Model-minimal calibration of the low-redshift Universe: model-independent curvature, absolute BAO ruler scale, and redshift-stability of the standard ruler

The 2014 'standard rulers, candles, and clocks' analysis of Heavens, Jimenez & Verde measured the baryon acoustic oscillation (BAO) ruler length and the recent expansion history from low-redshift data under only minimal assumptions — homogeneity and isotropy, a metric theory of gravity, a smooth expansion history, and the existence of standard candles and a standard ruler — and explicitly flagged what remained open: curvature was constrained only weakly, a test for a redshift-evolving distortion of the observed ruler gave a null result with uncertainty $\pm 8$ Mpc per unit redshift, and the absolute ruler scale required external calibrators. A decade of follow-up work — SNe+clocks, strong-lensing, and cosmographic curvature estimates; Lyman-alpha and DESI-era BAO; late-time determinations of the sound horizon — has improved each piece but left the cores open: model-independent curvature still sits at $\sigma(\Omega_k)\sim 0.15$–$0.3$ versus $\sim 0.002$ for model-dependent Planck analyses, late-time determinations of the ruler scale remain at 2.5–4% versus 0.4%, and the explicit redshift-stability test of the ruler has not been repeated with modern data. The problem asks for a complete model-minimal calibration of the low-redshift Universe — curvature, ruler stability, and absolute ruler scale — precise enough to test, rather than presume, the standard cosmological model.

Background

The baryon acoustic oscillation (BAO) feature is a preferred clustering scale in the large-scale distribution of galaxies, imprinted by sound waves in the pre-recombination photon–baryon plasma; its comoving length is normally identified with the sound horizon at radiation drag (the epoch $z_d$ at which baryons are released from the photon plasma), $r_d=\int_{\infty}^{z_d}c_s(z)/H(z)\,dz$, where $c_s$ is the plasma sound speed and $H(z)$ the expansion rate. Because $r_d$ is set by early-Universe physics at $z\gtrsim 1000$, the measured BAO scale, combined with Type Ia supernovae (SNe Ia; 'standard candles' whose luminosity distances calibrate the distance–redshift relation) and 'standard clocks' (differential ages of passively evolving galaxies, which yield $H(z)$), ties late-time geometry to early-Universe physics. Heavens, Jimenez & Verde (2014) asked how much can be measured from low-redshift data ($z\lesssim 1.3$) under only four minimal assumptions: the cosmological principle (homogeneity and isotropy, i.e., a Friedmann–Lemaître–Robertson–Walker metric), a metric theory of gravity (so luminosity and angular-diameter distances obey distance duality), a smooth expansion history $H(z)$, and the existence of standard candles and a standard BAO ruler — with the supernova absolute magnitude and the ruler length left as free parameters. Parametrizing $H(z)$ by piecewise-linear interpolation and fitting SNe Ia and BAO (optionally plus galaxy clocks and/or a local $H_0$ prior), they measured the ruler length $r_d=103.9\pm 2.3\,h^{-1}$ Mpc without a cosmological model for the first time (versus the model-dependent Planck-$\Lambda$CDM prediction $99.3\pm 2.1\,h^{-1}$ Mpc, where $\Lambda$CDM is the cold-dark-matter-plus-cosmological-constant model and $h\equiv H_0/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}})$), obtained the absolute scale $r_d=142.8\pm 3.7$ Mpc only when external anchors were added, and constrained curvature only weakly. Curvature enters through the inverse curvature radius $\kappa=c/(R_0H_0)$, where $R_0$ is the present-day radius of curvature of the FLRW metric; in general relativity $\Omega_k=\pm\kappa^2$ is the usual curvature density parameter, but the model-minimal analysis constrains $\kappa$ without assuming general relativity. The authors explicitly acknowledged the open limitations that define this problem: the curvature constraint is weak; a test for a redshift-evolving distortion of the observed ruler (from nonlinear structure growth, redshift-space distortions — apparent radial shifts of galaxy positions caused by their peculiar velocities — or other late-time physics) gave only $0\pm 8$ Mpc per unit redshift; some input BAO measurements use reconstruction — a procedure that sharpens the BAO peak by undoing large-scale bulk flows under an assumption of Newtonian gravity; the absolute ruler scale depends on external calibrators (clocks, local $H_0$); and the inference of the effective number of relativistic species $N_{\rm eff}$ (the count of relativistic degrees of freedom in the early Universe; the source obtained $N_{\rm eff}=3.53\pm 0.32$) assumes the measured ruler is the early-Universe sound horizon. Subsequent work sharpened pieces of this program but not its cores: a sequel with SDSS-III DR12 data measured the ruler at the 2% level yet still with $\sigma(\Omega_k)\sim 0.3$–$0.4$; model-independent curvature estimates from SNe plus clocks, strong-lensing time delays, and cosmography (expanding the distance–redshift relation as a power series in redshift rather than assuming a model) remain at $\sigma(\Omega_k)\sim 0.15$–$0.3$, roughly two orders of magnitude above the model-dependent Planck-$\Lambda$CDM constraint ($\sigma(\Omega_k)\approx 0.002$); late-time determinations of $r_d$ without a CMB anchor reach only 2.5–4% versus 0.4% for Planck; and the explicit redshift-gradient test of the ruler has not been repeated with modern data. Meanwhile the model-dependent side is itself contested — a 2.5–3σ 'curvature tension' between Planck CMB lensing and BAO, and the preference of DESI DR2 BAO combined with the CMB for evolving dark energy — which makes an assumption-lean geometric determination of the low-redshift Universe a benchmark rather than a redundant exercise.

