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Field theory in cosmology applies classical and quantum fields to the homogeneous universe and its perturbations.

This is the first-order sector of cosmological perturbation theory.

Linear cosmological perturbation theory expands the metric and matter fields to first order about a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric.

For an adiabatically expanding homogeneous universe, the entropy density satisfiesso the entropy $s{a}^3$ in a comoving volume is constant.

The baryon-to-photon ratio is the number density of baryons divided by the number density of photons. In the present universe it is of order $10^{-9}$.

Big Bang nucleosynthesis formed the light nuclei during the first minutes of cosmic expansion after weak freeze-out and the deuterium bottleneck.

The deceleration parameter is $q=-a\ddot{a}/{\dot{a}}^2$; negative values describe accelerated expansion.

In a curvature-dominated open FLRW regime, the scale factor is approximately linear in cosmic time and $tH$ approaches one.

The age is the proper cosmic time elapsed from the scale-factor singularity $a=0$ to the present epoch.

The age of the universe is the cosmological quantity; an FLRW model computes it by integrating proper cosmic time from its initial scale-factor singularity.

Energy conservation in FLRW gives $\dot{\rho }=-3H(\rho +P)$ when $\rho$ and $P$ are energy density and pressure.

The FLRW metric is the homogeneous and isotropic spacetime metric with scale factor $a(t)$ and constant spatial curvature $k$.

Luminosity distance is defined by $F=L/(4\pi d_L^2)$ and includes geometric dilution, photon redshift, and arrival-rate dilation.

A fluid with $w<-1$ grows in density during expansion and can produce a finite-time Big Rip singularity.

For pressureless subhorizon matter perturbations, a Fourier mode in conformal time obeyswhere $\mathcal{H}=a^{\prime }/a$.

These equations describe the small-amplitude phase of structure formation; nonlinear growth requires a more general treatment.

Cosmic inflation is an early period of accelerated expansion, often modeled by a scalar field whose potential energy dominates its kinetic energy.

A Big Crunch is a future finite-time singularity in which a recollapsing universe reaches vanishing scale factor.

Cosmological radiation has equation of state $P_r={\rho }_r/3$ and density scaling ${\rho }_r\propto a^{-4}$.

In a decelerating hot Big Bang model, widely separated regions of the observable universe have nearly equal conditions despite having no shared past light cone. Inflation supplies an earlier era with a shrinking comoving Hubble radius, allowing a formerly causal patch to grow beyond the later horizon.

At Hubble parameter $H$, the critical energy density is

This is the density threshold in the Friedmann equation for spatial flatness; thermodynamic and plasma critical densities are different concepts.

For a homogeneous fluid of energy density $\rho$, pressure $P$, and scale factor $a(t)$, adiabatic work in a comoving volume gives

A transfer function packages the response of a linear time-invariant system into a rule for turning a plant input into a plant output. Instead of solving a differential equation separately for every signal, take a Laplace transform, multiply by $G(s)$, then transform back. This also makes it possible to compare slow and fast oscillations through the continuous-time frequency response.

Here consider a continuous-time, causal, finite-dimensional linear time-invariant system with one plant input $u(t)$ and one plant output $y(t)$. With every initial state set to zero, its transfer function is defined byThe ratio $G(s)=Y(s)/U(s)$ is used where $U(s)\ne 0$ in a common Laplace convergence region; $G$ describes the system, independently of that particular plant input. Zero initial state matters: stored motion or another nonzero initial condition can contribute to the plant output even when the plant input vanishes. ^[ref]

The first-order low-pass filter below shows the same transfer function predicting both a step and a sustained sinusoid. Its pole gives a decay rate, its value at zero gives the eventual gain, and its values along the imaginary axis give the continuous-time frequency response. These descriptions make a useful bridge between a physical relaxation model and control theory calculations.