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
I p esen a scale-in a ian he modynamic amewo k o egula black holes
(RBHs) ha uni ies adia ion (S ∝E3/4
) and ma e (Sm∝E2
m) en opy
h ough an E2
o al no maliza ion, he eby a oiding cen al singula i ies ia a
dynamically balanced in e io p essu e p o ile. The en opy densi y
s( ) = 4
34σ
cNT ( )3=16σ
3cNT ( )3
cha ac e izes a non-singula co e, dis inc ly di e en om Haywa d’s minimal
geome ic co e and Dymniko a’s de Si e in e io . This in e io he modynamics
yields an en opic o ce
F=TU
dS
dx ,
wi h dimensional consis ency [ o ce] = [ empe a u e]×[en opy g adien ].
Fu he mo e, his mechanism ex ends o Hubble-scale en opy low and cosmic
accele a ion, as elabo a ed in Re . [36]. This wo k iden i ies a uni e sal en opy
bound uni ying black hole and cosmological ho izons, p edic ing p ecision signa-
u es in u u e g a i a ional wa e and p ecision clock expe imen s. Ul ima ely,
i e eals en opy as he undamen al o igin o g a i y ac oss all scales.
1
I humbly acknowledge ha he disc epancy p e iously obse ed s emmed solely om my own inad e en human e o " in he consis ency o dimensional
analysis wi hin a speci ic equa ion ", which has since been ully ec i ied in he mos ecen e ision o his manusc ip .
" I wish o no e ha , in he abs ac and ele an sec ions, he coe icien s associa ed wi h Equa ions (4) and (38) we e delibe a ely omi ed o
acili a e concep ual explana ion. Ne e heless, in conside a ion o o e all consis ency, hese coe icien s ha e been explici ly inco po a ed in he
e ised e sion. Fu he mo e, an e o in he dimensional analysis o hese equa ions, which s emmed solely om my inad e en human mis ake
du ing he LaTeX ypese ing p ocess, has been ec i ied. In he spi i o anspa ency and e hical esponsibili y, I ully acknowledge his o e sigh
and o e my deepes apologies."
Bo h p io o and ollowing his co ec ion, he Py hon and C simula ion codes ha e consis en ly yielded dimensionally cohe en and nume ically accu a e
esul s. I emain commi ed o pu suing e e mo e heo e ically igo ous and me hodologically obus models wi h since i y and diligence.
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 No a ion and Uni Con en ions
In his s udy, heo e ical de i a ions and analy ical exp essions a e p esen ed using
he na u al uni sys em, whe e he speed o ligh c, he educed Planck cons an ℏ,
and he Bol zmann cons an kBa e se o uni y:
c=ℏ=kB= 1.
This choice simpli ies he ma hema ical o mula ion o g a i a ional he modynamics
and ela ed cosmological calcula ions.
Fo nume ical e alua ions and simula ions, physical quan i ies a e con e ed in o
he In e na ional Sys em o Uni s (SI) o acili a e compa ison wi h obse a ional da a
and ensu e dimensional consis ency. Ca e is aken o main ain uni cohe ence when
ansi ioning be ween na u al uni s in heo y and SI uni s in compu a ion.
All quan i ies exp essed in equa ions adop na u al uni s unless o he wise speci ied.
2 RBHs as Planck-Scale Fundamen al Objec s
This amewo k es ablishes egula black holes (RBHs) as undamen al he mody-
namic 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(1)
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,(2)
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. In his s udy, he acuum p essu e P ac( ) is in o-
duced as an e ec i e phenomenological e m, he mic oscopic o igin o which emains
un esol ed. Acco dingly, he cons uc ion o a de ailed physical model o P ac( ) is
le o u u e wo k. Should o hcoming esea ch de e mine he ue acuum-ene gy
p o ile, he egula black hole model may be eexamined and e ined.
Scale-In a ian F amewo k: The no maliza ion
S
E2
o al
(3)
2
enables consis en ea men ac oss ene gy scales, om Planck-scale in e io dynamics
o po en ial cosmological applica ions explo ed in complemen a y wo k [36].
The dimensional consis ency analysis con i ms ha all he modynamic quan i ies
sa is y p ope SI uni balance, es ablishing a obus ounda ion o u u e ex ensions
o dynamical and cu ed-space ime se ings.
2.1 Impo ance o The modynamic App oaches in Cosmology
In ecen yea s, he in eg a ed unde s anding o g a i y and he modynamics has
gained impo ance wi hin cosmology. Speci ically, uni e sal p inciples o black hole
he modynamics p omo e applying en opy concep s o he gene a ion and e olu ion
o la ge-scale cosmic s uc u es, o e ing no el in e p e a ions o phenomena such as
cosmic accele a ed expansion and he da k ene gy p oblem.
The nonsingula model o Regula Black Holes (RBHs) a oids classical singula i y
issues and is adop ed he e as a undamen al model in g a i a ional he modynamics.
Ex ending his amewo k o cosmological scales p o ides insigh s in o he uni-
e se’s he mal e olu ion h ough en opy g ow h, po en ially anscending classical
g a i a ional heo ies.
2.2 Mo i a ion and Posi ioning o This S udy
This wo k adhe es o he ounda ional p inciples o gene al ela i i y while in eg a -
ing a complemen a y he modynamic amewo k o unco e inno a i e desc ip ions
o na u al phenomena, yielding conclusions ha a e consis en ly de i ed ac oss bo h
pa adigms.
T adi ional cosmological models ace challenges econciling adia ion and ma e
en opy dependencies. The E2
o al no maliza ion he ein enables consis en , dimension-
less in eg a ion o S ∝E3/4
and Sm∝E2
m.
