Friedmann–Lemaitre Cosmologies 85right-hand side, where they appear as corrections to the stress–energy tensor푇휇휈.Then
the physical interpretation is that of an ideal fluid with energy density휌휆=휆∕ 8 휋퐺and
negative pressure푝휆=−휌휆푐^2. When the cosmological constant is positive, the gravita-
tional effect of this fluid is a cosmic repulsion counteracting the attractive gravitation
of matter, whereas a negative휆corresponds to additional attractive gravitation.
The cosmology described by Equations (5.17) and (5.18) with a positive cosmolog-
ical constant is called theFriedmann–Lemaitre universeor the Concordance model.
Such a universe is now strongly supported by observations of a nonvanishing휆,so
the Einstein–de Sitter universe, which has휆=0, is a dead end.
In a Friedmann–Lemaitre universe the total density parameter is conveniently split
into a matter term, a radiation term and a cosmological constant term,
훺 0 =훺m+훺r+훺휆, (5.20)where훺rand훺휆are defined analogously to Equations (1.35) and (5.10) as
훺r=휌r
휌c,훺휆= 휆
8 휋퐺휌c= 휆
3 퐻 02
. (5.21)
훺m,훺rand훺휆are important dynamical parameters characterizing the Universe. If
there is a remainder훺푘≡훺 0 − 1 ≠0, this is called thevacuum-energyterm.
Using Equation (5.19) we can find the value of휆corresponding to the attractive
gravitation of the present mass density:
−휆= 8 휋퐺휌 0 = 3 훺 0 퐻 02 ≈ 1. 3 × 10 −^52 푐^2 m−^2. (5.22)No quantity in physics this small has ever been known before. It is extremely
uncomfortable that휆has to be fine-tuned to a value which differs from zero only
in the 52nd decimal place (in units of푐=1). It would be much more natural if휆were
exactly zero. This situation is one of the enigmas which will remain with us to the
end of this book. As we shall see, a repulsive gravitation of this kind may have been of
great importance during the first brief moments of the existence of the Universe, and
it appears that the present Universe is again dominated by a global repulsion.
Energy-Momentum Conservation. Let us study the solutions of Friedmann’s equa-
tions in the general case of nonvanishing pressure푝. Differentiating Equation (5.4)
with respect to time,
d
d푡(푎̇^2 +푘푐^2 )=^8 휋퐺
3
d
d푡(휌푎^2 ),
we obtain an equation of second order in the time derivative:
2 푎̈̇푎=8
3
휋퐺(휌푎̇^2 + 2 휌푎 ̇푎). (5.23)
Using Equation (5.6) to cancel the second-order time derivative and multiplying
through by푐^2 ∕푎^2 , we obtain a new equation containing only first-order time deriva-
tives:
휌푐̇^2 + 3 퐻(휌푐^2 +푝)= 0. (5.24)