Updated

Residual Space ime De o ma ions: A Geome ic
B idge o Quan um G a i y
Rhy hm
Sep embe 2025
Abs ac
Reconciling gene al ela i i y (GR) wi h quan um mechanics has emained one o he
mos p o ound challenges in mode n physics. Classical space ime canno ully accommo-
da e quan um fluc ua ions a Planck-scale cu a u es, and adi ional quan iza ion a emp s
o en lead o non- eno malizable infini ies. He e, I p opose he Residual Space ime
De o ma ion (RSD) amewo k, whe e ex eme quan um fluc ua ions e ch pe manen ,
coa se-g ained de o ma ions in o he classical geome y, encoded by a enso ∆
µν
. This
app oach allows quan um disc e eness o mani es mac oscopically as cu a u e mem-
o y, p o iding a alsifiable geome ic b idge be ween quan um heo y and g a i y. RSD
na u ally in eg a es semiclassical pa h-in eg al easoning, holog aphic duali ies, and scale-
dependen c i ical cu a u e h esholds, o e ing a no el ou e owa d unifica ion wi hou
in oking specula i e new pa icles o dimensions.
1 In oduc ion / P oblem S a emen
One o he cen al unsol ed p oblems in physics is he incompa ibili y be ween gene al el-
a i i y and quan um mechanics. GR desc ibes g a i y as smoo h space ime cu a u e, bu
a e y small scales, quan um heo y p edic s fluc ua ions ha make he classical geome ic
pic u e b eak down. T adi ional a emp s o quan ize g a i y—whe he ia pe u ba i e ap-
p oaches, s ing heo y, o loop quan um g a i y— ace challenges:
•Non- eno malizabili y: Quan um co ec ions o Eins eins equa ions di e ge uncon ol-
lably a high ene gies.
•Loss o classical in ui ion: A he Planck scale, space ime may become disc e e, oamy,
o opologically complex, making p edic ions di ficul .
•In o ma ion pa adoxes: Quan um in o ma ion seems o conflic wi h he smoo h, de-
e minis ic e olu ion o classical geome y, e.g., inside black holes.
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Despi e hese challenges, he e is g owing e idence ha some quan um e ec s may lea e
pe manen imp in s on space ime i sel , simila o how g a i a ional wa es p oduce esid-
ual memo y dis o ions. Could his be a pa hway o connec classical geome y and quan um
disc e eness?
2 Concep ual Explana ion
The cen al idea o RSD is simple bu powe ul:
When space ime cu a u e exceeds a c i ical h eshold Kc, quan um fluc ua ions pe -
manen ly de o m he classical geome y, p oducing a coa se-g ained imp in encoded in
a enso ∆
µν
.
2.1 Defining he Residual De o ma ion
In ui i ely, ∆
µν
ep esen s he memo y o quan um fluc ua ions:
g
µν
→g
µν
+∆
µν
,i K≥Kc
Whe e:
•g
µν
is he classical me ic.
•Kis a cu a u e in a ian (e.g., K e schmann scala ).
•∆
µν
encodes esidual, pe manen de o ma ions a ising om quan um fluc ua ions.
This is analogous o p essing a so ma e ial: he high-p essu e egion lea es a pe manen
den . He e, quan um fluc ua ions play he ole o he p essu e, and space ime cu a u e is he
ma e ial.
2.2 Connec ion o Quan um G a i y
1. Pa h-in eg al pic u e: In he sum-o e -geome ies, high-cu a u e configu a ions con-
ibu e significan ly. RSD in e p e s hese con ibu ions as lea ing pe manen imp in s
a he han a e aging ou en i ely.
2. Loop quan um g a i y: A ea and olume ope a o s a e disc e e; when hei fluc ua ions
exceed a h eshold, hey induce a mac oscopic cu a u e de o ma ion, cap u ed by ∆
µν
.
3. Holog aphic duali y: Bounda y co ela o s can encode he same in o ma ion as ∆
µν
,
connec ing bulk geome ic memo y o quan um in o ma ion on he bounda y.
2
2.3 Scale Dependence
The c i ical cu a u e h eshold Kcis scale-dependen , allowing RSD o uni y phenomena
ac oss scales:
Kc(ℓ)∼ℓ−
α
He e, ℓis he leng h scale and
α
is a posi i e exponen de e mined by eno maliza ion g oup
flow. This ensu es ha memo y is significan only in egimes whe e quan um e ec s domina e.
2.4 Pseudo-De i a ion
A simplified de i a ion in semiclassical e ms:
1. S a om he pa h in eg al o g a i y:
Z=∫D[g]eiSEH[g]/¯
h
2. Pa i ion configu a ions in o low- and high-cu a u e sec o s:
∫D[g] = ∫K<Kc
D[g]+∫K≥Kc
D[g]
3. Replace high-cu a u e fluc ua ions wi h hei coa se-g ained esidual,∆
µν
, con ibu -
ing a de e minis ic geome ic shi :
g
µν
→g
µν
+∆
µν
This b idges he gap: quan um fluc ua ions lea e classical finge p in s wi hou ully
quan izing geome y.
3 Implica ions / Mys e ies Sol ed
The RSD amewo k can po en ially add ess mul iple longs anding puzzles:
1. Quan um-classical ansi ion: Explains how disc e e quan um e ec s can appea as
smoo h bu de o med space ime mac oscopically.
2. Black hole in o ma ion pa adox: ∆
µν
could encode los quan um in o ma ion in he
classical geome y.
3. G a i a ional wa e memo y: Ex ends classical GW memo y e ec s o a gene al mech-
anism o cu a u e memo y.
3
4. Ea ly-uni e se quan um imp in s: Infla iona y quan um fluc ua ions could lea e pe -
manen geome ic sca s, possibly explaining anomalies in he cosmic mic owa e back-
g ound (CMB).
5. A oiding singula i ies: Pe manen esidual de o ma ions could egula e cu a u e blowups,
so ening classical singula i ies.
4 Po en ial Ex ensions / P edic ions
1. Obse able cu a u e memo y: Could RSD e ec s mani es as sub le de ia ions in
g a i a ional lensing, ame-d agging, o GW obse a ions?
2. Cosmological signa u es: La ge-scale CMB o s uc u e anomalies could be esidual
imp in s o quan um cu a u e memo y.
3. Holog aphic es s: Bounda y CFT co ela o s could e eal ∆
µν
finge p in s, es ing he
link o quan um in o ma ion.
4. Labo a o y-scale quan um g a i y analogues: Analog sys ems, e.g., condensed ma -
e geome ic de ec s, may emula e RSD-like memo y e ec s.
5 Conclusion
Residual Space ime De o ma ions p o ide a concep ually simple, ye po en ially e olu ion-
a y pa hway o link quan um fluc ua ions wi h classical geome y. By ea ing quan um e ec s
as sou ces o pe manen geome ic memo y, RSD bypasses he need o ull space ime quan-
iza ion, p o ides es able p edic ions, and o e s insigh in o deep puzzles such as black hole
in o ma ion, ea ly-uni e se anomalies, and singula i y a oidance.
Full ma hema ical o maliza ion, including explici dynamics o ∆
µν
, eno maliza ion g oup
flow de i a ions, and holog aphic mappings, will be de eloped in u u e wo k. Fo now,
he co e in ui ion and mo i a ion a e clea : quan um fluc ua ions lea e indelible ma ks on
space ime, b idging wo undamen al amewo ks o physics.
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Theo e ical Founda ions o Residual S ess De o ma ion (RSD)
Oc obe 6, 2025
1 Theo e ical Founda ions
1.1 No a ion, con en ions and s a ing ac ion
We wo k on a 4-dimensional mani old Mwi h me ic gµν (de e minan g), Le i-Ci i a con-
nec ion ∇, Riemann enso Rρσµν, Ricci enso Rµν =Rρµρν, and scala R=gµνRµν. The
Eins ein–Hilbe ac ion wi h ma e is
S[g, Ψ] = 1
16πG ZM
d4x√−g(R−2Λ) + Sm[g, Ψ],
whe e Ψdeno es ma e ields. The pa h in eg al o quan um g a i y ( o mal) is
Z=ZDgDΨeiS[g,Ψ].
We will use he K e schmann scala
K(x)≡RαβγδRαβγδ(x),
as a ep esen a i e cu a u e in a ian ; [K] = L−4.
In oduce a scale-dependen c i ical cu a u e Kc(ℓ)(see §3 below) such ha egions whe e
K ≥ Kca e ea ed as he high-cu a u e sec o esponsible o lea ing pe manen geome ic
esidues.
1.2 Mode / sec o decomposi ion and he in luence unc ional
Spli he ull me ic con igu a ions in o a low-cu a u e (coa se) componen gµν and a high-
cu a u e luc ua ion ξµν:
g o
µν =gµν +ξµν,
whe e ξis cons ained o suppo K ≥ Kc(mo e p ecisely, ξcon ains hose ield deg ees o
eedom whose local cu a u e con ibu ion exceeds he h eshold). Then o mally
Z=ZDgDΨlow ZK≥KcDξDΨhigh eiS[g+ξ,Ψlow,Ψhigh].
De ine he in luence unc ional (Feynman–Ve non s yle) by in eg a ing ou high-cu a u e
modes:
F[g, Ψlow]≡ZK≥KcDξDΨhigh eiS[g+ξ,Ψlow,Ψhigh]=eiSi [g,Ψlow],
so he esul ing e ec i e pa h in eg al is
Z=ZDgDΨlow ei(S[g,Ψlow]+Si [g,Ψlow]).
Si encodes he coa se-g ained de e minis ic and dissipa i e e ec s o high-cu a u e luc ua ions
on he low sec o . The cen al claim o RSD is ha Si con ains e ms ha p oduce a pe manen
geome ic shi ∆µν (a cu a u e memo y).
1

1.3 De ini ion and o mal exp ession o he esidual de o ma ion ∆µν
Decompose he expec a ion (mean) me ic in he low sec o a e in eg a ing ou ξ:
hg o
µν (x)ihigh =gµν(x)+∆µν(x),
whe e he pa h-in eg al de ini ion is
∆µν(x) = RK≥KcDξ ξµν(x)eiS[g+ξ]/ℏ
RK≥KcDξ eiS[g+ξ]/ℏ.
This is an exac o mal de ini ion. To p oceed we mus make con olled app oxima ions (saddle
poin / cumulan expansion / Gaussian app oxima ion) o compu e ∆µν as a unc ional o he
coa se ield g. Impo an limi ing p ope ies:
•Classical / low-cu a u e limi : as ℏ→0o when K  Kc,∆µν →0.
•Locali y:∆µν(x)is cons uc ed om local cu a u e enso s in he neighbou hood o x
when he coa se-g aining scale ℓis sho (nonlocal ke nels appea i he coa se-g aining
in eg a es o e ex ended egions).
1.4 E ec i e ac ion and eme gen s ess-ene gy T(∆)
µν
The e ec i e ac ion o he coa se ields is
Se [g, Ψlow] = S[g, Ψlow] + Si [g, Ψlow].
Va ying Se wi h espec o gµν yields he semiclassical ield equa ion
2
√−g
δSe
δgµν = 0 ⇒Gµν + Λgµν = 8πGTµν +T(∆)
µν ,
whe e we de ine he esidual (memo y) s ess-ene gy
T(∆)
µν (x)≡ − 2
p−g(x)
δSi [g]
δgµν(x).
Thus RSD p o ides a geome ic co ec ion ha appea s on he igh -hand side o Eins ein’s
equa ions and can be equi alen ly desc ibed by ei he (i) a me ic shi g7→ g+ ∆ o (ii) an
e ec i e T(∆)
µν . Bo h desc ip ions a e consis en and ela ed by:
T(∆)
µν =1
8πGGµν[g+ ∆] −Gµν[g]+O(ℏ2),
alid when ∆is he dominan ℏ-o de imp in p oduced by in eg a ing ou he high sec o .
1.5 Cumulan (Gaussian / saddle-poin ) app oxima ion — explici o mulae
To ge explici exp essions we expand S[g+ξ]abou ξ= 0 (backg ound g):
S[g+ξ] = S[g]+Zd4x√−g Jµν(x)ξµν(x)+ 1
2Zd4x d4y ξµν(x)D−1,µναβ(x, y)ξαβ(y)+Sin [ξ;g],
whe e
Jµν(x)≡1
p−g(x)
δS[g]
δgµν(x)
2
is he unc ional de i a i e (a sou ce o he high modes) and D−1is he in e se p opaga-
o /ope a o o he high-cu a u e modes on backg ound g. Neglec ing highe non-Gaussian
in e ac ions Sin (o ea ing hem pe u ba i ely), he Gaussian in eg al gi es he in luence
ac ion a leading o de :
Si [g]≃ −1
2Zd4x d4yp−g(x)p−g(y)Jµν(x)Dµναβ(x, y)Jαβ(y) + ···
whe e Dis he G een’s unc ion (p opaga o ) o he high-cu a u e sec o :
Zd4yD−1,µναβ(x, y)Dαβρσ(y, z) = δµνρσ
δ(4)(x−z)
√−g.
F om his,
T(∆)
µν (x)≃1
16πG
δ
δgµν(x)Zd4u d4 p−g(u)p−g( )Jαβ(u)Dαβγδ(u, )Jγδ( ).
Two impo an upsho s:
•Linea esponse o m. A leading o de he esidual is linea in he sou ce J ia he
p opaga o :
∆µν(x)≈ −Zd4yp−g(y)Dµναβ(x, y)Jαβ(y) + O(J2),
i.e., he high-cu a u e sec o esponds o he coa se backg ound and lea es a de e minis ic
mean shi .
•Conse a ion (consis ency) condi ion. The Bianchi iden i y ∇µGµν = 0 implies
∇µT(∆)
µν = 0
p o ided he in luence unc ional espec s di eomo phism in a iance (which i does i
he decomposi ion and egula o p ese e di eomo phisms). This ensu es consis ency o
he modi ied Eins ein equa ions.
The exp essions abo e a e o mal bu s anda d: he c ucial physics is ha in eg a ing
ou high-cu a u e luc ua ions p oduces nonlocal, gene ally highe -de i a i e bu de e minis ic
e ms in Si whose a ia ion appea s as T(∆) and/o a pe manen ∆µν.
1.6 Linea ized ela ion o he Eins ein ope a o (use ul explici o m)
When ∆µν is small we may linea ize:
Gµν[g+ ∆] = Gµν[g] + δGµν[g; ∆] + O(∆2),
wi h
δGµν =δRµν −1
2gµνδR −1
2∆µνR+1
2gµν∆αβRαβ,
and
δRµν =1
2−□∆µν −∇µ∇ν∆ + ∇µ∇α∆αν +∇ν∇α∆αµ,∆≡gαβ∆αβ.
Hence o linea o de
T(∆)
µν ≃1
8πG δGµν[g; ∆].
