What is modern cosmology

Almost all of cosmology consists of a series of applications of two fundamental equations of physics:

Modern cosmology is a mathematical physics subject, so we present the mathematics used in these studies.
However we try to clearly link this in to the underlying conceptual principles on the one hand, and the observational tests of the resulting models on the other.

Modern physical cosmology is dominated by the Big Bang theory, which attempts to bring together observational astronomy and particle physics; more specifically, a standard parameterization of the Big Bang with dark matter and dark energy, known as the Lambda-CDM model.

A leading theory, inflation, is an attractive addition to the framework, but it lacks cross bracing.
That is precisely what cosmologists are now seeking.
If measurements in progress agree with the unique signatures of inflation, then we will count them as a persuasive argument for this theory.