- Institut für Astrophysik, Universitätssternwarte Wien, Universität Wien, Vienna, Austria
In recent years, Bose-Einstein-condensed dark matter (BEC-DM) has become a popular alternative to standard, collisionless cold dark matter (CDM). This BEC-DM -also called scalar field dark matter (SFDM)- can suppress structure formation and thereby resolve the small-scale crisis of CDM for a range of boson masses. However, these same boson masses also entail implications for BEC-DM substructure within galaxies, especially within our own Milky Way. Observational signature effects of BEC-DM substructure depend upon its unique quantum-mechanical features and have the potential to reveal its presence. Ongoing efforts to determine the dark matter substructure in our Milky Way will continue and expand considerably over the next years. In this contribution, we will discuss some of the existing constraints and potentially new ones with respect to the impact of BEC-DM onto baryonic tracers. Studying dark matter substructure in our Milky Way will soon resolve the question, whether dark matter behaves classical or quantum on scales of
1 Introduction
According to the theme of this Research Topic article collection, we might imagine a fictitious conversation in the Parnassos of deceased scholars, involving Rubin, Einstein and Planck: while Rubin would elaborate on her observational findings of dark matter (DM) from the dynamics of galaxies, the question will arise whether we understand gravity sufficiently well to explain this phenomenology. Einstein would calmy point out that general relativity has passed all tests so far, including ever newer ones from the detection of gravitational waves from black hole mergers, to the direct observation of the central supermassive black hole of galaxy M87, and to tests of the equivalence principle, and so forth. As Einstein has convinced (almost) everyone, Planck may finally make a case that quantum mechanics may also have a say, after all, in this discussion. This is the framework of this paper: we believe in particle dark matter, consider general relativity the correct theory of gravity (at low energies), but quantum mechanics will also play a role. The wave-particle dualism applies also to DM particles, and it is appropriate to treat many DM candidates in the particle regime. In this paper, we consider DM whose wave nature cannot be neglected on astronomical or galactic scales anymore, i.e., quantum properties can affect astrophysical DM phenomenology. Ultralight bosonic DM with particle masses of
We will focus in this paper on the simplest model of a single bosonic species in a BEC, which has no self-interactions and no direct coupling to standard model particles. Also, this paper will focus on BEC-DM substructure within the Milky Way.
The mathematical framework described below applies to any DM bosons which can form a BEC, whether the underlying bosons be ultralight, appropriate for FDM, or whether they be QCD axions with
The lower the boson mass, the larger the expected substructure in BEC-DM and its impact onto small-scale structure formation. Consequently, cosmological probes have been used to infer lower bounds on that mass. Those bounds are weakened, if BEC-DM constitutes only part of the DM, but we are interested here in constraints that assume that all of the DM is in the form of BEC-DM. Previous constraints covered a broad range of lower-bound estimates. CMB anisotropies require
Now, this paper focuses on the study of DM substructure within the Milky Way, a topic which is being revolutionized by modern astrometry missions like GAIA, and even the near future holds promise in our quest to reveal the nature of DM. Unlike the measurement of the rotation curves of external galaxies, this time our inside-perspective of the Milky Way will come to our advantage. The high-precision big data from astrometry missions will help in multiple ways with respect to the DM problem, e.g., i) determine accurately the total mass of the Milky Way, ii) detect DM substructure indirectly from dynamical inferences and constrain it in the process, iii) reveal and disentangle the complicated astrophysics of baryonic tracers in the Milky Way so that we better understand the impact of different baryonic components and their interaction with DM, etc.
That the properties of many tracers such as open clusters in the disk, or the interstellar medium, have been studied and “understood” in all these decades without the need to account for DM substructure could be regarded somewhat as a mystery. The fact that it appears that many researchers of the Milky Way do not need to worry in their work about DM substructure should alert cosmologists. What shall that mean for the nature of DM? Standard CDM predicts a plentitude of small-scale structures and subhalos, e.g., for WIMPs (weakly interacting massive particles) all the way down to Earth-mass scales, and alternative DM models also predict enough substructure for it to “matter”. Sure enough, the quest to study DM substructure and its potential impact onto the disk, or tracers within it, dates back at least to the early days of CDM (some references will be given below). But somehow it seems that the overwhelming number of individual self-gravitating DM subhalos (we call them “drops” in this paper for reasons discussed below) should be either so tiny that their individual dynamical fingerprints are negligible, or else so large that a given single drop covers big stretches of the Milky Way disk or halo, in such a way that the encompassed stellar tracers hardly feel its presence over most of their lifetimes. Too many “intermediate size” DM drops might be prone to cause too much dynamical chaos in the Milky Way and to its stellar tracers1, such as giant molecular clouds (GMCs), large-scale gas filaments, open clusters or binaries. Thus, there is still work to be done to establish how much and what DM substructure is in accordance with the observed properties of the baryonic components of the Milky Way.
This paper is organized as follows. The equations of motion of BEC-DM, some basic concepts and halo structure will be discussed in Section 2. Novel aspects of BEC-DM, especially with respect to its substructure and the impact of quantum phenomena onto stellar tracers in the Milky Way will be reviewed in Section 3. In Section 4, we will discuss further quantum aspects of BEC-DM, which add even more distinction to the standard CDM model.
