Massive scalar field perturbations in Weyl black holes

In this work we consider the propagation of massive scalar fields in the background of Weyl black holes, and we study the effect of the scalar field mass in the spectrum of the quasinormal frequencies (QNFs) via the Wentzel–Kramers–Brillouin (WKB) method and the pseudo-spectral Chebyshev method. The spectrum of QNFs is described by two families of modes: the photon sphere and the de Sitter modes. Essentially, we show via the WKB method that the photon sphere modes exhibit an anomalous behaviour of the decay rate of the QNFs; that is, the longest-lived modes are the ones with higher angular numbers, and there is a critical value of the scalar field mass beyond which the anomalous behaviour is inverted. We also analyse the effect of the scalar field mass on each family of modes and on their dominance, and we give an estimated value of the scalar field mass where the interchange in the dominant family occurs.

In 1918 Hermann Weyl [1,2] attempted to unify the theory of General Relativity (GR) with electromagnetism.In this theory of gravity, the metric would transform under a conformal transformation g µν → Ω 2 (x)g µν whenever the electromagnetic field undergoes a gauge transformation A µ → A µ − ∂ µ log(Ω(x)), where Ω(x) is the local spacetime stretching, as a consequence the covariant derivative in Weyl's theory no longer preserve the metric, for that reason Weyl's theory of gravity never became a serious competitor for GR.However, Bach [3] derived a different theory of conformal gravity in 1921, whose action is constructed from contractions and squares of the Weyl tensor C µνρσ in four dimensions which is conformally invariant and it is usually called Weyl or Weyl-squared gravity.It is important to point out that in four dimensions the Weyl-squared action is the unique conformally invariant action constructed solely from the Weyl tensor.On the other hand, the action of the theory gives rise to fourth-order equations of motion for the gravitational field which make difficult to reconcile it with Newtonian gravity.One of the immediate consequences of postulating a gravitational theory with conformal invariance is that the artificially implanted cosmological constant, Λ, present in the Einstein-Hilbert action must be withdrawn, since not to do so would introduce a length scale that breaks the conformal symmetry of the theory.However, the same term naturally will emerge out of the metric, which provides further circumstantial evidence for the effectiveness of the principle under consideration.
Despite the absolute success of the GR theory, it fail to describe observations on scales much higher than the solar system without placing a large amount of dark matter; however, the absence of any direct experimental evidence for dark matter [4], has led to the consideration of various modified theories of gravity among which is conformal Weyl gravity, which may not require a dark matter component to explain the astrophysical data.The static and spherically symmetric vacuum solution describing a black hole was obtained by Mannheim and Kazanas [5], where a particular parameter of the solution can explain the flat rotation of galaxies without introducing dark matter.Also, the theory is intended to cover the dark energy related phenomena [6,7].Moreover, it was found three new exact solutions of this four-order theory, namely the Reissner-Nordström, Kerr and Kerr-Newmann solutions [8].Other solutions of the conformal Weyl gravity can be found in [9][10][11][12][13][14].
Moreover, in the context of the detection of gravitational waves [15], the detected signal is consistent with GR [16].However, there are possibilities for alternative theories of gravity due to the large uncertainties in mass and angular momenta of the ringing black hole [17].So, the study of the quasinormal modes (QNMs) and quasinormal frequencies (QNFs) [18][19][20][21][22][23] nowadays plays an important role.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, on the probe field parameters, and on the fundamental constants of the system.The QNFs are an infinite discrete spectrum of complex frequencies, in which the real part determines the oscillation timescale of the modes, while the complex part determines their exponential decaying timescale, for a review on QNMs see [20,23].
The tensor QNM spectrum for the Schwarzschild and Kerr black hole backgrounds show that the longest-lived modes are always the ones with lower angular number.This can be understood from the fact that the more energetic modes with high angular number would have faster decaying rates.However, for the propagation of massive scalar field a different behaviour was found [24][25][26][27], at least for the fundamental mode.If the mass of the scalar field is light, then the longest-lived QNMs are those with a high angular number, whereas if the mass of the scalar field is large the longest-lived modes are those with a low angular number.This behaviour, knowing as anomalous decay rate, is expected since if the probe scalar field is massive its fluctuations can maintain the QNMs to live longer even if the angular number is large.This behaviour of the QNMs introduces an anomaly of the decaying modes which depends on whether the mass of the scalar field exceeds a critical value or not.So, by introducing another scale in the theory through the presence of a cosmological constant an anomalous behaviour of QNMs was found in [28] as the result of the interplay of the mass of the scalar field and the value of the cosmological constant.Anomalous decay rate of the QNMs were also found if the background metric is the Reissner-Nordström and the probe scalar field is massive [29] or massive and charged [30] depending on critical values of the charge of the black hole, the charge of the scalar field and its mass.The presence of the anomalous behavior for a generalized Bronnikov-Ellis womhole and the Morris-Thorne wormhole was studied in Ref. [31].The anomalous decay rate of the QNMs has been studied in various setups, see [32][33][34][35][36][37].
The aim of this work is to study the propagation of massive scalar fields in Weyl black hole backgrounds, in order to study, the effect of the scalar field mass in such propagation.Some issues that we will address are the anomalous decay rate of QNMs, as well as, if there is a critical scalar field mass.Also, we will study the dominance between the family of modes, in order to analyze, if the dominant family suffers of such anomalous behaviour.It is worth mentioning that massless scalar field perturbations, dynamical evolution and Hawking radiation were recently studied for this spacetime, and it was shown that the propagation of massless scalar fields is stable.Also, the dominance between the two family of modes depending of the parameter λ was established, being the photon sphere (PS) modes dominant for small λ.However, as λ increases, the imaginary part of the PS mode decreases, whereas the de Sitter (dS) mode rises [38].The parameter λ has been considered in dark energy scenarios, and plays a role as the inverse proportion of the cosmological constant.Besides, the behaviour of the QNMs was used to study the thermal stability of black holes in conformal Weyl gravity, comparing this results with Schwarzschild black holes [39].In the particular case of nearly extreme black hole in Weyl gravity, it was shown in [40] the correspondence between the parameters of the circular null geodesic and the QNFs in the eikonal limit, and the QNMs of the gravitational and electromagnetic perturbations on a black hole in (exact) Weyl gravity was calculated and studied in [41].
The manuscript is organized as follows: In Sec.II we give a brief review about the Weyl black holes.Then, in Sec III, we study massive scalar perturbations in the background of Weyl black holes.In Sec.IV we consider the PS modes and we find the critical scalar field mass.Also, we show the anomalous behaviour of the decay rate by using the WKB method.Then, we analyze the dS modes via the pseudospectral Chebyshev method, and we study the dominance family modes.Finally, we conclude in Sec.V.

