GRB lensing parallax: Closing primordial black hole dark matter mass window

The primordial black hole (PBH) comprising full dark matter (DM) abundance is currently allowed if its mass lies between $10^{-16}M_{\odot} \lesssim M \lesssim 10^{-11} M_{\odot}$. This lightest mass range is hard to be probed by ongoing gravitational lensing observations. In this paper, we advocate that an old idea of the lensing parallax of Gamma-ray bursts (GRBs), observed simultaneously by spatially separated detectors, can probe the unconstrained mass range; and that of nearby stars can probe a heavier mass range. In addition to various good properties of GRBs, astrophysical separations achievable around us --- $r_\oplus \text{--}$ AU --- is just large enough to resolve the GRB lensing by lightest PBH DM.


I. INTRODUCTION
The PBH has become an interesting DM candidate. Possibly having been formed from overdensities in primordial density fluctuations during inflation, it might have evaporated too much by today if it is too light or could have disturbed CMB if too heavy. A wide mass range between those general constraints remains open as DM.
PBH DM have been probed mainly by two categories: gravitational lensing and PBH-induced dynamics. The latter includes dynamical disruption, capture, and friction of observed systems, but are subject to various assumptions and uncertainties [1,2]. Maybe the lensing is a more robust and direct probe. Among several proposals, basically only microlensing of nearby stars have been thoroughly measured, constraining 10 −11 M ∼ 10 M [3][4][5][6][7]; see also [8]. Heavier PBHs are expected to be probed by strong lensing of supernova [9]/FRB [10]/GRB and lensing fringes of GW at aLIGO [11]. On the other hand, lensing cannot efficiently probe lighter PBHs in the range 10 −16 M -10 −11 M , as lensing is typically too weak and fragile to be resolved; in particular, source size and wave-optics effects are limitations [12,13].
In this paper, we study and advocate lensing parallaxes of GRBs and stars as important probes of PBH DM, with a potential to probe the whole possible mass range with lensing.

II. LENSING PARALLAX
Lensing parallax allows the detection of lensing by correlating the brightnesses (or fluxes) of a source simultaneously measured by spatially separated detectors [14]. If the spatial separation is larger than about the Einstein radius of a PBH lens, the two detectors will observe different lensing magnifications. The observed magnification of unresolved images is and time-dealy between images ∆t d . Eq. (2) is applicable for a point-lens and source in the geometrical optics, but will be corrected by source-size effects in final results; see below Eq. (4) and Appendix B. It is this relative angle y of a source with respect to a lens (normalized by the lens Einstein angle) that can differ between spatially separated detectors by order one, causing lensing parallax. Since the interference fringe term quickly averages out, magnification becomes A µ 1 + µ 2 , frequency-independent for the majority of parameter space. Thus, different brightness with the same energy spectra between simultaneous measurements is a basic signal of lensing. However, the interference fringe (in the energy spectrum) [11,15,16] becomes relevant when the wavelength λ is about to be too long to probe small PBHs with mass M , i.e. λ ∼ O(GM/c 2 ) (equivalently, c∆t d /λ ∼ O(1)). In this regime, frequencydependent magnification (i.e., fringe and wave-optics effects) can further strengthen lensing-parallax detection. Our study is mainly based on the former feature (frequency-independent), but the latter is also relevant near the lower mass limit. Lensing parallax is similar to microlensing, in that brightness is measured at multiple relative angles to detect lensing. But lensing parallax can be detected even with short transients, based on simultaneous observations from well-separated detectors (whereas microlensing needs long-lasting sources). This allows us to use GRB pulse in lensing parallax [14]; as will be discussed, GRB is ideal to probe lightest unconstrained PBHs. Using pulses, we assume lensing effects do not change during measurements. The correlation can be made unambiguously between a whole time-series of pulse data, but longer temporal variation can also be cross-correlated to further strengthen the detection. The lensing parallax of long-lasting stars can also be detected in either way. We assume to use only short time-segment in our estimations.
Lensing parallax requires the following conditions to be met. First, the detector separation, ∆r, should be larger than about the Einstein radius, r E , of a PBH lens: where typical distance to the lens and source is denoted by D ∼ D S ∼ D L . A proportionality constant is not shown but varies orders of magnitudes depending on the fractional brightness resolution , determined by Eq. (4). This condition determines the maximum PBH mass that can be probed -M max .
Second, the apparent source angular size, θ S ∼ r S /D (where r S is the physical transverse emission size), must be smaller than about the Einstein angle θ E to induce sizable lensing; otherwise, only a small part of the source will be magnified. Requiring measured magnifications A i to be more different than the resolution, where each A i is given by Eq. (1), and we assume to compare only two measurements for simplicity. We take into account finite source size following Ref. [17] (see also Appendix B), where constant brightness profile and geometrical optics are used. The latter assumption is fine in our analysis as will be discussed; and the former, compared to profiles having a peak in the center, may slightly overestimate or underestimate the magnification depending on y. Source size r S taken into account, the maximum magnification is A max = 1 + 4/δ 2 ∼ δA (for y = 0 in the geometrical optics) with δ ≡ θ S /θ E ∼ r S /r E . Thus, we obtain (in the limit δ 1) Eq. (4) is the main condition that we solve numerically. Last but not least, the probe wavelength λ must be shorter than the Schwarzschild radius of a PBH lens; otherwise, the wave cannot see the lens. The geometrical optics is valid for where the approximate factor 0.1 is from Ref. M . We will discuss later how we treat wave optics effects.
The stronger of the last two conditions (apparent source size and wavelength) determines the minimum PBH mass that can be probed -M min . The two constraints are similar for GRBs, the latter is stronger for FRBs (as radio wavelength is much longer than gammaray's), and the former is somewhat stronger for stars (as they are closer, they appear larger, and also sources are assumed to be larger).

