Cosmic acceleration and the change of the Hubble parameter

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by  G. Y. Chee

Cosmic acceleration and the change of the Hubble parameter
G. Y. Chee1,∗
1College of Physics and Electronics, Liaoning Normal University, Dalian,
116029, China
Anewmodelofacceleratingexpansionoftheuniverseispresented. AuniversalvacuumdeSitter
solution is obtained. A new explanation of the acceleration of the cosmic expansion is given. It is
3
provedthatthechangingoftheexpansionfrom deceleratingtoaccelerating isanintrinsicproperty
1
0 of the universe without need of an exotic dark energy. The cosmological constant problem, the
2
coincidence problem and theproblem of phantom divideline crossing are naturally solved.

n
a
J PACSnumbers: 04.50.Kd,98.80.-k

5 Keywords: Modifiedgravity;Cosmicacceleration
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]
h
p
- I. Introduction
n
e
g
Very recently, there have been proposals for constructing generalizations of teleparallel gravity in Refs. [1], [2] and
.
s
c [3] which followed the spirit of f(R) gravity (see [4] for a review) as a generalization of general relativity. That is,
i
s the Lagrangians of the theories were generalized to the form f(T), where f is some suitably differentiable function
y
h and T is the Lagrangianof teleparallel gravity [5]. The interest in these theories was arousedby the claim that their
p
[ dynamicsdiffer fromthoseofgeneralrelativitybut theirequationsarestillsecondorderinderivativesand,therefore,

3 they might be able to account for the accelerated expansion of the universe and remain free of pathologies. It has
v
been shown, however, that this last expectation was unfounded: these theories are not locally Lorentz invariant and
5
6 appear to harbour extra degrees of freedom [6]. Even if one decides to give up teleparallelism, such actions would
8
0 not make sense as descriptions of the dynamics of gravity if local Lorentz symmetry was restored [7]. Based on the
.
1 definition given in these theories, the Weitzenbock connection, the torsion tensor and the contortion tensor are not
0
3 local Lorentz scalars (i.e. they do not remain invariant under a local Lorentz transformation in tangent space). This
1
is the root of the lack of Lorentz invariance in generalized teleparallel theories of gravity. The f(T) models behave
:
v
i very differently from the ΛCDM model on large scales,and are, therefore, very unlikely to be suitable alternatives to
X
it [8]. Since this theory is not invariant under local Lorentz transformations, the choice of tetrad plays a crucial role
r
a
in such models. Different tetrads will lead to different field equations which in turn have different solutions [9].
It is known that the Lagrangian T of teleparallel gravity only differs from the Riemann curvature scalar R by
a boundary term[7, 8]. Recall that in general relativity the Einstein field equation can be derived from a reduced
Lagrangian Γ = ( µ ν ν µ )gρσ which only differs from the Riemann curvature scalar R of the
σ ν ρ µ µ ν σ ρ
{ }{ }−{ }{ }
Christoffel symbol µ by a boundary term wµ with wµ = gνλ µ gνµ λ [10]. Following the spirit of
σ ν µ ν λ ν λ
{ } ∇ { }−
f(R) gravity we can propose a modified theory of gravity called reduced f(R) g(cid:8)ravi(cid:9)ty or f(Γ) gravity given by
∗Electronicaddress: [email protected]
2
a Lagrangian which is a function of Γ: = f(Γ). It will be seen in this letter that this theory is identified as
G
L
a metrical formulation of f(T) gravity. In contrast with f(T) gravity, the new theory, f(Γ) gravity respects local
Lorentz symmetry and harbour no extra degreesof freedom, since the dynamicalvariable is the metric instead of the
tetrad. At the same time it has the advantage over f(R) gravity that its field equations are second-order instead of
fourth-orderandthenisfreeofpathologies. Thedisadvantageoff(Γ)gravityisthatitsfieldequationsdonotrespect
general covariance. The significant pay-off that will be suggested in this letter is that f(Γ) gravity may provide an
satisfactory alternative to conventional dark energy in general relativistic cosmology and a new explanation of the
acceleration of the cosmic expansion. The changing of the expansion from decelerating to accelerating is an intrinsic
property of the universe without need of an exotic dark energy. The cosmological constant problem, the coincidence
problem and the problem of phantom divide line crossing are naturally solved.
