Solution of a problem of cosmological constant and superconductive cosmology

Abstract

The description of formation of primary vacuum on Planck scales, with the use of the microscopic theory of superconductivity, allows to solve a problem of a cosmological constant and to obtain the value of the vacuum density, coinciding with the observed one. It is offered the new model of exponential expansion and the hot stage of the Universe, in which the synchronization of physical processes and the maintenance of uniformity occurs during each moment of time, instead of single-pass, as in inflationary cosmology.

Full text

Physics of consciousness and life, космология and astrophysics
№ 1, 2011 17
COSMOLOGY AND ASTROPHYSICS
УДК 523.11:524.827:539.12:524.854:530.11
Bukalov A. V.
SOLUTION OF A PROBLEM OF COSMOLICAL CONSTANT
AND SUPERCONDUCTED COSMOLOGY
Physical Department of the International institute of Socionics, Kiev, Ukraine;
e-mail: boukalov@gmail.com
The description of formation of primary vacuum on Planck scales, using the
microscopic theory of superconductivity, allows to solve a problem of a cosmological constant
and to obtain the value of the vacuum density, coinciding with the observed one. It is offered
the new model of exponential expansion and the hot stage of the Universe, in which the
synchronization of physical processes and the maintenance of uniformity occur during each
moment of time, instead of inflationary scenario.
Keywords: cosmological constant, energy of vacuum, cosmology.
PACS numbers: 98.80.–k; 95.36. + x; 11.30.Rd; 42.40.-i
1. Introduction
As it is known, the standard estimations within the limits of the quantum field theory give val-
ues of vacuum energy density, which is in 10120 times more than the observed one:
34
0
P
E
vP
E dE E
≈ (1019GeV)4,
where
5 1/2
( / )
PN
E c G
is the Planck energy.
By this time the set of variants of the solution of the problem of the cosmological constant and
energy of vacuum is offered [1–4]. However the majority of such solutions require the use of specific
parameters, exotic variants of the field theory or modification of the gravitation theory. Therefore
those directions of the solution of this problem have advantage, which are characterized by maximum
physical naturalness and simplicity. It is possible to refer to such directions the attempts to solve the
cosmological constant problem by analogy to the superconductivity theory. However within the limits
of such approach the special assumptions on gravitational interaction between primary fermions,
which defines value of vacuum energy, are made [1–4].
The approach offered in this article does not use such additional hypotheses, and the law of in-
teraction between primary fermions follows from the theory and experimental data. It in turn gives the
value of the cosmological constant close to the observed ones. Thus it clears up not only the mecha-
nism of formation of modern vacuum, but also variety of the problems related to the theory of a Big
Bang and to the cosmological evolution of the Universe.
2. Vacuum structure on Planck scales and superconductivity
Usually it is supposed that the space-time on Planck scales has a foamy structure. However,
there are calculations of interactions on Planck distances which show that domains, with the masses
close to Planck mass, can form regular structures [5]. Such structures can form a regular crystal-like
lattice with the cells close to Planck scales
3 1/2
( / )
PN
L G c
.
As the domains are the quasi self-contained objects, their effective mass is close to zero. They
also can interact among themselves by quadrupole gravitational forces. At such description it is possi-
ble to consider the space-time on Planck scales as analogue of a solid body [5, 6]. Collective excita-
tions in such structure play a role of phonons, arising in a crystal-like lattice with the period close to an
Planck interval.
Let's consider such crystal-like structure not simply as analogue of a solid body, but as the
structure similar to structure of metal in which the free primary subfermions (b-fermions) exist. These

