Geome ic Res o a i i y o Space ime
Jianheng Huang
Linyi Uni e si y
No embe 7, 2025
Abs ac
We p opose a Geome ic Res o a i i y (GRS) amewo k in he weak, s a ic limi , (1
−
ℓ2∇2
)
∇2
Φ = 4
πGρ
, wi h an en i onmen -dependen eco e y leng h
ℓ
. Wi hou in oking
da k ma e , GRS uni ies disk o a ion cu es and me ging-clus e lensing: in he inne disk,
nea ly cons an
ℓ
yields a cen al analy ic ke nel and a solid-body ise
∝R
aligned wi h he
exponen ial-disk peak; in he ou e disk, se ing
ℓ
=
λR
p oduces a adial in a ian
Q
and
2
(
R
) =
Q
+
C Rn
wi h
n
= 2
−k+
(
λ
)
<
0, explaining la /mildly declining/ ising ypes and
enabling a di ec ex ac ion o
Q
and
λ
om de i a i es o
(
R
). When
ℓ→ℓsa
, he cu e
app oaches a Yukawa/Keple ail. P ojec ing o 2D gi es (1
−ℓ2
θ∇2
θ
)
κ
= Σ
/
Σ
c i
[
1
]: di e en ial
smoo hing (small
ℓ
o galaxies, la ge
ℓ
o gas) gene ically yields mass–gas peak o se s and
equipo en ial/shea b idges. The amewo k educes o New on/GR o
ℓ/ ≪
1and p o ides
alsi iable slope/scale ela ions ied o adius, di useness, and me ge phase.
Keywo ds: geome ic es o a i i y; nonlocal weak g a i y; uni ied o a ion cu es; ou e -disk
pla eau and slope; Yukawa ke nel; adial in a ian
Q
;
ℓ
=
λR
; Bulle Clus e ; peak o se ; equipo-
en ial/shea b idge
In oduc ion
In he weak, s a ic limi we p opose a geome ic es o a i i y amewo k go e ned by an en i onmen -
dependen eco e y leng h ℓ, wi h ield equa ion
(1 −ℓ2∇2)∇2Φ=4πGρ. (1)
Equa ion (1) is equi alen o a New onian ke nel d essed by an adap i e low-pass (Yukawa)
ope a o [
2
,
3
]:
ℓ
is small in high-cu a u e/dense egions and la ge in low-cu a u e/di use egions.
A uni ied o a ion-cu e na a i e hen ollows na u ally: in he inne disk wi h
ℓ≈cons
he
con ol ed ke nel is analy ic a he cen e , yielding a solid-body ise
∝R
and he s anda d
exponen ial-disk peak[4, 5]; in he ou e disk aking ℓ∝R ecas s he equa ion in o a conse a i e-
lux o m so ha
2
app oaches a cons an pla eau (allowing small powe -law de ia ions); a he
ou , i
ℓ
sa u a es o a cons an he solu ion ansi ions o a ini e-scale Yukawa ail ha smoo hly
ends o a Keple all-o [6].
P ojec ing (1) along he line o sigh gi es he modi ied lensing ela ion
(1 −ℓ2
θ∇2
θ)κ= Σ/Σc i ,(2)
1
so he same “ke nel as low-pass” mechanism p ese es dense galaxy peaks (small
ℓθ
) while sup-
p essing di use, am-s ipped gas (la ge
ℓθ
), he eby p edic ing bo h a mass–gas peak o se and a
shea /equipo en ial b idge along he me ge axis[
7
–
9
]. The amewo k au oma ically educes o
New on/GR in he
ℓ/ ≪
1 egime and uni ies disk o a ion cu es wi h me ging-clus e lensing in o
a single, obse a ion-quan i iable pic u e.
Con en s
1 Two Pos ula es o Geome ic Res o a i i y 3
1.1 P1 — Geome ic Res o a i i y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 P2 — Cu a u e–Reco e y Closu e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Immedia e co olla ies (no addi ional pos ula es). . . . . . . . . . . . . . . . . . . . . 3
2
Uni ied Ro a ion Cu es om Inne o Ou e Galac ic Disks: The Mainline o
Geome ic Res o a i i y 4
2.1 Inne disk: cen al analy ici y o he con ol ed ke nel and de e mina ion o he peak 4
2.2 Sola -sys em limi : why s ic ly New onian . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 T ansi ion zone: slowly a ying ℓ(R)and con inui y condi ions . . . . . . . . . . . . 5
2.4 Ou e pla eau (ρb≃0,ℓ=λR): conse ed quan i y and small de ia ions . . . . . . . 6
2.5 Fa he acuum: sa u a ion o ℓand he Yukawa/Keple ail . . . . . . . . . . . . . 6
2.6 Uni ied pic u e and es able p edic ions . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Me ging-Clus e Lensing: Bulle -Clus e Peak O se and Shea B idge 8
3.1 Se up&p ojec ion...................................... 8
3.2 Uni ied pic u e o he o se . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Two-componen , piecewise-cons an model. . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Implemen a ion & alsi iable ends. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Conclusions and Ou look 9
4.1 Conclusions.......................................... 9
4.2 Ou look............................................ 9
5 Re sec ion 10
6
Appendix A (De ailed): S ep-by-s ep de i a ion om he ac ion o he modi ied
Poisson equa ion 11
2
7
Appendix B — Weak-Field S a ic Limi and he New on–Yukawa G een Func ion
14
8
Appendix C: Uni ied De i a ion o Galac ic Ro a ion Cu es om Inne o Ou e
Disk in GRS 19
9
Appendix D:Bulle Clus e : Peak O se and Equipo en ial B idges (De i a ion)
27
1 Two Pos ula es o Geome ic Res o a i i y
Scope and egime. Th oughou his sec ion we wo k in he weak- ield, s a ic limi and adop
he New onian gauge wi h negligible aniso opic s ess so ha Φ
≃
Ψ. Sou ces a e non- ela i is ic
(T00 ≈ρ), and all ope a o s a e Euclidean in space.
