About pressure in model of the expanding Universe
Closed model of Universe on the earlier stage of its evolution is defined more precisely. It is considered here that after the Big Bang and De Sitter’s (exponential) expanding phase in which the pressure is p=-c²ρc (see [1-3]) the post De Sitter’s stage is beginning. State equation on this stage is...
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| Date: | 2007 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Cite this: | About pressure in model of the expanding Universe / S.S. Sannikov-Proskuryakov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 200-205. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859979329941798912 |
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| author | Sannikov-Proskuryakov, S.S. |
| author_facet | Sannikov-Proskuryakov, S.S. |
| citation_txt | About pressure in model of the expanding Universe / S.S. Sannikov-Proskuryakov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 200-205. — Бібліогр.: 12 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | Closed model of Universe on the earlier stage of its evolution is defined more precisely. It is considered here that after the Big Bang and De Sitter’s (exponential) expanding phase in which the pressure is p=-c²ρc (see [1-3]) the post De Sitter’s stage is beginning. State equation on this stage is written in the form p(R)=-c²ρ(R)A(R), where is some pure geometrical factor. Here this equation is investigated and boundary conditions for it are formulated. Explicit expression for A(R) in the class of almost periodical functions is found, that permits to integrate the conservation law. Outgoing from this we come to the following conclusion: at the post De-Sitter’s stage the configuration space of Universe is the Bohr’s compact. Manifold R on which acceleration is positive Rtt>0 are found. Due to the vibrant character of depending A(R) this manifold may advance far in future that is compatible with observed data.
Закрита модель Всесвіту на ранній стадії її розвитку визначена більш точно. Вважається, що після Вели-кого Вибуху й де-сітеровскої стадії, що (експоненціально) розширюється і на якій тиск дорівнює p=-c²ρc (див. [1-3]), починається пост-де-сітеровська стадія. Рівняння стану на цій стадії написано у вигляді p(R)=-c²ρ(R)A(R), де A(R) — якийсь чисто геометричний фактор. Досліджено це рівняння й сформульовані для нього граничні умови. Знайдено явний вираз для A(R) в класі майже періодичних функцій, що дозволяє проінтегрувати закон збереження. Виходячи із цього, приходимо до наступного висновку: на пост-де-сітеровскої стадії конфігураційний простір Всесвіту є компактним за Бором. Знайдено різноманіття R, на якому прискорення є позитивним, Rtt>0. Через вібруючий характер залежності A(R) це різноманіття може простиратися далі в майбутнє, що погоджується із даними спостереження.
Закрытая модель Вселенной на ранней стадии ее развития определена более точно. Считается, что после Большого Взрыва и де-ситтеровской (экспоненциально) расширяющейся стадии, на которой давление равно p=-c²ρc (см. [1-3]), начинается пост-де-ситтеровская стадия. Уравнение состояния на этой стадии написано в виде p(R)=-c²ρ(R)A(R), где A(R) – некий чистый геометрический фактор. Исследовано это уравнение и сформулированы для него граничные условия. Найдено явное выражение для A(R) в классе почти периодических функций, что позволяет проинтегрировать закон сохранения. Исходя из этого приходим к следующему выводу: на пост-де-ситтеровской стадии конфигурационное пространство Вселенной является боровски компактным. Найдено многообразие R, на котором ускорение является положительным, Rtt>0. Из-за вибрирующего характера зависимости A(R) это многообразие может простираться дальше в будущее, что согласуется с наблюдаемыми данными.
|
| first_indexed | 2025-12-07T16:25:34Z |
| format | Article |
| fulltext |
ABOUT PRESSURE IN MODEL OF THE EXPANDING UNIVERSE
S.S. Sannikov-Proskuryakov
A.I. Akhiezer Institute of Theoretical Physics
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: sanpros@kipt.kharkov.ua
Closed model of Universe on the earlier stage of its evolution is defined more precisely. It is considered here
that after the Big Bang and De Sitter’s (exponential) expanding phase in which the pressure is (see
[1-3]) the post De Sitter’s stage is beginning. State equation on this stage is written in the form
, where is some pure geometrical factor. Here this equation is investigated and bound-
ary conditions for it are formulated. Explicit expression for in the class of almost periodical functions is
found, that permits to integrate the conservation law. Outgoing from this we come to the following conclusion: at
the post De-Sitter’s stage the configuration space of Universe is the Bohr’s compact. Manifold ℜ on which
acceleration is positive are found. Due to the vibrant character of depending this manifold may
advance far in future that is compatible with observed data.
ccp ρ2−=
)()()( 2 RARcRp ρ−= )(RA
)(RA
0>R )(RA
PACS: 98.80.Cq, 98.80.Bp, 26.35.+c
1. INTRODUCTION
As is known (see for example [4]), the main evolution
equations in the Friedman’s model of Universe are writ-
ten in the form
ρ
πζ
2
2
2 3
8
c
G
cR
R
R
=
+ , Rpc
c
GR )3(
3
4 2
2
+−=
⋅⋅
ρ
π ,
232 3)( pRRc
dR
d
−=ρ (1)
(the third equation (conservation law) is a consequence
of the first two ones), where and are mass (en-
ergy) density and pressure in Universe (always ).