Physics of consciousness and life, cosmology and astrophysics
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subfermions can interreact with the spatial crystal-like lattice and, under certain conditions, can couple
by means of phonon interaction, it is similar to Bardeen-Cooper-Schrieffer model (BCS) [7] for elec-
trons in metal with the bose-condensate formation.
Thus coupling of subfermions can occur near ―quasi-crystal‖ Fermi-dimensional surface. Thus
the maximum oscillation frequency of crystal-like lattices, as an analogue of frequency of Debye, is
close to Planck frequency:
43 41
1.85 10 Hz 9.23 10 Hz
4 8 4 8
P
D
. (1)
The macroscopical equation, which describes interaction of fermions [8], looks like:
2( ) 0
2
itm
. (2)
At the use of pulsing representation of this equation [9] we found:
22
()
( ), ( )
22
F
FF
P P P
E E E
mm
(3)
2 2 2 2
( ) | |
FF
E E V P P
,
where
F
P
is an impulse of subfermions by Fermi’s surface,
2
is an energy gap, related to ef-
fect of coupling of fermions, which separates the basic and excited state. As it appears from theory
BCS, thus
2 2 2 2
()
1 (0)
2
V M d d
VM
. (4)
11
44
2 3.52
1
sinh (0)
DD
Bc
kT
ee
VM
, (5)
where
(0)VM
is a constant of fermion-phonon interactions,
V
is a potential,
(0)M
is a density
of fermion states on Fermi’s surface,
c
T
is a critical temperature [11].
Let’s consider now vacuum energy density, starting from (5). The initial density of the Uni-
verse is equal to Planck one:
0
44
3
3 5 3 3 5
3 3 3( )
8 (8 )
88
P P P
vP
P
m
cc
L
. (6)
In that case the vacuum energy density is defined by density of energy gaps
2v
as binding
energies of subfermion pairs, which form a condensate in a Planck crystal-like lattice:
11
1
28
P
vv
Eee
. (7)
Therefore
11
4
4
44
1
11
(2 ) sinh( )
88
PP
v v v
Eee
. (8)
In present period vacuum energy density
11
4
35
1 1 1 1
88
8
P
vv
Gcee
. (9)
At λ <<1
12
3
88
v
NP
G t e
. (10)
If the density of the condensate of energy gaps is close to the up-to-date critical density

Physics of consciousness and life, космология and astrophysics
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1
2
2
0
3 1 3
88
8
c
NN
P
H
GG
te
, (11)
then
1/2
1 5 1/2 2/
8 ( / ) 8
em
H N P
H t G c e t e
.
From here we can calculate a value of interaction constant . As
1H
Ht
≈1.4·1010years=
= 4.4·1017с,
2ln 137
8H
P
t
t
. (12)
Thus, the received value of coefficient of fermion-phonon interactions in spatial crystal-like
lattice is defined by value of electromagnetic fine-structure constant:
1
122
21
,22
em
em
cc
ee
. (13)
Let’s note that is defined also by a ratio of value of a charge of a Dirac magnetic monopole
to an electrical charge:
1/ge
. This result naturally confirms our point of view on character of
interaction of fermions in spatial crystal-like lattice. Thus, the mechanism of condensation of primary
fermions has not the gravitational nature, but is defined by special fermion-phonon interaction. The
coefficient of this interaction is the function of the electromagnetic fine-structure constant. The de-
scription of this fermion-phonon interaction will be given in separate work.
The vacuum energy density as condensate of energy gaps or energies interaction of subfermi-
ons in a primary crystal-like lattice is:
11
10
32 2
2
3 3 1
(8 )
8 (8 )
em em
v v v
N
NP
c
G
G t e e
. (14)
11 11
22 1
(3 ) 8 8
em em
v P v P
t e t e
. (15)
Thus the dynamic parameter, the Hubble parameter, is very close to a value of the constant or
is equal to it. Such exact coincidence, apparently, is related to affinity of values of density of substance
and vacuum in present period.
The fact, that the formula (11) gives the value of the up-to-date density of the Universe, allows
to assume that ―dark energy‖ and observable substance within the limits of superconducting model can
be described as a condensate and supercondensate of primary subfermions.
The definition of value of parameter
v
≈1.48 requires separate consideration.
Therefore, the existence of ―dark energy‖ of the Universe is defined by phase change pro-
cess — condensation of subfermions into superconducting electron pairs and formation of a Bose con-
densate of these pairs.
3. Dynamics of formation of modern density of vacuum
Let’s consider the process of formation of the up-to-date vacuum in the hot early Universe. As
it is known from quantum electrodynamics, a value of electromagnetic fine-structure constant is func-
tion of four-impulse
2
Q
:
2
1 ln
32
em
i
em
e
Q
m
,
or
2
1ln
32
i em
e
Q
m
, (16)
where
e
m
is the mass of electron,
2/
em ec
is the fine-structure constant.
Then the effective vacuum energy density as density of energy gaps will make:

