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:
3
•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)
.
4
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.
5
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.
The c osso e scale lceme ges om he equi emen ha he Un uh empe a-
u e associa ed wi h a local g a i a ional accele a ion becomes compa able o he
cosmological (Gibbons-Hawking) empe a u e:
TU(l)∼ℏ
2πkBc·c2
l≃TH=ℏH
2πkB
.(13)
Equa ing hese empe a u es yields l∼c/H =RH. A mo e p ecise ea men ,
accoun ing o geome ic p e ac o s and holog aphic deg ees o eedom, in oduces a
dimensionless coe icien αo o de uni y:
lc=RH
α,wi h α∼3–10.(14)
We adop α≈10 (lc≈0.1RH), which lies wi hin he heo e ically and obse a-
ionally mo i a ed ange [62?] while p o iding op imal in e pola ion o e 61 o de s
o magni ude om he Planck leng h o he Hubble adius.
The speci ic alue α≈10 is de e mined by ou physical consis ency equi emen s:
1. The modynamic consis ency (dS/d ≥0)
2. Obse a ional cons ain s (Planck 2018, DESI 2024–2025)
6
3. Nume ical s abili y (<10−15 e o ac oss 61 o de s)
4. Bounda y condi ion ma ching (TUand THlimi s)
Nume ical expe imen a ion shows ha α= 10±2p o ides op imal balance ac oss
hese c i e ia.
3.2 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, (15)
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].
3.3 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 .(16)
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.
3.4 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,(17)
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,(18)
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),(19)
p ese ing ˙
S > 0and Ve linde’s semiclassical limi , e i iable ia la ice QCD
holog aphic bounds [75,146].
7
3.4.1 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,(20)
wi h [Ts(l)·dS/dx] = [N].
3.5 Planck Fo ce De i a ion om Uni ied Scale-Dependen
En opic Fo ce
The Planck o ce ep esen s he undamen al o ce scale in quan um g a i y. Following
he uni ied en opic o ce amewo k, we de i e he Planck o ce a he Planck leng h
scale. A a Planck-scale in e ace wi h Planck empe a u e
FPl =TPl ×kB
lPl
(21)
=sℏc5
Gk2
B×kB× c3
ℏG(22)
=kBsℏc5
Gk2
B·c3
ℏG(23)
=kBsc8
G2k2
B
(24)
=kB×c4
GkB
(25)
=c4
G.(26)
Dimensional e i ica ion:
[TPl ×(kB/lPl)] = [K] ×[J ·K−1·m−1] = [J ·m−1] = [N].(27)
The nume ical alue is
FPl =c4
G≈1.21 ×1044 N.
8
Hea Capaci y a Planck Scale A he Planck scale:
CV=−8πkBGM2
ℏc.
The cha ac e is ic en opy g adien is ela ed o he undamen al en opy bound pe
Planck a ea. A he Planck scale whe e l∼LPlanck, he scale-dependen empe a u e
becomes app oxima ely he Planck empe a u e. The en opic o ce is:
FPl =TPl ·dσ
dxPlanck
,(28)
whe e he en opy g adien a Planck scales is se by undamen al in o ma ion densi y:
dσ
dxPlanck ∼kB
LPl
,(29)
wi h LPl =pℏG/c3as he Planck leng h [m]. Subs i u ing Planck empe a u e TPl =
pℏc5/(Gk2
B)and he en opy g adien :
FPl =sℏc5
Gk2
B·kB
LPl
(30)
= ℏc5
G·kB
pℏG/c3(31)
= ℏc5
G·kB· c3
ℏG(32)
=kB ℏc5
G·c3
ℏG(33)
=kB c8
G2(34)
=c4
G.(35)
This yields he undamen al Planck o ce:
FPl =c4
G≈1.21 ×1044 N.(36)
3.6 Cosmological Scale En opic Fo ce
Using Ve linde’s assump ions we ob ain F=TH·dS
dx =MHHc.
9
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. M
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.(62)
De ine cha ac e is ic imescales:
τdi =R2
D h
,(63)
τg a = R3
GM ,(64)
τexp =1
H.(65)
Then he Pécle numbe s
Pecosmo =τdi
τexp ≫1,(66)
Peg a =τdi
τg a ≫1(67)
indica e sus ained nonequilib ium s uc u es and enhanced s uc u e o ma ion.
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
16
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 . M
e lec ing he mic ophysical p ocesses inducing non-equilib ium en opy change
∂s
∂ +∇·Js=σs,(68)
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.
17
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 . M
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 ,(69)
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
18
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. M
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
(70)
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
19
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,(71)
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.(72)
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),(73)
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),(74)
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).
20
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.(75)
The cen al- o-edge densi y con as is
D=ρc
ρ(R)=eψ(η1),(76)
wi h ψ(η1)≈6.563 a he u ning poin , so
D=e6.563 ≈709.(77)
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 ,
21
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.(78)
The p imo dial densi y pe u ba ion ampli ude is app oxima ely:
δ≡pAs∼4.6×10−5.(79)
22
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.(80)
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).(81)
Gi en he edshi e olu ion o ene gy densi ies:
ρm(z) = ρm,0(1 + z)3,(82)
ρ (z) = ρ ,0(1 + z)4,(83)
he equali y condi ion yields:
1 + zeq =ρm,0
ρ ,0
=Ωm,0
Ω ,0
,(84)
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,(85)
Ω ,0= Ωγ,0+ Ων,0≈9.2×10−5,(86)
We ob ain:
zeq =0.315
9.2×10−5−1≈3424 ≈3400,(87)
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
,(88)
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.(89)
23
Subs i u ing Eq. (88) in o Eq. (89):
Dini ×1 + zeq
1 + z o m γ
=Dc i .(90)
Sol ing o z o m:
1 + zeq
1 + z o m γ
=Dc i
Dini
,(91)
1 + z o m = (1 + zeq)×Dini
Dc i 1/γ
.(92)
Subs i u ing nume ical alues wi h γ= 1:
1 + z o m = 3400 ×10−5
709 −1
= 3400 ×7.09 ×107
= 2.41 ×1011.(93)
The e o e:
z o m ≈2.41 ×1011.(94)
9.4 Physical In e p e a ion
The ex emely high edshi z o m ≈2.41 ×1011 signi ican ly exceeds he obse able
uni e se’s o ma ion epoch (z∼103). This esul indica es ha p imo dial densi y
luc ua ions cha ac e ized by Dini = 10−5a e insu icien o igge g a o he mal
ca as ophe (D > 709) h ough linea g ow h alone. In eali y, s uc u e o ma ion
p oceeds ia nonlinea g a i a ional ampli ica ion mechanisms, including:
•G a i a ional ins abili y and Jeans collapse,
•Da k ma e clus e ing and halo o ma ion,
•Ba yon-da k ma e eedback p ocesses.
These p ocesses enable densi y pe u ba ions o g ow nonlinea ly, eaching D∼709
a physically ealis ic edshi s (z∼10–100), he eby ini ia ing he g a o he mal
ins abili y and subsequen co e-halo s uc u e o ma ion obse ed in cosmological
simula ions.
10 Resul s
10.1 Summa y o Pa ame e s
Table 2summa izes he nume ical alues and obse a ional basis o he s uc u e o -
ma ion edshi calcula ion. This example demons a es he quan i a i e applica ion
24
Table 2 Pa ame e s o s uc u e o ma ion edshi calcula ion.
Pa ame e Value Obse a ional/Theo e ical Basis
Dini 10−5Planck 2018 CMB: As∼2.1×10−9[128]
zeq 3400 Ma e - adia ion equali y: Ωm,0/Ω ,0≈3424 [128]
Dc i 709 Lane-Emden equa ion solu ion: exp(ψ1)≈709 [102]
γ1Linea g ow h (Eins ein-de Si e app oxima ion)
z o m 2.41 ×1011 Calcula ed om D(z o m)=Dc i
o he g a o he mal ins abili y amewo k o cosmological s uc u e o ma ion, illus-
a ing he ansi ion om linea o nonlinea g ow h egimes. In a sel -g a i a ing
sys em, i hea ini ially lows om he ex e io o he in e io , he cen al egion
expands, and he ou e densi y inc eases. This educes he densi y con as , allowing
o dina y he modynamics o apply o he hypo he ical sphe e. The en opy s abilizes
a a maximum, esul ing in an iso he mal, uni o m-densi y he mal equilib ium s a e
Smaxi −Smaxj =Smaxk (95)
This di e ence, Smaxk, inc eases acco ding o he law o en opy inc ease. We adop s
a classical app oach, modeling he uni e se by conside ing a su icien ly la ge egion
ha expands wi h cosmic expansion. Acco ding o he cosmological p inciple, his
egion is assumed o be homogeneous and iso opic, so he ne in low and ou low o
hea o en opy in o his egion is ze o (equi alen o a sys em enclosed by adiaba ic
walls). O he wise, his egion would be a special egion, iola ing homogenei y and
iso opy, and he cosmological p inciple would no hold. E e y poin in he uni e se
can be conside ed a cen e , o al e na i ely, he uni e se can be hough o as ha ing no
cen e . The e o e, a su icien ly la ge subsys em wi hin he uni e se can be ea ed as a
closed, adiaba ic sys em. This simpli ied/modelled concep o ex ac ing a su icien ly
la ge egion om he uni e se is called he non- ela i is ic cosmic expansion model.
Fo he nume ical analysis and conside a ions in We, solu ions can be adequa ely
ob ained wi hou in oking gene al ela i i y. Fo ρ0< ρc in ini e expansion occu s.
Fo ρ0=ρc in ini e expansion occu s. Fo ρ0> ρc con ac ion occu s. De ining he
adius R(R≈a( he scale ac o )) and he mass densi y, he mass o his egion is
M=4π
3R3ρ(96)
Conside ing a pa icle o mass mplaced a a poin on R
md2R
d 2=−GMm
R2(97)
This simpli ies o
d2R
d 2=−GM
R2(98)
25
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
(155)
1 + Z= (0.025)−2∼1600, Z = 1600 −1 = 1599 (156)
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 (157)
ρma3=cons (158)
ρ =ρm,ρ
ρm∝(1 + Z)4
(1 + Z)3∼(1 + Z)
∼Z= 3400 ∼6380,(Ω ,0= 4.7∼8.4×10−5)
(159)
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 (160)
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
N
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
32
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(161)
The blackbody adia ion ene gy densi y
ρ=aT4=π2k4
BT4
15ℏ3c3(162)
The e o e, om he o mula o he c i ical densi y o he uni e se 109 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(163)
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(164)
RBHsIn e io T he modynamicsRadialen opydensi y :s ( ) = (4/3)aSBNT ( )3Radial empe a u e :T ( )Scale −Dependen Fo 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)
(165)
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],(166)
Scale-dependen empe a u e:
Ts(l) = TUexp −l2
l2
c+TH1−exp −l2
l2
c [K],(167)
33
Dimensional analysis:
[σ(l)] = J K−1m−3,(168)
[Ts(l)] = K,(169)
[l0, lc]=m.(170)
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)].(171)
This o mula p o ides a smoo h ansi ion be ween he wo egimes: in he limi
l→0(172)
(local scales),
Ts→TU,(173)
while o
l→ ∞ (174)
(cosmological scales),
Ts→TH.(175)
He e,
lc(176)
is a c i ical scale pa ame e , which can be associa ed wi h he Planck leng h
lc∼lpl (177)
o ela ed o he Hubble adius o mac oscopic ansi ions. This scale-dependen
e ec i e empe a u e
Ts(178)
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 (179)
34
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( ).(180)
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,(181)
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(182)
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
35
•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. (183)
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, (184)
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, (185)
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,(186)
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.
36
Nume ical Ve i ica ion.
Subs i u ing he obse able uni e se mass MHin o Eq. (185), The cosmological
en opic o ce:
FH=MHH0c=c4
G≈1.210 ×1044 N.(187)
This alue is iden ical o he Planck o ce, de ined as
FPlanck =c4
G≈1.210256 ×1044 N,(188)
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,(189)
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,(190)
[FPlanck]=[c4]/[G]=(ms−1)4/(m3kg−1s−2) = kg ·m·s−2=N.(191)
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.
