⚛️ general relativity

Detection of cosmic strings by gravitational wave lensing. Predictions for Einstein Telescope

This paper demonstrates that the Einstein Telescope could detect cosmic strings through gravitational wave lensing by applying wave optics to analyze amplification and waveform effects, ultimately using Bayesian inference on simulated merger data to estimate string tensions consistent with injected values.

Original authors: Jakub Szyndler, Tomasz Bulik

Published 2026-07-22

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Jakub Szyndler, Tomasz Bulik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean. Most of the time, we think of it as empty space, but in the world of physics, it's actually filled with ripples and wrinkles. Sometimes, massive objects like black holes crash into each other, sending out ripples called gravitational waves. Think of these like the sound waves from a drumbeat, but instead of traveling through air, they travel through the fabric of space and time itself. We have special "ears" on Earth, called detectors, that listen for these cosmic drumbeats.

Now, imagine that somewhere in this ocean, there are invisible, super-tight threads stretching across the universe. Scientists call these cosmic strings. They are like cosmic spaghetti that formed right after the Big Bang. If a gravitational wave passes near one of these strings, the string acts like a lens, bending the wave just like a glass lens bends light. This can make the wave look weird, creating echoes or interference patterns, almost like hearing a song played through a weird echo chamber. The big question is: Can we actually see these invisible threads by listening to the echoes they make in the gravitational waves?

This paper is a detective story about how a future super-detector, called the Einstein Telescope, might solve this mystery. The authors didn't go out and find a cosmic string yet; instead, they built a detailed computer simulation to see if the Einstein Telescope would be good enough to spot them. They took a group of fake black hole collisions, simulated them passing near a cosmic string, and then asked: "If the Einstein Telescope heard this, would it notice the weird echo?"

The answer is a very hopeful "yes." The researchers found that the Einstein Telescope is so sensitive that it could easily hear the "twist" a cosmic string puts on a gravitational wave. They looked at how the waves would change in strength and timing, and they saw clear, wiggly patterns that would act like a fingerprint for a cosmic string. Even better, they ran a test where they pretended to find eight different black hole collisions caused by a cosmic string. By using a clever math trick called Bayesian Inference (which is like a super-smart detective piecing together clues), they were able to guess the "tightness" of the string.

In their simulation, the true tightness of the string was set to a value of $-10$ (in a logarithmic scale). After running their detective work on the eight fake events, the Einstein Telescope's "guess" came out to be $-9.78$, with a small margin of error. This is incredibly close to the real value, proving that if these cosmic strings exist, the Einstein Telescope will likely be the tool that finally catches them. The paper concludes that while we haven't found them yet, the next generation of gravitational wave detectors is perfectly designed to hunt them down and measure their properties, turning a theoretical idea into a real discovery.

Technical Summary: Detection of Cosmic Strings by Gravitational Wave Lensing – Predictions for Einstein Telescope

Problem Statement
Cosmic strings (CS) are hypothetical one-dimensional topological defects formed during symmetry breaking in the early Universe. While their existence remains unconfirmed, they are predicted to act as gravitational lenses, potentially producing double images or characteristic interference patterns in signals from background sources. Previous searches for CS signatures have utilized electromagnetic observations (e.g., CMB anisotropies, optical lensing, FRB repetition), but these have yielded only upper limits or retracted candidates (e.g., the CSL-1 lens). While gravitational wave (GW) lensing by point masses has been identified, the specific detection of lensing by cosmic strings remains an open challenge. This paper investigates whether the third-generation Einstein Telescope (ET) possesses the sensitivity required to detect cosmic strings via GW lensing and to constrain their dimensionless tension ( $G\mu$ ).

Methodology
The study employs a wave optics formalism, necessary because the wavelength of gravitational waves can exceed the Schwarzschild radius of the lens, rendering geometric optics insufficient. The authors model the lensing of GWs by an infinitely long, straight cosmic string using the conical spacetime metric.

Theoretical Framework: The authors define the amplification factor $F(f)$ in the frequency domain, which relates the lensed waveform $\tilde{h}(f)$ to the unlensed waveform $\tilde{h}_0(f)$ . This factor depends on the dimensionless frequency $w$ and the dimensionless source position $y$ . The analysis incorporates phase shifts and time delays ( $\Delta t$ ) between the two lensed images, which are functions of the string tension and the effective comoving distance.
Wave Effects Analysis: The authors generate 2D heat maps and 1D slices of the amplification factor to visualize wave effects, including interference fringes and diffraction. They compare these wave-optics results against the geometric-optics approximation to demonstrate the necessity of the wave approach for cosmic string lensing.
Signal-to-Noise Ratio (SNR) and Detectability: Using the ET-D sensitivity curve, the authors simulate binary black hole (BBH) mergers with masses between 8 and 50 $M_\odot$ at various redshifts ( $z_S$ ). They calculate the SNR for lensed signals to determine if they exceed the detection threshold ( $S/N \geq 8$ ).
Mock Observations and Bayesian Inference: To estimate the string tension, the authors construct a mock dataset of eight BBH lensing events based on the StarTrack population synthesis simulation. They assume an injected string tension of $\log(G\mu) = -10$ $lo g (G μ) = - 10$ and a lens redshift of $z_L = 0.5$ $z_{L} = 0.5$ .
- Parameter Estimation: A multi-stage Bayesian inference pipeline is used. First, individual event parameters (chirp mass, mass ratio, source redshift, effective tension) are estimated using the Bilby package and UltraNest sampler, incorporating a time-delay likelihood to break mass-sheet degeneracy.
- Hyperinference: Finally, a hierarchical Bayesian analysis (hyperinference) using the emcee sampler combines the posterior distributions of the eight events to estimate the global string tension $\log(G\mu)$ .

Key Contributions and Results

Wave Optics Characterization: The study confirms that cosmic string lensing produces distinct wave effects, including characteristic beat patterns in the time-domain waveforms and oscillatory fringes in the frequency domain. These features are visible in the amplification factor and characteristic strain, particularly for dimensionless source positions $y \gtrsim 0.01$ .
Detectability with ET: Simulations indicate that lensed GW signals from BBH mergers are well within the sensitivity band of the Einstein Telescope. The SNR for these events consistently exceeds the detection threshold ( $S/N > 8$ ), even in regions where destructive interference suppresses the signal. The characteristic strains of these events lie well above the noise floor of ET, whereas they are often marginal for current detectors like aLIGO and Advanced Virgo.
Tension Estimation: Using the mock dataset of eight events, the authors successfully recovered the injected string tension. The final estimated value is $\overline{\log(G\mu)} = -9.78^{+0.44}_{-0.49}$ . This result is consistent with the injected value of $-10$ within the 90% credible interval. The low dispersion in the individual event estimates suggests that the method is robust for constraining cosmic string parameters.

Significance and Conclusions
The paper concludes that the Einstein Telescope represents a powerful tool for the detection and characterization of cosmic strings via gravitational wave lensing. Unlike previous electromagnetic searches, GW lensing allows for the direct probing of wave effects (interference and diffraction) that are unique to the conical geometry of cosmic strings. The study demonstrates that even for low tension values ( $G\mu \sim 10^{-10}$ ), which are currently allowed by pulsar timing array and CMB constraints, the ET can detect these events and statistically constrain the string tension.

The authors note that while their current analysis focuses on BBH mergers, future work could extend this methodology to other sources, such as neutron star mergers or the stochastic gravitational wave background. They also suggest that machine learning techniques could be explored to further optimize the detection and parameter estimation of these lensing events. The work establishes a viable pathway for using third-generation GW observatories to resolve the long-standing question of cosmic string existence.

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