Con inuous Uni e se: Ma e –An ima e Symme y and Alignmen o
E ec i e Dimension in Space-Time
Sóc a es Geo ges Pe akis
Physicis Licensed (UFMG), S.E.E- MG, Belo Ho izon e – MG, B azil
E-mail: [email p o ec ed].b
Abs ac
The ΛCDM model p edic s a con inuous expansion o 3D space, p o ided ha
da k ene gy (DE) emains cosmologically dominan . Wi h cons an densi y and
equa ion-o -s a e pa ame e w ≈ −1 (well below he h eshold w = −1/3), DE d i es
his expansion. In his scena io, he nega i e g a i a ional po en ial ene gy (Epg)
asymp o ically app oaches ze o. Wi h Planck-scale, |∂Epg / ∂ | can be in e up ed
by linea izing he e ec i e space. This s udy explo es a linea space ha modi ies
he Uni e se’s deg ees o eedom. The key mechanism associa ed wi h his
ansi ion is he symme ical annihila ion o ma e and an ima e . Assuming
spa ial la ness as in ΛCDM, and he absence o classical bounda ies, his wo k
aims o p esen a ma hema ical s uc u e ha cha ac e izes and ela es he
ex eme s a es o expansion and con ac ion h ough he con e sion be ween
ene gy and mass. Thus, he cause o an expansi e beginning is de ined. In his
con ex , na u al e ec i e dimensional bounda ies become possible wi hou
in oking classical spa ial edges. The esul is an e olução ha connec s he
cu en Uni e se o a b oade cosmological scena io.
Keywo ds: Uni e se
Linea iza ion o e ec i e space
Ma e -an ima e
En opy
Ene gy
Space
ΛCDM
1
I. In oduc ion
The ΛCDM model, cu en ly he leading pa adigm in mode n cosmology, p edic s
he con inuous and accele a ed expansion o a h ee-dimensional (3D Space)
Uni e se d i en by he dominance o da k ene gy.
Al hough his model success ully accoun s o a wide ange o obse a ions—
such as he cosmic mic owa e backg ound and he o ma ion o la ge-scale
s uc u es — emains open o ex ensions ha could cla i y undamen al aspec s
such as he o igin o he ma e –an ima e asymme y and he physical
cha ac e iza ion o ex eme bounda y s a es.
In his con ex , his wo k explo es he hypo hesis ha he Uni e se may unde go
a ansi ion in spa ial dimensionali y. This ansi ion educes he numbe o spa ial
deg ees o eedom.
A key elemen in his p ocess is he maximum annihila ion o ma e and
an ima e , which plays a decisi e ole in balancing he ene gy con en and allows
new opological con igu a ions.
The model enables a ein e p e a ion o cosmic e olu ion in oducing na u al
dimensional limi s in place o classical spa ial edges, in which linea expansion
and con ac ion a e seen as supe imposed and in e changeable s a es o a
b oade ecu en p ocess.
The main goal o his wo k is o cons uc a ma hema ical amewo k ha
cha ac e izes he mos ex eme po en ial s a e o cosmic expansion and
con ac ion h ough a uni ied physical mechanism.
Wi h a ocus on unco e ing answe s such as he de ini ion o he ini ial
mechanism o expansion, i is c ucial o highligh he need o analyze he po en ial
bounda y s a e, i.e., he mos ex eme one; his s a e is a well-de ined s a e o
maximum expansion, whe e he camou laged ma e consis s o wo minimal
pa icles sepa a ed o he maximum (maximum linea space), o ming a single
s uc u e: he ixed Uni e se, and whe e he only possible “quan um luc ua ion”
is he ans o ma ion in o he supe posed and adjacen s uc u e in he ime
con inuum o maximum con ac ion.
2
Thus, a e a p ocess o comple e annihila ion o 3D s uc u es (comple e
annihila ion be ween ma e and an ima e in a s a e o maximum expansion o
3D space), i is possible o eme ge a s a e o maximum s a ic 1D spa ial
expansion o minimum en opy o med by minimal 3D pa icles sepa a ed o he
maximum by a linea space illed wi h Da k Ene gy (DE) con ained in 1D space
( om annihila ion in 3D space) con e ing in o a s a e o maximum con ac ion
o med by ex ended masses illing he en i e same linea space sepa a ed by a
minimum cen al double space (Minimum DE) in mo ion, wi h he
in e changeabili y be ween mass and ene gy ((ρ DE = ρₘ), by Heisenbe g's
unce ain y p inciple and o he inc ease o en opy es a ing he 3D spa ial
expansion o a ecu ing Uni e se.
This p oposal is oo ed in gene al ela i i y, ene gy conse a ion, and he
possibili y o dimensional educ ion in ex eme g a i a ional o empo al egimes
I builds upon he idea ha opological ansi ions in space ime, coupled wi h
symme ic ene gy con e sions be ween da k ene gy and ma e , can gi e ise o
cosmic dynamics consis en wi h obse a ions.
II. E olu iona y Roadmap
The cu en s a e o accele a ed expansion, wi h da k ene gy densi y al eady
exceeding ha o ma e , indica es ha he Uni e se ends owa d inde ini e
expansion.
As he a ia ion in g a i a ional po en ial ene gy (Epg), which is nega i e,
becomes inc easingly small and app oaches a limi a he Planck scale, he
obse able 3D space o ma e s ops abso bing da k ene gy om a plana
in e ace ha sepa a es i om a 3D an ima e egion.
This da k ene gy in e ace is plana because he Uni e se is la . I s ex inc ion in
his cen al plana egion esul s om he ex inc ion o linea masses o ma e
and an ima e h ough a p og essi e encoun e ha has ed his da k ene gy
plane.
3
Wi hou he sepa a ing plane, he 3D space o ma e comes in o con ac wi h he
opposi e an ima e egion and unde goes symme ic annihila ion; i abso bs da k
ene gy and con inues in a plana expansion, pe pendicula o his annihila ion,
un il only wo minimal emnan pa icles emain, maximally dis an om each
o he . This con igu es an e ec i e space (which uly go e ns e olu ion) as linea ,
since i is he only ene ge ic s uc u e ha is always de ined be ween masses. As
he e a e only wo 3D masses, he e ec i e space is a 3D-linea con igu a ion,
because he minimal pa icles i sepa a es a e one o 3D ma e and one o 3D
an ima e . The e ec i e space is hus linea wi hin a 3D uni e sal space.
