5 Candidate selection

A microlensing event is characterized by specific features that distinguish it from other, much more common, types of luminosity variability, the main background to our search. In particular for a microlensing event the bump

• does not repeat;
• follows the (symmetric) Paczynski shape;
• is achromatic.

Variable stars usually show multiple flux variations and have an asymmetric chromatic shape. Moreover, different classes of variable stars are characterized by specific features, such as timescale variation and color, that can be used to distinguish them from real microlensing events.

In the following we devise selection techniques that make use of these characteristics while taking account of the specific features of our data set.

   
5.1 Bump detection

As a first step we select light curves showing a single flux variation. We begin by evaluating a baseline, i.e. the background flux ( $\bar{\Phi}_{\rm bkg}$) along each light curve, as defined in Sect. 4.

Once the baseline level has been fixed, we look for a significant bump on the light curve. This is identified whenever at least 3 consecutive points exceed the baseline by $3\,\sigma$. The variation is considered to be over when 2 consecutive points fall below the $3\,\sigma$ level. Under the hypothesis that the points follow a Gaussian distribution around the baseline, we use the estimator L, the likelihood function, to measure the statistical significance of a bump. We want to give more weight to points that are unlikely to be found, so that we define L as

$$L = -\ln\left(\Pi_{j\in {\rm bump}}P(\Phi\vert\Phi>\Phi_{j})\right) \;\;\mbox{given}\;\; \bar{\Phi}_{\rm bkg},\, \sigma_{j},$$ (16)

where

$$P\left(\Phi\vert\Phi>\Phi_{j}\right) = \! \int_{\Phi_{j}}^\in... ...ar{\Phi}_{\rm bkg}\right)^2}{2\sigma_{j}^{2}}}\right]\!\!\cdot$$ (17)

L is then a growing function of the unlikelihood that a given variation is the product of random noise. This estimator is different from the usual definition leading to a $\chi^{2}$ with n points, and has the advantage of giving weight only to the positive deviations above threshold, which are the ones of interest.

For each light curve we denote by L₁ and L₂ the two largest deviations, respectively. We fix a threshold $L_{\rm thresh}$, and we require $L_{1}> L_{\rm thresh}$ to distinguish real variations from noise. Moreover, we fix an upper limit to the ratio L₂/L₁ to exclude light curves with more than one significant variation. The shape analysis is then carried out on the superpixels that have the highest values of L in their immediate neighborhood since we find a cluster of pixels associated with each physical variation. This method suffers from a possible bias introduced by an underestimation of the baseline level (which we further analyse in the next section).

We have carried out a complete analysis selecting the pixels with the following criteria:

• exclusion of resolved stars;
• $L_{\rm tresh}=100$;
• L₂/L₁<0.1.

This selection is made only on R images in order to reduce contamination by variable sources. In this way we take advantage of the fact that most luminous variables (to which we are anyway sensitive) show stronger variations in the I than R band.

By using these peak detection criteria, the number of superpixels is reduced from $\sim$ $4\times 10^6$ to $\sim$ $5 \times 10^3$.

   
5.2 (Achromatic) shape analysis

As a second step we determine whether the selected flux variation is compatible with a microlensing event.

The light curve of a microlensing event with amplification A(t)due to a source star with unlensed flux $\phi^{*}$ (now to be evaluated in a superpixel) is

$$\Phi (t)=\Phi_{\rm bkg}+\left(A(t)-1\right) \, \phi^{*} ,$$ (18)

where $\Phi (t)$ represents the flux collected in the superpixel associated with a single pixel, as defined before, and A(t) is given by (3).

Actually, one cannot directly and easily measure $\phi^{*}$, the unamplified flux of the unresolved source star. Only a combination of the 5 parameters that characterize the light curve can be measured in a straightforward manner:

• $\Phi_{\rm bkg}$, the background level (which include the flux of the unamplified source);
• t₀, the time of maximum amplification;
• $t_{1/2}=t_{1/2}\left(t_{\rm E},\,u_{\min}\right)$, the time width of the bump at half-maximum;
• $\Delta \Phi_{\max} = \Delta \Phi_{\max}\left(\phi^{*},\,u_{\min}\right) = \phi^{*}\, (A_{\max}-1)$, the excess of the flux with respect to the background at maximum.

Whenever the amplification is high enough, one can approximate $A\left(t\right)\simeq 1/u\left(t\right)$ and $t_{1/2}\simeq 2\sqrt{3}\,t_{\textrm E}\,u_{\min}$. It is then possible to rewrite the expression (18) in terms of these 4 parameters, and a degeneracy arises among the parameters of the amplification u₀ and t_E, and the unknown flux of the unamplified source, $\phi^{*}$ ("degenerate'' Paczynski curve, Gould 1996). Because of this degeneracy it is in general difficult to get, without extra-information, a reliable insight into all the parameters that characterize the light curve, the Einstein time in particular. For this reason we can extract only the 4 aforementioned parameters even though we carry out a non-linear fit with the complete 5 parameters ("non-degenerate'') Paczynski curve.

