de Sitter Thin Brane Model

We discuss the large mass hierarchy problem in a braneworld model which represents our acceleratively expanding universe. The Randall-Sundrum (RS) model with warped one extra dimension added to flat 4-dimensional space-time cannot describe our expanding universe. Here, we study instead the de Sitter thin brane model. This is described by the same action as that for the RS model, but the 4-dimensional space-time on the branes is $\rm dS_4$. We study the model for both the cases of positive 5-dimensional cosmological constant $\Lambda_5$ and negative one. In the positive $\Lambda_5$ case, the 4-dimensional large hierarchy necessitates a 5-dimensional large hierarchy, and we cannot get a natural explanation. On the other hand, in the negative $\Lambda_5$ case, the large hierarchy is naturally realized in the 5-dimensional theory in the same manner as in the RS model. Moreover, another large hierarchy between the Hubble parameter and the Planck scale is realized by the ${\mathcal O}(10^2)$ hierarchy of the 5-dimensional quantities. Finally, we find that the lightest mass of the massive Kaluza-Klein modes and the intervals of the mass spectrum are of order $10^2\,\rm GeV$, which are the same as in the RS case and do not depend on the value of the Hubble parameter.


Introduction
In the last 15 years, it has been believed that braneworld models may give a solution to the problem of the extremely large hierarchy between the Planck scale, 10 19 GeV, and the weak scale, 10 2 GeV. In 1998, Arkani-Hamed, Dimopoulos and Dvali first tried resolving the hierarchy problem by using a braneworld model [1,2]. Their explanation is as the follows: First, they assume a flat (4 + d)-dimensional space-time, in which the extra d-dimensions are compactified by a common radius L. The relation between the (4 + d)-dimensional Planck mass M pl(4+d) and the 4-dimensional one M pl is given by Second, they assume that the fundamental scale is only the TeV scale, 1 TeV. Therefore, M pl(4+d) should be of order 1 TeV. This means that a too large M pl is specific to the 4-dimensional theory, and the large hierarchy does not exist in higher dimensions. For example, if we take the number of the extra dimensions d as 2, Eq. (1) implies that the radius L is approximately 1 mm. This does not contradict experiments, since the Newton's law has been verified only at distances larger than 1 cm. However, note that the energy scale of 1/L is much smaller than the weak scale. 1 In other words, the hierarchy between the Planck scale and the weak scale is just replaced by the one between the weak scale and the radius L.
In keeping with this problem, Randall and Sundrum proposed a new braneworld model (the RS model) [3,4]. 2 First, they assume a 5-dimensional space-time with negative 5dimensional cosmological constant Λ 5 and warped extra dimension y. Moreover, the extra dimension is compactified by S 1 /Z 2 with radius L. Under these assumptions, the metric is given by Hence, two branes are naturally introduced at the fixed points, y = 0, L. Second, they assume the energy scales (the vacuum expectation values of the Higgs boson) on the two branes to be 10 19 GeV (the Planck scale) and 10 2 GeV (the weak scale), respectively. This means that the hierarchy does not exist on the former brane (the Planck brane), since its energy scale is equal to the 4-dimensional Planck mass M pl . We live on the latter brane (the TeV brane), where the energy scale is equal to the weak scale. Then, the large hierarchy on the TeV brane is realized by the O(10) hierarchy, kL ∼ 39, between the 5-dimensional quantities k and 1/L. Moreover, we find that the 5-dimensional Planck mass M pl (5) is of the same order as k and 1/L. In this way, the 4-dimensional hierarchy is naturally explained in the 5-dimensional theory without any unnaturally large hierarchies among the 5-dimensional quantities.
However, the RS model does not represent our acceleratively expanding universe, since the 4-dimensional space-time on the branes is assumed to be flat. Thus, it is necessary to study the models in which the 4-dimensional space-time on our brane is dS 4 . Of course, there have been some papers studying such models, for example, [5,6,8,9]. In the later two papers [8,9], the author assumes one warped and uncompactified extra dimension y with negative Λ 5 . In addition, he introduces only a single brane with dS 4 space-time. Then, the 5-dimensional metric of the model is given by where H is the Hubble parameter and ξ is an arbitrary constant. In this model, the relation between M pl (5) and M pl is given by However, this is divergent at y = ±∞, and therefore, we cannot discuss the hierarchy problem in this model.

