\section{Introduction}
Einstein's general relativity is the most elegant and successful gravitational theory subjected to current experimental tests. However, this theory is still imperfect. One of the most important problems of Einstein's gravity is that singularity is inevitable in almost all of its solutions. Since the existence of the singularity in spacetime, causality will break down and predicability of the theory will be lost at this point thus to find the solution for the singularity problem becomes theoretically essential. There are many proposals devoted to solving this problem, one attempt is to construct modified gravitational theories which have the properties of conformal invariance~\cite{Englert:1976ep,narlikar:1977nf,Fradkin:1978yw,
Kaku:1982xu,tHooft:2009wdx,Mannheim:2011ds,Bars:2013yba,Prester:2013fia}.
A gravity theory is conformal invariant if its action is invariant under following transformation of metric
\begin{equation} \label{metrictransformation}
{g_{\mu \nu }} \to {g'_{\mu \nu }} = S \left( x \right){g_{\mu \nu }},
\end{equation}
where $S \left( x \right)$ is a regular conformal factor depending on spacetime coordinate. Obviously Einstein's gravity is not conformal invariant. There are several ways to construct a conformal invariant gravity theory, for example, by introducing an auxiliary field~\cite{Dirac:1973gk} or the conformal invariant Weyl curvature tensor in the action~\cite{Mannheim:2011ds,Mannheim:2016lnx}.

Einstein's gravity is invariant under coordinate transformation and the singularity at the event horizon can be removed by coordinate transformation therefore this singularity is not intrinsic as to Einstein's gravity. Similarly in the context of conformal gravity theory, since the singularity at $r=0$ can in principle be removed by suitable choice of conformal transformation, hence conformal gravity is free from the singularity.
However, the universe we are living in is not conformal invariant and if the conformal gravity theory can describe the real universe then there must be a phase transition at which the conformal symmetry is spontaneously broken. In the broken phase, there are singular and regular metrics with different physical properties due to that the conformal symmetry no longer exists and it was assumed that our nature has a kind of unknown mechanism to select among these different conformal invariant spacetimes~\cite{Toshmatov:2017kmw}. We believe conformal gravity theory is viable and actually, a class of non-singular black hole spacetime solutions from conformal gravity are compatible with the data from the X-ray observation \cite{Bambi:2017yoz,Zhou:2018bxk}.

In this paper, we will study the nonrotating singularity-free black hole which was found in~\cite{Modesto:2016max,Bambi:2016wdn}. The formation and evaporation process of this black hole has been analyzed in~\cite{Bambi:2016yne}, the quasinormal modes of various kinds of perturbations such as scalar, electromagnetic, axial, and gravitational perturbations in this black hole have been studied in~\cite{Toshmatov:2017bpx,Chen:2019iuo,Liu:2020ddo}. Besides, the violation of energy condition for this black hole has been investigated in~\cite{Toshmatov:2017kmw}.
However, the greybody factor of this black hole is barely explored. So far only the greybody factor of the massless scalar field from this black hole has been numerically studied in \cite{Lin:2020rvv} recently. In this paper, we give a rigorous lower bound for the greybody factor of the massless scalar field radiated from the black hole and study the sparsity of the Hawking radiation. Additionally, we investigate the effects of the variation of conformal factor parameters $N$ and $L$ (see eq.(\ref{S(r)})) on the greybody factor and the sparsity of the Hawking radiation and obtain values for conformal parameters which minimize the sparsity of the Hawking radiation numerically. Our study shows that the Hawking radiation of the massless scalar field is extremely sparse from the black hole and become even sparser for the large values of conformal parameters $N$ and $L$.

The paper is organized as follows. In section.\ref{Review} we briefly review the nonrotating singularity-free black hole solution obtained in~\cite{Modesto:2016max,Bambi:2016wdn} and the Regge-Wheeler equation for massless scalar field from this black hole derived in~\cite{Toshmatov:2017bpx}. In section.\ref{HG} we study the power spectrum of Hawking radiation and calculate the rigorous lower bound of the greybody factor for the massless scalar field. Moreover, we give the values for conformal parameters which maximize the lower bound on the greybody factor numerically. Various numerical results for the upper bound of dimensionless parameter $\eta$ used to measure the sparsity of Hawking radiation are given in section.\ref{Sparsity}. Finally, in section.\ref{Summary} we summarize the results of this paper and give a discussion. The geometric units, $\hbar {\rm{ = }}c = {k_{\rm{B}}} = G = 1$, are adopted in this paper.


