The Spacetime Resonance Theorem and the Origin of Dark Mass

The Space ime Resonance Theo em and
he O igin o Da k Mass
B yan Hennelly ∗1,2,3
1Depa men o Elec onic Enginee ing, Maynoo h Uni e si y,
Maynoo h, Co. Kilda e, I eland
2Depa men o Compu e Science, Maynoo h Uni e si y,
Maynoo h, Co. Kilda e, I eland
3The Hamil on Ins i u e, Maynoo h Uni e si y, Maynoo h, Co.
Kilda e, I eland
19 No embe 2025
Abs ac
We show ha he ull e a ded G een– unc ion s uc u e o linea ised
gene al ela i i y, when combined wi h he o dina y damped–oscilla o
suscep ibili y o ba yonic ma e , yields an e ec i e–medium desc ip ion
o he weak– ield g a i a ional esponse. Cohe en , ime– a ying ba y-
onic mo ion d i es a collec i e cu a u e ield whose delayed esca e ing
p oduces a small bu pe sis en cycle–a e aged componen . This s o ed
cu a u e ene gy appea s as an e ec i e da k–mass densi y and ep o-
duces he obse ed da k–mass phenomenology. No new ields, pa icles,
o modi ica ions o GR a e in oduced: all dynamics a ise om s anda d
linea ised g a i y once he usual quasis a ic unca ion o he e a ded
ke nel is a oided. The esul ing causal– esonan amewo k ep oduces
he e ec i e mass dis ibu ions o spi al galaxies, ellip icals, clus e s, and
ilamen s; yields na u al “local” and “homogeneous” silence; and emains
ully consis en wi h Sola –Sys em es s and wi h he CMB. In his pic u e,
da k–mass beha iou eme ges as cu a u e ene gy s o ed by he collec i e
esonan esponse o ba yonic s uc u e.
∗Email: [email p o ec ed]
1
Con en s
1 In oduc ion 5
2 Space ime Resonance Theo em 6
2.1 Cons i u ion o he E ec i e G a i a ional Medium . . . . . . . . 6
2.2
Rela ion o Known G a i a ional Oscilla ion Modes and P eceden s
10
3 Resonan Field Dynamics 11
3.1 ChannelSepa a ion.......................... 11
3.2 Space ime as an E ec i e Re e sible Medium . . . . . . . . . . . 12
3.3 Collec i e Mode Dynamics and he Masson . . . . . . . . . . . . . 13
3.4 Linea Spa ial Pe u ba ions and he P opaga ion Law . . . . . . 14
4 G a i a ional Sca e ing Theo y 16
4.1 Single–Sca e e Response . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Mul iple Sca e ing and he Foldy–Lax Hie a chy . . . . . . . . . 17
4.3 E ec i e–Medium and Ensemble A e aging . . . . . . . . . . . . . 19
4.4 T anspo and Di usion Limi s . . . . . . . . . . . . . . . . . . . 20
4.5 Uni ied In e p e a ion . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Spa io empo al Feedback and Resonan Cohe ence 22
5.1 Cons i uen En i ies & S a es o he Feedback Sys em . . . . . . . 22
5.2 Feedback Channels and In e ac ions . . . . . . . . . . . . . . . . . 24
5.3 Tempo al Cohe ence and Mu ual Suscep ibili y . . . . . . . . . . 27
5.3.1 Phase Locking and Mu ual Suscep ibili y . . . . . . . . . . 27
5.3.2 Global F equency Selec ion in Inhomogeneous Discs . . . . 29
5.3.3 Pola isa ion o Tempo al Cohe ence . . . . . . . . . . . . . 30
6 Spa ial Cohe ence o he Resonan Channel 31
6.1 Ki chho –Somme eld Rep esen a ion . . . . . . . . . . . . . . . . 32
6.2 E ec i e Po en ial and Ene gy S o age in he Resonan Medium . 34
6.3 Single Poin Sou ce in a Resonan Channel–1 Backg ound . . . . 35
6.4 In e e ence o Two Poin Sou ces . . . . . . . . . . . . . . . . . . 37
6.5 Tempo al Memo y and he S eady–S a e Reduc ion . . . . . . . . 42
7 S eady–S a e (A ac o ) Solu ions 43
7.1 Spi al Galaxies as Cohe en m= 1
(Lague e–Gaussian) Modes . . . . . . . . . . . . . . . . . . . . . 44
7.1.1 Obse ing he S eady–S a e Solu ion . . . . . . . . . . . . 44
7.1.2 The Fi s Momen . . . . . . . . . . . . . . . . . . . . . . 47
7.1.3
The Thin–Disk Model: Sol ing he 2D Helmhol z Equa ion
Re eals heHalo ....................... 49
7.1.4 Ex ending he Thin–Disk o Th ee Dimensions . . . . . . . 51
7.1.5 Gain om Incohe en Foldy–Lax Sca e ing . . . . . . . . 52
2
7.1.6
S o ed Ene gy, Da k–Mass Equi alence, and En i onmen-
alGain ............................ 53
7.1.7 Nume ical Resul s: 3D Thin–Disk Model . . . . . . . . . . 54
7.1.8 Physical In e p e a ion . . . . . . . . . . . . . . . . . . . . 58
7.2 Ellip icals as Random Resonan Media . . . . . . . . . . . . . . . 62
7.2.1 Obse ing he S a is ical S eady S a e . . . . . . . . . . . 62
7.2.2 Complex Ba yonic Fo cing wi h Memo y . . . . . . . . . . 64
7.2.3
Fou –dimensional speckle geome y o he g a i a ional ield
66
7.2.4 Ensemble educ ion: masson–speckle summa ion . . . . . . 67
7.2.5 T anspo Mean F ee Pa h and Collec i e Damping . . . . 70
7.2.6 Co e Regula iza ion in he Non–T anspo Regime . . . . 71
7.2.7
G adien –Ene gy Boos and Final T anspo –Regula ized
HaloLaw ........................... 73
7.2.8 Nume ical Resul s . . . . . . . . . . . . . . . . . . . . . . 74
7.2.9 Halo Fo ma ion om Cohe en Ene gy Flux . . . . . . . . 77
7.2.10 Physical In e p e a ion . . . . . . . . . . . . . . . . . . . . 78
7.3 Clus e s as Radia i e T anspo Ex ensions o Ellip icals . . . . . 81
7.3.1 Obse ing he S a is ical T anspo Equilib ium . . . . . . 81
7.3.2
Ma hema ical Fo mula ion: O e lapping Ellip ical T ans-
po En elopes ........................ 84
7.3.3 Nume ical Resul s . . . . . . . . . . . . . . . . . . . . . . 89
7.3.4 Physical In e p e a ion . . . . . . . . . . . . . . . . . . . . 92
8 Local Silence and he P ese a ion o Classical G a i y 94
9 P edic ion: Absence o Gain in Homogeneous Sys ems 96
9.1 Th ee Dis inc Regimes o Homogeneous Clouds . . . . . . . . . 96
9.2 Theo em: Homogeneous Media P oduce No Resonan Gain . . . . 97
9.3 In e p e a ion and Connec ion o Pe u ba ion Theo y . . . . . . 98
10
Eme gence o Mul iple Sca e ing om a Weakly Inhomogeneous
Con inuous Medium 98
10.1 A B ie Re iew o Ga i a ional Mul iple Sca e Theo y . . . . . 98
10.2 The Resonan F equency in a Con inuous Medium . . . . . . . . . 100
10.3 Weak Random Po en ial and he Ballis ic Regime . . . . . . . . . 100
10.4 Nonlinea G ow h Towa d he T anspo Th eshold . . . . . . . . 101
10.5 Eme gen Resonan Pa ches and E ec i e Disc e eness . . . . . . 101
10.6 Linea Limi and Connec ion o he CMB Powe Spec um . . . . 102
11 Conclusion 104
12 Dedica ion and Acknowledgemen 104
A Appendix: E ec i e–Medium De i a ion 105
3
B
Appendix: Bessel and Hankel Func ions Used in he Main Tex
107
B.1 Cylind ical Bessel and Hankel Func ions . . . . . . . . . . . . . . 107
B.2 Sphe ical Bessel Func ions . . . . . . . . . . . . . . . . . . . . . . 108
B.3 Physical In e p e a ion . . . . . . . . . . . . . . . . . . . . . . . . 108
C
Appendix: Code o Simula ing he 3D Thin–Disk Resonan
Halo 109
D
Appendix: Code o Simula ing Ellip ical Galaxies - T anspo
Theo y 118
E Appendix: Code o Simula ing Clus e Galaxies 128
4
1 In oduc ion
The empi ical success o New onian and Eins einian g a i y [
1
,
2
] on labo a o y,
Sola –Sys em, and s ella scales is beyond dispu e. Ye ac oss galaxies, clus e s,
and cosmological en i onmen s, he obse ed g a i a ional ield exceeds he
esponse p edic ed om ba yons alone. Ra he han pos ula ing a new pa icula e
componen , we e isi a well–known bu a ely exploi ed aspec o gene al
ela i i y: he ull e a ded G een– unc ion s uc u e o he linea ised ield.
In he weak– ield limi , cu a u e esponds h ough he causal ke nel
G e
, whose
suppo ex ends h oughou he pas ligh cone. These in e io con ibu ions— he
“ ails”—a e genuine ea u es o GR bu a e no mally disca ded when he ield
is app oxima ed by i s ins an aneous Poisson limi . Res o ing he ull e a ded
esponse exposes a small bu physically meaning ul e ec : cu a u e eac s o
ime– a ying ma e h ough a ini e–memo y, dispe si e ke nel al eady p esen
in linea ised GR once no quasis a ic unca ion is applied.
The cen al idea o his wo k is o desc ibe how ba yonic ma e in e ac s wi h
his causal ke nel. Each g a i a ing elemen esponds o pe iodic cu a u e
pe u ba ions as a damped g a i a ional oscilla o wi h a equency–dependen
linea suscep ibili y
χ⋆
(
ω
). When many such elemen s pa icipa e, hei sus-
cep ibili y–con olled delayed e-emission o cu a u e pe u ba ions p oduces a
g a i a ional analogue o classical mul iple sca e ing. Thus he e ec i e esponse
o an ex ended ba yonic sys em does no a ise om modi ying GR, bu om he
in e ac ion be ween (i) he e a ded G een unc ion o linea ised GR and (ii) he
damped-oscilla o suscep ibili y o o dina y ba yonic ma e .
F om hese wo ing edien s an e ec i e medium eme ges. Repea ed esca e ing
o weak cu a u e pe u ba ions eno malises he p opaga o in he s anda d
Dyson sense, p oducing a collec i e ield wi h a cha ac e is ic es o ing equency
Ω, damping a e Γ, and e ec i e p opaga ion speed
ce
. Mo i a ed by well–known
g a i a ional oscilla o s—s ella pulsa ion modes, Jeans modes, disk b ea hing
modes—i is na u al o ake Ω
2∝G¯ρb
, whe e
¯ρb
is a sui able coa se–g ained
ba yonic densi y. This iden i ica ion is no imposed by he ield equa ion; i
is a physically well–mo i a ed co espondence wi h es ablished g a i a ional
phenomena.
The esul ing collec i e cu a u e esponse, Φ
es
, is oscilla o y and dynamically
sus ained. Because he e a ded ke nel ca ies ini e memo y, his oscilla o y
ield does no cancel pe ec ly om cycle o cycle. I s cycle–a e aged cu a-
u e ene gy accumula es ex emely slowly, de ining an e ec i e mass densi y
ρe ∝ ⟨|
Φ
es|2⟩
. This s o ed componen con ibu es o he usual Poisson equa ion
while he oscilla o y pa emains locally silen . Thus he heo y p ese es all
labo a o y, Sola –Sys em, and s ella es s o g a i y while allowing ex ended sys-
ems—spi als, ellip icals, and clus e s— o acqui e subs an ial e ec i e da k–mass
en elopes.
S uc u ed ba yonic sys ems also ac as g a i a ional sca e e s. Cohe en mo ion
5

in o a ing disks, o bi ing co es, and clus e ed assemblies gene a es cons uc i e
esca e ing o cu a u e pe u ba ions. Depending on spa ial con as and
cohe ence scale, his yields wo obse a ionally dis inc egimes: la ge–scale
cohe en modes in spi als, and di usi e e ec i e–medium halos in ellip icals and
clus e s. In con as , homogeneous o weakly pe u bed en i onmen s possess
no spa ial con as and he e o e no esca e ing: hey emain esonan ly silen .
This homogeneous silence explains why he ea ly Uni e se and he CMB ollow
s anda d cosmology: no con as means no esonan channel.
The p esen wo k de elops his causal– esonan pic u e om i s p inciples.
Beginning wi h he e a ded solu ion o linea ised GR and he linea suscep ibil-
i y o ba yonic ma e , we de i e a sel –consis en e ec i e medium capable o
desc ibing spi al galaxies, ellip icals, clus e s, and cosmological s uc u e wi hou
new ields o non–ba yonic pa icles. The esul ing Space ime Resonance Theo em
o malises he eme gence o he eno malised esonan mode and p o ides he mi-
c oscopic ounda ion o he s o ed cu a u e ene gy ha beha es obse a ionally
as da k mass.
2 Space ime Resonance Theo em
2.1 Cons i u ion o he E ec i e G a i a ional Medium
The analysis in his pape emains en i ely wi hin he weak– ield limi o gene al
ela i i y and in oduces no modi ica ions o he Eins ein equa ions, no new
ields, and no physical p ope ies assigned o he acuum. The only e inemen
beyond he s anda d New onian app oxima ion is he decision o e ain he ull
e a ded G een– unc ion s uc u e o he linea ised g a i a ional ield, a he han
unca ing i o he ins an aneous Poisson o m.
In ex ended, inhomogeneous ba yonic sys ems his e a ded ke nel p oduces non-
i ial eedback: cohe en mass mo ions bo h sou ce and esca e weak cu a u e
pe u ba ions. When many ba yonic elemen s pa icipa e, epea ed esca e ing
gene a es a collec i e esponse analogous o e ec i e–medium beha iou in acous-
ics o elec omagne ic wa e anspo . In his sense he ma e dis ibu ion, no
he acuum, plays he ole o he medium. All dispe sion, memo y, a enua ion,
and esonan beha iou a ise solely om he ba yonic ensemble’s g a i a ional
suscep ibili y.
The e ec i e–medium iewpoin used h oughou his pape is he e o e en i ely
con en ional: i de i es om applying s anda d mul iple sca e ing, homogenisa-
ion, and G een– unc ion me hods o he weak– ield Eins ein p opaga o . The
ollowing s uc u al p inciples summa ise his in e p e a ion:
1.
Weak– ield GR wi h nonlocal p opaga o s. The linea ised po en ial is
a e a ded con olu ion o he ba yonic sou ce wi h he gene al– ela i is ic
G een unc ion. This nonlocali y in oduces ini e esponse ime and allows
6
mul iple sca e ing o accumula e, bu i does no endow space ime wi h
ma e ial p ope ies.
2.
Ba yonic suscep ibili y unde pe iodic o cing. A ime–pe iodic
componen o he ba yonic densi y,
ρb,1
(x
,
), ac s as he cohe en sou ce o
he cu a u e ield. The ba yonic ensemble esponds h ough a equency–
dependen g a i a ional suscep ibili y
χ⋆
(
ω
), which de e mines how e i-
cien ly cu a u e pe u ba ions a e egene a ed and esca e ed and he e-
o e p o ides he mic oscopic o igin o esonan beha iou .
3.
Lossless g a i a ional esca e ing by inhomogenei ies. Spa ial
a ia ions in he ba yonic densi y ac as weak g a i a ional sca e e s.
Collec i ely hey p oduce in e e ence, anspo , and pa ial cohe ence o
cu a u e pe u ba ions, di ec ly analogous o acous ic o elec omagne ic
mul iple sca e ing.
4.
Sel –ene gy and e ec i e p opaga ion. The ensemble o sca e e s
modi ies he p opaga o h ough a Dyson– ype ela ion,
G−1
e =G−1
0−Σ(ω, k),
whe e Σ encodes collec i e esca e ing. A long–wa eleng h, low– equency
expansion o Σ yields a local, second–o de PDE o he collec i e cu a u e
esponse wi h coe icien s (Ω,Γ, ce ).
5.
Resonan sel –o ganisa ion. A cen al wo king hypo hesis o his pape
is ha he e ec i e mac oscopic coe icien
a0
in he sel –ene gy expansion
yields a es o ing equency scaling as
Ω2∝G¯ρb,
whe e
¯ρb
is he mean ba yonic densi y in a cohe ence domain. This mi o s
he known
√Gρ
scaling o s ella and galac ic oscilla ion modes and leads
na u ally o long–li ed, phase–locked collec i e oscilla ions.
6.
E ec i e cu a u e ield and s o ed ene gy. The cohe en cu a u e
pe u ba ion Φ
es
gene a ed by mul iple sca e ing de ines a eno malised
weak– ield po en ial— he e ec i e cu a u e ield. I s cycle–a e aged in en-
si y s o es g a i a ional ene gy,
EΦ∝ |Φ es|2,
and his s o ed ene gy appea s in he Poisson channel as an e ec i e mass
densi y,
ρe ∝ ⟨|Φ es|2⟩.
No new g a i a ional deg ee o eedom is in oduced: he e ec i e cu a u e
ield is he mac oscopic, homogenised esponse o s anda d weak– ield GR in
he p esence o ba yonic suscep ibili y and esca e ing, and he associa ed
ene gy is en i ely a p ope y o ha ield.
7
These p inciples depic galaxies, clus e s, and o he ex ended ba yonic sys ems as
sel –consis en g a i a ional esona o s whose mac oscopic beha iou ollows om
s anda d weak– ield GR combined wi h mul iple sca e ing and homogenisa ion.
The ollowing heo em o malises his e ec i e–medium esonance and de i es
he local esonan e olu ion equa ion o he collec i e cu a u e esponse.
Theo em (E ec i e–Medium Resonance o he Weak—
Field G a i a ional Po en ial). Conside he linea ised weak– ield
g a i a ional po en ial
Φ(x) = −4πGZG e (x, x′)ρb(x′)d4x′,(1)
whe e
G e
is he ull GR e a ded G een unc ion. Decompose he ma e densi y
in o a s a ic and cohe en componen ,
ρb(x, ) = ρb,0(x)+ρb,1(x, ).(2)
Assume he ba yons o m a s a is ically inhomogeneous ensemble o weak g a i a-
ional sca e e s whose pe iodic in e nal ea angemen is desc ibed by a linea
suscep ibili y
χ⋆
(
ω
). Mul iple sca e ing eno malises he p opaga o acco ding o
he Dyson ela ion
G−1
e (ω, k) = G−1
0(ω, k)−Σ(ω, k),(3)
whe e Σis he ensemble sel –ene gy.
In he long–wa eleng h, low– equency limi , Σadmi s he local expansion
Σ(ω, k)≃a0+a1(iω)+a2ω2+b2k2,(4)
wi h coe icien s de e mined by χ⋆(ω)and he spa ial s a is ics o he sca e e s.
T ans o ming Eq.
(3)
back o con igu a ion space using
(4)
yields he homogenised
e olu ion equa ion o he cohe en oscilla o y ield,
∂2
Φ es + 2Γ ∂ Φ es + Ω2Φ es −c2
e ∇2Φ es = 4πG ρb,1(x, ),(5)
wi h iden i ica ions
Ω2=a0,Γ = a1, c2
e =c2(1 −b2).(6)
The esonan ield is pu ely dynamical: when he cohe en d i ing anishes,
ρb,1(x, )→0 =⇒Φ es(x, )→0,(7)
and he emaining g a i a ional po en ial is he usual Poisson esponse o ρb,0.
Thus, s anda d weak– ield GR, when applied o an inhomogeneous ba yonic
ensemble wi h g a i a ional suscep ibili y and mul iple sca e ing, suppo s a
mac oscopic esonan mode wi h pa ame e s (Ω
,
Γ
, ce
)de e mined en i ely by
he ma e dis ibu ion. No modi ica ions o gene al ela i i y o addi ional
g a i a ional ields a e equi ed.
8
Condensed de i a ion ( ull de ails in Appendix A).
We ou line he s eps connec -
ing he weak– ield e a ded po en ial o he local esonan equa ion (5).
The esonan componen o he po en ial is ob ained by es ic ing he e a ded
solu ion (1) o he cohe en sou ce:
Φ es(x, ) := −4πGZG e (x, x′)ρb,1(x′, ′)d4x′,(C1)
which anishes whene e ρb,1= 0, consis en wi h (7).
Fou ie ans o ming gi es he linea esponse ela ion
˜
Φ es(ω, k) = −4πG ˜
Ge (ω, k) ˜ρb,1(ω, k),(C2)
whe e he e ec i e p opaga o obeys he Dyson equa ion
˜
G−1
e =˜
G−1
0−Σ(ω, k).(C3)
In he long–wa eleng h, low– equency egime he sel –ene gy akes he local
o m
Σ(ω, k)≃a0+a1(iω)+a2ω2+b2k2,(C4)
wi h coe icien s se by he mic oscopic suscep ibili y
χ⋆
(
ω
) and he sca e e
s a is ics.
Subs i u ing (C4) in o (C3) yields he s anda d esonan s uc u e
˜
G−1
e ∝ −ω2−2iΓω+ Ω2+c2
e k2,(C5)
wi h
Ω2=a0,Γ = a1, c2
e =c2(1 −b2),(C6)
as in (6).
Inse ing
(C5)
in o
(C2)
and ans o ming back o con igu a ion space gi es he
homogenised esonan equa ion
(∂2
+ 2Γ ∂ + Ω2−c2
e ∇2)Φ es = 4πG ρb,1(x, ),(C7)
iden ical o Eq. (5).
Since
(C2)
implies
˜
Φ es →
0 whene e
˜ρb,1→
0, he esonan ield has no s a ic
limi , again ma ching
(7)
. The pa ame e s (Ω
,
Γ
, ce
) a ise solely om ba yonic
suscep ibili y and mul iple sca e ing; no modi ica ion o GR is in ol ed.
9
comp essible media, whe e equency esponse is con olled by he e ec i e bulk
modulus: high– equency densi y pe u ba ions canno d i e he medium.
Toge he , Eqs.
(13)
–
(19)
show ha he esonan ield beha es as a weakly damped,
ene gy–s o ing con inuum suppo ing dispe si e wa es wi h ini e bandwid h
∆ω= Ω/Q. The associa ed cohe ence ime,
τcoh ∼Q
Ω,(21)
links he mic oscopic phase–lag pa ame e Γ o he mac oscopic pe sis ence o
cu a u e oscilla ions. As demons a ed in Sec ion 5, he e ec i e memo y o he
sys em ex ends o
τmem ∼ Gen τcoh
, p o iding he basis o sus ained cohe en
ampli ica ion in Channel 1.
These linea p opaga ion and damping laws supply he ounda ion o he single–
and mul iple–sca e ing heo y de eloped nex in Sec ion 4.
4 G a i a ional Sca e ing Theo y
Ha ing es ablished ha he dynamical (Channel 1) sec o o g a i y is go e ned
by he e ec i e esonan wa e equa ion de i ed in he p e ious sec ion, we now
cons uc he co esponding g a i a ional sca e ing heo y. Th oughou his
sec ion we wo k exclusi ely wi h he oscilla o y ba yonic componen
ρb,1
, since
only empo ally a ying sou ces can exci e and sus ain he collec i e esonan
ield Φ
es
. The s a ic componen
ρb,0
gene a es he backg ound New onian ield
bu does no con ibu e o sca e ing.
The goal in wha ollows is o de elop he g a i a ional analogues o he Lippmann–
Schwinge , Foldy–Lax, and Dyson equa ions using he p opaga o
G e
associa ed
wi h he e ec i e-medium PDE. This p o ides he mic oscopic ounda ion o
he collec i e beha iou analysed la e —including cohe en spi al modes and
di usi e ellip ical/clus e egimes (Sec ion 7).
In his amewo k each ba yonic mass ac s as a g a i a ional sca e e cha ac-
e ised by a linea suscep ibili y
χ⋆
(Ω), which encodes i s phase-lagged esponse
o an oscilla o y po en ial a equency Ω. Mo e gene ally, he g a i a ional
esponse o a localized mass is a pola izabili y enso
χij
⋆
(Ω), whose symme y is
se by he local space ime en i onmen . In iso opic se ings he enso educes o
he scala
χ⋆
(Ω) used h oughou his sec ion, whe eas in aniso opic geome ies
he e ec i e medium de elops p e e ed di ec ions o esponse. This enso ial
s uc u e will become essen ial when compa ing plana spi al sys ems (wi h
la e al pola iza ion o cohe en gain) o he iso opic cohe ence cells o ellip icals,
bu o he p esen mic oscopic heo y he scala o m su ices.
An inciden pe u ba ion
δ
Φ
inc
p oduces a delayed e-emission desc ibed by he
e a ded G een unc ion
G e
, and he esul ing sca e ed ield p opaga es o
all o he masses. Repea ed e-sca e ing among many inclusions gene a es a
16

