Quel Nouveau Mond(e) nous attend à l'échelle de la Voie Lactée et au delà ?

A flash review on the Dark Matter versus Mond debate in cosmology and astrophysics

The achievements of both the standard model (or theory) of particle physics and of the standard (or concordance) model of cosmology ranging from yocto to hundreds of yotta meters must not make us forget that our understanding of physics is still patchy - particularly at intermediate astrophysical scales. Here is a brief but informative summary:

The Dark Matter (DM) paradigm has been remarkably successful at explaining various large-scale observations. The expansion history, the detailed shape of the peaks in the cosmic microwave background (CMB) anisotropy power spectrum, the growth history of linear perturbations and the shape of the matter power spectrum are all consistent with a non-baryonic, clustering component making up ∼ 25% of the total energy budget. Although this is usually hailed as evidence for weakly interacting particles, one should keep in mind that these large-scale observations only rely on the hydrodynamical limit of the dark component. Any perfect fluid with small equation of state (w ' 0) and sound speed (cs ' 0), and with negligible interactions with ordinary matter, would do equally well at fitting cosmological observations on linear scales. 

On non-linear scales, the evidence for DM particles is somewhat less convincing. N-body simulations reveal that DM particles self-assemble into halos with a universal density profile, the NFW profile [1]: ρ_NFW(r) = ρS/[(r/rS)(1+ r/rS) ²]. (1)
The density thus scales as ∼r^-1 in the interior, and asymptotes to ∼r^-3 on the outskirts. The regularity of DM self-assembly is certainly a welcome feature. Unfortunately, the NFW profile does not naturally account for flat rotation curves of spiral galaxies and the isothermality of galaxy clusters, both of which require ρ∼r^-2. The cold dark matter paradigm also faces challenges on small scales, for instance the cuspiness of galactic cores [2], the mass [4] and phase-space distributions [5–8] of satellite galaxies, and the internal dynamics of tidal dwarfs [9–11]. Of course, N-body simulations do not include baryons, so the NFW profile is not expected to hold exactly in the real universe. But the fact that the “zeroth-order” profile does not readily explain the coarse features of galaxies and clusters of galaxies should at least give us pause. The empirical success or failure of DM particles hinges ultimately on complex baryonic feedback processes. 

Quantifying the impact of baryonic physics is an area of active research, but simulations do not yet offer a clear picture. Even qualitative questions, such as whether baryons make the DM profile more cuspy or shallower in the core of galaxies, are still hotly debated [2]. In the absence of a precise answer, the best one can do when fitting data is incorporate baryonic expectations (e.g., adiabatic contraction [12, 13]) through empirical modifications of the NFW profile. Examples include the generalized NFW profile, cored NFW profile, Buckert profile [14], etc. See [15] for a recent comparison of how these fare at fitting galactic rotation curves. 

Meanwhile, despite the complexity of baryonic physics, actual structures in our universe show a remarkable level of regularity, embodied in empirical scaling relations. A famous example is the Tully-Fisher relation [16], which relates the luminosity of spiral galaxies to the asymptotic velocity v∞ of their rotation curves: L∼v_∞⁴... Another example is the Faber-Jackson relation [17] for elliptical galaxies L∼σ⁴, where σ is the stellar velocity dispersion. These relations are quite puzzling from the particle DM perspective — why should the rotational velocity in the galactic tail where DM completely dominates be so tightly correlated with the baryonic mass in the inner region? The hope is that these scaling relations will eventually emerge somehow from realistic simulations of coupled baryons and dark matter

MOdified Newtonian Dynamics (MOND) is a radical alternative proposal [18–20]. It attempts to replace dark matter entirely with a modified gravitational force law that kicks in once the acceleration drops to a critical value a₀ : a=a_N  if a₀≪a_N and a=√(a₀a_N) if a_N≪a₀ ... where a_N=G_NM(r)/r² is the standard Newtonian acceleration. By construction, the MOND force law accounts both for the flat rotation curves of spiral galaxies and the Tully-Fisher relation. Indeed, in the MOND regime the acceleration of a test particle orbiting a spiral galaxy satisfies v²/r = √(G_NMa₀/r²), hence v⁴= G_NMa₀ . This matches  the Tully-Fisher relation with M ∼ L... 

