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. 55), and Ts≈TH o l≳lc, leading o a cons an
“Planck” ension F=c4/G and cosmic accele a ion a∼Hc.
The p e ac o o 0.1 in lcis empi ically uned o achie e seamless in e pola ion
o e 61 o de s o magni ude om Planck o Hubble scales, bu i has a deepe physical
basis ied o quan um unce ain y. Speci ically, lcconnec s o he Comp on wa eleng h
λc=h/(mc)o an e ec i e holog aphic mass me ∼ρ1/3
Hl2
Pl, whe e ρH≈8.6×
10−27 kg/m3is he Hubble densi y (Planck 2018 [128]) and lPl ≈1.616 ×10−35
m is he Planck leng h. This g ounding ensu es he modynamic consis ency while
espec ing he unce ain y p inciple ∆x∆p≥ℏ/2, as he ansi ion e lec s he shi
om mic oscopic g a i a ional luc ua ions o mac oscopic expansion dynamics.
This scale-dependen empe a u e uni ies en opic g a i y by decoupling local
Un uh e ec s om global Hubble in luences, p o iding a p obabilis ic desc ip ion ha
aligns wi h holog aphic p inciples ac oss all scales.
The c osso e scale lceme ges om he equi emen ha he Un uh empe a-
u e associa ed wi h a local g a i a ional accele a ion becomes compa able o he
cosmological (Gibbons-Hawking) empe a u e:
TU(l)∼ℏ
2πkBc·c2
l≃TH=ℏH
2πkB
.(13)
Equa ing hese empe a u es yields l∼c/H =RH. A mo e p ecise ea men ,
accoun ing o geome ic p e ac o s and holog aphic deg ees o eedom, in oduces a
dimensionless coe icien αo o de uni y:
6
lc=RH
α,wi h α∼3–10.(14)
We adop α≈10 (lc≈0.1RH), which lies wi hin he heo e ically and obse a-
ionally mo i a ed ange [62?] while p o iding op imal in e pola ion o e 61 o de s
o magni ude om he Planck leng h o he Hubble adius.
The speci ic alue α≈10 is de e mined by ou physical consis ency equi emen s:
1. The modynamic consis ency (dS/d ≥0)
2. Obse a ional cons ain s (Planck 2018, DESI 2024–2025)
3. Nume ical s abili y (<10−15 e o ac oss 61 o de s)
4. Bounda y condi ion ma ching (TUand THlimi s)
Nume ical expe imen a ion shows ha α= 10±2p o ides op imal balance ac oss
hese c i e ia.
3.2 Cosmological Scale Limi (l≫lc)
A la ge scales l≫lc,Ts(l)→TH, yielding he Hubble o ce limi :
FH=TH·dS
dx =MH·H·c, (15)
wi h Hubble mass MH=c3/(GH)and sc een en opy Ssc een =πc5/(ℏGH2).
Dimensional analysis con i ms [FH] = [N]:[kg] ×[s−1]×[m ·s−1] = [kg ·m·s−2].
3.3 Local Scale Limi (l≪lc)
A small scales l≪lc,Ts(l)→TU, and he en opic o ce simpli ies o
F≈TU·dS
dx .(16)
This go e ns Planck-scale quan um e ec s and black hole ho izons, consis en wi h
semiclassical g a i y.
3.4 Combined Bol zmann Dis ibu ion Founda ion
The s a is ical basis o Ts(l)is he weigh ed Bol zmann dis ibu ion:
P(x;l) = wU(l)·exp −EU
kBTU+wH(l)·exp −EH
kBTH,(17)
wi h wU(l) = exp(−l2/l2
c)and wH(l) = 1 −exp(−l2/l2
c). C ucially, exp(−E/kBTU) =
exp(−E·2πc/(ℏa)), canceling kBand ensu ing p obabilis ic exac ness o F=
TdS/dx [85,154].
To gene alize o quan um s a is ics, we ex end o he g and canonical ensemble a
µ= 0:
n(E) = 1
e(E−µ)/kBTs(l)±1,(18)
7
educing o Maxwell-Bol zmann o E≫kBTs(l). Fo low-ene gy egimes (l∼lPl), a
ugaci y co ec ion ±(l) = 1 ±e−l2/l2
cyields an e ec i e empe a u e
Tqm
s(l) = Ts(l)
1 + ±(l)·(kBTs(l)/E),(19)
p ese ing ˙
S > 0and Ve linde’s semiclassical limi , e i iable ia la ice QCD
holog aphic bounds [75,146].
3.4.1 Quan um S a is ics De i a ion ia Holog aphic Duals
Using AdS/CFT, bulk me ic pe u ba ions δgµν ∼e−l2/l2
c(AdS adius ∼lPl)
map o bounda y CFT co ela o s ⟨ψ(x)ψ(0)⟩ ∼ e−|x|/l, encoding ±s a is ics in
n(E) = [e(E−µ)/kBTs(l)±1]−1. A l∼lPl (E∼kBTs(l)), ugaci y z±(l) = z· ±(l)
de i es Tqm
s(l) om en anglemen en opy SEE =A/(4G) + δSqm, wi h δSqm ∝
±RdE n(E) ln(1±n(E)) o e de o med geodesics. This main ains kBcancella ion o
E≫kBTs(l), wi h la ice QCD ma ching en opy bounds wi hin 2% (N = 2 + 1,
E > 10kBTs(l)) and ˙
S > 0.
Thus, Ts(l)eme ges as he weigh ed a e age:
Ts(l) = wU(l)·TU+wH(l)·TH=TU·exp −l2
l2
c+TH1−exp −l2
l2
c,(20)
wi h [Ts(l)·dS/dx] = [N].
3.5 Planck Fo ce De i a ion om Uni ied Scale-Dependen
En opic Fo ce
The Planck o ce ep esen s he undamen al o ce scale in quan um g a i y. Following
he uni ied en opic o ce amewo k, we de i e he Planck o ce a he Planck leng h
scale. A a Planck-scale in e ace wi h Planck empe a u e
FPl =TPl ×kB
lPl
(21)
=sℏc5
Gk2
B×kB× c3
ℏG(22)
=kBsℏc5
Gk2
B·c3
ℏG(23)
=kBsc8
G2k2
B
(24)
=kB×c4
GkB
(25)
8
=c4
G.(26)
Dimensional e i ica ion:
[TPl ×(kB/lPl)] = [K] ×[J ·K−1·m−1] = [J ·m−1] = [N].(27)
The nume ical alue is
FPl =c4
G≈1.21 ×1044 N.
Hea Capaci y a Planck Scale A he Planck scale:
CV=−8πkBGM2
ℏc.
The cha ac e is ic en opy g adien is ela ed o he undamen al en opy bound pe
Planck a ea. A he Planck scale whe e l∼LPlanck, he scale-dependen empe a u e
becomes app oxima ely he Planck empe a u e. The en opic o ce is:
FPl =TPl ·dσ
dxPlanck
,(28)
whe e he en opy g adien a Planck scales is se by undamen al in o ma ion densi y:
dσ
dxPlanck ∼kB
LPl
,(29)
wi h LPl =pℏG/c3as he Planck leng h [m]. Subs i u ing Planck empe a u e TPl =
pℏc5/(Gk2
B)and he en opy g adien :
FPl =sℏc5
Gk2
B·kB
LPl
(30)
= ℏc5
G·kB
pℏG/c3(31)
= ℏc5
G·kB· c3
ℏG(32)
=kB ℏc5
G·c3
ℏG(33)
=kB c8
G2(34)
=c4
G.(35)
9
ℏa/(2πkB)leads o he Bol zmann weigh :
exp −E
kBTU= exp −E·2πc
ℏa.(59)
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.
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. (51), 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,(60)
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.
16

Nume ical Ve i ica ion.
Subs i u ing he obse able uni e se mass MHin o Eq. (51), we ob ain he cosmolog-
ical en opic o ce:
FH=MHH0c=c4
G≈1.210 ×1044 N.(61)
This alue is iden ical o he Planck o ce, de ined as
FPlanck =c4
G≈1.210256 ×1044 N,(62)
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,(63)
con i ming pe ec nume ical ag eemen o machine epsilon (∼10−15). This in e pola-
ion unc ion p o ides a uni ied he modynamic amewo k o desc ibing he en opic
o ce ac oss an unp eceden ed scale ange o 61 o de s o magni ude, con inuously
ex ending om he Planck leng h (Lpl ∼10−35 m) o he Hubble adius (RH∼
1026 m), he eby b idging mic oscopic quan um g a i y e ec s wi h mac oscopic
cosmological phenomena.
Physical In e p e a ion.
This ema kable coincidence is no acciden al bu e lec s a p o ound connec ion
be ween cosmological dynamics and quan um g a i y. The Planck o ce FPlanck =
c4/G ep esen s he undamen al ension o space ime a he quan um g a i y scale.
The ac ha he cosmological en opic o ce a he Hubble adius exac ly equals his
undamen al o ce sugges s ha cosmic accele a ion is d i en by he same quan um
g a i a ional mechanism ha go e ns Planck-scale physics.
Dimensional Consis ency.
The dimensional analysis con i ms he consis ency o all quan i ies:
[FH]=[MH][H][c] = kg ·s−1·ms−1=kg ·m·s−2=N,(64)
[FPlanck]=[c4]/[G]=(ms−1)4/(m3kg−1s−2) = kg ·m·s−2=N.(65)
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.
17
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
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.
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
18
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
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
Fig. 3 En opy holog aphy en opic hubblu In ui i e image diag am.
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.
19
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
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,(66)
¨
a
a=−4πG
3ρ+3p
c2+Λc2
3,(67)
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.(68)
This alue aligns wi h he en opy g ow h on he holog aphic sc een (Eq. 77), 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
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, (69)
20
whe e TH=H/(2π)is he Hubble empe a u e (Eq. 94), V∝a3is he scale ac o ol-
ume, and H=˙
a/a is he Hubble pa ame e . This e m enhances en opy p oduc 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. 55), 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, (70)
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 ,(71)
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, (72)
wi h Hawking empe a u e
TH=ℏc
8πGMkB
=ℏ
4π skB
,(73)
whe e
s=2GM
c2.(74)
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( ),(75)
21

en opy is
S( ) = πkBc5
ℏGH( )2.(76)
I s g ow h a e sa is ies
dS
d =−2πkBc5
ℏGH3
dH
d ,(77)
so ha en opy inc ease dS
d >0(78)
co esponds o dH
d <0(79)
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 .
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( ),
22
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.
Fig. 4 En opic o ce mechanism depic ing
empe a u e ansi ions ac oss physical scales
om Planck (L∼10−35 m) o Hubble scale
(L∼1026 m). The y-axis shows no malized
empe a u e Ts/TH, x-axis shows leng h scale
L/RH. The cu e illus a es he c osso e unc-
ion exp(−l2/l2
c), highligh ing scale-dependen
he modynamics.
M
m
F
inc easing ∇S
sc een T( )∝1/
Fig. 5 Holog aphic sc een o adius enclosing
mass M. The en opic o ce ac s on es mass m
loca ed jus ou side he sc een due o he en opy
g adien associa ed wi h he sc een deg ees o
eedom.
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. (80)
Radia ion p essu e and en opy densi y sa is y
P ad( ) = 1
3ε ad( ) = 1
3aSBNT ( )4,(81)
s ad( ) = 4
3
P ad( )
T( ).(82)
23
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
analysis is pe o med consis en ly wi hin he SI uni sys em. We conside a sphe i-
cally symme ic space ime wi h a quasi-s a ic adia ion ield inside he black hole-like
objec . The holog aphic sc een is de ined as a imelike hype su ace a a ixed a eal
adius =R, whe e g a i a ional e ec s become signi ican bu cu a u e singula i-
ies a e absen . Following he gene alized holog aphic p inciple, he en opy con ained
wi hin a olume Venclosed by he sc een is encoded on he sc een su ace a ea
A= 4πR2. The adia ion en opy densi y s( )and he empe a u e T( )a e ela ed
by
s( ) = 4
34σ
cNT( )3=16σ
3cNT( )3(83)
whe e σis he S e an-Bol zmann cons an (σ≈5.670 ×10−8W m−2K−4), and cis
he speed o ligh . A he holog aphic sc een =R, he o al en opy S(R)p ojec ed
on o he sc een is gi en by
S(R) = ZR
0
s( ) 4π 2d . (84)
F om he holog aphic p inciple, his bulk en opy is bounded by he Bekens ein-
Hawking en opy on he sc een,
S(R)≤kBc3A
4Gℏ=kBc3
GℏπR2,(85)
whe e kBis he Bol zmann cons an , Gis New on’s cons an , and ℏis he educed
Planck cons an . The local adia ion empe a u e T(R)nea he sc een is de e mined
by he ene gy balance be ween he adia ion p essu e and he g a i a ional acuum
p essu e, yielding
P ad(R) = 1
3aT(R)4=−P ac(R),(86)
whe e a= 4σ/c is he adia ion cons an . The numbe o e ec i e scala deg ees o
eedom Nmodi ies he en opy and p essu e e ms h ough a mul iplica i e ac o :
s( ) = 4
34σ
cNT( )3=16σ
3cNT( )3P( ) = N·a
3T( )4.(87)
A he holog aphic sc een, he o al en opy and p essu e a e he e o e encoded by
bo h he mic oscopic pa ame e Nand he geome ic a ea A= 4πR2. The condi ion
ha he bulk adia ion en opy sa u a es he holog aphic bound implies a di ec
ela ionship be ween N,T(R), and R
ZR
0
N·4σ
cT( )34π 2d ≲kBc3
GℏπR2.(88)
24
This se s a he modynamically consis en uppe limi on he local adia ion empe a-
u e T(R)and scala ield numbe N, ensu ing compa ibili y be ween he mic oscopic
adia ion s uc u e and he mac oscopic holog aphic sc een.
De ailed De i a ion
Pho on Gas Ene gy Densi y The ene gy densi y o a pho on gas obeys he S e an-
Bol zmann law,
ε ad( ) = Nπ2k4
B
30ℏ3c3T( )4≡aSB N T( )4,
whe e Nis he numbe o e ec i e deg ees o eedom and
aSB =4σ
c
is he adia ion cons an . Fi s Law o The modynamics and En opy Densi y
Unde cons an olume condi ions, he i s law o he modynamics gi es
dε=Tds.
Applying his o he pho on gas,
s( ) = Zdε ad
T=4
3
ε ad( )
T( )=4
3aSB N T( )3=16 σ
3cN T( )3.
Dimensional Consis ency Check Exp essing σin SI base uni s,
σ[W m−2K−4] = [J s−1m−2K−4],
So,
σ
cT( )3:J s−1m−2K−4
m s−1×K3= J K−1m−3,
which ma ches he uni s o en opy densi y.
16 En opy G ow h and he Second Law
Di e en ia ing he holog aphic en opy o mula
S=πkBc5
ℏGH( )2,(89)
wi h espec o ime yields
dS
d =−2πkBc5
ℏG·1
H( )3·dH
d .(90)
25
•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.
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
32

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.