37
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
38
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 (192)
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
39
[20] and [79]
SBH =AkB
4L2
pl
=4πR2
SkB
4ℏGc−3=πkBc3R2
S
ℏG=4πkBGM2
BH
ℏc(193)
Bekens ein-Hawking En opy
The Bekens ein-Hawking en opy SBH o a black hole, when di ided by he Bol zmann
cons an kB, is in e p e ed as he en opy quan um numbe . Speci ically, he ollowing
ela ion holds SBH
kB
=4πGM2
ℏc(194)
He e, Gis he g a i a ional cons an , Mis he mass o he black hole, ℏis he
educed Planck cons an , and cis he speed o ligh . To con i m ha his quan i y
is dimensionless, I pe o m a dimensional analysis. The dimensions o he nume a o
and denomina o a e calcula ed as ollows
GM2=M−1L3T−2··M2=ML3T−2
[ℏc]= (ML2T−1)·(LT −1) = ML3T−2
Thus, he o e all dimension is
GM2
[ℏc]=ML3T−2
ML3T−2= 1
This esul con i ms ha SBH
kBis a dimensionless quan i y, in e p e ed as he en opy-
quan um numbe . O cou se, quan um mechanics is also e lec ed, as i inco po a es
he Planck cons an . I u he ex end he scope o calcula e he o al en opy S
based on nume ical analysis o he e olu ion equa ions o expansion du ing adia ion-
domina ed and ma e -domina ed e as, as ollows
S o al =Sm+S =AkB
4L2
pl
+4aT 3
3V =4πR2
SkB
4ℏGc−3+4aT 3
3V
=πkBc3R2
S
ℏG+4aT3
3V =4πkBGM2
m
ℏc+4aT3
3·4π 3
3(195)
The esul s o he nume ical analysis a e plo ed as a g aph, showing he en opy S
wi hin a egion as a unc ion o Z. The en opy Sinc eases sha ply om he Planck
scale, ollowing a powe -law inc ease on a double loga i hmic g aph. Thus, s anda d
he modynamics can be applied
dS o al =dSBH +dS =1
Ta−1
TbdQ (196)
40
indica ing ha he en opy Sinc eases. Since he expansion eloci y is less han c,
implying adiaba ic expansion, The alue
dQ(TdS) = dU +PdV = 0, dU =−P dV, dSBH =dQ
TBH
The ollowing Planck scale alues we e used
Planck ime pl = ℏG
c5= 5.391 ×10−44 s (197)
Planck leng h lpl = ℏG
c3= 1.616 ×10−35 m (198)
Planck mass mpl = ℏc
G= 2.176 ×10−8kg (199)
Planck empe a u e Tpl =mplc2
kB
= 1.417 ×1032 K (200)
Addi ionally, om he Hubble cons an H
1
H≥ℏ
2mHc2=ℏ
2c2(201)
1
H≥1
2mHc2=1
4mplc2(202)
Taking he ecip ocal
2mH= 4mpl (203)
mH= 2mpl (204)
I is pa icula ly in e es ing ha he Planck mass, he quan um mechanical mini-
mum uni , can be de i ed om he mac oscopic uni e se. Fu he mo e, he en opy
calcula ed om equa ion 195 is no malized by di iding by kB, and he esul ing
dimensionless g aph is shown below
S o al
kB
=Sm+S
kB
=
AkB
4L2
pl
+4aT 3
3V
kB
=
4πR2
SkB
4ℏGc−3+4aT 3
3V
kB
=
πkBc3R2
S
ℏG+4aT 3
3V
kB
=
4πkBGM2
m
ℏc+4aT 3
3·4π 3
3
kB
(205)
The adii o he pa icle ho izons, s a ing om Zco esponding o he Planck scale,
a e in eg a ed o he adia ion-domina ed, ma e -domina ed, and now accele a ed
expansion s ages, analyzed nume ically using he F iedmann model, and plo ed o he
Planck scale MplLplTpl. The nume ical esul s om equa ion 197 o 200 a e p esen ed
in he Appendix. A Z= 1.417 ×1032,S/kB≈2.754, sugges ing ha a Z=∞,
S/kB= 0. A Z= 0, he nume ical analysis yields S/kB≈2.756 ×10123. The
dimensionless en opy S/kBas a unc ion o Z, calcula ed using equa ion 205, is
41
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,(244)
p=1
3u=1
3a T4,(245)
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. (246)
Wi h dV = 0, one inds
dS =dE
T.(247)
17.3 Ene gy–Tempe a u e Rela ion
F om Eq. (244), he o al ene gy is
E =u V =a T4V. (248)
Sol ing o Tgi es
T=E
a V 1/4
.(249)
48
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
,(250)
hus es ablishing he scaling
S ∝E3/4
.(251)
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
.(252)
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.
49
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,(253)
50
¨
a
a=−4πG
3ρ+3p
c2+Λc2
3,(254)
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. 125), 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 N. 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,(255)
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. 149), 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. 256).
We ex end he en opy con inui y equa ion o include he Λ-d i en expansion:
∂s
∂ +∇·Js=σs+σΛ,(256)
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, (257)
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. 257) 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. 257). The modi ied equa ion o mo ion
51
o a es pa icle on he pa icle ho izon is
d2R
d 2=−4πG
3ρR +Λc2
3R. (258)
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. 195, 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 .
M
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
52
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.
M
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,
53
S o al ≈S wi h adia ion en opy
S ∝T3
(259)
and adia ion ene gy
E ∝T4
(260)
so
S ∝E3/4
(261)
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 (262)
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(263)
y∝Sm
E2
o al ≈Sm
E2
m≈Am(264)
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
(265)
Sm∝E2
m⇒˜
ym∝E2
m
E2
o al
(266)
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(267)
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
54
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.
N
Fig. 15 Absolu e alue o speci ic hea CV=−8πkBGM2
ℏcas a unc ion o Z.
N
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(268)
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:
55
Dimensional Consis ency.
By no malizing o he Planck ene gy scale, all en opy measu es become dimen-
sionless, enabling consis en ea men ac oss app oxima ely 80 o de s o magni ude
in ene gy—spanning om elemen a y pa icle physics (Ep o on ∼10−10 J) h ough
Planck-scale p ocesses (EPlanck ∼109J) o he o al ene gy con en o he obse able
uni e se (Euni e se =MHc2∼1070 J).
Holog aphic Connec ion.
The Planck-a ea no maliza ion connec s na u ally o he holog aphic bound S≤
A/(4L2
Planck), sugges ing ha ˜
y ep esen s a uni e sal measu e o holog aphic
e iciency ac oss all g a i a ional sys ems.
18.4 Ene gy Scale Hie a chy and Dimensional Consis ency
The Planck-no malized en opy amewo k ope a es ac oss an unp eceden ed ene gy
hie a chy, encompassing h ee dis inc physical egimes:
Pa icle Physics Scale.
The lowe bound is se by elemen a y pa icle es masses, exempli ied by he p o on
ene gy Ep o on =mpc2≈1.5×10−10 J. This scale ep esen s he h eshold o had onic
ma e and he s anda d model pa icle spec um.
Planck Scale.
The in e media e scale is de ined by he Planck ene gy EPlanck =pℏc5/G ≈1.96×109
J, ma king he quan um g a i y h eshold whe e space ime i sel becomes subjec o
quan um luc ua ions.
Cosmological Scale.
The uppe bound co esponds o he o al ene gy con en o he obse able uni e se,
Euni e se =MHc2≈1.66×1070 J, whe e MH=c3/(GH0)is he Hubble mass enclosing
he obse able cosmos.
Jus i ica ion o “80 O de s o Magni ude”.
The a io Euni e se/Ep o on ≈1080 de ines he p ac ical ene gy spec um accessible
o physical heo y and nume ical simula ion. This cha ac e iza ion b idges pa icle
physics, quan um g a i y, and cosmology wi hin a uni ied he modynamic amewo k,
ensu ing nume ical s abili y ac oss as ly dispa a e scales and p e en ing compu a-
ional o e low o unde low in simula ions ea ing black holes, adia ion, ma e , and
cosmological ho izons simul aneously.
Dis inc ion om Spa ial Scale F amewo k.
This ene gy-based hie a chy (80 o de s) di e s om he spa ial scale ange employed
in empe a u e in e pola ion o mulas, which spans om Planck leng h (Lpl ∼10−35
m) o Hubble adius (RH∼1026 m), co esponding o 61 o de s o magni ude. Bo h
56
pe spec i es a e complemen a y: he 80-o de ange ensu es uni e sali y in en opy
accoun ing ac oss all physical sys ems, while he 61-o de spa ial hie a chy go e ns
scale-dependen dynamical mechanisms such as he Un uh- o-Hubble o ce ansi ion
discussed in he en opic o ce amewo k.
19 Rela i e En opy Densi y
In e nal deg ees o eedom Na e assumed la ge (N≫100) [88]. Cu a u e scales as
In e nal deg ees o eedom
N
a e assumed la ge
(N≫100) (269)
Cu a u e scales as
RµνRµν ∼100
Nl2
p
.(270)
Ene gy adia ion densi y o
N
massless scala ields:
ε ad =Nπ2k4
BT4
30ℏ3c3,(271)
o e mions:
ε ad =N7π2k4
BT4
240ℏ3c3.(272)
Radia ion en opy densi y is
s ad( ) = 4
3
ε ad( )
T( )=4
3aSBNT( )3,(273)
wi h
aSB = 4σ/c = 7.5657 ×10−16 J·m−3·K−4
. In his sec ion, We analyze he ela ion be ween he adia i e en opy densi y s ad and
o he he modynamic quan i ies such as empe a u e T, p essu e P ad, and numbe
o in e nal deg ees o eedom N, unde he assump ion o local he mal equilib ium
inside a egula black hole (RBHs). We adop he S e an–Bol zmann o m o he
adia ion ene gy and en opy densi y, gene alized o accoun o Nscala deg ees o
eedom in he in e io
s ad( ) = 4
3·ϵ ad( )
T( )=4
3·aSB N T( )4
T( )=4
3aSB N T( )3,(274)
whe e aSB is he adia ion cons an in SI uni s gi en by
aSB =4σ
c=4π2k4
B
15c3ℏ3≈7.5657 ×10−16 J m−3K−4.(275)
57
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.(300)
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, (301)
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.(302)
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.(303)
The a io con i ms exac ag eemen o machine epsilon (∼10−15):
FH
FPlanck
=GMHH0
c3= 1.000.(304)
On cosmological scales, he en opic o ce is
F=TH·dS
dRH
=c4
G,(305)
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
64
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)],(306)
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( ).(307)
Bekens ein-Hawking en opy o mass Mwi hin he Hubble adius is:
SBH =4πkBGM2
ℏc.(308)
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,(309)
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
65
numbe s quan i y non-equilib ium dominance:
Pecosmo =τdi
τexp
=R2H
D h ≫1,(310)
Peg a =τdi
τg a
=R2
D h GM
R3≫1,(311)
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,(312)
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,(313)
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,
66
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( ),(314)
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).(315)
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,
67
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),(316)
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.
68
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.
69
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.
70
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.
71
I am deeply g a e ul o he many pionee ing esea che s whose p o ound insigh s
in o g a i a ional he modynamics, black hole physics, and cosmology ha e been a
g ea sou ce o inspi a ion. Thei con ibu ions no only o m he ounda ion o his
wo k bu also con inue o guide hose who seek o unde s and he deepe na u e o
ou uni e se. Humani y will ne e cease his endea o .
Abo e all, I exp ess my p o ound espec o Albe Eins ein. His gene al he-
o y o ela i i y emains he co ne s one o all mode n g a i a ional physics. This
well-es ablished and obus heo y is ne e con adic ed by his wo k. Ra he , I ha e
ound ha he esul s ob ained h ough en opic and g a i a ional he modynamic
app oaches a e consis en wi h he es ablished esul s by Eins ein.
Finally, I would like o exp ess my deepes g a i ude o Eme i us P o esso Dai-
ichi o Sugimo o, who augh me he essence o physics and guided me in o scien i ic
inqui y. P o esso Sugimo o augh me he u ili y and essence o en opy, g a i a-
ional he modynamics, and dimensional analysis. He ca e ully augh me o iew
phenomena om a comp ehensi e and simple pe spec i e h ough hese app oaches,
he eby e ealing he essence o he uni e se. P o esso Sugimo o’s men o ship con-
inues o be he d i ing o ce behind my in ellec ual cu iosi y o unde s and he
essence o he uni e se h ough he concep s o en opy, g a i a ional he modynam-
ics, and dimensional analysis.
Decla a ions
•Funding : No applicable
•Con lic o in e es : No applicable
•E hics app o al and consen o pa icipa e : Applicable
•Consen o publica ion : Applicable
•Da a a ailabili y : The da a ha suppo he indings o his a icle a e openly
a ailable below.
•Ma e ials a ailabili y : No applicable
•Code a ailabili y : Applicable
•Au ho con ibu ion : The au ho concei ed and designed he s udy, collec ed and
analyzed he da a, and w o e he manusc ip .
In o de o demons a e he heo e ical consis ency, igo , and obus ness o ou
amewo k and o ensu e ull anspa ency o he esea ch, and in acco dance wi h
he p inciples o open schola ly con ibu ion and academic e hics, we ha e decided o
make i publicly a ailable.
[Zenodo, Powe ed by CERN Da a Cen e and In enioRDM]
P ep in a ailable a Zenodo.
(P ep in DOI: 10.5281/zenodo.16143976)
72
Owing o i s ex ensi e leng h, he ollowing appendix has been
deposi ed in he a o emen ioned Zenodo eposi o y.
Fu he mo e, ex ended passages may be condensed and adjus ed as
equi ed.
Appendix A Da a Sou ces and Me hodological
F amewo k
The analy ical calcula ions p esen ed in his pape employ he Hubble cons an alue
om [66]. Fo he nume ical simula ions, we adop cosmological pa ame e s consis-
en wi h Planck 2018 da a [128] and undamen al physical cons an s om CODATA
2018 [48].
Appendix B S ∝E3/4
) and ma e (Sm∝E2
m)
De i a ion o en opy scaling
In his appendix, we p esen he de ailed de i a ion o he equa ions (Eq. ??) discussed
in Sec ion ??.
Appendix C En opy as a Func ion o Ene gy
Appendix D A Simple S a is ical De i a ion o he
Dimensionless In e pola ion Quan i y
y=S/E2
o al om he Law o La ge
Numbe s
We p esen a concise, h ee–s ep s a is ical de i a ion o he dimensionless a io
y=S
E2
o al
,
whe e Sdeno es he o al en opy and E o al he o al ene gy o a sys em o Niden ical
pa icles. U ilizing only he law o la ge numbe s and addi i i y o mic oscopic con i-
bu ions, We demons a e ha yscales in e sely wi h pa icle numbe , y∝1/N. This
app oach a oids a ia ional p inciples and u nishes immedia e in ui ion o ini e–size
e sus he modynamic–limi beha io .
D.1 De ailed Explana ion
In s a is ical mechanics, one o en encoun e s dimensionless measu es ha cap u e he
compe i ion be ween ene gy and en opy con ibu ions. A pa icula ly use ul quan i y
73
Appendix G 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
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. All app oaches a e g ounded in
he scale-dependen e ec i e empe a u e Ts(l) ha seamlessly in e pola es be ween
local Un uh e ec s and global Hubble in luences wi hou ul a iole cu o s.
G.1 Holog aphic Ene gy Densi y Fluc ua ions (S- ie )
The holog aphic sc een en opy associa ed wi h he Hubble ho izon is
Ssc een =πkBc5
ℏGH2=kBAH
4L2
pl
,(G28)
whe e AH= 4πc2/H2and Lpl =pℏG/c3. The numbe o deg ees o eedom is
N=πc5
ℏGH2≈2.26 ×10122 (H0= 2.1850 ×10−18 s−1).(G29)
In a ini e-N sys em, canonical ensemble luc ua ions (modula ed by Ts(l)) gi e
⟨δρ2⟩=ρ2
Λ
Nexp −l2
l2
c, lc≃0.1RH.(G30)
Fo w=−1,δP =−c2δρ, so
σholo =ρΛc2
√Nexp −l2
2l2
c≈5.10 ×10−71 Pa (G31)
(a cosmological scales l≳lc, exponen ial →1).
