Microlensing of fast and slow compact objects

This paper derives model-independent microlensing constraints on the density and mass of fast and slow compact objects with non-standard velocities, demonstrating how their distinct kinematic properties open new parameter spaces and significantly alter observational limits compared to conventional dark matter searches.

The Gist

Imagine the universe is a giant, dark forest at night. We can't see the trees (dark matter) directly, but we can guess where they are by watching how the light from distant stars bends and brightens as it passes through the forest. This phenomenon is called gravitational microlensing. It's like holding a magnifying glass up to a streetlamp; if a small object passes in front of the light, it briefly makes the light look brighter.

For decades, astronomers have used this trick to hunt for "compact objects"—dense, invisible things like black holes or neutron stars—that might make up the universe's missing mass (dark matter). But they've been playing by a very specific set of rules: they assumed these invisible objects were moving at a "normal" speed, drifting slowly like leaves in a gentle breeze, just like the rest of the dark matter in our galaxy.

The Big Twist: Fast and Slow Runners
This paper argues that we might be missing a huge part of the picture because we're only looking for the "leaves in the breeze." The authors suggest there could be two other types of invisible travelers:

1. The Speedsters: Objects moving incredibly fast, like rockets or bullets, perhaps born from violent cosmic collisions or exotic physics.
2. The Drifters: Objects moving very slowly, almost floating along with the current, perhaps ejected gently from their home systems.

The Speed-Mass Confusion
Here is the tricky part: In a microlensing event, we can measure how long the star stays bright (the duration), but we can't easily tell if that's because the object is heavy and slow or light and fast. It's like hearing a car zoom by; you can't tell if it's a heavy truck moving slowly or a tiny sports car moving at top speed just by the sound alone.

The paper shows that if we assume these objects are moving at "weird" speeds (much faster or slower than the standard dark matter), the rules change completely:

• Fast objects would create very short, flash-like brightenings.
• Slow objects would create very long, drawn-out brightenings.

The "Lensing Tube" Problem
The authors point out a major hurdle for finding the "Drifters" (slow objects). Imagine you are on a train (the Earth/Sun) looking out the window at a slow-moving car (the lens) on a parallel track. Even if the car is barely moving, the fact that your train is moving fast makes it look like the car is zooming past you.

In astronomy, the "train" is the motion of our solar system and the background stars. For very slow lenses, this "bulk motion" of our own system dominates the view. The paper notes that trying to find these slow objects is like trying to spot a snail on a highway while you are driving at 100 mph; the snail's own speed doesn't matter much compared to your speed. The authors still calculated limits for them, but they admit this is more of a theoretical exercise than a guaranteed discovery method right now.

The "Wall" of Finite Size
For the "Speedsters" (fast objects), there is a different problem. Usually, if an object is too small or moving too fast, the event is so brief and the magnification so weak that our telescopes miss it. This is often called a "wall" that stops us from finding very small, fast dark matter.

However, the authors found a loophole. Because these fast objects are moving so quickly, they pass in front of the star so fast that the "size" of the star (which usually blurs the signal) doesn't matter as much. This means that if we take pictures of the sky more frequently (higher "cadence"), we could actually see these tiny, fast-moving objects that standard searches would miss. It's like taking a video at 1000 frames per second instead of 30; you can catch the blur of a fast-moving insect that looks like a solid line in a normal video.

What They Did
The team took data from two major sky surveys (Subaru-HSC and OGLE) and ran new calculations. Instead of assuming the invisible objects move at the "standard" speed, they tested a wide range of speeds, from very slow to very fast.

The Results

• They found that for fast objects, we can rule out (exclude) certain densities and masses that were previously thought to be safe. In other words, if these fast objects existed in large numbers, we would have seen them by now.
• They showed that the "search space" for these objects is much bigger than we thought. By looking for events that are unusually short or long, we can find populations of objects that standard dark matter searches ignore.
• They emphasized that future surveys need to take pictures faster and more often to catch these "Speedsters" before they zip out of sight.