1.1 P1 — Geome ic Res o a i i y.
Space ime es o es local geome ic balance by smoo hing he New onian sou ce
u≡ ∇2
Φo e an
en i onmen -dependen eco e y leng h
ℓ
.Ope a ionally, his “ es o a i i y” is implemen ed by a
single-scale, linea , ansla ion- and o a ion-in a ian smoo hing o
u
ha : (i) educes o New onian
g a i y as
ℓ→
0; and (ii) emains bounded and causal a all scales. Unde hese minimal clauses,
he esponse is uniquely
ue = (1 −ℓ2∇2)−1u, ⇐⇒ (1 −ℓ2∇2)∇2Φ=4πG ρ,
equi alen ly a eal-space Yukawa- ype ke nel
Gℓ
(
s
)
∝
(1
−e−s/ℓ
)
/s
o a Fou ie mul iplie
µ
(
k
) =
1/(1+ℓ2k2).
1.2 P2 — Cu a u e–Reco e y Closu e.
The eco e y leng h
ℓ
is se by he local geome ic en i onmen (“cu a u e/compac ness”): i is
small in high-cu a u e o compac egions and la ge in low-cu a u e o di use egions. The
main ex lea es he unc ional o m
ℓ
=
ℓ
(x)unspeci ied; illus a i e closu es (e.g.
ℓ∝K−1/4
o
ℓ∝u−1/2
), as well as piecewise-cons an o
ℓ∝R
ansä ze o applica ions, a e examined in he
Appendices.
1.3 Immedia e co olla ies (no addi ional pos ula es).
•
New onian limi / Sola -sys em sa e y.
ℓ→
0
⇒
Poisson’s equa ion;
ℓ/ ≪
1 eco e s
Keple ian dynamics.
•
Galaxy disks (used la e ). Inne disk
ℓ≈cons ⇒
peak +quasi- la band; ou e disk
ℓ∝R⇒ 2→Q; a ou e egion whe e ℓsa u a es ⇒mild decline.
•
Me ging-clus e lensing. Line-o -sigh p ojec ion gi es (1
−ℓ2
θ∇2
θ
)
κ
= Σ
/
Σ
c i
; smalle
ℓ
(galaxies) is p ese ed while la ge
ℓ
(di use gas) is supp essed
⇒
peak o se +shea b idge.
3
2
Uni ied Ro a ion Cu es om Inne o Ou e Galac ic Disks:
The Mainline o Geome ic Res o a i i y
O e iew and claim. This chap e uses he eco e y leng h
ℓ
as he o ganizing h ead o p esen
asingle mechanism o o a ion cu es om he inne disk o he ou e mos acuum. The co e
conclusion is: in high-cu a u e egions (inne disk)
⇒ℓ
is nea ly cons an , he con ol ed ke nel
is analy ic a he cen e , p oducing a solid-body ise
∝R
; ac oss he mid–ou e ansi ion
⇒
ℓ
(
R
)inc eases smoo hly wi h adius and ma ches cu a u e sel -consis en ly, so he cu e changes
con inuously wi h no discon inui y; in he nea - acuum ou e egion
⇒
aking
ℓ≃λR
p oduces
a conse ed quan i y
Q
and a small de ia ion
2
=
Q
+
CRn
wi h
n <
0; and a he ou , i
ℓ
sa u a es o a cons an , he speed ansi ions o a Yukawa/Keple ail (
∝R−1/2
). The en i e
“ ise–pla eau–mild il –Keple ail” mo phology equi es no da k ma e and ollows solely om he
cu a u e dependence o ℓ oge he wi h he conse a ion s uc u e.
2.1
Inne disk: cen al analy ici y o he con ol ed ke nel and de e mina ion o
he peak
2.1.1 (i) The poin -sou ce nea ield is only a e e ence.
Fo a poin mass Mwi h cons an ℓ, he midplane esponse
2( ) = GM
1−(1 + x)e−x, x = /ℓ (C-1-1)
a
x≪
1gi es a
∝ 1/2
ise, ia (C-1-2)–(C-1-4). This is me ely a oy nea - ield o a poin sou ce
and does no desc ibe a eal hin disk.
2.1.2 (ii) A eal hin (exponen ial) disk necessa ily yields ∝Ra small R.
Con ol ing
K
(
s
) = (1
−e−s/ℓ0
)
/s
(C-1-5) wi h Σ(
R′
)(C-1-7), pe o ming he ec o Taylo expansion
o small R, and angle-a e aging he i s o de , he quad a ic coe icien is 1
2κ, leading o
g(R) = −Φ′(R) = κ R +O(R3), (R)∝R. (C-1-11)
Using he iden i y (C-1-12) and in eg a ion by pa s gi es
κ
as in (C-1-13). Fo an exponen ial disk
Σ = Σ
0e−R/Rd
we ha e
d
Σ
/dR <
0and
dK/dR <
0, hence
κ >
0
⇒g∝R, ∝R
(solid-body ise).
2.1.3 (iii) The peak ollows om he s a iona i y o he Hankel o m.
Wi h he k-space low-pass F(k) = 1/(1 + ℓ2
0k2)and
2(R) = 2πG RZ∞
0
˜
Σ(k)J1(kR)k dk
1+ℓ2
0k2[5],(C-1-16)
oge he wi h
˜
Σ
(
k
)o he exponen ial disk (C-1-20), one ob ains he ull- adius exp ession (C-1-22).
Di e en ia ing wi h espec o
R
and imposing
d 2/dR
= 0 gi es he peak condi ion (C-1-23). In
he limi β=ℓ0/Rd→0we eco e he F eeman peak
Rpeak ≈2.2Rd.
4
2.2 Sola -sys em limi : why s ic ly New onian
2.2.1 (i) k-space iew: modi ica ions a e supp essed a sho scales.
In he weak, s a ic limi he scala equa ion (C-2-1) shows ha he model equals he New onian
ke nel mul iplied by
F
(
k
) = 1
/
(1 +
ℓ2k2
). When he p oblem’s cha ac e is ic scale
L
sa is ies
kℓ ≪
1
(o ℓ≪L), he New onian limi (C-2-2) ollows.