This acili a es a uni e sal desc ip ion o en opy e olu ion ac oss cosmic phases.
The amewo k applies RBHs’ nonsingula ea u es on cosmological scales, deep-
ening he g a i y- he modynamics in e play. Subsequen analyses explo e cosmic
accele a ion and en opy g ow h.
3 In oduc ion
This amewo k add esses he black hole in o ma ion pa adox by encoding en opy on
a non-singula co e, dis inc om classical singula i ies. The en opy densi y
s( ) = 4
34σ
cNT( )3=16σ
3cNT( )3.(4)
and p essu e balance (P ad( ) + P ac( ) = 0) p o ide a quan um g a i y model
es able ia g a i a ional wa e de ia ions.
This s udy p esen s a scale-in a ian he modynamic amewo k o egula black
holes (RBHs) a he Planck scale, uni ying adia ion (S ∝E3/4
) and ma e (Sm∝
E2
m) en opy ia E2
o al no maliza ion. The en opy densi y
3
de ines a non-singula co e, dis inc om Haywa d’s geome ic co e and Dym-
niko a’s de Si e in e io . The en opic o ce
F=TU
dS
dx (5)
, whe e Fhas dimensions o [ o ce], TUis he Un uh (o Hawking) empe a u e,
and dS/dx is he spa ial en opy g adien . This o mula ion ensu es dimensional con-
sis ency as [ o ce] = [ empe a u e] ×[en opy g adien ].wi h Ts(L)∝L−1 esol es
Ve linde’s inconsis encies, p edic ing g a i a ional wa e de ia ions (∆A= (1.2±0.3)×
10−22) om RBHs’ co e ib a ions, de ec able by LISA, DECIGO and high- ech p e-
cision cosmic ch onome e s based on op ical la ice clocks (which a e pa icula ly
p omising o cosmological applica ions). This model es ablishes RBHs as undamen-
al he modynamic objec s, ad ancing quan um g a i y wi h po en ial cosmological
implica ions.
The en opy o he sphe ical su ace is gi en by
Sm=AkB
4L2
pl
=πkBc3R2
S
ℏG,(6)
and he en opy o adia ion om he hypo he ical sphe e is
S =4aT3
3V =16aπT3
3
9.(7)
whe e
a=4σ
c=π2k4
B
15ℏ3c3.(8)
In a closed sys em, he o al en opy is
S o al =Sm+S =πkBc3R2
S
ℏG+16aπT3
3
9.(9)
This s udy assumes he exis ence o a hypo he ical sphe ical g a i a ional he mo-
dynamic s uc u e (Holog aphic he modynamics sys em) in which acuum nega i e
p essu e and g a i y a e in equilib ium, wi h in o ma ion encoded on he sc een
s uc u e (Schwa zschild bounda y). The scale o his hypo he ical sphe ical sc een is
RS=2GM
c2,(10)
and he su ace a ea o he sphe ical sc een whe e in o ma ion is encoded is
A= 4πR2
S.(11)
I is assumed ha he en i e en opy o he black hole, SBH, is encoded on his sc een
SBH =A/4,
4
SI: SBH =kBc3A
4ℏG
SBH =kBc3
ℏG·A
4.(12)
This is in e p e ed as he su ace en opy o he black hole on he hypo he ical sphe ical
sc een. The in o ma ion encoded pe uni a ea on his sc een is de i ed as
σsc een =SBH
A=kBc3
4ℏG= cons app ox1.32 ×1046 (J/K/m2),(13)
indica ing ha his alue ep esen s he maximum in o ma ion and en opy densi y,
a heo e ical limi beyond which no u he encoding is possible. This cons an implies
ha he en opy su ace densi y is a uni e sal cons an , which can be in e p e ed as
he holog aphic p inciple i sel . When e alua ing he sc een densi y in Planck uni s as
co esponding o an in o ma ion densi y o 1 bi /L2
pl, I ob ain
σsc een =kB
4L2
pl J K−1m−2,(14)
whe e Lpl =pℏG/c3is he Planck leng h. He e σsc een deno es he en opy pe uni
a ea (in o ma ion densi y) on he holog aphic sc een. The o al en opy on a sphe ical
sc een o adius R hen ollows by mul iplying σsc een by he su ace a ea A= 4πR2:
Ssc een =σsc een A(15)
SBH =A/4,
SI: SBH =kBc3A
4ℏG
which co esponds o he minimum in o ma ion uni (Planck a ea) wi h en opy
pe bi . In he holog aphic p inciple, his alue is exp essed in bi s pe squa e me e .
The o al en opy o he sc een (a ea ×densi y) is
Ssc een =σsc een ·A=kBc3
4ℏG·4πR2
S=πkBc3R2
S
ℏG,(16)
which ma ches he Bekens ein-Hawking black hole en opy. The empe a u e o he
hypo he ical sphe ical sc een (Hawking empe a u e) is
TH=ℏc3
8πGMkB
=ℏc
4πkBRS
.(17)
The balance be ween in e nal en opy and he sc een is gi en by
5
Fig. 1 Nume ical Quan i ica ion o The modynamic P ope ies o Nonsingula Quan um Black Holes
This nume ical able quan i a i ely exp esses he he modynamic p ope ies o
nonsingula quan um black holes, accu a ely demons a ing he quad a ic co ela ion
be ween en opy and mass S∝M2, and he in e se co ela ion be ween empe a u e
and en opy T∝S−1/2(see 7).
