Table Of ContentINTRINSIC TIME IN FRIEDMANN – ROBERTSON –
WALKER UNIVERSE
Alexander Pavlov
Bogoliubov Laboratory for Theoretical Physics,
Joint Institute of Nuclear Research,
Joliot-Curie street 6, Dubna, 141980, Russia;
Institute of Mechanics and Energetics
Russian State Agrarian University –
6
Moscow Timiryazev Agricultural Academy, Moscow, 127550, Russia
1
0 [email protected]
2
n
a
PACS 04.20.Cv
J
4
] 1 Global intrinsic time in FRW universe
c
q
- In Geometrodynamics there is a many-fingered intrinsic time as a scalar
r
g
field [1, 2, 3]. A global time exists in homogeneous cosmological models
[
(see, for example, papers [4, 5, 6, 7, 8, 9]). The observational Universe
1
with high precision is homogeneous and isotropic [10]. So, it is natural to
v
2 choose a compact conformally flat space of modern scale a as a background
0
7
space. It is necessary to study an applicability of the intrinsic global time
4
0 chosen to nearest non-symmetric cases by taking into account linear metric
0
perturbations. The first quadratic form
.
1
0 f = f (x)dxi dxj
6 ij
⊗
1
: in spherical coordinates χ,θ,ϕ
v
i
X a2 dχ2 +sin2χ(dθ2 +sin2θdϕ2) a2(1f )dxidxj (1)
r 0 ≡ 0 ij
a
should be chose(cid:2)n for a background space of(cid:3)positive curvature. We intro-
duced the metric of invariant space (1f ) with a curvature = 1,0. For
ij
K ±
a pseudosphere in (1) instead of sinχ one should take sinhχ, and for a flat
space one should take χ. The spatial metric is presented as
2
a
(γ ) = a2(t)(1f ) = e−2D(f ) = γ˜ . (2)
ij ij ij ij
a
(cid:18) 0(cid:19)
Hence, one yields an equality: γ˜ = f , so, in our high symmetric case, the
ij ij
conformal metric is equal to the background one.
1
The energy-momentum tensor is of the form
N2ρ 0
(T ) = ,
µν 0 pγ
ij
(cid:18) (cid:19)
where ρ is a background energy density of matter, and p is its pressure. We
have for the matter contribution
√γT⊥⊥ = (1f)a3ρ, (3)
p
because of
1 1 Ni
T⊥⊥ = nµnνTµν = N2T00 = ρ, nµ = N,−N .
(cid:18) (cid:19)
The functional of action in the symmetric case considered here takes the
form
t0 t0
dD
W(0) = dt d3xπD dt d3xN ⊥. (4)
− dt − H
Z Z Z Z
tI Σt tI Σt
Let us present results of necessary calculations been executed
1 dγ 1 dD
ij
K = = γ ,
ij ij
−2N dt N dt
3 dD
K = γijK = ,
ij
N dt
2
6 dD
K Kij K2 = ,
ij − −N2 dt
(cid:18) (cid:19)
(cid:0) (cid:1) 2
dD dD 12 dD
π = 4√γK = fe−3D ,
D
dt dt N dt
(cid:18) (cid:19)
6 6 p
R = Ke2D = K.
a2 a2
0
The Hamiltonian constraint
⊥ = √γ KijKij K2 √γR+√γT⊥⊥, (5)
H − −
(cid:0) (cid:1)
takes the following form
2
1 dD 1
⊥ = 6 fe−3D + Ke2D ρ .
