wave-Dark Matter

The wave nature of ultra-light scalar dark matter particles, with only the boson mass as a free parameter, has been shown to be capable of resolving some small-scale problems.

The ψ-Dark Matter Halo

Forms of scalar field dark matter, such as axions, if sufficiently light, can satisfy the ground state condition of the coupled Schrödinger–Poisson equations. Pioneering high resolution cosmological simulation of this wavelike dark matter (ψDM) has been capable of revealing unpredicted small-scale structures on the de Broglie scale. Employing only one free parameter, the boson mass, the ψDM model predicts the formation of solitonic cores in the innermost region of each virialized halo, which accounts for the dark matter-dominated cores of dwarf spheroidal galaxies. Moreover, the central soliton is surrounded by an extended halo showing a granular texture on the de Broglie scale. The functional form of the soliton density profile can be well approximated by (more details can be found in [1] and reference within)

Pulsars arrival time

Since the occupation number of dark matter particle in the galactic halo is enormous dark matter can be well described by a classical oscillating scalar field with Compton frequency

Such a Compton oscillation leads to an oscillating gravitational potential which could produce a timing shift in the pulsar timing measurements opening up the possibility of detecting an imprint of scalar field dark matter with the next generation of radio telescope and Pulsar Timing Array (PTA) experiments. Fixing the boson mass to

We predicted the amplitudes of the residuals due to the oscillating gravitational potential. The figure is particularized to show the relative timing signal between local pulsars at 8 kpc, and pulsars close to the galactic center at a radius of 50 and 500 pc.

The most important case is represented in top-left panel where the amplitude of the residuals, that is dominated by the pulsar closer to the galactic center, reaches ∼ 600 ns. When pulsars are located at the same galactocentric radius, the predicted amplitude strongly depends on the phase of the pulse. The light blue shaded regions indicate how the density fluctuations of the dark matter distribution enhance or reduce the relative timing signal depending whether the pulsar is located in higher or lower density region with respect the average distribution. Finally, we depicted the characteristic amplitude predicted between pairs of pulsars and a comparison with sensitivity of the current and forthcoming PTA detectors [1].

Dynamics of stars in the Milky Way's Bulge

A wavelike solution for the non-relativistic universal dark matter (wave-DM) is rapidly gaining interest, following pioneering simulations of cosmic structure as an interference pattern of coherently oscillating bosons. A prominent solitonic standing wave is predicted at the center of every galaxy, representing the ground state solution of the coupled Schrödinger-Poisson equations, and it has been identified with the wide, kpc scale dark cores of common dwarf-spheroidal galaxies. A denser soliton is predicted for Milky Way sized galaxies where momentum is higher, so the de Broglie scale of the soliton is smaller, ≃ 100 pc, of mass ≃ 109M☉ . We have shown that the central motion of bulge stars in the Milky Way implies the presence of such a dark core, where the velocity dispersion rises inversely with radius to a maximum of ≃ 130 km/s, corresponding to an excess central mass of ≃ 1.5 × 109M☉ within ≃ 100 pc, favoring a boson mass of ≃ 10-22 eV . This quantitative agreement with such a unique and distinctive prediction is therefore strong evidence for a light bosonic solution to the long standing Dark Matter puzzle [2].

Figure: Predicted dispersion velocity as function of Galactic latitude for a fixed longitudel= 0◦. Again our baseline model (dotted magenta curve), with no soliton,underpredictsthe central velocity dispersion. Our prediction is in good agreementwith the independentprediction of Portail et al. (2017) shown as light blue that includes a detailed simulation ofthe stellar bulge, central black hole and a disk and NFW halo for the Galaxy, and who alsoconclude on this basis the need for a significant additional mass of≃109M☉. The solid lines are our baseline model, and the dashed lines show the effect of includinga Plummer density distribution for a small fraction of the visible starscontributing he tocentral dispersion due to the presence of a soliton. population

Dynamics of stars in Antlia II

The large dark cores of common dwarf galaxies are unexplained by the standard heavy particle interpretation of dark matter. This puzzle is exacerbated by the discovery of a very large but barely visible, dark matter dominated galaxy Antlia II orbiting the Milky Way, uncovered by tracking star motions with the Gaia satellite. Although Antlia II has a low mass, its visible radius is more than double any known dwarf galaxy, with an unprecedentedly low density core. We show that Antlia II favors dark matter as a Bose-Einstein condensate, for which the ground state is a stable soliton with a core radius given by the de Broglie wavelength. The lower the galaxy mass, the larger the de Broglie wavelength, so the least massive galaxies should have the widest soliton cores of lowest density. An ultralight boson of mψ∼1.1 ×10-22 eV [3] accounts well for the large size and slowly moving stars within Antlia II (left panel in the Figure below) and agrees with boson mass estimates derived from the denser cores of more massive dwarf galaxies (right panel in the Figure below).

