# Basic geometric and kinematic features of the Standard Cosmological Model

## TL;DR

This paper reviews the geometric and kinematic features of the Standard Cosmological Model, deriving key equations and analyzing cosmic evolution, horizons, and future scenarios based on current observational data.

## Contribution

It provides a comprehensive derivation of equations characterizing the $ ext{Lambda}$CDM model, including effects of radiation and neutrinos, and discusses cosmic evolution and future prospects.

## Key findings

- Derived equations for $ ext{Lambda}$CDM model with radiation and neutrinos.
- Identified key moments in cosmic evolution such as equality and acceleration points.
- Analyzed the universe's remote future and possibilities of extraterrestrial connection.

## Abstract

We present a brief history of the construction of models of the universe, followed by calculations of quantitative characteristics of basic geometric and kinematic properties of the Standard Cosmological Model ($\Lambda$CDM). Using the Friedmann equations of uniform space, we derive equations characterizing a $\Lambda$CDM model that describes a universe corresponding to current observational data. The equations take into account the effects of radiation and ultra-relativistic neutrinos. It is shown that the universe at very early and late stages can be described to sufficient accuracy by simple formulas. Certain important moments of cosmic evolution are determined: the times when densities of the gravitational components of the universe become equal, when they contribute equally to the gravitational force, when the accelerating expansion of space begins, and several others. The dependences of different distances on redshift and the scale factor of space are derived. The distance to the sphere that expands with the speed of light (the Hubble distance), and its current and future acceleration, are found. Concepts of a horizon, second inflation, and second horizon are discussed. We consider the remote future of the universe and the opportunity, in principle, of connection with extraterrestrial civilizations.