\section{A brief review on Bardeen-Kiselev black hole with cosmological constant} \label{secII} }

This section deals with a very brief introduction to the Bardeen-Kiselev BH following the work in Ref.\cite{ret28}, the corresponding action is given by

\begin{equation}  S=\int d^{4} x \sqrt{-g}[R+2 \lambda+L(F)],             \label{eq1} \end{equation}
here $R$, $\lambda$ and $L(F)=\frac{24 \sqrt{2} M q^{2}}{8\pi\left(\sqrt{\frac{2 q^{2}}{F}}+2 q^{2}\right)^{\frac{5}{2}}}-\frac{6 w c\left(\frac{2 F}{q^{2}}\right)^{\frac{3}{4}(w+1)}}{8\pi}$ denote the scalar curvature, the cosmological constant and the function of the field strength $F$ of the nonlinear Lagrangian of electromagnetic theory respectively. The field strength $F$ is given by
\begin{equation}  F=F^{\mu v} F_{\mu v},             \label{eq2} \end{equation}
where the scalar $F$ is $F=\frac{2 q^{2}}{r^{4}}$ and $F_{\mu \nu}$ is the electrodynamic field tensor that can be expressed in terms of a gauge potential as $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$. The electromagnetic field tensor implies that $F_{\mu \nu}$ is anti-symmetric and it has only six independent components, the only nonzero component of $F_{\mu \nu}$ for a spherically symmetric spacetime that is only magnetically charged is $F_{23}=qsin\theta$.

Here one can express the covariant equations of motion as
\begin{equation}
\begin{aligned}
 & R_{\mu v}-\frac{1}{2} g_{\mu v} R+\lambda g_{\mu v}=8 \pi T_{\mu v}, \\  &\nabla_{v}\left(L_{F} F^{\mu v}\right)=0, \label{eq3}
\end{aligned}   \end{equation}
if we consider the stress-energy tensor to nonlinear electrodynamics
\begin{equation} T_{\mu v}=g_{\mu v} L(F)-L_{F} F_{\mu}^{\alpha} F_{v \alpha},             \label{eq4} \end{equation}
with $L_{F}=\frac{\partial L(F)}{\partial F}$, then we will look for solutions which are static and spherically symmetric for Einstein equation
\begin{equation}  d s^{2}=f(r) d t^{2}-\frac{1}{f(r)} d r^{2}-r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right),             \label{eq5} \end{equation}
where
\begin{equation}  f(r)=1-\frac{2 c}{r^{3 w+1}}-\frac{2 M r^{2}}{\left(q^{2}+r^{2}\right)^{\frac{3}{2}}}-\frac{\lambda r^{2}}{3},    \label{eq6} \end{equation}
here $c$ is the the normalization factor,  $M$ is the total mass of the BH, $q$ is the magnetic charge and $w$ is the state parameter with $-1<w<-1/3$. Regarding this formalism, we find the Kiselev-(anti-)de Sitter solution with cosmological constant for the limit $q$ $\rightarrow$ $0$ and the Bardeen-(anti-)de Sitter solution for $c$ $\rightarrow$ $0$\cite{retBM}.


{\centering  \section{Gravitational perturbations and electromagnetic perturbations} \label{secIII} }

The study of black hole perturbations was first proposed by Regge and Wheeler in terms of the investigation for the odd parity type of the spherical harmonics\cite{ret29}, afterwards Zerilli generalized it to the even parity case\cite{ret30}. It is well known that generally there are two different kinds of perturbations of black holes in the context of the general theory of relativity: the test field in a black hole background and the perturbation of the metric. The former one is achieved through the method of solving the dynamical equation for the given test field in the background of the black hole, and the second one is obtained by linearising the Einstein equation to derived the evolution equations, i.e. the gravitational perturbation. In view of the fact that comparison with the strength of the external fields decaying in the vicinity of BH, the gravitational radiation is the strongest one. In this section, we firstly focus on the gravitational perturbation of the Bardeen-Kiselev black hole with cosmological constant. The general procedure for gravitational perturbation is to introduce a small perturbation into the static background metric. Considering the line element in Eq.\eqref{eq5}, the metric tensor can be written as

\begin{equation}  g_{\mu v}=\tilde{g}_{\mu v}+h_{\mu v},             \label{eq7} \end{equation}

The perturbations $h_{\mu v}$ can be decomposed as

\begin{equation}
h_{\mu v}=\left(\begin{array}{cccc}0 & 0 & 0 & h_{0}(t, r) \\ 0 & 0 & 0 & h_{1}(t, r) \\ 0 & 0 & 0 & 0 \\ h_{0}(t, r) & h_{1}(t, r) & 0 & 0\end{array}\right)(-\xi(\theta)), \label{eq8}
\end{equation}
this formalism is similar to the axial decomposition in the Regge-Wheeler gauge. Noted that it is not exactly the same gauge, we prefer to obtain the form of $\xi(\theta)$ through the
field equations rather than by imposition of some kind of expansion in terms of spherical harmonics.


