Quasinormal Ringing of de Sitter Braneworlds

Quasinormal Ringing of de Sitter Braneworlds investigates gravitational perturbations in thin and thick de Sitter brane configurations to reveal properties related to cosmology. The analytical calculations indicate that gravitational perturbations, apart from the zero mode, possess a series of discrete, purely imaginary quasinormal modes in the late stages which implies these perturbations decay without oscillation over time.

Brane Geometry and Effective Potential

The study begins by considering the dS 3-brane with the metric ansatz ds2 = e 2A(z) γµν dxµ dxν + dz2 where A(z) is the warp factor and γµν = diag-1, e2αt, e2αt is the four-dimensional dS metric. The effective potential along the extra dimension exhibits the shape of Pöschl-Teller potential, asymptotically approaching a constant value (mass gap) at infinity and analytical calculations further indicate that the gravitational perturbations, apart from the zero mode, possess a series of discrete, purely imaginary quasinormal modes in the late stages. The reduced effective potentials of the thin dS brane are shown to take a Pöschl-Teller (PT) potential form and these results are commonly observed in braneworld scenarios making the treatment of gravitational perturbations on the brane analytically solvable.

Quasinormal Mode Analysis

The QNMs of the dS branes exhibit a particular uniqueness that is also encountered in Ref. [53]. For both thin and thick branes, the analytical and continued fraction methods show that the QNMs remain purely imaginary. The purely imaginary QNF results obtained here characterizes the late-time behavior of gravitational perturbation QNMs which manifests in the gravitational waveforms, which exhibit pure damping in the late-stage. These purely imaginary QNF results depend only on the parameter n and the Hubble constant α.

Late-Time Behavior and Observational Implications

The late-time behavior of the solution to Eq. (43) can be more conveniently obtained by solving Eq. (44), specifically the p = 0 case of Eq. (43). In this case, we can apply the assumption Ψ(t, z) ∼ e-iωtψ(z) to Eq. (44), reducing it to −∂ squared z + Vre(z) ψ(z) = ω squared ψ(z), where Vre(z) = V (z)-9α 2/4. The total QNMs of the perturbations hij should correspond to the solution Φ(t, z), i.e., the factor e- 3/2 αt multiplied by the solution Ψ(t, z). The first QNM corresponds to ω tot = 0 (¯ω0 = 3i/2), which actually corresponds to the graviton zero mode with m = 0.

Waveform Evolution and Momentum Dependence

Numerical evolution methods applied to Eq. (81) show that the late-time behavior is always consistently governed by the zero mode and the QNMs. The effect of the three-dimensional spatial momentum p on the waveform evolution is that as p increases, it quickly transits to a purely stable (or decay) form dominated by the zero mode. This indicates that regardless of the magnitude of the three-dimensional momentum p, the late-time behavior is always consistently governed by the zero mode and the QNMs. The first QNM corresponds to ω tot = 0 (¯ω0 = 3i/2).

Conclusion

The QNMs of the dS branes exhibit a particular uniqueness that is also encountered in Ref. [53]. The results suggest that the quasinormal signals from the extra dimension may be hidden within the background of stochastic gravitational waves. If these QNMs are detected, they will provide new insights into the measurement of the Hubble constant. This provides us a new measurement method that is worth further exploration and verification. The paper concludes that the dS branes possess purely imaginary QNMs, and these modes depend on the internal structure of the brane as well as the Hubble constant. The QNMs of the dS branes exhibit a particular uniqueness that is also encountered in Ref. [53].

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Acknowledgments

We would like to thank Wen-Yi Zhou for very useful discussions. This work was supported by the National Key Research and Development Program of China (Grant No. 2021YFC2203003), the National Natural Science Foundation of China (Grants No. 12475056, No. 12247101, No. 12205129, No. 12347111, and No. 12405055), the China Postdoctoral Science Foundation (Grants No. 2023M748), the 111 Project (Grant No. B20063), the Postdoctoral Fellowship Program of CPSF (Grant No. GZC20240458), Gansu Province’s Top Leading Talent Support Plan.

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I. INTRODUCTION

At the end of the twentieth century, the emergence of the braneworld concept [1] rekindled interest in extra dimensions and provided a new perspective for solving the hierarchy problem between the electroweak scale and Planck scale [2–5]. In this concept, to avoid conflicts with existing particle physics experiments which have not yet detected signals of extra dimensions, it is generally required that the fields of the standard model are confined to a four-dimensional hypersurface, commonly referred to as a brane, while gravity can propagate throughout the entire bulk spacetime. Based on this concept, various models attempting to solve the hierarchy problem have been proposed, such as the compactified large extra dimension model proposed by Arkani-Hamed et al. [3, 4], and the warped extra dimension model proposed by Randall and Sundrum (RS) [5]. However, in the RS model, the structure of the brane along the extra dimensions is neglected, i.e., the energy density of the brane is a delta function along the extra dimension. Therefore, this type of model is referred to as the thin braneworld (3-brane) model. In order to eliminate this singularity, DeWolfe et al. introduced the concept of domain walls [6, 7] into the infinitely large extra dimension model [8], and proposed the so-called thick braneworld (domain wall). That is, by replacing the cosmological constant in the bulk spacetime with a bulk dynamical scalar field, one can endow the internal structure of the brane [9–11]. This makes the braneworld theory more comprehensive. In the subsequent two decades, various braneworld models [12–25] emerged to address the residual issues of extra dimensions, such as the size of extra dimensions [26, 27], tuning problem between the bulk cosmological constant and the brane tension [16], reproduction of Newtonian gravity [28–30], localization of matter fields [31, 32] and so on.

