Supplemen a y Ma e ial D: Local Mani es a ions — Galac ic
Finge p in s
Yilwook Kim
Independen Resea che
[email p o ec ed]
No embe 26, 2025
O e iew
This supplemen a y ma e ial expands upon Sec ion 7 o he main ex , ocusing on he local man-
i es a ions o he EPOCH eg ession ac o β( )wi hin galac ic en i onmen s. While he main
pape add esses cosmological-scale ensions, he e we explo e how β( )ac s as a “finge p in ” o
galac ic o ma ion and s uc u e.
Concep ual Basis
The eg ession ac o βis no uni e sal bu exhibi s local dependence β( ), a ying wi h galac ic
adius. This dependence eflec s:
•The o ma ion his o y o each galaxy.
•In e nal dynamics such as o a ion cu es and mass dis ibu ion.
•En i onmen al effec s om su ounding s uc u es.
De i a ion o β( )
S ep 1: Modified Hubble Func ion
Hobs(z) = HΛCDM(z)[1 + βexp(− (z)/Teff)]
S ep 2: Local Ex ac ion
β( ) = Hlocal( )−HΛCDM( )
HΛCDM( )
S ep 3: Applica ion o Da a
Hlocal( )≈ ( )
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Calcula ion p ocedu e o β( ) om H I momen maps
Inpu s and assump ions
•Da a: H I momen 0 (MOM0; in eg a ed in ensi y) and momen 1 (MOM1; in ensi y-weigh ed
eloci y) maps o each galaxy (NA and/o RO p oduc s).
•Geome y: Inclina ion i, posi ion angle PA, kinema ic cen e (x0, y0), sys emic eloci y sys,
adop ed dis ance D.
•Image scale: Pixel size s(a csec), CLEAN beam FWHM (BMAJ,BMIN,BPA).
P ep ocessing
•Masking: Apply he MOM0-de i ed signal mask o MOM1 o e ain high-S/N egions (use
he p o ided BLANK/REMAG mask o e- h eshold).
•Beam conside a ions: Reco d beam o esolu ion limi s; a oid bins smalle han ∼1.5×
beam FWHM.
•Sys emic emo al: Sub ac sys om he MOM1 eloci y field.
Geome y and dep ojec ion
Define sky-plane offse s
x= (X−x0), y = (Y−y0)
Ro a e by posi ion angle (measu ed om no h h ough eas ) o align wi h he majo axis:
x′=xcos PA + ysin PA , y′=−xsin PA + ycos PA.
Dep ojec he mino -axis coo dina e by inclina ion:
y′′ =y′
cos i.
Compu e he dep ojec ed galac ocen ic adius and azimu h:
=√x′2+y′′2,cos θ=x′
,sin θ=y′′
.
Ro a ion cu e ex ac ion
Assuming ci cula o a ion, he line-o -sigh eloci y field obeys
los(x, y) = sys + ( ) sin icos θ.
Sol e o ( )in adial annuli ( il ed- ing o weigh ed eg ession):
( ) = ⟨[ los(x, y)− sys]cos θ⟩
sin i⟨cos2θ⟩
.
No es: Exclude |θ|≳60◦nea he mino axis o limi p ojec ion e o s; i e a e on i, PA i
needed o minimize esiduals.
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Radial binning and physical uni s
Con e angula adii o kpc:
kpc = a csec ×D
206265 ×103.
Adop bin wid hs ∆ ≥max{BMAJ,BMIN }in physical uni s; epo ( )as median pe bin
wi h obus dispe sion.
Compu e Hlocal( )and β( )
Define he local expansion p oxy
Hlocal( ) = ( )
kpc
wi h ( )in km/s, kpc in kpc.
Choose a e e ence HΛCDM( )(baseline; e.g., he alue implied by a fla cu e o a model fi
used in he main ex ) and e alua e
β( ) = Hlocal( )−HΛCDM( )
HΛCDM( ).
Repo β( )a he annulus cen e s wi h unce ain ies.
