1046 Brazilian Journal of Physics, vol. 35, no. 4B, December, 2005
Toy Models Unifying Dark Energy and Dark Matter
Sandro Silva e Costa∗
Departamento de F´ısica – UFMT – Av. Fernando Correa da Costa, s/no– 78060-900 – Cuiab´a - MT - Brazil
(Received on 12 October, 2005)
It is very common to find numerical studies of dark energy and dark matter. Among these, one can find the interesting proposal of unifying dark matter and dark energy with the use of a single component with an ‘exotic’ equation of state. However, there is a certain shortage of analyses involving exact analytic models following this proposal. Therefore, here are presented examples of simple exact cosmological models which can reproduce some of the desired properties of an unified dark matter/energy fluid.
I. INTRODUCTION
Today, standard models of the universe are made of five ba-sic components: photons, barions, neutrinos, dark matter and dark energy. The question “what are dark energy and dark matter?” has a standard answer: weakly interacting massive particles (WIMPs) combined either to a cosmological con-stant (Λ-CDM models) or to some scalar field (quintessence models).
Another proposal consists of supposing that maybe both dark energy and dark matter could be in fact two different faces of a single component, named quartessence [1]. Ba-sic quartessential cosmological models possess a single fluid whose main characteristic is an exotic equation of state.
Numerical studies of this quartessential scenario can be eas-ily found in the recent literature [2]. But there is a certain shortage of analyses with exact analytical models. Therefore, a basic idea to overcome such deficiency is to choose mod-els where you can do the math first and then think about the physics later.
II. CLASSICAL SINGLE FLUID MODELS
Friedmann’s equation of General Relativity, 1
a2
da
dt 2
+ k a2=
8π
3 ρ+
Λ
3, (1)
plus conservation of energy in an expanding universe, d
ρa3+pda3=0, (2)
and an equation of state of the kindp=p(ρ)give the behavior of the scale factora(t).
The simplest models use the linear relation
p= (γ−1)ρ, (3)
whereγis a free parameter, and this yields the result
ρ=ρ0
a a0
−3γ
, (4)
∗The author thanks the support of FAPEMAT.
from where comes the general solutions
a(η) =
8π
3kρ0a 3γ 0
1/α
sin2/α
√ kα
2 η
, (5)
withα≡3γ−2 andγ=2/3, and
a(η) =exp
8π
3 ρ0a
2 0−k
1/2
η
, (6)
forγ=2/3, both valid forΛ=0 and whereηis a ‘conformal time’.
IfΛ>0 it is possible to obtain analytical solutions only for
some values ofγ. For example,
a=
8π
3 ρ0a
4 0
sinh 2t
Λ 3
Λ 3
−k
sinht
Λ 3
Λ 3
2
1/2
,
(7) forγ=4/3, and
a=
8π
3 ρ0a
2 0−k
sinht
Λ 3
Λ 3
, (8)
forγ=2/3.
A 0th order cosmological test for quartessential models is to compare them to such ‘classical’ single fluid cosmological solutions.
III. EXOTIC EQUATIONS OF STATE
The most usual exotic equation of state found in the litera-ture is a generalization of the Chaplygin gas,
p=−Mρ−α, (9)
withMandαparameters to be determined numerically and/or statistically. But why use such equation? From energy con-servation comes that
ρ=ρ0
1− M
ρ1+α
a a0
−6 + M
ρ1+α
1/(1+α)
Sandro Silva e Costa 1047
FIG. 1: Graphs ofa(t)for a classical single fluid model withp=ρ/3 andΛ=0.
FIG. 2: Graphs ofa(t)for a model withp=ρ/3−Mρ1/2andΛ=0.
and forα=1 one has
a→∞:ρ→M, a→0 :ρ→a−3. (11)
This kind of behavior can be implemented with another ap-proach: by adding non-linear terms to the usual linear equa-tion of state, i.e.,
p= (γ−1)ρ+f(ρ). (12) The non-linear termf(ρ)can be chosen to produce analytical
results and mimic both dark matter and energy. Therefore, the steps to be followed are:
1. firstly select exotic equations of state which give ana-lytical solutions;
2. then see which solutions can mimic dark matter and en-ergy, passing through cosmological tests of 0th, 1st, 2nd and higher orders.
One example of a generalized equation of state is
p= (γ−1)ρ−Mρ−α, (13) withα=−1. From this, one obtains, by energy conservation, withγ=0,
ρ=ρ0
1−M′
γ
a0
a 3γ(1+α)
+M′
γ
1+α1
, (14) whereM′≡Mρ0−(1+α). Choosing, for example, α=−1/2
andγ=43 one has the solution
a =
a20
4 3M′−1
e√6πρ0M′t−1
−k
sinh
3π 2ρ0M′t
3π 2ρ0M′
2
1 2
, (15)
while forγ=23 one has
a = 2πa0ρ0M′
1−3M′
2
sinh
3πρ0 2 M′t
3πρ0 2 M′
2
(16)
+
8πa20ρ0
3
1−3M2′ 2
−k 1/2
sinh6πρ0M′t
6πρ0M′
.
These results are comparable to the ones obtained in classical single fluid models (see Figures 1 and 2) and in models with
Λ=Λ(t)[3].
The search for functionsf(ρ)which produce analytical
re-sults can go on and on. But for all the proposals of exotic equations of state one must check what kind of energy/matter these equations can represent, their stability, growth of pertur-bations, etc [4].
IV. CONCLUSIONS
Since the hunting season for dark matter and dark energy models is still open, several proposals must be considered. Quartessence is a simple idea and as such can be used as food for thoughts (even if wrong). But does quartessence have a future? The answer may be on the next corner, and analytical results may help, providing forward steps.
[1] M. Makler, S.Q. Oliveira, and I. Waga. Phys. Lett. B555, 1 (2003).
[2] R. R. R. Reis, M. Makler, and I. Waga. Class. Quant. Grav.22, 353 (2005).