II. FOUR-DIMENSIONAL WEYL BLACK HOLES
The action of four-dimensional Weyl gravity is given by [5] where α is a dimensionless gravitational coupling constant which is usually chosen to be positive in order to satisfy the Newtonian lower limit, I M is the matter part of the action and C µνρσ is the Weyl tensor given by which satisfies the conformal invariance condition By using the definition of the Weyl tensor and making use of the Gauss-Bonnet theorem it is possible to express the action (1) in the following form This theory of gravity is governed by field equations, that can be derived by the functional variation of the action with respect to the metric g µν and take the following form: where W ρσ is the Bach tensor.It is important to note from (5) that in vacuum T µν = 0 (W µν = 0) every solution R µν = 0 in Einstein-Hilbert action also leads to a solution in Weyl gravity; however, not every vacuum solution from Weyl gravity implies a solution for GR.The first static and spherically symmetric vacuum solution describing a black hole in this theory was obtained by Mannheim and Kazanas [5].The lapse function, used in the line element, is given by: where the parameters β, γ and k are integration constants.In this solution, the parameter γ measures the departure of Weyl theory from GR, and so for small enough, both theories have similar predictions.On the other hand, it was argued that Weyl gravity can explain the flat rotation of galaxies without introducing dark matter, for which γ must be of the order of the inverse of the Hubble radius (γ ≈ 1 R H ) [5].Later, the solution (6) was generalized for rotating and charged solutions [8].It is important to point out that this solution reduces to Schwarzschild black holes when k = γ = 0 and to Schwarzschild-de Sitter black holes if γ = 0.
In [8] the authors extended their first work [5] and presented the exact solution to the Reissner-Nordström problem associated with static, spherically symmetric point electric and/or magnetic charge coupled to Weyl gravity.The metric function for the electric case looks like the following where Q is the electric charge.As pointed out by the authors, the first principal difference with the Reissner Nordström solution in standard Einstein theory is that the effect of the electromagnetic energy of a point electric charge is to produce a 1/r term in the exterior geometry of the black holes rather than the 1/r 2 term present in GR.
The second difference is that the geometry is not asymptotically conformally flat.
In this work we follow Refs.[42,43] in which the authors applied the background field method in the weak field limit, and it was possible to derive other Reissner-Nordström solutions.The authors in [42] found a general metric solution given by the line element: where dΩ 2 is the line elements of the 2-sphere and the lapse function: the coefficients c 1 and c 2 were found using the method named above.The last two terms of this f (r) can be seen as a perturbation to the Minkowski spacetime (h µν = g µν − η µν ), which was studied using the Poisson equation ∇ 2 h µν = 8πT µν .Now using the weak field limit (zero-zero component): Here, T 00 is the scalar part of the energy-momentum tensor of a source of mass m 0 , radius r 0 and E 00 is the another part of the energy-momentum tensor associated to the charge amount q 0 of the massive source.Using (9) in (10) it was obtained: and substituting this in (9) it was found where It should be noted its attractive inverse square potential due to the charged body, instead of the repulsive in Reissner-Nordström-de Sitter black hole.For λ > Q, the roots of the lapse function are The extremal black hole is obtained when λ = Q and both horizons coalesce to r ext = λ √ 2 .On the other hand, for λ < Q a naked singularity is encountered.In Fig. 1 we plot the lapse function, where for a fixed value of the black hole charge Q = 1, it is possible to observe the transition among a naked singularity λ = 0.5, the extremal black hole λ = 1, and a black hole with two horizons λ > 1.