hie a chy o coupled esponses, p ecisely analogous o classical mul iple-sca e ing
heo y in acous ics and elec omagne ism [6, 7, 8].
Thus ba yonic ma e plays a dual ole: i s oscilla o y componen
ρb,1
ac s as he
con inuous d i ing sou ce o he esonan ield, while he same masses also ac as
disc e e g a i a ional sca e e s ha e–emi cu a u e pe u ba ions h ough
hei suscep ibili y
χ⋆
(Ω). The i e a i e ne wo k o sca e ing and e-sca e ing
links he mic oscopic delayed esponse o indi idual ba yons o he eme gen ,
la ge-scale p opaga ion p ope ies o he collec i e esonan ield. I he e o e
supplies he necessa y b idge be ween he e ec i e wa e ope a o in oduced
ea lie and he mac oscopic g a i a ional beha iou de eloped in he emainde
o he pape .
4.1 Single–Sca e e Response
A disc e e ba yonic mass
m⋆
wi h linea g a i a ional suscep ibili y
χ⋆
(Ω) e-
sponds o an inciden oscilla o y po en ial
Ainc
h ough he s anda d Lipp-
mann–Schwinge ela ion
Asc(x)= ⋆(Ω) GΩ(x−x⋆)Ainc(x⋆), ⋆(Ω) = 4πG m⋆χ⋆(Ω),(22)
whe e
GΩ
is he equency–domain o m o he e a ded G een unc ion associa ed
wi h he e ec i e–medium p opaga ion equa ion. The quan i y
⋆
(Ω) de ines
he g a i a ional sca e ing s eng h o he inclusion, cap u ing i s phase–lagged
e–emission o he inciden cu a u e pe u ba ion.
In gene al, he suscep ibili y o a localized g a i a ing mass is a pola izabili y
enso
χij
⋆
(Ω). Local geome ic aniso opies ( o example, plana kinema ics o
cohe en s eaming) can p e e en ially ampli y pa icula enso componen s,
he eby pola izing he local g a i a ional esponse. In he iso opic app oxima ion
used he e he enso educes o i s ace,
χ⋆
(Ω), bu he ull enso s uc u e will
eappea in Sec ion 7.1.8 (spi al galaxies) and Sec ion 7.2.10 (ellip ical galaxies).
Because he esonan sec o is weakly damped (Γ
≪
Ω) and he unde lying
dynamics a e hose o linea ised GR, he in e ac ion is e e sible: no cu a u e
ene gy is i e e sibly abso bed by he sca e e . Ins ead, he inclusion s o es
cu a u e pe u ba ions only ansien ly, e–emi ing hem wi h a phase delay
de e mined by χ⋆(Ω) and he causal s uc u e o GΩ.
Equa ion
(22)
he e o e p o ides he mic oscopic building block o he g a i a-
ional mul iple–sca e ing heo y de eloped below, linking he indi idual ba yonic
esponse o he collec i e, ensemble–le el beha iou o he esonan ield.
4.2 Mul iple Sca e ing and he Foldy–Lax Hie a chy
In classical wa e physics, he in e ac ion o many sca e e s wi h a cohe en
inciden ield is desc ibed by he Foldy–Lax hie a chy [
6
,
7
,
8
], which p o ides
he s anda d sel –consis en o mula ion o mul iple sca e ing among disc e e
17
inclusions. O iginally de eloped o acous ics and la e gene alised o elec o-
magne ism, elas ici y, and pho onics, he same amewo k applies di ec ly o
he p esen g a i a ional p oblem once he e ec i e p opaga o
GΩ
has been
speci ied by he linea ised dynamics o Sec ion 3.4. Th oughou his subsec ion
he ield ampli ude
A
deno es he oscilla o y ( esonan ) cu a u e pe u ba ion
Φ es d i en by he ba yonic componen ρb,1.
Each ba yonic inclusion e–emi s he inciden cu a u e luc ua ion wi h a phase
delay de e mined by i s suscep ibili y
χ⋆
(Ω). These delayed e–emissions p opa-
ga e o all o he inclusions ia he G een unc ion
GΩ
, and he esul ing ne wo k
o mu ual couplings cons i u es a g a i a ional analogue o s anda d mul iple–
sca e ing heo y. When he spacing be ween inclusions is much smalle han he
esonan wa eleng h (
kΩd≪
1), hei phases emain locked o he inciden ield
and he sca e ed ampli udes add cohe en ly. In his ully cohe en Foldy–Lax
limi ,
Acoh ∝N|χ⋆(Ω)|Ainc,(23)
so ha he ensemble beha es as a single collec i e adia o wi h in ensi y
Icoh ∝N2|χ⋆(Ω)|2|Ainc|2,(24)
I is con enien o quan i y his as an en i onmen al gain
Gen ≡|A o |2
|Ainc|2,Gen =N2|χ⋆(Ω)|2(cohe en limi ).(25)
Fo an ensemble o
N
sca e e s, he o al ield sa is ies he disc e e Lippmann–
Schwinge sys em
A(x)=Ainc(x)+ (Ω) X
n
GΩ(x−xn)A(xn),(26)
wi h (Ω) = 4πGm⋆χ⋆(Ω). In ope a o no a ion his becomes
A= (1 − GΩ)−1Ainc,(27)
whose Neumann expansion,
A=Ainc + GΩAinc + ( GΩ)2Ainc +··· ,(28)
sums all o de s o mu ual g a i a ional e–sca e ing. In he limi Γ
→
0 (pu ely
eal
GΩ
), he esponse is s ic ly cohe en and Eq.
(27)
educes o he linea
scaling o Eq. (23).
I ins ead he ela i e phases o di e en inclusions become e ec i ely an-
dom—because o geome ic delays, ini e memo y (Γ
>
0), o dynamical phase
d i — hen sca e ed ampli udes no longe combine cohe en ly. They add in
in ensi y only:
I o =N|χ⋆(Ω)|2|Ainc|2,(29)
18
yielding he s anda d incohe en scaling
Gen =N|χ⋆(Ω)|2,pGen =√N|χ⋆(Ω)|.(30)
The spi al–disk egime discussed in Sec ion 7.1 lies in he long–wa eleng h, phase–
o de ed limi (
kΩd≪
1), whe e bo h spa ial phasing and delayed empo al esponse
o ganise cohe en ly in o a global
m
= 1 pa e n. As cohe ence dec eases—due o
ini e Γ, inc easing diso de , o la ge sepa a ions— he sys em ansi ions smoo hly
owa d he s a is ical egime ea ed in he nex subsec ion, whe e ensemble
a e aging o Eq.
(26)
p oduces an e ec i e–medium desc ip ion app op ia e o
ellip ical and clus e en i onmen s.
This subsec ion has ea ed mul iple sca e ing o a ixed a angemen o sca -
e e s, wi h cohe ence de e mined pu ely by he phase ela ions among hei
delayed e–emissions. The ollowing subsec ion ex ends he amewo k o s a-
is ically diso de ed o dynamically e ol ing ensembles, whe e a e aging o e
many such con igu a ions yields he eme gen e ec i e pa ame e s o he esonan
g a i a ional ield.
4.3 E ec i e–Medium and Ensemble A e aging
The Foldy–Lax hie a chy desc ibes he ully cohe en in e ac ion o sub-wa eleng h
ba yonic sca e e s wi h an inciden oscilla o y g a i a ional ield. In ealis ic
con igu a ions—pa icula ly in i egula , hick, o dynamically e ol ing sys-
ems—phase cohe ence among sca e e s is deg aded as sepa a ions app oach he
esonan wa eleng h o as di e en ial mo ions in oduce andom phase delays.
Unde such condi ions he local ields
A
(x
n
) a di e en inclusions can no longe
be ea ed as pe ec ly co ela ed, and he collec i e beha iou mus be desc ibed
s a is ically h ough an ensemble a e age o e possible con igu a ions.
A e aging he disc e e Lippmann–Schwinge equa ion
(26)
o e many ealisa ions
o sca e e posi ions yields he s anda d e ec i e–medium (Dyson) equa ion o
he cohe en mean ield:
(∇2+k2
e )⟨A⟩= 0, k2
e =k2+ Σ(Ω),(31)
whe e Σ(Ω) is he ensemble sel –ene gy. The sel –ene gy encapsula es he a e age
phase delay and ini e dwell ime associa ed wi h mul iple g a i a ional e-
sca e ings among ba yonic inclusions. I s eal pa modi ies he dispe sion
ela ion, while i s imagina y pa de e mines he ensemble a enua ion a e:
Γe =ce Im ke ,(32)
which is educed ela i e o he mic oscopic damping Γ whene e e e sible
esca e ing e ains phase cohe ence o e ex ended domains.
The deg ee o cohe en ampli ica ion p oduced by he en i onmen can be sum-
ma ised using he en i onmen al gain ac o
Gen
, de ined ea lie in Eq.
(30)
19
as he o al in ensi y gain. In he e ec i e–medium limi his quan i y can
be con enien ly w i en in e ms o he cohe ence olume
Vcell
o he eme gen
Bloch-like mode:
Gen ∝1
Vcell
=1
ℓ2
⊥ℓ∥
,(33)
whe e (
ℓ⊥, ℓ∥
) deno e he ans e se and longi udinal cohe ence leng hs o he
a e aged ield.
F om a mic oscopic pe spec i e, he local in ensi y gain in a cohe en pa ch scales
as he squa e o i s a e age coupling e iciency di ided by i s cohe ence olume:
Gen ∝|X|2
Vcell
=|X|2
ℓ2
⊥ℓ∥
,(34)
whe e
X
is he e ec i e (ensemble–a e aged) coupling ampli ude. La ge cohe ence
domains (la ge
ℓ⊥
and
ℓ∥
) hus imply s ong en i onmen al gain, while sh inking
cohe ence leng hs educe
Gen
by inc easing he numbe o independen sca e ing
cells.
Sys ems wi h s ong spa ial phase o de (e.g. hin, o a ing disks) possess la ge
cohe ence olumes and hence la ge en i onmen al gain. These sa is y
Γe ≃Γ,(35)
eco e ing he cohe en Foldy–Lax limi . Inc easing diso de o dynamical
andomness dec eases he cohe ence leng hs, lowe s
Gen
, and d i es he sys em
owa d egimes whe e ensemble damping is domina ed by phase andomisa ion
a he han he mic oscopic Γ.
Equa ion
(31)
he e o e p o ides he s a is ical ex ension o he cohe en mul i-
ple–sca e ing amewo k. O de ed sys ems co espond o eal, phase–aligned
sel –ene gies wi h la ge en i onmen al gain, whe eas diso de ed ensembles a e
desc ibed by complex, s ochas ic sel –ene gies whose imagina y pa s encode
di usi e loss o cohe ence. This e ec i e–medium o mula ion supplies he
mac oscopic link be ween he mic oscopic sca e ing in e ac ions and he global
in ensi y s uc u e analysed la e in Sec ion 7.2.
4.4 T anspo and Di usion Limi s
When he ela i e phases o indi idual ba yonic sca e e s become andom and
ime–dependen , he ensemble ield co ela ion unc ion obeys he s anda d
adia i e– ans e equa ion amilia om acous ic, elec omagne ic, and elas ic
mul iple–sca e ing heo y [8]:
ˆ
s·∇I(x,ˆ
s)=−I(x,ˆ
s)
ℓs
+1
ℓsZP(ˆ
s,ˆ
s′)I(x,ˆ
s′)dΩ′+S(x,ˆ
s),(36)
whe e
I
(x
,ˆ
s
) is he di ec ional in ensi y o he oscilla o y g a i a ional ield,
ℓs
is he sca e ing mean ee pa h, and
P
is he angula edis ibu ion ke nel
20
speci ying how g a i a ional pe u ba ion ene gy is edi ec ed by ba yonic inclu-
sions. Equa ion
(36)
is he exac g a i a ional analogue o he adia i e– ans e
o mula ion alida ed in esonan acous ic expe imen s by De ode, Tou in, and
Fink [9, 4, 5].
In he iso opic limi , Eq.
(36)
educes o he usual di usion equa ion o he
ensemble-a e aged in ensi y:
∇2I(x) = I(x)
L2
damp,e
,(37)
whe e he e ec i e damping (o cohe ence–decay) leng h is
Ldamp,e =ce
Γe
.(38)
This leng h cha ac e ises he scale o e which he cohe en componen o he
oscilla o y g a i a ional ield su i es epea ed andom-phase esca e ing. In o -
de ed o phase-aligned sys ems,
Ldamp,e →∞
, eco e ing he cohe en Foldy–Lax
egime, whe eas in diso de ed o dynamically luc ua ing ensembles i becomes
ini e, e lec ing he di usi e loss o cohe ence a ising solely om phase andomi-
sa ion in weak- ield GR.
Thus Eqs.
(36)
–
(38)
p o ide he s a is ical, anspo -le el closu e o he mul i-
ple–sca e ing amewo k: hey desc ibe how g a i a ional cu a u e pe u ba-
ions p opaga e once cohe en addi ions among sca e e s a e no longe main-
ained.
4.5 Uni ied In e p e a ion
The cohe en (Foldy–Lax) and di usi e ( anspo ) limi s de eloped abo e a ise
om he same go e ning ela ion,
A=1− (Ω) GΩ−1Ainc,(39)
and di e only in he s a is ical s uc u e o phase co ela ions among he ba yonic
sca e e s. In he mic oscopic desc ip ion he esponse o each inclusion is encoded
in i s g a i a ional suscep ibili y, which is in gene al a pola izabili y enso
χij
⋆
(Ω).
In iso opic en i onmen s his educes o he scala o m
χ⋆
(Ω) used h oughou
Sec ion 4, bu in aniso opic geome ies he enso de elops p e e ed di ec ions
o esponse ha eed di ec ly in o he la ge-scale cohe ence p ope ies.
When ela i e phases a e o ganised (as in o a ing disks), mul iple esca e ing
p oduces cohe en ampli ude addi ion and la ge en i onmen al gain. This co e-
sponds o a si ua ion in which he enso ial suscep ibili y is e ec i ely aligned:
a dominan in-plane pola isa ion sus ains la e al cohe ence o e la ge scales,
p oducing he global
m
= 1 spi al modes desc ibed in Sec ion 7.1. He e he
medium beha es as a phase-o de ed esona o , and he Foldy–Lax limi applies.
21

When phases a e andomized (as in i egula , hick, o p essu e-suppo ed sys-
ems), he enso ial esponse a e ages o i s iso opic ace and he same ope a-
o
(39)
gene a es a di usi e in ensi y ield wi h a ini e cohe ence leng h. This is
he egime analysed in he ellip ical and clus e cases, whe e ensemble a e aging
p oduces an iso opic e ec i e medium go e ned by a Dyson sel –ene gy Σ(Ω)
and a anspo damping a e Γe .
Thus he uni ied sca e ing equa ion couples seamlessly o bo h ex emes:
•
cohe en ein o cemen in spi al galaxies, whe e geome ic and kine-
ma ic aniso opies pola ize he suscep ibili y and s abilise la ge-scale phase
locking;
•
di usi e, anspo -domina ed beha iou in ellip icals and clus e s,
whe e andom h ee-dimensional mo ions iso opise he enso esponse
and educe he cohe ence leng h.
In bo h cases he eme gen mac oscopic beha iou ollows di ec ly om how he
mic oscopic pola izabili y, media ed by g a i a ional esca e ing, o ganises (o
ails o o ganise) phase co ela ions ac oss he ba yonic ensemble.
5
Spa io empo al Feedback and Resonan Co-
he ence
5.1
Cons i uen En i ies & S a es o he Feedback Sys em
Space ime in he weak– ield limi does no beha e as a pu ely passi e backg ound
bu as a causal medium whose cu a u e deg ees o eedom espond on wo
dis inc imescales. Channel 1 cap u es he dynamical, esonan esponse o
he me ic o he oscilla o y ba yonic componen
ρb,1
;Channel 2 cap u es he
quasi–s a ic, New onian esponse o he slowly a ying mass dis ibu ion and o
he cu a u e ene gy accumula ed om pas Channel 1 ac i i y.
Channel 1 is he e o e he dynamical d i e o he eedback sys em: i s esonan
cu a u e oscilla ions exchange ene gy and phase wi h he ba yons and deposi
cu a u e ene gy in o he long– e m s a e a iable
ρe
.Channel 2 p o ides
he eac i e, quasi–s a ic esponse o he me ic o he slowly a ying mass
dis ibu ion and he accumula ed s o ed cu a u e ene gy.
The in e ac ion o he wo channels o ms a coupled spa io empo al eedback
sys em: Channel 1 con inuously modi ies he long– e m me ic s a e, while
Channel 2 se s he backg ound po en ial wi hin which esonan oscilla ions occu .
The a chi ec u e hus consis s o wo physical en i ies—ba yonic ma e and
he space ime esponse ke nel—and wo cu a u e s a es ha e ol e on as
(Channel 1) and slow (Channel 2) imescales, summa ised in Table 1.
22
Table 1: Cons i uen en i ies and s a es in he wo–channel cu a u e–ma e
eedback sys em.
En i y / S a e Desc ip ion
Ba yonic
mass densi y
ρb=ρb,0+ρb,1
(En i y) The physical sou ce o cu a u e.
ρb,1
is
he oscilla o y componen ha d i es Channel 1 and
ac s as bo h emi e and sca e e o he esonan
me ic esponse.
ρb,0
is he slowly a ying componen
sou cing he New onian Channel 2 ield.
Space ime
esponse
pa ame e s
B= (ce ,Ω,Γ)
(En i y) The pa ame e s desc ibing he causal, e-
quency–dependen linea esponse o Channel 1:
ce
go e ns p opaga ion, Ω he es o ing e m, and Γ he
ini e memo y ime. Channel 2 uses he same me ic
bu esponds h ough he ins an aneous Poisson e-
la ion.
e ec i e cu -
a u e ield
Φ = Φ es + ΦN
(S a e o he space ime) Φ
es
is he esonan Chan-
nel 1 ield d i en by
ρb,1
, media ing phase–lagged
cu a u e exchange wi h he ba yons. Φ
N
is he
quasi–s a ic Channel 2 ield sou ced by
ρb,0
oge he
wi h he accumula ed s o ed cu a u e ene gy ρe .
S o ed cu a-
u e ene gy
(e ec i e ‘da k’
mass) ρe
(Long– e m s a e a iable) The slowly e ol ing cu -
a u e ene gy buil up om pas Channel 1 ac i i y.
I decays only on he imescale Γ
−1
and con ibu es o
he New onian po en ial ia
∇2
Φ
N
= 4
πG
(
ρb,0
+
ρe
).
23
5.2 Feedback Channels and In e ac ions
The cu a u e–ma e sys em e ol es h ough wo coupled eedback channels,
bo h a ising om he same weak– ield g a i a ional dynamics. Channel 1 is he
esonan , wa e–based esponse d i en by he oscilla o y ba yonic densi y
ρb,1
and go e ned by he causal, memo y–bea ing p opaga o de i ed in Sec ion 6.
Channel 2 is he New onian, e ec i ely ins an aneous esponse gene a ed by
he slowly a ying ba yonic densi y
ρb,0
oge he wi h he accumula ed e ec i e
(da k) mass
ρe
. Thei in e ac ion o ms he dual eedback a chi ec u e shown in
Fig. 1.
(1) Resonan Channel (Channel 1). The oscilla o y componen
ρb,1
exci es
he esonan cu a u e ield Φ
es
h ough he causal G een unc ion
GΩ
o he
damped esonan cu a u e equa ion. As de eloped in Sec ion 6,
GΩ
encodes
p opaga ion, phase delay, and ini e causal memo y a he e ec i e speed
ce
.
Because many ba yons ac as cohe en emi e s and sca e e s, he esonan ield
is s a is ically ampli ied by he en i onmen al gain Gen , gi ing
Φ es ∝pGen GΩ∗ρb,1,(40)
wi h
∗
deno ing causal con olu ion. The in e ac ion is e e sible: esonan cu a-
u e pe u ba ions exchange ene gy and phase wi h he ba yons ha gene a e
hem. O e he memo y ime
τmem
= Γ
−1
, his oscilla o y exchange accumula es
in o an e ec i e (da k) mass densi y,
ρe ∝ ⟨|Φ es|2⟩ ,(41)
which hen en e s he New onian esponse o Channel 2.
(2) New onian Channel (Channel 2). The backg ound ba yonic densi y
ρb,0
oge he wi h he s o ed e ec i e mass
ρe
sou ce he New onian po en ial Φ
N
ia he Poisson equa ion. This po en ial p opaga es e ec i ely ins an aneously a
speed
c
wi hin he weak– ield egime and es ablishes he local geodesic s uc u e.
The esul ing eloci y ield edis ibu es
ρb,0
and modula es he oscilla o y
componen
ρb,1
, he eby ese ing he pa e n o esonan sou ces ha eed
Channel 1. This cons i u es he s ess–ad ec ion loop: he accumula ed e ec i e
mass s abilizes he quasi–s a ic po en ial, while ba yonic mo ion con inually
eo ganizes he d i e s o he esonan ield.
Long– e m cohe ence and equilib ium. Channel 1 con inually deposi s
cu a u e ene gy in o
ρe
, while Channel 2 con inually edis ibu es ma e and
eshapes he sou ces o esonan exci a ion. Toge he hese p ocesses d i e he
sys em owa d a sel –o ganized spa io empo al equilib ium in which
ρb,1
, Φ
es
,
ρe
,
and Φ
N
emain mu ually consis en ac oss hei espec i e imescales. Figu e 1
illus a es hese causal links: ed a ows deno e he New onian s ess–ad ec ion
esponse; black and blue a ows deno e esonan exci a ion and slow memo y
accumula ion.
Toge he hese links o m he dual–channel a chi ec u e shown in Fig. 1. The
igu e comp esses he esonan eed and slow accumula ion in o a single a ow,
24
Channel 1
Resonan channel
eeding da k mass
Channel 2
New onian
channel
eeding con inui y
ρb,1Φ es ρe
GΩ
esca ./p op.
(∝ Gen )
ρb,0, ρe ΦN ρb,0, ρb,1
∇2−∇ Con inui y
Ad ec ion
long- e m
ene gy s o age
memo y ∝(τmem)
Figu e 1: Two-channel g a i a ional eedback ne wo k. Channel 1
( op, black →blue): he oscilla o y ba yonic componen
ρb,1
d i es he esonan
g a i a ional ield Φ
es
h ough he causal p opaga o
GΩ
. Mul iple sca e ing
p oduces an in ensi y gain
Gen
, and he esonan ield deposi s accumula ed
cu a u e in o he slow, long-li ed componen
ρe
o e he memo y ime
τmem
=
Γ
−1
.Channel 2 (bo om, ed): he slowly a ying densi y (
ρb,0, ρe
) sou ces he
New onian po en ial Φ
N
, which induces bulk eloci ies and ad ec s he ba yonic
dis ibu ion. Blue dashed a ows deno e slow memo y eedback; ed a ows
deno e ins an aneous New onian esponse and mass anspo . Toge he he wo
channels o m a closed spa io empo al eedback loop linking ba yonic mo ion,
esonan g a i a ional esponse, and long- e m cu a u e accumula ion.
25
The spa ial cohe ence ke nel
GΩ
(x
,
x
′
) implici ly de ines a spa ial pola izabili y
enso o he esonan channel: i selec s which di ec ions o he ba yonic luc u-
a ions couple mos e icien ly in o he cohe en ield. Disk–like cohe ence domains
ampli y la e al componen s o he esponse, whe eas sphe ical o ellipsoidal cohe -
ence cells weigh he componen s almos iso opically. This geome ic dependence
se s he ela i e spa ial esponse weigh s ha la e en e he suscep ibili y enso
χij
(Ω); i s explici consequences a e de eloped in he spi al and ellip ical sec ions.
As p oposed in Sec ion 5.3, cu a u e-media ed equency pulling d i es neigh-
bou ing egions owa d a common empo al equency, so a global ca ie Ω
is al eady selec ed. Spa ial cohe ence he e o e conce ns he geome y o he
esonan en elope a his locked equency, no he mechanism o empo al locking.
Wi h empo al cohe ence es ablished, he esonan ield admi s he decomposi ion
Φ es(x, ) = ReA(x)e−iΩ ,(53)
whe e he complex en elope
A
(x) sa is ies a s a iona y Helmhol z– ype equa ion.
Cons uc i e in e e ence among he phase–locked modes p oduces long–li ed,
slowly e ol ing esonan en elopes— he g a i a ional analogue o s anding–wa e
pa e ns in op ical o acous ic ca i ies. These en elopes e ol e only h ough slow
ba yonic ad ec ion and he long cu a u e–memo y ime o he esonan sec o .
Th oughou each cohe en domain we he e o e adop he locking app oxima ion
Ω≃ω es,(54)
neglec ing slow ampli ude a ia ions excep nea s eep densi y g adien s o s ong
ba yonic asymme y. The nex subsec ion de i es he co esponding Helmhol z
en elope equa ion and i s exac Ki chho –Somme eld ep esen a ion, iden i ying
he spa ial cohe ence scales ha go e n he in e nal s uc u e o he esonan
ield.
6.1 Ki chho –Somme eld Rep esen a ion
Assuming global synch oniza ion a equency Ω, all cu a u e modes ca y he
common empo al ac o
e−iΩ
. Subs i u ing he ansa z
(53)
in o he linea ized
ield equa ion
(13)
and neglec ing slow empo al modula ions gi es he s a iona y
Helmhol z en elope equa ion
∇2A(x) + Ω2+i2ΓΩ
c2
e
A(x) = 4πG
c2
e
δρb(x),(55)
which de e mines he spa ial p o ile o he phase–locked esonan ield.
The exac solu ion o Eq.
(55)
is he s anda d Ki chho –Somme eld ep esen a-
ion:
A(x) = G
c2
e Zd3x′eikΩ|x−x′|
|x−x′|δρb(x′),(56)
32

