% miko:type: complete
% miko:title: Complete Cosmology
% miko:summary: A standard introductory path through expansion, Friedmann dynamics, distance scales, and the thermal history of the universe.
% miko:tags: cosmology
% miko:difficulty: undergrad
% miko:related: friedmann-equations, scale-factor-redshift, hubble-law, critical-density, cosmological-constant, cosmic-microwave-background, complete-special-relativity
% miko:keywords: FLRW metric, Friedmann equation, redshift, CMB, Lambda CDM

\documentclass{article}
\usepackage{amsmath}
\title{Complete Cosmology}
\begin{document}
\maketitle

\section{Observational Starting Point}

Modern cosmology begins from two empirical facts. First, on sufficiently large
scales the universe is statistically homogeneous and isotropic. Second, distant
galaxies are redshifted in a way that is naturally described by expansion.

The homogeneous and isotropic spacetime ansatz is the FLRW metric. In units
where the spatial curvature parameter is $k=0,\pm 1$,

$$
ds^2 = -c^2dt^2 + a(t)^2\left[
\frac{dr^2}{1-kr^2}+r^2d\Omega^2
\right].
$$

The single function $a(t)$ is the scale factor. Much of introductory cosmology
is the study of how $a(t)$ evolves and how light propagates through this
geometry.

\section{Redshift and the Scale Factor}

For light emitted when the scale factor was $a_{\rm em}$ and observed when it
is $a_0$, the cosmological redshift is

$$
1+z = \frac{a_0}{a_{\rm em}}.
$$

This relation is kinematic: it follows from the stretching of wavelengths by
the expanding geometry. It is the bridge between observation and cosmic time.

\section{Hubble Expansion}

The expansion rate is defined by the Hubble parameter

$$
H(t) = \frac{\dot a(t)}{a(t)}.
$$

For nearby galaxies, the leading relation between recession velocity and
proper distance is Hubble's law,

$$
v \simeq H_0 d.
$$

The law is local and approximate. At large redshift, distances must be defined
more carefully because light travels through an evolving spacetime.

\section{Friedmann Dynamics}

Assuming general relativity and a perfect-fluid stress-energy tensor, the first
Friedmann equation is

$$
H^2 =
\frac{8\pi G}{3}\rho
-\frac{kc^2}{a^2}
+\frac{\Lambda c^2}{3}.
$$

The second Friedmann equation can be written as

$$
\frac{\ddot a}{a}
=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)
+\frac{\Lambda c^2}{3}.
$$

The first equation is an energy constraint; the second describes acceleration.
Together with energy conservation, they determine the scale-factor history for
specified matter, radiation, curvature, and cosmological constant.

\section{Critical Density}

For a spatially flat universe with $\Lambda=0$, the critical density is

$$
\rho_c = \frac{3H^2}{8\pi G}.
$$

Density parameters are defined by $\Omega_i=\rho_i/\rho_c$. They allow the
Friedmann equation to be written as a bookkeeping equation for the cosmic
contents.

\section{Thermal History}

In the early universe, radiation dominated. As the universe expanded, it cooled.
The cosmic microwave background is the relic radiation from the epoch when
photons decoupled from matter. Its near-blackbody spectrum and anisotropies
are central evidence for hot Big Bang cosmology.

\section{Standard Model of Cosmology}

The minimal successful model is often called Lambda CDM. It contains radiation,
baryonic matter, cold dark matter, and a cosmological constant. Its success is
not that it explains everything from first principles, but that it gives a
compact quantitative framework for the expansion history and large-scale
structure.

\end{document}