III. MASSIVE SCALAR FIELD PERTURBATIONS
In order to obtain the QNMs of scalar field perturbations in the background of the metric (12) we consider the Klein-Gordon equation with suitable boundary conditions, that is, only ingoing waves on the horizon, and on the cosmological horizon.In the expression above m is the mass of the scalar field ψ.It is worth mentioning that the Weyl tensor is traceless which implies that the stress-energy tensor of the matter fields must be traceless too.However, for a probe field it is not actually necessary to respect the same symmetries of the Weyl tensor.Now, by means of the ansatz ψ = e −iωt Y l,m (θ, φ)R(r) the Klein-Gordon equation ( 15) can be written as where κ 2 = −l(l + 1), with l = 0, 1, 2, ... that represent the eigenvalues of the Laplacian on the two-sphere and l is the multipole number or the angular momentum of the field.Now, defining R(r) = F (r) r and the tortoise coordinate dr * = dr f (r) the wave equation can be written as a one-dimensional Schrödinger-like equation given by with an effective potential V ef f (r), which is parametrically thought as V ef f (r * ), and it is given by where we have defined the dimensionless quantities r ≡ r/λ, Q ≡ Q/λ and m ≡ λm.In Fig. 2 we show the effective potential, for fixes values of the parameter Q, different values of the angular number l and for a mass m = 0.1 of the scalar field.It is possible to observe that for l = 0, part of the effective potential is negative, and when l increases the height of the potential barrier increases.However, when the parameter Q increases the height of the potential barrier decreases.It is worth noting that by means of the following identification λ = 3 Λ ef f the effective spacetime potential is asymptotically de Sitter and tends to − r 2 λ 4 (m 2 λ 2 − 2) for l = 0 reproducing the result obtained in [28].