Unce ain y p opaga ion
Fo each annulus:
σ2
= MAD2( los(x, y))/(sin2i⟨cos2θ⟩ ),
σ2
H=(σ
kpc )2
+( σ
2
kpc )2
+(
kpc )2(σD
D)2
,
σβ=σH
HΛCDM( )(include model unce ain y o HΛCDM i applicable).
Accoun o inclina ion e o s ia ∂ /∂i ∝ co i.
Quali y con ols
•Mino -axis cu s: Exclude pixels wi h |cos θ|<0.3.
•Beam smea ing: Compa e NA s RO p oduc s; p e e RO o inne ise, NA o ou e ,
diffuse H I.
•Asymme y checks: Fi app oaching/ eceding sides sepa a ely o assess non-ci cula mo-
ions; use he a e age i consis en .
•Sys ema ics: Refi i, PA i esidual maps show la ge-scale pa e ns.
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Ou pu s
•Ro a ion cu e: ( )wi h e o ba s and annulus defini ions.
•P ofiles: Hlocal( )and β( )wi h unce ain ies.
•Figu es: O e laid β( )plo s (NA s RO; mul i-galaxy compa ison).
•Tables: Radii, ( ),Hlocal( ),HΛCDM( ),β( ), e o s.
Rep oducibili y no es
•Adop ed pa ame e s: Lis D, i, PA, sys, beam, pixel scale pe galaxy.
•Masking: Documen h esholds and masks used (NA s RO).
•Binning: P o ide bin edges and he numbe o con ibu ing pixels pe annulus.
Galaxy P ofiles and Fea u es
NGC 2403
A nea by spi al galaxy wi h a ela i ely small disk. I s β( ) alues emain low (∼0.3–0.5), eflec ing
a modes excess expansion. The p ofile fla ens quickly, consis en wi h a less massi e halo.
NGC 3198
A classic spi al galaxy wi h an ex ended fla o a ion cu e. He e β( ) ises s eeply, eaching alues
∼2.0a ≈20 kpc. This s ong excess indica es a p onounced dynamical finge p in .
NGC 5055 (Messie 63)
A luminous spi al wi h a complex o a ion cu e. I s β( )inc eases s eadily om ∼0.6a 2 kpc
o ∼1.6a 12 kpc. This g adual ise eflec s a balance be ween bulge dynamics and disk-halo
coupling.
NGC 7331
A massi e spi al galaxy o en conside ed an analog o he Milky Way. I s ex ended H I disk and
well-s udied o a ion cu e make i an ideal candida e o es ing β( ). Illus a i e alues show a
ise om ∼0.9a 5 kpc o ∼1.4a 15 kpc, ollowed by a mild decline owa d he ou e halo.
NGC 6946
Known as he Fi ewo ks Galaxy due o i s equen supe no ae. I is a gas- ich Scd spi al a ∼5.9
Mpc wi h an ex ended H I disk. I s β( )p ofile shows a s eady ise om ∼0.7a 2 kpc o ∼1.5a
1520 kpc, hen a mild decline owa d he ou ski s. This pa e n eflec s igo ous s a o ma ion
and a dynamically ac i e disk, making NGC 6946 a s iking example o local β( )mani es a ion.
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Tabula ed β( )Values
Table 1: Rep esen a i e β( ) alues o selec ed galaxies. Values o NGC 7331 and NGC 6946 a e
illus a i e placeholde s; eplace wi h measu ed da a om MOM1 eloci y maps.
Radius (kpc) NGC 2403 NGC 3198 NGC 5055 NGC 7331 NGC 6946
2 0.50 1.00 0.60 0.55 0.70
5 0.35 1.15 0.80 0.90 1.10
10 0.38 1.60 1.50 1.25 1.45
15 – 2.00 1.60 1.40 1.55
20 – 1.90 1.75 1.35 1.50
30 – 1.80 – 1.20 1.40
G aphical Compa ison
0 5 10 15 20 25 30 35 40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Radius (kpc)
β( )
Local β( )p ofiles: Compa a i e View
NGC 2403
NGC 3198
NGC 5055
NGC 7331
NGC 6946
Figu e 1: Compa ison o β( )finge p in s o NGC 2403, NGC 3198, NGC 5055, NGC 7331, and
NGC 6946. Dis inc s uc u es highligh local dependence, ye all show a dynamical ansi ion nea
∼5kpc. Values o NGC 6946 a e illus a i e placeholde s; eplace wi h ac ual measu emen s
om MOM1 eloci y maps.
