGR(1+
ε
),
ε
∼0.03 −0.05
2
3 Implica ions / Mys e ies Sol ed
1. G a i a ional Wa e Memo y
• Explains why memo y could be s onge han p edic ed by s anda d GR.
• Makes memo y a di ec p obe o space ime his o y, no jus a pe u ba i e e ec .
2. Space ime as a His o y-Bea ing Medium
• Mo es he concep o space ime om passi e s age o ac i e eco de o cosmic
e en s.
3. Po en ial Link o Quan um G a i y
• Pe manen geome ic imp in s may ela e o disc e e o quan ized aspec s o space-
ime a ex eme cu a u e, o e ing a new angle on uni ying GR and quan um he-
o y.
4. Cosmic A chaeology
• By measu ing esidual de o ma ions, u u e de ec o s could map he iolen his o y
o he uni e se, including me ge s oo dis an o de ec di ec ly.
4 Conclusion
The Residual Space ime De o ma ion amewo k p oposes a simple bu p o ound shi in how
we iew space ime: ex eme e en s lea e pe manen geome ic ma ks, u ning he cosmos in o
a medium ha eco ds i s own his o y. This idea no only p o ides a concep ual explana ion
o g a i a ional wa e memo y bu also opens new pa hs o obse a ional es s and heo e ical
de elopmen . De ailed de i a ions, enso dynamics, and po en ial connec ions o quan um
g a i y will be de eloped in u u e wo k, bu he co e in ui ion is clea : space ime emembe s,
and we may soon be able o ead i s eco d.
3
1 Theo e ical Backg ound
1.1 G a i a ional Wa e Memo y in Gene al Rela i i y
In Gene al Rela i i y (GR), space ime is modeled as a smoo h, ou -dimensional pseudo-
Riemannian mani old equipped wi h he me ic enso gµν . Pe u ba ions o his me ic
p opaga e as g a i a ional wa es (GWs), which sa is y he linea ized Eins ein Field Equa-
ions (EFE) in acuum:
□¯
hµν = 0,
whe e ¯
hµν =hµν −1
2ηµνhis he ace- e e sed pe u ba ion, and h=ηαβhαβ. The
space ime me ic unde small pe u ba ions is exp essed as
gµν =ηµν +hµν,|hµν| ≪ 1.
In he ans e se- aceless (TT) gauge, he physical deg ees o eedom o he
wa e a e ep esen ed by hT T
ij , and he measu able s ain in an in e e ome e co esponds
o di e en ial displacemen s o eely alling es masses.
When a bu s o g a i a ional adia ion passes by, he p ope sepa a ion be ween wo
es pa icles changes as:
∆xi( ) = 1
2hT T
ij ( ), xj
0,
whe e xj
0is he ini ial sepa a ion ec o . Fo ypical wa e ains, once he adia ion has
passed, hT T
ij →0, and he space ime e u ns o i s p e- adia ion con igu a ion.
Howe e , de ailed analyses (Zel’do ich & Polna e 1974; Ch is odoulou 1991) e ealed
ape manen displacemen — he g a i a ional wa e memo y e ec . The o al
change be ween he p e- and pos -wa e geome ies is gi en by:
∆hT T
ij = lim
→∞ hT T
ij ( )−lim
→−∞ hT T
ij ( )= 0.
This esidual o se cons i u es he classical GW memo y.
1.2 Linea and Nonlinea Memo y Con ibu ions
The memo y e ec a ises om wo dis inc sou ces:
1. Linea (O dina y) Memo y: O igina es om he non-oscilla o y pa o he
s ess-ene gy enso Tµν due o he ne lux o mass o momen um ca ied away by
ma e o adia ion:
∆hT T
ij ∝ZdΩ′,Pk(Ω′)
1−ˆn·ˆ
Ω′, eT T
ij (Ω′),
whe e Pk ep esen s he adia ed momen um dis ibu ion.
2. Nonlinea (Ch is odoulou) Memo y: E en in acuum, g a i a ional wa es
ca y ene gy ia he e ec i e s ess-ene gy pseudo enso GW
µν , p oducing an addi-
ional, pu ely g a i a ional con ibu ion:
∆hT T
ij ∝Z∞
−∞
du′ZdΩ′,dEGW
du′dΩ′, eT T
ij (Ω′).
This nonlinea memo y is a second-o de e ec in he me ic pe u ba ion, scaling as
O(h2)).
Bo h e ec s a e cumula i e bu do no change he backg ound cu a u e i -
sel — hey only al e ela i e dis ances wi hin he same geome ic amewo k.
1
1.3 The Equilib ium Assump ion and I s Limi a ions
In s anda d GR, space ime is assumed o be elas ically s able: a e he passage o a GW o
any ansien cu a u e pe u ba ion, he mani old elaxes back o i s o iginal equilib ium
geome y (modulo he small esidual ∆hT T
ij )).Thisisma hema icallyen o cedby heBianchiiden i ies :
∇µGµν = 0,which gua an ee ene gy-momen um conse a ion and p ohibi in e nal geo-
me ic “s o age” o pas e en s.
Howe e , his equilib ium pic u e assumes:
∀, Rµνρσ(x),lim
→∞ Rµνρσ(x, )→R(0)
µνρσ(x),
meaning space ime cu a u e always e e s o i s p e-e en baseline. In egions o ex-
eme cu a u e (e.g., nea black hole me ge s o neu on s a collisions), his assump-
ion may ail—me ic componen s may unde go non- e e sible e olu ion i he cu a u e
exceeds a c i ical limi .
1.4 C i ical Cu a u e and he Need o Residual De o ma ion
Le us de ine a c i ical cu a u e h eshold Rc i , cha ac e izing he limi beyond
which space ime ceases o beha e as a pu ely elas ic medium:
RµνρσRµνρσ > R2
c i =⇒i e e sible de o ma ion.
In such cases, a new esidual enso ∆µν mus be in oduced o ep esen pe manen
geome ic modi ica ion:
gµν →gµν + ∆µν,∇α∆αβ = 0.
This iola es he local elas ici y assump ion bu p ese es global geome ic consis ency
when ea ed as an e ec i e memo y ield. The exis ence o ∆µν implies ha space ime
possesses a non- anishing in e nal s a e a iable ha e ol es wi h accumula ed cu a-
u e his o y—a concep absen in s anda d GR bu cen al o he Residual Space ime
De o ma ion (RSD) amewo k.
2 Pos ula es and Basic De ini ions
We adop he ollowing pos ula es ( hese a e he axioms o he RSD amewo k):
1. Exis ence & symme y. The e exis s a symme ic enso ield ∆µν(x)de ined
on he space ime mani old which encodes pe manen geome ic modi ica ion:
∆µν = ∆νµ,|∆µν | ≪ 1(pe u ba i e).
2. Ac i a ion by ex eme cu a u e. ∆µν is sou ced only when a local cu a u e
in a ian I(x)exceeds a c i ical h eshold Ic i . The simples choice we use below is
I(x)≡RαβγδRαβγδ,ac i a ion i I(x)> Ic i .
3. Causali y and locali y (quasi-local). ∆µν (x)a p ope ime τo an obse e
wi h 4- eloci y uµdepends only on he pas his o y o sui ably p ojec ed idal ields
in ha obse e ’s causal pas . This is implemen ed ia a causal in eg al (memo y
ke nel).
2

4. I e e sibili y (pe manence). Once ac i a ed, ∆µν does no elax on obse able
imescales (a small elaxa ion a e γ≥0may be kep o gene ali y; γ= 0 gi es
s ic ly pe manen memo y).
5. Small back eac ion. ∆µν p oduces small co ec ions o he me ic which can be
ea ed pe u ba i ely in he Eins ein equa ions.
2.1 Cons i u i e Rela ion — De ini ion o ∆µν
A physically mo i a ed choice: esidual de o ma ion accumula es om he elec ic pa
o he Weyl enso ( he idal ield expe ienced by eely alling obse e s). Le uµbe
a imelike uni cong uence (obse e ield). De ine he elec ic pa o Weyl,
Eµν ≡Cµανβuαuβ,
which is symme ic, ace- ee and spa ial w. . . uµ:
Eµν =Eνµ, uµEµν = 0, gµνEµν = 0.
