Revealing Short-period Exoplanets and Brown Dwarfs in the Galactic Bulge Using the Microlensing Xallarap Effect with the Nancy Grace Roman Space Telescope

Here we describe the xallarap effect observed by a single observatory. We follow the descriptions for the parallax effect observed by space-based observatories of B18 and then modify it for the case of xallarap effect.

In general, the observed flux of the microlensing event is

$$\begin{eqnarray}&&F={F}_{{\rm{S}}}A+{F}_{{\rm{B}}}=\overline{F}\left[(1-\nu )A+\nu \right],\end{eqnarray} \tag{ 5 }$$

where ${F}_{{\rm{S}}},{F}_{{\rm{B}}},\overline{F}(\equiv {F}_{{\rm{S}}}+{F}_{{\rm{B}}})$ are the fluxes of the source, blend, and baseline, respectively. ν ≡ F_B/(F_S + F_B) denotes the blend flux ratio. ⁶ For a single-lens single-source (1L1S) model, the source flux magnification A is described by

$$\begin{eqnarray}&&A(t)=\displaystyle \frac{{u}^{2}(t)+2}{u(t)\sqrt{{u}^{2}(t)+4}},\end{eqnarray} \tag{ 6 }$$

where u is the magnitude of the lens–source separation vector normalized by the angular Einstein radius θ_E, u . For uniform linear motions between the source, lens, and observers, $u(t)=\sqrt{{\tau }^{2}+{u}_{0}^{2}}$, where τ ≡ (t − t₀)/t_E, t₀ is the time of the magnification peak, and u₀ is the lens–source impact parameter normalized to θ_E.

Figure 1 gives a schematic view of the xallarap problem. Here we consider a planet in a circular orbit (e = 0) around a source star with orbital period P_ξ and mass M_P . Then the source also orbits around the barycenter of the source system. In this paper, we assume that the source companion contributes no flux to the event, i.e., it acts as a 1L1S event, not a binary-source event (Han & Jeong 1998). The displacement of the source position due to the orbital motion can be described by

$$\begin{eqnarray}&&{\boldsymbol{S}}(t)=\left(\begin{array}{c}{s}_{1}\\ {s}_{2}\end{array}\right)=\left(\begin{array}{c}\cos {\rm{\Omega }}-\cos {\phi }_{\xi }\\ \sin {\lambda }_{\xi }(\sin {\rm{\Omega }}-\sin {\phi }_{\xi })\end{array}\right),\end{eqnarray} \tag{ 7 }$$

where Ω = ω(t − t₀) + ϕ_ξ and ω = 2π/P_ξ . Here, λ_ξ denotes the inclination of the source orbital plane with respect to the observer and ϕ_ξ denotes the orbital phase at t₀. We define θ as the angle between the direction of the lens–source relative motion and the major axis of the source orbit projected on the sky. When we define the xallarap vector ${{\boldsymbol{\xi }}}_{{\rm{E}}}=({\xi }_{{\rm{E}},\parallel },{\xi }_{{\rm{E}},\perp })={\xi }_{{\rm{E}}}(\cos \theta ,\sin \theta )$, the displacement of the source position due to the orbit relative to the inertial source position is

$$\begin{eqnarray}&&\delta \tau ={{\boldsymbol{\xi }}}_{{\rm{E}}}\cdot {\boldsymbol{S}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\delta \beta ={{\boldsymbol{\xi }}}_{{\rm{E}}}\times {\boldsymbol{S}},\end{eqnarray} \tag{ 9 }$$

where ∣ ξ _E∣ = a_S/(D_S θ_E). The lens–source separation vector u (t) can be described by

$$\begin{eqnarray}&&{\boldsymbol{u}}(t)=\left(\begin{array}{c}{\tau }^{{\prime} }\cos \theta -{u}^{{\prime} }\sin \theta \\ {\tau }^{{\prime} }\sin \theta +{u}^{{\prime} }\cos \theta \end{array}\right),\end{eqnarray} \tag{ 10 }$$

where ${\tau }^{{\prime} }=\tau +\delta \tau$ and ${u}^{{\prime} }={u}_{0}+\delta \beta$. The xallarap model can be described by 10 parameters as

$$\begin{eqnarray}&&{\boldsymbol{\zeta }}=(\overline{F},\nu ,{t}_{0},{t}_{{\rm{E}}},{u}_{0},{\xi }_{{\rm{E}},\parallel },{\xi }_{{\rm{E}},\perp },{\phi }_{\xi },{\lambda }_{\xi },{P}_{\xi }).\end{eqnarray} \tag{ 11 }$$

