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Unbiased reconstruction of the mass function using
microlensing survey data
C. Alard
To cite this version:
Unbiased reconstruction of the mass function using microlensing
survey data
C. Alard
wInstitut d'Astrophysique de Paris, 98bis Boulevard Arago, F-75014 Paris, France Accepted 2000 August 9. Received 2000 July 24; in original form 2000 April 10
A B S T R A C T
The large number of microlensing events discovered towards the Galactic bulge bears the promise of reconstructing the stellar mass function, especially at low masses near the brown dwarf regime. However, because of the source confusion, even if the distribution and the kinematics of the lenses are known, the estimation of the mass function is extremely biased at low masses. The blending due to source confusion biases the duration of the event, which in turns dramatically biases the estimation of the lens mass. It is possible to overcome this problem by using differential photometry of the microlensing events. Differential photo-metry is free of any bias due to blending, but the baseline flux is unknown. In this paper it is shown that, even without knowledge of the baseline flux, pure differential photometry allows the estimation of the mass function without any bias. The basis of the method is that taking the scalar product of the microlensing light curves with a given function and taking its sum over all the microlensing events is equivalent to projecting the mass function on to another function. This method demonstrates that there is a direct correspondence between the space of the observations and the space of the mass function. The optimal functions on to which the light curves are projected are their principal components. There is no additional information about the distribution of the scalar products of the data beyond their sum (first-order moments). Higher-(first-order moments are only linear combinations of the first-(first-order moments. Thus the sum of the projections on the principal components contains all the information, and translates into an equal number of projections of the mass function with functions associated with the principal components. The method is illustrated with simulated data sets consistent with the microlensing experiments. With 1000 such simulations, we show that for instance the exponent of the mass function can be reconstructed without any bias.
Key words: stars: luminosity function, mass function ± dark matter ± gravitational lensing.
1 I N T RO D U C T I O N
The success of the microlensing experiments has been impressive, several collaborations having reported the detection of microlen-sing amplification of stars: DUO (Alard & Guibert 1997), EROS (Derue et al. 1999), VATT (Uglesich et al. 1999), OGLE (Udalski et al. 2000) and MACHO (Alcock et al. 2000). In particular, the recent release by the OGLE II collaboration of more than 200 microlensing events is of great interest. One of the obvious promises of such data sets is the possibility to explore the mass function near its low-mass end. However, the relation between the mass function and the microlensing observations is not straight-forward. In the classical scheme of analysis, one must first
estimate the duration of the events by fitting the theoretical light curves, then compute the lensing rates, and finally relate these lensing rates to the mass function. The trouble is that fitting the theoretical light curve to the microlensing data to obtain the duration of the event is a highly degenerated process (Alard 1997; Han 1997; Wozniak & PaczynÂski 1997). Because of the source confusion in crowded fields, it is almost impossible to estimate reliably the flux of the amplified source. The flux of the source is severely biased by blending with neighbouring sources. In such a case, an overestimation of the baseline flux will result in an underestimation of the duration of the event. This bias is severe for unresolved stars and will result in a reduction in the estimation of the duration by a factor of 2, 3 or even more. Since the bias on the mass of the lens goes like the square of the bias on the duration, the relevant bias on the mass function will be very large. It has
wE-mail: [email protected]
already been shown (Alard 1997; Han 1997) that the contribution of these unresolved, highly blended stars to the lensing rates was dominant at short durations. The consequence is that the estima-tion of the mass funcestima-tion at low mass will be largely over-estimated, suggesting the existence of a large number of brown dwarfs that actually do not exist. It is obvious that, as long as the bias due to the blended sources has not been solved, any correct estimation of the mass function is impossible.
