Constraints on Dark Energy and Modified Gravity Models from Fast Radio Bursts and Late-Time Geometric Probes

We investigate the impact of 104 localized FRBs on cosmological parameter estimation when combined with three established late-time probes: Cosmic Chronometers (CC), Type Ia Supernovae (SNe), and Baryon Acoustic Oscillations (BAO). By performing a Bayesian analysis of three dark energy models ($\Lambda$CDM, $w$CDM, and CPL) and three viable $f(R)$ gravity scenarios -- the Appleby-Battye (AB), Hu-Sawicki (HS), and Starobinsky (ST) models -- , we find that FRBs substantially improve the constraints on the baryon density $\Omega_{\rm b}$ by $25\%$--$43\%$, the Hubble constant $H_0$ by $12\%$--$35\%$, and the SNe absolute magnitude $M_B$ by $10\%$--$32\%$. Constraints on dark energy parameters show more modest improvements, with $w$ improving by $\sim 9\%$ in $w$CDM and $(w_0,w_a)$ improving by $\sim(8,22)\%$ in the CPL parametrization. Modified gravity parameters remain weakly constrained, with improvements of only $6\%$--$15\%$, indicating the limited sensitivity of current datasets to departures from $\Lambda$CDM. The Figure of Merit analysis shows overall improvements ranging from $\sim 48\%$ ($\Lambda$CDM) to $\sim 91\%$ (CPL), driven by enhanced precision in the $(H_0, \Omega_{\rm b})$ plane. Model comparison reveals moderate statistical preference for extensions beyond $\Lambda$CDM: AIC strongly favors $w$CDM, CPL, HS, and ST with $\Delta\mathrm{AIC} < -7$, and LRT yields $p \leq 0.004$, while BIC returns to positive evidence ($-3.2 < \Delta\mathrm{BIC} < -2.7$). These results show that FRBs may be useful as a complementary probe, particularly for constraining $\Omega_{\rm b}$ and alleviating key late-time degeneracies.