Author: tritonstation

Stacy McGaugh is an astrophysicist and cosmologist who studies galaxies, dark matter, and theories of modified gravity. He is an expert on low surface brightness galaxies, a class of objects in which the stars are spread thin compared to bright galaxies like our own Milky Way. He demonstrated that these dim galaxies appear to be dark matter dominated, providing unique tests of theories of galaxy formation and modified gravity. Professor McGaugh is currently the chair of the Department of Astronomy at Case Western Reserve University in Cleveland, Ohio, and director of the Warner and Swasey Observatory. Previously he was a member of the faculty at the University of Maryland, having also held research fellowships at Rutgers, the Department of Terrestrial Magnetism of the Carnegie Institution of Washington, and the Institute of Astronomy at the University of Cambridge after earning his Ph.D. from the University of Michigan.

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.

What is the Baryon Density, Anyway?

To continue… we had been discussing the baryon content of the universe, and the missing baryon problem. The problem exists because of a mismatch between the census of baryons locally and the density of baryons estimated from Big Bang Nucleosynthesis (BBN). How well do we know the latter? Either extremely well, or perhaps not so well, depending on which data we query.

At the outset let me say I do not doubt the basic BBN picture. BBN is clearly one of the great successes of early universe cosmology: it is pretty clear this is how the universe works. However, the absolute value we obtain for Ωb depends on the mutual agreement of independent measurements of the abundances of different isotopes. These agree well enough to establish the BBN paradigm, but not so well as to discount all debate about the exact value of Ωb – contrary to the impression one might get from certain segments of the literature.

BBN is thoroughly discussed elsewhere so I won’t belabor it here. In a nutshell, the primordial abundance of the isotopes of the light elements – especially deuterium, helium, and lithium relative to hydrogen – depends on the baryon density. Each isotope provides an independent constraint. This is perhaps the most (only?) over-constrained problem in cosmology.

It is instructive to look at the estimates of the baryon density over time. These are usually quoted as the baryon density multiplied by the square of the Hubble parameter normalized to 100 km/s/Mpc (Ωbh2). This is a hangover from the bad old days when we didn’t know H0 to a factor of 2.

bbnblog

The graph shows the baryon densities estimated by various methods by different people over the years. It starts with the compilation of Walker et al. (1991). By this time, BBN was already a mature subject, with an authoritative answer based on evidence from all the isotopes. Ωbh2 = 0.0125 ± 0.0025. It was Known, Khaleesi.

In the mid-90s there was a debate about the primordial deuterium abundance, largely between Hogan and Tytler. Deuterium (red tirangles) is a great isotope to measure for BBN because it is very sensitive to the baryon density, as it tends to get gobbled up into heavier isotopes like helium when there are lots of baryons around to react with. Moreover, one could measure it in the absorption along the line of sight to high redshift QSOs, presumably catching it before any nasty interstellar processing has polluted the primordial abundance. Unfortunately, Hogan found a high D/H (and hence a low baryon density) while Tytler found low D/H (hence high Ωbh2). This is a rare case when one side (Hogan) actually admitted error, and the standard density shot up to 0.019. At the time (1998) that seemed outrageously high, 2.6σ above the previous standard value. But we had bigger problems to wrap our head around (Λ) at the time, so this was accepted without much fuss.

The other elements (helium and lithium) preferred something in between at that time. Their uncertainties were large enough this didn’t seem a big deal. Helium in particular is notoriously hard to pin down. Not only is the measurement hard to make, but helium (unlike deuterium) is not particularly sensitive to the baryon density. You get about a quarter helium by mass out of BBN for any reasonable baryon density, so it is a great indicator that the basic picture is correct. But you really have to nail down the third decimal point to help distinguish between slightly lower or higher Ωb. So the new normal became Ωbh2 = 0.019 ± 0.001.

