Another Dark Matter Particle Model

Overview: An Improvement But Worse Than MOND

Another day, another dark matter particle model. 

This time, a spin-0 massive scalar boson and a spin-1 massive vector boson. Unsurprisingly with the additional degree of freedom that two bosonic dark matter particles can provide relative to single dark matter particle models, it can fit the data a little better than most one parameter dark matter particle models.

The authors "fix the vector boson mass µV = 9 × 10^−26 eV across all galaxies[.]" They allow "the scalar boson mass to vary in the range µS ∈ [10^−10, 10^−16] eV." [Ed. I have converted the MeV units used in the paper for the scalar boson mass to eV units for ease of comparison.]

The vector boson mass is (as is typical of ultralight bosonic dark matter models) of the same order of magnitude as the mass-energy of a typical graviton (for which there is also an obvious theoretical basis for gravitons to vary in mass-energy), suggesting a convergence towards the predictions of a gravitationally based explanation for dark matter phenomena with properties similar to a massless tensor (spin-2) graviton.

The second scalar bosonic dark matter particle found in dark matter sub-halos, however, has less of a clear analog for that mass scale, although the behavior of a scalar boson and a tensor boson have a lot of similarities. The authors of the paper note that:"The origin of this second DM source is unknown, which somehow points to a limitation of the model."

The model doesn't explain why the mass of the scalar bosonic dark matter candidate varies in average mass by a factor of a million from one galaxy to another, despite the fact that all of the bosonic dark matter of both types is assumed to be in its ground state in every galaxy, which is necessary for it to produce the assumed halos shapes. 

To be charitable, however, the number of sub-halos times the mass of each sub-halo could be addressed by varying the number of sub-halos rather than the mass of the dark matter particles in each sub-halo. 

But this would create a different problem. The size of each dark matter sub-halo in the model is basically a function of the mass of the scalar bosonic dark matter candidate (related to its reduced Compton wave length), but not all dark matter sub-halos that are inferred from rotation dynamics and gravitational lensing are the same size.

With three parameters: a fixed vector dark matter particle mass, a scalar dark matter particle mass which can be fit to the data on a galaxy by galaxy basis, and a factor to scale the total amount of dark matter to the total galaxy mass on a case by case basis, with a basically fixed and well-motivated formula for the halo and sub-halo shapes, this model it does almost as well at fitting a galactic rotation curve as a much simpler single fixed parameter MOND model with no parameters that vary from galaxy to galaxy, as shown in the figures below from the paper, although eyeballing it (I've seen the MOND fits to the rotation curves of many similar galaxies many times), it doesn't look like quite as tight a fit.

This model can't predict the total mass of the galaxy from the luminous matter distribution in the way that MOND does, without also resorting to the Tully-Fischer relationship to which MOND is equivalent. And, this model doesn't provide any theoretical explanation for the systemic variation in the mass-luminosity ratio from one galaxy to the next that MOND does.

Given the additional two degrees of freedom in this two type bosonic dark matter model, it's Chi-square fit should have a Chi-square fit two better than a MOND fit, which would be very tight indeed, instead of being sightly worse.

This model also, at least point, has been tested in a much narrower domain of applicability than MOND. It has basically only been tested in selected spiral galaxies in the SPARC sample, while MOND has been fit to essentially all galaxies of all shapes from smallest to largest. MOND doesn't work quite right in galaxy clusters, but this model hasn't been tested in galaxy clusters, so it provides nothing to compare to there. And, while simple extensions of MOND have been fit neatly to the cosmic microwave radiation background (CMB), and there are single particle type dark matter models that have been fit to the CMB, I have not yet seen a two particle type dark matter fit to the CMB and I'm not even really sure who that would work when one of the dark matter particle types has a mass that varies by a factor of a million from one galaxy to the next.

This is an improvement over models that took fifteen or so free parameters to fit galactic rotation curves as well as MOND, that I blogged about quite a few years ago, which still wasn't really any more predictive than this model. But, it is also definitely a work in progress that has multiple problems to solve before it is an attractive dark matter particle model that fits all of the data well.

