Category: Laws of Nature

What is Natural?

I have been musing for a while on the idea of writing about Naturalness in science, particularly as it applies to the radial acceleration relation. As a scientist, the concept of Naturalness is very important to me, especially when it comes to the interpretation of data. When I sat down to write, I made the mistake of first Googling the term.

The top Google hits bear little resemblance to what I mean by Naturalness. The closest match is specific to a particular, rather narrow concept in theoretical particle physics. I mean something much more general. I know many scientific colleagues who share this ideal. I also get the impression that this ideal is being eroded and cheapened, even among scientists, in our post-factual society.

I suspect the reason a better hit for Naturalness doesn’t come up more naturally in a Google search is, at least in part, an age effect. As wonderful a search engine as Google may be, it is lousy at identifying things B.G. (Before Google).  The concept of Naturalness has been embedded in the foundations of science for centuries, to the point where it is absorbed by osmosis by students of any discipline: it doesn’t need to be formally taught; there probably is no appropriate website.

In many sciences, we are often faced with messy and incomplete data. In Astronomy in particular, there are often complicated astrophysical processes well beyond our terrestrial experience that allow a broad range of interpretations. Some of these are natural while others are contrived. Usually, the most natural interpretation is the correct one. In this regard, what I mean by Naturalness is closely related to Occam’s Razor, but it is something more as well. It is that which follows – naturally – from a specific hypothesis.

An obvious astronomical example: Kepler’s Laws follow naturally from Newton’s Universal Law of Gravity. It is a trivial amount of algebra to show that Kepler’s third Law, P2 = a3, follows as a direct consequence of Newton’s inverse square law. The first law, that orbits are ellipses, follows with somewhat more math. The second law follows with the conservation of angular momentum.

It isn’t just that Newtonian gravity is the simplest explanation for planetary orbits. It is that all the phenomena identified by Kepler follow naturally from Newton’s insight. This isn’t obvious just by positing an inverse square law. But in exploring the consequences of such a hypothesis, one finds that one clue after another falls into place like the pieces of a jigsaw puzzle. This is what I mean by Naturalness.

I expect that this sense of Naturalness – the fitting together of the pieces of the puzzle – is what gave Newton encouragement that he was on the right path with the inverse square law. Let’s not forget that both Newton and his inverse square law came in for a lot of criticism at the time. Both Leibniz and Huygens objected to action at a distance, for good reason. I suspect this is why Newton prefaced his phrasing of the inverse square law with the modifier as if: “Everything happens… as if the force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.” He is not claiming that this is right, that it has to be so. Just that it sure looks that way.

The situation with the radial acceleration relation in galaxies today is the same. Everything happens as if there is a single effective force law in galaxies. This is true regardless of what the ultimate reason proves to be.

The natural explanation for the single effective force law indicated by the radial acceleration relation is that there is indeed a unique force law at work. In this case, such a force law has already been hypothesized: MOND. Often MOND is dismissed for other reasons, though reports of its demise have repeatedly been exaggerated. Perhaps MOND is just the first approximation of some deeper theory. Perhaps, like action at a distance, we simply don’t yet understand the underlying reasons for it.

Four Strikes

Four Strikes

So the radial acceleration relation is a new law of nature. What does it mean?

One reason we have posed it as a law of nature is that it is interpretation-free. It is a description of how nature works – in this case, a rule for how galaxies rotate. Why nature behaves thus is another matter.

Some people have been saying the RAR (I tire of typing out “radial acceleration relation”) is a problem for dark matter, while others seem to think otherwise. Lets examine this.

The RAR has a critical scale g† = 1.2 · 10-10 m s-2. At high acceleration, above this scale, we don’t need dark matter: systems like the solar system or the centers of high surface brightness galaxies are WYSIWYG. At low accelerations, below this scale, we begin to need dark matter. The lower the acceleration, the more dark matter we need.