\section{Non-singular black holes in conformal gravity}\label{Review}
In this section, we briefly review the nonrotating singularity-free black hole solution in conformal gravity~\cite{Modesto:2016max,Bambi:2016wdn}.

The construction of the black hole solution is straightforward and is obtained by simply rescaling the Schwarzschild metric with a conformal factor as follows
\begin{equation} \label{metric}
d{s^2} = S(r)ds_{Schw}^2,
\end{equation}
where $ds_{Schw}^2$ is the line element of Schwarzschild black hole
\begin{equation} \label{Schwarzschild black hole}
ds_{Schw}^2 =  - \left(1-\frac{r_{\text{H}}}{r}\right)d{t^2} + \frac{dr^2}{\left(1-\frac{r_{\text{H}}}{r}\right)} + {r^2}d{\Omega_2 ^2},
\end{equation}
in which $r_{\text{H}}$ is the horizon radius and $S(r)$ is the conformal factor
\begin{equation} \label{S(r)}
S(r) = {\left( {1 + \frac{{{L^2}}}{{{r^2}}}} \right)^{2N}}
\end{equation}
and $N$ is an arbitrary positive integer and $L$ is a new length scale\footnote{Evidently when $N=0$ or $L=0$, the metric (\ref{metric}) reduces to Schwarzschild metric.} which can be either of the order of Planck scale or of the order of the mass $M$ of the black hole. Due to the metric is conformal to the Schwarzschild black hole, the Hawking temperature of the black hole is still $T={1}/{(4\pi r_\text{H})}$. This metric can be the solution of a family of conformally invariant theories since usually solutions of Einstein's gravity theory are a subset of the wider class solutions of conformally invariant theories \cite{Grumiller:2013mxa,Sultana:2012qp,Said:2014lua} and indeed, metric in eq.(\ref{metric}) is obtained by the conformal transformation of Schwarzschild metric. It can be verified that the square of the Riemann tensor ${R_{\alpha \beta \gamma \delta }}{R^{\alpha \beta \gamma \delta }}$ of this spacetime is regular everywhere including $r=0$. Besides the spacetime is geodesically complete in the sense that it will take infinite proper time for a particle to reach $r=0$ (for more details see Ref.\cite{Bambi:2016wdn}).

The Klein-Gordon equation for a massless scalar field is
\begin{equation} \label{KG eq}
\frac{1}{{\sqrt { - g} }}{\partial _\mu }({g^{\mu \nu }}\sqrt { - g} {\partial _\nu }\Phi ) = 0.
\end{equation}
Decomposing $\Phi {\rm{ = }}{e^{ - i\omega t}}\frac{{\psi \left( r \right)}}{{r\sqrt {S(r)} }}Y_m^\ell (\theta ,\phi )$ and using the tortoise coordinate $d{r_*} = \frac{{dr}}{{f(r)}}$, the resulting Regge-Wheeler equation is~\cite{Toshmatov:2017bpx}
\begin{equation} \label{Master equation}
\left( {\frac{{{d^2}}}{{d{r_*}^2}} + {\omega ^2} - {V_{{\rm{eff}}}}} \right)\psi \left( r \right) = 0,
\end{equation}
where the effective potential ${{V_{{\rm{eff}}}}}$ is defined as
\begin{equation} \label{effective potential}
{V_{{\rm{eff}}}} \equiv f(r)\left[ {\frac{{\ell \left( {\ell  + 1} \right)}}{{{r^2}}} + \frac{{\frac{d}{{dr}}\left( {f(r)\frac{d}{{dr}}\left( {r\sqrt {S(r)} } \right)} \right)}}{{r\sqrt {S(r)} }}} \right].
\end{equation}

