G a i a ional Flux om Cosmic Expansion D i es Galac ic Dynamics
A o Annila∗
Depa men o Physics, Uni e si y o Helsinki, Helsinki, Finland †
(Da ed: No embe 27, 2025)
Ra he han pos ula ing da k ma e o modi ied g a i y, we a ibu e he accele a ion beyond he
local ield, seen in galaxy o a ion cu es and eloci y dispe sions, o g a i a ional lux om cosmic
expansion. This e ec , o o de 10-10 ms-2, ollows wi hou ee pa ame e s om applying Gauss’s
lux heo em o all eceding galaxies, ep oducing he gene alized Tully-Fishe and Fabe -Jackson
ela ions. Al hough iso opic on la ge scales, he eno mous e lux o space canno be uled ou by
he shell heo em— o mula ed o s a iona y a he han dispe sing ma e —no dismissed wi h
dis ance, since he quad a ic ise in galaxy numbe ou weighs he in e se allo o hei po en ials.
F om his pe spec i e, space is no me e abs ac geome y bu a dynamic ela i is ic subs ance:
a spin-2 boson, pai ed-pho on acuum wi h undula ing ene gy densi y consis en wi h gene al co-
a iance and es s o ela i i y. Acco dingly, bodies mo e nei he inwa d by inexplicable a ac ion
no ou wa d by unexplained expansion, bu by coupling, ia mass, o space in lux. The Type Ia
supe no a da a a e hen consis en wi h space expanding h ough he in lux o quan a eleased om
ma e in as ophysical p ocesses, a he han implying a da k-ene gy-d i en me ic accele a ion.
S uc u es eme ge on e e la ge scales because he i ial adii o e lux–in lux balance leng hen as
spa ial densi y hins om he ea ly uni e se o he p esen spa seness, gi ing ise o he obse ed
cosmic g a i a ional edshi .
I. INTRODUCTION
The conund um o why s a s do no spi al ou o galaxies, despi e hei eloci ies oo high o be bound by isible
mass, echoes he one- ime puzzle o why elec ons in hei cu ed pa hs do no adia i ely spi al in o he nucleus.
Back hen, Boh p oposed ha a oms a e s able because he angula momen um L=me =nh/2πo an elec on
o mass me, o bi ing wi h eloci y a a dis ance om he nucleus, is quan ized in mul iples no Planck’s cons an
h, educed by he 2πci cum e ence ac o . Howe e , he Boh model did no ac ually explain a omic s abili y by
associa ing he s anding elec on wa e wi h a quan ized physical subs ance.
In u n, galac ic s abili y is asc ibed o da k ma e ha ops up isible ma e o balance he angula momen um
L=m =m 2dφ/d =U , wi h he g a i a ional po en ial ene gy Uo e he o bi al pe iod . Then, in s able o bi ,
a s a o mass m, wi h eloci y a a dis ance om he galac ic cen e , sweeps ou equal a eas 2dφ in equal imes
d , by Keple ’s second law. Howe e , despi e ma ching obse a ions, he da k ma e hypo hesis does no explain
galac ic dynamics by associa ing he missing mass wi h a speci ic physical subs ance.
Al hough he ul ima e na u e o ma e and space lies beyond e ec i e heo ies, hei pa ame e s a e o en aken
o ep esen eal subs ances. Ye , a e ex ensi e s udies and su eys, da k ma e emains elusi e. The g a i y o he
p oblem cau ions agains dismissing pa adigm shi s ligh ly — could he da k ma e pa ame e in he me ic model
o space ime, in ac , ep esen he accele a ion a ising om he lux o spa ial physical subs ance?
In analogy wi h hyd odynamics, he accele a ion due o lux does no cancel ou : bodies a e d awn inwa d as
he spa ial subs ance escapes be ween hem o spa se su oundings. Con e sely, bodies a e d i en apa as he
spa ial subs ance en e s be ween hem om dense sou ces. Ine ia, oo, sugges s ha bodies a e coupled o a spa ial
subs ance. Thus, despi e iso opic cosmic expansion, he accele a ion om lux canno be uled ou by he shell
heo em o mula ed o s a iona y a he han dispe sing ma e , no dismissed wi h dis ance, since he quad a ic
inc ease in galaxy numbe ou weighs he in e se allo o hei g a i a ional po en ial.
To accoun o he e ec o expansion on galac ic dynamics, we apply Gauss’s heo em ΦR=RaR·dSR=−4πGM
o he whole uni e se o mass M, ea ing space i sel as a ela i is ic physical subs ance wi h densi y-dependen wa e
speed c. This spa ial subs ance expands h ough a sphe ical su ace SR= 4πR2a adius R=c , con ining causal
in luence wi hin cosmic age ≈13.8 billion yea s, no a he como ing adius de i ed om me ic expansion h ough
a model-dependen scale ac o . As he uni e se ages, he spa ial subs ance expands om he dense pas o he
spa se p esen along he g adien aR=−∇ϕin he g a i a ional po en ial ϕ=−GM/R =−c2o he undula ing
subs ance wi h cha ac e is ic wa e p opaga ion.