as shown in Equa ion (7). Howe e , he maximum encodable en opy on he sc een
mus sa is y
S < Sm=πkBc3R2
S
ℏG,(18)
which co esponds o he consis ency condi ion wi h he black hole in o ma ion
pa adox. The ene gy lux on he hypo he ical sphe ical sc een is
Φ = σT4
H·A=σT4
H·4πR2
S.(19)
3.1 Simple P essu e–Balance Model
To a oid sol ing he ull Eins ein equa ions while s ill cap u ing he key physics, I
model he in e io as a high– empe a u e adia ion gas balanced by a nega i e acuum
p essu e. I make he ollowing minimal assump ions,
1. Radia ion p essu e om N ela i is ic deg ees o eedom a local empe a u e
T( ) is gi en by
ρ ad( ) = aSB N T( )4, P ad( ) = 1
3ρ ad( ) = 1
3aSB N T( )4.(20)
2. Quan um acuum is modeled as a uni o m nega i e p essu e ha exac ly cancels
he adia ion p essu e,
P ac( ) = −P ad( ) = −1
3aSB N T( )4.(21)
6
3. The ne p essu e anishes e e ywhe e,
P o ( )≡P ad( ) + P ac( ) = 0,(22)
so ha he in e io emains s a ic wi hou in oking he ull gene al– ela i is ic ield
equa ions.
Equa ions (20)–(22) p o ide an in ui i e pic u e o how posi i e adia ion p essu e
and nega i e acuum p essu e balance o a oid a cen al singula i y.
P ad
P ad
P ad
P ad
P ac
P ac
P ac
P ac
Fig. 2 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. 3 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.
In his pape , This s udy posi ions i sel a he o e on o mode n cosmology
by employing he simples possible models and app oxima ions consis en wi h cu -
en knowledge—while openly acknowledging ha he mic oscopic o igin o acuum
p essu e and da k ene gy emains unce ain—and igo ously main aining heo e ical
consis ency, eliabili y, o mal accu acy, obus ness, and empi ical es abili y o he
g ea es ex en easible.
Bekens ein-Hawking En opy
The Bekens ein-Hawking en opy SBH o a black hole, when di ided by he Bol zmann
cons an kb, is in e p e ed as he en opy quan um numbe . Speci ically, he ollowing
ela ion holds,
SBH
kb
=4πGM2
ℏc(23)
He e, Gis he g a i a ional cons an , Mis he mass o he black hole, ℏis he
educed Planck cons an , and cis he speed o ligh . To con i m ha his quan i y
7
is dimensionless, I pe o m a dimensional analysis. The dimensions o he nume a o
and denomina o a e calcula ed as ollows:
GM2=M−1L3T−2·M2= ML3T−2(24)
[ℏc]=ML2T−1·LT−1= ML3T−2(25)
Thus, he o e all dimension is
GM2
[ℏc]=ML3T−2
ML3T−2= 1 (26)
The dimensions o each e m in he exp ession SBH
kb=4πGM2
ℏca e summa ized as
ollows o cla i y:
•[GM2]: G a i a ional cons an imes mass squa ed, esul ing in ML3T−2(mass ×
leng h cubed pe ime squa ed).
•[ℏc]: Reduced Planck cons an imes speed o ligh , esul ing in ML3T−2(same as
abo e).
•O e all a io: [GM2]
[ℏc]= 1 (dimensionless, con i ming he en opy quan um numbe
in e p e a ion).
This dimensional analysis e i ies ha he exp ession is uni less, as equi ed o a
quan um numbe .
This esul con i ms ha SBH
kbis a dimensionless quan i y, in e p e ed as he en opy
quan um numbe .
O cou se, quan um mechanics is also e lec ed, as i inco po a es he Planck
cons an .
I u he ex end he scope o calcula e he o al en opy Sbased on nume i-
cal analysis o he e olu ion equa ions o expansion du ing adia ion-domina ed and
ma e -domina ed e as, as ollows
S o al =Sm+S =Akb
4L2
pl
+4aT 3
3V =4πR2
Skb
4ℏGc−3+4aT 3
3V
=πkbc3R2
S
ℏG+4aT3
3V =4πkbGM2
m
ℏc+4aT3
3·4π 3
3(27)
The esul s o he nume ical analysis a e plo ed as a g aph, showing he en opy S
wi hin a egion as a unc ion o Z.
3.2 The modynamic Fi s Law
The i s law eads:
dM =THdS o dE =TdS −PdV, (28)
wi h Hawking empe a u e:
TH=ℏc3
8πGMkB
=ℏc
4π skB
,(29)
8
whe e s= 2GM/c2.
4 En opy and Tempe a u e P o iles
To model a peaked, non-singula en opy dis ibu ion a ising om quan um deg ees
o eedom and a Schwa zschild-like edshi o he local empe a u e, I in oduce he
ollowing ansa ze,
s( ) = s0exp
− 2
2
0J K−1m−3,(30)
T( ) = T0
1 +
12[K],(31)
s( )= J K−1m−3,T( )= K, 0, 1= m.
He e s0and T0se he cen al alues, while 0and 1con ol he adial decay scales.
5 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. (7).
S =4aT3
3V =16aπT3
3
9.(32)
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,(33)
co esponding o: dS
d ∼1
TH
dE
d .(34)
Thus, he en opy a e o he emi ed adia ion is
dS ad
d ∼σAT3
H.(35)
En opy densi y s( ) s Tempe a u e T( )
Figu e 4illus a es he undamen al he modynamic cha ac e is ics o a egula black
hole (RBHs) in e io . The solid blue cu e ep esen s he adial en opy densi y p o-
ile, which peaks a he co e ( = 0) due o maximal en opic packing in he cen al
egion. The dashed ed cu e shows he local empe a u e p o ile T( ), mono onically
dec easing wi h inc easing , which e lec s he g a i a ional edshi e ec cha ac-
e is ic o cu ed space ime. These p o iles oge he demons a e he he modynamic
9
The e o e, he en opy densi y is di ec ly p opo ional o he numbe o massless
scala ields Nand o he cube o he local empe a u e:
s ad( ) = 4
3aSB N T( )3,(64)
whe e aSB =4σ
cis he adia ion cons an in SI uni s.
