Tomographic Alcock-Paczynski Test with Marked Correlation Functions

Abstract

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 $w$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 $Ω_m$ 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 $\sim 40\%$ 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).

Figures (15)

• Figure 1: Comparison between the standard 2PCF and four MCFs with $\alpha = -0.3$, $0.3$, and $1$ at $z = 1$ and $z = 0.6069$, showing the radial (upper) and angular (middle) dependence. Besides, the lower panel shows the evolution of angular dependence of the standard 2PCF and these three MCFs. $W_{\Delta \mu}^{\alpha}(s) \equiv \int_{0}^{\mu_{\mathrm{cut}}} W^{\alpha}(s, \mu) \mathrm{d}\mu$, where $\mu_{\mathrm{max}} = 0.97$, while $\hat{W}_{\Delta s}^{\alpha}(\mu)$ is defined as Eq.\ref{['eq:w_normalization']}. The correspondence between 2PCF and MCFs remains consistent with the line styles and colors on each panel.
• Figure 2: Covariance matrices (top) and correlation coefficients (bottom) for the 1-dimensional standard 2PCF and MCFs with $\alpha = -0.3$, $0.3$, and $1$, respectively.
• Figure 3: Comparison of the posterior distributions of the cosmological parameters $\Omega_m$ and $w$ at the 68% confidence level, derived from the standard 2PCF and MCFs with $\alpha = -0.3$, $0.3$, and $1$, based on MCMC analysis, where $\delta \widehat{W}_{\Delta s}^{\alpha, \rm corr}$ is defined as Eq.\ref{['eq:correction_W']}, while $\delta \widehat{\xi}_{\Delta s}^{\rm corr}$ is defined as Eq.\ref{['eq:correlation_xi']}. The fiducial values $\Omega_m = 0.3071$ and $w = -1.0$ are indicated by the dashed lines.
• Figure 4: Same as in Fig. \ref{['fig:MCMC_results']}, but for combined MCFs with two or three $\alpha$s for illustration: $\boldsymbol{\alpha} = [0, 1]$, $\boldsymbol{\alpha} = [-0.3, -0.1]$, and $\boldsymbol{\alpha} = [-0.3, -0.1, 0.3]$. Clearly, the marginalized uncertainties are reduced relative to the 2PCF baseline.
• Figure 5: Convergence test based on the constraints on the $\Omega_m$--$w$ parameter space. From left to right: results for the 2PCF, and the combined MCFs with $\boldsymbol{\alpha} = [-0.3, 0.3]$ and $\boldsymbol{\alpha} = [-0.3, 0.3, 1]$, respectively. In each panel, we show results for $r_c = 0.99$, $0.995$, and $0.997$. Increasing $r_c$ from 0.99 to 0.995 noticeably affects the constraints, while further increasing $r_c$ yields negligible improvement.