A note on electromagnetic and gravitational perturbations of the Bardeen de Sitter black hole: quasinormal modes and greybody factors

We compute the quasi-normal (QN) frequencies for the regular Bardeen de Sitter (BdS) black hole due to electromagnetic and gravitational perturbations. We analyse the behaviour of both real and imaginary parts of QN frequencies by varying the black hole parameters. A study on the dynamics of the perturbation as well as the scattering from the BdS black holes using WKB approach is performed. Greybody factors and their variations with black hole parameters are investigated.


Introduction
It is very well known that general relativity is a theory which is plagued with the appearance of singularities. The invariant scalar curvature which necessarily tells about the gravitational field strength diverges at those spacetime singularities. Gravitational singularities appear in general relativity in the context of black holes. Black holes are objects which have singularities at the origin hidden by the event horizons. However, appearance of singularities in a theory means that the theory breaks down at the point where the singularity is present. Hence, the task of avoiding the singularities in general theory of relativity is one of the most fundamental ones and a set of solutions known as "regular black holes" play an important role in this context. As the name suggests, when the black hole does not have a spacetime singularity at the origin, it is termed as a "regular black hole". Bardeen [1] obtained the first solution of regular black holes with non-singular geometry satisfying the weak energy condition. The solutions is known as the Bardeen black hole in the literature. The solution Bardeen obtained was not a vacuum solution rather gravity was modified by introducing some form of matter. Therefore an energy momentum tensor was introduced in the Einstein's equation in order to achieve that goal. The introduction of the energy momentum tensor was done in an ad hoc manner and hence the Bardeen solution lacked physical motivation. After a long time, Ayón-Beato and García [2] showed that the energy momentum tensor necessary to obtain regular black hole solution is essentially the gravitational field of some magnetic monopole arising out of a specific form of non-linear electrodynamics. Many other solutions [3]- [13], motivating the avoidance of singularity was proposed thereafter. Stability properties [14,15,16] and quasinormal modes [17,18], thermodynamics [19] and geodesic structure [20] of such regular black holes were studied in detail. On another front, Fernando [21] has recently found out a de Sitter branch for the regular Bardeen black hole and corresponding grey body factors for such a black hole were calculated. The stability analysis and quasinormal modes due to scalar and Fermionic perturbations were also studied [22] for this background. The motivations for studying regular black holes in de Sitter space comes from the fact that our universe looks like asymptotically de Sitter at very early and late times. Observational data also indicates that our universe is going through a phase of accelerated expansion [23,24,25], which, along with many other explanations also indicates the existence of a positive cosmological constant. Hence, the study of black holes and its various features in de Sitter space is by itself an increasingly demanding area of research. In continuation of our earlier work [22], we will study the gravitational and electromagnetic perturbations of the regular Bardeen dS black hole in this paper.
The stability of a black hole spacetime is one of the most intriguing questions that one can ask in general relativity: the answer to the question of black hole stability under certain perturbation can answer many questions related to the black hole itself. The study of black hole perturbations is an active area of research and has immense effect on various important properties of black holes [26,27,28,29]. Generally one studies the evolution of a field (scalar, Fermionic, electromagnetic or gravitational) in a black hole background or in a black hole-black hole collision process in order to understand the stability of that particular black hole spacetime under the specific field perturbation. It is well know that the dynamical evolution of perturbations of a black hole background can be classified into three distinct stages, the first stage consists of an initial outburst of wave which depends completely on the initial perturbing field, the second one consists of damped oscillations, known in the literature as the quasinormal modes (QNM) whose frequencies are complex numbers. The real part of these frequencies represent the real oscillation frequency of the black hole under the perturbation and the imaginary part represents damping. The final stage is a power law tail behaviour at very late times. QN frequencies not only provide us with the information about the stability of the black hole spacetime, they are used to determine the black hole parameters (mass, charge and angular momentum) too. Numerical simulations depicting formation of a black holes in a gravitational collapses as well as that of collision of two black holes exclusively show that irrespective of the nature of the perturbations, the black hole's response will be dominated by the QNMs [30]. One important aspect of studying black hole stability is the fact that equations governing the black hole perturbations in most of the cases can be cast into a Schrödinger like equation. The QNMs are solutions to that Schrödinger like wave equation with complex frequencies for boundary conditions which are completely ingoing at the horizon and purely outgoing at asymptotic infinity (for the asymptotically flat or de Sitter black holes). It is to be noted that apart from the fact that the QN frequencies contain important information about the black hole parameters, they were also of importance from the point of view of AdS/CFT correspondence. It has been found [32,33] that QNMs in AdS space time appear naturally in the description of the dual conformal field theory on the boundary. This observation has motivated the study of QNMs towards asymptotically AdS black holes [34,35] too. On another front, despite their classical in origin, QNMs have been shown to provide glimpses to quantum nature of black holes [36,37,38].