Motivation for our work
The above divergence problem of (4) is from the infinitely large integration range, namely, the uncompactified extra dimension. If we compactify the extra dimension by S 1 /Z 2 similarly to the RS model, two branes are naturally introduced at the fixed points. Then, the integration in (4) becomes finite, and we can discuss the hierarchy problem. In other words, for making it possible to discuss the hierarchy problem, we need to introduce (at least) two branes. In addition, since we have no apriori reason to restrict ourselves to the Λ 5 < 0 case, we will discuss the hierarchy problem for both the cases Λ 5 > 0 and Λ 5 < 0. We call this model where the space-time on the branes is dS 4 "the de Sitter thin brane model" for brevity. Here, we have used the word "thin" to distinguish our model from thick brane models with smooth warp factors [10][11][12].
We should also examine the Kaluza-Klein modes for verifying the existence of the massless mode corresponding to the graviton, and non-existence of the light massive modes which could affect the Newton's law. For this purpose, we need the exact expression of the wave function of the Kaluza-Klein modes, and this was constructed, for example, in [6,7]. In the RS model, both the lightest mass of the massive Kaluza-Klein modes and the intervals of the mass spectrum are of order 10 2 GeV, which suggests the possibility of the massive Kaluza-Klein particles being observed. It is interesting to study the observability of the Kaluza-Klein modes in our model. Actually, in [5,6], the authors compactify the extra dimension and introduce two branes at the fixed points in the de Sitter thin brane model. They discuss the hierarchy problem and the Kaluza-Klein modes in the model. However, the Hubble parameter was not accurately determined at that time. The aim of this paper is to give a complete analysis of the hierarchy problem and the Kaluza-Klein modes in the de Sitter thin brane model by using the observed value of the Hubble parameter. Since our real universe has experienced much more complicated time evolution, namely, big-bang → inflation → reheating → deceleration → accelerative expansion (current), we should introduce the time-dependent Hubble parameter. However, in this paper, for simplicity, we focus only on the current universe with a constant Hubble parameter. 3 The parameter ξ appearing in the metric (3) plays an important role in our model. Before compactifying the y-direction to S 1 /Z 2 , ξ is merely the freedom of translation in the y-direction. However, after the compactification, if we put ξ equal to zero, we will find that the brane tension is divergent and the model turns out to be sick. Thus, for keeping the model sound, we must take non-zero ξ. Moreover, we will also find that the parameter ξ is important in naturally explaining the large hierarchy in our model, since, after the compactification, the integration (4) is finite and depends on ξ.

Outline of the results
We will find the following three important facts for our de Sitter thin brane model: Recall that two branes are naturally introduced in the present model similarly to the RS model. First, in the case with Λ 5 < 0, if we choose the energy scales on the two branes as 10 19 GeV and 10 2 GeV, respectively, the 5-dimensional quantities can be almost of the same order; namely, the 4-dimensional large hierarchy is naturally realized in the 5-dimensional theory similarly to the RS model. However, for Λ 5 > 0, we find that the large hierarchy in 4dimensions necessarily implies a large hierarchy in 5-dimensions. Thus, we conclude that we must choose Λ 5 < 0 to explain the hierarchy naturally.
Second, for Λ 5 < 0, the lightest non-zero mass of the Kaluza-Klein modes and the intervals of the mass spectrum are both of order 10 2 GeV, which are insensitive to the value of the Hubble parameter. Therefore, the lightest mass and the intervals in our model are the same as those in the RS model.
Finally, we find the importance of the parameter ξ as we have already mentioned above. Such a parameter certainly exists in the RS model. However, it only effects a constant multiplication to the warp factor, which can be offset by a rescaling of x µ . On the other hand, in our model, ξ has a physical meaning. Especially, ξ is related to the Hubble parameter H. Another 4-dimensional large hierarchy between H and the Planck scale is realized by the O(10 2 ) hierarchy, kξ ∼ 102, between the 5-dimensional quantities k and 1/ξ.
From the above results, we conclude that our thin brane model with Λ 5 < 0 can represent our acceleratively expanding universe, and at the same time naturally explain the large hierarchy.
The organization of this paper is as follows: In §2, we quickly review the RS model, with emphasis on the explanation of the hierarchy and the analysis of the Kaluza-Klein modes. In §3, we introduce the de Sitter thin brane model and study various properties of it: explanation of the hierarchy problem, analysis of the graviton modes, and consideration of the important parameter ξ. Finally, in §4, we discuss the possibility of making the Hubble parameter time-dependent in our model.