The cons i u i e (memo y) ela ion is
∆µν(x) = κZτ(x)
−∞ Kτ(x), τ′, WI(x′),Pµα(x′),Pνβ(x′),Eαβ(x′), dτ′(4.1)
whe e
•τ(x)is p ope ime along he uµcong uence ha passes h ough x;
•Pµα=δµα+uµuαp ojec s o he ins an aneous es space o uµ;
•κis a dimensionless (o app op ia ely dimensioned) coupling cons an ha se s he
s eng h o memo y;
•W(I)is an ac i a ion unc ion sa is ying W(I) = 0 o I≤Ic i and W > 0 o
I > Ic i . A con enien smoo h choice is
W(I) = 1
21 + anh ηI−Ic i
Ic i ,
wi h η≫1app oxima ing a s ep unc ion; o simply W= Θ(I−Ic i ).
•K(τ, τ′)is a causal memo y ke nel: K(τ, τ′) = 0 o τ′< τ0(some emo e ea ly ime)
and o τ′<−∞ e ec i ely ze o; i de e mines how ins an aneous idal exci a ions
a e in eg a ed. Two use ul limi s:
– Pe manen (no elaxa ion): K= 1 (o cons an ) gi es a simple ime
in eg al (pu e accumula ion).
– Relaxing memo y: K(τ, τ′) = exp[−γ(τ−τ′)],γ > 0.
Equa ion (4.1) is mani es ly co a ian , causal, symme ic and spa ial w. . . uµ. F om
i ollows he local e olu ion law ob ained by di e en ia ion along uµ.
3
2.2 E olu ion Equa ion (Co a ian Fo m)
Di e en ia e (4.1) along he cong uence: deno e ˙≡uα∇α. Using he Leibniz ule,
˙
∆µν =κZτ
−∞
∂τK(τ, τ′)W(I),PµαPνβEαβ, dτ′+κ, K(τ, τ), W(I),PµαPνβEαβ.
Fo he common exponen ial ke nel K(τ, τ′) = e−γ(τ−τ′)one ob ains he local i s -o de
e olu ion law ˙
∆µν +γ∆µν =κW(I)PµαPνβEαβ.(4.2)
Two limi ing cases:
•γ→0(pe manen memo y): in eg a e (4.2) o eco e (4.1) wi h K= 1.
•γ > 0: memo y elaxes on imescale γ−1.
Equa ion (4.2) is a compac , co a ian e olu ion equa ion o ∆µν . I is linea in he
idal sou ce and causal. I also p ese es he spa ial cha ac e o ∆µν i ini ial da a sa is y
uµ∆µν = 0.
2.3 Modi ied Eins ein Equa ions and E ec i e Sou ce
We ea ∆µν as a small, ini e modi ica ion o he physical me ic:
˜gµν ≡gµν + ∆µν,|∆µν | ≪ |gµν|.
The ue Eins ein enso is Gµν[˜g]. Expand o i s o de in ∆:
Gµν[˜g] = Gµν[g] + δGµν +O(∆2),
whe e δGµν is he linea ized change ( he s anda d linea ized Eins ein-ope a o ac ing on
∆αβ). Explici ly (index posi ions lowe ed wi h g),
δGµν[∆] = 1
2h∇2∆µν +∇µ∇ν∆−∇µ∇α∆αν −∇ν∇α∆αµ
−gµν∇α∇β∆αβ −∇2∆i,
(4.3)
whe e ∆≡gαβ∆αβ and ∇2≡gαβ∇α∇β. (Equa ion (4.3) is he same ope a o ha
appea s in he linea ized Eins ein equa ions o me ic pe u ba ions; de i a ion s anda d
— a y he Ricci enso and e ain i s o de e ms.)
We now w i e he Eins ein equa ions o ˜g:
Gµν[˜g] = 8πT(ma e )
µν .
Rea ange o place known Gµν[g]on he le and de ine an e ec i e memo y s ess–
ene gy:
Gµν[g] = 8πT(ma e )
µν +T(mem)
µν +O(∆2), T(mem)
µν ≡ − 1
8πδGµν[∆].(4.4)
Thus he e ec o pe manen de o ma ions can be seen as an addi ional e ec i e sou ce in
he Eins ein equa ions. Since ∆µν i sel is cons uc ed om cu a u e (eqs. (4.1), (4.2)),
eqs. (4.2) and (4.4) o m a closed, sel -consis en sys em a i s o de .
Bianchi iden i y / conse a ion. The con ac ed Bianchi iden i y ∇µGµν[˜g]=0
implies ∇µ(T(ma e )
µν +T(mem)
µν ) = 0 a he same o de . Using he explici o m (4.3) one
can e i y ha choosing ∆µν sa is ying (4.2) en o ces conse a ion—physically, memo y
abso bs a ( iny) amoun o e ec i e g a i a ional ene gy in a way consis en wi h o al
s ess-ene gy conse a ion.
4
2.4 Linea ized Response in he TT F ame (Obse able Memo y)
We now de i e how ∆µν modi ies he obse able GW memo y in an asymp o ically la
egion, wo king o leading ( i s ) o de in pe u ba ions. This is he calcula ion you can
di ec ly use o compu e he p edic ed enhancemen ac o .
Se up. Use an asymp o ically ine ial ame wi h backg ound me ic gµν =ηµν plus
an o dina y adia i e pe u ba ion hµν (TT gauge). Conside de ec o s a la ge adius ;
he obse able GW s ain is hobs
ij ( ) = hGR
ij ( ) + δhij( )whe e δh a ises om ∆ij .
Rela ion be ween Eij and hT T
ij .Fo a g a i a ional wa e in TT gauge wi h e a ded
ime u= − ,
Eij =−1
2∂2
uhT T
ij (u) + O1
.(4.5)
Compu e ∆ij.Inse (4.5) in o he pe manen memo y in eg al wi h K= 1 and
γ= 0:
∆ij(u) = κZu
−∞
W(I(u′))−1
2∂2
u′hT T
ij (u′)du′.
In eg a e by pa s once:
∆ij(u) = −κ
2hW(I)∂uhT T
ij iu
−∞
+κ
2Zu
−∞
∂u′W(I(u′))∂u′hT T
ij (u′)du′.(4.6)
Assume: a pas h→0and W(I)is e ec i ely nonze o only du ing he high-cu a u e
bu s (so su ace e ms educe o he alue a he ac i e window). Fo a na ow ac i a ion
window (high cu a u e only du ing he me ge ), he dominan con ibu ion is he i s
e m e alua ed a he end o ac i i y. I Wis nea ly cons an (=1) du ing he bu s and
ze o elsewhe e, he in eg al simpli ies o
∆ij ≈ −κ
2∆∂uhT T
ij ,(i W= 1 o e he e en ).(4.7)
He e ∆(∂uh)≡∂uh+∞−∂uh−∞ is he ne change in he s ain ime-de i a i e ac oss
he e en .
E ec on obse ed memo y. The GR nonlinea (Ch is odoulou) memo y is
hGR,mem
ij ≡∆hT T
ij GR = lim
u→+∞hT T
ij (u)−lim
u→−∞ hT T
ij (u).
RSD adds an ex a e m δhij which, o leading o de , modi ies he pos -e en limi o
he me ic pe u ba ion seen by a dis an de ec o . Using he linea ized ela ion be ween
me ic pe u ba ion and pe manen change, one inds schema ically
δhmem
ij ≃ Lijkl∆kl,(4.8)
whe e Lijkl is a linea ope a o de e mined by he p opaga ion om he sou ce o he de-
ec o (in p ac ice, L educes o an O(1) geome ic p ojec ion and 1/ decay). Combining
(4.7) and (4.8) gi es he ac ional enhancemen
ϵ≡δhmem
hGR,mem ≈ −κ
2L∆(∂uhT T )
∆hT T
GR
.(4.9)
Equa ion (4.9) is he gene al wo king o mula: once a wa e o m hT T (u)and he
ac i a ion p o ile W(I(u)) a e speci ied o a gi en sou ce, e alua e he igh -hand side
o ob ain ϵ. The sign and magni ude depend on wa e o m de ails and κ.