This paper proposes a solution to this problem. We will see that using differential photometry obtained using the image subtraction method (Alard & Lupton 1998; Alard 2000), it is possible to estimate the mass function, even without knowing the baseline flux of the source. This method assumes that the distribution and kinematics of the lenses are known with good accuracy. It is important to emphasize that the uncertainties related to the structure of the Galaxy in the bulge region are several orders of magnitude smaller than the uncertainties due to the blending bias. Many good models of the kinematics and structure of the Galactic bulge are already available (Zhao 1996; Fux 1997; Bissantz et al. 1997; Binney, Gerhard & Spergel 1997).
2 T H E M E T H O D 2.1 Introduction
When analysing microlensing observations one has to deal with the light curves of a number N of microlensing candidates. We will assume that we have pure differential photometry, with a baseline flux equal to zero. We emphasize that high-quality differential photometry can always be obtained after the events have been detected by classical methods by using the image subtraction method (Alard 1999). These light curves will be represented by the symbol Si t; i 1; ¼; N: Assuming that the unamplified flux of the source is F, the expression of Si(t) as a function of time will be given by the theoretical microlensing amplification formula: Sj t F u t 2 2 u tpu t2 42 1 " # : 1
For an event with an impact parameter u0and a duration tE, the expression for u(t) reads:
u t u2 0 t=tE2 q
:
(We recall that we are working on the assumption that we have at our disposal a differential flux only, with for convenience a baseline flux equal to 0.)
The most important issue is to find a relation between the observations and the microlensing parameters: the impact parameter, u0, and the time to cross the Einstein ring, tE. Also important is to relate the microlensing parameters to the mass function itself. The first serious difficulty we encounter is that u0 and tEcannot be extracted easily by fitting the light curve because of the extreme blending degeneracy (Alard 1997; Han 1997; Wozniak & PaczynÂski 1997). Thus our first and most essential step will be to derive a better set of parameters. This set of para-meters will have to be non-degenerate with respect to blending. The simplest and most natural way to deal with such a problem is to express the observations as a linear combination of a small number of vectors. The best way to make such a decomposition is known as principal components analysis, and is equivalent to an eigenvalue decomposition. Thus all we have to do is to look for
the first eigenfunctions, and to calculate the scalar product of the components with our vectors (time series) of observations.
2.2 Normalization
Before making the principal components analysis we have to consider that one of the parameters, the amplitude of the magnified sources, does not contain any information about the duration of the event, and the mass of the lens. The reason for the degeneracy of the fit is precisely the unknown amplitude of the source. The only relevant and meaningful parameters are u0 and tE. Thus it is important to derive an estimator that is independent of the amplitude. Since the amplitude is a linear parameter, it is sufficient to normalize the vector of observations in order to get rid of the amplitude. This normalized light curve SÅi(t) can be expressed as:
Si t 1a u t 2 2 u tpu t2 42 1 " # and a u t2 2 u tpu t2 42 1 " #2 dt v u u t :
Note that SÅi(t) no longer depends on the baseline flux, but only on u0and tE.
The simplest way to apply the previous normalization is to calculate the modulus of Sidirectly from the observations. How-ever, it is obvious that calculating jSij directly from the obser-vations may not be optimal. To estimate jSij we will use the following approach: First we fit a microlensing model to the observations. Even if the fit is very degenerated, one can always find a good solution that fits the data well. We will call this fitted model SÄi. Once we get this light-curve solution we estimate the modulus using the following formula:
jSij ~Si t2dt s :
We will use the same approach for all our other calculations. In the same way we will estimate the principal components not by taking the data directly, but by taking the models we have fitted to the light curves. Finally the scalar products with the principal components will also be estimated by cross-products with the theoretical models.