II. Field equations
We start from the reduced action
1
S = dΩ √ gf(Γ)+L , (1)
2κ2 − m
Z
(cid:0) (cid:1)
where κ2 =8πG with the bare gravitational constant G ,
N N
Γ=( µ ν ν µ )gρσ, (2)
σ ν ρ µ µ ν σ ρ
{ }{ }−{ }{ }
is the reduced gravitational Lagrangianof general relativity [10] with the Christoffel symbol µ . Γ is a quadratic
ν σ
{ }
formofthegravitationalstrength µ andidentifiedwiththekineticenergyofthegravitationalpotentialg . The
ν σ µν
{ }
variational principle yields the field equations for the metric g :
µν
1 1 ∂Γ
f R Rg + g (f Γ f(Γ)) f Γ =κ2T . (3)
Γ ρσ− 2 ρσ 2 ρσ Γ − − ΓΓ ,µ∂gρσ ρσ
(cid:18) (cid:19) ,µ
where
∂f ∂2f
f = ,f = , (4)
Γ ∂Γ ΓΓ ∂Γ2
R is the Ricci curvature tensor of µ and
ρσ σ ν
{ }
2 δL
m
T = , (5)
ρσ −√ gδgρσ
−
is the energy-momentum of the matter fields. These equations can be re-arrangedin the Einstein-like form
1 1 f ∂Γ κ2
ΓΓ
R Rg = (f f Γ)g + Γ + T . (6)
µν − 2 µν 2f − Γ µν f ,λ∂gµν f µν
(cid:18) (cid:19) Γ Γ ,λ Γ
III. Cosmological model
We now investigate the cosmologicaldynamics for the models based on f(Γ) gravity. In order to derive conditions
forthe cosmologicalviability off(Γ)models weshallcarryouta generalanalysiswithoutspecifying the formoff(Γ)
at first. We consider a flat Friedmann-Lemaitre-Robertson-Walker(FLRW) backgroundwith the metric
2 2 2
g =diag 1,a(t) ,a(t) ,a(t) , (7)
µν
−
(cid:16) (cid:17)
3
where a(t) is a scale factor. The non-vanishing components of the Christoffel symbol are
·
0 = 0, 0 = 0 =0, 0 =aaδ ,
0 0 0 i i 0 i j ij
(cid:8) i (cid:9) = 0,(cid:8) i (cid:9)=(cid:8) i (cid:9)=Hδ(cid:8)i, (cid:9)i =0,i,j,k,...=1,2,3. (8)
0 0 j 0 0 j j j k
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
and
Γ=( µ ν ν µ )gρσ = 6H2. (9)
σ ν ρ µ µ ν σ ρ
{ }{ }−{ }{ } −
·
Here H a /a is the Hubble parameter and a dot represents a derivative with respect to the cosmic time t. One
≡
can find that Γ is identified with the gravitational Lagrangian of the cosmological model of f(T) gravity given by
Bengochea and Ferraro [1]. The the field equations (6) take the forms
1 · f κ2
3H2 = f +6H2fΓ 18H2 H ΓΓ + ρ, (10)
−2f − f f
Γ Γ Γ
(cid:0) (cid:1)
· 1 · f κ2
2H 3H2 = f +6H2fΓ 18H2 H ΓΓ + p, (11)
− − 2f − f f
Γ Γ Γ
(cid:0) (cid:1)
which lead to
· κ2 36 ·
H=−2f ρ+p− κ2H2 H fΓΓ . (12)
Γ (cid:18) (cid:19)
It is easy to see that, in f(Γ) gravity, the gravitational constant κ2 is replaced by an effective (time dependent)
κ2 = κ2/f (Γ). On the other hand, it is reasonable to assume that the present day value of κ2 is the same as the
eff Γ eff
κ2 so that we get the simple constraint :
κ2 (z =0)=κ2 f (Γ )=1, (13)
eff → Γ 0
where z is the redshift.
