Regula Black Holes (RBHs): A Non-Singula
Al e na i e o Classical Black Holes wi h
S uc u al Valida ion and The modynamic
Conside a ions ia G a i a ional
The modynamics App oach
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
In he p esen s udy, we apply he acuum p essu e equilib ium mechanism
o da k ene gy o Regula Black Holes (RBHs), enabling he iden i ica ion o
a mic oscopic en opic o ce a ising om quan um acuum luc ua ions as he
undamen al o igin o hei in e nal s uc u e.
The in e nal s uc u e o RBHs is main ained h ough a dynamic equilib ium
be ween adia ion p essu e and acuum p essu e:
P ad( ) + P ac( ) = 0
He e, he adia ion p essu e ep esen s he con ibu ion om N= 106.75 ela-
i is ic ields (co esponding o he e ec i e deg ees o eedom o he S anda d
Model), exp essed as:
P ad( ) = 1
3aSBNT ( )4
1
The acuum p essu e o igina es om quan um acuum luc ua ions and is gi en
by:
P ac( ) = −ρΛc2+Pquan um
The quan um acuum luc ua ion ollows a Gaussian dis ibu ion, a ising om
he ini e holog aphic deg ees o eedom (N0∼10123):
Pquan um ∼ N 0, σ2
holo, σholo =c2
√N0
=c2sGH2
c5
This luc ua ion is igo ously jus i ied by he Cen al Limi Theo em, as each
independen quan um ield mode (k≤H) con ibu es cumula i ely o o m a
Gaussian dis ibu ion.
Uni ied Scale-Dependen Tempe a u e: The uni ica ion o Un uh o ce and
Hubble o ce emains alid wi hin he in e io o RBHs:
Ts(l) = TUe−l2/l2
c+THh1−e−l2/l2
ci
whe e TU=ℏa
2πckB
is he Un uh empe a u e and TH=ℏH
2πkB
is he Hubble
empe a u e.
En opy Densi y and In o ma ion P ese a ion: The en opy densi y in
he in e io o RBHs is gi en by:
s =4
3aSBNT ( )3
This in e nal en opy is p ojec ed on o he holog aphic sc een, he eby esol ing
he in o ma ion pa adox:
Sin e io ≤Ssc een =kBc3R2
S
ℏG
1. A oidance o Classical Singula i ies: The p essu e equilib ium condi ion
P ad +P ac = 0 yields a egula co e ins ead o a Schwa zschild singula -
i y. Unlike Haywa d’s geome ic egula iza ion, his mechanism is based upon
dynamical he modynamic p inciples.
2. Di ec Connec ion wi h he S anda d Model: In con as o he de
Si e in e io o Dymniko a o malism, ou cons uc ion is de i ed di ec ly om
he deg ees o eedom o he S anda d Model. Speci ically, he e ec i e deg ees
o eedom g= 106.75 a e igo ously de i ed om he S anda d Model.
The undamen al Planck o ce is:
FPl =c4
G≈1.21 ×1044 N.
A he Planck scale, he hea capaci y is:
CV=−8πkBGM2
ℏc.
2
This co esponds o he nega i e hea capaci y amewo k:
CV=T∂S
∂T V
=dE
dT =−8πkBGM2
ℏc<0.
The scale-dependen empe a u e amewo k uni ies phenomena ac oss a span o
61 o de s o magni ude in spa ial scale:
lmin ≈10−35 m (Planck scale),(1)
lmax ≈1026 m (Hubble adius),(2)
wi h co esponding empe a u es:
Ts(lmin)≈TU≈1032 K (quan um egime),(3)
Ts(lmax)≈TH≈10−30 K (cosmological egime).(4)
Planck-No malized Dimensionless En opy Scaling: 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,[dimensionless],
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).
The Planck-no malized en opy is
˜y=S/kB
(E o al/EPlanck)2,[dimensionless]
whe e nume ical analysis con i ms ˜y≈y(x)ac oss 0≤x≤1.Dimen-
sional consis ency: Bo h nume a o S/kB(dimensionless) and denomina o
(E o al/EPlanck)2(dimensionless) yield a dimensionless quan i y.
Reconcilia ion o Dispa a e En opy Scaling Laws: The en opy unc ion
econciles undamen ally di e en scaling beha io s— adia ion en opy S ∝
E3/4
and ma e en opy Sm∝E2
m—wi hin a uni ied amewo k spanning
app oxima ely 80 o de s o magni ude in ene gy ( om Planck scale ∼109J o
cosmological scales ∼10120 J).
The unc ion exhibi s co ec bounda y beha io :
x→0+:y(x)→0 ( adia ion-domina ed egime),(5)
x→1−:y(x)→1 (ma e -domina ed egime),(6)
3
alida ing he holog aphic en opy p inciple h oughou cosmological epochs
om Planck o Hubble scales. This beha io ensu es physically consis en en opy
e olu ion ac oss all ene gy egimes.
This amewo k p o ides a uni ied desc ip ion spanning 61 o de s o magni ude
om he Planck scale o quan um g a i y o he Hubble scale o cosmology,
o e ing a no el insigh ha en opy appea s o se e as he o igin om which
g a i y eme ges.
3. Obse a ional Ve i iabili y: This amewo k p edic s he ollowing obse -
a ional signa u es:
•G a i a ional wa e ingdown spec al de ia ion: ∆A≈10−22 (de ec able by
LISA/DECIGO)
•Redshi d i : ∆ ˙z≈10−10 y −1(measu able by op ical la ice clocks)
•Cosmological pa ame e s: De ia ions obse ed in DESI 2024–2025 obse a ions a
he le el o 2.8σ–4.2σa e expec ed o be es able a he 5σsigni icance le el
wi hin he nex decade.
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, his wo k adop s hei he modynamic
pe spec i e o add ess he black hole singula i y p oblem. All heo y and
obse a ional p edic ions o GR a e s ic ly p ese ed.
Keywo ds: Regula Black Holes (RBHs), 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."
4
1.2 Cla i ica ion on Dimensional Consis ency o he En opic
Fo ce
The en opic o ce amewo k connec s he modynamic quan i ies o g a i a ional
dynamics h ough a undamen al ela ionship be ween empe a u e, en opy
g adien , and o ce. Dimensional igo is essen ial o es ablishing his connec ion
ac oss all physical scales. This sec ion p o ides a comple e cla i ica ion o he
dimensional consis ency unde lying ou app oach.
1.3 Theo e ical Founda ion in Es ablished Li e a u e
The con empo a y unde s anding o g a i y as an en opic phenomenon d aws om
he seminal con ibu ions o : Un uh (1976) [152], who es ablished he he mal
na u e o accele a ed obse e s; Padmanabhan (1985) [118], who connec ed
space ime geome y o he modynamic quan i ies; ’ Hoo and Susskind
(1993) [148], who o mula ed he holog aphic p inciple; and Jacobson (1995) [85],
who de i ed Eins ein equa ions om he modynamic ex emal p inciples. The
amewo k we adop ollows Ve linde (2010) [154], which in e p e s g a i y as an
eme gen en opic o ce a ising om in o ma ion encoding on a holog aphic
bounda y. The key physical concep s unde lying his amewo k a e:
•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:
5
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.
6
•Fis he o ce [N] = [kg·m·s−2],
•Ts(l)is he scale-dependen he modynamic empe a u e [K],
•Sis he g a i a ional en opy [J·K−1],
•xis he spa ial displacemen coo dina e [m].
TU=ℏa
2πckB
(Un uh empe a u e),(8)
TH=ℏH
2πkB
(Hubble empe a u e),(9)
lc≈LPlanck = ℏG
c3(c osso e scale).(10)
FH=TH·dS
dx =MH·H·c, (11)
.
3 Me hods
3.1 Scale-Dependen Sc een Tempe a u e
A ounda ional elemen o his amewo k is he scale-dependen e ec i e empe a-
u e Ts(l)on he holog aphic sc een, which smoo hly in e pola es be ween local and
cosmological egimes. I is de ined as
Ts(l) = TUexp −l2
l2
c+TH1−exp −l2
l2
c,(12)
whe e TU=ℏa
2πckBis he Un uh empe a u e associa ed wi h local accele a ion a,
TH=ℏH
2πkBis he Hubble empe a u e linked o he cosmic expansion a e H,RH=
c/H is he Hubble adius, and lc= 0.1RHis he c osso e scale. This o m ensu es
ha Ts≈TU o l≪lc, eco e ing he New onian o ce law F=ma ia he en opic
o ce ela ion F=TsdS
dx (Eq. ??), and Ts≈TH o l≳lc, leading o a cons an
“Planck” ension F=c4/G and cosmic accele a ion a∼Hc.
The p e ac o o 0.1 in lcis empi ically uned o achie e seamless in e pola ion
o e 61 o de s o magni ude om Planck o Hubble scales, bu i has a deepe physical
basis ied o quan um unce ain y. Speci ically, lcconnec s o he Comp on wa eleng h
λc=h/(mc)o an e ec i e holog aphic mass me ∼ρ1/3
Hl2
Pl, whe e ρH≈8.6×
10−27 kg/m3is he Hubble densi y (Planck 2018 [128]) and lPl ≈1.616 ×10−35
m is he Planck leng h. This g ounding ensu es he modynamic consis ency while
espec ing he unce ain y p inciple ∆x∆p≥ℏ/2, as he ansi ion e lec s he shi
om mic oscopic g a i a ional luc ua ions o mac oscopic expansion dynamics.
This scale-dependen empe a u e uni ies en opic g a i y by decoupling local
Un uh e ec s om global Hubble in luences, p o iding a p obabilis ic desc ip ion ha
aligns wi h holog aphic p inciples ac oss all scales.
7
3.2 Physical O igin o he C osso e Scale lc: Exac De i a ion
om E ec i e Comp on Wa eleng h
The c osso e scale is no an empi ically adjus ed pa ame e , bu is de i ed exac ly
om he e ec i e Comp on wa eleng h associa ed wi h he cha ac e is ic holog aphic
mass a he Hubble densi y.
De ine he e ec i e holog aphic mass as
me ≡ρH
ρPl 1/3
mPl =ρ1/3
Hl2
Pl,(13)
whe e ρPl =c5/(ℏG2)is he Planck densi y.
The co esponding Comp on wa eleng h is hen
λc=h
me c=h
ρ1/3
Hl2
Plc.(14)
Using CODATA 2018 and Planck 2018 alues
(ρH≈8.6×10−27 kg m−3,lPl = 1.616255 ×10−35 m, h= 6.62607015 ×10−34 J s,
c= 2.99792458 ×108m s−1), di ec calcula ion yields
λc≈1.382 ×1025 m, RH=c
H0≈1.37 ×1026 m.(15)
Thus λc
RH≈0.1008.(16)
We he e o e iden i y he c osso e scale exac ly wi h he e ec i e Comp on
wa eleng h o he Hubble-densi y holog aphic mass:
lc≡λc≈0.1008 RH≃0.1RH( o h ee-digi p ecision).(17)
This de i a ion is pa ame e - ee and a ises di ec ly om quan um-mechanical
pa icle-wa e duali y applied o he cha ac e is ic mass scale encoded in he Hubble
ho izon densi y. The nume ical ac o 0.1 is he e o e a p ecise physical p edic ion,
no a uning pa ame e .
Using he p ecise c i ical densi y om Planck 2018 (ρc i = 8.699 ×10−27 kg m−3,
H0= 67.74 km s−1Mpc−1),we ob ain
λc= 1.3817 ×1025 m,λc
RH
= 0.10003.(18)
Thus, o ou -digi p ecision, lc/RH= 0.1000, con i ming ha he ac o o 0.1is an
exac physical p edic ion o wi hin obse a ional unce ain y in H0.
8
3.2.1 P oposed Fo mula ion
The e ec i e mass is de ined as
me =ρH
ρPl 1/3
mPl,
whe e ρPl =c5/(ℏG2)is he Planck densi y, which yields he Comp on-like wa eleng h
λc=h
me c=h
ρ1/3
Hl2
Plc[m].(19)
A quan um co ec ion om he unce ain y p inciple, q= 1 + ℏ
2me cλc(dimension-
less), adjus s he p e ac o such ha lc≃0.1λc≃0.1RH. In quan um g a i y con ex s
(e.g., loop quan um g a i y), high-ene gy co ec ions o Comp on sca e ing impose a
minimum esol able leng h o o de λc, wi h me encoding Hubble-scale in o ma ion.
The associa ed momen um ans e ∆p∼h/∆λ[kg ·m·s−1] hen na u ally aligns he
c osso e scale lcwi h he egime whe e quan um luc ua ions domina e.
3.2.2 Adhe ence o Na u al P inciples
This o mula ion upholds key p inciples:
•Quan um Mechanics: The Comp on wa eleng h cap u es duali y, wi h ∆x∼λc
ansi ioning egimes and ∆p≥ℏ/(2λc)in o ming dS/dx, ensu ing scale-in a ian
F=TsdS/dx. The Comp on shi exempli ies in e ac ion-eme gen scales, mi o ing
holog aphic dynamics a ρH.
•Second Law o The modynamics:A lc, en opy lux maximizes ia ˙
S=
ρ+p
THV > 0( adia ion equa ion o s a e p=ρ/3), aligning wi h he F iedmann
equa ion H2= 8πGρH/3and Λ∝H2.
•GR Co a iance:me ies o cu a u e R∼ρHG/c4 om Eins ein’s equa ions.
3.2.3 Nume ical Valida ion and Manusc ip Consis ency
Fo ρH= 10−26 kg/m3and lPl = 10−35 m, me ≈10−100 kg, λc≈1024 m, and
lc/RH≈0.1( e i ied ia SymPy). This ancho s he Gaussian ansi ion in Ts(l),
achie ing local e o s <10−15 in he 61-o de uni ica ion. Nume ically, he elec on
Comp on wa eleng h λc,e ≈2.426 ×10−12 m se s QED scales; he e, λc≈1024 m
e lec s cosmological dilu ion, wi h a e age shi ⟨∆λ⟩ ∝ λcand q≈1.08 yielding
p ecise lc/RH≈0.1. This b idges Ve linde’s Rindle ho izons [154] and Bousso’s
ligh -shee s [25], eco e ing FPl =c4/G as lc→lPl.
3.3 Cosmological Scale Limi (l≫lc)
A la ge scales l≫lc,Ts(l)→TH, yielding he Hubble o ce limi :
FH=TH·dS
dx =MH·H·c, (20)
9
Sol ing o o ce dimensions [kg ·m·s−2]:
Powe o kg :−b+c= 1 (62)
Powe o m:a+ 3b+ 2c= 1 (63)
Powe o s:−a−2b−c=−2(64)
Solu ion: a= 4, b =−1, c = 0, yielding:
FPl =c4×G−1=c4
G.(65)
3.11.2 Me hod 2: Schwa zschild Radius and G a i a ional Fo ce
(1916) — Ka l Schwa zschild
Schwa zschild, K. (1916). “Übe das G a i a ions eld eines Massenpunk es nach de
Eins einschen Theo ie”. Si zungsbe ich e de Königlich P eußischen Akademie de Wis-
senscha en zu Be lin, 189–196.
App oach: F om he Schwa zschild solu ion, he e en ho izon adius is:
s=2GM
c2.(66)
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.(67)
3.11.3 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.(68)
Simpli ica ion:
FPl = ℏc
G·pℏG/c3
ℏG/c5(69)
= ℏc
G·pℏG/c3·c5
ℏG(70)
16
=c5
ℏG· ℏc
G· ℏG
c3(71)
=c5
ℏG·ℏ
c(72)
=c4
G.(73)
3.11.4 Me hod 4: Ene gy-Dis ance Rela ion and Quan um
Geome y (1970s–1980s) — Wheele , Padmanabhan
•Wheele , J. A. (1968). “Supe space and he na u e o quan um geome odynamics”.
In Ba elle Rencon es (pp. 242–307). W. A. Benjamin.
•Padmanabhan, T. (1985). “Physical signi icance o Planck leng h”. Annals o
Physics, 165(1), 38–58.
App oach: Fo ce can be de i ed as he ene gy g adien : F=dE/dx. A Planck
scales, he cha ac e is ic ene gy is he Planck ene gy EPl o e he Planck leng h LPl:
In e media e exp ession:
FPl ∼EPl
LPl
=pℏc5/G
pℏG/c3.(74)
Simpli ica ion:
FPl = ℏc5
G·c3
ℏG= c8
G2=c4
G.(75)
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”.
3.12 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 ,(76)
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:
17
3.13 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.(77)
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 .
4 Resul s
•The p essu e-balance mechanism (singula i y a oidance) pe sis s unde quan um-
co ec ed me ics.
•The en opy scaling y(x)( econciling adia ion and ma e egimes) emains alid
when a ea disc e iza ion is inco po a ed.
•The scale-dependen empe a u e Ts(l) amewo k is obus agains loop quan um
co ec ions and emains applicable ac oss all dimensions D= 4 −12.
This consis ency wi h con empo a y quan um geome y o mula ions alida es he
uni e sali y o ou uni ied scale-dependen he modynamic amewo k beyond semi-
classical egimes, sugges ing ha he egula black hole s uc u e may eme ge as a
na u al p edic ion om quan um g a i y ab ini io.
4.1 Dimensional Uni ica ion Ac oss 61 O de s o Magni ude
(Planck leng h o Hubble adius)
The scale-dependen empe a u e amewo k, in eg a ed wi h he Planck o ce
de i a ion, uni ies phenomena ac oss 61 o de s o magni ude in spa ial scale:
lmin ≈10−35 m (Planck scale),(78)
lmax ≈1026 m (Hubble adius),(79)
wi h co esponding empe a u es and o ces:
Ts(lmin)≈TU≈1032 K (quan um egime),(80)
Ts(lmax)≈TH≈10−30 K (cosmological egime),(81)
F(lmin)≈FPl ≈1044 N (Planck o ce),(82)
F(lmax)≈FH=MHHc ≈10−10 N (cosmic o ce).(83)
This comp ehensi e amewo k enables uni ied desc ip ion o black hole he mo-
dynamics (local scales), egula black hole in e io dynamics (c osso e scales),
and cosmological ho izon dynamics (cosmological scales) wi hin a single heo e i-
cal s uc u e, wi h in e nal consis ency main ained h ough dimensional igo and
s a is ical-p obabilis ic ounda ion.
18
4.2 En opy Densi y and P essu e Balance in Regula Black
Holes
4.2.1 In e io En opy Densi y
The he modynamic s uc u e o egula black holes is cha ac e ized by a non-singula
co e con igu a ion undamen ally dis inc om classical Schwa zschild geome y. The
in e io en opy densi y is de ined as:
s( ) = 4
3aSBNT( )3,(84)
whe e:
•aSB = 7.5657 ×10−16 J·m−3·K−4is he adia ion ene gy densi y cons an ( ela ed
o S e an-Bol zmann cons an by aSB = 4σ/c), [J·m−3·K−4],
•N≈106.75 is he e ec i e deg ees o eedom om he S anda d Model
(dimensionless),
•T( )is he local empe a u e p o ile [K],
•The ac o 4/3a ises om he modynamic ela ions o adia ion.
Dimensional e i ica ion:
[s( )] = [J ·m−3·K−4]×[K3] = [J ·K−1·m−3],(85)
which co ec ly ep esen s en opy pe uni olume pe Kel in.
4.3 Holog aphic Sc een En opy Bound
In o ma ion in a egula black hole is encoded on a holog aphic sc een a he bounda y,
a he han los o a singula i y. The maximum en opy densi y on his sc een is gi en
by he undamen al bound:
σsc een =kB
4L2
Pl ≈1.32x1046 J·K−1·m−2.(86)
Dimensional e i ica ion:
[σsc een] = [J ·K−1]
[m2]= [J ·K−1·m−2],(87)
ep esen ing he maximum in o ma ion densi y pe uni a ea. Fo a sphe ical
holog aphic sc een o adius R, he o al en opy is:
Ssc een =σsc een ×4πR2=kBc3
4ℏG×4πR2=πkBc3R2
ℏG,(88)
which ma ches he Bekens ein-Hawking en opy.
19
4.4 P essu e Balance Condi ion
The non-singula co e is main ained h ough equilib ium be ween ou wa d adia ion
p essu e and inwa d acuum p essu e:
P ad( ) + P ac( ) = 0,(89)
whe e he adia ion p essu e is gi en by he adia ion equa ion o s a e:
P ad =1
3aSBNT( )4.(90)
Dimensional e i ica ion:
[P ad] = [J ·m−3·K−4]×[K4] = [J ·m−3] = [Pa] = [N ·m−2],(91)
co ec ly yielding p essu e dimensions. A he Planck scale, his p essu e equilib ium
de ines he cha ac e is ic s uc u e o he egula black hole co e, p e en ing classical
singula i y o ma ion.
4.5 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,(92)
which p o ides he holog aphic consis ency condi ion. The in e io adia ion en opy
is:
S =4aSBπT 3
3
9,(93)
whe e aSB =π2k4
B/(15ℏ3c3).Dimensional e i ica ion:
[S ] = [J ·m−3·K−4]×[K3]×[m3] = [J ·K−1],(94)
co ec ly ep esen ing en opy.
4.6 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.
20
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.
5 Uni ica ion o Radia ion and Ma e En opy
Ac oss Scales
Fundamen al Scaling Laws
Classical cosmology aces an essen ial challenge: econciling undamen ally di e en
en opy dependencies ac oss cosmic e as:
•Radia ion e a: En opy scales as S ∝E3/4
, a ising om ela i is ic pa icle
s a is ics.
•Ma e e a: En opy scales as Sm∝E2
m, e lec ing non- ela i is ic deg ees o
eedom.
These dispa a e scalings pose undamen al challenges o cons uc ing uni ied en opy
unc ions ac oss he cosmic e olu ion.
Dimensional Uni ica ion Ac oss 80 O de s o Magni ude
(pa icle o uni e se)
The en opy unc ion ha econciles bo h scaling laws ac oss app oxima ely 80 o de s
o magni ude in ene gy is:
y(x) = x2
1−(1 −x)3/4,(95)
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).
Physical In e p e a ion
The in e pola ion unc ion y(x)encodes he ansi ion om adia ion dominance
(small x) h ough ma e dominance (la ge x). The speci ic unc ional o m x2/(1 −
(1 −x)3/4)eme ges om combining:
S o al =Sm+S ∝E2
m+E3/4
,(96)
h ough Planck-ene gy no maliza ion, wi h Em=xE o al and E = (1−x)E o al. The
connec ion be ween local en opy scaling and dimensionless en opy is:
˜
S≈(xE o al)2+ ((1 −x)E o al)3/4
E2
o al
=x2+ (1 −x)3/4/E5/4
o al,(97)
21
which in he low-ene gy limi educes o he in e pola ion unc ion.
Planck-No malized Dimensionless En opy Scaling:
˜
y=S/kB
(E o al/EPlanck)2,[dimensionless] (98)
Planck-Ene gy-No malized Dimensionless En opy Func ion
We esol e his uni ica ion h ough dimensionless en opy a iables no malized by he
Planck ene gy scale. The Planck ene gy is:
EPl = ℏc5
G[J].(99)
De ine he dimensionless en opy as:
˜
S(x)≡S(x)/kB
(E o al/EPl)2,(100)
whe e x=Em/E o al is he dimensionless ma e ene gy ac ion [0,1], and he denom-
ina o (E o al/EPl)2p o ides he no maliza ion scale. Dimensional e i ica ion:
[˜
S] = [J ·K−1]/[J ·K−1]
1= [dimensionless],(101)
whe e he nume ical o m y ep esen s he dimensionless en opy ˜
S.Bounda y
beha io e i ica ion:
•Radia ion-domina ed limi (x→0+):
y(0) = 0
1−1= 0,(inde e mina e; L’Hopi al’s ule) ⇒y→0.(102)
This e lec s anishing en opy when ma e con ibu ion becomes negligible.
•Ma e -domina ed limi (x→1−):
y(1) = 1
1−0= 1,(103)
co ec ly ep esen ing en opy domina ed by ma e deg ees o eedom.
In e media e beha io : The unc ion exhibi s smoo h in e pola ion be ween bo h
egimes, main aining ma hema ical consis ency and physical sensibili y h oughou
cosmic e olu ion.
This comp ehensi e amewo k enables uni ied desc ip ion ac oss 61 o de s o magni-
ude in spa ial scale ( om Planck leng h LPl ∼10−35 m o Hubble adius RH∼1026
m) and 80 o de s o magni ude in ene gy scale ( om suba omic pa icles ∼10−10 J o
he obse able uni e se ∼1070 J), es ablishing dimensional consis ency in holog aphic
he modynamics ac oss all egimes.
22
5.1 The Non-Singula Co e S uc u e o Regula Black Holes
5.1.1 Dis inc ion om Al e na i e Models
•Haywa d’s geome ical co e: Haywa d’s egula black holes employ geome -
ic egula iza ion h ough modi ied me ic componen s. Ou p essu e-equilib ium
app oach p o ides a dynamical ( he modynamic) mechanism o singula i y a oid-
ance, wi hou ad hoc me ic modi ica ions.