IV. QUASINORMAL MODES A. Photon sphere modes
Anomalous decay rate and an approach to the critical scalar field mass.In order to get some analytical insight of the behaviour of the QNFs, and to determine the critical scalar field mass, we use the WKB method at third order [44][45][46][47][48][49].The WKB method can be used for effective potentials which have the form of a barrier potential, approaching to a constant value at the event horizon and at the cosmological horizon or spatial infinity [22].Here, we consider the eikonal limit l → ∞ to estimate the critical scalar field mass, by considering ω l I = ω l+1 I as a proxy for where the transition or critical behaviour occurs [50].The QNMs are determined by the behaviour of the effective potential near its maximum value r * max .The Taylor series expansion of the potential around its maximum is given by where corresponds to the i-th derivative of the potential with respect to r * evaluated at the location of the maximum of the potential.Using the WKB approximation up to third order the QNFs are given by the following expression [51] where and N = n P S + 1/2, with n P S = 0, 1, 2, . . ., is the overtone number.Now, defining L 2 = l(l + 1), we find that for large values of L, the maximum of the potential is approximately at and while the second derivative of the potential evaluated at r * max yields For the higher derivatives of the potential, we consider only the leading terms that are important in the limit considered.So, Now, by using these results we find that U evaluated at r * max is approximately given by where B ≡ . So, using these results together with Eq. ( 21) we obtain the following analytical QNFs that is valid for large values of L: where i(2n P S + 1) 1 − Q2 Now, the term proportional to 1/L 2 is zero at the value of the critical mass mc , which is given by mc ≡ λm c = 17n P S (n P S + 1) + 9 + (87 − 17n P S (n In Fig. 3, we show the behaviour of mc as a function of Q.We observe that mc decreases when Q increases, and for a fixed value of Q < 1, mc increases when the overtone number n P S increases.However, when Q → 1 (or r c → r h ) then mc → √ 2, and it does not depend on the overtone number n P S .Also, when Q → 0, mc → ∞.Anomalous decay rate.Here, we use the 6th order WKB method in order to show the anomalous decay rate by simplicity; however, at the end we compare the QNFs via the 6th order WKB method and the the pseudospectral Chebyshev method to show the accuracy of the 6th order WKB method.So, in Figs. 4, and 5, we show the behaviour of −Im(ω) as a function of m.We can observe an anomalous decay rate, i.e, for m < mc , the long-livest modes are the one with highest angular number l; whereas, for m > mc , the long-livest modes are the one with smallest angular number.Also, when the parameter Q increases the parameter mc decreases, and when the overtone number n P S increases the parameter mc increases.Now, in Fig. 6, and Fig. 7, we show the behaviour of Re(ω) as a function of m, we can observe that the frequency of oscillation increases when the scalar field mass increases, and the frequency of oscillation decreases when the overtone number increases.Also, when the parameter Q increases the frequency of oscillation decreases.Accuracy of the numerical techniques.Now, in order to check the correctness and accuracy of the 6th order WKB method, we will compare some QNFs with the pseudospectral Chebyshev method.Thus, we show in Table III, see appendix B, the QNFs by using the pseudospectral Chebyshev method and the WKB method.Also, we show the relative error, which is defined by where ω1 corresponds to the QNFs via the 6th order WKB method, and ω0 denotes the QNFs via the pseudospectral Chebyshev method.We can observed that the error does not exceed 117.964 (%) in the imaginary part, and 40.308 (%) in the real part.These maximum values occur for small values of l, where the 6th order WKB method does not provide a high accuracy, it is known that the WKB method provides better accuracy for larger l (and l > n).Note that the error increases for higher values of the scalar field mass and for higher values of the overtone number.Note also that the WKB method has a good accuracy, for l ≥ 20, where the error does not exceed 8.813 • 10 −6 (%) in the imaginary part, and 1.639 • 10 −7 (%) in the real part.So, this method with l ≥ 20 is appropriate in order to show the anomalous behaviour of the decay rate.