2 Dynamics, Characteristic Scales and Halos at Large
We assume that BEC-DM consists of fundamental spin-0 bosons, all with the same mass m. There are ways to create these bosons in the early Universe, either thermally, or via a vacuum realignment mechanism akin to the QCD axion. We will not consider these early times, but only start with the premise that these bosons are “cold” (i.e., nonrelativistic) by the time cosmic structure forms and ever since. In addition, we focus on scenarios where these bosons are (almost) all in the same quantum state, and they got there by a Bose-Einstein phase transition2. Once we have (almost) all N bosons in that same quantum state, we can describe their evolution through a single scalar field ψ (hence SFDM), i.e., the dynamics of the original N-particle wavefunction reduces to a 1-particle ”wavefunction” ψ of the Bose-Einstein condensate (BEC) itself. Its equation of motion is the Gross-Pitaevskii (GP) equation, a form of nonlinear Schrödinger equation, which is coupled to the Poisson equation in order to describe self-gravitating systems. The combined system of equations3 is abbreviated GPP,
where
The above equations are appropriate for isolated objects, but at cosmological scales one would need to add terms due to Hubble expansion. This model as decribed here is sometimes called the simplest ”vanilla” model of ultralight bosonic BEC-DM: it is modeled using a plain scalar field. Importantly, the vanilla types have no internal boson self-interactions, and in this context they have been called fuzzy DM, ψDM, wave DM or free SFDM. In this work, we will call the model often fuzzy dark matter (FDM), in observance of the early paper by (Hu et al., 2000), wherein that term was coined, although fiducial FDM models usually require ultralight bosons,
The GPP formalism assumes that all particles of the object under consideration are in the BEC and the latter requires that occupation numbers are high, and
assuming that the boson moves with nonrelativistic velocity v. It may not be so obvious what v should be in our context5. In a laboratory setting, where we think of cooling a thermal atomic gas, say, to become a BEC, we have the thermodynamic temperature T which refers to the average kinetic energy in a thermalized sample of atoms,
For BEC-DM models for which
In fact, early works such as (Membrado et al., 1989) or (Guzmán and Ureña-López, 2004) calculated already the numerical profile of an isolated, gravitationally bound ground-state system (“soliton”), as a result of solving the above differential equations as an eigenvalue problem. The corresponding mass-radius relationship reads as
where
Notice that Eq. 7 is valid for any boson mass m, ultralight or not, but the size of the soliton shrinks quickly for rising boson mass. Since we also discuss boson masses intermediate between ultralight and QCD axion-like, we will call self-gravitating objects, which follow the relation Eq. 7, “solitonic drops”, or “drops” for short.
However, these ground-state solitons not only describe individual BEC-DM drops, they also respresent the central parts of BEC-DM halos, which formed from mergers either in a static or expanding background, the so-called “solitonic cores”, whose size is of order Eq. 7. This has been first convincingly shown by the FDM simulations of (Schive et al., 2014a; Schive et al., 2014b), and the small boson mass of fiducial FDM models,
More simulations of FDM by (Schwabe et al., 2016) or (Mocz et al., 2017), following (Schive et al., 2014a), have confirmed that BEC-DM halo formation leads to the appearance of an “envelope” which can extend in size very much beyond
The smaller the boson mass, the more prominent the solitonic cores at the centers of halos, with observational consequences. Their impact on galactic rotation curves has been studied in (Bar et al., 2018) with the conclusion that masses of
Apart from the core-envelope structure of halos, some FDM simulations find large-scale coherence effects around and between halos (Schive et al., 2014a), (Mocz et al., 2017), extending all the way to cosmic filaments (Mocz et al., 2020), as well as finding signs of quantum turbulence within halo envelopes (Mocz et al., 2017). Unfortunately, current simulations cannot resolve the DM substructure expected in FDM and BEC-DM, in general. Nevertheless, we can infer some implications described in the next section.
Before that, we summarize a few more basic properties of BEC-DM. First, we state the fundamental scales of the bosons under question in fiducial units for an easier comparison later. We have
The time for a boson to cross the length scale
This time scale is related to the coherence properties of BEC-DM as a giant matter wave. The free-fall time of an object with mean density
Now in addition, by de Broglie’s wave-particle dualism, a frequency ν (an “internal clock”), is assigned to any particle with mass m by setting
The “clock time” associated with the oscillation frequency is thus
The quantities12 in Eqs. 8–11 are the characteristic time and length scales for condensed DM bosons without other microphysical properties, especially for those without self-interaction. The latter scales are much smaller than the former,
Notice that the “Compton scales” are natural constants for any given BEC-DM model, whereas the “de Broglie scales” depend upon the environment of bosons, i.e., for the latter it matters whether bosons belong to the uncollapsed fraction, or be part of a halo overdensity or substructure, respectively. This will be important to remember.
We stress that it is possible to rewrite the GP equation in a form that recovers hydrodynamical conservation equations of continuity and momentum (“quantum hydrodynamics”), which is just a different way of saying that we can use the de Broglie-Bohm formulation of quantum mechanics. Introducing a polar decomposition (also known as Madelung transformation in this context), according to
with the BEC-DM mass density
can be expressed in terms of the (bulk) velocity, if we write
and
We can see that Eq. 14 has the form of a continuity equation, while Eq. 13 has the form of an Euler-like fluid equation of motion. Hence, the original GP equation has been re-written as a set of quantum hydrodynamical equations13 with “hydro variables” ρ and v. The quantum potential Q gives rise to a so-called “quantum pressure” on the right-hand-side of the fluid momentum equation in Eq. 13. It stems basically from the quantum-mechanical uncertainty principle. The quantum hydrodynamical equations are then solved self-consistenly, along with the Poisson equation Eq. 2.
The astrophysics literature uses both formalisms, and the choice for one over the other is influenced by taste and numerical method at hand.
3 Aspects of Bose-Einstein-Condensed Dark Matter Substructure
3.1 Wave Interference
The question of DM substructure is a critical one, for CDM and for alternative DM models. In this paper, we discuss certain novel aspects of BEC-DM substructure, which have been increasingly studied in the literature. A review can be found in (Niemeyer, 2020), especially focusing on the cosmological perspective and axion-like particles. Estimating the minimum size of gravitationally bound objects in a given DM model can be done straightforwardly, in principle, by applying a Jeans analysis. Yet, the dynamical details depend upon the actual particle physics models and are complicated to work out case by case. The abundance of expected substructure is even harder to estimate because that depends upon the primordial structure formation on small scales, followed by the highly nonlinear dynamics of merging, gravitational heating, tidal stripping, etc. Many of these issues require future studies.
For BEC-DM, we encounter additional phenomena, relevant to the question of substructure, due to its quantum nature. Quantum wave interference phenomena arise on galactic scales, which have been studied for FDM using simulations and some analytic estimates. We will discuss some of these findings, and otherwise restrain to crude estimates to guide our discussion of substructure in the Milky Way.