If also to take into account so called cosmological con-
stant (which for Einstein’s ether is ), one
needs to make the substitution in equations (1)
ρ p
0>ρ
0λ >λ
GG
cpp
π
λ
ρρ
π
λ
8
,
8
2
+→−→ .
Formally (without clearing up of physical mean-
ing) it may be considered two limiting cases:
а) (the Friedman’s limit, herein
, if ) and b) (the
De Sitter’s limit, herein ). In the case b) the De
Sitter’s evolution equation is
λ
c/
Gcp //, 2 λρ >>
0< /2ρcp −>
⋅⋅
R 3 Gp /, 2 λρ <<
0>
⋅⋅
R
3
2
2
λ
=
+
ζ
cR
R
R
. In [5]
it is supposed that the case b) is realized in the begin-
ning of Universe evolution and is called there inflation
(hereby is considered as a large magnitude which is
connected with density energy of some hypothetic sca-
lar field). After this (exponential) expansion the Fried-
man’s stage is beginning. Nowadays there is just oppo-
site tendency [6]: it is supposed that the case b) takes
place in the end of Universe evolution (after finishing
Friedman’s expansion; hereby λ is considered as a
very small magnitude connected with energy density of
so called quintessence – some hypothetic media de-
scribed by state equation , where quantity
; for Einstein’s ether it is w ). Obviously
both these hypothesis except each other.
λ
wp ρ=
p =
0<
c2
3/1−<w 1−=
2ρc
λ
R
2. LAGRANGIAN APPROACH
In contemporary cosmology one considers that from
the dynamical point of view our Universe is always
pure Lagrangian system (so that its main characteristic
is some Lagrangian), and such a phenomenon as Big
Bang took place namely in Lagrangian system (i.e. Big
Bang is a Lagrangian explosion; as is known in such a
system total energy is conserved, it is the famous
E. Noether’s theorem). After Big Bang, Universe evolu-
tion is described by the equations (1), from which it
follows that for relativistic matter ( ), which
after the explosion predominated over the non-
relativistic one, and also for non-relativistic matter ac-
celeration was negative: . However at space ex-
pansion (in the end of Friedman’s evolution) density
and pressure of matter decreases so much that the quin-
tessence (or Einstein’s ether) begins to play the decisive
role (hereby it is considered that all time has a con-
stant magnitude). Apparently the only consequent
physical interpretation of cosmological constant λ is
connected with mass of quanta of metric field
(although notion of such a quanta is highly problematic,
see [7]). In result the De-Sitter’s (exponential) regime
of expansion with positive acceleration is com-
ing. So according to this idea initial (Friedman’s) phase
of expansion with is replaced by the phase with
. The observations of radiation emitted some bil-
lions years back (red shift parameter ) show [8]
3/
0>
10~z
R
m
0<
µνg
R
0>R
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 200-205. 200
that at that time our Universe was already expanding
with positive acceleration. However this result may be
interpreted by another way: in that time (after the Big
Bang and De Sitter’s stage) our Universe was still ex-
panding with positive acceleration. But nowadays it is
expanding with negative acceleration. For such a state-
ment there is a quite serious reason.
3. NON-LAGRANGIAN APPROACH. STATE
EQUATION ON POST DE SITTER’S STAGE
OF UNIVERSE EVOLUTION
According to the suggested in [1-3] model our Uni-
verse before the first Big Bang (in its zero cycle) has
been an ensemble of the bi-Hamiltonian dynamical rela-
tivistic systems. In [1-3] such an ensemble is called
ether (thus ether is the substance from which our Uni-
verse constituted before the first Big Bang). Mass of the
ensemble (mass of Universe in zero cycle) equaled
, [1-3]. Ether is some kind of two level
dynamical systems: its upper level is characterized
by positive energy and lower level by negative one.
The Big Bang is an irreversible total quantum transition
taken place at the ether collapse (or at its con-
densation in result of which the state arose). As a
result of such a transition (confluence with ) the
fundamental particles (Lagrangian matter) are arisen.