A lot of work [42]- [51] has been done on QNMs of scalar, electromagnetic, gravitational, Dirac perturbations, decay of charged fields, asymptotic QNMs and signature of quantum gravity etc in de Sitter space. However, the regular black holes in de Sitter space is comparatively a less studied regime. In this paper, we will try to fill up the gap in the literature by discussing the QNMs of the Bardeen de Sitter (henceforth BdS) black hole due to electromagnetic and gravitational perturbations. The plan of the paper is as follows: in the next section we give a brief discussion on the BdS black hole. In section 3 we present a discussion of WKB method for calculating the QNMs along with a study of the Electromagnetic QNMs of the BdS black holes. Section 4 deals with the Gravitational quasinormal modes of the BdS black hole. In section 5 we give a comparative discussion about the dynamics of the perturbations. Section 6 contains a discussion about the greybody factor and its variation with the black hole parameters. Finally, in section 7 we conclude the paper with a brief discussion on future directions.

A brief discussion on BdS black hole
This section deals with a very brief introduction to the Bardeen de Sitter (BdS) black hole following the works in [21]. The authors of [21] has modified the works of [2] to incorporate a positive cosmological constant in the action. The action therefore looks like: In the above, R is the Ricci Scalar and L(F ) = 3 is a function of the field strength tensor of the non-linear electrodynamics F µν = 2(∇ µ A ν − ∇ ν A µ ). The parameter α in L(F ) is related to the magnetic charge (q) and the mass (M ) of the space time as follows: α = q 2M . The equations of motion from the above action comes out to be [21]: A static spherically symmetric solution for the above set of equations exist [21]: gives the horizon. The BdS black hole there can have at most three horizons corresponding to three real roots of the function f (r): the black hole inner(r i ) and outer horizons(r h ) along with the cosmological horizon(r c ). It is to be noted that the BdS black hole is structurally similar to the Reissner-Nordström-de Sitter (RNdS) or Born-Infeld de Sitter (BIdS) black holes which also admits a possibility of three distinct horizons as well as a single or degenerate horizons too (corresponding to extremal case). However, the event horizon is much larger for RNdS black hole as compared to a BdS one [21]. The non-singular structure of the BdS geometry can be checked by direct calculation of the scalar curvatures R, R µν R µν , R µνλσ R µνλσ , which are finite everywhere as compared to divergences at r = 0 in case of Einstein black holes except the electromagnetic field invariant F which is singular at r = 0 [21].

Electromagnetic perturbations and QNMs of the BdS black hole
In this paper we focus our attention on the behaviour of the dynamical response of the spherically symmetric regular black hole in de Sitter space under electromagnetic and gravitational perturbations. In this section we will be discussing the electromagnetic field perturbations of the BdS black hole in order to study the behaviour of the QNMs in this background by varying a set of black hole parameters. Since our system is an open one, the black hole, after a small perturbation, relaxes to its equilibrium state by losing energy by emitting electromagnetic or gravitational radiation, depending on the nature of the underlying perturbations. As discussed in Section 2, BdS background metric is given by Eqn. (5) and in the Regge-Wheeler gauge, vector potential can be written in terms of spherical harmonics considering spherically symmetric BdS background.
The electromagnetic field tensor is defined in terms of the vector potential as follows and field equations are represented by ∇ µ F µν = 0.
