By P. Mannheim
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39 in the closed form dL = c z2 z+ H(t0 ) 2 . 01) and no other components, we see that while this model still gives a χ2 which is somewhat larger than the number of independent degrees of freedom, the model does not fare all that badly, suggesting that it may have captured some of the essence of the Hubble plot data. 44 Within a pressureless ΩM (t0 ), Ωk (t0 ) model, the only way to reduce the χ2 even lower than its ΩM (t0 ) = 0, Ωk (t0 ) = 1 value is to make either G or ρm negative – not that one of course can do so in the model, though it does show the needed trend, with the data favoring an even more reduced deceleration parameter, viz.
More naturally, it would have ΩM (t) fall to zero in the early universe itself. 2 Fine-tuning and the inflationary universe A beautiful solution to this conflict was proposed by Guth , the so-called inflationary universe, in which a de Sitter phase was to precede the current Robertson-Walker phase, with the rapid exponential expansion of the de Sitter phase precisely providing a dynamics which would lead to the needed initial condition for the subsequent Robertson-Walker phase, no matter what the value of k.
While the χ2 fitting itself is not at all sensitive to the value used for H(t0 ) which is required for the luminosity distance dL, the extraction of an actual value for M itself from the fitting is. 222. With an increasing value for the Hubble parameter leading to a decreasing 5log10 (c/H(t0)), an increasing value for the Hubble parameter thus also leads to an increasing derived M. In its turn an increased M leads to an increased (dimmer) value for m as extracted from a given distance modulus. On making this extraction, as we see from Fig.