Big Bang as a quantum jump is not described by any
equation (jump is described by transition matrix element
gM 45)0( 10~
ff →
f
f
f
f f
ff ,
gM 57)1( 10~
)0()1( / η=MM
ρ
30 /10~ cmgcρ
Rs
3/1η=
p =
, see [1-3]), but after explosion and forming
usual particles (in the first cycle) dynamical (bi-
Hamiltonian) type of Universe is changed: it becomes
Lagrangian or Hamiltonian one. And only after this
Universe is described by equations (1). In result of Big
Bang (transition , at which additional energy is
secreted) mass of Universe becomes equaled
, [1-3] (hereby dimensionless parameter
). Big Bang gives rise when
ether density is equaled (in result of collapse) to the
critical maximally possible magnitude
,
ff →
1210~
3
ρ =
)0(
min 10~
3Rscρ
cc ρ2−
[1-3]. At this time the radius of Uni-
verse (in zero cycle) was equaled [1-3].
At constant density Universe expands to the
size [1-3], that is determined
from the formula . At this stage pressure
is constant (it follows in particular from the
third equation in (1)), and Universe expands according
to the law , where
cm510
R
R
(
)0(
min
e)0
min
~
tH c
cρ
cm9
M=
R(
R
)1
) =
(
t
11110~ −s
3
8 G
cρ
π
=Hc , that follows from the first
equation in (1) (in which the member ς may be
omitted). Hereby mass of Universe changes according
to the law (only it is compatible
with 3-dimensionality of Universe space in zero cycle),
where . It is so called De Sitter’s
regime of expansion (in the end of it ). Thus
Universe expansion from the size to the is
accomplished by indispensable energy pumping charac-
terized by energy creation parameter
(at this part of evolution the
Universe is obviously non-Lagrangian system, see fur-
ther). After finishing De Sitter’s stage ( ) a new,
post De Sitter’s stage is beginning (it may not be mixed
with Friedman’s one on which pressure ). On this
stage (inertial expansion) energy is not pumped, mass of
Universe [Note that only the smaller part of total
mass (energy) of Universe corresponds
to the visible (charged) matter (for protons it is
2/ R
tHceMtM 3)0()( =
cs Ht 3/lnη=<
scmg ⋅340 /
)1(
1(M
t0 <
10~
M
csR ρρ =)(
)0(
minR
sRR =
0≡p
)0(Mη
sR
H ccρ
) =
g5110
3S
sR>
)()(2 RARc ρ−
ρ
)( =sRA
ρ2/ cp
ff →
f
3S
Mη
p −=
(p
A
2/ cp
)0( ~
cc ρ2
R
)R =
p
)(R
ρ
, for electrons ~ ,
see [1-3]). The larger part of it ( ) spends on
the expansion of space (it is dark energy)] does not
change and connection between pressure and mass den-
sity (state equation) is another. Taking into account that
on the previous stage this connection had the form
, we consider that on the post De Sitter’s
stage (at ) state equation has the form
g48
ρ2c
)R
M )0(4/1 ~η
g5710
0)(lim =
∞→
RA
R
p
A
75
R
=p
ρ2
10
/
(
0
~
1
. (2)
Here and are physical characteristics of Universe
and is some pure geometrical factor satisfying the
boundary conditions , . We
show further that the last condition playing important
role in Friedman’s model is not obligatory: ratio
may be small in another sense. Note, that the
mean value of such function (in particular of )
upon almost its period has a small magnitude; in this
sense the ratio is small only. Function as
a pure geometrical factor can not be connected with any
property of matter. In fact after finishing the De Sitter’s
stage and stopping energy pumping (the number of
transitions is finite, it is not bigger, than quan-
tity of quanta in zero cycle equaled 10 , see [1-3])
space of Universe expands on inertia. Hereby matter
particles continue to arise in every space point without
any relative motions giving rise to pressure: at expan-
sion (increasing of Universe radius ) particle coordi-
nates on the sphere stay unchanged, particles are in
rest. Therefore pressure of matter is zero (that
was taking into account in [1-3]). The formula (2) does
not describe the matter pressure (apparently necessity in
a new form of matter in connection with the equation
(2) is not, in fact for matter ratio is always 1 / cp 3/
201
[9] meanwhile in (2) may achieve the value − ,
i.e. maximal rate of is 1 , see (11)), it describes
joint reaction of space and matter on the Big Bang. Note
if at , so the De Sitter’s stage would be
continue beyond the (it follows from the conserva-
tion law at ), however it is impossible be-
cause energy pumping must stop (quantity of transitions
is terminated). Emphasize that configuration
space is not expanded at Lagrangian (or Hamiltonian)
explosion of matter: in Lagrangian (Hamiltonian) sys-
tem configuration space exists in already ready form.