An intriguing fact is that the best-fit value for the characteristic acceleration is comparable to the Hubble parameter: a₀^galaxies≈ 1/6×H₀≈1.2 × 10^-8 cm/s².
The MOND force law has been remarkably successful at explaining a wide range of galactic phenomena, from dwarf galaxies to ellipticals to spirals. See [21, 24] for comprehensive reviews. It explains the observed upper limit on the surface brightness of spirals, known as Freeman’s law [25], the characteristic surface brightness in ellipticals, known as the Fish law [26], as well as the FaberJackson law for ellipticals mentioned earlier. Even if DM particles do exist and gravity is standard, Milgrom’s scaling relation (3) should nonetheless be viewed on the same footing as the Tully-Fisher and Faber-Jackson relations. It is a powerful empirical relation that must be explained by standard theories of galaxy formation. 

Unfortunately, the empirical success of MOND is limited to galaxies. On cluster scales, the MOND force law fails miserably [27]. The baryonic component in clusters is dominated by gas, which to a good approximation is in hydrostatic equilibrium and in the MONDian regime. Hydrostatic equilibrium determines the temperature profile T(r) in terms of the observed density profile ρ(r) and the (MONDian) acceleration law a(r). The result does not match the observed isothermal profile of clusters... MOND proponents are forced to assume dark matter, usually in the form of massive neutrinos with mν∼2 eV [30–32] and/or cold (∼ 3K), dense gas clouds [33].  

On cosmological scales, the MOND law requires a relativistic completion. This was achieved just over ten years by Sanders and Bekenstein with a Tensor-Vector-Scalar (TeVeS) theory [34–36]. See [37] for an elegant reformulation of the theory, and [38, 39] for connections to Einstein-aether theories [40]. (Since TeVeS, other relativistic extensions have been proposed [41–44]. See [45] for a review.) First, some good news: perturbations in the vector field accelerate the growth of density perturbations, which allows for the formation of structures. More problematic is the CMB spectrum. An early analysis already revealed some tensions with the height of the third peak [46], and one would expect that the situation is now much worse with the exquisite data at higher multipoles from the Planck satellite [47] and ground-based experiments [48, 49] significant dark matter component, the baryonic oscillations in the matter power spectrum tend to be far too pronounced [46, 51]. Finally, numerical simulations of MONDian gravity with massive neutrinos fail to reproduce the observed cluster mass function [52, 53]. 

To summarize, the Cold Dark Matter (CDM) picture is very successful on linear scales, but the jury is still out as to whether it can explain the detailed structure of galaxies and their empirical scaling relations. MOND, on the other hand, is very successful on galactic scales, but it seems highly improbable that it can ever be made consistent with the detailed shape of the CMB and matter power spectra. 

An Alternative to Particle Dark Matter

Justin Khoury

(Submitted on 29 Aug 2014 (v1), last revised 11 Dec 2014 (this version, v3))

A snapshot of a hypothetical hybridization based on a superfluid analogy

Now let us make a risky step forward in these murky territories all filled with promises and traps. The following is of course highly speculative but I have found it interesting for its use of the superfluid analogy in a quite simple way (but sophisticated motivation) and most importantly for providing a rich list of potential observational implications. 

What we have learned is that MOND and CDM are each successful in almost mutually exclusive regimes... This has led various people to propose hybrid models that include both DM and MOND phenomena [75-79-82]. For instance, one of us recently proposed such a hybrid model, involving two scalar fields [83]: one scalar field acts as DM, the other mediates a MOND-like force law. This model enjoys a number of advantages compared to TeVeS and other relativistic MOND theories. For starters, it only requires two scalar fields, as opposed to the scalar and vector fields of TeVeS. Secondly, unlike TeVeS, its predictions on cosmological scales are consistent with observations, thanks to the DM scalar field. Finally, the model offers a better fit to the temperature profile of galaxy clusters. 