It has been known for a long time that DM substructure can potentially impact the amount of dynamical heating of the Milky Way thick disk, see e.g., (Lacey and Ostriker, 1985). There is a beautiful study by (Church et al., 2019), which is concerned with the impact of FDM substructure on the thickening of the Milky Way disk. The central part of their Galactic halo model comprises a solitonic core, outside of which the averaged density profile would follow a CDM-like profile, more precisely an NFW profile (Navarro et al., 1996). In the process, the authors discuss the qualitative differences between virialized FDM subhalos (“drops”, thus the collapsed fraction in substructure) vs. non-virialized, but overdense FDM “wavelets” (uncollapsed fraction in substructure). The latter arise as a result of linear interference patterns of the underlying velocity dispersion of bosons, which has been seen in simulations of idealized halo formation and virialization, e.g., in (Mocz et al., 2017), and can be associated with the quantum phenomena inherent to BEC-DM/FDM. As such, the abundance of these wavelets increases sharply with decreasing boson mass, as expected; by the same token wavelets are strongly suppressed by a factor
Apart from disk heating, FDM wavelets have been also constrained by their impact onto the thickening of thin stellar streams by (Amorisco and Loeb, 2018). Since stellar streams are even “colder” than the disk, the resulting bound on the boson mass is stricter, namely
Somewhat earlier dynamical studies of FDM substructure, including derivations of relaxation times and investigations of the impacts onto multiple baryonic tracers, can be found in the works of (Hui et al., 2017) and (Bar-Or et al., 2019). (Having at hand a copy of (Binney and Tremaine, 2008) helps in studying the papers). Hui et al. (2017) derive a relaxation time for what they call “FDM quasiparticles” of mass
using values appropriate for the solar circle,
with a dimensionless constant
Bar-Or et al. (2019) conduct an accurate study of relaxation of FDM substructure and dynamical heating, with an emphasis on the question how long it takes to bring infalling massive objects to acquire the velocity dispersion of the host halo. They adopt a singular isothermal sphere for the Galactic halo15, which explains some of the deviations between their’s and the results of others’. To a certain extent, they confirm the previous estimates by (Hui et al., 2017), e.g., the mass of the quasiparticle/wavelet, if evaluated using the mean density of the quasiparticle,
Detailed simulations including baryons will be required to see how the complicated wave interference in BEC-DM/FDM halos can be in accordance with observational findings regarding the dynamical heating of the disk. In this regard, widely extended stellar streams far from the disk will be a “cleaner” probe in order to study the impact of wave interference or solitonic drops onto them. The previously mentioned studies already have shown that DM in the Milky Way cannot be “too quantum” on scales
Another increasingly studied phenomenon is the nature and impact of dynamical friction of BEC-DM and how it differs from the classic approaches used for CDM. Previous estimates and considerations can be found in (Hui et al., 2017) applied to FDM, but the question has been taken up in (Lancaster et al., 2020), also for FDM, in much more analytic and numerical detail, including applications to various systems such as the globular clusters in the Fornax dwarf spheroidal, or the Sagittarius stream. The authors investigate dynamical friction acting on a satellite which moves through an FDM background. They identify basically three distinct regimes in FDM dynamical friction: i)
3.2 Solitons Abounding
Here, we turn to solitonic drops in anticipation that BEC-DM bosons may have to have higher masses than originally hoped. We have already mentioned that the importance of subhalos, i.e., drops, increases with mass m, while the importance of wavelets decreases. Therefore, we will put into perspective their potential impact onto baryonic tracers.
The size of gravitationally bound BEC-DM drops is determined by
and around the solar circle and beyond, with
At first sight, the above estimates seem counterintuitive, because from CDM we are used to thinking of the common hierarchical structure-formation scenario, where the smallest subhalos form first, merge, and build up ever bigger halos, albeit subhalos can be stripped in the process, as well, and any galaxy today, whether ultrafaint dwarf or Milky Way should have formed by hierarchical build-up17. This picture is still similar above the minimum Jeans scale of BEC-DM, as simulation results show, but the very mass-radius relationship of ground-state equilibria in Eq. 7,
The mass of the ground-state drops within the Milky Way can be estimated, using the mass-radius relationship in Eq. 7, and setting
where we re-introduced the m-dependence for an easier comparison later. Inserting
We may reconsider the relaxation time of (Hui et al., 2017) in Eq. 16, but now insert Eq. 19, instead of their
and the v-dependence dropped out, because of the relationship between
We proceed by deriving an admittedly crude upper bound
It is tempting to compare the upper bound on drops for smaller boson masses m; setting again
It has been pointed out that FDM drops are more easily tidally disrupted than CDM subhalos, if the tidal radius of a drop lies within the solitonic core of the primary halo. So, tidal stripping depends sensitively on the position of the drop within its host. However, the higher m, the smaller the solitonic core. Also, at higher m, drops within a halo background can be considered relatively isolated for most of their lifetime, because their size is small compared to their mutual distance. Since ground-state drops have undergone gravitational cooling (as described in Section 2), we might argue that these drops are little vulnerable against nonlinear “post-processing”. Of course, there ought to be unbound DM within the Milky Way halo, which arises from the “cooling” processes of these small-scale drops. Their expelled, unbound matter will undergo interference akin to the large-scale wave phenomena described above; however, the dynamical importance of that unbound DM becomes diminished, the higher m, as mentioned before.
The ever growing importance of drops, with their increasingly smaller size as the boson mass increases, gets them into a regime where they may eventually become MACHO-like (MACHO: massive, astrophysical, compact halo object). In the early days of CDM, theoretical studies investigated the dynamical impact of such objects in the Milky Way, whether they be of baryonic origin (e.g., dim stars such as brown dwarfs), or black holes (BHs), or other exotic compact objects which could account for the DM. Later, it has been shown that MACHOs could not be made of baryonic matter, see (Freese et al., 2000). Nevertheless, the issue of compact ”dark” halo objects is recurring and observational limits continue to be placed. Given that CDM substructure due to WIMPs, or other DM substructure, is expected to be plentiful, the issue of its impact in the solar neighborhood, or generally in the galactic disk and halo, has not gone away, only because certain MACHOs have come out of favor. And in fact, this impact may be an issue for alternative DM models, once they predict a lot of substructure. For BEC-DM, we will discuss some further potential constraints in the next subsection.
3.3 Impact Onto Stellar Tracers in the Disk and Halo and the GAIA Era
It has been long realized that the study of the impact of DM substructure onto stellar tracers, such as stellar clusters, wide binaries, or more recently stellar streams, has the potential to constrain the nature of DM. For instance, early work by (Bahcall et al., 1985) has shown that the mass of “dark halo objects” can be constrained by considerations of wide binary lifetimes, e.g., they found an upper bound of
We argue that a reconsideration of similar bounds such as placed on MACHOs, including the application of more accurate halo models, should be applied to other DM substructure, if the latter is compact enough. This is especially true for BEC-DM solitonic drops, since the inverse mass-radius relationship in Eq. 7 implies that the density of such drops grows rapidly with mass,
while
If a constraint as tight as required by Eq. 22 were to hold up in the future, it would suggest that BEC-DM bosons must have a mass similar to the QCD axion. In fact, the latter is in perfect agreement with such a tight constraint on substructure. While we do not discuss the QCD axion in this paper, its substructure including microlensing has been studied recently e.g., in (Davidson and Schwetz, 2016; Fairbairn et al., 2017; Schiappacasse and Hertzberg, 2018; Xiao et al., 2021).
