Fate of domain walls in 5D gravitational theory with compact extra dimension

We pursue the time evolution of the domain walls in 5D gravitational theory with a compact extra dimension by numerical calculation. In order to avoid a kink-antikink pair that decays into the vacuum, we introduce a topological winding in the field space. In contrast to the case of non-gravitational theories, there is no static domain-wall solution in the setup. In the case that the minimal value of the potential is non-negative, we find that both the 3D space and the extra dimension will expand at late times if the initial value of the Hubble parameter is chosen as positive. The wall width almost remains constant during the evolution. In other cases, the extra dimension diverges and the 3D space shrinks to zero at a finite time.


Introduction
The possibility that our four-dimensional (4D) spacetime is localized on a domain wall in the extra dimension has been extensively investigated as one of the simplest setups for the braneworld scenario [1]- [14]. Most of them considered the infinite extra dimension. This is because a kink configuration generically induces an antikink configuration due to the periodic boundary condition along the extra dimension, and such a field configuration is unstable and decays into the vacuum. However, this is not the case when the field space is compact and the gravity is neglected. In such a case, a stable kink solution can exist. The stability is ensured by the topological winding around the compact field space [3,4].
Such a compact field space appears in various effective theories as a phase of a complex scalar field that has a nonvanishing vacuum expectation value, just like the axion. Hence In order to discuss the cosmological evolution of the braneworld scenario, the gravity must be taken into account. When the gravity is turned on, the situation for the stability changes. The positive tension of the domain wall warps the ambient geometry, just like the Randall-Sundrum model [15]. Since each domain wall decreases the derivative of the warp factor [16], the periodic boundary condition for the warp factor cannot be satisfied. This indicates that there is no static domain-wall solution in the gravitational theory with the compact extra dimension. Still, we can introduce the topological winding around the field space even in such theories. In this paper, we consider a field configuration with nonzero winding number, and pursue its time evolution by numerical calculation.
The paper is organized as follows. In the next section, we briefly review the case of nongravitational theory, and provide an analytic expression for a static domain-wall solution in the compact extra dimension. In Sec. 3, we extend it to the gravitational theory, and give the field equations to solve. In Sec. 4, we show the numerical results for the time evolution of the domain walls. Sec. 5 is devoted to the summary. In Appendix A, we collect the definition and some properties of the Jacobi amplitude, which expresses the initial domain-wall configuration. In Appedix B, we provide a direct relation between the metric ansatze used in Secs. 3.1 and 3.2. Figure 1: A kink-antikink configuration is unstable, and will decay into the vacuum.
2 Case of non-gravitational theory 2.1 Topological winding in field space Throughout the paper, we consider a five-dimensional (5D) theory whose fifth dimension is compactified on S 1 , i.e., y ∼ y + L (y is the coordinate of S 1 ). Let us first consider a case of non-gravitational theory. Naively, the domain-wall configuration seems to be unstable because a kink configuration should be paired with an antikink configuration due to the periodic boundary condition, and the kink-antikink pair will decay into the vacuum (see Fig. 1). This problem can be solved by introducing a topological winding [3]. As an example, we consider the following model of a real scalar field Φ.
where M = 0, 1, · · · , 4, and where C 1 , C 2 and v are positive constants. The mass dimensions of the parameters are The periodic potential (2.2) has the following vacua.
Here we assume that the field space is compact, and take the following identification.
which is consistent with the scalar potential (2.2). In this case, n-domain-wall solution corresponds to the field configuration with a winding number n.

Domain-wall solution
From (2.1) with (2.2), the equation of motion is A background solution that is independent of the 4D coordinates x µ is found to be where am(u, k) denotes the Jacobi amplitude (see Appendix A). The integration constants are k and y 0 . In the following, we choose y 0 to be zero by shifting the origin of y. Namely, Φ bg (0) = 0. Using the property (A.4), the winding condition (2.6) is translated into where K(k) is the complete elliptic integral of the first kind. This is the equation that determines the value of k. The function kK(k) is approximated as for k ≃ 1 (see Fig. 2). Thus, the size of the extra dimension L is expressed as When L is large enough (i.e., k ≃ 1), the derivative of Φ bg (y) at the origin is determined where the prime denotes the y-derivative. Thus, the width of the domain wall w is insensitive to the parameter k, while the size of the extra dimension L depends on it as (2.11).
If we identify the internal region of the domain wall as the range of y where   the ratio of L to the width w is shown in Fig. 3 in the two-domain-wall case. The horizontal axis denotes s defined below (2.11).
In the limit of L → ∞ (i.e., k → 1), the solution Φ bg (y) approaches the domain wall in the non-compact extra dimension.
As (2.8) shows, Φ bg (y) connects the adjacent two vacua, and the domain walls are located at equal distances. This can be understood from the known fact that each kink configuration feels the repulsive force from other kink configurations.