wi h complex wa enumbe
kΩ=Ω+iΓ
ce
, λ es =2πce
Ω.(57)
The imagina y pa o
kΩ
p oduces a slow spa ial a enua ion o e he damping
leng h
Ldamp =ce
Γ=ce τmem,(58)
which measu es he spa ial ex en o cu a u e memo y. Fo weak damping
(Γ≪Ω) and o ≪Ldamp, one may ake kΩ≃2π/λ es as eal.
On galac ic and clus e scales he cha ac e is ic size
a
o a cohe en s uc u e
spans only a modes ac ion o he esonan wa eleng h,
a/λ es ∼
0
.
1–0
.
2. Thus
he usual a – ield di ac ion hie a chy does no apply: he e ec i e cu a u e
ield ne e samples many 2
π
cycles ac oss a s uc u e. Ins ead each galac ic disk,
ellip ical ensemble, o local speckle cell ac s as a phase–locked esonan domain: a
slowly a ying en elope main ained by ba yonic mo ion, cu a u e memo y, and
delayed eedback. All egions wi hin a cohe ence cell sha e a common empo al
phase and a spa ial co ela ion scale se by λ es and Ldamp.
Unlike op ical p opaga ion, his egime beha es as a quasi–s a ic, phase–cohe en
esona o whose geome y and memo y de e mine he spa ial p o ile o he ield.
Sec ion 6.2 shows how hese cohe en en elopes ansla e, h ough long– ime
a e aging, in o he s o ed cu a u e ene gy ha eeds he obse able New onian
channel.
In his ep esen a ion he sou ce e m
δρb
(x) in Eq.
(55)
should be unde s ood as
he cohe en Channel 1 d i e : he oscilla o y componen o he ba yonic densi y
ρb,1exp essed in he global esonan phase ame. Fo mally, one may w i e
δρb(x) = δρb(x)eiφd(x),(59)
whe e
φd
(x) encodes he local phase delay o he ba yonic mo ion ela i e o he
cu a u e ca ie
e−iΩ
. The complex ampli ude in Eq.
(55)
he e o e ep esen s
he phase–weigh ed mass densi y ha ac ually couples o he esonan mode, no
a s a ic geome ic o e densi y.
The s uc u e abo e is di ec ly amilia om scala Fou ie op ics (see Good-
man [
11
]). In a cohe en wa e sys em he physical sou ce is ep esen ed by
acomplex dis ibu ion, and he ield gene a ed a any poin is ob ained by
a phase–weigh ed supe posi ion o all emi e con ibu ions. In his language
he oscilla o y ba yonic sou ce
δρb
(x) =
|δρb|eiφd(x)
plays he ole o a complex
ape u e dis ibu ion, while he cu a u e ke nel
GΩ
supplies he p opaga ing
phase on . The ield en elope
A
(x) is he e o e ob ained by he s anda d
cohe en in eg al de ined in Eq.
(56)
, which is exac ly he Ki chho –Somme eld
ep esen a ion o he Helmhol z solu ion. The complex phase o
δρb
ensu es ha
each ba yonic emi e is au oma ically “ ephased” by he p opaga ion ke nel
be o e con ibu ing o he ield; he in eg al is simply he cohe en supe posi ion
o hese phase–weigh ed con ibu ions.
33
This o m makes clea ha he e ec i e d i e o he esonan channel is de e mined
no by he geome ic asymme y o he mass dis ibu ion bu by i s cohe en
p ojec ion on o he cu a u e phase on . In gene al, he G een unc ion
GΩ(x,x′) = eikΩ|x−x′|
|x−x′|(60)
ac s as he con inuous phase e e ence agains which he oscilla o y mass dis i-
bu ion is compa ed. The in eg al
A(x) = G
c2
e Zd3x′GΩ(x,x′)δρb(x′) (61)
can hus be in e p e ed as he phase–ma ched p ojec ion o he complex sou ce
δρb
on o he esonan e ec i e cu a u e ield. In egions whe e he ba yonic
phases
φd
(x
′
) a e aligned wi h he ke nel, con ibu ions add cons uc i ely; whe e
hey deco ela e, hey cancel s a is ically. A nea ly symme ic con igu a ion may
he e o e d i e he esonan ield s ongly i i s in e nal phase pa e n is cohe en ,
whe eas a highly asymme ic dis ibu ion may d i e i weakly i i s phases a e
andom.
This iewpoin applies equally o o de ed and diso de ed sys ems. In a spi al
disc, he phase pa e n
φd
(x) may align wi h a low–o de azimu hal s uc u e
( ea ed la e ia explici Fou ie momen s), while in an ellip ical galaxy he
same o malism desc ibes a andom ensemble o local “speckle” phaso s whose
ne d i e is go e ned by hei phase–weigh ed a e age unde Eq.
(61)
. In all
cases i is he cohe en ly p ojec ed mass, a he han he aw geome ic densi y,
ha con ols he s eng h o he Channel 1 ield and hence he a e a which
s o ed cu a u e ene gy ρe builds up in Sec ion 6.2.
6.2
E ec i e Po en ial and Ene gy S o age in he Reso-
nan Medium
Wi hin he wo–channel amewo k de eloped abo e, he physically obse able
g a i a ional ield is he Channel 2 po en ial Φ
N
. This po en ial is sou ced by he
ime–a e aged ba yonic densi y
ρb,0
oge he wi h he long– e m s o ed cu a u e
ene gy ρe ,
∇2ΦN= 4πGρb,0+ρe ,(62)
and ep esen s he quasi–s a ic geodesic s uc u e o he sys em. C ucially, Φ
N
does no con ain he ins an aneous oscilla o y con ibu ion om he esonan
channel. The oscilla o y e ec i e cu a u e ield Φ
es
(
), d i en solely by he
cohe en ba yonic componen
ρb,1
, a e ages o ze o o e a pe iod and is no
di ec ly obse able. Only i s s o ed cu a u e ene gy, accumula ed o e he
memo y ime τmem = Γ−1, eeds in o Channel 2.
Because he e ec i e esonan medium possesses ini e causal memo y, i s esponse
o he oscilla o y ield is phase–lagged. This delay p e en s pe ec cancella ion o
34
cu a u e s esses o e each cycle: pa o he esonan cu a u e ene gy injec ed
by
ρb,1
emains s o ed in he medium a he han being e u ned immedia ely.
The esul is a slowly e ol ing cu a u e bias— he quan i y
ρe
— ha pe sis s
on imescales long compa ed o 2
π/
Ω and sou ces he obse able Channel 2
po en ial.
The local ene gy densi y o he Channel 1 esonan ield, exp essed h ough he
complex spa ial en elope A(x) in oduced in Eq. (53), is
EΦ(x) = 1
8πGhc2
e |∇A|2+ Ω2|A|2i,(63)
ep esen ing he e e sible pa i ion be ween cu a u e–s ain g adien s and
empo al es o ing ene gy. By he equi alence o mass and ene gy, his s o ed
cu a u e ene gy beha es g a i a ionally as an e ec i e mass densi y,
ρe (x) = EΦ(x)
c2=1
8πGc2hc2
e |∇A|2+ Ω2|A|2i.(64)
This iden i ica ion ensu es ha all cu a u e ene gy e ained in he e ec i e
esonan medium en e s Channel 2 in p ecisely he same manne as con en ional
ma e .
Fo s a is ically incohe en sys ems—such as ellip ical galaxies o clus e en i on-
men s composed o many locally unco ela ed oscilla o s— he ele an obse able
is he ensemble a e age o Eq. (64),
ρe (x)=1
8πGc2hc2
e |∇A|2+ Ω2|A|2i,(65)
since bo h in ensi y and g adien luc ua ions con ibu e o he mac oscopic
cu a u e bias. In ei he he cohe en o s a is ical limi , he s o ed cu a u e
ene gy p oduces a g a i a ional ield iden ical in o m o ha a ibu ed in he
ΛCDM amewo k o a da k–ma e componen ,
∇2ΦN= 4πGρb,0+ρe ⇐⇒ ∇2ΦΛCDM = 4πGρb,0+ρDM.(66)
Thus
ρe
ac s obse a ionally as he “missing mass” equi ed o la o a ion
cu es and weak–lensing signa u es, e en hough i s o igin lies en i ely in he
esonan , ene gy–s o ing dynamics o Channel 1 a he han in pa icula e da k
ma e . The nex sec ion examines how his e ec i e esonan medium esponds
o a single ba yonic sou ce, whose localized oscilla ion cons i u es he undamen al
building block o all collec i e esonan s uc u es.
6.3
Single Poin Sou ce in a Resonan Channel–1 Back-
g ound
As in he p eceding subsec ions, all esul s he e apply exclusi ely o Channel 1.
The ields de i ed a e pu ely esonan , oscilla o y cu a u e esponses d i en by
35
he cohe en ba yonic componen
ρb,1
and a e he e o e no di ec ly obse able.
Thei only physical imp in a ises h ough he s o ed cu a u e ene gy hey
deposi in o he e ec i e esonan medium, which inc emen s he long– e m
e ec i e densi y ρe ha la e sou ces Channel 2.
Fo a esonan poin mass
mb
he Channel–1 d i e is ep esen ed as a Di ac–del a
oscilla o y sou ce [11]:
δρb,1(x)=mbδ(3)
x−xp.(67)
Inse ing his sou ce in o he Ki chho –Somme eld ep esen a ion (Eq.
(56)
)
yields he co esponding Channel–1 esonan ield δΦp( ):
δΦp(x) = Gmb
c2
e
eikΩ
, =|x−xp|.(68)
Equa ion
(68)
is simply he G een– unc ion e alua ion o he Channel–1 Helmhol z
ope a o wi h a del a– unc ion sou ce.
The eal, physical Channel–1 oscilla ion co esponds o he eal pa o Eq.
(68)
,
gi ing
δΦp( ) = Gmb
c2
e
e− /Ldamp j02π
λ es ,(69)
whe e
j0
desc ibes sphe ical sp eading and he exponen ial ac o ep esen s
a enua ion o e he cu a u e–memo y leng h
Ldamp
. This oscilla ion emains
con ined o Channel 1 and has no di ec dynamical e ec on ma e .
The quan i y o physical in e es is he s o ed cu a u e ene gy associa ed wi h
his oscilla ion, ob ained om he RMS magni ude
|δΦp( )|21/2=Gmb
c2
e
e− /Ldamp
√2j0(2π /λ es).(70)
This RMS ield is wha en e s he esonan ene gy densi y and ul ima ely modi ies
he e ec i e mass densi y ρe .
In he gene al case he e ec i e densi y con ains bo h g adien and cu a u e
con ibu ions, as in Eq.
(65)
. Fo he single, sphe ically symme ic poin –sou ce
mode conside ed he e we adop he long–wa eleng h, slowly a ying–en elope
app oxima ion (
|∇A|≪kΩ|A|
), so ha he g adien e m con ibu es only a small
co ec ion o he o al s o ed ene gy. Re aining only he dominan cu a u e
e m ∝Ω2⟨|A|2⟩yields he illus a i e exp ession below.
The con ibu ion o his single oscilla o y sou ce o he s o ed cu a u e ene gy
is he e o e
ρe ( ) = Ω2
4πGc2|δΦp( )|2
=G m2
bΩ2
8π c2c4
e
e−2 /Ldamp
2j2
02π
λ es .
(71)
36
This s o ed ene gy is he only mechanism by which a poin sou ce modi ies
obse able dynamics: i s con ibu ion adds o he o al
ρe
appea ing in he
Channel–2 Poisson equa ion. The ins an aneous oscilla o y ield ne e con ibu es
di ec ly o ΦN, bu he associa ed ρe does.
Thus he sphe ical Channel–1 esponse o a poin mass p o ides he elemen a y
“building block” om which collec i e esonan ields and he esul ing e ec i e
densi y
ρe
o ex ended sys ems a e cons uc ed. In ealis ic galac ic se ings he
ue
ρe
a ises om he in e e ence o many such poin –sou ce con ibu ions o
o m he global ampli ude en elope
A
(x). The nex subsec ion he e o e conside s
he in e e ence o wo oscilla o y poin sou ces as he simples non i ial example.
6.4 In e e ence o Two Poin Sou ces
We now examine he spa ial analogue o empo al cohe ence. Conside wo poin
masses,
m1
and
m2
, loca ed a ixed posi ions x
1
and x
2
wi hin he same esonan
domain. Bo h oscilla e cohe en ly a he common equency Ω, wi h a ixed
ela i e phase. A an a bi a y obse a ion poin x he dis ances o he sou ces
a e
i=|x−xi|, i = 1,2.
Because bo h sou ces adia e simul aneously, hei Channel–1 oscilla o y e ec i e
cu a u e ields o e lap h oughou he e ec i e esonan medium, p oducing
spa ial in e e ence pa e ns analogous o op ical di ac ion om wo cohe en
emi e s.
In he single–sou ce case, in e e ence is absen : he New onian backg ound
ca ies no phase and esponds only o ime–a e aged s o ed ene gy. Wi h
mul iple phase–locked emi e s, howe e , he oscilla o y componen s in e e e
cohe en ly wi h one ano he (bu ne e wi h he quasi–s a ic backg ound). This
mu ual in e e ence gene a es spa ial modula ion o he esonan en elope, and
hence o he s o ed cu a u e ene gy.
Because he Channel–1 equa ion is linea in he complex ampli ude, he o al
ield is he supe posi ion o he wo Helmhol z solu ions:
δΦ(x) = δΦp1(x)+δΦp2(x),(72)
wi h each con ibu ion gi en by
δΦpi(x) = Gmi
c2
e
eikΩ i
i
, i=|x−xi|, i = 1,2.(73)
In e e ence a ises en i ely om he complex phase ac o s eikΩ i; he physically
ele an s o ed ene gy depends on he RMS in ensi y |δΦ|2a e supe posi ion.
Figu es 2 and 3 illus a e he cu a u e in e e ence pa e n gene a ed by wo
equal, cohe en ly oscilla ing poin masses in he same esonan medium (using
he same pa ame e s as in he single–sou ce case:
ce
= 2
×
10
5
m s
−1
,
Q
= 5, Ω =
37

7
.
45
×
10
−16
s
−1
, and
m1,2
=
M⊙
). A a sepa a ion o 10 pc (Fig. 2) he spacing
is much smalle han he esonan wa eleng h
λ es
, so he indi idual ields me ge
in o a single smoo h en elope wi h only sligh modula ion. A 100 pc
2
(Fig. 3),
se e al wa eleng hs i be ween he sou ces and dis inc in e e ence inges
appea in bo h ampli ude and
ρe
. These al e na ing b igh and da k egions
co espond o cons uc i e and des uc i e cu a u e in e e ence—essen ially
space ime di ac ion pa e ns wi hin he e ec i e esonan medium.
The obse able cu a u e in ensi y is he ime–a e aged magni ude o he o al
ield:
|A|2=|δΦp1+δΦp2|2.(74)
Subs i u ing Eq. (73) gi es he explici in e e ence pa e n
|A|2=G2
c4
e
e−2( 1+ 2)/Ldamp "m2
1
2
1
j2
02π 1
λ es +m2
2
2
2
j2
02π 2
λ es 
+2m1m2
1 2
j0
2π 1
λ es j0
2π 2
λ es cos
2π∆
λ es #,
(75)
whe e ∆
=
2− 1
is he pa h–leng h di e ence. The cosine e m desc ibes
mu ual in e e ence; he exponen ial e m con eys he damping imposed by he
ini e cu a u e–memo y leng h Ldamp.
Cons uc i e in e e ence occu s o 2
π
∆
/λ es
= 2
nπ
, and des uc i e in e e -
ence o (2
n
+ 1)
π
. In he long–wa eleng h limi (
i≪λ es
), he ield educes
smoo hly o he New onian o m
G
(
m1
+
m2
)
/
. A in e media e scales, how-
e e , cohe en in e e ence edis ibu es s o ed cu a u e ene gy, o ming s able
esonan en elopes ha a e he building blocks o ex ended halo–like s uc u es.
In gene al he e ec i e densi y con ains bo h g adien and cu a u e con ibu ions,
as in Eq.
(65)
. Fo he p esen wo–sou ce con igu a ion we again adop he
long–wa eleng h, slow–en elope app oxima ion so ha he g adien e m p o ides
only a small co ec ion. Keeping only he dominan cu a u e con ibu ion yields
ρe (x) = Ω2
4πGc2|A(x)|2,(76)
showing di ec ly ha in e e ence be ween oscilla o y ields modula es he local
e ec i e g a i a ional densi y. The c oss– e m in Eq.
(75)
hus ep esen s a eal
edis ibu ion o s o ed cu a u e ene gy wi hin he e ec i e esonan medium; he
o al ene gy emains conse ed in he absence o damping, bu is shi ed be ween
egions o cons uc i e and des uc i e cu a u e ampli ude. The exponen ial
en elope is again con olled by he same damping a e Γ ha se s he empo al
cohe ence ime.
2
Subsequen simula ions (see Sec ion 7) indica e ha hese sepa a ions and wa eleng hs a e
no ealis ic o galaxy–scale esonances, bu he con igu a ion emains pedagogically use ul.
38
Figu e 2: Two–sou ce in e e ence (10 pc sepa a ion). Ampli ude, phase,
and e ec i e densi y
ρe
o wo cohe en poin sou ces. The sepa a ion is much
smalle han he esonan wa eleng h, so he wo ields me ge in o a single
en elope wi h gen le modula ion.
39
Figu e 3: Two–sou ce in e e ence (100 pc sepa a ion). Same se up
as Fig. 2, bu wi h he sou ces se e al wa eleng hs apa , p oducing s ong
cons uc i e and des uc i e in e e ence inges.
40
This wo–sou ce con igu a ion makes he phase s uc u e o he Ki chho –
Somme eld ep esen a ion (Sec ion 6.1) ully explici . The Channel–1 d i e
akes he o m
δρb(x) = δρb(x)eiφd(x),(77)
whe e
φd
(x) encodes he local phase delay o he ba yonic mo ion ela i e o
he global ca ie
e−iΩ
. The p opaga ion ke nel
GΩ
(x
,
x
′
) ephases each emi e
be o e summa ion, so ha
A
(x) is he phase–ma ched p ojec ion o he sou ce
on o he esonan ield.
Fo wo isola ed emi e s his becomes anspa en . Assign complex d i e
ampli udes
δρ1eiϕ1, δρ2eiϕ2,(78)
a x1and x2. Each gene a es a complex e ec i e cu a u e ield
δΦp1(x)eiϕ1, δΦp2(x)eiϕ2,(79)
so ha he o al Channel–1 en elope is
A(x)=δΦp1(x)eiϕ1+δΦp2(x)eiϕ2.(80)
The ela i e phase
∆ϕ=ϕ1−ϕ2(81)
de e mines whe he he supe posi ion is cons uc i e o des uc i e. I he
sou ces main ain a ixed con igu a ion o ela i e mo ion, ∆
ϕ
emains cons an
in he global esonan ame, p oducing a s a iona y in e e ence pa e n and a
ime–independen composi e en elope.
In his disc e e se ing he ole o he complex ampli udes is ully isible:
δρ1,2eiϕ1,2
desc ibes he phase–weigh ed ba yonic d i e, he ke nel applies he addi ional
p opaga ion phase
eikΩ i
, and Eq.
(80)
is he esul ing phaso sum. This is
p ecisely he wo–elemen analogue o he cohe en in eg al
A(x) = G
c2
e Zd3x′GΩ(x,x′)δρb(x′) (82)
in oduced in Eq. (56).
I many emi e s sa is y
∆ϕ(x)≈cons ,(83)
hei con ibu ions add cohe en ly and he en elope ampli ude g ows app oxi-
ma ely wi h he numbe o pa icipan s. I ins ead hei phases a e andom,
ei∆ϕ= 0,(84)
he c oss– e ms cancel s a is ically, lea ing only sel –in ensi y con ibu ions and
p oducing an incohe en , speckle–like backg ound.
This phase–locking beha iou is he disc e e p o o ype o he cohe en p ojec ion
mechanism go e ning ex ended ba yonic sys ems. I o ms he concep ual b idge
o he collec i e esonan modes de eloped in he ollowing sec ions, whe e he
in e e ence o many emi e s shapes he la ge–scale esonan en elope and he
eme gen e ec i e densi y ρe .
41
A mode emains phase–cohe en only i
ω( n)(1 −m)−Ω = 0,(108)
i.e.
ω( n) = Ω
1−m.(109)
Only m= 1 allows a physical solu ion,
ω( 0) = Ω,(110)
while all m= 1 dephase unde di e en ial o a ion.
I is con enien o w i e he sou ce in a o m ha makes he empo al equencies
explici ,
ρb( , θ, ) =
∞
X
m=−∞
ˆρm( ) exp−i[(Ω −ω( )) −(1 −m)θ],(111)
wi h mode equency
∆Ωm( )=Ω−(1 −m)ω( ).(112)
Only
m
= 1 sa is ies ∆Ω
1
(
) = 0 in a la – o a ion disk, and he e o e becomes
s ic ly ime–independen ,
ρb,1( , θ, ) = ˆρ1( )eiθ.(113)
All o he ha monics acqui e oscilla o y ac o s
e−i∆Ωm( )
and dephase. In a
high–
Q
medium he e ec i e cu a u e ield esponds only o he long– ime
a e age,
ρb( , θ, )T−−−→
T→∞ ˆρ1( )eiθ.(114)
Thus he cohe en d i e o he e ec i e cu a u e ield is p ecisely he
m
= 1
componen .
On scales la ge compa ed wi h he in e s ella spacing, he disc e e sou ce is
eplaced by a smoo h su ace densi y Σb( , θ) wi h e ical p o ile Z0(z),
ρb( , θ, z)≃Σb( , θ)Z0(z).(115)
The azimu hal a e age and i s momen a e
Σ0( ) = 1
2πZ2π
0
Σb( , θ)dθ, M1( ) =Z2π
0
Σb( , θ)e−iθ dθ. (116)
We pa ame ize he cohe en m= 1 componen as
Σb,1( , θ) = ϵ( ) Σ0( )eiθ,(117)
so ha
M1( )=2π ϵ( ) Σ0( ), ϵ( ) = M1( )
2πΣ0( ).(118)
48

This cohe en momen
M1
(
) is he unique long– ime d i e o he global
m
= 1
spi al esonance.
In Sec ion 7.1.3, he s a iona y en elope equa ion is educed o a 2D Helmhol z
p oblem o he midplane e ec i e cu a u e ield. The sou ce e m appea ing
he e is p ecisely
ρb,1
(
, θ
), o equi alen ly he adial momen
M1
(
). The e-
mainde o he analysis he e o e p oceeds by inse ing he cohe en i s momen
in o he Helmhol z equa ion o ob ain he adial en elope, he ou going G een
unc ion, and ul ima ely he ull spi al mo phology.
7.1.3
The Thin–Disk Model: Sol ing he 2D Helmhol z Equa ion
Re eals he Halo
The cohe en i s momen
M1
(
) ob ained in he p e ious subsec ion is he
unique long– ime d i e o he global
m
= 1 cu a u e mode. We now de e mine
he spa ial s uc u e o he co esponding ield by sol ing he midplane Helmhol z
p oblem. This cons i u es he p ojec ion o he ull h ee–dimensional eigenmode
es ablished in he Theo em in Sec ion 7.1.1 on o he hin disk.
3
The hin–disk
ea men ep esen s he limi ing case
ℓz/Ldamp →
0, p o iding an accu a e
desc ip ion o he galac ic midplane and cap u ing he dominan la e al s uc u e
o he cohe en
m
=1 space ime mode. The 3D model will be buil om his in
he sec ion ha ollows.
In his limi (
h≪
), he s a iona y en elope equa ion
(55)
educes in he
midplane o a wo–dimensional Helmhol z o m,
∂2
∂ 2+1
∂
∂ +1
2
∂2
∂θ2+k2
ΩA( , θ) = 4πG
c2
e
ρb( , θ),(119)
whe e ρb( , θ) deno es he p ojec ed ba yonic densi y o he disk.
Expanding bo h he cu a u e en elope and he ba yonic sou ce in azimu hal
Fou ie modes,
A( , θ) =
∞
X
m=−∞
Am( )eimθ, ρb( , θ) =
∞
X
m=−∞
ρm( )eimθ,(120)
is jus i ied because any squa e–in eg able unc ion on he ci cle (
θ∈
[0
,
2
π
))
admi s a comple e Fou ie ep esen a ion a each adius
. This decomposi ion
ollows om he in insic 2π–pe iodici y in θ.
The i s azimu hal coe icien ,
ρ1
(
), co esponds o he same phase–weigh ed
d i e
M1
(
) in oduced in Eq.
(118)
. Subs i u ing he expansions
(120)
in o
Eq. (119) yields, o each mode m,
d2Am
d 2+1
dAm
d +k2
Ω−m2
2Am=4πG
c2
e
ρm( ).(121)
3
In his p ojec ion he en elope
A
(
, θ
) ep esen s he la e al componen o he ull mode
A( , θ, z)=A( , θ)Z0(z), whe e Z0(z) is he lowes e ical eigen unc ion o scale heigh ℓz.
49
The homogeneous solu ions o Eq.
(121)
a e he cylind ical Bessel unc ions
Jm
(
kΩ
) and
Ym
(
kΩ
), o equi alen ly he ou going Hankel unc ion
H(1)
m
=
Jm
+
iYm
, which ep esen s a adia ing, causally damped wa e (see Appendix B
o de ini ions and asymp o ic o ms). The cohe en spi al a ac o co esponds
o he dominan
m
=1 azimu hal mode. Iden i ying he s a iona y en elope as
A
(
, θ
) =
eiθM1
(
)
/
(Γ +
i
Ω) in Eq.
(119)
yields he adial Helmhol z equa ion
o M1( ):
d2M1
d 2+1
dM1
d +k2
Ω−1
2M1=4πG(Γ+iΩ)
c2
e
ρ1( ).(122)
The unique ou going–wa e G een unc ion o he 2D Helmhol z ope a o , con-
sis en wi h he 3D no maliza ion (∇2+k2
Ω)G=−4πδ3( ), is
G1( , ′) = iπ J1(kΩ <)H(1)
1(kΩ >).
This no maliza ion ensu es ha he a – ield asymp o ic beha iou
A
(
)
∝
(
GM1/c2
e
)
eikΩ /
ma ches he h ee–dimensional ou going solu ion o Eq.
(56)
.
The co esponding G een– unc ion solu ion o Eq. (122) is
A( , θ) = eiθ 4π2iG
c2
e "J1(kΩ )Z∞
′M1( ′)H(1)
1(kΩ ′)d ′
+H(1)
1(kΩ )Z
0
′M1( ′)J1(kΩ ′)d ′#,
(123)
The i s in eg al ep esen s he in e io (sou ce–domina ed) egion ha emains
egula a
= 0, while he second desc ibes he ex e io esponse ha decays
ou wa d and sa is ies he ou going (Somme eld) bounda y condi ion.
I is con enien o de ine he complex adial ampli ude
A ( ) = 4π2iG
c2
e "J1(kΩ )Z∞
′M1( ′)H(1)
1(kΩ ′)d ′
+H(1)
1(kΩ )Z
0
′M1( ′)J1(kΩ ′)d ′#,
(124)
so ha he wo–dimensional ield can be w i en compac ly as
A( , θ) = eiθ A ( ).(125)
This sepa a ion makes explici ha all geome ic and esonan in o ma ion esides
in he adial en elope
A
(
), while he azimu hal dependence
eiθ
simply encodes
he
m
=1 o a ion. The same de ini ion o
A
(
) will be used in Sec ion 7.1.4 o
cons uc he h ee–dimensional ield
A
(
, θ, z
) and in Sec ion 7.1.6 o e alua e
he s o ed ene gy and e ec i e da k–mass dis ibu ion.
50
7.1.4 Ex ending he Thin–Disk o Th ee Dimensions
The wo–dimensional Helmhol z o mula ion de i ed abo e cap u es he dominan
adial and azimu hal s uc u e o he esonan halo, bu a comple e desc ip ion
o he cohe en
m
= 1 mode equi es inclusion o he ini e e ical ex en o
he disk. In cylind ical coo dina es (
, θ, z
) he s a iona y cu a u e en elope
sa is ies he ull h ee–dimensional Helmhol z equa ion,
∂2
∂ 2+1
∂
∂ +1
2
∂2
∂θ2+∂2
∂z2+k2
ΩA( , θ, z) = 4πG
c2
e
ρb( , θ, z),(126)
whe e
ρb
(
, θ, z
) is he ba yonic sou ce con ined o he hin midplane and i s
immedia e su oundings. The e ec i e cu a u e ield ex ends beyond his laye
in
z
, bu i s e ical p o ile is limi ed no by he empo al damping leng h
Ldamp
=
ce /
Γ, bu by a geome ic scale heigh
Lz
ha e lec s he ini e hickness
o he esonan laye i sel .
Assuming a sepa able o m
A
(
, θ, z
) =
A
(
, θ
)
Z
(
z
) leads o he e ical eigen-
alue equa ion
d2Z
dz2+k2
zZ= 0,(127)
in which he e ical wa enumbe
kz
is de e mined om he local dispe sion
ela ion
k2
Ω=k2
+m2
2+k2
z,so ha k2
z=k2
Ω−k2
−m2
2.(128)
He e
k
(
) deno es he local adial wa enumbe o he midplane solu ion, which
may be es ima ed in WKB o m as
k2
≃ |∂ ln A |2
. Bo h
k
and
kz
a y only
slowly wi h
, so he p oduc ansa z
A
(
, θ, z
)
≃A
(
, θ
)
Z0
(
z
;
) emains accu a e
in he adiaba ic ( hin–disk) limi .
The lowes –o de , e en e ical solu ion consis en wi h midplane symme y and
exponen ial decay a la ge |z|is
Z0(z; ) = e−|z|/Lz( )
p2Lz( ),(129)
whe e he e ical scale heigh ollows om he e anescen b anch o Eq.
(128)
as
L−1
z( ) = pℜ[−k2
z( )].(130)
The no maliza ion in Eq.
(129)
ensu es
R∞
−∞|Z0|2dz
= 1, so ha
R|A|2dV
is
iden ical o he midplane wo–dimensional in eg al o
|A |2
. Typically
Lz≫ℓz
bu
Lz≪Ldamp
, so he e ec i e cu a u e ield emains e ically s a i ied a he
han illing he ull damping leng h.
The same G een– unc ion no maliza ion used in he hin–disk solu ion - see
Eq.
(123)
- is p ese ed he e, ensu ing ha he h ee–dimensional ex ension
e ains he co ec a – ield ampli ude and ou going–wa e beha io .
51
The comple e h ee–dimensional eigenmode is he e o e
A( , θ, z) = eiθ e−|z|/Lz( )
p2Lz( )A ( ),(131)
whe e
A
(
) is de ined in Eq.
(124)
. Equa ion
(131)
hus de ines he p ope ly
no malized h ee–dimensional cu a u e en elope o he undamen al (
m, n
) =
(1
,
0) mode. The p e ac o combines he global phase and e ical con inemen ,
while he adial ampli ude
A
(
) encapsula es he ull ho izon al esponse o he
esonan disk h ough he ba yonic sou ce dis ibu ion
M1
(
). This e ically sel –
consis en , azimu hally o a ing Lague e–Gaussian mode he e o e ep esen s he
undamen al h ee–dimensional eigen unc ion o he ea lie Theo em, wi h
A
(
)
se ing he obse able halo mo phology and
Z0
(
z
;
) de ining a ini e, physically
de e mined scale heigh Lz( ) ha ollows om he local dispe sion ela ion.
7.1.5 Gain om Incohe en Foldy–Lax Sca e ing
The global
m
= 1 Lague e–Gaussian mode is selec ed cohe en ly a he le el o
he ba yonic sou ce h ough he i s azimu hal momen
ρ1
(
), which ex ac s
he
eiθ
o a ional ha monic associa ed wi h he one–a med spi al. Once launched,
howe e , he subsequen build–up o he cu a u e ield wi hin he galac ic
medium is no go e ned by ully cohe en in e e ence bu by he incohe en
mul iple–sca e ing dynamics desc ibed in Sec ion 4.2. The long p opaga ion
imes implied by he educed wa e speed
ce ≪c
in oduce o de –uni y phase
delays be ween di e en sca e ing pa hs a he ca ie equency Ω, placing he
sys em i mly in he egime whe e Foldy–Lax heo y p edic s in ensi y addi ion
a he han cohe en ampli ude summa ion. Acco dingly, he ne esonan
ampli ude obeys he scaling
|Ae | ∝ Q|X|pN⋆ϵ, (132)
a he han
Q|X|N⋆
, e lec ing he andomiza ion o empo al phases om
successi e esca e ing e en s.
C ucially, his empo al incohe ence does no des oy he
m
= 1 spa ial s uc u e
o he mode. The eason is he o a ional symme y o he Helmhol z G een
unc ion ha go e ns cu a u e p opaga ion in he hin–disk geome y. In pola
coo dina es he ke nel depends only on he sepa a ion angle,
GΩ( , θ; ′, θ′) = GΩ( , ′, θ −θ′),(133)
and is he e o e in a ian unde global o a ions (
θ, θ′
)
7→
(
θ
+
φ, θ′
+
φ
). This
in a iance implies ha he p opaga ion ope a o commu es wi h he azimu hal
gene a o
∂/∂θ
, so he Fou ie ha monics
eimθ
o m an o hogonal eigenbasis.
Consequen ly, a ield o he o m
A(0)( , θ) = A(0)
( )eiθ,(134)
52
sca e s acco ding o
A(1)( , θ) = ZGΩ( , ′, θ −θ′)A(0)
( ′)eiθ′ ′d ′dθ′.(135)
Pe o ming he
θ′
-in eg a ion isola es only he
m
= 1 ha monic o he ke nel,
yielding
A(1)( , θ) = eiθ A(1)
( ),(136)
wi h he angula dependence
eiθ
p ese ed exac ly. Since he ke nel con ains no
m
= 1 componen s, no azimu hal mixing can occu unless he medium i sel
b eaks axisymme y. I e a ing he Foldy–Lax expansion he e o e main ains he
m
= 1 dependence a e e y sca e ing o de , e en hough he empo al phases o
successi e gene a ions o he ield become unco ela ed.
The esul ing physical pic u e is ha he mode (
m
= 1) is selec ed cohe -
en ly by he s ella disk, bu he magni ude o ha mode is se by incohe en ,
phase– andomized mul iple sca e ing in he cu a u e medium. Ro a ional
symme y p ese es he angula s uc u e, while he long p opaga ion delays
en o ce he
√N⋆
Foldy–Lax scaling encoded in Eq.
(132)
. Thus he eme gen
one–a med spi al is a global esonan s a e whose geome y is ixed by cohe en
ba yonic o cing and symme y, bu whose ampli ude is de e mined by he s a-
is ical accumula ion o many incohe en sca e ing e en s wi hin he galac ic
en i onmen .
7.1.6
S o ed Ene gy, Da k–Mass Equi alence, and En i onmen al
Gain
The obse able (quasi–s a ic) cu a u e bias a ises om he s o ed oscilla o y
ene gy o he esonan ield. F om Sec ion 6.2, he local ield ene gy densi y and
i s mass equi alen a e
EΦ(x) = 1
8πGhc2
e |∇A|2+ Ω2|A|2i, ρe (x) = EΦ(x)
c2.(137)
Subs i u ing he no malized ield
A
(
, θ, z
) o Eq.
(131)
and sepa a ing a iables
gi es
|∇A|2=∂ A 
2|Z0|2+|A |2
2|Z0|2+|A |2∂zZ0
2.(138)
Fo he exponen ial e ical p o ile
Z0
(
z
;
) =
e−|z|/Lz( )/p2Lz( )
, we ha e
∂zZ0
2≃ |Z0|2/L2
z. The ene gy densi y he e o e becomes
ρe ( , θ, z) = |A( , θ, z)|2
8πGc2hΩ2+c2
e ∂ ln A 
2+1
2+1
L2
zi.(139)
In he adiaba ic WKB limi , he local dispe sion ela ion
k2
Ω
=
k2
+
m2/ 2
+
k2
z
(wi h he e ical e m eplaced by i s e anescen magni ude 1
/L2
z
) ende s he
53

