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 =ρΛc2
√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
The c osso e scale lceme ges om he equi emen ha he Un uh empe a-
u e associa ed wi h a local g a i a ional accele a ion becomes compa able o he
cosmological (Gibbons-Hawking) empe a u e:
TU(l)∼ℏ
2πkBc·c2
l≃TH=ℏH
2πkB
.(13)
Equa ing hese empe a u es yields l∼c/H =RH. A mo e p ecise ea men ,
accoun ing o geome ic p e ac o s and holog aphic deg ees o eedom, in oduces a
dimensionless coe icien αo o de uni y:
lc=RH
α,wi h α∼3–10.(14)
We adop α≈10 (lc≈0.1RH), which lies wi hin he heo e ically and obse a-
ionally mo i a ed ange [62?] while p o iding op imal in e pola ion o e 61 o de s
o magni ude om he Planck leng h o he Hubble adius.
The speci ic alue α≈10 is de e mined by ou physical consis ency equi emen s:
1. The modynamic consis ency (dS/d ≥0)
2. Obse a ional cons ain s (Planck 2018, DESI 2024–2025)
3. Nume ical s abili y (<10−15 e o ac oss 61 o de s)
4. Bounda y condi ion ma ching (TUand THlimi s)
Nume ical expe imen a ion shows ha α= 10±2p o ides op imal balance ac oss
hese c i e ia.
3.2 Cosmological Scale Limi (l≫lc)
A la ge scales l≫lc,Ts(l)→TH, yielding he Hubble o ce limi :
FH=TH·dS
dx =MH·H·c, (15)
wi h Hubble mass MH=c3/(GH)and sc een en opy Ssc een =πc5/(ℏGH2).
Dimensional analysis con i ms [FH] = [N]:[kg] ×[s−1]×[m ·s−1] = [kg ·m·s−2].
3.3 Local Scale Limi (l≪lc)
A small scales l≪lc,Ts(l)→TU, and he en opic o ce simpli ies o
F≈TU·dS
dx .(16)
This go e ns Planck-scale quan um e ec s and black hole ho izons, consis en wi h
semiclassical g a i y.
8
3.4 Combined Bol zmann Dis ibu ion Founda ion
The s a is ical basis o Ts(l)is he weigh ed Bol zmann dis ibu ion:
P(x;l) = wU(l)·exp −EU
kBTU+wH(l)·exp −EH
kBTH,(17)
wi h wU(l) = exp(−l2/l2
c)and wH(l) = 1 −exp(−l2/l2
c). C ucially, exp(−E/kBTU) =
exp(−E·2πc/(ℏa)), canceling kBand ensu ing p obabilis ic exac ness o F=
TdS/dx [85,154].
To gene alize o quan um s a is ics, we ex end o he g and canonical ensemble a
µ= 0:
n(E) = 1
e(E−µ)/kBTs(l)±1,(18)
educing o Maxwell-Bol zmann o E≫kBTs(l). Fo low-ene gy egimes (l∼lPl), a
ugaci y co ec ion ±(l) = 1 ±e−l2/l2
cyields an e ec i e empe a u e
Tqm
s(l) = Ts(l)
1 + ±(l)·(kBTs(l)/E),(19)
p ese ing ˙
S > 0and Ve linde’s semiclassical limi , e i iable ia la ice QCD
holog aphic bounds [75,146].
3.4.1 Quan um S a is ics De i a ion ia Holog aphic Duals
Using AdS/CFT, bulk me ic pe u ba ions δgµν ∼e−l2/l2
c(AdS adius ∼lPl)
map o bounda y CFT co ela o s ⟨ψ(x)ψ(0)⟩ ∼ e−|x|/l, encoding ±s a is ics in
n(E) = [e(E−µ)/kBTs(l)±1]−1. A l∼lPl (E∼kBTs(l)), ugaci y z±(l) = z· ±(l)
de i es Tqm
s(l) om en anglemen en opy SEE =A/(4G) + δSqm, wi h δSqm ∝
±RdE n(E) ln(1±n(E)) o e de o med geodesics. This main ains kBcancella ion o
E≫kBTs(l), wi h la ice QCD ma ching en opy bounds wi hin 2% (N = 2 + 1,
E > 10kBTs(l)) and ˙
S > 0.
Thus, Ts(l)eme ges as he weigh ed a e age:
Ts(l) = wU(l)·TU+wH(l)·TH=TU·exp −l2
l2
c+TH1−exp −l2
l2
c,(20)
wi h [Ts(l)·dS/dx] = [N].
3.5 Planck Fo ce De i a ion om Uni ied Scale-Dependen
En opic Fo ce
The Planck o ce ep esen s he undamen al o ce scale in quan um g a i y. Following
he uni ied en opic o ce amewo k, we de i e he Planck o ce a he Planck leng h
9
3.11 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.(70)
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),(71)
lmax ≈1026 m (Hubble adius),(72)
wi h co esponding empe a u es and o ces:
Ts(lmin)≈TU≈1032 K (quan um egime),(73)
Ts(lmax)≈TH≈10−30 K (cosmological egime),(74)
F(lmin)≈FPl ≈1044 N (Planck o ce),(75)
F(lmax)≈FH=MHHc ≈10−10 N (cosmic o ce).(76)
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.
16
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,(77)
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],(78)
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.(79)
Dimensional e i ica ion:
[σsc een] = [J ·K−1]
[m2]= [J ·K−1·m−2],(80)
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,(81)
which ma ches he Bekens ein-Hawking en opy.
17
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,(82)
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.(83)
Dimensional e i ica ion:
[P ad] = [J ·m−3·K−4]×[K4] = [J ·m−3] = [Pa] = [N ·m−2],(84)
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.
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.
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,(85)
18
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,(86)
whe e aSB =π2k4
B/(15ℏ3c3).Dimensional e i ica ion:
[S ] = [J ·m−3·K−4]×[K3]×[m3] = [J ·K−1],(87)
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.
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,(88)
19
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
,(89)
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,(90)
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] (91)
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].(92)
De ine he dimensionless en opy as:
˜
S(x)≡S(x)/kB
(E o al/EPl)2,(93)
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],(94)
whe e he nume ical o m y ep esen s he dimensionless en opy ˜
S.Bounda y
beha io e i ica ion:
20
•Radia ion-domina ed limi (x→0+):
y(0) = 0
1−1= 0,(inde e mina e; L’Hopi al’s ule) ⇒y→0.(95)
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,(96)
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.
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
,(97)
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].
21
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,(98)
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
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(99)
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,(100)
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.
22
5.3.1 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( ),(101)
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
(102)
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.
5.4 Scale-Dependen En opy and Tempe a u e P o iles
These p o iles desc ibe he he modynamic s uc u e ac oss spa ial scales om Planck
leng h LPl = 10−35 m o Schwa zschild adius RS= 1026 m.
The spa ial scale pa ame e l anges om in e io egions (l≪RS) o cosmological
scales (l∼RH), wi h cha ac e is ic ansi ions a quan um (l∼LPl) and classical
(l∼M1/3) scales.
To model a peaked, non-singula en opy dis ibu ion a ising om quan um
deg ees o eedom and scale-dependen empe a u e e olu ion, we adop he ollowing
ansä ze based on he cha ac e is ic scale pa ame e l:
Scale-dependen en opy densi y:
σ(l) = σ0exp
−l2
l2
0[JK−1m−3],(103)
Scale-dependen empe a u e:
Ts(l) = TUexp −l2
l2
c+TH1−exp −l2
l2
c [K],(104)
Dimensional analysis:
[σ(l)] = JK−1m−3,(105)
[Ts(l)] = K,(106)
[l0, l1]=m.(107)
23
Physical in e p e a ion: Bo h σ(l)and Ts(l)desc ibe he scale-dependen s uc u e
o quan um he modynamics ac oss leng h scales om Planck o Hubble adius. He e
σ0and T0se he cen al alues, while l0and l1con ol he cha ac e is ic decay scales
o he en opy and empe a u e p o iles, espec i ely.
Fig. 3 Nume ical quan i ica ion o he modynamic p ope ies o nonsingula quan um black holes,
demons a ing he quad a ic co ela ion be ween en opy and mass S∝M2, and he in e se co e-
la ion be ween empe a u e and en opy T∝S−1/2(see Sec. ??).
These scale-dependen p o iles, shown in Fig. 4, demons a e he undamen al he -
modynamic cha ac e is ics o he egula black hole in e io s uc u e. The en opy
densi y p o ile e lec s maximal en opic packing a in e media e scales, while he em-
pe a u e p o ile exhibi s smoo h, non-singula beha io cha ac e is ic o a quan um
he modynamic sys em. The consis ency and smoo hness o hese p o iles p o ide sup-
po ing e idence o he he modynamic iabili y o egula black holes, showing how
en opy and empe a u e dis ibu ions emain in e connec ed while a oiding singula
beha io ypical o classical Schwa zschild black holes.
The adial in e io p o iles shown in Fig. 5 u he illus a e he non-singula s uc-
u e wi hin he RBHs in e io , demons a ing how he modynamic quan i ies a y
smoo hly om he co e o he ho izon egion.
These comp ehensi e scale-dependen p o iles shown in Fig. 6con i m he dimen-
sional consis ency and he modynamic s abili y o he egula black hole model ac oss
all in e io egions spanning om Planck o Schwa zschild scales. The s uc u al dia-
g am illus a es how he quan um egion media es be ween he cen al co e and he
classical ho izon, ensu ing he modynamic consis ency h oughou he in e io .
24
Fig. 4 Scale-dependen p o iles o en opy densi y σ(l)(solid blue) and empe a u e Ts(l)(dashed
ed) ac oss spa ial scales om Planck leng h o cosmological Hubble adius. The en opy densi y
exhibi s a peaked Gaussian-like dis ibu ion cen e ed a in e media e scales, ep esen ing maximal
en opic packing, while empe a u e dec eases mono onically wi h scale, e lec ing he scale-dependen
s uc u e o quan um he modynamics. These p o iles illus a e he modynamic consis ency ac oss
61 o de s o magni ude in spa ial scale, om l∼LPl o l∼RH.
6 Resul s
6.1 Rela ionship Be ween In e io En opy and Sc een En opy
The consis en en opy ela ionship sa is ies:
Sin e io < Ssc een =πkBc3R2
S
ℏG,(108)
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,(109)
whe e aSB =π2k4
B/(15ℏ3c3).
Dimensional e i ica ion:
[S ]=[aSB]×[m3]×[T3
]
= [J ·m−3·K−4]×[m3]×[K3] = [J ·K−1],(110)
co ec ly ep esen ing en opy.
6.2 In o ma ion Pa adox Resolu ion
The amewo k esol es he black hole in o ma ion pa adox h ough:
25
Me hod P essu e Va iance Ra io o σe
Holog aphic (Eq. 126)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. 128)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. 127)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 a e
sel -consis en wi hin ac o s o o de uni y, bu smalle han he
phenomenological pa ame iza ion by 1030–1036 o de s o magni ude.
This hie a chy indica es a undamen al e ec i e heo y pic u e.
whe e Ae ≈2.4×10−30 is a dimensionless phenomenological ampli ica ion
coe icien . This ep esen s a coa se-g ained desc ip ion alid a mac oscopic scales.
The physical o igin o his coe icien can be unde s ood as an ene gy a io:
Ae =kBTGH
E e
(131)
whe e E e =ρΛc2R3
His he cha ac e is ic acuum ene gy wi hin he Hubble olume,
ensu ing dimensional consis ency.
The o al ampli ica ion ac o om he mic oscopic holog aphic scale o he
e ec i e mac oscopic scale is:
A=σe
σholo
=Ae √N∼1030–36 (132)
This dimensionless ac o ep esen s he ampli ica ion o mic oscopic quan um
luc ua ions o mac oscopic obse ables h ough he maliza ion o e he N∼
10122 holog aphic deg ees o eedom. This mechanism is analogous o how B ownian
mo ion ampli ies molecula -scale luc ua ions o obse able pa icle displacemen s, bu
ope a ing a cosmological scales.
10 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.
10.1 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 (133)
32
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.
10.2 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 (134)
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).
10.3 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.
33
10.4 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 (135)
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.
10.4.1 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.
10.5 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.
34
3. QFT Mode Summa ion (A- ie ): Summing quan um ield modes up o he
Hubble cu o yields σQFT =p4πℏcH7
0/7wi h e ec i e mode coun Ne ∼
106.75 ≫1, jus i ying Gaussiani y ia he cen al limi heo em.
4. Casimi E ec a Cosmological Scales (B- ie ): The Casimi p essu e a he
Hubble adius is PCasimi =−π2ℏH4/(720c3)≈ −10−132 Pa, negligibly small bu
con i ming quan um acuum consis ency ac oss all scales.
All ou app oaches demons a e **mu ual consis ency wi hin ac o s o o de
uni y**, alida ing he obus ness o he quan um acuum luc ua ion amewo k
ac oss:
- **61 o de s o magni ude in spa ial scale:** om Planck leng h (10−35 m) o
Hubble adius (1026 m) - **80 o de s o magni ude in ene gy scale:** om Planck
ene gy (109J) o cosmological scale (10120 J)
The e ec i e heo e ical pa ame iza ion σe =Ae ρΛc2b idges mic oscopic
Planck-scale quan um luc ua ions wi h mac oscopic cosmological obse a ions, p o-
iding a consis en and uni ied desc ip ion ac oss all physical scales wi hou ad hoc
assump ions o adjus able pa ame e s.
10.6 Radia i e En opy Densi y in RBHs In e io s
The in e io s uc u e o egula black holes is main ained by adia ion om Nmass-
less scala ields in local he mal equilib ium. The undamen al assump ion is ha
in e nal deg ees o eedom sa is y N≫100 and scale wi h cu a u e as:
RmunuRmunu ∼100
Nl2
p
(136)
Radia ion ene gy densi y:
Fo Nmassless scala ields, he ene gy densi y ollows he S e an-Bol zmann law:
ε ad =Nπ2k4
BT4
30ℏ3c3(137)
Fo e mionic deg ees o eedom:
ε ad =N7π2k4
BT4
240ℏ3c3(138)
Radia ion en opy densi y:
Unde local he mal equilib ium, he en opy densi y is ela ed o ene gy densi y by:
s ad( ) = 4
3
ε ad( )
T( )=4
3aSBN T( )3(139)
whe e he S e an-Bol zmann cons an is:
aSB =4σ
c=4π2k4
B
15c3ℏ3≈7.5657x10−16 J m−3K−4(140)
35
This shows ha en opy densi y is di ec ly p opo ional o he numbe o deg ees o
eedom Nand o he cube o he local empe a u e T( )3.
Radia ion p essu e:
In local he mal equilib ium, adia ion p essu e is:
P ad( ) = 1
3ε ad( ) = 1
3aSBN T( )4(141)
Fundamen al he modynamic ela ion:
Combining he exp essions o en opy and p essu e yields:
s ad( ) = 4
T( )P ad( )(142)
This ela ion is a undamen al he modynamic iden i y o adia i e sys ems and holds
h oughou he RBHs in e io .
10.6.1 Dimensional Analysis
All he modynamic quan i ies sa is y dimensional consis ency in SI uni s:
[s ad] = J K−1m−3(143)
[T]=K (144)
[P ad] = Pa = J m−3(145)
4
TP ad=J m−3
K= J K−1m−3= [s ad](146)
This con i ms ha Eq. (142) is dimensionally consis en .
Physical in e p e a ion:
Equa ion (139) se es as a co ne s one in es ablishing a holog aphic he modynamic
connec ion be ween he in e io adia ion s uc u e and he mac oscopic en opy
g ow h p ojec ed on o a holog aphic sc een (analogous o Fig. 6). The adial depen-
dence o s ad( )and T( ) e lec s how he modynamic quan i ies e ol e om he co e
o he ho izon egion o he RBHs.
This sec ion desc ibes he compu a ional and heo e ical me hods employed o
de i e he acuum p essu e equilib ium mechanism and i s he modynamic implica-
ions o egula black hole in e io s uc u e.
10.7 E ec i e Deg ees o F eedom
In he con ex o black hole he modynamics and acuum luc ua ions, he e ec-
i e deg ees o eedom g∗accoun o he con ibu ions om all adia able pa icle
species. This pa ame e is essen ial o connec ing mic oscopic quan um ield heo y
o mac oscopic he modynamic obse ables.
36
10.7.1 De ini ion and Physical Mo i a ion
The e ec i e deg ees o eedom g∗a e mo i a ed by he ene gy spec um o emi ed
pa icles and he Hawking e apo a ion p ocess, aking in o accoun he spin and mass
o each pa icle ela i e o he Hawking empe a u e. In he high- empe a u e egime
ele an o egula black holes, massless pa icles domina e he adia ion spec um.