Domain walls in gravitational theory
Now we extend the previous model (2.1) to the gravitational theory.
where κ 5 is the 5D gravitational coupling, G (5) is the determinant of the 5D metric g M N , and R (5) is the 5D Ricci scalar. The equations of motion are given by As we will show below, there is no static domain-wall solution, in contrast to the nongravitational theory.

Absence of static domain-wall solutions
In order to search for a static domain-wall solution, we take the following ansatz for the metric and the field.
Then, the equations in (3.2) becomes which come from the (µ, ν)-and (y, y)-components of the 5D Einstein equation, and the field equation for Φ, respectively. The prime denotes the y-derivative. The first two equations can be rewritten as When the widths of the domain walls are small enough compared to the size of the extra dimension, each wall can be regarded as a 3-brane with a positive tension. It is well-known that a periodic warp function σ(y) cannot be obtained by introducing only positive-tension branes [2,16]. A negative-tension brane is necessary for a static multi-brane solution in the compact extra dimension. However, such a brane cannot be obtained from any kinds of domain walls. Therefore, we expect that there is no static domain-wall solution in our setup. We can show that this is indeed the case as follows.
For our purpose, it is convenient to use the first-order formalism [17,18,19]. We introduce the function W = W (Φ), which satisfies where W Φ ≡ dW/dΦ. These are consistent with the first equation in (3.5). From the second equation in (3.5), the potential is expressed as By using this expression, we can show that (3.6) is also consistent with the last equation we can reproduce the potential (2.2), up to a constant term.
From the second equation in (3.6), we have 1 This indicates that W Φ (Φ) is a periodic function with the period v. 2 Besides, from the expression (3.7), |W (Φ)| must be bounded from above. Therefore, W (Φ) must satisfy the following conditions.
• W (Φ) is a periodic function of Φ with the period v. 1 We assume that Φ(y) is a monotonic function of y. 2 Φ(y) is unbounded due to the boundary condition (2.6).
• The following relation must be held Hence, the integral in the LHS of (3.10) diverges, and the requirement (3.10) cannot be satisfied. Namely, there is no static solution that has a non-zero winding number.

Non-static domain-wall solutions
As we showed, any domain walls must depend on time. In order to see this time-evolution, we take the following metric ansatz [6,8].
where the dot and the prime denote the derivatives with respect to t and z, respectively.
If the (M, N)-component of the 5D Einstein equation is denoted as G M N = 0, the first two equations in (3.13) are G ii + 2 3 (G tt − G zz ) = 0 and 1 3 (G tt − G zz ) = 0, while the equations in (3.14) are − 1 3 G tz = 0 and − 1 3 G tt = 0, respectively. The other components do not provide non-trivial equations. Note that the equations in (3.14) do not contain the second-order derivative with respect to time. Thus, they are treated as the constraints in the numerical calculation for the time evolution.
In the limit of decompactifying the extra dimension, the static solution is allowed. In fact, neglecting the time-dependences of the background functions, 4 and redefining the extra-dimensional coordinate as The physical size of the extra dimension at time t is For larger length scales thanL phys (t), the spacetime becomes 4D-like. From (3.12), the line elements along the time and the 3D space directions that are measured on the wall are where f ob (t, z) denotes the wave function of the observer in the extra dimension at time t.
Thus, the effective 4D metric is The cosmic time τ is thus given by In terms of τ , the 4D metric is rewritten as 21) 4 The two functions A(t, z) and B(t, z) should be reduced to the same functionσ(z) if we require the 4D Lorentz invariance. 5 Note that ∂ z = e σ(y) ∂ y .
where the scale factor a(τ ) is defined by Here, t(τ ) is the inverse function of (3.20). The Hubble parameter H is then expressed as In terms of τ , the size of the extra dimension (3.17) is rewritten as L phys (τ ) ≡L phys (t(τ )). (3.24)

