Charged dark matter and the $H_0$ tension

We consider cosmological models where dark matter is universally charged under a dark Abelian gauge field. This new interaction is repulsive and competes with gravity on large scales and in the dynamics of galaxies and clusters. We focus on nonlinear models of dark electrodynamics where the effects of the new force are screened within a K-mouflage radius that helps avoiding traditional constraints on charged dark matter models. We discuss the background cosmology of these models in a Newtonian approach and show the equivalence with relativistic Lemaître models where an inhomogeneous pressure due to the electrostatic interaction is present. In particular, after foliating the Universe using spherical shells, we find that dark matter shells with initially different radii do not evolve similarly as they exit their K-mouflage radii at different times, resulting in a breaking of the initial comoving evolution. In the large time regime, the background cosmology is described by a comoving but inhomogeneous model with a reduced gravitational Newton constant and a negative curvature originating from the electrostatic pressure. In this model, baryons do not directly feel the electrostatic interaction, but are influenced by the inhomogeneous matter distribution induced by the electric force. We find that shells of smaller radii evolve faster than the outer shells which feel the repulsive interaction earlier. This mimics the discrepancy between the large scale Hubble rate and the local one. Similarly, as galaxies and clusters are not screened by the new interaction, large scale global flows would result from the existence of the new dark electromagnetic interaction.

Figure 1

The following choice of values are only taken for illustration so that the effects can be visible. In the upper left panel we show the evolution of the shells size normalized to the radius of the innermost shell. The color coding has been chosen so that darker means inner shells. We have chosen ${\beta }^2=200$, $4\pi {\rho }_{\star }/3=1$ and $r_{\mathrm{s}}=500\sqrt{M}$ in units of $G=1$. In the upper right panel we show the shells scale factor evolution, where we can see the initial dust dominated evolution (dashed line) that transits to an open Universe (dotted line) with a spatial curvature generated by the electric pressure. We can also confirm the initial self-similar evolution that breaks as the different shells exit their corresponding screening radii. The lower left panel shows the time evolution of the function $\mathcal{U}$ which governs the evolution of the curvature in the effective Friedmann equation. Finally, in the lower right panel we plot the density (blue) and the pressure (orange) profiles at three different times: the initial profile (solid line), the profile at a time in the transition region, $t=3\times {10}^3$, (dashed) and the asymptotic profile (dotted). This confirms that the asymptotic state recovers the self-similar evolution but an inhomogeneous profile has been generated. Moreover we also see explicitly that the pressure acts on the outer shells at late times.

Figure 2

In this figure we show a case exhibiting shell crossing. We have chosen the same values as in Fig. 1 except for screening scale that has now been chosen as $r_{\mathrm{s}}=100\sqrt{M}$. The color coding is also the same. We can see how the initially innermost shell starts gaining mass when it exists its screening radius so it becomes the most massive shell in the asymptotic state, in agreement with the fact that it overtakes all the shells and it becomes the outermost one at late times. On the other hand, the outermost shells lose mass and eventually become the innermost shells.

Figure 3

The upper left panel shows the scale factor evolution for the DM (solid-green palette) and baryons (dashed-red palette) shells. In the upper right panel we plot the evolution of the shell’s size relative to the size of the innermost shell of each component. The lower left panel shows the expansion rates for both components normalized to the Hubble factor of the innermost shell of each component. Finally, in the lower right panel we can see how the relative size of the initially comoving shells of baryons and DM evolves.

Figure 4

These snapshots show, in arbitrary units, the numerical solution for the evolution of the shells without (top) and with (bottom) baryons together with the (log) mass distribution (insets), i.e., the mass inside the different shells from the inner most to the outer most. The upper snapshots show a case with shell-crossing and we can see how the mass distribution changes as the shells cross. Crucially, it is the very repulsive nature of the force what can lead to shell crossing in the expanding phase. This cannot happen for models with scalar fields due to their intrinsically attractive character. In the lower panels we can see how the charged DM shells separate from the baryons as they exit their screening radius. Animations are available in the Supplemental Material [55].

Figure 5

This figure shows the evolution of the expansion rates for the shells with the same parameters as in Fig. 1 and in the presence of a cosmological constant with $\frac{8\pi G{\rho }_{\mathrm{\Lambda }}}{3}={10}^{-4}$. Notice that the inner shells have a larger expansion rate than the outer shells. The figure on the left corresponds to the absolute expansion rates for all shells and the convergence to a Universe dominated by a cosmological constant can be seen. The right figure gives the normalised Hubble rate to the inner most shell. Baryons are represented in red while dark matter is in green.