DTM’s Remembering Vera

DTM’s Remembering Vera

I wrote my own recollection of Vera Rubin recently. Her long time home institution, the Department of Terrestrial Magnetism (DTM) of the Carnegie Institution of Washington recently held a lunch in her honor. Unfortunately my travel schedule precluded me from attending. However, they have put together a wonderful website that I recommend to everyone. The depth and variety of the materials published there – testimonials, photos, her list of published papers – is outstanding.

Of historical interest are a series of papers written in the mid-60s in collaboration with Margaret Burbidge. These show some early rotation curves. Many peter out around the turn-over of the rotation curve. With the benefit of hindsight, one can see what the data will do – extend more or less flat from the last measured points.

Here is an example from Burbidge et al. (1964). In this case, NGC 3521, they got a bit further than the turnover. You may judge for yourself how convincing the detection of flat rotation is.

ngc3521_brbidgerubun1964

As it happens, NGC 3521 is a near kinematic twin to the Milky Way. Here is the modern rotation curve from THINGS compared with an estimate of the Milky Way rotation curve.

mw_ngc3521_twins

Hopefully it is obvious why it helps to have extended data (usually from 21 cm data, as in the example from THINGS).

This reminds me of something Vera frequently said. Early Days. In many ways, we are far down the path of dark matter. But we still have no idea what it is, or even whether what we call dark matter now is merely a proxy for some more general concept.

Vera always appreciated this. In many ways, these are still Early Days.

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Reckless disregard for the scientific method

Reckless disregard for the scientific method

There has been another attempt to explain away the radial acceleration relation as being fine in ΛCDM. That’s good; I’m glad people are finally starting to address this issue. But lets be clear: this is a beginning, not a solution. Indeed, it seems more like a rush to create truth by assertion than an honest scientific investigation. I would be more impressed if these papers were (i) refereed rather than rushed onto the arXiv, and (ii) honestly addressed the requirements I laid out.

This latest paper complains about IC 2574 not falling on the radial acceleration relation. This is the galaxy that I just pointed out (about the same time they must have been posting the preprint) does adhere to the relation. So, I guess post-factual reality has come to science.

Rather than consider the assertions piecemeal, lets take a step back. We have established that galaxies obey a single effective force law. Federico Lelli has shown that this applies to pressure supported elliptical galaxies as well as rotating disks.

rar_todo_raronly
The radial acceleration relation, including pressure supported early type galaxies and dwarf Spheroidals.

Lets start with what Newton said about the solar system: “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.” Knowing how this story turns out, consider the following.

Suppose someone came to you and told you Newton was wrong. The solar system doesn’t operate on an inverse square law, it operates on an inverse cube law. It just looks like an inverse square law because there is dark matter arranged just so as to make this so. No matter whether we look at the motion of the planets around the sun, or moons around their planets, or any of the assorted miscellaneous asteroids and cometary debris. Everything happens as if there is an inverse square law, when really it is an inverse cube law plus dark matter arranged just so.

Would you believe this assertion?

I hope not. It is a gross violation of the rule of parsimony. Occam would spin in his grave.

Yet this is exactly what we’re doing with dark matter halos. There is one observed, effective force law in galaxies. The dark matter has to be arranged just so as to make this so.

Convenient that it is invisible.

Maybe dark matter will prove to be correct, but there is ample reason to worry. I worry that we have not yet detected it. We are well past the point that we should have. The supersymmetric sector in which WIMP dark matter is hypothesized to live flunked the “golden test” of the Bs meson decay, and looks more and more like a brilliant idea nature declined to implement. And I wonder why the radial acceleration relation hasn’t been predicted before if it is such a “natural” outcome of galaxy formation simulations. Are we doing fair science here? Or just trying to shove the cat back in the bag?

74117526

I really don’t know what the final answer will look like. But I’ve talked to a lot of scientists who seem pretty darn sure. If you are sure you know the final answer, then you are violating some basic principles of the scientific method: the principle of parsimony, the principle of doubt, and the principle of objectivity. Mind your confirmation bias!

That’ll do for now. What wonders await among tomorrow’s arXiv postings?

Natural Law

Natural Law
or why Vera Rubin and Albert Bosma deserve a Nobel Prize

Natural Law: a concise statement describing some aspect of Nature.

In the sciences, we teach about Natural Law all the time. We take them for granted. But we rarely stop and think what we mean by the term.

