WDS'10 Proceedings of Contributed Papers, Part III, 151–156, 2010.
ISBN 978-80-7378-141-5 © MATFYZPRESS
Inhomogeneous Cosmological Models
D. Vrba
Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.
Abstract. In this paper we describe briefly the averaging of scalars in relativistic
cosmology and its connection to Buchert equations. Then we focus on various
possible definitions of a deceleration parameter in inhomogeneous cosmology and
we show some particular examples in Lemaitre-Tolman-Bondi model.
Introduction
In the standard model the universe is assumed to be homogeneous and isotropic and the
spacetime is described by the FRW metric. But from observations we know that there are
inhomogeneities on scales less than 150 Mpc. In order to understand the influence of these
inhomogeneities on the expansion of the universe, it is important to study other solutions of
Einstein equations that don’t assume the homogeneity such as the Lemaitre-Tolman-Bondi
metric or a more general Szekeres metric. The Lemaitre-Tolman-Bondi metric is well uderstood
and it has been used to model inhomogeneity ever since it was discovered by Lemaitre [Lemaitre,
1933]. The Szekeres metric was studied quite recently by [Krasinski, 2008, 2007; Hellaby and
Krasinski, 2002] and it was used by [Bolejko, 2006, 2007] to model structure formation. Since
it has no symmetries it is possible to describe more structures within one model.
If the universe is not homogeneous the cosmological data may depend on the position
of the observer and it is important to perform proper averaging of physical quantities. One
important question in cosmology is whether or not the expansion of the universe is accelerated.
However, unlike in FRW model, in inhomogeneous cosmology the deceleration parameter has
an ambiguous meaning.
Lemaitre-Tolman-Bondi metric
Lemaitre-Tolman-Bondi (LTB) metric is an exact spherically symmetric inhomogeneous
dust solution of Einstein equations. In comoving coordinates (t, r, ϑ, ϕ) the spacetime element
may be written as
ds2 = −dt2 +
R′2 (r, t)
dr 2 + R2 (r, t) dϑ2 + sin2 ϑdϕ2 ,
1 + 2E (r)
(1)
where E is an arbitrary function of r. After plugging the metric (1) into Einstein equations we
get two constraints on the metric coefficients
M (r)
1 ˙2
R (r, t) −
= E (r)
2
R (r, t)
(2)
and
M ′ (r)
,
(3)
R′ (r, t) R2 (r, t)
where ρ is energy density, dot denotes the derivative with respect to t while prime is a derivative
with respect to r. M (r) is another arbitrary function of r which has the meaning of active
gravitational mass and it is in general different from the integral of density over the volume.
4πρ =
Averaging procedure and Buchert equations
The averaging problem in cosmology is an important question which has been investigated
for the past three decades [Ellis, 1984; Buchert, 1995, 1999, 2000, 2001; Buchert and Carfora,
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VRBA: INHOMOGENEOUS COSMOLOGICAL MODELS
2002, 2003; Ellis and Buchert, 2005]. One wants to know how to average physical quantities
in an inhomogeneous universe. It is not completely clear how to average tensorial quantities
although there have been some proposals [Zalaletdinov, 2007]. On the other hand, averaging
of scalar quantities is a fully covariant operation and has been developed by Buchert [Buchert,
1999, 2001]. Let us assume that we can foliate the spacetime with hypersurfaces of constant t.
We denote the spatial metric on the hypersurface by gij , θ is the expansion and σ is the shear.
An average hψi of a scalar quantity ψ over a spatial domain Ω of volume VΩ is defined as
hψi ≡
1
VΩ
Z
ψ
Ω
q
3 gd3 x,
(4)
where 3 g is the determinant of the spatial metric. The volume of the spatial domain is given by
VΩ =
Z q
3 gd3 x,
(5)
Ω
and we can also define an ”effective scale factor” of the domain
aΩ ≡
VΩ
V0
1
3
,
(6)
where V0 is a volume of the domain at some initial time that we may set up to be one. Buchert
also showed that averaging doesn’t commute with time evolution and derived a commutation
rule [Buchert, 1999]
D E
∂
hψi = hψi hθi − hψθi
ψ˙ −
(7)
∂t
which holds for any scalar ψ. Applying the commutation rule to the Raychaudhuri equation
2
θ
θ˙ = − − 2σ 2 + 2ω 2 − Rµν uµ uν
3
(8)
1
1 2
θ = 8πρ − R + σ 2 − ω 2
3
2
(9)
äΩ
= −4π hρi + Q,
aΩ
(10)
1
a˙Ω 2
1
= 8π hρi − hRi − Q,
2
2
a2Ω
(11)
and to Hamiltonian constraint
we obtain Buchert equations
3
3
where Q is called the backreaction and is defined as
Q≡
D E
2 D 2 E
θ − hθi2 − 2 σ 2 ,
3
(12)
ω is vorticity, Rµν is Ricci tensor, uµ is a timelike vector and R is the Ricci scalar on the spatial
hypersurface.
