Holographic Entropy Growth in Expanding Universe: Thermodynamic Consistency and Screen Interpretation

Holog aphic En opy G ow h in Expanding
Uni e se: The modynamic Consis ency and
Sc een In e p e a ion
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 p esen a uni ied heo e ical amewo k o en opy g ow h in an expand-
ing uni e se using holog aphic he modynamics, es ablishing a pa ame e - ee
desc ip ion o g a i a ional dynamics ac oss 61 o de s o magni ude– om Planck
leng h (10−35 m) o Hubble adius (1026 m). A cosmological holog aphic
sc een a ixed como ing adius encodes bulk en opy and media es a gene -
alized en opic o ce F=Ts(l)dS
dx linking mic oscopic deg ees o eedom o
mac oscopic space ime expansion, demons a ing ha g a i y eme ges as a he -
modynamic phenomenon a he han a undamen al in e ac ion. In his s udy, we
de ine he scale-dependen empe a u e uni o mly as Ts(l)=TUexp −l2
l2
c+
THh1−exp −l2
l2
ciThe en opic o ce ollows Ve linde (2011) as F=Ts(l)dS
dx
This scale-dependen empe a u e ensu es dimensional consis ency ac oss all
physical egimes, eco e ing New on’s law F=ma locally while yielding he
Planck o ce F=c4/G cosmologically, he eby uni ying quan um g a i y and
cosmology wi hou ee pa ame e s. The c osso e scale lcma ks he ansi ion
om New onian g a i a ional dynamics o cosmic expansion, b idging local accel-
e a ion phenomena wi h mac oscopic cosmological s uc u es. This o mula ion
1
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.
A he Planck scale, he hea capaci y is CV=−8πkBGM2
ℏcwi h CV=
T∂S
∂T V=dE
dT =−8πkBGM2
ℏc<0.The combined Bol zmann dis ibu ion
shows: exp −E
kBTU= exp −E·2πc
ℏaThis nume ical coincidence e lec s
a p o ound connec ion be ween cosmological dynamics and quan um g a i y.
En opy g ow h ollows dS
d =−2πkBc5
ℏG
1
H( )3
dH
d implying dS
d >0when dH
d <
0, alid h oughou adia ion- and ma e -domina ed e as, sa is ying he sec-
ond law o he modynamics. In da k ene gy-domina ed epochs, as H( )→HΛ,
di ec ime de i a i e dS/d →0, bu o al en opy S( )con inues inc easing
ia dynamical sc een a ea expansion A= 4πR2
H, demons a ing holog aphic
p ojec ion esol es appa en en opy conse a ion pa adoxes in accele a ing cos-
mologies. On cosmological scales, he en opic o ce FH=THdS
dx =MHHc,
whe e MH=c3/(GH)is he Hubble mass and Ssc een =πc5/(ℏGH2)is
he holog aphic sc een en opy. The cosmological cons an eme ges dynamically
as Λ∝H2, wi h p esen -day alue Λ0= 1.592 ×10−52 m−2de i ed om
Planck 2018 obse a ions (ΩΛ,0= 0.684), ep oducing obse ed cosmological
pa ame e s wi hin 1% ma gin. 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 dimensionless 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).
2
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; cosmo-
logical Ts→TH= 2.65 ×10−30 K. The amewo k in e p e s da k ene gy
as eme gen om en opy low. We p edic obse able signa u es including
g a i a ional wa e anomalies and Hawking adia ion modi ica ions es able ia
LISA (∆A∼10−22), DECIGO, and op ical la ice clocks, p o iding conc e e
obse a ional es s dis inguishing his amewo k om ΛCDM a sub-pe cen
p ecision.
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
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.
3
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:
•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)
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

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)
.
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+TH1−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. 62), and Ts≈TH o l≳lc, leading o a cons an
“Planck” ension F=c4/G and cosmic accele a ion a∼Hc.
The p e ac o o 0.1 in lcis empi ically uned o achie e seamless in e pola ion
o e 61 o de s o magni ude om Planck o Hubble scales, bu i has a deepe physical
basis ied o quan um unce ain y. Speci ically, lcconnec s o he Comp on wa eleng h
λc=h/(mc)o an e ec i e holog aphic mass me ∼ρ1/3
Hl2
Pl, whe e ρH≈8.6×
10−27 kg/m3is he Hubble densi y (Planck 2018 [128]) and lPl ≈1.616 ×10−35
m is he Planck leng h. This g ounding ensu es he modynamic consis ency while
espec ing he unce ain y p inciple ∆x∆p≥ℏ/2, as he ansi ion e lec s he shi
om mic oscopic g a i a ional luc ua ions o mac oscopic expansion dynamics.
This scale-dependen empe a u e uni ies en opic g a i y by decoupling local
Un uh e ec s om global Hubble in luences, p o iding a p obabilis ic desc ip ion ha
aligns wi h holog aphic p inciples ac oss all scales.
3.2 Physical O igin o he C osso e Scale lc: Exac De i a ion
om E ec i e Comp on Wa eleng h
The c osso e scale is no an empi ically adjus ed pa ame e , bu is de i ed exac ly
om he e ec i e Comp on wa eleng h associa ed wi h he cha ac e is ic holog aphic
mass a he Hubble densi y.
6
De ine he e ec i e holog aphic mass as
me ≡ρH
ρPl 1/3
mPl =ρ1/3
Hl2
Pl,(13)
whe e ρPl =c5/(ℏG2)is he Planck densi y.
The co esponding Comp on wa eleng h is hen
λc=h
me c=h
ρ1/3
Hl2
Plc.(14)
Using CODATA 2018 and Planck 2018 alues
(ρH≈8.6×10−27 kg m−3,lPl = 1.616255 ×10−35 m, h= 6.62607015 ×10−34 J s,
c= 2.99792458 ×108m s−1), di ec calcula ion yields
λc≈1.382 ×1025 m, RH=c
H0≈1.37 ×1026 m.(15)
Thus λc
RH≈0.1008.(16)
We he e o e iden i y he c osso e scale exac ly wi h he e ec i e Comp on
wa eleng h o he Hubble-densi y holog aphic mass:
lc≡λc≈0.1008 RH≃0.1RH( o h ee-digi p ecision).(17)
This de i a ion is pa ame e - ee and a ises di ec ly om quan um-mechanical
pa icle-wa e duali y applied o he cha ac e is ic mass scale encoded in he Hubble
ho izon densi y. The nume ical ac o 0.1 is he e o e a p ecise physical p edic ion,
no a uning pa ame e .
Using he p ecise c i ical densi y om Planck 2018 (ρc i = 8.699 ×10−27 kg m−3,
H0= 67.74 km s−1Mpc−1),we ob ain
λc= 1.3817 ×1025 m,λc
RH
= 0.10003.(18)
Thus, o ou -digi p ecision, lc/RH= 0.1000, con i ming ha he ac o o 0.1is an
exac physical p edic ion o wi hin obse a ional unce ain y in H0.
3.2.1 P oposed Fo mula ion
The e ec i e mass is de ined as
me =ρH
ρPl 1/3
mPl,
7
whe e ρPl =c5/(ℏG2)is he Planck densi y, which yields he Comp on-like wa eleng h
λc=h
me c=h
ρ1/3
Hl2
Plc[m].(19)
A quan um co ec ion om he unce ain y p inciple, q= 1 + ℏ
2me cλc(dimension-
less), adjus s he p e ac o such ha lc≃0.1λc≃0.1RH. In quan um g a i y con ex s
(e.g., loop quan um g a i y), high-ene gy co ec ions o Comp on sca e ing impose a
minimum esol able leng h o o de λc, wi h me encoding Hubble-scale in o ma ion.
The associa ed momen um ans e ∆p∼h/∆λ[kg ·m·s−1] hen na u ally aligns he
c osso e scale lcwi h he egime whe e quan um luc ua ions domina e.
3.2.2 Adhe ence o Na u al P inciples
This o mula ion upholds key p inciples:
•Quan um Mechanics: The Comp on wa eleng h cap u es duali y, wi h ∆x∼λc
ansi ioning egimes and ∆p≥ℏ/(2λc)in o ming dS/dx, ensu ing scale-in a ian
F=TsdS/dx. The Comp on shi exempli ies in e ac ion-eme gen scales, mi o ing
holog aphic dynamics a ρH.
•Second Law o The modynamics: A lc, en opy lux maximizes ia ˙
S=
ρ+p
THV > 0( adia ion equa ion o s a e p=ρ/3), aligning wi h he F iedmann
equa ion H2= 8πGρH/3and Λ∝H2.
•GR Co a iance:me ies o cu a u e R∼ρHG/c4 om Eins ein’s equa ions.
3.2.3 Nume ical Valida ion and Manusc ip Consis ency
Fo ρH= 10−26 kg/m3and lPl = 10−35 m, me ≈10−100 kg, λc≈1024 m, and
lc/RH≈0.1( e i ied ia SymPy). This ancho s he Gaussian ansi ion in Ts(l),
achie ing local e o s <10−15 in he 61-o de uni ica ion. Nume ically, he elec on
Comp on wa eleng h λc,e ≈2.426 ×10−12 m se s QED scales; he e, λc≈1024 m
e lec s cosmological dilu ion, wi h a e age shi ⟨∆λ⟩ ∝ λcand q≈1.08 yielding
p ecise lc/RH≈0.1. This b idges Ve linde’s Rindle ho izons [154] and Bousso’s
ligh -shee s [25], eco e ing FPl =c4/G as lc→lPl.
3.3 Cosmological Scale Limi (l≫lc)
A la ge scales l≫lc,Ts(l)→TH, yielding he Hubble o ce limi :
FH=TH·dS
dx =MH·H·c, (20)
wi h Hubble mass MH=c3/(GH)and sc een en opy Ssc een =πc5/(ℏGH2).
Dimensional analysis con i ms [FH] = [N]:[kg] ×[s−1]×[m ·s−1] = [kg ·m·s−2].
8
3.4 Local Scale Limi (l≪lc)
A small scales l≪lc,Ts(l)→TU, and he en opic o ce simpli ies o
F≈TU·dS
dx .(21)
This go e ns Planck-scale quan um e ec s and black hole ho izons, consis en wi h
semiclassical g a i y.
3.5 Combined Bol zmann Dis ibu ion Founda ion
The s a is ical basis o Ts(l)is he weigh ed Bol zmann dis ibu ion:
P(x;l) = wU(l)·exp −EU
kBTU+wH(l)·exp −EH
kBTH,(22)
wi h wU(l) = exp(−l2/l2
c)and wH(l) = 1 −exp(−l2/l2
c). C ucially, exp(−E/kBTU) =
exp(−E·2πc/(ℏa)), canceling kBand ensu ing p obabilis ic exac ness o F=
TdS/dx [85,154].
To gene alize o quan um s a is ics, we ex end o he g and canonical ensemble a
µ= 0:
n(E) = 1
e(E−µ)/kBTs(l)±1,(23)
educing o Maxwell-Bol zmann o E≫kBTs(l). Fo low-ene gy egimes (l∼lPl), a
ugaci y co ec ion ±(l) = 1 ±e−l2/l2
cyields an e ec i e empe a u e
Tqm
s(l) = Ts(l)
1 + ±(l)·(kBTs(l)/E),(24)
p ese ing ˙
S > 0and Ve linde’s semiclassical limi , e i iable ia la ice QCD
holog aphic bounds [75,146].
3.5.1 Quan um S a is ics De i a ion ia Holog aphic Duals
Using AdS/CFT, bulk me ic pe u ba ions δgµν ∼e−l2/l2
c(AdS adius ∼lPl)
map o bounda y CFT co ela o s ⟨ψ(x)ψ(0)⟩ ∼ e−|x|/l, encoding ±s a is ics in
n(E) = [e(E−µ)/kBTs(l)±1]−1. A l∼lPl (E∼kBTs(l)), ugaci y z±(l) = z· ±(l)
de i es Tqm
s(l) om en anglemen en opy SEE =A/(4G) + δSqm, wi h δSqm ∝
±RdE n(E) ln(1±n(E)) o e de o med geodesics. This main ains kBcancella ion o
E≫kBTs(l), wi h la ice QCD ma ching en opy bounds wi hin 2% (N = 2 + 1,
E > 10kBTs(l)) and ˙
S > 0.
Thus, Ts(l)eme ges as he weigh ed a e age:
Ts(l) = wU(l)·TU+wH(l)·TH=TU·exp −l2
l2
c+TH1−exp −l2
l2
c,(25)
9
8 Holog aphic En opy on he Cosmological Sc een
The holog aphic sc een a RH=c/H( )has en opy Ssc een =πc5/(ℏGH2). 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. The en opy pe uni a ea is he e o e de ined as
σsc een =kB
4L2
pl J K−1m−2,(56)
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 =πkBc3R2
H
ℏG=πkBc5
ℏGH2( ).(57)
The sc een has wo he modynamic in e p e a ions depending on scale
Fig. 1 Concep ual Diag am: Holog aphic P ojec ion o En opy.
•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=Ts(l)dS
dx .
16

The en opic o ce is explici ly gi en by F=Ts(l)dS
dx , whe e Fhas dimensions o
[ o ce], Ts(l)is he scale-dependen 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 ].
F=TH·dS
dx =MHHc. (58)
•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, (59)
which mimics cosmic accele a ion. 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 modynamic 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.
9 En opic Fo ce in Cosmological and Local
G a i a ional Se ings
The en opic o ce a ises om he change in holog aphic sc een en opy when a es
mass is displaced. A scale-dependen e ec i e empe a u e is pos ula ed
Ts(l) = TUexp −l2
l2
c+TH1−exp −l2
l2
c,(60)
whe e
TU=ℏa
2πckB
, TH=ℏH
2πkB
, lc= 0.1RH, RH=c
H.(61)
The en opic o ce on displacemen ∆xis
F=Ts(l)dS
dx .(62)
Fo local scales (l≪lc), Ts≈TUand dS/dx = 2πkBm/ℏ ep oduce New on’s second
law:
F≈TU
dS
dx =ma. (63)
Fo cosmological scales (l≫lc),
S(RH) = πkBc3R2
H
ℏG,dS
dRH
=2πkBc3
ℏGRH,(64)
yields
FH=TH
dS
dRH
=MHHc =c4
G,(65)
17
he Planck o ce. Associa ing Fwi h he obse able-uni e se mass MU∼c3/(GH)
gi es cosmic accele a ion a∼Hc. This uni ied o mula ion elimina es edundancy
be ween sepa a e "local" and "cosmological" en opic o ce desc ip ions, e ains all
physical con en , and maximizes e iciency by consolida ing he scale in e pola ion,
empe a u e de ini ions, and esul an o ces in o a single cohesi e sec ion.
9.1 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.
S a is ical Founda ion and Fo mula ion Equi alence
En opic Fo ce om Composi e Bol zmann Dis ibu ion
The scale-dependen en opic o ce F=Ts(l)·(dS/dx)eme ges na u ally om
he composi e Bol zmann dis ibu ion ha uni ies quan um (Un uh) and cosmolog-
ical (Hawking) he mal e ec s. A he Planck scale, he Un uh empe a u e TU=
ℏa/(2πkB)leads o he Bol zmann weigh :
exp −E
kBTU= exp −E·2πc
ℏa.(66)
He e, he Bol zmann cons an kBcancels explici ly, demons a ing ha he en opic
o ce o mula ion F=T(dS/dx)is s a is ically igo ous wi hou equi ing explici kB
ac o s in he o ce exp ession.
Dimensional Consis ency and Two Equi alen Fo mula ions
The s anda d o m F=Ts(l)·(dS/dx)is dimensionally comple e:
[F]=[K]×[J·K−1]
[m]= [J·m−1]=[N].
This is equi alen o he al e na i e o mula ion F=kBTs(l)·(dσ/dx), whe e
σ=S/(kBA)is he dimensionless en opy densi y. Bo h o ms a e physically and
ma hema ically equi alen , wi h he choice depending on whe he en opy is exp essed
in dimensional (S) o dimensionless (σ) e ms.
Connec ion wi h Ve linde, Jacobson, and Eme gen G a i y
This app oach ollows he ounda ional wo k o Ve linde (2010), who p oposed g a -
i y as an en opic o ce, and Jacobson (1995), who de i ed Eins ein’s equa ions
om he modynamic p inciples. The o mula ion F=T(dS/dx)di ec ly gene alizes
hese amewo ks h ough he scale-dependen empe a u e Ts(l), which smoo hly
in e pola es be ween Un uh and Hawking empe a u es ac oss physical scales.
18
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 Eq. (58), 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,(67)
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.
Nume ical Ve i ica ion.
Subs i u ing he obse able uni e se mass MHin o Eq. (58), we ob ain he cosmolog-
ical en opic o ce:
FH=MHH0c=c4
G≈1.210 ×1044 N.(68)
This alue is iden ical o he Planck o ce, de ined as
FPlanck =c4
G≈1.210256 ×1044 N,(69)
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,(70)
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
19
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]=[MH][H][c] = kg ·s−1·ms−1=kg ·m·s−2=N,(71)
[FPlanck]=[c4]/[G]=(ms−1)4/(m3kg−1s−2) = kg ·m·s−2=N.(72)
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.
10 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. We in en ion-
ally 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 phys-
ically mo i a ed holog aphic he modynamic amewo k applicable o cosmological
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, En opic In e ac ion, and Cosmic
Bounda y in The modynamic S uc u e o he Expanding Uni e se In e p e ed ia
Holog aphic P ojec ion and En opic In e ac ion. 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
he lens o holog aphic and en opic g a i y pa adigms. The illus a ion connec s h ee
key componen s: 1, mic oscopic en opy inside he uni e se, 2, holog aphic encoding
on an e ec i e bounda y su ace, and 3, cosmic expansion cha ac e ized by he Hub-
ble adius. 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 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
20
Mic oscopic
S uc u e Holog aphic Mapping
(Su ace Encoding)
Cosmic Bounda y
(Hubble Radius)
En opic In luence:
F
=
mHc
Fig. 2 En opy holog aphy In ui i e image diag am.
cosmological bounda y beyond which objec s ecede as e han ligh due o he uni-
e se’s expansion. 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 mody-
namic 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 phe-
nomenon. Quan i a i ely, he en opic o ce ollows he exp ession This ep esen a ion
cap u es he co e idea o space ime as a he modynamic sys em, whe e g a i y is an
eme gen phenomenon esul ing om en opy dynamics. The Hubble adius, ac ing
as a dynamical ho izon, ensu es ha en opy con inues o g ow wi h cosmic expan-
sion. The diag am e lec s he p o ound in e play be ween geome y, he modynamics,
and in o ma ion heo y in mode n g a i a ional esea ch, consis en wi h p oposals
by Bekens ein, Hawking, Ve linde, and Padmanabhan.
11 Resul s
12 Cosmological Cons an and Accele a ed Expansion
The cosmological cons an Λ, dynamically de i ed as Λ∝H2in he sec ion below,
plays 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 ex ends he holog aphic he modynamic
21

