Homogeneous isotropic cosmological models with pseudoscalar torsion function in Poincare gauge theory of gravity and accelerating Universe

0.09 MB·English

by  A. V. Minkevich

Homogeneous isotropic cosmological models with
pseudoscalar torsion function in Poincare gauge theory
of gravity and accelerating Universe
A.V. Minkevich1,2, A.S. Garkun1 and V.I. Kudin3
1Department of Theoretical Physics, Belarussian State University,
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0 2Department of Physics and Computer Methods, Warmia and Mazury University in
0
Olsztyn, Poland,
2
3Department of Technical Physics, Belarussian National Technic University
n
a
J Abstract. The ”dark energy” problem is investigated in the framework

3 of the Poincare gauge theory of gravity in 4-dimensional Riemann-Cartan

space-time. By using general expression for gravitational Lagrangian ho-
2
v mogeneous isotropic cosmological models with pseudoscalar torsion func-
6
tion are built and investigated. It is shown that by certain restrictions on
1
1 indefinite parameters of gravitational Lagrangian the cosmological equa-
2
tions at asymptotics contain effective cosmological constant and can lead
1
6 to observable acceleration of cosmological expansion. This effect has geo-
0 metrical nature and is connected with space-time torsion.
/
c
q
1 Introduction
-
r
g
:
v Cosmological researches fulfilled during several last decades were fruitful for theoretical and
i
X observational cosmology. The creation of inflationary paradigm, which permits to solve a
r number of problems of standard cosmological scenario, in particular, to explain the homo-
a
geneity and isotropy of the Universe at initial stages of cosmological expansion, is important
achievement of the early Universe cosmology [1]. The discovery of the acceleration of cos-
mological expansion at present epoch is the most principal achievement of observational
cosmology [2]. By using Friedmann cosmological equations in order to explain accelerating
cosmological expansion, the notion of dark energy (or quintessence) was introduced in cos-
mology. According to obtained estimations, approximately 70% energy in our Universe is
connected with some hypothetical form of gravitating matter — “dark energy” — of un-
known nature. Recently many investigations devoted to dark energy problem were carried
out (see review [3]). According to widely known opinion, the dark energy is associated with
cosmological term. If the cosmological term is connected with the vacuum energy density, it
is necessary to explain, why it has the value close to critical energy density at present epoch.
The present paper is devoted to investigation of the “dark energy” problem in the frame-
work of the Poincare gauge theory of gravity (PGTG), which is natural generalization of
Einsteinian general relativity theory (GR) by applying the gauge approach to theory of grav-
itational interaction. According to PGTG the physical space-time possesses the structure of
Riemann-Cartan continuum with curvature and torsion. By using gravitational Lagrangian
1
of PGTG in general form including both a scalar curvature and various invariants quadratic
in the curvature and torsion tensors (see below), isotropic cosmology was built and investi-
gated in a number of papers (see [4–6] and references herein). As it was shown, the PGTG
permits to solve the problem of cosmological singularity of GR: all solutions of generalized
cosmological Friedmann equations (GCFE) for homogeneous isotropic cosmological models
(HICM) deduced in the framework of PGTG are regular in metrics, Hubble parameter ,
its time derivative, if gravitating matter satisfies certain physically acceptable condition at
extremely high energy densities and pressures [4]. It is a consequence of gravitational re-
pulsion effect, which takes place at extreme conditions and is connected with space-time
torsion essentially [6]. Properties of discussed HICM in PGTG coincide practically with
that of GR at sufficiently small energy densities, which are much less in comparison with
limiting (maximum) energy density for such models. By including cosmological term of cor-
responding value to the GCFE, we can obtain regular cosmological solutions with observable
accelerating expansion stage. However, like GR, the problem of dark energy in such theory
is not solved.
