Dice Question Streamline Icon: https://streamlinehq.com

Ambiguity of AIC-based model preference when chi-squared per degree of freedom is below unity

Ascertain whether a parametrization with a lower chi-squared should be preferred according to the Akaike Information Criterion when the fits yield chi-squared per degree of freedom less than one for the DESI DR1 BAO, Pantheon+ supernovae, and Planck CMB shift-parameter combination.

Information Square Streamline Icon: https://streamlinehq.com

Background

The authors compare several parametrizations (constant w, CPL, and higher-order extensions) using chi-squared and AIC. Although CPL shows a lower chi-squared than ΛCDM, all tested models yield chi-squared values significantly below the number of degrees of freedom, implying chi-squared per dof < 1 across the board.

They explicitly note that in such a regime it is not clear that a lower chi-squared indicates a genuinely better model under AIC, potentially due to overestimated observational errors, thereby leaving open the question of how to interpret AIC preferences when chi-squared per dof is below unity.

References

It is perhaps appropriate at this point to note that all of the $\chi2$ values in \cref{tab:AICofCPLx} are significantly lower than the number of degrees of freedom (dof) $\simeq 1600$ minus the number of parameters, so it is not actually clear that a parametrization with a lower $\chi2$ is preferred, as the AIC would suggest.

— To CPL, or not to CPL? What we have not learned about the dark energy equation of state (2503.22529 - Nesseris et al., 28 Mar 2025) in Results, paragraph following Table I (AIC and chi-squared comparison)