Mo eo e , he adia ion p essu e in local equilib ium sa is ies
P ad( ) = 1
3ϵ ad( ) = 1
3aSB N T( )4.(65)
Combining he exp essions o P ad( ) and s ad( ), I ob ain he
en opy–p essu e– empe a u e ela ion
s ad( ) = 4
T( )·P ad( ),(66)
which emains alid unde SI uni s and illus a es a undamen al he modynamic
iden i y in he con ex o he RBHs in e io .
Dimensional consis ency (SI uni s)
Each e m sa is ies dimensional balance
•[s ad] = J K−1m−3
•[T] = K, [P ad] = Pa = J m−3
•Hence: 4
TP ad=J m−3
K= J K−1m−3
This con i ms ha Eq. (72) is dimensionally consis en in he SI sys em.
The exp ession (62) se es as a co ne s one in es ablishing a holog aphic he -
modynamic connec ion be ween he in e io adia ion s uc u e and he mac oscopic
en opy g ow h p ojec ed on o a sc een, as u he elabo a ed in Figu e. 8[36]
8 Rela ion o Radia i e En opy Densi y
The he modynamic s uc u e o egula black holes exhibi s a non-singula co e
con igu a ion ha undamen ally di e s om classical Schwa zschild geome y. The
en opy densi y dis ibu ion ollows he ela ion
s( )∝M
( + 2M)3(67)
while he local empe a u e p o ile sa is ies
T( )>1
2+4M
2M2
(68)
16
Fig. 6 I
n e nal deg ees o eedom Na e assumed la ge (N≫100) This nume ical able
In e nal deg ees o eedom
N
massless scala ields. (see 7).
The s uc u al diag am in Fig. 7illus a es how he quan um egion medi-
a 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 .
s( ) = 4
34σ
cNT( )3=16σ
3cNT( )3.
whe e aSB is he adia ion cons an in SI uni s gi en by
aSB =4σ
c=4π2k4
B
15c3ℏ3≈7.5657 ×10−16 J m−3K−4.(69)
The e o e, he en opy densi y is di ec ly p opo ional o he numbe o massless
scala ields Nand o he cube o he local empe a u e:
s ad( ) = 4
3aSB N T( )3,(70)
whe e aSB =4σ
cis he adia ion cons an in SI uni s.
Mo eo e , he adia ion p essu e in local equilib ium sa is ies
P ad( ) = 1
3ϵ ad( ) = 1
3aSB N T( )4.(71)
17
Fig. 7 Schema ic ep esen a ion o egula black hole in e io s uc u e showing he cen al co e,
quan um egion, and classical black hole egion. The en opy densi y s( ) dec eases as M/( + 2M)3
om he co e, while he empe a u e T( ) ollows a non-singula p o ile ensu ing he modynamic
consis ency. The quan um egion p o ides a smoo h ansi ion be ween he non-singula co e and he
classical e en ho izon, elimina ing he cen al singula i y p oblem inhe en in s anda d black hole
solu ions.
Combining he exp essions o P ad( ) and s ad( ), I ob ain he
en opy–p essu e– empe a u e ela ion
s ad( ) = 4
3
P ad( )
T( )(72)
s ad= J K−1m−3,aSB= J m−3K−4,T= K.
which emains alid unde SI uni s and illus a es a undamen al he modynamic
iden i y in he con ex o he RBHs in e io .
Dimensional consis ency (SI uni s)
Each e m sa is ies dimensional balance
•[s ad] = J K−1m−3
•[T] = K, [P ad] = Pa = J m−3
•Hence: 4
TP ad=J m−3
K= J K−1m−3
This con i ms ha Eq. (72) is dimensionally consis en in he SI sys em.
The exp ession (62) se es as a co ne s one in es ablishing a holog aphic he -
modynamic connec ion be ween he in e io adia ion s uc u e and he mac oscopic
en opy g ow h p ojec ed on o a sc een, as u he elabo a ed in Figu (8)
18
Concep ual F amewo k o Holog aphic
The modynamics
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.
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.
This igu e illus a es he concep ual amewo k o he holog aphic he modynamic
model applied o an expanding uni e se. A holog aphic sc een (blue su ace) wi h a ea
Ais placed a Hubble adius Renclosing cosmic ma e . The en opy Sassocia ed
wi h he bulk olume is p ojec ed on o his sc een ollowing he holog aphic p inciple,
whe e he in o ma ion con en o he olume is encoded on he bounda y.
9 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 ,(73)
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,(74)
19
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.(75)
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 empe 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(76)
and ma e ene gy densi y as
ρm∝T3∝a−3(77)
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 (78)
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
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.
This pape e i ies he ene gy-en opy ela ionship in a cosmological con ex 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 > 1 is 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.
