A Significant Theoretical Advance

A Significant Theoretical Advance

The missing mass problem has been with us many decades now. Going on a century if you start counting from the work of Oort and Zwicky in the 1930s. Not quite a half a century if we date it from the 1970s when most of the relevant scientific community started to take it seriously. Either way, that’s a very long time for a major problem to go unsolved in physics. The quantum revolution that overturned our classical view of physics was lightning fast in comparison – see the discussion of Bohr’s theory in the foundation of quantum mechanics in David Merritt’s new book.

To this day, despite tremendous efforts, we have yet to obtain a confirmed laboratory detection of a viable dark matter particle – or even a hint of persuasive evidence for the physics beyond the Standard Model of Particle Physics (e.g., supersymmetry) that would be required to enable the existence of such particles. We cannot credibly claim (as many of my colleagues insist they can) to know that such invisible mass exists. All we really know is that there is a discrepancy between what we see and what we get: the universe and the galaxies within it cannot be explained by General Relativity and the known stable of Standard Model particles.

If we assume that General Relativity is both correct and sufficient to explain the universe, which seems like a very excellent assumption, then we are indeed obliged to invoke non-baryonic dark matter. The amount of astronomical evidence that points in this direction is overwhelming. That is how we got to where we are today: once we make the obvious, imminently well-motivated assumption, then we are forced along a path in which we become convinced of the reality of the dark matter, not merely as a hypothetical convenience to cosmological calculations, but as an essential part of physical reality.

I think that the assumption that General Relativity is correct is indeed an excellent one. It has repeatedly passed many experimental and observational tests too numerous to elaborate here. However, I have come to doubt the assumption that it suffices to explain the universe. The only data that test it on scales where the missing mass problem arises is the data from which we infer the existence of dark matter. Which we do by assuming that General Relativity holds. The opportunity for circular reasoning is apparent – and frequently indulged.

It should not come as a shock that General Relativity might not be completely sufficient as a theory in all circumstances. This is exactly the motivation for and the working presumption of quantum theories of gravity. That nothing to do with cosmology will be affected along the road to quantum gravity is just another assumption.

I expect that some of my colleagues will struggle to wrap their heads around what I just wrote. I sure did. It was the hardest thing I ever did in science to accept that I might be wrong to be so sure it had to be dark matter – because I was sure it was. As sure of it as any of the folks who remain sure of it now. So imagine my shock when we obtained data that made no sense in terms of dark matter, but had been predicted in advance by a completely different theory, MOND.

When comparing dark matter and MOND, one must weigh all evidence in the balance. Much of the evidence is gratuitously ambiguous, so the conclusion to which one comes depends on how one weighs the more definitive lines of evidence. Some of this points very clearly to MOND, while other evidence prefers non-baryonic dark matter. One of the most important lines of evidence in favor of dark matter is the acoustic power spectrum of the cosmic microwave background (CMB) – the pattern of minute temperature fluctuations in the relic radiation field imprinted on the sky a few hundred thousand years after the Big Bang.

The equations that govern the acoustic power spectrum require General Relativity, but thankfully the small amplitude of the temperature variations permits them to be solved in the limit of linear perturbation theory. So posed, they can be written as a damped and driven oscillator. The power spectrum favors features corresponding to standing waves at the epoch of recombination when the universe transitioned rather abruptly from an opaque plasma to a transparent neutral gas. The edge of a cloud provides an analog: light inside the cloud scatters off the water molecules and doesn’t get very far: the cloud is opaque. Any light that makes it to the edge of the cloud meets no further resistance, and is free to travel to our eyes – which is how we perceive the edge of the cloud. The CMB is the expansion-redshifted edge of the plasma cloud of the early universe.

An easy way to think about a damped and a driven oscillator is a kid being pushed on a swing. The parent pushing the child is a driver of the oscillation. Any resistance – like the child dragging his feet – damps the oscillation. Normal matter (baryons) damps the oscillations – it acts as a net drag force on the photon fluid whose oscillations we observe. If there is nothing going on but General Relativity plus normal baryons, we should see a purely damped pattern of oscillations in which each peak is smaller than the one before it, as seen in the solid line here:

CMB_Pl_CLonly
The CMB acoustic power spectrum predicted by General Relativity with no cold dark matter (line) and as observed by the Planck satellite (data points).

As one can see, the case of no Cold Dark Matter (CDM) does well to explain the amplitudes of the first two peaks. Indeed, it was the only hypothesis to successfully predict this aspect of the data in advance of its observation. The small amplitude of the second peak came as a great surprise from the perspective of LCDM. However, without CDM, there is only baryonic damping. Each peak should have a progressively lower amplitude. This is not observed. Instead, the third peak is almost the same amplitude as the second, and clearly higher than expected in the pure damping scenario of no-CDM.

CDM provides a net driving force in the oscillation equations. It acts like the parent pushing the kid. Even though the kid drags his feet, the parent keeps pushing, and the amplitude of the oscillation is maintained. For the third peak at any rate. The baryons are an intransigent child and keep dragging their feet; eventually they win and the power spectrum damps away on progressively finer angular scales (large 𝓁 in the plot).

As I wrote in this review, the excess amplitude of the third peak over the no-CDM prediction is the best evidence to my mind in favor of the existence of non-baryonic CDM. Indeed, this observation is routinely cited by many cosmologists to absolutely require dark matter. It is argued that the observed power spectrum is impossible without it. The corollary is that any problem the dark matter picture encounters is a mere puzzle. It cannot be an anomaly because the CMB tells us that CDM has to exist.

Impossible is a high standard. I hope the reader can see the flaw in this line of reasoning. It is the same as above. In order to compute the oscillation power spectrum, we have assumed General Relativity. While not replacing it, the persistent predictive successes of a theory like MOND implies the existence of a more general theory. We do not know that such a theory cannot explain the CMB until we develop said theory and work out its predictions.

That said, it is a tall order. One needs a theory that provides a significant driving term without a large amount of excess invisible mass. Something has to push the swing in a universe full of stuff that only drags its feet. That does seem nigh on impossible. Or so I thought until I heard a talk by Pedro Ferreira where he showed how the scalar field in TeVeS – the relativistic MONDian theory proposed by Bekenstein – might play the same role as CDM. However, he and his collaborators soon showed that the desired effect was indeed impossible, at least in TeVeS: one could not simultaneously fit the third peak and the data preceding the first. This was nevertheless an important theoretical development, as it showed how it was possible, at least in principle, to affect the peak ratios without massive amounts of non-baryonic CDM.

At this juncture, there are two options. One is to seek a theory that might work, and develop it to the point where it can be tested. This is a lot of hard work that is bound to lead one down many blind alleys without promise of ultimate success. The much easier option is to assume that it cannot be done. This is the option adopted by most cosmologists, who have spent the last 15 years arguing that the CMB power spectrum requires the existence of CDM. Some even seem to consider it to be a detection thereof, in which case we might wonder why we bother with all those expensive underground experiments to detect the stuff.

Rather fewer people have invested in the approach that requires hard work. There are a few brave souls who have tried it; these include Constantinos Skordis and Tom Złosnik. Very recently, the have shown a version of a relativistic MOND theory (which they call RelMOND) that does fit the CMB power spectrum. Here is the plot from their paper:

CMB_RelMOND_2020

Note that black line in their plot is the fit of the LCDM model to the Planck power spectrum data. Their theory does the same thing, so it necessarily fits the data as well. Indeed, a good fit appears to follow for a range of parameters. This is important, because it implies that little or no fine-tuning is needed: this is just what happens. That is arguably better than the case for LCDM, in which the fit is very fine-tuned. Indeed, that was a large point of making the measurement, as it requires a very specific set of parameters in order to work. It also leads to tensions with independent measurements of the Hubble constant, the baryon density, and the amplitude of the matter power spectrum at low redshift.

As with any good science result, this one raises a host of questions. It will take time to explore these. But this in itself is a momentous result. Irrespective if RelMOND is the right theory or, like TeVeS, just a step on a longer path, it shows that the impossible is in fact possible. The argument that I have heard repeated by cosmologists ad nauseam like a rosary prayer, that dark matter is the only conceivable way to explain the CMB power spectrum, is simply WRONG.

A Philosophical Approach to MOND

A Philosophical Approach to MOND is a new book by David Merritt. This is a major development in the both the science of cosmology and astrophysics, on the one hand, and the philosophy and history of science on the other. It should be required reading for anyone interested in any of these topics.

For many years, David Merritt was a professor of astrophysics who specialized in gravitational dynamics, leading a number of breakthroughs in the effects of supermassive black holes in galaxies on the orbits of stars around them. He has since transitioned to the philosophy of science. This may not sound like a great leap, but it is: these are different scholarly fields, each with their own traditions, culture, and required background education. Changing fields like this is a bit like switching boats mid-stream: even a strong swimmer may flounder in the attempt given the many boulders academic disciplines traditionally place in the stream of knowledge to mark their territory. Merritt has managed the feat with remarkable grace, devouring the background reading and coming up to speed in a different discipline to the point of a lucid fluency.

