The recently unveiled deep-field images from the James Webb Space Telescope have renewed interest in what we can and cannot see of the universe. Answering these questions requires understanding the so-called “cosmological horizons” and the “Hubble sphere.” Here, we review the topic of cosmological horizons in a form that university physics teachers can use in their lessons, using the latest data about the so-called standard “Lambda cold dark matter” (ΛCDM) model. Graphical representations are plotted in terms of both conformal and proper coordinates as an aid to understand the propagation of light in the expanding universe at various epochs.

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15.
Observe that by virtue of Eq. (27), the uncertainties of Ωm and Ω Λ are equal.
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For a review of Gaussian hypergeometric functions, see, e.g., <https://mathworld.wolfram.com/HypergeometricFunction.html>.
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An educational presentation on numerical integration can be found at <https://mathworld.wolfram.com/NumericalIntegration.html>.
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