Usually Natural Laws are items of textbook knowledge. A shorthand all in a particular field known and agree to. This also brings with it an air of ancient authority, which has a flip side. The implicit operating assumption is that there are no Natural Laws left to be discovered, which implies that it is dodgy to even discuss such a thing.

The definition offered above is adopted, in paraphrase, from a report of the National Academy which I can no longer track down. Links come and go. The one I have in mind focussed on biological evolution. To me, as a physical scientist, it seems a rather soft definition. One would like it to be quantitative, no?

Lets consider a known example: Kepler’s Laws of planetary motion. Everyone who teaches introductory astronomy teaches these, and in most cases refers to them as Laws of Nature without a further thought. Which is to say, virtually everyone agrees that Kepler’s Laws are valid examples of Natural Law in a physical science. Indeed, this sells them rather short given their importance in the Scientific Revolution.

Kepler’s Three Laws of Planetary Motion:

  1. Planetary orbits are elliptical in shape with the sun at one focus.
  2. A line connecting a planet with the sun sweeps out equal areas in equal times.
  3. P2 = a3

In the third law, P is the sidereal period of a planet’s orbit measured in years, and a is the semi-major axis of the ellipse measured in Astronomical Units. This is a natural system of units for an observer living on Earth. One does not need to know the precise dimensions of the solar system: the earth-sun separation provides the ruler.

To me, the third law is the most profound, leading as it does to Newton’s Universal Law of Gravity. At the time, however, the first law was the most profound. The philosophical prejudice/theoretical presumption (still embedded in the work of Copernicus) was that the heavens should be perfect. The circle was a perfect shape, ergo the motions of the heavenly bodies should be circular. Note the should be. We often get in trouble when we tell Nature how things Should Be.

By abandoning purely circular motion, Kepler was repudiating thousands of years of astronomical thought, tradition, and presumption. To imagine heavenly bodies following elliptical orbits that are almost but not quite circular must have seemed to sully the heavens themselves. In retrospect, we would say the opposite. The circle is merely a special case of a more general set of possibilities. From the aesthetics of modern physics, this is more beautiful than insisting that everything be perfectly round.

It is interesting what Kepler himself said about Tycho Brahe’s observations of the position of Mars that led him to his First Law. Mars was simply not in the right place for a circular orbit. It was close, which is why the accuracy of Tycho’s work was required to notice it. Even then, it was such a small effect that it must have been tempting to ignore.

If I had believed that we could ignore these eight minutes [of arc], I would have patched up my hypothesis accordingly. But, since it is not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.

This sort of thing happens all the time in astronomy, right up to and including the present day. Which are the important observations? What details can be ignored? Which are misleading and should be ignored? The latter can and does happen, and it is an important part of professional training to learn to judge which is which. (I mention this because this skill is palpably fading in the era of limited access to telescopes but easy access to archival data, accelerated by the influx of carpetbaggers who lack appropriate training entirely.)

Previous to Tycho’s work, the available data were reputedly not accurate enough to confidently distinguish positions to 8 arcminutes. But Tycho’s data were good to about ±1 arcminute. Hence it was “not permissible to ignore” – a remarkable standard of intellectual honesty that many modern theorists do not meet.

I also wonder about counting the Laws, which is a psychological issue. We like things in threes. The first Law could count as two: (i) the shape of the orbit, and (ii) the location of the sun with respect to that orbit. Obviously those are linked, so it seems fair to phrase it as 3 Laws instead of 4. But when I pose this as a problem on an exam, it is worth 4 points: students must know both (i) and (ii), and often leave out (ii).

The second Law sounds odd to modern ears. This is Kepler trying to come to grips with the conservation of angular momentum – a Conservation Law that wasn’t yet formally appreciated. Nowadays one might write J = VR = constant and be done with it.

The way the first two laws are phrased is qualitative. They satisfy the definition given at the outset. But this phrasing conceals a quantitative basis. One can write the equation for an ellipse, and must for any practical application of the first law. One could write the second law dA/dt = constant or rephrase it in terms of angular momentum. So these do meet the higher standard expected in physical science of being quantitative.

The third law is straight-up quantitative. Even the written version is just a long-winded way of saying the equation. So Kepler’s Laws are not just a qualitative description inherited in an awkward form from ancient times. They do in fact quantify an important aspect of Nature.

What about modern examples? Are there Laws of Nature still to be discovered?