Physics of consciousness and life, cosmology and astrophysics
20 № 1, 2011
1
2
1
4
55
3
22
2
32
2 ln
34
32
3 1 3 3
82
(8 )
(8 )
em
em e
cond v v v
E
N H e
N
m
N
c c Q
G t m
Ge
Ge
, (17)
where
/Q kT c
is an impulse of quanta of radiation and substance in the early Universe:
1
4
43
2
(2 )
82
8em
vP
v
e
Q
m
e
.
Thus, the energy density of a vacuum condensate is driven by the energy density of radiation
and substances. Thus
v
reaches a minimum and becomes a stationary value at
2
2e
Qc m c
= 1.022
MeV. For energies of Major integrating
15
10
GUT
eV
11
00
( ) ( ) ln
2i GUT
GUT
b
,
10
2
4
32 ( ) 0
8
i
b
pGUT
Ve
.
Let's consider now a question on quantity of
1
em
change. At the Planck energies the value for
vacuum is
vP
and
11
i
. Thus the value of an electrical charge equally primary one:
22
00
e c Q
. (18)
For definition of the law of
1
i
change we will consider some aspects of the Universe for-
mation. If the Universe started from Planck density
4
~
PP
M
, it extended in a vacuum-like state till
the moment of phase transition under the law
1
24
2
(2 )
1 1 1
88
8
i
v
v P i
NNV
GG
e
,
and the radius of the Universe at the moment of transition from vacuum-like state in a hot state was
CMBR
U v H
GUT
T
R R R T
1
1/ 2 8i
iP
te
, where
H
c
RH
is the up-to-date Hubble radius,
CMBR
T
is the
temperature of relict radiation (temperature of cosmic microwave background radiation). Thus
V
plays a role of parameter of time for phase transition.
Thus the value
1
i
could change from 1 to
180
GUT
. At the moment of phase transition we
can estimate the radius of the Universe
U
R
and, accordingly, value
1
i
at
2
3
32 GUT
GUT
N
Gt
≈ (1015 GeV)4
depending on value of energy of a vacuum energy gap
v
.
So, at
2v
3.22·103 GeV and
B GUT
kT
2.11·1016 GeV the Universe radius is
U
R
=2.32·10-2 sm at
11
/2
i em
68.518.
At
0
*
2v
vH
m
246.3 GeV and
B GUT
kT
1.35·1015 GeV the Universe radius is
U
R
2.292 sm
at
1
i
73.1.
At
0
*
2Zv
vm
91.18 GeV and
B GUT
kT
5.02·1014 GeV the Universe radius is
U
R
16.7 sm
at
1
i
75.
At
*
2v
vW
m
80.4 GeV and
B GUT
kT
4.43·1014 GeV the Universe radius is
U
R
21.49 sm