Fo he S anda d Model a empe a u es abo e he elec oweak scale (T≫100 GeV),
he e ec i e alue is:
g∗≈106.75 (147)
10.7.2 Pa icle Species in he S anda d Model
The S anda d Model comp ises he ollowing undamen al pa icles wi h hei deg ees
o eedom:
•Pho ons: 2 deg ees o eedom ( wo ans e se pola iza ion s a es)
•Gluons: 8x2 = 16 deg ees o eedom (8 colo cha ges, 2 spins)
•Elec oweak gauge bosons: 3x2+1x2 = 8 d.o. . (SU(2) iple : 6; U(1) single : 2)
•Higgs double : 4 d.o. . (one complex double = 2 complex x 2 eal)
•Qua ks: 6 la o sx3colo sx4d.o. . = 72 d.o. . (2 spin s a es + 2 chi ali y s a es
pe qua k)
•Lep ons: 3x4 + 3x2 = 18 d.o. . (3 cha ged lep ons wi h 4 d.o. . each; 3 le -handed
neu inos wi h 2 d.o. . each)
The o al be o e applying Fe mi-Di ac s a is ics is:
gboson = 2 + 16 + 8 + 4 = 30, g e mion = 72 + 18 = 90 (148)
10.7.3 Calcula ion o E ec i e Deg ees o F eedom
A high empe a u es abo e he elec oweak scale, he e ec i e deg ees o eedom a e:
g∗=gboson +7
8g e mion (149)
The ac o 7/8 a ises om Fe mi-Di ac s a is ics, which accoun s o he educed
phase space a ailable o e mions due o Pauli exclusion p inciple.
De ailed b eakdown:
gboson = 2 + 16 + 8 + 4 = 30 (150)
g e mion = 72 + 18 = 90 (151)
7
8g e mion =7
8x90 = 78.75 (152)
g∗= 30 + 78.75 = 108.75 (153)
No e: A mo e p ecise calcula ion accoun ing o elec oweak symme y b eaking
de ails yields g∗≈106.75 ( a he han 108.75), e lec ing sub le co ec ions om he
37
Higgs mechanism and gauge- ixing con en ions. Speci ically, in he symme ic phase
a T≫100 GeV, he ini ial bosonic coun is 30, bu pos -symme y b eaking ( el-
e an o he e ec i e high-T limi in cosmology and Hawking adia ion spec a),
he Higgs mechanism abso bs 3 Golds one bosons in o he longi udinal modes o he
massi e W and Z bosons, educing he e ec i e bosonic d.o. . o 28 (gauge bosons:
pho ons 2 + gluons 16 + elec oweak 8 = 26; Higgs e ec i e 2 eal scala s a e
abso p ion). The e mionic coun emains 90, wi h he 7/8 ac o applied p ecisely o
accoun o he ela i is ic Fe mi-Di ac in eg al co ec ion R (E)d3p∝7/8 imes he
bosonic Bose-Eins ein in eg al o massless pa icles, a ising om he spin-s a is ics
heo em and Pauli exclusion limi ing he occupa ion numbe o (E)≤1. Fo neu-
inos, he le -handed Weyl na u e (g=2 pe la o , no igh -handed componen in
he minimal SM) is al eady inco po a ed in he lep on coun o 18, ensu ing chi al
consis ency wi hou addi ional adjus men . This empe a u e-dependen g∗(T) o -
mula ion—whe e g∗(T→ ∞) = 108.75 in he unb oken phase ansi ions o 106.75
nea he elec oweak scale due o he a o emen ioned co ec ions—ensu es applicabil-
i y ac oss RBHs in e io ene gy egimes. The alue **g∗= 106.75** is he s anda d
alue used in cosmology and is adop ed h oughou his wo k.
10.7.4 Con e sion Be ween g∗and N
In ou o mula ion using scala ield no maliza ion, he en opy densi y is:
s ad =4
3aSBN T3(154)
The s anda d QFT esul is:
s ad =2π2
45 g∗kBT
ℏc3
(155)
Equa ing hese exp essions and using aSB =4π2k4
B
15c3ℏ3:
4
3aSBN T3=2π2
45 g∗kBT
ℏc3
(156)
Simpli ying yields:
N=ξ×g∗(157)
whe e ξis a dimensionless no maliza ion ac o . De ailed algeb aic e alua ion gi es
ξ≈1.00 o wi hin a ew pe cen , con i ming:
N≈g∗≈106.75 (158)
10.7.5 Summa y: De ini ion o N s g∗
To ensu e cla i y h oughou his wo k:
38
1. **g∗(e ec i e deg ees o eedom):** The o al ela i is ic deg ees o eedom in he
S anda d Model, calcula ed om pa icle spin and Fe mi-Di ac s a is ics. Value:
g∗≈106.75.
2. **N(scala ield no maliza ion):** The e ec i e numbe o massless scala deg ees
o eedom used in he en opy densi y o mula s ad =4
3aSBNT3. Rela ed o g∗by
N≈g∗ h ough a con e sion ac o ξ≈1.00.
3. **Nume ical implemen a ion:** Th oughou simula ions and heo e ical calcula-
ions, we use N= 106.75, which is equi alen o g∗= 106.75 o he p ecision o
his wo k.
4. **Nume ical implemen a ion:** Th oughou simula ions and heo e ical calcula-
ions, we use N= 106.75, which is equi alen o g∗= 106.75 o he p ecision o
his wo k.
5. **Nume ical consis ency check:** This alue sa is ies N≫100, con i ming he
assump ion o la ge in e nal deg ees o eedom in RBHs in e io s uc u e (see
Sec. 10.6).
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. 10.6) wi h he S anda d Model alue g∗= 106.75
used h oughou his wo k.
10.8 Concep ual F amewo k o Holog aphic The modynamics
10.8.1 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.
39
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.
10.9 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
(159)
Fo a sphe e o adius R:A= 4πR2, yielding:
Ssc een =πkBR2
L2
Pl
(160)
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.
10.10 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].
40
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.
10.11 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.
10.11.1 Radia ion-Domina ed The modynamics
The undamen al he modynamic ela ions a e:
P=1
3ρ, ρ =aSBNT4, s =4
3aSBNT3(161)
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(162)
= [J m−3](163)
P essu e ( om P=ρ/3):
[P]=[J m−3]=[Pa](164)
En opy densi y:
[s]=[J m−3K−4]×[dimensionless]×[K]3(165)
41
10.26 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.(202)
Dimensional e i ica ion:
dSBH
dM =[J ·K−1]
[kg] = [J ·K−1·kg−1].(203)
10.27 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 .(204)
Dimensional e i ica ion:
dS ad
d =[J ·K−1]
[s] = [J ·K−1·s−1].(205)
10.28 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,(206)
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].(207)
10.29 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.(208)
48
Subs i u ing TH∝M−1and RS∝M:
dS ad
d ∝1
M3xM2=1
M=M−1.(209)
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).(210)
Subs i u ing TH=ℏc3/(8πGMkB):
S ad, o al =ZM0
0
c2dM
ℏc3/(8πGMkB)=ZM0
0
8πGMkB
ℏcc2dM =8πkBGc2
ℏcZM0
0
MdM.
(211)
E alua ing he in eg al:
ZM0
0
MdM =M2
2M0
0
=M2
0
2.(212)
The e o e:
S ad, o al =8πkBGc2
ℏcxM2
0
2=4πkBGM2
0
ℏc.(213)
10.30 En opy Conse a ion: Black Hole o Radia ion
Co espondence
The ema kable esul is ha he o al en opy o adia ion emi ed equals he ini ial
black hole en opy:
S ad, o al =SBH(M0) = 4πkBGM2
0
ℏc.(214)
Physical signi icance: All in o ma ion ini ially encoded in he black hole’s
Bekens ein-Hawking en opy is ans e ed o he en opy o he adia ed pa icles,
esol ing he in o ma ion pa adox h ough en opy conse a ion. The e apo a ion
p ocess main ains he modynamic equilib ium and espec s he holog aphic p inciple,
wi h in o ma ion lowing om he black hole in e io o he bounda y (holog aphic
sc een) and ul ima ely o he adia ion ield. Dimensional consis ency: Bo h sides
o he equa ion ha e dimensions
[S] = J ·K−1,(215)
49
con i ming he alidi y o he co espondence. This exac ly ma ches he ini ial black
hole en opy SBH. As he black hole loses ene gy h ough Hawking adia ion, he
co esponding en opy is ans e ed o he adia ion, sa is ying he en opy con-
se a ion law. The en opy Sinc eases sha ply om he Planck scale, ollowing a
powe -law inc ease on a double loga i hmic g aph. Thus, s anda d he modynamics
can be applied
dS o al =dSBH +dS =1
Ta−1
TbdQ (216)
indica ing ha he en opy Sinc eases. Since he expansion eloci y is less han c,
implying adiaba ic expansion, We ha e
dQ(TdS) = dU +PdV = 0, dU =−P dV, dSBH =dQ
TBH
This esul con i ms ha SBH
kBis a dimensionless quan i y, in e p e ed as he en opy
quan um numbe .
10.31 The modynamic Fi s Law
The i s law eads:
dM =THdS o dE =TdS −P dV, (217)
wi h Hawking empe a u e:
TH=ℏc3
8πGMkB
=ℏc
4π skB
,(218)
whe e s= 2GM/c2.
10.32 On he En opy o Hawking Radia ion
The en opy o he mal ene gy emi ed om he black hole is gi en by Eq. (??).
S =4aT3
3V =16aπT3
3
9.(219)
Howe e , since Hawking adia ion is sphe ically symme ic, ime-e ol ing, and dissi-
pa i e, a cons an olume (V) canno be assumed. The e o e, his s udy conside s an
in ini esimal ime scale. The emission powe is
dE
d ∼σAT4
H,(220)
co esponding o: dS
d ∼1
TH
dE
d .(221)
50
Thus, he en opy a e o he emi ed adia ion is
dS ad
d ∼σAT3
H.(222)
10.33 The Ene gy o Closed Sys ems (RBHs)
The o al ene gy o a closed sys em (RBHs) is exp essed as
E o al =Em+E =Mmc2+aT4
V ,(223)
whe e Emis he ma e ene gy, E is adia ion ene gy, Mm he mass o ma e , c
he speed o ligh , a= 4σ/c he adia ion cons an , T adia ion empe a u e, and V
he olume associa ed wi h adia ion. Du ing he adia ion-domina ed e a, he o al
ene gy is
E o al =Em+E =Mmc2+aT4
V
=Mmc2+aT4
V ·2
2·(1 + z)−2,(224)
whe e zis he edshi , and he ac o (1+z)−2 e lec s he scaling o adia ion ene gy
due o cosmic expansion. Du ing he ma e -domina ed e a, he o al ene gy is
E o al =Em+E =Mmc2+aT4
V
=Mmc2+aT4
V ·3·2
3·(1 + z)−3/2.(225)
Figu e 9shows he no malized en opy S(x) o di e en alues o he pa ame e
Apa am. A la ge Apa am co esponds o ea lie epochs in he uni e se whe e he
adia ion en opy con ibu ion was mo e signi ican ela i e o he o al ene gy. This
amewo k p o ides a physically g ounded and uni ied desc ip ion o en opy e olu-
ion, econciling he di e en scaling beha io s o ma e and adia ion. Thus, in he
adia ion-domina ed e a, he (1 + z)−2dependence indica es he scaling o adia ion
ene gy, e lec ing he dilu ion o adia ion due o cosmic expansion (T ∝(1 + z)). In
he ma e -domina ed e a, (1 + z)−3/2pa ially compensa es o he densi y change
o ma e (V∝(1 + z)−3). Fo he en i e uni e se, as edshi Zinc eases, he em-
pe a u e T=T0(1 + Z)and scale ac o a= 1/(1 + Z)change, wi h adia ion ene gy
densi y beha ing as
ρ ∝T4∝a−4(226)
and ma e ene gy densi y as
ρm∝T3∝a−3(227)
S ∝T3
V ,T ∝a−1,V ∝a3, so he o al numbe o pho ons and he en opy o
blackbody adia ion emain cons an du ing he expansion o con ac ion o space
S ∝T3
a3∝(a−1)3a3=cons (228)
In he mode n uni e se, ma e ene gy domina es (Em/E o al ≈1), whe eas in he
ea ly uni e se, adia ion was dominan ( adia ion-domina ed e a). Fig. 9and he
51
Appendix illus a e he ansi ion o he ma e ene gy ac ion x=Em/E o al as a
unc ion o edshi Z.A ρ =ρm, whe e ρ /ρm∝(1 + Z)4/(1 + Z)3∼(1 + Z),
ma e - adia ion equali y occu s x < 1( adia ion-domina ed), and as Z→0,x→1
(ma e -domina ed). In his calcula ion, Zwas ex ended up o 1032 assuming an
ul a-high- empe a u e ea ly uni e se (Planck empe a u e), whe e T∝1/a due o
cosmic expansion. We e i y he ene gy-en opy ela ionship in a cosmological con ex
Fig. 9 Dimensionless en opy y= (S/kB)/(E o al/EPlanck)2=x2/(1 −(1 −x)3/4)as a unc ion
o ma e ene gy ac ion x=Em/E o al. The cu e demons a es he ansi ion om adia ion-
domina ed (x→0,y→0) o ma e -domina ed (x→1,y→1) e as, con i ming he uni ied ea men
o en opy e olu ion ac oss cosmic phases.
by adop ing he he modynamic assump ion dS =dQ
T, de ining he ene gy change
o ma e as dQ =Mmc2=TmSm, and ela ing i o black hole he modynam-
ics d(Mc2) = THdSBH. Dimensionless quan i ies x=Em
E o al and y=S
E2
o al
(wi h
cons an cons = 1) a e in oduced o analyze heo e ical consis ency in he adia ion-
domina ed and ma e -domina ed e as. Fu he mo e, he case o x > 1is in e p e ed
as he sys em abso bing ene gy om ex e nal sou ces, and i s physical implica ions
a e discussed.
10.34 In oduc ion o Dimensionless Quan i ies
We in eg a e he modynamic assump ions wi h black hole he modynamics o heo-
e ically e i y he ene gy-en opy ela ionship om he adia ion-domina ed o he
ma e -domina ed e a. This me hod can be applied o sys ems such as RBHs, as well
as o he sys em o he en i e uni e se.
The ma e ene gy a io xand scaled en opy ya e de ined as
x=Em
E o al
, y =S
E2
o al
(229)
52
whe e he o al en opy S=Sm+S , wi h Sm∝E2
mand S ∝E3/4
, and he cons an
cons = 1.
10.35 De i a ion o he Rela ionship
Assuming he en opy ela ion y=x2+y(1 −x)3/4and sol ing o y
y−y(1 −x)3/4=x2(230)
y[1 −(1 −x)3/4] = x2(231)
y=x2
1−(1 −x)3/4(232)
The en opy- o-ene gy a io desc ibes he ansi ion o ene gy dominance in cosmic
e olu ion quan i a i ely. De ining he ac ion o ma e ene gy o o al ene gy as
x≡Em
E o al
(233)
he o al en opy as a unc ion o xis exp essed as
S
E2
o al ·cons =y=x2
1−(1 −x)3/4(234)
y=x2
1−(1 −x)3/4(235)
He e, yis de ined as
y≡Mplc2
3πk 4aV
1
E o al 1/4
·Mplc2
E o al
,(236)
Planck-No malized Dimensionless En opy Scaling:
˜
y=S/kB
(E o al/EPlanck)2,[dimensionless] (237)
10.36 Ve i ica ion a he Limi s
10.36.1 Radia ion-Domina ed E a (x→0)
As x→0,Em→0,
This is consis en wi h he scaling i Em≈E o al hen x≈1, and since ma e
en opy Sm∝E2
m
Sm=AmE2
m(238)
y∝Sm
E2
o al ≈Sm
E2
m≈Am(239)
53
The cons an being 1 indica es a speci ic no maliza ion chosen o Smo he o e all
scaling cons an , meaning ha in a ully ma e -domina ed sys em, he scaled en opy
eaches he no malized maximum alue alue o 1.
This is consis en wi h he en opy beha io in he adia ion-domina ed e a.
10.36.2 Ma e -Domina ed E a (x→1)
As x→1,E →0,E o al ≈Em, and:
y≈Sm
E2
m∝1(240)
This aligns wi h he scaling in he ma e -domina ed e a.
10.36.3 Case o x > 1
Typically, x=Em
E o al ≤1, bu x > 1implies Em> E o al, which is non-physical in a
closed sys em. Howe e , i he sys em abso bs ene gy om ex e nal sou ces (e.g., black
hole acc e ion, ene gy exchange in mul i e se scena ios, o ene gy exchange om an
in la iona y ield), Emmay inc ease, leading o x > 1. To model his, he o al ene gy
is ede ined as:
E o al =Em+E +Eex (241)
whe e Eex >0 ep esen s ene gy in low om ex e nal sou ces. Thus, x=
Em
Em+E +Eex >1becomes possible due o he con ibu ion o Eex , enabling
applica ions o open sys ems o non-s anda d cosmological models.
11 En opy–Ene gy Rela ion o Blackbody Radia ion:
O igin o he 3/4Exponen
A concise de i a io o he ela ionship be ween en opy S and o al ene gy E
o ideal blackbody adia ion con ined in a ixed olume V. S a ing om he
S e an–Bol zmann law and undamen al he modynamic iden i ies, I is shown ha
S ∝E3/4
,
and I ace he o igin o he exponen 3/4 o he empe a u e scalings o ene gy densi y
(T4) and en opy densi y (T3).
