Observational viability of Herglotz $f(R,T)$ gravity: A multi-probe Bayesian analysis
We investigate the observational viability of the linear Herglotz-type $f(R,T)$ gravity model, $f(R,T)=R+αT$, which incorporates both geometry--matter coupling and non-conservative gravitational dynamics. Unlike previous analyses\cite{Wazny:2025jth} based on illustrative parameter choices, we perform a systematic Bayesian estimation of the four-dimensional parameter space $\{H_0,A,w,Φ_0\}$, where $A$ characterizes the matter--geometry coupling, $w$ is the effective equation-of-state parameter, and $Φ_0$ denotes the present-day Herglotz field. The background evolution is obtained by numerically integrating the coupled Herglotz cosmological equations at every point in the parameter space. We employ Cosmic Chronometer (CC), DESI DR2 baryon acoustic oscillation (BAO), and Union3 Type-Ia supernova data, both independently and in combination. The joint analysis yields $H_0=66.67^{+1.32}_{-1.28}\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, $A=1.57^{+0.66}_{-0.51}$, $w=-0.992^{+0.136}_{-0.119}$, and $Φ_0=-0.062^{+0.185}_{-0.157}$ at $68\%$ credibility. The reconstructed Hubble expansion closely follows the flat $Λ$CDM prediction over the redshift range probed by the CC data. However, the deceleration parameter, effective equation of state, $Om(z)$ diagnostic, and statefinder $\{r,s\}$ trajectories exhibit appreciable departures from the concordance model, with the magnitude and redshift evolution depending on the observational dataset. These results demonstrate that Herglotz-type $f(R,T)$ gravity can provide an observationally viable description of the late-time expansion while retaining distinguishable cosmological signatures beyond the background Hubble history.