Why the search for particle dark matter is really hopeless, part 2

Last time, I discussed the requirement for dark matter to naturally explain the observed MONDian phenomenology. That is a high standard that no dark matter model currently meets. But it is at least conceivable, and the widespread attitude in the community is that whatever needs to happen will eventually emerge from the effects of feedback in ever-higher resolution simulations. Here I attempt to explain why that is an unobtainable chimera. There will be math.

We’ve been here before. Newton proposed his universal law of gravity over three centuries ago to explain the orbits of the moon and planets. For our purposes here, it is useful to write what Newton taught us in terms of the acceleration the gravity of a mass M causes on a test mass at distance r:

$g_N = \frac{GM}{r^2}$.

This is for a point mass; applying this to an extended mass distribution like a galaxy requires solving the Poisson equation numerically. Such details are not important here; all that really matters is that Newton gave us an equation that relates the kinematic acceleration to the mass distribution. We can observe both of these quantities.

An analogous situation occurs in galaxies. To paraphrase Newton^&, everything happens as if MOND is the effective force law. There is a one-to-one relation between what we see in the distribution of stars and gas and what we get for the acceleration. This is such a direct connection that one can write down a simple equation for it:

$a = \nu(g_N/a_0)\, g_N$.

Here, g_N has exactly the same meaning as above. It is the acceleration that Newton predicts for the observed mass distribution. MOND amplifies this by the function $\nu$ to become the acceleration a. This is the line in the plot below:

The function $\nu$ has the properties that $\nu \rightarrow 1$ at high accelerations $a \gg a_0$ and $\nu \rightarrow \sqrt{a_0 / g_N}$ at low accelerations $a \ll a_0$ where Milgrom’s constant is observed to have the value $a_0 = 1.2 \times 10^{-10}\;\mathrm{m}\,\mathrm{s}^{-2}$. Possible interpolation functions* are discussed by Milgrom & Sanders, but the details are unimportant here. What is important is that the acceleration depends only on the observed distribution of mass as encoded in the Newtonian acceleration g_N. The excess acceleration that we normally attribute to dark matter is determined by the normal matter that we see. This is generally true for galaxies spanning properties over many orders of magnitude in luminosity, surface brightness, gas fraction, etc.

This behavior is in the data. We’ve bent over backwards to demonstrate that the radial acceleration relation is present in the data independent of any considerations specific to MOND. It is not my fault that the data look exactly like what MOND predicts. Dark matter does not naturally do that. If we entertain unnatural models, what does it need to do?

The requirement for a dark matter halo reduces to a simple equation. The observed acceleration is attributed to a dark matter halo g_DM in addition to the observed baryonic material g_N. Consequently, the ordinate of the RAR is $a = g_{DM}+g_N$. We know empirically that $a = \nu(g_N/a_0)\, g_N$, so we can solve for what the dark matter is doing:

$g_{DM} = [\nu(g_N/a_0)-1]\, g_N$.

That’s it^. The acceleration^# due to dark matter can be written entirely in terms of that due to the observed baryons. Light tells mass where to go, with spooky precision. The baryonic tail wags the dark matter dog.

Here’s an example, using the same^$ galaxies from before:

The chief lesson of this is that we don’t need dark matter; we just need to know about the RAR [$\nu(g_N/a_0)$] and measure the baryon distribution that informs $g_N$. If we insist on calling the rest dark matter, then if follows from the difference via $g_{DM} = [\nu(g_N/a_0)-1]\, g_N$. The baryon distribution determines the dark matter distribution^!. Note that the the reverse is not true: the dark matter distribution appears only as a quantity derived from the baryon distribution.

There is nothing in conventional dark matter theory that explains this. Indeed, an important aspect of viable dark matter candidates is that they not interact directly with baryons, lest they be readily detected in the laboratory. And yet we have here a simple equation that tells us the two are intimately connected. So, if you want to claim that a dark matter model explains the observed phenomenology, you need to be able to derive this equation from it. This has not been done, and there is every reason to think it cannot be done.