All functional form of field components are discussed in detail in [52]. Now, one can introduce Q(t, r * ) in terms of vector potential component, Using Eqn. (6) in Eqn. (7), one has the standard Schrödinger-like wave equation for the perturbation of the BdS metric due to an electromagnetic field given by where the potential V (r) has the form, The coordinate r * in Eqn. (9) is the standard tortoise coordinate related to the radial coordinate r by the relation dr * = dr f (r) . The advantage of using the tortoise coordinate lies in the fact that the range of the coordinate now extends between −∞ to ∞, whereas in the old radial coordinate r, the physically accessible region lies only between the black hole's outer horizon(r h ) and the cosmological horizon(r c ). Note also that the potential V (r) → 0 as r * → ±∞ which can also be seen in Fig.1 . In Fig.1 , we have examined nature of the Regge-Wheeler potential V (r) with tortoise coordinate r * for Λ = 0.002 and q = 0.2. As it is clear from the nature of the potential (10) that V (r) vanishes for = 0, we will always consider = 0. In this domain potential is always positive and finite which increases its height with increasing . Our target is to solve the wave equation with proper boundary conditions for complex QN frequencies semi-analytically, using the sixth order WKB method developed in [54]. It is already established in the literature that sixth order WKB method is more accurate than the third order one and the former in fact gives results coinciding with those obtained from full numerical integration of the wave equation [54] for low overtones, i.e. for modes with small imaginary parts, and for all multipole numbers ≥ 1. The sixth order formula for a general black hole potential V (r) is given by , r 0 is the value of the radial coordinate corresponding to the maximum of the potential V (r) and n is the overtone number. In general QN frequencies ω take the form ω = ω R − iω I , where, as mentioned earlier, the real part of ω represents actual field oscillation and imaginary part corresponds to damping of the perturbation.
In eqn. (11), Λ 2 and Λ 3 are given by [53] In the above expression b = n + 1 2 , V (n) 0 = d n V /dr n * at r = r 0 and Λ 4 , Λ 5 and Λ 6 can be found in the Appendix of [54]. The above method also works extremely well in the eikonal limit of large corresponding to large quality factors.
In Fig.2 , the QNMs are plotted as a function of multipole index for Λ = 0.003, magnetic charge q = 0.4 and overtone number n = 0. It is found that Re ω increases linearly with , while magnitude of Im ω initially increases rapidly with and later on, it saturates.  Utilizing the master eqn (11), we have determined the QNMs for different set of parameters in this work. One can define the quality factor (Q.F.) to look at the strength of the field oscillation over damping as follows: Q.F. = Re(ω) 2|Im(ω)| . As it is well known that the quality factor is essentially a dimensionless parameter that describes how underdamped an oscillator is. In Fig.3 , we have plotted the Q.F. versus the charge q and cosmological constant Λ. It is easy to check that field oscillation increases with q for = 2 and n = 0 but it remains almost constant throughout the variation of Λ. This implies that the BdS black hole system becomes underdamped with the increase of the magnetic charge while the cosmological constant has little or no effect on the quality factor. Next, we plot QN frequencies vs cosmological constant Λ and magnetic charge q for Here, Re ω increases almost linearly with q. The variation although is very slow with the magnetic charge. But -Im ω declines abruptly with increasing q and finally approaches to take a single value. It therefore implies that for BdS black holes with higher magnetic charge, the damping factor tends to become equal for different values of the multipole number ( ). Whereas, Fig.5 demonstrates linear decrement in both real and imaginary part of QNMs with increasing Λ for all sets of multipole numbers =1, 2 and 3. Finally, in Table[1], we have listed the numerical values of QN frequencies which are obtained by sixth order WKB approach for the parameter Λ = 0.007 and q = 0.57. As it is well known that WKB method is accurate for n < , we have tabulated the QN frequencies considering this condition. Data of Table [1] shows as increases both Re ω and -Im ω increase for a fixed overtone number (n). Another aspect of listed QNMs is that real oscillation frequency and imaginary part of the frequency representing damping are decreasing and increasing respectively with increasing overtone number n for fixed values. It is worth mentioning here that by computing inverse of the instability timescale which is associated with the geodesic motion, it is possible to show that in the eikonal limit, parameters of the circular null geodesics can determine the QNMs of black holes in any dimensions [55]. This is a very important and strong result since the parameters of null geodesics can throw some light on the stability of a black hole. It has also been shown to be independent of the field equations. The only assumption which went into the consideration of the authors of [55], is the fact that the black hole spacetime is static, spherically symmetric and asymptotically flat. However as a non-trivial example, they have discussed non-asymptotically flat near extremal Schwarzschild de Sitter black hole space time in this context. Therefore, the same analysis can be applied for BdS black holes in the limit of near extremal regime (Nariai or cold black holes) where either the black hole horizon and the cosmological horizon coincides or the inner and outer horizon merges.

