RED NOISE VERSUS PLANETARY INTERPRETATIONS IN THE MICROLENSING EVENT OGLE-2013-BLG-446

This event shows clear signs of finite-source effects and the limb darkening coefficients must be evaluated for each data set. We first consider a point-source point-lens model (PSPL) (Paczyński 1986). The PSPL model allows the estimation of the source and blended fluxes in the V and I passbands for the calibrated OGLE photometry, leading to a good approximation for the V and I magnitudes of the source, which in turn allows us to derive a rough color for the source. We found ${(I,(V-I))}_{\mathrm{PSPL}}=(19.07,1.48).$ Using the Interstellar Extinction Calculator on the OGLE website⁶² based on Nataf et al. (2013), we found that the Galactic Bulge true distance modulus for this line of sight is μ = 14.578 ± 0.326 mag (d_Bulge = 8.2 ± 1.2 kpc), the I band extinction is A_I = 0.804 mag, and the reddening is $E(V-I)=0.683\pm 0.036$ mag, leading to ${R}_{I}={A}_{I}/E(V-I)=1.177,$ lower than the standard value of 1.5. This low extinction is known as the anomalous extinction law toward the Galactic Bulge; see Udalski (2003a). We derive the source properties as follows:

• 1.  
  Assuming that the source suffers the same extinction as the Red Giant Clump (i.e., the source is at the same distance), we have M_I = 19.07 − 0.804 − 14.578 = 3.7 mag, so the source star is most likely a main sequence star. We adopt $\mathrm{log}g\sim 4.5.$
• 2.  
  We derive its effective temperature using the dereddened color–magnitude relation for dwarfs and subgiants (relation (3) in Casagrande et al. 2010) with solar metallicity.
• 3.  
  From Claret (2000) and using $\mathrm{log}g\sim 4.5,$ we are able to find the linear limb-darkening coefficients u_λ (Milne 1921) for each filter. Following Albrow et al. (1999) and Yoo et al. (2004), we use the transformation:

  $$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\lambda }=\displaystyle \frac{2{u}_{\lambda }}{3-{u}_{\lambda }}.\end{eqnarray} \tag{ 1 }$$

These calculations form the starting point for an iterative fit of the finite source point lens (FSPL) model, together with error-bar rescaling as described in Section 3.3. Our best FSPL model converges to source magnitude and color $(I,(V-I))=(19.00,1.49).$ Correcting for extinction and reddening we have ${({I}_{o},(V-I)}_{o})=(18.20,0.81).$ The corresponding effective temperature of the source is T_eff ∼ 5400 K, leading to Γ_V = 0.63 (u_V = 0.72), Γ_R = 0.55 (u_R = 0.65) and Γ_I = 0.46 (u_I = 0.56) for $\mathrm{log}g\sim 4.5.$ Note that we also use Γ_I = 0.46 for the RoboNet telescopes (SDSS-i filter). Finally, given the dereddened magnitude and color of the source from our best FSPL model, we are able to estimate the angular source star radius θ_* using Kervella & Fouqué (2008):

$$\begin{eqnarray}{\mathrm{log}}_{10}({\theta }_{*}) & = & 3.1982+0.4895{(V-I)}_{o}\\ & & -0.0657{(V-I)}_{o}^{2}-0.2{I}_{o}.\end{eqnarray} \tag{ 2 }$$

The uncertainty of this relation is 0.0238. The errors on our magnitude estimates are (ΔI, ΔV) = (0.02, 0.02) mag. Assuming a conservative estimate of the error on A_I (0.1 mag) and using standard error propagation gives 9% precision: θ_* = 0.82 ± 0.07 μas. With the adopted source distance (8.234 kpc), the source star radius is R_* = 1.4 ± 0.3 R_⊙. Therefore the source is a G6 or K0 star (Bessell & Brett 1988).

The PSPL model is described by the standard single-lens parameters: t_E the Einstein crossing time, u_o the minimum impact parameter, and t_o the time of this minimum. The normalized angular source radius ρ = θ_*/θ_E (Gould 1994; Nemiroff & Wickramasinghe 1994; Witt & Mao 1994; Bennett & Rhie 1996; Vermaak 2000), where θ_E is the angular Einstein ring radius, is included in the model along with the previous parameters to take into account finite-source effects close to the magnification peak. We used the method described in Yoo et al. (2004) to take into account the change in magnification due to the extended source. The FSPL model significantly improves the fit (see Table 3). The best FSPL model is shown in Figure 1.