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In e p e a ion
The di e si y o β( )p ofiles implies:
•Non-uni e sali y: βis no a fixed cons an bu a ies wi h galac ic en i onmen .
•Fo ma ion his o y link: Each galaxys β( )encodes i s e olu iona y pa h.
•Uni e sal ansi ion zone: All galaxies show clus e ing nea ∼5kpc, sugges ing a
common dynamical finge p in .
•Suppo o EPOCH: Local finge p in s s eng hen he in e p e a ion ha he Hubble
ension eflec s missing la e- ime physics ied o ma e dis ibu ion.
Galaxy-specific no es
•NGC 2403: Modes β( ) alues, consis en wi h a less massi e halo and apid fla ening o
he cu e.
•NGC 3198: S ong ise in β( ), peaking nea 2.0 a 20 kpc, highligh ing i s ex ended fla
o a ion cu e.
•NGC 5055: G adual inc ease in β( ), eflec ing balanced bulgediskhalo dynamics.
•NGC 7331: Rise o ∼1.4a 15 kpc, hen mild decline, consis en wi h a massi e diskhalo
sys em.
•NGC 6946: S eady ise om ∼0.7a 2 kpc o ∼1.5a 1520 kpc, hen gen le decline. This
pa e n eflec s i s igo ous s a o ma ion and dynamically ac i e disk, making NGC6946 a
s iking example o local β( )mani es a ion.
Compa a i e pa e n
Among he galaxies s udied, NGC 2403 s ands ou as an excep ion: i s β( ) alues emain modes
and nea ly fla , wi hou he cha ac e is ic ise-and-decline seen in NGC 3198, NGC 5055, NGC
7331, and NGC 6946. This con as highligh s he ole o galaxy mass and o a ion cu e shape in
shaping he local eg ession ac o , wi h mo e massi e, gas- ich sys ems exhibi ing a p onounced
peak ollowed by decline, while low-mass sys ems like NGC 2403 show apid fla ening and no
s ong maximum.
No es on in e p e a ion
The clus e ing we emphasize e e s o he adius o dynamical ansi ion (o en nea ∼5kpc),
no o any uni e sal β alue. The peak ampli ude o β( )is galaxy-specific: i encodes each
sys em’s mass dis ibu ion (bulge/disk/halo), non-ci cula mo ions, and en i onmen al his o y.
This di e si y is he co e o he “galac ic finge p in ” idea and is consis en wi h he EPOCH
amewo k’s local mani es a ion.
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Conclusion
This supplemen a y ma e ial demons a es ha he eg ession ac o βmani es s locally as β( ),
p o iding independen e idence o he physical mechanism unde lying he EPOCH amewo k.
The galac ic finge p in s highligh he b eakdown o iso opy a small scales and ein o ce he
unified cosmological in e p e a ion p esen ed in he main ex .
By compa ing NGC 2403, NGC 3198, NGC 5055, NGC 7331, and NGC 6946, we see ha
each galaxy exhibi s a dis inc β( )p ofile, eflec ing i s unique mass dis ibu ion, o a ion cu e,
and e olu iona y his o y. Despi e diffe ences in peak ampli udes, all galaxies e eal a common
dynamical ansi ion zone nea ∼5kpc, suppo ing he idea o a uni e sal scale whe e local
physics imp in s i sel on galac ic s uc u e.
The inclusion o NGC 6946, wi h i s igo ous s a o ma ion and dynamically ac i e disk,
s eng hens he case ha β( )encodes he finge p in s o galac ic o ma ion and e olu ion. To-
ge he , hese esul s p o ide compelling local suppo o he b oade EPOCH in e p e a ion o he
Hubble ension.
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