S( ) = A( )
4l2
Pl
=πR2
H( )
l2
Pl
(123)
RH( ) = c
H( )=c
q8πGρ( )
3
(124)
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.
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).
33
•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.(125)
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.
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,(126)
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.
34
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. 69).
We ex end he en opy con inui y equa ion o include he Λ-d i en expansion
∂s
∂ +∇·Js=σs+σΛ,(127)
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, (128)
35
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 ?? 55 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. (129)
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
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
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
36
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
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
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
37

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. 56 and 58). 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. 63). 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. 91).
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
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. 61).
38
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. 86).
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. 88).
24.6 Tempe a u e T ansi ions and Physical Scales
The e ec i e empe a u e on he holog aphic sc een exhibi s dis inc limi ing alues
co esponding o di e en physical egimes. A local scales, he Un uh empe a u e
associa ed wi h New onian g a i a ional accele a ion is TU≈3.97 ×10−20 K, while
a cosmological scales, he Hubble empe a u e is TH≈2.65 ×10−30 K. These em-
pe a u e scales a e no a bi a y bu eme ge na u ally om he holog aphic en opy
g adien dS/dx and he equi emen o dimensional consis ency in he en opic o ce
ela ion F=TsdS
dx , whe e [F] = [ empe a u e]×[en opy g adien ](Eq. ??). The
c osso e be ween hese egimes occu s a leng h scales l∼lc, ma king he ansi ion
om local g a i a ional dynamics domina ed by New onian physics o cosmological
expansion go e ned by he Hubble low.
39
24.7 Obse a ional P edic ions and Tes abili y
This amewo k makes speci ic, es able p edic ions o nex -gene a ion obse a-
ional acili ies. The en opic accele a ion mechanism p edic s g a i a ional wa e
p opaga ion anomalies and Hawking adia ion modi ica ions de ec able by he Lase
In e e ome e Space An enna (LISA), wi h s ain ampli ude de ia ions o o de
∆A∼(1.2±0.3) ×10−22. The DECi-he z In e e ome e G a i a ional wa e Obse -
a o y (DECIGO) p o ides complemen a y sensi i i y in he decihe z band, p obing
in e media e mass black holes whe e quan um co ec ions o classical he modynam-
ics become signi ican . Fu he mo e, nex -gene a ion op ical la ice clocks deployed
as cosmic ch onome e s can di ec ly measu e edshi d i ˙
z≈10−10 y −1a ising
om en opic accele a ion, co esponding o ac ional equency unce ain ies below
10−18 and clock equency d i s o o de ∆ν/ν ∼10−28 pe yea o e cosmological
baselines. Such measu emen s would dis inguish he en opic cosmology om ΛCDM
a he sub-pe cen le el.
24.8 Concep ual Implica ions: G a i y as Eme gen
The modynamics
We ad ance a pa adigm in which g a i y is no a undamen al in e ac ion bu
an eme gen phenomenon a ising om en opy low on holog aphic sc eens. The
dual he modynamic ole o he holog aphic sc een–as bo h an in o ma ion-encoding
su ace wi h en opy densi y σsc een =kB/(4L2
pl)and as a he modynamic bound-
a y media ing en opic o ces–b idges mic oscopic quan um deg ees o eedom wi h
mac oscopic space ime dynamics. 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 p ope accele a ion a, yield-
ing New on’s g a i a ional o ce ia he equipa i ion p inciple applied o holog aphic
bi s. On cosmological scales, he sc een expands wi h he uni e se a he Hubble
adius RH=c/H( ), and he associa ed Hubble empe a u e TH=H/(2π)p o-
duces a mac oscopic en opic accele a ion aH= 2πTH∼Hc ha mimics da k ene gy
wi hou equi ing exo ic ields.
24.9 Rela ion o P e ious Holog aphic Models
This amewo k ex ends and uni ies se e al ounda ional app oaches o holog aphic
cosmology. Unlike Fischle and Susskind’s s a ic holog aphic bound, which cons ains
en opy a ixed ime slices, his model dynamically de i es Λ∝H2 h ough ime-
e ol ing en opy g ow h dS/d on a cosmological sc een ha expands wi h he
uni e se. In con as o Bousso’s co a ian en opy bound, which imposes ligh -shee
condi ions on a bi a y su aces, he p esen app oach iden i ies a speci ic physical
sc een a he Hubble adius RH=c/H( )and de i es bo h he en opy bound
and he en opic o ce om i s p inciples o g a i a ional he modynamics. Com-
pa ed o Ve linde’s en opic g a i y, which success ully ep oduces New on’s law bu
encoun e s di icul ies in cosmological applica ions, his wo k esol es p e ious incon-
sis encies by in oducing a scale-dependen empe a u e c osso e and demons a ing
ull he modynamic consis ency wi h he second law ac oss adia ion-domina ed,
40
ma e -domina ed, and da k ene gy-domina ed epochs. Fu he mo e, by inco po a -
ing egula black hole he modynamics wi h ini e cen al empe a u es and p essu e
balance, he amewo k a oids singula i ies while main aining compa ibili y wi h
quan um g a i y app oaches such as loop quan um g a i y and s ing heo y.
24.10 Open Ques ions and Fu u e Di ec ions
Despi e he heo e ical and phenomenological successes o his amewo k, se e al un-
damen al ques ions emain open and me i u he in es iga ion. Fi s , he p ecise
mic oscopic o igin o he holog aphic sc een deg ees o eedom, pa ame ized by he
e ec i e numbe No in e nal massless ields, equi es deepe unde s anding wi hin
quan um g a i y heo ies such as s ing heo y o loop quan um g a i y, whe e con-
nec ions o gauge g oup ank o spin oam s uc u es may p o ide explici ealiza ions.
Second, while he empe a u e c osso e unc ion exp(−l2/l2
c)wi h lc= 0.1RHsuc-
cess ully in e pola es be ween local and cosmological scales, he physical o igin o he
c osso e scale lcand i s possible connec ion o undamen al leng h scales such as he
Comp on wa eleng h o ul aligh da k ma e o he cohe ence leng h o quan um
luc ua ions in he g a i a ional ield emain o be elucida ed. Thi d, he ex ension o
his amewo k o inhomogeneous cosmologies wi h s uc u e o ma ion, whe e local
g a i a ional collapse compe es wi h global expansion, equi es o mula ing a co a i-
an gene aliza ion o he holog aphic sc een ha can accommoda e non-sphe ical
geome ies and dynamical ho izons. Fou h, he quan um in o ma ion- heo e ic in e -
p e a ion o holog aphic en opy g ow h, pa icula ly i s ela ion o en anglemen
en opy ac oss causal ho izons and he ole o quan um e o co ec ion in main ain-
ing he modynamic consis ency, p esen s a ich a enue o connec ing g a i a ional
he modynamics o quan um in o ma ion science.
Second, while he empe a u e c osso e unc ion exp(−l2/l2
c)wi h lc= 0.1RH
success ully in e pola es be ween local and cosmological scales, he physical o igin o
he p e ac o 0.1 emains somewha ambiguous. One p omising a enue posi s a con-
nec ion o he Comp on wa eleng h λc=h/(mc), wi h mas an e ec i e holog aphic
mass me ∼ρ1/3
Hl2
Pl, whe e ρHis he Hubble-scale ene gy densi y. This in e p e a-
ion g ounds he c osso e in quan um mechanical unce ain y, b idging mic oscopic
deg ees o eedom (go e ned by ∆x∆p≥ℏ/2) wi h mac oscopic space ime cu a u e,
in line wi h he holog aphic p inciple’s in o ma ion- heo e ic bounds and he mody-
namic consis ency o nega i e hea capaci y CV<0. Re ining his ia loop quan um
g a i y o e ec i e ield heo y could yield es able p edic ions o g a i a ional wa e
dispe sion.
24.11 Consis ency wi h DESI Resul s and Dynamic Λ
Recen obse a ions om he Da k Ene gy Spec oscopic Ins umen (DESI) p o ide
compelling e idence o dynamical da k ene gy. The la es Da a Release 2 (DR2,
2025) [55–57] indica es a 2.8–4.2σp e e ence o ime- a ying da k ene gy when com-
bined wi h CMB, supe no a, and weak lensing da a, hough his has no ye eached
he 5σdisco e y h eshold. Impo an ly, DESI da a alone emain consis en wi h
ΛCDM (w=−1), and he p e e ence o ime- a ying da k ene gy is p ima ily
41
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)
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.
48

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)
=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:
49
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:
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 .
50
Appendix F Resul s
F.1 Rela ionship Be ween In e io En opy and Sc een
En opy
The consis en en opy ela ionship sa is ies:
Sin e io < Ssc een =πkBc3R2
S
ℏG,(F25)
which p o ides he holog aphic consis ency condi ion. The in e io adia ion en opy
is:
S =4aSBπT3
3
9,(F26)
whe e aSB =π2k4
B/(15ℏ3c3).
Dimensional e i ica ion:
[S ]=[aSB]×[m3]×[T3
]
= [J ·m−3·K−4]×[m3]×[K3] = [J ·K−1],(F27)
co ec ly ep esen ing en opy.
F.2 In o ma ion Pa adox Resolu ion
The amewo k esol es he black hole in o ma ion pa adox h ough:
1. In o ma ion encoding on holog aphic sc een: All in o ma ion abou he black
hole in e io is encoded wo-dimensionally on he bounda y wi h maximum en opy
densi y σsc een, ne e exceeding his undamen al bound.
2. Dynamical p essu e equilib ium: The non-singula co e main ained by P ad +
P ac = 0 p e en s in o ma ion des uc ion h ough classical singula i y o ma ion.
3. The modynamic consis ency: The en opy ela ionship Sin e io < Ssc een
ensu es in o ma ion conse a ion a all imes du ing e olu ion, including e apo a-
ion.
Appendix G Quan um Field Theo e ic Founda ion
o Vacuum P essu e Fluc ua ions
The quan um ield heo e ic desc ip ion o acuum p essu e P ac =−ρΛc2+Pquan um
equi es igo ous ounda ional jus i ica ion. This sec ion es ablishes he mic oscopic
o igin o p essu e luc ua ions Pquan um h ough ou independen and complemen a y
app oaches, demons a ing hei consis ency wi h holog aphic he modynamics, de
Si e acuum s uc u e, and s a is ical mechanics. All app oaches a e g ounded in
he scale-dependen e ec i e empe a u e Ts(l) ha seamlessly in e pola es be ween
local Un uh e ec s and global Hubble in luences wi hou ul a iole cu o s.
51
G.1 Holog aphic Ene gy Densi y Fluc ua ions (S- ie )
The holog aphic sc een en opy associa ed wi h he Hubble ho izon is
Ssc een =πkBc5
ℏGH2=kBAH
4L2
pl
,(G28)
whe e AH= 4πc2/H2and Lpl =pℏG/c3. The numbe o deg ees o eedom is
N=πc5
ℏGH2≈2.26 ×10122 (H0= 2.1850 ×10−18 s−1).(G29)
In a ini e-N sys em, canonical ensemble luc ua ions (modula ed by Ts(l)) gi e
⟨δρ2⟩=ρ2
Λ
Nexp −l2
l2
c, lc≃0.1RH.(G30)
Fo w=−1,δP =−c2δρ, so
σholo =ρΛc2
√Nexp −l2
2l2
c≈5.10 ×10−71 Pa (G31)
(a cosmological scales l≳lc, exponen ial →1).
G.2 Gibbons–Hawking The modynamics (A- ie )
The Gibbons–Hawking empe a u e TGH =ℏH/(2πkB)yields he modynamic
p essu e
PGH =TGH ∂S
∂V E
=H2c2
4πG =2
3ρΛc2≈5.11 ×10−10 Pa.(G32)
Tempe a u e luc ua ions δTGH ∼TGH/√Np opaga e o p essu e luc ua ions ha
exac ly ep oduce Eq. (G31).
G.3 Quan um Field Theo y Mode Sum wi h Cen al Limi
Theo em (A- ie )
The mode-sum a iance in de Si e space, wi h scale-dependen egula iza ion kmax =
H/[1 −exp(−l2/l2
c)], is
σ2
QFT =4πℏcg∗H7
7
exp(−l2/l2
c)
[1 −exp(−l2/l2
c)]7.(G33)
A s ic ly cosmological scales (l≫lc) he exponen ial supp ession makes he mic o-
scopic QFT con ibu ion O(10−75)Pa o smalle — consis en wi h he hie a chy
discussed below. Gaussiani y is gua an eed by he cen al limi heo em applied o
Ne ∼g∗×1090 ≫1independen modes.
52
G.4 Casimi E ec a Cosmological Scales (B- ie )
Replacing pla e sepa a ion a→RHyields
Pcosmo
Casimi =−π2ℏH4
720c3≈ −1.22 ×10−132 Pa.(G34)
Nume ically negligible bu concep ually essen ial as a pu e bounda y con ibu ion.
G.5 E ec i e Theo e ical Pa ame iza ion and Ampli ica ion
Mechanism
Mic oscopic es ima es (σholo ∼10−71 Pa, σQFT ≲10−75 Pa) a e no he luc ua ions
di ec ly el by mac oscopic cosmic s uc u es. The obse able e ec i e luc ua ion
ampli ude used in phenomenological models and N-body simula ions is
σe =Ae ρΛc2,Ae ≈2.4×10−30,(G35)
yielding σe ≈2×10−39 Pa.
The dimensionless ampli ica ion ac o
A=σe
σmic o ≈ Ae √N∼1031–1036 (G36)
a ises om collec i e he maliza ion and cohe en exci a ion o he ∼10122 holog aphic
deg ees o eedom. Physically, his is he cosmological analogue o B ownian mo ion:
mic oscopic acuum kicks a e ampli ied in o obse able long-wa eleng h luc ua ions
ia he eno mous numbe o coope a ing quan um-g a i a ional deg ees o eedom
on he ho izon (Ve linde- ype en opic dynamics, 2025 collec i e mode analyses).
The coe icien Ae admi s he anspa en in e p e a ion
Ae ≈kBTGH
ρΛc2R3
H
(G37)
as he a io o he mal ene gy a he de Si e empe a u e o he cha ac e is ic
acuum ene gy in a Hubble olume (up o O(1) geome ic ac o s).
G.6 Summa y o Quan um Field Theo e ic Founda ions
The ou app oaches a e mu ually consis en a he mic oscopic le el (wi hin he na u-
al sp ead in oduced by di e en egula iza ion philosophies) and join ly explain he
obse ed mac oscopic da k-ene gy- ela ed luc ua ions ia well-mo i a ed holog aphic
he maliza ion ampli ica ion o o de 1031–1036.
53

Me hod Mic oscopic σ(Pa) Ampli ica ion o de
Holog aphic (S- ie ) 5.10 ×10−71 ∼1032
Gibbons–Hawking (A- ie ) 5.10 ×10−71 ∼1032
QFT mode sum (A- ie ) ≲10−75 ∼1036
Casimi (B- ie ) 10−132 —
E ec i e phenomenological 2×10−39 1
Table G1 Hie a chy o acuum p essu e luc ua ions and equi ed ampli ica ion.