•Dymniko a’s de Si e in e io : Dymniko a’s models inco po a e a de Si e
in e io ma ching smoo hly o he ex e io . Ou amewo k uses ealis ic adia ion-
ma e p essu e balance, mo e di ec ly connec ed o undamen al physics.
The physical basis o singula i y a oidance in ou model is he balance P ad+P ac = 0,
which main ains a non-singula he modynamic s uc u e encoding in o ma ion on
he holog aphic sc een.
5.2 Cosmological Ex ension and En opy G ow h
5.2.1 En opic Fo ce Ac oss Cosmological Scales
Ex ending he RBHs he modynamic amewo k o cosmological scales e eals en opy
as he undamen al d i ing o ce o cosmic accele a ion:
Fcosmic =THubble
dSuni e se
dxcosmic
,(104)
whe e THubble is an e ec i e empe a u e a he Hubble ho izon [K], and xcosmic
ep esen s a cha ac e is ic cosmological leng h scale [m].
5.2.2 Uni e sal Desc ip ion o En opy E olu ion
The Planck-ene gy-no malized en opy unc ion enables a uni e sal desc ip ion span-
ning om Planck scales o he obse able uni e se:
y(x, ) = x2
1−(1 −x)3/4,(105)
whe e x( )e ol es wi h cosmic ime, e lec ing he dynamical ansi ion om adia ion
o ma e domina ion. The he modynamic consis ency ensu es ha :
•In o ma ion is conse ed h oughou cosmic e olu ion,
•En opy ne e exceeds he holog aphic bound a any scale,
•The amewo k na u ally inco po a es quan um e ec s a Planck scales and classical
e ec s a mac oscopic scales.
5.2.3 Da k Ene gy In e p e a ion
The amewo k sugges s ha da k ene gy phenomena may a ise om he en opic
endency o maximize in o ma ion densi y while espec ing holog aphic bounds. This
23
p o ides an al e na i e in e p e a ion complemen a y o Lambda-CDM phenomenol-
ogy wi hou con adic ing Gene al Rela i i y.
5.3 RBHs as Planck-Scale Fundamen al Objec s
We es ablish egula black holes (RBHs) as undamen al he modynamic en i ies a
he Planck scale, dis inc om phenomenological modi ica ions o classical black holes.
The key inno a ions include: Mic oscopic Founda ion: The en opy densi y ela ion
s( )∝N T( )3(106)
p o ides a mic oscopic basis o en opy e olu ion, whe e N ep esen s he e ec i e
numbe o scala deg ees o eedom in he in e io . Ene gy Balance Mechanism:
Unde he model’s in e io equilib ium condi ion
P ad( ) + P ac( ) = 0,(107)
ensu es he modynamic s abili y while a oiding singula i ies, undamen ally di e en
om geome ic-co e app oaches. Scale-In a ian No maliza ion: The no maliza-
ion S
E2
o al
is mani es ly dimensionless, p ese ing dimensional consis ency ac oss ene gy scales
om Planck-scale in e io dynamics o po en ial cosmological applica ions. This
scale-in a iance p ope y elimina es he need o a bi a y dimension ul pa ame e s,
es ablishing a ounda ion obus o ex ensions o dynamical and cu ed-space ime
se ings.
5.4 Simple P essu e-Balance Model
To a oid sol ing he ull Eins ein equa ions while s ill cap u ing he key physics, The
in e io is modeled as a high- empe a u e adia ion gas balanced by a nega i e acuum
p essu e. This wo k adop s he ollowing minimal assump ions, 1. Radia ion p essu e
om N ela i is ic deg ees o eedom a local empe a u e T( )is gi en by
ρ ad( ) = aSB N T( )4, P ad( ) = 1
3ρ ad( ) = 1
3aSB N T( )4.(108)
2. Quan um acuum is modeled as a uni o m nega i e p essu e ha exac ly cancels
he adia ion p essu e,
P ac( ) = −P ad( ) = −1
3aSB N T( )4.(109)
3. The ne p essu e anishes e e ywhe e,
P o ( )≡P ad( ) + P ac( ) = 0,(110)
24
so ha he in e io emains s a ic wi hou in oking he ull gene al- ela i is ic ield
equa ions. Equa ions (108)–(110) p o ide an in ui i e pic u e o how posi i e adia ion
p essu e and nega i e acuum p essu e balance o a oid a cen al singula i y.
P ad
P ad
P ad
P ad
P ac
P ac
P ac
P ac
Fig. 1 Schema ic o adia ion p essu e and ac-
uum p essu e balancing inside he egula black
hole co e. A (0, -1.2) In ui i e p essu e-balance
model inside he co e, showing P ad ( ed ou wa d
a ows) balanced by P ac (blue inwa d a ows).
P ac
P ac
P ac
P ac
Fig. 2 Schema ic illus a ing he in ui i e pic-
u e in which many quan um modes each con-
ibu e ze o-poin ene gy, and hei collec i e
a e age e ec p oduces a uni o m nega i e p es-
su e ( acuum p essu e) inside he sphe ical co e.
This nega i e acuum p essu e hen balances he
ou wa d adia ion p essu e o a oid a cen al sin-
gula i y.
5.4.1 Dis inc ion om Exis ing Regula Black Hole Models
The p esen amewo k di e s undamen ally om exis ing egula black hole mod-
els in h ee key aspec s: 1. In e io S uc u e: While Haywa d’s model [81] elies
on pu ely geome ic modi ica ions wi h minimal he modynamic con en , and Dym-
niko a’s app oach [61] employs a s a ic de Si e co e, The p esen RBHs model
ea u es a dynamically balanced he modynamic in e io sa is ying
P ad( ) = −P ac( ),(111)
which a oids singula i ies h ough local p essu e equilib ium. 2. En opy Fo mula-
ion: Unlike he con en ional S∝Ascaling in Haywa d and Dymniko a models, We
E2
o al no maliza ion
y=S
E2
o al
(112)
enables a uni ied dimensionless ea men o adia ion (S ∝E3/4
) and ma e
(Sm∝E2
m) con ibu ions. 3. Physical Founda ion: We model es ablishes RBHs as
undamen al he modynamic objec s a he Planck scale, wi h in e io en opy den-
si y p o iding a mic oscopic ounda ion o mac oscopic en opy e olu ion, in con as
o pu ely geome ic in e io s o p e ious models.
25
This con i ms ha he he modynamically de i ed p essu e is p opo ional o he
magni ude o he canonical da k ene gy p essu e |PΛ|=ρΛc2, wi h a coe icien o 2/3
a ising om he holog aphic en opy- olume ela ionship, consis en wi h Ts(l)≈TH
o l≳lc.
Nume ical Ve i ica ion:
PGH ≈5.11 ×10−10 Pa,PGH
ρΛc2= 0.6667 ≈2
3✓(136)
6.2.3 Tempe a u e Fluc ua ions and P essu e Va iance
The Gibbons-Hawking empe a u e i sel exhibi s he mal luc ua ions in a ini e
holog aphic sys em, scaled by he in e pola ion:
δTGH ∼TGH 1
Nexp −l2
2l2
c(137)
The p essu e’s empe a u e dependence, de i ed om Eq. (135):
∂P
∂T ∼ρΛc2
TGH
(138)
yields p essu e luc ua ions:
δPGH =∂P
∂T δTGH ∼ρΛc2
TGH ×TGH 1
Nexp −l2
2l2
c=ρΛc2
√Nexp −l2
2l2
c(139)
This ep oduces Eq. (124), con i ming consis ency be ween holog aphic ene gy
luc ua ions and Gibbons-Hawking he modynamics ia he en opic uni ica ion.
6.2.4 Non-Equilib ium Ex ensions in de Si e Space
In non-equilib ium de Si e he modynamics [60], he GH empe a u e acqui es a
ime-dependen co ec ion TGH →TGH(1 + γ˙
H/H2), wi h γ∼1 om en opy
p oduc ion ˙
S > 0, u he modula ed by Ts(l). This yields p essu e luc ua ions:
δPGH =ρΛc2
√Nexp −l2
2l2
c 1 + γ˙
H
H2!,(140)
ensu ing second-law compliance du ing slow- oll in la ion. Dimensional analysis
(SymPy) upholds [δP ] = [Pa], b idging equilib ium GH o dynamic cosmology and
esol ing ho izon pa adoxes in 2025 analyses [59] h ough scale-dependen en opy
g adien s.
32
6.3 Quan um Field Theo y Mode Sum and Cen al Limi
Theo em
The Gaussian o m o p essu e luc ua ions Pquan um ∼ N(0, σ2)is igo ously jus-
i ied by he cen al limi heo em applied o quan um ield heo y modes, wi h
scale-dependen egula iza ion om Ts(l).
6.3.1 Vacuum Fluc ua ions in de Si e Space
In de Si e space, each quan um ield mode kcon ibu es o acuum ene gy and
p essu e. Fo a massless scala ield ( ep esen ing he dominan con ibu ion om
pho ons and g a i ons), he p essu e luc ua ion pe mode is:
⟨δP 2
k⟩ ∼ ℏω4
k
c3exp −l2
l2
c(141)
whe e ωk=c|k|is he mode equency, and he exponen ial ensu es consis ency wi h
local Un uh e ec s a small l.
6.3.2 Hubble Cu o and Mode In eg a ion
The Hubble ho izon imposes a na u al in a ed cu o , wi h he c osso e lcmodula ing
high-mode con ibu ions:
kmax ∼H
1−exp −l2
l2
c(142)
In eg a ing o e all modes in momen um space:
σ2
QFT =Zkmax
0⟨δP 2
k⟩d3k= exp −l2
l2
cZkmax
0
ℏc4k4
c3×4πk2dk = 4πℏcexp −l2
l2
cZkmax
0
k6dk.
(143)
To e alua e he in eg al exac ly, pe o m he subs i u ion k=ukmax,dk =kmax du,
whe e u∈[0,1]. This yields
Zkmax
0
k6dk =Z1
0
(ukmax)6kmax du =k7
max Z1
0
u6du =k7
max
7.(144)
Thus,
σ2
QFT =4πℏcg∗
7k7
max exp −l2
l2
c,(145)
whe e he ac o g∗accoun s o he S anda d Model e ec i e deg ees o eedom,
ensu ing he mode sum inco po a es all ela i is ic ield con ibu ions. Subs i u ing
33
kmax =H
1−exp−l2
l2
cp o ides he closed- o m scale-dependen exp ession
σ2
QFT =4πℏcg∗
7H7exp −l2
l2
c
h1−exp −l2
l2
ci7,(146)
which aligns he H7scaling wi h he ρΛscale ia en opic bounds, whe e he in e po-
la ion in Ts(l) unes he p e ac o o ma ch holog aphic luc ua ions wi hou ex e nal
egula iza ion. This o m enhances mode con ibu ions a small scales (l≪lc, whe e
kmax ≫H) consis en wi h local quan um e ec s and supp esses hem a la ge scales
(l≳lc, eco e ing ini e holog aphic a iance).
Dimensional Analysis:
[ℏcH7]=(J·s)(m/s)(s−7)
=J·s−6=kg ·m2·s−4=Pa2✓(147)
Nume ical Es ima e:
σQFT = 4πℏcg∗H7
0
7exp −l2
2l2
c≈3.67 ×10−75 Pa (148)
6.3.3 Cen al Limi Theo em Jus i ica ion
Since Pquan um =PkδPkis a sum o independen andom a iables (each mode
con ibu es independen ly), he cen al limi heo em gua an ees:
Pquan um
Nmodes→∞
−−−−−−−→ N(0, σ2)(149)
The numbe o independen modes up o kmax ∼His:
Nmodes ∼RH
λmin 3
∼1090 (150)
When conside ing all ield species wi h g∗= 106.75 s anda d model deg ees o eedom,
he e ec i e mode coun becomes:
Ne ∼g∗Nmodes ≫1(151)
This igo ously jus i ies he Gaussian app oxima ion o p essu e luc ua ions, wi h
he scale-dependen weigh ing om Ts(l)p ese ing kBcancella ion and en opic o ce
exac ness.
34
6.3.4 Inco po a ing S anda d Model Fields and G a i ons
Ex ending he mode sum o ull SM ields (g∗= 106.75) and g a i ons [145], he
a iance becomes σ2
QFT =4πℏcg∗
7H7exp−l2
l2
c
1−exp−l2
l2
c7, wi h CLT con e gence accele -
a ed by Ne ≫1090. Fo cosmology, he c osso e scale lc egula izes con ibu ions
ia en opic in e pola ion, aligning wi h bounds om ρc i h ough e ec i e ield
con ibu ions:
σQFT ≈
u
u
u
u
4πℏcg∗H7
7
exp −l2
l2
c
h1−exp −l2
l2
ci7≈3.67 ×10−75 Pa,(152)
yielding σQFT ∼10−75 Pa. This 2025 holog aphic in e play [13] alida es Gaussiani y
o da k ene gy luc ua ions, wi h he g∗co ec ion aligning he scale o ρΛ h ough
he weigh ed Bol zmann dis ibu ion ounda ion o Ts(l).
6.4 Casimi E ec a Cosmological Scales
The Casimi e ec , a ising om bounda y condi ions on quan um ields, p o ides an
addi ional pe spec i e on acuum p essu e a cosmological scales.
6.4.1 Casimi P essu e Gene aliza ion
The Casimi p essu e be ween pa allel pla es sepa a ed by dis ance ais:
PCasimi =−π2ℏc
720a4(153)
Ex ending his o cosmological scales by eplacing a→RH=c/H:
Pcosmo
Casimi =−π2ℏc
720(c/H)4=−π2ℏH4
720c3(154)
Dimensional Analysis:
[ℏH4/c3]=(J·s)(s−4)/(m3·s−3)
=J/m3=Pa ✓(155)
Nume ical Es ima e:
Pcosmo
Casimi ≈ −1.22 ×10−132 Pa (156)
While his con ibu ion is negligibly small compa ed o ρΛc2∼10−9Pa, i ep e-
sen s a genuine quan um acuum e ec a ising om he ini e size o he obse able
uni e se. The nega i e sign indica es an a ac i e con ibu ion, consis en wi h he
in e p e a ion o acuum ene gy as a o m o ension in space ime.
35
6.4.2 Casimi as Da k Ene gy Mechanism
The cosmological Casimi p essu e links o da k ene gy ia nega i e acuum en-
sion [39], wi h b ane-wo ld co ec ions Pcosmo
Casimi → −π2ℏH4
720c3(1 + δρDM
ρΛ), whe e δ∼0.1
om DM- acuum coupling. This gene a es w≈ −1equa ion-o -s a e:
PDE
Casimi ≈ −1.22 ×10−132 Pa 1+0.1ρDM
ρΛ,(157)
consis en wi h Planck ΩΛ= 0.684 (SymPy: [Pa] exac ). 2025 b ane models [52]
posi ion Casimi as a iable da k ene gy sou ce, esol ing he acuum ene gy
disc epancy.
6.5 E ec i e Theo e ical Pa ame iza ion
The mic oscopic es ima es om holog aphic luc ua ions (Eq. 124), QFT mode sums
(Eq. 145), and Gibbons-Hawking he modynamics (Eq. 139) all yield p essu e a i-
ances ha a e sys ema ically ela ed o he e ec i e heo e ical pa ame iza ion
σe =TGHρΛc2used in mac oscopic simula ions:
Me hod Va iance Ra io o σe
Holog aphic (Eq. 124)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. 145)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. 139)5.10 ×10−71 Pa 2.50 ×10−32
E ec i e Theo e ical 2.04 ×10−39 Pa 1.00
Table 2 Compa ison o acuum p essu e luc ua ion magni udes om
di e en heo e ical app oaches. All mic oscopic es ima es a e
sel -consis en wi hin ela i e de ia ions o o de uni y, bu di e om
he e ec i e heo e ical pa ame iza ion by 1030–1036 o de s o
magni ude due o ampli ica ion h ough he maliza ion o e holog aphic
deg ees o eedom.
6.5.1 In e p e a ion as E ec i e Theo y
The e ec i e heo e ical pa ame iza ion:
σe =TGHρΛc2=ℏH
2πkB×3H2c2
8πG =3ℏH3c2
16π2kBG(158)
ep esen s a coa se-g ained desc ip ion alid a mac oscopic scales ℓ≫Lpl. The
empe a u e ac o TGH ac s as an e ec i e ampli ica ion pa ame e , cap u ing he
he mal p ope ies o he de Si e acuum a scales whe e holog aphic in o ma ion is
a e aged o e many Planck-scale cells.
36
6.5.2 Ampli ica ion Mechanism and Scale B idge
The ampli ica ion ac o om mic oscopic o mac oscopic scales is quan i ied by:
A=σe
σholo
=TGH√N∼ℏH
2πkB× πc5
ℏGH2∼1030–36 (159)
This ampli ica ion ep esen s he he maliza ion o mic oscopic quan um luc ua-
ions o e he ini e numbe o holog aphic deg ees o eedom, analogous o how
B ownian mo ion ampli ies molecula -scale he mal luc ua ions o obse able pa i-
cle displacemen s in mac oscopic sys ems. The e ec i e heo e ical amewo k hus
b idges Planck-scale quan um acuum luc ua ions wi h mac oscopically obse able
cosmic dynamics h ough holog aphic he modynamics.
6.6 Summa y: Quan um Field Theo e ic Founda ions o
Vacuum P essu e
The p esen wo k es ablishes he quan um ield heo e ic ounda ions o acuum
p essu e luc ua ions h ough ou independen and mu ually alida ing heo e ical
app oaches:
1. Holog aphic Ene gy Fluc ua ions (S- ie ): The ini e numbe o holog aphic
deg ees o eedom N∼10122 implies quan um s a is ical luc ua ions:
σholo =ρΛc2
√N(160)
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(161)
The p essu e luc ua ions de i ed om his he modynamic analysis ep oduce he
holog aphic esul , con i ming undamen al he modynamic consis ency.
3. QFT Mode Summa ion wi h Cen al Limi Theo em (A- ie ): Summing
quan um ield modes up o he Hubble cu o wi h p ope no maliza ion yields:
σQFT = 4πℏcH7
7(162)
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.
37
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 (163)
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.
6.6.1 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.
6.6.2 E ec i e Theo e ical F amewo k
The e ec i e heo e ical pa ame iza ion
σe =TGHρΛc2(164)
is jus i ied as a coa se-g ained desc ip ion alid a mac oscopic scales. The empe a-
u e ac o TGH =ℏH/(2πkB)ac s as an e ec i e coupling pa ame e , cap u ing how
he mal deg ees o eedom a he Hubble scale b idge Planck-scale quan um luc u-
a ions wi h cosmologically obse able e ec s. This amewo k p o ides a consis en
desc ip ion wi hou ad hoc pa ame e s, o e ing p edic i e powe o u u e obse a-
ional es s h ough edshi d i measu emen s, g a i a ional wa e obse a ions, and
p ecision cosmology.
7 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.
7.1 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 (165)
38
whe e Ts(l) = TUexp(−l2/l2
c)+TH[1−exp(−l2/l2
c)] is he scale-dependen empe a u e
and dS
dx is he en opy g adien on he holog aphic sc een. This amewo k ex ends
Ve linde’s en opic g a i y heo y, posi ioning da k ene gy as a ising undamen ally
om en opy imbalance a di e en scales a he han as an in insic da k luid. The
en opic o ce d i es he uni e se’s accele a ed expansion h ough non-equilib ium
he modynamic p ocesses encoded in holog aphic deg ees o eedom.
7.2 Vacuum Ene gy and E ec i e Theo e ical P essu e Balance
In his e ec i e heo e ical amewo k, acuum p essu e is d i en by en opy g adien s:
P ac =−ρΛc2+Pquan um (166)
whe e he quan um p essu e e m a ises om scale-dependen empe a u e luc ua-
ions. This acuum ene gy de i es om h ee undamen al sou ces:
•Scale-Dependen Tempe a u e T ansi ion: The e olu ion om Un uh em-
pe a u e (TU∼3.97 ×10−20 K a local Planck scales) o Hubble empe a u e
(TH∼2.65 ×10−30 K a cosmological scales), cap u ed by he scale-dependen
o mula ion Ts(l).
•En opy Densi y and Deg ees o F eedom: En opy densi y scaling s( )∝
NT( )3, whe e N∼10122 is he e ec i e holog aphic deg ees o eedom and T( )
is he local scale-dependen empe a u e.
•Pa ame e -F ee Desc ip ion: Da k ene gy is explained en i ely h ough he
e ec i e heo e ical amewo k wi hou pa ame e uning, aligning p ecisely wi h
Planck 2018 obse a ions (ΩΛ= 0.684,H0= 67.36 ±0.54 km/s/Mpc).
7.3 Nume ical Simula ion Ve i ica ion o En opic Dynamics
In he N-body simula ion code (using Ba nes-Hu oc ee accele a ion), he mo-
dynamic o cing e ms based on en opy g adien s a e inco po a ed in o pa icle
in e ac ions o simula e en opic o ce dynamics. The simula ions con i m:
•Ene gy Conse a ion: Nume ical simula ions e i y ene gy conse a ion wi h
d i less han 0.1% o e 10,000 ime s eps, con i ming he consis ency and s abili y
o he en opic o ce implemen a ion.
•En opy G ow h and Second Law: Mono onic inc ease in sys em en opy is
demons a ed, con i ming ha he dynamics a e undamen ally consis en wi h he
second law o he modynamics.
•Scale-Dependen Ampli ica ion: The scale-dependen empe a u e o mula ion
success ully ep oduces bo h local quan um e ec s (Un uh empe a u e a Planck
scales) and cosmological dynamics (Hubble empe a u e a ho izon scales), spanning
61 o de s o magni ude in spa ial scale.
39
7.4 Da k Ene gy as Dynamic The modynamic P ocess
Ra he han a s a ic cosmological cons an , da k ene gy eme ges as a dynamic en opic
p ocess:
˙
Eda k =Ts(l)dS
d (167)
This dynamic in e p e a ion based on en opy e olu ion econciles h ee key aspec s
o con empo a y cosmology:
1. Consis ency wi h Gene al Rela i i y: Gene al ela i i y is no nega ed bu
ein e p e ed as he mac oscopic he modynamic mani es a ion o mic oscopic
quan um en opy g adien s on he holog aphic sc een. Eins ein’s ield equa ions
eme ge as he hyd odynamic limi o he e ec i e heo e ical amewo k.
2. Pa ame e Economy: All cha ac e is ic ene gy and leng h scales de i e om
undamen al physics cons an s (Planck leng h Lpl, s anda d model deg ees o ee-
dom g∗= 106.75, holog aphic en opy bounds) wi hou in oducing addi ional ee
pa ame e s o da k ene gy.
3. Obse a ional P edic ions: Fu u e high-p ecision es s di ec ly p obe he
en opic o igin o da k ene gy:
•Redshi d i measu emen s (∆˙
z≈4.0×10−11 y −1) using nex -gene a ion
op ical la ice clocks.
•G a i a ional wa e obse a ions wi h LISA/DECIGO de ec ing ingdown de i-
a ions a ∼10−22 le el.
•P ecision cosmological cons ain s om DESI 2024-2025 and Planck legacy da a.
7.4.1 En opy as Fundamen al O ganizing P inciple
The hypo hesis ha en opy cons i u es he undamen al "sou ce" o cosmic dynam-
ics, wi h gene al ela i i y eme ging as i s mac oscopic he modynamic mani es a ion,
ep esen s a concep ual pa adigm shi in heo e ical physics. By uni ying quan um
and cosmological egimes h ough holog aphic p inciples while main aining consis-
ency wi h Eins ein’s ield equa ions and Planck obse a ions wi hou addi ional ee
pa ame e s, his en opy-cen ic amewo k o e s a comp ehensi e unde s anding o
da k ene gy as undamen ally he modynamic in o igin, po en ially b idging quan um
g a i y and cosmology h ough he modynamic p inciples.
7.4.2 P essu e Scale Uni ica ion ia The modynamic Analysis
The mic oscopic es ima es om holog aphic luc ua ions, QFT mode sums, and
Gibbons-Hawking he modynamics yield p essu e a iances σmic o ha di e by many
o de s o magni ude om he e ec i e phenomenological scale σholonomic =TGHρΛc2
used in simula ions and obse a ions. Table 4compa es hese es ima es.
40
Me hod P essu e Va iance Ra io o σholonomic
Holog aphic (Eq. 124)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. 145)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. 139)5.10 ×10−71 Pa 2.50 ×10−32
Phenomenological 2.04 ×10−39 Pa 1.00
Table 3 Compa ison o acuum p essu e luc ua ion magni udes om di e en
heo e ical app oaches. All mic oscopic es ima es (Holog aphic, QFT, and
Gibbons-Hawking) a e sel -consis en wi h each o he wi hin ac o s o o de
uni y, bu smalle han he phenomenological pa ame iza ion by 1030–1036
o de s o magni ude. This hie a chy indica es a undamen al e ec i e heo y
pic u e.
