1 Introduction

Predictability is a fundamental principle in classical physics. In order to ensure the predictability of gravitational theory, ensuring a sound causal structure, Roger Penrose proposed the strong cosmic censorship (SCC) conjecture [1]. It demands that our spacetime is globally hyperbolic, enabling us to predict the universe’s evolution by specifying physically reasonable initial data on a spacelike hypersurface, also known as a Cauchy hypersurface. To ensure the global hyperbolicity of spacetime, the absence of naked singularities and Cauchy horizons [2, 3] is required. The absence of naked singularities is guaranteed by the weak cosmic censorship (WCC) conjecture [1]. Recently, Sorce and Wald introduced a new version of the gedanken experiment using the Noether charge method to demonstrate the validity of WCC in Kerr–Newman black holes under second-order perturbation approximations [4]. Subsequently, discussions were extended to various gravity and black hole models, most of which indicate the integrity of black holes, preventing the existence of naked singularities in spacetime [5,6,7,8,9,10,11,12,13,14,15,16,17]. With the establishment of WCC, verifying the efficacy of SCC requires examining the stability of the Cauchy horizon. For an asymptotically flat spacetime, the Cauchy horizon experiences a blueshift instability, leading disturbances towards it, eventually rendering it singular [18,19,20,21,22,23].

However, subsequent investigations have presented a more nuanced perspective on the validity of the SCC within asymptotic de-Sitter (dS) spacetime \((\Lambda > 0).\) The emergence of a cosmological horizon in this context alters the decay characteristics of our matter field perturbations, transitioning from a power-law decay to an exponential one. This modification changes the relationship between the exponential blueshift at the Cauchy horizon and the later-stage power-law decay of perturbations [24]. Detailed analyses of perturbation dynamics with various spin properties within dS spacetime suggest that the quasi-normal modes (QNMs) of perturbations outside the event horizon decay exponentially faster, inducing a redshift effect that counteracts the blueshift effect [25,26,27,28,29,30]. This scenario implies that the disturbance decay rate might surpass the SCC threshold. Observations of such violations were noted specifically in the linear perturbation of massless neutral scalar fields within the near-extremal region of the Reissner–Nordström–dS (RNdS) black hole [31]. Subsequent investigations across various types of dS black holes have revealed diverse violations of the SCC through a range of perturbations, including massless charged scalar fields, massless Dirac fields, gravitational perturbations, and non-minimally coupled scalar fields, while also examining SCC effectiveness across different black hole geometries and exploring the influences of initial data, nonlinear effects, quantum effects, and environmental impacts, among others [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62].

Recent observational advancements have significantly enhanced our comprehension of the accelerated expansion phase of the universe, as evidenced by phenomena such as Type Ia supernovae [63], anomalies in cosmic microwave background radiation [64, 65], and large-scale cosmic structures [66]. These findings have led to the recognition of an enigmatic, repulsive force termed dark energy. To elucidate the nature of this dark energy, a plethora of theoretical models have been proposed. Among these, the simplest is the vacuum energy or Einstein’s cosmological constant, characterized by a state parameter \( \omega _f = -1 \). However, aligning this model with empirical data necessitates meticulous fine-tuning. Furthermore, dynamic dark energy models, typically formulated through scalar field mechanisms like quintessence [67,68,69], phantom fields [70], and models investigating the interplay between dark energy and dark matter [71, 72], have emerged. These models are distinguished by their state equations, which delineate the relationship between pressure and energy density. In the case of quintessence, the state equation spans \(-1 \le \omega _f \le -1/3\). In this theoretical framework, Kiselev [73] successfully derived the exact solutions of Einstein’s field equations for quintessential dark energy in the vicinity of black holes, catalyzing subsequent research into the evolution of fields in such environments and their thermodynamic characteristics. As a fundamental assumption ensuring the causality of spacetime, a key question regarding the SCC arises: is SCC satisfied when dark energy is present? In Ref. [52], calculations of QNMs within the charged black holes surrounded by quintessence reveal that for these dark energy models with a dS-like horizon, SCC is violated as long as the black hole approaches extremality. Moreover, they also analyze the impact of energy density amplitude of dark energy on SCC. Our study plans to further analyze the SCC by deepening the precision and accuracy of QNM calculations, discussing the effects of changes in black hole asymptotic properties due to variations in quintessential state parameters on the SCC.

