Anomalous decay rate of quasinormal modes in Schwarzschild-dS and Schwarzschild-AdS black holes

The quasinormal modes of a massive scalar field in Schwarzschild black holes backgrounds present an anomalous decay rate, which was recently reported. In this work, we extend the study to other asymptotic geometries, such as, Schwarzschild-de Sitter and Schwarzschild-AdS black holes. Mainly, we found that such behaviour is present in the Schwarzschild-de Sitter background, i.e, the absolute values of the imaginary part of the quasi normal frequencies decay when the angular harmonic numbers increase if the mass of the scalar field is smaller than the critical mass, and they grow when the angular harmonic numbers increase, if the mass of the scalar field is larger than the critical mass. Also, the value of the critical mass increases when the cosmological constant increases and also the overtone number is increasing. On the other hand, the anomalous behaviour is not present in Schwarzschild-AdS black holes backgrounds.


I. INTRODUCTION
The quasinormal modes (QNMs) and quasinormal frequencies (QNFs) [1][2][3][4][5] have recently acquired great interest due to the detection of gravitational waves [6]. Despite the detected signal is consistent with the Einstein gravity [7], there are possibilities for alternative theories of gravity due to the large uncertainties in mass and angular momenta of the ringing black hole [8]. The QNMs and QNFs give information about the stability of matter fields that evolve perturbatively in the exterior region of a black hole without backreacting on the metric. Also, the QNMs are characterized by a spectrum that is independent of the initial conditions of the perturbation and depends on the black hole parameters and probe field parameters, and on the fundamental constants of the system. The QNM infinite discrete spectrum consists of complex frequencies, ω = ω R + iω I , in which the real part ω R determines the oscillation timescale of the modes, while the complex part ω I determines their exponential decaying timescale (for a review on QNM modes see [3,9]).
The QNFs have been calculated by means of numerical and analytical techniques; some well known numerical methods are: the Mashhoon method, Chandrasekhar-Detweiler method, WKB method, Frobenius method, method of continued fractions, Nollert, asymptotic iteration method (AIM) and improved AIM, among others. In the case of a probe massless scalar field it was found that for the Schwarzschild and Kerr black hole background the longest-lived modes are always the ones with lower angular number . This is expected in a physical system because the more energetic modes with high angular number would have faster decaying rates. In the case of a massive probe scalar field it was found [10][11][12][13], at least for the overtone n = 0, that if we have a light scalar field, then the longest-lived quasinormal modes are those with a high angular number , whereas for a heavy scalar field the longest-lived modes are those with a low angular number . This behaviour can be understood because for the case of massive scalar field even if its mass is small its fluctuations can maintain the quasinormal modes to live longer even if the angular number is large. This anomalous behaviour is depending on whether the mass of the scalar field exceeds a critical value or not. This anomalous decay rate for small mass scale of the scalar field was recently discussed in [14].
Extensive study of QNMs of black holes in asymptotically flat spacetimes have been performed for the last few decade mainly due to the potential astrophysical interest. Considering the case when the black hole is immersed in an expanding universe, the QNMs of black holes in de Sitter (dS) space have been investigated [15,16]. The AdS/CFT correspondence [17,18] stimulated the interest in calculating the QNMs and QNFs of black holes in anti-de Sitter (AdS) spacetimes. It was shown in [19] that this principle leads to a correspondence of the QNMs of the gravity bulk to the decay of perturbations in the dual conformal field theory.
The aim of this work is to study the propagation of scalar fields in the Schwarzschild-dS and Schwarzschild-AdS black hole backgrounds in order to see if there is an anomalous decay rate of quasinormal modes. We carry out this study by using the pseudospectral Chebyshev method [20] which is an effective method to find high overtone modes [21][22][23][24][25]. The gravitational QNMs of Schwarzschild-de Sitter black hole were studied in [26][27][28]. The QNMs for this geometry was calculated [29] by using the sixth order WKB formula and the approximation by the Pöschl-Teller potential. Also, it was shown the frequencies all have a negative imaginary part, which means that the propagation of scalar field is stable in this background. The presence of the cosmological constant leads to decrease of the real oscillation frequency and to a slower decay and high overtones was studied in Ref. [30]. Also, a novel infinite set of purely imaginary modes was found [31], which depending on the black hole mass may even be the dominant mode.
