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For an expanding pressureless matter universe with zero cosmological constant, fixed current Hubble parameter $H_0$ and current matter cosmological density parameter $\Omega >0$, the age of a dust universe without a cosmological constant is $H_0{t}_0={\int }_0^1\sqrt{x}/\sqrt{\Omega +(1-\Omega )x}dx$. For every $0<x<1$ the integrand decreases strictly as $\Omega$ increases. Thus open models with $0<\Omega <1$ are older than the flat model, and closed expanding models with $\Omega >1$ are younger. The flat value is $2/3$ and the empty Milne universe limit is $1$. Specifying what is held fixed matters: at fixed present scale factor and energy density, the integral $t_0={\int }_0^{a_0}da/\sqrt{C/a-k}$ instead increases with $k$, provided all three models reach $a_0$ on their expanding branches.

Alan Guth proposed a period of rapid early expansion as a physical explanation of the horizon problem and flatness problem. His 1981 model used a high-energy state whose nearly constant energy density could drive accelerated expansion before ordinary hot Big Bang evolution.

The causal argument is that a region initially small enough to equilibrate can expand to encompass the subsequently observable region. Rapid expansion also suppresses the relative importance of spatial curvature. Ending that stage and obtaining a sufficiently uniform hot state is a separate dynamical requirement; solving the motivating geometric problems does not by itself supply a successful exit.^[ref]

Alexander Friedmann derived evolving homogeneous cosmological solutions of the Einstein field equations. The Friedmann equations relate cosmic expansion to energy density, pressure, spatial curvature and the cosmological constant.

The conceptual change is that general relativity permits the scale of a homogeneous universe to evolve, rather than requiring a static geometry. Observational evidence and the choice of matter content are additional inputs needed to identify which mathematical solution describes the Universe.^[ref]

Black-body radiation is the electromagnetic radiation in thermal equilibrium with an ideal blackbody. It can be described as a photon gas: a cavity whose walls are all at the same temperature contains radiation with the universal Planck's law spectrum. A small opening in such a cavity approximates a blackbody emitter because incoming radiation undergoes repeated absorption inside. The radiation's equilibrium spectrum depends on temperature, rather than the wall material. ^[ref]

There are three useful quantities to keep distinct: directional spectral radiance $B_{\nu }$ or $B_{\lambda }$, total energy density $u$ in the cavity, and outward radiant exitance $j^{\ast }$ from its opening. Planck's law supplies the first; integrals over frequency and directions supply the others. For isotropy, every direction has the same $B_{\nu }$, so the full $4\pi$ solid angle givesHere $\hslash =h/(2\pi )$ is the reduced Planck constant, and the radiation constant is $a_{\mathrm{rad}}\approx 7.565733250\times 10^{-16}\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-4}$. The explicit Boltzmann constant makes this an SI formula; adopting natural units with $k_B=1$ recovers $u={\pi }^2{T}^4/(15{\hslash }^3{c}^3)$.

For a short integral outline, substitute $x=h\nu /(k_{B}T)$ and useThe expansion is a geometric series with nonnegative terms; repeated integration by parts evaluates each integral, and $\zeta (4)={\pi }^4/90$ is a value of the Riemann zeta function. This gives the coefficient above.

The photon gas also exerts radiation pressure, $p=u/3$. The factor $1/3$ comes from the angular average of the squared directional component of photon momentum, $⟨{\cos }^2\theta ⟩=1/3$. The thermodynamic derivation of blackbody radiation pressure gives the same result. In contrast, outgoing power involves one projected-direction factor over an outward hemisphere, yielding radiant exitanceThis is the Stefan–Boltzmann law with unit emissivity; $\sigma$ is the Stefan-Boltzmann constant. The factor $\pi$ follows from Lambert's cosine law, rather than the $4\pi$ full-sphere factor used for energy density.

For example, an ideal $6000\mathrm{K}$ cavity has $u\approx 0.980519\mathrm{J}{\mathrm{m}}^{-3}$, radiation pressure $p\approx 0.326840\mathrm{Pa}$, and outward radiant exitance $j^{\ast }\approx 7.34881\times 10^7\mathrm{W}{\mathrm{m}}^{-2}$. A stipulated $3000\mathrm{K}$ cavity has one-sixteenth each value because all three scale as $T^4$. These describe stored energy, mechanical pressure, and outgoing power per area, respectively. In a closed equilibrium cavity, equal opposing radiation balances the directional power flow even though $u$ and $p$ are nonzero.

