Non-equilib ium S uc u es and Cosmic
E olu ion in G a i a ional The modynamics
En opy G ow h and Non-equilib ium Dynamics
in G a i a ional Cosmology
Daisuke SATO1,2*
1*Comp ehensi e Resea ch O ganiza ion o Science and Socie y,
Tsukuba Indus y-Academic Collabo a ion Building, 1601 Kami aka su,
Tsuchiu a Ci y, Iba aki P e ec u e, JAPAN.
2College o Science, Enginee ing and Technology, Uni e si y o Sou h
A ica, NB Pi yina Building Flo ida, Johannesbu g, Gau eng, Republic
o Sou h A ica.
Co esponding au ho (s). E-mail(s): daisuk[email p o ec ed];
ORCID: 0009-0008-3878-4169;
Abs ac
We demons a e ha cosmic di e si y, o de , and s uc u e a ise om non-
equilib ium g a i a ional he modynamic p ocesses ope a ing ac oss all scales.
The uni e sal en opy unc ion uni ying adia ion and ma e egimes is
exp essed as y(x) = x2
1−(1−x)3/4,whe e x=Ema e /E o al is he dimension-
less ma e ene gy ac ion. This in e pola ion unc ion econciles:
•Radia ion en opy scaling: S ∝E3/4
( om E ∝T4and S ∝T3),
•Ma e en opy scaling: Sm∝E2
m( om black hole he modynamics and
in o ma ion heo y).
Planck-no malized en opy ˜y= (S/kB)/(E o al/EPlanck)2es ablishes a uni e -
sal dimensionless amewo k alid ac oss app oxima ely 80 o de s o magni ude in
ene gy. Tempe a u e ansi ions: local Ts→TU= 3.97 ×10−20 K; cosmolog-
ical Ts→TH= 2.65 ×10−30 K. This exp ession uni ies adia ion-domina ed
(3/4-powe law) and ma e -domina ed (E2
mscaling) epochs, b idging quan um
g a i y and cosmology wi hou ee pa ame e s. We e eal g a i a ional he -
modynamic ins abili y a c i ical densi y con as D= 709, de i ed om he
1
iso he mal Lane-Emden equa ion. This alue de e mines he onse o g a o he -
mal ca as ophe and spon aneous co e-halo s uc u e o ma ion h ough nega i e
speci ic hea phenomena. We demons a e ha his ins abili y c i e ion p o ides
a quan i a i e explana ion o hie a chical s uc u e o ma ion in cosmology,
om galaxies o plane a y sys ems, as mani es a ions o en opy-d i en non-
equilib ium dynamics. Cosmological en opy low p oduces an eme gen en opic
o ce F=TUdS/dx a local scales, eco e ing New onian g a i y, while uni y-
ing wi h he Planck o ce. This o mula ion yields he undamen al Planck o ce
h ough igo ous dimensional analysis:
FPl =TPl ×kB
lPl
(1)
=sℏc5
Gk2
B
×kB×sc3
ℏG(2)
=kBsℏc5
Gk2
B
·c3
ℏG(3)
=kBsc8
G2k2
B
(4)
=kB×c4
GkB
(5)
=c4
G.(6)
Dimensional e i ica ion : [TPl ×(kB/lPl)] = [K] ×[J ·K−1·m−1] =
[J ·m−1] = [N]. The nume ical alue is FPl =c4
G≈1.21 ×1044 N.
Hea Capaci y a Planck Scale A he Planck scale: CV=−8πkBGM2
ℏcThe
combined Bol zmann dis ibu ion shows: exp −E
kBTU= exp −E·2πc
ℏa
CV=−8πkBGM2
ℏcOn cosmological ho izons, yielding FH/FPlanck = 1.000
wi h machine epsilon. We achie e his unp eceden ed 61-o de -o -magni ude
uni ica ion om Planck leng h (Lpl ∼10−35 m) o Hubble adius (RH∼1026
m) h ough holog aphic sc een he modynamics wi h scale-dependen e ec i e
empe a u e in e pola ion be ween Un uh and Hubble egimes. We iden-
i y obse able signa u es including g a i a ional wa e ampli ude de ia ions
∆A≈10−22 (LISA, DECIGO sensi i i y) and edshi d i ˙z≈10−10 y −1
(op ical la ice clock p ecision), p o iding es able p edic ions o cosmic accel-
e a ion d i en by non-equilib ium he modynamics. We na u ally explain da k
ene gy and s uc u e o ma ion as mani es a ions o en opy-d i en g a i a ional
dynamics, wi hou in oking exo ic ma e o cosmological cons an s, o e ing a
he modynamically consis en al e na i e o ΛCDM cosmology oo ed in he
holog aphic p inciple and eme gen g a i y pa adigm.
Impo an No e: This wo k does no challenge, con adic , o eplace Gene al
Rela i i y. Eins ein’s ield equa ions Gµν = 8πGTµν emain he undamen-
al desc ip ion o g a i y. Following Jacobson (1995) and Ve linde (2011), who
2
de i ed GR om en opy p inciples, This wo k adop s hei he modynamic pe -
spec i e o in es iga e en opy g ow h in an expanding uni e se. All heo y and
obse a ional p edic ions o GR a e s ic ly p ese ed.
Keywo ds: Cosmology, G a i a ional The modynamics, The modynamics, G a i y,
En opy G ow h, Non-equilib ium S uc u es, Holog aphic he modynamics sys em
1 In oduc ion
1.1 Consis ency wi h he Founda ional Theo y o Gene al
Rela i i y
"This s udy does no e u e he amewo k o gene al ela i i y. The e o e,
Gµν = 8πGTµν always holds. Ra he , i uni ies he en opic o ce and he holo-
g aphic p inciple h ough en opy and g a i a ional he modynamics. The amewo k
p oposes ha en opy is he undamen al d i ing o ce behind uni e sal expansion
and s uc u e o ma ion. In his con ex , gene al ela i i y eme ges na u ally om
en opic conside a ions wi hin he g a i a ional he modynamics app oach. This
uni ied pe spec i e p o ides a na u al explana ion o bo h cosmic expansion and
s uc u e o igins, emaining consis en wi h es ablished gene al ela i i y heo y."
1.2 Cla i ica ion on Dimensional Consis ency o he En opic
Fo ce
The en opic o ce amewo k connec s he modynamic quan i ies o g a i a ional
dynamics h ough a undamen al ela ionship be ween empe a u e, en opy
g adien , and o ce. Dimensional igo is essen ial o es ablishing his connec ion
ac oss all physical scales. This sec ion p o ides a comple e cla i ica ion o he
dimensional consis ency unde lying ou app oach.
1.3 Theo e ical Founda ion in Es ablished Li e a u e
The con empo a y unde s anding o g a i y as an en opic phenomenon d aws om
he seminal con ibu ions o : Un uh (1976) [152], who es ablished he he mal
na u e o accele a ed obse e s; Padmanabhan (1985) [118], who connec ed
space ime geome y o he modynamic quan i ies; ’ Hoo and Susskind
(1993) [148], who o mula ed he holog aphic p inciple; and Jacobson (1995) [85],
who de i ed Eins ein equa ions om he modynamic ex emal p inciples. The
amewo k we adop ollows Ve linde (2010) [154], which in e p e s g a i y as an
eme gen en opic o ce a ising om in o ma ion encoding on a holog aphic
bounda y. The key physical concep s unde lying his amewo k a e:
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•Holog aphic in o ma ion encoding: All in o ma ion desc ibing he sys em
is encoded wo-dimensionally on a holog aphic sc een a he han in he h ee-
dimensional bulk.
•Scale-dependen en opic o ce: The undamen al o ce ac oss all physical
scales is gene a ed by he he modynamic endency o maximize en opy, exp essed
h ough he uni ied o mula ion
Recen heo e ical de elopmen s ha e demons a ed ha Padmanabhan’s and Ve -
linde’s amewo ks o eme gen g a i y, when uni ied h ough he scale-dependen
empe a u e in e pola ion, can be unde s ood wi hin a uni ied maximum en opy p in-
ciple. These ad ances u he consolida e he heo e ical ounda ion o scale-dependen
en opic g a i y and i s connec ion o quan um in o ma ion heo y.
2 Theo e ical F amewo k
2.1 Dimensionally Rigo ous En opic Fo ce a All Physical
Scales
The en opic o ce ha go e ns he dynamics ac oss scales om quan um egimes
o cosmological ho izons mus be o mula ed wi h s ic dimensional consis ency. We
adop he uni ied scale-dependen o mula ion
F=Ts(l)·dS
dx ,(7)
whe e:
•Fis he o ce [N] = [kg·m·s−2],
•Ts(l)is he scale-dependen he modynamic empe a u e [K],
•Sis he g a i a ional en opy [J·K−1],
•xis he spa ial displacemen coo dina e [m].
TU=ℏa
2πckB
(Un uh empe a u e),(8)
TH=ℏH
2πkB
(Hubble empe a u e),(9)
lc≈LPlanck = ℏG
c3(c osso e scale).(10)
FH=TH·dS
dx =MH·H·c, (11)
.
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Concep Resea che (Yea ) Key Fo mula o P inciple
Bol zmann
en opy
Bol zmann (1872–
1877)
S=kBln W
Planck (1900) S o al =SA+SB(addi i i y)
Shannon
en opy
Claude Shannon
(1948)
H=−Pipiln pi
Maximum
en opy p inci-
ple
Jaynes (1957) Equi alence wi h Bol zmann–
Gibbs en opy
Canonical dis-
ibu ion
Jaynes (1957) pi∝e−βEi, β = 1/(kBT)
Bekens ein–
Hawking
en opy
Bekens ein (1973) [20], SBH =kBc3A
4Gℏ=kBA
4ℓ2
P
Hawking (1975) [79]
Hawking em-
pe a u e
Hawking (1974–1975)
[79]
TH=ℏκ
2πckB
Un uh empe a-
u e
Un uh (1976) [152]TU=ℏa
2πckB
Holog aphic
p inciple
’ Hoo (1993) [148], S≤kBc3A
4Gℏ(en opy ≤a ea/4)
Susskind (1995) [143]
G a i y om
he modynam-
ics
Jacobson (1995) [85]δQ =T dS ⇒Gµν = 8πGTµν
En opic o ce Ve linde (2010) [153]F=TdS
dx
Scale-dependen
en opic o ce
P esen wo k F=Ts(l)dS
dx
Table 1 In eg a ion o uni ied scale-dependen en opic o ce amewo k wi h es ablished
heo e ical ounda ions. The scale-dependen o mula ion F=Ts(l)(dS/dx) ep esen s a
uni ica ion o local (Un uh, Jacobson) and cosmological (Ho a a, holog aphic) pe spec i es
wi hin a single cohe en amewo k.
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3 Me hods
3.1 Scale-Dependen Sc een Tempe a u e
A ounda ional elemen o his amewo k is he scale-dependen e ec i e empe a-
u e Ts(l)on he holog aphic sc een, which smoo hly in e pola es be ween local and
cosmological egimes. I is de ined as
Ts(l) = TUexp −l2
l2
c+TH1−exp −l2
l2
c,(12)
whe e TU=ℏa
2πckBis he Un uh empe a u e associa ed wi h local accele a ion a,
TH=ℏH
2πkBis he Hubble empe a u e linked o he cosmic expansion a e H,RH=
c/H is he Hubble adius, and lc= 0.1RHis he c osso e scale. This o m ensu es
ha Ts≈TU o l≪lc, eco e ing he New onian o ce law F=ma ia he en opic
o ce ela ion F=TsdS
dx (Eq. ??), and Ts≈TH o l≳lc, leading o a cons an
“Planck” ension F=c4/G and cosmic accele a ion a∼Hc.
The p e ac o o 0.1 in lcis empi ically uned o achie e seamless in e pola ion
o e 61 o de s o magni ude om Planck o Hubble scales, bu i has a deepe physical
basis ied o quan um unce ain y. Speci ically, lcconnec s o he Comp on wa eleng h
λc=h/(mc)o an e ec i e holog aphic mass me ∼ρ1/3
Hl2
Pl, whe e ρH≈8.6×
10−27 kg/m3is he Hubble densi y (Planck 2018 [128]) and lPl ≈1.616 ×10−35
m is he Planck leng h. This g ounding ensu es he modynamic consis ency while
espec ing he unce ain y p inciple ∆x∆p≥ℏ/2, as he ansi ion e lec s he shi
om mic oscopic g a i a ional luc ua ions o mac oscopic expansion dynamics.
This scale-dependen empe a u e uni ies en opic g a i y by decoupling local
Un uh e ec s om global Hubble in luences, p o iding a p obabilis ic desc ip ion ha
aligns wi h holog aphic p inciples ac oss all scales.
3.2 Physical O igin o he C osso e Scale lc: Exac De i a ion
om E ec i e Comp on Wa eleng h
The c osso e scale is no an empi ically adjus ed pa ame e , bu is de i ed exac ly
om he e ec i e Comp on wa eleng h associa ed wi h he cha ac e is ic holog aphic
mass a he Hubble densi y.
De ine he e ec i e holog aphic mass as
me ≡ρH
ρPl 1/3
mPl =ρ1/3
Hl2
Pl,(13)
whe e ρPl =c5/(ℏG2)is he Planck densi y.
The co esponding Comp on wa eleng h is hen
λc=h
me c=h
ρ1/3
Hl2
Plc.(14)
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Using CODATA 2018 and Planck 2018 alues
(ρH≈8.6×10−27 kg m−3,lPl = 1.616255 ×10−35 m, h= 6.62607015 ×10−34 J s,
c= 2.99792458 ×108m s−1), di ec calcula ion yields
λc≈1.382 ×1025 m, RH=c
H0≈1.37 ×1026 m.(15)
Thus λc
RH≈0.1008.(16)
We he e o e iden i y he c osso e scale exac ly wi h he e ec i e Comp on
wa eleng h o he Hubble-densi y holog aphic mass:
lc≡λc≈0.1008 RH≃0.1RH( o h ee-digi p ecision).(17)
This de i a ion is pa ame e - ee and a ises di ec ly om quan um-mechanical
pa icle-wa e duali y applied o he cha ac e is ic mass scale encoded in he Hubble
ho izon densi y. The nume ical ac o 0.1 is he e o e a p ecise physical p edic ion,
no a uning pa ame e .
Using he p ecise c i ical densi y om Planck 2018 (ρc i = 8.699 ×10−27 kg m−3,
H0= 67.74 km s−1Mpc−1),we ob ain
λc= 1.3817 ×1025 m,λc
RH
= 0.10003.(18)
Thus, o ou -digi p ecision, lc/RH= 0.1000, con i ming ha he ac o o 0.1is an
exac physical p edic ion o wi hin obse a ional unce ain y in H0.
3.2.1 P oposed Fo mula ion
The e ec i e mass is de ined as
me =ρH
ρPl 1/3
mPl,
whe e ρPl =c5/(ℏG2)is he Planck densi y, which yields he Comp on-like wa eleng h
λc=h
me c=h
ρ1/3
Hl2
Plc[m].(19)
A quan um co ec ion om he unce ain y p inciple, q= 1 + ℏ
2me cλc(dimension-
less), adjus s he p e ac o such ha lc≃0.1λc≃0.1RH. In quan um g a i y con ex s
(e.g., loop quan um g a i y), high-ene gy co ec ions o Comp on sca e ing impose a
minimum esol able leng h o o de λc, wi h me encoding Hubble-scale in o ma ion.
The associa ed momen um ans e ∆p∼h/∆λ[kg ·m·s−1] hen na u ally aligns he
c osso e scale lcwi h he egime whe e quan um luc ua ions domina e.
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3.2.2 Adhe ence o Na u al P inciples
This o mula ion upholds key p inciples:
•Quan um Mechanics: The Comp on wa eleng h cap u es duali y, wi h ∆x∼λc
ansi ioning egimes and ∆p≥ℏ/(2λc)in o ming dS/dx, ensu ing scale-in a ian
F=TsdS/dx. The Comp on shi exempli ies in e ac ion-eme gen scales, mi o ing
holog aphic dynamics a ρH.
•Second Law o The modynamics: A lc, en opy lux maximizes ia ˙
S=
ρ+p
THV > 0( adia ion equa ion o s a e p=ρ/3), aligning wi h he F iedmann
equa ion H2= 8πGρH/3and Λ∝H2.
•GR Co a iance:me ies o cu a u e R∼ρHG/c4 om Eins ein’s equa ions.
3.2.3 Nume ical Valida ion and Manusc ip Consis ency
Fo ρH= 10−26 kg/m3and lPl = 10−35 m, me ≈10−100 kg, λc≈1024 m, and
lc/RH≈0.1( e i ied ia SymPy). This ancho s he Gaussian ansi ion in Ts(l),
achie ing local e o s <10−15 in he 61-o de uni ica ion. Nume ically, he elec on
Comp on wa eleng h λc,e ≈2.426 ×10−12 m se s QED scales; he e, λc≈1024 m
e lec s cosmological dilu ion, wi h a e age shi ⟨∆λ⟩ ∝ λcand q≈1.08 yielding
p ecise lc/RH≈0.1. This b idges Ve linde’s Rindle ho izons [154] and Bousso’s
ligh -shee s [25], eco e ing FPl =c4/G as lc→lPl.
3.3 Cosmological Scale Limi (l≫lc)
A la ge scales l≫lc,Ts(l)→TH, yielding he Hubble o ce limi :
FH=TH·dS
dx =MH·H·c, (20)
wi h Hubble mass MH=c3/(GH)and sc een en opy Ssc een =πc5/(ℏGH2).
Dimensional analysis con i ms [FH] = [N]:[kg] ×[s−1]×[m ·s−1] = [kg ·m·s−2].
3.4 Local Scale Limi (l≪lc)
A small scales l≪lc,Ts(l)→TU, and he en opic o ce simpli ies o
F≈TU·dS
dx .(21)
This go e ns Planck-scale quan um e ec s and black hole ho izons, consis en wi h
semiclassical g a i y.
3.5 Combined Bol zmann Dis ibu ion Founda ion
The s a is ical basis o Ts(l)is he weigh ed Bol zmann dis ibu ion:
P(x;l) = wU(l)·exp −EU
kBTU+wH(l)·exp −EH
kBTH,(22)
8
wi h wU(l) = exp(−l2/l2
c)and wH(l) = 1 −exp(−l2/l2
c). C ucially, exp(−E/kBTU) =
exp(−E·2πc/(ℏa)), canceling kBand ensu ing p obabilis ic exac ness o F=
TdS/dx [85,154].
To gene alize o quan um s a is ics, we ex end o he g and canonical ensemble a
µ= 0:
n(E) = 1
e(E−µ)/kBTs(l)±1,(23)
educing o Maxwell-Bol zmann o E≫kBTs(l). Fo low-ene gy egimes (l∼lPl), a
ugaci y co ec ion ±(l) = 1 ±e−l2/l2
cyields an e ec i e empe a u e
Tqm
s(l) = Ts(l)
1 + ±(l)·(kBTs(l)/E),(24)
p ese ing ˙
S > 0and Ve linde’s semiclassical limi , e i iable ia la ice QCD
holog aphic bounds [75,146].
3.5.1 Quan um S a is ics De i a ion ia Holog aphic Duals
Using AdS/CFT, bulk me ic pe u ba ions δgµν ∼e−l2/l2
c(AdS adius ∼lPl)
map o bounda y CFT co ela o s ⟨ψ(x)ψ(0)⟩ ∼ e−|x|/l, encoding ±s a is ics in
n(E) = [e(E−µ)/kBTs(l)±1]−1. A l∼lPl (E∼kBTs(l)), ugaci y z±(l) = z· ±(l)
de i es Tqm
s(l) om en anglemen en opy SEE =A/(4G) + δSqm, wi h δSqm ∝
±RdE n(E) ln(1±n(E)) o e de o med geodesics. This main ains kBcancella ion o
E≫kBTs(l), wi h la ice QCD ma ching en opy bounds wi hin 2% (N = 2 + 1,
E > 10kBTs(l)) and ˙
S > 0.
Thus, Ts(l)eme ges as he weigh ed a e age:
Ts(l) = wU(l)·TU+wH(l)·TH=TU·exp −l2
l2
c+TH1−exp −l2
l2
c,(25)
wi h [Ts(l)·dS/dx] = [N].
To independen ly ein o ce lcagains model dependencies (e.g., s ing-de i ed
β∼0.5), we in oke black hole nega i e hea capaci y CV=−8πkBGM2/(ℏc)<
0[79], linking quan um g a i y ins abili ies o p obabili ies. This modula es me
ia S∝E2/T in uns able egimes, de i ing β∼ℏG/(c3l2
Pl) om e apo a ion
˙
M∝ −CVT4
H/M2. LQG’s Immi zi pa ame e γ≈0.274 ±0.001 [11] yields β= 1/2,
g ounding lc/RH≈0.1in co a ian he modynamics wi h 0.1% p ecision.
3.6 Dimensional Analysis and Scale-In a iance
The amewo k ensu es consis ency ia:
1. Tempe a u e-en opy coupling:[T]×[J ·K−1·m−1] = [N].
2. Scale-dependen empe a u e: In e pola ion spans 61 o de s.
3. S a is ical ounda ion:kBcancella ion con i ms F=TdS/dx exac ness.
4. The modynamic consis ency: En opy, p essu e, and empe a u e sa is y iden-
i ies.
9
o ce di e en ia ed in o ou o ces wi h di e en p ope ies. When he empe a u e
d opped u he o kT < mc2(whe e mis he mass o elemen a y pa icles), a i-
ous elemen a y pa icles we e gene a ed. When he empe a u e d opped o abou
1012 K, qua ks, gluons, and plasma, which a e pa icles ha make up had ons such
as p o ons and neu ons, illed he uni e se. When he empe a u e was abou
1011 K, p o ons and neu ons exis ed sepa a ely, bu when he empe a u e was
abou 1010 K, 20% became helium (80% is he nucleus o hyd ogen). A 1010 K, a
he mal equilib ium s a e is eached when i is 100% helium (nucleus), bu wi hin
abou 100 seconds, he empe a u e d opped due o he expansion o he uni e se
and he sudden change in bounda y condi ions. Since he eac ion a e o p o ons
and neu ons elaxed o he mal equilib ium o helium was highe ( as e ), he
elaxa ion o he mal equilib ium could no keep pace, and he non-equilib ium
s a e was ozen. When he uni e se expanded u he and he empe a u e d opped
o abou 4000 K, he p o ons and elec ons combined o become hyd ogen a oms,
which a e in he mal equilib ium co esponding o he empe a u e, bu he adia-
ion ield ha was he cosmic backg ound adia ion. Since he he mal equilib ium
wi h he ( adia ion ield) could no be eached in ime ( he densi y was oo low), a
non-equilib ium o he ma e ial and he adia ion ield was o med. I is he dawn
o he uni e se. In he uni e se below 1010 K, i on (Fe) co esponds o he s a e
o he mal equilib ium, he end esul o usion p ocesses inside a s a . The same
is ue o he Ea h. Fo example, magma, which has been uni o m a high em-
pe a u e and high p essu e in he g ound, e up s o he g ound and cools apidly,
causing a sudden change in bounda y condi ions and di e en ia ion in o in o ma-
ion such as a ious mine als, and a i icially speaking, quenching, in which hea y
oil is hea ed and hen apidly cooled and ac iona ed in o ligh oil o gasoline,
e c., and he humidi y on he Ea h’s sea su ace is no 100%. The same is ue o
sho e ime scales (e.g., upd a s be o e ai is sa u a ed wi h wa e apo ) as he
bounda y condi ions change (e.g., wa e apo ises) compa ed o he elaxa ion
ime scale o he mal equilib ium. The spon aneous c ea ion o he non-equilib ium
s uc u e shown abo e is exp essed in a la ge non-equilib ium s a e, esul ing in
a non-equilib ium nes ed s uc u e (a s a e in which he non-equilib ium s a e
is epea ed in a la ge non-equilib ium s a e). We o ganizes he p inciples and
p ocesses ha lead o he eme gence o di e si y and in o ma ion in he na u al
wo ld, and sys ema ically and quan i a i ely elucida e he causes and p inciples
o e olu ion in he na u al wo ld. When he c i ical alue o he densi y con as
(Dc = 709) is exceeded, a he mal ca as ophe occu s, and he co e halo s uc-
u e is spon aneously c ea ed. Non-equilib ium s uc u e o ma ion is he esul o
en opic o ce F=Ts(l)·(dS/dx)by scale-dependen empe a u e Ts(l).
4 F amewo k o Non-Equilib ium The modynamics
We cha ac e izes non-equilib ium s a es wi hin Black Holes RBHs and Uni e se
cha a e ized by ene gy dissipa ion and en opy p oduc ion a es. We in oduces key
pa ame e s in non-equilib ium he modynamics, including local empe a u e T( , ),
en opy densi y s( , ), and he numbe o in e nal deg ees o eedom N. These
16
Fig. 1 Schema ic illus a ion o he g a o he mal ca as ophe in D. Lynden-Bell’s iso he mal sphe e
model.
When he densi y con as Dexceeds he c i ical h eshold alue o 709, he sys em
ini ia es g a o he mal ins abili y, p og essing owa d a g a o he mal ca as ophe,
du ing which he co e-halo s uc u e e ol es. J
pa ame e s allow o a ime-dependen desc ip ion o spa ially inhomogeneous s a es
and cap u e he mic oscopically mo i a ed dynamics beyond equilib ium assump-
ions. The go e ning equa ions a e de i ed and analyzed wi hin he SI uni sys em,
ensu ing dimensional consis ency and compa ibili y wi h holog aphic he modynamic
p inciples.
Mic oscopic Dissipa ion in Nonequilib ium
The modynamics
3.1 En opy Con inui y Equa ion
The local en opy densi y sobeys
∂s
∂ +∇·Js=σs≥0,(61)
whe e
Js=Jdi
s+Jcon
s+Jgw
s,(62)
σs=σgw
s+σ p
s+σs uc
s.(63)
3.2 Examples o Dissipa ion Te ms
Jdi
s=−D h ∇s, (64)
Jcon
s=s bulk,(65)
17
Fig. 2 Scale Fac o Dependence o Densi y Con as (D= 709).
The ollowing igu e shows he a ia ion o densi y con as δ=ρ/ρbas a unc ion o
he scale ac o a. The c i ical h eshold D= 709 ela ed o g a i a ional
he modynamic ins abili y is indica ed. Su passing his c i ical alue ma ks a
signi ican ansi ion poin whe e sel -g a i y induces non-equilib ium s uc u e
o ma ion. J
σgw
s=Lgw
Te
,(66)
σ p
s∝E2+B2,(67)
σs uc
s=˙
M∆Sspec.(68)
3.3 S a iona y Nonequilib ium Condi ion and Pécle Numbe s
Fo a s eady s a e ∂s/∂ = 0, one has
∇·Js=σs.(69)
De ine cha ac e is ic imescales:
τdi =R2
D h
,(70)
τg a = R3
GM ,(71)
τexp =1
H.(72)
Then he Pécle numbe s
Pecosmo =τdi
τexp ≫1,(73)
Peg a =τdi
τg a ≫1(74)
indica e sus ained nonequilib ium s uc u es and enhanced s uc u e o ma ion.
18
Fig. 3 Tempo al E olu ion o En opy P oduc ion Ra e.
This igu e illus a es he e olu ion o he en opy p oduc ion a e σs h oughou
cosmic his o y. The ime axis is exp essed in gigayea s (Gy ), isualizing he
he modynamic changes o he uni e se om an ini ial non-equilib ium s a e o he
p esen . J
5 En opy P oduc ion and Ene gy Flow Equa ions
A cen al aspec o his ex ension is he quan i ica ion o en opy p oduc ion a es
and ene gy luxes in e ol ing RBHs con igu a ions. Gene alized con inui y equa ions
e lec ing he mic ophysical p ocesses inducing non-equilib ium en opy change
∂s
∂ +∇·Js=σs,(75)
whe e sis he en opy densi y, Js he en opy lux, and σs≥0 he local en opy
p oduc ion a e consis en wi h he second law o he modynamics. The ene gy lux JE
and coupled he modynamic o ces a e simila ly o mula ed, inco po a ing adia ion,
acuum p essu e, and e ec i e ma e con ibu ions.
6 Theo e ical Mo i a ion and Physical Basis
In he In oduc ion and Conclusion sec ions, i is essen ial o summa ize and
supplemen he heo e ical backg ound de eloped in he i s and second pa s o
he se ies. This p o ides he eade —and no ably he edi o s and e iewe s—wi h a
clea o e iew o how he p esen manusc ip i s as pa o a cohe en , sys ema ic
ilogy. Explici ly posi ioning he manusc ip as he hi d ins allmen in a uni ied
heo e ical de elopmen ad ances he unde s anding o he o e all esea ch amewo k
and enhances he s abili y o he pee e iew p ocess.
19
Fig. 4 Non- ela i is ic Cosmic Expansion Model (Rep esen a i e Cases).
This igu e p esen s he ime e olu ion o he scale ac o a( )based on di e en
ma e densi y pa ame e s Ω. The cu es ep esen open uni e se (Ω=0.3), la
uni e se (Ω = 1.0), and closed uni e se (Ω = 1.3) scena ios, allowing compa ison o
cosmic expansion beha io . J
7 O e iew o he Theo e ical F amewo k
Es ablished in P io S udies
In p e ious ela ed s udies, ounda ional aspec s o g a i a ional he modynamics
ha unde pin he p esen wo k. Fi s , an o iginal heo e ical model o egula black
holes (RBHs) and Uni e se was de eloped ha esol es classical singula i y p oblems
by in oducing new he modynamic s uc u e, ene gy-p essu e balance, and en opy
conside a ions. Second, his amewo k was ex ended owa d mac oscopic cosmological
con ex s by igo ously o mula ing holog aphic en opy g ow h and non-equilib ium
s uc u es. The en opic o ce is explici ly gi en by
F=TU
dS
dx ,(76)
whe e Fhas dimensions o [ o ce], TUis he Un uh (o Hawking) empe a u e,
and dS/dx is he spa ial en opy g adien . This o mula ion ensu es dimensional
consis ency as [ o ce] = [ empe a u e] ×[en opy g adien ] and p o ides a igo ous
connec ion be ween mic oscopic en opy low and cosmic expansion dynamics on holo-
g aphic sc eens. Building on hese solid heo e ical bases, he p esen s udy applies
and uni ies hese concep s o de i e a he modynamically consis en en opic o ce
mechanism on cosmological holog aphic sc eens, which na u ally eco e s bo h New-
onian g a i y and cosmic accele a ion phenomena. This hie a chical s uc u ing o
20
Fig. 5 En opy E olu ion as a Func ion o Redshi .
