Cosmology without cosmic variance

TL;DR

This paper shows that combining a large-scale redshift survey with a coincident weak-lensing survey enables sample-variance-free measurements of the growth rate $f$ and growth function $G$ in each Fourier mode, by leveraging the McDonald–Seljak bias-modulation method alongside bias calibration from lensing. Through a Fisher-matrix forecast, it demonstrates that such a joint survey can outperform significantly larger high-redshift redshift surveys, achieving precise tests of general relativity (e.g., constraining $oldsymbol{\\gamma}$ in $f=\,\Omega_m^{\gamma}$) with far fewer redshifts. The key result is that bias information from lensing breaks degeneracies and reduces the effective survey volume required, delivering growth constraints comparable to a tenfold increase in survey size, especially at low redshift where cosmic variance is most limiting. The analysis also highlights practical caveats, such as non-linear redshift-space effects and stochasticity in halo bias, which must be addressed to realize these gains in real data.

Abstract

We examine the improvements in constraints on the linear growth factor G and its derivative f=d ln G / dln a that are available from the combination of a large-scale galaxy redshift survey with a weak gravitational lensing survey of background sources. In the linear perturbation theory limit, the bias-modulation method of McDonald & Seljak allows one to distinguish the real-space galaxy clustering from the peculiar velocity signal in each Fourier mode. The ratio of lensing signal to galaxy clustering in transverse modes yields the bias factor b of each galaxy subset (as per Pen 2004), hence calibrating the conversion from galaxy real-space density to matter density in every mode. In combination these techniques permit measure of the growth rate f in each Fourier mode. This yields a measure of the growth rate free of sample variance, i.e. the uncertainty in f can be reduced without bound by increasing the number of redshifts within a finite volume. In practice, the gain from the absence of sample variance is bounded by the limited range of bias modulation among dark-matter halos. Nonetheless, the addition of background weak lensing data to a redshift survey increases information on G and f by an amount equivalent to a 10-fold increase in the volume of a standard redshift-space distortion measurement---if the lensing signal can be measured to sub-percent accuracy. This argues that a combined lensing and redshift survey over a common low-redshift volume is a more powerful test of general relativity than an isolated redshift survey over larger volume at high redshift. An example case is that a survey of ~10^6 redshifts over half the sky in the redshift range $z=0.5\pm 0.05$ can determine the growth exponent γfor the model $f=Ω_m^γ$ to an accuracy of $\pm 0.015$, using only modes with k<0.1h/Mpc, but only if a weak lensing survey is conducted in concert. [Abridged]

Figures (3)

• Figure 1: Schematic illustration of the McDonald-Seljak technique: in a chosen Fourier mode with real-space mass density fluctuation $\delta$, we observe the redshift-space amplitude $\delta^s_i$ of galaxies with different biases $b_i$. If the Kaiser formula (\ref{['deltasi']}) holds, then linear regression of the $\delta^s_i$ data points against bias will yield the $y$-intercept value $f\mu^2\delta$ and the $x$-intercept value $f\mu^2$. The former is one sample from a Gaussian with variance $f^2\mu^4P$. The latter gives $f$ without sample variance. However the uncertainty in $f$ from this mode's data is amplified if $\mu\ll1$, or if the range $\Delta b$ of galaxy biases is small compared to the typical $b+f\mu^2$ value.
• Figure 2: Forecasted uncertainty in growth parameters are plotted against the number of halo redshifts obtained in our fiducial survey: $z=0.5$, $\Delta z=0.1$, $f_{\rm sky}=0.5$. The survey is assumed to reach all halos above a minimum mass, marked on the top axis. Left and right panels assume different wavenumbers $k_{\rm max}$ to which linear perturbation theory is sufficiently accurate. The legend marks the different types of analyses used to extract redshift-space distortion information: blue and purple curves use single-bin or McDonald-Seljak bias binning to analyze pure galaxy-redshift information and extract the degenerate combination $fG$. With weak-lensing enables measurements of galaxy bias, $f$ and $G$ can be measured separately to the plotted accuracy, for perfectly known biases (red) and for a single weighted bias uncertain to the marked levels (green). Combination of weak lensing data with the galaxy redshift survey not only enables a direct measure of $f$, but a distinct measure of $G$ with substantially better precision. The upper horizontal dotted lines mark the cosmic-variance limits for the standard RSD measure, and the lower dotted lines are the sample variance limits for $\ln fG$ (MS method) or $\ln G$ (lensing$+$redshift methods).
• Figure 3: Forecasted constraints on deviations of the growth parameter $\gamma$ in Equation (\ref{['fgamma']}) from the General Relativity value of $\gamma=0.55$. The axes and line colors apply to the same fiducial survey and types of analyses as in Figure \ref{['sigtheta']}. In the case of $k_{\rm max}=0.03h\,{\rm Mpc}^{-1}$ (0.1), a lensing-based bias measurement to accuracy 0.01 (0.001) provides improves constraints on $\gamma$ over a standard redshift-space-distortion experiment by an amount equivalent to a 10-fold increase in survey size.