Forecasting Constraints on Cosmology and Modified Gravitational-wave Propagation by Combining Strongly Lensed Gravitational Waves and Galaxy Surveys

This paper presents the first analysis of time-delay cosmography using doubly lensed gravitational wave events, forecasting that while current and near-future detector networks (LVK O5/post-O5) will yield few qualified events, the future Einstein Telescope and Cosmic Explorer network will provide enough detections to precisely constrain the Hubble constant and dynamical dark energy parameters while simultaneously testing modified gravitational-wave propagation.

The Gist

The Big Picture: Listening to the Universe's Echoes

Imagine the universe is a giant, dark concert hall. Usually, we can only hear the music (gravitational waves) from the instruments (colliding black holes) if they are loud enough to reach our ears directly. But sometimes, the music gets caught in a "hall of mirrors" made by massive galaxies. This is called gravitational lensing.

When a galaxy sits between us and a colliding black hole, it bends space-time like a giant magnifying glass. This can split the sound of the collision into multiple "echoes" that arrive at Earth at slightly different times.

This paper is about a new way to use these echoes to solve two of the biggest mysteries in physics:

1. How fast is the universe expanding? (The Hubble Constant).
2. Is gravity behaving exactly as Einstein predicted, or is there something weird going on? (Modified Gravity).

The Problem: The "Rare Bird" vs. The "Common Bird"

Scientists have known for a while that if they catch a black hole collision that has been split into four distinct echoes (a "quadruply lensed" event), they can measure the universe's expansion rate with incredible precision. It's like having four different maps of the same terrain; comparing them gives you a perfect picture.

However, finding four echoes is like finding a four-leaf clover. It's extremely rare. Most of the time, the galaxy only splits the sound into two echoes (a "doubly lensed" event).

• The Old View: Scientists thought, "Two echoes aren't enough. We can't get a good map from just two points. Let's wait for the rare four-echo events."
• The New Idea (This Paper): The authors say, "Wait! What if we treat those two echoes like a pair of shoes? If we know how they fit together, we can still measure the terrain very well."

How They Did It: The "SIS" Map and the "Galaxy Database"

The researchers created a computer simulation to see if this "two-echo" strategy would work with future telescopes. Here is the step-by-step process they imagined:

1. The Sound (Gravitational Waves): They simulated black holes colliding. They used the "Singular Isothermal Sphere" (SIS) model. Think of this as a simplified, perfect round lens (like a smooth, round marble) to represent the galaxy bending the light. It's not a perfect description of every galaxy, but it's a good starting point for a first guess.
2. The Echoes: They simulated the two echoes arriving at different detectors (like LIGO, Virgo, and KAGRA).
3. The Visual Match (The Key Step): This is the clever part. The gravitational wave detectors tell us where the sound came from, but not very precisely. However, the paper assumes we will soon have giant galaxy surveys (like the LSST or Euclid) that have taken pictures of millions of galaxies.
   • The Analogy: Imagine you hear a siren echo off a building, but you aren't sure which building it was. But you have a photo album of every building in the city. If you can match the siren's location to a specific building in your photo album, you know exactly which "mirror" bent the sound.
4. The Measurement: Once they matched the sound to the galaxy, they could measure:
   • How far apart the two echoes are (the angle).
   • How much time passed between the echoes.
   • How far away the galaxy is.

By combining the time delay (how long the echoes took) with the distance (from the standard "siren" method), they could calculate the expansion rate of the universe.

The Results: From "Maybe" to "Definitely"

The team ran their simulation 1,000 times to see how many "two-echo" events they could catch with different generations of detectors.

• Current/Next-Gen Detectors (LVK O5): These are like listening with a slightly better microphone. The result? They found very few events (about 0.2 per simulation). It's like trying to find a needle in a haystack with a weak magnet. They could get a rough idea of the universe's expansion (about 14% error), but it wasn't precise enough to solve the big mysteries.
• Future Super-Detectors (ET + CE): These are the "Einstein Telescope" and "Cosmic Explorer." Imagine these are super-sensitive ears that can hear a whisper from across the galaxy.
  • The Result: They found 80.9 events on average per simulation!
  • The Impact: With this many events, they could measure the universe's expansion rate with 0.42% error. That is incredibly precise! It's precise enough to finally settle the argument between different methods of measuring the universe's speed.
  • Dark Energy: They also found they could start to measure how "dark energy" (the force pushing the universe apart) changes over time, though the measurements were a bit fuzzier than the expansion rate.
  • Modified Gravity: They could also check if gravity behaves differently than Einstein predicted. The two-echo method allowed them to test these theories alongside the expansion rate.