The Einstein equations are given by

\begin{equation}  G_{\mu v}+\lambda g_{\mu v}=8\pi T_{\mu v},             \label{eq9} \end{equation}
and
\begin{equation}
\begin{aligned}
&\delta R_{\mu v}=-\nabla_{\alpha} \delta \Gamma_{\mu v}^{\alpha}+\nabla_{v} \delta \Gamma_{\mu \alpha}^{\alpha} \\
&\delta \Gamma_{\beta \gamma}^{\alpha}=\frac{1}{2} \tilde{g}^{\alpha v}\left(\partial_{\gamma} h_{\beta v}+\partial_{\beta} h_{\gamma v}-\partial_{v} h_{\beta \gamma}\right).
 \label{eq10}      \end{aligned}      \end{equation}
We obtain coupled second order partial differential equations from $\theta\phi$ and $r\phi$ components of Einstein equations as Eq.\eqref{eq11} with $T_{\theta\phi}=0$ and $T_{r\phi}=-h_{1} L \xi(\theta)$
\begin{equation}
\begin{aligned}
&-\frac{1}{f} \frac{\partial}{\partial t} h_{0}+\frac{\partial}{\partial r}\left(f h_{1}\right)=0, \\
&\frac{1}{f}\left(\frac{\partial^{2}}{\partial t^{2}} h_{1}-\frac{\partial{ }^{2} h_{0}}{\partial t \partial r}+\frac{2}{r} \frac{\partial}{\partial t} h_{0}\right)+\left[\frac{\ell(\ell+1)-2}{r^{2}}+\frac{2}{r} f^{\prime}+f^{\prime \prime}+2\left(k^{2} L+\lambda\right)\right] h_{1}=0,       \label{eq11}      \end{aligned}      \end{equation}
here $\xi(\theta)=P_{\ell}(\cos \theta)$.
By using the definition $\varphi(t, r)=\frac{f}{r} h_{1}(t, r)$, Eq.\eqref{eq11} can be expressed as
\begin{equation}  {\left[\frac{\partial^{2}}{\partial t^{2}}-f^{2} \frac{\partial^{2}}{\partial r^{2}}-f f^{\prime} \frac{\partial}{\partial r}+f\left(\frac{\ell(\ell+1)+2(f-1)}{r^{2}}+\frac{f^{\prime}}{r}+f^{\prime \prime}+2\left(k^{2} L+\lambda\right)\right)\right] \varphi(t, r)=0},  \label{eq12} \end{equation}
in order to facilitate this procedure, we change the variable $r$ to the tortoise coordinate $r_{*}$ with the defination $d r=f(r) d r_{*}$, which leads to
 \begin{equation}  {\left[\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial r_{*}^{2}}+V(r)\right] \varphi\left(t, r_{*}\right)=0 },  \label{eq13} \end{equation}
with the effective potential
 \begin{equation}  V(r)=f\left(\frac{\ell(\ell+1)+2(f-1)}{r^{2}}+\frac{f^{\prime}}{r}+f^{\prime \prime}+2\left(k^{2} L+\lambda\right)\right),  \label{eq14} \end{equation}

here if we consider the temporal dependence as $\varphi \sim \varphi e^{-i{\omega}t}$, we can get the master equation for gravitational perturbations of the BH as
\begin{equation}\frac{d^{2}\Psi (r_{*}) }{dr^{2}_{*}}+(\omega^{2}-V(r))\Psi(r_{*})=0. \label{eq15} \end{equation}
By looking for the solution of Eq.\eqref{eq15}, we are likely to construct the perturbed metric for detecting the features of spacetime. The parameter $\omega$ denotes the dissipative modes in time, thus the BH will oscillate after the perturbation and then go back to a stable state, which is known as quasinormal frequencies.

Next, the behaviour of the dynamical response of the spherically symmetric BH for electromagnetic perturbations is our topic. We decompose 4-vector potential of the electromagnetic field as unperturbed background potential $\widetilde{A}_{\mu}$ and perturbed part $\delta A_{\mu}$
\begin{equation}A_{\mu}=\widetilde{A}_{\mu}+\delta A_{\mu},\label{eq16} \end{equation}
in view of the static and spherically symmetric background, we consider the unperturbed 4-vector potential of magnetically charged black hole as
\begin{equation}\widetilde{A}_{\mu}=-q \cos \theta \delta_{\mu}^{\phi}.\label{eq17} \end{equation}
In spherically symmetric background, the perturbation in vector potential can be expressed as a superposition of vector spherical harmonics, i.e.
\begin{equation}\delta A_{\mu}=\sum_{\ell, m}\left[\begin{array}{c}0 \\ 0 \\ \psi(t, r) \frac{\partial_{\phi} Y_{\ell m}(\theta, \phi)}{\sin \theta} \\ -\psi(t, r) \sin \theta \partial_{\theta} Y_{\ell m}(\theta, \phi)\end{array}\right],\label{eq18} \end{equation}
with $Y_{\ell m}(\theta, \phi)$ denotes scalar spherical harmonics.

The nonvanishing covariant components of the electromagnetic field tensor of the 4-potential Eq.\eqref{eq16} with perturbation Eq.\eqref{eq18} are given by
\begin{equation}
\begin{aligned}
&F_{t \theta}=\frac{1}{\sin \theta} \partial_{t} \psi(t, r) \partial_{\phi} Y_{\ell m}(\theta, \phi), \\
&F_{t \phi}=-\sin \theta \partial_{t} \psi(t, r) \partial_{\theta} Y_{\ell m}(\theta, \phi), \\
&F_{r \theta}=\frac{1}{\sin \theta} \partial_{r} \psi(t, r) \partial_{\phi} Y_{\ell m}(\theta, \phi), \\
&F_{r \phi}=-\sin \theta \partial_{r} \psi(t, r) \partial_{\theta} Y_{\ell m}(\theta, \phi),    \\
&F_{\theta \phi}=\sin \theta\left(q+\ell(\ell+1) \psi(t, r) Y_{\ell m}(\theta, \phi)\right),
 \label{eq19}      \end{aligned}      \end{equation}
all non zero contravariant components of electromagnetic field tensor are following:

