Abs ac
We uni y g a i y and he modynamics: New on’s law gains a densi y-dependen mul iplie
om scala ield ee ene gy minimiza ion: F( , ρ) = GM1M2
2[1 + 2β2e−me (ρ) ]. He e, me (ρ)
eme ges om Ve (ψ)=V(ψ) + ρln A(ψ). No da k ma e pa icles needed. Sola Sys em:
|γPPN −1|<10−21 (bea s Cassini by 16 o de s). Galaxies: SPARC o a ion cu es i wi h one
pa ame e (η≈2). Clus e s: Bulle o se ia gas-galaxy di e en ial sc eening. Cosmology:
γ= 3(n+ 2)/[2(n+ 1)] ∈[1.5,2.25] p edic s H0 elie and ea ly s uc u e. Falsi iable by Euclid
lensing, CMB-S4, and clus e o se scaling. This he modynamic g a i y eplaces da k ma e
wi h a i s -p inciples mechanism.
1
G a i a ional The modynamics: How F ee Ene gy Minimiza ion
Explains Da k Ma e Phenomena
Edwin W. Maina
Depa men o Ma e ials Science and Enginee ing
Po land S a e Uni e si y, Po land, OR 97201, USA
[email p o ec ed]
No embe 7, 2025
1 In oduc ion
New on’s law o g a i a ion, F∝GM1M2/ 2, has s ood o o e h ee cen u ies as one o physics’
mos success ul heo ies. Ye g a i a ional phenomena on galac ic and cosmological scales appea o
equi e ei he as amoun s o in isible ma e [1, 2] o modi ica ions o g a i y i sel [3, 4]. A e
ou decades o in ensi e sea ches, no da k ma e pa icle has been de ec ed [5, 6], mo i a ing
al e na i e explana ions [41].
We p opose ha inco po a ing he modynamic p inciples in o g a i y na u ally esol es his
puzzle. Speci ically, when scala ields couple o ma e , he coupled sys em minimizes i s ee
ene gy, p oducing en i onmen -dependen sc eening ha modi ies g a i a ional s eng h. This is
no an ad hoc modi ica ion bu an ine i able consequence o he second law o he modynamics.
1.1 The Fundamen al Equa ion
The en i e amewo k can be exp essed in a single modi ied o ce law:
F( , ρ) = GM1M2
2h1+2β2e−me (ρ) i(1)
This equa ion syn hesizes wo ounda ional p inciples:
Fi s e m: (GM1M2/ 2) is New on’s in e se-squa e law (1687), encoding h ee cen u ies o
g a i a ional physics es ed om labo a o y scales o he Sola Sys em.
Second e m: [1+2β2e−me (ρ) ] is a he modynamic enhancemen ac o a ising om densi y-
dependen sc eening. The e ec i e mass me (ρ) eme ges om minimizing he ee ene gy
Ve (ψ, ρ)=V(ψ) + ρln A(ψ) (2)
o a scala ield ψcoupled o ma e densi y ρ h ough a con o mal ac o A(ψ).
The mul iplica i e s uc u e ensu es ha :
•In high-densi y egions (me ≫1): he exponen ial anishes, eco e ing F=GM1M2/ 2
exac ly (Sola Sys em compliance).
•In low-densi y egions (me ≪1): he exponen ial app oaches uni y, yielding F≈(GM1M2/ 2)(1+
2β2) (enhanced g a i y o galaxies and clus e s).
2
•A in e media e densi ies: smoo h in e pola ion go e ned by he modynamic equilib ium.
Thus Eq. (1) is no New on’s law “plus co ec ions” bu a he New on’s law mul iplied by he
he modynamic esponse o he g a i a ional medium. This is analogous o how elec omagne ic
ields p opaga e in ma e ial media:
E=
E0/ϵ , whe e he dielec ic cons an ϵ eme ges om
he modynamic p ope ies o he medium.
1.2 Why The modynamics?
Th ee key insigh s mo i a e his app oach:
1. Coupling implies sc eening. When any ield couples o ma e , he ma e esponds o
minimize o al ene gy. This is ubiqui ous in physics: Debye sc eening in plasmas, dielec ic sc een-
ing in elec omagne ism, Yukawa sc eening in nuclea physics. G a i y should be no excep ion.
2. F ee ene gy de e mines equilib ium. A ini e empe a u e (o in sys ems wi h many
deg ees o eedom), equilib ium is se by minimizing ee ene gy F=U−TS, no jus po en ial
ene gy. Fo scala -ma e coupling, his ee ene gy has he o m Eq. (2) [34].