We now refine the selection based on the likelihood estimator in order to remove unwanted light curves with low S/N and for which the available data do not allow us to well characterize the bump. To this end we perform a Paczynski 5-parameters fit and we study

• the signal to noise ratio for the R flux variation;
• the sampling of the data on the bump.

We define the S/N estimator as

$$Q\equiv \frac{\chi^2_{\rm const}-\chi^2_{\rm ml}}{\chi^2_{\rm ml}/n.d.f},$$ (19)

where $\chi^2_{\rm const}$ is the $\chi^{2}$ with respect to a constant flux and $\chi^2_{\rm ml}$ is the $\chi^{2}$ with respect to the Paczynski fit.

The ratio Q is actually correlated with the likelihood estimator L we used in the previous step. In parallel with the cut L₁>100 we then keep only light curves with Q>100.

We do not ask for the I bump to be significant.

The second point concerns the necessity to well characterize the bump shape in order to recognize it as a microlensing event in the presence of highly irregular time sampling of data (see Fig. 2). For this purpose we require at least 4 points on both sides of the maximum, and at least 2 points inside the interval $t_0\pm t_{1/2}/2$.

After this selection we are left with 1356 flux variations.

From now on we work with the data in both colors (R and I) and we carry out a shape analysis of the light curve based on a two step procedure as follows:

• $\chi^{2}$ selection criterion;
• Durbin-Watson test on residuals;

which we now discuss in some detail.

 

 

Table 1: Summary of selection criteria.

criterion	$\%$ pixel excluded	pixel left
exclusion of resolved stars	$\sim$$10\%$	$\sim$ $3.6\times 10^6$
mono bump likelihood analysis	$\sim$$99.8\%$	5269
signal to noise ratio (Q>100)	$\sim$$69\%$	1650
sampling of the data on the bump	$\sim$$18\%$	1356
$\chi^2< 1.5$	$\sim$$98\%$	27
1.54<dwR(I)<2.46	$\sim$$59\%$	11
t1/2<40 d or R-I<1.0	$\sim$$55\%$	5

 

 

Table 2: Characteristics of microlensing candidates.

id	$\alpha$(J2000)	$\delta$(J2000)	t1/2 (d)	t0 (J-2449624.5)	$R_{\max}$	R-I	$\tilde{\chi}^2$	dwR	dwI
1	00$^{\rm h}$43$^{\rm m}$27.4$^{\rm s}$	$41^\circ 13' 11''$	$32\pm 6$	$1506\pm 1$	$22.6\pm 0.1$	$0.3\pm 0.2$	1.25	1.78	1.65
2	00$^{\rm h}$43$^{\rm m}$26.5$^{\rm s}$	$41^\circ 13' 16''$	$22\pm 7$	$1508\pm 1$	$22.7\pm 0.1$	$0.2\pm 0.2$	1.37	1.57	1.65
3	00$^{\rm h}$43$^{\rm m}$39.9$^{\rm s}$	$41^\circ 18' 41''$	$39\pm 9$	$1505\pm 1$	$22.2\pm 0.1$	$0.8\pm 0.2$	1.48	1.97	1.67
4	00$^{\rm h}$42$^{\rm m}$39.3$^{\rm s}$	$41^\circ 6' 53''$	$15\pm 1$	$1470\pm 1$	$22.3\pm 0.1$	$0.5\pm 0.2$	1.42	1.68	1.95
5	00$^{\rm h}$42$^{\rm m}$39.1$^{\rm s}$	$41^\circ 11' 26''$	$25\pm 3$	$1501\pm 1$	$21.7\pm 0.1$	$0.5\pm 0.2$	1.16	1.82	1.99

The first point is taken into account by performing the non-degenerate Paczynski fit in both colors simultaneously, so that we check also for achromaticity of the selected luminosity variations. In particular, we require that the three geometrical parameters that characterize the amplification ( $t_{\textrm E},\,u_0$ and t₀) be the same in both colors. We get, therefore, a 7 parameters least $\chi^{2}$ non linear fit:

$$\!\chi^2\!=\!\!\sum_{j=1}^{2}\!\sum_{i=1}^{N_{j}}\!\!\frac{\l... ...t_{\rm E},u_0\!\right) \!\right]}{\sigma_{i,j}^2}^{\!2}\!\cdot$$ (20)

To retain a light curve as a candidate microlensing event, we require that the reduced $\chi^{2}$

$$\frac{\chi^2}{N-7}\equiv \tilde\chi^{2}< 1.5$$ (21)

where N = N_R+N_I is the total number of points in I and R.