\begin{equation}
\begin{aligned}
&F^{t \theta}=-\frac{1}{f(r) r^{2} \sin \theta} \partial_{t} \psi(t, r) \partial_{\phi} Y_{\ell m}(\theta, \phi), \\
&F^{t \phi}=\frac{1}{f(r) r^{2} \sin \theta} \partial_{t} \psi(t, r) \partial_{\theta} Y_{\ell m}(\theta, \phi),  \\
&F^{r \theta}=\frac{f(r)}{r^{2} \sin \theta} \partial_{r} \psi(t, r) \partial_{\phi} Y_{\ell m}(\theta, \phi),  \\
&F^{r \phi}=-\frac{f(r)}{r^{2} \sin \theta} \partial_{r} \psi(t, r) \partial_{\theta} Y_{\ell m}(\theta, \phi),   \\
&F^{\theta \phi}=\frac{1}{r^{4} \sin \theta}\left(q+\ell(\ell+1) \psi(t, r) Y_{\ell m}(\theta, \phi)\right).
 \label{eq20}      \end{aligned}      \end{equation}
In view of the fact that the field strength remains same at zeroth order but has components in first order for the total 4-vector potential, combining Eq.\eqref{eq19} and Eq.\eqref{eq20}, we present the electromagnetic field strength $F$ as
\begin{equation}        F \approx \tilde{F}+\delta F,        \label{eq21} \end{equation}
where $\tilde{F}=\frac{2 q^{2}}{r^{4}}$ and $\delta F=\frac{4 q}{r^{4}} \ell(\ell+1) \psi(t, r) Y_{\ell m}(\theta, \varphi)$. Near $\tilde{F}$, by using Taylor series up to first order term for expanding $L_{F}$, we have
\begin{equation}
 L_{F} \approx \tilde{L}_{\tilde{F}}(\tilde{F})+\tilde{L}_{\tilde{F} \tilde{F}} \delta F,  \label{eq22}   \end{equation}
where $\tilde{L}_{\tilde{F} \tilde{F}}=\partial_{\tilde{F}}^{2}\tilde{L}=\partial_{\tilde{F}}\tilde{L}_{\tilde{F}}$.
By combining Eq.\eqref{eq20} into Eq.\eqref{eq3} we have
\begin{equation}
 \frac{\partial\left(L_{F} F^{\mu t}\right)}{\partial t}+\frac{1}{r^{2}} \frac{\partial\left(r^{2} L_{F} F^{\mu r}\right)}{\partial r}+\frac{1}{\sin \theta} \frac{\partial\left(\sin \theta L_{F} F^{\mu \theta}\right)}{\partial \theta}+\frac{\partial\left(L_{F} F^{\mu \phi}\right)}{\partial \phi}=0.  \label{eq23}   \end{equation}
For $u=\theta$ and $u=\phi$, the above equation transforms into
\begin{equation}
[-\frac{\partial^{2}}{\partial t^{2}}+(\frac{f(r)^{2}}{\tilde{L}_{\tilde{F}}} \tilde{L}_{\tilde{F}}^{\prime}+f(r) f^{\prime}(r) ) \frac{\partial}{\partial r}+f(r)^{2} \frac{\partial^{2}}{\partial r^{2}}-\frac{\ell(\ell+1)}{r^{2}} f(r) (1+\frac{4 q^{2} \tilde{L}_{\tilde{F} \tilde{F}}}{r^{4} \tilde{L}_{\tilde{F}}})] \psi(t, r)=0,
\label{eq24}   \end{equation}
where $\tilde{L}_{\tilde{F}}^{\prime}$ denotes the first derivative of $\tilde{L}_{\tilde{F}}$ with respect to $r$, for the sake of convenience, we take a transformation $\psi(t, r)=\left(\widetilde{L}_{\tilde{F}}\right)^{-\frac{1}{2}} \varphi(t, r)$, and considering the tortoise coordinate $d r=f(r) d r_{*}$, we obtain the Schr\"odinger-like wave equation
\begin{equation}
\left[\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial r_{*}^{2}}+V(r)\right] \varphi\left(t, r_{*}\right)=0,    \label{eq25}     \end{equation}
with the effective potential
\begin{equation}
 V(r)=-\frac{f^{2}}{4}\left(\frac{\widetilde{L}_{\tilde{F}}^{\prime}}{\widetilde{L}_{\tilde{F}}}\right)^{2}+\frac{f f^{\prime}}{2} \frac{\tilde{L}_{\tilde{F}}^{\prime}}{\widetilde{L}_{\tilde{F}}}+\frac{f^{2}}{2} \frac{\widetilde{L}_{\tilde{F}}^{\prime \prime}}{\widetilde{L}_{\tilde{F}}}+\frac{\ell(\ell+1)}{r^{2}} f(r)\left(1+\frac{4 q^{2} \widetilde{L}_{\tilde{F} \tilde{F}}}{r^{4} \widetilde{L}_{\tilde{F}}}\right),              \label{eq26}         \end{equation}
thus we also possibly construct the perturbed metric for detecting the features of spacetime.

\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{1.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{2.pdf}}
\centerline{(b)}
\end{minipage}
\end{tabular}
\caption{(a)The effective potential $V(r)$ for Bardeen-Kiselev BH with cosmological constant for $M=0.5$, $W=-2/3$, $q=0.05$ and $c=\lambda=0.001$ in gravitational perturbations;
(b)The effective potential $V(r)$ for Bardeen-Kiselev BH with cosmological constant for $M=1$, $W=-2/3$, $q=0.5$ and $c=\lambda=0.001$ in electromagnetic perturbations.}
\label{fig1}
\end{figure}

Moreover, for the gravitational perturbations and electromagnetic perturbations, we present the effective potentials $V(r)$ for Bardeen-Kiselev BH with cosmological constant in  Fig.\ref{fig1}. Roughly speaking, the overall nature of both potentials are the same, i.e. both potentials are positive definite between the event and cosmological horizons, and they all have a single maxima, which increases its height with increasing multipole number $\ell$. Besides, by taking a fixed $r$ for both potentials, one can see that the bigger multipole number $\ell$ is responsible for the bigger height of potential, which leads to a smaller absorption coefficient.