G.2 Gibbons–Hawking The modynamics (A- ie )
The Gibbons–Hawking empe a u e TGH =ℏH/(2πkB)yields he modynamic
p essu e
PGH =TGH ∂S
∂V E
=H2c2
4πG =2
3ρΛc2≈5.11 ×10−10 Pa.(G32)
Tempe a u e luc ua ions δTGH ∼TGH/√Np opaga e o p essu e luc ua ions ha
exac ly ep oduce Eq. (G31).
80
G.3 Quan um Field Theo y Mode Sum wi h Cen al Limi
Theo em (A- ie )
The mode-sum a iance in de Si e space, wi h scale-dependen egula iza ion kmax =
H/[1 −exp(−l2/l2
c)], is
σ2
QFT =4πℏcg∗H7
7
exp(−l2/l2
c)
[1 −exp(−l2/l2
c)]7.(G33)
A s ic ly cosmological scales (l≫lc) he exponen ial supp ession makes he mic o-
scopic QFT con ibu ion O(10−75)Pa o smalle — consis en wi h he hie a chy
discussed below. Gaussiani y is gua an eed by he cen al limi heo em applied o
Ne ∼g∗×1090 ≫1independen modes.
G.4 Casimi E ec a Cosmological Scales (B- ie )
Replacing pla e sepa a ion a→RHyields
Pcosmo
Casimi =−π2ℏH4
720c3≈ −1.22 ×10−132 Pa.(G34)
Nume ically negligible bu concep ually essen ial as a pu e bounda y con ibu ion.
G.5 E ec i e Theo e ical Pa ame iza ion and Ampli ica ion
Mechanism
Mic oscopic es ima es (σholo ∼10−71 Pa, σQFT ≲10−75 Pa) a e no he luc ua ions
di ec ly el by mac oscopic cosmic s uc u es. The obse able e ec i e luc ua ion
ampli ude used in phenomenological models and N-body simula ions is
σe =Ae ρΛc2,Ae ≈2.4×10−30,(G35)
yielding σe ≈2×10−39 Pa.
The dimensionless ampli ica ion ac o
A=σe
σmic o ≈ Ae √N∼1031–1036 (G36)
a ises om collec i e he maliza ion and cohe en exci a ion o he ∼10122 holog aphic
deg ees o eedom. Physically, his is he cosmological analogue o B ownian mo ion:
mic oscopic acuum kicks a e ampli ied in o obse able long-wa eleng h luc ua ions
ia he eno mous numbe o coope a ing quan um-g a i a ional deg ees o eedom
on he ho izon (Ve linde- ype en opic dynamics, 2025 collec i e mode analyses).
The coe icien Ae admi s he anspa en in e p e a ion
Ae ≈kBTGH
ρΛc2R3
H
(G37)
81
as he a io o he mal ene gy a he de Si e empe a u e o he cha ac e is ic
acuum ene gy in a Hubble olume (up o O(1) geome ic ac o s).
Me hod Mic oscopic σ(Pa) Ampli ica ion o de
Holog aphic (S- ie ) 5.10 ×10−71 ∼1032
Gibbons–Hawking (A- ie ) 5.10 ×10−71 ∼1032
QFT mode sum (A- ie ) ≲10−75 ∼1036
Casimi (B- ie ) 10−132 —
E ec i e phenomenological 2×10−39 1
Table G1 Hie a chy o acuum p essu e luc ua ions and equi ed
ampli ica ion.
G.6 Summa y o Quan um Field Theo e ic Founda ions
The ou app oaches a e mu ually consis en a he mic oscopic le el (wi hin he na u-
al sp ead in oduced by di e en egula iza ion philosophies) and join ly explain he
obse ed mac oscopic da k-ene gy- ela ed luc ua ions ia well-mo i a ed holog aphic
he maliza ion ampli ica ion o o de 1031–1036.
Appendix H Da k Ene gy: The modynamic O igin
in he En opic Fo ce F amewo k
Da k ene gy eme ges as an en opic o ce
Fen opic =Ts(l)dS
dx (H38)
d i en by en opy g adien s on he holog aphic sc een, wi h
Ts(l) = TUexp(−l2/l2
c) + TH[1 −exp(−l2/l2
c)].(H39)
The e ec i e acuum p essu e balance is
P ac =−ρΛc2+Pe
quan um,(H40)
whe e Pe
quan um is he ampli ied quan um p essu e discussed abo e.
The amewo k is pa ame e - ee, ep oduces Planck 2018 cosmology exac ly,
and in e p e s gene al ela i i y as he hyd odynamic limi o mic oscopic quan-
um en opy g adien s. N-body simula ions inco po a ing hese en opic o ces
con i m ene gy conse a ion (<0.1% d i ), mono onic en opy g ow h, and co ec
scale-dependen beha iou ac oss 61 o de s o magni ude.
82
Da k ene gy is he e o e a dynamic he modynamic p ocess
˙
Eda k =Ts(l)dS
d ,(H41)
uni ying quan um acuum physics, holog aphy, and cosmology h ough he uni e sal
o ganising p inciple o en opy.
Appendix I Heu is ic Mo i a ion o he C osso e
Scale
I.1 Physical O igin o he C osso e Scale lc: Heu is ic
Mo i a ion om Holog aphic Physics
The c osso e scale lc≈0.1RHis a phenomenological pa ame e whose alue is
cons ained by he modynamic consis ency, obse a ional da a,
I.1.1 E ec i e Holog aphic Mass
De ine he e ec i e holog aphic mass as
me ≡ρH
ρPl 1/3
mPl =ρ1/3
Hℓ2
Pl,(I42)
whe e ρPl =c5/(ℏG2)≈5.16 ×1096 kg/m3is he Planck densi y. This mass scale
ep esen s he cha ac e is ic mass associa ed wi h a holog aphic cell a he Hubble
densi y, embodying he collec i e beha io o Ndo ∼(RH/ℓPl)2∼10122 deg ees o
eedom.
I.2 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(I43)
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.
83
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(I44)
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.
3. QFT Mode Summa ion wi h Cen al Limi Theo em (A- ie ): Summing
quan um ield modes up o he Hubble cu o wi h p ope no maliza ion yields:
σQFT = 4πℏcH7
7(I45)
Gaussiani y is igo ously jus i ied by he cen al limi heo em applied o Nmodes ∼
1090 independen quan um ield con ibu ions, p o iding mic oscopic s a is ical
jus i ica ion.
4. Casimi E ec a Cosmological Scales (B- ie ): The Casimi p essu e o a
ca i y o size equal o he Hubble adius is:
PCasimi =−π2ℏH4
720c3≈ −10−132 Pa (I46)
Though nume ically negligible, his quan um acuum bounda y e ec is concep-
ually impo an and p o ides consis ency wi h he comple e quan um acuum
ene gy budge o he ini e obse able uni e se.
I.2.1 Consis ency and Robus ness
All ou independen mic oscopic es ima es a e mu ually consis en wi hin ac o s o
o de uni y, wi h ela i e de ia ions spanning app oxima ely 1030–36 in he ampli i-
ca ion ac o . This ema kable ag eemen con i ms he heo e ical obus ness o he
quan um acuum luc ua ion amewo k ac oss all ene gy scales om Planck leng h
o Hubble adius.
I.2.2 P essu e Scale Uni ica ion ia The modynamic Analysis
The mic oscopic es ima es om holog aphic luc ua ions (σholo), QFT mode sums
(σQFT), and Gibbons-Hawking he modynamics yield p essu e a iances ha di e
by many o de s o magni ude om he e ec i e phenomenological scale σe used in
simula ions and obse a ions. Table I2 compa es hese es ima es.
In e p e a ion as e ec i e heo y:
The phenomenological pa ame iza ion is de ined as:
σe =Ae ρΛc2(I47)
84
Me hod P essu e Va iance Ra io o σe
Holog aphic (Eq. I43)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. I45)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. I44)5.10 ×10−71 Pa 2.50 ×10−32
Phenomenological 2.04 ×10−39 Pa 1.00
Table I2 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 ac o s o o de uni y, bu smalle han he
phenomenological pa ame iza ion by 1030–1036 o de s o magni ude.
This hie a chy indica es a undamen al e ec i e heo y pic u e.
whe e Ae ≈2.4×10−30 is a dimensionless phenomenological ampli ica ion
coe icien . This ep esen s a coa se-g ained desc ip ion alid a mac oscopic scales.
The physical o igin o his coe icien can be unde s ood as an ene gy a io:
Ae =kBTGH
E e
(I48)
whe e E e =ρΛc2R3
His he cha ac e is ic acuum ene gy wi hin he Hubble olume,
ensu ing dimensional consis ency.
The o al ampli ica ion ac o om he mic oscopic holog aphic scale o he
e ec i e mac oscopic scale is:
A=σe
σholo
=Ae √N∼1030–36 (I49)
This dimensionless ac o ep esen s he ampli ica ion o mic oscopic quan um
luc ua ions o mac oscopic obse ables h ough he maliza ion o e he N∼
10122 holog aphic deg ees o eedom. This mechanism is analogous o how B ownian
mo ion ampli ies molecula -scale luc ua ions o obse able pa icle displacemen s, bu
ope a ing a cosmological scales.
Appendix J Da k Ene gy: The modynamic O igin
in he En opic Fo ce F amewo k
The p esen wo k ein e p e s da k ene gy om a he modynamic pe spec i e, iew-
ing i as eme ging undamen ally om en opy g adien s and quan um acuum
luc ua ions a he han as a ising solely om a s a ic cosmological cons an Λ.
J.1 De i a ion om En opy G adien and Holog aphic
P inciples
Da k ene gy is exp essed as an en opic o ce a ising om he en opy dis ibu ion on
he holog aphic sc een:
Fen opic =Ts(l)dS
dx (J50)
85
whe e Ts(l) = TUexp(−l2/l2
c)+TH[1−exp(−l2/l2
c)] is he scale-dependen empe a u e
and dS
dx is he en opy g adien on he holog aphic sc een. This amewo k ex ends
Ve linde’s en opic g a i y heo y, posi ioning da k ene gy as a ising undamen ally
om en opy imbalance a di e en scales a he han as an in insic da k luid. The
en opic o ce d i es he uni e se’s accele a ed expansion h ough non-equilib ium
he modynamic p ocesses encoded in holog aphic deg ees o eedom.
J.2 Vacuum Ene gy and E ec i e Theo e ical P essu e
Balance
In his e ec i e heo e ical amewo k, acuum p essu e is d i en by en opy g adien s:
P ac =−ρΛc2+Pquan um (J51)
whe e he quan um p essu e e m a ises om scale-dependen empe a u e luc ua-
ions. This acuum ene gy de i es om h ee undamen al sou ces:
•Scale-Dependen Tempe a u e T ansi ion: The e olu ion om Un uh em-
pe a u e (TU∼3.97 ×10−20 K a local Planck scales) o Hubble empe a u e
(TH∼2.65 ×10−30 K a cosmological scales), cap u ed by he scale-dependen
o mula ion Ts(l).
•En opy Densi y and Deg ees o F eedom: En opy densi y scaling s( )∝
NT ( )3, whe e N∼10122 is he e ec i e holog aphic deg ees o eedom and T( )
is he local scale-dependen empe a u e.
•Pa ame e -F ee Desc ip ion: Da k ene gy is explained en i ely h ough he
e ec i e heo e ical amewo k wi hou pa ame e uning, aligning p ecisely wi h
Planck 2018 obse a ions (ΩΛ= 0.684,H0= 67.36 ±0.54 km/s/Mpc).
J.3 Nume ical Simula ion Ve i ica ion o En opic Dynamics
In he N-body simula ion code (using Ba nes-Hu oc ee accele a ion), he mo-
dynamic o cing e ms based on en opy g adien s a e inco po a ed in o pa icle
in e ac ions o simula e en opic o ce dynamics. The simula ions con i m:
•Ene gy Conse a ion: Nume ical simula ions e i y ene gy conse a ion wi h
d i less han 0.1% o e 10,000 ime s eps, con i ming he consis ency and s abili y
o he en opic o ce implemen a ion.
•En opy G ow h and Second Law: Mono onic inc ease in sys em en opy is
demons a ed, con i ming ha he dynamics a e undamen ally consis en wi h he
second law o he modynamics.
•Scale-Dependen Ampli ica ion: The scale-dependen empe a u e o mula ion
success ully ep oduces bo h local quan um e ec s (Un uh empe a u e a Planck
scales) and cosmological dynamics (Hubble empe a u e a ho izon scales), spanning
61 o de s o magni ude in spa ial scale.
86
J.4 Da k Ene gy as Dynamic The modynamic P ocess
Ra he han a s a ic cosmological cons an , da k ene gy eme ges as a dynamic en opic
p ocess:
˙
Eda k =Ts(l)dS
d (J52)
This dynamic in e p e a ion based on en opy e olu ion econciles h ee key aspec s
o con empo a y cosmology:
1. Consis ency wi h Gene al Rela i i y: Gene al ela i i y is no nega ed bu
ein e p e ed as he mac oscopic he modynamic mani es a ion o mic oscopic
quan um en opy g adien s on he holog aphic sc een. Eins ein’s ield equa ions
eme ge as he hyd odynamic limi o he e ec i e heo e ical amewo k.
2. Pa ame e Economy: All cha ac e is ic ene gy and leng h scales de i e om
undamen al physics cons an s (Planck leng h Lpl, s anda d model deg ees o ee-
dom g∗= 106.75, holog aphic en opy bounds) wi hou in oducing addi ional ee
pa ame e s o da k ene gy.
3. Obse a ional P edic ions: Fu u e high-p ecision es s di ec ly p obe he
en opic o igin o da k ene gy:
•Redshi d i measu emen s (∆˙
z≈4.0×10−11 y −1) using nex -gene a ion
op ical la ice clocks.
