248 Dark Energy
universe푓RR>0for푅≫푓RR. Further requirements are 1+푓푅>0forallfinite푅(this
prevents the graviton from becoming ghost-like),푓(푅)∕푅→0and푓푅→0as푅→∞
(GR must be recovered early before the BBN and the CMB), and푓푅must be small at
recent epochs [13]. The condition for an accelerated expansion is푎(푡)∝푡푑with푑>1,
suitable also for the exponential quintessence potential.
An example of a good potential is
푓(푅)=푅
⎡
⎢
⎢
⎣
1 +훼
(
−
푅
퐻 02
)훽− (^1) ⎤
⎥
⎥
⎦
(11.33)
with훼and훽constants, and which describes well the radiation-dominated era, the
matter-dominated era, and the present accelerated era, but not the inflation. In the
limit푅=0 one has flat geometry and standard GR.
Higher Order Invariants. Nothing forbids one to complement the Ricci scalar푅
with invariants of higher order in curvature. The advantage is that they introduce
extra degrees of freedom, but they may come at a cost. They can lead to instabilities
and conflicts with local tests of gravity. The invariants of lowest mass dimension are
푃≡푅휇휈푅휇휈 푄≡푅훼훽훾훿푅훼훽훾훿. (11.34)which have actions similar to Equation (11.27),
푆∝[
∫
d^4 푥√
−푔[푅+푓(푅, 푃 , 푄)] +
∫
d^4 푥√
−푔푚(푔휇휈,훹)
]
. (11.35)
The equations of motion can again be found by varying this action with respect
to푔휇휈.
The combination of invariants that seems most promising and is most studied is
푅^2 − 4 푃+푄, called theGauss–Bonnet term.
11.4 Extra Dimensions
Already the generalized form of the Einstein–Hilbert action [Equation (5.85)] allowed
for the possibility of a spacetime of푛>4 dimensions. Let us assume that the observ-
able physics occurs on a four-dimensional brane embedded in a five-dimensional
Minkowski bulk universe. Gravitation occurs everywhere in this bulk, but at cosmo-
logically late times (now) it gets weaker on the brane, it leaks out into the bulk outside
our brane, causing the present accelerated expansion.
The DGP model. A simple and well-studied model of modified gravity in five space-
time dimensions is theDvali–Gabadadze–Porrati (DGP)braneworld model. The model
is characterized by a cross-over scale푟푐such that gravity is a four-dimensional theory
at scales푎≪푟푐퐻 0 where matter behaves as pressureless dust. In theself-accelerating
DGP branch, gravity ‘leaks out’ into the bulk when푎≈푟푐퐻 0 , and at scales푎≫푟푐퐻 0 the
model approaches the behavior of a cosmological constant. To explain the accelerated