# Mixed Dark Matter and Galaxy Clustering: The Importance of Relative Perturbations

## TL;DR

This paper develops a perturbative model for large-scale structure in cosmologies with mixed cold and warm dark matter, highlighting the importance of relative perturbations for accurate galaxy clustering analysis and constraints.

## Contribution

It introduces linear solutions for relative perturbations in mixed dark matter models and demonstrates their impact on galaxy bias and power spectrum analysis.

## Key findings

- Neglecting relative perturbations biases warm dark matter constraints.
- Upcoming surveys can detect signatures of mixed dark matter scenarios.
- Light WDM masses significantly affect galaxy clustering measurements.

## Abstract

We develop a perturbative model to describe large-scale structure in cosmologies where dark matter consists of a mixture of cold (CDM) and warm (WDM) components. In such mixed dark matter (MDM) scenarios, even a subdominant warm component can introduce distinctive signatures via its free-streaming effects, altering the evolution of density and velocity perturbations. We present linear-order solutions for both total and relative perturbations in the two-fluid system, identifying novel contributions to galaxy bias caused by the relative density and velocity modes between the components. Incorporating these effects into the galaxy bias expansion, we compute the linear galaxy power spectrum in both real and redshift space. Using Fisher matrix forecasts, we assess the sensitivity of upcoming surveys such as DESI and PFS to MDM scenarios. Our results demonstrate that neglecting relative perturbations can lead to significant biases in inferred constraints on the warm dark matter fraction, particularly for lighter WDM masses ($\lesssim 150~\mathrm{eV}$ and $\lesssim 80~\mathrm{eV}$) for PFS and DESI, respectively. This framework provides a consistent and generalizable approach for incorporating multi-component dark matter dynamics into galaxy clustering analyses.