ΛCDM and MOND compared

One of the frustrating things about ΛCDM and MOND as competing scientific paradigms is that where one is elegant and predictive, the other tends to be mute. This makes a straightforward comparison difficult. Nevertheless, I make a stab at it in the table below.

What conclusions one draws about the virtues and flaws of each theory depends on how one chooses to weigh the data. Roughly speaking, the large scale data - clusters of galaxies, large scale structure formation, and the cosmic microwave background - favor ΛCDM, while galaxy scale data generally favor MOND. Put like this, ΛCDM sounds like the better bet, with galaxy scale "problems" being minor details that will eventually get sorted out. Having worked on these problems for over a decade, I can only say it aint that easy.

The table below gives a summary of the situation as I write a review article with Benoit Famaey in late 2011. Serious flaws can be found with both theories, even in their regime of presumptive superority. Still, the more important issue is how we choose to weigh the various lines of evidence, which is an entirely human and subjective exercise.

Observational tests of ΛCDM and MOND
Observational TestSuccessfulPromisingUnclearProblematic
Spiral Galaxies
galaxy rotation curve shapesXX
surface brightness ~ Σ ~ a2 X X
galaxy rotation curve fits X X
fitted M*/L X X
Tully-Fisher Relation
baryon based X X
slope X X
normalization X X
no size nor Σ dependence X X
no intrinsic scatter X X
Galaxy Disk Stability
maximum surface density X X
spiral structure in LSBGs X X
thin & bulgeless disks X X
Interacting Galaxies
tidal tail morphology X X
dynamical friction X X
tidal dwarfs X X
Spheroidal Systems
star clusters X X
ultrafaint dwarfs X X
dwarf Spheroidals X X
ellipticals X X
Faber-Jackson relation X X
Clusters of Galaxies
dynamical mass X X
mass-temperature slope X X
velocity (bulk & collisional) X X
Gravitational Lensing
strong lensing X X
weak lensing X X
expansion history X X
geometry X X
big bang nucleosynthesis X X
Structure Formation
galaxy power spectrum X X
BAO scale X X
empty voids X X
early structure X X
Background Radiation
first:second acoustic peak X X
second:third acoustic peak X X
detailed fit X X
early re-ionization X X


As I said above, one's take on the evidence is subjective, so here I give a brief discussion of each subject in the table. I don't expect everyone to agree with my evaluation of the evidence, but I do hope you will allow the evidence to challenge any preconceived notions you may have.
Spiral Galaxy Rotation Curves
The rotation curves of spiral galaxies are a clear success of MOND. They are also problematic for ΛCDM, which predicts none of the observed systematics. One can always fit rotation curves with dark matter, because one is using three parameters (a minimum of two to describe the dark matter halo plus the stellar M*/L) to describe data that only requires one parameter (MOND with fixed acceelration constant a0 but M*/L differing from galaxy to galaxy). Dark matter fits, with their extra parameters, are inevitably degenerate. They can always be made, but do nothing to explain what is observed.

The Tully-Fisher Relation
MOND has strong requirements for the quantitative details of the Tully-Fisher Relation. So far, the observations are bang on these expectations. In contrast, ΛCDM does not predict the Tully-Fisher relation per se, but rather an analogous relation for total (predominantly dark) mass and rotation velocity as the virial radius, well beyond the reach of observations. Mapping that to the data can be done, but not in a satisfactory, predictive way. Tortuous might be more descriptive.

Disk Stability
The stability of dynamically cold, rotationally supported disk galaxies like the Milky Way was one of the launching points of dark matter. Dark matter halos were needed to stabilize disks, as a bare Newtonian disk was subject to rapid self destruction through the bar instability. Problem is, too much dark halo suppresses disk instabilities too much. In low surface brightness galaxies (LSBGs), which are clearly dark matter dominated, we nevertheless see bars and spiral structure - features that occur naturally as the result of disk self gravity. That makes more sense in MOND, which provides the required stability, but doesn't overdo it - disks of all surface brightnesses are self gravitating. MOND also provides a natural explanation for the onserved maximum in the surface brightnesses of disks: it can only provide stability in the MOND regime defined by a0 = G Σmax. In contrast, we could stabilize disks of arbitrary surface density with dark matter simply by increasing the dark matter density as needed. The observed scale Σmax is not native to ΛCDM and has to be inserted into models by hand.