2D marginalized contours of the model parameters [mψ,Mhalo] obtained from our MCMC analysis of Antlia II.The 68%, 95% and 99% confidence levels are shown with the 1D marginalized likelihood distribution.

Joint likelihood constraint on the boson mass fromour Antlia II analysis (solid blue line), together with previous published work on Fornax (dotted red), Sextans (dashed cyan), Sculptor (dot dashed magenta), Draco (triple dotteddashed green). The final joint probability distribution for theboson mass is shown as the black solid curve

The ψ-Dark Matter and the Ultra Diffuse Galaxies

We extend these ``wave dark matter" (ψDM) predictions to the newly discovered class of ``Ultra Diffuse Galaxies" that resemble dwarf spheroidal galaxies but with more extended stellar profiles. Currently the best studied example, DF44, has a uniform velocity dispersion of about 33 km/s, extending to at least 3 kpc, that we show is reproduced by our $\psi$DM simulations with a soliton radius of about 0.5 kpc. In the ψDM context, we show the relatively flat dispersion profile of DF44 lies between massive galaxies with compact dense solitons [4], as may be present in the Milky Way on a scale of 100 pc and lower mass galaxies where the velocity dispersion declines centrally within a wide, low density soliton, like Antlia II, of radius 3 kpc.

The best fit soliton + NFW model and its one-sigma deviation iscalculated by randomly sampling from the distributions of velocity disper-sion at each radius.

Correlated distributions of inferred parameters: core mass, half-light mass and halo mass from MCMC simulation. Note, the 1D and 2D posterior distributions of four UFDs taken from MCMC chains using emcee

Heating of the ψ-Dark Matter halo

We investigate whether the oblate, spheroidal morphology of common dwarf spheroidal galaxies (dSph) may result from the slow relaxation of stellar orbits within a halo of Wave Dark Matter (ψDM) when starting from an initial disk of stars [5]. Stellar orbits randomly walk over a Hubble time, perturbed by the pervasive "granular" interference pattern of ψDM, that fully modulates the dark matter density on the de Broglie scale. Our simulations quantify the level of stellar disk thickening over the Hubble time, showing that the distribution of stars is predicted to become an oblate spheroid of increasing radius, which plausibly accounts for the morphology of dSph galaxies.

Figure: The density field that we obtained as a result of our merger simulation. Left panel: we report a slice on the plane 𝑧 = 𝐿/2 of the squared norm of the wavefunction 𝜓. The colour map corresponds to the density values in the logarithmic scale. The red and green circles are centered on the central solitonic overdensity of the halo and their radii correspond to the core radius 𝑟𝑐 (red circle) and 3.5𝑟𝑐. The latter coincides with the points where the solitonic profile breaks and the halo is dominated by 𝜓DM fluctuations. Right panel: We report the radial density profile of the halo in a logarithmic scale (green triangles). The dashed black line reports the best-fitting soliton profile in the inner regions of the halo. Vertical dotted lines report the best-fitting core radius 𝑟𝑐 and 3.5 times this value. For greater radii, the density profile of our simulation departs from the solitonic profile and follows an NFW-like (red dotted line).

Video: The animation corresponding the simulation N = 512 X 512 X 512 used in the paper. Credits by R. Della Monica https://github.com/rdellamonica/wavedm-dwarfs

psidm.mp4

We predict a low level of residual rotation remains after a Hubble time at the 1-3 km/s level, depending on orientation, that compares with recent claims of rotation for some well-studied local dSph galaxies. This steady internal dynamical evolution may be witnessed directly with JWST for well-resolved dwarf galaxies, appearing more oblate with look back time and tending to small disks of young stars at high redshift.