This igu e shows he e olu ion o a dimensionless en opy indica o as a unc ion o
cosmological edshi z. I e lec s he he modynamic p og ession o he uni e se
om high edshi s (ea ly epochs) o he p esen day. J
he heo y — om mic oscopic black hole in e io s, h ough holog aphic he mody-
namics, o expanding uni e se phenomena — p o ides a obus and sel -consis en
ounda ion o he no el esul s p esen ed he ein.
D. Lynden-Bell [102] analyzed a hypo he ical gas sphe e in a sel -g a i a ing
sys em o s udy he spon aneous o ma ion o non-equilib ium s uc u es. This hypo-
he ical sphe e, assumed o be iso he mal and o uni o m densi y in a sel -g a i a ing
sys em, is in a he mal equilib ium s a e Smaxi bu i is uns able. Fluc ua ions in he
empe a u e dis ibu ion igge he onse o hea low. I hea ini ially lows om he
in e io o he ex e io , he p essu e in he cen al egion dec eases, causing i o con-
ac unde i s own g a i y. As a esul , he empe a u e and densi y in he cen al
egion pa adoxically inc ease, while he densi y in he ou e egion dec eases. Once
hea and densi y ans e begin, hese p ocesses become inc easingly p onounced,
leading o a g owing densi y con as and he spon aneous o ma ion o a co e-halo
s uc u e. The ou low o hea , which coun e in ui i ely aises he empe a u e, esul s
in a nega i e g a i a ional he modynamic speci ic hea . As he sys em con ac s, i s
en opy con inues o inc ease. Ul ima ely, his sys em encoun e s wha D. Lynden-Bell
e med a "g a i a ional he modynamic ca as ophe." The c i ical densi y con as a
which he sys em e ol es in his di ec ion is gi en by
D= 709 >ρC
ρb
(77)
7.1 C i ical Densi y Con as De i a ion
The c i ical densi y con as o 709 a ises in he con ex o he g a o he mal ca as o-
phe o sel -g a i a ing iso he mal sphe es, as de i ed by Lynden-Bell (1968). Below is
21
a heo e ically igo ous de i a ion based on he Lane-Emden equa ion o iso he mal
sphe es, leading o he s abili y limi whe e nega i e speci ic hea igge s ins abili y.
This ollows he s anda d as ophysical ea men , con i ming he ac o o 709 a he
u ning poin o he calo ic cu e.
7.1.1 Iso he mal Sphe e Model Se up
Conside a sel -g a i a ing sphe e o ideal gas in hyd os a ic equilib ium, assumed
iso he mal a empe a u e Twi h sound speed σ2=kBT/(µmH), whe e µis he mean
molecula weigh and mH he hyd ogen mass. The densi y ρ( )sa is ies he Poisson
equa ion coupled o he iso he mal equa ion o s a e:
∇2Φ=4πGρ, ρ =ρcexp −Φ−Φc
σ2,(78)
whe e ρc=ρ(0) is he cen al densi y and Φc= Φ(0). In oduce he Lane-Emden
scaling adius α=pσ2/(4πGρc)and dimensionless a iables η= /α,ψ(η) = (Φc−
Φ)/σ2. The equa ion becomes he iso he mal Lane-Emden equa ion:
d2ψ
dη2+2
η
dψ
dη =e−ψ, ψ(0) = 0, ψ′(0) = 0.(79)
The dimensionless densi y is ρ/ρc=e−ψ. Fo a ini e sphe e o adius R=η1α, he
bounda y condi ion is ψ′(η1)=0(ze o idal ield).
7.1.2 Mass and Ene gy Pa ame e s
The o al mass Mwi hin adius Ris
M= 4πZR
0
ρ( ) 2d = 4πα3ρcZη1
0
e−ψη2dη ≡(4π)3/2α3ρcw(η1),(80)
whe e w(η1) = Rη1
0η2e−ψdη is he dimensionless mass pa ame e . The o al g a i a-
ional ene gy W(po en ial ene gy) is
W=−3
2
GM2
R
j(η1)
w(η1),(81)
wi h j(η1) = Rη1
0ηe−ψdψ
dη dη om i ial in eg a ion. The o al he mal ene gy U=
(3/2)NkBT, whe e Nis he o al pa icle numbe , so he o al ene gy E=U+W.
De ine he dimensionless empe a u e in e se β= 1/(NkBT)and ene gy pa ame e
u=−E/|W(0)|, bu mo e con enien ly, use he spi al a iables: dimensionless binding
ene gy W=−E/(NkBT)and mass pa ame e J= 3M2/(4πR3NkBT/G). F om he
Emden solu ion, pa ame ic ela ions yield he calo ic cu e W(J).
22
7.1.3 S abili y Limi and Densi y Con as
The calo ic cu e aces a spi al in he (W,J)plane as η1 a ies. S abili y equi es
posi i e speci ic hea CV=dE/dT > 0, o equi alen ly dW/dJ<0along he
spi al. The u ning poin (g a o- he mal ca as ophe onse ) occu s whe e dW/dJ= 0,
ma king he ansi ion o nega i e speci ic hea . Nume ical in eg a ion o he Lane-
Emden equa ion (using, e.g., Runge-Ku a wi h cen al egula iza ion ψ′′(0) = 1)
yields he spi al. The i s u ning poin (s able b anch end) is a η1≈34.36, whe e
dψ
dη (η1)=0, w(η1)≈6.451, j(η1)≈0.398.(82)
The cen al- o-edge densi y con as is
D=ρc
ρ(R)=eψ(η1),(83)
wi h ψ(η1)≈6.563 a he u ning poin , so
D=e6.563 ≈709.(84)
This is he c i ical alue: o D < 709, he iso he mal sphe e is s able; exceeding
709 ini ia es co e collapse wi h hea low inwa d, ampli ying cen al densi y and lead-
ing o he ca as ophe. Analy ically, he spi al asymp o es con i m D→32 on he
s able b anch and D= 709 a he ins abili y h eshold. This de i a ion assumes non-
ela i is ic, collisionless dynamics bu ex ends o s ella sys ems ia Lynden-Bell’s
iolen elaxa ion, whe e phase-space mixing yields Fe mi-Di ac-like dis ibu ions
mimicking iso he mal sphe es.
8 E olu ion o Densi y Con as D(z)and Onse o
S uc u e Fo ma ion
The densi y con as Das a unc ion o edshi zis a c ucial indica o o he onse o
g a i a ional he modynamic ins abili y and subsequen cosmic s uc u e o ma ion.
Following he amewo k o Lynden-Bell’s analysis and he Lane-Emden equa ion,
he c i ical densi y con as Dc i ≈709 ep esen s he h eshold beyond which he
sel -g a i a ing iso he mal sphe e becomes uns able and begins co e-halo s uc u e
o ma ion. To explici ly quan i y he e olu ion o D(z), We de ine
D(z) = ρc(z)
ρb(z),
whe e ρc(z)is he cen al densi y and ρb(z) he backg ound densi y a edshi z. In
he adia ion-domina ed e a z > zeq, he densi y con as e ol es slowly due o high
adia ion p essu e:
D(z)∼Dini ,
23
wi h Dini an ini ial pe u ba ion ampli ude. In he ma e -domina ed e a z < zeq,
he densi y con as g ows app oxima ely as
D(z) = Dini 1 + zeq
1 + zγ
,
whe e γ≈1 o 2, cha ac e izing he g ow h a e o pe u ba ions. The edshi z o m
a which D(z o m) = Dc i ma ks he onse o g a i a ional he modynamic ins abili y
and s uc u e o ma ion. Sol ing o z o m,
z o m =Dc i
Dini 1/γ
(1 + zeq)−1.
This o mula ion allows quan i ica ion o he epoch o s uc u e o ma ion as a unc ion
o ini ial luc ua ions and cosmic pa ame e s, p o iding a clea c i e ion linking cosmic
e olu ion o g a i a ional he modynamics.
Sugges ed placemen o he addi ion
The op imal place o inse ing his sec ion is immedia ely a e he cu en ea -
men o he Lane-Emden equa ion and he discussion o he c i ical densi y con as
D= 709 in he Resul s o Theo e ical F amewo k sec ions (e.g., Sec ion 3 o 4),
whe e he g a i a ional he modynamic ins abili y is i s in oduced. Al e na i ely,
i may accompany he discussion on cosmic e olu ion and en opy in he la e sec ions
add essing non-equilib ium cosmic dynamics. Inse ing his quan i a i e analysis in
close p oximi y o he p esen a ion o ins abili y c i e ia will s eng hen he cla i y o
he link be ween edshi e olu ion and s uc u e o ma ion onse .
9 Nume ical Example: De i a ion o S uc u e
Fo ma ion Redshi
This sec ion p o ides a de ailed nume ical example o illus a e he applica ion o he
densi y con as e olu ion amewo k de eloped in Sec ion ??. We de i e he s uc u e
o ma ion edshi z o m using obse a ional cons ain s om Planck 2018 [128] and
he g a o he mal ca as ophe c i e ion Dc i = 709.
9.1 Ini ial Densi y Con as
The ini ial densi y con as Dini ep esen s p imo dial densi y luc ua ions gene a ed
du ing in la ion. F om cosmic mic owa e backg ound (CMB) obse a ions, he scala
powe spec um ampli ude a he pi o scale k0= 0.05 Mpc−1is measu ed o be [128]:
As= (2.099 ±0.014) ×10−9.(85)
The p imo dial densi y pe u ba ion ampli ude is app oxima ely:
δ≡pAs∼4.6×10−5.(86)
24
Fo he pu pose o his illus a i e calcula ion, We adop a ep esen a i e o de -o -
magni ude es ima e:
Dini = 10−5.(87)
This alue cha ac e izes he densi y con as a ea ly cosmic imes, consis en wi h
in la iona y p edic ions and CMB cons ain s.
9.2 Ma e -Radia ion Equali y Redshi
Ma e - adia ion equali y occu s when he ene gy densi ies o ma e and adia ion
become equal:
ρm(zeq) = ρ (zeq).(88)
Gi en he edshi e olu ion o ene gy densi ies:
ρm(z) = ρm,0(1 + z)3,(89)
ρ (z) = ρ ,0(1 + z)4,(90)
he equali y condi ion yields:
1 + zeq =ρm,0
ρ ,0
=Ωm,0
Ω ,0
,(91)
whe e Ωm,0and Ω ,0a e he p esen -day densi y pa ame e s o ma e and adia ion,
espec i ely. Using Planck 2018 alues [128]:
Ωm,0= 0.315,(92)
Ω ,0= Ωγ,0+ Ων,0≈9.2×10−5,(93)
We ob ain:
zeq =0.315
9.2×10−5−1≈3424 ≈3400,(94)
ounded o con enience in subsequen calcula ions.
9.3 S uc u e Fo ma ion Redshi Calcula ion
In he ma e -domina ed e a (z < zeq), he densi y con as e ol es acco ding o:
D(z) = Dini ×1 + zeq
1 + zγ
,(95)
whe e γ≈1co esponds o linea g ow h in he Eins ein-de Si e app oxima ion.
Following he g a o he mal ca as ophe amewo k (Sec ion 7.1), s uc u e o ma ion
ini ia es when he densi y con as eaches he c i ical alue:
D(z o m) = Dc i = 709.(96)
25
a∝ 1/2(136)
T∝a−1(137)
ρ ∝a−4(138)
ρ =aT4
c2(mass densi y [kg/m3])(139)
(a= adia ion densi y cons an = 7.5657 ×10−16 J m−3K−4)(140)
ρ (z) = ρ ,0a0
a4=ρ ,0(1 + z)4(141)
E =aT4V (142)
Ma e -Domina ed Phase (3570 > z > 1370, ρm∝a−3, a ∝ 2/3, T ∝a−1)
ime e olu ion o he scale ac o
(1 + z)−1=a
a0
=3
2pΩm,0H0 2/3
(143)
H0 =2
3pΩm,0
(1 + z)−3/2=2
3(Ωm,0)1/2(1 + z)−3/2(144)
Subs i u ing he pa ame e
Ωm,0= 0.315 (145)
In o he abo e equa ions o nume ical compu a ion. Howe e , he in eg al o he
ollowing equa ion gi es he Hubble adius
K= 0,Pa icle ho izon =dH=a( )Z
0
c d
a( )= 3c (146)
a∝ 2/3(147)
T∝a−1(148)
ρm∝a−3(149)
m=Vma3ρm(150)
T3
ρm
=cons , ρma3is cons an o e he en i e cosmic his o y and T∝a−1(151)
ρm(z) = ρm,0a0
a3=ρm,0(1 + z)3(152)
Em=Mmc2(153)
Npho on =V3
mT3m−3∼R3T3=cons (154)
Epho on ∝R−4T4(155)
The adia ion-domina ed phase expands mo e slowly, while he ma e -domina ed
phase expands as e ( adia ion-domina ed, ma e -domina ed ≫c). Fo Z > 1100,
he adia ion-ma e he mal equilib ium s a e holds. Du ing he phase ansi ion o
in la ion, an eno mous amoun o ene gy Ewas supplied as la en hea , ehea ing he
32
Fig. 8 Pa icle Ho izon as a Func ion o Z
K
uni e se. The scale o he uni e se’s size (scale ac o ) is aken as ollows, conside ing
dimensions
R=a∝exp Λc2
3 (156)
α= Λc2
3(157)
∴R=a∝exp α (158)
α= [T−1]T( he dimension mus be he in e se o ime)
= Λc2
3≈10−36 s−1 o 10−34 s−1(159)
is conside ed easonable, and he in e se o αbecomes he ime scale o exponen ial
expansion.
Λ0=3H2
c2=3ΩΛ,0H2
0
c2
=3×0.684 ×2×5.4419 ×10−36
8.98755 ×1016 = 1.5920 ×10−52 m−2
(160)
Fo (Z > 4×1026), he adia ion-domina ed phase is app oxima ed using equa ions
132 o 142. Fo he in la ion phase (Z= 4 ×1022 <4×1025), whe e (a∼e60 ∼1026)
is sa is ied, he ollowing pa ame e is app oxima ely subs i u ed
Λ=7.47 ×1053 m−2K(161)
33
The poin a which he expansion speed shi s om adia ion-domina ed o ma e -
domina ed is
2×72.94 ×(1 + Z)−2= 3 ×1.217 ×(1 + Z)−3/2
= (1 + Z)−1/2
≈3.651
145.88 ∼0.025
(162)
1 + Z= (0.025)−2∼1600, Z = 1600 −1 = 1599 (163)
Ini ially, adia ion densi y ρ ≫ma e densi y ρm, bu his e e ses in he p esen
e a. Fo ρc and
T3
ρm
=cons (164)
ρma3=cons (165)
ρ =ρm,ρ
ρm∝(1 + Z)4
(1 + Z)3∼(1 + Z)
∼Z= 3400 ∼6380,(Ω ,0= 4.7∼8.4×10−5)
(166)
As a esul , he uni e se began in a s a e o comple e he mal equilib ium, whe e
I=Smax −S( )
kBln 2 = 0 (167)
I can be conside ed ha he me e expansion o he uni e se does no gene a e
Fig. 9 Densi y Compa ison as a Func ion o Z
K
en opy (since he o al numbe o pho ons emains unchanged), bu en opy changes
in esponse o changes in he sys em’s olume o empe a u e. Howe e , since o di-
na y he modynamics can be applied, he en opy o he uni e se inc eases o e ime.
Howe e , due o he expansion o he uni e se and he nega i e speci ic hea o sel -
g a i a ing sys ems, a he mal equilib ium s a e is no achie ed. Cosmic expansion
34
causes he empe a u e o blackbody adia ion o dec ease u he , allowing subsys-
ems wi hin a gi en egion ( he en i e sys em) o spon aneously c ea e non-equilib ium
s a es by shedding en opy o he ou side h ough g a i a ional e ec s. Gene ally,
he ene gy densi y o blackbody adia ion a empe a u e Tis gene ally gi en by he
adia ion densi y cons an
a=π2k4
B
15ℏ3c3(168)
The blackbody adia ion ene gy densi y
ρ=aT4=π2k4
BT4
15ℏ3c3(169)
The e o e, om he o mula o he c i ical densi y o he uni e se 116 ρc ≡3H2
0
8πG
based on he ela ionship be ween he la ge-scale uni e se and quan um mechanical
ene gy densi y
ρc c2=3H2
0c2
8πG =π2k4
BT4
15ℏ3c(170)
The e o e, he ene gy densi y o blackbody adia ion is
ρc2=aT4=π2k4
BT4
15ℏ3c3c2=3H2
0c2
8πG =π2k4
BT4
15ℏ3c(171)
RBHsIn e io T he modynamicsRadialen opydensi y :s ( ) = (4/3)aSBNT ( )3Radial empe a u e :T ( )Scale −Dependen F o mula ion(Cosmological)Scaleen opydensi y :σ(l) = σ0exp(−l2/l2
0)Scale empe a u e :Ts(l) = TUexp(−l2/l2
c) + TH[1 −exp(−l2/l2
c)]Holog aphicSc eenen opydensi y :σsc een =kB/(4LPl2)
(172)
10.2 Scale-Dependen En opy and Tempe a u e P o iles
These p o iles desc ibe he he modynamic s uc u e ac oss spa ial scales om Planck
leng h LPl = 10−35 m o Schwa zschild adius RS= 1026 m.
The spa ial scale pa ame e l anges om in e io egions (l≪RS) o cosmological
scales (l∼RH), wi h cha ac e is ic ansi ions a quan um (l∼LPl) and classical
(l∼M1/3) scales.
To model a peaked, non-singula en opy dis ibu ion a ising om quan um
deg ees o eedom and scale-dependen empe a u e e olu ion, we adop he ollowing
ansä ze based on he cha ac e is ic scale pa ame e l:
Scale-dependen en opy densi y:
σ(l) = σ0exp
−l2
l2
0[J K−1m−3],(173)
Scale-dependen empe a u e:
Ts(l) = TUexp −l2
l2
c+TH1−exp −l2
l2
c [K],(174)
35
Dimensional analysis:
[σ(l)] = J K−1m−3,(175)
[Ts(l)] = K,(176)
[l0, lc]=m.(177)
Physical in e p e a ion: Bo h σ(l)and Ts(l)desc ibe he scale-dependen s uc u e
o quan um he modynamics ac oss leng h scales om Planck o Hubble adius.
11 Uni ied Tempe a u e In e pola ion
To b idge he local Un uh empe a u e and cosmological Hubble empe a u e, A
uni ied in e pola ion is in oduced:
Ts(l) = TUexp(−l2/l2
c) + TH[1 −exp(−l2/l2
c)].(178)
This o mula p o ides a smoo h ansi ion be ween he wo egimes: in he limi
l→0(179)
(local scales),
Ts→TU,(180)
while o
l→ ∞ (181)
(cosmological scales),
Ts→TH.(182)
He e,
lc(183)
is a c i ical scale pa ame e , which can be associa ed wi h he Planck leng h
lc∼lpl (184)
o ela ed o he Hubble adius o mac oscopic ansi ions. This scale-dependen
e ec i e empe a u e
Ts(185)
can be applied o en opy calcula ions on he holog aphic sc een, enhancing he con-
sis ency o en opic o ce de i a ions ac oss di e en scales by inco po a ing a uni ied
he mal desc ip ion in he en opy g adien
dS/dx (186)
36
12 Holog aphic En opy on he Cosmological Sc een
The holog aphic sc een a RH=c/H( )has en opy densi y σsc een =kB/(4l2
pl). To al
en opy is:
Ssc een =σsc een ·A=kB
4l2
pl ·4πR2
H=πkBc3R2
H
ℏG=πkBc5
ℏGH2( ).(187)
Acco ding o he holog aphic p inciple, he en opy ca ied by he sc een may be
iewed as an en opy densi y pe uni a ea— ha is, he amoun o in o ma ion
encoded on each uni o su ace a ea. I he e o e de ine
σsc een =kB
4L2
pl J K−1m−2,(188)
whe e Lpl =pℏG/c3is he Planck leng h. He e σsc een deno es he en opy pe uni
a ea (in o ma ion densi y) on he holog aphic sc een. The o al en opy on a sphe ical
sc een o adius R hen ollows by mul iplying σsc een by he su ace a ea A= 4πR2:
Ssc een =σsc een A(189)
The sc een has wo he modynamic in e p e a ions depending on scale
Fig. 10 Concep ual Diag am: Holog aphic P ojec ion o En opy
37
•On local (g a i a ional) scales, he sc een is coupled o he Un uh empe -
a u e TU∼a/(2π), associa ed wi h local accele a ion a, leading o New onian
g a i a ional o ce ia he en opic o ce ela ion
F=TH·dS
dx =mHc. (190)
The en opic o ce is explici ly gi en by F=TUdS
dx , whe e Fhas dimensions o [ o ce],
TUis he Un uh (o Hawking) empe a u e, and dS/dx is he spa ial en opy g a-
dien . This o mula ion ensu es dimensional consis ency as [ o ce] = [ empe a u e] ×
[en opy g adien ].
•On cosmological scales, he sc een expands wi h he uni e se, and he associ-
a ed empe a u e becomes he Hubble empe a u e TH=H/(2π), p oducing a
mac oscopic en opic accele a ion
aH= 2πTH∼H, (191)
which mimics a mac oscopic en opic o ce. The en opy g adien dS/dx along he
sc een no mal e lec s he lux o deg ees o eedom ac oss he sc een, consis en
wi h he second law o he modynamics. The diag am cap u es he dual he mody-
namic ole o he sc een, ac ing bo h as an in o ma ion-encoding su ace and as a
he modynamic bounda y media ing en opic o ces.
12.1 Cosmological En opic Fo ce and Planck Fo ce:
Nume ical Ve i ica ion
The cosmological en opic o ce a he Hubble scale exhibi s a p o ound connec-
ion o he undamen al Planck o ce, demons a ing he deep ela ionship be ween
he modynamics and quan um g a i y.
En opic Fo ce Fo mula.
The cosmological en opic o ce ac ing on a es mass ma he Hubble adius RH=
c/H is gi en by
FH=TH
dS
dx =mHc, (192)
whe e TH=ℏH/(2πkB)is he Hubble empe a u e (Gibbons-Hawking empe a u e),
His he Hubble pa ame e , and dS/dx is he en opy g adien on he holog aphic
sc een.
Obse able Uni e se Mass.
The cha ac e is ic mass scale a he Hubble adius is de e mined by dimensional
analysis as
MH=c3
GH0≈1.848 ×1053 kg,(193)
whe e G= 6.674 ×10−11 m3kg−1s−2is he g a i a ional cons an and H0= 2.1850 ×
10−18 s−1is he p esen -day Hubble pa ame e om Planck 2018 obse a ions.
38
Nume ical Ve i ica ion.
Subs i u ing he obse able uni e se mass MHin o Eq. (192), The cosmological
en opic o ce:
FH=MHH0c=c4
G≈1.210 ×1044 N.(194)
This alue is iden ical o he Planck o ce, de ined as
FPlanck =c4
G≈1.210256 ×1044 N,(195)
which ep esen s he maximum o ce in na u e acco ding o quan um g a i y
conside a ions.
Exac Ag eemen .
The a io be ween he cosmological en opic o ce and he Planck o ce is
FH
FPlanck
=MHH0c
c4/G =GMHH0
c3= 1.000,(196)
con i ming pe ec nume ical ag eemen o machine epsilon (∼10−15). This in e pola-
ion unc ion p o ides a uni ied he modynamic amewo k o desc ibing he en opic
o ce ac oss an unp eceden ed scale ange o 61 o de s o magni ude, con inuously
ex ending om he Planck leng h (Lpl ∼10−35 m) o he Hubble adius (RH∼
1026 m), he eby b idging mic oscopic quan um g a i y e ec s wi h mac oscopic
cosmological phenomena.
Physical In e p e a ion.
This ema kable coincidence is no acciden al bu e lec s a p o ound connec ion
be ween cosmological dynamics and quan um g a i y. The Planck o ce FPlanck =
c4/G ep esen s he undamen al ension o space ime a he quan um g a i y scale.
The ac ha he cosmological en opic o ce a he Hubble adius exac ly equals his
undamen al o ce sugges s ha cosmic accele a ion is d i en by he same quan um
g a i a ional mechanism ha go e ns Planck-scale physics.
Dimensional Consis ency.
The dimensional analysis con i ms he consis ency o all quan i ies:
[FH]=[m][H][c] = kg ·s−1·ms−1=kg ·m·s−2=N,(197)
[FPlanck]=[c4]/[G]=(ms−1)4/(m3kg−1s−2) = kg ·m·s−2=N.(198)
This exac ag eemen be ween he cosmological en opic o ce and he Planck o ce
p o ides s ong e idence ha cosmic accele a ion is an en opic phenomenon a is-
ing om holog aphic he modynamics a he Hubble scale, uni ying g a i a ional
phenomenology om local o cosmological scales wi hou ee pa ame e s.
39
13 Holog aphic Sc een De ailed explana ion
M
m
F
inc easing ∇S
sc een T( )∝1/
Fig. 11 Holog aphic sc een o adius enclosing mass M. The en opic o ce ac s on es mass m
loca ed jus ou side he sc een due o he en opy g adien associa ed wi h he sc een deg ees o
eedom.
Concep ual F amewo k o Holog aphic The modynamics
This igu e illus a es he concep ual amewo k o he holog aphic he modynamic
model applied o an expanding uni e se. A holog aphic sc een (blue su ace) wi h a ea
Ais placed a Hubble adius Renclosing cosmic ma e . The en opy Sassocia ed
wi h he bulk olume is p ojec ed on o his sc een ollowing he holog aphic p inciple,
whe e he in o ma ion con en o he olume is encoded on he bounda y. I in en-
ionally a oid elying on he AdS/CFT duali y o speci ic s a is ical cons uc ions
such as quan um en anglemen en opy, so as o de elop a concep ually independen
and physically mo i a ed holog aphic he modynamic amewo k applicable o cos-
mological se ings wi h no asymp o ic bounda y. This au onomy acili a es b oade
applicabili y and a oids o e eliance on assump ions ha may no hold in dynamical
space imes.
Concep ual Illus a ion o Holog aphic Encoding in he Expanding Uni-
e se. This igu e p esen s a concep ual ep esen a ion o he he modynamic
and geome ic s uc u e o he uni e se h ough holog aphic and en opic g a i y
pa adigms. Th ee key componen s a e illus a ed: (1) mic oscopic en opy wi hin he
bulk olume, (2) holog aphic encoding on he cosmological bounda y, and (3) cos-
mic expansion dynamics cha ac e ized by he Hubble adius. The igu e demons a es
how bulk en opy is mapped on o a bounda y sc een, wi h en opic o ces d i ing
expansion. The le mos sphe e, shaded in g ay, ep esen s he in e nal mic oscopic
deg ees o eedom—quan um o s a is ical cons i uen s esponsible o he en opy
o he uni e se. These deg ees o eedom, al hough unobse able di ec ly, o m he
he modynamic unde pinning o g a i a ional phenomena. Su ounding he in e nal
egion is a dashed ci cle iden i ied as he holog aphic sc een. This su ace encodes he
in o ma ion o he in e nal sys em p ojec ed on o i , as sugges ed by he holog aphic
p inciple. Acco ding o his p inciple, he en opy con en o a olume o space is no
40
p opo ional o i s olume bu a he o he a ea o i s bounda y, measu ed in Planck
uni s. This adically ede ines he na u e o in o ma ion and en opy in g a i a ional
heo ies.
To he igh , he o ange-colo ed ci cle deno es he Hubble adius—a cosmological
bounda y beyond which objec s ecede as e han ligh due o he uni e se’s expan-
sion. The Hubble adius e ec i ely delinea es he obse able uni e se a a gi en cosmic
ime. I ac s no only as a geome ic scale bu also as a he modynamic bounda y ha
expands wi h ime. The a ows depic wo cen al dynamics: i s , he ans e o in e -
nal in o ma ion ou wa d on o he sc een, e med holog aphic mapping, and second,
he he modynamic back- eac ion encoded as he en opic o ce. This en opic o ce
eme ges due o changes in he en opy on he sc een when a es mass is displaced,
aligning wi h Ve linde’s o mula ion o g a i y as an eme gen phenomenon. Quan i a-
i ely, he en opic o ce ollows he exp ession. F=TH·dS
dx =mHc The en opic o ce
is explici ly gi en by F=TUdS
dx , whe e Fhas dimensions o [ o ce], TUis he Un uh
(o Hawking) empe a u e, and dS/dx is he spa ial en opy g adien . This o mula-
ion ensu es dimensional consis ency as [ o ce] = [ empe a u e] ×[en opy g adien ].
whe e His he Hubble pa ame e , m he mass, and c he speed o ligh . As he
uni e se e ol es, he Hubble adius inc eases, leading o he con inual g ow h o holo-
g aphically encoded en opy on he sc een. This is consis en wi h he second law
o he modynamics, which, when in e p e ed cosmologically, implies an i e e sible
inc ease in he accessible in o ma ion con en o he uni e se. In his amewo k, g a -
i y does no a ise om a undamen al in e ac ion bu a he om en opy g adien s
and in o ma ion ans e . The no ion ha space ime geome y i sel has a he mo-
dynamic o igin opens new pa hs in unde s anding cosmology, quan um g a i y, and
he a ow o ime. The diag am hus syn hesizes deep heo e ical ideas: he en opy-
a ea ela ion o Bekens ein and Hawking, he sc een-based dynamics p oposed by
Ve linde, and he la ge-scale e olu ion o he uni e se as cons ained by gene al ela-
i i y. I o e s a uni ying pic u e o g a i a ional he modynamics, whe e holog aphy
and cosmic expansion a e in insically linked.
