How to measure redshift-space distortions without sample variance

TL;DR

This work introduces a multi-tracer strategy to measure redshift-space distortions with dramatically reduced sample variance by using two galaxy populations with different biases. Through a Fisher-matrix treatment of the two-tracer perturbations, it shows that $\beta=f/b$ and the velocity-divergence spectrum $P_{\theta\theta}=f^2P_m$ can be constrained far beyond single-tracer limits, potentially matching the precision of directly observing the velocity field in the low-noise limit. The authors quantify substantial improvements in the Dark Energy FoM when including $P_{\theta\theta}$ measurements across various planned surveys (e.g., EUCLID, SNAP, SDSS-BOSS), and they extend the approach to non-Gaussianity $(f_{\rm NL})$, showing powerful constraints with unbiased tracers. They discuss practical considerations for survey design, the Alcock-Paczynski effect, and the synergy with weak lensing, arguing for higher tracer densities to fully exploit the no-cosmic-variance potential of multi-tracer RSD measurements.

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.

Figures (13)

• Figure 1: Projected fractional error on the normalization of ${P_{\theta\theta}}\equiv f^2 P_m$, for 30000 square degrees, in redshift bins with width $dz=0.2$. The upper and lower green (short-dashed) lines show the constraints from single tracers with $b=2$ and $b=1$, respectively. Black (solid) lines show the two tracers together, each with, from top to bottom, S/N=1, 3, 10, 30, 100 at $k=0.4\, h\, {\rm Mpc}^{-1}$. The red (long-dashed) line shows the case where both tracers are perfectly sampled. For the left panel we assume $k_{\rm max}(z)= 0.1~\left[D\left(z\right)/D\left(0\right)\right]^{-1}\, h\, {\rm Mpc}^{-1}$, while the right assumes $k_{\rm max}(z)= 0.05~\left[D\left(z\right)/D\left(0\right)\right]^{-1}\, h\, {\rm Mpc}^{-1}$.
• Figure 2: Dependence of ${P_{\theta\theta}}\left(k=0.1\, h\, {\rm Mpc}^{-1},z\right)$ on cosmological parameters. The derivatives are at fixed values of the other parameters, including not-shown parameters $\omega_b$, $\theta_s$, and $n_s$. The derivatives are normalized by typical projected errors on the parameters, for the scenario described in Fig. \ref{['figallparamsvsz']}, with $z_{\rm max}=1$. Lines correspond to: black (solid): $w_0$, green (long-dashed): $w^\prime$, red (dot-short-dashed): $\Omega_k$, magenta (dot-long-dashed): $\omega_m$, cyan (dotted): $\log A$. Error bars show the projected errors on ${P_{\theta\theta}}$ for $k_{\rm max}(z)= 0.1~\left[D\left(z\right)/D\left(0\right)\right]^{-1}\, h\, {\rm Mpc}^{-1}$, and $S/N=10$.
• Figure 3: Improvement in Dark Energy constraints (quantified by the DETF FoM), relative to Planck + DETF Stage II, when the ${P_{\theta\theta}}$ constraints from Fig. \ref{['figf2Pvsz']} are added. The lines refer to the same cases as in Fig. \ref{['figf2Pvsz']}, and again the left panel is $k_{\rm max}(z)= 0.1~\left[D\left(z\right)/D\left(0\right)\right]^{-1}\, h\, {\rm Mpc}^{-1}$ while the right is $k_{\rm max}(z)= 0.05~\left[D\left(z\right)/D\left(0\right)\right]^{-1}\, h\, {\rm Mpc}^{-1}$.
• Figure 4: Improvement in Dark Energy constraints (quantified by the DETF FoM), relative to Planck + DETF Stage II, when the ${P_{\theta\theta}}$ constraints from Fig. \ref{['figf2Pvsz']} are added, and the BAO distance measurements from the same survey is added. The lines refer to the same cases as in Fig. \ref{['figf2Pvsz']}, except the blue (dotted) line is the case with BAO alone. The left and right panels again show the stronger and weaker values of $k_{\rm max}$, respectively.
• Figure 5: Improvements in constraints on various parameters, vs. $z_{\rm max}$, for the case including BAO, $k_{\rm max}(z)= 0.1~\left[D\left(z\right)/D\left(0\right)\right]^{-1}\, h\, {\rm Mpc}^{-1}$, and $S/N=10$ for the ${P_{\theta\theta}}$ measurement. Lines are: black (solid): $w_0$, green (long-dashed): $w^\prime$, red (dot-short-dashed): $\Omega_k$, blue (dashed): $\Omega_\Lambda$, magenta (dot-long-dashed): $\omega_m$, cyan (dotted): $\log A$.