2.2.2 (ii) Real-space iew: he co ec ion ac o is exponen ially small.
Fo a poin mass,
g( ) = GM
21−(1 + x)e−x=gN( ) [1 −δ(x)], δ(x) = (1 + x)e−x, x = /ℓ, (C-2-3)
and a
x≫
1we ha e
g
(
)
≃gN
(
)(C-2-4). I one equi es
|δ|≲
10
−9∼−10
, i su ices o ha e
x≳
25, i.e.
ℓ≲ /
25. On AU scales he high cu a u e d i es
ℓ≪AU
, making he co ec ion
unobse able.
2.2.3 (iii) Conclusion.
The sola sys em adhe es o New on/GR because
ℓ
is a smalle han o bi al scales in a high-
cu a u e en i onmen , whe eas only in low-cu a u e ou e disks do we see he pla eau and small
slopes (quan i ied by (C-2-5)).
2.3 T ansi ion zone: slowly a ying ℓ(R)and con inui y condi ions
2.3.1 (i) Slow a iable and conse a i e o m.
De ine
ε(R)≡dln ℓ/d ln R(≪1,≲O(0.1)).(C-3-1)
The go e ning equa ion can be ecas on he midplane in o he conse a i e o m
∇·∇Φ−∇(ℓ2u)= 4πGρ, u ≡ ∇2Φ.(C-3-2)
2.3.2 (ii) Ring-in eg a ion con inui y condi ions.
In eg a ing o e a hin coaxial ing o adius R gi es
[RΦR]+
−= 0,[R(ℓ2u)R]+
−= 0,(C-3-3)
ensu ing seamless ma ching o he ou e -zone in a ian s.
5
2.4 Ou e pla eau (ρb≃0,ℓ=λR): conse ed quan i y and small de ia ions
2.4.1 (i) Sou ce- ee equa ion and lux conse a ion.
In he ou e disk he ba yonic sou ce is exhaus ed (
ρb≃
0) and
ℓ
is se by he en i onmen scale
( ake ℓ=λR). The equa ion becomes
∇·∇Φ−∇(ℓ2u)= 0,(C-4-1)
so ha he adial in a ian
Q≡RΦR−(ℓ2u)R=cons , 2(R) = RΦR=Q+R(ℓ2u)R(C-4-3)
holds.
2.4.2 (ii) La ge- adius ansa z and admissible exponen .
Assuming
ℓ( ) = λ , u( ) = A −k, λ > 0,(C-4-4)
and subs i u ing in o he ou e equa ion yields
λ2(k−2)(k−3) = 1, k±(λ) = 5±p1+4/λ2
2, k+(λ)>3,(C-4-6, C-4-7)
so ha
2(R) = Q+C Rn, n = 2 −k+<0.(C-4-13)
The sign (and ampli ude) se by he ma ching cons an
A
gi es wo mo phologies:
A <
0
⇒
posi i e
co ec ion and gen le decline o
Q
;
A >
0
⇒
nega i e co ec ion and gen le ise o
Q
. These
co espond o he obse ed “ la /mildly declining/mildly ising” ypes.
2.4.3 (iii) In e ing Qand λ om obse a ions.
Le
W(R)≡ 2(R) = RΦR.(1)
Then using (C-4-11) one ob ains
Q=W−2λ2R W′−λ2R2W′′ ≈cons , la =pQ. (C-4-12)
This p o ides a di ec ou e -disk ecipe: om obse ed
(
R
), cons uc
Q
using smoo hed
W′
and
W′′, and es ima e λ(hence k+and n).
2.5 Fa he acuum: sa u a ion o ℓand he Yukawa/Keple ail
2.5.1 (i) Sa u a ion case and local slope.
I a la ge adii
ℓ
s ops g owing and sa u a es o
ℓsa
(beyond
Rsa
), he ex e io solu ion e u ns
o he cons an -ℓYukawa o m. De ine
y≡R/ℓsa , B(y)≡1−(1+y)e−y,(2)
6
and by con inui y a Rsa wi h 2(Rsa )=Qwe ix an e ec i e enclosed mass Me , gi ing
(R) = sGMe
RB(y).(C-5-2)
The local loga i hmic slope is
sV(R) = 1
2 y2e−y
B(y)−1!.(C-5-4)
2.5.2 (ii) Asymp o ic Keple limi and a 1% c i e ion.
As y→ ∞,
sV(R) = −1
2+1
2y2e−y+O(e−2y)→ −1
2.(C-5-7)
Requi ing
|sV
+ 1
/
2
|< δ
gi es
1
2y2e−y< δ
. Fo
δ
= 0
.
01 we ob ain
R≳
8
.
1
ℓsa
, whe e he ac ional
speed e o is ∼1
2(1+y)e−y≲1.5×10−3—p ac ically a Keple egime.
2.6 Uni ied pic u e and es able p edic ions
2.6.1 (i) Uni ied pic u e.
High cu a u e
⇒
sho
ℓ
, cen al analy ici y
⇒ ∝R
;dec easing cu a u e
⇒ℓ∝R
, gi ing
2
=
Q
+
CRn
wi h a powe -law de ia ion ha decays wi h adius; nea acuum
⇒ℓ
sa u a es and
he speed app oaches a Yukawa/Keple ail. The chain is closed by (C-1-1)–(C-5-7).
2.6.2 (ii) Tes able p edic ions and da a wo k low.
•
Peak–scale alignmen : In he
β→
0limi ,
Rpeak ≈
2
.
2
Rd
cons ains he uppe bound o
ℓ0/Rd;
•
Pla eau alue
Q
:Cons uc
Q
om ou e -disk
(
R
) ia smoo hed de i a i es using (C-4-12),
and es i s cons ancy wi h R;
•
Mild- il index
n
:Fi he ou e -disk slope
s
(
R
) =
dln /d ln R≃
(
n/
2)
ε
(
R
) o in e
λ
and
k+;
•
Fa - egion c i e ion: Use
|sV
+ 1
/
2
|
oge he wi h
1
2
(1 +
y
)
e−y
om (C-5-4)–(C-5-7) o
ma k he onse o he Keple zone.