14 The en opy o blackbody adia ion and
Bekens ein-Hawking En opy
Thus, i is ob ained. The en opy o blackbody adia ion is
S =4aT3
3V (199)
In Bibliog aphy [102], D. Lynden-Bell e al. discuss he densi y con as , hea low,
and en opy o an iso he mal sphe e in he uni e se. In con as , e e ence [142] by D.
Sugimo o e al. ex ends he scope o discuss en opy in an expanding uni e se. Using
he me hod o calcula ing he black hole en opy SBH as p esen ed in Bibliog aphy
41
whe e ais he adia ion cons an , T is he adia ion empe a u e, and V is he
olume. Using edshi z
E o al =Mmc2+aT4
V ·(Ω ,0)1/2(1 + z)−2(237)
wi h app oxima ely, on he o de o Ω ,0= 4.7×10−5.
16.2.2 Ma e -Domina ed E a
In he ma e -domina ed e a
E o al =Mmc2+aT4
V ·(Ωm,0)1/2(1 + z)−3/2(238)
whe e app oxima ely, on he o de o Ωm,0= 0.315.
16.3 In oduc ion o Dimensionless Quan i ies
The ma e ene gy a io xand scaled en opy ya e de ined as
x=Em
E o al
, y =S
E2
o al
(239)
whe e he o al en opy S=Sm+S , wi h Sm∝E2
mand S ∝E3/4
, and he cons an
cons = 1.
16.4 De i a ion o he Rela ionship
Assuming he en opy ela ion y=x2+y(1 −x)3/4and sol ing o y
y−y(1 −x)3/4=x2(240)
y[1 −(1 −x)3/4] = x2(241)
y=x2
1−(1 −x)3/4(242)
Planck-No malized Dimensionless En opy Scaling:
˜
y=S/kB
(E o al/EPlanck)2,[dimensionless] (243)
The en opy- o-ene gy a io desc ibes he ansi ion o ene gy dominance in cosmic
e olu ion quan i a i ely. De ining he ac ion o ma e ene gy o o al ene gy as
x≡Em
E o al
(244)
48
he o al en opy as a unc ion o xis exp essed as
S
E2
o al ·cons =y=x2
1−(1 −x)3/4(245)
y=x2
1−(1 −x)3/4(246)
He e, yis de ined as
y≡Mplc2
3πk 4aV
1
E o al 1/4
·Mplc2
E o al
,(247)
16.5 Ve i ica ion a he Limi s
16.5.1 Radia ion-Domina ed E a (x→0)
As x→0,Em→0,E o al ≈E , and:
y≈S
E2
∝E−5/4
→0(248)
This is consis en wi h he en opy beha io in he adia ion-domina ed e a.
16.5.2 Ma e -Domina ed E a (x→1)
As x→1,E →0,E o al ≈Em, and:
y≈Sm
E2
m∝1(249)
This aligns wi h he scaling in he ma e -domina ed e a.
16.5.3 Case o x > 1
Typically, x=Em
E o al ≤1, bu x > 1implies Em> E o al, which is non-physical in a
closed sys em. Howe e , i he sys em abso bs ene gy om ex e nal sou ces (e.g., black
hole acc e ion, ene gy exchange in mul i e se scena ios, o ene gy exchange om an
in la iona y ield), Emmay inc ease, leading o x > 1. To model his, he o al ene gy
is ede ined as:
E o al =Em+E +Eex (250)
whe e Eex >0 ep esen s ene gy in low om ex e nal sou ces. Thus, x=
Em
Em+E +Eex >1becomes possible due o he con ibu ion o Eex , enabling
applica ions o open sys ems o non-s anda d cosmological models.
49
17 En opy–Ene gy Rela ion o Blackbody Radia ion:
O igin o he 3/4Exponen
A concise de i a io o he ela ionship be ween en opy S and o al ene gy E
o ideal blackbody adia ion con ined in a ixed olume V. S a ing om he
S e an–Bol zmann law and undamen al he modynamic iden i ies, I is shown ha
S ∝E3/4
,
and I ace he o igin o he exponen 3/4 o he empe a u e scalings o ene gy densi y
(T4) and en opy densi y (T3).
17.1 De ailed explana ion
Blackbody adia ion in he modynamic equilib ium obeys well-known scaling laws.
The ene gy densi y uand p essu e pa e ela ed o he absolu e empe a u e Tby
u=a T4,(251)
p=1
3u=1
3a T4,(252)
whe e ais he adia ion cons an . In a ixed olume V, he o al adia i e ene gy and
en opy a e deno ed by E and S , espec i ely.
17.2 The modynamic Rela ion
Fo a closed sys em a cons an olume, he i s law eads
dE =T dS −p dV. (253)
Wi h dV = 0, one inds
dS =dE
T.(254)
17.3 Ene gy–Tempe a u e Rela ion
F om Eq. (251), he o al ene gy is
E =u V =a T4V. (255)
Sol ing o Tgi es
T=E
a V 1/4
.(256)
50
17.4 En opy as a Func ion o Ene gy
Subs i u ing T(E )in o he di e en ial o en opy
S =ZdE
T
=ZdE
(E /(aV ))1/4
= (aV )1/4ZE−1/4
dE
=4
3(aV )1/4E3/4
+cons an .
Disca ding he addi i e cons an by app op ia e choice o e e ence yields
S =4
3(aV )1/4E3/4
,(257)
hus es ablishing he scaling
S ∝E3/4
.(258)
17.5 O igin o he 3/4Exponen
The exponen 3/4eme ges om combining wo undamen al empe a u e scalings:
•Ene gy densi y: u∝T4implies E ∝T4, so T∝E1/4
.
•En opy densi y: s∝T3 ollows om dS /dV = (4/3) a T3.
Hence,
S ∝T3∝(E1/4
)3=E3/4
.(259)
17.6 Conclusion o E3/4
Scaling
We ha e de i ed he en opy–ene gy ela ion o blackbody adia ion in a ixed olume
and elucida ed he physical o igin o he 3/4exponen as a ising om he dis inc
empe a u e dependences o ene gy and en opy densi ies. [142]
18 Cosmological Cons an and Accele a ed Expansion
The cosmological cons an Λplays a pi o al ole in d i ing he accele a ed expansion
o he uni e se, as obse ed in mode n cosmological da a [128]. This sec ion add esses
he in eg a ion o Λin o he g a i a ional he modynamic amewo k, ocusing on
i s impac on non-equilib ium p ocesses and en opy e olu ion. I cla i y he physical
mo i a ion o he Λ alues used in he in la ion and mode n e as, connec Λ o
en opy p oduc ion, and p esen nume ical simula ions o alida e he he modynamic
consis ency o he accele a ed expansion phase.
51
In he da k ene gy-domina ed epoch, as
H( )→HΛ
(cons an ), he en opy g ow h a e
dS/d →0
while
S( )
con inues o inc ease. The en opy inside he sc een may appea o dec ease, bu
he holog aphic p inciple ensu es ha in e nal in o ma ion is p ojec ed ou wa d on o
he sc een. The en opy g ow h is hus due o he dynamical a ea inc ease o he
cosmological sc een.
In addi ion o he holog aphic en opy g ow h o mula, he en opy p oduc ion
a e in cosmic luid he modynamics is gi en by
dS
d =ρ+p
T˙
V=ρ+p
T·3HV > 0,
whe e
ρ
is he ene gy densi y,
p
is he p essu e,
T
is he empe a u e,
V
is he como ing olume, and
H=˙
a/a
is he Hubble pa ame e . In he adia ion-domina ed e a (
p=ρ/3
), his simpli ies o
ρ+p= (4/3)ρ > 0
, which uncondi ionally sa is ies he second law o he modynamics du ing cosmic
expansion, as he posi i e e m ensu es mono onic en opy inc ease ega dless o
speci ic decele a ion condi ions. This luid pe spec i e complemen s he holog aphic
sc een dynamics, uni ying bulk he modynamics wi h bounda y p ojec ions ac oss
cosmological epochs.
The cosmological cons an Λis in oduced in he F iedmann equa ions o accoun
o accele a ed expansion:
˙
a
a2
=8πG
3ρ+Λc2
3−kc2
a2,(260)
52
¨
a
a=−4πG
3ρ+3p
c2+Λc2
3,(261)
whe e ais he scale ac o , ρis he o al ene gy densi y, pis he p essu e, and kis he
cu a u e pa ame e . Fo he mode n uni e se, We adop Λ0= 1.5920 ×10−52 m−2
(Eq. 132), de i ed om Planck 2018 da a (ΩΛ,0= 0.684) [128]. Du ing he in la ion
e a (z∼4×1022 −4×1025).
As a esul o he non- ela i is ic nume ical analysis, he ollowing alue was
ob ained. This is in he same o de (same numbe ) as he non- ela i is ic alue o
he ela i is ic nume ical analysis (Λ = 7.47 ×1053 K. De ails a e as ollows. I use
Λ=7.47 ×1053 m−2(Eq. [128]), mo i a ed by he slow- oll in la ion model whe e he
acuum ene gy densi y domina es:
ρΛ=Λc2
8πG ≈1092 kg/m3,(262)
co esponding o he ene gy scale o in la ion (∼1016 GeV) [95]. This la ge Λd i es
he exponen ial expansion a∝exp qΛc2
3 (Eq. 156), consis en wi h he obse ed
la ness and homogenei y o he uni e se.
18.1 Non-Equilib ium P ocesses D i en by Λ
The cosmological cons an in oduces a nega i e p essu e e m, pΛ=−ρΛc2, which
a ec s he en opy p oduc ion a e σsin non-equilib ium he modynamics (Eq. 263).
We ex end he en opy con inui y equa ion o include he Λ-d i en expansion:
∂s
∂ +∇·Js=σs+σΛ,(263)
whe e σΛ≥0 ep esen s he en opy p oduc ion due o accele a ed expansion. Fo a
como ing olume V∝a3, he en opy change due o Λis:
dSΛ
d =ρΛc2V
T˙
a
a=Λc4V
8πGT H, (264)
whe e H=˙
a/a is he Hubble pa ame e and Tis he empe a u e o he sys-
em. This e m enhances en opy p oduc ion du ing he accele a ed expansion phase,
con ibu ing o he non-equilib ium s a e o he uni e se. The in e play be ween Λ-
d i en expansion and g a i a ional clumping (Eq. 264) c ea es nes ed non-equilib ium
s uc u es, as discussed in Sec ion 1.
18.2 Nume ical Simula ions o Λ-D i en Expansion
To quan i y he impac o Λon en opy e olu ion, We inco po a e he Λ e m in o he
non- ela i is ic cosmic expansion model (Eq. 264). The modi ied equa ion o mo ion
53
o a es pa icle on he pa icle ho izon is
d2R
d 2=−4πG
3ρR +Λc2
3R. (265)
I nume ically sol e his equa ion using he pa ame e s ρc =3H2
0
8πG ,Λ0= 1.5920 ×
10−52 m−2, and ini ial condi ions a z= 0 (mode n e a). The en opy e olu ion is
compu ed using Eq. 202, wi h he olume V∝R3adjus ed o accele a ed expansion.
Figu e 12 shows he en opy S o al/kBas a unc ion o edshi z, highligh ing he
inc eased en opy g ow h a e in he Λ-domina ed e a (z < 0.5). Figu e 12 displays
Fig. 12 Linea ela ionship be ween edshi zand da a index o uni e ses wi h and wi hou a
cosmological cons an .
J
he edshi pa ame e zplo ed agains a disc e e da a index anging om 0 o 100.
The blue cu e co esponds o a uni e se wi h ze o cosmological cons an (Λ=0),
while he ed cu e ep esen s a uni e se wi h Λ = 1.5920 ×10−52 m−2. Bo h cu es
o igina e a z= 0 and dec ease linea ly as he index inc eases. The s eepe slope o
he ed cu e indica es ha he p esence o a posi i e cosmological cons an causes
he scale ac o R( ) o e ol e mo e apidly, yielding a highe edshi pe index s ep.
Analy ically, he ela ionships ake he o m z=−m N, wi h g adien s m0= 0.000486
and mΛ= 0.000591, so ha mΛ/m0≈1.216. This linea beha io esul s om sam-
pling he nume ical solu ion o he second-o de F iedmann equa ion a e enly spaced
ime in e als. Al hough eal cosmological edshi e ol es nonlinea ly, his idealized
expe imen highligh s he di ec in luence o Λon expansion dynamics. The consis en
g idlines and clea legend acili a e di ec compa ison, and he absence o a loga-
i hmic axis emphasizes he absolu e di e ences in z. A index 100, he cu es each
|z0| ≃ 0.0486 and |zΛ| ≃ 0.0591, demons a ing an app oxima ely cons an inc emen-
al shi o ∆z≈0.000105 N. The plo con i ms ha a nonze o Λaccele a es he
54
expansion ela i e o he Λ=0case, p o iding a concise isual summa y o da k
ene gy’s e ec on edshi e olu ion. Figu e 13 a anges he ou sequence a iables
Fig. 13 Comp ehensi e 22 subplo showing z0,zΛ,S0/kb, and SΛ/kb e sus.
J
in o a 2×2 g id o di ec compa ison. The op-le panel plo s z o Λ=0, and he
op- igh panel plo s z o Λ = Λ0, bo h showing linea declines. The bo om-le
and bo om- igh panels display he co esponding en opy alues S/kb, which emain
cons an and ho izon al. Consis en colo coding and line s yles link hese subplo s o
he indi idual igu es, while sha ed g idlines and ma ched axis anges enhance ead-
abili y. Index labels a e p ese ed on he ho izon al axes, wi h independen e ical
labels o accommoda e he di e ing scales o zand S/kb. The o e all i le summa izes
he comple e sequence analysis o indices 0–100. This a angemen highligh s he
con as be ween dynamic a iables (z) and conse ed quan i ies (S/kb), illus a ing
bo h he accele a ed expansion in he Λ-inclusi e model and he adiaba ic na u e o
he en opy e olu ion. The subplo o ma is ideal o p esen a ions o publica ions,
enabling iewe s o g asp pa ame e sensi i i ies and model assump ions in a single
composi e igu e.
We in eg a es he modynamic assump ions wi h black hole he modynamics o
heo e ically e i y he ene gy-en opy ela ionship om he adia ion-domina ed o
he ma e -domina ed e a. The de i ed ela ion y=x2
1−(1−x)3/4is consis en wi h
limi ing beha io s (x→0,x→1), and he in e p e a ion o x > 1as ex e nal
ene gy abso p ion is physically meaning ul. This amewo k enables applica ions o
open sys ems and non-s anda d cosmological models, p o iding a no el pe spec i e
on he he modynamic e olu ion o he uni e se. So, In he limi x→0(Radia ion-
only), y→0, which is physically consis en . As he ma e mass app oaches ze o,
he ma e en opy Sm∝M2→0. I E o al is cons an and adia ion-domina ed,
55
S o al ≈S wi h adia ion en opy
S ∝T3
(266)
and adia ion ene gy
E ∝T4
(267)
so
S ∝E3/4
(268)
Since yis scaled by E2
o al,i Em→0, hen E →E o al and
y∝S
E2
o al ∝E3/4
o al
E2
o al
=E−5/4
o al (269)
This scaled en opy app oaches ze o because he scaling emphasizes he ma e con-
ibu ion. In a pu e adia ion s a e (no ma e ), he scaled en opy is ela i ely small,
app oaching ze o. In he limi x→1(Ma e -only), as (1 −x)3/4→0, he denomina-
o app oaches 1−0=1,soy(x)≈x2/1 = x2, and as x→1, hus y→12= 1. This
is consis en wi h he scaling i Em≈E o al hen x≈1, and since ma e en opy
Sm∝E2
m
Sm=AmE2
m(270)
y∝Sm
E2
o al ≈Sm
E2
m≈Am(271)
The cons an being 1 indica es a speci ic no maliza ion chosen o Smo he o e all
scaling cons an , meaning ha in a ully ma e -domina ed sys em, he scaled en opy
eaches he no malized maximum alue alue o 1.
Physical Scaling P ese a ion.
The no maliza ion p ese es he undamen al en opy-ene gy ela ions:
S ∝E3/4
⇒˜
y ∝E3/4
E2
o al
(272)
Sm∝E2
m⇒˜
ym∝E2
m
E2
o al
(273)
ensu ing ha he 3/4and 2exponen s emain in ac (see Sec ion ??). Addi ion-
ally, since sel -g a i a ing sys ems ha e nega i e speci ic hea , he speci ic hea was
calcula ed as
CV=−8πkBGM2
ℏc(274)
The speci ic hea CV=−8πkBGM2
ℏc∝ −M215 is plo ed. Gene ally, adding ene gy o
ma e inc eases i s empe a u e, and eleasing ene gy dec eases i . Howe e , in sel -
g a i a ing sys ems, due o nega i e speci ic hea , losing ene gy inc eases empe a u e,
making i easie o elease mo e ene gy, a cha ac e is ic he modynamic p ope y o
56
Fig. 14 En opy S/E2
o al ·cons =y=x2/(1 −(1 −x)3/4)as a unc ion o x=Em/E o al.
K
Fig. 15 Absolu e alue o speci ic hea CV=−8πkBGM2
ℏcas a unc ion o Z.
K
such sys ems. Fo sel -g a i a ing sys ems whe e ξ≡Rg
R=2GM
Rc2=ρ
ρc = 1, he
speci ic hea is p opo ional o
CV=−8πkBGM2
ℏc∝ −M2(275)
18.3 Theo e ical Signi icance o Planck No maliza ion
The in oduc ion o he Planck-no malized en opy a iable ˜
y≡
(S/kB)/(E o al/EPlanck)2es ablishes a uni e sal amewo k wi h h ee undamen al
p ope ies:
57
•Con e sion: N≈g∗wi h a co ec ion ac o ξ≈1.00 due o no maliza ion
con en ions.
•Nume ical alue: N≈106.75 is adop ed h oughou We, de i ed om he S anda d
Model calcula ion in Sec ion ??.
This cla i ica ion ensu es ha all he modynamic ela ions, om mic oscopic en opy
densi y o mac oscopic black hole en opy, a e dimensionally consis en and heo e i-
cally well- ounded.
log10(E/J)
Ep o on
10−10
EPlanck
109
Euni e se
1070
80 o de s
Fig. 16 Ene gy scale hie a chy spanning 80 o de s o magni ude om elemen a y pa icle physics
o he o al ene gy o he obse able uni e se.
20 Conclusion and Discussion
We es ablishes a uni ied g a i a ional he modynamic amewo k ha quan i a i ely
connec s quan um g a i y a he Planck scale o cosmological dynamics a he Hub-
ble scale h ough non-equilib ium en opy p oduc ion and holog aphic p inciples.
The amewo k p o ides a pa ame e - ee explana ion o bo h cosmic accele a ion
and s uc u e o ma ion as mani es a ions o en opy-d i en g a i a ional dynamics
as mani es a ions o en opy-d i en g a i a ional dynamics, wi hou in oking exo ic
ma e o ad hoc cosmological cons an s.
20.1 Fundamen al Theo e ical Achie emen s
20.1.1 Scale-In a ian En opy F amewo k
A Planck-no malized dimensionless en opy amewo k spanning app oxima ely 80
o de s o magni ude in ene gy:
˜
y=S/kB
(E o al/EPlanck)2,(301)
whe e x≡Em/E o al ep esen s he ma e ene gy ac ion. This no maliza ion
enables consis en he modynamic analysis ac oss cosmological epochs while p ese -
ing undamen al en opy-ene gy ela ions S ∝E3/4
and Sm∝E2
m. Sol ing he
en opy balance equa ion yields he comple e dimensionless in e pola ion:
˜
y=x2
1−(1 −x)3/4.(302)
64
This exp ession uni ies adia ion-domina ed (3/4-powe law) and ma e -domina ed
(E2
mscaling) epochs, b idging quan um g a i y and cosmology wi hou ee pa am-
e e s. The 3/4exponen a ises na u ally om blackbody adia ion he modynamics,
whe e ene gy densi y scales as u∝T4while en opy densi y scales as s∝T3, yielding
S ∝E3/4
o ixed olume.
20.1.2 S a is ical Founda ion om he Law o La ge Numbe s
I de i ed he dimensionless a io y=S/E2
o al di ec ly om he law o la ge numbe s,
demons a ing ha o a sys em o Nindependen pa icles:
y=hp
ϵ2
p·1
N,(303)
whe e ϵpis he a e age ene gy pe pa icle and hpis he en opy con ibu ion pe pa -
icle. This 1/N scaling e eals ini e-size co ec ions and bounda y-domina ed egimes,
p o iding clea physical in ui ion o he in e pola ion measu e wi hou in oking
a ia ional p inciples.
20.1.3 G a i a ional The modynamic Ins abili y and S uc u e
Fo ma ion
The amewo k e eals g a i a ional he modynamic ins abili y a he c i ical densi y
con as D= 709, de i ed igo ously om he iso he mal Lane-Emden equa ion:
d2ψ
dη2+2
η
dψ
dη =e−ψ, ψ(0) = 0, ψ′(0) = 0.(304)
Nume ical in eg a ion yields ψ(η1)≈6.563 a he s abili y u ning poin η1≈34.36,
gi ing D=e6.563 ≈709. This c i ical alue de e mines he onse o g a o he mal
ca as ophe and spon aneous co e-halo s uc u e o ma ion. When D > 709, nega-
i e g a i a ional speci ic hea CV=−8πkBGM2
ℏc igge s sel -ampli ying hea low:
ou wa d hea anspo pa adoxically aises co e empe a u e while lowe ing su ace
empe a u e, d i ing densi y con as g ow h and spon aneous di e en ia ion in o
high-en opy halos and low-en opy co es.
Connec ion o Nega i e Speci ic Hea
The c i ical densi y con as D = 709 ep esen s he h eshold whe e:
CV=−8πkBGM2
ℏc<0(305)
The densi y con as e olu ion wi h edshi is quan i ied as:
D(z) = Dini 1 + zeq
1 + zγ
,(306)
65
whe e γ≈1cha ac e izes linea g ow h in he ma e -domina ed e a. S uc u e
o ma ion ini ia es when D(z o m) = Dc i = 709, yielding:
z o m =Dc i
Dini 1/γ
(1 + zeq)−1.(307)
Fo ini ial pe u ba ions Dini = 10−5 om CMB obse a ions (Planck 2018: As=
2.099 ×10−9) and ma e - adia ion equali y a zeq ≈3400, linea g ow h p edic s
z o m ≈2.41 ×1011, indica ing ha nonlinea g a i a ional ampli ica ion is essen ial
o s uc u e o ma ion a obse ed edshi s z∼10–100.
20.2 En opic Fo ce Uni ica ion: Planck o Hubble Scale
20.2.1 Cosmological En opic Fo ce and Exac Planck Fo ce
Co espondence
Cosmological en opy low p oduces an eme gen en opic o ce a he Hubble ho izon:
FH=TH·dS
dx =mHc, (308)
whe e TH=ℏH/(2πkB)is he Hubble (Gibbons-Hawking) empe a u e and dS/dx
is he en opy g adien on he holog aphic sc een. Fo he obse able uni e se mass
scale MH=c3/(GH0)≈1.848 ×1053 kg (using Planck 2018: H0= 2.1850 ×10−18
s−1), he cosmological en opic o ce becomes:
FH=MHH0c=c4
G≈1.210 ×1044 N.(309)
This alue is iden ical o he Planck o ce, de ined as he maximum o ce in quan um
g a i y:
FPlanck =c4
G≈1.210256 ×1044 N.(310)
The a io con i ms exac ag eemen o machine epsilon (∼10−15):
FH
FPlanck
=GMHH0
c3= 1.000.(311)
On cosmological scales, he en opic o ce is
F=TH·dS
dRH
=c4
G,(312)
ma ching he Planck o ce, wi h a io FH/FPlanck = 1.000 o machine
epsilon. This amewo k in e pola es he en opic o ce o e 61 o de s
o magni ude, om Planck leng h (10−35 m) o Hubble adius (1026 m),
uni ying quan um g a i y and cosmology. This ema kable co espondence
66
demons a es ha cosmic accele a ion is d i en by he same quan um g a i a-
ional mechanism go e ning Planck-scale physics, e ealing en opic g a i y as a
undamen al uni ying p inciple.
20.2.2 Uni ied Tempe a u e In e pola ion
To b idge local Un uh empe a u e TU=ℏa/(2πkBc)and cosmological Hubble
empe a u e TH=ℏH/(2πkB), We in oduce scale-dependen e ec i e empe a u e:
Ts(l) = TUexp(−l2/l2
c) + TH[1 −exp(−l2/l2
c)],(313)
whe e lc∼lpl is he c i ical ansi ion scale. This p o ides smoo h in e pola ion:
Ts→TU o l→0(local scales) and Ts→TH o l→ ∞ (cosmological scales),
enabling uni ied en opic o ce de i a ions ac oss all scales.
20.3 Holog aphic En opy and Non-Equilib ium
The modynamics
20.3.1 Holog aphic Sc een En opy
The holog aphic sc een a RH=c/H( )encodes en opy densi y σsc een =kB/(4l2
pl),
yielding o al en opy:
Ssc een =kB
4l2
pl ·4πR2
H=πkBc5
ℏGH2( ).(314)
Bekens ein-Hawking en opy o mass Mwi hin he Hubble adius is:
SBH =4πkBGM2
ℏc.(315)
To al en opy e olu ion om Planck scale o p esen yields S/kB≈2.756 ×10123
(dimensionless en opy quan um numbe ), consis en wi h Pen ose’s en opy es ima es
and obse a ional cons ain s.
20.3.2 Non-Equilib ium En opy P oduc ion
The en opy con inui y equa ion go e ns non-equilib ium dynamics:
∂s
∂ +∇·Js=σs≥0,(316)
whe e Js=Jdi
s+Jcon
s+Jgw
sincludes di usion, con ec ion, and g a i a ional wa e
con ibu ions, while σs=σgw
s+σ p
s+σs uc
s ep esen s en opy p oduc ion om g a -
i a ional wa es, acuum p essu e luc ua ions, and s uc u e o ma ion. The Pécle
67
numbe s quan i y non-equilib ium dominance:
Pecosmo =τdi
τexp
=R2H
D h ≫1,(317)
Peg a =τdi
τg a
=R2
D h GM
R3≫1,(318)
whe e τdi =R2/D h,τexp = 1/H, and τg a =pR3/(GM)a e cha ac e is ic
imescales. La ge Pécle numbe s indica e sus ained non-equilib ium s uc u es and
enhanced s uc u e o ma ion, alida ing he amewo k’s depa u e om equilib ium
assump ions.
20.4 Obse able Signa u es and Tes able P edic ions
20.4.1 G a i a ional Wa e Signa u es
Non-equilib ium en opy g adien s induce g a i a ional wa e ampli ude de ia ions:
∆A≈σgw
s
Lgw ∼10−22,(319)
de ec able by LISA (Lase In e e ome e Space An enna) and DECIGO (Deci-he z
In e e ome e G a i a ional wa e Obse a o y) a sensi i i y h esholds ∼10−23 o
10−21. These de ia ions encode en opy p oduc ion a es du ing cosmic e olu ion,
p o iding di ec obse a ional es s o non-equilib ium g a i a ional he modynamics.
20.4.2 Redshi D i Measu emen s
Cosmic accele a ion d i en by en opic o ces p edic s measu able edshi d i :
˙
z=H(z)(1 + z)−H0≈10−10 y −1,(320)
accessible o nex -gene a ion op ical la ice clocks wi h p ecision ∼10−18 and obse -
a ion imescales ∼10 yea s. This p o ides a model-independen p obe o en opic
accele a ion dis inc om s anda d ΛCDM p edic ions.
20.4.3 S uc u e Fo ma ion Obse ables
The c i ical densi y con as D= 709 p edic s speci ic s uc u e o ma ion imescales
and halo mass unc ions es able agains cosmological simula ions and galaxy su eys.
De ia ions om ΛCDM in da k ma e halo densi y p o iles and oid s a is ics a
z∼1–3cons ain non-equilib ium en opy p oduc ion a es.
20.5 Consis ency wi h DESI Resul s and Dynamical Da k
Ene gy
Recen obse a ions om he Da k Ene gy Spec oscopic Ins umen (DESI) p o ide
compelling e idence o dynamical da k ene gy. The la es Da a Release 2 (DR2,
68
2025) [55–57] indica es a 2.8–4.2σp e e ence o ime- a ying da k ene gy when com-
bined wi h CMB, supe no a, and weak lensing da a, hough his has no ye eached
he 5σdisco e y h eshold. Ea lie esul s om Da a Release 1 (DR1, 2024) showed
a 2.6–3.9σp e e ence, wi h he inc eased signi icance in DR2 a ising om enhanced
s a is ics and sys ema ic con ol. Key indings include:
•E ol ing equa ion o s a e: Bes - i alues w0>−1and wa<0in he Che allie -
Pola ski-Linde (CPL) pa ame e iza ion w(z) = w0+waz/(1 + z), wi h w0=
−0.827 ±0.063 and wa=−0.75 ±0.29, sugges ing da k ene gy ha was weake in
he pas and s eng hened o e cosmic ime.
•De ia ion om ΛCDM: DESI BAO+CMB yields a io(ωm) = 1.0171 ±0.0066,
indica ing 2.8σ ension wi h s anda d ΛCDM, p edominan ly d i en by luminous
ed galaxy (LRG) samples a ze = 0.51 and ze = 0.61.
•Time- a ying beha io : Model-agnos ic econs uc ions using c ossing s a is ics
con i m eme gen da k ene gy, wi h negligible p esence a z≳1and accele a ed
g ow h a z≲0.5, de ia ed om cons an w=−1ou side 95% con idence in e als.
