Microlensing of x-ray pulsars: A method to detect primordial black hole dark matter

Primordial black holes (PBHs) with a mass from ${10}^{-16}$ to ${10}^{-11}{M}_{\odot }$ may comprise 100% of dark matter. Due to a combination of wave and finite source size effects, the traditional microlensing of stars does not probe this mass range. In this paper, we point out that x-ray pulsars with higher photon energies and smaller source sizes are good candidate sources for microlensing for this mass window. Among the existing x-ray pulsars, the Small Magellanic Cloud (SMC) X-1 source is found to be the best candidate because of its apparent brightness and long distance from Earth. We have analyzed the existing observation data of SMC X-1 by the RXTE telescope (around 10 days) and found that PBH as 100% of dark matter is close to but not yet excluded. Future longer observation of this source by x-ray telescopes with larger effective areas such as AstroSat, Athena, Lynx, and eXTP can potentially close the last mass window where PBHs can make up all of dark matter.

Figure 1

The averaged magnification factor $\overline{\mu }$ for a range of photon energies as a function of $y$ defined as the ratio of the tangential source-lens separation in the lens plane over the Einstein radius. The source is assumed to be pointlike for this plot. The black and dashed line is in the infinite mass limit.

Figure 2

The allowed parameter space (below the curves) in $x\text{−}y$ after including both wave and finite source size effects for two different energy-averaged magnification factors ${\overline{\mu }}_T=2.0$ (left panel) and 1.3 (right panel).

Figure 3

The averaged time interval between two lensing events as a function of the minimum of event timescale, $t_{\min }$. Various source sizes are displayed with different line textures; some of these source-size curves overlap, indicating that source size is not important for that particular $M$ and ${\overline{\mu }}_T$.

Figure 4

Example light curve for a 600-second portion of one observation period (observation ID P10139 with all five PCUs added) binned in 0.1 s intervals after removing the intrinsic pulsation frequency. Red crosses indicate where there are two consecutive events exceeding the mean by at least $3\sigma$. There are no three consecutive events with $3\sigma$ deviation, which is close to the requirement for a lensing event.

Figure 5

Constraints on the PBH dark matter fraction $f_{\mathrm{PBH}}$ and mass $M$ at 95% C.L. using around 10 days observation of SMC X-1 by RXTE. Also shown are the projected limits from future observations of SMC X-1 by AstroSat, Athena and Lynx, and eXTP. The SMC X-1 source size with the x-ray energy above 2 keV is fixed to be 20 km for all thicker curves (the thinner green dot-dashed line for eXTP has a source size of 100 km for illustration purposes). The flux measurements are optimistically assumed to be uncorrelated during persistent emission. Finite source size effects are unimportant due to the optimization of $t_{\mathrm{bin}}$ (see text for details). Also shown are extragalactic gamma-ray bounds from BH evaporation [4] and Subaru/HSC microlensing bounds [7] (truncated at $f_{\mathrm{PBH}}=1$).