A Chi-Squared Analysis of the Measurements of Two Cosmological Parameters over Time

Abstract

The aim of this analysis was to determine whether or not the given error bars truly represented the dispersion of values in a historical compilation of two cosmological parameters: the amplitude of mass fluctuations (σ8) and Hubble’s constant (H0) parameters in the standard cosmological model. For this analysis, a chi-squared test was executed on a compiled list of past measurements. It was found through analysis of the chi-squared (χ2) values of the data that for σ8 (60 data points measured between 1993 and 2019 and χ2 between 182.4 and 189.0) the associated probability Q is extremely low, with Q=1.6×10−15 for the weighted average and Q=8.8×10−15 for the best linear fit of the data. This was also the case for the χ2 values of H0 (163 data points measured between 1976 and 2019 and χ2 between 480.1 and 575.7), where Q=1.8×10−33 for the linear fit of the data and Q=1.0×10−47 for the weighted average of the data. The general conclusion was that the statistical error bars associated with the observed parameter measurements have been underestimated or the systematic errors were not properly taken into account in at least 20% of the measurements. The fact that the underestimation of error bars for H0 is so common might explain the apparent 4.4σ discrepancy formally known today as the Hubble tension.

Hubble tensions: a historical statistical analysis

“…We obtain 0 = 68.26 ± 0.40 km s −1 Mpc −1 and 2 = 575.7 (Faerber & López-Corredoira 2020). For the value of 2 calculated using the weighted average, the probability that the observed trend is due to chance is = 1.0 × 10 −47 .…”

“…The correlation factor of 0 with time 2 is = 0.027 ± 0.013, a 2 significant correlation (Faerber & López-Corredoira 2020). Two sigma correlation is not a high significant detection, it is possibly a statistical fluctuation.…”

“…We have examined the trend and dispersion of values of the meaurements of 0 during 43 years . The probabilities for the distribution of different measurements of 0 and their errors are extremely low, both with respect to a constant value (weighted average of all the measurements) or with a linear fit (Faerber & López-Corredoira 2020). We need removing 24-27 measurements to reach a statistically significant dataset ( ≥ 0.05).…”

“…We use the compililation of 163 values for 0 between the years of 1976 and 2019 by Faerber & López-Corredoira (2020), where 120 measurements between 1990-2010 were taken from the previous compilation by Croft & Dailey (2015), The latest measurement we use is the value given by Riess et al (2019), which marked the origin of the present-day Hubble tension. Our goal is investigating the historical records previous to 2019, and not the most recent ones, because we want to investigate how common the kind of tension pointed out by Riess et al (2019) was until then.…”

Post-Editorial of “Universe: 5th Anniversary” Special Volume

“…Finally, he calculates some orbital consequences of his model which could be put to the test in the Solar System with future planetary ephemerides. At cosmological scales, T. Faerber and M. López-Corredoira [5] make a compilation of the measurements performed so far in the literature of the amplitude of mass fluctuations σ 8 and of the Hubble parameter H 0 investigating if the reported error bars are truly representative of the dispersion of the measured values of such two cosmological parameters. Their outcome is that the statistical error bars have been underestimated, or systematic errors were not properly taken into account in at least 20% of the measurements analyzed, suggesting that, in the case of the Hubble parameter, this may be the source of the recently observed H 0 tension among different kinds of measurements.…”