Scale factor

In Cosmology, the scale factor measurement the way in which the distance between two objects, taken in practice between two distant celestial objects, varies with time because of Expansion of the universe. The concept is used when one considers a cosmological Modèle satisfying the cosmological Principe i.e. isotropic Homogène and . More precisely if one calls $a (T)$ this scale factor (where T corresponds to the cosmic Temps, i.e. at the ordinary time, called clean Temps, measured by an observer according to the general movement of expansion) and that at one moment $t_0$ the physical distance between two objects (whose coordinated comobiles is supposed to be fixed) is $L_0$ then any other moment $t$ the physical distance between these two same objects has will be given by

L (T) = \ frac {has (T)}{has (t_0)}L_0 . In such a cosmological model, the temporal variation of the scale factor is primarily determined by the properties of the various forms of energy which fill up the universe, via the equations of Einstein (or their form adapted to the problem, in general the equations of Friedmann). The measurable quantity observationnellement is not EM itself, but its rate scale factor of variation, which in this context is called the Constante of Hubble. The latter is in general not constant during time even if its variations are slow on a human scale. By noting it H the relationship to the scale factor is written:

H = \ frac {1} {has} \ frac

The standardization of the scale factor is arbitrary. It is determined by a given reference length. One can for example standardize it while imposing that it has value 1 today.

Examples

The standard cosmological model revealing the scale factor is the model known as of Friedmann-Lemaître-Robertson-Walker (FLRW). In the simplest case, with a space Curve null, the Métrique of the Espace-temps is written

{\ rm D} s^2= c^2 {\ rm D} t^2 - has (T) ^2 ({\ rm D} x^2+ {\ rm D} y^2+ {\ rm D} z^2) \,

where

x, there, z

is the coordinated comobiles.

t

represents well the clean time measured by an observer whose coordinates comobiles are fixed and the scale factor is well the quantity making it possible to pass from the Distance comobile at the physical distance.

In the general case where space can be curved, the metric one is written, in a frame of reference given:

${\ rm D} s^2= c^2 {\ rm D} t^2 - has (T) ^2 \ gamma_ {ij} {\ rm D} x^i {\ rm D} x^j \,$ ,

where $\ gamma_ {ij}$ expresses the local space curve of space.

In practice, the celestial objects are subjected to the gravitational attraction of the other surrounding objects and are not strictly motionless compared to those. Their coordinates comobiles are then not fixed. Nevertheless if the latter sufficiently distant one from the other one can neglect these own movement, which seldom do not exceed the thousand of Kilomètre S by second. The scale factor can thus represent the variation of distance between two sufficiently distant objects so that their speed compared to the Référentiel of the cosmological diffuse Fond is negligible in front of their speed of distancing via the Loi of Hubble. The order of magnitude of the Constante of Hubble being the hundred kilometer a second and by Mégaparsec, the scale factor describes the relative variation of distance between two objects now separate that a few hundreds of mégaparsecs or more.

See too

Equations of Metric Friedmann
of Friedmann-Lemaître-Robertson-Walker
Distance comobile
Law of Hubble

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