b acke app oxima ely cons an :
Ω2+c2
e k2
+m2
2+1
L2
z≃Ω2+c2
e k2
Ω≈2 Ω2,(140)
since kΩ= Ω/ce and Γ≪Ω. Thus, o good app oxima ion,
ρe ( , θ, z)≃2 Ω2
8πGc2|A( , θ, z)|2.(141)
To include he cumula i e e ec o mul iple sca e ing in a s a is ically homoge-
neous ensemble, he ield ampli ude is scaled by he en i onmen al ein o cemen
ac o ,
A( , θ, z)−→ Ae ( , θ, z) = QpGen A( , θ, z),(142)
whe e
Q
is he empo al quali y ac o o he cu a u e esonance, and
Gen
ep esen s s a is ical ampli ica ion o ield ampli ude h ough epea ed sca e ing
and e–emission. Fo a long–wa eleng h ield (
λΩ≫
mean in e –sca e e spacing),
indi idual sca e e s espond addi i ely in ampli ude, yielding
Gen =N⋆|X|2.(143)
whe e
N⋆
is he numbe o ba yonic sca e e s wi hin he e ec i e cohe ence
olume and
|X| ≤
1 quan i ies hei mean coupling e iciency o he global
oscilla ion equency Ω. The ac o
|X|
hus measu es how s ongly each sca e e
pa icipa es in he collec i e esonance.
Subs i u ing Eq.
(142)
in o Eq.
(141)
gi es he inal exp ession o he e ec i e
ene gy densi y:
ρe ( , θ, z) = 2 Ω2
8πGc2Gen Q2|A( , θ, z)|2,Gen =N⋆|X|2.(144)
The esul
(144)
he e o e p o ides he comple e desc ip ion o he s a is ically
ampli ied da k–mass densi y in e ms o he in insic mode ampli ude, esonance
quali y ac o , and he e ec i e numbe o coupled ba yonic sca e e s.
7.1.7 Nume ical Resul s: 3D Thin–Disk Model
The comple e nume ical esul s ob ained om he h ee–dimensional hin–disk
model o Sec ion 7.1.4 a e shown in Figu es 4–6. All quan i ies and algo i hms
a e implemen ed exac ly as speci ied in he simula ion code gi en in Appendix C.
The o a ion equency is ixed by he la ci cula speed
0
= 230
km s−1
a
a e e ence adius
0
= 8
.
5
kpc
, gi ing Ω = 8
.
77
×
10
−16 s−1
and a esonan
wa eleng h
λ es ≃
46
.
44
kpc
. An e ec i e p opaga ion speed
ce
= 2
×
10
5m s−1
is adop ed, while he quali y ac o
Q
= 12
.
6 is chosen such ha he esul ing
damping leng h
Ldamp
=
ce /
Γ
≃
192
.
17
kpc
(whe e Γ = Ω
/
(2
Q
)) ep oduces
54
he obse ed halo ex en and he asymp o ic la ening o ypical spi al–galaxy
o a ion cu es.
In his o mula ion he o al esonan ampli ude a ises di ec ly om he combina-
ion o he empo al cohe ence
Q
, coupling e iciency
X
, and geome ic p ojec ion
ac o
ϵ
, gi ing an o e all scaling p opo ional o
Q|X|√N⋆ϵ
. Unlike ea lie
pa ame e iza ions in which ex e nal gain was inse ed by hand, hese pa ame e s
now en e sel –consis en ly h ough he in eg al Helmhol z solu ion. The esul ing
ield s eng hs, densi y con as s, and o a ion cu es emain iden ical in o m o
hose o he o iginal model, con i ming he in e nal no maliza ion o he upda ed
implemen a ion.
Figu e 4 p esen s he one–dimensional adial solu ions de i ed om he cohe en
m
= 1 in eg al o mula ion. Pa (a) shows he ield ampli ude
|A
(
)
|
, which
ises om ze o a he o igin o a well–de ined maximum nea
≃
6
kpc
, be o e
declining exponen ially on scales compa able o he damping leng h. Pa (b)
displays he loga i hmic mass densi ies: he e ec i e cu a u e densi y
ρe
(blue)
and he ba yonic midplane densi y
ρb
(o ange dashed). The e ec i e componen
dec eases almos linea ly in log–space and app oaches he backg ound le el nea
≃
650
kpc
. Pa (c) compa es he ba yonic, esonan , and o al o a ion cu es.
The o al eloci y emains app oxima ely la beyond
∼
10
kpc
, consis en
wi h obse ed spi al–galaxy kinema ics. Pa (d) shows he enclosed–mass a io
Me /Mb
, which inc eases om
∼
10 a 7
kpc
o
∼
14 a 100
kpc
,
∼
17 a 150
kpc
,
and app oaches i s asymp o ic alue o
≃
20 by
≃
200–250
kpc
. These esul s
con i m ha he hin–disk model eco e s bo h he ou e la ening o he o a ion
cu e and he canonical da k– o–ba yon mass a io in e ed o Milky–Way–like
galaxies.
Figu e 5 illus a es he wo–dimensional midplane s uc u e o he same
m
= 1
esonan mode o e a 100
×
100
kpc
domain. Pa (a) shows he no malized
ampli ude
|A|
, e ealing he expec ed o oidal mo phology peaking a
≃
6
kpc
. Pa (b) displays he no malized azimu hal phase
a g
(
A
), which o ms a
con inuous one–a med spi al wi h a single 2
π
winding, indica ing a cohe en global
m
= 1 mode. Pa (c) o e lays he ba yonic ( ed) and esonan (g een) densi ies,
showing hei pa ial spa ial displacemen wi hin he disk plane. Pa (d) shows
he o al (ba yonic + esonan ) densi y on a linea scale; only he b igh ba yonic
co e and nea by esonan en elope a e isible, as he ex ended halo alls below
he display con as .
Figu e 6 shows he co esponding h ee–dimensional e ec i e–densi y dis ibu ion.
Pa (a) p esen s he ho izon al (XY) midplane slice, and pa (b) he e ical
(XZ) c oss–sec ion, each o e a 200
×
200
kpc
domain. The XY p ojec ion
again displays he o oidal halo s uc u e, whe eas he XZ p ojec ion e eals a
e ically s a i ied en elope ex ending o heigh s o o de
|z|∼Lz≃
57
.
65
kpc
,
compa able o he esonan wa eleng h
λ es ≃
46
.
44
kpc
. This scale ollows
om he e anescen e ical b anch o he disk dispe sion ela ion and indica es
ha he e ec i e cu a u e ield emains con ined wi hin a ew wa eleng hs
o he midplane a he han illing he en i e damping egion. The esul ing
55
Figu e 4: Radial p o iles and o a ion cu es. (a) Field ampli ude
|A
(
)
|
;
(b) e ec i e and ba yonic mass densi ies (loga i hmic); (c) o a ion cu es o
ba yonic, esonan , and o al componen s; (d) enclosed–mass a io
Me /Mb
.
Resul s om he 3D hin–disk Helmhol z model (Sec ion 7.1.4); nume ical
pa ame e s as de ined in Appendix C.
56
Figu e 5: Midplane ampli ude and densi y maps. (a) No malized ampli-
ude
|A|
showing he o oidal esonan s uc u e; (b) no malized phase
a g
(
A
)
exhibi ing he spi al
m
= 1 winding; (c) combined ba yonic ( ed) and esonan
(g een) densi ies; (d) o al no malized densi y on a linea scale. Spa ial domain:
100 ×100 kpc.
57
In he sou ce– ee ex e io egion, whe e he e ec i e cu a u e ield becomes
s a is ically iso opic and di usi e, he s eady–s a e solu ion implies an ensemble–
a e aged densi y p o ile
ρe ( )∝exp[−2 /Ldamp,e ]
2,(150)
wi h anspo damping leng h
Ldamp,e =ce
Γe
,(151)
whe e ce is he e ec i e cu a u e–wa e speed.
In e p e a ion. Equa ion
(148)
exp esses he condi ion o s a is ical s a iona i y:
he e ec i e g a i a ing densi y gene a ed by andom s ella mo ion balances
he densi y emo ed by anspo damping. Equa ions
(150)
–
(151)
hen gi e
he esul ing spa ial o m—a anspo – egula ed cu a u e halo wi h an
−2
in ensi y ail and an exponen ial cu o se by Ldamp,e .
7.2.2 Complex Ba yonic Fo cing wi h Memo y
A key esul o his pape is ha he cu a u e–en elope ield
A
(x
,
) is he
phononic exci a ion o he esonan e ec i e cu a u e ield o space ime, sou ced
and sca e ed by ba yons.
4
Th oughou wha ollows we he e o e in e p e
A
as a con inuous ield o cu a u e phaso s whose in e e ence ca ies s ess and
ene gy h ough he sys em. In egions whe e hese phaso s ein o ce cohe en ly,
he ield o ms localized packe s o comp ession and a e ac ion—phonon-like
cu a u e wa epacke s which we e e o as massons (Sec ion 3.3). These a e no
ma e ial quan a bu localized solu ions o he same linea , causal PDE de i ed
ea lie .
The go e ning dynamics a e elas ic, obeying he same o ced–damped Helmhol z
equa ion and anspo iden i ies as in acous ic mul iple sca e ing. One may
pic u e a as ensemble o andomly dis ibu ed esonan sca e e s, each ac ing
as a kinema ic sou ce while simul aneously obeying a common o ced–damped
oscilla o esponse ha de ines
χ
(Ω), and he e o e bo h emi ing and esonan ly
sca e ing cu a u e phonons. This is di ec ly analogous o he esonan – od
expe imen s o De ode e al. [
4
,
5
], in which mul iple sca e ing o di use ul-
asound p oduces cohe en ampli ica ion and long– ange in e e ence wi hin a
andom elas ic ne wo k [16, 17, 18]. 5
4
Recall ha
A
(x
,
) is he slowly a ying complex ampli ude (en elope) o he esonan
e ec i e cu a u e ield, Φ
es
=
A e−iΩ
+
c.c.
. I ep esen s a con inuous ield o cu a u e
phaso s a ising om he same o ced–damped oscilla o equa ion ha go e ns single-mass
suscep ibili y χ(Ω).
5
In he acous ic case he ods a e d i en by an ex e nal sou ce; in he g a i a ional sys em
each mass elemen is a sel –d i en kinema ic emi e , while he same causal o ced–damped
PDE de e mines how i s emission p opaga es, sca e s, and damps.
64

In such a sca e ing en i onmen , he ini e anspo mean ee pa h educes he
spa ial cohe ence o he esonan e ec i e cu a u e ield while simul aneously
ex ending i s e ec i e damping leng h. These wo e ec s (sho e cohe ence,
longe di usi e decay) go e n how cu a u e ene gy is edis ibu ed h ough
he medium and ul ima ely de ine he smoo h, aniso opic en elope o he
e ec i e densi y ield. As shown below, his ansi ion om cohe en o di usi e
p opaga ion is p ecisely wha gi es ise o he ex ended, ellip ical mo phology o
he da k–mass halo.
In spi al galaxies, he low–en opy
m
=1 o bi al o de en o ced a nea ly uni o m
geome ic phase. The complex sou ce e m
mneiϕn
was he e o e a cohe en
phaso sum: he d i ing was essen ially ins an aneous and spa ially phase–locked
(see Sec ion 7.1). By con as , ellip ical galaxies hos a phase–mixed s ella
popula ion. The geome ic phase
ϕn
e ol es s ochas ically due o andom s ella
eloci ies, des oying long– ange cohe ence. Cohe ence su i es only locally in
space ( ans e se scale ℓ⊥) and in ime (memo y scale τcoh).
The co ec sou ce o cu a u e phaso s in an ellip ical is he e o e he phaso
ep esen a ion o he ba yonic o cing e m in he same causal PDE:
ρb(x, ) =
N⋆
X
n=1
m⋆eiϕn( )δ(3)
x−xn( ),(152)
which d i es he o ced–damped Helmhol z ope a o
∇2+k2
ΩA(x, ) = 4πG
c2
e
ρb(x, ),(153)
whe e
m⋆
is he mass o a ep esen a i e emi e , x
n
(
) i s ins an aneous posi ion,
and
ϕn
(
) i s geome ic phase ela i e o he local ield. This is he same d i en,
damped equa ion used in he cohe en (spi al) case; only he phase s a is ics
di e .
The ield ampli ude
A
(x
,
) may be w i en using he causal (single–sca e e )
G een unc ion:
A(x, ) = 4πG
c2
e
N⋆⋆
X
n=1
m⋆eiϕn( )GΩ
x−xn( ),(154)
The causal G een unc ion
GΩ
( ) desc ibes he esponse o a single esonan ly
d i en s ella sou ce. I s ampli ude decays in he a ield as
|GΩ( )|2=e−2| |/Ldamp
(4π)2| |2, Ldamp =ce
Γ,(155)
ep esen ing he causal damping o an indi idual cu a u e phaso . Ensemble
a e aging o e many such phaso s will la e p oduce he anspo – eno malized
ke nel.
65
In Eq. (154), N⋆⋆ deno es he ull popula ion o pas emi e s wi hin he causal
memo y window
−τmem < ′≤ ,
whe e
τmem
= Γ
−1
is he memo y ime o
he unde lying o ced–damped oscilla o esponse and
τcoh
= (2Γ)
−1
i s sho e
phase–deco ela ion ime. Equa ion
(154)
he e o e ep esen s a ou –dimensional
ensemble o phaso s, each ca ying i s causal phase d i ac oss he memo y
window. No all o he
N⋆⋆
phaso s a e s a is ically independen : co ela ions
pe sis o e ini e egions o space and ime, de ining a ou –dimensional speckle
scale de e mined by he cohe ence o he same unde lying PDE. The ollowing
subsec ion he e o e de ines he cha ac e is ic size o a ou –dimensional speckle
cell, wi hin which phaso s emain phase–co ela ed. De e mining his cohe ence
olume and imescale allows he numbe o s a is ically independen phaso s
wi hin
N⋆⋆
—and he co esponding ba yonic mass— o be quan i ied. These
quan i ies a e hen used in Sec ion 7.2.4 o ew i e Eq.
(154)
in e ms o he
independen phaso ensemble and o compu e ensemble a e ages such as
⟨|A|2⟩
and ⟨ρe ⟩.
7.2.3 Fou –dimensional speckle geome y o he g a i a ional ield
To ex ac s a is ical p edic ions om Eq.
(154)
, we mus de e mine o e wha
egions o space and ime he g a i a ional ield
A
(x
,
) e ains phase cohe ence.
Because ellip ical galaxies lack global o bi al o de , cohe ence is nei he disk–wide
no s eady in ime: i is localized and ansien . The ield he e o e o ms a
g anula , ou –dimensional speckle pa e n— he g a i a ional analogue o di use
ul asound speckle.
T ans e se cohe ence. Random s ella mo ion p oduces geome ic phase d i .
When he phase di e ence be ween wo emi e s eaches one adian, hei con i-
bu ions become s a is ically unco ela ed. This de ines he ans e se cohe ence
leng h
ℓ⊥=λ es
2π,(156)
so phaso s launched om wi hin a ans e se pa ch o adius
ℓ⊥
in e e e cohe -
en ly.
Longi udinal cohe ence. Along a s a ’s ajec o y, o bi al mo ion also sc ambles
phase in ime. Du ing he cohe ence in e al
τcoh
he ypical o bi al excu sion is
ℓ∥= τcoh =
2Γ,(157)
so emissions sepa a ed by mo e han ℓ∥along an o bi a e deco ela ed.
Space– ime cohe ence cell. The smalles egion o e which phaso s e ain a
common phase is a ans e se disk o adius
ℓ⊥
ex ended o e a longi udinal
dis ance ℓ∥. I s spa ial olume is
Vcell =α ℓ2
⊥ℓ∥=αλ es
2π2
2Γ,(158)
66
whe e
α
is a geome ic ac o o o de uni y. The co esponding empo al ex en
is
∆ cell =τcoh =1
2Γ.(159)
This pai (
Vcell,
∆
cell
) is he g a i a ional analogue o he isoplana ic pa ch and
cohe ence ime o di use acous ics.
Popula ion s a is ics. Le
Vgal
deno e he ac i e ba yonic olume. The numbe o
disjoin cohe ence cells a any momen is
N3D =Vgal
Vcell
,(160)
and each cell con ains on a e age
Ncell =n⋆Vcell =N⋆Vcell
Vgal
,(161)
s a s, whe e n⋆is he mean s ella numbe densi y.
E ec i e phaso mass. Wi hin a single cell all s a s sha e, o leading o de , a
common ield phase. Thei masses he e o e add cohe en ly as one composi e
phaso o e ec i e mass
Mcell =Ncell m⋆.(162)
The o iginal s ella sum in Eq.
(154)
may hus be e-exp essed as a sum o e
s a is ically independen ou –dimensional cohe ence cells, each weigh ed by i s
complex mass Mcell and he app op ia e G een ke nel.
In summa y, cohe ence in ellip icals is local in space and ini e in ime. The
g a i a ional ield is d i en no by a global mode bu by a ou –dimensional
mosaic o s a is ically independen phaso ensembles. Rew i ing Eq.
(154)
in his
s a is ical basis enables he e alua ion o ensemble quan i ies such as
⟨|A|2⟩
and
he mean e ec i e densi y ⟨ρe ⟩, as de eloped in he ollowing subsec ion.
7.2.4 Ensemble educ ion: masson–speckle summa ion
We now ew i e he o cing in Eq.
(154)
in e ms o he s a is ically independen
ou –dimensional cohe ence cells de i ed in Sec ion 7.2.3. Each spa ial cohe ence
olume
Vcell
=
α ℓ2
⊥ℓ∥
de ines an elemen a y uni o he ensemble en i onmen ,
wi hin which he s ella phaso s emain phase–locked and he local e ec i e
cu a u e ield e ol es cohe en ly o e he memo y ime
τcoh ≃
1
/
(2Γ). Cells
sepa a ed by mo e han ℓ⊥in space o ∆ cell in ime a e ully deco ela ed.
S a s wi hin a common cell he e o e ac as a single cohe en sou ce o e ec i e
mass
Mcell
, while he ull galaxy o ms a andom, ime– a ying mosaic o such
locally cohe en “masson” exci a ions. The e ec i e cu a u e ield can hus be
exp essed as a supe posi ion o s a is ically independen phaso s:
A(x, ) = 4πG
c2
e
N3D
X
c=1
NT
X
j=1
M(c,j)
cell GΩ
x−xc,(163)
67
whe e
c
indexes spa ial cells and
j
indexes independen empo al ealiza ions
wi hin he ield’s memo y window.
Tempo al mul iplici y: cu a u e memo y. Each spa ial cell s o es a ini e numbe
o s a is ically dis inc phaso s wi hin i s cohe ence ime. Using he de ini ions
o
τcoh
and he oscilla ion pe iod 2
π/
Ω, he numbe o independen empo al
ealiza ions pe cell is
NT=τcoh
∆ cell
=
1
2Γ
2π
Ω
=Ω
4πΓ=Q
2π,(164)
so ha each cell con ibu es
NT
independen empo al samples o i s masson
exci a ion.
In ensi y expansion. Taking he squa e modulus o Eq. (163) gi es
|A(x, )|2=4πG
c2
e 2
X
c,j X
c′,j′
M(c,j)
cell M(c′,j′)
cell ∗GΩx−xcG∗
Ωx−xc′,(165)
om which ensemble a e ages may be compu ed. S a is ical independence o
dis inc cells and empo al windows gi es
DM(c,j)
cell M(c′,j′)
cell ∗E=(|Mcell|2, c =c′, j =j′,
0,o he wise, (166)
so all c oss e ms anish. The mean ield in ensi y is he e o e
D|A(x, )|2E=4πG
c2
e 2
NT
N3D
X
c=1 |Mcell|2GΩ(x−xc)
2.(167)
Fo a Poisson s ella dis ibu ion,
|Mcell|2=N2
cellm2
⋆+NcellVa (m⋆),(168)
and o Ncell ≫1 his simpli ies o
|Mcell|2≃M2
cell =N2
cellm2
⋆.(169)
Using N3D =Vgal/Vcell and Ncell =N⋆Vcell/Vgal,we ob ain
N3D M2
cell =Vgal
Vcell N⋆Vcell
Vgal 2
m2
⋆=N2
⋆m2
⋆
Vgal
Vcell.(170)
I is now con enien o collec he ensemble coupling e ms in o a single di-
mensionless en i onmen al gain ac o
Gen
. In Sec ion 4.1 his quan i y was
de ined mic oscopically as he o al in ensi y gain ela i e o he inciden ield
educing in he ully cohe en Foldy–Lax limi o
Gen
=
N2|χ⋆|2
[Eq.
(25)
]. In
68
he e ec i e–medium pic u e o Sec ion 4.3, he same quan i y was shown o
scale in e sely wi h he cohe ence olume
Vcell
h ough
Gen ∝
1
/Vcell
e lec ing
he ac ha la ge cohe ence cells p oduce s onge local phase alignmen .
In he p esen , ully explici cell–coun ing pic u e we include bo h he local
coupling pe cell and he numbe o cells illing he galaxy. To keep ack o
he mean s ella coupling e iciency in oduced ea lie , we w i e he ensemble
en i onmen al gain as
Gen := |X|2N3D |Mcell|2
m2
⋆≃ |X|2N3D N2
cell
=|X|2N2
⋆Vcell
Vgal
=|X|2N2
⋆
Vgal
α ℓ2
⊥ℓ∥
=|X|2N2
⋆
Vgal
αλ es
2π2
2Γ,(171)
whe e
Vcell
=
α ℓ2
⊥ℓ∥
is he spa ial cohe ence olume om Eq.
(158)
,
ℓ⊥
=
λ es/
(2
π
)
is he ans e se cohe ence leng h (Eq.
(156)
), and
ℓ∥
=
/
(2Γ) is he longi udinal
cohe ence leng h (Eq.
(157)
). The dimensionless ac o
|X|2≤
1 ep esen s he
mean s ella coupling e iciency al eady in oduced in he sca e ing heo y: in
he cohe en Foldy–Lax limi i mul iplies he in ensi y as
Gen ∝|X|2N2
, while
he e i weigh s he o al cell–ensemble gain in exac ly he same way. These
successi e equali ies make explici how he ensemble en i onmen al gain e ains
he same concep ual s uc u e as in he sca e ing heo y. The i s line shows
he s a is ical de ini ion, whe e he cohe en mass con en o each cell,
|Mcell|2
,
is weigh ed by he mean coupling e iciency
|X|2
and mul iplied by he numbe o
spa ial and empo al ealiza ions. The second line e o mula es his in geome ic
e ms, in oducing he o al s ella popula ion
N⋆
and he a io
Vcell/Vgal
, which
connec s he mic oscopic cell s a is ics o he mac oscopic galac ic olume. The
hi d line hen exp esses he same quan i y in explici physical scales, eplacing
he cohe ence leng hs (
ℓ⊥, ℓ∥
) wi h he esonan wa eleng h
λ es
and damping
ime 1
/
Γ, he eby e ealing how
Gen
depends on he unde lying causal and
geome ic pa ame e s o he medium.
Thus he ea lie local ela ion
Gen ∝ |X|2/Vcell
o a single cohe en pa ch is
na u ally ex ended he e o he global ensemble gain: he pe –cell esponse scales
as
|X|2/Vcell
, while he o al numbe o cohe en ly con ibu ing sou ces scales as
N2
⋆Vcell/Vgal
. Thei p oduc yields he dimensionless
Gen
ele an o ellip icals.
The eplacemen s Γ
→
Γ
e
and
Ldamp →Ldamp,e
will be in oduced in he
ollowing subsec ion when anspo e ec s a e included.
Subs i u ing Eqs. (164), (170), and (171) in o Eq. (167) inally gi es
D|A(x, )|2E=4πG
c2
e 2Q
2πGen GΩ(x−xc)
2,(172)
69