In e p e a ion as e ec i e heo y:
The phenomenological pa ame iza ion:
σholonomic =TGHρΛc2=ℏH
2πkB×3H2c2
8πG =3ℏH3c2
16π2kBG(168)
should be unde s ood as an e ec i e coa se-g ained desc ip ion alid a mac oscopic
scales ℓ≫LPl. The empe a u e ac o TGH ac s as an e ec i e ampli ica ion
pa ame e , cap u ing he he mal p ope ies o he de Si e acuum a scales whe e
holog aphic in o ma ion is a e aged o e many Planck-scale deg ees o eedom. The
ampli ica ion a io is:
σholonomic
σholo
=TGHpN0=ℏH
2πkB× πc5
ℏGH2∼1030–36 (169)
This ep esen s he **ampli ica ion o mic oscopic quan um luc ua ions o mac o-
scopic obse ables** h ough he maliza ion o e he holog aphic deg ees o eedom.
This mechanism is analogous o how B ownian mo ion ampli ies molecula -scale
luc ua ions o obse able pa icle displacemen s, bu ope a ing a cosmological scales.
7.5 Summa y and Consis ency
This wo k es ablishes he quan um ield heo e ic ounda ions o acuum p essu e
luc ua ions h ough ou complemen a y and mu ually alida ing app oaches:
1. Holog aphic Fluc ua ions (S- ie ): The ini e holog aphic deg ees o eedom
N0≈2.26 ×10122 yield p essu e luc ua ions σholo =ρΛc2/√N0, p o iding he
mos di ec connec ion o en opy bounds.
2. Gibbons-Hawking The modynamics (A- ie ): Applying he i s law o he
de Si e ho izon yields he mal p essu e PGH = (2/3)ρΛc2and ep oduces he
holog aphic p essu e luc ua ions, con i ming he modynamic consis ency.
3. QFT Mode Summa ion (A- ie ): Summing quan um ield modes up o he
Hubble cu o yields σQFT =p4πℏcH7
0/7wi h e ec i e mode coun Ne ∼
106.75 ≫1, jus i ying Gaussiani y ia he cen al limi heo em.
41
Fig. 7 Nume ical da a showing in e nal deg ees o eedom Nand he modynamic p ope ies (T,
s ad) o egula black hole in e io s wi h N≫100 massless scala ields. This able con i ms he
consis ency o he en opy densi y o mula ion (Sec. 7.7) wi h he S anda d Model alue g∗= 106.75
used h oughou his wo k.
7.9 Concep ual F amewo k o Holog aphic The modynamics
7.9.1 Holog aphic Sc een Illus a ion
This o mula ion ex ends na u ally o quasi-s a ic o cosmological se ings when g ( )
is gene alized o FLRW me ics.
M
m
F
inc easing ∇S
sc een T( )∝1/
Fig. 8 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.
48
7.10 Holog aphic The modynamic F amewo k
The holog aphic p inciple connec s he in o ma ion con en o a bulk olume o
he en opy encoded on i s bounda y su ace. This sec ion applies he holog aphic
amewo k o egula black hole in e io s and he cosmological ho izon.
Holog aphic sc een concep :
A holog aphic sc een is a wo-dimensional su ace (a adius Ro Hubble adius RH)
wi h a ea A ha encodes he en opy o all ma e and adia ion enclosed wi hin.
Acco ding o he holog aphic p inciple, he en opy Sassocia ed wi h he bulk olume
is p ojec ed on o his sc een, whe e he in o ma ion con en o he olume is encoded
on he bounda y acco ding o:
Ssc een =kBA
4L2
Pl
(195)
Fo a sphe e o adius R:A= 4πR2, yielding:
Ssc een =πkBR2
L2
Pl
(196)
This ela ionship ensu es ha he mac oscopic he modynamic s uc u e (in e io
en opy, empe a u e, p essu e) emains consis en wi h he mic oscopic cons ain s
imposed by quan um g a i y and holog aphy.
7.11 Dimensional Consis ency and Scaling Rela ions
To cla i y he mu ual consis ency o all he modynamic quan i ies used in his wo k,
we p esen a comp ehensi e dimensional analysis. All quan i ies a e exp essed in SI
base uni s [kg, m, s, K].
Dimensional summa y:
•Deg ees o F eedom (N): [dimensionless]
E ec i e numbe o massless scala ields (N≈106.75).
•Tempe a u e (T): [K]
Local Hawking-like empe a u e in he in e io ame.
•Radia ion P essu e (P): [Pa] = [J·m−3] = [kg·m−1·s−2]
Scaling: P∝NT4. Physical in e p e a ion: ou wa d p essu e om ela i is ic
adia ion.
•Ene gy Densi y (ρ): [J·m−3] = [kg·m−1·s−2]
Scaling: ρ∝NT4(same as p essu e by equa ion o s a e P=ρ/3).
•En opy Densi y (s): [J·K−1·m−3]
Scaling: s∝NT3. Physical in e p e a ion: in o ma ion densi y pe uni olume.
7.12 The modynamic S uc u e o Black Hole In e io s
The he modynamic s uc u e o a egula black hole in e io illed wi h Nmassless
ela i is ic ields in local he mal equilib ium is go e ned by s anda d adia ion he -
modynamics, app op ia ely ans o med acco ding o he Tolman edshi ela ion.
49
7.12.1 Radia ion-Domina ed The modynamics
The undamen al he modynamic ela ions a e:
P=1
3ρ, ρ =aSBNT4, s =4
3aSBNT3(197)
whe e:
•aSB =4π2k4
B
15c3ℏ3= 7.5657 ×10−16 J·m−3·K−4is he adia ion densi y cons an ,
•N≈106.75 is he e ec i e deg ees o eedom,
•T[K] is he local empe a u e,
•P[Pa], ρ[J·m−3], s[J·K−1·m−3].
Dimensional e i ica ion:
Ene gy densi y:
[ρ]=[J m−3K−4]×[dimensionless]×[K]4(198)
= [J m−3](199)
P essu e ( om P=ρ/3):
[P]=[J m−3]=[Pa](200)
En opy densi y:
[s]=[J m−3K−4]×[dimensionless]×[K]3(201)
= [J K−1m−3](202)
All ela ions exhibi co ec dimensional s uc u e consis en wi h ela i is ic s a is-
ical mechanics.
7.12.2 Tolman Redshi Rela ion
All he modynamic quan i ies abo e a e e alua ed in he local p ope ame o
obse e s a coo dina e posi ion . These quan i ies ans o m be ween di e en adial
posi ions acco ding o he **Tolman ela ion**:
T( )p−g ( ) = T∞=cons an (203)
whe e:
•T( )[K] is he local empe a u e a adius ,
•p−g ( )[dimensionless] is he edshi ac o (me ic componen ),
•T∞[K] is he empe a u e a spa ial in ini y ( e e ence ame).
50
Physical in e p e a ion:
The Tolman ela ion e lec s ha local empe a u e combines bo h in insic he mal
ene gy and g a i a ional edshi . In a s onge g a i a ional ield (la ge |g |), he
local empe a u e T( )mus be highe o main ain cons an e ec i e empe a u e
T∞a in ini y. This ensu es he modynamic consis ency ac oss he cu ed space ime
in e io .
7.12.3 Fi s Law o The modynamics
Fo a ixed mass elemen in he RBHs in e io , he i s law o he modynamics in
di e en ial o m is:
dU =δQ −P dV (204)
Fo e e sible (adiaba ic equilib ium) p ocesses:
dU =T dS −P dV (205)
whe e:
•dU [J] is he change in in e nal ene gy,
•δQ [J] is hea added o he sys em,
•T dS [J] is he e e sible hea e m,
•P dV [J] is wo k done by he sys em.
This ensu es ha empe a u e imes en opy g adien d i es he modynamic e olu-
ion, es ablishing he undamen al connec ion be ween en opy g ow h and he mal
dynamics in he RBHs in e io .
Consis ency wi h adia ion domina ed equa ion o s a e:
Fo adia ion wi h P=ρ/3, he in e nal ene gy pe uni olume is u=ρ, and en opy
pe uni olume sa is ies s= (4/3)ρ/T. These ela ions a e au oma ically sa is ied by
Eq. (197), con i ming ull he modynamic consis ency.
7.12.4 P essu e Balance Condi ion
In equilib ium, he p essu e g adien balances g a i a ional o ces:
dP
d =−ρg( ),(206)
whe e g( )[m s−2] is he local g a i a ional accele a ion. All e ms ha e consis en
dimensions [Pa m−1].
Ene gy Conse a ion
To al ene gy conse a ion is sa is ied h ough:
dE o al
d =−dE adia ion
d −dEg a i a ional
d = 0,(207)
51
ensu ing ha ene gy changes in di e en o ms balance [J s−1].
7.13 Summa y: Dimensional Comple eness
The he modynamic amewo k is dimensionally comple e and in e nally consis en :
•P essu e (ene gy densi y): [J m−3],
•En opy densi y: [J K−1m−3],
•Tempe a u e: [K],
•All equa ions p ese e dimensional s uc u e ac oss coo dina e ans o ma ions.
The ole o N(e ec i e ield coun ) as a dimensionless mul iplie p o ides he oun-
da ion o en opy-a ea co espondence h ough he local equilib ium scheme adop ed
in holog aphic he modynamics.
7.14 Bekens ein-Hawking En opy and In o ma ion Encoding
7.14.1 Bekens ein-Hawking En opy Fo mula
The en opy o a black hole is desc ibed by he Bekens ein-Hawking o mula:
SBH =4πkBGM2
ℏc,(208)
whe e:
•SBH is black hole en opy [J * K−1],
•kB= 1.380649 ×10−23 J*K−1is Bol zmann cons an ,
•G= 6.67430 ×10−11 m3·kg−1·s−2is New on’s g a i a ional cons an ,
•M[kg] is black hole mass,
•ℏ= 1.054571817 ×10−34 J * s is educed Planck cons an ,
•c= 2.99792458 ×108m * s−1is speed o ligh .
7.15 Dimensional Analysis: En opy Quan um Numbe
In e p e a ion
When he Bekens ein-Hawking en opy is di ided by Bol zmann cons an , he esul
is in e p e ed as an en opy quan um numbe (dimensionless coun o in o ma ion
uni s):
N=SBH
kB
=4πGM2
ℏc.(209)
We e i y dimensional consis ency h ough explici dimensional b eakdown:
Componen : GM2
[GM2] = [m3·kg−1·s−2]×[kg]2(210)
= [m3·kg ·s−2].(211)
52
Componen : ℏc
[ℏc] = [J ·s] ×[m ·s−1](212)
= [kg ·m2·s−2·s] ×[m ·s−1](213)
= [kg ·m2·s−1]×[m ·s−1](214)
= [kg ·m3·s−2].(215)
Ra io:
[GM2]
[ℏc]=[m3·kg ·s−2]
[kg ·m3·s−2]= [dimensionless].(216)
Conclusion: The quan i y N=SBH/kBis igo ously dimensionless and ep esen s
he undamen al quan um numbe encoding black hole in o ma ion. The p esence o
ℏ(Planck cons an ) e lec s quan um mechanical na u e o his in o ma ion bound.
7.16 Nume ical Value
Fo a sola -mass black hole (M=M⊙= 1.989x1030 kg), he en opy quan um numbe
is:
N⊙=SBH(M⊙)
kB≈1.37x1067 [dimensionless quan um numbe ].(217)
This eno mous quan um numbe demons a es ha mac oscopic black holes encode
an as onomically la ge amoun o in o ma ion on hei bounda ies.
7.17 To al En opy E olu ion Ac oss Cosmic E as
7.18 Ma e -Domina ed and Radia ion-Domina ed En opy
We ex end he amewo k o compu e o al en opy in a cosmological con ex , com-
bining ma e su ace en opy on a holog aphic sc een wi h adia ion in e io en opy.
The o al en opy in a olume egion is:
S o al( ) = Sm( ) + S ( ),(218)
whe e:
•Smis ma e /su ace en opy [J K−1],
•S is adia ion in e io en opy [J K−1].
Ma e (Su ace) En opy on Holog aphic Sc een
The ma e en opy encoded on he holog aphic sc een is:
Sm=AkB
4L2
Pl
,(219)
whe e:
53
•A= 4πR2
S[m2] is he Schwa zschild su ace a ea,
•LPl =pℏG/c3≈1.616x10−35 m is he Planck leng h.
Dimensional e i ica ion:
[Sm] = [m2]×[J ·K−1]
[m2]= [J ·K−1].(220)
Exp essed in e ms o Schwa zschild adius RS= 2GM/c2:
Sm=4πR2
SkB
4L2
Pl
=πkBc3R2
S
ℏG.(221)
This ma ches he Bekens ein-Hawking en opy, con i ming holog aphic co espon-
dence.
7.19 Radia ion In e io En opy
The adia ion en opy illing he in e io olume is:
S =ZV
s( , )d3x≈4
3aSBN⟨T3⟩V o al,(222)
whe e:
•s( , )[J * K−1·m−3] is local en opy densi y,
•V o al [m3] is o al olume,
•⟨T3⟩[K3] is olume-weigh ed a e age o T3.
Fo a sphe ical egion o adius :
S =4
3aSBNT3
·4π 3
3=16πaSBNT 3
3
9.(223)
Dimensional e i ica ion:
[S ] = [J ·m−3·K−4]×[K]3×[m]3= [J ·K−1].(224)
7.20 Combined To al En opy Exp ession
The comple e exp ession o o al en opy is:
S o al =πkBc3R2
S
ℏG+16πaSBNT 3
3
9,(225)
whe e all quan i ies main ain dimensional consis ency:
[J K−1]+[J K−1]=[J K−1].(226)
54
7.21 Nume ical E olu ion Analysis
Nume ical in eg a ion o e olu ion equa ions o adia ion-domina ed and ma e -
domina ed e as yields he en opy S o al(Z)as a unc ion o edshi pa ame e Z.
The esul s demons a e:
1. Radia ion e a (Z≫1): En opy scales dominan ly as S ∝a3T3∝a3/a =a2,
e lec ing adia ion en opy densi y e olu ion,
2. Ma e e a (Z≲1): En opy app oaches holog aphic bound Sm, demons a ing
he ansi ion o ma e -domina ed s uc u e,
3. T ansi ion egion: Smoo h c osso e be ween egimes ensu es physical con inui y
ac oss cosmic e olu ion.
7.22 The modynamic De i a ion o Black Hole E apo a ion
and En opy Co espondence
7.23 Ene gy Conse a ion in Black Hole E apo a ion
When a black hole adia es h ough Hawking emission, ene gy conse a ion ela es
he ene gy loss o en opy changes:
dE ad =−dMc2,(227)
whe e:
•dE ad [J] is ene gy eleased as Hawking adia ion,
•dM [kg] is mass loss (nega i e o e apo a ing black hole),
•c2[m2·s−2] con e s mass o ene gy.
Dimensional e i ica ion:
[dE ad] = [kg] ×[m2·s−2] = [J].(228)
7.24 Black Hole En opy Change
The en opy dec ease o he black hole is ela ed o ene gy elease h ough he Hawking
empe a u e:
dSBH =−1
TH
dE ad,(229)
whe e TH[K] is he Hawking empe a u e. The nega i e sign e lec s en opy dec ease
as he black hole sh inks. Dimensional e i ica ion:
[dSBH] = [K]−1x[J] = [J ·K−1].(230)
7.25 Radia ion En opy Inc ease
The emi ed Hawking adia ion ca ies en opy:
dS ad =−dSBH =1
TH
dE ad.(231)
55
This ensu es ha o al en opy inc ease (o conse a ion) is main ained:
dS o al =dSBH +dS ad = 0 ( e e sible p ocess).(232)
7.26 Hawking Tempe a u e and I s De i a ion
The Hawking empe a u e is:
TH=ℏc3
8πGMkB
=ℏc
4πkBRS
,(233)
whe e RS= 2GM/c2is he Schwa zschild adius. Dimensional e i ica ion:
[TH] = [J ·s]x[m ·s−1]3
[m3·kg−1·s−2]x[kg]x[J ·K−1](234)
=[J ·s·m3·s−3]
[m3·s−2·J·K−1](235)
=[J ·s−2]
[s−2·J·K−1](236)
= [K].(237)
7.27 En opy Change wi h Black Hole Mass
Taking he de i a i e o Bekens ein-Hawking en opy wi h espec o mass:
dSBH
dM =d
dM 4πkBGM2
ℏc=8πkBGM
ℏc.(238)
Dimensional e i ica ion:
dSBH
dM =[J ·K−1]
[kg] = [J ·K−1·kg−1].(239)
7.28 Radia ion En opy Ra e
The a e o en opy gene a ion in adia ed Hawking adia ion is:
dS ad
d =−dSBH
d =−d
d 4πkBGM( )2
ℏc=−8πkBGM
ℏc
dM
d .(240)
Dimensional e i ica ion:
dS ad
d =[J ·K−1]
[s] = [J ·K−1·s−1].(241)
56
7.29 Hawking E apo a ion Powe
The ene gy emission a e (luminosi y) o a black hole is:
dE
d =σAT4
H=−ϵM−2,(242)
whe e:
•σ= 5.670 ×10−8W * m−2·K−4is S e an-Bol zmann cons an [W*m−2·K−4],
•A[m2] is su ace a ea,
•ϵ[J·m2·s−1] is he e ec i e adia ion coe icien .
Dimensional e i ica ion:
dE
d = [W ·m−2·K−4]×[m2]×[K]4= [W] = [J ·s−1].(243)
7.30 Radia ion En opy Gene a ion Scaling
The adia ion en opy gene a ion a e scales as:
dS ad
d ∝T3
HR2
S.(244)
Subs i u ing TH∝M−1and RS∝M:
dS ad
d ∝1
M3xM2=1
M=M−1.(245)
Physical in e p e a ion: Smalle black holes e apo a e as e and gene a e en opy
a accele a ing a es, e lec ing he he modynamic ins abili y o Hawking adia ion.
To al Radia ed En opy: In eg a ion O e E apo a ion
The o al en opy emi ed as a black hole e apo a es om ini ial mass M0 o ze o is
ob ained by in eg a ing he en opy lux o e he e apo a ion ime:
S ad, o al =ZM0
0
dE ad
TH
=ZM0
0
c2dM
TH(M).(246)
Subs i u ing TH=ℏc3/(8πGMkB):
S ad, o al =ZM0
0
c2dM
ℏc3/(8πGMkB)=ZM0
0
8πGMkB
ℏcc2dM =8πkBGc2
ℏcZM0
0
MdM.
(247)
E alua ing he in eg al:
ZM0
0
MdM =M2
2M0
0
=M2
0
2.(248)
57
8.4 En opy as a Func ion o Ene gy
Subs i u ing T(E )in o he di e en ial o en opy
S =ZdE
T
=ZdE
(E /(aV ))1/4
= (aV )1/4ZE−1/4
dE
=4
3(aV )1/4E3/4
+cons an .
Disca ding he addi i e cons an by app op ia e choice o e e ence yields
S =4
3(aV )1/4E3/4
,(284)
hus es ablishing he scaling
S ∝E3/4
.(285)
8.5 O igin o he 3/4Exponen
The exponen 3/4eme ges om combining wo undamen al empe a u e scalings:
•Ene gy densi y: u∝T4implies E ∝T4, so T∝E1/4
.
•En opy densi y: s∝T3 ollows om dS /dV = (4/3) a T3.
Hence,
S ∝T3∝(E1/4
)3=E3/4
.(286)
8.6 Conclusion o E3/4
Scaling
We de i e he en opy–ene gy ela ion o blackbody adia ion in a ixed olume
and elucida ed he physical o igin o he 3/4exponen as a ising om he dis inc
empe a u e dependences o ene gy and en opy densi ies. [142]
9 Conclusion and Discussion
This wo k es ablishes egula black holes (RBHs) as undamen al he modynamic
objec s a he Planck scale h ough a scale-in a ian amewo k ha uni ies g a i a-
ional he modynamics ac oss all ene gy egimes. The key achie emen s demons a e
how en opy eme ges as he undamen al o igin o g a i y, b idging mic oscopic
quan um s uc u e wi h mac oscopic cosmological phenomena.
64
9.1 Co e Theo e ical Ad ances
Non-Singula In e io ia P essu e Equilib ium. Unlike geome ic egula iza-
ion schemes such as Haywa d’s co e o Dymniko a’s de Si e in e io , singula i y
a oidance is ealized h ough physically well-de ined dynamic p essu e balance
P ad( ) + P ac( ) = 0,(287)
whe e adia ion p essu e om N≈106.75 S anda d Model deg ees o eedom bal-
ances acuum nega i e p essu e. This mic oscopic ounda ion, exp essed h ough
en opy densi y s( ) = 4
3aSBNT( )3, p o ides he modynamic s abili y while encoding
in o ma ion on a non-singula co e dis inc om classical singula i ies.
Uni e sal En opy No maliza ion. The Planck-no malized dimensionless
en opy
˜
y≡S/kB
(E o al/EPlanck)2=x2
1−(1 −x)3/4(288)
econciles undamen ally dis inc scaling laws- adia ion en opy S ∝E3/4
and ma -
e en opy Sm∝E2
m-wi hin a uni ied amewo k. The ma e ene gy ac ion
x≡Em/E o al in e pola es con inuously be ween adia ion-domina ed (x→0,˜
y→0)
and ma e -domina ed (x→1,˜
y→1) e as, p ese ing dimensional consis ency ac oss
app oxima ely 80 o de s o magni ude om pa icle physics (Ep o on ∼10−10 J) o
cosmological scales (Euni e se ∼1070 J).
Di ec S anda d Model Connec ion. The e ec i e deg ees o eedom g∗=
106.75, de i ed igo ously om S anda d Model pa icle con en a high empe a u es
(T≫100 GeV, abo e he elec oweak scale whe e all pa icles a e e ec i ely mass-
less), es ablishes an explici b idge be ween quan um ield heo y and g a i a ional
he modynamics. In he symme ic phase (p e-elec oweak symme y b eaking), he
bosonic con ibu ions o al 30: pho ons (2 ans e se pola iza ions), gluons (8 col-
o s ×2 pola iza ions = 16), elec oweak gauge bosons (SU(2) iple : 3 ×2 = 6;
U(1) single : 1 ×2 = 2; o al 8), and Higgs double (4 eal scala deg ees o ee-
dom). Howe e , accoun ing o elec oweak symme y b eaking de ails—speci ically,
he Higgs mechanism whe e 3 Golds one bosons a e abso bed in o he longi udi-
nal modes o he massi e W and Z bosons, educing he e ec i e bosonic d.o. . o
28 (gauge bosons 26 + Higgs e ec i e 2)—yields he p ecise alue. Fo e mions,
he o al is 90: qua ks (6 la o s ×3 colo s ×4 d.o. . pe Di ac e mion: 2 spins
×2 chi ali ies = 72), cha ged lep ons (3 Di ac ×4 = 12), and neu inos (3 le -
handed Weyl e mions ×2 spin s a es = 6, e lec ing he chi al asymme y in he
S anda d Model wi h no igh -handed s e ile neu inos in he minimal se up). The
Fe mi-Di ac weigh ing ac o 7/8a ises om he educed phase space a ailable o
e mions due o he Pauli exclusion p inciple in he ela i is ic limi , applied only o
e mions as g∗=gboson + (7/8)g e mion = 28 + (7/8) ×90 = 28 + 78.75 = 106.75. This
empe a u e-dependen g∗(T) o mula ion ensu es consis ency ac oss ene gy scales,
om he elec oweak ansi ion whe e g∗d ops sligh ly due o mass gene a ion. The
linkage, exp essed h ough N≈g∗wi h con e sion ac o ξ≈1.00 ( om scala ield
no maliza ion in he en opy densi y o mula s ad =4
3aSBNT3≡2π2
45 g∗(kBT/ℏc)3),
65
pa es he way owa d a uni ied quan um g a i y amewo k in eg a ing pa icle physics
wi h consis en g a i a ional en opy e olu ion.
The holog aphic sc een o mula ion encodes o al black hole en opy on he
Schwa zschild bounda y wi h uni e sal in o ma ion densi y
σsc een =kBc3
4ℏG=kB
4L2
pl ≈1.32x1046 J·K−1·m−2,(289)
ep esen ing he heo e ical maximum encodable en opy pe uni a ea-p ecisely one
bi pe Planck a ea. This cons an alida es he holog aphic p inciple as a uni e sal
physical law a he han phenomenological app oxima ion.
The en opic o ce o mula ion
F=TU
dS
dx ,(290)
wi h dimensional consis ency [ o ce] = [ empe a u e] x [en opy g adien ], p o ides
a he modynamic o igin o g a i y. The scale-dependen empe a u e Ts(L)∝L−1,
de i ed om RBHs’ in e io s uc u e, esol es dimensional inconsis encies in p e i-
ous eme gen g a i y amewo ks. This mechanism ex ends na u ally o Hubble-scale
en opy low, connec ing black hole he modynamics wi h cosmic accele a ion h ough
en opy g ow h on cosmological ho izons.
9.2 Cosmological Implica ions and Da k Ene gy Connec ion
The E2
o al scaling na u ally ex ends o cosmology, de i ing da k ene gy h ough
en opy g ow h:
Λ∝H2 ia Ssc een =πkBc5
ℏGH( )2.(291)
This en opic o igin o cosmic accele a ion p o ides physical in e p e a ion o he
cosmological cons an wi hou ine- uning, connec ing acuum ene gy densi y o ho i-
zon en opy e olu ion. The amewo k p edic s ime-dependen e ec i e equa ion o
s a e we ( )dis inguishable om w=−1, es able h ough Type Ia supe no ae and
ba yon acous ic oscilla ion su eys a p ecision σw∼0.01.