Quick review of the RS model
In this section, we briefly summarize the RS model for use in later sections.

Setup and the classical solution
Let us consider the 5-dimensional space-time x M = (x µ , y) described by the following bulk action with negative cosmological constant Λ 5 : where M pl(5) is 5-dimensional Planck mass. Under the metric assumption, the solution to the Einstein equation with the condition that the extra dimension y is compactified by S 1 /Z 2 with radius L is given by where the constant k is defined by 4 The absolute value in (7) is due to the compactification, and it causes the delta functions, δ(y) and δ(y − L), in A (y). Because of these extra delta function terms, the Einstein equation is in fact not satisfied at y = 0, L. To canceling the extra terms, we introduce the following two rigid brane actions located at y = 0, L: Here, y a is the brane position, and g a is the induced metric on the brane, From the requirement that the Einstein equation of the whole system S + S 1 + S 2 holds at y = 0, L, the tension λ a is determined as In this way, the compactification to S 1 /Z 2 naturally introduces the branes.

Exponential hierarchy
In this subsection, we describe how the hierarchy is naturally explained in the RS model. We introduce the Higgs field H with symmetry breaking Mexican hat like potential on each of the two branes located at y = 0, L: Since the coefficient of the kinetic term e −2k|ya| (see (10)) is not equal to 1 for the second brane, we redefine H to normalize it. Then, the vacuum expectation value v a of the redefined fieldH on the brane a is expressed as and their ratio is given by Note that v a is regarded as the energy scale on the brane a. Now, we assume that the energy scale v 1 on the first brane is of the order of the Planck scale, v 1 = M pl ∼ 10 19 GeV.
And we assume also that our universe is on the second brane and therefore that v 2 = M w ∼ 10 2 GeV (the weak scale). Then, (14) implies that which is neither too large nor too small. From now on, we call the brane at y = 0 (y = L) "the Planck brane" ("the TeV brane").
To understand how the 4-dimensional hierarchy is naturally realized in the 5-dimensional theory, we need to drive the relation between the 4-dimensional Planck mass M pl and the 5-dimensional one M pl (5) . For this purpose, we add to the metric (6) the following perturbation which does not depend on y: 5 where we have chosen the RS gauge with Then the 5-dimensional action (5) (without the cosmological constant term) is reduced to where the 4-dimensional metricf µν is given bỹ andR andR 4D are made fromg M N andf µν , respectively. From (18), we get the following relation between the 4-dimensional Planck mass and the 5-dimensional one: Note that we can neglect e −2kL since we are taking kL 39. The relation (20) is, for example, realized by taking In this case, all the 5-dimensional quantities, M pl(5) , k, and 1/L, are approximately equal to M pl , and this is a welcome result. Thus, the 4-dimensional hierarchy M pl /v 2 ∼ 10 17 is realized without introducing unnatural 5-dimensional hierarchies.
Then, is the Planck scale the unique 5-dimensional fundamental scale? The answer is no. For example, let us adopt a new coordinatex µ related to the original x µ by the scale transformationx The warp factor for the coordinatex µ , is normalized at y = L (the TeV brane position). Since d 4 x andR 4D (x) are equal to e +4kL d 4x and e −2kLR4D (x), respectively, the 5-dimensional action (18) is now given by This implies that the relation (20) is modified to In this case, M pl ∼ 10 19 GeV is realized by taking in which all the 5-dimensional quantities are approximately of the weak scale. In this way, we can arbitrarily change the energy scale of all the 5-dimensional quantities through the transformation. In the above two examples with the coordinates x µ andx µ , all the 5dimensional quantities are of the scale of the brane at which the warp factor is normalized.

