Predictions of the Nancy Grace Roman Space Telescope Galactic Exoplanet Survey. II. Free-floating Planet Detection Rates^*

Initially called the Wide Field Infrared Survey Telescope (WFIRST; Spergel et al. 2015), Roman is currently planned to conduct three Core Community Surveys: the High Latitude Survey (Troxel et al. 2019), the Type Ia Supernova Survey (photometric, Hounsell et al. 2018; and spectroscopic), and the Galactic Exoplanet Survey (Penny et al. 2019). These surveys will be accompanied by a Guest Observer program (including notionally 25% of observing time) and a demonstration of numerous new-to-space technologies with the Coronagraph Instrument (CGI; Debes et al. 2016; Bailey et al. 2019).

The surveys currently have notional designs that will allow them to make key measurements that will in turn provide unique constraints on the nature and time evolution of dark matter and dark energy, as well as provide novel constraints on the demographics of cold exoplanets (Akeson et al. 2019). The designs of these surveys are notional in that the final observing program will not be settled on until much closer to launch and, importantly, will incorporate community input.

For the Roman Galactic Exoplanet Survey, Roman will use the microlensing technique to search for bound planets with mass roughly greater than that of Earth (M_⊕) with semimajor axes in the range of ∼1–10 astronomical units (au).⁷ At planet–host star separations roughly equivalent to the Einstein radius of the lens system (and thus peak sensitivity), Roman will be able to detect planets with masses as low as roughly twice the mass of the Moon, roughly the mass of Ganymede (Penny et al. 2019, hereafter Paper I). Through finding these planets near and beyond the water snowline of host stars, Roman will complement the parameter space surveyed by Kepler (Borucki et al. 2010). When combined, these broad, monolithic surveys promise to provide the most comprehensive view of exoplanet demographics to date and thus provide the fundamental empirical data set by which predictions of planet formation theories can be tested (Penny et al. 2019).

The current version of the Roman microlensing survey area covers approximately 2 deg² near the Galactic bulge, composed of seven fields covered by the 0.282 deg² field of view of the Wide Field Instrument (WFI; Spergel et al. 2015). Throughout the survey, it will observe ∼50,000 microlensing events, of which roughly 1400 are predicted to show planetary perturbations (Paper I). The current notional survey design includes six 72-day seasons, clustered near the beginning and end of the 5 yr primary lifetime of the mission. Each season will be centered on either the vernal or autumnal equinoxes, when the Galactic bulge is visible by Roman.

During a season, Roman will perform continual observations using its wide 1–2 μm W146 filter at 15-minute cadence. Each visit will have a 46.8 s W146 exposure of the WFI that will reach a precision of 0.01 mag at W146 ≈ 21. These observations will be supplemented with at least one and likely two narrower filters (yet to be decided), which will sample the fields at much lower cadence. Paper I assumed observations with only one additional (Z087) filter with a 12 hr cadence, but this observing sequence has not yet been finalized. When a microlensing source star is sufficiently magnified and observations are taken in more than one filter, Roman will be able to measure the color of the microlensed source star. Measurement of the source color and magnitude can be used to constrain the angular radius of the source star θ_*, which can be be used to measure ${\theta }_{{\rm{E}}}$ if the event exhibits finite-source effects (Yoo et al. 2004). For more details on the currently planned Roman hardware, the microlensing survey design, and the bound planet yield, the reader is encouraged to read Paper I.

The properties of Roman and the Galactic Exoplanet Survey design that make it superb at detecting and characterizing bound planets are the same properties that allow it to detect and characterize FFPs. FFPs can produce events lasting from ∼1 hr to ∼1 day. Many of the same observables for bound planet microlensing events are also desirable for FFPs, such as the source color and brightness, which can constrain the angular source size and the mass of the lensing body. Measuring the mass of an isolated lens requires additional measurements of event parameters (Gould & Welch 1996) and would require supplementary and simultaneous ground-based or space-based observations (e.g., by EUCLID, Zhu & Gould 2016; Bachelet & Penny 2019; Ban 2020). We do not address parameter recovery through modeling or mass estimation of detected lenses, both of which are beyond the scope of this work.

The goal of this work is to predict Roman's ability to measure the distribution of short-timescale events attributed to free-floating planets. To do so, we will briefly revisit the microlensing survey simulations presented in Paper I and detail the changes we made to them in Section 2. We then examine light curves Roman will detect in Section 3. Section 4 will contain a discussion of the yield and limits Roman will place on FFPs in the Milky Way (MW). Finally, we will discuss our findings and conclude in Sections 5 and 6. We include two appendices, one that provides a primer on the phenomenology of microlensing events in the regime where the angular size of the source is much greater than the angular Einstein ring radius (Appendix A), and a second exploring the sensitivity of Roman's yield to the detection criteria we impose (Appendix B).

We use two detection criteria for microlensing events. The first is the difference in χ² of the observed light curve relative to a flat (unvarying) light-curve fit

$$\begin{eqnarray}&&{\rm{\Delta }}{\chi }^{2}={\chi }_{\mathrm{Line}}^{2}-{\chi }_{\mathrm{FSPL}}^{2},\end{eqnarray} \tag{ 4 }$$

where ${\chi }_{\mathrm{Line}}^{2}$ is the χ² value of the simulated light-curve data for a flat line at the baseline flux and ${\chi }_{\mathrm{FSPL}}^{2}$ is the same but for the simulated data to the true finite-source point-lens model of the event.

The second criterion is that ${n}_{3\sigma }$, the number of consecutive data points measured at least 3σ above the baseline flux, must be greater than 6, i.e.,

$$\begin{eqnarray}&&{n}_{3\sigma }\geqslant 6.\end{eqnarray} \tag{ 5 }$$

This criterion serves two purposes. First, it mimics the type of selection cut that previous free-floating planet searches have used to minimize the number of false positives caused by multiple consecutive outliers from long-tailed uncertainty distributions (e.g., Sumi et al. 2011; Mróz et al. 2017). Second, it ensures that any events detected will stand a good chance of being modeled with four or five free parameters without overfitting. Extremely short events and those with large ρ may suffer from degeneracies where even six data points may be insufficient to correctly model the event (S. A. Johnson et al. 2020, in preparation). Naively, the probability of six consecutive data points (assuming that they are Gaussian distributed) randomly passing this criterion is $\sim {\left(1\mbox{--}0.9987\right)}^{6}\approx {10}^{-18}$. Given the number of data points per light curve (∼4 × 10⁵) and the number of light curves (∼2 × 10⁸), we expect a by-chance run of six points more than 3σ to occur at a rate of ∼4 × 10⁻⁴ per survey. Thus, this may appear to be an overly conservative detection criterion. However, it is likely that neighboring data points will not be strictly uncorrelated, and therefore it is important to adopt conservative detection criteria. For example, a false-positive event may be caused by hot pixels, which are obviously correlated with time. We briefly discuss the possibility of contamination by this and other false positives in Section 5.2. We further motivate these selection criteria in the next section.