Physics of consciousness and life, космология and astrophysics
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at
1
i
77.22.
The energy of phonon interaction between primary subfermions can be considered as the spe-
cial form of energy which differs from observable forms of energy. In the observable Universe it is
displayed by means of (anti) gravitational interaction.
Let’s note that the value of initial exponential expansion of the Universe corresponds to that
which arises also in the inflation theory. Therefore the superconducting mechanism of the Universe's
expansion provides causality and homogeneity of the Universe. Moreover, while in the inflationary
theory maintenance of the Universe homogeneity is a one-time event, in the superconducting scenario
existence of a quantum vacuum condensate continuously provides homogeneity of the Universe. It
resolves the Penrose’s paradox: R. Penrose repeatedly underlined the lack of mechanism of sync of
electroweak phase transition in various points of space, which occurs much more after inflationary
dilating [10].
Thus, the energy density of vacuum at
GUT
E
is equivalent to energy density of vacuum elec-
troweak interactions:
*4
v
v EW
m
(102 GeV)4.
It means that the birth of substance during a Big Bang is defined by presence of special vacu-
um vector bosons and, possibly, the Higgs vacuum field. The energies of prospective usual X, Y- bos-
ons with
E
≈ 1015 Gev correspond to these vacuum bosons. Therefore the higher energies
3
2 10
v
GeV and
16
10
RP
GeV, seemingly, are not realized because the Universe warming up at
transition from a vacuum state to radiation-dominating state begins with
GUT
E
≈ 1.35·1015Gev.
At the same time there is a possibility of Universe expansion not from Planck, but from bigger
volume, for example from volume with radius
1/2
8P
r L e
2.32·10-2 sm. Such volume corre-
sponds to the greatest possible packaging of the Planck domains which number is
1
3
2
P
Ne
1.86·1089.
As far as the quantity of fundamental particles in the Universe is close to this number, it is ob-
vious that phonon oscillations of the lattices, formed by domains of spatial quasi-crystal, at phase tran-
sition have served as the cause of formation of these particles. Let’s note that the presence of a vacuum
condensate with
v
(3.22·103 GeV)4 provides homogeneity of such volume and there is no necessity
to care of existence of 1089 causally unrelated Planck volumes: all of them prove to be related to a
vacuum coherent subfermion condensate.
From this it follows that the Universe birth could occur in 2 stages: the first was the formation
of quasi-stable vacuum-like state with
1/28
vP
r e l
and the second was the expansion of this volume
for
37
10
GUT
t
in 90–100 times.
At exterior energy action the vacuum-like Universe could begin transition into radiation-
dominated state with an energy liberation
GUT
EE
. It caused the phase transition and the beginning
of the hot Universe. Thus, according to (16) and (17), a reorganisation of structure of vacuum hap-
pened because of
1
change. At
11
0i em
a radius of vacuum curvature
1
2
becomes close to
H
R
:
1
2~H
R
at
02e
Em
.
At the second scenario phase transition with condensate formation looks natural, because the
certain value of parameter of interaction
2
already exist.
4. Physical time as phase change function
Within the limits of superconducting cosmology the radius
1/2
and the Hubble radius are
nearly equal:
1
1/2 1
8P
R l e
. (19)
The cosmological time is

