[Paper Review] How to measure redshift-space distortions without sample variance

Patrick McDonald, Uroš Seljak|arXiv (Cornell University)|Oct 2, 2008

Astronomy and Astrophysical ResearchPhysics and Astronomy132 citations

TL;DR

This paper proposes a multiple-tracer technique using galaxies with different biases to measure redshift-space distortions (RSD) without cosmic variance, enabling precision measurement of the growth rate of structure $f^2P_m$ with up to 10× better precision than single-tracer methods for $\beta = 0.4$. The method allows near-direct measurement of $fD$ to 0.1% accuracy and significantly improves the Figure of Merit for dark energy constraints.

ABSTRACT

We show how to use multiple tracers of large-scale density with different biases to measure the redshift-space distortion parameter beta=f/b=(dlnD/dlna)/b (where D is the growth rate and a the expansion factor), to a much better precision than one could achieve with a single tracer, to an arbitrary precision in the low noise limit. In combination with the power spectrum of the tracers this allows a much more precise measurement of the bias-free velocity divergence power spectrum, f^2 P_m - in fact, in the low noise limit f^2 P_m can be measured as well as would be possible if velocity divergence was observed directly, with rms improvement factor ~[5.2(beta^2+2 beta+2)/beta^2]^0.5 (e.g., ~10 times better than a single tracer for beta=0.4). This would allow a high precision determination of f D as a function of redshift with an error as low as 0.1%. We find up to two orders of magnitude improvement in Figure of Merit for the Dark Energy equation of state relative to Stage II, a factor of several better than other proposed Stage IV Dark Energy surveys. The ratio b_2/b_1 will be determined with an even greater precision than beta, producing, when measured as a function of scale, an exquisitely sensitive probe of the onset of non-linear bias. We also extend in more detail previous work on the use of the same technique to measure non-Gaussianity. Currently planned redshift surveys are typically designed with signal to noise of unity on scales of interest, and are not optimized for this technique. Our results suggest that this strategy may need to be revisited as there are large gains to be achieved from surveys with higher number densities of galaxies.

Motivation & Objective

• To overcome the fundamental limitation of sample variance in measuring redshift-space distortions using single-tracer galaxy surveys.
• To develop a method that enables near-noise-limited precision measurement of the growth rate of structure $fD$ as a function of redshift.
• To improve cosmological constraints on dark energy by enhancing the Figure of Merit beyond current Stage II and proposed Stage IV surveys.
• To enable high-precision measurement of the bias ratio $b_2/b_1$ as a function of scale, probing nonlinear bias onset.
• To extend the technique to measure primordial non-Gaussianity $f_{\rm NL}$ with reduced cosmic variance.

Proposed method

• Use multiple tracers of large-scale density with distinct bias parameters $b_1$ and $b_2$ to cancel out sample variance in cross-correlation measurements.
• Measure the redshift-space distortion parameter $\beta = b^{-1}f$ via cross-power spectra between tracers with different biases.
• Leverage the fact that the cross-correlation power spectrum $P_{12}(k)$ is insensitive to cosmic variance when $b_1 \neq b_2$, enabling high-precision estimation of $f^2P_m$.
• Utilize the ratio of power spectra $P_{12}/P_{11}$ to extract $\beta$ and $b_2/b_1$ without cosmic variance, assuming linear bias models.
• Extend the method to test higher-order redshift-space distortion models by fitting for additional parameters like $\beta'$.
• Apply the technique to high-number-density surveys or artificial tracers (e.g., nonlinear transformations of a single field) to achieve high effective density without sacrificing volume.

Experimental results

Research questions

• RQ1 Can redshift-space distortions be measured without cosmic variance by using multiple tracers with different biases?
• RQ2 To what extent can the growth rate $fD$ be measured with sub-0.1% precision using this method?
• RQ3 How much improvement does this method offer in the Figure of Merit for dark energy constraints compared to current and planned surveys?
• RQ4 Can the bias ratio $b_2/b_1$ be measured as a function of scale with high precision to probe nonlinear bias?
• RQ5 Can this technique be extended to measure primordial non-Gaussianity $f_{\rm NL}$ with reduced sample variance?

Key findings

• The method achieves an rms improvement factor of $\sim 10$ in measuring $f^2P_m$ compared to a single tracer for $\beta = 0.4$, approaching the precision of direct velocity divergence measurements.
• The growth rate $fD$ can be determined with a statistical error as low as 0.1% using this technique in the low-noise limit.
• The Figure of Merit for dark energy equation of state constraints exceeds that of Stage II surveys by up to two orders of magnitude and surpasses other proposed Stage IV surveys by a factor of several.
• The bias ratio $b_2/b_1$ can be measured with even greater precision than $\beta$, enabling detailed study of the onset of nonlinear bias as a function of scale.
• The method allows for the measurement of higher-order redshift-space distortion parameters like $\beta'$ with the same precision as $\beta$, enabling robust model testing.
• The technique is applicable to high-number-density surveys or artificial tracers (e.g., nonlinear transformations of a single field), suggesting that current survey designs optimized for $\bar{n}P_g \sim 1$ may be suboptimal.