A comparison of Bayesian and frequentist confidence intervals in the presence of a late Universe degeneracy

Abstract Hubble tension is a problem in one-dimensional (1D) posteriors, since local $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> determinations are only sensitive to a single parameter. Projected 1D posteriors for $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> 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 $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> minimum at $$68\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>68</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> 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 $$z \gtrsim 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>≳</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> cosmic chronometer (CC) data currently prefer a non-evolving (constant) Hubble parameter over a Planck- $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM cosmology at $$\sim 2 \sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>2</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> . Within the Hubble tension debate, it is imperative that subsamples of data sets with differing redshifts yield similar $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> values. In addition, we confirm that MCMC degeneracies observed in 2D posteriors are not due to curves of constant $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . Finally, on the assumption that the Planck- $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM cosmological model is correct, using profile likelihoods we confirm a $$&gt;2 \sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> discrepancy with Planck- $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM in a combination of CC and baryon acoustic oscillations (BAO) data beyond $$z \sim 1.5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>1.5</mml:mn> </mml:mrow> </mml:math> . This confirms a discrepancy reported earlier with fresh methodology.