The usefulness of black-body radiation is this common reference: Planck's law, Wien's displacement law, and the Stefan–Boltzmann law consistently connect spectral measurements, temperature, and total emission. The three illustrative curves in Planck's law are ideal equilibrium spectra; applying them to a material requires considering its emissivity and whether thermal equilibrium is a suitable approximation.

The canonical stress-energy tensor is the Noether current associated with translation symmetry. For a Lagrangian density with no explicit coordinate dependence,

The Euler-Lagrange field equations imply ${\partial }_{\mu }{T}^{\mu }{}_{\nu }=0$. Thus $T^0{}_{\nu }$ are conserved densities and $T^i{}_{\nu }$ their fluxes; integrated charges are conserved when the boundary flux vanishes. For a real scalar field with $\mathcal{L}=\frac{1}{2}(\partial \Phi {)}^2-\frac{1}{2}{m}^2{\Phi }^2$, the energy density is $T^{00}=\frac{1}{2}{\dot{\Phi }}^2+\frac{1}{2}|\nabla \Phi {|}^2+\frac{1}{2}{m}^2{\Phi }^2$ in natural units. The general canonical stress-energy tensor need not be symmetric. ^[ref]

The capacitor energy is the electrostatic energy stored when electric charge is separated. For a linear capacitor of fixed positive capacitance $C$, plate electric charge magnitude $Q$, and terminal electric potential difference $V=Q/C$, it isThe zero is the uncharged state. Thus the capacitor energy is positive for nonzero $Q$, even though the total electric charge of the two plates is zero. ^[ref].

For the parallel-plate capacitor example, capacitance $C=8.854\times 10^{-11}\mathrm{F}$ and terminal electric potential difference $V=10\mathrm{V}$ giveThe plate electric charge magnitude is $Q=0.8854\mathrm{nC}$. Doubling terminal electric potential difference with capacitance unchanged doubles the plate electric charge magnitude and quadruples the capacitor energy.

The electric charge is a linear function of terminal electric potential difference, while capacitor energy is a quadratic function. The second curve in each plot uses twice the capacitance: at a given terminal electric potential difference, both plate electric charge magnitude and capacitor energy double. These relations let us calculate how much energy an electrical circuit can receive when a capacitor discharges.

A brief charging argument explains the factor $1/2$. If the plate electric charge magnitude has reached $q$, the terminal electric potential difference is $q/C$. Transferring a further amount $dq$ of electric charge quasistatically requires work $dW=(q/C)dq$. With the geometry and permittivity fixed, the integral isUsing $Q=CV$ gives the other two forms. This integral accounts for the gradually rising terminal electric potential difference; it is the work stored in the capacitor's electric field, rather than a claim that all energy drawn from a fixed-voltage battery during charging ends up stored.

The field picture gives the same result. In the vacuum gap of a parallel-plate capacitor, the electric field has energy density $u={ϵ}_0{E}^2/2$. Multiplying by the gap volume $Ad$ gives the electrostatic energyFor the worked example, $u=4.427\times 10^{-4}J/{\mathrm{m}}^3$ and $Ad=10^{-5}{\mathrm{m}}^3$ reproduce $U=4.427\mathrm{nJ}$. The electrostatic energy resides in the electric field throughout the gap, despite overall charge neutrality.

When capacitance changes, specify the constraint. At fixed plate electric charge magnitude $Q$, increasing capacitance decreases capacitor energy $Q^2/(2C)$; doubling capacitance halves it. At fixed terminal electric potential difference $V$, increasing capacitance increases capacitor energy $C{V}^2/2$; doubling capacitance doubles it. In the latter case a connected battery supplies additional electric charge and energy, and there may also be mechanical work. Use the fixed-voltage capacitor energy balance to account for all transfers before inferring a force from changing capacitor energy.

In an FLRW metric with zero separate cosmological constant, $\varrho +3p\ge 0$ makes the Friedmann acceleration equation imply $\ddot{a}\le 0$. If $a_0=a(t_0)>0$ and $v_0=\dot{a}(t_0)>0$, then the scale factor is a concave function andthroughout its regular past interval. Thus a regular positive scale factor cannot persist earlier than $t_0-a_0/v_0$. If the solution continues regularly whenever $a>0$, its first past zero occurs within $a_0/v_0=H(t_0{)}^{-1}$, where $H$ is the Hubble parameter.