For the most part, practicing scientists have little interaction with philosophers and historians of science. Worse, we tend to have little patience for them. The baseline presumption of many physical scientists is that we know what we’re doing; there is nothing the philosophers can teach us. In the daily practice of what Kuhn called normal science, this is close to true. When instead we are faced with potential paradigm shifts, the philosophy of science is critical, and the absence of training in it on the part of many scientists becomes glaring.

In my experience, most scientists seem to have heard of Popper and Kuhn. If that. Physical scientists will almost always pay lip service to Popper’s ideal of falsifiablity, and that’s pretty much the extent of it. Living up to applying that ideal is another matter. If an idea that is near and dear to their hearts and careers is under threat, the knee-jerk response is more commonly “let’s not get carried away!”

There is more to the philosophy of science than that. The philosophers of science have invested lots of effort in considering both how science works in practice (e.g., Kuhn) and how it should work (Popper, Lakatos, …) The practice and the ideal of science are not always the same thing.

The debate about dark matter and MOND hinges on the philosophy of science in a profound way. I do not think it is possible to make real progress out of our current intellectual morass without a deep examination of what science is and what it should be.

Merritt takes us through the methodology of scientific research programs, spelling out what we’ve learned from past experience (the history of science) and from careful consideration of how science should work (its philosophical basis). For example, all scientists agree that it is important for a scientific theory to have predictive power. But we are disturbingly fuzzy on what that means. I frequently hear my colleagues say things like “my theory predicts that” in reference to some observation, when in fact no such prediction was made in advance. What they usually mean is that it fits well with the theory. This is sometimes true – they could have predicted the observation in advance if they had considered that particular case. But sometimes it is retroactive fitting more than prediction – consistency, perhaps, but it could have gone a number of other ways equally well. Worse, it is sometimes a post facto assertion that is simply false: not only was the prediction not made in advance, but the observation was genuinely surprising at the time it was made. Only in retrospect is it “correctly” “predicted.”

The philosophers have considered these situations. One thing I appreciate is Merritt’s review of the various takes philosophers have on what counts as a prediction. I wish I had known these things when I wrote the recent review in which I took a very restrictive definition to avoid the foible above. The philosophers provide better definitions, of which more than one can be usefully applicable. I’m not going to go through them here: you should read Merritt’s book, and those of the philosophers he cites.

From this philosophical basis, Merritt makes a systematic, dare I say, scientific, analysis of the basic tenets of MOND and MONDian theories, and how they fare with regard to their predictions and observational tests. Along the way, he also considers the same material in the light of the dark matter paradigm. Of comparable import to confirmed predictions are surprising observations: if a new theory predicts that the sun will rise in the morning, that isn’t either new or surprising. If instead a theory expects one thing but another is observed, that is surprising, and it counts against that theory even if it can be adjusted to accommodate the new fact. I have seen this happen over and over with dark matter: surprising observations (e.g., the absence of cusps in dark matter halos, the small numbers of dwarf galaxies, downsizing in which big galaxies appear to form earliest) are at first ignored, doubted, debated, then partially explained with some mental gymnastics until it is Known and of course, we knew it all along. Merritt explicitly points out examples of this creeping determinism, in which scientists come to believe they predicted something they merely rationalized post-facto (hence the preeminence of genuinely a priori predictions that can’t be fudged).

Merritt’s book is also replete with examples of scientists failing to take alternatives seriously. This is natural: we have invested an enormous amount of time developing physical science to the point we have now reached; there is an enormous amount of background material that cannot simply be ignored or discarded. All too often, we are confronted with crackpot ideas that do exactly this. This makes us reluctant to consider ideas that sound crazy on first blush, and most of us will rightly display considerable irritation when asked to do so. For reasons both valid and not, MOND skirts this bondary. I certainly didn’t take it seriously myself, nor really considered it at all, until its predictions came true in my own data. It was so far below my radar that at first I did not even recognize that this is what had happened. But I did know I was surprised; what I was seeing did not make sense in terms of dark matter. So, from this perspective, I can see why other scientists are quick to dismiss it. I did so myself, initially. I was wrong to do so, and so are they.

A common failure mode is to ignore MOND entirely: despite dozens of confirmed predictions, it simply remains off the radar for many scientists. They seem never to have given it a chance, so they simply don’t pay attention when it gets something right. This is pure ignorance, which is not a strong foundation from which to render a scientific judgement.

Another common reaction is to acknowledge then dismiss. Merritt provides many examples where eminent scientists do exactly this with a construction like: “MOND correctly predicted X but…” where X is a single item, as if this is the only thing that [they are aware that] it does. Put this way, it is easy to dismiss – a common refrain I hear is “MOND fits rotation curves but nothing else.” This is a long-debunked falsehood that is asserted and repeated until it achieves the status of common knowledge within the echo chamber of scientists who refuse to think outside the dark matter box.

This is where the philosophy of science is crucial to finding our way forward. Merritt’s book illuminates how this is done. If you are reading these words, you owe it to yourself to read his book.

The Hubble Constant from the Baryonic Tully-Fisher Relation

The Hubble Constant from the Baryonic Tully-Fisher Relation

The distance scale is fundamental to cosmology. How big is the universe? is pretty much the first question we ask when we look at the Big Picture.

The primary yardstick we use to describe the scale of the universe is Hubble’s constant: the H0 in

v = H0 D

that relates the recession velocity (redshift) of a galaxy to its distance. More generally, this is the current expansion rate of the universe. Pick up any book on cosmology and you will find a lengthy disquisition on the importance of this fundamental parameter that encapsulates the size, age, critical density, and potential fate of the cosmos. It is the first of the Big Two numbers in cosmology that expresses the still-amazing fact that the entire universe is expanding.

Quantifying the distance scale is hard. Throughout my career, I have avoided working on it. There are quite enough, er, personalities on the case already.

AliceMadPeople

No need for me to add to the madness.

Not that I couldn’t. The Tully-Fisher relation has long been used as a distance indicator. It played an important role in breaking the stranglehold that H0 = 50 km/s/Mpc had on the minds of cosmologists, including myself. Tully & Fisher (1977) found that it was approximately 80 km/s/Mpc. Their method continues to provide strong constraints to this day: Kourkchi et al. find H0 = 76.0 ± 1.1(stat) ± 2.3(sys) km s-1 Mpc-1. So I’ve been happy to stay out of it.

Until now.

d8onl2_u8aetogk

I am motivated in part by the calibration opportunity provided by gas rich galaxies, in part by the fact that tension in independent approaches to constrain the Hubble constant only seems to be getting worse, and in part by a recent conference experience. (Remember when we traveled?) Less than a year ago, I was at a cosmology conference in which I heard an all-too-typical talk that asserted that the Planck H0 = 67.4 ± 0.5 km/s/Mpc had to be correct and everybody who got something different was a stupid-head. I’ve seen this movie before. It is the same community (often the very same people) who once insisted that H0 had to be 50, dammit. They’re every bit as overconfident as before, suffering just as much from confirmation bias (LCDM! LCDM! LCDM!), and seem every bit as likely to be correct this time around.

So, is it true? We have the data, we’ve just refrained from using it in this particular way because other people were on the case. Let’s check.

The big hassle here is not measuring H0 so much as quantifying the uncertainties. That’s the part that’s really hard. So all credit goes to Jim Schombert, who rolled up his proverbial sleeves and did all the hard work. Federico Lelli and I mostly just played the mother-of-all-jerks referees (I’ve had plenty of role models) by asking about every annoying detail. To make a very long story short, none of the items under our control matter at a level we care about, each making < 1 km/s/Mpc difference to the final answer.

In principle, the Baryonic Tully-Fisher relation (BTFR) helps over the usual luminosity-based version by including the gas, which extends application of the relation to lower mass galaxies that can be quite gas rich. Ignoring this component results in a mess that can only be avoided by restricting attention to bright galaxies. But including it introduces an extra parameter. One has to adopt a stellar mass-to-light ratio to put the stars and the gas on the same footing. I always figured that would make things worse – and for a long time, it did. That is no longer the case. So long as we treat the calibration sample that defines the BTFR and the sample used to measure the Hubble constant self-consistently, plausible choices for the mass-to-light ratio return the same answer for H0. It’s all relative – the calibration changes with different choices, but the application to more distant galaxies changes in the same way. Same for the treatment of molecular gas and metallicity. It all comes out in the wash. Our relative distance scale is very precise. Putting an absolute number on it simply requires a lot of calibrating galaxies with accurate, independently measured distances.

