Geodesic motion of S2 and G2 as a test of the fermionic dark matter nature of our Galactic core

The motion of S-stars around the Galactic center implies that the central gravitational potential is dominated by a compact source, Sagittarius A* (Sgr A*), which has a mass of about 4 x 10(6)M(circle dot) and is traditionally assumed to be a massive black hole (BH). The explanation of the multiyear accurate astrometric data of the S2 star around Sgr A*, including the relativistic redshift that has recently been verified, is particularly important for this hypothesis and for any alternative model. Another relevant object is G2, whose most recent observational data challenge the scenario of a massive BH: its post-pericenter radial velocity is lower than expected from a Keplerian orbit around the putative massive BH. This scenario has traditionally been reconciled by introducing a drag force on G2 by an accretion flow. As an alternative to the central BH scenario, we here demonstrate that the observed motion of both S2 and G2 is explained in terms of the dense core - diluted halo fermionic dark matter (DM) profile, obtained from the fully relativistic Ruffini-Arguelles-Rueda (RAR) model. It has previously been shown that for fermion masses 48-345 keV, the RAR-DM profile accurately fits the rotation curves of the Milky Way halo. We here show that the solely gravitational potential of such a DM profile for a fermion mass of 56 keV explains (1) all the available time-dependent data of the position (orbit) and line-of-sight radial velocity (redshift function z) of S2, (2) the combination of the special and general relativistic redshift measured for S2, (3) the currently available data on the orbit and z of G2, and (4) its post-pericenter passage deceleration without introducing a drag force. For both objects, we find that the RAR model fits the data better than the BH scenario: the mean of reduced chi-squares of the time-dependent orbit and z data are chi (2)(S2,RAR) approximate to 3.1 chi overbar 2 S 2 , RAR approximate to 3.1 and chi (2)(S2,BH) approximate to 3.3 chi overbar 2 S 2 , BH approximate to 3.3 for S2 and chi (2)(G2,RAR) approximate to 20 chi overbar 2 G 2 , RAR approximate to 20 and chi (2)(G2,BH) approximate to 41 chi overbar 2 G 2 , BH approximate to 41 for G2. The fit of the corresponding z data shows that while for S2 we find comparable fits, that is, chi (2)(z,RAR) approximate to 1.28 chi overbar over z , RAR 2 approximate to 1.28 and chi (2)(z,BH) approximate to 1.04 chi overbar over z , BH 2 approximate to 1.04 , for G2 the RAR model alone can produce an excellent fit of the data, that is, chi (2)(z,RAR) approximate to 1.0 . chi overbar over z , RAR 2 approximate to 1.0 and chi (2)(z,BH) approximate to 26 . chi overbar over z , BH 2 approximate to 26 . In addition, the critical mass for gravitational collapse of a degenerate 56 keV-fermion DM core into a BH is similar to 10(8)M(circle dot). This result may provide the initial seed for the formation of the observed central supermassive BH in active galaxies, such as M 87.