Constraining dark energy models using Jackknife and Bootstrap resampling

TL;DR

This work investigates how non-parametric resampling techniques can constrain dark energy models using the PPS dataset, by applying Generalised Least Squares and evaluating bias/variance with Jackknife and Bootstrap, while also comparing to Bayesian constraints from MCMC and nested sampling. It analyzes multiple models, including $Λ$CDM, flat $Λ$CDM, $w$CDM, flat $w$CDM, and flat CPL $w_0\,w_a$CDM, highlighting how parameter biases and uncertainties depend on model complexity. A key finding is that the Jackknife method reveals a strong positive correlation between $h$ and $M$ and larger uncertainties for these parameters, with implications for the Hubble tension, while Bootstrap can show multimodalities in some parameters for higher-dimensional models. Overall, the study demonstrates the value of resampling as a diagnostic tool for assessing data limitations and parameter identifiability, and it underscores the need for improved data to robustly distinguish among dynamical dark energy scenarios.

Abstract

Analyses of type Ia supernovae have helped us shed light on the existence and nature of dark energy. Most of these analyses have relied on Bayesian techniques. In this work, we rely on resampling techniques to analyse supernova data. In particular, we use the generalised least squares method together with Jackknife and Bootstrap techniques to estimate parameters of $Λ$CDM, flat $Λ$CDM, $w$CDM, flat $w$CDM, and flat $w_0\,w_a$CDM models from the recent PantheonPlus and SH0ES data. For completeness, we also perform Bayesian analysis using Markov chain Monte Carlo (MCMC) and nested sampling algorithms, and compare the results. We note that resampling techniques can help highlight the limitations of the data. For instance, we see that the Jackknife method estimates a strong positive correlation between $h$ and $M$ and higher standard deviations for both. This may have significant implications for the Hubble tension. We conclude with a discussion of our results.

Figures (5)

• Figure 1: Figure compares the marginalised confidence contours between different pairs of cosmological parameters in flat $w_0\,w_a$CDM model (left) and $w$CDM model (right) obtained with emcee (blue contours) and pypolychord (red contours). These two models have been chosen because they have the most parameters. Plots show that both methods largely lead to similar results.
• Figure 2: Corner plots for $\Lambda$CDM (left) and flat $\Lambda$CDM models (right). Differently coloured contours are obtained by using different methods. Bayesian contours are obtained using pypolychord. From two-dimensional confidence contours, we see that the Bayesian contours are more conservative and that all three frequentist methods are consistent with them. An exception is the confidence contour between parameters $h$ and $M$. We find that the Jackknife confidence contour between $h$ and $M$ shows a high correlation. With Jackknife, we also find a large standard deviation for these parameters. For the flat $\Lambda$CDM model, this alleviates the Hubble tension. In fact, Planck's estimate of $H_0$ lies well within $3-\sigma$ of the estimate using the Jackknife Planck:2018vyg. Further, in the flat $\Lambda$CDM model, we note that the bias-corrected Jackknife estimate of $\Omega_m$ is about $2\sigma$ lower than the GLS estimate.
• Figure 3: Same as Figure \ref{['fig:lcdm_flcdm']}, but for $w$CDM (left) and flat $w$CDM (right) models. As observed in Figure \ref{['fig:lcdm_flcdm']}, in general, Bayesian confidence contours are more conservative. An exception is the Jackknife contours for $h$ and $M$, which show high correlation. We also note that the standard deviations of $h$ and $M$ are large, allowing a smaller value of $H_0$. Above statements hold for both models. For flat $w$CDM, results obtained using all methods are consistent. However, for $w$CDM model, some of the Jackknife contours, associated with $\Omega_m$, $\Omega_{\rm DE}$, and $w_0$ deviate from other methods. We also note the multimodal behaviour in Bootstrap contours associated with these parameters.
• Figure 4: Same as Figure \ref{['fig:lcdm_flcdm']}, but for flat $w_0\,w_a$CDM models. In this figure also, we note that Bayesian contours are, in general, broader. An exception is the case of Jackknife contours in the $h$ and $M$ plane, which exhibit high correlation. Further, we note that standard deviation of $h$ and $M$ are broader allowing lower values of $H_0$. In addition, we note that as in the case of $w$CDM model, which also has five parameters, certain jackknife contours involving $\Omega_m$ and $w_0$ are not consistent with other methods. We also note that Bootstrap contours and probability distributions for $\Omega_m$ and $w_a$ exhibit multimodality.
• Figure 5: Figure compares the marginalised confidence contours between different pairs of cosmological parameters for the dark energy models considered in this work, obtained with pypolychord nested sampling algorithm. Note that different models have different sets of parameters. Further, since we have not investigated $w_0\,w_a$CDM model, which has six parameters, there are no contours in $w_a$ - $\Omega_{\rm DE}$ plane. We note with interest that $\Omega_m$ is best constrained for flat $\Lambda$CDM model and that constraints on $h$ and $M$ are largely independent of the choice of model.