The application of the $\chi^{2}$ criterion test reduces the sample of light curves from 1356 to 27.

As a further step we apply the Durbin-Watson (Durbin & Watson 1951) test to the residuals with respect to the 7-parameters non-degenerate Paczynski fit. With the DW test we check the null hypothesis that the residuals are not timely - correlated by studying possible correlation effects between each residual and the next one against type I error (i.e. against the error to reject the null hypothesis although it is correct, e.g. Babu & Feigelson 1996). We require a significance level of 10%. As time plays a fundamental role in the DW test, we perform this test on each color separately.

We call dw_R and dw_I the coefficients for the DW test on the full data set. In order to retain a light curve we require 1.54<dw_R(I) < 2.46, appropriate for 40-42 points along the light curve.

This statistical analysis reduces our sample of light curves from 27 to 11.

It is worthwhile to note that some light curves, showing a real microlensing event superimposed on a signal due to some nearby variable source, and passing the previous selection criteria, could be excluded by the DW test, sensitive to timely correlated residuals.

In order to test our efficiency with respect to the introduction of the DW test we have done a study on "flat'' light curves performing a "constant flux'' fit, and selecting light curves requiring a reduced $\tilde{\chi}^2<1.5$. By applying the DW test we then reject about 50% of these light curves, i.e., more than the 10% we could expect if we had just random fluctuations. In the discussion of the Monte Carlo simulation we duly take into account this effect.

   
5.3 Color and timescale selection

By far, the most efficient way to get rid of multiple flux variations due to variable stars is to acquire data that are distributed regularly for a sufficiently long period of time. Unfortunately, at present, the data cover with regularity only the first three months of observation, and the total baseline is less than 2 years long.

For this reason, in addition to the analytical treatment that looks for the compatibility with an achromatic Paczynski light curve (efficient, for instance, to reject nova-like events), we introduce another criterion based on the study of some physical characteristics of the selected flux variations.

In particular we note that long period red variable stars (such as Miras) could not be completely excluded by the selection procedure applied so far. By contrast, short period variable stars are eliminated thanks to the cut on the second bump of the likelihood function. A preliminary analysis of the period, the color and the light curves of long period red variable stars, taken from de Laverny et al. (1997), lead us, with a rather conservative approach, to exclude those light curves that present at the same time a duration t_1/2>40 days and a color $(R-I)_{\rm C}>1.0$. A more detailed analysis aimed at a better estimation of that background noise is currently underway.

We are aware that this last selection criterion could eliminate some real microlensing events. The probability that this might happen is however low because the microlensing timescales are expected to be uncorrelated with the source color. In fact, a combination of a large t_1/2 and color $(R-I)_{\rm C}>1.0$ is extremely unlikely for microlensing events, but quite common for red variables.

This last criterion further reduces the number of candidates from 11 down to 5.

   
5.4 Results of microlensing search

We now summarize (Table 1) the different steps in the selection, give the number of surviving pixels after the application of the indicated criteria and the details of our set of microlensing candidate events.

We take these 5 light curves, whose characteristics we are now going to discuss, as our final selection of microlensing candidate events.

In the following table (Table 2) we give their position, the estimated t_1/2 in days, the time of maximum amplification t₀ (J-2449624.5), the magnitude at maximum $R_{\max}$ and the color $(R-I)_{\rm C}$. We then give the values of the fit: the reduced $\tilde{\chi}^2$ and the values of the Durbin-Watson dw_R(I) coefficients.

The corresponding light curves are shown in Fig. 7.

Figure 7: Light curves of the 5 candidate microlensing events. On the y axis, $\Delta \Phi \equiv \Phi -\Phi _{\rm bkg}$. On the x axis, the origin of time is in J-2449624.5. The dashed line represent the result of the 7-parameters Paczynski fit.

   
5.5 Variable stars

Our data contain many more varying light curves that are due not to microlensing events but to other variable sources. The study of these variable stars is an interesting task in itself. Clearly, pixel lensing is well suited for this research. We give here (see Fig. 8) only the light curve of an event characterized by a very strong and chromatic flux variation,

Figure 8: The R and I light curve of the nova event in $\alpha =00^{\rm h}43^{\rm m}1.7^{\rm s}$, $\delta =41^\circ 15' 37''$ (J2000).

that has already been considered as due to a nova (Modjaz & Li 1999) and whose light curve is also shown in Riffeser et al. (2001). We note that our points, even if with a poorer sampling in the descent, are in good agreement with those of this collaboration. This nova is found in the region of the bulge of M 31 and we localize it in $\alpha =00^{\rm h}43^{\rm m}1.7^{\rm s}$, $\delta =41^\circ 15' 37''$ (J2000). We evaluate the magnitude at maximum amplification as $R_{\rm c}=17.0$ and $I_{\rm c}=16.8$.