Summa y. This chap e s i ches oge he “cen al analy ici y o he con ol ed ke nel — con inui y
ac oss he ansi ion — ou e in a ian wi h a powe -law de ia ion — sa u a ed Yukawa/Keple ail”
in o a single mechanism ha needs no da k ma e . The key quan i ies (
ℓ0, λ, Q, n
)a e di ec ly
ex ac able om obse ed cu es and ma ch one- o-one o he analy ic s uc u e (C-1-1)–(C-5-7).
7
3
Me ging-Clus e Lensing: Bulle -Clus e Peak O se and Shea
B idge
3.1 Se up & p ojec ion.
S a ing om he weak, s a ic ield equa ion o geome ic es o a i i y,
(1 −ℓ2∇2)∇2Φ(x) = 4πG ρ(x),(3)
line-o -sigh (LOS) p ojec ion gi es he modi ied 2D lensing ela ion
(1 −ℓ2
θ∇2
θ)κ(θ) = Σ(θ)
Σc i
, ℓθ≡ℓ/DL,Σc i ≡c2
4πG
DS
DLDLS
.(4)
Equi alen ly, lensing is a Yukawa- ype smoo hing o Σ/Σc i :
κ(θ) = Zd2θ′
2π ℓ2
θ
K0|θ−θ′|
ℓθΣ(θ′)
Σc i
,(5)
o , in Fou ie space, a s ic low-pass il e ,
˜κ(k) = 1
1+ℓ2
θk2
˜
Σ(k)
Σc i
.(6)
3.2 Uni ied pic u e o he o se .
Du ing a me ge , galaxy-domina ed componen s a e dense (highe cu a u e) and hus ha e smalle
ℓ
; am-p essu e–s ipped X- ay gas is di use and has la ge
ℓ
. The low-pass ope a o he e o e
p ese es sha p galaxy peaks (small
ℓ
) and supp esses di use gas peaks (la ge
ℓ
), na u ally pulling
he lensing peak owa d he galaxies and away om he gas.
3.3 Two-componen , piecewise-cons an model.
Spli Σ=Σgal + Σgas and assume ℓgal ≪ℓgas in hei dominan egions:
κ(θ) = Σgal/Σc i ∗Hℓgal +Σgas/Σc i ∗Hℓgas , Hℓ(∆θ)∝K0(|∆θ|/ℓ).(7)
Along he me ge axis (1D cu ), a su icien condi ion o he lensing peak o shi owa d he galaxies
is
Ag
K1(a/ℓg)
ℓ3
g
> Ab
K1(a/ℓb)
ℓ3
b
,(ℓb≫ℓg),(8)
whe e
a
is he angula sepa a ion o he cen oids. The b idge ampli ude a he midpoin scales as
he sum o wo Yukawa ails,
κb idge ≈Ag
2πℓ2
g
K0
a
ℓg+Ab
2πℓ2
b
K0
a
ℓb,(9)
decaying app oxima ely as e−a/ℓ/√a.
8
3.4 Implemen a ion & alsi iable ends.
•
Real-space: con ol e Σ
/
Σ
c i
wi h
Hℓθ
; o spa ially a ying
ℓ
(
θ
), i e a e a closu e (guess
ℓ
,
compu e κ, upda e ℓ[κ]).
•Fou ie -space: apply he mul iplie (1 + ℓ2
θk2)−1pa chwise, hen in e se- ans o m.
•
T ends: la ge gas di useness/s onge shocks
⇒
la ge
ℓgas
, la ge peak o se , and a
b oade /s onge b idge; la e- ime elaxa ion (smalle ℓ) diminishes bo h.
4 Conclusions and Ou look
4.1 Conclusions.
We sys ema ize Geome ic Res o a i i y (GRS) in he weak, s a ic egime ia
(1 −ℓ2∇2)∇2Φ = 4πG ρ, (10)
whe e he en i onmen -dependen eco e y leng h
ℓ
implemen s a con olled low-pass il e ing o he
New onian sou ce
u≡ ∇2
Φ. This single mechanism yields a uni ied accoun o wo os ensibly sepa a e
phenomena: (i) Uni ied disk o a ion cu es. In inne disks whe e
ℓ
a ies slowly, aligning he
ke nel’s in insic ex emal scale wi h he exponen ial-disk peak gi es an “up u n–peak–quasi- la ”
pa e n; in ou e disks,
ℓ∝R
p oduces a conse ed lux and d i es
2
(
R
)
→Q
; in a ou ski s
a sa u a ed
ℓ
eco e s a ini e-leng h Yukawa ail, yielding a mild nega i e slope and asymp o ic
Keple ian allo . (ii) Me ging-clus e lensing (Bulle Clus e ). LOS p ojec ion leads o he
modi ied 2D lensing equa ion
(1 −ℓ2
θ∇2
θ)κ(θ) = Σ(θ)
Σc i
,(11)
equi alen o an
ℓθ
-scale Yukawa smoo hing o Σ
/
Σ
c i
. Since galaxies a e dense (smalle
ℓ
) while
am-p essu e–s ipped gas is di use (la ge
ℓ
), he lensing map p ese es galaxy peaks and supp esses
gas peaks, na u ally p oducing he obse ed mass–gas peak o se [
7
,
8
] and an equipo en ial/shea
b idge[
8
] along he me ge axis. Analy ical o se c i e ia and b idge scalings ollow om he same
ke nel. The amewo k eco e s he New onian limi o
ℓ/ ≪
1and emains compa ible wi h
i s -o de cosmological cons ain s unde a con olled
k
-dependence. O e all, a single en i onmen al
scale
ℓ
and one Yukawa- ype ke nel o ganize disk kinema ics and me ging-clus e lensing in o a
cohe en pic u e[9].