B. de Sitter modes
These modes are associated with the presence of the cosmological horizon, in this spacetime the effective cosmological constant Λ ef f = 3/λ 2 provides the asymptotically de Sitter solution, and resemble those of a pure de Sitter spacetime [52], which are given by where Λ is the (positive) cosmological constant of pure de Sitter spacetime.It it worth to notice that for m 2 ≤ 3Λ/4 the QNFs of pure de Sitter spacetime are purely imaginary whereas for m 2 > 3Λ/4 the QNFs acquire a real part.
We can observe in Table I that for Q = 1 and for large values of the parameter λ (or small values of Q = Q/λ) the modes resemble those of the pure de Sitter spacetime (36).Also, note that for massless scalar field with n dS = 0, l = 0, there is a branch where ω dS = 0, the zero mode.By considering this fact one could say that the zero mode is associated with the dS family.So, in the following we will consider the zero mode as a dS mode.Here, the QNFs are obtained via the pseudospectral Chebyshev method using a number of Chebyshev polynomials in the range 95-100, the values inside the quotation marks "...", means that the QNF converges for a number of polynomials in the range 165-170, and ... means that there is not convergence until 170 polynomials with nine decimals places of accuracy for the QNF.The values between parenthesis are obtained via Eq.(36).In Fig. 8, we plot the behaviour of the decay rate as a function of the parameter Q, for massless scalar fields, we can observe that for n dS = 0, left panel, the decay rate is not sensitive to the increment of Q, giving a null slope approximately.However, it is possible to observe a positive slope for n dS > 0, and l > 0, see central and right panels.IV .
Now, in order to analyze the effect of the scalar field mass on the decay rate, we consider Q = 0.50 in Table II, and m = 0, 1, 1.5, 1.6, 1.7, 1.8.We can observe that for l = 0, 1, 2, n dS = 0, and purely imaginary QNFs, the decay rate increases when the scalar field mass increases.However, in general this is not true for higher overtone numbers.Also, note that the dS modes also can acquire a real part if the mass of the scalar field increases enough, which is similar to what happens to the modes of pure de Sitter spacetime.
TABLE II: de Sitter modes ωdS for massive scalar fields in the background of Weyl black holes, with Q = 0.5.Here, the QNFs are obtained via the pseudospectral Chebyshev method using a number of Chebyshev polynomials in the range 50-60, with eight decimals places of accuracy for the QNFs.

C. Dominance family modes
As we mentioned the purely imaginary modes belong to the family of dS modes, and they continuously approach those of pure de Sitter space in the limit that Q vanishes.However, the dS modes also can acquire a real part if the mass of the scalar field increases enough, which is similar to what happens to the modes of pure de Sitter spacetime.The other family corresponds to complex modes, for massless and massive scalar field, with a non null real part namely PS modes.Thus, in order to analyze the dominance between the family modes, we plot in Fig. 9 both families, black points correspond to dS modes and red points to PS modes.So, for massless scalar fields, we can observe that, the dS family is dominant for l = 0. Also, the dS modes are dominant for l = 1, 2, and small values of the parameter Q.But, for higher values of Q the PS modes are dominant for massless scalar field.Therefore, there is a critical value of Qc , where for Q < Qc the dominant family is the dS; otherwise the dominant family is the PS, for l > 0 and massless scalar field.Then, in order to analyze the effect of the scalar field mass on the dominance, we plot in Fig. 10 and Fig. 11 both families for Q = 0.5, and 0.75 respectively.Black points correspond to dS modes with a purely imaginary QNF, blue points correspond to dS modes with a complex QNF, and red points to PS modes.Interestingly, for l = 0, the dominance of the dS modes depends on the scalar fields mass, for small values of m the dS family with purely imaginary QNF is dominant.Otherwise, the PS family is dominant, see left panels.Therefore, there is a critical value of m = µ c , such that, for m < µ c , the dS modes with purely imaginary QNF are the dominant; otherwise the PS modes are dominant.Note that the dS family with complex QNF does not dominate.Also, note that the same behaviour is observed for l = 1 and small values of Q, see central panel of Fig. 10.Remarkably, for higher values of Q, and l > 0, the dominance of the PS family does not depend on the scalar field mass, see central and right panel of Fig. 11, and the PS is the dominant family.Now, in order to give an approximate value of m = µ c , where there is an interchange in the family dominance, we consider Im(ω dS ) = Im(ω P S ) as a proxy for where the interchange in the family dominance occurs, where for ω P S we consider the analytical expression given by Eq. ( 30), which yields the QNFs at third order beyond the eikonal limit, and for ω dS we consider the analytical expression given by Eq. ( 36) for the pure de Sitter spacetime ω pure−dS with Λ ef f = 3/λ 2 , which yields well-approximated QNFs for the dS family for high values of l.It is important because allows discern if the dominant family is able to suffers the anomalous behaviour of the decay rate.So, the equality of Im(ω pure−dS ) with Im(ω P S ) for n P S = 0 and n dS = 0 yields for µ c ≤ 3/2, and for µ c > 3/2.The mass where the transition of dominance occurs µ c depends on the parameters Q and l.
Note that the imaginary part of the QNFs of the pure de Sitter depends on m for m 2 ≤ 3Λ ef f /4 ( m < 3/2), while that in the opposite case it does not depend on m and the QNFs acquire a real part.This is reflected in the behaviour of µ c , which is different for µ c ≤ 3/2 and µ c > 3/2 as is shown in Fig. 12.Also, in that figure we show numerical results using the pseudospectral Chebyshev method (red and blue points) where there is a change of dominance of the families.We observe that the analytical values of µ c , represented by the solid line, is more accurate for high values of l. 12: The solid line corresponds to µc as a function of Q for l = 1 (left panel) and l = 8 (right panel), and separates regions in the parameters space where a family of QNFs dominates according to the analytical approximation.In the region to the left of the line always the de Sitter modes dominate, while that in the region to the right the PS modes dominate.The numerical results using the pseudospectral Chebyshev method are represented by points.The points are close to the frontier where the change of dominance occurs.The black points are in the side where the de Sitter modes dominate, while that the red points are in the side where the PS modes dominate.The coincidence of the line with the points for l = 1 is more accurate for low values of µc while that for bigger values of µc the difference between the analytical and the numerical results increases.However, for l = 8 the analytical approximation is accurate even for high values of µc.

