The fine-tuning problem encountered by dark matter models that I talked about last time is generic. The knee-jerk reaction of most workers seems to be “let’s build a more sophisticated model.” That’s reasonable – if there is any hope of recovery. The attitude is that dark matter has to be right so something has to work out. This fails to even contemplate the existential challenge that the fine-tuning problem imposes.

Perhaps I am wrong to be pessimistic, but my concern is well informed by years upon years trying to avoid this conclusion. Most of the claims I have seen to the contrary are just specialized versions of the generic models I had already built: they contain the same failings, but these go unrecognized because the presumption is that something has to work out, so people are often quick to declare “close enough!”

In my experience, fixing one thing in a model often breaks something else. It becomes a game of cosmic whack-a-mole. If you succeed in suppressing the scatter in one relation, it pops out somewhere else. A model that seems like it passes the test you built it to pass flunks as soon as you confront it with another test.

Let’s consider a few examples.

Squeezing the toothpaste tube

Our efforts to evade one fine-tuning problem often lead to another. This has been my general experience in many efforts to construct viable dark matter models. It is like squeezing a tube of toothpaste: every time we smooth out the problems in one part of the tube, we simply squeeze them into a different part. There are many published claims to solve this problem or that, but they frequently fail to acknowledge (or notice) that the purported solution to one problem creates another.

One example is provided by Courteau and Rix (1999). They invoke dark matter domination to explain the lack of residuals in the Tully-Fisher relation. In this limit, Mb/R ​≪ ​MDM/R and the baryons leave no mark on the rotation curve. This can reconcile the model with the Tully-Fisher relation, but it makes a strong prediction. It is not just the flat rotation speed that is the same for galaxies of the same mass, but the entirety of the rotation curve, V(R) at all radii. The stars are just convenient tracers of the dark matter halo in this limit; the dynamics are entirely dominated by the dark matter. The hypothesized solution fixes the problem that is addressed, but creates another problem that is not addressed, in this case the observed variation in rotation curve shape.

The limit of complete dark matter domination is not consistent with the shapes of rotation curves. Galaxies of the same baryonic mass have the same flat outer velocity (Tully-Fisher), but the shapes of their rotation curves vary systematically with surface brightness (de Blok & McGaugh, 1996; Tully and Verheijen, 1997; McGaugh and de Blok, 1998a,b; Swaters et al., 2009, 2012; Lelli et al., 2013, 2016c). High surface brightness galaxies have steeply rising rotation curves while LSB galaxies have slowly rising rotation curves (Fig. 6). This systematic dependence of the inner rotation curve shape on the baryon distribution excludes the SH hypothesis in the limit of dark matter domination: the distribution of the baryons clearly has an impact on the dynamics.

Fig. 6. Rotation curve shapes and surface density. The left panel shows the rotation curves of two galaxies, one HSB (NGC 2403, open circles) and one LSB (UGC 128, filled circles) (de Blok & McGaugh, 1996; Verheijen and de Blok, 1999; Kuzio de Naray et al., 2008). These galaxies have very nearly the same baryonic mass (~ 1010 ​M), and asymptote to approximately the same flat rotation speed (~ 130 ​km ​s−1). Consequently, they are indistinguishable in the Tully-Fisher plane (Fig. 4). However, the inner shapes of the rotation curves are readily distinguishable: the HSB galaxy has a steeply rising rotation curve while the LSB galaxy has a more gradual rise. This is a general phenomenon, as illustrated by the central density relation (right panel: Lelli et al., 2016c) where each point is one galaxy; NGC 2403 and UGC 128 are highlighted as open points. The central dynamical mass surface density (Σdyn) measured by the rate of rise of the rotation curve (Toomre, 1963) correlates with the central surface density of the stars (Σ0) measured by their surface brightness. The line shows 1:1 correspondence: no dark matter is required near the centers of HSB galaxies. The need for dark matter appears below 1000 ​M pc−2 and grows systematically greater to lower surface brightness. This is the origin of the statement that LSB galaxies are dark matter dominated.