The continuation condition matters: without restrictions on the equation of state, an earlier curvature singularity can occur at positive $a$. For example, in fixed units $a(t)=1+\sqrt{t}$, $t>0$, with $k=0$, defines energy density $\varrho =3(\dot{a}/a{)}^2/(8\pi G)$ and pressure $p=-\varrho /3-\ddot{a}/(4\pi Ga)$. These satisfy the Friedmann equations and $\varrho +3p>0$, yet $a\to 1$ at the singular past boundary. Conversely the Milne model reaches zero without divergent curvature. A finite duration, a zero of the scale factor, and a curvature singularity are distinct conclusions.

For a perfect fluid in general relativity with constant equation-of-state parameter $w$, use conformal time $dt=ad\tau$ and the conformal Hubble parameter $\mathcal{H}=a^{\prime }/a=aH$. The Friedmann acceleration equation gives ${\mathcal{H}}^{\prime }=a\ddot{a}=-4\pi G{a}^2(1+3w)\rho /3$. The Friedmann equation eliminates $\rho$ and gives this Riccati equation. For positive energy density, $\alpha \ne 0$ and appropriately shifted conformal time, its usual branches are $\cot (\alpha \tau )$ for $k=1$, $1/(\alpha \tau )$ for $k=0$, and $\coth (\alpha \tau )$ for $k=-1$. Their scale factors are proportional to ${\sin }^{1/\alpha }(\alpha \tau )$, $(\alpha \tau {)}^{1/\alpha }$ and ${\sinh }^{1/\alpha }(\alpha \tau )$ respectively on intervals where the bases are positive. At $\alpha =0$ the conformal Hubble parameter is constant and the separate solution is $a=A{e}^{h\tau }$.

A deliberately restricted forced harmonic oscillator shows two effects of the baryon loading parameter without duplicating the full photon Boltzmann hierarchy. Use conformal time $d\eta =dt/a$ in seconds, a comoving wavenumber $k$ in inverse metres, constant $R=3{\rho }_b/(4{\rho }_{\gamma })$, and equal constant dimensionless Newtonian gauge gravitational potentials $\Phi =\Psi \ne 0$. The energy densities ${\rho }_b$ and ${\rho }_{\gamma }$ use the same units. Neglect expansion drag, Silk damping and evolution of the potentials. The photon-baryon sound speed is $c_s=c/\sqrt{3(1+R)}$. Writing ${\Theta }_0$ for the photon monopole temperature perturbation, the photon-baryon acoustic oscillator reduces toFor $y=({\Theta }_0+\Psi )/\Psi$ this gives the displayed forced equation. Constant loading and the neglected expansion drag make this a local teaching model, not a self-consistent expanding cosmology with $R$ fixed forever.

Choose $y(0)=1/3$ and $y^{\prime }(0)=0$, the static matter-era Sachs-Wolfe combination normalized by $\Psi$. With $q=k{c}_s\eta$, direct substitution givesAt $R=0$, successive extrema have equal magnitudes $1/3$. At the stipulated $R=0.6$, $y(\pi )=-23/15$ whereas $y(2\pi )=1/3$. The shifted equilibrium $y=-R$ makes alternating extrema unequal. This is an elementary illustration of how the baryon loading parameter changes acoustic amplitudes; these extrema are neither measured peak heights nor a complete Cosmic microwave background power spectrum. Comparing curves at fixed $q$ isolates the response; at fixed $k$ and $\eta$, increasing $R$ also reduces the photon-baryon sound speed and hence the phase.

For example, holding $c{\eta }_{\ast }=300\mathrm{Mpc}$ gives sound horizons $r_s=173.2\mathrm{Mpc}$ at $R=0$ and $136.9\mathrm{Mpc}$ at $R=0.6$. Their ratio is $1/\sqrt{1.6}$. Each term of the forced equation has units of inverse seconds squared because $y$ and $R$ are dimensionless and $k{c}_s$ is an inverse time. In the actual photon-baryon fluid, $R$ grows with the scale factor and the sound horizon is the integral of the evolving photon-baryon sound speed.

Nonrelativistic matter dilutes as $a^{-3}$, while a cosmological constant has constant energy density. The ratio of vacuum energy density to matter density consequently scales as $a^3$, so comparable contributions occur during a limited epoch. The cosmic coincidence problem asks why observers find these very different components comparable near their own epoch. This timing puzzle is distinct from the cosmological constant problem, which concerns the tiny absolute vacuum-energy scale.

