 achieved by binning the mass intervals logarithmically, going from $m = 0.2 \ M_{\odot}$ to $10 \ M_{\odot}$ in 20 bins. We determined the slope of the mass function by fitting a power law to the points with $0.3 M_{\odot} < m < 0.8 M_{\odot}$, because this is the same fitting area as used for the observational data. The result is shown in Figure \ref{fig:ref}.b. As expected, the slope of the mass function increases with the age of the cluster because the cluster loses low-mass stars through two-body relaxation. As with the bound mass fraction, there are no significant differences between models with different particle numbers other than the time scaling (when these values are plotted as a function of the fraction of the total lifetime $t/t_{diss}$ all three models agree). 
   
Plots \ref{fig:ref}.c and d depict the evolution of the two Lagrangian radii that include 10~\% and 50~\% of the cluster mass, respectively. The core collapse is indicated by a drop in the 10~\% Lagrangian radius followed by an immediate rise due to binary heating. Surprisingly, the exact occurrence of that event does not seem to depend  on the particle number. While the 32k model undergoes core collapse first, the 128k core collapse still occurs before the 64k model reaches this point. We assume the reason for the absence of a clear trend for the occurrence of core collapse with particle number arises from stochastic effects caused by the low number of massive stars, which mainly drive the dynamical evolution of the cluster even for the 128k models.


The last two plots (Figure \ref{fig:ref}.e and f) show the evolution of the stellar remnants in the clusters. In Figure \ref{fig:ref}.e the remnant mass fraction increases with time. This is because low-mass stars are ejected more efficiently than high-mass stars. Consequently, the fraction of high-mass stars to low-mass stars constantly increases. The steep rise at the beginning is due to initially rapid stellar evolution and the consequent formation of stellar remnants. The evolution of the stellar-mass black hole numbers in the clusters in Figure \ref{fig:ref}.f shows a roughly  exponential behavior. For a model with 128k stars, the number of black holes already drops to half of the initial value within 1 Gyr. After 5 Gyr all black holes have been ejected from the cluster. To lead the eye, we applied rough logarithmic fits to the evolution of black-hole numbers (dashed lines). This agrees well with previous results \citep[e.g.][]{sigurdsson_1993,kulkarni_1993,miller_2002,oleary_2006}, which show that stellar mass-black holes eject themselves from the cluster core by building a dynamically decoupled subcore through mass segregation \citep{spitzer_1969}. A recent study by \citet{sippel_2013} has shown, however, that it is possible to retain a large number of stellar-mass black holes in the cluster after a Hubble time with even small retention fractions of $10\%$. The initial half-mass radius in their simulations was a factor of 6 larger than our values. This leads to longer crossing and relaxation times and consequently to a delayed dynamical evolution of the cluster. This is shown by the fact that the models of \citet{sippel_2013} do not undergo core collapse after 10 Gyr. Therefore, the ejection rate of the stellar-mass black holes is also low. We note, however, that models with initial half-mass radii of several parsec end up with physical sizes too large compared with those of the majority of galactic GCs after their dynamical evolution. A comparison of panel c) and f) indicates that the core collapse in all models occurs after the cluster has lost its stellar-mass black holes. 


   \begin{figure}
  \centering
   \includegraphics[width=0.49\textwidth]{tdiss2.eps}
      \caption{Lifetimes of all clusters as a function of their initial particle number.}
         \label{fig:tdiss}
   \end{figure}



\subsection{Model families} \label{subsec:fam}

We ran models with different IMBH masses, black-hole retention fractions, and binary fractions, which we sorted to three different families. To use realistic values for the IMBH masses we considered several scaling relations such as the $M_{\bullet} - M_{\rm{bulge}}$ relation \citep{magorrian_1998, haering_2004} and the $M_{\bullet} - \sigma$ relation \citep{ferrarese_2000,gebhardt_2000,gultekin_2009}. When these empirical correlations are extrapolated, they predict IMBH masses of $M_{\bullet} \sim 0.001 - 0.03 \ M_{\rm{tot}}$ in GCs \citep{Goswami_2012}. This corresponds to black-hole masses  $M_{\bullet} \sim 10^2 - 5 \times 10^3 \ M_{\odot}$  in an average GC with a mass of $M_{\rm{tot}} = 1.5 \times 10^5 \ M_{\odot}$. These masses were found by \cite{portegies_2004}, who used $N$-body simulations to model the runaway merging of massive stars in the dense star cluster MGG-11. Their values, however, should be treated as upper limits, since stellar mass loss of the 'runaway' star was not included \citep[see]{glebbeek_2009}. Taking into account these predictions, we used two different black-hole masses: $1\%$ and $3\%$ of the total cluster mass. For models with black-hole masses $< 1\%$ the actual black-hole mass is $< 800 \ M_{\odot}$ even in simulations of N = 128k stars and the central black hole has a high chance of being ejected by encounters with massive stars.
   
   \begin{figure*}
  \centering
   \includegraphics[width=0.33\textwidth]{imbh_rmn.eps}
      \includegraphics[width=0.33\textwidth]{ret_rmn.eps}
      \includegraphics[width=0.33\textwidth]{bf_rmn.eps}
      \caption{Remnant mass fractions as a function of the completed lifetime fraction for simulations with different a) IMBH masses, b) black-hole retention fractions, and c) primordial binary fractions. All models contain 128k particles, except for the binary-fraction models, which are limited to 64k stars.}
         \label{fig:remn}
   \end{figure*}   
   
The retention fraction gives the fraction of stellar remnants that remain in the cluster after their formation. It is assumed that very compact objects such as black holes and neutron stars suffer a kick velocity during their formation that is often higher than the escape velocity of the cluster \citep{lyne_1994}. Theoretical estimates for the retention fraction of neutron stars range from 5~\% to 20~\% \citep{drukier_1996}. For black holes, however, the range of possible values is not well-constrained since the physics of the formation of black holes is still not fully understood.  For this reason, we assumed a constant retention fraction of $10~\%$ for neutron stars and varied the retention fractions of the black holes from 0~\% (no black hole remains in the cluster) to 100~\% (all black holes that are formed remain in the cluster). For white dwarfs we always assumed a retention fraction of $100 \%$ since they do not suffer any kick velocities \citep{davis_2008}. 

Binaries are important for the dynamical evolution of GCs and need to be considered in our models. To test how primordial binaries affect the models, we introduced a third family, in which we varied the fraction of binaries in the initial conditions of the model from 10~\% to 30~\%. This covers the current typical binary fraction in GCs, which is about 10~\% \cite[e.g.][]{rubenstein_1997,davis_2008,dalessandro_2011}. Since calculating the individual time steps of the binary systems is very time-consuming, we limited this exercise to a maximum particle number of 64k for the models with primordial binary fractions.




\section{Observables}\label{sec:obs}

The main goal of this study is finding observables in GCs that give information about the presence of a possible intermediate-mass black hole at their centers. In this section we study the effects of IMBHs on their environment to pin down important tracers for central black holes in GCs. We especially concentrate on mimicking observations, that is, measuring cluster properties in the same way as observers would do to compare our results with observational studies.

\subsection{Lifetimes} \label{sec:time}

The lifetime, or dissolution time $t_{\rm{diss}}$, gives information about the dynamical evolution of the model as a function of its internal properties. In this work, we define the dissolution time as the time when the cluster has lost 95\% of its initial mass. 

In Figure \ref{fig:tdiss} we compare the lifetimes of our model families with the reference models (green bars). As expected, there is a clear trend with particle number and slight differences (of about $\sim 10 -20 \%$) between models with different initial conditions. The higher the mass of the central IMBH, the shorter the lifetime