Graviton modes
Since the extra dimension y is compactified, there appear the towers of the Kaluza-Klein modes. We must verify the existence of the massless graviton corresponding to the zeromode. We must also verify the non-existence of light massive modes which could affect the Newton's law. Let us consider the perturbation (16) with h M N now depending on y as well as on x µ . Under the assumption that h µν can be expanded as with φ (n) µν (x) being the 4-dimensional field with mass m n , the modes ψ n (y) excluding the points |y| = 0, L are given by Here, J α and Y α are the Bessel functions of the first and second kind, respectively, and a n and b n are constants. In particular, ψ 0 (y) is the zero-mode with m 0 = 0. Note that the y dependences of the n = 0 term in (27) cancel.
The 4-dimensional mass m n and the ratio a n /b n are determined by the boundary conditions at the brane positions. These conditions are derived by integrating the differential equation for ψ n (y) in infinitesimal small regions containing the brane positions. We find that m n for the massive modes are determined by the following equation: For a very small m n with m n /k 1/e, 6 the first term of (29) can be neglected, since we have Therefore, m n is determined as where j n is the n-th zero of J 1 . Since the intervals of adjacent zeroes of J 1 are approximately equal to π, the mass difference ∆m n is given by This result implies that the Newton's law remains unmodified for a scale larger than 10 −18 m. However, the first massive Kaluza-Klein particle could be observed in the near future.

de Sitter thin brane model
The RS model assumes that the 4-dimensional space-time on the branes is static. However, we know that our real universe is acceleratively expanding. In this section, we construct a 5-dimensional braneworld model where the 4-dimensional space-time on the branes is dS 4 describing our expanding universe. The warp factor in our model has cusp singularities at the brane positions as in the RS model. Therefore, we call our model "the de Sitter thin brane model" in contrast to "the de Sitter thick brane model" where the warp factor is smooth and has non-singularities [10][11][12]. As mentioned in §1.1, we consider for simplicity only our current universe with a constant Hubble parameter.

Setup and the classical solution
Let us consider the 5-dimensional action (5) and the following metric: Here, we consider both the cases of Λ 5 > 0 and Λ 5 < 0. The main difference from the RS model of §2 is that the 4-dimensional space-time is not static, but is the FLRW metric with flat space, which is the simplest metric describing our acceleratively expanding universe. Under these assumptions, the Einstein tensor G M N is expressed as follows: where the overdots and the primes denote derivatives with respect to t and y, respectively. Thus, from the Einstein equation, we get the differential equations for A(y) and a(t): Plugging the solution of the last equation of (35), where H is an arbitrary constant (the Hubble parameter), into the rest of the equations of (35), we obtain Two differential equations in (37) are not independent; the first is obtained by differentiating the second. In any case, the general solution is given by where ξ is an arbitrary constant, and k in the present model is defined by 7 In (38), we have introduced a new function Sin(h)(x) defined by for treating the both cases Λ 5 ≷ 0 by a single equation. Later we will also introduce Cot(h) defined similarly. Now, to introduce two branes naturally, let us compactify the 5th-dimension by S 1 /Z 2 with radius L. Then, the expressions of A(y) and its derivatives are altered as follows: A (y) = k sgn(y) Cot(h)(k(|y| + ξ)), Due to the delta function term in A (y), the Einstein equation is not satisfied at y = 0 and y = L. To compensate, we must introduce the following two brane actions S a (a = 1, 2): where y 1,2 = 0, L are the brane positions, and g a are the induced metrics on the branes: 7 Since (38) is invariant under the replacement k → −k, we choose k to be positive.