{\centering  \section{The QNMs of the Bardeen-Kiselev black hole with cosmological constant for gravitational perturbations and electromagnetic perturbations } \label{secIV} }

In this section, our purpose is to discuss the QNMs of the Bardeen-Kiselev black hole with cosmological constant for gravitational perturbations and electromagnetic perturbations. Until then, it's necessary for us to review the sixth order WKB method. As seen in Ref.\cite{ret32}, this method is more accurate than the third order one, and for the low overtones, the results of the sixth order WKB method coincides with those obtained from full numerical integration of the wave equation. The BH potential $V(r)$ in the present of the sixth order formula is
\begin{equation}
\frac{i(\omega^{2}-V_{0})}{\sqrt{-2V_{0}^{''}}}-\sum_{i=2}^{6}\varLambda_{i}=n+\frac{1}{2}, n=0,1,2,..., \label{eq27}    \end{equation}
among them, $V_{0}$ is the maximum effective potential of $V(r)$ at the tortoise coordinate $r_{*}$, $n$ is the overtone number and the correction term $\varLambda_{i}$ can be obtained in Ref.\cite{ret20}. And generally speaking, the quasinormal frequencies $\omega$ take the form $\omega=\omega_{R}-i\omega_{I}$, where the real part and the imaginary part of $\omega$ denote actual field oscillation and damping of the perturbation respectively.

\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{3.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{4.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{5.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{6.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a)Variation of Re $\omega$ with the magnetic charge $q$ for the state parameter $w=-1/3$; (b) Variation of -Im $\omega$ with the magnetic charge $q$ for the state parameter $w=-1/3$; (c)Variation of Re $\omega$ with the magnetic charge $q$ for the state parameter $w=-2/3$; (d) Variation of -Im $\omega$ with the magnetic charge $q$ for the state parameter $w=-2/3$. In both cases we take $M=1$, $c=0.01$ and $\lambda=0.001$. }
\label{fig2}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{7.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{8.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{9.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{10.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a)Variation of Re $\omega$ with the normalization factor $c$ for the state parameter $w=-1/3$; (b) Variation of -Im $\omega$ with the normalization factor $c$ for the state parameter $w=-1/3$; (c)Variation of Re $\omega$ with the normalization factor $c$ for the state parameter $w=-2/3$; (d) Variation of -Im $\omega$ with the normalization factor $c$ for the state parameter $w=-2/3$. In both cases we take $M=1$, $q=0.5$ and $\lambda=0.001$.}
\label{fig3}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{11.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{12.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{13.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{14.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a)Variation of Re $\omega$ with the cosmological constant $\lambda$ for the state parameter $w=-1/3$; (b) Variation of -Im $\omega$ with the cosmological constant $\lambda$ for the state parameter $w=-1/3$; (c)Variation of Re $\omega$ with the cosmological constant $\lambda$ for the state parameter $w=-2/3$; (d) Variation of -Im $\omega$ with the cosmological constant $\lambda$ for the state parameter $w=-2/3$. In both cases we take $M=1$, $q=0.5$ and $c=0.001$.}
\label{fig4}
\end{figure}


\begin{table}[htbp]
\caption{QNM frequencies of electromagnetic field perturbation for Bardeen-Kiselev BH with cosmological constant.}
\label{tab1}
\begin{tabular}{lllll}
\hline
\multicolumn{5}{l}{M=1, c=0.01, $\lambda=0.001$,  w=-1/3}
  \\ \hline
\multicolumn{2}{c}{}             & $\ell=2$                    &             &  $\ell=3$
 \\ \hline
\multicolumn{1}{c}{} &3th order WKB                             &6th order WKB                        & 3th order WKB                    & 6th order WKB
\\  \hline
q=0.1         & 0.36819   -0.119833 i           & 0.36724   -0.119367 i           & 0.535596   -0.121087 i         & 0.535244   -0.121049 i         \\
q=0.2         & 0.496109   -0.112492 i          & 0.495555   -0.112316 i          & 0.69374   -0.11101 i           & 0.693571   -0.110998 i         \\
q=0.3         & 0.53998   -0.104159 i           & 0.53953   -0.104102 i           & 0.744926   -0.102517 i         & 0.744857   -0.10253 i          \\
q=0.4         & 0.557773   -0.0977236 i         & 0.557533   -0.0977495 i         & 0.764098   -0.0965005 i        & 0.76412 -0.0965164 i           \\
q=0.5         & 0.56459   -0.0929047 i          & 0.564534   -0.0929304 i         & 0.769958   -0.0921448 i        & 0.770024   -0.0921494 i        \\
q=0.6         & 0.565673   -0.08927 i           & 0.565716   -0.089275 i          & 0.768601   -0.0889021 i        & 0.771675   -0.0888982 i        \\
 \hline
\multicolumn{5}{l}{M=1, c=0.01, $\lambda=0.001$, w=-2/3}                                                                                                      \\ \hline
\multicolumn{2}{c}{}   & $\ell=2$                               &                 &$\ell=3$                                                                       \\ \hline
\multicolumn{1}{c}{} &3th   order WKB                    &6th order WKB                     &3th order WKB                &6th order WKB                \\ \hline
q=0.1         & 0.19961 -0.151224 i             & 0.198681 -0.152195 i            & 0.306687 -0.148405 i           & 0.306161 -0.14864 i            \\
q=0.2         & 0.369781 -0.12888 i             & 0.369186 -0.129174 i            & 0.53293 -0.12605 i             & 0.532669 -0.126129 i           \\
q=0.3         & 0.43796 -0.118469 i             & 0.437717 -0.118541 i            & 0.619565 -0.115036 i           & 0.619425 -0.115049 i           \\
q=0.4         & 0.471901 -0.110704 i            & 0.471818 -0.110363 i            & 0.660318 -0.106929 i           & 0.619565 -0.115036 i           \\
q=0.5         & 0.490147 -0.104453 i            & 0.489898 -0.103874 i            & 0.680427 -0.100545 i           & 0.68026 -0.100461 i            \\
q=0.6         & 0.499561 -0.0993155 i           & 0.498972 -0.0987989 i           & 0.689009 -0.0954784 i          & 0.688793 -0.0954421 i          \\
q=0.7         & 0.50321 -0.0952474 i            & 0.502352 -0.0949737 i           & 0.689909 -0.0917494 i          & 0.689691 -0.091777 i           \\
 \hline
\end{tabular}
\end{table}