A more recent example of this toothpaste tube problem for SH-type models is provided by the EAGLE simulations (Schaye et al., 2015). These are claimed (Ludlow et al., 2017) to explain one aspect of the observations, the radial acceleration relation (McGaugh et al., 2016), but fail to explain another, the central density relation (Lelli et al., 2016c) seen in Fig. 6. This was called the ‘diversity’ problem by Oman et al. (2015), who note that the rotation velocity at a specific, small radius (2 kpc) varies considerably from galaxy to galaxy observationally (Fig. 6), while simulated galaxies show essentially no variation, with only a small amount of scatter. This diversity problem is exactly the same problem that was pointed out before [compare Fig. 5 of Oman et al. (2015) to Fig. 14 of McGaugh and de Blok (1998a)].

There is no single, universally accepted standard galaxy formation model, but a common touchstone is provided by Mo et al. (1998). Their base model has a constant ratio of luminous to dark mass md [their assumption (i)], which provides a reasonable description of the sizes of galaxies as a function of mass or rotation speed (Fig. 7). However, this model predicts the wrong slope (3 rather than 4) for the Tully-Fisher relation. This is easily remedied by making the luminous mass fraction proportional to the rotation speed (md ​∝ ​Vf), which then provides an adequate fit to the Tully-Fisher4 relation. This has the undesirable effect of destroying the consistency of the size-mass relation. We can have one or the other, but not both.

Fig. 7. Galaxy size (as measured by the exponential disk scale length, left) and mass (right) as a function of rotation velocity. The latter is the Baryonic Tully-Fisher relation; the data are the same as in Fig. 4. The solid lines are Mo et al. (1998) models with constant md (their equations 12 and 16). This is in reasonable agreement with the size-speed relation but not the BTFR. The latter may be fit by adopting a variable md ​∝ ​Vf (dashed lines), but this ruins agreement with the size-speed relation. This is typical of dark matter models in which fixing one thing breaks another.

This failure of the Mo et al. (1998) model provides another example of the toothpaste tube problem. By fixing one problem, we create another. The only way forward is to consider more complex models with additional degrees of freedom.


It has become conventional to invoke ‘feedback’ to address the various problems that afflict galaxy formation theory (Bullock & Boylan-Kolchin, 2017; De Baerdemaker and Boyd, 2020). It goes by other monikers as well, variously being called ‘gastrophysics’5 for gas phase astrophysics, or simply ‘baryonic physics’ for any process that might intervene between the relatively simple (and calculable) physics of collisionless cold dark matter and messy observational reality (which is entirely illuminated by the baryons). This proliferation of terminology obfuscates the boundaries of the subject and precludes a comprehensive discussion.

Feedback is not a single process, but rather a family of distinct processes. The common feature of different forms of feedback is the deposition of energy from compact sources into the surrounding gas of the interstellar medium. This can, at least in principle, heat gas and drive large-scale winds, either preventing gas from cooling and forming too many stars, or ejecting it from a galaxy outright. This in turn might affect the distribution of dark matter, though the effect is weak: one must move a lot of baryons for their gravity to impact the dark matter distribution.

There are many kinds of feedback, and many devils in the details. Massive, short-lived stars produce copious amounts of ultraviolet radiation that heats and ionizes the surrounding gas and erodes interstellar dust. These stars also produce strong winds through much of their short (~ 10 Myr) lives, and ultimately explode as Type II supernovae. These three mechanisms each act in a distinct way on different time scales. That’s just the feedback associated with massive stars; there are many other mechanisms (e.g., Type Ia supernovae are distinct from Type II supernovae, and Active Galactic Nuclei are a completely different beast entirely). The situation is extremely complicated. While the various forms of stellar feedback are readily apparent on the small scales of stars, it is far from obvious that they have the desired impact on the much larger scales of entire galaxies.

For any one kind of feedback, there can be many substantially different implementations in galaxy formation simulations. Independent numerical codes do not generally return compatible results for identical initial conditions (Scannapieco et al., 2012): there is no consensus on how feedback works. Among the many different computational implementations of feedback, at most one can be correct.