Writing the “conservation” law (third equation in (1)) in
the form
)(RA
p
sRR >
2c−=
1
ρ2/ c
sR
ρ
1)( =RA
f→
p
f
R
dR)RA(3=Rd (ln ρ )3
)R
3
s
(3
3
)1(
Be
s RR )(ρ
)(R =ρ
1M )(
R
M
=
∫
R
Rs
′
′
′ dRA )(=RB )(
R
R
dR
dBR=
( sRB
(∞B
1()1(~ M=
UT −=
GM4π−=
TH =
)(/ )1( tM
ciH
A
M
L
U
L
)R
=ρ
(M
3/)
1(M
4π
(B
0
0
)(B ≠∞
e RB(3)2
U ζ−=
)()1 t
2/2 +
) =
)
)e
(
+
M
=
≠
3
1
(
R
∞
/ R
=
/2
)R
)(R
∫ ′
R
Rs
RC )( ′Rd
R
1
C
RB(
C
A+
R, .
C,
(3)
and integrating it we get
, (4) C
where , and
. (5)
It is comfortable to write the function in the form )(RA
where satisfies the boundary condi-
tions (that is obviously) and . In
fact if so at density would be
equaled , where
. However in Lagrangian
system it is impossible: total energy (mass) of Universe
must be conserved. Note on the post De Sitter’s stage
Universe is Lagrangian system: there is Lagrangian
, where T and
from which motion equa-
tion (second equation in (1)) and conserving Hamilto-
nian (first equation in (1))
follow. At depending on time (on the De Sit-
ter’s stage), Universe is not Lagrangian. In connection
with this we note that there is a “Lagrangian”
, concerning mass
unite which describes the oscillator with image fre-
quency . Automatically both conditions for
may be satisfied if we write in the form of
0)( =∞
2
B
B
ρ
/2)1( R
→R
)1(~M
)1
R
22 /)c
RG cρ
(B
3
M
3
A
=) , (6)
where is a summed function on interval
Hereby all three functions are connected by the
relation (it is needed only to differentiate formula (6))
)(R [Rs
BA,
[
B = (7) ,[ A
hereat satisfies the boundary conditions
(however the last condition is in-
deed not obligatory, see below Eq. (8)). It is essential to
emphasize hereby that contribution in (in pressure)
from quite large class of singular and step functions
vanishes without living a trace because for such
functions we have almost everywhere
C
,1 C 0)()( =∞=RC s
)(RB
A
0=
dR
dB , i.e.
, and therefore formula (4) is written in the form 0=A
3
)1(
R
M)(R =ρ (pressure and integral 0=p
∫ dR
dR
dB
B
= 0 ). These functions are not restored on their
derivatives, see [10]. Therefore further cosmological
problem is considered solely in the class of absolutely
continuous functions. Considering only continuous
functions (without step functions) note that (7) is the
well-known Lebesgue formula representing continuous
function ( ) as a sum of absolutely continuous ( )
and singular ( ) functions (continuous functions is
called singular one if its derivative equals zero almost
everywhere, see [10]; the well known example of such a
function gives Cantor’s staircase). If in (7) there is no
absolutely continuous function ( ), so C and
formula (6) leads to the integral equation for
A
0=A B=
B
∫ ′′=
R
Rs
RdRB
R
RB )(1)( . (6.1)
It is the degenerate case of the homogeneous Volterra
equation. Formula (6.1) may be considered as an inte-
gral characteristic of singular function; from it the dif-
ferential characteristic follows (note that
(6.1) does not take place for step functions). Actually
consideration of cosmological problem based on singu-
lar functions (zero pressure) is given in [1-3]. We may
indicate the suitable functions for the problem. Note
first of all if the function C is given so the functions
are given too according to (6). To choose the suit-
able class of functions the following circumstance
helps us. Formula (6) determines function as the
mean value of C on the interval [ (so that
0)( =′ RB
C
B,
B
[)(R , RRs
)(RC=)(RB
R
). In expanding Universe the upper limit
in (6) depends on time and grows with growing .
