Tomographic Alcock-Paczynski test with marked correlation functions

The tomographic Alcock-Paczynski (AP) method, developed over the past decade, exploits redshift evolution for cosmological determination, aiming to mitigate contamination from redshift distortions and capture nonlinear scale information. Marked correlation functions (MCFs) extend information beyond the two-point correlation. For the first time, this study integrated the tomographic AP test with MCFs to constrain the flat $w$ cold dark matter ($w\mathrm{CDM}$) cosmology model. Our findings show that multiple density weights in MCFs outperform the traditional two-point correlation function, reducing the uncertainties of the matter density parameter ${\mathrm{\ensuremath{\Omega}}}_{m}$ and dark energy equation of state $w$ by 48% and 45%, respectively. Furthermore, we introduce a novel principal component analysis (PCA) compression scheme that efficiently projects high-dimensional statistical measurements into a compact set of eigenmodes while preserving most of the cosmological information. This approach retains significantly more information than traditional coarse binning methods, which simply average adjacent bins in a lossy manner, yielding an additional $\ensuremath{\sim}40%$ reduction in error margins. To assess robustness, we incorporate realistic redshift errors expected in future spectroscopic surveys. While these errors modestly degrade cosmological constraints, our combined framework, which utilizes MCFs and PCA compression within tomographic AP tests, is less affected and always yields tight cosmological constraints. This scheme remains highly promising for upcoming slitless spectroscopic surveys, such as the Chinese Space Station Telescope (CSST).