OK, so this means there is little to no dark matter when the baryons are dense (high gbar), but progressively more as gbar becomes smaller than the critical scale g†. Low gbar happens when the surface density of baryons is low. So the amount of dark matter scales inversely with baryonic surface density.

That’s weird.

This is weird for a number of reasons. First, there is no reason for the dark matter to care what the baryons are doing when dark matter dominates. When gobs ≫ gbar the dark matter greatly outweighs the baryons, which simply become tracer particles in the gravitational potential of the dark matter halo. There is no reason for the dark matter to know or care about what the baryonic tracer particles are doing. And yet the RAR persists as a tight correlation well into this regime. It is as if the baryonic tail wags the dark matter dog.

Second, there should be more dark matter where there are more baryons. Galaxies form by baryons falling into dark matter halos. As they do so, they dissipate energy and sink to the center of the halo. In this process, the drag some of the dark matter along with them in a process commonly referred to as “adiabatic compression.” In practice, the process need not be adiabatic, but the dark matter must respond to the rearrangement of the gravitational potential caused by the dissipative infall of the baryons.

These topics have been discussed at great length in the galaxy formation literature. Great arguments have erupted time and again about how best to implement the compression in models, and how big the effect is in practice. These details need not concern us here. What matters is that they are non-negotiable fundamentals of the dark matter paradigm.

Galaxies form by baryonic infall within dark matter halos. The halos form first while the baryons are still coupled to the photons prior to last scattering. This is one of the fundamental reasons we need non-baryonic cold dark matter that does not interact with photons: to get a jump on structure formation. Without it, we cannot get from the smooth initial condition observed in the cosmic microwave background to the rich amount of structure we see today.

As the baryons fall into halos, they must sink to the center to form galaxies. Why? Dark matter halos are much bigger than the galaxies that reside within them. All tracers of the gravitational potential say so. Initially, this might seem odd, as the baryons might to just track the dominant dark matter. But baryons are different: they can dissipate energy. By so doing, they can sink to the center – not all baryons need to sink to the centers of their dark matter halos, but enough to make a galaxy. This they must do in order to form the galaxies that we observe – galaxies that are more centrally condensed than their dark matter halos.

That’s enough, in return, to affect the dark matter. As the baryons dissipate, the gravitational potential is non-stationary. The dark matter distribution must respond to this change in the total gravitational potential. The net result is a further concentration of the dark matter towards the center of the halo: in effect, the baryons drag some dark matter along with them.

I have worked on adiabatic compression myself, but a nice illustration is given by this figure from Elbert et al. (2016):

compressedhalos_cdmonly
Dark matter halos formed in numerical simulations illustrating the effect of adiabatic compression. One the left is a pristine halo without baryons. In the middle is a halo after formation of a disk galaxy. On right is a halo after formation of a more compact disk.

One can see by eye the compression caused by the baryons. The more dense the baryons become, the more dark matter they drag towards the center with them.

The fundamental elements of the dark matter paradigm, galaxy formation by baryonic infall and dissipation accompanied by the inevitable compression of the dark matter halo, inevitably lead us to expect that more baryons in the center means more dark matter as well. We observe the exact opposite in the RAR. As baryons become denser, they become the dominant component, to the point where they are the only component. Rather than more dark matter as we expect, more baryons means less dark matter in reality.

Third, the RAR correlation is continuous and apparently scatter-free over all accelerations. The data map from the regime of no dark matter at high accelerations to lots of dark matter at low accelerations in perfect 1:1 harmony with the distribution of the baryons. If we observe the distribution of baryons, we know the corresponding distribution of dark matter. The tail doesn’t just wag the dog. It tells it to sit, beg, and roll over.

Fourth, there is a critical scale in the data, g†. That’s the scale where the mass discrepancy sets in. This is a purely empirical statement.

Cold dark matter is scale free. Being scale free is fundamental to its nature. It is essential to fitting the large scale structure, which it does quite well.

