4 The inverse problem to estimate $\vec{(\alpha, f)}$

We show now how it is possible to use the simple technique of the inverse problem to estimate the slope $\alpha$ of the lens MF and the dark halo mass fraction f composed by MACHOs for each one of the models labelled by the codes previously explained. We consider data coming from the first 5.7 years of observations towards LMC by the MACHO collaboration (Alcock et al. 2000a), limiting ourselves to the thirteen events selected according to the so-called selection criteria A. These are high S/N events and are spatially distributed in a way which is consistent with the hypothesis that they are due to lenses belonging to our halo and not to LMC self lensing; we will discuss of this problem later on. For this set of events we have

$$N_{\rm ev}^{\rm obsd} = 13, \ \ \tau_{\rm obsd} = 3.5 \times 10^{-8}, \ \ < t_{\rm E} >_{\rm obsd}~= 76.2 \ {\rm d},$$

where $< t_{\rm E} >_{\rm obsd}$ is simply the average value of the duration of the observed events. As a test of the correctness of this estimate we may note that $(\pi/4E) < t_{\rm E} >_{\rm obsd}$ (with $E = 6.12 \times 10^{7}$ star-years) is exactly equal to $\tau_{\rm obsd}/N_{\rm ev}^{\rm obsd}$, as it has to be for the reasons we are going to explain later. We have then to estimate the uncertainties on the observed quantities. First, we consider the directly observed optical depth, and we simply use a method as similar as possible to the one proposed by the MACHO group itself for a very conservative estimate of the error on $\tau_{\rm meas}$ (Alcock et al. 1997a): we divide the observed events according to their duration $t_{\rm E}$ in bin of 10 days; in such a way $\tau_{\rm obsd}$ is more or less the same for events in the same bin and the errors are approximately poissonian. For each bin we estimate $N_{\rm low} = N_{\rm ev}^{\rm obsd}{\rm (bin)} - \sqrt{N_{\rm ev}^{\rm obsd}{\rm (bin)}}$ and $N_{\rm up} = N_{\rm ev}^{\rm obsd}{\rm (bin)} + \sqrt{N_{\rm ev}^{\rm obsd}{\rm (bin)}}$ and define $\tau_{1}^{\rm min}$ and $\tau_{1}^{\rm max}$ as the minimum and maximum value of $\tau_{1}$ for the events in that bin (being $\tau_1 = \pi t_{\rm E} / 4 E$). Then we estimate

$$\tau_{\rm obsd}^{\rm low} = \sum_{\rm bin} N_{\rm low} \tau_{... ...sd}^{\rm max} = \sum_{\rm bin} N_{\rm up} \tau_{1}^{\rm max} .$$

At the end we find

$$\tau_{\rm obsd} = (3.5 \pm 2.5) \times 10^{-8} .$$

The error on the optical depth turns out to be so large ( ${\sim} 70\%$) because of the limited number of events. Let us turn now to the error on the number of observed events. This simply comes from the low statistics and may be assumed to be poissonian, i.e. $\delta N_{\rm ev}^{\rm obsd}/ N_{\rm ev}^{\rm obsd} = 1/\sqrt{N_{\rm ev}^{\rm obsd}} \simeq 28\%$. Finally, the error on $< t_{\rm E} >$ is obtained by propagating the error on $N_{\rm ev}^{\rm obsd}$ which gives us

$$\frac{\delta < t_{\rm E} >_{\rm obsd}}{< t_{\rm E} >_{\rm obs... ...frac{\delta N_{\rm ev}^{\rm obsd}}{N_{\rm ev}^{\rm obsd}}\cdot$$

The observed quantities may be now compared to the theoretical ones evaluated in the previous section, which also take into account the detection efficiency in order to make the comparison meaningful. To do this we must first remember that the theoretical quantities have been calculated under the hypothesis that the dark halo is totally made out by MACHOs, i.e. with f = 1. Actually, the exact value of f is not well known: from the more recent observational constraints it is quite unlikely that f = 1. However to take into account of f is not very difficult: we have simply to multiply by f the expression of the differential rate ${\rm d}\Gamma/{\rm d}t_{\rm E}$ and consequently the ones obtained for $N_{\rm ev}^{\rm oble}$ and $\tau_{\rm oble}$. Note that $< t_{\rm E} >$ is independent on the value of f. Then we have the following relations between observable and observed quantities:

$$% \left \{ \begin{array}{ll} N_{\rm ev}^{\rm oble} = N_{\rm e... ... \tau_{\rm oble} = \tau_{\rm obsd}/f \\ \end{array}\right . .$$ (27)

Dividing these two equations and using the relation $\tau_{\rm oble}/N_{\rm ev}^{\rm oble} = (\pi/4E)< t_{\rm E} >$, one gets: $\tau_{\rm obsd}/N_{\rm ev}^{\rm obsd} = (\pi/4E)< t_{\rm E} >_{\rm obsd}$, which may be used to test the correctness of our previous estimate of $< t_{\rm E} >_{\rm obsd}$. For each model, we may solve the system (27) in the unknowns $\alpha$ and f and estimate them toghether with the errors connected to our analysis. Solving Eqs. (27) is not really useful since the very high error on $\tau_{\rm obsd}$ leads us to get no constraints at all on the parameters $(\alpha , f)$. We may then use a third relation we have at our disposal, given by

$$% < t_{\rm E} >(\alpha) =~< t_{\rm E} >_{\rm obsd} ,$$ (28)

which is quite easy to solve numerically to get an estimate of the slope $\alpha$ of the MF. Solving Eq. (28), taking into account also the uncertainties, will give us in general more than one solution compatible with the microlensing data. A further selection can be done imposing that $\alpha$ must be in the range (0.0, 5.0) since values outside this range are reasonably quite unlikely. In this way, for each model, we have estimated a range $(\alpha_{\rm l}, \alpha_{\rm u})$ for the slope of the MF simply requiring that

