Interpretation of LMC Events

Using our sample of microlensing events, there are two

complementary analyses which can be performed. First, we can set

a conservative limit on the Macho contribution to the dark halo.

Since we know our efficiencies, and we have certainly not seen

more than 3 microlensing events from halo objects, any halo model

which predicts more than 7.75 events can be ruled out at the 95%

C.L. This result will be independent of whether or not all three

candidate events are due to microlensing, and independent of

whether or not the lenses are in the dark halo. Second, if we make

the further assumption that all three events are due to microlensing

of halo objects, we can estimate the mass of the Machos and their

contribution to the mass of the dark halo.

In order to do either analysis we need a model of the dark halo. We

need to know the total mass of the halo, and we need the density

and velocity distribution to calculate an expected microlensing rate.

The main constraints on the halo come from the Milky Way

rotation curve, which is not as well determined as rotation curves in

other galaxies. Constraints from the orbits of satellite galaxies also

exist, but there is considerable uncertainty in both the total halo

mass and the expected microlensing rate coming from uncertainty

in the size and shape of the Milky Way halo [50, 51, 52]. Using a

very simple, but commonly used halo model [49], we can calculate

the number of expected events as described above, and the results

are shown in Figure 9 ([fig], [captions]).

If the Milky Way has a standard halo consisting entirely

of Machos of mass then we should have seen more

than 20 events, with fewer events at larger or smaller masses.

However, even if the halo dark matter consists of Machos, it is

very unlikely that they all have the same mass. Fortunately, it can

be shown [49] that if one rules out all halos made of unique Macho

mass between masses and , then one has ruled out a halo

consisting of ANY distribution of masses as long as only masses

between and are included. Thus we can make the powerful

conclusion that a standard halo consisting of any objects with

masses between and has been ruled out by our

first year LMC data.

As mentioned above, there is no strong reason to believe that the

Milky Way halo is precisely as specified in the standard halo, and we

would like to test the robustness of the important results above by

considering a wider range of viable halo models. To this end, we

have investigated a class of halo models due to Evans [53]. These

models have velocity distributions which are consistent with their

density profiles, and allow for halos which are non-spherical, and

which have rotation curves which gently rise or fall. A description

of the parameters that specify these models, along with

microlensing formulas can be found in Alcock et al. [50]. Basically

we consider models which give rotation velocities within 15% of the

IAU standard value of 220 km/sec, at the solar circle (8.5 kpc) and

twice the solar circle. As pointed out by Evans and Jijina [54], the

contribution of the stellar disk plays an important role in the

predicted microlensing rate. This is because much (or even most) of

the rotation speed could be due to material in the disk, so we

consider various size disks, as well.

Using these models, we find strong limits are found on heavy halo

models, while only very weak limits are found on light halo models.

This is because microlensing is sensitive not to the total mass in the

halo, but only to the mass in Machos. So one can get a much more

model independent limit on the Macho content of the halo by

limiting the total mass in Machos, rather than the Macho fraction

of the halo. A more robust statement of our first year LMC

microlensing results is thus that objects in the

range can contribute no more than to the dark halo,

where we consider the halo to extend out to 50 kpc. The standard

halo has out to this radius, and so is ruled out as before,

but much smaller all-Macho halos, would be allowed. It should be

clear that in order to get good information on the Macho fraction

of the halo, more work is needed on the total mass of the halo. This

requires better measurement of the Milky Way parameters and

rotation curve. Microlensing measurements themselves may also be

able to help [52, 51, 50].

The limits above are valid whether or not the three events shown in

Figure 7 ([fig], [captions]) are due to microlensing of halo objects.

However, if we make the additional assumption that they are, we

can go beyond limits and estimate the Macho contribution to the

halo, and also the masses of the Machos. The results obtained,

especially on the lens masses, depend strongly on the halo

model used, so keep in mind that it is not clear that all three events

are microlensing, and it is certainly not known that they are due to

objects residing in the galactic halo.

Proceeding anyway, we can construct a likelihood function as the

product of the Poisson probability of finding 3 events when

expecting and the probabilities of drawing the observed 's

from the calculated model duration distribution [50, 37, 38]. The

resulting likelihood contours can be found in references [37] and

[38]. We find that for a standard halo, a Macho fraction of %

is most likely, with Macho masses in the range likely.

Note that the errors in these estimates are very large due to the

small number statistics, and that there is an enormous additional

uncertainty due to the halo model. However, once again, the

maximum likelihood estimate of the total mass in Machos is quite

model independent and is about . Since the mass in

known stars, gas, etc. is only about , we see this would be

a major new component of the Milky Way if it is confirmed to exist.

BIBLIOGRAPHY