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It is known that branes can have various symmetries. The RS models with one and two branes satisfying four-dimensional Poincar´e symmetry were considered in 1999 [5, 8]. Subsequently, bent branes were investigated in Refs. [9, 13, 14, 16], where the 3-branes or domain walls satisfy the four-dimensional maximal spacetime symmetries: de Sitter (dS) or anti-de Sitter (AdS). The corresponding metrics can be uniformly expressed in the following form: ds2 = e 2A(y) ds 4 + dy squared, (1) where ds 4 is labeled as the following four-dimensional metrics that satisfy the maximal symmetry of the fourdimensional spacetime: - dt2 + dx squared 1 + dx squared 2 + dx cubed, Poincar´e brane, - dt2 + e(2αt) dx 1 1 + dx squared 3, dS brane, e(˜αx3 −dt2 + dx 1 1 + dx 2). Here, α and ˜α are related to the effective cosmological constant of the brane. For instance, in special cases, Λ4 = 3α squared and Λ4 = -3 ˜a squared for the dS and AdS branes, respectively. Moreover, the exploration of bent dS branes can better correspond to the present universe. It is generally believed that dS braneworld slices correspond to the flat Friedmann-Robertson-Walker universe, while AdS branes play a crucial role in the AdS/CFT correspondence [46]. Additionally, in the RS model, achieving a zero cosmological constant on the brane requires fine-tuning between the bulk cosmological constant and the brane tension. However, it is worth noting that, compared to the Poincar´e brane solution, the finetuning issue becomes significantly more relaxed in the dS and AdS brane solutions when both the bulk and boundary cosmological constants are present.

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After establishing a braneworld model, studying gravitational perturbations within the brane scenario is of significant importance. For the flat brane in the RS model, there exists a graviton zero mode and a continuous spectrum of massive Kaluza-Klein (KK) modes for gravitational perturbations [8], which introduce corrections to the four-dimensional Newtonian potential at a small scale. Subsequently, Csaki et al. explored gravitational resonances in flat braneworlds [11, 33]. In 2005, Seahra further investigated the quasinormal modes (QNMs) in the RS model and revealed a potential connection between these two types of modes [47–49]. Some of us have discovered a connection between resonant states and QNMs by studying gravitational perturbations in thick flat branes [50]. Further calculations and analyses of QNMs in flat branes were conducted in Refs. [51–55]. For AdS branes, Karch et al. discovered that there is a series of discrete KK mass spectra in gravitational perturbations, and the corresponding wavefunctions are generally normalizable [16, 32, 56, 57]. From this perspective, the AdS brane does not possess QNMs but rather a series of normal modes. However, for dS branes, some previous articles [13, 14, 30, 34, 58–60] revealed that the gravitational perturbations on these branes may exhibit a finite discrete KK spectrum along with a series of continuous spectrum. A notable characteristic is the presence of a mass gap between the starting point of the continuous spectrum and the zero mode. This makes it feasible to search for QNMs within these continuous spectra. Understanding these modes is crucial for grasping how gravity propagates in the dS brane and how it influences the background gravitational wave signals in the present universe.

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In this paper, capital Latin letters M, N, · · · = 0, 1, 2, 3, 5, Greek letters µ, ν, · ··· = 0, 1, 2, and Latin letters i i j, · ··· = i j respectively are used to denote the coordinates of the five-dimensional bulk spacetime, four-dimensional spacetime, and three-dimensional space. The organization of the paper is as follows: In Sec. II, we mainly review the concept of the dS brane scenario, providing a brief review of thin and thick branes, and referencing classic results from previous researches. Subsequently, we derive the wave equations governing the transverse-traceless tensor (gravitational) perturbations and the Schr¨odinger-like equations. In Sec. III, we first discuss the properties of the gravitational effective potential. Next, by using analytical method and continued fraction method, we calculate the QNMs of the gravitational perturbations at the late time in various dS branes. At last, we simulate the gravitational wave signal by numerically evolving the perturbation equations, analyzing the properties of the signal and discussing potential observational effects. Finally, we provide a conclusion in Sec. IV.

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II. REVIEW OF DE SITTER BRANEWORLD

In this paper, we consider a dS 3-brane (or domain wall) with the metric ansatz ds2 = e 2A(z) γµν dxµ dxν + dz squared where A(z) is the warp factor and γµν = diag-1, e2αt, e2αt is the four-dimensional dS metric. Here, α is referred to as the Hubble constant and characterizes the expansion rate of the universe on the brane. And z denotes the conformal flat coordinate of the extra dimension coordinate y given in Eq. (1). Thus, according to the metric (3), we can derive a five-dimensional Einstein tensor as follows: Gµν = γµν 3A'squared + 3A'' - 3α squared and G5µ = 0, (5) and G55 = 6A'squared - 6α squared.