2.3 Principal components
As we have already explained, we will first fit a microlensing model SÄito the light curves, normalize this series of N vectors to obtain the series of vectors SÄÅi, and then calculate the principal components of these SÄÅivectors. Usually it is possible to express almost all the information with a reduced number of principal components with the following linear decomposition:
~Si X
j aijPj t;
where aijis the projection of time series number i, SÄi(u0, tE, t), on the principal component number j:
aij u0; tE
~Si u0; tE; tPj t dt:
Note that aijis a function of u0and tE: aijb u0; tE:
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For instance, a typical set of 100 microlensing light curves can be expressed as the combination of only four components with an accuracy good to 1 per cent. The other principal components contain very little additional information, but mostly noise. 2.4 Statistical distribution of the projections on the principal components
We consider that most of the information is contained in a number NCof principal components. Thus almost all the information can be extracted from the distribution of the projection of the N time series on the NC principal components. The first meaningful quantity concerning the distribution of the aij is the first-order moment of the distribution:
kajl X
i aij:
The second quantity of interest is the second-order moment of the distribution: ka2 jl X i a2 ij:
It is important to note that, according to the Fischer±Riesz theorem, one can always find a function F(t) such that:
a2
ij b u0; tE2
~Si u0; tE; tF t dt; 2 so a2
ij b u0; tE2can be written as a scalar product of the data with F(t). Finally, using the principal components decomposition of SÄÅi, equation (2) reads: a2 ij. XNC j1 Pj tF t dt aij:
Thus, we see that the distribution of the second moment is just a linear combination of the NCfirst moments (mean). Consequently the mean of the second moment will not bring any additional information with respect to the mean. With a similar reasoning it is possible to show that the same property is true for the Nth moment. Consequently we can conclude that all the information concerning the distribution of the aijis contained in the mean of these components. No additional information (uncorrelated infor-mation) will be found by looking at higher-order moments. 2.5 From the principal components to the mass function It is possible to re-express the previous definition of kajl by using the number densityr(u0, tE, M) to observe a given amplification of parameters (u0,tE) with a lens of mass M. In such a case, the sum can be approximated very closely with an integral expression: kajl .
r u0; tE; Maj u0; tE du0dtEdM:
If we define the efficiency function of the experiment, Q(u0, tE), the lensing rates for one solar mass lenses, G0(t), and the mass function,f(M), the formulae for r reads:
r u0; tE; M G0 ptEM
f MQ u0; tE; leading to the following expression for kajl : kajl . G0 tE M p
f MQ u0; tEaj u0; tE du0dtEdM:
It is easy to rearrange this integral in the following way: kajl . c Mf M dM with c M G0 tE M p
Q u0; tEaj u0; tE du0dtE: 3 Thus we see that basically projecting the microlensing light curve on to a basis of functions and taking the statistical sum is equivalent to projecting the mass function itself on to another function. Consequently, we see that a projection in the space of the observations (the light curve) is directly equivalent to a projection in the space of the mass function. The problem of finding the optimal set of projections for the observations is solved by using the principal component method. Then all we have to do is to calculate the `image' of these principal components in the space of the mass function.
3 M O N T E C A R L O S I M U L AT I O N S 3.1 Introduction
To illustrate this method and show how the mass function can be reconstructed, we will use a series of Monte Carlo simulations. We will simulate microlensing events by selecting the parameters of the events according to the density distribution, r(u0, tE, M). Basicallyr(u0, tE, M) has three parts. One additional parameter we will have to select is the amplitude of the source. To get random variables that reproduce the distributionr, we need to decompose the problem.
3.1.1 The rates, GM(tE) We have
GM tE G0 ptEM
;
and for G0we will adopt the analytical expression given in Mao & PaczynÂski (1996).