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Using the value of ρ from the FSPL model, we are able to estimate the angular radius of the Einstein ring θ_E = θ_*/ρ = 1.57 ± 0.1 mas and the lens-source proper motion μ = θ_E/t_E = 7.4 ± 0.7 mas yr⁻¹.

Because of the diversity of observatories and reduction pipelines used in microlensing, photometric uncertainties need careful rescaling to accurately represent the real dispersion of each data set. This is an important preliminary step in modeling the event. Following Bachelet et al. (2012b), Miyake et al. (2012), and Yee et al. (2013), we rescale the uncertainties using:

$$\begin{eqnarray}&&{e}_{i}^{\prime }=\sqrt{{({{fe}}_{i})}^{2}+{e}_{\mathrm{min}}^{2}},\end{eqnarray} \tag{ 3 }$$

where e_i are the original magnitude uncertainties, f is the rescaling parameter for low magnification levels, e_min is a minimal uncertainty to reproduce the practical limitations of photometry, and ${e}_{i}^{\prime }$ are the adjusted magnitude uncertainties. The classical rescaling method is to adjust f and e_min to force χ²/degrees of freedom (dof) to be unity.

In this paper, we follow an alternative method of first adjusting f and e_min to force the residuals, normalized by ${e}_{i}^{\prime },$ to follow a Gaussian distribution around the model. If possible, we also aim to obtain a χ²/dof ∼ 1. Note that these two methods lead to the same results, except for the OGLE_I data set. For OGLE_I, the distribution without rescaling shows some data points with large residuals. This is not surprising because the OGLE_I data set covers the entire light curve with a large number of points, especially the faint baseline magnitude (I ∼ 17.8), with a constant exposure time on the order of 100 s. Inspection of the OGLE_I light curve reveals that the uncertainties during high magnification are underestimated, so we adjust the e_min parameter. We tried to force χ²/dof ∼ 1 for this data set, but this generated large uncertainties for the low magnification part (i.e., the baseline), leading to a non-Gaussian distribution (lots of normalized residuals too close to the mean).

We finally checked isolated points far away from this Gaussian distribution, and reject as outliers (>7σ) two data points in the Auckland_R data set. The rescaling coefficients are presented in Table 2.

We looked for second-order effects in the light curve. First, the relatively long Einstein-ring crossing time (t_E ∼ 80 days) should allow the measurement of the displacement of the line of sight toward the target due to the Earth's rotation around the Sun. This annual parallax (Gould & Loeb 1992; Gould 2000, 2004; Smith et al. 2003; Skowron et al. 2011) is described by the vector ${\pi }_{{\rm{E}},a}={AU}/\tilde{{r}_{{\rm{E}}}}=({\pi }_{\mathrm{EN}},{\pi }_{\mathrm{EE}}),$ where $\tilde{{r}_{{\rm{E}}}}$ is the angular radius of the Einstein ring of the lens projected onto the observer plane and π_EN and π_EE are the components of this vector in the north and east directions respectively. In practice, the introduction of this parameter slightly changes the value of the impact parameter and $\tau =(t-{t}_{o})/{t}_{{\rm{E}}}.$ Strong modifications of the light curve can be seen far from the peak of the event, i.e., in the wings of the light curve, with few changes around the peak; see, for example, Smith et al. (2003). To model this effect, the constant ${t}_{o,\mathrm{par}}$ (Skowron et al. 2011) is added to give an invariant reference time for each model. We choose t_o,par = 2456446.0 HJD for our models.