which makes explici how he en i onmen al gain
Gen
, he empo al memo y
Q/
(2
π
), and he G een– unc ion geome y combine o de e mine he obse able
cu a u e– ield in ensi y in a s a is ically mixed s ella sys em. This esul o ms
he s a ing poin o he anspo –enhanced desc ip ion de eloped in he nex
subsec ion.
7.2.5 T anspo Mean F ee Pa h and Collec i e Damping
The ensemble o mula ion de eloped abo e ea s he e ec i e cu a u e ield as
a supe posi ion o kinema ically d i en s ella sou ces, each o which con ibu es
phase-modula ed cu a u e phaso s wi hin i s local cohe ence window. In ealis ic
galac ic en i onmen s, howe e , each s a ac s simul aneously as (i) a kinema ic
sou ce o cu a u e phonons gene a ed by i s o bi al mo ion, and (ii) a compac
elas ic sca e e go e ned by he same o ced–damped Helmhol z ope a o ha
p opaga es and damps he ield. Th ough con inual emission, sca e ing, and
eabso p ion, hese mass elemen s gene a e a di usi e in e e ence ne wo k whose
s a is ics mi o hose o esonan ul asound and di use ield co ela ions in
complex ma e ials [
13
,
9
,
4
,
5
,
16
,
17
,
18
]. The go e ning dynamics a e elas ic,
obeying he same damped Helmhol z equa ion and anspo iden i ies as acous ic
phonons in a di use medium. Following he o malism o mul iple–sca e ing
heo y [
8
,
19
], he cohe en ield p opaga ion is no de e mined by he mic oscopic
damping a e Γ bu by an e ec i e anspo wa enumbe ha inco po a es he
ensemble sel –ene gy Σ(Ω):
k2
e =k2
0+ Σ(Ω), k0=Ω+iΓ
ce
.(173)
The sel –ene gy Σ(Ω) encodes he cumula i e phase delay and dwell ime p oduced
by he esonan mul iple-sca e ing ne wo k, no by he emi e s hemsel es. I s
imagina y pa de ines he anspo damping a e
Γe =ce Im
ke ,(174)
and he associa ed e ec i e damping leng h
Ldamp,e =ce
Γe
.(175)
In he di usi e egime,
Γe ≪Γ, Ldamp,e ≫Ldamp =ce
Γ,(176)
exp essing he key anspo esul : esonan mul iple sca e ing sho ens spa ial
cohe ence bu g ea ly ex ends he e ec i e damping leng h. Cu a u e phonons
unde go epea ed dwell– ime in e ac ions be o e deco ela ing, con inuously
ecycling pa o hei ene gy back in o he cohe en mode.
This ecycling p oduces an e ec i e quali y ac o
Qe =Ω
2 Γe ≫Q=Ω
2 Γ,(177)
70
so ha he obse ed la ge
Q
alues in ellip ical halos a e no mic oscopic p ope -
ies o single emi e s bu collec i e anspo e ec s o he s ella ensemble. In
his egime, all ensemble ela ions o Sec ion 7.2.4 e ain hei o mal s uc u e
unde he eplacemen s
Γ→Γe , Q →Qe , Ldamp →Ldamp,e ,(178)
and he en i onmen al gain i sel becomes anspo – eno malized h ough he
e ec i e longi udinal cohe ence leng h ℓ∥,e = /(2Γe ), yielding
Gen ,e =|X|2N2
⋆
Vgal
αλ es
2π2
2Γe .(179)
This o m p ese es he same mic oscopic s uc u e as Eq.
(171)
, bu wi h Γ
eplaced e e ywhe e by i s anspo alue Γ
e
o accoun o he collec i e
damping educ ion o he ensemble medium.
The angula ly a e aged G een unc ion o he mul iply sca e ed cu a u e phonon
ield de ines he causal anspo ke nel
Ke ( ) = exp−2 /Ldamp,e 
(4π)2 2,(180)
which ep esen s he sphe ically symme ic ene gy en elope o he cohe en
cu a u e mode. I p o ides he physical basis o he closed halo law de i ed in
he nex subsec ion: a long– ange, slowly decaying e ec i e cu a u e ield whose
ex ended ellip ical mo phology a ises na u ally om he di usi e ecycling o
cu a u e ene gy wi hin he anspo medium.
Subs i u ing Eqs.
(179)
and
(178)
in o he ensemble in ensi y
(172)
, and eplacing
he mic oscopic p opaga o
|GΩ|2
by he anspo ke nel
Ke
(
), yields he
compac anspo –enhanced exp ession
D|A( )|2E=4πG
c2
e 2Qe
2πGen ,e Ke ( ).(181)
This o m makes he anspo eno maliza ion o bo h empo al and spa ial
cohe ence explici :
Gen ,e
now embodies he educed damping a e Γ
e
, while
Qe
and
Ke
(
) encode he ex ended memo y ime and spa ial anspo leng h.
I p o ides he physical closu e o he ensemble halo law, om which he co e
egula iza ion and g adien –ene gy co ec ion will ollow.
7.2.6 Co e Regula iza ion in he Non–T anspo Regime
Be o e in oducing he ull ene gy con e sion in Sec ion 7.2.7, we i s es ablish
he spa ial mo phology implied by he anspo solu ion o he ield in ensi y
⟨|A
(
)
|2⟩
. A his s age, only he ela i e scaling o he en elope is needed: he
absolu e no maliza ion and physical p e ac o s om he cu a u e–ene gy law
71
will be applied in he nex subsec ion. Acco dingly, he discussion below ea s
ρe
(
) as a quan i y p opo ional o
⟨|A
(
)
|2⟩
, wi h he unde s anding ha he
p opo ionali y cons an will be speci ied la e .
The anspo heo y desc ibed in he p e ious sec ion p edic s a a – ield p o ile
p opo ional o
exp
(
−
2
/Ldamp,e
)
/ 2
. This exp ession is exac o he ensem-
ble–a e aged cohe en mode o a mul iply sca e ed cu a u e–phonon ield in
he di usi e egime; ha is, a adii beyond se e al cohe ence leng hs whe e
angula iso opy and s a is ical deco ela ion hold [
19
,
8
]. Howe e , he same
o m canno apply a bi a ily close o he cen e. A small adii: (i) s ella
sca e e s a e closely packed, (ii) di ec (ballis ic) p opaga ion domina es o e
di usion, and (iii) he local speckle ield becomes domina ed by he cohe en
mean– ield cu a u e a he han by ensemble s a is ics. In his inne egion, he
di usion app oxima ion implici in he ensemble– anspo ke nel
Ke
(
)
∝
1
/ 2
b eaks down. As
→
0, Eq.
(181)
would he e o e di e ge unphysically, signalling
he ansi ion om he s a is ical (Dyson) egime back o he cohe en Foldy–Lax
limi .
Cu a u e lux mus emain ini e a he cen e o any bound sys em. Fo a
sphe ically symme ic, s a iona y en elope, he ne cu a u e ene gy lux c ossing
a sphe e o adius mus anish smoo hly as →0; in pa icula , he e can be
no singula poin sou ce o sink a he o igin. W i ing he lux schema ically as
F( )∝4π 2ρe ( ),(182)
egula i y equi es ha F( ) dec ease a leas as as as 3,
F( )∝ 3,( →0),(183)
which implies he small– scaling
ρe ( )∝ , ( ≪ℓ∥),(184)
whe e
ℓ∥
=
/
(2Γ
e
) is he longi udinal cohe ence leng h o he ensemble–a e aged
ield. This linea –co e beha iou is he e o e a na u al, nonsingula choice: i
egula izes he 1
/ 2
di e gence o he di usi e ke nel while ensu ing ini e cen al
lux and a smoo h app oach o he cohe en (inne ) egime.
We choose a ma ching adius
Rma ch
o o de he ou e s ella scale adius, whe e
he ield has unde gone many sca e ings and he anspo desc ip ion becomes
alid:
Rma ch ∼ O( ew Re)∼5 kpc o ypical ellip icals.(185)
The egula ized en elope hen akes he gene ic o m
ρe ( )∝


, < Rma ch,
exp[−2 /Ldamp,e ]/ 2, ≥Rma ch.
(186)
Bo h he densi y and i s e ec i e lux 4
π 2ρe
a e con inuous a
=
Rma ch
by
cons uc ion.
72
The inne linea law is hus no an ad hoc empi ical pa ch bu a egula ized limi
o he anspo solu ion: i emo es he unphysical cen al di e gence o he
di usi e ke nel, keeps he cen al lux ini e, and ma ches smoo hly on o he
exponen ial–o e – 2 anspo ail. I p edic s:
•co ed densi y p o iles in ellip icals and clus e s,
•a la ci cula – eloci y con ibu ion a small adii, and
•a smoo h ansi ion o he exponen ial–o e – 2 anspo ail.
The co e adius is ixed by he cohe ence scale
ℓ∥
=
/
(2Γ
e
), while he ou e
halo is con olled by he anspo damping leng h
Ldamp,e
. Toge he , hese
wo scales de ine a wo–zone s uc u e—a cohe en co e joined con inuously o
a di usi e halo— ha emains ully consis en wi h he anspo hie a chy. In
wha ollows, we in oduce he cu a u e–ene gy con e sion and g adien –ene gy
co ec ion ha comple e he physical no maliza ion o his en elope, yielding he
inal quan i a i e halo law.
7.2.7
G adien –Ene gy Boos and Final T anspo –Regula ized Halo
Law
Ha ing es ablished he spa ial mo phology o he cohe en cu a u e en elope,
we now comple e i s physical no maliza ion by es o ing he cu a u e–ene gy
ela ion and including he g adien –ene gy con ibu ion. This s ep connec s he
ensemble ield in ensi y
⟨|A
(
)
|2⟩
o he obse able e ec i e densi y
ρe
(
) and
yields he inal closed halo law.
The local cu a u e–ene gy densi y is
ρe (x) = 1
8πGc2Ω2|A(x)|2+c2
e |∇A(x)|2,(187)
whe e he i s e m ep esen s he po en ial ene gy o he global cu a u e
oscilla ion and he second i s elas ic cu a u e (g adien ) ene gy.
De ining a co ela ion leng h ℓcby
ℓ−2
c:= |∇A|2
|A|2,(188)
A e aging Eq. (187) and using he de ini ion ℓ−2
c=⟨|∇A|2⟩/⟨|A|2⟩gi es
ρe ( )=Ω2
8πGc2D|A( )|2E1 + c2
e
Ω2ℓ2
c≡Ξ∇
Ω2
8πGc2D|A( )|2E,(189)
whe e he dimensionless g adien –ene gy boos is
Ξ∇= 1 + 2π Ldamp,e
λ es 2
,(190)
73
Because he speckle ba h is con inually e eshed, he ensemble–a e aged e ec i e
densi y beha es as a causal mo ing a e age o he ins an aneous ield:
⟨ρe ⟩ +∆ = (1 −β)⟨ρe ⟩ +β ρins ( ), β = 1 −e−∆ /τcoh,e ,(200)
wi h
τcoh,e
de e mining he memo y ime o he anspo p ocess. In s a iona y
egions he a e age con e ges o a cons an alue; in e ol ing en i onmen s i
go e ns how he ensemble ield acks ansien cu a u e luc ua ions. Th ough
his con inual enewal on he cohe ence imescale, he sys em main ains a pe -
sis en backg ound o s ochas ic cu a u e s esses ha bo h mi o and sus ain
he andom s ella mo ions hemsel es— hus closing he eedback loop be ween
ba yonic agi a ion and cu a u e di usion.
Al hough he mean e ec i e cu a u e ield is smoo h and ene ge ically sub-
dominan , i s ini e a iance implies a nonze o p obabili y o localized cu a u e
in ensi ica ion. I such peaks pe sis o se e al cohe ence imes hey may weakly
bias he nea by s ella dis ibu ion, p oducing ansien clus e ing o ain sub-
s uc u e. These ea u es a ise na u ally om he eedback loop i sel : he
e y luc ua ions ha p ese e equilib ium occasionally imp in b ie , localized
cu a u e enhancemen s wi hin an o he wise elaxed halo.
The pola isa ion p ope ies o he cu a u e–ma e esonance in ellip icals ollow
om he suscep ibili y enso
χij
(Ω) in oduced in Sec ion 5.3.3. Al hough
indi idual ou –dimensional speckle cells a e locally aniso opic—each ha ing
ans e se and longi udinal scales (
ℓ⊥, ℓ∥
) se by he s ella eloci y dis ibu ion—
he ensemble o cells is andomly o ien ed and s a is ically well mixed. Because
he s ella eloci y ellipsoid in p essu e–suppo ed ellip icals is nea ly iso opic,
his andom o ien a ion washes ou he aniso opy o indi idual cells in he
ensemble a e age. The linea esponse he e o e educes o he iso opic o m
χij(Ω) = χ(Ω) δij,(201)
so all enso eigenmodes sha e he same suscep ibili y, χ(x)=χ(y)=χ(z).
No p e e ed pola isa ion axis exis s in his limi . Th ee–dimensional s ella
agi a ion exci es all eigen ec o s
i
(α)
wi h compa able s eng h, and apid phase
mixing emo es any ansien aniso opy. Ellip icals a e he e o e s a is ically
unpola ised: hei cu a u e–phonon ield popula es all enso eigenmodes equally,
consis en wi h he nea ly sphe ical anspo en elopes o p essu e–suppo ed
sys ems.
This amewo k na u ally ex ends o len icula , iaxial, and i egula sys ems.
Whene e he s ella eloci y dis ibu ion depa s om iso opy, he ensemble–
a e aged suscep ibili y acqui es a small bu ini e eigen alue spli ing,
χxx =χyy =χzz,(202)
wi h he p incipal axes aligned wi h hose o he s ella eloci y ellipsoid. The
cohe ence ke nel emains h ee–dimensional bu becomes mildly obla e o iax-
ial, impa ing a co espondingly weak pola isa ion bias o he global e ec i e
80

cu a u e ield. Such sys ems lie be ween he iso opic (gian ellip ical) and
s ongly plana (spi al) limi s.
In S0 galaxies, he esidual disk componen enhances he in–plane suscep ibili ies,
χxx ≃χyy > χzz,(203)
p oducing an obla e cohe ence enso and a la ened anspo en elope, hough
s ill wi hou he s ong phase locking o la ge eigen alue con as cha ac e is ic
o spi al galaxies. T iaxial ellip icals exhibi h ee dis inc suscep ibili ies, each
aligned wi h a p incipal ine ial axis, na u ally yielding he obse ed E3–E7
sequence as a con inuous amily o aniso opic anspo –cohe ence enso s. Thus
he p ojec ed halo ellip ici y aces he eigen alue hie a chy o χij(Ω).
I egula galaxies ep esen he pa ially cohe en limi . Low ba yonic densi ies
and u bulen , s ochas ic en i onmen s p e en long–li ed phase locking, ye
sho – ange cu a u e co ela ions a ise wi hin ansien s ella associa ions.
Each sys em is he e o e a luc ua ing ensemble o small cohe ence pa ches whose
c oss–co ela ions a e weak bu nonze o,
0<⟨AiA∗
j⟩
⟨|A|2⟩<1,(204)
con inually o ming, dissol ing, and me ging wi hou es ablishing a global mode.
This pic u e ag ees wi h hei pa chy mo phology, i egula kinema ics, and
di use halos: i egula s a e u bulence–d i en cu a u e ensembles ha occupy
he ansi ion be ween he cohe en and ully andom ex emes o he esonan –
speckle hie a chy.
7.3
Clus e s as Radia i e T anspo Ex ensions o Ellip i-
cals
7.3.1 Obse ing he S a is ical T anspo Equilib ium
In his sec ion we ex end he adia i e anspo o malism de eloped o isola ed
ellip icals and spi als o he clus e scale. Whe eas he ellip ical case equi ed
a anspo ea men o cap u e long-li ed di usi e ails, and he spi al case
was ully desc ibed by cohe en Foldy–Lax sca e ing, he clus e en i onmen
in ol es he supe posi ion o many independen ellip ical anspo en elopes.
These en elopes o e lap and pa ially in e e e, p oducing a collec i e in ensi y
ield ha mus be modelled s a is ically.
The p ima y goal o his sec ion is o demons a e ha he esul ing e ec i e
anspo densi y ep oduces, in bo h o m and ampli ude, he obse ed da k-
ma e dis ibu ion o clus e s as desc ibed by he Na a o–F enk–Whi e (NFW)
p o ile [21, 22].
The NFW model, de i ed empi ically om cosmological
N
-body simula ions,
desc ibes he da k-ma e halo as
ρNFW( ) = ρs
x(1+x)2, x =
s
,(205)
81
wi h
s
and
ρs
as cha ac e is ic scale adius and densi y, espec i ely. Recen
ex ensions include he in oduc ion o a “splashback” adius
Rsp
, beyond which
ρ( ) apidly s eepens [23].
The heo e ical basis o he clus e -scale ea men p esen ed in his sec ion, lies
in he long-s anding adia i e anspo amewo k de eloped o di use acous ic
and op ical p opaga ion. Wea e [
13
,
14
] es ablished ha acous ic ene gy in
he e ogeneous solids obeys a adia i e ans e equa ion o he in ensi y ield,
wi h dis inc ballis ic, cohe en , and di usi e egimes. These wo ks p o ided he
i s quan i a i e o mula ion o acous ic di usion and localiza ion, de ining he
s a is ical ke nel ha ou own model gene alises o as ophysical scales.
The subsequen expe imen al alida ion was achie ed by De ode, Tou in, and
Fink [
9
,
4
,
5
], who in es iga ed ul asonic p opaga ion h ough andom ensembles
o esonan sca e e s. Thei expe imen s con i med he adia i e anspo
egime p edic ed by Wea e , demons a ing he sepa a ion o cohe en , ballis ic,
and di use componen s, as well as he eme gence o e e be a ion gain in mul iply
sca e ed ields. These esonan - od sys ems ep esen di ec analogues o ou
ellip ical galaxies: indi idual mesoscopic esona o s embedded in a di usi e hos
ha collec i ely gene a e ex ended ene gy-densi y ails.
La e de elopmen s by Wea e and Lobkis [
16
,
17
] es ablished ha di use
ields om independen andom ensembles can exhibi weak c oss-cohe ence,
eco e able h ough c oss-co ela ion measu emen s. This phenomenon—now
widely used in seismology and acous ics— o malised he concep o o e lapping
di use anspo ields, p o iding a igo ous physical basis o he small nonlinea
coupling pa ame e ηused in ou clus e model.
Finally, La ose e al. [
18
] uni ied hese concep s ac oss acous ics, op ics, and
geophysics, showing ha he supe posi ion o di use ields om independen
sou ces can ep oduce e ec i e G een’s unc ions h ough e e be an co ela ions.
This p o ides he closes expe imen al and heo e ical p eceden o ou ea men
o o e lapping ellip ical anspo en elopes wi hin clus e s: andom, ex ended
esona o s (galaxies) embedded in a di usi e medium (in aclus e ba yons)
ha gene a e pa ially cohe en , o e lapping anspo ails and a measu able
ensemble e e be a ion gain.
The ensemble beha iou demons a ed in he acous ic and op ical anspo
expe imen s o Wea e and De ode p o ides he di ec me hodological ounda ion
o he p esen clus e model. He e we apply he same adia i e– anspo
o malism—p e iously shown o desc ibe o e lapping di use ields in andom
media— o he collec i e cu a u e– anspo egime o space ime i sel . Each
ellip ical galaxy sus ains i s own di usi e en elope, gene a ed by he andom
s ella mo ions ha de ine i s in e nal anspo equilib ium, and hese en elopes
coexis wi hin a sha ed e ec i e cu a u e ield ( he common anspo con inuum
o space ime) ha suppo s hei mu ual o e lap. The esul ing ensemble o
o e lapping anspo in ensi ies o ms a s a is ically smoo h supe posi ion whose
82
weak c oss–cohe ence p oduces he modes e e be a ion gain
G e =1+Ngalη, (206)
which, when applied a he clus e scale, ep oduces he NFW–like cu a u e o
he e ec i e densi y p o ile wi hou in oking any addi ional physical assump ions.
In he sec ions ha ollow, we adop iden ical me hodologies o hose used in he
acous ic and op ical anspo s udies, applying hem di ec ly o he as ophysical
anspo egime o cons uc and no malize he clus e -scale e ec i e densi y
ield.
Each ellip ical o len icula ac s as amesoscale d i e o he e ec i e cu a u e
ield,
ρe ,i( )∝exp(− /Lmean),(207)
whe e
Lmean =ce τ (208)
is he mean anspo dis ance de e mined by he e ec i e anspo ime
τ
.
Wi hin a clus e , hese en elopes o e lap and in e e e weakly. Thei mu ual
co ela ion de ines a small bu ini e e e be a ion pa ame e η,
⟨AiA∗
j⟩=ηq⟨|Ai|2⟩⟨|Aj|2⟩,(i=j),(209)
yielding he s a is ical ampli ica ion
G e =1+Ngalη, (210)
whe e
Ngal
is he numbe o con ibu ing esona o s. This weak collec i e gain
ep oduces he mac oscopic cu a u e excess associa ed wi h he obse ed clus e
halo.
Theo em (S a is ical T anspo Theo em o Galaxy Clus-
e s). Le a clus e con ain
Ngal
ellip ical o len icula galaxies, each p oducing
a s a iona y di usi e en elope
ρe ,i
(
)
∝exp
(
− /Lmean
)wi hin a sha ed e ec i e
anspo ield o speed
ce
and damping a e Γ
. I he ensemble sa is ies he
s a is ical balance condi ion
dρe
d =G −L ≃ 0,(211)
hen he ensemble–a e aged e ec i e densi y obeys
ρe ( )∝G e
exp[−2 /Lmean]
2,(212)
wi h
Lmean =ce
Γ
, G e =1+Ngalη. (213)
The con olu ion o hese o e lapping exponen ial en elopes app oaches he obse ed
NFW o m ρ( )∝ −1(1+ / s)−2 o Lmean ∼ s.
83
In e p e a ion. Equa ion
(230)
de ines he s eady anspo equilib ium: he e -
ec i e cu a u e ene gy densi y, go e ned by anspo , is con inually eplenished
by embedded ellip ical and len icula galaxies. The ele an leng h scale
Lmean
depends on he anspo p ope ies o he medium, no on he ba yonic densi y
o esonan equency. Ellip icals ac as iso opic di usi e emi e s; len icula s,
wi h hei obla e mo phology, gene a e asphe ical anspo ails p e e en ially
elonga ed owa d he clus e cen e—a beha iou consis en wi h X– ay and
weak–lensing obse a ions indica ing ha S0s a e adially aligned and concen-
a ed owa d clus e co es. Spi al galaxies can con ibu e negligibly o he
clus e e ec i e cu a u e ield because hey ha e no anspo ails and a e
co espondingly obse ed o popula e only he pe iphe y.
The same anspo equa ions ex end na u ally o ilamen s, which ep esen he
cylind ical con inua ion o he clus e ensemble. In his geome y, o e lapping
di usi e en elopes align along a ilamen a y axis, and he mean gain is educed
by geome ic dilu ion:
G il ≃G e Lmean
R il ,(214)
whe e
R il
is he ilamen adius. Al hough weake in ampli ude, hese linea
anspo ensembles p ese e he same physical mechanism—cu a u e ene gy di -
usion and e e be a ion coupling among ex ended esona o s—and hus na u ally
explain he eme gence and pe sis ence o cosmic ilamen s.
Finally, he e ec i e cu a u e ield ha sus ains he clus e halo is sel –main aining
in he same sense as in smalle sys ems. The appa en da k mass p oduced by
he collec i e ampli ica ion
G e
is also he mechanism ha enhances he coupling
among membe galaxies: he inc eased cu a u e ene gy densi y ocuses ans-
po back owa d dense egions, s eng hening o e lap and main aining nonlinea
equilib ium. The clus e is he e o e a sel – egula ed, adia i e– anspo sys-
em—a s a is ically sus ained e ec i e cu a u e ield whose ensemble dynamics
ep oduce bo h he ampli ude and mo phology o obse ed da k–ma e halos.
7.3.2
Ma hema ical Fo mula ion: O e lapping Ellip ical T anspo
En elopes
Each ellip ical galaxy ac s as an e ec i e sou ce o he cu a u e–phonon ield
ampli ude
Ai
( ), whose ensemble–a e aged in ensi y
⟨|Ai|2⟩
is gi en by he
single–galaxy anspo solu ion de i ed in Eq.
(181)
. A he clus e scale, he
o al ield ampli ude is he cohe en sum
A o ( ) =
Ngal
X
i=1
Ai( ),(215)
so ha he ensemble–a e aged in ensi y becomes
|A o ( )|2=X
i⟨|Ai|2⟩+X
i=j⟨AiA∗
j⟩.(216)
84
The c oss–co ela ion e m
⟨AiA∗
j⟩
ep esen s he weak mu ual cohe ence be ween
o e lapping anspo ails. Following he s anda d adia i e– anspo ea men
o di use ield supe posi ion [13, 9, 4, 16, 18], we w i e
⟨AiA∗
j⟩=ηijq⟨|Ai|2⟩⟨|Aj|2⟩,(217)
whe e 0
≤ηij ≪
1 measu es he ac ional cohe ence o he pai . Fo s a is ically
independen galaxies,
ηij
luc ua es andomly abou a small mean alue
η
. The
o al in ensi y he e o e eads
|A o |2≃X
i⟨|Ai|2⟩(1+Ngalη),(218)
iden i ying he dimensionless e e be a ion gain
G e =1+Ngalη. (219)
This ampli ude–le el o mula ion main ains he same causal hie a chy as he
ellip ical case: he ield in ensi y a ises om he quad a ic combina ion o s a-
is ically independen ampli ude ealiza ions, and he weak cohe ence ac o
η
p o ides a small mul iplica i e enhancemen due o o e lapping anspo e-
gions. Analogous ensemble ea men s appea in adia i e–acous ic anspo
and ime– e e sal acous ics [
13
,
9
,
5
,
17
,
18
], whe e he e e be an ene gy o
independen di use sou ces p oduces a measu able in ensi y gain.
The co esponding e ec i e ene gy–densi y ield o he clus e ollows di ec ly
om he ensemble in ensi y law o Eq.
(181)
. Each ellip ical con ibu es a
anspo – eno malized ampli ude en elope
⟨|Ai|2⟩ ∝ Ke
(
|
−
i|
), including
he g adien –ene gy boos ac o Ξ
∇
de ined in Eq.
(190)
. The clus e -scale
supe posi ion is he e o e
ρe ( ) = G e Ξ∇
Ngal
X
i=1 Ke (| − i|),(220)
whe e
G e
accoun s o he weak e e be an coupling be ween o e lapping
anspo ails. This disc e e o m will subsequen ly be eplaced by i s con inuous
analogue h ough con olu ion wi h he galaxy numbe –densi y p o ile
ngal
(
) in
he ollowing subsec ion.
This o mula ion di ec ly pa allels he in ensi y–based adia i e anspo models
de eloped o mul iply sca e ing acous ic and elas ic media. Wea e [
13
,
14
]
i s exp essed he di use ene gy ield as an ensemble a e age o e incohe en
ampli ude ealiza ions, while De ode, Tou in, and Fink demons a ed expe imen-
ally ha weak mu ual cohe ence among independen esonan clus e s p oduces
a measu able e e be a ion gain [
9
,
4
,
5
]. Subsequen co ela ion analyses by
Wea e and Lobkis [
16
,
17
] and by La ose [
18
] showed ha such di use ields
e ain small bu ini e c oss–cohe ence, eco e able h ough c oss–co ela ion o
85

