Gravitational microlensing is a phenomenon that allows us to observe the dark remnants of stellar evolution even if these bodies are no longer emitting electromagnetic radiation. In particular it can be useful to observe solitary neutron stars or stellarmass black holes providing a unique window through which to understand stellar evolution. Obtaining direct mass measurements with this technique requires precise observations of both the change in brightness and the position of the microlensed star. The European Space Agencys Gaia satellite can provide both. Using publicly available data from different surveys we analysed events published in the Gaia Data Release 3 Gaia DR3 microlensing catalogue. Here we describe our selection of candidate dark lenses where we suspect the lens is a white dwarf WD a neutron star NS a black hole BH or a massgap object with a mass in the range between the heaviest NS and the least massive BH. We estimated the mass of the lenses using information obtained from the bestfitting microlensing models source star Galactic model and the expected parameter distributions. We found eleven candidates for dark remnants one WDs three NSs three massgap objects and four BHs.
['gravitational lensing: micro', 'techniques: photometric', 'stars: black holes', 'stars: neutron', 'white dwarfs', 'Astrophysics - Astrophysics of Galaxies']
Dark lens candidates from Gaia Data Release 3
{'Dark lens candidates from Gaia Data Release 3': "K. Kruszy'nska 1 , 2, ⋆ , Ł. Wyrzykowski 1 , K. A. Rybicki 3 , K. Howil 1 , M. Jabło'nska 1 , 4 , Z. Kaczmarek 5 , N. Ihanec 1 , M. Maskoli¯unas 6 , M. Bronikowski 7 , U. Pylypenko 1 , A. Udalski 1 , P. Mróz 1 , R. Poleski 1 , J. Skowron 1 , M. K. Szyma'nski 1 , I. Soszy'nski 1 , P. Pietrukowicz 1 , S. Kozłowski 1 , K. Ulaczyk 8 , P. Iwanek 1 , M. Wrona 1 , M. Gromadzki 1 , M. J. Mróz 1 , F. Abe 9 , K. Bando 10 , R. Barry 11 , D. P. Bennett 11 , 12 , A. Bhattacharya 11 , 12 , I. A. Bond 13 , A. Fukui 14 , R. Hamada 10 , S. Hamada 10 , N. Hamasaki 10 , Y. Hirao 15 , S. Ishitani Silva 11 , 16 , Y. Itow 9 , N. Koshimoto 10 , Y. Matsubara 9 , S. Miyazaki 17 , Y. Muraki 9 , T. Nagai 10 , K. Nunota 10 , G. Olmschenk 11 , C. Ranc 18 , N. J. Rattenbury 19 , Y. Satoh 10 , T. Sumi 10 , D. Suzuki 10 , P. J. Tristram 20 , A. Vandorou 11 , 12 , and H. Yama 10  \n(A ffi liations can be found after the references)  \nReceived xxxx; accepted xxxx", 'ABSTRACT': "Gravitational microlensing is a phenomenon that allows us to observe dark remnants of stellar evolution even if they no longer emit electromagnetic radiation. In particular, it can be useful to observe solitary neutron stars or stellar-mass black holes, providing a unique window through which to understand stellar evolution. Obtaining direct mass measurements with this technique requires precise observations of both the change in brightness and the position of the microlensed star. The European Space Agency's Gaia satellite can provide both. Using publicly available data from di ff erent surveys, we analysed events published in the Gaia Data Release 3 ( Gaia DR3) microlensing catalogue. Here we describe our selection of candidate dark lenses, where we suspect the lens is a white dwarf (WD), a neutron star (NS), a black hole (BH), or a mass-gap object, with a mass in a range between the heaviest NS and the least massive BH. We estimated the mass of the lenses using information obtained from the best-fitting microlensing models, the source star, the Galactic model and the expected distribution of the parameters. We found eleven candidates for dark remnants: one WDs, three NS, three mass-gap objects, and four BHs.  \nKey words. Gravitational lensing: micro - Techniques: photometric - white dwarfs - Stars: neutron - Stars: black holes", '1. Introduction': "There are still many outstanding questions connected with the remnants of stellar evolution. The most common stellar remnant is a white dwarf, and more than 95 per cent will become one by the end of their life (Fontaine et al. 2001). Our understanding of white dwarfs was expanded in recent years by Gaia and its superb parallaxes. The largest catalogue consists of over 350'000 high-confidence WD candidates, expanding almost ten times the amount of known WDs before Gaia (Gentile Fusillo et al. 2021). The best catalogue of known pulsars is two orders of magnitude smaller than the one we have for WDs in our Galaxy (Manchester et al. 2005). Our most limited observational material is on BHs, in particular solitary ones. Most of the known BHs are linked to binary systems found either through X-ray emission due to accretion of their companions (e.g. Corral-Santana et al. 2016) or as gravitational wave sources due to their merger (e.g. Abbott et al. 2019). Additionally, gravitational wave mergers are detected most frequently in distant galaxies. Recently, Shenar et al. (2022), El-Badry et al. (2023b), and El-Badry et al. (2023a); Chakrabarti et al. (2023) have reported on BH candidates also detected as non-interacting binary systems. However, the only known direct mass measurement for a solitary stellar-mass BH was recently presented for OGLE-2011-BLG0462 / MOA-2011-BLG-191 (Lam et al. 2022; Sahu et al. 2022; Mróz et al. 2022; Lam & Lu 2023) using the gravitational microlensing phenomenon.  \nGravitational microlensing is an e ff ect of Einstein's General Relativity, which occurs when a massive object passes in front  \nof a distant star within the Milky Way or its neighbourhood (Einstein 1936; Paczynski 1986). In contrast to strong gravitational lensing, here the separated, deformed images of the source are typically impossible to spatially resolve unless the world's largest telescopes are used, and only in case of very bright events (Dong et al. 2019; Cassan et al. 2022). Instead, what can be observed is a brightening of the source occurring during the event. Images of the source, though di ffi cult to resolve, are unequally magnified and change position. This causes a distinctive shift in the centroid of light called astrometric microlensing (Dominik & Sahu 2000; Belokurov & Evans 2002). This e ff ect can be measured with precise enough observatories like Hubble Space Telescope (HST) (Sahu et al. 2017), or Gaia 1 .  \nCombining both e ff ects allows the mass of the lens M L to be measured following Gould (2000):  \nM L = θ E κπ E , (1)  \nwhere κ = 4 G / c 2 au ≈ 8 . 144mas / M ⊙ , θ E is the angular Einstein Radius, which can be measured with astrometric microlensing, and π E is the microlensing parallax obtained from modelling the time-series photometry. A combination of these two e ff ects was used to detect a stellar-mass BH for the first time in Lam et al. (2022) and Sahu et al. (2022), who used astrometric observations from HST and photometric observations from the ground.  \nHowever, even without a measurement of the angular Einstein radius, we can still estimate the mass of the lens by employing the Galactic model and expected distributions of lens  \nparameters. We can obtain a posterior distribution for the lens mass and distance using the microlensing parallax, proper motion measurements, estimated distance to the source, the Galactic model and assumed mass function of stellar remnants (e.g. Wyrzykowski et al. 2016; Mróz & Wyrzykowski 2021). This method was used for objects observed by the OGLE survey where no Einstein radius information was available, where events exhibited clear parallax signal (e.g. Wyrzykowski & Mandel 2020; Mróz et al. 2021). The same technique could also be applied for microlensing events seen by Gaia (Gaia Collaboration et al. 2016), using both archival data and transients detected as part of Gaia Science Alerts (GSA) system (Hodgkin et al. 2021). This paper presents a similar analysis of Gaia Data Release 3 ( Gaia DR3) microlensing catalogue (Wyrzykowski et al. 2023).  \nThis work is split into six sections. Section 2 presents the microlensing models compared in this work, the criteria of event pre-selection and the sources of data used for this analysis. Section 3 explains the criteria to select events for detailed analysis, while section 4 summarises those results. Section 6 shows how we estimated the masses and distances to the lenses. Section 7 discusses the obtained results and summarises this work.", '2.1. Compared models': "In this paper, we focused only on events that could exhibit the microlensing parallax e ff ect, which occurs when the observer changes position during the event. There are three types of microlensing parallax: annual, terrestrial and space. The annual microlensing parallax is connected to the Earth's movement around the Sun. The observer on Earth changes their position during the entire year, which creates distinctive asymmetry and, in some cases, wobbles in the light curve (Gould 1992; Alcock et al. 1995; Maskoli¯unas et al. 2023). The terrestrial parallax is connected to the di ff erent positions of the observatories on Earth. It is measurable only in the most extreme cases, such as catching a caustic crossing with telescopes on two sites distant from each other (Hardy & Walker 1995; Holz & Wald 1996; Gould et al. 2009). Finally, space parallax occurs when the event is observed from observatories located on Earth and in space. When the space observatory is located as far as one au from the Earth, it can cause a significant di ff erence in the amplification and time of the peak of the lens (Refsdal 1966; Specht et al. 2023). It can be also measured if the space observatory is closer but during a caustic crossing (Wyrzykowski et al. 2020) or if the event is densely covered. This is the main mechanism behind the way that the Nancy Grace Roman Space Telescope is going to be used for mass measurements of the observed lenses (Penny et al. 2019).  \nGaia DR3 microlensing events catalogue contains events which were most likely caused by a single object as an outcome of the used pipeline (Wyrzykowski et al. 2023). All of the events within this catalogue were detected in the Galactic plane, which is a dense field, especially within the Galactic bulge. This means that we had to include blending when some of the light is coming from the stars near the line of sight towards the source and lens. In the case of microlensing, blending also factors in that the lens is luminous in the majority of cases.  \nWe used the following models in our analysis with these parameters:  \n- -point source - point lens (PSPL) model without blending, parameterised by t 0, u 0, t E, I 0;  \n- -PSPL with blending, parameterised by t 0, u 0, t E, I 0, fb ;\n- -PSPL model with parallax e ff ect without blending, parameterised by t 0, u 0, t E, I 0, π EN, π EE;\n- -PSPL model with a parallax e ff ect with blending, parameterised by t 0, u 0, t E, π EN, π EE, I 0, fb ;  \nwhere t 0 is the time of the peak of brightness, u 0 is the impact parameter at t 0, and t E is the Einstein timescale when the source is crossing the angular Einstein ring. Microlensing parallax is described by its northern and eastern components π EN and π EE. The baseline magnitude of the event is denoted by I 0 and the blending parameter is defined as fb = Fb Fs + Fb , where Fs is the source flux and Fb is the blend flux.  \nIn this work, we used models without blending in the preselection stage, and for each event, we fitted models with and without parallax. We used models with blending when we fitted each event individually. At this stage, we also fitted models with and without parallax. Each event should have at least two bestfitting solutions: PSPL without blending, and PSPL with parallax and without blending.", '2.2. Pre-selection of the candidate events': "The Gaia Data Release 3, or alternatively table vari\\_microlensing of the Gaia DR3, contains 363 candidate events. Many of them do not exhibit second-order e ff ects and are best described by the standard Paczy'nski model. We suspected that events with short Einsten timescales are less likely to be a ff ected by the annual movement of the Earth around the Sun. Thus we selected events with paczynski\\_0\\_te timescale larger than 50 days. This was an arbitrary cut, based on the fact that Gaia produces on average one point per month per source. An event with an Einstein timescale of 50 days would last more than 100 days, allowing for at least 3 observations during the event. Additionally, previous studies of candidate parallax events show that in most cases parallax is not detectable for shorter events (see for example Rodriguez et al. (2022) and Zhai et al. (2023)). After applying this cut, we were left with 204 candidate events to analyse.", '2.3. Data': 'The Wyrzykowski et al. (2023) catalogue was built using only Gaia G , G BP and G RP photometry, but for this work, we utilised data available from other surveys. In particular, we wanted to include information from microlensing surveys which have better cadence, especially in the Galactic bulge. We have crossmatched the Gaia sources with the OGLE survey (Udalski et al. 1992, 2015a). We have found 145 events in common with the public OGLE events. 130 events were published as a part of the OGLE-IV analysis of microlensing optical depth in the Galactic plane (Mróz et al. 2019, 2020). 78 events were published were also published as OGLE Early Warning System alerts (Udalski et al. 2015b) 2 , overlapping with the 130 events coming from the OGLE-IV papers. We downloaded all publicly available data. If the event was published in OGLE-EWS, Mróz et al. (2019) or Mróz et al. (2020), we used the data shared with the article. We performed a similar search with MOA survey (Abe et al. 1997; Bond et al. 2001) using its alert stream and we found 20 events in common. We found 32 events in common with KMTNet survey public alerts (Lee et al. 2014; Kim et al. 2016) 3 . Six events  \nwere published by Gaia Science Alerts 4 . These events were published with preliminary photometry and without errors. We simulated the errors using the following formula, (Wyrzykowski et al. 2023):  \nσ G , i = ( √ 30 × 10 0 . 17 × 13 . 5 -5 . 1 , for Gi < 13 . 5 mag , √ 30 × 10 0 . 17 × Gi -5 . 1 , for Gi ≥ 13 . 5 mag , (2)  \nwhere Gi is the i -th point in the GSA light curve. Since the error bars and photometric data had di ff erent properties, they came from di ff erent pipelines, and GSA data was created using raw photometric data. Gaia DR3 light curves were created by the photometric pipeline that was used on all data used for this Data Release and produced the most accurate light curves we have. We decided to treat them as a di ff erent dataset. Seven events were found in the publicly available data of the ASAS-SN survey (Shappee et al. 2014). Two were published as alerts: ASASSN16li and ASASSN-16oe (Strader et al. 2016; Munari et al. 2016), and one was published as an ATEL (Jayasinghe et al. 2017). The rest was found in the ASAS-SN Photometric Database (Jayasinghe et al. 2019). We did not include Zwicky Transient Facility (ZTF) (Bellm et al. 2019) while cross-matching events, because this survey started after May 2017, which was the end of Gaia DR3 timespan. We did however check for sources appearing in the 9th Data Release of ZTF if a given source brightened only once. The list of all 204 sources with their names in other surveys is available in Table A.1.  \nIn the case of MOA and KMTNet, we have used the available photometry published in fluxes, instead of magnitudes. KTMNet is a network of three robotic telescopes, located in Australia, South Africa, and Chile. These sites have di ff erent weather conditions, and when we used KMTNet DIA photometry, we separated each light curve by the observatory. For Gaia photometry, we followed Wyrzykowski et al. (2023), and modified the available uncertainties to match the method used to find candidate events.  \nAll data sources listed above were then used either at the preliminary or the detailed event modelling stages or both. We provide data used for this stage in a machine-readable online archive 5 .', '3. Selection of candidate events for further analysis': "To find preliminary models, we used the MulensModel package Poleski & Yee (2019) to generate microlensing models, and the pyMultiNest package (Feroz et al. 2009; Buchner et al. 2014) to find the best fitting solutions. To simplify the parameter space explored by the pyMultiNest package, we calculated models without blending. pyMultiNest provides a Python interface for a nested sampling algorithm which returns the best solutions for probability densities containing multiple modes and degeneracies. This made it a perfect tool for comparing models including microlensing parallax. For the parallax model, we included both the annual and space e ff ects. Gaia is located in space, and there may be an o ff set between observatories.  \nWe have recorded the four best solutions for models with and without parallax and compared their χ 2 values. These solutions are available in machine-readable format. Using preliminary models we selected events, that:  \n- -had Einstein timescale of the best PSPL solution larger than 50 days, and  \n- -the di ff erence of χ 2 per degrees-of-freedom of the best PSPL model and the best parallax model should be larger than one ( χ 2 PSPL / dof -χ 2 Par / dof > 1).  \nThis way we selected 34 events. We removed two events from this sample. For the first one ( Gaia DR3-ULENS-024), we didn't have a full light curve, and the event did not finish before the end of the Gaia DR3 period. The second one ( Gaia DR3ULENS-178) turned out to be a binary event, when we inspected the MOA light curve. We decided to add three additional events, that had ASAS-SN data ( Gaia DR3-ULENS-023, Gaia DR3-ULENS-032, and Gaia DR3-ULENS-118). In these cases, the automatic algorithm struggled to find a correct solution, and we concluded that was caused by the vastly di ff erent pixel size of the ASAS-SN, compared to Gaia and other surveys, and the exclusion of blending in fitted models.", '4. Detailed analysis of selected events': 'We conducted a case-by-case analysis of the 35 events selected in the previous step. We used MulensModel to generate the microlensing models and emcee (Foreman-Mackey et al. 2013) to explore the parameter space. In this step, we used the KMTNet pySIS photometric data in magnitudes. For KMTNet data, when possible, we pre-processed data, removing any points that had a negative value of FWHM column or with photometric uncertainty larger than 1 magnitude. For OGLE and MOA events, we used re-processed data coming from the end-of-the-season DIA photometric reduction pipelines (Udalski et al. 2015a; Bond et al. 2001). We also applied a correction procedure for uncertainties following Skowron et al. (2016) to OGLE data from bulge fields. Finally, we re-scaled the photometric uncertainties using this formula:  \nσ \' n , i = kn σ n , i , (3)  \nwhere σ \' n , i is the re-scaled i -th uncertainty of the n -th telescope\'s light-curve, kn is the scale factor for the n -th telescope, and σ n , i is the original i -th uncertainty of the n -th telescope\'s light-curve. We obtained the scale factors in the following manner:  \n- 1. we fitted preliminary PSPL models with and without parallax;\n- 2. we selected a model with the smallest χ 2 value;\n- 3. using this model we used it as a starting point, we ran an MCMCfit with scale factors as one of the fit parameters.  \nIf the scale factor of the median solution found in the final step exceeded 1.0 for a given telescope, we used this value and then we re-fitted the PSPL models with and without parallax. For events Gaia DR3-ULENS-003, Gaia DR3-ULENS-032, Gaia DR3-ULENS-118, Gaia DR3-ULENS-196, and Gaia DR3ULENS-284 we had to use the PSPL without parallax model instead of the best model to find the scale factors. For event Gaia DR3-ULENS-025, Gaia DR3-ULENS-143, we couldn\'t find scale factors due to poor event coverage. We report the values for each scale factor in A.2. For some events, we had to perform an outlier removal procedure. We have done this before applying the uncertainty re-scaling. We used the best-fitting preliminary model (step 2 of the uncertainty re-scaling procedure). Then we removed all data points outside the 3 σ range of the residuals from the preliminary model for a given light curve. We marked those light curves in bold in Table A.2. In some cases, we had to remove certain light curves, because they were too noisy, or carried little information about the event.  \nWe marked those data sets with a strike-through text in Table A.2. Table A.2 provides the name of the fitted event and a list of data sets with the amount of data points for each light curve. Table A.6 presents the median values of the posterior distributions (PDFs) obtained for best-fitting solutions. KMTNet data for event Gaia DR3-ULENS-047 / KMT-2015-BLG-0157 revealed features around the peak, for which a PSPL model was insu ffi cient in characterising, so it was excluded from further analysis. Gaia DR3-ULENS-284 had a large parallax value, which means other e ff ects should be included. We excluded that event from further analysis. Gaia DR3-ULENS-057 had only three points at magnification, so we could only find a non-parallax solution. We named the solutions with the following convention: " Gaia DR3-AAA-BC", where AAA is the number assigned to the event in Wyrzykowski et al. (2023), B is a string of letters denoting which type of Gaia data was used: "G" for events where we used the Gaia DR3 photometry, and "GSA" for events where we opted for a Gaia Science Alerts lightcurve, finally C is a sign of the u 0 of the solution (" + " for positive and "-" for negative). If there was more than one solution with the same u 0 sign, we numbered them starting from one.  \nTo select the dark lens candidate sample, we applied the following criteria to the modelled events:  \n- -the blending parameter in G band was smaller than 0.3,\n- -the π E was not consistent with zero in the three-sigma range,\n- -the χ 2 of the parallax solution was smaller than the χ 2 of the non-parallax solution.  \nThis way we obtained 14 events, where at least one solution passed those criteria. For these events, we then estimated the lens distance and mass. From the remaining events, a non-parallax model better described two, seven didn\'t pass the blending parameter criterion, and six did not pass the π E criterion. We found four events that passed the blending and χ 2 criterion but failed the π E criterion in the calculated three sigma, however, their π E distribution was inconsistent with 0. We analysed these solutions further but displayed their results in a di ff erent table.', '5. Source stars': "To determine the properties of the lens, we have to determine the properties of the source. Ideally, we would obtain the distance to the source, but that isn't always possible. Instead, we decided to find the angular stellar radius of the source θ ∗ and use it as a prior during lens mass and distance estimation. We followed different procedures, depending on the event location and available information.  \nWe could use the colour-magnitude diagrams (CMDs) calibrated to the OGLE-III data for events with MOA data ( Gaia DR3-ULENS-035, Gaia DR3-ULENS-069, Gaia DR3ULENS-073, Gaia DR3-ULENS-088, Gaia DR3-ULENS-155, Gaia DR3-ULENS-343, Gaia DR3-ULENS-353). First, we determined the red clump centre (RCG) location following the procedure outlined in Nataf et al. (2013). We use the de-reddened RCG distance modulus determined in Nataf et al. (2013) for each event 6 , and find the reddened distance modulus of the RCG using fitted position on the CMD and absolute magnitude of the RCG MI , RCG = ( -0 . 12 , 1 . 06) mag Nataf et al. (2013); Bensby et al. (2013). Then we calculate the extinction in AI and AV and use it to find the de-reddened magnitude and colour of the source. In the case of Gaia DR3-ULENS-353, the blending parameter was negative, so instead of using  \ncalculated source magnitude, we used the baseline magnitudes to determine source brightness and colour. Finally, we use these values to determine the angular stellar radius of the source star using relations from Adams et al. (2018).  \nOther sources were more di ffi cult. If the source was located towards the Galatic Bulge, we assumed that the Bulge is located 8.1 kpc and extends for 2.4 kpc. We used this value to determine the de-reddened distance modulus to the red clump centre. We constructed a CMD in V and I data using Gaia DR3 sources. We selected sources within 30' of the event, that had the Renormalized Unit Weight Error (RUWE) parameter smaller than 1.4, astrometric parallax error not larger than 20% of the measured value, and that had available GSP-Phot solutions (Andrae et al. 2023). Then, we transformed their colours into V and I bands following relations from Busso et al. (2022) Section 5.5.1 7 . We determined the position of the RGC following the procedure in Nataf et al. (2013), and used the calculated distance modulus to find the extinction in I and V bands. Finally, we found the θ ∗ using relation from Adams et al. (2018). We used this method for events Gaia DR3-ULENS-025, Gaia DR3-ULENS089, Gaia DR3-ULENS-142, and Gaia DR3-ULENS-270. For many of these events, some information was missing. When baseline magnitude and blending in G BP and / or G RP filter, we used blending parameter in G (for G BP) or in I (for G RP) bands and Gaia DR3 entry for this source to determine the missing brightness for the θ ∗ estimation. We couldn't use this method for events Gaia DR3-ULENS-331 and Gaia DR3-ULENS-363, because they were too dim compared to the data with GSP-Phot entries to infer the extinction.  \nFinally, for events located towards the Galactic disc, we couldn't determine the de-reddened distance modulus towards the RCG, and therefore θ ∗ . This a ff ected events Gaia DR3ULENS-103, Gaia DR3-ULENS-118, and Gaia DR3-ULENS259.  \nAll determined values can be found in Table A.3.", '6. Estimating lens parameters of the candidate dark events': "We used the same approach as in Wyrzykowski et al. (2016), Mróz & Wyrzykowski (2021) and Kruszy'nska et al. (2022) to estimate the mass and distance to the lens. We dubbed it the DarkLensCode 8 . and we explained this method in greater detail in Howil et al. (2024). The DarkLensCode was used to find the posterior distribution of lens distance and lens mass, using PDFs of the photometric model parameters, and the Galactic model. The final estimates are the median values of obtained mass and distance PDFs. Here we will focus on the presentation and the resulting mass and distance estimates. We present the results in Tables A.4 and, Table A.5.  \nIf we found more than one solution passed the criteria outlined in Section 4, we analysed them separately providing mass estimates for each solution.  \nThe extinction AG was calculated following a method similar to one outlined in Fukui et al. (2019), but we used Gaia DR3 data instead. We selected all sources within a 30' radius with a Renormalised Unit Weight Error parameter smaller than 1.4, parallax uncertainty smaller than 20% of the measured value, and available GSP-Phot solutions (Andrae et al. 2023). Then we  \ncalculated the mean and standard deviation of the ag\\_gspphot in 50 pc bins and fitted a fourth-order polynomial. We used the fitted polynomial as a function of extinction depending on the distance towards the lens or the source. If the distance to the lens or source was larger than 8 kpc, we used the calculated extinction value AG at 8kpc.  \nWe did not know the distance to the source, so we assumed di ff erent maximum and minimum ranges. For events located towards the bulge, we initially assumed that the distance can be between 1 kpc and 12 kpc. For events located towards the disc, we chose the distance between 0.1 kpc and 8 kpc ( Gaia DR3ULENS-118) or 10 kpc ( Gaia DR3-ULENS-259). When available, we used θ ∗ derived in Section 5. First, we randomly selected a distance to the source from the range described above and calculated the source radius R S. In the next step, we found the absolute magnitude of the source star. The procedure depended on the position of the source star in the CMD.  \nIf the source star position on the CMD was located in the main sequence, we assumed the star was a dwarf. Using the tables from Pecaut & Mamajek (2013) provided on the author's website 9 we found the corresponding value of absolute magnitude in G band.  \nIf the source star position on the CMD was located in the red giant clump, we assumed the star was a giant. For the absolute magnitude determination, we followed the information contained in van Belle et al. (2021). First, we found the ( V 0 -K 0) value corresponding to the radius based on the inverted relation presented in the paper with coe ffi cients coming from Table 16. Then we used Equation 4 from van Belle et al. (2021) to find the e ff ective temperature T e ff of the source star. We used the well-known relation L bol = 4 π R 2 S σ SB T 4 e ff to find the bolometric luminosity of the source star. To find the bolometric correction BC G in G band, we followed the recipe provided in Manteiga et al. (2018), Chapter 8.3.3. Finally, we could derive the absolute magnitude in G following Equation 8.6 from Manteiga et al. (2018).  \nWethen found the extinction AG and the observed magnitude of the source star at a selected distance. If the absolute value of the di ff erence between the calculated source magnitude and the observed source magnitude from the microlensing model was smaller than the sum of the source magnitude uncertainty from the model and the derived source magnitude uncertainty, we accepted that source distance value.  \nWe used three mass functions as lens mass priors: Kroupa (2001) function that describes stars:  \nf ( M ) ∼            M -0 . 3 , M ≤ 0 . 08 M ⊙ , M -1 . 3 , 0 . 08 M ⊙ < M ≤ 0 . 5 , M -2 . 3 , 0 . 5 M ⊙ < M < 150 M ⊙ , (4)  \nMróz et al. (2021) function that describes solitary dark remnants in our Galaxy:  \nf ( M ) ∼ ( M 0 . 