Interacting Galaxies
At this juncture, simulations in both paradigms have demonstrated the ability to form tidal tails as the result of galaxy interactions. However, I would give ΛCDM clear preference in terms of the dynamical friction required to merge galaxies. Dark matter halos act like big catcher's mitts to absorb all the orbital energy and angular momentum that has to be transferred for colliding galaxies to stick. It is not clear to me how this works in MOND: where does the energy go? On the other hand, MOND gives a natural explanation for the mass discrepancies observed in tidal dwarfs formed in galaxy collisions. In contrast, in ΛCDM the collision should be very efficient at segregating dark and baryonic mass, and tidal dwarfs should be devoid of dark matter.

Note added 8/18:
One must keep pace with the data, which can change. As of 2018, I would swap the above: tidal friction has become problematic for ΛCDM while tidal dwarfs now appear to disfavor MOND. One could include those UDGs that appear to be dark matter free in that category, but I think the jury is still out on those.

Spheroidal Galaxies
Star clusters and giant elliptical galaxies are both made mostly of stars and reside predominantly in the Newtonian regime, so neither make particularly good probes of either ΛCDM or MOND. It would have been very natural for globular clusters to have formed in the first minihalos, but apparently it didn't happen that way, and purely baryonic processes were involved. There are more subtle effects that can be endlessly debated (and are). The dwarf spheroidal satellites to the Milky Way and Andromeda, on the other hand, provide strong tests of both theories. It is unclear whether the mass distribution in these dwarfs is consistent with ΛCDM. Similarly, these objects fall close to where they should in MOND, but detailed fits are more problematic, requiring a rather specific variation of anisotropy. The ultrafaint dwarfs (including Ursa Minor and Draco) are particularly problematic for MOND. If current data are correct, then these systems must be in the throes of tidal distruption by their host. The situation is also weird in ΛCDM, where it is natural for such systems to be subsumed into the host, but where detailed simulations show that the stellar content of the dwarfs is well protected by its dark matter coccoon and resist disruption till the bitter end. The situation is further complicated by the missing satellite problem and the various proffered solutions.

Note added:
This subject has grown dramatically with the discovery of many new dwarfs around Andromeda and in the vicinity of the LMC. I see little prospect for a useful test from ultrafaints until some are found well away from the tidal influence of giant galaxies. Previously problematic dwarfs (e.g., Draco) remain problematic (for MOND. ΛCDM does not make a comparably testable prediction: any amount of dark matter will do) while many of MOND's a priori predictions for the velocity dispersions of newly discovered dwarfs around Andromeda have been corroborated. Intriguingly, there is evidence among pairs of Andromeda dwarfs for the external field effect, which is unique to MOND. The extraordinarily low velocity dispersion of Crater 2 provides further evidence for the external field effect, as might the case of NGC 1052-DF2.

Something that disturbs me deeply is the ability of MOND to predict what you get from what you see in objects for which the equivalent prediction simply can't be made with ΛCDM. There has to be large scatter in the relation between halo mass and stellar mass at the low masses of dwarf spheroidals; this should make it impossible to predict the velocity dispersion from the luminous mass. Yet MOND has succeeded repeatedly in doing just that.

Clusters of Galaxies
Clusters of galaxies are a clear win for ΛCDM in terms of their overall mass. MOND fails (by a factor of 2 to 3) to explain the masses of clusters. That is, one needs some form of unseen mass in clusters, even with MOND. The famous bullet cluster is merely a special case of a general rule (see, e.g., the review by Sanders & McGaugh). However, it is an overinterpretation to suppose that this falsifies MOND: there is nothing about clusters that requires the unseen mass to be non-baryonic cold dark matter. (Epistomology is terrible here - any unseen mass - "dark matter" that might be anything - often gets conflated with the non-baryonic cold dark matter that we require in ΛCDM.) In the context of MOND, it is perhaps more accurate to say that MOND suffers a missing baryon problem in clusters (though conceivably massive neutrinos might also play a role). I don't like that idea, but it is a logical possibility. Indeed, it has happened in the past - we used to think the mass discrepancy in clusters was of order ~100, not the current ~6. That was before we realized most of the baryons in clusters were in the intracluster medium rather than the stars in galaxies. So there were, historically, at least two missing mass problems in clusters, one of which was the missing baryons that are now known in the X-ray gas. Indeed, ΛCDM suffers a missing baryon problem of its own, which is more severe in amplitude if less troubling philosophically for a theory that has already embraced Darkness.

