Dark Matter: Evidence, Theory, and Constraints

by David J.E. Marsh, David Ellis, and Viraf M. Mehta
Princeton University Press, 2024, 328 pages
reviewed by T. Nelson

How do you write a scientific book on something that may or may not be real? In many ways, it's like starting a new religion: your first task is to convince people the entity exists. And so nearly half of this book (37.6% if you want to be precise) is a series of proofs intended to show that dark matter really exists.

We've known for almost a century that galaxies rotate too fast. In 1933, Fritz Zwicky measured the Doppler velocity dispersion in the Coma galaxy Cluster. Instead of 80 km/s, as he calculated from the visible mass, the dispersion was 12.5 times faster, indicating that the total mass was 200 times more than the visible mass. Similar but smaller discrepancies were found in other celestial bodies, including Andromeda and the Milky Way.

For a long time it was assumed that something was wrong with the models. The most popular model for the density profile of a galaxy is the Navarro-Frenk-White (NFW) model, which defines its “halo” or visible profile as having a slope of 1/r at small radius, 1/r² at medium radius, 1/r³ at a large radius. The easiest thing to do was to tweak the NFW model by adding dark matter. The density of dark matter you'd need, in terms of particles per cubic centi­meter, would be infinitesimal: a grand total of one, creating a density about 1 / 3.346×10²² that of water. Those few particles over galactic distances add up to an astronom­ic­ally big mass. (The authors calculate it wrong in the book, but for sure it's tiny. There are other minor errors: one TeV is 10¹² eV, not 10⁹ as they say on page 7.)

Alternative theories predate even Zwicky, originating with Einstein's cosmological constant. The strongest evidence for the dominant theory today, called ΛCDM, where Λ (lambda) is the cosmological constant and CDM means cold dark matter, comes from measurements of the cosmic microwave background (CMB). The authors write:

The existence of a new, very cold, and almost pressureless degree of freedom is required to explain the detailed shape of the CMB anisotropies and to achieve consistency with the observed distribution of galaxies. Cosmological measurements are extremely statistically accurate and measure the required density of DM [dark matter] to within 1%.

At redshift (z) below 1100, after recombination happens, baryons behave like dark matter: they're decoupled from photons and are non-relativistic. At earlier times, where z is above 1100, baryons are tightly coupled to photons. This is important to the CMB because the sub-horizon modes oscillate in baryon acoustic oscillations. The term sub-horizon refers to length scales within the physical reach of some perturbation in the CMB. There is also a damping term called Silk damping or diffusion damping, which comes from Thomson scattering and the non-zero photon quadrupole. These factors go into elaborate computer simulations to calculate the CMB power spectrum, which displays the amount of correlation in the temperature as a function of angular scale.

The CMB power spectrum, shown in the graph on page 102, has many peaks and valleys. Small-scale modes (i.e., peaks) create galaxies. Computer simulations with and without CDM show huge differences in the spectrum, particularly at an angular scale (called a multipole) of 200, which is the angular size of the acoustic horizon as measured at the redshift where decoupl­ing occurs.

The authors say that logarithmic growth of CDM perturb­ations makes CDM necessary for galaxy formation. In the absence of dark matter (DM), acoustic oscillations would have large amplitude. With DM they're smaller because DM provides gravitational potential wells for the baryon-photon fluid. Thus, the authors say, dark matter is needed to reconcile the theory of structural growth of galaxies due to gravitational instability with the observed fluctuation amplitude in the CMB.

Dark energy

This is not to be confused with dark energy. Dark energy is a hypo­thet­ical force thought to accelerate expan­sion of the universe and is totally different from dark matter. In some ways they're opposites. Dark matter is thought to make up 27% of the matter-energy content of the universe, while dark energy takes 69%, leaving only four to five percent as what we can see.

The news media are saying a new paper has proved dark energy is not real. The ‘new’ theory is David Wiltshire's “timescape” model, which has actually been floating around for a number of years. The new paper isn't proof, but another model based on supernova data that tweaks the FLRW model used by ΛCDM. Dark energy was also recently challenged in a 109-author article based on data of early supernovae.

Pop science sites all over the Internet have suddenly decided that Wiltshire's most recent paper is a break­through: “Huge if true” says one. “Debunked” says another. NASA still says they're sure dark energy exists. It's remarkable how fast “It's just a placeholder for something we don't understand” changed into “We're sure it's real.” If two-thirds of the universe was all an artifact, people were too confident in the old model. It's reasonable to ask how we can trust the new one. Maybe if there is no dark energy, the model is still wrong and there's no dark matter either.