In the case of massless scalar field in the background of a Schwarzschild-dS black hole we find two types of QNMs, the complex modes and the pure imaginary ones. These modes have a different behaviour as the cosmological constant is changing. First of all for the complex modes all the frequencies have a negative imaginary part, which means that the propagation of scalar field is stable in this background. However the presence of a larger cosmological constant leads to decrease the real oscillation frequency and to a slower decay. On the contrary in the case of pure imaginary modes we find the cosmological constant leads to a fast decay, when it increases, that is, contrary to the complex QNFs.
In the case of massive scalar field in the background of a Schwarzschild-dS black hole we find that the imaginary part of these frequencies has an anomalous behaviour, i.e, the QNFs either grow or decay when the angular harmonic numbers increase, depending on whether the mass of the scalar field is small or large than a critical mass. We also find that as the value of the cosmological constant increases the value of the critical mass also increases. As we will discuss in the following the mass of the scalar field redefines the cosmological constant to Λ ef f and at the critical value of the mass of the scalar field at which the anomalous behaviour appears, Λ ef f goes to zero. In the case of Schwarzschild-AdS black hole background we find that we do not have an anomalous behaviour of the QNMs i.e we have a faster decay when the mass of the scalar field increases and when the angular harmonic numbers decrease. We also show that in the case of massive scalar field the pure imaginary quasinormal frequencies acquire a real part which depends on the scalar fields mass.
The manuscript is organized as follows: In Sec. II, we study the scalar field stability by calculating the QNFs of scalar perturbations numerically of a massless and massive scalar field in the background of Schwarzschild-dS and Schwarzschild-AdS black hole background by using the pseudospectral Chebyshev method. We conclude in Sec. III.

II. SCALAR PERTURBATIONS
The Schwarzschild-(dS)AdS black holes are maximally symmetric solutions of the equations of motion that arise from the action where G is the Newton constant, R is the Ricci scalar and Λ the cosmological constant. The Schwarzschild-dS and Schwarzschild-AdS black holes are described by the metric where f (r) = 1 − 2M r − Λr 2 3 , M is the black hole mass, Λ > 0 in the metric represents the Schwarzschild-dS black hole, while Λ < 0 represents the Schwarzschild-AdS black hole. In Fig. 1 we plot the behavior of f (r), where we observe that for Schwarzschild-dS black holes (left figure) the difference between the event horizon r H and the cosmological horizon r Λ decreases when the cosmological constant increases, while for the Schwarzschild-AdS black holes (right figure) there is one event horizon that decreases when the absolute value of the cosmological constant increases. The QNMs of scalar perturbations in the background of the metric (2) are given by the scalar field solution of the Klein-Gordon equation with suitable boundary conditions for a black hole geometry. In the above expression m is the mass of the scalar field ϕ. Now, by means of the following ansatz the Klein-Gordon equation reduces to where we defined κ = − ( + 1), with = 0, 1, 2, ..., which represents the eigenvalue of the Laplacian on the two-sphere and is the multipole number. Now, defining R(r) = F (r) r and by using the tortoise coordinate r * given by dr * = dr f (r) , the Klein-Gordon equation can be written as a one-dimensional Schrödinger-like equation with an effective potential V ef f (r), which parametrically thought, V ef f (r * ), is given by In Fig. 2 we plot the effective potential for massless scalar fields in the background of Schwarzschild-dS black holes and in Fig. 3 we plot a small zone for Λ = 0.11, where we can observe that the effective potential is positive in the zone between the event horizon and the cosmological horizon. Also, in Fig. 4 we plot the effective potential for   In Fig. 6, we plot the effective potential for massless scalar fields in the background of a Schwarzchild-AdS black hole, which is positive outside the event horizon.  