# Fisher matrix for multiple tracers: model independent constraints on the redshift distortion parameter

## TL;DR

This paper presents a model-independent method using multiple tracers to constrain the redshift distortion parameter eta, overcoming cosmic variance limitations and improving measurement precision in large-scale structure surveys.

## Contribution

It introduces an analytic Fisher matrix framework for multiple tracers to measure eta independently of cosmological models, with forecasts for Euclid-like surveys.

## Key findings

- Cosmic variance cancels out when using multiple tracers, improving eta constraints.
- Analytic formulas for Fisher matrix enable precise error forecasts.
- Effective multi-tracer analysis can significantly enhance redshift distortion measurements.

## Abstract

We show how to obtain constraints on $\beta=f/b$, the ratio of the matter growth rate and the bias that quantifies the linear redshift-space distortions, that are independent of the cosmological model, using multiple tracers of large-scale structure. For a single tracer the uncertainties on $\beta$ are constrained by the uncertainties in the amplitude and shape of the power spectrum, which is limited by cosmic variance. However, for two or more tracers this limit does not apply, since taking the ratio of power spectra cosmic variance cancels out, and in the linear (Kaiser) approximation one measures directly the quantity $(1+ \beta_1 \mu^2)^2/(1+ \beta_2 \mu^2)^2$, where $\mu$ is the angle of a given mode with the line of sight. We provide analytic formulae for the Fisher matrix for one and two tracers, and quantify the signal-to-noise ratio needed to make effective use of the multiple-tracer technique. We also forecast the errors on $\beta$ for a survey like Euclid.