This paper investigates the effects of dark energy on the SCC in a charged dS-like black hole surrounded by quintessential dark energy. Section 2 introduces the black hole geometry influenced by quintessential dark energy, followed by Sect. 3 where we analyze the massless scalar field equation in this context and assess the SCC. Section 4 applies a detailed numerical method to examine the QNM frequency and potential SCC violations. The paper concludes with a synthesis of the findings.

2 Reissner–Nordstrom black hole in the quintessence

In our study, we explore the charged black hole within the framework of quintessential dark energy, focusing on the SCC. The foundational action for our model is expressed as:

$$\begin{aligned} S=\frac{1}{16\pi }\int d^4 x\sqrt{-g}\left[ R-F_{ab}F^{ab}+\mathcal {L}_{\text {DE}}\right] , \end{aligned}$$
(1)

with R denoting the Ricci scalar, while \(F_{ab}=\nabla _a A_b-\nabla _b A_a\) describes the electromagnetic field tensor, and \(\mathcal {L}_{\text {DE}}\) stands for the Lagrangian density of the quintessential dark energy. The derived field equations are:

$$\begin{aligned} R_{ab}-\frac{1}{2}R g_{ab}=T_{ab}^{\text {DE}}+T_{ab}^{\text {EM}},\quad \nabla _a F^{ab}=0\, \end{aligned}$$
(2)

from the action variation, where \(T^{\text {DE}}_{ab}\) as well as

$$\begin{aligned} T_{ab}^{\text {EM}}=2F_a{}^{c}F_{bc}-\frac{1}{2}F_{cd}F^{cd}g_{ab} \end{aligned}$$
(3)

represent the stress–energy tensors of the dark matter and electromagnetic components respectively.

In line with earlier research, we adopt a static and spherically symmetric solution with an electric charge and it can be expressed as [73]:

$$\begin{aligned}{} & {} ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+ r^{2}(d\theta ^{2} + \sin ^{2}\theta d\phi ^{2}), \nonumber \\{} & {} A_a=-\frac{Q}{r}(dt)_a, \nonumber \\{} & {} (T^{\text {DE}})_t^t=(T^{\text {DE}})_r^r=\rho _f=\frac{\lambda }{r^{3(1+\omega _f)}}, \nonumber \\{} & {} (T^{\text {DE}})_\theta ^\theta = (T^{\text {DE}})_\phi ^\phi =-\frac{\sigma }{2}p_f \end{aligned}$$
(4)

with blackening factor

$$\begin{aligned} f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\lambda }{r^\sigma }, \end{aligned}$$
(5)

and the equation of state \(p_f=\omega _f \rho _f\) of the quintessential dark energy with the anisotropic fluid state parameter \( \omega _{f} \) satisfying \( \sigma =3 \omega _f+1 \). Here, M and Q as the mass and charge of black holes, while \( \lambda \) and \( \sigma \) are scaling and dark energy parameters.

The fluid dynamics, particularly for \( \omega _{f} < 0 \), refer to X-matter or X-cold-dark-matter [74], has intriguing properties. Vacuum energy is represented by \( \omega _{f} = -1 \), while cosmic textures or strings are denoted by \( \omega _{f} = -{1}/{3} \). In cosmology, such matter, especially at \( \omega _{f} = -{2}/{3} \), aligns with theories that account for cosmic acceleration and galactic rotations without invoking dark matter, akin to the Mannheim–Kazanas solution in conformal Weyl gravity [75, 76], and is also applicable in braneworld contexts [77].

In cases where \( \lambda > 0 \) and \( -2 \le \sigma \le 1 \), the anisotropic fluid satisfies the null energy condition, as detailed in [78, 79]. Our study, however, primarily focuses on the range \( -2 \le \sigma < 0 \) (\(-1\le \omega _f<-1/3\)), where the spacetimes exhibit dS-like characteristics, diverging from asymptotically flat black holes, which are not within the purview of our research. The causal structures revealed by our solutions bear resemblance to those of the Reissner–Nordström–dS black hole, yet they exhibit unique attributes beyond the cosmological-like horizon \( r > r_{c} \). When \( -2 \le \sigma < 0 \), the state parameter \(\sigma \) can fully characterize the asymptotic behavior of the spacetime, meaning that when r is effectively large, the blackening function f(r) scales as \(1/r^\sigma \). In our paper, we mainly aim to analyze the impact on the validity of SCC when there is a change in the asymptotic behavior of the black hole.

