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27
UDC 3937
On Multidimensional Cosmology with Anisotropic Fluid: Asymptotical Acceleration and Zero Variation of G
A. G. Pakhomov
Institute of Gravitation and Cosmology Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, Russia, 117198
A multidimensional cosmological model describing the dynamics of n +1 flat factor-spaces Mi in the presence of a one-component anisotropic fluid is offered. The pressures in all spaces are proportional to the density: pi = Wip, i = 0,. .., n. Solutions with accelerated expansion of our 3-space M0 and zero variation of the gravitational constant G are studied. These solutions exist for two branches of the parameter w0: The first branch describes the matter with wo > 1, the second one may contain phantom matter with w0 < -1. It is shown that these solutions are special case of more general solutions with accelerated expansion of our 3-space M0 and asymptotically zero variation of the gravitational constant G.
The model of an ideal many-dimensional substance with three isotropic dimensions of our space, additional dimensions and time is considered. Spacelike dimensions are presented by the power metric depending on parameters of an equation of state. It is shown, that association of dynamic parameters of our three-dimensional space on additional dimensions in the open view may be expressed through coefficient of anisotropy of additional dimensions. Dependence from parameter of an equation of state of our isotropic 3-dimensional space to coefficient of anisotropy of the additional dimensions, requiring the accelerated expansion of the Universe is received in an explicit aspect. The received association is presented pictorially.
Key words and phrases: multidimensional gravitation, anisotropic fluid, accelerated expansion, variation of G.
1. Introduction
This paper deals with a possible temporal variation of gravitational constant G. This problem arose due to papers of Milne (1935) and Dirac (1937). In Russia, these ideas were developed in the 60s and 70s by K.P. Staniukovich [1,2], who was the first to consider simultaneous variations of several fundamental constants.
The first calculations based on general relativity with a perfect fluid and a confor-mal scalar field [3] gave G/G at the level of 10-11 — 10-13 per year. The calculations in string-like [4] and multidimensional models with perfect fluid [5] led to the level 10-12, those based on a general class of scalar-tensor theories [6] and simple multidimensional model with branes [7, 8] led for the present values of cosmological parameters 10-13 — 10-14 and 10-13 per year, respectively. Similar estimations were made by Miyazaki within Machian theories [9] giving for G/G the estimate 10-13 per year and by Fujii — on the level 10-14 — 10-15 per year [10]. Analysis of one more multidimensional model with two curvatures in different factor spaces gave an estimate on
the level 10-12 .
Here we continue our studies (see [11]) on variation of G in multidimensional cosmological model with 1-component "perfect-fluid" matter. We show that the class of solutions from [11] may be enlarged by more general solutions with accelerated expansion of 3-dimensional space M0 and asymptoticaly zero variation of the gravitational constant G.
2. The Model
We consider a cosmological model describing the dynamics of n flat spaces in the presence of a 1-component "perfect-fluid" matter [12]. The metric of the model
n
g = - exp[27(t)]dt ® di + ^ exp[2^(i)]^ (1)
¿=0
is defined on the manifold
M = R x M0 x Mi x ... x Mn, (2)
where M¿ with the metric (f is a flat space of dimension d¿, i = 0,... ,n; n ^ 2. The multidimensional Hilbert-Einstein equations have the form
tjM 1 XM TJ 2rpM KN - ^ dN K = K 1N ,
where k2 is the gravitational constant, and the energy-momentum tensor is adopted as (TM) = diag (-p, p0^k°,..., Pn^k™), describing, in general, an anisotropic fluid.
We put pressures of this "perfect" fluid in all spaces to be proportional to the density,
p¿(t) = (1 - Ui/<h)p(t), (3)
where ^ = const, i = 0,... ,n. We also put p > 0.
We impose also the following restriction on the vector u = £ R n:
(u,u) = 0. (4)
Here, the bilinear form (.,.) : Rn x Rn ^ R is defined by the relation
(u,v) = G^^Vj, (5)
u,v £ Rn+i, where
1
G" = * + 2-D ■ <6>
are components of the matrix inverse to the matrix of the minisuperspace metric [13,14]
(^¿j = d¿5¿j — ¿¿dj. (7)
In (6), D = 1 + n=0 ^ is the total dimension of the manifold M from (2). The inequality (4) reads
n i 1 / n \2
M = £ ^ - cM ^ ") = °' <8)
=0 - =0
3. Solutions with Power-Law Scale Factors
Here, we consider a special family of "power-law" solutions from [12,15] with the metric written in the synchronous time parametrization
n
g = —dis » dis + ^ al {ts)g\ (9)
¿=0
Here ts is cosmological (synchronous) time variable.
A special class of solutions with a power-law behaviour of the scale factors take place for
( u(A) -u,u) = 0. (10)
Here and below the vector
4A) = 2di (11)
corresponds to the A-term fluid with pi = —p.
In this case, the solutions are determined by the metric (9) with the scale factors
ai = ai(ta) = Aitl , (12)
and the density
_ _
(u(A) — u,u)212 .