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To estimate the expected uncertainty of each parameter (ζ_i ) by the Roman microlensing survey, we calculate the Fisher matrix of the light-curve model F(t_k ,  ζ ) with given parameter set ζ . Under the assumption of independent errors, the Fisher matrix elements b_i,j can be written as

$$\begin{eqnarray}&&{b}_{i,j}=\displaystyle \sum _{k=1}^{N}\displaystyle \frac{1}{{\sigma }_{k}^{2}}\displaystyle \frac{\partial F({t}_{k})}{\partial {\zeta }_{i}}\displaystyle \frac{\partial F({t}_{k})}{\partial {\zeta }_{j}},\end{eqnarray} \tag{ 12 }$$

where N is the total number of the data points and σ_k is the photometric error on data point at t_k . Once the Fisher matrix is calculated, the covariance matrix for parameters, C , is given by its inverse matrix, i.e.,

$$\begin{eqnarray}&&{\boldsymbol{C}}={{\boldsymbol{b}}}^{-1}.\end{eqnarray} \tag{ 13 }$$

We follow the logic of M16 and discard (negligible) the contribution of $\bar{F}$ from our Fisher matrix analysis. ⁷ In principle, the uncertainties for the xallarap amplitude depend on the event geometry, i.e., ${\sigma }_{{\xi }_{{\rm{E}}}}^{2}(\theta ,{\phi }_{\xi })$. To produce the results which are independent from the geometric conditions, M16 analytically found the minimum uncertainty on the parallax measurement, ${\sigma }_{{\pi }_{{\rm{E}}},\min }^{2}$, which is independent of θ. We modify it for xallarap measurements ${\sigma }_{{\xi }_{{\rm{E}}},\min }^{2}({\phi }_{\xi })$ as

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{\xi }_{{\rm{E}}},\pm }^{2}({\phi }_{\xi }) & = & \displaystyle \frac{{\sigma }_{{\xi }_{{\rm{E}},\parallel }}^{2}+{\sigma }_{{\xi }_{{\rm{E}},\perp }}^{2}}{2}\\ & \pm & \displaystyle \frac{\sqrt{{\left({\sigma }_{{\xi }_{{\rm{E}},\parallel }}^{2}-{\sigma }_{{\xi }_{{\rm{E}},\perp }}^{2}\right)}^{2}+4\mathrm{cov}{\left({\xi }_{{\rm{E}},\parallel },{\xi }_{{\rm{E}},\perp }\right)}^{2}}}{2},\\ {\sigma }_{{\xi }_{{\rm{E}}},\min }^{2} & \equiv & \mathop{\min }\limits_{{\phi }_{\xi }\in [0,2\pi ]}{\sigma }_{{\xi }_{{\rm{E}}},-}^{2}({\phi }_{\xi }).\end{array}\end{eqnarray} \tag{ 14 }$$

As M16 noted, for P_ξ  ≪ u₀ t_E, the covariance between the xallarap vector components ξ_E,∥ and ξ_E,⊥ disappear so that ${\sigma }_{{\xi }_{{\rm{E}}}}$ becomes independent of ϕ_ξ .

In this work, we assume continuous observations of 72 days with a 15 minute cadence and Gaussian photometric errors that are consistent with simulated photometric error bars shown in Figure 4 of P19. Periodic correlated noise is expected to be mainly produced by spacecraft systematics and stellar pressure-driven (p-mode) oscillations. Detailed photometric simulations on all systematic errors are computationally expensive. P19 applied suboptimal aperture photometry in their simulated photometric pipeline and added a Gaussian systematic error floor of 1 mmag in quadrature into their photometric results to compensate for unmodeled systematic errors. Timescales of xallarap signals, which are typically >0.1 day, are much longer than the expected timescale of p-mode oscillations of main-sequence stars (Broomhall et al. 2009). Moreover, Gilliland et al. (2015) found that most Kepler stars have a median photometric variability of ∼0.2 mmag, with a timescale of <0.6 hr. This is somewhat smaller than the error floor of P19 of 1 mmag.