Fromgeometricalpointofview, thestructureofHICMinPGTGcanbemorecomplicated
incomparison with models describing by GCFE.Really, inthecase of homogeneous isotropic
models the torsion tensor Sλ = Sλ can have the following non-vanishing components
µν νµ
−
R3r2
[7, 8]: S1 = S2 = S3 = S (t), S = S = S = S (t) sinθ, where S and
10 20 30 1 123 231 312 2 1
√1 kr2
S are two functions of time, spatial spherical coordinates are used.−The functions S and S
2 1 2
have different properties with respect to transformations of spatial inversions, namely, the
function S (t) has pseudoscalar character. The GCFE follow from gravitational equations
2
of PGTG for HICM together with S = 0. Obtained physical consequences of GCFE have
2
principal character. However, it is necessary to note that gravitational equations of PGTG
for HICM have also other solution with non-vanishing function S . The HICM with two
2
torsionfunctions arestudied below inthe frameofPGTG inconnection with thedark energy
problem. In Section 2 gravitational equations of PGTG for HICM with two torsion functions
areobtainedandinvestigated. InSection3theasymptotics ofcosmologicalsolutionsforsuch
models is analyzed.
2 Homogeneous isotropic models with two torsion func-
tions in PGTG
At fist let us mention some general relations of the PGTG. Gravitational field is described
in the frame of PGTG by means of the orthonormalized tetrad hi and anholonomic Lorentz
µ
connection Aik (tetrad and holonomic indices are denoted by latin and greek respectively);
µ
corresponding field strengths are torsion Si and curvature Fik tensors defined as
µν µν
Si = ∂ hi h Aik,
µν [ν µ] − k[µ ν]
Fik = 2∂ Aik +2AilAk .
µν [µ ν] [µ |l|ν]
2
We will consider the PGTG based on the following general form of gravitational Lagrangian
= h f F +Fαβµν (f F +f F +f F )+Fµν (f F +f F )+f F2
G 0 1 αβµν 2 αµβν 3 µναβ 4 µν 5 νµ 6
L
+Sαµν (a S +a S )+a Sα S µβ , (1)
(cid:2) 1 αµν 2 νµα 3 µα β
where h = det(hi ), F = Fα , F = Fµ , f (i = 1,2,...,6), a (k = 1,2,3) are ind(cid:3)efinite
µ µν µαν µ i k
parameters, f = (16πG)−1, G is Newton’s gravitational constant. Gravitational equations
0
ofPGTGobtainedfromtheactionintegralI = ( + )d4x, where istheLagrangian
g m m
L L L
of matter, contain the system of 16+24 equations corresponding to gravitational variables
R
hi and Aik .
µ µ
Any homogeneous isotropic gravitating system in PGTG is characterized in general case
by three functions of time: the scale factor of Robertson-Walker metrics R and two torsion
functions S and S . Below by using spherical coordinate system, the tetrad is taken in
1 2
diagonalform. Then thecurvaturetensor hasthefollowing non-vanishing tetradcomponents
denoted by means of the sign ˆ :
Fˆ0ˆ1 = F ˆ0ˆ2 = F ˆ0ˆ3 A , F ˆ1ˆ2 = F ˆ1ˆ3 = F ˆ2ˆ3 A ,
ˆ0ˆ1 ˆ0ˆ2 ˆ0ˆ3 ≡ 1 ˆ1ˆ2 ˆ1ˆ3 ˆ2ˆ3 ≡ 2
F ˆ2ˆ3 = F ˆ3ˆ1 = F ˆ1ˆ2 A , F ˆ0ˆ1 = F ˆ0ˆ2 = F ˆ0ˆ3 A ,
ˆ0ˆ1 ˆ0ˆ2 ˆ0ˆ3 ≡ 3 ˆ3ˆ2 ˆ1ˆ3 ˆ2ˆ1 ≡ 4
with
A = H˙ +H2 2HS 2S˙ ,
1 1 1
− −
k
A = +(H 2S )2 S2,
2 R2 − 1 − 2 (2)
A = 2(H 2S )S ,
3 1 2
−
A = S˙ +HS ,
4 2 2
where H = R˙/R is the Hubble parameter and a dot denotes the differentiation with respect
to time.