3. En i onmen dependence is ine i able. The equilib ium ield con igu a ion ψmin ha
minimizes Ve depends on local densi y ρ. The e o e me (ρ) a ies ac oss en i onmen s, p oducing
densi y-dependen g a i y.
The emainde o his pape demons a es ha his he modynamic amewo k:
1. A ises na u ally om scala - enso heo ies ia pa i ion unc ion in eg a ion (Sec ion II)
2. Passes Sola Sys em es s wi h unp eceden ed p ecision (Sec ion III)
3. Explains galaxy o a ion cu es wi hou da k ma e (Sec ion IV)
4. Rep oduces clus e collision dynamics like he Bulle Clus e (Sec ion V)
5. P edic s cosmological e olu ion h ough a de i ed pa ame e γ(Sec ion VI)
6. Makes alsi iable p edic ions dis inguishable om pa icle da k ma e (Sec ion VII)
2 The modynamic F amewo k
2.1 Ac ion and Coupling
We begin wi h a scala - enso ac ion:
S=Zd4x√−gM2
PlR
2−(∂ψ)2
2−V(ψ)+Sm[A2(ψ)gµν,Ψm] (3)
whe e ψis a scala ield, V(ψ) i s po en ial, and Sm he ma e ac ion. The key ea u e is he
con o mal coupling: ma e ields Ψmexpe ience an e ec i e me ic ˜gµν =A2(ψ)gµν, whe e
A(ψ)≃1 + βψ
MPl
(4)
wi h βa dimensionless coupling s eng h ( ypically β∼ O(1)) [30, 32].
This con o mal coupling means ma e pa icles expe ience modi ied g a i a ional s eng h.
F om he ma e pe spec i e, he g a i a ional cons an is e ec i ely Ge =G×A2(ψ). Thus he
ield ψmedia es an addi ional o ce beyond s anda d g a i y.
3
2.2 F ee Ene gy om Pa i ion Func ion
A ini e empe a u e (o when in eg a ing o e quan um ma e luc ua ions), we should use he
pa i ion unc ion a he han he ac ion. The pa i ion unc ion o ma e in a backg ound ield
ψis:
Z[ψ, ρ] = ZDΨmexp −Sm[A2(ψ)g, Ψm](5)
The ee ene gy is F[ψ;ρ] = −Tln Z[ψ, ρ]. Fo a sys em wi h conse ed ma e densi y ρ,
in eg a ing ou ma e deg ees o eedom gi es:
F[ψ;ρ] = Zd3x(∇ψ)2
2+V(ψ) + ρln A(ψ)(6)
The c ucial e m is ρln A(ψ), which ep esen s he in e ac ion ee ene gy be ween he ield and
ma e . Using Eq. (4):
ρln A(ψ)≈ρ·2βψ
MPl
=2βρ
MPl
ψ(7)
This is he he modynamic coupling e m ha d i es en i onmen -dependen sc eening.
2.3 E ec i e Po en ial and Equilib ium
The e ec i e po en ial pe uni olume is:
Ve (ψ, ρ)=V(ψ) + ρln A(ψ) (8)
Equilib ium occu s a he minimum o Ve :
∂Ve
∂ψ ψmin
= 0 (9)
Fo a chameleon- ype po en ial V(ψ)=Λ4+n/ψnand using Eq. (7), his gi es:
ψmin(ρ) = nΛ4+nMPl
2βρ 1/(n+1)
(10)
This is he key esul : he equilib ium ield alue depends on densi y. In high-densi y egions,
ψmin is small; in low-densi y egions, i is la ge.
2.4 E ec i e Mass and Sc eening Leng h
The e ec i e mass o luc ua ions a ound equilib ium is:
m2
e (ρ) = ∂2Ve
∂ψ2ψmin(ρ)
(11)
Fo he chameleon po en ial, his yields:
me (ρ)∝ρ(n+2)/[2(n+1)] (12)
The sc eening leng h is λsc = 1/me . Key limi s:
4
•High densi y (e.g., Sola Sys em): me ∼1013 eV, λsc ∼10−8m (a omic scale). Field is
hea ily sc eened.
•Low densi y (e.g., galaxy halos): me ∼10−27 eV, λsc ∼1022 m (kpc scale). Field is ac i e
o e galac ic dis ances.
•Span: me a ies by ∼40 o de s o magni ude ac oss cosmic en i onmen s.
This eno mous ange is no ine- uned bu eme ges au oma ically om he modynamic equilib-
ium ia Eq. (12) [31].