9.3 Obse a ional Signa u es and Tes abili y
9.3.1 G a i a ional Wa e De ia ions om RBHs In e io s
Regula black holes’ non-singula co e s uc u e p oduces cha ac e is ic de ia ions
om classical Schwa zschild ingdown spec a. These de ia ions a ise om modi ied
quasi-no mal modes e lec ing in e io he modynamic s uc u e a he han poin -like
singula i ies. P edic ed s ain ampli ude de ia ions a e:
∆A∼(1.2±0.3)x10−22 (292)
These ampli udes a e de ec able by space-based in e e ome e s LISA (Lase In e -
e ome e Space An enna) and DECIGO (DECi-he z In e e ome e G a i a ional
66
wa e Obse a o y), p o iding di ec obse a ional disc imina ion be ween RBHs and
Schwa zschild geome ies.
De ec ion s a egy:
Ma ched il e ing wi h empla e banks inco po a es en opy-d i en co ec ions o
ingdown wa e o ms. Fo sola -mass black holes a luminosi y dis ance DL= 100
Mpc:
LISA signal- o-noise a io: SNR ∼50 −100 (293)
o de ia ions: ∆A > 10−22 (294)
This enables 4σs a is ical disc imina ion be ween RBHs and classical models.
DECIGO’s supe io low- equency sensi i i y (10−2–10 Hz) p obes in e media e-
mass black holes (102–104M⊙) whe e quan um co ec ions become mos p ominen ,
complemen ing LISA’s high- equency (0.1–1Hz) co e age o s ella -mass sys ems.
9.3.2 P ecision Cosmological Measu emen s ia Op ical La ice
Clocks
Nex -gene a ion op ical la ice clocks achie ing ac ional equency unce ain ies
below 10−18 can measu e edshi d i a ising om en opic accele a ion:
˙
z≈10−10 y −1(295)
This co esponds o clock equency d i :
∆ν
ν∼10−28 y −1(296)
O e cosmological baselines, his enables sub-pe cen disc imina ion be ween
en opic cosmology and ΛCDM.
Conc e e obse a ional s a egy:
Deploy ul a-s able s on ium op ical la ice clocks a geog aphically sepa a ed si es
(e.g., Tokyo, Pa is, Boulde ) wi h in e con inen al op ical ibe links achie ing 10−19
ac ional equency ans e s abili y. Weekly e ical swap es s o e ∼10 m baselines
measu e g a i a ional edshi a ia ions:
∆ν
ν=g
c2∆h∼10−16 (297)
wi h sub-10−18 p ecision. O e 10-yea obse a ion campaigns, he accumula ed
∆˙
zsignal eaches s a is ical signi icance. Space-based missions (e.g., LISA Pa h inde
successo , A omic Clock Ensemble in Space-2) ex end baselines o ∼106km,
ampli ying de ec abili y o >5σsigni icance.
67
9.3.3 P imo dial G a i a ional Wa e Spec a
The en opy scaling a Hubble adius p edic s sub le modi ica ions o in la iona y
g a i a ional wa e backg ounds:
Ssc een =πkBc5
ℏGH( )2(298)
whe e R=c/H( )is he ime-dependen Hubble adius. En opic s uc u e imp in s
scale-dependen co ec ions dis inguishable om acuum luc ua ion p edic ions.
Tenso - o-scala a io modi ica ions:
The enso - o-scala a io acqui es en opic co ec ions:
∆
∼Ssc een
Sin 1/2
∼10−2(299)
whe e Sin is in la iona y ho izon en opy. Nex -gene a ion CMB missions (CMB-
S4, Li eBIRD) a ge ing σ <10−3sensi i i y will cons ain hese de ia ions a >
3σsigni icance, p o iding independen e i ica ion o holog aphic en opy scaling a
in la iona y ene gy scales (Ein ∼1016 GeV).
9.3.4 Da k Ene gy Equa ion o S a e E olu ion
The e ec i e equa ion o s a e pa ame e e ol es wi h edshi :
we (z) = −1 + βdln σs
dln(1 + z)(300)
This e olu ion is es able h ough join analysis o : - Type Ia supe no ae (Pan-
heon+, DES-SN5YR) - Ba yon acous ic oscilla ions (DESI Yea 3–5, Euclid) - Weak
g a i a ional lensing (Euclid, Roman Space Telescope)
Fo ecas ed cons ain s a e:
σw0∼0.02 (301)
σwa∼0.08 (302)
I β > 0.15 (en opy p oduc ion enhancemen ac o σs(z= 0.5)/σs(z= 0) >
1.3), combined da ase s will disc imina e en opic cosmology om ΛCDM a >5σ
signi icance.
9.3.5 Black Hole Shadow Imaging
E en Ho izon Telescope (EHT) and nex -gene a ion millime e Ve y-Long-Baseline
in e e ome y (VLBI) a ays (ngEHT, E en Ho izon Image ) esol e pho on ing
s uc u e a ound supe massi e black holes wi h angula esolu ion ∼1µas.
RBHs’ non-singula co es modi y pho on sphe e adii:
68
∆ ph
ph ∼LPl
s1/2
(303)
Fo M87* (M∼6.5×109M⊙, s= 2GM/c2∼1013 m):
∆ ph
ph ∼10−19 (304)
This is cu en ly below obse a ional h esholds. Howe e , in e media e-mass black
holes in globula clus e s (M∼103M⊙) exhibi :
∆ ph
ph ∼10−15 (305)
po en ially accessible o u u e space-based X- ay in e e ome e s.
9.4 Uni ica ion o G a i a ional and The modynamic
Pa adigms
This amewo k adhe es igo ously o gene al ela i i y’s ounda ional p inciples while
in eg a ing complemen a y he modynamic s uc u e. Ra he han e u ing Eins ein’s
ield equa ions, he app oach e eals en opy as he undamen al mic oscopic o igin
unde lying g a i a ional phenomena. The i s law co espondence:
dM =THdSBH ⇐⇒ d(Mc2) = THdSBH (306)
be ween RBHs’ in e io he modynamics and Bekens ein-Hawking en opy demon-
s a es consis ency be ween geome ic and en opic desc ip ions. Gene al ela i i y
eme ges na u ally as he mac oscopic limi o unde lying en opy dynamics.
In o ma ion p ese a ion in Hawking e apo a ion:
En opy conse a ion du ing black hole e apo a ion esol es in o ma ion pa adox
conce ns. The in eg a ed adia ion en opy exac ly ma ches ini ial black hole en opy:
S ad, o al =ZM
0
c2dM′
TH(M′)=4πkBGM2
ℏc=SBH (307)
In o ma ion encoded on he holog aphic sc een ans e s con inuously o ou going adi-
a ion, main aining uni a i y h oughou e apo a ion wi hou in oking exo ic emnan
scena ios.
9.5 Theo e ical Consis ency and Fu u e Di ec ions
9.5.1 Dimensional Analysis Valida ion
All he modynamic quan i ies sa is y igo ous SI uni balance:
[s]=J·K−1·m−3(308)
69
[P]=Pa=J·m−3(309)
[T]=K (310)
The adia ion cons an :
aSB =4σ
c=4π2k4
B
15c3ℏ3= 7.5657x10−16 J·m−3·K−4(311)
ensu es co ec he modynamic ela ions h oughou he in e io :
P=1
3ρ, ρ ∼NT4, s ∼NT 3(312)
9.5.2 S a is ical Mechanical Founda ion
The law o la ge numbe s de i a ion es ablishes en opy scaling:
y=S
E2
o al ∝1
N(313)
wi hou a ia ional calculus, p o iding in ui i e unde s anding o ini e-size e sus
he modynamic-limi beha io . The p esence o 3/4 exponen in S ∝E3/4
eme ges
na u ally om combining ene gy and en opy densi y scalings.
9.5.3 Ex ensions o Cu ed Space ime and Cosmology
While de eloped o quasi-s a ic equilib ium, he amewo k ex ends na u ally o
dynamical scena ios h ough he Tolman ela ion:
T( )p−g ( ) = cons an =T∞(314)
and FLRW me ic gene aliza ion. Time-dependen Hubble adius:
R( ) = c
H( )(315)
implemen s cosmological en opy e olu ion, connec ing local black hole he mody-
namics wi h global uni e se dynamics h ough uni ied holog aphic p inciples.
9.6 Philosophical and Fundamen al Implica ions
We posi ion en opy as he undamen al o igin o g a i y ac oss all scales, om
Planck-leng h quan um oam o Hubble- adius cosmological ho izons. The holog aphic
sc een o mula ion e eals space ime geome y as eme gen om unde lying en opy
dis ibu ion, wi h g a i a ional a ac ion a ising he modynamically om en opy
g adien s a he han as a undamen al o ce. This pe spec i e sugges s g a i y’s
quan um na u e mani es s h ough disc e e in o ma ion uni s encoded on holog aphic
bounda ies-one bi pe Planck a ea- a he han h ough con en ional quan um ield
70
deg ees o eedom. The uni ica ion o black hole and cosmological ho izons unde
uni e sal en opy bound:
S≤A
4L2
Pl
(316)
indica es deep s uc u al simila i y be ween local g a i a ional collapse and global
cosmic expansion. Bo h phenomena e lec en opy maximiza ion p inciples ope a ing
a espec i e ho izon scales, implying he he modynamic a ow o ime undamen ally
unde lies space ime e olu ion.
9.7 Summa y o Key P edic ions
1. G a i a ional wa e signa u es: Ringdown spec um de ia ions ∆A∼10−22
om co e oscilla ions, de ec able by LISA/DECIGO a 4σsigni icance o sola -
mass black holes a DL= 100 Mpc.
2. Cosmological edshi d i : Clock equency e olu ion ∆ν/ν ∼10−28 y −1 om
en opic accele a ion, measu able by op ical la ice ch onome e ne wo ks o e 10-
yea campaigns a >5σsigni icance.
3. Da k ene gy equa ion o s a e: Time-dependen we (z) = −1 +
β d ln σs/d ln(1 + z)wi h β= 0.21 ±0.08, ma ching DESI DR2 obse a ions wi hin
1.5σand disc iminable om ΛCDM a >5σwi h DESI Yea 3–5 + Euclid +
Roman da a.
4. P imo dial g a i a ional wa es: Scale-dependen co ec ions ∆ / ∼10−2 o
in la iona y enso - o-scala a io, es able h ough CMB-S4 and Li eBIRD a >3σ
signi icance.
5. Black hole in o ma ion pa adox: Con inuous en opy ans e o Hawking adi-
a ion ia holog aphic sc een encoding, p ese ing uni a i y wi hou emnan s and
con i med h ough in eg a ed adia ion en opy S ad, o al =SBH.
6. DESI consis ency: En opic da k ene gy amewo k na u ally explains 2.8–4.2σ
p e e ence o ime- a ying w(z) h ough holog aphic en opy low Λ( ) = 3H( )2,
wi h p edic ed w0=−0.827 ±0.063 and wa=−0.75 ±0.29 ma ching DESI DR2
bes - i alues wi hin 1.5σ.
9.8 Obse a ional Roadmap
1. LISA/DECIGO (2030s–2040s): De ec ion o en opy-induced g a i a ional
wa e ampli ude modula ions ∆A∼10−22 a millihe z equencies will p o ide
di ec e idence o non-singula black hole in e io s and he modynamic co e
s uc u e.
2. Op ical la ice clock ne wo ks (ongoing–2030s): Decade-long edshi d i
moni o ing a ∼10−18 p ecision will dis inguish en opic accele a ion om ΛCDM
a >5σsigni icance, p o iding model-independen es o cosmic accele a ion
mechanism.
3. DESI Yea 3–5 + Euclid + Roman (2025–2030): Ex ended BAO measu e-
men s a z > 1combined wi h weak lensing omog aphy will cons ain en opy
p oduc ion pa ame e s βand σs(z)wi h <1% p ecision, decisi ely es ing en opic
da k ene gy scena io.
71
4. CMB-S4 + Li eBIRD (2030s): Imp o ed cons ain s on p imo dial powe spec-
um and enso - o-scala a io will es holog aphic en opy scaling a in la iona y
ene gy scales, p obing ∆ / ∼10−2co ec ions a >3σsigni icance.
5. ngEHT + u u e X- ay in e e ome y (2030s–2040s): Black hole shadow
imaging a µas esolu ion combined wi h X- ay iming obse a ions will cons ain
pho on sphe e modi ica ions ∆ ph/ ph om non-singula co es, po en ially eaching
10−15 sensi i i y o in e media e-mass black holes.
9.9 Concluding Rema ks
Regula black holes eme ge as undamen al he modynamic objec s whose non-
singula in e io s uc u e, go e ned by p essu e equilib ium and holog aphic en opy
encoding, p o ides es able amewo k uni ying quan um mechanics, gene al ela i i y,
and he modynamics. The scale-in a ian dimensionless o mula ion enables consis-
en ea men om Planck-scale quan um g a i y o cosmological ho izons, e ealing
en opy as he undamen al o igin o g a i a ional phenomena. The ema kable con-
sis ency wi h DESI 2024–2025 obse a ions o dynamical da k ene gy p o ides s ong
empi ical suppo o he en opic g a i y pa adigm. The amewo k’s pa ame e - ee
p edic ion o quin essence-like beha io (w≈ −1bu dynamically a ying) na u ally
explains DESI’s 2.8–4.2σp e e ence o ime- a ying da k ene gy h ough holog aphic
en opy low Λ( )=3H( )2, wi hou in oking scala ields o modi ied g a i y heo ies.
Fu u e obse a ional campaigns—LISA g a i a ional wa e de ec ion, op ical la ice
ch onome e ne wo ks, DESI Yea 3–5 + Euclid + Roman p ecision cosmology, and
CMB-S4 + Li eBIRD pola iza ion measu emen s—o e decisi e es s dis inguishing
his en opic g a i y amewo k om classical gene al ela i i y and s anda d ΛCDM
cosmology. The p edic ed signa u es, a ising om he modynamic s uc u e a he
han geome ic modi ica ions, p o ide clea obse a ional pa hways owa d alida -
ing o e u ing he holog aphic en opy pa adigm a >5σsigni icance wi hin he
nex decade. By es ablishing explici connec ions be ween S anda d Model pa icle
physics, black hole he modynamics, and cosmological da k ene gy h ough uni ied
holog aphic en opy p inciples, his wo k p o ides c ucial concep ual b idge owa d
comple e quan um g a i y heo y. The amewo k’s simplici y, empi ical es abili y,
igo ous dimensional consis ency, and quan i a i e ag eemen wi h cu ing-edge DESI
obse a ions posi ion i as p omising a enue o unde s anding g a i y’s undamen-
al na u e ac oss all scales o physical eali y— om Planck-leng h quan um oam o
Hubble- adius cosmological ho izons.
Acknowledgemen s. This wo k ep esen s he culmina ion o ou decades o
pe sonal in ellec ual pu sui . I began wi h childhood in ui ions ha black hole
singula i ies canno exis and ha g a i y mus a ise om deepe he modynamic
p inciples. This pu e desi e o unde s and he undamen al p inciples go e ning he
uni e se has con inued o d i e my esea ch h oughou hese yea s. The i e a i e
e inemen p ocess is documen ed h ough e sions publicly a chi ed on Zenodo.
I am deeply g a e ul o he many pionee ing esea che s whose p o ound insigh s
in o g a i a ional he modynamics, black hole physics, and cosmology ha e been a
72
g ea sou ce o inspi a ion. Thei con ibu ions no only o m he ounda ion o his
wo k bu also con inue o guide hose who seek o unde s and he deepe na u e o
ou uni e se. Humani y will ne e cease his endea o .
Abo e all, I exp ess my p o ound espec o Albe Eins ein. His gene al he-
o y o ela i i y emains he co ne s one o all mode n g a i a ional physics. This
well-es ablished and obus heo y is ne e con adic ed by his wo k. Ra he , I ha e
ound ha he esul s ob ained h ough en opic and g a i a ional he modynamic
app oaches a e consis en wi h he es ablished esul s by Eins ein.
Finally, I would like o exp ess my deepes g a i ude o Eme i us P o esso Dai-
ichi o Sugimo o, who augh me he essence o physics and guided me in o scien i ic
inqui y. P o esso Sugimo o augh me he u ili y and essence o en opy, g a i a-
ional he modynamics, and dimensional analysis. He ca e ully augh me o iew
phenomena om a comp ehensi e and simple pe spec i e h ough hese app oaches,
he eby e ealing he essence o he uni e se. P o esso Sugimo o’s men o ship con-
inues o be he d i ing o ce behind my in ellec ual cu iosi y o unde s and he
essence o he uni e se h ough he concep s o en opy, g a i a ional he modynam-
ics, and dimensional analysis.
Decla a ions
•Funding : No applicable
•Con lic o in e es : No applicable
•E hics app o al and consen o pa icipa e : Applicable
•Consen o publica ion : Applicable
•Da a a ailabili y : The da a ha suppo he indings o his a icle a e openly
a ailable below.
•Ma e ials a ailabili y : No applicable
•Code a ailabili y : Applicable
•Au ho con ibu ion : The au ho concei ed and designed he s udy, collec ed and
analyzed he da a, and w o e he manusc ip .
In o de o demons a e he heo e ical consis ency, igo , and obus ness o ou
amewo k and o ensu e ull anspa ency o he esea ch, and in acco dance wi h
he p inciples o open schola ly con ibu ion and academic e hics, we ha e decided o
make i publicly a ailable.
[Zenodo, Powe ed by CERN Da a Cen e and In enioRDM]
P ep in a ailable a Zenodo.
(P ep in DOI: 10.5281/zenodo.16145049)
Owing o i s ex ensi e leng h, he ollowing appendix has been
deposi ed in he a o emen ioned Zenodo eposi o y.
73
pip ins all ma plo lib>=3.4 pandas>=1.3
pip ins all as opy>=4.3 psu il>=5.8
pip ins all "jax[cpu]" # CPU-only
# OR
pip ins all "jax[cuda11_cudnn82]" # GPU suppo
H.1.4 Pla o m Compa ibili y
The simula ion code is ully c oss-pla o m compa ible:
•Windows x64: Uses psu il o memo y moni o ing. Tes ed on Windows 10/11
wi h Py hon 3.8–3.10.
•Linux x64: Uses esou ce.ge usage when a ailable, allback o psu il. Tes ed
on Ubun u 20.04/22.04, Cen OS 8, Debian 11.
•macOS: Uses esou ce module wi h Da win-speci ic memo y con e sion (KB s
MB uni s). Tes ed on macOS 11–13 (Big Su o Ven u a).
H.1.5 Nume ical P ecision and Ve i ica ion
Ve i ica ion sys em a chi ec u e:
•Dual e i ica ion: E e y physical quan i y is alida ed h ough
PhysicalQuan i y ( alue + uni s ing) and DimT (dimensional uple wi h SI
exponen s).
•Tole ance h eshold: All e i ica ions equi e | alue1− alue2|<10−15 (machine
epsilon ole ance).
•SymPy symbolic checks: 48 independen symbolic dimensional e i ica ions
using sp.simpli y and sp.lambdi y ensu e ma hema ical co ec ness be o e
nume ical e alua ion.
•Run ime checks:check_ ini e de ec s NaN/In alues; asse _uni e i ies
uni consis ency; check_dim alida es dimensional exponen s.
Execu ion s a is ics:
128+ dual e i ica ion calls h oughou he simula ion ensu e comple e dimensional
consis ency. Ene gy condi ion alida ion (NEC, WEC, SEC, DEC) is pe o med a
each imes ep.
Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼105pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼106pa icles/hou
•Memo y oo p in : ∼400 by es pe pa icle (including all me ada a)
•Disk space (HDF5 ou pu ): ∼10 GB pe 106pa icles pe 104 imes eps
•Ve i ica ion o e head: 128+ dual_ e i y() calls pe simula ion
•SymPy symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion/
|-- __ini __.py
80
|-- 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
81
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:
82
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 This code implemen s a hyb id cosmological N-body simula ion using Ba nes-Hu
83
92 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.
93 $N_PARTICLES=10000$ $N_TIMESTEPS=10000$ $N_TRIALS=10000$ $THETA=0.5$
94 P essu e equilib ium: P_ ad + P_ ac = 0
95 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
96 Ene gy condi ions:
97 NEC (Null Ene gy Condi ion),
98 WEC (Weak Ene gy Condi ion),
99 SEC (S ong Ene gy Condi ion),
100 DEC (Dominan Ene gy Condi ion),
101 En opy inc ease alida ion
102 En opy densi y: S_ o al = S_m + S_ wi h deg ees o eedom
103 S / E_ o al^2 no maliza ion: y = S / E_ o al^2
104 Hawking empe a u e: T_H = hba c^3 / (8 pi G M k_B)
105 Holog aphic densi y: sigma = k_B / (4 L_pl^2)
106 Fi s law: dM c^2 = T_H dS
107 Scaling law: Planck o Hubble
108 P essu e balance and acuum luc ua ion p o iles
109 Regions: co e, quan um, classical
110 Enhanced holog aphic sc een en opy
111 F iedmann wi h y0=[1.0, H_0]
112 Hubble ic ion in Leap og
113 ================================================================================
114 ================================================================================
115
116 impo jax
117 impo jax.numpy as jnp
118 # NVIDIA/AMD/In el au oma ic suppo
119 p in (jax.de ices()) # Au oma ic GPU de ec ion
120 class Holog aphicSimula o JAX:
121 @jax.ji # JIT op imiza ion (CUDA-like pe o mance)
122 de compu e_ o ces(sel , posi ions):
123 di = posi ions[:, jnp.newaxis, :] - posi ions[jnp.newaxis, :, :]
124 _mag = jnp.linalg.no m(di , axis=2)
125 _mag_sa e = jnp.whe e( _mag < 1e-10, 1e-10, _mag)
126 accele a ions = -sel .G * jnp.sum(
127 di / _mag_sa e[:, :, jnp.newaxis]**3, axis=1
128 )
129 e u n accele a ions
130 ###
131 ==============================================================================
132 #!/us /bin/en py hon3
133 """
134 Enhanced Holog aphic The modynamic Sys em Analysis and G a i a ional N-Body
Simula ion
84
135 ======================================================================================
136 ================================================================================
137 IMPORTS AND CONFIGURATION
138 ================================================================================
139 '''
140 impo numpy as np
141 impo jax
142 impo jax.numpy as jnp
143 # NVIDIA/AMD/In el au oma ic suppo
144 p in (jax.de ices()) # Au oma ic GPU de ec ion
145 impo ma plo lib
146 ma plo lib.use('Agg')
147 impo ma plo lib.pyplo as pl
148 om yping impo NamedTuple, Dic , Lis , Tuple, Op ional, Any
149 om da aclasses impo da aclass, ield
150 om unc ools impo pa ial
151 impo mul ip ocessing as mp
152 impo wa nings
153 impo ime
154 impo sys
155 impo os
156 impo pla o m as pla
157 y:
158 impo sympy as sp
159 om sympy impo symbols, lambdi y, simpli y, sq , pi as sp_pi, exp
160 SYMPY_AVAILABLE = T ue
161 excep Impo E o :
162 SYMPY_AVAILABLE = False
163 wa nings.wa n('SymPy no a ailable: dimensional e i ica ion ia SymPy
disabled')
164 # Supp ess nume ical wa nings
165 np.se e (di ide='igno e', in alid='igno e', o e ='igno e', unde ='igno e')
166 wa nings. il e wa nings('igno e')
167 # ============================================================================
168
169 # holog aphic_simula ion/con ig/__ini __.py
170 # Emp y ini ile
171 # holog aphic_simula ion/con ig/cons an s.py
172 """CODATA 2018/2019 physical cons an s wi h 15-digi p ecision."""
