A Brief Review of the Singularities in 4D and 5D Viscous Cosmologies Near the Future Singularity
ABSTRACT Analytic properties of physical quantities in the cosmic fluid such as energy density \rho(t) and Hubble parameter H(t) are investigated near the future singularity (Big Rip). Both 4D and 5D cosmologies are considered (the Randall-Sundrum II model in the 5D case), and the fluid is assumed to possess a bulk viscosity \zeta. We consider both Einstein gravity and modified gravity, where in the latter case the Lagrangian contains a term R^\alpha with \alpha a constant. If \zeta is proportional to the power (2\alpha-1) of the scalar expansion, the fluid can pass from the quintessence region into the phantom region as a consequence of the viscosity. A property worth noticing is that the 4D singularity on the brane becomes carried over to the bulk region.
arXiv:0811.1129v1 [gr-qc] 7 Nov 2008
A BRIEF REVIEW OF THE SINGULARITIES IN 4D AND 5D
VISCOUS COSMOLOGIES NEAR THE FUTURE
Department of Energy and Process Engineering, Norwegian University of
Science and Technology, N-7491 Trondheim, Norway
Tomsk State Pedagogical University, Tomsk, Russia
Analytic properties of physical quantities in the cosmic fluid such
as energy density ρ(t) and Hubble parameter H(t) are investigated
near the future singularity (Big Rip). Both 4D and 5D cosmologies
are considered (the Randall-Sundrum II model in the 5D case), and
the fluid is assumed to possess a bulk viscosity ζ. We consider both
Einstein gravity and modified gravity, where in the latter case the
Lagrangian contains a term Rαwith α a constant. If ζ is proportional
to the power (2α −1) of the scalar expansion, the fluid can pass from
the quintessence region into the phantom region as a consequence of
the viscosity. A property worth noticing is that the 4D singularity on
the brane becomes carried over to the bulk region.
The possibility of crossing the w = −1 barrier in dark energy cosmology has
recently become a topic of considerable interest. One usually assumes that
the equation of state for the cosmic fluid can be written in the form
p = wρ, (1)
where w is a constant. If w = −1 the fluid is called a ”vacuum fluid”,
with peculiar thermodynamic properties such as negative entropy . More
general forms for the equation of state can be envisaged, such as
p = w(ρ)ρ = −ρ − f(ρ), (2)
which is a form that we shall consider below. As is know, cosmological obser-
vations indicate that the present universe is accelerating. Recent discussions
on the actual value of w can be found, for instance, in refs. [2, 3, 4]. Perhaps,
w is even an oscillating function in time. For discussions on time-dependent
values of w, one may consult Refs. [5, 6, 7]. The possibility of crossing from
the quintessence region (−1 < w < −1/3) into the phantom region w < −1,
is obviously of physical interest. It may be noted that both quintessence and
phantom fluids lead to the inequality ρ + 3p ≤ 0, thus breaking the strong
Once being in the phantom region, the cosmic fluid will inevitably be led
into a future singularity, called the Big Rip [8, 9, 10, 11]. And this brings
us to the main theme of the present paper, namely to give an overview of
the behavior of central physical quantities near the future singularity. This
is the case of main interest. We think that such an exposition should be
useful, not least so because the situation is rather complex. Namely, there is
a variety of different factors at play here: (i) the thermodynamic parameter
w(ρ), (ii) the possible time dependence of the bulk viscosity, ζ = ζ(t), and
(iii) the adoption of Einstein’s gravity, or a version of the so-called modified
gravity. (For an introduction to modified gravity theories, one may consult
Refs. [12, 13].)
To begin with, it is convenient to quote from Ref.  the classification
of possible future singularities:
(i) Type I (”Big Rip”): For t → ts, a → ∞, ρ → ∞, and |p| → ∞, or p
and ρ are finite at t = ts.
(ii) Type II (”sudden”): For t → ts, a → as, ρ → ρs, and |p| → ∞,
(iii) Type III: For t → ts, a → as, ρ → ∞, and |p| → ∞,
(iv) Type IV: For t → ts, a → as, ρ → 0, |p| → 0, or p and ρ are finite.
Higher order derivatives of H diverge.
Here the notation is standard, a meaning the scale factor and tsreferring
to the instant of the singularity. The above classification was introduced in
the context of ideal, i.e., nonviscous, cosmology. We can however make use
of the same classification also in the viscous case.