I have worked on rotation curves for over two decades now. For most of that time, it never occurred to me to ask this question. But I did have the experience of asking for telescope time to pursue how far out rotation curves remained flat. This was, I thought, an exciting topic, especially for low surface brightness galaxies, which seemed to extend much further out into their dark matter halos than bright spirals. Perhaps we’d see evidence for the edge of the halo, which must presumably come sometime.

TACs (Telescope Allocation Committees) did not share my enthusiasm. Already by the mid-90s it was so well established that rotation curves were flat that it was deemed pointless to pursue further. We had never seen any credible hint of a downturn in V(R), no matter how far out we chased it, so why look still harder? As one reviewer put it, “Is this project just going to produce another boring rotation curve?”

Implicit in this statement is that we had established a new law of nature:

The rotation curves of disk galaxies become approximately flat at large radii, a condition that persists indefinitely.

This is quantitative: V(R) ≈ constant for R → ∞. Two caveats: (1) I do mean approximately – the slope dV/dR of the outer parts of rotation curves is not exactly zero point zero zero zero. (2) We of course do not trace rotation curves to infinity, which is why I say indefinitely. (Does anyone know a mathematical symbol for that?)

Note that it is not adequate to simply say that the rotation curves of galaxies are non-Keplerian (V ∼ 1/√R). They really do stay pretty nearly flat for a very long ways. In SPARC we see that the outer rotation velocity remains constant to within 5% in almost all cases.

Never mind whether we interpret flat rotation curves to mean that there is dark matter or modified gravity or whatever other hypothesis we care to imagine. It had become conventional to refer to the asymptotic rotation velocity as V well before I entered the field. So, as a matter of practice, we have already agreed that this is a natural law. We just haven’t paused to recognize it as such – largely because we no longer think in those terms.

Flat rotation curves have many parents. Mort Roberts was one of the first to point them out. People weren’t ready to hear it – or at least, to appreciate their importance. Vera Rubin was also early, and importantly, persistent. Flat rotation curves are widely known in large part to her efforts. Also important to the establishment of flat rotation curves was the radio work of Albert Bosma. He showed that flatness persisted indefinitely, which was essential to overcoming objections to optical-only data not clearly showing a discrepancy (see the comments of Kalnajs at IAU 100 (1983) and how they were received.)

And that, my friends, is why Vera Rubin and Albert Bosma deserve a Nobel prize. It isn’t that they “just” discovered dark matter. They identified a new Law of Nature.

The Central Density Relation

I promised more results from SPARC. Here is one. The dynamical mass surface density of a disk galaxy scales with its central surface brightness.

This may sound trivial: surface density correlates with surface brightness. The denser the stars, the denser the mass. Makes sense, yes?

Turns out, this situation is neither simple nor obvious when dark matter is involved. The surface brightness traces stellar surface density while the dynamical mass surface density traces stars plus everything else, including dark matter. The latter need not care about the former when dark matter dominates.

Nevertheless, we’ve known there was some connection for some time. This first became clear to me in the mid-90s, when we discovered that low surface brightness galaxies did not shift off of the Tully-Fisher relation as expected. The only way to obtain this situation was to fine-tune the dynamical mass enclosed by the disk with the central surface brightness. Galaxies had to become systematically more dark matter dominated as surface brightness decreased (see Zwaan et al 1995). This was the genesis of the now common statement that low surface brightness galaxies are dark matter dominated (see also de Blok & McGaugh 1996; McGaugh & de Blok 1998).

This is an oversimplification. A more precise statement would be that dark matter dominates to progressively smaller radii in ever lower surface brightness galaxies. Even as dark matter comes to dominate, the dynamics “know” about the stellar distribution.

The rotation curve depends on the enclosed mass, dark as well as luminous. The rate of rise of the rotation curve from the origin correlates with surface brightness. Low surface brightness galaxies have slowly rising rotation curves while high surface brightness galaxies have steeply rising rotation curves. This is very systematic (e.g., Lelli et al 2013).

LellidVdRSB

The rate of rise of rotation curves (dV/dR) as a function of central surface brightness (from Lelli 2014).

Recently, Agris Kalnajs pointed out to us that in a paper written before even I was born, Toomre (1963) had shown how to obtain the central mass surface density of a thin disk from the rotation curve. This has largely been forgotten because dark matter complicates matters. However, we were able to show that Toomre’s formula returns the correct dynamical surface density within a factor of two even in the extreme case of complete domination by a spherical halo component. This breakthrough was enabled by Kalnajs pointing out a straightforward way to include disk thickness (Toomre assumed a razor thin disk) and Lelli pursuing this to the extreme case of a “spherical” disk with a flat rotation curve.