The horizons of interest are the inner/Cauchy horizon \( r_- \), the event horizon \( r_+ \), and the cosmological horizon \( r_c \), where \( r_-<r_+<r_c \), all of theme are obtained from \(f(r)=0\). The surface gravity for each horizon is given by:

$$\begin{aligned} \kappa _i=\frac{1}{2} \big | f'(r_i) \big | \end{aligned}$$
(6)

for i in \(\{-,+,c\}\).

3 Quasinormal modes and strong cosmic censorship

We now consider a minimally coupled massless scalar field perturbation in the aforementioned spacetime background. The field equation is governed by the Klein–Gordon equation,

$$\begin{aligned} \nabla _a\nabla ^a\Psi = 0. \end{aligned}$$
(7)

Given the spherical symmetry and time translational invariance, the scalar field is decomposed as

$$\begin{aligned} \Psi (t,r,\theta ,\phi )=\sum _{l,m}e^{-i \omega t}Y_{lm}(\theta ,\phi )\frac{\psi (r)}{r} \end{aligned}$$
(8)

with \(Y_{lm}(\theta ,\phi )\) denoting the spherical harmonics. This decomposition allows the scalar perturbation field equation to be articulated as

$$\begin{aligned} \frac{d^2 \psi (r)}{d r_*^2}+[\omega ^2-V(r)] \psi (r)=0 \end{aligned}$$
(9)

where

$$\begin{aligned} V(r)=\frac{f(r)}{r^2}\left[ l(l+1)+\frac{2M}{r}-\frac{2Q^2}{r^2}+\frac{\lambda \sigma }{r^\sigma }\right] \end{aligned}$$
(10)

and the tortoise coordinate is defined as

$$\begin{aligned} r_*=\int _{r_0}^r\frac{d r}{f(r)} \end{aligned}$$
(11)

with \(r_+< r_0 < r_c\). This definition implies that \(r_* \rightarrow \pm \infty \) as \(r \rightarrow r_{c,+}\).

Typically, the initial values of our matter field are set within the physical spacetime region, that is, between \( r_+ \) and \( r_c \) [80]. Under these circumstances, the scalar field must adhere to specific boundary conditions that ensure only ingoing waves near the event horizon and outgoing waves near the cosmological horizon. These conditions are precisely defined as:

$$\begin{aligned}&\psi \sim e^{-i\omega r_{*}}, \quad r \rightarrow r_+ \quad (r_*\rightarrow -\infty ),\nonumber \\&\psi \sim e^{i\omega r_{*}}, \quad r \rightarrow r_c \quad (r_*\rightarrow \infty ). \end{aligned}$$
(12)

To satisfy these boundary conditions, certain restrictions on the scalar field frequency \( \omega \) are necessary. The solution \( \Psi \) obtained under these conditions is known as the QNM, and the corresponding \( \omega \) is referred to as the QNM frequency. We will next explore the relationship between these QNM frequencies and the validity of the SCC in dS-like spacetimes. The validity of the SCC necessitates that our spacetime is inextendible at the Cauchy horizon. In our context, we require the spacetime to be inextendible within the realm of weak solutions at the Cauchy horizon. If \(\Psi \) and \(g_{\mu \nu }\) are considered as weak solutions of the Einstein equation in a small neighborhood \({\mathcal {V}}\) of spacetime, then by multiplying each term in the field equation by any smooth and compactly supported test function \(\Phi \), the integrals of each term within \({\mathcal {V}}\) are finite and satisfy the following integral form of the field equation [31, 40, 44, 50]:

$$\begin{aligned} \int _{\mathcal {V}}d^4x\sqrt{-g}\left( R_{\mu \nu }-\frac{1}{2}R g_{\mu \nu }-8\pi T_{\mu \nu }\right) \Phi =0. \end{aligned}$$
(13)