The Paper

The introduction to the paper explains the model more fully:

Observations of galaxy and galaxy cluster rotation curves reveal a striking deviation from classical expectations. Instead of exhibiting a Keplerian decline, the measured velocities remain unexpectedly flat, extending far beyond the visible boundaries of galaxies. This persistent flatness, commonly known as the Rotation Curves (RC) problem, constitutes a critical argument in favor of non-baryonic dark matter (DM). Various theories have been proposed to explain these anomalies. Some authors have suggested modifications to Newtonian gravity, while others advocate for the existence of invisible, non-interacting DM. The early observations of the Coma cluster together with more precise measurements of galaxy RC during the 1970s reinforced the DM hypothesis. Freeman’s model of spherical halos introduced the concept of a linearly increasing mass function, and subsequent studies have mapped DM halos exceeding quantitatively the observable galactic regions. 

The possibility that galactic halos could be composed of bosonic DM has also been investigated in several works. In particular, models incorporating an ultralight axion-like particle have attracted much attention, as they naturally give rise to DM halos modeled as Newtonian Bose-Einstein Condensates. The Scalar Field Dark Matter model, which is consistent with the ΛCDM paradigm, predicts large-scale phenomena that align with linear-order perturbations. By employing ground state solutions of the Schrödinger-Poisson (SP) system—the only stable configuration where all bosonic particles reside in the lowest energy state—these models successfully reproduce the observed RC. Stability analyses further confirm that while the ground state is robust against gravitational perturbations, excited state configurations remain inherently unstable. 

Models in which a bosonic field minimally coupled to gravity acts as a source of DM are directly linked with bosonic stars. These can be primarily classified into scalar boson stars (BS) and Proca stars (PS), which are localized, regular, horizonless solutions modeled by massive, free or self-interacting, complex scalar and vector fields bound by gravity, respectively. Recent advances have expanded this framework to include ProcaHiggs stars (PHS), where complex vectors interact with real scalars to yield richer dynamics. Moreover, investigations into multi-field configurations have led to the development of multi-state boson stars and ℓ-bosonic stars, thereby broadening the spectrum of viable models. 

Ed. Proca theories were initially devised as massive photon theories. They don't actually describe the behavior of photons well, but does provide the propagators for massive spin-1 bosons, which include the W and Z bosons of the Standard Model.

The vector boson dark matter candidate is essentially a sterile Z boson (i.e. one that unlike the Z boson doesn't interact via the weak force) that is 35 orders of magnitude less massive than a Z boson. The scalar bosonic dark matter candidate is a massive spin-0 boson, much like a sterile Higgs boson (i.e. one that doesn't interact via any non-gravitational force), but 21 to 27 orders of magnitude less massive than the Standard Model Higgs boson. The vector dark matter candidate is about 9 to 15 orders of magnitude less massive than the scalar dark matter candidate.

The primary dark matter halo component has to be less massive than the subhalo dark matter component, because the size of the Bose-Einstein condensate boson star distribution of a less massive bosonic dark matter candidate is large enough to extend across the entire galaxy, while the size of the Bose-Einstein condensate boson start distribution of a more massive bosonic dark matter candidate is only large enough to extend across a dark matter subhalo which is much smaller than a galaxy. 

The theoretical feasibility of bosonic stars has been extensively examined, particularly regarding their formation mechanisms, stability conditions, and dynamical behavior. While initially conceptually conceived as static and spherically symmetric objects, modern studies now routinely explore rotating configurations that manifest as axisymmetric, spinning solutions in both scalar and vector forms. Their versatility in emulating a range of astrophysical objects—including neutron stars, black holes, and intermediate-mass bodies—renders them a powerful tool in astrophysical modeling, allowing to explore the effect of purely gravitational entities. 

This paper explores the modeling of galactic DM halos using bosonic fields, extending the study initiated in [68] and addressing two open issues identified in that work. First, the previous analysis changed the properties of the vector field for each galaxy instead of treating it as a single DM candidate; and second, it introduced an additional dark component without a clear physical justification. As in [68] we employ bosonic vector fields coupled to gravity to represent the primary galactic halo, thus enhancing RC fits when combined with ordinary matter contributions. 

