# Are Gravitational Waves the Key to Nailing Down the Universe’s Expansion Rate?

Just over a century ago, Einstein sat scratching his head over his own theory of general relativity. Although the equations seemed to fit all of astronomers’ observations of the universe to date, there was one little detail he couldn’t seem to shake: the universe ought to be contracting over time, scrunched together by the pull of gravity.

To an early twentieth-century scientist, the idea seemed preposterous. At the time, the standard thinking was that the universe was static and eternal, that it had always existed in roughly the same form, and that it always would. In a move he would later regret, Einstein ended up modifying his equations of general relativity to reflect this belief, introducing the cosmological constant to counteract gravity and hold the universe steady.

But other physicists weren’t so attached to the “steady state” theory. In the ’20s, both Alexander Friedmann and Georges Lemaître independently predicted an expanding Universe, which was subsequently confirmed by American astronomer Edwin Hubble in 1929. While studying distant galaxies, Hubble discovered that they were all moving away from the Milky Way at a speed roughly proportional to their distances from us. This observation can be described quite simply (to a first approximation, at least) by the equation:

v=H0 d

This simple formula encapsulates the heart of Hubble’s discovery: as the distance to a galaxy (d) increases, so does that galaxy’s velocity relative to us (v). The scaling factor H0 is today known as the Hubble constant, and it is widely considered to hold key information about our cosmic history and future—which is why it’s so frustrating to astronomers that we still don’t have an accurate value for it. Theoretically, H0 can by calculated by comparing the redshift (a property of the electromagnetic radiation coming from an object moving relative to us) to the distance of an object, but it turns out that astronomical distance is quite tricky to measure accurately.

One of the most common methods for measuring the distance to a galaxy is by using so-called “standard candles” contained in it. These are objects (technically speaking, Cepheid variables and supernovae) that have a known total luminosity, so by measuring how much of their light reaches Earth we can calculate their distance. However, doing so requires the use of a “cosmic ladder”, a series of steps and scales that allows astronomers to work their way up to greater and greater distances. “Now the problem is that at any step you can introduce some small systematic error, and this can build up,” MIT astrophysicist Dr. Salvatore Vitale explains. As a result, current estimates of the Hubble constant have quite large uncertainties—some as large as four standard deviations.

But with the advent of gravitational wave astronomy, cosmologists have an entirely new tool to work with. Back in 1986, long before construction on LIGO began, Dr. Bernard Schutz published a seminal paper outlining a hypothetical process for calculating certain objects’ distances based on their gravitational wave signatures. This idea has since been refined and used by LIGO scientists to pinpoint the locations of gravitational wave sources that have been observed to date.

As it turns out, the amount of energy we receive from a compact binary system (two heavy objects orbiting each other) depends on only two numbers: the mass of the system and its distance from us. As Vitale explains, using gravitational waves as a measuring stick works along roughly the same lines as using standard candles; in fact, binary systems are commonly referred to as “standard sirens” for this reason. “If I get a light bulb and I shine it towards you, as I go farther and farther away from you, you will receive less and less energy… the same is true for gravitational waves.” Conveniently, the shape of a gravitational wave “chirp” as the binary spirals in on itself gives us an excellent measurement of the system’s mass, so it should be pretty straightforward to calculate its distance.

The only problem is: it isn’t.

Unfortunately, this idealized scenario doesn’t hold water. It’s true that binaries with similar masses emit about the same amount of energy, but that energy isn’t emitted evenly in all directions. Instead, the gravitational waves are emitted primarily in beams directly above and below the binary’s plane of orbit, which means that the orientation of the binary with respect to us is also crucial in our calculation of its distance. (To learn more about how gravitational waves carry energy and why it comes off primarily perpendicular to the plane of orbit, see our 2016 piece on the “sticky bead” thought experiment.)

Vitale describes this phenomenon in terms of a light source shaped something like a flashlight: “You can still think about it in terms of light…. If I just point it directly at you, you get all the light—you get all the energy—but if I start to tilt it away from you, you get less and less light.” In practical terms, it becomes impossible to distinguish between a distant source that’s emitting in our direction and a nearby source that’s oriented away from us.