Hierarchy problem
Let us consider whether the above time-dependent model can explain the hierarchy problem. We introduce the same Higgs action as (12) on each of the two branes a = 1, 2. For the present metric (33), it is given by where we have introducedH = e A(ya) H to normalize the kinetic term. Thus, the vacuum expectation value v a of the fieldH on the brane a is given by which can be regarded as the energy scale on the brane a. Using (45), the ratio v 2 /v 1 is given by Since the numerator and the denominator of (46) is exchanged under the replacement ξ → −(L + ξ), we can restrict ourselves to the case v 1 < v 2 without loss of generality. Now, let us consider the situation, v 1 = M w ∼ 10 2 GeV and v 2 = M pl ∼ 10 19 GeV, which means that our universe is on the first brane and the 4-dimensional hierarchy dose not exist on the second brane. Hereafter, we call the branes at y = 0 and y = L "the TeV brane" and "the Planck brane", respectively. In the following, we consider whether the large hierarchy v 2 /v 1 ∼ 10 17 can be realized without introducing any unnatural hierarchies among the 5-dimensional quantities k, L and ξ, for both the cases Λ 5 > 0 and Λ 5 < 0.
In this case, our problem is how the condition sin(k(L + ξ)) sin(kξ) ∼ 10 17 can be naturally realized. Examination of this condition for both the cases of kL = O(1) and kL 1 (modulo integer multiples of π) leads to a single requirement namely, we need a fine tuning.
To verify whether the present model can explain the large hierarchy, we must calculate the relationship between the 5-dimensional Planck mass M pl(5) and the 4-dimensional one M pl . For this purpose, let us add a perturbation to the metric (33) as follows: 9 where we have taken the RS gauge (17). The Ricci scalarR(x, y) made from the metric (50) is calculated asR whereR 4D (x) is the Ricci scalar made fromf µν (x) := f µν (t) + h µν (x). Thus, the Ricci scalar part of the 5-dimensional action (5) is given as follows: From this, we get the following relationship between M pl and M pl (5) : In the following, we would like to take as the Hubble parameter H the observed value. However, the Hubble parameter H can be varied by a rescaling of the 4-dimensions x µ . 8 In deriving (49), we used the formula sinh −1 x = log(x + √ x 2 + 1) to rewrite (46) as follows: k(L + ξ) ∼ log 10 17 sinh(k|ξ|) + {10 17 sinh(k|ξ|)} 2 + 1 k|ξ| 10 −17 log(2 · 10 17 sinh(k|ξ|)) 39 + log e kξ − e −kξ .
Note that we have not made any restrictions on kL. 9 As we will see in the next subsection §3.3, the zero-mode h When the warp factor is normalized at y = 0 (the TeV brane position), the Hubble parameter H should be equal to the observed value on the TeV brane H 0 ∼ 10 −42 GeV. Therefore, we adopt a new 4-dimensional coordinatex µ defined bȳ Then, the metric (33) is modified to withf Accordingly, the relation between M pl and M pl(5) (53) is modified to namely, the rescaling is just equivalent to replacing H with H 0 . Hereafter, when we use the observed value of the Hubble parameter, we rescale x µ as (54) and use H 0 . Now, we impose that M pl(5) ∼ k, namely, a requirement of the absence of the 5dimensional hierarchy. Therefore, (57) is rewritten as where we have used M pl ∼ 10 19 GeV and H 0 ∼ 10 −42 GeV. We will examine (58) for the cases of Λ 5 > 0 and Λ 5 < 0.
In this case, the second term inside the curly brackets of (58) is at most 1. Therefore, we must take as kL an extremely large value, kL ∼ 10 122 .
In this case, from (49) and (58), we obtain and solving this equation numerically, we get two solutions: (kξ, kL) ∼ (+102, 39), (−102, 243). (60) Both of these values are consistent with our assumption k|ξ| 10 −17 , and at the same time show that kL is neither too large nor too small. As we will see in §3.4.1, we must exclude the negative ξ case, since the action of fluctuation diverges. However, we will continue our argument without restricting ourselves to the positive ξ case. At this point, we conclude that the case Λ 5 < 0 is a candidate for solving the hierarchy problem.
In the above discussion, we did not mention the absolute values of the 5-dimensional quantities M pl (5) , k, 1/L and 1/ξ. However, we can fix the value of k from the expression of H 0 (56) and (60) to obtain k ∼ 10 2 GeV. Consequently, all the absolute values of the 5-dimensional quantities are uniquely fixed to be almost of the same order 10 2 GeV, which is equal to the weak scale M w (the energy scale on the TeV brane). In this respect, the de Sitter thin brane model is largely different from the RS model (see §2.2); the RS model lacks information that can uniquely fix the absolute values of the 5-dimensional quantities.