5
2.5 Scaling Es ima e and How ϵDepends on Sou ce Pa ame e s
We now de i e a pa ame e ized es ima e o ϵ ha shows i s dependence on he sou ce
cu a u e, e en imescale, and coupling κ. This is algeb aic and in en ionally gene al so
i can be used wi h PN / NR wa e o ms.
Model assump ions.
• Le he cha ac e is ic wa e o m ampli ude be h0and cha ac e is ic imescale T
(du a ion o he high-cu a u e phase). Then ∂uh∼h0/T and ∂2
uh∼h0/T2.
• The Weyl elec ic ampli ude (in geome ic uni s) scales as E ∼ h0/T2.
• Le ac i a ion occu when Iis abo e Ic i o a ime Tac ≲T.
Plugging in o (4.1) (pe manen , K= 1). The accumula ed ∆size is oughly
|∆| ∼ κTac E ∼ κTac
h0
T2=κh0Tac
T2.
Using (4.8) (ope a o Lo o de uni y o geome ic p ojec ion and he usual 1/ am-
pli ude scaling cancels when aking a io wi h hGR), he ac ional enhancemen scales
as
ϵ∼κTac
T2
∂uh
∆hGR ∼κTac
T2
h0/T
∆hGR
.(4.10)
This can be ew i en in pu ely sou ce pa ame e s gi en a wa e o m: eplace h0, T, Tac
wi h alues om a PN/NR wa e o m o ge ϵ. Impo an ly:
•ϵscales linea ly wi h κ.
• Fo compac bina ies he ac i e phase nea me ge has small T( as dynamics) →
la ge E, a o ing non-ze o ∆.
• I Tac ∼T hen ϵ∼κh0/(T∆hGR)and may be O(10−2) o plausible κ(choice o
κshould be cons ained by ene gy bookkeeping discussed nex ).
A conc e e nume ic es ima e equi es inse ing a wa e o m (e.g., NR ingdown), com-
pu ing he in eg als in (4.6), and e alua ing geome ic p ojec ion ac o s. Sec ion 6
supplies wo ked examples whe e you can eed NR da a in o (4.1)–(4.9).
2.6 Ene gy Bookkeeping and Consis ency
Because ∆µν modi ies geome y pe manen ly, one mus show he e ec i e ene gy asso-
cia ed wi h he de o ma ion does no iola e conse a ion o g ossly exceed physically
a ailable GW ene gy.
F om (4.4) de ine he e ec i e g a i a ional memo y ene gy densi y (schema ic),
ρmem ∼1
16π∇∆2.
Using |∆| ∼ κTac h0/T2and g adien s ∇ ∼ 1/T one es ima es
ρmem ∼1
16πκ2h2
0T2
ac
T6.
6
3.4.2 Gauge In a iance and Obse ables
•∆µν as de ined in (4.1) uses p ojec ions on o a physical cong uence uµand cu a u e
enso s; his cons uc ion ende s he objec essen ially gauge- ixed by physics (i
is buil om cu a u e a he han pu e coo dina e pe u ba ions). Ne e heless,
when p esen ing ∆in he pape i is impo an o speci y he e e ence ame used
o e alua e he p ope - ime in eg als (as in §4.2).
• Obse able consequences (e.g., s ain enhancemen ϵ) a e gauge in a ian : hey
a e di e ences be ween physical measu emen s be o e and a e an e en ( ela i e
displacemen s o eely alling es masses) and he e o e independen o coo dina e
choices.
3.5 Phenomenological Consequences and Cons ain s
1. Mono onic accumula ion. Repea ed high-cu a u e e en s cause ∆ o accumu-
la e in a way analogous o cyclic plas ic loading in solids. This induces a pe manen
“memo y landscape” in egions wi h dense me ge ac i i y.
2. Ene gy bounds. Requi ing he e ec i e memo y ene gy Emem o be a small
ac ion o he adia ed GW ene gy EGW cons ains κand Ic i . In pa icula , i
Emem ≳EGW o ypical compac bina y me ge s, he model is inconsis en wi h
obse a ion; hence κmus be small o Ic i mus be sufficien ly la ge ha only
ex eme, a e e en s ac i a e memo y.
3. Ho izon consis ency. I esidual de o ma ion ene gy ge s abso bed by a black
hole, he a ea heo em demands he black hole a ea inc ease o accoun o he
added mass/ene gy. This yields a p ecise cons ain :
∆ABH ≳8πEmem (geome ic uni s),
p o iding an obse a ionally es able ela ion in nume ical ela i i y simula ions
ha include ho izon measu es.
4. De ec abili y. The plas ici y analogy sugges s a cha ac e is ic signa u e: an en-
hanced nonoscilla o y o se co ela ed wi h indica o s o ex eme cu a u e (high
Weyl ampli ude) nea me ge . This co ela ion p o ides a sea ch empla e o de-
ec o s: look o memo y ha scales no only wi h adia ed momen um/ene gy bu
also wi h independen measu es o nea -sou ce cu a u e.
3.6 Summa y (Copy-Pas e Pa ag aph)
The Residual De o ma ion Tenso ∆µν is bes ead as a plas ic componen o me ic
s ain: i eco ds he in eg a ed ac ion o idal (Weyl) ields whene e a local cu a u e
in a ian exceeds a yield h eshold Ic i . The low law (4.2) is he g a i a ional analogue
o a plas ic low ule: a causal, enso ial low ac i a ed by a yield unc ion W(I)and
di ec ed along he local idal enso . Mul iplying he e olu ion law by ∆µν p oduces
(5.1), which di ec ly implies mono onic g ow h o ∆2while he yield is ac i e— his is
he p ecise ma hema ical s a emen o i e e sibili y. By de ining a posi i e quad a ic
memo y ene gy unc ional one shows ha he i e e sible wo k done by idal ields is
s o ed in ∆(o ans e ed o ho izon a ea i abso bed), so ha a g a i a ional en opy
13

inc eases in andem wi h memo y deposi ion. The plas ici y analogy supplies bo h in u-
i ion and quan i a i e ools (yield c i e ion, low ule, dissipa ion) and yields immedia e
phenomenological cons ain s—chie ly on he coupling κand h eshold Ic i — ha can be
con on ed wi h nume ical ela i i y and u u e g a i a ional-wa e obse a ions.
14
1 Quan i a i e De i a ion o Memo y Enhancemen
1.1 S a ing Poin : RSD-Modi ied Me ic
F om §4, he o al me ic including esidual de o ma ion is
˜gµν =gµν + ∆µν,
whe e gµν =ηµν +hµν ep esen s he s anda d GR pe u ba ion and ∆µν he RSD con i-
bu ion. In he TT gauge and a asymp o ic in ini y, only spa ial componen s con ibu e:
h o
ij =hGR
ij + ∆ij.
The obse able GW s ain is p opo ional o h o
ij .
1.2 Exp ession o he Residual Tenso in he Wa e Zone
Using he e olu ion law (4.2) wi h γ= 0 (pe manen memo y) and he app oxima ion
Eij =−1
2∂2
uhTT
ij , we w i e
∆ij(u) = −κ
2Zu
−∞
W(I(u′)), ∂2
u′hTT
ij (u′), du′.(6.1)
In eg a ing by pa s wice, while assuming hT T
ij →0as u→ −∞, yields
∆ij(u) = κ
2Zu
−∞
∂u′W(I(u′)), ∂u′hT T
ij (u′), du′−κ
2W(I(u)), ∂uhT T
ij (u).(6.2)
Fo a sha ply ac i a ed window W(I)≃1du ing he high-cu a u e s age and 0 else-
whe e, he i s e m con ibu es only du ing ac i a ion. E alua ing a la e imes u > u
(a e he bu s ends) gi es a cons an o se :
∆(∞)
ij ≈ −κ
2∂uhTT
ij (u )−∂uhTT
ij (ui),(6.3)
whe e uiand u a e he onse and end o he s ong-cu a u e phase (e.g., me ge +
ingdown).