In this equation, \(T_{ab} = T_{ab}^{\text {SC}} + T_{ab}^\text {EM} + T_{ab}^{\text {DE}}\) represents the total stress–energy–momentum tensor, which encompasses contributions from the scalar field, electromagnetic field, and dark matter. Specifically, the scalar field’s contribution to the tensor is expressed as

$$\begin{aligned} T^{\text {SC}}_{ab} = 2 \nabla _a \Psi \nabla _b \Psi - g_{ab} \nabla _c\Psi \nabla ^c\Psi . \end{aligned}$$
(14)

The first term of Eq. (13) can be represented as

$$\begin{aligned}{} & {} \int _{\mathcal {V}}d^4x\sqrt{-g}(R_{\mu \nu }-1/2 R g_{\mu \nu })\Phi \nonumber \\{} & {} \sim \int _{\mathcal {V}}d^4x\sqrt{-g}(\partial \Phi )\Gamma + \int _{\mathcal {V}}d^4x\sqrt{-g} \Gamma ^2\Phi \end{aligned}$$
(15)

where \(\Gamma \) denotes the Christoffel symbols. For brevity, most indices are omitted in our discuss. To ensure the boundedness of the integral, \(\Gamma \) must belong to the space of locally square-integrable functions in \({\mathcal {V}}\), i.e., \(\Gamma \in L^2_{\text {loc}}\) [40].

For the contribution from the electromagnetic field, to ensure the weak solution is extendable beyond Cauchy horizon,

$$\begin{aligned} \int _{\mathcal {V}} d^4x\sqrt{-g}T^{\text {EM}}_{\mu \nu }\Phi \sim \int _{\mathcal {V}}d^4x\sqrt{-g} (\partial A)^2\Phi . \end{aligned}$$
(16)

is integrable, i.e., we have \(A_\mu \in H^1_{\text {loc}}\).

Table 1 Utilizing various numerical methods, we computed the lowest-lying QNMs, characterized by \(\beta = -{\text {Im}(\omega )}/{\kappa _-}\), across different l values for the parameters \(\lambda /M^\sigma = 0.01, \sigma = -1.5\) and \(Q = 0.8\)
Full size table
Table 2 Utilizing various numerical methods, we computed the lowest-lying QNMs, characterized by \(\beta = -{\text {Im}(\omega )}/{\kappa _-}\), across different l values for the parameters \(\lambda /M^\sigma = 0.01, \sigma = -1.5\) and \(Q = 0.99\)
Full size table
Table 3 Utilizing various numerical methods, we computed the lowest-lying QNMs, characterized by \(\beta = -{\text {Im}(\omega )}/{\kappa _-}\), across different l values for the parameters \(\lambda /M^\sigma = 0.01, \sigma = -1.5\) and \(Q = 0.999 Q_{\text {ext}}\)
Full size table

The contribution from the stress–energy–momentum tensor of the scalar field can be represented by:

$$\begin{aligned} \int _{\mathcal {V}} d^4x\sqrt{-g}T^{\text {SC}}_{\mu \nu }\Phi \sim \int _{\mathcal {V}}d^4x\sqrt{-g} (\partial \Psi )^2\Phi . \end{aligned}$$
(17)

Then, the boundedness of this integral leads to the requirement that the derivative of the scalar field \(\Psi \) must be locally square integrable, signifying \(\Psi \in H^1_{\text {loc}}\).

The asymptotic solution near the Cauchy horizon is given by [40, 44]

$$\begin{aligned} \Psi \sim e^{i\frac{\omega }{\kappa _-}\ln |r-r_-|}\propto |r-r_-|^\beta \end{aligned}$$
(18)

with

$$\begin{aligned} \beta \equiv -\frac{\text {Im} (\omega )}{\kappa _-}. \end{aligned}$$
(19)

For the weak solution to extend beyond the Cauchy horizon, it requires the local square integrability of the derivative of the scalar field (i.e., \(\Psi \in H^1_{\text {loc}}\)), necessitating

$$\begin{aligned} \beta > 1/2. \end{aligned}$$
(20)

Similarly, for the DE components, to ensure the weak solution extends beyond the Cauchy horizon, we also require \(T^{\text {DM}}_{\mu \nu }\) to be integrable within \(\mathcal {V}\), i.e., \(T^{\text {DM}}_{\mu \nu }\in L^1_{\text {loc}}\). This condition imposes certain constraints on dark energy.