Now, in order to compute the QNFs, we will solve numerically the differential equation (5) by using the pseudospectral Chebyshev method, see for instance [20]. First, it is convenient to perform a change of variable in order to limit the values of the radial coordinate to the range [0, 1]. Thus, we define the change of variable y = (r − r H )/(r Λ − r H ). So, the event horizon is located at y = 0 and the cosmological horizon at y = 1. Also, the radial equation (5) becomes In the vicinity of the horizon (y → 0) the function R(y) behaves as Here, the first term represents an ingoing wave and the second represents an outgoing wave near the black hole horizon. So, imposing the requirement of only ingoing waves at the horizon, we fix C 2 = 0. On the other hand, at the cosmological horizon the function R(y) behaves as Here, the first term represents an outgoing wave and the second represents an ingoing wave near the cosmological horizon. So, imposing the requirement of only ingoing waves on the cosmological horizon requires D 1 = 0. Taking the behaviour of the scalar field at the event and cosmological horizons we define the following ansatz Then, by inserting the above ansatz for R(y) in Eq. (15), it is possible to obtain an equation for the function F (y). The solution for the function F (y) is assumed to be a finite linear combination of the Chebyshev polynomials, and it is inserted in the differential equation for F (y). Also, the interval [0, 1] is discretized at the Chebyshev collocation points. Then, the differential equation is evaluated at each collocation point. So, a system of algebraic equations is obtained, and it corresponds to a generalized eigenvalue problem, which is solved numerically to obtain the QNFs (ω). In Table I we show some fundamental QNFs, in order to check the correctness and accuracy of the numerical technique used. Also, we show the relative error, which is defined by where ω 1 corresponds to the result from [29], and ω 0 denotes our result. The complex QNFs for this geometry was determined in Ref. [29] by using the WKB and Pöschl-Teller method. We can observed that error does not exceed 0.37% when we compare our results with the WKB method and 2.198% with the P-T method. As it was observed, the frequencies all have a negative imaginary part, which means that the propagation of scalar field is stable in this background. Also, we observe that the presence of a bigger cosmological constant leads to decrease the real oscillation frequency and to a slower decay.  [29]. On the other hand, in [31] another branch of purely imaginary QNFs was found for this geometry by using the pseudospectral Chebyshev method, with the metric expressed in Eddington-Finkelstein coordinates. Here, we consider the coordinates given by the metric. Eq. (2) along with the change of variables y = (r − r H )/(r Λ − r H ). In the next, we will show that these quasinormal frequencies acquire a real part which depends on the scalar field mass. Now, in order to check the correctness and accuracy of the numerical techniques used, we show the purely imaginary and fundamental QNFs in Table II, where the relative error vanishes. As it was observed, the frequencies all are negative, which means that the propagation of scalar field is stable in this background. However, the presence of the cosmological constant leads to a fast decay, when it increases, that is, contrary to the complex QNFs. Also, it was shown that depending on the black hole mass may even be the dominant modes [31]. II: Purely imaginary quasinormal frequencies (n = 0) for massless scalar fields with = 1 in the background of Schwarzchild-de Sitter black holes with M = 1. The values of ωIm appear in Ref. [31].
Now, in order to show the existence of anomalous decay rate of quasinormal modes, we plot in Fig. 7 the behaviour of the complex fundamental QNFs, for different values of the parameter , and different values of the mass m of the scalar field. The numerical values are in appendix A Table III. It is possible to observe that the imaginary part of these frequencies has an anomalous behaviour, i.e, the QNFs either grow or decay when the angular harmonic numbers increase, depending on whether the mass of the scalar field is smaller or larger than the critical mass, where Im(ω) = Im(ω) +1 . Also, the behaviour of the real and imaginary part of the QNFs is smooth, and there is a slower decay of the mode when the mass of the scalar field increases. In order to show the same anomalous behaviour for other overtone numbers, we plot in Fig. 8, the imaginary and real part of the complex QNFs. Note that, the critical mass value increases when the overtone number n increases, for ≥ n. The numerical values are in appendix A Table V. The behaviour of the other branch is showed in Fig. 9. We can observe that this branch acquires a real part depending on the scalar field mass, see Table IV. Thus, as it was observed all the frequencies are negative, which means that the propagation of scalar field is stable in this background. Moreover, we can observe a faster decay when the parameter increases, and a faster decay when the scalar field mass increases until the QNFs acquire a real part after which the decay is stabilized. Also, the real part increases when the scalar field mass increases. However, there are other behaviours when we consider higher overtone numbers, see Fig. 10, where we plot the behaviour of the imaginary part of the QNFs as a function of the scalar