Old Math Model Aids Search For Gravitational Waves

The waves in question come from small black holes or neutron stars in extremely elongated orbits around vastly larger black holes, says Dr. Lior Burko, an assistant physics professor at The University of Alabama in Huntsville (UAH). "This reopens an area of research that was closed several years ago."

The exotic gravitational waves are generated (as predicted by general relativity theory) when an orbiting compact object changes speed, accelerating as it approaches the larger black hole and slowing as it moves away.

"Just as an accelerating electric charge emits electromagnetic waves, so mass emits gravitational waves as the speed changes," Burko said.

The time-domain formulas that Burko and Khanna studied had been abandoned because they produced error rates of ten percent or more, compared to the more accurate frequency-domain formulae.

The problem, said Burko, is that the frequency-domain math doesn’t work especially well with objects in extremely elliptic orbits without burning up tons of computer time (and still getting large errors). And objects in elliptic or parabolic orbits are reasonably common in astronomy, as stars and small black holes wander or are pulled into orbits around massive black holes.

What to do, what to do?

The solution started with a change of perspective. To save computer time, early time-domain experiments looked at gravity waves inside a relatively small grid — only 100 times the radius of the large black hole.

Burko and Khanna, however, found that the larger you make the grid, the smaller your error bar shrinks. At 500 radii the error had dropped from ten percent to one percent. At 1,500 radii the error shrinks to 0.1 percent. The goal is an error of not more than 0.01 percent.

The tradeoff, of course, is that the larger grid means substantially more grid points and longer computing times. With 40 data points per radius, enlarging the grid from 100 to 1,500 radii increases the number of grid points from 320 million to 72 billion. And each grid point requires hundreds of calculations to simulate how gravity waves might form and evolve. That’s a lot of calculations, even for a super computer. A single run using the new model can take at least two weeks on a fast workstation.

The second big step had to do with how you calculate the size of the small black hole or neutron star. Mathematically, it is considered a point source compared to the much larger black hole. Making calculations based on a single point in the grid introduces new errors so astronomers use a Gaussian formula to simulate the gravity well of the orbiting object.Burko and Khanna found that how big you make that Gaussian spot also influences how big your error will be.

"Too small and you under sample," said Burko. "Too big and you start bringing in finite size effects. We found a sweet spot that would give you the least errors."

The goal is to accurately model what these special gravity waves look like when they leave their home galaxies. Astronomers use that data to calculate (and estimate) how those waves might look after they travel a few billion light years to Earth and various gravity wave detectors in the U.S. or in orbit following the Earth around the Sun. Knowing what to expect will help scientists recognize the "signal" from these gravitational waves.

The results of this research by Burko and Dr. Gaurav Khanna at the University of Massachusetts at Dartmouth are published on-line by EuroPhysics Letters.