Correlations Between kHz QPOs and Spectral Parameters from Time-Resolved Spectro-Temporal Analysis of 4U 1728-34

We present a time-resolved analysis of the persistent emission in 4U 1728--34 using AstroSat observations from 2016 to 2019. We detect kilohertz quasi-periodic oscillations (kHz QPOs) during all epochs, with centroid frequencies ranging from $\sim 350$ to $1180~\mathrm{Hz}$, although some detections are of lower significance ($< 3\sigma$). We model the simultaneous spectra from the Soft X-ray Telescope and the Large Area X-ray Proportional Counter using a combination of an absorbed disk component (diskbb), a blackbody component (bbodyrad), a thermal Comptonization model (thcomp), and a broad Gaussian line. From the diskbb parameters, we estimate the accretion rate and find that all observations fall into two accretion regimes, namely AR1 and AR2, with accretion rates of $\sim 3 \times 10^{16}~\mathrm{g\,s^{-1}}$ and $\sim 7 \times 10^{16}~\mathrm{g\,s^{-1}}$, respectively. Interestingly, we find that for AR1, the lower kHz QPO frequency ($\nu_{\mathrm{L}}$) is always $< 500~\mathrm{Hz}$, while for AR2 it is $\gtrsim 500~\mathrm{Hz}$. We found that the spectral index showed no clear correlation with $\nu_{\mathrm{L}}$. For AR1, the coronal electron temperature ($kT_{\mathrm{e}}$) and optical depth ($\tau$) are $\sim 10~\mathrm{keV}$ and $\sim 5$, respectively. In contrast, for AR2, $kT_{\mathrm{e}}$ decreases to $\sim 3~\mathrm{keV}$ and $\tau$ increases to $\sim 12$, showing correlations with $\nu_{\mathrm{L}}$, with Spearman's rank correlation coefficients of $-0.78$ and $0.71$, respectively. The transition of the spectral parameters at $\nu_{\mathrm{L}} \sim 500~\mathrm{Hz}$ indicates the existence of a critical QPO frequency governed or influenced by the accretion state of the source.