However, regarding dark matter, research is primarily based on assumptions made about the large-scale structure of the universe and corresponding symmetries. Its dynamical information remains unknown; hence, the specific form of the action for dark matter is also unknown. Fortunately, due to the absence of coupling between the scalar field and dark matter in our consideration, \(T^{\text {DM}}_{\mu \nu }\) does not contain the scalar field. Therefore, the integrability condition for \(T^{\text {DM}}_{\mu \nu }\) only imposes specific limitations on dark matter without introducing additional conditions for the scalar field.

In summary of the above results, to extend the weak solution beyond the Cauchy horizon, we need to ensure that the conditions \(\Gamma \in L^2_{\text {loc}}\), \(A \in H^1_{\text {loc}}\), \(\beta >1/2\) and \(T^{\text {DM}}_{\mu \nu }\in L^1_{\text {loc}}\) are all satisfied. Therefore, when considering scalar field perturbations, as long as there exists a QNM satisfying the condition

$$\begin{aligned} \beta \le 1/2, \end{aligned}$$
(21)

the Cauchy horizon will become inextendible within the realm of weak solutions, thereby obeying the SCC. That is to say, for evaluating the validity of the SCC, attention should be directed to the QNM with the minimal \(\beta \), specifically, the lowest-lying mode.

Fig. 1
figure 1

The temporal evolution of perturbations in the scalar field within the context of a Reissner–Nordström black hole coupled with quintessence. The parameters under consideration include \(\lambda /M^\sigma =0.02\), \(Q=0.995 Q_{\text {ext}}\), and variations in the angular momentum quantum number l, as well as the state parameter \(\sigma \)

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4 Numerical methods and results

4.1 Numerical methods

In our paper, we will primarily use the pseudospectral method [81, 82] for calculating the QNM frequencies. This method necessitates the domain of differential equation to be within a finite closed interval and requires the variable to be analytic throughout this interval. However, it is easy to see that the field equation (9) does not meet these criteria. By employing the boundary conditions (12), we introduce a new variable y(r), defined as

$$\begin{aligned} \psi (r)=\left( x+1\right) ^{\frac{i \omega }{2\kappa _+}} \left( x-1\right) ^{-\frac{i\omega }{2\kappa _c}}y(x), \end{aligned}$$
(22)

where

$$\begin{aligned} r=\frac{r_c-r_+}{2}x+\frac{r_c+r_+}{2}, \end{aligned}$$
(23)

thus transforming the field equation into a regular form in the interval \([-1,1]\) and ensuring that y(r) is analytic throughout. Consequently, the field equation (9) can be simplified to

$$\begin{aligned} a_0(\omega ,x)y(x)+a_1(\omega ,x)y'(x)+a_2(\omega ,x)y''(x)=0. \end{aligned}$$
(24)

Given that y(x) is an analytic function, we can expand it using an nth-order cardinal function \(C_i(x)\), i.e.,

$$\begin{aligned} y(x)=\sum _{i=0}^n z_i C_i(x). \end{aligned}$$
(25)

This allows us to reformulate the equation as

$$\begin{aligned} \sum _{i=0}^n E_i(x) z_i=0, \end{aligned}$$
(26)

where

$$\begin{aligned} E_i(x)=a_0(x)C_i(x)+a_1(x)C_i'(x)+a_2(x)C_i''(x). \end{aligned}$$
(27)

Evaluating this equation at \(n+1\) grid points \(x_j\), we can transform it into a matrix equation. Ultimately, using the matrix augmentation method (details in Refs. [81, 82]), this matrix equation simplifies to \((M_0+\omega M_1) z=0\). The QNM frequencies can then be obtained by solving for the eigenvalues of the matrix \((-M_1^{-1}M_0)\).

In addition to the pseudospectral method [81, 82], we will also employ the direct integration method [83, 84] to validate our results. Moreover, due to the challenges in resolving high-angular-momentum (large l) cases with numerical methods, we will apply the WKB approximation [85] to determine the photo-sphere modes, which correspond to the QNMs in the large-l limit.