This leads o he phase o maximum linea expansion, wi h wo s a ic minimal
pa icles sepa a ed by a maximally ex ended linea space illed wi h da k ene gy.
Since da k ene gy is con ined wi hin his linea i y — ha is, i is linea da k ene gy
— i s densi y becomes equal o he local densi y o each mass, making da k
ene gy and ma e in e changeable. By ene gy conse a ion, his linea s a e o
maximum expansion becomes in e changeable wi h a s a e o maximum linea
con ac ion: a mi o ed con igu a ion whe e he p esence o da k ene gy con e s
in o he p esence o ma e , and he ma e a he ex emi ies becomes double
cen al da k ene gy (a oiding in ini e-densi y singula i ies and enabling he 3D
accele a ed expansion s a e obse ed cu en ly).
Due o he unce ain y p inciple, he inc ease in en opy, and ene gy conse a ion,
he s a ic linea s a e o maximum expansion ansi ions in o a s a e o maximum
linea con ac ion, which al eady eme ges in mo ion owa d he o ma ion o 3D
space.
The only possible mo ion o linea masses is owa d he cen al space illed wi h
dense da k ene gy. As hese linea masses o ma e om one side and an ima e
om he o he mo e owa d each o he , in he ini ial in e ac ion he da k ene gy
b eaks hem in o pa icles and dispe ses hem, o ming 3D space o ma e on
one side and 3D space o an ima e on he o he . As his encoun e con inues,
he cen al da k ene gy densi y dec eases wi h he o ma ion o 3D space on each
side. This no only dis ibu es he masses in o 3D space, bu —due o he
weakening densi y—also allows pa s o he linea masses o mee and
annihila e.
4
Thus, each egion o ma e and an ima e expands while emaining sepa a ed
by he cen al da k ene gy plane, which i sel expands wi h pa ial annihila ion.
When he densi y o his cen al da k ene gy plane becomes su icien ly low, ou
ma e -domina ed egion and he opposi e an ima e egion cease o gain mass,
as he linea masses can only annihila e wi hin he plana egion. This is he
phase cu en o he obse able Uni e se, whe e he 3D Uni e se gains only da k
ene gy (i.e., spa ial olume), while p ese ing a cons an amoun o ma e .
Annihila ion adia ion is no cu en ly obse ed because adia ion equi es a
h ee-dimensional medium o p opaga e, which he da k ene gy plane does no
p o ide. In he u u e, when he linea masses a e ex inguished, he cen al da k
ene gy plane will also anish; da k ene gy will be abso bed, and he Uni e se will
comple e i s cycle, e u ning o a linea s a e o maximum expansion, wi h
maximal annihila ion, once again consis ing only o wo minimal pa icles—one o
ma e and one o an ima e —maximally sepa a ed.
Due o he a ia ion in e ec i e spa ial geome y, linea da k ene gy has a
di e en densi y han he da k ene gy abso bed by obse able 3D space, which
co esponds o he obse ed cha ac e is ics and densi y alues in ou Uni e se.
In he s a ic bounda y s a e o maximum linea expansion, he o al ene gy
sa is ies: E0 + Epg = 0 , wi h Epg < 0 , and Ek = 0
This co esponds o a camou lage o mass, ye i implies a po en ial o mo ion—
as i a s a e o "no hing" could gi e ise o a s a e o "e e y hing".
III. Dimensional Equilib ium P inciple o Da k Ene gy
This wo k explo es he hypo hesis ha he Uni e se may unde go linea iza ion in
i s e ec i e spa ial dimensionali y.
A key elemen in his p ocess is he symme ic annihila ion o ma e and
an ima e , which plays a decisi e ole in balancing he ene gy con en and
enabling new e ec i e opological con igu a ions.
5
The model enables a ein e p e a ion o cosmic e olu ion in oducing na u al
e ec i e dimensional limi s in place o classical spa ial edges, in which maximum
expansion and con ac ion a e seen as in e changeable linea s a es o a b oade
ecu en p ocess.
The main goal o his wo k is o cons uc a ma hema ical amewo k ha
cha ac e izes and ela es he mos ex eme po en ial s a e o cosmic expansion
and con ac ion in he linea e ec i e space h ough a uni ied physical
mechanism.
This s udy conside s ha , h ough a u u e p ocess o maximum annihila ion o
3D s uc u es — occu ing be ween ma e and an ima e in a s a e o maximum
expansion o 3D space (whe e he deg ees o eedom a e minimized and only
ans o ma ions a e possible, as ull 3D expansion has been eached) — i is
possible o an e ec i e s a e o eme ge: a maximally ex ended, s a ic, linea
spa ial con igu a ion o minimum en opy, o med by mass s uc u es (a posi on
and an elec on) sepa a ed o he g ea es possible ex en by a linea space illed
wi h con ained Da k Ene gy (DE).
In he u u e, a e he maximum annihila ion be ween ma e and an ima e , he
Uni e se eaches a phase in which i is composed o wo minimal pa icles a he
ex emi ies, sepa a ed maximally by a linea space illed wi h da k ene gy. In his
s a e, he e will be no o he pa icles a ailable o annihila e and inc ease he
amoun o da k ene gy o space. When maximally ex ended, he uni e sal
(global) da k ene gy densi y mus be much highe han he ma e densi y; only
locally, a he ex emi ies, could he ma e densi y be highe .
Acco ding o he p inciple o linea momen um conse a ion, wi h no u he
physical p ocesses capable o gene a ing new da k ene gy, he Uni e se ends
o hal i s expansion in his asymp o ic s a ic o bounda y s a e. Howe e , i he
uni e sal (global) da k ene gy densi y is highe han he ma e densi y, he
Uni e se would ha e o con inue expanding, since he p essu e o da k ene gy
would be much g ea e han he p essu e o ma e .