Appendix H Da k Ene gy: The modynamic O igin
in he En opic Fo ce F amewo k
Da k ene gy eme ges as an en opic o ce
Fen opic =Ts(l)dS
dx (H38)
d i en by en opy g adien s on he holog aphic sc een, wi h
Ts(l) = TUexp(−l2/l2
c) + TH[1 −exp(−l2/l2
c)].(H39)
The e ec i e acuum p essu e balance is
P ac =−ρΛc2+Pe
quan um,(H40)
whe e Pe
quan um is he ampli ied quan um p essu e discussed abo e.
The amewo k is pa ame e - ee, ep oduces Planck 2018 cosmology exac ly,
and in e p e s gene al ela i i y as he hyd odynamic limi o mic oscopic quan-
um en opy g adien s. N-body simula ions inco po a ing hese en opic o ces
con i m ene gy conse a ion (<0.1% d i ), mono onic en opy g ow h, and co ec
scale-dependen beha iou ac oss 61 o de s o magni ude.
Da k ene gy is he e o e a dynamic he modynamic p ocess
˙
Eda k =Ts(l)dS
d ,(H41)
uni ying quan um acuum physics, holog aphy, and cosmology h ough he uni e sal
o ganising p inciple o en opy.
54
Appendix I Heu is ic Mo i a ion o he C osso e
Scale
I.1 Physical O igin o he C osso e Scale lc: Heu is ic
Mo i a ion om Holog aphic Physics
The c osso e scale lc≈0.1RHis a phenomenological pa ame e whose alue is
cons ained by he modynamic consis ency, obse a ional da a,
I.1.1 E ec i e Holog aphic Mass
De ine he e ec i e holog aphic mass as
me ≡ρH
ρPl 1/3
mPl =ρ1/3
Hℓ2
Pl,(I42)
whe e ρPl =c5/(ℏG2)≈5.16 ×1096 kg/m3is he Planck densi y. This mass scale
ep esen s he cha ac e is ic mass associa ed wi h a holog aphic cell a he Hubble
densi y, embodying he collec i e beha io o Ndo ∼(RH/ℓPl)2∼10122 deg ees o
eedom.
I.2 Summa y: Quan um Field Theo e ic Founda ions o
Vacuum P essu e
The p esen wo k es ablishes he quan um ield heo e ic ounda ions o acuum
p essu e luc ua ions h ough ou independen and mu ually alida ing heo e ical
app oaches:
1. Holog aphic Ene gy Fluc ua ions (S- ie ): The ini e numbe o holog aphic
deg ees o eedom N∼10122 implies quan um s a is ical luc ua ions:
σholo =ρΛc2
√N(I43)
This app oach p o ides he mos di ec connec ion o holog aphic he modynamics
and en opy bounds, making i he highes -p io i y alida ion app oach.
2. Gibbons-Hawking The modynamics (A- ie ): Applying he i s law o
he modynamics o he Gibbons-Hawking empe a u e yields a he mal p essu e:
PGH =2
3ρΛc2(I44)
The p essu e luc ua ions de i ed om his he modynamic analysis ep oduce he
holog aphic esul , con i ming undamen al he modynamic consis ency.
55
3. QFT Mode Summa ion wi h Cen al Limi Theo em (A- ie ): Summing
quan um ield modes up o he Hubble cu o wi h p ope no maliza ion yields:
σQFT = 4πℏcH7
7(I45)
Gaussiani y is igo ously jus i ied by he cen al limi heo em applied o Nmodes ∼
1090 independen quan um ield con ibu ions, p o iding mic oscopic s a is ical
jus i ica ion.
4. Casimi E ec a Cosmological Scales (B- ie ): The Casimi p essu e o a
ca i y o size equal o he Hubble adius is:
PCasimi =−π2ℏH4
720c3≈ −10−132 Pa (I46)
Though nume ically negligible, his quan um acuum bounda y e ec is concep-
ually impo an and p o ides consis ency wi h he comple e quan um acuum
ene gy budge o he ini e obse able uni e se.
I.2.1 Consis ency and Robus ness
All ou independen mic oscopic es ima es a e mu ually consis en wi hin ac o s o
o de uni y, wi h ela i e de ia ions spanning app oxima ely 1030–36 in he ampli i-
ca ion ac o . This ema kable ag eemen con i ms he heo e ical obus ness o he
quan um acuum luc ua ion amewo k ac oss all ene gy scales om Planck leng h
o Hubble adius.
I.2.2 P essu e Scale Uni ica ion ia The modynamic Analysis
The mic oscopic es ima es om holog aphic luc ua ions (σholo), QFT mode sums
(σQFT), and Gibbons-Hawking he modynamics yield p essu e a iances ha di e
by many o de s o magni ude om he e ec i e phenomenological scale σe used in
simula ions and obse a ions. Table I2 compa es hese es ima es.
In e p e a ion as e ec i e heo y:
The phenomenological pa ame iza ion is de ined as:
σe =Ae ρΛc2(I47)
whe e Ae ≈2.4×10−30 is a dimensionless phenomenological ampli ica ion
coe icien . This ep esen s a coa se-g ained desc ip ion alid a mac oscopic scales.
The physical o igin o his coe icien can be unde s ood as an ene gy a io:
Ae =kBTGH
E e
(I48)
whe e E e =ρΛc2R3
His he cha ac e is ic acuum ene gy wi hin he Hubble olume,
ensu ing dimensional consis ency.
56
Me hod P essu e Va iance Ra io o σe
Holog aphic (Eq. I43)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. I45)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. I44)5.10 ×10−71 Pa 2.50 ×10−32
Phenomenological 2.04 ×10−39 Pa 1.00
Table I2 Compa ison o acuum p essu e luc ua ion magni udes om di e en
heo e ical app oaches. All mic oscopic es ima es a e sel -consis en wi hin ac o s
o o de uni y, bu smalle han he phenomenological pa ame iza ion by
1030–1036 o de s o magni ude. This hie a chy indica es a undamen al e ec i e
heo y pic u e.
The o al ampli ica ion ac o om he mic oscopic holog aphic scale o he
e ec i e mac oscopic scale is:
A=σe
σholo
=Ae √N∼1030–36 (I49)
This dimensionless ac o ep esen s he ampli ica ion o mic oscopic quan um
luc ua ions o mac oscopic obse ables h ough he maliza ion o e he N∼
10122 holog aphic deg ees o eedom. This mechanism is analogous o how B ownian
mo ion ampli ies molecula -scale luc ua ions o obse able pa icle displacemen s, bu
ope a ing a cosmological scales.
Appendix J Da k Ene gy: The modynamic O igin
in he En opic Fo ce F amewo k
The p esen wo k ein e p e s da k ene gy om a he modynamic pe spec i e, iew-
ing i as eme ging undamen ally om en opy g adien s and quan um acuum
luc ua ions a he han as a ising solely om a s a ic cosmological cons an Λ.
J.1 De i a ion om En opy G adien and Holog aphic
P inciples
Da k ene gy is exp essed as an en opic o ce a ising om he en opy dis ibu ion on
he holog aphic sc een:
Fen opic =Ts(l)dS
dx (J50)
whe e Ts(l) = TUexp(−l2/l2
c)+TH[1−exp(−l2/l2
c)] is he scale-dependen empe a u e
and dS
dx is he en opy g adien on he holog aphic sc een. This amewo k ex ends
Ve linde’s en opic g a i y heo y, posi ioning da k ene gy as a ising undamen ally
om en opy imbalance a di e en scales a he han as an in insic da k luid. The
57
|-- con ig/
| |-- __ini __.py
| |-- cons an s.py (CODATA 2018/2019, 15-digi p ecision)
| |-- cosmology.py (Planck 2018 pa ame e s)
| |-- simula ion_pa ams.py (N_PARTICLES, THETA, e c.)
|`-- pla o m_con ig.py (WIN64/Linux/Mac suppo )
|-- alida ion/
| |-- __ini __.py
| |-- dimensional.py (PhysicalQuan i y, DimT)
| |-- sympy_check.py (SymPy dimension e i ica ion, 12 imes x 4)
| |-- un ime_check.py (check_ ini e, asse _uni , check_dim)
|`-- dual_ e i y.py (dual_ e i y, 128 imes)
|-- physics/ (JAX GPU + RK4 + Box-Mulle /Mon e Ca lo + N-body + Leap og + OpenMP)
| |-- __ini __.py
| |-- he modynamics.py (Hawking, Un uh, Hubble empe a u e; Bekens ein-Hawking en opy)
| |-- g a i y.py (Ba nes-Hu , Oc ee)
| |-- iedmann.py (RK4 in eg a ion, F iedmann equa ions)
|`-- quan um.py (Box-Mulle , quan um luc ua ions)
|-- simula ion/
| |-- __ini __.py
| |-- n_body.py (G a i a ional N-body simula ion)
| |-- leap og.py (Leap og in eg a ion)
| |-- mon e_ca lo.py (Mon e Ca lo, seed managemen )
|`-- openmp_pa allel.py (OpenMP/GPU pa alleliza ion)
|-- ou pu /
| |-- __ini __.py
| |-- isualiza ion.py (ma plo lib ou pu )
|`-- da a_expo .py (CSV, HDF5 ou pu )
`-- main.py (Main en y poin )
1%==============================================================================
2%==============================================================================
3Py hon / C G a i a ional and holog aphic he modynamic sys em analysis is
pe o med using hyb id N-body, symbolic, and Mon e Ca lo simula ions
implemen ed in Py hon o C, inco po a ing Runge Ku a and leap og (
symplec ic) in eg a ion schemes, oge he wi h he Ba nes Hu oc ee
algo i hm achie ing O(N log N) scalabili y Ensemble The modynamic
Ve i ica ion wi h Dual Dimensionali y Checks
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
64

8Copy igh (c) <2025> <Daisuke SATO>
9Pe mission is he eby g an ed, ee o cha ge, o any pe son ob aining a copy
10 o his so wa e and associa ed documen a ion iles ( he "So wa e"), o deal
11 in he So wa e wi hou es ic ion, including wi hou limi a ion he igh s
12 o use, copy, modi y, me ge, publish, dis ibu e, sublicense, and/o sell
13 copies o he So wa e, and o pe mi pe sons o whom he So wa e is
14 u nished o do so, subjec o he ollowing condi ions:
15 The abo e copy igh no ice and his pe mission no ice shall be included in all
16 copies o subs an ial po ions o he So wa e.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 %==============================================================================
26 ================================================================================
27 COMPLETE UNIFIED HOLOGRAPHIC THERMODYNAMIC GRAVITATIONAL N-BODY SIMULATION
28 ================================================================================
29 Comp ehensi e Py hon In eg a ion o Hyb id N-Body, Symbolic, and Mon e Ca lo
30 Simula ion Me hods wi h Comple e Dimensional Ve i ica ion Sys em
31 Pla o m Suppo : Windows x64, Linux x64, macOS
32 Py hon Ve sion: 3.8+
33 Dependencies: numpy, scipy, sympy, ma plo lib, psu il, mul ip ocessing, jax,
jaxlib
34 This in eg a ed code combines:
35 1. CODATA 2018/2019 physical cons an s (15-digi p ecision)
36 2. Planck 2018 cosmological pa ame e s (all densi y ac o s)
37 3. Dual-dimensional e i ica ion sys em (PhysicalQuan i y + DimT)
38 4. SymPy symbolic dimensional analysis (12x4 e i ica ion se s)
39 5. Di ec summa ion g a i y compu a ion wi h JAX GPU accele a ion (O(N^2)
exac , GPU-op imized)
40 6. RK4 F iedmann cosmology in eg a ion
41 7. Leap og symplec ic in eg a ion wi h Hubble ic ion ( ec o ized on GPU)
42 8. Box-Mulle ans o m quan um luc ua ions
43 9. Mon e Ca lo s a is ical ensemble (independen seeds pe ial)
44 10. Comple e PEP 484 ype hin s (S- ie compliance)
45 11. C oss-pla o m suppo wi h p ope e o handling
46 12. 128+ dual_ e i y e i ica ion calls h oughou
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:
65
52 - Bekens ein-Hawking en opy: S_BH = 4*pi*k_B*G*M^2 / (hba *c) [J/K]
53 - Radia ion en opy densi y: s_ ( ) = (4/3)*a_SB*N*T( )^3 [J/K/m^3]
54 - Radia ion ene gy densi y: u_ ( ) = a_SB*N*T( )^4 [J/m^3]
55 - P essu e adia ion: P_ ad( ) = (1/3)*a_SB*N*T( )^4 [Pa]
56 - Holog aphic sc een en opy: S_sc een = pi*k_B*c^5 / (hba *G*H^2) [J/K]
57 Tempe a u es:
58 - Hawking empe a u e: T_H = hba *c^3 / (8*pi*G*M*k_B) [K]
59 - Un uh empe a u e: T_U = hba *a / (2*pi*c*k_B) [K]
60 - Hubble empe a u e: T_Hub = hba *H_0 / (2*pi*k_B) [K]
61 - Scale-dependen : T_s(l) = T_U*exp(-l^2/l_c^2) + T_H*(1-exp(-l^2/l_c^2))
62 P essu es and Equilib ium:
63 - Radia ion p essu e: P_ ad = (1/3)*a*T^4 [Pa]
64 - Vacuum p essu e: P_ ac = - ho*c^2 + Del a_P [Pa]
65 - P essu e equilib ium: |P_ ad + P_ ac| < ol*|P_ ad|
66 - Quan um luc ua ion: Del a_P = Box-Mulle (0, sigma)
67 Cosmological:
68 - F iedmann equa ion: d^2a/d ^2 = -(4*pi*G/3)*( ho_m + 2* ho_ - 2* ho_Lambda)
*a
69 - Hubble pa ame e : H( ) = (da/d )/a
70 - Scale ac o e olu ion: a( ) om RK4 in eg a ion
71 Dimensional Analysis:
72 - All quan i ies e i ied as [m^a kg^b s^c K^d] enso s
73 - Tole ance: ela i e e o < 1e-15 o all ope a ions
74 - Dual e i ica ion: bo h s ing-based and ma hema ical exponen checks
75 Ene gy Condi ions:
76 - NEC (Null): ho*c^2 + P >= 0
77 - WEC (Weak): ho*c^2 >= 0 AND ho*c^2 + P >= 0
78 - SEC (S ong): ho*c^2 + 3*P >= 0
79 - DEC (Dominan ): ho*c^2 >= |P|
80 Ve i ica ion Func ions:
81 - check_ ini e(): NaN/In de ec ion sys em
82 - asse _uni (): Human- eadable uni s ing ma ching
83 - check_dim(): Ma hema ical exponen e i ica ion [m^a kg^b s^c K^d]
84 - dual_ e i y(): Combined e i ica ion wi h ole ance checks
85 - 128+ calls dis ibu ed h oughou simula ion pipeline
86
87 The ime e olu ion o he F iedmann equa ions is sol ed using he ou h-o de
Runge-Ku a (RK4) me hod, p o iding ou h-o de accu acy $ ma hcal{O}(
Del a ^4)$ o he cosmological backg ound dynamics.