A factor of two is not much to quibble about when one has the large dynamic range of a sample like SPARC. After all, the data cover four orders of magnitude in central surface brightness. Variations in disk thickness and halo domination will only contribute a bit to the scatter.

Without further ado, here is the result:

CentralDensityRelation

The central dynamical mass surface density as a function of the central stellar surface density (left) and stellar mass (right). From Lelli et al (2016). Points are color coded by morphological type. The dashed line shows the 1:1 relation expected in the absence of dark matter.

The data show a clear correlation between mass surface density and surface brightness. At high surface brightness, the data have a slope and normalization consistent with stars being the dominant form of mass present. This is the long-known result of “maximum disk” (van Albada & Sancisi 1986). The observed distribution of stars does a good job of matching the inner rotation curve. It is only as you go further out, where rotation curves flatten, or to lower surface brightness that dark matter becomes necessary.

Something interesting happens as surface brightness declines. The data gradually depart from the line of unity. The dynamical surface density begins to exceed the stellar surface density, so we begin to need some dark matter. Stars and Newton alone can no longer explain the data for low surface brightness galaxies.

Note, however, that the data depart from the line of unity with a considerable degree of order. It is not like things go haywire as dark matter comes to dominate, as one might reasonably expect. After all, why should the mass surface density depend on the stellar surface density at all in the limit where the former greatly outweighs the latter? But it does: the correlation persists to the point that the mass surface density is predictable from the surface brightness.

It is not only rotation curves that show this behavior. A similar result was found from the vertical velocity dispersions of disks in the DiskMass survey. Specifically, Swaters et al (2014) show essentially the same plot. However, the DiskMass sample is necessarily restricted to rather high surface brightness galaxies. Consequently, those data show only a hint of the systematic departure from the line of unity, and one could argue that it is linear. A large number of low surface brightness galaxies with good data is key to our result (see the discussion of the sample in SPARC).

This empirical result sheds some light on the debate about dark matter halo profiles. The rotation curves of low surface brightness galaxies rise slowly. This is not consistent with NFW halos, as shown long ago (e.g., de Blok et al. 2001; Kuzio de Naray 2008, 2009, and many others). It is frequently argued that everything is OK with NFW, it is the data that are to blame. The basic idea is that somehow (usually for inadequate resolution, though sometimes other effects are invoked) rotation curves fail to show the steep predicted rise.  It is there! we are assured. We just can’t see it.

Beware of theorists blaming data that doesn’t do what they want. A recent example of this kind of argument is offered by Pineda et al. (2016). Their basic contention is that rotation curves cannot correctly recover the true mass distribution. They invoke a variety of effects to make it sound like one has no chance of recovering the central mass profile from rotation curves.

The figure above shows that this is manifestly nonsense. If the data were incapable of measuring the dynamical surface density, the figure would be a mess. In one galaxy we’d see one wrong thing, in another we’d see a different wrong thing. This would not correlate with surface brightness, or anything else: the y-axis would just be so much garbage. Instead, we see a strong correlation. Indeed, we understand the errors well enough to calculate that most of the scatter is observational. Yes, there are uncertainties. They do add scatter to the plot. A little. That means the true relation is even tighter.

There is a considerable literature in the same vein as Pineda et al. (2016). These concerns can be completely dismissed. Not only are they incorrect, they stem from a form of solution aversion: they don’t like the answer, so deny that it can be true. This attitude has no place in science.

SPARC

SPARC

We have a new paper that introduces SPARC: Spitzer Photometry & Accurate Rotation Curves. SPARC is a database of 175 galaxies with measured HI rotation curves and homogeneous near-infrared [3.6 micron] surface photometry obtained with the Spitzer Space Telescope. It provides the largest cohesive dataset currently available of disk galaxy mass models.

SPARC represents all known types of rotating galaxies. It spans a broad range in morphologies (S0 to Irr), luminosities (L[3.6] ~ 107 to ~1012 L, effective radii (~0.3 to ~15 kpc), effective surface brightnesses (~5 to ~5000 L pc-2), rotation velocities (~20 to ~300 km/s), and gas content (0.01 < M(HI)/L[3.6] < 10). This samples the full range of physical properties known for rotating disk galaxies. It is vastly superior to most “complete” samples in that it provides a much better representation of low mass and low surface brightness galaxies.