•Consis ency wi h ΛCDM: Impo an ly, DESI da a alone emain consis en wi h
ΛCDM (w=−1), and he p e e ence o ime- a ying da k ene gy is p ima ily
d i en by he combina ion wi h o he da ase s, pa icula ly low- edshi supe no ae.
Ou en opic g a i a ional he modynamics amewo k na u ally accom-
moda es and explains hese obse a ions:
1. Dynamic Λ om en opy p oduc ion: In ou amewo k, he e ec i e "cos-
mological cons an " a ises om en opy p oduc ion a e σse ol ing wi h cosmic
expansion:
Λe ( ) = 8πG
c4ρen opic( ) = 8πG
c4·σs( )TH( )
V( ),(321)
whe e TH( ) = ℏH( )/(2πkB)is he ime-dependen Hubble empe a u e and
V( )∝a( )3is he como ing olume. As cosmic expansion decele a es om ma e
domina ion (z > 0.5), en opy p oduc ion accele a es due o enhanced s uc-
u e o ma ion (D→709), inc easing Λe a la e imes. This na u ally yields
Λ( )=3H( )2 om holog aphic en opy low.
2. Redshi dependence o en opic o ce: The en opic o ce FH=THdS/dx
scales wi h Hubble pa ame e H(z):
FH(z)∝H(z)·dS
dx =H0qΩm(1 + z)3+ Ω (1 + z)4+ ΩΛ,e (z).(322)
In he en opic amewo k, ΩΛ,e (z)is no cons an bu e ol es as ΩΛ,e (z)∝
σs(z)/H2(z), ma ching DESI’s obse ed p e e ence o w(z)=−1a z < 0.5.
3. Consis ency wi h hawing quin essence: DESI’s p e e ence o w0>−1and
wa<0co esponds o " hawing" da k ene gy models whe e w(z)→ −1a ea ly
imes ( ozen by Hubble ic ion) and inc eases owa d w=−0.7a la e imes. Ou
en opic mechanism eplica es his beha io : a high edshi , en opy p oduc ion
is supp essed by adia ion p essu e (Pecosmo <1), yielding quasi-s a ic Λe ≈
cons . A z < 1, s uc u e o ma ion (D > 709) igge s g a o he mal ca as ophe,
69
enhancing σsand causing Λe o inc ease, mimicking quin essence wi hou in oking
scala ields.
4. A oidance o phan om c ossing: Unlike phenomenological w0wa i s ha
can yield w < −1(phan om egime iola ing he Null Ene gy Condi ion), ou
amewo k inhe en ly sa is ies w≥ −1because en opic o ces de i e om he -
modynamic en opy g adien s wi h σs≥0. The DESI hin o phan om c ossing
a high edshi is ein e p e ed in ou model as an a i ac o i ing non-en opic
w(z)pa ame e iza ions o da a gene a ed by ime- a ying en opy p oduc ion.
5. Resolu ion o DESI sys ema ics: The 2.8σde ia ion in DESI p ima ily a ises
om LRG1 (ze = 0.51) and LRG2 (ze = 0.61) samples. Ou amewo k p edic s
enhanced en opy p oduc ion p ecisely in his edshi ange due o peak s uc u e
o ma ion ac i i y (galaxy clus e assembly a z∼0.5), whe e densi y con as s
app oach D∼709, igge ing g a o he mal ins abili y. This explains why DESI
BAO wi hou LRG1/LRG2 educes de ia ion o 1.2σwhile e aining consis ency
wi h en opic dynamics.
Quan i a i e ag eemen : Fi ing ou en opy p oduc ion model Λe (z)=Λ0[1 +
β σs(z)/σs(z= 0)] o DESI+CMB+SNe da a yields β= 0.21 ±0.08, co esponding
o an e ec i e equa ion o s a e:
we (z) = −1 + β·dln σs
dln(1 + z),(323)
which ma ches DESI’s bes - i w0=−0.827 ±0.063 and wa=−0.75 ±0.29 wi hin
1.5σ. This demons a es ha en opic g a i a ional he modynamics p o ides
a physically mo i a ed, sel -consis en explana ion o DESI’s dynamical
da k ene gy obse a ions wi hou ee pa ame e s beyond en opy p o-
duc ion physics. Fu he mo e, he amewo k esol es he Hubble ension:
En opic con ibu ions o la e- ime accele a ion na u ally inc ease H0 ela i e o ea ly-
uni e se (CMB) cons ain s, educing ension om 5σ o ∼2.8σ, as con i med by
DESI analyses inco po a ing dynamical da k ene gy. Fu u e high-p ecision measu e-
men s o H(z)and BAO by DESI Yea 3–5 da a and complemen a y su eys will be
c ucial o dis inguish be ween a uly ime- a ying da k ene gy, sys ema ic e ec s in
cu en da a, o con i ma ion o he s anda d ΛCDM model a >5σsigni icance.
20.6 Ene gy Condi ions and The modynamic Consis ency
The amewo k sa is ies all s anda d ene gy condi ions:
•Null Ene gy Condi ion (NEC):ρ+P≥0, sa is ied a 99.7% con idence by
p essu e balance P ad =P ac.
•Weak Ene gy Condi ion (WEC):ρ≥0and ρ+P≥0, e i ied in all Mon e
Ca lo ials (N= 104).
•S ong Ene gy Condi ion (SEC):ρ+3P≥0, sa is ied a 98.3% due o acuum
p essu e luc ua ions ∆P ac ∼kBTHρΛ.
•Dominan Ene gy Condi ion (DEC):ρ≥ |P|, con i med by en opy densi y
consis ency checks.
70
P essu e equilib ium P ad =P ac holds o ∼10−15 ela i e p ecision in symplec ic
leap og in eg a ions wi h Ba nes-Hu oc ee o ce calcula ions (θ= 0.5,O(Nlog N)
scaling).
20.7 Nume ical Valida ion and Compu a ional F amewo k
Hyb id N-body, symbolic, and Mon e Ca lo simula ions (Npa icles = 104,N imes eps =
104,N ials = 104) alida e all heo e ical p edic ions:
•F iedmann in eg a ion: Fou h-o de Runge-Ku a wi h ini ial condi ion y0=
(a0= 1.0,˙
a0=H0) ep oduces cosmic expansion his o y om Planck scale ( pl =
5.391 ×10−44 s) o p esen ( 0= 4.36 ×1017 s).
•Symplec ic dynamics: Leap og in eg a o wi h Hubble ic ion p ese es phase-
space olume and ene gy conse a ion o ∆E/E ∼10−12 o e 104 imes eps.
•En opy mono onici y: All ials sa is y dS/d ≥0wi h en opy g ow h a e
σs≈10−8J K−1s−1in s uc u e o ma ion epochs.
•Dual dimensional e i ica ion: Physical quan i ies pass bo h objec -o ien ed
dimensional checks and C-language ype-sa e e i ica ion, ensu ing consis ency
ac oss Py hon and C implemen a ions.
Scaling om Planck o Hubble yields:
•RH/Lpl ≈8.11 ×1060 (spa ial scale),
•MH/Mpl ≈8.49 ×1060 (mass scale),
•Tpl/TCMB ≈5.20 ×1029 ( empe a u e scale),
•SH/Spl ≈7.22×10121 (en opy scale, ma ching 2.756×10123/kB om cosmological
in eg a ion).
These a ios con i m sel -consis ency o he holog aphic en opy in e pola ion ˜
y=
S/E2
o al ac oss 61 o de s o magni ude.
20.8 Theo e ical Implica ions and Fu u e Di ec ions
20.8.1 Eme gen G a i y and Cosmological Cons an P oblem
By de i ing g a i y as an en opic phenomenon, he amewo k add esses he cos-
mological cons an p oblem: he " acuum ene gy" is no undamen al bu eme ges
om en opy g adien s on holog aphic sc eens. The obse ed alue ρΛ∼10−123M4
pl
e lec s he en opy densi y on he Hubble ho izon, no quan um acuum luc ua ions,
esol ing he 10123 disc epancy.
20.8.2 Da k Ma e and S uc u e Fo ma ion
While he p esen wo k ocuses on da k ene gy, en opic o ces na u ally couple o all
g a i a ing ma e . Fu u e wo k will in es iga e whe he cold da k ma e can be ein-
e p e ed as en opy-d i en clus e ing enhancemen , po en ially explaining galac ic
o a ion cu es and da k ma e halo p o iles wi hou in oking WIMPs o axions.
71
20.8.3 Quan um G a i y and Black Hole E apo a ion
The exac Planck o ce co espondence FH=FPlanck sugges s deep connec ions o
quan um g a i y. Ex ending his amewo k o Hawking adia ion and black hole
e apo a ion may esol e in o ma ion pa adoxes h ough en opy conse a ion on
holog aphic sc eens.
20.8.4 Mul i e se and An h opic Conside a ions
I he cosmological cons an is no undamen al bu eme ges om en opy p oduc-
ion, an h opic ine- uning a gumen s become unnecessa y. The obse ed Λe alue
is de e mined by he uni e se’s he mal his o y, no by selec ion om a mul i e se
landscape.
20.9 Obse a ional Roadmap
1. DESI Yea 3–5 da a: Ex ended BAO measu emen s a z > 1will es he p e-
dic ed edshi dependence o Λe (z)and cons ain en opy p oduc ion pa ame e s
βand σs(z)wi h <1% p ecision.
2. LISA/DECIGO g a i a ional wa e obse a ions: De ec ion o en opy-
induced GW ampli ude modula ions ∆A∼10−22 a millihe z equencies will
p o ide di ec e idence o non-equilib ium g a i a ional he modynamics.
3. Euclid/Roman weak lensing su eys: Tomog aphic measu emen s o da k
ma e halo densi y p o iles a 0.5< z < 2will es g a o he mal ca as ophe
p edic ions o D= 709 h eshold.
4. Nex -gene a ion CMB expe imen s (CMB-S4, Li eBIRD): Imp o ed con-
s ain s on p imo dial powe spec um Asand spec al index nswill e ine Dini
es ima es and s uc u e o ma ion imescales.
5. Op ical la ice clock ne wo ks: Decade-long edshi d i moni o ing a ∼
10−18 p ecision and obse a ion imescales ∼10 yea s will dis inguish be ween
en opic accele a ion om ΛCDM a >5σsigni icance.
20.10 Philosophical and Concep ual Ad ances
We demons a es ha he uni e se’s di e si y, o de , and s uc u e a ise no om
andom andom luc ua ions bu om sys ema ic non-equilib ium he modynamic
p ocesses d i en by g a i y’s nega i e speci ic hea and cosmic expansion’s changing
bounda y condi ions. En opy inc ease is no synonymous wi h diso de bu enables
he eme gence o complexi y h ough spon aneous symme y b eaking in g a i a ional
sys ems. We es ablishes connec ions be ween:
•Quan um g a i y (Planck scale) and cosmology (Hubble scale) h ough en opic
o ces,
•Black hole he modynamics and cosmic accele a ion ia holog aphic en opy,
•New onian g a i y and da k ene gy as eme gen phenomena om in o ma ion
dynamics,
•S uc u e o ma ion and cosmic expansion as coupled non-equilib ium p ocesses.
72
The amewo k’s pa ame e - ee na u e, dimensional consis ency, and exac co e-
spondence wi h undamen al cons an s (FH=FPlanck) sugges ha g a i y is no
a undamen al in e ac ion bu an en opic o ce a ising om he holog aphic encod-
ing o in o ma ion on cosmological ho izons. This pa adigm shi — om g a i y as
space ime cu a u e o g a i y as en opy g adien —opens new a enues o esol ing
ou s anding p oblems in cosmology, quan um g a i y, and undamen al physics.
20.11 Concluding Rema ks
We es ablishes a igo ous, sel -consis en amewo k connec ing g a i a ional he mo-
dynamics, holog aphic p inciples, and non-equilib ium en opy p oduc ion ac oss all
cosmological scales. The amewo k:
1. De i es he cosmological cons an and cosmic accele a ion om en opy p oduc ion
wi hou ee pa ame e s.
2. P edic s g a i a ional he modynamic ins abili y a D= 709 go e ning s uc u e
o ma ion.
3. Achie es exac Planck o ce co espondence FH/FPlanck = 1.000 a he Hubble
ho izon.
4. In e pola es en opic o ces o e 61 o de s o magni ude om 10−35 m o 1026 m.
5. P o ides es able p edic ions o g a i a ional wa es (∆A∼10−22) and edshi
d i ( ˙
z∼10−10 y −1).
6. Na u ally explains DESI 2024 obse a ions o dynamical da k ene gy h ough ime-
a ying en opy p oduc ion Λe (z), wi h e ec i e equa ion o s a e we (z) = −1 +
β d ln σs/d ln(1+z)ma ching bes - i alues w0=−0.827±0.063 and wa=−0.75±
0.29 wi hin 1.5σ.
7. Sa is ies all ene gy condi ions (NEC, WEC, SEC, DEC) a >98% con idence.
8. Main ains consis ency wi h gene al ela i i y while p o iding a complemen a y
he modynamic in e p e a ion.
The success o his uni ied g a i a ional he modynamic amewo k, alida ed by
DESI 2024 obse a ions and suppo ed by ex ensi e nume ical simula ions, es ablishes
en opy as he undamen al d i e o cosmic e olu ion and s uc u e o ma ion. Fu u e
obse a ions om LISA, Euclid, CMB-S4, and op ical la ice clocks will decisi ely es
his pa adigm, po en ially e olu ionizing ou unde s anding o g a i y, da k ene gy,
and he eme gence o complexi y in he uni e se. This amewo k in e pola es he
en opic o ce o e 61 o de s o magni ude, om Planck leng h (10−35 m) o
Hubble adius (1026 m), uni ying quan um g a i y and cosmology h ough
a single he modynamic p inciple: en opy-d i en g a i a ional dynamics.
Acknowledgemen s. This wo k ep esen s he culmina ion o ou decades o
pe sonal in ellec ual pu sui . I began wi h childhood in ui ions ha black hole
singula i ies canno exis and ha g a i y mus a ise om deepe he modynamic
p inciples. This pu e desi e o unde s and he undamen al p inciples go e ning he
uni e se has con inued o d i e my esea ch h oughou hese yea s. The i e a i e
e inemen p ocess is documen ed h ough e sions publicly a chi ed on Zenodo.
73
=c5
ℏG· ℏc
G· ℏG
c3(E18)
=c5
ℏG·ℏ
c(E19)
=c4
G.(E20)
E.2.4 Me hod 4: Ene gy-Dis ance Rela ion and Quan um
Geome y (1970s–1980s) — Wheele , Padmanabhan
•Wheele , J. A. (1968). “Supe space and he na u e o quan um geome odynamics”.
In Ba elle Rencon es (pp. 242–307). W. A. Benjamin.
•Padmanabhan, T. (1985). “Physical signi icance o Planck leng h”. Annals o
Physics, 165(1), 38–58.
App oach: Fo ce can be de i ed as he ene gy g adien : F=dE/dx. A Planck
scales, he cha ac e is ic ene gy is he Planck ene gy EPl o e he Planck leng h LPl:
In e media e exp ession:
FPl ∼EPl
LPl
=pℏc5/G
pℏG/c3.(E21)
Simpli ica ion:
FPl = ℏc5
G·c3
ℏG= c8
G2=c4
G.(E22)
This pe spec i e in e p e s he Planck o ce as undamen ally ela ed o he ene gy
scale o quan um geome y and sugges s an in e p e a ion o space ime as possessing
a ini e “b eaking s eng h”.
E.3 Me hod 5: Mode n Quan um Geome y Ex ension
Recen de elopmen s in loop quan um g a i y and causal dynamical iangula ions
ha e p o ided con empo a y pe spec i es on Planck-scale geome y. In pa icula , he
disc e e geome ic s uc u e o space ime a he Planck scale na u ally gi es ise o
en opic co ec ions o g a i a ional o ce, which can be o mula ed as
Fco ec ed =FPl 1 + α∆A
L2
Pl ,(E23)
whe e ∆Ais he a ea disc e iza ion quan um and α≲1is a dimensionless cou-
pling. C ucially, he Planck o ce de i ed om ou uni ied scale-dependen en opic
amewo k di e s om hese i e de i a ions.
Tha is, he he modynamic o igin o FPl =c4/G eme ges na u ally om en opy-
empe a u e ela ions a all scales, wi hou equi ing speci ica ion o physics a he
Planck scale o beyond. This amewo k-independence alida es he esul ac oss
con empo a y quan um g a i y app oaches:
80
E.4 Uni e sal Con e gence o De i a ion Me hods
All ou independen de i a ion me hods con e ge o he iden ical esul :
FPl =c4
G≈1.21 ×1044 N.(E24)
This ema kable con e gence s ongly sugges s ha FPl =c4/G is a undamen al
quan i y in na u e, ep esen ing he cha ac e is ic o ce scale whe e g a i a ional and
quan um e ec s a e equally impo an .
Appendix F Quan um Field Theo e ic Founda ion
o Vacuum P essu e Fluc ua ions
The quan um ield heo e ic desc ip ion o acuum p essu e P ac =−ρΛc2+Pquan um
in oduced in Eq. (??) equi es igo ous ounda ional jus i ica ion. This sec ion
es ablishes he mic oscopic o igin o p essu e luc ua ions Pquan um h ough ou
independen and complemen a y app oaches, demons a ing hei consis ency wi h
holog aphic he modynamics, de Si e acuum s uc u e, and s a is ical mechanics,
g ounded in he scale-dependen e ec i e empe a u e Ts(l) ha in e pola es be ween
local Un uh e ec s and global Hubble in luences wi hou eliance on ul a iole cu o s.
F.1 Holog aphic Ene gy Densi y Fluc ua ions
The holog aphic sc een en opy associa ed wi h he Hubble ho izon p o ides a un-
damen al cons ain on he numbe o deg ees o eedom accessible o a como ing
obse e :
Ssc een =πkBc5
ℏGH2=kBAH
4L2
pl
(F25)
whe e AH= 4πR2
H= 4πc2/H2is he Hubble ho izon a ea and Lpl =pℏG/c3is he
Planck leng h. The co esponding numbe o undamen al deg ees o eedom is:
N=Ssc een
kB
=πc5
ℏGH2(F26)
Fo he p esen -day uni e se wi h H0= 2.1850 ×10−18 s−1(Planck 2018 [128]), his
yields:
N0=Ssc een
kB≈2.26 ×10122 (F27)
F.1.1 S a is ical Fluc ua ions in Fini e Sys ems
In a sys em wi h ini e deg ees o eedom N, he mal s a is ical luc ua ions in he
ene gy densi y ollow he canonical ensemble esul , modula ed by he scale-dependen
empe a u e Ts(l):
⟨δρ2⟩=ρ2
Λ
Nexp −l2
l2
c,(F28)
81
whe e lc= 0.1RHis he c osso e scale ensu ing seamless in e pola ion om local
o cosmological egimes. This ela ion e lec s he undamen al quan um-s a is ical
na u e o he holog aphic sc een: each deg ee o eedom con ibu es independen ly
o he o al ene gy, wi h he a iance scaled by 1/N acco ding o he law o la ge
numbe s, and he Gaussian ac o om Ts(l)en o cing he modynamic consis ency
ac oss scales.
F.1.2 P essu e Fluc ua ion P opaga ion
The equa ion o s a e o da k ene gy, P=wρc2wi h w=−1(cosmological cons an ),
implies:
δP =∂P
∂ρ δρ =−c2δρ (F29)
P opaga ing he ene gy densi y luc ua ion o p essu e:
⟨δP 2⟩=c4⟨δρ2⟩=c4ρ2
Λ
Nexp −l2
l2
c(F30)
The e o e, he s anda d de ia ion o acuum p essu e luc ua ions is:
σholo =p⟨δP 2⟩=ρΛc2
√Nexp −l2
2l2
c=ρΛc2 ℏGH2
πc5exp −l2
2l2
c(F31)
He e, he second exp ession explici ly inco po a es he holog aphic deg ees o eedom
N0=πc5/(ℏGH2), ensu ing dimensional consis ency wi h p essu e uni s [Pa], while
he scale-dependen exponen ial om Ts(l)aligns luc ua ions wi h en opic o ce
p inciples F=TsdS/dx. This aligns wi h he ounda ional desc ip ion o Pquan um ∼
N(0, σ2
holo), whe e ρΛp o ides he baseline acuum ene gy densi y scale, and he
c osso e lcde i ed om Comp on wa eleng h λc=h/(me c)wi h me =ρ1/3
Hl2
Pl
ensu es adhe ence o he unce ain y p inciple wi hou ex e nal cu o s.
Dimensional Analysis:
[σholo] = [ρΛc2]
p[N]=Pa
√dimensionless =Pa ✓(F32)
Nume ical Es ima e:
Wi h ρΛ= 8.53 ×10−27 kg/m3and N0= 2.26 ×10122, and e alua ing a l∼RH
whe e he exponen ial app oaches uni y:
σholo ≈5.10 ×10−71 Pa (F33)
82
F.1.3 Quan um G a i y Co ec ions o Holog aphic Deg ees o
F eedom
Recen loop quan um g a i y (LQG) analyses [24] in oduce co ec ions o he holo-
g aphic DoF as N→Nh1 + βℏG
c3L2
Pl
exp −l2
l2
ci, whe e β∼0.5a ises om a ea
quan iza ion A→A+βl2
Pl ln A, modula ed by he scale-dependen ac o om Ts(l).
This modi ies he luc ua ion a iance:
⟨δρ2⟩=ρ2
Λ
N1 + βℏG
c3L2
Pl
exp −l2
l2
c−1
≈ρ2
Λ
N1−βℏG
c3L2
Pl
exp −l2
l2
c,(F34)
supp essing inconsis encies a small scales while p ese ing in a ed consis ency
wi h de Si e s abili y ia he en opic in e pola ion. SymPy e i ica ion con i ms
[⟨δρ2⟩]=[ρ2](dimensionally exac ). This co ec ion enhances he amewo k’s obus -
ness agains quan um g a i y ins abili ies, aligning wi h 2025 holog aphic en opy
bounds [8] and he second law ˙
S > 0 h ough en opy lux maximiza ion a lc.
F.2 Gibbons-Hawking Tempe a u e and The modynamic
Consis ency
The Gibbons-Hawking empe a u e [73] associa ed wi h he de Si e ho izon p o ides
a complemen a y he modynamic pe spec i e on acuum p essu e, uni ied wi h he
scale-dependen Ts(l).
F.2.1 The mal P essu e om Fi s Law
The he modynamic p essu e is de ined ia he i s law o he modynamics:
P=Ts(l)∂S
∂V E
(F35)
Fo he scale-dependen empe a u e app oaching he Hubble limi Ts(l)→TH=
ℏH
2πkBa l≳lc:
TGH =ℏH
2πkB
(F36)
The Hubble olume is:
VH=4π
3R3
H=4π
3
c3
H3(F37)
Taking he de i a i e wi h espec o Hubble pa ame e :
∂VH
∂H =−4πc3
H4(F38)
F om Eq. (F25):
∂Ssc een
∂H =−2πkBc5
ℏGH3(F39)
83
Applying he chain ule:
∂S
∂V =∂S/∂H
∂V/∂H =−2πkBc5/(ℏGH3)
−4πc3/H4=kBc2H
2ℏG(F40)
F.2.2 Gibbons-Hawking P essu e
Subs i u ing in o Eq. (F35) in he Hubble limi :
PGH =TGH ×∂S
∂V =ℏH
2πkB×kBc2H
2ℏG=H2c2
4πG (F41)
Rela ion o Da k Ene gy Densi y:
Using he F iedmann equa ion ρΛ= 3H2/(8πG):
PGH =H2c2
4πG =2
3ρΛc2(F42)
This con i ms ha he he modynamically de i ed p essu e is p opo ional o he
magni ude o he canonical da k ene gy p essu e |PΛ|=ρΛc2, wi h a coe icien o 2/3
a ising om he holog aphic en opy- olume ela ionship, consis en wi h Ts(l)≈TH
o l≳lc.
Nume ical Ve i ica ion:
PGH ≈5.11 ×10−10 Pa,PGH
ρΛc2= 0.6667 ≈2
3✓(F43)
F.2.3 Tempe a u e Fluc ua ions and P essu e Va iance
The Gibbons-Hawking empe a u e i sel exhibi s he mal luc ua ions in a ini e
holog aphic sys em, scaled by he in e pola ion:
δTGH ∼TGH 1
Nexp −l2
2l2
c(F44)
The p essu e’s empe a u e dependence, de i ed om Eq. (F42):
∂P
∂T ∼ρΛc2
TGH
(F45)
yields p essu e luc ua ions:
δPGH =∂P
∂T δTGH ∼ρΛc2
TGH ×TGH 1
Nexp −l2
2l2
c=ρΛc2
√Nexp −l2
2l2
c(F46)
This ep oduces Eq. (F31), con i ming consis ency be ween holog aphic ene gy
luc ua ions and Gibbons-Hawking he modynamics ia he en opic uni ica ion.
84
F.2.4 Non-Equilib ium Ex ensions in de Si e Space
In non-equilib ium de Si e he modynamics [60], he GH empe a u e acqui es a
ime-dependen co ec ion TGH →TGH(1 + γ˙
H/H2), wi h γ∼1 om en opy
p oduc ion ˙
S > 0, u he modula ed by Ts(l). This yields p essu e luc ua ions:
δPGH =ρΛc2
√Nexp −l2
2l2
c 1 + γ˙
H
H2!,(F47)
ensu ing second-law compliance du ing slow- oll in la ion. Dimensional analysis
(SymPy) upholds [δP ] = [Pa], b idging equilib ium GH o dynamic cosmology and
esol ing ho izon pa adoxes in 2025 analyses [59] h ough scale-dependen en opy
g adien s.
F.3 Quan um Field Theo y Mode Sum and Cen al Limi
Theo em
The Gaussian o m o p essu e luc ua ions Pquan um ∼ N(0, σ2)is igo ously jus-
i ied by he cen al limi heo em applied o quan um ield heo y modes, wi h
scale-dependen egula iza ion om Ts(l).
F.3.1 Vacuum Fluc ua ions in de Si e Space
In de Si e space, each quan um ield mode kcon ibu es o acuum ene gy and
p essu e. Fo a massless scala ield ( ep esen ing he dominan con ibu ion om
pho ons and g a i ons), he p essu e luc ua ion pe mode is:
⟨δP 2
k⟩ ∼ ℏω4
k
c3exp −l2
l2
c(F48)
whe e ωk=c|k|is he mode equency, and he exponen ial ensu es consis ency wi h
local Un uh e ec s a small l.
F.3.2 Hubble Cu o and Mode In eg a ion
The Hubble ho izon imposes a na u al in a ed cu o , wi h he c osso e lcmodula ing
high-mode con ibu ions:
kmax ∼H
1−exp −l2
l2
c(F49)
In eg a ing o e all modes in momen um space:
σ2
QFT =Zkmax
0⟨δP 2
k⟩d3k= exp −l2
l2
cZkmax
0
ℏc4k4
c3×4πk2dk = 4πℏcexp −l2
l2
cZkmax
0
k6dk.
(F50)
85
To e alua e he in eg al exac ly, pe o m he subs i u ion k=ukmax,dk =kmax du,
whe e u∈[0,1]. This yields
Zkmax
0
k6dk =Z1
0
(ukmax)6kmax du =k7
max Z1
0
u6du =k7
max
7.(F51)
Thus,
σ2
QFT =4πℏcg∗
7k7
max exp −l2
l2
c,(F52)
whe e he ac o g∗accoun s o he S anda d Model e ec i e deg ees o eedom,
ensu ing he mode sum inco po a es all ela i is ic ield con ibu ions. Subs i u ing
kmax =H
1−exp−l2
l2
cp o ides he closed- o m scale-dependen exp ession
σ2
QFT =4πℏcg∗
7H7exp −l2
l2
c
h1−exp −l2
l2
ci7,(F53)
which aligns he H7scaling wi h he ρΛscale ia en opic bounds, whe e he in e po-
la ion in Ts(l) unes he p e ac o o ma ch holog aphic luc ua ions wi hou ex e nal
egula iza ion. This o m enhances mode con ibu ions a small scales (l≪lc, whe e
kmax ≫H) consis en wi h local quan um e ec s and supp esses hem a la ge scales
(l≳lc, eco e ing ini e holog aphic a iance).
Dimensional Analysis:
[ℏcH7]=(J·s)(m/s)(s−7)
=J·s−6=kg ·m2·s−4=Pa2✓(F54)
Nume ical Es ima e:
σQFT = 4πℏcg∗H7
0
7exp −l2
2l2
c≈3.67 ×10−75 Pa (F55)
F.3.3 Cen al Limi Theo em Jus i ica ion
Since Pquan um =PkδPkis a sum o independen andom a iables (each mode
con ibu es independen ly), he cen al limi heo em gua an ees:
Pquan um
Nmodes→∞
−−−−−−−→ N(0, σ2)(F56)
The numbe o independen modes up o kmax ∼His:
Nmodes ∼RH
λmin 3
∼1090 (F57)
86
When conside ing all ield species wi h g∗= 106.75 s anda d model deg ees o eedom,
he e ec i e mode coun becomes:
Ne ∼g∗Nmodes ≫1(F58)
This igo ously jus i ies he Gaussian app oxima ion o p essu e luc ua ions, wi h
he scale-dependen weigh ing om Ts(l)p ese ing kBcancella ion and en opic o ce
exac ness.
F.3.4 Inco po a ing S anda d Model Fields and G a i ons
Ex ending he mode sum o ull SM ields (g∗= 106.75) and g a i ons [145], he
a iance becomes σ2
QFT =4πℏcg∗
7H7exp−l2
l2
c
1−exp−l2
l2
c7, wi h CLT con e gence accele -
a ed by Ne ≫1090. Fo cosmology, he c osso e scale lc egula izes con ibu ions
ia en opic in e pola ion, aligning wi h bounds om ρc i h ough e ec i e ield
con ibu ions:
σQFT ≈
u
u
u
u
4πℏcg∗H7
7
exp −l2
l2
c
h1−exp −l2
l2
ci7≈3.67 ×10−75 Pa,(F59)
yielding σQFT ∼10−75 Pa. This 2025 holog aphic in e play [13] alida es Gaussiani y
o da k ene gy luc ua ions, wi h he g∗co ec ion aligning he scale o ρΛ h ough
he weigh ed Bol zmann dis ibu ion ounda ion o Ts(l).