Gravitational perturbations and QNMs of the BdS black hole
It has to be mentioned here that generally there are two different categories of perturbations of black holes that are considered within the regime of general theory of relativity. In the first method, one adds a test field in a black hole background and the system is studied by solving the dynamical equation for the particular test field in the background of the black hole. The second one is to perturb the metric itself and in order to find the evolution equations, one linearises the Einstein's equations. This is the gravitational perturbation and is the most important one amongst all types of perturbations since the gravitational radiation is much stronger than strength of any external fields decaying near the black hole. It is also important because the metric perturbations gives us tools to study about the gravitational stability of a black hole. The investigation of black hole perturbations was first carried out by Regge and Wheeler [31] for the odd parity type of the spherical harmonics and was extended to the even parity type by Zerilli [56]. A brief discussion about the calculations involved in gravitational perturbation is given in the appendix. The form l=3 l=4 r * V Figure 6: Variation of gravitational potential V against r * for different values of with Λ = 0.002 and q = 0.2.
of the potential due to gravitational perturbation is given by where, f (r) denotes the metric elements. In Fig.6 , we have shown the nature of the effective potential V for BdS black hole with different spherical harmonic for fixed values of the parameters Λ = 0.002 and q = 0.2. As before, the value of the mass of the black hole is taken as unity throughout this paper. In this domain potential is always positive and finite which increases its height with increasing . With = 0, V has more than one extremum which prevents us to apply the WKB approach. Therefore, like electromagnetic perturbation, here also we will be considering = 0. In Fig.(7), the QNMs are plotted as a function of multipole index for Λ = 0.003, magnetic charge q = 0.4 and overtone number n = 0. It is found that Re ω increases linearly with while magnitude of Im ω initially increases rapidly with and later on, it saturates and becomes almost a constant. Although the behaviour of the frequencies remains similar to those of electromagnetic ones as we vary the multipole index, the rapidity with which the imaginary parts of the frequency change with in case of the gravitational perturbation is much higher than that of the electromagnetic case. To understand the strength of the gravitational perturbation, we  Fig.8 . It is clear from the plot that field oscillation increases in a regular manner with Λ making a significant difference with it's electromagnetic counterpart for q = 0.40. At the same time variation of Q.F. with magnetic charge(q) shows us a non linear increment behaviour as we increase q for a fixed value of Λ = 0.007. Next, we plot QN frequencies of the BdS black hole due to gravitational perturbation vs cosmological constant (Λ) and magnetic charge parameter (q) for different and n values. Fig.9 precisely suggests the nature of QNMs as a function of q. Here, Re ω almost follows linear relation with q keeping similarity with its Electromagnetic counterpart (see Fig.4). However, the magnitude of Im ω falls rapidly with increasing magnetic charge q having fixed separation among different curves correspond to different values. It therefore, clearly shows difference in nature of dependence from the electromagnetic perturbation case (see Fig.4 ).  Table 2: The list of gravitational QNMs in the Bardeen de-Sitter black hole space time as a function of and n for q = 0.57 and Λ = 0.007. Fig.10 shows Re ω and |Im ω| decreases steadily for increasing Λ for q = 0.4. Finally, In Table 2, we have listed the numerical values of QN frequencies with corresponding parameters considering n < . Like electromagnetic class (see Table 1), tabulated QNMs indicate that as increases both real and modulus of imaginary ω increase for a fixed overtone number (n). Another feature of listed QNMs is that oscillation frequency and damping of perturbation are decreasing and increasing respectively when we increase n keeping fixed.