V. CONCLUSIONS
In this work we studied the propagation of massive scalar fields in the Weyl black hole as a background, and we analyzed their QNFs.Mainly, we showed that two families of modes are present.One of them is a family of complex QNFs, and the other one is a family of purely imaginary modes for massless scalar fields (for massive scalar fields, the dS modes also can be complex).We showed that the purely imaginary modes belong to the family of de Sitter modes, and they continuously approach those of pure de Sitter space in the limit that the black hole parameter Q vanishes, and the complex ones corresponds to the photon sphere modes.Both families of modes show that the propagation of massless and massive scalar fields is stable in Weyl black hole backgrounds, for the cases considered.
For the PS modes and by using the WKB method at third order beyond the eikonal limit, we were able to estimate the value of the critical scalar field mass, and we found their dependence on Q, and on the overtone number n P S in the eikonal limit.Mainly, we found that the critical scalar field mass decreases when Q increases, and it increases when the overtone number n P S increases.Interestingly, at the extremal limit Q → 1 or (r c → r h ), mc → √ 2, and it does not depend on the overtone number n P S .Also, when Q → 0, mc → ∞.Then, we showed the anomalous decay rate of the QNMs via the WKB method at sixth order beyond the eikonal limit, where both methods i.e, the WKB method and the pseudospectral Chebyshev, show a high accuracy.Also, we showed that the frequency of oscillation increases when the scalar field mass increases, and such frequency decreases when the overtone number n P S or the parameter Q increases.
For the dS modes we found that for massless scalar fields, and l null, the decay rate is not sensitive to the increment of Q.However, the decay rate increases when Q increases n dS > 1, and l > 0. Also, when the scalar field acquires mass, we showed that the decay rate increases when the scalar field mass increases for l = 0, 1, 2, n dS = 0, and purely imaginary QNFs; however, in general this is not true for higher overtone numbers.
Finally, using the pseudospectral Chebyshev method we studied the dominance between the families of modes.Mainly, we showed that for massless scalar field the dS modes are dominant for l = 0.However, for l = 1, 2, there is a critical value of Qc , so that, if Q < Qc the dominant family is the dS; otherwise the dominant family is the PS.Interestingly, when the scalar field acquires mass, and for l = 0, the dominance of the dS modes depends on the scalar fields mass, and there is a critical value of m = µ c , such that, for m < µ c , the dS modes with purely imaginary QNFs are the dominant; otherwise the PS modes are dominant.The same behaviour was observed for l = 1 and small values of Q. Remarkably, for higher values of Q, and l > 0, the dominance of the PS family does not depend on the scalar field mass.Then, by considering as a proxy Im(ω dS ) = Im(ω P S ) we were able to estimate the value of µ c , where there is an interchange in the family dominance for a null overtone number, this value depends on the parameters Q and l.
It is worth to mention that despite the effective potential is negative for a range of values of r for l = 0, the propagation of massive scalar field is stable.However, it would be interesting to extent this work to the case of charged massive scalar field, and to study the superradiance, as well as, the existence of bound states which could to trigger an instability for l = 0.
TABLE IV: de Sitter modes ωdS for massless scalar fields in the background of Weyl black holes, with Q = 0.25, 0.50, 0.60, 0.65, 0.75, and 0.95.Here, the QNFs are obtained via the pseudospectral Chebyshev method using a number of Chebyshev polynomials in the range 95-100, with eight decimals places of accuracy for the QNF.Here, the QNFs are obtained via the pseudospectral Chebyshev method using a number of Chebyshev polynomials in the range 95-100, with eight decimals places of accuracy for the QNF.