In fact, there is a recent body of literature which takes up the question of microlensing by extended DM substructure/MACHOs, such as NFW subhalos and boson stars18, see (Croon et al., 2020a; Croon et al., 2020b), as the latter serve as lenses within the Milky Way and M31, magnifying extended stellar sources in M31. While the fraction of point-mass DM structures as lenses can be constrained in a broad range of
On the other hand, dMACHOs (composite objects that are made up of dark-sector elementary particles) have been considered in (Bai et al., 2020), but we will not discuss them here.
Now, what is the size of solitonic drops of mass
In the era of high-precision astrometry and photometry, many previously derived observational bounds should be reconsidered, in general, especially those which attempt to determine the impact of DM substructure (MACHO-like, or other) onto open clusters, wide binaries and stellar streams. The GAIA mission is currently revolutionizing our census of stellar populations and clusters in the Milky Way, revealing a highly complicated phase space structure within stretches of the Milky Way, devoid of symmetries or isotropy so cherished by theoreticians. New open clusters are being found and previous would-be clusters are being debunked. Many young open clusters are found with large tidal tails
In the spirit of earlier theoretical studies, however (Hui et al., 2017), infer some rough estimates on the disruption time of open clusters due to FDM quasiparticles/wavelets, assuming the diffusive regime for extended perturbers. For chosen nominal cluster values of mass
Now (Hui et al., 2017), draw the same conclusion of negligible disruption rates by fiducial FDM wavelets, when it comes to wide binaries (semi-major axis of
To reiterate, it will be important in the future to reconsider previous bounds on DM substructure, especially in light of new GAIA data. While it had seemed that fiducial FDM with
4 Other Novelties
4.1 Scalar Field Oscillations
The dynamical differences of BEC-DM which behaves as a quantum fluid, compared to CDM, arise from the astronomically large de Broglie scales Eqs. 8, 9. However, the Compton scales in Eqs. 10, 11 give also rise to effects. While these scales are much smaller than the de Broglie scales, they can still be prominent, especially for BEC-DM with small m (see Section 2;
However, no signal from oscillating BEC-DM has yet been detected. The Parkes Pulsar Timing Array (PPTA) has placed upper limits on the local DM density for ultralight BEC-DM/FDM with
However, the analysis in (Khmelnitsky and Rubakov, 2014) applies to real scalar fields, appropriate for axion-like particles. A similar analysis, but for complex scalar fields, reveals that the expected signal is very much smaller22 than that for the real case (Steininger, 2018), placing the observational limits far beyond any future measurement capabilities.
4.2 Coherence and Quantum Measurement in Astronomy
In this last subsection, we touch upon some issues that have implications for our understanding of quantum mechanics and quantum gravity.
We have mentioned quantum interference phenomena of BEC-DM, which is supposed to be in a coherent state on cosmic scales, so let us introduce some notions first. Coherence describes all the properties of the correlation between physical quantities of a single wave, or between several waves or wave packets. The degree of coherence is mathematically described via correlation functions. A perfectly coherent state has a density matrix that is a projection onto the pure coherent state, and it is equivalent to a wave function, while a mixed state is described by a classical probablity function for the pure states that make up the mixture. Now, the “wave function” of the BEC-DM condensate (i.e., the single scalar field that describes an N-particle system) is supposed to have this property of coherence. It has been shown that there is a correspondence to quantum entanglement, i.e., quantum coherence is equivalent to quantum entanglement in the sense that coherence can be described as an entanglement measure, see (Tan and Jeong, 2018), and conversely each entanglement measure corresponds to a coherence measure. With this in mind, cosmic BEC-DM should be quantum-entangled, but that means cosmic BEC-DM is not “separable”: any parts can occupy any of their possible states and in order to explain the behavior as a whole, one has to consider all possible states. It is only once a quantum measurement is performed on a part that its state is determined, but so will instantly be the “rest”. This is common lore in a laboratory setting, but lifting this up to the cosmic scale, it would imply that a local measurement, e.g. within the Milky Way halo, would instantly determine the “outcome” or state of BEC-DM in the rest of the Universe!
What would qualify as a (quantum) measurement ? The indirect dynamical inference of BEC-DM onto baryonic tracers? Do baryonic tracers somehow act as an environment for BEC-DM, which makes the latter decohere23? If not, can cosmic BEC-DM be regarded as an “isolated system”, for it pervades the whole Universe anyway? Would the direct detection of a boson (axion-like or other) constitute that particular measurement that would determine the cosmic state of coherent, entangled BEC-DM? Is quantum measurement questioned by this problem?
This topic has been addressed by the beautiful paper of (Helfer, 2018). It takes FDM with boson mass of order
Quantum measurements generally violate conservation laws. In a laboratory setting, this problem can be put under the rug, because any failure of conservation in the observed system can be thought to be absorbed by the much bigger measurement apparatus, i.e., the failures can be made small, if not eliminated within conventional quantum theory. But in the case of BEC-DM/FDM with large Compton length, the energies involved in the measuring devices can be small, compared to the system which is e.g., the local DM density. Thus, a DM candidate such as BEC-DM challenges our (non-)understanding of quantum measurement theory. Since DM interacts gravitationally, the observables to probe quantum measurement derive from a stress-energy tensor, i.e., they are related to geometry and, in essence, entail implications for quantum gravity, too. Now (Helfer, 2018), considers quantum fields described by a special-relativistic Klein-Gordon equation (upon “second quantization”). A renormalized stress-energy operator is calculated from which energy density and pressure of FDM can be derived. These quantites are all averaged over a length scale a few times the Compton scale. The interference of individual modes and their energy is determined. By defining a quantum-coherent state of “many possibilities”, the measurement of, say, the averaged energy density will project that state onto an eigenspace, since the measurement “picks” one possibility. It can be shown that for both energy density and pressure, the quantum measurement will excite relativistic modes (even if initially nonrelativistic), i.e., particles will depopulate out of their states, become gravitationally unbound, and escape the Milky Way (or any galaxy or cluster, if the measurement would be made there). In order to show that such a measurement could be performed in principle (if not by humans at this time) (Helfer, 2018), moves on to present a particular scenario. It is conceptually similar to the gravitational-wave detection via pulsar timing arrays, but cannot rely on pulsars because the measurement has to be performed within a Compton scale of
This proposal highlights that there are issues with quantum measurement that have become exposed with BEC-DM. An obvious resolution could be that quantum measurement theory, or our understanding of conventional quantum mechanics is incomplete (which certainly is). Of course, another resolution would be to say there ought to be no BEC-DM, which would seem a “bad excuse” argument, in our opinion (Helfer, 2018), adds a discussion, one of which concerns alternative approaches to quantum theory, e.g., the de Broglie-Bohm formulation, whose formalism was also sketched in Section 2. However, the latter still lacks a generally accepted relativistic extension. It is certainly true that by using the de Broglie-Bohm formulation with its quantum hydrodynamical equations, astrophysicists have put certain aspects under the rug. On the other hand, if this approach allows us to infer limits on DM in the form of BEC-DM, axion-like particles, etc. by studying its impact on real-world baryons, there is hope that this endevaour may shed more light onto fundamental questions than originally expected, i.e. not only on the DM conundrum, but also on issues in quantum mechanics and quantum gravity.