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22 № 1, 2011
1
18i
HP
cH ct R l e
. (20)
In present period, at
0z
,
11
em j
. Then it is possible to present the evolution of the criti-
cal density of the Universe energy also in the form of
1
j
evolution:
1
2
3
22
33
2
888 em
j
c
NH NH
Q
me
Gt G t e
, (21)
where
2
j
Q
is a 4-impulse of quanta of radiation and substance, which are so far unknown and, proba-
bly, related to other dimensions.
Thus, current cosmological time
H
t
and observable evolution of the Universe can be described
as process of phase transition with change of parameter of interaction between fermions and phonons
1ln
jt
, similar to
em
. So a hierarchy of parameters of evolution or times appears, it relates to the
hierarchy of the energies of the fermions condensates, each of them plays a role of vacuum for overly-
ing level.
Occurrence and destruction of condensates in the hot Universe correspond to the law (21).
Therefore usually calculated vacuum of electroweak interactions and a QCD vacuum, which arose
during phase transitions in the early Universe [2], are distinct from real vacuum. They, possibly, repre-
sent false vacuums in relation to the vacuum condensate, considered above. During cooling of the hot
Universe they disappear.
It is easy to understand if to take into consideration that the hypothetical vacuum should stop
evolution at level of energy density of gluon condensate
~
v
(0.2 GeV) 4 or π-mesons (0.14 GeV)4
[2]. Then up to modern vacuum density
v
= (2·10-3 эВ)4 more 44 orders are necessary, but there are
no observable phase transitions to the modern vacuum density.
However the Universe cooling to the up-to-date temperature gives the chance to display the
true vacuum condensate. It is thus obvious that the density of real vacuum energy of other fields of the
observable Universe does not exceed the energy of this vacuum condensate:
, ,...pv
. (22)
Apparently, the true vacuum of fields both in hot and in the up-to-date Universe coincides
with a vacuum evolving condensate as the inferior energy level.
5. The conclusion
1. Within the limits of the theory of superconductivity for crystal-like space on Planck distances the
real value of density of vacuum energy as density of energy gaps
4
(2 )
of phonon interactions of
a primary subfermions condensate is received. Subfermions, apparently, do not give a contribution
to observable energy density. The contribution to observable forms of energy is given only by
phonon excitation of a primary lattice — boze - and fermi-phonons.
2. Initial exponential expansion of the vacuum-like Universe within the limits of superconducting
cosmology allows to provide the birth of the hot Universe and solves the same problems which are
solved by an inflationary cosmology. Thus in the initial Universe at least two components exist:
the first component is, probably, the supercondensat, it breaks up and generates the hot Universe
with temperature
GUT
T
1.35·1015GeV; the second component with much lower energy plays a
role of vacuum for the hot Universe.
3. Origin of cosmological time
H
t
becomes clear: in the observable Universe time is a consequence
of proceeding phase transition of II kind, which is similar to the phase transition, which has creat-
ed the up-to-date vacuum energy density with change and fixing of a fine-structure constant
ln
jH
t
.
4. Transition into a superconducting state at
c
TT
is accompanied by entropy reduction, because
entropy of a superconducting state
S
S
is less than entropy of a normal state
N
S
:
0
SN
SS
. It

Physics of consciousness and life, космология and astrophysics
№ 1, 2011 23
means that during the Universe expansion process both entropy of a vacuum condensate
(
0
V
S
t
) and entropy of the Universe (
00
t
S
t
) at its cooling decrease. Let’s note that it corre-
sponds to Nernst’s theorem, because
0S
at
0T
. It explains forming of complex structures,
including live beings, together with observers, during the Universe evolution, contrary to the
―thermal death‖ concept.
5. If the observer is in a point close to the phase transition termination, he fixes coincidence of some
dynamic and static quantities, such as the Dirac Great numbers etc., as it occurs in a reality [7].
If occurrence of the observer of terrestrial type is the indicator of the termination of phase transi-
tion and the beginning of a new stage of the Universe evolution, it explains the cause of applicabil-
ity of Anthropic principle: the determined phase transition of the Universe in a certain state gener-
ates the observable Universe with observers.
6. Existence of a coherent condensate of primary fermions with one wave function
c
, interreacting
with the crystal-like space lattice on Planck scales, eliminates a problem of homogeneity of the
Universe at all stages of its evolution.
Thus, the vacuum condensate, defining the primary Universe expansion, defines also modern
dynamics of its evolution.
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Quantum Gravity, May 25–29, Moskow / Ed. by M.A.Markov. — Singapore: World Scientific, 1988. —
P. 813
6. Fomin P. I. On the crystal-like structure of physical vacuum on Planck distances // Probl. phys. kinetics and
physics of solid body. — Kiev: Naukova dumka, 1990. — P. 387–398.
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Букалов А. В.
Решение проблемы космологической постоянной и свехпроводящая космология
Описание формирования первичного вакуума на планковских масштабах с использованием мик-
роскопической теории сверхпроводимости позволяет решить проблему космологической постоянной и
получить значение плотности вакуума, совпадающее с наблюдаемым. Предложена новая модель экспо-
ненциального расширения и горячей стадии Вселенной, в которой синхронизация физических процессов
и поддержание однородности происходит в каждый момент времени, а не однократно, как в инфляцион-
ном сценарии.
Ключевые слова: космологическая постоянная, энергия вакуума, космология.