The Bianchi identities in this case are reduced to two following relations:
A˙ +2H(A A )+4S A +2S A = 0,
2 2 1 1 1 2 4
− (3)
A˙ +2H(A A )+4S A 2S A = 0.
3 3 4 1 4 2 1
− −
By using the gravitational Lagrangian (1) and the Bianchi identities (3), we obtain the
following system of gravitational equations for HICM
ρ
a(H S )S 2bS2 2f A +4f A2 A2 +2q A2 A2 = , (4)
− 1 1 − 2 − 0 2 1 − 2 2 3 − 4 −3
(cid:0) (cid:1) (cid:0) (cid:1)
a S˙ +2HS S2 2bS2 2f (2A +A ) 4f A2 A2 2q A2 A2 = p, (5)
1 1 − 1 − 2 − 0 1 2 − 1 − 2 − 2 3 − 4
(cid:16) ˙ ˙ (cid:17) (cid:0) (cid:1) (cid:0) a (cid:1)
f A +A +4S (A +A ) +q S A +(2f q )S A + f + S = 0, (6)
1 2 1 1 2 2 2 3 1 2 4 0 1
− 8
h(cid:16) (cid:17) i (cid:16) (cid:17)
q A˙ +A˙ +4S (A +A ) 4fS A 2(q +q )S A (f b)S = 0, (7)
2 3 4 1 3 4 2 2 1 2 2 1 0 2
− − − −
h(cid:16) (cid:17) i
3
where
a = 2a +a +3a , b = a a ,
1 2 3 2 1
−
f
2
f = f + +f +f +f +3f ,
1 3 4 5 6
2
q = f 2f +f +f +6f , q = 2f f ,
1 2 3 4 5 6 2 1 2
− −
ρ is the energy density, p is the pressure and the average of spin distribution of gravitating
matter is supposed to be equal to zero. Equations (4)–(5) do not contain high derivatives
for the scale factor R, if a = 0, moreover, equations (6)–(7) take more symmetric form, if
2f = q +q . Then the system of gravitational equations for HICM take the following form:
1 2
1
2bS2 2f A +4f A2 A2 +2q A2 A2 = ρ, (8)
− 2 − 0 2 1 − 2 2 3 − 4 −3
(cid:0) (cid:1) (cid:0) (cid:1)
2bS2 2f (2A +A ) 4f A2 A2 2q A2 A2 = p, (9)
− 2 − 0 1 2 − 1 − 2 − 2 3 − 4
f A˙ +A˙ +4S (A +A )(cid:0) +q S ((cid:1)A +A(cid:0))+f S (cid:1)= 0, (10)
1 2 1 1 2 2 2 3 4 0 1
h(cid:16) (cid:17) i
q A˙ +A˙ +4S (A +A ) 4f S (A +A ) (f b)S = 0. (11)
2 3 4 1 3 4 2 1 2 0 2
− − −
h(cid:16) (cid:17) i
The system of equations (8)–(11) together with definition of curvature functions (2) is the
base of our investigation of HICM below. Note also that the conservation law for spinless
matter has usual form:
ρ˙ +3H(ρ+p) = 0. (12)
From (8)–(9) follows that
1 b
A +A = (ρ 3p) S2. (13)
1 2 12f − − f 2
0 0
By using (13) and the formula following from definition (2) of curvature functions
k 2
A2 A2 = 4A S2 4 S2 S2 S˙ +HS ,
3 − 4 2 2 − R2 − 2 2 − 2 2
(cid:18) (cid:19)
(cid:16) (cid:17)
we find from gravitational equations (8)–(9) the following expressions for A and A :
1 2
1 α αε
A = ρ+3p ρ 3p 12bS2 2 ρ 3p 12bS2 S2
1 −12f Z − 2 − − 2 − Z − − 2 2
0
h i
3αεf(cid:0) 2(cid:1) k(cid:0) (cid:1)
+ 0 HS +S˙ +4 S2 S2 , (14)
Z 2 2 R2 − 2 2
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:16) (cid:17)
1 α
A = ρ 6bS2 + ρ 3p 12bS2 2
2 6f Z − 2 4 − − 2
0
h i
3αε(cid:0)f (cid:1)2 k
0 HS +S˙ +4 S2 S2 , (15)
− Z 2 2 R2 − 2 2
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:16) (cid:17)
4
q q
where Z 1 + α[(ρ 3p) 12(b+εf )S2], α f , ε 2, 2 = 3αεf . By using the
≡ − − 0 2 ≡ 3f02 ≡ f f0 0
conservation law (12), the formula (13) and the following relation obtained from definition
of A and A
3 4
A +A = S˙ +3HS 4S S , (16)
3 4 2 2 1 2
−
we obtain from (10) the following expression for the torsion function S :
1
3α
S = 4(2b εf )S S˙ HY , (17)
1 0 2 2
4Z − −
h i
where
dp
Y (ρ+p) 3 1 +12εf S2.
