(Guest post by Tobias Mistele, December 1st, 2021)
Tobias Mistele is a PhD student at the Frankfurt Institute for Advanced Studies studying hybrid MOND-dark-matter models. Besides his physics research he also works on Scimeter.
Hybrid models, which combine dark matter and modified gravity, were long neglected. In this post, I explain why such models are now attracting attention as a path out of a stalemate.
There is observational evidence for missing baryonic mass on both cosmological and galactic scales. Most notably, the fluctuations in the cosmological microwave background (CMB) on cosmological scales and rotation curves on galactic scales. Traditionally, this is explained by non-relativistic dark matter particles (cold dark matter, CDM) that do not interact much except gravitationally. These CDM particles form a pressureless fluid on cosmological scales and later accumulate around galaxies, forming a dark matter halo. The pressureless fluid explains the fluctuations in the CMB. The mass of the halo around galaxies explains galactic rotation curves. An alternative paradigm is modified gravity. Instead of postulating particles that produce additional mass, modified gravity postulates a different gravitational force. Modified Newtonian Dynamics (MOND) is a modified gravity model that is quite successful on galactic scales. For example, consider the so-called Radial Acceleration Relation (RAR) shown in Figure 1. This is a relation between the standard Newtonian gravitational acceleration due to the stars and gas in a galaxy, gbar = GMb/r2, and the total acceleration gobs we infer from observed rotation curves. In a world without dark matter and without modified gravity, these two are the same, gbar = gobs. In the real world they are not. This is the missing mass problem in galaxies. More importantly, this relation between gobs and gbar has little scatter. Thus, the total acceleration gobs can be predicted just from the baryonic mass distribution, i.e. from gbar.
Figure 1: The Radial Acceleration Relation (RAR). Credits: One Law to Rule Them All: The Radial Acceleration Relation of Galaxies (Lelli et al. 2017). With kind permission by Federico Lelli.
This is non-trivial in DM models. In principle, two galaxies with the same baryonic mass distribution (the same gbar) can have different dark matter halos and thus a different gobs, but this doesn’t happen. In contrast, MOND naturally explains this. In fact, the RAR shows precisely what MOND postulates. At Newtonian accelerations gbar larger than a0 ≈ 10-10 m/s2 nothing new happens. Newton’s gravitational force law remains. But at accelerations gbar smaller than a0, the total acceleration changes to (a0 gbar)1/2.
Unfortunately, both CDM and MOND remain unsatisfactory when considered individually. MOND, for example, cannot explain all the missing mass on galaxy cluster scales. And, so far, no MOND-based models have been able to explain the fluctuations in the CMB, at least not without introducing some type of dark matter after all. CDM, on the other hand, has its own problems. For example, there is so far no convincing explanation for MOND-like scaling relations like the RAR. There’s just no reason why the dominant dark matter halo should be predictable from the visible baryonic mass in such a simple way. Another problem is that observed galactic bars tend to rotate faster than what CDM predicts. The dynamical friction of a CDM fluid slows down galactic bars. Then there’s the plane-of-satellites problem. Satellites of the Milky Way co-orbit in thin, planar structures. A natural explanation would be that these satellites were created from the tidal tails of interacting galaxies. But then they would not have their own dark matter halo which contradicts their high internal velocity dispersion. Also, CDM seems to be too slow to grow large structures. Massive clusters at high redshift like El Gordo are very unlikely to form so early in CDM.
So if not MOND or CDM – then what? One answer is both! That’s what hybrid MOND-dark-matter models are about. These are models that have both a pressureless fluid on cosmological scales (to explain the CMB) and a MOND-like force in galaxies (to explain e.g. the RAR).
Let me illustrate the general ideas behind hybrid models with an example – a model called superfluid dark matter (SFDM) proposed by Berezhiani & Khoury. This model has various problems, but it serves as a good illustration of the general features of hybrid models. SFDM postulates a specific new type of particle that behaves like standard CDM on cosmological scales and therefore explains the CMB in the standard way. But around galaxies, these particles condense to form a superfluid. The collective excitations of this superfluid, called phonons, then mediate a MOND-like force in galaxies. This MOND-like force is an emergent property of these particles in the superfluid phase. This is how this model explains MOND-like scaling relations like the RAR.
Of course, the superfluid itself has a mass. This produces a standard gravitational force that affects stars and gas. That is to say the superfluid also acts as dark matter in galaxies. So then we have both a MOND-like force and dark matter around galaxies. But does this not solve the missing mass problem twice? So that rotation velocities end up even larger than what we observe?
Figure 2: The Milky Way rotation curve in superfluid dark matter. The dark matter contribution is small, but becomes more important at larger radii. Credits: Tobias Mistele
The answer to that is that the superfluid DM component is usually subdominant within galaxies because the superfluid halo is very cored. Its mass becomes relevant only at larger radii. This is illustrated in Figure 2 for the Milky Way rotation curve.