51 , M ≤ 1 . 0 M ⊙ , M -0 . 83 , 1 . 0 M ⊙ < M < 100 M ⊙ , (5)  \nand a f ( M ) ∼ M -1 corresponding to applying no prior on the lens function. The reported values of the lens mass and distance are median values of the posterior distribution, while their uncertainty is represented by the 16-th and 84-th quantiles.  \nWe noticed that the mass function greatly a ff ects the lens mass estimate. Using the Kroupa (2001) mass function results  \nin lighter lenses at greater distances, and in turn more likely MS stars. In contrast, the Mróz et al. (2021) mass function produced more massive lenses at closer distances. This is because the Kroupa (2001) mass function is steeper and less likely to produce massive lenses.  \nWecompared the brightness of the blend from the microlensing model to the brightness of an MS star at an estimated distance from the lens. We summed the number of solutions where the brightness of the blend was smaller than the MS brightness and divided this number by the number of all solutions. The resulting number was interpreted as the probability that the lens is dark and not an MS star.  \nAll input parameters are available in machine-readable form in an online archive attached to this paper.", '7. Discussion and conclusions': "We found in total 11 lenses for which the probability for the dark lens scenario for at least one solution exceeded 80% when we looked only at the Kroupa (2001) mass function. Eight of them passed all of the criteria imported in Section 4 ( Gaia DR3-ULENS-025, Gaia DR3-ULENS-035, Gaia DR3ULENS-069, Gaia DR3-ULENS-073, Gaia DR3-ULENS-088, Gaia DR3-ULENS-155, Gaia DR3-ULENS-343, and Gaia DR3ULENS-353). Among the solutions that didn't pass the π E criterion, we found three more candidates ( Gaia DR3-ULENS103, Gaia DR3-ULENS-212, Gaia DR3-ULENS-331, and Gaia DR3-ULENS-155, but we accounted for this event in the first group). All but one of these events have Galactic coordinates towards the Galactic Centre. The estimated distance for 16 solutions (eight events) would suggest that they are located in the Galactic disc, rather than the Galactic bulge. For three solutions (two events), the estimated lens distance suggests a Galactic bulge lens. Three events seem to have the lens located within one kiloparsec from Earth. One event located towards the Galactic disc seems to have a lens no closer than 3.1 kpc, which means it would belong to the Scutum-Centaurus Arm of the Milky Way.  \nIf we assumed that all of those 11 events are dark lenses, and instead follow a Mróz et al. (2021) mass function, we would end up with the majority of events with masses in the range of massgap objects and black holes. One event, Gaia DR3-ULENS-035, has mass consistent with a white dwarf, and three ( Gaia DR3ULENS-025, Gaia DR3-ULENS-155, Gaia DR3-ULENS-212) are consistent with a neutron star. Three events, Gaia DR3ULENS-069, Gaia DR3-ULENS-103 and Gaia DR3-ULENS343 overlap with the first mass gap in the one-sigma range. Four objects have larger masses ( Gaia DR3-ULENS-073, Gaia DR3ULENS-088, Gaia DR3-ULENS-331, Gaia DR3-ULENS-353) are consistent with BHs. All BH candidates have large uncertainties. There are issues with mass estimates for two of them: for Gaia DR3-ULENS-073 we struggled with finding the correct position of the RGC, which could result in the wrong source distance, leading to the wrong lens distance. Gaia DR3-ULENS331 belongs to the group which didn't pass the π E criterion in Section 4.  \nWe present these estimates, comparing them to dark remnant mass estimates found through other methods in Figure 3.  \nIn the case of events where Gaia reported a proper motion, we calculated the transverse velocity. We present results for selected nine events in Figure 4. There is one high-velocity events: Gaia DR3-ULENS-212. This event has a relatively short Einstein time scale of 56 days. The estimated mass is consistent with an  \nNS. The proper motion found in Gaia DR3 is not out of the ordinary, so this means that the lens would have to move fast to justify the relative proper motion and the resulting Einstein time scale.  \nAll nine objects seem to separate into two groups: highvelocity, similar to the NS velocity from Hobbs et al. (2005), and low-velocity, similar to the velocity of a solitary BH from Lam & Lu (2023). The velocity doesn't seem to depend on the lens mass, but the error bars are large and it is hard to draw conclusive statements. Moreover, here we are most likely getting an estimate of a velocity coming from a Galactic prior. More accurate estimates will be possible once we obtain an astrometric time series for these events.  \nHere we also measure only the most likely mass of candidate events. We cannot, however, confirm their nature as WDs, NS or BHs until we perform additional observations in other ranges of electromagnetic radiation, especially X-rays and UV. They could be some unusual objects, such as quark stars, or products of primordial black hole (PBH) mergers with other objects. Abramowicz et al. (2018) and Abramowicz et al. (2022) show a mechanism, that could produce a low-mass BH from a moonmass PBH (10 25 g > M PBH > 10 17 g) collision with an NS. Such low-mass BH would be in the mass range of an NS.  \nIt is worth noting, that these are only candidates for dark lenses, and their mass measurement will be possible only when we will include astrometric time series. This data will be available only with Gaia DR4, no sooner than the end of 2025. This is an exciting prospect, as Gaia may allow us to observe previously unseen stellar populations. This becomes even more promising with the approaching start of the Vera C. Rubin Observatory and its Legacy Survey of Space and Time (Ivezi'c et al. 2019), as well as the launch of the Roman mission (Spergel et al. 2015; Akeson et al. 2019). Rubin will allow us to select long-duration microlensing events from the entire Galactic plane, while Roman can provide us with high-cadence astrometric and photometric observations in the Galactic bulge.  \nAcknowledgements. KK would like to thank Monika Sitek, Drs. Etienne Bachelet, Mariusz Gromadzki, Przemek Mróz, Radek Poleski, Milena Ratajczak, Rachel Street, Paweł Zieli'nski, Przemysław Mikołajczyk, as well as Profs. Michał Bejger, Wojciech Hellwing, Katarzyna Małek, and Łukasz Stawarz. This work was supported from the Polish NCN grants: Harmonia No. 2018 / 30 / M / ST9 / 00311, Daina No. 2017 / 27 / L / ST9 / 03221, and NCBiR grant within POWER program nr POWR.03.02.00-00-l001 / 16-00. LW acknowledges MNiSW grant DIR / WK / 2018 / 12 and funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 101004719 (OPTICON-RadioNET Pilot, ORP). The MOA project is supported by JSPS KAKENHI Grant Number JP24253004, JP26247023, JP16H06287 and JP22H00153. This work has made use of data from the European Space Agency (ESA) mission Gaia (https: // www.cosmos.esa.int / gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https: // www.cosmos.esa.int / web / gaia / dpac / consortium). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. We acknowledge ESA Gaia , DPAC and the Photometric Science Alerts Team (http: // gsaweb.ast.cam.ac.uk / alerts). 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Ujazdowskie 4, 00-478 Warszawa, Poland\n- 2 Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117, USA\n- 3 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel\n- 4 Research School of Astronomy and Astrophysics, Australian National University, Mount Stromlo Observatory, Cotter Road Weston Creek, ACT 2611, Australia\n- 5 Zentrum für Astronomie der Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14, 69120 Heidelberg, Germany\n- 6 Institute of Theoretical Physics and Astronomy, Vilnius University, Saul˙etekio al. 3, Vilnius, LT-10257, Lithuania\n- 7 Center for Astrophysics and Cosmology, University of Nova Gorica, Vipavska 11c, SI-5270 Ajdovšˇcina, Slovenia\n- 8 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK\n- 9 Institute for Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601, Japan\n- 10 Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan\n- 11 Code 667, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n- 12 Department of Astronomy, University of Maryland, College Park, MD20742, USA\n- 13 Institute of Natural and Mathematical Sciences, Massey University, Auckland 0745, New Zealand\n- 14 Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\n- 15 Institute of Astronomy, Graduate School of Science, The University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan\n- 16 Oak Ridge Associated Universities, Oak Ridge, TN 37830, USA\n- 17 Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo, Sagamihara, Kanagawa 252-5210, Japan\n- 18 Sorbonne Universitè, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France\n- 19 Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand\n- 20 University of Canterbury Mt. John Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand  \ngdr3-ulens-343  \nFig. 1. Top panel: light curve of the event known as GaiaDR3-ULENS-343, BLG502.29.100629 Mróz et al. (2019), OGLE-2017-BLG-0095, MOA-2017-BLG-160, KMT-2017-BLG-1123. Gaia G data is shown in green, G RP in red, KMTNet data is shown in dark red, violet and aqua for the South African Astronomical Observatory (KMTA), Cerro Tololo Inter-American Observatory (KMTC), and Siding Springs Observatory (KMTS), MOA I and V band data are shown in light green and dark red respectively, and OGLE in light-blue. The four solutions are marked: PSPL without parallax G0 with a black dashed line, and two PSPLs with parallax, G + and G- with red and grey continuous lines respectively. The bracket in the top left shows the χ 2 of di ff erent solutions. Bottom panel: Residuals of the G + model. A Black dashed line marks the G0 and G + models di ff erence, while the dark grey continuous line marks the G + and G- models di ff erence respectively.  \n<!-- image -->  \nFig. 2. Colour-magnitude diagram based using MOA data calibrated to OGLE-III catalogue for event Gaia DR3-ULENS-343. MOA data is displayed in light blue dots. The red dot marks the source position, black dot marks the blended baseline source colour and magnitude. Dashed lines mark the region that we used for estimating RGC, and the dark blue star represents the found RGC position.  \n<!-- image -->  \nFig. 3. Distribution of known masses of WDs, NS and BHs. Light pink marks WDs known from Gaia Gentile Fusillo et al. (2021). In light red we marked NS with known masses coming from John Antoniadis's catalogue (Lattimer 2012; Antoniadis 2013). Objects found by gravitational wave detectors were marked in yellow (Abbott et al. 2019; Abbott et al. 2021a, 2024, 2023, 2021b). In red we marked high mass x-ray binaries (Orosz et al. 2007; Val-Baker et al. 2007; Orosz et al. 2009, 2014; Corral-Santana et al. 2016; Miller-Jones et al. 2021). In light blue, we marked candidates for dark remnants found by microlensing (Sahu et al. 2017; Kaczmarek et al. 2022; Kruszy'nska et al. 2022; Jabło'nska et al. 2022; McGill et al. 2023), including Lam & Lu (2023). In olive, we marked non-interacting dark remnants (Shenar et al. 2022; El-Badry et al. 2023b,a; Chakrabarti et al. 2023; El-Badry et al. 2024; Gaia Collaboration et al. 2024). Black dots mark masses of objects known from this work. Each solution for the event is shown separately. We marked solutions with positive u 0 with circles, and negative with diamonds. Dashed, vertical lines mark di ff erent mass thresholds: the leftmost is the Chandrasekhar mass limit, second to the left is the Tolman-Volko ff -Oppenheimer limit, and two rightmost mark the limits for the theoretical pair-instability supernovae region (Farmer et al. 2020). A solid vertical line marks the conventional limit of 5 M ⊙ for the lightest BHs. Data used to create this plot can be found in https: // github.com / KKruszynska / dark\\_lens\\_plots / tree / main.  \n<!