field mass for different overtone numbers and = 0, and Fig. 11 shows the real part. In these figures we can recognize two branches: For zero mass, a branch of complex QNFs given by the black curves, and a purely imaginary branch given by the blue dashed curves. Also, we observe the behavior of the branches when the scalar field mass increases. We see that the black curves remains complex for all the values of m considered. Interestingly, the purely imaginary QNFs for zero mass can combine yielding complex QNFs, given by the continuous colored curves, when the mass increases, and then they split into purely imaginary QNFs which combine into new complex QNFs. As we have mentioned, it was shown that depending on the black hole mass the purely imaginary branch may even be the dominant mode [31]. Here, we can observe that for a fixed value of the black hole mass, the purely imaginary QNFs can be dominant depending on the scalar field mass and the angular harmonic numbers. Note that for the fundamental QNFs n = 0 and a small scalar field mass the dominant branch is the purely imaginary branch however for a scalar field mass m ≥ 0.15 the dominant branch is the complex one. Now, in order to see the influence of the cosmological constant in the critical mass, we plot in Fig. 12 the behaviour of the complex fundamental QNFs, for different values of the parameter , and different values of the mass m of the scalar field, but for a cosmological constant greater than the previous case Λ = 0.11. The numerical values are in appendix B Table VI. We can observe that for a greater cosmological constant the value of the critical mass increases. It is convenient to compare our result with those of [19], so we express the mass M as a function of the event horizon r H where the cosmological constant is taken as Λ = − 3 R 2 , with R being the AdS radius. Now, under the change of variable y = 1 − r H /r the radial equation (5) becomes where the prime denotes derivative with respect to y. In the new coordinate the event horizon is located at y = 0 and the spatial infinity at y = 1. In the neighborhood of the horizon (y → 0) the function R(y) behaves as where the first term represents an ingoing wave and the second represents an outgoing wave near the black hole horizon. Imposing the requirement of only ingoing waves on the horizon, we fix C 2 = 0. Also, at infinity the function R(y) behaves as So, imposing the scalar field vanishes at infinity requires D 2 = 0. Therefore, by considering the behaviour at the event horizon and at infinity of the scalar field, it is possible to define the following ansatz Then, by inserting this last expression in Eq. (15) we obtain an equation for the function F (y), which we solve numerically employing the pseudospectral Chebyshev method, as in the previous case. Now, in order to check our results, we can see that for r H = 10/R and massless scalar fields, we recover the QNF found in Ref. [19], see Appendix C Table VII, the QNFs were also studied in Ref. [32]. Also, in order to see if there is an anomalous decay rate of quasinormal modes, we plot in Fig. 13 the behaviour of the fundamental QNFs, for different values of the parameter , and different values of the mass m of the scalar field. The numerical values are in Appendix C Table VII. We can observe that the anomalous behaviour in the QNMs is not present in Schwarzchild-AdS black holes, for the cases considered. Also, they present a faster decay when the scalar field mass increases and when the angular harmonic numbers decrease. The frequency of the oscillations increases sightly when the scalar field mass increases and also when the angular harmonic numbers decrease.

III. CONCLUSIONS
In this work, we considered the Schwarzschild-dS and the Schwarzschild-AdS black hole as backgrounds and we studied the propagation of massive scalar fields through the QNFs by using the pseudospectral Chebyshev method in order to determine if there is an anomalous decay behaviour in the QNMs as it was observed in the asymptotically flat Schwarzschild black hole background.
The QNMs in the background of a Schwarzschild-dS black hole are characterized by one branch of QNFs which is complex and another one consisting of purely imaginary QNFs. The pure imaginary QNFs are generated for small scalar field mass and eventually this branch acquires a real part, and it is worth to mention that to our knowledge, this is the first time that this behaviour has been reported. All the frequencies have a negative imaginary part, which means that the propagation of scalar field is stable in this background. For the complex branch, the presence of the cosmological constant leads to decrease the real oscillation frequency and to a slower decay [29]. We showed that for the fundamental QNFs there is a slower decay rate when the mass of the scalar field increases for a fixed angular harmonic number .