4.2 Ou look.
We ou line a da a– heo y p og am o ende he abo e mechanism alsi iable:
•
Disk-le el i s and diagnos ics: Hie a chical i s o
ℓ
(
R
)closu es; di ec cons uc ion o he
ou e -disk in a ian
Q
om de i a i es o
V
(
R
)and popula ion s a is ics o quasi- la adii
and ou e slopes[10].
•
Me ging-clus e mo phology: Reg ess he o se –di useness (o Mach-numbe ) co ela ion
and b idge wid h s.
ℓ
ac oss a sample; alida e analy ic scalings using join X- ay/lensing
econs uc ions.
9
B.2 Ve i ica ion: (1 −ℓ2∇2)∇2ˆ
G( )=−δ(3)
W i e
ˆ
G( ) = −1
4π +e−β
4π , β =1
ℓ>0, =| |.(B-2-1)
Use he dis ibu ional iden i ies
∇21
4π =−δ(3)( ),(B-2-2)
∇2 e−β
4π !=β2e−β
4π −δ(3)( ).(B-2-3)
Thus
∇2ˆ
G=−−δ(3)+β2e−β
4π −δ(3)=β2e−β
4π −2δ(3).(B-2-4)
I is con enien o in oduce
Y
(
) :=
e−β
4π
. F om he Fou ie de i a ion we also ha e he compac
iden i y
∇2ˆ
G=−β2Y. (B-2-5)
Applying he ope a o (1 −ℓ2∇2)and using ℓ2β2= 1:
(1 −ℓ2∇2)∇2ˆ
G=−β2Y+ℓ2β2∇2Y=−β2Y+∇2Y=−δ(3)( ),(B-2-6)
since ∇2Y=β2Y−δ(3). This con i ms Lˆ
G=−δ(3).
B.3 Angula a e aging o sphe ical symme y and ex e io solu ion
S a ing om he con olu ion
Φ(x) = −GZd3x′ρ(x′)1−e−|x−x′|/ℓ
|x−x′|,(B-3-1)
le
=
|
|
,
′
=
|
′|
, and assume
ρ
=
ρ
(
′
). Decompose Φ = Φ
N
+ Φ
Y
in o New on and Yukawa
pa s:
ΦN( ) = −GZd3 ′ρ( ′)1
| − ′|,(B-3-2)
ΦY( ) = + GZd3 ′ρ( ′)e−| − ′|/ℓ
| − ′|.(B-3-3)
Choose he pola axis along . Wi h µ= cos θ′,
| − ′|=q 2+ ′2−2 ′µ. (B-3-4)
and he angula a e age o he Yukawa ke nel is
1
4πZdΩ′e−| − ′|/ℓ
| − ′|=1
2Z1
−1
dµ e−√ 2+ ′2−2 ′µ/ℓ
p 2+ ′2−2 ′µ
=ℓ
2 ′e−| − ′|/ℓ −e−( + ′)/ℓ.(B-3-5)
16
Simila ly, o he New on ke nel,
1
4πZdΩ′1
| − ′|=1
max( , ′).(B-3-6)
Subs i u ing and using d3 ′= 4π ′2d ′, one ob ains he uni ied 1D ke nel o m
Φ( ) = −GZ∞
0
d ′4π ′2ρ( ′)1
max( , ′)−ℓ
2 ′e−| − ′|/ℓ −e−( + ′)/ℓ.(B-3-7)
De ine he angle-a e aged New on–Yukawa ke nel
K( , ′;ℓ) := 1
max( , ′)−ℓ
2 ′e−| − ′|/ℓ −e−( + ′)/ℓ.(B-3-8)
The i s e m (New on) sa is ies he shell heo em; he second (Yukawa) does no .
B.4 Piecewise exp essions and ex e io solu ion
B.4.1 New on pa .
Spli ing ′a ,
ΦN( ) = −G1
Z
0
4π ′2ρ( ′)d ′+Z∞
4π ′ρ( ′)d ′.(B-4-1)
Fo > R when he sou ce is ully inside adius R,ΦN( ) = −GM/ .
B.4.2 Yukawa pa .
Using he a e aged Yukawa ke nel,
ΦY( ) = GZ
0
d ′4π ′2ρ( ′)ℓ
2 ′e−( − ′)/ℓ −e−( + ′)/ℓ
+GZ∞
d ′4π ′2ρ( ′)ℓ
2 ′e−( ′− )/ℓ −e−( + ′)/ℓ.(B-4-2)
Fac o izing yields
ΦY( ) = G ℓ
e− /ℓ Z
0
d ′4π ′ρ( ′) sinh ′
ℓ+ sinh
ℓZ∞
d ′4πρ( ′)e− ′/ℓ.(B-4-3)
Fo he ex e io egion > R wi h ρ( ′)=0 o ′> R, he second in eg al anishes and
ΦY( ) = G ℓ
e− /ℓ ZR
0
d ′4π ′ρ( ′) sinh ′
ℓ,( > R).(B-4-4)
The ull ex e io solu ion is he e o e
Φ( ) = −GM
+G ℓ
e− /ℓ ZR
0
d ′4π ′ρ( ′) sinh ′
ℓ,( > R).(B-4-5)
17
Rema k (no a poin -sou ce equi alence). Because he Yukawa ke nel iola es he shell heo em, he
ex e io po en ial depends in gene al on he in e io mass dis ibu ion. Only in he compac -sou ce
limi R≪ℓ, whe e sinh( ′/ℓ)≃ ′/ℓ, one inds
ZR
0
4π ′ρ( ′) sinh ′
ℓd ′≃1
ℓZR
0
4π ′2ρ( ′)d ′=M
ℓ,(B-4-6)
hence
ΦY( )≃GM
e− /ℓ,(B-4-7)
Φ( )≃ −GM
h1−e− /ℓi,(R≪ℓ, > R).(B-4-8)
B.5 Ro a ion cu es
B.5.1 Ex e io egion > R (compac -sou ce app oxima ion)
Wi h R≪ℓ, he poin -sou ce app oxima ion gi es
Φ( )≃ −GM
h1−e− /ℓi,(B-5-1)
g( ) = −Φ′( ) = GM
21−1 +
ℓe− /ℓ,(B-5-2)
2( ) = g( ) = GM
1−1 +
ℓe− /ℓ,(B-5-3)
consis en wi h he main- ex beha iou (New onian a la ge
, nea - la wi h a mild nega i e slope
a in e media e adii).