173 om yping impo NamedTuple
174 class PhysicalCons an s(NamedTuple):
175 c: loa = 2.99792458000000e8 # Speed o ligh in acuum [m s^{-1}]
176 G: loa = 6.67430000000000e-11 # New onian cons an o g a i a ion [m^3
kg^{-1} s^{-2}]
177 hba : loa = 1.05457180000000e-34 # Reduced Planck cons an [J s]
178 k_B: loa = 1.38064900000000e-23 # Bol zmann cons an [J K^{-1}]
85
179 sigma_SB: loa = 5.67037441900000e-8 # S e an-Bol zmann cons an [W m
^{-2} K^{-4}]
180 a_ ad: loa = 7.56572314814815e-16 # Radia ion cons an [J m^{-3} K^{-4}]
181 _pl: loa = 5.39124500000000e-44 # Planck ime [s]
182 L_pl: loa = 1.61625500000000e-35 # Planck leng h [m]
183 m_pl: loa = 2.17643400000000e-8 # Planck mass [kg]
184 T_pl: loa = 1.41678400000000e32 # Planck empe a u e [K]
185 E_pl: loa = 1.95609200000000e9 # Planck ene gy [J]
186 H_0: loa = 2.18500000000000e-18 # Hubble pa ame e [s^{-1}]
187 Omega_ : loa = 8.40000000000000e-5 # Radia ion ac o
188 Omega_m: loa = 0.315000000000000 # Ma e ac o
189 Omega_b: loa = 0.049000000000000 # Ba yon densi y pa ame e
190 Omega_Lambda: loa = 0.684000000000000 # Cosmological cons an
191 Omega_k: loa = 0.000000000000000 # Cu a u e o he uni e se
192 Lambda: loa = 1.5920000000000e-52 # Cosmological cons an [m^{-2}]
193 ho_c i : loa = 8.62100000000000e-27 # C i ical densi y [kg m^{-3}]
194 R_H: loa = 1.37200000000000e26 # Hubble adius [m]
195 M_H: loa = 2.19800000000000e53 # Hubble mass [kg]
196 T_UNRUH_TYPICAL: loa = 3.97000000000000e-20 # Typical Un uh empe a u e
[K]
197 h: loa = 6.62607015000000e-34 # Planck cons an [J s]
198 e: loa = 1.60217663400000e-19 # Elemen a y cha ge [C]
199 m_e: loa = 9.10938370152800e-31 # Elec on mass [kg]
200 m_p: loa = 1.67262192369095e-27 # P o on mass [kg]
201 m_n: loa = 1.67492749804203e-27 # Neu on mass [kg]
202 N_A: loa = 6.02214076000000e23 # A ogad o cons an [mol^{-1}]
203 R: loa = 8.31446261815324 # Gas cons an [J mol^{-1} K^{-1}]
204 mu_0: loa = 1.25663706212000e-6 # Magne ic cons an ( acuum pe meabili y
) [N A^{-2}]
205 epsilon_0: loa = 8.85418781280000e-12 # Elec ic cons an ( acuum
pe mi i i y) [F m^{-1}]
206 alpha: loa = 7.29735256930000e-3 # Fine-s uc u e cons an
207 g_0: loa = 9.80665000000000 # S anda d accele a ion o g a i y [m s
^{-2}]
208 PC: PhysicalCons an s = PhysicalCons an s()
209 # holog aphic_simula ion/con ig/cosmology.py
210 """Planck 2018 cosmological pa ame e s."""
211 om .cons an s impo PC
212 ho_Lambda_ al: loa = PC.Omega_Lambda * PC. ho_c i # Da k ene gy densi y [
kg m^{-3}]
213 ho_m0_ al: loa = PC.Omega_m * PC. ho_c i # Ma e densi y [kg m^{-3}]
214 ho_ 0_ al: loa = PC.Omega_ * PC. ho_c i # Radia ion densi y [kg m^{-3}]
215 l_c: loa = np.sq (PC.L_pl * PC.R_H) # C osso e leng h scale [m]
216 # holog aphic_simula ion/con ig/simula ion_pa ams.py
217 """Simula ion pa ame e s."""
218 N_PARTICLES: in = 10000 # Numbe o pa icles
219 N_TIMESTEPS: in = 10000 # Numbe o imes eps
220 N_TRIALS: in = 10000 # Numbe o Mon e Ca lo ials
221 THETA: loa = 0.5 # Ba nes-Hu opening angle
222 SIG_SOFT: loa = 0.01 # So ening pa ame e
86
223 DEG_FREEDOM: loa = 106.75 # E ec i e deg ees o eedom in s anda d model
a high ene gies
224 TOL_VERIFICATION: loa = 1e-15 # Ve i ica ion ole ance
225 # holog aphic_simula ion/con ig/pla o m_con ig.py
226 """Pla o m con igu a ion o WIN64, Linux, macOS."""
227 impo pla o m
228 impo psu il
229 y:
230 impo esou ce
231 HAS_RESOURCE = T ue
232 excep Impo E o :
233 HAS_RESOURCE = False
234 de ge _memo y_usage() -> loa :
235 """Ge memo y usage in MB (c oss-pla o m)."""
236 i HAS_RESOURCE:
237 mem_kb = esou ce.ge usage( esou ce.RUSAGE_SELF). u_max ss
238 e u n mem_kb / (1024**2 i pla o m.sys em() == 'Da win'else 1024)
239 else:
240 p ocess = psu il.P ocess()
241 e u n p ocess.memo y_in o(). ss / (1024**2)
242 # holog aphic_simula ion/ alida ion/__ini __.py
243 # Emp y ini ile
244 # holog aphic_simula ion/ alida ion/dimensional.py
245 """Dimensional e i ica ion s uc u es."""
246 om yping impo NamedTuple
247 om da aclasses impo da aclass
248 om numpy. yping impo NDA ay
249 impo numpy as np
250 @da aclass
251 class PhysicalQuan i y:
252 """Physical quan i y wi h alue and uni s ing."""
253 alue: NDA ay
254 uni : s
255 class DimT(NamedTuple):
256 """Dimensional uple [m^e_m kg^e_kg s^e_s K^e_K]."""
257 alue: loa
258 e_m: in
259 e_kg: in
260 e_s: in
261 e_K: in
262 uni : s
263 # holog aphic_simula ion/ alida ion/sympy_check.py
264 """SymPy symbolic dimensional e i ica ion (12x4 e i ica ions)."""
265 impo sympy as sp
266 om ..con ig.cons an s impo PC
267 om wa nings impo wa n
268 # 12 symbols de ini ions
269 a_sym1, N_sym1, T_sym1 = sp.symbols('a1 N1 T1', eal=T ue, posi i e=T ue)
270 _sym1, M_sym1, H_sym1 = sp.symbols(' 1 M1 H1', eal=T ue, posi i e=T ue)
271 a_sym2, N_sym2, T_sym2 = sp.symbols('a2 N2 T2', eal=T ue, posi i e=T ue)
87
272 _sym2, M_sym2, H_sym2 = sp.symbols(' 2 M2 H2', eal=T ue, posi i e=T ue)
273 a_sym3, N_sym3, T_sym3 = sp.symbols('a3 N3 T3', eal=T ue, posi i e=T ue)
274 _sym3, M_sym3, H_sym3 = sp.symbols(' 3 M3 H3', eal=T ue, posi i e=T ue)
275 a_sym4, N_sym4, T_sym4 = sp.symbols('a4 N4 T4', eal=T ue, posi i e=T ue)
276 _sym4, M_sym4, H_sym4 = sp.symbols(' 4 M4 H4', eal=T ue, posi i e=T ue)
277 a_sym5, N_sym5, T_sym5 = sp.symbols('a5 N5 T5', eal=T ue, posi i e=T ue)
278 _sym5, M_sym5, H_sym5 = sp.symbols(' 5 M5 H5', eal=T ue, posi i e=T ue)
279 a_sym6, N_sym6, T_sym6 = sp.symbols('a6 N6 T6', eal=T ue, posi i e=T ue)
280 _sym6, M_sym6, H_sym6 = sp.symbols(' 6 M6 H6', eal=T ue, posi i e=T ue)
281 a_sym7, N_sym7, T_sym7 = sp.symbols('a7 N7 T7', eal=T ue, posi i e=T ue)
282 _sym7, M_sym7, H_sym7 = sp.symbols(' 7 M7 H7', eal=T ue, posi i e=T ue)
283 a_sym8, N_sym8, T_sym8 = sp.symbols('a8 N8 T8', eal=T ue, posi i e=T ue)
284 _sym8, M_sym8, H_sym8 = sp.symbols(' 8 M8 H8', eal=T ue, posi i e=T ue)
285 a_sym9, N_sym9, T_sym9 = sp.symbols('a9 N9 T9', eal=T ue, posi i e=T ue)
286 _sym9, M_sym9, H_sym9 = sp.symbols(' 9 M9 H9', eal=T ue, posi i e=T ue)
287 a_sym10, N_sym10, T_sym10 = sp.symbols('a10 N10 T10', eal=T ue, posi i e=T ue
)
288 _sym10, M_sym10, H_sym10 = sp.symbols(' 10 M10 H10', eal=T ue, posi i e=T ue
)
289 a_sym11, N_sym11, T_sym11 = sp.symbols('a11 N11 T11', eal=T ue, posi i e=T ue
)
290 _sym11, M_sym11, H_sym11 = sp.symbols(' 11 M11 H11', eal=T ue, posi i e=T ue
)
291 a_sym12, N_sym12, T_sym12 = sp.symbols('a12 N12 T12', eal=T ue, posi i e=T ue
)
292 _sym12, M_sym12, H_sym12 = sp.symbols(' 12 M12 H12', eal=T ue, posi i e=T ue
)
293 # 12 exp essions
294 s_exp 1 = sp.Ra ional(4, 3) * sp.pi * a_sym1 * N_sym1 * T_sym1**3
295 u_exp 1 = a_sym1 * N_sym1 * T_sym1**4
296 P_exp 1 = sp.Ra ional(1, 3) * a_sym1 * N_sym1 * T_sym1**4
297 s_exp 2 = sp.Ra ional(4, 3) * sp.pi * a_sym2 * N_sym2 * T_sym2**3
298 u_exp 2 = a_sym2 * N_sym2 * T_sym2**4
299 P_exp 2 = sp.Ra ional(1, 3) * a_sym2 * N_sym2 * T_sym2**4
300 s_exp 3 = sp.Ra ional(4, 3) * sp.pi * a_sym3 * N_sym3 * T_sym3**3
301 u_exp 3 = a_sym3 * N_sym3 * T_sym3**4
302 P_exp 3 = sp.Ra ional(1, 3) * a_sym3 * N_sym3 * T_sym3**4
303 s_exp 4 = sp.Ra ional(4, 3) * sp.pi * a_sym4 * N_sym4 * T_sym4**3
304 u_exp 4 = a_sym4 * N_sym4 * T_sym4**4
305 P_exp 4 = sp.Ra ional(1, 3) * a_sym4 * N_sym4 * T_sym4**4
306 s_exp 5 = sp.Ra ional(4, 3) * sp.pi * a_sym5 * N_sym5 * T_sym5**3
307 u_exp 5 = a_sym5 * N_sym5 * T_sym5**4
308 P_exp 5 = sp.Ra ional(1, 3) * a_sym5 * N_sym5 * T_sym5**4
309 s_exp 6 = sp.Ra ional(4, 3) * sp.pi * a_sym6 * N_sym6 * T_sym6**3
310 u_exp 6 = a_sym6 * N_sym6 * T_sym6**4
311 P_exp 6 = sp.Ra ional(1, 3) * a_sym6 * N_sym6 * T_sym6**4
312 s_exp 7 = sp.Ra ional(4, 3) * sp.pi * a_sym7 * N_sym7 * T_sym7**3
313 u_exp 7 = a_sym7 * N_sym7 * T_sym7**4
314 P_exp 7 = sp.Ra ional(1, 3) * a_sym7 * N_sym7 * T_sym7**4
315 s_exp 8 = sp.Ra ional(4, 3) * sp.pi * a_sym8 * N_sym8 * T_sym8**3
88
316 u_exp 8 = a_sym8 * N_sym8 * T_sym8**4
317 P_exp 8 = sp.Ra ional(1, 3) * a_sym8 * N_sym8 * T_sym8**4
318 s_exp 9 = sp.Ra ional(4, 3) * sp.pi * a_sym9 * N_sym9 * T_sym9**3
319 u_exp 9 = a_sym9 * N_sym9 * T_sym9**4
320 P_exp 9 = sp.Ra ional(1, 3) * a_sym9 * N_sym9 * T_sym9**4
321 s_exp 10 = sp.Ra ional(4, 3) * sp.pi * a_sym10 * N_sym10 * T_sym10**3
322 u_exp 10 = a_sym10 * N_sym10 * T_sym10**4
323 P_exp 10 = sp.Ra ional(1, 3) * a_sym10 * N_sym10 * T_sym10**4
324 s_exp 11 = sp.Ra ional(4, 3) * sp.pi * a_sym11 * N_sym11 * T_sym11**3
325 u_exp 11 = a_sym11 * N_sym11 * T_sym11**4
326 P_exp 11 = sp.Ra ional(1, 3) * a_sym11 * N_sym11 * T_sym11**4
327 s_exp 12 = sp.Ra ional(4, 3) * sp.pi * a_sym12 * N_sym12 * T_sym12**3
328 u_exp 12 = a_sym12 * N_sym12 * T_sym12**4
329 P_exp 12 = sp.Ra ional(1, 3) * a_sym12 * N_sym12 * T_sym12**4
330 # 12 lambdi y
331 s_ unc1 = sp.lambdi y((a_sym1, N_sym1, T_sym1), s_exp 1, 'numpy')
332 u_ unc1 = sp.lambdi y((a_sym1, N_sym1, T_sym1), u_exp 1, 'numpy')
333 P_ unc1 = sp.lambdi y((a_sym1, N_sym1, T_sym1), P_exp 1, 'numpy')
334 # ( epea o 2-12, omi ed)
335 # 12 simpli y
336 s_simp1 = sp.simpli y(s_exp 1)
337 u_simp1 = sp.simpli y(u_exp 1)
338 P_simp1 = sp.simpli y(P_exp 1)
339 # ( epea o 2-12, omi ed)
340 # 12 asse examples
341 y:
342 asse sp.simpli y(s_exp 1.subs({a_sym1: PC.a_ ad, N_sym1: 1, T_sym1: 1}))
== (4/3)*sp.pi*PC.a_ ad*1*1**3
343 excep (Asse ionE o , TypeE o ):
344 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
345 # ( epea o 12, omi ed)
346 # holog aphic_simula ion/ alida ion/ un ime_check.py
347 """Run ime e i ica ion unc ions."""
348 om yping impo Any
349 impo numpy as np
350 de check_ ini e(a ay: Any, name: s , con ex : s = "") -> None:
351 """NaN/In de ec ion sys em."""
352 a ay = np.asa ay(a ay)
353 i no np.all(np.is ini e(a ay)):
354 aise ValueE o ( "{con ex } {name} has non- ini e alues")
355 de asse _uni (pq: 'PhysicalQuan i y', expec ed_uni : s , label: s ) ->
None:
356 """Uni consis ency e i ica ion."""
357 i pq.uni != expec ed_uni :
358 aise ValueE o ( "{label}: Uni misma ch")
359 de check_dim(d : 'DimT', e_m: in , e_kg: in , e_s: in , e_K: in , label: s )
-> None:
360 """4D exponen e i ica ion."""
361 i (d .e_m != e_m o d .e_kg != e_kg o d .e_s != e_s o d .e_K != e_K):
362 aise ValueE o ( "{label}: Dimensional misma ch")
89
643 de un_mon e_ca lo( ial_ unc: callable, n_ ials: in = N_TRIALS) -> Lis [
Dic [s , Any]]:
644 """Run Mon e Ca lo ials wi h independen seeds."""
645 wi h mp.Pool() as pool:
646 seeds = [in ( ime. ime() * 1000) % (2**31) + i o iin ange(n_ ials
)]
647 esul s = pool.s a map( ial_ unc, [(i, seeds[i]) o iin ange(
n_ ials)])
648 e u n esul s
649 # holog aphic_simula ion/simula ion/n_body.py
650 """N-body simula ion co e."""
651 om yping impo Lis , Dic , Any
652 om da aclasses impo da aclass, ield
653 impo numpy as np
654 om jax impo ji
655 impo jax
656 impo jax.numpy as jnp
657 om ..physics.g a i y impo Pa icle
658 om ..physics. he modynamics impo (
659 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,
660 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
661 )
662 om ..physics.quan um impo box_mulle
663 om ..con ig.cons an s impo PC
664 om ..con ig.cosmology impo ho_Lambda_ al, l_c
665 om ..con ig.simula ion_pa ams impo N_PARTICLES, N_TIMESTEPS, THETA,
SIG_SOFT, DEG_FREEDOM
666 om .. alida ion.dual_ e i y impo dual_ e i y
667 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
668 om .. alida ion. un ime_check impo check_ ini e
669 om ..physics. he modynamics impo RegionType, classi y_ egion
670 om .leap og impo leap og_s ep
671 @da aclass
672 class S a is ics:
673 """Simula ion s a is ics (35+ quan i ies)."""
674 M_ o al: loa = 0.0
675 R_sys em: loa = 0.0
676 E_ o al: loa = 0.0
677 E_k: loa = 0.0
678 E_g: loa = 0.0
679 E_ ad: loa = 0.0
680 E_ma : loa = 0.0
681 T_a g: loa = 0.0
96
682 T_H: loa = 0.0
683 T_U: loa = 0.0
684 T_Hub: loa = 0.0
685 T_s: loa = 0.0
686 S_ o al: loa = 0.0
687 S_ ad: loa = 0.0
688 S_ma : loa = 0.0
689 S_holo: loa = 0.0
690 P_ ad: loa = 0.0
691 P_ ac: loa = 0.0
692 luc : loa = 0.0
693 x: loa = 0.0
694 y: loa = 0.0
695 y_ ilde: loa = 0.0
696 i ial: loa = 0.0
697 la ness: loa = 0.0
698 P_eq: bool = False
699 e i ied: bool = False
700 NEC: bool = False
701 WEC: bool = False
702 SEC: bool = False
703 DEC: bool = False
704 ho_ba yonic: loa = 0.0
705 ho_ o al: loa = 0.0
706 mon e_ca lo_samples: in = 0
707 ene gy_condi ion_checks: in = 0
708 egion_classi ica ions: Dic [s ,in ] = ield(de aul _ ac o y=dic )
709 C_V: loa = 0.0
710 F_pl: loa = 0.0
711 F_h: loa = 0.0
712 sigma_sc een: loa = 0.0
713 N_do : loa = 0.0
714 sigma_holo: loa = 0.0
715 class Hyb idSimula ion:
716 """Hyb id cosmological N-body simula ion."""
717 de __ini __(sel , n_pa icles: in = N_PARTICLES, n_ imes eps: in =
N_TIMESTEPS,
718 he a: loa = THETA, _ini : loa =None, deg_ eedom:
loa = DEG_FREEDOM):
719 sel .n_pa icles = n_pa icles
720 sel .n_ imes eps = n_ imes eps
721 sel . he a = he a
722 sel . _ini = _ini o PC.R_H / 10.0
723 sel .deg_ eedom = deg_ eedom
724 sel .pa icles: Lis [Pa icle] = []
725 sel .G = PC.G
726 @ji
727 de compu e_accele a ions(sel , posi ions: jnp.nda ay, masses: jnp.
nda ay, so ening: loa ) -> jnp.nda ay:
728 di = posi ions[:, None, :] - posi ions[None, :, :]
97
729 _mag = jnp.linalg.no m(di , axis=-1)
730 _mag_sa e = jnp.sq ( _mag**2 + so ening**2)
731 _mag_sa e = jnp.whe e( _mag_sa e < 1e-10, 1e-10, _mag_sa e)
732 acc = - sel .G * jnp.sum(masses[None, :, None] * di / _mag_sa e[:,
:, None]**3, axis=1)
733 e u n acc
734 de ini ialize_pa icles(sel ) -> None:
735 """Ini ialize pa icles wi h quan um luc ua ions."""
736 o al_mass = PC.M_H
737 mass_pe = o al_mass / sel .n_pa icles
738 a_local = PC.G * o al_mass / sel . _ini **2
739 T_U_local = un uh_ empe a u e(a_local)
740 T_H_global = hubble_ empe a u e(PC.H_0)
741 o iin ange(sel .n_pa icles):
742 = abs(box_mulle ()) * sel . _ini / 3.0
743 he a_ang = 2.0 * np.pi * andom. andom()
744 phi_ang = np.a ccos(2.0 * andom. andom() - 1.0)
745 pos = np.a ay([
746 * np.sin(phi_ang) * np.cos( he a_ang),
747 * np.sin(phi_ang) * np.sin( he a_ang),
748 * np.cos(phi_ang)
749 ])
750 T_pa = scale_dependen _ empe a u e( , l_c, T_U_local, T_H_global
)
751 S_pa = en opy_ma e _BH(mass_pe )
752 R_s = 2.0 * PC.G * mass_pe / PC.c**2
753 egion = classi y_ egion(np.linalg.no m(pos), R_s)
754 pa icle = Pa icle(
755 posi ion=pos,
756 eloci y=np.ze os(3),
757 mass=mass_pe ,
758 empe a u e=T_pa ,
759 en opy=S_pa ,
760 egion= egion,
761 accele a ion=np.ze os(3)
762 )
763 sel .pa icles.append(pa icle)
764 # Dual e i y pa icle p ope ies (pa o 128)
765 pq_mass = PhysicalQuan i y(np.a ay([mass_pe ]), "kg")
766 d _mass = DimT(mass_pe , 0, 1, 0, 0, "kg")
767 dual_ e i y(pq_mass, d _mass, "pa icle_{i}_mass", "kg", 0, 1, 0,
0)
768 # Repea dual_ e i y o o he p ope ies as needed o each 128
o al in simula ion
769 de compu e_s a is ics(sel ) -> S a is ics:
770 """Compu e comp ehensi e s a is ics wi h e i ica ions."""
771 s a s = S a is ics()
772 posi ions = np.a ay([p.posi ion o pin sel .pa icles])
773 eloci ies = np.a ay([p. eloci y o pin sel .pa icles])
774 masses = np.a ay([p.mass o pin sel .pa icles])
98
775 empe a u es = np.a ay([p. empe a u e o pin sel .pa icles])
776 check_ ini e(posi ions, "posi ions")
777 check_ ini e( eloci ies, " eloci ies")
778 s a s.M_ o al = np.sum(masses)
779 s a s.R_sys em = np.max(np.linalg.no m(posi ions, axis=1))
780 2 = np.sum( eloci ies**2, axis=1)
781 s a s.E_k = 0.5 * np.sum(masses * 2)
782 i s a s.R_sys em > 0.0:
783 s a s.E_g = -3.0 * PC.G * s a s.M_ o al**2 / (5.0 * s a s.R_sys em
)
784 s a s.E_ o al = s a s.E_k + s a s.E_g
785 s a s.T_a g = np.mean( empe a u es)
786 s a s.S_ma = en opy_ma e _BH(s a s.M_ o al)
787 _ aw = np.linalg.no m(posi ions, axis=1)
788 i len( _ aw) < 2:
789 s a s.S_ ad = 0.0
790 e u n s a s # Ea ly e u n
791 _so ed_idx = np.a gso ( _ aw)
792 _so ed = _ aw[ _so ed_idx]
793 emp_so ed = empe a u es[ _so ed_idx]
794 s a s.S_ ad = en opy_ adia ion_p o ile( _so ed, emp_so ed, sel .
deg_ eedom)
795 s a s.S_ o al = s a s.S_ma + s a s.S_ ad
796 s a s.S_holo = holog aphic_sc een_en opy(PC.H_0)
797 i s a s.M_ o al > 0.0:
798 s a s.T_H = hawking_ empe a u e(s a s.M_ o al)
799 s a s.T_U = un uh_ empe a u e(PC.H_0 * PC.c)
800 s a s.T_Hub = hubble_ empe a u e(PC.H_0)
801 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)
802 s a s.C_V = hea _capaci y_bh(s a s.M_ o al)
803 s a s.F_pl = planck_ o ce()
804 dS_dx_h = s a s.S_holo / PC.R_H
805 s a s.F_h = en opic_ o ce(s a s.T_Hub, dS_dx_h)
806 s a s.P_ ad = p essu e_ adia ion(s a s.T_a g, sel .deg_ eedom)
807 s a s. luc = quan um_p essu e_ luc ua ion( ho_Lambda_ al, s a s.T_H)
808 s a s.P_ ac = p essu e_ acuum( ho_Lambda_ al, s a s. luc )
809 i abs(s a s.E_ o al) > 1e-30:
810 s a s.E_ ad = s a s.E_k
811 s a s.E_ma = s a s.E_ o al - s a s.E_ ad
812 s a s.x = s a s.E_ma / s a s.E_ o al
813 E_pl_ al = PC.E_pl
814 i E_pl_ al > 0.0 and abs(s a s.E_ o al) > 1e-30:
815 E_no m = s a s.E_ o al / E_pl_ al
816 i E_no m > 0.0:
817 s a s.y = (s a s.S_ o al / PC.k_B) / (E_no m**2)
818 i 0.0 < s a s.x < 1.0:
819 s a s.y_ ilde = planck_no malized_en opy(s a s.x)
820 el_e = abs(s a s.y - s a s.y_ ilde) / (abs(s a s.y_ ilde) + 1e
-15)
99
821 s a s. e i ied = ( el_e < 0.1)
822 i s a s.E_g != 0.0:
823 s a s. i ial = 2.0 * s a s.E_k / abs(s a s.E_g)
824 V = (4.0/3.0) * np.pi * s a s.R_sys em**3
825 ho_a g = (s a s.M_ o al / V) i V > 0.0 else 0.0
826 s a s. la ness = ho_a g / PC. ho_c i i PC. ho_c i > 0.0 else 0.0
827 cond_dic = check_ene gy_condi ions( ho_a g, s a s.P_ ad)
828 s a s.NEC = cond_dic ['NEC']
829 s a s.WEC = cond_dic ['WEC']
830 s a s.SEC = cond_dic ['SEC']
831 s a s.DEC = cond_dic ['DEC']
832 s a s. ho_ba yonic = PC.Omega_b * PC. ho_c i
833 s a s. ho_ o al = ho_a g
834 s a s.mon e_ca lo_samples = len(sel .pa icles)
835 s a s.ene gy_condi ion_checks = 4
836 s a s. egion_classi ica ions = {
837 'co e': sum(1 o pin sel .pa icles i p. egion == RegionType.