III. GRB PARAMETERS
We turn to estimate the sensitivities of GRB lensing parallax. We describe source parameters, how we treat wave-optics effects, and derive upper limits on the PBH abundance; see Appendix E for more details.
The physical transverse size r S of emission region of GRB sources is typically correlated with the measured minimum variability time scale t var as r S ∼ c t var · Γ/(1 + z S ) ∼ c T 90 /(1 + z S ) [19], with the observed burst duration T 90 , Lorentz boost Γ and source redshift z S . Based on GRBOX database of ∼ 2000 GRBs detected so far [20], 10% of GRBs have r S r ; 3% have smaller r S 0.1 r ; and the remaining 90% have r S r . Although constituting a minor fraction, these smallest GRBs with r S r are the ones that allow to probe lightest unconstrained PBH mass range. It is also notable that actual GRB sizes could be smaller; a recent work estimated smaller t var based on broader GRB spectrum [21], low sampling frequency and limited photon statistics might have overestimated t var [19], theoretical minimum from emission region being optically thin can also be small [12]. As a concrete example, we use the GRBOX distribution.
The GRB frequency spectrum is also relevant, as the highest frequency can probe the lightest PBHs. Rather than modeling the frequency spectrum of photon counts (which varies among GRBs) and carrying out frequency-dependent lensing analysis (needed only when Eq. (6) is not satisfied), we can simply consider a highest-frequency range where assumed sensitivity = 0.1 (conservative scenario) or 0.01 (optimistic) can be typically achieved. This is a good simplification to estimate detection prospects; if the same magnification is shown at a lower-energy range (with higher flux) or if frequency-dependent magnification is observed consistent with fringe, the lensing (fringe) detection will be strengthened. But if this highest frequency range does not show evidence of lensing, detectable lensing did not occur.
Given the typical spectrum measured by Fermi GBM+LAT [22], we assume to use ∼ 0.5 MeV as the highest frequency with = 0.1 (conservative); this is the highest frequency where O(100) photon counts per MeV is typically achieved. This frequency range allows us to use geometrical optics down to M DM 10 −16 M , according to Eq. (6). In the final results, we show results only down to this mass range and defer a dedicated waveoptics study below this range. But our geometrical-optics calculation can already cover a whole unconstrained mass range. For an optimistic scenario, we assume = 0.01 up to ∼ 5 MeV (hence, M DM 10 −17 M ). Although we use geometrical optics everywhere in this simplified analysis, we take into account source-size effects as discussed. GRBs are at cosmological distances, observed up to z S ≤ 10 and majority at z S = 0.5 ∼ 3. We use the redshift distribution from the GRBOX database [20] to calculate lensing probability. But we do not correlate distances with observed intensities; rather, for all GRBs, we will use common intensity resolution assumed above.
Benchmark detector separations are ∆r = 2r ⊕ 0.04 sec, r L2 5 sec, and 2 AU 10 3 sec. The first choice is based on telescopes on the ground and low Earth orbits such as Fermi and Swift; the second may be possible as IR-band space missions such as Gaia, WFIRST and Euclid are operating or planned on orbits around the Lagrange point L 2 of the Sun-Earth system; and the AU is a characteristic scale of the solar system, as is realized in Kepler mission. The detection benchmarks are summarized in Table. I.