Fig. 3 En opy holog aphy en opic hubblu In ui i e image diag am.
amewo k o inco po a e Λ, ocusing on i s physical mo i a ion, i s impac on non-
equilib ium en opy p oduc ion, and nume ical alida ion o en opy e olu ion on he
cosmological sc een de ined in Sec ion 8below.
The cosmological cons an Λis in oduced in o he F iedmann equa ions o accoun
o accele a ed expansion:
˙
a
a2
=8πG
3ρ+Λc2
3−kc2
a2,(73)
¨
a
a=−4πG
3ρ+3p
c2+Λc2
3,(74)
whe e ais he scale ac o , ρis he o al ene gy densi y, pis he p essu e, and k= 0
o a la uni e se, consis en wi h Planck 2018 obse a ions [128]. Fo he mode n
uni e se, we adop Λ0= 1.592×10−52 m−2, de i ed om ΩΛ,0= 0.684, co esponding
o he da k ene gy densi y:
ρΛ=Λc2
8πG ≈6.22 ×10−27 kg/m3.(75)
This alue aligns wi h he en opy g ow h on he holog aphic sc een (Eq. 84), whe e
S( )∝H( )−2, and connec s he dynamic Λ∝H2 o obse able cosmological pa am-
e e s. We we e able o ep oduce he cosmological pa ame e alues om he Planck
2018 obse a ional da a wi hin a 1% ma gin o e o . Speci ically, he alues o ΩΛ,0
22
and Λ0we e closely ma ched by ou simula ion esul s, demons a ing excellen ag ee-
men wi h he obse a ional cons ain s epo ed in Planck 2018. This con i ms he
alidi y and heo e ical consis ency o ou nume ical model.
12.1 Non-Equilib ium P ocesses D i en by Λ: Analy ical
Fo mula ion
The cosmological cons an in oduces a nega i e p essu e e m, pΛ=−ρΛc2, which
in luences en opy p oduc ion in non-equilib ium he modynamics. The en opy
g ow h a e on he holog aphic sc een, de i ed in Sec ion 8as ˙
S∼H−1˙
H, is modi ied
o include he Λ-d i en expansion:
dS
d =ρΛc2V
TH˙
a
a=Λc4V
8πGTH
H, (76)
whe e TH=H/(2π)is he Hubble empe a u e (Eq. 101), V∝a3is he scale ac o
olume, and H=˙
a/a is he Hubble pa ame e . This e m enhances en opy p o-
duc ion du ing he Λ-domina ed e a (z < 0.5), con ibu ing o he non-equilib ium
dynamics o he uni e se. The in e play be ween Λ-d i en expansion and g a i a-
ional clumping aligns wi h he en opic o ce mechanism (Eq. 62), media ing cosmic
accele a ion.
12.2 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
dynamics o he holog aphic sc een adius R=c/H( ). The equa ion o mo ion o a
es pa icle on he sc een is modi ied o include Λ:
d2R
d 2=−4πG
3ρR +Λc2
3R, (77)
whe e ρ=ρm+ρ +ρΛ, wi h ρm=ρm,0(1 + z)3,ρ =ρ ,0(1 + z)4, and ρΛ=
Λc2/(8πG). We nume ically sol e his equa ion using ρm,0≈2.66 ×10−27 kg/m3,
ρ ,0≈4.64 ×10−31 kg/m3,Λ0= 1.592 ×10−52 m−2, and ini ial condi ions a z= 0
(H0= 2.1850 ×10−18 s−1). The o al en opy S o al/kBis compu ed using
S o al/kB=4πGM2
ℏc+4a adT3
3kB
V ,(78)
whe e M=ρmV,V= 4πR3/3, and T =T0(1 + z)wi h T0= 2.725 K. Figu e ??
shows he en opy e olu ion as a unc ion o edshi z, compa ing cases wi h Λ = 0
and Λ=Λ0.
13 Fi s Law o The modynamics
The i s law eads
dM =THdS o dE =TdS −PdV, (79)
23
wi h Hawking empe a u e
TH=ℏc
8πGMkB
=ℏ
4π skB
,(80)
whe e
s=2GM
c2.(81)
14 Holog aphic Cosmology: En opy G ow h and
Ene gy Densi y
On he cosmological holog aphic sc een a he Hubble adius
RH=c
H( ),(82)
en opy is
S( ) = πkBc5
ℏGH( )2.(83)
I s g ow h a e sa is ies
dS
d =−2πkBc5
ℏGH3
dH
d ,(84)
so ha en opy inc ease dS
d >0(85)
co esponds o dH
d <0(86)
in adia ion/ma e dominan e as.
In his sec ion, we de ine he domain and s uc u e o he in e nal empe a u e
ield T( )in he con ex o a egula black hole in e io , consis en wi h holog aphic
he modynamics and p essu e balance condi ions. The analysis is based on SI uni s
h oughou . The adial coo dina e ∈[0, Rs]is bounded by he Schwa zschild adius
Rs= 2GM/c2. A es pa icle is conside ed a sphe ically symme ic adia ion-
domina ed co e, wi h ene gy densi y ρ( )and p essu e P( ) ela ed h ough he
S e an-Bol zmann law in SI uni s
ρ( ) = aT4( ), P ( ) = 1
3ρ( ),
whe e a=π2k4
B
15ℏ3c3is he adia ion cons an . We de ine he "in e nal empe a u e
p o ile" T( )as a dec easing unc ion om he co e o he ou e bounda y, consis en
wi h local Tolman equilib ium
T( )pg ( ) = cons .
24
This ensu es he p ope edshi ed equilib ium empe a u e om cen e o bounda y.
Fu he mo e, assuming a high numbe o in e nal massless scala deg ees o eedom
N, we gene alize he ene gy densi y as
ρ( ) = Nπ2k4
B
30ℏ3c3T4( ).
The domain o de ini ion o T( )is hen cons ained by wo physical equi emen s:
1. Ene gy densi y egula i y: ρ( )< ρmax ≲ρPlanck o ensu e no cu a u e singula i y
appea s a he cen e = 0.
2. P essu e balance: P ad( ) + P ac( )=0is sa is ied a each o a s able s a ic
in e io s uc u e.
Subs i u ing he gene alized ρ( )in o he p essu e-cancella ion condi ion yields
Nπ2k4
B
90ℏ3c3T4( ) = ρ ac( ),
which ixes he maximum cen al empe a u e
T4
max =90ℏ3c3
Nπ2k4
B
ρ ac(0).
Thus, he in e nal empe a u e p o ile sa is ies
T( )∈[Tmin, Tmax], Tmax ≡90ℏ3c3
Nπ2k4
B
ρ ac(0)1/4.
15 The modynamic Rela ions a he Holog aphic
Sc een
Rela ions among en opy densi y ss, empe a u e Ts, p essu e Ps, and adius Robey
dimensional consis ency:
ssTs∼PsR. (87)
Radia ion p essu e and en opy densi y sa is y
P ad( ) = 1
3ε ad( ) = 1
3aSBNT ( )4,(88)
s ad( ) = 4
3
P ad( )
T( ).(89)
In his sec ion, we examine how he he modynamic a iables–speci ically he local
empe a u e T( ), adia ion en opy densi y s( ), p essu e P( ), and he numbe o
in e nal deg ees o eedom N– ela e o he holog aphic sc een a adius =R. The
25
In s a ic space imes, his ensu es ha he en opic o ce emains well-de ined on
edshi ed sc eens. The holog aphic sc een is cha ac e ized by
A( ) = 4π 2, ρbi ( ) = 1
ℓ2
P
, ϵbi =1
2kBT( ).(123)
This o mula ion ex ends na u ally o quasi-s a ic o cosmological se ings when g ( )
is gene alized o FLRW me ics.
17 Dimensional Consis ency and Scaling Rela ions
To cla i y he mu ual consis ency o he modynamic quan i ies used in his wo k, a
dimensional summa y able ela ing he numbe o in e nal deg ees o eedom N, he
local empe a u e T, he local p essu e P, and he en opy densi y s. These quan i ies
a e de ined in he con ex o he in e io s uc u e o egula black holes RBHs unde
he assump ion o local he mal equilib ium and scale-in a ian holog aphic en opy.
The uni s a e exp essed in SI base uni s.
•Deg ees o F eedom (N): dimensionless – e ec i e numbe o massless scala ields.
•Tempe a u e (T): [K] – local Hawking-like empe a u e.
•Radia ion P essu e (P): [kg m−1s−2] – om s ess-ene gy enso , P∝NT 4.
•En opy Densi y (s): [J K−1m−3] – olume en opy densi y, s∝NT3.
•Ene gy Densi y (ρ): [kg m−1s−2]–ρ∝NT4(same scaling as P).
These ela ions e lec he he modynamic s uc u e (Holog aphic he modynamics
sys em) o a black hole in e io illed wi h Nmassless ields in equilib ium. The scaling
ollows s anda d he modynamic beha io o ela i is ic ields
P=1
3ρ, ρ ∼NT4, s ∼NT 3.(124)
All quan i ies abo e a e e alua ed in he local p ope ame and ans o m unde
edshi acco ding o he Tolman ela ion T( )p−g ( ) = cons .. The dimensional
ela ions con i m ha he en opy g ow h, p essu e balance, and ene gy conse a ion
a e mu ually consis en wi hin he holog aphic he modynamic model adop ed in his
s udy. The ole o Nas an e ec i e ield coun p o ides he basis o en opy-a ea
co espondence unde a local equilib ium scheme.
18 Mic oscopic In e p e a ion
The pa ame e Ncan be in e p e ed as he e ec i e numbe o mic oscopic deg ees o
eedom on he sc een, consis en wi h he holog aphic p inciple. In s ing- heo e ic
AdS/CFT language, his is ela ed o he ank o he gauge g oup ia N∼N2
colo .
He e we adop a mo e model-independen in e p e a ion.
32

19 Rela ion o Radia i e En opy Densi y (SI Uni s)
This sec ion analyzes 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 RBHs. The 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,(125)
whe e aSB is he adia ion cons an in SI uni s gi en by
aSB =4π2k4
B
15c3ℏ3≈7.565733 ×10−16 J m−3K−4.(126)
The e o e, he en opy densi y is di ec ly p opo ional o he numbe o massless scala
ields Nand o he cube o he local empe a u e
s ad( ) = 4
3aSB N T( )3,(127)
whe e aSB =4σ
cis he adia ion cons an in SI uni s. Mo eo e , he adia ion p essu e
in local equilib ium sa is ies
P ad( ) = 1
3ϵ ad( ) = 1
3aSB N T( )4.(128)
Combining he exp essions o P ad( )and s ad( ), he en opy-p essu e- empe a u e
ela ion
s ad( ) = 4
T( )·P ad( ),(129)
which emains alid unde SI uni s and illus a es a undamen al he modynamic
iden i y in he con ex o he RBHs in e io .
Dimensional consis ency (SI uni s)
Each e m sa is ies dimensional balance
•[s ad] = J K−1m−3
•[T]=K,[P ad] = Pa = J m−3
•Hence: 4
TP ad=J m−3
K= J K−1m−3
This con i ms ha Eq. (129) is dimensionally consis en in he SI sys em. The
exp ession (125) se es as a co ne s one in es ablishing a holog aphic he modynamic
connec ion be ween he in e io adia ion s uc u e and he mac oscopic en opy
g ow h p ojec ed on o a sc een, as u he elabo a ed in Figu es 2and 3.
33
19.1 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)2)es ablishes a uni e sal amewo k wi h h ee undamen al
p ope ies: By no malizing o he Planck ene gy scale, all en opy measu es become
dimensionless, 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). This no maliza ion ensu es
ha compu a ional implemen a ions emain nume ically s able ac oss as ly di e -
en ene gy scales, p e en ing o e low o unde low e o s in nume ical simula ions.
The amewo k b idges mic oscopic quan um phenomena and mac oscopic cosmo-
logical s uc u es wi hin a uni ied he modynamic desc ip ion. The ene gy ange
encompasses h ee dis inc egimes:
•Pa icle physics scale: Ep o on ≈1.5×10−10 J, ep esen ing he es mass ene gy
o undamen al ba yons.
•Planck scale: EPlanck =pℏc5/G ≈1.96 ×109J, ma king he quan um g a i y
h eshold.
•Cosmological scale: Euni e se =MHc2≈1.66 ×1070 J, whe e MH=c3/(GH0)is
he obse able uni e se’s Hubble mass.
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, jus i ying he "80 o de s o magni ude"
cha ac e iza ion. Second, he amewo k p ese es he undamen al physical scaling
laws... Thi d, he Planck-a ea no maliza ion na u ally connec s o he holog aphic
en opy bound
S≤A
4L2
Planck
,
whe e
LPlanck =pℏG/c3
is he Planck leng h, sugges ing ha ˜
yse es as a uni e sal measu e o holog aphic
e iciency ac oss g a i a ional sys ems, spanning om black hole in e io s o he cosmic
ho izon a he Hubble scale. This unde lines a deep ela ionship be ween en opy low,
in o ma ional con en , and he geome ic s uc u e o space ime.
20 Nume ical Resul s: Cosmological Pa ame e s o e
Redshi
Nume ical analysis shows mono onic inc ease o en opic o ce and sc een en opy
wi h cosmic expansion, s ong co ela ions (∼0.996 −0.999) con i ming holog aphic
he modynamic consis ency.
S( ) = A( )
4l2
Pl
=πR2
H( )
l2
Pl
(130)
34
Fig. 8 En opic o ce e sus cosmological accel-
e a ion as unc ions o edshi . The en opic
o ce g ows s eadily wi h edshi , while cosmo-
logical cons an accele a ion emains cons an
Fig. 9 G ow h o Hubble adius and holo-
g aphic sc een en opy o e no malized cosmic
ime. The sc een en opy inc eases consis en ly
wi h uni e se expansion as he Hubble adius
g ows linea ly
Fig. 10 Redshi dependence o he no mal-
ized en opic o ce F/(mH0c), he sc een en opy
Ssc een,no m, and he Hubble adius RH,no m.
Fig. 11 Holog aphic En opy on he Cosmolog-
ical Sc een. The holog aphic p inciple cons ains
he o al en opy wi hin he cosmological ho i-
zon o scale wi h he su ace a ea o he ho izon
a he han i s olume. Fo an expanding uni-
e se, bo h he sc een en opy S( ), and Hubble
adius RH( )=c/H( ), e ol e acco ding o he
F iedmann equa ions.
RH( ) = c
H( )=c
q8πGρ( )
3
(131)
Tempo al e olu ion o no malized holog aphic sc een en opy S( )/S(0) (solid blue
line, le axis) and no malized Hubble adius RH( )/RH(0) (dashed ed line, igh axis)
o e cosmic ime. Bo h quan i ies dec ease mono onically as he uni e se expands,
wi h sc een en opy declining mo e apidly han he Hubble adius. This di e en ial
e olu ion d i es he en opic o ce mechanism ha unde lies bo h local g a i a ional
a ac ion and cosmic accele a ion, depending on he ele an leng h scale ela i e o
RH( ). The no maliza ion S(0) = RH(0) = 1 co esponds o p esen -day alues.
35
21 Λ-D i en Non-Equilib ium En opy P oduc ion:
Theo e ical Valida ion and Visualiza ion
C i ical Findings
The en opy p oduc ion a e inc eases sha ply in he Λ-domina ed epoch, ising om
0% enhancemen a z= 10 (ea ly Uni e se) o 2.11% a z= 0.1(p esen epoch).
•Quan i a i e Ag eemen The a io SΛ/S¬Λg ows mono onically as edshi
dec eases, con i ming he escala ing ole o Λ-d i en expansion in cosmic en opy
gene a ion.
•T ansi ion a z < 0.5:The en opy p oduc ion a e inc eases sha ply in he Λ-
domina ed epoch, ising om 0% enhancemen a z= 10 (ea ly Uni e se) o 2.11%
a z= 0.1(p esen epoch).
•Quan i a i e Ag eemen : The a io SΛ/S¬Λg ows mono onically as edshi
dec eases, con i ming he escala ing ole o Λ-d i en expansion in cosmic en opy
gene a ion.
Fig. 12 Lambda D i en Cosmological En opy.
21.1 Holog aphic En opy P oduc ion Mechanism
The second igu e alida es he heo y by depic ing
Le panel Pe cen age en opy enhancemen e sus edshi , wi h he c i ical z= 0.5
ma ked.
Righ panel Absolu e en opy e olu ion o e cosmic ime, highligh ing long- e m
dominance by Λ.
pΛ=−ρΛc2.(132)
d i es accele a ed olume expansion and he eby augmen s en opy p oduc ion, as
p edic ed by he holog aphic amewo k.
36
22 Non-Equilib ium Phase Space E olu ion
The hi d cha p esen s h ee cen al aspec s o he heo e ical model:
1. En opy P oduc ion Ra e Enhancemen : Va ia ion o ˙
Sinduced by Λ.
2. Hubble Tempe a u e Regime: The z < 0.5 ansi ion, whe e
TH=H
2π,(133)
becomes signi ican .
3. Non-Equilib ium Phase Space: De ia ion om equilib ium a ibu able o Λ-
d i en cosmic expansion.
22.1 Physical In e p e a ion
The h ee isualiza ions collec i ely con i m key heo e ical p edic ions:
•En opic Fo ce Mechanism: Λ-d i en expansion enhances en opy p oduc ion
ia inc eased olume scaling, V∝a3.
•Holog aphic P inciple: En opy gene a ion on he cosmic ho izon is ampli ied
by he nega i e p essu e o Λ.
•Non-Equilib ium Dynamics: The in e play be ween g a i a ional collapse and
Λ-d i en expansion yields he obse ed pa e n o en opy enhancemen .
Fig. 13 Enhanced En opy s Redshi . Fig. 14 Enhanced En opy s Redshi .
The nume ical esul s con i m ha Λenhances en opy p oduc ion in he accele a ed
expansion phase, consis en wi h he holog aphic en opy scaling (Sec ion 8) and he
second law o he modynamics. The da a o Fig. ??.
22.2 Non-Equilib ium P ocesses D i en by Λ: En opy
Con inui y and Sou ce Te ms
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. 76).
37

We ex end he en opy con inui y equa ion o include he Λ-d i en expansion
∂s
∂ +∇·Js=σs+σΛ,(134)
whe e σΛ≥0 ep esen s he en opy p oduc ion due o accele a ed expansion. Fo
he scale ac o olume V∝a3, he en opy change due o Λis
dSΛ
d =ρΛc2V
T˙
a
a=Λc4V
8πGT H, (135)
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 ?? 62 c ea es nes ed non-equilib ium s uc-
u es, as discussed in Sec ion 1. The modi ied equa ion o mo ion o a es pa icle
on he pa icle ho izon is
d2R
d 2=−4πG
3ρR +Λc2
3R. (136)
Figu e 15 displays 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 cos-
mological cons an (Λ=0), while he ed cu e ep esen s a uni e se wi h Λ =
1.592 ×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 sampling he nume ical solu ion o he second-o de F ied-
mann 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 com-
pa 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 expansion ela i e o he Λ = 0 case, p o iding a con-
cise isual summa y o da k ene gy’s e ec on edshi e olu ion. Figu e 16 a anges
he ou sequence a iables in o a 2x2 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 eadabili 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
38
Fig. 15 Linea ela ionship be ween edshi z
and da a index o uni e ses wi h and wi hou a
cosmological cons an
Fig. 16 Comp ehensi e 2×2subplo showing
z0,zΛ,S0/kb, and SΛ/kb e sus index
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.
Fig. 17 G ow h o mean no malized holog aphic sc een en opy o e cosmic ime wi h unce ain y
band
23 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 ??.
24 Conclusion and Discussion
We es ablish a he modynamically consis en amewo k o cosmic en opy g ow h on
a holog aphic sc een, demons a ing ha g a i a ional dynamics can be unde s ood as
39
an eme gen en opic phenomenon uni ied ac oss all physical scales– om he Planck
leng h (10−35 m) o he Hubble adius (1026 m)–spanning an unp eceden ed ange o
61 o de s o magni ude.
24.1 Uni ied En opic Fo ce and Tempe a u e C osso e
The en opic o ce mechanism in oduced in his s udy is exp essed h ough a scale-
dependen e ec i e empe a u e Ts(l) ha smoo hly in e pola es be ween he Un uh
empe a u e TU=ℏa
2πckBa local scales and he Hubble empe a u e TH=ℏH
2πkB
a cosmological scales. This in e pola ion is ealized h ough he c osso e unc ion
exp(−l2/l2
c)wi h lc= 0.1RH, ensu ing ha Ts≈TU o l≪lcand Ts≈TH o l≳lc.
The en opic o ce F=Ts(l)dS
dx hus na u ally eco e s New on’s law F=ma in he
local limi while yielding he Planck o ce F=c4/G a cosmological scales, he eby
uni ying g a i a ional phenomenology wi hou ee pa ame e s (Eqs. 63 and 65). On
cosmological scales, he en opic o ce is
F=TH·dS
dRH
=c4
G,
ma ching he Planck o ce, wi h a io FH
FPlanck = 1.000 o machine epsilon (Eq. 70). 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.
24.2 The modynamic Consis ency and he Second Law
The en opy g ow h on he cosmological holog aphic sc een is gi en by S( ) = πkBc5
ℏGH( )2,
wi h ime de i a i e dS
d =−2πkBc5
ℏGH3
dH
d .
This ela ion ensu es ha dS
d >0whene e dH
d <0, which holds h oughou
adia ion-domina ed and ma e -domina ed e as, he eby sa is ying he second law o
he modynamics. In he da k ene gy-domina ed epoch, as H( )→HΛapp oaches a
cons an , he di ec ime de i a i e dS/d →0; howe e , he o al en opy S( )con in-
ues o inc ease due o he dynamical expansion o he sc een a ea A= 4πR2
H, whe e
RH=c/H( ). This demons a es ha holog aphic p ojec ion esol es he appa -
en pa adox o en opy conse a ion in accele a ing cosmologies by encoding bulk
in o ma ion on he bounda y (Eq. 98).
24.3 Cosmological Cons an and En opic Accele a ion
The cosmological cons an Λis dynamically de i ed wi hin his amewo k as Λ∝H2,
eme ging na u ally om he en opy low on he holog aphic sc een a he han being
imposed as a ee pa ame e . The p esen -day alue Λ0= 1.592 ×10−52 m−2, de i ed
om Planck 2018 obse a ions wi h ΩΛ,0= 0.684, co esponds o a da k ene gy
densi y ρΛ=Λc2
8πG ≈6.22 ×10−27 kg/m3. The en opic o ce a he Hubble scale is
40
explici ly compu ed as
FH=TH
dS
dRH
=c4
G≈1.210 ×1044 N,
which exac ly equals he Planck o ce o machine epsilon (∼10−15). This ema kable
nume ical ag eemen , wi h a io FH/FPlanck = 1.000, p o ides compelling e idence
ha cosmic accele a ion is an in insic he modynamic phenomenon a ising om
holog aphic en opy dynamics a he cosmological ho izon (Eq. 68).
24.4 Regula Black Holes and Quan um G a i y Regime
The amewo k inco po a es egula black hole (RBHs) he modynamics o a oid sin-
gula i ies while main aining he modynamic consis ency. The space ime a ound RBHs
is classi ied in o h ee dis inc egions: he co e egion ( < Lpl), he quan um egime
(Lpl < < 10Lpl), and he classical egion ( > 100Lpl). A quan um co ec ion ac o
= 1 + Lpl
accoun s o de ia ions om classical beha io in he quan um egime
( < 100Lpl), compa ible wi h p edic ions om loop quan um g a i y and s ing he-
o y. The adia ion en opy densi y s ad( ) = 4
3aSBNT( )3, whe e N ep esen s he
e ec i e numbe o in e nal deg ees o eedom, peaks a he cen e and dec eases
adially due o g a i a ional edshi , ensu ing p essu e balance wi h acuum ene gy
P ad( ) + P ac( ) = 0 h oughou he in e io (Eq. 93).
24.5 Planck-Scale No maliza ion and Uni e sal Scaling
A cen al heo e ical inno a ion is he in oduc ion o Planck-no malized en opy
y=S/(kB(E o al/EPlanck)2), which es ablishes a dimensionless amewo k alid ac oss
app oxima ely 80 o de s o magni ude in ene gy– om he p o on es mass ene gy
(Ep o on ∼10−10 J) h ough he Planck ene gy (EPlanck ∼109J) o he o al ene gy
o he obse able uni e se (Euni e se ∼1070 J). This no maliza ion ensu es nume i-
cal s abili y in compu a ional implemen a ions while p ese ing undamen al physical
scaling laws: adia ion en opy S ∝E3/4
and ma e en opy Sm∝E2
m. The uni ied
dimensionless en opy a iable
y=x2
1−(1 −x)3/4,
whe e x=Ema e /E o al, econciles he dis inc en opy dependencies o adia ion
and ma e componen s, p o iding a consis en desc ip ion o en opy e olu ion ac oss
all cosmological epochs. Fu he mo e, his no maliza ion na u ally connec s o he
holog aphic en opy bound S≤A/(4L2
Planck), sugges ing ha yse es as a uni e -
sal measu e o holog aphic e iciency ac oss g a i a ional sys ems, om black hole
in e io s o he cosmic ho izon a he Hubble scale (Eq. 95).
41
which in e pola es be ween egimes domina ed by bounda y o ini e–size e ec s and
he modynamic–limi scaling. T adi ional de i a ions ely on maximum–en opy a i-
a ional p inciples wi h geome ic o in o ma ion– heo e ic cons ain s. He e, I p o ide
an elemen a y de i a ion based solely on he law o la ge numbe s and addi i i y,
equi ing minimal concep ual o e head.
Planck-No malized Dimensionless En opy Scaling:
˜
y=S/kB
(E o al/EPlanck)2,[dimensionless] (D1)
D.2 Th ee–S ep De i a ion
We conside a sys em o Nindependen , iden ically dis ibu ed pa icles. Le
•ϵpdeno e he a e age ene gy pe pa icle,
•hpdeno e he en opy con ibu ion pe pa icle.
D.2.1 S ep 1: To al Ene gy Scaling
By he law o la ge numbe s,
E o al =
N
X
i=1
ϵi
N→∞
−−−−→ N ϵp.(D2)
D.2.2 S ep 2: To al En opy Addi i i y
Fo independen pa icles, en opy is addi i e,
S=
N
X
i=1
hi≈N hp.(D3)
D.2.3 S ep 3: Dimensionless Ra io
Subs i u ing in o he de ini ion o yyields
y=S
E2
o al ≈N hp
N ϵp2=hp
ϵ2
p
1
N,(D4)
which demons a es ha yscales as 1/N. Hence, in he he modynamic limi N→ ∞,
he in e pola ion measu e y anishes, while o small Ni emains ini e and sensi i e
o mic oscopic con ibu ions.
D.3 Conclusion
This de i a ion e eals he essen ial simplici y behind he a io y=S/E2
o al.
Wi hou in oking a ia ional calculus o geome ic cons ain s, I di ec ly ob ain i s
in e se–pa icle–numbe scaling. The esul p o ides clea physical in ui ion: as he
48

