Could the high-mass black holes from gravitational-wave observations be explained by lensing?

This paper demonstrates that gravitational lensing cannot explain the high-mass black holes observed by LIGO and Virgo, as the proposed lensing model fails to simultaneously satisfy constraints from merger counts, mass and distance distributions, the absence of strongly lensed events, and the stochastic gravitational-wave background.

The Gist

The Big Mystery: "Too Heavy" Black Holes

Imagine you are a detective looking at a lineup of suspects. In the past, you knew that most "black hole suspects" (the ones we see in our own galaxy) were relatively light, weighing about as much as a small car (10 times the mass of our Sun).

But recently, the LIGO and Virgo detectors (our cosmic "ears") started hearing crashes from black holes that were massive—weighing as much as a large truck or even a bus (30+ times the Sun's mass). This was a shock. It was like finding a fleet of semi-trucks in a neighborhood where everyone drives compact cars.

The Suspect's Alibi: "It's a Trick of the Light!"

A group of scientists (Broadhurst, Diego, and Smoot, or "BDS") came forward with a clever alibi. They argued: "Wait a minute! Those black holes aren't actually heavy. They just look heavy because of a cosmic optical illusion called Gravitational Lensing."

The Analogy:
Think of gravitational lensing like a funhouse mirror or a magnifying glass.

• If you look at a small toy through a magnifying glass, it looks huge.
• The BDS team argued that the LIGO detectors were looking at normal-sized black holes through a "cosmic magnifying glass" created by massive galaxies in between.
• Because the signal was magnified, the detectors thought the black holes were much closer and much heavier than they really were. They claimed the "heavy" black holes were actually just the normal "compact car" sized ones, distorted by the universe's lens.

The Investigation: Putting the Alibi to the Test

The authors of this paper (Harshe, Prasad, and Ajith) decided to play the role of the skeptical detective. They said, "That's a cool theory, but does it hold up if we check the other evidence?"

They built a massive computer simulation to see if the "Magnifying Glass Theory" could explain four different things at the same time. Think of it like trying to solve a puzzle where you have to fit four different pieces together perfectly.

1. The Body Count (How many crashes did we hear?)

• The Theory: To make the magnifying glass work, you need a lot of black holes happening far away in the universe.
• The Reality Check: If you have that many black holes out there, LIGO should have heard way more crashes than it actually did.
• The Verdict: The theory predicts too many events. It's like saying, "I only saw one car drive by," but your theory requires a traffic jam of 10,000 cars to explain the noise. Fail.

2. The Double Vision (Did we see the same crash twice?)

• The Theory: Strong gravitational lensing doesn't just magnify; it often splits the image. You should see the same black hole crash twice, arriving a few minutes or hours apart, like an echo.
• The Reality Check: LIGO has never seen a confirmed "echo" or a double event.
• The Verdict: The theory predicts we should see these double events frequently, but we haven't found a single one. Fail.

3. The Weight Distribution (Are the masses right?)

• The Theory: If you take normal black holes and magnify them, the math should line up perfectly with the heavy ones we see.
• The Reality Check: When the authors ran the numbers, the "magnified" weights didn't match the actual distribution of weights LIGO found. The shapes of the data didn't fit.
• The Verdict: The puzzle pieces don't align. Fail.

4. The Background Noise (The Cosmic Hum)

• The Theory: If there are billions of these black holes crashing in the distance (as the theory requires), they should create a constant, low-level "hum" or background noise across the whole universe, similar to the static on an old radio.
• The Reality Check: LIGO has listened very carefully and has not heard this hum.
• The Verdict: The theory predicts a loud hum that isn't there. Fail.

The Final Conclusion

The authors drew a map of all the possible settings for the "Magnifying Glass Theory." They found that:

• If you tune the theory to fix the Body Count, it breaks the Double Vision rule.
• If you tune it to fix the Double Vision, it breaks the Background Noise rule.
• If you try to fix the Mass Distribution, it breaks the Body Count.

The Bottom Line:
There is no single setting on the dial that makes the "Magnifying Glass" theory work for all the evidence. The theory is like a key that fits one lock but breaks the other three.

Therefore, the paper concludes that gravitational lensing is NOT the reason we see these massive black holes. The heavy black holes are real. They are likely the result of stars forming in environments with very little "metal" (heavy elements), allowing them to grow much larger than the ones in our own galaxy.

In short: The universe isn't playing a trick on us with a magnifying glass. We really have found a new, heavier population of black holes!

Technical Summary

1. Problem Statement

Gravitational-wave (GW) observations by LIGO and Virgo (LVK) have revealed a population of high-mass black holes (BHs) with masses $M \gtrsim 30 M_\odot$. This contrasts with galactic X-ray binary observations, which typically show BHs with $M \lesssim 10 M_\odot$.

• Conventional Explanation: These high-mass BHs are remnants of metal-poor massive stars or potentially primordial BHs.
• The Lensing Hypothesis (BDS Model): Broadhurst, Diego, and Smoot (BDS) proposed that the observed high masses are an artifact of gravitational lensing magnification. They argued that the true intrinsic mass distribution of extragalactic binary black holes (BBHs) matches the galactic distribution, but a significant fraction of detected events are strongly lensed. Lensing magnifies the signal ($\mu > 1$), causing the inferred luminosity distance to be underestimated and the inferred source-frame mass to be overestimated.
• The Conflict: To explain the observed mass distribution, the BDS model requires a merger rate at high redshifts that is significantly higher than standard astrophysical predictions. This paper aims to test the consistency of the BDS model against multiple independent GW observables to determine if lensing is a viable explanation.