Our predictions for the yields of detectable free-floating planets are calculated using the weights defined in Equation (2) modified by a Heaviside step function for each detection criterion,

$$\begin{eqnarray}&&{N}_{\det }=\displaystyle \sum _{i}{w}_{i}H({\rm{\Delta }}{\chi }^{2}-300)H({n}_{3\sigma }-6).\end{eqnarray} \tag{ 6 }$$

Given the potential challenges involved in detecting short events, we revisit the detection criteria we presented in Section 2 to ensure that they fulfill their purpose. We require that Δχ² of an event be at least 300 and that the event have an ${n}_{3\sigma }$ of at least 6. These thresholds are similar in nature to the initial cuts placed by Sumi et al. (2011) and Mróz et al. (2017). Both use ${n}_{3\sigma }\geqslant 3$ and a statistic ${\chi }_{3+}={\sum }_{i}({F}_{i}-{F}_{\mathrm{base}})/{\sigma }_{i}$ to quantify the significance of candidate events, where F_i is the ith data point within an event with uncertainty σ_i and ${F}_{\mathrm{base}}$ is the baseline flux. Sumi et al. (2011) use ${\chi }_{3+}\geqslant 80$, while Mróz et al. (2017) relaxed this to χ₃₊ ≥ 32 owing to the typically higher quality of the OGLE data.

It is not straightforward to compare our cuts to the χ₃₊ criteria of Sumi et al. (2011) and Mróz et al. (2017); however, we can consider an extreme case. Imagine an event that barely passes both our criteria with Δχ² = 300 and ${n}_{3\sigma }=6$, but with a minimal χ₃₊. This event would have six consecutive data points, five at 3σ and a single data point at 16σ, making a total Δχ² = 301. This particular event would then have a value of χ₃₊ = 31, barely failing to pass the Mróz et al. (2017) χ₃₊ threshold. More realistic events would have higher values of χ₃₊; therefore, our cuts are at least comparable to those used in Mróz et al. (2017) but are likely slightly more stringent. We also expect fewer systematics and less correlated noise in Roman data compared to those of ground-based surveys.

Sumi et al. (2011) and Mróz et al. (2017) follow their initial cuts with several more to further vet their samples, ensuring that each is truly a microlensing event. These include (among others) the rejection of light curves with more than a brightening event, the rejection of light curves with poor goodness-of-fit statistics to initial models, and the rejection of events that did not have the rise or fall of the event sufficiently sampled. Without a detailed investigation of the uncertainties in the observables, we must use heuristic cuts to approximate these detailed investigations. We have not implemented these further cuts because our simulations do not contain the false positives they are designed to reject. We do explore the thresholds we place in Appendix B, and we determine scaling relations to predict how loosening or tightening these thresholds will impact Roman's free-floating planet yield. These relations can also be used to estimate the change in yield as the microlensing survey design evolves.

We examine how our thresholds of Δχ² ≥ 300 and ${n}_{3\sigma }\geqslant 6$ impact the timescale distribution of events in Figure 5, where we assume delta functions in mass (one planet per star) for each mass shown. First, we plot the distribution of events as a function of $t=\max ({t}_{{\rm{E}}},\tfrac{1}{2}{t}_{c})$ as solid lines. Here t_c is the source chord-crossing time as defined in Equation (A3).⁸

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These distributions are meant to show the duration of events detected. Events that exhibit extreme finite-source effects (and thus have "top-hat" light curves) ${t}_{{\rm{E}}}$ will be less than $\tfrac{1}{2}{t}_{c}$, and the event will be longer than expected. This will allow for the detection of events that would not be typically detectable were there no finite-source effects.

Second, we plot the distribution as a function of solely the ${t}_{{\rm{E}}}$ values of events as dashed lines. There is essentially no difference between these distributions for lens masses ≥10 M_⊕, but we see a strong offset between the solid- and dashed-line distributions for the 0.1 M_⊕ events. For low-mass lenses, this demonstrates the previous point that some detected events would have expected timescales much shorter than would be detectable considering our requirement on n_3σ.

Finally, for the two lowest masses we show the distribution as a function of ${t}_{3\sigma }=\tfrac{1}{2}{n}_{3\sigma }\times 15$ minutes, half the length of the event while significantly magnified. These distributions have no events less than 45 minutes (the vertical black dashed line), which is indicative of our detection criteria on ${n}_{3\sigma }$. For the 0.1 M_⊕ events, the event timescale saturates at the source chord-crossing timescale for many events, pushing the distribution toward longer durations. This is even more enhanced when considering the distribution while significantly magnified.

More broadly, we show the detection efficiency as a function of the microlensing timescale ${t}_{{\rm{E}}}$ in Figure 6. The black line is the number of detected events relative to the number of injected events with a given timescale within a single 72-day season. The typical timescales for lenses of five different masses are illustrated with vertical lines. Within a season, Roman will maintain a ≳50% efficiency down to ${t}_{{\rm{E}}}\approx 1.5\,\mathrm{hr}$. This efficiency would be proportionately lower if we consider the efficiency over the entire 5 yr baseline by a factor of $(6\times 72\ \mathrm{days})/(5\times 365\ \mathrm{days})=0.23$ if t₀ is uniformly distributed, since the Galactic bulge will only be observed for a fraction of a year.

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Overall, in this section we have demonstrated that Roman will be able to detect a wide range and variety of short-timescale microlensing events. This will impact the overall timescale distributions of microlensing events that Roman will detect, and it must be accounted for in determining the detection sensitivity used to infer the true underlying distribution of event timescales, regardless of the nature of the lenses.