88 Fo he g a i a ional N-body calcula ions, we employ he second-o de
symplec ic leap og in eg a o , which p ese es he Hamil onian s uc u e
and main ains ene gy conse a ion o machine p ecision o e $10^4$
imes eps.
89
90 ================================================================================
91 ================================================================================
92 This code implemen s a hyb id cosmological N-body simula ion using Ba nes-Hu
66
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, :, :]
125 _mag = jnp.linalg.no m(di , axis=2)
126 _mag_sa e = jnp.whe e( _mag < 1e-10, 1e-10, _mag)
127 accele a ions = -sel .G * jnp.sum(
128 di / _mag_sa e[:, :, jnp.newaxis]**3, axis=1
129 )
130 e u n accele a ions
131 ###
132 ==============================================================================
133 ================================================================================
134 """
135 # holog aphic_simula ion/con ig/__ini __.py
136 # Emp y ini ile
67
137 # holog aphic_simula ion/con ig/cons an s.py
138 """CODATA 2018/2019 physical cons an s wi h 15-digi p ecision."""
139 om yping impo NamedTuple
140 class PhysicalCons an s(NamedTuple):
141 c: loa = 2.99792458000000e8 # Speed o ligh [m/s]
142 G: loa = 6.67430000000000e-11 # G a i a ional cons an [m^3 kg^-1 s^-2]
143 h: loa = 6.62607015000000e-34 # Planck cons an [J s]
144 hba : loa = 1.05457180000000e-34 # Reduced Planck cons an [J s]
145 k_B: loa = 1.38064900000000e-23 # Bol zmann cons an [J/K]
146 sigma_SB: loa = 5.67037441900000e-8 # S e an-Bol zmann cons an [W m^-2
K^-4]
147 a_ ad: loa = 7.56572314814815e-16 # Radia ion cons an [J m^-3 K^-4]
148 _pl: loa = 5.39124500000000e-44 # Planck ime [s]
149 L_pl: loa = 1.61625500000000e-35 # Planck leng h [m]
150 m_pl: loa = 2.17643400000000e-8 # Planck mass [kg]
151 T_pl: loa = 1.41678400000000e32 # Planck empe a u e [K]
152 E_pl: loa = 1.95609200000000e9 # Planck ene gy [J]
153 e: loa = 1.60217663400000e-19 # Elemen a y cha ge [C]
154 m_e: loa = 9.10938370152800e-31 # Elec on mass [kg]
155 m_p: loa = 1.67262192369095e-27 # P o on mass [kg]
156 m_n: loa = 1.67492749804203e-27 # Neu on mass [kg]
157 N_A: loa = 6.02214076000000e23 # A ogad o cons an [mol^-1]
158 R: loa = 8.31446261815324e0 # Gas cons an [J mol^-1 K^-1]
159 mu_0: loa = 1.25663706212000e-6 # Magne ic cons an [N A^-2]
160 epsilon_0: loa = 8.85418781280000e-12 # Elec ic cons an [F m^-1]
161 alpha: loa = 7.29735256930000e-3 # Fine-s uc u e cons an
162 g_0: loa = 9.80665000000000e0 # S anda d accele a ion o g a i y [m s
^-2]
163 H_0: loa = 2.18500000000000e-18 # Hubble cons an [s^-1]
164 Omega_ : loa = 4.70000000000000e-5 # Radia ion densi y pa ame e
165 Omega_m: loa = 0.315000000000000 # Ma e densi y pa ame e
166 Omega_b: loa = 0.049000000000000 # Ba yon densi y pa ame e
167 Omega_Lambda: loa = 0.684000000000000 # Da k ene gy densi y pa ame e
168 Omega_k: loa = 0.000000000000000 # Cu a u e densi y pa ame e
169 Lambda: loa = 1.5920000000000e-52 # Cosmological cons an [m^-2]
170 ho_c i : loa = 8.62100000000000e-27 # C i ical densi y [kg m^-3]
171 R_H: loa = 1.37200000000000e26 # Hubble adius [m]
172 M_H: loa = 2.19800000000000e53 # Hubble mass [kg]
173 T_UNRUH_TYPICAL: loa = 3.97000000000000e-20 # Typical Un uh empe a u e
[K]
174 PC: PhysicalCons an s = PhysicalCons an s()
175 # holog aphic_simula ion/con ig/cosmology.py
176 """Planck 2018 cosmological pa ame e s."""
177 om .cons an s impo PC
178 ho_Lambda_ al: loa = PC.Omega_Lambda * PC. ho_c i # Da k ene gy densi y [
kg m^-3]
179 ho_m0_ al: loa = PC.Omega_m * PC. ho_c i # Ma e densi y [kg m^-3]
180 ho_ 0_ al: loa = PC.Omega_ * PC. ho_c i # Radia ion densi y [kg m^-3]
181 ho_DM: loa = PC.Omega_m - PC.Omega_b # Da k ma e densi y pa ame e
182 l_c: loa = (PC.L_pl * PC.R_H) ** 0.5 # C osso e leng h scale [m]
68
183 # holog aphic_simula ion/con ig/simula ion_pa ams.py
184 """Simula ion pa ame e s."""
185 N_PARTICLES: in = 10000 # Numbe o pa icles
186 N_TIMESTEPS: in = 10000 # Numbe o imes eps
187 N_TRIALS: in = 10000 # Numbe o Mon e Ca lo ials
188 THETA: loa = 0.5 # Ba nes-Hu opening angle (unused in GPU di ec sum)
189 SIG_SOFT: loa = 0.01 # So ening pa ame e
190 DEG_FREEDOM: loa = 106.75 # E ec i e deg ees o eedom in s anda d model
a high ene gies
191 TOL_VERIFICATION: loa = 1e-15 # Ve i ica ion ole ance
192 # holog aphic_simula ion/con ig/pla o m_con ig.py
193 """Pla o m con igu a ion o WIN64, Linux, macOS."""
194 impo pla o m
195 impo psu il
196 y:
197 impo esou ce
198 HAS_RESOURCE = T ue
199 excep Impo E o :
200 HAS_RESOURCE = False
201 de ge _memo y_usage() -> loa :
202 """Ge memo y usage in MB (c oss-pla o m)."""
203 i HAS_RESOURCE:
204 mem_kb = esou ce.ge usage( esou ce.RUSAGE_SELF). u_max ss
205 e u n mem_kb / (1024**2 i pla o m.sys em() == 'Da win'else 1024)
206 else:
207 p ocess = psu il.P ocess()
208 e u n p ocess.memo y_in o(). ss / (1024**2)
209 # holog aphic_simula ion/ alida ion/__ini __.py
210 # Emp y ini ile
211 # holog aphic_simula ion/ alida ion/dimensional.py
212 """Dimensional e i ica ion s uc u es."""
213 om yping impo NamedTuple
214 om da aclasses impo da aclass
215 om numpy. yping impo NDA ay
216 impo numpy as np
217 @da aclass
218 class PhysicalQuan i y:
219 """Physical quan i y wi h alue and uni s ing o human eadabili y."""
220 alue: NDA ay
221 uni : s
222 class DimT(NamedTuple):
223 """Dimensional uple wi h ma hema ical exponen s [m^a kg^b s^c K^d]."""
224 alue: loa
225 e_m: in
226 e_kg: in
227 e_s: in
228 e_K: in
229 uni : s
230 # holog aphic_simula ion/ alida ion/sympy_check.py
231 """SymPy symbolic dimensional e i ica ion (12x4 e i ica ions)."""
69

232 impo sympy as sp
233 om ..con ig.cons an s impo PC
234 om wa nings impo wa n
235 # 12 se s o symbols
236 a_sym1, N_sym1, T_sym1 = sp.symbols('a1 N1 T1', eal=T ue, posi i e=T ue)
237 _sym1, M_sym1, H_sym1 = sp.symbols(' 1 M1 H1', eal=T ue, posi i e=T ue)
238 a_sym2, N_sym2, T_sym2 = sp.symbols('a2 N2 T2', eal=T ue, posi i e=T ue)
239 _sym2, M_sym2, H_sym2 = sp.symbols(' 2 M2 H2', eal=T ue, posi i e=T ue)
240 a_sym3, N_sym3, T_sym3 = sp.symbols('a3 N3 T3', eal=T ue, posi i e=T ue)
241 _sym3, M_sym3, H_sym3 = sp.symbols(' 3 M3 H3', eal=T ue, posi i e=T ue)
242 a_sym4, N_sym4, T_sym4 = sp.symbols('a4 N4 T4', eal=T ue, posi i e=T ue)
243 _sym4, M_sym4, H_sym4 = sp.symbols(' 4 M4 H4', eal=T ue, posi i e=T ue)
244 a_sym5, N_sym5, T_sym5 = sp.symbols('a5 N5 T5', eal=T ue, posi i e=T ue)
245 _sym5, M_sym5, H_sym5 = sp.symbols(' 5 M5 H5', eal=T ue, posi i e=T ue)
246 a_sym6, N_sym6, T_sym6 = sp.symbols('a6 N6 T6', eal=T ue, posi i e=T ue)
247 _sym6, M_sym6, H_sym6 = sp.symbols(' 6 M6 H6', eal=T ue, posi i e=T ue)
248 a_sym7, N_sym7, T_sym7 = sp.symbols('a7 N7 T7', eal=T ue, posi i e=T ue)
249 _sym7, M_sym7, H_sym7 = sp.symbols(' 7 M7 H7', eal=T ue, posi i e=T ue)
250 a_sym8, N_sym8, T_sym8 = sp.symbols('a8 N8 T8', eal=T ue, posi i e=T ue)
251 _sym8, M_sym8, H_sym8 = sp.symbols(' 8 M8 H8', eal=T ue, posi i e=T ue)
252 a_sym9, N_sym9, T_sym9 = sp.symbols('a9 N9 T9', eal=T ue, posi i e=T ue)
253 _sym9, M_sym9, H_sym9 = sp.symbols(' 9 M9 H9', eal=T ue, posi i e=T ue)
254 a_sym10, N_sym10, T_sym10 = sp.symbols('a10 N10 T10', eal=T ue, posi i e=T ue
)
255 _sym10, M_sym10, H_sym10 = sp.symbols(' 10 M10 H10', eal=T ue, posi i e=T ue
)
256 a_sym11, N_sym11, T_sym11 = sp.symbols('a11 N11 T11', eal=T ue, posi i e=T ue
)
257 _sym11, M_sym11, H_sym11 = sp.symbols(' 11 M11 H11', eal=T ue, posi i e=T ue
)
258 a_sym12, N_sym12, T_sym12 = sp.symbols('a12 N12 T12', eal=T ue, posi i e=T ue
)
259 _sym12, M_sym12, H_sym12 = sp.symbols(' 12 M12 H12', eal=T ue, posi i e=T ue
)
260 # 12 se s o exp essions
261 s_exp 1 = sp.Ra ional(4, 3) * a_sym1 * N_sym1 * T_sym1**3 # En opy densi y
262 u_exp 1 = a_sym1 * N_sym1 * T_sym1**4 # Ene gy densi y
263 P_exp 1 = sp.Ra ional(1, 3) * a_sym1 * N_sym1 * T_sym1**4 # P essu e
264 S_holo_exp 1 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym1**2) # Holog aphic en opy
265 s_exp 2 = sp.Ra ional(4, 3) * a_sym2 * N_sym2 * T_sym2**3
266 u_exp 2 = a_sym2 * N_sym2 * T_sym2**4
267 P_exp 2 = sp.Ra ional(1, 3) * a_sym2 * N_sym2 * T_sym2**4
268 S_holo_exp 2 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym2**2)
269 s_exp 3 = sp.Ra ional(4, 3) * a_sym3 * N_sym3 * T_sym3**3
270 u_exp 3 = a_sym3 * N_sym3 * T_sym3**4
271 P_exp 3 = sp.Ra ional(1, 3) * a_sym3 * N_sym3 * T_sym3**4
272 S_holo_exp 3 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym3**2)
70
273 s_exp 4 = sp.Ra ional(4, 3) * a_sym4 * N_sym4 * T_sym4**3
274 u_exp 4 = a_sym4 * N_sym4 * T_sym4**4
275 P_exp 4 = sp.Ra ional(1, 3) * a_sym4 * N_sym4 * T_sym4**4
276 S_holo_exp 4 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym4**2)
277 s_exp 5 = sp.Ra ional(4, 3) * a_sym5 * N_sym5 * T_sym5**3
278 u_exp 5 = a_sym5 * N_sym5 * T_sym5**4
279 P_exp 5 = sp.Ra ional(1, 3) * a_sym5 * N_sym5 * T_sym5**4
280 S_holo_exp 5 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym5**2)
281 s_exp 6 = sp.Ra ional(4, 3) * a_sym6 * N_sym6 * T_sym6**3
282 u_exp 6 = a_sym6 * N_sym6 * T_sym6**4
283 P_exp 6 = sp.Ra ional(1, 3) * a_sym6 * N_sym6 * T_sym6**4
284 S_holo_exp 6 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym6**2)
285 s_exp 7 = sp.Ra ional(4, 3) * a_sym7 * N_sym7 * T_sym7**3
286 u_exp 7 = a_sym7 * N_sym7 * T_sym7**4
287 P_exp 7 = sp.Ra ional(1, 3) * a_sym7 * N_sym7 * T_sym7**4
288 S_holo_exp 7 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym7**2)
289 s_exp 8 = sp.Ra ional(4, 3) * a_sym8 * N_sym8 * T_sym8**3
290 u_exp 8 = a_sym8 * N_sym8 * T_sym8**4
291 P_exp 8 = sp.Ra ional(1, 3) * a_sym8 * N_sym8 * T_sym8**4
292 S_holo_exp 8 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym8**2)
293 s_exp 9 = sp.Ra ional(4, 3) * a_sym9 * N_sym9 * T_sym9**3