The improved consistency with data does come at the price of having two a priori distinct components — a DM-like component and a modified-gravity component. It would be much more compelling if these two components somehow had a common origin. Furthermore, the theory must be adjusted such as to avoid co-existence of DM-like and MOND-like behavior. This requires that the parameters of the theory be mildly scale or mass dependent, which adds another layer of complexity 

In this paper, along with its shorter companion [84], we propose a unified framework for the DM and MOND phenomena. The DM and MOND components have a common origin, representing different phases of a single underlying substance. This is achieved through the rich and well-studied physics of superfluidity.  

Our central idea is that DM forms a superfluid inside galaxies, with a coherence length of order the size of galaxies. As a back-of-the-envelope calculation, we can estimate the condition for the onset of superfluidity by ignoring interactions among DM particles. With this simplifying approximation, the requirement for superfluidity amounts to demanding that the de Broglie wavelength λdB ∼ 1/mv of DM particles should overlap. Using the typical velocity v and density of DM particles.in galaxies, this translates into an upper bound m <2 eV on the DM particle mass.  

Another requirement for Bose-Einstein condensate is that DM thermalize within galaxies. We assume that DM particles interact through contact repulsive interactions. Demanding that the interaction rate be larger than the galactic dynamical time places a lower bound of σ/m>0.1cm²/g. This is just below the most recent constraint <0.5cm²/g from galaxy cluster mergers [85], though we will argue such constraints must be carefully reanalyzed in the superfluid context.  

Again ignoring interactions, the critical temperature for DM superfluidity is Tc∼mK, which intriguingly is comparable to known critical temperatures for cold atom gases, e.g., ⁷Li atoms have Tc≈0.2 mK. We will see that cold atoms provide more than just a useful analogy — in many ways, our DM component behaves exactly like cold atoms. In cold atom experiments, atoms are trapped using magnetic fields; in our case, it is gravity that attracts DM particles in galaxies.  

The superfluid nature of DM dramatically changes its macroscopic behavior in galaxies. Instead of behaving as individual collisionless particles, the DM is more aptly described as collective excitations: phonons and massive quasi-particles. Phonons, in particular, play a key role by mediating a long-range force between ordinary matter particles. As a result, a test particle orbiting the galaxy is subject to two forces: the (Newtonian) gravitational force and the phonon-mediated force. Our postulate is that the phonon-mediated force is MONDian, such that the DM superfluid reproduces the empirical success of MOND in galaxies.   

Specifically, it is well-known that the effective field theory (EFT) of phonon excitations at lowest order in derivatives is in general a P(X) theory [86]. Our postulate is that DM phonons are described by the non-relativistic MOND scalar action, 

P(X) ∼ ΛX √|X|;    X = θ' − mΦ − (∇θ)²/2m . (4) 

where Λ ∼ meV to reproduce the MOND critical acceleration, and Φ is the gravitational potential [The possible connection between MOND and superfluidity was mentioned briefly by Milgrom in [87]...]. To mediate a force between ordinary matter, θ must couple to the baryon density: 

L_int ∼ (Λ/ M_Pl) θρ_b . (5) 

From a particle physics standpoint, such a coupling is fairly innocuous — it represents a soft explicit breaking of the global U(1) symmetry. In the superfluid interpretation, however, where θ is the phase of a wavefunction, this coupling picks out a preferred phase, which seems unphysical. One possibility is that (5) follows from baryons coupling to the vortex sector of the superfluid... , 