Hereby in open models (
t
0;1−=
t
ζ ; in these cases kinetic
energy of expansion is always more than potential en-
ergy of gravitational interaction) tends to at
(Universe behaves like a white hole). At the
limit integral
R ∞
∞
∞→R
→t
∫∞→
R
Rs
RC
R
(1lim
)
,∞sR
′R
A′
′ d)
[
R
R
determines so
called the Bohr’s mean value of function C . Therefore
it is natural to take from the class of almost peri-
odical functions on the ray [ which is denoted
. Strictly speaking [1-3] this class is factor-ring
, where is ring of continuous functions on ray
and is maximal ideal in consisting of
(C
A′
BA
AA /′
∞sR [
202
functions obeying the condition C . Note,
the solution with dissipation belongs to (hereby we
have to consider not only this ray but all three dimen-
sional space of Universe to be Bohr’s compact, see
[1-3]). Any such function is written in the form of gen-
eralized Fourier series [11]
)(RC 0)( =∞
A
∑=
n
ip
n
n
ec(
−
R
RR
s
s
R)C ,
where is real and c is complex. Writing
we will have for real function satisfying
the required boundary conditions (see above)
n
nib+
p
n
n
n ac =
∑=
n
a(
2/ cp
1=
−
s
n R
RR
psin
nb
ρ2/ cp
−
s
s
R
R
∑
n
RC( )
R
2)1 3/ c
Rg 10~
gR
8=
n +
gR
nκ
n arctg=
2
nn b+a
∑=
n
C
/π≤n
+ n
sR
)ϕ
=n 1cosϕ
kcos()
ϕ
nk
,]
n
s
k
C
R ∑ )n
s
s
R
R
ϕ+
−sin(
s
n
R
)nϕ
)(
n
R
R)(
neC
RC
nκ
=
−
)
− s
nnn b
R
p )cos (8)
(in our case all , sum and may be
arbitrary. Note, mean value of such a function (in par-
ticular of ) upon its almost period is a small
magnitude. In this sense the ratio is small
only). However we are interested in another, closed
model (
0≠np
ρ
=na 1
ζ ; note, only this value of ζ is compatible
with conditions of Universe creation). Here is re-
stricted from above: (however
is indeed very large magnitude ;
Universe behaves like black hole, because its radius is
always under the Schwarzschild’s radius ). It is es-
sential that in this model potential energy is more than
kinetic one. Due to this not only Universe expansion is
stopped but also momentum dissipation takes place in
result of that the wave vectors are smearing (they be-
come complex), i.e. (dimensionless
and we call universal wave vectors and decrement
parameters). So in closed model real function is
written in the form (
(GMπ
niκ
Rg
k=
R ≤
np
cm29
nk
C
n
n
a
b
ϕ ,
2
nC = )
−
−
−
s
n
R
RR
n R
R
eRC s
s
n
(
κ
(9)
with boundary condition and
. At the obligatory condition
we have
∑
n
nc
22/π ≤
n <<κ
sin
[)( nR
RR
n
n RkeRB s
n
ϕ
κ
−
≈
−
−
,(10)
and also
cos(
)((
s
sR
RR
n R
Rk
CRBRA
s
s
+
−
=
≈−
−
∑
. (11)
The solution (11) of the equation (2) we call the press
(shock) wave in Universe. Further keeping only one
item in sum (9) (under the simplifying condition
), let us address to the inequality 1
(see the second equation in (1)), at fulfillment of which
acceleration is positive . At this inequal-
ity is equivalent to the condition
0,1 == ϕc 03 <− A
0>R sRR >>
s
R
R
R
Rke s cos
3
1 κ−
< . (12)
Inequality (12) has a solution in the form of non-
connected manifold ℜ [ ]gs RR ,⊂ . At (without
dissipation) ℜ consists of segments
0=κ
3
1arccos
3
1arccos2 <−
R
Rkm
s
π
...2,1=m 0≠
2 +< mπ
m
where
At κ numbers are bounded from
above because it must be at all 3ln<
sR
Rκ
0=k
(it follows
from (12) at ). Hence 3ln
κ
sR
R < . Denoting
Mk
=
κ
(it is quantity of connected components in ℜ)
we may approximately write
k
R
M s
M
R < . As is small
and is large the length
k
k
R
M s
1=n
4=
5
1−=
gR
may be very large.
So components of ℜ on which acceleration may
progress far in the future. A question about spectra of
and is arisen. We consider that the spectrum of is
connected with various phases (eras) in Universe evolu-
tion. At least there are a few different evidential phases
distinguished by various value of Universe radius
[1-3]. So for hadron era (it begins
at and finishes at ), for lepton one ,
for atomic , for galaxy n , for final era before
stop of expansion (in our numeration the De Sit-
ter’s era begins at and finishes at ). So
interval from to we divide onto six eras (like
octave consists of six big seconds: as, b, c, d, e, fis).
