A Prediction for DESI Full-Shape: Increasing Tomographic $Ω_m(z)$ Trend

Abstract: To confirm $Λ$CDM deviations are due to missing physics (not systematics), one should demonstrate that the model fitting parameters exhibit qualitatively similar redshift drift across independent observables. This is the only way one guarantees new physics. Here, we show that a recent Dark Energy Spectroscopic Instrument (DESI) DR2 Full-Shape (FS) modelling Lyman-$α$ constraint at $z_{\rm eff} = 2.33$ combined with earlier DR1 FS modelling constraints with $0.295 \leq z_{\rm eff} \leq 1.491$ leads to a straight line $Ω_m(z) = m z + c$ with slope $m = 0.022 \pm 0.012$, $1.8 σ$ removed from constant $Ω_m$. Akaike Information Criterion and Bayesian evidence confirm that constant $Ω_m$ and increasing $Ω_m(z)$ are statistically indistinguishable. Through the $Om(z)$ diagnostic, we review how increasing and decreasing $Om(z)$ map to phantom and quintessence dark energy (DE) regimes, respectively. While FS modelling constraints map to phantom DE, the decreasing and increasing $Ω_m(z)$ trends in DESI BAO and DESI with external data make a phantom crossing inevitable. Since dynamical DE is but one interpretation for $Ω_m(z)$ trends, it is imperative that different datasets converge on their $Ω_m(z)$ trends before one jumps to physical conclusions. We forecast how DESI FS modelling $Ω_m$ constraints will improve up to the final data release and explore the implications for model selection.