For the direct integration method, we also employ the regularized field equation (2427). For a given \(\omega \), we use the series expansion of y(x) at \(x=\pm 1\) as the boundary condition of y(x) near \(x=\pm 1\) and solve the equation individually in the intervals \((-1, 0]\) and [0, 1) using Mathematica. Finally, to ensure the smoothness of the two solutions at \(x=0\), we determine the acceptable frequency \(\omega \).

Fig. 2
figure 2

At \(\lambda /M^\sigma = 0.015\), the graph displays the lowest-lying QNMs as \(\beta =-{\text {Im}(\omega )}/{\kappa _-}\) for various \(\sigma \), relative to the black hole’s charge ratio \(Q/Q_{\text {ext}}\) at a fixed l. The SCC critical charge ratio is indicated by the horizontal dashed line at \(\beta =1/2\)

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Fig. 3
figure 3

The highlighted red area on the \(\sigma -Q/Q_{\text {ext}}\) diagram indicates the region where SCC violations occur for \(\lambda /M^\sigma =0.005\), \(\lambda /M^\sigma =0.01\), \(\lambda /M^\sigma =0.015\), and \(\lambda /M^\sigma =0.02\), respectively

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4.2 Numerical results

In our study, we mainly focus on the scenario where \(-1\le \omega _f<-1/3\), indicating that the black hole exhibits a cosmological-like horizon and it is a dS-like spacetime. We know that the instability of the Cauchy horizon primarily stems from the exponential blueshift of matter near it. For black holes in asymptotically flat spacetime, such as RN or Kerr black holes, the final stage of QNMs transitions into an inverse power-law tail [18,19,20,21,22,23]. Consequently, the corresponding decay cannot counterbalance the exponential blueshift near the Cauchy horizon, thus maintaining the instability of the Cauchy horizon. However, in dS-like black holes (e.g., RNdS black holes), scalar perturbations also exhibit exponential decay in the physical spacetime region, capable of offsetting the exponential blueshift near the Cauchy horizon, potentially leading to the failure of the SCC [25,26,27,28,29,30]. Therefore, discussing the effectiveness of SCC based on solving QNMs is only valid when perturbations exhibit exponential decay.

After defining

$$\begin{aligned} \Psi (t,r,\theta ,\phi )=\sum _{l,m}Y_{lm}(\theta ,\phi )\frac{u(t, r)}{r}, \end{aligned}$$
(28)

we have

$$\begin{aligned} -\frac{d^2 u(t, r)}{d t^2}+\frac{d^2 u(t, r)}{d r_*^2}-V(r) u(t, r)=0. \end{aligned}$$
(29)

By applying the finite difference method at \(r^*\), the Runge–Kutta method in the time-direction, and Gaussian initial condition:

$$\begin{aligned} u(t=0, r)\simeq \exp \left[ \frac{-(r_*-r_*^{0})^2}{2 w^2}\right] , \end{aligned}$$
(30)

we solve the field equation (29) for u(t, r) numerically to obtain the time-evolution of the scalar field. In Fig. 1, we investigate the temporal evolution of perturbations in the scalar field within the context of a Reissner–Nordström black hole coupled with quintessence. Under the chosen parameters, the corresponding results indicate that the scalar perturbations exhibit exponential decay in the late-time regime, rather than an inverse power-law behavior. In other words, in the scenario we consider, the stability of the Cauchy horizon needs to be discussed through specific calculations of QNM.

In Tables 1, 2, and 3, we calculated \(\beta = -\text {Im} (\omega )/\kappa _-\) of the lowest-lying modes, using both the pseudospectral method and direct integration method for various black hole parameters and different values of l. In the large-l limit, we employed the WKB approximation to estimate \(\beta \) for the lowest-lying modes. Our results demonstrate the consistency of different numerical methods in calculating QNMs, thus confirming the reliability of our numerical computations.

Furthermore, our numerical findings reveal that for black holes far from extremality, where \(\kappa _-\) is relatively large and \(-\text {Im} (\omega )\) is typically small, the lowest \(\beta \) values are generally less than 1/2. Only when black holes approach extremality, and \(\kappa _-\) becomes significantly small, is there a possibility of violating the SCC, where the lowest-lying mode’s \(\beta \) exceeds 1/2. Therefore, in the subsequent results, we only present cases near extremality.