This p o ides an explici linea (ellip ic/hype bolic) PDE ela ing ∆ o an e ec i e sou ce. In
pa icula , i one posi s a cons i u i e ela ion
Lg∆µν =Sµν[g],
whe e Lgis he Lichne owicz (o ela ed) ope a o on symme ic ank-2 enso s and Sis con-
s uc ed om cu a u e in a ian s and he p opaga o ke nel, hen ∆is sol ed by in e ing Lg
(subjec o bounda y/ egula i y condi ions).
3
1.7 Ansä ze & cons i u i e modelling o ∆µν
To ob ain ac able models we can posi a causally-local/nonlocal cons i u i e ke nel K ha
ies ∆ o cu a u e scala s. A gene al co a ian ansa z is
∆µν(x) = Zd4yp−g(y)Kµναβ(x, y;ℓ) ΘK(y)−Kc(ℓ)Sαβ[g](y)
whe e
•Kµναβ(x, y;ℓ)is a esponse ke nel (decays o |x−y|  ℓ),
•Θimplemen s he h esholding (only high cu a u e con ibu es), and
•Sαβ[g]is a local cu a u e-buil sou ce ( o example Sαβ ∝ ∇α∇βK, o combina ions o
Rαβ, R, gαβ, e c).
This ansa z makes explici he coa se-g aining scale ℓ, he nonlocali y and he h eshold
mechanism.
1.8 Reno maliza ion-g oup a gumen o he scaling o Kc(ℓ)
Dimensional analysis: Khas canonical mass dimension 4 ([K] = leng h−4). Allowing an anoma-
lous dimension γ om unning yields
Kc(ℓ)∼ℓ−(4+γ)⇒ Kc(ℓ)∼ℓ−α, α ≡4 + γ.
Mo e o mally, de ine a eno maliza ion scale µ∼ℓ−1and a be a unc ion o he composi e
ope a o K:
µdK
dµ =βK(K, G, . . .).
A ixed-poin o powe -law solu ion implies K(µ)∝µ4+γ. The RSD hypo hesis uses a scale-
dependen h eshold o his o m o ensu e ha memo y imp in s appea only when local cu -
a u e uns in o he nonpe u ba i e egime.
1.9 Illus a i e analy ic example: Schwa zschild cu a u e adius whe e RSD
ac i a es
Fo a Schwa zschild black hole (mass M) he K e schmann scala is
KSchw( ) = 48G2M2
6.
De ine he adius cwhe e KSchw( c) = Kc. Then
c=48G2M2
Kc1/6.
Thus RSD imp in s o m inside ≲ c. I Kcis Planckian (∼ℓ−4
P) hen cis pa ame ically
small bu la ge han he Planck leng h o mac oscopic M, gi ing a mac oscopic co e whe e ∆
mus be sol ed sel -consis en ly and whe e classical singula beha iou can be smoo hed.
4
1.10 Singula - esolu ion mechanism (ske ch)
Assume inside c he e ec i e ield equa ions a e
Gµν[g+ ∆] = 8πGT (phys)
µν .
I ∆con ibu es an e ec i e s ess-ene gy whose adial p essu e/ene gy densi y egula ize cu -
a u e ( o example a co e wi h ini e ene gy densi y ρ∆( )as →0), hen in a ian s such as
Kbecome bounded. Conc e ely, wi h a simple iso opic co e ansa z
T(∆) ≡ −ρ∆( ), ρ∆( )∼ρ0
1+( / 0)n,
one can sol e he (modi ied) Tolman–Oppenheime –Volko –like equa ions o show ha cu a-
u e does no di e ge as →0i 0≳ Planck and pa ame e s sa is y egula i y condi ions.
The de ails educe o sol ing o ∆ om he cons i u i e ela ion and checking boundedness o
K[g+ ∆].
1.11 Ene gy condi ions and physical in e p e a ion
Because ∆de i es om in eg a ing ou quan um luc ua ions, T(∆)
µν can iola e classical ene gy
condi ions (e.g., he null ene gy condi ion) in limi ed egions. This is no a bug bu a ea u e:
such iola ions a e a mechanism o a oid singula i y heo ems while p ese ing di eomo phism
in a iance. Howe e , causali y and s abili y mus be checked o any chosen ke nel Kµναβ o
a oid unaway modes.
1.12 Summa y o he ma hema ical p og am (wha o compu e nex )
To make he abo e ully igo ous and sui able o a publica ion-g ade Theo e ical Founda ions
sec ion you should (and we can do):
1. Choose a egula o / mode decomposi ion (e.g., p ope - ime cu o , spec al cu o
w. . . some ellip ic ope a o ) ha implemen s he K ≥ Kcspli while keeping di eomo -
phism in a iance mani es .
2. Compu e he p opaga o Do he high-cu a u e modes on a chosen backg ound
(Schwa zschild, FRW) o in pe u ba ion heo y abou la /AdS.
3. E alua e he cumulan expansion o Si o a leas second o de (Gaussian) and
ob ain explici ke nels Kµναβ(x, y).
4. In e he linea ope a o Lg(Lichne owicz-s yle) nume ically/analy ically o compu e
∆ o examples (Schwa zschild co e, cosmology).
5. Ve i y conse a ion and s abili y (show ∇µT(∆)
µν = 0 and absence o ghos s/ achyons
o he chosen ke nel).
6. P oduce obse able es ima es (magni ude o ∆in as ophysical GW e en s, CMB
imp in s e c).
1.13 Ready- o-pas e ma hema ical s a emen ( o he pape )
Theo e ical Founda ions (RSD). Le K ≡ RαβγδRαβγδ and ix a scale pa ame e
ℓ. De ine he high-cu a u e unc ional sec o by {ξ:K[ξ+g]≥ Kc(ℓ)}. In eg a ing
ou ξyields an in luence ac ion Si [g]and a esidual de o ma ion ∆µν gi en by
∆µν(x) = RK≥KcDξ ξµν(x)eiS[g+ξ]
RK≥KcDξ eiS[g+ξ].
5
Residual S ess De o ma ion (RSD) in Schwa zschild Backg ound
1 Goal and simple cons i u i e ansa z
We now pick he Schwa zschild me ic as a conc e e backg ound and compu e he esidual
de o ma ion ∆
µν
a leading o de using a simple local ke nel ansa z. We hen show how a
con olled nonlinea i y (sa u a ion) p oduces a egula de Si e –like co e and gi e o de -o -
magni ude nume ical es ima es.
Cons i u i e (local) ansa z (dimensionally consis en )
∆
µν
(x) =
α
Kc
Θ(K(x)−Kc)H
µν
[g](x),
whe e
•K≡R
αβγδ
R
αβγδ
is he K e schmann scala ,
•Kcis he c i ical cu a u e h eshold (scale-dependen in gene al; he e aken ixed o he
example),
•
α
is a dimensionless coupling (expec ed O(1)),
•H
µν
is a symme ic, cu a u e–quad a ic sou ce- enso wi h he same mass dimension as
K(so ha ∆
µν
is dimensionless). A con enien and physically-mo i a ed choice is he
ace- ee quad a ic enso
H
µν
≡R
µαβγ
R
ναβγ
−1
4g
µν
K,
which is local, co a ian and buil om he same cu a u e in a ian s ha di e ge in
Schwa zschild.
Rema ks:
• The p e ac o 1/Kchas dimensions L4, so ∆
µν
is dimensionless.
• The Hea iside Θen o ces ha only egions whe e K≥Kc( he high-cu a u e sec o )
con ibu e o he esidual.
This ansa z is he leading (linea ) model; la e we in oduce a minimal nonlinea i y (sa u-
a ion) which is physically necessa y o a oid unaway di e gences as →0.
2 Schwa zschild cu a u e scalings (analy ic)
Fo a Schwa zschild black hole o mass M(geome ic mass Mgeom =GM/c2in SI uni s) he
K e schmann scala is he s anda d exac exp ession
K( ) = R
αβγδ
R
αβγδ
=48M2
geom
6,
1

( alid in any coo dina e ep esen a ion because Kis a scala ). In ou H
µν
choice each compo-
nen o H
µν
scales ∝K(up o dimensionless angula /coo dina e-dependen coefficien s). Hence,
a he le el o scaling,
∆
µν
( )∼
α
Kc
Θ(K−Kc)K( ) =
α
Θ(K−Kc)K( )
Kc
.
Thus he dimensionless magni ude o ∆is app oxima ely he dimensionless a io K/Kc
( imes an O(1)cons an
α
).
De ine he ac i a ion adius cby K( c) = Kc. Sol ing gi es
c=(48M2
geom
Kc)1/6,Mgeom ≡GM
c2;
(geome ic-uni s de i a ion inse ed back o SI h ough Mgeom).
3 Linea model beha iou and necessi y o sa u a ion
Unde he linea ansa z ∆ ∝ K/Kc, o ≲ c he me ic co ec ion quickly g ows because
K( )∝ −6. Two obse a ions:
1. A = cwe ha e ∆∼
α
(o de -one me ic de o ma ion).
2. Fo  c he nai e linea model p oduces ∆1, iola ing he small-pe u ba ion as-
sump ion used o de i e he Gaussian esul . Physically we expec nonlinea back- eac ion
(highe cumulan s, sel -consis en sol ing o g7→g+∆) o sa u a e he g ow h and p oduce
a ini e co e.
A minimal, physically mo i a ed sa u a ion model is he a ional o m
∆
µν
( ) = −
γ
K( )
Kc
1+K( )
Kc
g
µν
,
γ
∼O(1),
which has he p ope ies:
• o KKc:∆≈−
γ
(K/Kc)(linea egime);
• o KKc:∆→−
γ
g
µν
(sa u a ed co e, ini e).
This o m models he in ui i e expec a ion: la ge quan um luc ua ions p oduce a bounded
geome ic memo y (co e de o ma ion), no an unbounded me ic blowup.
4 Eme gence o a de Si e co e — analy ic ma ching
Assume sa u a ion yields ( o leading app oxima ion) a cons an iso opic de o ma ion o ≲ c
∆
µν
≃−
γ
g
µν
( ≲ c),
wi h 0<
γ
<1so ha g+∆= (1−
γ
)g emains Lo en zian. In his co e egion he induced
esidual s ess-ene gy beha es like a acuum ene gy (iso opic):
T(∆)
µν
≈− Λe
8
π
Gg
µν
,
2
i.e., an e ec i e cosmological cons an Λe ( he sign and magni ude depend on de ails o he
nonlinea i y; he e we assume i akes he sign ha p oduces a egula de Si e in e io ).
Fo a cons an -cu a u e in e io (de Si e ) he K e schmann scala is
KdS =8
3Λ2
e =⇒Λe =√3
8√KdS.
Ma ching he de Si e cu a u e o he ac i a ion h eshold Kc(physically na u al: he co e
cu a u e sa u a es a he h eshold scale) gi es he o de -o -magni ude iden i ica ion
Λe ∼√3
8√Kc;(co e)
and he e o e he adius 0o he de Si e co e (by equa ing he mass M o he in e io de
Si e mass wi hin 0) is
0=(6GM
Λe )1/3=(6GM
√3
8√Kc)1/3.
Compa e 0 o c. Using he o mula o cand algeb a, one ob ains he simple o de -one
a io
0
c
=(6/√3/8)1/3
481/6≈1.12246,
i.e., he de Si e co e adius 0is he same o de as he ac i a ion adius c(wi hin ∼10% o
his minimal model). Thus he pic u e is consis en : memo y imp in s ac i a e a ∼ cand
nonlinea ly sel -o ganize in o a ini e, de Si e –like co e o adius 0∼ c.
5 Nume ical, o de -o -magni ude es ima es (SI uni s)
Adop he na u al assump ion Kc≃ℓ−4
P(c i ical cu a u e se by he Planck scale). Use
ℓP=√¯
hG
c3≈1.616255 ×10−35 m.
Ac i a ion adius c(exac o mula used: c= (48M2
geom/Kc)1/6wi h Mgeom =GM/c2):
• Fo a sola -mass black hole (M=1.98847 ×1030 kg):
c≈1.39 ×10−22 m(≈8.6×1012 ℓP).
• Fo a Planck mass (mP≈2.17644 ×10−8kg):
c≈3.08 ×10−35 m(≈1.9ℓP).
De Si e co e adius 0(using Λe ∼√3/8√Kc):
• Fo he sola mass:
0≈1.56 ×10−22 m, 0/ c≈1.12.
• Fo he Planck mass:
0∼a ew×ℓP.
In e p e a ion. Fo mac oscopic black holes (sola -mass and abo e) he ac i a ion adius c
and co e adius 0a e eno mously la ge han he Planck leng h bu ex emely small compa ed
wi h he Schwa zschild adius s:
3
• Sola ( s∼2GM/c2≈2.95 ×103m). The a io
0
s∼10−22 m
103m∼10−25,
i.e., he co e is deep inside he ho izon and mic oscopically iny on as ophysical scales.
6 Obse a ional consequences (o de -o -magni ude)
Using he linea (unsa u a ed) ansa z gi es a local dimensionless me ic de o ma ion o o de
∆∼
α
K( )
Kc
.
A he Schwa zschild adius s his a io is as onomically small o mac oscopic M. Fo a sola
mass one inds K( s)
Kc∼10−152,
so esidual de o ma ions a he scale o he ho izon a e u e ly negligible in his model (i.e., no
la ge modi ica ions o he ex e io me ic a adii ≳ s). De ec able as ophysical signa u es
he e o e equi e ei he :
• quan um memo y imp in s ha a ec dynamics du ing ex emely nonlinea ansien
phases (me ge , ingdown) and p oduce iny obse a ionally ampli ied e ec s like echoes
(model-dependen and gene ically highly supp essed), o
• obse a ional access o physics a scales ≲ 0(no possible o ex e nal obse e s wi hou
ho izon-pene a ing p obes).
In sho : RSD p oduces a ini e, Planckian (o nea -Planckian) cu a u e co e whose adius
is small bu pa ame ically la ge han ℓP o mac oscopic mass; he ex e io me ic is essen ially
classical a obse able adii. This is a s ong, alsi iable p edic ion: RSD egula izes singula i ies
while lea ing ex e io , low-cu a u e space ime nea ly unchanged.
7 Rema ks on obus ness and nex s eps
1. Choice o H
µν
— I used a physically na u al quad a ic cu a u e enso ; o he choices
(Bel–Robinson ype enso s, nonlocal ke nels, TT-p ojec ed ke nels) change nume ic p e -
ac o s bu no he quali a i e conclusion: linea g ow h nea he classical singula i y and
nonlinea sa u a ion a e gene ic.
2. Nonlinea modelling — he a ional sa u a ion ansa z is phenomenological; a comple e
ea men equi es sol ing he ull nonlinea in eg o-di e en ial sel -consis ency p oblem
(G[g+∆] = 8
π
G(T+T(∆)[g])) wi h ∆de i ed om Si . The Gaussian esul ( his sec ion)
p o ides a con olled s a ing poin and sugges s he scale and o m o he nonlinea i y.