Here is the absolute calibration of the BTFR that we obtain:

btf_cep_trgb
The Baryonic Tully-Fisher relation calibrated with 50 galaxies with direct distance determinations from either the Tip of the Red Giant Branch method (23) or Cepheids (27).

In constructing this calibrated BTFR, we have relied on distance measurements made or compiled by the Extragalactic Distance Database, which represents the cumulative efforts of Tully and many others to map out the local universe in great detail. We have also benefited from the work of Ponomareva et al, which provides new calibrator galaxies not already in our SPARC sample. Critically, they also measure the flat velocity from rotation curves, which is a huge improvement in accuracy over the more readily available linewidths commonly employed in Tully-Fisher work, but is expensive to obtain so remains the primary observational limitation on this procedure.

Still, we’re in pretty good shape. We now have 50 galaxies with well measured distances as well as the necessary ingredients to construct the BTFR: extended, resolved rotation curves, HI fluxes to measure the gas mass, and Spitzer near-IR data to estimate the stellar mass. This is a huge sample for which to have all of these data simultaneously. Measuring distances to individual galaxies remains challenging and time-consuming hard work that has been done by others. We are not about to second-guess their results, but we can note that they are sensible and remarkably consistent.

There are two primary methods by which the distances we use have been measured. One is Cepheids – the same type of variable stars that Hubble used to measure the distance to spiral nebulae to demonstrate their extragalactic nature. The other is the tip of the red giant branch (TRGB) method, which takes advantage of the brightest red giants having nearly the same luminosity. The sample is split nearly 50/50: there are 27 galaxies with a Cepheid distance measurement, and 23 with the TRGB. The two methods (different colored points in the figure) give the same calibration, within the errors, as do the two samples (circles vs. diamonds). There have been plenty of mistakes in the distance scale historically, so this consistency is important. There are many places where things could go wrong: differences between ourselves and Ponomareva, differences between Cepheids and the TRGB as distance indicators, mistakes in the application of either method to individual galaxies… so many opportunities to go wrong, and yet everything is consistent.

Having  followed the distance scale problem my entire career, I cannot express how deeply impressive it is that all these different measurements paint a consistent picture. This is a credit to a large community of astronomers who have worked diligently on this problem for what seems like aeons. There is a temptation to dismiss distance scale work as having been wrong in the past, so it can be again. Of course that is true, but it is also true that matters have improved considerably. Forty years ago, it was not surprising when a distance indicator turned out to be wrong, and distances changed by a factor of two. That stopped twenty years ago, thanks in large part to the Hubble Space Telescope, a key goal of which had been to nail down the distance scale. That mission seems largely to have been accomplished, with small differences persisting only at the level that one expects from experimental error. One cannot, for example, make a change to the Cepheid calibration without creating a tension with the TRGB data, or vice-versa: both have to change in concert by the same amount in the same direction. That is unlikely to the point of wishful thinking.

Having nailed down the absolute calibration of the BTFR for galaxies with well-measured distances, we can apply it to other galaxies for which we know the redshift but not the distance. There are nearly 100 suitable galaxies available in the SPARC database. Consistency between them and the calibrator galaxies requires

H0 = 75.1 +/- 2.3 (stat) +/- 1.5 (sys) km/s/Mpc.

This is consistent with the result for the standard luminosity-linewidth version of the Tully-Fisher relation reported by Kourkchi et al. Note also that our statistical (random/experimental) error is larger, but our systematic error is smaller. That’s because we have a much smaller number of galaxies. The method is, in principle, more precise (mostly because rotation curves are more accurate than linewidhts), so there is still a lot to be gained by collecting more data.

Our measurement is also consistent with many other “local” measurements of the distance scale,

hubbletension1but not with “global” measurements. See the nice discussion by Telescoper and the paper from which it comes. A Hubble constant in the 70s is the answer that we’ve consistently gotten for the past 20 years by a wide variety of distinct methods, including direct measurements that are not dependent on lower rungs of the distance ladder, like gravitational lensing and megamasers. These are repeatable experiments. In contrast, as I’ve pointed out before, it is the “global” CMB-fitted value of the Hubble parameter that has steadily diverged from the concordance region that originally established LCDM.

So, where does this leave us? In the past, it was easy to dismiss a tension of this sort as due to some systematic error, because that happened all the time – in the 20th century. That’s not so true anymore. It looks to me like the tension is real.

 

Tracing the baryons in star forming galaxies

Tracing the baryons in star forming galaxies

Galaxies are big. Our own Milky Way contains about fifty billion solar masses of stars, and another ten billion of interstellar gas, roughly speaking. The average star is maybe half a solar mass, so crudely speaking, that’s one hundred billion stars. Give or take. For comparison, the population of the Earth has not quite reached eight billion humans. So if you gave each one of us our own personal starship, in order to visit every star in the Galaxy, each one of us would have to visit a dozen stars. Give or take. I’m getting old, so I call dibs on Proxima Centauri through Procyon.

Figure 1 shows a picture of NGC 628, a relatively nearby spiral galaxy. What you see here is mostly stars, along with some interstellar dust and ionized gas. In addition to those components, there are also stellar remnants left behind by dead stars (mostly white dwarfs, some neutron stars, and the occasional black hole). In the space between the stars resides colder forms of interstellar gas, including both atomic gas (individual atoms in space) and molecular gas (the cold, dense material from which new stars form). How much is there of each component?

ngc628_final
Fig 1. The spiral galaxy NGC 628. The continuum light of stars in contrasted by dark dust lanes and highlighted by pink pinpoints of Balmer line emission. These are regions of interstellar gas illuminated by the UV emission of short-lived, massive O stars. Not visible here is the interstellar atomic and molecular gas from which stars form.

The bulk of the normal mass (excluding dark matter) in big spiral galaxies like the Milky Way is stars and their remnants. But there is also diffuse material in the vast interstellar medium – the ample space between the stars. This includes dust and several distinct phases of gas – molecular, atomic, and ionized (plasma). The dust and plasma are easy to see, but don’t add up to much – a mere millions of solar masses each for the whole Milky Way. The atomic and molecular gas add up to a lot more, but cannot be seen optically.

Atomic gas can be traced by 21 cm emission from the spin-flip transition of atomic hydrogen using radio telescopes. This is commonly referred to with the spectroscopic notation “HI”. The HI mass – the mass of atomic hydrogen – is usually the second largest mass component in spirals, after stars. In dwarf galaxies, the atomic gas often outweighs the stars (Fig. 2).

MgMst
Fig 2. Gas mass vs. stellar mass for galaxies in the SPARC database (blue) and an independent sample selected from SDSS (grey) by Bradford. The line is the line of equality where gas mass and stellar mass are equal. The red point is the Milky Way. Like other bright spirals, it is more stars than gas. Among lower mass dwarf galaxies, the reverse is commonly true: those in the field have more gas than stars.

Stars and atomic (HI) gas are the big two. When it comes to star forming galaxies, more massive spirals are usually star dominated while less massive dwarfs are usually dominated by atomic gas. But what about molecular gas?

Molecular gas is important to the star formation process. It is the densest (a very relative term!) material in the interstellar medium, the place where cold gas can condense into the nuggets that sometimes form stars. How much of this necessary ingredient is there?

The bulk of the mass of molecular gas is in molecular hydrogen, H2. Spectroscopically, H2 is a really boring molecule. It has no transitions in wavelength regimes that are readily accessible to observation. So, unlike atomic hydrogen, which brazenly announces its presence throughout the universe via the 21 cm line, molecular hydrogen is nigh-on invisible.

So we use proxies. The most commonly employed proxy for tracing molecular gas mass is carbon monoxide. CO is one of many molecules that accompany the much more abundance molecular hydrogen, and CO produces emission features that are more readily accessible observationally in the mm wavelength range. That has made it the tracer of choice.

CO is far from an ideal tracer of mass. Carbon and oxygen are both trace elements compared to hydrogen, so the correspondence between CO emission and molecular gas mass depends on the relative abundance of both. If that sounds dodgy, it gets worse. It also depends on the interstellar radiation field, the opacity thereto (molecular gas is inevitably associated with dense clumps of dust that shield it from the ambient radiation), and the spatial overlap of the two components – CO and H2 thrive in similar but not identical regions of space. Hence, converting the observed intensity of CO into a molecular hydrogen mass is a highly sensitive procedure that we typically bypass by assuming it is a universal constant.

It’s astronomy. We do what we can.

People have obsessed long and hard about the CO-to-H2 conversion, so we do have a reasonable idea what it is. While many debates can be had over the details, we have a decent idea of what the molecular gas mass is in some galaxies, at least to a first approximation. Molecular gas is usually outweighed by atomic gas, but sometimes it is comparable. So we’d like to keep track of it for the mass budget.