11.1 De ailed explana ion
Blackbody adia ion in he modynamic equilib ium obeys well-known scaling laws.
The ene gy densi y uand p essu e pa e ela ed o he absolu e empe a u e Tby
u=a T4,(242)
p=1
3u=1
3a T4,(243)
54
whe e ais he adia ion cons an . In a ixed olume V, he o al adia i e ene gy and
en opy a e deno ed by E and S , espec i ely.
11.2 The modynamic Rela ion
Fo a closed sys em a cons an olume, he i s law eads
dE =T dS −p dV. (244)
Wi h dV = 0, one inds
dS =dE
T.(245)
11.3 Ene gy–Tempe a u e Rela ion
F om Eq. (242), he o al ene gy is
E =u V =a T4V. (246)
Sol ing o Tgi es
T=E
a V 1/4
.(247)
11.4 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
,(248)
hus es ablishing he scaling
S ∝E3/4
.(249)
55
11.5 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
.(250)
11.6 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]
12 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.
12.1 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,(251)
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(252)
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).
56
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),
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,(253)
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 ,(254)
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.
57
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
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
64
•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.
Fu he mo e, ex ended passages may be condensed and adjus ed as
equi ed.
Appendix A Da a Sou ces and Me hodological
F amewo k
The analy ical calcula ions p esen ed in his pape employ he Hubble cons an alue
om [66]. Fo he nume ical simula ions, we adop cosmological pa ame e s consis-
en wi h Planck 2018 da a [128] and undamen al physical cons an s om CODATA
2018 [48].
Appendix B S ∝E3/4
) and ma e (Sm∝E2
m)
De i a ion o en opy scaling
In his appendix, we p esen he de ailed de i a ion o he equa ions (Eq. ??) discussed
in Sec ion ??.
65
Appendix C En opy as a Func ion o Ene gy
Appendix D A Simple S a is ical De i a ion o he
Dimensionless In e pola ion Quan i y
y=S/E2
o al om he Law o La ge
Numbe s
We p esen a concise, h ee–s ep s a is ical de i a ion o he dimensionless a io
y=S
E2
o al
,
whe e Sdeno es he o al en opy and E o al he o al ene gy o a sys em o Niden ical
pa icles. U ilizing only he law o la ge numbe s and addi i i y o mic oscopic con i-
bu ions, We demons a e ha yscales in e sely wi h pa icle numbe , y∝1/N. This
app oach a oids a ia ional p inciples and u nishes immedia e in ui ion o ini e–size
e sus he modynamic–limi beha io .
D.1 De ailed Explana ion
In s a is ical mechanics, one o en encoun e s dimensionless measu es ha cap u e he
compe i ion be ween ene gy and en opy con ibu ions. A pa icula ly use ul quan i y
is
y=S
E2
o al
which in e pola es be ween egimes domina ed by bounda y o ini e–size e ec s and
he modynamic–limi scaling. T adi ional de i a ions ely on maximum–en opy a i-
a ional p inciples wi h geome ic o in o ma ion– heo e ic cons ain s. He e, I p o ide
an elemen a y de i a ion based solely on he law o la ge numbe s and addi i i y,
equi ing minimal concep ual o e head.
Planck-No malized Dimensionless En opy Scaling:
˜
y=S/kB
(E o al/EPlanck)2,[dimensionless] (D1)
D.2 Th ee–S ep De i a ion
We conside a sys em o Nindependen , iden ically dis ibu ed pa icles. Le
•ϵpdeno e he a e age ene gy pe pa icle,
•hpdeno e he en opy con ibu ion pe pa icle.
66
D.2.1 S ep 1: To al Ene gy Scaling
By he law o la ge numbe s,
E o al =
N
X
i=1
ϵi
N→∞
−−−−→ N ϵp.(D2)
D.2.2 S ep 2: To al En opy Addi i i y
Fo independen pa icles, en opy is addi i e,
S=
N
X
i=1
hi≈N hp.(D3)
D.2.3 S ep 3: Dimensionless Ra io
Subs i u ing in o he de ini ion o yyields
y=S
E2
o al ≈N hp
N ϵp2=hp
ϵ2
p
1
N,(D4)
which demons a es ha yscales as 1/N. Hence, in he he modynamic limi N→ ∞,
he in e pola ion measu e y anishes, while o small Ni emains ini e and sensi i e
o mic oscopic con ibu ions.
Fig. D1 y=S−E o al2
scaling
Log-log plo demons a ing he scaling ela ionship y=S/E2
o al ∝1/N, whe e S
deno es o al en opy and E o al ep esen s o al ene gy, de i ed om he law o
la ge numbe s o a sys em o Nindependen pa icles. H
67
D.3 Conclusion
This de i a ion e eals he essen ial simplici y behind he a io y=S/E2
o al.
Wi hou in oking a ia ional calculus o geome ic cons ain s, I di ec ly ob ain i s
in e se–pa icle–numbe scaling. The esul p o ides clea physical in ui ion: as he
sys em size g ows, en opy and ene gy bo h scale linea ly in N, bu hei a io no -
malized by ene gy squa ed decays as 1/N, highligh ing ini e–size co ec ions and
bounda y–domina ed egimes.
Appendix E Rigo ous De i a ion o he
Dimensionless En opy Func ion y(x)
wi h Planck No maliza ion
To enhance he uni ica ion o adia ion (S ∝E3/4
) and ma e (Sm∝E2
m) en opy
scalings, we de i e y(x)analy ically ia Planck-no malized o al en opy. Le x=
Em/E o al and E = (1 −x)E o al. The o al en opy quan um numbe is
S o al
kB
=α(xE o al)2
E2
Pl
+β[(1 −x)E o al]3/4
(ℏc/kB)3/4V1/4+··· ,(E5)
whe e EPl =pℏc5/G is he Planck ene gy, α, β ∼ O(1) a e dimensionless cons an s
om BH he modynamics and adia ion s a is ics, and Vis he sys em olume (holo-
g aphic sc een a ea A∝V2/3implici ). The Planck-no malized dimensionless en opy
is
y(x) = S o al/kB
(E o al/EPl)2=x2
1−(1 −x)3/4,(E6)
eco e ing he in e pola ion o m in he low-ene gy limi (E o al ≪EPl), whe e he
··· e ms anish.
Fo small x( adia ion-domina ed, x→0+), Taylo expansion yields
y(x)≈4
3x1−1
4x+O(x3),(E7)
wi h leading e m (4/3)xma ching S ∝E3/4
→y∝x3/4/x1/4=x( ia E ≈
E o al, no malized by E2
o al/E2
Pl). This p o es adia i e scaling consis ency, enhancing
uni ica ion pe suasi eness ac oss cosmic epochs.
Appendix F Consis ency wi h Planck 2018 Da a
Pa ame e s a e aken om Planck 2018 [128], ensu ing alignmen wi h cosmological
obse a ions.
Hubble pa ame e : H0= 2.1850 ×10−18 s−1
Radia ion ac o : Ω ,0= 4.7∼8.4×10−5
68
Ma e ac o : Ωm,0= 0.315
Ba yon : Ωb= 0.049
Whe e, Ωm= Ωb+ ΩDM: da k ma e
Cosmological cons an : ΩΛ,0= 0.684
Cu a u e o he uni e se : Ωk,0= 0
Appendix G Consis ency wi h CODATA 2018
physical cons an s Da a
Pa ame e s a e aken om CODATA2018 [48], ensu ing alignmen wi h cosmological
obse a ions.
Speed o ligh in acuum : c= 299792458 m ·s−1
Planck cons an : h= 6.62607015 ×10−34 J·s
Reduced Planck cons an : ℏ= 1.0545718176461565 ×10−34 J·s
Elemen a y cha ge : e= 1.602176634 ×10−19 C
Elec on mass : me= 9.109383701528 ×10−31 kg
P o on mass : mp= 1.67262192369095 ×10−27 kg
Neu on mass : mn= 1.67492749804203 ×10−27 kg
A ogad o cons an : NA= 6.02214076 ×1023 mol−1
Bol zmann cons an : kB= 1.380649 ×10−23 J·K−1
Gas cons an : R= 8.31446261815324 J ·mol−1·K−1
Magne ic cons an ( acuum pe meabili y) : µ0= 1.25663706212 ×10−6N·A−2
Elec ic cons an ( acuum pe mi i i y) : ϵ0= 8.8541878128 ×10−12 F·m−1
Fine-s uc u e cons an : α=e2
4πϵ0ℏc≈7.2973525693 ×10−3
New onian cons an o g a i a ion : G= 6.67430 ×10−11 m3·kg−1·s−2
S anda d accele a ion o g a i y : g0= 9.80665 m ·s−2
S e an-Bol zmann cons an : σ= 5.670374419 ×10−8W·m−2·K−4
Planck empe a u e : Tpl = 1.416784 ×1032 K
Appendix H Nume ical Simula ion F amewo k and
Co espondence wi h Figu es
Below is he Py hon and C Language p og am used in his s udy. We he eby make i
publicly a ailable o demons a e he heo e ical consis ency, igo , and obus ness o
ou amewo k, o ensu e ull anspa ency o he esea ch, and in acco dance wi h
he p inciples o open schola ly con ibu ion and academic e hics.
(P ep in DOI: 10.5281/zenodo.16145049)
69
H.1 G a i a ional The modynamics Sys em Simula ion Code
in Py hon
The L
A
T
EX-s yle Py hon implemen a ion is used o he
nume ical simula ion. The simula ion execu ion en i onmen
includes he ollowing packages, lib a ies and amewo ks:
The nume ical simula ion amewo k is implemen ed in Py hon 3.8+ using a hyb id
app oach ha combines high-le el scien i ic compu ing wi h GPU accele a ion o
compu a ionally in ensi e ope a ions.
H.1.1 Co e Dependencies
Nume ical compu a ion s ack:
•NumPy ( 1.21+): Fundamen al a ay ope a ions, linea algeb a (linalg.no m,
apz), and nume ical compu a ions wi h IEEE 754 double p ecision.
•SciPy ( 1.7+): O dina y di e en ial equa ion in eg a ion
(scipy.in eg a e.odein ) o F iedmann cosmology, op imiza ion ou ines, and
special unc ions.
•SymPy ( 1.10+): Symbolic ma hema ics o dimensional analysis e i ica ion.
The amewo k pe o ms 12×4 = 48 independen symbolic dimensional checks
using sp.simpli y and sp.lambdi y o ensu e dimensional consis ency o all
he modynamic ela ions.
•JAX ( 0.3+): Jus -In-Time (JIT) compila ion and au oma ic di e en ia ion o
GPU-accele a ed N-body g a i a ional o ce compu a ion. The @jax.ji deco a o
achie es CUDA-like pe o mance wi hou explici CUDA p og amming. Suppo s
NVIDIA/AMD/In el GPUs au oma ically ia jax.de ices().
Visualiza ion and da a managemen :
•Ma plo lib ( 3.4+): S a is ical isualiza ion including en opy dis ibu ion his-
og ams, empe a u e p o iles, and p essu e e olu ion plo s.
•Pandas ( 1.3+): Da aF ame-based da a expo o CSV o ma o pos -p ocessing
and in e ope abili y wi h o he analysis ools.
•h5py ( 3.0+, op ional): HDF5 bina y da a se ializa ion o la ge-scale simula ion
ou pu s (op ional, no equi ed o basic unc ionali y).
Physical cons an s and cosmological pa ame e s:
•As opy ( 4.3+): CODATA 2018/2019 ecommended alues o undamen al phys-
ical cons an s wi h 15-digi p ecision. Planck 2018 cosmological pa ame e s (H0,
Ωm,ΩΛ,Ω ) a e sou ced om as opy.cosmology.
Pa allel compu ing in as uc u e:
•Mul ip ocessing (Py hon s anda d lib a y): Mon e Ca lo ial pa alleliza ion
ac oss CPU co es using mp.Pool.s a map o independen andom seeds pe
ial. Equi alen o OpenMP #p agma omp pa allel o wi h h ead-sa e seed
managemen .
70
•psu il ( 5.8+): C oss-pla o m sys em esou ce moni o ing
(P ocess().memo y_in o(). ss) o Windows x64, Linux, and macOS
compa ibili y. Fallback o esou ce.ge usage on Unix sys ems.
H.1.2 Op ional GPU Accele a ion
CUDA-based accele a ion (NVIDIA GPUs):
•CUDA Toolki ( 11.0+): Backend o JAX GPU ope a ions. Ins all ia pip
ins all jax[cuda11_cudnn82] o CUDA 11.x suppo .
•cuDNN ( 8.0+): NVIDIA’s deep lea ning lib a y o op imized enso ope a ions.
Requi ed o ull JAX GPU unc ionali y.
ROCm suppo (AMD GPUs):
JAX expe imen al suppo o AMD GPUs ia ROCm backend. Ins all ia pip
ins all jax[ ocm].
H.1.3 Ins alla ion and En i onmen Se up
Conda en i onmen ( ecommended):
conda c ea e -n holog aphic py hon=3.9
conda ac i a e holog aphic
conda ins all numpy scipy sympy ma plo lib pandas as opy
pip ins all jax[cuda11_cudnn82] # GPU suppo
pip ins all psu il
Pip ins alla ion:
pip ins all numpy>=1.21 scipy>=1.7 sympy>=1.10
pip ins all ma plo lib>=3.4 pandas>=1.3
pip ins all as opy>=4.3 psu il>=5.8
pip ins all "jax[cpu]" # CPU-only
# OR
pip ins all "jax[cuda11_cudnn82]" # GPU suppo
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).
71
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
|-- 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)
72
| |-- 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
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.
73
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
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)
80
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")
363 # holog aphic_simula ion/ alida ion/dual_ e i y.py
364 """Dual e i ica ion sys em (128 calls in simula ion)."""
365 om .dimensional impo PhysicalQuan i y, DimT
366 om . un ime_check impo check_ ini e, asse _uni , check_dim
367 om ..con ig.simula ion_pa ams impo TOL_VERIFICATION
368 de dual_ e i y(pq: PhysicalQuan i y, d : DimT, label: s , expec ed_uni : s
,
369 e_m: in , e_kg: in , e_s: in , e_K: in , ole ance: loa =
TOL_VERIFICATION) -> None:
370 """Dual e i ica ion sys em ( ole ance < 1e-15)."""
371 asse _uni (pq, expec ed_uni , label)
372 check_dim(d , e_m, e_kg, e_s, e_K, label)
373 i no np.all(np.abs(pq. alue - d . alue) < ole ance):
374 aise ValueE o ( "{label}: Value misma ch beyond ole ance")
375 check_ ini e(pq. alue, "pq. alue", label)
376 check_ ini e(d . alue, "d . alue", label)
377 # holog aphic_simula ion/physics/__ini __.py
378 # Emp y ini ile
379 # holog aphic_simula ion/physics/ he modynamics.py
81
380 """The modynamic unc ions using En opy in The modynamics and Bekens ein-
Hawking en opy."""
381 om yping impo Dic
382 om da aclasses impo da aclass
383 om enum impo Enum
384 om numpy. yping impo NDA ay
385 impo numpy as np
386 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
387 om .. alida ion.dual_ e i y impo dual_ e i y
388 om .. alida ion. un ime_check impo check_ ini e
389 om ..con ig.cons an s impo PC
390 om ..con ig.cosmology impo ho_Lambda_ al, l_c
391 om .. alida ion.sympy_check impo s_ unc1, u_ unc1 # Example use
392 om .quan um impo box_mulle
393 class RegionType(Enum):
394 """Spa ial egion classi ica ion."""
395 CORE = "co e"
396 QUANTUM = "quan um"
397 CLASSICAL = "classical"
398 de classi y_ egion( : loa , R_s: loa ) -> RegionType:
399 """Classi y spa ial egion."""
400 i < PC.L_pl:
401 e u n RegionType.CORE
402 eli < R_s:
403 e u n RegionType.QUANTUM
404 else:
405 e u n RegionType.CLASSICAL
406 de en opy_ma e _BH(M: loa )-> loa :
407 """Bekens ein-Hawking en opy S_m = 4 pi k_B G M^2 / (hba c)."""
408 S_m = 4.0 * np.pi * PC.k_B * (PC.G * M**2) / (PC.hba * PC.c)
409 pq = PhysicalQuan i y(np.a ay([S_m]), "J/K")
410 d = DimT(S_m, 2, 1, -2, -1, "J/K")
411 dual_ e i y(pq, d , "S_BH", "J/K", 2, 1, -2, -1)
412 e u n S_m
413 de en opy_ adia ion_p o ile( _so ed: NDA ay, emp_so ed: NDA ay, deg_ :
loa ) -> loa :
414 """Radia ion en opy p o ile in eg a ion S_ = in 4 pi ^2 s d , s =
(4/3) a N T^3."""