∗Con ac au ho : [email p o ec ed]
†h ps://www.m .helsinki. i/home/aannila/a o
2
By Bi kho ’s heo em, he de ailed dynamics o expansion is smoo hed ou on la ge scales, lea ing an asymp o ically
la uni e sal ield aR=GM/R2=c/ ≈7.0×10−10 ms−2— o he same o de as he h eshold accele a ion ∼10−10
ms−2whe e galaxy o a ion cu es and eloci y dispe sions begin o de ia e om local New onian dynamics [1–9].
The Machian de ini ion GM/c2R= 1 o he g a i a ional cons an G[10–13], essen ially he mass–ene gy equi alence
o he whole uni e se Mc2=GM2/R, yields M=c2R/G ≈1.7×1053 kg. Then, he co esponding es ima e o he
ene gy densi y in ma e ρMc2≈0.6×10−9Jm−3is na u ally in balance wi h he spa ial ene gy densi y, consis en
wi h obse a ions o a spa ially la uni e se [14], hough dis inc om he much smalle ba yonic ma e densi y ρb
de i ed om he me ic model.
In he ollowing sec ions, we i s a gue ha cosmic expansion se s space in lux, gi ing ise o a ield ha pe mea es
he uni e se. Nex , we p o ide a physical basis o his lux in he o m o a anspa en , quan ized, ela i is ic,
and weakly in e ac ing spa ial subs ance, whe e pho ons a e pai ed in opposing phases, consis en wi h e es ial
measu emen s and as onomical obse a ions. Then, we show ha galac ic dynamics esul s om he e lux o space
in o spa se su oundings, while expansion i sel esul s om he in lux o quan a eleased om ma e h ough
as ophysical p ocesses. Lea ing no need o da k ma e o da k ene gy, we conclude ha galaxy o a ion cu es,
eloci y dispe sions, la ge-scale s uc u e o ma ion, and Type Ia supe no a da a a e mo e na u ally unde s ood as
mani es a ions o space in lux han as e idence o a da k sec o .
II. THE UNIVERSAL GRAVITATIONAL FLUX
As p oposed by Mach and o malized by Sciama [10], he g a i a ional po en ial ϕo all eceding ma e gi es ise
o ine ia. The change ∂ /∂ in he eloci y o a body ela i e o he ela i is ic spa ial subs ance, cha ac e ized
by wa e speed c, induces a ield o −(ϕ/c2)∂ /∂ . This all-pe mea ing e ec o dis an eceding ma e a ises om
he geome ic s uc u e o he uni e se: while he g a i a ional po en ial o each galaxy alls as 1/ , he numbe o
galaxies ises as 2wi h dis ance. Consequen ly, he mos dis an galaxies, such as hose imaged by he James Webb
Space Telescope, con ibu e he mos o ϕ, ende ing ine ia insensi i e o local a ia ions [10].
We a e no he i s o cons uc cosmology om Mach’s p inciple o add ess da k ma e and da k ene gy p oblems
[15, 16]; howe e , we desc ibe he in luence o dis an ma e ia a angible spa ial subs ance. All o dina y ma e ,
dispe sing om he dense o igin o he uni e se a = 0 o i s spa se p esen s a e a =R/c, gene a es a g adien
−∇ϕ=aR. Acco ding o Gauss’s heo em o g a i a ional lux, his g adien d i es space in o lux
ΦR=ZaR·dSR=−4πG ZρMdV =−4πGM (1)
h ough he su ace SRa adius R=c . The o al mass M=RρMdV sums all sou ces wi hin he expanding
olume V. Ra he han he c i ical densi y ρc= 3H2/8πG de i ed om he i s F iedmann equa ion o me ic
expansion, he mass densi y he e is gi en by ρM= 1/4πG 2≈6.3×10−27 kgm−3, dec easing o e cosmic ime
h ough ans o ma ions o ma e in o spa ial subs ance, as i is consis en wi h Poisson’s equa ion o g a i y,
∇2ϕ= 4πGρM= 1/ 2, he di e en ial o m o Gauss’s heo em.
The spa ial subs ance is cha ac e is ed by he wa e speed c≈3.0×108ms−1 ha inc eases as he densi y ρ=
ρMc2≈6×10−10 Jm−3dec eases in balance wi h he dec easing mass densi y ρM[14]. This dynamic balance
equi es spa ial con inui y, gi en by ∂ρ/∂ +∇·(ρ ) = σ, whe e σdeno es he a e o ma e - o-space ans o ma ions
h ough as ophysical p ocesses [17–19]. In in eg al o m, his becomes RVσdV =d/d RVρdV +R∂V ρ ·dS, which is
he Reynolds anspo heo em gene alized wi h a sou ce e m. Fo an iso opically expanding uni e se wi h adial
eloci y =cˆ , he su ace e m educes o he ou wa d lux ac oss a sphe ical bounda y o S= 4πR2.