3. Obse a ional signa u es — compu e wa e o ms o inspi al/me ge including an in e nal
(RSD) co e and de e mine whe he (and when) small de ia ions could p oduce de ec able
echoes o phase shi s ( his equi es nume ical ela i i y wi h a modi ied in e io s ess
enso ; I can d a ha model nex ).
4
8 Ready- o-pas e summa y pa ag aph ( o he pape )
Wo ked example (Schwa zschild). — Using he local cons i u i e ansa z ∆
µν
= (
α
/Kc)Θ(K−
Kc)H
µν
wi h H
µν
=R
µαβγ
R
ναβγ
−1
4g
µν
K, one inds ∆∼
α
(K/Kc). Fo Schwa zschild K=
48M2
geom/ 6, so he ac i a ion adius whe e K=Kcis c= (48M2
geom/Kc)1/6. Linea g ow h
equi es nonlinea sa u a ion; a minimal a ional model ∆
µν
∝−(K/Kc)/(1+K/Kc)g
µν
yields
a ini e, de Si e –like co e wi h Λe ∼√3/8√Kcand co e adius 0= (6GM/Λe )1/3∼1.12 c.
Taking Kc∼ℓ−4
Pgi es 0∼10−22 m o a sola mass black hole (many o de s o magni ude la ge
han ℓPbu mic oscopically small compa ed o s), demons a ing ha RSD can egula ize he
singula i y while lea ing he ex e io essen ially classical.
9 Assump ions and s a egy
1. Wo k in sphe ical symme y and acuum ou side ma e (no o dina y ma e inside he
co e): he physical s ess Tphys
µν
=0 o he egion conside ed.
2. RSD sa u a es in he high-cu a u e egion p oducing an e ec i e iso opic memo y which
beha es as an e ec i e acuum ene gy (cosmological cons an ) in he co e. We he e o e
model he esidual s ess by an e ec i e cosmological e m
T(∆)
µν
(x) = −Λe
8
π
Gg(in )
µν
(x) ( ≤ 0),
wi h Λe >0de e mined by he sa u a ion physics (a simple model: Λe ∼
β
√Kcwi h
β
=√3/8as in he p e ious sec ion).
3. Ou side he co e ( > 0) he me ic is Schwa zschild:
ds2
+=− +( )d 2+ −1
+( )d 2+ 2dΩ2, +( ) = 1−2GM
.
4. Inside he co e ( ≤ 0) he sel -consis en solu ion o he Eins ein equa ions wi h pu e
acuum ene gy is de Si e :
ds2
−=− −( )d 2+ −1
−( )d 2+ 2dΩ2, −( ) = 1−Λe
3 2.
(We choose he same ime coo dina e so ha me ic ma ching is simple; gene aliza ions
wi h ime escaling a e s aigh o wa d.)
5. We will impose con inui y o he induced me ic a = 0( i s undamen al o m). I
he second undamen al o ms a e discon inuous he e will be a hin shell; we compu e i s
su ace s ess-ene gy using he Is ael junc ion condi ions.
10 Ma ching o he me ic (con inui y o he i s undamen al o m)
Con inui y o g and he angula pa a = 0gi es he single non i ial scala condi ion
−( 0) = +( 0).
Explici ly,
1−Λe
3 2
0=1−2GM
0
.
5
Cancel he ones and sol e o Λe :
Λe =6GM
3
0
.(Me ic con inui y)
This is he cen al algeb aic ela ion be ween he co e adius 0, he black-hole mass M,
and he e ec i e memo y cosmological cons an Λe . I is equi alen o he ma ching o mula
quo ed ea lie in in ui i e o m ( 3
0=6GM/Λe ).
11 Is ael junc ion condi ions — su ace s ess-ene gy a 0
I he ex insic cu a u e Kab jumps ac oss = 0 he e is a hin shell ca ying su ace s ess-
ene gy Sab=diag(−
σ
,p,p)de e mined by he Lanczos–Is ael condi ion
Sab=−1
8
π
G([Kab]−
δ
ab[K]),
whe e [X]≡X+−X−is he jump ac oss he shell and indices a,b un o e (
τ
,
θ
,
ϕ
)(wi h
τ
he p ope ime on he shell). Fo he s a ic sphe ically symme ic me ics abo e he ele an
ex insic cu a u e componen s e alua ed on a sphe e o adius a e
K
ττ
= 0( )
2√ ( ),
K
θθ
=√ ( )
(and K
ϕϕ
=K
θθ
).
(These ollow om n =1/√ and he nonze o Ch is o el symbols; sign con en ion chosen so
n
µ
poin s ou wa d.)
Applying he junc ion o mulas yields he well-known closed o ms (pu ±= ±( 0), 0
±=
0
±( 0)):
11.1 Su ace ene gy densi y
σ
=−1
4
π
G 0(√ +−√ −)(su ace ene gy densi y)
11.2 Su ace ( angen ial) p essu e
p=1
8
π
G( 0
+
2√ +− 0
−
2√ −
+√ +−√ −
0)(su ace p essu e)
Now subs i u e he explici de i a i es and me ic unc ions:
• Ex e io : +( ) = 1−2GM
, so a 0
+=1−2GM
0
, 0
+=2GM
2
0
.
• In e io (de Si e ): −( ) = 1−Λe
3 2, so a 0
−=1−Λe
3 2
0, 0
−=−2Λe
3 0.
6

Using he me ic-con inui y ela ion Λe =6GM/ 3
0one may simpli y hese exp essions u -
he algeb aically i desi ed; o example −= +by cons uc ion and he exp essions o
σ
,p
educe o explici algeb aic unc ions o (M, 0).
Rema ks.
• I one desi es no hin shell hen he ex insic cu a u es mus ma ch: [Kab] = 0. This would
equi e bo h √ +=√ −and 0
+= 0
−a 0. Gene ically his is no possible o a bi a y
Mand Λe — hence a hin shell is expec ed unless he pa ame e s sa is y a special uning.
The hin shell is a na u al locus whe e he RSD sa u a ion egion ansi ions o he
classical ex e io .
• One can e alua e
σ
and pand check ene gy condi ions ( hey can iola e classical ene gy
condi ions — ha is allowed because hey o igina e om in eg a ed quan um luc ua-
ions).
12 Explici closed- o m exp ession o he in e io g+∆and he esidual
∆
µν
Inside he co e we ha e he me ic (sel -consis en solu ion)
g(in )
µν
=diag(−(1−Λe
3 2),(1−Λe
3 2)−1, 2, 2sin2
θ
).
The o al esidual de o ma ion wi h espec o he ex e io Schwa zschild backg ound g(Schw)
µν
is simply
∆
µν
( ) = g(in )
µν
( )−g(Schw)
µν
( ),( ≤ 0).
In componen s (using he common coo dina e cha ( , ,
θ
,
ϕ
)):
∆ ( ) = −(1−Λe
3 2)+(1−2GM
)=Λe
3 2−2GM
,
∆ ( ) = (1−Λe
3 2)−1
−(1−2GM
)−1,
∆
θθ
( ) = 0,∆
ϕϕ
( ) = 0,
which is algeb aically closed and eady o pas e. (One may expand ∆ as a powe se ies i
needed.)
13 Cu a u e egula iza ion (explici )
Fo he de Si e in e io he K e schmann scala is ini e and cons an :
KdS ≡R
αβγδ
R
αβγδ
=8
3Λ2
e .
Because Λe is ini e (se by sa u a ion), KdS is ini e a =0. Thus he RSD co e eplaces he
classical →0singula i y o Schwa zschild by a ini e-cu a u e de Si e in e io — singula i y
esolu ion is explici .
Using he ma ching ela ion Λe =6GM/ 3
0:
Kco e =8
3(6GM
3
0)2=8·36
3
G2M2
6
0
=96G2M2
6
0
.
Compa e wi h he Schwa zschild K e schmann KSchw =48G2M2/ 6: he co e alue is ini e and
o he same scaling bu ixed by 0.
7
14 Linking 0 o he RSD h eshold Kc(sel -consis ency)
I he sa u a ion p esc ip ion ixes Λe in e ms o he c i ical cu a u e Kc( o example he
sa u a ion es ima e used ea lie ),
Λe ≃
β
√Kc(
β
=√3/8in he simple model),
combine wi h Λe =6GM/ 3
0 o ob ain an explici exp ession o he co e adius:
0=(6GM
β
√Kc)1/3.(co e adius in e ms o M,Kc)
Using he ac i a ion ( h eshold) adius cde ined by K( c) = Kcin Schwa zschild (i.e.,
K( ) = 48G2M2/ 6⇒ c= (48G2M2/Kc)1/6) one inds he analy ic a io (se
β
=√3/8i you
wan he model nume ic)
0
c
=(6/
β
)1/3
481/6
β
=√3/8
=1.122462...
Thus in he na u al uni sys em used in he pape he sel -consis en co e adius is o he
same o de as he ac i a ion adius: 0∼1.12 c. This con i ms he in e nal consis ency o he
sa u a ion pic u e: he egion whe e Kexceeds Kcsel -o ganizes in o a ini e de Si e co e o
he same scale.
15 Physical in e p e a ion & eady- o-pas e conclusion pa ag aph
Sel -consis en RSD co e (analy ic esul ). Assume RSD sa u a es in high cu a u e so ha he
esidual imp in ac s as an e ec i e acuum ene gy inside a adius 0. The sel -consis en in e io
is he de Si e me ic −( ) = 1−Λe
3 2while he ex e io is Schwa zschild +( ) = 1−2GM/ .
Con inui y o he i s undamen al o m a 0gi es Λe =6GM/ 3
0and in gene al a hin shell
a 0ca ies su ace s ess-ene gy wi h densi y
σ
and p essu e pgi en in closed o m by
σ
=−1
4
π
G 0(√ +−√ −),p=1
8
π
G( 0
+
2√ +− 0
−
2√ −
+√ +−√ −
0).
The co e cu a u e is ini e, Kco e =8
3Λ2
e , so he classical singula i y is eplaced by a egula
de Si e in e io . I one models Λe ia he RSD sa u a ion scale (e.g., Λe ∼
β
√Kc) hen
he closed ela ion 0= (6GM/(
β
√Kc))1/3holds and one inds 0 o be he same o de as he
RSD ac i a ion adius c(nume ically 0/ c≈1.122 o he simple model), gi ing a ully sel -
consis en , analy ic nonlinea solu ion sui able o publica ion.
16 Sugges ed immedia e ollowups you can pas e nex
• Pu he shell ene gy densi y/p essu e in o he pape as a sho pa ag aph and discuss
whe he he shell iola es o sa is ies ene gy condi ions (compu e sign o
σ
and pwi h
Λe =6GM/ 3
0).
• Show a sho plo (nume ical) o K( ) o gSchw and o gSchw +∆(de Si e in e io ) o
isualize egula iza ion. I can p oduce a eady- o- un sc ip o ha .
• Gene alize his ma ching o cha ged o o a ing backg ounds (Reissne –No ds öm, Ke )
— I can w i e he Ke ma ching ske ch nex .
8
Ma hema ical De i a ion o he Residual De o ma ion Tenso Δ𝜇𝜈
1 Se up — Ac ion, Pa h In eg al, and he C i ical Cu a u e Sec o
The de i a ion begins wi h he Eins ein–Hilbe ac ion coupled o ma e ields, exp essed in na u al uni s
(𝑐= ℏ = 1) wi h he me ic signa u e (−,+,+,+):
𝑆[𝑔, Ψ]=1
16𝜋𝐺 ∫𝑑4𝑥√−𝑔(𝑅−2Λ) +𝑆m[𝑔, Ψ],(1)
whe e 𝑅is he Ricci scala , Λ he cosmological cons an , and 𝑆m he ma e ac ion. The g a i a ional
pa h in eg al is o mally de ined as:
𝑍=∫D𝑔DΨ𝑒𝑖𝑆[𝑔,Ψ].(2)
To iden i y high-cu a u e egions, we in oduce he K e schmann in a ian :
K(𝑥) ≡ 𝑅𝛼𝛽𝛾 𝛿 𝑅𝛼𝛽𝛾 𝛿 (𝑥),(3)
and de ine a scale-dependen h eshold K𝑐(ℓ)such ha egions wi h K ≥ K𝑐cons i u e he high-
cu a u e sec o . The objec i es a e o pa i ion he pa h in eg al in o low- and high-cu a u e sec o s,
in eg a e ou he high-cu a u e sec o o ob ain an in luence unc ional, and de i e he esidual de o -
ma ion enso Δ𝜇𝜈 a ec ing he low-cu a u e e ec i e geome y.
2 Sec o Decomposi ion (Coa se + Fluc ua ions)
Me ic con igu a ions a e decomposed in o a coa se backg ound 𝑔𝜇𝜈 and luc ua ions 𝜉𝜇𝜈:
𝑔 o
𝜇𝜈 (𝑥)=𝑔𝜇𝜈 (𝑥) +𝜉𝜇𝜈 (𝑥).(4)
The pa h in eg al is spli acco dingly:
∫D𝑔 o =∫D𝑔∫K≥K𝑐(𝑔)D𝜉, (5)
whe e he inne in eg al is es ic ed o luc ua ions 𝜉 ha , when added o 𝑔, yield K ≥ K𝑐. This es ic-
ion may be implemen ed using a unc ional Hea iside Θ[K −K𝑐].
The ull pa h in eg al becomes:
𝑍=∫D𝑔DΨlow [∫K≥K𝑐D𝜉DΨhigh 𝑒𝑖𝑆[𝑔+𝜉 ,Ψlow,Ψhigh ]].(6)
The in luence unc ional is de ined by in eg a ing ou he high-cu a u e sec o :
F[𝑔, Ψlow] ≡ ∫K≥K𝑐D𝜉DΨhigh 𝑒𝑖𝑆[𝑔+𝜉 ,Ψlow,Ψhigh ]=𝑒𝑖𝑆i [𝑔,Ψlow ],(7)
yielding he low-sec o e ec i e pa h in eg al:
𝑍=∫D𝑔DΨlow 𝑒𝑖(𝑆[𝑔,Ψlow ]+𝑆i [𝑔,Ψlow ]).(8)
1
3 De ini ion o he Residual De o ma ion Δ𝜇𝜈
The esidual de o ma ion Δ𝜇𝜈 ep esen s he mean imp in o high-cu a u e luc ua ions on he coa se
geome y:
Δ𝜇𝜈 (𝑥)=⟨𝜉𝜇𝜈 (𝑥)⟩K≥K𝑐
=∫K≥K𝑐D𝜉 𝜉𝜇𝜈 (𝑥)𝑒𝑖𝑆[𝑔+𝜉]
∫K≥K𝑐D𝜉 𝑒𝑖𝑆[𝑔+𝜉](9)
Equi alen ly, he mean o al me ic a e in eg a ing ou he high sec o is:
⟨𝑔 o
𝜇𝜈 (𝑥)⟩high =𝑔𝜇𝜈 (𝑥) +Δ𝜇𝜈 (𝑥).(10)
This is a o mal exp ession; con olled app oxima ions (e.g., saddle poin o cumulan expansion) a e
equi ed o explici compu a ions.