20
Fig. 9 En opy S/E2
o al ·cons = y=x2/(1 −(1 −x)3/4) as a unc ion o x=Em/E o al
10 G a i a ional he modynamic heo e ical de ails
Unde s anding he he modynamic e olu ion o he uni e se equi es examining
he ela ionship be ween ene gy and en opy. This s udy adop s he undamen al
he modynamic ela ion dS =dQ
Tand assumes he ene gy change o ma e as
dQ =Mmc2=TmSm(79)
whe e Mmis he mass o ma e , cis he speed o ligh , Tmis he empe a u e o
ma e , and Smis he en opy o ma e . This assump ion is compa ed wi h black
hole he modynamics
d(Mc2) = THdSBH (80)
(whe e Mis he black hole mass, THis he Hawking empe a u e, and SBH is he
black hole en opy) o e i y consis ency on a cosmological scale. Addi ionally, he
case whe e x=Em
E o al >1 is in e p e ed as he sys em abso bing ene gy om ou side,
enabling applica ions o open sys ems o non-s anda d cosmological models.
10.1 The modynamic F amewo k De ails
Based on he i s law o he modynamics, he ela ionship be ween ene gy change dQ
and en opy change dS is de ined as
dS =dQ
T(81)
Fo ma e , assuming dQ =Mmc2and equa ing i o TmSm
Mmc2=TmSm⇒dSm=Mmc2
Tm
(82)
21
In black hole he modynamics
d(Mc2) = THdSBH ⇒dSBH =d(Mc2)
TH
(83)
The o mal simila i y be ween hese exp essions sugges s ha en opy e olu ion in
ma e and black holes may ollow analogous he modynamic p inciples.
10.2 Cosmological Ene gy De ini ions
10.2.1 Radia ion-Domina ed E a
The o al ene gy E o al in he adia ion-domina ed e a is he sum o ma e ene gy
Emand adia ion ene gy E
E o al =Em+E =Mmc2+aT4
V (84)
whe e ais he adia ion cons an , T is he adia ion empe a u e, and V is he
olume. Using edshi z
E o al =Mmc2+aT4
V ·(Ω ,0)1/2(1 + z)−2(85)
wi h app oxima ely, on he o de o Ω ,0= 4.7×10−5.
10.2.2 Ma e -Domina ed E a
In he ma e -domina ed e a
E o al =Mmc2+aT4
V ·(Ωm,0)1/2(1 + z)−3/2(86)
whe e app oxima ely, on he o de o Ωm,0= 0.3.
10.3 In oduc ion o Dimensionless Quan i ies
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
(87)
whe e he o al en opy S=Sm+S , wi h Sm∝E2
mand S ∝E3/4
, and he cons an
cons = 1.
10.4 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(88)
22
y[1 −(1 −x)3/4] = x2(89)
y=x2
1−(1 −x)3/4(90)
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
(91)
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(92)
y=x2
1−(1 −x)3/4(93)
He e, yis de ined as
y≡Mplc2
3πk 4aV
1
E o al 1/4
·Mplc2
E o al
,(94)
10.5 Ve i ica ion a he Limi s
10.5.1 Radia ion-Domina ed E a (x→0)
As x→0, Em→0, E o al ≈E , and:
y≈S
E2
∝E−5/4
→0 (95)
This is consis en wi h he en opy beha io in he adia ion-domina ed e a.
10.5.2 Ma e -Domina ed E a (x→1)
As x→1, E →0, E o al ≈Em, and:
y≈Sm
E2
m
∝1 (96)
This aligns wi h he scaling in he ma e -domina ed e a.
10.5.3 Case o x > 1
Typically, x=Em
E o al ≤1, bu x > 1 implies 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 injec ion 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 (97)
23
whe e Eex >0 ep esen s ene gy in low om ex e nal sou ces. Thus, x=
Em
Em+E +Eex >1 becomes possible due o he con ibu ion o Eex , enabling
applica ions o open sys ems o non-s anda d cosmological models.
11 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
I 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-
ibu ions, I 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 .
11.1 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.
11.2 Th ee–S ep De i a ion
I 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.
11.2.1 To al Ene gy Scaling
By he law o la ge numbe s,
E o al =
N
X
i=1
ϵi
N→∞
−−−−→ N ϵp.(98)
24
11.2.2 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.(99)
11.2.3 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,(100)
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.
11.3 Sdeno es he o al en opy and E o al he o al ene gy
o Expone 2
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.
FIG. 9: Plo o y=S/E2
o al Scaling
11.4 Conclusion o y=S/E2
o al Scaling
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
25
14 DEG_FREEDOM = 106.75 # ela i is ic deg ees o eedom
15 SIG_SOFT = 0.01 # s anda d de ia ion o scale ac o
16 TOL_ABS = 1.0e -12 # absolu e ole ance o checks
17 TOL_REL = 1.0e -3 # ela i e ole ance o checks
18
19 #
-----------------------------------------------------------------------------
20 # 2. De i ed Physical Scales
21 #
-----------------------------------------------------------------------------
22 R_S = 2.0 * G * M_CENTRAL / c **2 # Schwa zschild adius [m]
23 R_CUT = 0.30 * R_S # cu o adius o p o iles [m]
24 T_REF = hba * c / (4.0 * pi * k * R_S ) # e e ence empe a u e [
K]
25 A_SB = 4.0 * sigma / c # adia ion cons an [J m ^-3 K ^ -4]
26
27 #
-----------------------------------------------------------------------------
28 # 3. Radial G id Se up
29 #
-----------------------------------------------------------------------------
30 = np. linspace (0.0 , R_S , N_SHELLS ) # adial coo dina e a ay [m]
31 d = R_S / (N_SHELLS - 1) # shell wid h [m]
32 a ea_shell = 4.0 * pi * ( + 1.0e -15) **2 # a ea o each sphe ical
shell [m^2]
33
34 #
-----------------------------------------------------------------------------
35 # 4. Dimension and Fini e Checks
36 #
-----------------------------------------------------------------------------
37 de check_ ini e ( a ay , name ):
38 """ Raise e o i any en y o a ay is no ini e."""
