Dynamical Domain Wall and Localization

Based on the previous works (arXiv:1202.5375 and 1402.1346), we investigate the localization of the fields on the dynamical domain wall, where the four dimensional FRW universe is realized on the domain wall in the five dimensional space-time. Especially we show that the chiral spinor can localize on the domain wall, which has not been succeeded in the past works as the seminal work in arXiv:0810.3746.


Domain wall model with two scalar fields
General FRW in the five dimensional space-time, the metric is given by { dr 2 1 − kr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 } .
Action with two scalar fields ϕ and χ: } .
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Field equations read
We now choose ϕ = t and χ = w.
We can construct a model to realize the arbitrary metric in the form, ) .
Einstein equation is satisfied. Field equations are nothing but the Bianchi identities.
If any eigenvalue of the matrix is negative, there appears a ghost.

Example without ghost
· · · No ghost. General FRW universe can be realized by the Brans-Dicke type model.

Localization of graviton
Perturbation g µν → g µν + h µν . Assume h µν = 0 if µ or ν ̸ = 1, 2, 3. Einstein eq. ⇒ equation for graviton in five dimensions: If u goes to minus infinity sufficiently rapidly for large |w|, h ij (w, x) is normalized in the direction of w and therefore there occurs the localization of graviton.

Graviton localized on the domain wall
In the four dimensional FRW space-time,

Dirac equation in five dimensions
f χ: a function of the scalar field χ = w, general yukawa interaction. ∇ M : covariant derivative: LetC 1 (w) and D 1 (t) be arbitrary functions.

Condition of localization
-Typeset by FoilT E X - It was assumed that the warp factor does not depend on the time but we have used the time-dependent warp factor.

Localization of vector field
Action of vector field

Variation of
Last two eqs. : nothing but the field equations of the vector field in four dimensions. First eq. : a gauge condition, which is a generalization of the Landau gauge, ∂ µ A µ = 0.

Summary
• We have constructed the dynamical domain wall, where arbitrary four dimensional FRW universe is realized on the domain wall in the five dimensional space-time.
• Graviton, Chiral Spinor, and Vector can localize on the brane.
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Graviton in FRW Universe in Four Dimensions
Perturbation consider scalar field theory

Imposing the gauge condition
Because we are now interested in the graviton, we may assume h µν = 0 except the components with µ, ν = 1, 2, 3.

Explicit expressions of connections and curvatures in five dimensions
Metric -Typeset by FoilT E X -

Explicit form of spin connection
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