The length of the low-redshift standard ruler

TL;DR

This work addresses measuring the length of the low-redshift standard ruler in a model-independent way by combining $H(z)$ information from Type Ia supernovae and cosmic clocks with BAO data. It reconstructs the expansion history using a node-based representation of ${ m h^{-1}}(z)$ and derives both the relative ruler $r_{ m s}^h$ in $h^{-1}$ Mpc and the absolute ruler $r_{ m s}$ in Mpc by anchoring with $H_0$ or clocks. The key findings show $r_{ m s}^h=101.0 \pm 2.3$ $h^{-1}$ Mpc from BAO+SN, $r_{ m s}=150.0 \pm 4.7$ Mpc with clocks, $r_{ m s}=141.0 \pm 5.5$ Mpc with local $H_0$, and $r_{ m s}=143.9 \pm 3.1$ Mpc when all data are combined, with only mild sensitivity to curvature and strong consistency with Planck CMB results within ΛCDM for the combined data. The study demonstrates robustness to prior choices and interpolation schemes, and forecasts substantial improvements from upcoming BAO surveys (e.g., DESI) and refined $H_0$ probes, which will tighten the ruler measurements and test early-universe physics more stringently.

Abstract

Assuming the existence of standard rulers, standard candles and standard clocks, requiring only the cosmological principle, a metric theory of gravity, a smooth expansion history, and using state-of-the-art observations, we determine the length of the "low-redshift standard ruler". The data we use are a compilation of recent Baryon acoustic oscillation data (relying on the standard ruler), Type 1A supernovæ (as standard candles), ages of early type galaxies (as standard clocks) and local determinations of the Hubble constant (as a local anchor of the cosmic distance scale). In a standard $Λ$CDM cosmology the "low-redshift standard ruler" coincides with the sound horizon at radiation drag, which can also be determined --in a model dependent way-- from CMB observations. However, in general, the two quantities need not coincide. We obtain constraints on the length of the low-redshift standard ruler: $r^h_{\rm s}=101.0 \pm 2.3 h^{-1}$ Mpc, when using only Type 1A supernovæ and Baryon acoustic oscillations, and $r_{\rm s}=150.0\pm 4.7 $ Mpc when using clocks to set the Hubble normalisation, while $r_{\rm s}=141.0\pm 5.5 $ Mpc when using the local Hubble constant determination (using both yields $r_{\rm s}=143.9\pm 3.1 $ Mpc). The low-redshift determination of the standard ruler has an error which is competitive with the model-dependent determination from cosmic microwave background measurements made with the {\em Planck} satellite, which assumes it is the sound horizon at the end of baryon drag.

Figures (4)

• Figure 1: At glance: comparison of central values and 1$\sigma$ errors on the $r_{\rm s}^h$ (left) and $r_{\rm s}$ (right) measurements for flat geometry (blue) and marginalizing over the curvature (black). Note the change of the scale in the x-axis in each figure.
• Figure 2: Reconstructed expansion history $H(z)$ (95% confidence envelop) for two representative dataset combinations: SBH (left) CSBH (right). The last redshift nodes (one on the left, two on the right) are not shown as there $H(z)$ is poorly constrained. The jagged shape of the envelop is due to the linear interpolation being performed in $1/H$ while the quantity plotted is $H(z)$. Symbols represent the best fit values for the reconstruction.
• Figure 3: Effects of prior assumptions and MCMC sampling method. We show the comparison of the posterior distributions of $\Omega_k$ (left) and in the $\Omega_k$-$r_{\rm s}^h$ plane (right) obtained from the same data (SBH) with different methodologies: this work (blue), using an Affine Invariant sampler instead of Metropolis Hastings (red) with two choices for the redshift sampling, the one form this work (solid) and the one from BVR (dashed), and the approach of BVR (green), which uses Affine Invariant sampler, $r_{\rm s}$ and $H(z_i)$ as variables and a spline interpolation of $H(z)$.
• Figure 4: Effect of combining the low-redhisft standard ruler measurement (interpreted as the sound horizon at radiation drag) with CMB Planck observations. The transparent contours show the joint $r_{\rm s}$ vs $N_{\rm eff}$ 68% and 95% marginalised confidence regions obtained from the posterior sample provided by the Planck CMB mission. On the left, all temperature and polarisation data are used, on the right, high $\ell$ polarisation data are not included. The filled contours result from importance sampling this with our CSB measurement (top row), SBH (middle row) and CSBH (bottom row).