Introduction to Cosmology

Extra Dimensions 249

expansion which is of recent date (푧≈ 0 .7or푎≈ 0 .6),푟푐퐻 0 must be of the order
of unity. In theself-decelerating DGP branch, gravity ‘leaks in’ from the bulk at
scales푎≈푟푐퐻 0 , in conflict with the observed dark energy acceleration. Note that
the self-accelerating branch has a ghost, whereas the self-decelerating branch is
ghost-free.
On the four-dimensional brane the action of gravity is proportional to푀Pl^2 whereas
in the bulk it is proportional to the corresponding quantity in five dimensions,푀 53.
The cross-over length is defined as

푟푐=푀^2 Pl∕ 2 푀 53. (11.36)

It is customary to associate a density parameter to this,

훺푟푐=( 2 푟푐퐻 0 )−^2 , (11.37)

such that푟푐퐻 0 is a length scale (similar to푎).
The Friedmann–Lemaitre equation may be written

퐻^2 +

푘

푎^2

−휖

1

푟푐

√

퐻^2 +

푘

푎^2

=휅휌, (11.38)

where푎=( 1 +푧)−^1 ,휅= 8 휋퐺∕3, and휌is the total cosmic fluid energy density with
components휌푚for baryonic and dark matter, and휌휑for whatever additional dark
energy may be present.
Clearly the standard FLRW cosmology is recovered in the limit푟푐→∞or퐻≪푟푐.
When퐻⩾푟푐the root term becomes important. In flat-space geometry푘=0, and at
late times when휌∝ 1 ∕푎^3 →0and퐻→퐻∞this becomes a de Sitter acceleration,푎(푡)∝
exp푡∕푟푐.
Theself-accelerating branchcorresponds to휖=+1, theself-decelerating branchto
휖=−1. In DGP geometry the continuity equations for ideal fluids have the same form
as in FLRW geometry [Equation (11.28)].
Pressureless dust with푝=0 then evolves as휌푚(푎)∝푎−^3. The free parameters in the
DGP model are훺푟푐and훺푚=휅휌푚∕퐻 02. Note that there is no curvature term훺푘since
we have assumed flatness by setting푘=0 in Equation (11.33).
In the space of the parameters훺k,훺mand훺푟푐≡ 1 ∕ 4 푟^2 푐퐻 02 one can generalize the
DGP model to

퐻^2 −푘

푎^2

−푟−푐^2

(

푟푐

√

퐻^2 − 푘

푎^2

) 2 −푛

=휅휌, (11.39)

where푛defines a parametric family. Comparisons with data show that푛=1and푛> 3
give poor fits,푛=2 corresponds to the휆CDM concordance model, and푛=3 gives
good fits.

The Chaplygin–DGP Model. Both the self-accelerating DGP model and the stan-
dard Chaplygin gas model have problems fitting present observational data, both
cause too much acceleration. They have at least one parameter more than휆CDM,
yet they fit data best in the limit where they reduce to휆CDM.