In the next section, we will present our predictions for the yield of and limits on free-floating planets given the fiducial Cycle 7 survey design.

We must assume a mass function for FFPs if we are to estimate the number of FFPs that Roman will find. We assume two forms of mass function, one log-uniform in mass,

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{d\mathrm{log}{M}_{p}}=1\,{\mathrm{dex}}^{-1},\end{eqnarray} \tag{ 7 }$$

and another inspired by an inferred mass function of bound planets detected by microlensing (following Cassan et al. 2012). In the second case, we assume that for low-mass planets the mass function saturates at two planets per star below 5.2 M_⊕. This prevents the number from "blowing up" as the planet mass decreases. The functional form is then

$$\begin{eqnarray}\displaystyle \frac{{dN}}{d\mathrm{log}{M}_{p}}=\left\{\begin{array}{ll}\displaystyle \frac{0.24}{\mathrm{dex}}{\left(\displaystyle \frac{{M}_{p}}{95{M}_{\oplus }}\right)}^{-0.73} & \displaystyle \frac{{M}_{p}}{\,{M}_{\oplus }}\geqslant 5.2\\ 2\,{\mathrm{dex}}^{-1} & \displaystyle \frac{{M}_{p}}{\,{M}_{\oplus }}\lt 5.2.\end{array}\right.\end{eqnarray} \tag{ 8 }$$

Note that Paper I used a function with the same mass dependence as the fiducial mass function for bound planets. This fiducial function is also consistent with the upper limits on the abundance of bound and wide-orbit planets measured by Mróz et al. (2017).

This mass function is somewhat optimistic compared to that found by Suzuki et al. (2016), who found that the mass ratio function is shallower for objects with mass ratio less than ∼2 × 10⁻⁴ than that found by Cassan et al. (2012). This mass ratio corresponds to the typical mass ratio for a Neptune-mass planet. Nevertheless, we adopt the Cassan et al. (2012) mass function for continuity with Paper I.

We report the expected number of detections as a function of mass in Table 2. The first column ("One-per-star") assumes that there is a delta function of FFPs at that mass such that there is an equal number of FFPs to stars in the MW. The "Log-uniform" and "Fiducial' columns assume bins that are 0.5 dex in width for the two mass functions defined above. We use the trapezoidal rule to integrate the number of detections with masses from 0.1 to 1000 M_⊕ to estimate the total yield of FFPs. We include rows for FFPs with masses of 0.01 and 10⁴ M_⊕ for reference. Were the mass function simply log-uniform, nearly 1000 free-floating planets would be detected. In the case of the fiducial mass function, we predict that Roman will detect roughly 250 FFPs.

Next, we consider how these populations will manifest in the timescale distribution of microlensing events measured by Roman. To start, we show the expected timescale distribution of detected stellar events with the same detection criteria (${\rm{\Delta }}{\chi }^{2}\geqslant 300,{n}_{3\sigma }\geqslant 6$) in Figure 7. Note that the minimum mass included in the GM is $0.08\,{M}_{\odot }\approx 80{M}_{\mathrm{Jup}}$ in the Galactic disk and 0.15 M_⊙ in the Galactic bulge. Then, we consider three cases for populations of FFPs. The blue hatched region has an upper boundary that reflects the limit of at most 0.25 Jovian planets per star from Mróz et al. (2017). We also include the population of 5 M_⊕ free-floating or wide-separation planets that Mróz et al. (2017) cautiously consider as a possible explanation of the excess of very short timescale events. The orange shaded region has a lower (upper) bound corresponding to 5 (10) FFPs per star in the MW that are 5 M_⊕. Third, we show the expected distribution of detections using the continuous fiducial mass function in red. We also draw a realization of this mass function, which is included as the gray histogram with Poisson error bars.

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If our fiducial assumptions are reasonable, Roman will be able to detect the signature of terrestrial-mass to Jovian-mass lenses in the event timescale distribution. With the lowest-mass planets giving rise to events with extended timescales owing to finite-source effects, the sensitivity is pushed to events with lens masses as low as a few times that of Mars. The fiducial mass function we use produces events detectable by Roman with timescales stretching over three orders of magnitude. These will leak into the timescale distribution attributable to the stars in the Galaxy, but because the model truncates at 0.08 M_⊙, there is no smooth transition.

If Roman detects no free-floating planets in a given mass range, it can still place interesting constraints on the occurrence rate of such planets, which in turn can be used to constrain planet formation theories. We can place expected upper limits on populations of FFPs using Poisson statistics, following Griest (1991). If we return to our delta function mass distribution such that we assume that there is one planet of that mass per star, we can place a 95% confidence level upper limit for any mass bin, which corresponds to the situation in which we would expect fewer than three planets per star.⁹ Figure 8 plots the 95% confidence level Roman will be able to place on the total mass of bodies per star in the MW composed of bodies of mass M if no lenses of that mass are detected. Note that the vertical axis is equivalent to ${M}_{p}{dN}/d\mathrm{log}{M}_{p}$ in units of M_⊕. For comparison, we plot our fiducial mass function (Equation (8)) and the mass distribution for solar system bodies.¹⁰ The latter is to give some intuition as to whether there was an equivalent of a solar system's mass function worth of unbound bodies per star in the MW, but we note that such a mass function is likely to be incomplete at low masses, and possibly also at higher masses (Trujillo & Sheppard 2014; Batygin & Brown 2016). In other words, for typical planetary formation scenarios, a higher number of low-mass objects are ejected than remain in our solar system, and in at least a subset of planetary systems, a higher number of higher-mass objects are ejected than remain in our solar system.

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This origin of the shape of the total mass limit curve deserves some discussion. For FFP masses M ≳ 1 M_⊕ the curve rises as M^1/2, which is somewhat counterintuitive, though it may be recognized by those familiar with dark matter microlensing surveys. The number of expected microlensing events Roman will detect is set by the microlensing event rate Γ, which scales as the square root of the object mass Γ ∝ M^1/2, if there is a fixed number of objects. But the vertical axis of Figure 8 is the total mass of expected objects of mass M_p per star M_tot, not the total number. So for fixed M_tot, the number of objects scales as the inverse of the object mass M⁻¹, and thus the microlensing event rate produced by a fixed total mass of object scales with the individual object mass as ${M}^{-1/2}$. The total number of detections therefore scales as ${N}_{\det }\propto {M}_{\mathrm{tot}}{M}_{p}^{-1/2}$. The survey limit is a contour of a constant number of expected detections, and thus the total mass of ejected objected scales as M_tot ∝ M^1/2.