IV. GRB RESULTS
The optical depth, τ , is obtained from the volume of the desired PBH locations (for the given GRB) that can lead to appreciable lensing parallax; then the expected number of PBHs within this volume is the optical depth. PBHs are assumed to be uniformly distributed with the comoving number density n = ρ crit,0 Ω DM f /M , where f ≡ Ω PBH /Ω DM is the PBH abundance; the clustering of PBH does not matter as long as their Einstein angles do not overlap, which is true for most regions of galaxies, as discussed in Appendix D. As shown in Fig. 2, τ is usually smaller than 1, but τ 1 is sizable for GRBs at far distances and for small . In this case, multiple lensing can occur. Although multi-lensing can still lead to lensing parallax, we (conservatively) focus on singlelensing cases which are much easier to calculate. The single-lensing probability is P 1 = τ e −τ for the Poisson distribution of lenses, and it becomes its maximum value 0.37 at τ = 1, which is still sizable.
In Fig. 1, we obtain 95% CL upper limits on the PBH abundance by assuming that no lensing parallax will be detected among total N GRB = 1000 (as already more than this) or 10 4 GRBs simultaneously observed by ∆r-separated detectors. The Poisson distribution of the number of lenses is used. In each of conservative and optimistic projections, three benchmark results with ∆r are shown. As discussed, we plot results down to the lowest mass where geometrical optics is valid for the highest-frequency range that we consider.  Table I. The region with f > 1 is not possible but is shown for reference. Also shown are existing (shaded) and proposed (non-shaded) lensing constraints from microlensing (SUB-ARU [3], Kepler [5], OGLE/EROS/MACHO [4,6,7]), strong lensing of FRB [10]/supernova [9], GW fringe (aLIGO) [11], and GRB fringe [12]; evaporation of too light PBHs [23], and CMB distortion from too heavy ones [24]. In all, the full range of possible PBH DM mass can be probed by lensing.
One can first understand overall behaviors of the results. The three ∆r lines have similar M min as it is determined dominantly by and r S ; M min ∼ 10 −15 −10 −16 M is indeed expected from Eq. (5) with = 0.1, r S 0.1 r , and D ∼ 1 Gpc. Optimistic results improve M min ∝ with higher frequencies available. But the three lines have different M max ∝ (∆r) 2 (Eq. (3)). The constraints on f is also improved roughly with the increase of N GRB / , yielding higher statistics.
As a result, the unconstrained PBH DM mass range -10 −16 M M 10 −11 M -can be potentially probed by lensing. Existing telescopes (having conservative specs while running on the ground and low Earth orbits) alone can barely achieve sensitivities yet; see ∆r = 2r ⊕ result. But, remarkably, one space telescope at a larger orbit (L 2 or AU), paired with existing ones, can be enough to probe a whole unconstrained mass range; see ∆r = r L2 and 2AU results. With optimistic benchmarks, any combinations of well-separated detectors (even low-orbit ones) can probe the whole unconstrained mass range.
Confidently probing the lightest mass range near 10 −16 M likely requires improvements from the conservative scenario; in particular, better resolution < 0.1. Such can perhaps be achieved with higher photon counting rates from larger detector area, higher sampling frequency, and better detection efficiency. GRB source size is also a key factor (Eq. (5)) as recently emphasized for GRB fringes [12] (their result with 10 4 ×3% = 300 GRBs with r S = 10 9 cm [12] is shown in Fig. 1). If sources turn out to be larger than the estimate used here, much better is needed. Toward the heaviest mass range, GRB lensing parallax can probe potentially up to M 0.1M . In the heavier mass range, various strong and microlensing observables are available, as shown in Fig. 1. Supernova lensing magnification can be detected as a deviation in the brightness-redshift relation (result with current data is shown [9]); FRB and GRB pulses can be strongly time-delayed or angular separated (10 4 FRBs up to z ≤ 0.5 [10]); and GW lensing fringes in chirping waveforms can be detected at aLIGO (∼ 10 3 mergers for 1 yr [11]) probing the mass range corresponding to the LIGO frequency band.