We also assume the source mass and distance to be M_S = 1 M_⊙ and D_S = 8 kpc, respectively, and the angular Einstein radius to be θ_E = 0.3 mas, which approximately leads to

$$\begin{eqnarray}&&{\xi }_{{\rm{E}}}\sim 4.6\times {10}^{-5}\left(\displaystyle \frac{{M}_{P}}{{M}_{\mathrm{Jup}}}\right){\left(\displaystyle \frac{{P}_{\xi }}{\mathrm{day}}\right)}^{2/3}.\end{eqnarray} \tag{ 15 }$$

In our analysis, we consider the orbital inclination of λ_ξ  = 45° and t₀ to be at the center of the 72 day Roman observing window. Our result is hardly dependent on θ because we adopt ${\sigma }_{{\xi }_{{\rm{E}},\min }}$ which is independent of the event geometry. Here we set θ = 45°. For each W146_S , we adopt the value of the blend flux ratio ν to be the same with the median value of the ν distribution that is obtained by the P19 simulation. Note that we that assume the microlens parallax effect does not affect the measurement of the xallarap effect because the xallarap period we focus on here is much shorter than 365 days for the parallax, and is thus likely distinguishable. We do not consider the finite source effect (Witt & Mao 1994) because the effect can be easily modeled and is distinguishable from xallarap. We also do not consider binary-lens events. Although the binary-lens event would be more sensitive to the xallarap effect than the single-lens event, it is outside the scope of this work. We also ignore any accelerations induced by Roman's orbit around L2.

The nominal expected fractional error on the companion mass can be derived from Equation (3) as

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{{M}_{P}}}{{M}_{P}}=\sqrt{{\left(\displaystyle \frac{{\sigma }_{{\xi }_{{\rm{E}},\min }}}{{\xi }_{{\rm{E}}}}\right)}^{2}+\displaystyle \frac{4}{9}{\left(\displaystyle \frac{{\sigma }_{{P}_{\xi }}}{{P}_{\xi }}\right)}^{2}}\end{eqnarray} \tag{ 16 }$$

by assuming the uncertainties on θ_E, D_S, and M_S are negligible. We set the detection threshold for the xallarap companions as $\left({\sigma }_{{M}_{P}}/{M}_{P}\right)=0.3$ in the following analysis. The impacts of uncertainties of θ_E, D_S, and M_S on the measurements of companion's masses is discussed in Appendix B.

Figure 2 shows two samples of model light curves with and without the xallarap effect in the top panels. The residuals of the light curves between that with and without xallarap are shown in the second panels. The left figure represents an event with t_E = 30 days, u₀ = 0.1, M_P  = 2 M_Jup, and P_ξ  = 5 days. In this case, the maximum deviations from the standard model is about 0.5% of the source flux, which is comparable to the Roman photometric noise level for W146 ∼ 20 mag (see Figure 4 of P19). This indicates that brighter and/or high-magnification events are promising targets for Roman to detect xallarap features induced by a Jupiter-mass planet around the source star. In the three bottom panels, we plot the components of integrands of the Fisher matrix ∂F(t_k )/∂ζ_i at a given time t_k . These panels imply that the observations during t₀ ± P_ξ are most important to determine the xallarap parameters (ζ_i ) related to the mass of companion M_P . However, note that observations further into the wings continue to add to the precision of the period estimate (see the bottommost panels).

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For understanding what parts of light curves are important to constrain the parameters, we derive the parameter uncertainties using the Fisher matrix analysis each time a Roman observation is conducted. Figure 3 represents the cumulative precision on ξ_E and P_ξ as a function of time for the Roman light curve with W146_S = 18 mag (corresponding to Figure 2). We found that the parameter uncertainties are gradually constrained with increasing data points from the wing of the light curves. Moreover, we also derive the cumulative precision on the parameters as a function of the coverage duration for the light-curve peak (Figure 4). We found that the resultant parameter precision strongly depend on how long the light curves cover around the event peaks. Figures 3 and 4 indicate that the xallarap parameters can be measured with incomplete light curves that cover around t₀ ± P_ξ . This is particularly important for Roman, which has a short observing window of 72 days.