In each subplot of Fig. 2, we illustrate the relationship between the lowest-lying QNMs’ \(\beta \) and the extremal charge ratio \(Q/Q_{\text {ext}}\) for different state parameters \(\sigma =3\omega _f+1\) of dark energy, considering \(l=0\), \(l=1\), and large l cases. In all the plots, we fix \(\lambda /M^\sigma =0.15\), and \(Q_{\text {ext}}\) represents the charge value when the black hole reaches extremality \(r_+=r_-\). All non-extremal black holes have \(Q<Q_{\text {ext}}\). For all cases, we find that as long as spacetime is sufficiently close to extremal black holes, the values of \(\beta \) for the lowest-lying QNMs are greater than 1/2, resulting in a violation of SCC. Since the chosen state parameters satisfy \(-2\le \sigma <0\), our black holes exhibit a dS-like structure, making this result clearly reasonable. This finding is consistent with the conclusion drawn in Ref. [52]. However, within the parameter range we have chosen, we observe that as \(|\sigma |\) becomes sufficiently small, indicating an asymptotic behavior approaching asymptotically flatness, the violation range of the charge ratio becomes exceedingly small, i.e., \(Q_{\text {crit}}/Q_{\text {ext}} \rightarrow 1\). Due to certain limitations in computational accuracy, the analyses in Ref. [52] did not explore situations very close to extremality, and thus this particular outcome was not present in their results.

From Fig. 2, it is easy to observe that when we fix the parameter \(\lambda /M^\sigma =0.15\), for \(\sigma \) values close to \(-2\), such as \(\sigma =-2\) and \(-1.8\), in the \(Q/Q_{\text {ext}}-\beta \) diagram, the lowest-lying QNMs are primarily determined by the \(l=0\) mode (nearly extremal modes) and the large-l mode (photo-sphere modes). This implies that for non-nearly extremal black holes, the lowest-lying QNMs are influenced by large-l modes, whereas for nearly extremal black holes, the lowest-lying QNMs are determined by the \(l=0\) mode (as seen in cases with \(\lambda /M^\sigma =-2,-1.8\), etc.). However, as \(\sigma \) increases, the lowest-lying QNMs in the \(Q/Q_{\text {ext}}-\beta \) diagram gradually transition to being governed by the \(l=0\) mode (nearly extremal modes) and the \(l=1\) mode (dS modes) (as seen in cases with \(\lambda /M^\sigma =-1.6,-1.4,-1.2\), etc.). Furthermore, as \(\sigma \) continues to decrease, the lowest-lying QNMs are entirely determined by the \(l=0\) lowest-lying modes, as seen in cases with \(\lambda /M^\sigma =-1, -0.8,-0.6\), etc. The corresponding results are depicted in Fig. 3, where the boundary of our violation interval may consist of several discontinuous curves. In Fig. 2, we mark the position of the critical charge ratio \(Q_{\text {crt}}/Q_{\text {ext}}\) using a red dotted line. From these figures, it is not hard to see that with the increase in the state parameter \(\sigma \), particularly after \(\sigma \ge -1.6\), the interval of violation for SCC decreases, indicating a significant impact of the state parameter \(\sigma \) of Quintessential dark energy on the effectiveness of the SCC.

