Light from a distant star travels to us for billions of years, and during that time the universe has time to expand. How can we check that we correctly understand this journey? Astronomers use two 'rulers': supernova explosions, which shine with a known brightness, and fast radio bursts — short, powerful pulses of radio waves from other galaxies. If both rulers show the same distance, then cosmic geometry works like clockwork.

In this study, scientists took supernova data from the Pantheon+ survey and compared them with distances calculated from fast radio bursts using neural networks. The neural network helped reconstruct how radio waves are 'delayed' along their path due to intervening matter, and convert that into distances. The main result: no deviations from the expected relationship between brightness and size were found. The cosmos is honest.

Additionally, they managed to estimate how much matter lies directly near the sources of radio bursts — in their host galaxies. This number is important for future cosmological studies. For now, the precision is modest, but the method promises to become a powerful tool as more data accumulate.