The optical depth is widely used in Galactic microlensing studies as a means to determine the density of MACHOs , since in theory it depends only upon their spatial distribution and is therefore less model-dependent than other microlensing observables .
However , since the measured optical depth is restricted to that of events with measurable timescales , inferences of total MACHO density are dependent on the assumed timescale distribution .
Using the 2-year LMC results of the MACHO collaboration , we illustrate this point by showing how , for an assumed isothermal halo , the inferred MACHO halo fraction as determined from optical depth estimates depends upon MACHO mass .

The analysis highlights the following conclusions : ( 1 ) The MACHO density inferred from optical depth measurements depends on the assumed MACHO mass function for a given Galactic distribution function ; ( 2 ) without extra information on the MACHO mass function , such as can be obtained from the rate-timescale distribution , optical depth measurements can provide a lower limit but not an upper limit on the MACHO density for a given Galactic distribution function ; ( 3 ) a comparison between the inferred total optical depth of different Galactic models , or of different components in a multi-component Galactic model , requires knowledge of the underlying timescale distribution for each model or component .

For our assumed Galactic halo model we find excellent agreement between our model-dependent lower-limit halo fraction of f > 0.3 ( 84 \% confidence ) or f > 0.15 ( 97.5 \% confidence ) , and the MACHO collaboration ’ s own model-independent optical depth lower-limit estimates .
MACHO ’ s optical depth upper limits are consistent with the minimum value of the upper limit derived for our assumed model .