That was the summation of decades of work, but it wasn’t to last long. In 2000, cosmic microwave background (CMB) experiments like BOOMERanG and MAXIMA began to resolve the acoustic power spectrum. A funny thing emerged: the second peak was lower than expected. (At least by other people. I totally nailed this prediction.) In order to explain the low second peak conventionally (in the context of ΛCDM), one had to crank up the baryon density. This first point from the CMB (blue in plot above) was well above previously suspected levels.

Note the dotted lines in the figure. These denote the maximum baryon density (horizontal line) before the first relevant CMB data (vertical line). No isotope of any light element had ever suggested Ωbh2 > 0.02 prior to CMB constraints. Once those became available, this changed.

The change happened first to deuterium, which has not suggested Ωbh2 < 0.02 since the CMB said so. Helium was slower to respond, but it has also drifted slowly upward. Lithium has remained put. This is a serious problem that has not been satisfactorily resolved. The general presumption seems to be that this is a detail to be blamed on stellar rotation or some similarly obscure mechanism.

Different communities work on each of these elements. Deuterium is the subject of high redshift astronomy, a field closely coupled to cosmology. Helium is the subject of nearby galaxies, a field aware of cosmology but less strongly tied to it. Lithium is measured in stars, a field that is not coupled to cosmology. Given the long history of confirmation bias in cosmology, it is hard not to be suspicious of the temporal variation in BBN baryon density estimates. The isotope most closely associated with cosmology, deuterium, quickly fell in line with the “right” result from the CMB. Helium has more gradually followed suit, while lithium continues to prefer lower baryon densities.

I do not doubt the sincerity of any particular measurement. But people talk. They have arguments about what is right and why. The communities that are closest are most likely to influence each other. Those further apart are less likely to be swayed. If we were suffering from confirmation bias, this is what it would look like.

The ΛCDM picture requires us to believe the CMB value, currently  Ωbh2 = 0.02230 ± 0.00023 (Planck 2015). You simply cannot fit the acoustic power spectrum with a number much different. Modern deuterium measurements are consistent with that, within the errors, so that has to be right, no? Lets just ignore lithium.

If instead we ignore the CMB and its associated baggage, this is not at all obvious. Perhaps the pre-CMB deuterium measurement is the one to trust. That is a bit higher than lithium, but consistent within the errors. Helium can go either way. So from a pure BBN (no CMB) perspective, maybe it is lithium and the other isotopes that are right and it is CMB fits that are misleading.

Where does this leave us with the missing baryons? The figure below shows the time evolution of the baryon density. The area is proportional to Ωbh2. This has grown over time, by an amount greater than the stated uncertainties (the circles show the change in area allowed). The baryon density has nearly doubled, being now ~4σ above the Known value of Walker.

baryoncontentblog

As the baryon density has grown, the missing baryon problem has grown worse. If we still had the classical Walker baryon density, there would be no missing baryon problem at al.  Indeed, Shull’s inventory is a bit too large, though it is consistent within the errors. If we go up to the pre-CMB deuterium value, then there is a missing baryon problem. It is big enough to solve the cluster problem in MOND, but without a lot left over. If we insist on the CMB-fitted baryon density, then the missing baryon problem is severe, at a level where it is hard to figure where else they could be.

IF ΛCDM is the right picture, then I think a high baryon density is unavoidable. Accepting this, there must then be something wrong with lithium. There is no lack of papers motivated by this line of reasoning, though the most common approach seems to be to ignore lithium entirely. I’ve heard a lot of talks bragging about the excellent agreement between BBN and the CMB, but this  really only applies to post-CMB deuterium.

IF BBN, as originally posed, is correct so that lithium and the other pre-CMB measurements are not misleading, then it becomes impossible to fit the CMB with pure General Relativity. This is the case even if we spot it non-baryonic cold dark matter and dark energy. This situation might be considered a motivation to seek extensions of the theory.

Regardless of where the right answer ultimately lies, there is real tension between primordial lithium measurements and the ΛCDM interpretation of the CMB. Something is fishy in the state of the early universe.