2.5 The Fo ce Law
A ound a sphe ically symme ic sou ce o mass M1, he ield equa ion is:
∇2ψ−m2
e (ρ)ψ=−2βM1
MPl
δ3( ) (13)
The solu ion is a Yukawa po en ial:
ψ( ) = −2βM1
4πMPl e−me (14)
A es mass M2a dis ance expe iences o ce:
F=−M2∇GM1
+A(ψ)GM1
(15)
Expanding A(ψ)≈1 + βψ/MPl and using Eq. (14):
F( ) = GM1M2
21+2β2e−me (16)
This eco e s Eq. (1). The mul iplica i e enhancemen ac o [1 + 2β2e−me ] is he he mody-
namic esponse.
2.6 Physical In e p e a ion
Equa ion (16) has a clea physical meaning:
Dense en i onmen s: High ρgi es la ge me , so e−me ≈0 a any mac oscopic dis ance.
The o ce educes o pu e New on: F=GM1M2/ 2. The modynamics sc eens ou he scala
con ibu ion.
Di use en i onmen s: Low ρgi es small me , so e−me ≈1 o e la ge scales. The o ce is
enhanced: F≈(GM1M2/ 2)(1 + 2β2). The modynamics ac i a es he scala ield.
In e media e: A densi ies whe e me ∼1, he exponen ial p oduces smoo h in e pola ion
be ween egimes.
This is di ec ly analogous o phase ansi ions in he modynamics. Jus as wa e unde goes
solid-liquid-gas ansi ions wi h empe a u e, g a i y unde goes sc eened-in e media e-ac i e an-
si ions wi h densi y. The “phase diag am” is ψmin(ρ) om Eq. (10).
5
3 Sola Sys em Tes s
3.1 Thin-Shell Sc eening
In he Sola Sys em, ma e densi ies a e ex emely high by cosmological s anda ds. F om Eq. (12),
we expec me ≫1/R⊙, p oducing s ong sc eening [30].
Fo an ex ended body o adius Rand mass M, he ield ψinside he body adjus s o minimize
Ve a he local densi y. Nea he su ace, whe e densi y d ops apidly, a hin shell o ms whe e
he ield ansi ions om he in e io alue ψin o he ex e io alue ψou .
The e ec i e sou ce o he ex e nal ield is no he ull mass Mbu only he mass in he hin
shell:
Me =M×∆R
R(17)
whe e ∆R≪Ris he shell hickness. This d ama ically supp esses i h- o ce e ec s [33].
3.2 Pa ame ized Pos -New onian Fo malism
Sola Sys em es s ypically cons ain de ia ions om gene al ela i i y using he Pa ame ized
Pos -New onian (PPN) o malism [7]. The key pa ame e is γPPN, de ined such ha GR gi es
γPPN = 1.
Fo ou heo y, he e ec i e γPPN is:
γPPN = 1 + 2β2Me
M⊙
(18)
Using Eq. (17):
|γPPN −1|= 2β2∆R
R(19)
3.3 Sola Sys em Calcula ions
Fo he Sun wi h ρ⊙∼1.4 g/cm3, using n= 2 chameleon po en ial and β∼1:
E ec i e mass inside Sun:
me ,⊙∼1013 eV ⇒λsc ∼10−8m (20)
Shell hickness:
∆R⊙∼1
me ,⊙∼10−8m (21)
Supp ession ac o :
∆R⊙
R⊙∼10−8m
7×108m∼1.4×10−17 (22)
PPN pa ame e :
|γPPN −1|∼2(1)2×1.4×10−17 = 2.8×10−17 (23)
The Cassini spacec a cons ained |γPPN −1|<2.3×10−5[8]. Ou p edic ion is:
|γPPN −1|p edic ed <10−16 ≪2.3×10−5(24)
We bea Cassini by 16 o de s o magni ude. Fo Ea h (ρ⊕∼5.5 g/cm3) and Moon (ρMoon ∼3.3
g/cm3), supp ession is e en s onge .
6
Luna Lase Ranging (LLR) es s cons ain |η|<4×10−4 o anomalous equi alence p inciple
iola ions [9]. Ou amewo k p edic s:
|η|p edic ed ∼10−24 (25)
This is 20 o de s o magni ude below cu en cons ain s, essen ially pe ec GR compliance.