6
The only way o econcile he p essu e exe ed by da k ene gy (DE) wi h he s a ic
condi ion o he Uni e se is h ough Uni e sal (global) equali y be ween he
densi ies: ρ DE = ρ m.
Th ough he p ocess o linea iza ion, which con ines DE o an inc easingly linea
space wi h ewe deg ees o eedom, i becomes possible o achie e such an
in ense concen a ion o DE ha i s densi y equals he ma e densi y a each
loca ion in space, eaching a ully linea bounda y s a e.
Da k ene gy ceases o d i e he expansion o he Uni e se when i s global and
local densi y equals ha o he emaining ma e , which is only possible in a linea
bounda y s a e, wi h educed spa ial deg ees o eedom and a s a ic me ic.
In his s a e o equal da k ene gy and ma e densi y, due o he e en ion o a
linea space be ween he wo maximally sepa a ed pa icles, da k ene gy
becomes in e changeable wi h mass.
The bounda y s a e, ep esen ed by he s a e o maximum linea expansion, in
which space is maximally illed by he maximum amoun o da k ene gy, is
cha ac e ized by a minimum es ene gy (E₀), co esponding o wo minimal
masses loca ed a he ex emi ies; a maximum (i.e., leas nega i e) g a i a ional
po en ial ene gy (Epg), ending owa d ze o due o he maximally ex ended linea
space illed wi h da k ene gy; ze o kine ic ene gy (Ek), as he Uni e se is s a ic;
and a maximum da k ene gy (DE) con en .
The o al ene gy in his s a e (ET me) is gi en by:
ET me = [(E0 min + (− Epg max)) + Ek = 0] + DE max
Since E0 + (−Epg) = 0 and Ek = 0, i ollows ha :
ET me = DE max
7
Conside ing he maximal con e sion o da k ene gy in o ma e , along wi h he
co esponding inc ease in en opy, his s a e e ol es in o a s a e o maximum
linea con ac ion, ep esen ed by a mi o -symme ic s uc u e. Tha is, ins ead
o wo minimal masses a he ex emi ies, he con igu a ion now p esen s wo
minimal cen al gaps ( illed wi h da k ene gy) be ween wo maximally ex ended
linea masses, o ming an e ec i e g a i a ional well and enabling he onse o
mo ion owa d a non-linea 3D space.
The o al ene gy in his s a e (ET mc) is composed o :
ET mc = [(E0 max + (− Epg min)) + Ek] + DE min
F om he s a e o maximum linea expansion o ha o maximum linea
con ac ion, he con e sion o da k ene gy in o ma e esul s in he massi e illing
o space, leading o an inc ease in E0 and a co esponding a ia ion in Epg ,
which con inues o cancel E0 unde he linea con igu a ion.
Thus, he s a e o maximum linea expansion e ol es in o a s a e o maximum
linea con ac ion plus mo ion in o a non-linea 3D space, d i en by en opy
g ow h associa ed wi h he di e ence in da k ene gy be ween hese wo s a es:
ET me = DE max = ET mc = DE min + Ek.
In o he wo ds:
Wi h a minimum amoun o ma e loca ed a he ex emi ies, in a maximally
dis ibu ed con igu a ion and sepa a ed by he g ea es linea olume, he ma e
densi y (ρₘ) becomes maximally educed and uni e sally (globally) much smalle
han ha o da k ene gy, despi e being locally g ea e .
In his bounda y s a e, all o he masses (ma e and an ima e ) ha e al eady
been annihila ed. As a esul , he emaining minimal pa icles (posi on and
elec on) a e al eady ixed a he ex emi ies (Ek = 0) and ep esen he minimal
es ene gy (E₀).
8
The g a i a ional po en ial ene gy (Epg) eaches i s maximum, leas nega i e
alue (close o ze o), su icien o cancel E₀. The o al ene gy {ET = [E₀ + (−Epg)
= 0] + (Ek = 0) + da k ene gy (DE)}, ha is, ET = DE (maximum). Howe e , upon
becoming s a ic— h ough he annihila ion o masses and p og essi e
alignmen — he da k ene gy densi y mus equal he ma e densi y uni e sally
(globally).
The alignmen p ocess, e en wi h annihila ion, when comple ed, maximally
con e s da k ene gy in o ma e (E₀ = m₀ c ²), p og essi ely illing he oid
be ween he masses, no by b inging hem close h ough mo ion, bu by making
space i sel hicke o mo e concen a ed in DE, o ming mass.
This cons i u es a ype o massi e expansi e con inui y o he only con igu ed
space ha exis s ( he in e nal space be ween he masses), al hough i is
cha ac e ized as a s uc u al expansion, since da k ene gy is epulsi e.
This p ocess con inues un il he ma e densi y equals he da k ene gy densi y
globally and locally (excep in a minimal dual cen al space a he linea cen e , a
speci ic conse a ion mus be p ese ed).
Thus, a s a ic linea s a e o maximum expansion eaches a s a e o maximum
linea con ac ion, which eme ges al eady in mo ion, o p ese e a ce ain
conse a ion p inciple.
To o malize he s uc u al con e sion o da k ene gy in o ma e in his linea
bounda y s a e, a scala ield Φ( ) ha go e ns he alignmen dynamics and
egula es he e ec i e deg ee o dimensional condensa ion is p esen ed.
As he sys em e ol es owa d a globally s a ic con igu a ion, wi h kine ic ene gy
anishing and he spa ial geome y educed o a single dimension, he ma e
densi y ρm ( ) inc eases p og essi ely, sou ced di ec ly om he s uc u al
hickening o da k ene gy. This con e sion is no media ed by mo ion bu by a
local econ igu a ion o he acuum s a e.
9
As he annihila ion be ween ma e and an ima e ini ially occu s in he cen al
zones and p opaga es symme ically ou wa d, ene gy is eleased again o a new
cycle, igge ing he ansi ion o he non-linea h ee-dimensional Uni e se back
in o an e ec i e linea s a e.
A he end o he p ocess, a e he spa ial plane is comple ely annihila ed, all
olume ic mass is ei he con e ed o annihila ed, lea ing only wo undamen al
pa icles — a posi on and an elec on — posi ioned a opposi e ex emi ies o
he new linea space, ma king he comple e es o a ion o he s a e o maximum
linea expansion and es a ing he cosmological egene a ion cycle.