LCOMHIMst
Fig 3. The mass in molecular hydrogen gas as a function of atomic hydrogen (left) and stellar mass (right) from xGASS. The dotted line is the line of equality; molecular gas is usually outweighed by both atomic gas and stars. The red line at right is where the molecular gas mass is 7% of the stellar mass.

Obtaining CO observations is expensive, and often impossible: there are a lot of star forming galaxies where it simply isn’t detected. So we presume there is molecular gas there – that’s where the stars form, but we can’t always see it. So it would be handy to have another proxy besides CO.

Atomic gas is a lousy proxy for molecular gas. The mass of one hardly correlates with the other (Fig. 3). The two phases coexist in a complex and ever-changing variable quasi-equilibrium, with the amount of each at any given moment subject to change so that a snapshot of many galaxies provides a big mess.

Fortunately, the molecular gas mass correlates better with other properties – notably star formation. This makes sense, because stars form from molecular gas. So in some appropriately averaged sense, one follows the other. Star formation can be traced in a variety of ways, like the Balmer emission in Fig. 1. We can see the stars forming and infer the amount of molecular gas required to fuel that star formation even if we can’t detect the gas directly (Fig. 4).

MH2SFRMst
Fig 4. The current star formation rate (left) and molecular gas mass (right) as a function of stellar mass. The grey and black points are from xGASS, with the black points being those where the current star formation rate is within a factor of two of the past average (i.e., the stellar mass divided by the age of the universe). Blue points are low surface brightness galaxies. These extend the relation at left to much lower mass, but are generally not detected in CO. The molecular gas at right (open squares) is inferred by the amount required to sustain the observed star formation.

I’ve done a lot of work on low surface brightness galaxies, a class of objects that have proven particularly difficult to detect in CO. They have low dust contents, low oxygen abundances, relatively hard interstellar radiation fields – all factors that mitigate against CO. Yet we do see them forming stars, sometimes just one O star at a time, and we know how much molecular gas it takes to do that. So we can use star formation as a proxy for molecular gas mass. This is probably no worse than using CO, and perhaps even better – or would be, if we didn’t have to rely on CO to calibrate it in the first place.

Accurate tracers of star formation are also somewhat expensive to obtain. There are situations in which we need an estimate for the molecular gas mass where we don’t have either CO or a measurement of the star formation rate. So… we need a proxy for the proxy. Fortunately, that is provided by the stellar mass.

The stellar mass of a star-forming galaxy correlates with both its molecular gas mass and its star formation rate (Figs. 3 and 4). This is not surprising. It takes molecules to form stars, and it takes star formation to build up stellar mass. Indeed, the stellar mass is the time-integral of the star formation rate, so a correlation between the two (as seen in the left panel of Fig. 4) is mathematically guaranteed.

This brings us to the seven percent solution. Going through all the calibration steps, the molecular gas mass is, on average, about 7% of the stellar mass (the red lines in Figs. 3 and 4). The uncertainties in this are considerable. I’ve tried to work this out previously, and typically came up with numbers in the 5 – 10% range. So it seems to be in there somewhere.

This is adequate for some purposes, but not for others. One thing I want it for is to keep track of the total mass budget of baryons in galaxies so that we can calibrate the Baryonic Tully-Fisher relation. For this purpose it is adequate since molecular gas ranks behind both stars and atomic gas in the mass budget of almost every rotating galaxy. If it is 5% or 10% instead of 7%, this is a difference of a few percent of something that is itself typically < 10% of the total, and often less. A few percent of a few percent is a good working definition of negligible – especially in astronomy.

On top of all that, we also have to keep track of the stuff that isn’t hydrogen – helium and everything else in the periodic table, which astronomers often refer to collectively as “metals.” This makes for all sorts of partially-deserved jokes – oxygen isn’t a metal! but it is number 3 in cosmic abundance after hydrogen and helium. Like many anachronisms, the practice has good historical precedent. Early efforts to measure the abundances of the chemical elements in stars first gave results for iron. As other elements were probed, their abundances followed a pattern that scaled pretty well with the abundance of iron relative to hydrogen. So once again we have a proxy – this time, the iron abundance being a stand-in for that of everything else. Hence the persistence of the terminology – the metallicity of a star is a shorthand for the fraction of its mass that is not hydrogen and helium.

And that fraction is small. We usually write the mass fractions of hydrogen, helium, and everything else (metals) as

X + Y + Z = 1

where X is the fraction of mass in hydrogen, Y that in helium, and Z is everything else. For the sun, Lodders gives X = 0.7389, Y = 0.2463, and Z = 0.0148. Do I believe all those significant digits? No. Is there a good reason for them to be there? Yes. So without delving into those details, let’s just note that the universe is about 3 parts hydrogen, one part helium, with a sprinkling of everything else. Everything else being all the elements in the periodic table that aren’t hydrogen or helium – all the carbon and nitrogen and oxygen and silicon and magnesium and noble gases and actual metals – these all add up to about 1.5% of the mass of the sun, which is typical of nearby stars. So you can see why they’re all just metals to many astronomers.

For the mass of gas in galaxies, we need to correct what we measure in hydrogen for the presence of helium and metals. We measure the mass of atomic hydrogen using the 21 cm line, but that’s just the hydrogen. There is a corresponding amount of helium and metals that goes along with it. So we estimate the mass fraction in hydrogen, X, and use divide by that to get the total mass: Mgas = MHI/X. We do the same for molecular gas, etc.

There are measurements of the metallicities of entire galaxies, but – you guessed it – this isn’t observationally cheap, and isn’t always available. So we need another proxy. Luckily for us, it turns out that once again there is a pretty good correlation of metallicity with stellar mass: galaxies with lots of stars have made lots of supernovae that have processed lots of material into metals. Most of it is still hydrogen, so this is a very subtle effect: 1/X = 1.34 for the tiniest dwarf, going up to about 1.4 for a galaxy like the Milky Way. Still, we know this happens, so we can account for it, at least in a statistical way.

For those who are curious about the details, or want the actual formulae to use, please refer to this AAS research note. Next time, I hope to discuss an application for all this.

The Other

The Other

I am a white American male. As such, I realize that there is no way for me to grasp and viscerally appreciate all the ways in which racism afflicts black Americans. Or, for that matter, all the ways in which sexism afflicts women. But I can acknowledge that these things exist. I can recognize when it happens. I’ve seen it happen to others, both friends and strangers. I can try not to be part of the problem.

It isn’t just black and white or male and female. There are so many other ways in which we classify and mistreat each other. Black Americans were enslaved; Native Americans were largely eradicated. It is easy to think of still more examples – religious heretics, colonized peoples, members of the LGBT community – anything that sets one apart as the Other. Being the Other makes one less than human and more akin to vermin that should be controlled or exterminated: clearly the attitude taken by Nazis towards Jews in occupied Europe.

When I was a child, my family moved around a lot. [It doesn’t matter why; there was no good reason.] We moved every other year. I was born in Oklahoma, but my only memory of it is from visiting relatives later: we moved to central Illinois when I was still a baby. We lived in a series of small towns – Decatur, Sullivan, made a brief detour to Escondido, California, then back to Shelbyville. My earliest memories are of the rich smell of the fertile Illinois landscape coming to life in springtime as my consciousness dawned in a beautiful wooded landscape about which I was infinitely curious. The shady forests and little creeks were as much my classrooms as the brick schoolhouses inhabited by teachers, friends, and bullies.

I was painfully, cripplingly shy as a child. It took a year to start to make new friends, and another to establish them. Then we would move away.

Bullies came more quickly than friends. Every bully wants to pick on others, but especially if they are different – the Other. I was different in so many ways. I was from somewhere else, an alien immigrant to each parochial little town. I was small for my age and young for my grade, having skipped first grade. I was an egghead, a nerd in a time where the only thing society seemed to value was size and strength. Worst of all, I did not attend the same little church that they did, so I was going to hell, and many  illiterate bible-thumping bullies seemed to take it as their religious duty to speed me on my way.

When I was 13, we moved to Flint, Michigan. We went from a tiny farm town to an urban industrial area the epitomizes “rust belt.” I could no longer see the stars at night because the sky was pink – a lurid, poisonous pink – from the lights of the nearby AC Spark Plugs factory (then an active facility in which I briefly worked; now a vast empty slab of concrete). I still wandered in the limited little woods wedged between the freeway and a golf course, but the creek there ran thick with the sheen of petrochemical runoff.

I became a part of the 1970s effort at desegregation. The white religious bigot bullies were replaced with black ghetto bullies. Some seemed to think it to be their duty to return the shit white people had given them by being shitty to white people whenever they could. I didn’t really get that at the time. To me, they were just bullies. Same old, same old. Their hatred for the Other was palpably the same.