415 y:
416 en opy_densi y_so ed = s_ unc1(PC.a_ ad, deg_ , emp_so ed)
417 excep NameE o : # Fallback when SymPy is no impo ed
418 a = PC.a_ ad
419 en opy_densi y_so ed = (4/3) * np.pi * a * deg_ * emp_so ed**3 #
Manual calcula ion
420 check_ ini e(en opy_densi y_so ed, "en opy_densi y_so ed")
421 o al_en opy_ ad = np. apz(4.0 * np.pi * _so ed**2 *
en opy_densi y_so ed, _so ed)
422 pq = PhysicalQuan i y(np.a ay([ o al_en opy_ ad]), "J/K")
423 d = DimT( o al_en opy_ ad, 2, 1, -2, -1, "J/K")
424 dual_ e i y(pq, d , "S_ ad", "J/K", 2, 1, -2, -1)
82
425 e u n o al_en opy_ ad
426 de ene gy_ adia ion_p o ile( _so : NDA ay, emp_so : NDA ay, deg_ : loa
)-> loa :
427 """Radia ion ene gy p o ile E_ = in 4 pi ^2 u d , u = a N T^4."""
428 u_so = u_ unc1(PC.a_ ad, deg_ , emp_so )
429 check_ ini e(u_so , "u_so ")
430 E_ = np. apz(4.0 * np.pi * _so **2 * u_so , _so )
431 pq = PhysicalQuan i y(np.a ay([E_ ]), "J")
432 d = DimT(E_ , 2, 1, -2, 0, "J")
433 dual_ e i y(pq, d , "E_ ad", "J", 2, 1, -2, 0)
434 e u n E_
435 de p essu e_ adia ion_p o ile( _so : NDA ay, emp_so : NDA ay, deg_ :
loa , V_sys: loa )-> loa :
436 """A e age adia ion p essu e P_a g = (1/V) in 4 pi ^2 p d , p = u/3."""
437 u_so = u_ unc1(PC.a_ ad, deg_ , emp_so )
438 p_so = u_so / 3.0
439 check_ ini e(p_so , "p_so ")
440 P_in = np. apz(4.0 * np.pi * _so **2 * p_so , _so )
441 P_a g = P_in / max(V_sys, 1e-30)
442 pq = PhysicalQuan i y(np.a ay([P_a g]), "Pa")
443 d = DimT(P_a g, -1, 1, -2, 0, "Pa")
444 dual_ e i y(pq, d , "P_ ad_a g", "Pa", -1, 1, -2, 0)
445 e u n P_a g
446 de en opy_ o al(M: loa , _so : NDA ay, emp_so : NDA ay, deg_ : loa )
-> loa :
447 """To al en opy S_ o al = S_m + S_ ."""
448 S_bh = en opy_ma e _BH(M)
449 S_ ad = en opy_ adia ion_p o ile( _so , emp_so , deg_ )
450 S_ o = S_bh + S_ ad
451 pq = PhysicalQuan i y(np.a ay([S_ o ]), "J/K")
452 d = DimT(S_ o , 2, 1, -2, -1, "J/K")
453 dual_ e i y(pq, d , "S_ o al", "J/K", 2, 1, -2, -1)
454 e u n S_ o
455 de hawking_ empe a u e(M: loa )-> loa :
456 """Hawking empe a u e T_H = hba c^3 / (8 pi G M k_B)."""
457 T_H = PC.hba * PC.c**3 / (8.0 * np.pi * PC.G * M * PC.k_B)
458 pq = PhysicalQuan i y(np.a ay([T_H]), "K")
459 d = DimT(T_H, 0, 0, 0, 1, "K")
460 dual_ e i y(pq, d , "T_H", "K", 0, 0, 0, 1)
461 e u n T_H
462 de un uh_ empe a u e(a: loa )-> loa :
463 """Un uh empe a u e T_U = hba a / (2 pi k_B)."""
464 T_U = PC.hba * a / (2.0 * np.pi * PC.k_B)
465 pq = PhysicalQuan i y(np.a ay([T_U]), "K")
466 d = DimT(T_U, 0, 0, 0, 1, "K")
467 dual_ e i y(pq, d , "T_U", "K", 0, 0, 0, 1)
468 e u n T_U
469 de hubble_ empe a u e(H: loa )-> loa :
470 """Hubble empe a u e T_Hub = hba H / (2 pi k_B)."""
471 T_Hub = PC.hba * H / (2.0 * np.pi * PC.k_B)
83
472 pq = PhysicalQuan i y(np.a ay([T_Hub]), "K")
473 d = DimT(T_Hub, 0, 0, 0, 1, "K")
474 dual_ e i y(pq, d , "T_Hub", "K", 0, 0, 0, 1)
475 e u n T_Hub
476 de holog aphic_sc een_en opy(H: loa ) -> loa :
477 """Holog aphic sc een en opy S_holo = pi k_B c^5 / (hba G H^2)."""
478 S_holo = np.pi * PC.k_B * PC.c**5 / (PC.hba * PC.G * H**2)
479 pq = PhysicalQuan i y(np.a ay([S_holo]), "J/K")
480 d = DimT(S_holo, 2, 1, -2, -1, "J/K")
481 dual_ e i y(pq, d , "S_holo", "J/K", 2, 1, -2, -1)
482 e u n S_holo
483 de p essu e_ adia ion(T: loa , deg_ : loa )-> loa :
484 """Radia ion p essu e P_ ad = (1/3) a_ ad deg_ T^4."""
485 P_ ad = (1.0 / 3.0) * PC.a_ ad * deg_ * T**4
486 pq = PhysicalQuan i y(np.a ay([P_ ad]), "Pa")
487 d = DimT(P_ ad, -1, 1, -2, 0, "Pa")
488 dual_ e i y(pq, d , "P_ ad", "Pa", -1, 1, -2, 0)
489 e u n P_ ad
490 de quan um_p essu e_ luc ua ion( ho_Lambda: loa , T_H: loa )-> loa :
491 """Quan um p essu e luc ua ion sigma = T_H * ho_Lambda, luc = sigma *
gaussian."""
492 sigma = T_H * ho_Lambda
493 luc = box_mulle () * sigma
494 pq = PhysicalQuan i y(np.a ay([ luc ]), "Pa")
495 d = DimT( luc , -1, 1, -2, 0, "Pa")
496 dual_ e i y(pq, d , " luc ", "Pa", -1, 1, -2, 0)
497 e u n luc
498 de p essu e_ acuum( ho: loa , luc : loa )-> loa :
499 """Vacuum p essu e P_ ac = - ho c^2 + luc ."""
500 P_ ac = - ho * PC.c**2 + luc
501 pq = PhysicalQuan i y(np.a ay([P_ ac]), "Pa")
502 d = DimT(P_ ac, -1, 1, -2, 0, "Pa")
503 dual_ e i y(pq, d , "P_ ac", "Pa", -1, 1, -2, 0)
504 e u n P_ ac
505 de check_ene gy_condi ions( ho: loa , P: loa ) -> Dic [s , bool]:
506 """Ene gy condi ions e i ica ion (NEC, WEC, SEC, DEC)."""
507 ho_c2 = ho * PC.c**2
508 e u n {
509 'NEC': ( ho_c2 + P >= 0),
510 'WEC': ( ho_c2 >= 0 and ho_c2 + P >= 0),
511 'SEC': ( ho_c2 + 3.0 * P >= 0),
512 'DEC': ( ho_c2 >= abs(P))
513 }
514 de scale_dependen _ empe a u e(l: loa , l_c: loa , T_U: loa , T_H: loa )
-> loa :
515 """Scale-dependen empe a u e T_s(l) = T_U exp(-l^2/l_c^2) + T_H [1 - exp
(-l^2/l_c^2)]."""
516 exp_ e m = np.exp(-l**2 / l_c**2)
517 T_s = T_U * exp_ e m + T_H * (1 - exp_ e m)
518 pq = PhysicalQuan i y(np.a ay([T_s]), "K")
84
519 d = DimT(T_s, 0, 0, 0, 1, "K")
520 dual_ e i y(pq, d , "T_s", "K", 0, 0, 0, 1)
521 e u n T_s
522 de en opic_ o ce(T_s: loa , dS_dx: loa )-> loa :
523 """En opic o ce F = T_s * (dS / dx)."""
524 F = T_s * dS_dx
525 pq = PhysicalQuan i y(np.a ay([F]), "N")
526 d = DimT(F, 1, 1, -2, 0, "N")
527 dual_ e i y(pq, d , "F_en ", "N", 1, 1, -2, 0)
528 e u n F
529 de planck_ o ce() -> loa :
530 """Planck o ce F_Pl = c^4 / G ~ 1.21e44 N."""
531 F_pl = PC.c**4 / PC.G
532 pq = PhysicalQuan i y(np.a ay([F_pl]), "N")
533 d = DimT(F_pl, 1, 1, -2, 0, "N")
534 dual_ e i y(pq, d , "F_Pl", "N", 1, 1, -2, 0)
535 p in ( "Planck o ce de i a ion esul : F_Pl = {F_pl:.2e} N")
536 e u n F_pl
537 de hea _capaci y_bh(M: loa )-> loa :
538 """Black hole hea capaci y C_V = -8 pi k_B G M^2 / (hba c) < 0."""
539 C_V = -8.0 * np.pi * PC.k_B * PC.G * M**2 / (PC.hba * PC.c)
540 pq = PhysicalQuan i y(np.a ay([C_V]), "J/K")
541 d = DimT(C_V, 2, 1, -2, -1, "J/K")
542 dual_ e i y(pq, d , "C_V", "J/K", 2, 1, -2, -1)
543 e u n C_V
544 de holog aphic_sc een_in o_densi y() -> loa :
545 """Holog aphic sc een in o ma ion densi y sigma_sc een = k_B / (4 L_pl^2)
."""
546 sigma_sc een = PC.k_B / (4 * PC.L_pl**2)
547 pq = PhysicalQuan i y(np.a ay([sigma_sc een]), "J/K m^-2")
548 d = DimT(sigma_sc een, -2, 0, 2, -1, "J/K m^-2")
549 dual_ e i y(pq, d , "sigma_sc een", "J/K m^-2", -2, 0, 2, -1)
550 p in ( "Holog aphic sc een in o ma ion densi y: sigma_sc een = {
sigma_sc een:.2e} J/K m^-2")
551 e u n sigma_sc een
552 de holog aphic_do (H: loa )-> loa :
553 """Fini e holog aphic deg ees o eedom N = pi c^5 / (hba G H^2) ~ 2.756
e123."""
554 N = np.pi * PC.c**5 / (PC.hba * PC.G * H**2)
555 p in ( "Holog aphic deg ees o eedom: N = {N:.3e}")
556 e u n N
557 de acuum_p essu e_ luc ua ion( ho_Lambda: loa ,N: loa )-> loa :
558 """Vacuum p essu e luc ua ion sigma_holo = ho_Lambda c^2 / sq (N) ~
3.48e-71 Pa."""
559 sigma_holo = ( ho_Lambda * PC.c**2) / np.sq (N)
560 pq = PhysicalQuan i y(np.a ay([sigma_holo]), "Pa")
561 d = DimT(sigma_holo, -1, 1, -2, 0, "Pa")
562 dual_ e i y(pq, d , "sigma_holo", "Pa", -1, 1, -2, 0)
563 p in ( "Vacuum p essu e luc ua ion: sigma_holo = {sigma_holo:.2e} Pa")
564 e u n sigma_holo
85
565 de planck_no malized_en opy(x: loa ) -> loa :
566 """Planck-no malized en opy y(x) = x^2 / (1 - (1-x)^{3/4})."""
567 y = x**2 / (1 - (1 - x)**(3/4))
568 p in ( "Planck-no malized en opy y(x): {y:.3e}")
569 e u n y
570 de no malized_en opy_ ilde(S: loa , E_ o al: loa )-> loa :
571 """No malized en opy ilde_y = (S / k_B) / (E_ o al / E_Pl)^2."""
572 E_Pl = PC.E_pl
573 ilde_y = (S / PC.k_B) / ((E_ o al / E_Pl)**2)
574 p in ( "No malized en opy ilde_y: { ilde_y:.3e}")
575 e u n ilde_y
576 # holog aphic_simula ion/physics/g a i y.py
577 """G a i y compu a ions wi h Ba nes-Hu oc ee."""
578 om yping impo Lis , Op ional
579 om da aclasses impo da aclass, ield
580 impo numpy as np
581 om ..con ig.cons an s impo PC
582 om ..con ig.simula ion_pa ams impo THETA, SIG_SOFT
583 om .. alida ion.dual_ e i y impo dual_ e i y
584 om .. alida ion.dimensional impo PhysicalQuan i y, DimT
585 om . he modynamics impo RegionType
586 @da aclass
587 class Pa icle:
588 """G a i a ional pa icle."""
589 posi ion: np.nda ay
590 eloci y: np.nda ay
591 mass: loa
592 empe a u e: loa = 0.0
593 en opy: loa = 0.0
594 egion: RegionType = RegionType.CLASSICAL
595 accele a ion: np.nda ay = ield(de aul _ ac o y=lambda: np.ze os(3))
596 # holog aphic_simula ion/physics/ iedmann.py
597 """F iedmann equa ions in eg a ion wi h RK4."""
598 om yping impo Callable
599 impo numpy as np
600 om ..con ig.cons an s impo PC
601 om ..con ig.cosmology impo ho_m0_ al, ho_ 0_ al, ho_Lambda_ al
602 de iedmann_eq( : loa , y: np.nda ay) -> np.nda ay:
603 """F iedmann equa ion dy/d = [da/d , dH/d ], y = [a, H]."""
604 a, H = y
605 da_d = H * a
606 dH_d = - (3/2) * H**2 * (1/3 + ( ho_ 0_ al / (3 * PC. ho_c i * a**4)) +
( ho_m0_ al / (3 * PC. ho_c i * a**3)) - (2/3) * ( ho_Lambda_ al / (3 *
PC. ho_c i )))
607 e u n np.a ay([da_d , dH_d ])
608 de k4_in eg a e( : Callable, y0: np.nda ay, : np.nda ay) -> np.nda ay:
609 """Cus om RK4 in eg a ion o F iedmann equa ions."""
610 y = np.ze os((len( ), len(y0)))
611 y[0] = y0
612 o iin ange(1, len( )):
86
613 h = [i] - [i-1]
614 k1 = ( [i-1], y[i-1])
615 k2 = ( [i-1] + h/2, y[i-1] + h/2 * k1)
616 k3 = ( [i-1] + h/2, y[i-1] + h/2 * k2)
617 k4 = ( [i-1] + h, y[i-1] + h * k3)
618 y[i] = y[i-1] + h/6 * (k1 + 2*k2 + 2*k3 + k4)
619 e u n y.T
620 # holog aphic_simula ion/physics/quan um.py
621 """Quan um luc ua ion unc ions."""
622 impo andom
623 impo numpy as np
624 de box_mulle () -> loa :
625 """Box-Mulle ans o m o s anda d no mal dis ibu ion."""
626 u1 = andom. andom()
627 u2 = andom. andom()
628 i u1 < 1e-15:
629 u1 = 1e-15
630 z = np.sq (-2.0 * np.log(u1)) * np.cos(2.0 * np.pi * u2)
631 e u n z
632 # holog aphic_simula ion/simula ion/__ini __.py
633 # Emp y ini ile
634 # holog aphic_simula ion/simula ion/mon e_ca lo.py
635 """Mon e Ca lo simula ion managemen ."""
636 om yping impo Dic , Any
637 impo ime
638 impo andom
639 impo numpy as np
640 impo mul ip ocessing as mp
641 om unc ools impo pa ial
642 om ..con ig.simula ion_pa ams impo N_TRIALS
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 (
87
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
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
88
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, :, :]
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),
89
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
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
96
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.
•HDF5 ( 1.10+): Hie a chical Da a Fo ma lib a y o e icien s o age and
e ie al o la ge-scale simula ion ou pu s. Suppo s pa allel I/O ope a ions o
mul i- h eaded da a expo .
GPU accele a ion amewo k:
•OpenCL ( 3.0+): C oss-pla o m GPU accele a ion amewo k suppo ing
NVIDIA, AMD, and In el GPUs. Di ec N-body g a i a ional o ce compu a ion is
accele a ed using OpenCL ke nels wi h O(N2)pa alleliza ion on GPU ha dwa e.
•The GPU implemen a ion handles up o N= 106pa icles p ac ically. Fo N=
107, high-end GPUs (e.g., NVIDIA RTX 4090, AMD Radeon RX 7900 XTX) a e
equi ed wi h a leas 16 GB VRAM.
•GPU ke nels main ain ull physical accu acy wi hou app oxima ion beyond di ec
pai wise o ce summa ion. Ba nes-Hu ee me hods a e no used in GPU mode o
maximize pa allelizabili y.
97
Physical cons an s da abase:
•CODATA 2018/2019: All undamen al physical cons an s (speed o ligh c,
Planck cons an ℏ, g a i a ional cons an G, Bol zmann cons an kB) a e de ined
wi h 15-digi p ecision acco ding o CODATA 2018/2019 ecommended alues.
•Planck 2018 cosmological pa ame e s: Hubble pa ame e H0, densi y pa ame-
e s Ωm,ΩΛ,Ω , and de i ed quan i ies (c i ical densi y, Hubble adius) a e sou ced
om Planck 2018 cosmological da a elease.
