Galactic microlensing by acoustic Schwarzschild black holes

Technical Summary: Galactic Microlensing by Acoustic Schwarzschild Black Holes

Problem Statement
The study addresses the challenge of distinguishing between classical vacuum black holes (Schwarzschild solutions) and "acoustic black holes," which are analogue gravity systems arising from fluid dynamics in condensed matter or high-energy physics contexts. While acoustic black holes share features like event horizons and Hawking radiation with their gravitational counterparts, their underlying field equations differ fundamentally. The authors investigate whether these theoretical distinctions manifest in observable astrophysical signatures, specifically through Galactic microlensing. The core problem is determining if the tuning parameter $\xi$, which characterizes the radial fluid velocity in acoustic Schwarzschild metrics, produces detectable deviations in microlensing light curves, event rates, and optical depths compared to standard Schwarzschild black holes.

Methodology
The authors employ a theoretical framework based on the acoustic Schwarzschild metric derived by Ge et al., which describes a gravitational black hole embedded in a flowing superfluid medium. The metric is defined by a function $f(r)$ incorporating a dimensionless tuning parameter $\xi \geq 0$.

1. Deflection Angle and Lensing Geometry: In the weak-field limit, the authors derive the light deflection angle for the acoustic metric, expanding it to second order in $M/r$. This angle is used to construct the lens equation relating the source angle $\beta$, image angle $\theta$, and impact parameter.
2. Analytical Solutions: The lens equation is reduced to a cubic equation in terms of normalized variables. The authors solve for the two real image positions and calculate the total magnification $A$ as the sum of the magnifications of the individual images.
3. Light Curve Modeling: The temporal evolution of the magnification is modeled using the standard Paczyński formalism, incorporating the Einstein radius crossing time $t_E$, which depends on the transverse velocity of the lens.
4. Statistical Analysis: The study calculates the microlensing optical depth ($\tau$) and the event rate ($\Gamma$) for a population of $10^6$ monitored stars.
5. Case Studies: To provide a concrete context, the authors apply these calculations to three known black hole candidates: Cygnus X-1, A0620-00 (V616 Mon), and GRO J1655-40. They assume specific distances and masses for these objects and evaluate the observables for $\xi$ values ranging from 0 (standard Schwarzschild) to 6.

Key Contributions and Results
The paper demonstrates that the tuning parameter $\xi$ significantly alters microlensing observables:

• Einstein Ring Radius: An increase in $\xi$ leads to a larger Einstein ring radius ($R_E$) and angular Einstein radius ($\theta_E$). For example, for Cygnus X-1, increasing $\xi$ from 0 to 6 increases $R_E$ from $\approx 1.99 \times 10^9$ km to $\approx 5.27 \times 10^9$ km.
• Event Duration: Consequently, the Einstein radius crossing time ($t_E$) increases with $\xi$. The study notes that $t_E$ is significantly shorter for unbound systems (high transverse velocity) compared to bound systems.
• Light Curve Magnification: The peak magnification of the Paczyński light curve increases as $\xi$ increases. However, the authors find that the position of the peak in the first-order approximation remains unchanged, consistent with the standard profile.
• Second-Order Deviations: While the first-order light curves appear similar, a second-order analysis reveals subtle deviations in the peak brightness. The difference in peak magnification between $\xi=0$ and $\xi=6$ is on the order of $10^{-9}$ in flux variations. The authors note that detecting such minute deviations in individual events is beyond current ground-based or space-based photometric capabilities.
• Event Rates and Optical Depth: The study calculates that both the optical depth ($\tau$) and the event rate ($\Gamma$) are enhanced for acoustic black holes compared to Schwarzschild black holes of equivalent mass. For the Cygnus X-1 system, the predicted event rate increases from approximately 37 events/year at $\xi=0$ to 100 events/year at $\xi=6$.

Significance and Claims
The paper claims that Galactic microlensing serves as a promising, albeit indirect, astrophysical channel for constraining analogue gravity metrics. The primary significance lies in the statistical detectability of these objects rather than the resolution of individual light curve anomalies.

• Statistical Detection: The authors argue that while the second-order deviations ($10^{-9}$) are too small for individual detection, the cumulative effect of higher peak magnifications and, crucially, the enhanced event rates ($\Gamma$) and optical depths ($\tau$) could be identified through the statistical analysis of large-scale microlensing surveys (e.g., OGLE, MOA).
• Environmental Parameter: The tuning parameter $\xi$ is interpreted as an effective environmental parameter related to the ratio of fluid velocity to the speed of sound. The paper suggests that different astrophysical environments (e.g., accretion disks vs. stellar winds) correspond to different ranges of $\xi$, with "fast-fluid" environments potentially yielding the strongest statistical signatures.
• Distinction from Standard GR: The work establishes that if dark matter halo objects or compact lenses were acoustic black holes, they would exhibit a systematically higher probability of detection and longer event durations compared to standard Schwarzschild black holes. The findings provide quantitative predictions that can be tested against existing and future microlensing survey data to distinguish between vacuum and non-vacuum (analogue) black hole metrics.