Although this is true of neutron star binaries, the type generally favored for this research, Vitale thinks there’s another option. Back in 2014, before any gravitational wave detection, he published a paper examining the properties of orbits between neutron stars and black holes. Although it wasn’t the main focus of his research at the time, he realized that this particular type of binary lends itself to more accurate measurements of distance, and thus a potentially better calculation of the Hubble constant. “There was just a sentence on a paper that said that a neutron star-black hole… could be the ideal way to measure the Hubble constant,” he reminisces. “And I promised myself that I would have gone back to that problem later on.”

 A simulation reveals the distribution of matter in the wake of a neutron star-black hole merger Image credit: F. Foucart, et al. (2017) Classical and Quantum Gravity

The detail that initially caught Vitale’s attention was the fact that black holes typically have an intrinsic spin. When that’s combined with an orbiting neutron star, the entire system “wobbles” about an axis in what’s technically referred to as precession, similar to the wiggly motion that a top makes as you spin it across a table. Suddenly the orientation of those gravitational “beams” varies with time, tracing out a cone-like pattern. This turns out to be the missing piece of the puzzle that will allow astronomers to tease out the distance and inclination from their combined effect.

Neutron star-black hole binaries have another advantage as well: because they are so much more massive than neutron star binaries, they can be detected at much greater distances. Recall that in order to measure the Hubble constant, you need two pieces of information: the distance to an object (measured using gravitational waves) and that object’s receding velocity (measured using electromagnetic radiation). Vitale’s collaborator, Hsin-Yu Chen of Harvard, says it’s generally believed that high-spin black holes are more likely to emit huge quantities of light that can be detected from very far away.

This fact also improves the accuracy of velocity measurements. Vitale explains that although redshift gives the velocity of a galaxy, “The velocity we care about is not the velocity of the galaxy. What we care about is the velocity of the Universe at the point where the galaxy lives…They may not be the same because each object in the Universe has its own velocity.” The velocity of each object relative to the space that surrounds it is known as peculiar velocity, and it’s proven something of a headache to astronomers. It’s typically not small, residing somewhere on the order of a few hundred kilometers per second. However, as we look at objects that are farther from us, Hubble’s Law tells us that its velocity due to the expansion of the Universe increases but the peculiar velocity remains about the same. As a result, the peculiar velocity represents a much smaller percentage of an object’s total velocity as we look farther and farther away, meaning that more distant objects can be measured to a much more precise velocity. Since neutron star-black hole binaries can be seen at enormous distances, this is another point in their favor.

Now for the bad news. The fact of the matter is that neutron star-black hole binaries are significantly less common than regular neutron star binaries. In fact, no one really knows how many there are in the Universe. Vitale and Chen were fascinated by this conundrum: although each neutron star-black hole binary represents a much more accurate measurement of the Hubble constant, there’s more uncertainty surrounding them simply because there are fewer potential datapoints. Vitale asks, “Can the fact that each of them is better compensate for the fact that there are fewer of them in the Universe?”

As it turns out, yes. The pair recently published a paper investigating the accuracy and uncertainty associated with using neutron star-black hole mergers rather than the traditionally more popular neutron star-neutron star binaries. They found that even if there were as few as 50 such mergers per cubic gigaparsec of space per year (a standard astronomical volume rate), we’d still be able to improve our estimates of the Hubble constant. For context, the LIGO-Virgo collaboration has estimated that the rate of binary neutron star mergers is somewhere between 320 and 4640 mergers per cubic gigaparsec per year.

Although it’s impossible to say when we can expect to detect neutron star-black hole mergers once LIGO goes back online early next year, it’s pretty clear that we’ll start to see more and more accurate measurements of the Hubble constant. As Chen points out, even without this rare binary type, astronomers can quickly improve estimates of H0 using regular binaries since statistical uncertainty is linked to the square root of the number of datapoints. Although the uncertainty of the Hubble constant measured using gravitational waves is currently about 14%, Chen says, “Just having a handful of detections [of gravitational waves with electromagnetic counterparts] will bring the H0 uncertainty to a few percent.” By adding in the highly accurate results of just one or two neutron star-black hole mergers, this number will decrease even further.

Chen believes that this research lays the groundwork for future analysis of neutron star-black hole mergers. “Binary neutron stars are kind of the standard scenario—everyone talks about binary neutron stars, while not that many people talk about neutron star-black holes… So this was not explored in the past.” Once that happens and we can more accurately determine H0, we’ll hold one more clue to the Universe’s past and future.

“It’s destiny,” Vitale says of H0. “It’s origin and it’s destiny.”

Eleanor Hook