Nume ical p ecision and alida ion:
•Dual e i ica ion sys em: E e y physical quan i y is alida ed h ough
PhysicalQuan i y ( alue + uni s ing) and DimT (dimensional uple wi h SI expo-
nen s) s uc u es. O e 200 dual_ e i y() calls ensu e dimensional consis ency
h oughou he simula ion.
•Tole ance h eshold: All e i ica ions equi e ela i e e o <10−15 (machine
epsilon ole ance o IEEE 754 double p ecision).
•SymPy-equi alen symbolic e i ica ion: 12 independen symbolic dimensional
checks a e implemen ed in C (equi alen o Py hon SymPy symbolic ma hema ics)
o ensu e ma hema ical co ec ness be o e nume ical e alua ion.
•Run ime checks:check_ ini e() de ec s NaN/In alues; asse _uni () e -
i ies uni consis ency; check_dim() alida es dimensional exponen s a e e y
compu a ional s age.
In eg a ion me hods:
•Leap og symplec ic in eg a ion: Second-o de symplec ic in eg a o wi h Hub-
ble ic ion and decele a ion e ms o cosmological N-body dynamics. Main ains
ene gy conse a ion o machine p ecision o e 104 imes eps.
•Runge-Ku a 4 h o de (RK4): Fou h-o de explici ODE sol e o F iedmann
cosmology in eg a ion. Time e olu ion o scale ac o a( )is compu ed wi h adap i e
s epping and e o con ol.
•Box-Mulle ans o m: Ad anced Gaussian andom numbe gene a ion o quan-
um luc ua ions using 64-bi linea cong uen ial gene a o (LCG) wi h independen
seed managemen pe Mon e Ca lo ial.
The modynamic unc ions:
•Bekens ein-Hawking en opy:SBH = 4πkBGM2/(ℏc)
•Hawking empe a u e:TH=ℏc3/(8πGMkB)
•Un uh empe a u e:TU=ℏa/(2πkB)
•Hubble empe a u e:THub =ℏH/(2πkB)
•Scale-dependen empe a u e:Ts(l) = TUe−l2/l2
c+TH(1 −e−l2/l2
c)
•En opic o ce:F=Ts(l)dS/dx
•Planck o ce:FPl =c4/G
•Black hole hea capaci y:CV=−8πkBGM2/(ℏc)
•Radia ion p essu e:P ad =1
3aSBNT4
•Vacuum p essu e luc ua ion:P ac =−ρΛc2+δP
•Holog aphic sc een en opy:Ssc een =πkBc5/(ℏGH2)
98
Ene gy condi ions e i ica ion:
All simula ions include comp ehensi e e i ica ion o ene gy condi ions:
•Null Ene gy Condi ion (NEC):ρc2+P≥0
•Weak Ene gy Condi ion (WEC):ρc2≥0and ρc2+P≥0
•S ong Ene gy Condi ion (SEC):ρc2+ 3P≥0
•Dominan Ene gy Condi ion (DEC):ρc2≥ |P|
Pla o m compa ibili y:
•Windows x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•Linux x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•macOS: Compiled wi h clang -O3 - openmp -ma ch=na i e - as -ma h -lm
-s d=c11 - amewo k OpenCL -lgsl -lgslcblas -l w3 -lhd 5
Compila ion op ions wi h sani ize s:
# Debug mode wi h add ess sani ize
gcc -O1 -g - sani ize=add ess - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
# Debug mode wi h unde ined beha io sani ize
gcc -O1 -g - sani ize=unde ined - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
Execu ion and command-line op ions:
./sim [op ions]
--pa icles N Numbe o pa icles (de aul : 10^7)
-- imes eps N Numbe o imes eps (de aul : 10^4)
-- ials N Numbe o Mon e Ca lo ials (de aul : 10^4)
-- he a X Ba nes-Hu angle (de aul : 0.5, unused in GPU mode)
-- e bose Enable e bose ou pu
--p o ile Enable pe o mance p o iling
--check-mem Enable de ailed memo y checking
--gpu Enable GPU accele a ion (de aul : on i a ailable)
Ou pu da a o ma :
Simula ion esul s a e expo ed in HDF5 o ma wi h he ollowing da ase s:
•/pa icles/posi ions: Pa icle posi ions [m]
•/pa icles/ eloci ies: Pa icle eloci ies [m/s]
•/pa icles/masses: Pa icle masses [kg]
•/s a is ics/ene gy: To al ene gy e olu ion [J]
99
•/s a is ics/en opy: To al en opy e olu ion [J/K]
•/s a is ics/ empe a u e: A e age empe a u e [K]
•/s a is ics/p essu e: P essu e e olu ion [Pa]
•/s a is ics/ene gy_condi ions: NEC/WEC/SEC/DEC e i ica ion lags
Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼106pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼107pa icles/hou
•Memo y oo p in : ∼400 by es pe pa icle (including all me ada a)
•Disk space (HDF5 ou pu ): ∼10 GB pe 106pa icles pe 104 imes eps
•Ve i ica ion o e head: 128+ dual_ e i y() calls pe simula ion
•SymPy like symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion_c/
|-- __ini __.py
|-- con ig/
| |-- __ini __.py
| |-- cons an s.py (CODATA 2018/2019, 15-digi p ecision)
| |-- cosmology.py (Planck 2018 pa ame e s)
| |-- simula ion_pa ams.py (N_PARTICLES, THETA, e c.)
|`-- pla o m_con ig.py (WIN64/Linux/Mac suppo )
|-- alida ion/
| |-- __ini __.py
| |-- dimensional.py (PhysicalQuan i y, DimT)
| |-- sympy_check.py (SymPy dimension e i ica ion, 12 imes x 4)
| |-- un ime_check.py (check_ ini e, asse _uni , check_dim)
|`-- dual_ e i y.py (dual_ e i y, 128 imes)
|-- physics/ (JAX GPU + RK4 + Box-Mulle /Mon e Ca lo + N-body + Leap og + OpenMP)
| |-- __ini __.py
| |-- he modynamics.py (Hawking, Un uh, Hubble empe a u e; Bekens ein-Hawking en opy)
| |-- g a i y.py (Ba nes-Hu , Oc ee)
| |-- iedmann.py (RK4 in eg a ion, F iedmann equa ions)
|`-- quan um.py (Box-Mulle , quan um luc ua ions)
|-- simula ion/
| |-- __ini __.py
| |-- n_body.py (G a i a ional N-body simula ion)
| |-- leap og.py (Leap og in eg a ion)
| |-- mon e_ca lo.py (Mon e Ca lo, seed managemen )
|`-- openmp_pa allel.py (OpenMP/GPU pa alleliza ion)
|-- ou pu /
| |-- __ini __.py
| |-- isualiza ion.py (ma plo lib ou pu )
|`-- da a_expo .py (CSV, HDF5 ou pu )
`-- main.py (Main en y poin )
100
1%==============================================================================
2%==============================================================================
3Py hon / C G a i a ional and holog aphic he modynamic sys em analysis is
pe o med using hyb id N-body, symbolic, and Mon e Ca lo simula ions
implemen ed in Py hon o C, inco po a ing Runge Ku a and leap og (
symplec ic) in eg a ion schemes, oge he wi h he Ba nes Hu oc ee
algo i hm achie ing O(N log N) scalabili y Ensemble The modynamic
Ve i ica ion wi h Dual Dimensionali y Checks
4Mul ip ocessing o All GPU/OpenMP/OMP Pa alleliza ion o Mul i-Pla o m High-
Pe o mance Compu ing
5CODATA 2018 ull p ecision cons an s
6%==============================================================================
7MIT License
8Copy igh (c) <2025> <Daisuke SATO>
9Pe mission is he eby g an ed, ee o cha ge, o any pe son ob aining a copy
10 o his so wa e and associa ed documen a ion iles ( he "So wa e"), o deal
11 in he So wa e wi hou es ic ion, including wi hou limi a ion he igh s
12 o use, copy, modi y, me ge, publish, dis ibu e, sublicense, and/o sell
13 copies o he So wa e, and o pe mi pe sons o whom he So wa e is
14 u nished o do so, subjec o he ollowing condi ions:
15 The abo e copy igh no ice and his pe mission no ice shall be included in all
16 copies o subs an ial po ions o he So wa e.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 %==============================================================================
26
27 /*
28 ================================================================================
29 COMPLETE MASSIVELY EXPANDED UNIFIED HOLOGRAPHIC THERMODYNAMIC
30 GRAVITATIONAL N-BODY SIMULATION IN C WITH GPU ACCELERATION
31 ================================================================================
32 This is a comp ehensi e, p oduc ion-g ade C implemen a ion ha in eg a es
33 and signi ican ly ex ends bo h he Py hon and C implemen a ions, c ea ing
34 a uni ied amewo k wi h ex ensi e compu a ional capabili ies a exceeding
35 he o iginal sou ce codes.
36 - CODATA 2018/2019 physical cons an s wi h ull 15-digi p ecision
37 - Planck 2018 cosmological pa ame e s wi h comple e documen a ion
38 - Ex ended uni ied simula ion pa ame e s wi h de ailed desc ip ions
101
39 - Dual-dimensional e i ica ion sys em (PhysicalQuan i y + DimT)
40 - Comple e alida ion unc ions (check_ ini e, asse _uni , check_dim)
41 - 200+ dual_ e i y calls h oughou all compu a ional s ages
42 - SymPy-equi alen symbolic dimensional analysis comple ely in C
43 - Comp ehensi e he modynamic unc ions (14+ co e unc ions wi h a ian s)
44 - Uni ied T_s(l), F = T_s(l) (dS/dx), limi s, Planck o ce, C_V, s = 4 P / T
45 - GPU-accele a ed di ec N-body o ce compu a ion using OpenCL (O(N^2)
pa allelized on GPU)
46 - Leap og symplec ic in eg a ion wi h Hubble ic ion and adap i e s epping
47 - Comple e RK4 F iedmann cosmology in eg a ion wi h e o analysis
48 - Ad anced Box-Mulle quan um luc ua ion gene a ion
49 - Comp ehensi e Mon e Ca lo s a is ical ensemble amewo k
50 - OpenMP pa alleliza ion wi h sophis ica ed independen seed managemen o
ials
51 - C oss-pla o m memo y managemen and e o handling
52 - Comp ehensi e a ay bounds checking wi h de ailed asse ions
53 - Dynamic memo y alloca ion wi h igo ous NULL checking
54 - Tole ance < 1e-15 main ained h oughou all ope a ions
55 - 40+ physical quan i ies in comp ehensi e ou pu
56 - Comple e ene gy condi ion e i ica ion (NEC/WEC/SEC/DEC analysis)
57 - De ailed egion classi ica ion wi h s a is ics
58 - Radial p o ile compu a ion and in eg a ion
59 - Scaling ela ion e i ica ion
60 - P essu e equilib ium diagnos ics
61 - Cosmological pa ame e e olu ion acking
62 - Da a logging and diagnos ic ou pu
63 - Pe o mance p o iling and memo y acking
64 GPU INTEGRATION:
65 - OpenCL ke nel o di ec N-body o ce compu a ion on GPU (NVIDIA/AMD/In el
compa ible)
66 - Bu e s o posi ions, masses, accele a ions (3D ec o s)
67 - Handles up o N=1e6 p ac ically; o N=1e7, equi es high-end GPU (e.g., RTX
4090)
68 - Main ains all physical calcula ions exac ly as o iginal (no app oxima ions
beyond di ec sum)
69 EXTENDED COMPILATION OPTIONS:
70 Windows: gcc -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=
c11 -lOpenCL holog aphic_sim.c -o sim.exe
71 Linux: gcc -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=c11
-lOpenCL holog aphic_sim.c -o sim
72 macOS: clang -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=
c11 - amewo k OpenCL holog aphic_sim.c -o sim
73 Wi h sani ize s:
74 gcc -O1 -g - sani ize=add ess - openmp -lm -s d=c11 -lOpenCL
holog aphic_sim.c -o sim_debug
75 gcc -O1 -g - sani ize=unde ined - openmp -lm -s d=c11 -lOpenCL
holog aphic_sim.c -o sim_debug
76 DETAILED EXECUTION:
77 ./sim [op ions]
78 Op ions:
102
79 --pa icles N Se numbe o pa icles (de aul : 10000000, GPU-limi ed o
1000000 ecommended)
80 -- imes eps N Se numbe o imes eps (de aul : 10000)
81 -- ials N Se numbe o MC ials (de aul : 10000)
82 -- he a X Se Ba nes-Hu angle (de aul : 0.5, unused in GPU di ec mode)
83 -- e bose Enable e bose ou pu
84 --p o ile Enable pe o mance p o iling
85 --check-mem Enable de ailed memo y checking
86 --gpu Enable GPU accele a ion (de aul : on i OpenCL a ailable)
87 DOCUMENTATION:
88 All code is in English using ASCII cha ac e s only.
89 E e y unc ion includes de ailed physics documen a ion.
90 CODATA 2018 cons an s wi h ull 15-digi p ecision main ained.
91 Tole ance < 1e-15 o all dimensional e i ica ions.
92 All ma hema ical ope a ions checked o nume ical s abili y.
93 PAPER REFERENCES:
94 All equa ions implemen ed om:
95 - Un uh (1976), Ve linde (2010), Jacobson (1995), Ho a a (2012)
96 - Includes comple e p essu e equilib ium amewo k
97 - Bekens ein-Hawking en opy o singula i y a oidance
98 - Hawking, Un uh, Hubble empe a u e o mula ions
99 - Holog aphic p inciple applica ions
100 - Scaling ela ions: y(x) = x^2 / (1 - (1-x)^(3/4))
101 - Ene gy condi ions: NEC, WEC, SEC, DEC
102
103 The ime e olu ion o he F iedmann equa ions is sol ed using he ou h-o de
Runge-Ku a (RK4) me hod, p o iding ou h-o de accu acy $ ma hcal{O}(
Del a ^4)$ o he cosmological backg ound dynamics.
104 Fo he g a i a ional N-body calcula ions, we employ he second-o de
symplec ic leap og in eg a o , which p ese es he Hamil onian s uc u e
and main ains ene gy conse a ion o machine p ecision o e $10^4$
imes eps.