Du ing he dilu ing expansion, he g a i a ional lux e ol es as
dΦR
d =−4πd
d ZGρMdV =−4πd
d GM =−4πc3,(2)
whe e he balance condi ion Mc2=GM2/R, analogous o he i ial heo em applied o he uni e se [11–13], e-
a anges h ough R=c o c2R=c3 =GM, and hus di e en ia es o c3. This olume ic change h ough
ans o ma ions o ma e in o space in he leas ime na u ally esul s in an iso opic expansion ha conse es o al
momen um dp/d = 0. The o al powe o ans o ma ions
P=Z4πR2dR c2dρM/d =Z4πc2 2cd c2(d/d )(1/4πG 2) = c5/G (3)
is equal o he Planck luminosi y LP=c5/G ≈3.6×1052 W [20].
3
The seemingly classical equa ions do no educe o New onian g a i y, whe e he mass is conse ed. While consis en
wi h mass-ene gy equi alence, he g a i a ional lux does no ha e a one- o-one co espondence in gene al ela i i y
because space ime does no ully geome ize he spa ial subs ance—i s low along densi y g adien s, i s eme gence
om ma e , i s e olu ion in ene ge ic balance wi h dec easing ma e , and i s quan ized cha ac e .
G a i a ional lux is analogous o elec omagne ic lux. Sha ing he same undula ing spa ial subs ance, g a i a ion
and elec omagne ism p opaga e wi h he same wa e speed c2=GM/R = 1/εoµo, de e mined by he g a i a ional
po en ial and he acuum pe mi i i y εoand pe meabili y µo[21–23]. Con inui y in he g a i a ional po en ial man-
i es s i sel as ine ia, jus as con inui y in he elec omagne ic po en ial mani es s i sel as displacemen cu en and
induc ance. As an icipa ed by Fa aday and Eins ein [24, 25], g a i a ion is in eg al o space whe e e densi y exis s,
whe eas elec omagne ism is con igu a ional o space whe e phase cohe ence pe sis s. The wo emain uncoupled
because elec omagne ic componen s a e ans e se o he null geodesic — he leas - ime pa h o a pho on.
The no ion o space as a subs ance endowed wi h physical p ope ies once ini ia ed a pa adigm shi om ac ion-
a -a-dis ance o causal in e ac ions p opaga ing a ini e speed [21, 25]. Building on his iew, g a i y is no longe an
inexplicable A is o elian a ac ion; ins ead, bodies mo e because hey couple, ia mass, o an all-pe ading spa ial
subs ance ha lows owa d he modynamic balance along pa hs o leas ac ion.
III. THE PAIRED-PHOTON VACUUM
To g ound he physics o g a i a ion in a spa ial subs ance, we e i e ae he , howe e , as a anspa en , ela i is ic,
quan ized, and locally Lo en z-in a ian quin essence comp ising quan a o ligh . Since he pho on has ne e been
obse ed o di ide o decay, i s ands as he indi isible and indes uc ible building block o bo h acuum and pa icles
[26]. This a omis ic p emise could be alsi ied, bu o he bes o ou knowledge, all e idence emains consis en wi h
i . Elemen a y pa icle eac ions, including annihila ion and pai p oduc ion, con inue o be consis en wi h he ene
ha bo h space and ma e comp ise pho ons.
The Michelson–Mo ley expe imen disp o ed a ligh -ca ying medium, no he ligh -comp ising subs ance we p o-
pose: ays o pho ons, eal a he han i ual, pai ed wi h opposi e phases (Fig. 1) [18, 27, 28]. This neu al spin-2
composi e sha es he p ope ies a ibu ed o he g a i on. As cons i uen s o he acuum, pho ons in a iably p opa-
ga e a he cha ac e is ic wa e speed c=λ/τ, de ined by he wa eleng h λpe pe iod τ. Consequen ly, ligh a e ses
a ixed dis ance Lin equal ime bo h pa allel ( ∥= 2L/c) and pe pendicula ( ⊥= 2L/c) o Ea h’s eloci y [23],
consis en wi h ela i i y. Pu simply, he pai ed-pho on acuum is Lo en z in a ian , consis en wi h he null esul
o he Michelson–Mo ley expe imen .
FIG. 1. Two cop opaga ing wa ele s wi h opposing elec omagne ic componen s (colo ed da k and ligh ) illus a e a pai o
pho ons ou -o -phase, o ming a massless, spin-2 composi e pa icle wi hou a ne elec omagne ic ield.
The e y exis ence o g a i a ional wa es sugges s ha space is a physical subs ance wi h densi y a ia ions p op-
aga ing a he cha ac e is ic wa e speed c. Then, he chi ps de ec ed a he Lase In e e ome e G a i a ional Wa e
Obse a o y (LIGO) a e be e unde s ood as changes in acuum wa eleng h a he han in e e ome e a m leng h
— a hypo hesis ha once oubled Lo en z himsel [29].
