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.
Highlights
The standard cosmological model consists of 12 parameters [1]: Ω M is the ratio of the current matter density to the critical density, ΩΛ is the cosmological constant as a fraction of the critical density, H0 is Hubble’s constant, σ8 is the amplitude of mass fluctuations, Ωb is the baryon density as a fraction of the critical density, n is the primordial spectral index, β is the redshift distortion, mv is the neutrino mass, Γ is Ωm H0 /100 kms−1 Mpc−1, Ω0.6 m σ8 is a combination of two other parameters that is useful in some peculiar velocity and lensing measurements, Ωk is the curvature, and w0 is the equation of state for the dark energy parameter [1]
The original Q values for both the weighted average and best fit calculations of the probability of the data for both parameters are extremely low before the removal of bad values
Even though this is the case, a rather large discrepancy can be seen in how many bad values need removing to reach a statistically significant dataset (Q ≥ 0.05)
Summary
The standard cosmological model is a model that aims to describe the evolution and structure of the Universe that we live in. The amplitude of mass fluctuations (σ8 ) is a parameter in the standard cosmological model that is concerned with the respective distributions of mass and light in the Universe [2]. It is important for cosmologists to study and understand the distribution tendencies of mass and light in the Universe through σ8 because large-scale differences in distribution of matter and energy in the present-day Universe tell us about density fluctuations in the early Universe on the cluster mass scale of R = 8 h−1 Mpc [2]. Knowing the exact value of H0 is important to cosmologists, as H0 can be used to roughly calculate the age of the Universe
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