\begin{table}[htbp]
\caption{QNM frequencies of electromagnetic field perturbation for Bardeen-Kiselev BH with cosmological constant.}
\label{tab2}
\begin{tabular}{lllll}
\hline
\multicolumn{5}{l}{M=1, q=0.5, $\lambda=0.001$,  w=-1/3}
  \\ \hline
\multicolumn{2}{c}{}             & $\ell=2$                   &             & $\ell=3$
 \\ \hline
\multicolumn{1}{c}{} &3th order WKB                             &6th order WKB                        & 3th order WKB                    & 6th order WKB
\\  \hline
c=0     & 0.606212   -0.0855342 i & 0.60626   -0.0854858 i  & 0.820997   -0.0871244 i & 0.821065   -0.0871122 i    \\
c=0.001 & 0.60202   -0.0865691 i  & 0.602076   -0.086518 i  & 0.81579   -0.087828 i   & 0.81586   -0.0878153 i         \\
c=0.002 & 0.597816   -0.0875378 i & 0.597886   -0.0874856 i & 0.8106   -0.0884822 i   & 0.810674   -0.0884689 i          \\
c=0.003 & 0.593609   -0.0884371 i & 0.593691   -0.0883888 i & 0.805429   -0.0890882 i & 0.805508   -0.0890751 i          \\
c=0.004 & 0.589407   -0.0892662 i & 0.589495   -0.0892268 i & 0.800282   -0.0896477 i & 0.800365 -0.0896358 i           \\
c=0.005 & 0.585218   -0.0900266 i & 0.585301   -0.0899992 i & 0.795159   -0.0901626 i & 0.795244   -0.0901527 i        \\
c=0.006 & 0.581046   -0.0907207 i & 0.581113   -0.0907068 i & 0.790062   -0.0906351 i & 0.790147   -0.0906279 i         \\
c=0.007 & 0.576894   -0.0913518 i & 0.576938   -0.0913509 i & 0.784993   -0.0910675 i & 0.785076   -0.0910633 i        \\
c=0.008 & 0.572766   -0.0919237 i & 0.57278   -0.0919342 i  & 0.779953   -0.0914617 i & 0.780031 -0.0914606 i        \\ \hline
\multicolumn{5}{l}{M=1, q=0.5, $\lambda=0.001$,  w=-2/3}                                                                                                      \\ \hline
\multicolumn{2}{c}{}   &  $\ell=2$                                &                 &$\ell=3$                                                                       \\ \hline
\multicolumn{1}{c}{} &3th   order WKB                    &6th order WKB                     &3th order WKB                &6th order WKB                \\ \hline
c=0     & 0.606212 -0.0855342 i   & 0.60626 -0.0854858 i    & 0.820997 -0.0871244 i   & 0.821065 -0.0871122 i                \\
c=0.001 & 0.594888 -0.0916488 i   & 0.595604 -0.0916735 i   & 0.806481 -0.0909698 i   & 0.806694 -0.0909355 i             \\
c=0.002 & 0.582465 -0.0966386 i   & 0.582242 -0.0972697 i   & 0.806481 -0.0909698 i   & 0.806481 -0.0909698 i           \\
c=0.003 & 0.569857 -0.0999979 i   & 0.568667 -0.100549 i    & 0.776993 -0.0964762 i   & 0.777026 -0.0966223 i           \\
c=0.004 & 0.557464 -0.102165 i    & 0.556067 -0.102324 i    & 0.762461 -0.0981574 i   & 0.762317 -0.0982896 i          \\
c=0.005 & 0.545413 -0.103528 i    & 0.544186 -0.103322 i    & 0.748178 -0.0993171 i   & 0.747943 -0.0993956 i           \\
c=0.006 & 0.533724 -0.104337 i    & 0.532764 -0.103887 i    & 0.734156 -0.100076 i    & 0.733896 -0.100096 i          \\
c=0.007 & 0.522383 -0.104748 i    & 0.521673 -0.104168 i    & 0.720386 -0.100522 i    & 0.720137 -0.100496 i             \\
c=0.008 & 0.511359 -0.104863 i    & 0.510851 -0.104236 i    & 0.706854 -0.10072 i     & 0.70663 -0.100662 i          \\ \hline
\end{tabular}
\end{table}


