more, known as cosmic inflation. The early universe remained hot (above 10,000 K) for several hundred thousand years, a state that is detectable as a residual cosmic microwave background, or CMB, a very low energy radiation emanating from all parts of the sky. The "Big Bang" scenario, with cosmic inflation and standard particle physics, is the only current cosmological model consistent with the observed continuing expansion of space, the observed distribution of lighter elements in the universe (hydrogen, helium, and lithium), and the spatial texture of minute irregularities (anisotropies) in the CMB radiation. Cosmic inflation also addresses the "horizon problem" in the CMB; indeed, it seems likely that the universe is larger than the observable particle horizon. The model uses the Friedmann–Lemaître–Robertson–Walker metric, the Friedmann equations and the cosmological equations of state to describe the observable universe from right after the inflationary epoch to present and future. Cosmic expansion history The expansion of the universe is parameterized by a dimensionless scale factor (with time counted from the birth of the universe), defined relative to the present day, so ; the usual convention in cosmology is that subscript 0 denotes present-day values, so is the current age of the universe. The scale factor is related to the observed redshift of the light emitted at time by The expansion rate is described by the time-dependent Hubble parameter, , defined as where is the time-derivative of the scale factor. The first Friedmann equation gives the expansion rate in terms of the matter+radiation density the curvature and the cosmological constant where as usual is the speed of light and is the gravitational constant. A critical density is the present-day density, which gives zero curvature , assuming the cosmological constant is zero, regardless of its actual value. Substituting these conditions to the Friedmann equation gives where is the reduced Hubble constant. If the cosmological constant were actually zero, the critical density would also mark the dividing line between eventual recollapse of the universe to a Big Crunch, or unlimited expansion. For the Lambda-CDM model with a positive cosmological constant (as observed), the universe is predicted to expand forever regardless of whether the total density is slightly above or below the critical density; though other outcomes are possible in extended models where the dark energy is not constant but actually time-dependent. It is standard to define the present-day density parameter for various species as the dimensionless