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At a given (t_E, u₀, W146_S), we conducted the Fisher matrix analysis on a grid of points over the ranges of $-2\leqslant \mathrm{log}({M}_{P}/{M}_{\mathrm{Jup}})\leqslant 2$ and $-1\leqslant \mathrm{log}({P}_{\xi }/\mathrm{day})\leqslant 3$ with 20 × 20 grid points, respectively. In Figure 5, we present samples of xallarap sensitivity maps in the mass–orbital period plane for the Roman event with W146_S = 18 mag. The color maps in each panel represent the distributions of ${\sigma }_{{M}_{P}}/{M}_{P}$ and the black lines correspond to contour lines of our detection threshold of $\left({\sigma }_{{M}_{P}}/{M}_{P}\right)=0.3$. The row and column of a panel correspond to the labeled values of u₀ and t_E. For example, in the case of $\mathrm{log}({t}_{{\rm{E}}})=2,\mathrm{log}({u}_{0})=-1.5$ (bottom right panel), the Roman light curve has the potential to precisely measure the mass of a warm Jupiter with an orbital period of 10–30 days via xallarap. We found that M_P is well constrained when t_E is longer and u₀ is smaller in a given (M_P , P_ξ ) grid. One also can find that there are sharp cutoffs of the sensitivity with the orbital period of a few dozen days. This might be because the Roman observing window of 72 days could not cover the full orbital period of the events beyond the cutoff and thus is insufficient to constrain the xallarap parameters. ⁸ This can be expected from the results of Figures 3 and 4.

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In this section, we estimate the detection number of close-in planets and brown dwarfs (BDs) in source systems via xallarap during the Roman mission. To simulate the Roman microlensing survey, we employ the single stellar lens module of the GULLS microlensing simulator (Penny et al. 2013, 2019) which uses version 1106 of the Besançon Galactic population synthesis model (Robin et al. 2003, 2012) to generate pairs of lens and source stars. In Table 1, we summarize the survey parameters for the Cycle 7 design that we use in our simulation. The full survey details are described in P19 and Johnson et al. (2020). Note that we consider only single-lens events whose peaks are within the Roman observing window.

We classified the simulated Roman events from GULLS by the values of (u₀, t_E, and W146_S) into a bin of 7 × 10 × 10 over the ranges of 14 < W146_S < 28 mag, −2 < logu₀ < 0, and $0\lt \mathrm{log}({t}_{{\rm{E}}}/\mathrm{day})\lt 2.5$, respectively. We generated the sensitivity maps with the parameters at the center of each bin to be used for all events in each bin. Then we counted the number of detected events using the corresponding sensitivity maps at each (M_P , P_ξ ) grid. Figure 6 shows the resultant detections, i.e., the expected planet yields during the Roman survey mission if all source stars were to have a planet at each (M_P , P_ξ ) grid point. The black solid lines represent the contours of the planet yields for 1, 10, 100, and 1000. The detection sensitivity peaks around P_ξ  = 20 ∼ 30 days and there it reaches to sub-Jovian or Saturn masses. In Figure 6, we also plotted observed exoplanets (open dots) from the NASA Exoplanet Archive ⁹ (Akeson et al. 2013). Roman's sensitivity to planets via the xallarap effect largely covers the parameter spaces of hot and warm Jupiters with M_P  > 0.5 M_Jup and 0.1 < P < 100 days, which suggests that this method could be useful to probe the hot and warm Jupiter populations in the Galactic bulge. Note that we used only a single 72 day season for each event.

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In order to estimate the planet yields, we assume the secondary mass and period distributions. We describe the distribution function f as a double power law,

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}f}{\partial \mathrm{ln}M\,\partial \mathrm{ln}P}={C}_{\mathrm{norm}}{\left(\displaystyle \frac{{M}_{P}}{{M}_{\mathrm{Jup}}}\right)}^{{\alpha }_{M}}{\left(\displaystyle \frac{P}{\mathrm{day}}\right)}^{{\beta }_{P}},\end{eqnarray} \tag{ 17 }$$

where C_norm is a normalization factor. In this work, we adopted the power-law indexes of α_M  = −0.31 ± 0.2 and β_P  = 0.26 ± 0.1 derived by Cumming et al. (2008, hereafter C08). These values are derived using 48 radial-velocity-detected planets ranging 0.3 < M_P  < 10 M_Jup and 2 < P < 2000 days that are around Sun-like stars. We adopted C_norm = 0.036 dex⁻² star⁻¹ to be consistent with a planet frequency of 10.5% around Sun-like stars in these ranges derived by C08. Note that we simply extrapolate this power law of C08 to the ranges 0.01 < M_P  < 100 M_Jup and 0.1 < P < 1000 days because it is still uncertain. The extrapolation below 0.3 M_Jup hardly affects the final result because the sensitivities to low-mass planets with <0.3 M_Jup is very low and the Kepler survey suggested that the occurrence rate does not significantly rise until below Neptune size of ∼5 M_⊕ (e.g., Fressin et al. 2013). Extrapolations to other ranges require caution as discussed below. We also estimate the yields assuming a simple frequency model of (α_M , β_P ) = (0,0) and C_norm = 0.208 dex⁻² star⁻¹, which corresponds to single planet per star over the ranges. This can be considered as the reference yields.