In a Nutshell
This paper is a wake-up call to astronomers: "Don't just look for the slow, drifting dark matter. There might be a whole population of fast-moving or ultra-slow invisible objects hiding in plain sight, but you need to change your camera settings (look for different time durations and take pictures faster) to find them." They have drawn new maps showing where these objects can't be, which helps narrow down the search for the universe's hidden mass.

Technical Summary

1. Problem Statement

Gravitational microlensing is a primary method for constraining the abundance of compact objects (such as Primordial Black Holes or MACHOs) that could constitute dark matter. Conventional analyses rely on a critical set of assumptions:

• Velocity Distribution: Lenses are assumed to trace the Milky Way's dark matter halo, following a Maxwell-Boltzmann velocity distribution with a characteristic circular speed of $\sim 10^{-3}c$ (approx. 220 km/s).
• Spatial Distribution: Lenses are assumed to follow the dark matter density profile (e.g., NFW).

The Gap: Many theoretical scenarios predict populations of compact objects with velocities that deviate drastically from virialized dark matter.

• Fast Lenses ($v \gg 10^{-3}c$): Examples include ultrarelativistic primordial black holes from cosmic string collapse, black hole merger remnants with "superkicks" ($\sim 5000$ km/s), or mirror neutron stars. These may exceed galactic escape velocities.
• Slow Lenses ($v \ll 10^{-3}c$): Examples include free-floating planets, objects ejected from stellar disks, or structures in a "dark disk" with dispersion speeds $\sim 10^{-4}c$.

The Consequence: Standard analyses suffer from a mass-velocity degeneracy. The Einstein crossing time ($t_E$) depends on both mass ($M$) and transverse velocity ($v_\perp$) as $t_E \propto \sqrt{M}/v_\perp$. If the velocity distribution is unknown or non-standard, a given $t_E$ cannot be uniquely mapped to a mass. Fast lenses mimic high-mass objects (longer $t_E$ for a given mass), while slow lenses mimic low-mass objects. This opens up qualitatively new regions of parameter space that standard searches miss.

2. Methodology

The authors derive model-independent constraints on the density of these non-standard lens populations using data from major microlensing surveys (Subaru-HSC, OGLE, EROS-2, MACHO).

Key Steps:

1. Model-Independent Event Rate Limits:

   • They first establish upper limits on the microlensing event rate ($\Gamma_{excl}$) as a function of the Einstein diameter crossing time ($t_E$) without assuming specific masses or velocities.
   • This is calculated using observed event counts ($N_{obs}$), expected limits at 95% C.L. ($N_{excl}$), the number of source stars ($N_\star$), and observation time ($T_{obs}$), corrected for detection efficiency $\epsilon(t_E)$.

2. Phase Space Modeling:

   • Velocity Distributions: They test two benchmarks:
     • Maxwell-Boltzmann (MB): Represents a virialized system with velocity dispersion.
     • Dirac Delta (DD): Represents a mono-velocity population (zero dispersion).
   • Spatial Distributions: They test two benchmarks:
     • Uniform: Constant density (relevant for extragalactic or unbound populations).
     • NFW: Tracing the dark matter halo profile.
   • Bulk Motion Correction: Crucially, they account for the transverse motion of the "lensing tube" (the relative motion of the source and observer, e.g., Sun vs. LMC/M31). For slow lenses ($v \ll 10^{-3}c$), this bulk motion dominates the event rate, creating a "floor" for detectable velocities.

3. Calculation of Constraints:

   • They calculate the expected number of events ($N_{ev}$) by integrating the differential rate over the survey's sensitivity window ($t_{E,min}$ to $t_{E,max}$).
   • They define a ratio $\tilde{r} = \rho_L / \rho_{DM}$, where $\rho_L$ is the lens density and $\rho_{DM}$ is a benchmark dark matter density (either local solar neighborhood density or NFW profile).
   • They exclude regions in the $(\tilde{r}, M)$ plane where the predicted event rate exceeds the observed limits.

