Abstract
The luminosity distance in the standard cosmology as given by ΛCDM and, consequently, the distance modulus for supernovae can be defined by the Padé approximant. A comparison with a known analytical solution shows that the Padé approximant for the luminosity distance has an error of 4 % at redshift = 10 . A similar procedure for the Taylor expansion of the luminosity distance gives an error of 4 % at redshift = 0 . 7 ; this means that for the luminosity distance, the Padé approximation is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg–Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Padé approximant. A comparison of ΛCDM with other cosmologies is done adopting a statistical point of view.
Highlights
In order to obtain astronomical observables, such as the distance modulus and the absolute magnitude for supernovae (SN) of Type Ia in the standard cosmological approach, as given by the ΛCDM model, we need the evaluation of the luminosity distance, which is derived from the comoving distance
The statistical parameters for the different cosmologies here analyzed can be found in Table 5 in the case of the Union 2.1 compilation and in Table 6 for the joint light-curve analysis (JLA) compilation
It is generally thought that in the case of the luminosity distance, the Padé approximant is more accurate than the Taylor expansion
Summary
In order to obtain astronomical observables, such as the distance modulus and the absolute magnitude for supernovae (SN) of Type Ia in the standard cosmological approach, as given by the ΛCDM model, we need the evaluation of the luminosity distance, which is derived from the comoving distance. Once an approximate solution is obtained for the luminosity distance, we can evaluate the distance modulus and the absolute magnitude for SNs. the minimax rational approximation can provide a compact formula for the two above astronomical observables as functions of the redshift. The approximate magnitude here derived is applied to parametrize a new luminosity function for galaxies at high redshift; see Section 3. Galaxies 2016, 4, 4 cosmologies is reviewed, and the main statistical parameters connected with the distance modulus are derived; see Section 4
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