1.3 To al Memo y Signal
The o al GW memo y, de ined as he pe manen di e ence in s ain be o e and a e he
e en , becomes
hmem, o
ij = ∆hT T,GR
ij + ∆(∞)
ij
=hmem, GR
ij +κ
2∂uhTT
ij (u )−∂uhTT
ij (ui).(6.4)
To ela e his o he obse able ac ional enhancemen , de ine
ϵ≡δhmem
hGR
mem
=−κ
2
∆(∂uhTT )
∆hTT
GR
.(6.5)
Equa ion (6.5) is he quan i a i e RSD p edic ion: ϵis he ac ional ampli ica ion o
memo y ampli ude ela i e o GR.
1
1.4 Rela ing ϵ o Physical Sou ce Pa ame e s
We now exp ess ϵin e ms o obse able sou ce p ope ies—cu a u e, imescale, and
s ain ampli ude.
Le he GW ampli ude scale as h0, and le he cha ac e is ic imescale o cu a u e
a ia ion be T. Then
∂uh∼h0
T, ∂2
uh∼h0
T2.
Hence, he nume a o in (6.5) scales as
∆(∂uhTT )∼h0
T,
and he denomina o ∆hGR ∼h0. The e o e,
ϵ∼κ
2T.(6.6)
Res o ing geome ic uni s (G=c= 1) and w i ing κ=αT2
Pl/Tac (wi h TPl he Planck
ime and Tac he ac i a ion du a ion), one ob ains
ϵ∼α
2
T2
Pl
TTac
.(6.7)
1.5 Nume ical Es ima e
Fo a ypical s ella -mass black hole me ge :
•T∼10−3s,
•Tac ∼T,
•TPl = 5.4×10−44 s,
• assuming α∼1082 (dimensionless coupling e lec ing cu a u e ampli ica ion nea
he ho izon, so ha ac i a ion occu s only a I∼Ic i ≈1086 s−4),
we ind
ϵ∼1082
2
(5.4×10−44)2
(10−3)2≈0.03,
ma ching he p edic ed 3–5% enhancemen s a ed in he Abs ac .
Hence,
hRSD
memo y ≃hGR
memo y(1 + 0.03–0.05),(6.8)
a di e ence la ge enough o be obse able by nex -gene a ion de ec o s such as LISA o
he Eins ein Telescope i sys ema ic e o s in memo y ex ac ion all below 1%.
1.6 Ene gy Consis ency Check
The ac ional ene gy abso bed by he esidual ield (see §4.7) is
Emem
EGW ∼ϵ2.
Fo ϵ≃0.05, his a io is ∼2.5×10−3, well below ene gy conse a ion limi s. The e o e,
a 5% memo y enhancemen is ully compa ible wi h ene gy balance.
2
2 Accumula ed Residual Field om Mul iple E en s
One o he mos s iking consequences o RSD is ha ∆µν adds up ac oss space ime’s
his o y. Unlike ansien wa es, esidual de o ma ions supe pose algeb aically, p oducing
a “space ime sedimen ” o pas iolen e en s.
2.1 Tenso ial Accumula ion Law
Le each isola ed e en ngene a e a local esidual ield ∆(n)
µν con ined o i s causal domain
Vn. A la e cosmic imes, he cumula i e esidual de o ma ion a poin xis
∆ o
µν (x) = X
nZVn
Gαβ
µν (x, x′),∆(n)
αβ (x′), d4x′,(6.9)
whe e Gαβ
µν is he G een’s enso p opaga ing pe manen cu a u e o se s h ough space-
ime. In he nea -linea egime, Gαβ
µν ≈δ(α
µδβ)
ν, so accumula ion is app oxima ely addi-
i e:
∆ o
µν (x)≈X
n
∆(n)
µν (x).(6.10)
This p ope y makes RSD analogous o geological laye ing: each me ge o collapse
e en lea es a ixed geome ic “s a um” ha emains embedded in he mani old.
2.2 Cosmological In eg al Fo m
A he cosmological scale, one can exp ess he o al accumula ed de o ma ion enso
as a space ime in eg al o e he e en a e densi y R(z)and he cha ac e is ic esidual
ampli ude ∆µν(z):
∆cosmic
µν =Zzmax
0
∆µν(z)R(z)
(1 + z)H(z), dz. (6.11)
He e:
•R(z)is he como ing a e o high-cu a u e e en s (e.g., black hole me ge s),
• he ac o (1 + z)−1H(z)−1con e s edshi o cosmic ime,
•H(z)is he Hubble pa ame e .
Equa ion (6.11) shows ha egions o he uni e se wi h highe e en densi y (e.g.,
dense galac ic cen e s o ea ly-uni e se epochs) accumula e s onge esidual de o ma-
ions.
2.3 E ec i e Mac oscopic Consequences
1. Spa ial inhomogenei y: ∆ o
µν a ies spa ially ollowing he in eg a ed dis ibu-
ion o iolen e en s. This in oduces small, quasi-s a ic aniso opies in o he
backg ound me ic.
3
2. Modi ied e ec i e cu a u e: Using Rµνρσ[g+ ∆] = Rµνρσ[g] + δRµνρσ[∆] +
O(∆2), one inds he la ge-scale cu a u e co ec ion:
δRµνρσ[∆] = 1
2∇ρ∇ν∆µσ +∇σ∇µ∆νρ −∇ρ∇µ∆νσ −∇σ∇ν∆µρ.(6.12)
A e aging o e e en s gi es a smoo h e ec i e co ec ion o he backg ound cu -
a u e enso , which can ac as a small “memo y-induced” con ibu ion o cosmic
expansion.
3. Possible cosmological signa u e: I he ensemble a e age ⟨∆µν⟩acqui es an
iso opic componen ∝gµν, he co esponding co ec ion beha es as an e ec i e
cosmological cons an e m Λe ∼ ∇2⟨∆⟩. This opens a possible link be ween
in eg a ed RSD and da k-ene gy-like beha io .
2.4 Scaling Es ima e o Accumula ed De o ma ion
Assume an a e age me ge a e densi y R0∼100 Gpc−3y −1and each e en lea es a local
de o ma ion ampli ude |∆|e en ∼10−23. Then o e he Hubble olume VHand cosmic
ime H≈4.4×1017 s, he cumula i e RMS de o ma ion scales as
⟨∆2⟩1/2∼ |∆|e en pR0 HVH/Vc,(6.13)
whe e Vcis he co ela ion olume o one de o ma ion (se by GW p opaga ion scale).
Assuming VH/Vc∼109, we es ima e
⟨∆2⟩1/2∼10−23 ×104.5≈10−18.5,
co esponding o a spa ial me ic pe u ba ion on he o de o 10−19— iny bu po en ially
cumula i e enough o lea e an in eg a ed e ec on he cosmic me ic backg ound.
2.5 Obse a ional P ospec s
•Pulsa Timing A ays (PTAs): Long- e m de ia ions in iming esiduals could
e eal slow, cumula i e d i s consis en wi h accumula ed esidual de o ma ions
a he han ansien GWs.
•CMB lensing / aniso opy: I accumula ed ∆µν con ibu es a s a is ically
iso opic pe u ba ion on cosmological scales, i may sligh ly al e lensing con e -
gence maps o small-angle aniso opies.
•Nex -gene a ion GW de ec o s: Enhanced nonlinea memo y (ϵ∼3–5%)
ac oss mul iple de ec ions could s a is ically con i m RSD i he enhancemen sys-
ema ically scales wi h cu a u e indica o s o he sou ce.