One might be tempted to adjust one’s models so that the DM contribution becomes even smaller. Just to reproduce the MOND-like scaling relations even more cleanly. But one must be careful with this. Some amount of dark matter is needed in hybrid models to explain the missing mass on galaxy cluster scales for which MOND cannot fully account and, in some models, also to explain gravitational lensing.
Superfluid dark matter is not the only hybrid model. For example, recently Skordis and Złosnik proposed a model that reproduces MOND in galaxies (SZ model). This is also a hybrid model and has, deservedly, received a lot of attention since it is fully-relativistic and it was demonstrated explicitly that this model fits the CMB. Like in SFDM, the MOND and DM components are related to each other in the SZ model.
Such a common origin for the cosmological and galactic phenomena is theoretically appealing. But not all hybrid models have such a common origin. For example, the so-called νHDM model does not. Moreover, such a common origin often brings about internal tensions that must be carefully avoided.
In SFDM, for example, this common origin means that the phonon field is involved both in providing the DM and the MOND components. One technical consequence is that the usual U(1) symmetry of the superfluid must be explicitly broken which has various non-technical implications. For example, the superfluid equilibrium state might not be valid on timescales longer than galactic timescales.
The common origin for the DM and MOND components complicates things also for the SZ model. In this model, there is a kind of mass term for the static gravitational field in galaxies. Mass terms generally make forces short-range. To keep the gravitational force in galaxies long-range, the mass term must be chosen small. But a smaller mass term in galaxies means a larger pressure of the DM-like fluid in cosmology. Observations indicate a very small pressure of the DM fluid. So the galactic and cosmological phenomena push the model in different directions. This has forced the authors to include certain non-linearities as a counter.
Besides these model-specific constraints, there is also a new type of phenomenon that quite generally constrains models with a common origin for the MOND and DM components. Namely, stars often lose energy just by moving through a galaxy. Let me explain.
Accelerated charges produce electromagnetic waves. Accelerated masses produce gravitational waves. In general, whenever matter is coupled to a force carrier (e.g. the electromagnetic or the gravitational field), matter that accelerates produces waves corresponding to that force carrier. But even non-accelerated matter objects can produce waves. Namely if they move faster than the speed with which these waves propagate. For example, in a medium, electromagnetic waves propagate slower than the vacuum speed of light. Charged particles in such a medium emit electromagnetic waves if they move faster than this reduced speed of light. These waves are then called Cherenkov radiation. Such charged particles lose energy and slow down. A similar phenomenon occurs frequently in modified gravity theories whenever gravitational waves propagate at less than the vacuum speed of light. This is called gravitational Cherenkov radiation. Usually, only highly relativistic matter objects emit Cherenkov radiation, both in modified gravity theories and in electromagnetism. This is because the propagation speed of waves is usually relativistic, so that only relativistic particles are fast enough.
But this is different in hybrid MOND-DM models with a common origin for the MOND and DM components. Such models usually contain a force carrier (for the MOND-like force) whose associated waves propagate with non-relativistic speed (because this force is related to the non-relativistic dark matter fluid). Thus, even non-relativistic objects like stars might move faster than the wave propagation speed associated with the MOND force. Such stars will then lose energy and slow down, because they emit a special type of gravitational Cherenkov radiation. For example, in SFDM stars that move faster than the superfluid’s speed of sound will lose energy by emitting sound waves and slow down until they are slower than the superfluid’s speed of sound. This is illustrated in Figure 3. A star may be on a standard circular orbit when it is sufficiently slow, but will otherwise lose energy and circle towards the center of a galaxy.
Figure 3: The orbit of a star in the plane Z = 0 of a galaxy with (dotted orange line) and without (straight blue line) the Cherenkov radiation typical of hybrid MOND-DM models with a common origin for the MOND and DM components. The two cases are labeled as “With friction” and “Without friction” because in the specific approximation used, the Cherenkov radiation acts like an effective friction force on the star. Credits: Tobias Mistele
This reasoning applies only to models with a common origin for the DM and MOND components. So it does apply to SFDM and the SZ model, but not to the νHDM model. When actually doing the calculation one needs to be careful because of the non-linearities that are inherent in any MOND model. Still, it is possible to rule out part of the parameter space of SFDM using the observed Milky Way rotation curve. Basically, one requires that stars that orbit around the Milky Way with the rotation curve velocity do not lose much of their energy during the Milky Way’s lifetime. The SZ model avoids such constraints due to a special property. The coupling to matter is much larger in the static limit than in dynamical situations, which suppresses the energy emitted by Cherenkov radiation. Though I should say that the calculation for this model was done in a simplified setup so that the result should be taken with a grain of salt.
To sum up, the observational evidence for both MOND-like scaling relations on galactic scales and a DM-like fluid on cosmological scales has only become more convincing in recent years. This motivates hybrid MOND-DM models. We may not yet have a completely satisfactory model and much remains to be explored. Still, this general type of model will likely become ever more relevant in the future.
In The Dark Matter Crisis by Moritz Haslbauer, Marcel Pawlowski and Pavel Kroupa. A listing of contents of all contributions is available here.
The Dark Matter Crisis 
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