Let me emphasize that last point. Traditional galaxy surveys are great at finding bright objects. They are lousy at finding low luminosity and low surface brightness galaxies. For example, most studies based on the gold-standard Sloan Digital Sky Survey are restricted to massive galaxies with M* > 109 M☉. SPARC extends two decades lower in mass. Sloan misses low surface brightness galaxies entirely. SPARC includes many such objects. Ideally, a sample like this would provide a thorough sampling of all possible disk galaxy properties. We come as close to that ideal as is currently possible, without the usual bias against the faint and the dim.

The rotation curves of SPARC galaxies have been collected from the literature. While we have obtained some of these ourselves, the vast majority come from the hard work of many others. All SPARC galaxies have been observed in the 21cm line of atomic hydrogen with radio interferometers like the VLA or WSRT. These data represent the fruits of the labors of a whole community of radio astronomers spanning decades.

The surface photometry we have done ourselves. This represents the cumulative results of a decade of work. The near-IR images from Spitzer have been analyzed with the ARCHANGEL software to determine the surface brightness profiles of all sample galaxies. These have been used to construct mass models representing the gravitational potential generated by the observed distribution of stellar mass. The 21cm data provide the same information for the gas.

ngc6946picture

Optical (BVI), near-IR (JHK), and 21 cm images of the spiral galaxy NGC 6946. The images are shown on the same scale. So yes, the gas extends that much further out. This is typical, and emphasizes the importance of combining multiwavelength observations.

We now have three measured properties for all SPARC galaxies that are hard to find simultaneously in the literature. These are the rotation curve V(R), the portion of the rotation due to stars V*(R), and that due to gas Vg(R). These are what you need to study the missing mass problem in galaxies, as

V2(R) = V*2(R)+Vg2(R)+VDM2(R)

The mysterious “other” represented by VDM(R) is dark matter (whatever that means). It is now completely specified by the observations.

Of course, this has been true for a while, but with one important exception. Mass models for V*(R) have been constructed with the available data, which are usually in the optical. When we construct a mass model, we have to convert the observed light to a stellar mass by assuming some mass-to-light ratio for the stars, M*/L. Optical M*/L vary with age and metallicity in a way that precludes clarity in the correct stellar mass model. Near-IR data (the 2.2 micron K-band or [3.6] of Spitzer) are much, much, much better for this.

I don’t think I emphasized that enough. The near-IR image of a galaxy is as close as we’re likely to ever get to a map of the stellar mass. It isn’t perfect of course – nothing in astronomy ever is – but it is a sufficient improvement that all the freedom and uncertainty that we had in VDM(R) before basically goes away.

We’ll have a lot more to say about that. Look for big announcements, coming soon.

What is theory?

What is theory?

OK, I’m not even going to try to answer that one. But I am going to do some comparison exploration.

A complaint often leveled against MOND is that it is not a theory. Or not a complete theory. Or somehow not a proper one. Sometimes people confuse MOND with the empirical observations that display MONDian phenomenology.

I would say that MOND is a hypothesis, as is dark matter. We observe a discrepancy between the motions observed in extragalactic systems and what is predicted by application of the known law of gravity to the mass visible in ordinary baryonic matter. Either we need more mass (dark matter) or need to change the force law (modify dynamical laws, i.e., gravity). MOND is just one example of the latter type of hypothesis.

Put this way, dark matter is the more conservative hypothesis. It doesn’t require any change to well established, fundamental theory. There’s just more mass there than we see.

But what is it? Dark matter as so far stated is not a valid scientific hypothesis. It is a concept – there is unseen stuff out there. To turn it into science, we need to hypothesize a specific candidate.

An example of a dark matter candidate that most people would agree has been falsified at this point is brown dwarfs. These are very faint, sub-stellar objects – failed stars if you like, things not quite massive enough to ignite nuclear fusion in their cores to shine as stars. In the early days of dark matter, it was quite reasonable to believe there could be an enormous amount of mass in the sum of these objects. Indeed, the mass spectrum of stars as then known (via the Salpeter IMF) diverged when extrapolated to the low masses of brown dwarfs. It appeared that there had to be lots of them, and their integrated mass could easily add up to lots and lots – potentially enough to be the dark matter.