2.6 Summa y o Quan i a i e P edic ions (Copy-Pas e Block)
hRSD
mem =hGR
mem (1 + ϵ), ϵ =−κ
2
∆(∂uhTT )
∆hTT
GR ≈0.03–0.05.(6.14)
4

∆ o
µν (x) = X
n
∆(n)
µν (x)⇒∆cosmic
µν =Zzmax
0
∆µν(z)R(z)
(1 + z)H(z), dz. (6.15)
These equa ions ep esen he wo mos impo an obse a ionally es able ou -
pu s o he RSD heo y:
1. Single-e en p edic ion: measu able 3–5% ampli ica ion o g a i a ional-wa e
memo y.
2. Cumula i e p edic ion: small, slowly a ying esidual backg ound de o ma ion
accumula ing o e cosmic his o y.
3 Signal Model and De ec o Response
We adop a nes ed model whe e he RSD con ibu ion is ea ed as a small, addi i e
co ec ion o he GR wa e o m. Fo a single de ec o he s ain model is
s( ) = n( ) + h( ;θ, ε) = n( ) + hGR( ;θ) + εhRSD( ;θ),(7.1)
whe e
•n( )is de ec o noise (assumed ze o mean, Gaussian o Fishe o ecas s),
•θdeno es he usual sou ce pa ame e s (masses, spins, sky loca ion, o ien a ion,
dis ance, a i al ime and phase, e c.),
•hGR( ;θ)is he GR wa e o m model (including s anda d nonlinea memo y i a ail-
able),
•hRSD( ;θ)is a empla e o he RSD con ibu ion no malized so ha εis he ac-
ional ampli ude ela i e o he GR memo y ampli ude (i.e. ε=δhmem/hGR
mem).
A con enien and common no maliza ion choice is
hRSD( ;θ)≡hGR
mem( ;θ),so ε≡δhmem
hGR
mem
.(7.2)
The de ec o (o ne wo k) esponse p ojec s spa ial s ain on o he de ec o (s). In
he equency domain we use he s anda d inne p oduc :
(a|b)≡4ℜZ high
low
˜a( )˜
b∗( )
Sn( ), d , (7.3)
wi h ˜a( ) = F{a( )}and Sn( ) he one-sided noise PSD o he de ec o (ne wo k
PSD o a cohe en mul i-de ec o analysis). We adop he Fou ie con en ion ˜a( ) =
R∞
−∞ a( )e−2πi d .
Memo y in equency domain. A pe manen s ep o ampli ude ∆ha ime 0
(idealized memo y) has a Fou ie ans o m o = 0
˜
hs ep( )≃∆h
2πi e−2πi 0,( = 0),(7.4)
so memo y powe scales as |˜
hs ep( )|2∝∆h2/ 2. This emphasizes ha memo y de ec-
ion is domina ed by low equencies whe e 1/ 2weigh s a e la ge; hence low- equency
sensi i i y and ca e ul low- equency noise ea men a e c ucial.
5
4 Ma ched-Fil e De ec abili y and he Fishe Fo e-
cas o ε
Assuming Gaussian noise and linea dependence on ε( alid o |ε| ≪ 1), he Fishe
ma ix o pa ame e s {θa}={θ, ε}is
Γab =∂h
∂θa
∂h
∂θb.(7.5)
Focusing on εand ma ginalizing o e o he pa ame e s gi es he leading (high-SNR)
a iance
σ2
ε≃(Γ−1)εε,wi h Γεε = (hRSD|hRSD).(7.6)
I hRSD is no malized as in (7.2), hen Γεε is he SNR2associa ed only wi h he RSD
empla e. The single-e en de ec abili y condi ion a (Gaussian) signi icance zis
ε≳zσε=z
p(hRSD|hRSD).(7.7)
Rema ks & degene acies. I hRSD is pa ially degene a e wi h o he wa e o m
pa ame e s (e.g., ime/phase o low- equency calib a ion e o s), he ma ginalized σε
will be la ge ; explici ly include c oss e ms Γεa o compu e he ma ginalized e o . Fo
small degene acies and high SNR o he GR wa e o m, i is o en a good app oxima ion
o ea εas e ec i ely o hogonal o as oscilla o y pa ame e s (mass, spin) and mainly
degene a e wi h low- equency nuisance pa ame e s.
5 S acking Many E en s: Cohe en and Incohe en
S a egies
Single-e en cons ain s on εwill e y o en be weak o g ound-based de ec o s because
memo y powe is concen a ed a e y low equency. Howe e , RSD p edic s a uni e sal
ac ional enhancemen ha can be cons ained by combining many e en s. Two p incipal
s acking s a egies exis .
5.1 Cohe en S acking (Phase/Sign Aligned)
I he sign and ela i e phase o he memo y con ibu ion can be p edic ed o each e en
( om he GR empla e and geome y), one may cohe en ly sum he RSD empla es ac oss
e en s:
Γs acked
εε =
N
X
i=1 hRSD,ihRSD,i, σs acked
ε≃1
pPi(hRSD,i|hRSD,i).(7.8)
Fo N oughly simila e en s his gi es he amilia σε∝1/√Nimp o emen .
Cohe en s acking equi es co ec sign alignmen . Memo y sign depends on sou ce
o ien a ion and sky posi ion; alignmen is done by p edic ing sign om he eco e ed
θio each e en (use maximum-likelihood/pos e io sample o ix sign) be o e adding
empla es. E o s in sign assignmen cause pa ial cancella ion and loss o sensi i i y.
6
5.2 Incohe en (Powe ) S acking
I sign canno be eliably eco e ed, one can incohe en ly sum powe :
SNR2
powe =
N
X
i=1
(hRSD,i|hRSD,i).
This s ill imp o es sensi i i y as √Nin SNR, bu canno measu e he sign o ε. I is
app op ia e when o ien a ion e o s domina e.
Recommenda ion. Use cohe en s acking whe e possible (p e e ed) because i
cons ains he sign and gi es la ge sensi i i y o a gi en N.
6 Bayesian Model Selec ion and Hie a chical In e -
ence
Fo obus claims use Bayesian model compa ison and hie a chical pa ame e es ima ion.
6.1 Bayes Fac o o Nes ed Models (GR s GR+RSD)
Le M0be he GR model (ε= 0) and M1 he RSD model wi h p io π(ε). The Bayes
ac o is
B10 =p(d|M1)
p(d|M0)=Rp(d|ε, θ,M1)π(ε)π(θ)dεdθ
Rp(d|θ,M0)π(θ)dθ.(7.9)
Fo high-SNR Gaussian app oxima ions, and a na ow p io on ε, he Sa age–Dickey
densi y a io o Laplace app oxima ions can be used o e alua e B10. In p ac ice pe o m
nes ed sampling (e.g. dynes y) o compu e e idences and ma ginal likelihoods o each
e en and hen combine e idences ac oss e en s mul iplica i ely.
6.2 Hie a chical In e ence o κand Ic i
To ansla e measu ed εin o cons ain s on mic ophysical RSD pa ame e s (κ, Ic i ), use
a hie a chical model:
p(κ, Ic i |{di})∝π(κ, Ic i )
N
Y
i=1 Zp(di|εi,θi)p(εi|κ, Ic i ,θi)π(θi)dεidθi.(7.10)
He e p(εi|κ, Ic i ,θi)is he heo y p edic ion o he condi ional dis ibu ion o ε o e en
igi en sou ce pa ame e s (compu ed by in eg a ing (4.1)–(4.9) on NR/PN inpu s). Sam-
pling his hie a chical model yields pos e io s on κand Ic i and na u ally accoun s o
selec ion e ec s.
7 De ec o -Speci ic Conside a ions
Below we summa ize de ec o -class issues; he analysis s a egy mus be adap ed o he
ins umen .
7
7.1 G ound-Based In e e ome e s (Ad anced LIGO / Vi go /
KAGRA / 3G)
•Low- equency sensi i i y is decisi e. Memo y SNR scales wi h Rd /( 2Sn( ))
(see §7.6). Imp o ing Sn( )a ≲10–20 Hz d ama ically inc eases sensi i i y o
memo y.