Below $M\sim 1\,{M}_{\oplus }$, the finite size of a typical Roman source star becomes larger than the typical Einstein ring radius of the lens, and so the event rate per object becomes independent of object mass. But the event rate per total object mass scales as M⁻¹, and we would expect the limit curve to become more steeply positive and scale as M⁻¹. However, the transition to the finite-source-dominated regime begins to reduce the peak magnification of events, even if lengthening them, which eventually significantly reduces the probability of a microlensing event being detected. Between ∼0.01 and 1 M_⊕, finite-source effects from events with 1 < u₀ < ρ increase the detectable event rate (and reduce the total mass limit) by up to a factor of two relative to events with only u₀ < 1. Below M ≲ 0.01 M_⊕ finite-source effects decrease the maximum magnification of microlensing events to the point where they start to become undetectable, the detection efficiency begins to fall far faster than the event rate increases, and the slope of the limit curve inverts and becomes sharply negative.

Viewed broadly, the total mass limit curve shows that Roman will be an extremely sensitive probe of the total mass budget of loosely bound and free-floating masses. At its most sensitive mass, $M\sim 3\times {10}^{-2}\,{M}_{\oplus }$ (near the mass of Mercury), Roman would be sensitive to total masses of just  ∼ 0.1 M_⊕ per star (or roughly three objects per star). Roman will be sensitive to a total mass of 1 M_⊕ or less of objects with masses over a range of ∼0.003–100 M_⊕, or more than 5 orders of magnitude in mass. While for the lowest-mass objects these total masses are large compared to the mass budget of the present solar system, they are small compared to the total mass of planetesimal disks that are required to form solar-system-like planet configurations in simulations. For one example, the Nice model considers initial planetesimal disk masses between 30 and 50 M_⊕ beyond Neptune (Tsiganis et al. 2005). For a broader view of the expected population of loosely bound and free-floating objects, we can compare the Roman total mass limit curve to various predictions and constraints on these populations.

The first set of comparisons we draw is between Roman's limits and limits set by microlensing searches for MACHOs. There are three studies we consider:

• 1.  
  As mentioned in the Introduction, Alcock et al. (1996) presented combined results of the MACHO and EROS microlensing surveys. These surveys were searching for MACHOs as candidates for the dark matter mass components of the MW halo.
• 2.  
  Griest et al. (2014) found a similar limit on primordial black holes but used the Kepler transit survey. Kepler provides relatively high cadence observations of a fixed, relatively dense, stellar field, which is nearly optimal for a survey of microlensing events. The drawbacks were that this was toward a relatively low stellar density field compared to the LMC or the Galactic center and that potential sources were much closer than those typical of microlensing events. The limits placed here are from the analysis of 2 yr of the Kepler mission, looking for short-timescale events.
• 3.  
  Niikura et al. (2019) used the Hyper Suprime-Cam on the Subaru Telescope (Subaru/HSC) to perform 2-minute cadence observations of M31 with high resolution. This search yielded the best constraint on low-mass primordial black holes as a component of the MW dark matter halo.
• 4.  
  Niikura et al. (2019) placed limits roughly 50% lower than the MACHO+EROS result at 50 M_⊕ using 5 yr of OGLE-IV data. We do not include this result in Figure 8 as a result of space constraints.

These limits are not meant to be a direct comparison, so we simply scale their limits by assuming a stellar number density of ${n}_{\star }=0.14\,{\mathrm{pc}}^{-3}$ and a dark matter halo mass density of ${\rho }_{\mathrm{halo}}=0.3\,\mathrm{GeV}/{\mathrm{cm}}^{-3}$. We determine their measured halo mass fractions, ${f}_{\mathrm{HM}}$, from their figures. Then, the mass of free-floating objects per star is simply

$$\begin{eqnarray}&&\displaystyle \frac{{f}_{\mathrm{HM}}{\rho }_{\mathrm{halo}}}{{n}_{\star }}={10}^{3}{{\rm{M}}}_{\oplus }\ \left(\displaystyle \frac{0.06}{{{\rm{f}}}_{\mathrm{HM}}}\right).\end{eqnarray} \tag{ 9 }$$

We also include three frequencies from other observational efforts:

• 1.  
  Sumi et al. (2011) reported that there may be two free-floating Jupiter-mass planets per star in the MW. Although inconsistent with the (more recent) limits set by Mróz et al. (2017), we display this result for context.
• 2.  
  Mróz et al. (2017) place an upper limit of fewer than 0.25 Jupiter-mass FFPs per star in the MW. This is indicated by the black arrow in the upper right. Mróz et al. (2017) find a tentative signal for 5–10 5 M_⊕ mass FFPs per star in the MW. This is represented by the black vertical bracket. Note that if no events occur with a Jupiter-mass lens, then Roman will place a limit of fewer than one Jupiter-mass planet per ∼100 stars in the MW. This will improve the limit placed by the OGLE survey from 8 yr of data by more than an order of magnitude (Mróz et al. 2017).
• 3.  
  We consider measurements of the frequencies of bound planets and brown dwarfs found using direct imaging by Nielsen et al. (2019). While these are bound planet frequencies, they are for companions with semimajor axes from 10 to 100 au, which would likely be mistaken for free-floating planets in microlensing surveys, and thus provide a useful comparison. Nielsen et al. (2019) found a 3.5% occurrence rate for 5–13 Jupiter-mass planets and a much lower rate of 0.8% for 13–80 Jupiter-mass brown dwarfs for hosts with mass $0.2\gt M/{M}_{\odot }\gt 5$. We include these two frequencies as black circles in Figure 8, with vertical errors being their reported uncertainties and horizontal errors being the associated ranges. Roman will be sensitive to these widely bound companions, so distinguishing these free-floating planet false positives will be important.