[In the quasi-static limit (θ'=0) our action ∼X^3/2 becomes invariant under time-dependent spatial Weyl transformations: h_ij→Ω²(x,t)h_ij [96, 97]. At lowest order in derivatives it is the unique action with this property. Intriguingly, the SO(4,1) global part of the 3d Weyl group coincides with the de Sitter isometry group, which hints at a deep connection between the MOND phenomenon and dark energy [97]]. The fractional 3/2 power would be strange if (4) described a fundamental scalar field. As a theory of phonons, however, it is not uncommon to see fractional powers in cold atom systems. For instance, the Unitary Fermi Gas (UFG) [91, 92], which has generated much excitement recently in the cold atom community, describes a gas of cold fermionic atoms tuned such that their scattering length diverges [93, 94]. The effective action for the UFG superfluid is uniquely fixed by 4d scale invariance at lowest-order in derivatives, L_UFG(X) ∼ X^5/2 , which is also non-analytic [95]...

As is familiar from liquid helium, a superfluid at finite temperature (but below the critical temperature) is best described phenomenologically as a mixture of two fluids [98–100]: i) the superfluid, which by definition has vanishing viscosity and carries no entropy; ii) the “normal” component, comprised of massive particles, which is viscous and carries entropy. The fraction of particles in the condensate decreases with increasing temperature. Thus our framework naturally distinguishes between galaxies (where MOND is successful) and galaxy clusters (where MOND is not). Galaxy clusters have a higher velocity dispersion and correspondingly higher DM temperature. For m∼ eV we find that galaxies are almost entirely condensed, whereas galaxy clusters are either in a mixed phase or entirely in the normal phase.  

Assuming hydrostatic equilibrium with P∼ρ³, the resulting DM halo density profile is cored, not surprisingly, and therefore avoids the cusp problem of CDM. Remarkably, for our parameter values (m∼eV, Λ∼ meV) the size of the condensate halo is ∼100 kpc for a galaxy of Milky-Way mass. In the inner region of galaxies where rotation curves are probed, the DM condensate has a negligible effect on baryonic particles, and their motion is dominated by the phonon-mediated MOND force. In the outer region probed by gravitational lensing, the DM condensate gives the dominant contribution to the force on a test particle.  

In the vicinity of individual stars the phonon effective theory breaks down and the correct description is in terms of normal DM particles. This is good news on two counts. First, it is well-known that the MONDian acceleration, while giving a small correction to Newtonian gravity in the solar system, is typically too large to conform to planetary orbital constraints. This usually requires introducing additional complications to the theory [101]. In our case, the MONDian behavior is avoided entirely in the solar system, as DM behaves as ordinary particles. The second piece of good news pertains to experimental searches of axion-like particles. By allowing the usual axion-like couplings to Standard Model operators, our DM particles can be detected through the suite of standard axion experiments, e.g., [102].

The idea of a Bose-Einstein DM condensate (BEC) in galaxies has been studied before [103, 104, 113–125].5 There are important differences with the present work. In BEC DM galactic dynamics are caused by the condensate density profile, similar to what happens in CDM, with phonons being irrelevant. In our case, phonons play a key role in generating flat rotation curves and explaining the BTFR. Moreover, the equation of state is different: the BEC DM is governed by two-body interactions and hence has P∼ρ², compared to ∼ρ³ in our case. This difference only has a minor effect on the condensate density profiles, but it does imply a different phonon sound speed. In particular, for the Bullet Cluster the sound speed in BEC DM is only cs ∼<100 km/s, i.e., more than an order of magnitude smaller than the bullet infall velocity. As a result dissipation is important, which puts BEC DM in tension with observations [129]. 

A Theory of Dark Matter Superfluidity

Lasha Berezhiani, Justin Khoury

(Submitted on 3 Jul 2015)

We conclude with some astrophysical implications of our DM superfluid: 

Vortices: When spun faster than a critical velocity, a superfluid develops vortices. The typical angular velocity of halos is well above critical [24], giving rise to an array of DM vortices permeating the disc [51]. It will be interesting to see whether these vortices can be detected through substructure lensing, e.g., with ALMA [52]. 