4 Semiclassical Field Equa ions and he Role o 𝑆i
The e ec i e ac ion is de ined as:
𝑆eff [𝑔, Ψlow]=𝑆[𝑔, Ψlow] +𝑆i [𝑔, Ψlow].(11)
Va ying wi h espec o 𝑔𝜇𝜈 yields he semiclassical Eins ein equa ions:
2
√−𝑔(𝑥)
𝛿𝑆eff
𝛿𝑔𝜇𝜈 (𝑥)=0⇒𝐺𝜇𝜈 (𝑥) +Λ𝑔𝜇𝜈 (𝑥)=8𝜋𝐺 (𝑇𝜇𝜈 (𝑥) +𝑇(Δ)
𝜇𝜈 (𝑥)),(12)
whe e he esidual (memo y) s ess enso is:
𝑇(Δ)
𝜇𝜈 (𝑥) ≡ − 2
√−𝑔(𝑥)
𝛿𝑆i [𝑔]
𝛿𝑔𝜇𝜈 (𝑥).(13)
The esidual de o ma ion can be ep esen ed as ei he a me ic shi (𝑔𝜇𝜈 ↦→ 𝑔𝜇𝜈 +Δ𝜇𝜈) o an e ec i e
sou ce 𝑇(Δ)
𝜇𝜈 , ela ed o leading o de by:
𝑇(Δ)
𝜇𝜈 ≃1
8𝜋𝐺 (𝐺𝜇𝜈 [𝑔+Δ] −𝐺𝜇𝜈 [𝑔])+O(ℏ2).(14)
Conse a ion (∇𝜇𝑇(Δ)
𝜇𝜈 =0) holds i he coa se-g aining p ese es di eomo phism in a iance.
5 Cumulan / Gaussian App oxima ion: Explici Compu a ion o 𝑆i and
Δ
To compu e 𝑆i and Δ, we expand he ac ion o second o de in 𝜉(Gaussian app oxima ion), alid when
high-mode in e ac ions a e pe u ba i e o a saddle poin domina es:
𝑆[𝑔+𝜉]=𝑆[𝑔]+∫𝑑4𝑥√−𝑔(𝑥)𝐽𝜇𝜈 (𝑥)𝜉𝜇𝜈 (𝑥)+1
2∫𝑑4𝑥 𝑑4𝑦√−𝑔(𝑥)√−𝑔(𝑦)𝜉𝜇𝜈 (𝑥)D−1,𝜇𝜈𝛼𝛽 (𝑥, 𝑦)𝜉𝛼𝛽 (𝑦)+O(𝜉3),
(15)
whe e he linea sou ce is:
𝐽𝜇𝜈 (𝑥) ≡ 1
√−𝑔(𝑥)
𝛿𝑆[𝑔]
𝛿𝑔𝜇𝜈 (𝑥).(16)
Fo he Eins ein–Hilbe plus ma e ac ion, 𝐽𝜇𝜈 =1
16𝜋𝐺 (𝐺𝜇𝜈 +Λ𝑔𝜇𝜈)− 1
2𝑇𝜇𝜈. The ope a o D−1is he
in e se p opaga o , wi h he G een’s unc ion Dsa is ying:
∫𝑑4𝑦√−𝑔(𝑦)D−1,𝜇𝜈𝛼𝛽 (𝑥, 𝑦)D𝛼𝛽𝜌𝜎 (𝑦, 𝑧)=𝛿𝜇𝜈
𝜌𝜎
𝛿(4)(𝑥−𝑧)
√−𝑔(𝑧).(17)
2
1 RSD Applied o FLRW Cosmology — De i a ion, Cons i u i e Models,
and a Nume ical Example
1.1 Se up
We wo k in na u al uni s (𝑐= ℏ = 1) wi h he me ic signa u e (−,+,+,+). Conside a spa ially homo-
geneous and iso opic F iedmann–Lemaî e–Robe son–Walke (FLRW) me ic:
𝑑𝑠2=−𝑑𝑡2+𝑎(𝑡)2(𝑑𝑟2
1−𝑘𝑟2+𝑟2𝑑Ω2),(1)
wi h he Hubble a e 𝐻(𝑡) ≡ ¤𝑎/𝑎. The semiclassical ield equa ions, including he esidual s ess 𝑇(Δ)
𝜇𝜈
om in eg a ing ou he high-cu a u e sec o , a e:
𝐺𝜇𝜈 +Λ𝑔𝜇𝜈 =8𝜋𝐺 (𝑇𝜇𝜈 +𝑇(Δ)
𝜇𝜈 ),(2)
whe e 𝑇𝜇𝜈 is he o dina y ma e s ess enso , and he esidual (memo y) s ess is de ined as:
𝑇(Δ)
𝜇𝜈 (𝑥)=−2
√−𝑔(𝑥)
𝛿𝑆i [𝑔]
𝛿𝑔𝜇𝜈 (𝑥).(3)
Due o he maximal spa ial symme y o he FLRW backg ound, 𝑇(Δ)
𝜇𝜈 mus espec homogenei y and
iso opy, aking he pe ec - luid o m in como ing coo dina es:
𝑇(Δ)𝜇𝜈=diag (−𝜌Δ(𝑡), 𝑝Δ(𝑡), 𝑝Δ(𝑡), 𝑝Δ(𝑡)).(4)
The modi ied F iedmann and accele a ion equa ions ollow:
Modi ied F iedmann equa ion:
𝐻2+𝑘
𝑎2=8𝜋𝐺
3(𝜌(𝑡) + 𝜌Δ(𝑡))+Λ
3,(5)
Modi ied accele a ion equa ion:
¥𝑎
𝑎=¤
𝐻+𝐻2=−4𝜋𝐺
3(𝜌+𝜌Δ+3(𝑝+𝑝Δ))+Λ
3.(6)
I he coa se-g aining p ese es di eomo phism in a iance, he o al s ess-ene gy is co a ian ly
conse ed, implying:
¤𝜌+3𝐻(𝜌+𝑝) + ¤𝜌Δ+3𝐻(𝜌Δ+𝑝Δ)=0.(7)
I he o dina y ma e is sepa a ely conse ed ( ¤𝜌+3𝐻(𝜌+𝑝)=0), he esidual sec o sa is ies:
¤𝜌Δ+3𝐻(𝜌Δ+𝑝Δ)=0.(8)
1.2 Cons i u i e Models o 𝑇(Δ)
𝜇𝜈
The pa h-in eg al o malism gi es Δ𝜇𝜈 (𝑥)and 𝑇(Δ)
𝜇𝜈 as nonlocal unc ionals:
Δ𝜇𝜈 (𝑥)=−∫𝑑4𝑦√−𝑔(𝑦)D𝜇𝜈 𝛼𝛽 (𝑥, 𝑦)𝐽𝛼𝛽 (𝑦) +··· ,(9)
𝑇(Δ)
𝜇𝜈 (𝑥)=−2
√−𝑔
𝛿
𝛿𝑔𝜇𝜈 (𝑥)(−1
2∫𝐽D𝐽+𝑖
2T ln D−1+···).(10)
These a e in eg odi e en ial ela ions, wi h 𝜌Δ(𝑡)and 𝑝Δ(𝑡)depending nonlocally on he cu a u e his-
o y ia Dand 𝐽. We in oduce wo physically mo i a ed cons i u i e models:
1

1.2.1 (A) Local Vacuum-Like Sa u a ion Model (Simple & Analy ic)
Assume RSD ac s as a local, cu a u e- h esholded acuum ene gy:
𝑇(Δ)
𝜇𝜈 (𝑡)=−Λeff
8𝜋𝐺 𝐹(K(𝑡)
K𝑐)𝑔𝜇𝜈,(11)
whe e:
•K(𝑡) ≡ 𝑅𝛼𝛽𝛾 𝛿 𝑅𝛼𝛽𝛾 𝛿 is he K e schmann scala on FLRW,
•K𝑐is he ac i a ion h eshold,
•Λeff se s he sa u a ion ampli ude (dimension [𝐿−2]),
•𝐹(𝑧)is a smoo h sa u a ion unc ion.
A con enien choice is:
𝐹(𝑧)=𝑧
1+𝑧(linea o 𝑧1,sa u a es o 1 o 𝑧1),(12)
yielding a small imp in o K  K𝑐and a acuum ene gy −Λeff𝑔𝜇𝜈/(8𝜋𝐺) o K  K𝑐. The esidual
ene gy densi y and p essu e a e:
𝜌Δ(𝑡)=Λeff
8𝜋𝐺 𝐹(K(𝑡)
K𝑐), 𝑝Δ(𝑡)=−𝜌Δ(𝑡).(13)
Subs i u ing in o (5)–(6) gi es an implici equa ion o 𝐻(𝑡), as K(𝑡)depends on 𝐻and ¤
𝐻.
K e schmann scala on FLRW ( la 𝑘=0):
K(𝑡)=12 ((¥𝑎
𝑎)2
+(¤𝑎2
𝑎2)2)=12 ((¤
𝐻+𝐻2)2+𝐻4).(14)
1.2.2 (B) Causal Nonlocal Ke nel Model
A mo e ai h ul ep esen a ion o he Gaussian in luence ac ion is:
𝜌Δ(𝑡)=∫𝑡
𝑡0
𝑑𝑡0G𝜌(𝑡, 𝑡0)𝐽(𝑡0), 𝑝Δ(𝑡)=∫𝑡
𝑡0
𝑑𝑡0G𝑝(𝑡, 𝑡0)𝐽(𝑡0),(15)
whe e 𝐽(𝑡)is a scala sou ce (e.g., con ac ions o 𝐺𝜇𝜈 [𝑔]) and G𝜌, 𝑝 (𝑡, 𝑡0)a e causal esponse ke nels
de i ed om D. This o m cap u es nonlocal memo y e ec s, sui able o de ailed heo e ical compu a-
ions bu equi ing nume ical speci ica ion o G.
1.3 Adiaba ic (Slow-Time) App oxima ion — Explici ODE o 𝐻(𝑡)
In he adiaba ic app oxima ion (|¤
𝐻|  𝐻2), he K e schmann scala simpli ies:
K(𝑡) ≈ 24𝐻(𝑡)4( la 𝑘=0,adiaba ic).(16)
Using he local acuum-like model (11) and (16), he esidual ene gy densi y is:
𝜌Δ(𝐻)=Λeff
8𝜋𝐺 𝐹(24𝐻4
K𝑐).(17)
Fo adia ion (𝑝=1
3𝜌), he con inui y equa ion gi es ¤𝜌=−4𝐻𝜌, and he accele a ion equa ion (6)
becomes:
¤
𝐻=−𝐻2−8𝜋𝐺
3(𝜌−𝜌Δ(𝐻))+Λ
3.(18)
The sys em is closed by:
¤𝜌=−4𝐻𝜌, ¤𝑎=𝑎𝐻. (19)
2
1.4 A Minimal Nume ical Demons a ion (Illus a i e Pa ame e Choice)
We in eg a e (18)–(19) nume ically wi h:
• Radia ion equa ion o s a e (𝑝=𝜌/3),
• Sa u a ion unc ion 𝐹(𝑧)=𝑧/(1+𝑧),
• Adiaba ic app oxima ion K ≃ 24𝐻4,
• Pa ame e s (Planck uni s 𝐺=1): K𝑐=1,Λeff =1,Λ = 0,
• Ini ial condi ions: 𝐻(𝑡0)=𝐻0=1.0,𝑎(𝑡0)=1,
• Ini ial adia ion densi y sa is ying he F iedmann cons ain :
𝜌(RSD)
0=3
8𝜋𝐺 (𝐻2
0−Λ
3−Λeff
3𝐹(24𝐻4
0
K𝑐)),(20)
𝜌(no RSD)
0=3
8𝜋𝐺 (𝐻2
0−Λ
3).(21)
The ODEs we e in eg a ed using an adap i e Runge–Ku a sol e . The esul s a e quali a i e bu obus :
Nume ical ou come:
• The RSD e m, ini ially non-negligible due o Planckian 𝐻0, ac s as an ex a acuum ene gy, slow-
ing he decay o 𝐻(𝑡)compa ed o he no-RSD case.
•𝜌Δ(𝐻(𝑡)) is ini ially compa able o he adia ion densi y and decays mo e slowly.
• A inal ime 𝑡final =200 (Planck uni s):
–𝐻(𝑡final)RSD ≈5.21 ×10−3,
–𝐻(𝑡final)no RSD ≈2.51 ×10−3,
–𝜌(𝑡final)RSD ≈1.57 ×10−15,𝜌(𝑡final)no RSD ≈7.50 ×10−7.
The esul s a e isualized in wo PNG iles:
• ‘/mn /da a/ sd𝑣𝑠𝑛𝑜𝑟𝑠𝑑𝐻.𝑝𝑛𝑔‘(𝐻𝑢𝑏𝑏𝑙𝑒𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛),‘/𝑚𝑛𝑡/𝑑𝑎𝑡𝑎/𝑟𝑠𝑑𝑒𝑛𝑒𝑟𝑔𝑦𝑑𝑒𝑛𝑠𝑖𝑡𝑖𝑒𝑠.𝑝𝑛𝑔‘(𝑒𝑛𝑒𝑟𝑔𝑦𝑑𝑒𝑛𝑠𝑖𝑡𝑖𝑒𝑠, 𝑙𝑜𝑔𝑠𝑐𝑎𝑙𝑒).
1.5 Discussion, Limi s, and Nex S eps
•1. Validi y o he adiaba ic app oxima ion. The app oxima ion K ≃ 24𝐻4 equi es |¤
𝐻|  𝐻2. Fo
apid ansi ions, he ull K(14) mus be used, sol ing he implici in eg odi e en ial sys em.
2. Physical in e p e a ion. The acuum-like RSD model (11) sa u a es a high cu a u e, ac ing as a
cosmological e m ha can egula ize singula i ies o d i e accele a ion. The nonlocal model (15)
cap u es memo y e ec s, po en ially dissipa ing o eeding ene gy in o long-wa eleng h modes.
3. Ene gy condi ions. 𝑇(Δ)may iola e classical ene gy condi ions, which is physically allowed due
o quan um back eac ion bu equi es case-by-case assessmen .
4. Obse a ional implica ions. Fo Planckian K𝑐, RSD co ec ions o la e- ime expansion and p i-
mo dial spec a a e negligible. Lowe K𝑐o nea -Planckian 𝐻could p oduce obse able e ec s.
5. How o use his sec ion. Use equa ions (5)–(8) o he amewo k, (11) and (15) o cons i u i e
models, and (16)–(19) o he adiaba ic educ ion, ollowed by he nume ical demons a ion.
