Rotation curves, dark matter, and modified gravity

Astronomers like to plot pairs of observables on xy planes. Sometimes the result is a blobby mess of uncorrelated points. Sometimes the plotter is thrilled to discover a correlation that turns out to be astronomically useful, astrophysically significant, or both.

Edwin Hubble’s galactic redshift–distance relation is perhaps the most famous astronomical xy plot. Ejnar Hertzsprung and Henry Norris Russell’s diagram of stellar luminosity and spectral type is perhaps the most beautiful .

In 1977 Brent Tully and Richard Fisher discovered another important correlation. For a sample of galaxies that have disks, they plotted each galaxy’s intrinsic luminosity against each galaxy’s total emission in atomic hydrogen’s 21-cm spectral line. On a log–log plot, the points arrange themselves on a straight line with surprising neatness.

Tully and Fisher assembled their relation using disk galaxies whose distances from Earth had already been determined. But if you don’t know the distance to a disk galaxy, you can use the Tully–F isher relation to calculate it: Just measure the 21-cm linewidth, look up what the galaxy’s intrinsic luminosity should be, and then work out the distance from the galaxy’s apparent brightness.

The Tully–Fisher relation is not only astronomically useful, it’s also astrophysically significant. The 21-cm linewidth reflects how fast stars are orbiting a galaxy’s center of mass; the intrinsic luminosity reflects how much luminous mass a galaxy contains. The more massive the galaxy is, the faster the stars orbit.

Galaxies consist of more than just stars. Besides interstellar gas, they also contain—or seem to contain—dark matter. How the luminous mass is distributed varies considerably from galaxy to galaxy. Some disk galaxies have wide, flat disks, small central bulges, and relatively little interstellar gas. Others, like M102 shown here, have big, gas-filled central bulges. Remarkably, the Tully–Fisher relation is valid not only over several orders of magnitude in total luminosity, but also over several orders of magnitude in bulge size.

Hubble’s diagram directly reflects how the universe is expanding. The Hertzsprung–Russell diagram, on the other hand, reflects how the physics of gravity, nuclear interactions, and fluid dynamics interact and play out in stars of different mass and chemical composition. Regardless of whether the Tully–Fisher corresponds to a direct physical correlation or to something more involved, if you think you understand how matter and gravity interact to shape galaxies, your theory should be able to reproduce it.

But to connect the theories (which deal with mass) with the observations (which deal with light), you have to make assumptions about the mass distributions of stars—at least in the case of disk galaxies whose nondark, baryonic mass is dominated by stars. Fortunately, some galaxies’ interstellar media contain a lot more baryonic matter than their stars do. From a sample of such gassy galaxies, you can create an accurate baryonic Tully–Fisher relation (BTFR), in which total baryonic mass replaces intrinsic luminosity.

MOND the gap

That’s what the University of Maryland’s Stacy McGaugh has done in a paper in press at Physical Review Letters. McGaugh’s paper is causing quite a stir. According to his calculations, a form of modified gravity known as MOND can reproduce the gassy galaxies’ BTFR without invoking dark matter. “Gassy Galaxies Defy Dark Matter” is the catchy headline that drew readers to a story on the BBC’s homepage.

The MOND modification is straightforward in form. At the high values of gravity-induced acceleration found in our solar system, gravitational force is proportional to the acceleration, just as Isaac Newton specified. But in the outskirts of galaxies, where gravity-induced acceleration is far weaker, gravitational force is proportional to acceleration squared.

McGaugh also finds that the reigning cosmological paradigm, ΛCDM , can also yield the gassy galaxies’ BTFR, but only with seemingly contrived fine-tuning.

Moreover, in a galaxy where MOND prevails, the stellar rotation is determined only by the amount of baryonic matter. The BTFR’s neat correlation is accounted for by the robustness of the measurements. But in a galaxy where ΛCDM prevails, stellar rotation is determined by the amount of dark matter and baryonic matter. The BTFR’s observed neatness becomes harder to explain because it would seem to require two values to fluctuate conspiratorially.

What do McGaugh’s results mean for astronomy and physics? First, they don’t refute the existence of dark matter. As McGaugh acknowledges in his paper, whereas ΛCDM has difficulty reproducing the gassy galaxies’ BTFR, MOND has difficulty reproducing its equivalent for clusters of galaxies. Both theories fall short.

You should also keep in mind what’s required to calculate a BTFR. In a MOND galaxy, you can integrate the gas mass, calculate the gravitational potential, and obtain the stellar rotation directly. In a ΛCDM galaxy, you can’t see the dark matter, nor do you know how the baryonic and dark matter interacted in the past to create today’s galaxies. To me, the surprise and significance of McGaugh’s paper is not that ΛCDM struggles, but that MOND succeeds.

But even if MOND or some other form of modified gravity supplants dark matter, it would amount, for the most part, to swapping one embodiment of our ignorance (a new form of matter) for another (a new type of gravity). Neither dark matter nor modified gravity connect directly to fundamental physics. Until they do, we remain in the dark.

Charles Day