An isotropic cosmic string network averages the directional stress and energy density of many long cosmic strings. For an ideal smooth string whose tension equals its rest energy per unit length, a network frozen into the expansion has negative averaged pressure,A comoving string segment then has physical length proportional to $a$, while its enclosing volume grows as $a^3$, giving $\rho \propto a^{-2}$. This is the nonmoving, noninteracting limit, not the general evolution law of a mobile network.

In units $c=1$, let $v^2$ be the energy-weighted mean squared transverse string velocity. For an isotropic smooth-string network the averaged equation of state isThe kinetic contribution raises the averaged pressure; for example $v^2=1/2$ gives $w=0$. Motion, reconnection and production of smaller loops therefore matter even when the initial network is statistically isotropic.

The corresponding long-string balance, neglecting friction against matter, can be writtenwhere $\mathcal{Q}\ge 0$ is energy removed from the long-string population into loops or other radiation channels. For $\mathcal{Q}=0$ this agrees with $\dot{\rho }+3H(\rho +P)=0$ and proves the equation-of-state relation. If $v$ is constant and losses vanish, the solution is $\rho \propto a^{-2(1+v^2)}$. The primary velocity-dependent model defines $\rho =\mu /L^2$ and evolves both its characteristic length $L$ and its velocity.^[ref]

A horizon-scaling cosmic string network is consequently distinct from a frozen fluid with fixed $w=-1/3$. The ideal frozen equation of state can be used in a specified cosmological toy model, but it cannot be inferred from isotropy alone.

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.

For an expanding Friedmann-Lemaître-Robertson-Walker metric with positive energy density, put $\Omega =8\pi G\rho /(3H^2)$ and $w=P/\rho$, in units $c=1$. The Friedmann equation, Friedmann acceleration equation and cosmological perfect-fluid continuity equation giveFor constant $w$, linearizing at $\Omega =1$ gives $\Omega -1\propto a^{1+3w}$. For variable $w$, replace the power by $\exp \int (1+3w(a))d\log a$. This describes the flatness problem for decelerating expansion, away from epochs at which $H=0$.

At a specified Hubble parameter $H$, the critical energy density isThe Friedmann equation sets the spatial-curvature term to zero when the total energy density, including any cosmological constant contribution, equals this value. Dividing by $c^2$ gives the corresponding critical mass density $3H^2/(8\pi G)$. Confusing these two conventions loses a factor of $c^2$.^[ref]

The cosmological density parameter ${\Omega }_i={\rho }_i/{\rho }_{\mathrm{crit}}$ compares one component with this geometric normalization at the same epoch. Spatial flatness requires the sum of the component cosmological density parameters to be one, rather than each component separately. Since ${\rho }_{\mathrm{crit}}$ changes with $H$, a constant energy density need not have a constant cosmological density parameter.

The criterion concerns spatial geometry in an FLRW metric. It does not alone determine the future lifetime of an expanding universe: pressure and dark energy affect its subsequent dynamics. It also differs from critical densities defined by phase transitions or plasma processes.

Neglect radiation in cosmology and take pressureless matter plus a separately conserved dark energy component with $p_{\mathrm{DE}}=w{\rho }_{\mathrm{DE}}{c}^2$, where $\rho$ here denotes mass-equivalent energy density. The Friedmann acceleration equation givesThus $w<-1/3$ for a positive-density component is necessary for that component to accelerate expansion, but the total must also overcome pressureless matter. For a cosmological constant, $w=-1$ and $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}$, so acceleration requires ${\Omega }_{\Lambda ,0}>{\Omega }_{m,0}/2$. Spatial curvature of an FLRW universe enters the distance relation but has no separate term in this acceleration criterion. A Hubble diagram alone does not identify a microscopic dark energy mechanism.

For one massless neutrino flavor with one helicity per member and $|{\mu }_{\nu }|\gg T_{\nu }$, the Fermi-Dirac distribution fills momentum states up to the positive chemical potential. If ${\mu }_{\nu }>0$, the neutrino density is ${\mu }_{\nu }^4/(8{\pi }^2)$ and the antineutrino contribution is approximately $3T_{\nu }^4{e}^{-{\mu }_{\nu }/T_{\nu }}/{\pi }^2$. For negative chemical potential the roles reverse. Thus the leading total energy density is even in ${\mu }_{\nu }$ and has no factor of two from the two species. Extra populated internal degrees of freedom do multiply it. This distinction prevents using a positive-degeneracy approximation for an exponentially dilute negative-chemical-potential distribution.