Numerical results
In this section, we show our numerical results. We focus on the case of two domain walls, as an example. As an initial configuration, we choose the static solution in the non-gravitational case. Namely, A = B = 0 and Φ = Φ bg , which is shown in (2.8) with (2.9) for n = 2. Then, since we work in the gravitational theory, the t-derivatives of the fields must have nontrivial profiles due to the constraints in (3.14). Specifically, the initial configuration is given by where b 0 is a real constant, and the constant k is determined by the model parameters At the second equality forȦ(0, z), we have used that In order to calculate the effective 4D metric, we choose the wave function of the observer 4) where N ob (t) is the normalization factor that satisfies which lead to v = 0.685. As we will show below, the time evolution of the configuration depends on the minimal value of the potential, First we consider a case of V min ≥ 0 (C 1 ≥ 0.1). Since the qualitative behavior of the background configuration does not depend much on a specific value of V min as long as V min ≥ 0, we mainly focus on the case of V min = 0 in this subsection. Fig. 4 shows the profile of Φ at various times. We can see that the scalar configuration once loses the kink shape and approaches a linear function of z. Then, after some time, it starts to form the kink configuration again, and approaches the (periodic) step function as τ → ∞. From Fig. 4, it seems that the scalar configuration reaches the singular stepfunction profile at a finite time t ≃ 32. However, Fig. 5 indicates that it takes infinite time for the cosmic time τ . Thus, the step-function profile is just an asymptotic configuration.
Here note that a distance measured by the coordinate z is not the physical one. It should be measured by the proper length. As we will see below, the above-mentioned behavior   to the Randall-Sundrum model [15]. The positive tensions of the walls warp the ambient geometry. In contrast, the "3D scale factor" B monotonically increases with time. Its z-dependence also grows, and it has peaks at the wall positions, which is similar to the behavior of A at late times.    As mentioned above, the width of the domain wall must be measured by the proper length. Here we identify the domain wall region as the range of z in which Then, the physical wall width w phys is defined by where z w is determined by Φ(z w ) = 4v 10 . (4.10) From Fig. 7, we can see that the width almost remains constant during the evolution.
Therefore, the behavior of Φ that approaches the singular step function shown in Fig. 4 is understood as a result of the expansion of the extra dimension. Namely, although w phys itself does not decrease, the ratio w phys /L phys approaches zero because of the linear increase of L phys . indicates that our 3D space will shrink (see Fig. 11).  and V min = −0.05.
As we can see from Fig. 11, the size of the extra dimension L phys diverges at a finite value of τ . So we cannot continue the numerical calculation beyond this time. This is before the scalar configuration reaches the step function profile. Beyond this time, the theory should be treated as 5D theory with non-compact extra dimension. The wall width w phys roughly stays constant during the evolution. In our calculations, we have assumed that A and B have constant profiles in the extra dimension at the initial time, just for simplicity. They will develop nontrivial profiles at late times. In the case of V min ≥ 0, they will have peaks at the positions of the domain walls ( Fig. 10). This is similar to the Randall-Sundrum model [15], in which the 3-brane with a positive tension warps the ambient geometry and the warp factor has a peak at the brane.
In the case of V min < 0, on the other hand, A will have peaks at the middle points between the walls. This indicates that, in the extra-dimensional direction, the region between the walls will expand faster than around the walls. As for the 3D space, B will have mimima at the middle points (Fig. 10). Thus, the non-compact 3D space directions will shrinks faster there compared to those at the wall positions. Recalling that the scale factor a(τ ) is mainly affected by the geometry around the walls due to the wave function of the observer (see (3.18) and (3.22)), the 3D space in the bulk region shrinks faster than that shown in Fig. 11, and will collapse.
Although Fig. 4 shows that the scalar profile approaches the singular (periodic) step function, this just means that the ratio of the wall width to the size of the extra dimension becomes small due to the expansion of the latter. In fact, the physical wall width w phys , which determines the mass scale of the excited modes localized at the wall, does not decrease. It almost remains constant during the evolution.
If we choose the initial condition such that the initial value ofḂ is negative, the behavior of the configuration is similar to those at late time in the case of Sec. 4.2. Thus, our calculations fails at a finite time.
In summary, the background configuration evolves without the collapse of the 3D space only when V min is non-negative and b 0 is positive. Otherwise, the setup will be destabilized at a finite cosmic time. The extra dimension always expands at late times while whether the 3D space expands or shrinks depends on the sign of V min . The former property is related to the fact that the kink configuration feels the repulsive force from other kinks.
For the purpose of constructing a realistic model, the extra dimension must be stabilized at some finite value. Inspired by the Goldberger-Wise mechanism [20], an extra 5D scalar field might be necessary. The introduction of the extra scalar that induces an attractive force between the kinks makes it possible to stabilize the extra dimension. Therefore, an extension of our analysis to a model that has multi scalar fields with nontrivial topological windings is an intriguing subject. We will discuss this issue in a subsequent paper.

H.A. is supported by Institute for Advanced Theoretical and Experimental Physics, Waseda
University.