The Catch (Limitations)

The authors are honest about the hurdles:

• The "Two-Image" Blur: Using only two echoes is harder than using four. It's like trying to draw a perfect circle with only two points; you have to make some assumptions (like the galaxy being a perfect sphere). If the galaxy is actually an oval or weird shape, the math gets messy.
• Finding the Match: You have to be sure you matched the sound to the right galaxy in the photo album. If the sound is fuzzy, you might pick the wrong building.
• The Future: While this method works well with the future super-detectors, it's not quite ready for the current detectors.

The Bottom Line

This paper proposes a new strategy: Don't wait for the rare four-echo events. Instead, use the more common two-echo events, combine them with giant galaxy photo albums, and use a simplified model to measure the universe.

With the next generation of super-sensitive gravitational wave detectors, this method could turn "two echoes" into a powerful tool, giving us a precise map of the universe's expansion and helping us understand the mysterious forces shaping our cosmos.

Technical Summary

Technical Summary: Forecasting Constraints on Cosmology and Modified Gravitational-Wave Propagation by Combining Strongly Lensed Gravitational Waves and Galaxy Surveys

Problem Statement
The standard $\Lambda$CDM cosmology model faces significant challenges, including the Hubble tension (discrepancies between early-universe CMB measurements and local universe $H_0$ measurements) and emerging evidence favoring dynamical dark energy models over a cosmological constant. While gravitational wave (GW) "standard sirens" offer an independent probe of cosmology, current constraints are limited by event statistics and systematic uncertainties. Furthermore, strongly lensed GW events, which can provide precise time-delay cosmography, are traditionally thought to require quadruply lensed systems (four images) to reconstruct the Fermat potential necessary for cosmological inference. However, such events are predicted to be extremely rare, with approximately 70% of strongly lensed events expected to produce only two images (doubly lensed). This rarity severely limits the potential of time-delay cosmography using current and near-future detector networks (LVK O5/post-O5).

Methodology
The authors propose a novel framework to utilize doubly lensed GW events in conjunction with strong lensing systems identified in large-scale galaxy surveys (specifically the Vera C. Rubin Observatory's LSST) to constrain cosmological parameters and modified gravity.

1. Theoretical Framework:

   • Lens Model: The study employs the Singular Isothermal Sphere (SIS) model to approximate galaxy lenses. Under this model, the Einstein radius ($\theta_E$) can be directly derived from the angular separation of the two images ($\theta_+ - \theta_-$) without needing the complex reconstruction of the Fermat potential required for quadruply lensed systems.
   • Observables: The method combines two distinct measurements:
     • Standard Siren: The GW luminosity distance ($d_L^{GW}$) and source redshift ($z_S$).
     • Time-Delay Cosmography: The observed time delay ($\Delta t$) between the two lensed images, combined with the lens redshift ($z_L$), source redshift, and the impact parameter ($y$).
   • Modified Gravity: The framework incorporates modifications to GW propagation, specifically damping effects parameterized by $(\Xi_0, n)$ or the $\alpha$-basis scalar-tensor parameters ($c_M$), which alter the relationship between GW and electromagnetic luminosity distances.

2. Simulation and Data Generation:

   • Mock Events: The authors generated mock binary black hole (BBH) merger events using the "Power-law + Double Peak" mass distribution and Madau-Dickinson redshift evolution models, consistent with GWTC-4 data.
   • Detector Networks: Simulations were performed for three scenarios:
     • LVK O5 network (LIGO-Hanford, LIGO-Livingston, Virgo, KAGRA).
     • LVK post-O5 network (including LIGO-India and A# upgrades).
     • Next-generation network: Einstein Telescope (ET) + Cosmic Explorer (CE).
   • Selection Criteria: Events were filtered based on:
     • Lensing Probability: Calculated via optical depth $\tau$, considering the network SNR threshold (including sub-threshold searches down to SNR $\approx 6$).
     • Resolvability: Only events where the angular separation of the lensed images exceeds the LSST resolution limit ($0.2''$) and fall within the LSST sky footprint were retained.
   • Parameter Estimation: For LVK events, a joint Bayesian re-analysis of the two lensed images was performed to recover the true $d_L^{GW}$ and impact parameter $y$, correcting for magnification bias. For ET+CE events, a Fisher matrix analysis was used due to computational constraints.

