Topological Signatures of Heating and Dark Matter in the 21 cm Forest

We show that persistence-based topology of the 21 cm forest encodes information about Cosmic Dawn that is complementary to traditional amplitude- or correlation-based statistics. Applying topological data analysis to simulated one-dimensional forest spectra over a grid of X-ray heating efficiencies $f_X$ and warm-dark-matter masses $m_{\rm WDM}$ (which set the free-streaming scale), we construct persistence diagrams and Betti-0 curves that track the birth-merger hierarchy of absorption troughs under sublevel filtrations. From these summaries we define three interpretable descriptors: the trough line density $\lambda(t_\star)$, the total squared persistence $M_2=\sum_{j\in I_{\rm long}}\tau_j^2$, and the Betti-curve asymmetry $A_{\rm skew}$. In a Fisher forecast around a fiducial WDM model, $\lambda(t_\star)$ and $A_{\rm skew}$ provide strong local leverage on the heating axis, while $M_2$ retains appreciable sensitivity to the free-streaming scale and supplies an inclined constraint direction that reduces the remaining degeneracy in the $(f_X,m_{\rm WDM})$ plane. We further demonstrate that, under an SKA1-Low-like uncorrelated thermal-noise model, noise predominantly produces short-lived fluctuations that are removed by a uniform persistence cut, leaving the topology of long-lived troughs and the gross Betti-curve morphology largely intact. These results establish persistence-based descriptors as a robust non-Gaussian probe of small-scale structure and heating during Cosmic Dawn, naturally complementing power-spectrum and wavelet-based analyses.