\begin{table}[htbp]
\caption{QNM frequencies of electromagnetic field perturbation for Bardeen-Kiselev BH with cosmological constant.}
\label{tab3}
\begin{tabular}{lllll}
\hline
\multicolumn{5}{l}{M=1, q=0.5, c=0.01, w=-1/3}
  \\ \hline
\multicolumn{2}{c}{}             & $\ell=2$                   &             & $\ell=3$
 \\ \hline
\multicolumn{1}{c}{} &3th order WKB                             &6th order WKB                        & 3th order WKB                    & 6th order WKB
\\  \hline
$\lambda=0$      & 0.56755   -0.0933325 i  & 0.567498   -0.0933613 i & 0.773648   -0.0925694 i & 0.773717   -0.0925742 i   \\
$\lambda=0.001$  & 0.56459   -0.0929047 i  & 0.564534   -0.0929304 i & 0.769958   -0.0921448 i & 0.770024   -0.0921494 i          \\
$\lambda=0.002$  & 0.561624   -0.092475 i  & 0.561566   -0.0924977 i & 0.766256   -0.0917183 i & 0.76632   -0.0917227 i          \\
$\lambda=0.003$  & 0.558654   -0.0920432 i & 0.558593   -0.0920629 i & 0.762543   -0.0912899 i & 0.762605   -0.0912941 i          \\
$\lambda=0.004$  & 0.555678   -0.0916094 i & 0.555615 -0.0916263 i   & 0.758818   -0.0908596 i & 0.758877   -0.0908635 i           \\
$\lambda=0.005$  & 0.552696   -0.0911736 i & 0.552631 -0.0911876 i   & 0.755081   -0.0904273 i & 0.755138   -0.090431 i       \\
$\lambda=0.006$  & 0.549708   -0.0907356 i & 0.549642   -0.0907469 i & 0.751331   -0.089993 i  & 0.751386   -0.0899965 i          \\
$\lambda=0.007$  & 0.546715   -0.0902955 i & 0.546647   -0.0903042 i & 0.747569   -0.0895568 i & 0.747622   -0.08956 i        \\
$\lambda=0.008$  & 0.543715   -0.0898533 i & 0.543646   -0.0898594 i & 0.743795   -0.0891185 i & 0.743845   -0.0891215 i        \\ \hline
\multicolumn{5}{l}{M=1, q=0.5, c=0.01, w=-2/3}                                                                                                      \\ \hline
\multicolumn{2}{c}{}   &  $\ell=2$                                &                 &$\ell=3$                                                                       \\ \hline
\multicolumn{1}{c}{} &3th   order WKB                    &6th order WKB                     &3th order WKB                &6th order WKB                \\ \hline
$\lambda=0$       & 0.492897 -0.105212 i    & 0.492648 -0.104614 i    & 0.684039 -0.101229 i    & 0.683868 -0.101142 i              \\
$\lambda=0.001$  & 0.490147 -0.104453 i    & 0.489898 -0.103874 i    & 0.680427 -0.100545 i    & 0.68026 -0.100461 i              \\
$\lambda=0.002$  & 0.487388 -0.103693 i    & 0.487139 -0.103133 i    & 0.676799 -0.099861 i    & 0.676636 -0.0997798 i            \\
$\lambda=0.003$  & 0.48462 -0.102934 i     & 0.484371 -0.102392 i    & 0.673157 -0.0991758 i   & 0.672998 -0.0990974 i          \\
$\lambda=0.004$  & 0.481842 -0.102174 i    & 0.481594 -0.10165 i     & 0.669499 -0.0984899 i   & 0.669344 -0.0984141 i         \\
$\lambda=0.005$  & 0.479055 -0.101415 i    & 0.478808 -0.100907 i    & 0.665825 -0.097803 i    & 0.665674 -0.0977298 i           \\
$\lambda=0.006$  & 0.476257 -0.100655 i    & 0.476012 -0.100164 i    & 0.662136 -0.0971152 i   & 0.661989 -0.0970446 i          \\
$\lambda=0.007$  & 0.47345 -0.0998944 i    & 0.473206 -0.0994197 i   & 0.658431 -0.0964265 i   & 0.658287 -0.0963583 i           \\
$\lambda=0.008$  & 0.470633 -0.0991339 i   & 0.470391 -0.0986748 i   & 0.654709 -0.0957368 i   & 0.654569 -0.095671 i         \\ \hline
\end{tabular}
\end{table}

We have numerically obtained the quasinormal frequencies for the gravitational and electromagnetic perturbations. In Fig.\ref{fig2}, Fig.\ref{fig3} and Fig.\ref{fig4}, we have exploited the 6rd order WKB approximation for calculating the frequencies of the gravitational perturbations of Bardeen-Kiselev BH with cosmological constant, these frequencies can be obtained by individually varying $q$, $c$ and $\lambda$. In Fig.\ref{fig2}, one can see that the oscillation frequency increases with the increasing magnetic charge $q$, which corresponds to the decreasing damping rate, also implies that the decay of the modes is slower. Compared with the Fig.\ref{fig3} and Fig.\ref{fig4}, we observed similar qualitative behaviours, i.e. both the oscillation frequency and the damping rate decreases with increasing values of $c$ and $\lambda$. In table\ref{tab1}, table\ref{tab2} and table\ref{tab3}, the quasinormal frequencies in electromagnetic perturbations for different parameters are reported, we have exploited the 6rd order WKB approximation and 3rd order WKB approximation for calculating the frequencies of the electromagnetic perturbations of Bardeen-Kiselev BH with cosmological constant. As mentioned before, by using the sixth order WKB method, our target in this paper is to solve the wave equation with proper boundary conditions for corresponding complex quasinormal frequencies, and it is well known that the sixth order WKB method is more accurate than the third order one, for all multipole numbers $\ell$ and for modes with small imaginary parts, the results of sixth order WKB method are agree with those obtained from full numerical integration of the wave equation, and in fact these results can be verified in table\ref{tab1}, table\ref{tab2} and table\ref{tab3}. Concretely speaking, in table\ref{tab1} we present the QNM frequencies of electromagnetic field perturbation for Bardeen-Kiselev BH with cosmological constant in terms of the changeable magnetic charge $q$, it shows that the oscillation frequency will increase with the increasing $q$ and the damping rate decreases with increasing values of $q$. We present the QNM frequencies of electromagnetic field perturbation for Bardeen-Kiselev BH with cosmological constant in terms of the changeable normalization factor $c$ in table\ref{tab2}, which shows that the oscillation frequency will decrease with the increasing values of $c$, while the damping rate increases with the increasing values of $c$, this implies the decay of the modes is faster. In table\ref{tab3} we present the QNM frequencies of electromagnetic field perturbation for Bardeen-Kiselev BH with cosmological constant in terms of the changeable cosmological constant $\lambda$, which shows that both the oscillation frequency and the damping rate decrease with the increasing values of $\lambda$.

Compared with the results of the quasinormal frequencies for the gravitational and electromagnetic perturbations, except for the effect of the normalization factor $c$ on the damping rate, the gravitational perturbations and have similar qualitative behavior to electromagnetic perturbations in terms of the other parameters.