Furthermore, we showed the existence of anomalous decay rate of QNMs, i.e, the absolute values of the imaginary part of the QNFs decay when the angular harmonic numbers increase if the mass of the scalar field is smaller than a critical mass. On the contrary they grow when the angular harmonic numbers increase, if the mass of the scalar field is larger than the critical mass and they also increase with the overtone number n, for ≥ n. We also showed that the effect of the cosmological constant is to shift the values of the critical masses i.e when the cosmological constant increases the value of the critical mass also increases. It is worth to mention here that the critical mass is an interesting quantity, because it shows that it is possible to have a scalar field with a critical mass and the decay rate does not depend on the angular harmonic numbers ; however, its frequency of oscillation depends on the angular harmonic numbers , increasing when increases.
It is interesting to note that, despite that the spacetime is asymptotically dS, where the boundary conditions are imposing at the event horizon and at the cosmological horizon the effective potential tends to −Λ(3m 2 − 2Λ)r 2 /9 for = 0, at infinity, and it can diverge positively, negatively, or be null, specifically, it vanishes for m c = ± 2Λ/3. So, for Λ = 0.04, and for a scalar field with mass m = m c ≈ 0.163, and also for Λ = 0.11 and for a scalar field with mass m = m c ≈ 0.278, the effective potential vanishes at infinity. These are the critical masses we have considered with n = 0. So, for a scalar field with critical mass, and = 0 the effective potential at infinity is not divergent. Also, for = 0 and m = m c , the effective potential tends to a negative constant at infinity given by − ( + 1)Λ/3 and the scalar field does not generate such divergence.
Another simple way to understand the appearance of the anomalous behaviour of the QNMs in the Schwarzschild-dS black hole background, is to define an effective cosmological constant through the relation Λ ef f = Λ(3m 2 − 2Λ). Then as we already discussed, the critical mass of the scalar field to have the anomalous behaviour and the corresponding value of the cosmological constant satisfy the relation m c = 2Λ/3. But then the effective cosmological constant becomes Λ ef f = 0 leading to the anomalous behaviour of QNMs at that critical mass. The physical picture behind it is that there is a specific critical scale of the scalar field that cancels out the scale introduced by the cosmological constant.
For the other branch, it was shown, depending on the black hole mass it may even be the dominant mode [31]. We found that for a fixed value of the black hole mass, the purely imaginary QNFs can be dominant depending on the scalar field mass and the angular harmonic numbers. Also, a faster decay is observed when the parameter increases, as well as, when the scalar field mass increases until that the QNFs acquire a real part, after it the decay is stabilized, and the frequency of the oscillations increases when the scalar field mass increases. Furthermore, we showed that this branch does not present an anomalous behaviour of the QNFs, for the range of scalar field mass analyzed.
In the case of a Schwarzschild-AdS black hole background we have shown that the QNMs of massive scalar fields do not present an anomalous behaviour. In this case, and according to the previous analysis, the effective potential at infinity always diverge, due to the fact that the cosmological constant is negative, and consequently the scalar field can probe the divergence of the effective potential at infinity. Also, they present a faster decay when the scalar field mass increases and when the angular harmonic numbers decrease. The frequency of the oscillations increases slightly when the scalar field depends on the curvature at infinity, i.e, such anomalous behaviour is possible in asymptotically flat and in asymptotically dS spacetimes and it is not present in asymptotically AdS spacetimes. The anomalous behaviour also could depend if the scalar field probes the divergence of the effective potential at infinity, despite that the boundary conditions can be imposed in a different point. It is worth to mention that for a Schwarzschild black hole, the effective potential tends to m 2 , so the scalar field does not probe the divergence, and consequently the anomalous behaviour in the QNMs can be observed.
It would be interesting to extent this work to the case the background black hole to be charged and study the behaviour of QNMs in this background and in different asymptotic spacetimes. If for example there is an anomalous QNMs decay for massive scalar perturbations in the background of Reissner-Nordström black hole then this behaviour of the QNMs may have important consequences on the Strong Cosmic Censorship [33,34]