3. Inference:

   • A log-likelihood function was constructed using $\chi^2$ statistics over the ensemble of lensed events, incorporating uncertainties in time delay, angular separation, and redshifts.
   • Markov Chain Monte Carlo (MCMC) sampling was used to derive posterior distributions for cosmological parameters ($H_0, \Omega_{m,0}$) and dark energy/modified gravity parameters ($w_0, w_a, \Xi_0, n, c_M$) across 1000 realizations for LVK and 100 for ET+CE.

Key Contributions and Results

• Event Yield: The study forecasts the number of resolvable doubly lensed events associated with galaxy surveys:
  • LVK O5: Average of $\sim 0.17$ events per realization.
  • LVK post-O5: Average of $\sim 2.3$ events per realization.
  • ET+CE: Average of $\sim 80.9$ events per realization (over 100 realizations).
• Cosmological Constraints ($\Lambda$CDM):
  • LVK O5/Post-O5: Constraints on $H_0$ are modest, with relative uncertainties of $\sim 14\%$ and $10\%$, respectively. Constraints on $\Omega_{m,0}$ are uninformative ($>70\%$ uncertainty).
  • ET+CE: The method yields a precise constraint on $H_0$ with a relative uncertainty of 0.42% and $\Omega_{m,0}$ with 2.6%. This precision is comparable to Planck and local universe measurements, offering a competitive independent probe.
• Dynamical Dark Energy ($w_0w_a$CDM):
  • LVK networks provide uninformative constraints on $w_0$ and $wa$.
  • ET+CE yields moderate constraints: $w_0 = -1.02^{+0.31}_{-0.25}$ and $w_a = 0.39^{+1.01}_{-1.55}$.
• Modified Gravity:
  • The method jointly constrains $H_0$ and modified propagation parameters.
  • For the $(\Xi_0, n)$ model, ET+CE constrains $\Xi_0$ with $\sim 8\%$ uncertainty.
  • For the $\alpha$-basis model, ET+CE constrains $c_M$ with an uncertainty of $\sim 0.12$.
  • The study notes that while LVK constraints on modified gravity are weaker than dark siren-only forecasts, the GW+EM strong-lensing approach provides complementary systematics.

Significance and Claims
The paper claims that combining doubly lensed GWs with strong lensing galaxy surveys is a viable and powerful strategy for precision cosmology, particularly for next-generation detectors.

• Overcoming Rarity: The authors demonstrate that one does not need rare quadruply lensed events to perform time-delay cosmography; the more common doubly lensed events, when paired with high-resolution galaxy surveys (like LSST) and the SIS model, suffice to break degeneracies and measure $H_0$.
• Hubble Tension: The projected 0.42% uncertainty on $H_0$ from ET+CE suggests this method could play a critical role in resolving the Hubble tension by providing a high-precision, independent measurement that is free from the systematics affecting both CMB and local distance ladder methods.
• Complementarity: The work highlights that while standard siren (dark siren) methods rely on galaxy catalog completeness and population models, the strong-lensing approach relies on accurate lens identification and reconstruction. These two approaches are complementary, and the strong-lensing method can achieve high precision with significantly fewer events (e.g., $\sim 80$ doubly lensed events vs. hundreds of unlensed events for similar $H_0$ precision).
• Limitations: The authors modestly acknowledge limitations, including the difficulty of localizing doubly lensed events for LVK (making host identification hard), the potential bias introduced by using the SIS model for elliptical lenses, and the degeneracy between lens parameters and cosmology. They suggest that future work should explore more complex lens models (e.g., SIE) and the combination of doubly and quadruply lensed events to further enhance constraints.