•G a i a ional wa e obse a ions wi h LISA/DECIGO de ec ing ingdown de i-
a ions a ∼10−22 le el.
•P ecision cosmological cons ain s om DESI 2024-2025 and Planck legacy da a.
J.4.1 En opy as Fundamen al O ganizing P inciple
The hypo hesis ha en opy cons i u es he undamen al "sou ce" o cosmic dynam-
ics, wi h gene al ela i i y eme ging as i s mac oscopic he modynamic mani es a ion,
ep esen s a concep ual pa adigm shi in heo e ical physics. By uni ying quan um
and cosmological egimes h ough holog aphic p inciples while main aining consis-
ency wi h Eins ein’s ield equa ions and Planck obse a ions wi hou addi ional ee
pa ame e s, his en opy-cen ic amewo k o e s a comp ehensi e unde s anding o
da k ene gy as undamen ally he modynamic in o igin, po en ially b idging quan um
g a i y and cosmology h ough he modynamic p inciples.
J.5 Summa y and Consis ency
This wo k es ablishes he quan um ield heo e ic ounda ions o acuum p essu e
luc ua ions h ough ou complemen a y and mu ually alida ing app oaches:
1. Holog aphic Fluc ua ions (S- ie ): The ini e holog aphic deg ees o eedom
N0≈2.26 ×10122 yield p essu e luc ua ions σholo =ρΛc2/√N0, p o iding he
mos di ec connec ion o en opy bounds.
2. Gibbons-Hawking The modynamics (A- ie ): Applying he i s law o he
de Si e ho izon yields he mal p essu e PGH = (2/3)ρΛc2and ep oduces he
holog aphic p essu e luc ua ions, con i ming he modynamic consis ency.
87
3. QFT Mode Summa ion (A- ie ): Summing quan um ield modes up o he
Hubble cu o yields σQFT =p4πℏcH7
0/7wi h e ec i e mode coun Ne ∼
106.75 ≫1, jus i ying Gaussiani y ia he cen al limi heo em.
4. Casimi E ec a Cosmological Scales (B- ie ): The Casimi p essu e a he
Hubble adius is PCasimi =−π2ℏH4/(720c3)≈ −10−132 Pa, negligibly small bu
con i ming quan um acuum consis ency ac oss all scales.
All ou app oaches demons a e **mu ual consis ency wi hin ac o s o o de
uni y**, alida ing he obus ness o he quan um acuum luc ua ion amewo k
ac oss:
- **61 o de s o magni ude in spa ial scale:** om Planck leng h (10−35 m) o
Hubble adius (1026 m) - **80 o de s o magni ude in ene gy scale:** om Planck
ene gy (109J) o cosmological scale (10120 J)
The e ec i e heo e ical pa ame iza ion σe =Ae ρΛc2b idges mic oscopic
Planck-scale quan um luc ua ions wi h mac oscopic cosmological obse a ions, p o-
iding a consis en and uni ied desc ip ion ac oss all physical scales wi hou ad hoc
assump ions o adjus able pa ame e s.
Appendix K Consis ency wi h Planck 2018 Da a
Pa ame e s a e aken om Planck 2018 [128], ensu ing alignmen wi h cosmological
obse a ions.
Hubble pa ame e : H0= 2.1850 ×10−18 s−1
Radia ion ac o : Ω ,0= 4.7∼8.4×10−5
Ma e ac o : Ωm,0= 0.315
Ba yon : Ωb= 0.049
Whe e, Ωm= Ωb+ ΩDM: da k ma e
Cosmological cons an : ΩΛ,0= 0.684
Cu a u e o he uni e se : Ωk,0= 0
Appendix L Consis ency wi h CODATA 2018
physical cons an s Da a
Pa ame e s a e aken om CODATA2018 [48], ensu ing alignmen wi h cosmological
obse a ions.
Speed o ligh in acuum : c= 299792458 m ·s−1
Planck cons an : h= 6.62607015 ×10−34 J·s
Reduced Planck cons an : ℏ= 1.0545718176461565 ×10−34 J·s
Elemen a y cha ge : e= 1.602176634 ×10−19 C
Elec on mass : me= 9.109383701528 ×10−31 kg
P o on mass : mp= 1.67262192369095 ×10−27 kg
88
Neu on mass : mn= 1.67492749804203 ×10−27 kg
A ogad o cons an : NA= 6.02214076 ×1023 mol−1
Bol zmann cons an : kB= 1.380649 ×10−23 J·K−1
Gas cons an : R= 8.31446261815324 J ·mol−1·K−1
Magne ic cons an ( acuum pe meabili y) : µ0= 1.25663706212 ×10−6N·A−2
Elec ic cons an ( acuum pe mi i i y) : ϵ0= 8.8541878128 ×10−12 F·m−1
Fine-s uc u e cons an : α=e2
4πϵ0ℏc≈7.2973525693 ×10−3
New onian cons an o g a i a ion : G= 6.67430 ×10−11 m3·kg−1·s−2
S anda d accele a ion o g a i y : g0= 9.80665 m ·s−2
S e an-Bol zmann cons an : σ= 5.670374419 ×10−8W·m−2·K−4
Planck empe a u e : Tpl = 1.416784 ×1032 K
Appendix M Nume ical Simula ion F amewo k and
Co espondence wi h Figu es
Below is he Py hon and C Language p og am used in his s udy. We he eby make i
publicly a ailable o demons a e he heo e ical consis ency, igo , and obus ness o
ou amewo k, o ensu e ull anspa ency o he esea ch, and in acco dance wi h
he p inciples o open schola ly con ibu ion and academic e hics.
(P ep in DOI: 10.5281/zenodo.16143976)
M.1 G a i a ional The modynamics Sys em Simula ion Code
in Py hon
The L
A
T
EX-s yle Py hon 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:
The nume ical simula ion amewo k is implemen ed in Py hon 3.8+ using a hyb id
app oach ha combines high-le el scien i ic compu ing wi h GPU accele a ion o
compu a ionally in ensi e ope a ions.
M.1.1 Co e Dependencies
Nume ical compu a ion s ack:
•NumPy ( 1.21+): Fundamen al a ay ope a ions, linea algeb a (linalg.no m,
apz), and nume ical compu a ions wi h IEEE 754 double p ecision.
•SciPy ( 1.7+): O dina y di e en ial equa ion in eg a ion
(scipy.in eg a e.odein ) o F iedmann cosmology, op imiza ion ou ines, and
special unc ions.
89
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 ###
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)
96
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 )
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
97
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)
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
98
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
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:
99
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
317 de ge _memo y_usage_mb() -> loa :
318 # Re u ns cu en memo y usage in MB
319 # Pla o m-speci ic implemen a ion:
320 # - Linux/macOS: use esou ce module
321 # - Windows: e u n 0.0 (no implemen ed)
322 y:
323 impo esou ce
324 mem_kb: in = esou ce.ge usage( esou ce.RUSAGE_SELF). u_max ss
325 i PLATFORM_NAME == 'Da win':# macOS epo s in by es
326 e u n mem_kb / (1024.0 ** 2)
327 else:# Linux epo s in KB
328 e u n mem_kb / 1024.0
329 excep Impo E o :
330 # Windows o esou ce module no a ailable
331 e u n 0.0
332 # File pa h sepa a o (c oss-pla o m)
333 PATH_SEP: s ='/'i PLATFORM_NAME != 'Windows'else ' '
334 # Ini ialize mul ip ocessing on impo
335 con igu e_mul ip ocessing()
336 ================================================================================
337 FILE: alida ion/__ini __.py
338 ================================================================================
339 # Valida ion Package
340 # P o ides dimensional analysis, un ime checks, and dual e i ica ion sys em
341 om .dimensional impo PhysicalQuan i y, DimT
342 om . un ime_check impo check_ ini e, asse _uni , check_dim
343 om .dual_ e i y impo dual_ e i y
344 om .sympy_check impo ini ialize_sympy_ e i ica ion, SYMBOLIC_FUNCTIONS
345 __all__ = [
346 'PhysicalQuan i y','DimT',
347 'check_ ini e','asse _uni ','check_dim',
348 'dual_ e i y',
349 'ini ialize_sympy_ e i ica ion','SYMBOLIC_FUNCTIONS'
350 ]
351 ================================================================================
352 FILE: alida ion/dimensional.py
353 ================================================================================
354 # Dimensional Analysis S uc u es
355 # De ines PhysicalQuan i y and DimT o dual e i ica ion sys em
100
356 om yping impo Any, NamedTuple
357 om jax.numpy. yping impo NDA ay
358 impo jax.numpy as jnp
359 class DimT(NamedTuple):
360 # Dimensional acking s uc u e
361 # T acks SI base dimensions: [m^e_m kg^e_kg s^e_s K^e_K]
362 alue: loa
363 e_m: in # Exponen o me e (leng h)
364 e_kg: in # Exponen o kilog am (mass)
365 e_s: in # Exponen o second ( ime)
366 e_K: in # Exponen o Kel in ( empe a u e)
367 uni : s
368 class PhysicalQuan i y:
369 # Physical quan i y wi h alue and uni
370 # Human- eadable uni ep esen a ion o cla i y
371 de __ini __(sel , alue: Any, uni : s )->None:
372 # Ini ialize PhysicalQuan i y
373 # A gs:
374 # alue: Nume ical alue
375 # uni : Uni s ing
376 sel . alue: NDA ay = jnp.asa ay( alue, d ype=jnp. loa 64)
377 sel .uni : s = uni
378 # Check o NaN/In on ini ializa ion
379 sel ._check_ ini e_in e nal()
380 de _check_ ini e_in e nal(sel ) -> None:
381 # In e nal check o ini e alues
382 # Raises ValueE o i alue con ains NaN o In
383 i isins ance(sel . alue, jnp.nda ay):
384 i no jnp.all(jnp.is ini e(sel . alue)):
385 nan_coun : in =in (jnp.sum(jnp.isnan(sel . alue)))
386 in _coun : in =in (jnp.sum(jnp.isin (sel . alue)))
387 aise ValueE o (
388 "PhysicalQuan i y ini : non- ini e alues de ec ed: "
389 "{nan_coun } NaNs, {in _coun } In s"
390 )
391 else:
392 i no jnp.is ini e(sel . alue):
393 s a us: s ='NaN'i jnp.isnan(sel . alue) else 'In '
394 aise ValueE o (
395 "PhysicalQuan i y ini : non- ini e alue: {s a us}"
396 )
397 de __ ep __(sel ) -> s :
398 # S ing ep esen a ion
399 e u n "PhysicalQuan i y( alue={sel . alue}, uni ='{sel .uni }')"
400 ================================================================================
401 FILE: alida ion/ un ime_check.py
402 ================================================================================
403 # Run ime Valida ion Func ions
101
404 # P o ides check_ ini e, asse _uni , and check_dim o un ime checks
405 om yping impo Any
406 impo jax.numpy as jnp
407 om jax.numpy. yping impo NDA ay
408 om .dimensional impo PhysicalQuan i y, DimT
409 de check_ ini e( alue: Any, name: s , con ex : s )->None:
410 # Check i alue is ini e (no NaN o In )
411 # A gs:
412 # alue: Value o check
413 # name: Va iable name o e o message
414 # con ex : Con ex s ing o e o message
415 # Raises:
416 # ValueE o : I alue con ains NaN o In
417 i isins ance( alue, jnp.nda ay):
418 i no jnp.all(jnp.is ini e( alue)):
419 nan_coun : in =in (jnp.sum(jnp.isnan( alue)))
420 in _coun : in =in (jnp.sum(jnp.isin ( alue)))
421 aise ValueE o (
422 "{con ex }: {name} has non- ini e alues: "
423 "{nan_coun } NaNs, {in _coun } In s"
424 )
425 else:
426 i no jnp.is ini e( alue):
427 s a us: s ='NaN'i jnp.isnan( alue) else 'In '
428 aise ValueE o (
429 "{con ex }: {name} is non- ini e: {s a us}"
430 )
431 de asse _uni (pq: PhysicalQuan i y, expec ed_uni : s , label: s ) -> None:
432 # Asse ha PhysicalQuan i y has expec ed uni
433 # A gs:
434 # pq: PhysicalQuan i y ins ance
435 # expec ed_uni : Expec ed uni s ing
436 # label: Label o e o message
437 # Raises:
438 # ValueE o : I uni s do no ma ch