3
2 Quick Summa y o RSD
RSD posi s ha in eg a ing ou high-cu a u e con igu a ions (K ≡ 𝑅𝛼𝛽𝛾 𝛿 𝑅𝛼𝛽𝛾 𝛿 ≥ K𝑐(ℓ)) lea es a
de e minis ic esidual de o ma ion Δ𝜇𝜈 and s ess-ene gy 𝑇(Δ)
𝜇𝜈 . A Gaussian o de :
𝑆i [𝑔] ≃ −1
2∫𝐽D𝐽+𝑖
2T ln D−1,(22)
Δ𝜇𝜈 (𝑥) ≃ −∫𝑑4𝑦√−𝑔(𝑦)D𝜇𝜈 𝛼𝛽 (𝑥, 𝑦)𝐽𝛼𝛽 (𝑦),(23)
wi h e ec i e ield equa ions:
𝐺𝜇𝜈 [𝑔+Δ] +Λ(𝑔𝜇𝜈 +Δ𝜇𝜈)=8𝜋𝐺𝑇𝜇𝜈,(24)
𝐺𝜇𝜈 +Λ𝑔𝜇𝜈 =8𝜋𝐺(𝑇𝜇𝜈 +𝑇(Δ)
𝜇𝜈 ).(25)
He e, 𝐽𝛼𝛽 =1
√−𝑔
𝛿𝑆
𝛿𝑔𝛼𝛽 , and Dis he high-mode p opaga o .
3 Compa ison wi h O he App oaches
3.1 Loop Quan um G a i y (LQG)
Co e idea. LQG quan izes g a i y wi h disc e e a ea and olume ope a o s, using holonomies and luxes
o spin- oam ampli udes.
Singula i y esolu ion. Singula i ies a e esol ed ia disc e e a iables o holonomy co ec ions,
p oducing bounces in symme y- educed models.
Con as wi h RSD.
•Mic oscopic pic u e: LQG assumes undamen al disc e eness; RSD ope a es in he con inuum
wi h Δ𝜇𝜈 om high-cu a u e modes.
•Ma hema ical o igin: LQG uses modi ied commu a o s; RSD uses an in luence unc ional.
•Obse ables: LQG p edic s disc e e spec a; RSD p edic s co e adii and small CMB/GW co ec-
ions.
Complemen a i y. RSD could be an e ec i e desc ip ion o LQG’s la ge-scale e ec s i LQG p o-
duces a simila in luence unc ional.
3.2 S ing Theo y
Co e idea. S ing heo y uni ies g a i y and ma e ia ex ended objec s, wi h highe -de i a i e co ec-
ions in he e ec i e ac ion.
Singula i y esolu ion. S ingy e ec s (e.g., T-duali y, uzzballs) smoo h ce ain singula i ies.
Con as wi h RSD.
•New deg ees o eedom: S ing heo y in oduces new exci a ions; RSD uses only g a i a ional
con igu a ions.
•Co ec ions: S ing heo y yields highe -de i a i e e ms; RSD p oduces nonlocal e ms ia D.
•P edic i i y: S ing heo y’s landscape complica es p edic ions; RSD’s pa ame e s (K𝑐,D) enable
alsi iable p edic ions.
Complemen a i y. RSD could mani es s ingy UV physics i he in luence unc ional aligns.
4
3.3 Asymp o ic Sa e y
Co e idea. G a i y is a quan um ield heo y wi h a UV ixed poin , wi h unning couplings like 𝐺(𝜇).
Singula i y esolu ion. Running couplings so en high-cu a u e beha io .
Con as wi h RSD.
•Mechanism: Asymp o ic Sa e y uses RG low; RSD uses h esholded high-cu a u e in eg a ion.
•Scale dependence: Bo h a e scale-dependen , bu RSD uses K𝑐(ℓ) ∼ ℓ−𝛼.
Complemen a i y. Asymp o ic Sa e y could p oduce an RSD-like in luence unc ional a he ixed
poin .
4 Why RSD is Simple, Geome ic, and Falsi iable
Simplici y. RSD a oids new ields o dimensions, using he g a i a ional pa h in eg al o de i e Δ𝜇𝜈.
Geome ic na u e. Δ𝜇𝜈 and 𝑇(Δ)
𝜇𝜈 a e co a ian , p ese ing GR’s geome ic s uc u e.
Falsi iabili y.
1. Co e adius scaling:
𝑟0=(6𝐺𝑀
𝛽√K𝑐)1/3
.(26)
Incompa ible 𝑟0wi h Planckian K𝑐 alsi ies RSD.
2. QNM/ ingdown. Negligible modi ica ions unless 𝑟0app oaches he pho on sphe e.
3. CMB/p imo dial spec a. Planckian K𝑐p edic s 𝛿𝑃𝜁/𝑃𝜁10−10.
4. Theo e ical alsi ica ion. A ze o Δ𝜇𝜈 o pu ely dissipa i e 𝑆i would in alida e RSD.
5. Consis ency. Nonconse a ion o 𝑇(Δ)would alsi y RSD.
5 Limi a ions, Possible Objec ions, and Responses
1. Regula o dependence. Use co a ian egula o s o ensu e egula o -insensi i e obse ables.
2. Nonuniqueness o D.Pa ame e ize and cons ain Dwi h heo y and da a.
3. Ene gy condi ions. Allow con olled NEC iola ions and check s abili y.
4. Mic oscopic o igin. RSD is an e ec i e amewo k; mic oscopic de i a ions ix Dand K𝑐.
6 Conc e e P og am o S eng hen o Falsi y RSD
1. Compu e Don key backg ounds wi h a co a ian egula o .
2. Pe o m s abili y analyses o ep esen a i e ke nels.
3. Inse 𝑇(Δ)in o nume ical ela i i y simula ions.
4. Ve i y bulk- o-bounda y mappings in holog aphic se ups.
5. Use obse a ional da a o cons ain K𝑐,Λeff, and ke nel pa ame e s.
5
7 Closing Rema k
RSD p ese es GR’s geome ic language, a oids new mic odeg ees o eedom, and educes UV com-
plexi y o K𝑐(ℓ),D, and Λeff. This enables conc e e, alsi iable p edic ions, making RSD a obus sci-
en i ic p oposal.
6

Residual S ess De o ma ion (RSD) in In la iona y and
Holog aphic Con ex s
1 Se up and no a ion
Backg ound FRW (con o mal ime
η
):
ds2=a2(
η
)−d
η
2+dx2,H≡a0
a2,
p ime (0) deno es d/d
η
. Single scala ield
φ
(
η
)d i es in la ion. De ine
z(
η
)≡a(
η
)
φ
0(
η
)
H(
η
),H≡a0
a.
Mukhano –Sasaki (MS) a iable (
η
,x)≡z(
η
)
ζ
(
η
,x)sa is ies
00−∇2 −z00
z =0.
In Fou ie space (mode k),
00
k(
η
)+k2−z00(
η
)
z(
η
) k(
η
) = 0.(1)
Powe spec um o cu a u e pe u ba ion
ζ
a la e ime (
η
→0) (a e ho izon exi ):
P
ζ
(k) = k3
2
π
2|
ζ
k(
η
→0)|2=k3
2
π
2| k(
η
→0)|2
z(
η
→0)2.(2)
RSD en e s by p oducing a small esidual s ess T(∆)
µν
ha pe u bs he backg ound and
he e o e changes z(
η
). To linea o de he e ec on MS is encapsula ed by a small pe u ba ion
o he MS po en ial:
z00
z(
η
) = z00
0
z0
(
η
)+
δ
z00
z(
η
)≡z00
0
z0
(
η
)+
δ
Q(
η
),
whe e z0is he unpe u bed backg ound alue and
δ
Q(
η
)is he (small) pe u ba ion induced
by T(∆). A p ac ical ela ion (used la e ) is
δ
Q
a2H2∼O
ρ
∆
ρ
o ≡∆e (
η
),
i.e.,
δ
Qis o o de he dimensionless ac ional backg ound pe u ba ion p oduced by he esid-
ual s ess.
1
2 In e ac ion Hamil onian and in–in i s o de exp ession
T ea ing
δ
Q(
η
)as an in e ac ion, he quad a ic in e ac ion Hamil onian (densi y) in con o mal
ime is
HI(
η
) = 1
2Zd3x
δ
Q(
η
) (
η
,x)2=1
2Zd3p
(2
π
)3
δ
Q(
η
) p(
η
) −p(
η
).
Using he in–in o malism, he i s -o de co ec ion o he wo-poin unc ion is
δ
h k(
η
) k0(
η
)i=−iZ
η
−∞
d
η
0h[ k(
η
) k0(
η
),HI(
η
0)]i0,
whe e h·i0is expec a ion in he ee Bunch–Da ies acuum. E alua e he commu a o using he
s anda d mode expansion
(
η
,x) = Zd3q
(2
π
)3haq q(
η
)eiq·x+a†
q ∗
q(
η
)e−iq·xi,
wi h [aq,a†
q0] = (2
π
)3
δ
(3)(q−q0)and k(
η
) he posi i e- equency solu ion (Bunch–Da ies). A
s anda d e alua ion (s aigh o wa d bu algeb aic; d op he i ial momen um del a) yields
he compac esul
δ
h k(
η
) −k(
η
)i=−2Imh k(
η
)2Z
η
−∞
d
η
0
δ
Q(
η
0) ∗
k(
η
0)2i.(3)
This o mula is exac a i s o de in
δ
Q(i ollows om e alua ing he commu a o and
using Wick’s heo em).
3 F om
δ
h k −ki o
δ
P
ζ
/P
ζ
F om (2) and (3) he ac ional change in he cu a u e powe spec um a la e ime (
η
) is
δ
P
ζ
(k)
P
ζ
(k)=
δ
| k(
η
)|2
| k(
η
)|2−2
δ
z(
η
)
z(
η
).
Use
δ
| k|2=
δ
h k −ki. Subs i u ing (3) gi es
δ
P
ζ
(k)
P
ζ
(k)=−2
Imh k(
η
)2R
η
−∞d
η
0
δ
Q(
η
0) ∗
k(
η
0)2i
| k(
η
)|2−2
δ
z(
η
)
z(
η
).(4)
Commen s on he wo e ms.
• The i s e m is he di ec e ec o he pe u bed MS po en ial (quad a ic in e ac ion)
on he mode unc ions.
• The second e m is he explici change o he no maliza ion ac o z(
η
)a he e alua ion
ime (la e ime);
δ
z/zis i sel o o de he ac ional backg ound change (∼∆e ) p oduced
by T(∆). One mus include i o a consis en i s -o de esul .
Equa ion (4) is he main, exac i s -o de (in
δ
Q) esul exp essed wi h mode unc ions. I
is eady o e alua e once k(
η
)and
δ
Q(
η
)a e speci ied.
4 Use ul app oxima ions and a compac pa ame ic es ima e
We now simpli y (4) unde s anda d in la iona y app oxima ions o ob ain an in ui i e and
obus scaling es ima e.
2
4.1 (i) La e ime and supe ho izon limi
Take
η
→0(la e ime) and conside modes ha exi he ho izon du ing in la ion. In slow oll
he unpe u bed mode unc ion o Bunch–Da ies in quasi-de Si e is
(0)
k(
η
) = 1
√2k1−i
k
η
e−ik
η
,
so a la e imes | (0)
k(
η
)|2≃1
2k3
η
2
and (0)
k(
η
)∝z(
η
)( his is he s a emen ha
ζ
k eezes ou ).
The phase o k(
η
) a ies apidly o subho izon imes and slows a e ho izon exi .
4.2 (ii) Ke nel domina ed nea ho izon c ossing
The ime in eg al in (4) ypically ecei es i s dominan con ibu ion a ound he epoch when
he mode is lea ing he ho izon, |k
η
|∼O(1). Thus e alua e he in eg and in he neighbo hood
η
0∼
η
k≡−1/k. Fo pe u ba ions
δ
Q(
η
) ha a y on Hubble imescales o a e localized (e.g.,
ac i a ion when cu a u e c osses h eshold), he app oxima ion
Z0
−∞
d
η
0
δ
Q(
η
0) ∗
k(
η
0)2∼ ∗
k(
η
k)2Zd
η
0
δ
Q(
η
0)·O(1)
is conse a i e; he complex phase gi es an O(1)oscilla o y ac o whose imagina y pa is also
O(1)in magni ude.
4.3 (iii) Pa ame ic scaling
Using he las wo poin s and ha | k(
η
k)|2∼O(1/k3)while | k(
η
)|2∼O(1/k3
η
2
)bu hese
η
-dependences la gely cancel in he a io in (4), one ob ains he obus pa ame ic scaling
δ
P
ζ
(k)
P
ζ
(k)∼Ck
δ
Q(
η
k)
a(
η
k)2H(
η
k)2+O
δ
z
z∼O∆e (
η
k),(5)
whe e Ckis an o de -one complex numbe (model / k-dependen ke nel ac o whose magni ude
is ∼1 o slow- a ying
δ
Q). In wo ds:
> The ac ional change in he p imo dial powe pe mode is o he same o de as he
ac ional change in he backg ound ene gy densi y p oduced by he esidual s ess a he epoch
when ha mode c osses he ho izon.
This s a emen is physically in ui i e:
δ
Qmodi ies he e ec i e mass/po en ial o he luc-
ua ions du ing ho izon c ossing, p oducing O(∆e ) ac ional changes in mode ampli ude.
5 Exp essing
δ
Qin e ms o T(∆)
µν
To connec wi h he RSD o malism, exp ess
δ
Q h ough he backg ound Eins ein equa ions
pe u bed by T(∆). A backg ound le el he F iedmann equa ion in con o mal ime is
3H2=8
π
Ga2(
ρ
infl +
ρ
∆),
so a small esidual ene gy densi y
ρ
∆induces ac ional changes
δ
(H)
H∼1
2
ρ
∆
ρ
o ∼1
2∆e .
Because z00/z∼a2H2×(slow oll combina ions), a ia ions p oduce
δ
Q
a2H2∼O∆e ,∆e (
η
)≡
ρ
∆(
η
)
ρ
o (
η
).
3
Thus (5) becomes
δ
P
ζ
P
ζ
(k)∼O∆e (
η
k).(6)
6 Conc e e nume ical es ima e ( e-using ea lie RSD scaling)
Recall om he ea lie wo ked example a simple es ima e o an in la iona y de Si e pa ch:
KdS =24H4,∆e ∼
α
KdS
Kc
=24
α
(HℓP)4(wi h Kc∼ℓ−4
P).
Take a ep esen a i e (op imis ic) in la iona y scale H∼1014 GeV. Using HℓP∼H/EPl ≈8.2×
10−6and
α
∼1we ind
∆e ∼24(8.2×10−6)4∼10−19,
and he e o e (by (6))
δ
P
ζ
P
ζ
∼10−19 (Planckian h eshold, ypical in la iona y H).
This ag ees wi h he pa ame ic conclusion in §5 o he main ex : i Kcis Planckian, RSD
p oduces u e ly negligible p imo dial imp in s.