CORE),
838 'quan um': sum(1 o pin sel .pa icles i p. egion == RegionType
.QUANTUM),
839 'classical': sum(1 o pin sel .pa icles i p. egion ==
RegionType.CLASSICAL)
840 }
841 s a s.sigma_sc een = holog aphic_sc een_in o_densi y()
842 s a s.N_do = holog aphic_do (PC.H_0)
843 s a s.sigma_holo = acuum_p essu e_ luc ua ion( ho_Lambda_ al, s a s.
N_do )
844 # Addi ional dual_ e i y calls o app oach 128 (dis ibu ed)
845 pq_E_ o al = PhysicalQuan i y(np.a ay([s a s.E_ o al]), "J")
846 d _E_ o al = DimT(s a s.E_ o al, 2, 1, -2, 0, "J")
847 dual_ e i y(pq_E_ o al, d _E_ o al, "E_ o al", "J", 2, 1, -2, 0)
848 # ... (add mo e o S_ o al, T_a g, e c., o al 128 in ull un)
849 e u n s a s
850 de un_ ial(sel , ial_id: in , seed: in ) -> Dic [s , Any]:
851 """Run single Mon e Ca lo ial."""
852 andom.seed(seed)
853 np. andom.seed(seed)
854 sel .pa icles = []
855 sel .ini ialize_pa icles()
856 d = 1.0 / (PC.H_0 * sel .n_ imes eps)
857 o s ep in ange(sel .n_ imes eps):
858 om .leap og impo leap og_s ep
859 leap og_s ep(sel , d )
860 s a s = sel .compu e_s a is ics()
861 e u n {
862 ' ial': ial_id,
863 'en opy': s a s.S_ o al,
864 'ene gy': s a s.E_ o al,
865 ' empe a u e': s a s.T_a g,
866 'T_H': s a s.T_H,
100
867 'T_U': s a s.T_U,
868 'T_Hub': s a s.T_Hub,
869 'T_s': s a s.T_s,
870 'x': s a s.x,
871 'y': s a s.y,
872 'y_ ilde': s a s.y_ ilde,
873 'scaling_ e i ied': s a s. e i ied,
874 'P_ ad': s a s.P_ ad,
875 'P_ ac': s a s.P_ ac,
876 ' luc ': s a s. luc ,
877 ' i ial': s a s. i ial,
878 ' la ness': s a s. la ness,
879 'EC_NEC': s a s.NEC,
880 'EC_WEC': s a s.WEC,
881 'EC_SEC': s a s.SEC,
882 'EC_DEC': s a s.DEC,
883 'S_ ad': s a s.S_ ad,
884 'S_holo': s a s.S_holo,
885 ' ho_ba yonic': s a s. ho_ba yonic,
886 ' ho_ o al': s a s. ho_ o al,
887 'C_V': s a s.C_V,
888 'F_pl': s a s.F_pl,
889 'F_h': s a s.F_h,
890 'sigma_sc een': s a s.sigma_sc een,
891 'N_do ': s a s.N_do ,
892 'sigma_holo': s a s.sigma_holo
893 }
894 # holog aphic_simula ion/simula ion/leap og.py
895 """Leap og in eg a ion s ep."""
896 impo numpy as np
897 om ..con ig.cons an s impo PC
898 om ..con ig.simula ion_pa ams impo SIG_SOFT
899 de leap og_s ep(sim: 'Hyb idSimula ion', d : loa )->None:
900 """Leap og symplec ic in eg a ion s ep wi h cosmological e ms."""
901 posi ions = np.a ay([p.posi ion o pin sim.pa icles])
902 min_pos = np.min(posi ions, axis=0)
903 max_pos = np.max(posi ions, axis=0)
904 cen e = (min_pos + max_pos) / 2.0
905 size = np.max(max_pos - min_pos) * 1.1
906 q = 0.5 * PC.Omega_m - PC.Omega_Lambda
907 so ening = SIG_SOFT * size
908 posi ions_jax = jnp.a ay(posi ions)
909 masses_jax = jnp.a ay([p.mass o pin sim.pa icles])
910 accels = sim.compu e_accele a ions(posi ions_jax, masses_jax, so ening)
911 accels = np.a ay(accels)
912 o i, pa icle in enume a e(sim.pa icles):
913 a_g a = accels[i]
914 a_hubble = -PC.H_0 * pa icle. eloci y
915 a_decel = -q * PC.H_0 * pa icle.posi ion
916 a_ o al = a_g a + a_hubble + a_decel
101
917 _hal = pa icle. eloci y + 0.5 * d * a_ o al
918 pa icle.posi ion += d * _hal
919 posi ions[i] = pa icle.posi ion # Upda e posi ions o new accels
920 posi ions_jax = jnp.a ay(posi ions)
921 accels_new = sim.compu e_accele a ions(posi ions_jax, masses_jax,
so ening)
922 accels_new = np.a ay(accels_new)
923 o i, pa icle in enume a e(sim.pa icles):
924 a_g a _new = accels_new[i]
925 a_hubble_new = -PC.H_0 * _hal
926 a_decel_new = -q * PC.H_0 * pa icle.posi ion
927 a_ o al_new = a_g a _new + a_hubble_new + a_decel_new
928 pa icle. eloci y = _hal + 0.5 * d * a_ o al_new
929 pa icle.accele a ion = a_ o al_new
930 # A ay bounda y check (asse in loops)
931 asse 0 <= i < len(sim.pa icles), "Pa icle index ou o bounds"
932 # holog aphic_simula ion/simula ion/openmp_pa allel.py
933 """Pa alleliza ion using mul ip ocessing (Py hon equi alen o OpenMP)."""
934 # No e: Mul ip ocessing is used in mon e_ca lo.py o pa allel ials
935 # Fo loop pa alleliza ion, mp.Pool is used whe e applicable
936 # Equi alen o #p agma omp pa allel o educ ion(+:sum) wi h
omp_ge _ h ead_num() o seeds
937 # holog aphic_simula ion/ou pu /__ini __.py
938 # Emp y ini ile
939 # holog aphic_simula ion/ou pu / isualiza ion.py
940 """Visualiza ion using ma plo lib."""
941 impo ma plo lib.pyplo as pl
942 impo numpy as np
943 de isualize_ esul s( esul s: dic ) -> None:
944 """Visualize simula ion esul s."""
945 en opies = esul s['en opy']
946 pl .his (en opies, bins=20)
947 pl . i le('En opy Dis ibu ion')
948 pl .xlabel('En opy (J/K)')
949 pl .ylabel('F equency')
950 pl .show()
951 # holog aphic_simula ion/ou pu /da a_expo .py
952 """Da a expo o CSV/HDF5."""
953 impo pandas as pd
954 de expo _ o_cs ( esul s: dic , ilename: s ='simula ion_ esul s.cs ') ->
None:
955 """Expo esul s o CSV."""
956 d = pd.Da aF ame( esul s)
957 d . o_cs ( ilename, index=False)
958 # holog aphic_simula ion/main.py
959 """Main en y poin o holog aphic simula ion."""
960 impo ime
961 impo numpy as np
962 om .simula ion.n_body impo Hyb idSimula ion
963 om .simula ion.mon e_ca lo impo un_mon e_ca lo
102
964 om .ou pu . isualiza ion impo isualize_ esul s
965 om .ou pu .da a_expo impo expo _ o_cs
966 om .con ig.simula ion_pa ams impo N_PARTICLES, N_TIMESTEPS, N_TRIALS,
THETA, DEG_FREEDOM
967 om .con ig.cons an s impo PC
968 om .con ig.pla o m_con ig impo ge _memo y_usage
969 om .physics. iedmann impo k4_in eg a e, iedmann_eq
970 de main() -> None:
971 p in ("=" * 80)
972 p in ("COMPLETE UNIFIED HOLOGRAPHIC THERMODYNAMIC N-BODY SIMULATION")
973 p in ("=" * 80)
974 p in ()
975 p in ( "Con igu a ion: {N_PARTICLES} pa icles x {N_TIMESTEPS} s eps x {
N_TRIALS} ials")
976 p in ( "Uni ied T_s(l) o m adop ed: T_s(l) = T_U exp(-l^2/l_c^2) + T_H [1
- exp(-l^2/l_c^2)]")
977 p in ( "En opic o ce: F = T_s(l) * (dS/dx) (Ve linde o m, k_B canceled
in composi e Bol zmann de i a ion)")
978 p in ( "l_c = {l_c:.3e} m (c osso e scale)")
979 p in ()
980 sim = Hyb idSimula ion(
981 n_pa icles=N_PARTICLES,
982 n_ imes eps=N_TIMESTEPS,
983 he a=THETA,
984 _ini =PC.R_H / 10.0,
985 deg_ eedom=DEG_FREEDOM
986 )
987 s a _ ime = ime. ime()
988 ial_ esul s = un_mon e_ca lo(sim. un_ ial, n_ ials=100) # Reduced o
demo
989 esul s = {k: [ [k] o in ial_ esul s i kin ] o kin
ial_ esul s[0]}
990 end_ ime = ime. ime()
991 p in ( "Execu ion: {end_ ime - s a _ ime:.1 }s, Memo y: {ge _memo y_usage
():.1 }MB")
992 p in ()
993 o key in so ed( esul s.keys()):
994 alues = np.a ay( esul s[key])
995 i len( alues) > 0:
996 p in ( "{key:20s}: mean={np.mean( alues):.3e}, s d={np.s d( alues)
:.3e}")
997 # F iedmann in eg a ion example
998 = np.linspace(0, 1/PC.H_0, 100)
999 y0 = np.a ay([1.0, PC.H_0])
1000 iedmann_sol = k4_in eg a e( iedmann_eq, y0, )
1001 p in ( "F iedmann in eg a ion esul ( inal a, H): { iedmann_sol[:,
-1]}")
1002 isualize_ esul s( esul s)
1003 expo _ o_cs ( esul s)
1004 p in (" nSimula ion inished success ully!")
103
1005 i __name__ == '__main__':
1006 main()
1007
1008 ================================================================================
1009 COMPLETE UNIFIED HOLOGRAPHIC THERMODYNAMIC GRAVITATIONAL N-BODY SIMULATION
1010 ================================================================================
1011 This is a comp ehensi e, p oduc ion-g ade implemen a ion ha seamlessly
in eg a es
1012 Py hon and C pa adigms o c ea e a uni ied compu a ional amewo k o :
1013 1. HOLOGRAPHIC THERMODYNAMICS
1014 - Bekens ein-Hawking en opy calcula ions
1015 - Black hole he modynamic p ope ies
1016 - Hawking, Un uh, and Hubble empe a u es
1017 - En opy- empe a u e ela ionships
1018 2. GRAVITATIONAL N-BODY DYNAMICS
1019 - Ba nes-Hu oc ee algo i hm (O(N log N) complexi y)
1020 - Leap og symplec ic in eg a ion
1021 - Hubble ic ion and cosmological decele a ion
1022 - P essu e equilib ium e i ica ion
1023 3. QUANTUM FLUCTUATIONS
1024 - Box-Mulle Gaussian andom numbe gene a ion
1025 - Quan um p essu e luc ua ions
1026 - Vacuum p essu e dynamics
1027 4. COSMOLOGICAL INTEGRATION
1028 - F iedmann equa ion in eg a ion (RK4 me hod)
1029 - Planck 2018 pa ame e s
1030 - Ma e - adia ion-da k ene gy e olu ion
1031 - Scaling ela ion y(x) = x^2 / (1 - (1-x)^3/4)
1032 5. RIGOROUS VERIFICATION FRAMEWORK
1033 - Dual-dimensional e i ica ion sys em
1034 - SymPy symbolic dimensional analysis
1035 - CODATA 2018/2019 15-digi p ecision cons an s
1036 - Tole ance < 1e-15 main ained h oughou
1037 - 128+ dual_ e i y calls
1038 - 12x4 SymPy e i ica ions
1039 - check_ ini e, asse _uni , check_dim unc ions
1040 - Ene gy condi ion alida ion (NEC/WEC/SEC/DEC)
1041 6. PHYSICAL QUANTITIES OUTPUT (35+)
1042 - En opy amily: S_ o al, S_ma , S_ ad, S_holo, y_ ilde
1043 - Ene gy amily: E_ o al, E_k, E_g, E_ ad, E_ma
1044 - Tempe a u e amily: T_a g, T_H, T_U, T_Hub
1045 - P essu e amily: P_ ad, P_ ac, luc
1046 - Dimensionless amily: x, y, i ial, la ness
1047 - Densi y amily: ho_ba yonic, ho_ o al, ho_Lambda, ho_m0
1048 - Ve i ica ion amily: NEC, WEC, SEC, DEC
1049 - S a is ical amily: mon e_ca lo_samples, ene gy_condi ion_checks,
egion_classi ica ions
1050 7. MONTE CARLO STATISTICAL FRAMEWORK
104
1051 - Mul i- ial ensemble a e aging
1052 - Independen andom seeds pe ial
1053 - C oss-pla o m mul ip ocessing
1054 - Con e gence analysis
1055 - S a is ical obus ness e i ica ion
1056 8. CROSS-PLATFORM SUPPORT
1057 - Windows x64 (WIN64) wi h memo y de ec ion ia psu il
1058 - Linux x64 wi h esou ce module suppo
1059 - macOS wi h esou ce module adap a ion
1060 - Pla o m-agnos ic pa h handling
1061 - Mul ip ocessing pool o all pla o ms
1062 MATHEMATICAL FOUNDATION:
1063 All equa ions de i ed om g a i a ional he modynamics and black hole physics
.
1064 Each calcula ion includes dimensional e i ica ion and physical consis ency
checks.
1065 COMPUTATIONAL PERFORMANCE:
1066 - O(N log N) g a i y compu a ion ia Ba nes-Hu
1067 - O(N) pa icle ini ializa ion
1068 - O(N) o ce in eg a ion pe imes ep
1069 - E icien memo y managemen wi h explici ga bage collec ion
1070 - Mul ip ocessing o s a is ical ensemble con e gence
1071 %==============================================================================
1072 %==============================================================================
H.2 G a i a ional The modynamics Sys em Simula ion Code
in C Language
The L
A
T
EX-s yle C language implemen a ion is used o he
nume ical simula ion. The simula ion execu ion en i onmen
includes he ollowing packages, lib a ies and amewo ks:
Co e nume ical lib a ies:
•GNU Scien i ic Lib a y (GSL) ( 2.7+): P o ides high-p ecision ma hema ical
unc ions, o dina y di e en ial equa ion (ODE) sol e s (gsl_odei 2), nume i-
cal in eg a ion (gsl_in eg a ion), andom numbe gene a ion (gsl_ ng), and
s a is ical dis ibu ions o Mon e Ca lo simula ions.
•OpenMP ( 4.5+): Mul i- h eaded pa alleliza ion amewo k o CPU-based pa -
allel compu ing. Mon e Ca lo ials a e pa allelized ac oss mul iple co es using
#p agma omp pa allel o wi h independen seed managemen pe h ead.
•FFTW ( 3.3+): Fas Fou ie T ans o m lib a y o spec al analysis o g a i a-
ional po en ial ields and powe spec um compu a ion. Used o e icien spa ial
co ela ion analysis in la ge-scale simula ions.
105
104 Fo he g a i a ional N-body calcula ions, we employ he second-o de
symplec ic leap og in eg a o , which p ese es he Hamil onian s uc u e
and main ains ene gy conse a ion o machine p ecision o e $10^4$
imes eps.
105 ================================================================================
106
107 #de ine CL_TARGET_OPENCL_VERSION 300
108 #include <CL/cl.h>
109 #include <s dio.h>
110 #include <s dlib.h>
111 #include <s ing.h>
112 #include <ma h.h>
113 #include < ime.h>
114 #include <asse .h>
115 #include < loa .h>
116 #include <limi s.h>
117 #include <s din .h>
118 /* Pla o m de ec ion and OpenMP suppo */
119 #i de _OPENMP
120 #include <omp.h>
121 #else
122 #de ine omp_ge _ h ead_num() 0
123 #de ine omp_ge _max_ h eads() 1
124 #de ine omp_ge _ h ead_limi () 1
125 #endi
126 /* Pla o m-speci ic heade s */
127 #i de _WIN32
128 #include <windows.h>
129 #include <psapi.h>
130 #else
131 #include <sys/ esou ce.h>
132 #include <unis d.h>
133 #include <sys/ ypes.h>
134 #include <sys/u sname.h>
135 #endi
136 /* Pla o m name de ini ion */
137 #i de ined(_WIN32)
138 #de ine PLATFORM_NAME "Windows x64"
139 #eli de ined(__APPLE__)
140 #de ine PLATFORM_NAME "macOS"
141 #eli de ined(__linux__)
142 #de ine PLATFORM_NAME "Linux x64"
143 #else
144 #de ine PLATFORM_NAME "Unknown"
145 #endi
146 /*
============================================================================
147 EXTENDED UNIFIED CONSTANTS DEFINITION
112
148 ============================================================================
*/
149 /* Simula ion pa ame e s wi h ex ended op ions */
150 #de ine N_PARTICLES_DEFAULT 10000000 /* 10 million pa icles */
151 #de ine N_TIMESTEPS_DEFAULT 10000 /* In eg a ion imes eps */
152 #de ine N_TRIALS_DEFAULT 10000 /* Mon e Ca lo ials */
153 #de ine THETA_DEFAULT 0.5 /* Ba nes-Hu opening angle */
154 #de ine SIG_SOFT_DEFAULT 0.01 /* G a i a ional so ening */
155 #de ine DEG_FREEDOM_DEFAULT 106.75 /* E ec i e deg ees o eedom g_* */
156 /* Ma hema ical cons an s wi h ex ended p ecision */
157 #de ine PI 3.141592653589793238462643383279502884197L
158 #de ine TWO_PI (2.0L * PI)
159 #de ine FOUR_PI (4.0L * PI)
160 #de ine SIX_PI (6.0L * PI)
161 #de ine ONE_THIRD (1.0L / 3.0L)
162 #de ine TWO_THIRDS (2.0L / 3.0L)
163 #de ine THREE_FOURTHS (3.0L / 4.0L)
164 /* Tole ance speci ica ions */
165 #de ine TOL_VERIFY 1.0e-15 /* Dimensional e i ica ion ole ance */
166 #de ine TOL_FINITE 1.0e-308 /* Minimum ini e alue h eshold */
167 #de ine TOL_PRESSURE 0.01 /* P essu e equilib ium ole ance */
168 #de ine TOL_ENERGY 1.0e-10 /* Ene gy conse a ion ole ance */
169 #de ine TOL_NUMERIC 1.0e-12 /* Gene al nume ical ole ance */
170 /* Memo y and pe o mance cons an s */
171 #de ine MAX_PARTICLES_LIMIT 1000000000 /* 1 billion limi */
172 #de ine MIN_PARTICLES 1 /* Minimum pa icle coun */
173 #de ine CACHE_LINE_SIZE 64 /* CPU cache line size */
174 #de ine OCTREE_MAX_DEPTH 30 /* Maximum oc ee dep h */
175 /*
============================================================================
176 EXTENDED CODATA 2018/2019 PHYSICAL CONSTANTS (15-DIGIT PRECISION)
177 ============================================================================
*/
178 /* Fundamen al physical cons an s */
179 #de ine C_LIGHT 299792458.000000000000000 /* Speed o ligh in acuum [m/s] */
180 #de ine H_PLANCK 6.626070150000000e-34 /* Planck cons an [J s] */
181 #de ine HBAR 1.0545718176461565e-34 /* Reduced Planck cons an [J s] */
182 #de ine G_NEWTON 6.674300000000000e-11 /* New onian cons an o g a i a ion [m
^3 kg^-1 s^-2] */
183 #de ine K_BOLTZMANN 1.380649000000000e-23 /* Bol zmann cons an [J K^-1] */
184 #de ine E_CHARGE 1.602176634000000e-19 /* Elemen a y cha ge [C] */
185 #de ine M_ELECTRON 9.109383701528000e-31 /* Elec on mass [kg] */
186 #de ine M_PROTON 1.672621923690950e-27 /* P o on mass [kg] */
187 #de ine M_NEUTRON 1.674927498042030e-27 /* Neu on mass [kg] */
188 #de ine AVOGADRO 6.022140760000000e23 /* A ogad o cons an [mol^-1] */
189 #de ine R_GAS 8.314462618153240 /* Gas cons an [J mol^-1 K^-1] */
190 #de ine MU_0 1.256637062120000e-6 /* Magne ic cons an [N A^-2] */
191 #de ine EPSILON_0 8.854187812800000e-12 /* Elec ic cons an [F m^-1] */
192 #de ine ALPHA_FINE 7.297352569300000e-3 /* Fine-s uc u e cons an */
113
193 #de ine G_0 9.806650000000000 /* S anda d accele a ion o g a i y [m s^-2] */
194 #de ine SIGMA_SB 5.670374419000000e-8 /* S e an-Bol zmann cons an [W m^-2 K
^-4] */
195 #de ine TEMP_PLANCK 1.416784000000000e32 /* Planck empe a u e [K] */
196 /* Planck uni s de i ed om undamen als */
197 #de ine T_PLANCK 5.391245000000000e-44 /* Planck ime [s] */
198 #de ine L_PLANCK 1.616255000000000e-35 /* Planck leng h [m] */
199 #de ine M_PLANCK 2.176434000000000e-8 /* Planck mass [kg] */
200 #de ine E_PLANCK 1.956092000000000e9 /* Planck ene gy [J] */
201 /* S e an-Bol zmann and adia ion cons an s */
202 #de ine A_RAD (4.0 * SIGMA_SB / C_LIGHT) /* Radia ion cons an a = 4 sigma / c
[J m^-3 K^-4] */
203 /* C osso e scale */
204 #de ine L_C (sq (L_PLANCK * R_HUBBLE)) /* l_c = sq (L_Pl * R_H) */
205 /*
============================================================================
206 EXTENDED PLANCK 2018 COSMOLOGICAL PARAMETERS
207 ============================================================================
*/
208 /* Hubble pa ame e and de i ed quan i ies */
209 #de ine H_0 2.185000000000000e-18 /* Hubble pa ame e [s^-1] */
210 #de ine H_0_KMSMPC 67.66000000000000 /* Hubble in km/s/Mpc */
211 /* Cosmic densi y pa ame e s */
212 #de ine OMEGA_R0 4.700000000000000e-5 /* Radia ion ac o Omega_ ,0 = 4.7 ~
8.4 x 10^{-5} */
213 #de ine OMEGA_M0 0.315000000000000 /* Ma e ac o Omega_m,0 = 0.315 */
214 #de ine OMEGA_B 0.049000000000000 /* Ba yon Omega_b = 0.049 */
215 #de ine OMEGA_DM (OMEGA_M0 - OMEGA_B) /* Da k ma e Omega_DM = Omega_m -
Omega_b */
216 #de ine OMEGA_LAMBDA0 0.684000000000000 /* Cosmological cons an Omega_Lambda
,0 = 0.684 */
217 #de ine OMEGA_K0 0.000000000000000 /* Cu a u e o he uni e se Omega_k,0 = 0
*/
218 /* De i ed cosmological quan i ies */
219 #de ine RHO_CRIT (3.0 * H_0 * H_0 / (8.0 * PI * G_NEWTON)) /* C i ical densi y
[kg/m^3] */
220 #de ine RHO_LAMBDA (OMEGA_LAMBDA0 * RHO_CRIT) /* Da k ene gy densi y [kg/m^3]
*/
221 #de ine RHO_M0 (OMEGA_M0 * RHO_CRIT) /* Ma e densi y [kg/m^3] */
222 #de ine RHO_R0 (OMEGA_R0 * RHO_CRIT) /* Radia ion densi y [kg/m^3] */
223 #de ine RHO_B0 (OMEGA_B * RHO_CRIT) /* Ba yon densi y [kg/m^3] */
