... flat.¹

Or at least that it should be extremely close to flat: $\Omega = 1\pm\epsilon$ with $\epsilon$ exponentially small. I use $\Omega =1$ interchangeably with $\Omega = 1\pm\epsilon$ in this document. In particular, that means that when I refer to delta functions (infinitely narrow spikes) in the probability densities, those are really spikes with positive but extremely tiny widths.

Incidentally, there are inflationary models that don't predict $\Omega$ close to 1. Like many other people, I regard those models as very ill-motivated and unlikely, so I will assume that inflation implies $\Omega =1$.

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... find.²

Suppose, for instance, that someone in 1998 completely discounted the evidence suggesting that $\Omega = 0.2$. Then they might adopt a completely constant prior for $\Omega$, to indicate that they were completely ignorant of the value of $\Omega$. In that case the value of $\beta$ would be $(1-p_i)/ \Delta\Omega$, where $\Delta\Omega$ is the range of values of $\Omega$ considered possible. The range $\Delta\Omega$ would probably be significantly greater than one in that case, so $\beta$ would not be very large. Moreover, a good case can be made that the correct ``non-informative prior'' on $\Omega$ would be one in which the probability density of $\ln\Omega$ was flat rather than the probability density of $\Omega$. That would lead to a prior probability density on $\Omega$ proportional to $\Omega^{-1}$ and would generally end up giving a value of $\beta$ smaller than one would find for a flat prior.

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