Dynamics of perturbation
Our initial motivation was to study black hole stability under an external perturbation. C. V. Vishveshwara was the first person to realize that we may observe a solitary black hole by observation of scattering of radiation from the black hole, provided the black hole left its fingerprint on the scattered wave [57]. Realising this, he started pelting the black hole with Gaussian wave packets and found that the black hole responds by ringing in a very unique decay mode: the lowest damped one of the black hole QNMs. Following this, here we will demonstrate a complete evolution picture of the BdS space time from a single master equation (see Eqn. 36 discussed in the appendix). This is a wave equation with a schrödinger like form.
We have used finite difference method to numerically integrate this wave equation (15). As a boundary condition we have Eqn. (16) which describes asymptotic behaviour with pure ingoing and outgoing waves at r * → ∓∞ respectively. We assume a solution of Eqn. (15) with oscillatory factor e −iωt in time. lim r * →±∞ ψe ∓iωr * = 1 (16) To give the first external "kick" in the field we use two Gaussian waves (17) and (18) as initial conditions for the second order differential equation (15): where σ 1 and σ 2 represents width of the Gaussian packet and µ denotes position of the peak of the curves. It is found that broad Gaussian waves can not excite the background sufficiently to observe the dynamical features [59]. On the other hand, using sharp localised packet, one can get maximum number of extrema in field oscillation. First we have discretized the domain of integration (r * − t) plane by using t j = j∆t, j = 0, 1, 2, 3...
Here x is same as r * . ∆t and ∆x are grid size of y axis and x axis respectively. x 0 is a point on boundary of x axis. To determine the perturbation ψ in advanced time, we use Taylor theorem in Eqn. (15) and get a discretized version of it: where, in general, F, C, E, A, D points are defined as : With the initial conditions equ. (17) and equ. (18), we also specify all values of ψ at t = 0 and t = ∆t grid line of Fig.11 . To determine the perturbation in one step advance (in time) at the point F, we need to know value of the perturbation in four neighbourhood points of F, which are represented by A, C, D and E. By applying this procedure repeatedly, one can determine the dynamics of perturbation over a complete domain. During this numerical integration scheme, one dimensional version of Courant-Friedrichs-Lewy (CFL) condition is satisfied which is a necessary condition for convergence of an explicit finite difference method of a hyperbolic PDE. Other parameters on which convergence depends are discussed in [52]. In Fig.12 we show the evolution of ψ for two integral spin (s = 1 and s = 2 representing electromagnetic and gravitational perturbations respectively) perturbations with different values.
It is quiet evident from plots that one of our motivations of studying stability feature of the scattered wave is fulfilled as it shows characteristic decay modes in late time. On the same time Fig.13 exhibits evolution of perturbation in log scale. It is clear from the applied numerical scheme that to find late time dynamics of ψ, we need initial conditions (17) , (18) on more number of grid points. As our integration domain is limited between finite values of −r * and r * because of finite value of cosmological horizon r c , we were unable to generate numerical values of ψ at very late time and therefore any power law tail is absent in the dynamics. In real world, it is well known that Λ is very small, of the order of 10 −52 . Use of this small value of Λ in numerical computation has its own challenges. So we choose a small, finite value for Λ = 0.002 here, which, of course is not as small as the cosmological constant itself. To capture feature of small value of cosmological constant, one can decrease Λ and recalculate it further. Using Eqn. (5) it is found that as Λ → 0, r c → ∞. Therefore numerically domain of integration also becomes large enough and finally one can get very late time dynamics. Fig.14 is a spectrum of Fourier transformation for the same perturbing wave of Fig.12 . Here we find independently oscillation frequency of perturbing wave from the frequency ω 0 corresponding to maximum of |G(ω)|. In s = 1 condition, these frequencies are 0.6080 and 0.8107 for = 3 and = 4 respectively. In s = 2 condition, these frequencies are same as s = 1 for = 3 and = 4. Although there is no way to find specific overtone no(n) of oscillation from the variation of ψ but in late time it is expected that system will oscillate in fundamental mode [58]. By that time all higher modes will be damped out. With s = 1 and n = 0, oscillation frequencies (ω 0 ) from the WKB method are 0.6557 and 0.8514 for = 3 and = 4. With s = 2 and n = 0 frequencies are 0.6000 and 0.8088. These slight differences between two methods are due to large grid size (∆ ω = 0.1044) in fourier space, for which we are unable to resolve the exact mode. In Fourier transformation technique, variation of frequency with also depicts same nature as the one which we have shown in Table 1 and 2 .