5 Conclusion
We have reviewed certain novel aspects of dark matter in the form of a Bose-Einstein condensate (BEC-DM), and how its new quantum phenomena may be studied and constrained within our own Milky Way. While the literature has continuously placed more and tighter limits on the mass of the boson of BEC-DM, there are still many probes left which should be studied in the future, on the theory side and also on the observational side, given the ongoing GAIA revolution. Quantum phenomena will be increasingly harder to infer, the higher the boson mass, and ever smaller-scale probes within the Milky Way will have to be used. In this paper, we have disregarded any microphysical extensions, notably boson self-interactions, and so similar studies than those conducted for BEC-DM without self-interaction will be required, in order to see whether any BEC-DM model will reveal itself one day as the alternative solution to the CDM paradigm.
Author Contributions
The author confirms being the sole contributor of this work and has approved it for publication.
Funding
The author is supported by the Austrian Science Fund FWF through an Elise Richter fellowship, grant no. V656-N28.
Conflict of Interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Acknowledgments
The author thanks Stefan Meingast for helpful discussions.
Footnotes
1Not to speak of the greater impact onto the spiral structure, or the central galactic bar, which can be also used to constrain DM; we will not discuss these here.
2There is still confusion in the literature, which does not distinguish a squeezed state from the ground state of a Bose-Einstein-condensed system, but we will not enter this discussion here and presume that condensation actually happened at some point in the early Universe.
3In models without self-interaction, as described here, these equations of motion are also known as Schrödinger-Poisson (SP) system.
4For instance, adding a 2-boson self-interaction to the equations of motion expands the parameter space of models, because in addition to m we gain another parameter to describe this self-interaction. For example, an effective contact interaction in the limit of vanishing energy can be modeled with a constant coupling strength g, yielding a term
5In de Broglie’s wave-particle dualism, v is understood as the group velocity of the stationary wave “assigned” to a particle, i.e., the velocity of this particle is the group velocity of that wave.
6Just like for CDM, the “dynamic temperature” of BEC-DM increases, once originally uncollapsed matter finds its way into post-collapse halos.
7It would be more meaningful to assign
8Analytical approximations to the numerically calculated soliton profile, like a Gaussian profile or an
9Indeed, even in gravity-free laboratory settings, it has been known that the BEC state remains robust, in spite of splitting the condensate, or upon expansion of the condensate that increases even the interparticle separation beyond
10See (Rindler-Daller and Shapiro, 2012), where
11De Broglie was aware of the fact that this marriage was not relativistically invariant, prompting his actual conception of the particle’s stationary wave. Since
12Sometimes these quantities are defined in terms of
13The equivalence of these pictures has been questioned in (Wallstrom, 1994), but that claim has been shown to be invalid in (Hushwater, 2010).
14See e.g., (Bovy and Tremaine, 2012) who estimate
15(Bovy and Rix, 2013) estimate the profile for the DM density at the solar circle
16The results by (Marsh and Niemeyer, 2019) and (Nadler et al., 2021) suggest that our fiducial choice is already ruled out, but detailed contraints of factors of a few are not so decisive for the general arguments made here.
17It is just that the Jeans scale of CDM is much smaller; the smallest CDM subhalos form near the thermal cutoff scale of density perturbations via monolithic collapse, while bigger halos have experienced mergers subsequently.
18Boson stars are the relativistic brethren of solitonic drops; they have higher densities and were originally conceived as potential compact objects to fill the gap between BHs and neutron stars. The limiting mass for solitons made up of bosons, above which they collapse to a BH has been derived in (Ruffini and Bonazzola, 1969), and it is given by
19The most extreme MACHOs are BHs: The Schwarzschild radius of a BH with mass M in fiducial units is
20However, the point-mass approximation becomes worse for solitons of larger size, not just for obvious reasons, but also for the fact that solitonic drops have no compact support; this is why
21All papers discussed in this section share the assumption that plane-wave approximations for the wave function can be used. This is justified, so long as the spacetime curvature does not vary significantly over a region of the size of the spatial spread of the (quantum) particle.
22In the past, such differences with respect to oscillation amplitudes have been also noticed for complex boson stars versus real oscillons.
23A related problem has been considered in (Allali and Hertzberg, 2020; Allali and Hertzberg, 2021), where two DM particles form a Schrödinger-cat-like system, interacting with an incoming baryonic particle. However, the case of a macroscopic BEC-DM system is not covered by their analysis, though a rough estimate suggests that the decoherence rate for boson stars is very rapid, for all particle masses of interest here. This reinforces earlier work by (Guth et al., 2015), which concludes that such boson stars behave “classical enough” to justify an effective field description.
References
Alcock, C., Allsman, R. A., Alves, D. R., Axelrod, T. S., Becker, A. C., Bennett, D. P., et al. (2001). MACHO Project Limits on Black Hole Dark Matter in the 1-30 M⊙ Range. ApJ 550, L169–L172. doi:10.1086/319636
Allali, I., and Hertzberg, M. P. (2020). Gravitational Decoherence of Dark Matter. J. Cosmol. Astropart. Phys. 2020, 056. doi:10.1088/1475-7516/2020/07/056
Allali, I. J., and Hertzberg, M. P. (2021). General Relativistic Decoherence with Applications to Dark Matter Phenomenology. arXiv e-prints: arXiv:2103.15892.