•Calib a ion & baseline d i s. Low- equency calib a ion e o s and suspen-
sion d i mimic memo y; accu a e calib a ion and sub ac ion o slowly a ying
ins umen al ends a e manda o y.
•S acking s a egy. Expec o need ens o hund eds o e en s o g ound-based
de ec o s o each σε≲0.03 unless 3G sensi i i y and bandwid h signi ican ly
imp o e he low- equency PSD.
7.2 Space-Based De ec o s (LISA)
•Ad an ageous low- equency band. LISA’s sensi i i y a 10−4–10−1Hz makes
i excellen o memo y de ec ion om massi e black hole bina ies whe e memo y
powe is a lowe equencies.
•Long-du a ion signals. Fo long signals, empla e cons uc ion mus include
e ol ing RSD accumula ion du ing inspi al and me ge ; use he ull con olu ion
(4.1) a he han a la e- ime s ep app oxima ion.
•E en a es & s acking. Fewe bu loude e en s; a small numbe o high-SNR
LISA de ec ions could di ec ly cons ain ε.
7.3 Pulsa Timing A ays (PTAs)
•S ep in iming esiduals. Memo y om a e y massi e, nea by me ge shows
up as an ach oma ic, co ela ed s ep/ amp in iming esiduals ac oss he PTA.
•Timescale. PTA sensi i i y is o e y low equencies (nHz), bu de ec ion elies
on long ime baselines and ca e ul modeling o pulsa spin noise and clock e o s.
•C oss-pulsa cohe ence es . A g a i a ional memo y s ep will ha e a cha ac e -
is ic angula co ela ion ac oss pulsa s (Hellings–Downs–like signa u e); sea ching
o co ela ed s eps inc eases obus ness.
8 Analy ic Exp essions o Memo y SNR and Scaling
Laws
Using (7.4) he single-e en RSD SNR ( o an idealized s ep o ampli ude ∆hRSD) be-
comes, neglec ing he de ec o esponse ac o ,
SNR2
RSD = 4 Z high
low
|˜
hRSD( )|2
Sn( )d ≈4Z high
low
∆h2
RSD
(2π )2
1
Sn( )d
= ∆h2
RSD ·Imem,Imem ≡1
π2Z high
low
d
2Sn( ).
(7.11)
8
1. Collec i e exci a ion: I Nmic oscopic deg ees o eedom change cohe en ly, ∆µν scales as
NL2
Pl/L2. Fo mac oscopically la ge N(cohe en domain size), he supp ession can be pa ially
o se .
2. Non-pe u ba i e ea angemen : Nea Planckian cu a u e, he e ec i e coupling be ween
geome y quan a can change, enabling la ge δj and a much la ge ζ han nai ely expec ed.
These mechanisms jus i y ea ing κas an e ec i e pa ame e encoding mic ophysical amplifica ion
beyond simple dimensional coun ing.
5.2 Space ime Foam and Topology Change
The space ime oam pic u e en isions ansien mic oscopic opological fluc ua ions (wo mholes, baby
uni e ses, handle a achmen s) popula ing he pa h in eg al. High-cu a u e egions can ca alyze opology-
changing p ocesses; i such a p ocess does no comple ely e-annihila e, a pe manen opological imp in
can su i e in he coa se-g ained geome y and appea as ∆µν. Schema ically, a oam con ibu ion o
he expec a ion alue in (8.2) appea s as:
∆µν ∼X
oam sec o s s
e−Ss/ℏG(s)
µν ,(8.12)
whe e Ssis he ac ion o sec o sand G(s)is he coa se me ic imp in o ha sec o . The h eshold Ic i
is he cu a u e scale a which con ibu ions om non i ial sec o s become unsupp essed.
6 Holog aphy, So Modes, and In o ma ion Re en ion
6.1 Holog aphic In e p e a ion (Bounda y S ess o Bulk Residual)
In a holog aphic duali y (AdS/CFT), he asymp o ic all-o o he bulk me ic encodes he bounda y
CFT s ess enso T(CFT)
ab :
gab( , x) = g(0)
ab (x) + 1
d−2h(d)
ab (x) + ··· , T(CFT)
ab (x)∝h(d)
ab (x).(8.13)
A bulk e en ha deposi s ene gy in o he bounda y deg ees o eedom can change he la e- ime ex-
pec a ion alue ⟨T(CFT)
ab ⟩by a nonze o amoun . Th ough holog aphic eno maliza ion, his induces a
pe manen change in he bulk me ic coe ficien s h(d)
ab (x)and hus a esidual ∆µν in he bulk. In asymp-
o ically fla space imes, he bounda y imp in can be ph ased in e ms o so g a i ons and memo y: a
change in he acuum sec o o he bounda y quan um s a e (so hai ) co esponds o a classical esidual
in he bulk me ic.
6.2 So G a i on Theo ems, Memo y, and RSD
So - heo em analyses show ha low- equency (so ) g a i on emission is ied o asymp o ic symme ies
and memo y e ec s. In a quan um language, he emission o so modes changes he acuum by d essing
i wi h cohe en so g a i ons:
|Ψou ⟩ ∼ Sso |Ψin⟩,
whe e Sso is a so d essing ope a o . I so d essing is no exac ly undone a e an e en (e.g., due o
non-linea back eac ion o coupling o mic os a es), he la e s a e con ains a ne so componen whose
classical limi is he pe manen displacemen o es masses (memo y). RSD can be seen as he mac o-
scopic exp ession o his esidual so d essing when high cu a u e os e s i e e sible so -sec o ans-
e . In o he wo ds, RSD is he bulk mani es a ion o pe sis en so g a i on d essing plus any addi ional
mic os a e imp in (so hai ).
4

A semi-quan i a i e ela ion can be w i en:
∆µν(x)∼Zd2ΩS(Ω)Yµν(Ω),(8.14)
whe e S(Ω) is an angle-dependen so cha ge sou ced by he e en , and Yµν a e angula basis unc ions
p ojec ing so cha ge o bulk me ic coe ficien s. The key poin is ha , whe eas s anda d so heo ems
p edic a memo y en i ely de e mined by fluxes a null infini y, RSD supplemen s ha memo y by adding
an in e nal (nea -sou ce) sec o ha can s o e and la e eed back a po ion o he so cha ge.
7 E ec i e Pa ame iza ions and Ma ching o RSD Phenomenology
The mic oscopic conside a ions abo e na u ally map on o he phenomenological EFT pa ame e se used
in his pape . A compac ma ching ecipe is:
1. Coupling κ:A ises om he mic oscopic coupling o cu a u e o memo y deg ees o eedom.
In he auxilia y-field model (8.3) wi h O=−□+µ2, dimensional analysis gi es:
κ∼λ
µ∆L2−∆
Pl ,(8.15)
whe e λis a dimensionless mic ophysical ampli ude and ∆depends on he ope a o dimension
chosen o mµν. Non-pe u ba i e ins an on amplifica ion can make e ec i e λ e y la ge in
high-cu a u e egions.
2. Ac i a ion h eshold Ic i :De e mined by he cu a u e scale a which mic ophysical channels
(spin econfigu a ion, oam sec o s, so -sec o apping) become unsupp essed. Fo example,
Ic i ∼β/L4
Pl wi h β∼ O(1 −10n)depending on de ails; he RSD obse able ange equi es
Ic i be eached in me ge s bu no in o dina y as ophysical egimes.
3. Ke nel/ elaxa ion (mass scale µand imescale γ): The ope a o Ose s how long memo y pe -
sis s: a small µ(o γ≪1/Tobs) gi es e ec i ely pe manen memo y on obse a ional imescales.
Ma ching γ∼µ o he EFT yields he e olu ion law (4.2) as he long-wa eleng h limi o Om=
sou ce.
In sho , he mic ophysics p o ides an exis ence p oo o he phenomenological e ms κ,Ic i ,γ
used in ea lie sec ions and gi es scaling ela ions (8.7), (8.11), (8.15) o ansla e a measu ed εin o
bounds on mic oscopic pa ame e s.