6 Greybody factors and Absorption coefficients 6.1 Nature of the greybody factor In this subsection we will discuss Reflection coefficients R(ω) and Transmission coefficients T (ω) for different parameter spaces as well as in different type of perturbations. In [21] coefficients for scalar type perturbation is discussed for Bds black holes in detail. Here, in order to fill the gap in the present literature, we will concentrate on the electromagnetic and gravitational perturbation part for this black hole. The use of WKB method to compute reflection and transmission coefficients(greybody factors) are not new. It has already been employed in various scenarios [60]- [63], which includes the calculation of greybody factors of black holes in braneworld models, in the context of calculations of these coefficients for wormholes. By another analytical approach which was originally proposed by Unruh [64], greybody factors can also be calculated [65]. Here we have already seen that for both s = 1 and s = 2, finite potential barrier (Fig.1 , Fig.6 ) exists between cosmological horizon(r c ) and event horizon(r h ). Now, any wave travelling past the cosmological horizon will face these finite positive potential barriers as obstacles. Therefore some part of the wave will be reflected back towards r c and some parts will be transmitted towards r h . Following [21], we can represent them as ψ(r * ) = e −iωr * + R(ω)e iωr * ; r * → +∞ In general the reflection and transmission coefficients are functions of oscillation frequency (ω) of the wave. The reflection coefficient R(ω) in the WKB approximation is defined as, where α is given by and expression for Λ 2 , Λ 3 can be found out from Eqns. (12) and (13) respectively. Conservation of probability requires: Finally, the greybody factor is defined as Depending on the frequency and height of the potential barrier, there may be different cases which can arise: when ω 2 V (r 0 ), i.e. when a wave with frequency larger than the height of the barrier comes, it will not be reflected by the barrier classically. In this case, one should expect the reflection coefficient to be close to zero, because the frequency of the wave is large enough to cross the barrier. Therefore we expect that under this conditions, |T | 2 will be close to 1. When ω 2 V (r 0 ), i.e. square of the frequency is very small compared to barrier height, wave will be reflected back from the barrier and some part may be transmitted through the barrier by tunnelling effect depending on the values of ω and V (r 0 ). We should get exactly opposite behaviour of R(ω) and T (ω) compared to previous case. In this case, the WKB method does not have very high accuracy. When ω 2 ≈ V (r 0 ), we have to take help of numerical techniques to understand the nature of R(ω) and T (ω). Here we can apply WKB approximation method because of the small distance between the turning points. Fig.15 shows variation of |R(ω)| 2 with ω for different class of perturbations with different values. In electromagnetic perturbations, i.e. (s = 1), it is almost one for low frequency and for high frequency it is close to zero. For a fixed frequency, |R| 2 is larger for multipole number = 4 than = 3. It can be explained easily from the dependency of effective potential V (r) on . Gravitational perturbation(s = 2) has also same nature and features like that of the electromagnetic type (s = 1). On the contrary Fig.16 shows variation of |T (ω)| 2 with ω for both (s = 1) and (s = 2) type. Following Eqn. (26), it shows exactly opposite nature to Fig.15 for both the limits. Next we study the behaviour of |R(ω)| 2 with ω by varying black hole magnetic charge q and keeping other parameters fixed. This is shown in Fig.17 . Larger q values increases reflection coefficient compared to smaller q values. Although s = 1 and s = 2 have same nature but for gravitational perturbation |R(ω)| 2 is slightly larger than electromagnetic perturbation which makes the chances of detection larger. Fig.18 shows |T (ω)| 2 with ω with the same parameter values. Next we plot |R(ω)| 2 vs ω by varying the cosmological constant Λ in Fig.19 . For both s = 1 and s = 2, we have exactly the same |R(ω)| 2 values for both small and large ω values. They start to differ in moderate ω values and again matches for lower ω. Larger Λ values decreases the reflection coefficient compared to smaller Λ values. Fig.20 shows the variation of |T (ω)| 2 with ω with the same set of parameter values following Eqn. (26).