Reference [52] also conducted calculations on the relationship between QNMs and the state parameter \(\omega _f\), as illustrated in Fig. 5 of their paper. Their findings revealed that as \(|\omega _f|\) diminishes (i.e., as the asymptotic properties of the black hole approach asymptotic flatness), the width of the violation region of the charge ratio consistently expands. This appears somewhat contradictory to the observations depicted in Fig. 2 and the intuitive expectation that” as the black hole approaches asymptotically flat spacetime, SCC should become more difficult to violate.” Therefore, here we employ more refined numerical calculations to investigate the influence of the dark-energy state parameter \(\sigma \) on the validity of the SCC, specifically analyzing its impact on the violation region. We plot the SCC violation region in the \(\sigma -Q/Q_{\text {ext}}\) diagram (see Fig. 3), where the shaded area represents the parameter region where SCC is violated, with the lower boundary determined by the critical charge rate \(Q_{\text {crit}}/Q_{\text {ext}}\). In Fig. 3, we consider four different values of \(\lambda /M^\sigma \): 0.005, 0.01, 0.015, and 0.02. The figure shows that regardless of the values of \(\sigma \) within the range \([-2, 0),\) SCC is consistently violated whenever the black hole approaches the extremal limit. However, as the dark energy parameter \(\sigma \) increases and approaches zero, the width of the violation interval also tends to zero. This result is intuitive from a physical perspective. The parameter \(\sigma \) characterizes the asymptotic behavior of the black hole. As \(\sigma \) approaches zero, the asymptotic behavior of the black hole gradually approaches asymptotically flat spacetime, where the SCC holds. Overall, as the parameter \(\sigma \) increases and approaches zero, the width of the violation interval decreases, making it more challenging for SCC to be violated. However, it’s worth noting that the length of the violation interval concerning the parameter \(\sigma \) doesn’t exhibit a monotonic change, as observed from the last three figures of Fig. 3. At this point, there exists a critical value of the dark energy state parameter, \(\sigma _{\text {crt}}\), marked by an asterisk in Fig. 3. When \(\sigma < \sigma _{\text {crt}}\), the width of the violation interval remains almost constant, even showing a slight increase with the increment of \(\sigma \). However, once \(\sigma \) surpasses its critical value, the violation interval of SCC rapidly decreases until it approaches nearly zero.

This result differs somewhat from the findings in Ref. [52], which could stem from numerical errors. In Ref. [52], they mainly used a time-domain method to compute QNMs. For situations very close to extremal and where the asymptotic properties closely approach asymptotic flatness, extracting the influence of asymptotic properties and extremal black hole properties on QNMs requires long time-domain evolutions. Therefore, in these parameter regimes, the results may suffer from some numerical inaccuracies, as also mentioned in the conclusion of their paper.

5 Conclusion

Based on current observational results, it is possible that dark energy exists in our universe. In Ref. [52], calculations of QNMs within charged black holes surrounded by quintessence reveal that for these dark energy models with a dS-like horizon, the SCC is violated as long as the black hole approaches extremality. Additionally, they primarily analyze the impact of the energy density amplitude of dark energy on the SCC. However, their calculations did not account for situations when the black hole is extremely close to extremality, and their computations may also miss information about the asymptotic properties when the black hole closely approaches asymptotic flatness.

Therefore, to comprehensively investigate the asymptotic properties of black holes, namely the influence of the dark energy state parameter \(\sigma \) on the effectiveness of SCC, we employ various numerical methods to compute the QNMs of a massless neutral scalar field in charged black holes with a dS-like asymptotic behavior, where the state parameter \(\omega _f\) falls within the range \(-1\le \omega _f<-1/3\). We then explored the influence of the dark energy state parameter \(\omega _f\) on the effectiveness of SCC.

However, as the dark energy parameter \(\sigma = 3\omega _f + 1\) increases and approaches zero, there is an overall decreasing trend in the width of the violation interval within the entire range. Consequently, increasing the dark energy state parameter \(\sigma \) seems to partially alleviate the violation of SCC. When \(\sigma \) approaches zero, the width of the violation interval tends toward zero. From a physical perspective, this result is intuitive. The parameter \(\sigma \) characterizes the asymptotic behavior of the black hole. As \(\sigma \) approaches zero, the asymptotic behavior of the black hole gradually approaches asymptotically flat spacetime, preserving the SCC.

However, it’s worth noting that increasing the state parameter \(\sigma \) of the dark matter doesn’t necessarily always alleviate the violation of SCC. For a given parameter \(\lambda /M^\sigma \), there exists a critical state parameter \(\sigma _{\text {crt}}\), and only when the state parameter surpasses this critical value does the width of the violation interval rapidly decrease. Therefore, we can draw the following conclusion: once the state parameter exceeds its critical value, increasing its magnitude can effectively mitigate the violation of SCC.

There are several promising directions for future research to build upon the findings of this study. Firstly, exploring the effects of different types of dark energy on SCC would be intriguing. Additionally, considering that dark energy is a dynamical field, investigating its influence on SCC as a perturbation in spacetime is crucial. Furthermore, previous research has shown that rotating black holes satisfy SCC, so it’s essential to discuss the impact of dark energy on the validity of SCC in the context of rotating black holes as well.