«V =,.2";-"',,,. (13)
Here
v* = 2 ui/{ u(A) —u,u), (14)
where u% = G^u and Ai are positive constants, i = 0,... ,n.
The model under consideration was integrated in [12] for { u, u) < 0. The solutions from [12] were generalized in [15] to the case when a massless minimally coupled scalar field was added.
4. Acceleration and Variation of G
In this section, the metric g° is assumed to be 3-dimensional flat metric, i.e. d0 = 3. The subspace (M0, g°) describes "our" 3-dimensional space and (Mi, gl) internal factor-spaces. One may consider the special case di = 1, i = 1,... ,n, as it was done in [11]. See also [16].
We are interested in solutions with accelerated expansion of our space and small enough variations of the gravitational constant obeying the present experimental constraints, see [17]:
|G/(GH)l(tso) < 0.001, (15)
where
H = ao (16)
ao
is the Hubble parameter. We suppose that the internal spaces are compact. Hence our 4-dimensional constant is (see [5])
n
G = const ■ H(a-dl). (17)
i=1
We will use the following explicit formulae for the contravariant components:
1 n
« = a""l = I + 2—D .
3 = 0
and the scalar product reads
n I "\2 o n 1 ( n \2
—«.«) =—£^ + ^+ TT-2 • <18>
¿=0 1 i=0 \i=0 /
4.1. Power-Law Expansion with Acceleration
For solutions with power-law expansion, an accelerated expansion of our space takes place for
p0 > 1. (19)
For D = 4, when internal spaces are absent, we get
v0 = 2/(6 - uo), (20)
(u(A) - u,u) = 1(u- - 6)uo = 0, (21)
6
which implies u0 = 0 and u0 = 6 (here (u,u)* = — 6u0 < 0). The condition v0 > 1 is equivalent to 4 < u0 < 6, or, equivalently, — p < p < —p/3, which agrees with the well-known result for D = 4. (We note that special 5-dimensional power-law solutions (e.g., with acceleration) were considered in [18]). For power-law solutions we get
n
C E-, Vldl d vo
G 3=1 H = * = ^, (22)
G ts ao t,
and hence
1 n
G/(GH) = - = 5. (23)
V 3=1
The constant parameter 5 describes variation of the gravitational constant and, according to (15),
|£| < 0.001. (24)
It follows from the definition (14) that
1 n
5 = - ^V u%d%. (25)
u0 ^
%=i
or, in terms of covariant components
n
(D - 4)uo - 2 E ui
5 =--^n^. (26)
3(5 - D)uo + X] ui
i=i
4.1.1. The Case of Constant G
Consider the most important case 5 = 0, i.e., when the variation of G is absent: G = 0.
We note that according to arguments of [19], 5 < 10-4. This constraint just follows from the identity
G/G = a/a, (27)
where a is the fine structure constant.
Isotropic case. Let us consider the isotropic case when the pressures coincide in all internal spaces. This takes place when
Ui = vdi, i = 1,...,n. (28)
For pressures in internal spaces we get from (3)
Pi = (1 - v)p, i = 1,...,n. (29)
Then we get from (8) and (18)
{u, u) =
2-D
— -(d — 1)uo + 2du0v — 2dv2 3
(30)
( u(A) -u,u) = —^ [2uo + 2dv + 1(d — 1)u0 - 2du0v + 2dv2]. (31) 2 — D 3
Here we denote d = D — 4.
For <5 = 0, we get in the isotropic case
v = uo/2, (32)
or, in terms of pressures,
Pi = (3po — p)/2, i = 1,...,n. (33)
Substituting (32) into (30) and (31) we get ( u,u) = — u0/6, (u(A) —u,u) = u0(u0 — 6)/6.
Thus, we obtain the same relations as in D = 4 case (see the remark above). For our solution, we should put u0 = 0 and u0 = 6.
Using (28) and (32) we get u0 = —u0/6 and u* = 0 for i > 0, hence Vi = 0 for i = 1,... ,n, i.e., all internal spaces are static.
The metric (9) reads in our case
n
g = — dis ® dis + Altg0 + £ A*g', (34)
i=1
where Ai are positive constants, and
v0 = 2/(6 — u0). (35)
We see that the power v0 is the same as in case D = 4. For the density we get from (13)
12
n2p = (u0 — 6)2*2 . (36)
Thus the equations of state (3), with relations (28) and (32) imposed, lead to the solution (34)-(36) with flat 3-metric and n static internal flat spaces. For
4 < u0 < 6 (37)
or, equivalently, — p < p0 < — p/3, we get an accelerated expansion of "our" 3-dimensional flat space.
Anisotropic case. Consider the anisotropic (w.r.t. internal spaces) case with = 0, or, equivalently (see (26)),
n
( D — 4)u0 = 2 (38)
i=1
This implies
{u(A) - u,u) = 1 u°(u° - 6) - A, (39)
6
{u,u) = -1 ul + A, (40)
6
where
n 2 1 / n \ 2
A = E | -1 E *) > 0 « = * -4. №
The inequality in (41) can be readily proved using the well-known Cauchy-Schwarz inequality:
(L »2)(L «*) > g »4- (42)
A = 0 only in the isotropic case (28). In the anisotropic case we get A > 0. Thus A is anisotropy parameter.