Corresponding time of finishing of n-era may be calcu-
lated from the formula
0>
n
0
R
1=n
n
k
2
κ
3/n
sn RR η=
0=n
3=n
=n
n
)0(
minR
k
=
=
6
5−3n
1−= nn Ht
s1
== n
g
n
c
R
ns kR /=
/1 nn k−
R
R
Rn
κ
τη
1~
where . Putting we will
have . Let us consider when the n-era be-
gins the preceding finishing (n-1)-era already decreased
at least in e time. It means that , i.e.
c 10~/
3/n
Rs
/1 η
−=τ
nk =
3
1+n
R
/1=n ηκ and hence . Moreover
considering n-era we may neglect all previous ones be-
cause they are described by fast oscillating (singular)
items in (9) which do not give visible contribution in
pressure at . Further considering n-era we
4103 ~/1η=M
ns kR /~
203
may put the contribution from the next (n+1)-era and
following ones equaled C if
. So eras as if take part in
relay race in which momentum from short wave lengths
transfers onto long ones. The magnitude (frequency of
pulsations)
0...cos 11 =+++ nn ϕ
1cos...cos 11 =++ nnCC ϕϕ
6
23
/
−n
cH η
s st 5
1 10~ − t ~2
3/4
4 /1,1 η== k
)/4 sRR
4k
GR
ss RR <<
/ ==Ω nn tM
k
11
0 10~ −
st 7
3 10~
st 13
4 10~
ηϕ /1,1,0 333 === kC
44 ,2/π= C∓
sin()/cos()( 3 s kRRkRC ±=
k 16
4 10~ −k
cmRs
253/4 10~η
3/13/4 ηη gs RR =
1210~
R −
t
is important
characteristic of n-era. As is known usually connection
between and κ is given by dispersion relations (cf.
with [12]) however this approach is not utilized here. If
to pay attention that the first four eras are very quick
(they finish in t , , ,
; in these eras microstructure of Universe is
formed) and only following era is long (it finished in
) we may approximately unit all first four
eras in one that will be characterized by
. Then the next era is charac-
terized by ϕ . In this
approximation neglecting dissipation we may use
(hereby we con-
sider that sign plus corresponds to the closed model and
minus – to open one). In our consideration the smallest
value of is . We consider that galaxy era
(galaxy structure of Universe) is connected with i.e.
is maximal size of galaxy. As
so we may conclude that our Uni-
verse consists of η galaxies. Conversely we may
consider that galaxy structure of Universe testifies the
Bohr’s compact structure of configuration space of Uni-
verse. Now address to the first equation in (1). Not
difficult to show that at dependence
on is given by the formula
s10
R
t
R
GM
s3
8 )1(πRtR )( = s +
(where s
et
cm /1020
kRs /<<
3/1)1( )GM
3/2t
c
R
GM
s
~
3
8 6/5
)1(
η
π
=
RRs <<
6()( RtR s π+=
gs RRkR <<<</
3/1)1( )6()( GMtR π=
gRR <<<<λ
is expanding
velocity on this segment (emphasize it is analytical de-
pendence on t ). At we have another
dependence , and at
(it is considered that e )
solution goes out on the
Friedman’s regime (acceleration is negative). At
we approximately have
3/2
1~)(3 RB
∫ −
λ/
0
)sin
2
31(
x
xdxx λ
Tcg =/
=
λ /3
Rc
t
2 R
g
(here ). De-
noting we get from here
ks /R=
R
]cos
λ
R
4
9
2
8
9 2/32/3
λ
π
R
+
1[ −
3
2
T
t
=
2
λ
RR
R
g
.
As is seen our expanding Universe pulsates (it as if
breathes) but these pulsations of in t are small.
Practically depends on like in the Friedman’s
model: . However pulsations of pressure is very
important. Undoubtedly they together with gravity at-
tractive affect played important role in forming of gal-
axy kernel and spacing between them. In order to con-
sider a question about energy spectrum of we
have to apply the Fourier transform to the (how-
ever we do not study this problem, note only the spec-
tral density behaves like θ ).