H − N2 dt a2 − 6
" (cid:18) (cid:19) 0 #
p
2
The action (4) presents a model of the classical mechanics [1]
t0
2
6 dD 6
W(0) = V dt e−3D + KNe−D Ne−3Dρ (6)
0 −N dt a2 −
Z " (cid:18) (cid:19) 0 #
tI
after integration over the slice Σ , where
0
V := f d3x (7)
0
Z
Σ0 p
is a volume of the space with a scale a . We rewrite it in Hamiltonian form
0
t0
dD
W(0) = dt pD NH⊥ , (8)
dt −
Z (cid:20) (cid:21)
tI
where
1 6 V
H⊥ = −24V e3Dp2D − Ka2 0e−D +V0e−3Dρ (9)
0 0
is the Hamiltonian constraint as in classical mechanics. Here D(t) is a gen-
eralized coordinate, and p is its canonically conjugated momentum
D
δW(0) 12V dD
p = = 0e−3D . (10)
D δD˙ − N dt
(cid:18) (cid:19)
Resolving the constraint (9), we get the energy
6
p2 = E2(D), E(D) := 2√6V e−2D e−2Dρ K, (11)
D 0 s − a20
that was lost in Standard cosmology [11].
The equation of motion is obtained from the action (8):
e3D 6 V ∂ρ
p˙ = N p2 K 0e−D +3V e−3Dρ V e−3D . (12)
D 8V D − a2 0 − 0 ∂D
(cid:20) 0 0 (cid:21)
From the Hamiltonian constraint (9), the equation of motion (12), and
the energy continuity equation
a˙
ρ˙ = 3(ρ+p) , (13)
− a
3
we get Einstein’s equations in standard form:
2
a˙ 1
+ K = ρ,
a a2 6
(cid:18) (cid:19)
2
a¨ a˙ 1
2 + + K = p.
a a a2 −2
(cid:18) (cid:19)
Here we have used the normal slicing condition N = 1.
Substituting p from (10) into (11), we obtain the famous Friedmann
D
equation
2
1 dD 1
+ K = e−2Dρ. (14)
N2e2D dt a2 6
(cid:18) (cid:19) 0
From the Standard cosmology point of view, it connects the expansion rate
of the Universe (Hubble parameter)
a˙ dD
H := = (15)
a − dt
(cid:18) (cid:19)
with an energy density of matter ρ and a spatial curvature
2
dD 1
H2 = (ρ +ρ +ρ +ρ ). (16)
M rigid rad curv
≡ dt 6
(cid:18) (cid:19)
In the right side of equation (16), ρ is an energy density of nonrelativistic
M
matter
a 3
0
ρ = ρ ,
M M,0
a
(cid:16) (cid:17)
ρ is an energy density of matter
rigid
a 6
0
ρ = ρ ,
rigid rigid,0
a
(cid:16) (cid:17)
with a rigid state equation [12]
p = ρ,
ρ is an energy density of radiation
rad
a 4
0
ρ = ρ ,
rad rad,0
a
(cid:16) (cid:17)
ρ , which is defined as
curv
1
ρ := Ke2D = K,
6 curv −a2 −a2
0
4
is a contribution from the spatial curvature. In the above formulae ρ ;
M,0
ρ ; ρ ; ρ are modern values of the densities. Subsequently, the
rigid,0 rad,0 curv,0
equation for the energy (11) can be rewritten as
E(D) = 2√6V e−3D√ρ+ρ .
0 curv
The CDM model considered has not dynamical degrees of freedom. Accord-
ing to the Conformal cosmology interpretation, the Friedmann equation (16)
has a following sense: it ties the intrinsic time interval dD with the coor-
dinate time interval dt, or the conformal interval dη = dt/a. Let us note,
that in the left side of the Friedmann equation (16) we see the square of the
extrinsic York’s time which is proportional to the square of the Hubble pa-
rameter. If we choose the extrinsic time, the equation (16) becomes algebraic
of the second order, and the connection between temporal intervals should
be lost1.
In an observational cosmology, the density can be expressed in terms of
the present-day critical density ρ :
c
ρ (a) 6H2,
c ≡ 0
whereH isamodernvalueoftheHubbleparameter. Further,itisconvenient
0
to use density parameters as ratio of present-day densities
ρ ρ ρ ρ
M,0 rigid,0 rad,0 curv,0
Ω , Ω , Ω , Ω ,
M rigid rad curv
≡ ρ ≡ ρ ≡ ρ ≡ ρ
c c c c
satisfying the condition
Ω +Ω +Ω +Ω = 1.