2. Methodology

The authors performed astrophysical simulations of lensed BBH populations using the BDS model parameters and compared the results against four distinct observational constraints.

A. The BDS Model Parameters

The model assumes:

1. Intrinsic Mass Distribution: A log-normal distribution identical to galactic BHs (Corral-Santana et al. 2016), with $\mu_m = \log_{10}(8)$ and $\sigma_m = 0.27$.
2. Merger Rate ($R(z)$): A specific redshift-dependent rate function defined by two parameters:
   • $A_1$: The peak merger rate at $z_{peak} = 1.8$.
   • $t_h$: The "half-life" time characterizing the decay of the rate from the peak to the present epoch.
   • The original BDS claim used $A_1 \approx 30,000 \text{ yr}^{-1} \text{ Gpc}^{-3}$ and $t_h = 1.25 \text{ Gyr}$.

B. Simulation Framework

• Cosmology: Flat $\Lambda$CDM (Planck 2018).
• Lensing Scheme: Sources are lensed by Singular Isothermal Sphere (SIS) lenses.
  • Strong Lensing: Occurs when the source is within the Einstein radius ($y \leq 1$), producing multiple images with magnifications $\mu_\pm = 1 \pm 1/y$.
  • Weak Lensing: Occurs for $y > 1$, producing a single magnified image.
• Detection Criteria: Events are simulated for LVK runs (O1, O2, O3a, O3b). Detection requires a Signal-to-Noise Ratio (SNR) $\geq 8$. Strongly lensed pairs are counted only if both images exceed the SNR threshold.

C. Four Observational Constraints

The study evaluates the model against:

1. Total Number of Detectable Events ($\Lambda$): Comparing the simulated count against the observed 83 BBH events (O1-O3b).
2. Fraction of Strongly Lensed Pairs ($u$): Comparing the predicted fraction of detectable double-image events against the non-detection of such pairs in LVK data (using Posterior Overlap statistics).
3. Distribution of Redshifted Mass ($M_z$) and Apparent Luminosity Distance ($\tilde{d}_L$): Using hierarchical Bayesian inference to check if the model reproduces the observed joint distribution of these parameters.
4. Stochastic GW Background (SGWB): Calculating the SGWB signal-to-noise ratio. A high merger rate at high redshifts (required by BDS) should produce a detectable SGWB, which has not been observed.

3. Key Results

The authors systematically ruled out the entire parameter space of the BDS model ($A_1$ vs. $t_h$) by showing that no single parameter choice satisfies all four constraints simultaneously.

• Constraint 1: Event Count:

  • The observed number of events ($N_{det}=83$) constrains the 3$\sigma$ credible range of the Poisson mean to $\Lambda \in [60, 115]$.
  • Large regions of the $A_1-t_h$ parameter space predict either too many or too few events, leaving only a narrow viable strip.

• Constraint 2: Strong Lensing Fraction:

  • The BDS model predicts a strong lensing fraction ($u$) ranging from 2% to 30% depending on parameters.
  • LVK data shows no confirmed strongly lensed pairs. The 3$\sigma$ upper limit on the fraction is 6.3%.
  • This rules out the left side of the parameter space (lower $t_h$), where the lensing fraction exceeds the observational limit.

• Constraint 3: Mass and Distance Distribution:

  • Hierarchical inference was used to compare the model's predicted distribution of $M_z$ and $\tilde{d}_L$ with LVK posteriors.
  • The posterior peaks at $t_h \approx 0.95$ Gyr. However, even at this "best fit," the model fails to satisfactorily reproduce the observed distribution of redshifted masses and apparent distances (see Appendix E).

• Constraint 4: Stochastic GW Background:

  • The high merger rates required by the BDS model generate a significant SGWB.
  • For the original BDS parameters ($A_1=30,000, t_h=1.25$), the predicted SGWB SNR exceeds 2 during the O3a run.
  • Since no SGWB has been detected, the region of parameter space predicting an SNR $>3$ (3$\sigma$ confidence) is ruled out. This eliminates the high-$A_1$ region.

• Synthesis (Figure 6):

  • The viable regions for each constraint are disjoint.
  • The region consistent with event counts is inconsistent with the lensing fraction.
  • The region consistent with the lensing fraction is inconsistent with the SGWB non-detection.
  • Conclusion: There is no point in the $A_1-t_h$ parameter space that is consistent with all four observables simultaneously.

4. Significance and Conclusion

• Refutation of the Lensing Hypothesis: The paper definitively concludes that gravitational lensing magnification cannot explain the high-mass BHs observed by LIGO and Virgo. The specific model proposed by BDS is untenable because it creates a "tension" between the required high merger rate (to produce enough high-mass events via lensing) and the non-detection of the resulting SGWB and strongly lensed pairs.
• Robustness: The authors note that while they used the SIS lens model, alternative models (like NFW) would have even lower optical depths, making it even harder to explain the high-mass population via lensing.
• Implications for Astrophysics: The results support the conventional astrophysical explanation: the high-mass BHs are likely genuine astrophysical objects formed in low-metallicity environments, rather than artifacts of lensing.
• Future Work: The authors highlight that lensing will become increasingly important for future detectors (e.g., Einstein Telescope, Cosmic Explorer) and are developing formalisms to account for lensing effects in population inference without relying on the specific BDS hypothesis.

In summary, this work provides a rigorous multi-messenger consistency check that invalidates the hypothesis that the LVK high-mass black hole population is an illusion caused by gravitational lensing.