There are other aspects of the cluster data that actually sit better with MOND than with ΛCDM. The observed slope of the cluster mass-X-ray temperature relation is closer to the M ~ T2 expected in MOND than the M ~ T3/2 of ΛCDM. This difference is what leads us to the uncomfortable need to "preheat" the gas in clusters - an awkard situation that goes away in MOND. Additionally, the high bulk velocity of clusters and the high collision velocity of the bullet cluster itself is natural in MOND, while problematic for ΛCDM. How problematic seems to depend on who you ask and even how you pose the question.

This subject is a good example of what I mean about how we weigh the data. Whenever MOND comes up, there are people who say "bullet cluster!" as if it clearly falsified MOND and confirmed ΛCDM. A tall order for one object! But all they really mean is the need for unseen mass, which I agree is problematic for MOND. Somehow the people who are obsessed with the bullet cluster in this context are unconcerned with its problematic collision velocity. I.e., the part of the observation that agrees with our pre-existing belief gets 100% of the weight while the part of the observation that challenges it gets 0% even for the same object. Psychologists have a term for this: cognitive dissonance.

Gravitational Lensing
Lensing was for a long time a challenge to the creation of a generally covariant theory that contained MOND. Early theories tended to predict less lensing rather than more. TeVeS broke this barrier, which is a good demonstration of principle, but it also makes it hard to distinguish between dark matter and MOND. A system whose dynamics is described by MOND will have a lensing signal identical to the equivalent Newtoinan dark matter distribution.

I have little to say about the evidence here. Lensing statistics used to be considered problematic for ΛCDM, as you'd get lots of lenses as the volume of the universe increased with increasing Λ. That seems to have gone away (selection effects were particularly pernicious) or at least it no longer seems to be a source of active concern. There have been claims that lensing is problematic for MOND, ranging from credible to absurd. [In the latter category was a paper claiming to falsify MOND with error bars so huge that a more appropriate interpretation was that they failed to detect the need for dark matter in any system.] A recent (2013) analysis of weak lensing returns much more favorable results for MOND than some previous claims.

ΛCDM is cosmology, while MOND doesn't have a cosmology. Given the history of cosmology, I'm not sure that should be considered a shortcoming. But certainly ΛCDM fits the expansion history and geometry of the universe, by construction. The only objection is to the particular parameters (Ω, Λ) we're stuck with. Totally bizarre, though familiarity seems to be taking the edge off of that. The one place where it is not so obvious that ΛCDM is better is in the baryon density from big bang nucleosynthesis (BBN). BBN was arguably the first example of precision cosmology, and for decades before WMAP, it was known that Ωb h2 = 0.0125. WMAP says it is 0.02258. That's right - the baryon density basically doubled while gaining an extra significant digit. This creates a tention between the baryon density measured by fitting WMAP and that found from the 7Li abundance in low metallicity stars. Since it is widely presumed that ΛCDM is correct and that WMAP is the definitive word on everything, the 7Li abundances must somehow be wrong (most likely through some sort of mixing that exposes 7Li to thermonuclear processing). This tension does not exist in MOND - the CMB behaved as predicted for the BBN baryon density that is consistent with all of the relevant isotopes. As an added bonus, the missing baryon problem disappears in MOND.