≡ dρ − 0 2
(cid:18) (cid:19)
Then by using formulas (13) and (16) we find from (11) the following differential equation
of second order for the torsion function S :
2
ε S¨ +3HS˙ + 3H˙ 4S˙ S +4S S (3H 4S )
2 2 1 2 1 2 1
− −
h (cid:16) (cid:17) 1 i (f b)
ρ 3p 12bS2 + 0 − S = 0. (18)
− 3f − − 2 f 2
(cid:20) 0 (cid:21)
(cid:0) (cid:1)
The obtained expressions (14)–(15) for curvature functions A and A together with their
2 1
definition (2) give the generalization of cosmological Friedmann equations for HICM:
1 α
k/R2 +(H 2S )2 = ρ+6(f Z b)S2 + ρ 3p 12bS2 2
− 1 6f Z 0 − 2 4 − − 2
0
h i
3αεf (cid:0) 2 (cid:1)k
0 HS +S˙ +4 S2 S2 , (19)
− Z 2 2 R2 − 2 2
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:16) (cid:17)
1 α αε
H˙ +H2 2HS 2S˙ = ρ+3p ρ 3p 12bS2 2 ρ 3p 12bS2 S2
− 1− 1 −12f Z − 2 − − 2 − Z − − 2 2
0
h i
3αε(cid:0)f (cid:1)2 k(cid:0) (cid:1)
+ 0 HS +S˙ +4 S2 S2 . (20)
Z 2 2 R2 − 2 2
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:16) (cid:17)
These equations contain the torsion function S determined by (17) and the torsion function
1
S , satisfying the equation (18). Besides indefinite parameter α determining the scale of
2
extremely high energy densities [4], obtained equations contain additionally two indefinite
parameters: b with dimension of parameter f and the parameter ǫ without dimension. We
0
have to analyze all these equations in order to investigate HICM with pseudoscalar torsion
function in the frame of PGTG.