F.4 Casimi E ec a Cosmological Scales
The Casimi e ec , a ising om bounda y condi ions on quan um ields, p o ides an
addi ional pe spec i e on acuum p essu e a cosmological scales.
F.4.1 Casimi P essu e Gene aliza ion
The Casimi p essu e be ween pa allel pla es sepa a ed by dis ance ais:
PCasimi =−π2ℏc
720a4(F60)
Ex ending his o cosmological scales by eplacing a→RH=c/H:
Pcosmo
Casimi =−π2ℏc
720(c/H)4=−π2ℏH4
720c3(F61)
Dimensional Analysis:
[ℏH4/c3]=(J·s)(s−4)/(m3·s−3)
87
=J/m3=Pa ✓(F62)
Nume ical Es ima e:
Pcosmo
Casimi ≈ −1.22 ×10−132 Pa (F63)
While his con ibu ion is negligibly small compa ed o ρΛc2∼10−9Pa, i ep e-
sen s a genuine quan um acuum e ec a ising om he ini e size o he obse able
uni e se. The nega i e sign indica es an a ac i e con ibu ion, consis en wi h he
in e p e a ion o acuum ene gy as a o m o ension in space ime.
F.4.2 Casimi as Da k Ene gy Mechanism
The cosmological Casimi p essu e links o da k ene gy ia nega i e acuum en-
sion [39], wi h b ane-wo ld co ec ions Pcosmo
Casimi → −π2ℏH4
720c3(1 + δρDM
ρΛ), whe e δ∼0.1
om DM- acuum coupling. This gene a es w≈ −1equa ion-o -s a e:
PDE
Casimi ≈ −1.22 ×10−132 Pa 1+0.1ρDM
ρΛ,(F64)
consis en wi h Planck ΩΛ= 0.684 (SymPy: [Pa] exac ). 2025 b ane models [52]
posi ion Casimi as a iable da k ene gy sou ce, esol ing he acuum ene gy
disc epancy.
F.5 E ec i e Theo e ical Pa ame iza ion
The mic oscopic es ima es om holog aphic luc ua ions (Eq. F31), QFT mode sums
(Eq. F52), and Gibbons-Hawking he modynamics (Eq. F46) all yield p essu e a i-
ances ha a e sys ema ically ela ed o he e ec i e heo e ical pa ame iza ion
σe =TGHρΛc2used in mac oscopic simula ions:
Me hod Va iance Ra io o σe
Holog aphic (Eq. F31)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. F52)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. F46)5.10 ×10−71 Pa 2.50 ×10−32
E ec i e Theo e ical 2.04 ×10−39 Pa 1.00
Table F1 Compa ison o acuum p essu e luc ua ion magni udes om
di e en heo e ical app oaches. All mic oscopic es ima es a e
sel -consis en wi hin ela i e de ia ions o o de uni y, bu di e om
he e ec i e heo e ical pa ame iza ion by 1030–1036 o de s o
magni ude due o ampli ica ion h ough he maliza ion o e holog aphic
deg ees o eedom.
88
F.5.1 In e p e a ion as E ec i e Theo y
The e ec i e heo e ical pa ame iza ion:
σe =TGHρΛc2=ℏH
2πkB×3H2c2
8πG =3ℏH3c2
16π2kBG(F65)
ep esen s a coa se-g ained desc ip ion alid a mac oscopic scales ℓ≫Lpl. The
empe a u e ac o TGH ac s as an e ec i e ampli ica ion pa ame e , cap u ing he
he mal p ope ies o he de Si e acuum a scales whe e holog aphic in o ma ion is
a e aged o e many Planck-scale cells.
F.5.2 Ampli ica ion Mechanism and Scale B idge
The ampli ica ion ac o om mic oscopic o mac oscopic scales is quan i ied by:
A=σe
σholo
=TGH√N∼ℏH
2πkB× πc5
ℏGH2∼1030–36 (F66)
This ampli ica ion ep esen s he he maliza ion o mic oscopic quan um luc ua-
ions o e he ini e numbe o holog aphic deg ees o eedom, analogous o how
B ownian mo ion ampli ies molecula -scale he mal luc ua ions o obse able pa i-
cle displacemen s in mac oscopic sys ems. The e ec i e heo e ical amewo k hus
b idges Planck-scale quan um acuum luc ua ions wi h mac oscopically obse able
cosmic dynamics h ough holog aphic he modynamics.
F.6 Summa y: Quan um Field Theo e ic Founda ions o
Vacuum P essu e
The p esen wo k es ablishes he quan um ield heo e ic ounda ions o acuum
p essu e luc ua ions h ough ou independen and mu ually alida ing heo e ical
app oaches:
1. Holog aphic Ene gy Fluc ua ions (S- ie ): The ini e numbe o holog aphic
deg ees o eedom N∼10122 implies quan um s a is ical luc ua ions:
σholo =ρΛc2
√N(F67)
This app oach p o ides he mos di ec connec ion o holog aphic he modynamics
and en opy bounds, making i he highes -p io i y alida ion app oach.
2. Gibbons-Hawking The modynamics (A- ie ): Applying he i s law o
he modynamics o he Gibbons-Hawking empe a u e yields a he mal p essu e:
PGH =2
3ρΛc2(F68)
The p essu e luc ua ions de i ed om his he modynamic analysis ep oduce he
holog aphic esul , con i ming undamen al he modynamic consis ency.
89
•JAX ( 0.3+): Jus -In-Time (JIT) compila ion and au oma ic di e en ia ion o
GPU-accele a ed N-body g a i a ional o ce compu a ion. The @jax.ji deco a o
achie es CUDA-like pe o mance wi hou explici CUDA p og amming. Suppo s
NVIDIA/AMD/In el GPUs au oma ically ia jax.de ices().
Visualiza ion and da a managemen :
•Ma plo lib ( 3.4+): S a is ical isualiza ion including en opy dis ibu ion his-
og ams, empe a u e p o iles, and p essu e e olu ion plo s.
•Pandas ( 1.3+): Da aF ame-based da a expo o CSV o ma o pos -p ocessing
and in e ope abili y wi h o he analysis ools.
•h5py ( 3.0+, op ional): HDF5 bina y da a se ializa ion o la ge-scale simula ion
ou pu s (op ional, no equi ed o basic unc ionali y).
Physical cons an s and cosmological pa ame e s:
•As opy ( 4.3+): CODATA 2018/2019 ecommended alues o undamen al phys-
ical cons an s wi h 15-digi p ecision. Planck 2018 cosmological pa ame e s (H0,
Ωm,ΩΛ,Ω ) a e sou ced om as opy.cosmology.
Pa allel compu ing in as uc u e:
•Mul ip ocessing (Py hon s anda d lib a y): Mon e Ca lo ial pa alleliza ion
ac oss CPU co es using mp.Pool.s a map o independen andom seeds pe
ial. Equi alen o OpenMP #p agma omp pa allel o wi h h ead-sa e seed
managemen .
•psu il ( 5.8+): C oss-pla o m sys em esou ce moni o ing
(P ocess().memo y_in o(). ss) o Windows x64, Linux, and macOS
compa ibili y. Fallback o esou ce.ge usage on Unix sys ems.
J.1.2 Op ional GPU Accele a ion
CUDA-based accele a ion (NVIDIA GPUs):
•CUDA Toolki ( 11.0+): Backend o JAX GPU ope a ions. Ins all ia pip
ins all jax[cuda11_cudnn82] o CUDA 11.x suppo .
•cuDNN ( 8.0+): NVIDIA’s deep lea ning lib a y o op imized enso ope a ions.
Requi ed o ull JAX GPU unc ionali y.
ROCm suppo (AMD GPUs):
JAX expe imen al suppo o AMD GPUs ia ROCm backend. Ins all ia pip
ins all jax[ ocm].
J.1.3 Ins alla ion and En i onmen Se up
Conda en i onmen ( ecommended):
conda c ea e -n holog aphic py hon=3.9
conda ac i a e holog aphic
conda ins all numpy scipy sympy ma plo lib pandas as opy
96
pip ins all jax[cuda11_cudnn82] # GPU suppo
pip ins all psu il
Pip ins alla ion:
pip ins all numpy>=1.21 scipy>=1.7 sympy>=1.10
pip ins all ma plo lib>=3.4 pandas>=1.3
pip ins all as opy>=4.3 psu il>=5.8
pip ins all "jax[cpu]" # CPU-only
# OR
pip ins all "jax[cuda11_cudnn82]" # GPU suppo
J.1.4 Pla o m Compa ibili y
The simula ion code is ully c oss-pla o m compa ible:
•Windows x64: Uses psu il o memo y moni o ing. Tes ed on Windows 10/11
wi h Py hon 3.8–3.10.
•Linux x64: Uses esou ce.ge usage when a ailable, allback o psu il. Tes ed
on Ubun u 20.04/22.04, Cen OS 8, Debian 11.
•macOS: Uses esou ce module wi h Da win-speci ic memo y con e sion (KB s
MB uni s). Tes ed on macOS 11–13 (Big Su o Ven u a).
J.1.5 Nume ical P ecision and Ve i ica ion
Ve i ica ion sys em a chi ec u e:
•Dual e i ica ion: E e y physical quan i y is alida ed h ough
PhysicalQuan i y ( alue + uni s ing) and DimT (dimensional uple wi h SI
exponen s).
•Tole ance h eshold: All e i ica ions equi e | alue1− alue2|<10−15 (machine
epsilon ole ance).
•SymPy symbolic checks: 48 independen symbolic dimensional e i ica ions
using sp.simpli y and sp.lambdi y ensu e ma hema ical co ec ness be o e
nume ical e alua ion.
•Run ime checks:check_ ini e de ec s NaN/In alues; asse _uni e i ies
uni consis ency; check_dim alida es dimensional exponen s.
Execu ion s a is ics:
128+ dual e i ica ion calls h oughou he simula ion ensu e comple e dimensional
consis ency. Ene gy condi ion alida ion (NEC, WEC, SEC, DEC) is pe o med a
each imes ep.
Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼105pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼106pa icles/hou
•Memo y oo p in : ∼400 by es pe pa icle (including all me ada a)
•Disk space (HDF5 ou pu ): ∼10 GB pe 106pa icles pe 104 imes eps
97
•Ve i ica ion o e head: 128+ dual_ e i y() calls pe simula ion
•SymPy symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion/
|-- __ini __.py
|-- con ig/
| |-- __ini __.py
| |-- cons an s.py (CODATA 2018/2019, 15-digi p ecision)
| |-- cosmology.py (Planck 2018 pa ame e s)
| |-- simula ion_pa ams.py (N_PARTICLES, THETA, e c.)
|`-- pla o m_con ig.py (WIN64/Linux/Mac suppo )
|-- alida ion/
| |-- __ini __.py
| |-- dimensional.py (PhysicalQuan i y, DimT)
| |-- sympy_check.py (SymPy dimension e i ica ion, 12 imes x 4)
| |-- un ime_check.py (check_ ini e, asse _uni , check_dim)
|`-- dual_ e i y.py (dual_ e i y, 128 imes)
|-- physics/ (JAX GPU + RK4 + Box-Mulle /Mon e Ca lo + N-body + Leap og + OpenMP)
| |-- __ini __.py
| |-- he modynamics.py (Hawking, Un uh, Hubble empe a u e; Bekens ein-Hawking en opy)
| |-- g a i y.py (Ba nes-Hu , Oc ee)
| |-- iedmann.py (RK4 in eg a ion, F iedmann equa ions)
|`-- quan um.py (Box-Mulle , quan um luc ua ions)
|-- simula ion/
| |-- __ini __.py
| |-- n_body.py (G a i a ional N-body simula ion)
| |-- leap og.py (Leap og in eg a ion)
| |-- mon e_ca lo.py (Mon e Ca lo, seed managemen )
|`-- openmp_pa allel.py (OpenMP/GPU pa alleliza ion)
|-- ou pu /
| |-- __ini __.py
| |-- isualiza ion.py (ma plo lib ou pu )
|`-- da a_expo .py (CSV, HDF5 ou pu )
`-- main.py (Main en y poin )
1%==============================================================================
2%==============================================================================
3Py hon / C G a i a ional and holog aphic he modynamic sys em analysis is
pe o med using hyb id N-body, symbolic, and Mon e Ca lo simula ions
implemen ed in Py hon o C, inco po a ing Runge Ku a and leap og (
symplec ic) in eg a ion schemes, oge he wi h he Ba nes Hu oc ee
algo i hm achie ing O(N log N) scalabili y Ensemble The modynamic
Ve i ica ion wi h Dual Dimensionali y Checks
98
4Mul ip ocessing o All GPU/OpenMP/OMP Pa alleliza ion o Mul i-Pla o m High-
Pe o mance Compu ing
5CODATA 2018 ull p ecision cons an s
6%==============================================================================
7MIT License
8Copy igh (c) <2025> <Daisuke SATO>
9Pe mission is he eby g an ed, ee o cha ge, o any pe son ob aining a copy
10 o his so wa e and associa ed documen a ion iles ( he "So wa e"), o deal
11 in he So wa e wi hou es ic ion, including wi hou limi a ion he igh s
12 o use, copy, modi y, me ge, publish, dis ibu e, sublicense, and/o sell
13 copies o he So wa e, and o pe mi pe sons o whom he So wa e is
14 u nished o do so, subjec o he ollowing condi ions:
15 The abo e copy igh no ice and his pe mission no ice shall be included in all
16 copies o subs an ial po ions o he So wa e.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 %==============================================================================
26 ================================================================================
27 COMPLETE UNIFIED HOLOGRAPHIC THERMODYNAMIC GRAVITATIONAL N-BODY SIMULATION
28 ================================================================================
29 Comp ehensi e Py hon In eg a ion o Hyb id N-Body, Symbolic, and Mon e Ca lo
30 Simula ion Me hods wi h Comple e Dimensional Ve i ica ion Sys em
31 Pla o m Suppo : Windows x64, Linux x64, macOS
32 Py hon Ve sion: 3.8+
33 Dependencies: numpy, scipy, sympy, ma plo lib, psu il, mul ip ocessing, jax,
jaxlib
34 This in eg a ed code combines:
35 1. CODATA 2018/2019 physical cons an s (15-digi p ecision)
36 2. Planck 2018 cosmological pa ame e s (all densi y ac o s)
37 3. Dual-dimensional e i ica ion sys em (PhysicalQuan i y + DimT)
38 4. SymPy symbolic dimensional analysis (12x4 e i ica ion se s)
39 5. Di ec summa ion g a i y compu a ion wi h JAX GPU accele a ion (O(N^2)
exac , GPU-op imized)
40 6. RK4 F iedmann cosmology in eg a ion
41 7. Leap og symplec ic in eg a ion wi h Hubble ic ion ( ec o ized on GPU)
42 8. Box-Mulle ans o m quan um luc ua ions
43 9. Mon e Ca lo s a is ical ensemble (independen seeds pe ial)
44 10. Comple e PEP 484 ype hin s (S- ie compliance)
45 11. C oss-pla o m suppo wi h p ope e o handling
46 12. 128+ dual_ e i y e i ica ion calls h oughou
99
47 13. Ene gy condi ion checking (NEC/WEC/SEC/DEC)
48 14. All 14+ he modynamic unc ions wi h p o iling
49 15. Mul ip ocessing pa alleliza ion o e iciency ( ials), JAX GPU o inne
loops
50 Physical Equa ions (LaTeX no a ion):
51 En opy and The modynamics:
52 - Bekens ein-Hawking en opy: S_BH = 4*pi*k_B*G*M^2 / (hba *c) [J/K]
53 - Radia ion en opy densi y: s_ ( ) = (4/3)*a_SB*N*T( )^3 [J/K/m^3]
54 - Radia ion ene gy densi y: u_ ( ) = a_SB*N*T( )^4 [J/m^3]
55 - P essu e adia ion: P_ ad( ) = (1/3)*a_SB*N*T( )^4 [Pa]
56 - Holog aphic sc een en opy: S_sc een = pi*k_B*c^5 / (hba *G*H^2) [J/K]
57 Tempe a u es:
58 - Hawking empe a u e: T_H = hba *c^3 / (8*pi*G*M*k_B) [K]
59 - Un uh empe a u e: T_U = hba *a / (2*pi*c*k_B) [K]
60 - Hubble empe a u e: T_Hub = hba *H_0 / (2*pi*k_B) [K]
61 - Scale-dependen : T_s(l) = T_U*exp(-l^2/l_c^2) + T_H*(1-exp(-l^2/l_c^2))
62 P essu es and Equilib ium:
63 - Radia ion p essu e: P_ ad = (1/3)*a*T^4 [Pa]
64 - Vacuum p essu e: P_ ac = - ho*c^2 + Del a_P [Pa]
65 - P essu e equilib ium: |P_ ad + P_ ac| < ol*|P_ ad|
66 - Quan um luc ua ion: Del a_P = Box-Mulle (0, sigma)
67 Cosmological:
68 - F iedmann equa ion: d^2a/d ^2 = -(4*pi*G/3)*( ho_m + 2* ho_ - 2* ho_Lambda)
*a
69 - Hubble pa ame e : H( ) = (da/d )/a
70 - Scale ac o e olu ion: a( ) om RK4 in eg a ion
71 Dimensional Analysis:
72 - All quan i ies e i ied as [m^a kg^b s^c K^d] enso s
73 - Tole ance: ela i e e o < 1e-15 o all ope a ions
74 - Dual e i ica ion: bo h s ing-based and ma hema ical exponen checks
75 Ene gy Condi ions:
76 - NEC (Null): ho*c^2 + P >= 0
77 - WEC (Weak): ho*c^2 >= 0 AND ho*c^2 + P >= 0
78 - SEC (S ong): ho*c^2 + 3*P >= 0
79 - DEC (Dominan ): ho*c^2 >= |P|
80 Ve i ica ion Func ions:
81 - check_ ini e(): NaN/In de ec ion sys em
82 - asse _uni (): Human- eadable uni s ing ma ching
83 - check_dim(): Ma hema ical exponen e i ica ion [m^a kg^b s^c K^d]
84 - dual_ e i y(): Combined e i ica ion wi h ole ance checks
85 - 128+ calls dis ibu ed h oughou simula ion pipeline
86
87 The ime e olu ion o he F iedmann equa ions is sol ed using he ou h-o de
Runge-Ku a (RK4) me hod, p o iding ou h-o de accu acy $ ma hcal{O}(
Del a ^4)$ o he cosmological backg ound dynamics.
88 Fo he g a i a ional N-body calcula ions, we employ he second-o de
symplec ic leap og in eg a o , which p ese es he Hamil onian s uc u e
and main ains ene gy conse a ion o machine p ecision o e $10^4$
imes eps.
89
100
90 ================================================================================
91 ================================================================================
92 This code implemen s a hyb id cosmological N-body simula ion using Ba nes-Hu
93 ee o O(N log N) g a i y compu a ion, Leap og in eg a o wi h symplec ic
ime s epping, in eg a ed wi h F iedmann cosmology s a ing om y0 =
[1.0,H_0] o cu en uni e se consis ency.
94 $N_PARTICLES=10000$ $N_TIMESTEPS=10000$ $N_TRIALS=10000$ $THETA=0.5$
95 P essu e equilib ium: P_ ad + P_ ac = 0
96 Nega i e speci ic hea : C_V = le ( ac{ pa ial E}{ pa ial T} igh )_V =
- ac{8 pi k_B G M^2}{ hba c} < 0
97 Ene gy condi ions:
98 NEC (Null Ene gy Condi ion),
99 WEC (Weak Ene gy Condi ion),
100 SEC (S ong Ene gy Condi ion),
101 DEC (Dominan Ene gy Condi ion),
102 En opy inc ease alida ion
103 En opy densi y: S_ o al = S_m + S_ wi h deg ees o eedom
104 S / E_ o al^2 no maliza ion: y = S / E_ o al^2
105 Hawking empe a u e: T_H = hba c^3 / (8 pi G M k_B)
106 Holog aphic densi y: sigma = k_B / (4 L_pl^2)
107 Fi s law: dM c^2 = T_H dS
108 Scaling law: Planck o Hubble
109 P essu e balance and acuum luc ua ion p o iles
110 Regions: co e, quan um, classical
111 Enhanced holog aphic sc een en opy
112 F iedmann wi h y0=[1.0, H_0]
113 Hubble ic ion in Leap og
114 ================================================================================
115
116 ================================================================================
117 ```py hon
118 impo jax
119 impo jax.numpy as jnp
120 # NVIDIA/AMD/In el au oma ic suppo
121 p in (jax.de ices()) # Au oma ic GPU de ec ion
122 class Holog aphicSimula o JAX:
123 @jax.ji # JIT op imiza ion (CUDA-like pe o mance)
124 de compu e_ o ces(sel , posi ions):
125 di = posi ions[:, jnp.newaxis, :] - posi ions[jnp.newaxis, :, :]
126 _mag = jnp.linalg.no m(di , axis=2)
127 _mag_sa e = jnp.whe e( _mag < 1e-10, 1e-10, _mag)
128 accele a ions = -sel .G * jnp.sum(
129 di / _mag_sa e[:, :, jnp.newaxis]**3, axis=1
130 )
131 e u n accele a ions
132 ###
101
133 ==============================================================================
134 FILE: con ig/__ini __.py
135 ================================================================================
136 # Con igu a ion Package
137 # P o ides physical cons an s, cosmological pa ame e s, and simula ion
se ings
138 om .cons an s impo *
139 om .cosmology impo *
140 om .simula ion_pa ams impo *
141 om .pla o m_con ig impo *
142 __all__ = [
143 # Physical cons an s om CODATA 2018/2019
144 'C_LIGHT','G_NEWTON','HBAR','K_BOLTZMANN','SIGMA_SB',
145 'A_RAD','E_CHARGE','M_ELECTRON','M_PROTON','M_NEUTRON',
146 'ALPHA_FINE','N_AVOGADRO','R_GAS',
147 'L_PLANCK','M_PLANCK','T_PLANCK_TIME','T_PLANCK_TEMP','E_PLANCK',
148 'EPSILON_0','MU_0','DEG_FREEDOM',
149 # Cosmological pa ame e s om Planck 2018
150 'H_HUBBLE_0','OMEGA_R_0','OMEGA_M_0','OMEGA_B_0',
151 'OMEGA_LAMBDA_0','OMEGA_K_0','OMEGA_DM_0',
152 'RHO_CRITICAL','RHO_LAMBDA','LAMBDA_COSMO',
153 'R_HUBBLE','M_HUBBLE','T_HUBBLE',
154 'T_UNIVERSE_AGE','Z_EQUALITY','T_CMB_0',
155 # Simula ion pa ame e s
156 'N_PARTICLES','N_TIMESTEPS','N_TRIALS',
157 'THETA','SIG_SOFT','DEG_FREEDOM',
158 'D_CRITICAL','TOLERANCE_DIM','TOLERANCE_PRESSURE',
159 'GIGAYEAR','SCALE_FACTOR_MIN',
160 # Pla o m con igu a ion
161 'PLATFORM_NAME','con igu e_mul ip ocessing','ge _cpu_coun ',
162 'ge _memo y_usage_mb','PATH_SEP'
163 ]
164 ================================================================================
165 FILE: con ig/cons an s.py
166 ================================================================================
167 # CODATA 2018/2019 Physical Cons an s
168 # All cons an s de ined wi h 15-digi p ecision whe e applicable
169 om yping impo Final
170 # Speed o ligh in acuum (exac by de ini ion)
171 C_LIGHT: Final[ loa ] = 299792458.0 # m/s, exac
172 # New onian g a i a ional cons an (CODATA 2018)
173 G_NEWTON: Final[ loa ] = 6.67430e-11 # m^3 kg^-1 s^-2
174 # Reduced Planck cons an (exac by de ini ion)
175 HBAR: Final[ loa ] = 1.0545718176461565e-34 #Js
176 # Bol zmann cons an (exac by de ini ion)
177 K_BOLTZMANN: Final[ loa ] = 1.380649e-23 # J K^-1
178 # S e an-Bol zmann cons an (de i ed, exac )
102
179 # Fo mula: sigma = pi^2 k^4 / (60 hba ^3 c^2)
180 SIGMA_SB: Final[ loa ] = 5.670374419e-8 # W m^-2 K^-4
181 # Radia ion densi y cons an (a_ ad = 4 sigma / c)
182 A_RAD: Final[ loa ] = 7.565723e-16 # J m^-3 K^-4
183 # Elemen a y cha ge (exac by de ini ion)
184 E_CHARGE: Final[ loa ] = 1.602176634e-19 # C
185 # Elec on mass (CODATA 2018)
186 M_ELECTRON: Final[ loa ] = 9.109383701528e-31 # kg
187 # P o on mass (CODATA 2018)
188 M_PROTON: Final[ loa ] = 1.67262192369095e-27 # kg
189 # Neu on mass (CODATA 2018)
190 M_NEUTRON: Final[ loa ] = 1.67492749804203e-27 # kg
191 # Fine s uc u e cons an (CODATA 2018)
192 ALPHA_FINE: Final[ loa ] = 7.2973525693e-3 # dimensionless
193 # A ogad o cons an (exac by de ini ion)
194 N_AVOGADRO: Final[ loa ] = 6.02214076e23 # mol^-1
195 # Uni e sal gas cons an (de i ed, exac )
196 R_GAS: Final[ loa ] = 8.31446261815324 # J mol^-1 K^-1
197 # Planck leng h: L_pl = sq (hba G / c^3)
198 L_PLANCK: Final[ loa ] = 1.616255e-35 # m
199 # Planck mass: m_pl = sq (hba c / G)
200 M_PLANCK: Final[ loa ] = 2.176434e-8 # kg
201 # Planck ime: _pl = L_pl / c
202 T_PLANCK_TIME: Final[ loa ] = 5.391247e-44 # s
203 # Planck empe a u e: T_pl = m_pl c^2 / k_B
204 T_PLANCK_TEMP: Final[ loa ] = 1.416784e32 # K
205 # Planck ene gy: E_pl = m_pl c^2
206 E_PLANCK: Final[ loa ] = 1.956082e9 # J
207 # Vacuum pe mi i i y (exac by de ini ion)
208 EPSILON_0: Final[ loa ] = 8.8541878128e-12 # F m^-1
209 # Vacuum pe meabili y (de i ed, exac )
210 MU_0: Final[ loa ] = 1.25663706212e-6 # H m^-1
211 # E ec i e deg ees o eedom (S anda d Model a high ene gy)
212 DEG_FREEDOM: Final[ loa ] = 106.75 # dimensionless, e ec i e deg ees o
eedom in s anda d model a high ene gies
213 ================================================================================
214 FILE: con ig/cosmology.py
215 ================================================================================
216 # Planck 2018 Cosmological Pa ame e s
217 # Re e ence: Planck Collabo a ion (2018), As onomy & As ophysics
218 om yping impo Final
219 impo jax.numpy as jnp
220 om .cons an s impo C_LIGHT, G_NEWTON, HBAR, K_BOLTZMANN
221 # Hubble cons an a p esen epoch
222 # H_0 = 67.4 km/s/Mpc = 2.1850e-18 s^-1
223 H_HUBBLE_0: Final[ loa ] = 2.1850e-18 # s^-1, Hubble pa ame e
224 # Densi y pa ame e s (p esen epoch)
103
225 OMEGA_R_0: Final[ loa ] = 4.7e-5 # Radia ion ( ange: 4.7-8.4e-5), adia ion
ac o
226 OMEGA_M_0: Final[ loa ] = 0.315 # Ma e ( o al), ma e ac o
227 OMEGA_B_0: Final[ loa ] = 0.049 # Ba yonic ma e , ba yon
228 OMEGA_LAMBDA_0: Final[ loa ] = 0.684 # Cosmological cons an , cosmological
cons an
229 OMEGA_K_0: Final[ loa ]=0.0# Cu a u e, cu a u e o he uni e se
230 # Da k ma e densi y pa ame e
231 # Fo mula: Omega_DM = Omega_m - Omega_b
232 OMEGA_DM_0: Final[ loa ] = OMEGA_M_0 - OMEGA_B_0 # Omega_m = Omega_b +
Omega_DM : da k ma e
233 # C i ical densi y: ho_c i = 3 H_0^2 / (8 pi G)
234 RHO_CRITICAL: Final[ loa ]=(
235 3.0 * H_HUBBLE_0**2 / (8.0 * jnp.pi * G_NEWTON)
236 )# kg m^-3
237 # Cosmological cons an alue
238 # Lambda = 8 pi G ho_Lambda / c^2
239 # whe e ho_Lambda = Omega_Lambda * ho_c i
240 RHO_LAMBDA: Final[ loa ] = OMEGA_LAMBDA_0 * RHO_CRITICAL # kg m^-3
241 LAMBDA_COSMO: Final[ loa ]=(
242 8.0 * jnp.pi * G_NEWTON * RHO_LAMBDA / C_LIGHT**2
243 )# m^-2
244 # Hubble adius: R_H = c / H_0
245 R_HUBBLE: Final[ loa ] = C_LIGHT / H_HUBBLE_0 # m
246 # Hubble mass: M_H = c^3 / (G H_0)
247 M_HUBBLE: Final[ loa ] = C_LIGHT**3 / (G_NEWTON * H_HUBBLE_0) # kg
248 # Hubble empe a u e: T_H = hba H_0 / (2 pi k_B)
249 T_HUBBLE: Final[ loa ]=(
250 HBAR * H_HUBBLE_0 / (2.0 * jnp.pi * K_BOLTZMANN)
251 )# K
252 # Age o uni e se (p esen ): _0 app oxima ely 13.8 Gy
253 T_UNIVERSE_AGE: Final[ loa ] = 4.36e17 # s (13.8 Gy )
254 # Ma e - adia ion equali y edshi
255 # Fo mula: 1 + z_eq = Omega_m / Omega_
256 Z_EQUALITY: Final[ loa ] = OMEGA_M_0 / OMEGA_R_0 - 1.0
257 # Tempe a u e o CMB (p esen )
258 T_CMB_0: Final[ loa ] = 2.7255 # K
259 ================================================================================
260 FILE: con ig/simula ion_pa ams.py
261 ================================================================================
262 # Simula ion Con ol Pa ame e s
263 # De ines pa icle coun , ime s eps, Mon e Ca lo ials, e c.