Deceleration parameter in inhomogeneous cosmology
Unlike in Friedmann model, in inhomogeneous cosmology, it is not completely clear how
the deceleration parameter should be defined. In the literature there have been several different
definitions introduced and they usually lead to different answers to the question whether or not
the expansion of the universe is accelerated. In this section we explore some of these definitions.
Let us first start with the standard deceleration parameter in Friedmann model,
q≡−
1
äa
= (ΩM − 2ΩΛ ).
2
ȧ
2
152
(13)
VRBA: INHOMOGENEOUS COSMOLOGICAL MODELS
The second equality in the last equation can be derived by using the Friedmann equation and we
can see, that in the case when the cosmological constant is zero, the deceleration parameter has
to be positive. Alternative definition for the deceleration parameter can be given by expanding
the luminosity distance redshift relation
1
(1 + z)
DL (z) =
H
Z
0
in a Taylor series
DL =
dDL
dz
z+
z=0
1 d2 DL
2 dz 2
!
dz ′
z
q
z 2 + ... =
(14)
Ωm (1 + z ′ )3 + ΩΛ
1
1
z+
(2 − ΩM + 2ΩΛ ) z 2 + ...
H
4H
(15)
z=0
Now comparing the equation (13) with the equation (15) we can see that the deceleration
parameter can be written as
!
d2 DL
.
(16)
q =1−H
dz 2 z=0
In inhomogeneous cosmology, one possible option how to define a deceleration parameter
is by using Buchert equations. This definition is in the literature called a volume deceleration
[Bolejko and Anderson, 2008]. It is defined similarly to the deceleration parameter (13) in
Friedmann model [Bolejko, 2008]
a¨Ω aΩ
,
(17)
qΩ ≡ −
a˙Ω 2
here aΩ is the effective scale factor as it is defined in Buchert equations. We can now substitute
from Buchert equations for the effective scale factor and its derivatives and we get
qΩ ≡ −
−4π hρi + Q
äΩ aΩ
=−
.
2
ȧΩ
8π hρi − 12 hRi − 12 Q
(18)
We can now calculate the backreaction and the average of Ricci curvature in some specific
model. Suppose for instance a LTB model with the energy function chosen to be constant. This
choice is convenient, because various integrals can be now calculated analytically. We get for
the backreaction and Ricci curvature
Q=
8π
VΩ
Z
rΩ
0
hRi = −
2RR˙ R˙ ′ + R˙ 2 R′
2
√
dr − hθi2 = ... = 0,
3
1 + 2E
16π
VΩ
Z
rΩ
0
(ER)′
E
√
dr = −12 2
,
R (rΩ , t)
1 + 2E
(19)
(20)
where rΩ is the radial coordinate of the boundary of the region that we take the average over.
Now inserting (19) and (20) into (18) leads to the volume deceleration
qΩ =
M
1
.
2 ER (rΩ , t) + M
(21)
The sign of qΩ depends on the sign and value of E and in the case when E = 0 it reduces to
1
2 . We can approach the volume acceleration in a different way. We can substitute into the
definition (18) from the definition of the scale factor (6) in terms of the volume. The first and
second derivative of the effective scale factor are
ȧΩ =
1 − 23 ˙
V VΩ ,
3 Ω
2 − 5 2 1 − 2
äΩ = − VΩ 3 V˙ Ω + VΩ 3 V¨Ω
9
3
153
(22)
VRBA: INHOMOGENEOUS COSMOLOGICAL MODELS
and for the volume acceleration we obtain
qΩ ≡ −
VΩ V¨Ω
äΩ aΩ
= 2 − 3 2 .