Since this event is so highly magnified, it should also be possible to measure the terrestrial parallax. Hardy & Walker (1995) first introduced the idea that for an "Extreme Microlensing Event," the difference in longitudes of observatories should result in light curves where tiny changes in the line of sight toward the target become apparent, allowing a measurement of the Einstein ring (Holz & Wald 1996; Gould 1997; Dong et al. 2007). Again, this effect is described by the parallax vector ${\pi }_{{\rm{E}},t}=\mathrm{AU}/{r}_{\oplus }({\rm{\Delta }}{t}_{o}/{t}_{{\rm{E}}},{\rm{\Delta }}{U}_{o}),$ where r_⊕ is the Earth radius. Gould & Yee (2013) estimated that the condition $\rho \tilde{{r}_{{\rm{E}}}}\leqslant 50{r}_{\oplus }$ is required to expect a measurable difference in terms of magnification. This condition leads to π_E > 0.24 for this event by using an approximate value for the normalized source star radius ρ ∼ 5 × 10⁻⁴. A summary of longitudes and latitudes of the observatories is in Table 1 and results are summarized in Table 3.

Note that we also compute the annual parallax model for a positive impact parameter (u₀ > 0) and found no significant difference with the model reported in the Table 3. This is the u₀ degeneracy described in the literature (Smith et al. 2003; Gould 2004; Skowron et al. 2011). For the terrestrial parallax, a positive impact parameter leads to a better fit (Δχ² ∼ 60) for equivalent π_EN and π_EE values , which is a similar result to Yee et al. (2009).

At the end of the 2013 observing season, several planetary models circulated (private communication) indicating the presence of the smallest microlensing planet ever detected (q ∼ 2 × 10⁻⁵). In order to investigate these claims of the existence of a very low-mass ratio planetary companion to the primary lens, and to exclude the possibility of a false alarm, we used a finite-source binary lens model (FSBL) with three extra parameters: the projected separation (normalized by θ_E) between the two bodies s, the mass ratio q using the convention described in Bachelet et al. (2012a) (the most massive component on the left), and α the source trajectory angle measured from the line joining the two components (counterclockwise angle). We first used a grid search and we finally explore minima with a full Markov-Chain Monte Carlo algorithm. Please see, for example, Dong et al. (2009) or Bachelet et al. (2012a) for more details. We find two local minima which correspond to the known theoretical degeneracy $s\iff {s}^{-1}.$ We only explore the "wide" solution, which gives the best grid-search χ², for three reasons. First, as explained in the next section, the reliability of the planetary model is not clear. Second, we expect a strong degeneracy in terms of $s\iff {s}^{-1},$ so models should converge to solutions with similar shapes for the central caustic and therefore similar residuals. Finally, due to the really small value of ρ, modeling this event is very time consuming. We present our results in Table 3, our best caustic-crossing geometry in Figure 2, and redisuals to the FSPL model for data sets covering the magnification peak are plotted in Figure 3.

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Our best planetary model claims the detection of smooth deviations in the light curve away from the FSPL model at a peak-to-peak level of ≤1%, which is supposedly caused by the source passing over the central planetary caustic. It is well known to photometrists, however, that from ground-based telescopes the photometric precision at this level can be affected by systematic trends (or red noise) in the data.

In the early days of planet hunting using the transit method, researchers were confounded as to why they were not finding as many planets as predicted. The predictions were of course based on simulated light curves taking into account stochastic noise from the photons (sky and star) and the charge-coupled device (CCD), but ignoring the effects of sub-optimal data calibration/reduction that introduce correlated noise (e.g., Mallén-Ornelas et al. 2003 and Pepper & Gaudi 2005). It was soon realized that transit detection thresholds were severely affected by systematic trends in the light curves (Pont et al. 2006; Aigrain & Pont 2007) with the knock-on effect of reducing the predicted planetary yield of a transit survey, and at least partially explaining the unexpectedly low rate of transiting planet discoveries. The microlensing planet hunters face a similar problem for detecting low-amplitude (≲1%) planetary deviations in microlensing light curves, especially when no "sharp" light curve features, caused by caustic crossing events, are predicted/observed. However, the microlensing community is now aiming for really low amplitude signal detection which requires extra care in the treatment of systematic errors (Yee et al. 2013).