In summa y:
• Tμν (ma e )= 0 ( o al ene gy o ma e — es mass — canceled by Epg).
• Tμν ( o al) = Tμν (DE).
• This leads o Gμν ≠ 0, Rμν ≠ 0, and gμν ≠ gμν.
• E en wi h ze o spa ial cu a u e, space- ime is accele a ed by da k
ene gy.
The esul is a spa ially con igu ed, s a ic linea Uni e se, dynamically uns able
and eady o ini ia e he nex cycle.
In he bounda y s a e:
• The es ene gy o he masses, E₀, is canceled by (− Epg).
• Kine ic ene gy is ze o (Ek = 0).
Thus, Tμν (ma e ) = 0 globally, and Tμν ( o al) = Tμν (DE) — pu e, cons an ,
and dominan da k ene gy.
• The linea i y o space is pe ec in his s a e (pu ely linea space).
• Da k ene gy is spa ially cons an bu ac s by cu ing ime, no space.
The e o e: Gμν ≠ 0, bu he cu a u e is pu ely empo al, associa ed wi h he
expansion/accele a ion o space, no spa ial de o ma ion.
16
Ricci enso and scala (Rμν and R):
• Rμν ≠ 0 solely due o da k ene gy, no due o esidual masses since hey
ha e Tμν = 0.
• The Ricci scala R ≠ 0 because space- ime has empo al cu a u e.
The space is pe ec ly linea (spa ial cu a u e = 0), bu space- ime is no la due
o he con ibu ion o da k ene gy, which a ec s empo al dynamics (ins abili y:
“expansion” [ illing] in o he only con igu ed space — he in e nal space exis ing
in he bounda y s a e / accele a ion).
The space is linea (by cons uc ion o he bounda y s a e).
The e is no spa ial de o ma ion due o masses, since hei ene ge ic
con ibu ion has been canceled.
Howe e , g μν ≠ η μν because o empo al cu a u e induced by da k ene gy.
→ The me ic can ake he o m:
ds² = – d ² + a( )² dx²
E en hough dx² ep esen s a 1D linea space, he scale ac o a( ) can a y due
o he nega i e p essu e o da k ene gy, gene a ing empo al cu a u e.
The me ic is nea ly Minkowskian, bu wi h empo al de o ma ion caused by da k
ene gy. This de o ma ion is no local (no a ising om ma e ) bu a he global
and smoo h.
Wi h a ocus on unco e ing answe s such as he de ini ion o he ini ial
mechanism o expansion, i is c ucial o highligh he need o analyze he po en ial
bounda y s a e, i.e., he mos ex eme one.
This s a e is a well-de ined linea space o maximum expansion, whe e he
“camou laged” ma e consis s o wo pa icles sepa a ed o he maximum (linea
space), o ming a single s uc u e: he ixed Uni e se, and whe e he only
possible luc ua ion by Heisenbe g's unce ain y p inciple is he ans o ma ion
in o he adjacen s uc u e o he ime con inuum o maximum con ac ion.
17
In linea space, adia ion ceases o p opaga e; he emaining pa icles cease o
abso b adia ion and emain s a ic in posi ion.
Wi h he in e changeabili y be ween mass and ene gy [ρm = ρΛ (ρ DE)], a ises a
s a e o maximum con ac ion o med by wo ex ended masses (ma e and
an ima e ) illing he en i e linea space sepa a ed by a minimum cen al double
space (Minimum DE) in mo ion, by Heisenbe g's unce ain y p inciple and o he
inc ease o en opy es a ing he 3D spa ial expansion o a ecu ing Uni e se.
This p oposal is oo ed in gene al ela i i y, ene gy conse a ion, and he
possibili y o dimensional linea iza ion in ex eme g a i a ional o empo al
egimes [1,2].
IV. F om Accele a ed Expansion o Uni e sal Regene a ion
1. E olu ion and Founda ions
P esen ing a ecu ing Uni e se ha sol es he p oblem o hea dea h h ough
e ec i e dimensional linea iza ion and ene gy egene a ion p ocesses, his
model p oposes ha he Uni e se al e na es be ween e ec i e s a es o
maximum h ee-dimensional expansion and linea con igu a ions, main aining
pe manen exis ence h ough mass-ene gy con e sions in ex eme s a es.
As will be clea ly explained, he cu en 3D space Uni e se is composed o
dis inc egions o ma e and an ima e , well isola ed om each o he by a egion
o space (wi h da k ene gy), which sus ains bo h egions in expansion, and a he
same ime is sus ained by a p og essi e deg ada ion o a p imo dial s a e o linea
space.
In he u u e o an ex emely expanded Uni e se, wi h he comple e ex inc ion o
his linea space, he in e media e egion will p og essi ely disappea om i s
cen e ou wa d. This occu s because he cen al pa s o bo h he ma e -
domina ed and an ima e -domina ed egions will ha e a highe mass
concen a ion, leading o s onge g a i a ional a ac ion.
18
This will p omo e a mo e in ense con e gence be ween he dis inc egions,
culmina ing in he maximum annihila ion be ween ma e and an ima e , and
gene a ing he ene ge ic ans o ma ion ha esul s in he ansi ion om he
cu en 3D space o an e ec i e linea space.
1.1 Cu en Cosmological Con ex
The ΛCDM model, suppo ed by galac ic edshi obse a ions, cosmic
mic owa e backg ound (CMB), and ype Ia supe no ae, desc ibes a spa ially la
Uni e se (Ω ≈ 1) domina ed by da k ene gy (w ≈ – 1) ha d i es accele a ed
expansion. Wi h cons an da k ene gy densi y al eady exceeding ma e densi y,
he Uni e se ends o expand inde ini ely, leading o hea dea h in he s anda d
model.
1.2 Fundamen al appa en p oblems
Du ing accele a ed expansion, he global g a i a ional po en ial ene gy, being
nega i e, ends asymp o ically owa d ze o, dec easing p og essi ely. When his
a ia ion becomes smalle han he Planck scale, i becomes impe cep ible,
sugges ing he need o new physical mechanisms o p ese e a a ia ion in
subsequen e olu ion.