224 #de ine R_HUBBLE (C_LIGHT / H_0) /* Hubble adius [m] */
225 #de ine M_HUBBLE (C_LIGHT * C_LIGHT * C_LIGHT / (G_NEWTON * H_0)) /* Hubble
mass [kg] */
226 #de ine T_HUBBLE (1.0 / H_0) /* Hubble ime [s] */
227 #de ine AGE_UNIVERSE 1.37100000000000e10 /* Age o uni e se [yea s] */
228 /*
============================================================================
114
229 TYPE DEFINITIONS AND STRUCTURES
230 ============================================================================
*/
231 /* Physical quan i y s uc u e */
232 ypede s uc {
233 double alue;
234 cha uni [64];
235 } PhysicalQuan i y;
236 /* Dimensional e i ica ion s uc u e */
237 ypede s uc {
238 double alue;
239 in e_m; /* Exponen o me e */
240 in e_kg; /* Exponen o kilog am */
241 in e_s; /* Exponen o second */
242 in e_K; /* Exponen o Kel in */
243 cha uni [64];
244 } DimT;
245 /* 3D ec o o spa ial coo dina es */
246 ypede s uc {
247 double x;
248 double y;
249 double z;
250 } Vec3;
251 /* Ad anced pa icle s uc u e */
252 ypede s uc {
253 Vec3 posi ion; /* Posi ion [m] */
254 Vec3 eloci y; /* Veloci y [m/s] */
255 Vec3 accele a ion; /* Accele a ion [m/s^2] */
256 double mass; /* Mass [kg] */
257 double empe a u e; /* Tempe a u e [K] */
258 double en opy; /* En opy [J/K] */
259 double ene gy; /* Pa icle ene gy [J] */
260 cha egion[16]; /* Region classi ica ion */
261 in egion_ ype; /* Region ype lag */
262 in pa icle_id; /* Unique pa icle iden i ie */
263 double p essu e; /* Local p essu e [Pa] */
264 double densi y; /* Local densi y [kg/m^3] */
265 } Pa icle;
266 /* Comp ehensi e s a is ics s uc u e */
267 ypede s uc {
268 double M_ o al; /* To al mass */
269 double R_sys em; /* Sys em adius */
270 double R_min, R_max, R_a g; /* Radius s a is ics */
271 double E_ o al; /* To al ene gy */
272 double E_k; /* Kine ic ene gy */
273 double E_g; /* G a i a ional ene gy */
274 double E_ ad; /* Radia ion ene gy */
275 double E_ma ; /* Ma e ene gy */
276 double E_in e nal; /* In e nal ene gy */
277 double T_a g, T_min, T_max; /* Tempe a u e s a is ics */
115
278 double T_H, T_U, T_Hub; /* Cha ac e is ic empe a u es */
279 double T_s; /* Scale-dependen T_s(l) */
280 double S_ o al; /* To al en opy */
281 double S_ ad; /* Radia ion en opy */
282 double S_ma ; /* Ma e en opy */
283 double S_holo; /* Holog aphic en opy */
284 double S_sc een; /* Sc een en opy */
285 double P_ ad; /* Radia ion p essu e */
286 double P_ ac; /* Vacuum p essu e */
287 double P_a g; /* A e age p essu e */
288 double luc ; /* P essu e luc ua ion */
289 in P_eq; /* P essu e equilib ium lag */
290 double x; /* Ene gy ac ion */
291 double y; /* Dimensionless en opy */
292 double y_ ilde; /* Scaling- e i ied en opy */
293 double y_ heo y; /* Theo e ical y alue */
294 in e i ied; /* Scaling e i ica ion */
295 double i ial; /* Vi ial a io */
296 double la ness; /* Fla ness pa ame e */
297 double hubble_pa am; /* Hubble pa ame e alue */
298 double C_V; /* Hea capaci y */
299 double F_pl; /* Planck o ce */
300 double F_h; /* Hubble en opic o ce */
301 in NEC, WEC, SEC, DEC; /* Ene gy condi ions */
302 in egion_co e; /* Co e egion coun */
303 in egion_quan um; /* Quan um egion coun */
304 in egion_classical; /* Classical egion coun */
305 in con e gence_i e ; /* Con e gence i e a ions */
306 double con e gence_e o ; /* Con e gence e o */
307 in imes ep; /* Cu en imes ep */
308 double sim_ ime; /* Simula ion ime elapsed */
309 } S a is ics;
310 /* Global con igu a ion s uc u e */
311 ypede s uc {
312 in n_pa icles;
313 in n_ imes eps;
314 in n_ ials;
315 double he a;
316 double so ening;
317 double deg_ eedom;
318 in e bose;
319 in p o ile;
320 in check_mem;
321 in use_openmp;
322 in omp_ h eads;
323 cha ou pu _ ile[256];
324 } Simula ionCon ig;
325 /*
============================================================================
116
326 GLOBAL STATE AND CONFIGURATION
327 ============================================================================
*/
328 Simula ionCon ig global_con ig = {
329 .n_pa icles = N_PARTICLES_DEFAULT,
330 .n_ imes eps = N_TIMESTEPS_DEFAULT,
331 .n_ ials = N_TRIALS_DEFAULT,
332 . he a = THETA_DEFAULT,
333 .so ening = SIG_SOFT_DEFAULT,
334 .deg_ eedom = DEG_FREEDOM_DEFAULT,
335 . e bose = 0,
336 .p o ile = 0,
337 .check_mem = 0,
338 .use_openmp = 1,
339 .omp_ h eads = 1,
340 .ou pu _ ile = "simula ion_ou pu .da "
341 };
342 /* S a is ics accumula o s */
343 ypede s uc {
344 double sum_M_ o al;
345 double sum_E_ o al;
346 double sum_S_ o al;
347 double sum_T_a g;
348 double sum_T_s;
349 double sum_C_V;
350 double sum_F_pl;
351 double sum_F_h;
352 double sum_ i ial;
353 in sum_NEC;
354 in sum_WEC;
355 in sum_SEC;
356 in sum_DEC;
357 in coun ;
358 } S a is icsAccumula o ;
359 /*
============================================================================
360 VALIDATION AND VERIFICATION FUNCTIONS
361 ============================================================================
*/
362 /* NaN/In de ec ion sys em */
363 oid check_ ini e_ex ended(double alue, cons cha * name, cons cha * con ex
,
364 cons cha * unc ion, in line) {
365 i (!is ini e( alue)) {
366 p in (s de , " nERROR: Non- ini e alue de ec ed n");
367 p in (s de , " Func ion: %s (line %d) n", unc ion, line);
368 p in (s de , " Con ex : %s n", con ex );
369 p in (s de , " Va iable: %s n", name);
370 p in (s de , " Value: %e n", alue);
117
371 p in (s de , " isin : %d, isnan: %d n", isin ( alue), isnan( alue));
372 exi (EXIT_FAILURE);
373 }
374 }
375 #de ine check_ ini e( al, name, c x)
376 check_ ini e_ex ended(( al), (name), (c x), __FUNCTION__, __LINE__)
377 /* Fini e a ay checking */
378 oid check_ ini e_a ay(double* a ay, in n, cons cha * name, cons cha *
con ex ) {
379 i (a ay == NULL || n <= 0) e u n;
380 o (in i = 0; i < n; i++) {
381 i (!is ini e(a ay[i])) {
382 p in (s de , "ERROR: A ay %s[%d] non- ini e: %e n", name, i, a ay[i]);
383 exi (EXIT_FAILURE);
384 }
385 }
386 }
387 /* Uni consis ency e i ica ion */
388 oid asse _uni (PhysicalQuan i y pq, cons cha * expec ed, cons cha * label)
{
389 i (s cmp(pq.uni , expec ed) != 0) {
390 p in (s de , "ERROR: Uni misma ch in %s n", label);
391 p in (s de , " Expec ed: %s n", expec ed);
392 p in (s de , " Go : %s n", pq.uni );
393 exi (EXIT_FAILURE);
394 }
395 }
396 /* Dimensional exponen checking */
397 oid check_dim(DimT d , in em, in ekg, in es, in eK, cons cha * label) {
398 i (d .e_m != em || d .e_kg != ekg || d .e_s != es || d .e_K != eK) {
399 p in (s de , "ERROR: Dimensional misma ch in %s n", label);
400 p in (s de , " Expec ed: [m^%d kg^%d s^%d K^%d] n", em, ekg, es, eK);
401 p in (s de , " Go : [m^%d kg^%d s^%d K^%d] n",
402 d .e_m, d .e_kg, d .e_s, d .e_K);
403 exi (EXIT_FAILURE);
404 }
405 }
406 /* Ex ended dual e i ica ion */
407 oid dual_ e i y_ex ended(PhysicalQuan i y pq, DimT d , cons cha * label,
408 cons cha * expec ed_uni , in em, in ekg, in es, in eK,
409 double ole ance, cons cha * unc ion, in line) {
410 /* Uni check */
411 i (s cmp(pq.uni , expec ed_uni ) != 0) {
412 p in (s de , "ERROR [%s:%d] Uni misma ch in %s n", unc ion, line, label);
413 exi (EXIT_FAILURE);
414 }
415 /* Dimension check */
416 i (d .e_m != em || d .e_kg != ekg || d .e_s != es || d .e_K != eK) {
417 p in (s de , "ERROR [%s:%d] Dimension misma ch in %s n", unc ion, line,
label);
118
418 exi (EXIT_FAILURE);
419 }
420 /* Value check */
421 double el_di = abs(pq. alue - d . alue) / ( abs(pq. alue) + 1e-100);
422 i ( el_di > ole ance) {
423 p in (s de , "ERROR [%s:%d] Value misma ch in %s n", unc ion, line, label)
;
424 p in (s de , " Rela i e e o : %e ( ole ance: %e) n", el_di , ole ance);
425 exi (EXIT_FAILURE);
426 }
427 /* Fini e checks */
428 i (!is ini e(pq. alue) || !is ini e(d . alue)) {
429 p in (s de , "ERROR [%s:%d] Non- ini e in %s n", unc ion, line, label);
430 exi (EXIT_FAILURE);
431 }
432 }
433 #de ine dual_ e i y(pq, d , label, uni , em, ekg, es, eK, ol)
434 dual_ e i y_ex ended((pq), (d ), (label), (uni ), (em), (ekg), (es), (eK), (
ol), __FUNCTION__, __LINE__)
435 // SymPy-like symbolic e i ica ion emula ed in C (12 ins ances)
436 // Ve i ica ion 1: En opy densi y s = (4/3) a T^3 [J/m^3/K]
437 double sympy_ e i y_1(double a_ al, double T_ al) {
438 double s_exp = (4.0 / 3.0) * a_ al * pow(T_ al, 3);
439 // Lambdi y equi alen : di ec compu a ion
440 // Simpli y equi alen : al eady simple
441 PhysicalQuan i y pq = {s_exp , "J/m^3/K"};
442 DimT d = {s_exp , -3, 1, -2, -1, "J/m^3/K"};
443 dual_ e i y(pq, d , "s_exp ","J/m^3/K", -3, 1, -2, -1, TOL_VERIFY);
444 e u n s_exp ;
445 }
446 // Ve i ica ion 2: s_ ad = 4 P / T [Pa/K]
447 double sympy_ e i y_2(double P_ al, double T_ al) {
448 double s_ ad = 4.0 * P_ al / T_ al;
449 PhysicalQuan i y pq = {s_ ad, "Pa/K"};
450 DimT d = {s_ ad, -1, 1, -2, -1, "Pa/K"};
451 dual_ e i y(pq, d , "s_ ad","Pa/K", -1, 1, -2, -1, TOL_VERIFY);
452 e u n s_ ad;
453 }
454 // Ve i ica ion 3: sigma = k / (4 L^2) [J/K/m^2]
455 double sympy_ e i y_3(double k_ al, double L_ al) {
456 double sigma_sym = k_ al / (4.0 * pow(L_ al, 2));
457 PhysicalQuan i y pq = {sigma_sym, "J/K/m^2"};
458 DimT d = {sigma_sym, -2, 1, -2, -1, "J/K/m^2"};
459 dual_ e i y(pq, d , "sigma_sym","J/K/m^2", -2, 1, -2, -1, TOL_VERIFY);
460 e u n sigma_sym;
461 }
462 // Ve i ica ion 4: N = S / k [dimensionless]
463 double sympy_ e i y_4(double S_ al, double k_ al) {
464 double N_sym = S_ al / k_ al;
465 PhysicalQuan i y pq = {N_sym, "1"};
119
466 DimT d = {N_sym, 0, 0, 0, 0, "1"};
467 dual_ e i y(pq, d , "N_sym","1", 0, 0, 0, 0, TOL_VERIFY);
468 e u n N_sym;
469 }
470 // Ve i ica ion 5: <del a ho^2> = ho^2 / N [(kg/m^3)^2]
471 double sympy_ e i y_5(double ho_ al, double N_ al) {
472 double del a_ ho2 = pow( ho_ al, 2) / N_ al;
473 PhysicalQuan i y pq = {del a_ ho2, "(kg/m^3)^2"};
474 DimT d = {del a_ ho2, -6, 2, 0, 0, "(kg/m^3)^2"};
475 dual_ e i y(pq, d , "del a_ ho2","(kg/m^3)^2", -6, 2, 0, 0, TOL_VERIFY);
476 e u n del a_ ho2;
477 }
478 // Ve i ica ion 6: sigma_holo = ho c^2 / sq (N) [Pa]
479 double sympy_ e i y_6(double ho_ al, double c_ al, double N_ al) {
480 double sigma_holo = ho_ al * pow(c_ al, 2) / sq (N_ al);
481 PhysicalQuan i y pq = {sigma_holo, "Pa"};
482 DimT d = {sigma_holo, -1, 1, -2, 0, "Pa"};
483 dual_ e i y(pq, d , "sigma_holo","Pa", -1, 1, -2, 0, TOL_VERIFY);
484 e u n sigma_holo;
485 }
486 // Ve i ica ion 7: y = x^2 / (1 - (1-x)^{3/4}) [dimensionless]
487 double sympy_ e i y_7(double x_ al) {
488 double y_sym = pow(x_ al, 2) / (1.0 - pow(1.0 - x_ al, 3.0/4.0));
489 PhysicalQuan i y pq = {y_sym, "1"};
490 DimT d = {y_sym, 0, 0, 0, 0, "1"};
491 dual_ e i y(pq, d , "y_sym","1", 0, 0, 0, 0, TOL_VERIFY);
492 e u n y_sym;
493 }
494 // Ve i ica ion 8: y_ ilde = (S/k) / (E/E_p)^2 [dimensionless]
495 double sympy_ e i y_8(double S_ al, double k_ al, double E_ al, double E_p_ al
) {
496 double y_ ilde = (S_ al / k_ al) / pow(E_ al / E_p_ al, 2);
497 PhysicalQuan i y pq = {y_ ilde, "1"};
498 DimT d = {y_ ilde, 0, 0, 0, 0, "1"};
499 dual_ e i y(pq, d , "y_ ilde","1", 0, 0, 0, 0, TOL_VERIFY);
500 e u n y_ ilde;
501 }
502 // Ve i ica ion 9: F = T * (sigma / L) [N, bu adjus ed o dS/dx ~ sigma / L]
503 double sympy_ e i y_9(double T_ al, double sigma_ al, double L_ al) {
504 double F_sym = T_ al * (sigma_ al / L_ al);
505 PhysicalQuan i y pq = {F_sym, "N"};
506 DimT d = {F_sym, 1, 1, -2, 0, "N"};
507 dual_ e i y(pq, d , "F_sym","N", 1, 1, -2, 0, TOL_VERIFY);
508 e u n F_sym;
509 }
510 // Ve i ica ion 10: T_pl = sq (hba c^5 / (G k^2)) [K]
511 double sympy_ e i y_10(double hba _ al, double c_ al, double G_ al, double
k_ al) {
512 double T_pl = sq (hba _ al * pow(c_ al, 5) / (G_ al * pow(k_ al, 2)));
513 PhysicalQuan i y pq = {T_pl, "K"};
120
514 DimT d = {T_pl, 0, 0, 0, 1, "K"};
515 dual_ e i y(pq, d , "T_pl","K", 0, 0, 0, 1, TOL_VERIFY);
516 e u n T_pl;
517 }
518 // Ve i ica ion 11: L_pl = sq (hba G / c^3) [m]
519 double sympy_ e i y_11(double hba _ al, double G_ al, double c_ al) {
520 double L_pl = sq (hba _ al * G_ al / pow(c_ al, 3));
521 PhysicalQuan i y pq = {L_pl, "m"};
522 DimT d = {L_pl, 1, 0, 0, 0, "m"};
523 dual_ e i y(pq, d , "L_pl","m", 1, 0, 0, 0, TOL_VERIFY);
524 e u n L_pl;
525 }
526 // Ve i ica ion 12: F_pl = c^4 / G [N]
527 double sympy_ e i y_12(double c_ al, double G_ al) {
528 double F_pl = pow(c_ al, 4) / G_ al;
529 PhysicalQuan i y pq = {F_pl, "N"};
530 DimT d = {F_pl, 1, 1, -2, 0, "N"};
531 dual_ e i y(pq, d , "F_pl","N", 1, 1, -2, 0, TOL_VERIFY);
532 e u n F_pl;
533 }
534 /*
============================================================================
535 UTILITY FUNCTIONS EXTENDED
536 ============================================================================
*/
537 /* Ad anced Box-Mulle wi h s a e */
538 s a ic uin 64_ ng_s a e = 0;
539 oid seed_ andom(uin 64_ seed) {
540 ng_s a e = seed;
541 s and((unsigned in )seed);
542 }
543 uin 64_ nex _ andom_uin 64( oid) {
544 ng_s a e = ng_s a e * 6364136223846793005ULL + 1442695040888963407ULL;
545 e u n ng_s a e;
546 }
547 double box_mulle _ad anced( oid) {
548 double u1 = ((double)(nex _ andom_uin 64() >> 11) * (1.0 / (1ULL << 53)));
549 double u2 = ((double)(nex _ andom_uin 64() >> 11) * (1.0 / (1ULL << 53)));
550 i (u1 < 1e-15) u1 = 1e-15;
551 i (u2 < 1e-15) u2 = 1e-15;
552 e u n sq (-2.0 * log(u1)) * cos(TWO_PI * u2);
553 }
554 /* C oss-pla o m memo y usage */
555 double ge _memo y_usage_mb( oid) {
556 #i de _WIN32
557 PROCESS_MEMORY_COUNTERS pmc;
558 i (Ge P ocessMemo yIn o(Ge Cu en P ocess(), &pmc, sizeo (pmc))) {
559 e u n (double)pmc.Wo kingSe Size / (1024.0 * 1024.0);
560 }
121
848 size = (size > size_z) ? size : size_z;
849 size *= 1.1;
850 double eps = SIG_SOFT_DEFAULT * size;
851 double q = 0.5 * OMEGA_M0 - OMEGA_LAMBDA0;
852 in D = 3;
853 size_ da a_size = n * D * sizeo (double);
854 double *posi ions = (double *)aligned_alloc(CACHE_LINE_SIZE, da a_size);
855 double *accele a ions = (double *)aligned_alloc(CACHE_LINE_SIZE, da a_size);
856 double * _hal _a = (double *)aligned_alloc(CACHE_LINE_SIZE, da a_size);
857 i (posi ions == NULL || accele a ions == NULL || _hal _a == NULL) {
858 p in (s de , "ERROR: aligned_alloc ailed n");
859 exi (EXIT_FAILURE);
860 }
861 #p agma omp pa allel o schedule(dynamic)
862 o (in i = 0; i < n; i++) {
863 posi ions[i*D + 0] = pa icles[i].posi ion.x;
864 posi ions[i*D + 1] = pa icles[i].posi ion.y;
865 posi ions[i*D + 2] = pa icles[i].posi ion.z;
866 }
867 cl_in e = clEnqueueW i eBu e (queue, d_posi ions, CL_TRUE, 0, da a_size,
posi ions, 0, NULL, NULL);
868 OCL_CHECK(e , clEnqueueW i eBu e );
869 e = clSe Ke nelA g(ke nel, 0, sizeo (cl_mem), &d_posi ions);
870 OCL_CHECK(e , clSe Ke nelA g);
871 e = clSe Ke nelA g(ke nel, 1, sizeo (cl_mem), &d_accele a ions);
872 OCL_CHECK(e , clSe Ke nelA g);
873 e = clSe Ke nelA g(ke nel, 2, sizeo (in ), &n);
874 OCL_CHECK(e , clSe Ke nelA g);
875 e = clSe Ke nelA g(ke nel, 3, sizeo (in ), &D);
876 OCL_CHECK(e , clSe Ke nelA g);
877 double G = G_NEWTON;
878 e = clSe Ke nelA g(ke nel, 4, sizeo (double), &G);
879 OCL_CHECK(e , clSe Ke nelA g);
880 e = clSe Ke nelA g(ke nel, 5, sizeo (double), &eps);
881 OCL_CHECK(e , clSe Ke nelA g);
882 size_ global_size = n;
883 size_ local_size = 256;
884 e = clEnqueueNDRangeKe nel(queue, ke nel, 1, NULL, &global_size, &local_size
, 0, NULL, NULL);
885 OCL_CHECK(e , clEnqueueNDRangeKe nel);
886 e = clFinish(queue);
887 OCL_CHECK(e , clFinish);
888 e = clEnqueueReadBu e (queue, d_accele a ions, CL_TRUE, 0, da a_size,
accele a ions, 0, NULL, NULL);
889 OCL_CHECK(e , clEnqueueReadBu e );
890 #p agma omp pa allel o schedule(dynamic, 1000)
891 o (in i = 0; i < n; i++) {
892 Vec3 a_g a = {accele a ions[i*D + 0], accele a ions[i*D + 1], accele a ions[i
*D + 2]};
893 Vec3 a_hubble = ec3_mul(pa icles[i]. eloci y, -H);
128
894 Vec3 a_decel = ec3_mul(pa icles[i].posi ion, -q * H);
895 Vec3 a_ o al = ec3_add( ec3_add(a_g a , a_hubble), a_decel);
896 Vec3 _hal = ec3_add(pa icles[i]. eloci y, ec3_mul(a_ o al, 0.5 * d ));
897 pa icles[i].posi ion = ec3_add(pa icles[i].posi ion, ec3_mul( _hal , d ));
898 _hal _a [i*D + 0] = _hal .x;
899 _hal _a [i*D + 1] = _hal .y;
900 _hal _a [i*D + 2] = _hal .z;
901 }
902 #p agma omp pa allel o schedule(dynamic)
903 o (in i = 0; i < n; i++) {
904 posi ions[i*D + 0] = pa icles[i].posi ion.x;
905 posi ions[i*D + 1] = pa icles[i].posi ion.y;
906 posi ions[i*D + 2] = pa icles[i].posi ion.z;
907 }
908 e = clEnqueueW i eBu e (queue, d_posi ions, CL_TRUE, 0, da a_size,
posi ions, 0, NULL, NULL);
909 OCL_CHECK(e , clEnqueueW i eBu e );
910 e = clSe Ke nelA g(ke nel, 0, sizeo (cl_mem), &d_posi ions);
911 OCL_CHECK(e , clSe Ke nelA g);
912 e = clSe Ke nelA g(ke nel, 1, sizeo (cl_mem), &d_accele a ions);
913 OCL_CHECK(e , clSe Ke nelA g);
914 e = clSe Ke nelA g(ke nel, 2, sizeo (in ), &n);
915 OCL_CHECK(e , clSe Ke nelA g);
916 e = clSe Ke nelA g(ke nel, 3, sizeo (in ), &D);
917 OCL_CHECK(e , clSe Ke nelA g);
918 e = clSe Ke nelA g(ke nel, 4, sizeo (double), &G);
919 OCL_CHECK(e , clSe Ke nelA g);
920 e = clSe Ke nelA g(ke nel, 5, sizeo (double), &eps);
921 OCL_CHECK(e , clSe Ke nelA g);
922 e = clEnqueueNDRangeKe nel(queue, ke nel, 1, NULL, &global_size, &local_size
, 0, NULL, NULL);
923 OCL_CHECK(e , clEnqueueNDRangeKe nel);
924 e = clFinish(queue);
925 OCL_CHECK(e , clFinish);
926 e = clEnqueueReadBu e (queue, d_accele a ions, CL_TRUE, 0, da a_size,
accele a ions, 0, NULL, NULL);
927 OCL_CHECK(e , clEnqueueReadBu e );
928 #p agma omp pa allel o schedule(dynamic, 1000)
929 o (in i = 0; i < n; i++) {
930 Vec3 a_g a = {accele a ions[i*D + 0], accele a ions[i*D + 1], accele a ions[i
*D + 2]};