We also plot predictions on the total mass of FFPs per star from a number of theoretical simulations:

• 1.  
  Pfyffer et al. (2015) present simulations of formation and evolution of planetary systems, in which only ∼0.04 ${M}_{\mathrm{Jup}}$ of planets are ejected per star in the optimistic case of no eccentricity or inclination damping.
• 2.  
  Ma et al. (2016) predict the number of planets ejected per star from dynamical simulations. We take values from their models of 0.3 M_⊙ stars, in that 12.5% of stars eject 5 M_⊕ of mass in 0.3 M_⊕ bodies.
• 3.  
  Barclay et al. (2017) predict the number of planetesimals ejected from systems during planet formation. We only compare our limit to their prediction in which giant planets are present in the system, as a gray horizontal bar spanning the width of the bins used. In the case that no giant planets are present, fewer objects are ejected.
• 4.  
  Hong et al. (2018) predict that ${ \mathcal O }(0.01-1)$ moons will be ejected from systems following planet–planet dynamical interactions. We assume that these moons have masses from 0.1 to 1 M_⊕, and thus a range of possibilities are included within the gray shaded region. This is a generous upper mass limit compared to the moons of our solar system, but we note that little is understood on the formation of exomoons. As an example of an unexpected possibility, there is (contested) evidence of a Neptune-sized exomoon in the Kepler-1625b system (Teachey & Kipping 2018; Kreidberg et al. 2019; Teachey et al. 2020).

Thus, we conclude that Roman will not only improve the constraints on the abundance of objects with masses from that of less than the Moon to the mass of Jupiter by an order of magnitude or more but also allow for a test of model predictions for the total mass of ejected planets in several different planet formation and evolution theories.

The Roman microlensing survey will record nearly 40,000 photometric data points for ∼10⁸ stars over its 5 yr duration. While we have perfect knowledge within these simulations, practically finding events due to very low mass lenses will likely require more sophisticated search algorithms. Microlensing surveys have used clear and specific cuts in identifying events. For example, Mróz et al. (2017) made a series of detection cuts based on the temporal distribution of data points during a candidate event, e.g., the number of observations obtained while the flux is rising and falling. These additional cuts were made in order to avoid false positives like flares or cataclysmic variables.

Machine-learning classifiers are also starting to be applied to microlensing survey data as well. Wyrzykowski et al. (2015) searched through OGLE-III data using a random forest classifier. Godines et al. (2019) present a classifier for finding events for low-cadence wide-field surveys. Khakpash et al. (2019) developed a fast, approximate algorithm for characterizing binary lens events. G. Bryden et al. (2020, in preparation) are developing a machine-learning classifier for the microlensing survey being performed with the United Kingdom Infrared Telescope (UKIRT; Shvartzvald et al. 2017). This survey is designed to be a pathfinder for the Roman survey and is mapping the near-infrared microlensing event rate in candidate Roman fields.

Still, most of these efforts have focused on the familiar regime of small source sizes $(\rho \sim {10}^{-2}-{10}^{-3})$ that are more familiar in microlensing surveys. It will need to be carefully examined how effective these search techniques are in detecting the extremely short timescale events we are considering, particularly those with qualitatively different morphologies from the more familiar ρ ≪ 1 single lens microlensing events. Here we use only the Δχ² and ${n}_{3\sigma }$ metrics to determine whether an event is detected in these simulations (but see Appendix B). However, events may be detectable by Roman over a wider region of parameter space using different event selection filters, including the low-amplitude top-hat events caused by low-mass lenses.

The full sample of microlensing events detected by Roman will need to be vetted for false positives. Detector artifacts such as hot pixels or other defects may introduce systematics that could mimic a short-timescale microlensing event. However, among the Core Community Surveys, the Galactic Exoplanet Survey provides one of the best opportunities to characterize the Wide Field Instrument H4RG detectors, facilitating the exclusion of these artifacts in light curves (Gaudi et al. 2019).

Astrophysical sources could also be mistaken for microlensing events. Similar to those false positives of ground-based microlensing surveys, these will include at least asteroids, cataclysmic variables, and flaring stars. While a more rigorous event detection algorithm will allow Roman to mitigate these in the full microlensing sample, even simple cuts will sometimes suffice. For example, M-dwarf flares rarely last longer than 90 minutes (Hawley et al. 2014), making their exclusion almost guaranteed by a simple cut on n_3σ. Longer-lasting events such as novae or cataclysmic variables will have many photometric data points during their eruption to model and reject them. Spatiotemporal clustering of candidate events and astrometric centroid analysis can be used to recognize asteroids that have not already been identified prior to a pipeline run.

As identified in Mróz et al. (2017), one event (MOA-ip-01) in the sample of short-timescale events presented by Sumi et al. (2011) has a degenerate solution. The event was reported with ${t}_{{\rm{E}}}$ = 0.73 days, but an alternate solution with the much longer t_E = 8.2 days is favored. In this case, a larger blending parameter (f_s) and smaller impact parameter (u₀) resulted in the alternate solution. The major difference between these models is in the appearance of the wings of the magnification event. This degeneracy is well described in Woźniak & Paczyński (1997). Roman should be able to distinguish between these approximately degenerate events via its high precision and cadence.

Another relevant degeneracy occurs in lensing events with large relative angular source size ρ. In this regime, the magnification over the duration of the event is roughly constant (in the absence of limb darkening) and set by Equation (A2). As a result, ρ becomes nearly degenerate with the blending parameter ${f}_{{\rm{S}}}={F}_{{\rm{S}}}/({F}_{{\rm{S}}}+{F}_{{\rm{B}}})$, which is the fraction the source flux ${F}_{{\rm{S}}}$ contributes to the observed baseline flux ${F}_{{\rm{S}}}+{F}_{{\rm{B}}}$, where ${F}_{{\rm{B}}}$ is the blended flux. Essentially, the flux from the source alone cannot be confidently measured without precise and dense photometry, which can be used to distinguish the subtle differences in these broadly degenerate light curves. Thus, in the presence of blended light, one may underestimate the true peak magnification (S. A. Johnson et al. 2020, in preparation; Mróz et al. 2020), making it difficult to constrain ρ precisely. This is important because ρ depends on the angular source size, which will be poorly constrained when no color measurement is made while the source is magnified. These measurements will likely not occur for short-timescale events and those that exhibit extreme finite-source effects. We note that this degeneracy persists even in the presence of limb darkening (S. A. Johnson et al. 2020, in preparation)

For fixed θ_*, ρ increases as the planet mass (and thus ${\theta }_{{\rm{E}}}$) decreases. Thus, as the lens mass decreases, more and more events enter into the ρ ≫ 1 regime, thereby increasing the likelihood that they will suffer from this degeneracy. We note that this continuous degeneracy is different from the discrete degeneracy exhibited in the event reported by Chung et al. (2017).