Figure: Stellar population positions on the 𝑥 − 𝑧 plane after 1 (blue), 3 (green) and 10 Gyrs (red) of evolution. The dots correspond to the positions of the 104 stars while the solid lines represent the contours enclosing 90% of the stars at that epoch. The puffing-up of the stellar disk due to the orbital perturbation induced by the 𝜓DM granularity is clearly shown in our plot. The inset plot reports the growth over the Hubble time of the vertical spread of the stellar population and of the ellipticity of the population

Figure: Rotation curve for an observer with inclination 𝑖 = 45 degrees at the inital epoch (blue dashed line) and at 3 Gyr (green dotted line) and 10 Gyr (red dash-dotted line). The shaded area around the 10 Gyr profile corresponds to the statistical dispersion of the stellar velocity around the mean computed using a bootstrapping technique.

Dwarf spheroidal galaxies with a radially varying anisotropy

Differently to a previous analysis, we adopted an anisotropy parameter that varies with the distance from the centre of the galaxy to assess whether this assumption would help to resolve, or at least alleviate, the well-known tension with the value of the boson mass favored by the cosmological analysis. Under this assumption, we predicted the velocity dispersion along the line of sight, and we fit it to the kinematic data of eight dSph galaxies employing an MCMC algorithm. Our results indicate that, differently to what happens in ultrafaint dwarf galaxies, such a tension cannot be lifted introducing a variable anisotropy parameter, leaving as a possible solution the existence of additional axion or axionlike particles with higher masses as naturally predicted in the axiverse [6].

Figure. The figure depicts the radial velocity dispersion profile of the 8 dwarf galaxy Carina with a β(r) model. In green are reported the data and their error bar, while the blue solid lines represent the median profiles obtained from the best fit. Finally, shaded regions show the 68% confidence interval around the median value.

Measuring the boson mass of fuzzy dark matter with stellar proper motions

We investigated the capability of future astrometric Theia-like missions to detect the properties of such a halo within the FDM model, namely the boson mass and the core radius. To this aim, we built mock catalogs containing three-dimensional positions and velocities of stars within a target dwarf galaxy. We exploited these catalogs using a Markov Chain Monte Carlo algorithm and found that measuring the proper motion of at least 2000 stars within the target galaxy, with uncertainty less than 3 km/s on the velocity components, will constrain the boson mass and the core radius with 3% accuracy. Furthermore, the transition between the solitonic core and the outermost NFW-like density profile could be detected with an uncertainty of 7%. Such results would not only help to confirm the existence of FDM, but they would also be useful for alleviating the current tension between galactic and cosmological estimations of the boson mass, or demonstrating the need for multiple particles with a broad mass spectrum as naturally arising in the String Axiverse.

Bibliography:

I. De Martino, T. Broadhurst, H. Tye, T. Chiueh, Hsi-Yu Schive, R. Lazkoz, ’Recognising Axionic Dark Matter by Compton and de-Broglie Scale Modulation of Pulsar Timing.’, 2017, Phys. Rev. Lett, 119, 22110
I. De Martino, T. Broadhurst, H. Tye, T. Chiueh, Hsi-Yu Schive, ’Dynamical Evidence of a Solitonic Core of 109M☉ in the Milky Way'., 2020, Physics of Dark Universe, 28, p.100503
T. Broadhurst, I. De Martino, H. N. Luu, G. F. Smoot III, S.-H. H. Tye, ’Ghostly Galaxies as Solitons of Bosonic Dark Matter', 2020, Phys. Rev. D, 101, 083012
A. Pozo, T. Broadhurst, I. De Martino, H. N. Luu, G. F. Smoot, J. Lim, M. Neyrinck, 'Wave Dark Matter and Ultra Diffuse Galaxies', 2021, Mon. Not. R. Astron. Soc., 504, 2868-2876
R. Della Monica, I. De Martino, T. Broadhurst, "Explaining the oblate morphology of dwarf spheroidals with Wave Dark Matter perturbations", 2023, under review
I. De Martino, 2023, 'Constraining ultralight bosons in dwarf spheroidal galaxies with a radially varying anisotropy', Phys. Rev. D, 108, 123044
G. d'Andrade Furlanetto, R. Della Monica, I. De Martino, "Measuring the boson mass of fuzzy dark matter with stellar proper motions", 2025, Classical and Quantum Gravity, 42, 075011

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