M rigid rad curv
Then, one yields
1 a 3 a 6 a 4 a 2
H2 = ρ Ω 0 +Ω 0 +Ω 0 +Ω 0 .
c M rigid rad curv
6 a a a a
(cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
According to NASA diagram, 25% of the Universe is dark matter, 70% of
the Universe is dark energy about which practically nothing is known.
After transition to conformal variables
Ndt = a e−Ddη, ρ˜= e−4Dρ, (17)
0
the Friedmann equation takes the form
2
dD 1
+ = a N2e2Dρ˜. (18)
0
dη K 6
(cid:18) (cid:19)
1“The time is out of joint”. William Shakespeare: Hamlet. Act 1. Scene V. Longman,
London (1970).
5
2 Global time in perturbed FRW universe
Let us consider additional corrections to the Friedmann equation from non
ideal FRW model, taking into account metric perturbations. The metric of
perturbed FRW universe can be presented as
g = a2(t) 1f +h , (19)
µν µν µν
where (1f ) is the metric of spacetim(cid:0)e with the(cid:1)spatial metric (1f ) consid-
µν ij
ered above, deviations h are assumed small. The perturbation h is not a
µν µν
tensor in the perturbed universe, nonetheless we define
hν (1fνρ)h , hµν (1fµρ)(1fνσ)h .
µ ≡ ρµ ≡ ρσ
Foracoordinatesystem chosen inthebackgroundspace, therearevarious
possible coordinate systems in the perturbed spacetime. In GR perturbation
theory, a gauge transformation means a coordinate transformation between
coordinate systems in the perturbed spacetime. The coordinates of the back-
ground spacetime are kept fixed, the correspondence between the points in
the background and perturbed spacetime is changing. A manifestly gauge
invariant cosmological perturbation theory was built by James Bardeen [13],
and analyzed in details by Hideo Kodama and Misao Sasaki [14]. Now, keep-
ing the gauge chosen, id est the correspondence between the background
and perturbed spacetime points, we implement coordinate transformation
in the background spacetime. Because of our background coordinate sys-
tem was chosen to respect the symmetries of the background, we do not
want to change our slicing. Eugene Lifshitz made decomposition of pertur-
bations of metric and energy – momentum tensor into scalar, vortex, and
tensor contributions refer to their transformation properties under rotations
in the background space in his pioneer paper [15]. The scalar perturbations
couple to density and pressure perturbations. Vector perturbations couple
to rotational velocity perturbations. Tensor perturbations are nothing but
gravitational waves.
In a flat case ( = 0) the eigenfunctions of the Laplace – Beltrami
• K
operator are plane waves. For arbitrary perturbation f(t,x) we can make an
expansion
∞
f(t,x) = fk(t)eık·x
k=0
X
over Fourier modes. We can consider them in future as a particular case of
models with constant curvature. Let us proceed the harmonic analysis of
6
linear geometric perturbations using irreducible representations of isometry
group of the corresponding constant curvature space [16, 17].
Scalar harmonic functions.
•
Let us define for a space of positive curvature = 1 an invariant space
K
with the first quadratic form
(1f )dxidxj = dχ2 +sin2χ(dθ2 +sin2θdϕ2). (20)
ij
The eigenfunctions ofthe Laplace– Beltrami operator forma basis of unitary
representations of the group of isometries of the three-dimensional space Σ
(20) of constant unit curvature. In particular, the eigenfunctions on a sphere
S3 are the following [18]:
1 λ(λ+l)!