Note that cannot be done is explicitly physics reasoning, following the reductionist principle: if you can write down a simple equation to explain an observed phenomenon, then that’s what you should do. The task then is to build a theory that yields that equation, not to ignore Occam’s razor by building Rube Goldberg simulations with dark matter and AGN feedback and star formation feedback and other ugly details we’d prefer not to discuss to sorta kinda but not really mimic what the simple equation already told you. In short,

The only way to reconcile dark matter with observations is by very precisely mimicking the predictions of MOND.

Again, this is explicitly physics reasoning. I realized this a long time ago – in the previous century – but have been too polite to say it so explicitly. I’m not the first to notice; Bob Sanders made the same point that gas-rich galaxies amount to a practical falsification of dark matter. Gas rich galaxies, like NGC 1560 above, are particularly important because their stellar mass is sub-dominant, so unimportant to the MOND fit. The one free parameter, the mass-to-light ratio, is effectively irrelevant. The predicted rotation curve follows directly from the observed gas distribution. We can’t do that with dark matter^%.

Returning to Newton’s as if, we can make an analogy to the solar system. We know the solar system looks like an inverse square force law, but what if it is really an inverse cube law? The sun could have a dark matter halo that makes the force law look like inverse square because there is dark matter arranged just so as to always produce that result. And not just the sun – each and every planet would have to have its own dark matter subhalo so that its moons would also appear to obey as if the inverse square law.

Absurd? Yes, very! And yet that is exactly what we have to believe to sustain the dark matter interpretation of galaxy rotation curves. This must hold for the particular mass distribution of each and every individual galaxy. Dark matter somehow always manages to be arranged just so as to mimic MOND.

So what kinds of dark matter does this line of reasoning exclude? Basically all of them:

The vast majority of ideas currently being contemplated simply come up with a new particle that could provide the extra mass needed by cosmology. That might suffice for cosmology, but just adding mass does not explain the strong coupling with the observed distribution of baryons in galaxies. A viable dark matter candidate needs to do both, but the vast majority of them have been constructed in ignorance (or denial) of the need to explain the MONDian phenomenology.

It might be possible to conceive of a type of dark matter that satisfies both cosmological and galaxy dynamical requirements. Examples of such possibilities include superfluid dark matter and bipolar dark matter. These hybrid solutions are well outside the main stream of thought on dark matter candidates at present, but are more viable than any of the more commonly discussed dark sector candidates.

Of course, Milgrom already built a theory that gives us the right equation, so that seems like the obvious avenue to pursue. The remaining challenge is to build a deeper theory that encompasses both MOND and General Relativity. Searching for new particles invented in ignorance (or denial) of the observed phenomenology will inevitably prove fruitless.

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^&Newton preceded his Universal Law of Gravity with as if as a caution: he didn’t claim to know why it worked, and it seemed crazy at the time that one could have action at a distance, but everything happened as if the force of gravity was proportional to the product of the masses and the inverse square of the distance between them. The same holds today: we may not know why MOND works, but everything in galaxies happens as if MOND is the effective force law.

*One can write MOND as $a = \nu(g_N/a_0)\, g_N$ or as $\mu(a/a_0)a = g_N$. This choice is important to developing deeper theories. For our purposes here, it is a distinction without a difference.

^In McGaugh (2004), I generalized this to include a parameter Q that enabled a dark matter model to have a different stellar mass-to-light ratio from that preferred by MOND. That freedom has practically disappeared with the progress we’ve made with near-infrared observations and stellar population modeling, so I don’t bother to write the slightly more complicated math here – see the paper if you’re interested. It makes no difference to the argument here beyond adding the appearance of a tiny bit of flexibility where there isn’t really any.

^#The circular velocity of a test particle is related to the centripetal acceleration by $a = \frac{V^2}{r}$.

^$One can always quibble with any one galaxy for this reason or that, but we’ve shown this works in well over a hundred galaxies, so these examples suffice to illustrate the general rule whatever the quibble.

^!Sadly, the formula for $g_{DM}$ does not translate into a simple function for the dark matter density $\rho_{DM}$ because it depends on the details of the baryon distribution, including their bumps and wiggles (as reflected in the red lines for the dark matter), which is unique to each galaxy.

^%At present, nearly all conventional models relate the dark matter halo of a galaxy to its stellar mass but ignore the gas.