7 Ready- o-pas e concise conclusion ( o you pape )
Pe u ba i e e ec o RSD on p imo dial powe . T ea ing he esidual s ess T(∆)
µν
as a pe u -
ba ion o he backg ound, he Mukhano –Sasaki po en ial acqui es
δ
Q(
η
) =
δ
(z00/z). A i s
(linea ) o de he in–in exp ession o he wo-poin unc ion yields
δ
h k −ki=−2Imh k(
η
)2Z
η
−∞
d
η
0
δ
Q(
η
0) ∗
k(
η
0)2i,
and hence he ac ional change o he cu a u e spec um is (e alua ed a la e ime)
δ
P
ζ
(k)
P
ζ
(k)=−2Im k(
η
)2R
η
−∞d
η
0
δ
Q(
η
0) ∗
k(
η
0)2
| k(
η
)|2−2
δ
z(
η
)
z(
η
).
Fo modes whose ho izon c ossing ime domina es he in eg al his educes o he obus pa a-
me ic es ima e
δ
P
ζ
/P
ζ
∼∆e (
η
k), whe e ∆e (
η
)≡
ρ
∆/
ρ
o is he ac ional backg ound pe u ba-
ion induced by RSD. Wi h a Planckian ac i a ion h eshold Kc∼ℓ−4
P his gi es
δ
P
ζ
/P
ζ
∼10−19
o ep esen a i e H∼1014 GeV — comple ely negligible obse a ionally.
8 Holog aphic & quan um-in o ma ion link — igo ous de i a ion
hT(CFT)
ab (x)i=CdZbulk
dd+1YpG(Y)Kab
µν
(x;Y)∆
µν
(Y),
and hen show he commonly quo ed educ ion hT(CFT)
ab i∼∆
µν
n
µ
n
ν
as a con olled app oxima ion
o localized bulk memo y. All s eps a e explici and eady o pas e.
4
1.4 Echoes: Time Delay and Ampli ude Es ima e
I he RSD co e p oduces an inne pa ially- e lec ing su ace a ci cum e en ial adius 𝑟0(o an e ec i e
po en ial ba ie / ca i y), apped adia ion be ween he pho on sphe e and ha inne s uc u e will leak
ou in a sequence o echoes. The ime delay be ween successi e echoes is app oxima ely he ound-
ip ligh a el ime be ween he ou e sca e ing egion (pho on sphe e) and he inne e lec ing adius
measu ed in o oise coo dina e:
Δ𝑡echo ≃2𝑟∗(𝑟ph) −𝑟∗(𝑟0),(11)
wi h he o oise coo dina e:
𝑟∗(𝑟)=∫𝑑𝑟
𝑓(𝑟), 𝑓 (𝑟)=1−2𝐺𝑀
𝑟(Schwa zschild).(12)
Fo 𝑟nea he ho izon (𝑟→2𝐺𝑀), he o oise coo dina e beha es as 𝑟∗≃2𝐺𝑀 ln 𝑟
2𝐺𝑀 −1+cons .
I 𝑟0is jus inside he ho izon (i.e., 𝑟0=2𝐺𝑀(1+𝜖)wi h |𝜖|  1), one ob ains he commonly used
app oxima ion:
Δ𝑡echo ≃ −4𝐺𝑀 ln |𝜖| +(cons ),(13)
so ha small displacemen s om he ho izon p oduce a loga i hmically la ge ime delay.
Echo ampli ude. A c ude es ima e o he echo ampli ude 𝐴echo ela i e o he main ingdown am-
pli ude 𝐴RD is:
𝐴echo ∼ Rinne Tou e 𝑒−𝜏leak/𝜏damp ,(14)
whe e Rinne is he e lec i i y o he inne objec (se by he magni ude o he misma ch he e, and hence
by Δ), Tou e is he ansmission coe icien h ough he pho on-sphe e ba ie , and 𝜏leak is he ime o
leakage ou o he ca i y. In quan i a i e models, Rinne ∝Δ0 o small Δ, so he echo ampli ude scales
linea ly wi h Δ0.
De ec abili y o echoes depends on: (i) ampli ude 𝐴echo/𝐴RD, (ii) Δ𝑡echo ela i e o de ec o sensi-
i i y window, and (iii) cohe en s acking ac oss mul iple e en s (see §9.7).
1.5 Inspi al Phasing — How ΔModi ies Accumula ed Phase
Du ing inspi al, he GW phase 𝜙(𝑡)is de e mined by he ene gy balance equa ion:
¤
𝐸bind(𝑟)=−F∞(𝑟) − Fabs(𝑟),(15)
whe e 𝐸bind is he bina y binding ene gy, F∞is he lux adia ed o in ini y, and Fabs is he lux ab-
so bed/modi ied by he ho izons. I RSD modi ies he nea -ho izon abso p ion p ope ies (o he local
g a i a ional po en ial el by he bina y componen s), hen bo h 𝐸bind and Facqui e small co ec ions
𝛿𝐸 and 𝛿F. The i s -o de co ec ion o he GW phase accumula ed up o equency 𝑓is:
𝛿𝜙(𝑓)=−2𝜋∫𝑓𝛿𝐸0(𝑓0) −𝛿F(𝑓0)/(2𝜋)
¤
E0(𝑓0)𝑑𝑓 0,(16)
whe e ¤
E0is he unpe u bed ene gy- lux de i a i e and p imes deno e de i a i es w. . . equency. To
leading PN o de and o small, localized RSD co e ha does no each o bi al adii, 𝛿𝜙 is supp essed by
he smallness o Δa he o bi al adius, so i is gene ically iny. Howe e , e en iny 𝛿𝜙 can be measu able
i i accumula es cohe en ly o e Ncycles: he c i e ion o de ec ion o a phase pe u ba ion is oughly
|𝛿𝜙|≳1/𝜌(single e en ) o |𝛿𝜙|≳1/𝜌 o (s acked e en s), whe e 𝜌is he SNR.
3

1.6 Templa e Misma ch and De ec abili y — Rigo ous C i e ion
Fo wo wa e o ms ℎ(no-RSD) and ℎ+𝛿ℎ (wi h RSD), he noise-weigh ed inne p oduc is:
(ℎ1|ℎ2) ≡ 4 Re ∫∞
0
˜
ℎ1(𝑓)˜
ℎ∗
2(𝑓)
𝑆𝑛(𝑓)𝑑𝑓 , (17)
whe e 𝑆𝑛(𝑓)is he de ec o noise spec al densi y and ˜
ℎ(𝑓)a e Fou ie ans o ms. The (minimal) ma ch
is:
M ≡ max
𝑡0,𝜙0
(ℎ|ℎ+𝛿ℎ)
p(ℎ|ℎ)(ℎ+𝛿ℎ|ℎ+𝛿ℎ)
.(18)
Fo small pe u ba ions 𝛿ℎ, he leading misma ch is:
1−M ≃ 1
2h𝛿ℎ|𝛿ℎi
hℎ|ℎi≡1
2|𝛿ℎ|2
𝜌2,(19)
wi h 𝜌2≡ (ℎ|ℎ). A de ec able de ia ion a pai wise signi icance co esponds oughly o:
1−M ≳1
2𝜌2⇐⇒ |𝛿ℎ|≳1.(20)
Thus, he h eshold misma ch o de ec ion is abou 1/(2𝜌2). Fo a single LIGO-like e en wi h 𝜌∼20,
he de ec able misma ch is 1.25 ×10−3; o 𝜌∼50, i is 2×10−4. S acking 𝑁simila e en s (cohe en o
incohe en s acking s a egies) e ec i ely inc eases 𝜌by √𝑁(cohe en ) o yields an imp o emen ∝𝑁1/4
(semi-cohe en ), so he misma ch h eshold sh inks as 1/(2𝜌2
o ).
Connec ing |𝛿ℎ| o Δ.Fo small Δand linea esponse:
𝛿ℎ ∼ S[Δ],(21)
so |𝛿ℎ|2∝Δ2
0. Hence de ec ion equi es:
Δ0≳1
𝜌.(22)
This is consis en wi h he ea lie ule o humb 𝛿 𝑓 /𝑓∼Δ0and misma ch c i e ion (20).
1.7 O de -o -Magni ude Nume ical Es ima es & As ophysical Realism
Baseline (Planckian h eshold). Using he simple local ansa z Δ∼𝛼K/K𝑐wi h K𝑐∼ℓ−4
𝑃and 𝛼∼
𝑂(1), e alua e he dimensionless de o ma ion whe e wa es a e gene a ed (nea he pho on sphe e 𝑟ph ≈
3𝐺𝑀/𝑐2 o Schwa zschild). In geome ic uni s:
K(𝑟ph)=48𝑀2
(3𝑀)6=48
36𝑀−4≈0.0659𝑀−4.(23)
Compa ing o Planck cu a u e (ℓ−4
𝑃) gi es:
K
ℓ−4
𝑃∼0.066 ℓ𝑃
𝑀4
.(24)
Fo a sola -mass black hole (𝑀≈1.477 ×103m,ℓ𝑃≈1.616 ×10−35 m), his a io is:
K
ℓ−4
𝑃∼9.4×10−154.(25)
The e o e, o Planckian K𝑐, he local dimensionless de o ma ion a he pho on sphe e is:
Δ0∼𝛼K
K𝑐∼10−153 (sola mass).(26)
4
By (10)–(22), he co esponding ac ional QNM shi s and wa e o m misma ches a e ∼10−153, u e ly
unde ec able by any o eseeable GW de ec o .
Requi emen o de ec abili y. Se ing he de ec able ac ional e ec o ∼10−3(LIGO single-e en
h eshold) and using Δ0∼ K/K𝑐gi es he equi ed h eshold:
K𝑐≲103K(𝑟ph) ∼ 103·9.4×10−154ℓ−4
𝑃∼9.4×10−151ℓ−4
𝑃.(27)
In o he wo ds, o make RSD p oduce a ∼10−3 ac ional GW e ec wi h a sola -mass BH, he ac i a ion
cu a u e K𝑐would ha e o be smalle han Planck cu a u e by ∼10151 o de s o magni ude — physi-
cally ex emely implausible. The same conclusion holds o echo ampli udes: o ha e 𝐴echo/𝐴RD ∼10−3,
he co e e lec i i y (hence Δ0) mus be compa ably la ge, which again equi es K𝑐ℓ−4
𝑃.
Conclusion ( ealis ic as ophysics). Fo any model whe e K𝑐is Planckian, RSD e ec s on as o-
physical GWs a e e ec i ely ze o. Obse able GW signa u es equi e one o mo e o :
•K𝑐many (100) o de s o magni ude below Planck cu a u e ( heo y-challenging), o
• an ampli ica ion mechanism ha maps a iny local Δon o an 𝑂(1)e ec i e e lec i i y / nonlocal
obse able ( equi es a speci ic nonlocal ke nel Dwi h la ge spec al weigh nea obse a ional
equencies), o
• s acking an eno mous numbe o high-SNR e en s so ha 𝜌 o becomes as onomically la ge.
1.8 P ac ical Obse a ional S a egy (Recommended Analysis Pipeline)
1. Theo y → empla es: Compu e pa ame ic wa e o m modi ica ions 𝛿ℎ(𝑝;Δ0, 𝑟0, . . .) o a small
se o phenomenological RSD pa ame e s (ampli ude Δ0, co e adius 𝑟0, e lec i i y model R(𝜔),
ke nel leng hscale ℓ). P o ide bo h ime-domain and equency-domain models.
2. Ma ched- il e sea ch o QNM shi s: Use exis ing ingdown pipelines o pe o m Bayesian
pa ame e es ima ion including 𝛿 𝑓 , 𝛿𝜏 as ee pa ame e s and place uppe limi s on Δ0.
3. Echo sea ch wi h model p io s: Sea ch o echoes using he p edic ed ime delay amily (11) and
empla e amilies o echo wa ele s; use bo h cohe en s acking and model-agnos ic ime- equency
me hods o de ec low-ampli ude echoes.
4. Inspi al pa ame ic es s: Include low- equency modi ica ions o he phase (16) as ex a PN
coe icien s o as pa ame e ized pos -Eins einian (ppE) e ms; pe o m join in e ence on Δ0and
s anda d sou ce pa ame e s.
5. Popula ion s acking: Combine pos e io dis ibu ions om many e en s; uppe limi s scale oughly
as 1/√𝑁 o incohe en s acking and as 1/𝑁(op imis ic) o ully cohe en s acking o consis en
wa e o ms.
6. Null es s & sys ema ics: Accoun o wa e o m modeling sys ema ics, calib a ion unce ain y,
and unce ain as ophysical p io s; equi e consis en de ec ion ac oss de ec o s and independen
channels ( ingdown + inspi al).
Deli e able o obse e s: A small se o eady- o-use empla es and a p io ange o Δ0, 𝑟0,R(I
can p oduce hese nume ically o he ansä ze used in he pape ).
1.9 Limi a ions, Ca ea s, and Recommended Theo e ical Wo k
•No maliza ion o QNMs. Use o (9) equi es ca e ul QNM no maliza ion (Lea e / complex-
equency no maliza ion). Fo publica ion-g ade numbe s, one should compu e 𝛿𝜔 using con ou
me hods o nume ical eigen alue shi s a he han he nai e in eg al i 𝛿𝑉 is nonlocal o i wa e-
unc ions di e ge a he bounda ies.
5
•Nonlinea consis ency. I Δis no pa ame ically small, linea pe u ba ion b eaks down; hen
one mus sol e he ull linea ized equa ions on he sel -consis en backg ound 𝑔+Δ(nume ically)
a he han using i s -o de o mulae.
•Degene acies. QNM equency shi s may be degene a e wi h spin/mass es ima ion e o s; join
in e ence wi h inspi al pa ame e s educes alse posi i es.
•Model dependence. All es ima es depend on he chosen cons i u i e ela ion o Δand on he
ke nel D. Obse a ional uppe limi s should he e o e be epo ed bo h o speci ic model amilies
and as conse a i e, model-agnos ic bounds on Δ0in he egion nea he pho on sphe e.
1.10 Ready- o-Pas e Concluding Pa ag aph ( o You Pape )
G a i a ional-wa e signa u es (summa y). — Residual space ime de o ma ions Δ𝜇𝜈 imp in hem-
sel es on g a i a ional wa es by (i) shi ing quasi-no mal mode equencies, (ii) p oducing delayed echoes
i an inne e lec ing s uc u e o ca i y o ms, and (iii) p oducing iny inspi al phase/abso p ion co ec-
ions i he de o ma ion eaches o bi al scales. To i s o de , he complex equency shi is gi en o mally
by equa ion (9), echo delays by (11)–(13), and he de ec ion c i e ion is go e ned by he ma ch/misma ch
condi ion (19) wi h de ec abili y h eshold ∼1/(2𝜌2). Fo physically mo i a ed Planckian ac i a ion
h esholds K𝑐∼ℓ−4
𝑃, he esul ing dimensionless de o ma ion a as ophysical pho on-sphe e adii is
Δ0≲10−150 o s ella -mass black holes, ende ing GW signa u es e ec i ely ze o; obse a ionally
in e es ing e ec s he e o e equi e ei he (i) a non-Planckian ac i a ion scale K𝑐ℓ−4
𝑃o (ii) an ampli-
ica ion mechanism buil in o he nonlocal ke nel D. We ecommend p oducing a small se o pa ame ic
wa e o m empla es (Δ0, 𝑟0,R(𝜔)), implemen ing Bayesian sea ches o 𝛿 𝑓 , 𝛿𝜏 and echoes, and epo -
ing uppe limi s on Δ0as di ec , alsi iable cons ain s on he RSD pa ame e space.
