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The Friedmann equations, named for Alexander Friedmann, apply the Einstein field equations to an FLRW metric. Using natural units with $c=1$, energy density $\rho$, pressure $p$ and cosmological constant $\Lambda$, they readHere $a$ is the scale factor, $H=\dot{a}/a$ the Hubble parameter, and $k$ specifies spatial curvature. The cosmological continuity equation completes the fluid description. For a flat universe with $\Lambda =0$, pressureless matter gives $a\propto t^{2/3}$ and radiation in cosmology gives $a\propto t^{1/2}$.^[ref]

An inflaton is a hypothetical scalar field whose energy density drives cosmic inflation. In the canonical single-field description, a spatially homogeneous value $\varphi (t)$ haswhere $V$ is the inflaton potential. In units $c=\hslash =1$, its inflaton equation of motion iswith Hubble parameter $H$. Potential domination makes pressure negative. When this scalar field dominates the Friedmann equations, accelerating expansion requires $V>{\dot{\varphi }}^2$. Stronger potential domination gives $P_{\varphi }\simeq -{\rho }_{\varphi }$ and nearly exponential expansion.^[ref]

The slow-roll approximation supplies a tractable evolution in this regime. Couplings to other fields can transfer the inflaton's energy into particles through reheating. The identity of the inflaton and the form of its inflaton potential are model assumptions to be constrained by primordial perturbations, rather than an identification of a presently known particle. Its quantum fluctuations supply the scalar seeds in canonical single-field models of cosmic inflation.

Quintessence models dark energy through an evolving scalar field. For a homogeneous canonical field in natural units,A positive potential dominating the kinetic term gives negative pressure. The Friedmann acceleration equation requires $w=p_{\varphi }/{\rho }_{\varphi }<-1/3$ for scalar-dominated accelerated expansion. A nonnegative canonical kinetic term prevents $w<-1$ when ${\rho }_{\varphi }>0$.

Unlike a strict cosmological constant, the field can evolve, so $w$ need not remain $-1$. This is a model class, not an identified substance.^[ref]

The slow-roll approximation describes a canonical inflaton whose potential energy dominates its kinetic energy and whose field acceleration is small compared with Hubble friction:If the inflaton also dominates the energy density and cosmological spatial curvature is negligible, the Friedmann equation and inflaton equation of motion reduce towhere $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$ is the reduced Planck mass and $c=\hslash =1$. The second equation describes a damping-controlled drift down the inflaton potential.^[ref]

The potential slow-roll parameters give useful consistency checks, ${ϵ}_V\ll 1$ and $|{\eta }_V|\ll 1$, along the traversed field interval. Small initial kinetic energy alone does not guarantee that these inequalities persist. For $V=m^2{\varphi }^2/2$ and $\varphi >0$, the two approximate equations give $\dot{\varphi }\simeq -\sqrt{2/3}m{M}_{\mathrm{Pl}}$, so the field decreases nearly linearly while the Hubble parameter slowly changes. Cosmic inflation ends at ${ϵ}_H=1$, where the first Hubble slow-roll parameter equals unity; estimating that endpoint by ${ϵ}_V\simeq 1$ extrapolates the approximation to its boundary. Cosmic inflation can also occur outside ordinary slow roll, so the two terms are not interchangeable.

Motion can weigh a system whose mass is otherwise inaccessible. For a self-gravitating isolated system in virial equilibrium, let $K$ and $U$ denote equilibrium averages and use its gravitational radius $R_g=-G_N{M}^2/U$ and a mass-weighted mean squared speed relative to its centre of mass. Then $K=M⟨v^2⟩/2$ gives $M=R_g⟨v^2⟩/G_N$. If random velocities are isotropic and mean streaming has been removed, $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$, where ${\sigma }_{\mathrm{los}}$ is the one-dimensional line-of-sight velocity dispersion. Observations sample many objects at one epoch; replacing a time average by that sample requires a representative, approximately stationary population, beyond the trajectory identity itself.^[ref]

A practical measurement route uses an optical telescope to locate members and a spectrograph to measure Doppler shifts of identified spectral lines. Their shifts relative to the system's mean give radial velocities; angular separations plus a distance estimate constrain its size. Subtracting instrumental broadening and measurement variance, testing membership, and looking for coherent rotation or subgroups help control the inference. The projected luminous size need not equal $R_g$, and luminous tracers need not have the same distribution as all gravitating mass. Unknown velocity-anisotropy parameters, interlopers, mergers, external tidal forces and appreciable boundary stresses can bias a naive estimate. A large velocity dispersion alone does not establish virial equilibrium.

The virial theorem relates the average value of a function along a trajectory to the forces that bind it. For an isolated system of particles interacting through Newtonian gravitational potential energy, it gives $2⟨K⟩+⟨U⟩=0$: the average kinetic energy is half the magnitude of the average negative potential energy. An eccentric Kepler orbit makes the distinction vivid: kinetic energy and potential energy vary throughout the orbit, although their sum is constant. The relation concerns their time averages, rather than their values at every instant. It underlies the virial mass estimate and the negative heat capacity of a virialized gravitational system.

Use constant positive particle masses $m_i$, positions ${\mathbf{r}}_i$, mechanical momenta ${\mathbf{p}}_i=m_i{\dot{\mathbf{r}}}_i$, and total kinetic energy $K={\sum }_i{\mathbf{p}}_i^2/(2m_i)$. The auxiliary quantity $G(t)$ is a sum of dot products, not the gravitational constant, which we denote here by $G_N$:This finite-$T$ identity is exact wherever Newton's second law holds along a regular trajectory; ${\mathbf{F}}_i$ must include every force. The endpoint term is what a finite observation window leaves behind. For the infinite-time theorem, suppose the averages on the left exist and $(G(T)-G(0))/T\to 0$. A sufficient condition is uniformly bounded positions and velocities for finitely many particles. More generally $G(T)=o(T)$ suffices, even without those individual bounds. Bounded positions alone do not supply a bound on momenta, especially near a singular potential energy; nor does a small endpoint term prove thermodynamic equilibrium. Periodic motion with $G(T)=G(0)$ gives the identity without an endpoint term at that finite $T$. For an isolated system, use an inertial centre of mass frame so translational drift does not contribute bulk kinetic energy.^[ref]

For a differentiable potential energy that is a homogeneous function of degree $q$, choose its additive constant so that the scaling law really holds. With conservative forces, the Euler theorem for homogeneous functions givesThus a harmonic oscillator has $q=2$ and $⟨K⟩=⟨U⟩$. An isolated system with Newtonian gravitational potential energy hasEach interacting pair occurs once, and $U=0$ at infinite separation. Here the total energy $E=K+U$ is conserved. An external field or a mixture of different scaling powers requires its own force term; one cannot assign the whole potential energy a single degree $q$. The virial theorem for a homogeneous pair potential states the corresponding general energy relation.