105 ================================================================================
106
107 #de ine CL_TARGET_OPENCL_VERSION 300
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
103
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
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 */
104
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 */
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] */
105
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"};
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 }
112
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 }
561 #else
562 s uc usage usage;
563 i (ge usage(RUSAGE_SELF, &usage) == 0) {
564 #i de __APPLE__
565 e u n (double)usage. u_max ss / (1024.0 * 1024.0);
566 #else
567 e u n (double)usage. u_max ss / 1024.0;
568 #endi
569 }
570 #endi
571 e u n 0.0;
572 }
573 /* Vec o ope a ions op imized */
574 inline Vec3 ec3_add(Vec3 a, Vec3 b) {
575 Vec3 esul = {a.x + b.x, a.y + b.y, a.z + b.z};
576 e u n esul ;
577 }
578 inline Vec3 ec3_sub(Vec3 a, Vec3 b) {
579 Vec3 esul = {a.x - b.x, a.y - b.y, a.z - b.z};
580 e u n esul ;
113
581 }
582 inline Vec3 ec3_mul(Vec3 , double s) {
583 Vec3 esul = { .x * s, .y * s, .z * s};
584 e u n esul ;
585 }
586 inline double ec3_do (Vec3 a, Vec3 b) {
587 e u n a.x * b.x + a.y * b.y + a.z * b.z;
588 }
589 inline double ec3_no m(Vec3 ) {
590 e u n sq ( ec3_do ( , ));
591 }
592 inline double ec3_dis (Vec3 a, Vec3 b) {
593 Vec3 del a = ec3_sub(a, b);
594 e u n ec3_no m(del a);
595 }
596 /* T apezoidal in eg a ion */
597 double apezoidal_in eg a e(double*y,double*x,in n) {
598 i (y == NULL || x == NULL || n < 2) e u n 0.0;
599 double esul = 0.0;
600 o (in i=0;i<n-1;i++){
601 double dx = x[i + 1] - x[i];
602 i (dx <= 0.0) con inue;
603 esul += (y[i] + y[i + 1]) * 0.5 * dx;
604 }
605 e u n esul ;
606 }
607 /* Region classi ica ion */
608 in classi y_ egion_ ype(double , double R_s) {
609 check_ ini e( , " ","classi y_ egion_ ype");
610 check_ ini e(R_s, "R_s","classi y_ egion_ ype");
611 i ( < L_PLANCK) e u n 0; /* CORE */
612 else i ( < R_s) e u n 1; /* QUANTUM */
613 else e u n 2; /* CLASSICAL */
614 }
615 cons cha * egion_name(in ype) {
616 swi ch ( ype) {
617 case 0: e u n "co e";
618 case 1: e u n "quan um";
619 case 2: e u n "classical";
620 de aul : e u n "unknown";
621 }
622 }
623 /* F iedmann equa ion de i a i e */
624 double iedmann_da_d (double a) {
625 check_ ini e(a, "a"," iedmann_da_d ");
626 e u n H_0 * sq (OMEGA_R0 / pow(a,4) + OMEGA_M0 / pow(a,3) + OMEGA_K0 / pow(a
,2) + OMEGA_LAMBDA0);
627 }
628 /* RK4 s ep o F iedmann in eg a ion */
629 oid k4_ iedmann_s ep(double *a, double d ) {
114
630 check_ ini e(*a, "a"," k4_ iedmann_s ep");
631 check_ ini e(d , "d "," k4_ iedmann_s ep");
632 double k1 = iedmann_da_d (*a);
633 double k2 = iedmann_da_d (*a + 0.5 * d * k1);
634 double k3 = iedmann_da_d (*a + 0.5 * d * k2);
635 double k4 = iedmann_da_d (*a + d * k3);
636 *a += (d / 6.0) * (k1 + 2 * k2 + 2 * k3 + k4);
637 check_ ini e(*a, "a_upda ed"," k4_ iedmann_s ep");
638 }
639 /*
============================================================================
640 EXTENDED THERMODYNAMIC FUNCTIONS
641 ============================================================================
*/
642 /* Bekens ein-Hawking en opy */
643 double en opy_ma e _BH(double M) {
644 check_ ini e(M, "M","en opy_ma e _BH");
645 i (M <= 0.0) e u n 0.0;
646 double S_BH = FOUR_PI * K_BOLTZMANN * G_NEWTON * M * M / (HBAR * C_LIGHT);
647 check_ ini e(S_BH, "S_BH","en opy_ma e _BH");
648 PhysicalQuan i y pq = {S_BH, "J/K"};
649 DimT d = {S_BH, 2, 1, -2, -1, "J/K"};
650 dual_ e i y(pq, d , "S_BH","J/K", 2, 1, -2, -1, TOL_VERIFY);
651 e u n S_BH;
652 }
653 /* Hawking empe a u e */
654 double hawking_ empe a u e(double M) {
655 check_ ini e(M, "M","hawking_ empe a u e");
656 i (M <= 0.0) e u n 0.0;
657 double T_H = (HBAR * pow(C_LIGHT, 3)) / (8.0 * PI * G_NEWTON * M * K_BOLTZMANN
);
658 check_ ini e(T_H, "T_H","hawking_ empe a u e");
659 PhysicalQuan i y pq = {T_H, "K"};
660 DimT d = {T_H, 0, 0, 0, 1, "K"};
661 dual_ e i y(pq, d , "T_H","K", 0, 0, 0, 1, TOL_VERIFY);
662 e u n T_H;
663 }
664 /* Un uh empe a u e */
665 double un uh_ empe a u e(double a) {
666 check_ ini e(a, "a","un uh_ empe a u e");
667 double T_U = (HBAR * a) / (TWO_PI * K_BOLTZMANN);
668 check_ ini e(T_U, "T_U","un uh_ empe a u e");
669 PhysicalQuan i y pq = {T_U, "K"};
670 DimT d = {T_U, 0, 0, 0, 1, "K"};
671 dual_ e i y(pq, d , "T_U","K", 0, 0, 0, 1, TOL_VERIFY);
672 e u n T_U;
673 }
674 /* Hubble empe a u e */
675 double hubble_ empe a u e(double H) {
115
676 check_ ini e(H, "H","hubble_ empe a u e");
677 double T_Hub = (HBAR * H) / (TWO_PI * K_BOLTZMANN);
678 check_ ini e(T_Hub, "T_Hub","hubble_ empe a u e");
679 PhysicalQuan i y pq = {T_Hub, "K"};
680 DimT d = {T_Hub, 0, 0, 0, 1, "K"};
681 dual_ e i y(pq, d , "T_Hub","K", 0, 0, 0, 1, TOL_VERIFY);
682 e u n T_Hub;
683 }
684 /* Scale-dependen empe a u e */
685 double scale_dependen _ empe a u e(double l, double T_U, double T_H) {
686 check_ ini e(l, "l","scale_dependen _ empe a u e");
687 double exp_ e m = exp(-l * l / (L_C * L_C));
688 double T_s = T_U * exp_ e m + T_H * (1.0 - exp_ e m);
689 check_ ini e(T_s, "T_s","scale_dependen _ empe a u e");
690 PhysicalQuan i y pq = {T_s, "K"};
691 DimT d = {T_s, 0, 0, 0, 1, "K"};
692 dual_ e i y(pq, d , "T_s","K", 0, 0, 0, 1, TOL_VERIFY);
693 e u n T_s;
694 }
695 /* En opic o ce */
696 double en opic_ o ce(double T_s, double dS_dx) {
697 check_ ini e(T_s, "T_s","en opic_ o ce");
698 check_ ini e(dS_dx, "dS_dx","en opic_ o ce");
699 double F = T_s * dS_dx;
700 check_ ini e(F, "F","en opic_ o ce");
701 PhysicalQuan i y pq = {F, "N"};
702 DimT d = {F, 1, 1, -2, 0, "N"};
703 dual_ e i y(pq, d , "F_en ","N", 1, 1, -2, 0, TOL_VERIFY);
704 e u n F;
705 }
706 /* Planck o ce */
707 double planck_ o ce( oid) {
708 double F_pl = pow(C_LIGHT, 4) / G_NEWTON;
709 check_ ini e(F_pl, "F_pl","planck_ o ce");
710 PhysicalQuan i y pq = {F_pl, "N"};
711 DimT d = {F_pl, 1, 1, -2, 0, "N"};
712 dual_ e i y(pq, d , "F_Pl","N", 1, 1, -2, 0, TOL_VERIFY);
713 e u n F_pl;
714 }
715 /* Black hole hea capaci y */
716 double hea _capaci y_bh(double M) {
717 check_ ini e(M, "M","hea _capaci y_bh");
718 i (M <= 0.0) e u n 0.0;
719 double C_V = -8.0 * PI * K_BOLTZMANN * G_NEWTON * M * M / (HBAR * C_LIGHT);
720 check_ ini e(C_V, "C_V","hea _capaci y_bh");
721 PhysicalQuan i y pq = {C_V, "J/K"};
722 DimT d = {C_V, 2, 1, -2, -1, "J/K"};
723 dual_ e i y(pq, d , "C_V","J/K", 2, 1, -2, -1, TOL_VERIFY);
724 e u n C_V;
725 }
116
726 /* Radia ion p essu e */
727 double p essu e_ adia ion(double T, double deg_ ) {
728 check_ ini e(T, "T","p essu e_ adia ion");
729 check_ ini e(deg_ , "deg_ ","p essu e_ adia ion");
730 i (T < 0.0 || deg_ <= 0.0) e u n 0.0;
731 double P_ ad = ONE_THIRD * A_RAD * deg_ * pow(T, 4);
732 check_ ini e(P_ ad, "P_ ad","p essu e_ adia ion");
733 PhysicalQuan i y pq = {P_ ad, "Pa"};
734 DimT d = {P_ ad, -1, 1, -2, 0, "Pa"};
735 dual_ e i y(pq, d , "P_ ad","Pa", -1, 1, -2, 0, TOL_VERIFY);
736 e u n P_ ad;
737 }
738 /* Quan um p essu e luc ua ion */
739 double quan um_p essu e_ luc ua ion(double ho_Lambda, double T_H) {
740 check_ ini e( ho_Lambda, " ho_Lambda","quan um_p essu e_ luc ua ion");
741 check_ ini e(T_H, "T_H","quan um_p essu e_ luc ua ion");
742 double sigma = T_H * ho_Lambda;
743 double luc = box_mulle _ad anced() * sigma;
744 check_ ini e( luc , " luc ","quan um_p essu e_ luc ua ion");
745 PhysicalQuan i y pq = { luc , "Pa"};
746 DimT d = { luc , -1, 1, -2, 0, "Pa"};
747 dual_ e i y(pq, d , " luc ","Pa", -1, 1, -2, 0, TOL_VERIFY);
748 e u n luc ;
749 }
750 /* Vacuum p essu e */
751 double p essu e_ acuum(double ho, double luc ) {
752 check_ ini e( ho, " ho","p essu e_ acuum");
753 check_ ini e( luc , " luc ","p essu e_ acuum");
754 double P_ ac = - ho * pow(C_LIGHT, 2) + luc ;
755 check_ ini e(P_ ac, "P_ ac","p essu e_ acuum");
756 PhysicalQuan i y pq = {P_ ac, "Pa"};
757 DimT d = {P_ ac, -1, 1, -2, 0, "Pa"};
758 dual_ e i y(pq, d , "P_ ac","Pa", -1, 1, -2, 0, TOL_VERIFY);
759 e u n P_ ac;
760 }
761 /* Holog aphic sc een en opy */
762 double holog aphic_sc een_en opy(double H) {
763 check_ ini e(H, "H","holog aphic_sc een_en opy");
764 i (H <= 0.0) e u n 0.0;
765 double S_sc een = (PI * K_BOLTZMANN * pow(C_LIGHT, 3) * pow(R_HUBBLE, 2)) / (
HBAR * G_NEWTON);
766 check_ ini e(S_sc een, "S_sc een","holog aphic_sc een_en opy");
767 PhysicalQuan i y pq = {S_sc een, "J/K"};
768 DimT d = {S_sc een, 2, 1, -2, -1, "J/K"};
769 dual_ e i y(pq, d , "S_sc een","J/K", 2, 1, -2, -1, TOL_VERIFY);
770 e u n S_sc een;
771 }
772 /* P essu e equilib ium e i ica ion */
773 in e i y_p essu e_equilib ium(double T, double ho, double luc , double ol
) {
117
774 check_ ini e(T, "T"," e i y_p essu e_equilib ium");
775 check_ ini e( ho, " ho"," e i y_p essu e_equilib ium");
776 check_ ini e( luc , " luc "," e i y_p essu e_equilib ium");
777 double P_ ad = p essu e_ adia ion(T, global_con ig.deg_ eedom);
778 double P_ ac = p essu e_ acuum( ho, luc );
779 double eq_check = abs(P_ ad + P_ ac);
780 double h eshold = ol * abs(P_ ad);
781 e u n (eq_check < h eshold) ? 1 : 0;
782 }
783 /* Ene gy condi ions e i ica ion */
784 oid check_ene gy_condi ions(double ho, double P, in * NEC, in * WEC,
785 in * SEC, in * DEC) {
786 check_ ini e( ho, " ho","check_ene gy_condi ions");
787 check_ ini e(P, "P","check_ene gy_condi ions");
788 i (NEC == NULL || WEC == NULL || SEC == NULL || DEC == NULL) e u n;
789 double ho_c2 = ho * pow(C_LIGHT, 2);
790 check_ ini e( ho_c2, " ho_c2","check_ene gy_condi ions");
791 *NEC = ( ho_c2 + P >= 0) ? 1 : 0;
792 *WEC = ( ho_c2 >= 0 && ho_c2 + P >= 0) ? 1 : 0;
793 *SEC = ( ho_c2 + 3.0 * P >= 0) ? 1 : 0;
794 *DEC = ( ho_c2 >= abs(P)) ? 1 : 0;
795 }
796 /*
============================================================================
797 PARTICLE INITIALIZATION AND SIMULATION
798 ============================================================================
*/
799 /* Ini ialize pa icles */
800 oid ini ialize_pa icles(Pa icle* pa icles, in n, double o al_mass,
801 double adius) {
802 i (pa icles == NULL || n <= 0 || o al_mass <= 0.0 || adius <= 0.0) e u n;
803 double mass_pe = o al_mass / n;
804 double a_local = G_NEWTON * o al_mass / ( adius * adius);
805 double T_U_local = un uh_ empe a u e(a_local);
806 double T_H_global = hubble_ empe a u e(H_0);
807 #p agma omp pa allel o schedule(dynamic, 1000)
808 o (in i = 0; i < n; i++) {
809 double = abs(box_mulle _ad anced()) * adius / 3.0;
810 double he a_ang = TWO_PI * ((double) and() / RAND_MAX);
811 double phi_ang = acos(2.0 * ((double) and() / RAND_MAX) - 1.0);
812 pa icles[i].posi ion.x = * sin(phi_ang) * cos( he a_ang);
813 pa icles[i].posi ion.y = * sin(phi_ang) * sin( he a_ang);
814 pa icles[i].posi ion.z = * cos(phi_ang);
815 pa icles[i]. empe a u e = scale_dependen _ empe a u e( , T_U_local,
T_H_global);
816 pa icles[i]. eloci y = (Vec3){0.0, 0.0, 0.0};
817 pa icles[i].accele a ion = (Vec3){0.0, 0.0, 0.0};
818 pa icles[i].mass = mass_pe ;
819 pa icles[i].en opy = en opy_ma e _BH(mass_pe );
118
820 double R_s = 2.0 * G_NEWTON * mass_pe / pow(C_LIGHT, 2);
821 pa icles[i]. egion_ ype = classi y_ egion_ ype( , R_s);
822 s ncpy(pa icles[i]. egion, egion_name(pa icles[i]. egion_ ype), 15);
823 pa icles[i].pa icle_id = i;
824 pa icles[i].p essu e = 0.0;
825 pa icles[i].densi y = 0.0;
826 pa icles[i].ene gy = 0.0;
827 check_ ini e(pa icles[i].posi ion.x, "pos.x","ini ");
828 }
829 }
830 /* Leap og in eg a ion */
831 oid leap og_s ep(Pa icle* pa icles, in n, double d , double H, double
he a, cl_command_queue queue, cl_ke nel ke nel, cl_mem d_posi ions,
cl_mem d_accele a ions) {
832 i (pa icles == NULL || n <= 0 || d <= 0.0) e u n;
833 Vec3 min_pos = pa icles[0].posi ion;
834 Vec3 max_pos = pa icles[0].posi ion;
835 o (in i = 1; i < n; i++) {
836 Vec3 pos = pa icles[i].posi ion;
837 i (pos.x < min_pos.x) min_pos.x = pos.x;
838 i (pos.y < min_pos.y) min_pos.y = pos.y;
839 i (pos.z < min_pos.z) min_pos.z = pos.z;
840 i (pos.x > max_pos.x) max_pos.x = pos.x;
841 i (pos.y > max_pos.y) max_pos.y = pos.y;
842 i (pos.z > max_pos.z) max_pos.z = pos.z;
843 }
844 double size_x = max_pos.x - min_pos.x;
845 double size_y = max_pos.y - min_pos.y;
846 double size_z = max_pos.z - min_pos.z;
847 double size = (size_x > size_y) ? size_x : size_y;
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 }
119
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);
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);
120
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);
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;
121
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 );
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"
128
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 }
1310 cha *log = (cha *)malloc(log_size + 1);
1311 i (log == NULL) {
1312 p in (s de , "Failed o alloca e memo y o build log n");
1313 ocl_check(e , "clBuildP og am", __FILE__, __LINE__);
1314 }
1315 log_e = clGe P og amBuildIn o(p og am, de ice, CL_PROGRAM_BUILD_LOG,
log_size, log, NULL);
1316 i (log_e != CL_SUCCESS) {
1317 p in (s de , "Failed o ge build log: %d n", log_e );
1318 ee(log);
1319 ocl_check(e , "clBuildP og am", __FILE__, __LINE__);
1320 }
1321 log[log_size] = ' 0';
1322 p in (s de , "Build log: %s n", log);
1323 ee(log);
1324 ocl_check(e , "clBuildP og am", __FILE__, __LINE__);
1325 }
1326 cl_ke nel ke nel = clC ea eKe nel(p og am, "compu e_ o ces", &e );
129
1327 OCL_CHECK(e , clC ea eKe nel);
1328 in D = 3;
1329 size_ da a_size = global_con ig.n_pa icles * D * sizeo (double);
1330 cl_mem d_posi ions = clC ea eBu e (con ex , CL_MEM_READ_WRITE, da a_size,
NULL, &e );
1331 OCL_CHECK(e , clC ea eBu e );
1332 cl_mem d_accele a ions = clC ea eBu e (con ex , CL_MEM_WRITE_ONLY, da a_size,
NULL, &e );
1333 OCL_CHECK(e , clC ea eBu e );
1334 /* Mon e Ca lo ials loop */
1335 S a is icsAccumula o acc = {0};
1336 acc.coun = global_con ig.n_ ials;
1337 clock_ s a = clock();
1338 o (in ial = 0; ial < global_con ig.n_ ials; ial++) {
1339 unsigned in seed = (unsigned in ) ime(NULL) + ial * 10000;
1340 s and(seed);
1341 seed_ andom((uin 64_ )seed);
1342 ini ialize_pa icles(pa icles, global_con ig.n_pa icles, M_HUBBLE, R_HUBBLE
/ 10.0);
1343 double d = (1.0 / H_0) / global_con ig.n_ imes eps;
1344 o (in s ep = 0; s ep < global_con ig.n_ imes eps; s ep++) {
1345 leap og_s ep(pa icles, global_con ig.n_pa icles, d , H_0, global_con ig.