Missing baryons in LCDM and MOND

Missing baryons in LCDM and MOND

People often ask for a straight up comparison between ΛCDM and MOND. This is rarely possible because the two theories are largely incommensurable. When one is eloquent the other is mute, and vice-versa.

It is possible to attempt a comparison about how bad the missing baryon problem is in each. In CDM, we expect a relation between dynamical mass and rotation speed of the form Mvir ∝ Vvir3. In MOND the equivalent relation has a different power law, Mb ∝ Vf4.

In CDM we speak of virial quantities – the total mass of everything, including dark matter, and the circular speed way out at the virial radius (typically far outside the luminous extent of a galaxy). In MOND, we use the observed baryonic mass (stars and gas) and the flat rotation speed. These are not the same, so strictly speaking, still incommensurable. But they provide a way to compare the baryonic mass with the total inferred mass.

missingbaryonsinLCDMMOND

This plot shows the detected baryon fraction as a function of mass. The top panel is identical to last time. In ΛCDM we see most of the baryons in the most massive systems, but progressively less in smaller systems. In MOND the situation is reversed. The check-sum is complete in galaxies, but falls short in clusters of galaxies. (Note that the error bars have been divided by an extra power of velocity in the lower panel, which amplifies their appearance.) The reader may judge for himself which of these problems is more serious.

Critics of MOND frequently cite the bullet cluster as having falsified MOND. Period. No room for debate. See the linked press release from NASA: dark_matter_proven.

OK, what kind of dark matter? As discussed previously, we need at least two kinds of dark matter in ΛCDM: non-baryonic cold dark matter (some entirely novel particle) and dark baryons (normal matter not yet detected). Unfortunately, “dark matter” is a rather generic, catch-all term that allows these two distinct problems to be easily confused. We see the need for unseen mass in objects like the bullet cluster, and make the natural leap to conclude that we are seeing the non-baryonic cold dark matter that we expect in cosmology. There it is, case closed.

This is an example of a logical fallacy. There is nothing about the missing mass problem suffered by MOND in clusters that demands the unseen mass be non-baryonic. Indeed, even in ΛCDM we suffer some missing baryon problem on top of the need for non-baryonic cold dark matter. In both theories, there is a missing baryon problem in clusters. In both cases, this missing baryon problem is more severe at small radii, suggestive of a connection with the also-persistent cooling flow problem. Basically, the X-ray emitting gas observed in the inner 200 kpc or so of clusters have time to cool, so it ought to be condensing into – what? Stars? MACHOs? Something normal but as yet unseen.

It is not obvious that cooling flows can solve MOND’s problem in clusters. The problem is both serious and persistent. It was first pointed out in 1988 by The & White, and is discussed in this 2002 Annual Review. A factor or two (or even a bit more) of the expected baryons in clusters are missing (the red portion of the plot above). Note, however, that this problem was known long before the bullet cluster was discovered. From this perspective, it would have been very strange had the bullet cluster not shown the same discrepancy as every other cluster in the sky.

I do not know if the missing mass in clusters is baryonic. I am at a loss to suggest a plausible form that the missing baryons might be lurking in. Certainly others have tried. But lets take a step back and as if it is plausible.

As seen above, we have a missing baryon problem in both theories. It just manifests in different places. Advocates of ΛCDM do not, by and large, seem to consider the baryon discrepancy in galaxies to be a problem. The baryons were blown out, or are there but just not detected yet. No Problem. I’m not as lenient, but if we are to extend that grace to ΛCDM, why not also to MOND?

Recall that Shull et al. found that about 30% of baryons remain undetected in the local universe. In order to solve the problem MOND suffers in clusters, we need a mass in baryons about equal to the ICM wedge in this pie chart:

GlobalMissingBaryons

Note that the missing wedge is much larger than the ICM wedge. There are more than enough baryons out there to solve this problem. Indeed, it hardly makes a dent in the global missing baryon problem. Those baryons “must” be somewhere, so why not some in clusters of galaxies?