But you know what? Most people aren’t bullies. Bullies are just the first in line to greet Others onto whom they hope to unload their own self-loathing. Given time, I met better people in each and every place I lived. And what I found, over and over again, is that people are people. There are craven, nasty people and their are extraordinary, wonderful people, and everything else you can imagine in between. I’ve lived in all-white neighborhoods and mostly black neighborhoods and pretty well integrated neighborhoods. I’ve seen differences in culture but zero evidence that one race is better or worse or even meaningfully different from the other. Both have a tendency to mistrust the Other that seems deeply ingrained in human nature. We aren’t quite human to each other until we’re personally known. Once you meet the Other, they cease to be the Other and become an individual with a name and a personality. I suspect that’s what people mean when they claim not to see color – it’s not that they cease to see it, but for the people they’ve actually met, it ceases to be their defining characteristic.

And yet we persist in making implicitly racist assumptions. To give just one tiny example, a few years back a friend was helping to organize the Larchmere Porch Fest, and asked my wife and I to help. This is a wonderful event in which people in the Larchmere neighborhood offer their porches as stages for musical performances. One can wander up and down and hear all manner of music. On this occasion, I wound up helping to set up one porch for a performance by Obnox. I realized that some electricity would be needed, so knocked on the homeowner’s door. A woman appeared, and after a brief discussion, she provided an extension cord with a pink, Barbie-themed power strip that we threaded through an open window. Lamont Thomas and his drummer arrived, and set up went fairly smoothly, but he thought of something else, so also knocked on the door. I don’t remember what he was looking for, but I remember the reaction of the woman upon opening the door. Lamont is a tall, imposing black man. Her eyes got as big as saucers. She closed the door without a word. We heard the -snick- of the lock and her retreating footsteps. Lamont looked at the door that had been shut in his face, then looked at me and spoke softly: “My lyrics are kinda… raw. Is that going to be a problem?” I could only shrug. “She signed up for this,” I replied.

I don’t know what went through her mind. I would guess that like a lot of white people in the U.S., she had conveniently forgotten that black people exist – or at least, weren’t a presence in her regular circle of life. So when she chose to participate in a positive civic activity, in this case porch fest, it simply hadn’t occurred to her that black people might be involved. Who would have guessed that some musicians might be black!

That episode is but one tiny example of the pervasive, reflexive fear of the Other that still pervades American culture. More generally, I marvel at the human potential that we must have wasted in this way. The persecution of minorities, both ethnic and religious, the suppression of novel thought outside the mainstream, the utter disregard for women in far too many societies… For every Newton, for every Einstein, for every brilliant person who became famous for making a positive impact on the world, how many comparably brilliant people found themselves in circumstances that prevented them from making the contributions that they might otherwise have made? Einstein happened to be visiting the U.S. when Hitler came to power, and wisely declined to return home to Germany. He was already famous, so it was possible to financially arrange to keep him on. How might it have gone if the timing were otherwise? How many were less fortunate? What have we lost? Why do we continue to throw away so much human potential?

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Obnox performs at the Larchmere Porch Fest in 2014.

Black Lives Matter

Black Lives Matter

I started this blog as a place to discuss science, and have refrained from discussing overtly political matters. This is no longer possible. Today is June 10, 2020 – the date set to strike for black lives. I want to contribute in a tiny way by writing here. If that seems inappropriate to you or otherwise makes you uncomfortable, then that probably means that you need to read it and reflect on the reasons for your discomfort.

To start, I quote the statement made by my colleagues and myself:

The CWRU Department of Astronomy stands in solidarity with our Black colleagues and fellow citizens across the United States in expressing what should be a clear moral absolute: that people of color should enjoy the same freedoms as other Americans to life, liberty, and the pursuit of happiness. We condemn the de facto system of racial oppression that leads to pervasive police brutality up to and including the extrajudicial murders of Black Americans like George Floyd and far too many others.

We strive to build an academic community that welcomes, encourages, and supports students and scientists of color. To achieve this goal, we recognize that we must continually reflect on the injustices faced by under-represented and marginalized people, and repair the institutional structures that place them at a disadvantage. We encourage our colleagues in astronomy, throughout academia, and more broadly across society to do the same.

We will participate in the Strike for Black Lives this Wednesday, June 10, and encourage others to join us.

As the current chairperson of the CWRU Department of Astronomy, I was initially reluctant to post something about the Black Lives Matter movement on the department website. It is a different thing to make a statement on behalf of an organization of many people than it is to do so for oneself. Moreover, we are a science entity, not a political one. But we are also people, and cannot separate our humanity from our vocation. There comes a point when way too much is ever so much more than more than enough. We have reached such a point. So when I contacted my colleagues about doing this, there was unanimous agreement and eager consent to do so among all the faculty and scientific staff.

I value the freedom of speech enshrined in the first amendment of the constitution of the United States of America. I think it is worth reproducing here:

Amendment I

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.

Freedom of speech is often construed to mean the right to espouse whatever opinion one might hold, and I think that is indeed an essential personal freedom that Americans take for granted in a way that is rather special in the history of humankind. Note also that the first amendment explicitly includes “the right of the people peaceably to assemble” – a right that Americans sometimes exercise but also frequently attempt to deny to each other.

Why does this come up now? Well, if you haven’t been keeping up with current events, George Floyd died in custody after being arrested in Minneapolis, sparking protests – peaceable assemblages – across the country and around the world.

In the last sentence, I intentionally use a misleading structure common in both the press and in police reports: “George Floyd died…”, as if it were something that just happened, like a butterfly happening to pass by. Indeed, the initial police report on the incident stated that Floyd “seemed to be in medical distress” while omitting mention of any causal factor for that distress. Similarly, the medical examiner’s report exonerated the police, attributing Floyd’s death to “underlying medical conditions.”

That is some major league bullshit.

The cause of Floyd’s death is not mysterious. Officer Derek Chauvin crushed Floyd’s windpipe by kneeling on his neck for eight minutes and forty six seconds. That is considerably longer than the longest TV commercial break you have ever been modestly annoyed by. Who among us has never raged WILL THESE COMMERCIALS NEVER END? Now imagine feeling the life being crushed out of you for a considerably longer period while lying flat on your belly with your hands already cuffed behind your back. That’s right – George Floyd was already handcuffed and on the ground while being pinned by the neck. In no way can this be construed as resisting arrest. He was already under police control and in no position to resist anything, up to and including being murdered.

A more accurate statement using the active voice would be “Police arrested George Floyd, then brutally murdered him as he lay helplessly handcuffed on the ground.” There was an obvious  cause for his “medical distress:” Derek Chauvin’s knee and body weight. “Underlying conditions” played no role. Before being pinned and crushed, Floyd was alive. After, he was dead. It didn’t matter if he had been suffering from terminal cancer: that’s not what killed him. Officer Chauvin did. There is no alleged about it: we can all personally witness this heinous act through now-ubiquitous video recordings.

The more puritanical grammarians might object that I am not merely using the active voice that the police and coroner’s report (and some press accounts) take care to avoid. I am also using pejorative adverbs: brutally and helplessly. Yes. Yes I am. Because those words apply. If you want an illustration to go along with the dictionary definition of these words, then go watch all 8:46 of the execution of George Floyd.

As egregious as this case is, it is not an isolated incident. That both the police and coroner’s reports whitewash the incident with intentionally vague and passive language is a dead give away that this is standard operating procedure. They’ve done it before. Many times. So many times that there is a well-rehearsed language of obfuscation to subvert the plain facts of the matter.

This event has sparked protests around the country because it illustrates an all too familiar pattern of police behavior in black communities. I’ve heard various people say things like “It can’t be that bad.” Yet this systematic police brutality is what protesters are saying is their life experience of being black in America. Are you in a position to know better than they?

I’ve heard people say worse things. Like blaming the victim. Floyd was a career criminal, so he deserved what he got. This is such a common sentiment, apparently, that it affected a Google search I did the other day. I was trying to look up a geology term, and got as far as typing “geo” when Google auto-suggested

IMG_6892

Really? This is such a common conceit that the mere three letters g e o leads Google to think I’m searching on George Floyd’s criminal past? I can think of a lot of more likely things to follow from g e o. Given the timing, I can see how his name would come up quickly. Just his name. Why add on “criminal past”? How many people must be doing that search for this to be Google’s top hit? 

News flash: people are supposed to be innocent until proven guilty. It is the purpose of police to apprehend suspects and that of the courts and a jury of citizens to decide guilt or innocence. Whatever the alleged crime, the punishment is not summary execution by the police on the spot. As much as some few of them seem to want to be, the police are not and should not be Judge Dredd.