B.5.2 Gene al sphe ical ρ( )(no > R assump ion)
F om
Φ( ) = −GZ∞
0
d ′4π ′2ρ( ′)1
max( , ′)−ℓ
2 ′e−| − ′|/ℓ −e−( + ′)/ℓ,(B-5-4)
di e en ia e wi h espec o and use he angle a e ages o ob ain he 1D ke nel o m
g( ) = −Φ′( ) = GM(< )
2
| {z }
New on
−G ℓ
2e− /ℓZ
0
d ′4π ′ρ( ′)ℓcosh ′
ℓ− ′sinh ′
ℓ−cosh
ℓZ∞
d ′4π ρ( ′)e− ′/ℓ
| {z }
Yukawa
,
(B-5-5)
whe e
M
(
<
) =
R
0
4
π ′2ρ
(
′
)
d ′
. Then
2
(
) =
g
(
) ollows di ec ly. Nume ically, his educes o
a single 1D quad a u e.
18
8
Appendix C: Uni ied De i a ion o Galac ic Ro a ion Cu es
om Inne o Ou e Disk in GRS
C.1 Inne Disk: Poin -Sou ce Limi and Cons an ℓApp oxima ion
C.1.1 Nea - ield expansion o a poin mass wi h cons an ℓ
In he high-cu a u e cen al egion o a galac ic disk, he geome ic eco e y leng h
ℓ
can be
ega ded as cons an . Fo a poin mass M, he geome ic- es o a i e po en ial gi es
2( ) = GM
1−(1 + x)e−x, x ≡
ℓ.(C-1-1)
Fo x≪1,
e−x= 1 −x+x2
2−x3
6+O(x4),(C-1-2)
1−(1 + x)e−x=x2
2−x3
3+O(x4).(C-1-3)
Subs i u ing back yields
2( ) = GM
2ℓ2 −GM
3ℓ3 2+···, ( )≃sGM
2ℓ2 1/21−
3ℓ+···.(C-1-4)
This nea - ield esul indeed p oduces a
∝ 1/2
ise, bu i co esponds only o he esponse o a
poin sou ce. Real galaxies a e dis ibu ed hin disks; o desc ibe a eal disk one mus con ol e he
abo e ke nel wi h he su ace densi y Σ(R′)in an axisymme ic manne .
C.1.2 Con olu ion o an exponen ial hin disk and cen al expansion
Le he obse a ion poin be Rwi h
R
=
|
R
|
, and a sou ce poin on he disk be R
1
wi h
R1
=
|
R
1|
.
W i e he poin -sou ce midplane ke nel as he angle-a e aged o m
K(s) = 1−e−s/ℓ0
s, s =|R−R1|,(ℓ≃ℓ0).(C-1-5)
I s small-sexpansion is ini e and analy ic:
K(s) = 1
ℓ0−s
2ℓ2
0
+s2
6ℓ3
0
+O(s3).(C-1-6)
The po en ial is he con olu ion
Φ(R) = −GZd2R1Σ(R1)K(|R−R1|).(C-1-7)
Fo small
R
, he ec o Taylo expansion ( he i s -o de e m angle-a e ages o ze o by axisymme y)
gi es o second o de
K(|R1−R|)=K(R1) + RaRb
2∂a∂bK(R1)+O(R4).(C-1-8)
19
Fo any adial unc ion ( ),
RaRb∂a∂b ( )=R2
2 ′′( ) + ′( )
=R1
.(C-1-9)
Subs i u ing back yields
Φ(R) = Φ(0) −GR2
4Zd2R1Σ(R1) d2K
dR2
1
+1
R1
dK
dR1!+O(R4).(C-1-10)
Deno ing he quad a ic coe icien by 1
2κ, we ob ain
g(R) = −Φ′(R) = κR +O(R3), (R)∝R. (C-1-11)
Using he iden i y
d2K
dR2
1
+1
R1
dK
dR1
=1
R1
d
dR1R1
dK
dR1,(C-1-12)
an in eg a ion by pa s ( anishing bounda y e m) gi es
κ=πGZ∞
0
dΣ
dR1
R1
dK
dR1
dR1.(C-1-13)
Fo an exponen ial disk Σ(
R1
)=Σ
0e−R1/Rd
we ha e
d
Σ
/dR1<
0and
dK/dR1<
0, hence
κ > 0⇒g∝R, ∝R(solid-body ise).
C.1.3 Fou ie (Hankel) o m and he inne peak
In he inne disk ake ℓ≃ℓ0(cons an ). The poin -ke nel in k-space ac s as a low-pass il e
F(k) = 1
1+ℓ2
0k2.