Figure 7 represents the expected yields of the Roman observations assuming the extended C08 distribution. The three left panels show the distributions of t_E, u₀, and W146_S for events in which the planet/BD companion around the source is detected. The yields are expected to largely come from the events with 18 < W146_S < 22 mag, which mostly consist of main-sequence source stars. Note that although the histogram with u₀ indicates that the detections increase toward larger u₀, as shown in Figure 5, the events with large u₀ are only sensitive to massive companions. The right panel in Figure 7 shows the distribution for the number of planet and BD detections over the parameter space of masses and periods assuming the frequency from C08. Table 2 summarizes the expected yields assuming the two different distribution functions. Adopting the extended C08 distribution, we found that ∼10 planets with M ≤ 10 M_Jup would be detected by xallarap. We can expect ∼30 companions with 10 < M_P  < 100 M_Jup if the extrapolation of C08 is correct. However, due to the "BD desert" (Grether & Lineweaver 2006), BD discoveries may by much less common than predicted using the C08 frequencies. ¹⁰ We can test the BD desert in the Galactic bulge by applying this method to the upcoming Roman light curves.

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If the lensing body is in a binary system, lens orbital motion (LOM) will provide a similar effect to that of xallarap, which has been pointed out in several papers (Rahvar & Dominik 2009; Penny et al. 2011). In a planetary system with an orbital period of P ≤ 30 days, a projected angular separation between a lensing host star and its planet in units of θ_E, s = a/θ_E, is expected to be an order of s ≤ 0.06. It is unlikely that a lens companion with such a small s provides noticeable light-curve deviations by their extremely small caustics. For example, Penny et al. (2011) found that periodic (caustic) features of a light curve due to LOM would be most detectable for binary lens with semimajor axes of ∼1 au. Thus it is difficult to distinguish between xallarap and LOM. However the degeneracy may be resolved when the following additional high-order effects are observed in the light curves. The following effects can be observed only in the xallarap events, not in the LOM events.

• Magnified planet flux. Planets orbiting source stars produce reflected light from their host and/or emit their own flux from thermal emission. If flux from these companions are magnified, we can observe these contributions in the light curve, as a so-called binary -source microlensing event (Graff & Gaudi 2000; Sajadian & Rahvar 2010). Typical flux ratios between solar-type stars and hot Jupiters in the W146 band is on the order of ∼10⁻³–10⁻⁴. Using this detection channel, Bagheri et al. (2019) estimated that Roman will discover ∼70 exoplanets from single-lens events and ∼3 exoplanets from binary-lens events. ¹¹ We expect that the binary-source effect might be observed in the Roman xallarap single-lens events of P < 5 days, which corresponds to ∼50% of the total yields. It would also help us to constrain the orbital parameters of source systems because it provides the geometric relation between the host and planet at the time when the planet is magnified. Full binary-source modeling including the xallarap effect (Miyazaki et al. 2020) is more realistic and might have more sensitivity, but is out of the scope of this work.
• Transiting source stars. If the planet transits the source star, we can also simultaneously observe the transiting signal during the microlensing event (Lewis 2001; Rybicki & Wyrzykowski 2014). Typical amplitudes of the transit signal for a Jupiter-sized planet would be ∼1% of the source brightness, which will be easily detectable for most xallarap planetary events. For example, Roman is expected to detect thousands on transiting hot Jupiters, including in the Galactic bulge (McDonald et al. 2014; Montet et al. 2017). The geometric transit probability of warm Jupiters is ∼5%, so that ∼5% of the xallarap planetary events could be distinguishable from LOM events by the transit signals. With the measurement of the planet radius by the transit, we can estimate the density of the planet in combination with the mass measurement by xallarap. And, this potentially can test how chemistry affects giant planet structures (e.g., Cabral et al. 2019).
• Ellipsoidal variations. Ellipsoidal variations can be caused by tidal effects on the source from the companion (Morris 1985). The amplitude of the ellipsoidal variation is approximated by

