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Cosmic microwave background anisotropy is the direction dependence of the relic radiation's temperature and polarization. A temperature map can be written $T(\hat{\mathbf{n}})=\bar{T}[1+\Theta (\hat{\mathbf{n}})]$, separating a mean from its fractional angular variation. Comparing directions with a Differential Microwave Radiometer measures these variations without requiring the instrument to determine the absolute mean temperature.^[ref]

Intrinsic plasma fluctuations, motions and gravitational frequency shifts all contribute. The Cosmic microwave background power spectrum organizes the signal by angular scale; the Cosmic microwave background acoustic peaks record the evolution of the photon-baryon fluid. Frequency comparisons and instrumental checks help distinguish the relic signal from galactic foregrounds and noise before the map is used to infer cosmological parameters.

The large-scale structure of the universe is the spatial organization of matter into dark matter halos, galaxy clusters, sheets, filaments and cosmic voids. Galaxy clustering traces this pattern, while the underlying matter also includes material that does not shine. Gravity amplifies initial linear cosmological density perturbations, but expansion and the matter's dynamics determine which scales grow efficiently. Comparing observed statistics with models of cosmological structure formation turns this pattern into a test of the initial conditions and subsequent evolution.

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

Linear cosmological perturbation theory describes small departures from a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric. Introduce a bookkeeping parameter $ϵ$ and expand the metric tensor and stress-energy tensor asSubstituting into the Einstein field equations separates the background equations from linear equations for the perturbations; products of two first-order perturbations are discarded. Smallness of $h_{ab}$ is assessed relative to the background metric tensor, not by its determinant alone.

Spatial homogeneity lets one analyze each Fourier mode separately. The fields and particle species at a given wavevector can remain coupled, but different wavevectors do not exchange amplitude through the linear equations. Products retained in nonlinear cosmological perturbation theory instead generate convolutions between modes. The linear density equations help connect initially small fluctuations to the growth of cosmic structure.^[ref]

A cosmological gauge transformation changes how the perturbed and background spacetimes are identified. For example, under $\tilde{\eta }=\eta +T$, a density perturbation transforms aswhere the prime is a conformal time derivative. A nonzero coordinate-dependent density perturbation can therefore represent a time relabelling of unchanged matter. One fixes a gauge or forms invariant combinations before relating perturbations to observations.

Tensor cosmological perturbations describe gravitational-wave degrees of freedom, while density and velocity perturbations describe other responses. A weak gravitational potential does not guarantee a small density contrast; collapsed structures can require a nonlinear matter treatment even when their metric potentials remain small.

Let $s$ be proper entropy density in a homogeneous FLRW metric and $u^{\mu }$ the comoving fluid four-velocity. Reversible adiabatic expansion has a conserved entropy current,Since a fixed comoving volume has physical volume proportional to $a^3$, the covariant divergence givesThus the conserved quantity is comoving entropy; the proper entropy density falls as $a^{-3}$. This dilution does not require a decrease of the total entropy in the selected material volume.

For equilibrium relativistic matter at negligible chemical potentials, $s$ is proportional to $g_{\ast s}{T}^3$, where $g_{\ast s}$ is the effective number of relativistic entropy degrees of freedom. Conservation gives $aT{g}_{\ast s}^{1/3}=\text{constant}$, so $T\propto a^{-1}$ only while $g_{\ast s}$ is constant. Annihilation into a still-coupled bath can change its temperature relative to a decoupled species while conserving the bath's comoving entropy.^[ref]

Irreversible entropy production replaces the zero divergence by a nonnegative source. Separately decoupled sectors need their own entropy balances; imposing one common temperature after decoupling would add an assumption.

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 forms light atomic nuclei during the first minutes of the expanding, cooling universe. Weak interactions initially maintain neutron-proton chemical equilibrium. After cosmological weak freeze-out, free-neutron decay after freeze-out reduces the available neutron supply.

The deuterium bottleneck delays efficient assembly: energetic photons break up newly formed deuterium until sufficient cooling occurs. Most surviving neutrons then enter helium-4, with smaller residual abundances of other light nuclei. If $r=n_n/n_p$ at assembly and nearly all neutrons enter helium, counting two neutrons per helium nucleus gives the approximate helium mass fractionThis links nuclear reactions and particle decay to an astronomical abundance measurement. Comparing primordial deuterium and helium with the independently inferred cosmological baryon density tests the thermal history; the cosmological lithium problem remains an important qualification to the comparison.^[ref]

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.

In natural units, conservation of the stress-energy tensor for a homogeneous perfect fluid in an FLRW metric gives the displayed equation for its energy density and pressure. It is equivalent to $d(\rho a^3)=-Pd(a^3)$. For a constant equation-of-state parameter $w=P/\rho$, separation gives $\rho \propto a^{-3(1+w)}$. If $\rho$ instead denotes mass density, the pressure term is $P/c^2$.

This is local stress-energy conservation; it does not in general define a globally conserved total energy for an expanding universe. The adiabatic work in a comoving volume makes this distinction explicit.

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 a proposed early period of accelerating expansion of the universe. In general relativity, a homogeneous component with sufficiently negative pressure can drive this behaviour. In units $c=1$, the Friedmann acceleration equation gives acceleration when the total source satisfies $\rho +3P<0$. A slowly evolving canonical scalar field can meet this condition when its potential energy dominates its kinetic energy.

During acceleration the comoving Hubble radius decreases. Perturbations can move from scales smaller than the Hubble radius to larger scales, linking the early dynamics with later Cosmic microwave background anisotropy and structure formation. Different inflaton potentials predict different fluctuation spectra and correlations, so inflation is a class of models with testable assumptions. Ending acceleration and producing a hot particle bath also require an explanation through a graceful exit from inflation and reheating.^[ref]

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 ${\rho }_r\propto a^{-4}$.