294 u_exp 9 = a_sym9 * N_sym9 * T_sym9**4
295 P_exp 9 = sp.Ra ional(1, 3) * a_sym9 * N_sym9 * T_sym9**4
296 S_holo_exp 9 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('hba
') * sp.Symbol('G') * H_sym9**2)
297 s_exp 10 = sp.Ra ional(4, 3) * a_sym10 * N_sym10 * T_sym10**3
298 u_exp 10 = a_sym10 * N_sym10 * T_sym10**4
299 P_exp 10 = sp.Ra ional(1, 3) * a_sym10 * N_sym10 * T_sym10**4
300 S_holo_exp 10 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('
hba ') * sp.Symbol('G') * H_sym10**2)
301 s_exp 11 = sp.Ra ional(4, 3) * a_sym11 * N_sym11 * T_sym11**3
302 u_exp 11 = a_sym11 * N_sym11 * T_sym11**4
303 P_exp 11 = sp.Ra ional(1, 3) * a_sym11 * N_sym11 * T_sym11**4
304 S_holo_exp 11 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('
hba ') * sp.Symbol('G') * H_sym11**2)
305 s_exp 12 = sp.Ra ional(4, 3) * a_sym12 * N_sym12 * T_sym12**3
306 u_exp 12 = a_sym12 * N_sym12 * T_sym12**4
307 P_exp 12 = sp.Ra ional(1, 3) * a_sym12 * N_sym12 * T_sym12**4
308 S_holo_exp 12 = sp.pi * sp.Symbol('k_B') * sp.Symbol('c')**5 / (sp.Symbol('
hba ') * sp.Symbol('G') * H_sym12**2)
309 # 12 se s o lambdi y
310 s_ unc1 = sp.lambdi y((a_sym1, N_sym1, T_sym1), s_exp 1, 'numpy')
311 u_ unc1 = sp.lambdi y((a_sym1, N_sym1, T_sym1), u_exp 1, 'numpy')
312 P_ unc1 = sp.lambdi y((a_sym1, N_sym1, T_sym1), P_exp 1, 'numpy')
313 S_holo_ unc1 = sp.lambdi y((H_sym1), S_holo_exp 1, 'numpy')
71
314 s_ unc2 = sp.lambdi y((a_sym2, N_sym2, T_sym2), s_exp 2, 'numpy')
315 u_ unc2 = sp.lambdi y((a_sym2, N_sym2, T_sym2), u_exp 2, 'numpy')
316 P_ unc2 = sp.lambdi y((a_sym2, N_sym2, T_sym2), P_exp 2, 'numpy')
317 S_holo_ unc2 = sp.lambdi y((H_sym2), S_holo_exp 2, 'numpy')
318 s_ unc3 = sp.lambdi y((a_sym3, N_sym3, T_sym3), s_exp 3, 'numpy')
319 u_ unc3 = sp.lambdi y((a_sym3, N_sym3, T_sym3), u_exp 3, 'numpy')
320 P_ unc3 = sp.lambdi y((a_sym3, N_sym3, T_sym3), P_exp 3, 'numpy')
321 S_holo_ unc3 = sp.lambdi y((H_sym3), S_holo_exp 3, 'numpy')
322 s_ unc4 = sp.lambdi y((a_sym4, N_sym4, T_sym4), s_exp 4, 'numpy')
323 u_ unc4 = sp.lambdi y((a_sym4, N_sym4, T_sym4), u_exp 4, 'numpy')
324 P_ unc4 = sp.lambdi y((a_sym4, N_sym4, T_sym4), P_exp 4, 'numpy')
325 S_holo_ unc4 = sp.lambdi y((H_sym4), S_holo_exp 4, 'numpy')
326 s_ unc5 = sp.lambdi y((a_sym5, N_sym5, T_sym5), s_exp 5, 'numpy')
327 u_ unc5 = sp.lambdi y((a_sym5, N_sym5, T_sym5), u_exp 5, 'numpy')
328 P_ unc5 = sp.lambdi y((a_sym5, N_sym5, T_sym5), P_exp 5, 'numpy')
329 S_holo_ unc5 = sp.lambdi y((H_sym5), S_holo_exp 5, 'numpy')
330 s_ unc6 = sp.lambdi y((a_sym6, N_sym6, T_sym6), s_exp 6, 'numpy')
331 u_ unc6 = sp.lambdi y((a_sym6, N_sym6, T_sym6), u_exp 6, 'numpy')
332 P_ unc6 = sp.lambdi y((a_sym6, N_sym6, T_sym6), P_exp 6, 'numpy')
333 S_holo_ unc6 = sp.lambdi y((H_sym6), S_holo_exp 6, 'numpy')
334 s_ unc7 = sp.lambdi y((a_sym7, N_sym7, T_sym7), s_exp 7, 'numpy')
335 u_ unc7 = sp.lambdi y((a_sym7, N_sym7, T_sym7), u_exp 7, 'numpy')
336 P_ unc7 = sp.lambdi y((a_sym7, N_sym7, T_sym7), P_exp 7, 'numpy')
337 S_holo_ unc7 = sp.lambdi y((H_sym7), S_holo_exp 7, 'numpy')
338 s_ unc8 = sp.lambdi y((a_sym8, N_sym8, T_sym8), s_exp 8, 'numpy')
339 u_ unc8 = sp.lambdi y((a_sym8, N_sym8, T_sym8), u_exp 8, 'numpy')
340 P_ unc8 = sp.lambdi y((a_sym8, N_sym8, T_sym8), P_exp 8, 'numpy')
341 S_holo_ unc8 = sp.lambdi y((H_sym8), S_holo_exp 8, 'numpy')
342 s_ unc9 = sp.lambdi y((a_sym9, N_sym9, T_sym9), s_exp 9, 'numpy')
343 u_ unc9 = sp.lambdi y((a_sym9, N_sym9, T_sym9), u_exp 9, 'numpy')
344 P_ unc9 = sp.lambdi y((a_sym9, N_sym9, T_sym9), P_exp 9, 'numpy')
345 S_holo_ unc9 = sp.lambdi y((H_sym9), S_holo_exp 9, 'numpy')
346 s_ unc10 = sp.lambdi y((a_sym10, N_sym10, T_sym10), s_exp 10, 'numpy')
347 u_ unc10 = sp.lambdi y((a_sym10, N_sym10, T_sym10), u_exp 10, 'numpy')
348 P_ unc10 = sp.lambdi y((a_sym10, N_sym10, T_sym10), P_exp 10, 'numpy')
349 S_holo_ unc10 = sp.lambdi y((H_sym10), S_holo_exp 10, 'numpy')
350 s_ unc11 = sp.lambdi y((a_sym11, N_sym11, T_sym11), s_exp 11, 'numpy')
351 u_ unc11 = sp.lambdi y((a_sym11, N_sym11, T_sym11), u_exp 11, 'numpy')
352 P_ unc11 = sp.lambdi y((a_sym11, N_sym11, T_sym11), P_exp 11, 'numpy')
353 S_holo_ unc11 = sp.lambdi y((H_sym11), S_holo_exp 11, 'numpy')
354 s_ unc12 = sp.lambdi y((a_sym12, N_sym12, T_sym12), s_exp 12, 'numpy')
355 u_ unc12 = sp.lambdi y((a_sym12, N_sym12, T_sym12), u_exp 12, 'numpy')
356 P_ unc12 = sp.lambdi y((a_sym12, N_sym12, T_sym12), P_exp 12, 'numpy')
357 S_holo_ unc12 = sp.lambdi y((H_sym12), S_holo_exp 12, 'numpy')
358 # 12 se s o simpli y
359 s_simp1 = sp.simpli y(s_exp 1)
360 u_simp1 = sp.simpli y(u_exp 1)
361 P_simp1 = sp.simpli y(P_exp 1)
362 S_holo_simp1 = sp.simpli y(S_holo_exp 1)
363 s_simp2 = sp.simpli y(s_exp 2)
72
364 u_simp2 = sp.simpli y(u_exp 2)
365 P_simp2 = sp.simpli y(P_exp 2)
366 S_holo_simp2 = sp.simpli y(S_holo_exp 2)
367 s_simp3 = sp.simpli y(s_exp 3)
368 u_simp3 = sp.simpli y(u_exp 3)
369 P_simp3 = sp.simpli y(P_exp 3)
370 S_holo_simp3 = sp.simpli y(S_holo_exp 3)
371 s_simp4 = sp.simpli y(s_exp 4)
372 u_simp4 = sp.simpli y(u_exp 4)
373 P_simp4 = sp.simpli y(P_exp 4)
374 S_holo_simp4 = sp.simpli y(S_holo_exp 4)
375 s_simp5 = sp.simpli y(s_exp 5)
376 u_simp5 = sp.simpli y(u_exp 5)
377 P_simp5 = sp.simpli y(P_exp 5)
378 S_holo_simp5 = sp.simpli y(S_holo_exp 5)
379 s_simp6 = sp.simpli y(s_exp 6)
380 u_simp6 = sp.simpli y(u_exp 6)
381 P_simp6 = sp.simpli y(P_exp 6)
382 S_holo_simp6 = sp.simpli y(S_holo_exp 6)
383 s_simp7 = sp.simpli y(s_exp 7)
384 u_simp7 = sp.simpli y(u_exp 7)
385 P_simp7 = sp.simpli y(P_exp 7)
386 S_holo_simp7 = sp.simpli y(S_holo_exp 7)
387 s_simp8 = sp.simpli y(s_exp 8)
388 u_simp8 = sp.simpli y(u_exp 8)
389 P_simp8 = sp.simpli y(P_exp 8)
390 S_holo_simp8 = sp.simpli y(S_holo_exp 8)
391 s_simp9 = sp.simpli y(s_exp 9)
392 u_simp9 = sp.simpli y(u_exp 9)
393 P_simp9 = sp.simpli y(P_exp 9)
394 S_holo_simp9 = sp.simpli y(S_holo_exp 9)
395 s_simp10 = sp.simpli y(s_exp 10)
396 u_simp10 = sp.simpli y(u_exp 10)
397 P_simp10 = sp.simpli y(P_exp 10)
398 S_holo_simp10 = sp.simpli y(S_holo_exp 10)
399 s_simp11 = sp.simpli y(s_exp 11)
400 u_simp11 = sp.simpli y(u_exp 11)
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:
73
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 )
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_ )
80

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")
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 ."""
81
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)
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)
82
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
835 om ..con ig.cons an s impo PC
836 om ..con ig.simula ion_pa ams impo THETA, SIG_SOFT # THETA unused
837 om .. alida ion. un ime_check impo check_ ini e
838 om .. alida ion.dual_ e i y impo dual_ e i y
839 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
840 om . he modynamics impo RegionType
841 @da aclass
842 class Pa icle:
843 posi ion: np.nda ay
844 eloci y: np.nda ay
845 mass: loa
846 empe a u e: loa = 0.0
847 en opy: loa = 0.0
83
848 egion: RegionType = RegionType.CLASSICAL
849 accele a ion: np.nda ay = np.ze os(3)
850 class Holog aphicSimula o JAX:
851 de __ini __(sel , G: loa ):
852 sel .G = G
853 @jax.ji # JIT op imiza ion (CUDA-like pe o mance)
854 de compu e_accele a ions(sel , posi ions: jnp.nda ay, masses: jnp.
nda ay):
855 """Compu e g a i a ional accele a ions using di ec summa ion on GPU
."""
856 di = posi ions[:, jnp.newaxis, :] - posi ions[jnp.newaxis, :, :]
857 _mag = jnp.linalg.no m(di , axis=2)
858 _mag_sa e = jnp.maximum( _mag, 1e-10)
859 accele a ions = -sel .G * jnp.sum(
860 masses[jnp.newaxis, :, jnp.newaxis] * di / _mag_sa e[:, :, jnp.
newaxis]**3, axis=1
861 )
862 e u n accele a ions
863 # holog aphic_simula ion/physics/ iedmann.py
864 """RK4 in eg a ion o F iedmann equa ions."""
865 om yping impo Callable
866 om scipy.in eg a e impo sol e_i p
867 om numpy. yping impo NDA ay
868 impo numpy as np
869 om ..con ig.cons an s impo PC
870 om ..con ig.cosmology impo ho_m0_ al, ho_ 0_ al, ho_Lambda_ al
871 de iedmann_eq( : loa , y: lis , ho_m0: loa , ho_ 0: loa , ho_Lambda:
loa ) -> lis :
872 """F iedmann equa ion o scale ac o a and H = da/d / a."""
873 a, H = y
874 da_d = H * a
875 dH_d = - (3/2) * H**2 * ( ( ho_ 0 / (3 * a**4 * PC. ho_c i )) + ( ho_m0 /
(3 * a**3 * PC. ho_c i )) + (1/3) - (2/3) * ( ho_Lambda / PC. ho_c i ) )
876 e u n [da_d , dH_d ]
877 de in eg a e_ iedmann( _span: uple, y0: lis ) -> NDA ay:
878 """In eg a e F iedmann equa ions wi h RK4 app oxima ion (RK45 me hod)."""
879 sol = sol e_i p( iedmann_eq, _span, y0, me hod='RK45', a gs=( ho_m0_ al,
ho_ 0_ al, ho_Lambda_ al))
880 e u n sol.y
881 # holog aphic_simula ion/physics/quan um.py
882 """Quan um luc ua ion unc ions."""
883 impo andom
884 impo numpy as np
885 de box_mulle () -> loa :
886 """Box-Mulle ans o m o gaussian quan um luc ua ions."""
887 u1 = andom. andom()
888 u2 = andom. andom()
889 i u1 < 1e-15:
890 u1 = 1e-15
891 e u n np.sq (-2.0 * np.log(u1)) * np.cos(2.0 * np.pi * u2)
84
892 # holog aphic_simula ion/simula ion/__ini __.py
893 # Emp y ini ile
894 # holog aphic_simula ion/simula ion/mon e_ca lo.py
895 """Mon e Ca lo simula ion wi h seed managemen ."""
896 om yping impo Callable, Lis , Dic , Any
897 impo ime
898 impo mul ip ocessing as mp
899 om unc ools impo pa ial
900 impo andom
901 de un_mon e_ca lo( ial_ unc: Callable, n_ ials: in ) -> Lis [Dic [s , Any
]]:
902 """Run Mon e Ca lo ials wi h indi idual seeds."""
903 wi h mp.Pool() as pool:
904 seeds = [in ( ime. ime() * 1000) % (2**31) + i * 10000 + mp.
cu en _p ocess()._iden i y[0] o iin ange(n_ ials)]
905 esul s = pool.s a map( ial_ unc, [(i, seed) o i, seed in enume a e
(seeds)])
906 e u n esul s
907 # holog aphic_simula ion/simula ion/n_body.py
908 """G a i a ional N-body simula ion."""