The ze o h-o de Hankel ans o m is de ined by
˜
Σ(k) = Z∞
0
Σ(R1)J0(kR1) 2πR1dR1.(C-1-14)
The midplane angen ial speed o e all adii is
2(R) = 2πGRZ∞
0
˜
Σ(k)J1(kR)k dk
1+ℓ2
0k2.[5](C-1-15)
Fo small R, wi h J1(z)=z/2+O(z3), one ge s
2(R) = πGR2Z∞
0
˜
Σ(k)k2dk
1+ℓ2
0k2+O(R4), κ =πGZ∞
0
˜
Σ(k)k2dk
1+ℓ2
0k2.(C-1-16)
Key in eg al (s a ed and p o ed):
Z∞
0
e−a J0(b ) d =a
(a2+b2)3/2, a > 0.(C-1-17)
20
Se ing a= 1/Rd, b =kgi es
Z∞
0
e−R1/RdJ0(kR1)R1dR1=R2
d
(1 + (kRd)2)3/2.(C-1-18)
The e o e
˜
Σ(k) = 2πΣ0R2
d
(1 + (kRd)2)3/2.[11](C-1-19)
Subs i u ing in o κ,
κ=2π2GΣ0
RdZ∞
0
q2dq
(1+q2)3/2(1+β2q2), β =ℓ0
Rd
.(C-1-20)
The ull o a ion cu e is
2(R)
2πGΣ0Rd
= 2yZ∞
0
J1(2yq)
(1+q2)3/2(1+β2q2)q dq, y =R
2Rd
.(C-1-21)
Di e en ia ing wi h espec o
R
and se ing
d 2/dR
= 0 (using
J′
1
=
1
2
(
J0−J2
)) gi es he peak
condi ion:
Z∞
0
J1(2yq)
(1+q2)3/2(1+β2q2)q dq +yZ∞
0
J0(2yq)−J2(2yq)
(1+q2)3/2(1+β2q2)q2dq = 0.(C-1-22)
When β=ℓ0/Rd→0(i.e., ℓ0≪Rd), he nume ical solu ion yields Rpeak ≈2.2Rd[4].
C.2 Sola -sys em limi : e i ica ion o New onian eco e y
C.2.1 Weak- ield equa ion and k-space il e ing
In he s a ic weak- ield limi , he scala po en ial obeys
(1 −ℓ2∇2)∇2Φ=4πGρ, ˜
Φ(k) = −4πG
k2(1+ℓ2k2)˜ρ(k).(C-2-1)
When he p oblem’s cha ac e is ic scale L≫ℓ(equi alen ly kℓ ≪1),
1
1+ℓ2k2≃1−ℓ2k2+···,
so he model immedia ely educes o he New onian ke nel ∝1/k2.
C.2.2 Poin -mass esponse and he supp ession ac o
Fo a poin mass M,
g( ) = GM
21−(1 + x)e−x, x =
ℓ=gN( ) [1 −δ( /ℓ)] , δ = (1 + x)e−x.(C-2-2)
When x≫1(i.e. ℓ≪ ), he co ec ion ac o δis exponen ially supp essed, hence
g( )≃gN( ) [1 −O(e− /ℓ)] ≃gN( ).(C-2-3)
To equi e
|δ|<
10
−9∼−10
, one needs (1 +
x
)
e−x≪
10
−9∼−10
, implying
x≳
25, i.e.
ℓ≲ /
25. A
∼1AU, an ℓo o de 106km o smalle ende s he de ia ion p ac ically unobse able.
21
C.2.3 Cu a u e dependence and New onian eco e y
By he second pos ula e, he eco e y leng h
ℓ
dec eases in s onge cu a u e and inc eases in weake
cu a u e. In he co ona and plane a y-o bi scales o he sola sys em,
ℓ
is d i en a below AU,
yielding he New onian/GR weak- ield limi ; in he low-cu a u e ou e disks o galaxies,
ℓ
inc eases
wi h adius (app oxima ely
ℓ∝R
), p oducing he obse ed la /mildly a ying o a ion cu es.
In
k
-space, he model mul iplies he New onian ke nel by
F
(
k
) = 1
/
(1 +
ℓ2k2
). Fo sola -sys em
wa enumbe s
k∼
1
/
, i
ℓ≪
hen
kℓ ≪
1
⇒F
(
k
)
≃
1, so o bi al, geode ic and ada - ime
obse ables ag ee wi h New onian/GR p edic ions o e y high p ecision.
C.2.4 Rema k on he nea - ield oy limi
The
x≪
1expansion leading o
g≃GM/
(2
ℓ2
)and
∝ 1/2
is me ely a poin -sou ce, cons an -
ℓ
ul a-nea - ield oy limi . In he sola sys em o in e es we ha e
≳AU
and ac ually
/ℓ ≫
1,
en e ing he New onian limi . Fo a dis ibu ed hin disk, he con ol ed cen al po en ial is quad a ic,
gi ing g∝ and ∝ , independen o any cons an -gassump ion.
C.2.5 Summa y
The sola sys em s ic ly ollows New onian g a i y because in a high-cu a u e en i onmen
ℓ
is a below o bi al scales, wi h he ac ional co ec ion (1 +
/ℓ
)
e− /ℓ ≪
1. In
k
-space his is
kℓ ≪
1
⇒F
(
k
)
≃
1. Thus he model na u ally eco e s he New onian/GR weak- ield limi in he
sola sys em, while exhibi ing la /weakly a ying beha io only in low-cu a u e galac ic ou e
disks.
C.3 F om Inne Disk o T ansi ion Zone: Slowly Va ying
ℓ
and Geome ic
Res o a i i y
C.3.1 Slowly a ying ℓand conse a i e o m
As cu a u e weakens ou wa d, ℓg ows slowly wi h adius. De ine he ela i e a ia ion a e
ε(R)≡dln ℓ
dln R, ε ≪1 ( ypically O(0.1)).(C-3-1)
The go e ning equa ion
(1 −ℓ2∇2)∇2Φ=4πGρ
can be ecas in o a lux-conse a i e di e gence o m:
∇·∇Φ−∇(ℓ2u)= 4πGρ, u ≡ ∇2Φ.(C-3-2)
When
ρ
is al eady small while
ε
emains mild, he e ec o
ℓ′
is an
O
(
ε
)co ec ion; he o a ion
cu e a ies smoo hly ac oss he ansi ion and connec s o he ou e zone wi hou any discon inui y.
22
C.3.2 Con inui y condi ions
In eg a ing o e a hin coaxial ing a adius
R
and using he di e gence heo em yields he
con inui y condi ions
[RΦR]+
−= 0,[R(ℓ2u)R]+
−= 0,(C-3-3)
ensu ing smoo h ma ching o he in a ian s o he ou e -zone equa ion.