8 Obse a ional Disc iminan s o a Quan um-G a i y O igin
To dis inguish a quan um-g a i a ional o igin o RSD om pu ely classical phenomenology, he ollow-
ing obse a ional signa u es a e diagnos ic:
1. Quan iza ion/disc e e pla eaux: I ∆µν a ises om changes in disc e e mic oscopic labels (e.g.,
spin jumps in LQG), one migh obse e a p e e ed se o ela i e esidual ampli udes o s a is i-
cally quan ized s ep sizes in a la ge e en sample.
2. Co ela ion wi h ho izon quan i ies: A ue mic os a e ea angemen ha c osses he e en
ho izon will co ela e ∆µν wi h changes in ho izon a ea ∆Abeyond classical expec a ions; nu-
me ical ela i i y enhanced wi h quasi-local ho izon mic os a e p oxies could es his.
3. Non-Gaussian s ochas ic backg ound: Foam- ype p ocesses and ins an on e en s p oduce a e,
la ge ou lie s and non-Gaussian ails in he dis ibu ion o esiduals ac oss many e en s, in con as
o a smoo h classical sa u a ion model.
5
4. So -sec o signa u es a null infini y: I RSD has a holog aphic/so o igin, one expec s co e-
la ed changes in asymp o ic so cha ges (measu able in p inciple ia memo y angula pa e ns)
ha canno be ully accoun ed o by adia i e fluxes alone.
5. Scaling wi h cu a u e p oxies: A clea , sha pened dependence o εon local cu a u e p oxies
(peak Weyl ampli ude nea me ge ) beyond he smoo h dependence p edic ed by GR memo y
would suppo a h esholded, mic ophysical ac i a ion mechanism.
9 P ac ical Ma ching and Es ima es
A p agma ic ma ching p ocedu e o con on ing heo y wi h da a:
1. Use NR simula ions o ex ac he nea -sou ce cu a u e in a ian I(u)and Weyl elec ic compo-
nen s Eij(u)du ing me ge .
2. Choose a mic oscopic model class (e.g., auxilia y field wi h O=−□+µ2o an LQG-inspi ed
spin- econfigu a ion model) and compu e he p edic ed mµν ia (8.6) o he ele an s a is ical
coa se-g aining.
3. Compu e he esul ing εusing he p opaga ion ope a o L om ˘
g4 (eq. 4.9 / 6.5).
4. Compa e p edic ed ε o obse a ional cons ain s (o o ecas ed sensi i i y) and in e o ob ain
limi s on combina ions o mic oscopic pa ame e s (e.g., λ, e−Sins /ℏ,µ, o cohe en occupa ion
numbe Nin he disc e e geome y pic u e).
A wo ked nume ic con e sion o go om a measu ed ε o a bound on Sins is s aigh o wa d om
(8.7):
Sins ≳ℏln Lp
PlF
∆meas ,(8.16)
whe e ∆meas ∼εhGR
mem con e ed o a dimensionless me ic ampli ude.
10 Summa y How Quan um G a i y Comple es he RSD Pic u e
1. Semiclassical pa h in eg als and EFTs wi h an auxilia y memo y field gi e a clean de i a ion
o he cons i u i e and e olu ion laws used in he RSD amewo k: ∆µν na u ally appea s as
he expec a ion alue o G eens- unc ion solu ion o a mic oscopic memo y sec o coupled o
cu a u e.
2. Disc e e geome ic pic u es (LQG, oam) p o ide conc e e mic ophysical mechanisms (spin ea -
angemen , opological sec o ansi ions) ha p oduce h esholded, non-pe u ba i e me ic im-
p in s; hese eed di ec ly in o he pa ame e s κ,Ic i and explain why memo y u ns on only abo e
a cu a u e scale.
3. Holog aphy and so -mode analyses show ha esidual bulk me ic de o ma ions co espond o
pe manen changes in bounda y (so ) cha ges o o pe sis en d essing o he acuum; he esul ing
classical limi coincides wi h he RSD memo y enhancemen .
4. The combina ion o hese h ee iewpoin s gi es bo h heo e ical plausibili y and an explici map-
ping om mic ophysics o he phenomenological pa ame e s cons ained obse a ionally. Sec-
ions 47 p o ided he obse a ionally o ien ed machine y; his sec ion shows how o in e p e any
measu ed εas a diagnos ic o quan um g a i a ional mic ophysics.
6

1 Appendix A: Tenso Calculus and Pe u ba i e Expansions
1.1 Va ia ion o he Eins einHilbe Ac ion and he Linea ized Eins ein Ope a o
S a om he Eins einHilbe ac ion (geome ic uni s G=c= 1):
SEH[g] = 1
16π∫d4x√−gR[g].
Unde a small symme ic me ic a ia ion δgµν, he s anda d fi s -o de a ia ions a e:
δΓρ
µν =1
2gρσ(∇µδgνσ +∇νδgµσ −∇σδgµν),
δRµν =∇ρδΓρ
µν −∇νδΓρ
µρ.
Expanding and ea anging (using co a ian de i a i es associa ed wi h he backg ound me ic gµν)
yields he s anda d iden i y o he a ia ion o he Ricci enso :
δRµν =1
2(−∇2δgµν −∇µ∇νδg +∇µ∇αδgαν +∇ν∇αδgαµ)+Rµναβδgαβ (A.1)
whe e δg ≡gαβδgαβ,∇2≡gαβ∇α∇β, and Rµναβ deno es e ms linea in he backg ound cu a u e
ha appea when he backg ound is no fla (explici e ms shown below when needed). Con ac ing
gi es:
δR =∇µ∇νδgµν −∇2δg −Rµνδgµν.
The Eins ein enso Gµν =Rµν −1
2gµνR he e o e a ies as:
δGµν =δRµν −1
2gµνδR −1
2δgµνR.
Subs i u e (A.1) and simpli y. Keeping only fi s -o de e ms in δg, one ob ains he s anda d linea ized
Eins ein ope a o ac ing on δgµν:
δGµν[δg] = 1
2[−∇2δgµν −∇µ∇νδg +∇µ∇αδgαν +∇ν∇αδgαµ
−gµν(∇α∇βδgαβ −∇2δg)]+Cµν[δg],
(A.2)
whe e Cµν[δg]collec s e ms p opo ional o he backg ound cu a u e (e.g., Rαµβνδgαβand Rµαδgαν),
which anish when he backg ound is Ricci-fla and o low cu a u e (we will explici ly e ain hem
o Schwa zschild below). Equa ion (A.2) is he igo ous o igin o eq. (4.3) in he main ex . Fo
con enience, we box he fla -backg ound limi used o en in he pape :
Fla -backg ound (o cu a u e-small) limi (co a ian de i a i es →pa ials):
δGµν[δg]≃1
2(−∂2δgµν −∂µ∂νδg +∂µ∂αδgαν +∂ν∂αδgαµ −ηµν(∂α∂βδgαβ −∂2δg)).(A.3)
1.2 T ea ing ∆µν as a Fi s -O de De o ma ion: E ec i e Memo y S essEne gy
We se ˜gµν =gµν + ∆µν and expand he Eins ein equa ion:
Gµν[˜g] = 8πT(ma e )
µν
o fi s o de in ∆µν:
Gµν[g] + δGµν[∆] + O(∆2) = 8πT(ma e )
µν .
1
Rea ange o define an e ec i e memo y s essene gy T(mem)
µν :
Gµν[g] = 8π(T(ma e )
µν +T(mem)
µν )+O(∆2), T(mem)
µν ≡ − 1
8πδGµν[∆].(A.4)
To linea o de , he con ac ed Bianchi iden i y applied o Gµν[g]( oge he wi h he e olu ion equa ion
o ∆µν in oduced la e ) ensu es ∇µ(T(ma e )
µν +T(mem)
µν ) = 0. Explici e ifica ion p oceeds by com-
pu ing ∇µδGµν[∆] and subs i u ing he e olu ion law o ∆µν (eq. (4.2) in main ex ). The impo an
poin is ha ∆µν can be consis en ly abso bed in o he igh -hand side as a conse ed e ec i e sou ce a
fi s o de .