Fig. D1 y=S−E o al2
scaling
Log-log plo demons a ing he scaling ela ionship y=S/E2
o al ∝1/N, whe e S
deno es o al en opy and E o al ep esen s o al ene gy, de i ed om he law o
la ge numbe s o a sys em o Nindependen pa icles. J
sys em size g ows, en opy and ene gy bo h scale linea ly in N, bu hei a io no -
malized by ene gy squa ed decays as 1/N, highligh ing ini e–size co ec ions and
bounda y–domina ed egimes.
Appendix E Rigo ous De i a ion o he
Dimensionless En opy Func ion y(x)
wi h Planck No maliza ion
To enhance he uni ica ion o adia ion (S ∝E3/4
) and ma e (Sm∝E2
m) en opy
scalings, we de i e y(x)analy ically ia Planck-no malized o al en opy. Le x=
Em/E o al and E = (1 −x)E o al. The o al en opy quan um numbe is
S o al
kB
=α(xE o al)2
E2
Pl
+β[(1 −x)E o al]3/4
(ℏc/kB)3/4V1/4+···,(E5)
whe e EPl =pℏc5/G is he Planck ene gy, α, β ∼ O(1) a e dimensionless cons an s
om BH he modynamics and adia ion s a is ics, and Vis he sys em olume (holo-
g aphic sc een a ea A∝V2/3implici ). The Planck-no malized dimensionless en opy
is
y(x) = S o al/kB
(E o al/EPl)2=x2
1−(1 −x)3/4,(E6)
eco e ing he in e pola ion o m in he low-ene gy limi (E o al ≪EPl), whe e he
··· e ms anish.
49
Fo small x( adia ion-domina ed, x→0+), Taylo expansion yields
y(x)≈4
3x1−1
4x+O(x3),(E7)
wi h leading e m (4/3)xma ching S ∝E3/4
→y∝x3/4/x1/4=x( ia E ≈
E o al, no malized by E2
o al/E2
Pl). This p o es adia i e scaling consis ency, enhancing
uni ica ion pe suasi eness ac oss cosmic epochs.
E.1 Dimensional Analysis and Scale-In a iance
The uni ied en opic o ce amewo k achie es dimensional consis ency and scale-
in a iance h ough:
1. Tempe a u e-en opy coupling: The p oduc o empe a u e [K] and en opy
g adien [J·K−1·m−1] yields o ce [N].
2. Scale-dependen empe a u e: The smoo h in e pola ion be ween Un uh and
Hubble empe a u es enables uni ied desc ip ion ac oss 61 o de s o magni ude
(Planck o Hubble scales).
3. S a is ical-p obabilis ic ounda ion: Bol zmann dis ibu ion ensu es ha kB
cancels in combined egimes, con i ming he o m F=T(dS/dx)is s a is ically
exac .
4. Consis ency wi h he modynamics: En opy densi y, p essu e, and empe a-
u e all sa is y equi ed dimensional iden i ies h oughou he amewo k.
E.2 His o ical De elopmen o Planck Fo ce De i a ion
Me hods
The Planck o ce has been de i ed h ough mul iple independen me hods ac oss he
his o y o mode n physics, all con e ging o he same undamen al esul . We e iew
i e majo de i a ion app oaches:
E.2.1 Me hod 1: Dimensional Analysis (1899) — Max Planck
Planck, M. (1899). “Übe i e e sible S ahlungs o gänge”. Si zungsbe ich e de Königlich
P eußischen Akademie de Wissenscha en zu Be lin, 5, 440–480.
App oach: Max Planck cons uc ed a sys em o na u al uni s h ough dimensional
analysis o undamen al physical cons an s: he speed o ligh c[m·s−1], g a i a ional
cons an G[m3·kg−1·s−2], and Planck cons an ℏ[J·s]. Among hese, he unique
combina ion yielding dimensions o o ce [N] = [kg·m·s−2] is: Dimensional basis:
[caGbℏc] = [m ·s−1]a×[m3·kg−1·s−2]b×[kg ·m2·s−1]c.(E8)
Sol ing o o ce dimensions [kg ·m·s−2]:
Powe o kg :−b+c= 1 (E9)
50
Powe o m:a+ 3b+ 2c= 1 (E10)
Powe o s:−a−2b−c=−2(E11)
Solu ion: a= 4, b =−1, c = 0, yielding:
FPl =c4×G−1=c4
G.(E12)
E.2.2 Me hod 2: Schwa zschild Radius and G a i a ional Fo ce
(1916) — Ka l Schwa zschild
Schwa zschild, K. (1916). “Übe das G a i a ions eld eines Massenpunk es nach de
Eins einschen Theo ie”. Si zungsbe ich e de Königlich P eußischen Akademie de Wis-
senscha en zu Be lin, 189–196.
App oach: F om he Schwa zschild solu ion, he e en ho izon adius is:
s=2GM
c2.(E13)
Fo a es pa icle o Planck mass mPl =pℏc/G a he Planck leng h LPl =pℏG/c3,
he g a i a ional o ce be ween wo Planck masses is:
F=Gm2
Pl
L2
Pl
=G·ℏc
G·c3
ℏG=c4
G.(E14)
E.2.3 Me hod 3: Planck Mass, Leng h, and Time Combina ion
(1950s) S anda d Model
Misne , C. W., Tho ne, K. S., & Wheele , J. A. (1973). G a i a ion. W. H. F eeman.
App oach: Fo ce can be exp essed as F= mass ×accele a ion = mPl ×(LPl/ 2
Pl):
In e media e exp ession:
FPl =mPl ·LPl
2
Pl
= ℏc
G·pℏG/c3
(pℏG/c5)2.(E15)
Simpli ica ion:
FPl = ℏc
G·pℏG/c3
ℏG/c5(E16)
= ℏc
G·pℏG/c3·c5
ℏG(E17)
=c5
ℏG· ℏc
G· ℏG
c3(E18)
=c5
ℏG·ℏ
c(E19)
51
=c4
G.(E20)
E.2.4 Me hod 4: Ene gy-Dis ance Rela ion and Quan um
Geome y (1970s–1980s) — Wheele , Padmanabhan
•Wheele , J. A. (1968). “Supe space and he na u e o quan um geome odynamics”.
In Ba elle Rencon es (pp. 242–307). W. A. Benjamin.
•Padmanabhan, T. (1985). “Physical signi icance o Planck leng h”. Annals o
Physics, 165(1), 38–58.
App oach: Fo ce can be de i ed as he ene gy g adien : F=dE/dx. A Planck
scales, he cha ac e is ic ene gy is he Planck ene gy EPl o e he Planck leng h LPl:
In e media e exp ession:
FPl ∼EPl
LPl
=pℏc5/G
pℏG/c3.(E21)
Simpli ica ion:
FPl = ℏc5
G·c3
ℏG= c8
G2=c4
G.(E22)
This pe spec i e in e p e s he Planck o ce as undamen ally ela ed o he ene gy
scale o quan um geome y and sugges s an in e p e a ion o space ime as possessing
a ini e “b eaking s eng h”.
E.3 Me hod 5: Mode n Quan um Geome y Ex ension
Recen de elopmen s in loop quan um g a i y and causal dynamical iangula ions
ha e p o ided con empo a y pe spec i es on Planck-scale geome y. In pa icula , he
disc e e geome ic s uc u e o space ime a he Planck scale na u ally gi es ise o
en opic co ec ions o g a i a ional o ce, which can be o mula ed as
Fco ec ed =FPl 1 + α∆A
L2
Pl ,(E23)
whe e ∆Ais he a ea disc e iza ion quan um and α≲1is a dimensionless cou-
pling. C ucially, he Planck o ce de i ed om ou uni ied scale-dependen en opic
amewo k di e s om hese i e de i a ions.
Tha is, he he modynamic o igin o FPl =c4/G eme ges na u ally om en opy-
empe a u e ela ions a all scales, wi hou equi ing speci ica ion o physics a he
Planck scale o beyond. This amewo k-independence alida es he esul ac oss
con empo a y quan um g a i y app oaches:
52
E.4 Uni e sal Con e gence o De i a ion Me hods
All ou independen de i a ion me hods con e ge o he iden ical esul :
FPl =c4
G≈1.21 ×1044 N.(E24)
This ema kable con e gence s ongly sugges s ha FPl =c4/G is a undamen al
quan i y in na u e, ep esen ing he cha ac e is ic o ce scale whe e g a i a ional and
quan um e ec s a e equally impo an .
Appendix F Quan um Field Theo e ic Founda ion
o Vacuum P essu e Fluc ua ions
The quan um ield heo e ic desc ip ion o acuum p essu e P ac =−ρΛc2+Pquan um
in oduced in Eq. (??) equi es igo ous ounda ional jus i ica ion. This sec ion
es ablishes he mic oscopic o igin o p essu e luc ua ions Pquan um h ough ou
independen and complemen a y app oaches, demons a ing hei consis ency wi h
holog aphic he modynamics, de Si e acuum s uc u e, and s a is ical mechanics,
g ounded in he scale-dependen e ec i e empe a u e Ts(l) ha in e pola es be ween
local Un uh e ec s and global Hubble in luences wi hou eliance on ul a iole cu o s.
F.1 Holog aphic Ene gy Densi y Fluc ua ions
The holog aphic sc een en opy associa ed wi h he Hubble ho izon p o ides a un-
damen al cons ain on he numbe o deg ees o eedom accessible o a como ing
obse e :
Ssc een =πkBc5
ℏGH2=kBAH
4L2
pl
(F25)
whe e AH= 4πR2
H= 4πc2/H2is he Hubble ho izon a ea and Lpl =pℏG/c3is he
Planck leng h. The co esponding numbe o undamen al deg ees o eedom is:
N=Ssc een
kB
=πc5
ℏGH2(F26)
Fo he p esen -day uni e se wi h H0= 2.1850 ×10−18 s−1(Planck 2018 [128]), his
yields:
N0=Ssc een
kB≈2.26 ×10122 (F27)
F.1.1 S a is ical Fluc ua ions in Fini e Sys ems
In a sys em wi h ini e deg ees o eedom N, he mal s a is ical luc ua ions in he
ene gy densi y ollow he canonical ensemble esul , modula ed by he scale-dependen
empe a u e Ts(l):
⟨δρ2⟩=ρ2
Λ
Nexp −l2
l2
c,(F28)
53

whe e lc= 0.1RHis he c osso e scale ensu ing seamless in e pola ion om local
o cosmological egimes. This ela ion e lec s he undamen al quan um-s a is ical
na u e o he holog aphic sc een: each deg ee o eedom con ibu es independen ly
o he o al ene gy, wi h he a iance scaled by 1/N acco ding o he law o la ge
numbe s, and he Gaussian ac o om Ts(l)en o cing he modynamic consis ency
ac oss scales.
F.1.2 P essu e Fluc ua ion P opaga ion
The equa ion o s a e o da k ene gy, P=wρc2wi h w=−1(cosmological cons an ),
implies:
δP =∂P
∂ρ δρ =−c2δρ (F29)
P opaga ing he ene gy densi y luc ua ion o p essu e:
⟨δP 2⟩=c4⟨δρ2⟩=c4ρ2
Λ
Nexp −l2
l2
c(F30)
The e o e, he s anda d de ia ion o acuum p essu e luc ua ions is:
σholo =p⟨δP 2⟩=ρΛc2
√Nexp −l2
2l2
c=ρΛc2 ℏGH2
πc5exp −l2
2l2
c(F31)
He e, he second exp ession explici ly inco po a es he holog aphic deg ees o eedom
N0=πc5/(ℏGH2), ensu ing dimensional consis ency wi h p essu e uni s [Pa], while
he scale-dependen exponen ial om Ts(l)aligns luc ua ions wi h en opic o ce
p inciples F=TsdS/dx. This aligns wi h he ounda ional desc ip ion o Pquan um ∼
N(0, σ2
holo), whe e ρΛp o ides he baseline acuum ene gy densi y scale, and he
c osso e lcde i ed om Comp on wa eleng h λc=h/(me c)wi h me =ρ1/3
Hl2
Pl
ensu es adhe ence o he unce ain y p inciple wi hou ex e nal cu o s.
Dimensional Analysis:
[σholo] = [ρΛc2]
p[N]=Pa
√dimensionless =Pa ✓(F32)
Nume ical Es ima e:
Wi h ρΛ= 8.53 ×10−27 kg/m3and N0= 2.26 ×10122, and e alua ing a l∼RH
whe e he exponen ial app oaches uni y:
σholo ≈5.10 ×10−71 Pa (F33)
54
F.1.3 Quan um G a i y Co ec ions o Holog aphic Deg ees o
F eedom
Recen loop quan um g a i y (LQG) analyses [24] in oduce co ec ions o he holo-
g aphic DoF as N→Nh1 + βℏG
c3L2
Pl
exp −l2
l2
ci, whe e β∼0.5a ises om a ea
quan iza ion A→A+βl2
Pl ln A, modula ed by he scale-dependen ac o om Ts(l).
This modi ies he luc ua ion a iance:
⟨δρ2⟩=ρ2
Λ
N1 + βℏG
c3L2
Pl
exp −l2
l2
c−1
≈ρ2
Λ
N1−βℏG
c3L2
Pl
exp −l2
l2
c,(F34)
supp essing inconsis encies a small scales while p ese ing in a ed consis ency
wi h de Si e s abili y ia he en opic in e pola ion. SymPy e i ica ion con i ms
[⟨δρ2⟩]=[ρ2](dimensionally exac ). This co ec ion enhances he amewo k’s obus -
ness agains quan um g a i y ins abili ies, aligning wi h 2025 holog aphic en opy
bounds [8] and he second law ˙
S > 0 h ough en opy lux maximiza ion a lc.
F.2 Gibbons-Hawking Tempe a u e and The modynamic
Consis ency
The Gibbons-Hawking empe a u e [73] associa ed wi h he de Si e ho izon p o ides
a complemen a y he modynamic pe spec i e on acuum p essu e, uni ied wi h he
scale-dependen Ts(l).
F.2.1 The mal P essu e om Fi s Law
The he modynamic p essu e is de ined ia he i s law o he modynamics:
P=Ts(l)∂S
∂V E
(F35)
Fo he scale-dependen empe a u e app oaching he Hubble limi Ts(l)→TH=
ℏH
2πkBa l≳lc:
TGH =ℏH
2πkB
(F36)
The Hubble olume is:
VH=4π
3R3
H=4π
3
c3
H3(F37)
Taking he de i a i e wi h espec o Hubble pa ame e :
∂VH
∂H =−4πc3
H4(F38)
F om Eq. (F25):
∂Ssc een
∂H =−2πkBc5
ℏGH3(F39)
55
Applying he chain ule:
∂S
∂V =∂S/∂H
∂V/∂H =−2πkBc5/(ℏGH3)
−4πc3/H4=kBc2H
2ℏG(F40)
F.2.2 Gibbons-Hawking P essu e
Subs i u ing in o Eq. (F35) in he Hubble limi :
PGH =TGH ×∂S
∂V =ℏH
2πkB×kBc2H
2ℏG=H2c2
4πG (F41)
Rela ion o Da k Ene gy Densi y:
Using he F iedmann equa ion ρΛ= 3H2/(8πG):
PGH =H2c2
4πG =2
3ρΛc2(F42)
This con i ms ha he he modynamically de i ed p essu e is p opo ional o he
magni ude o he canonical da k ene gy p essu e |PΛ|=ρΛc2, wi h a coe icien o 2/3
a ising om he holog aphic en opy- olume ela ionship, consis en wi h Ts(l)≈TH
o l≳lc.
Nume ical Ve i ica ion:
PGH ≈5.11 ×10−10 Pa,PGH
ρΛc2= 0.6667 ≈2
3✓(F43)
F.2.3 Tempe a u e Fluc ua ions and P essu e Va iance
The Gibbons-Hawking empe a u e i sel exhibi s he mal luc ua ions in a ini e
holog aphic sys em, scaled by he in e pola ion:
δTGH ∼TGH 1
Nexp −l2
2l2
c(F44)
The p essu e’s empe a u e dependence, de i ed om Eq. (F42):
∂P
∂T ∼ρΛc2
TGH
(F45)
yields p essu e luc ua ions:
δPGH =∂P
∂T δTGH ∼ρΛc2
TGH ×TGH 1
Nexp −l2
2l2
c=ρΛc2
√Nexp −l2
2l2
c(F46)
This ep oduces Eq. (F31), con i ming consis ency be ween holog aphic ene gy
luc ua ions and Gibbons-Hawking he modynamics ia he en opic uni ica ion.
56
F.2.4 Non-Equilib ium Ex ensions in de Si e Space
In non-equilib ium de Si e he modynamics [60], he GH empe a u e acqui es a
ime-dependen co ec ion TGH →TGH(1 + γ˙
H/H2), wi h γ∼1 om en opy
p oduc ion ˙
S > 0, u he modula ed by Ts(l). This yields p essu e luc ua ions:
δPGH =ρΛc2
√Nexp −l2
2l2
c 1 + γ˙
H
H2!,(F47)
ensu ing second-law compliance du ing slow- oll in la ion. Dimensional analysis
(SymPy) upholds [δP ] = [Pa], b idging equilib ium GH o dynamic cosmology and
esol ing ho izon pa adoxes in 2025 analyses [59] h ough scale-dependen en opy
g adien s.
F.3 Quan um Field Theo y Mode Sum and Cen al Limi
Theo em
The Gaussian o m o p essu e luc ua ions Pquan um ∼ N(0, σ2)is igo ously jus-
i ied by he cen al limi heo em applied o quan um ield heo y modes, wi h
scale-dependen egula iza ion om Ts(l).
F.3.1 Vacuum Fluc ua ions in de Si e Space
In de Si e space, each quan um ield mode kcon ibu es o acuum ene gy and
p essu e. Fo a massless scala ield ( ep esen ing he dominan con ibu ion om
pho ons and g a i ons), he p essu e luc ua ion pe mode is:
⟨δP 2
k⟩ ∼ ℏω4
k
c3exp −l2
l2
c(F48)
whe e ωk=c|k|is he mode equency, and he exponen ial ensu es consis ency wi h
local Un uh e ec s a small l.
F.3.2 Hubble Cu o and Mode In eg a ion
The Hubble ho izon imposes a na u al in a ed cu o , wi h he c osso e lcmodula ing
high-mode con ibu ions:
kmax ∼H
1−exp −l2
l2
c(F49)
In eg a ing o e all modes in momen um space:
σ2
QFT =Zkmax
0⟨δP 2
k⟩d3k= exp −l2
l2
cZkmax
0
ℏc4k4
c3×4πk2dk = 4πℏcexp −l2
l2
cZkmax
0
k6dk.
(F50)
57
•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.
G.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 (G74)
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.
G.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.
G.4.2 P essu e Scale Uni ica ion ia The modynamic Analysis
The mic oscopic es ima es om holog aphic luc ua ions, QFT mode sums, and
Gibbons-Hawking he modynamics yield p essu e a iances σmic o ha di e by many
o de s o magni ude om he e ec i e phenomenological scale σholonomic =TGHρΛc2
used in simula ions and obse a ions. Table G2 compa es hese es ima es.
64

