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Fisher matrix for multiple tracers: model independent constraints on the redshift distortion parameter
Cosmology and Nongalactic Astrophysics · Submitted 1 Apr 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 1.00 | 0.93 | 0.82 | asserted 1.00 |
| s1 | 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. | background 0.98 | 0.10 | 0.04 | asserted 1.00 |
| s2 | 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. | method 0.74 | 0.25 | 0.36 | asserted 1.00 |
| s3 | 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. | contribution 0.93 | 0.96 | 0.12 | asserted 1.00 |
| s4 | We also forecast the errors on $\beta$ for a survey like Euclid. | contribution 0.72 | 0.91 | 0.08 | asserted 0.84 |
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Original 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.
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2 arXiv papers cite this in 2 sentences
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Cites it as background 1 paper
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Considerable effort has been applied to exploit information from multiple tracers with the goal of measuring RSD, including: improving power spectrum estimators [ref] ; exploring the parameter space to better constrain the growth rate [ref] ; and adding further velocity corrections [ref] .
Constraining the growth rate by combining multiple future surveys · Introduction · 0.87
What it cites · 3 sentences without a confident label
Cites as background 1 paper
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This quantity enters the linear galaxy power spectrum in the so-called Kaiser term [ref] , which converts the real-space spectrum $P^(x)(k) = b^2 P_m (k)$ ($P_m$ is the matter power spectrum) into its observed, redshift-space version, $P^(z)(k,μ) = (1+ βμ^2)^2 P^(x) (k)$, where the angle of the Fourier mode $k⃗$ with the line of sight $r̂$ is encoded in $μ= r̂ ·k̂$ — [ref] , see [ref] for a review.
Linear Redshift Distortions: A Review · Introduction · 0.98
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