If f(Γ) has the form
f(Γ)=Γ+Φ(Γ), (14)
then the equations (10), (11) and (12) become
3H2 =κ2(ρ+ρ ), (15)
de
·
2H 3H2 =κ2(p+pde), (16)
− −
and
··
a κ2
= ((ρ+ρ )+3(p+p )), (17)
de de
a − 6
where
1 6 18 ·
ρde =−2κ2Φ− κ2H2ΦΓ− κ2H2 H ΦΓΓ, (18)
4
and
1 6 2 ·
pde = 2κ2Φ+ κ2H2ΦΓ+ κ2 H ΦΓ−9H2ΦΓΓ , (19)
(cid:0) (cid:1)
are the density and the pressure of the ”dark energy” Φ(Γ), respectively. The state equation of the ”dark energy” is
then
·
pde 2H ΦΓ 18H2ΦΓΓ
w = = 1 − . (20)
de ·
ρde − − 12Φ+6H(cid:0)2ΦΓ+18H2 H(cid:1)ΦΓΓ
·
Let N =lna. For any function Φ(a) we have Φ′ =: dΦ =H Φ. The the equation (20) becomes
dN
′
1Γ Φ +3ΓΦ
Γ ΓΓ
w == 1+ . (21)
de − 3 Γ Φ/Γ 2Φ + 1Γ′Φ
− Γ 2 ΓΓ
The equations (15) and (16) become
κ2 1 1 ·
H2 = ρ Φ+ ΓΦΓ+ΓH ΦΓΓ, (22)
3 − 6 3
H2 ′ = 2κ2p−Γ+Φ−2ΓΦΓ. (23)
− 3ΓΦ +2Φ +2
ΓΓ Γ
(cid:0) (cid:1)
We see that a constant Φ acts just like a cosmologicalconstant (dark energy), and Φ linear in Γ (i.e. Φ =constant)
Γ
is simply a redefinition of gravitationalconstant κ2. The the equation (23) can be written as
1 3 Γ 1
1+Φ ΓΦ Γ′ = + Φ ΓΦ +κ2p. (24)
Γ ΓΓ Γ
6 − 2 −2 2 −
(cid:18) (cid:19)
Taking a universe with only dust matter so
p=0, (25)
we have
1+Φ 3ΓΦ
Γ− 2 ΓΓ dΓ=dN, (26)
− 3Γ(1 Φ/Γ+2Φ )
Γ
−
and then we find the solution Γ(a) in closed form:
1 Γ dx1+Φ (x) 3xΦ (x)
a(Γ)=exp x − 2 xx . (27)
(−3Z−6H02 x 1−Φ(x)/x+2Φx(x) )
The equations (22), (23) and (27) take the same forms as the ones for f(T) gravity given by Linder [2]. In a sense
our model can be considered as a metric formulation of the f(T) model.
The equations (22) and (23) i.e.
1 ·
3H2 = Φ 6H2ΦΓ 18H2 H ΦΓΓ+κ2ρ. (28)
−2 − −
· 1 ·
2H 3H2 = Φ+6H2ΦΓ+2H ΦΓ 9H2ΦΓΓ +κ2p. (29)
− − 2 −
(cid:0) (cid:1)
yield
· κ2
H= (ρ+p). (30)
−2(1+ΦΓ 18H2ΦΓΓ)
−
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So, κ2 is simply a redefinition of gravitationalconstant κ2.
1+ΦΓ−18H2ΦΓΓ
In the vacuum (30) gives a universal de Sitter solution
·
H=0, (31)
for any function Φ(Γ). Then (20) gives
w = 1, (32)
de
−
which means that the function Φ(Γ) plays the role of the cosmological constant or dark energy.
