6 Discussion of microlensing search

In order to gain insight into the results obtained, we compare our 5 candidate microlensing events with the prediction of a Monte Carlo simulation that takes into account the experimental set up and the time sampling of the observations. We assume a standard model (isothermal sphere with a core radius of 5 kpc) for the haloes of both M 31 and the Milky Way. The total mass of M 31 is assumed to be twice that of our Galaxy. MACHOs can be located in either haloes. Moreover, we consider also self-lensing due to stars in the M 31 bulge or disk. We fix the lens masses at different values for MACHOs in the halo and stars in the bulge or disk. The model of the bulge is taken in Kent (1989), the luminosity function in Han et al. (1998). The luminosity function of the disk is determined considering two models: the one developed in Devriendt et al. (1999) and the model obtained considering the data of the solar neighborhood taken in Allen (1973) corrected for high luminosities (Hodge et al. 1988). The results we obtain are almost insensitive to the particular choice between these two models.

We choose the mass of the lenses in the bulge to be $m_{\rm MACHO}^{\rm bulge}=0.4~M_\odot$, and the mass for MACHOs in the haloes equal either to $m_{\rm MACHO}^{\rm halo}=0.5~M_\odot$ or to $m_{\rm MACHO}^{\rm halo}=0.01~M_\odot$. In both cases about $90\%$ of the expected events are due to lenses located in M 31. Taking into account our selection criteria, the results for the expected number of events for a halo fully composed of MACHOs, including a contribution due to lensing by stars of the bulge and disk of M 31 of $\sim$1 event independent of halo parameters, is $\sim$4and $\sim$9, respectively. The Monte Carlo simulation does not yet include the effect of secondary bumps due to artifacts of the image processing (alignment, seeing stabilization) and to underlying variable objects. From the data, we estimate that these effects reduce the number of observed events by at most 30%, and this for the shortest events.

We expect, and this is confirmed by simulations, that the sources of most detectable events are red giants and have very large radii. Therefore, finite size and limb darkening effects are important, in particular for low mass lenses. These effects are included in the simulations. However, in both the real and simulated analysis, we do not include finite size and limb darkening effects in the amplification light curve fitted to the events. This results in a loss of detection efficiency smaller than 5%, which is taken into account in the simulations.

Locating our candidates in the parameter space predicted by the simulation is more meaningful. We report the expected values of t_1/2 and on the R magnitude at maximum. In Figs. 9 and 10 we give plots of their functional relation and of their projected distributions.

Figure 9: Results of Monte Carlo simulations, $m_{\rm MACHO}^{\rm halo}=10^{-2}~M_\odot$ case. The scale on the y coordinate of the two distributions on the right are in arbitrary units. The dots (left) give the position of the candidates as labelled in Table 2.

Figure 10: Results of Monte Carlo simulations, $m_{\rm MACHO}^{\rm halo}=0.5~M_\odot$ case. The scale on the y coordinate of the two distributions on the right are in arbitrary units. The dots (left) give the position of the candidates as labelled in Table 2.

On these same plots we give the position of our selected light curves in this parameter space.

Looking at the distributions we notice that for the $m_{\rm MACHO}^{\rm halo}=0.5~M_\odot$ and $m_{\rm MACHO}^{\rm halo}=0.01~M_\odot$ cases, $80\%$ of the light curves are expected to have a time width at half maximum t_1/2<24 and t_1/2<10 days, respectively (of course, shorter events are expected when the MACHO mass is smaller). In both cases, $\sim$$80\%$ of the events are predicted with a magnitude at maximum $R_{\max}<21.7$. Our candidates have $t_{1/2}\ge 15$ days and $R_{\max}\ge 21.7$ (somewhat at the limit of the expected distributions) and therefore most of them are probably not microlensing events. Still, we expect $\sim$1 self-lensing event and it is possible that one or two of them are true microlensing events. In any case, from the t_1/2distribution, we are led to exclude that the microlensing candidate events are due to MACHOs of very small mass (only $\sim$$10\%$ of events with $m_{\rm MACHO}^{\rm halo}=0.01~M_\odot$ are expected to have t_1/2>15 days). This is indeed in agreement with the results found by the MACHO and EROS collaborations: they find lens masses in the halo within the range 0.2-0.6 $M_\odot$(Alcock et al. 2000; Lasserre et al. 2000).

We are not yet in a position to tell what kind of varying objects generate our events if they are not due to microlensing. They may be irregular or long period variable giants, but only a longer time baseline, and/or observations of the object at minimum light, will allow us to conclude.

The MDM analysis is not yet complete. The results from the analysis of data acquired in the other field (located on the opposite side with respect to the major axis of M 31 of the Target field analysed here) and results from new observations scheduled for October and November 2001 will give us the opportunity to gain further insight into the still open question of the composition of dark haloes. At the present time, with the caution suggested by the just mentioned problems, the analysis discussed in this paper tends to confirm that only a minor fraction of dark matter in the galactic haloes is in the form of MACHOs within the mass range 0.01-0.5 $M_\odot$ under the assumption of a standard model of the halo and given source luminosity functions.

Acknowledgements

We thank M. Crézé, S. Droz, L. Grenacher, G. Marmo, G. Papini and N. Straumann for useful discussions and suggestions. Work by AG was supported by NSF grant AST $\sim$97-27520 and by a grant from Le Centre Français pour l'Accueil et les Échanges Internationaux.