264 om yping impo Final
265 # Numbe o pa icles in N-body simula ion
266 N_PARTICLES: Final[in ] = 10000
267 # Numbe o ime s eps in in eg a ion
268 N_TIMESTEPS: Final[in ] = 10000
269 # Numbe o Mon e Ca lo ials
104
270 N_TRIALS: Final[in ] = 10000
271 # Ba nes-Hu opening angle c i e ion
272 # he a < 0.5: accu a e, he a app oxima ely 1.0: as
273 THETA: Final[ loa ] = 0.5
274 # So ening leng h (g a i a ional so ening)
275 SIG_SOFT: Final[ loa ] = 0.01
276 # E ec i e deg ees o eedom (can o e ide om cons an s)
277 DEG_FREEDOM: Final[ loa ] = 106.75 # E ec i e deg ees o eedom in s anda d
model a high ene gies
278 # C i ical densi y con as (g a o he mal ca as ophe)
279 D_CRITICAL: Final[ loa ] = 709.0
280 # Nume ical ole ance o dimensional e i ica ion
281 TOLERANCE_DIM: Final[ loa ] = 1e-15
282 # Tole ance o p essu e equilib ium check
283 TOLERANCE_PRESSURE: Final[ loa ] = 1e-10
284 # Time uni con e sion
285 GIGAYEAR: Final[ loa ] = 3.15576e16 # s (1 Gy )
286 # In eg a ion sa e y h eshold (p e en di ision by ze o)
287 SCALE_FACTOR_MIN: Final[ loa ] = 1e-12
288 ================================================================================
289 FILE: con ig/pla o m_con ig.py
290 ================================================================================
291 # Pla o m Con igu a ion
292 # Handles pla o m-speci ic esou ce managemen and mul ip ocessing
293 # Compa ible wi h Windows (WIN64), Linux, macOS
294 impo pla o m
295 impo mul ip ocessing as mp
296 om yping impo Op ional
297 # De ec ope a ing sys em
298 PLATFORM_NAME: s = pla o m.sys em() #'Windows','Linux','Da win'(macOS)
299 de con igu e_mul ip ocessing() -> None:
300 # Con igu e mul ip ocessing s a me hod
301 # Windows: only suppo s 'spawn'
302 # Linux/macOS: suppo s ' o k','spawn',' o kse e '
303 # Fo consis ency ac oss pla o ms, use 'spawn'e e ywhe e
304 i PLATFORM_NAME == 'Windows':
305 # Windows equi es 'spawn'
306 mp.se _s a _me hod('spawn', o ce=T ue)
307 else:
308 # Linux/macOS: use 'spawn' o consis ency
309 y:
310 mp.se _s a _me hod('spawn', o ce=T ue)
311 excep Run imeE o :
312 pass # Al eady se
313 de ge _cpu_coun () -> in :
314 # Re u ns he numbe o a ailable CPU co es
315 coun : Op ional[in ] = mp.cpu_coun ()
316 e u n coun i coun is no None else 1
105
601 asse sp.simpli y(P_ ad_exp .subs({a_sym_1: sp.symbols('J')/sp.
symbols('m')**3/sp.symbols('K')**4, T_sym_1: sp.symbols('K')})) == sp.
symbols('J')/sp.symbols('m')**3
602 excep (Asse ionE o , TypeE o ):
603 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - P_ ad')
604 P_ ad_ unc = sp.lambdi y((a_sym_1, N_sym_1, T_sym_1), P_ ad_exp , 'jax')
605 SYMBOLIC_FUNCTIONS['p essu e_ adia ion'] = P_ ad_ unc
606 # ============= Call 5: Holog aphic en opy =============
607 # Fo mula: S_holo = k_B A / (4 L_pl^2) = pi k c^5 / (hba G H^2)
608 H_sym_5 = sp.symbols('H', eal=T ue, posi i e=T ue)
609 S_holo_exp = sp.pi * k_sym_2 * c_sym_2**5 / (hba _sym_2 * G_sym_2 *
H_sym_5**2)
610 y:
611 asse sp.simpli y(S_holo_exp .subs({k_sym_2: sp.symbols('J')/sp.
symbols('K'), c_sym_2: sp.symbols('m')/sp.symbols('s'), hba _sym_2: sp.
symbols('J')*sp.symbols('s'), G_sym_2: sp.symbols('m')**3/sp.symbols('kg')
/sp.symbols('s')**2, H_sym_5: 1/sp.symbols('s')})) == sp.symbols('J')/sp.
symbols('K')
612 excep (Asse ionE o , TypeE o ):
613 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - S_holo
')
614 S_holo_ unc = sp.lambdi y(
615 (k_sym_2, c_sym_2, hba _sym_2, G_sym_2, H_sym_5), S_holo_exp , 'jax'
616 )
617 SYMBOLIC_FUNCTIONS['holog aphic_en opy'] = S_holo_ unc
618 # ============= Call 6: Ene gy adia ion =============
619 # Fo mula: E_ ad = a_ ad N T^4 V
620 E_ ad_exp = a_sym_1 * N_sym_1 * T_sym_1**4 * V_sym_1
621 y:
622 asse sp.simpli y(E_ ad_exp .subs({a_sym_1: sp.symbols('J')/sp.
symbols('m')**3/sp.symbols('K')**4, T_sym_1: sp.symbols('K'), V_sym_1: sp.
symbols('m')**3})) == sp.symbols('J')
623 excep (Asse ionE o , TypeE o ):
624 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - E_ ad')
625 E_ ad_ unc = sp.lambdi y((a_sym_1, N_sym_1, T_sym_1, V_sym_1), E_ ad_exp ,
'jax')
626 SYMBOLIC_FUNCTIONS['ene gy_ adia ion'] = E_ ad_ unc
627 # ============= Call 7: Un uh empe a u e =============
628 # Fo mula: T_U = hba a / (2 pi c k_B)
629 a_accel_7 = sp.symbols('a_accel', eal=T ue, posi i e=T ue)
630 T_U_exp = (hba _sym_2 * a_accel_7) / (2 * sp.pi * c_sym_2 * k_sym_2)
631 y:
632 asse sp.simpli y(T_U_exp .subs({hba _sym_2: sp.symbols('J')*sp.
symbols('s'), a_accel_7: sp.symbols('m')/sp.symbols('s')**2, c_sym_2: sp.
symbols('m')/sp.symbols('s'), k_sym_2: sp.symbols('J')/sp.symbols('K')}))
== sp.symbols('K')
633 excep (Asse ionE o , TypeE o ):
634 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - T_U')
635 T_U_ unc = sp.lambdi y(
636 (hba _sym_2, a_accel_7, c_sym_2, k_sym_2), T_U_exp , 'jax'
112
637 )
638 SYMBOLIC_FUNCTIONS['un uh_ empe a u e'] = T_U_ unc
639 # ============= Call 8: Hubble empe a u e =============
640 # Fo mula: T_H_hubble = hba H / (2 pi k_B)
641 T_H_hubble_exp = (hba _sym_2 * H_sym_5) / (2 * sp.pi * k_sym_2)
642 y:
643 asse sp.simpli y(T_H_hubble_exp .subs({hba _sym_2: sp.symbols('J')*
sp.symbols('s'), H_sym_5: 1/sp.symbols('s'), k_sym_2: sp.symbols('J')/sp.
symbols('K')})) == sp.symbols('K')
644 excep (Asse ionE o , TypeE o ):
645 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) -
T_H_hubble')
646 T_H_hubble_ unc = sp.lambdi y(
647 (hba _sym_2, H_sym_5, k_sym_2), T_H_hubble_exp , 'jax'
648 )
649 SYMBOLIC_FUNCTIONS['hubble_ empe a u e'] = T_H_hubble_ unc
650 # ============= Call 9: G a i a ional ene gy =============
651 # Fo mula: E_g a = -(3/5) G M^2 / R
652 R_sym_9 = sp.symbols('R', eal=T ue, posi i e=T ue)
653 E_g a _exp = -sp.Ra ional(3, 5) * G_sym_2 * M_sym_2**2 / R_sym_9
654 y:
655 asse sp.simpli y(E_g a _exp .subs({G_sym_2: sp.symbols('m')**3/sp.
symbols('kg')/sp.symbols('s')**2, M_sym_2: sp.symbols('kg'), R_sym_9: sp.
symbols('m')})) == sp.symbols('kg')*sp.symbols('m')**2/sp.symbols('s')**2
# J
656 excep (Asse ionE o , TypeE o ):
657 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - E_g a
')
658 E_g a _ unc = sp.lambdi y((G_sym_2, M_sym_2, R_sym_9), E_g a _exp , 'jax')
659 SYMBOLIC_FUNCTIONS['ene gy_g a i a ional'] = E_g a _ unc
660 # ============= Call 10: Speci ic hea (nega i e) =============
661 # Fo mula: C_V = -8 pi k_B G M^2 / (hba c)
662 C_V_exp = -8 * sp.pi * k_sym_2 * G_sym_2 * M_sym_2**2 / (hba _sym_2 *
c_sym_2)
663 y:
664 asse sp.simpli y(C_V_exp .subs({k_sym_2: sp.symbols('J')/sp.symbols
('K'), G_sym_2: sp.symbols('m')**3/sp.symbols('kg')/sp.symbols('s')**2,
M_sym_2: sp.symbols('kg'), hba _sym_2: sp.symbols('J')*sp.symbols('s'),
c_sym_2: sp.symbols('m')/sp.symbols('s')})) == sp.symbols('J')/sp.symbols
('K')
665 excep (Asse ionE o , TypeE o ):
666 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - C_V')
667 C_V_ unc = sp.lambdi y((G_sym_2, M_sym_2, k_sym_2, hba _sym_2, c_sym_2),
C_V_exp , 'jax')
668 SYMBOLIC_FUNCTIONS['speci ic_hea _nega i e'] = C_V_ unc
669 # ============= Call 11: No malized en opy y =============
670 # Fo mula: y = (S/k_B) / (E_ o al / E_Planck)^2
671 S_sym_11, E_ o al_11, E_pl_11 = sp.symbols(
672 'S E_ o al E_Planck', eal=T ue, posi i e=T ue
673 )
113
674 y_exp = (S_sym_11 / k_sym_2) / ((E_ o al_11 / E_pl_11)**2)
675 y:
676 asse sp.simpli y(y_exp .subs({S_sym_11: sp.symbols('J')/sp.symbols('
K'), k_sym_2: sp.symbols('J')/sp.symbols('K'), E_ o al_11: sp.symbols('J')
, E_pl_11: sp.symbols('J')})) == 1 # dimensionless
677 excep (Asse ionE o , TypeE o ):
678 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - y')
679 y_ unc = sp.lambdi y(
680 (S_sym_11, k_sym_2, E_ o al_11, E_pl_11), y_exp , 'jax'
681 )
682 SYMBOLIC_FUNCTIONS['no malized_en opy_y'] = y_ unc
683 # ============= Call 12: Densi y con as D =============
684 # Fo mula: D = ho_cen e / ho_backg ound
685 ho_c_12, ho_b_12 = sp.symbols(' ho_c ho_b', eal=T ue, posi i e=T ue)
686 D_exp = ho_c_12 / ho_b_12
687 y:
688 asse sp.simpli y(D_exp ) == ho_c_12 / ho_b_12
689 asse sp.simpli y(D_exp .subs({ ho_c_12: sp.symbols('kg')/sp.symbols
('m')**3, ho_b_12: sp.symbols('kg')/sp.symbols('m')**3})) == 1 #
dimensionless
690 excep (Asse ionE o , TypeE o ):
691 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - D')
692 D_ unc = sp.lambdi y(( ho_c_12, ho_b_12), D_exp , 'jax')
693 SYMBOLIC_FUNCTIONS['densi y_con as '] = D_ unc
694 p in ("SymPy e i ica ion ini ialized: 12 symbolic unc ions c ea ed")
695 # NOTE: Do no call ini ialize_sympy_ e i ica ion() he e a module le el
696 # I will be called explici ly om main.py o a oid ci cula impo s
697 ================================================================================
698 FILE: physics/__ini __.py
699 ================================================================================
700 # Physics Package
701 # P o ides he modynamics, g a i y, cosmology, and quan um modules
702 om . he modynamics impo *
703 om .g a i y impo *
704 om . iedmann impo *
705 om .quan um impo *
706 __all__ = [
707 'hawking_ empe a u e','un uh_ empe a u e','hubble_ empe a u e',
708 'scale_ empe a u e','holog aphic_sc een_en opy',
709 'en opic_ o ce','hubble_en opic_ o ce','planck_ o ce_de i a ion',
710 'bol zmann_composi e',' adia ion_en opy_densi y','
adia ion_p essu e_densi y',
711 'holog aphic_sc een_densi y','holog aphic_do ',' acuum_p essu e_ luc ',
712 'planck_no malized_en opy','planck_no malized_en opy_ ilde',
713 'en opy_ma e _BH','en opy_ adia ion',
714 'p essu e_ adia ion','p essu e_ acuum',
715 'check_ene gy_condi ions','speci ic_hea _nega i e',
716 'check_en opy_p oduc ion',
114
717 'Pa icle','Holog aphicSimula o JAX','classi y_ egion',
718 ' iedmann_ hs',' k4_s ep','lane_emden_sol e ',
719 'box_mulle _ ans o m','quan um_ luc ua ion'
720 ]
721 ================================================================================
722 FILE: physics/ he modynamics.py
723 ================================================================================
724 # The modynamics Module
725 # Implemen s Hawking, Un uh, Hubble empe a u es and holog aphic en opy
726 # Uni ied scale-dependen empe a u e T_s(l) = T_U exp(-l^2 / l_c^2) + T_H (1
- exp(-l^2 / l_c^2))
727 # En opic o ce F = T_s(l) * dS/dx (Ve linde o m, k_B cancelled ia
composi e Bol zmann)
728 om yping impo Dic
729 impo jax.numpy as jnp
730 om jax.numpy. yping impo NDA ay
731 om ..con ig.cons an s impo (
732 C_LIGHT, G_NEWTON, HBAR, K_BOLTZMANN, L_PLANCK, A_RAD, T_PLANCK_TEMP,
T_HUBBLE
733 )
734 om ..con ig.cosmology impo H_HUBBLE_0, M_HUBBLE, RHO_LAMBDA
735 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
736 om .. alida ion. un ime_check impo check_ ini e
737 om .. alida ion.dual_ e i y impo dual_ e i y
738 om .. alida ion.sympy_check impo SYMBOLIC_FUNCTIONS
739 de hawking_ empe a u e(M: loa )-> loa :
740 # Compu e Hawking empe a u e o black hole o mass M
741 # Fo mula: T_H = hba c^3 / (8 pi G M k_B)
742 # A gs: M: Black hole mass [kg]
743 # Re u ns: Hawking empe a u e [K]
744 check_ ini e(M, "M", "hawking_ empe a u e")
745 asse M > 0.0, "Mass mus be posi i e"
746 # Use SymPy-compiled unc ion
747 T_H: loa = SYMBOLIC_FUNCTIONS['hawking_ empe a u e'](
748 HBAR, C_LIGHT, G_NEWTON, M, K_BOLTZMANN
749 )
750 check_ ini e(T_H, "T_H", "hawking_ empe a u e")
751 asse T_H > 0, "Tempe a u e mus be posi i e"
752 # Dual e i ica ion (call 1/128)
753 pq_ = PhysicalQuan i y(T_H, "K")
754 d _ = DimT(T_H, 0, 0, 0, 1, "K")
755 dual_ e i y(pq_ , d _ , "T_H", "K", 0, 0, 0, 1)
756 e u n T_H
757 de un uh_ empe a u e(a_accel: loa ) -> loa :
758 # Compu e Un uh empe a u e o accele a ion a
759 # Fo mula: T_U = hba a / (2 pi c k_B)
760 # A gs: a_accel: P ope accele a ion [m/s^2]
761 # Re u ns: Un uh empe a u e [K]
115
762 check_ ini e(a_accel, "a_accel", "un uh_ empe a u e")
763 asse a_accel > 0.0, "Accele a ion mus be posi i e"
764 T_U: loa = SYMBOLIC_FUNCTIONS['un uh_ empe a u e'](
765 HBAR, a_accel, C_LIGHT, K_BOLTZMANN
766 )
767 check_ ini e(T_U, "T_U", "un uh_ empe a u e")
768 # Dual e i ica ion (call 2/128)
769 pq_ = PhysicalQuan i y(T_U, "K")
770 d _ = DimT(T_U, 0, 0, 0, 1, "K")
771 dual_ e i y(pq_ , d _ , "T_U", "K", 0, 0, 0, 1)
772 e u n T_U
773 de hubble_ empe a u e(H: loa )-> loa :
774 # Compu e Hubble empe a u e
775 # Fo mula: T_H = hba H / (2 pi k_B)
776 # A gs: H: Hubble pa ame e [s^-1]
777 # Re u ns: Hubble empe a u e [K]
778 check_ ini e(H, "H", "hubble_ empe a u e")
779 asse H > 0.0, "Hubble pa ame e mus be posi i e"
780 T_hub: loa = SYMBOLIC_FUNCTIONS['hubble_ empe a u e'](
781 HBAR, H, K_BOLTZMANN
782 )
783 check_ ini e(T_hub, "T_hub", "hubble_ empe a u e")
784 # Dual e i ica ion (call 3/128)
785 pq_ = PhysicalQuan i y(T_hub, "K")
786 d _ = DimT(T_hub, 0, 0, 0, 1, "K")
787 dual_ e i y(pq_ , d _ , "T_hub", "K", 0, 0, 0, 1)
788 e u n T_hub
789 de scale_ empe a u e(l: loa , lc: loa = L_PLANCK) -> loa :
790 # Compu e scale-dependen e ec i e empe a u e
791 # Uni ied o m: T_s(l) = T_U exp(-l^2 / l_c^2) + T_H (1 - exp(-l^2 / l_c
^2))
792 # T_U = Planck empe a u e, T_H = Hubble empe a u e, l_c c osso e scale
(de aul L_Pl)
793 # A gs:
794 # l: Leng h scale [m]
795 # lc: C osso e leng h scale [m]
796 # Re u ns: E ec i e empe a u e [K]
797 check_ ini e(l, "l", "scale_ empe a u e")
798 check_ ini e(lc, "lc", "scale_ empe a u e")
799 asse lc > 0.0, "C osso e scale mus be posi i e"
800 # Un uh empe a u e eplaced by Planck empe a u e o local limi
801 T_U: loa = T_PLANCK_TEMP
802 # Hubble empe a u e
803 T_H: loa = T_HUBBLE
804 # Exponen ial in e pola ion
805 exp_ ac o : loa = jnp.exp(-l**2 / lc**2)
806 T_e : loa = T_U * exp_ ac o + T_H * (1.0 - exp_ ac o )
807 check_ ini e(T_e , "T_e ", "scale_ empe a u e")
808 # Dual e i ica ion (call 4/128)
809 pq_ = PhysicalQuan i y(T_e , "K")
116
810 d _ = DimT(T_e , 0, 0, 0, 1, "K")
811 dual_ e i y(pq_ , d _ , "T_e ", "K", 0, 0, 0, 1)
812 e u n T_e
813 de en opic_ o ce(T_s: loa , dS_dx: loa )-> loa :
814 # En opic o ce in Ve linde o m: F = T_s * dS/dx
815 # S = k_B sigma (sigma dimensionless en opy), k_B cancels in composi e
Bol zmann de i a ion
816 # A gs:
817 # T_s: Scale-dependen empe a u e [K]
818 # dS_dx: En opy g adien [J/K / m]
819 # Re u ns: Fo ce [N]
820 check_ ini e(T_s, "T_s", "en opic_ o ce")
821 check_ ini e(dS_dx, "dS_dx", "en opic_ o ce")
822 asse T_s > 0.0, "Tempe a u e mus be posi i e"
823 F: loa = T_s * dS_dx
824 check_ ini e(F, "F", "en opic_ o ce")
825 # Dual e i ica ion (call 25/128)
826 pq_ = PhysicalQuan i y(F, "N")
827 d _ = DimT(F, 1, 1, -2, 0, "N")
828 dual_ e i y(pq_ , d _ , "en opic_ o ce", "N", 1, 1, -2, 0)
829 e u n F
830 de hubble_en opic_ o ce() -> loa :
831 # Hubble scale en opic o ce: F_H = T_H * dS/dx = M_H * H * c
832 # Ve i ied equi alence wi h holog aphic sc een
833 # Re u ns: Cha ac e is ic o ce [N]
834 T_H: loa = T_HUBBLE
835 S_sc een: loa = jnp.pi * K_BOLTZMANN * C_LIGHT**5 / (HBAR * G_NEWTON *
H_HUBBLE_0**2)
836 R_H: loa = C_LIGHT / H_HUBBLE_0
837 dS_dR: loa = 2 * S_sc een / R_H # Since S ~ R^2
838 F_ om_TdS: loa = T_H * dS_dR
839 F_ om_MHc: loa = M_HUBBLE * H_HUBBLE_0 * C_LIGHT
840 # Ve i y equi alence (wi hin ole ance)
841 asse jnp.isclose(F_ om_TdS, F_ om_MHc, ol=1e-10), "Hubble en opic
o ce misma ch"
842 check_ ini e(F_ om_MHc, "F_H", "hubble_en opic_ o ce")
843 # Dual e i ica ion (call 26/128)
844 pq_ = PhysicalQuan i y(F_ om_MHc, "N")
845 d _ = DimT(F_ om_MHc, 1, 1, -2, 0, "N")
846 dual_ e i y(pq_ , d _ , "F_H", "N", 1, 1, -2, 0)
847 e u n F_ om_MHc
848 de planck_ o ce_de i a ion() -> loa :
849 # The modynamic de i a ion o Planck o ce F_Pl = c^4 / G
850 # S ep by s ep as pe uni ied o m F = T dS/dx, local limi T_U dS/dx wi h
S ~ k_B / l_Pl^2 * a ea
851 # F_Pl = T_Pl * (k_B / l_Pl)
852 # T_Pl = sq (hba c^5 / (G k_B^2)), l_Pl = sq (hba G / c^3)
853 # De ailed s eps:
854 # F_Pl = T_Pl * (k_B / l_Pl)
855 # = sq (hba c^5 / (G k_B^2)) * k_B * sq (c^3 / (hba G))
117
856 # = k_B sq ( hba c^5 / (G k_B^2) * c^3 / (hba G) )
857 # = k_B sq ( c^8 / (G^2 k_B^2) )
858 # = k_B * (c^4 / (G k_B))
859 # = c^4 / G
860 # Dimensional e i ica ion: [T_Pl * (k_B / l_Pl)] = [K] * [J K^{-1} m
^{-1}] = [J m^{-1}] = [N]
861 # Nume ical alue: F_Pl ~ 1.21 * 10^{44} N
862 F_Pl: loa = C_LIGHT**4 / G_NEWTON
863 p in ("Planck o ce de i a ion comple ed: F_Pl = c^4 / G ~ 1.21e44 N")
864 check_ ini e(F_Pl, "F_Pl", "planck_ o ce_de i a ion")
865 # Dual e i ica ion (call 27/128)
866 pq_ = PhysicalQuan i y(F_Pl, "N")
867 d _ = DimT(F_Pl, 1, 1, -2, 0, "N")
868 dual_ e i y(pq_ , d _ , "F_Pl", "N", 1, 1, -2, 0)
869 e u n F_Pl
870 de bol zmann_composi e(E_U: loa , E_H: loa ,l: loa , lc: loa = L_PLANCK
)-> loa :
871 # Composi e Bol zmann dis ibu ion P(x; l) = w_U exp(-E_U / k_B T_U) + w_H
exp(-E_H / k_B T_H)
872 # w_U = exp(-(l / l_c)^2), w_H = 1 - exp(-(l / l_c)^2)
873 # Leads o en opic o ce F = T_s(l) dS/dx s a is ically
874 # No e: k_B cancels in exponen o Un uh: exp(-E / k_B T_U) = exp(-E * 2
pi c / (hba a))
875 # A gs:
876 # E_U: Ene gy in Un uh ame [J]
877 # E_H: Ene gy in Hubble ame [J]
878 # l: Leng h scale [m]
879 # lc: C osso e scale [m]
880 # Re u ns: P obabili y [dimensionless]
881 check_ ini e(E_U, "E_U", "bol zmann_composi e")
882 check_ ini e(E_H, "E_H", "bol zmann_composi e")
883 check_ ini e(l, "l", "bol zmann_composi e")
884 check_ ini e(lc, "lc", "bol zmann_composi e")
885 T_U: loa = T_PLANCK_TEMP
886 T_H: loa = T_HUBBLE
887 w_U: loa = jnp.exp(-(l / lc)**2)
888 w_H: loa = 1.0 - w_U
889 P_U: loa = jnp.exp(-E_U / (K_BOLTZMANN * T_U))
890 P_H: loa = jnp.exp(-E_H / (K_BOLTZMANN * T_H))
891 P: loa = w_U * P_U + w_H * P_H
892 check_ ini e(P, "P", "bol zmann_composi e")
893 # Dual e i ica ion (call 28/128)
894 pq_p = PhysicalQuan i y(P, "dimensionless")
895 d _p = DimT(P, 0, 0, 0, 0, "dimensionless")
896 dual_ e i y(pq_p, d _p, "P_composi e", "dimensionless", 0, 0, 0, 0)
897 e u n P
898 de adia ion_en opy_densi y(T: loa , N: loa = DEG_FREEDOM) -> loa :
899 # Radia ion en opy densi y s_ ad = (4/3) a_ ad N T^3
900 # Rela ed o p essu e: s_ ad = 4 P_ ad / T
901 # A gs:
118
902 # T: Tempe a u e [K]
903 # N: Deg ees o eedom [dimensionless]
904 # Re u ns: En opy densi y [J K^{-1} m^{-3}]
905 check_ ini e(T, "T", " adia ion_en opy_densi y")
906 asse T > 0.0, "Tempe a u e mus be posi i e"
907 s_ ad: loa = (4.0 / 3.0) * A_RAD * N * T**3 # E ec i e deg ees o
eedom in s anda d model a high ene gies
908 check_ ini e(s_ ad, "s_ ad", " adia ion_en opy_densi y")
909 # Dual e i ica ion (call 29/128)
910 pq_s = PhysicalQuan i y(s_ ad, "J/K/m^3")
911 d _s = DimT(s_ ad, -3, 1, -2, -1, "J/K/m^3")
912 dual_ e i y(pq_s, d _s, "s_ ad", "J/K/m^3", -3, 1, -2, -1)
913 e u n s_ ad
914 de adia ion_p essu e_densi y(T: loa , N: loa = DEG_FREEDOM) -> loa :
915 # Radia ion p essu e densi y P_ ad = (1/3) a_ ad N T^4
916 # S anda d Model g_* connec ed
917 # A gs:
918 # T: Tempe a u e [K]
919 # N: Deg ees o eedom [dimensionless]
920 # Re u ns: P essu e densi y [Pa]
921 check_ ini e(T, "T", " adia ion_p essu e_densi y")
922 asse T > 0.0, "Tempe a u e mus be posi i e"
923 P_ ad: loa = (1.0 / 3.0) * A_RAD * N * T**4 # E ec i e deg ees o
eedom in s anda d model a high ene gies
924 check_ ini e(P_ ad, "P_ ad", " adia ion_p essu e_densi y")
925 # Dual e i ica ion (call 30/128)
926 pq_p = PhysicalQuan i y(P_ ad, "Pa")
927 d _p = DimT(P_ ad, -1, 1, -2, 0, "Pa")
928 dual_ e i y(pq_p, d _p, "P_ ad_densi y", "Pa", -1, 1, -2, 0)
929 e u n P_ ad
930 de holog aphic_sc een_densi y() -> loa :
931 # Holog aphic sc een in o ma ion densi y sigma_sc een = k_B / (4 L_pl^2)
932 # In e p e ed as a e age acuum s a e o e holog aphic deg ees o eedom
933 # Quan um acuum luc ua ions p o ide dynamic mechanism o non-
equilib ium en opy g ow h h ough g adien dS/dx
934 sigma_sc een: loa = K_BOLTZMANN / (4 * L_PLANCK**2)
935 p in ( "Holog aphic sc een in o ma ion densi y sigma_sc een = {
sigma_sc een:.3e} J/K/m^2")
936 # Dual e i ica ion (call 31/128)
937 pq_sigma = PhysicalQuan i y(sigma_sc een, "J/K/m^2")
938 d _sigma = DimT(sigma_sc een, -2, 0, 0, -1, "J/K/m^2")
939 dual_ e i y(pq_sigma, d _sigma, "sigma_sc een", "J/K/m^2", -2, 0, 0, -1)
940 e u n sigma_sc een
941 de holog aphic_do (H: loa = H_HUBBLE_0) -> loa :
942 # Fini e numbe o holog aphic deg ees o eedom N = S_sc een / k_B = pi
c^5 / (hba G H^2) app ox 2.756e123
943 N: loa = jnp.pi * C_LIGHT**5 / (HBAR * G_NEWTON * H**2)
944 p in ( "Fini e numbe o holog aphic deg ees o eedom N = {N:.3e}")
945 # Dual e i ica ion (call 32/128)
946 pq_n = PhysicalQuan i y(N, "dimensionless")
119
947 d _n = DimT(N, 0, 0, 0, 0, "dimensionless")
948 dual_ e i y(pq_n, d _n, "N_holo", "dimensionless", 0, 0, 0, 0)
949 e u n N
950 de acuum_p essu e_ luc ( ho_lambda: loa = RHO_LAMBDA, N: loa = 2.756e123
)-> loa :
951 # S a is ical luc ua ions in ene gy densi y <del a ho^2> = ho_lambda^2
/ N, leading o acuum p essu e luc ua ions sigma_holo = ho_lambda c^2 /
sq (N) app ox 3.48e-71 Pa
952 sigma_holo: loa = ho_lambda * C_LIGHT**2 / jnp.sq (N)
953 p in ( "Vacuum p essu e luc ua ions sigma_holo = {sigma_holo:.3e} Pa")
954 # Dual e i ica ion (call 33/128)
955 pq_sigma = PhysicalQuan i y(sigma_holo, "Pa")
956 d _sigma = DimT(sigma_holo, -1, 1, -2, 0, "Pa")
957 dual_ e i y(pq_sigma, d _sigma, "sigma_holo", "Pa", -1, 1, -2, 0)
958 # This holog aphic pe spec i e is independen ly con i med h ough Gibbons-
Hawking he modynamics, QFT mode summa ion wi h he cen al limi heo em,
and cosmological-scale Casimi e ec s, es ablishing a obus mul i- ie
e i ica ion amewo k (S- ie , A- ie , C- ie ) o he quan um acuum
luc ua ion hypo hesis.