Fu he mo e, s ella abe a ion θ= a c an(cy/n ÷cx/n) = a c an(cy÷cx), in Ca esian componen s, esul ing
om Ea h’s mo ion wi h eloci y =cx, is inhe en ly independen o he e ac i e index n, no because space
lacks subs ance. Simila ly, he Sagnac ime di e ence, ∆ = 2( /n ÷c/n)(L/c), is ully consis en wi h a dynamic,
ela i is ic, pai ed-pho on subs ance. Following F esnel’s easoning, he Fizeau expe imen is unde s ood o show
pho ons p opaga ing a a lowe wa e speed, c′=c/[n−(n2−1) /c]≈c/n + (1 −1/n2) , due o he o e densi y
(n2−1) in he acuum ha pe mea es he wa e lowing a a eloci y /c ≪1. In sho , al hough hese expe imen s
and hei ei e a ions uled ou he lumini e ous ae he , hey did no ule ou he possibili y ha he acuum i sel is
composed o pho ons.
4
The pai ed-pho on acuum also makes sense o as onomical obse a ions by ela ing a local po en ial GMo/
o he uni e sal po en ial c2=GM/R h ough n2=c2/c′2= (1 −GMo/c2 )−1≈1 + GMo/c2 , as obse ed in
he pe ihelion p ecession, geode ic p ecession, ame d agging, escape eloci ies, g a i a ional lensing, edshi , ime
delay, g a i a ional wa es, decay imes, and clock a es [27, 30–33]. The dissipa ion a e o pho ons, o ime dila ion
′=γ , likewise depends on g a i y and eloci y h ough n2=c2/c′2= ′2/ 2= (1−GMo/c2 )−1= (1− 2/c2)−1, as
demons a ed by he Pound–Rebka, Vesso –Le ine, Rossi–Hall, and Ha ele–Kea ing expe imen s. In u n, he geodesic
su ace o he obla e Ea h, wi h adius oand spinning a eloci y ω o= ≪c, shows ha he g a i a ional and
kinema ic e ec s balance as GMo(1/ o−1/ ′
o) = 2.
Analogous o g a i y in gene al ela i i y, he pai ed-pho on acuum can be geome ized, cons ained by he in insic
p ope ies o he pho on. The pho on is massless because i canno couple o he acuum i cons i u es, which de ines
he pho on’s pa h h ough he null geodesic condi ion
gµν
dxµ
dλ
dxν
dλ = 0,(4)
encoded in he me ic gµν, whe e xµ(λ) a e coo dina es along he geodesic wi h he a ine pa ame e λp ese ing he
p opo ionali y o in e als. The ini e wa e speed cse s he causal s uc u e, gi en by he null space ime in e al
ds2=gµνdxµdxν= 0.
The ull geodesic equa ion, accoun ing o bo h cu a u e and o sion o he pho on pa h,
d2xµ
dλ2+ Γµ
νρ
dxν
dλ
dxρ
dλ −1
2Tµ
νρ
dxν
dλ
dxρ
dλ = 0
⇒d2xµ
dλ2+ Γµ
νρkνkρ−1
2kµSνρkνkρ= 0,(5)
wi h kµ≡dxµ/dλ, geome izes he acuum wi h a gene alized connec ion ˜
Γµ
νρ = Γµ
νρ −1/
2Tµ
νρ, whe e he symme ic
pa Γµ
νρ accoun s o a ia ions in acuum densi y ia he me ic gµν, while he an isymme ic pa Tµ
νρ =kµSνρ
encodes he acuum phase h ough Sνρ =−Sρν, akin o he elec omagne ic enso Fνρ.
In he p oposed pai ed-pho on acuum, he e is no ee es pho on mo ing h ough an ex e nal medium; a he ,
e e y pho on is an in eg al pa o he pai ed-pho on medium, a non- i ial acuum s a e. Consequen ly, o sion
aligns wi h he di ec ion o pho on p opaga ion and encodes he ans e se phase co ela ions o his sel -sus aining
s uc u e. The ela ion Tµ
νρ =kµSνρ hus na u ally desc ibes a pho on as pa o he backg ound pho on sea. Because
he phase (pola iza ion) is ans e se o he p opaga ion di ec ion, he elec omagne ic ield emains uncoupled om
g a i a ion, i.e., he densi y o he pai ed-pho on subs ance.
Al hough geome iza ion o g a i y ia he Eins ein ield equa ions ecas ins an aneous ac ion as mo ion in cu ed
space ime, i also edi ec ed a en ion om explaining g a i a ion i sel o modeling i s e ec s. Acco dingly, F iedmann
in oduced he scale ac o a( ) o accoun o ea ly cosmological obse a ions wi h he Hubble pa ame e H≡˙a/a.
La e , da k ma e and da k ene gy pa ame e s we e added o i u he da a, ye wi hou explaining he cause o
expansion.
IV. GALACTIC DYNAMICS
Galaxies a e so ex ended ha , a om hei dense cen e s, he e lux o space d awn along wi h expansion e eals
i sel in s ella eloci y dispe sions and ci cula eloci ies. Simila ly, galaxy clus e s a e so ou sp ead ha he e lux
is appa en in hei eloci y dispe sions. On e en la ge scales, beyond he adii o lux balance, a - lung galaxies a e
lown a he apa by he eno mous in lux eme ging om ma e - o-space ans o ma ions h oughou he uni e se.