Most galaxy formation codes do not resolve the scale of single stars where stellar feedback occurs. They rely on some empirically calibrated, analytic approximation to model this ‘sub-grid physics’ — which is to say, they don’t simulate feedback at all. Rather, they simulate the accumulation of gas in one resolution element, then follow some prescription for what happens inside that unresolved box. This provides ample opportunity for disputes over the implementation and effects of feedback. For example, feedback is often cited as a way to address the cusp-core problem — or not, depending on the implementation (e.g., Benítez-Llambay et al., 2019; Bose et al., 2019; Di Cintio et al., 2014; Governato et al., 2012; Madau et al., 2014; Read et al., 2019). High resolution simulations (Bland-Hawthorn et al., 2015) indicate that the gas of the interstellar medium is less affected by feedback effects than assumed by typical sub-grid prescriptions: most of the energy is funneled through the lowest density gas — the course of least resistance — and is lost to the intergalactic medium without much impacting the galaxy in which it originates.

From the perspective of the philosophy of science, feedback is an auxiliary hypothesis invoked to patch up theories of galaxy formation. Indeed, since there are many distinct flavors of feedback that are invoked to carry out a variety of different tasks, feedback is really a suite of auxiliary hypotheses. This violates parsimony to an extreme and brutal degree.

This concern for parsimony is not specific to any particular feedback scheme; it is not just a matter of which feedback prescription is best. The entire approach is to invoke as many free parameters as necessary to solve any and all problems that might be encountered. There is little doubt that such models can be constructed to match the data, even data that bear little resemblance to the obvious predictions of the paradigm (McGaugh and de Blok, 1998a; Mo et al., 1998). So the concern is not whether ΛCDM galaxy formation models can explain the data; it is that they can’t not.

One could go on at much greater length about feedback and its impact on galaxy formation. This is pointless. It is a form of magical thinking to expect that the combined effects of numerous complicated feedback effects are going to always add up to looking like MOND in each and every galaxy. It is also the working presumption of an entire field of modern science.

6 thoughts on “Cosmic whack-a-mole

  1. Basically, it is about explaining Newton and Tully-Fisher.
    If there would be only one relationship, one could accept this,
    as has been done with Newton for centuries.
    But here something very exciting happens.

    Is the acceleration with Newton in the 3-dimensional space
    and proportional to the mass and the gravitational constant,
    Tully-Fisher looks like it’s happening in 2-dimensional space.
    and the mass and gravitational constant only enter with the square root.
    Very exciting, very interesting.
    Fascinating, the man with the pointy ears would say.

    The cat reminds me of


  2. The cat meme and squeezed toothpaste tube analogies are so perfect for the frustration faced by dark matter modelers as they ceaselessly try to match their theories with data. Looking at Fig. 6 reinforces my confidence in a simple, amateur idea which has a straightforward explanation for the rotation curves, though not worked out in fine detail, more of a broad brush approach. Section 6 which addresses the model’s link to the complex math (sqrt of -1), that is integral to QM, is looking soon to be complete. Am debating whether to bring up the topic of UAP’s in a final section 11, even though it’s relevant in a peripheral way with respect to the model’s possible technological implications. In any case the model is so wildly speculative it will be a good fit for the viXra platform, and inclusion of section 11 probably won’t make much difference.


  3. could a combination of MOND plus primordial black hole from Higgs inflation hypothesis
    explain why both hypothesis seems correct but incomplete ?

    MOND for galaxy rotation and primordial black hole for galaxies clusters CMB and gravitational lenses


    1. That may seem appealing, especially for clusters. For the CMB and structure formation, it could be problematic. If you make enough primordial black holes to do that, then there are enough of them to congregate in galaxies and act as dark matter. It is very hard to have it both ways.


      1. thanks for replying,

        the reason I ask is that you had mentioned earlier of sterile neutrinos plus MOND

        Are sterile neutrinos consistent with clusters, the CMB and MOND?
        Garry W. Angus

        If a single sterile neutrino exists such that mνs∼11eV, it can serendipitously solve all outstanding issues of the Modified Newtonian Dynamics

        this paper claims otherwise

        MOND plus classical neutrinos are not enough for cluster › paper › MOND-plus-cl…
        Jun 18, 2008 — 35 Citations ; Equilibrium configurations of 11 eV sterile neutrinos in MONDian galaxy clusters · G. Angus, B. Famaey, A. Diaferio

        so what about also adding primordial black holes, or planck mass remnants of evaporating primordial black holes + MOND

        some combination of sterile neutrinos, primordial black holes, or planck mass remnants of evaporating primordial black holes + MOND

        since some form of dark matter is needed in MOND for galaxy clusters


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