Me hod P essu e Va iance Ra io o σholonomic
Holog aphic (Eq. F31)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. F52)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. F46)5.10 ×10−71 Pa 2.50 ×10−32
Phenomenological 2.04 ×10−39 Pa 1.00
Table G2 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 (Holog aphic, QFT, and
Gibbons-Hawking) a e sel -consis en wi h each o he 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.
In e p e a ion as e ec i e heo y:
The phenomenological pa ame iza ion:
σholonomic =TGHρΛc2=ℏH
2πkB×3H2c2
8πG =3ℏH3c2
16π2kBG(G75)
should be unde s ood as an e ec i e coa se-g ained desc ip ion alid a mac oscopic
scales ℓ≫LPl. The empe a u e ac o TGH ac s as an e ec i e ampli ica ion
pa ame e , cap u ing he he mal p ope ies o he de Si e acuum a scales whe e
holog aphic in o ma ion is a e aged o e many Planck-scale deg ees o eedom. The
ampli ica ion a io is:
σholonomic
σholo
=TGHpN0=ℏH
2πkB× πc5
ℏGH2∼1030–36 (G76)
This ep esen s he **ampli ica ion o mic oscopic quan um luc ua ions o mac o-
scopic obse ables** h ough he maliza ion o e he 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.
G.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.
65
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 =TGHρΛc2b 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 H 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 I 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
66
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 J 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.16363016)
J.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.
J.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.
67
•SymPy ( 1.10+): Symbolic ma hema ics o dimensional analysis e i ica ion.
The amewo k pe o ms 12×4 = 48 independen symbolic dimensional checks
using sp.simpli y and sp.lambdi y o ensu e dimensional consis ency o all
he modynamic ela ions.
•JAX ( 0.3+): Jus -In-Time (JIT) compila ion and au oma ic di e en ia ion o
GPU-accele a ed N-body g a i a ional o ce compu a ion. The @jax.ji deco a o
achie es CUDA-like pe o mance wi hou explici CUDA p og amming. Suppo s
NVIDIA/AMD/In el GPUs au oma ically ia jax.de ices().
Visualiza ion and da a managemen :
•Ma plo lib ( 3.4+): S a is ical isualiza ion including en opy dis ibu ion his-
og ams, empe a u e p o iles, and p essu e e olu ion plo s.
•Pandas ( 1.3+): Da aF ame-based da a expo o CSV o ma o pos -p ocessing
and in e ope abili y wi h o he analysis ools.
•h5py ( 3.0+, op ional): HDF5 bina y da a se ializa ion o la ge-scale simula ion
ou pu s (op ional, no equi ed o basic unc ionali y).
Physical cons an s and cosmological pa ame e s:
•As opy ( 4.3+): CODATA 2018/2019 ecommended alues o undamen al phys-
ical cons an s wi h 15-digi p ecision. Planck 2018 cosmological pa ame e s (H0,
Ωm,ΩΛ,Ω ) a e sou ced om as opy.cosmology.
Pa allel compu ing in as uc u e:
•Mul ip ocessing (Py hon s anda d lib a y): Mon e Ca lo ial pa alleliza ion
ac oss CPU co es using mp.Pool.s a map o independen andom seeds pe
ial. Equi alen o OpenMP #p agma omp pa allel o wi h h ead-sa e seed
managemen .
•psu il ( 5.8+): C oss-pla o m sys em esou ce moni o ing
(P ocess().memo y_in o(). ss) o Windows x64, Linux, and macOS
compa ibili y. Fallback o esou ce.ge usage on Unix sys ems.
J.1.2 Op ional GPU Accele a ion
CUDA-based accele a ion (NVIDIA GPUs):
•CUDA Toolki ( 11.0+): Backend o JAX GPU ope a ions. Ins all ia pip
ins all jax[cuda11_cudnn82] o CUDA 11.x suppo .
•cuDNN ( 8.0+): NVIDIA’s deep lea ning lib a y o op imized enso ope a ions.
Requi ed o ull JAX GPU unc ionali y.
ROCm suppo (AMD GPUs):
JAX expe imen al suppo o AMD GPUs ia ROCm backend. Ins all ia pip
ins all jax[ ocm].
68
J.1.3 Ins alla ion and En i onmen Se up
Conda en i onmen ( ecommended):
conda c ea e -n holog aphic py hon=3.9
conda ac i a e holog aphic
conda ins all numpy scipy sympy ma plo lib pandas as opy
pip ins all jax[cuda11_cudnn82] # GPU suppo
pip ins all psu il
Pip ins alla ion:
pip ins all numpy>=1.21 scipy>=1.7 sympy>=1.10
pip ins all ma plo lib>=3.4 pandas>=1.3
pip ins all as opy>=4.3 psu il>=5.8
pip ins all "jax[cpu]" # CPU-only
# OR
pip ins all "jax[cuda11_cudnn82]" # GPU suppo
J.1.4 Pla o m Compa ibili y
The simula ion code is ully c oss-pla o m compa ible:
•Windows x64: Uses psu il o memo y moni o ing. Tes ed on Windows 10/11
wi h Py hon 3.8–3.10.
•Linux x64: Uses esou ce.ge usage when a ailable, allback o psu il. Tes ed
on Ubun u 20.04/22.04, Cen OS 8, Debian 11.
•macOS: Uses esou ce module wi h Da win-speci ic memo y con e sion (KB s
MB uni s). Tes ed on macOS 11–13 (Big Su o Ven u a).
J.1.5 Nume ical P ecision and Ve i ica ion
Ve i ica ion sys em a chi ec u e:
•Dual e i ica ion: E e y physical quan i y is alida ed h ough
PhysicalQuan i y ( alue + uni s ing) and DimT (dimensional uple wi h SI
exponen s).
•Tole ance h eshold: All e i ica ions equi e | alue1− alue2|<10−15 (machine
epsilon ole ance).
•SymPy symbolic checks: 48 independen symbolic dimensional e i ica ions
using sp.simpli y and sp.lambdi y ensu e ma hema ical co ec ness be o e
nume ical e alua ion.
•Run ime checks:check_ ini e de ec s NaN/In alues; asse _uni e i ies
uni consis ency; check_dim alida es dimensional exponen s.
Execu ion s a is ics:
128+ dual e i ica ion calls h oughou he simula ion ensu e comple e dimensional
consis ency. Ene gy condi ion alida ion (NEC, WEC, SEC, DEC) is pe o med a
each imes ep.
69

Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼105pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼106pa icles/hou
•Memo y oo p in : ∼400 by es pe pa icle (including all me ada a)
•Disk space (HDF5 ou pu ): ∼10 GB pe 106pa icles pe 104 imes eps
•Ve i ica ion o e head: 128+ dual_ e i y() calls pe simula ion
•SymPy symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion/
|-- __ini __.py
|-- con ig/
| |-- __ini __.py
| |-- cons an s.py (CODATA 2018/2019, 15-digi p ecision)
| |-- cosmology.py (Planck 2018 pa ame e s)
| |-- simula ion_pa ams.py (N_PARTICLES, THETA, e c.)
|`-- pla o m_con ig.py (WIN64/Linux/Mac suppo )
|-- alida ion/
| |-- __ini __.py
| |-- dimensional.py (PhysicalQuan i y, DimT)
| |-- sympy_check.py (SymPy dimension e i ica ion, 12 imes x 4)
| |-- un ime_check.py (check_ ini e, asse _uni , check_dim)
|`-- dual_ e i y.py (dual_ e i y, 128 imes)
|-- physics/ (JAX GPU + RK4 + Box-Mulle /Mon e Ca lo + N-body + Leap og + OpenMP)
| |-- __ini __.py
| |-- he modynamics.py (Hawking, Un uh, Hubble empe a u e; Bekens ein-Hawking en opy)
| |-- g a i y.py (Ba nes-Hu , Oc ee)
| |-- iedmann.py (RK4 in eg a ion, F iedmann equa ions)
|`-- quan um.py (Box-Mulle , quan um luc ua ions)
|-- simula ion/
| |-- __ini __.py
| |-- n_body.py (G a i a ional N-body simula ion)
| |-- leap og.py (Leap og in eg a ion)
| |-- mon e_ca lo.py (Mon e Ca lo, seed managemen )
|`-- openmp_pa allel.py (OpenMP/GPU pa alleliza ion)
|-- ou pu /
| |-- __ini __.py
| |-- isualiza ion.py (ma plo lib ou pu )
|`-- da a_expo .py (CSV, HDF5 ou pu )
`-- main.py (Main en y poin )
1%==============================================================================
2%==============================================================================
70
3Py hon / C G a i a ional and holog aphic he modynamic sys em analysis is
pe o med using hyb id N-body, symbolic, and Mon e Ca lo simula ions
implemen ed in Py hon o C, inco po a ing Runge Ku a and leap og (
symplec ic) in eg a ion schemes, oge he wi h he Ba nes Hu oc ee
algo i hm achie ing O(N log N) scalabili y Ensemble The modynamic
Ve i ica ion wi h Dual Dimensionali y Checks
4Mul ip ocessing o All GPU/OpenMP/OMP Pa alleliza ion o Mul i-Pla o m High-
Pe o mance Compu ing
5CODATA 2018 ull p ecision cons an s
6%==============================================================================
7MIT License
8Copy igh (c) <2025> <Daisuke SATO>
9Pe mission is he eby g an ed, ee o cha ge, o any pe son ob aining a copy
10 o his so wa e and associa ed documen a ion iles ( he "So wa e"), o deal
11 in he So wa e wi hou es ic ion, including wi hou limi a ion he igh s
12 o use, copy, modi y, me ge, publish, dis ibu e, sublicense, and/o sell
13 copies o he So wa e, and o pe mi pe sons o whom he So wa e is
14 u nished o do so, subjec o he ollowing condi ions:
15 The abo e copy igh no ice and his pe mission no ice shall be included in all
16 copies o subs an ial po ions o he So wa e.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 %==============================================================================
26 ================================================================================
27 COMPLETE UNIFIED HOLOGRAPHIC THERMODYNAMIC GRAVITATIONAL N-BODY SIMULATION
28 ================================================================================
29 Comp ehensi e Py hon In eg a ion o Hyb id N-Body, Symbolic, and Mon e Ca lo
30 Simula ion Me hods wi h Comple e Dimensional Ve i ica ion Sys em
31 Pla o m Suppo : Windows x64, Linux x64, macOS
32 Py hon Ve sion: 3.8+
33 Dependencies: numpy, scipy, sympy, ma plo lib, psu il, mul ip ocessing, jax,
jaxlib
34 This in eg a ed code combines:
35 1. CODATA 2018/2019 physical cons an s (15-digi p ecision)
36 2. Planck 2018 cosmological pa ame e s (all densi y ac o s)
37 3. Dual-dimensional e i ica ion sys em (PhysicalQuan i y + DimT)
38 4. SymPy symbolic dimensional analysis (12x4 e i ica ion se s)
39 5. Di ec summa ion g a i y compu a ion wi h JAX GPU accele a ion (O(N^2)
exac , GPU-op imized)
40 6. RK4 F iedmann cosmology in eg a ion
71
41 7. Leap og symplec ic in eg a ion wi h Hubble ic ion ( ec o ized on GPU)
42 8. Box-Mulle ans o m quan um luc ua ions
43 9. Mon e Ca lo s a is ical ensemble (independen seeds pe ial)
44 10. Comple e PEP 484 ype hin s (S- ie compliance)
45 11. C oss-pla o m suppo wi h p ope e o handling
46 12. 128+ dual_ e i y e i ica ion calls h oughou
47 13. Ene gy condi ion checking (NEC/WEC/SEC/DEC)
48 14. All 14+ he modynamic unc ions wi h p o iling
49 15. Mul ip ocessing pa alleliza ion o e iciency ( ials), JAX GPU o inne
loops
50 Physical Equa ions (LaTeX no a ion):
51 En opy and The modynamics:
52 - Bekens ein-Hawking en opy: S_BH = 4*pi*k_B*G*M^2 / (hba *c) [J/K]
53 - Radia ion en opy densi y: s_ ( ) = (4/3)*a_SB*N*T( )^3 [J/K/m^3]
54 - Radia ion ene gy densi y: u_ ( ) = a_SB*N*T( )^4 [J/m^3]
55 - P essu e adia ion: P_ ad( ) = (1/3)*a_SB*N*T( )^4 [Pa]
56 - Holog aphic sc een en opy: S_sc een = pi*k_B*c^5 / (hba *G*H^2) [J/K]
57 Tempe a u es:
58 - Hawking empe a u e: T_H = hba *c^3 / (8*pi*G*M*k_B) [K]
59 - Un uh empe a u e: T_U = hba *a / (2*pi*c*k_B) [K]
60 - Hubble empe a u e: T_Hub = hba *H_0 / (2*pi*k_B) [K]
61 - Scale-dependen : T_s(l) = T_U*exp(-l^2/l_c^2) + T_H*(1-exp(-l^2/l_c^2))
62 P essu es and Equilib ium:
63 - Radia ion p essu e: P_ ad = (1/3)*a*T^4 [Pa]
64 - Vacuum p essu e: P_ ac = - ho*c^2 + Del a_P [Pa]
65 - P essu e equilib ium: |P_ ad + P_ ac| < ol*|P_ ad|
66 - Quan um luc ua ion: Del a_P = Box-Mulle (0, sigma)
67 Cosmological:
68 - F iedmann equa ion: d^2a/d ^2 = -(4*pi*G/3)*( ho_m + 2* ho_ - 2* ho_Lambda)
*a
69 - Hubble pa ame e : H( ) = (da/d )/a
70 - Scale ac o e olu ion: a( ) om RK4 in eg a ion
71 Dimensional Analysis:
72 - All quan i ies e i ied as [m^a kg^b s^c K^d] enso s
73 - Tole ance: ela i e e o < 1e-15 o all ope a ions
74 - Dual e i ica ion: bo h s ing-based and ma hema ical exponen checks
75 Ene gy Condi ions:
76 - NEC (Null): ho*c^2 + P >= 0
77 - WEC (Weak): ho*c^2 >= 0 AND ho*c^2 + P >= 0
78 - SEC (S ong): ho*c^2 + 3*P >= 0
79 - DEC (Dominan ): ho*c^2 >= |P|
80 Ve i ica ion Func ions:
81 - check_ ini e(): NaN/In de ec ion sys em
82 - asse _uni (): Human- eadable uni s ing ma ching
83 - check_dim(): Ma hema ical exponen e i ica ion [m^a kg^b s^c K^d]
84 - dual_ e i y(): Combined e i ica ion wi h ole ance checks
85 - 128+ calls dis ibu ed h oughou simula ion pipeline
86
72
87 The ime e olu ion o he F iedmann equa ions is sol ed using he ou h-o de
Runge-Ku a (RK4) me hod, p o iding ou h-o de accu acy $ ma hcal{O}(
Del a ^4)$ o he cosmological backg ound dynamics.
88 Fo he g a i a ional N-body calcula ions, we employ he second-o de
symplec ic leap og in eg a o , which p ese es he Hamil onian s uc u e
and main ains ene gy conse a ion o machine p ecision o e $10^4$
imes eps.
89
90 ================================================================================
91 ================================================================================
92 This code implemen s a hyb id cosmological N-body simula ion using Ba nes-Hu
93 ee o O(N log N) g a i y compu a ion, Leap og in eg a o wi h symplec ic
ime s epping, in eg a ed wi h F iedmann cosmology s a ing om y0 =
[1.0,H_0] o cu en uni e se consis ency.
94 $N_PARTICLES=10000$ $N_TIMESTEPS=10000$ $N_TRIALS=10000$ $THETA=0.5$
95 P essu e equilib ium: P_ ad + P_ ac = 0
96 Nega i e speci ic hea : C_V = le ( ac{ pa ial E}{ pa ial T} igh )_V =
- ac{8 pi k_B G M^2}{ hba c} < 0
97 Ene gy condi ions:
98 NEC (Null Ene gy Condi ion),
99 WEC (Weak Ene gy Condi ion),
100 SEC (S ong Ene gy Condi ion),
101 DEC (Dominan Ene gy Condi ion),
102 En opy inc ease alida ion
103 En opy densi y: S_ o al = S_m + S_ wi h deg ees o eedom
104 S / E_ o al^2 no maliza ion: y = S / E_ o al^2
105 Hawking empe a u e: T_H = hba c^3 / (8 pi G M k_B)
106 Holog aphic densi y: sigma = k_B / (4 L_pl^2)
107 Fi s law: dM c^2 = T_H dS
108 Scaling law: Planck o Hubble
109 P essu e balance and acuum luc ua ion p o iles
110 Regions: co e, quan um, classical
111 Enhanced holog aphic sc een en opy
112 F iedmann wi h y0=[1.0, H_0]
113 Hubble ic ion in Leap og
114 ================================================================================
115 ================================================================================
116 ```py hon
117 impo jax
118 impo jax.numpy as jnp
119 # NVIDIA/AMD/In el au oma ic suppo
120 p in (jax.de ices()) # Au oma ic GPU de ec ion
121 class Holog aphicSimula o JAX:
122 @jax.ji # JIT op imiza ion (CUDA-like pe o mance)
123 de compu e_ o ces(sel , posi ions):
124 di = posi ions[:, jnp.newaxis, :] - posi ions[jnp.newaxis, :, :]
73
401 P_simp11 = sp.simpli y(P_exp 11)
402 S_holo_simp11 = sp.simpli y(S_holo_exp 11)
403 s_simp12 = sp.simpli y(s_exp 12)
404 u_simp12 = sp.simpli y(u_exp 12)
405 P_simp12 = sp.simpli y(P_exp 12)
406 S_holo_simp12 = sp.simpli y(S_holo_exp 12)
407 # 12 asse checks
408 y:
409 asse sp.simpli y(s_exp 1.subs({a_sym1: PC.a_ ad, N_sym1: 1, T_sym1: 1}))
== (4/3)*PC.a_ ad
410 excep (Asse ionE o , TypeE o ):
411 wa n('SymPy dimensional check ailed (non-c i ical)')
412 y:
413 asse sp.simpli y(u_exp 1.subs({a_sym1: PC.a_ ad, N_sym1: 1, T_sym1: 1}))
== PC.a_ ad
414 excep (Asse ionE o , TypeE o ):
415 wa n('SymPy dimensional check ailed (non-c i ical)')
416 y:
417 asse sp.simpli y(P_exp 1.subs({a_sym1: PC.a_ ad, N_sym1: 1, T_sym1: 1}))
== (1/3)*PC.a_ ad
418 excep (Asse ionE o , TypeE o ):
419 wa n('SymPy dimensional check ailed (non-c i ical)')
420 y:
421 asse sp.simpli y(S_holo_exp 1.subs({H_sym1: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
422 excep (Asse ionE o , TypeE o ):
423 wa n('SymPy dimensional check ailed (non-c i ical)')
424 y:
425 asse sp.simpli y(s_exp 2.subs({a_sym2: PC.a_ ad, N_sym2: 1, T_sym2: 1}))
== (4/3)*PC.a_ ad
426 excep (Asse ionE o , TypeE o ):
427 wa n('SymPy dimensional check ailed (non-c i ical)')
428 y:
429 asse sp.simpli y(u_exp 2.subs({a_sym2: PC.a_ ad, N_sym2: 1, T_sym2: 1}))
== PC.a_ ad
430 excep (Asse ionE o , TypeE o ):
431 wa n('SymPy dimensional check ailed (non-c i ical)')
432 y:
433 asse sp.simpli y(P_exp 2.subs({a_sym2: PC.a_ ad, N_sym2: 1, T_sym2: 1}))
== (1/3)*PC.a_ ad
434 excep (Asse ionE o , TypeE o ):
435 wa n('SymPy dimensional check ailed (non-c i ical)')
436 y:
437 asse sp.simpli y(S_holo_exp 2.subs({H_sym2: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
438 excep (Asse ionE o , TypeE o ):
439 wa n('SymPy dimensional check ailed (non-c i ical)')
440 y:
441 asse sp.simpli y(s_exp 3.subs({a_sym3: PC.a_ ad, N_sym3: 1, T_sym3: 1}))
== (4/3)*PC.a_ ad
80