Graviton modes
Next, let us study the Kaluza-Klein graviton modes in the present model. In particular, we are interested in whether the massless graviton exists, 10 and the effect of the massive modes on the Newton's law. Then, we consider the perturbed metric (50) with h µν now having the y-dependence as well as the x µ -dependence, h µν = h µν (x, y). Moreover, we assume that h µν can be expanded as where φ (n) µν (x) is the 4-dimensional field with mass m n . The modes ψ n (y) have to satisfy the following differential equation [6,7]: Here, we have defined M n as The general solution to (62) excluding |y| = 0, L is given by ψ n (y) = a n P Mn 3/2 cosh(k(|y| + ξ)) + b n Q Mn 3/2 cosh(k(|y| + ξ)) , where P µ ν and Q µ ν are the associated Legendre functions of the first and second kind, respectively. Similarly to the case of the RS model of §2.3, the 4-dimensional mass m n and the ratio a n /b n are determined by the boundary conditions at the brane positions obtained by integrating (62) in the infinitesimal small regions containing y = 0 and y = L. These conditions are given by namely, 11 a n P Mn 1/2 cosh(k(y a + ξ) + b n Q Mn 1/2 cosh(k(y a + ξ)) = 0.
The masses m n are determined by the condition of the existence of non-trivial (a n , b n ):

Zero-mode
For the zero-mode with m 0 = 0 (M 0 = 3/2), we see that (67) is realized owing to the following relations: Moreover, from (65) and the following relations, we see that the zero-mode ψ 0 (y) is given by Hence, from (61), we find that the zero-mode part of h µν (x, y) does not depend on y: 11 In deriving (66), we have used the following recursion relation: and the same relation for Q µ ν (w) [13].

Massive modes
Now, let us consider the left-hand side of (67) with M n replaced with M , and denote it by B(M ; kξ, k(L + ξ)). We seek the zero points of B(M ; kξ, k(L + ξ)) as a function of M . for the value of k|ξ| given by (60), k|ξ| ∼ 102 (kL is related to k|ξ| by (49)), and the result is shown in Fig. 1.
For pure imaginary M (m 2 > 9 4 H 2 ), this analysis is impossible to carry out with Mathematica due to overflow and underflow problems. However, since we have we can approximately determine the masses m n by solving For convenience, we denote the left-hand side of (74) by Q(M ; kξ, k(L + ξ)). Then, we can numerically analyze Q M ; kξ, k(L + ξ) = 39 + log e kξ − e −kξ (75) for k|ξ| ∼ 102, and the result is shown in Fig. 2. 12 From Figs. 1 and 2, we realize that only the zero-mode, M 0 = 3/2, exists in the range 0 ≤ m ≤ 3 2 H, and the zeroes in the range m > 3 2 H appear at almost even intervals ∆M n = |M n+1 | − |M n |. To be exact, ∆M n depends on n, and for larger n, it seems to converge to a constant (see Fig. 3). The value of M 1 corresponding to the mass m 1 of the first massive mode and that of ∆M n (n 1) are determined as and accordingly, m 1 and the mass intervals ∆m n = m n+1 − m n are as