C.4 Ou e Pla eau (wi h
ρb≃
0,
ℓ
=
λR
): Conse ed Quan i y and Small De ia ions
C.4.1 Equa ion and lux conse a ion
Once in he ou e disk, he ba yonic sou ce is essen ially exhaus ed (
ρb≃
0), and
ℓ
is se by he
en i onmen scale ( ake ℓ=λR). The equa ion becomes
(1 −ℓ2∇2)∇2Φ=0 ⇔ ∇·(∇Φ−∇(ℓ2u)) = 0.(C-4-1)
De ining he composi e lux
F≡ ∇Φ−∇(ℓ2u),
i s di e gence anishes. Fo a coaxial cylinde o adius
R
and heigh
H
, s a iona i y and symme y
lea e only he la e al lux, hence
Q≡RΦR−(ℓ2u)R=cons , 2(R) = RΦR=Q+R(ℓ2u)R.(C-4-2)
C.4.2 La ge- adius ansa z and cons ain
A la ge adii, adop
ℓ(R) = λR, u(R)=A R−k, λ > 0.(C-4-3)
Unde sphe ical symme y o any adial unc ion ( ),
∇ = ′( ) ˆ , ∇2 =1
2
d
d 2 ′( ).
Thus
u′( ) = −kA −k−1,∇2u=k(k−1)A −k−2, ℓ2( ) = λ2 2,∇2ℓ2= 6λ2.(C-4-4)
Subs i u ing in o
−u+ℓ2∇2u+ 2∇ℓ2·∇u+u∇2ℓ2= 0
and ac o ing ou A −kgi es
λ2(k−2)(k−3) = 1,(C-4-5)
wi h solu ions
k±(λ) = 5±p1+4/λ2
2, k+>3.(C-4-6)
23
C.4.3 Ou e speed and powe -law co ec ion
Since ℓ2u=λ2A 2−k,
(ℓ2u) =λ2A(2 −k) 1−k, R(ℓ2u)R=λ2A(2 −k) 2−k.(C-4-7)
When k=k+>3, he exponen 2−k+<0, so o any sign o A,
2−k+−−−→
→∞ 0,
implying
2(R)−−−→
→∞ Q. (C-4-8)
The sign o
A
de e mines he mo phology:
A <
0gi es a posi i e co ec ion and a gen le decline
owa d
Q
(“sligh ly declining”);
A >
0gi es a nega i e co ec ion and a gen le ise owa d
Q
(“sligh ly ising”). These co espond o he h ee obse ed ou e mo phologies: la , mildly declining,
and mildly ising.
C.4.4 De e mining Qand λ om obse a ions
De ine
2(R) = RΦR≡W(R), Q =W−Rd
dR(ℓ2u)=W−R2ℓℓ′u+ℓ2uR,(C-4-9)
using d(ℓ2)
dR = 2ℓℓ′. In he nea - acuum ou e zone wi h nea ly sphe ical equipo en ials,
u=∇2Φ = 1
R2
d
dR R2ΦR,ΦR=W/R ⇒u=W+RW′
R2, uR=2W′+RW′′
R2−2(W+RW′)
R3.
(C-4-10)
I ℓ=λR, hen ℓ′=λ,2ℓℓ′/R = 2λ2, and ℓ2=λ2R2. Subs i u ing in o (C-4-9),
Q=W−2λ2RW′−λ2R2W′′ ≈cons , la =pQ. (C-4-11)
Hence he ou e speed can be w i en as
2(R) = Q+C Rn, n = 2 −k+<0,(C-4-12)
and wi h
ε(R)≡C
QRn(|ε|≪1), (R) = pQ√1+ε≃pQ1 + ε
2,(C-4-13)
he speed de ia ion is
∆ (R) = −pQ≃√Q
2ε(R) = C
2√QRn.(C-4-14)
The dimensionless local slope
s(R)≡dln
dln R=1
2
n ε
1+ε≃n
2ε(R).(C-4-15)
Examples:
λ
= 1
⇒k+≈
3
.
618
, n ≈ −
1
.
618 (decay
∼R−1.62
);
λ
= 0
.
5
⇒k+≈
4
.
56
, n ≈ −
2
.
56
( as e decay ∼R−2.56).
24
C.5 Fa Vacuum: Sa u a ion o
ℓ
and a Mild Down u n (Yukawa/Keple Ap-
p oach)
C.5.1 Veloci y o m upon sa u a ion
When he adius g ows u he in o he a ou ski s, he eco e y leng h
ℓ
ceases o g ow and
sa u a es o a cons an
ℓsa
. The ex e io solu ion hen e u ns o he cons an -
ℓ
Yukawa o m.
De ine
y≡R
ℓsa
, B(y)≡1−(1+y)e−y.(C-5-1)
By con inui y a Rsa wi h 2(Rsa )=Q, an e ec i e enclosed mass Me is ixed, gi ing
(R) = sGMe
RB(y).(C-5-2)
C.5.2 Local loga i hmic slope: closed o m
De ine he local loga i hmic slope:
sV(R)≡dln
dln R.(C-5-3)
F om ln =1
2(ln GMe −ln R+ ln B(y)), we ge
sV(R) = 1
2−1 + dln B(y)
dln R.
By he chain ule (wi h ℓsa cons an ),
dln B(y)
dln R=B′(y)
B(y)
dy
dln R=B′(y)
B(y)y,
and since
B′(y) = d
dy[1 −(1 + y)e−y]=y e−y,
we ob ain dln B(y)
dln R=y2e−y
B(y).(C-5-3)
The e o e
sV(R) = 1
2 y2e−y
B(y)−1!.(C-5-4)
Example: a
y
= 2,
e−2≈
0
.
135335,
B
(2) = 1
−
3
e−2≈
0
.
593994, so
y2e−y
B(y)≈
0
.
9114 and
sV≈ −
0
.
044
(a mild nega i e slope jus a e sa u a ion).
C.5.3 Fa - acuum asymp o ics
As y→ ∞,
B(y)=1−(1+y)e−y= 1 −ε, ε := (1 + y)e−y→0,
25