2. Model De elopmen
2.1 Ba yon Asymme y Ques ion
Ini ial Objec ion: The s anda d model assumes global ba yon asymme y based
only on local obse a ions.
P oposed Solu ion: All collide expe imen s and con olled obse a ions show
symme ic ma e -an ima e pai p oduc ion. This undamen al symme y
sugges s ha locally obse ed asymme y indica es he exis ence o dis inc
egions wi h an ima e p edominance, p ese ing global balance. The bu den o
p oo alls on explaining asymme y " om no hing" e sus na u al spa ial
sepa a ion.
19
2.2 Sepa a ion and Isola ion
Objec ion: Absence o obse a ional e idence o an ima e egions and gamma
adia ion a bounda ies.
Solu ion: Ma e and an ima e egions a e e ec i ely sepa a ed by a da k
ene gy–domina ed plane ha does no in e ac elec omagne ically.
Thus, in he cu en s age o e olu ion, wi h he p og essi e ex inc ion o his
cen al plane ( egion) — abso bed by ou obse able Uni e se and p omo ing he
cu en accele a ed expansion — he annihila ion ene gy esul ing om he
encoun e be ween ma e and an ima e , o he eme gen esidual adia ion in
his plane ha sepa a es ou ma e Uni e se om he an ima e Uni e se, since
i occu s wi hin his plane (due o lack o deg ees o eedom) emains con ined,
as i is no capable o p opaga ing in o he 3D space (non-plana and non-linea );
hus, i is abso bed as da k ene gy (in space), and becomes inco po a ed in o ou
obse able Uni e se, mani es ing as accele a ed expansion a he han as
adia ion.
In an ea lie s age o e olu ion, s ill supplied by he linea masses, his sepa a ion
p e en s di ec con ac , a oiding bounda y annihila ion, which also explains he
absence o de ec able la ge-scale annihila ion signals.
The adia ion om pa ial annihila ion a he cen e , due o i s linea na u e, is no
able o p opaga e in o non-linea space, and i s ene gy is he e o e con e ed in o
da k ene gy (DE), which in u n uels he expansion o he la sepa a ion
bounda y.
3. T ansi ion Mechanisms
3.1 Annihila ion and Abso p ion P ocess
Du ing ex eme expansion, when a ia ions in g a i a ional po en ial ene gy a e
minimized on he Planck scale, linea space, which main ains cen al space (DE),
disappea s; all DE is abso bed by dis inc egions o ma e and an ima e
p edominance, and hus, hese egions a ac each o he g a i a ionally. Thus,
he ma e and an ima e o hese p e iously sepa a e egions g adually
annihila e each o he .
20
The esul ing adia ion :
1. P og essi e sepa a ion o pa icles
2. C ea ion o g ea e spa ial olume
3. P opo ional inc ease in o al da k ene gy
3.2 Final s a e o maximum expansion
The p ocess con inues un il only wo undamen al pa icles emain—an elec on
and a posi on. These pa icles, being he las wo emaining, con igu e he linea
space; hus, hey emain in his s a ic s a e and as a apa om each o he as
possible, c ea ing an e ec i ely one-dimensional spa ial con igu a ion.
Explana ion o why elec ons and posi ons a e he mos likely pa icles in
he ex eme egime:
• Smalles cha ged pa icles.
• S able undamen al lep ons.
• Rep esen pe ec ma e -an ima e symme y.
• Do no decay spon aneously.
3.3 E ec i e Dimensional T ansi ion
Objec ion: No mechanism exis s o 3D → linea space ansi ion.
Solu ion: The concep o "e ec i e space" dis inguishes be ween undamen al
space ime geome y and ma e con igu a ion. Wi h only wo pa icles sepa a ed
by maximum possible dis ance, ele an dynamics become e ec i ely one-
dimensional along he connec ing line, wi hou iola ing 3D+1 space ime
s uc u e.
Cla i ica ion on E ec i e Dimensional T ansi ion
3.4 E ec i e Dimensional T ansi ion: F om 3D o linea Sys em Dynamics
I is c ucial o cla i y wha is mean by "dimensional ansi ion" in his model o
a oid misin e p e a ion and guide p ope ma hema ical o mula ion.
21
3.4.1 Na u e o he T ansi ion
The p oposed dimensional ansi ion is no a change in he undamen al
geome y o space ime, which emains 3D+1 h oughou he en i e p ocess.
Ra he , i ep esen s an e ec i e educ ion in he deg ees o eedom o he
physical sys em, leading o a undamen al change in sys em dynamics.
3.4.2 Ini ial S a e: 3D E ec i e Sys em
In he ini ial con igu a ion:
• Mul iple masses a e dis ibu ed h oughou 3D space.
• The sys em has h ee spa ial deg ees o eedom (x, y, z)
• Pa icles can mo e and in e ac in all h ee spa ial dimensions.
• The e ec i e space is de ined as he egion be ween masses whe e
ene gy ans o ma ions occu .
• Sys em dynamics a e genuinely h ee-dimensional.
3.4.3 Final S a e: linea E ec i e Sys em
A e he annihila ion and abso p ion p ocess:
• Only wo 3D masses a e s ill (elec on and posi on)
• These masses occupy disc e e 3D poin s in space.
• Howe e , he sys em's deg ees o eedom a e educed o one.
• The only possible mo ion/ ans o ma ion occu s along he linea di ec ion
connec ing he wo masses.
• The e ec i e space becomes he one-dimensional line segmen be ween
he pa icles.
• All ene gy and dynamics a e cons ained o his linea con igu a ion.
3.4.4 Ma hema ical In e p e a ion
F om a ma hema ical pe spec i e:
• Space ime me ic: Is s ill 3D+1 (no geome ic dimensional change)
• Mass objec s: Con inue o be 3D en i ies.
• Sys em dynamics: Reduce om 3D o 1D e ec i e beha io .
• Deg ees o eedom: T ansi ion om 3 → 1 spa ial deg ee
• Ene gy dis ibu ion: Becomes con ined o linea geome y.
