246 Dark Energy
Chaplygin Gas. Another simple and well-studied model of dark energy introduces
into푇휇휈the density휌휑and pressure푝휑of an ideal fluid with a constant potential
푉(휑)=퐴>0 calledChaplygin gasfollowing Chaplygin’s historical work in aerodynam-
ics. Here퐴has the dimensions of energy density squared. Ordinary matter is assumed
not to interact with Chaplygin gas, therefore one has separate continuity equations for
the energy densities휌푚and휌휑of the same form as in FLRW geometry, respectively,
휌̇+ 3 퐻(휌+푝)= 0. (11.23)Pressureless dust with푝=0 then evolves as휌푚(푎)∝푎−^3.
The barotropic equation of state is
푝휑=−퐴∕휌휑. (11.24)The continuity Equation (11.23) is then
휌̇휑+^3 퐻(휌휑−퐴∕휌휑)=^0 ,which integrates to
휌휑(푎)=√
퐴+퐵∕푎^6 , (11.25)
where퐵is an integration constant. Thus this model has two free parameters.
Obviously the limiting behavior of the energy density is
휌휑(푎)∝
√
퐵
푎^3
for푎≪(퐵
퐴
) 1 ∕ 6
,휌휑(푎)∝
√
퐴for푎≫(퐵
퐴
) 1 ∕ 6
. (11.26)
At early times this gas behaves like pressureless dust, like CDM, at late times it
behaves like the cosmological constant, causing accelerated expansion. The problem
is, that when the dust effect disappears but CDM remains, this causes a strong ISW
effect and loss of CMB power, thus the model is a poor fit to data (SNe, BAO, CMB).
To remedy this one has complicated the model by adding a new parameter,훽,tothe
barotropic equation of state:
푝휑=−퐴∕휌훽휑. (11.27)Unfortunately, the fit then approaches the standard cosmological constant solution.
11.3 풇(푹)Models
In Section 5.5 we have already met models of modified gravity which are candidates
both for inflation and for the present accelerated expansion.
In Equation (5.85) we replaced the curvature scalar푅by a general function푓(푅, 휑),
and enlarged the dimensionality of space-time from 4 to푛.
The Einstein–Hilbert action is not renormalizable, therefore standard general rel-
ativity cannot be properly quantized. However, renormalizability can be cured by
adding higher order terms in curvature invariants. This is similar to the situation
in quantum field theory whererenormalizationis some procedure to remove infinities
in calculations. If the Lagrangian contains combinations of field operators of high