Measuring the cosmological constant with redshift surveys

TL;DR

The paper addresses measuring the cosmological constant $\Lambda$ through geometric distortions in large-scale structure, but shows that the detectable squashing factor $F$ remains modest (typically $F \lesssim 1.3$ for realistic $\Omega_m$) and can be confounded with redshift-space distortions. It develops a maximum-likelihood framework that jointly fits geometry and velocity distortions in the anisotropic power spectrum, incorporating nonlinear damping and survey weighting. The work finds that distinguishing geometry from redshift-space effects hinges on large, high-fidelity datasets and careful control of systematics, with $\beta$-dependent degeneracies and evolution requiring partitioned redshift analysis. Applying the method to next-generation galaxy and quasar surveys suggests a real, though challenging, potential to detect a cosmologically significant $\Lambda$, especially when combining multiple tracers and exploiting high-redshift information.

Abstract

It has been proposed that the cosmological constant $Λ$ might be measured from geometric effects on large-scale structure. A positive vacuum density leads to correlation-function contours which are squashed in the radial direction when calculated assuming a matter-dominated model. We show that this effect will be somewhat harder to detect than previous calculations have suggested: the squashing factor is likely to be $<1.3$, given realistic constraints on the matter contribution to $Ω$. Moreover, the geometrical distortion risks being confused with the redshift-space distortions caused by the peculiar velocities associated with the growth of galaxy clustering. These depend on the density and bias parameters via the combination $β\equiv Ω^{0.6}/b$, and we show that the main practical effect of a geometrical flattening factor $F$ is to simulate gravitational instability with $β_{\rm eff}\simeq 0.5(F-1)$. Nevertheless, with datasets of sufficient size it is possible to distinguish the two effects; we discuss in detail how this should be done. New-generation redshift surveys of galaxies and quasars are potentially capable of detecting a non-zero vacuum density, if it exists at a cosmologically interesting level.

Figures (8)

• Figure 1: (a) Flattening factor $F$ as a function of redshift for flat models ($\Omega_{m} + \Omega_{\Lambda} = 1$). The curves range from $\Omega_{\Lambda} = 1$ (top curve) to $\Omega_{\Lambda} = -1$ in steps of $0.1$. (b) $F$ against redshift for models with a fixed mass density $\Omega_{m} = 0.2$ and a cosmological constant. $\Omega_{\Lambda}$ varies as for (a). (c) $F$ against redshift for models with no cosmological constant. $\Omega_{m}$ varies from 0 (top) to 2, again in steps of 0.1.
• Figure 2: Contours of the power spectrum in the $k_{\parallel}$ and $k_{\perp}$ plane, assuming a power-law index $n=-1.5$ for the true power spectrum. The contour interval is $\Delta {\ln P} = {1/2}$. The models are linear (Kaiser) redshift distortion only with $\beta = 0.5$ (full contours) and geometric squashing effect only, with $F=1.5$ (dotted contours).
• Figure 3: Contours for both linear and nonlinear redshift distortions with $\beta = 0.5$, and $\sigma_{\rm p} = 350\; {\rm kms}^{-1}$ (again for $n = -1.5$). The solid contours are for exponential small scale velocity distribution, the dotted Gaussian. The contour interval is $\Delta {\ln P} = {1/2}$. The stretching of contours of the correlation function along the line of sight leads to a reduction in line of sight power on small scales (large k) -- see also Cole, Fisher & Weinberg (1994).
• Figure 4: Solid contours for a model with $F=1.1$, $\beta = 0.5$ and $\sigma_{\rm p} = 350 \;{\rm kms}^{-1}$. Dotted contours are for the best fit to this model with redshift distortions only ($F=1$).
• Figure 5: Expected contours of likelihood in the $F-\beta$ plane, for the case of a survey with negligible shot noise, for three values of $\beta$. The scaling is set so that $\sigma(\beta)=0.1$ for $\beta=1$. At each $(\beta,F)$ point, the maximum-likelihood value of the power-spectrum amplitude has been chosen. The contour interval is $\Delta{\ln {\cal L}} = 1/2$. The position of the true values of the parameters is marked by the cross; contours are shown for $\beta = 0.3$, $0.6$ and $1.0$.