Modelling cosmic rays flux with Pierre Auger and Telescope Array data in f(R) and f(Q) theories of gravity

Abstract

We investigate the effects of the magnetic field and the redshift on the propagation of galactic and extragalactic cosmic rays (CRs) in a modified theory of gravity (MTG) and an alternative theory of gravity (ATG) framework. For this purpose, we consider the $f\left(R\right)$ gravity and the $f\left(Q\right)$ gravity theories. We utilise these two MTG and ATG to compute the density enhancement factor of CRs as a function of the magnetic field and the redshift. For this work, we take the magnetic field strength from $1-100$ nG, while $0-2.5$ for the redshift. For each of these parameters, we take 100 bins within their considered range for the computation. The enhancement parameter for the mixed composition of heavy nuclei up to Fe is also taken into account for this work. Further, we compute the $E^3$ magnified diffusive flux of ultra high-energy cosmic rays (UHECRs) for 150 sources separated by a distance $d_{\text{s}}$ for the different cosmological models. For the fitting with observational data from the Pierre Auger Observatory (PAO) and Telescope Array (TA), we parameterised some set that consists of the redshift $z$, the separation distances $d_{\text{s}}$ between the 150 sources, and the maximum cutoff energy $E_{\text{max}}$. For each case, a residue plot and ${\chi }^2$ value are also added to check the goodness of fit. A comparative analysis of the considered models has been performed in each of the cases along with the $\Lambda$CDM model. The $f\left(Q\right)$ model shows the highest CR density enhancement and the lowest reduced ${\chi }^2$ value when fitted to PAO and TA data of the UHECRs flux. The uncertainty calculations in flux have been performed in this work with PAO and TA data, also supporting the validation of the considered MTG and ATG in UHECR studies.