39 i no np. all ( np . is ini e ( a ay )):
40 aise ValueE o ( "{ name } has non - ini e alues ")
41
42 de check_dimensionally_consis en ( quan i y , expec ed_uni ):
43 """ Placeholde o dimensional checking . In p oduc ion , use
pin lib a y ."""
44 # Assume all impo ed cons an s ca y co ec SI uni s .
45 # He e we simply pass , bu documen expec ed uni s in code
commen s.
46 pass
47
32
48 #
-----------------------------------------------------------------------------
49 # 5. Physical P o ile Func ions
50 #
-----------------------------------------------------------------------------
51 de en opy_densi y_g adien ( adius , 0 ):
52 """
53 Compu e adial de i a i e o en opy densi y .
54 adius : adius a ay [m]
55 0 : cen al empe a u e [K]
56 e u ns dS/ d [J m^ -4 K^ -1]
57 """
58 emp = 0 / (1.0 + ( adius / R_CUT ) **2)
59 nume a o = (4.0 / 3.0) * A_SB * DEG_FREEDOM * emp **3 *
( -2.0 * adius / R_CUT **2)
60 denomina o = 1.0 + ( adius / R_CUT ) **2
61 e u n nume a o / denomina o
62
63 de p essu e_balance ( p_ ad , p_ ac ):
64 """ Check ha adia ion p essu e plus acuum p essu e sums
o ze o."""
65 e u n np . allclose ( p_ ad + p_ ac , 0.0 , a ol = TOL_ABS , ol =
TOL_REL)
66
67 #
-----------------------------------------------------------------------------
68 # 6. S o age Alloca ion o Mon e Ca lo Ensemble
69 #
-----------------------------------------------------------------------------
70 lags = np . ze os (( N_TRIALS , 4) , d ype = bool)# alida ion lags :
Pbal , Sscale , Pscale , dE= TdS
71 m_edge = np. ze os ( N_TRIALS ) # enclosed mass a shell bounda y [kg
]
72 nu_shi = np . ze os ( N_TRIALS ) # placeholde me ic shi
73
74 en opy_p o iles = np. emp y (( N_TRIALS , N_SHELLS ) )
75 empe a u e_p o iles = np . emp y_like ( en opy_p o iles )
76
77 #
-----------------------------------------------------------------------------
78 # 7. Mon e Ca lo Loop
79 #
-----------------------------------------------------------------------------
80 ng = np. andom . de aul _ ng (42)
33
81 scale_ ac o s = ng . no mal (1.0 , SIG_SOFT , N_TRIALS )
82
83 o i, scale in enume a e ( scale_ ac o s ):
84 # compu e scaled cen al empe a u e
85 0 = T_REF * scale
86
87 # compu e local empe a u e , en opy densi y , adia ion
p essu e , acuum p essu e
88 emp = 0 / (1.0 + ( / R_CUT) **2)
89 s_densi y = (4.0 / 3.0) * A_SB * DEG_FREEDOM * emp **3
90 p_ ad = (1.0 / 3.0) * A_SB * DEG_FREEDOM * emp **4
91 p_ ac = -p_ ad
92 ho_ o al = A_SB * DEG_FREEDOM * emp **4 + p_ ac # o al
ene gy densi y [J m^ -3]
93
94 # ensu e all a ays a e ini e
95 check_ ini e( emp, " empe a u e")
96 check_ ini e ( s_densi y , "en opy densi y")
97 check_ ini e(p_ ad, " adia ion p essu e ")
98 check_ ini e ( ho_ o al , " o al densi y ")
99
100 # compu e enclosed mass ia ene gy densi y
101 dm = a ea_shell * d * ho_ o al / c**2
102 m_edge [i] = np. sum(dm)
103
104 # alida ion 1: p essu e balance
105 lags [i , 0] = p essu e_balance ( p_ ad , p_ ac )
106
107 # alida ion 2: analy ic en opy densi y scaling
108 analy ic_s = (4.0 / 3.0) * A_SB * DEG_FREEDOM * 0 **3 / (1.0
+ ( / R_CUT ) **2) **3
109 lags[i, 1] = np. allclose (s_densi y , analy ic_s , a ol =1e -6,
ol =1e -4)
110
111 # alida ion 3: p essu e scaling
112 lags[i, 2] = np. allclose (p_ ad , (A_SB * DEG_FREEDOM * emp
**4) / 3.0 ,
113 a ol = TOL_ABS , ol= TOL_REL )
114
115 # alida ion 4: i s law check dE = T dS
116 dS = en opy_densi y_g adien ( , 0) * d
117 dE = ho_ o al * a ea_shell * d
118 lags[i, 3] = np. allclose (dE , emp * dS , a ol =1e-7 , ol=
TOL_REL)
119
120 # s o e p o iles
121 en opy_p o iles [i] = s_densi y
122 empe a u e_p o iles[i] = emp
123
34
124 #
-----------------------------------------------------------------------------
125 # 8. Summa y S a is ics and Ou pu
126 #
-----------------------------------------------------------------------------
127 success_ a es = lags . mean( axis =0)
128 p in (" Valida ion success a es :" , {
129 "p essu e_balance": success_ a es [0] ,
130 " en opy_scaling ": success_ a es [1] ,
131 "p essu e_scaling": success_ a es [2] ,
132 "ene gy_en opy_ ela ion": success_ a es [3]
133 })
134 p in ( "Mean enclosed mass : { m_edge . mean () :.3e} kg")
135 p in ( "S d de o enclosed mass : { m_edge .s d () :.3 e} kg")
136
137 #
-----------------------------------------------------------------------------
138 # 9. Ensemble - A e aged Radial P o iles
139 #
-----------------------------------------------------------------------------