Tachyonic modes
From Fig. 1, we see that another zero point M = 5/2 exists in the range m 2 ≤ 9 4 H 2 . In addition, more zero points exist at M = 7/2, 9/2, 11/2, ..., though not shown in Fig. 1. The masses m corresponding to these M 's are pure imaginary and "tachyonic"! If these tachyonic modes really existed, the present model would fail. Fortunately, these modes do not actually exist. The origin of the problem is that P (2r+1)/2 3/2 (cosh w) is proportional to Q (2r+1)/2 3/2 (cosh w) for an integer r ≥ 2, and (67) is automatically satisfied. For these exceptional values of M , we must prepare two special independent solutions to (62). For M −1 = 5/2 (m −1 = 2iH), the general solution to (62) is given by However, this cannot satisfy the boundary conditions (65) except for the trivial case a −1 = b −1 = 0, implying that the tachyonic mode with M −1 = 5/2 does not exist. The same is expected to be true for other possible tachyonic modes.

Parameter ξ
In this last subsection, we discuss the importance of the parameter ξ. As we saw in §3.1, it came from the Einstein equation as an integration constant. Such a parameter can also appear in the RS model, though we did not consider it. If it is included, the solution A(y) (7) is modified as and the warp factor e 2A(y) as This implies that the parameter ξ causes only a constant multiplication to the warp factor, which can be offset by a rescaling of x µ . Therefore, we do not need to consider the parameter ξ in the RS model.
On the other hand, in our model, we cannot offset the parameter ξ. Moreover, for Λ 5 < 0, kξ is related to the Hubble parameter H. Restoring H and M pl in (59), the relation between H and kξ is given by From this relation, we find that another 4-dimensional large (∼ 10 122 ) hierarchy between the Hubble parameter H 0 and the Planck scale M pl is realized by the O(10 2 ) hierarchy, k|ξ| ∼ 102, between the 5-dimensional quantities k and 1/ξ.
From the above argument, we see that k|ξ| must not be zero. This is consistent with the requirement from (42); ξ = 0 implies that the tension of brane 1 located at y = y 1 = 0 becomes infinite. Hence, the non-zero ξ keeps the model sound. In the above discussion, there is no way to determine the sign of ξ. In other words, we can carry out the analysis of the Kaluza-Klein modes for the both solutions of (60). For negative ξ, the warp factor e 2A(y) has a zero at y = −ξ, namely, the 4-dimensional spacetime shrinks to a point there. From the geodesic equation, we can show that particles go through this point. However, we can also show that the action of the fluctuation h µν (x, y) is divergent for negative ξ. In fact, using (62) for ψ n (y), we find that the quadratic part of the 4-dimensional fields φ This integral is divergent at y = −ξ, since both e −A(y) = 1/ |sinh(k(|y| + ξ)| and ψ 2 n (y) are divergent at y = −ξ. (The mode ψ n (y) (64) can be chosen to be real.) Thus, we conclude that the negative ξ case must be excluded, and, among the two candidates of (60), only (kξ, kL) ∼ (+102, 39) is the acceptable one. In §3.3.2, we determined the mass m 1 of the first massive Kaluza-Klein mode and the intervals of the mass spectrum ∆m n for kξ ∼ 102, which is determined by the observed value of the Hubble parameter H 0 ∼ 10 −42 GeV. Here, let us consider how m 1 and ∆m n depend on kξ or equivalently on H 0 . This is to consider the RS limit of H 0 → 0. Keeping the 4-dimensional Planck mass M pl a constant, M pl ∼ 10 19 GeV, the relation between kξ and H 0 is given by (81). In Fig. 4 (Fig. 5), we give the plots of log |M 1 | and log ∆M n (m 1 and ∆m n ) as functions of kξ.
From Fig. 4, we realize that |M 1 | and ∆M n are both monotonically increasing functions of kξ. However, from Fig. 5, we also see that m 1 and ∆m n are both almost independent of kξ. This means that the product of H 0 and |M 1 | and that of H 0 and ∆M n are constants independent of kξ: where we have made the large m n approximation of (63), and have written explicitly that H 0 , |M 1 |, and ∆M n depend on kξ. For large kξ (e kξ 1), the relation (81)   and H 0 (kξ) is approximately expressed as Here, we have used the second equation of (56). Now, we keep M pl a constant, and then, from (83) together with (84), we obtain Fig. 6 shows |M 1 |(kξ) · e −kξ (left figure) and ∆M n (kξ) · e −kξ for large n (n 200) (right figure) at kξ = 10, 20, ..., 100, 102. From this figure, we reconfirm that |M 1 |(kξ) and ∆M n (kξ) are almost proportional to e kξ , and moreover, we find that ∆M n (kξ) · e −kξ is almost equal to π/2. Hence, for pure imaginary M and large kξ, we expect that the function B(M ; kξ, k(L + ξ)) can be expressed as