Amorisco, N. C., and Loeb, A. (2018). First Constraints on Fuzzy Dark Matter from the Dynamics of Stellar Streams in the Milky Way. arXiv e-prints: arXiv:1808.00464.
Armengaud, E., Palanque-Delabrouille, N., Yèche, C., Marsh, D. J. E., and Baur, J. (2017). Constraining the Mass of Light Bosonic Dark Matter Using SDSS Lyman-α forest. MNRAS 471, 4606–4614. doi:10.1093/mnras/stx1870
Arzoumanian, Z., Baker, P. T., Blumer, H., Bécsy, B., Brazier, A., Brook, P. R., et al. (2020). The NANOGrav 12.5 Yr Data Set: Search for an Isotropic Stochastic Gravitational-Wave Background. ApJ 905, L34. doi:10.3847/2041-8213/abd401
Bahcall, J. N., Hut, P., and Tremaine, S. (1985). Maximum Mass of Objects that Constitute Unseen Disk Material. ApJ 290, 15–20. doi:10.1086/162953
Bai, Y., Long, A. J., and Lu, S. (2020). Tests of Dark MACHOs: Lensing, Accretion, and Glow. J. Cosmol. Astropart. Phys. 2020, 044. doi:10.1088/1475-7516/2020/09/044
Bar, N., Blas, D., Blum, K., and Sibiryakov, S. (2018). Galactic Rotation Curves versus Ultralight Dark Matter: Implications of the Soliton-Host Halo Relation. Phys. Rev. D 98, 083027. doi:10.1103/PhysRevD.98.083027
Bar-Or, B., Fouvry, J.-B., and Tremaine, S. (2019). Relaxation in a Fuzzy Dark Matter Halo. ApJ 871, 28. doi:10.3847/1538-4357/aaf28c
Binney, J., and Tremaine, S. (2008). Galactic Dynamics. Second Edition. Princeton: Princeton University Press.
Bovy, J., and Rix, H.-W. (2013). A Direct Dynamical Measurement of the Milky Way's Disk Surface Density Profile, Disk Scale Length, and Dark Matter Profile at 4 Kpc ≲R≲ 9 Kpc. ApJ 779, 115. doi:10.1088/0004-637X/779/2/115
Bovy, J., and Tremaine, S. (2012). On the Local Dark Matter Density. ApJ 756, 89. doi:10.1088/0004-637X/756/1/89
Braaten, E., Mohapatra, A., and Zhang, H. (2018). Classical Nonrelativistic Effective Field Theories for a Real Scalar Field. Phys. Rev. D 98, 096012. doi:10.1103/PhysRevD.98.096012
Brandt, T. D. (2016). Constraints on MACHO Dark Matter from Compact Stellar Systems in Ultra-faint Dwarf Galaxies. ApJ 824, L31. doi:10.3847/2041-8205/824/2/L31
Brown, A. G. A. (2021). Microarcsecond Astrometry: Science Highlights from Gaia. arXiv e-prints: arXiv:2102.11712.
Calabrese, E., and Spergel, D. N. (2016). Ultra-light Dark Matter in Ultra-faint dwarf Galaxies. Mon. Not. R. Astron. Soc. 460, 4397–4402. doi:10.1093/mnras/stw1256
Chavanis, P.-H. (2019a). Derivation of the Core Mass-Halo Mass Relation of Fermionic and Bosonic Dark Matter Halos from an Effective Thermodynamical Model. Phys. Rev. D 100, 123506. doi:10.1103/PhysRevD.100.123506
Chavanis, P.-H. (2011). Mass-radius Relation of Newtonian Self-Gravitating Bose-Einstein Condensates with Short-Range Interactions. I. Analytical Results. Phys. Rev. D 84, 043531. doi:10.1103/PhysRevD.84.043531
Chavanis, P.-H. (2019b). Predictive Model of BEC Dark Matter Halos with a Solitonic Core and an Isothermal Atmosphere. Phys. Rev. D 100, 083022. doi:10.1103/PhysRevD.100.083022
Church, B. V., Mocz, P., and Ostriker, J. P. (2019). Heating of Milky Way Disc Stars by Dark Matter Fluctuations in Cold Dark Matter and Fuzzy Dark Matter Paradigms. MNRAS 485, 2861–2876. doi:10.1093/mnras/stz534
Corianò, C., and Frampton, P. H. (2020). Does CMB Distortion Disfavour Intermediate Mass Dark Matter?. arXiv e-prints: arXiv:2012.13821.
Croon, D., McKeen, D., and Raj, N. (2020a). Gravitational Microlensing by Dark Matter in Extended Structures. Phys. Rev. D 101, 083013. doi:10.1103/PhysRevD.101.083013
Croon, D., McKeen, D., Raj, N., and Wang, Z. (2020b). Subaru-HSC through a Different Lens: Microlensing by Extended Dark Matter Structures. Phys. Rev. D 102, 083021. doi:10.1103/PhysRevD.102.083021
Davidson, S., and Schwetz, T. (2016). Rotating Drops of Axion Dark Matter. Phys. Rev. D 93, 123509. doi:10.1103/PhysRevD.93.123509
Dawoodbhoy, T., Shapiro, P. R., and Rindler-Daller, T. (2021). Core-Envelope Haloes in Scalar Field Dark Matter with Repulsive Self-Interaction: Fluid Dynamics Beyond the de Broglie Wavelength. arXiv e-prints. arXiv:2104.07043.
De Martino, I., Broadhurst, T., Tye, S.-H. H., Chiueh, T., Schive, H.-Y., and Lazkoz, R. (2017). Recognizing Axionic Dark Matter by Compton and de Broglie Scale Modulation of Pulsar Timing. Phys. Rev. Lett. 119, 221103. doi:10.1103/PhysRevLett.119.221103
Erken, O., Sikivie, P., Tam, H., and Yang, Q. (2012). Cosmic Axion Thermalization. Phys. Rev. D 85, 063520. doi:10.1103/PhysRevD.85.063520
Fairbairn, M., Marsh, D. J. E., and Quevillon, J. (2017). Searching for the QCD Axion with Gravitational Microlensing. Phys. Rev. Lett. 119, 021101. doi:10.1103/PhysRevLett.119.021101
Freese, K., Fields, B., and Graff, D. (2000). Limits on Stellar Objects as the Dark Matter of Our Halo: Nonbaryonic Dark Matter Seems to Be Required. Nucl. Phys. B Proc. Supplements 80, 3.