{\centering  \section{Greybody factors and absorption coefficients} \label{secV} }

By using the WKB method to obtain reflection and transmission coefficients(greybody factors) has been seen in the literatures\cite{ret33,ret34,ret35}, such as in the context of  braneworld models and wormholes so on. In this section we intend to discuss the frequency dependent reflection $R(\omega)$ and transmission coefficient $T(\omega)$ for a scattering process of the gravitational and electromagnetic perturbations from the Bardeen-Kiselev BH with cosmological constant. In view of the fact that we expand the boundary condition of no-incoming wave from infinity, the calculation of this method is different from the quasi normal frequency calculation. The formalism of the asymptotic behaviour of the wave after scattering off of the effective potential can be expressed in tortoise coordinate as
\begin{equation}
\Psi(r_{*})=T(\omega))e^{-i\omega r_{*}}, r_{*} \rightarrow -\infty                \label{eq28}    \end{equation}
\begin{equation}
\Psi(r_{*})=e^{-i\omega r_{*}}+R(\omega)e^{i\omega r_{*}}.  r_{*} \rightarrow +\infty                       \label{eq29}    \end{equation}
As seen in Eq.\eqref{eq28} and Eq.\eqref{eq29}, the reflection and transmission coefficients are functions of oscillation frequency $\omega$ of the wave. The reflection coefficient in the presence of the WKB approximation is given by
\begin{equation}
R(\omega)=(1+e^{-2\pi i \eta})^{-\frac{1}{2}},               \label{eq30}    \end{equation}
where $\eta$ is
\begin{equation}
\eta= \frac{i(\omega^{2}-V(r_{0}))}{\sqrt{-2V^{''}(r_{0})}}-\varLambda_{i}, i=2,3            \label{eq31}    \end{equation}
noted that the values of $\varLambda_{i}$ can be found out from Eq.\eqref{eq27}. Thus conserving probability we have
\begin{equation}
\gamma_{l}= \left| T{(\omega)} \right|^{2}=1-\left| R{(\omega)} \right|^{2}, \label{eq32}
\end{equation}
where $\gamma_{l}$ is the greybody factor.

\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{15.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{16.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{17.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{18.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $q=0.25$,  $\lambda=0.0025$  and $c=0.02$.}
\label{fig5}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{19.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{20.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{21.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{22.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $\ell=3$,  $\lambda=0.0025$  and $c=0.02$.}
\label{fig6}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{23.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{24.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{25.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{26.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $\ell=3$,  $\lambda=0.0025$  and $q=0.25$.}
\label{fig7}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{27.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{28.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{29.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{30.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $\ell=3$,  $q=0.25$  and $c=0.02$.}
\label{fig8}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{31.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{32.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{33.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{34.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $q=0.25$,  $\lambda=0.0025$  and $c=0.02$.}
\label{fig9}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{35.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{36.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{37.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{38.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $\ell=3$,  $\lambda=0.0025$  and $c=0.02$.}
\label{fig10}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{39.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{40.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{41.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{42.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $\ell=3$,  $\lambda=0.0025$  and $q=0.25$.}
\label{fig11}
\end{figure}


\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{43.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{44.pdf}}
\centerline{(b)}
\end{minipage}
\\
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{45.pdf}}
\centerline{(c)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{46.pdf}}
\centerline{(d)}
\end{minipage}
\end{tabular}
\caption{(a) $|R|^{2}$ vs $\omega$ for electromagnetic perturbations; (b) $|T|^{2}$ vs $\omega$ for electromagnetic perturbations; (c) $|R|^{2}$ vs $\omega$ for gravitational perturbations; (d) $|T|^{2}$ vs $\omega$ for gravitational perturbations. In both cases we take $M=1$, $\ell=3$,  $q=0.25$  and $c=0.02$.}
\label{fig12}
\end{figure}

By using the Regge-Wheeler potential from Sec.\ref{secII}, we numerically plot the variation of the greybody factor with various parameters($\ell$, $q$, $c$, $\lambda$) for the certain  state parameter $w$ from Fig.\ref{fig5} to Fig.\ref{fig12}. These figures show that the value of greybody bound is zero when the frequency is minimal, and the value of greybody bound turns out to be 1 when the frequency is large enough, which implies that the wave is basically totally reflected when the frequency is small, while in view of the tunneling effect, part wave could pass through the potential barrier when the frequency increases, or the wave will not be reflected when the frequency reaches a certain level. Moreover, noted that according to the Eq.\eqref{eq32}, for the high frequency and low frequency limits, the properties of Reflection coefficients and Transmission coefficients should be opposite, in fact these results can be verified from Fig.\ref{fig5} to Fig.\ref{fig12}.

Fig.\ref{fig5} and Fig.\ref{fig9} have similar behaviors, i.e.for a fixed frequency, the greybody factor will decrease with the increasing multipole number $\ell$, this result can be verified from these two figures, which also can be explained easily from the dependency of effective potential $V(r)$ on $\ell$ in the presence of Fig.\ref{fig1}. Also Fig.\ref{fig6} and Fig.\ref{fig10} have similar behaviors, for a certain frequency, the greybody factor will decrease with the increasing magnetic charge $q$, and for the same spacetime parameters, regarding the electromagnetic perturbations, response of greybody factor under different $q$ is much larger than gravitational perturbations. Therefore, regarding the Bardeen-Kiselev BH with cosmological constant for large magnetic charge, the BH scatters the incident wave greatly.

Fig.\ref{fig7} shows that for electromagnetic perturbations and gravitational perturbations, for a fixed frequency, the greybody factor will increase with the increasing normalization factor $c$. In Fig.\ref{fig11}, for electromagnetic perturbations, the greybody factor will decrease the increasing normalization factor $c$, while for gravitational perturbations, the greybody factor will increase with the increasing normalization factor $c$. Compared with Fig.\ref{fig7} and Fig.\ref{fig11}, regarding the electromagnetic perturbations, response of greybody factor under different $c$ is much larger than gravitational perturbations.