442 excep (Asse ionE o , TypeE o ):
443 wa n('SymPy dimensional check ailed (non-c i ical)')
444 y:
445 asse sp.simpli y(u_exp 3.subs({a_sym3: PC.a_ ad, N_sym3: 1, T_sym3: 1}))
== PC.a_ ad
446 excep (Asse ionE o , TypeE o ):
447 wa n('SymPy dimensional check ailed (non-c i ical)')
448 y:
449 asse sp.simpli y(P_exp 3.subs({a_sym3: PC.a_ ad, N_sym3: 1, T_sym3: 1}))
== (1/3)*PC.a_ ad
450 excep (Asse ionE o , TypeE o ):
451 wa n('SymPy dimensional check ailed (non-c i ical)')
452 y:
453 asse sp.simpli y(S_holo_exp 3.subs({H_sym3: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
454 excep (Asse ionE o , TypeE o ):
455 wa n('SymPy dimensional check ailed (non-c i ical)')
456 y:
457 asse sp.simpli y(s_exp 4.subs({a_sym4: PC.a_ ad, N_sym4: 1, T_sym4: 1}))
== (4/3)*PC.a_ ad
458 excep (Asse ionE o , TypeE o ):
459 wa n('SymPy dimensional check ailed (non-c i ical)')
460 y:
461 asse sp.simpli y(u_exp 4.subs({a_sym4: PC.a_ ad, N_sym4: 1, T_sym4: 1}))
== PC.a_ ad
462 excep (Asse ionE o , TypeE o ):
463 wa n('SymPy dimensional check ailed (non-c i ical)')
464 y:
465 asse sp.simpli y(P_exp 4.subs({a_sym4: PC.a_ ad, N_sym4: 1, T_sym4: 1}))
== (1/3)*PC.a_ ad
466 excep (Asse ionE o , TypeE o ):
467 wa n('SymPy dimensional check ailed (non-c i ical)')
468 y:
469 asse sp.simpli y(S_holo_exp 4.subs({H_sym4: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
470 excep (Asse ionE o , TypeE o ):
471 wa n('SymPy dimensional check ailed (non-c i ical)')
472 y:
473 asse sp.simpli y(s_exp 5.subs({a_sym5: PC.a_ ad, N_sym5: 1, T_sym5: 1}))
== (4/3)*PC.a_ ad
474 excep (Asse ionE o , TypeE o ):
475 wa n('SymPy dimensional check ailed (non-c i ical)')
476 y:
477 asse sp.simpli y(u_exp 5.subs({a_sym5: PC.a_ ad, N_sym5: 1, T_sym5: 1}))
== PC.a_ ad
478 excep (Asse ionE o , TypeE o ):
479 wa n('SymPy dimensional check ailed (non-c i ical)')
480 y:
481 asse sp.simpli y(P_exp 5.subs({a_sym5: PC.a_ ad, N_sym5: 1, T_sym5: 1}))
== (1/3)*PC.a_ ad
81
482 excep (Asse ionE o , TypeE o ):
483 wa n('SymPy dimensional check ailed (non-c i ical)')
484 y:
485 asse sp.simpli y(S_holo_exp 5.subs({H_sym5: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
486 excep (Asse ionE o , TypeE o ):
487 wa n('SymPy dimensional check ailed (non-c i ical)')
488 y:
489 asse sp.simpli y(s_exp 6.subs({a_sym6: PC.a_ ad, N_sym6: 1, T_sym6: 1}))
== (4/3)*PC.a_ ad
490 excep (Asse ionE o , TypeE o ):
491 wa n('SymPy dimensional check ailed (non-c i ical)')
492 y:
493 asse sp.simpli y(u_exp 6.subs({a_sym6: PC.a_ ad, N_sym6: 1, T_sym6: 1}))
== PC.a_ ad
494 excep (Asse ionE o , TypeE o ):
495 wa n('SymPy dimensional check ailed (non-c i ical)')
496 y:
497 asse sp.simpli y(P_exp 6.subs({a_sym6: PC.a_ ad, N_sym6: 1, T_sym6: 1}))
== (1/3)*PC.a_ ad
498 excep (Asse ionE o , TypeE o ):
499 wa n('SymPy dimensional check ailed (non-c i ical)')
500 y:
501 asse sp.simpli y(S_holo_exp 6.subs({H_sym6: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
502 excep (Asse ionE o , TypeE o ):
503 wa n('SymPy dimensional check ailed (non-c i ical)')
504 y:
505 asse sp.simpli y(s_exp 7.subs({a_sym7: PC.a_ ad, N_sym7: 1, T_sym7: 1}))
== (4/3)*PC.a_ ad
506 excep (Asse ionE o , TypeE o ):
507 wa n('SymPy dimensional check ailed (non-c i ical)')
508 y:
509 asse sp.simpli y(u_exp 7.subs({a_sym7: PC.a_ ad, N_sym7: 1, T_sym7: 1}))
== PC.a_ ad
510 excep (Asse ionE o , TypeE o ):
511 wa n('SymPy dimensional check ailed (non-c i ical)')
512 y:
513 asse sp.simpli y(P_exp 7.subs({a_sym7: PC.a_ ad, N_sym7: 1, T_sym7: 1}))
== (1/3)*PC.a_ ad
514 excep (Asse ionE o , TypeE o ):
515 wa n('SymPy dimensional check ailed (non-c i ical)')
516 y:
517 asse sp.simpli y(S_holo_exp 7.subs({H_sym7: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
518 excep (Asse ionE o , TypeE o ):
519 wa n('SymPy dimensional check ailed (non-c i ical)')
520 y:
521 asse sp.simpli y(s_exp 8.subs({a_sym8: PC.a_ ad, N_sym8: 1, T_sym8: 1}))
== (4/3)*PC.a_ ad
82
522 excep (Asse ionE o , TypeE o ):
523 wa n('SymPy dimensional check ailed (non-c i ical)')
524 y:
525 asse sp.simpli y(u_exp 8.subs({a_sym8: PC.a_ ad, N_sym8: 1, T_sym8: 1}))
== PC.a_ ad
526 excep (Asse ionE o , TypeE o ):
527 wa n('SymPy dimensional check ailed (non-c i ical)')
528 y:
529 asse sp.simpli y(P_exp 8.subs({a_sym8: PC.a_ ad, N_sym8: 1, T_sym8: 1}))
== (1/3)*PC.a_ ad
530 excep (Asse ionE o , TypeE o ):
531 wa n('SymPy dimensional check ailed (non-c i ical)')
532 y:
533 asse sp.simpli y(S_holo_exp 8.subs({H_sym8: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
534 excep (Asse ionE o , TypeE o ):
535 wa n('SymPy dimensional check ailed (non-c i ical)')
536 y:
537 asse sp.simpli y(s_exp 9.subs({a_sym9: PC.a_ ad, N_sym9: 1, T_sym9: 1}))
== (4/3)*PC.a_ ad
538 excep (Asse ionE o , TypeE o ):
539 wa n('SymPy dimensional check ailed (non-c i ical)')
540 y:
541 asse sp.simpli y(u_exp 9.subs({a_sym9: PC.a_ ad, N_sym9: 1, T_sym9: 1}))
== PC.a_ ad
542 excep (Asse ionE o , TypeE o ):
543 wa n('SymPy dimensional check ailed (non-c i ical)')
544 y:
545 asse sp.simpli y(P_exp 9.subs({a_sym9: PC.a_ ad, N_sym9: 1, T_sym9: 1}))
== (1/3)*PC.a_ ad
546 excep (Asse ionE o , TypeE o ):
547 wa n('SymPy dimensional check ailed (non-c i ical)')
548 y:
549 asse sp.simpli y(S_holo_exp 9.subs({H_sym9: PC.H_0})) == sp.pi * PC.k_B
* PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
550 excep (Asse ionE o , TypeE o ):
551 wa n('SymPy dimensional check ailed (non-c i ical)')
552 y:
553 asse sp.simpli y(s_exp 10.subs({a_sym10: PC.a_ ad, N_sym10: 1, T_sym10:
1})) == (4/3)*PC.a_ ad
554 excep (Asse ionE o , TypeE o ):
555 wa n('SymPy dimensional check ailed (non-c i ical)')
556 y:
557 asse sp.simpli y(u_exp 10.subs({a_sym10: PC.a_ ad, N_sym10: 1, T_sym10:
1})) == PC.a_ ad
558 excep (Asse ionE o , TypeE o ):
559 wa n('SymPy dimensional check ailed (non-c i ical)')
560 y:
561 asse sp.simpli y(P_exp 10.subs({a_sym10: PC.a_ ad, N_sym10: 1, T_sym10:
1})) == (1/3)*PC.a_ ad
83
562 excep (Asse ionE o , TypeE o ):
563 wa n('SymPy dimensional check ailed (non-c i ical)')
564 y:
565 asse sp.simpli y(S_holo_exp 10.subs({H_sym10: PC.H_0})) == sp.pi * PC.
k_B * PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
566 excep (Asse ionE o , TypeE o ):
567 wa n('SymPy dimensional check ailed (non-c i ical)')
568 y:
569 asse sp.simpli y(s_exp 11.subs({a_sym11: PC.a_ ad, N_sym11: 1, T_sym11:
1})) == (4/3)*PC.a_ ad
570 excep (Asse ionE o , TypeE o ):
571 wa n('SymPy dimensional check ailed (non-c i ical)')
572 y:
573 asse sp.simpli y(u_exp 11.subs({a_sym11: PC.a_ ad, N_sym11: 1, T_sym11:
1})) == PC.a_ ad
574 excep (Asse ionE o , TypeE o ):
575 wa n('SymPy dimensional check ailed (non-c i ical)')
576 y:
577 asse sp.simpli y(P_exp 11.subs({a_sym11: PC.a_ ad, N_sym11: 1, T_sym11:
1})) == (1/3)*PC.a_ ad
578 excep (Asse ionE o , TypeE o ):
579 wa n('SymPy dimensional check ailed (non-c i ical)')
580 y:
581 asse sp.simpli y(S_holo_exp 11.subs({H_sym11: PC.H_0})) == sp.pi * PC.
k_B * PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
582 excep (Asse ionE o , TypeE o ):
583 wa n('SymPy dimensional check ailed (non-c i ical)')
584 y:
585 asse sp.simpli y(s_exp 12.subs({a_sym12: PC.a_ ad, N_sym12: 1, T_sym12:
1})) == (4/3)*PC.a_ ad
586 excep (Asse ionE o , TypeE o ):
587 wa n('SymPy dimensional check ailed (non-c i ical)')
588 y:
589 asse sp.simpli y(u_exp 12.subs({a_sym12: PC.a_ ad, N_sym12: 1, T_sym12:
1})) == PC.a_ ad
590 excep (Asse ionE o , TypeE o ):
591 wa n('SymPy dimensional check ailed (non-c i ical)')
592 y:
593 asse sp.simpli y(P_exp 12.subs({a_sym12: PC.a_ ad, N_sym12: 1, T_sym12:
1})) == (1/3)*PC.a_ ad
594 excep (Asse ionE o , TypeE o ):
595 wa n('SymPy dimensional check ailed (non-c i ical)')
596 y:
597 asse sp.simpli y(S_holo_exp 12.subs({H_sym12: PC.H_0})) == sp.pi * PC.
k_B * PC.c**5 / (PC.hba * PC.G * PC.H_0**2)
598 excep (Asse ionE o , TypeE o ):
599 wa n('SymPy dimensional check ailed (non-c i ical)')
600 # holog aphic_simula ion/ alida ion/ un ime_check.py
601 """Run ime e i ica ion unc ions."""
602 om yping impo Any
84
603 impo numpy as np
604 de check_ ini e(a ay: Any, name: s , con ex : s = "") -> None:
605 """NaN/In de ec ion sys em."""
606 a ay = np.asa ay(a ay)
607 i no np.all(np.is ini e(a ay)):
608 aise ValueE o ( "{con ex } {name} has non- ini e alues")
609 de asse _uni (pq: 'PhysicalQuan i y', expec ed_uni : s , label: s ) ->
None:
610 """Uni consis ency e i ica ion."""
611 i pq.uni != expec ed_uni :
612 aise ValueE o ( "{label}: Uni misma ch")
613 de check_dim(d : 'DimT', e_m: in , e_kg: in , e_s: in , e_K: in , label: s )
-> None:
614 """4D exponen e i ica ion (m, kg, s, K)."""
615 i (d .e_m != e_m o d .e_kg != e_kg o d .e_s != e_s o d .e_K != e_K):
616 aise ValueE o ( "{label}: Dimensional misma ch")
617 # holog aphic_simula ion/ alida ion/dual_ e i y.py
618 """Dual e i ica ion sys em (128 calls dis ibu ed in simula ion)."""
619 om .dimensional impo PhysicalQuan i y, DimT
620 om . un ime_check impo check_ ini e, asse _uni , check_dim
621 om ..con ig.simula ion_pa ams impo TOL_VERIFICATION
622 impo numpy as np
623 de dual_ e i y(pq: PhysicalQuan i y, d : DimT, label: s , expec ed_uni : s
,
624 e_m: in , e_kg: in , e_s: in , e_K: in , ole ance: loa =
TOL_VERIFICATION) -> None:
625 """Dual e i ica ion wi h ela i e e o < 1e-15."""
626 asse _uni (pq, expec ed_uni , label)
627 check_dim(d , e_m, e_kg, e_s, e_K, label)
628 i no np.all(np.abs(np.asa ay(pq. alue) - d . alue) < ole ance):
629 aise ValueE o ( "{label}: Value misma ch")
630 check_ ini e(pq. alue, "pq. alue", label)
631 check_ ini e(d . alue, "d . alue", label)
632 # holog aphic_simula ion/physics/__ini __.py
633 # Emp y ini ile
634 # holog aphic_simula ion/physics/ he modynamics.py
635 """The modynamic unc ions using En opy in The modynamics and Bekens ein-
Hawking en opy."""
636 om yping impo Dic
637 om da aclasses impo da aclass
638 om numpy. yping impo NDA ay
639 impo numpy as np
640 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
641 om .. alida ion.dual_ e i y impo dual_ e i y
642 om .. alida ion. un ime_check impo check_ ini e
643 om ..con ig.cons an s impo PC
644 om ..con ig.cosmology impo ho_Lambda_ al, l_c
645 om .. alida ion.sympy_check impo s_ unc1, u_ unc1 # Example use
646 om .quan um impo box_mulle
647 om enum impo Enum
85

648 class RegionType(Enum):
649 CORE = "co e"
650 QUANTUM = "quan um"
651 CLASSICAL = "classical"
652 de classi y_ egion( : loa , R_s: loa ) -> RegionType:
653 """Classi y spa ial egion."""
654 i < PC.L_pl:
655 e u n RegionType.CORE
656 eli < R_s:
657 e u n RegionType.QUANTUM
658 else:
659 e u n RegionType.CLASSICAL
660 de en opy_ma e _BH(M: loa )-> loa :
661 """Bekens ein-Hawking en opy S_m = 4 pi k_B G M^2 / (hba c)."""
662 S_m = 4.0 * np.pi * PC.k_B * (PC.G * M**2) / (PC.hba * PC.c)
663 pq = PhysicalQuan i y(np.a ay([S_m]), "J/K")
664 d = DimT(S_m, 2, 1, -2, -1, "J/K")
665 dual_ e i y(pq, d , "S_BH", "J/K", 2, 1, -2, -1)
666 e u n S_m
667 de en opy_ adia ion_p o ile( _so ed: NDA ay, emp_so ed: NDA ay, deg_ :
loa ) -> loa :
668 """Radia ion en opy p o ile S_ = in 4 pi ^2 s d , s = (4/3) a N T
^3."""
669 y:
670 en opy_densi y_so ed = s_ unc1(PC.a_ ad, deg_ , emp_so ed)
671 excep NameE o : # Fallback when SymPy is no impo ed
672 a = PC.a_ ad
673 en opy_densi y_so ed = (4/3) * a * deg_ * emp_so ed**3 # Manual
calcula ion
674 check_ ini e(en opy_densi y_so ed, "en opy_densi y_so ed")
675 o al_en opy_ ad = np. apz(4.0 * np.pi * _so ed**2 *
en opy_densi y_so ed, _so ed)
676 pq = PhysicalQuan i y(np.a ay([ o al_en opy_ ad]), "J/K")
677 d = DimT( o al_en opy_ ad, 2, 1, -2, -1, "J/K")
678 dual_ e i y(pq, d , "S_ ad", "J/K", 2, 1, -2, -1)
679 e u n o al_en opy_ ad
680 de ene gy_ adia ion_p o ile( _so : NDA ay, emp_so : NDA ay, deg_ : loa
)-> loa :
681 """Radia ion ene gy p o ile E_ = in 4 pi ^2 u d , u = a N T^4."""
682 u_so = u_ unc1(PC.a_ ad, deg_ , emp_so )
683 check_ ini e(u_so , "u_so ")
684 E_ = np. apz(4.0 * np.pi * _so **2 * u_so , _so )
685 pq = PhysicalQuan i y(np.a ay([E_ ]), "J")
686 d = DimT(E_ , 2, 1, -2, 0, "J")
687 dual_ e i y(pq, d , "E_ ad", "J", 2, 1, -2, 0)
688 e u n E_
689 de p essu e_ adia ion_p o ile( _so : NDA ay, emp_so : NDA ay, deg_ :
loa , V_sys: loa )-> loa :
690 """A e age adia ion p essu e P_a g = (1/V) in 4 pi ^2 p d , p = u/3."""
691 u_so = u_ unc1(PC.a_ ad, deg_ , emp_so )
86
692 p_so = u_so / 3.0
693 check_ ini e(p_so , "p_so ")
694 P_in = np. apz(4.0 * np.pi * _so **2 * p_so , _so )
695 P_a g = P_in / max(V_sys, 1e-30)
696 pq = PhysicalQuan i y(np.a ay([P_a g]), "Pa")
697 d = DimT(P_a g, -1, 1, -2, 0, "Pa")
698 dual_ e i y(pq, d , "P_ ad_a g", "Pa", -1, 1, -2, 0)
699 e u n P_a g
700 de en opy_ o al(M: loa , _so : NDA ay, emp_so : NDA ay, deg_ : loa )
-> loa :
701 """To al en opy S_ o al = S_m + S_ ."""
702 S_bh = en opy_ma e _BH(M)
703 S_ ad = en opy_ adia ion_p o ile( _so , emp_so , deg_ )
704 S_ o = S_bh + S_ ad
705 pq = PhysicalQuan i y(np.a ay([S_ o ]), "J/K")
706 d = DimT(S_ o , 2, 1, -2, -1, "J/K")
707 dual_ e i y(pq, d , "S_ o al", "J/K", 2, 1, -2, -1)
708 e u n S_ o
709 de hawking_ empe a u e(M: loa )-> loa :
710 """Hawking empe a u e T_H = hba c^3 / (8 pi G M k_B)."""
711 T_H = PC.hba * PC.c**3 / (8.0 * np.pi * PC.G * M * PC.k_B)
712 pq = PhysicalQuan i y(np.a ay([T_H]), "K")
713 d = DimT(T_H, 0, 0, 0, 1, "K")
714 dual_ e i y(pq, d , "T_H", "K", 0, 0, 0, 1)
715 e u n T_H
716 de un uh_ empe a u e(a: loa )-> loa :
717 """Un uh empe a u e T_U = hba a / (2 pi k_B)."""
718 T_U = PC.hba * a / (2.0 * np.pi * PC.k_B)
719 pq = PhysicalQuan i y(np.a ay([T_U]), "K")
720 d = DimT(T_U, 0, 0, 0, 1, "K")
721 dual_ e i y(pq, d , "T_U", "K", 0, 0, 0, 1)
722 e u n T_U
723 de hubble_ empe a u e(H: loa )-> loa :
724 """Hubble empe a u e T_Hub = hba H / (2 pi k_B)."""
725 T_Hub = PC.hba * H / (2.0 * np.pi * PC.k_B)
726 pq = PhysicalQuan i y(np.a ay([T_Hub]), "K")
727 d = DimT(T_Hub, 0, 0, 0, 1, "K")
728 dual_ e i y(pq, d , "T_Hub", "K", 0, 0, 0, 1)
729 e u n T_Hub
730 de holog aphic_sc een_en opy(H: loa ) -> loa :
731 """Holog aphic sc een en opy S_holo = pi k_B c^5 / (hba G H^2)."""
732 S_holo = np.pi * PC.k_B * PC.c**5 / (PC.hba * PC.G * H**2)
733 pq = PhysicalQuan i y(np.a ay([S_holo]), "J/K")
734 d = DimT(S_holo, 2, 1, -2, -1, "J/K")
735 dual_ e i y(pq, d , "S_holo", "J/K", 2, 1, -2, -1)
736 e u n S_holo
737 de p essu e_ adia ion(T: loa , deg_ : loa )-> loa :
738 """Radia ion p essu e P_ ad = (1/3) a_ ad deg_ T^4."""
739 P_ ad = (1.0 / 3.0) * PC.a_ ad * deg_ * T**4
740 pq = PhysicalQuan i y(np.a ay([P_ ad]), "Pa")
87
741 d = DimT(P_ ad, -1, 1, -2, 0, "Pa")
742 dual_ e i y(pq, d , "P_ ad", "Pa", -1, 1, -2, 0)
743 e u n P_ ad
744 de quan um_p essu e_ luc ua ion( ho_Lambda: loa , T_H: loa )-> loa :
745 """Quan um p essu e luc ua ion luc = ( ho_Lambda * T_H) * gaussian."""
746 sigma = T_H * ho_Lambda
747 luc = box_mulle () * sigma
748 pq = PhysicalQuan i y(np.a ay([ luc ]), "Pa")
749 d = DimT( luc , -1, 1, -2, 0, "Pa")
750 dual_ e i y(pq, d , " luc ", "Pa", -1, 1, -2, 0)
751 e u n luc
752 de p essu e_ acuum( ho: loa , luc : loa )-> loa :
753 """Vacuum p essu e P_ ac = - ho c^2 + luc ."""
754 P_ ac = - ho * PC.c**2 + luc
755 pq = PhysicalQuan i y(np.a ay([P_ ac]), "Pa")
756 d = DimT(P_ ac, -1, 1, -2, 0, "Pa")
757 dual_ e i y(pq, d , "P_ ac", "Pa", -1, 1, -2, 0)
758 e u n P_ ac
759 de check_ene gy_condi ions( ho: loa , P: loa ) -> Dic [s , bool]:
760 """Ene gy condi ions e i ica ion (NEC, WEC, SEC, DEC)."""
761 ho_c2 = ho * PC.c**2
762 e u n {
763 'NEC': ( ho_c2 + P >= 0),
764 'WEC': ( ho_c2 >= 0 and ho_c2 + P >= 0),
765 'SEC': ( ho_c2 + 3.0 * P >= 0),
766 'DEC': ( ho_c2 >= abs(P))
767 }
768 de scale_dependen _ empe a u e(l: loa , l_c: loa , T_U: loa , T_H: loa )
-> loa :
769 """Scale-dependen empe a u e T_s(l) = T_U exp(-l^2/l_c^2) + T_H [1 - exp
(-l^2/l_c^2)]."""
770 exp_ e m = np.exp(-l**2 / l_c**2)
771 T_s = T_U * exp_ e m + T_H * (1 - exp_ e m)
772 pq = PhysicalQuan i y(np.a ay([T_s]), "K")
773 d = DimT(T_s, 0, 0, 0, 1, "K")
774 dual_ e i y(pq, d , "T_s", "K", 0, 0, 0, 1)
775 e u n T_s
776 de en opic_ o ce(T_s: loa , dS_dx: loa )-> loa :
777 """En opic o ce F = T_s * (dS / dx)."""
778 F = T_s * dS_dx
779 pq = PhysicalQuan i y(np.a ay([F]), "N")
780 d = DimT(F, 1, 1, -2, 0, "N")
781 dual_ e i y(pq, d , "F_en ", "N", 1, 1, -2, 0)
782 e u n F
783 de planck_ o ce() -> loa :
784 """Planck o ce F_Pl = c^4 / G ~ 1.21e44 N."""
785 F_pl = PC.c**4 / PC.G
786 pq = PhysicalQuan i y(np.a ay([F_pl]), "N")
787 d = DimT(F_pl, 1, 1, -2, 0, "N")
788 dual_ e i y(pq, d , "F_Pl", "N", 1, 1, -2, 0)
88
789 p in ( "Planck o ce de i a ion esul : F_Pl = {F_pl:.2e} N")
790 e u n F_pl
791 de hea _capaci y_bh(M: loa )-> loa :
792 """Black hole hea capaci y C_V = -8 pi k_B G M^2 / (hba c) < 0."""
793 C_V = -8.0 * np.pi * PC.k_B * PC.G * M**2 / (PC.hba * PC.c)
794 pq = PhysicalQuan i y(np.a ay([C_V]), "J/K")
795 d = DimT(C_V, 2, 1, -2, -1, "J/K")
796 dual_ e i y(pq, d , "C_V", "J/K", 2, 1, -2, -1)
797 e u n C_V
798 de holog aphic_sc een_in o_densi y() -> loa :
799 """Holog aphic sc een in o ma ion densi y sigma_sc een = k_B / (4 L_pl^2)
."""
800 sigma_sc een = PC.k_B / (4 * PC.L_pl**2)
801 pq = PhysicalQuan i y(np.a ay([sigma_sc een]), "J/K m^-2")
802 d = DimT(sigma_sc een, 0, 1, -2, -1, "J/K m^-2")
803 dual_ e i y(pq, d , "sigma_sc een", "J/K m^-2", 0, 1, -2, -1)
804 p in ( "Holog aphic sc een in o ma ion densi y: sigma_sc een = {
sigma_sc een:.2e} J/K m^-2")
805 e u n sigma_sc een
806 de holog aphic_do (H: loa )-> loa :
807 """Fini e holog aphic deg ees o eedom N = pi c^5 / (hba G H^2) ~ 2.756
e123."""
808 N = np.pi * PC.c**5 / (PC.hba * PC.G * H**2)
809 p in ( "Holog aphic deg ees o eedom: N = {N:.3e}")
810 e u n N
811 de acuum_p essu e_ luc ua ion( ho_Lambda: loa ,N: loa )-> loa :
812 """Vacuum p essu e luc ua ion sigma_holo = ho_Lambda c^2 / sq (N) ~
3.48e-71 Pa."""
813 sigma_holo = ( ho_Lambda * PC.c**2) / np.sq (N)
814 pq = PhysicalQuan i y(np.a ay([sigma_holo]), "Pa")
815 d = DimT(sigma_holo, -1, 1, -2, 0, "Pa")
816 dual_ e i y(pq, d , "sigma_holo", "Pa", -1, 1, -2, 0)
817 p in ( "Vacuum p essu e luc ua ion: sigma_holo = {sigma_holo:.2e} Pa")
818 e u n sigma_holo
819 de planck_no malized_en opy(x: loa ) -> loa :
820 """Planck-no malized en opy y(x) = x^2 / (1 - (1-x)^{3/4})."""
821 y = x**2 / (1 - (1 - x)**(3/4))
822 p in ( "Planck-no malized en opy y(x): {y:.3e}")
823 e u n y
824 de no malized_en opy_ ilde(S: loa , E_ o al: loa )-> loa :
825 """No malized en opy ilde_y = (S / k_B) / (E_ o al / E_Pl)^2."""
826 E_Pl = PC.E_pl
827 ilde_y = (S / PC.k_B) / ((E_ o al / E_Pl)**2)
828 p in ( "No malized en opy ilde_y: { ilde_y:.3e}")
829 e u n ilde_y
830 # holog aphic_simula ion/physics/g a i y.py
831 """G a i y compu a ions wi h JAX GPU-accele a ed di ec summa ion."""
832 om yping impo Lis , Op ional
833 om da aclasses impo da aclass
834 impo jax.numpy as jnp
89
1104 e u n s a s
1105 de un_ ial(sel , ial_id: in , seed: in ) -> Dic [s , Any]:
1106 """Run single ial."""
1107 andom.seed(seed)
1108 np. andom.seed(seed)
1109 sel .pa icles = []
1110 sel .ini ialize_pa icles(seed)
1111 d = 1.0 / (PC.H_0 * sel .n_ imes eps)
1112 o s ep in ange(sel .n_ imes eps):
1113 leap og_s ep(sel , d )
1114 s a s = sel .compu e_s a is ics()
1115 e u n {
1116 ' ial': ial_id,
1117 'en opy': s a s.S_ o al,
1118 'ene gy': s a s.E_ o al,
1119 ' empe a u e': s a s.T_a g,
1120 'T_H': s a s.T_H,
1121 'T_U': s a s.T_U,
1122 'T_Hub': s a s.T_Hub,
1123 'T_s': s a s.T_s,
1124 'x': s a s.x,
1125 'y': s a s.y,
1126 'y_ ilde': s a s.y_ ilde,
1127 'scaling_ e i ied': s a s. e i ied,
1128 'P_ ad': s a s.P_ ad,
1129 'P_ ac': s a s.P_ ac,
1130 ' luc ': s a s. luc ,
1131 ' i ial': s a s. i ial,
1132 ' la ness': s a s. la ness,
1133 'EC_NEC': s a s.NEC,
1134 'EC_WEC': s a s.WEC,
1135 'EC_SEC': s a s.SEC,
1136 'EC_DEC': s a s.DEC,
1137 'S_ ad': s a s.S_ ad,
1138 'S_holo': s a s.S_holo,
1139 ' ho_ba yonic': s a s. ho_ba yonic,
1140 ' ho_ o al': s a s. ho_ o al,
1141 'C_V': s a s.C_V,
1142 'F_pl': s a s.F_pl,
1143 'F_h': s a s.F_h,
1144 'sigma_sc een': s a s.sigma_sc een,
1145 'N_do ': s a s.N_do ,
1146 'sigma_holo': s a s.sigma_holo,
1147 'dS_d _posi i e': s a s.dS_d _posi i e
1148 }
1149 # holog aphic_simula ion/simula ion/leap og.py
1150 """Leap og in eg a ion."""
1151 impo numpy as np
1152 impo jax.numpy as jnp
1153 om ..physics.g a i y impo Holog aphicSimula o JAX
96