The equation (29) gives
· κ2p+3H2+ 1Φ+6H2Φ
H= 2 Γ. (33)
− 2(1+Φ 9H2Φ )
Γ ΓΓ
−
Substituting (33) into (20) yields
Φ+6H2Φ +54H4Φ +2κ2 9H2Φ Φ p
Γ ΓΓ ΓΓ Γ
w = − . (34)
de −(1+Φ )(Φ+12H2Φ ) 18H2Φ (12H2Φ +3H2+Φ) 18H2Φ κ2p
Γ Γ − ΓΓ (cid:0)Γ (cid:1)− ΓΓ
In the case
Φ(Γ)=α( Γ)n =α6nH2n, (35)
−
(33) and (34) become
· κ2p+3H2+ 1 n α6nH2n
H= 2 − , (36)
−2(1+3n(3n 5)α6n−2H2n−2)
−(cid:0) (cid:1)
and
3(3n 2)(n 1)H2+n(3n 1)κ2p
wde =− 3(n+1)(3n 2)H−2+6nn−(2n 1)(3n 2−)αH2n 3n(n 1)κ2p. (37)
− − − − − −
For dust matter
p=0,
we have
· 3H2+ 1 n α6nH2n
H= 2 − , (38)
−2(1+3n(3n 5)α6n−2H2n−2)
(cid:0) − (cid:1)
3(3n 2)(n 1)H2
wde =− 3(n+1)(3n 2)H−2+6nn−(2n 1)(3n 2)αH2n. (39)
− − − −
When
n=2,
i.e. for the gravitationalLagrangian
L =√ g Γ+αΓ2 , (40)
g
−
(cid:0) (cid:1)
6
(39) gives
1
w = . (41)
de −3(24αH2 1)
−
Letting
w = 1
de
−
and
H =74km/sec/Mpc 2.4 10−18sec−1
0
≃ ×
one can compute
α=1.0145 10−5(km/sec/Mpc)−2 =9.6451 1033sec2. (42)
× ×
Then when
1
w = , (43)
de
−3
we obtain
H =90.631km/sec/Mpc. (44)
Observations of type Ia supernovae at moderately large redshifts (z 0.5 to 1) have led to the conclusion that the
∼
Hubbleexpansionoftheuniverseisaccelerating[11]. Thisisconsistentalsowithmicrowavebackgroundmeasurements
[12]. According to the result of [13], H =90.631km/sec/Mpccorresponds to
z 0.88,
∼
which is consistent with the observations.
The equation (38) now becomes
· 3H2 18αH2 1
H= − . (45)
2(6αH2+1)
(cid:0) (cid:1)
Using the formula
·
a 1 ·
H = = z, (46)
a −1+z
(45) can be rewritten as
2 6αH2+1 dH dz
= , (47)
− 3H(18αH2 1) (1+z)
(cid:0) (cid:1)−
and integrated
H2 1/3 18αH2 1 4/9
z = 0 − 1. (48)
H2 18αH2 1 −
(cid:18) 0(cid:19) (cid:18) − (cid:19)
This function is illustrated in Fig. 1, which is roughly consistent with the observations [13].
7
1.5
1.25
1
z0.75
0.5
0.25
0
80 100 120 140 160 180
H
FIG. 1: The function of z=z(H).
The equation (15) indicates that during the evolution of the universe H2 decreases owing to decreasing of the
matter density ρ. This makes w descend during the evolution of the universe. The evolution of the function
de
w =w αH2 given by (41) is illustrated in Fig. 2. According to (41) when H2 = 1 , w = 1, Φ(Γ) changes
de de 12α de −3
from ”visib(cid:0)le” to(cid:1)dark as indicated by (17). If H2 > 1 , it decelerates the expansion, if H2 < 1 , it accelerates the
12α 12α
expansion. When H2 = 1 , w crosses the phantom divide line 1. In other words, the expansion of the universe
18α de −
naturallyincludes a deceleratingandanacceleratingphase. αgivenby(42)canbe seenas a new constantdescribing
the evolution of the universe.
0
-0.25
-0.5
wde-0.75
-1
-1.25
-1.5
0.05 0.06 0.07 0.08 0.09 0.1
ΑH2
FIG. 2: The evolution of wde.
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