\section{Hawking radiation and Greybody factor}\label{HG}
As discussed in \cite{Gray:2015pma,Miao:2017jtr,Chowdhury:2020bdi}, the energy emitted per unit time for a 4-dimensional black hole is
\begin{equation} \label{energy emit}
\frac{{dE\left( \omega  \right)}}{{dt}} = \sum\limits_\ell  {{\Gamma _\ell }\left( \omega  \right)} \frac{\omega }{{{e^{{\omega  \mathord{\left/
 {\vphantom {\omega  T}} \right.
 \kern-\nulldelimiterspace} T}}} - 1}}\hat{\vec k} \cdot \hat{\vec n}\frac{{{d^3}kdA}}{{{{\left( {2\pi } \right)}^3}}},
\end{equation}
where $dA$ is the infinitesimal surface area, $\hat{\vec k}$ is the unit tangent vector of the momentum of the scalar field, $\hat{\vec{n}}$ is the unit normal vector of $dA$ and ${{\Gamma _\ell }\left( \omega  \right)}$ is the greybody factor. Since the scalar field we consider is massless thus $| {\vec k} | = \omega$, then the energy emitted per unite time on finite surface $A$ is

\begin{equation} \label{energy emit2}
P = \sum\limits_\ell  {\int {{P_\ell }\left( \omega  \right)d\omega } }
\end{equation}
with
\begin{equation} \label{energy emit3}
{P_\ell }\left( \omega  \right) = \frac{A}{{8{\pi ^2}}}\frac{{{\Gamma _\ell }\left( \omega  \right){\omega ^3}}}{{{e^{{\omega  \mathord{\left/
 {\vphantom {\omega  T}} \right.
 \kern-\nulldelimiterspace} T}}} - 1}}.
\end{equation}
It was shown that $A$ can be taken as 27/4 times the horizon area for Schwarzschild black hole to match high frequency results \cite{Page:1976df,Gray:2015pma} and here we can also take this value since the cross section in geometrical optical limit for massless particle in this model is the same as that of Schwarzschild black hole~\cite{Lin:2020rvv}. Besides, the Hawking temperature $T$ is independent of conformal parameters $N$ and $L$ (see section.\ref{Review}). Therefore, the dependence of power spectrum ${P_\ell }\left( \omega  \right)$ for Hawking radiation on conformal parameters $N$ and $L$ is as same as that of greybody factor ${\Gamma _\ell }\left( \omega  \right)$.