A dielectric is a material in which an electric field produces electric polarization while transport of electric charge is sufficiently small for the intended use. An insulating solid in a capacitor is an example; real dielectrics can still have dielectric loss and nonzero electrical conductivity.

The important distinction is between rearranging bound electric charge and transporting mobile charge carriers through the specimen. Bound electric charges can move a little within a chemical atom or molecule, or an existing electric dipole can turn. Their averaged electric dipole moments produce electric polarization even if there is negligible steady electric current between the electrodes. A changing electric polarization does give a bound current; the absence of steady conduction does not mean that the material is electrically unresponsive.^[ref]

For uniform electric polarization $\mathbf{P}=P\hat{\mathbf{x}}$ in a rectangular slab, polarization density gives zero bound volume charge density in the interior and opposite bound surface charge densities $-P,+P$ on the left and right faces. The electric charge has been redistributed, not created. More generally, the divergence theorem makes the cancellation explicit:Here $\mathbf{n}$ is the outward normal vector on the material boundary. This accounts for both bound volume charge density and bound surface charge density; omitting either can falsely suggest that an initially neutral dielectric acquired net electric charge.

An electrical insulator between capacitor plates supports a useful combination: appreciable electric polarization, little leakage electric current, and tolerance of the intended electric field. This lets a small parallel-plate capacitor store separated electric charge and recoverable capacitor energy. The comparison in dielectric capacitor response explains why specifying fixed electric charge or fixed electric potential difference changes the answer. High relative permittivity alone does not establish good insulation or a large usable energy density; dielectric loss and dielectric breakdown matter as well.

Dielectric strength characterizes the electric field at dielectric breakdown under specified test conditions. In an ideal uniform gap, it sets a voltage scale $V_{\mathrm{bd}}\simeq E_{\mathrm{bd}}d$.^[ref] The parallel-plate capacitor model consequently gives the stored energy density scaleThis is a model scale, not an operating rating: fringing field of a capacitor make local electric fields differ from $V/d$, while leakage and dielectric loss spend some supplied energy as heat. Comparing usable capacitor energy therefore requires insulation and breakdown performance as well as permittivity.

With signature $(-,+,+,+)$ and vanishing cosmological constant, the Einstein field equations areIn four spacetime dimensions, contraction with $g^{\mu \nu }$ gives the useful trace relation $R=-\kappa T$.

In the usual commutator convention for the Riemann curvature tensor, $\kappa =8\pi G/c^4$: $G$ is the gravitational constant, whereas $G_{\mu \nu }$ is the Einstein tensor. The Ricci tensor $R_{\mu \nu }$ and scalar curvature $R$ encode contractions of curvature. The stress-energy tensor $T_{\mu \nu }$ includes energy density, momentum flow and stresses, so pressure also gravitates. A cosmological constant adds $\Lambda g_{\mu \nu }$ to the left-hand side. See Tong's Einstein equations and matter coupling.

For a concrete consequence, set $T_{\mu \nu }=0$ and $\Lambda =0$. The trace gives $R=0$, and substitution gives $R_{\mu \nu }=0$. Yet the exterior Schwarzschild spacetime still has the tidal forces computed under geodesic deviation: vanishing Ricci tensor does not imply vanishing Riemann curvature tensor. Geometry carries gravitational effects through empty space. Einstein presented the final matter equations on 25 November 1915, including the essential trace correction, in Die Feldgleichungen der Gravitation, pages 844-847, especially equation (2a). His symbols and sign conventions differ from modern ones; the displayed formula uses modern notation.

An energy flux describes transport of energy across a surface. A flux density ${\mathbf{j}}_E$ has units of power per area; the total outward power through $S$ is ${\int }_S{\mathbf{j}}_E\cdot \mathbf{n}dS$, with outward unit normal $\mathbf{n}$. A local conservation law relates the divergence of ${\mathbf{j}}_E$ to the time derivative of the energy density and any sources.

For a one-dimensional inviscid gas with velocity $u$, pressure $p$, mass density $\rho$ and internal energy density $W$, the total mechanical-plus-internal energy flux isThe first two terms advect internal energy and kinetic energy. The term $pu$ is pressure work: a moving section of area $A$ transmits force $pA$ at velocity $u$, giving power $pAu$. This formula assumes negligible viscous stress and heat conduction; those processes contribute their own energy-transfer terms when retained.