Introduction to Cosmology

Extra Dimensions 251

genuine alternative to the cosmological constant model because it does not reduce to
it in any limit of the parameter space.

The Gauss–Bonnet Model. Consider an action in a six-dimensional spacetime of
the form

푆∝

[

∫

d^6 푥

√

−푔(푅+휖GB)

]

(11.44)

where휖is theGauss–Bonnet (GB)parameter. Only휖>0 yields accelerating solutions.
The metric consists of two pieces, the FRW metric and the GB metric,

d푠^2 =푐^2 d푡^2 −푎(푡)^2 d푙^2 FRW−푏(푡)^2 [(d푥^4 )^2 +(d푥^5 )^2 ]. (11.45)
The FRW Hubble flow and acceleration are the standard ones,퐻=푎̇∕푎and퐻̇,
respectively. The GB Hubble flow isℎ=푏̇∕푏and the GB acceleration isℎ̇. The model
has (at least) two interesting solutions:

(i) In GB space there is decreasing deceleration,ℎ<̇ 0, towards a stop whenℎ̇=0.

With time, the FRW behavior starts to dominate. In FRW space the expansion

퐻and acceleration퐻>̇ 0 continue until a future singularity as in de Sitter

expansion.

(ii)퐻andℎoscillate around future values of expansion and contraction. In FRW

space there is almost no acceleration,퐻̇ ≈0. In GB spaceℎ̇oscillates between

acceleration and deceleration, finally arriving at a future fixed point.

Lemaitre–Tolman-Bondi Models. A way to explain the dimming of distant super-
novae without dark energy may be an inhomogeneous universe described by the
Lemaˆıtre–Tolman–Bondi solution (LTB) to Einstein’s equation. We could be located
near the center of a low-density void which would distort our measurements of the age
of the Universe, the CMB acoustic scale, and the Baryon Acoustic Oscillations (BAO).
The LTB model describes general radially symmetric spacetimes in four dimen-
sions. The metric can be described by

푑푠^2 =−훼^2 푑푡^2 +푋^2 (푟, 푡)푑푟^2 +퐴^2 (푟, 푡)푑훺^2 , (11.46)

where푑훺^2 =푑휃^2 +si푛^2 휃푑휑^2 ,퐴(푟, 푡)and푋(푟, 푡)are scale functions, and훼(푡, 푟)>0isthe
lapse function.
Assuming a spherically symmetric matter source with baryonic+dark matter den-
sity휌푀, dark energy density휌DE, negligible matter pressure푝푀=0 and dark energy
pressure density푝DE, the stress-energy tensor is

푇휈휇=푝DE푔휇휈+(휌푀+휌DE+푝DE)푢휇푢휈. (11.47)

The( 0 ,푟)components of Einstein’s equations,퐺^0 푟=0, imply

푘̇(푡, 푟)

2 [ 1 −푘(푡, 푟)]

+훼

′퐴̇

훼퐴′

= 0 (11.48)

with an arbitrary function푘(푡, 푟)playing the rôle of the spatial curvature parameter.
Here an overdot is designating휕푡and an apostrophe휕푟.