Calculating the analytical expression for greybody factor is generally impractical and usually, it is obtained from numerical computations~\cite{Harris:2004mf,Harris:2005jx,Kanti:2005ja,Duffy:2005ns}. The analytical calculations are available only in some special cases, such as the low frequency limit~\cite{Unruh:1976fm,Rocha:2009xy,Harmark:2007jy,Kanti:2014dxa,
Anderson:2015bza,Sporea:2015hla}, the near extremal limit~\cite{Cvetic:2009jn,Bredberg:2009pv,Chen:2010bsa,Chen:2010as,Chen:2010ywa,Chen:2010yu}, or the large spacetime dimension limit~\cite{Emparan:2013xia,Emparan:2013moa,Guo:2015swu,Emparan:2020inr}. In this paper we follow \cite{Visser:1998ke,Boonserm:2008zg,Boonserm:2014rma} to give a rigorous lower bound for the greybody factor. The bound for greybody factor can be written as follow
\begin{equation} \label{Bound 0}
{{\Gamma _\ell }\left( \omega  \right)} \geqslant {\rm{sec}}{{\rm{h}}^2}\left\{ {\int_{ - \infty }^\infty  {\vartheta d{r_*}} } \right\}
\end{equation}
with
\begin{equation} \label{vartheta1 }
\vartheta  = \frac{{\sqrt {{{\left( {h'\left( r \right)} \right)}^2} + {{\left( {{\omega ^2} - {V_{{\rm{eff}}}} - h{{\left( r \right)}^2}} \right)}^2}} }}{{2h\left( r \right)}},
\end{equation}
where $h(r)$ is an arbitrary positive definite function and satisfy the limits $h( - \infty ) = h( + \infty ) = \omega $. A particularly simple
choice of $h(r)$ is $h(r) = \omega$,
which leads to
\begin{equation} \label{Integrate of vartheta 1 }
\int_{ - \infty }^\infty  {\vartheta d{r_*}}  = \int_{2M}^\infty  {\frac{{{V_{{\rm{eff}}}}}}{{2\omega f(r)}}dr}.
\end{equation}
Combining eq.(\ref{effective potential}) and eq.(\ref{Integrate of vartheta 1 }) we have
\begin{equation} \label{Integrate of vartheta 2}
\int_{ - \infty }^\infty  {\vartheta d{r_*}}  = \frac{{2NM(2N - 1)}}{{\omega {L^2}}}\ln \left( {1 + \frac{{{L^2}}}{{4{M^2}}}} \right) + \frac{{N\left( {3N - 2} \right)}}{{{\rm{2}}\omega L}}\left[ {2\arctan \left( {\frac{{2M}}{L}} \right) - \pi } \right] + \frac{{{{\left( {2N - 1} \right)}^2}}}{{8\omega M}} + \frac{{\ell \left( {\ell  + 1} \right)}}{{4\omega M}},
\end{equation}
where $M$ is the mass of the black hole. Substituting the above equation into eq.(\ref{Bound 0}), we obtain the bound as
\begin{equation} \label{Bound 1}
{{\Gamma _\ell }\left( \omega  \right)} \geqslant{\rm{sec}}{{\rm{h}}^2}\left\{ {\frac{{2NM(2N - 1)}}{{\omega {L^2}}}\ln \left( {1 + \frac{{{L^2}}}{{4{M^2}}}} \right) + \frac{{N\left( {3N - 2} \right)}}{{{\rm{2}}\omega L}}\left[ {2\arctan \left( {\frac{{2M}}{L}} \right) - \pi } \right] + \frac{{{{\left( {2N - 1} \right)}^2}}}{{8\omega M}} + \frac{{\ell \left( {\ell  + 1} \right)}}{{4\omega M}}} \right\}.
\end{equation}
Now let us analyze the dependence of this bound on the parameters $N$ and $L$ in various situations. At first,
for $N=0$ we have
\begin{equation} \label{Schwarzschild bound}
{{\Gamma _\ell }\left( \omega  \right)}\geqslant{\rm{sec}}{{\rm{h}}^2}\left\{ {\frac{{\ell \left( {\ell  + 1} \right)}}{{4\omega M}} + \frac{1}{{8\omega M}}} \right\},
\end{equation}
which is exactly the result for Schwarzschild black hole as it should be. Note that this result has been obtained in \cite{Boonserm:2008zg}.
In addition, when $L$ is large, the bound in eq.(\ref{Bound 1}) can be simplified as follows
\begin{equation} \label{Schwarzschild bound1}
{{\Gamma _\ell }\left( \omega  \right)}\geqslant{\text{sec}}{{\text{h}}^2}\left\{ {\frac{{\ell \left( {\ell  + 1} \right)}}{{4\omega M}} + \frac{{{{\left( {2N - 1} \right)}^2}}}{{8\omega M}}} \right\},
\end{equation}
while in small $L$ limit eq.(\ref{Bound 1}) reduces to eq.(\ref{Schwarzschild bound}).

Furthermore, it is not hard to find out specific values for the conformal parameters $N$ and $L$ that can maximize the lower bound obtained in eq.(\ref{Bound 1}). Specifically it can be done as follows. We first rewrite eq.(\ref{Bound 1}) as
\begin{equation} \label{Bound 2}
{\Gamma _\ell }\left( \omega  \right)\geqslant{\text{sec}}{{\text{h}}^2}\left\{ {{{J\left( {L,N,\ell } \right)} \mathord{\left/
 {\vphantom {{h\left( {L,N,\ell } \right)} {\left( {\omega {r_{\text{H}}}} \right)}}} \right.
 \kern-\nulldelimiterspace} {\left( {\omega {r_{\text{H}}}} \right)}}} \right\},
\end{equation}
where we have defined
\begin{equation} \label{h function}
J\left( {L,N,\ell } \right) \equiv \frac{{N(2N - 1)}}{{{{\left( {L/{r_{\text{H}}}} \right)}^2}}}\ln \left( {1 + \frac{{{L^2}}}{{r_{\text{H}}^{\text{2}}}}} \right) + \frac{{N\left( {3N - 2} \right)}}{{{\text{2}}\left( {L/{r_{\text{H}}}} \right)}}\left[ {2\arctan \left( {\frac{{{r_{\text{H}}}}}{L}} \right) - \pi } \right] + \frac{{{{\left( {2N - 1} \right)}^2}}}{4} + \frac{{\ell \left( {\ell  + 1} \right)}}{2}.
\end{equation}
Note the function $y(x) = {\operatorname{sech} ^2}(x)$ is even, monotonously decreasing when $x>0$, and reaches its maximum at $x=0$, thus at fixed $\omega$, the values of $N$ and $L$ that minimize  $\left| {J\left( {L,N,\ell } \right)} \right|$ will maximize ${\text{sec}}{{\text{h}}^2}\left\{ {{{J\left( {L,N,\ell } \right)} \mathord{\left/
 {\vphantom {{h\left( {L,N,\ell } \right)} {\left( {\omega {r_{\text{H}}}} \right)}}} \right.
 \kern-\nulldelimiterspace} {\left( {\omega {r_{\text{H}}}} \right)}}} \right\}$. Therefore to obtain the values for $N$ and $L$ that can maximize the greybody factor we only need to find out the values that can minimize $\left| {J\left( {L,N,\ell } \right)} \right|$.