3. Key Contributions

• Breaking the Degeneracy: The paper systematically maps how varying lens velocities shifts the excluded mass ranges. It demonstrates that for a fixed $t_E$, faster lenses imply higher masses, and slower lenses imply lower masses, effectively expanding the search window.
• Finite Source & Wave Optics Effects: The authors analyze how finite source effects (which suppress magnification for small lenses) and wave optics effects act as a "wall" for standard dark matter searches. They show that for fast lenses, increasing the observation cadence allows probing smaller masses without being suppressed by these effects, because the high velocity shortens the event duration, keeping it within the cadence window while maintaining a large Einstein radius relative to the source.
• Slow Lens "Unphysical" Limits: They explicitly discuss the limitation of searching for very slow lenses ($10^{-4}c$). They show that unless the lenses are co-moving with the lensing tube (a specific, unlikely scenario), the event rate is dictated by the bulk motion of the source/observer, making standard limits for these speeds unphysical.
• Benchmark Comparisons: By comparing Maxwell-Boltzmann and Dirac Delta distributions, they show that the mean velocity is the dominant factor, with the shape of the distribution affecting limits by less than a factor of 2.

4. Key Results

• Mass-Dependent Constraints: Using Subaru-HSC (M31) and OGLE (LMC) data, the authors provide exclusion limits for lens densities across a velocity range of $10^{-4}c$ to $10^{-1}c$.
  • Fast Lenses ($v \sim 10^{-2}c - 10^{-1}c$): The constraints on mass shift to significantly higher values compared to standard dark matter searches. For example, a lens moving at $10^{-1}c$ is constrained at masses orders of magnitude larger than a $10^{-3}c$ lens for the same $t_E$.
  • Slow Lenses ($v \sim 10^{-4}c$): The limits are shown primarily as an educational exercise. The authors note that for isotropic slow lenses, the bulk motion of the lensing tube ($\sim 10^{-3}c$) dominates, rendering the derived limits unphysical unless the lenses are co-moving.
• Survey Performance:
  • Subaru-HSC: Shows unique behavior for fast lenses. As velocity increases, the finite source effect becomes less dominant (because the Einstein radius grows with mass), allowing Subaru-HSC to probe deeper into the high-mass, high-velocity regime than OGLE.
  • OGLE: Provides the tightest constraints for standard and moderately fast lenses due to its long baseline and large number of stars.
• Density Exclusions: The study excludes lens densities that differ from standard dark matter constraints by orders of magnitude for specific mass-velocity combinations. For instance, a population of fast-moving compact objects cannot constitute a significant fraction of dark matter in the mass ranges probed.

5. Significance and Future Outlook

• New Discovery Space: The paper argues that the microlensing discovery space is far richer than conventionally assumed. Exotic compact objects with non-standard velocities (e.g., from cosmic strings or dark disks) could evade standard analyses but are detectable if the velocity distribution is accounted for.
• Observational Strategy: A major finding is that higher cadence (more frequent observations) is beneficial for detecting fast lenses. Unlike standard searches where higher cadence hits a wall due to finite source effects, high-velocity lenses produce short-duration events that remain detectable at smaller masses with higher cadence. This incentivizes surveys like Subaru-HSC and OGLE to maximize their observing frequency.
• Astrometric Prospects: The authors note that astrometric microlensing (measuring the shift in the source position) can break the mass-velocity degeneracy entirely, offering a path to determine lens mass and velocity independently.
• Context: The work addresses recent debates regarding Subaru-HSC data re-analyses (which found fewer events than initially thought) and emphasizes the need for robust, model-independent limits that can be updated as data and selection criteria evolve.

In summary, this work provides a comprehensive framework for interpreting microlensing data beyond the standard dark matter paradigm, revealing that the universe may harbor populations of compact objects with extreme velocities that have previously been overlooked.