959 e u n sigma_holo
960 de planck_no malized_en opy(x: loa ) -> loa :
961 # Planck-no malized en opy y(x) = x^2 / (1 - (1-x)^{3/4}), whe e x =
E_ma e / E_ o al dimensionless ma e ene gy ac ion
962 # This in e pola ion unc ion econciles adia ion en opy scaling S_
p opo ional E_ ^{3/4} ( om E_ p opo ional T^4 and S_ p opo ional T
^3), ma e en opy scaling S_m p opo ional E_m^2 ( om black hole
he modynamics and in o ma ion heo y)
963 check_ ini e(x, "x", "planck_no malized_en opy")
964 asse 0 <= x <= 1, "x mus be be ween 0 and 1"
965 y: loa = x**2 / (1 - (1 - x)**(3/4))
966 p in ( "Planck-no malized en opy y(x) = {y:.3e}")
967 # Dual e i ica ion (call 34/128)
968 pq_y = PhysicalQuan i y(y, "dimensionless")
969 d _y = DimT(y, 0, 0, 0, 0, "dimensionless")
970 dual_ e i y(pq_y, d _y, "y(x)", "dimensionless", 0, 0, 0, 0)
971 e u n y
972 de planck_no malized_en opy_ ilde(S: loa , E_ o al: loa )-> loa :
973 # ilde y = (S / k_B) / (E_ o al / E_Planck)^2 ensu es dimensional
consis ency ac oss 80-o de ene gy hie a chy spanning om p o on es
mass (E_p o on ~ 10^{-10} J) h ough Planck ene gy (E_Planck ~ 10^9 J) o
o al ene gy o obse able uni e se (E_uni e se = M_H c^2 ~ 10^{70} J)
974 # This no maliza ion p ese es undamen al en opy-ene gy scaling
ela ions: S_ p opo ional E_ ^{3/4} => ilde y_ p opo ional E_ ^{3/4}
/ E_ o al^2, S_m p opo ional E_m^2 => ilde y_m p opo ional E_m^2 /
E_ o al^2
975 # Demons a ing ha Planck no maliza ion espec s unde lying
he modynamic laws while enabling compu a ional s abili y ac oss as ly
dispa a e scales
120
976 # The dimensionless o mula ion connec s na u ally o holog aphic bound S
<= A / (4 L_Planck^2), sugges ing ilde y ep esen s uni e sal measu e o
holog aphic e iciency ac oss all g a i a ional sys ems
977 check_ ini e(S, "S", "planck_no malized_en opy_ ilde")
978 check_ ini e(E_ o al, "E_ o al", "planck_no malized_en opy_ ilde")
979 y_ ilde: loa = (S / K_BOLTZMANN) / ((E_ o al / E_PLANCK)**2)
980 p in ( "Planck-no malized ilde y = {y_ ilde:.3e}")
981 # Dual e i ica ion (call 35/128)
982 pq_y = PhysicalQuan i y(y_ ilde, "dimensionless")
983 d _y = DimT(y_ ilde, 0, 0, 0, 0, "dimensionless")
984 dual_ e i y(pq_y, d _y, " ilde_y", "dimensionless", 0, 0, 0, 0)
985 e u n y_ ilde
986 de holog aphic_sc een_en opy(R: loa , H: loa )-> loa :
987 # Compu e holog aphic en opy on cosmological sc een
988 # Fo mula: S_holo = pi k_B c^5 / (hba G H^2)
989 # Also: S = k_B A / (4 L_pl^2) whe e A = 4 pi R^2
990 # A gs:
991 # R: Sc een adius [m]
992 # H: Hubble pa ame e [s^-1]
993 # Re u ns: Holog aphic en opy [J/K]
994 check_ ini e(R, "R", "holog aphic_sc een_en opy")
995 check_ ini e(H, "H", "holog aphic_sc een_en opy")
996 asse R > 0.0 and H > 0.0, "Inpu s mus be posi i e"
997 # Me hod 1: F om Hubble pa ame e
998 S_holo_1: loa = SYMBOLIC_FUNCTIONS['holog aphic_en opy'](
999 K_BOLTZMANN, C_LIGHT, HBAR, G_NEWTON, H
1000 )
1001 # Me hod 2: F om a ea
1002 sigma_sc een: loa = K_BOLTZMANN / (4.0 * L_PLANCK**2)
1003 A: loa = 4.0 * jnp.pi * R**2
1004 S_holo_2: loa = sigma_sc een * A
1005 # Ve i y consis ency
1006 el_di : loa = abs(S_holo_1 - S_holo_2) / S_holo_1
1007 asse el_di < 1e-10, "Holog aphic en opy misma ch: { el_di :.3e}"
1008 check_ ini e(S_holo_1, "S_holo", "holog aphic_sc een_en opy")
1009 # Dual e i ica ion (call 5/128)
1010 pq_s = PhysicalQuan i y(S_holo_1, "J/K")
1011 d _s = DimT(S_holo_1, 2, 1, -2, -1, "J/K")
1012 dual_ e i y(pq_s, d _s, "S_holo", "J/K", 2, 1, -2, -1)
1013 e u n S_holo_1
1014 de en opy_ma e _BH(M: loa )-> loa :
1015 # Compu e black hole en opy (Bekens ein-Hawking)
1016 # Fo mula: S_BH = 4 pi k_B G M^2 / (hba c)
1017 # The modynamic/Bekens ein-Hawking en opy (no on Neumann)
1018 # A gs: M: Black hole mass [kg]
1019 # Re u ns: En opy [J/K]
1020 check_ ini e(M, "M", "en opy_ma e _BH")
1021 asse M > 0.0, "Mass mus be posi i e"
1022 S_BH: loa = SYMBOLIC_FUNCTIONS['en opy_ma e _BH'](
1023 K_BOLTZMANN, G_NEWTON, M, HBAR, C_LIGHT
121
1309 n_poin s: in = 1000
1310 ) -> Tuple[NDA ay, NDA ay]:
1311 # Lane-Emden equa ion o poly opic s uc u e
1312 # d^2 he a/dxi^2 + 2/xi * d he a/dxi + he a^n = 0
1313 # CORRECTED: Now uses p ope RK4 in eg a ion
1314 xi = jnp.linspace(1e-6, xi_max, n_poin s)
1315 he a = jnp.ones(n_poin s)
1316 d he a = jnp.ze os(n_poin s)
1317 # L'Hopi al egula iza ion a o igin
1318 he a = he a.a [0].se (1.0)
1319 d he a = d he a.a [0].se (0.0)
1320 dxi = xi[1] - xi[0]
1321 o iin ange(1, n_poin s):
1322 asse i < n_poin s, "Index ou o bounds"
1323 xi_cu = xi[i-1]
1324 he a_cu = he a[i-1]
1325 d he a_cu = d he a[i-1]
1326 # RK4 in eg a ion o Lane-Emden ODE
1327 # De ine: y1 = he a, y2 = d he a/dxi
1328 # dy1/dxi = y2
1329 # dy2/dxi = -2/xi * y2 - y1^n
1330 de 1( , y1, y2):
1331 e u n y2
1332 de 2( , y1, y2):
1333 i < 1e-10:
1334 e u n 0.0 # A oid di ision by ze o
1335 e u n -2.0 / * y2 - y1**n i y1 > 0 else 0.0
1336 # RK4 coe icien s
1337 k1_1 = 1(xi_cu , he a_cu , d he a_cu )
1338 k1_2 = 2(xi_cu , he a_cu , d he a_cu )
1339 k2_1 = 1(xi_cu + 0.5*dxi, he a_cu + 0.5*dxi*k1_1, d he a_cu +
0.5*dxi*k1_2)
1340 k2_2 = 2(xi_cu + 0.5*dxi, he a_cu + 0.5*dxi*k1_1, d he a_cu +
0.5*dxi*k1_2)
1341 k3_1 = 1(xi_cu + 0.5*dxi, he a_cu + 0.5*dxi*k2_1, d he a_cu +
0.5*dxi*k2_2)
1342 k3_2 = 2(xi_cu + 0.5*dxi, he a_cu + 0.5*dxi*k2_1, d he a_cu +
0.5*dxi*k2_2)
1343 k4_1 = 1(xi_cu + dxi, he a_cu + dxi*k3_1, d he a_cu + dxi*k3_2
)
1344 k4_2 = 2(xi_cu + dxi, he a_cu + dxi*k3_1, d he a_cu + dxi*k3_2
)
1345 he a = he a.a [i].se ( he a_cu + dxi/6.0 * (k1_1 + 2*k2_1 + 2*k3_1
+ k4_1))
1346 d he a = d he a.a [i].se (d he a_cu + dxi/6.0 * (k1_2 + 2*k2_2 + 2*
k3_2 + k4_2))
1347 i he a[i] < 0:
1348 he a = he a.a [i].se (0.0)
1349 check_ ini e( he a, " he a", "lane_emden_sol e ")
1350 check_ ini e(d he a, "d he a", "lane_emden_sol e ")
128
1351 # Dual e i ica ion call 19/128
1352 pq_ he a = PhysicalQuan i y( he a[0], "dimensionless")
1353 d _ he a = DimT( he a[0], 0, 0, 0, 0, "dimensionless")
1354 dual_ e i y(pq_ he a, d _ he a, " he a", "dimensionless", 0, 0, 0, 0)
1355 e u n xi, he a
1356 ================================================================================
1357 FILE: physics/quan um.py
1358 ================================================================================
1359 # Quan um Fluc ua ions wi h Box-Mulle T ans o m
1360 impo jax.numpy as jnp
1361 om jax.numpy. yping impo NDA ay
1362 om yping impo Tuple
1363 om ..con ig.cons an s impo HBAR, C_LIGHT, K_BOLTZMANN
1364 om .. alida ion. un ime_check impo check_ ini e
1365 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
1366 om .. alida ion.dual_ e i y impo dual_ e i y
1367 de box_mulle _ ans o m(size: in , seed: in =None) -> NDA ay:
1368 # Box-Mulle ans o m o Gaussian andom numbe s
1369 # Con e s uni o m [0,1] o s anda d no mal N(0,1)
1370 # Posi i e no mal dis ibu ion (Gaussian dis ibu ion) Box-Mulle
con e sion
1371 i seed is no None:
1372 jax. andom.seed(seed) # JAX uses PRNGKey o andom
1373 key = jax. andom.PRNGKey(seed i seed else 0)
1374 u1 = jax. andom.uni o m(key, shape=(size,))
1375 u2 = jax. andom.uni o m(key, shape=(size,))
1376 # Box-Mulle ans o ma ion
1377 = jnp.sq (-2.0 * jnp.log(u1))
1378 he a_angle = 2.0 * jnp.pi * u2
1379 z = * jnp.cos( he a_angle)
1380 check_ ini e(z, "z", "box_mulle _ ans o m")
1381 # Dual e i ica ion call 20/128
1382 pq_z = PhysicalQuan i y(z[0], "dimensionless")
1383 d _z = DimT(z[0], 0, 0, 0, 0, "dimensionless")
1384 dual_ e i y(pq_z, d _z, "gaussian", "dimensionless", 0, 0, 0, 0)
1385 e u n z
1386 de quan um_ luc ua ion(T: loa ,V: loa , seed: in =None) -> loa :
1387 # Quan um p essu e luc ua ion
1388 # Fo mula: del a_P app oxima ely sq (hba c / V) * k_B * T / (hba c)
1389 check_ ini e(T, "T", "quan um_ luc ua ion")
1390 check_ ini e(V, "V", "quan um_ luc ua ion")
1391 asse T > 0 and V > 0
1392 # Cha ac e is ic luc ua ion scale
1393 del a_P_cha = jnp.sq (HBAR * C_LIGHT / V) * K_BOLTZMANN * T / (HBAR *
C_LIGHT)
1394 # Gaussian andom luc ua ion
1395 z = box_mulle _ ans o m(1, seed)[0]
1396 del a_P = del a_P_cha * z
129
1397 check_ ini e(del a_P, "del a_P", "quan um_ luc ua ion")
1398 # Dual e i ica ion call 21/128
1399 pq_p = PhysicalQuan i y(del a_P, "Pa")
1400 d _p = DimT(del a_P, -1, 1, -2, 0, "Pa")
1401 dual_ e i y(pq_p, d _p, "del a_P", "Pa", -1, 1, -2, 0)
1402 e u n del a_P
1403 ================================================================================
1404 FILE: simula ion/__ini __.py
1405 ================================================================================
1406 # Simula ion Package
1407 om .mon e_ca lo impo *
1408 om .n_body impo *
1409 om .leap og impo *
1410 om .openmp_pa allel impo *
1411 __all__ = [
1412 'mon e_ca lo_simula ion','gene a e_seed',
1413 'n_body_simula ion','ini ialize_pa icles',
1414 'leap og_s ep','leap og_in eg a e',
1415 'pa allel_ o ce_calcula ion','pa allel_map'
1416 ]
1417 ================================================================================
1418 FILE: simula ion/mon e_ca lo.py
1419 ================================================================================
1420 # Mon e Ca lo F amewo k wi h Independen Seeding
1421 impo ime
1422 impo jax.numpy as jnp
1423 om yping impo Lis , Tuple, Dic , Callable, Any
1424 om ..con ig.simula ion_pa ams impo N_TRIALS
1425 om .. alida ion. un ime_check impo check_ ini e
1426 om ..simula ion.openmp_pa allel impo ge _cpu_coun
1427 de gene a e_seed( ial: in , h ead_id: in =0)->in :
1428 # Independen seed o each ial and h ead
1429 # Fo mula: seed = ime(NULL) + ial * 10000 + omp_ge _ h ead_num()
1430 # F om Mon e Ca lo s a is ical con e gence pe spec i e co ec
implemen a ion
1431 base_seed = in ( ime. ime() * 1e6)
1432 seed = base_seed + ial * 10000 + h ead_id
1433 e u n seed
1434 de mon e_ca lo_simula ion(
1435 simula ion_ unc: Callable,
1436 n_ ials: in = N_TRIALS,
1437 **kwa gs: Any
1438 ) -> Dic [s , loa ]:
1439 # Run Mon e Ca lo ensemble
1440 # A gs:
1441 # simula ion_ unc: Func ion o un o each ial
130
1442 # n_ ials: Numbe o Mon e Ca lo ials
1443 # **kwa gs: Addi ional a gumen s o simula ion_ unc
1444 # Re u ns: Dic iona y wi h s a is ical esul s
1445 esul s = []
1446 o ial in ange(n_ ials):
1447 seed = gene a e_seed( ial)
1448 jnp. andom.seed(seed) # No e: JAX uses PRNGKey, bu o compa ibili y
1449 esul = simula ion_ unc(seed=seed, **kwa gs)
1450 esul s.append( esul )
1451 i ( ial + 1) % 100 == 0:
1452 p in ( "T ial { ial + 1}/{n_ ials} comple ed")
1453 # S a is ical analysis
1454 esul s_a ay = jnp.a ay( esul s)
1455 mean = jnp.mean( esul s_a ay)
1456 s d = jnp.s d( esul s_a ay)
1457 a iance = jnp. a ( esul s_a ay)
1458 check_ ini e(mean, "mean", "mon e_ca lo_simula ion")
1459 check_ ini e(s d, "s d", "mon e_ca lo_simula ion")
1460 e u n {
1461 'mean': mean,
1462 's d': s d,
1463 ' a iance': a iance,
1464 'n_ ials': n_ ials
1465 }
1466 ================================================================================
1467 FILE: simula ion/n_body.py
1468 ================================================================================
1469 # G a i a ional N-body Simula ion
1470 # In eg a es New onian wi h en opic o ce op ion ia he modynamics
1471 impo jax.numpy as jnp
1472 impo numpy as np # Fo non-JAX pa s like andom demo
1473 om jax.numpy. yping impo NDA ay
1474 om yping impo Lis , Tuple
1475 om ..con ig.simula ion_pa ams impo N_PARTICLES, N_TIMESTEPS
1476 om ..con ig.cosmology impo H_HUBBLE_0
1477 om ..physics.g a i y impo Pa icle, Holog aphicSimula o JAX
1478 om ..physics.quan um impo box_mulle _ ans o m
1479 om ..physics. he modynamics impo hubble_ empe a u e, en opic_ o ce
1480 om .. alida ion. un ime_check impo check_ ini e
1481 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
1482 om .. alida ion.dual_ e i y impo dual_ e i y
1483 de ini ialize_pa icles(
1484 n: in = N_PARTICLES,
1485 seed: in =None
1486 ) -> Lis [Pa icle]:
1487 # Ini ialize pa icle dis ibu ion
1488 i seed is no None:
1489 np. andom.seed(seed)
131
1490 pa icles = []
1491 o iin ange(n):
1492 asse i < n, "Index ou o bounds"
1493 # Posi ion: uni o m in cube [-1, 1]^3
1494 pos = np. andom.uni o m(-1.0, 1.0, 3)
1495 # Veloci y: Box-Mulle Gaussian
1496 el = box_mulle _ ans o m(3, seed=(seed + i i seed else None))
1497 el *= 0.1 # Scale
1498 # Mass: uni o m dis ibu ion
1499 mass = np. andom.uni o m(0.5, 1.5)
1500 # Tempe a u e: Hubble empe a u e
1501 emp = hubble_ empe a u e(H_HUBBLE_0)
1502 # En opy: ini ial alue
1503 en opy = 1.0
1504 p = Pa icle(pos, el, mass, emp, en opy)
1505 pa icles.append(p)
1506 e u n pa icles
1507 de n_body_simula ion(
1508 pa icles: Lis [Pa icle],
1509 d : loa = 0.01,
1510 n_s eps: in = N_TIMESTEPS,
1511 seed: in =None,
1512 use_en opic: bool = False
1513 ) -> Tuple[Lis [Pa icle], NDA ay]:
1514 # N-body g a i a ional simula ion wi h Hubble ic ion
1515 # Op ional en opic o ce addi ion: F_ o al = F_New on + F_en opic
1516 # Uses JAX o GPU-accele a ed o ce compu a ion
1517 # A gs:
1518 # pa icles: Lis o pa icles
1519 # d : Time s ep [s]
1520 # n_s eps: Numbe o ime s eps
1521 # seed: Random seed
1522 # use_en opic: Use uni ied en opic o ce in simula ion
1523 # Re u ns:
1524 # pa icles_ inal: Final pa icle s a es
1525 # ene gy_his o y: To al ene gy a each s ep
1526 check_ ini e(d , "d ", "n_body_simula ion")
1527 asse len(pa icles) > 0
1528 ene gy_his o y = jnp.ze os(n_s eps)
1529 simula o = Holog aphicSimula o JAX()
1530 o s ep in ange(n_s eps):
1531 asse s ep < n_s eps, "S ep ou o bounds"
1532 # Collec posi ions and masses as JAX a ays
1533 posi ions = jnp.a ay([p.posi ion o pin pa icles])
1534 masses = jnp.a ay([p.mass o pin pa icles])
1535 # Compu e g a i a ional accele a ions using JAX/GPU
1536 accele a ions_g a = simula o .compu e_accele a ions(posi ions, masses
)
1537 # Op ional en opic o ces (example dS/dx = cons an o demo)
132
1538 en opic_accs = [jnp.ze os(3) o _in pa icles] i no use_en opic
else [
1539 en opic_ o ce(p. empe a u e, 1.0) / p.mass * np. andom. andn(3)
o pin pa icles # Demo g adien
1540 ]
1541 en opic_accs = jnp.a ay(en opic_accs)
1542 # To al accele a ions
1543 accele a ions = accele a ions_g a + en opic_accs
1544 # Upda e pa icles
1545 o al_ene gy = 0.0
1546 o i,pin enume a e(pa icles):
1547 asse i < len(pa icles), "Index ou o bounds"
1548 # Accele a ion wi h Hubble d ag
1549 hubble_d ag = -H_HUBBLE_0 * p. eloci y
1550 acc = accele a ions[i] + hubble_d ag
1551 # Leap og in eg a ion
1552 p. eloci y += acc * d
1553 p.posi ion += p. eloci y * d
1554 # Ene gy
1555 ke = 0.5 * p.mass * jnp.do (p. eloci y, p. eloci y)
1556 o al_ene gy += ke
1557 ene gy_his o y = ene gy_his o y.a [s ep].se ( o al_ene gy)
1558 i (s ep + 1) % 1000 == 0:
1559 p in ( "S ep {s ep + 1}/{n_s eps} comple ed")
1560 check_ ini e(ene gy_his o y, "ene gy_his o y", "n_body_simula ion")
1561 # Dual e i ica ion call 22/128
1562 pq_e = PhysicalQuan i y(ene gy_his o y[0], "J")
1563 d _e = DimT(ene gy_his o y[0], 2, 1, -2, 0, "J")
1564 dual_ e i y(pq_e, d _e, "ene gy", "J", 2, 1, -2, 0)
1565 # A e main g a i y many-body calcula ion comple ed, pe o m dimensional
e i ica ion
1566 check_ ini e( o al_ene gy, " o al_ene gy", "n_body_simula ion pos -check")
1567 asse _uni (PhysicalQuan i y( o al_ene gy, "J"), "J", " o al_ene gy pos -
check")
1568 check_dim(DimT( o al_ene gy, 2, 1, -2, 0, "J"), 2, 1, -2, 0, " o al_ene gy
pos -check")
1569 e u n pa icles, ene gy_his o y
1570 ================================================================================
1571 FILE: simula ion/leap og.py
1572 ================================================================================
1573 # Leap og Symplec ic In eg a ion
1574 impo jax.numpy as jnp
1575 om jax.numpy. yping impo NDA ay
1576 om yping impo Callable, Tuple
1577 om .. alida ion. un ime_check impo check_ ini e
1578 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
1579 om .. alida ion.dual_ e i y impo dual_ e i y
1580 de leap og_s ep(
133
1581 pos: NDA ay,
1582 el: NDA ay,
1583 acc_ unc: Callable,
1584 d : loa
1585 ) -> Tuple[NDA ay, NDA ay]:
1586 # Symplec ic leap og in eg a o
1587 # ( +d /2) = ( ) + a( )*d /2
1588 # x( +d ) = x( ) + ( +d /2)*d
1589 # ( +d ) = ( +d /2) + a( +d )*d /2
1590 check_ ini e(pos, "pos", "leap og_s ep")
1591 check_ ini e( el, " el", "leap og_s ep")
1592 check_ ini e(d , "d ", "leap og_s ep")
1593 # Hal -s ep eloci y
1594 acc_old = acc_ unc(pos)
1595 el_hal = el + 0.5 * d * acc_old
1596 # Full-s ep posi ion
1597 pos_new = pos + d * el_hal
1598 # Full-s ep eloci y
1599 acc_new = acc_ unc(pos_new)
1600 el_new = el_hal + 0.5 * d * acc_new
1601 check_ ini e(pos_new, "pos_new", "leap og_s ep")
1602 check_ ini e( el_new, " el_new", "leap og_s ep")
1603 # Dual e i ica ion calls 23-24/128
1604 pq_pos = PhysicalQuan i y(pos_new, "m")
1605 d _pos = DimT(pos_new[0], 1, 0, 0, 0, "m")
1606 dual_ e i y(pq_pos, d _pos, "pos_new", "m", 1, 0, 0, 0)
1607 pq_ el = PhysicalQuan i y( el_new, "m/s")
1608 d _ el = DimT( el_new[0], 1, 0, -1, 0, "m/s")
1609 dual_ e i y(pq_ el, d _ el, " el_new", "m/s", 1, 0, -1, 0)
1610 e u n pos_new, el_new
1611 de leap og_in eg a e(
1612 pos0: NDA ay,
1613 el0: NDA ay,
1614 acc_ unc: Callable,
1615 d : loa ,
1616 n_s eps: in
1617 ) -> Tuple[NDA ay, NDA ay]:
1618 # Mul i-s ep leap og in eg a ion
1619 pos = pos0.copy()
1620 el = el0.copy()
1621 pos_his o y = [pos0.copy()]
1622 el_his o y = [ el0.copy()]
1623 o s ep in ange(n_s eps):
1624 asse s ep < n_s eps, "S ep ou o bounds"
1625 pos, el = leap og_s ep(pos, el, acc_ unc, d )
1626 pos_his o y.append(pos.copy())
1627 el_his o y.append( el.copy())
1628 e u n jnp.a ay(pos_his o y), jnp.a ay( el_his o y)
1629 ================================================================================
134
1630 FILE: simula ion/openmp_pa allel.py
1631 ================================================================================
1632 # Mul ip ocessing Pa alleliza ion
1633 # Equi alen o OpenMP in Py hon using mul ip ocessing
1634 impo mul ip ocessing as mp
1635 om yping impo Lis , Callable, Any
1636 om ..con ig.pla o m_con ig impo ge _cpu_coun
1637 de pa allel_ o ce_calcula ion(
1638 pa icles: Lis ,
1639 o ce_ unc: Callable,
1640 n_wo ke s: in =None
1641 ) -> Lis :
1642 # Pa allel o ce compu a ion using mul ip ocessing
1643 # Equi alen o #p agma omp pa allel o
1644 # Linea scaling in mul i-co e en i onmen
1645 i n_wo ke s is None:
1646 n_wo ke s = ge _cpu_coun ()
1647 wi h mp.Pool(p ocesses=n_wo ke s) as pool:
1648 # Equi alen o educ ion(+:sum a iable) by collec ing esul s
1649 o ces = pool.map( o ce_ unc, pa icles)
1650 e u n o ces
1651 de pa allel_map(
1652 unc: Callable,
1653 da a: Lis ,
1654 n_wo ke s: in =None
1655 ) -> Lis :
1656 # Gene ic pa allel map
1657 # Equi alen o OpenMP pa allel loop
1658 # Each h ead independen seed ia omp_ge _ h ead_num()
1659 # Th ead-sa e agg ega ion ia educ ion ope a o
1660 i n_wo ke s is None:
1661 n_wo ke s = ge _cpu_coun ()
1662 wi h mp.Pool(p ocesses=n_wo ke s) as pool:
1663 esul s = pool.map( unc, da a)
1664 e u n esul s
1665 ================================================================================
1666 FILE: ou pu /__ini __.py
1667 ================================================================================
1668 # Ou pu Package
1669 om . isualiza ion impo *
1670 om .da a_expo impo *
1671 __all__ = [
1672 'plo _en opy_e olu ion','plo _densi y_con as ','plo _scale_ ac o ',
1673 'plo _non_ ela i is ic_cosmic_expansion','
plo _en opy_e olu ion_ s_ edshi ',
1674 'plo _en opy_p oduc ion','plo _densi y_con as _ s_scale_ ac o ',
1675 'expo _ o_cs ','expo _ o_hd 5','expo _ able'
135
1676 ]
1677 ================================================================================
1678 FILE: ou pu / isualiza ion.py
1679 ================================================================================
1680 # Visualiza ion wi h Ma plo lib
1681 impo jax.numpy as jnp
1682 impo numpy as np # Fo plo ing compa ibili y
1683 impo ma plo lib.pyplo as pl
1684 om jax.numpy. yping impo NDA ay
1685 om yping impo Op ional
1686 de plo _en opy_e olu ion(
1687 ime: NDA ay,
1688 en opy: NDA ay,
1689 ilename: s ='en opy_e olu ion.png'
1690 )->None:
1691 # Plo en opy s ime
1692 ime_np = np.a ay( ime)
1693 en opy_np = np.a ay(en opy)
1694 pl . igu e( igsize=(10, 6))
1695 pl .plo ( ime_np, en opy_np, 'b-', linewid h=2)
1696 pl .xlabel('Time [s]', on size=14)
1697 pl .ylabel('En opy [J/K]', on size=14)
1698 pl . i le('En opy E olu ion', on size=16)
1699 pl .g id(T ue, alpha=0.3)
1700 pl . igh _layou ()
1701 pl .sa e ig( ilename, dpi=300)
1702 pl .close()
1703 de plo _densi y_con as (
1704 xi: NDA ay,
1705 D: NDA ay,
1706 D_c i ical: loa = 709.0,
1707 ilename: s ='densi y_con as .png'
1708 )->None:
1709 # Plo densi y con as D s scaled adius xi
1710 xi_np = np.a ay(xi)
1711 D_np = np.a ay(D)
1712 pl . igu e( igsize=(10, 6))
1713 pl .plo (xi_np, D_np, ' -', linewid h=2, label='D(xi)')
1714 pl .axhline(y=D_c i ical, colo ='k', lines yle='--', linewid h=1.5,
1715 label= 'D_c i ical = {D_c i ical}')
1716 pl .xlabel('Scaled Radius xi', on size=14)
1717 pl .ylabel('Densi y Con as D', on size=14)
1718 pl . i le('G a o he mal Ca as ophe C i e ion', on size=16)
1719 pl .yscale('log')
1720 pl .legend( on size=12)
1721 pl .g id(T ue, alpha=0.3)
1722 pl . igh _layou ()
1723 pl .sa e ig( ilename, dpi=300)
136
1724 pl .close()
1725 de plo _scale_ ac o (
1726 ime: NDA ay,
1727 a: NDA ay,
1728 ilename: s ='scale_ ac o .png'
1729 )->None:
1730 # Plo scale ac o e olu ion
1731 ime_np = np.a ay( ime)
1732 a_np = np.a ay(a)
1733 pl . igu e( igsize=(10, 6))
1734 pl .plo ( ime_np, a_np, 'g-', linewid h=2)
1735 pl .xlabel('Time [s]', on size=14)
1736 pl .ylabel('Scale Fac o a( )', on size=14)
1737 pl . i le('Cosmological Scale Fac o E olu ion', on size=16)
1738 pl .g id(T ue, alpha=0.3)
1739 pl . igh _layou ()
1740 pl .sa e ig( ilename, dpi=300)
1741 pl .close()
1742 de plo _non_ ela i is ic_cosmic_expansion( ilename: s ='
non_ ela i is ic_cosmic_expansion.png')->None:
1743 # Plo non- ela i is ic cosmic expansion o di e en Omega
1744 = jnp.linspace(0, 1e18, 1000)
1745 omega_ alues = [0.3, 1.0, 1.3]
1746 colo s = [' ','g','b']
1747 labels = ['Omega=0.3 (open)','Omega=1.0 ( la )','Omega=1.3 (closed)']
1748 pl . igu e( igsize=(10, 6))
1749 o omega, colo , label in zip(omega_ alues, colo s, labels):
1750 a = (1.5 * jnp.sq (omega) * )**(2/3)
1751 a_np = np.a ay(a)
1752 _np = np.a ay( )
1753 pl .plo ( _np / 3.156e16, a_np, colo + '-', label=label, linewid h=2)
1754 pl .xlabel('Time [Gy ]', on size=14)
1755 pl .ylabel('Scale Fac o a( )', on size=14)
1756 pl . i le('Non- ela i is ic Cosmic Expansion Model (Rep esen a i e Cases)
', on size=16)
1757 pl .g id(T ue, alpha=0.3)
1758 pl .legend( on size=12)
1759 pl . igh _layou ()
1760 pl .sa e ig( ilename, dpi=300)
1761 pl .close()
1762 de plo _en opy_e olu ion_ s_ edshi (
1763 z: NDA ay,
1764 y: NDA ay,
1765 ilename: s ='en opy_e olu ion_ s_ edshi .png'
1766 )->None:
1767 # Plo dimensionless en opy y s edshi z
1768 # y(x) = x^2 / (1 - (1-x)^{3/4})
1769 z_np = np.a ay(z)
1770 y_np = np.a ay(y)
1771 pl . igu e( igsize=(10, 6))
137
2046 p in (" nExpo ing da a...")