5
3 Asymptotics of cosmological solutions for HICM with
pseudoscalar torsion function
The structure of obtained equations describing HICM with pseudoscalar torsion function is
essentially more complicated in comparison with the case of HICM with S = 0. By using
2
the conservation law (12), the cosmological equation (19) can be written in the following
form:
2
3α 3α
H2 Z + Y +3αεf S2Z 6αH 2 Z + Y (2b εf )S S˙ εf S S˙ Z
2 0 2 − 2 − 0 2 2 − 0 2 2
" #
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:21)
k k
+36α2(2b εf )2S2S˙2 +3αεf S˙2 +4 S2 S2 Z + S2 Z2
− 0 2 2 0 2 R2 − 2 2 R2 − 2
(cid:20) (cid:18) (cid:19) (cid:21) (cid:18) (cid:19)
1 α
= ρ 6bS2 + ρ 3p 12bS2 2 Z. (21)
6f − 2 4 − − 2
0
h i
From this equation follows, that cosmological solutions possess(cid:0)extreme points (cid:1)for the scale
factor R. In fact, if we put that at t = 0 the Hubble parameter vanishes H = 0, then from
(21) follows the relation for S , its time derivative and energy density, which has to be valid
2
at a bounce:
k k
36α2(2b εf )2S2 S˙2 +3αεf S˙2 +4 S2 S2 Z + S2 Z2
− 0 20 20 0 20 R2 − 20 20 0 R2 − 20 0
(cid:20) (cid:18) 0 (cid:19) (cid:21) (cid:18) 0 (cid:19)
1 α
= ρ 6bS2 + ρ 3p 12bS2 2 Z . (22)
6f 0 − 20 4 0 − 0 − 20 0
0
h i
(cid:0) (cid:1)
(Valuesofvariousquantitiesatextremepointaredenotedbyindex0). Byusingthefollowing
equation of state p = ρ and by choosing some initial values for S , S˙ , ρ at a bounce,
20 20 0
particular cosmological solution for flat model (k = 0) presented in Fig. 1 - Fig. 2 was
obtained numerically. We see that the pseudoscalar torsion function S and the Hubble
2
parameterH havesomenon-vanishing valuesatasymptotics. Belowweanalyzethefollowing
question: by what restrictions on indefinite parameters b and ε asymptotical value for the
Hubble parameter corresponds to observable accelerating cosmological expansion?
By taking into account that various parameters of HICM have to be small at asymp-
totics, we see from (18), that if ǫ 1, the pseudoscalar torsion function has the following
≪
asymptotics:
f (f b) ρ 3p
S2 = 0 0 − + − . (23)
2 4fb 12b
Note, that at asymptotics one uses for gravitating matter the equation of state for dust
p = 0; then it follows from (23) that b f and we obtain at asymptotics: Z (b/f ),
0 0
≤ →
S 0. The cosmological equations (19)–(20) at asymptotics take the form of cosmological
1
→
Friedmann equations with cosmological constant:
k 1 3 (f b)2
+H2 = ρ+ 0 − , (24)
R2 6b 4f
" #
6
H S2
2
6
1.8
4
1.6
2
1.4
t
-2 -1 1 2 1.2
-2 t
-3 -2 -1 1 2 3
-4 0.8
-6 0.6
Figure 1: The Hubble parameter H(t) in di- Figure 2: The pseudoscalar torsion function
mensionless units. S (t) in dimensionless units.
2
1 3(f b)2
H˙ +H2 = ρ+3p 0 − . (25)
−12b − 2f
" #
FromEqs.(24)–(25)wesee, thatparameterbhastobevery closetof . Thevalueofbleading
0
to observable acceleration of cosmological expansion depends on the scale of extremely high
energy density defined by α−1. If we take into account that the value of 3(f b)2/f =
4 0 −
1α−1(1 b/f )2 has to be equal approximately to 0.7ρ (ρ = 6f H2, H is the value of
4 − 0 cr cr 0 0 0
the Hubble parameter at present epoch), then we obtain that b = [1 (2.8ρ α)1/2]f . If
cr 0
−
we suppose that the value of α−1 is greater than the energy density for quark-gluon matter,
but smaller than the Planckian one, then we can obtain the corresponding estimation for b,
which is extremely close to f .
0
4 Conclusion
As it was shown, the presence of pseudo-scalar torsion function in HICM built in the frame-
work of PGTG can lead to effective cosmological constant in asymptotics of cosmological
solutions and to observable accelerating cosmological expansion. The effect of acceleration
of cosmological expansion in PGTG has the geometrical nature and is connected with space-
time torsion. From the point of view of considered theory hypothetical form of gravitating
matter — dark energy or quintessence — is fiction. The further investigation of HICM with
pseudoscalar torsion function will be to continued.
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