22
This is analogous o a pendulum in 3D space: he pendulum bob is a 3D objec
in 3D space, bu i s dynamics a e e ec i ely one-dimensional due o he
cons ain imposed by he s ing leng h.
3.4.5 Physical Signi icance
This e ec i e dimensional educ ion has p o ound physical consequences:
1. Elec omagne ic p opaga ion: In he linea e ec i e space be ween he
pa icles, elec omagne ic wa es canno p opaga e no mally, as hey
equi e pe pendicula ield componen s ha need a leas wo spa ial
dimensions.
2. Ene gy con inemen : All a ailable ene gy becomes concen a ed along
he linea axis, maximizing he sepa a ion be ween he wo emaining
masses.
3. Sys em s abili y: The educ ion o one deg ee o eedom c ea es a
con igu a ion whe e u he ene gy abso p ion is impossible.
4. En opy conside a ions: The sys em eaches i s minimum en opy s a e
gi en he cons ain o wo emaining pa icles (by annihila ion).
3.4.6 Dis inc ion om Geome ic Dimensional Change
I is impo an o emphasize ha his model does no p opose:
• Change in he undamen al dimensionali y o space ime
• Al e a ion o he 3D+1 me ic s uc u e
• Modi ica ion o gene al ela i is ic geome y
• C ea ion o genuinely lowe -dimensional space egions
Ins ead, i desc ibes:
• Con igu a ion-induced dimensional educ ion o sys em dynamics
• E ec i e space is de ined by ma e dis ibu ion and ene gy low.
• Cons ain -imposed educ ion in deg ees o eedom by massi e educ ion
• Physical sys em e olu ion wi hin unchanged space ime geome y
23
3.4.7 Implica ions o Ma hema ical Fo mula ion
This cla i ica ion guides ma hema ical de elopmen by showing ha :
• The me ic enso is s ill 3D+1.
• Mass-ene gy enso s desc ibe 3D objec s.
• Sys em dynamics can be educed o 1D e ec i e equa ions.
• Ene gy low and ans o ma ions ollow linea e ec i e geome y.
• Conse a ion laws apply in he con ex o educed deg ees o eedom.
This e ec i e dimensional linea iza ion p o ides he ounda ion o he cyclical
egene a ion mechanism while main aining ull consis ency wi h es ablished
space ime physics.
4. S a ic S a e and Heisenbe g P inciple
4.1 Ene ge ic Equilib ium
In he linea maximum expansion s a e, da k ene gy densi y equals ma e densi y
(ρΛ = ρm). Howe e , such s a e would iola e Heisenbe g's unce ain y p inciple
(ΔE · Δ ≥ ℏ / 2), which equi es con inuous ene gy a ia ion com E0 + (−Epg) = 0
and Ek = 0.
4.2 Obliga o y T ansi ion
To sa is y Heisenbe g, he sys em mus ansi ion o a supe posed s a e o
opposi e s uc u e - maximum con ac ion wi h mo ion. Da k ene gy con e s o
mass (E₀ = m₀c²), using elec on and posi on as nuclea ion cen e s o ex ensi e
ma e and an ima e egion g ow h.
5. Regene a ion and Re u n
5.1 Massi e Region Fo ma ion
The ρΛ → ρm con e sion p oduces:
• Ex ensi e ma e mass nuclea ed om he elec on.
• Ex ensi e an ima e mass nuclea ed om he posi on.
• Sepa a ion is suppo ed by cen al double space ha ing esidual da k
ene gy.
24
5.2 De agmen a ion and Re-expansion
The inc ease in en opy o ces he illing o he only emp y space (wi hou mass,
wi h DE) con igu ed cen ally. The da k ene gy p esen p og essi ely
de agmen s he linea masses, sending agmen s o ma e and an ima e in o
3D space, es a ing h ee-dimensional expansion ollowing he s anda d
cosmological e olu ion.
5.3 Isola ion o da k ene gy
Du ing e-expansion, pa o he ma e and an ima e annihila e a he in e ace,
gene a ing adia ion abso bed (DE). This abso p ion causes sepa a ion,
p og essi ely c ea ing mo e space ha inc eases in ex en (and, consequen ly,
mo e da k ene gy) be ween and p opo ionally o he inc ease o expansion o
he dis inc egions, which e ec i ely isola es hese egions, explaining why we
do no cu en ly obse e la ge-scale annihila ion.
6. Comple e Cycle and Conse a ion
6.1 Recu ence Mechanism
When all 3D ma e /an ima e s a i ies om linea space in o dis inc 3D egions,
isola ing da k ene gy p oduc ion ceases. Regions g adually abso b he
sepa a ing space un il g a i a ional a ac ion p edomina es, causing new
annihila ion and e u n o linea space. This annihila ion begins a he cen e and
sp eads o he edges, ending wi h he wo emaining pa icles (posi on and
elec on) dis an and aligned wi h he linea space ha was ini ially ins inc .
6.2 Pe pe ual Ene gy Conse a ion
E ec i e ene gy ne e deple es because E↔mc² con e sions in ex eme s a es
con inuously es uc u e ene gy wi h wo k capaci y:
• Maximum expansion: Mass → ene gy (annihila ion)
• Maximum con ac ion: Ene gy → mass ( egene a ion)
• Cosmic engine: Ex eme s a es as enewal igge s
25
9) The o al ene gy in s a e o linea maximum expansion wi h (ρm = ρΛ) is
ep esen ed by he ollowing ene gy:
TEME (linea space)=[Ene gy inhe en in mass]+[Ene gy inhe en o space]
TE ME (linea space) = [(E0 + (− Epg) = 0) + (Ek = 0)] + [DE (Da k ene gy)]
TE ME (linea space) = DE (maximum)
10) Wi h ρm = ρΛ (in e changeable mass and ene gy),
TE ME (linea space) = [(E0 + (− Epg) = 0) + (Ek = 0)], and
TE ME (linea space) = DE (maximum).