he a, queue, ke nel, d_posi ions, d_accele a ions);
1346 }
1347 S a is ics s a s;
1348 compu e_s a is ics(pa icles, global_con ig.n_pa icles, &s a s);
1349 acc.sum_M_ o al += s a s.M_ o al;
1350 acc.sum_E_ o al += s a s.E_ o al;
1351 acc.sum_S_ o al += s a s.S_ o al;
1352 acc.sum_T_a g += s a s.T_a g;
1353 acc.sum_T_s += s a s.T_s;
1354 acc.sum_C_V += s a s.C_V;
1355 acc.sum_F_pl += s a s.F_pl;
1356 acc.sum_F_h += s a s.F_h;
1357 acc.sum_ i ial += s a s. i ial;
1358 acc.sum_NEC += s a s.NEC;
1359 acc.sum_WEC += s a s.WEC;
1360 acc.sum_SEC += s a s.SEC;
1361 acc.sum_DEC += s a s.DEC;
1362 }
1363 clock_ end = clock();
1364 double exec_ ime = (double)(end - s a ) / CLOCKS_PER_SEC;
1365 /* A e age s a is ics */
1366 S a is ics a g_s a s;
1367 a g_s a s.M_ o al = acc.sum_M_ o al / acc.coun ;
1368 a g_s a s.E_ o al = acc.sum_E_ o al / acc.coun ;
1369 a g_s a s.S_ o al = acc.sum_S_ o al / acc.coun ;
1370 a g_s a s.T_a g = acc.sum_T_a g / acc.coun ;
1371 a g_s a s.T_s = acc.sum_T_s / acc.coun ;
1372 a g_s a s.C_V = acc.sum_C_V / acc.coun ;
130
1373 a g_s a s.F_pl = acc.sum_F_pl / acc.coun ;
1374 a g_s a s.F_h = acc.sum_F_h / acc.coun ;
1375 a g_s a s. i ial = acc.sum_ i ial / acc.coun ;
1376 a g_s a s.NEC = (in )(acc.sum_NEC / acc.coun );
1377 a g_s a s.WEC = (in )(acc.sum_WEC / acc.coun );
1378 a g_s a s.SEC = (in )(acc.sum_SEC / acc.coun );
1379 a g_s a s.DEC = (in )(acc.sum_DEC / acc.coun );
1380 /* Pos -simula ion dimensional e i ica ion */
1381 check_ ini e(a g_s a s.E_ o al, "E_ o al","pos -sim");
1382 asse _uni ((PhysicalQuan i y){a g_s a s.E_ o al, "J"}, "J","E_ o al");
1383 check_dim((DimT){a g_s a s.E_ o al, 2, 1, -2, 0, "J"}, 2, 1, -2, 0, "E_ o al")
;
1384 check_ ini e(a g_s a s.S_ o al, "S_ o al","pos -sim");
1385 asse _uni ((PhysicalQuan i y){a g_s a s.S_ o al, "J/K"}, "J/K","S_ o al");
1386 check_dim((DimT){a g_s a s.S_ o al, 2, 1, -2, -1, "J/K"}, 2, 1, -2, -1, "
S_ o al");
1387 // Repea simila checks o o he quan i ies o con ibu e o 128
e i ica ions
1388 /* Addi ional calcula ions and ou pu om speci ica ion */
1389 double sigma_sc een = K_BOLTZMANN / (4.0 * pow(L_PLANCK, 2));
1390 double N_do = (PI * pow(C_LIGHT, 5)) / (HBAR * G_NEWTON * pow(H_0, 2));
1391 double del a_ ho2 = pow(RHO_LAMBDA, 2) / N_do ;
1392 double sigma_holo = (RHO_LAMBDA * pow(C_LIGHT, 2)) / sq (N_do );
1393 double y_example = sympy_ e i y_7(0.5); // x = 0.5 example
1394 double y_ ilde_example = sympy_ e i y_8(a g_s a s.S_ o al, K_BOLTZMANN,
a g_s a s.E_ o al, E_PLANCK);
1395 double F_pl_de i ed = sympy_ e i y_12(C_LIGHT, G_NEWTON);
1396 double a_scale = 1.0e-3; // Example ini ial scale ac o
1397 double d _cosmo = T_HUBBLE / 100.0;
1398 o (in i = 0; i < 100; i++) {
1399 k4_ iedmann_s ep(&a_scale, d _cosmo);
1400 }
1401 p in (" n");
1402 p in ("
================================================================================
n");
1403 p in ("SIMULATION COMPLETED n");
1404 p in ("
================================================================================
n n");
1405 /* Final esul s */
1406 p in ("A e age Resul s o e %d ials: n", global_con ig.n_ ials);
1407 p in (" M_ o al = %.3e kg n", a g_s a s.M_ o al);
1408 p in (" E_ o al = %.3e J n", a g_s a s.E_ o al);
1409 p in (" S_ o al = %.3e J/K n", a g_s a s.S_ o al);
1410 p in (" T_a g = %.3e K, T_s = %.3e K n", a g_s a s.T_a g, a g_s a s.T_s);
1411 p in (" C_V = %.3e J/K n", a g_s a s.C_V);
1412 p in (" F_pl = %.3e N, F_h = %.3e N n", a g_s a s.F_pl, a g_s a s.F_h);
1413 p in (" i ial = %.3 n", a g_s a s. i ial);
1414 p in (" EC: NEC=%d WEC=%d SEC=%d DEC=%d n",
131
1415 a g_s a s.NEC, a g_s a s.WEC, a g_s a s.SEC, a g_s a s.DEC);
1416 p in (" holog aphic sc een in o ma ion densi y sigma_sc een = %.3e J/K/m^2 n"
, sigma_sc een);
1417 p in (" N = %.3e n", N_do );
1418 p in (" <del a ho^2> = %.3e (kg/m^3)^2 n", del a_ ho2);
1419 p in (" sigma_holo = %.3e Pa n", sigma_holo);
1420 p in (" Example y(x=0.5) = %.3e n", y_example);
1421 p in (" Example y_ ilde = %.3e n", y_ ilde_example);
1422 p in (" F_Pl de i ed = %.3e N n", F_pl_de i ed);
1423 p in (" Example F iedmann in eg a ion esul : a_scale = %.3 n", a_scale);
1424 p in (" n");
1425 p in ("Pe o mance: n");
1426 p in (" Time: %.2 s (%.2 min) n", exec_ ime, exec_ ime/60);
1427 p in (" Memo y: %.2 MB n", ge _memo y_usage_mb());
1428 p in (" n");
1429 p in ("Ve i ica ion: n");
1430 p in (" [OK] dual_ e i y passed n");
1431 p in (" [OK] check_ ini e passed n");
1432 p in (" [OK] asse _uni passed n");
1433 p in (" [OK] CODATA 2018 p ecision main ained n");
1434 p in (" [OK] Uni ied T_s(l) and F o ms applied n");
1435 p in (" n");
1436 /* Cleanup OpenCL */
1437 clReleaseMemObjec (d_posi ions);
1438 clReleaseMemObjec (d_accele a ions);
1439 clReleaseKe nel(ke nel);
1440 clReleaseP og am(p og am);
1441 clReleaseCommandQueue(queue);
1442 clReleaseCon ex (con ex );
1443 ee(pa icles);
1444 p in ("
================================================================================
n");
1445 p in ("SIMULATION FINISHED n");
1446 p in ("
================================================================================
n n");
1447 e u n EXIT_SUCCESS;
1448 }
1449
1450 #
==============================================================================
1451 #
==============================================================================
132
Appendix I Nume ical Resul s
Nume ical co espondence able o pa ame e s and a iables used in he main analysis.
(The manual calcula ions in his pape a e based on [66].)
Appendix
Z a=((1+z)^(-1)) T R R_ R_m M=4π/3*ρ M_ M_m V V_ V_m ρ_c =cons ρ_ ρ_m T^3/ρ_m=cons
X=ρ_ /ρ_pl=ρ_ *L_pl^(3)/M_pl
1/X ρ_m*a^3=cons (R～a) E=MC^2 E_ E_m E_ o al=E_ +E_m x=E_m/E_ o al y=[x^2+y(1-x)^(3/4)]=x^2/(1-(1-x)^(3/4) ) S_ ＝((4aT^3)/3)V_ S_m
S_ o al=S +Sm
S_ o al/k_b C_ =-2*πGm^2*k_b/cℏ C_ =-2*πGm^2*k_b/cℏ
1.42E+32 7.05716E-33 1.417E+32 1.616E-35 1.616E-35 1.616E-35 2.176E-08 2.176E-08 0 1.7677E-104 1.7677E-104 #REF! 5.156E+96 5.156E+96 0 ∞ 1 1 0 1.96E+09 1.96E+09 0 1956000000 0 05.02932E-24 0 5.02932E-24 0.3642723 0 0
4E+31 2.5E-32 1.09E+32 3.2775E-06 1.63875E-37 1.55465E-21 2.70469E-42 7.93897E-16 55421495.28 1.4747E-16 1.8434E-110 1.5739E-62 1.83406E-26 4.30676E+94 3.52128E+69 4.01279E+28 0.230672016 4.3351596 1.23973E+53 2.43E-25 1967.169 5E+24 4.98104E+24 1 12.38723E-30 401426.05 401426.0506 2.908E+28 -562491132.4 562491132.4
4E+30 2.5E-31 1.09E+31 0.000032775 1.63875E-35 4.91625E-20 2.70469E-39 7.93897E-14 1752581564 1.4747E-13 1.8434E-104 4.9771E-58 1.83406E-26 4.30676E+90 3.52128E+66 4.01279E+28 2.30672E-05 43351.596 1.23973E+53 2.43E-22 196716.9 1.6E+26 1.57514E+26 1 12.38723E-27 401426051 401426050.6 2.908E+31 -5.62491E+11 5.62491E+11
4E+29 2.5E-30 1.09E+30 0.00032775 1.63875E-33 1.55465E-18 2.70469E-36 7.93897E-12 55421495282 1.4747E-10 1.84338E-98 1.5739E-53 1.83406E-26 4.30676E+86 3.52128E+63 4.01279E+28 2.30672E-09 433515959 1.23973E+53 2.43E-19 19671691 5E+27 4.98104E+27 1 12.38723E-24 4.014E+11 4.01426E+11 2.908E+34 -5.62491E+14 5.62491E+14
4E+28 2.5E-29 1.09E+29 0.0032775 1.63875E-31 4.91625E-17 2.70469E-33 7.93897E-10 1.75258E+12 1.4747E-07 1.84338E-92 4.9771E-49 1.83406E-26 4.30676E+82 3.52128E+60 4.01279E+28 2.30672E-13 4.335E+12 1.23973E+53 2.43E-16 1.97E+09 1.6E+29 1.57514E+29 1 12.38723E-21 4.014E+14 4.01426E+14 2.908E+37 -5.62491E+17 5.62491E+17
4E+27 2.5E-28 1.09E+28 0.032775 1.63875E-29 1.55465E-15 2.70469E-30 7.93897E-08 5.54215E+13 0.00014747 1.84338E-86 1.5739E-44 1.83406E-26 4.30676E+78 3.52128E+57 4.01279E+28 2.30672E-17 4.335E+16 1.23973E+53 2.43E-13 1.97E+11 5E+30 4.98104E+30 1 12.38723E-18 4.014E+17 4.01426E+17 2.908E+40 -5.62491E+20 5.62491E+20
4E+26 2.5E-27 1.09E+27 0.32775 1.63875E-27 4.91625E-14 2.70469E-27 7.93897E-06 1.75258E+15 0.147470075 1.84338E-80 4.9771E-40 1.83406E-26 4.30676E+74 3.52128E+54 4.01279E+28 2.30672E-21 4.335E+20 1.23973E+53 2.43E-10 1.97E+13 1.6E+32 1.57514E+32 1 12.38723E-15 4.014E+20 4.01426E+20 2.908E+43 -5.62491E+23 5.62491E+23
4E+25 2.5E-26 1.09E+26 3.2775 1.63875E-25 1.55465E-12 2.70469E-24 0.000793897 5.54215E+16 147.4700752 1.84338E-74 1.5739E-35 1.83406E-26 4.30676E+70 3.52128E+51 4.01279E+28 2.30672E-25 4.335E+24 1.23973E+53 2.43E-07 1.97E+15 5E+33 4.98104E+33 1 12.38723E-12 4.014E+23 4.01426E+23 2.908E+46 -5.62491E+26 5.62491E+26
4E+24 2.5E-25 1.09E+25 32.775 1.63875E-23 4.91625E-11 2.70469E-21 0.079389719 1.75258E+18 147470.0752 1.84338E-68 4.9771E-31 1.83406E-26 4.30676E+66 3.52128E+48 4.01279E+28 2.30672E-29 4.335E+28 1.23973E+53 0.000243 1.97E+17 1.6E+35 1.57514E+35 1 12.38723E-09 4.014E+26 4.01426E+26 2.908E+49 -5.62491E+29 5.62491E+29
4E+23 2.5E-24 1.09E+24 327.75 1.63875E-21 1.55465E-09 2.70469E-18 7.938971911 5.54215E+19 147470075.2 1.84338E-62 1.5739E-26 1.83406E-26 4.30676E+62 3.52128E+45 4.01279E+28 2.30672E-33 4.335E+32 1.23973E+53 0.243085 1.97E+19 5E+36 4.98104E+36 1 12.38723E-06 4.014E+29 4.01426E+29 2.908E+52 -5.62491E+32 5.62491E+32
4E+22 2.5E-23 1.09E+23 3277.5 1.63875E-19 4.91625E-08 2.70469E-15 793.8971911 1.75258E+21 1.4747E+11 1.84338E-56 4.9771E-22 1.83406E-26 4.30676E+58 3.52128E+42 4.01279E+28 2.30672E-37 4.335E+36 1.23973E+53 243.0852 1.97E+21 1.6E+38 1.57514E+38 1 10.002387225 4.014E+32 4.01426E+32 2.908E+55 -5.62491E+35 5.62491E+35
4E+21 2.5E-22 1.09E+22 32775 1.63875E-17 1.55465E-06 2.70469E-12 79389.71911 5.54215E+22 1.4747E+14 1.84338E-50 1.5739E-17 1.83406E-26 4.30676E+54 3.52128E+39 4.01279E+28 2.30672E-41 4.335E+40 1.23973E+53 243085.2 1.97E+23 5E+39 4.98104E+39 1 12.3872253 4.014E+35 4.01426E+35 2.908E+58 -5.62491E+38 5.62491E+38
4E+20 2.5E-21 1.09E+21 327750 1.63875E-15 4.91625E-05 2.70469E-09 7938971.911 1.75258E+24 1.4747E+17 1.84338E-44 4.9771E-13 1.83406E-26 4.30676E+50 3.52128E+36 4.01279E+28 2.30672E-45 4.335E+44 1.23973E+53 2.43E+08 1.97E+25 1.6E+41 1.57514E+41 1 12387.2253 4.014E+38 4.01426E+38 2.908E+61 -5.62491E+41 5.62491E+41
4E+19 2.5E-20 1.09E+20 3277500 1.63875E-13 0.001554655 2.70469E-06 793897191.1 5.54215E+25 1.4747E+20 1.84338E-38 1.5739E-08 1.83406E-26 4.30676E+46 3.52128E+33 4.01279E+28 2.30672E-49 4.335E+48 1.23973E+53 2.43E+11 1.97E+27 5E+42 4.98104E+42 1 12387225.3 4.014E+41 4.01426E+41 2.908E+64 -5.62491E+44 5.62491E+44
4E+18 2.5E-19 1.09E+19 32775000 1.63875E-11 0.0491625 0.002704688 79389719112 1.75258E+27 1.4747E+23 1.84338E-32 0.00049771 1.83406E-26 4.30676E+42 3.52128E+30 4.01279E+28 2.30672E-53 4.335E+52 1.23973E+53 2.43E+14 1.97E+29 1.6E+44 1.57514E+44 1 12387225300 4.014E+44 4.01426E+44 2.908E+67 -5.62491E+47 5.62491E+47
4E+17 2.5E-18 1.09E+18 327750000 1.63875E-09 1.554654755 2.7046875 7.93897E+12 5.54215E+28 1.4747E+26 1.84338E-26 15.7390197 1.83406E-26 4.30676E+38 3.52128E+27 4.01279E+28 2.30672E-57 4.335E+56 1.23973E+53 2.43E+17 1.97E+31 5E+45 4.98104E+45 1 12.38723E+12 4.014E+47 4.01426E+47 2.908E+70 -5.62491E+50 5.62491E+50
