The objective of the present paper is not to propose an alternative method to compute the age of the universe or the luminosity distance in the situations relevant for the observed universe in which these expressions are already known (the computation method remains the same in these cases). The truncation of the hypergeometric series, or the equivalent Chebyshev theorem, were used in the 1960s, and were recently rediscovered, to derive two- and three-fluid (or effective fluid) analytical solutions of the Einstein–Friedmann equations (see for a review). ![]() Equivalently, it happens when the integral expressing lookback time, age, or luminosity distance is of a special form contemplated by the Chebyshev theorem of integration. This simplification happens when the hypergeometric series expressing them truncate. Of course, lookback time, age, and luminosity distance can always be computed numerically in a given cosmological model, however one would also like to know when they can be computed analytically in terms of a finite number of elementary functions. Lookback time, age of the universe, and redshift-luminosity distance relation, of crucial importance for modern cosmology, are expressed by integrals taking the form of infinite hypergeometric series. ![]() ![]() When can we compute exactly lookback time, age of the universe, and redshift-luminosity distance relation D L ( z ) in Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology? The age of the universe t 0 sets an upper bound on the present value H 0 of the Hubble function, with implications for the current Hubble tension.
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