439 i pq.uni != expec ed_uni :
440 aise ValueE o (
441 "{label}: uni misma ch - expec ed '{expec ed_uni }', "
442 "go '{pq.uni }'"
443 )
444 de check_dim(
445 d : DimT,
446 expec ed_e_m: in ,
447 expec ed_e_kg: in ,
448 expec ed_e_s: in ,
449 expec ed_e_K: in ,
450 label: s
451 )->None:
452 # Check dimensional exponen s ma ch expec ed alues
453 # Ve i ies ha DimT has co ec exponen s o [m^a kg^b s^c K^d]
102
454 # A gs:
455 # d : DimT ins ance
456 # expec ed_e_m: Expec ed me e exponen
457 # expec ed_e_kg: Expec ed kilog am exponen
458 # expec ed_e_s: Expec ed second exponen
459 # expec ed_e_K: Expec ed Kel in exponen
460 # label: Label o e o message
461 # Raises:
462 # ValueE o : I dimensions do no ma ch
463 i (d .e_m != expec ed_e_m o
464 d .e_kg != expec ed_e_kg o
465 d .e_s != expec ed_e_s o
466 d .e_K != expec ed_e_K):
467 aise ValueE o (
468 "ERROR: Dimensional misma ch in {label} n"
469 "Expec ed: [m^{expec ed_e_m} kg^{expec ed_e_kg} "
470 "s^{expec ed_e_s} K^{expec ed_e_K}] n"
471 "Go : [m^{d .e_m} kg^{d .e_kg} s^{d .e_s} K^{d .e_K}]"
472 )
473 ================================================================================
474 FILE: alida ion/dual_ e i y.py
475 ================================================================================
476 # Dual Ve i ica ion Func ion
477 # Combines PhysicalQuan i y and DimT e i ica ion wi h ole ance checking
478 # Called 128 imes h oughou he codebase
479 impo jax.numpy as jnp
480 om .dimensional impo PhysicalQuan i y, DimT
481 om . un ime_check impo asse _uni , check_dim
482 de dual_ e i y(
483 pq: PhysicalQuan i y,
484 d : DimT,
485 label: s ,
486 expec ed_uni : s ,
487 e_m: in ,
488 e_kg: in ,
489 e_s: in ,
490 e_K: in ,
491 ole ance: loa = 1e-15
492 )->None:
493 # Dual e i ica ion: checks bo h uni s ings and dimensional exponen s
494 # Ensu es ela i e e o be ween pq. alue and d . alue is < ole ance
495 # A gs:
496 # pq: PhysicalQuan i y ins ance
497 # d : DimT ins ance
498 # label: Label o e o messages
499 # expec ed_uni : Expec ed uni s ing
500 # e_m: Expec ed me e exponen
501 # e_kg: Expec ed kilog am exponen
103
502 # e_s: Expec ed second exponen
503 # e_K: Expec ed Kel in exponen
504 # ole ance: Rela i e e o ole ance (de aul 1e-15)
505 # Raises:
506 # Asse ionE o : I alues di e by mo e han ole ance
507 # ValueE o : I uni s o dimensions do no ma ch
508 # Check uni s ing
509 asse _uni (pq, expec ed_uni , label)
510 # Check dimensional exponen s
511 check_dim(d , e_m, e_kg, e_s, e_K, label)
512 # Check alue ag eemen wi h ole ance
513 pq_ al = jnp.asa ay(pq. alue)
514 d _ al = jnp.asa ay(d . alue)
515 di = jnp.abs(pq_ al - d _ al)
516 i jnp.all(di < ole ance):
517 # Absolu e di e ence check passed
518 pass
519 else:
520 # Check ela i e e o
521 max_ al = jnp.maximum(jnp.abs(pq_ al), jnp.abs(d _ al))
522 el_e = di / (max_ al + 1e-100) # A oid di ision by ze o
523 i no jnp.all( el_e < ole ance):
524 aise Asse ionE o (
525 "{label}: alue misma ch exceeds ole ance { ole ance} n"
526 "Max ela i e e o : {jnp.max( el_e ):.3e}"
527 )
528 # Repea checks o edundancy (as speci ied)
529 asse _uni (pq, expec ed_uni , label + " ( epea )")
530 check_dim(d , e_m, e_kg, e_s, e_K, label + " ( epea )")
531 ================================================================================
532 FILE: alida ion/sympy_check.py
533 ================================================================================
534 # SymPy Symbolic Dimensional Ve i ica ion
535 # Pe o ms symbolic dimensional analysis using SymPy
536 # Includes 12 calls each o sp.symbols, sp.lambdi y, sp.simpli y, dual_ e i y
537 impo wa nings
538 om yping impo Any, Callable, Dic , Lis
539 impo jax.numpy as jnp
540 impo sympy as sp
541 om jax.numpy. yping impo NDA ay
542 # Impo cons an s only when needed o a oid ci cula impo s
543 # These will be impo ed in he unc ions ha use hem
544 # Global dic iona y o s o e symbolic exp essions and unc ions
545 SYMBOLIC_FUNCTIONS: Dic [s , Any] = {}
546 de ini ialize_sympy_ e i ica ion() -> None:
547 # Ini ialize SymPy symbolic e i ica ion sys em
548 # C ea es symbolic exp essions and compiles hem in o jax unc ions
549 # Pe o ms 12 calls o sp.symbols, sp.lambdi y, sp.simpli y, dual_ e i y
104
550 # Impo cons an s he e o a oid ci cula impo
551 om ..con ig.cons an s impo (
552 A_RAD, K_BOLTZMANN, G_NEWTON, HBAR, C_LIGHT, SIGMA_SB
553 )
554 # ============= Call 1: En opy adia ion =============
555 # Fo mula: S_ = (4/3) a_ ad N T^3 V
556 a_sym_1, N_sym_1, T_sym_1, V_sym_1 = sp.symbols(
557 'a_ ad N T V', eal=T ue, posi i e=T ue
558 )
559 S_ _exp = sp.Ra ional(4, 3) * a_sym_1 * N_sym_1 * T_sym_1**3 * V_sym_1
560 # Dimensional simpli ica ion check
561 y:
562 # Expec ed: [J/K] = [J m^-3 K^-4] * [K^3] * [m^3]
563 asse sp.simpli y(S_ _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')/sp.symbols('K')
564 excep (Asse ionE o , TypeE o ):
565 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - S_ ')
566 S_ _ unc = sp.lambdi y((a_sym_1, N_sym_1, T_sym_1, V_sym_1), S_ _exp , '
jax')
567 SYMBOLIC_FUNCTIONS['en opy_ adia ion'] = S_ _ unc
568 # ============= Call 2: En opy ma e (black hole) =============
569 # Fo mula: S_m = 4 pi k G M^2 / (hba c)
570 k_sym_2, G_sym_2, M_sym_2, hba _sym_2, c_sym_2 = sp.symbols(
571 'k G M hba c', eal=T ue, posi i e=T ue
572 )
573 S_m_exp = 4 * sp.pi * k_sym_2 * G_sym_2 * M_sym_2**2 / (hba _sym_2 *
c_sym_2)
574 y:
575 simpli ied = sp.simpli y(S_m_exp )
576 asse simpli ied
577 excep (Asse ionE o , TypeE o ):
578 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - S_m')
579 S_m_ unc = sp.lambdi y(
580 (k_sym_2, G_sym_2, M_sym_2, hba _sym_2, c_sym_2), S_m_exp , 'jax'
581 )
582 SYMBOLIC_FUNCTIONS['en opy_ma e _BH'] = S_m_ unc
583 # ============= Call 3: Hawking empe a u e =============
584 # Fo mula: T_H = hba c^3 / (8 pi G M k_B)
585 T_H_exp = (hba _sym_2 * c_sym_2**3) / (
586 8 * sp.pi * G_sym_2 * M_sym_2 * k_sym_2
587 )
588 y:
589 simpli ied_TH = sp.simpli y(T_H_exp )
590 asse simpli ied_TH
591 excep (Asse ionE o , TypeE o ):
592 wa nings.wa n('SymPy dimensional check ailed (non-c i ical) - T_H')
593 T_H_ unc = sp.lambdi y(
594 (hba _sym_2, c_sym_2, G_sym_2, M_sym_2, k_sym_2), T_H_exp , 'jax'
595 )
105
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))
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)
112
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:
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)
113
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")
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)
114
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
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)
115
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
1024 )
1025 check_ ini e(S_BH, "S_BH", "en opy_ma e _BH")
1026 # Dual e i ica ion (call 6/128)
1027 pq_s = PhysicalQuan i y(S_BH, "J/K")
1028 d _s = DimT(S_BH, 2, 1, -2, -1, "J/K")
1029 dual_ e i y(pq_s, d _s, "S_BH", "J/K", 2, 1, -2, -1)
1030 e u n S_BH
1031 de en opy_ adia ion(T: loa ,V: loa , deg_ : loa = DEG_FREEDOM) -> loa
:
1032 # Compu e adia ion en opy
1033 # Fo mula: S_ = (4/3) a_ ad deg_ T^3 V
1034 # The modynamic en opy o adia ion
1035 # A gs:
1036 # T: Tempe a u e [K]
1037 # V: Volume [m^3]
1038 # deg_ : Deg ees o eedom [dimensionless]
1039 # Re u ns: Radia ion en opy [J/K]
1040 check_ ini e(T, "T", "en opy_ adia ion")
1041 check_ ini e(V, "V", "en opy_ adia ion")
1042 asse T > 0.0 and V > 0.0, "Inpu s mus be posi i e"
1043 S_ : loa = SYMBOLIC_FUNCTIONS['en opy_ adia ion'](
1044 A_RAD, deg_ , T, V
1045 )
1046 check_ ini e(S_ , "S_ ", "en opy_ adia ion")
1047 # Dual e i ica ion (call 7/128)
1048 pq_s = PhysicalQuan i y(S_ , "J/K")
1049 d _s = DimT(S_ , 2, 1, -2, -1, "J/K")
1050 dual_ e i y(pq_s, d _s, "S_ ", "J/K", 2, 1, -2, -1)
1051 e u n S_
1052 de p essu e_ adia ion(T: loa , deg_ : loa = DEG_FREEDOM) -> loa :
1053 # Compu e adia ion p essu e
1054 # Fo mula: P_ ad = (1/3) a_ ad deg_ T^4
1055 # A gs:
1056 # T: Tempe a u e [K]
1057 # deg_ : Deg ees o eedom [dimensionless]
1058 # Re u ns: P essu e [Pa]
1059 check_ ini e(T, "T", "p essu e_ adia ion")
1060 asse T > 0.0, "Tempe a u e mus be posi i e"
1061 P_ ad: loa = SYMBOLIC_FUNCTIONS['p essu e_ adia ion'](
1062 A_RAD, deg_ , T
1063 )
1064 check_ ini e(P_ ad, "P_ ad", "p essu e_ adia ion")
1065 # Dual e i ica ion (call 8/128)
116
1066 pq_p = PhysicalQuan i y(P_ ad, "Pa")
1067 d _p = DimT(P_ ad, -1, 1, -2, 0, "Pa")
1068 dual_ e i y(pq_p, d _p, "P_ ad", "Pa", -1, 1, -2, 0)
1069 e u n P_ ad
1070 de p essu e_ acuum( ho_ ac: loa , luc : loa = 0.0) -> loa :
1071 # Compu e acuum p essu e wi h quan um luc ua ions
1072 # Fo mula: P_ ac = - ho_ ac c^2 + luc ua ion
1073 # A gs:
1074 # ho_ ac: Vacuum ene gy densi y [kg/m^3]
1075 # luc : Quan um p essu e luc ua ion [Pa]
1076 # Re u ns: Vacuum p essu e [Pa]
1077 check_ ini e( ho_ ac, " ho_ ac", "p essu e_ acuum")
1078 check_ ini e( luc , " luc ", "p essu e_ acuum")
1079 P_ ac: loa = - ho_ ac * C_LIGHT**2 + luc
1080 check_ ini e(P_ ac, "P_ ac", "p essu e_ acuum")
1081 # Dual e i ica ion (call 9/128)
1082 pq_p = PhysicalQuan i y(P_ ac, "Pa")
1083 d _p = DimT(P_ ac, -1, 1, -2, 0, "Pa")
1084 dual_ e i y(pq_p, d _p, "P_ ac", "Pa", -1, 1, -2, 0)
1085 e u n P_ ac
1086 de check_ene gy_condi ions( ho: loa , P: loa ) -> Dic [s , bool]:
1087 # Check ene gy condi ions (NEC, WEC, SEC, DEC)
1088 # NEC: ho c^2 + P >= 0
1089 # WEC: ho c^2 >= 0 and ho c^2 + P >= 0
1090 # SEC: ho c^2 + 3P >= 0
1091 # DEC: ho c^2 >= |P|
1092 # A gs:
1093 # ho: Ene gy densi y [kg/m^3]
1094 # P: P essu e [Pa]
1095 # Re u ns: Dic iona y wi h boolean lags o each condi ion
1096 check_ ini e( ho, " ho", "check_ene gy_condi ions")
1097 check_ ini e(P, "P", "check_ene gy_condi ions")
1098 ho_c2: loa = ho * C_LIGHT**2
1099 nec: bool = ( ho_c2 + P) >= -1e-15
1100 wec: bool = ( ho_c2 >= 0) and (( ho_c2 + P) >= -1e-15)
1101 sec: bool = ( ho_c2 + 3.0 * P) >= -1e-15
1102 dec: bool = ho_c2 >= abs(P)
1103 e u n {
1104 'NEC': nec,
1105 'WEC': wec,
1106 'SEC': sec,
1107 'DEC': dec
1108 }
1109 de speci ic_hea _nega i e(M: loa )-> loa :
1110 # Compu e nega i e speci ic hea o g a i a ional sys em
1111 # Fo mula: C_V = -8 pi k_B G M^2 / (hba c) < 0
1112 # A gs: M: To al mass [kg]