The same victim-blaming is going on with the protests. People have assembled in communities all over the country to protest – a right guaranteed by the first amendment. As near as I can tell, most of these assemblies have been peaceable. Given the righteous, raw anger over the arbitrary state-abetted murder of American citizens, it is hardly surprising that some of these assemblies devolve into riots. The odds of this happening are seen time and again to be greatly enhanced when the police show up to “keep order.” All too often we have seen the police act as the aggressors and instigators of violence. If you haven’t seen that, then you are not paying attention – or not following a credible news source. Fox, OANN, Breitbart, the Sinclair broadcasting network – these are not credible new sources. They are propaganda machines that are keen on focusing attention on the bad behavior of a minority of protesters in the hopes that you’ll be distracted from the police brutality that sparked the demonstrations in the first place.

Victim-blaming is an excuse closet racists use to dodge engagement with the real issue of police misconduct. “He was a career criminal! He deserved it!” and “Riots are bad! Police must keep order and protect property!” These are distractions from the real issue. Property is not as important as life, liberty, and the pursuit of happiness. Black Americans are not assured of any of those. When they peacefully assemble to petition the government for a redress of grievances, they are met with masses of police in riot gear hurling flash-bangs and teargas. Even if a few of these assemblages lead to riots and some looting, so what? That is nothing in comparison with existential threat to life and liberty suffered by all too many Americans because of the color of their skin.

An old friend tried to make the case to me that, basically, “mobs are bad.” I reacted poorly to his clueless but apparently sincere buy-in to the misdirection of victim-blaming, and felt bad about it afterwards. But he was wrong, in an absolute moral sense, and I have no patience left for blaming the victim. Yes. Mobs are bad. Duh. But going straight to that willfully misses the point. This didn’t start with mob violence out nowhere. It started with the systematic oppression of an entire group of American citizens defined in literally the most superficial way possible –  the pigmentation of their skin. The police have many roles in our society, some for the good, some not. One of the bad roles has been as enforcers of a de facto system of white supremacy – a system so deeply ingrained that most white people aren’t even aware that it exists.

I would like to believe, as many white folk apparently do, that white supremacy is a thing of the past. An ugly chapter in our past now relegated to the dustbin of history. Yet I look around and see that it is alive and well all around us.

We – all of us who are American citizens – have an obligation to make things better for our fellow citizens. At a very minimum, that means listening to their concerns, not denying their experience. Just because it is horrible doesn’t make it untrue. So don’t try to tell me about the evils of riots and mobs until you first engage with the underlying causes therefore. These are mere symptoms of the societal cancer that is white supremacy. They are natural, inevitable reactions to decades upon decades of degradation and disenfranchisement heaped on top of centuries of dehumanization through slavery and lynchings. Until you acknowledge and engage meaningfully with these brutal aspects of history and modern-day reality, you have zero credibility to complain about any of their toxic offspring. Doing so is a clear sign that you are part of the problem.

IMG_6882

Predictive Power in Science

Predictive Power in Science

“Winning isn’t everything. It’s the only thing.”

Red Sanders

This is a wise truth that has often been poorly interpreted. I despise some of the results that this sports quote has had in American culture. It has fostered a culture of bad sportsmanship in some places: an acceptance, even a dictum, that the ends justify the means – up to and including cheating, provided you can get away with it.

Winning every time is an impossible standard. In any competitive event, someone will win a particular game, and someone else will lose. Every participant will be on the losing side some of the time. Learning to lose gracefully despite a great effort is an essential aspect of sportsmanship that must be taught and learned, because it sure as hell isn’t part of human nature.

But there is wisdom here. The quote originates with a football coach. Football is a sport where there is a lot of everything – to even have a chance of winning, you have to do everything right. Not just performance on the field, but strategic choices made before and during the game, and mundane but essential elements like getting the right personnel on the field for each play. What? We’re punting? I thought it was third down!

You can do everything right and still lose. And that’s what I interpret the quote to really mean. You have to do everything to compete. But people will only judge you to be successful if you win.

To give a recent example, the Kansas City Chiefs won this year’s Superbowl. It was only a few months ago, though it seems much longer in pandemic time. The Chiefs dominated the Superbowl, but they nearly didn’t make it past the AFC Championship game.

The Tennessee Titans dominated the early part of the AFC Championship game. They had done everything right. They had peaked at the right time as a team in the overly long and brutal NFL season. They had an excellent game plan, just as they had had in handily defeating the highly favored New England Patriots on the way to the Championship game. Their defense admirably contained the high octane Chiefs offense. It looked like they were going to the Superbowl.

Then one key injury occurred. The Titans lost the only defender who could match up one on one with tight end Travis Kelce. This had an immediate impact on the game, as they Chiefs quickly realized they could successfully throw to Kelce over and over after not having been able to do so at all. The Titans were obliged to double-cover, which opened up other opportunities. The Chief’s offense went from impotent to unstoppable.

I remember this small detail because Kelce is a local boy. He attended the same high school as my daughters, playing on the same field they would (only shortly later) march on with the marching band during half times. If it weren’t for this happenstance of local interest, I probably wouldn’t have noticed this detail of the game, much less remember it.

The bigger point is that the Titans did everything right as a team. They lost anyway. All most people will remember is that the Chiefs won the Superbowl, not that the Titans almost made it there. Hence the quote:

“Winning isn’t everything. It’s the only thing.”

The hallmark of science is predictive power. This is what distinguishes it from other forms of knowledge. The gold standard is a prediction that is made and published in advance of the experiment that tests it. This eliminates the ability to hedge: either we get it right in advance, or we don’t.

The importance of such a prediction depends on how surprising it is. Predicting that the sun will rise tomorrow is not exactly a bold prediction, is it? If instead we have a new idea that changes how we think about how the world works, and makes a prediction that is distinct from current wisdom, then that’s very important. Judging how important a particular prediction may be is inevitably subjective.

RedQueen
That’s very important!

It is rare that we actually meet the gold standard of a priori prediction, but it does  happen. A prominent example is the prediction of gravitational lensing by General Relativity. Einstein pointed out that his theory predicted twice the light-bending that Newtonian theory did. Eddington organized an expedition to measure this effect during a solar eclipse, and claimed to confirm Einstein’s prediction within a few years of it having been made. This is reputed to have had a strong impact that led to widespread acceptance of the new theory. Some of that was undoubtedly due to Eddington’s cheerleading: it does not suffice merely to make a successful prediction, that it has happened needs to become widely known.

It is impossible to anticipate every conceivable experimental result and publish a prediction for it in advance. So there is another situation: does a theory predict what is observed? This has several standards. The highest standard deserves a silver medal. This happens when you work out the prediction of a theory, and you find that it gives exactly what is observed, with very little leeway. If you had had the opportunity to make the prediction in advance, it would have risen to the gold standard.

Einstein provides another example of a silver-standard prediction. A long standing problem in planetary dynamics was the excess precession of the perihelion of Mercury. The orientation of the elliptical orbit of Mercury changes slowly, with the major axis of the ellipse pivoting by 574 arcseconds per century. That’s a tiny rate of angular change, but we’ve been keeping very accurate records of where the planets are for a very long time, so it was well measured. Indeed, it was recognized early that precession would be cause by torques from other planets: it isn’t just Mercury going around the sun; the rest of the solar system matters too. Planetary torques are responsible for most of the effect, but not all. By 1859, Urbain Le Verrier had worked out that the torques from known planets should only amount to 532 arcseconds per century. [I am grossly oversimplifying some fascinating history. Go read up on it!] The point is that there was an excess, unexplained precession of 43 arcseconds per century. This discrepancy was known, known to be serious, and had no satisfactory explanation for many decades before Einstein came on the scene. No way he could go back in time and make a prediction before he was born! But when he worked out the implications of his new theory for this problem, the right answer fell straight out. It explained an ancient and terrible problem without any sort of fiddling: it had to be so.

The data for the precession of the perihelion of Mercury were far superior to the first gravitational lensing measurements made by Eddington and his colleagues. The precession was long known and accurately measured, the post facto prediction clean and irresolute. So in this case, the silver standard was perhaps better than the gold standard. Hence the question once posed to me by a philosopher of science: why we should care if the prediction came in advance of the observation? If X is a consequence of a theory, and X is observed, what difference does it make which came first?

In principle, none. In practice, it depends. I made the hedge above of “very little leeway.” If there is zero leeway, then silver is just as good as gold. There is no leeway to fudge it, so the order doesn’t matter.

It is rare that there is no leeway to fudge it. Theorists love to explore arcane facets of their ideas. They are exceedingly clever at finding ways to “explain” observations that their theory did not predict, even those that seem impossible for their theory to explain. So the standard by which such a post-facto “prediction” must be judged depends on the flexibility of the theory, and the extent to which one indulges said flexibility. If it is simply a matter of fitting for some small number of unknown parameters that are perhaps unknowable in advance, then I would award that a bronze medal. If instead one must strain to twist the theory to make it work out, then that merits at best an asterisk: “we fit* it!” can quickly become “*we’re fudging it!” That’s why truly a priori prediction is the gold standard. There is no way to go back in time and fudge it.