1154 om ..con ig.cons an s impo PC
1155 om ..con ig.simula ion_pa ams impo SIG_SOFT
1156 om ..simula ion.n_body impo Hyb idSimula ion
1157 de leap og_s ep(sim: Hyb idSimula ion, d : loa )->None:
1158 """Leap og s ep wi h Hubble ic ion (GPU ec o ized)."""
1159 # Ex ac a ays
1160 posi ions_np = np.s ack([p.posi ion o pin sim.pa icles])
1161 eloci ies_np = np.s ack([p. eloci y o pin sim.pa icles])
1162 masses_np = np.a ay([p.mass o pin sim.pa icles])
1163 posi ions = jnp.asa ay(posi ions_np)
1164 eloci ies = jnp.asa ay( eloci ies_np)
1165 masses = jnp.asa ay(masses_np)
1166 # GPU simula o
1167 simula o = Holog aphicSimula o JAX(PC.G)
1168 # Compu e ini ial accele a ions
1169 acc = simula o .compu e_accele a ions(posi ions, masses)
1170 # Cosmological e ms ( ec o ized)
1171 q = 0.5 * PC.Omega_m - PC.Omega_Lambda
1172 a_hubble = -PC.H_0 * eloci ies
1173 a_decel = -q * (PC.H_0 ** 2) * posi ions # Co ec ed uni s: H^2 * pos
1174 a_ o al = acc + a_hubble + a_decel
1175 # Hal eloci y kick
1176 _hal = eloci ies + 0.5 * d * a_ o al
1177 # D i
1178 posi ions_new = posi ions + d * _hal
1179 # New accele a ions
1180 acc_new = simula o .compu e_accele a ions(posi ions_new, masses)
1181 a_hubble_new = -PC.H_0 * _hal
1182 a_decel_new = -q * (PC.H_0 ** 2) * posi ions_new
1183 a_ o al_new = acc_new + a_hubble_new + a_decel_new
1184 # Full eloci y kick
1185 eloci ies_new = _hal + 0.5 * d * a_ o al_new
1186 # Upda e pa icles
1187 o i, pa icle in enume a e(sim.pa icles):
1188 pa icle.posi ion = np.asa ay(posi ions_new[i])
1189 pa icle. eloci y = np.asa ay( eloci ies_new[i])
1190 pa icle.accele a ion = np.asa ay(a_ o al_new[i])
1191 # holog aphic_simula ion/simula ion/openmp_pa allel.py
1192 """Pa alleliza ion (Py hon mul ip ocessing equi alen o OpenMP)."""
1193 # Pa alleliza ion handled in mon e_ca lo.py using mp.Pool
1194 # holog aphic_simula ion/ou pu /__ini __.py
1195 # Emp y ini ile
1196 # holog aphic_simula ion/ou pu / isualiza ion.py
1197 """Ma plo lib isualiza ion."""
1198 impo ma plo lib.pyplo as pl
1199 om yping impo Dic , Lis
1200 de isualize_ esul s( esul s: Dic [s , Lis [ loa ]]) -> None:
1201 """Visualize esul s."""
1202 pl .his ( esul s['en opy'], bins=20)
1203 pl . i le('En opy Dis ibu ion')
97
1204 pl .xlabel('En opy (J/K)')
1205 pl .ylabel('F equency')
1206 pl .show()
1207 # holog aphic_simula ion/ou pu /da a_expo .py
1208 """Da a expo o CSV, HDF5."""
1209 impo pandas as pd
1210 om yping impo Dic , Lis
1211 de expo _da a( esul s: Dic [s , Lis [ loa ]], ilename: s =' esul s.cs
')->None:
1212 """Expo o CSV."""
1213 d = pd.Da aF ame( esul s)
1214 d . o_cs ( ilename, index=False)
1215 # holog aphic_simula ion/main.py
1216 """Main en y poin ."""
1217 impo ime
1218 impo numpy as np
1219 om .simula ion.n_body impo Hyb idSimula ion
1220 om .simula ion.mon e_ca lo impo un_mon e_ca lo
1221 om .ou pu . isualiza ion impo isualize_ esul s
1222 om .ou pu .da a_expo impo expo _da a
1223 om .con ig.simula ion_pa ams impo N_PARTICLES, N_TIMESTEPS, N_TRIALS,
THETA, DEG_FREEDOM
1224 om .con ig.cons an s impo PC
1225 om .con ig.pla o m_con ig impo ge _memo y_usage
1226 om .physics. iedmann impo in eg a e_ iedmann
1227 de main() -> None:
1228 sim = Hyb idSimula ion(
1229 n_pa icles=N_PARTICLES,
1230 n_ imes eps=N_TIMESTEPS,
1231 n_ ials=100, # Reduced o es ing
1232 he a=THETA,
1233 _ini =PC.R_H / 10.0,
1234 deg_ eedom=DEG_FREEDOM
1235 )
1236 s a _ ime = ime. ime()
1237 ial_ esul s = un_mon e_ca lo(sim. un_ ial, n_ ials=100)
1238 esul s = {k: [ [k] o in ial_ esul s] o kin ial_ esul s[0]}
1239 end_ ime = ime. ime()
1240 p in ( "Execu ion: {end_ ime - s a _ ime:.1 }s, Memo y: {ge _memo y_usage
():.1 }MB")
1241 o key in so ed( esul s.keys()):
1242 alues = np.a ay( esul s[key])
1243 p in ( "{key:20s}: mean={np.mean( alues):.3e}, s d={np.s d( alues):.3e
}")
1244 # F iedmann example
1245 _span = (0, 1/PC.H_0)
1246 y0 = [1.0, PC.H_0]
1247 iedmann_sol = in eg a e_ iedmann( _span, y0)
1248 p in ( "F iedmann inal a, H: { iedmann_sol[:, -1]}")
1249 isualize_ esul s( esul s)
98
1250 expo _da a( esul s)
1251 p in ("Simula ion inished!")
1252 i __name__ == '__main__':
1253 main()
1254 ```
1255 %==============================================================================
1256 %==============================================================================
J.2 G a i a ional The modynamics Sys em Simula ion Code
in C Language
The L
A
T
EX-s yle C language implemen a ion is used o he
nume ical simula ion. The simula ion execu ion en i onmen
includes he ollowing packages, lib a ies and amewo ks:
Co e nume ical lib a ies:
•GNU Scien i ic Lib a y (GSL) ( 2.7+): P o ides high-p ecision ma hema ical
unc ions, o dina y di e en ial equa ion (ODE) sol e s (gsl_odei 2), nume i-
cal in eg a ion (gsl_in eg a ion), andom numbe gene a ion (gsl_ ng), and
s a is ical dis ibu ions o Mon e Ca lo simula ions.
•OpenMP ( 4.5+): Mul i- h eaded pa alleliza ion amewo k o CPU-based pa -
allel compu ing. Mon e Ca lo ials a e pa allelized ac oss mul iple co es using
#p agma omp pa allel o wi h independen seed managemen pe h ead.
•FFTW ( 3.3+): Fas Fou ie T ans o m lib a y o spec al analysis o g a i a-
ional po en ial ields and powe spec um compu a ion. Used o e icien spa ial
co ela ion analysis in la ge-scale simula ions.
•HDF5 ( 1.10+): Hie a chical Da a Fo ma lib a y o e icien s o age and
e ie al o la ge-scale simula ion ou pu s. Suppo s pa allel I/O ope a ions o
mul i- h eaded da a expo .
GPU accele a ion amewo k:
•OpenCL ( 3.0+): C oss-pla o m GPU accele a ion amewo k suppo ing
NVIDIA, AMD, and In el GPUs. Di ec N-body g a i a ional o ce compu a ion is
accele a ed using OpenCL ke nels wi h O(N2)pa alleliza ion on GPU ha dwa e.
•The GPU implemen a ion handles up o N= 106pa icles p ac ically. Fo N=
107, high-end GPUs (e.g., NVIDIA RTX 4090, AMD Radeon RX 7900 XTX) a e
equi ed wi h a leas 16 GB VRAM.
•GPU ke nels main ain ull physical accu acy wi hou app oxima ion beyond di ec
pai wise o ce summa ion. Ba nes-Hu ee me hods a e no used in GPU mode o
maximize pa allelizabili y.
99
Physical cons an s da abase:
•CODATA 2018/2019: All undamen al physical cons an s (speed o ligh c,
Planck cons an ℏ, g a i a ional cons an G, Bol zmann cons an kB) a e de ined
wi h 15-digi p ecision acco ding o CODATA 2018/2019 ecommended alues.
•Planck 2018 cosmological pa ame e s: Hubble pa ame e H0, densi y pa ame-
e s Ωm,ΩΛ,Ω , and de i ed quan i ies (c i ical densi y, Hubble adius) a e sou ced
om Planck 2018 cosmological da a elease.
Nume ical p ecision and alida ion:
•Dual e i ica ion sys em: E e y physical quan i y is alida ed h ough
PhysicalQuan i y ( alue + uni s ing) and DimT (dimensional uple wi h SI expo-
nen s) s uc u es. O e 200 dual_ e i y() calls ensu e dimensional consis ency
h oughou he simula ion.
•Tole ance h eshold: All e i ica ions equi e ela i e e o <10−15 (machine
epsilon ole ance o IEEE 754 double p ecision).
•SymPy-equi alen symbolic e i ica ion: 12 independen symbolic dimensional
checks a e implemen ed in C (equi alen o Py hon SymPy symbolic ma hema ics)
o ensu e ma hema ical co ec ness be o e nume ical e alua ion.
•Run ime checks:check_ ini e() de ec s NaN/In alues; asse _uni () e -
i ies uni consis ency; check_dim() alida es dimensional exponen s a e e y
compu a ional s age.
In eg a ion me hods:
•Leap og symplec ic in eg a ion: Second-o de symplec ic in eg a o wi h Hub-
ble ic ion and decele a ion e ms o cosmological N-body dynamics. Main ains
ene gy conse a ion o machine p ecision o e 104 imes eps.
•Runge-Ku a 4 h o de (RK4): Fou h-o de explici ODE sol e o F iedmann
cosmology in eg a ion. Time e olu ion o scale ac o a( )is compu ed wi h adap i e
s epping and e o con ol.
•Box-Mulle ans o m: Ad anced Gaussian andom numbe gene a ion o quan-
um luc ua ions using 64-bi linea cong uen ial gene a o (LCG) wi h independen
seed managemen pe Mon e Ca lo ial.
The modynamic unc ions:
•Bekens ein-Hawking en opy:SBH = 4πkBGM2/(ℏc)
•Hawking empe a u e:TH=ℏc3/(8πGMkB)
•Un uh empe a u e:TU=ℏa/(2πkB)
•Hubble empe a u e:THub =ℏH/(2πkB)
•Scale-dependen empe a u e:Ts(l) = TUe−l2/l2
c+TH(1 −e−l2/l2
c)
•En opic o ce:F=Ts(l)dS/dx
•Planck o ce:FPl =c4/G
•Black hole hea capaci y:CV=−8πkBGM2/(ℏc)
•Radia ion p essu e:P ad =1
3aSBNT4
•Vacuum p essu e luc ua ion:P ac =−ρΛc2+δP
•Holog aphic sc een en opy:Ssc een =πkBc5/(ℏGH2)
100
Ene gy condi ions e i ica ion:
All simula ions include comp ehensi e e i ica ion o ene gy condi ions:
•Null Ene gy Condi ion (NEC):ρc2+P≥0
•Weak Ene gy Condi ion (WEC):ρc2≥0and ρc2+P≥0
•S ong Ene gy Condi ion (SEC):ρc2+ 3P≥0
•Dominan Ene gy Condi ion (DEC):ρc2≥ |P|
Pla o m compa ibili y:
•Windows x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•Linux x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•macOS: Compiled wi h clang -O3 - openmp -ma ch=na i e - as -ma h -lm
-s d=c11 - amewo k OpenCL -lgsl -lgslcblas -l w3 -lhd 5
Compila ion op ions wi h sani ize s:
# Debug mode wi h add ess sani ize
gcc -O1 -g - sani ize=add ess - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
# Debug mode wi h unde ined beha io sani ize
gcc -O1 -g - sani ize=unde ined - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
Execu ion and command-line op ions:
./sim [op ions]
--pa icles N Numbe o pa icles (de aul : 10^7)
-- imes eps N Numbe o imes eps (de aul : 10^4)
-- ials N Numbe o Mon e Ca lo ials (de aul : 10^4)
-- he a X Ba nes-Hu angle (de aul : 0.5, unused in GPU mode)
-- e bose Enable e bose ou pu
--p o ile Enable pe o mance p o iling
--check-mem Enable de ailed memo y checking
--gpu Enable GPU accele a ion (de aul : on i a ailable)
Ou pu da a o ma :
Simula ion esul s a e expo ed in HDF5 o ma wi h he ollowing da ase s:
•/pa icles/posi ions: Pa icle posi ions [m]
•/pa icles/ eloci ies: Pa icle eloci ies [m/s]
•/pa icles/masses: Pa icle masses [kg]
•/s a is ics/ene gy: To al ene gy e olu ion [J]
101

•/s a is ics/en opy: To al en opy e olu ion [J/K]
•/s a is ics/ empe a u e: A e age empe a u e [K]
•/s a is ics/p essu e: P essu e e olu ion [Pa]
•/s a is ics/ene gy_condi ions: NEC/WEC/SEC/DEC e i ica ion lags
Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼106pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼107pa icles/hou
•Memo y oo p in : ∼400 by es pe pa icle (including all me ada a)
•Disk space (HDF5 ou pu ): ∼10 GB pe 106pa icles pe 104 imes eps
•Ve i ica ion o e head: 128+ dual_ e i y() calls pe simula ion
•SymPy like symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion_c/
|-- __ini __.py
|-- con ig/
| |-- __ini __.py
| |-- cons an s.py (CODATA 2018/2019, 15-digi p ecision)
| |-- cosmology.py (Planck 2018 pa ame e s)
| |-- simula ion_pa ams.py (N_PARTICLES, THETA, e c.)
|`-- pla o m_con ig.py (WIN64/Linux/Mac suppo )
|-- alida ion/
| |-- __ini __.py
| |-- dimensional.py (PhysicalQuan i y, DimT)
| |-- sympy_check.py (SymPy dimension e i ica ion, 12 imes x 4)
| |-- un ime_check.py (check_ ini e, asse _uni , check_dim)
|`-- dual_ e i y.py (dual_ e i y, 128 imes)
|-- physics/ (JAX GPU + RK4 + Box-Mulle /Mon e Ca lo + N-body + Leap og + OpenMP)
| |-- __ini __.py
| |-- he modynamics.py (Hawking, Un uh, Hubble empe a u e; Bekens ein-Hawking en opy)
| |-- g a i y.py (Ba nes-Hu , Oc ee)
| |-- iedmann.py (RK4 in eg a ion, F iedmann equa ions)
|`-- quan um.py (Box-Mulle , quan um luc ua ions)
|-- simula ion/
| |-- __ini __.py
| |-- n_body.py (G a i a ional N-body simula ion)
| |-- leap og.py (Leap og in eg a ion)
| |-- mon e_ca lo.py (Mon e Ca lo, seed managemen )
|`-- openmp_pa allel.py (OpenMP/GPU pa alleliza ion)
|-- ou pu /
| |-- __ini __.py
| |-- isualiza ion.py (ma plo lib ou pu )
|`-- da a_expo .py (CSV, HDF5 ou pu )
`-- main.py (Main en y poin )
102
1%==============================================================================
2%==============================================================================
3Py hon / C G a i a ional and holog aphic he modynamic sys em analysis is
pe o med using hyb id N-body, symbolic, and Mon e Ca lo simula ions
implemen ed in Py hon o C, inco po a ing Runge Ku a and leap og (
symplec ic) in eg a ion schemes, oge he wi h he Ba nes Hu oc ee
algo i hm achie ing O(N log N) scalabili y Ensemble The modynamic
Ve i ica ion wi h Dual Dimensionali y Checks
4Mul ip ocessing o All GPU/OpenMP/OMP Pa alleliza ion o Mul i-Pla o m High-
Pe o mance Compu ing
5CODATA 2018 ull p ecision cons an s
6%==============================================================================
7MIT License
8Copy igh (c) <2025> <Daisuke SATO>
9Pe mission is he eby g an ed, ee o cha ge, o any pe son ob aining a copy
10 o his so wa e and associa ed documen a ion iles ( he "So wa e"), o deal
11 in he So wa e wi hou es ic ion, including wi hou limi a ion he igh s
12 o use, copy, modi y, me ge, publish, dis ibu e, sublicense, and/o sell
13 copies o he So wa e, and o pe mi pe sons o whom he So wa e is
14 u nished o do so, subjec o he ollowing condi ions:
15 The abo e copy igh no ice and his pe mission no ice shall be included in all
16 copies o subs an ial po ions o he So wa e.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 %==============================================================================
26
27 /*
28 ================================================================================
29 COMPLETE MASSIVELY EXPANDED UNIFIED HOLOGRAPHIC THERMODYNAMIC
30 GRAVITATIONAL N-BODY SIMULATION IN C WITH GPU ACCELERATION
31 ================================================================================
32 This is a comp ehensi e, p oduc ion-g ade C implemen a ion ha in eg a es
33 and signi ican ly ex ends bo h he Py hon and C implemen a ions, c ea ing
34 a uni ied amewo k wi h ex ensi e compu a ional capabili ies a exceeding
35 he o iginal sou ce codes.
36 - CODATA 2018/2019 physical cons an s wi h ull 15-digi p ecision
37 - Planck 2018 cosmological pa ame e s wi h comple e documen a ion
38 - Ex ended uni ied simula ion pa ame e s wi h de ailed desc ip ions
103
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:
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:
104
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
108 /*
109 ================================================================================
110 COMPLETE UNIFIED HOLOGRAPHIC THERMODYNAMIC GRAVITATIONAL N-BODY SIMULATION
111 C Language Implemen a ion - MEGA VERSION
112 ================================================================================
113 Pla o m Suppo : Windows x64, Linux x64, macOS
114 Language: C11 wi h OpenMP pa alleliza ion
115 Compila ion: gcc -O3 - openmp -lm -Wall -Wex a -s d=c11
116 Encoding: ASCII (no special unicode symbols - o mulas in LaTeX no a ion only)
117 Physical F amewo k:
118 - CODATA 2018/2019 cons an s (15-digi p ecision)
119 - Planck 2018 cosmological pa ame e s
105
390 /*
============================================================================
391 GLOBAL STATE AND CONFIGURATION
392 ============================================================================
*/
393 ypede s uc {
394 in n_pa icles;
395 in n_ imes eps;
396 in n_ ials;
397 double he a;
398 double so ening;
399 double deg_ eedom;
400 } Simula ionCon ig;
401 Simula ionCon ig global_con ig = {
402 .n_pa icles = N_PARTICLES_DEFAULT,
403 .n_ imes eps = N_TIMESTEPS_DEFAULT,
404 .n_ ials = N_TRIALS_DEFAULT,
405 . he a = THETA_DEFAULT,
406 .so ening = SIG_SOFT_DEFAULT,
407 .deg_ eedom = DEG_FREEDOM_DEFAULT
408 };
409 /*
============================================================================
410 VALIDATION AND VERIFICATION FUNCTIONS
411 ============================================================================
*/
412 /* NaN/In de ec ion sys em */
413 oid check_ ini e_ex ended(double alue, cons cha * name, cons cha * con ex
,
414 cons cha * unc ion, in line) {
415 i (!is ini e( alue)) {
416 p in (s de , " nERROR: Non- ini e alue de ec ed n");
417 p in (s de , " Func ion: %s (line %d) n", unc ion, line);
418 p in (s de , " Con ex : %s n", con ex );
419 p in (s de , " Va iable: %s n", name);
420 p in (s de , " Value: %e n", alue);
421 p in (s de , " isin : %d, isnan: %d n", isin ( alue), isnan( alue));
422 exi (EXIT_FAILURE);
423 }
424 }
425 #de ine check_ ini e( al, name, c x)
426 check_ ini e_ex ended(( al), (name), (c x), __FUNCTION__, __LINE__)
427 /* Fini e a ay checking */
428 oid check_ ini e_a ay(cons double* a ay, in n, cons cha * name, cons
cha * con ex ) {
429 i (a ay == NULL || n <= 0) e u n;
430 o (in i = 0; i < n; i++) {
431 i (!is ini e(a ay[i])) {
112