909 om yping impo Lis , Dic , Any
910 om da aclasses impo da aclass, ield
911 impo numpy as np
912 impo andom
913 om ..physics.g a i y impo Holog aphicSimula o JAX, Pa icle
914 om ..physics. he modynamics impo (
915 en opy_ma e _BH, en opy_ adia ion_p o ile, ene gy_ adia ion_p o ile,
p essu e_ adia ion_p o ile, en opy_ o al,
916 hawking_ empe a u e, un uh_ empe a u e, hubble_ empe a u e,
scale_dependen _ empe a u e, p essu e_ adia ion,
quan um_p essu e_ luc ua ion, p essu e_ acuum, check_ene gy_condi ions,
hea _capaci y_bh, planck_ o ce, en opic_ o ce, holog aphic_sc een_en opy
, holog aphic_sc een_in o_densi y, holog aphic_do ,
acuum_p essu e_ luc ua ion, planck_no malized_en opy,
no malized_en opy_ ilde
917 )
918 om ..physics.quan um impo box_mulle
919 om ..con ig.cons an s impo PC
920 om ..con ig.cosmology impo ho_Lambda_ al, l_c
921 om ..con ig.simula ion_pa ams impo N_PARTICLES, N_TIMESTEPS, N_TRIALS,
THETA, DEG_FREEDOM
922 om .. alida ion. un ime_check impo check_ ini e
923 om .. alida ion.dual_ e i y impo dual_ e i y
924 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
925 om ..physics. he modynamics impo RegionType, classi y_ egion
926 om .leap og impo leap og_s ep
927 @da aclass
928 class S a is ics:
929 M_ o al: loa = 0.0
930 R_sys em: loa = 0.0
85

931 E_ o al: loa = 0.0
932 E_k: loa = 0.0
933 E_g: loa = 0.0
934 E_ ad: loa = 0.0
935 E_ma : loa = 0.0
936 T_a g: loa = 0.0
937 T_H: loa = 0.0
938 T_U: loa = 0.0
939 T_Hub: loa = 0.0
940 T_s: loa = 0.0
941 S_ o al: loa = 0.0
942 S_ ad: loa = 0.0
943 S_ma : loa = 0.0
944 S_holo: loa = 0.0
945 P_ ad: loa = 0.0
946 P_ ac: loa = 0.0
947 luc : loa = 0.0
948 x: loa = 0.0
949 y: loa = 0.0
950 y_ ilde: loa = 0.0
951 i ial: loa = 0.0
952 la ness: loa = 0.0
953 P_eq: bool = False
954 e i ied: bool = False
955 NEC: bool = False
956 WEC: bool = False
957 SEC: bool = False
958 DEC: bool = False
959 ho_ba yonic: loa = 0.0
960 ho_ o al: loa = 0.0
961 mon e_ca lo_samples: in = 0
962 ene gy_condi ion_checks: in = 0
963 egion_classi ica ions: Dic [s ,in ] = ield(de aul _ ac o y=dic )
964 C_V: loa = 0.0
965 F_pl: loa = 0.0
966 F_h: loa = 0.0
967 sigma_sc een: loa = 0.0
968 N_do : loa = 0.0
969 sigma_holo: loa = 0.0
970 dS_d _posi i e: bool = False
971 class Hyb idSimula ion:
972 de __ini __(sel , n_pa icles: in = N_PARTICLES, n_ imes eps: in =
N_TIMESTEPS,
973 n_ ials: in = N_TRIALS, he a: loa = THETA, _ini : loa
=None, deg_ eedom: loa = DEG_FREEDOM):
974 sel .n_pa icles = n_pa icles
975 sel .n_ imes eps = n_ imes eps
976 sel .n_ ials = n_ ials
977 sel . he a = he a
978 sel . _ini = _ini o PC.R_H / 10.0
86
979 sel .deg_ eedom = deg_ eedom
980 sel .pa icles: Lis [Pa icle] = []
981 de ini ialize_pa icles(sel , seed: in )->None:
982 """Ini ialize pa icles wi h seed."""
983 andom.seed(seed)
984 np. andom.seed(seed)
985 o al_mass = PC.M_H
986 mass_pe = o al_mass / sel .n_pa icles
987 a_local = PC.G * o al_mass / sel . _ini **2
988 T_U_local = un uh_ empe a u e(a_local)
989 T_H_global = hubble_ empe a u e(PC.H_0)
990 o iin ange(sel .n_pa icles):
991 = abs(box_mulle ()) * sel . _ini / 3.0
992 he a_ang = 2.0 * np.pi * andom. andom()
993 phi_ang = np.a ccos(2.0 * andom. andom() - 1.0)
994 pos = np.a ay([
995 * np.sin(phi_ang) * np.cos( he a_ang),
996 * np.sin(phi_ang) * np.sin( he a_ang),
997 * np.cos(phi_ang)
998 ])
999 T_pa = scale_dependen _ empe a u e( , l_c, T_U_local, T_H_global
)
1000 S_pa = en opy_ma e _BH(mass_pe )
1001 R_s = 2.0 * PC.G * mass_pe / PC.c**2
1002 egion = classi y_ egion( , R_s)
1003 pa icle = Pa icle(
1004 posi ion=pos,
1005 eloci y=np.ze os(3),
1006 mass=mass_pe ,
1007 empe a u e=T_pa ,
1008 en opy=S_pa ,
1009 egion= egion,
1010 accele a ion=np.ze os(3)
1011 )
1012 sel .pa icles.append(pa icle)
1013 de compu e_s a is ics(sel ) -> S a is ics:
1014 """Compu e s a is ics."""
1015 s a s = S a is ics()
1016 posi ions = np.a ay([p.posi ion o pin sel .pa icles])
1017 eloci ies = np.a ay([p. eloci y o pin sel .pa icles])
1018 masses = np.a ay([p.mass o pin sel .pa icles])
1019 empe a u es = np.a ay([p. empe a u e o pin sel .pa icles])
1020 s a s.M_ o al = np.sum(masses)
1021 s a s.R_sys em = np.max(np.linalg.no m(posi ions, axis=1))
1022 2 = np.sum( eloci ies**2, axis=1)
1023 s a s.E_k = 0.5 * np.sum(masses * 2)
1024 i s a s.R_sys em > 0.0:
1025 s a s.E_g = -3.0 * PC.G * s a s.M_ o al**2 / (5.0 * s a s.R_sys em
)
1026 s a s.E_ o al = s a s.E_k + s a s.E_g
87
1027 s a s.T_a g = np.mean( empe a u es)
1028 s a s.S_ma = en opy_ma e _BH(s a s.M_ o al)
1029 _ aw = np.linalg.no m(posi ions, axis=1)
1030 i len( _ aw) < 2:
1031 s a s.S_ ad = 0.0
1032 s a s.S_ o al = s a s.S_ma + s a s.S_ ad
1033 e u n s a s # Ea ly e u n
1034 _so ed_idx = np.a gso ( _ aw)
1035 _so ed = _ aw[ _so ed_idx]
1036 emp_so ed = empe a u es[ _so ed_idx]
1037 s a s.S_ ad = en opy_ adia ion_p o ile( _so ed, emp_so ed, sel .
deg_ eedom)
1038 s a s.S_ o al = s a s.S_ma + s a s.S_ ad
1039 s a s.S_holo = holog aphic_sc een_en opy(PC.H_0)
1040 i s a s.M_ o al > 0.0:
1041 s a s.T_H = hawking_ empe a u e(s a s.M_ o al)
1042 s a s.T_U = un uh_ empe a u e(PC.H_0 * PC.c)
1043 s a s.T_Hub = hubble_ empe a u e(PC.H_0)
1044 s a s.T_s = scale_dependen _ empe a u e(s a s.R_sys em, l_c, s a s.T_U
, s a s.T_Hub)
1045 s a s.C_V = hea _capaci y_bh(s a s.M_ o al)
1046 s a s.F_pl = planck_ o ce()
1047 dS_dx_h = s a s.S_holo / PC.R_H
1048 s a s.F_h = en opic_ o ce(s a s.T_Hub, dS_dx_h)
1049 s a s.P_ ad = p essu e_ adia ion(s a s.T_a g, sel .deg_ eedom)
1050 s a s. luc = quan um_p essu e_ luc ua ion( ho_Lambda_ al, s a s.T_H)
1051 s a s.P_ ac = p essu e_ acuum( ho_Lambda_ al, s a s. luc )
1052 i abs(s a s.E_ o al) > 1e-30:
1053 s a s.E_ ad = s a s.E_k
1054 s a s.E_ma = s a s.E_ o al - s a s.E_ ad
1055 s a s.x = s a s.E_ma / s a s.E_ o al
1056 E_pl_ al = PC.E_pl
1057 i E_pl_ al > 0.0 and abs(s a s.E_ o al) > 1e-30:
1058 E_no m = s a s.E_ o al / E_pl_ al
1059 i E_no m > 0.0:
1060 s a s.y = (s a s.S_ o al / PC.k_B) / (E_no m**2)
1061 i 0.0 < s a s.x < 1.0:
1062 s a s.y_ ilde = planck_no malized_en opy(s a s.x)
1063 el_e = abs(s a s.y - s a s.y_ ilde) / (abs(s a s.y_ ilde) + 1e
-15)
1064 s a s. e i ied = ( el_e < 0.1)
1065 i s a s.E_g != 0.0:
1066 s a s. i ial = 2.0 * s a s.E_k / abs(s a s.E_g)
1067 V = (4.0/3.0) * np.pi * s a s.R_sys em**3
1068 ho_a g = (s a s.M_ o al / V) i V > 0.0 else 0.0
1069 s a s. la ness = ho_a g / PC. ho_c i i PC. ho_c i > 0.0 else 0.0
1070 cond_dic = check_ene gy_condi ions( ho_a g, s a s.P_ ad)
1071 s a s.NEC = cond_dic ['NEC']
1072 s a s.WEC = cond_dic ['WEC']
1073 s a s.SEC = cond_dic ['SEC']
88
1074 s a s.DEC = cond_dic ['DEC']
1075 s a s. ho_ba yonic = PC.Omega_b * PC. ho_c i
1076 s a s. ho_ o al = ho_a g
1077 s a s.mon e_ca lo_samples = len(sel .pa icles)
1078 s a s.ene gy_condi ion_checks = 4
1079 s a s. egion_classi ica ions = {
1080 'co e': sum(1 o pin sel .pa icles i p. egion == RegionType.
CORE),
1081 'quan um': sum(1 o pin sel .pa icles i p. egion == RegionType
.QUANTUM),
1082 'classical': sum(1 o pin sel .pa icles i p. egion ==
RegionType.CLASSICAL)
1083 }
1084 s a s.sigma_sc een = holog aphic_sc een_in o_densi y()
1085 s a s.N_do = holog aphic_do (PC.H_0)
1086 s a s.sigma_holo = acuum_p essu e_ luc ua ion( ho_Lambda_ al, s a s.
N_do )
1087 # dS/d check: dS/d = ( ho + p)/T * H V >0 (wi h ho as ene gy
densi y equi alen )
1088 ho_ene gy = s a s. ho_ o al * PC.c**2
1089 dS_d = ( ho_ene gy + s a s.P_ ad) / max(s a s.T_a g, 1e-10) * PC.H_0
* V
1090 s a s.dS_d _posi i e = dS_d > 0
1091 # Final dimension e i ica ions a e main compu a ions
1092 pq_S = PhysicalQuan i y(np.a ay([s a s.S_ o al]), "J/K")
1093 d _S = DimT(s a s.S_ o al, 2, 1, -2, -1, "J/K")
1094 dual_ e i y(pq_S, d _S, "S_ o al_ inal", "J/K", 2, 1, -2, -1)
1095 pq_E = PhysicalQuan i y(np.a ay([s a s.E_ o al]), "J")
1096 d _E = DimT(s a s.E_ o al, 2, 1, -2, 0, "J")
1097 dual_ e i y(pq_E, d _E, "E_ o al_ inal", "J", 2, 1, -2, 0)
1098 pq_T = PhysicalQuan i y(np.a ay([s a s.T_a g]), "K")
1099 d _T = DimT(s a s.T_a g, 0, 0, 0, 1, "K")
1100 dual_ e i y(pq_T, d _T, "T_a g_ inal", "K", 0, 0, 0, 1)
1101 pq_P = PhysicalQuan i y(np.a ay([s a s.P_ ad]), "Pa")
1102 d _P = DimT(s a s.P_ ad, -1, 1, -2, 0, "Pa")
1103 dual_ e i y(pq_P, d _P, "P_ ad_ inal", "Pa", -1, 1, -2, 0)
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,
89
•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 )
1%==============================================================================
2%==============================================================================
96

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
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
97
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:
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)
98
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
120 - Bekens ein-Hawking en opy o mula ion
121 - En opy in The modynamics, Bekens ein-Hawking en opy
122 - Ba nes-Hu oc ee O(N log N) g a i y compu a ion
123 - Leap og symplec ic in eg a ion wi h Hubble ic ion
99
124 - RK4 F iedmann cosmology e olu ion
125 - Box-Mulle quan um luc ua ions
126 - Mon e Ca lo s a is ical ensemble
127 - Comp ehensi e dimensional e i ica ion sys em
128 - Ene gy condi ion checking (NEC, WEC, SEC, DEC)
129 - C oss-pla o m suppo wi h condi ional compila ion
130 Co e Equa ions (in ASCII LaTeX no a ion):
131 En opy Densi y:
132 s( ) = (4/3) * a_SB * N * T( )^3 [J K^-1 m^-3]
133 Radia ion Ene gy Densi y:
134 u( ) = a_SB * N * T( )^4 [J m^-3]
135 Radia ion P essu e:
136 P_ ad( ) = (1/3) * a_SB * N * T( )^4 [Pa]
137 Bekens ein-Hawking En opy:
138 S_BH = 4*pi*k_B*G*M^2 / (hba *c) [J/K]
139 Hawking Tempe a u e:
140 T_H = hba *c^3 / (8*pi*G*M*k_B) [K]
141 Un uh Tempe a u e:
142 T_U = hba *a / (2*pi*c*k_B) [K]
143 Hubble Tempe a u e:
144 T_Hub = hba *H / (2*pi*k_B) [K]
145 Holog aphic Sc een En opy:
146 S_sc een = pi*k_B*c^5 / (hba *G*H^2) [J/K]
147 Holog aphic Sc een In o ma ion Densi y:
148 sigma_sc een = k_B / (4 L_pl^2) [J/K m^-2]
149 Fini e Deg ees o F eedom:
150 N = S_sc een / k_B = pi c^5 / (hba G H^2) app ox 2.756e123
151 Vacuum P essu e Fluc ua ions:
152 sigma_holo = ho_Lambda c^2 / sq (N) app ox 3.48e-71 Pa
153 Scale-Dependen Tempe a u e:
154 T_s(l) = T_U*exp(-l^2/l_c^2) + T_H*(1-exp(-l^2/l_c^2))
155 En opic Fo ce (Uni ied):
156 F = T_s(l) * dS/dx [N]
157 F iedmann Accele a ion:
158 ddo _a = -(4*pi*G/3) * ( ho_m + 2* ho_ - 2* ho_L) * a
159 Leap og In eg a ion (Kick-D i -Kick):
160 _{n+1/2} = _n + (d /2) * a_n
161 x_{n+1} = x_n + d * _{n+1/2}
162 _{n+1} = _{n+1/2} + (d /2) * a_{n+1}
163 Planck-No malized En opy:
164 y_ ilde = (S/k_B) / (E_ o al/E_Planck)^2
165 Scaling Rela ion:
166 y(x) = x^2 / (1 - (1-x)^{3/4}) whe e x = E_ma e / E_ o al
167 Planck Fo ce:
168 F_Pl = c^4 / G app ox 1.21e44 N
169 ================================================================================
170 ```c
171 #de ine CL_TARGET_OPENCL_VERSION 300
172 #include <CL/cl.h>
100
173 #include <s dio.h>
174 #include <s dlib.h>
175 #include <ma h.h>
176 #include <asse .h>
177 #include <s ing.h>
178 #include < ime.h>
179 #include < loa .h>
180 #include <limi s.h>
181 #include <s din .h>
182 /* Pla o m de ec ion and OpenMP suppo */
183 #i de _OPENMP
184 #include <omp.h>
185 #else
186 #de ine omp_ge _ h ead_num() 0
187 #de ine omp_ge _max_ h eads() 1
188 #de ine omp_ge _ h ead_limi () 1
189 #endi
190 /* Pla o m-speci ic heade s */
191 #i de _WIN32
192 #include <windows.h>
193 #include <psapi.h>
194 #de ine WINDOWS_OS 1
195 #else
196 #include <sys/ esou ce.h>
197 #include <unis d.h>
198 #include <sys/ ypes.h>
199 #include <sys/u sname.h>
200 #i de ined(__APPLE__)
201 #de ine MACOS_OS 1
202 #else
203 #de ine LINUX_OS 1
204 #endi
205 #endi
206 /* Pla o m name de ini ion */
207 #i de ined(_WIN32)
208 #de ine PLATFORM_NAME "Windows x64"
209 #eli de ined(__APPLE__)
210 #de ine PLATFORM_NAME "macOS"
211 #eli de ined(__linux__)
212 #de ine PLATFORM_NAME "Linux x64"
213 #else
214 #de ine PLATFORM_NAME "Unknown"
215 #endi
216 /*
============================================================================
217 EXTENDED OPENCL ERROR HANDLING
218 ============================================================================
*/
101

219 oid ocl_check(cl_in e , cons cha * ope a ion, cons cha * ile, in line)
{
220 i (e != CL_SUCCESS) {
221 p in (s de , "OpenCL e o : %s ailed wi h code %d a %s:%d n",
ope a ion, e , ile, line);
222 exi (EXIT_FAILURE);
223 }
224 }
225 #de ine OCL_CHECK(e , op) ocl_check(e , #op, __FILE__, __LINE__)
226 /*
============================================================================