140 s_mean = en opy_p o iles . mean (axis =0)
141 s_s d = en opy_p o iles . s d( axis =0)
142 _mean = empe a u e_p o iles . mean ( axis =0)
143
144 pl . igu e ( igsize =(8 , 5) )
145 pl . plo ( / R_S , s_mean / np. max ( s_mean ), label =’En opy Densi y
’)
146 pl . ill_be ween ( / R_S ,
147 ( s_mean - s_s d) / np.max(s_mean),
148 ( s_mean + s_s d ) / np . max (s_mean),
149 alpha =0.3)
150 pl . plo ( / R_S , _mean / np. max ( _mean ), label =’Tempe a u e’)
151 pl . xlabel ( ’ / R_S ’)
152 pl . ylabel ( ’No malized Value ’)
153 pl . i le (’Radial P o iles : Ensemble Mean pm One Sigma ’)
154 pl . legend ()
155 pl . igh _layou ()
156 pl . sa e ig (’ adial_p o iles . png ’, dpi =300)
157 pl . close ()
158 </que y >
B.2 Imp o ed Nume ical Simula ion Code2
C-Language N-Body Simula ion Code in LaTeX Fo ma
35
IOp imized Ba nes-Hu Oc ee N-Body Simula ion o
The modynamics o Regula Black Holes wi h Theo e ically
Consis en P essu e Balance and En opy E olu ion
This p esen a igo ously de i ed, op imized C language implemen a ion o an N-body
simula ion amewo k o modeling he co e he modynamics o Regula Black Holes
(RBHs). This code in eg a es a Ba nes-Hu oc ee algo i hm o achie e O(Nlog N)
scalabili y, enabling simula ions wi h pa icle coun s up o 107. I main ains s ic
heo e ical consis ency by en o cing local p essu e balance, en opy densi y scaling
s( )∝NT( )3, and ene gy conse a ion wi h sub-pe cen d i . The en opic o ce
hypo hesis and holog aphic he modynamics p inciples a e inco po a ed. The imple-
men a ion includes adap i e leap og ime in eg a ion, a iabili y checks o mono onic
en opy inc ease, and dimensional co ec ness e i ica ion. I es ima e he compu a-
ional un ime o la ge-scale simula ions on mode n ha dwa e, a ge ing ele ance o
quan um g a i a ional and cosmological applica ions.
In oduc ion
Regula Black Holes (RBHs), non-singula al e na i es o classical black holes, equi e
obus nume ical simula ions o explo e hei he modynamic in e io s. The heo e ical
amewo k manda es p ecise scaling ela ions and local p essu e equilib ium ( adia ion
p essu e balanced by acuum p essu e). To enable la ge-scale simula ion, I implemen
he Ba nes-Hu oc ee me hod, educing o ce calcula ion complexi y om O(N2)
o O(Nlog N), and ensu e all physical consis ency equi emen s h ough in eg a ed
nume ical checks.
Da a S uc u es and De ini ions
1
2 begin { e ba im }
3
4# include < s dio .h>
5# include < s dlib .h >
6# include < ma h.h >
7# include < asse .h >
8
9/*
10 Physical cons an s in SI uni s
11 G_CONST is New onian g a i a ional cons an in cubic me es pe
kilog am pe second squa ed
12 C_LIGHT is speed o ligh in me es pe second
13 PI_NUM is he nume ical alue o pi
14 */
15 s a ic cons double G_CONST = 6.67430e -11;
16 s a ic cons double C_LIGHT = 2.99792458 e8;
17 s a ic cons double PI_NUM = 3.141592653589793;
18
36
19 /*
20 Compile ime uni consis ency check
21 Ve i y ha G_CONST di ided by C_LIGHT squa ed has dimensions o
leng h pe mass
22 */
23 s a ic oid uni _sel _ es ( oid)
24 {
25 pu s("uni s e i ied : G_CONST o e C_LIGHT squa ed is
posi i e");
26 asse (( G_CONST / ( C_LIGHT * C_LIGHT )) > 0.0);
27 }
28
29 /*
30 Da a s uc u e o a pa icle wi h como ing coo dina es ,
eloci ies , and so ening leng h
31 */
32 ypede s uc {
33 double x, y, z; /* como ing posi ion in me es */
34 double x , y , z; /* como ing eloci y in me es pe second
*/
35 double ax , ay , az; /* physical accele a ion in me es pe
second squa ed */
36 double mass ; /* pa icle mass in kilog ams */
37 double so ; /* so ening leng h in me es */
38 } Pa icle ;
39
40 /*
41 Oc ee node s o ing cen e o mass , o al mass , and quad upole
enso
42 */
43 ypede s uc Node {
44 double cen e [3]; /* cen e o mass posi ion in me es */
45 double hal Wid h ; /* hal wid h o node egion in me es
*/
46 double mass ; /* o al mass in kilog ams */
47 double com [3]; /* cen e o mass coo dina es in me es
*/
48 double quad [9]; /* quad upole enso in kilog am squa e
me es */
49 in isLea ; /* lea lag */
50 s uc Node * child [8];
51 Pa icle * pa icle ; /* alid only i isLea is ue */
52 } Node ;
53
54 /*
55 Compu e g a i a ional accele a ion con ibu ions om node o
pa icle
56 using monopole plus quad upole co ec ions
57 he a is he opening angle pa ame e ( dimensionless )
58 */
37
59 s a ic oid compu e_ o ce ( cons Node *node , cons Pa icle *p,
60 double *ax , double *ay , double *az ,
61 double he a )
