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Calibrating the cosmic distance scale ladder: the role of the sound horizon scale and the local expansion rate as distance anchors

Antonio J. Cuesta, Licia Verde, Adam Riess, Raul Jimenez

TL;DR

The paper constructs a cosmology-model independent cosmic distance ladder by combining Type Ia supernovae and BAO measurements, anchored either by the local expansion rate $H_0$ or by the sound horizon $r_d$ from the CMB. Through cosmology fits across several expansion histories, it yields a precise inverse-ladder $H_0 \approx 67.7 \pm 1.1$ km s$^{-1}$ Mpc$^{-1}$ and derives $r_d$ values from the ladder that are consistent with Planck within uncertainties, illustrating a broad agreement between direct and inverse calibrations. The expansion history shape $E(z)$ is tightly constrained by SN1a, with BAO providing additional, albeit sparser, leverage, and the results show no strong need for non-standard cosmologies. Overall, the work demonstrates a robust cross-validation of the cosmic distance scale and highlights how future tensions could signal new physics rather than methodological issues.

Abstract

We exploit cosmological-model independent measurements of the expansion history of the Universe to provide a cosmic distance ladder. These are supernovae type Ia used as standard candles (at redshift between 0.01 and 1.3) and baryon acoustic oscillations (at redshifts between 0.1 and 0.8) as standard rulers. We calibrate (anchor) the ladder in two ways: first using the local $H_0$ value as an anchor at $z$ = 0 (effectively calibrating the standard candles) and secondly using the cosmic microwave background-inferred sound-horizon scale as an anchor (giving the standard ruler length) as an inverse distance ladder. Both methods are consistent, but the uncertainty in the expansion history $H(z)$ is smaller if the sound horizon scale is used. We present inferred values for the sound horizon at radiation drag $r_d$ which do not rely on assumptions about the early expansion history nor on cosmic microwave background measurements but on the cosmic distance ladder and baryon acoustic oscillations measurements. We also present derived values of $H_0$ from the inverse distance ladder and we show that they are in very good agreement with the extrapolated value in a $Λ$CDM model from Planck cosmic microwave background data.

Calibrating the cosmic distance scale ladder: the role of the sound horizon scale and the local expansion rate as distance anchors

TL;DR

The paper constructs a cosmology-model independent cosmic distance ladder by combining Type Ia supernovae and BAO measurements, anchored either by the local expansion rate or by the sound horizon from the CMB. Through cosmology fits across several expansion histories, it yields a precise inverse-ladder km s Mpc and derives values from the ladder that are consistent with Planck within uncertainties, illustrating a broad agreement between direct and inverse calibrations. The expansion history shape is tightly constrained by SN1a, with BAO providing additional, albeit sparser, leverage, and the results show no strong need for non-standard cosmologies. Overall, the work demonstrates a robust cross-validation of the cosmic distance scale and highlights how future tensions could signal new physics rather than methodological issues.

Abstract

We exploit cosmological-model independent measurements of the expansion history of the Universe to provide a cosmic distance ladder. These are supernovae type Ia used as standard candles (at redshift between 0.01 and 1.3) and baryon acoustic oscillations (at redshifts between 0.1 and 0.8) as standard rulers. We calibrate (anchor) the ladder in two ways: first using the local value as an anchor at = 0 (effectively calibrating the standard candles) and secondly using the cosmic microwave background-inferred sound-horizon scale as an anchor (giving the standard ruler length) as an inverse distance ladder. Both methods are consistent, but the uncertainty in the expansion history is smaller if the sound horizon scale is used. We present inferred values for the sound horizon at radiation drag which do not rely on assumptions about the early expansion history nor on cosmic microwave background measurements but on the cosmic distance ladder and baryon acoustic oscillations measurements. We also present derived values of from the inverse distance ladder and we show that they are in very good agreement with the extrapolated value in a CDM model from Planck cosmic microwave background data.

Paper Structure

This paper contains 10 sections, 10 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. State-of the art and background
  3. Datasets
  4. BAO data
  5. SNe data
  6. Calibrating the Cosmic Ladder
  7. The Hubble constant and the inverse distance ladder
  8. The sound horizon from the distance ladder and $H_0$
  9. Expansion history between $0<z\lesssim1$
  10. Conclusions

Figures (8)

  • Figure 1: Schematic diagram of the redshift ranges covered by each of the datasets used in this paper. From low to high redshift we show the local measurement of the expansion rate $H_0$, the luminosity distances of type Ia supernovae, the distance determinations from the baryon acoustic oscillations in galaxy clustering, and the sound horizon scale in the cosmic microwave background.
  • Figure 2: Left panel: Comparison between local measurements (using Riess2011Humphreys2013 (dark blue, upper bar) and its reinterpretation by Efstathiou2014 (light blue, lower bar)) and CMB-derived measurements of the Hubble constant from Planck+WP data (labelled as Planck) for several assumed cosmologies. The error-bars correspond to 68 per cent confidence. The tension between the local and the CMB determinations is evident for some models ($\Lambda$CDM and O$\Lambda$CDM ) but not for others ($w$CDM or $N_{\rm eff}\Lambda$CDM). The two measurements labelled "Planck $\Lambda$CDM" refer to the Planck collaboration measurement (dark red, lower bar,Planck2013) and the re-analysis of Spergel2013 (light red, upper bar). The bars in faded out colors represent reinterpretations of the original datasets represented in solid colors. Right panel: the sound horizon scale: its determination is virtually cosmology-independent for cosmologies that differ on late-time history of the universe, but the determination is extremely sensitive to uncertainties in the early (pre-recombination) history.
  • Figure 3: Constraints in the $\Omega_m$--$H_0$ plane from BAO only, SN1a only, and the combinations BAO+SN1a+$H_0$ and BAO+SN1a+$r_d$. A $\Lambda$CDM model is assumed. Here $r_d$ is considered a derived parameter which depends on the densities of matter, baryons, and radiation, but we will drop that assumption in our analysis. The contours represent the 1 $\sigma$ and 2 $\sigma$ regions.
  • Figure 4: Left: The distance-redshift relation as probed by current BAO measurements. The quantity plotted is $D_V(z)/r_d=((1+z)^2D_A(z)^2cz/H(z))^{1/3}/r_d$ normalised by the values for our fiducial cosmology given by the best-fitting parameters from the Planck analysis for a $\Lambda$CDM model BAO measurements shown in black are not used here, but are included in this plot for completeness. Right: the luminosity distance-redshift relation from SN1a measurements normalised by the fiducial cosmology values. Here the JLA sample has been binned using 31 nodes equally separated in log(1+$z$). We remind the reader that these bins are correlated, therefore their full covariance matrix is included in our analysis and required to establish concordance with Planck.
  • Figure 5: Constraints on the sound horizon $r_d$ derived from SN1a+BAO+$H_0$ chains. Dashed lines show the constraints from CMB only, whereas solid lines show our results.
  • ...and 3 more figures