-- image -->  \nFig. 4. Transverse velocities v t , L estimated for nine candidate events using DarkLensCode and Gaia proper motion measurements. M L is the mass of the lens. Black dots represent estimates of the masses and velocities of the nine candidates with error bars as lines, and each solution is plotted separately. We marked solutions with positive u 0 with circles, and negative with diamonds. The red line represents the median transverse velocity of NS from Hobbs et al. (2005), with a light red rectangle representing their dispersion. The dark blue rectangle represents the mass and transverse velocity of the BH from Lam & Lu (2023).  \n<!-- image -->", 'Appendix A: Additional tables': 'In this appendix, we provide five tables:  \n- -Table A.1, which is the result of the cross-match between the preliminary sample of 204 Gaia DR3 microlensing events and other surveys;\n- -Table A.2, which is the list of data sets used to obtain the final models for each of the 35 analysed events;\n- -Table A.3, which contains information about the source star;\n- -Table A.4 with the lens mass and distance estimates of the 14 candidate dark lens microlensing events;\n- -Table A.5 with the lens mass and distance estimates of the five candidate dark lens microlensing events, that didn\'t pass the π E criterion, but were chosen to be analysed further.\n- -Table A.6, which is the list of parameters of best-fitting solutions of the 35 analysed events. This paper only provides the baseline magnitude and blending parameter for the G -band. A full, machine-readable version of this table, with all parameters, is available online.  \nTable A.1. Table of the results of cross-match between 204 analysed Gaia DR3 events with other surveys. A full, machine-readable version of this table is available at the CDS.  \nTable A.2. The list of data sets used to obtain the final models for each of the 35 analysed events. The description of the columns is at the end of the table.  \nTable A.2. continued.  \nTable A.2. continued.  \nNotes: The table contains the event name, the name of the survey and the filter, the number of points used to obtain the best models, and the scale factor k used to re-scale the photometric uncertainties. Light curves not used in the modelling process are marked with a strike-through. Light curves that were cleaned using the procedure outlined in Section 4 are marked in bold.  \nTable A.3. Source star properties. The description of the columns is at the end of the table.  \nNotes: The table contains the event name, its equatorial coordinates (RA, dec), proper motion in Right Ascension µ RA and declination µ dec from Gaia DR3 and their correlation corr( µ RA, µ dec), as well as the Renormalized Unit Weight Error (RUWE) (if available), and angular stellar radius θ ∗ found for the event solution. Column "Flag" marks if the event\'s colour is within the applicable range for the relation form Adams et al. (2018). Column "Type" marks the assumed source star type for further analysis.  \nTable A.4. Table of the lens mass and distance estimates of the 14 candidate dark lens microlensing events. The description of the columns is at the end of the table.  \ncontinued.  \nTable A.4.  \ncontinued.  \nTable A.4.  \nNotes: The table contains the event name, its equatorial coordinates (RA, dec), proper motion in Right Ascension µ RA and declination µ dec from Gaia DR3 and their correlation corr( µ RA, µ dec), as well as the Renormalized Unit Weight Error (RUWE) (if available), and angular stellar radius θ ∗ found for the event solution. Column "Flag" marks if the event\'s colour is within the applicable range for the relation form Adams et al. (2018). Column "Type" marks the assumed source star type for further analysis.  \nTable A.5. Table of the lens mass and distance estimates of the five candidate dark lens microlensing events, that didn\'t pass the π E criterion, but were chosen to be analysed further. The description of the columns is at the end of the table.  \nArticle number, page 16 of 20  \ncontinued.  \nTable A.5.  \nNotes: The columns are event name, mass function MF used as a prior for lens mass, the mass of the lens M L in solar masses, distance to the lens D L in kpc, the brightness of an MS star with M L mass at a distance D L, the brightness of the blend G blend obtained from the microlensing model, Einstein radius θ E, maximal astrometric displacement δ max, proper motion of the lens µ L, transverse velocity v t, and probability for a dark lens. In the MF column, K2001 refers to Kroupa (2001), and M2021 to Mróz et al. (2021).  \nTable A.6. Parameters of all best-fitting solutions of the 35 analysed events. The description of the columns is at the end of the table.  \ncontinued.  \nTable A.6.  \nArticle number, page 18 of 20  \ncontinued.  \nTable A.6.  \nKruszy\'nska et al.: Dark lens candidates from Gaia Data Release 3', 'A & A proofs: manuscript no. main': 'Table A.6. continued.  \nNotes: The table contains parameters for two types of microlensing point source-point lens models: with and without parallax. The non-parallax PSPL model parameters are: t 0 - the time of the peak of brightness, u 0 - corresponding separation of the lens and source at t 0, t E - Einstein timescale of the event, I 0 , G - brightness in baseline in the G -band, fb , G - a fraction of the total flux at baseline belonging to the blend in the G -band. The parallax model adds two additional parameters: π EN and π EE, which are north and east components of the microlensing parallax vector. t 0 , par is a non-fitted parameter, which defines the coordinate system for parallax measurement. Both t 0 and t 0 , par are in HJD\' = HJD - 2450000. B is a string of letters denoting which type of Gaia data was used: "G" for events where we used the Gaia DR3 photometry, and "GSA" for events where we opted for a Gaia Science Alerts lightcurve, finally C is a sign of the u 0 of the solution (" + " for positive and "-" for negative). If there was more than one solution with the same u 0 sign, we numbered them starting from one.'}