Testing $f(R)$ gravity models with quasar x-ray and UV fluxes

Recently, active galactic nuclei (AGN) have been proposed as “standardizable candles,” thanks to an observed nonlinear relation between their x-ray and optical-ultraviolet (UV) luminosities, which provides an independent measurement of their distances. In this paper, we use these observables for the first time to estimate the parameters of $f(R)$ gravity models (specifically, the Hu-Sawicki and the exponential models) together with the cosmological parameters. The importance of these types of modified gravity theories lies in the fact that they can explain the late time accelerated expansion of the universe without the inclusion of a dark energy component. We have also included other observable data to the analyses such as estimates of the Hubble parameter $H(z)$ from cosmic chronometers (CCs), the Pantheon Type Ia (SnIa) supernovae compilation, and baryon acoustic oscillation (BAO) measurements. The $1\sigma$ inferred constraints using all datasets are $b\le 0.276$, ${\mathrm{\Omega }}_m=0.304_{-0.011}^{+0.010}$, and $H_0=67.553_{-0.936}^{+1.242}$ for the Hu-Sawicki model, and $b=0.785_{-0.606}^{+0.409}$ ${\mathrm{\Omega }}_m=0.305_{-0.010}^{+0.011}$ and $H_0=68.348_{-0.760}^{+0.959}$ for the exponential one, but we stress that for both $f(R)$ models results within $2\sigma$ are consistent with the $\mathrm{\Lambda }$ cold dark matter (CDM) model. Our results show that the allowed space parameter is restricted when both AGN and BAO data are added to CC and SnIa data, with the BAO dataset being the most restrictive one. We can conclude that both the $\mathrm{\Lambda }\mathrm{CDM}$ model and small deviations from general relativity given by the $f(R)$ models studied in this paper are allowed by the considered observational datasets.

Figure 1

Results of the statistical analysis for the $f(R)$ Hu-Sawicki model (left) and the exponential model (right). The darker and brighter regions correspond to 65% and 95% confidence regions, respectively. The plots in the diagonal show the posterior probability density for each of the free parameters of the model.

Figure 2

Results for the matter density ${\mathrm{\Omega }}_m$ and the Hubble parameter $H_0$ using all the datasets ($\mathrm{CC}+\mathrm{S}\mathrm{n}\mathrm{I}\mathrm{a}+\mathrm{BAO}+\mathrm{AGN}$). The plots show the 68% and 95% confidence region together with the posterior probability density for each parameter obtained for the two $f(R)$ models considered in this paper and the $\mathrm{\Lambda }\mathrm{CDM}$ model.

Figure 3

Constraints on the free parameters of the Hu-Sawicki model. Comparison between the $1\sigma$ confidence intervals obtained in this work and the ones reported by other authors.

Figure 4

Constraints on the free parameters of the exponential model. Comparison between the $1\sigma$ confidence intervals obtained in this work and the ones reported by other authors.