Reconstruction of domain wall universe and localization of gravity

Abstract

We construct a four-dimensional domain wall universe using the Brans–Dicke type gravity with two scalar fields. We give a formulation where for arbitrarily given warp and scale factors, we construct an action which reproduces both the warp and scale factors as a solution of the Einstein equation and the field equations given by the action. This formulation could be described as a reconstruction. We show that the model does not contain ghosts with negative kinetic term and that the localization of gravity occurs as in the Randall–Sundrum model. It should be noted that in the equation of the graviton, there appear extra terms related to the extra dimension, which might affect the fluctuations in the inflation epoch.

Appendix: Explicit expressions of connections and curvatures in five dimensions

In this “Appendix”, we give explicit expressions for the connections and curvatures in five dimensional spacetime, whose metric is given by

$$\begin{aligned} g_{AB}=\begin{pmatrix} -L^2 \mathrm {e}^{u(w,t)}&{}&{}&{}&{}\\ &{} L^2 \mathrm {e}^{u(w,t)}\frac{a(t)^2}{1-kr^2} &{} &{} &{} \\ &{} &{} L^2 \mathrm {e}^{u(w,t)}a(t)^2r^2 &{} &{} \\ &{} &{} &{} L^2 \mathrm {e}^{u(w,t)}a(t)^2r^2\sin ^2\theta &{} \\ &{} &{} &{} &{} 1 \end{pmatrix}. \end{aligned}$$

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The connections are given by

$$\begin{aligned} \Gamma ^t_{tt}&= \frac{1}{2}\dot{u},\quad \Gamma ^w_{tt}=\frac{1}{2}L^2e^uu^\prime , \quad \Gamma ^r_{rt}=\Gamma ^\theta _{\theta t}=\Gamma ^\phi _{\phi t}=\frac{\dot{a}}{a}+\frac{1}{2}\dot{u}, \nonumber \\ \Gamma ^t_{tw}&= \Gamma ^r_{rw}=\Gamma ^\theta _{\theta w}=\Gamma ^\phi _{\phi w}=\frac{1}{2}u^\prime , \nonumber \\\Gamma ^t_{ij}&= L^{-2} \mathrm {e}^{-u}\left( \frac{\dot{a}}{a}+\frac{1}{2}\dot{u}\right) g_{ij}, \quad \Gamma ^r_{rr}=\frac{kr}{1-kr^2}, \quad \Gamma ^w_{ij}=-\frac{1}{2}u^\prime g_{ij} ,\nonumber \\ \Gamma ^\theta _{\theta r}&= \Gamma ^\phi _{\phi r}=\frac{1}{r} , \nonumber \\\Gamma ^r_{\theta \theta }&= -r(1-kr^2) , \quad \Gamma ^\phi _{\phi \theta }=\cot \theta , \quad \Gamma ^r_{\phi \phi }=-r(1-kr^2)\sin ^2\theta ,\nonumber \\ \Gamma ^\theta _{\phi \phi }&= -\cos \theta \sin \theta . \end{aligned}$$

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The Ricci curvatures have the following forms:

$$\begin{aligned} R_{tt}&=\left[ -\frac{1}{2}u^{\prime \prime }-u^{\prime 2}+\frac{3}{2}L^{-2} \mathrm {e}^{-u}\left( \ddot{u}+\frac{\dot{a}\dot{u}}{a} +2\frac{\ddot{a}}{a}\right) \right] g_{tt} ,\nonumber \\R_{ij}&=\left[ -\frac{1}{2}u^{\prime \prime }-u^{\prime 2}+\frac{1}{2}L^{-2} \mathrm {e}^{-u}\left( \ddot{u}+5\frac{\dot{a}\dot{u}}{a} +2\frac{\ddot{a}}{a}+4\frac{\dot{a}^2}{a^2}+\dot{u}^2+4\frac{k}{a^2}\right) \right] g_{ij} ,\nonumber \\R_{ww}&=-2u^{\prime \prime }-u^{\prime 2} \, \nonumber \\R_{tw}&=-\frac{3}{2}\dot{u}^\prime . \end{aligned}$$

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The scalar curvatures is

$$\begin{aligned} R=-4u^{\prime \prime }-5u^{\prime 2}+3L^{-2} \mathrm {e}^{-u}\left( \ddot{u}+\frac{1}{2}\dot{u}^2 +3\frac{\dot{a}\dot{u}}{a}+2\frac{\ddot{a}}{a}+2\frac{\dot{a}^2}{a^2}+2\frac{k}{a^2}\right) . \end{aligned}$$

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