In the following, we will present salient features of 4D, respective 5D,
viscous cosmology theory, and thereafter focus on the classification of the
2 Viscous 4D theory: Basics
We include this basic material mainly for reference purposes. We consider
the standard FRW metric,
ds2= −dt2+ a2(t)dx2, (3)
and set the spatial curvature k, as well as the 4D cosmological constant Λ4,
equal to zero. The Hubble parameter is H = ˙ a/a, the scalar expansion is
θ = Uµ;µ= 3H with Uµthe four-velocity of the fluid, and κ2
gravitational coupling. Of main interest are the (tt) and (rr) components of
the Friedmann equations. They are
4= 8πG4is the
a+ H2= −κ2
4˜ p, (5)
where ˜ p = p −ζθ is the effective pressure. From the differential equation for
energy, T0ν;ν= 0, we get
˙ ρ + (ρ + p)θ = ζθ2. (6)
As a consequence of positive entropy change in an irreversible process, we
must require the value of ζ to be non-negative. From the equations above
we obtain the following differential equation for the scalar expansion,
4ζθ = 0. (7)
In view of the relationship˙θ = (√3κ4/2)˙ ρ/√ρ we can alternatively reformu-
late this equation as an equation for the density,
˙ ρ − κ4
3ρf(ρ) − 3κ2
4ζρ = 0. (8)
The solution is (cf. Eq. (9) in )
√ρf(ρ)[1 + κ4ζ√3ρ/f(ρ)].(9)
We here let t = 0 be the initial (present) time, and let the correspond-
ing initial density be ρ∗. The functional form of the bulk viscosity ζ is so
far unspecified. The shear viscosity is omitted, due to the assumed spatial
isotropy in the cosmic fluid. Note the dimensions: [κ2
[ρ] = cm−4, [ζ] = cm−3. Viscous cosmology are treated at various places, for
instance, in Refs. [3, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
4] = cm2, [f(ρ)] =
3 Specific cases in 4D
We are now in a position to discuss various cases in 4D explicitly. We have
to distinguish between several alternatives: (i) use of Einstein or modified
gravity; (ii) possible density dependence of the thermodynamic parameter
w = w(ρ); and (iii) possible time dependence of the bulk viscosity ζ(t).
3.1 Einstein gravity, w and ζ being constants
Let f(ρ) = αρ, with α a constant. The equation of state is then
p = wρ = −(1 + α)ρ.(10)
We can now solve explicitly for the Hubble parameter [14, 26],
where H∗is the present-time value of H and tcis the ’viscosity time’,
From Eq. (11) it is seen that H(t) becomes singular when the denominator
vanishes. Let us first for reference purposes set ζ = 0:
The nonviscous case. If ts0designates the singularity time, we have
The viscous case. If now tsζ denotes the singularity time, we get from
Close to the singularity we thus obtain the same singular behavior as in the
nonviscous case. Moreover, we get the following forms,
tsζ= tcln1 +
t → tsζ. (18)
a(t) ∼ (tsζ− t)−2/3α,
ρ(t) ∼ (tsζ− t)−2,
t → tsζ,
t → tsζ.
The viscosity tends to shorten the singularity time,
tsζ< ts0, (21)
but it does not modify the exponents in the singularity. The singularity is of
Type I if α > 0, and of Type II if α < 0.
3.2Einstein gravity, f(ρ) = Aρβ, and ζ being constant
We shall assume that β ≥ 1. From Eq. (9) it is apparent that the last term
in the denominator dominates for large ρ. Near the singularity we obtain the
ρ(t) ∼ (tsζ− t)
which generalizes Eq. (20) and reduces to it when β = 1. Thus ρ → ∞
implying, according to Eq. (2), that also |p| → ∞. The Hubble parameter
H(t) ∼ (tsζ− t)
If β > 1, a → as(a finite value) when t → tsζ. The singularity is of Type
III. If β = 1, the singularity is of Type I.
The material of this subsection was discussed also in Ref. , whereas
the equation of state corresponding to (22) and(23) was discussed by Nojiri
and Odintsov .
2β−1,t → tsζ, (22)
3.3 Modified gravity, w being constant, and ζ = τθ2α−1
Consider now the following gravity model,
d4x√−g(f0Rα+ Lm), (24)
where f0and α are constants, Lmbeing the matter Lagrangian. This model
has been considered before; cf., for instance, Refs. [28, 29, 30, 23]. The case
f0= 1 and α = 1 yields Einstein’s gravity. The equations of motion following
from the action above are
where Tµν corresponds to the term Lmin the Lagrangian. For the cosmic
fluid we have
Tµν= ρUµUν+ ˜ phµν, (26)
where hµν= gµν+UµUνis the projection tensor and ˜ p = p−ζθ the effective
pressure. In comoving coordinates, U0= 1, Ui= 0. We assume now the
simple equation of state given in Eq. (1).