In TABLE.\ref{tab:table1} and TABLE.\ref{tab:table2}, for given values of ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}$($N$), we find out numerically the corresponding $N$(${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}$) which can maximize the lower bound obtained in eq.(\ref{Bound 1}).
\begin{table}[hbp]
  \begin{center}
 \caption{For different values of $L/r_{\text{H}}$ the corresponding values of $N$ which maximize the lower bound on greybody factor with $\ell=1$.}
    \label{tab:table1}
    \begin{tabular}{c|c|c|c|c|c|c|c|c|c}
     \hline \hline
     ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}$&$0.1$&$0.3$&$0.5$&$0.7$&$0.9$&$1.5$&$2$&$2.5$&$3$\\
      \hline
      $N$&8&8&6&3&2&1&1&1&1\\
      \hline\hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}[h!]
  \begin{center}
 \caption{For different values of $N$ the corresponding values of $L/{r_{\text{H}}}$ which maximize the lower bound on greybody factor with $\ell=1$.}
    \label{tab:table2}
    \begin{tabular}{c|c|c|c|c|c|c|c|c}
     \hline \hline
     $N$&$1$&$3$&$5$&$7$&$9$&$12$&$15$&$20$\\
      \hline
      ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}$&2.893&0.785&0.558&0.337&0.456&0.395&0.299&0.257\\
 \hline\hline
    \end{tabular}
  \end{center}
\end{table}

To see the dependence of greybody factor and power spectrum of Hawking radiation on conformal parameters $N$, $L$ and the frequency $\omega$, intuitively, we plot the lower bound of greybody factor and power spectrum of Hawking radiation as a function of frequency for different values of $N$ and $L$. FIG.\ref{figure.1} shows that, when the value of ${L \mathord{\left/{\vphantom {L {{r_{\rm{H}}}}}} \right.\kern-\nulldelimiterspace} {{r_{\rm{H}}}}}$ is fixed, the increase of $N$ leads greybody factor to increase at first (top left) and then to decrease (top right). Similarly the increase of  ${L \mathord{\left/{\vphantom {L {{r_{\rm{H}}}}}} \right.\kern-\nulldelimiterspace} {{r_{\rm{H}}}}}$ will enhance the greybody factor at the beginning (bottom left) and suppress the greybody factor later (bottom right) for a given value of $N$. Obviously FIG.\ref{figure.1} is consistent with TABLE.\ref{tab:table1} and TABLE.\ref{tab:table2}.

