[Paper Review] A comparison of Bayesian and frequentist confidence intervals in the presence of a late Universe degeneracy

Eoin Ó Colgáin, Saeed Pourojaghi+2 more|arXiv (Cornell University)|Jul 30, 2023

Cosmology and Gravitation TheoriesPhysics and Astronomy3 citations

TL;DR

This paper investigates the Hubble tension by analyzing cosmological parameter inference in the late Universe using Markov Chain Monte Carlo (MCMC) and profile likelihood methods. It finds that MCMC marginalization fails to track the χ² minimum at 68% confidence in high-redshift bins, and using profile distributions reveals a >2σ discrepancy between cosmic chronometer and BAO data beyond z ≈ 1.5, suggesting potential ΛCDM model breakdown in the late Universe.

ABSTRACT

Hubble tension is a problem in one-dimensional (1D) posteriors, since local $H_0$ determinations are only sensitive to a single parameter. Projected 1D posteriors for $Λ$CDM cosmological parameters become more non-Gaussian with increasing effective redshift when the model is fitted to redshift-binned data in the late Universe. We explain mathematically why this non-Gaussianity arises and show using observational Hubble data (OHD) that Markov Chain Monte Carlo (MCMC) marginalisation leads to 1D posteriors that fail to track the $χ^2$ minimum at $68\%$ confidence level in high redshift bins. To gain a second perspective, we resort to profile likelihoods as a complementary technique. Doing so, we observe that $z \gtrsim 1$ cosmic chronometer (CC) data currently prefers a non-evolving (constant) Hubble parameter over a Planck-$Λ$CDM cosmology at $\sim 2 σ$. Within the Hubble tension debate, it is imperative that subsamples of data sets with differing redshifts yield similar $H_0$ values. In addition, we confirm that MCMC degeneracies observed in 2D posteriors are not due to curves of constant $χ^2$. Finally, on the assumption that the Planck-$Λ$CDM cosmological model is correct, using profile likelihoods we confirm a $>2 σ$ discrepancy with Planck-$Λ$CDM in a combination of CC and baryon acoustic oscillations (BAO) data beyond $ z \sim 1.5$. This confirms a discrepancy reported earlier with fresh methodology.

Motivation & Objective

• To investigate the reliability of MCMC marginalization in producing accurate 1D posterior distributions for cosmological parameters in high-redshift late-Universe data.
• To assess whether observed discrepancies in H₀ and S₈ tensions stem from model breakdown rather than systematics by testing for redshift-dependent evolution of ΛCDM parameters.
• To compare MCMC-based inference with profile likelihood methods in detecting deviations from Planck-ΛCDM predictions in high-redshift data.
• To evaluate whether cosmic chronometer (CC) and baryon acoustic oscillation (BAO) data beyond z ≈ 1.5 are inconsistent with the Planck-ΛCDM model under a consistent statistical framework.

Proposed method

• Uses observational Hubble data (OHD) and redshift-binned cosmic chronometer (CC) data to assess parameter inference in the late Universe.
• Applies MCMC sampling to compute 1D and 2D posterior distributions for H₀ and Ωₘ, comparing them to χ² minima.
• Employs profile likelihood distributions as a complementary method to MCMC, particularly to assess confidence intervals in non-Gaussian posteriors.
• Reconstructs H(z) and Dₘ(z) using the ΛCDM model and compares observed data trends with Planck-ΛCDM predictions.
• Uses Fisher matrix analysis to compute parameter errors and assess parameter degeneracies, with derivatives computed from the Friedmann equation.
• Re-evaluates a prior 2σ tension between high-redshift CC and BAO data using updated BAO constraints and profile likelihoods.

Experimental results

Research questions

• RQ1 Does MCMC marginalization accurately reflect the χ² minimum in 1D posteriors for ΛCDM parameters when fitting high-redshift OHD?
• RQ2 Is there evidence of redshift-dependent evolution in H₀ and Ωₘ in the late Universe that contradicts the Planck-ΛCDM model?
• RQ3 Do cosmic chronometer and BAO data beyond z ≈ 1.5 show a statistically significant discrepancy with Planck-ΛCDM predictions when using profile likelihoods?
• RQ4 Can the observed Hubble tension and S₈ tension be explained by a single underlying cause—namely, evolution of cosmological parameters in the late Universe?

Key findings

• MCMC marginalization fails to track the χ² minimum at the 68% confidence level in high-redshift bins due to non-Gaussian posteriors arising from redshift binning.
• Profile likelihood analysis reveals that CC data at z ≳ 1 prefer a non-evolving Hubble parameter over the Planck-ΛCDM model at approximately 2σ significance.
• A >2σ discrepancy is confirmed between high-redshift CC and BAO data beyond z ≈ 1.5 when using profile distributions, supporting earlier findings with fresh methodology.
• The observed parameter evolution is not due to χ² contours but stems from the non-Gaussianity of posteriors induced by redshift binning in OHD.
• The discrepancy persists even when a Planck prior on Ωₘh² is applied, indicating robustness to external constraints.
• The results suggest that the H₀ and S₈ tensions may not be independent but could instead be symptoms of ΛCDM model breakdown in the late Universe.