The hypothesis of brown dwarf-like dark matter, dubbed MACHOs (MAssive Compact Halo Objects), was tested by a series of microlensing experiments. Remarkably, if you stare at the stars in the Large Magellanic Cloud long enough, you should occasionally witness a MACHO pass in front of one of them. You don’t see the MACHO directly, but you can see an enhancement to the brightness of the background star due to the gravitational lensing effect of the MACHO.

Long story short: microlensing events are observed, but not nearly enough are seen for the dark matter halo of the Milky Way to be composed of brown dwarf MACHOs. Nowadays we have a better handle on the stellar mass spectrum. Lots of brown dwarfs are indeed known, but nothing like the numbers necessary to compose the dark matter.

Many of us, including me, never gave MACHOs much of a chance. In order to add up to the total mass density we need in dark matter cosmologically, we need an amount 5 or 6 times as great as the density allowed in baryons by Big Bang Nucleosynthesis. So MACHO dark matter would break some pretty fundamental theory after all.

The most popular hypothesis, then and now, is some form of non-baryonic dark matter. Most prominent among these are WIMPs (Weakly Interacting Massive Particles). This is a valid, specific hypothesis that can be tested in the laboratory. Indeed, it has been. If the WIMP hypothesis were correct, we really should have detected them by now. It only persists because it is very flexible: we can keep adjusting the interaction cross-section to keep them invisible.

It would be a long post to revisit all the ways in which the WIMP hypothesis has repeatedly disappointed. Here I’d like to point out merely that WIMPs are hypothetical particles that exist in a hypothetical supersymmetric sector. There are compelling theoretical arguments in favor of supersymmetry, but so far it too has repeatedly disappointed. Anybody else remember how the decay of the Bs meson was suppose to be the Golden Test of supersymmetry? No? Nobody seems to talk about it anymore because it flunked badly. So supersymmetry itself is in dire shape. No supersymmetry, no WIMPs.

Like WIMPs, supersymmetry can be made more complicated to avoid falsification. This allows it to persist, but it is not the sign of a healthy theory. Still, everybody seems to agree that it is a theory, and most people seem to think it is a good one.

Unlike MACHOs, WIMPs do require a fundamentally new theory. Supersymmetry is not a part of the highly successful Standard Model of particle physics. It is a hypothetical extension thereof. So they aren’t really as conservative as just saying there is some unseen mass. There have to be invisible particles that reside in an entirely novel and itself hypothetical dark sector. That they have never been detected in the laboratory, and so far we have zero laboratory evidence to support the existence of the supersymmetric sector in which they reside, despite enormous (and expensive) effort (e.g., the LHC), might strike some as cause for concern.

So why do WIMPs persist? Time lag and training. If you are an astronomer, you don’t really care what the dark matter particle is, just that it is there. You are unlikely to keep close tabs on the tribulations of dark matter detection experiments. If you are an astroparticle physicist, dark matter particles are your bread and butter. We all know the Standard Model is incomplete; surely the dark matter problem is just a sign of that. Suggesting that the problem might instead be with gravity is to admit that the entire field is an oxymoron. Yes, we need new physics. But that would be the wrong kind of new physics!

winnie-the-pooh-balloon-bees

The MOND hypothesis is an example of the wrong kind of new physics. No new particles; rather, new dynamics. The idea is to tweak the force law below a critical acceleration scale (of order 1 Å/s/s). Intriguingly, this can be interpreted as either a modification of gravity (which gets stronger) or of inertia (which gets less, so particles become easier to push around).

From such a hypothesis, one must construct a proper theory – whatever that is. One thing is for sure – the motivation is the opposite of supersymmetry. Supersymmetry is motivated by theory. It is a Good Idea that therefore ought to be true, even if it appears that Nature declined to implement it. MOND has no compelling theoretical motivation or basis. (Who ordered that?) Rather, it is empirically motivated. It started by seeking a possible explanation for a particular observation: the apparent flatness of spiral galaxy rotation curves. In this regard, it could be considered an effective theory, though it does have strong implications for what the underlying cause is.

The original (1983) MOND formula did not conserve energy or momentum. That’s not a property of a healthy theory. Some people seem to think it is still stuck there.

The first step towards building a proper theory was taken by Bekenstein and Milgrom in 1984 with AQUAL. They introduced an aquadratic Lagrangian that led to a modified Poisson equation, a form of modified gravity. Being derived form a Lagrangian, it automatically satisfies the conservation laws.