Structure Formation
The formation of large scale strucure is one of the success stories of ΛCDM. Simulations in this context do a nice job of growing the sort of filamentary structure that is observed in large redshift surveys. MOND has less mass to work with, but a stronger long range force. The result isn't terribly different - MOND is just a tweak of Newton after all - but it is much harder to quantify. Unlike ΛCDM, MOND structure formation is completely non-linear, and gastrophysics cannot be avoided. It is quite possible that MOND may overshoot, in the sense that it may produce too much structure for the same initial condition. That is not entirely clear as of yet, but two salient difference seem to be that MOND is more efficient at emptying out the voids and more likely to form structure early: both testable predictions.

The Cosmic Microwave Background
The CMB provides another frustrating example of irreconcilable merits. ΛCDM fits the acoustic power spectrum of the CMB very well. It did not correctly predict its shape, however. The only successful a priori predicion concerning that was by MOND, which nailed the first-to-second peak amplitude ratio. (Fitting for this is what drove up the baryon density in ΛCDM.) However, the same model that correctly predicted the second peak also predicted a third peak that is much lower than subsequently observed. ΛCDM is flexible enough to fit the observations, though if you look at the history of WMAP releases it was pretty lousy at predicting itself. (That is to say, the fit to the year 1 data did a poor job of predicting the new parts of the year three data, year three doesn't extrapolate well to year 5, and so on.) On the one hand, that is a good thing: it was worthwhile to keep operating WMAP as there was new information to be gained (proivided that systematic effects like foreground subtraction and PSF corrections don't dominate the signal at large ℓ). On the other hand, the most that should be claimed here is that ΛCDM "wins ugly" by virtue of its greater flexibility in fitting the data.

Note added 11/12:
Looking again at this table a year later, I have been quite generous to ΛCDM. For example, I grade early reionization as "promising." Really? Early reionization was a huge surprise. "Certainy not before redshift 7" multiple prominent cosmologists assured us shortly before the data contradicted them. Concern over early reionization launched a thousand papers about Pop. III stars. But now nobody worries about it, because ΛCDM is so flexible it can be adjusted to account for just about anything. Maybe it is true, but is it still science?

Note added 4/13:
We now have Planck, which basically sees the same power spectrum as WMAP just with smaller error bars. So nothing above changes significantly. It will be interesting to see if the small error bars lead to any tension with independent data, or if "independent" measurements simply start migrating towards what Planck says they should be, as has happened with the baryon density from BBN. We've seen that happen historically with cosmic parameters like the Hubble constant. As of this writing, Planck reports 67.3 ± 1.2 while Riess et al. (2011) give 73.8 ± 2.4 km/s/Mpc.

Note added 5/21:
My concerns about the Hubble constant tension seem not to have been misplaced, at least insofar as I encounter many cosmologists who simply assume those on the other side of this debate are wrong. So far, the tension has grown mostly thanks to improving accuracy: the error bars are shrinking, but the basic results are not changing.

One of the predictions that was made around the turn of the century was that structure would form early in MOND. This should lead to more gravitational lensing of the CMB, causing the amplitude of fluctuations at high multipoles to be a tad larger (there is more fine structure than there should be). Fitting this seems to be what is driving the tension in the Hubble constant and the normalization of the power spectrum, which is also in tension between Planck and local measurements (e.g., from the Dark Energy Survey).

Looking back at the table above, it remains unclear to me that one or the other theory is clearly better. I can, of course, choose to apply a weighting scheme to the various observations that yields one or the other answer. That's because the table bears out the dischotomy I suspected before constructing it - some observations clearly favor ΛCDM while others clearly favor MOND. For example, if I place lots of weight on the bullet cluster (which would seem to be the only relevant object in the universe, to hear the more dogmatic opponents of MOND tell it), then MOND seems impossible. On the other hand, the probability (in the Bayesian sense) of obtaining the observed MONDian phenomenology in galaxies given the prior that we live in a ΛCDM universe is practically zero.

There's whats right, and there's whats right, and never the twain shall meet.

Use your own judgement, but please don't ignore observational facts. We can argue over interpretation indefinitely, but we don't get to pick and choose what data to believe. Ultimately, a valid theory must explain all data. That includes the mass discrepancy-acceleration relation as much as it does the CMB. I would think this an obvious admonition, but I have seen many otherwise brilliant scientists do the opposite, falling back on a very simple algorithm for evaluating the evidence discussed above.