Goodman, J. (2000). Repulsive Dark Matter. New Astron. 5, 103–107. doi:10.1016/S1384-1076(00)00015-4
Grin, D., Amin, M. A., Gluscevic, V., Hlozek, R., Marsh, D. J. E., Poulin, V., et al. (2019). Gravitational Probes of Ultra-light Axions. BAAS 51, 567.
Guth, A. H., Hertzberg, M. P., and Prescod-Weinstein, C. (2015). Do dark Matter Axions Form a Condensate with Long-Range Correlation?. Phys. Rev. D 92, 103513. doi:10.1103/PhysRevD.92.103513
Guzmán, F. S., and Ureña-López, L. A. (2004). Evolution of the Schrödinger-Newton System for a Self-Gravitating Scalar Field. Phys. Rev. D 69, 124033. doi:10.1103/PhysRevD.69.124033
Hartman, Z. D., and Lépine, S. (2020). The SUPERWIDE Catalog: A Catalog of 99,203 Wide Binaries Found in Gaia and Supplemented by the SUPERBLINK High Proper Motion Catalog. ApJS 247, 66. doi:10.3847/1538-4365/ab79a6
Helfer, A. D. (2018). Quantum Measurement and Fuzzy Dark Matter. Phys. Rev. D 98, 065015. doi:10.1103/PhysRevD.98.065015
Hlozek, R., Grin, D., Marsh, D. J. E., and Ferreira, P. G. (2015). A Search for Ultralight Axions Using Precision Cosmological Data. Phys. Rev. D 91, 103512. doi:10.1103/PhysRevD.91.103512
Hu, W., Barkana, R., and Gruzinov, A. (2000). Fuzzy Cold Dark Matter: The Wave Properties of Ultralight Particles. Phys. Rev. Lett. 85, 1158–1161. doi:10.1103/PhysRevLett.85.1158
Hui, L., Ostriker, J. P., Tremaine, S., and Witten, E. (2017). Ultralight Scalars as Cosmological Dark Matter. Phys. Rev. D 95, 043541. doi:10.1103/PhysRevD.95.043541
Hushwater, V. (2010). Comment on “Inequivalence between the Schrodinger Equation and the Madelung Hydrodynamic Equations”. arXiv e-prints: arXiv:1005.2420.
Iršič, V., Viel, M., Haehnelt, M. G., Bolton, J. S., and Becker, G. D. (2017). First Constraints on Fuzzy Dark Matter from Lyman-α forest Data and Hydrodynamical Simulations. Phys.Rev.Lett. 119, 031302.
Khmelnitsky, A., and Rubakov, V. (2014). Pulsar Timing Signal from Ultralight Scalar Dark Matter. J. Cosmol. Astropart. Phys. 2014, 019. doi:10.1088/1475-7516/2014/02/019
Lacey, C. G., and Ostriker, J. P. (1985). Massive Black Holes in Galactic Halos? ApJ 299, 633–652. doi:10.1086/163729
Lancaster, L., Giovanetti, C., Mocz, P., Kahn, Y., Lisanti, M., and Spergel, D. N. (2020). Dynamical Friction in a Fuzzy Dark Matter Universe. J. Cosmol. Astropart. Phys. 2020, 001. doi:10.1088/1475-7516/2020/01/001
Lesgourgues, J., Arbey, A., and Salati, P. (2002). A Light Scalar Field at the Origin of Galaxy Rotation Curves. New Astron. Rev. 46, 791–799. doi:10.1016/s1387-6473(02)00247-6
Li, Z., Shen, J., and Schive, H.-Y. (2020). Testing the Prediction of Fuzzy Dark Matter Theory in the Milky Way center. ApJ 889, 88. doi:10.3847/1538-4357/ab6598
Lora, V., Magaña, J., Bernal, A., Sánchez-Salcedo, F. J., and Grebel, E. K. (2012). On the Mass of Ultra-light Bosonic Dark Matter from Galactic Dynamics. J. Cosmol. Astropart. Phys. 2012, 011. doi:10.1088/1475-7516/2012/02/011
Magaña, J., and Matos, T. (2012). A Brief Review of the Scalar Field Dark Matter Model. J. Phys. Conf. Ser. 378, 012012. doi:10.1088/1742-6596/378/1/012012
Marsh, D. J. E., and Niemeyer, J. C. (2019). Strong Constraints on Fuzzy Dark Matter from Ultrafaint Dwarf Galaxy Eridanus II. Phys. Rev. Lett. 123, 051103. doi:10.1103/PhysRevLett.123.051103
Matos, T., Guzmán, F. S., and Ureña-López, L. A. (2000). Scalar Field as Dark Matter in the Universe. Class. Quan. Grav. 17, 1707–1712. doi:10.1088/0264-9381/17/7/309
Meingast, S., and Alves, J. (2019). Extended Stellar Systems in the Solar Neighborhood. A&A 621, L3. doi:10.1051/0004-6361/201834622
Membrado, M., Pacheco, A. F., and Sañudo, J. (1989). Hartree Solutions for the Self-Yukawian Boson Sphere. Phys. Rev. A. 39, 4207–4211. doi:10.1103/PhysRevA.39.4207
Mocz, P., Fialkov, A., Vogelsberger, M., Becerra, F., Shen, X., Robles, V. H., et al. (2020). Galaxy Formation with BECDM - II. Cosmic Filaments and First Galaxies. MNRAS 494, 2027–2044. doi:10.1093/mnras/staa738
Mocz, P., Lancaster, L., Fialkov, A., Becerra, F., and Chavanis, P.-H. (2018). Schrödinger-Poisson-Vlasov-Poisson Correspondence. Phys. Rev. D 97, 083519. doi:10.1103/PhysRevD.97.083519
Mocz, P., Vogelsberger, M., Robles, V. H., Zavala, J., Boylan-Kolchin, M., Fialkov, A., et al. (2017). Galaxy Formation with BECDM - I. Turbulence and Relaxation of Idealized Haloes. MNRAS 471, 4559–4570. doi:10.1093/mnras/stx1887
Nadler, E. O., Drlica-Wagner, A., Bechtol, K., Mau, S., Wechsler, R. H., Gluscevic, V., et al. (2021). Constraints on Dark Matter Properties from Observations of Milky Way Satellite Galaxies. Phys. Rev. Lett. 126, 091101. doi:10.1103/PhysRevLett.126.091101