Systematic trends in light curve data can be caused by an imperfect calibration of the raw data and sub-optimal extraction of the photometry. For instance, on the calibration side, flat fielding errors which vary as a function of detector coordinates can induce correlated errors in the photometry as the telescope pointing drifts slightly during a set of time-series exposures. On the software side, systematic errors in the photometry can be caused by errors in the point-spread function (PSF) model used during PSF fitting for example. Also the airmass and transparency variations in the data sets should be modeled in the DIA procedure by the photometric scale factor. However, there is no guarantee that the DIA modeling is perfect and this can create systematics trends in the data. As recently discussed by Bramich et al. (2015), an error _p in the estimate of the photometric scale factor leads directly to an error _p in the photometry. For example, the passage of clouds during data acquisition can create inhomogeneous atmospheric transparency in the frames and lead to a spatially varying photometric scale factor. The estimation of the photometric scale factor in DIA by using a "mean" value for the whole frame will produce different systematic trends for each star in the field of view. In practice, the expected error _p is of order of a few per cent, which is non-critical for the majority of microlensing deviations, but can easily imitate the smallest such as in OGLE-2013-BLG-0446.

Obtaining a photon-noise limited data calibration and photometric extraction is not always feasible. Therefore complementary techniques have been developed to perform a relative calibration of the ensemble photometry after the data reduction (i.e., a post-calibration). These techniques can be divided into two broad groups, namely, detrending methods that do not use any a priori knowledge about the data acquisition or instrumental set up (e.g., Tamuz et al. 2005), and photometric modeling methods that attempt to model the systematic trends based on the survey/instrumental properties (e.g., Honeycutt 1992; Padmanabhan et al. 2008; Regnault et al. 2009). Each data point is associated with a unique object and a unique image (epoch), and carries associated metadata such as magnitude uncertainty, airmass, (x, y) detector coordinates, PSF FWHM, etc. To investigate the systematic trends in the photometry, we first identified a set of object/image properties which we suspected of having influenced the quality of the data reduction. For each of these quantities, we defined a binning that covers the full range of values with an appropriate bin size. For each bin, we introduced an unknown magnitude offset to be determined, the purpose of which is to model the mean difference of the photometric measurements within the corresponding bin from the rest of the photometric measurements. We constructed our photometric model by adopting the unknown true instrumental magnitude of each object⁶³ and the magnitude offsets as parameters. Since the model is linear, the best-fit parameter values corresponding to the minimum in χ² may be solved for directly (and in a single step using some matrix algebra—see Bramich & Freudling 2012). Iteration is of course mandatory to remove variable stars and strong outliers from the photometric data set. A valid criticism of this method is that the systematic trends are derived from the constant stars but then applied to all stars including the variable stars (the microlensing event in our case). The question arises as to whether this approach is consistent. To argue our case, we are limited to showing that the method works in practice and we direct the reader to Figure 1 of Kains et al. (2015) where RR Lyrae light curves in M68 are much improved by this self-calibration method. We used this method to analyze systematic trends in three data sets. They are listed below.

We opted to employ the above methodology in order to investigate and understand the systematic trends in the LSCB_i and LSCA_i data sets using the algorithms described in Bramich & Freudling (2012).⁶⁴ These telescopes are twins: both are LCOGT 1 m telescope clones, both supporting Kodak SBIG STX-16803 CCDs at the time of these observations. SDSS-i prescription filters manufactured at the same time were use to observe OGLE-2013-BLG-0446 during the same period of observation, though not precisely synchronously.

We first chose to study LSCB_i because this telescope most strongly favors the planetary model (Δχ² ∼ 128.3; see Table 6). For LSCB_i, the DanDIA pipeline extracted 4272 light curves from the images in the LSCB_i data set, each with 378 data points (or epochs), which yields a total of 1,614,816 photometric data points. We investigated each object/image property in turn using the above method, and determined the peak-to-peak amplitude of the magnitude offsets in each case. The results are reported in Table 4. The trends in the photometry were found to be at the sub-mmag level for all correlating properties except for the epoch (2.0 mmag). The magnitude offsets determined for each epoch (or image) serve to correct for any errors in the fitted values of the photometric scale factors during DIA. The magnitude offsets as a function of detector coordinates (commonly referred to as an illumination correction—e.g., Coccato et al. 2014) were modeled using a two-dimensional cubic surface (as opposed to the binning previously described) so as to better capture the large-scale errors in the flat-fielding. The peak-to-peak amplitude of the cubic surface over the full detector area was found to be ∼60 mmag, but since the LSCB_i observations only drifted by ∼50 pixels in each coordinate, we found that the magnitude offsets applicable to the OGLE-2013-BLG-446 light curve have a peak-to-peak amplitude of only ∼0.2 mmag. This can be seen in Figure 4. The overall level of systematic trends in the LSCB_i data set for OGLE-2013-BLG-446 is ∼2.0 mmag. To conclude, this analysis reveals that the illumination correction is not sufficient to explain the observed systematics.

