$\Lambda$CDM Cosmology for Astronomers

TL;DR

This paper provides simplified derivations and accurate equations connecting cosmological observables with intrinsic properties of distant sources within the $\\Lambda$CDM model, facilitating easier analysis for astronomers.

Contribution

It introduces straightforward derivations and highly accurate approximations for key cosmological equations used in observational astronomy within the $\\Lambda$CDM framework.

Findings

An analytic lookback time equation accurate within 0.1% for all redshifts.

A simple approximation for comoving distance with errors less than 0.2% up to redshift 50.

Clarification that the universe can be described with Euclidean geometry and basic relativity principles.

Abstract

The homogeneous, isotropic, and flat $\Lambda$CDM universe favored by observations of the cosmic microwave background can be described using only Euclidean geometry, locally correct Newtonian mechanics, and the basic postulates of special and general relativity. We present simple derivations of the most useful equations connecting astronomical observables (redshift, flux density, angular diameter, brightness, local space density,...) with the corresponding intrinsic properties of distant sources (lookback time, distance, spectral luminosity, linear size, specific intensity, source counts,...). We also present an analytic equation for lookback time that is accurate within 0.1% for all redshifts $z$. The exact equation for comoving distance is an elliptic integral that must be evaluated numerically, but we found a simple approximation with errors $< 0.2$% for all redshifts up to $z \approx 50$.