 

$$% (1 - \delta t_{\rm E} / t_{\rm E} ) < t_{\rm E} >_{\rm obsd}~\l... ...{\rm obsd} \ \iff 54.86 \ {\rm d} \le~< t_{\rm E} >(\alpha) \le 97.54 \ {\rm d}$$ (29)

being $\delta t_{\rm E}/ t_{\rm E}$ the fractional error on $< t_{\rm E} >_{\rm obsd}$ which we have previously estimated to be of order of 28%. These values are summarised in Table 4, where we report also the value of $\alpha _0$ corresponding to the average value $< t_{\rm E} >(\alpha_0) = 76.20$d.

 

 

Table 4: Estimates of the slope $\alpha$ and the dark halo mass fraction f composed by MACHOs for different models labelled as explained before. $M_{50}^{\rm bar}$ is the mass in MACHOs inside 50 kpc (with $M_{50,(0)}^{\rm bar}$ the value corresponding to $\alpha _0$ and f₀) measured in units of $10^{10}~M_{\odot}$. Note that the value of $\alpha _{\rm l}$ for model B2a differs considerably from the others since the maximum value of $< t_{\rm E} >(\alpha )$ is only 96.2 d, which is lower than the upper limit on the $< t_{\rm E} >_{\rm obsd}$ given in Eq. (29).

Code	$(\alpha_{\rm l}, \alpha_{\rm u})$	$(f_{\rm min}, f_{\rm max})$	$M_{50}^{\rm bar}$	$(\alpha_0, f_0, M_{50,(0)}^{\rm bar})$
A2a	(0.50, 1.48)	(0.12, 0.33)	4.68 $\div$ 12.87	(1.10, 0.21, 8.19)
A2b	(0.73, 1.50)	(0.10, 0.30)	4.02 $\div$ 12.06	(1.15, 0.19, 7.63)
A2c	(0.80, 1.50)	(0.11, 0.32)	4.63 $\div$ 13.47	(1.17, 0.20, 8.42)
A2d	(0.84, 1.56)	(0.12, 0.35)	5.18 $\div$ 15.12	(1.21, 0.22, 9.50)
B2a	(0.27, 1.69)	(0.11, 0.33)	4.29 $\div$ 12.87	(1.12, 0.24, 9.36)
B2b	(0.63, 1.80)	(0.10, 0.30)	4.02 $\div$ 12.06	(1.25, 0.18, 7.24)
B2c	(0.77, 1.89)	(0.10, 0.32)	4.21 $\div$ 13.47	(1.34, 0.19, 8.00)
B2d	(0.77, 1.90)	(0.11, 0.34)	4.75 $\div$ 14.79	(1.34, 0.21, 9.07)

Having estimated $\alpha$ and being $N_{\rm ev}^{\rm obsd}$ known, we may now get a constraint also on f: on the plot $f N_{\rm ev}^{\rm oble}$ as a function of $\alpha$ and f we make the contour levels for $f N_{\rm ev}^{\rm oble} = N_{\rm ev}^{\rm obsd} - 28\% = 9.39$ and $f N_{\rm ev}^{\rm oble} = N_{\rm ev}^{\rm obsd} + 28\,\% = 16.61$. On this graph one has to add also the vertical lines corresponding to $\alpha = \alpha_{\rm l}$ and $\alpha = \alpha_{\rm u}$ (see Fig. 2). The region $\cal {R}$ of the parameter space $(\alpha , f)$ delimited by the two level curves and these two vertical lines is that which is consistent with the constraints on $\alpha$ and the ones coming from microlensing observations towards LMC, i.e. the number of observed events and their mean duration. From Fig. 2 one sees that it is possible to define $f_{\rm min}$ and $f_{\rm max}$ as the minimum and maximum value of f in the region $\cal {R}$ and a value f₀ such that

$$\left \{ \begin{array}{ll} f_0 N_{\rm ev}^{\rm oble}(\alpha_0... ...m E} >(\alpha_0) = 76.20 \ {\rm d} \\ \end{array}\right . \ .$$

One has then that for each value of $f \in (f_{\rm min}, f_{\rm max})$ there exists a value of $\alpha \in (\alpha_{\rm l}, \alpha_{\rm u})$ such that

$$% \left \{ \begin{array}{ll} 9.39 \le f N_{\rm ev}^{\rm oble}... ...\rm E} >(\alpha) \le 97.54 \ {\rm d} \\ \end{array}\right . .$$ (30)

The values of $(f_{\rm min}, f_{\rm max})$ and f₀ are summarised in Table 4 for the different models considered. We also give in the same table the range for the mass in MACHOs inside 50 kpc (indicated as $M_{50}^{\rm bar}$) which is easily estimated as $f \times M_{50}$ (with M₅₀ given in Table 1) since it is this quantity which is most strongly constrained by microlensing observations.

Figure 2: In the upper panel we plot $f \times N_{\rm ev}^{\rm oble}$ as a function of $\alpha$ and f; in the lower one, we determine the region $\cal {R}$ of the parameter space $(\alpha , f)$ consistent with microlensing observations towards LMC: the upper line in the second panel is the level curve $f \times N_{\rm ev}^{\rm oble} = 16.61$, the lower one for $f \times N_{\rm ev}^{\rm oble} = 9.39$, whilst the two dashed lines have been drawn intersecting the level curves with the vertical lines $\alpha = 0.80$ and $\alpha = 1.50$. The plot is for model A2c; similar plots are obtained for the other models.