An important corollary is that if a theory gets its predictions right in advance, then we are obliged to acknowledge the efficacy of that theory. The success of a priori predictions is the strongest possible sign that the successful theory is a step in the right direction. This is how we try to maintain objectivity in science: it is how we know when to suck it up and say “OK, my favorite theory got this wrong, but this other theory I don’t like got its prediction exactly right. I need to re-think this.” This ethos has been part of science for as long as I can remember, and a good deal longer than that. I have heard some argue that this is somehow outdated and that we should give up this ethos. This is stupid. If we give up the principle of objectivity, science would quickly degenerate into a numerological form of religion: my theory is always right! and I can bend the numbers to make it seem so.

Hence the hallmark of science is predictive power. Can a theory be applied to predict real phenomena? It doesn’t matter whether the prediction is made in advance or not – with the giant caveat that “predictions” not be massaged to fit the facts. There is always a temptation to massage one’s favorite theory – and obfuscate the extent to which one is doing so. Consequently, truly a priori prediction must necessarily remain the gold standard in science. The power to make such predictions is fundamental.

Predictive power in science isn’t everything. It’s the only thing.

 

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As I was writing this, I received email to the effect that these issues are also being discussed elsewhere, by Jim Baggot and Sabine Hossenfelder. I have not yet read what they have to say.

The halo mass function

The halo mass function

I haven’t written much here of late. This is mostly because I have been busy, but also because I have been actively refraining from venting about some of the sillier things being said in the scientific literature. I went into science to get away from the human proclivity for what is nowadays called “fake news,” but we scientists are human too, and are not immune from the same self-deception one sees so frequently exercised in other venues.

So let’s talk about something positive. Current grad student Pengfei Li recently published a paper on the halo mass function. What is that and why should we care?

One of the fundamental predictions of the current cosmological paradigm, ΛCDM, is that dark matter clumps into halos. Cosmological parameters are known with sufficient precision that we have a very good idea of how many of these halos there ought to be. Their number per unit volume as a function of mass (so many big halos, so many more small halos) is called the halo mass function.

An important test of the paradigm is thus to measure the halo mass function. Does the predicted number match the observed number? This is hard to do, since dark matter halos are invisible! So how do we go about it?

Galaxies are thought to form within dark matter halos. Indeed, that’s kinda the whole point of the ΛCDM galaxy formation paradigm. So by counting galaxies, we should be able to count dark matter halos. Counting galaxies was an obvious task long before we thought there was dark matter, so this should be straightforward: all one needs is the measured galaxy luminosity function – the number density of galaxies as a function of how bright they are, or equivalently, how many stars they are made of (their stellar mass). Unfortunately, this goes tragically wrong.

Galaxy stellar mass function and the predicted halo mass function
Fig. 5 from the review by Bullock & Boylan-Kolchin. The number density of objects is shown as a function of their mass. Colored points are galaxies. The solid line is the predicted number of dark matter halos. The dotted line is what one would expect for galaxies if all the normal matter associated with each dark matter halo turned into stars.

This figure shows a comparison of the observed stellar mass function of galaxies and the predicted halo mass function. It is from a recent review, but it illustrates a problem that goes back as long as I can remember. We extragalactic astronomers spent all of the ’90s obsessing over this problem. [I briefly thought that I had solved this problem, but I was wrong.] The observed luminosity function is nearly flat while the predicted halo mass function is steep. Consequently, there should be lots and lots of faint galaxies for every bright one, but instead there are relatively few. This discrepancy becomes progressively more severe to lower masses, with the predicted number of halos being off by a factor of many thousands for the faintest galaxies. The problem is most severe in the Local Group, where the faintest dwarf galaxies are known. Locally it is called the missing satellite problem, but this is just a special case of a more general problem that pervades the entire universe.

Indeed, the small number of low mass objects is just one part of the problem. There are also too few galaxies at large masses. Even where the observed and predicted numbers come closest, around the scale of the Milky Way, they still miss by a large factor (this being a log-log plot, even small offsets are substantial). If we had assigned “explain the observed galaxy luminosity function” as a homework problem and the students had returned as an answer a line that had the wrong shape at both ends and at no point intersected the data, we would flunk them. This is, in effect, what theorists have been doing for the past thirty years. Rather than entertain the obvious interpretation that the theory is wrong, they offer more elaborate interpretations.

Faced with the choice between changing one’s mind and proving that there is no need to do so, almost everybody gets busy on the proof.

J. K. Galbraith

Theorists persist because this is what CDM predicts, with or without Λ, and we need cold dark matter for independent reasons. If we are unwilling to contemplate that ΛCDM might be wrong, then we are obliged to pound the square peg into the round hole, and bend the halo mass function into the observed luminosity function. This transformation is believed to take place as a result of a variety of complex feedback effects, all of which are real and few of which are likely to have the physical effects that are required to solve this problem. That’s way beyond the scope of this post; all we need to know here is that this is the “physics” behind the transformation that leads to what is currently called Abundance Matching.

Abundance matching boils down to drawing horizontal lines in the above figure, thus matching galaxies with dark matter halos with equal number density (abundance). So, just reading off the graph, a galaxy of stellar mass M* = 108 M resides in a dark matter halo of 1011 M, one like the Milky Way with M* = 5 x 1010 M resides in a 1012 M halo, and a giant galaxy with M* = 1012 M is the “central” galaxy of a cluster of galaxies with a halo mass of several 1014 M. And so on. In effect, we abandon the obvious and long-held assumption that the mass in stars should be simply proportional to that in dark matter, and replace it with a rolling fudge factor that maps what we see to what we predict. The rolling fudge factor that follows from abundance matching is called the stellar mass–halo mass relation. Many of the discussions of feedback effects in the literature amount to a post hoc justification for this multiplication of forms of feedback.

This is a lengthy but insufficient introduction to a complicated subject. We wanted to get away from this, and test the halo mass function more directly. We do so by use of the velocity function rather than the stellar mass function.

The velocity function is the number density of galaxies as a function of how fast they rotate. It is less widely used than the luminosity function, because there is less data: one needs to measure the rotation speed, which is harder to obtain than the luminosity. Nevertheless, it has been done, as with this measurement from the HIPASS survey:

Galaxy velocity function
The number density of galaxies as a function of their rotation speed (Zwaan et al. 2010). The bottom panel shows the raw number of galaxies observed; the top panel shows the velocity function after correcting for the volume over which galaxies can be detected. Faint, slow rotators cannot be seen as far away as bright, fast rotators, so the latter are always over-represented in galaxy catalogs.

The idea here is that the flat rotation speed is the hallmark of a dark matter halo, providing a dynamical constraint on its mass. This should make for a cleaner measurement of the halo mass function. This turns out to be true, but it isn’t as clean as we’d like.

Those of you who are paying attention will note that the velocity function Martin Zwaan measured has the same basic morphology as the stellar mass function: approximately flat at low masses, with a steep cut off at high masses. This looks no more like the halo mass function than the galaxy luminosity function did. So how does this help?

To measure the velocity function, one has to use some readily obtained measure of the rotation speed like the line-width of the 21cm line. This, in itself, is not a very good measurement of the halo mass. So what Pengfei did was to fit dark matter halo models to galaxies of the SPARC sample for which we have good rotation curves. Thanks to the work of Federico Lelli, we also have an empirical relation between line-width and the flat rotation velocity. Together, these provide a connection between the line-width and halo mass:

Halo mass-line width relation
The relation Pengfei found between halo mass (M200) and line-width (W) for the NFW (ΛCDM standard) halo model fit to rotation curves from the SPARC galaxy sample.

Once we have the mass-line width relation, we can assign a halo mass to every galaxy in the HIPASS survey and recompute the distribution function. But now we have not the velocity function, but the halo mass function. We’ve skipped the conversion of light to stellar mass to total mass and used the dynamics to skip straight to the halo mass function:

Empirical halo mass function
The halo mass function. The points are the data; these are well fit by a Schechter function (black line; this is commonly used for the galaxy luminosity function). The red line is the prediction of ΛCDM for dark matter halos.

The observed mass function agrees with the predicted one! Test successful! Well, mostly. Let’s think through the various aspects here.

First, the normalization is about right. It does not have the offset seen in the first figure. As it should not – we’ve gone straight to the halo mass in this exercise, and not used the luminosity as an intermediary proxy. So that is a genuine success. It didn’t have to work out this well, and would not do so in a very different cosmology (like SCDM).