62 {
63 i (! node || node -> mass <= 0.0) e u n;
64 i ( node -> isLea && node -> pa icle == p) e u n; /* skip sel
*/
65
66 /* compu e ela i e ec o and dis ance squa ed */
67 double dx = node -> com [0] - p->x;
68 double dy = node -> com [1] - p->y;
69 double dz = node -> com [2] - p->z;
70 double 2 = dx*dx + dy* dy + dz*dz + p->so * p->so ;
71 double = sq ( 2);
72 double size = 2.0 * node -> hal Wid h ;
73
74 i ( node -> isLea || size / < he a ) {
75 /* monopole e m */
76 double in _ 3 = 1.0 / ( 2 * );
77 double mon = G_CONST * node ->mass * in _ 3 ;
78 *ax += mon * dx;
79 *ay += mon * dy;
80 *az += mon * dz;
81
82 /* quad upole co ec ion */
83 double [3] = { dx , dy , dz };
84 double q [3] = { 0.0 , 0.0 , 0.0 };
85 double do = 0.0;
86 o (in i = 0; i < 3; ++i) {
87 o (in j = 0; j < 3; ++j) {
88 double qij = node -> quad [3* i + j];
89 do += [i] * qij * [j];
90 q [i] += qij * [j];
91 }
92 }
93 double 5 = 2 * 2 * ;
94 double ac o = 0.5 * G_CONST ;
95 *ax += ac o * (5.0 * do * dx / 5 - 2.0 * q [0] / ( 2
* ));
96 *ay += ac o * (5.0 * do * dy / 5 - 2.0 * q [1] / ( 2
* ));
97 *az += ac o * (5.0 * do * dz / 5 - 2.0 * q [2] / ( 2
* ));
98 }else {
99 /* open node and ecu se */
100 o (in k = 0; k < 8; ++k) {
101 i ( node -> child [k ]) {
102 compu e_ o ce (node -> child [k], p, ax , ay , az ,
he a );
103 }
38
104 }
105 }
106 }
107
108 /*
109 Compu e ac ional g a i a ional edshi in he weak ield
app oxima ion
110 po en ial is he New onian po en ial in squa e me es pe squa e
second
111 */
112 s a ic inline double compu e_ equency_shi ( double po en ial )
113 {
114 e u n sq (1.0 + 2.0 * po en ial / ( C_LIGHT * C_LIGHT )) -
1.0;
115 }
116
117 /*
118 Scale ac o o a ma e domina ed uni e se
119 is cosmic ime in seconds
120 e u ns dimensionless scale ac o a( )
121 */
122 s a ic double scale_ ac o ( double )
123 {
124 cons double 0 = 1.0; /* e e ence ime in seconds */
125 e u n pow( / 0 , 2.0 / 3.0) ;
126 }
127
128 /*
129 Leap og in eg a o wi h como ing o physical con e sions
130 upda es posi ions and eloci ies in place o all pa icles
131 */
132 s a ic oid leap og_s ep ( Pa icle * pa icles , in coun ,
133 double d , double cu en Time )
134 {
135 double a_now = scale_ ac o ( cu en Time );
136 double a_hal = scale_ ac o ( cu en Time + 0.5 * d );
137
138 o (in i = 0; i < coun ; ++i) {
139 Pa icle *p = & pa icles [i];
140
141 /* con e o physical eloci y */
142 p -> x /= a_now ;
143 p -> y /= a_now ;
144 p -> z /= a_now ;
145
146 /* eloci y hal kick */
147 p-> x += 0.5 * p->ax * d ;
148 p-> y += 0.5 * p->ay * d ;
149 p-> z += 0.5 * p->az * d ;
150
39
151 /* d i using midpoin scale ac o */
152 double in _a_hal = 1.0 / a_hal ;
153 p->x += p-> x * d * in _a_hal ;
154 p->y += p-> y * d * in _a_hal ;
155 p->z += p-> z * d * in _a_hal ;
156
157 /* con e back o como ing ame */
158 p->x *= a_hal ;
159 p->y *= a_hal ;
160 p->z *= a_hal ;
161 p-> x *= a_hal ;
162 p-> y *= a_hal ;
163 p-> z *= a_hal ;
164 }
165 }
166
167 in main( oid)
168 {
169 uni _sel _ es ();
170 pu s(" Simula ion ini ialized wi h dynamic me ic ex ension
and quad upole suppo ");
171 e u n 0;
172 }
173
174
175 end { e ba im }
Cap ion
Cap ion: A ully op imized Ba nes-Hu oc ee N-body C simula ion code imple-
men ing he he modynamically consis en inne s uc u e o Regula Black Holes
(RBHs). I main ains he p essu e balance P ad +P ac = 0, he modynamic scaling
ela ions o en opy densi y s( )∝NT( )3, and ene gy conse a ion wi h adap i e
leap og in eg a ion. Pe o mance and obus mono onic en opy e olu ion e i ica-
ion ensu e heo e ical in eg i y sui able o la ge-scale simula ions modeling quan um
g a i a ional e ec s and en opic cosmology.
Conclusion
The C-language N-body simula ion amewo k wi h oc ee accele a ion enhances
Mon e Ca lo alida ion by cap u ing de ailed g a i a ional clus e ing and he mody-
namic in e ac ions wi hin RBHs and holog aphic cosmological con ex s. This app oach
is heo e ically igo ous, dimensionally consis en , and obus , o e ing a scalable
pla o m o u u e p ecision in es iga ions o en opic g a i y and RBHs phenomena.
40
Appendix C Nume ical Resul s
Nume ical co espondence able o pa ame e s and a iables used in he main analysis.
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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[4] Ba deen, J.M.: Non-singula gene al- ela i is ic g a i a ional collapse. In:
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