independen ealiza ions. Ou ea men o he clus e in ensi y ollows his same
logic: he o al ield is he incohe en sum o independen ellip ical anspo
en elopes, augmen ed by a small global cohe ence ac o
η
ep esen ing he
ensemble–a e aged e e be an coupling be ween hem.
In p inciple, one migh a emp o model he o al clus e ield di ec ly om
he ensemble o o e lapping galaxy ampli udes, ully accoun ing o all phase
co ela ions. In p ac ice, howe e , his app oach is bo h compu a ionally p o-
hibi i e and concep ually agile, as he indi idual phases a e apidly andomized
by mul iple sca e ing and anspo e ec s. We he e o e p oceed by employing
he s anda d p ac ice o adia i e anspo heo y o model phenomenon o his
ype: by cons uc ing he e ec i e in ensi y (o ene gy–densi y) ield ins ead,
e aining he co ec ensemble s a is ics while a e aging o e he andom phases.
This yields a ac able and physically anspa en o mula ion, which na u ally
leads o he s epwise de elopmen ou lined below.
Con inuous Galaxy Dis ibu ion and NFW Analogy
Obse a ional s udies indica e ha he adial numbe –densi y p o iles o clus e
galaxies a e well desc ibed by an NFW–like o m, albei wi h sys ema ically lowe
concen a ions han he co esponding da k–ma e halos [
24
]. Fo analy ical
con enience and physical ealism, we he e o e model he spa ial dis ibu ion o
membe galaxies as an NFW– ype p o ile smoo hly unca ed a la ge adii by a
“splashback” [23] e m:
ngal( )=n0
Wsp( )
x(1+x)2, Wsp( ) = 1
1 + exp[( −Rsp)/∆sp],(221)
whe e
x
=
/ s
, and
Rsp
and ∆
sp
ep esen he splashback adius and i s
cha ac e is ic ansi ion wid h, espec i ely. This con inuous pa ame e isa ion
eplaces he
∼
100–500 disc e e galaxies o a ypical ich clus e wi h a smoo h
numbe –densi y ield app op ia e o con olu ion wi h he single–galaxy anspo
ke nel.
A key in e p e a i e poin is ha hese galaxies a e no s a ic pa icles. Thei long
di usi e decay imes ensu e ha , o e many dynamical pe iods, he anspo
en elopes o indi idual membe s o e lap h oughou he clus e olume, o ming
an e ec i e con inuum o esonan sou ces.
The Necessi y o 2.5D Con olu ion
A di ec h ee–dimensional con olu ion o he o m
ρe ( ) = ZK(| − ′|)ngal( ′)d3 ′(222)
does no yield physically consis en esul s because
K
(
) desc ibes an an-
gle–in eg a ed in ensi y a he han a G een’s unc ion. This dis inc ion is
undamen al: a G een’s unc ion p ese es ull phase and di ec ional in o ma ion,
whe eas an in ensi y ke nel embodies an ensemble a e age o e hose angles. The
86
issue is well known in anspo heo y and has been ex ensi ely discussed in
he con ex s o op ical di usion, ul asound p opaga ion, and acous ic mul iple
sca e ing [8, 4, 5, 13, 19].
The app op ia e o malism—long es ablished in bo h adia i e and acous ic ans-
po —is o employ an angula ly a e aged, educed–dimensional ep esen a ion:
ρe ( )=2πZ∞
0
ngal( ′) ′⟨K(s)⟩µd ′, s2= 2+ ′2−2 ′µ, (223)
whe e
⟨K
(
s
)
⟩µ
deno es he Legend e–weigh ed a e age o e
µ
=
cos θ
. This
“2.5D” o mula ion emo es edundan angula deg ees o eedom and gua an ees
ma hema ical s abili y while p ese ing he co ec physical o m o he anspo
ield. I omi s only a single geome ic leng h ac o , eco e able h ough explici
no maliza ion.
Such dimensional educ ions a e s anda d p ac ice in adia i e ans e [
8
],
ul asound di usion [
4
,
5
,
13
], and op ical mul iple sca e ing [
19
]. In all o hese
con ex s, he educed o malism omi s a single geome ic leng h ac o , p oducing
an appa en scale ambigui y ha is con en ionally esol ed by in oducing a
mac oscopic no maliza ion pa ame e , Λ
geom
, de e mined h ough global ene gy
o mass conse a ion. In ou case, because he con olu ion ep esen s a di ec
supe posi ion o in ensi ies a he han ields, he app op ia e no maliza ion
ollows na u ally om he o al enclosed mass o he cons i uen galaxies. W i ing
ρe ( )=Λgeom ρshape( ), we impose
Z∞
0
4π 2ρe ( )d =Ngal M(ellip ical)
DM ,(224)
which yields he explici exp ession
Λgeom =Ngal M(ellip ical)
DM
Z∞
0
4π 2ρshape( )d
.(225)
This ensu es ha he in eg a ed e ec i e anspo densi y eco e s he combined
da k–mass con ibu ion o all ellip ical membe s, ancho ing he 2.5D con olu-
ion o a physically measu able mass scale wi hou in oducing addi ional ee
pa ame e s.
Re e be a ion Gain and Pa ial Cohe ence
The pa ame e
η
quan i ies weak s a is ical co ela ions among o e lapping
anspo ails, ep esen ing he esidual cohe ence ha emains a e ensemble
a e aging. Such pa ial cohe ence is a well–es ablished ea u e o adia i e
and acous ic anspo sys ems con aining mul iple sca e ing and long–li ed
esonances. In ul asonic and op ical analogues [
4
,
5
,
13
,
19
,
17
], he e ec a ises
om ecu en sca e ing loops and ime–delayed “ e e be an ” e u ns ha
couple neighbou ing in ensi y ields. These p ocesses lead o a small, posi i e
excess in he ensemble–a e aged in ensi y ela i e o he pu ely incohe en sum.
87
We model his e ec phenomenologically h ough a nonlinea ampli ica ion,
ρe , e ( )=G e Λgeom ρshape( ), G e =1+Ngalη, (226)
whe e
ρshape
(
) is he no malized 2.5D con olu ion esul and Λ
geom
en o ces he
global mass no maliza ion. The e e be a ion gain
G e
hus cap u es he small
bu ini e c oss–cohe ence among he o e lapping anspo ields o neighbou ing
galaxies.
Labo a o y measu emen s in acous ics and op ics ypically ind
η∼
10
−3
–10
−2
,
co esponding o o al ampli ica ion ac o s G e ≃3–10 o sys ems wi h many
independen sca e e s. This ange coincides ema kably well wi h he ampli ica-
ion equi ed o clus e s: he a io be ween he summed da k–mass con ibu ion
o all ellip icals and he o al mass in e ed om NFW i s is o p ecisely he
same o de . In his sense, he “ e e be a ion gain” o malism p o ides a na u al,
physically mo i a ed b idge be ween he mesoscopic anspo beha iou o indi-
idual galaxies and he mac oscopic g a i a ional po en ial o he clus e as a
whole.
Logno mal Dis ibu ion o T anspo Leng hs
In many s udies o adia i e, op ical, and acous ic anspo h ough diso de ed
o he e ogeneous media, i is common p ac ice o ep esen a iabili y in local
sca e ing o damping leng hs using a logno mal ensemble a he han a single
mean alue. This e lec s he mul iplica i e na u e o sca e ing pa hs and
he in insic he e ogenei y o local en i onmen s, which na u ally p oduce a
logno mal dis ibu ion o e ec i e anspo leng hs. Such o mula ions a e well
es ablished in wa e-di usion and mul iple-sca e ing analyses ac oss acous ics [
4
,
13], op ics [19, 15], and mesoscopic anspo heo y [8].
Following his con en ion, we ex end he ke nel o an ensemble-a e aged o m,
Kens( ) = ZK( ;L)P(L;Lmean, σln L)dL, (227)
whe e
P
(
L
) is a logno mal p obabili y densi y wi h loga i hmic dispe sion
σln L
.
This p ocedu e ep oduces he empi ically obse ed hea y ails in in ensi y
anspo and egula izes he la ge-scale o e lap beha iou wi hou in oducing
any ad hoc smoo hing.
Typical acous ic o op ical di usion s udies employ 0
.
3
≲σln L≲
1
.
0 o mode -
a ely he e ogeneous media, and occasionally
σln L∼
2 in s ongly sca e ing o
po ous sys ems whe e anspo pa hs a y o e se e al o de s o magni ude [
19
,
8
].
In he p esen con ex , a alue o
σln L
= 2
.
5 yields a physically consis en and
obse a ionally well-ma ched p o ile: i p ese es he co ec NFW-like slope a
la ge adii while cap u ing he equi ed con ex cu a u e nea he co e. The
long- ail ou lie s in his dis ibu ion ca y di ec physical meaning— hey ep e-
sen a e ins ances in which a neighbou ing galaxy’s anspo ail pene a es
a ba yonic egion, e-ini ia ing a seconda y adia i e- anspo p ocess wi h an
ex ended e ec i e damping leng h.
88
While labo a o y s udies ypically in e 0
.
5
≲σln L≲
2
.
0 o single-phase di usi e
media [
4
,
13
], s ongly esonan o dual-channel sys ems exhibi e ec i e sp eads
app oaching
σln L∼
2–3 [
8
], ully consis en wi h he b oad he e ogenei y expec ed
in he clus e -scale double- anspo egime conside ed he e.
E ec i e unca ion o he ensemble. In p ac ice, adia i e- anspo models
a ely in eg a e he logno mal dis ibu ion o in ini y. Ra he , hey app oxima e
he ensemble a e age by sampling o e a ini e ange o
L/Lmean
, ypically wi hin
a ew s anda d de ia ions o he loga i hmic mean. This supp esses he s a is ical
weigh o ex eme ou lie s wi hou al e ing he in insic shape o K( ;L). Such
e ec i e unca ion is s anda d p ac ice in acous ic and op ical di usion analyses,
whe e he physical cohe ence leng h o he medium imposes a na u al limi on
alid anspo pa hs [
4
,
13
,
8
]. Ou i e-poin sampling ac oss
±
2
σln L
ollows
his con en ion: i e ains he ull logno mal cu a u e while s ongly supp essing
con ibu ions om exceedingly long pa hs (
L≫Lmean
). This p ese es he
physical b oadness o he ensemble and he ideli y o he ke nel’s o m, exac ly
as in es ablished adia i e and ul asonic anspo ea men s.
7.3.3 Nume ical Resul s
The nume ical implemen a ion ollows di ec ly om he o malism de eloped
abo e. All calcula ions employ he educed 2.5D con olu ion o e alua e he
anspo –o e lap densi y
ρe
(
), ollowed by applica ion o he geome ic no -
maliza ion Λ
geom
o en o ce o al mass conse a ion, and he e e be a ion gain
G e
= 1 +
Ngalη
o inco po a e pa ial cohe ence be ween o e lapping ails. The
comple e algo i hm and pa ame e de ini ions a e gi en in Appendix E, oge he
wi h he Py hon code used o gene a e he igu es.
Fo a iducial 10
15 M⊙
clus e , we adop
Ngal
= 300 ellip icals,
Lmean
= 636
kpc
,
σln L
= 2
.
5, and
η
= 8
.
3
×
10
−3
, co esponding o
G e ≃
3
.
5. The esul ing
one-dimensional anspo p o iles a e p esen ed in Figu e 10, which compa es
he model’s e ec i e densi y
ρe
(
) and enclosed mass
Me
(
) wi h a s anda d
NFW benchma k. All quan i ies we e compu ed on a loga i hmic g id spanning
0
.
2
kpc ≤ ≤
3
Mpc
, using he 2.5D con olu ion o
ρshape
(
) and no malized
by Λ
geom
as de ined in Eq. (225) o p ese e o al enclosed mass. The e e ence
NFW halo co esponds o M200 = 1015 M⊙and c200 = 4.
Panels (a) and (b) o Figu e 10 show he e ec i e densi y in linea and loga i hmic
scales, espec i ely. The anspo –o e lap p o ile ep oduces he NFW slope
and ampli ude ac oss h ee o de s o magni ude in adius, main aining close
ag eemen om
∼
30
kpc
o 3
Mpc
. The sligh ly inc eased conca i y o
ρe
(
) a
small adii (
≲
50
kpc
) possibly a ises om he omission o he cen al b igh es
clus e galaxy (BCG), whose ex ended s ella en elope would na u ally la en
he inne slope. The shallow dip nea
→
0 o igina es om he disc e e angula
in eg a ion and is pu ely nume ical.
Panels (c) and (d) display he co esponding enclosed-mass p o iles. The o al
mass de i ed om he anspo model ollows he NFW p edic ion almos
89
A use ul analogy is supplied by acous ics. The acous ic ield oscilla es apidly
ela i e o he slow buildup o ime–a e aged p essu e; wha one measu es is no
he sound ield i sel bu he quasi–s a ic p essu e i lea es behind. Simila ly, he
esonan cu a u e ield Φ
es
is no obse able in isola ion. Only i s s o ed cycle–
a e aged ene gy densi y
ρe
, ac ing g a i a ionally h ough Channel 2, en e s
measu able dynamics. This cap u es p ecisely why he esonan channel emains
locally silen e en while shaping he la ge–scale g a i a ional en i onmen .
9
P edic ion: Absence o Gain in Homogeneous
Sys ems
This sec ion es ablishes a cen al and alsi iable p edic ion o he wo–channel
causal– esonan amewo k: a pe ec ly homogeneous ba yonic medium canno
gene a e cu a u e gain in Channel 1 unde any ci cums ances. Inhomogene-
i y is equi ed o o m he sou ce– esca e loops, Foldy–Lax coupling, and
e e be a ion p ocesses ha de ine he e ec i e da k densi y
ρe
. A spa ially
smoo h sys em exci es only Channel 2 ( he New onian Channel) and is he e o e
esonan ly silen .
To a oid con usion, we i s cla i y how he linea pe u ba ion heo y o Sec-
ion 3.4 ela es o homogeneous media. The pe u ba i e equa ion, epea ed he e
o e e ence,
∇2δΦ + 1
c2
e −∂2
−2Γ ∂ + Ω2δΦ = 4πG
c2
e
δρb,(237)
p esupposes ha mul iple sca e ing has al eady p oduced an e ec i e medium
wi h eno malised pa ame e s Ω, Γ, and
ce
. A homogeneous con igu a ion canno
gene a e such a medium and he e o e canno sa is y Eq. (237) in isola ion.
9.1 Th ee Dis inc Regimes o Homogeneous Clouds
A homogeneous cloud may a ise in h ee physically dis inc se ings:
[label=(1)]
1.
Isola ed homogeneous cloud. The cloud possesses a na u al equency
ωna ∝√ρ0
, bu gene a es no sca e ing, no sel –ene gy, and he e o e no
eno malised pa ame e s (Ω,Γ, ce ). Thus Channel 1 does no o m:
Φ(isola ed)
es = 0.(238)
2.
Cloud embedded wi hin an ex e nally gene a ed e ec i e medium
bu no phase–locked. The su ounding medium possesses (Ω
,
Γ
, ce
),
bu he cloud oscilla es a i s own
ωna
and canno couple cohe en ly h ough
he causal G een unc ion. Hence no Channel 1 ield is gene a ed:
Φ(embedded, ee)
es = 0.(239)
96

3.
Cloud embedded wi hin an ex e nally gene a ed e ec i e medium
and o ced a he global esonance Ω.In his a i icial si ua ion he
homogeneous cloud may oscilla e in empo al phase wi h he backg ound
because he d i ing ield imposes he phase ex e nally. The cloud hen
sa is ies he d i en Helmhol z ela ion
∇2+k2
ΩδΦ = 4πG
c2
e
δρb, k2
Ω=Ω2
c2
e
,(240)
bu he sca e ing s eng h emains
σ(x)≡0,(241)
because empo al cohe ence does no c ea e spa ial con as . Thus no Foldy–
Lax loops o m, no sel -ene gy is gene a ed, and he esponse emains a
s ic ly single–pass, non-ampli ying Helmhol z ield.
The heo em below applies o all h ee egimes.
9.2
Theo em: Homogeneous Media P oduce No Resonan
Gain
Theo em (Absence o Resonan Gain in Homogeneous
Sys ems). Le
ρb
(x) =
ρ0
be a pe ec ly homogeneous ba yonic dis ibu ion
wi h no spa ial con as . Then, in each o he egimes in Sec ion 9.1, he Channel 1
cu a u e esponse exhibi s no esonan ampli ica ion:
1. Isola ed case: No e ec i e medium o ms, so Channel 1 does no exis :
Φ es = 0.(242)
2.
Embedded bu ee- unning: Wi hou phase–locking, he cloud does no
couple o he esonan ield:
Φ es = 0.(243)
3.
Embedded and phase–locked: E en when d i en a he global eso-
nance Ω, he esponse is single–pass and non–ampli ying because
σ
(x)
≡
0:
A=GΩ∗(σA) =⇒A= 0.(244)
The e e be a ion gain anishes,
G e =1+Nη −→ 1, η = 0,(245)
and he s o ed cu a u e ene gy emains a he single–pass le el:
ρ(hom)
e ∼1
Nρ(inhom)
e ,(246)
o sys ems wi h he same o al mass, whe e
N
coun s he disc e e subs uc-
u es a ailable in he inhomogeneous case.
The e o e a pe ec ly homogeneous ba yonic medium canno gene a e measu able
cu a u e ampli ica ion in Channel 1 unde any physical con igu a ion.
97
9.3
In e p e a ion and Connec ion o Pe u ba ion The-
o y
The heo em shows ha spa ial con as is essen ial o Channel 1. Wi hou
i , no Foldy–Lax loops o m, no sel –ene gy accumula es, and he collec i e
esonan mode does no a ise. This explains why he pe u ba i e amewo k
o Sec ion 3.4 canno be applied o a homogeneous sys em in isola ion: he
pa ame e s (Ω
,
Γ
, ce
) exis only a e esca e ing has gene a ed an e ec i e
medium.
In he idealised ci cums ance o egime (3), whe e a homogeneous cloud is
ex e nally o ced o oscilla e a he global esonance wi hin an al eady-exis ing
e ec i e medium, he esponse educes o he d i en Helmhol z con olu ion
δΦ(x) = ZGΩ(|x−x′|)δρb(x′)d3x′,(247)
bu i s ampli ude emains ain , being limi ed o he single-pass le el because no
spa ial con as exis s o sus ain cohe en eedback.
The causal– esonan model he e o e p edic s ha appa en da k s uc u e
co ela es wi h ba yonic con as and empo al cohe ence, no wi h o al ba yonic
mass:
ρe ∝(ba yonic con as ) ×(cohe ence ime).(248)
In clus e collisions such as he Bulle Clus e (1E 0657
−
56), whe e dense, colli-
sionless galaxies e ain hei spa ial con as while he di use plasma is s ipped
away, he cu a u e esonance mus emain ancho ed o he galaxy componen
a he han o he displaced gas. This beha iou p ecisely ma ches obse a ions
wi hou in oking non–ba yonic ma e , and yields a decisi e es : i lensing
peaks consis en ly coincide wi h egions o g ea es ba yonic con as , he causal–
esonan in e p e a ion gains s ong empi ical suppo .
10
Eme gence o Mul iple Sca e ing om a
Weakly Inhomogeneous Con inuous Medium
10.1
A B ie Re iew o Ga i a ional Mul iple Sca e
Theo y
Be o e a emp ing o analyse he ea ly Uni e se, i is use ul o summa ise he
assump ions ha unde lie he e ec i e–medium amewo k de eloped hus a .
Th oughou p e ious sec ions we ha e conside ed a ba yonic mass dis ibu ion
ρb(x) =
N
X
j=1
mjδ(x−xj),(249)
98
consis ing o spa ially sepa a ed s ella o galac ic cons i uen s. Wi hin a cohe en
domain hese cons i uen s oscilla e a a common cu a u e equency
Ω2∝G¯ρb,(250)
whe e ¯ρbis he oo –mean ba yonic densi y o e he cohe en olume.
Each ba yonic elemen esponds o he esonan cu a u e ield as a damped
oscilla o , ¨
Xj+ 2Γ ˙
Xj+ Ω2Xj=αΦ es(xj),(251)
which einjec s cu a u e wa es back in o he medium. Subs i u ing he induced
oscilla o s in o he G een– unc ion ep esen a ion yields he sel –consis en Foldy–
Lax equa ion,
Φ es(x)=Φinc(x) + ZGΩ(|x−x′|)σ(x′) Φ es(x′)d3x′,(252)
whe e he con as –de ined sca e ing s eng h is
σ(x)∝δρb(x).(253)
Applying he Foldy–Lax ope a o o bo h sides o Eq.
(252)
p oduces he eno -
malised e ec i e–medium wa e equa ion,
∇2Φ es +1
c2
e −∂2
−2Γ ∂ + Ω2Φ es =4πG
c2
e
ρb,1,(254)
whe e
ρb,1
deno es he slowly a ying componen o he ba yonic dis ibu ion
ha d i es he cohe en mode. All eno malised pa ame e s (Ω
,
Γ
, ce
) a ise sel –
consis en ly om epea ed esca e ing encoded in Eq.
(252)
. The cycle–a e aged
s o ed cu a u e ene gy de ines he e ec i e densi y,
ρe (x)∝ ⟨|Φ es(x)|2⟩.(255)
C ucially, no hing in Eqs.
(250)
–
(255)
equi es
ρb
(x) o be disc e e. Any ba yonic
dis ibu ion wi h su icien spa ial con as —e en a con inuous densi y wi h
in e nal s uc u e— p oduces a nonze o
σ
(x) and he e o e pa icipa es in he
mul iple–sca e ing mechanism. This obse a ion is essen ial o cosmology: a
con inuous medium can become a esonan , sca e ing e ec i e medium once i
de elops spa ial inhomogenei y.
Howe e , he ea ly Uni e se does no begin in ha egime. The p imo dial
ba yon–pho on plasma is ex emely smoo h, wi h
δρb
ρb∼10−5,(256)
and he e o e ealises Regime 1 o Sec ion 9.2, namely an isola ed, homogeneous
sys em wi h
σ(x)≡0, G e = 1,Φ es = 0.(257)
99
In his s a e no Foldy–Lax eedback ope a es, no e ec i e sel –ene gy can o m,
and he pe u ba ion heo y o Sec ion 3.4 does no apply. The e is he e o e
ρe (x, ) = 0 (ea ly homogeneous e a).(258)
Only when g a i a ional ins abili y ampli ies densi y luc ua ions o
δ∼ O
(1)
does
σ
(x) become nonze o and he Foldy–Lax anspo mechanism u n on.
The emainde o his sec ion aces his ansi ion: how a con inuous, nea ly
homogeneous plasma dynamically e ol es in o he esonan , mul iply–sca e ing
en i onmen equi ed o he e ec i e da k densi y desc ibed in he es o his
wo k.
10.2 The Resonan F equency in a Con inuous Medium
Be o e an e ec i e medium o ms, he local (ba e) cu a u e esonance is de e -
mined solely by he coa se–g ained ba yon densi y:
Ω2(x, )∝G¯ρb(x, ),(259)
In he ea ly plasma, he coa se–g ained densi y akes he o m
¯ρb(x, ) = ¯ρb( ) [1 + δ(x, )],|δ| ≪ 1,(260)
so ha
Ω(x, ) = Ω0( )1 + δ(x, )
2+O(δ2).(261)
E en iny densi y pe u ba ions he e o e imp in a weak spa ial modula ion on
he local cu a u e esonance. Howe e , as he homogenei y heo em s a es, such
luc ua ions do no p oduce a nonze o sca e ing s eng h:
σ(x)∝δρb(x)⇒σ≡0 (δ∼10−5).(262)
Thus he ea ly medium lies s ic ly in he non-sca e ing egime o Regime 1.
10.3 Weak Random Po en ial and he Ballis ic Regime
Fo mally inse ing
(261)
in o he esonan ope a o yields a Helmhol z equa ion
o he o m
∇2+k2
0+V(x, )Φ es =S, k0=Ω0( )
ce
,(263)
wi h weak andom po en ial
V(x, )=2k0
δΩ(x, )
ce
+O(δ2)∝δ(x, ).(264)
100
Fo a con inuum wi h RMS ampli ude
σV
and co ela ion leng h
ℓc
, andom–
medium heo y gi es he single–sca e ing mean ee pa h
ℓ−1
s( )∝k4
0( )σ2
V( )ℓ3
c( ).(265)
In he p imo dial plasma one has σV∼δ∼10−5, gi ing
k0ℓs≫1,(266)
so cu a u e p opaga ion is en i ely ballis ic. The e is no esca e ing and hence
no possibili y o o ming an e ec i e medium. This esol es he appa en ension
wi h Sec ion 3.4: since no e ec i e medium exis s, he pe u ba i e heo y he e
does no apply a ea ly imes.
10.4
Nonlinea G ow h Towa d he T anspo Th eshold
As g a i a ional ins abili y ampli ies densi y luc ua ions,
δ→ O(1), σV→ O(1), ℓc→galac ic scales,(267)
he combina ion in Eq.
(265)
g ows by many o de s o magni ude. The e exis s
a cosmological epoch ∗de ined implici ly by
k0( ∗)ℓs( ∗)∼1,(268)
which is he analogue o he Io e–Regel c i e ion o wa es in andom media. A
∗
cu a u e–phonon p opaga ion ansi ions om he ballis ic o he mul iple–
sca e ing egime. Beyond his poin , he densi y ield is no longe a weak
pe u ba ion, and Channel 1 becomes dynamically ac i a ed.
10.5
Eme gen Resonan Pa ches and E ec i e Disc e e-
ness
When
δ∼
1, he medium agmen s in o nonlinea o e densi ies. Wi hin each
pa ch,
Ω(x)≃Ωpa ch ∝√ρpa ch,(269)
so cu a u e ene gy becomes locally apped. The cycle–a e aged e ec i e densi y
g ows as
ρe (x)∝|Φ es|2,(270)
leading o posi i e eedback: cu a u e apping enhances mass in low ia Chan-
nel 2, sha pening con as un il each egion beha es as a disc e e, mesoscale
esonan sca e e . These “ esonan pa ches” cons i u e he p ogeni o s o ellip i-
cals and he basic uni s equi ed o he Foldy–Lax and anspo o malisms o
Sec ions 4–7.3.4.
This mechanism closely pa allels modula ional ins abili y in plasmas, whe e
spa ial a ia ions in
ωp
(
x
)
∝pne(x)
p oduce a e ac i e index landscape ha
101