The short answer is cognitive dissonance. If one comes to the problem sure of the answer, then one sees in the data what one expects to see. MOND fits rotation curves? That’s just a fluke: it bounces off the wall of cognitive dissonance without serious consideration. MOND needs dark matter in clusters? Well of course – we knew that it had to be wrong in the first place.

I understand this perspective exceedingly well. It is where I started from myself. But the answer I wanted is not the conclusion that a more balanced evaluation of the evidence leads one to. The challenge is not in the evidence – it is to give an unorthodox idea a chance in the first place.

Missing Baryons

A long standing problem in cosmology is that we do not have a full accounting of all the baryons that we believe to exist. Big Bang Nucleosynthesis (BBN) teaches us that the mass density in normal matter is Ωb ≈ 5%. One can put a more precise number on it, but that’s close enough for our purposes here.

Ordinary matter fails to account for the closure density by over an order of magnitude. To make matters worse, if we attempt an accounting of where these baryons are, we again fall short. As well as the dynamical missing mass problem, we also have a missing baryon problem.

For a long time, this was also an order of magnitude problem. The stars and gas we could most readily see added up to < 1%, well short of even 5%. More recent work has shown that many, but not all, of the missing baryons are in the intergalactic medium (IGM).  The IGM is incredibly diffuse – a better vacuum than we can make in the laboratory by many orders of magnitude – but it is also very, very, very, well, very big. So all that nothing does add up to a bit of something.

GlobalMissingBaryons

A thorough accounting has been made by Shull et al. (2012). A little over half of detected baryons reside in the IGM, in either the Lyman alpha forest (Ly a in the pie chart above) or in the so-called warm-hot intergalactic medium (WHIM). There are also issues of double-counting, which Shull has taken care to avoid.

Gravitationally bound objects like galaxies and clusters of galaxies contain a minority of the baryons. Stars and cold (HI) gas in galaxies are small wedges of the pie, hence the large problem we initially had. Gas in the vicinity of galaxies (CGM) and in the intracluster medium of clusters of galaxies (ICM) also matter. Indeed, in the most massive clusters, the ICM outweighs all the stars in the galaxies there. This situation reverses as we look at lower mass groups. Rich clusters dominated by the ICM are rare; objects like our own Local Group are more typical. There’s no lack of circum-galactic gas (CGM), but it does not obviously outweigh the stars around L* galaxies.

There are of course uncertainties, so one can bicker and argue about the relative size of each slice of the pie. Even so, it remains hard to make their sum add up to 5% of the closure density. It appears that ~30% of the baryons that we believe to exist from BBN are still unaccounted for in the local universe.

The pie diagram only illustrates the integrated totals. For a long time I have been concerned about the baryon budget in individual objects. In essence, each dark matter halo should start with a cosmically equal share of baryons and dark matter. Yet in most objects, the ratio of baryons to total mass falls well short of the cosmic baryon fraction.

The value of the cosmic baryon fraction is well constrained by a variety of data, especially the cosmic microwave background. The number we persistently get is

fb = Ωb/Ωm = 0.17

or maybe 0.16, depending on which CMB analysis you consult.  But it isn’t 0.14 nor 0.10 nor 0.01. For sticklers, note that this the fraction of the total gravitating mass in baryons, not the ratio of baryons to dark matter: Ωm includes both. For numerologists, note that within the small formal uncertainties, 1/fb = 2π.

This was known long before the CMB experiments provided constraints that mattered. Indeed, one of the key findings that led us to repudiate standard SCDM in favor of ΛCDM was the recognition that clusters of galaxies had too many baryons for their dynamical mass. We could measure the baryon fraction in clusters. If we believe that these are big enough chunks of the universe to be representative of the whole, and we also believe BBN, then we are forced to conclude that Ωm ≈ 0.3.