−l−1/2
Y (χ,θ,ϕ) = P (cosχ)Y (θ,ϕ). (21)
λlm √sinχs(λ l +1)! λ−1/2 lm
−
Here, Pν(z) are attached Legendre functions, Y (θ,ϕ) are spherical func-
µ lm
tions, indices run the following values
λ = 1,2,...; l = 0,1,...,λ 1; m = l, l +1,...,l.
− − −
There is a condition of orthogonality and normalization for functions (21)
π π 2π
dχsin2χ dθsinθ dϕYλ∗lm(χ,θ,ϕ)Yλ′l′m′(χ,θ,ϕ) = δλλ′δll′δmm′. (22)
Z Z Z
0 0 0
In a hyperbolic case with negative curvature = 1 the eigenfunctions are
K −
following [18]:
1 Γ(ıλ+l +1) −l−1/2
Y (χ,θ,ϕ) = | |P (coshχ)Y (θ,ϕ), (23)
λlm √sinhχ Γ(ıλ) ıλ−1/2 lm
| |
where 0 λ < ;l = 0,1,2,...;m = l, l +1,...,l. There is a condition
≤ ∞ − −
of orthogonality and normalization for functions (23)
Λ π 2π
dχsinh2χ dθsinθ dϕYλ∗lm(χ,θ,ϕ)Yλ′l′m′(χ,θ,ϕ) = δ(λ−λ′)δll′δmm′,
Z Z Z
0 0 0
where Λ is some cut-off limit.
7
The equation on eigenvalues can be presented in the following symbolic
form
((1∆¯)+k2)Y(s) = 0, (24)
k
where k2 is an eigenvalue of the Laplace – Beltrami operator (1∆¯) on Σ.
The co−nnection (1¯) is associated with the metric of invariant space (20).
∇
For a flat space ( = 0) the eigenvectors of the equation (24) are flat waves
K
as the unitary irreducible representations of the Euclidean translation group.
For a positive curvature space ( > 0) we have k2 = l(l+2), and for a space
K
of negative curvature ( < 0) we have k2 > 1. The scalar contributions of
K
the vector and the symmetric tensor fields can be expanded in terms of
1
Y(s) : = (1¯ )Y(s), (25)
k,i −k ∇i k
1 1
Y(s) : = (1¯ )(1¯ )Y(s) + (1f )Y(s). (26)
k,ij k2 ∇i ∇j k 3 ij k
Different modes do not couple in the linearized approximation. So we are
able to consider a contribution of a generic mode.
Let us start with considering scalar modes because of their main con-
tribution to galaxy formation. Linear perturbations of the four-metric in
terms of the Bardeen gauge invariant potentials Ψ(t) and Φ(t) in conformal–
Newtonian gauge are of the
2 2
ds2 = N2 1+ΨY(s) dt2 +a2 1 ΦY(s) (1f )dxidxj, (27)
− D k − k ij
(cid:16) (cid:17) (cid:16) (cid:17)
where a summation symbol over harmonics is omit, N is the Dirac’s lapse
D
function defined as
(s)
N N 1+ΨY . (28)
D k
≡
(cid:16) (cid:17)
The spatial metric is presented as a sum of the Friedmann metric (2) con-
sidered in the previous section and a perturbed part
(γ ) = a2(t)(1f ) 2a2(t)ΦY(s)(1f ). (29)
ij ij k ij
−
The determinant of the spatial metric (29) in the first order of accuracy is
det(γ ) = a6(t) 1 6ΦY(s) det(1f ). (30)
ij k ij
−
(cid:16) (cid:17)
For the high symmetric Friedmann case, the background metric (1) coincides
with the conformal one: f = γ˜ because of
ij ij
1/3 2
γ a(t)
= 1 2ΦY(s) = e−2D(t) 1 2ΦY(s) . (31)
k k
f a − −
(cid:18) (cid:19) (cid:18) 0 (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
8
One obtains from relations (29), (31) the connection between components of
perturbed metric tensor and the background one
(γ ) = e−2D(t) 1 2ΦY(s) (f ), (γij) = e2D(t) 1+2ΦY(s) (fij).