Galaxy mergers: A key difference with ΛCDM is the merger rate of galaxies. Applying Landau’s criterion, we find two possible outcomes. If the infall velocity vinf is less than the phonon sound speed cs (of order the viral velocity [24]), then halos will pass through each other with negligible dissipation, resulting in multiple encounters and a longer merger time. If vinf >cs, however, the encounter will excite DM particles out of the condensate, resulting in dynamical friction and rapid merger. 

Bullet Cluster: For merging galaxy clusters, the outcome also depends on the relative fraction of superfluid vs normal components in the clusters. For subsonic mergers, the superfluid cores should pass through each other with negligible friction (consistent with the Bullet Cluster), while the normal components should be slowed down by self interactions. Remarkably this picture is consistent with the lensing map of the Abell 520 “train wreck” [53– 56], which show lensing peaks coincident with galaxies (superfluid components), as well as peaks coincident with the X-ray luminosity peaks (normal components). 

Dark-bright solitons: Galaxies in the process of merging should exhibit interference patterns (so-called darkbright solitons) that have been observed in BECs counterflowing at super-critical velocities [57]. This can potentially offer an alternative mechanism to generate the spectacular shells seen around elliptical galaxies [58]. 

Globular clusters: Globular clusters are well-known to contain negligible amount of DM, and as such pose a problem for MOND [59]. In our case the presence of a significant DM component is necessary for MOND. If whatever mechanism responsible for DM removal in ΛCDM is also effective here, our model would predict DM-free (and hence MOND-free) globular clusters

Dark Matter Superfluidity and Galactic Dynamics

Lasha Berezhiani, Justin Khoury

(Submitted on 25 Jun 2015)

A last reminder on the experimental physics of the unitary Fermi gas 
After such wild quantum speculations let's landing on down to Earth tabletop experiments but still fascinating physics:

Ultracold Fermi gases are dilute systems with interparticle interactions that can be controlled through Feshbach resonances, which allow the access of strongly interacting regimes. Until recently, superfluids were classified as either Bardeen-Cooper-Schrieffer (BCS) states or the Bose-Einstein Condensate (BEC). In fact they are limit cases of a continuum of the interaction strength. The possibility of tuning the parameters to observe changes from one regime to the other is conceptually interesting, but real enthusiasm came from the experimental realization of the BCS-BEC crossover [1]. 

The three dimensional unitary Fermi gas is a strongly interacting system with short-range interactions of remarkable properties. When the scattering length a diverges, 1/ak_F →0 (k_F is the Fermi momentum of the system), the low-energy s-wave scattering phase shift is δ₀=π/2. The ground state energy per particle E₀ is proportional to the one of the noninteracting Fermi gas E_FG in a box: 

E₀=ξE_FG= ξ(3/10) ℏ²k_F²/M,  (1)  

where the constant ξ is known as the Bertsch parameter and M is the mass of the fermion. In the limit ak_F →∞, quantum Monte Carlo (QMC) results give the exact value of ξ = 0.372(5) [2], in agreement with experiments [3, 4]. 

One signature of superfluidity is the formation of quantized vortices. Since their first observations in superfluid ⁴He a large body of experimental and theoretical work has been carried out concerning bosonic systems [5–8]. On the other hand, the discovery of vortex lattices in a strongly interacting rotating Fermi gas of ⁶Li [9] was a milestone in the study of superfluidity in cold Fermi gases. 

A vortex line consists of an extended irrotational flow field, with a core region where the vorticity is concentrated. The quantization of the flow manifests itself in the quantized units h/2M of circulation. There is no evidence for quantized vortices with more than one unit of circulation. Many questions remain to be answered concerning the structure of the vortex core for fermions...

A vortex line in the unitary Fermi gas

Lucas Madeira, Silvio A. Vitiello, Stefano Gandolfi, Kevin E. Schmidt

(Submitted on 27 Oct 2015)