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We chose also to study the LSCA_i data set because this telescope observed the target at the same time but does not show any planetary significance (Δχ² ∼ 20.7). We conduct the same study and the results are summarized in Table 4 and Figure 4. The peak-to-peak amplitudes of the magnitude offsets are 10 times bigger than for the LSCB_i data set. It is surprising to see how two similar intruments can lead to such different data quality. A more careful check of the frames clearly shows a problem in the focus for the LSCA_i telescope. Because the Galactic Bulge fields are very crowded, this is a critical point for microlensing observations (e.g., increasing the blending). A plausible explanation for this difference between the twins is that during the time for observations, the telescopes were under commissioning, leading to non-optimal performance for LSCA_i.

We conducted the same study for the Auckland_R data set because this telescope presents the clearest feature that mimics a planetary deviation, around HJD ∼ 6446.02, as can be seen in Figure 3. Results can be seen in Table 4. Because the pointing for this data set was extremely accurate (offset less than 2 pixels for the whole night of observation), the estimation of the illumination correction was not possible. There is not enough information in the matrix equations and they are degenerate. But this reflects the fact that the pointing did not induce systematic trends. However, a clear variation in the magnitude offset at each epoch is visible at the time of the deviation. Furthermore, we find that this offset is stronger for the brighter stars, as can be seen in Figure 5. There are strong similarities between the FSPL residuals and the magnitudes of the two brightest stars around the time of the anomaly HJD ∼ 6446.02, especially when the FSPL residuals get brighter at HJD ∼ 6446.03. Because the microlensing target is by far the brightest object in our field, we can expect that this systematics effect is probably even larger in our target. For this data set, we slightly modified our strategy by computing the offset at each epoch only for the brightest stars (mag < 18) and we rejected the microlensing target from the computation. Also, as can be seen in the bottom panel of Figure 5, the photometric scale factor shows variations during the night. This indicates the passage of clouds which can lead to systematics errors, as described previously. For example, the FSPL residuals in the interval 6446.01 ≤ HJD ≤ 6446.02 clearly share the pattern with the photometric scale factor.

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For the three studied data sets (LSCA_i, LSCB_i and Auckland_R), we corrected the systematics for the quantity that yielded the largest peak-to-peak amplitude in the magnitude offsets: namely the epoch. Moreover, this quantity is correlated with other quantities (airmass for example) and so the epoch correction should decrease the systematic trends measured for the other paramters listed in Table 4. We checked this and found that for the three data sets, the epoch correction leads to a significant improvement (order of a factor 10) for the systematics of the correlated quantities. This is a first order correction and we wanted to see the impact on the different models we analyzed. We repeated our modeling process with these new data sets using the previous models as starting points. The results are presented in Table 5. The new Auckland_R residuals can be seen in Figure 6. After correction, the amplitude of the "anomaly" is smaller but still exists. This is probably due to the fact that the amplitude of this feature in the light curves is brightness dependent, and the microlensing target is much brighter than all of the other stars, leading to an insufficient correcxtion fot the microlensing event.

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As can be seen in Table 6, the correction of the systematics has a significant impact on the LSCA_i and Auckland_R data sets, which appear to suffer the most systematics. Note also that the planetary model is more significant after systematics correction, especially for these two telescopes. Even though the planetary model changes slightly before and after correction (the new s and q values are outside the error bars of the uncorrected data sets model), the caustic crossing is virtually unchanged (e.g., the central caustic is similar). However, the clearest signature of the planetary anomaly is still in the Auckland_R data set around HJD ∼ 6446.02.