aps Langmui wa es. He e, luc ua ions in Ω(x) ap cu a u e–phonons, gi ing
ise o long–li ed esonan en elopes ha unc ion as da k–ma e analogues
wi hin he p esen heo y.
The p edic ed cosmological p og ession is he e o e
homogeneous plasma (Regime 1) −→ weak andom medium (ballis ic) −→
nonlinea esonan pa ches −→ mul iple–sca e ing esona o s (ellip ical
p ogeni o s) −→ cohe en disks (spi als) −→ anspo ensembles (clus e s,
ilamen s).
The onse o mul iple sca e ing is no assumed bu a he eme ges dynamically
when he mean ee pa h alls o he esonan scale, enabling he same e ec i e–
medium physics ha go e ns he p esen -day Uni e se.
10.6
Linea Limi and Connec ion o he CMB Powe
Spec um
The de elopmen s o Sec ions 4–7.3.4 demons a e ha mul iple sca e ing,
eno malised p opaga ion, and
ρe
a ise only once he con as
σ
(x) becomes
nonze o. In he p imo dial plasma, howe e , Eqs.
(256)
–
(258)
imply ha he
Uni e se lies s ic ly in Regime 1, so ha
σ(x) = 0,Φ es = 0,(Ω,Γ, ce ) unde ined,(271)
and no e ec i e medium exis s.
Despi e he absence o a sca e ing medium, he unde lying ba e oscilla o —
he cu a u e–phonon deg ee o eedom wi h equency Ω
0
(
) om Eq.
(259)
—
emains a well-de ined linea esponse ield. In Regime 1 all eno malised pa am-
e e s e e o hei ba e alues:
ce →c, Γ→0,Ω→Ω0( ),(272)
and he e ec i e–medium PDE
(254)
consis en ly educes o he ballis ic Helmhol z
equa ion
∇2+1
c2−∂2
+ Ω2
0( )Φ es(x, ) = 4πG
c2¯ρb( )δb(x, ),(273)
which is simply he linea cu a u e esponse o he ba yon o e densi y.
Fou ie ans o ming Eq. (273) gi es
¨
Φ es(k, ) + Ω2
0( ) Φ es(k, )=4πG ¯ρb( )δb(k, ),(274)
showing ha Φ
es
isa o ced, linea oscilla o sou ced by he ba yonic o e densi y.
Impo an ly, because
σ
= 0, he e is no sel -ene gy, no eno malisa ion, and no
eedback loop: Eq. (274) adds no nonlinea g a i a ional e ec .
102
I is impo an o emphasise ha Eq.
(274)
does no in oduce an addi ional g a -
i a ional po en ial beyond he usual New onian me ic pe u ba ions (Φ
N,
Ψ
N
).
Ra he , Φ
es
is a linea cu a u e esponse ield d i en by he same ba yon luc ua-
ions ha also gene a e he New onian po en ials. The e is no independen sou ce
e m and no modi ica ion o he Eins ein equa ions in Regime 1. Consequen ly,
he combina ion
ΦN(k, )−→ ΦN(k, ) + ϵ( ) Φ es(k, ) (275)
should be iewed me ely as a bookkeeping de ice o how cu a u e–phonon
oscilla ions p oduce a iny co ec ion o he e ec i e d i ing o he pho on
Bol zmann hie a chy. I does no ep esen new g a i a ional physics o an
addi ional sou ce e m.
The dimensionless p e ac o
ϵ( )≡Ω2
0( )
k2c2(276)
quan i ies he ela i e size o his co ec ion. I is possible o pe o m a basic
pa ame ic es ima e a ecombina ion. Using ¯ρb(z∼1100) ∼10−21 kg m−3gi es
Ω0( ∗)∼pG¯ρb∼10−17 s−1.(277)
The como ing acous ic scale co esponds o k∼0.01–0.1 Mpc−1, i.e.
k c ∼10−13–10−12 s−1.(278)
Thus
ϵ( ∗) = Ω2
0
k2c2∼10−8–10−10,(279)
many o de s o magni ude below uni y.
The e o e:
ϵΦ es ≪ΦN,( ecombina ion e a),(280)
so he cu a u e–phonon con ibu ion is a oo small o a ec he acous ic
oscilla ions o he esul ing CMB empe a u e and pola isa ion spec a. I is
au oma ically consis en wi h Planck-p ecision cons ain s and canno lead o
double coun ing o me ic pe u ba ions.
In summa y, he Regime 1 limi o he model maps seamlessly on o he s anda d
linea CMB hie a chy. The cu a u e–phonon ield ac s only as a iny, s ic ly
linea esponse d i en by he same ba yonic luc ua ions al eady p esen in Φ
N
,
and i s con ibu ion is
O
(10
−8
) ela i e o he dominan g a i a ional po en ial.
Only once
δ→ O
(1) does
σ
(x) become nonze o and he eno malised e ec i e
medium o Sec ions 4–7.3.4 begin o o m.
103
11 Conclusion
This wo k has shown ha a small bu physically essen ial ea u e o linea ised gen-
e al ela i i y— he ull e a ded G een– unc ion esponse— combined wi h he o -
dina y damped–oscilla o suscep ibili y o ba yonic ma e , yields a sel –consis en
and obse a ionally success ul ex ension o weak– ield g a i a ional dynamics.
No new ields, pa icles, o modi ica ions o GR a e in oduced. Once he usual
quasis a ic unca ion o he e a ded ke nel is a oided, he delayed in e ac ion
be ween ba yons and cu a u e gene a es a collec i e oscilla o y ield whose
cycle–a e aged ene gy beha es as an e ec i e da k–mass densi y.
The esul ing causal– esonan amewo k p oduces a eno malised esponse cha -
ac e ised by h ee e ec i e pa ame e s (Ω
,
Γ
, ce
) de e mined by ba yonic sus-
cep ibili y and mul iple sca e ing. A single e olu ion equa ion o he collec i e
cu a u e ield ep oduces he obse ed e ec i e mass dis ibu ions o spi al galax-
ies, ellip ical galaxies, clus e s, and ilamen s. Cohe en mo ion in s uc u ed
sys ems d i es cons uc i e esca e ing and la ge–scale o ganised modes, while
homogeneous o weakly pe u bed en i onmen s p o ide no spa ial con as and
he e o e emain esonan ly silen . This dicho omy accoun s na u ally o la
o a ion cu es, exponen ial ellip ical en elopes, clus e –scale halos, ilamen a y
cohe ence, and he absence o esonan e ec s in he ea ly Uni e se and in he
CMB.
The unde lying mechanism is uni ied and minimal: ime– a ying ba yonic s uc-
u e sou ces cu a u e oscilla ions h ough i s suscep ibili y; mul iple esca e ing
shapes hei la ge–scale beha iou ; and he ini e memo y o he e a ded ke nel
allows a slow accumula ion o cycle–a e aged cu a u e ene gy. This s o ed
componen en e s he Poisson channel as an e ec i e mass densi y while he
oscilla o y ield emains locally silen , ensu ing ull consis ency wi h labo a o y,
Sola –Sys em, and s ella es s o g a i y.
In summa y, he causal– esonan amewo k e eals ha he obse ed da k–mass
phenomena need no signal new g a i a ional laws o unseen pa icles. They
eme ge na u ally om he in e play be ween he e a ded esponse o gene al ela-
i i y and he o dina y dynamical beha iou o ba yonic ma e . The la ge–scale
g a i a ional s uc u e o he Uni e se—i s halos, clus e s, and ilamen s—can
hus be unde s ood as a mani es a ion o he pe sis en memo y and collec i e
esonan dynamics al eady p esen wi hin weak– ield GR.
12 Dedica ion and Acknowledgemen
I would like o dedica e his pape o he la e P o . John T. She idan, my o me
PhD supe iso and iend, who in oduced me o wa e op ics and is sadly missed.
I acknowledge wi h g a i ude he wo k o P o . Joseph Goodman, whose seminal
ex s Fou ie Op ics and S a is ical Op ics ha e emained page- u ne s o me o
104
mo e han wen y yea s. His con ibu ions unde pin much o my unde s anding
o wa e heo y and cohe ence.
I am indeb ed o he wo k o Sheng, De ode, Roux, Fink, Wea e , an Tiggelen,
and o he s in wa e anspo and mul iple-sca e ing heo y—an a ea I was
no e y amilia wi h when I began his p ojec , ye essen ial o de i ing
he beha iou s o ellip ical galaxies and clus e s. I since ely hope I ha e no
misin e p e ed hei heo ies.
I g a e ully acknowledge he suppo o Resea ch I eland / Science Founda ion
I eland and he Eu opean Union, whose unding o e he pas wen y yea s
enabled my esea ch in op ics and allowed me o deepen my unde s anding o
physics.
Las and mos impo an ly, I hank my wi e Ka en and my daugh e s o hei
unwa e ing suppo as I w es led wi h his heo y o e hese pas se e al mon hs.
A Appendix: E ec i e–Medium De i a ion
Fo comple eness we ske ch he s anda d weak–sca e ing e ec i e–medium
educ ion leading o he homogenised esonan equa ion used in he main ex .
No modi ica ion o gene al ela i i y is in oduced; all nonlocali y a ises om
he o dina y e a ded esponse o he linea ised g a i a ional ield and om he
esca e ing o weak pe u ba ions by ba yonic inhomogenei ies.
Re a ded solu ion and decomposi ion o he sou ce
In he weak– ield limi he g a i a ional po en ial sa is ies
Φ(x) = −4πGZG e (x−x′)ρb(x′)d4x′,(A.1)
whe e
G e
is he s anda d e a ded G een unc ion in a slowly a ying backg ound.
Decomposing he densi y in o a slowly a ying pa and a cohe en oscilla o y
pa ,
ρb(x, ) = ρb,0(x)+ρb,1(x, ),(A.2)
we isola e he oscilla o y esponse Φ es d i en by ρb,1.
Mul iple sca e ing and Dyson eno malisa ion
T ea ing he ba yons as weak g a i a ional sca e e s wi h linea equency–dependen
suscep ibili y
χ⋆
(
ω
), he ensemble eno malises he p opaga o h ough he usual
Dyson se ies,
G−1
e (ω, k) = G−1
0(ω, k)−Σ(ω, k),(A.3)
whe e
G0
is he ee weak– ield p opaga o and Σ is he sel –ene gy desc ibing
cohe en esca e ing by he inhomogeneous ba yonic dis ibu ion. No new
dynamical ields appea : Σ is de e mined ully by he s a is ics and suscep ibili y
o he sca e e s.
105
94 h_z = 0.30 * kpc
95 Sigma0 = M_disk_ o al / (2.0 * np.pi * R_d**2)
96
97 de Sigma_ ( ): e u n Sigma0 * np.exp(- / R_d)
98 de ho_b_midplane( ): e u n Sigma_ ( ) / (2.0 * h_z)
99
100 # ---------- Radial g id ----------
101 _max = 1000.0 * kpc
102 N = 1800
103 = np.linspace(1.0e-3 * kpc, _max, N)
104
105 # ---------- Special unc ions ----------
106 de J1(z): e u n mp.besselj(1, z)
107 de H1(z): e u n mp.hankel1(1, z)
108
109 J1_ als = np.a ay([complex(J1(k_complex * i)) o i in ])
110 H1_ als = np.a ay([complex(H1(k_complex * i)) o i in ])
111
112 # ---------- Sou ce momen ----------
113 M1_ ol = 2.0 * np.pi * eps * ho_b_midplane( )
114
115 # ---------- Complex apezoidal in eg a ion ----------
116 de cum apz_complex(y, x):
117 ou = np.ze os_like(y, d ype=complex)
118 o iin ange(1, len(x)):
119 ou [i] = ou [i-1] + 0.5 * (y[i] + y[i-1]) * (x[i] - x[i-1])
120 e u n ou
121
122 in eg and_J = * M1_ ol * J1_ als
123 in eg and_H = * M1_ ol * H1_ als
124
125 I_J_cum = cum apz_complex(in eg and_J, )
126 I_H_cum = cum apz_complex(in eg and_H, )
127 I_H_ o al = I_H_cum[-1]
128
129 # ---------- En elope ield A( ) ----------
130 p e ac = 4.0 * np.pi**2 * 1j * G / (c_e **2)
131 A_ = p e ac * (J1_ als * (I_H_ o al - I_H_cum) + H1_ als * I_J_cum)
132
133 # ---------- Apply cohe ence gain ----------
134 # En i onmen al ampli ica ion (Q (X N_s a ))
135 gain_ ield = Q * G_en _ oo
136 A_ *= gain_ ield
137
138 # ---------- E ec i e and ba yonic densi ies ----------
139 # Field ene gy densi y |A|, al eady dep h-no malized.
140 ho_e _mid = (2*Omega**2 / (8.0 * np.pi * G * c_ligh **2)) * np.abs(A_ )**2
141 ho_b = ho_b_midplane( )
142
143 # ---------- Ve ically a e aged su ace densi ies ----------
144 # The e ical column dep h is ep esen ed by L_damp (no ex a L_z
,→no maliza ion)
145 Sigma_e = L_damp * ho_e _mid
146
112

147 # ---------- Enclosed masses ----------
148 M_b, M_e = np.ze os_like( ), np.ze os_like( )
149 o iin ange(1, len( )):
150 dM_b = 2 * np.pi * 0.5 * ( [i]* ho_b[i] + [i-1]* ho_b[i-1]) *
,→( [i]- [i-1]) * (2*h_z)
151 dM_e = 2 * np.pi * 0.5 * ( [i]*Sigma_e [i] + [i-1]*Sigma_e [i-1]) *
,→( [i]- [i-1])
152 M_b[i] = M_b[i-1] + dM_b
153 M_e [i] = M_e [i-1] + dM_e
154
155 p in ( "Check: in eg a ed ba yonic mass = {M_b[-1]/Msun:.2e} Msun ( a ge
,→{M_disk_ o al/Msun:.2e})")
156
157
158 # ================================================================
159 # FIGURE 1 Radial P o iles and Ro a ion Cu es
160 # ================================================================
161
162 _kpc = / kpc
163 ig, axs = pl .subplo s(2, 2, igsize=(11, 9))
164 pl .subplo s_adjus (wspace=0.35, hspace=0.35)
165
166 pl . cPa ams.upda e({
167 ’ on .size’: 21,
168 ’axes. i lesize’: 23,
169 ’axes.labelsize’: 21,
170 ’x ick.labelsize’: 19,
171 ’y ick.labelsize’: 19,
172 ’legend. on size’: 19
173 })
174
175 # (a) Ampli ude |A|
176 axs[0,0].plo ( _kpc, np.abs(A_ ), colo =’black’)
177 axs[0,0].se _ i le("(a) Ampli ude $|A|$")
178 axs[0,0].se _xlabel(" [kpc]")
179 axs[0,0].se _ylabel( "$|A( )|$")
180
181 # (b) Mass densi y
182 Msun_pc3 = (pc**3) / Msun
183 axs[0,1].plo ( _kpc, ho_e _mid*Msun_pc3, label= "$ ho_{ m e }$",
,→colo =’blue’)
184 axs[0,1].plo ( _kpc, ho_b*Msun_pc3, label= "$ ho_{ m b}$",
,→colo =’o ange’, lines yle=’--’)
185 axs[0,1].se _yscale(’log’)
186 axs[0,1].se _ylim(1e-8, 1e2)
187 axs[0,1].se _ i le("(b) Mass densi y")
188 axs[0,1].se _xlabel(" [kpc]")
189 axs[0,1].se _ylabel( "Densi y [$M_ odo ,{ m pc}^{-3}$]")
190 axs[0,1].legend()
191
192 # (c) Ro a ion cu es
193 limi = _kpc <= 50
194 axs[1,0].plo ( _kpc[limi ], np.sq (G*M_b[limi ]/ [limi ])/1e3, ’--’,
,→label= "$ _{ m b}$", colo =’o ange’)
113
195 axs[1,0].plo ( _kpc[limi ], np.sq (G*M_e [limi ]/ [limi ])/1e3,
,→label= "$ _{ m e }$", colo =’blue’)
196 axs[1,0].plo ( _kpc[limi ],
,→np.sq (G*(M_b[limi ]+M_e [limi ])/ [limi ])/1e3, label= "$ _{ m
,→ o }$", colo =’black’)
197 axs[1,0].se _ i le("(c) Ro a ion cu es (50 kpc)")
198 axs[1,0].se _xlabel(" [kpc]")
199 axs[1,0].se _ylabel("Veloci y [km s$^{-1}$]")
200 axs[1,0].legend(loc=’cen e igh ’, ameon=T ue, acecolo =’whi e’,
,→ amealpha=0.3, edgecolo =’none’, on size=16)
201
202 # (d) Mass a io
203 axs[1,1].plo ( _kpc, M_e / np.maximum(M_b, 1e-30), colo =’pu ple’)
204 axs[1,1].se _ i le("(d) Enclosed mass a io")
205 axs[1,1].se _xlabel(" [kpc]")
206 axs[1,1].se _ylabel( "$M_{ m e }/M_{ m b}$")
207
208 ig.sup i le("3D Helmhol z Halo Model (m = 1 Mode)",
209 on size=26, on weigh =’bold’, y=0.98)
210 pl . igh _layou ( ec =[0, 0, 1, 0.96])
211 pl .sa e ig("C:/Use s/B yanH/Downloads/ ig1_spi al_p o iles.png", dpi=300,
,→ acecolo =’whi e’)
212 pl .show()
213
214 # ================================================================
215 # FIGURE 2 2D Midplane Maps (Ampli ude, Phase, and Densi ies)
216 # ================================================================
217
218 Nmap = 400
219 x = np.linspace(-50, 50, Nmap) * kpc
220 y = np.linspace(-50, 50, Nmap) * kpc
221 X, Y = np.meshg id(x, y)
222 R = np.sq (X**2 + Y**2)
223 TH = np.a c an2(Y, X)
224
225 A_ adial = np.in e p(R, , A_ )
226 ho_b_2D = np.in e p(R, , ho_b)
227 # --- co ec ed ac o o 2 o consis ency wi h ho_e _mid ---
228 ho_e _2D = (2 * Omega**2 / (8.0 * np.pi * G * c_ligh **2)) *
,→np.abs(A_ adial)**2
229 A_2D = A_ adial * np.exp(1j * TH)
230
231 amp_map = np.abs(A_2D)
232 phase_map = (np.angle(A_2D) + np.pi) / (2.0 * np.pi)
233 ho_ o _2D = ho_b_2D + ho_e _2D
234
235 amp_no m = amp_map / np.max(amp_map)
236 phase_no m = phase_map
237 ho_b_no m = ho_b_2D / np.max( ho_b_2D)
238 ho_e _no m = ho_e _2D / np.max( ho_e _2D)
239 ho_ o _no m = ho_ o _2D / np.max( ho_ o _2D)
240
241 ig, axs = pl .subplo s(2, 2, igsize=(10, 10))
242
114
243 axs[0,0].imshow(amp_no m, ex en =[-50, 50, -50, 50], o igin=’lowe ’,
,→cmap=’ i idis’)
244 axs[0,0].se _ i le("(a) |A| ampli ude n(no malized)", on size=20)
245 axs[0,0].axis(’o ’)
246
247 axs[0,1].imshow(phase_no m, ex en =[-50, 50, -50, 50], o igin=’lowe ’,
,→cmap=’g ay’)
248 axs[0,1].se _ i le("(b) Phase a g(A) n(no malized)", on size=20)
249 axs[0,1].axis(’o ’)
250
251 gb = np.ze os((* ho_e _no m.shape, 3))
252 gb[..., 0] = ho_b_no m
253 gb[..., 1] = ho_e _no m
254 axs[1,0].imshow( gb, ex en =[-50, 50, -50, 50], o igin=’lowe ’, min=0,
,→ max=1)
255 axs[1,0].se _ i le("(c) Ba yonic ( ed) n and esonan (g een)",
,→ on size=20)
256 axs[1,0].axis(’o ’)
257
258 axs[1,1].imshow( ho_ o _no m, ex en =[-50, 50, -50, 50], o igin=’lowe ’,
,→cmap=’g ay’)
259 axs[1,1].se _ i le("(d) Combined densi y n (no malized)", on size=20)
260 axs[1,1].axis(’o ’)
261
262 ig.sup i le("Spi al Helmhol z Solu ions nNo malized Resul s (100 kpc 100
,→kpc)",
263 on size=22, on weigh =’bold’, y=0.93)
264 pl . igh _layou ( ec =[0, 0, 1, 0.95])
265 pl .subplo s_adjus (wspace=-0.3, hspace=0.25)
266 pl .sa e ig("C:/Use s/B yanH/Downloads/ ig2_spi al_maps.png", dpi=300,
,→ acecolo =’whi e’)
267 pl .show()
268
269 # ================================================================
270 # FIGURE 3 Da k Mass Densi y Slices (XY and XZ)
271 # ================================================================
272 # The e ical XZ slice shows exponen ial decay exp(-2|z|/L_z),
273 # whe e L_z ep esen s he geome ic con inemen scale. The e ical
274 # con as he e o e illus a es he halos s a i ica ion ypically
275 # a ew 10 kpc hick and is no escaled by any no maliza ion ac o .
276 # ================================================================
277
278 Nxy = 600
279 WIDTH = 200
280 ex en _xy = [-WIDTH/2, WIDTH/2, -WIDTH/2, WIDTH/2]
281 ex en _xz = [-WIDTH/2, WIDTH/2, -WIDTH/2, WIDTH/2]
282
283 x = np.linspace(-WIDTH/2, WIDTH/2, Nxy) * kpc
284 y = np.linspace(-WIDTH/2, WIDTH/2, Nxy) * kpc
285 z = np.linspace(-WIDTH/2, WIDTH/2, Nxy) * kpc
286
287 # --- Ho izon al (XY) slice a z = 0 ---
288 X, Y = np.meshg id(x, y)
289 R_xy = np.sq (X**2 + Y**2)
115
290 ho_e _xy = np.in e p(R_xy, np. eal( ), np. eal( ho_e _mid))
291 ho_e _xy_no m = ho_e _xy / np.max( ho_e _xy)
292
293 # --- Ve ical (XZ) slice using ue L_z( ) p o ile ---
294 Xz, Z = np.meshg id(x, z)
295 R_xz = np.abs(Xz)
296
297 # --- Ensu e L_z and a e eal 1D a ays ---
298 _ eal = np.a leas _1d(np. eal( ))
299 ho_mid_ eal = np.a leas _1d(np. eal( ho_e _mid))
300
301 # --- Case 1: L_z is an a ay ---
302 i np.ndim(L_z) > 0 and len(np.a leas _1d(L_z)) > 1:
303 Lz_ eal = np.a leas _1d(np. eal(L_z))
304 ho_mid_map = np.in e p(R_xz. a el(), _ eal,
,→ ho_mid_ eal). eshape(R_xz.shape)
305 Lz_map = np.in e p(R_xz. a el(), _ eal, Lz_ eal). eshape(R_xz.shape)
306 else:
307 # --- Case 2: L_z is scala ---
308 Lz_scala = loa (np. eal(L_z)) i np.ndim(L_z) == 0 else
,→ loa (np. eal(L_z[0]))
309 ho_mid_map = np.in e p(R_xz. a el(), _ eal,
,→ ho_mid_ eal). eshape(R_xz.shape)
310 Lz_map = np. ull_like(R_xz, Lz_scala )
311
312 # --- Apply e ical exponen ial en elope exp(-2|z|/L_z( )) ---
313 ho_e _xz = ho_mid_map * np.exp(-2.0 * np.abs(Z) / Lz_map)
314 ho_e _xz_no m = ho_e _xz / np.max( ho_e _xz)
315
316 # --- Plo esul s ---
317 ig, axs = pl .subplo s(1, 2, igsize=(12, 6), cons ained_layou =False)
318
319 # (a) Ho izon al slice XY
320 axs[0].imshow( ho_e _xy_no m, ex en =ex en _xy, o igin=’lowe ’,
,→cmap=’g ay’, min=0, max=1)
321 axs[0].se _ i le("(a) Da k mass densi y nHo izon al slice XY", on size=20)
322 axs[0].axis(’o ’)
323 axs[0].se _aspec (’equal’, adjus able=’box’)
324
325 # (b) Ve ical slice XZ ( ue o scala L_z)
326 axs[1].imshow( ho_e _xz_no m, ex en =ex en _xz, o igin=’lowe ’,
,→cmap=’g ay’, min=0, max=1)
327 axs[1].se _ i le("(b) Da k mass densi y nVe ical slice XZ", on size=20)
328 axs[1].axis(’o ’)
329 axs[1].se _aspec (’equal’, adjus able=’box’)
330
331 ig.sup i le("Spi al 3D Halo No malized |A| (200 kpc 200 kpc)",
332 on size=22, on weigh =’bold’, y=0.97)
333 ig.subplo s_adjus (le =0.04, igh =0.96, bo om=0.05, op=0.80,
,→wspace=0.10)
334
335 pl .sa e ig("C:/Use s/B yanH/Downloads/ ig3_da kmass_xy_xz.png",
336 dpi=300, acecolo =’whi e’)
337 pl .show()
116
Lis ing 1: Py hon code o he ull 3D Helmhol z esonan –halo simula ion.
117

D
Appendix: Code o Simula ing Ellip ical
Galaxies - T anspo Theo y
This appendix p o ides he exac nume ical implemen a ion used o gene a e
he ensemble–a e aged esonan –halo p edic ions shown in Sec ion 7.2.8. The
simula ion ealizes he closed anspo –law cu a u e–phonon halo de i ed
analy ically in Sec ion 7.2.4 onwa ds, including he mul iplica i e g adien –ene gy
con ibu ion o Sec ion 7.2.7.
The s ella mass dis ibu ion is modeled as a He nquis sphe e wi h o al s ella
popula ion
N⋆
, indi idual mass
m⋆
, and scale leng h
a
=
Re/
1
.
8153, gi ing he
e ec i e ba yonic olume
Vgal = 2πa3.(D.1)
The ellip ical anspo closu e is go e ned by ou con ol pa ame e s:
• he anspo damping leng h
Ldamp,e =ce
Γe
,(D.2)
ixed di ec ly om he obse ed halo ex en ;
• he esonan cu a u e wa eleng h
λ es,(D.3)
which se s bo h he ca ie equency
Ω = 2πce
λ es
(D.4)
and he e ec i e quali y ac o
Qe =Ω
2Γe
; (D.5)
•
he speckle–shape ac o
α∼
1, which con ols he mean geome ic
cohe ence o he s ella sou ces; and
•
he mean coupling e iciency
|X|2∼
1, which se s he ela i e weigh ing
o indi idual sou ces in he cohe en ensemble ield.
He e Γ
e
is he anspo –damping a e associa ed wi h mul iple s ella esca e -
ing, and
ce
is he cu a u e–phonon p opaga ion speed; bo h a e de ined exac ly
as in he main ex .
118
The ensemble–a e aged e ec i e g a i a ing densi y implemen ed in he code
ollows he closed anspo –halo law,
ρe ( )=4πG
c2
e 2Qe
2πΞ∇Gen ,e 