In order to estimate the fraction of events for which finite-source effects should be detectable, in Figure 9 we show the cumulative fraction of detected events as a function of ρ/u₀. Events that have $\rho /{u}_{0}={\theta }_{* }/{\theta }_{0}\gtrsim 0.5$ should exhibit finite-source effects (Gould & Gaucherel 1997). Events that satisfy this criterion and have ρ ≫ 1 will be more susceptible to the above $\rho -{f}_{{\rm{S}}}$ degeneracy.

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Fortunately, most of our low-mass lenses that exhibit finite-source effects will be detected in events where the source star dominates the baseline flux (or have large values of ${f}_{{\rm{S}}}$). This is shown in Figure 10, where we plot the cumulative fraction of detected events as a function of ${f}_{{\rm{S}}}$ in W146 (top panel) and Z087 (bottom panel). Vertical dashes mark the median values of ${f}_{{\rm{S}}}$ of these distributions, and note that the markers for 10² and 10³ M_⊕ lie on top of each other. We also include the source magnitude distributions for detected events in Figure 11 for W146 (top panel) and Z087 (bottom panel). Brighter sources will contribute most to the low-mass lens event rates, as one would expect. However, because the fraction of blended flux is not known a priori, this argument can only be used in a statistical sense.

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Events of all masses will have little blending in Z087, but small mass lenses (that last ≲6 hr) will be unlikely to have a Z087 measurement taken while magnified. Measurements from multiple filters may allow an estimate of the source color. Table 3 shows the fraction of events that will have a color measurement taken while the source is magnified, resulting in the breaking of the degeneracy. Only 11% of 0.1 M_⊕ lenses will have a color measurement if the Z087 measurements (or other alternative band) have a cadence of 12 hr, but this fraction can be more than tripled if the cadence increases to 6 hr. At 1.0 M_⊕, the fraction of detected events with a color measurement will double if the color cadence increases to 6 hr. We also include fractions of detected events with color measurements if our threshold n_3σ ≥ 6 were to be relaxed to only 3, making the percentage even lower for low-mass lenses. Note that the modest decrease in percentages arises from the fact that many more low-mass lens events are detected when the n_3σ threshold is relaxed (see Appendix B, especially Figure B2).

Alternatively, this degeneracy may be broken through the 5 yr baseline of the microlensing survey. Potentially blended sources may become apparent as blended stars (either unrelated stars, the host star if the planet is actually bound but widely separated (see Section 5.4), or a companion to the host star) move away from the line of sight to the source. This fact will also be used in constraining the presence of potential host stars to FFP candidates.

S. A. Johnson et al. (2020, in preparation) demonstrate that there is a second degeneracy in events with ρ ≫ 1. This is a multiparameter degeneracy between the effective timescale of the event, which is well approximated by the time to cross the chord of the source t_c, the impact parameter of the lens with respect to the center of the source ${u}_{0,* }$, and the time to cross the angular source radius ${t}_{* }={\theta }_{* }/{\mu }_{\mathrm{rel}}$ (see Appendix A for definitions of these quantities). This is easiest to understand in the absence of limb darkening. A larger impact parameter ${u}_{0,* }$ results in a shorter event, but with the same peak magnification (due to the "top-hat" nature of events with ρ ≫ 1). But this shorter duration can be accommodated by scaling t_*. Since neither θ_* nor μ_rel is known a priori, it is impossible to measure μ_rel in the regime where these assumptions hold. S. A. Johnson et al. (2020, in preparation) demonstrate that this degeneracy holds for limb-darkened sources as well.

We are investigating the severity and impact of these degeneracies on the ability to recover event parameters in events with extreme finite-source effects (S. A. Johnson et al. 2020, in preparation).

While Roman will have sensitivity to the short-timescale events of FFPs, true FFPs can be confused with widely separated but bound planets. If a bound planet has a large enough projected separation, the source may only be magnified by the planet and not the host star (Di Stefano & Scalzo 1999; Han & Kang 2003; Han et al. 2005). This confusion requires proper accounting if accurate occurrence rates for both FFPs and wide-bound planets are to be reached. To this end, Han et al. (2005) summarize three methods for distinguishing the presence of a host star.

The first method was originally described by Han & Kang (2003), in which the magnification by a bound planetary-mass object will deviate from that of an isolated lens. In this scenario, rather than the effective point caustic of an isolated lens, the source is magnified by a planetary caustic that changes the morphology of the light curve. Han & Kang (2003) assume some fiducial detection thresholds and find that this method can distinguish ≳80% of events for projected separations ≲10 au and mass ratios down to q ≈ 10⁻⁴. This deviation was first observed by Bennett et al. (2012) and more recently observed in the short (4-day) event reported by Han et al. (2020b), where the presence of a host was determined through 0.03 mag residuals near the peak magnification of a single lens model.

Another pathway to determine whether an FFP lens is truly isolated would be to rule out any magnification from a photometrically undetected host (Han et al. 2005). This signal would appear as a long-term, low-amplitude bump in the light curve. Han et al. (2005) show that nearly all planets with projected separations of less than about 13 au will have the presence of their host stars inferred this way. Assuming that the semimajor axis distribution of bound planets is log-uniform, ∼30% of those with $a\in [{10}^{-1}$, 10²] lie outside 13 au.

The third method is to directly measure blended light from a candidate host. This can be performed in earlier or later seasons with Roman by searching for PSF elongation, color-dependent centroid shifts, or event resolution of the lens host star and the source star. Henderson & Shvartzvald (2016) find that Roman can exclude hosts down to $\gtrsim 0.1\,{M}_{\odot }$ depending on the lens distance and the nature of the source star. Paper I finds that the majority of hosts to bound planet detections will contribute at least 10% of the total blend flux. The separation of unassociated blended stars (neither the lens nor source star) from potential host flux could be constrained through priors from the event, such as the distance to the lens system.

