A cosmic string is a hypothetical one-dimensional topological defect formed by symmetry breaking in the early universe. A cosmic string network consists of many such defects and requires additional assumptions to infer an averaged equation of state.

The scale factor $a(t)>0$ measures how spatial lengths in an FLRW metric change with cosmic time $t$. Choose $a$ to be a dimensionless quantity with $a(t_0)=1$ at a reference epoch, usually today. For two comoving observers, their fixed radial comoving distance $\chi$ and their proper distance in cosmology $d(t)$ satisfyThe proper distance in cosmology is measured along a spatial slice at one cosmic time; it is not the distance that a travelling photon covers between emission and reception. Only ratios of the scale factor have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi /C$ leaves $d$ unchanged.

For example, take $\chi =100\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million parsecs. At $a=0.5,1,1.5$, the same pair of comoving observers has proper distance in cosmology $50,100,150\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal comoving observers; internal motions in galaxies need not follow this background scaling.

The beauty of the scale factor is that one function describes the changing spatial ruler for every comoving observer. Spatial homogeneity gives no preferred centre of cosmic expansion: in a flat spatial patch, relative positions obey $\mathbf{r}-{\mathbf{r}}_O=a(\mathbf{x}-{\mathbf{x}}_O)$, so recentering on any comoving observer preserves Hubble's law. The evolution of the scale factor is determined by the Friedmann equations, while its ratio between emission and reception determines cosmological redshift.

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]

Cosmological recombination is the epoch in which cooling allowed free electrons and ions to form neutral atoms, sharply reducing the free-electron density.

Cosmological perturbation theory studies departures of the metric tensor and matter fields from a homogeneous and isotropic FRW metric, order by order. Linear cosmological perturbation theory keeps only first-order terms. Higher orders describe nonlinear coupling between perturbations. Nonlinear cosmological perturbation theory also uses expansions that allow finite amplitudes, such as the long-wavelength approximation in cosmology.

The cosmological principle assumes that the universe is statistically homogeneous and isotropic on sufficiently large spatial scales. It applies to a coarse-grained description, not to each galaxy or local void. Combined with the Copernican principle, cosmological isotropy around us supports a homogeneous description rather than a specially centred model. It motivates an FRW metric, whose expansion is governed by a few time-dependent quantities. Observations sample only a finite past light cone, so they support this hypothesis without proving global spatial homogeneity beyond the observed region.

Structure formation is the growth of cosmological matter inhomogeneities into structures such as galaxies through gravitational instability and gas dynamics. Linear cosmological density perturbations describe the early, small-amplitude regime; nonlinear evolution is required once the density contrast becomes large.

The cosmological monopole problem is the excessive relic density of stable magnetic monopoles produced in some early-universe symmetry-breaking scenarios. Cosmic inflation dilutes their number density as $a^{-3}$. Successful later reheating must also avoid recreating an excessive abundance.

The cosmological spectral background is received radiant power per detector area, received frequency interval and solid angle, summed over a cosmological source population. Its intensity depends on emitted spectra, their cosmic evolution, the expansion history and absorption. A finite radial comoving distance alone does not guarantee a finite background when high-redshift sources are sampled at increasingly high emitted frequencies.

Feedback control selects a control input using observations of the actual controlled plant. In a tracking loop, compare the reference signal $r$ with the measured output $y_m$ to form the tracking error $e=r-y_m$, then let the controller change $u$. The control input changes the plant output, which is measured again: that return path closes the loop. A controller can use earlier observations as well as the latest one. ^[ref]

For the ideal controlled plant, proportional control $u=2(1-x)$ initially commands $u(0)=2$. As $x$ approaches $1$, the command shrinks. When a positive control disturbance pushes $x$ above $1$, the tracking error becomes negative and the control input pushes back. The result is a bounded offset rather than the continuing drift of open-loop control. Integral action can remove that offset by remembering past tracking errors.

A control selected from the observed current state, rather than specified only as a function of the initial state and time. For $x_{t+1}=x_t+u_t$ and stage cost $x_t^2+2x_t{u}_t+7u_t^2$, the infinite-horizon feedback $u_t=-x_t/3$ gives $x_t=(2/3{)}^t{x}_0$ and total cost $2x_0^2$. The finite-horizon Bellman coefficients satisfy ${\Pi }_0=0$, ${\Pi }_s=6(1+{\Pi }_{s-1})/(7+{\Pi }_{s-1})$ and increase to two, proving optimality.

Quantum control chooses externally adjustable interactions to steer a quantum state or implement a desired unitary operator. Its achievable transformations depend on the Hamiltonian or Lindblad equation, the available control inputs, and physical constraints such as bandwidth and amplitude. Quantum optimal control optimizes performance within those constraints.

An open-loop control is prescribed as a function of time, possibly after calculation from an initial state and a dynamical model. It is executed without using observations of the actual current state to change the input. A simulated feedback control law can generate an open-loop waveform, but the resulting laboratory control does not automatically correct errors in the actual state or model.

For the ideal controlled plant with $w=0$, plan the control inputIntegration gives $x(t)=\min (t,1)$, so the planned command reaches the reference signal $r=1$ at $t=1$. If an unplanned constant control disturbance $w=0.2$ begins at $t=2$, the same open-loop control gives $x(t)=1+0.2(t-2)$ for $t\ge 2$: $x(4)=1.4$ and $x(8)=2.2$. Knowing the nominal response did not provide a way to correct the actual plant output.