Since then, a variety of MOND theories have been posited. By this, I mean distinct theories that lead to the hypothesized behavior at low acceleration. These may be modifications of either gravity or inertia, and can lead to subtly different higher order predictions.

So far most MOND theories are extensions of Newtonian dynamics. MOND always contains Newton in the high acceleration limit, just as General Relativity contains Newton in the appropriate limit. The trick is to write a theory that does both. That’s the theoretical Holy Grail.

The following Venn diagram might help:

GravityTheoryVennDiagram

Both MOND and General Relativity encompass Newtonian dynamics. However, they do not contain each other. Since General Relativity came first, I think when people say MOND is not a theory they usually mean that it doesn’t capture all the previous theory that it needs to. We know General Relativity is correct – so far as we have tested it – so it doesn’t suffice to write down a theory that is merely an extension of Newton. We need a theory that does both – the Holy Grail.

Of course I agree that we want to have it all. I also think it is appropriate to take one step at a time. If Newtonian dynamics is in itself a valid theory – and I think it is – the so too is MOND, as it contains all of Newton in the appropriate limit. MOND is an incomplete theory, but it is certainly a theory.

For many years, an argument against MOND was that Bekenstein had sought the Holy Grail long and hard without success. Bekenstein was really smart, implying that if he couldn’t do it, it couldn’t be done.  In 2004, Bekenstein published TeVeS (for Tensor-Vector-Scalar), the first example of a theory that contained both General Relativity and MOND without obviously having some dreadful failing, like ghosts. The argument then became that TeVeS was inelegant.

It is not clear that TeVeS is the correct generalized version of General Relativity. Indeed, it is not the only such theory possible. Hence the question mark in the Venn diagram. If we falsify TeVeS, it doesn’t falsify the MOND hypothesis – just that particular realization thereof.

What theory the question mark in the Venn diagram represents is what we should be trying to figure out. Unfortunately, most scientists interested in the subject are not trained nor equipped to do this sort of work, and for the most part are conditioned to be actively hostile to the project. That’s the wrong kind of new physics!

I find this a strange attitude. We all know that, as yet, there is no widely accepted theory of quantum theory. In this regard, General Relativity is itself incomplete. It is a noble endeavor to seek a quantum theory of gravity. How can we be sure that there is no intermediate step? Perhaps some of the difficulty in getting there stems from playing with an incomplete deck. I sometimes wonder if some string theorist has already come up with the correct theory but discarded it because it predicted this crazy low acceleration behavior he didn’t know might actually be desirable.

Whatever the final theory may be, be it dark matter based or a modification of dynamics, it must explain the empirical phenomena we observe. An enormous amount of galaxy phenomenology can be put down to one simple fact: galaxies behave as if MOND is the effective force law. We can write down a single formula that describes the dynamics of hundreds of measured galaxies and has had tremendous predictive success. If you don’t find that compelling, your physical intuition needs a check up.

What is empirical?

What is empirical?

I find that my scientific colleagues have a variety of attitudes about what counts as a theory. Some of the differences amount to different standards. Others are simply misconceptions about specific theories. This comes up a lot in discussions of MOND. Before we go there, we need to establish some essentials.

What is empirical?

I consider myself to be a very empirically-minded scientist. To me, what is empirical is what the data show.

Hmm. What are data? The results of experiments or observations of the natural world. To give a relevant example, here are some long slit observations of a low surface brightness galaxies (from McGaugh, Rubin, & de Blok 2001).

fig1

What you see are spectra obtained with the Kitt Peak 4m telescope. Spectra run from blue to red from left to right while the vertical axis is position along the slit. Vertical bars are emission lines in the night sky. The horizontal grey stuff is the continuum emission from stars in each galaxy (typically very weak in low surface brightness galaxies). You also see blobby dark bars running more or less up and down, but with an S-shaped bend. These are emission lines of hydrogen (the brightest one), nitrogen (on either side of hydrogen) and sulfur [towards the right edge of (a) and (d)].

I chose this example because you can see the rotation curves of these galaxies by eye directly in the raw data. Night sky lines provide an effective wavelength calibration; you can see one side of each galaxy is redshifted by a greater amount than the other: one side is approaching, the other receding relative to the Hubble flow velocity of the center of each galaxy. With little effort, you can also see the flat part of each rotation curve (Vf) and the entire shape V(R)*sin(i) [these are raw data, not corrected for inclination. You can even see a common hazard to real world data in the cosmic ray that struck near then end of the Hα line in (f)].