Ro a ion cu es, eloci y dispe sions, and ecession all poin o expansion as he common cause o adial and ci cula
accele a ions aR=c2/R =u2/ = 2π 2/ ≈10−10 ms−2, on op o local accele a ion ao=GMo/ 2. Consequen ly,
he mass e sus he adial u=√aR and o bi al =paR /2π eloci y lines un pa allel in he log-log plo s, o se
by a ac o o √2π≈2.5 [34], and he single-pa ame e ela ions a=ao+aR=u2/ o eloci y dispe sions and
a=ao+aR/2π= 2/ o o a ion cu es i well, almos as i New onian dynamics we e modi ied [9, 35, 36].
To ocus on he dynamics o a galaxy, a g oup, o a clus e , we single ou a local lux Φo om he uni e sal lux
(Eq. 1)
ΦR= Φo+ ΦR−=Z(ao+aR−)·dS +ZaR−·(dSR−dS )
=−4πGMo−4πG(M−Mo) (6)
5
by pa i ioning aRin o a=ao+aR−, h eading h ough he local su ace, S = 4π 2, enclosing he local ba yonic mass,
Mo, wi hin he adius, , and he emaining uni e sal ield, aR−, h eading h ough SR−S . Assuming symme ic
mass dis ibu ions locally (by he shell heo em) and globally (by he cosmological p inciple), aoand aR−a e adial
and hence pa allel, allowing hei magni udes o be di ec ly compa ed. The sou ce o he local ield, ao=GMo/ 2, is
Mo, and he sou ces o aR−=G(M−Mo)/R2a e all o he o dina y masses, M−Mo. When conside ing a galaxy, a
g oup o galaxies, o e en a clus e o galaxies, Mo≪M, and hence aR−≈aR.
A. Radial accele a ion
When he local ρMoexceeds he uni e sal ρMdensi y, e lux domina es, and he ela i e s eng hs o he accele a ion
componen s
a≈ao+aR=ao1 + aR
ao(7)
de ine h ee egimes. In dense sys ems whe e aR/ao<1, as in la ge ellip ical galaxies, local accele a ion ules [37].
Sys ems whe e aR/ao≈1, such as ich clus e s, show mo e complex beha io . In spa se sys ems whe e aR/ao≫1,
he uni e sal accele a ion go e ns; dwa galaxies, poo clus e s, and pe iphe al objec s display in hei eloci y
dispe sions essen ially only aR[34, 36].
To de i e he ull eloci y dispe sion, Eq. 7 is mul iplied by he lux ela ion σ2 ∝GMo, h ough he sphe ical
su ace enclosing mass Mo, yielding he gene alized Fabe –Jackson ela ion σ4∝aGMo[36, 38, 39], app op ia e o
he sys em in global dynamical balance wi h he whole uni e se a he han in isola ion om i (Fig. 2).
The accele a ion due o he e lux o space is easily mis ead h ough he dynamical mass balance σ2∝G(Mo+
MDM)/ , as i he e we e da k ma e (DM) wi hin he adius o in e es alongside ba yonic ma e . The smalle
he ba yonic mass Mo, he g ea e his misin e p e a ion, pa icula ly e iden in dwa galaxies [40]. Con e sely, he
low eloci y dispe sions seen in globula clus e s o ul a-di use galaxies do no imply a lack o da k ma e [41, 42],
bu a he e lec minimal e lux om hei own low densi ies, while hei peculia eloci ies e eal he e lux om
hei hos ing sys em. Likewise, dwa sa elli es show low eloci y dispe sion and high o bi al eloci ies.
B. Ci cula accele a ion
Simila ly o adial accele a ion, a s a o bi ing in a spi al galaxy expe iences o al accele a ion
a= 2
≈ao+aR
2π=ao1+ 1
2π
aR
ao=GMo
21+ 1
2π
M 2
MoR2,(8)
a ising om wo con ibu ions: (i) he local ield ao=GMo/ 2, due o he ba yonic mass Mowi hin he o bi al
ci cum e ence 2π , (ii) he uni e sal ield aR/2π=GM/2πR2, scaled down by he 2πci cum e ence ac o because
he ield ac s o e he ull cycle a he han o e he linea uni adius, due o he e lux o space d awn along wi h
all eceding mass M−Mo≈M[18, 43].
Nea he galac ic cen e , he local ield ao≫aR/2πexceeds he uni e sal ield, so he g a i a ional lux Φo≈
−4πGMoscales di ec ly wi h he enclosed mass. In hyd odynamic e ms, he e lux o space, pe cei ed as g a i a ional
a ac ion, is in ense due o he s eep densi y g adien be ween he local ρMoand he su ounding cosmic ρMdensi y.
A la ge adii, whe e aR/2π≈ao, he uni e sal ield becomes disce nible on op o he local ield [44]. Fu he ou ,
s a s and gas clouds expe ience essen ially only aR/2π≫ao.
To de i e he ull o a ion cu e, Eq. 8 is mul iplied by he lux ela ion 2 =GMo h ough he sphe ical su ace
enclosing mass Mo, yielding he gene alized Tully–Fishe ela ion 4=aGMo[36, 39, 45], cha ac e is ic o o bi al
sys ems in dynamical balance wi h he uni e sal g a i a ional ield a he han isola ed om i (Fig. 2).