Furthermore, as shown in FIG.\ref{figure.2}, for a fixed value of $L/{r_{\rm H}}$, the peak value of power spectrum of Hawking radiation increases as the increase of $N$ in the first place and decreases if $N$ continue to grow. While from the bottom of FIG.\ref{figure.2} one can find that for a relatively small value of $L/{r_{\rm H}}$ the growth of  $L/{r_{\rm H}}$ will cause the peak value of Hawking radiation power spectrum to increase. Once $L/{r_{\rm H}}$ becomes relatively larger the increase of $L/{r_{\rm H}}$ will give rise to the decrease of the peak value of power spectrum when $N$ is fixed. Also, we find that the values of $L/{r_{\rm H}}$ and $N$ which maximize the power spectrum of Hawking radiation are exactly those values for $L/{r_{\rm H}}$ and $N$ that maximize the greybody factor, which is in agreement with our analysis about the dependence of Hawking radiation's power spectrum on $N$ and $L$ above.
\begin{figure}
\centering
\subfigure{
\includegraphics[width=5cm]{ImBDN.eps}
}
\subfigure{
\includegraphics[width=5cm]{ImBDN2.eps}
}
\\
\subfigure{
\includegraphics[width=5cm]{ImBDL.eps}
}
\subfigure{
\includegraphics[width=5cm]{ImBDL2.eps}
}
\DeclareGraphicsExtensions.
\caption{(Top) Plot of the bound of Greybody factor ${{\Gamma _\ell }\left( \omega  \right)}$ as a function of $\omega r_\text{H}$ for different values of $N$ with $\ell=1$ and ${L}/{r_\text{H}}=0.7$; (Bottom) Plot of the bound of Greybody factor ${{\Gamma _\ell }\left( \omega  \right)}$ as a function of $\omega r_\text{H}$ for different values of ${L}/{r_\text{H}}$ with $\ell=1$ and $N=20$.}
\label{figure.1}
\end{figure}
\begin{figure}
\centering
\subfigure{
\includegraphics[width=5.5cm]{ImPSN.eps}
}\subfigure{
\includegraphics[width=5.5cm]{ImPSN2.eps}
}
\subfigure{
\includegraphics[width=5.5cm]{ImPSN3.eps}
}
\\
\subfigure{
\includegraphics[width=5.5cm]{ImPSL.eps}
}
\subfigure{
\includegraphics[width=5.5cm]{ImPSL2.eps}
}
\subfigure{
\includegraphics[width=5.5cm]{ImPSL3.eps}
}
\DeclareGraphicsExtensions.
\caption{(Top) Plots of power spectrum  ${r_{\rm{H}}}{P_\ell }\left( \omega  \right)$ as a function of $\omega r_\text{H}$ for different values of $N$ with $\ell=1$ and ${L}/{r_\text{H}}=0.7$; (Bottom) Plots of power spectrum ${r_{\rm{H}}}{P_\ell }\left( \omega  \right)$ as a function of $\omega r_\text{H}$ for different values of ${L}/{r_\text{H}}$ with $\ell=1$ and $N=20$.}
\label{figure.2}
\end{figure}

\section{The sparsity of Hawking radiation}\label{Sparsity}
The sparsity of Hawking radiation can be measured by the following quantity~\cite{Gray:2015pma}
\begin{equation} \label{Sparsity parameter}
\eta  = \frac{{{\tau _{{\rm{gap}}}}}}{{{\tau _{{\rm{emission}}}}}},
\end{equation}
where ${{\tau _{{\rm{gap}}}}}$ is the average time interval between the emission of two Hawking quanta, which is proportional to inverse of the total Hawking radiation power $P$ in eq.(\ref{energy emit2}). More specifically, it can be computed by~\cite{Gray:2015pma,Paul:2016xvb}
\begin{equation} \label{Average time}
{\tau _{{\rm{gap}}}} = \frac{{{\omega _{{\rm{peak}}}}}}{P}.
\end{equation}
Here ${{\omega _{{\rm{peak}}}}}$ is the value at which ${{{\omega ^3}} \mathord{\left/
 {\vphantom {{{\omega ^3}} {\left( {{e^{\omega T}} - 1} \right)}}} \right.
 \kern-\nulldelimiterspace} {\left( {{e^{\omega T}} - 1} \right)}}$ is maximized, i.e.
\begin{equation} \label{Peak value}
{\omega _{{\rm{peak}}}} = T\left[ {3 + W\left( { - 3{e^{ - 3}}} \right)} \right],
\end{equation}
where $W\left( x \right)$ is Lambert W-function which is the inverse function of $F\left( W \right) = W{e^W}$. ${{\tau _{{\rm{emission}}}}}$ is the characteristic time for an individual quantum emitted from black hole, which can be taken equal to the period of the quantum emitted, since the quanta emitted at peak value can only be temporarily localized within a few oscillation periods. Therefore $\tau _{{\text{emission}}}$ obeys the bound
\begin{equation} \label{characteristic time}
{\tau _{{\text{emission}}}} \geqslant \frac{{2\pi }}{{{\omega _{{\text{peak}}}}}},
\end{equation}
which leads to an upper bound for sparsity
\begin{equation} \label{etamax}
\eta  \leqslant \frac{{\omega _{{\text{peak}}}^2}}{{2\pi P}} \equiv {\eta _{\max }}.
\end{equation}
In summary, the sparsity of radiation in eq.(\ref{Sparsity parameter}) tells us that if  $\eta  \gg 1$, i.e. the average interval time $\tau _{{\rm{gap}}}$ between the emission of two quanta is much larger than the time $\tau _{{\text{emission}}}$  taken by an individual quantum emitted from the black hole, then naturally the Hawking radiation would be very sparse. On the contrary, if $\eta  \ll 1$  we will have almost continuous Hawking radiation.