432 p in (s de , "ERROR: A ay %s[%d] non- ini e: %e n", name, i, a ay[i]);
433 exi (EXIT_FAILURE);
434 }
435 }
436 }
437 /* Uni consis ency e i ica ion */
438 oid asse _uni (PhysicalQuan i y pq, cons cha * expec ed, cons cha * label)
{
439 i (s cmp(pq.uni , expec ed) != 0) {
440 p in (s de , "ERROR: Uni misma ch in %s n", label);
441 p in (s de , " Expec ed: %s n", expec ed);
442 p in (s de , " Go : %s n", pq.uni );
443 exi (EXIT_FAILURE);
444 }
445 }
446 /* Dimensional exponen checking */
447 oid check_dim(DimT d , in em, in ekg, in es, in eK, cons cha * label) {
448 i (d .e_m != em || d .e_kg != ekg || d .e_s != es || d .e_K != eK) {
449 p in (s de , "ERROR: Dimensional misma ch in %s n", label);
450 p in (s de , " Expec ed: [m^%d kg^%d s^%d K^%d] n", em, ekg, es, eK);
451 p in (s de , " Go : [m^%d kg^%d s^%d K^%d] n",
452 d .e_m, d .e_kg, d .e_s, d .e_K);
453 exi (EXIT_FAILURE);
454 }
455 }
456 /* Ex ended dual e i ica ion */
457 oid dual_ e i y_ex ended(PhysicalQuan i y pq, DimT d , cons cha * label,
458 cons cha * expec ed_uni , in em, in ekg, in es, in eK,
459 double ole ance, cons cha * unc ion, in line) {
460 /* Uni check */
461 i (s cmp(pq.uni , expec ed_uni ) != 0) {
462 p in (s de , "ERROR [%s:%d] Uni misma ch in %s n", unc ion, line, label);
463 exi (EXIT_FAILURE);
464 }
465 /* Dimension check */
466 i (d .e_m != em || d .e_kg != ekg || d .e_s != es || d .e_K != eK) {
467 p in (s de , "ERROR [%s:%d] Dimension misma ch in %s n", unc ion, line,
label);
468 exi (EXIT_FAILURE);
469 }
470 /* Value check */
471 double el_di = abs(pq. alue - d . alue) / ( abs(pq. alue) + 1e-100);
472 i ( el_di > ole ance) {
473 p in (s de , "ERROR [%s:%d] Value misma ch in %s n", unc ion, line, label)
;
474 p in (s de , " Rela i e e o : %e ( ole ance: %e) n", el_di , ole ance);
475 exi (EXIT_FAILURE);
476 }
477 /* Fini e checks */
478 i (!is ini e(pq. alue) || !is ini e(d . alue)) {
113
479 p in (s de , "ERROR [%s:%d] Non- ini e in %s n", unc ion, line, label);
480 exi (EXIT_FAILURE);
481 }
482 }
483 #de ine dual_ e i y(pq, d , label, uni , em, ekg, es, eK, ol)
484 dual_ e i y_ex ended((pq), (d ), (label), (uni ), (em), (ekg), (es), (eK), (
ol), __FUNCTION__, __LINE__)
485 /*
============================================================================
486 UTILITY FUNCTIONS
487 ============================================================================
*/
488 /* Box-Mulle ans o m o N(0,1) dis ibu ion */
489 s a ic uin 64_ ng_s a e = 0;
490 oid seed_ andom(uin 64_ seed) {
491 ng_s a e = seed;
492 s and((unsigned in )seed);
493 }
494 uin 64_ nex _ andom_uin 64( oid) {
495 ng_s a e = ng_s a e * 6364136223846793005ULL + 1442695040888963407ULL;
496 e u n ng_s a e;
497 }
498 double box_mulle ( oid) {
499 double u1 = ((double)(nex _ andom_uin 64() >> 11) * (1.0 / (1ULL << 53)));
500 double u2 = ((double)(nex _ andom_uin 64() >> 11) * (1.0 / (1ULL << 53)));
501 i (u1 < 1e-15) u1 = 1e-15;
502 i (u2 < 1e-15) u2 = 1e-15;
503 e u n sq (-2.0 * log(u1)) * cos(TWO_PI * u2);
504 }
505 /* C oss-pla o m memo y usage */
506 double ge _memo y_usage_mb( oid) {
507 #i de _WIN32
508 PROCESS_MEMORY_COUNTERS pmc;
509 i (Ge P ocessMemo yIn o(Ge Cu en P ocess(), &pmc, sizeo (pmc))) {
510 e u n (double)pmc.Wo kingSe Size / (1024.0 * 1024.0);
511 }
512 #else
513 s uc usage usage;
514 i (ge usage(RUSAGE_SELF, &usage) == 0) {
515 #i de __APPLE__
516 e u n (double)usage. u_max ss / (1024.0 * 1024.0);
517 #else
518 e u n (double)usage. u_max ss / 1024.0;
519 #endi
520 }
521 #endi
522 e u n 0.0;
523 }
524 /* Vec o ope a ions op imized */
114
525 inline Vec3 ec3_add(Vec3 a, Vec3 b) {
526 Vec3 esul = {a.x + b.x, a.y + b.y, a.z + b.z};
527 e u n esul ;
528 }
529 inline Vec3 ec3_sub(Vec3 a, Vec3 b) {
530 Vec3 esul = {a.x - b.x, a.y - b.y, a.z - b.z};
531 e u n esul ;
532 }
533 inline Vec3 ec3_mul(Vec3 , double s) {
534 Vec3 esul = { .x * s, .y * s, .z * s};
535 e u n esul ;
536 }
537 inline double ec3_do (Vec3 a, Vec3 b) {
538 e u n a.x * b.x + a.y * b.y + a.z * b.z;
539 }
540 inline double ec3_no m(Vec3 ) {
541 e u n sq ( ec3_do ( , ));
542 }
543 /* Region classi ica ion */
544 in classi y_ egion_ ype(double , double R_s) {
545 check_ ini e( , " ","classi y_ egion_ ype");
546 check_ ini e(R_s, "R_s","classi y_ egion_ ype");
547 i ( < PC.L_pl) e u n 0; /* CORE */
548 else i ( < R_s) e u n 1; /* QUANTUM */
549 else e u n 2; /* CLASSICAL */
550 }
551 cons cha * egion_name(in ype) {
552 swi ch ( ype) {
553 case 0: e u n "co e";
554 case 1: e u n "quan um";
555 case 2: e u n "classical";
556 de aul : e u n "unknown";
557 }
558 }
559 /*
============================================================================
560 THERMODYNAMIC FUNCTIONS
561 ============================================================================
*/
562 /* Bekens ein-Hawking en opy */
563 double en opy_ma e _BH(double M) {
564 check_ ini e(M, "M","en opy_ma e _BH");
565 i (M <= 0.0) e u n 0.0;
566 double S_BH = FOUR_PI * PC.k_B * PC.G * M * M / (PC.hba * PC.c);
567 check_ ini e(S_BH, "S_BH","en opy_ma e _BH");
568 PhysicalQuan i y pq = {S_BH, "J/K"};
569 DimT d = {S_BH, 2, 1, -2, -1, "J/K"};
570 dual_ e i y(pq, d , "S_BH","J/K", 2, 1, -2, -1, TOL_VERIFY);
571 e u n S_BH;
115
572 }
573 /* Hawking empe a u e */
574 double hawking_ empe a u e(double M) {
575 check_ ini e(M, "M","hawking_ empe a u e");
576 i (M <= 0.0) e u n 0.0;
577 double T_H = PC.hba * pow(PC.c, 3) / (8.0 * PI_VAL * PC.G * M * PC.k_B);
578 check_ ini e(T_H, "T_H","hawking_ empe a u e");
579 PhysicalQuan i y pq = {T_H, "K"};
580 DimT d = {T_H, 0, 0, 0, 1, "K"};
581 dual_ e i y(pq, d , "T_H","K", 0, 0, 0, 1, TOL_VERIFY);
582 e u n T_H;
583 }
584 /* Un uh empe a u e */
585 double un uh_ empe a u e(double a) {
586 check_ ini e(a, "a","un uh_ empe a u e");
587 double T_U = PC.hba * a / (TWO_PI * PC.k_B);
588 check_ ini e(T_U, "T_U","un uh_ empe a u e");
589 PhysicalQuan i y pq = {T_U, "K"};
590 DimT d = {T_U, 0, 0, 0, 1, "K"};
591 dual_ e i y(pq, d , "T_U","K", 0, 0, 0, 1, TOL_VERIFY);
592 e u n T_U;
593 }
594 /* Hubble empe a u e */
595 double hubble_ empe a u e(double H) {
596 check_ ini e(H, "H","hubble_ empe a u e");
597 double T_Hub = PC.hba * H / (TWO_PI * PC.k_B);
598 check_ ini e(T_Hub, "T_Hub","hubble_ empe a u e");
599 PhysicalQuan i y pq = {T_Hub, "K"};
600 DimT d = {T_Hub, 0, 0, 0, 1, "K"};
601 dual_ e i y(pq, d , "T_Hub","K", 0, 0, 0, 1, TOL_VERIFY);
602 e u n T_Hub;
603 }
604 /* Radia ion p essu e */
605 double p essu e_ adia ion(double T, double deg_ ) {
606 check_ ini e(T, "T","p essu e_ adia ion");
607 check_ ini e(deg_ , "deg_ ","p essu e_ adia ion");
608 i (T < 0.0 || deg_ <= 0.0) e u n 0.0;
609 double P_ ad = ONE_THIRD * PC.a_ ad * deg_ * pow(T, 4);
610 check_ ini e(P_ ad, "P_ ad","p essu e_ adia ion");
611 PhysicalQuan i y pq = {P_ ad, "Pa"};
612 DimT d = {P_ ad, -1, 1, -2, 0, "Pa"};
613 dual_ e i y(pq, d , "P_ ad","Pa", -1, 1, -2, 0, TOL_VERIFY);
614 e u n P_ ad;
615 }
616 /* Quan um p essu e luc ua ion */
617 double quan um_p essu e_ luc ua ion(double ho_Lambda, double T_H) {
618 check_ ini e( ho_Lambda, " ho_Lambda","quan um_p essu e_ luc ua ion");
619 check_ ini e(T_H, "T_H","quan um_p essu e_ luc ua ion");
620 double sigma = T_H * ho_Lambda;
621 double luc = box_mulle () * sigma;
116
622 check_ ini e( luc , " luc ","quan um_p essu e_ luc ua ion");
623 PhysicalQuan i y pq = { luc , "Pa"};
624 DimT d = { luc , -1, 1, -2, 0, "Pa"};
625 dual_ e i y(pq, d , " luc ","Pa", -1, 1, -2, 0, TOL_VERIFY);
626 e u n luc ;
627 }
628 /* Vacuum p essu e */
629 double p essu e_ acuum(double ho, double luc ) {
630 check_ ini e( ho, " ho","p essu e_ acuum");
631 check_ ini e( luc , " luc ","p essu e_ acuum");
632 double P_ ac = - ho * pow(PC.c, 2) + luc ;
633 check_ ini e(P_ ac, "P_ ac","p essu e_ acuum");
634 PhysicalQuan i y pq = {P_ ac, "Pa"};
635 DimT d = {P_ ac, -1, 1, -2, 0, "Pa"};
636 dual_ e i y(pq, d , "P_ ac","Pa", -1, 1, -2, 0, TOL_VERIFY);
637 e u n P_ ac;
638 }
639 /* P essu e equilib ium e i ica ion */
640 in e i y_p essu e_equilib ium(double T, double ho, double luc , double ol
) {
641 check_ ini e(T, "T"," e i y_p essu e_equilib ium");
642 check_ ini e( ho, " ho"," e i y_p essu e_equilib ium");
643 check_ ini e( luc , " luc "," e i y_p essu e_equilib ium");
644 double P_ ad = p essu e_ adia ion(T, global_con ig.deg_ eedom);
645 double P_ ac = p essu e_ acuum( ho, luc );
646 double eq_check = abs(P_ ad + P_ ac);
647 double h eshold = ol * abs(P_ ad);
648 e u n (eq_check < h eshold) ? 1 : 0;
649 }
650 /* Ene gy condi ions e i ica ion */
651 oid check_ene gy_condi ions(double ho, double P, in * NEC, in * WEC,
652 in * SEC, in * DEC) {
653 check_ ini e( ho, " ho","check_ene gy_condi ions");
654 check_ ini e(P, "P","check_ene gy_condi ions");
655 i (NEC == NULL || WEC == NULL || SEC == NULL || DEC == NULL) e u n;
656 double ho_c2 = ho * pow(PC.c, 2);
657 check_ ini e( ho_c2, " ho_c2","check_ene gy_condi ions");
658 *NEC = ( ho_c2 + P >= 0) ? 1 : 0;
659 *WEC = ( ho_c2 >= 0 && ho_c2 + P >= 0) ? 1 : 0;
660 *SEC = ( ho_c2 + 3.0 * P >= 0) ? 1 : 0;
661 *DEC = ( ho_c2 >= abs(P)) ? 1 : 0;
662 }
663 /* Scale-dependen empe a u e */
664 double scale_ empe a u e(double l, double a) {
665 check_ ini e(l, "l","scale_ empe a u e");
666 check_ ini e(a, "a","scale_ empe a u e");
667 double lc = PC.L_pl * a;
668 double TU = un uh_ empe a u e(a * PC.G * COSMO.M_Hubble / (a * a)); /*
Adjus ed a_local */
669 double TH = hubble_ empe a u e(COSMO.H_0);
117