Simila ly o he ci cula eloci ies o dis an s a s, he eloci ies o sa elli e galaxies, 4≈aRGMo/2π, emain
la because a≈aR/2π > aois cons an , no because 2( ) = G(Mo+MDM( ))/ would be la , as i he e we e
a da k ma e halo MDM( )∝ ⇒ρDM ∝ −2. A e en g ea e dis ances, he e lux ades as he local o e densi y
δρ≡(1 −ρM/ρMo) ends o ze o, and he eloci y
4=aGMo=δρ(ao+aR/2π)GMo(9)
dec eases, as i he da k-ma e halo had s eepened [46, 47]. The cubic all-o o he o e densi y δρ owa d he lux
balance b = (Mo/M)1/3R aces obse a ions. Milky Way sa elli e galaxies lie wi hin b ≈1.4×106ly, gi en a
ba yonic mass MMW ≈1.0×1011M⊙, and clus e s o MCL ≈1.0×1014M⊙span b ≈14 ×106ly.
6
FIG. 2. Veloci y dispe sion σand ci cula eloci y as unc ions o adial dis ance , shown on linea (le ) and loga i h-
mic ( igh ) scales, om he cen e o an ellip ical galaxy (black do ed lines) and a spi al galaxy (black solid lines). Thei
ba yonic masses, Mo= 1011M⊙, a e dis ibu ed acco ding o He nquis ( e= 10,000 ly) and S´e sic ( e= 20,000 ly, n= 2)
p o iles. Veloci y dispe sions a e compu ed using he gene alized Fabe –Jackson ela ion, σ4=δρ(ao+aR)GMo, assuming
iso opic s ella mo ion along he line o sigh . Ci cula eloci ies a e compu ed using he gene alized Tully–Fishe ela-
ion, 4=δρ(ao+aR/2π)GMo, assuming ci cula o bi s (Eq. 8). As long as he local densi y exceeds he uni e sal densi y
δρ≡(1 −ρM/ρMo)>0, he uni e sal accele a ion aR=c/ a ises om he e lux o space in o he expanding uni e se. Fo
compa ison, he New onian accele a ion ao= 2/ =GMo( )/ 2is also plo ed (g ay do ed and solid lines). The Colab
no ebook GalaxyDynamiX is a ailable o a y he pa ame e s.
Al hough eminiscen o MOdi ied New onian Dynamics (MOND), he accele a ion ela ions in Eqs. 7 and 8 do
no modi y New on’s law. In MOND, a cosmic-scale h eshold accele a ion, on he o de o 10−10 ms−2, is used as a
pa ame e o model aRo he e lux as i g a i y i sel we e modi ied. Consequen ly, his app oach ails pa icula ly o
dwa sphe oidals, galaxy clus e s, and ce ain i egula o gas- ich sys ems. Since he luxes o space along he local
and uni e sal densi y g adien s mani es in de ailed dynamics, he p ecise o m o he MOND in e pola ion unc ion,
(ao/aR)ao=a, o accoun o he mass disc epancy 2/ 2
b= 1 + aR/2πaobe ween he obse ed eloci y and he
ba yonic expec a ion b, is no he main issue [34, 36, 44].
C. Recession
Beyond he adius, whe e he local ρMo( ) alls below he uni e sal ρM( , ) densi y, he in lux o space om
as ophysical p ocesses h oughou he uni e se supe sedes he local e lux, and he Hubble low ca ies bodies apa .
As desc ibed by Hubble’s law, he adial eloci y u=√aR =cp /R ollows om aR=c/ =c2/R, causing he
mos dis an galaxies o ecede a speeds app oaching c.
The di e en ial o m o Gauss’s lux heo em, Poisson’s equa ion o g a i y,
∇2ϕ=∇·aR= 4πGρM=c2
R2=1
2,(10)
gi es he decele a ing a e o expansion dH/d =−1/ 2wi h H= 1/ , as he ma e densi y ρMdeclines. Acco dingly,
he lux densi y 4πGρMdec eases as i s sou ces M ans o m in o spa ial subs ance (Eq. 2). I is wo h s essing ha
R=c is no a pa ame e bu he ex en o spa ial subs ance, de e mined by i s densi y-dependen wa e speed c, and
by cosmic ime , summing he pho on pe iods since he onse o expansion. Thus, R=c should no be mis aken
o a s a emen o linea expansion; a he , he expansion is decele a ing because as ophysical p ocesses ans o m
ma e in o spa ial subs ance.
7
Sol ing ρMo=ρMde ines he adius o lux balance,
b =Mo
M1
/3
R=Mo
4πρM1
/3
=GMo 21
/3,(11)
which leng hens as he uni e se ages (Fig. 3). This scale- ee g ow h co esponds o he powe -law au oco ela ion
o galaxies and oids [48, 49]. The cosmic web becomes coa se no because da k ma e concen a es, bu because
space a e ies o e ime. Consequen ly, s uc u es o m ea lie and on la ge scales han p edic ed by ΛCDM [50, 51].