2
ȧΩ
V˙
(23)
Ω
Calculating the volume and its first and second derivatives in the LTB model with constant
energy function lead to the same result as (21) and we didn’t have to use the Buchert equations.
This approach was used by [Chuang, 2008; Paranjape and Singh, 2006].
In inhomogeneous cosmology, we can think of the acceleration also in a different way. We
may for instance calculate the proper distance between two points in spacetime and investigate
how this distance changes in time. In analogy to definition (13) we may define
qd ≡ −
¨
dd
,
d˙2
(24)
where the proper distance d between two points is given by
Z rΩ
√
grr dr,
d (t) =
(25)
0
assuming one of the points is at the center of symmetry and the other one is at the coordinate rΩ .
The value of the deceleration will of course depend on the choice of these points. In LTB model
with constant energy function the deceleration parameter (24) reduces to the same formula as
(21) . However in a more general case when the energy function is not constant the definitions
(17) and (24) were analyzed numerically by [Chuang, 2008] and they give different answers.
Another possible definition of the deceleration parameter in inhomogeneous cosmology is
by using the luminosity distance. Similarly as in Friedmann model we can calculate the second
derivative of the luminosity distance function with respect to the redshift at the origin. In LTB
model this leads to
!
¨′
R′ R
R′′
R˙ ′′
d2 DL
=
−
−
+
.
(26)
qL = 1 − H
2
dz 2
R′ R˙ ′ R˙ ′ 2
R˙ ′
z=0
This definition of the deceleration parameter was studied numerically by [Bolejko and
Anderson, 2008] and by [Vanderveld et al., 2006] who showed that it needs to be positive in
order to avoid a singularity.
More general definition of deceleration parameter independent of the cosmological model
can be based on investigation of a timelike geodesic congruence. Suppose uµ is a tangent vector
of a congruence of timelike geodesics. We can define
1
q ≡ − 1 + 2 H,µuµ ,
H1
1
1
H ≡ θ ≡ uµ ;µ ,
3
3
(27)
where H is the Hubble parameter and θ is the expansion of the congruence. Using the Raychaudhuri equation we can now show that this deceleration parameter has to be positive in the
universe where the strong energy condition is satisfied. From the definitions (27) we have
1 ˙ θ2
θ+
qH = − H + H˙ = −
3
3
2
2
!
.
(28)
Now substituting last formula into Raychaudhuri equation (8) we obtain an equation which
relates the deceleration parameter with the shear, vorticity and the Ricci tensor
1 2
2σ − 2ω 2 + Rµν uµ uν .
(29)
qH 2 =
3
So we can see that if the vorticity is zero and the strong energy condition holds, the deceleration
parameter defined via equation (27) has to be positive. This definition of the deceleration parameter and three more are discussed in [Hirata and Seljak, 2005]. Notice that in the Friedmann
model the definition (27) is equivalent to the definition (13) or (16).
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Conclusion
In FRW cosmology the deceleration parameter has to be positive unless we introduce a
medium with negative pressure. In inhomogeneous cosmology the situation is different, the
deceleration parameter has an ambiguous meaning. There are several papers in the literature
where the authors investigate the sign and the value of the deceleration parameter and they
obtain different answers. The reason for that is, that different authors refer to different definitions. According to the local equation (29) which is model independent it is obvious that the
expansion of the universe has to be decelerating everywhere if the strong energy condition is
satisfied. On the other hand from Buchert equations it follows that the expansion of a spatial
region might be on average accelerating. The explanation for this was given by Rsnen [Rsnen,
2008]. Since in inhomogeneous universe different regions expand at different rates, the volume
of faster expanding region increases faster. This may lead to the situation that the average
expansion rate increases.
We would like to emphasize that deceleration parameter is not an observable quantity, so
we can’t directly compare any of the deceleration parameters with measurements. What is
usually measured in practice is the luminosity distance curve, so the parameter derived from
the luminosity distance function seems to be the closest to observations.
The studying of exact inhomogeneous solutions of Einstein equations may give us, among
other things, a better understanding of the effects of inhomogeneities on the expansion of the
universe.
Acknowledgments. I would like to thank to Daniel Finley, Rouzbeh Allahverdi and Otakar
Svtek for interesting and helpful discussions. Present work was supported from the project SVV 261301
of the Charles University in Prague, GACR-205/09/H033 and GAUK-86508.
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