, < Rma ch,
e−2 /Ldamp,e
(4π)2 2, ≥Rma ch.
(D.6)
whe e
Gen ,e
is he ensemble en i onmen al gain,
Qe
= Ω
/
(2Γ
e
) is he e ec i e
quali y ac o , and Ξ
∇
is he anspo –enhanced g adien –ene gy boos all
de ined in Sec ion 7.2.7. Fo nume ical egula i y, he small– adius linea co e
⟨ρe ( )⟩∝ is en o ced o < Rma ch (see Sec ion 7.2.6).
Fo nume ical consis ency and physical egula i y, a small– adius linea co e,
⟨ρe ( )⟩∝ ( < Rma ch),(D.7)
is en o ced as in Sec ion 7.2.6.
All symbols in Eqs.
(D.1)
–
(D.7)
e ain he exac same meaning and no a ion as in
he main ex ; no new a iables a e in oduced in he nume ical implemen a ion.
The code uses α=|X|2=1 o he baseline models.
The code below:
•cons uc s ρe ( ), enclosed Me ( ), and ci cula eloci ies;
•compa es di ec ly o he s ella mass dis ibu ion; and
•
p oduces bo h midplane composi e maps and ull–scale XY/XZ slices o
he esonan da k halo.
1
2# =====================================================================
3# Ellip ical Galaxy T anspo -Law Cu a u e-Phonon Speckle Halo
4# W i en by: B yan Hennelly 28 Oc obe 2025
5# =====================================================================
6#
7# This sc ip compu es he ensemble-a e aged e ec i e halo densi y
8# gene a ed by cu a u e-phonon anspo in a densely popula ed,
9# phase-mixed s ella medium (ellip ical galaxy).
10 #
11 # Physics summa y:
12 #
13 # S a s a e bo h sou ces and sca e e s o cu a u e phonons.
14 # Mul iple esonan sca e ing inc eases he cohe en pa h leng h
15 # a beyond single-pass damping: _e << .
16 # The cohe en anspo ail decays as exp(-2 /L_damp_e )/ .
17 # The ensemble-s o ed cu a u e ene gy mani es s as an
18 # e ec i e g a i a ing densi y _e ( ).
19 #
20 # The e a e FOUR p incipal con ol pa ame e s:
119
21 #
22 # (1) L_damp_e anspo damping leng h o he cohe en ail
23 # (DATA-DRIVEN om obse ed halo ex en )
24 #
25 # (2) _ es s uc u al esonan wa eleng h o cu a u e
26 # (PHYSICAL MICRO-INPUT con olling ampli ude)
27 #
28 # (3) speckle-shape ac o (~1), con ols phase-a e aged
29 # geome ic cohe ence o s ella sou ces
30 #
31 # (4) |X| mean s ella coupling e iciency (~1), se s
32 # ela i e weigh ing o each sou ce in he cohe en
,→ ield
33 #
34 # F om _ es we ob ain:
35 #
36 # = 2 c_e / _ es (ca ie equency)
37 # Q_e = / (2 _e ) (e ec i e quali y ac o )
38 #
39 # whe e _e = c_e / L_damp_e is implied by anspo heo y,
40 # and c_e is he cu a u e-phonon p opaga ion speed we ake o be 2e5 m/s
41 #
42 # FINAL anspo -enhanced halo law:
43 #
44 # _e ( ) =
45 # _nabla *
46 # [ G /(2 c c_e ) ] *
47 # L_damp_e * Q_e *
48 # (N_* m_* / V_gal) * |X| *
49 # { , <R_ma ch ; exp[-2 /L_damp_e ]/ , R_ma ch }
50 #
51 # wi h:
52 # _e = c_e / L_damp_e
53 # Q_e = / (2 _e )
54 # _nabla = 1 + (2 L_damp_e / _ es)
55 # G_en ,e = |X| (N_*/V_gal) (_ es/2) ( / 2_e )
56 #
57 # In e p e a ion:
58 # Tail SHAPE is dic a ed solely by L_damp_e (obse ed).
59 # Tail AMPLITUDE is dic a ed solely by _ es (mic ophysics).
60 # and |X| en e only as weak, O(1) no maliza ion ac o s.
61 #
62 # Pa ame e -economical closu e:
63 # One obse a ional inpu halo size ixed
64 # One physical pa ame e halo mass ixed
65 #
66 # A small- linea co e is en o ced by egula i y o cohe en lux.
67 #
68 # Ou pu s:
69 # _e ( ), ci cula eloci ies, enclosed masses
70 # 2D composi e ba yonic s da k mass maps
71 # la ge XY/XZ slices o he ull halo ield
72 #
73 # All a iable names ollow he pape s no a ion o 1:1 co espondence
120
74 # be ween analy ic exp essions and nume ical quan i ies.
75 # =====================================================================
76
77
78
79 # =====================================================================
80 # Ellip ical Galaxy T anspo -Law S a is ical Resonan Halo
81 # Wi h Mul iple-Sca e ing E ec i e Damping + G adien Boos
82 # =====================================================================
83
84 impo numpy as np
85 impo ma plo lib.pyplo as pl
86
87 # ---------- Cons an s ----------
88 G = 6.67430e-11
89 c = 2.99792458e8
90 Msun = 1.98847e30
91 pc = 3.085677581491367e16
92 kpc = 1.0e3 * pc
93 Mpc = 1.0e6 * pc
94 pi = np.pi
95
96 # ============================================================
97 # TRANSPORT MEDIUM (e ec i e) PARAMETERS [DATA-DRIVEN]
98 # ============================================================
99
100 L_damp_e = 636.0 * kpc # [m] SET by obse ed halo e- olding
101 c_e = 2.0e5 # [m/s] TUNABLE wi hin plausible ange
102 Gamma_e = c_e / L_damp_e # [1/s]
103 au_coh_e = 1/(2 * Gamma_e ) # [s]
104
105 p in (" n--- T anspo Medium (e ec i e) ---")
106 p in ( "L_damp_e = {L_damp_e /kpc:.1 } kpc [SET]")
107 p in ( "c_e = {c_e :.3e} m/s [TUNE]")
108 p in ( "Gamma_e = {Gamma_e :.3e} s^-1 [IMPLIED]")
109 p in ( " au_coh_e = { au_coh_e /3.15576e13:.2 } My ")
110
111 # ============================================================
112 # RESONANT MICROPHYSICS
113 # ============================================================
114
115 lambda_ es = 27.0 * pc # [m]
116 Omega = 2*np.pi*c_e /lambda_ es # [1/s]
117 Q_e = Omega/(2*Gamma_e ) # dimensionless
118 Xi_nabla = 1.0 + (2.0*pi*L_damp_e / lambda_ es)**2 # g adien -ene gy
,→boos
119
120 p in (" n--- Resonan Mic ophysics ---")
121 p in ( "lambda_ es = {lambda_ es/pc:.2 } pc")
122 p in ( "Omega = {Omega:.3e} s^-1")
123 p in ( "Q_e = {Q_e :.3e}")
124 p in ( "Xi_nabla = {Xi_nabla:.3e}")
125
126 # ============================================================
121
E
Appendix: Code o Simula ing Clus e Galax-
ies
This appendix p o ides he comple e Py hon implemen a ion used o gene a e
Figu es 10–12. The code ollows di ec ly om he adia i e– anspo o mal-
ism de eloped in Sec ion 7.3.1, implemen ing he o e lapping ellip ical–halo
con olu ion in educed “2.5D” o m o nume ical s abili y.
Each galaxy is ep esen ed by an exponen ial anspo ke nel
K(s;L) = e−2s/L
(4π)2s2,(E.1)
ensemble–a e aged o e a logno mal dis ibu ion o damping leng hs
L
wi h
loga i hmic dispe sion
σln L
. The angula ly a e aged 2.5D con olu ion is e alua ed
as
ρshape( )=2πZ∞
0
ngal( ′) ′⟨K(s)⟩µd ′, s2= 2+ ′2−2 ′µ, (E.2)
using Gauss–Legend e quad a u e o he µ–in eg a ion.
The galaxy numbe –densi y p o ile adop s an NFW-like o m wi h a smoo h
splashback ape ,
ngal( )=n0
Wsp( )
x(1+x)2, Wsp( ) = 1
1 + exp( −Rsp)/∆sp,(E.3)
ensu ing a ini e ou e ex en wi hou discon inui ies.
The educed con olu ion esul ρshape( ) is no malized by he geome ic ac o
Λgeom =Ngal M(ellip ical)
DM
Z∞
0
4π 2ρshape( )d
,(E.4)
and scaled by he nonlinea e e be a ion gain
G e =1+Ngalη, (E.5)
o yield he inal e ec i e densi y
ρe ( )=G e Λgeom Sbase ρshape( ).(E.6)
Fo he iducial 1015 M⊙clus e , he adop ed pa ame e s a e
Lmean = 636 kpc, σln L= 2.5, Ngal = 300, η = 8.3×10−3,
co esponding o
G e ≃
3
.
5. The code p oduces h ee diagnos ic igu es: (i) clus-
e densi y and enclosed mass, (ii) ci cula eloci y p o ile, and (iii) 2D dis ibu-
ions o ρe and c, ep oducing he esul s p esen ed in Figu es 10–12.
128

1
2# =====================================================================
3# Galaxy Clus e S a ic T anspo -O e lap s. NFW Benchma k
4#
5# Pu pose:
6# Compu e he clus e -scale e ec i e anspo densi y _e ( )
7# a ising om he collec i e o e lap o mesoscopic anspo halos
8# su ounding indi idual galaxies, and compa e i s eme gen p o ile
9# wi h he canonical NFW da k-ma e dis ibu ion.
10 #
11 # Model O e iew:
12 # 1. K_ ail(s) de ines he dimensionless single-galaxy anspo ke nel,
13 # ep esen ing he exponen ial in ensi y ail o an ellip ical halo.
14 #
15 # 2. The ellip ical-halo mic ophysics de e mine he absolu e scaling
16 # h ough Sba _base, which se s he no maliza ion o _e ( )
17 # once he e ec i e damping leng h L_damp_e is chosen.
18 # He e, we adop he same L_damp_e as used in he single-ellip ical
19 # simula ion o in e nal consis ency.
20 #
21 # 3. The spa ial con olu ion is pe o med in a educed (~2.5D) o m
22 # o nume ical s abili y. One geome ic leng h ac o is omi ed
23 # and la e es o ed by _geom, de i ed ia global mass conse a ion.
24 #
25 # 4. A nonlinea e e be a ion gain ac o ,
26 # G_ e = 1 + N_gal * ,
27 # accoun s o weak c oss-cohe ence be ween o e lapping anspo
28 # ields, whe e is he ac ional coupling coe icien .
29 #
30 # Key Con ol Pa ame e s:
31 # _lnL Loga i hmic dispe sion o anspo leng hs (he e ogenei y):
32 # b oadens he e ec i e o e lap and go e ns he cu a u e
33 # o _e ( ). C ucial o ma ching he NFW slope.
34 #
35 # Re e be a ion coupling coe icien :
36 # con ols he deg ee o collec i e ampli ica ion among
37 # o e lapping galaxy halos. Values in he ange 1010
38 # yield physically plausible cohe ence gains.
39 #
40 # Once he galaxy mic ophysics and damping scale a e ixed,
41 # _lnL and become he wo p ima y uning pa ame e s de e mining
42 # he clus e -scale densi y and eloci y p o iles.
43 #
44 # Final Exp ession:
45 # _e ( ) = G_ e * _geom * Sba _base * _shape( )
46 # =====================================================================
47
48 impo numpy as np
49 impo ma plo lib.pyplo as pl
50
51 # ---------------------------------------------------------------------
52 # Fundamen al cons an s and uni s
53 # ---------------------------------------------------------------------
129
54 G = 6.67430e-11 # m^3 kg^-1 s^-2
55 c = 2.99792458e8 # m/s
56 Msun= 1.98847e30 # kg
57 pc = 3.085677581491367e16 # m
58 kpc = 1.0e3 * pc
59 Mpc = 1.0e6 * pc
60 pi = np.pi
61
62 # ---------------------------------------------------------------------
63 # Clus e -scale pa ame e s
64 # ---------------------------------------------------------------------
65 L_ba = 636.0 * kpc # main anspo leng h [m]
66 sigma_lnL = 2.5 # logno mal sca e in L
67 R_cl = 2.0 * Mpc # clus e adius [m]
68 N_gal = 300 # numbe o ellip ical galaxies
69
70 # S uc u al (clus e -scale) pa ame e s
71 R_sp = 2.5 * Mpc # splashback adius [m]
72 Del a_sp = 0.5 * Mpc # splashback ansi ion wid h [m]
73 _s = 1.0 * Mpc # sa elli e NFW scale adius [m]
74 n0_sa = 2.0e-63 # sa elli e numbe -densi y scale [m^-3]
75
76 # ---------------------------------------------------------------------
77 # Ellip ical-halo mic ophysics Sba _base
78 # ---------------------------------------------------------------------
79 L_damp_e = L_ba
80 c_e _E = 2.0e5
81 Gamma_e = c_e _E / L_damp_e
82 lambda_ es = 27.0 * pc
83 Omega_E = 2.0 * pi * c_e _E / lambda_ es
84 Q_e = Omega_E / (2.0 * Gamma_e )
85 Xi_nabla = 1.0 + (2.0 * pi * L_damp_e / lambda_ es)**2
86
87 M_ o _E = 2.0e11 * Msun # o al mass o one ellip ical
88 m_s a _E = Msun
89 N_s a _E = M_ o _E / m_s a _E
90 Re_E = 5.0 * kpc
91 a_h_E = Re_E / 1.8153
92 V_gal_E = 2.0 * pi * a_h_E**3
93 _E = 200.0e3
94 alpha_E = 1.0
95
96 K_base = (G * alpha_E * _E) / (2.0 * c**2 * c_e _E**3)
97 * L_damp_e * Q_e * (N_s a _E**2 * m_s a _E**2 / V_gal_E)
98 Sba _base = Xi_nabla * K_base
99
100 p in ("=== Single Ellip ical Ke nel Pa ame e s ===")
101 p in ( "L_ba = {L_ba /kpc:.1 } kpc, Q_e = {Q_e :.3e}, Xi_nabla =
,→{Xi_nabla:.3e}")
102 p in ( "Sba _base = {Sba _base:.3e} [kg/m^3] n")
103
104 # ---------------------------------------------------------------------
105 # Dimensionless anspo ke nel (shape only)
106 # ---------------------------------------------------------------------
130
107 de K_ ail(s, L):
108 """Dimensionless anspo ke nel."""
109 s_eps = 1.0 * pc
110 s_clp = np.maximum(s, s_eps)
111 e u n np.exp(-2.0 * s_clp / L) / (((4.0 * pi)**2) * s_clp**2)
112
113 de K_ ail_ensemble(s, L_mean, sigma_ln):
114 """Logno mal-a e aged ke nel ensemble."""
115 i sigma_ln <= 0.0:
116 e u n K_ ail(s, L_mean)
117 xi = np.a ay([-2, -1, 0, 1, 2]) * sigma_ln
118 w = np.a ay([1, 4, 10, 4, 1], d ype= loa )
119 w /= w.sum()
120 ou = np.ze os_like(s)
121 o wi, Li in zip(w, L_mean * np.exp(xi)):
122 ou += wi * K_ ail(s, Li)
123 e u n ou
124
125 K un = lambda s: K_ ail_ensemble(s, L_ba , sigma_lnL)
126
127 # ---------------------------------------------------------------------
128 # Galaxy numbe -densi y p o ile (sa elli e popula ion)
129 # ---------------------------------------------------------------------
130 de W_splash( ):
131 """Smoo h splashback ape ."""
132 e u n 1.0 / (1.0 + np.exp(( - R_sp) / Del a_sp))
133
134 de n_sa _NFW( ):
135 """Sa elli e popula ion ~ 1/[x(1+x)^2]."""
136 x = np.maximum( / _s, 1e-9)
137 e u n W_splash( ) * n0_sa / (x * (1.0 + x)**2)
138
139 # ---------------------------------------------------------------------
140 # Reduced sphe ical con olu ion (~2.5D o m)
141 # ---------------------------------------------------------------------
142 de angula _a e age_K( , p, K un):
143 """Angula ly a e aged ke nel in eg al K(s) d."""
144 mu, w = np.polynomial.legend e.leggauss(48)
145 , p = np.a leas _1d( ). eshape(-1,1), np.a leas _1d( p). eshape(1,-1)
146 mu3, w3 = mu[:,None,None], w[:,None,None]
147 s = np.sq ( **2 + p**2 - 2.0 * * p * mu3)
148 e u n np.sum(w3 * K un(s), axis=0)
149
150 de ho_ om_popula ion(n_pop):
151 """Compu e educed con olu ion _shape( )."""
152 I = angula _a e age_K( , , K un)
153 in eg and = n_pop *
154 e u n 2.0 * pi * (I @ (in eg and * d ))
155
156 # ---------------------------------------------------------------------
157 # Radial g id and con olu ion
158 # ---------------------------------------------------------------------
159 = np.geomspace(0.2 * kpc, 3.0 * Mpc, 480)
160 d = np.g adien ( )
131
161 ho_e _shape = ho_ om_popula ion(n_sa _NFW( ))
162
163 # ---------------------------------------------------------------------
164 # (1) Geome ic no maliza ion _geom by mass conse a ion
165 # ---------------------------------------------------------------------
166 _single = np.geomspace(0.1 * kpc, 100.0 * kpc, 400)
167 ho_single = Sba _base * K_ ail( _single, L_ba )
168 M_DM_ellip = 4.0 * np.pi * np. apz( ho_single * _single**2, _single)
169
170 sum_M_DM_ellip icals = N_gal * M_DM_ellip
171 M_clus e _shape = 4.0 * np.pi * np. apz( ho_e _shape * **2, )
172 Lambda_geom = sum_M_DM_ellip icals / (Sba _base * M_clus e _shape)
173
174 p in ( "Geome ic no maliza ion _geom = {Lambda_geom:.3e}")
175
176 # ---------------------------------------------------------------------
177 # (2) Re e be a ion gain ac o G_ e = 1 + N_gal *
178 # ---------------------------------------------------------------------
179 e a = 0.00825 # ac ional cohe ence (~1e-31e-2 ypical)
180 G_ e = 1.0 + N_gal * e a
181 p in ( "Re e be a ion gain G_ e = {G_ e :.2 } n")
182
183 # ---------------------------------------------------------------------
184 # Final physical densi y
185 # ---------------------------------------------------------------------
186 ho_e = G_ e * Lambda_geom * Sba _base * ho_e _shape
187
188 # ---------------------------------------------------------------------
189 # Enclosed mass and ci cula eloci y
190 # ---------------------------------------------------------------------
191 M_e = 4.0 * pi * np.cumsum( ho_e * **2 * d )
192 _c = np.sq (G * np.maximum(M_e , 0.0) / ) / 1.0e3 # [km/s]
193
194 # ---------------------------------------------------------------------
195 # NFW benchma k
196 # ---------------------------------------------------------------------
197 H0 = 70.0 * 1000.0 / (1.0e6 * pc)
198 ho_c i = 3.0 * H0**2 / (8.0 * pi * G)
199 M200 = 1.0e15 * Msun
200 c200 = 4.0
201 R200 = (3.0 * M200 / (4.0 * pi * 200.0 * ho_c i ))**(1/3)
202 _s_DM = R200 / c200
203 ho_s = (200.0/3.0) * ho_c i * (c200**3) / (np.log(1+c200) -
,→c200/(1+c200))
204
205 de ho_n w( i):
206 x = np.maximum( i / _s_DM, 1e-12)
207 e u n ho_s / (x * (1.0 + x)**2)
208
209 ho_n w_ als = ho_n w( )
210 _n w = np.sq (G * (4.0*pi*np.cumsum( ho_n w_ als* **2*d )) / ) / 1.0e3
211
212 # ---------------------------------------------------------------------
213 # Diagnos ics
132
214 # ---------------------------------------------------------------------
215 p in ("=== Diagnos ics ===")
216 p in ( "L_ba = {L_ba /kpc:.1 } kpc")
217 p in ( "sigma_lnL = {sigma_lnL:.2 } (logno mal wid h)")
218 p in ( " (e a) = {e a:.4 } ( e e be a ion coupling s eng h)")
219 p in ( "G_ e = {G_ e :.2 } (1 + N_gal , nonlinea e e be a ion
,→gain)")
220 p in ( "_geom = {Lambda_geom:.3e} (mass-conse ing no maliza ion)")
221 p in ( "max _e = { ho_e .max():.3e} kg/m a =
,→{ [np.a gmax( ho_e )]/kpc:.2 } kpc")
222 p in ( "max _c = { _c.max():.2 } km/s a =
,→{ [np.a gmax( _c)]/kpc:.2 } kpc n")
223
224
225 _Mpc = / Mpc
226 # ================================================================
227 # FIGURE 1 Densi y and Enclosed Mass (Linea + Log, aised i les)
228 # ================================================================
229
230 om ma plo lib. icke impo Scala Fo ma e
231 o ma e = Scala Fo ma e (useMa hTex =T ue)
232 o ma e .se _powe limi s((-2, 4))
233
234 _Mpc = / Mpc
235 ho_e _msun_pc3 = ho_e * (pc**3 / Msun)
236 ho_n w_msun_pc3 = ho_n w_ als * (pc**3 / Msun)
237
238 # ---------- Figu e se up ----------
239 ig1, axs1 = pl .subplo s(2, 2, igsize=(13, 11))
240 pl .subplo s_adjus (wspace=0.35, hspace=0.35)
241
242 # ---------- Fon sizes ----------
243 i le_ s = 26
244 label_ s = 22
245 ick_ s = 18
246 legend_ s = 20
247 sup i le_ s = 32
248 i le_pad = 28 # aised i les o all panels
249
250 # (a) Densi y (linea )
251 axs1[0,0].plo ( _Mpc, ho_e _msun_pc3, colo =’na y’, lw=2,
,→label= "$ ho_{ m e }$")
252 axs1[0,0].plo ( _Mpc, ho_n w_msun_pc3, ’--’, colo =’da ko ange’, lw=2,
,→label="NFW")
253 axs1[0,0].se _ i le("(a) Densi y (linea )", on size= i le_ s, pad= i le_pad)
254 axs1[0,0].se _xlabel(" [Mpc]", on size=label_ s)
255 axs1[0,0].se _ylabel( "Densi y [$M_ odo ,{ m pc^{-3}}$]",
,→ on size=label_ s)
256 axs1[0,0].legend( on size=legend_ s)
257 axs1[0,0]. ick_pa ams(axis=’bo h’, which=’majo ’, labelsize= ick_ s)
258 axs1[0,0].xaxis.se _majo _ o ma e ( o ma e )
259 axs1[0,0].yaxis.se _majo _ o ma e ( o ma e )
260
261 # (b) Densi y (loglog)
133

262 axs1[0,1].loglog( _Mpc, ho_e _msun_pc3, colo =’na y’, lw=2)
263 axs1[0,1].loglog( _Mpc, ho_n w_msun_pc3, ’--’, colo =’da ko ange’, lw=2)
264 axs1[0,1].se _ i le("(b) Densi y (loglog)", on size= i le_ s, pad= i le_pad)
265 axs1[0,1].se _xlabel(" [Mpc]", on size=label_ s)
266 axs1[0,1].se _ylabel( "Densi y [$M_ odo ,{ m pc^{-3}}$]",
,→ on size=label_ s)
267 axs1[0,1]. ick_pa ams(axis=’bo h’, which=’majo ’, labelsize= ick_ s)
268
269 # (c) Enclosed mass (linea )
270 M_n w = 4.0 * np.pi * np.cumsum( ho_n w_ als * **2 * d )
271 axs1[1,0].plo ( _Mpc, M_e /Msun, colo =’c imson’, lw=2, label= "$M_{ m
,→e }$")
272 axs1[1,0].plo ( _Mpc, M_n w/Msun, ’--’, colo =’o ange’, lw=2, label="NFW")
273 axs1[1,0].se _ i le("(c) Enclosed mass (linea )", on size= i le_ s,
,→pad= i le_pad)
274 axs1[1,0].se _xlabel(" [Mpc]", on size=label_ s)
275 axs1[1,0].se _ylabel( "$M(< ) ,[M_ odo ]$", on size=label_ s)
276 axs1[1,0].legend( on size=legend_ s)
277 axs1[1,0]. ick_pa ams(axis=’bo h’, which=’majo ’, labelsize= ick_ s)
278 axs1[1,0].xaxis.se _majo _ o ma e ( o ma e )
279 axs1[1,0].yaxis.se _majo _ o ma e ( o ma e )
280 axs1[1,0].yaxis.ge _o se _ ex ().se _ on size( ick_ s)
281
282 # (d) Enclosed mass (loglog)
283 axs1[1,1].loglog( _Mpc, M_e /Msun, colo =’c imson’, lw=2)
284 axs1[1,1].loglog( _Mpc, M_n w/Msun, ’--’, colo =’o ange’, lw=2)
285 axs1[1,1].se _ i le("(d) Enclosed mass (loglog)", on size= i le_ s,
,→pad= i le_pad)
286 axs1[1,1].se _xlabel(" [Mpc]", on size=label_ s)
287 axs1[1,1].se _ylabel( "$M(< ) ,[M_ odo ]$", on size=label_ s)
288 axs1[1,1]. ick_pa ams(axis=’bo h’, which=’majo ’, labelsize= ick_ s)
289
290 # ---------- Main i le ----------
291 ig1.sup i le("Clus e Densi y and Enclosed Mass",
292 on size=sup i le_ s, on weigh =’bold’, y=0.98)
293
294 # ---------- Sa e & display ----------
295 pl . igh _layou ( ec =[0, 0, 1, 0.96])
296 pl .sa e ig("C:/Use s/B yanH/Downloads/ ig1_densi y_mass.png",
297 dpi=300, acecolo =’whi e’)
298 pl .show()
299
300 # ================================================================
301 # FIGURE 2 Ci cula Veloci y P o ile (Linea Only, ma ching Figu e 1 on
,→sizes)
302 # ================================================================
303
304 ig2, ax2 = pl .subplo s( igsize=(14, 6))
305
306 # Ma ch Figu e 1 on sizes
307 i le_ s = 26
308 label_ s = 22
309 ick_ s = 18
310 legend_ s = 20
134
311 sup i le_ s = 32
312
313 ax2.plo ( _Mpc, _c, colo =’c imson’, lw=2, label="T anspo o e lap")
314 ax2.plo ( _Mpc, _n w, ’--’, colo =’o ange’, lw=2, label="NFW")
315
316 ax2.se _ i le("(a) Ci cula eloci y (linea )", on size= i le_ s, pad=20)
317 ax2.se _xlabel(" [Mpc]", on size=label_ s)
318 ax2.se _ylabel( "$ _c$[km s$^{-1}$]", on size=label_ s)
319 ax2.legend( on size=legend_ s, loc=’bes ’)
320 ax2. ick_pa ams(axis=’bo h’, which=’majo ’, labelsize= ick_ s)
321 ax2.xaxis.se _majo _ o ma e ( o ma e )
322 ax2.yaxis.se _majo _ o ma e ( o ma e )
323
324 ig2.sup i le("Ci cula Veloci y P o ile T anspo s NFW",
325 on size=sup i le_ s, on weigh =’bold’, y=0.97)
326
327 pl . igh _layou ( ec =[0.02, 0, 0.98, 0.95])
328 pl .sa e ig("C:/Use s/B yanH/Downloads/ ig2_ eloci y.png",
329 dpi=300, acecolo =’whi e’)
330 pl .show()
331
332
333 # ================================================================
334 # FIGURE 3 2D Dis ibu ions o e _max ( inal wi h ex a op space)
335 # ================================================================
336
337 # ---------- Cons uc 2D g id ----------
338 n = 700
339 _max_Mpc = .max() / Mpc
340 ex en = _max_Mpc
341 x2 = np.linspace(-ex en , ex en , n)
342 y2 = np.linspace(-ex en , ex en , n)
343 X, Y = np.meshg id(x2, y2)
344 R2 = np.sq (X**2 + Y**2) * Mpc # con e o me e s
345
346 # ---------- In e pola e 1D adial p o iles ----------
347 ho_2D = np.in e p(R2, , ho_e , igh =np.nan)
348 _2D = np.in e p(R2, , _c, igh =np.nan) # [km/s]
349
350 # ---------- No malize e ec i e densi y ----------
351 ho_log = np.log10( ho_2D + 1e-40)
352 ho_no m = ( ho_log - np.nanmin( ho_log)) / (np.nanmax( ho_log) -
,→np.nanmin( ho_log))
353
354 # ---------- Figu e se up ----------
355 ig3, axs3 = pl .subplo s(1, 2, igsize=(10, 6))
356 i le_ s = 22
357
358 # (a) E ec i e densi y
359 im0 = axs3[0].imshow(
360 ho_no m, ex en =[- _max_Mpc, _max_Mpc, - _max_Mpc, _max_Mpc],
361 o igin=’lowe ’, cmap=’in e no’, min=0, max=1
362 )
363 axs3[0].se _ i le( "(a) $ log_{10} ho_{ ma h m{e }}$" " n(no malized)",
135
364 on size= i le_ s, pad=10)
365 axs3[0].axis(’o ’)
366 ig3.colo ba (im0, ax=axs3[0], ac ion=0.046, pad=0.04)
367
368 # (b) Veloci y ield (linea )
369 im1 = axs3[1].imshow(
370 _2D, ex en =[- _max_Mpc, _max_Mpc, - _max_Mpc, _max_Mpc],
371 o igin=’lowe ’, cmap=’ i idis’
372 )
373 axs3[1].se _ i le( "(b) Ci cula eloci y $ _c$" " n(linea )",
374 on size= i le_ s, pad=10)
375 axs3[1].axis(’o ’)
376 cba 1 = ig3.colo ba (im1, ax=axs3[1], ac ion=0.046, pad=0.04)
377 cba 1.se _label( "$ _c$[km s$^{-1}$]", on size=14)
378
379 # ---------- Main i le ----------
380 ig3.sup i le(
381 "2D Dis ibu ions o e { _max_Mpc:.1 } Mpc",
382 on size=28, on weigh =’bold’, y=0.995
383 )
384
385 # ---------- Layou & spacing ----------
386 pl .subplo s_adjus (le =0.03, igh =0.97, op=0.86, bo om=0.05,
387 wspace=0.20, hspace=0.25)
388
389 # ---------- Sa e & display ----------
390 pl .sa e ig("C:/Use s/B yanH/Downloads/ ig3_2D_dis ibu ions.png",
391 dpi=300, acecolo =’whi e’, bbox_inches=’ igh ’)
392 pl .show()
Lis ing 3: Py hon code o he qua e –wa e sphe ical clus e esona o .
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