A. Nearby-Stars Results
We also estimate the lensing parallax of stars in the Milky-Way and nearby galaxies. Although it cannot probe the unconstrained mass range, star's lensing parallax can perhaps be observed with ongoing and near-future missions.
We simply assume that all stars have r S = r and we use geometrical optics. We consider 10 7 (as in current microlensings) or 10 9 stars observed within a 100 kpc radius around us. Such a large number of sources can compensate the small optical depth (τ 10 −5 shown in Fig. 2, similar to microlensing values [3]). DM comprises the total galaxy mass 10 12 M and are isotropically distributed around us.
Although stars are long-lasting, we assume to consider short time-segments as discussed; but annual parallaxes in microlensing observations have been studied, e.g. [25,26]. We simply require the same lensingdetection criteria and detector benchmarks as in the GRB case; but prospects for realizing various ∆r and good resolution are higher here, as various IR-band space missions are already operating with such specs or planned to be achieving them.
The sensitivities are overlapped in Fig. 1. Stars are sensitive to heavier PBHs; as they are closer than GRBs, they appear larger and the Einstein radius of a PBH is smaller. Compared to GRBs, M max is larger by roughly (1 Gpc/100 kpc) ∼ 10 4 (see Eq. (3)) while M min is larger by ∼ 10 5 −10 6 as no stars are assumed to be smaller than the solar size. Notably, ongoing and proposed missions on the ground and in large orbits (L 2 or AU) can already achieve any of conservative results, probing 10 −9 M M 10 −2 M . Optimistically, 10 −12 M M 10 3 M can all be probed, complementary to microlensing and strong lensing.

V. DISCUSSION
The GRB has unique advantages in completing the lensing probe of PBH DM. The large distance and compact source make it appear small compared to the Einstein angle of the lightest PBH. High-frequency gamma rays not only probe small PBHs but make the lensing effect insusceptible to intergalactic scattering dispersion, one of the limiting factors for FRB fringes [27]. Compared with GRB fringes (also known as femtolensing, shown in Fig. 1), intensity correlation relaxes the sourcesize constraint, hence probing a much wider mass range. Lastly, the lensing parallax enables the detection of lensing with short pulses of GRBs.
In all, the full range of PBH DM mass can be probed by lensing. GRB lensing parallax can uniquely probe the unconstrained lightest mass range, while star's lensing parallax can probe a heavier range complementary to existing observations. The maximum mass range is determined by detector separation, while the minimum depends crucially on GRB source size r , high-energy spectrum 1-100 keV, and brightness resolution 0.1. It is also important to have a wide field of view such as in GBM in order to obtain high event rate. All these technologies are likely already available, and one (paired with existing ones) or two such space telescopes will be good enough to accomplish this probe. Fortunately, the astrophysical scale accessible to us -from r ⊕ to AU -is just right to probe the lightest PBH mass range, which cannot be lensing probed otherwise.

ACKNOWLEDGMENTS
Authors would like to thank Han Gil Choi, Myungshin Im, Ji-hoon Kim, Chang Dong Rho, Il H. Park, Sascha Trippe and Eunil Won for useful discussions. Authors are supported by Grant Korea NRF-2019R1C1C1010050, 2015R1A4A1042542, and SJ also by POSCO Science Fellowship.

Appendix A: Useful numbers and formulas
Here we collect useful numbers and formulas, with c = 1.
Cosmological constants we assumed are The Einstein radius of a PBH is The Schwarzschild radius is The time-delay is Appendix B: Finite source-size effect We present the equation for the magnification A that we used in the calculation, with finite source-size considered.

Appendix C: Computing optical depth and constraints
Here we explain our computational procedure for the optical depths and the bounds on the PBH abundance.
We assumed a uniform distribution of PBHs with number density n, so the optical depth for a source of physical transverse size r S at comoving distance χ S is given by where V L is the desired comoving volume for PBHs and ∆r ⊥ = ∆r cos θ is the component of ∆r perpendicular to the line-of-sights (LOSs). The volume V L is given by where σ is the comoving cross-section for the lensingdetection of PBH at comoving distance χ L . Since the universe is flat, we do not distinguish the transverse comoving distance from the usual LOS comoving distance.
To calculate σ, we first draw a region on the lens plane that at least one of the A 1 or A 2 exceeds 1 + (the necessary condition for Eq. (4) of the main paper). Then we draw a rectangular region circumscribing the previous one, and divide into 6400 equal-sized rectangular sectors (80 divisions for each edge). Next, we check whether Eq. (4) of the main paper is satisfied at the center of each sector, and summing up the area of the relevant sectors gives σ.
Going back to Eq. (C2), we equally divided the χ L range into 24 sectors and used the midpoint quadrature for the integral. Finally, multiplying it by n gives the optical depth τ . Note that the only dependence on f comes from n ∝ f , so single evaluation of τ at f = 1 immediately gives τ (f ). Now we move onto the bound calculation. The probability of having no detection for observing N sources is given by P null (f ) = N i=1 (1 − P 1,i (f )), where i is the source index. The bound of f is the one which gives P null (f ) = 0.05. In the actual calculation, we sample 1000 sources, with distance and source size randomly selected according to the distributions given by Fig. 4 and with purely random angular positions. We approximated P null by assuming that each sampled source represents N/1000 observations, giving P null (f ) ≈ 1000 i=1 (1 − P 1,i (f )) N/1000 .