(Heisenbe g's unce ain y p inciple and he endency owa d
inc eased en opy, he s a e o maximum con ac ion eme ges:
S a e o Maximum Expansion (linea ) ≡ S a e o Maximum Con ac ion + Mo ion
11) Maximum expansion s uc u e: ET (ME) = DE (max.) →
Maximum con ac ion s uc u e: ET (MC) = DE (min.) + ene gy a ia ion.
ET (ME) = DE (max.) = ET (MC) = (E0 + (− Epg)) ≠ 0) + Ek (≠ 0) + DE (min.)
The g a i a ional camou lage (maximum expansion) implies an e ec i e
anishing o he ma e ene gy–momen um enso , T μν ma e = 0. Consequen ly,
he o al ene gy–momen um enso is domina ed by he da k ene gy con ibu ion:
Tμν ( o al) = Tμν (ma e ) + Tμν (Λ)
Tμν ( o al) = 0 + Tμν (Λ)
Tμν ( o al) = Tμν (Λ)
Subs i u ing his in o Eins ein's ield equa ions yields:
Gμν + Λ gμν = κ Tμν(Λ)
Gμν + Λ gμν = − Λ gμν
Gμν = − 2 Λ gμν
32
In summa y, he amewo k is buil on six concep ual pilla s:
1. G a i a ional Camou lage : E0 + (− Epg) = 0 hides he mass om classical
g a i y.
2. Da k Ene gy as Fuel: ΔE = DE max – DE min d i es he ansi ion.
3. Dimensional T ansi ion (linea space → 3D) ini ia es a new expanding
phase o he Uni e se.
This ex ension lea es Eins ein's equa ions unal e ed in egula domains,
p ese es co a iance and ene gy conse a ion, and is consis en wi h gene al
ela i i y a i s ex eme geome ic bounda y.
VII. Mechanism o he Onse o Expansion
Maximum expansion (linea space) = Maximum con ac ion (linea space) +
mo ion (T ansi ion o 3D space).
En opy is con olled geome ically and he modynamically, allowing o ansi ion
wi hou physical iola ions.
Wi h he con e sion Expansion Maximum (linea space) → Con ac ion Maximum
+ Mo ion, e en wi h he change in spa ial dimensionali y (e ec i e 3D →linea
space), he numbe o physical cons i uen s (quan ized by he Planck scale)
emains cons an — only hei geome ic mani es a ion changes, edis ibu ing all
uni e sal cons i uen s be ween space- ime and mo ion con igu a ions.
This di e gence in da k ene gy: DE(ME) − DE(MC) > 0, gene a ed by he
in e changeabili y be ween mass and ene gy wi h he inc ease in en opy in linea
space, no andom quan um luc ua ions, is he d i e o he p imo dial
dimensional ansi ion. E ec i e dimensional educ ion has been sugges ed in
se e al app oaches o quan um g a i y, including causal dynamical iangula ions
and spon aneous dimensional educ ion.
This linea spa ial con igu a ion p o ides a na u al se ing o ex eme cosmic
e olu ion. The Uni e se hus becomes ecu en , wi h a na u al cause o he
onse o expansion and he egene a ion o 3D space.
33
Thus, he s uc u e p esen ed shows ha he linea space → 3D spa ial ansi ion
i sel gene a es a p imo dial expansion.
This occu s because:
(i) The DEME > DEMC asymme y c ea es a ‘cosmological imbalance.
(ii) The g a i a ional po en ial (Φ) eaches a c i ical h eshold.
(iii) Geome ic en opy o ces he es o a ion o dimensions.
A he same ime:
• Adap s he ΛCDM scena io by add essing he asymp o ic limi s o ΔEpg
and Planck ime (Tₚ).
• In oduces a physically mo i a ed dimensional ansi ion as a na u al
con inua ion beyond ΛCDM.
The model da a a e consis en wi h:
Es ablishes a de e minis ic, con inuous model o he Uni e se.
VIII. Towa d a Comple e E olu iona y Dynamics
The Uni e se, in i s p imo dial s a e, consis s o wo in e changeable s uc u es:
Maximum Expansion: wo s a ic pa icles (2 m0), sepa a ed by he maximum
dis ance allowed by he e ec i e linea space hey de ine. This a angemen
o ms an ex ensi e linea s uc u e, ep esen ing he en i e Uni e se.
• Da k Ene gy: adia ion om p io 3D pa icle annihila ions canno
p opaga e in he s a ic linea space; he da k ene gy densi y equilib a es
wi h he ma e densi y in linea space.
• T ansi ion o he Con ac i e S a e: En opy inc ease ede ines he sys em
as: Maximum expansion s a e ≡ Maximum con ac ion s a e + mo ion.
34
All adia ion / da k ene gy a ian s con e o mass (E0 = m0 .c2), yielding he
maximum possible ma e con en . A minimal double space eme ges a he
cen e , ini ia ing he ansi ion o a 3D Uni e se.
T ansi ion o 3D space:
• F agmen a ion and Clus e ing: Linea masses a emp o ill he cen al
space bu a e agmen ed by minimal da k ene gy.
• Th ough g a i a ional and kine ic in e ac ions, symme ic egions a e
o med:
3D Ma e (one side)
3D An ima e (opposi e side).
Da k ma e : Ung ouped pa icles (0D) a e as non-ba yonic componen s.
• Region Expansion: As ma e and an ima e accumula e, hei egions
expand p opo ionally abso bing he cen al spa ial plane be ween hem.
They a e sepa a ed by a da k ene gy space.
This space does no p opaga e adia ion (ene gy is con ined o a da k
ese oi ).
No p e e ed di ec ion: The space is symme ically p essed by adia ion
om bo h 3D egions.
Collapse and Recu ence Mass Deple ion: When linea masses cease o
eed ma e /an ima e egions, he da k ene gy space will begin o
dissipa e.
Final Annihila ion: Wi hou he spa ial ba ie , ma e and an ima e
annihila e symme ically, con e ing all 3D mass o “ adia ion” non-
p opagable ( = DE).
Residue and Recu ence: The las annihila ions lea e wo emnan
pa icles, one a each ex eme.
35
Residual adia ion, unable o p opaga e o be abso bed = da k ene gy in space;
he pa icles a e sepa a ed as much as possible, and he cycle es a s.