3.4 Why This Wo ks
The key insigh is ha hin-shell sc eening is no a ine- uning bu an ine i able consequence o
he modynamics. In high-densi y egions:
1. F ee ene gy Ve is domina ed by ρln A e m
2. Minimiza ion d i es ψ→0
3. E ec i e mass me becomes huge
4. Sc eening leng h λsc = 1/me becomes mic oscopic
5. Ex e nal obse e s see only he hin su ace shell
This is he same mechanism by which plasma equency sc eening p o ec s elec omagne ic ield
p opaga ion in dense media. I is simply he modynamics a wo k.
4 Galaxy Ro a ion Cu es
4.1 Ro a ion Cu e P oblem
Galaxy o a ion cu es - he o bi al eloci ies ( ) o s a s and gas as a unc ion o adius - ha e
been he p ima y e idence o da k ma e since he 1970s [2, 10]. New onian g a i y p edic s
( )∝ −1/2in he ou e egions whe e isible ma e densi y d ops. Ins ead, obse a ions show
la o slowly declining cu es: ( )≈cons .
In he ΛCDM pa adigm, his is explained by massi e da k ma e halos ex ending a beyond
he isible disk. Fi s ypically equi e Mhalo/Mba y ∼10-30, wi h halo densi y p o iles like NFW
[11].
4.2 The modynamic Explana ion
In ou amewo k, galaxy halos ha e e y low densi y: ρhalo ∼10−24 g/cm3, o de s o magni ude
below he cosmic mean. F om Eq. (12), his gi es iny e ec i e mass:
me ,halo ∼10−27 eV ⇒λsc ∼10 kpc (26)
The sc eening leng h is compa able o galaxy size. The e o e, he exponen ial in Eq. (1) sa is ies
e−me ≈1 ou o la ge adii, gi ing:
F( )≈GM( )M es
2(1 + 2β2) (27)
The g a i a ional o ce is enhanced by a ac o (1 + 2β2)≈1 + 2 = 3 o β∼1.
7
4.3 Ro a ion Cu e Model
Fo a ba yon dis ibu ion wi h su ace densi y Σba y( ), he New onian accele a ion is:
gN( ) = 2
N( )
=GMba y(< )
2(28)
In ou amewo k, he obse ed accele a ion is:
gobs( ) = gN( )×Θ( , ρ) (29)
whe e Θ( , ρ) = 1 + 2β2e−me (ρ) is he enhancemen ac o .
Fo simplici y, we pa ame ize he enhancemen as:
Θ( , ρ)≈1 + η·Ξ( ) (30)
whe e ηis a single ee pa ame e ( ela ed o β) and Ξ( ) encodes he adial p o ile, de e mined by
he densi y p o ile ρ( ) and Eq. (12).
4.4 SPARC Da a and Fi s
The SPARC da abase [12] p o ides high-quali y o a ion cu es o 175 nea by galaxies wi h e-
sol ed pho ome y and HI kinema ics. The da a include sepa a e con ibu ions om s a s, gas,
and da k ma e (in he s anda d in e p e a ion).
We i SPARC galaxies using:
2
model( ) = 2
disk( )+ 2
gas( )
+η·Ξ( )· 2
disk( )+ 2
gas( )(31)
Resul s o a ep esen a i e sample [36]:
Galaxy η χ2/do
NGC 2403 1.8 ±0.2 1.2
NGC 3198 2.1 ±0.3 0.9
DDO 154 2.4 ±0.4 1.1
Table 1: SPARC i s. Full sample: η= 2.0±0.3, χ2/do = 1.0.
The single pa ame e η∼2 explains o a ion cu es ac oss he ull galaxy mass ange om
107M⊙(dwa s) o 1011M⊙(spi als). No in isible ma e is needed.
4.5 Tully-Fishe Rela ion
An independen success is he ba yonic Tully-Fishe ela ion [13], an empi ical powe law:
Mba y ∝ α
la (32)
whe e la is he la o a ion eloci y and α≈4.
In ou amewo k, his eme ges na u ally. Fo a disk wi h scale leng h 0and o al mass Mba y:
2
la ∼GMba y
0
(1+η) (33)
8
I 0∝M1/2
ba y (as obse ed o disk galaxies), hen:
4
la ∝M2
ba y ⇒Mba y ∝ 4
la (34)
The Tully-Fishe slope α= 4 is p edic ed, no i . This is a pa ame e - ee success.
4.6 Radial Accele a ion Rela ion
McGaugh e al. (2016) [14] disco e ed a igh co ela ion be ween obse ed accele a ion gobs and
ba yonic accele a ion gba y in SPARC galaxies:
gobs =gba y "1 + gba y
g†−1#(35)
wi h cha ac e is ic scale g†= 1.2×10−10 m/s2and sca e ∼0.1 dex.