RS limit
Finally, we consider a limit where we can obtain the results of the RS model from those of our model. (This limit is often called "the RS limit".) Here, we consider the situation where the warp factor is normalized at the TeV brane position, and all the 5-dimensional quantities are almost of the same order 10 2 GeV. Taking the limit of H 0 → +0 in the metric ansatz (33) of our model, we obtain the metric (6)   sign of A(y). In taking this limit, we must fix H 0 e kξ a constant as seen from (84), namely, we must take the limit ξ → ∞. Then, let us check whether this limit, H 0 → +0, ξ → ∞ with fixed H 0 e kξ = 2k, is just the RS limit. Applying this limit to the warp factor of (55), we find that sinh(k(|y| + ξ)) sinh(kξ) which is equal to the warp factor of the RS model (see (6) and (7)). 13 4 Possibility of making the Hubble parameter timedependent In this paper, we focused for simplicity only on our current universe with the observed Hubble parameter, and did not consider the complicated time evolution of the universe. A possible way to make the Hubble parameter H time-dependent would be to promote the brane interval L, which is a constant in the present model, to a dynamical variable. This also makes the weak scale, namely, the scale v 1 appearing in (46), time-dependent. 13 As stated in footnote 4 on page 5, reversing the sign of k is essentially equivalent to replacing y with L − y, which means exchanging the positions of the Planck brane and the TeV brane. Actually, in the RS model, we located the Planck brane (the TeV brane) at y = 0 (y = L). On the contrary, in our model, the positions of the two branes are exchanged. To exchange the brane positions in our model, we should do the same replacement of y → L − y. Then, (87) is modified to sinh(k(−|y| + L + ξ)) sinh(kξ) 2 ξ→∞ −→ e 2k(L−|y|) , which is just equal to (23).
Furthermore, to stabilize the model, we must introduce a scalar field (radion) in the bulk [14,15], and the analyses of the Einstein equation, the hierarchy problem, the Kaluza-Klein modes, and so on, will become more complicated. (Then, the model must be called "the thick brane model".) On the other hand, one might think that another way to make the Hubble parameter H time-dependent is to allow the warp factor to depend both on t and y. Under the metric assumption, ds 2 = e 2B(y,t) −dt 2 + a 2 (t)η ij dx i dx j + dy 2 , the Einstein equation leads to B(y, t) = A(y) + ω(t), ω(t) + log a(t) = H t dt e ω(t ) , where H is a constant, A(y) is given by (38), and ω(t) is an arbitrary function of t. Then, let us introduce a new coordinate τ defined by and realize dτ 2 = e 2ω(t) dt 2 . In this manner, the metric (88) becomes ds 2 = e 2A(y) −dτ 2 + e 2Hτ η ij dx i dx j + dy 2 , which is equivalent to the metric of our model with a constant Hubble parameter. Hence, we realize that it is meaningless to make the warp factor time-dependent.