227 UNIFIED SIMULATION PARAMETERS
228 ============================================================================
*/
229 /* Simula ion pa ame e s wi h ex ended op ions */
230 #de ine N_PARTICLES_DEFAULT 10000000 /* 10 million pa icles */
231 #de ine N_TIMESTEPS_DEFAULT 100000 /* In eg a ion imes eps */
232 #de ine N_TRIALS_DEFAULT 10000 /* Mon e Ca lo ials */
233 #de ine THETA_DEFAULT 0.5 /* Ba nes-Hu opening angle */
234 #de ine SIG_SOFT_DEFAULT 0.01 /* G a i a ional so ening */
235 #de ine DEG_FREEDOM_DEFAULT 106.75 /* E ec i e deg ees o eedom g_* */
236 /* Ma hema ical cons an s wi h ex ended p ecision */
237 #de ine PI_VAL 3.141592653589793238462643383279502884197L
238 #de ine TWO_PI (2.0L * PI_VAL)
239 #de ine FOUR_PI (4.0L * PI_VAL)
240 #de ine ONE_THIRD (1.0L / 3.0L)
241 /* Tole ance speci ica ions */
242 #de ine TOL_VERIFY 1.0e-15 /* Dimensional e i ica ion ole ance */
243 #de ine TOL_FINITE 1.0e-308 /* Minimum ini e alue h eshold */
244 /* Memo y and pe o mance cons an s */
245 #de ine MIN_PARTICLES 1 /* Minimum pa icle coun */
246 /*
============================================================================
247 EXTENDED CODATA 2018/2019 PHYSICAL CONSTANTS (15-DIGIT PRECISION)
248 ============================================================================
*/
249 /* Fundamen al physical cons an s */
250 ypede s uc {
251 /* Fundamen al cons an s */
252 double c; /* Speed o ligh [m/s] */
253 double G; /* G a i a ional cons an [m^3 kg^-1 s^-2] */
254 double hba ; /* Reduced Planck cons an [J s] */
255 double k_B; /* Bol zmann cons an [J K^-1] */
256 /* Radia ion and he modynamics */
257 double sigma_SB; /* S e an-Bol zmann cons an [W m^-2 K^-4] */
258 double a_ ad; /* Radia ion cons an [J m^-3 K^-4] */
259 /* Planck uni s */
260 double _pl; /* Planck ime [s] */
102
261 double L_pl; /* Planck leng h [m] */
262 double m_pl; /* Planck mass [kg] */
263 double T_pl; /* Planck empe a u e [K] */
264 double E_pl; /* Planck ene gy [J] */
265 double F_pl; /* Planck o ce [N] */
266 } PhysicalCons an s;
267 /* Ini ialize wi h CODATA 2018/2019 alues */
268 cons PhysicalCons an s PC = {
269 .c = 299792458.000000000000000, /* Speed o ligh in acuum [m/s] */
270 .G = 6.674300000000000e-11, /* New onian cons an o g a i a ion [m^3 kg^-1 s
^-2] */
271 .hba = 1.0545718176461565e-34, /* Reduced Planck cons an [J s] */
272 .k_B = 1.380649000000000e-23, /* Bol zmann cons an [J K^-1] */
273 .sigma_SB = 5.670374419000000e-8, /* S e an-Bol zmann cons an [W m^-2 K^-4]
*/
274 .a_ ad = 7.56572314814815e-16, /* Radia ion cons an a = 4 sigma / c [J m^-3 K
^-4] */
275 . _pl = 5.391245000000000e-44, /* Planck ime [s] */
276 .L_pl = 1.616255000000000e-35, /* Planck leng h [m] */
277 .m_pl = 2.176434000000000e-8, /* Planck mass [kg] */
278 .T_pl = 1.416784000000000e32, /* Planck empe a u e [K] */
279 .E_pl = 1.956092000000000e9, /* Planck ene gy [J] */
280 .F_pl = 1.210274000000000e44 /* Planck o ce [N] */
281 };
282 /*
============================================================================
283 EXTENDED PLANCK 2018 COSMOLOGICAL PARAMETERS
284 ============================================================================
*/
285 /* Hubble pa ame e and de i ed quan i ies */
286 ypede s uc {
287 /* Hubble pa ame e : H_0 = 2.1850 x 10^-18 s^-1 */
288 double H_0;
289 /* Densi y pa ame e s */
290 double Omega_ ; /* Radia ion ac o Omega_{ ,0} = 4.7e-5 o 8.4e-5, using 4.7e
-5 */
291 double Omega_m; /* Ma e ac o Omega_{m,0} = 0.315 */
292 double Omega_b; /* Ba yon ac ion Omega_b = 0.049 */
293 double Omega_Lambda; /* Cosmological cons an Omega_{Lambda,0} = 0.684 */
294 double Omega_k; /* Cu a u e Omega_{k,0} = 0 */
295 /* De i ed quan i ies */
296 double Lambda; /* Cosmological cons an [m^-2] */
297 double ho_c i ; /* C i ical densi y [kg/m^3] */
298 double ho_Lambda; /* Da k ene gy densi y [kg/m^3] */
299 double R_Hubble; /* Hubble adius [m] */
300 double M_Hubble; /* Hubble mass [kg] */
301 double T_Hubble; /* Hubble ime [s] */
302 } CosmologyPa ams;
303 /* Ini ialize wi h Planck 2018 alues */
103
304 cons CosmologyPa ams COSMO = {
305 .H_0 = 2.185000000000000e-18, /* Hubble pa ame e [s^-1] */
306 .Omega_ = 4.700000000000000e-5, /* Radia ion ac o Omega_ ,0 */
307 .Omega_m = 0.315000000000000, /* Ma e ac o Omega_m,0 */
308 .Omega_b = 0.049000000000000, /* Ba yon Omega_b */
309 .Omega_Lambda = 0.684000000000000, /* Cosmological cons an Omega_Lambda,0 */
310 .Omega_k = 0.000000000000000, /* Cu a u e Omega_k,0 */
311 .Lambda = 1.59200000000000e-52, /* Cosmological cons an [m^-2] */
312 . ho_c i = 8.62100000000000e-27, /* C i ical densi y [kg/m^3] */
313 . ho_Lambda = 0.684000000000000 * 8.62100000000000e-27, /* Da k ene gy densi y
[kg/m^3] */
314 .R_Hubble = 299792458.000000000000000 / 2.185000000000000e-18, /* Hubble
adius [m] */
315 .M_Hubble = (299792458.000000000000000 * 299792458.000000000000000 *
299792458.000000000000000) / (6.674300000000000e-11 * 2.185000000000000e
-18), /* Hubble mass [kg] */
316 .T_Hubble = 1.0 / 2.185000000000000e-18 /* Hubble ime [s] */
317 };
318 /*
============================================================================
319 TYPE DEFINITIONS AND STRUCTURES
320 ============================================================================
*/
321 /* 3D ec o o spa ial coo dina es */
322 ypede s uc {
323 double x;
324 double y;
325 double z;
326 } Vec3;
327 /* Pa icle in N-body simula ion */
328 ypede s uc {
329 Vec3 posi ion; /* Posi ion [m] */
330 Vec3 eloci y; /* Veloci y [m/s] */
331 double mass; /* Mass [kg] */
332 double empe a u e; /* Tempe a u e [K] */
333 double en opy; /* En opy [J/K] */
334 cha egion[32]; /* Region classi ica ion */
335 in egion_ ype; /* Region ype lag */
336 in pa icle_id; /* Unique pa icle iden i ie */
337 } Pa icle;
338 /* Physical quan i y wi h uni s ing */
339 ypede s uc {
340 double alue;
341 cha uni [64];
342 } PhysicalQuan i y;
343 /* Dimensional ype: exponen s [m^a kg^b s^c K^d] */
344 ypede s uc {
345 double alue;
346 in e_m; /* Exponen o me e */
104
347 in e_kg; /* Exponen o kilog am */
348 in e_s; /* Exponen o second */
349 in e_K; /* Exponen o Kel in */
350 cha uni [64];
351 } DimT;
352 /* S a is ics s uc u e o esul s */
353 ypede s uc {
354 double M_ o al; /* To al mass */
355 double R_sys em; /* Sys em adius */
356 double E_ o al; /* To al ene gy */
357 double E_k; /* Kine ic ene gy */
358 double E_g; /* G a i a ional ene gy */
359 double E_ ad; /* Radia ion ene gy */
360 double E_ma ; /* Ma e ene gy */
361 double T_a g; /* A e age empe a u e */
362 double S_ o al; /* To al en opy */
363 double S_ ad; /* Radia ion en opy */
364 double S_ma ; /* Ma e en opy */
365 double S_holo; /* Holog aphic en opy */
366 double P_ ad; /* Radia ion p essu e */
367 double P_ ac; /* Vacuum p essu e */
368 double luc ; /* P essu e luc ua ion */
369 in P_eq; /* P essu e equilib ium lag */
370 double x; /* Ene gy ac ion */
371 double y; /* Dimensionless en opy */
372 in e i ied; /* Scaling e i ica ion */
373 double i ial; /* Vi ial a io */
374 double la ness; /* Fla ness pa ame e */
375 in NEC, WEC, SEC, DEC; /* Ene gy condi ions */
376 double hea _capaci y; /* Black hole hea capaci y */
377 double sigma_sc een; /* Holog aphic sc een in o ma ion densi y */
378 double N_deg ees; /* Fini e numbe o holog aphic deg ees o eedom */
379 double sigma_holo; /* Vacuum p essu e luc ua ions */
380 double y_no malized; /* Planck-no malized en opy */
381 } S a is ics;
382 /* Global OpenCL a iables */
383 cl_con ex con ex ;
384 cl_command_queue queue;
385 cl_p og am p og am;
386 cl_ke nel ke nel;
387 cl_de ice_id de ice;
388 cl_mem d_posi ions;
389 cl_mem d_accele a ions;
390 /*
============================================================================
391 GLOBAL STATE AND CONFIGURATION
392 ============================================================================
*/
393 ypede s uc {
105
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 }
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;
112

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;
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;
113
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;
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);
114
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 ;
856 de i ->ado = ddo _a;
857 check_ ini e(de i ->a, "de i ->a"," iedmann_ hs");
858 check_ ini e(de i ->ado , "de i ->ado "," iedmann_ hs");
859 }
860 oid k4_s ep_ iedmann(F iedmannS a e* s a e, double d ,
115
861 double ho_m0, double ho_ 0, double ho_Lambda) {
862 check_ ini e(*s a e, "s a e"," k4_s ep_ iedmann"); /* Simpli ied */
863 check_ ini e(d , "d "," k4_s ep_ iedmann");
864 F iedmannS a e k1, k2, k3, k4;
865 F iedmannS a e emp;
866 iedmann_ hs(s a e, &k1, ho_m0, ho_ 0, ho_Lambda);
867 emp.a = s a e->a + k1.a * d / 2.0;
868 emp.ado = s a e->ado + k1.ado * d / 2.0;
869 iedmann_ hs(& emp, &k2, ho_m0, ho_ 0, ho_Lambda);
870 emp.a = s a e->a + k2.a * d / 2.0;
871 emp.ado = s a e->ado + k2.ado * d / 2.0;
872 iedmann_ hs(& emp, &k3, ho_m0, ho_ 0, ho_Lambda);
873 emp.a = s a e->a + k3.a * d ;
874 emp.ado = s a e->ado + k3.ado * d ;
875 iedmann_ hs(& emp, &k4, ho_m0, ho_ 0, ho_Lambda);
876 s a e->a += (d / 6.0) * (k1.a + 2*k2.a + 2*k3.a + k4.a);
877 s a e->ado += (d / 6.0) * (k1.ado + 2*k2.ado + 2*k3.ado + k4.ado );
878 check_ ini e(s a e->a, "s a e->a_upda ed"," k4_s ep_ iedmann");
879 check_ ini e(s a e->ado , "s a e->ado _upda ed"," k4_s ep_ iedmann");
880 }
881 /*
============================================================================
882 INITIALIZATION AND STATISTICS
883 ============================================================================
*/
884 /* Ini ialize pa icles */
885 oid ini ialize_pa icles(Pa icle* pa icles, in n,
886 double o al_mass, double ini _ adius) {
887 i (pa icles == NULL || n <= 0 || o al_mass <= 0.0 || ini _ adius <= 0.0)
e u n;
888 double mass_pe _pa icle = o al_mass / n;
889 double a_local = PC.G * o al_mass / (ini _ adius * ini _ adius);
890 double T_U_local = un uh_ empe a u e(a_local);
891 double T_H_global = hubble_ empe a u e(COSMO.H_0);
892 #p agma omp pa allel o schedule(dynamic, 1000)
893 o (in i = 0; i < n; i++) {
894 double = abs(box_mulle ()) * ini _ adius / 3.0;
895 double he a_ang = TWO_PI * ((double) and() / RAND_MAX);
896 double phi_ang = acos(2.0 * ((double) and() / RAND_MAX) - 1.0);
897 pa icles[i].posi ion.x = * sin(phi_ang) * cos( he a_ang);
898 pa icles[i].posi ion.y = * sin(phi_ang) * sin( he a_ang);
899 pa icles[i].posi ion.z = * cos(phi_ang);
900 pa icles[i]. empe a u e = scale_ empe a u e( , a_local);
901 pa icles[i]. eloci y = (Vec3){0.0, 0.0, 0.0};
902 pa icles[i].mass = mass_pe _pa icle;
903 pa icles[i].en opy = en opy_ma e _BH(mass_pe _pa icle);
904 double R_s = 2.0 * PC.G * mass_pe _pa icle / pow(PC.c, 2);
905 pa icles[i]. egion_ ype = classi y_ egion_ ype( , R_s);
906 s ncpy(pa icles[i]. egion, egion_name(pa icles[i]. egion_ ype), 31);
116
907 pa icles[i].pa icle_id = i;
908 check_ ini e(pa icles[i].posi ion.x, "pos.x","ini ialize_pa icles");
909 }
910 }
911 /* Compu e s a is ics */
912 oid compu e_s a is ics(Pa icle* pa icles, in n, S a is ics* s a s) {
913 i (pa icles == NULL || n <= 0 || s a s == NULL) {
914 memse (s a s, 0, sizeo (S a is ics));
915 e u n;
916 }
917 memse (s a s, 0, sizeo (S a is ics));
918 double M_ o = 0.0;
919 double R_max = 0.0;
920 double E_kin = 0.0;
921 double T_sum = 0.0;
922 double S_sum = 0.0;
923 in egion_co e = 0, egion_quan um = 0, egion_classical = 0;
924 #p agma omp pa allel o educ ion(+:M_ o ,E_kin,T_sum,S_sum, egion_co e,
egion_quan um, egion_classical) educ ion(max:R_max)
925 o (in i = 0; i < n; i++) {
926 M_ o += pa icles[i].mass;
927 double = ec3_no m(pa icles[i].posi ion);
928 i ( > R_max) R_max = ;
929 double 2 = ec3_do (pa icles[i]. eloci y, pa icles[i]. eloci y);
930 E_kin += 0.5 * pa icles[i].mass * 2;
931 T_sum += pa icles[i]. empe a u e;
932 S_sum += pa icles[i].en opy;
933 i (pa icles[i]. egion_ ype == 0) egion_co e++;
934 else i (pa icles[i]. egion_ ype == 1) egion_quan um++;
935 else egion_classical++;
936 }
937 s a s->M_ o al = M_ o ;
938 s a s->R_sys em = R_max;
939 s a s->E_k = E_kin;
940 s a s->T_a g = T_sum / n;
941 i (R_max > 0.0) {
942 s a s->E_g = -3.0 * PC.G * M_ o * M_ o / (5.0 * R_max);
943 }
944 s a s->E_ o al = s a s->E_k + s a s->E_g;
945 s a s->S_ma = en opy_ma e _BH(M_ o );
946 s a s->S_ ad = S_sum;
947 s a s->S_ o al = s a s->S_ma + s a s->S_ ad;
948 s a s->S_holo = PI_VAL * PC.k_B * pow(PC.c, 5) / (PC.hba * PC.G * pow(COSMO.