MOND

One possible alternative, which the authors strongly oppose, is modified Newtonian dynamics or MOND. Instead of changing NFW, which most people have never heard of, MOND changes the famous Einstein-Hilbert action, which is S = ∫ d⁴x √−gR (where R is the Ricci scalar), by adding two new fields. Physicists regard making ad hoc changes to a formula for every new thing that comes along as too risky. But either our fundamental equations of gravity need tweaking, or the total amount of matter we can detect is only a small fraction of what's out there.

There's considerable resurging interest in MOND (see here and here). A recent article by Stacy S. McGaugh et al. claims that MOND predicted the early emergence of massive galaxies and clusters of galaxies, while ΛCDM did not.

There are many other formulations of modified gravity besides MOND, including a “post-Friedmann” frame­work and Hořava-Lifshitz gravity. Philip Bull et al. summarize their status. Giving fair treatment to these blasphemers rather than focusing exclusively on ΛCDM would have strengthened the book.

If you reject the idea of modifying the theory of gravity, you have an even worse problem: there are no known particles with the right properties.

The Standard Model

Before speculating on the particles, the authors provide a very nice overview of the Standard Model (SM) and Higgs mechan­ism. They break down each of the terms in the formulas such as the SM Lagrangian and explain individually what they do. They also explain spinors and why they're important for particles with left or right handedness; and what a gauge transformation is and why there can be no such a thing as a right-handed neutrino.

If you've been looking for a clear description of particle physics math and don't have any strength left to lift a copy of Weinberg's excellent two-volume set, get this book and a copy of Penrose's Road To Reality today.

Neutrinos

At this point the book turns to categorizing the particles in the ΛCDM gospel. Neutrinos with a mass greater than 12 eV would be too heavy to account for dark matter. But they must be less than 1 eV to be consistent with the CMB. So ordinary neutrinos are ruled out. The authors drop a big hint about what they're really thinking. The theory says there's no such thing as a right-handed neutrino, but if there were, it wouldn't interact with photons or with any known particles. (This is heresy and they'd be crucified if they proposed it, but . . . hint hint!)

WIMPs

Some physicists suggest that dark matter could be weakly-interacting massive particles or WIMPs. The “wimp miracle” is that the numbers calculated for WIMPs are all at the exact scale needed for relic density. Sadly, WIMPS, like right-handed neutrinos, face the challenge of non-existence, which may be an insurmountable obstacle: in order for WIMPs to exist, we must invoke supersymmetry, which postulates a large number of gigantic hypothetical particles (see Astrophysics At Very High Energies reviewed at left).

Axions and primordial black holes

Axions are particles with such a low mass they'd be extremely hard to detect if even they existed, and—even worse—the theory doesn't favor any particular mass, so nobody knows where in the spectrum to look. Primordial black holes, which are much smaller than ordinary everyday black holes (which are ruled out), likewise face the challenge of non-existence. If they existed, they'd have to form by some unknown ‘exotic mechanism’ (such as divine intervention).

People have searched for primordial black holes and concluded that their mass is most likely between 10⁻¹⁵ and 10⁻¹¹ solar masses, or at least two billion metric tons. The authors say such tiny black holes are still big enough to cause a Richter 4 earthquake if they smashed into the Earth, so they should be fairly easy to detect if they exist.

So the authors return to their favorite candidate: the sterile neutrino. These hypothetical uncharged, invisible particles are only affected by gravity, so even if they exist we might never detect them. But if not, that would mean we're running out of candidates. Astrophysicists are almost desperate enough to propose spin-2 gravitons from the Randall-Sundrum model of a five-dimensional brane in string theory. As usual, the science popularizers are misinterp­ret­ing that as matter from “another dimension.” That's not exactly what it means. But string theory? That's the work of the Devil.

Unless someone finds the particle, there's a good chance that the premise of this book—that dark matter exists—will have to be abandoned. If so, it wouldn't be a failure, but a good example of self-correction in science. There are Big Clues: why would some galaxies have more dark matter than others? The theory doesn't say. Simulations can prove a theory wrong, but they can never prove a theory is correct, nor can they prove that something is real. Maybe more radio observations are needed, or maybe quantum gravity will come to the rescue. This book gives you the background to understand the science when somebody figures out the answer.

jan 05, 2025. last updated jan 11, 2025