FIG. 3 :
FIG.3:The behaviour of mc as a function of Q for different values of the overtone number nP S = 0, 1, 2 and 5.

FIG. 6 :FIG. 7 :
FIG. 6: The behaviour of Re(ω) as a function of m, with Q = 0.5.Left panel for l = 20, central panel for l = 40, and right panel for l = 60.Solid lines for nP S = 0, dashed lines for nP S = 1, and dotted lines for nP S = 2.

FIG. 8 :
FIG. 8: de Sitter modes ωdS for massless scalar fields in the background of Weyl black holes for several values of Q in the range 0.25-0.95.Black points for l = 0, blue points for l = 1, and red points for l = 2. Left panel for n dS = 0, central panel for n dS = 1, and right panel for n dS = 2.Some numerical values are shown in appendix B TableIV.

FIG. 9 :
FIG. 9: −Im(ω) as a function of Q, for massless scalar fields in the background of Weyl black holes.Here, the QNFs are obtained via the pseudospectral Chebyshev method.Black points correspond to dS modes while that red points correspond to PS modes.Left panel for l = 0, central panel for l = 1, and right panel for l = 2.Some numerical values are shown in appendix B TableV.

FIG. 10 :FIG. 11 :
FIG.10: −Im(ω) as a function of m, for scalar fields in the background of Weyl black hole with Q = 0.5.Here, the QNFs are obtained via the pseudospectral Chebyshev method.Black points correspond to dS modes with a purely imaginary QNF, blue points correspond to dS modes with a complex QNF, and red points to PS modes.Left panel for l = 0, central panel for l = 1, and right panel for l = 2.Some numerical values are shown in appendix B TableVI.
FIG.12:The solid line corresponds to µc as a function of Q for l = 1 (left panel) and l = 8 (right panel), and separates regions in the parameters space where a family of QNFs dominates according to the analytical approximation.In the region to the left of the line always the de Sitter modes dominate, while that in the region to the right the PS modes dominate.The numerical results using the pseudospectral Chebyshev method are represented by points.The points are close to the frontier where the change of dominance occurs.The black points are in the side where the de Sitter modes dominate, while that the red points are in the side where the PS modes dominate.The coincidence of the line with the points for l = 1 is more accurate for low values of µc while that for bigger values of µc the difference between the analytical and the numerical results increases.However, for l = 8 the analytical approximation is accurate even for high values of µc.

TABLE V :
Quasinormal frequencies ω for massless scalar fields in the background of Weyl black holes, with Q = 0.25, 0.50, 0.75.

TABLE VI :
Quasinormal frequencies ω for massive scalar fields in the background of Weyl black holes, with Q = 0.5.Here, the QNFs are obtained via the pseudospectral Chebyshev method using a number of Chebyshev polynomials in the range 95-100, with eight decimals places of accuracy for the QNF.

TABLE VII :
Quasinormal frequencies ω for massive scalar fields in the background of Weyl black holes, with Q = 0.75.Here, the QNFs are obtained via the pseudospectral Chebyshev method using a number of Chebyshev polynomials in the range 95-100, with eight decimals places of accuracy for the QNF.