3.1.2 The efficiency function, Q(u0, tE)
For the efficiency we will adopt the following criterion: an event is detected if it has a minimum number (Nm) of data points above a 5s threshold. For a given duration tE this criterion is simply equivalent to a threshold on the impact parameter u0. All we have to do is to search for the maximum value uTof u0such that the amplification of at least Nmdata points is above 5s. If the sampling is even, with a time stepdt, it just means that the amplification must be larger than 5s in a window of time around the maximum DT Nmdt: Provided the maximum is at the origin of the time axis, this will finally result in the condition:
SjÿuT; tE;12DT. 5s: 4 The distribution of u0is uniform; thus to get a set of values for u0 we will use a random generator, which will provide a uniform distribution between 0 and the maximum value for u0, uM(tE). This value uM(tE) corresponds to the impact parameter for the brightest source given by equation (4). Once we have selected tEfrom the distribution GM(tE), we will calculate u0,threshusing equation (4). If
our random u0 value is above u0,thresh, then Q u0; tE 0; otherwise Q u0; tE 1: To summarize:
Q u0; tE
1 u0, uT; 0 u0. uT: (
3.1.3 The mass function,f(M)
In all our simulations we will adopt a pure power-law expression for the mass function, with a lower cut-off (MI) and an upper cut-off (MS):
f M M2a MI, M , MS; 0 otherwise: (
3.1.4 The amplitude
We will assume that the flux of the unamplified source amplitude distribution is a power law with an exponent of 22 (Zhao, Spergel
& Rich 1995), in a given range of amplitude Fmaxand Fmin. In our simulations, the amplitudes will be generated by the Monte Carlo method, using this power law.
3.2 Description of the implementation
To implement our Monte Carlo simulation we will select the random variables in the following order:
(i) The baseline flux of the source, according to the probability law F2n.
(ii) The mass of the lens, according to the probability law M2a. (iii) The duration of the event tEusing the distribution GM(tE). (iv) The impact parameter u0using a uniform distribution in the range 0 to uM(tE).
(v) Once tEand u0are selected, we apply the efficiency cut-off, using the function Q(u0,tE).This procedure produces a complete set of variables, F, tE, u0, for the events passing the efficiency cut-off. From these variables we compute the microlensing light curve
Figure 1. Examples of light curves as obtained using the Monte Carlo simulation procedure described in the text.
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using equation (1). We add noise to the light curves according to the Poisson statistics.
3.3 Example
In this example we will take an amplitude range that is typical for microlensing images towards the Galactic bulge:
Fmax 106; Fmin 10:
For the mass function we take the following parameters:
f M M
22 0:05 , M , 10:0; 0 otherwise: (
A numerical simulation can be constructed by generating a fixed number, NG, of parameter sets, (F, tE, u0). The number NG has been adjusted in order that the mean number of events above the detection efficiency threshold be close to 100. For illustration, we extracted a sample of light curves from one of these
simulations (see Fig. 1). The next step is to perform the decom-position in principal components. The orthonormal set of principal components was calculated by using a singular value decomposi-tion. The first four principal components are presented in Fig. 2. The functionsc(M) corresponding to the principal components in the space of the mass function can be computed using equation (3). Projecting the data on the principal components is equivalent to projecting the mass function on c(M). To illustrate our discussion, an example of c(M) functions calculated using the settings of the previous Monte Carlo simulation are presented in Fig. 3.
3.4 Estimating the mass function
To illustrate the ability of the method to reconstruct the mass function, we will assume that the exponent of the mass function, a, is unknown. For each simulation we will calculate the four principal components of the set of microlensing light curves we have simulated. We will compute the sum of the projection on a
Figure 2. The first four principal components corresponding to the simulation presented in Fig. 1.
principal component of all the light curves kajl: Using equation (3) we will calculate the function that corresponds to each principal component in the space of the mass function. Then all we have to do is to compare the projection of a trial mass function with kajl:
The trial mass function that matches the four kajl as closely as possible (using a least-squares criterion) will be the best mass function. This procedure was applied for 1000 simulations; each time the program derived the best value of the exponenta. The histogram of the values ofa for the 1000 simulations is presented in Fig. 4.
R E F E R E N C E S
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This paper has been typeset from a TEX/LATEX file prepared by the author.
Figure 3. The functionc(M) that corresponds to the principal components. In the upper panel the original four functions are presented, whereas in the lower panel an orthonormal linear combination of these four functions is shown. (Note that in the upper panel the functions were scaled to have the same amplitude).
Figure 4. Histogram of the values ofa(the exponent of the mass function) found by fitting the Monte Carlo simulation by projecting the data in the space of the mass function.
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