The distance that converts a small angular separation into a transverse comoving distance at a specified cosmological redshift. It equals the radial comoving distance in spatially flat geometry; cosmological spatial curvature changes that equality. A comoving acoustic ruler has angular size $r_d/D_M$.

In metric light propagation with photon conservation and the usual reciprocity assumptions, luminosity distance and angular diameter distance satisfy $D_L=(1+z{)}^2{D}_A$. Comparing received flux with apparent angular area then gives the bolometric cosmological surface-brightness dimming factor $(1+z{)}^{-4}$. Absorption, photon conversion or an inconsistent definition of emitted/observed bands must be treated separately.

A source-frame transverse physical speed $v$ produces an observed angular rate $\dot{\theta }$. cosmological time dilation converts source time to observer time, so this operational distance equals $(1+z)$ times the angular diameter distance. In an FLRW geometry it is the transverse comoving distance $R_{0}S(\chi )$. It should be derived with the time convention stated, rather than interpreting the source speed and angular rate as measurements made by the same clock.

The look-back time is the cosmic-time interval between emission at redshift $z$ and reception today. Differentiating $1+z=R_0/R(t)$ gives $dt/dz=-1/[(1+z)H]$ and hence the displayed integral. The radial comoving-distance integral instead has denominator $H$ alone; elapsed time and present spatial separation answer different operational questions.

For a homogeneous cosmic fluid, the physical volume bounded by fixed comoving coordinates obeys $V\propto a^3$. The cosmological continuity equation therefore gives $d(\rho c^{2}V)=-PdV$. This is the first law of thermodynamics with conserved entropy and no additional chemical work. The internal energy of that volume need not be constant: a component with negative pressure gains internal energy as the universe expands. Local stress-energy conservation does not imply a global conserved energy for an arbitrary expanding spacetime.

A network whose characteristic long-string separation grows with the horizon scale can lose energy into loops and radiation. Its density can scale as $\mu /t^2$ in natural units. This is distinct from the $a^{-2}$ dilution of an ideal frozen noninteracting string network.

A space mission that mapped the temperature and polarization of the cosmic microwave background. Its nine-year analysis separates foreground emission from the microwave sky signal and fits cosmological models to the resulting maps. Combined-data fits additionally use other observations; they are not measurements from microwave maps alone.^[ref]

In flat general relativity with no separate cosmological constant, a canonical massive scalar has $\rho +3P=2{\dot{\varphi }}^2-m^2{\varphi }^2$. Accelerated expansion therefore requires ${\dot{\varphi }}^2<m^2{\varphi }^2/2$. Initial potential dominance is an instantaneous condition and does not alone fix the duration.

A family of inflationary scenarios in which a scalar initially away from the minimum of a potential can drive accelerated expansion without an initial thermal equilibrium near that minimum. For the quadratic example, a large slowly rolling field gives ${ϵ}_V=1/(4\pi G{\varphi }^2)$; its predictions depend on the potential and subsequent trajectory.

The leading relations $r=16{ϵ}_V$ and $n_s-1=-6{ϵ}_V+2{\eta }_V$ assume a canonical single slow-roll scalar, Einstein gravity and the usual vacuum initial fluctuations. More general inflation models need not obey these relations.

Primordial-tensor foreground separation is the attempt to distinguish a polarization signal from early-universe gravitational waves from other signals in microwave observations. A measured B-mode polarization pattern can contain polarized Galactic emission and conversion of other Cosmic microwave background polarization by gravitational lensing, as well as a possible primordial contribution.

Frequency coverage helps because the cosmic microwave background, dust and synchrotron radiation do not share one emission spectrum. Spatial correlations and instrumental tests provide additional information. A joint component fit must propagate uncertainty in the foreground model; a B-mode excess at one frequency is not by itself a detection of cosmic inflation.^[ref]

An integral coordinate is a coordinate value belonging to the integers. A point has integral coordinates when all its components do, relative to a specified coordinate system. In standard Cartesian coordinates, these points form the integer lattice: $(2,-3)$ qualifies, whereas $(1/2,0)$ does not. Integrality depends on the axes and units; a general rotation need not preserve it.

Sigma donation transfers electron density from an occupied ligand orbital into a metal acceptor orbital with the appropriate symmetry along the bond axis. CO uses an occupied carbon-centered $5\sigma$ orbital. Backdonation can occur simultaneously.

Pi backbonding is donation from occupied metal orbitals into ligand acceptor orbitals with pi symmetry about the metal–ligand axis. In a carbonyl ligand, occupation of CO ${\pi }^{\ast }$ weakens C–O while strengthening metal–carbon bonding. Lower CO stretching spectroscopic wavenumber is therefore a useful comparative probe, with electrostatic and mode-coupling qualifications.

Donation from an occupied ligand orbital into a metal orbital of pi symmetry relative to the metal–ligand axis. It differs from metal-to-ligand pi backbonding.

An electron-pair interaction represented as donation from a metal-base fragment into an orbital of a metal-acid fragment. electron counting must avoid adding a new pair twice to an already occupied donor orbital.

An empirical occupancy relation, with half-occupancy concentration $K_{1/2}$ and Hill coefficient $n$. A fitted $K_{1/2}$ need not be an individual site's microscopic dissociation constant.

In evolutionary analysis, an action with a direct fitness cost to its actor and a benefit to another. The comparison depends on the specified alternative and timescale.

Selection of interaction partners in a way that can favour useful contributions through access to future benefits. Its effects depend on available alternatives, information and switching costs.

Conditional cooperation based on previous interaction with the same partner. Repeated-game support for cooperation depends on encounter continuation, observation and the competing strategies.