R
R
/2t
t
/)
3~R
)(tR
)(tR
3/5( εε
4. CONCLUSION
Here it is demonstrated how the recent cosmological
observations [6, 8] might be coordinated with the rela-
tivistic bi-Hamiltonian dynamical system and Fried-
man’s background of closed model of Universe (in (1)
1=ζ ) without acceptance of supplementary hypothesis
about quintessence (and k-essence) interpreted hereby
quite freely. In fact, according to Aristoteles, quintes-
sence and ether are synonyms, aren’t they? Meaning of
ether Aristoteles saw in its primogeniture: ether is the
entity underlain our Worldbuilding. In connection with
this we consider that in real cosmology it is quite
enough only one hypothesis concerning Big Bang as an
explosion taking place in the bi-Hamiltonian system (in
ether): energy emitted in irreversible quantum transition
originates in this system but not in Lagrangian one. In
quantum Lagrangian system all processes are princi-
pally reversible (there is no time arrow) and go under
the condition of strict fulfillment of conservation law of
initial energy. On the contrary in bi-Hamiltonian system
there exists time arrow: it is connected with non-unitary
character of quantum theory of the system but here
there is no energy conservation (hence such a system is
an energy source) [1-3]. We also demonstrated here that
in R-inhomogeneous Universe configuration space
(Universe atlas on the post De Sitter’s stage) is en-
dowed by special topology in which it is so called
Bohr’s compact. Hereby micro-maps of this atlas where
wave mechanics must be used are endowed by this to-
pology too (one and the same boundary condition con-
nected with the Bohr’s compactification of configura-
tion space, therefore pressure in cosmology and wave
function in wave mechanics are almost periodical func-
tions; from our point of view wave functions play the
role of singular functions). A new kind of wave me-
chanics is connected with such a space structure, it is
well adopted for description of special (living) form of
matter [1-3].
REFERENCES
1. S.S. Sannikov-Proskuryakov. Cosmology and a Liv-
ing Cell //Russian Physics Journal. 2004, v. 47,
p. 500-511.
2. S.S. Sannikov-Proskuryakov. About processes of
dispersion in the new quantum theory //Ukr. Journ.
of Phys. 1995, v. 40, p. 650-658.
204
3. S.S. Sannikov-Proskuryakov. Spacetime & sub-
stance //Ukr. Journ. of Phys. 1995, v. 40, p. 901-
908.
4. S. Weinberg. Gravitation and Cosmology, N.-Y.,
1972, p. 72.
5. A.D. Linde. Physics of Elementary Particles and
Inflationary Cosmology. M.: “Nauka”,1990, p. 36
(in Russian).
6. S. Perlmutter Supernovae, dark energy, and the ac-
celerating Universe: The status of the cosmological
//International Journal of Modern Physics A. 2000,
v. 15, p. 715-739.
7. S.S. Sannikov-Proskuryakov. On the physical prin-
ciples of gravitation //Russian Physics Journal.
2004, v. 47, p. 42-52.
8. W. Zimdahl, D. Pavon, L.P. Chimento. Interacting
Quintessence //Phys.Lett B. 2001, v. 521, p. 133.
9. L.D. Landau, E.M. Lifshitz. Field Theory. M.:
“Nauka”, 1967, p. 437 (in Russian).
10. A.N. Kolmogorov, S.V. Fomin. Elements of Theory
of Functions and Functional Analysis. M.:
“Nauka”,1981, p. 341 (in Russian)
11. H. Bohr. Fastperiodische Functionen. Berlin, 1932,
p. 93.
12. L.D. Landau, E.M. Lifshitz. Electrodynamics of
Solid Media. M.: “Nauka”, 1982, p. 324 (in Rus-
sian).
О ДАВЛЕНИИ В МОДЕЛИ РАСШИРЯЮЩЕЙСЯ ВСЕЛЕННОЙ
С.С. Санников-Проскуряков
Закрытая модель Вселенной на ранней стадии ее развития определена более точно. Считается, что после
Большого Взрыва и де-ситтеровской (экспоненциально) расширяющейся стадии, на которой давление равно
(см. [1-3]), начинается пост-де-ситтеровская стадия. Уравнение состояния на этой стадии напи-
сано в виде ) , где ) – некий чистый геометрический фактор. Исследовано это урав-
нение и сформулированы для него граничные условия. Найдено явное выражение для ) в классе почти
периодических функций, что позволяет проинтегрировать закон сохранения. Исходя из этого приходим к
следующему выводу: на пост-де-ситтеровской стадии конфигурационное пространство Вселенной является
боровски компактным. Найдено многообразие ℜ, на котором ускорение является положительным, .
Из-за вибрирующего характера зависимости ) это многообразие может простираться дальше в будущее,
что согласуется с наблюдаемыми данными.
ccp ρ2−=
()()( 2 RARcRp ρ−= (R
A
A
(RA
0>R
(R
(RA
0>R (RA
ТИСК У МОДЕЛІ ВСЕСВІТУ, ЩО РОЗШИРЮЄТЬСЯ
С.С. Санников-Проскуряков
Закрита модель Всесвіту на ранній стадії її розвитку визначена більш точно. Вважається, що після Вели-
кого Вибуху й де-сітеровської стадії, що (експоненціально) розширюється і на якій тиск дорівнює
(див. [1-3]), починається пост-де-сітеровська стадія. Рівняння стану на цій стадії написано у ви-
гляді ) , де ) — якийсь чисто геометричний фактор. Досліджено це рівняння й
сформульовані для нього граничні умови. Знайдено явний вираз для ) в класі майже періодичних
функцій, що дозволяє проінтегрувати закон збереження. Виходячи із цього, приходимо до наступного
висновку: на пост-де-сітеровскої стадії конфігураційний простір Всесвіту є компактним за Бором. Знайдено
різноманіття ℜ, на якому прискорення є позитивним, . Через вібруючий характер залежності ) це
різноманіття може простиратися далі в майбутнє, що погоджується із даними спостереження.