Absorption Cross-section
In this subsection we will discuss partial and total absorption cross section in the context of electromagnetic and gravitational perturbation for different parameter spaces in the BdS background. Partial (σ ) and total absorption cross sections (σ) are defined respectively as: In Fig.21, variation of σ are plotted with different q and Λ values where individual peak represents σ . For both s = 1 and s = 2, Total absorption cross sections have similar feature. Variation of σ in Fig.21 can be classified into three distinct regions, the first region consists of a growing phase which is the signature of increasing |T (ω)| with ω. The second region shows oscillations in σ which comes considering different modes. In this particular example, we have added upto = 10 modes to determine σ. The last part is a power law fall-off. The reason behind this fall-off is the following: after certain critical frequency (ω 0 ), the transmission coefficient attains maximum value to 1. Afterwards, with further increase in ω, σ becomes proportional to

Summary and Conclusion
In this paper, we have focused on two most important types of black hole perturbations: electromagnetic and gravitational, for a regular BdS black hole. We have used sixth order WKB approximation method to compute the QN frequencies of the BdS black hole under these perturbations and found out the response of the black hole to these perturbations by varying different parameters of the space-time. It is easy to see that from one type to another type of perturbation only the potential profile changes in the Schrödinger-like wave equations, while keeping forms of all the relevant equations intact. We studied how the frequencies vary as a function of multipole number ( ) as well as with the parameters like the cosmological constant (Λ), magnetic charge (q) and overtone number (n). It was observed that the real part of the QN frequency increases monotonically with the multipole number whereas the imaginary part at the beginning starts to increase but then it saturates after reaching for a certain value. This behaviour of the frequency with the multipole number is a common feature of both the electromagnetic as well as gravitational perturbations, although the rapidity with which the imaginary part of the frequency increases is more in the case of gravitational perturbations as compared to the electromagnetic one. We have also conducted a study to find out the Q-factor of the BdS black hole system and found that both the magnetic charge and cosmological constant has less influence on the Q-factor for the electromagnetic perturbation than the gravitational ones. While the Q-factor changes slowly with magnetic charge q and remains almost unchanged with the variation of Λ for electromagnetic perturbations, it increases linearly with q and shows an increasing trend with Λ for gravitational perturbation. In both the cases of electromagnetic and gravitational perturbations, the real part shows small or no variations with q and Λ, but there were significant impact of variations of q and Λ on the imaginary part of the frequencies. We have further studied the dynamics of the perturbations using a standardised numerical integration method. Finally, we investigate the reflection and transmission coefficients from the BdS black hole due to electromagnetic and gravitational perturbations. In both the cases, behaviour of the greybody factor were studied by varying the black hole parameters. The total absorption cross section for different multipole values (upto = 10) was studied.
For future directions, it would be interesting to study whether isospectrality of the QN spectrum holds in BdS spacetime or not. It is already known that both the axial and polar perturbations (electromagnetic as well as gravitational) gives rise to same QN frequencies. The well known example being the Reissner-Nordström black hole arising out of Einstein's general theory of relativity coupled to Maxwell's electrodynamics. However, to our knowledge, no such works in case of regular black holes exist. It would therefore be interesting to study the isospectrality of different types of perturbations in regular black holes in de Sitter space. Another important area of study would be to look at the circular null geodesics of the near extremal BdS black holes and find out whether there can be any relation between the Lyapunov exponents and the QNMs of the BdS background. Finally we would like to mention that, although many works have been done on the electromagnetic and gravitational perturbations of black holes in the regime of Einstein's general theory of relativity, not many examples are present in the context of regular black holes. Particularly, the stability and QN properties of regular black holes in de Sitter universe remains a very less studied area in the literature so far. We believe that this work will fill the gap.

A Appendix: The gravitational perturbation
In this appendix we will briefly discuss about the gravitational perturbation in general. The spherically symmetric, static background metric is represented by g µν and the small perturbation to the background metric is denoted by h µν . In order to perform the calculation to linearise the Einstein equation, we follow |h µν | 1. Then R µν is evaluated from g µν and R µν + δRµν from g µν + h µν .