So why is there this ridiculous acceleration scale in the data?!? Who ordered this?! It should not be there.

So yes, the radial acceleration relation is a problem for the cold dark matter paradigm.

The Third Law of Galactic Rotation

Flat rotation curves were the first clear evidence that the dynamics of galaxies do not follow the same rules as planetary systems. But they do follow rules. These include asymptotic flatness, Tully-Fisher, the luminosity-size-rotation curve shape relation (aka the `universal‘ rotation curve), Renzo’s rule, and the central density relation.

vrsd_pop
Rotation curves color coded by the characteristic surface density of stars and gas, ranging from low surface brightness galaxies (blue) to those  of high surface brightness (red).

These various relations sound like a hodge-podge of random astronomical effects. This is misleading. There is a great deal of organization in the data. The surface density of stars and gas, and the acceleration (gbar) determined by their gravitational potential, plays a defining role. Indeed, the known relations are all manifestations of a single, more fundamental relation, the radial acceleration relation.

rar
The radial acceleration relation. The centripetal acceleration measured by the rotation curve (gobs) correlates with that predicted by the observed distribution of stars and gas (gbar). The data consist of 2,693 resolved points along the rotation curves of 153 rotating galaxies from the SPARC database. All galaxies fall along the same relation, within the uncertainties. Red squares are binned data. The lower panel shows residuals from a fit to the data. The dashed lines are the scatter in the data; the red lines are the amount of scatter expected from measurement uncertainties.

The radial acceleration relation connects what you see in galaxies with what you get for the gravitational force. This would be a trivial statement if galaxies behave as planetary systems do. They would follow the 1:1 dotted line in the figure. Instead, they bend away from that line.

Indeed, the data are consistent with a single effective force law, which can be written

rareqn

Other functional forms could also work. But they would all necessarily have a critical acceleration scale g† ≈ 10-10 m s-2. This is an important scale that is ubiquitous in extragalactic astronomy. It seems to be a new fundamental scale in physics.

The critical acceleration marks the onset of the missing mass problem. Above this scale, there is no need for dark matter. Below it, the difference between the 1:1 line and the data is what we attribute to dark matter. The more the observed acceleration exceeds that which can be explained by the stars and gas, the larger the mass discrepancy.

Irrespective of interpretation, the data establish the radial acceleration relation in a purely empirical way. There is nothing but data here. The axes are independent: one is measured from rotation curves, the other from photometry. These need not be well connected – the dark matter could cause any sort of acceleration independently of the stars and gas. But they are intimately coupled.

There are no deviations from the radial acceleration relation beyond those attributable to experimental error. The residuals do not correlate with mass, size, surface brightness, color, environment, how many intelligent civilizations a galaxy hosts, or anything else. The scale that matters is not luminosity or halo mass or size. It is the acceleration determined from the surface density of stars and gas.

The radial acceleration relation is a fundamental relation. In effect, it is a law of nature. Third in our counting, but first in importance, as both flat rotation curves and the Tully-Fisher relation follow from it. It must be explained by any theory that claims to provide a satisfactory description of galaxy dynamics.

Tully-Fisher: the Second Law

Tully-Fisher: the Second Law

Previously I noted how we teach about Natural Law, but we no longer speak in those terms. All the Great Laws are already know, right? Surely there can’t be such things left to discover!

That rotation curves tend towards asymptotic flatness is, for all practical purposes, a law of nature. It is tempting to leap straight to the interpretation (dark matter!), but it is worth appreciating the discovery for itself. It isn’t like rotation curves merely exceed what can be explained by the stars and gas, nor that they rise and fall willy-nilly. The striking, ever-repeated observation is an indefinitely extended radial range with near-constant rotation velocity.

flatrcillwpictures
The rotation curves of galaxies over a large dynamic range in mass, from the most massive spiral with a well measured rotation curve (UGC 2885) to tiny, low mass, low surface brightness, gas rich dwarfs.