Second, it breaks down at high mass. The data shows the usual Schechter cut-off at high mass, while the predicted number of dark matter halos continues as an unabated power law. This might be OK if high mass dark matter halos contain little neutral hydrogen. If this is the case, they will be invisible to HIPASS, the 21cm survey on which this is based. One expects this, to a certain extent: the most massive galaxies tend to be gas-poor ellipticals. That helps, but only by shifting the turn-down to slightly higher mass. It is still there, so the discrepancy is not entirely cured. At some point, we’re talking about large dark matter halos that are groups or even rich clusters of galaxies, not individual galaxies. Still, those have HI in them, so it is not like they’re invisible. Worse, examining detailed simulations that include feedback effects, there do seem to be more predicted high-mass halos that should have been detected than actually are. This is a potential missing gas-rich galaxy problem at the high mass end where galaxies are easy to detect. However, the simulations currently available to us do not provide the information we need to clearly make this determination. They don’t look right, so far as we can tell, but it isn’t clear enough to make a definitive statement.

Finally, the faint-end slope is about right. That’s amazing. The problem we’ve struggled with for decades is that the observed slope is too flat. Here a steep slope just falls out. It agrees with the ΛCDM down to the lowest mass bin. If there is a missing satellite-type problem here, it is at lower masses than we probe.

That sounds great, and it is. But before we get too excited, I hope you noticed that the velocity function from the same survey is flat like the luminosity function. So why is the halo mass function steep?

When we fit rotation curves, we impose various priors. That’s statistics talk for a way of keeping parameters within reasonable bounds. For example, we have a pretty good idea of what the mass-to-light ratio of a stellar population should be. We can therefore impose as a prior that the fit return something within the bounds of reason.

One of the priors we imposed on the rotation curve fits was that they be consistent with the stellar mass-halo mass relation. Abundance matching is now part and parcel of ΛCDM, so it made sense to apply it as a prior. The total mass of a dark matter halo is an entirely notional quantity; rotation curves (and other tracers) pretty much never extend far enough to measure this. So abundance matching is great for imposing sense on a parameter that is otherwise ill-constrained. In this case, it means that what is driving the slope of the halo mass function is a prior that builds-in the right slope. That’s not wrong, but neither is it an independent test. So while the observationally constrained halo mass function is consistent with the predictions of ΛCDM; we have not corroborated the prediction with independent data. What we really need at low mass is some way to constrain the total mass of small galaxies out to much larger radii that currently available. That will keep us busy for some time to come.

Fuzzy Thing!

Fuzzy Thing!

I was contacted today by a colleague at NASA’s Goddard Space Flight Center who was seeking to return some photographic plates of Halley’s comet that had been obtained with the Burrell Schmidt telescope. I at first misread the email – I get so many requests for data, I initially assumed that he was looking for said plates. That sent me into a frenzy of where the heck are they? about data obtained by others well before my time as the director of the Warner & Swasey Observatory. Comet Halley last came by in 1986.

Fortunately, reading comprehension kicked in, and I realized that all I really needed to figure out was where they should go. The lower pressure version of where the heck are they? That would be the Pisgah Astronomical Research Institute, which has had the good sense to archive the vast treasury of astronomical plates that many observatories obtained in the pre-digital era but don’t always have the ability to preserve. But this post isn’t about that; it is just a spark to the memory.

In 1986, I was a first-year graduate student in the Princeton physics department. As such, I had at that time little more competence in observing the sky than any other physicist (practically none). Nevertheless, I traipsed out into an open field at the dark edge of town on a clear night with a pair of binoculars and a vague knowledge of what part of the sky Comet Halley should be in. How hard could it be to spot the most famous comet in history?

Impossibly hard. There was nothing to see, so far as I could find. The apparition of 1986 was a bust. This informed in me a bad attitude towards comets. There had never been a good apparition in my lifetime (all of 22 years at that point), and Halley certainly wasn’t one. I accepted that decent comets must be a rare occurrence.

Flash forward a decade to 1996, by which time I was an accomplished observer with a good working knowledge of the celestial sphere. A new comet was discovered – Hyakutake – and with it came much hype. Yeah, yeah, I’d heard it all before. Boring. Comets were always a flop.

Comet Hyakutake made a close approach to Earth in March of 1996. Its wikipedia page is pretty good, with a nice illustration of its orbit and its path on the sky as perceived from the Earth. I was working at DTM at the time, where there were lots of planetary scientists as well as a few astronomers. Someone posted an ephemeris, so despite my distrust of comets I found myself peeking at what its trajectory would be. Nevertheless, we had a long period of cloudy weather, so there was nothing to see even if there was something to see, which I expected there wasn’t.

At this time, my elder daughter Caitlyn was two years old. I made a habit of taking her out and pointing things out in the sky. We watched the sunset, the moon set after it near new moon, and the moon rise near full moon. She seemed content to listen to her old man babble about the lights in the sky. Apparently more of that sank in than I realized.

My wife Anne was teaching at Loyola, and her department chair had invited us over for a party around the vernal equinox. We enjoyed the adult company and Caitlyn put up well with it – up to a point. It got dark and we bid our farewells and headed out. We had parked across the street, and on the way out Betsy (our hostess) said “Stacy – you’re an astronomer. Where’s the comet?”

I got this pained expression. Stupid comets. But it had cleared up for the first time in nearly a week, and looking up from the front door, I could quickly orient myself on the sky. Doing so, I realize that the comet was behind the house. So I pointed up and over, towards the back yard and through the roof: “Over there.” I continued across the street to the car with the toddler cradled in my left arm, fiddling with the keys with my right hand.

We did not have a nice car: one had to insert the key manually into the door to unlock it. As I went around the car to get to the driver’s side, I was focused on this mundane task. It did not occur to me to look up in the direction I had just pointed. I felt Caitlyn stretch her arm to point at the sky, exclaiming “Fuzzy thing!”

I looked up. There is was: a big, bright, fuzzy ball. A brilliant cometary apparition, the coma easily visible even in Baltimore. My two-year old daughter spotted it and accurately classified it before I even looked up.

Comet Hyakutake on March 22, 1996.

Comet Hyakutake was an amazing event. Not only spectacular to look at, but it drove home celestial mechanics in a visceral way. It was at this time very close to Earth (by the scale of such things). That meant it made noticeable progress in its orbit from night to night. You couldn’t see it moving just staring at it, but one night is was here, the next night it was there, the following night over there. It was skipping through the constellations at a dizzying speed for an object that takes c. 70,000 years to complete one orbit. But we were close enough that one could easily see the progress it made across the sky from night to night, if not minute to minute. If you wanted to take a picture with a telescope, you had to track the telescope to account for this – hence the star trails in the image above: the stars appear as streaks because the telescope is moving with the comet, not with the sky.

The path of Comet Huyakutake across the sky.

This figure (credit: Tom Ruen) shows the orbital path of Comet Huyakutake projected on the sky (constellations outlined in blue). Most of the time, the comet is far away near the aphelion of its orbit. As it fell in towards the sun, its path made annual ellipses due to the reflex motion of the Earth’s own orbit – the parallax. These grew in size until the comet came sweeping by in the month of March, 1996. Think about it: it spent tens of thousands of years spiraling down towards us, only to shoot by, transitioning well across the sky in only a couple of weeks. Celestial mechanics made visible.

Not long after Hyakutake started to fade, Comet Hale-Bopp became visible. Hale-Bopp did not pass as close to the Earth as Hyakutake, so it didn’t leap across the sky like Tom Bombadil. But Hale-Bopp was a physically larger comet. As such, it got bright and stayed bright for a long time, remaining visible to the naked eye for a record year and half. In the months after Hyakutake’s apparition, we could see Hale-Bopp chasing the sunset from the balcony of our apartment. Caitlyn and I would sit there and watch it as the twilight faded into dark. Her experience of comets had been the opposite of mine: where in my thirty years (before that point) they had been rare and disappointing, in her (by then) three years they had been common and spectacular.

The sky is full of marvels. You never know when you might get to see one.

New web domain

I happened to visit this blog as a visitor from a computer not mine. Seeing it that way made me realize how obnoxious the ads had become. So WordPress’s extortion worked; I’ve agreed to send them a few $ every month to get rid of the ads. With it comes a new domain name: tritonstation.com. Bookmarks to the previous website (tritonstation.wordpress.com) should redirect here. Let me know if a problem arises, or the barrage of ads fails to let up. I may restructure the web page so there is more here than just this blog, but that will have to await my attention in my copious spare time.

As it happens, I depart soon to attend an IAU meeting on galaxy dynamics. This is being held in part to honor the career of Prof. Jerry Sellwood, with whom I had the pleasure to work while a postdoc at Rutgers. He hosted a similar meeting at Rutgers in 1998; I’m sure that some of the same issues discussed then will be debated again next week.