Tomohiro Harada - Theoretical Astrophysics Group

218 Cosmic acceleration and higher-dimensional gravity
intrinsic curvature terms on the 3-brane and 4-brane. As the first step to this direction, we will look for
the solutions which contain a de Sitter 3-brane. They may give rise to the self-accelerating cosmological
solutions in the simplest six-dimensional cascading DGP model.
The system of our interest is that our 3-brane Universe Σ 4 is placed on a 4-brane Σ 5 , embedded into
the six-dimensional bulk M 6 . For simplicity, we suppress the matter terms in the bulk and on the branes.
The total action is given by
∫
∫
∫
S = M 4 6
2
d 6 X √ −G (6) R + M 5
3
M 6
2
d 5 y √ −q (5) R + M 4
2
Σ 5
2
Σ 4
d 4 x √ −g (4) R, (1)
where G AB , q ab and g µν represent metrics in M 6 , on Σ 5 and Σ 4 , respectively. (i) R (i = 6, 5, 4) are
Ricci scalar curvature terms associated with respect to G AB , q ab and g µν . For the later discussion, it is
useful to introduce the crossover mass scales m 5 := M5 3 /M4
2 and m 6 := M6 4 /M5 3 , which determines the
energy scale where the five-dimensional and six-dimensional physics appear, respectively. We assume that
m 5 > m 6 . Then, it is natural to expect that the effective gravitational theory becomes four-dimensional
for H > m 5 , five-dimensional for m 5 > H > m 6 , and finally six-dimensional for H < m 6 , where H is the
cosmic expansion rate.
We consider the six-dimensional Minkowski spacetime, which is covered by the following choice of the
coordinates
ds 2 6 = G AB dX A dX B = dr 2 + dθ 2 + H 2 r 2 γ µν dx µ dx ν , (2)
where γ µν is the metric of the four-dimensional de Sitter spacetime with the expansion rate H. The r
and θ coordinates represent two extra dimensions and x µ (µ = 0, 1, 2, 3) do the ordinary four-dimensional
spacetime. The surface of r = 0 corresponds to a (Rindler-like) horizon and only the region of r ≥ 0 is
considered. Note that the boundary surface of r = 0 does not cause any pathological effect because it
is not a singularity. We consider a 4-brane located along the trajectory (r(|ξ|), θ(|ξ|)), where the affine
parameter ξ gives the proper coordinate along the 4-brane. The 3-brane is placed at ξ = 0, and for
decreasing value of |ξ| one approaches the 3-brane. We assume the Z 2 -symmetry across the 4-brane and
hence an identical copy is glued to the opposite side. Along the trajectory of the 4-brane ṙ 2 + ˙θ 2 = 1,
where the dot represents the derivative with respect to ξ. The induced metric on the 4-brane is given by
ds 2 5 = q ab dy a dy b = dξ 2 + H 2 r(|ξ|) 2 γ µν dx µ dx ν . (3)
The point where r(ξ) = 0 on the 4-brane corresponds to a horizon and the 4-brane is not extended beyond
it. The 3-brane geometry is exactly de Sitter spacetime with the normalization condition Hr(0) = 1,
ds 2 4 = g µν dx µ dx ν = γ µν dx µ dx ν . (4)
The nonvanishing components of the tangential and normal vectors to the 4-brane are given by u r =
ṙ,u θ = ˙θ,n r = ϵ ˙θ,and n θ = −ϵṙ. We restrict that the region to be considered is to be r > 0 and the
3-brane is sitting on r-axis (θ = 0). The 4-brane trajectory is Z 2 -symmetric across the 3-brane. ˙θ > 0
for increasing ξ. In the case of ϵ = 1, the bulk space is in the side of increasing r, while in the case of
ϵ = −1, the bulk space is the side of decreasing r.
The extrinsic curvature tensor is defined by K ab := ∇ a n b . The junction condition is given by
M 4 6
[
]
K ab − q ab K
(
)
= M5 3(5) G ab + M4 2(4) G µν δ a µ δb ν δ(ξ) . (5)
where the square bracket denotes the jump of a bulk quantity across the 4-brane.By taking the Z 2 -
symmetry across the 4-brane into consideration, the matching condition becomes
−M6 4 ϵ 4 (
1 − ṙ
2 ) 1/2
= −3M
3 1 − ṙ 2 (
r
5
r 2 , M6 4 ϵ − 3 (
1 − ṙ
2 ) 1/2 ¨r
)
+ = 3M 5
3 ṙ 2 + r¨r − 1
r
(1 − ṙ 2 ) 1/2 2 r 2 .(6)
The way to construct the solution is essentially the same as the case of a tensional 3-brane on a tensional
4-brane (See Appendix).