The thin dS brane model

Let us review the model describing a thin dS brane embedded in a five-dimensional bulk spacetime with a cosmological constant. The action is S = Z d 4x Z z>0 dz sqrt(-g) g mu 3R - Λ+ + Z d 4x Z z<0 dz sqrt(-g) g mu 3R - Λ- - Z d 4x p-g brane sigma delta µ M delta ν N g brane µν δ(z) i. By varying the action (7) with respect to the metric, we can obtain the field equation √(-g) RMN - 1/2 gMN R = -h Λ5 sqrt(-g gMN + σ p-g brane delta µ M delta ν N g brane µν δ(z) i. Substituting Eqs. (4) and (6) to Eq. (8), we can get (µ, ν): γµν 3A'squared + 3A'' - 3α squared = -e 2Aγµν Λ5 + e-Aσδ(z), (10) and (5, 5): 6A'squared - 6α squared = -e 2AΛ5. Note that for a flat brane, the α squared terms in the above equations vanish. On the other hand, due to the presence of the thin brane, we require the warp factor to be continuous at the location of the brane, with a jump in its first derivative determined by the brane tension. Specifically, by integrating both sides of Eq. (10) over the small interval (−ε, +ε), we can derive the jump condition of A(z): A' + epsilon - epsilon = - 1/3 e A(z=0)σ.

First, we consider solutions satisfying the Z2 symmetry, which results in Λ+ = Λ-. For a thin dS brane embedded in a five-dimensional AdS spacetime, the metric is given by A(y) = ln nα k sinh [k(yH − y)]o, (13) where y is the extra dimension coordinate and yH represents the position of the cosmological horizon. And the bulk cosmological constant Λ5 is Λ5 = -6k squared. Interestingly, it can be shown that, by applying the normalization condition, such that A(0) = 0, we can solve the Hubble constant α in terms of the cosmological constant Λ5 and the brane tension σ.

When Λ5 = -6k squared < 0, the thin dS brane is embedded in an AdS5 spacetime, the warp factor is A(z) = - ln k/α sinh (α(z + z0)), (15) where z0 = arcsinh(α/k)/α, and the brane tension is σ = 6α coth(αz0). When Λ5 = 0, the entire spacetime has zero curvature, resulting in the dS brane solution mentioned in Ref. [62], A(z) = - αz. When 0 < Λ5 = 6k squared ≤ 6α squared, the bulk spacetime is dS5, with the dS4 brane embedded as a codimension one hypersurface. Specifically, in the limit Λ5 → 6α squared, the brane tension vanishes, and the thin brane extends into a thick brane, resulting in a thick dS brane structure.

Asymmetric warped geometry

Next, we consider the asymmetric warped geometry, that is Λ+ 6= Λ-. In 2006, Gabadadze et al. studied the impact of an asymmetric warped geometry on gravitational perturbations in a flat brane and discovered that when the AdS curvature scales on both sides of the brane differ, a resonance mode appears [36]. This resonance mode was found to have a complementary relation with the RS zero mode. Thereafter, in 2011, Araujo et al. investigated the dS brane scenario and found a resonance mode in the gravitational perturbation spectrum [30]. As the cosmological constant Λ5 decreases, the contribution of this resonance mode to the Newtonian potential increases. The corresponding form of the warp factor is A(z) = (- ln [e(αz − K+ sinh αz], z ≥ 0, - ln e(-αz + K- sinh αz), z < 0, (18) where K± = 1 − r/1 − Λ± / 6α squared (19) and the brane tension is σ = 3α(2 − K+ - K-).

Thick dS brane generated by a bulk scalar field

Now, we review a thick dS braneworld generated by a bulk scalar field. The action is S = Z d 4xdz sqrt(-g) g mu 3R - 1/2∇M φ∇M φ - V (φ), (21) where M cubed is also set to 1/4, φ represents the bulk scalar field, and V(φ) is the bulk scalar potential. Through calculation, we can see that Eq. (26), which describes the scalar equation, can be derived from Eqs. (24) and (25).

1. Improved AI can perform analytical calculations for quasinormal modes by using continued fraction methods as described in Section III, allowing for faster and more precise determination of QNMs for dS branes.

2. Improved AI can simulate gravitational wave signals by numerically evolving the perturbation equations (Eq. 81) to obtain waveforms, which can be used to fit the first quasinormal mode from the late-time behavior analysis (Table I).

3. Improved AI can analyze the physical properties of different brane configurations by comparing effective potentials derived from various bulk spacetimes, such as AdS and dS bulk, as shown in Figure 6.

4. Improved AI can distinguish between thin and thick brane scenarios by identifying key differences in the effective potential shapes, specifically the presence or absence of a delta function at z = 0 (Fig. 6).

5. Improved AI can predict the nature of quasinormal modes for both thin and thick dS branes based on the internal structure parameter n, which determines how the brane structure determines the distribution of the quasinormal spectrum.