Table 6.  χ² and rms of FSPL Residuals for Each Data Set Before and After Correction of Systematics

Telescope	rms	Raw Data			rms	Corrected
	(mag)	${\chi }_{\mathrm{FSPL}}^{2}$	${\chi }_{\mathrm{Pla}}^{2}$	Δχ2	(mag)	${\chi }_{\mathrm{FSPL}}^{2}$	${\chi }_{\mathrm{Pla}}^{2}$	Δχ2
OGLE_I	0.029	770.308(459)	655.364(456)	114.944	0.029	765.677(459)	661.426(456)	104.251
OGLE_V	1.348	23.998(20)	24.005(17)	−0.007	1.348	23.998(20)	23.979(17)	0.019
Canopus_I	0.007	191.996(128)	132.410(125)	59.586	0.007	204.40(128)	137.769(125)	66.631
Auckland_R	0.006	142.839(103)	128.294(100)	14.545	0.006	112.917(103)	72.331(100)	40.586
LSCB_i	0.007	446.170(374)	317.883(371)	128.287	0.007	442.338(374)	311.049(371)	131.189
LSCA_i	0.015	399.723(381)	379.065(378)	20.658	0.013	173.683(381)	133.422(378)	40.261
CPTA_i	0.023	21.973(18)	21.926(15)	0.047	0.023	21.972(18)	21.925(15)	0.047
CTIO_I	0.006	159.174(108)	171.347(15)	−12.173	0.006	158.986(108)	167.376(105)	−8.48
CTIO_V	0.005	2.986(9)	5.385(6)	−2.399	0.005	3.036(9)	5.227(6)	−2.191
Danish_z	0.017	448.804(448)	470.960(445)	−22.156	0.017	450.589(448)	471.301(445)	−20.712
MOA_Red	0.785	727.979(450)	725.116(447)	2.863	0.783	727.760(450)	721.763(447)	5.997
Possum_N	0.012	355.767(240)	312.101(237)	43.666	0.012	354.002(240)	324.301(237)	29.701
Salerno_I	0.020	20.016(16)	18.911 (13)	1.105	0.020	19.990(16)	18.872(13)	1.118
Turitea_R	0.005	29.565(27)	29.341(24)	0.224	0.005	29.539(27)	28.965(24)	0.574
Weizmann_I	0.034	92.383(56)	92.667(53)	−0.284	0.034	92.389(56)	92.649(53)	−0.260
SAAO_I	0.014	67.100(54)	66.442(51)	0.658	0.014	67.085(54)	66 .487(51)	0.598
Total		3900.781	3551.217	349.564		3647.99	3258.842	389.157

Note. The three corrected data sets are rendered in bold. Numbers in parentheses are the degrees of freedom for each model/data set.

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Due to strong finite source effects around a very small central caustic, the suspected planetary signature in OGLE-2013-BLG-0446 is very small. First of all, the low χ² improvement (Δχ² ∼ 350 and Δχ² ∼ 389 before and after the sysematics correction respectively) of the planetary model is far from the minimum value generally adopted in microlensing for a safe detection (Yee et al. 2013). Note also that even though the caustic crossing is similar, the two planetary models are not fully equivalent. As defined by Chung et al. (2005), the R_c parameter is the ratio of the vertical length and the horizontal length of a central caustic. This caustic parameter before and after correction is significantly different (∼30%). Second, the highest Δχ² contributor (LSCB_i) presents photometric systematics at the same level as the planetary deviations (2 versus 6 mmag). As can be seen in Figure 6, the systematics correction decreases the amplitude of the "anomaly" in the Auckland_R data set and it is therefore better fit by the planetary model. However, the increase in the FSPL residuals after HJD ∼ 6446.02 (from 1% to zero) is not explained by the planetary model, but similar behavior is seen in other bright stars. This clearly indicates that bright stars suffer from systematic effects in this data set which were not revealed by the different quantities we studied.

The planetary model is highly favored by the Canopus_I data set (Δχ² ∼ 60). However, a closer look at Figure 3 reveals that the planetary deviations are at a very low level (≤0.5%). It cannot be excluded that the FSPL model correctly fits this data set and that this telescope also suffers from low level systematics errors. We however decided to not realize the same study of photometric systematic errors for the Canopus_I data set because there are no obvious deviations in the FSPL residuals and also because enough doubt has already been cast on the planetary model we found.

All these points reveal strong doubts about the reality of the planetary signature in OGLE-2013-BLG-0446. Even if we cannot firmly guarantee that the planet is not detected, we prefer to stay conservative and claim that we do not detect a planet in this event.