Ou Eq. (29) na u ally p oduces such a ela ion. The ansi ion occu s whe e me (ρ) ∼1,
co esponding o a cha ac e is ic accele a ion scale. The p ecise o m and sca e equi e de ailed
modeling o ρ( ) p o iles, bu he exis ence o a uni e sal ela ion is a p edic ion, no a i .
5 Clus e Collisions
5.1 Bulle Clus e Challenge
The Bulle Clus e (1E 0657-56) has been called he “smoking gun” o da k ma e [15]. This
me ging clus e sys em shows:
1. X- ay gas (ba yons) concen a ed a he collision in e ace
2. Weak lensing mass (“da k ma e ”) cen e ed on he galaxies, ahead o he gas
3. Spa ial o se o ∼720 kpc be ween ba yons and lensing peaks
The s anda d in e p e a ion: collisionless da k ma e pa icles pass h ough unimpeded, while
gas unde goes am p essu e and lags behind. Modi ied g a i y heo ies like MOND s uggle o
explain his sepa a ion [16, 39].
5.2 The modynamic Explana ion: Di e en ial Response
In ou amewo k, he key is ha he scala ield ψis sou ced by ma e densi y ρ, no mass alone.
Gas and galaxies ha e e y di e en densi y p o iles:
In aclus e gas:
•Densi y: ρgas ∼10−27 g/cm3(ho , di use plasma)
•Tempe a u e: T∼107K
•E ec i e mass: me ,gas ∼10−29 eV
•Sc eening leng h: λsc ∼100 kpc
Galaxy in e io s:
•Densi y: ρgal ∼10−21 g/cm3(s a s + ISM)
9
•Known in e ac ion: Con o mal coupling (s anda d in beyond-SM physics)
The choice is be ween:
1. In isible pa icle ne e de ec ed, wi h ine- uned p ope ies
2. The modynamic sc eening o a scala ield
We a gue (2) is mo e pa simonious.
8.2 Compa ison o MOND
Modi ied New onian Dynamics [3] p oposes an accele a ion scale a0∼10−10 m/s2below which
g a i y de ia es om 1/ 2. The in e pola ion unc ion µ(a/a0) is phenomenological.
Ou amewo k has se e al ad an ages:
1. De i a ion: MOND’s µ unc ion is i o da a. Ou enhancemen ac o eme ges om
he modynamic i s p inciples.
2. Sola Sys em: MOND s uggles wi h igh Sola Sys em cons ain s. We na u ally pass ia
hin-shell sc eening.
3. Clus e s: MOND equi es ∼2×neu ino mass o o he adjus men s o clus e s. We explain
clus e dynamics and collisions wi hou addi ions [39, 40, 4].
4. Cosmology: MOND has di icul ies wi h CMB and la ge-scale s uc u e. Ou γpa ame e
allows consis en cosmological e olu ion.
MOND’s key success (Tully-Fishe ela ion, RAR) a e also explained by ou heo y, plus addi-
ional p edic ions (en i onmen dependence, ime e olu ion) ha MOND lacks.
8.3 Compa ison o (R)G a i y
(R) heo ies eplace he Eins ein-Hilbe ac ion RRwi h R (R) o some unc ion . This modi ies
ield equa ions a he le el o GR i sel [32].
Ou app oach is mo e conse a i e:
•We keep Eins ein g a i y: SEH =RM2
PlR/2
•We add a scala sec o : Sψ=R[(∂ψ)2/2+V(ψ)]
•Coupling is h ough ma e ac ion: Sm[A2(ψ)gµν]
This is a scala - enso heo y, well-s udied and heo e ically cleane han (R). Mo eo e :
1. (R) o en has ins abili ies o acausali y [26, 42]
2. Ou chameleon mechanism au oma ically ensu es s abili y
3. Obse a ional cons ain s on (R) a e o en igh e due o cosmological issues
16
8.4 Why Mul iplica ion, No Addi ion
The s uc u e o Eq. (1) is c ucial:
F=FNew on ×[1 + he modynamic ac o ] (51)
This is no F=FNew on +Fex a. Mul iplica i e modi ica ion is how physics ac ually wo ks:
•Special ela i i y: E=mc2γwhe e γ= 1/p1− 2/c2
•Quan um elec odynamics: α(q2)=α0[1+α0ln(q2/m2
e)+···]
•Elec omagne ic wa es in media: E=E0/√ϵ
Addi i e modi ica ions (“New on + new o ce”) sound like epicycles. Mul iplica i e modi ica-
ions (“New on ×medium esponse”) a e how undamen al physics ex ends o new egimes.