Problem Statement

Determine, from low-redshift observations — BAO, SNe Ia, and standard clocks or other late-time calibrators such as strong-lensing time delays (arrival-time differences between the multiple images of a gravitationally lensed source) or standard-siren distances (luminosity distances from gravitational-wave events) — under only the minimal assumptions of the source analysis (the cosmological principle, a metric theory of gravity, a smooth expansion history, and the existence of standard candles and a standard ruler) the geometry and calibration of the low-redshift Universe, to the point where each result tests rather than presumes the standard cosmological model. The problem comprises three deliverables sharing this single objective. (1) Curvature: a determination of the inverse curvature radius $\kappa=c/(R_0H_0)$, with $R_0$ the present-day radius of curvature (equivalently $\Omega_k=\pm\kappa^2$ if general relativity is assumed), from low-redshift data alone, at a precision $\sigma(\Omega_k)\le 0.05$ — a substantial improvement on the current model-independent constraints at the $\sigma(\Omega_k)\sim 0.2$–$0.4$ level, sufficient to test the near-flatness inferred from model-dependent CMB analyses without matching the $\approx 0.002$ precision of those analyses, which low-redshift data alone cannot reach. (2) Ruler stability: a direct test of whether the observed BAO ruler length is stable in redshift — bounding, or detecting, a redshift-evolution of the effective ruler, for example a linear gradient $dr_d/dz$ — improving on the 2014 null result of $0\pm 8$ Mpc per unit redshift, with the candidate late-time distortion mechanisms (nonlinear structure growth; redshift-space distortions, i.e. apparent radial shifts of galaxy positions caused by peculiar velocities; reconstruction assumptions) quantified and controlled. (3) Absolute calibration: a determination of the ruler length $r_d$ in physical units (Mpc) without a CMB-derived or otherwise model-dependent anchor, using only late-time calibrators (passive-galaxy clocks, local $H_0$, strong-lensing time delays, standard sirens), improving on the current 2.5–4% late-time determinations of $r_d$. Together these deliverables secure the source's stated aim: a self-consistent, assumption-lean calibration of the low-redshift distance–expansion relation that would survive if $\Lambda$CDM is wrong, so that agreement or disagreement between the low-redshift measurements of $r_d$, $\Omega_k$, and the expansion history and the early-Universe-calibrated predictions becomes a genuine test of the cosmological model rather than a consequence of assuming it.

References

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