A significant modification in the present work, which addresses the first open issue, is that we now fix the vector boson mass to a specific scale relevant to the problem, allowing only for the field frequency to vary, thereby identifying constraints on this parameter. Regarding the second issue, in [68] the extra component was introduced through an ad-hoc mathematical adjustment, lacking physical motivation and coherence across the configurations. 

Here, we provide a physically meaningful explanation by employing a subhalo model consisting of a scalar bosonic field coupled to gravity to represent intermediate galactic structures. An illustration of our model is depicted in Fig. 1. 

It consists of the following components: the luminous matter contribution, represented by a yellow ellipsoid (the galaxy); a quasispherical main halo formed by rotating vector bosonic matter extending beyond the galaxy, shown in dark grey; and a set of spherical subhalos modeled as scalar boson star-like structures, depicted in light grey. As we show in this paper, this model significantly improves the RC fits presented in [68], in addition to provide a physically justified framework for them.

The structure of the paper is as follows: In Section II we introduce the theoretical framework for the bosonic fields we employ in our model, along with key clarifications regarding the rescalings and physical units used. Section III details how the individual contributions to the RC are obtained from both luminous and DM systems. In particular, this section discusses the model for DM subhalos under the assumption that they could be formed by a distribution of boson stars. Next, in Section IV we compare our model predictions with observational data, using the same sample of galaxies employed in [68]. This section also provides a quantitative assessment of our new model against the fits reported in [68]. A discussion of this study is presented in Section V along with our conclusions. Additional information on the equations of motion for the scalar and vector bosonic models is provided in Appendix A.

Are these models converging on the massless spin-2 graviton, with varying energies, as a dark matter particle that is really just a gravitational explanation for dark matter phenomena?

One wonders what a model with a single massive tensor dark matter particle with continuous range of masses from about 10^-26 eV to 10^-10 eV with masses distributed in something like a power law (or just an empirical estimate of the frequency of gravitational waves at each frequency which might have a lumpy and gappy distribution) fitted to the distribution of gravitational wave strengths observed by gravitational wave detectors would look like. This wide variation of graviton mass-energies is natural and expected in a graviton as dark matter candidate model.

A galaxy length wave-length would be something on the order of 10^-24 Hertz (which is basically undetectable with current gravitational wave detectors), while some rare phenomena could generate gravitational waves with much shorter wave lengths as indicated in the chart shown below from Wikipedia:

The sensitivity rang of existing gravitational wave observatories is shown in this chart from Wikipedia:

This would put the stochastic background gravitational wave wavelengths in the same vicinity as the scalar dark matter particles in the paper's model, while the vector dark matter particles would be far, far below the frequencies that existing gravitational wave observatories can detect in a far noisier background.

If the frequency range of gravitational waves has a very low floor value similar to that of the vector dark matter candidate in the paper, and there is a big gap between that floor value and the low frequency stochastic background, however, this could be a reasonable fit to the two type bosonic dark matter model in the paper.

But, Einstein's Field Equations are structured in a manner that does not depict gravitational waves and/or graviton which have mass-energy (but not rest mass) as possible sources on the right hand side in the stress-energy tensor, and also obscures their self-interactions which are hidden within the many non-linear differential equations on the left hand side. 

So, while the cumulative impact of graviton mass-energy and non-perturbative gravitational self-interactions is potentially significant in galaxy scale or larger systems, it is systemically ignored because it is very hard to extract from Einstein's Field Equations and is negligible in the strong field systems relative to first order general relative effects, like those seen in mergers of compact objects like stars and black holes, in inspiraling binary systems, where purely perturbative approximations like the Post-Newtonian approximations work well.

One way to overcome this would be to use a massive graviton model with a range of masses comparable to the range of graviton mass-energies, which have been better developed theoretically, instead, and to consider qualitatively and in order of magnitude quantities, how that model would behave differently if all of the gravitons were traveling at exactly the speed of light, rather than the slightly below the speed of light speed of a true massive graviton with a slight rest mass with that speed itself varying slightly based upon graviton mass.

Deur's gravitational work has used somewhat similar modeling, although using massless scalar gravitons, and ignoring the differences between scalar and tensor fields (which are quite plausibly comparatively small). And, I suspect that Deur's model would closely approximate this massive graviton model (and the real world data). But, I don't have the general relativity and mathematical expertise necessary to test this myself.