Acco ding o Eq. 11, Milky Way-like galaxies, wi h = 150 ×103ly and Mo= 1011M⊙, could ha e eme ged as ea ly
as ≈300 ×106yea s (Fig. 3). Ea lie objec s we e mo e compac . In u n, galaxy g oups began o o m a ound
one billion yea s and clus e s a ound h ee billion yea s. The KBC Void, he Local Hole, is na u ally he leas dense,
being among he oldes egions in he uni e se.
FIG. 3. The age o he uni e se as a unc ion o dec easing densi y ρ(Eq. 10) ou lines he cha ac e is ic g a i a ional imescale
o s uc u e o ma ion, om he ea ly sup anuclea densi ies compa able o s ella -mass black hole densi ies o he p esen -day
densi ies o supe clus e s.
The me ic expansion i ed o he Type Ia supe no a da a wi h da k ene gy implies an accele a ed expansion. In
con as , he decele a ing physical expansion na u ally aligns wi h he da a wi hou ee pa ame e s because ligh
is unde s ood o shi owa d ed as i a els om he dense pas o he spa se p esen [18, 31, 32]. In o he
wo ds, in addi ion o climbing ou o local g a i a ional wells, hose o a s a , galaxy, and clus e , a pho on also
climbs he b oade g a i a ional po en ial o he aging uni e se. The obse ed cosmological edshi is he e o e no
undamen ally di e en om local g a i a ional shi s.
Acco dingly, his physical cosmology, whe e ma e ans o ms in o space esul ing in an iso opic expansion, is
consis en wi h obse a ions in e p e ed h ough cosmic-scale a e aging, unlike imescape cosmology [52, 53] o back-
eac ion due o local inhomogenei ies [54, 55], and s ands in con as o he lawed i ed-ligh hypo hesis. In e ing
om he o ce o expansion F=c4/G, pe SR, he nega i e p essu e −p=F/SR=ρMc2≈10−9Jm−3(Eq. 10)
a ises om ba yonic ma e ans o ming in o space a he han om da k ene gy.
8
V. DISCUSSION
Despi e ex ensi e sea ches, da k ma e and da k ene gy emain unsubs an ia ed, ye a e p esumed o accoun o
95% o he uni e se’s ene gy con en o econcile cosmological models wi h obse a ions — a disc epancy ha may
signal a undamen al misunde s anding o g a i a ion.
Recognizing ha pe cep ions a e heo y-laden [56], we ask: Could g a i a ional a ac ion and cosmic expansion
be wo sides o he same phenomenon? F equency shi s only show ha dis an galaxies ecede and nea by ones
app oach, bu no why. A common h ead is he cha ac e is ic accele a ion, on he o de o 10−10 ms−2, consis en ly
in e ed om galaxy o a ion cu es, eloci y dispe sions, and ecession eloci ies — hough in opposi e di ec ions.
I s magni ude, no ably close o c/ , poin s o a uni e sal ield aR=c/ =c2/R =GM/R2, ela ed o all o dina y
ma e Mdispe sing h oughou he uni e se, expanding i s adius R=c a speed co e ime ≈13.8×109yea s.
By con as , assuming me ic expansion, he calcula ed accele a ion o app oxima ely 0.2c/ due o dis an ma e
alls sho by a ac o o i e [57].
By Poisson’s equa ion (Eq. 10), he decele a ing expansion −dH/d = 1/ 2= 4πGρMis ela ed o he dec easing
densi y o ma e ρM, sugges ing ha space eme ges om ma e , a he han galaxies d i ing apa wi hou cause.
A e all, he amoun o ma e consumed by s a s, supe no ae, and ac i e galac ic nuclei is no negligible. Ex ap-
ola ing om dense o igins o hea dea h ρM( → ∞)→0, he ans o ma ion o ma e in o space na u ally en ails
Mc2=GM2/R and de ines he ma e densi y ρMequal o he c i ical densi y ρc≡1/4πG 2, hus add essing he
Hubble ension and ende ing he cosmological ine- uning p oblem o la ness null and oid. F om his pe spec i e,
he ela ion ρ= Λc4/8πG be ween he da k ene gy densi y and he cosmological cons an Λ models he balance
ρ=ρMc2=c4/4πGR2≈0.6×10−9Jm−3be ween he spa ial ene gy densi y and he ma e densi y wi hin he
causal bounds o he uni e se 4πR2.
A i s glance, he uni ying iew o g a i a ional a ac ion and cosmic expansion as a spa ial subs ance in lux,
ca ying bodies bo h inwa d and ou wa d, may seem adical. Howe e , his o ically, he no ion ha space embodies
g a i y is no new [25, 58, 59]. The no el y he e is ha pho ons make he medium, cha ac e ized by wa e speed c,
a he han mo e h ough some p e-exis ing medium a speed c. This anspa en , ela i is ic acuum, comp ising
ligh quan a pai ed ou o phase [18], is consis en wi h he classical expe imen s, such as he Michelson-Mo ley null
esul , and mode n ones, such as he dynamic Casimi e ec [60], as well as wi h heo e ical ounda ions: Bose-Eins ein
s a is ics, Maxwell’s equa ions, Lo enz gauge, and g a i on p ope ies. Con e sely, e en ual empi ical e idence con-
adic ing his physical plenum, o example, i a pho on we e obse ed o spli , would alsi y he p oposed explana ion
o galaxy o a ion, eloci y dispe sion, and cosmic expansion.
