Although gravitational waves were recently detected for the first time by LIGO/Virgo, the understanding of the universe will still be based on electromagnetic waves for a long time to come. 96% of the universe being unknown to us (dark matter and dark energy), the role of fundamental Physics is therefore to question the Maxwellian linear electromagnetism of the 19th century (and its bosons, photons), which remains the current framework of astrophysics. Among the proposed non-linear theories are Born-Infeld formalism which introduces electron regularization and Heisenberg-Euler formalism with quantum light interactions and very intense electromagnetic fields. In magnetars, the magnetic field would cause a shift, towards blue or red depending on the polarization, of the frequency of emitted photons. This shift would reach up to one tenth of the gravitational shift due to the mass of the magnetar.
Another aspect of non-Maxwellian electromagnetism concerns the massive photon proposed by de Broglie-Proca and pursued by Schrödinger, Born and many others to this day. The mass of the photon could manifest itself through a deviation from Ampere’s law. We used the Cluster data on the solar wind and looked for the difference between the rotational current of the magnetic field, measured by magnetometers, and the current of ions and electrons. Another manifestation of the massive photon could be the arrival delay of the low-frequency photons of pulsars. Several tests have been published in the radio (GHz), optical and gamma domains. However, the spectrum for f < 15 MHz is unknown. Here we assume exotic sources, other than pulsars. To escape the atmosphere, a swarm of nano-satellites, called OLFAR, is being studied. LPC2E is a member of the scientific team but the involvement could be extended because of the laboratory's interest in nanosatellites (plasma chamber, pedagogy). The delays due to mass and plasma dispersion are both proportional to the inverse of the frequency square, but it is hoped that the two effects can be separated through superdispersion (excess dispersion) analysis. Another opportunity to bypass the dispersion is to look for extragalactic sources. FRBs (Fast Radio Bursts) could give this opportunity if the host galaxy, and therefore the distance (i.e. the redshift), were identified. A different redshift dependence between the effect of electron density and that of photon mass would allow this decoupling. A future step could also consider different values of cosmological parameters. The question of the massive photon could also be approached from the low-frequency analysis of storm discharges in connection with the work of the "atmosphere" team. Finally, another opportunity would be the use of the S and K-band signal of atomic clocks (ACES-PHARAO) on the space station. The Standard Model (MS) unifies nuclear, electromagnetic, weak and strong interactions. Although the MS has shown enormous success and provides experimental predictions, it leaves some phenomena unexplained: the gravitation of general relativity, the accelerated expansion of the universe, dark matter, oscillations and neutrino masses. Finally, it predicts the mass of the Higgs boson to be greater than the mass detected at CERN. Theories that go beyond the MS have therefore been proposed. They are called MS Extension (EMS), including SUper SYmmetry (SUSY). The latter guarantees Lorentz invariance (the independence of experiments in orientation, position and speed). The consequences of the breakage of SUSY and LI during the primordial universe would also manifest themselves at our current energies: among these consequences, the massive photon - gauge invariant - detectable with the means already described. Finally, among all these questions, the most relevant for cosmology is to understand whether non-Maxwellian electromagnetism could cause a large-scale frequency shift.