In what follows we will use the relation
(m(a) - u,u) = \(u° - M+)(«0 - u°), (43)
6
where u± = 3 ± V9 + 6A are roots of the quadratic polynomial (39) obeying u° < 0, u+ > 6 for A > 0. It follows from (38) that u0 = -u0/6 and hence
=---r (44)
uo(uo - 6) - 6A v ;
(here u° = u±).
The accelerated expansion of our space takes place when v0 > 1, or, equivalently, when either
(A) u° e (u°*,u°), or (B) u° e (u+*,u+), (45)
where u±* =2 ± V4 + 6A. In terms of the parameter w°,
p° = w°p, w° = 1 - u°/3, (46)
these two branches read:
2 12/3
(A) w- = ^/1 + 3A <w0 < 3 + Wl + 2A = (47)
(B) = -/1 + |A <w0 < 1 - 3 Jl + 3A = <. (48)
The range w° when the accelerated expansion (A, B) is realized is shown in Figure 1.
The first branch (A) describes a matter with w° > 1 and p < 0 (it follows from (13) and {u, u) > 0.
The second branch (B) corresponds to matter with a broken weak energy condition (since w° < -3) and p > 0 (since {u, u) < 0). This matter is phantom (i.e., w° < -1) when A > 2.
Accelerated expansion of the Universe
Figure 1. Dependence from parameter of an equation of state of our isotropic 3-dimensional space w0 to coefficient of anisotropy of the additional dimensions A. Two branches of ranges of the solutions (A, B), realizing accelerated expansion are shown
5. Solutions with Acceleration and Asymptotically
Constant G
It may be shown that power-law solutions from section 2 obeying restrictions (4), (10), {u, u) < 0 and
( u,uA)/{u,u) > 1 (49)
are attractor solutions for more general class of solutions from [12] (for solutions with a massless scalar field see [15]).
So, in the case when restrictions (4), (10) and (49) take place we get
ai = ai(ts) , (50)
and
2 -2{ u,u)
KP ~{u(A) — u, u)211 , (51)
as ts ^ instead of (12) and (13).
When restriction on u parameters (38) is valid we get from (17)
G ~Go = const, (52)
as ts ^
Moreover, it may be shown that for any solution obeying restrictions (4), (10) and (49) there exists T > 0 such that for ts G (T, +ro) we get an accelerated expansion of our 3-space M0 and small enough variation of the effective gravitational constant G(ts), obeying the observational restriction (15).
6. Conclusions
We have considered multidimensional cosmological models describing the dynamics of n + 1 flat factor spaces Mi in the presence of a one-component anisotropic fluid with pressures in all spaces proportional to the density: pi = Wip, i = 0,... ,n. Solutions with accelerated expansion of our 3-dimensional space M0 and zero variation of the gravitational constant G were considered. These solutions exist for two branches of
the parameter w0: (A) and (B). Branch (A) describes superstiff matter with w0 > 1 while branch (B) may contain phantom matter with w0 < -1.
We have shown that these solutions are special case of more general solutions with accelerated expansion of our 3-space M0 and asymptotically zero variation of the gravitational constant G, i.e. G(ts) ^ const for ts ^ (ts is a synchronous time variable).
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УДК 3937
Многомерная космологическая модель с анизотропной жидкостью: ассимптотическое ускорение и нулевая
вариация С
А. Г. Пахомов
Учебно-научный институт гравитации и космологии Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198
Предлагается многомерная космологическая модель, описывающая динамику п + 1 плоских пространств Мг в присутствии однокомпонентной анизтропной жидкости. Давление во всех пространствах пропорциональны плотности: р^ = "Шгр, г = 0,... ,п. Изучаются решения с ускоренным расширением нашего трёхмерного пространства М0 и нулевой вариацией гравитационной постоянной С. Эти решения существуют для двух ветвей параметра ю0: первая ветвь описывает материю с ю0 > 1, вторая может содержать фантомную материю с ю0 < — 1. Показано, что эти решения являются частным случаем более общих решениий с ускоренным расширением нашего трёхмерного пространства М0 и асимптотически нулевой вариацией гравитационной постоянной С.
Рассмотрена модель идеальной многомерной субстанции с тремя изотропными измерениями нашего пространства, дополнительными измерениями и временем. Пространственные измерения представлены степенной метрикой, зависящей от параметров уравнения состояния. Изучаются плоские фактор-пространства с однокомпонентной идеальной субстанцией. Получена в явном виде зависимость параметра уравнения состояния нашего изотропного 3-мерного пространства от коэффициента анизотропии дополнительных измерений, требующая ускоренного расширения Вселенной. Полученная зависимость представлена графически.
Ключевые слова: многомерная гравитация, анизотропная жидкость, ускоренное расширение, вариация С.