New Laws of Nature aren’t discovered every day. This discovery should have warranted a Nobel prize for Vera Rubin and Albert Bosma. If only we were able to see it in those terms three decades ago. Instead, we phrased it in terms of dark matter, and that was a radical enough idea it has to await verification in the laboratory. Now the prize will go to some experimental group (should there ever be a successful detection) while the new law of nature goes unrecognized. That’s OK – there should be a Nobel prize for a verified laboratory detection of non-baryonic dark matter, should that ever occur – but there should also be a Nobel prize for flat rotation curves, and it should have been awarded a long time ago.

It takes a while to appreciate these things. Another well known yet unrecognized Law of Nature is the Tully-Fisher relation. First discovered as a relation between luminosity and line-width (figure from Tully & Fisher 1977), this relation is most widely known for its utility in measuring the cosmic distance scale.

tforig
The original Tully-Fisher relation.

At the time, it gave the “wrong” answer (H0 ≠ 50), and Sandage is reputed to have suppressed its publication for a couple of years. This is one reason astronomy journals have, and should have, a high acceptance rate – too many historical examples of bad behavior to protect sacred cows.

Besides its utility as a distance indicator, the Tully-Fisher relation has profound implications for physical theory. It is not merely a relation between two observables of which only one is distance-dependent. It is a link between the observed mass and the physics that sets the flat velocity.

btf_mst_mb_2009
The stellar mass Tully-Fisher relation (left) and the baryonic Tully-Fisher relation (right). In both cases, the x-axis is the flat rotation velocity measured from resolved rotation curves. In the right panel, the y-axis is the baryonic mass – the sum of observed stars and gas. The latter appears to be a law of nature from which galaxies never stray.

The original y-axis of the Tully-Fisher relation, luminosity, was a proxy for stellar mass. The line-width was a proxy for rotation velocity, of which there are many variants. At this point it is clear that the more fundamental variables are baryonic mass – the sum of observed stars and gas – and the flat rotation velocity.

I had an argument – of the best scientific sort – with Renzo Sancisi in 1995. I was disturbed that our then-new low surface brightness galaxies were falling on the same Tully-Fisher relation as previously known high surface brightness galaxies of comparable luminosity. The conventional explanation for the Tully-Fisher relation up to that point invoked Freeman’s Law – the notion (now deprecated) that all spirals had the same central surface brightness. This had the effect of suppressing the radius term in Newton’s

V2 = GM/R.

Galaxies followed a scaling between luminosity (mass) and velocity because they all had the same R at a given M.

By construction, this was not true for low surface brightness galaxies. They have larger radii at fixed luminosity (representing the mass M). That’s what makes them low surface brightness – their stars are more spread out. Yet they fall smack on the same Tully-Fisher relation!

Renzo and I looked at the result and argued up and down, this way and that about the data, the relation, everything. We were getting no closer to understanding it, or agreeing on what it meant. Finally he shouted “TULLY-FISHER IS GOD!” to which I retorted “NEWTON IS GOD!”

It was a healthy exchange of viewpoints.

Renzo made his assertion because, in his vast experience as an observer, galaxies always fell on the Tully-Fisher relation. I made mine, because, well, duh. The problem is that the observed Tully-Fisher relation does not follow from Newton.

But Renzo was right. Galaxies do always fall on the Tully-Fisher relation. There are no residuals from the baryonic Tully-Fisher relation. Neither size nor surface brightness are second parameters. The relation cares not whether a galaxy disk has a bar or not. It does not matter whether a galaxy is made of stars or gas. It does not depend on environment or pretty much anything else one can imagine. Indeed, there is no intrinsic scatter to the relation, as best we can tell. If a galaxy rotates, it follows the baryonic Tully-Fisher relation.

The baryonic Tully-Fisher relation is a law of nature. If you measure the baryonic mass, you know what the flat rotation speed will be, and vice-versa. The baryonic Tully-Fisher relation is the second law of rotating galaxies.