Fig.\ref{fig8} shows that in electromagnetic perturbations, for a fixed frequency, the greybody factor will slightly increase with the increasing cosmological constant $\lambda$, this result is contrary to the gravitational perturbations. Fig.\ref{fig12} shows that the electromagnetic perturbations and gravitational perturbations have similar behaviors, i.e. for a fixed frequency, the greybody factor will slightly increase with the increasing cosmological constant $\lambda$, and for the same parameters, regarding the gravitational perturbations, response of greybody factor under different $\lambda$ is much larger than electromagnetic perturbations. As mentioned above, the properties of Reflection coefficients and Transmission coefficients are opposite, then we will not present specific description for Reflection coefficients.


Finally, we will study partial and total absorption cross section in the context of electromagnetic and gravitational perturbation for different parameter spaces in the Bardeen-Kiselev BH with cosmological constant background. The partial and total absorption cross sections are given by
\begin{equation}
\begin{aligned}
\sigma_{\ell} &=\frac{\pi(2 \ell+1)}{\omega^{2}}\left|T_{\ell}(\omega)\right|^{2}, \\
\sigma &=\sum_{\ell} \frac{\pi(2 \ell+1)}{\omega^{2}}\left|T_{\ell}(\omega)\right|^{2} .   \label{eq33}
\end{aligned}
\end{equation}

\begin{figure}[htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{47.pdf}}
\centerline{(a)}
\end{minipage}
\hspace{7mm}
\begin{minipage}[t]{0.45\linewidth}
\centerline{\includegraphics[width=7.0cm]{48.pdf}}
\centerline{(b)}
\end{minipage}
\end{tabular}
\caption{(a) Total absorption cross section($\sigma$) vs $\omega$ for $w=-1/3$, $M=1$, $q=0.25$,  $\lambda=0.0025$  and $c=0.02$; (b) Total absorption cross section($\sigma$) vs $\omega$ for $w=-2/3$, $q=0.5$, $M=1$, $c=0.001$ and $\lambda=0.0004$.}
\label{fig13}
\end{figure}

Fig.\ref{fig13} shows that variation of $\sigma$ are expressed with different parameters where the individual peak denotes $\sigma_{\ell}$. As seen in the figure, the electromagnetic ($s = 1$) and the gravitational ($s = 2$) parts have a similar influence on the absorption cross section of the Bardeen-Kiselev BH with cosmological constant.


{\centering  \section{conclusions} \label{secVI} }

In this work, we evoted to study the gravitational and electromagnetic perturbations for the Bardeen-Kiselev BH with cosmological constant in terms of quasinormal modes and greybody factors. We present the effective potentials under the two perturbations, it shows that  both potentials are positive definite between the event and cosmological horizons, and they all have a single maxima, which increases its height with increasing multipole number $\ell$, and by taking a fixed $r$ for both potentials, we find that the bigger multipole number $\ell$ is responsible for the bigger height of potential, which will lead to a smaller absorption coefficient.  Afterwards, by using the sixth order WKB approximation method, we obtain the quasinormal frequencies of the Bardeen-Kiselev BH with cosmological constant under gravitational perturbations, it shows that the oscillation frequency increases with the increasing magnetic charge $q$, which implies that the decay of the modes is slower. We also see that both the oscillation frequency and the damping rate decreases with increasing values of normalization factor $c$ and cosmological constant $\lambda$. By using the sixth order WKB approximation method and the third order WKB approximation method, we compute the quasinormal frequencies of the BH under electromagnetic perturbations. It shows that the oscillation frequency will increase with the increasing magnetic charge $q$ and the damping rate decreases with increasing values of magnetic charge $q$,  the oscillation frequency will decrease with the increasing values of normalization factor $c$, the damping rate increases with the increasing values of c, and both the oscillation frequency and the damping rate decrease with the increasing values of cosmological constant $\lambda$. Moreover, we discussed how the frequencies vary as a function of multipole number, the magnetic charge, the normalization factor and the cosmological constant, compared with these results, we find the response of the BH to the two perturbations by varying different parameters of the spacetime and these results show that the electromagnetic perturbations and gravitational perturbations for different state parameters, the greybody factor will decrease with the increasing multipole number $\ell$ for a fixed frequency, and this result also can be found for the magnetic charge $q$. Regarding the state parameter $w=-1/3$, for a fixed frequency, the greybody factor will increase with the increasing normalization factor $c$ in terms of the two perturbations. While for the state parameter $w=-2/3$, the greybody factor will decrease the increasing normalization factor $c$ for electromagnetic perturbations, and the greybody factor will increase with the increasing normalization factor $c$ for gravitational perturbations. Regarding the state parameter $w=-1/3$, the greybody factor will slightly increase with the increasing cosmological constant $\lambda$ for a fixed frequency, this result is contrary to the gravitational perturbations. While for the state parameter $w=-2/3$, the greybody factor will slightly increase with the increasing cosmological constant $\lambda$ for a fixed frequency in terms of the two perturbations. The characteristics shown from Fig.\ref{fig5} to Fig.\ref{fig12}, we find that the value of the greybody bound will be lower when the energy $\omega$ of the particle is smaller, which implies that the smaller energy, the less energy the excited particle can acquire to complete the tunneling effect, and eventually will be reflected back by the effective potential, while as the energy increases, some particles could possibly transmit to infinity. Finally, we studied partial and total absorption cross section in the context of electromagnetic and gravitational perturbations for different parameter spaces in the Bardeen-Kiselev BH with cosmological constant background, which show that the two perturbations have a similar influence on the absorption cross section of the BH.

\begin{acknowledgments}
 This research was funded by the Guizhou Provincial Science and Technology Project(Guizhou Scientific Foundation-ZK[2022] General 491) and National Natural Science Foundation of China (Grant No.11465006).
\end{acknowledgments}

