Where does expansion collide?

The tendency of a material to change its dimensions when its temperature changes, quantified by a coefficient of thermal expansion that relates the fractional change in length or volume to the change in temperature. The effect arises from the anharmonicity of the interatomic potential, so that the mean spacing between atoms grows as thermal vibration increases. Drebushchak's 2020 review Thermal expansion of solids: review on theories surveys the lattice-dynamical accounts of this behaviour. The referent is a bulk mechanical response of matter to heat, measured per kelvin, not a cosmic stretching of space, a phase of national output, or a sum of terms in mathematics.[1]

Unit

A material's fractional change in size per degree, the coefficient of thermal expansion in reciprocal kelvin

Recovery means

On cooling the material contracts back along its expansion curve, reversibly in the elastic regime absent a phase change or hysteresis

Measured by

Determined by dilatometry, optical interferometry, or X-ray lattice-parameter measurement as strain versus temperature

The steady increase of proper distances between distant galaxies as the universe ages: on large scales space itself stretches, so recession velocity grows with distance according to the Hubble-Lemaitre law, and the change is carried by a time-dependent scale factor in the cosmological metric. Hubble's 1929 A relation between distance and radial velocity among extra-galactic nebulae reported the linear velocity-distance relation that first evidenced this expansion. The referent is a global geometric property of the cosmos, tracked by redshift and the scale factor, not the swelling of a heated solid, a rise in economic output, or an algebraic series.[2]

Unit

The rate at which cosmic distances grow with distance, the Hubble constant in kilometres per second per megaparsec

Recovery means

Not applicable

Measured by

Inferred from galaxy redshifts against distance indicators such as Cepheids and supernovae, calibrating the rate of expansion

The phase of the business cycle during which aggregate real activity rises broadly across the economy, running from a cyclical trough up to the following peak, before a contraction sets in. Burns and Mitchell's 1946 Measuring Business Cycles defined the reference cycle in just these terms, with expansion and contraction as its two phases, dated from turning points read across many series at once. The referent is a datable interval of rising national output and employment, not a material swelling with heat, a stretching of the cosmos, or a mathematical series.[3]

Unit

The interval of generally rising output from a trough to the next peak, dated in months

Recovery means

The expansion ends at a peak, after which a contraction reverses the rise until a trough begins the next expansion

Measured by

Dated by a committee from the co-movement of output, income, employment, and sales, marking the trough and peak that bound the phase

The representation of a function as an infinite sum of simpler terms, such as a power series or Taylor series whose coefficients come from the function's derivatives, valid within a radius of convergence. Rudin's Principles of Mathematical Analysis develops power series and Taylor's theorem, giving conditions under which such an expansion converges to the function it represents. A related asymptotic expansion approximates a function without needing to converge at all. The referent is a formal identity between a function and a sum of terms, not the swelling of matter, the stretching of space, or a phase of the economy.[4]

Unit

A function rewritten as an infinite sum of terms, with a radius or region of validity

Recovery means

Not applicable

Measured by

Assessed by the radius of convergence, the remainder or truncation error, and the rate at which the partial sums approach the function