2 Black-Hole Shadow Memo y
Summa y (in ui i e). In he Residual Space ime De o ma ion (RSD) pic u e, high-cu a u e quan um
luc ua ions lea e a pe manen coa se-g ained imp in Δ𝜇𝜈 on he classical me ic. Tha imp in modi ies
null geodesics nea he pho on sphe e and he e o e shi s (and possibly de o ms) he black-hole shadow
seen by a dis an obse e . The obse able ”shadow memo y” is he pe sis en change o he c i ical
impac pa ame e (and o he shadow bounda y shape) p oduced by Δ𝜇𝜈. The ollowing de i a ion gi es
he linea esponse o he shadow bounda y o a small Δ𝜇𝜈, explici diagnos ic o mulae, and o de -o -
magni ude cons ain s. (Backg ound and no a ion o RSD a e in he main ex and in he accompanying
RSD no e.)
2.1 Geome ic Se -Up and No a ion
Wo k in geome ic uni s (𝐺=𝑐=1) o he de i a ion and es o e SI uni s in nume ical es ima es when
needed. Conside a s a ic, sphe ically symme ic backg ound me ic (Schwa zschild o he unpe u bed
geome y):
𝑑𝑠2=−𝑓(𝑟)𝑑𝑡2+1
𝑔(𝑟)𝑑𝑟2+𝑟2𝑑Ω2,(28)
wi h he unpe u bed Schwa zschild unc ions 𝑓(𝑟)=𝑔(𝑟)=1−2𝑀/𝑟. The coa se-g ained, RSD-
modi ied geome y is:
˜𝑔𝜇𝜈 (𝑥)=𝑔𝜇𝜈 (𝑥) +Δ𝜇𝜈 (𝑥).(29)
Fo sphe ically symme ic esiduals, we may w i e ( o leading o de ):
˜𝑔𝑡𝑡 =−𝑓(𝑟) +𝛿 𝑓 (𝑟),˜𝑔𝑟𝑟 =1
𝑔(𝑟)+𝛿ℎ(𝑟).(30)
The leading shadow e ec is con olled by 𝛿 𝑓 (𝑟)(equi alen ly Δ𝑡𝑡 =−𝛿 𝑓 in ou sign con en ion).
6
Null geodesics wi h conse ed ene gy 𝐸and angula momen um 𝐿sa is y he adial equa ion:
𝑑𝑟
𝑑𝜆 2
+𝑉eff (𝑟)=𝐸2, 𝑉eff (𝑟) ≡ 𝐿2
𝑟2𝑓(𝑟).(31)
The c i ical (uns able ci cula ) pho on o bi adius 𝑟ph is de ined by:
𝑉eff (𝑟ph)=𝐸2, 𝑉0
eff (𝑟ph)=0=⇒𝑑
𝑑𝑟 𝑓(𝑟)
𝑟2𝑟ph
=0.(32)
The c i ical impac pa ame e (shadow adius in geome ic uni s) is:
𝑏𝑐≡𝐿
𝐸c i
=𝑟ph
p𝑓(𝑟ph).(33)
A dis an obse e a coo dina e dis ance 𝐷𝑀sees an angula shadow adius:
𝛼≃𝑏𝑐
𝐷(𝛼small).(34)
Thus, o compu e he shadow imp in , we need he linea esponse 𝛿𝑏𝑐induced by a small 𝛿 𝑓 (o
Δ𝑡𝑡 ).
2.2 Linea ized Calcula ion: Shi o he Pho on Sphe e and o he C i ical Impac Pa-
ame e
De ine he unc ion:
𝐹(𝑟) ≡ 𝑟 𝑓 0(𝑟) −2𝑓(𝑟).(35)
The pho on adius 𝑟ph is he oo o 𝐹(𝑟)=0. Unde a small pe u ba ion 𝑓→𝑓+𝛿 𝑓 , he pe u bed
equa ion becomes 𝐹(𝑟) + 𝛿𝐹(𝑟)=0. Expanding o i s o de abou he unpe u bed oo 𝑟ph yields:
𝛿𝑟ph =−𝛿𝐹(𝑟ph)
𝐹0(𝑟ph)=−𝑟ph𝛿 𝑓 0(𝑟ph) −2𝛿 𝑓 (𝑟ph)
𝑟ph 𝑓00(𝑟ph) − 𝑓0(𝑟ph).(36)
(He e p imes deno e 𝑑/𝑑𝑟.)
F om (33), we ob ain he ac ional change in he c i ical impac pa ame e (linea ized):
𝛿𝑏𝑐
𝑏𝑐
=𝛿𝑟ph
𝑟ph −1
2
𝛿 𝑓 (𝑟ph)
𝑓(𝑟ph).(37)
Combining (36) and (37) and w i ing 𝛿 𝑓 =−Δ𝑡𝑡 (since Δ𝑡𝑡 is he change o 𝑔𝑡𝑡 ) gi es he compac
linea esul :
𝛿𝑏𝑐
𝑏𝑐
=
Δ0
𝑡𝑡 (𝑟ph) − 2
𝑟ph
Δ𝑡𝑡 (𝑟ph)
𝑟ph 𝑓00(𝑟ph) − 𝑓0(𝑟ph)+1
2
Δ𝑡𝑡 (𝑟ph)
𝑓(𝑟ph).(38)
Equa ion (38) is he main p ac ical o mula: any model ha supplies Δ𝑡𝑡 (𝑟)can be e alua ed a 𝑟ph and
inse ed o ob ain he ac ional shi o he shadow adius.
Rema ks.
• Equa ion (38) is comple ely gene al o s a ic, sphe ically symme ic backg ounds and does no
ely on a pa icula choice o Δ. Fo non-sphe ically symme ic Δ𝜇𝜈, he same logic applies, bu
he shadow becomes angle-dependen , and one mus sol e he Hamil on–Jacobi equa ions o null
geodesics in he pe u bed me ic (see §4 below o a p ac ical expansion).
• The wo e ms on he igh o (38) ha e clea meanings: he i s e m encodes he adial g adien
o he me ic pe u ba ion (how he pho on sphe e mo es), he second e m is a local g a i a ional
edshi (e alua ed a he o iginal pho on sphe e).
7
2.3 E alua e (38) o Schwa zschild + Typical RSD Ansa z
Fo Schwa zschild ( 𝑓(𝑟)=1−2𝑀/𝑟), one has:
𝑟ph =3𝑀, 𝑓 (𝑟ph)=1
3, 𝑟 𝑓 00 −𝑓0𝑟ph
=−2
3𝑀.(39)
Using hese, (38) becomes:
𝛿𝑏𝑐
𝑏𝑐Schw
=−3𝑀
2Δ0
𝑡𝑡 (3𝑀) − 2
3𝑀Δ𝑡𝑡 (3𝑀)+3
2Δ𝑡𝑡 (3𝑀).(40)
This o mula is pa icula ly simple when he pe u ba ion inhe i s he adial scaling o cu a u e-
d i en ansä ze used in RSD models. A common phenomenological model (used ea lie in he pape )
is:
Δ𝜇𝜈 (𝑟) ∝ K(𝑟)
K𝑐
,wi h K(𝑟)=48𝑀2
𝑟6,(41)
so ha Δ𝑡𝑡 (𝑟) ∝ 𝑟−6. Fo a powe law Δ𝑡𝑡 (𝑟) ∝ 𝑟−𝑝, one inds Δ0
𝑡𝑡 =−𝑝Δ𝑡𝑡 /𝑟, hence:
Δ0
𝑡𝑡 −2
𝑟Δ𝑡𝑡 =−𝑝+2
𝑟Δ𝑡𝑡 .(42)
Pu ing 𝑝=6(RSD ∝K e schmann) in o (40) yields he simple es ima e:
𝛿𝑏𝑐
𝑏𝑐Schw,p=6≃4+3
2Δ𝑡𝑡 (3𝑀)=5.5Δ𝑡𝑡 (3𝑀).(43)
Thus, o he RSD (𝑝=6) model, he ac ional change o he shadow adius is an O(1) ac o imes he
local dimensionless me ic de o ma ion Δ𝑡𝑡 e alua ed a he pho on sphe e. No ine unings a e hidden:
he shadow ela i e change is linea in he local de o ma ion.
2.4 Angle-Dependen (Shape) De o ma ions and Mul ipole Expansion
Fo non-sphe ical Δ𝜇𝜈 (𝜃, 𝜑)(e.g., aniso opic sa u a ion a e an asymme ic high-cu a u e e en ), he
c i ical impac pa ame e becomes angle-dependen , 𝑏𝑐(𝜑), and he shadow bounda y in he obse e ’s
sky can be pa ame e ized by:
𝑏𝑐(𝜑)=𝑏0"1+Õ
𝑛≥1𝑎𝑛cos 𝑛𝜑 +𝑏𝑛sin 𝑛𝜑#.(44)
To linea o de , he mul ipole coe icien s a e linea unc ionals o he me ic pe u ba ion:
𝑎𝑛, 𝑏𝑛=∫VK𝜇𝜈
𝑛(𝑟, 𝜃)Δ𝜇𝜈 (𝑟, 𝜃)𝑑3𝑥, (45)
whe e K𝑛is a compu able bulk- o-bounda y ke nel de e mined by he unpe u bed pho on geodesic con-
g uence (explici exp essions ollow om in eg a ing he pe u bed Hamil on–Jacobi equa ions; he ke -
nel is peaked nea he pho on shell). Fo small, localized aniso opic RSD, he lowes nonze o mul ipole
(𝑛=1) (dipole) con ols he cen oid o se and (𝑛=2) he ellip ici y; hese a e di ec ly cons ained by
image model i s (see §6).
8

2.5 Nume ical Examples — Requi ed Ampli ude o De ec abili y and RSD P edic ions
Key scaling. F om (38)–(43), he ac ional shadow shi scales oughly as:
𝛿𝛼
𝛼≃𝛿𝑏𝑐
𝑏𝑐∼𝐶Δ0,(wi h 𝐶=𝑂(1–10)depending on p o ile),(46)
whe e Δ0deno es he dimensionless ampli ude o Δin he pho on-sphe e egion.
Obse ed shadow scales (M87* example). Fo a dis an obse e , he angula shadow adius o
Schwa zschild is:
𝛼=𝑏𝑐
𝐷=3√3𝑀geom
𝐷,(47)
whe e 𝑀geom =𝐺𝑀/𝑐2and 𝐷is he sou ce dis ance. Fo he pa ame e s used by EHT:
𝑀M87 ≃6.5×109𝑀, 𝐷M87 ≃16.8Mpc,(48)
one inds (nume ically):
𝛼M87 ≃19.84 𝜇as (angula adius) ⇒diame e ≃39.7𝜇as,(49)
consis en wi h he EHT esul . A ac ional change 𝛿𝛼/𝛼maps o an angula change 𝛿𝛼 as:
𝛿𝛼(𝜇as) ≃ 19.84Δ0.(50)
De ec abili y c i e ion (EHT). The EHT quo ed unce ain y on he M87 diame e is ∼3𝜇as (o de
7
Δ0≳𝛿𝛼
𝛼min ∼3/2
19.84 ∼0.075,(51)
i.e., Δ0∼ O(10−1)is needed o p oduce a ∼3𝜇as adius change o M87*. E en o an op imis ic
de ec abili y a he ∼1𝜇as le el, one needs Δ0∼5×10−2.This se s he obse a ional ampli ude scale
equi ed o di ec shadow de ec ion.
RSD p edic ion (Planckian h eshold). Using he simple RSD scaling ansa z Δ∼𝛼dim(K/K𝑐)
wi h 𝛼dim ∼𝑂(1)and K𝑐∼ℓ−4
𝑃(Planck cu a u e), he local dimensionless de o ma ion a he pho on
sphe e is:
Δ0∼K(𝑟ph)
ℓ−4
𝑃
=48𝑀2
(3𝑀)6ℓ4
𝑃=48
729 𝑀−4ℓ4
𝑃.(52)
E alua e his o M87* (mass in Planck uni s 𝑀M87 ≃5.94 ×1047𝑚𝑃):
•K(𝑟ph) ≃ 5.29 ×10−193 (Planck uni s),
• hence Δ0M87 ≃5.3×10−193.
Fo Sg A* (mass ∼4.3×106𝑀), one inds Δ0Sg A ≃2.8×10−180.
Conclusion om numbe s. The RSD ampli ude p edic ed by a Planckian ac i a ion h eshold is
as onomically iny (Δ0≲10−180–193) o as ophysical black holes, so he induced shadow shi :
𝛿𝛼
𝛼∼𝐶Δ0(53)
is comple ely negligible (many o de s o magni ude below EHT sensi i i y). Con e ing he EHT de-
ec abili y c i e ion o a cons ain on he ac i a ion cu a u e:
K𝑐≲K(𝑟ph)
Δobs
,(54)
gi es nume ically o M87* and Δobs ∼0.07:
K𝑐≲5×10−191ℓ−4
𝑃,(55)
i.e., he ac i a ion cu a u e would need o be ∼10191 imes smalle han he Planck cu a u e o p oduce
a po en ially obse able 7
9
2.6 P ac ical Obse a ional Recipe & In e se P oblem
1. Model building. Choose a small se o phenomenological RSD models deli e ing Δ𝑡𝑡 (𝑟, 𝜃)(ex-
amples: iso opic sa u a ion ∝ K/K𝑐, causal ke nel models, o localized shell imp in s).
2. Compu e ke nel in eg als. Use eq. (38) (sphe ical) o he Hamil on–Jacobi linea ized ke nel
(axisymme ic / Ke ) o compu e 𝑏𝑐(𝜑). P o ide empla es 𝐼(𝜑;pa ams)gi ing he bounda y
cu e.
3. Image modelling. Fi EHT isibili y da a wi h he amily o blu ed ing empla es including
he RSD pe u ba ion pa ame e s (Δ0, 𝑟0,aniso opy coe icien s). The usual EHT model- i ing
machine y ( ing diame e , wid h, asymme y, cen oid o se ) can be epu posed o bound Δ0.
4. S acking / mul i-epoch. Because RSD is a pe sis en geome ic memo y, he same sign and mag-
ni ude should appea ac oss epochs (unless subsequen high-cu a u e e en s change he imp in ).