8.5 The Role o The modynamics
The modynamics is o en iewed as an eme gen , s a is ical desc ip ion applicable only o la ge
sys ems. Why should i a ec undamen al o ces?
Recen wo k sugges s deepe connec ions:
•G a i y as he modynamics [27, 28, 34]
•En anglemen en opy in QFT ela es o he modynamic en opy
•Black hole he modynamics shows g a i a ional sys ems ha e in insic empe a u e and en-
opy
Ou amewo k adds o his: g a i a ional ield con igu a ions a e de e mined by ee ene gy
minimiza ion, no jus po en ial ene gy. This is a gene al p inciple ha should apply o any ield
coupled o ma e .
The success o chameleon sc eening in passing Sola Sys em es s while explaining da k ma e
phenomena sugges s he modynamics is no jus “s a is ical app oxima ion” bu a undamen al
o ganizing p inciple.
9 Conclusion
We ha e demons a ed ha g a i a ional dynamics ac oss all scales can be desc ibed by a single
equa ion:
F( , ρ) = GM1M2
2h1+2β2e−me (ρ) i(52)
whe e he e ec i e mass me (ρ) eme ges om minimizing he ee ene gy o a scala ield coupled
o ma e . This syn hesis o New on’s 300-yea -old law wi h 150 yea s o s a is ical mechanics
explains:
1. Sola Sys em p ecision: Thin-shell sc eening yields |γPPN −1|<10−16, bea ing Cassini
by 16 o de s o magni ude.
2. Galaxy o a ion cu es: Single pa ame e η∼2 explains SPARC da a om dwa s o
spi als; Tully-Fishe and RAR eme ge na u ally.
17
3. Clus e collisions: Di e en ial scala coupling p oduces gas-lensing o se s in Bulle Clus e
and simila sys ems.
4. Cosmological e olu ion: Time- a ying me (a)∝a−γwi h γ= 3(n+ 2)/[2(n+ 1)] con-
s ains po en ial choice and add esses JWST high-zanomalies.
5. Falsi iable p edic ions: En i onmen -dependen lensing, ou -o -equilib ium e ec s, em-
pe a u e dependence, and cons ained γ ange.
The key insigh is ha sc eening is no an ad hoc ix bu an ine i able consequence o he -
modynamics. When scala ields couple o ma e , he coupled sys em minimizes ee ene gy
Ve =V(ψ) + ρln A(ψ), p oducing densi y-dependen ield con igu a ions. In high-densi y egions
(Sola Sys em), he modynamics demands s ong sc eening; in low-densi y egions (galaxy halos),
he modynamics ac i a es he ield. The ma hema ics is he same as phase ansi ions in condensed
ma e .
This amewo k makes no e e ence o da k ma e pa icles. The g a i a ional anomalies
a ibu ed o da k ma e a e ins ead mani es a ions o he modynamic enhancemen in di use
en i onmen s. A e 40 yea s wi hou pa icle de ec ion, his al e na i e dese es se ious consid-
e a ion. The null esul s o WIMP/axion sea ches now a o he modynamic o igins o e pa icle
da k ma e [35].
Fu u e obse a ions will es ou p edic ions:
•Euclid/LSST: En i onmen -dependen lensing in oids s. clus e s
•CMB-S4/DESI: P ecision γmeasu emen o ∆γ∼0.05
•JWST/Roman: High-zgalaxy e olu ion consis en wi h enhanced ea ly g a i y
•Chand a legacy su eys: Mul i-clus e gas-lensing o se s a is ics
I γ alls ou side he p edic ed ange [1.5,2.25] o i en i onmen -dependen lensing is no
obse ed, he heo y is alsi ied.
We conclude wi h a philosophical poin . Physics has wo g ea uni ie s: geome y (which
ga e us ela i i y) and he modynamics (which ga e us s a is ical mechanics). This wo k sugges s
hey uni e again: g a i y is geome y, bu i s s eng h is he modynamics. The equa ion F=
(New on) ×[1 + he modynamics] encapsula es his syn hesis.
The da k ma e puzzle may be esol ed no by disco e ing new pa icles bu by ecognizing
ha he modynamic p inciples apply o g a i y jus as hey do o e e y o he ield in na u e.
Acknowledgmen s
This wo k builds on he comp ehensi e amewo k de eloped in Re . [29].
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