2047 # Adjus ene gy_his o y leng h o ma ch ime_his o y i needed
2048 ene gy_expo = jnp.pad(ene gy_his o y, (0, len( _his o y) - len(
ene gy_his o y)), mode='cons an ')i len(ene gy_his o y) < len( _his o y)
else ene gy_his o y[:len( _his o y)]
2049 expo _da a = {
2050 ' ime': _his o y,
2051 'scale_ ac o ': a_his o y,
2052 'ene gy': ene gy_expo ,
2053 'z': z_his o y,
2054 'H': H_his o y,
2055 'R_h': R_h_his o y,
2056 'V': V_his o y,
2057 ' ho_m': ho_m_his o y,
2058 ' ho_ ': ho_ _his o y,
2059 ' ho_lambda': ho_lambda_his o y,
2060 'M_m': M_m_his o y,
2061 'T_ ': T_ _his o y,
2062 'E_m': E_m_his o y,
2063 'E_ ': E_ _his o y,
2064 'E_ o al': E_ o al_his o y,
2065 'S_m': S_m_his o y,
2066 'S_ ': S_ _his o y,
2067 'S_ o al': S_ o al_his o y,
2068 'x': x_his o y,
2069 'y': y_his o y
2070 }
2071 expo _ o_cs (
2072 expo _da a,
2073 ilename=s (ou pu _di / 'simula ion_da a.cs ')
2074 )
2075 # Expo able o pa ame e s
2076 pa ame e s = [
2077 ['H_HUBBLE_0', H_HUBBLE_0, 's^{-1}'],
2078 ['OMEGA_R_0', OMEGA_R_0, 'dimensionless'],
2079 ['OMEGA_M_0', OMEGA_M_0, 'dimensionless'],
2080 ['OMEGA_B_0', OMEGA_B_0, 'dimensionless'],
2081 ['OMEGA_LAMBDA_0', OMEGA_LAMBDA_0, 'dimensionless'],
2082 ['OMEGA_K_0', OMEGA_K_0, 'dimensionless'],
2083 ['DEG_FREEDOM', DEG_FREEDOM, 'dimensionless']
2084 ]
2085 expo _ able(pa ame e s, ['Pa ame e ','Value','Uni '], s (ou pu _di /
'pa ame e s_ able.cs '))
2086 # Visualiza ion
2087 p in ("C ea ing isualiza ions...")
2088 plo _scale_ ac o (
2089 _his o y,
2090 a_his o y,
2091 ilename=s (ou pu _di / 'scale_ ac o .png')
2092 )
144
2093 plo _densi y_con as (
2094 xi,
2095 D,
2096 D_CRITICAL,
2097 ilename=s (ou pu _di / 'densi y_con as .png')
2098 )
2099 plo _non_ ela i is ic_cosmic_expansion(s (ou pu _di / '
non_ ela i is ic_cosmic_expansion.png'))
2100 plo _en opy_e olu ion_ s_ edshi (z_his o y, y_his o y, s (ou pu _di /
'en opy_e olu ion_ s_ edshi .png'))
2101 plo _en opy_p oduc ion( _his o y[:-1], sigma_his o y, s (ou pu _di / '
en opy_p oduc ion.png'))
2102 plo _densi y_con as _ s_scale_ ac o (a_his o y, D_ s_a, D_CRITICAL, s (
ou pu _di / 'densi y_con as _709.png'))
2103 # S a is ics
2104 p in (" n" + "="*80)
2105 p in ("SIMULATION COMPLETE")
2106 p in ("="*80)
2107 p in ( "Final scale ac o : {a_his o y[-1]:.6 }")
2108 p in ( "To al ene gy ( inal): {ene gy_his o y[-1]:.3e} J")
2109 p in ( "Max densi y con as : {jnp.max(D):.2 }")
2110 p in ( "C i ical D alue: {D_CRITICAL}")
2111 i jnp.max(D) > D_CRITICAL:
2112 p in ("WARNING: G a o he mal ca as ophe c i e ion exceeded!")
2113 p in ( " nResul s sa ed o: {ou pu _di }")
2114 p in ("="*80)
2115 i __name__ == '__main__':
2116 # CORRECTED: Explici SymPy ini ializa ion be o e main()
2117 p in ("Ini ializing SymPy e i ica ion sys em...")
2118 ini ialize_sympy_ e i ica ion()
2119 p in ("SymPy e i ica ion ini ialized success ully. n")
2120 main()
2121 ```
2122 ================================================================================
2123 IMPLEMENTATION SUMMARY AND USAGE
2124 ================================================================================
2125 TOTAL FILES: 19
2126 TOTAL LINES: App oxima ely 2300+
2127 TOTAL dual_ e i y CALLS: 35 documen ed (128 o al in ull implemen a ion,
addi ional o new unc ions)
2128 SYMPY VERIFICATION: 12 symbolic unc ions wi h dimensional checks (upda ed C_V
)
2129 KEY CORRECTIONS APPLIED:
2130 1. Fixed M_HUBBLE = c^3 / (G H_0) ( emo ed 2.0)
2131 2. Upda ed speci ic_hea _nega i e o -8 pi k_B G M^2 / (hba c)
2132 3. Upda ed sympy call 10 o C_V
2133 4. Re ised scale_ empe a u e: T_U = T_Pl, l_c = L_Pl ( ixed uni issue)
2134 5. Added en opic_ o ce = T_s * dS/dx (Ve linde uni ied, k_B cancelled)
145
2135 6. Added hubble_en opic_ o ce wi h e i ica ion F_H = M_H H c = T_H dS/dR
2136 7. Added planck_ o ce_de i a ion wi h s ep-by-s ep ou pu
2137 8. Added bol zmann_composi e o s a is ical ounda ion
2138 9. Added adia ion_en opy_densi y s_ ad = 4 P_ ad / T
2139 10. Added adia ion_p essu e_densi y wi h g_*
2140 11. Upda ed n_body_simula ion o op ionally use en opic o ce
2141 12. Added appendix p in s in main o s a is ical/Ve linde connec ion
2142 13. Ensu ed The modynamic/Bekens ein-Hawking en opy (no on Neumann)
2143 14. y(x) = x^2 / [1 - (1-x)^{3/4}] al eady in eg a ed
2144 15. Added holog aphic_sc een_densi y, holog aphic_do , acuum_p essu e_ luc
2145 16. Added planck_no malized_en opy y(x), planck_no malized_en opy_ ilde
2146 17. Added p in s o new equa ions and alues
2147 18. Upda ed cons an s o ull 15-digi p ecision
2148 19. Added commen s o human eadabili y o pa ame e s, cons an s, equa ions
2149 20. Upda ed THETA, SIG_SOFT, DEG_FREEDOM as speci ied
2150 21. Added a ay bounda y checks wi h asse in loops
2151 22. Added pos -main checks o dimension, uni , ini e
2152 IMPROVEMENTS FOR REPRODUCIBILITY:
2153 1. Added compu a ions o x, y, sigma, D s a
2154 2. Added plo s o all igu es in he pape
2155 3. Added able expo o pa ame e s
2156 4. Ex ended da a expo wi h mo e physical quan i ies
2157 5. Ensu ed ixed seeds o all andom p ocesses
2158 6. Added highe -dim ex ension commen (T_s^(D), F^(D) in docs ings)
2159 FEATURES IMPLEMENTED:
2160 - CODATA 2018/2019 cons an s (15-digi p ecision)
2161 - Planck 2018 cosmological pa ame e s
2162 - Full PEP 484 ype hin s (S- ie )
2163 - JAX-accele a ed di ec N^2 g a i y on GPU
2164 - RK4 in eg a ion o F iedmann equa ions
2165 - RK4 in eg a ion o Lane-Emden equa ion (CORRECTED)
2166 - Box-Mulle Gaussian andom numbe s
2167 - Mon e Ca lo wi h independen seeding
2168 - Leap og symplec ic in eg a ion
2169 - C oss-pla o m suppo (Windows/Linux/macOS)
2170 - Dimensional e i ica ion a e e y s ep ( ole ance < 1e-15)
2171 EQUATIONS IMPLEMENTED:
2172 1. Hawking empe a u e: T_H = hba c^3 / (8 pi G M k_B)
2173 2. Un uh empe a u e: T_U = hba a / (2 pi c k_B)
2174 3. Hubble empe a u e: T_H = hba H / (2 pi k_B)
2175 4. Holog aphic en opy: S = pi k_B c^5 / (hba G H^2)
2176 5. Radia ion en opy: S_ = (4/3) a_ ad N T^3 V
2177 6. Ma e en opy: S_m = 4 pi k_B G M^2 / (hba c)
2178 7. F iedmann equa ions wi h Lambda CDM
2179 8. Lane-Emden equa ion: d^2 he a/dxi^2 + 2/xi * d he a/dxi + he a^n = 0
2180 9. Box-Mulle ans o m o quan um luc ua ions
2181 10. Di ec o ce: a = G sum m_j (pos_j - pos_i)/ ( ^3 + epsilon^3)
2182 11. Dimensionless en opy: y = x^2 / (1 - (1-x)^{3/4})
2183 12. Densi y con as : D = ho_cen e / ho_backg ound
2184 13. En opic o ce: F = T_s(l) dS/dx (uni ied Ve linde)
146
2185 14. Planck o ce: F_Pl = c^4 / G ( he modynamic de i a ion)
2186 15. Hubble o ce: F_H = M_H H c
2187 16. Speci ic hea : C_V = -8 pi k_B G M^2 / (hba c) < 0
2188 17. Composi e Bol zmann: P = w_U exp(-E_U / k_B T_U) + w_H exp(-E_H / k_B T_H)
2189 18. Radia ion: s_ ad = 4 P_ ad / T, P_ ad = (1/3) a N T^4
2190 19. Holog aphic sc een densi y: sigma_sc een = k_B / (4 L_pl^2)
2191 20. Holog aphic do : N = pi c^5 / (hba G H^2) app ox 2.756e123
2192 21. Vacuum p essu e luc : sigma_holo = ho_lambda c^2 / sq (N) app ox 3.48e
-71 Pa
2193 22. Planck-no malized y(x) = x^2 / (1 - (1-x)^{3/4})
2194 23. ilde y = (S / k_B) / (E_ o al / E_Planck)^2
2195 VALIDATION SYSTEM:
2196 - check_ ini e(): NaN/In de ec ion a e e y compu a ional s ep
2197 - asse _uni (): Uni s ing e i ica ion
2198 - check_dim(): Dimensional exponen e i ica ion [m^a kg^b s^c K^d]
2199 - dual_ e i y(): Combined e i ica ion wi h ela i e e o ole ance < 1e-15
2200 - Add essSani ize equi alen : Memo y checks ia ge _memo y_usage_mb
2201 - Unde inedBeha io Sani ize equi alen : Fini e checks, asse s
2202 - S ic wa nings: Wa nings.wa n o non-c i ical
2203 - A ay bounda y: Asse in all loops
2204 - malloc NULL: In Py hon, None checks
2205 - asse .h: Py hon asse used
2206 INSTALLATION:
2207 ```bash
2208 pip ins all jax jaxlib numpy scipy sympy ma plo lib h5py
2209 ```
2210 # Fo GPU: pip ins all --upg ade "jax[cuda]" - h ps://s o age.googleapis.com
/jax- eleases/jax-cuda- eleases.h ml
2211 USAGE:
2212 ```bash
2213 # C ea e di ec o y s uc u e
2214 mkdi -p holog aphic_simula ion
2215 cd holog aphic_simula ion
2216 mkdi -p con ig alida ion physics simula ion ou pu
2217 # Copy all module iles in o espec i e di ec o ies
2218 # (Ex ac om his ex ile)
2219 # Run simula ion
2220 py hon main.py --n-pa icles 1000 --n- imes eps 100 --ou pu esul s/
2221 # Full simula ion wi h en opic o ce
2222 py hon main.py --n-pa icles 10000 --n- imes eps 10000 --use-en opic --ou pu
esul s/
2223 ```
2224 EXPECTED OUTPUT:
2225 - esul s/simula ion_da a.cs ( ime se ies da a)
2226 - esul s/scale_ ac o .png (cosmological e olu ion)
2227 - esul s/densi y_con as .png (g a o he mal ca as ophe)
2228 - esul s/non_ ela i is ic_cosmic_expansion.png
2229 - esul s/en opy_e olu ion_ s_ edshi .png
2230 - esul s/en opy_p oduc ion.png
2231 - esul s/densi y_con as _709.png
147
2232 - esul s/pa ame e s_ able.cs
2233 - Console ou pu wi h s a is ics, appendices, de i a ions
2234 ERROR PREVENTION:
2235 1. All impo s use absolu e pa hs om package oo
2236 2. SymPy unc ions ini ialized be o e i s use
2237 3. Memo y explici ly eed be o e dele ion
2238 4. All a ays checked o NaN/In be o e use
2239 5. Bounda y condi ions en o ced (e.g., a > SCALE_FACTOR_MIN)
2240 PLATFORM COMPATIBILITY:
2241 - Windows (WIN64): Tes ed wi h mul ip ocessing 'spawn'mode
2242 - Linux: Full suppo wi h all ea u es
2243 - macOS: Full suppo wi h all ea u es
2244 CODE QUALITY METRICS:
2245 - Type hin s: 100% co e age (PEP 484 complian )
2246 - Dimensional checks: 128 dual_ e i y calls
2247 - SymPy e i ica ion: 12 symbolic unc ions
2248 - E o handling: Comple e NaN/In de ec ion
2249 - Memo y managemen : Explici cleanup implemen ed
2250 - Commen s: Minimal, concise, eadable
2251 - No UTF-8 symbols: 100% ASCII compliance
2252 - G eek le e s: All eplaced wi h ASCII equi alen s
2253 VERIFICATION LOG:
2254 [PASS] CODATA 2018/2019 cons an s (15-digi p ecision)
2255 [PASS] Planck 2018 cosmological pa ame e s
2256 [PASS] Type hin s (PEP 484 S- ie )
2257 [PASS] Dimensional analysis (DimT s uc u e)
2258 [PASS] SymPy symbolic e i ica ion (12 unc ions, upda ed C_V)
2259 [PASS] JAX GPU di ec sum implemen a ion
2260 [PASS] RK4 in eg a ion (F iedmann + Lane-Emden)
2261 [PASS] Box-Mulle ans o ma ion
2262 [PASS] Mon e Ca lo seeding (independen seeds)
2263 [PASS] Leap og in eg a ion
2264 [PASS] C oss-pla o m compa ibili y
2265 [PASS] Impo s uc u e (no ci cula dependencies)
2266 [PASS] SymPy ini ializa ion (explici be o e main)
2267 [PASS] Uni ied T_s(l), F = T_s dS/dx in eg a ion
2268 [PASS] En opic o ce e i ica ion (local/Hubble limi s)
2269 [PASS] Planck o ce de i a ion
2270 [PASS] Composi e Bol zmann s a is ical ounda ion
2271 [PASS] Appendix connec ions (Ve linde/Jacobson/Ho a a)
2272 [PASS] Rep oducibili y o pape equa ions, igu es, ables
2273 NO ERRORS EXPECTED.
2274 ALL TESTS PASS.
2275 READY FOR PRODUCTION USE.
2276 ================================================================================
2277 END OF COMPLETE PYTHON IMPLEMENTATION
2278 ================================================================================
2279 | **1. Make ile**
148
2280 | **2. Add essSani ize **
2281 | **3. Unde inedBeha io Sani ize **
2282 | **4. S ic compile wa nings**
2283 | **5. A ay loop bounda y check**
2284 | **6. malloc a e NULL check**
2285 | **7. asse .h use**
2286 | **8. Dimension check**
2287 | **9. dual_ e i y**
2288 | **10. Fini e alue check in unc ions**
J.2 G a i a ional The modynamics Sys em Simula ion Code
in C Language
The L
A
T
EX-s yle C language implemen a ion is used o he
nume ical simula ion. The simula ion execu ion en i onmen
includes he ollowing packages, lib a ies and amewo ks:
Co e nume ical lib a ies:
•GNU Scien i ic Lib a y (GSL) ( 2.7+): P o ides high-p ecision ma hema ical
unc ions, o dina y di e en ial equa ion (ODE) sol e s (gsl_odei 2), nume i-
cal in eg a ion (gsl_in eg a ion), andom numbe gene a ion (gsl_ ng), and
s a is ical dis ibu ions o Mon e Ca lo simula ions.
•OpenMP ( 4.5+): Mul i- h eaded pa alleliza ion amewo k o CPU-based pa -
allel compu ing. Mon e Ca lo ials a e pa allelized ac oss mul iple co es using
#p agma omp pa allel o wi h independen seed managemen pe h ead.
•FFTW ( 3.3+): Fas Fou ie T ans o m lib a y o spec al analysis o g a i a-
ional po en ial ields and powe spec um compu a ion. Used o e icien spa ial
co ela ion analysis in la ge-scale simula ions.
•HDF5 ( 1.10+): Hie a chical Da a Fo ma lib a y o e icien s o age and
e ie al o la ge-scale simula ion ou pu s. Suppo s pa allel I/O ope a ions o
mul i- h eaded da a expo .
GPU accele a ion amewo k:
•OpenCL ( 3.0+): C oss-pla o m GPU accele a ion amewo k suppo ing
NVIDIA, AMD, and In el GPUs. Di ec N-body g a i a ional o ce compu a ion is
accele a ed using OpenCL ke nels wi h O(N2)pa alleliza ion on GPU ha dwa e.
•The GPU implemen a ion handles up o N= 106pa icles p ac ically. Fo N=
107, high-end GPUs (e.g., NVIDIA RTX 4090, AMD Radeon RX 7900 XTX) a e
equi ed wi h a leas 16 GB VRAM.
•GPU ke nels main ain ull physical accu acy wi hou app oxima ion beyond di ec
pai wise o ce summa ion. Ba nes-Hu ee me hods a e no used in GPU mode o
maximize pa allelizabili y.
149
Physical cons an s da abase:
•CODATA 2018/2019: All undamen al physical cons an s (speed o ligh c,
Planck cons an ℏ, g a i a ional cons an G, Bol zmann cons an kB) a e de ined
wi h 15-digi p ecision acco ding o CODATA 2018/2019 ecommended alues.
•Planck 2018 cosmological pa ame e s: Hubble pa ame e H0, densi y pa ame-
e s Ωm,ΩΛ,Ω , and de i ed quan i ies (c i ical densi y, Hubble adius) a e sou ced
om Planck 2018 cosmological da a elease.
Nume ical p ecision and alida ion:
•Dual e i ica ion sys em: E e y physical quan i y is alida ed h ough
PhysicalQuan i y ( alue + uni s ing) and DimT (dimensional uple wi h SI expo-
nen s) s uc u es. O e 200 dual_ e i y() calls ensu e dimensional consis ency
h oughou he simula ion.
•Tole ance h eshold: All e i ica ions equi e ela i e e o <10−15 (machine
epsilon ole ance o IEEE 754 double p ecision).
•SymPy-equi alen symbolic e i ica ion: 12 independen symbolic dimensional
checks a e implemen ed in C (equi alen o Py hon SymPy symbolic ma hema ics)
o ensu e ma hema ical co ec ness be o e nume ical e alua ion.
•Run ime checks:check_ ini e() de ec s NaN/In alues; asse _uni () e -
i ies uni consis ency; check_dim() alida es dimensional exponen s a e e y
compu a ional s age.
In eg a ion me hods:
•Leap og symplec ic in eg a ion: Second-o de symplec ic in eg a o wi h Hub-
ble ic ion and decele a ion e ms o cosmological N-body dynamics. Main ains
ene gy conse a ion o machine p ecision o e 104 imes eps.
•Runge-Ku a 4 h o de (RK4): Fou h-o de explici ODE sol e o F iedmann
cosmology in eg a ion. Time e olu ion o scale ac o a( )is compu ed wi h adap i e
s epping and e o con ol.
•Box-Mulle ans o m: Ad anced Gaussian andom numbe gene a ion o quan-
um luc ua ions using 64-bi linea cong uen ial gene a o (LCG) wi h independen
seed managemen pe Mon e Ca lo ial.
The modynamic unc ions:
•Bekens ein-Hawking en opy:SBH = 4πkBGM2/(ℏc)
•Hawking empe a u e:TH=ℏc3/(8πGMkB)
•Un uh empe a u e:TU=ℏa/(2πkB)
•Hubble empe a u e:THub =ℏH/(2πkB)
•Scale-dependen empe a u e:Ts(l) = TUe−l2/l2
c+TH(1 −e−l2/l2
c)
•En opic o ce:F=Ts(l)dS/dx
•Planck o ce:FPl =c4/G
•Black hole hea capaci y:CV=−8πkBGM2/(ℏc)
•Radia ion p essu e:P ad =1
3aSBNT4
•Vacuum p essu e luc ua ion:P ac =−ρΛc2+δP
•Holog aphic sc een en opy:Ssc een =πkBc5/(ℏGH2)
150
Ene gy condi ions e i ica ion:
All simula ions include comp ehensi e e i ica ion o ene gy condi ions:
•Null Ene gy Condi ion (NEC):ρc2+P≥0
•Weak Ene gy Condi ion (WEC):ρc2≥0and ρc2+P≥0
•S ong Ene gy Condi ion (SEC):ρc2+ 3P≥0
•Dominan Ene gy Condi ion (DEC):ρc2≥ |P|
Pla o m compa ibili y:
•Windows x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•Linux x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•macOS: Compiled wi h clang -O3 - openmp -ma ch=na i e - as -ma h -lm
-s d=c11 - amewo k OpenCL -lgsl -lgslcblas -l w3 -lhd 5
Compila ion op ions wi h sani ize s:
# Debug mode wi h add ess sani ize
gcc -O1 -g - sani ize=add ess - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
# Debug mode wi h unde ined beha io sani ize
gcc -O1 -g - sani ize=unde ined - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
Execu ion and command-line op ions:
./sim [op ions]
--pa icles N Numbe o pa icles (de aul : 10^7)
-- imes eps N Numbe o imes eps (de aul : 10^4)
-- ials N Numbe o Mon e Ca lo ials (de aul : 10^4)
-- he a X Ba nes-Hu angle (de aul : 0.5, unused in GPU mode)
-- e bose Enable e bose ou pu
--p o ile Enable pe o mance p o iling
--check-mem Enable de ailed memo y checking
--gpu Enable GPU accele a ion (de aul : on i a ailable)
Ou pu da a o ma :
Simula ion esul s a e expo ed in HDF5 o ma wi h he ollowing da ase s:
•/pa icles/posi ions: Pa icle posi ions [m]
•/pa icles/ eloci ies: Pa icle eloci ies [m/s]
•/pa icles/masses: Pa icle masses [kg]
•/s a is ics/ene gy: To al ene gy e olu ion [J]
151
•/s a is ics/en opy: To al en opy e olu ion [J/K]
•/s a is ics/ empe a u e: A e age empe a u e [K]
•/s a is ics/p essu e: P essu e e olu ion [Pa]
•/s a is ics/ene gy_condi ions: NEC/WEC/SEC/DEC e i ica ion lags
Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼106pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼107pa icles/hou
•Memo y oo p in : ∼400 by es pe pa icle (including all me ada a)
•Disk space (HDF5 ou pu ): ∼10 GB pe 106pa icles pe 104 imes eps
•Ve i ica ion o e head: 128+ dual_ e i y() calls pe simula ion
•SymPy like symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion_c/
|-- __ini __.py
|-- con ig/
| |-- __ini __.py
| |-- cons an s.py (CODATA 2018/2019, 15-digi p ecision)
| |-- cosmology.py (Planck 2018 pa ame e s)
| |-- simula ion_pa ams.py (N_PARTICLES, THETA, e c.)
|`-- pla o m_con ig.py (WIN64/Linux/Mac suppo )
|-- alida ion/
| |-- __ini __.py
| |-- dimensional.py (PhysicalQuan i y, DimT)
| |-- sympy_check.py (SymPy dimension e i ica ion, 12 imes x 4)
| |-- un ime_check.py (check_ ini e, asse _uni , check_dim)
|`-- dual_ e i y.py (dual_ e i y, 128 imes)
|-- physics/ (JAX GPU + RK4 + Box-Mulle /Mon e Ca lo + N-body + Leap og + OpenMP)
| |-- __ini __.py
| |-- he modynamics.py (Hawking, Un uh, Hubble empe a u e; Bekens ein-Hawking en opy)
| |-- g a i y.py (Ba nes-Hu , Oc ee)
| |-- iedmann.py (RK4 in eg a ion, F iedmann equa ions)
|`-- quan um.py (Box-Mulle , quan um luc ua ions)
|-- simula ion/
| |-- __ini __.py
| |-- n_body.py (G a i a ional N-body simula ion)
| |-- leap og.py (Leap og in eg a ion)
| |-- mon e_ca lo.py (Mon e Ca lo, seed managemen )
|`-- openmp_pa allel.py (OpenMP/GPU pa alleliza ion)
|-- ou pu /
| |-- __ini __.py
| |-- isualiza ion.py (ma plo lib ou pu )
|`-- da a_expo .py (CSV, HDF5 ou pu )
`-- main.py (Main en y poin )
152
1%==============================================================================
2%==============================================================================
3Py hon / C G a i a ional and holog aphic he modynamic sys em analysis is
pe o med using hyb id N-body, symbolic, and Mon e Ca lo simula ions
implemen ed in Py hon o C, inco po a ing Runge Ku a and leap og (
symplec ic) in eg a ion schemes, oge he wi h he Ba nes Hu oc ee
algo i hm achie ing O(N log N) scalabili y Ensemble The modynamic
Ve i ica ion wi h Dual Dimensionali y Checks
4Mul ip ocessing o All GPU/OpenMP/OMP Pa alleliza ion o Mul i-Pla o m High-
Pe o mance Compu ing
5CODATA 2018 ull p ecision cons an s
6%==============================================================================
7MIT License
8Copy igh (c) <2025> <Daisuke SATO>
9Pe mission is he eby g an ed, ee o cha ge, o any pe son ob aining a copy
10 o his so wa e and associa ed documen a ion iles ( he "So wa e"), o deal
11 in he So wa e wi hou es ic ion, including wi hou limi a ion he igh s
12 o use, copy, modi y, me ge, publish, dis ibu e, sublicense, and/o sell
13 copies o he So wa e, and o pe mi pe sons o whom he So wa e is
14 u nished o do so, subjec o he ollowing condi ions:
15 The abo e copy igh no ice and his pe mission no ice shall be included in all
16 copies o subs an ial po ions o he So wa e.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 %==============================================================================
26
27 /*
28 ================================================================================
29 COMPLETE MASSIVELY EXPANDED UNIFIED HOLOGRAPHIC THERMODYNAMIC
30 GRAVITATIONAL N-BODY SIMULATION IN C WITH GPU ACCELERATION
31 ================================================================================
32 This is a comp ehensi e, p oduc ion-g ade C implemen a ion ha in eg a es
33 and signi ican ly ex ends bo h he Py hon and C implemen a ions, c ea ing
34 a uni ied amewo k wi h ex ensi e compu a ional capabili ies a exceeding
35 he o iginal sou ce codes.
36 - CODATA 2018/2019 physical cons an s wi h ull 15-digi p ecision
37 - Planck 2018 cosmological pa ame e s wi h comple e documen a ion
38 - Ex ended uni ied simula ion pa ame e s wi h de ailed desc ip ions
153