1113 # Re u ns: Speci ic hea [J/K] (nega i e alue)
1114 check_ ini e(M, "M", "speci ic_hea _nega i e")
1115 asse M > 0.0, "Mass mus be posi i e"
117
1116 C_V: loa = SYMBOLIC_FUNCTIONS['speci ic_hea _nega i e'](
1117 G_NEWTON, M, K_BOLTZMANN, HBAR, C_LIGHT
1118 )
1119 check_ ini e(C_V, "C_V", "speci ic_hea _nega i e")
1120 asse C_V < 0, "Speci ic hea should be nega i e"
1121 # Dual e i ica ion (call 10/128)
1122 pq_c = PhysicalQuan i y(C_V, "J/K")
1123 d _c = DimT(C_V, 2, 1, -2, -1, "J/K")
1124 dual_ e i y(pq_c, d _c, "C_V", "J/K", 2, 1, -2, -1)
1125 e u n C_V
1126 de check_en opy_p oduc ion( ho: loa , T: loa ,H: loa , V: loa )->None
:
1127 # Check dS/d = ( ho + p)/T * H V >0 wi h p= ho/3 adia ion EOS
1128 # A gs:
1129 # ho: Ene gy densi y [kg/m^3]
1130 # T: Tempe a u e [K]
1131 # H: Hubble pa ame e [s^-1]
1132 # V: Volume [m^3]
1133 check_ ini e( ho, " ho", "check_en opy_p oduc ion")
1134 check_ ini e(T, "T", "check_en opy_p oduc ion")
1135 check_ ini e(H, "H", "check_en opy_p oduc ion")
1136 check_ ini e(V, "V", "check_en opy_p oduc ion")
1137 asse T > 0.0 and H > 0.0 and V > 0.0 and ho > 0.0, "Inpu s mus be
posi i e"
1138 p = ho / 3.0
1139 ds_d = ( ho + p) / T * H * V
1140 check_ ini e(ds_d , "ds_d ", "check_en opy_p oduc ion")
1141 asse ds_d > 0.0, "En opy p oduc ion mus be posi i e o adia ion EOS
"
1142 # Dual e i ica ion (call 36/128)
1143 pq_ds = PhysicalQuan i y(ds_d , "J/K/s")
1144 d _ds = DimT(ds_d , 2, 1, -3, -1, "J/K/s")
1145 dual_ e i y(pq_ds, d _ds, "ds_d ", "J/K/s", 2, 1, -3, -1)
1146 ================================================================================
1147 FILE: physics/g a i y.py
1148 ================================================================================
1149 # G a i y Module - Di ec N^2 o ce calcula ion wi h JAX o GPU
pa alleliza ion
1150 # Replaced Ba nes-Hu wi h JAX-accele a ed di ec sum o GPU compa ibili y
1151 # En opic o ce in eg a ion: New onian as base, en opic ia he modynamics
module
1152 om yping impo Lis
1153 om da aclasses impo da aclass
1154 impo jax
1155 impo jax.numpy as jnp
1156 om jax.numpy. yping impo NDA ay
1157 om ..con ig.cons an s impo G_NEWTON
1158 om ..con ig.simula ion_pa ams impo SIG_SOFT
118
1159 om .. alida ion. un ime_check impo check_ ini e
1160 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
1161 om .. alida ion.dual_ e i y impo dual_ e i y
1162 om . he modynamics impo scale_ empe a u e, en opic_ o ce
1163 de classi y_ egion( : loa , _co e: loa = 1.0, _quan um: loa = 10.0) ->
s :
1164 # Classi y spa ial egion
1165 # A gs:
1166 # : Radial dis ance [m]
1167 # _co e: Co e bounda y [m]
1168 # _quan um: Quan um egime bounda y [m]
1169 # Re u ns: Region classi ica ion s ing
1170 check_ ini e( , " ", "classi y_ egion")
1171 asse >= 0.0, "Radius mus be non-nega i e"
1172 i < _co e:
1173 e u n "co e"
1174 eli < _quan um:
1175 e u n "quan um"
1176 else:
1177 e u n "classical"
1178 @da aclass
1179 class Pa icle:
1180 # Pa icle ep esen a ion o N-body simula ion
1181 posi ion: NDA ay
1182 eloci y: NDA ay
1183 mass: loa
1184 empe a u e: loa
1185 en opy: loa
1186 egion: s = "classical"
1187 de __pos _ini __(sel ) -> None:
1188 # Valida e pa icle da a on ini ializa ion
1189 check_ ini e(sel .posi ion, "posi ion", "Pa icle.__pos _ini __")
1190 check_ ini e(sel . eloci y, " eloci y", "Pa icle.__pos _ini __")
1191 check_ ini e(sel .mass, "mass", "Pa icle.__pos _ini __")
1192 check_ ini e(sel . empe a u e, " empe a u e", "Pa icle.__pos _ini __
")
1193 check_ ini e(sel .en opy, "en opy", "Pa icle.__pos _ini __")
1194 asse sel .mass > 0, "Mass mus be posi i e"
1195 asse sel . empe a u e > 0, "Tempe a u e mus be posi i e"
1196 asse sel .en opy >= 0, "En opy mus be non-nega i e"
1197 # Dual e i ica ion o mass (call 11/128)
1198 pq_m = PhysicalQuan i y(sel .mass, "kg")
1199 d _m = DimT(sel .mass, 0, 1, 0, 0, "kg")
1200 dual_ e i y(pq_m, d _m, "mass", "kg", 0, 1, 0, 0)
1201 # Dual e i ica ion o empe a u e (call 12/128)
1202 pq_ = PhysicalQuan i y(sel . empe a u e, "K")
1203 d _ = DimT(sel . empe a u e, 0, 0, 0, 1, "K")
1204 dual_ e i y(pq_ , d _ , " empe a u e", "K", 0, 0, 0, 1)
1205 # Dual e i ica ion o en opy (call 13/128)
1206 pq_s = PhysicalQuan i y(sel .en opy, "J/K")
119
1207 d _s = DimT(sel .en opy, 2, 1, -2, -1, "J/K")
1208 dual_ e i y(pq_s, d _s, "en opy", "J/K", 2, 1, -2, -1)
1209 # Classi y egion based on posi ion
1210 _dis : loa = jnp.linalg.no m(sel .posi ion)
1211 sel . egion = classi y_ egion( _dis )
1212 class Holog aphicSimula o JAX:
1213 de __ini __(sel , G: loa = G_NEWTON, sig_so : loa = SIG_SOFT):
1214 sel .G = G
1215 sel .sig_so = sig_so
1216 @jax.ji # JIT op imiza ion (CUDA-like pe o mance)
1217 de compu e_accele a ions(sel , posi ions: NDA ay, masses: NDA ay) ->
NDA ay:
1218 # Compu e accele a ions using di ec summa ion on GPU
1219 # a_i = G * sum_j m_j * (pos_j - pos_i) / ( _ij^3 + sig_so ^3)
1220 # posi ions: (N, 3), masses: (N,)
1221 # Re u ns: accele a ions (N, 3)
1222 di = posi ions[jnp.newaxis, :, :] - posi ions[:, jnp.newaxis, :]
1223 _mag = jnp.linalg.no m(di , axis=-1)
1224 _mag_sa e = jnp.whe e( _mag < 1e-10, 1e-10, _mag)
1225 denomina o = _mag_sa e[:, :, jnp.newaxis]**3 + sel .sig_so **3
1226 accele a ions = sel .G * jnp.sum(
1227 masses[:, jnp.newaxis, jnp.newaxis] * di / denomina o ,
1228 axis=0
1229 )
1230 e u n accele a ions
1231 ================================================================================
1232 FILE: physics/ iedmann.py
1233 ================================================================================
1234 # F iedmann Equa ions and RK4 In eg a ion
1235 impo jax.numpy as jnp
1236 om jax.numpy. yping impo NDA ay
1237 om yping impo Tuple
1238 om ..con ig.cons an s impo C_LIGHT, G_NEWTON
1239 om ..con ig.cosmology impo H_HUBBLE_0, OMEGA_M_0, OMEGA_LAMBDA_0,
OMEGA_R_0
1240 om ..con ig.simula ion_pa ams impo SCALE_FACTOR_MIN, D_CRITICAL
1241 om .. alida ion. un ime_check impo check_ ini e
1242 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
1243 om .. alida ion.dual_ e i y impo dual_ e i y
1244 de iedmann_ hs(a: loa , a_do : loa , : loa ) -> Tuple[ loa , loa ]:
1245 # F iedmann equa ions
1246 # da/d = a_do
1247 # d^2a/d ^2 = -4piG/3 * a * ( ho + 3P/c^2) + Lambda*c^2/3 * a
1248 check_ ini e(a, "a", " iedmann_ hs")
1249 check_ ini e(a_do , "a_do ", " iedmann_ hs")
1250 i a < SCALE_FACTOR_MIN:
1251 a = SCALE_FACTOR_MIN
1252 H = a_do / a
120
1253 ho_m = OMEGA_M_0 * H_HUBBLE_0**2 / a**3
1254 ho_ = OMEGA_R_0 * H_HUBBLE_0**2 / a**4
1255 ho_lambda = OMEGA_LAMBDA_0 * H_HUBBLE_0**2
1256 # Second de i a i e
1257 a_ddo = (
1258 -4.0 * jnp.pi * G_NEWTON / 3.0 * a * ( ho_m + 2.0 * ho_ ) +
1259 OMEGA_LAMBDA_0 * H_HUBBLE_0**2 * a / 3.0
1260 )
1261 check_ ini e(a_ddo , "a_ddo ", " iedmann_ hs")
1262 # Dual e i ica ion calls 16-17/128
1263 pq_h = PhysicalQuan i y(H, "s^-1")
1264 d _h = DimT(H, 0, 0, -1, 0, "s^-1")
1265 dual_ e i y(pq_h, d _h, "H", "s^-1", 0, 0, -1, 0)
1266 pq_ ho = PhysicalQuan i y( ho_m, "kg/m^3")
1267 d _ ho = DimT( ho_m, -3, 1, 0, 0, "kg/m^3")
1268 dual_ e i y(pq_ ho, d _ ho, " ho_m", "kg/m^3", -3, 1, 0, 0)
1269 e u n a_do , a_ddo
1270 de k4_s ep(a: loa , a_do : loa , : loa , d : loa ) -> Tuple[ loa ,
loa , loa ]:
1271 # RK4 in eg a ion o coupled ODEs
1272 check_ ini e(a, "a", " k4_s ep")
1273 check_ ini e(a_do , "a_do ", " k4_s ep")
1274 check_ ini e(d , "d ", " k4_s ep")
1275 # k1
1276 k1_a, k1_ = iedmann_ hs(a, a_do , )
1277 # k2
1278 k2_a, k2_ = iedmann_ hs(
1279 a + 0.5 * d * k1_a,
1280 a_do + 0.5 * d * k1_ ,
1281 + 0.5 * d
1282 )
1283 # k3
1284 k3_a, k3_ = iedmann_ hs(
1285 a + 0.5 * d * k2_a,
1286 a_do + 0.5 * d * k2_ ,
1287 + 0.5 * d
1288 )
1289 # k4
1290 k4_a, k4_ = iedmann_ hs(
1291 a + d * k3_a,
1292 a_do + d * k3_ ,
1293 + d
1294 )
1295 # Upda e
1296 a_new = a + d / 6.0 * (k1_a + 2.0 * k2_a + 2.0 * k3_a + k4_a)
1297 a_do _new = a_do + d / 6.0 * (k1_ + 2.0 * k2_ + 2.0 * k3_ + k4_ )
1298 _new = + d
1299 check_ ini e(a_new, "a_new", " k4_s ep")
1300 check_ ini e(a_do _new, "a_do _new", " k4_s ep")
1301 # Dual e i ica ion call 18/128
121
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(
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):
128
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 ================================================================================
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 *
129
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'
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)
130
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)
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,
131
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))
1772 pl .plo (z_np, y_np, 'b-', linewid h=2)
1773 pl .xlabel('Redshi z', on size=14)
1774 pl .ylabel('Dimensionless En opy y', on size=14)
1775 pl . i le('En opy E olu ion as a Func ion o Redshi ', on size=16)
1776 pl .xscale('log')
1777 pl .yscale('log')
1778 pl .g id(T ue, alpha=0.3)
1779 pl . igh _layou ()
1780 pl .sa e ig( ilename, dpi=300)
1781 pl .close()
1782 de plo _en opy_p oduc ion(
1783 : NDA ay,
1784 sigma: NDA ay,
1785 ilename: s ='en opy_p oduc ion.png'
1786 )->None:
1787 # Plo en opy p oduc ion a e sigma s ime
1788 _np = np.a ay( )
1789 sigma_np = np.a ay(sigma)
1790 pl . igu e( igsize=(10, 6))
1791 pl .plo ( _np / 3.156e16, sigma_np, ' -', linewid h=2)
1792 pl .xlabel('Time [Gy ]', on size=14)
1793 pl .ylabel('En opy P oduc ion Ra e sigma [J/K/s/m^3]', on size=14)
1794 pl . i le('Tempo al E olu ion o En opy P oduc ion Ra e', on size=16)
1795 pl .xscale('log')
1796 pl .yscale('log')
1797 pl .g id(T ue, alpha=0.3)
1798 pl . igh _layou ()
1799 pl .sa e ig( ilename, dpi=300)
1800 pl .close()
1801 de plo _densi y_con as _ s_scale_ ac o (
1802 a: NDA ay,
1803 D: NDA ay,
1804 D_c i ical: loa = 709.0,
1805 ilename: s ='densi y_con as _709.png'
1806 )->None:
1807 # Plo densi y con as D s scale ac o a
1808 a_np = np.a ay(a)
1809 D_np = np.a ay(D)
1810 pl . igu e( igsize=(10, 6))
1811 pl .plo (a_np, D_np, 'b-', linewid h=2, label='D(a)')
1812 pl .axhline(y=D_c i ical, colo ='k', lines yle='--', linewid h=1.5,
1813 label= 'D_c i ical = {D_c i ical}')
1814 pl .xlabel('Scale Fac o a', on size=14)
132
1815 pl .ylabel('Densi y Con as D', on size=14)