Cohe en mul i-epoch cons ain s s eng hen uppe limi s on Δ0.
5. Degene acies con ol. Join ly i o he space ime pa ame e s (mass, spin, inclina ion, plasma
sca e ing) and Δpa ame e s; pe o m Bayesian model selec ion o a oid alse de ec ions due o
sca e ing / plasma sys ema ic e o s.
2.7 In e p e a ion and Implica ions
•I a signi ican , pe sis en (non-plasma) shadow o se o shape de o ma ion is obus ly de-
ec ed and shown o be inconsis en wi h as ophysical/plasma explana ions, RSD p o ides one
concei able geome ic o igin: an aniso opic Δ𝜇𝜈 in he pho on-shell egion. The claimed de-
ec ion would hen imply an unexpec edly small ac i a ion cu a u e K𝑐o a non i ial nonlocal
ampli ica ion ke nel D.
•I no such de o ma ion is ound (cu en si ua ion), cu en EHT da a place a di ec uppe bound
o o de Δ0≲10−1(o de o ens o pe cen ) o M87*. T ansla ing o RSD pa ame e space
excludes only unna u ally la ge de ia ions om Planck scale; in pa icula , he Planckian h eshold
RSD p edic ion is al eady a below he obse a ional loo .
•S onge cons ain s will be possible wi h nex -gene a ion mm-VLBI (imp o ed baseline co -
e age and sensi i i y), and wi h join ing+pola iza ion+ a iabili y analyses: hese can push he
de ec able Δ0down by o de s o magni ude, bu eaching he Planckian RSD p edic ion would e-
qui e physically implausible de ec o pe o mance o heo y changes ha ampli y memo y e ec s
d ama ically.
2.8 Ready- o-Pas e Concluding Pa ag aph ( o he Pape )
Black-hole shadow memo y. — The RSD amewo k p edic s ha pe sis en me ic imp in s Δ𝜇𝜈 p o-
duced by high-cu a u e quan um luc ua ions modi y he uns able pho on o bi and he e o e he ob-
se ed black-hole shadow. Fo s a ic, sphe ically symme ic backg ounds, he ac ional change o he
c i ical impac pa ame e is gi en in closed o m by eq. (38) and o Schwa zschild educes o (40). Fo
cu a u e-d i en RSD models ha scale wi h he K e schmann scala , he shadow ac ional shi is o
o de a small nume ical ac o imes he local dimensionless de o ma ion Δ0a he pho on sphe e (eq.
(43)). Nume ical e alua ion shows ha wi h a Planckian ac i a ion h eshold K𝑐∼ℓ−4
𝑃, he expec ed
de o ma ion o as ophysical black holes is anishingly small (Δ0≲10−180–10−193), so cu en EHT
cons ain s do no p obe minimal RSD. Obse a ional de ec ion would equi e ei he a non-Planckian
ac i a ion scale o an ampli ica ion mechanism in he nonlocal ke nel; con e sely, imp o ed shadow
measu emen s will place di ec , model-dependen uppe limi s on K𝑐and he ke nel D.
10
Residual Space ime De o ma ion (RSD) and CMB Signa u es
1 Summa y (one-pa ag aph)
RSD p oduces a de e minis ic esidual de o ma ion ∆
µν
(x)o he coa se me ic a e in eg a ing
ou high-cu a u e con igu a ions. I RSD is ac i e du ing (o p io o) he gene a ion o
p imo dial pe u ba ions, i modi ies he p imo dial cu a u e wo-poin unc ion P
ζ
(k)and
he eby he obse ed CMB angula co ela ions CXY
ℓ( empe a u e/pola iza ion). The leading
e ec s a e (i) an iso opic ac ional escaling
δ
P
ζ
/P
ζ
(a nea ly scale-independen ampli ude shi
i ∆modi ies backg ound H); (ii) scale-dependen ea u es (oscilla ions, s eps) i RSD ac i a ion
is localized in ime; (iii) s a is ical aniso opy / o -diagonal co ela ions i ∆is aniso opic;
and (i ) highe -o de co ela ions (bispec um, ispec um) i non-Gaussian componen s o he
in luence unc ional a e impo an . Below I de i e hese esul s om i s p inciples, gi e closed
o mulae mapping ∆→
δ
Cℓ, p o ide oy analy ic models (including a sha p ac i a ion model ha
p oduces oscilla o y ea u es), de i e de ec ion- h eshold o mulae (cosmic- a iance limi ed),
and inally gi e ealis ic nume ical es ima es showing ha a Planckian ac i a ion h eshold
p oduces u e ly negligible CMB signals (bu also show how o u n da a in o cons ain s on
RSD pa ame e s).
2 Con en ions and basic mapping (P
ζ
(k)→Cℓ)
Fou ie con en ion
ζ
(x) = ∫d3k
(2
π
)3
ζ
(k)eik·x,h
ζ
(k)
ζ
(k0)i= (2
π
)3
δ
(3)(k+k0)P
ζ
(k).
Angula powe spec a ( empe a u e/pola iza ion) a e ob ained by he usual line-o -sigh
ans e unc ion ∆X
ℓ(k)(X=T,E,B):
CXY
ℓ=4
π
∫∞
0
dk
kP
ζ
(k)∆X
ℓ(k)∆Y
ℓ(k).(1)
A small change
δ
P
ζ
(k)p oduces
δ
CXY
ℓ=4
π
∫∞
0
dk
k
δ
P
ζ
(k)∆X
ℓ(k)∆Y
ℓ(k).(2)
I he ac ional change is scale independen o e he window ha domina es ℓ(i.e.,
δ
P
ζ
/P
ζ
≡
cons ), hen
δ
CXY
ℓ
CXY
ℓ≃
δ
P
ζ
P
ζ
.(3)
(Use (2) and (1) and he cons ancy o pull
δ
P
ζ
/P
ζ
ou side he in eg al.)
Thus he cen al heo e ical ask is o compu e
δ
P
ζ
(k)/P
ζ
(k)p oduced by RSD.
1
3 Leading-o de o mula o
δ
P
ζ
/P
ζ
om RSD (in–in linea esponse)
S a ing om he Mukhano –Sasaki esul (see §A in he pape ), he i s -o de (in he RSD-
induced pe u ba ion o he Mukhano –Sasaki po en ial
δ
Q(
η
)≡
δ
(z00/z)) in–in exp ession is
δ
h k(
η
) −k(
η
)i=−2Im[ k(
η
)2∫
η
−∞
d
η
0
δ
Q(
η
0) ∗
k(
η
0)2],
wi h k≡z
ζ
k. E alua ing a la e ime
η
and di iding by he unpe u bed powe yields (see
ea lie de i a ion)
δ
P
ζ
(k)
P
ζ
(k)=−2
Im[ k(
η
)2∫
η
−∞d
η
0
δ
Q(
η
0) ∗
k(
η
0)2]
| k(
η
)|2−2
δ
z(
η
)
z(
η
).(4)
In e p e a ion / app oxima ions.
• The i s e m is he di ec e ec o he pe u bed MS po en ial du ing mode e olu ion;
i is domina ed by he epoch nea ho izon c ossing |k
η
|∼1.
• The second e m accoun s o he change in he no maliza ion z(
η
)a la e ime (back-
g ound shi ). In mos slow- oll scena ios bo h e ms a e o o de he dimensionless
backg ound ac ional pe u ba ion ∆e ≡
ρ
∆/
ρ
o p oduced by RSD. Hence, gene ically
δ
P
ζ
P
ζ
(k)∼O(∆e (
η
k)),(5)
whe e
η
kis he ho izon-c ossing ime o mode k.
4 Two oy models (analy ic) o
δ
Q(
η
)and he esul ing
δ
P/P
Below I gi e wo analy ic oy cases ha a e di ec ly usable in a pape : (A) slow, scale–
independen backg ound shi ; (B) sha p, ime-localized ac i a ion (p oduces oscilla o y ea-
u es).
4.1 Model A — adiaba ic (nea ly scale–independen ) shi
I RSD ac s as an app oxima ely cons an ac ional backg ound ene gy densi y du ing he ele-
an epoch, hen
δ
Q/a2H2∼∆e =cons . Inse in o (4) and use ha he in eg al is domina ed
a ound ho izon c ossing; one ob ains he simple scaling ( o slow oll)
δ
P
ζ
(k)
P
ζ
(k)≃Ck∆e ≈O(∆e ),(6)
wi h Ckan O(1)complex ke nel (depends weakly on slow- oll pa ame e s). Fo a scale-
independen ∆e his leads di ec ly o he esul (3) o
δ
Cℓ/Cℓ.
Ready- o-pas e s a emen : “I RSD p oduces a slow, app oxima ely cons an ac ional
backg ound co ec ion ∆e du ing ho izon c ossing, hen he CMB powe spec a a e escaled
by
δ
Cℓ/Cℓ≈∆e o leading o de .”
2
– De i e a Källén–Lehmann spec al ep esen a ion o he nonlocal ke nels; impose
posi i i y cons ain s on he spec al measu e.
– P o ide sufficien condi ions (e.g., Dis e a ded and i s Fou ie ans o m ˜
D(
ω
,k)
has no ze os o Im
ω
≥0) gua an eeing linea s abili y.
• Deli e able. A compac lemma gi ing spec al condi ions and hei physical in e p e a-
ion.
13.6 Non–Gaussian co ec ions (highe cumulan s)
• Objec i e. Compu e he hi d and ou h cumulan s o he high–cu a u e sec o and
quan i y co ec ions o ∆and o T(∆)beyond Gaussian o de .
• Me hod. Diag amma ic cumulan expansion (in–in o malism) o saddle–poin plus loop
co ec ions; de i e scaling o co ec ions and es ima e he egime o alidi y o he Gaussian
unca ion.
• Pape i em. An appendix calcula ing he i s non i ial cubic co ec ion and gi ing an
uppe bound on i s e ec ela i e o he Gaussian esul .
13.7 RG– low o he ac i a ion h eshold Kc(ℓ)
• P oblem. Make he heu is ic scaling Kc(ℓ)∼ℓ−
α
p ecise by de i ing
α
=4+
γ
(anomalous
dimension
γ
) om an FRG o ope a o p oduc expansion.
• Plan. Use unc ional RG echniques (We e ich equa ion) o composi e ope a o s, o
pe u ba i e RG in a backg ound ield me hod, o compu e anomalous dimensions o
cu a u e composi es.
13.8 Holog aphic ealiza ion and ke nel compu a ion
• Conc e e ask. In asymp o ically AdS se ups compu e he g a i on bulk→bounda y ke nel
Kab
µν
(x;Y)explici ly (linea ized Eins ein equa ions, known p opaga o ) and ela e ∆ o
hTabiCFT wi h p ecise p e ac o s. P o ide he explici AdSd+1exp ession o p ac ical d.
• Deli e able. A wo king example in AdS5(o AdS4) wi h a localized ∆and he esul ing
bounda y s ess change.
13.9 Nume ical implemen a ion and asymp o ic ma ching
• P ac ical p og am.
– Implemen spec al/PDE sol e s o he in eg odi e en ial sel –consis ency equa ion
in sphe ical symme y (use spec al adial basis, New on–K ylo i e a ions).
– Tes con e gence and s abili y; pe o m pa ame e scans in (Kc,Λe ).
– In eg a e wi h nume ical ela i i y codes o collapse/me ge wi h a phenomenolog-
ical T(∆)inse ed.
• Deli e able. A nume ical appendix plus open–sou ce code package ( eady o ep oducibil-
i y).
9

13.10 Obse a ional in e se p oblems and pa ame e es ima ion
• Task. Build pipelines ha map obse a ional cons ain s (QNM shi s, echo ampli udes,
shadow mul ipoles,
δ
P/P) in o pos e io bounds on RSD pa ame e s. Use Bayesian in e -
ence wi h o wa d models gene a ed om he abo e ma hema ical cons uc ion.
• Deli e able. A companion da a-analysis no ebook and a likelihood module o exis ing
GW / EHT / CMB codes.
14 Sugges ed heo em / p oposi ion (example o include in he pape )
P oposi ion (local exis ence o ∆ o small sou ce). Le gbe a smoo h backg ound me ic on
a globally hype bolic mani old M. Suppose he high–mode p opaga o D[g]de ines a bounded
linea map L2(M)→L2(M)and J[g]∈L2(M). Then he e exis s
ε
>0such ha i |J|L2<
ε
he
nonlinea ixed–poin map T[∆]:=−D[g+∆]J[g+∆]is a con ac ion in a small ball o L2(M)and
he e o e admi s a unique solu ion ∆∈L2(M). Ske ch o p oo . Expand D[g+∆] = D[g]+O(∆);
bound |T[∆1]−T[∆2]|≤C|∆1−∆2|wi h C<1 o small |J|, apply Banach ixed–poin . (De ails:
es ima e nonlinea emainde using ope a o no ms and Sobole embeddings; use causali y o
con ol suppo s.)
Including a s a emen like his — wi h he igo ous hypo heses and clea e cons an s —
s eng hens he ma hema ical c edibili y o RSD.
15 P ac ical “nex s eps” o inish a 10/10 pape (copy-pas e checklis )
1. Appendix A: Hea –ke nel de i a ion o Si o wo o de s (w i e ou a0,a1,a2and ini e
emainde ).
2. Appendix B: Exis ence/uniqueness p oposi ion o ∆( ull p oo ske ch, unc ion spaces,
cons an s).
3. Appendix C: Linea ized ke nel compu a ion o Schwa zschild (explici
δ
V[∆] o Regge–
Wheele / Ze illi).
4. Nume ical supplemen : elease code sol ing he sel –consis ency ODE/PDE o sphe ical
co e and compu ing obse able quan i ies.
5. Da a appendix: p oduce o ecas ed cons ain s / uppe limi s on Kc,Λe om LIGO/EHT/Planck
using he pipeline desc ibed.
16 Closing pa ag aph
Conclusion. Residual Space ime De o ma ion (RSD) is a minimal, geome ic mechanism by
which quan um/high–cu a u e sec o s can in luence classical g a i y: ins ead o in oducing
new low–ene gy ields, high–cu a u e modes lea e a de e minis ic geome ic memo y ∆
µν
en-
coded by an in luence ac ion Si [g]. Ma hema ically his leads o a ac able se o p oblems—
co a ian cons uc ion o Si , exis ence and s abili y o sel –consis en ∆, spec al condi ions
ensu ing causali y and ghos – eedom, and he explici compu a ion o ke nels ha map bulk
memo y o bounda y obse ables. Physically i yields alsi iable p edic ions (co e adii, QNM
shi s, shadow de o ma ions, CMB empla es) ha educe he UV ambigui y o a small, es able
pa ame e space (Kc,D,Λe ). The p og am we ha e ou lined — igo ous analy ic esul s, con-
olled nume ics, and di ec da a cons ain s — p o ides a conc e e and balanced pa h o ele a e
RSD om an appealing idea o a ully quan i ied b idge be ween gene al ela i i y and quan um
g a i y.
10