Navarro, J. F., Frenk, C. S., and White, S. D. M. (1996). The Structure of Cold Dark Matter Halos. ApJ 462, 563. doi:10.1086/177173
Niemeyer, J. C. (2020). Small-scale Structure of Fuzzy and Axion-like Dark Matter. Prog. Part. Nucl. Phys. 113, 103787. doi:10.1016/j.ppnp.2020.103787
Padilla, L. E., Rindler-Daller, T., Shapiro, P. R., Matos, T., and Vázquez, J. A. (2021). Core-halo Mass Relation in Scalar Field Dark Matter Models and its Consequences for the Formation of Supermassive Black Holes. Phys. Rev. D 103, 063012. doi:10.1103/PhysRevD.103.063012
Porayko, N. K., Zhu, X., Levin, Y., Hui, L., Hobbs, G., Grudskaya, A., et al. (2018). Parkes Pulsar Timing Array Constraints on Ultralight Scalar-Field Dark Matter. Phys. Rev. D 98, 102002. doi:10.1103/PhysRevD.98.102002
Rindler-Daller, T., and Shapiro, P. R. (2012). Angular Momentum and Vortex Formation in Bose-Einstein-Condensed Cold Dark Matter Haloes. MNRAS 422, 135–161. doi:10.1111/j.1365-2966.2012.20588.x
Rindler-Daller, T., and Shapiro, P. R. (2014). Complex Scalar Field Dark Matter on Galactic Scales. Mod. Phys. Lett. A. 29, 1430002. doi:10.1142/S021773231430002X
Robles, V. H., Bullock, J. S., and Boylan-Kolchin, M. (2019). Scalar Field Dark Matter: Helping or Hurting Small-Scale Problems in Cosmology?. MNRAS 483, 289–298. doi:10.1093/mnras/sty3190
Ruffini, R., and Bonazzola, S. (1969). Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State. Phys. Rev. 187, 1767–1783. doi:10.1103/PhysRev.187.1767
Safarzadeh, M., and Spergel, D. N. (2020). Ultra-light Dark Matter Is Incompatible with the Milky Way's Dwarf Satellites. ApJ 893, 21. doi:10.3847/1538-4357/ab7db2
Schiappacasse, E. D., and Hertzberg, M. P. (2018). Analysis of Dark Matter Axion Clumps with Spherical Symmetry. J. Cosmol. Astropart. Phys. 2018, 037. doi:10.1088/1475-7516/2018/01/037
Schive, H.-Y., Chiueh, T., and Broadhurst, T. (2014a). Cosmic Structure as the Quantum Interference of a Coherent Dark Wave. Nat. Phys 10, 496–499. doi:10.1038/nphys2996
Schive, H.-Y., Liao, M.-H., Woo, T.-P., Wong, S.-K., Chiueh, T., Broadhurst, T., et al. (2014b). Understanding the Core-Halo Relation of Quantum Wave Dark Matter from 3D Simulations. Phys. Rev. Lett. 113, 261302. doi:10.1103/PhysRevLett.113.261302
Schobesberger, S. O., Rindler-Daller, T., and Shapiro, P. R. (2021). Angular Momentum and the Absence of Vortices in the Cores of Fuzzy Dark Matter Haloes. MNRAS (1), 505, 802–829. doi:10.1093/mnras/stab1153
Schwabe, B., Niemeyer, J. C., and Engels, J. F. (2016). Simulations of Solitonic Core Mergers in Ultralight Axion Dark Matter Cosmologies. Phys. Rev. D 94, 043513. doi:10.1103/PhysRevD.94.043513
Seidel, E., and Suen, W.-M. (1994). Formation of Solitonic Stars through Gravitational Cooling. Phys. Rev. Lett. 72, 2516–2519. doi:10.1103/PhysRevLett.72.2516
Sikivie, P., and Yang, Q. (2009). Bose-Einstein Condensation of Dark Matter Axions. Phys. Rev. Lett. 103, 111301. doi:10.1103/PhysRevLett.103.111301
Sin, S.-J. (1994). Late-time Phase Transition and the Galactic Halo as a Bose Liquid. Phys. Rev. D 50, 3650–3654. doi:10.1103/PhysRevD.50.3650
Steininger, F. (2018). Pulsar Timing Signal from Complex Scalar Field Dark Matter. Vienna: University of Vienna: Bachelor Thesis, supervised by T.Rindler-Daller.
Tan, K. C., and Jeong, H. (2018). Entanglement as the Symmetric Portion of Correlated Coherence. Phys. Rev. Lett. 121, 220401. doi:10.1103/PhysRevLett.121.220401
Tian, H.-J., El-Badry, K., Rix, H.-W., and Gould, A. (2020). The Separation Distribution of Ultrawide Binaries across Galactic Populations. ApJS 246, 4. doi:10.3847/1538-4365/ab54c4
Tisserand, P., Le Guillou, L., Afonso, C., Albert, J. N., Andersen, J., Ansari, R., et al. (2007). Limits on the Macho Content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds. A&A 469, 387–404. doi:10.1051/0004-6361:20066017
Ureña-López, L. A. (2019). Brief Review on Scalar Field Dark Matter Models. Front. Astron. Space Sci. 6, 47. doi:10.3389/fspas.2019.00047
Keywords: cosmology, Bose-Einstein-condensed dark matter, galactic halos, Milky Way, dark matter substructure, quantum measurement
Citation: Rindler-Daller T (2021) To Observe, or Not to Observe, Quantum-Coherent Dark Matter in the Milky Way, That is a Question. Front. Astron. Space Sci. 8:697140. doi: 10.3389/fspas.2021.697140
Received: 18 April 2021; Accepted: 15 June 2021;
Published: 02 July 2021.
Edited by:
Paolo Salucci, International School for Advanced Studies (SISSA), ItalyReviewed by:
Luis Arturo Urena-Lopez, University of Guanajuato, MexicoPierre-Henri Chavanis, Laboratoire de Physique Theorique Toulouse, France
Copyright © 2021 Rindler-Daller. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Tanja Rindler-Daller, tanja.rindler-daller@univie.ac.at