1816 pl . i le('Scale Fac o Dependence o Densi y Con as (D=709)', on size
=16)
1817 pl .xscale('log')
1818 pl .yscale('log')
1819 pl .legend( on size=12)
1820 pl .g id(T ue, alpha=0.3)
1821 pl . igh _layou ()
1822 pl .sa e ig( ilename, dpi=300)
1823 pl .close()
1824 ================================================================================
1825 FILE: ou pu /da a_expo .py
1826 ================================================================================
1827 # Da a Expo (CSV, HDF5)
1828 impo jax.numpy as jnp
1829 impo numpy as np # Fo CSV compa ibili y
1830 om jax.numpy. yping impo NDA ay
1831 impo cs
1832 om yping impo Dic , Lis
1833 de expo _ o_cs (
1834 da a: Dic [s , NDA ay],
1835 ilename: s ='simula ion_da a.cs '
1836 )->None:
1837 # Expo da a o CSV ile
1838 heade s = lis (da a.keys())
1839 da a_np = {k: np.a ay( ) o k, in da a.i ems()}
1840 ows = zip(*[da a_np[key] o key in heade s])
1841 wi h open( ilename, 'w', newline='')as :
1842 w i e = cs .w i e ( )
1843 w i e .w i e ow(heade s)
1844 w i e .w i e ows( ows)
1845 de expo _ o_hd 5(
1846 da a: Dic [s , NDA ay],
1847 ilename: s ='simula ion_da a.h5'
1848 )->None:
1849 # Expo da a o HDF5 ile
1850 y:
1851 impo h5py
1852 wi h h5py.File( ilename, 'w')as :
1853 o key, alue in da a.i ems():
1854 .c ea e_da ase (key, da a=np.a ay( alue))
1855 excep Impo E o :
1856 p in ("h5py no a ailable, skipping HDF5 expo ")
1857 de expo _ able(
1858 da a: Lis [Lis [Any]],
1859 heade s: Lis [s ],
1860 ilename: s =' able.cs '
1861 )->None:
133
1862 # Expo able da a o CSV
1863 wi h open( ilename, 'w', newline='')as :
1864 w i e = cs .w i e ( )
1865 w i e .w i e ow(heade s)
1866 w i e .w i e ows(da a)
1867 ================================================================================
1868 FILE: main.py (CORRECTED VERSION)
1869 ================================================================================
1870 # Main En y Poin
1871 # CORRECTED: Fixed impo s a emen s and SymPy ini ializa ion
1872 # In eg a ed uni ied co ec ions: T_s(l), F = T_s dS/dx, appendices,
de i a ions
1873 impo sys
1874 om pa hlib impo Pa h
1875 impo jax
1876 impo jax.numpy as jnp
1877 impo a gpa se
1878 # Add pa en di ec o y o pa h o impo s
1879 sys.pa h.inse (0, s (Pa h(__ ile__).pa en ))
1880 # Impo om con ig
1881 om con ig.cons an s impo *
1882 om con ig.cosmology impo *
1883 om con ig.simula ion_pa ams impo *
1884 om con ig.pla o m_con ig impo *
1885 # Impo om alida ion
1886 om alida ion.dimensional impo PhysicalQuan i y, DimT
1887 om alida ion. un ime_check impo check_ ini e
1888 om alida ion.dual_ e i y impo dual_ e i y
1889 om alida ion.sympy_check impo ini ialize_sympy_ e i ica ion,
SYMBOLIC_FUNCTIONS
1890 # Impo om physics
1891 om physics. he modynamics impo (
1892 hawking_ empe a u e, un uh_ empe a u e, hubble_ empe a u e,
1893 scale_ empe a u e, holog aphic_sc een_en opy,
1894 en opy_ma e _BH, en opy_ adia ion,
1895 p essu e_ adia ion, p essu e_ acuum,
1896 check_ene gy_condi ions, speci ic_hea _nega i e,
1897 check_en opy_p oduc ion,
1898 en opic_ o ce, hubble_en opic_ o ce, planck_ o ce_de i a ion,
1899 bol zmann_composi e, adia ion_en opy_densi y, adia ion_p essu e_densi y
,
1900 holog aphic_sc een_densi y, holog aphic_do , acuum_p essu e_ luc ,
1901 planck_no malized_en opy, planck_no malized_en opy_ ilde
1902 )
1903 om physics.g a i y impo Pa icle, Holog aphicSimula o JAX, classi y_ egion
1904 om physics. iedmann impo iedmann_ hs, k4_s ep, lane_emden_sol e
1905 om physics.quan um impo box_mulle _ ans o m, quan um_ luc ua ion
1906 # Impo om simula ion
134
1907 om simula ion.mon e_ca lo impo mon e_ca lo_simula ion, gene a e_seed
1908 om simula ion.n_body impo n_body_simula ion, ini ialize_pa icles
1909 om simula ion.leap og impo leap og_s ep, leap og_in eg a e
1910 om simula ion.openmp_pa allel impo pa allel_ o ce_calcula ion,
pa allel_map
1911 # Impo om ou pu
1912 om ou pu . isualiza ion impo plo _en opy_e olu ion, plo _densi y_con as
, plo _scale_ ac o , plo _non_ ela i is ic_cosmic_expansion,
plo _en opy_e olu ion_ s_ edshi , plo _en opy_p oduc ion,
plo _densi y_con as _ s_scale_ ac o
1913 om ou pu .da a_expo impo expo _ o_cs , expo _ o_hd 5, expo _ able
1914 de main():
1915 # Pa se command line a gumen s
1916 pa se = a gpa se.A gumen Pa se (
1917 desc ip ion='Holog aphic The modynamics Simula ion'
1918 )
1919 pa se .add_a gumen ('--n-pa icles', ype=in , de aul =N_PARTICLES,
1920 help='Numbe o pa icles')
1921 pa se .add_a gumen ('--n- imes eps', ype=in , de aul =N_TIMESTEPS,
1922 help='Numbe o imes eps')
1923 pa se .add_a gumen ('--n- ials', ype=in , de aul =N_TRIALS,
1924 help='Numbe o Mon e Ca lo ials')
1925 pa se .add_a gumen ('--ou pu ', ype=s , de aul =' esul s/',
1926 help='Ou pu di ec o y')
1927 pa se .add_a gumen ('--use-en opic', ac ion='s o e_ ue',
1928 help='Use uni ied en opic o ce in simula ion')
1929 a gs = pa se .pa se_a gs()
1930 # C ea e ou pu di ec o y
1931 ou pu _di = Pa h(a gs.ou pu )
1932 ou pu _di .mkdi (pa en s=T ue, exis _ok=T ue)
1933 p in ("="*80)
1934 p in ("HOLOGRAPHIC THERMODYNAMICS SIMULATION")
1935 p in ("="*80)
1936 p in ( "Pla o m: {PLATFORM_NAME}")
1937 p in ( "CPU co es: {ge _cpu_coun ()}")
1938 p in (jax.de ices()) # Au oma ically check a ailable GPUs
1939 p in ( "N_PARTICLES: {a gs.n_pa icles}")
1940 p in ( "N_TIMESTEPS: {a gs.n_ imes eps}")
1941 p in ( "N_TRIALS: {a gs.n_ ials}")
1942 p in ( "Use en opic o ce: {a gs.use_en opic}")
1943 p in ("="*80)
1944 # Appendix A: Bol zmann dis ibu ion co espondence
1945 p in (" nAppendix A: Bol zmann dis ibu ion wi h co espondence")
1946 p in ("En opic o ce F = T_s(l) * dS/dx de i ed om composi e Bol zmann
:")
1947 p in ("P(x; l) = w_U(l) exp(-E_U / k_B T_U) + w_H(l) exp(-E_H / k_B T_H)")
1948 p in ("w_U(l) = exp(-(l/l_c)^2), w_H(l) = 1 - exp(-(l/l_c)^2)")
1949 p in ("k_B cancels in Un uh exponen : exp(-E / k_B T_U) = exp(-E * 2 pi c
/ (hba a))")
135
1950 p in ("Thus F = T dS/dx s a is ically exac (The modynamic/Bekens ein-
Hawking en opy used)")
1951 # Appendix B: Ve linde connec ion
1952 p in (" nAppendix B: Connec ion o Ve linde (2010), Jacobson (1995),
Ho a a (2012)")
1953 p in ("Adop s F = T dS/dx o m; S = k_B sigma (sigma dimensionless en opy
)")
1954 p in ("Local limi : F ~ T_U dS/dx (Planck o ce c^4/G)")
1955 p in ("Hubble limi : F_H = T_H dS/dx = M_H H c")
1956 # Run Planck o ce de i a ion
1957 F_Pl = planck_ o ce_de i a ion()
1958 p in ( "Planck o ce F_Pl = {F_Pl:.3e} N")
1959 # Run Hubble en opic o ce e i ica ion
1960 F_H = hubble_en opic_ o ce()
1961 p in ( "Hubble en opic o ce F_H = {F_H:.3e} N (ma ches Planck o ce in
limi )")
1962 # New speci ica ions: Holog aphic quan i ies
1963 sigma_sc een = holog aphic_sc een_densi y()
1964 N_holo = holog aphic_do ()
1965 sigma_holo = acuum_p essu e_ luc ()
1966 # Example planck_no malized_en opy
1967 x_example = 0.315 # Omega_m,0 as example
1968 y = planck_no malized_en opy(x_example)
1969 # Example ilde y
1970 S_example = 1e100 # App oxima e uni e se en opy
1971 E_ o al_example = M_HUBBLE * C_LIGHT**2
1972 y_ ilde = planck_no malized_en opy_ ilde(S_example, E_ o al_example)
1973 # Ene gy hie a chy examples
1974 E_p o on = M_PROTON * C_LIGHT**2
1975 p in ( "E_p o on ~ {E_p o on:.3e} J")
1976 E_planck = E_PLANCK
1977 p in ( "E_planck ~ {E_planck:.3e} J")
1978 E_uni e se = M_HUBBLE * C_LIGHT**2
1979 p in ( "E_uni e se ~ {E_uni e se:.3e} J")
1980 # Run simula ion
1981 p in (" nIni ializing pa icles...")
1982 pa icles = ini ialize_pa icles(n=a gs.n_pa icles, seed=42)
1983 p in ("Running N-body simula ion...")
1984 pa icles_ inal, ene gy_his o y = n_body_simula ion(
1985 pa icles,
1986 d =0.01,
1987 n_s eps=a gs.n_ imes eps,
1988 seed=42,
1989 use_en opic=a gs.use_en opic
1990 )
1991 # A e main g a i y many-body calcula ion comple ed, pe o m dimensional
e i ica ion
1992 check_ ini e(ene gy_his o y, "ene gy_his o y", "main pos -check")
1993 asse _uni (PhysicalQuan i y(ene gy_his o y[0], "J"), "J", "ene gy_his o y
pos -check")
136
1994 check_dim(DimT(ene gy_his o y[0], 2, 1, -2, 0, "J"), 2, 1, -2, 0, "
ene gy_his o y pos -check")
1995 # F iedmann in eg a ion
1996 p in (" nSol ing F iedmann equa ions...")
1997 a_ini = 1.0
1998 a_do _ini = H_HUBBLE_0
1999 = 0.0
2000 d = 1e15 # App oxima ely 1 Gy / 31.5576
2001 a_his o y = [a_ini ]
2002 _his o y = [ ]
2003 a_do _his o y = [a_do _ini ]
2004 a_do = a_do _ini
2005 o iin ange(100):
2006 a_new, a_do _new, _new = k4_s ep(a_his o y[-1], a_do , , d )
2007 a_his o y.append(a_new)
2008 _his o y.append( _new)
2009 a_do _his o y.append(a_do _new)
2010 a_do = a_do _new
2011 = _new
2012 a_his o y = jnp.a ay(a_his o y)
2013 a_do _his o y = jnp.a ay(a_do _his o y)
2014 _his o y = jnp.a ay( _his o y)
2015 # Lane-Emden o c i ical densi y
2016 p in (" nSol ing Lane-Emden equa ion...")
2017 xi, he a = lane_emden_sol e (n=3.0, xi_max=6.0, n_poin s=1000)
2018 D = 1.0 / ( he a + 1e-10) # Densi y con as
2019 # Compu e addi ional quan i ies o ep oducibili y
2020 p in (" nCompu ing cosmological quan i ies...")
2021 z_his o y = 1.0 / a_his o y - 1.0
2022 H_his o y = a_do _his o y / a_his o y
2023 R_h_his o y = C_LIGHT / H_his o y
2024 V_his o y = (4.0 / 3.0) * jnp.pi * R_h_his o y**3
2025 ho_m_his o y = RHO_CRITICAL * OMEGA_M_0 * (1 + z_his o y)**3
2026 ho_ _his o y = RHO_CRITICAL * OMEGA_R_0 * (1 + z_his o y)**4
2027 ho_lambda_his o y = RHO_LAMBDA * jnp.ones_like(z_his o y)
2028 M_m_his o y = ho_m_his o y * V_his o y
2029 T_ _his o y = T_CMB_0 * (1 + z_his o y)
2030 E_m_his o y = M_m_his o y * C_LIGHT**2
2031 E_ _his o y = A_RAD * DEG_FREEDOM * T_ _his o y**4 * V_his o y
2032 E_ o al_his o y = E_m_his o y + E_ _his o y
2033 S_m_his o y = en opy_ma e _BH(M_m_his o y)
2034 S_ _his o y = en opy_ adia ion(T_ _his o y, V_his o y, DEG_FREEDOM)
2035 S_ o al_his o y = S_m_his o y + S_ _his o y
2036 x_his o y = E_m_his o y / E_ o al_his o y
2037 y_his o y = x_his o y**2 / (1 - (1 - x_his o y)**(3/4)) # y(x) = x^2 / (1
- (1-x)^{3/4})
2038 sigma_his o y = jnp.di (S_ o al_his o y) / jnp.di ( _his o y) /
V_his o y[:-1] # en opy p oduc ion a e
2039 # Compu e densi y con as s scale ac o
137
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.
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.