Why stop with clusters? One can do this accounting in every gravitationally bound object. The null hypothesis is that every object should be composed of the universal composition, roughly 1 part baryons for every 5 parts dark matter. This almost works in rich clusters of galaxies. It fails in small clusters and groups of galaxies, and gets worse as you examine progressively smaller systems. So: not only are we missing baryons in the cosmic sum, there are some missing in each individual object.

missingbaryonsinLCDM

The figure shows the ratio of detected baryons to those expected in individual systems. I show the data I compiled in McGaugh et al. (2010), omitting the tiniest dwarfs for which the baryon content becomes imperceptible on a linear scale. By detected baryons I mean all those seen to exist in the form of stars or gas in each system (Mb = M*+Mg), such that

fd = Mb/(fbMvir)

where Mvir is the total mass of each object. This `virial’ mass is a rather uncertain quantity, but in this plot it can only slide the data up and down a little bit. The take-away is that not a single, gravitationally bound object appears to contain its fair share of cosmic baryons. There is a missing baryon problem not just globally, but in each and every object.

This halo-by-halo missing baryon problem is least severe in the most massive systems, rich clusters. Indeed, the baryon fraction of clusters is a rising function of radius, so a case could be made that the observations simply don’t reach far enough out to encompass a fair total. This point has been debated at great length in the literature, and I have little to add to it, except to observe that rich clusters are perhaps like horseshoes – close enough.

Irrespective of whether we consider the most massive clusters to be close enough to the cosmic baryon fraction or not, no other system comes close to close enough. There is already a clear discrepancy among smaller clusters, and an apparent trend with mass. This trend continues smoothly and continuously over many decades in baryonic mass through groups, then individual L* galaxies, and on to the tiniest dwarfs.

A respectively massive galaxy like the Milky Way has many tens of billions of solar masses in form of stars, and another ten billion or so in the form of cold gas. Yet this huge mass represents only a 1/4 or so of the baryons that should reside in the halo of the Milky Way. As we look at progressively smaller galaxies, the detected baryon fraction decreases further. For a galaxy with a mere few hundred million stars, fd ≈ 6%. It drops below 1% for M* < 107 solar masses.

That’s a lot of missing baryons. In the case of the Milky Way, all those stars and cold gas are within a radius of 20 kpc. The dark matter halo extends out to at least 150 kpc. So there is plenty of space in which the missing baryons might lurk in some tenuous form. But they have to remain pretty well hidden. Joel Bregman has spent a fair amount of his career searching for such baryonic reservoirs. While there is certainly some material out there, it does not appear to add up to be enough.

It is still harder to see this working in smaller galaxies. The discrepancy that is a factor of a few in the Milky Way grows to an order of magnitude and more in dwarfs. A common hypothesis is that these baryons do indeed lurk there, probably in a tenuous, hot gas. If so, direct searches have yet to see them. Another common idea is that the baryons get expelled entirely from the small potential wells of dwarf galaxy dark matter halos, driven by winds powered by supernovae. It that were the case, I’d expect to see a break at a critical mass where the potential well was or wasn’t deep enough to prevent this. If there is any indication of this, it is at still lower mass than shown above, and begs the question as to where those baryons are now.

So we don’t have a single missing mass problem in cosmology. We have at least two. One is the need for non-baryonic dark matter. The other is the need for unseen normal matter: dark baryons. This latter problem has at least two flavors. One is that the global sum of baryons comes up short. The other is that each and every individual gravitationally bound object comes up short in the number of baryons it should have.

An obvious question is whether accounting for the missing baryons in individual objects helps with the global problem. The wedges in the pie chart represent what is seen, not what goes unseen. Or do they? The CGM is the hot gas around galaxies, the favored hiding place for the object-by-object missing baryon problem.

Never mind the potential for double counting. Lets amp up the stars wedge by the unseen baryons indicated in red in the figure above. Just take for granted, for the moment, that these baryons are there in some form, associated in the proper ratio. We can then reevaluate the integrated sum and… still come up well short.

Low mass galaxies appear to have lots of missing baryons. But they are low mass. Even when we boost their mass in this way, they still contribute little to the integral.

This is a serious problem. Is it hopeless? No. Is it easily solved? No. At a minimum, it means we have at least two flavors of dark matter: non-baryonic [cosmic] dark matter, and dark baryons.