Of main interest is the (00)-component of Eq. (25). Using that R =
6(˙H + 2H2), T00= ρ, as well as the energy conservation equation (6) which
in turn follows from ∇νTµν= 0, we obtain
2γf0Rα+3αf0[2˙H −3γ(˙H +H2)]Rα−1+3α(α−1)f0[(3γ −1)H˙R+¨R]Rα−2
+ 3α(α − 1)(α − 2)f0˙R2Rα−3= 9κ2
with γ = w + 1. The important point now is that this complicated equation
for H(t) is satisfied with the following form
H = H∗/X,whereX = 1 − BH∗t,(28)
B being a nondimensional parameter. For Big Rip to occur, B has to be
Taking the bulk viscosity to have the form
ζ = τθ2α−1= τ(3H)2α−1
with τ a positive constant, the time-dependent factors in Eq. (27) drop out.
There remains an algebraic equation, determining B.
Of main interest is the time-dependent forms
ζ = τ(3H∗/X)2α−1,ρ = ρ∗/X2α. (30)
As an example, the case α = 2 turns out to yield a cubic equation for B.
There is one positive root (assuming f0 positive), leading to a viscosity-
generated Big Rip. If α < 0, typically α = −1, there may still be positive
solutions for B implying that H = H∗/X is diverging. By contrast, ζ ∝
X−(2α−1)and ρ ∝ X−2αgo to zero.
4Relationship to 5D viscous theory
Let us investigate the possible link between the 4D theory above and the
analogous viscous theory in 5D space. To this end we consider a spatially
flat (k = 0) brane located at the fifth dimension y = 0, surrounded by an
anti-de-Sitter (AdS) space. If the 5D cosmological constant, called Λ, is
negative, the configuration is that of the Randall-Sundrum II model (RSII)
. The 5D coordinates are denoted xA= (t,x,y), and the 5D gravitational
coupling is κ2
5= 8πG5. The Einstein equations are
2gABR + gABΛ = κ2
and the metric is
ds2= −n2dt2+ a2δijdxidxj+ dy2, (32)
where n(t,y) and a(t,y) are to be determined from the Einstein equations.
Of main interest are the (tt) and (yy) components of the field equations.
− Λn2= κ2
+ Λ = κ2
(cf. for instance, Refs. [32, 33, 34, 19, 26]). Overdots and primes mean deriva-
tives with respect to t and y respectively. On the brane y = 0 we assume
there is a constant tension σ, and an isotropic fluid with time-dependent
energy density ρ = ρ(t). The energy-momentum tensor is now
TAB= δ(y)(−σδµν+ ρUµUν+ ˜ phµν)δµ
Applying the junction conditions across the brane we obtain, for arbitrary y,
after integration with respect to y ,
36(σ + ρ)2. (37)
We here let subscript zero refer to the brane. On the brane, n0(t) = 1. Recall
that Λ and σ are constants, and that Eq. (37) is a 5D, not a 4D, equation.
Its essential new feature is that it contains a ρ2term. The equation functions
as a bridge between 4D and 5D cosmologies.
We observe the solution for a0(t) if ρ = 0:
a0(t) = e
normalized such that a0(0) = 1.
Inserting ρ = ρ∗/X2αinto Eq. (37) we get
(1 − BH∗t)2α
Near the Big Rip, ts= 1/(BH∗), the quantities Λ and σ become unimportant,
and we get
(2α − 1)(BH∗)2α(ts− t)2α−1
showing that if α > 1/2, a0(t) has an essential singularity. Einstein’s gravity
corresponds to α = 1. The singularity becomes stronger, the higher is the
value of α. If α < 1/2, a0(t) does not diverge at ts. From Eq. (36),
a0(t) ∼ exp
with µ =
on the brane becomes transferred to the bulk. The bulk scale factor a(t,y)
diverges for arbitrary y at t = tsif a0(t) diverges at ts. There is no funda-
mental difference between an Einstein fluid and a modified gravity fluid in
this respect; their behavior is essentially the same.
In summary, we have discussed viscous dark energy as a particular rep-
resentative of inhomogeneous equation-of-state fluids and the appearance of
finite-time future singularities for such energies. It is of interest to note
that due to the relationship between modified gravity and inhomogeneous
equation-of-state ideal fluids , our findings may be useful in the study of
future singularities in modified gravity .
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