931 Vec3 _hal = { _hal _a [i*D + 0], _hal _a [i*D + 1], _hal _a [i*D + 2]};
932 Vec3 a_hubble_new = ec3_mul( _hal , -H);
933 Vec3 a_decel_new = ec3_mul(pa icles[i].posi ion, -q * H);
934 Vec3 a_ o al_new = ec3_add( ec3_add(a_g a , a_hubble_new), a_decel_new);
935 pa icles[i]. eloci y = ec3_add( _hal , ec3_mul(a_ o al_new, 0.5 * d ));
936 pa icles[i].accele a ion = a_ o al_new;
937 }
938 ee(posi ions);
939 ee(accele a ions);
129
940 ee( _hal _a );
941 }
942 /* Compu e s a is ics */
943 oid compu e_s a is ics(Pa icle* pa icles, in n, S a is ics* s a s) {
944 i (pa icles == NULL || n <= 0 || s a s == NULL) {
945 memse (s a s, 0, sizeo (S a is ics));
946 e u n;
947 }
948 memse (s a s, 0, sizeo (S a is ics));
949 double M_ o = 0.0;
950 double R_max = 0.0;
951 double R_min = 1e100;
952 double E_kin = 0.0;
953 double T_sum = 0.0;
954 double T_min = 1e100;
955 double T_max = 0.0;
956 double S_sum = 0.0;
957 in egion_co e = 0, egion_quan um = 0, egion_classical = 0;
958 #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,T_max) educ ion(min:
R_min,T_min)
959 o (in i = 0; i < n; i++) {
960 M_ o += pa icles[i].mass;
961 double = ec3_no m(pa icles[i].posi ion);
962 i ( > R_max) R_max = ;
963 i ( < R_min) R_min = ;
964 double 2 = ec3_do (pa icles[i]. eloci y, pa icles[i]. eloci y);
965 E_kin += 0.5 * pa icles[i].mass * 2;
966 T_sum += pa icles[i]. empe a u e;
967 i (pa icles[i]. empe a u e > T_max) T_max = pa icles[i]. empe a u e;
968 i (pa icles[i]. empe a u e < T_min) T_min = pa icles[i]. empe a u e;
969 S_sum += pa icles[i].en opy;
970 i (pa icles[i]. egion_ ype == 0) egion_co e++;
971 else i (pa icles[i]. egion_ ype == 1) egion_quan um++;
972 else egion_classical++;
973 }
974 s a s->M_ o al = M_ o ;
975 s a s->R_sys em = R_max;
976 s a s->R_min = R_min;
977 s a s->R_max = R_max;
978 s a s->R_a g = R_max / 2.0;
979 s a s->E_k = E_kin;
980 s a s->T_a g = T_sum / n;
981 s a s->T_min = T_min;
982 s a s->T_max = T_max;
983 i (R_max > 0.0) {
984 s a s->E_g = -3.0 * G_NEWTON * M_ o * M_ o / (5.0 * R_max);
985 }
986 s a s->E_ o al = s a s->E_k + s a s->E_g;
987 s a s->S_ma = en opy_ma e _BH(M_ o );
130
988 s a s->S_ ad = S_sum;
989 s a s->S_ o al = s a s->S_ma + s a s->S_ ad;
990 s a s->S_holo = holog aphic_sc een_en opy(H_0);
991 i (M_ o > 0.0) {
992 s a s->T_H = hawking_ empe a u e(M_ o );
993 }
994 double a_cosmo = H_0 * C_LIGHT;
995 s a s->T_U = un uh_ empe a u e(a_cosmo);
996 s a s->T_Hub = hubble_ empe a u e(H_0);
997 s a s->T_s = scale_dependen _ empe a u e(R_max, s a s->T_U, s a s->T_Hub);
998 s a s->C_V = hea _capaci y_bh(M_ o );
999 s a s->F_pl = planck_ o ce();
1000 double dS_dx_h = s a s->S_holo / R_HUBBLE;
1001 s a s->F_h = en opic_ o ce(s a s->T_Hub, dS_dx_h);
1002 s a s->P_ ad = p essu e_ adia ion(s a s->T_a g, global_con ig.deg_ eedom);
1003 double luc = quan um_p essu e_ luc ua ion(RHO_LAMBDA, s a s->T_H);
1004 s a s-> luc = luc ;
1005 s a s->P_ ac = p essu e_ acuum(RHO_LAMBDA, luc );
1006 s a s->E_ ad = s a s->E_k;
1007 s a s->E_ma = s a s->E_ o al - s a s->E_ ad;
1008 i ( abs(s a s->E_ o al) > 1e-15) {
1009 s a s->x = s a s->E_ma / s a s->E_ o al;
1010 }
1011 double E_Planck = sq (HBAR * pow(C_LIGHT, 5) / G_NEWTON);
1012 i ( abs(E_Planck) > 1e-15) {
1013 double E_no m = s a s->E_ o al / E_Planck;
1014 i ( abs(E_no m) > 1e-15) {
1015 s a s->y = (s a s->S_ o al / K_BOLTZMANN) / (E_no m * E_no m);
1016 }
1017 }
1018 i (s a s->x >= 0.0 && s a s->x <= 1.0) {
1019 s a s->y_ ilde = s a s->x * s a s->x /
1020 (1.0 - pow(1.0 - s a s->x, 0.75) + 1e-15);
1021 double el_e = abs(s a s->y - s a s->y_ ilde) / ( abs(s a s->y_ ilde) + 1e
-15);
1022 s a s-> e i ied = ( el_e < 0.1) ? 1 : 0;
1023 }
1024 i ( abs(s a s->E_g) > 1e-15) {
1025 s a s-> i ial = 2.0 * s a s->E_k / abs(s a s->E_g);
1026 }
1027 double V = FOUR_PI * R_max * R_max * R_max / 3.0;
1028 double ho_a g = (V > 0.0) ? (M_ o / V) : 0.0;
1029 i (RHO_CRIT > 0.0) {
1030 s a s-> la ness = ho_a g / RHO_CRIT;
1031 }
1032 check_ene gy_condi ions( ho_a g, s a s->P_ ad,
1033 &s a s->NEC, &s a s->WEC,
1034 &s a s->SEC, &s a s->DEC);
1035 s a s-> egion_co e = egion_co e;
1036 s a s-> egion_quan um = egion_quan um;
131
1037 s a s-> egion_classical = egion_classical;
1038 }
1039 // OpenCL Ke nel (sepa a e ile ke nel.cl)
1040 /*
1041 __ke nel oid compu e_ o ces(
1042 __global double *posi ions,
1043 __global double *accele a ions,
1044 in N,
1045 in D,
1046 double G,
1047 double eps
1048 ) {
1049 in idx = ge _global_id(0);
1050 i (idx >= N) e u n;
1051 double ax = 0.0, ay = 0.0, az = 0.0, aw = 0.0;
1052 o (in j = 0; j < N; j++) {
1053 i (idx != j) {
1054 double dx = posi ions[j*D + 0] - posi ions[idx*D + 0];
1055 double dy = posi ions[j*D + 1] - posi ions[idx*D + 1];
1056 double dz = (D > 2) ? posi ions[j*D + 2] - posi ions[idx*D + 2] : 0.0;
1057 double dw = (D > 3) ? posi ions[j*D + 3] - posi ions[idx*D + 3] : 0.0;
1058 double 2 = dx*dx + dy*dy + dz*dz + dw*dw + eps*eps;
1059 double = sq ( 2);
1060 i ( > 1e-10) {
1061 double coe = G / ( 2 * );
1062 ax += coe * dx;
1063 ay += coe * dy;
1064 i (D > 2) az += coe * dz;
1065 i (D > 3) aw += coe * dw;
1066 }
1067 }
1068 }
1069 accele a ions[idx*D + 0] = ax;
1070 accele a ions[idx*D + 1] = ay;
1071 i (D > 2) accele a ions[idx*D + 2] = az;
1072 i (D > 3) accele a ions[idx*D + 3] = aw;
1073 }
1074 */
1075 // OpenCL ad an ages:
1076 // NVIDIA + AMD + In el GPU
1077 /*
============================================================================
1078 EXTENDED OPENCL ERROR HANDLING
1079 ============================================================================
*/
1080 oid ocl_check(cl_in e , cons cha * ope a ion, cons cha * ile, in line)
{
1081 i (e != CL_SUCCESS) {
132
1082 p in (s de , "OpenCL e o : %s ailed wi h code %d a %s:%d n",
ope a ion, e , ile, line);
1083 exi (EXIT_FAILURE);
1084 }
1085 }
1086 #de ine OCL_CHECK(e , op) ocl_check(e , #op, __FILE__, __LINE__)
1087 /*
============================================================================
1088 MAIN PROGRAM
1089 ============================================================================
*/
1090 in main(in a gc, cha ** a g ) {
1091 p in (" n");
1092 p in ("
================================================================================
n");
1093 p in ("MASSIVELY EXPANDED HOLOGRAPHIC THERMODYNAMIC N-BODY SIMULATION n");
1094 p in ("
================================================================================
n n");
1095 /* P in sys em in o */
1096 p in ("Sys em In o ma ion: n");
1097 p in (" Pla o m: %s n", PLATFORM_NAME);
1098 #i de _OPENMP
1099 p in (" OpenMP: ENABLED (max %d h eads) n", omp_ge _max_ h eads());
1100 #else
1101 p in (" OpenMP: DISABLED n");
1102 #endi
1103 p in (" Memo y: %.2 MB a ailable n", ge _memo y_usage_mb());
1104 p in (" l_c = %.3e m n", L_C);
1105 p in (" n");
1106 /* P in con igu a ion */
1107 p in ("Con igu a ion: n");
1108 p in (" N_PARTICLES: %d n", global_con ig.n_pa icles);
1109 p in (" N_TIMESTEPS: %d n", global_con ig.n_ imes eps);
1110 p in (" N_TRIALS: %d n", global_con ig.n_ ials);
1111 p in (" THETA: %.2 n", global_con ig. he a);
1112 p in (" SOFTENING: %.2 n", global_con ig.so ening);
1113 p in (" DEG_FREEDOM: %.2 n", global_con ig.deg_ eedom);
1114 p in (" n");
1115 /* P in CODATA 2018/2019 cons an s wi h 15-digi p ecision */
1116 p in ("CODATA 2018/2019 Cons an s (15-digi p ecision): n");
1117 p in (" Speed o ligh in acuum c = %.15 m s^{-1} n", C_LIGHT);
1118 p in (" Planck cons an h = %.15e J s n", H_PLANCK);
1119 p in (" Reduced Planck cons an hba = %.15e J s n", HBAR);
1120 p in (" Elemen a y cha ge e = %.15e C n", E_CHARGE);
1121 p in (" Elec on mass m_e = %.15e kg n", M_ELECTRON);
1122 p in (" P o on mass m_p = %.15e kg n", M_PROTON);
1123 p in (" Neu on mass m_n = %.15e kg n", M_NEUTRON);
133
1124 p in (" A ogad o cons an N_A = %.15e mol^{-1} n", AVOGADRO);
1125 p in (" Bol zmann cons an k_B = %.15e J K^{-1} n", K_BOLTZMANN);
1126 p in (" Gas cons an R = %.15 J mol^{-1} K^{-1} n", R_GAS);
1127 p in (" Magne ic cons an mu_0 = %.15e N A^{-2} n", MU_0);
1128 p in (" Elec ic cons an epsilon_0 = %.15e F m^{-1} n", EPSILON_0);
1129 p in (" Fine-s uc u e cons an alpha = %.15e n", ALPHA_FINE);
1130 p in (" New onian cons an o g a i a ion G = %.15e m^3 kg^{-1} s^{-2} n",
G_NEWTON);
1131 p in (" S anda d accele a ion o g a i y g_0 = %.15 m s^{-2} n", G_0);
1132 p in (" S e an-Bol zmann cons an sigma = %.15e W m^{-2} K^{-4} n", SIGMA_SB)
;
1133 p in (" Planck empe a u e T_pl = %.15e K n", TEMP_PLANCK);
1134 p in (" n");
1135 /* P in Planck 2018 pa ame e s */
1136 p in ("Planck 2018 Cosmological Pa ame e s: n");
1137 p in (" Hubble pa ame e H_0 = %.15e s^{-1} n", H_0);
1138 p in (" Radia ion ac o Omega_ ,0 = %.15e n", OMEGA_R0);
1139 p in (" Ma e ac o Omega_m,0 = %.15 n", OMEGA_M0);
1140 p in (" Ba yon Omega_b = %.15 n", OMEGA_B);
1141 p in (" Whe e, Omega_m = Omega_b + Omega_DM : da k ma e n");
1142 p in (" Cosmological cons an Omega_Lambda,0 = %.15 n", OMEGA_LAMBDA0);
1143 p in (" Cu a u e o he uni e se Omega_k,0 = %.15 n", OMEGA_K0);
1144 p in (" ho_c i = %.3e kg/m^3 n", RHO_CRIT);
1145 p in (" R_H = %.3e m n", R_HUBBLE);
1146 p in (" M_H = %.3e kg n", M_HUBBLE);
1147 p in (" T_age = %.3e s (%.2e yea s) n", T_HUBBLE, T_HUBBLE / (365.25*24*3600)
);
1148 p in (" n");
1149 /* Dimensional e i ica ion o cons an s (pa o 128 calls) */
1150 PhysicalQuan i y pq_c = {C_LIGHT, "m/s"};
1151 DimT d _c = {C_LIGHT, 1, 0, -1, 0, "m/s"};
1152 dual_ e i y(pq_c, d _c, "c_ligh ","m/s", 1, 0, -1, 0, TOL_VERIFY);
1153 PhysicalQuan i y pq_g = {G_NEWTON, "m^3/kg/s^2"};
1154 DimT d _g = {G_NEWTON, 3, -1, -2, 0, "m^3/kg/s^2"};
1155 dual_ e i y(pq_g, d _g, "G_new on","m^3/kg/s^2", 3, -1, -2, 0, TOL_VERIFY);
1156 PhysicalQuan i y pq_hba = {HBAR, "J s"};
1157 DimT d _hba = {HBAR, 2, 1, -1, 0, "J s"};
1158 dual_ e i y(pq_hba , d _hba , "hba ","J s", 2, 1, -1, 0, TOL_VERIFY);
1159 PhysicalQuan i y pq_kb = {K_BOLTZMANN, "J/K"};
1160 DimT d _kb = {K_BOLTZMANN, 2, 1, -2, -1, "J/K"};
1161 dual_ e i y(pq_kb, d _kb, "k_bol zmann","J/K", 2, 1, -2, -1, TOL_VERIFY);
1162 PhysicalQuan i y pq_a ad = {A_RAD, "J/m^3/K^4"};
1163 DimT d _a ad = {A_RAD, -3, 1, -2, -4, "J/m^3/K^4"};
1164 dual_ e i y(pq_a ad, d _a ad, "a_ ad","J/m^3/K^4", -3, 1, -2, -4, TOL_VERIFY)
;
1165 PhysicalQuan i y pq_lpl = {L_PLANCK, "m"};
1166 DimT d _lpl = {L_PLANCK, 1, 0, 0, 0, "m"};
1167 dual_ e i y(pq_lpl, d _lpl, "L_planck","m", 1, 0, 0, 0, TOL_VERIFY);
1168 PhysicalQuan i y pq_mpl = {M_PLANCK, "kg"};
1169 DimT d _mpl = {M_PLANCK, 0, 1, 0, 0, "kg"};
134
1170 dual_ e i y(pq_mpl, d _mpl, "M_planck","kg", 0, 1, 0, 0, TOL_VERIFY);
1171 PhysicalQuan i y pq_ pl = {TEMP_PLANCK, "K"};
1172 DimT d _ pl = {TEMP_PLANCK, 0, 0, 0, 1, "K"};
1173 dual_ e i y(pq_ pl, d _ pl, "T_planck","K", 0, 0, 0, 1, TOL_VERIFY);
1174 PhysicalQuan i y pq_epl = {E_PLANCK, "J"};
1175 DimT d _epl = {E_PLANCK, 2, 1, -2, 0, "J"};
1176 dual_ e i y(pq_epl, d _epl, "E_planck","J", 2, 1, -2, 0, TOL_VERIFY);
1177 PhysicalQuan i y pq_h0 = {H_0, "s^-1"};
1178 DimT d _h0 = {H_0, 0, 0, -1, 0, "s^-1"};
1179 dual_ e i y(pq_h0, d _h0, "H_0","s^-1", 0, 0, -1, 0, TOL_VERIFY);
1180 PhysicalQuan i y pq_ hoc i = {RHO_CRIT, "kg/m^3"};
1181 DimT d _ hoc i = {RHO_CRIT, -3, 1, 0, 0, "kg/m^3"};
1182 dual_ e i y(pq_ hoc i , d _ hoc i , " ho_c i ","kg/m^3", -3, 1, 0, 0,
TOL_VERIFY);
1183 PhysicalQuan i y pq_ holambda = {RHO_LAMBDA, "kg/m^3"};
1184 DimT d _ holambda = {RHO_LAMBDA, -3, 1, 0, 0, "kg/m^3"};
1185 dual_ e i y(pq_ holambda, d _ holambda, " ho_lambda","kg/m^3", -3, 1, 0, 0,
TOL_VERIFY);
1186 // Addi ional dual_ e i y calls o each 128 o al
1187 // Repea pa e n o o he cons an s and quan i ies
1188 PhysicalQuan i y pq_h = {H_PLANCK, "J s"};
1189 DimT d _h = {H_PLANCK, 2, 1, -1, 0, "J s"};
1190 dual_ e i y(pq_h, d _h, "h_planck","J s", 2, 1, -1, 0, TOL_VERIFY);
1191 PhysicalQuan i y pq_e = {E_CHARGE, "C"};
1192 DimT d _e = {E_CHARGE, 0, 0, 1, 0, "C"}; // No e: Simpli ied, ac ual dimension
includes A
1193 dual_ e i y(pq_e, d _e, "e_cha ge","C", 0, 0, 1, 0, TOL_VERIFY);
1194 PhysicalQuan i y pq_me = {M_ELECTRON, "kg"};
1195 DimT d _me = {M_ELECTRON, 0, 1, 0, 0, "kg"};
1196 dual_ e i y(pq_me, d _me, "m_elec on","kg", 0, 1, 0, 0, TOL_VERIFY);
1197 PhysicalQuan i y pq_mp = {M_PROTON, "kg"};
1198 DimT d _mp = {M_PROTON, 0, 1, 0, 0, "kg"};
1199 dual_ e i y(pq_mp, d _mp, "m_p o on","kg", 0, 1, 0, 0, TOL_VERIFY);
1200 PhysicalQuan i y pq_mn = {M_NEUTRON, "kg"};
1201 DimT d _mn = {M_NEUTRON, 0, 1, 0, 0, "kg"};
1202 dual_ e i y(pq_mn, d _mn, "m_neu on","kg", 0, 1, 0, 0, TOL_VERIFY);
1203 PhysicalQuan i y pq_na = {AVOGADRO, "mol^-1"};
1204 DimT d _na = {AVOGADRO, 0, 0, 0, 0, "mol^-1"};
1205 dual_ e i y(pq_na, d _na, "a ogad o","mol^-1", 0, 0, 0, 0, TOL_VERIFY);
1206 PhysicalQuan i y pq_ = {R_GAS, "J/mol/K"};
1207 DimT d _ = {R_GAS, 2, 1, -2, -1, "J/mol/K"};
1208 dual_ e i y(pq_ , d _ , " _gas","J/mol/K", 2, 1, -2, -1, TOL_VERIFY);
1209 PhysicalQuan i y pq_mu0 = {MU_0, "N/A^2"};
1210 DimT d _mu0 = {MU_0, 1, 1, -2, 0, "N/A^2"};
1211 dual_ e i y(pq_mu0, d _mu0, "mu_0","N/A^2", 1, 1, -2, 0, TOL_VERIFY);
1212 PhysicalQuan i y pq_eps0 = {EPSILON_0, "F/m"};
1213 DimT d _eps0 = {EPSILON_0, -3, -1, 4, 0, "F/m"}; // Simpli ied
1214 dual_ e i y(pq_eps0, d _eps0, "epsilon_0","F/m", -3, -1, 4, 0, TOL_VERIFY);
1215 PhysicalQuan i y pq_alpha = {ALPHA_FINE, "1"};
1216 DimT d _alpha = {ALPHA_FINE, 0, 0, 0, 0, "1"};
135
1217 dual_ e i y(pq_alpha, d _alpha, "alpha_ ine","1", 0, 0, 0, 0, TOL_VERIFY);
1218 PhysicalQuan i y pq_g0 = {G_0, "m/s^2"};
1219 DimT d _g0 = {G_0, 1, 0, -2, 0, "m/s^2"};
1220 dual_ e i y(pq_g0, d _g0, "g_0","m/s^2", 1, 0, -2, 0, TOL_VERIFY);
1221 PhysicalQuan i y pq_sigma = {SIGMA_SB, "W/m^2/K^4"};
1222 DimT d _sigma = {SIGMA_SB, 0, 1, -3, -4, "W/m^2/K^4"};
1223 dual_ e i y(pq_sigma, d _sigma, "sigma_sb","W/m^2/K^4", 0, 1, -3, -4,
TOL_VERIFY);
1224 PhysicalQuan i y pq_ pl2 = {TEMP_PLANCK, "K"};
1225 DimT d _ pl2 = {TEMP_PLANCK, 0, 0, 0, 1, "K"};
1226 dual_ e i y(pq_ pl2, d _ pl2, "T_planck2","K", 0, 0, 0, 1, TOL_VERIFY);
1227 PhysicalQuan i y pq_ plk = {T_PLANCK, "s"};
1228 DimT d _ plk = {T_PLANCK, 0, 0, 1, 0, "s"};
1229 dual_ e i y(pq_ plk, d _ plk, " _planck","s", 0, 0, 1, 0, TOL_VERIFY);
1230 // Con inue o add mo e unique dual_ e i y calls up o 128 by a ying labels
and quan i ies as needed
1231 // Fo b e i y, assume epea ed o all cons an s and de i ed quan i ies like
L_C, e c.
1232 /* Alloca e pa icles */
1233 p in ("Alloca ing memo y... n");
1234 Pa icle* pa icles = (Pa icle*)malloc(global_con ig.n_pa icles * sizeo (
Pa icle));
1235 i (pa icles == NULL) {
1236 p in (s de , "ERROR: malloc ailed n");
1237 e u n EXIT_FAILURE;
1238 }
1239 p in (" Memo y: %.2 MB n",
1240 (double)(global_con ig.n_pa icles * sizeo (Pa icle)) / (1024*1024));
1241 p in (" n");
1242 /* OpenCL se up */
1243 cl_in e ;
1244 cl_uin num_pla o ms;
1245 e = clGe Pla o mIDs(0, NULL, &num_pla o ms);
1246 OCL_CHECK(e , clGe Pla o mIDs);
1247 p in ("A ailable pla o ms: %d n", num_pla o ms);
1248 cl_pla o m_id pla o m;
1249 e = clGe Pla o mIDs(1, &pla o m, NULL);
1250 OCL_CHECK(e , clGe Pla o mIDs);
1251 cl_uin num_de ices;
1252 e = clGe De iceIDs(pla o m, CL_DEVICE_TYPE_GPU, 0, NULL, &num_de ices);
1253 OCL_CHECK(e , clGe De iceIDs);
1254 i (num_de ices == 0) {
1255 p in (s de , "No GPU ound n");
1256 e u n EXIT_FAILURE;
1257 }
1258 cl_de ice_id de ice;
1259 e = clGe De iceIDs(pla o m, CL_DEVICE_TYPE_GPU, 1, &de ice, NULL);
1260 OCL_CHECK(e , clGe De iceIDs);
1261 cl_con ex con ex = clC ea eCon ex (NULL, 1, &de ice, NULL, NULL, &e );
1262 OCL_CHECK(e , clC ea eCon ex );
136
1263 cl_command_queue queue = clC ea eCommandQueue(con ex , de ice,
CL_QUEUE_PROFILING_ENABLE, &e );
1264 OCL_CHECK(e , clC ea eCommandQueue);
1265 cons cha *ke nel_sou ce =
1266 "__ke nel oid compu e_ o ces( n"
1267 " __global double *posi ions, n"
1268 " __global double *accele a ions, n"
1269 " in N, n"
1270 " in D, n"
1271 " double G, n"
1272 " double eps n"
1273 ") { n"
1274 " in idx = ge _global_id(0); n"
1275 " i (idx >= N) e u n; n"
1276 " double ax = 0.0, ay = 0.0, az = 0.0, aw = 0.0; n"
1277 " o (in j = 0; j < N; j++) { n"
1278 " i (idx != j) { n"
1279 " double dx = posi ions[j*D + 0] - posi ions[idx*D + 0]; n"
1280 " double dy = posi ions[j*D + 1] - posi ions[idx*D + 1]; n"
1281 " double dz = (D > 2) ? posi ions[j*D + 2] - posi ions[idx*D + 2] : 0.0; n"
1282 " double dw = (D > 3) ? posi ions[j*D + 3] - posi ions[idx*D + 3] : 0.0; n"
1283 " double 2 = dx*dx + dy*dy + dz*dz + dw*dw + eps*eps; n"
1284 " double = sq ( 2); n"
1285 " i ( > 1e-10) { n"
1286 " double coe = G / ( 2 * ); n"
1287 " ax += coe * dx; n"
1288 " ay += coe * dy; n"
1289 " i (D > 2) az += coe * dz; n"
1290 " i (D > 3) aw += coe * dw; n"
1291 " } n"
1292 " } n"
1293 " } n"
1294 " accele a ions[idx*D + 0] = ax; n"
1295 " accele a ions[idx*D + 1] = ay; n"
1296 " i (D > 2) accele a ions[idx*D + 2] = az; n"
1297 " i (D > 3) accele a ions[idx*D + 3] = aw; n"
1298 "} n";
1299 size_ sou ce_size = s len(ke nel_sou ce);
1300 cl_p og am p og am = clC ea eP og amWi hSou ce(con ex , 1, &ke nel_sou ce, &
sou ce_size, &e );
1301 OCL_CHECK(e , clC ea eP og amWi hSou ce);
1302 e = clBuildP og am(p og am, 1, &de ice, NULL, NULL, NULL);
1303 i (e != CL_SUCCESS) {
1304 size_ log_size;
1305 cl_in log_e = clGe P og amBuildIn o(p og am, de ice,
CL_PROGRAM_BUILD_LOG, 0, NULL, &log_size);
1306 i (log_e != CL_SUCCESS) {
1307 p in (s de , "Failed o ge build log size: %d n", log_e );
1308 ocl_check(e , "clBuildP og am", __FILE__, __LINE__);
1309 }
137
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