In what follows we will study the dependence of sparsity on the conformal parameters $N$ and $L$ numerically.
Firstly, let us investigate the behavior of $\eta_{max}$ with different $N$ while keeping $L$ fixed. In TABLE.\ref{tab:table3}  the values of ${\eta _{\max }}$ for different values of $N$ with $L$ fixed are listed.
The TABLE.\ref{tab:table3} shows that given ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}=0.7$ and $\ell=1$, as $N$ increasing, the values of ${\eta _{\max }}$ will decreases at first, and begin to increase at $N=3$.
In the next step, we go on to study the case with fixed $N=20$ (also $\ell=1$) and letting $L/r_H$ vary. The results are presented in TABLE.\ref{tab:table4}, from which it is clear that the situation in TABLE.\ref{tab:table4} is similar to that of TABLE.\ref{tab:table3}.  TABLE.\ref{tab:table4} shows that ${\eta _{\max }}$ takes its minimum value at ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}=0.257$, and as ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}$ getting larger than this values, ${\eta _{\max }}$  will increase as the increase of ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}$.

In TABLE.\ref{tab:table3} and TABLE.\ref{tab:table4} the sparsity of $N=0$ or ${L \mathord{\left/
 {\vphantom {L {{r_{\text{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\text{H}}}}}=0$ corresponds to Schwarzschild black hole, we show that when the $N$ and $L$ is relatively small (comparing with according values which minimize the sparsity) one can have nonrotating singularity-free black holes whose Hawking radiation are less sparse than that of Schwarzschild black hole with the same mass.

Also note that, as shown in TABLE.\ref{tab:table3} and TABLE.\ref{tab:table4}, the values for conformal parameters that minimize the sparsity of Hawking radiation are those values which maximize the greybody factor listed in TABLE.\ref{tab:table1} and TABLE.\ref{tab:table2}.
 This can be understood as follows,
as mentioned in section.\ref{HG} the dependence of the power spectrum (see eq.(\ref{energy emit3})) on $N$ and $L$ are totally coming from the greybody factor, hence, the values of $N$ and $L$ that maximize the greybody factor as presented in TABLE.\ref{tab:table1} and TABLE.\ref{tab:table2} will also maximize the power spectrum for Hawking radiation, which in turns minimize the upper bound ${\eta _{\max }}$ in eq.(\ref{etamax}).

The large values of ${\eta _{\max }}$ from TABLE.\ref{tab:table3} and TABLE.\ref{tab:table4} implies that the Hawking radiation from this black hole is extremely sparse. Moreover, the Hawking radiation becomes even more sparse for the increase of relatively large values of conformal parameters $N$ and $L$.

