250 Dark Energy
Because of the similarities in the asymptotic properties of the two-parametric DGP
model and the two-parametric Chaplygin gas model. I have proposed [13] to combine
theself-deceleratingbranch of the DGP model with the accelerating Chaplygin gas
model. There are then four free parameters,훺푟푐,훺푚,퐴,and퐵, one of which shall be
eliminated shortly.
We now choose the length scales in the two models,푟푐퐻 0 and(퐵∕퐴)^1 ∕^6 , to be pro-
portional by a factor푥,sothat
(퐵
퐴
) 1 ∕ 6
=푥푟푐퐻 0 = 푥
2
√
훺푟푐
. (11.40)
The proportionality constant subsequently disappears because of a normalizing con-
dition at푧=0. Then the model has only one parameter more than the standard훬CDM
model.
It is convenient to replace the parameters퐴and퐵in Equation (11.40) by two new
parameters,훺퐴=퐻 0 −^2 휅
√
퐴and푥= 2√
훺푟푐(퐵∕퐴)^1 ∕^6. The dark energy density can thenbe written
휌휑(푎)=퐻 02 휅−^1 훺퐴√
1 +푥^6 ( 4 훺푟푐푎^2 )−^3. (11.41)
Let us now return to Equation (11.38) and solve it for the expansion history퐻(푎).
Substituting훺푟푐from Equation (11.37),휌휑(푎)from Equation (11.41), and using훺푚=
훺^0 푚푎−^3 , it becomes
퐻(푎)
퐻 0=−
√
훺푟푐+
[
훺푟푐+훺^0 푚푎−^3 +훺^0 푟푎−^4 +훺퐴
√
1 +푥^6 ( 4 훺푟푐푎^2 )−^3
] 1 ∕ 2
. (11.42)
Note that훺푟푐and훺퐴do not evolve with푎,justlike훺휆in the the휆CDM model. In the
limitofsmall푎this equation reduces to two terms which evolve as푎−^3 ∕^2 , somewhat
similarly to dust with density parameter
√
훺^0 푚+훺퐴푥^3 ( 4 훺푟푐)−^3 ∕^2.
In the limit of large푎, Equation (11.42) describes a de Sitter acceleration with acosmological constant훺휆=−
√
훺푟푐+
√
훺푟푐+훺퐴.
A closer inspection of Equation (11.42) reveals that it is not properly normalized
at푎=1to퐻( 1 )∕퐻 0 =1, because the right-hand-side takes different values at different
points in the space of the parameters훺^0 푚,훺푟푐,훺퐴,and푥. This gives us a condition: at
푎=1werequirethat퐻( 1 )=푥퐻 0 so that Equation (11.42) takes the form of a sixth
order algebraic equation in the variable푥
푥=−√
훺푟푐+
[
훺푟푐+훺^0 푚+훺퐴
√
1 +푥^6 ( 4 훺푟푐)−^3
] 1 ∕ 2
. (11.43)
This condition shows that푥is a function푥=푓(훺^0 푚,훺푟푐,훺퐴). Finding real, positive
roots푥and substituting them into Equation (11.42) would normalize the equation
properly. The only problem is that the function cannot be expressed in closed form,
so one has to resort to numerical iterations. The average value of푥is found to be푥≈
0 .956; it varies over the interesting part of the parameter space, but only by≈ 0 .002.
This model [13] fits SNeIa data with the same goodness of fit as the the cosmological
constant model, it also fits the CMB shift parameter푅well, and notably, it offers a