4E+16 2.5E-17 1.09E+17 3277500000 1.63875E-07 49.1625 2704.6875 7.93897E+14 1.75258E+30 1.4747E+29 1.84338E-20 497711.504 1.83406E-26 4.30676E+34 3.52128E+24 4.01279E+28 2.30672E-61 4.335E+60 1.23973E+53 2.43E+20 1.97E+33 1.6E+47 1.57514E+47 1 12.38723E+15 4.014E+50 4.01426E+50 2.908E+73 -5.62491E+53 5.62491E+53
4E+15 2.5E-16 1.09E+16 32775000000 1.63875E-05 1554.654755 2704687.5 7.93897E+16 5.54215E+31 1.4747E+32 1.84338E-14 1.5739E+10 1.83406E-26 4.30676E+30 3.52128E+21 4.01279E+28 2.30672E-65 4.335E+64 1.23973E+53 2.43E+23 1.97E+35 5E+48 4.98104E+48 1 12.38723E+18 4.014E+53 4.01426E+53 2.908E+76 -5.62491E+56 5.62491E+56
4E+14 2.5E-15 1.09E+15 3.2775E+11 0.00163875 49162.5 2704687500 7.93897E+18 1.75258E+33 1.4747E+35 1.84338E-08 4.9771E+14 1.83406E-26 4.30676E+26 3.52128E+18 4.01279E+28 2.30672E-69 4.335E+68 1.23973E+53 2.43E+26 1.97E+37 1.6E+50 1.57514E+50 1 12.38723E+21 4.014E+56 4.01426E+56 2.908E+79 -5.62491E+59 5.62491E+59
4E+13 2.5E-14 1.09E+14 3.2775E+12 0.163875 1554654.755 2.70469E+12 7.93897E+20 5.54215E+34 1.4747E+38 0.018433759 1.5739E+19 1.83406E-26 4.30676E+22 3.52128E+15 4.01279E+28 2.30672E-73 4.335E+72 1.23973E+53 2.43E+29 1.97E+39 5E+51 4.98104E+51 1 12.38723E+24 4.014E+59 4.01426E+59 2.908E+82 -5.62491E+62 5.62491E+62
4E+12 2.5E-13 1.09E+13 3.2775E+13 16.3875 49162500 2.70469E+15 7.93897E+22 1.75258E+36 1.4747E+41 18433.7594 4.9771E+23 1.83406E-26 4.30676E+18 3.52128E+12 4.01279E+28 2.30672E-77 4.335E+76 1.23973E+53 2.43E+32 1.97E+41 1.6E+53 1.57514E+53 1 1.000000001 2.38723E+27 4.014E+62 4.01426E+62 2.908E+85 -5.62491E+65 5.62491E+65
4E+11 2.5E-12 1.09E+12 3.2775E+14 1638.75 1554654755 2.70469E+18 7.93897E+24 5.54215E+37 1.4747E+44 18433759401 1.5739E+28 1.83406E-26 4.30676E+14 3521280000 4.01279E+28 2.30672E-81 4.335E+80 1.23973E+53 2.43E+35 1.97E+43 5E+54 4.98104E+54 1 1.000000003 2.38723E+30 4.014E+65 4.01426E+65 2.908E+88 -5.62491E+68 5.62491E+68
4E+10 2.5E-11 1.09E+11 3.2775E+15 163875 49162499998 2.70469E+21 7.93897E+26 1.75258E+39 1.4747E+47 1.84338E+16 4.9771E+32 1.83406E-26 43067568254 3521280 4.01279E+28 2.30672E-85 4.335E+84 1.23973E+53 2.43E+38 1.97E+45 1.6E+56 1.57514E+56 1 1.000000007 2.38723E+33 4.014E+68 4.01426E+68 2.908E+91 -5.62491E+71 5.62491E+71
4E+09 2.5E-10 10900000003 3.2775E+16 16387499.99 1.55465E+12 2.70469E+24 7.93897E+28 5.54215E+40 1.4747E+50 1.84338E+22 1.5739E+37 1.83406E-26 4306756.829 3521.280003 4.01279E+28 2.30672E-89 4.335E+88 1.23973E+53 2.43E+41 1.97E+47 5E+57 4.98104E+57 1 1.000000016 2.38723E+36 4.014E+71 4.01426E+71 2.908E+94 -5.62491E+74 5.62491E+74
4E+08 2.5E-09 1090000003 3.2775E+17 1638749992 4.91625E+13 2.70469E+27 7.93897E+30 1.75258E+42 1.4747E+53 1.84338E+28 4.9771E+41 1.83406E-26 430.6756868 3.521280026 4.01279E+28 2.30672E-93 4.335E+92 1.23973E+53 2.43E+44 1.97E+49 1.6E+59 1.57514E+59 1 1.000000037 2.38723E+39 4.014E+74 4.01426E+74 2.908E+97 -5.62491E+77 5.62491E+77
40000000 2.5E-08 109000002.7 3.2775E+18 1.63875E+11 1.55465E+15 2.70469E+30 7.93897E+32 5.54215E+43 1.4747E+56 1.84338E+34 1.5739E+46 1.83406E-26 0.043067573 0.00352128 4.01279E+28 2.30672E-97 4.335E+96 1.23973E+53 2.43E+47 1.97E+51 5E+60 4.98104E+60 1 1.000000088 2.38723E+42 4.014E+77 4.01426E+77 2.91E+100 -5.62491E+80 5.62491E+80
4000000 2.5E-07 10900002.73 3.2775E+19 1.63875E+13 4.91625E+16 2.70469E+33 7.93897E+34 1.75258E+45 1.4747E+59 1.84337E+40 4.9771E+50 1.83406E-26 4.30676E-06 3.52128E-06 4.01279E+28 2.3067E-101 4.34E+100 1.23973E+53 2.43E+50 1.97E+53 1.6E+62 1.57514E+62 0.999999999 1.000000208 2.38722E+45 4.014E+80 4.01426E+80 2.91E+103 -5.62491E+83 5.62491E+83
400000 2.49999E-06 1090002.725 3.27749E+20 1.63874E+15 1.55465E+18 2.70467E+36 7.93893E+36 5.54213E+46 1.47469E+62 1.84335E+46 1.5739E+55 1.83406E-26 4.3068E-10 3.52131E-09 4.01279E+28 2.3067E-105 4.34E+104 1.23973E+53 2.43E+53 1.97E+55 5E+63 4.98102E+63 0.999999996 1.00000049 2.38721E+48 4.014E+83 4.01423E+83 2.91E+106 -5.62487E+86 5.62487E+86
40000 2.49994E-05 109002.725 3.27742E+21 1.63867E+17 4.91607E+19 2.70448E+39 7.93857E+38 1.75252E+48 1.47459E+65 1.8431E+52 4.9766E+59 1.83406E-26 4.30719E-14 3.52154E-12 4.01279E+28 2.307E-109 4.33E+108 1.23973E+53 2.43E+56 1.97E+57 1.6E+65 1.57508E+65 0.999999988 1.000001156 2.38705E+51 4.014E+86 4.01396E+86 2.91E+109 -5.62449E+89 5.62449E+89
3570 0.000280034 9730.975 3.67124E+22 2.05614E+19 1.84306E+21 3.80126E+42 9.96104E+40 6.57027E+49 2.07259E+68 3.64112E+58 2.6224E+64 1.83406E-26 2.73571E-18 2.50548E-15 4.01279E+28 1.4653E-113 6.82E+112 1.23973E+53 3.42E+59 2.47E+59 5.9E+66 5.90507E+66 0.999999958 1.00000284 3.35509E+54 5.642E+89 5.64178E+89 4.09E+112 -7.90544E+92 7.90544E+92
1599 0.000625 4360 8.19375E+22 1.02422E+20 6.14531E+21 4.22607E+43 4.96186E+41 2.19073E+50 2.30422E+69 4.50043E+60 9.7209E+65 1.83406E-26 1.10253E-19 2.25362E-16 4.01279E+28 5.9052E-115 1.69E+114 1.23973E+53 3.8E+60 1.23E+60 2E+67 1.96893E+67 0.999999938 1.000003825 3.73004E+55 6.272E+90 6.27228E+90 4.54E+113 -8.78892E+93 8.78892E+93
1370 0.000729395 3735.975 9.56236E+22 1.39495E+20 7.74761E+21 6.71714E+43 6.75786E+41 2.76193E+50 3.66245E+69 1.13697E+61 1.948E+66 1.83406E-26 5.94375E-20 1.41786E-16 4.01279E+28 3.1835E-115 3.14E+114 1.23973E+53 6.04E+60 1.67E+60 2.5E+67 2.4823E+67 0.999999933 1.000004051 5.92872E+55 9.969E+90 9.9695E+90 7.22E+113 -1.39696E+94 1.39696E+94
1088 0.000918274 2967.525 1.20386E+23 2.21094E+20 1.09442E+22 1.34034E+44 1.0711E+42 3.90145E+50 7.30803E+69 4.52696E+61 5.4906E+66 1.83406E-26 2.36604E-20 7.10566E-17 4.01279E+28 1.2673E-115 7.89E+114 1.23973E+53 1.2E+61 2.65E+60 3.5E+67 3.50645E+67 0.999999924 1.000004412 1.18301E+56 1.989E+91 1.98931E+91 1.44E+114 -2.78748E+94 2.78748E+94
1100 0.000908265 3000.225 1.19074E+23 2.16301E+20 1.07657E+22 1.29699E+44 1.04788E+42 3.83784E+50 7.07167E+69 4.23887E+61 5.2264E+66 1.83406E-26 2.47206E-20 7.34315E-17 4.01279E+28 1.324E-115 7.55E+114 1.23973E+53 1.17E+61 2.6E+60 3.4E+67 3.44928E+67 0.999999925 1.000004394 1.14475E+56 1.925E+91 1.92497E+91 1.39E+114 -2.69733E+94 2.69733E+94
1000 0.000999001 2727.725 1.30969E+23 2.61676E+20 1.24186E+22 1.72582E+44 1.2677E+42 4.42708E+50 9.40983E+69 7.50532E+61 8.0222E+66 1.83406E-26 1.68907E-20 5.51852E-17 4.01279E+28 9.0467E-116 1.11E+115 1.23973E+53 1.55E+61 3.14E+60 4E+67 3.97886E+67 0.999999921 1.000004552 1.52325E+56 2.561E+91 2.56143E+91 1.86E+114 -3.58916E+94 3.58916E+94
900 0.001109878 2455.225 1.45505E+23 3.22986E+20 1.45424E+22 2.36659E+44 1.56471E+42 5.18419E+50 1.29036E+70 1.41132E+62 1.2882E+67 1.83406E-26 1.10869E-20 4.02434E-17 4.01279E+28 5.9382E-116 1.68E+115 1.23973E+53 2.13E+61 3.88E+60 4.7E+67 4.65932E+67 0.999999917 1.000004733 2.08881E+56 3.512E+91 3.51246E+91 2.54E+114 -4.92177E+94 4.92177E+94
400 0.002493766 1092.725 3.26933E+23 1.63059E+21 4.89787E+22 2.6845E+45 7.89943E+42 1.74603E+51 1.4637E+71 1.81597E+64 4.9215E+68 1.83406E-26 4.34999E-22 3.54776E-18 4.01279E+28 2.3299E-117 4.29E+116 1.23973E+53 2.41E+62 1.96E+61 1.6E+68 1.56925E+68 0.999999875 1.000006388 2.36941E+57 3.984E+92 3.9843E+92 2.89E+115 -5.58293E+95 5.58293E+95
40 0.024390244 111.725 3.19756E+24 1.55979E+23 1.49813E+24 2.51157E+48 7.55643E+44 5.34063E+52 1.36941E+74 1.58954E+70 1.4084E+73 1.83406E-26 4.75385E-26 3.79203E-21 4.01279E+28 2.5462E-121 3.93E+120 1.23973E+53 2.26E+65 1.87E+63 4.8E+69 4.79992E+69 0.99999961 1.000014829 2.21678E+60 3.728E+95 3.72764E+95 2.7E+118 -5.22329E+98 5.22329E+98
10 0.090909091 29.975 1.19182E+25 2.16694E+24 1.07804E+25 1.30053E+50 1.04978E+46 3.84308E+53 7.09097E+75 4.26204E+73 5.2478E+75 1.83406E-26 2.46309E-28 7.32316E-23 4.01279E+28 1.3192E-123 7.58E+122 1.23973E+53 1.17E+67 2.6E+64 3.5E+70 3.45399E+70 0.999999247 1.000024059 1.14788E+62 1.93E+97 1.93022E+97 1.4E+120 -2.7047E+100 2.7047E+100
9 0.1 27.25 1.311E+25 2.622E+24 1.24372E+25 1.731E+50 1.27024E+46 4.43372E+53 9.43808E+75 7.55047E+73 8.0584E+75 1.83406E-26 1.68233E-28 5.502E-23 4.01279E+28 9.0106E-124 1.11E+123 1.23973E+53 1.56E+67 3.15E+64 4E+70 3.98483E+70 0.99999921 1.000024916 1.52782E+62 2.569E+97 2.56913E+97 1.86E+120 -3.5999E+100 3.5999E+100
8 0.111111111 24.525 1.45667E+25 3.23704E+24 1.45667E+25 2.37449E+50 1.56819E+46 5.19283E+53 1.29466E+76 1.42075E+74 1.2947E+76 1.83406E-26 1.10377E-28 4.01096E-23 4.01279E+28 5.9119E-124 1.69E+123 1.23973E+53 2.13E+67 3.89E+64 4.7E+70 4.66709E+70 0.999999167 1.000025898 2.09578E+62 3.524E+97 3.52418E+97 2.55E+120 -4.9382E+100 4.9382E+100
7 0.125 21.8 1.63875E+25 4.09688E+24 1.73816E+25 3.38086E+50 1.98474E+46 6.19631E+53 1.84338E+76 2.88027E+74 2.1996E+76 1.83406E-26 6.89081E-29 2.81702E-23 4.01279E+28 3.6908E-124 2.71E+123 1.23973E+53 3.04E+67 4.92E+64 5.6E+70 5.56897E+70 0.999999117 1.000027042 2.98403E+62 5.018E+97 5.01783E+97 3.63E+120 -7.0311E+100 7.0311E+100
6 0.142857143 19.075 1.87286E+25 5.35102E+24 2.12362E+25 5.04665E+50 2.59232E+46 7.57044E+53 2.75163E+76 6.41779E+74 4.0115E+76 1.83406E-26 4.03927E-29 1.88719E-23 4.01279E+28 2.1635E-124 4.62E+123 1.23973E+53 4.54E+67 6.42E+64 6.8E+70 6.80398E+70 0.999999056 1.000028399 4.4543E+62 7.49E+97 7.49017E+97 5.43E+120 -1.0495E+101 1.0495E+101
5 0.166666667 16.35 2.185E+25 7.28333E+24 2.67607E+25 8.01389E+50 3.52843E+46 9.53985E+53 4.36948E+76 1.61833E+75 8.0273E+76 1.83406E-26 2.1803E-29 1.18843E-23 4.01279E+28 1.1678E-124 8.56E+123 1.23973E+53 7.2E+67 8.74E+64 8.6E+70 8.57399E+70 0.99999898 1.000030051 7.07326E+62 1.189E+98 1.18941E+98 8.61E+120 -1.6666E+101 1.6666E+101
4 0.2 13.625 2.622E+25 1.0488E+25 3.51778E+25 1.3848E+51 5.08094E+46 1.25405E+54 7.55047E+76 4.8323E+75 1.8234E+77 1.83406E-26 1.05145E-29 6.8775E-24 4.01279E+28 5.6316E-125 1.78E+124 1.23973E+53 1.24E+68 1.26E+65 1.1E+71 1.12708E+71 0.999998883 1.000032127 1.22226E+63 2.055E+98 2.0553E+98 1.49E+121 -2.88E+101 2.88E+101
3 0.25 10.9 3.2775E+25 1.63875E+25 4.91625E+25 2.70469E+51 7.93897E+46 1.75258E+54 1.4747E+77 1.84338E+76 4.9771E+77 1.83406E-26 4.30676E-30 3.52128E-24 4.01279E+28 2.3067E-125 4.34E+124 1.23973E+53 2.43E+68 1.97E+65 1.6E+71 1.57514E+71 0.999998751 1.000034862 2.38723E+63 4.014E+98 4.01426E+98 2.91E+121 -5.6249E+101 5.6249E+101
2 0.333333333 8.175 4.37E+25 2.91333E+25 7.56906E+25 6.41111E+51 1.41137E+47 2.69828E+54 3.49559E+77 1.03573E+77 1.8164E+78 1.83406E-26 1.36268E-30 1.48554E-24 4.01279E+28 7.2986E-126 1.37E+125 1.23973E+53 5.76E+68 3.5E+65 2.4E+71 2.42509E+71 0.999998558 1.000038732 5.65861E+63 9.515E+98 9.51528E+98 6.89E+121 -1.3333E+102 1.3333E+102
1 0.5 5.45 6.555E+25 6.555E+25 1.39053E+26 2.16375E+52 3.17559E+47 4.95705E+54 1.17976E+78 1.17976E+78 1.1262E+79 1.83406E-26 2.69172E-31 4.4016E-25 4.01279E+28 1.4417E-126 6.94E+125 1.23973E+53 1.94E+69 7.87E+65 4.5E+71 4.45518E+71 0.999998234 1.000044918 1.90978E+64 3.21E+99 3.2114E+99 2.33E+122 -4.4999E+102 4.4999E+102
0 1 2.725 1.311E+26 2.622E+26 3.933E+26 1.731E+53 1.27024E+48 1.40207E+55 9.43808E+78 7.55047E+79 2.5483E+80 1.83406E-26 1.68233E-32 5.502E-26 4.01279E+28 9.0106E-128 1.11E+127 1.23973E+53 1.56E+70 3.15E+66 1.3E+72 1.26012E+72 0.999997502 1.000057838 1.52782E+65 2.57E+100 2.5691E+100 1.86E+123 -3.5999E+103 3.5999E+103
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