Unveiling the Interaction between Dark Matter and Gravity: from Galaxy Kinematics to Cosmology

The Cold Dark Matter paradigm emerged as the prevailing solution to the cosmic missing mass conundrum. While highly successful on cosmological scales, this model is partially inconsistent with galaxy-scale observations. This thesis addresses such a challenge by pursuing a dual approach. The first part presents a model featuring a non-local interaction between cold dark matter and gravity. The model, referred to as fractional gravity, in its Newtonian setting is based on a modified Poisson equation featuring the fractional Laplacian and two new parameters: the fractional index s and length scale ℓ. To begin with, I will provide the solution to this equation for the gravitational potential generated by a Navarro-Frenk-White distribution of cold dark matter. Remarkably, the result can be interpreted in terms of an almost cored effective density distribution, similar in shape to a cored isothermal sphere, thus solving the cusp-core problem of cold dark matter. Moreover, the predicted profiles are consistent with the core-surface density relation observed for dwarf galaxies. A Bayesian analysis will then test fractional gravity against stacked rotation curves of a broad sample of spiral galaxies. In terms of reduced chi-squared and BIC, the fits yielded by this model are always superior to the standard, cold dark matter Navarro–Frenk–White profile. On the other hand, I will show how this model can explain the observed interplay between dark matter and baryons in late-type galaxies, embodied by tight dynamical scaling relations such as the Radial Acceleration Relation. The analysis predicts a scaling between the parameters s and ℓ and the virial mass of dark matter haloes, with non-local effects being stronger in smaller systems. After using the fractional gravity model to fit galaxy clusters’ pressure profiles from the X-COP collaboration, I will show that such scaling holds consistently from the dwarf-galaxy regime up to galaxy clusters’ virial masses. Overall, fractional gravity shows a rich phenomenology in a comprehensive set of scales capable of addressing long-standing issues of the cold dark matter paradigm. To go beyond the Newtonian setting, I will extend fractional gravity to a relativistic framework, deriving it from a suitable action. I will show that the resulting field equations can be interpreted as Einstein equations sourced by an effective stress-energy tensor for the dark matter component, with pressure, heat vectors, and anisotropic stresses emerging dynamically from the coupling with gravity. I will then study propagating degrees of freedom in this model, from scalar fields to gravitational waves, showing that non-locality introduces memory effects which extend the propagation inside the past light-cone. The second part of this thesis is devoted to study a relativistic model, named non-minimally coupled dark matter. Although models of this kind had been already studied in other contexts [1, 2, 3, 4], in the present case this provides a localization of fractional gravity. The greater simplicity of this model enables to do explicit calculations in the strong gravity regime, giving a taste of the corresponding solutions in the non-local theory. I will solve the equations of non-minimally coupled dark matter in three relevant scenarios. First, I will show how the dynamically generated pressure is able to balance gravity by supporting spherically-symmetric, static equilibrium configurations of dark matter. The resulting objects are regular and horizonless, but ultra-compact, so that, if formed in the early universe, could have provided the seeds for supermassive black holes observed at high redshifts by the JWST collaboration. Next, I will turn the attention to cosmology, computing the evolution of the universe in the non-minimally coupled dark matter framework. Remarkably, at early times a phase of accelerated expansion driven by the non-minimal coupling emerges, so that inflation is a natural prediction of the model. At even earlier times the fate of the universe depends on the spatial curvature, with the remarkable possibility of avoiding the initial singularity in favour of a bouncing universe in the case of negative curvature. Since no inflationary model can call itself complete without discussing the formation and growth of linear perturbations, in the final chapter I will provide a complete analysis. In particular I will compute the linear power spectrum of scalar, vector, and tensor perturbations, and show that in the case of a bouncing cosmology one obtains near scale-invariant power spectra, in agreement with observations of the Cosmic Microwave Background. All in all, the non-minimally coupled dark matter model presents a rich and interesting phenomenology, providing seeds for supermassive black holes, naturally predicting inflation, and generating a power spectrum of perturbations compatible with observations.