ccp ρ2−=
)(Rp ()(2 RARc ρ−= (RA
205
|
| id | nasplib_isofts_kiev_ua-123456789-110964 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:25:34Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Sannikov-Proskuryakov, S.S. 2017-01-07T15:07:11Z 2017-01-07T15:07:11Z 2007 About pressure in model of the expanding Universe / S.S. Sannikov-Proskuryakov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 200-205. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 98.80.Cq, 98.80.Bp, 26.35.+c https://nasplib.isofts.kiev.ua/handle/123456789/110964 Closed model of Universe on the earlier stage of its evolution is defined more precisely. It is considered here that after the Big Bang and De Sitter’s (exponential) expanding phase in which the pressure is p=-c²ρc (see [1-3]) the post De Sitter’s stage is beginning. State equation on this stage is written in the form p(R)=-c²ρ(R)A(R), where is some pure geometrical factor. Here this equation is investigated and boundary conditions for it are formulated. Explicit expression for A(R) in the class of almost periodical functions is found, that permits to integrate the conservation law. Outgoing from this we come to the following conclusion: at the post De-Sitter’s stage the configuration space of Universe is the Bohr’s compact. Manifold R on which acceleration is positive Rtt>0 are found. Due to the vibrant character of depending A(R) this manifold may advance far in future that is compatible with observed data. Закрита модель Всесвіту на ранній стадії її розвитку визначена більш точно. Вважається, що після Вели-кого Вибуху й де-сітеровскої стадії, що (експоненціально) розширюється і на якій тиск дорівнює p=-c²ρc (див. [1-3]), починається пост-де-сітеровська стадія. Рівняння стану на цій стадії написано у вигляді p(R)=-c²ρ(R)A(R), де A(R) — якийсь чисто геометричний фактор. Досліджено це рівняння й сформульовані для нього граничні умови. Знайдено явний вираз для A(R) в класі майже періодичних функцій, що дозволяє проінтегрувати закон збереження. Виходячи із цього, приходимо до наступного висновку: на пост-де-сітеровскої стадії конфігураційний простір Всесвіту є компактним за Бором. Знайдено різноманіття R, на якому прискорення є позитивним, Rtt>0. Через вібруючий характер залежності A(R) це різноманіття може простиратися далі в майбутнє, що погоджується із даними спостереження. Закрытая модель Вселенной на ранней стадии ее развития определена более точно. Считается, что после Большого Взрыва и де-ситтеровской (экспоненциально) расширяющейся стадии, на которой давление равно p=-c²ρc (см. [1-3]), начинается пост-де-ситтеровская стадия. Уравнение состояния на этой стадии написано в виде p(R)=-c²ρ(R)A(R), где A(R) – некий чистый геометрический фактор. Исследовано это уравнение и сформулированы для него граничные условия. Найдено явное выражение для A(R) в классе почти периодических функций, что позволяет проинтегрировать закон сохранения. Исходя из этого приходим к следующему выводу: на пост-де-ситтеровской стадии конфигурационное пространство Вселенной является боровски компактным. Найдено многообразие R, на котором ускорение является положительным, Rtt>0. Из-за вибрирующего характера зависимости A(R) это многообразие может простираться дальше в будущее, что согласуется с наблюдаемыми данными. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники QED processes in strong fields About pressure in model of the expanding Universe Тиск у моделі Всесвіту, що розширюється О давлении в модели расширяющейся Вселенной Article published earlier |
| spellingShingle | About pressure in model of the expanding Universe Sannikov-Proskuryakov, S.S. QED processes in strong fields |
| title | About pressure in model of the expanding Universe |
| title_alt | Тиск у моделі Всесвіту, що розширюється О давлении в модели расширяющейся Вселенной |
| title_full | About pressure in model of the expanding Universe |
| title_fullStr | About pressure in model of the expanding Universe |
| title_full_unstemmed | About pressure in model of the expanding Universe |
| title_short | About pressure in model of the expanding Universe |
| title_sort | about pressure in model of the expanding universe |
| topic | QED processes in strong fields |
| topic_facet | QED processes in strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110964 |
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