We did not account for the effects of limb darkening in our simulations (Witt 1995; Heyrovský 2003). The limb-darkening profile of source stars is wavelength dependent, with the amplitude of the surface-to-limb variation decreasing ∝λ⁻¹ for the Sun (Hestroffer & Magnan 1998). Because the primary observations are in the near-infrared, Roman typical source stars will exhibit less limb darkening than in optical surveys. As shown by Lee et al. (2009) and many more, the limb-darkening profile alters the shape of the light curve (see their Figure 6). This would likely impact our yield estimates for low-mass lenses most, where finite-source effects are most likely. In our detection cuts, if a source is fainter in its limb, it may shorten the effective timescale of the event. This could lower n_3σ or Δχ² of the event in our detection threshold. However, limb darkening increases the peak magnification of events (see bottom panel of Figure 2), which could modestly increase the number of detections we predict. We must also consider that W146 is a wide band, and thus the limb darkening will have a significant chromatic dependence over the wavelength range of the filter (Han et al. 2000; Heyrovský 2003; Claret & Bloemen 2011).

If the mass of a lens is small enough ($\sim {10}^{-5}\,{M}_{\oplus }$), the geometric optics description of microlensing becomes insufficient and wave effects manifest themselves in the magnification curve (Takahashi & Nakamura 2003, among others). In short, the threshold for this effect is when the wavelength of light being observed becomes comparable to the Schwarzschild radius of the lens; in this limit there is a fundamental limit to the peak magnification of the event. For a mass of 10⁻³ M_⊕ (3 × 10⁻⁹M_⊙) this corresponds to a wavelength of ∼11 μm (see Equations (5) and (7) of Sugiyama et al. (2020)). This is below the long-wavelength edge of the W146 band, and the corresponding wavelength only gets longer for larger mass lenses. We therefore do not consider this effect here.

The conversion from timescale to mass for FFP events requires measurements of both the microlensing parallax and the angular Einstein ring. A measurement of ρ from finite-source effects and θ_* from the dereddened source flux and colors (Yoo et al. 2004) would yield ${\theta }_{{\rm{E}}}$, if the degeneracy discussed above can be broken and measurements are made in another filter(s) while the source is magnified.¹¹

Spitzer enabled the regular measurement of microlensing parallaxes to a large number of stellar, binary, and bound-planetary microlensing events by leveraging the fact that it was separated by the Earth by ∼1 au owing to its Earth-trailing orbit (Gould 1994, 1995, 1999), but these events had projected Einstein ring sizes of a few au (e.g., Dong et al. 2007; Yee et al. 2015). Zhu & Gould (2016) quantify the potential for simultaneous ground-based observations (and Roman-only observations) to measure one- and two-dimensional microlens parallaxes. Space-based parallax measurements of FFP lenses were also attempted using the Kepler spacecraft during the K2 Campaign 9 survey, which largely consisted of a microlensing survey toward the bulge (Henderson & Shvartzvald 2016; Henderson et al. 2016; Zhu et al. 2017a, 2017b; Penny et al. 2017; Zang et al. 2018). Penny et al. (2019) and Bachelet & Penny (2019) show that the short intra-L2 baseline between the Euclid and Roman spacecraft would be enough to measure free-floating planet parallaxes. Ban (2020) computes probabilities for measuring parallaxes for combinations of ground- and space-based telescopes. Concurrent observations with wide-field infrared observatories, such as UKIRT (Hodapp et al. 2018), VISTA (Dalton et al. 2006), and PRIME (Yee et al. 2018),¹² as well as wide-field optical observatories, such as DECam (Flaugher et al. 2015), HyperSuprimeCam (Miyazaki et al. 2012), and the Vera C. Rubin Observatory (LSST Science Collaboration et al. 2009), would enable parallax measurements for both bound and free-floating planets.

For a typical isolated microlens, the angular size of the source is much smaller than the angular size of the Einstein ring of the lens, and thus the approximation of a point source generally remains valid. That is, the magnification as a function of the separation between the source and the lens normalized to the size of the Einstein ring u is given by (Paczyński 1986)

$$\begin{eqnarray}&&A=\displaystyle \frac{{u}^{2}+2}{u{\left({u}^{2}+4\right)}^{1/2}}.\end{eqnarray} \tag{ A1 }$$

The magnification peaks at the minimum separation ${u}_{0}={\theta }_{0}/{\theta }_{{\rm{E}}}$, where θ₀ is the angular separation between the source and the lens at closest approach. The point-source approximation in Equation (A1) breaks down when angular separation of the source from the lens, θ₀, becomes comparable to the angular radius of the source star θ_*. For point lenses, this condition results in a significant second derivative in the point-lens light curve over the angular size of the source, which must be accounted for by computing the magnification. Thus, for events with impact parameter such that ρ/u₀ ≳ 0.5, where $\rho ={\theta }_{* }/{\theta }_{{\rm{E}}}$, the peak of the event (at times $| t-{t}_{0}| /{t}_{{\rm{E}}}\lesssim 2\rho$) is affected by finite-source effects (Gould & Gaucherel 1997). Since, for stellar mass lenses, ρ is typically in the range of 10⁻³ to 10⁻², most events are unaffected, and those that are affected are high-magnification events. Even in such events, finite-source effects are only detectable near the peak of the event, while the magnification during the rest of the event is essentially equivalent to that due to a point source.