Reachability in control theory asks about a particular initial state $x_0$ and target $x_1$: does some admissible control input make $x(T)=x_1$ at the declared time $T$, or at some allowed duration? Controllability requires this for every pair in the declared state space. This is a property of state evolution, distinct from graph reachability.

For a linear time-invariant system, solving the ordinary differential equation by a matrix exponential givesThe first term is the unforced endpoint. Reachability in control theory asks whether the control input can supply exactly the displacement $x_1-e^{AT}{x}_0$. The timing matters because the same command is carried by different amounts of subsequent dynamics. ^[ref]

Here is a contrasting mathematical example with one shared control input:The controllability matrix has rank of a matrix one. Since $\frac{d}{dt}(z_1-z_2)=0$, the difference is invariant under every allowed control input. From $(0,0)$ the reachable set is the diagonal $z_1=z_2$ when input amplitudes are unrestricted: $(1,1)$ is reachable at $T=2$ using $u=1/2$, while $(1,0)$ is impossible at every duration. Having a control input appear in both state equations did not give independent control of both states. This reachable set is pictured alongside the bounded double integrator below.

Controllability asks whether the available control inputs can move a dynamical system between every pair of states in its declared state space. Fixing a duration $T>0$ gives the precise condition ${\mathcal{R}}_T(x_0)=X$ for every $x_0\in X$, where $X$ is the state space and ${\mathcal{R}}_T(x_0)$ its reachable set. With freely chosen duration, every pair must instead be connectable in some permitted finite time, which may depend on the pair. Always specify admissible control inputs, deadlines and any constraints along the path. ^[ref]

The surprising part is that one control input can control several state variables by acting at different times. For the double integrator, acceleration changes velocity directly, while the accumulated velocity changes position. Starting from rest, accelerate during the first half of the journey and brake during the second: the final velocity can vanish although the final position has changed. The figure is an analytic mathematical example in dimensionless quantities, with no experimental measurements.

Controllability is useful before trying to design a controller: a requested endpoint outside the reachable set cannot be attained by clever choice of feedback control. The Kalman rank criterion makes this an exact linear algebra calculation for the stated linear time-invariant system model. It concerns whole states, so attaining one plant output is not enough to establish controllability. In quantum control, the declared state space may be an orbit constrained by conservation laws; see quantum controllability and unitary orbit of a density operator.

A controller is the rule or mechanism that chooses a control signal. An open-loop control can depend on a precomputed schedule; a feedback control also uses a measured output and can have its own memory, as in integral action.

The controlled plant is the part of a control theory model that receives the control input and produces the plant output, including the mechanism that applies the command. The controller is specified separately. In the illustrative ordinary differential equation $\dot{x}=u+w$, the controlled plant accumulates the control input and control disturbance; it does not itself choose $u$.

Sound intensity is the time-averaged acoustic energy flux, a vector measured in $\mathrm{W}{\mathrm{m}}^{-2}$. In linear acoustics, $\mathbf{I}=⟨p^{\prime }{\mathbf{u}}^{\prime }⟩$, where $p^{\prime }$ is acoustic pressure and ${\mathbf{u}}^{\prime }$ particle velocity. Its direction identifies net energy transport, and integrating its normal component over a surface gives the acoustic power crossing that surface.

A progressive plane acoustic wave satisfies $I=p_{\mathrm{rms}}^2/(\rho c)$, with mean mass density $\rho$ and sound speed $c$. This formula uses the relation between pressure and velocity for that particular travelling field. Two opposite equal waves instead form a standing wave with zero net sound intensity, despite nonzero root mean square pressure. A microphone alone measures pressure; inferring transported power from it therefore requires assumptions about the field, especially near reflecting boundaries.

A Falkner-Skan boundary layer is a similarity boundary-layer flow driven by an outer velocity proportional to a power of distance along the wall. Its streamfunction reduction gives the Falkner-Skan equation, with no-slip wall and outer-velocity matching conditions. It generalizes the Blasius flat-plate boundary layer.

Elasticity is the reversible mechanical response of a material to a deformation: removing the applied stress allows recovery of its original configuration. Stretching a metal wire slightly, twisting a shaft and bending a thin sheet probe different components of that response. Reversibility does not require a straight stress-strain curve; a material can be elastic while requiring finite elasticity to describe a large change of shape.

A useful experiment measures both loading and unloading. Apply a known axial force to a specimen with initial cross-sectional area $A_0$, and measure its extension over initial length $L_0$. In the small-deformation regime, the engineering stress $F/A_0$ and engineering strain $\Delta L/L_0$ determine Young's modulus. Recovery tests the elastic limit, while a proportional loading curve tests the proportional limit. Lateral contraction and twisting instead reveal Poisson's ratio and the shear modulus. These material parameters let the same constitutive equation predict the response of specimens with different dimensions.

Deformation changes a body's shape or size. Strain quantifies relative changes in lengths and angles, whereas a rigid translation or rotation alone produces no strain. The continuum approximation describes these changes using displacement fields and their spatial variation.