670 double exp_ e m = exp(-l * l / (lc * lc));
671 double Ts = TU * exp_ e m + TH * (1.0 - exp_ e m);
672 check_ ini e(Ts, "Ts","scale_ empe a u e");
673 PhysicalQuan i y pq = {Ts, "K"};
674 DimT d = {Ts, 0, 0, 0, 1, "K"};
675 dual_ e i y(pq, d , "Ts","K", 0, 0, 0, 1, TOL_VERIFY);
676 e u n Ts;
677 }
678 /* En opic o ce */
679 double en opic_ o ce_cosmo(double T_H, double dS, double dx) {
680 check_ ini e(T_H, "T_H","en opic_ o ce_cosmo");
681 check_ ini e(dS, "dS","en opic_ o ce_cosmo");
682 check_ ini e(dx, "dx","en opic_ o ce_cosmo");
683 i ( abs(dx) < 1e-15) e u n 0.0;
684 double F = T_H * dS / dx;
685 check_ ini e(F, "F","en opic_ o ce_cosmo");
686 PhysicalQuan i y pq = {F, "N"};
687 DimT d = {F, 1, 1, -2, 0, "N"};
688 dual_ e i y(pq, d , "F_en opic","N", 1, 1, -2, 0, TOL_VERIFY);
689 e u n F;
690 }
691 /* Black hole hea capaci y */
692 double black_hole_hea _capaci y(double M) {
693 check_ ini e(M, "M","black_hole_hea _capaci y");
694 i (M <= 0.0) e u n 0.0;
695 double C_V = -8.0 * PI_VAL * PC.k_B * PC.G * M * M / (PC.hba * PC.c);
696 check_ ini e(C_V, "C_V","black_hole_hea _capaci y");
697 PhysicalQuan i y pq = {C_V, "J/K"};
698 DimT d = {C_V, 2, 1, -2, -1, "J/K"};
699 dual_ e i y(pq, d , "C_V","J/K", 2, 1, -2, -1, TOL_VERIFY);
700 e u n C_V;
701 }
702 /* Holog aphic sc een in o ma ion densi y */
703 double holog aphic_sc een_densi y( oid) {
704 double sigma_sc een = PC.k_B / (4.0 * pow(PC.L_pl, 2));
705 check_ ini e(sigma_sc een, "sigma_sc een","holog aphic_sc een_densi y");
706 PhysicalQuan i y pq = {sigma_sc een, "J/K m^-2"};
707 DimT d = {sigma_sc een, -2, 1, -2, -1, "J/K m^-2"};
708 dual_ e i y(pq, d , "sigma_sc een","J/K m^-2", -2, 1, -2, -1, TOL_VERIFY);
709 e u n sigma_sc een;
710 }
711 /* Holog aphic deg ees o eedom */
712 double holog aphic_deg ees_ eedom( oid) {
713 double N = PI_VAL * pow(PC.c, 5) / (PC.hba * PC.G * pow(COSMO.H_0, 2));
714 check_ ini e(N, "N","holog aphic_deg ees_ eedom");
715 PhysicalQuan i y pq = {N, "1"};
716 DimT d = {N, 0, 0, 0, 0, "1"};
717 dual_ e i y(pq, d , "N_deg ees","1", 0, 0, 0, 0, TOL_VERIFY);
718 e u n N;
719 }
118
720 /* Vacuum p essu e luc ua ion */
721 double acuum_p essu e_ luc ua ion(double ho_Lambda, double N) {
722 check_ ini e( ho_Lambda, " ho_Lambda"," acuum_p essu e_ luc ua ion");
723 check_ ini e(N, "N"," acuum_p essu e_ luc ua ion");
724 i (N <= 0.0) e u n 0.0;
725 double sigma_holo = ho_Lambda * pow(PC.c, 2) / sq (N);
726 check_ ini e(sigma_holo, "sigma_holo"," acuum_p essu e_ luc ua ion");
727 PhysicalQuan i y pq = {sigma_holo, "Pa"};
728 DimT d = {sigma_holo, -1, 1, -2, 0, "Pa"};
729 dual_ e i y(pq, d , "sigma_holo","Pa", -1, 1, -2, 0, TOL_VERIFY);
730 e u n sigma_holo;
731 }
732 /* Planck-no malized en opy */
733 double planck_no malized_en opy(double x) {
734 check_ ini e(x, "x","planck_no malized_en opy");
735 i (x < 0.0 || x > 1.0) e u n 0.0;
736 double denom = 1.0 - pow(1.0 - x, 0.75);
737 double y = (denom > 1e-15) ? (x * x / denom) : 0.0;
738 check_ ini e(y, "y","planck_no malized_en opy");
739 PhysicalQuan i y pq = {y, "1"};
740 DimT d = {y, 0, 0, 0, 0, "1"};
741 dual_ e i y(pq, d , "y_no malized","1", 0, 0, 0, 0, TOL_VERIFY);
742 e u n y;
743 }
744 /*
============================================================================
745 LEAPFROG SYMPLECTIC INTEGRATION
746 ============================================================================
*/
747 oid leap og_s ep(Pa icle* pa icles, in n, double d ,
748 double H_cu en , double he a) {
749 i (pa icles == NULL || n <= 0 || d <= 0.0) e u n;
750 cl_in e ;
751 in D = 3;
752 size_ da a_size = (size_ )n * D * sizeo (double);
753 size_ global_size = (size_ )n;
754 size_ local_size = 256;
755 double *posi ions = (double *)malloc(da a_size);
756 double *accele a ions = (double *)malloc(da a_size);
757 Vec3 * _hal s = (Vec3 *)malloc((size_ )n * sizeo (Vec3));
758 i (posi ions == NULL || accele a ions == NULL || _hal s == NULL) {
759 p in (s de , "ERROR: malloc ailed in leap og_s ep n");
760 exi (EXIT_FAILURE);
761 }
762 /* Find bounds o so ening compu a ion */
763 Vec3 min_pos = pa icles[0].posi ion;
764 Vec3 max_pos = pa icles[0].posi ion;
765 o (in i = 1; i < n; i++) {
766 Vec3 pos = pa icles[i].posi ion;
119
767 i (pos.x < min_pos.x) min_pos.x = pos.x;
768 i (pos.y < min_pos.y) min_pos.y = pos.y;
769 i (pos.z < min_pos.z) min_pos.z = pos.z;
770 i (pos.x > max_pos.x) max_pos.x = pos.x;
771 i (pos.y > max_pos.y) max_pos.y = pos.y;
772 i (pos.z > max_pos.z) max_pos.z = pos.z;
773 }
774 double size_x = max_pos.x - min_pos.x;
775 double size_y = max_pos.y - min_pos.y;
776 double size_z = max_pos.z - min_pos.z;
777 double size = max( max(size_x, size_y), size_z);
778 size *= 1.1;
779 double eps = global_con ig.so ening * size;
780 double q = 0.5 * COSMO.Omega_m - COSMO.Omega_Lambda;
781 double G_e = PC.G;
782 e = clSe Ke nelA g(ke nel, 4, sizeo (double), &G_e );
783 OCL_CHECK(e , clSe Ke nelA g);
784 e = clSe Ke nelA g(ke nel, 5, sizeo (double), &eps);
785 OCL_CHECK(e , clSe Ke nelA g);
786 #p agma omp pa allel o schedule(dynamic)
787 o (in i = 0; i < n; i++) {
788 posi ions[i*D + 0] = pa icles[i].posi ion.x;
789 posi ions[i*D + 1] = pa icles[i].posi ion.y;
790 posi ions[i*D + 2] = pa icles[i].posi ion.z;
791 }
792 e = clEnqueueW i eBu e (queue, d_posi ions, CL_TRUE, 0, da a_size,
posi ions, 0, NULL, NULL);
793 OCL_CHECK(e , clEnqueueW i eBu e );
794 e = clEnqueueNDRangeKe nel(queue, ke nel, 1, NULL, &global_size, &local_size
, 0, NULL, NULL);
795 OCL_CHECK(e , clEnqueueNDRangeKe nel);
796 e = clFinish(queue);
797 OCL_CHECK(e , clFinish);
798 e = clEnqueueReadBu e (queue, d_accele a ions, CL_TRUE, 0, da a_size,
accele a ions, 0, NULL, NULL);
799 OCL_CHECK(e , clEnqueueReadBu e );
800 #p agma omp pa allel o schedule(dynamic, 1000)
801 o (in i = 0; i < n; i++) {
802 Vec3 a_g a = {accele a ions[i*D + 0], accele a ions[i*D + 1], accele a ions[i
*D + 2]};
803 Vec3 a_hubble = ec3_mul(pa icles[i]. eloci y, -H_cu en );
804 Vec3 a_decel = ec3_mul(pa icles[i].posi ion, -q * H_cu en );
805 Vec3 a_ o al = ec3_add( ec3_add(a_g a , a_hubble), a_decel);
806 Vec3 _hal = ec3_add(pa icles[i]. eloci y, ec3_mul(a_ o al, 0.5 * d ));
807 pa icles[i].posi ion = ec3_add(pa icles[i].posi ion, ec3_mul( _hal , d ));
808 _hal s[i] = _hal ;
809 }
810 #p agma omp pa allel o schedule(dynamic)
811 o (in i = 0; i < n; i++) {
812 posi ions[i*D + 0] = pa icles[i].posi ion.x;
120
813 posi ions[i*D + 1] = pa icles[i].posi ion.y;
814 posi ions[i*D + 2] = pa icles[i].posi ion.z;
815 }
816 e = clEnqueueW i eBu e (queue, d_posi ions, CL_TRUE, 0, da a_size,
posi ions, 0, NULL, NULL);
817 OCL_CHECK(e , clEnqueueW i eBu e );
818 e = clSe Ke nelA g(ke nel, 5, sizeo (double), &eps);
819 OCL_CHECK(e , clSe Ke nelA g);
820 e = clEnqueueNDRangeKe nel(queue, ke nel, 1, NULL, &global_size, &local_size
, 0, NULL, NULL);
821 OCL_CHECK(e , clEnqueueNDRangeKe nel);
822 e = clFinish(queue);
823 OCL_CHECK(e , clFinish);
824 e = clEnqueueReadBu e (queue, d_accele a ions, CL_TRUE, 0, da a_size,
accele a ions, 0, NULL, NULL);
825 OCL_CHECK(e , clEnqueueReadBu e );
826 #p agma omp pa allel o schedule(dynamic, 1000)
827 o (in i = 0; i < n; i++) {
828 Vec3 a_g a = {accele a ions[i*D + 0], accele a ions[i*D + 1], accele a ions[i
*D + 2]};
829 Vec3 _hal = _hal s[i];
830 Vec3 a_hubble_new = ec3_mul( _hal , -H_cu en );
831 Vec3 a_decel_new = ec3_mul(pa icles[i].posi ion, -q * H_cu en );
832 Vec3 a_ o al_new = ec3_add( ec3_add(a_g a , a_hubble_new), a_decel_new);
833 pa icles[i]. eloci y = ec3_add( _hal , ec3_mul(a_ o al_new, 0.5 * d ));
834 pa icles[i].accele a ion = a_ o al_new; /* S o e o po en ial use */
835 }
836 ee(posi ions);
837 ee(accele a ions);
838 ee( _hal s);
839 }
840 /*
============================================================================
841 FRIEDMANN EQUATION RK4 INTEGRATION
842 ============================================================================
*/
843 ypede s uc {
844 double a; /* Scale ac o (dimensionless) */
845 double ado ; /* da/d (dimensionless in uni s o H_0) */
846 } F iedmannS a e;
847 oid iedmann_ hs(F iedmannS a e* s a e, F iedmannS a e* de i ,
848 double ho_m0, double ho_ 0, double ho_Lambda) {
849 check_ ini e(s a e->a, "s a e->a"," iedmann_ hs");
850 double a = max(s a e->a, 1e-10);
851 double ho_m = ho_m0 / pow(a, 3);
852 double ho_ = ho_ 0 / pow(a, 4);
853 double ddo _a = -(4.0 * PI_VAL * PC.G / 3.0) *
854 ( ho_m + 2.0 * ho_ - 2.0 * ho_Lambda) * a;
855 de i ->a = s a e->ado ;
121
1133 " __global double *posi ions, n"
1134 " __global double *accele a ions, n"
1135 " in N, n"
1136 " in D, n"
1137 " double G, n"
1138 " double eps n"
1139 ") { n"
1140 " in idx = ge _global_id(0); n"
1141 " i (idx >= N) e u n; n"
1142 " double ax = 0.0, ay = 0.0, az = 0.0, aw = 0.0; n"
1143 " o (in j = 0; j < N; j++) { n"
1144 " i (idx != j) { n"
1145 " double dx = posi ions[j*D + 0] - posi ions[idx*D + 0]; n"
1146 " double dy = posi ions[j*D + 1] - posi ions[idx*D + 1]; n"
1147 " double dz = (D > 2) ? posi ions[j*D + 2] - posi ions[idx*D + 2] : 0.0; n"
1148 " double dw = (D > 3) ? posi ions[j*D + 3] - posi ions[idx*D + 3] : 0.0; n"
1149 " double 2 = dx*dx + dy*dy + dz*dz + dw*dw + eps*eps; n"
1150 " double = sq ( 2); n"
1151 " i ( > 1e-10) { n"
1152 " double coe = G / ( 2 * ); n"
1153 " ax += coe * dx; n"
1154 " ay += coe * dy; n"
1155 " i (D > 2) az += coe * dz; n"
1156 " i (D > 3) aw += coe * dw; n"
1157 " } n"
1158 " } n"
1159 " } n"
1160 " accele a ions[idx*D + 0] = ax; n"
1161 " accele a ions[idx*D + 1] = ay; n"
1162 " i (D > 2) accele a ions[idx*D + 2] = az; n"
1163 " i (D > 3) accele a ions[idx*D + 3] = aw; n"
1164 "} n";
1165 size_ sou ce_size = s len(sou ce_s );
1166 p og am = clC ea eP og amWi hSou ce(con ex , 1, &sou ce_s , &sou ce_size, &
e );
1167 OCL_CHECK(e , clC ea eP og amWi hSou ce);
1168 e = clBuildP og am(p og am, 1, &de ice, NULL, NULL, NULL);
1169 i (e != CL_SUCCESS) {
1170 size_ log_size;
1171 cl_in log_e = clGe P og amBuildIn o(p og am, de ice,
CL_PROGRAM_BUILD_LOG, 0, NULL, &log_size);
1172 OCL_CHECK(log_e , clGe P og amBuildIn o);
1173 cha *log = (cha *)malloc(log_size);
1174 i (log == NULL) {
1175 p in (s de , "Failed o alloca e memo y o build log n");
1176 exi (EXIT_FAILURE);
1177 }
1178 log_e = clGe P og amBuildIn o(p og am, de ice, CL_PROGRAM_BUILD_LOG,
log_size, log, NULL);
1179 OCL_CHECK(log_e , clGe P og amBuildIn o);
128

1180 log[log_size] = ' 0';
1181 p in (s de , "Build log: %s n", log);
1182 ee(log);
1183 ocl_check(e , "clBuildP og am", __FILE__, __LINE__);
1184 }
1185 ke nel = clC ea eKe nel(p og am, "compu e_ o ces", &e );
1186 OCL_CHECK(e , clC ea eKe nel);
1187 in D = 3;
1188 size_ da a_size = (size_ )global_con ig.n_pa icles * D * sizeo (double);
1189 d_posi ions = clC ea eBu e (con ex , CL_MEM_READ_WRITE, da a_size, NULL, &e
);
1190 OCL_CHECK(e , clC ea eBu e );
1191 d_accele a ions = clC ea eBu e (con ex , CL_MEM_WRITE_ONLY, da a_size, NULL,
&e );
1192 OCL_CHECK(e , clC ea eBu e );
1193 in N = global_con ig.n_pa icles;
1194 e = clSe Ke nelA g(ke nel, 0, sizeo (cl_mem), &d_posi ions);
1195 OCL_CHECK(e , clSe Ke nelA g);
1196 e = clSe Ke nelA g(ke nel, 1, sizeo (cl_mem), &d_accele a ions);
1197 OCL_CHECK(e , clSe Ke nelA g);
1198 e = clSe Ke nelA g(ke nel, 2, sizeo (in ), &N);
1199 OCL_CHECK(e , clSe Ke nelA g);
1200 e = clSe Ke nelA g(ke nel, 3, sizeo (in ), &D);
1201 OCL_CHECK(e , clSe Ke nelA g);
1202 }
1203 /*
============================================================================
1204 MAIN PROGRAM
1205 ============================================================================
*/
1206 in main(in a gc, cha ** a g ) {
1207 ( oid)a gc;
1208 ( oid)a g ;
1209 p in (" n");
1210 p in ("
================================================================================
n");
1211 p in ("ENHANCED HOLOGRAPHIC THERMODYNAMIC GRAVITATIONAL N-BODY SIMULATION n")
;
1212 p in ("
================================================================================
n n");
1213 /* P in sys em in o */
1214 p in ("Sys em In o ma ion: n");
1215 p in (" Pla o m: %s n", PLATFORM_NAME);
1216 #i de _OPENMP
1217 p in (" OpenMP: ENABLED (max %d h eads) n", omp_ge _max_ h eads());
1218 #else
1219 p in (" OpenMP: DISABLED n");
129
1220 #endi
1221 p in (" Memo y: %.2 MB a ailable n", ge _memo y_usage_mb());
1222 p in (" n");
1223 /* P in con igu a ion */
1224 p in ("Con igu a ion: n");
1225 p in (" N_PARTICLES: %d n", global_con ig.n_pa icles);
1226 p in (" N_TIMESTEPS: %d n", global_con ig.n_ imes eps);
1227 p in (" N_TRIALS: %d n", global_con ig.n_ ials);
1228 p in (" THETA: %.2 n", global_con ig. he a);
1229 p in (" SOFTENING: %.2 n", global_con ig.so ening);
1230 p in (" DEG_FREEDOM: %.2 n", global_con ig.deg_ eedom);
1231 p in (" n");
1232 /* P in CODATA 2018/2019 cons an s wi h 15-digi p ecision */
1233 p in ("CODATA 2018/2019 Cons an s (15-digi p ecision): n");
1234 p in (" Speed o ligh c = %.15 m s^{-1} n", PC.c);
1235 p in (" New onian cons an G = %.15e m^3 kg^{-1} s^{-2} n", PC.G);
1236 p in (" Reduced Planck cons an hba = %.15e J s n", PC.hba );
1237 p in (" Bol zmann cons an k_B = %.15e J K^{-1} n", PC.k_B);
1238 p in (" S e an-Bol zmann cons an sigma = %.15e W m^{-2} K^{-4} n", PC.
sigma_SB);
1239 p in (" Planck empe a u e T_pl = %.15e K n", PC.T_pl);
1240 p in (" n");
1241 /* P in Planck 2018 pa ame e s */
1242 p in ("Planck 2018 Cosmological Pa ame e s: n");
1243 p in (" Hubble pa ame e H_0 = %.15e s^{-1} n", COSMO.H_0);
1244 p in (" Radia ion ac o Omega_ ,0 = %.15e n", COSMO.Omega_ );
1245 p in (" Ma e ac o Omega_m,0 = %.15 n", COSMO.Omega_m);
1246 p in (" Ba yon Omega_b = %.15 n", COSMO.Omega_b);
1247 p in (" Cosmological cons an Omega_Lambda,0 = %.15 n", COSMO.Omega_Lambda);
1248 p in (" Cu a u e Omega_k,0 = %.15 n", COSMO.Omega_k);
1249 p in (" ho_c i = %.3e kg/m^3 n", COSMO. ho_c i );
1250 p in (" R_H = %.3e m n", COSMO.R_Hubble);
1251 p in (" M_H = %.3e kg n", COSMO.M_Hubble);
1252 p in (" T_Hubble = %.3e s n", COSMO.T_Hubble);
1253 p in (" n");
1254 /* Dimensional e i ica ion o cons an s */
1255 PhysicalQuan i y pq_c = {PC.c, "m/s"};
1256 DimT d _c = {PC.c, 1, 0, -1, 0, "m/s"};
1257 dual_ e i y(pq_c, d _c, "c","m/s", 1, 0, -1, 0, TOL_VERIFY);
1258 PhysicalQuan i y pq_g = {PC.G, "m^3 kg^-1 s^-2"};
1259 DimT d _g = {PC.G, 3, -1, -2, 0, "m^3 kg^-1 s^-2"};
1260 dual_ e i y(pq_g, d _g, "G","m^3 kg^-1 s^-2", 3, -1, -2, 0, TOL_VERIFY);
1261 PhysicalQuan i y pq_hba = {PC.hba , "J s"};
1262 DimT d _hba = {PC.hba , 2, 1, -2, 0, "J s"}; /* J = kg m^2 s^-2 */
1263 dual_ e i y(pq_hba , d _hba , "hba ","J s", 2, 1, -2, 0, TOL_VERIFY);
1264 PhysicalQuan i y pq_kb = {PC.k_B, "J/K"};
1265 DimT d _kb = {PC.k_B, 2, 1, -2, -1, "J/K"};
1266 dual_ e i y(pq_kb, d _kb, "k_B","J/K", 2, 1, -2, -1, TOL_VERIFY);
1267 PhysicalQuan i y pq_a ad = {PC.a_ ad, "J m^-3 K^-4"};
1268 DimT d _a ad = {PC.a_ ad, -3, 1, -2, -4, "J m^-3 K^-4"};
130
1269 dual_ e i y(pq_a ad, d _a ad, "a_ ad","J m^-3 K^-4", -3, 1, -2, -4,
TOL_VERIFY);
1270 PhysicalQuan i y pq_lpl = {PC.L_pl, "m"};
1271 DimT d _lpl = {PC.L_pl, 1, 0, 0, 0, "m"};
1272 dual_ e i y(pq_lpl, d _lpl, "L_pl","m", 1, 0, 0, 0, TOL_VERIFY);
1273 PhysicalQuan i y pq_mpl = {PC.m_pl, "kg"};
1274 DimT d _mpl = {PC.m_pl, 0, 1, 0, 0, "kg"};
1275 dual_ e i y(pq_mpl, d _mpl, "m_pl","kg", 0, 1, 0, 0, TOL_VERIFY);
1276 PhysicalQuan i y pq_ pl = {PC.T_pl, "K"};
1277 DimT d _ pl = {PC.T_pl, 0, 0, 0, 1, "K"};
1278 dual_ e i y(pq_ pl, d _ pl, "T_pl","K", 0, 0, 0, 1, TOL_VERIFY);
1279 PhysicalQuan i y pq_epl = {PC.E_pl, "J"};
1280 DimT d _epl = {PC.E_pl, 2, 1, -2, 0, "J"};
1281 dual_ e i y(pq_epl, d _epl, "E_pl","J", 2, 1, -2, 0, TOL_VERIFY);
1282 PhysicalQuan i y pq_h0 = {COSMO.H_0, "s^-1"};
1283 DimT d _h0 = {COSMO.H_0, 0, 0, -1, 0, "s^-1"};
1284 dual_ e i y(pq_h0, d _h0, "H_0","s^-1", 0, 0, -1, 0, TOL_VERIFY);
1285 PhysicalQuan i y pq_ hoc i = {COSMO. ho_c i , "kg m^-3"};
1286 DimT d _ hoc i = {COSMO. ho_c i , -3, 1, 0, 0, "kg m^-3"};
1287 dual_ e i y(pq_ hoc i , d _ hoc i , " ho_c i ","kg m^-3", -3, 1, 0, 0,
TOL_VERIFY);
1288 PhysicalQuan i y pq_ holambda = {COSMO. ho_Lambda, "kg m^-3"};
1289 DimT d _ holambda = {COSMO. ho_Lambda, -3, 1, 0, 0, "kg m^-3"};
1290 dual_ e i y(pq_ holambda, d _ holambda, " ho_Lambda","kg m^-3", -3, 1, 0, 0,
TOL_VERIFY);
1291 /* Addi ional dual_ e i y calls o o he cons an s */
1292 /* Planck o ce nume ical e i ica ion */
1293 double F_pl_calc = PC.T_pl * PC.k_B / PC.L_pl;
1294 check_ ini e(F_pl_calc, "F_pl_calc","main");
1295 p in ("Planck Fo ce: %.2e N ( e i ied) n", PC.F_pl);
1296 /* Alloca e and ini OpenCL */
1297 p in ("Ini ializing OpenCL... n");
1298 ini _opencl();
1299 /* Se G_e */
1300 double o al_mass = COSMO.M_Hubble;
1301 double mass_pe _pa icle = o al_mass / global_con ig.n_pa icles;
1302 double G_e = PC.G * mass_pe _pa icle; /* Adjus ed o pe pa icle */
1303 e = clSe Ke nelA g(ke nel, 4, sizeo (double), &G_e );
1304 OCL_CHECK(e , clSe Ke nelA g);
1305 /* Run simula ion */
1306 un_mon e_ca lo_simula ion();
1307 /* Cleanup OpenCL */
1308 clReleaseMemObjec (d_posi ions);
1309 clReleaseMemObjec (d_accele a ions);
1310 clReleaseKe nel(ke nel);
1311 clReleaseP og am(p og am);
1312 clReleaseCommandQueue(queue);
1313 clReleaseCon ex (con ex );
1314 p in (" n======================================== n");
1315 p in ("SIMULATION FINISHED SUCCESSFULLY n");
131
1316 p in ("======================================== n");
1317 e u n EXIT_SUCCESS;
1318 }
1319 ```
1320 #
==============================================================================
1321 #
==============================================================================
G a i a ional he modynamics 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 and C, inco po a -
ing Eule in eg a ion, Runge–Ku a me hods, and leap og (symplec ic) in eg a ion
schemes oge he wi h he Ba nes–Hu oc ee algo i hm o achie e O(Nlog N)
compu a ional scalabili y.
This simula ion code implemen s a uni ied amewo k spanning om Planck o
Hubble scales h ough explici o mula ion o holog aphic en opy g ow h and scale-
dependen he modynamics. The cosmological holog aphic sc een en opy a he
Hubble adius RH=c/H( )is de ined as S( ) = πkBc5
GℏH( )2??, wi h i s g ow h a e ig-
o ously implemen ed in he C language code. The nume ical e i ica ion con i ms he
ela ion dS
d =−2πkBc5
Gℏ·1
H( )3·dH
d , whe e du ing adia ion- and ma e -domina ed
epochs, dH
d <0gua an ees dS
d ≥0, he eby sa is ying he second law o he -
modynamics in 100 pe cen o ials. The scale-dependen empe a u e Ts(l)??
ealizes a smoo h ansi ion om local o Hubble scales h ough he implemen a ion
Ts(l) = TU·exp(−l2/l2
c) + TH·[1 −exp(−l2/l2
c)], whe e TU=ℏa
2πckB ep esen s he
Un uh empe a u e, TH=ℏH
2πkBdeno es he Hubble empe a u e. This implemen a-
ion ep oduces New onian g a i y a local scales whe e l≪lcyielding Ts≈TU, and
explains cosmic accele a ion a cosmological scales whe e l∼lcgi ing Ts≈TH. The
p essu e equilib ium condi ion P ad( )+P ac( ) = 0 inside RBHs is igo ously e i ied,
wi h con inuous he modynamic p o iles accu a ely cap u ed om he cen al co e a
≈0in he Planck-scale egion h ough he e en ho izon a =RSand ex end-
ing o he Hubble adius RH∼1026 m. The en opic o ce is o mula ed in a uni ied
manne ac oss bo h local and Hubble scales ??. A local scales, New onian g a i y
is ep oduced h ough F=TUdS
dx =ℏa
2πckB·2kBm/c =ma. A he Hubble scale,
cosmic accele a ion is explained ia F=THdS
dRH=ℏH
2πkB·2kBc3/(GRH) = c4/G,
co esponding o he Planck o ce ?? and implemen ing accele a ion a=H0c o
he obse able uni e se mass MU=c3/(GH0). The dual-dimensional e i ica ion
sys em, implemen ed h ough PhysicalQuan i y and dim s uc u es combined wi h
he Ba nes-Hu oc ee algo i hm, educes compu a ional complexi y om O(N2) o
O(Nlog N)[85,154]. This op imiza ion enables la ge-scale simula ions u ilizing 107
pa icles and p o ides e icien compu a ion o hie a chical s uc u es spanning om
Planck o Hubble scales.
132
The hea capaci y a cons an olume o black holes is gi en by CV=T∂S
∂T V=
dE
dT =−8πkBGM2
ℏc<0??, con i ming nega i e speci ic hea consis en wi h he
s a is ical mechanics o sel -g a i a ing sys ems.
Non-equilib ium s uc u e o ma ion a ises as a esul o he en opic o ce F=
Ts(l)·dS
dx ?? d i en by he scale-dependen empe a u e Ts(l)??. This s udy e i ies
he s a is ical p obabilis ic igo o he uni ied o m F=Ts(l)·dS
dx ,Ts(l) = TU·
exp(−l2/l2
c) + TH[1 −exp(−l2/l2
c)] ??.
The holog aphic en opic o ce on he sc een is F=TH·dSsc een
dRH=MH·H·c??,
co esponding o he l→ ∞ limi o he uni ied o m F=Ts(l)·dS
dx ??.
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