ij k ij k
−
(cid:16) (cid:17) (cid:16) (cid:17) (32)
Now, we calculate components of the tensor of extrinsic curvature
1 dD dD dΦ
(s)
K = (Ψ+2Φ) Y γ , (33)
ij k ij
N dt − dt − dt
D (cid:20) (cid:18) (cid:19) (cid:21)
and its trace
3 dD dD dΦ
K = γijK = (Ψ+2Φ) Y(s) . (34)
ij k
N dt − dt − dt
D (cid:20) (cid:18) (cid:19) (cid:21)
Then we obtain
2
6 dD dD dD dΦ
(K Kij K2) = 2 (Ψ+2Φ) Y(s) .
ij − −N2 dt − dt dt − dt k
D "(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) #
From the relation between metric tensors (32) one obtains the relation be-
tween the corresponding Jacobians
√γ = e−3D(t) 1 3ΦY(s) f.
k
−
(cid:16) (cid:17)p
The calculation gives
2
dD dD 12 dD
π = 4√γK = e−3D f
D
dt dt N dt −
D (cid:18) (cid:19)
p
12 dD dD dΦ
e−3D f (Ψ+5Φ) Y(s).
k
− N dt dt − dt
D (cid:18) (cid:19)
p
We need to the perturbation of the Ricci scalar additionally. According to
the Palatini identity [15] from the differential geometry, a variation of the
Ricci tensor is set by the formula
1
δR = δγn + δγn δγn ∆δγ . (35)
ij 2 ∇n∇j i ∇n∇i j −∇j∇i n − ij
(cid:0) (cid:1)
Here the Levi–Civita connection is associated with the Friedmann metric
i
∇
γ (2). The metric variations are expressed through harmonics (29)
ij
δγ = 2a2(t)ΦY(s)(1f ), (36)
ij k ij
−
9
the indices of the metric variations are moved up with respect to the back-
ground metric
δγn = γnjδγ = 2ΦY(s)δn, δγn = 6ΦY(s). (37)
i ji − k i n − k
Substituting them into the (35), one obtains
(s)
δR = Φ[( )+ +γ ∆]Y .
ij j i i j j i ij k
∇ ∇ −∇ ∇ ∇ ∇
Using the Laplace – Beltrami equation (24), commutativity of the covariant
differentiation operators, and properties of harmonics (25), (26), one gets
4
δR = Φk2 Y(s) 1f Y(s) . (38)
ij k,ij − 3 ij k
(cid:20) (cid:21)
(cid:0) (cid:1)
Remark the connection between the operators, used above (∆) = 1∆¯ /a2.
For getting the variation of the Ricci scalar we make summation
(cid:0) (cid:1)
4
δR = γijδR = Φk2Y(s),
ij −a2 k
i(s)
where we used the property of traceless of the tensor Y = 0. Finally, we
k,i
yield
6 4
√γR = K fe−D(t) fe−D(t)Φk2Y(s). (39)
a2 − a2 k
0 0
p p
Letusconsiderthefirstordercorrectionstotermsoftheaction(4), taking
into account the connection between lapse functions (28),
2
6 dD
N√γ K Kij K2 = e−3D f (40)
ij
− −N dt −
D (cid:18) (cid:19)
(cid:0) (cid:1) 2 p
6 dD dD dΦ
e−3D f (Ψ+7Φ) +2 Y(s).
k
− N − dt dt dt
D " (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)#
p
The curvature term after the correction is the following
2
N√γR = N fe−D 3 +(3 Ψ 2k2Φ)Y(s) . (41)
a2 D K K − k
0
p (cid:16) (cid:17)
The matter term with the correction has a form
N√γT⊥⊥ = NDe−3D fρ 1+(Ψ 3Φ)Yk(s) . (42)
−
p (cid:16) (cid:17)
10