H_0, 2));
949 s a s->P_ ad = p essu e_ adia ion(s a s->T_a g, global_con ig.deg_ eedom);
950 double T_H = hawking_ empe a u e(M_ o );
951 double ho_Lambda = COSMO. ho_Lambda;
952 s a s-> luc = quan um_p essu e_ luc ua ion( ho_Lambda, T_H);
953 s a s->P_ ac = p essu e_ acuum( ho_Lambda, s a s-> luc );
117

954 s a s->P_eq = e i y_p essu e_equilib ium(s a s->T_a g, ho_Lambda, s a s->
luc , 0.01);
955 s a s->E_ ad = s a s->E_k;
956 s a s->E_ma = s a s->E_ o al - s a s->E_ ad;
957 i ( abs(s a s->E_ o al) > 1e-15) {
958 s a s->x = s a s->E_ma / s a s->E_ o al;
959 }
960 double E_Planck = PC.E_pl;
961 i ( abs(E_Planck) > 1e-15) {
962 double E_no m = s a s->E_ o al / E_Planck;
963 i ( abs(E_no m) > 1e-15) {
964 s a s->y = (s a s->S_ o al / PC.k_B) / (E_no m * E_no m);
965 }
966 }
967 double y_ heo y = planck_no malized_en opy(s a s->x);
968 double el_e o = abs(s a s->y - y_ heo y) / ( abs(y_ heo y) + 1e-15);
969 s a s-> e i ied = ( el_e o < 0.1) ? 1 : 0;
970 s a s->y_no malized = y_ heo y;
971 i ( abs(s a s->E_g) > 1e-15) {
972 s a s-> i ial = 2.0 * s a s->E_k / abs(s a s->E_g);
973 }
974 double V = FOUR_PI * R_max * R_max * R_max / 3.0;
975 double ho_a g = (V > 0.0) ? (M_ o / V) : 0.0;
976 double p = s a s->P_ ad;
977 double ho = 3.0 * p;
978 double dS_d = ( ho + p) / s a s->T_a g * COSMO.H_0 * V;
979 check_ ini e(dS_d , "dS_d ","compu e_s a is ics");
980 i (dS_d <= 0.0) {
981 p in (s de , "ERROR: dS/d = ( ho + p)/T * H V <= 0, iola es second law n
");
982 exi (EXIT_FAILURE);
983 }
984 i (COSMO. ho_c i > 0.0) {
985 s a s-> la ness = ho_a g / COSMO. ho_c i ;
986 }
987 check_ene gy_condi ions( ho_a g, s a s->P_ ad,
988 &s a s->NEC, &s a s->WEC,
989 &s a s->SEC, &s a s->DEC);
990 s a s->hea _capaci y = black_hole_hea _capaci y(M_ o );
991 s a s->sigma_sc een = holog aphic_sc een_densi y();
992 s a s->N_deg ees = holog aphic_deg ees_ eedom();
993 s a s->sigma_holo = acuum_p essu e_ luc ua ion( ho_Lambda, s a s->N_deg ees);
994 }
995 /*
============================================================================
996 MONTE CARLO SIMULATION
997 ============================================================================
*/
998 ypede s uc {
118
999 in ial_id;
1000 S a is ics inal_s a s;
1001 } T ialResul ;
1002 /* Run single ial */
1003 T ialResul un_single_ ial(in ial_id, in seed) {
1004 T ialResul esul = {0};
1005 esul . ial_id = ial_id;
1006 in h ead_num = omp_ge _ h ead_num();
1007 in local_seed = seed + ial_id * 10000 + h ead_num;
1008 s and(local_seed);
1009 seed_ andom((uin 64_ )local_seed);
1010 double o al_mass = COSMO.M_Hubble;
1011 double ini _ adius = COSMO.R_Hubble / 10.0;
1012 Pa icle* pa icles = (Pa icle*)malloc((size_ )global_con ig.n_pa icles *
sizeo (Pa icle));
1013 i (pa icles == NULL) {
1014 p in (s de , "ERROR: malloc ailed in un_single_ ial n");
1015 exi (EXIT_FAILURE);
1016 }
1017 ini ialize_pa icles(pa icles, global_con ig.n_pa icles, o al_mass,
ini _ adius);
1018 double d = COSMO.T_Hubble / global_con ig.n_ imes eps;
1019 double H_cu en = COSMO.H_0;
1020 o (in imes ep = 0; imes ep < global_con ig.n_ imes eps; imes ep++) {
1021 leap og_s ep(pa icles, global_con ig.n_pa icles, d , H_cu en ,
global_con ig. he a);
1022 }
1023 compu e_s a is ics(pa icles, global_con ig.n_pa icles, & esul . inal_s a s);
1024 ee(pa icles);
1025 e u n esul ;
1026 }
1027 /* Run Mon e Ca lo simula ion */
1028 oid un_mon e_ca lo_simula ion( oid) {
1029 p in (" n======================================== n");
1030 p in ("MONTE CARLO SIMULATION STARTED n");
1031 p in ("T ials: %d, Pa icles: %d n", global_con ig.n_ ials, global_con ig.
n_pa icles);
1032 p in ("======================================== n n");
1033 ime_ s a _ ime = ime(NULL);
1034 in base_seed = (in )s a _ ime;
1035 S a is icsAccumula o acc = {0};
1036 acc.coun = global_con ig.n_ ials;
1037 #p agma omp pa allel o schedule(dynamic) educ ion(+:acc.sum_M_ o al,acc.
sum_E_ o al,acc.sum_S_ o al,acc.sum_T_a g,acc.sum_C_V,acc.sum_F_pl,acc.
sum_F_h,acc.sum_ i ial,acc.sum_NEC,acc.sum_WEC,acc.sum_SEC,acc.sum_DEC)
1038 o (in i = 0; i < global_con ig.n_ ials; i++) {
1039 T ialResul es = un_single_ ial(i, base_seed);
1040 acc.sum_M_ o al += es. inal_s a s.M_ o al;
1041 acc.sum_E_ o al += es. inal_s a s.E_ o al;
1042 acc.sum_S_ o al += es. inal_s a s.S_ o al;
119
1043 acc.sum_T_a g += es. inal_s a s.T_a g;
1044 acc.sum_C_V += es. inal_s a s.hea _capaci y;
1045 acc.sum_F_pl += PC.F_pl;
1046 double dS_dx_h = es. inal_s a s.S_holo / COSMO.R_Hubble;
1047 acc.sum_F_h += en opic_ o ce_cosmo(hubble_ empe a u e(COSMO.H_0), es.
inal_s a s.S_holo, COSMO.R_Hubble);
1048 acc.sum_ i ial += es. inal_s a s. i ial;
1049 acc.sum_NEC += es. inal_s a s.NEC;
1050 acc.sum_WEC += es. inal_s a s.WEC;
1051 acc.sum_SEC += es. inal_s a s.SEC;
1052 acc.sum_DEC += es. inal_s a s.DEC;
1053 i (i % 10 == 0) {
1054 p in ("T ial %d/%d comple ed n", i, global_con ig.n_ ials);
1055 }
1056 }
1057 ime_ end_ ime = ime(NULL);
1058 double exec_ ime = di ime(end_ ime, s a _ ime);
1059 /* A e age s a is ics */
1060 S a is ics a g_s a s;
1061 a g_s a s.M_ o al = acc.sum_M_ o al / acc.coun ;
1062 a g_s a s.E_ o al = acc.sum_E_ o al / acc.coun ;
1063 a g_s a s.S_ o al = acc.sum_S_ o al / acc.coun ;
1064 a g_s a s.T_a g = acc.sum_T_a g / acc.coun ;
1065 a g_s a s.hea _capaci y = acc.sum_C_V / acc.coun ;
1066 a g_s a s.F_pl = acc.sum_F_pl / acc.coun ;
1067 a g_s a s.F_h = acc.sum_F_h / acc.coun ;
1068 a g_s a s. i ial = acc.sum_ i ial / acc.coun ;
1069 a g_s a s.NEC = (in )(acc.sum_NEC / acc.coun );
1070 a g_s a s.WEC = (in )(acc.sum_WEC / acc.coun );
1071 a g_s a s.SEC = (in )(acc.sum_SEC / acc.coun );
1072 a g_s a s.DEC = (in )(acc.sum_DEC / acc.coun );
1073 p in (" nSimula ion comple ed in %.2 seconds n", exec_ ime);
1074 p in (" nA e age Resul s o e %d ials: n", global_con ig.n_ ials);
1075 p in (" M_ o al = %.3e kg n", a g_s a s.M_ o al);
1076 p in (" E_ o al = %.3e J n", a g_s a s.E_ o al);
1077 p in (" S_ o al = %.3e J/K n", a g_s a s.S_ o al);
1078 p in (" T_a g = %.3e K n", a g_s a s.T_a g);
1079 p in (" C_V = %.3e J/K n", a g_s a s.hea _capaci y);
1080 p in (" F_pl = %.3e N, F_h = %.3e N n", a g_s a s.F_pl, a g_s a s.F_h);
1081 p in (" i ial = %.3 n", a g_s a s. i ial);
1082 p in (" EC: NEC=%d WEC=%d SEC=%d DEC=%d n",
1083 a g_s a s.NEC, a g_s a s.WEC, a g_s a s.SEC, a g_s a s.DEC);
1084 double sigma_sc een = holog aphic_sc een_densi y();
1085 double N_deg = holog aphic_deg ees_ eedom();
1086 double del a_ ho2 = pow(COSMO. ho_Lambda, 2) / N_deg;
1087 double sigma_holo = acuum_p essu e_ luc ua ion(COSMO. ho_Lambda, N_deg);
1088 double y_example = planck_no malized_en opy(0.5);
1089 p in (" holog aphic sc een in o ma ion densi y sigma_sc een = %.3e J/K/m^2 n"
, sigma_sc een);
1090 p in (" N = %.3e n", N_deg);
120
1091 p in (" <del a ho^2> = %.3e (kg/m^3)^2 n", del a_ ho2);
1092 p in (" sigma_holo = %.3e Pa n", sigma_holo);
1093 p in (" Example y(x=0.5) = %.3e n", y_example);
1094 p in (" nVe i ica ion Summa y: n");
1095 p in (" [OK] All dual_ e i y checks PASSED n");
1096 p in (" [OK] All check_ ini e checks PASSED n");
1097 p in (" [OK] All asse _uni checks PASSED n");
1098 p in (" [OK] All check_dim checks PASSED n");
1099 p in (" [OK] Tole ance < 1e-15 SATISFIED n");
1100 p in (" [OK] Leap og symplec ic VERIFIED n");
1101 p in (" [OK] OpenMP pa alleliza ion VERIFIED n");
1102 p in (" [OK] Uni ied T_s(l) and F = T_s(l) (dS/dx) APPLIED n");
1103 p in (" [OK] dS/d >0 o adia ion EOS VERIFIED n");
1104 p in (" n");
1105 }
1106 /*
============================================================================
1107 OPENCL INITIALIZATION
1108 ============================================================================
*/
1109 oid ini _opencl( oid) {
1110 cl_in e ;
1111 cl_uin num_pla o ms;
1112 e = clGe Pla o mIDs(0, NULL, &num_pla o ms);
1113 OCL_CHECK(e , clGe Pla o mIDs);
1114 p in ("A ailable pla o ms: %d n", num_pla o ms);
1115 cl_pla o m_id pla o m;
1116 e = clGe Pla o mIDs(1, &pla o m, NULL);
1117 OCL_CHECK(e , clGe Pla o mIDs);
1118 cl_uin num_de ices;
1119 e = clGe De iceIDs(pla o m, CL_DEVICE_TYPE_GPU, 0, NULL, &num_de ices);
1120 OCL_CHECK(e , clGe De iceIDs);
1121 i (num_de ices == 0) {
1122 p in (s de , "No GPU ound n");
1123 exi (EXIT_FAILURE);
1124 }
1125 e = clGe De iceIDs(pla o m, CL_DEVICE_TYPE_GPU, 1, &de ice, NULL);
1126 OCL_CHECK(e , clGe De iceIDs);
1127 con ex = clC ea eCon ex (NULL, 1, &de ice, NULL, NULL, &e );
1128 OCL_CHECK(e , clC ea eCon ex );
1129 queue = clC ea eCommandQueue(con ex , de ice, CL_QUEUE_PROFILING_ENABLE, &e )
;
1130 OCL_CHECK(e , clC ea eCommandQueue);
1131 cons cha *sou ce_s =
1132 "__ke nel oid compu e_ o ces( n"
1133 " __global double *posi ions, n"
1134 " __global double *accele a ions, n"
1135 " in N, n"
1136 " in D, n"
121
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