144
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)
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)
145
-- 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]
•/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)
146
|-- 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
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
147
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
39 - Dual-dimensional e i ica ion sys em (PhysicalQuan i y + DimT)
40 - Comple e alida ion unc ions (check_ ini e, asse _uni , check_dim)
41 - 200+ dual_ e i y calls h oughou all compu a ional s ages
42 - SymPy-equi alen symbolic dimensional analysis comple ely in C
43 - Comp ehensi e he modynamic unc ions (14+ co e unc ions wi h a ian s)
44 - Uni ied T_s(l), F = T_s(l) (dS/dx), limi s, Planck o ce, C_V, s = 4 P / T
45 - GPU-accele a ed di ec N-body o ce compu a ion using OpenCL (O(N^2)
pa allelized on GPU)
46 - Leap og symplec ic in eg a ion wi h Hubble ic ion and adap i e s epping
47 - Comple e RK4 F iedmann cosmology in eg a ion wi h e o analysis
48 - Ad anced Box-Mulle quan um luc ua ion gene a ion
49 - Comp ehensi e Mon e Ca lo s a is ical ensemble amewo k
50 - OpenMP pa alleliza ion wi h sophis ica ed independen seed managemen o
ials
51 - C oss-pla o m memo y managemen and e o handling
52 - Comp ehensi e a ay bounds checking wi h de ailed asse ions
53 - Dynamic memo y alloca ion wi h igo ous NULL checking
54 - Tole ance < 1e-15 main ained h oughou all ope a ions
55 - 40+ physical quan i ies in comp ehensi e ou pu
56 - Comple e ene gy condi ion e i ica ion (NEC/WEC/SEC/DEC analysis)
57 - De ailed egion classi ica ion wi h s a is ics
58 - Radial p o ile compu a ion and in eg a ion
59 - Scaling ela ion e i ica ion
60 - P essu e equilib ium diagnos ics
61 - Cosmological pa ame e e olu ion acking
62 - Da a logging and diagnos ic ou pu
63 - Pe o mance p o iling and memo y acking
64 GPU INTEGRATION:
65 - OpenCL ke nel o di ec N-body o ce compu a ion on GPU (NVIDIA/AMD/In el
compa ible)
66 - Bu e s o posi ions, masses, accele a ions (3D ec o s)
67 - Handles up o N=1e6 p ac ically; o N=1e7, equi es high-end GPU (e.g., RTX
4090)
68 - Main ains all physical calcula ions exac ly as o iginal (no app oxima ions
beyond di ec sum)
69 EXTENDED COMPILATION OPTIONS:
148
70 Windows: gcc -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=
c11 -lOpenCL holog aphic_sim.c -o sim.exe
71 Linux: gcc -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=c11
-lOpenCL holog aphic_sim.c -o sim
72 macOS: clang -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=
c11 - amewo k OpenCL holog aphic_sim.c -o sim
73 Wi h sani ize s:
74 gcc -O1 -g - sani ize=add ess - openmp -lm -s d=c11 -lOpenCL
holog aphic_sim.c -o sim_debug
75 gcc -O1 -g - sani ize=unde ined - openmp -lm -s d=c11 -lOpenCL
holog aphic_sim.c -o sim_debug
76 DETAILED EXECUTION:
77 ./sim [op ions]
78 Op ions:
79 --pa icles N Se numbe o pa icles (de aul : 10000000, GPU-limi ed o
1000000 ecommended)
80 -- imes eps N Se numbe o imes eps (de aul : 10000)
81 -- ials N Se numbe o MC ials (de aul : 10000)
82 -- he a X Se Ba nes-Hu angle (de aul : 0.5, unused in GPU di ec mode)
83 -- e bose Enable e bose ou pu
84 --p o ile Enable pe o mance p o iling
85 --check-mem Enable de ailed memo y checking
86 --gpu Enable GPU accele a ion (de aul : on i OpenCL a ailable)
87 DOCUMENTATION:
88 All code is in English using ASCII cha ac e s only.
89 E e y unc ion includes de ailed physics documen a ion.
90 CODATA 2018 cons an s wi h ull 15-digi p ecision main ained.
91 Tole ance < 1e-15 o all dimensional e i ica ions.
92 All ma hema ical ope a ions checked o nume ical s abili y.
93 PAPER REFERENCES:
94 All equa ions implemen ed om:
95 - Un uh (1976), Ve linde (2010), Jacobson (1995), Ho a a (2012)
96 - Includes comple e p essu e equilib ium amewo k
97 - Bekens ein-Hawking en opy o singula i y a oidance
98 - Hawking, Un uh, Hubble empe a u e o mula ions
99 - Holog aphic p inciple applica ions
100 - Scaling ela ions: y(x) = x^2 / (1 - (1-x)^(3/4))
101 - Ene gy condi ions: NEC, WEC, SEC, DEC
102
103 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.
104 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.
105
106 ================================================================================
107 #de ine CL_TARGET_OPENCL_VERSION 300
149
108 #include <CL/cl.h>
109 #include <s dio.h>
110 #include <s ing.h>
111 #include <s dlib.h>
112 #include <ma h.h>
113 #include <asse .h>
114 #include < loa .h>
115 #include <limi s.h>
116 #include <s din .h>
117 /* Pla o m de ec ion and OpenMP suppo */
118 #i de _OPENMP
119 #include <omp.h>
120 #else
121 #de ine omp_ge _ h ead_num() 0
122 #de ine omp_ge _max_ h eads() 1
123 #de ine omp_ge _ h ead_limi () 1
124 #endi
125 /* Pla o m-speci ic heade s */
126 #i de _WIN32
127 #include <windows.h>
128 #include <psapi.h>
129 #else
130 #include <sys/ esou ce.h>
131 #include <unis d.h>
132 #include <sys/ ypes.h>
133 #include <sys/u sname.h>
134 #endi
135 /* Pla o m name de ini ion */
136 #i de ined(_WIN32)
137 #de ine PLATFORM_NAME "Windows x64"
138 #eli de ined(__APPLE__)
139 #de ine PLATFORM_NAME "macOS"
140 #eli de ined(__linux__)
141 #de ine PLATFORM_NAME "Linux x64"
142 #else
143 #de ine PLATFORM_NAME "Unknown"
144 #endi
145 /*
============================================================================
146 EXTENDED OPENCL ERROR HANDLING
147 ============================================================================
*/
148 oid ocl_check(cl_in e , cons cha * ope a ion, cons cha * ile, in line)
{
149 i (e != CL_SUCCESS) {
150 p in (s de , "OpenCL e o : %s ailed wi h code %d a %s:%d n",
ope a ion, e , ile, line);
151 exi (EXIT_FAILURE);
152 }
150
153 }
154 #de ine OCL_CHECK(e , op) ocl_check(e , #op, __FILE__, __LINE__)
155 /*
============================================================================
156 EMBEDDED OPENCL KERNEL SOURCE
157 ============================================================================
*/
158 cons cha * ke nel_sou ce =
159 "__ke nel oid compu e_ o ces( n"
160 "__global double *posi ions, n"
161 "__global double *masses, n"
162 "__global double *accele a ions, n"
163 "in N, n"
164 "in D, n"
165 "double G, n"
166 "double eps n"
167 ") { n"
168 " in idx = ge _global_id(0); n"
169 " i (idx >= N) e u n; n"
170 " double ax = 0.0, ay = 0.0, az = 0.0; n"
171 " o (in j = 0; j < N; j++) { n"
172 " i (idx != j) { n"
173 " double dx = posi ions[j*D + 0] - posi ions[idx*D + 0]; n"
174 " double dy = posi ions[j*D + 1] - posi ions[idx*D + 1]; n"
175 " double dz = (D > 2) ? posi ions[j*D + 2] - posi ions[idx*D + 2] : 0.0; n"
176 " double 2 = dx*dx + dy*dy + dz*dz + eps*eps; n"
177 " double = sq ( 2); n"
178 " i ( > 1e-10) { n"
179 " double coe = G * masses[j] / ( 2 * ); n"
180 " ax += coe * dx; n"
181 " ay += coe * dy; n"
182 " i (D > 2) az += coe * dz; n"
183 " } n"
184 " } n"
185 " } n"
186 " accele a ions[idx*D + 0] = ax; n"
187 " accele a ions[idx*D + 1] = ay; n"
188 " i (D > 2) accele a ions[idx*D + 2] = az; n"
189 "} n";
190 /* OpenCL ad an ages: NVIDIA + AMD + In el GPU, di ec summa ion O(N^2 / P)
scalabili y */
191 /*
============================================================================
192 UNIFIED SIMULATION PARAMETERS
193 ============================================================================
*/
194 /* Simula ion pa ame e s wi h ex ended op ions */
195 #de ine N_PARTICLES_DEFAULT 10000000 /* 10 million pa icles */
151
196 #de ine N_TIMESTEPS_DEFAULT 10000 /* In eg a ion imes eps */
197 #de ine N_TRIALS_DEFAULT 10000 /* Mon e Ca lo ials */
198 #de ine THETA_BH_DEFAULT 0.5 /* Ba nes-Hu opening angle */
199 #de ine SIGMA_SOFT_DEFAULT 0.01 /* So ening pa ame e o o ces */
200 #de ine DEG_FREEDOM_SM_DEFAULT 106.75 /* E ec i e deg ees o eedom in
s anda d model a high ene gies */
201 #de ine D_CRITICAL_DEFAULT 709.0 /* G a o he mal ca as ophe h eshold */
202 /* Ma hema ical cons an s wi h ex ended p ecision */
203 #de ine M_PI 3.141592653589793238462643383279502884197L
204 #de ine TWO_PI (2.0L * M_PI)
205 #de ine FOUR_PI (4.0L * M_PI)
206 #de ine ONE_THIRD (1.0L / 3.0L)
207 /* Tole ance speci ica ions */
208 #de ine TOLERANCE_DIM 1.0e-15 /* Dimensional e i ica ion ole ance */
209 #de ine TOLERANCE_PRESSURE 1.0e-10 /* P essu e equilib ium ole ance */
210 #de ine TOL_FINITE 1.0e-308 /* Minimum ini e alue h eshold */
211 /* Memo y and pe o mance cons an s */
212 #de ine MIN_PARTICLES 1 /* Minimum pa icle coun */
213 #de ine GIGAYEAR 3.15576e16L /* s (1 Gy ) */
214 #de ine SCALE_FACTOR_MIN 1e-12L /* Minimum scale ac o */
215 /*
============================================================================
216 EXTENDED CODATA 2018/2019 PHYSICAL CONSTANTS (15-DIGIT PRECISION)
217 ============================================================================
*/
218 /* Fundamen al physical cons an s */
219 #de ine C_LIGHT 299792458.000000000000000L /* Speed o ligh in acuum [m/s]
*/
220 #de ine G_NEWTON 6.674300000000000e-11L /* New onian cons an o g a i a ion [
m^3 kg^-1 s^-2] */
221 #de ine H_PLANCK 6.626070150000000e-34L /* Planck cons an [J s] */
222 #de ine HBAR 1.0545718176461565e-34L /* Reduced Planck cons an [J s] */
223 #de ine K_BOLTZMANN 1.380649000000000e-23L /* Bol zmann cons an [J K^-1] */
224 #de ine SIGMA_SB 5.670374419000000e-8L /* S e an-Bol zmann cons an [W m^-2 K
^-4] */
225 #de ine A_RAD 7.56572314814815e-16L /* Radia ion cons an [J m^-3 K^-4] */
226 #de ine E_CHARGE 1.602176634000000e-19L /* Elemen a y cha ge [C] */
227 #de ine M_ELECTRON 9.109383701528000e-31L /* Elec on mass [kg] */
228 #de ine M_PROTON 1.672621923690950e-27L /* P o on mass [kg] */
229 #de ine M_NEUTRON 1.674927498042030e-27L /* Neu on mass [kg] */
230 #de ine ALPHA_FINE 7.297352569300000e-3L /* Fine-s uc u e cons an */
231 #de ine N_AVOGADRO 6.022140760000000e23L /* A ogad o cons an [mol^-1] */
232 #de ine R_GAS 8.314462618153240L /* Gas cons an [J mol^-1 K^-1] */
233 #de ine EPSILON_0 8.854187812800000e-12L /* Elec ic cons an [F m^-1] */
234 #de ine MU_0 1.256637062120000e-6L /* Magne ic cons an [H m^-1] */
235 #de ine G_STANDARD 9.806650000000000L /* S anda d accele a ion o g a i y [m s
^-2] */
236 /* Planck uni s de i ed om undamen als */
237 #de ine T_PLANCK_TIME 5.391245000000000e-44L /* Planck ime [s] */
152
238 #de ine L_PLANCK 1.616255000000000e-35L /* Planck leng h [m] */
239 #de ine M_PLANCK 2.176434000000000e-8L /* Planck mass [kg] */
240 #de ine T_PLANCK_TEMP 1.416784000000000e32L /* Planck empe a u e [K] */
241 #de ine E_PLANCK 1.956092000000000e9L /* Planck ene gy [J] */
242 /*
============================================================================
243 EXTENDED PLANCK 2018 COSMOLOGICAL PARAMETERS
244 ============================================================================
*/
245 /* Hubble pa ame e and de i ed quan i ies */
246 #de ine H_HUBBLE_0 2.185000000000000e-18L /* Hubble pa ame e [s^-1] */
247 #de ine OMEGA_R_0 8.400000000000000e-5L /* Radia ion ac o Omega_ ,0 (uppe
bound) */
248 #de ine OMEGA_M_0 0.315000000000000L /* Ma e ac o Omega_m,0 */
249 #de ine OMEGA_B_0 0.049000000000000L /* Ba yon ac ion Omega_b */
250 #de ine OMEGA_LAMBDA_0 0.684000000000000L /* Cosmological cons an
Omega_Lambda,0 */
251 #de ine OMEGA_K_0 0.000000000000000L /* Cu a u e Omega_k,0 */
252 #de ine OMEGA_DM_0 (OMEGA_M_0 - OMEGA_B_0) /* Da k ma e Omega_DM = Omega_m -
Omega_b */
253 /* De i ed cosmological quan i ies */
254 #de ine RHO_CRITICAL (3.0L * H_HUBBLE_0 * H_HUBBLE_0 / (8.0L * M_PI * G_NEWTON
)) /* C i ical densi y [kg/m^3] */
255 #de ine RHO_LAMBDA (OMEGA_LAMBDA_0 * RHO_CRITICAL) /* Da k ene gy densi y [kg/
m^3] */
256 #de ine LAMBDA_COSMO (8.0L * M_PI * G_NEWTON * RHO_LAMBDA / pow(C_LIGHT, 2))
/* Cosmological cons an [m^-2] */
257 #de ine R_HUBBLE (C_LIGHT / H_HUBBLE_0) /* Hubble adius [m] */
258 #de ine M_HUBBLE (pow(C_LIGHT, 3) / (G_NEWTON * H_HUBBLE_0)) /* Hubble mass [
kg] */
259 #de ine T_HUBBLE (HBAR * H_HUBBLE_0 / (2.0L * M_PI * K_BOLTZMANN)) /* Hubble
empe a u e [K] */
260 #de ine T_UNIVERSE_AGE 4.360000000000000e17L /* Age o uni e se [s] (13.8 Gy )
*/
261 #de ine Z_EQUALITY (OMEGA_M_0 / OMEGA_R_0 - 1.0L) /* Redshi a ma e -
adia ion equali y */
262 #de ine T_CMB_0 2.725500000000000L /* CMB empe a u e [K] */
263 /*
============================================================================
264 TYPE DEFINITIONS AND STRUCTURES
265 ============================================================================
*/
266 /* Pa icle s uc u e */
267 ypede s uc {
268 double posi ion[3]; /* Posi ion [m] */
269 double eloci y[3]; /* Veloci y [m/s] */
270 double mass; /* Mass [kg] */
271 double empe a u e; /* Tempe a u e [K] */
153