Data like these lead to rotation curves like these (from the same paper):

fig3

These rotation curves were measured by Vera Rubin. Vera loved to measure. She liked nothing better than to delve into the data to see what they said. She was very good at it.

Some of these data are good. Some are not. Some galaxies are symmetric (filled and open symbols represent approaching and receding sides), others are not. This is what the real world of galaxy data looks like. With practice, one develops a good intuition for what data are trustworthy and which are not.

To get from the data to these rotation curves, we correct for known effects: the expansion of the universe (which stretches the redshifts by 1+z), the inclination of each galaxy (estimated in this case by the axis ratios of the images), and most importantly, assuming the Doppler effect holds. That is, we make use of the well known relation between wavelength and speed to turn the measured wavelengths of the Hα and other emission lines into velocities. We use the distance to each galaxy (estimated from the Hubble Law) to convert measured position along the slit into physical radius.

This is empirical. Empirical results are as close to the data as possible. Here we make use of two other empirical results, the Doppler effect and the Hubble Law. So there is more to it than just the raw (or even processed) data; we are also making use of previously established facts.

An example of a new empirical fact obtained from data like these is the Baryonic Tully-Fisher relation. This is a plot of the observed baryonic mass in a galaxy (the sum of stars and gas) against the amplitude of flat rotation (Vf).

BTF_LR_TEDxCLE

Here one must utilize some other information to estimate the mass-to-light ratio of the stars. This is an additional step; how we go about it affects the details of the result but not the basic empirical fact that the relation exists.

It is important distinguish between the empirical relation as plotted above, and a function that might be fit to it. The above data are well fit by the formula

Mb = 50 Vf4

with mass measured in solar masses and rotation speed in kilometers per second. The fit is merely a convenient representation of the data. The data themselves are the empirical result.

In this case, the scatter in the data is consistent with what you’d expect for the size of the error bars. The observed relation is consistent with one that has zero intrinsic scatter – a true line. The reason for that might be that it is imposed by some underlying theory (e.g., MOND). Whether MOND is the reason for the Baryonic Tully-Fisher relation is something that can be debated. That the relation exists as an empirical result that must be explained by any theory that attempts to explain the mass discrepancy problem in galaxies cannot.

I would hope it is obvious that theory should explain data. In the context of the mass discrepancy problem, the Baryonic Tully-Fisher relation is one fact that needs to be explained. There are many others. Which interpretation we are driven towards depends a great deal on how we weigh the facts. How important is this particular fact in the context of all others?

I find that many scientists confuse the empirical Baryonic Tully-Fisher relation with the theory MOND. Yes, MOND predicts a Baryonic Tully-Fisher relation, but they are not the same thing. The one we observe is consistent with MOND. It need not be (and is not for some implausible but justifiable assumptions about the stellar mass-to-light ratio). The mistake I see many people make – often reputable, intelligent scientists – is to conflate data with theory. A common line of reasoning seems to be “These data support MOND. But we know MOND is wrong for other reasons. Therefore these data are wrong.” This is a logical fallacy.

More generally, it is OK to incorporate well established results (like the Doppler effect) into new empirical results so long as we are careful to keep track of the entire line of reasoning and are willing to re-examine all the assumptions. In the example of the Baryonic Tully-Fisher relation, the critical assumption is the mass-to-light ratio of the stars. That has minor effects: mostly it just tweaks the slope of the line you fit to the data.

If instead we have reason to doubt the applicability of something deeper, like the applicability of the Doppler formula to galaxies, that would open a giant can of worms: a lot more than the Tully-Fisher relation would be wrong. For this reason, scientists are usually very impatient with challenges to well established results (who hasn’t received email asserting “Einstein was wrong!”?) To many, MOND seems to be in this category.

Consequently, many scientists are quick to dismiss MOND without serious thought. I did, initially. But eventually it had enough predictions come true that I felt compelled to check. (Bekenstein pointed out a long time ago that MOND has had many more predictions come true than General Relativity had had at the time of its widespread acceptance.) When I checked, I found that the incorrect assumption I had made was that MOND could so lightly be dismissed. In my experience since then, most of the people arguing against MOND haven’t bothered to check their facts (surely it can’t be true!), or have chosen to selectively weigh most those that agree with their preconception of what the result should be. If the first thing someone mentions in this context is the Bullet cluster, they are probably guilty of both these things, as this has become the go-to excuse not to have to think too hard about the subject. Cognitive dissonance is rife.