Primordial black hole lensing

Take 1E-13 solar masse as target regime. Einstein radius is 3E=7 smaller than solar mass MACHO. 

Quasar ulensing paper from Kochanik has this table: 

Lens R_e is for 0.3 solar masses. We are down by sqrt(1E-13/0.2) = 5 e-7. So for a favorable source, lowest mass lensed QSO, we have R_e=7.8e9 cm, and Rs=2e13. Ouch. 

We need a more compact source. So use lensed SNe? Typical timescale is dominated by lens motion not source motion, and is 10 years * 5e-7 = 5e-6 yers x 3e7 sec/year = 150 sec. 

Now we need an estimate of event rate. Note that apparent angular size is increased by lensed magnification so we don't want highly magnified lensed SNe. 

Diego et al (2022) in paper about lensed type Ia say that photosphere radius after 1 mo is 10^-3 parsecs, or 3E15 cm. Too big! In retrospect this had to be true. 
If these objects have comparable spectra, then to have comparable magnitudes their emitting areas have to be comparable. rats. 

EHT observed a source 5 microarcsec in size at z=0.9

1 arcsec is 7.35 kpc at that distance. 1 microarcsec is 7.5e-3 parsecs, and 5 uas is 3.7e-3 parsecs. Also too large. rats. 

diameter of our sun is 1.4e11 cm. So single stars, esp variables in frame subtraction, are interesting?

diameter of a white dwarf star is 5000 km or 5e6m or 5e8 cm. But they're faint! There are HST-detected white dwarf stars in the LMC. What's Einstein radius for PBH in our halo? 
Assume 10 kpc distance, and mass of 1E-12 solar masses. MACHO project pooped out at 1E-7 solar masses for solar type stars. white dwarf is 3.5E-3 times smaller. 
That should gain us 5 orders of magnitude in mass, going down to 1E-12 solar masses. OK, that's in the right regime, finally. 
Event duration for Jupiter mass MW halo lenses is around 3 days, at 1E-3 solar masses. If we drop nine orders in mass to 10^-12 M_solar, then time scale is 3E4 shorter, or 1E-4 days or 10 seconds. 
For an optical depth of 5E-7 need to look at a single source for 2E7 seconds, or 200 days. If, as this paper says, there are 10 of them in the cluster then we need to observe
for 20 days, continuously.

If LSST can do a fast-cadence survey of the LMC using 15 sec images in u to get down to 5 sigma sensitivity of u=24 in 30 sec, we're in business. An amplification of 10 is an
increase of 2.5 magnitudes, and should bring these sources above threshold even in u band, which is centered at 350nm. see https://www.lsst.org/scientists/keynumbers  

So: a u band imaging survey with 10-15 sec exposures on a 6-8m class telescope, pointed at LMC globular clusters, could potentially set an interesting limit for 
PBHs in the 10^-12 to 10^-13 solar mass range. What about WD stars in M31? Angular size is down by distance ratio, 760/50 = 15, apparent magnitudes are down by
difference in distance modulus, which is 24.4-18.5 ~ 6 magnitudes so WD at M31 is 30th magnitude. 

Could also look for young hot WDs in other globular clusters, some closer than the LMC. 47 Tuc is at 4 kpc, for example. Galactic coords are 305, -44. 
Much brighter: 

There are Chandra xray observations of WDs in 47 Tuc: From Ferraro et al "core of 47 Tuc', 2001:

One advantage of this approach is to use other globular clusters to probe the halo. 

Wave optics

If diffraction becomes important on Schwarzchild radius rather than Einstein radius scale, what is that? GM/c^2 for M=10^20 gm = 10^17 kg = 6.67E-11*1E17/(3e8)^2 = 7 E-11 m 

Sure enough, that's way less than a micron. This might be because the entire Einstein radius does not make a thin lens, light only arrives from the marginal rays. 

For 100 km Eistein radius at 10 kpc = 3e17 km, angle is 100/3e17 = 30 e-15 = 3e-14. Path length difference is (1-cos(theta))*L ~ 

pulsars

what about pulsars? They're only 20 km or so in diameter! That's 2e4 m or 2e6 cm! yup, found a good paper on that.... 

collision rate

How many hit the Earth? If R_E for LMC distance is 100 km, surface area of the microlensing sausage is 2piR*L = 600km * 1.5E18 km ~ 10^21 km^2. We get about one PBH per year
through that, at 10^12 solar masses. Cross section of the Earth is pi*(6500km)^2 = 1.3e8 km^2. So we get hit once in every 10^21/10^8 = 10^13 years, or  >> a Hubble time. rats.