Appendix D: Effects of PBH clustering
We now demonstrate that assuming a uniform distribution of PBHs is decent in calculating lensing parallax, even though nearly half of DM is thought to be clustered around galaxies. As discussed in the paper, we (conservatively) consider only single-lensing events; multi-lensing can still lead to parallax, but is more complicated to calculate and analyze. Thus, we show that the single lensing probability P 1 does not change significantly for clustered PBHs in most regions of galaxies.
First of all, the clustering does not change the optical depth τ , as the average number of lenses within the V L (the desired volume of PBH locations for lensing) remains the same. But P 1 may still change because once a lens is within V L it is more likely that there are other clustered lenses within the same V L so that multi-lensing occurs more often than single lensing. More quantitatively, the number of lenses within V L no longer follows the Poisson distribution.
Suppose a LOS passes through N halos (the clustered PBH) and the optical depth within each halo is denoted by τ H (the more clustered within a halo, the larger τ H ); then the expectation value of N = τ /τ H . The singlelensing probability for the case of N halos is P 1|N halos = (number of halos) × (probability for one halo to give single lensing) × (probability for other halos to give no lensing) where we assume that PBH DM is clustered but uniformly distributed within each halo. Summing N with its own Poisson distribution, we obtain the total probability for single lensing Fig. 3 shows P 1 (τ ) for three selected values of τ H . The deviation from the uniform distribution is sizable for τ H 1 irrespective of τ ; this is the manifestation of non-Poissonian properties in Eq. (D2). This is understandable because τ H 1 directly means that clustering is so high that there are likely multiple lenses within the V L of a single halo. The τ H is related to the halo radius R H with the mass M H . The PBH number density within the halo is n |H = (M H f /M )/(4/3 × πR 3 H ). The V L within the halo, denoted by V L|H , is approximately the cylinder with a length R H and a (Einstein) cross-section σ = k · π(D L θ E ) 2 , where k is determined by detection criteria and detector sensitivities (see below for realistic values of k). Thus we obtain τ H as a function of R H (D3) This allows us to interpret Fig. 3 for Milky-Way-like galaxies. For M H = 10 12 M , τ H becomes 1 for R H 4.4 kpc. The Milky-Way is thought to have a much larger halo ∼ 100 kpc. Thus, the PBH clustering is small enough not to affect our calculations based on the uniform distribution of PBH. For reference, the Milky-Way gives τ H 2 × 10 −3 with R H = 100 kpc. In all these estimations, we use k = 10 and f = 10 −2 ; the values of k in our results are usually 10 but grow with improvement so that the clustering and multi-lensing can become more relevant in the future.
Appendix E: GRB source parameter distribution Finally, we present the distributions for GRB redshifts and transverse sizes that we use.
The GRB data on [20] were used to obtain the redshift and the transverse size distributions for observed GRBs. Out of 2105 GRB observations, 487 have redshift information and 1944 have T 90 information which is converted to the transverse size by r S ∼ c × T 90 [19].   4 shows the redshift and transverse size distribution of observed GRBs. For the redshift distribution, the population extends to very high redshifts of z ≤ 10, but the majority of the population lies in z 3. For the transverse size distribution, the bimodal distribution is shown while the majority of the population lies in r S ≥ r . Out of 1944 GRBs with transverse size converted from T 90 , 226 (12%) have r S ≤ r and only 58 (3%) among them satisfies r S ≤ 0.1r . These fractions are smaller than the analysis done in [12,21], estimating about 10% of observed GRBs have r S ≤ 0.1r . Although the proper transverse size should be estimated by T 90 /(1 + z S ), we omitted the redshift factor in calculating the size distribution to be conservative for unknown redshifts of the majority of GRBs used in obtaining the transverse size distribution.