\begin{table}[h!]
  \begin{center}
 \caption{Numerical values of dimensionless parameter ${\eta _{\max }}$ for different values of $N$  with $\ell=1$ and ${L}/{{{r_{\rm{H}}}}}=0.7$.}
    \label{tab:table3}
    \begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c}
     \hline \hline
     $N$&$0$&$1$&$2$&$3$&$4$&$5$&$6$&$7$&$8$&$9$&$10$&$15$&$20$\\
      \hline
      ${\eta _{\max }}$&2506.84&2068.86&1823.34&1719.36&1736.27&1877.39&2170.99&2679.09&3517.53&4895.76&7193.68&95611.3&273617\\
      \hline\hline
    \end{tabular}
  \end{center}
\end{table}
\begin{table}[h!]
  \begin{center}
 \caption{Numerical values of dimensionless parameter ${\eta _{\max }}$ for different values of ${L}/{{{r_{\rm{H}}}}}$ with $\ell=1$ and $N=20$.}
    \label{tab:table4}
    \begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c}
     \hline \hline
     ${L \mathord{\left/
 {\vphantom {L {{r_{\rm{H}}}}}} \right.
 \kern-\nulldelimiterspace} {{r_{\rm{H}}}}}$&0&0.05&0.1&0.15&0.2&0.257&0.3&0.35&0.4&0.5&0.7&1&1.5\\
      \hline
      ${\eta _{\max }}$&2506.84&2441.15&2268.87&2051.76&1862.66&1768.19&1846.84&2225.15&3233.5&13293.1&$2.7{\text{4}} \times {\text{1}}{{\text{0}}^{\text{6}}}$&$2.4{\text{8}} \times {10^{11}}$&$1.2{\text{8}} \times {10^{20}}$\\
      \hline\hline
    \end{tabular}
  \end{center}
\end{table}

\section{Summary}\label{Summary}
We studied the sparsity of Hawking radiation from a nonrotating singularity-free black hole in conformal gravity proposed in \cite{Modesto:2016max,Bambi:2016wdn}. We mainly focused on analyzing the dependence of sparsity of Hawking radiation on the conformal parameters $N$ and $L$ (see eq.(\ref{S(r)})).

We first obtained a rigorous lower bound on the greybody factor (see eq.(\ref{Bound 1})) for the massless scalar field radiated from the black hole and analyzed the dependence of the greybody factor on the conformal parameters $N$ and $L$ via the bound. We found that there exist values for conformal parameters $N$ and $L$ that can maximize the lower bound on the greybody factor. For fixed $L$ (or $N$), the values of $N$ (or $L$)  maximizing this bound are given numerically (see TABLE.\ref{tab:table1} and TABLE.\ref{tab:table2}). Furthermore, we investigated the power spectrum of Hawking radiation (see eq.(\ref{energy emit3})) and showed that the dependence of power spectrum for Hawking radiation on conformal parameters $N$ and $L$ is the same as that of the greybody factor.

Finally, with the rigorous lower bound on greybody factor and power spectrum for Hawking radiation in hand, we studied the sparsity of Hawking radiation by the upper bound ${\eta _{\max }}$ (see eq.(\ref{etamax})) of dimensionless parameter $\eta$ (see eq.(\ref{Sparsity parameter})). Since the dependence of  ${\eta _{\max }}$ on conformal parameters $N$ and $L$ is completely coming from the greybody factor thus those values for conformal parameters $N$ and $L$ that maximize the lower bound on greybody factor in TABLE.\ref{tab:table1} and TABLE.\ref{tab:table2} can also minimize  ${\eta _{\max }}$ (sparsity). We discovered that when the conformal parameters $N$ and $L$ are relatively small comparing with the values that minimize the sparsity one can have nonrotating singularity-free black holes with Hawking radiation less sparse than that of Schwarzschild black hole with the same mass (see TABLE.\ref{tab:table3} and TABLE.\ref{tab:table4}). Also, we found that generally ${\eta _{\max }} \gg {\text{1}}$ as listed in TABLE.\ref{tab:table3} and TABLE.\ref{tab:table4}, which means the Hawking radiation is extremely sparse.  When the values of conformal parameters surpassing the values that minimize the sparsity of Hawking radiation, the further increase of conformal parameters will enhance ${\eta _{\max }}$ rapidly, which implies the Hawking radiation becomes even more sparse.

In future work, we plan to extend this study to the case of rotating singularity-free black hole which has also been proposed in \cite{Modesto:2016max,Bambi:2016wdn} since rotating black holes are more likely to exist in our universe.



\section*{Acknowledgement}
We would like to thank Jia-Rui Sun, Bin Wu, and Zhen-Ming Xu for useful discussions. This work was supported by the National Natural Science Foundation of China (No.~11675272).


