252 Dark Energy
The푇 00 component of Einstein’s equations gives the Friedmann–Lemaˆıtre equation
in the LTB metric [Equation (11.46)]
퐻푇^2
훼^2+
2 퐻푇퐻퐿
훼^2
+ 푘
퐴^2
+
푘′(푡, 푟)
퐴퐴′
= 8 휋퐺(휌푀+휌DE). (11.49)
Here퐻푇=퐴̇∕퐴is the transversal Hubble expansion,퐻퐿=퐴̇′∕퐴′the longitudinal Hub-
ble expansion.
The푇푟푟components of Einstein’s equations give
2 퐻̇푇+ 3 퐻푇^2 +푘
퐴^2
− 8 휋퐺 푝DE= 0. (11.50)
Multiplying each term with퐴^3 퐻푇this can be integrated over time to give
퐻푇^2 =퐹(푟)
퐴^3
−
푘(푟)
퐴^2
−
8 휋퐺
퐴^3 ∫
dt퐴^3 퐻푇푝DE. (11.51)Here퐹(푟)is an arbitrary time-independent function which arose as an integration
constant.
We now assume that dark energy does not interact with matter. There are then sep-
arate continuity equations for ordinary matter and dark energy. The one for matter is
휌̇푀(푟, 푡)+[ 2 퐻푇(푟, 푡)+퐻퐿(푟, 푡)]휌푀(푟, 푡)= 0 , (11.52)and it can be integrated to give
휌푀(푟, 푡)=퐹(푟)
퐴^2 퐴′
(11.53)
The continuity equation for dark energy is
휌̇DE(푟, 푡)+[^2 퐻푇(푟, 푡)+퐻퐿(푟, 푡)] [휌DE(푟, 푡)+푝DE(푟, 푡)] =^0. (11.54)
To proceed we should now specify several arbitrary functions, so we leave the sub-
ject here unfinished.
Problems
- Derive Equation (5.76).
- What should푡eqbe for K-essence휌kto drop precisely to the magnitude of the
present-day휌휆≈ 2. 9 × 10 −^47 GeV^4? - Suppose that dark energy is described by an equation of state푤=− 0 .9whichis
constant in time. At what redshift did this dark energy density start to dominate
over matter density? What was the radiation density at that time? - Derive this expression for the K-essence energy density:
휌k=[ 2 푋퐾′(푋)−퐾(푋)] +푉(휑)- Derive the field equations for an action of the form in Equation (11.28) with
푓(푅)=푅^2. The ordinary matter part푚can be ignored.