Dark Matter Meets Einstein’s Equivalence Principle: A Tale Of Two Particles

Dark matter is one of the most puzzling challenges of present-day physics. It is a mysterious entity that presents itself only through the gravitational interaction and corresponds to about 27% of the energy content of the Universe at present. Its existence has been known since 1933, however, its nature and origin is yet to be understood.

Dark matter is observed through the rotation curves of galaxies, which reveal that the velocity of peripheric stars becomes fairly constant away from the center and through the cohesion of galaxies as components of clusters of galaxies. The stability of the galactic disks suggests that dark matter is spread in spheroidal halos surrounding the disk and galactic bulge. Observations show that the dynamics of the Milky Way is also dominated by dark matter. There have been different attempts to mathematically describe the distribution of dark matter in halos using N-body simulations. One of the most discussed descriptions is so-called Navarro-Frenk-White profile whose density of dark matter in the halo is essentially inversely proportional to the cube of the distance from the center of the halo (remember that Newton’s universal law of gravity establishes that the force is inversely proportional to the square of the distance, which leads to a density of matter inversely proportional to the distance).

In our work, we have shown that by using the distribution of dark matter in the halo, it is possible to obtain quite stringent limits on the Weak Equivalence Principle (WEP), one of the fundamental basic tenets of the General Relativity (GR). The WEP states that the trajectory of a point mass particle in a gravitational field depends only on its initial position and velocity and is independent of its composition and structure. In other words, the WEP is a restatement of the equality of gravitational and inertial masses of Newtonian mechanics.

GR is the best theory we have so far to describe gravitational interaction, and it has been tested through several repeated experiments and observations since a few years after the appearance of the final version of Einstein’s field equations in 1915. However, although its success and accuracy at the Solar System level to provide a comprehensive understanding of several gravitational phenomena, at larger scales, the necessity to introduce dark matter (and dark energy) encourages the search for deviations from GR at astrophysical and cosmological scales.

A fairly general framework to compare GR with other alternative theories of gravity is the so-called parametrized post-Newtonian formalism, in which one considers either two different particles (photons and neutrinos, for instance) or just one type of particle with different energies, and compares their trajectories through the same gravitational field. In this framework, each particle is labeled with a parameter “γ” (γ1 for a particle 1 and γ2 for a particle 2) and thus the WEP predicts that γ12 given that both particles have the same velocity, the speed of light, travel the same distance and their trajectories should be independent of any internal structure and other features.

This line of reasoning provides an experimental procedure to obtain constraints on the WEP, namely through the measurement of the time delay of two different particles (such as photons and neutrinos, or two photons with different energies) arising from astrophysical phenomena, such as gamma-ray bursts or fast radio bursts. These phenomena consist of bursts of photons with different frequencies (and thus with different energies) emitted from an astrophysical object or system, such as, for instance, a binary neutron star, which may reveal, if the WEP is not respected, a time delay on their arrival when observed on Earth. Given that the time of travel of each particle is proportional to γ, a putative violation of the WEP will show itself through a difference between γ1 and γ2. Given the success of GR, this difference is expected to be very small.

The difference γ1 – γ2 also depends inversely on the matter density through which the particles travels. Hence, since the particles measured on Earth are predominantly affected by the gravitational potential of the Milky Way and our galaxy is dominated by dark matter, it is natural to ask how the presence of the latter affects the bounds on the WEP discussed above. This effect was considered for the first time in our work. As the contribution of dark matter makes the mass of the galaxy bigger, it follows that it allows for more stringent limits for the difference γ1 – γ2.

Therefore, taking into account the presence of dark matter in our galaxy, it is possible to obtain more stringent constraints on the difference of the parameter γ. Considering the Navarro-Frenk-White profile and the time delay measurement of a polarized gamma-ray burst, we have obtained the most stringent limit to date, γ1 – γ2 < 10-28,  indicating that the WEP is an extremely well-grounded fundamental principle and that dark matter plays an important role in this discussion.

These findings are described in the article entitled Cosmic transients, Einstein’s Equivalence Principle and dark matter halos published in Physics of the Dark Universe (Volume 21, September 2018, Pages 16-20). This work was conducted by O. Bertolami and R. Landim at the Universidade do Porto.

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