Diagnostic of $f(R)$ under the $Om(z)$ function

We perform the two-point diagnostic for the $Om(z)$ function proposed by Sahni et al. in 2014 for the Starobinsky and the Hu and Sawicki models in $f(R)$ gravity. We show that the observed values of the $Om{h}^2$ function can be explained in $f(R)$ models, while in $\mathrm{\Lambda }$CDM the $Om{h}^2$ function is expected to be a redshift independent number. We perform the analysis for some particular values of ${\mathrm{\Omega }}_m^0$, finding a cumulative probability ($P({\chi }^2\le {\chi }_{\text{model}}^2)$) $P\sim 0.16$ or $\sim 0.09$ for the better cases versus a cumulative probability of $P\sim 0.98$ in the $\mathrm{\Lambda }\mathrm{CDM}$ scenario. We also show that these models present a characteristic signature around the interval between $z\sim 2$ and $z\sim 4$ that could be confronted with future observations using the same test.

Figure 1

$Om(z)$ function for the Starobinsky model assuming (top to bottom) ${\mathrm{\Omega }}_M^0=0.286$ (red), 0.26 (blue), 0.25 (orange), 0.24 (gray), and 0.23 (black).

Figure 2

$Om(z)$ function for the Hu & Sawicki model assuming (top to bottom) ${\mathrm{\Omega }}_M^0=0.286$ (red), 0.26 (blue), 0.25 (orange), 0.24 (gray), and 0.23 (black).

Figure 3

Evolution of the $Om(z)$ function. Starobinsky and Hu and Sawicki models, top and bottom, respectively, assuming ${\mathrm{\Omega }}_M^0=0.25$ in both cases.