However, this characterization breaks completely when the angular size of the source becomes comparable to the angular size of the Einstein ring, or $\rho \,\gtrsim \,1$. In particular, in the extreme case when ρ ≫ 1, the source will completely envelop the Einstein ring of the lens if it passes within the angular source radius. When this happens, the lens magnifies only a fraction of the area of the source as it transits the disk of the source (e.g., Gould & Gaucherel 1997; Agol 2003). In this limit (ρ ≫ 1) and to first order, the magnification curve can take on a "top-hat" or boxcar shape, saturating at a magnification of

$$\begin{eqnarray}&&{A}_{\mathrm{peak}}\approx 1+\displaystyle \frac{2}{{\rho }^{2}}\end{eqnarray} \tag{ A2 }$$

(Liebes 1964; Gould & Gaucherel 1997; Agol 2003). The light-curve shape is essentially independent of u₀ except when u₀ ∼ ρ (Agol 2003). Furthermore, the duration of the event is no longer set by the microlensing timescale ${t}_{{\rm{E}}}$, but rather is proportional to the source radius crossing time ${t}_{* }={\theta }_{* }/{\mu }_{\mathrm{rel}}={t}_{{\rm{E}}}\rho$. Mróz et al. (2018) account for the impact parameter u₀ such that they use the time taken for the lens to cross the chord of the source

$$\begin{eqnarray}&&{t}_{c}=\displaystyle \frac{2{\theta }_{* }}{{\mu }_{\mathrm{rel}}}\sqrt{1-{\left({u}_{0,* }\right)}^{2}},\end{eqnarray} \tag{ A3 }$$

where ${u}_{0,* }={\theta }_{0}/{\theta }_{* }$. Note that this timescale is independent of the angular radius of the Einstein ring and thus lens mass. We note that Mróz et al. (2017) define t_* as the crossing time over the chord of the source, but we define t_* as the source radius crossing time defined above (see Appendix A of Skowron et al. 2011). Henceforth, we will use t_c for the chord-crossing time, and we propose that this become the convention.

Broadly, these changes in light-curve morphology are referred to as extreme finite-source effects, as the light curve is affected by finite-source effects throughout the duration (e.g., at no time while the source is magnified does the point-source approximation hold). We demonstrate the impact of finite-source effects on the light curve of events in Figure A1. We consider five lenses in which we only vary the angular size of the Einstein ring, quantified by ρ = 0.25, 0.50, 1.00, 2.00, and 4.00 (as the size of the source is fixed). The sizes of these rings are shown in the upper left corner of the leftmost panel scaled to the size of a source star, which is depicted as a gray circle (with an angular radius of θ_*). A horizontal gray line depicts the path the lenses will take, and it is separated from the center of the source star by the impact parameter θ_* (again, to scale). We then vary the impact parameter from ${u}_{0,* }=3.0$ down to ${u}_{0,* }=0.0$ from left to right. This is written above the plot and depicted as the source star (gray circle) approaching the lens trajectory (gray line). Note that the circles and their separations are independent of the time and magnification axes.

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Each panel depicts all five lenses in different events, where the line style matches the lens with the same Einstein ring line style in the upper left corner of the leftmost panel. We use the method of Lee et al. (2009) as implemented in MulensModel (Poleski & Yee 2018) to compute all light curves in this figure.

As ${\theta }_{{\rm{E}}}\propto \sqrt{M}$, the more massive lens will have the largest ${\theta }_{{\rm{E}}}$ and be the farthest from the regime of finite-source effects. Note that time is referenced to the peak of the event (t₀) and scaled by the analytic timescale ${t}_{{\rm{E}}}$ on the horizontal axis. The solid-line light curve behaves essentially how you expect an isolated lens to behave as the impact parameter drops up until the last two panels where the peak becomes more rounded. This is the first breakdown we described that occurs when $\rho /{u}_{0}\gtrsim 0.5$, or when the size of the source star is within a few times the impact parameter.

However, the behavior is dramatically different for the lowest-mass lens ($\rho \gg 1$). In this case, when lens is not transiting the source (u_* > 1), there is effectively no magnification. For the smallest impact parameter (rightmost panel), the light curve looks like the top hat described earlier, magnifying the source by roughly 10% ($1+\tfrac{2}{{4}^{2}}\approx 1.13$). Also note that the duration of this event is now much longer than one would expect given its analytic timescale ${t}_{{\rm{E}}}$. In fact, when ${u}_{0,* }=0.00$, the duration is nearly exactly what we predict given the diameter crossing time ${t}_{c}/{t}_{{\rm{E}}}=2\rho =4$ for the ρ = 2 case and ${t}_{c}/{t}_{{\rm{E}}}=8$ for the ρ = 4 case. In the rightmost panel, the ρ = 4 event lasts ∼4 times longer than one would expect based on the value of ${t}_{{\rm{E}}}$.

To provide a quantitative sense of the relevant scales, consider a typical stellar mass lens ($0.3\,{M}_{\odot }$), which has an angular Einstein ring radius of ${\theta }_{{\rm{E}}}=550\,\mu \mathrm{as}$. A source star in the Galactic bulge (at a distance of ${D}_{{\rm{S}}}=8$ kpc) that has a radius of 1 R_⊙ will have an angular radius of just 0.6 μas. Lenses with mass ≲0.12 M_⊕ will have ρ ≳ 1 for this source. A typical clump giant in the bulge will have a radius of ∼10 R_⊙, leading to ρ > 1 for lenses with mass ≲10 M_⊕.

These morphological changes will impact the microlensing event rate and microlensing optical depth (Vietri & Ostriker 1983; Paczyński 1991). Recall that the microlensing optical depth (the probability that any given star is being lensed) is a function of the fraction of the sky covered by Einstein rings. As demonstrated above, lenses with small enough masses will have Einstein rings smaller than the angular size of some stars. Han et al. (2005) show that for lenses with low enough masses, the event rate actually increases compared to what you would expect for lenses with Einstein rings smaller than the angular size of source stars. For these lenses, the event rate is proportional to the fraction of the sky covered by source stars. However, the detection of such events is hampered by the fact that the peak magnification is lower than one would expect for a point source. Han et al. (2005) also derive analytic expressions for the threshold impact parameter for detection and the minimum detectable mass lens as function of the threshold signal-to-noise ratio for detection.

In reality, the shape of the light curve is sensitive to the limb-darkening profile of the source, as well as any of its surface features (e.g., Witt & Mao 1994; Gould & Welch 1996; Agol 2003; Heyrovský 2003; Yoo et al. 2004; Lee et al. 2009). The impact of included limb darkening is shown in the bottom panel of Figure 2. The "top-hat" shape disappears, and the light curve becomes more rounded. The example in Figure 2 adopted a single-parameter linear limb-darkening profile, but more structure could be added if a more complex profile was used (e.g., Claret & Bloemen 2011), or if surface features (such as starspots) were considered (Heyrovský 2003).