Ma e /An ima e Symme y:
• The egions a e iden ical in con en bu sepa a ed by a cen al space.
• No global asymme y: "Ou side" (ma e dominance) is a opological esul
o he ini ial clus e ing p ocess.
Dimensions as E ec i e Deg ees o F eedom:
• Da k ma e consis s o 0D pa icles ha ailed o clus e in o 3D s uc u es.
• The da k ene gy space is a collec i e phenomenon (no a undamen al
objec ).
Ene gy Conse a ion and Da k Ene gy Flow,
Ini ial S a e (3D Regions Fo ma ion), mass clus e s o o m ixed quan i ies o :
• Ba yonic ma e (3D).
• Residual adia ion.
• Da k ma e
Ini ial da k ene gy is s o ed in he cen al 3D space om bounda y annihila ions.
Cen al space Dynamics: Tempo a y ene gy ese oi ed by:
• Residual annihila ions a ma e /an ima e bounda ies.
• Radia ion decay. G adual abso p ion: Da k ene gy is symme ically
abso bed by bo h egions, inc easing da k ene gy bu no c ea ing new
ma e / adia ion.
Cycle End: Cen al space anishes when deple ed → Ma e -an ima e
annihila ion → Cycle es a s.
36
IX. Conclusion:
A Con inuous Uni e se Go e ned by linea Ex emal S a es. T adi ional
cosmological models, con ined o h ee-dimensional (3D) Space non-linea , ail
o desc ibe he ue physical limi s o cosmic expansion and con ac ion.
This model conside s he opposi e, ha he Uni e se e ol es be ween wo
ex eme s a es — maximum con ac ion and maximal expansion [wi h (ρm = ρΛ)]
— bo h e ec i ely linea space.
These linea con igu a ions de ine he na u al dimensional bounda ies o cosmic
e olu ion, elimina ing he need o specula i e mechanisms while p ese ing he
o al sum: ene gy + mass + mo ion.
Key A gumen s Suppo ing he Model
1. Dimensional Necessi y:
• The linea limi p o ides a physically meaning ul bounda y condi ion o
cosmic e olu ion, unlike unde ined singula i ies o ad hoc assump ions.
2. Cyclical Dynamics:
• The ansi ion be ween linea and 3D non- linea s a es ensu es sel -
consis en e olu ion wi hou equi ing ex e nal in luences o "uncaused"
e en s. .
3. Pa simony & Physical Robus ness:
• The model adhe es o Occam’s azo , elying only on g a i y, kine ic
ene gy, and da k ene gy—no specula i e cons uc s.
• I a oids he concep ual pi alls o in ini e ene gy o ex e nal in e ac ions
by ea ing he Uni e se as a closed, sel - ans o ming sys em.
37
4. Implica ions o Cosmology
Any comple e cosmological model mus accoun o :
• The dimensional ansi ion be ween linea and 3D Space is a undamen al
p ocess.
• The in e nal conse a ion laws ha d i e pe pe ual e olu ion wi hou
ex e nal dependencies. By aming cosmic e olu ion as a cycle be ween
linea ex emal s a es, his model p o ides a minimalis ye igo ous
al e na i e o complemen he s anda d cosmological model — one ha is
consis en wi h obse a ions while a oiding unphysical assump ions.
This model is alsi iable due o i s in insic connec ion o he asymme y obse ed
be ween ma e and an ima e — a phenomenon ha s ongly con adic s he
uni e sal symme y o physical laws. In all con olled expe imen s and
as ophysical obse a ions, ma e and an ima e a e c ea ed and annihila ed in
pe ec balance, highligh ing cosmic asymme y as a cen al mys e y a he han
a mino anomaly.
The main e idence o his symme y includes:
• Pai p oduc ion in pa icle accele a o s (e.g., LHC), whe e pa icles and
an ipa icles (such as elec on-posi on, qua k-an iqua k) a e always
p oduced in equal numbe s.
• Annihila ion e en s in he labo a o y, whe e ma e and an ima e a e
comple ely con e ed in o pho ons, con i ming ime e e sal symme y.
• Spec oscopy o an ihyd ogen (e.g., ALPHA expe imen a CERN),
showing iden ical spec al lines o hyd ogen and an ihyd ogen, ein o cing
CPT in a iance.
• Composi ion o cosmic ays, whe e he nea absence o p imo dial
an ima e (such as an i-helium nuclei) sugges s a global, a he han local,
asymme y.
38
This model p oposes ha he cu en obse able Uni e se domina ed by ma e
is no he esul o an unexplained ini ial bias, bu he emnan o a sepa a ion o
dis inc egions ha exis and a e physically e ec i e o he appa en b eaking o
s anda d symme y.
I u u e heo y o da a we e o p o e ha no such symme y-b eaking p ocesses
can occu , he co e mechanism o his model would be alsi ied. Finally, he model
explains he su i al o his esidual ma e h ough he p io expansion o da k
ene gy, which spa ially sepa a ed ma e and an ima e domains.
Re e ences
[1] Ca lip, S. Dimension and Dimensional Reduc ion in Quan um G a i y. Classical and Quan um
G a i y, 34(19), 193001. (2017).
[2] Ambjø n, J., Ju kiewicz, J., & Loll, R. Recons uc ing he uni e se. Physical Re iew D, 72(6),
064014 (2005).
[3] Misne , C. W., Tho ne, K. S.,Wheele , J. A. G a i a ion. W. H. F eeman (1973).
[4] Oli e, K. A., e al. (Pa icle Da a G oup). Re iew o Pa icle Physics. Chin. Phys. C 38, 090001
(2014).
[5] Tanabashi, M., e al. (Pa icle Da a G oup). Re iew o Pa icle Physics. Phys. Re . D 98,
030001 (2018).
[6] Vilenkin, A., & Shella d, E. P. S. Cosmic S ings and O he Topological De ec s. Camb idge
Uni e si y P ess (2000).
[7] Gu h, A. H. E e nal in la ion and i s implica ions. J. Phys. A 40, 6811 (2007).
[8] Lew, H., Foo , R. & Volkas, R. R. Possible consequences o pa i y conse a ion. Phys. Le . B
272, 67 (1991).
39