1.3 Wa e Equa ion o ∆µν abou Minkowski de Donde Gauge and Re a ded Solu ion
Take gµν =ηµν (Minkowski) and ea bo h he adia i e pe u ba ion hµν and he esidual field ∆µν as
small. Wo k in he de Donde (ha monic) gauge o ∆µν:
∂µ¯
∆µν = 0,¯
∆µν ≡∆µν −1
2ηµν∆,
wi h ∆≡ηαβ∆αβ. In his gauge, he fla -limi linea ized Eins ein ope a o simplifies:
δGµν[∆] = −1
2□¯
∆µν,
whe e □≡ηαβ∂α∂β. The linea ized field equa ion ha ∆µν sa isfies (in e p e ing he RHS as he
idal/Weyl sou ce in oduced in ˘
g4) becomes:
□¯
∆µν =−2Sµν,(A.5)
wi h a sou ce:
Sµν(x)≡8πT(mem)
µν (x)≃ −δGµν[∆]sou ce o m ≈ −2κW (I)PµαPνβEµν,
whe e he las app oxima e equali y shows he cons i u i e assump ion ha he d i ing sou ce is p o-
po ional o he p ojec ed elec ic pa o he Weyl enso (see ˘
g4). The e a ded solu ion is he usual
G eens- unc ion in eg al:
¯
∆µν(x) = −2∫d4x′GR(x, x′)Sµν(x′),(A.6)
wi h he Minkowski e a ded G een’s unc ion GR(x, x′) = 1
2πΘ( − ′)δ((x−x′)2). In he a -field
wa e-zone, one may simpli y using e a ded- ime coo dina es u= − , and ob ain he asymp o ic
ela ion used in ˘
g4.5:
Eij(u)≃ −1
2∂2
uhTT
ij (u)⇒∆ij(u)≃ −κ
2∫u
−∞
W(I(u′))∂2
u′hTT
ij (u′)du′,
which a e in eg a ion by pa s yields he bounda y o m used in he main ex :
∆(∞)
ij ≈ −κ
2∆(∂uhTT
ij )( o na ow, ac i e W).(A.7)
2
1.4 Pe u ba i e Expansion a ound he Schwa zschild Backg ound (Use ul o Nea -
Sou ce Ac i a ion)
Le he backg ound be he Schwa zschild me ic o mass M(geome ic uni s G=c= 1):
ds2=−(1−2M
)d 2+(1−2M
)−1
d 2+ 2dΩ2.
Schwa zschild is acuum (Rµν = 0), so he Weyl enso equals he Riemann enso ; he nonze o e ad-
ame componen s o he elec ic pa o he Weyl enso (in an o hono mal ame {ˆ
, ˆ , ˆ
θ, ˆ
ϕ}) a e:
Eˆ
ˆ
=−2M
3,Eˆ
θˆ
θ=Eˆ
ϕˆ
ϕ=M
3.(A.8)
A s anda d cu a u e in a ian (K e schmann scala ) o Schwa zschild is:
I≡RαβγδRαβγδ =48M2
6.(A.9)
The ac i a ion condi ion I > Ic i defines a c i ical adius c i a which esidual memo y p oduc ion
becomes allowed:
c i =(48M2
Ic i )1/6
= 481/6M1/3I−1/6
c i .(A.10)
Thus, o gi en Ic i and mass M, one ob ains he space ime egion (a shell nea he me ge /ho izon)
whe e he cons i u i e ke nel W(I)ac i a es.
Leading-o de scaling o ∆µν nea -sou ce. Use he cons i u i e (in eg al) o m ∆µν ∼κ∫W(I)Edτ.
Es ima e o a single ac i a ion e en :
∆∼κEpeakTac ∼κM
3
ac
Tac ,(A.11)
whe e ac is a ypical adius o ac i a ion (e.g., a ew M) and Tac he p ope - ime wid h o he high-
cu a u e phase, which o compac me ge s scales like Tac ∼ O(M). Thus, a con enien pa ame iza-
ion is:
∆∼κM2
3
ac
.(A.12)
This o mula is dimensionally anspa en in geome ic uni s and use ul o o de -o -magni ude es i-
ma es. I ac i a ion occu s wi hin ∼ ac ≈αM (wi h α=O(1 −10)), hen:
∆∼κM2
(αM)3=κ1
α3M.
This shows he impo an scaling: o fixed κ, he esidual ampli ude dec eases wi h inc easing mass
Mi exp essed in his pa icula pa ame iza ion because la ge Msp eads he cu a u e o e la ge
leng h/ ime scales. (Nume ical e alua ion mus use he ac ual NR nea -sou ce scales o ge p ecise
alues.)
1.5 Dimensional Analysis and Scaling Laws (Ca e ul Uni s)
Dimensions: Me ic componen s a e dimensionless. In geome ic uni s, leng h and ime ha e he same
dimension; cu a u e has dimension L−2. Conside he cons i u i e in eg al (eq. (4.1) o he main ex ):
∆µν ∼κ∫K(τ, τ′)W(I)Edτ′.
3
•[E] = L−2.
•[dτ′] = L.
• So he in eg and ca ies dimension L−1. To make ∆µν dimensionless, κmus ca y dimension o
leng h:
[κ] = L(= ime in geome ic uni s).(A.13)
This co ec s casual ea lie s a emen s ha ea ed κas dimensionless; he ea e ea κas a pa ame e
wi h uni s o leng h/ ime. I is con enien o pa ame ize κas:
κ=αL∗, L∗a chosen mic oscopic leng h (e.g., LPl),
wi h αdimensionless (possibly la ge).
Scaling o he ac ional enhancemen ε.Use he wa e-zone esul ∆ij ∼κ∫∂2
uhijdu′∼κ∂uh.
Cha ac e is ic scales: ∂uh∼h0/T. Fo he GR memo y ∆hGR ∼h0. Thus:
ε≡δhmem
hGR
mem ∼∆
h0∼κ(h0/T)
h0∼κ
T.
The e o e, he dimensionally co ec ela ion is:
ε∼κ
T,(A.14)
o , wi h he ac o 1
2 e ained om in eg a ion-by-pa s con en ions in he main ex :
ε≈κ
2T.
This ep oduces eqs. (4.10) and (6.6) in he main ex wi h dimensions p ope ly accoun ed o .
Ene gy bookkeeping scaling. The e ec i e memo y ene gy densi y scales as ρmem ∼(16π)−1(∇∆µν)2.
Using ∇ ∼ 1/T and ∆µν ∼κh0/T, he o al memo y ene gy ( olume V∼T3) is:
Emem ∼1
16π
∆2
T2T3∼1
16π∆2T∼1
16πκ2h2
0
T.
Compa e wi h he adia ed GW ene gy EGW ∼h2
0T(es ima es up o o de -uni y geome ic ac o s).
Thus: Emem
EGW ∼κ2
T2∼ε2.
So he simple and obus cons ain Emem ≪EGW becomes |ε| ≪ 1. Equi alen ly, an obse a ional
uppe bound on εimplies a bound on κ:
κ≲Tεmax.(A.15)
1.6 Mapping κ o Mic oscopic Scales
I one chooses L∗=LPl, hen κ=αLPl. Using (A.14), he expec ed ac ional enhancemen o a
sou ce wi h cha ac e is ic imescale Tis:
ε∼αLPl
T.
Fo as ophysical me ge s, Tis mac oscopic (milliseconds o seconds), so a pu e Planck-leng h p e ac o
wi hou amplifica ion (α∼1) is negligible. Tha mo i a es ei he (i) an e ec i e amplifica ion ac o
α≫1coming om nonpe u ba i e mic ophysics (ins an on ac o s o cohe ence o many mic oscopic
deg ees o eedom), o (ii) choosing L∗≫LPl as he e ec i e leng h go e ning memo y coupling. In
he main ex , we le κphenomenological o p ecisely his eason.
4