F om a hyd odynamic pe spec i e, space in lux beha es like any o he subs ance lowing along geodesics owa d
he modynamic balance, whe e o ces e en ou . As he uni e se la ens owa d a s a e o leas cu a u e, bodies
shape in o sphe oids, se le in o planes [38, 39], and galaxies align [61]. Sa elli e galaxies a e no missing a ound hei
hos s; ins ead, uni e sal accele a ion, ac ing as a cen al o ce, has ze oed hem in on s able o bi s. As he expanding
space hins, e lux and in lux balance e e a he ou , and hence e e la ge s uc u es s e ch ac oss widening oids.
Al hough pe u ba ions o he F iedmann-Lemaˆı e-Robe son-Walke (FLRW) me ic ep oduce he peaks in he
angula powe spec um o he cosmic mic owa e backg ound (CMB), hey canno accoun o la ge-scale s uc u e
o ma ion wi hou in oking da k ma e . Since a se ies o peaks can a ise om gene al p inciples, no solely om
acous ic oscilla ions [62], he need o da k ma e and da k ene gy pa ame e s sugges s ha a s e ching me ic
may no ai h ully model he physics o expansion. Fu he mo e, in he FLRW me ic, he angula diame e dis ance
is non-mono onic wi h edshi , in con as o he in ui i e expec a ion ha he a he an objec is, he smalle i
appea s [32].
The e en dispe sion o dis an galaxies and he uni o mi y o CMB na u ally ollow om ma e ans o ming in o
space in he leas ime [17, 18], a he han om imp omp u cosmic in la ion. Consis en wi h Gauss’s lux heo em,
New on’s second law F=dp/d s a es ha he g ea e he o ce, he as e he change: massi e s a s bu n b igh es
bu ade as es ; ea ly- ype galaxies decline in numbe densi y wi h edshi as e han la e- ype galaxies. In essence,
he highe he local densi y, he as e he a e o ma e - o-space ans o ma ions, i.e., he as e expansion. Thus,
ega dless o ini ial i egula i ies, he leas - ime p inciple d i es he uni e se owa d iso opy.
Unlike da k ma e , he uni e sal ield o all o dina y ma e is insepa able om he ma e i sel . Thus, abundance
ma ching, which co ela es galaxy luminosi ies wi h da k-ma e halos, is nei he heo e ically mo i a ed no empi -
ically subs an ia ed [63, 64]. Fu he mo e, while da k ma e s uggles wi h he cusp-co e p oblem, he uni e sal
g a i a ion o all o dina y ma e is na u ally la , ea u eless, and a -ex ending, lending i sel o single-pa ame e
modeling (MOND).
Despi e he a gumen s p esen ed, he explana ion o galaxy o a ion and eloci y dispe sion h ough g a i a ional
lux om cosmic expansion may be dismissed on he g ounds ha lensing by o dina y ma e appea s oo li le by a
la ge ma gin. Howe e , i is wo h ecalling ha he magni ude o de lec ion θ, o a known lens mass M⊙, is de i ed
solely om he angula di e ence be ween he ligh ays ha g aze he eclipsed Sun and hose om he nigh sky.
9
Al hough i may seem i ial, he pa allel displacemen o he wo ays, nea ly equal in magni ude o he de lec ion
i sel [65], has been neglec ed in de e mining θ[31]. Since he de lec ed pho on also ge s delayed, one migh a gue
ha he Shapi o ime delay ∆ , o a adio signal g azing he sola limb o, p o ides an exac calib a ion be ween
he lensing powe and he sola mass ee om da k ma e . In ac , he ound- ip delay 2∆ ≈200 µs [66, 67]
co esponds by oθ≈c∆ o a de lec ion θabou i e imes g ea e han θGR = 4GM⊙/c2 o[31]. Acco dingly, i is no
coincidence ha da k ma e is es ima ed o be oughly i e imes mo e abundan han o dina y ma e . Gi en his
disc epancy be ween g a i a ional bending and ime delay, lensed images o backg ound galaxies do no , con a y o
common belie , subs an ia e da k ma e ; e en he g a i a ional lensing o he Bulle Clus e , o en ci ed as decisi e
e idence, can be in e p e ed h ough he leas - ime pa hs o ligh wi hou da k ma e [18, 31].
In conclusion, he lux o space o e s a na u al explana ion o he phenomena a ibu ed o da k ma e and da k
ene gy. This is ha dly su p ising, since da k ma e is me ely a monike o missing mass, and da k ene gy is an alias
o expanding quin essence. E ec i e heo ies a e uned o i obse a ions bu emain un alsi iable abs ac ions wi h
unsubs an ia ed pa ame e s. Thus, he eal issue lies in disca ding he subs ance o space i sel — he e y essence o
g a i a ion—by misin e p e ing 19 h-cen u y classical expe imen s as i he acuum embodied no physical subs ance
wha soe e .
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