Does this confuse things immensely? Oh my yes.

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 – then 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 – that would be like saying Newton is wrong because Yilmaz gravity isn’t the right version of general relativity. There are many such theories that are possible; TeVeS is just one 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.

 

Structure Formation Mythology

Do not be too proud of this technological terror you’ve constructed. The ability to simulate the formation of large scale structure is insignificant next to the power of the Force.

– Darth Vader, Lord of the Sith

The now standard cosmology, ΛCDM, has a well developed cosmogony that provides a satisfactory explanation of the formation of large scale structure in the universe. It provides a good fit to both the galaxy power spectrum at low redshift and that of the cosmic microwave background (CMB) at z=1080. This has led to a common misconception among cosmologists that this is only way it can be.

The problem is this: the early universe was essentially homogeneous, while the current universe is not. At the time of recombination, one patch of plasma had the same temperature and density as the next patch to 1 part in 100,000. Look around at the universe now, and you see something very different: galaxies strung along a vast web characterized chiefly by empty space and enormous voids. Trouble is, you can’t get here from there.

Gravity will form structure, making the over-dense patches grow ever denser, in a classic case of the rich getting richer. But gravity is extraordinarily weak. There simply is not enough time in the ~13 Gyr age of the universe for it to make the tiny density variation observed in the CMB into the rich amount of structure observed today.

We need something to goose the process. This is where non-baryonic cold dark matter (CDM) comes in. It outweighs the normal matter, and does not interact with the photons of the CMB. This latter part is critical, as the baryons are strongly coupled to the photons, which don’t let them clump up enough early on. The CDM can. So it starts to form structure early which the baryons subsequently trace. Since structure formed, CDM must exist.

This is a sound line of reasoning. It convinced many of us, including myself, that there had to be some form of non-baryonic mass made of some particle outside the standard model of particle physics. The other key fact was that the gravitating mass density was inferred to outweigh the amount of baryons indicated by Big Bang Nucleosynthesis (Ωm ≫ Ωb).

Does anyone spot the problem with this line of thinking?

It took me a long time to realize what it was. Both the structure formation argument and the apparent fact that Ωm ≫ Ωb implicitly assume that gravity is normal. All we need to know to approach either problem is what Newton and Einstein taught us. Once we make that assumption, we are absolutely locked into the line of reasoning that leads us to CDM.

I worry that CDM is a modern æther. Given our present understanding of physics, it has to exist. In the nineteenth century, so too did æther. Had to. Only problem was, it didn’t.

If, for a moment, we let go of our implicit assumption, then we may realize that what structure formation needs is an extra push (or pull, to make overdensities collapse faster). That extra push may come from CDM, or it may come from an increase in the strength of the effective force law. Rather than being absolute proof of the existence of CDM, the rapid formation of structure might also be another indication that we need to tweak for force law.

I have previously outlined how structure might form in a modified force law like MOND. Early efforts do not provide as good a fit to the power spectrum as ΛCDM. But they provide a much better approximation than did the predecessor of ΛCDM, SCDM.

Indeed, there have been some striking predictive successes. As we probe to ever higher redshift, we see time and again more structure than had been anticipated by ΛCDM. Galaxies form early in MOND, so this is quite natural. So too does the cosmic web, which I predict to be more developed in MOND at redshifts of 3 and even 5. By low redshift, MOND does a much better job of emptying out the voids than does ΛCDM. Ultimately, I expect we may get a test from 21 cm reverberation mapping in the dark ages, where I predict we may find evidence of strong baryonic oscillations. (These predictions were made, and published in refereed journals, in the previous millennium.)

I would not claim that MOND provides a satisfactory description of large scale structure. The subject requires a lot more work.  Structure formation in MOND is highly non-linear. It is a tougher problem than standard perturbation theory. Yet we have lavished tens of thousands of person-years of effort on ΛCDM, and virtually no effort on the harder problem in the case of MOND. Having failed to make an effort does not suffice as evidence.