  $$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{ellip}} & \simeq & {\alpha }_{\mathrm{ellip}}\displaystyle \frac{{M}_{P}\,\sin {\lambda }_{\xi }}{{M}_{{\rm{S}}}}{\left(\displaystyle \frac{{R}_{{\rm{S}}}}{{a}_{{SC}}}\right)}^{3}\sin {\lambda }_{\xi }\\ & = & 13\,\mathrm{ppm}\,\left(\displaystyle \frac{{M}_{P}\sin {\lambda }_{\xi }}{{M}_{\mathrm{Jup}}}\right)\,{\left(\displaystyle \frac{{R}_{{\rm{S}}}}{{R}_{\odot }}\right)}^{3}\\ & & \times \,{\left(\displaystyle \frac{{M}_{{\rm{S}}}}{{M}_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{P}_{\xi }}{\mathrm{day}}\right)}^{-2}{\alpha }_{\mathrm{ellip}}\sin {\lambda }_{\xi },\end{array}\end{eqnarray} \tag{ 18 }$$

  where R_S is the radius of the source star. The coefficient α_ellip accounts for the stellar limb darkening and gravity darkening,

  $$\begin{eqnarray}&&{\alpha }_{\mathrm{ellip}}=0.15\displaystyle \frac{(15+u)(1+g)}{3-u},\end{eqnarray} \tag{ 19 }$$

  where g and u are the coefficients of the gravity darkening and linear limb darkening, respectively (Shporer 2017). It would be difficult to observe this effect for the xallarap planetary events. However, it might be possible for events of substellar source companions.
• Doppler beaming. It is known that a line-of-sight motion due to the orbit causes a periodic variation in the light curve, also known as Doppler beaming (Loeb & Gaudi 2003; Shporer 2017). The photometric amplitude of the beaming effect A_beam is described as

  $$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{beam}} & = & 2.8\times {10}^{-3}\,{\alpha }_{\mathrm{beam}}{\left(\displaystyle \frac{{P}_{\xi }}{\mathrm{day}}\right)}^{-1/3}\\ & & \times \,{\left(\displaystyle \frac{{M}_{{\rm{S}}}+{M}_{P}}{{M}_{\odot }}\right)}^{-2/3}\left(\displaystyle \frac{{M}_{P}\sin {\lambda }_{\xi }}{{M}_{\odot }}\right),\end{array}\end{eqnarray} \tag{ 20 }$$

  where α_beam is the coefficient accounting of the variation of photon amounts in a specific band, e.g., α_beam ≡ 1 in bolometric light. For a longer orbital period of P_ξ  > 10 days, the amplitude of this effect is much stronger than that of the ellipsoidal variations and it is more promising to observe.

These effects may be able to resolve the degeneracy between the xallarap and LOM interpretations in some fraction of xallarap events and provide additional information for source systems.

Some observational results have suggested that there are possible differences in planetary populations between the local neighborhood and the distant region of our Milky Way. Radial velocity surveys around the local region close to the Sun indicate an occurrence rate of hot Jupiters of an order of 1% (Cumming et al. 2008). On the other hand, the Kepler transit survey toward the Cygnus region suggested the occurrence rate of 0.4% ± 0.1% (Howard et al. 2012), which is approximately half of that in the local neighborhood. Penny et al. (2016) used a sample of 31 microlensing exoplanetary systems and suggested that the abundance of planets might be less in the Galactic bulge than the disk. Such studies would be very important to help understand whether the planetary formations could depend on their surrounding environments in our Galaxy.

Montet et al. (2017) predicted that Roman is expected to discover ∼100,000 transiting planets and enable us to directly confirm several thousands hot Jupiters via its secondary eclipses in the light curves. A large fraction of the transiting planets will belong to the Galactic bulge. However, in general, most host stars are too faint to conduct follow-up radial velocity observations to constrain their planetary mass and avoid false positives. Montet et al. (2017) also suggested some feasibility for the validation for transiting planets and the estimation of their masses using phase curve variations, although it requires high photometric precision to observe. Applying our xallarap method to the Roman microlensing light curves, we can expect to discover some dozens of hot and warm Jupiters and close-in BDs with measured masses. These samples will help our understanding of the exoplanet demographics at short orbital periods in the Galactic bulge. For larger masses, they are also useful to probe the BD desert around main-sequence stars and study the stellar binary distribution in the Galactic bulge.