Estimation of systematic error from bulk deformation of end test mass induced by photon calibrator for LIGO post-O5 gravitational wave projects

To hear the universe speak, scientists have built instruments of staggering sensitivity: giant laser interferometers that can detect a change in distance smaller than a proton. These machines, scattered across the globe, listen for ripples in spacetime caused by cataclysmic events like colliding black holes or merging neutron stars. To trust what they hear, researchers must first be certain that the instruments themselves are not distorting the message. This requires a process called calibration, where scientists apply a known, tiny push to the massive mirrors inside the detectors to see how the machine responds. If the mirror moves exactly as predicted, the detector is working correctly. If it moves differently, the data collected from the cosmos could be misinterpreted, leading to wrong conclusions about the mass or distance of the cosmic events being observed.

A new study by Daiki Tanabe and his colleagues addresses a subtle but critical flaw in how these mirrors react to calibration pushes. The mirrors, which weigh dozens of kilograms, are not perfectly rigid blocks of glass. When a laser beam pushes on their surface, the force does not just move the entire mirror forward; it also causes the solid body of the mirror to flex and deform, much like a drumhead vibrating when struck. This internal squirming, known as bulk deformation, creates a displacement that differs from the ideal, rigid motion the scientists expect. The researchers focused on how this deformation behaves at high frequencies, specifically in the range of thousands of cycles per second, which is crucial for studying the final moments of merging neutron stars. They found that if the calibration lasers are not positioned with extreme precision, this flexing can introduce errors large enough to skew the scientific results.

The team investigated this problem using detailed computer simulations of the mirrors used in four major gravitational wave experiments: Advanced LIGO, Advanced Virgo, KAGRA, and the future LIGO A# project. They modeled the mirrors as solid cylinders and simulated the effect of pushing them with laser beams at various locations. The researchers discovered that the mirrors have specific vibration patterns, or modes, that are excited depending on where the laser hits. One such pattern, called the drumhead mode, involves the center of the mirror moving in the opposite direction to its edges. Another, known as the butterfly mode, creates a pattern where the center remains relatively still while the outer regions vibrate. The study showed that the ideal spot to push the mirror to avoid these unwanted vibrations is not always at the center, nor is it always exactly at the theoretical "nodal" points where the mirror does not move. Instead, the optimal position depends on the specific shape, material, and size of the mirror, as well as the exact location of the main laser beam used to detect gravitational waves.

Using two different types of simulation software to ensure their results were robust, the team calculated the best positions for the calibration lasers for each of the four detector designs. For the Advanced LIGO mirror, which is made of fused silica and weighs nearly 40 kilograms, the ideal position was found to be roughly 111.6 millimeters from the center. This spot is slightly different from the theoretical nodal radius of the drumhead mode, a difference caused by the complex interaction between the drumhead and butterfly vibration patterns. The researchers also tested what happens when the lasers are not perfectly aligned, a scenario that reflects real-world conditions where mechanical imperfections cause slight shifts. They found that even a tiny misalignment of just two millimeters could cause the calibration to drift by more than one percent at frequencies above 2,100 hertz. This level of error is significant because it occurs in the frequency range where the "ringdown" signal of a merging neutron star is expected, potentially obscuring vital clues about the nature of these dense stellar remnants.

The study also looked at how different materials and sizes affect these vibrations. The KAGRA detector in Japan uses mirrors made of sapphire, a material that is much stiffer than the fused silica used in other detectors. This stiffness pushes the problematic vibration frequencies much higher, well above the range of interest for current observations, giving KAGRA a natural advantage in this specific area. However, the proposed LIGO A# upgrade plans to use much larger, 100-kilogram mirrors to improve sensitivity. The simulations showed that these larger mirrors would have vibration modes that fall directly within the critical frequency band for neutron star studies. Without careful adjustment of the calibration laser positions, the larger mirrors would introduce substantial errors, making it difficult to interpret the data accurately. The researchers concluded that simply scaling up the mirror size is not enough; the calibration strategy must be re-engineered to account for these new vibration characteristics.

In the end, this work provides a roadmap for the next generation of gravitational wave observations. By mapping out exactly how the mirrors deform under the pressure of calibration lasers, the team has identified the precise locations where the lasers should be aimed to minimize errors. They demonstrated that while the mirrors are massive and heavy, they are not perfectly rigid, and their internal flexibility must be accounted for to hear the universe clearly. The findings suggest that future detectors will need to employ multiple calibration beams or more sophisticated control systems to suppress these vibrations. As the field moves toward more sensitive instruments capable of detecting fainter signals from deeper in the cosmos, understanding and correcting for the subtle flexing of the mirrors themselves will be just as important as the lasers and optics that make the detection possible.