Theexpansion of the universe is parametrized by adimensionlessscale factor${\displaystyle a}$. Also known as thecosmic scale factor or sometimes theRobertson–Walker scale factor,^[1] this is a key parameter of theFriedmann equations.

In the early stages of theBig Bang, most of the energy was in the form of radiation, and that radiation was the dominant influence on the expansion of the universe. Later, with cooling from the expansion the roles of matter and radiation changed and the universe entered a matter-dominated era. Recent results suggest that we have already entered an era dominated bydark energy, but examination of the roles of matter and radiation are most important for understanding the early universe.

Using the dimensionless scale factor to characterize the expansion of the universe, the effective energy densities of radiation and matter scale differently. This leads to aradiation-dominated era in the very early universe but a transition to amatter-dominated era at a later time and, since about 4 billion years ago, a subsequentdark-energy–dominated era.^{[2][notes 1]}

By itself the scale factor in cosmology is ageometrical scaling factor conventionally set to be 1.0 at the present time ("now" or${\displaystyle t_0}$). At earlier times the factor is less than one. For three galaxies, their relative positions,${\displaystyle r_{ij}}$, over time are related through the scale factor${\displaystyle a(t)}$:$${\displaystyle r_{12}=a(t)r_{12}(t_0),r_{23}=a(t)r_{23}(t_0),r_{31}=a(t)r_{31}(t_0).}$$ Models of the universe specify the value of the scale factor as a function ofcosmic time. The scale factor is independent of location and direction.^[3]: 43

Some insight into the expansion can be obtained from a Newtonian expansion model which leads to a simplified version of the Friedmann equation. It relates the proper distance (which can change over time, unlike thecomoving distance${\displaystyle d_C}$ which is constant and set to today's distance) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contractingFLRW universe at any arbitrary time${\displaystyle t}$ to their distance at some reference time${\displaystyle t_0}$. The formula for this is:$${\displaystyle d(t)=a(t)d_0,}$$where${\displaystyle d(t)}$ is the proper distance at epoch${\displaystyle t}$,${\displaystyle d_0}$ is the distance at the reference time${\displaystyle t_0}$, usually also referred to as comoving distance, and${\displaystyle a(t)}$ is the scale factor.^[4] Thus, by definition,${\displaystyle d_0=d(t_0)}$ and${\displaystyle a(t_0)=1}$.

The scale factor is dimensionless, with${\displaystyle t}$ counted from the birth of the universe and${\displaystyle t_0}$ set to the presentage of the universe:13.799±0.021 Gyr^[5] giving the current value of${\displaystyle a}$ as${\displaystyle a(t_0)}$ or${\displaystyle 1}$.

The evolution of the scale factor is a dynamical question, determined by the equations ofgeneral relativity, which are presented in the case of a locally isotropic, locally homogeneous universe by theFriedmann equations.

TheHubble parameter is defined as:

$${\displaystyle H(t)\equiv \frac{\dot{a}(t)}{a(t)}}$$

where the dot represents a timederivative. The Hubble parameter varies with time, not with space, with the Hubble constant${\displaystyle H_0}$ being its current value.

From the previous equation${\displaystyle d(t)=d_{0}a(t)}$ one can see that${\displaystyle \dot{d}(t)=d_0\dot{a}(t)}$, and also that${\displaystyle d_0=\frac{d(t)}{a(t)}}$, so combining these gives${\displaystyle \dot{d}(t)=\frac{d(t)\dot{a}(t)}{a(t)}}$, and substituting the above definition of the Hubble parameter gives${\displaystyle \dot{d}(t)=H(t)d(t)}$ which is justHubble's law.

Current evidence suggests thatthe expansion of the universe is accelerating, which means that the second derivative of the scale factor${\displaystyle \ddot{a}(t)}$ is positive, or equivalently that the first derivative${\displaystyle \dot{a}(t)}$ is increasing over time.^[6] This also implies that any given galaxy recedes from us with increasing speed over time, i.e. for that galaxy${\displaystyle \dot{d}(t)}$ is increasing with time. In contrast, the Hubble parameter seems to be decreasing with time, meaning that if we were to look at some fixed distance d and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.^[7]

According to theFriedmann–Lemaître–Robertson–Walker metric which is used to model the expanding universe, if at present time we receive light from a distant object with aredshift ofz, then the scale factor at the time the object originally emitted that light is${\displaystyle a(t)=\frac{1}{1+z}}$.^[8][9]

AfterInflation, and until about 47,000 yearsafter the Big Bang, the dynamics of theearly universe were set byradiation (referring generally to the constituents of the universe which movedrelativistically, principallyphotons andneutrinos).^[10]

For a radiation-dominated universe the evolution of the scale factor in theFriedmann–Lemaître–Robertson–Walker metric is obtained solving theFriedmann equations:^[11]$${\displaystyle a(t)\propto t^{1/2}.}$$

Between about 47,000 years and 9.8 billion yearsafter the Big Bang,^[12] theenergy density of matter exceeded both the energy density of radiation and the vacuum energy density.^[13]

When theearly universe was about 47,000 years old (redshift 3600),mass–energy density surpassed theradiation energy, although the universe remainedoptically thick to radiation until the universe was about 378,000 years old (redshift 1100). This second moment in time (close to the time ofrecombination), at which the photons which compose thecosmic microwave background radiation were last scattered, is often mistaken^{[neutrality isdisputed]} as marking the end of the radiation era.

For a matter-dominated universe the evolution of the scale factor in theFriedmann–Lemaître–Robertson–Walker metric is easily obtained solving theFriedmann equations:$${\displaystyle a(t)\propto t^{2/3}}$$

Inphysical cosmology, thedark-energy–dominated era is proposed as the last of the three phases of the known universe, beginning when the Universe was about 9.8 billion years old.^[14] In the era ofcosmic inflation, the Hubble parameter is also thought to be constant, so the expansion law of the dark-energy–dominated era also holds for the inflationary prequel of the big bang.

Thecosmological constant is given the symbol Λ, and, considered as a source term in theEinstein field equation, can be viewed as equivalent to a "mass" of empty space, ordark energy. Since this increases with the volume of the universe, the expansion pressure is effectively constant, independent of the scale of the universe, while the other terms decrease with time. Thus, as the density of other forms of matter – dust and radiation – drops to very low concentrations, the cosmological constant (or "dark energy") term will eventually dominate the energy density of the Universe. Recent measurements of the change in Hubble constant with time, based on observations of distantsupernovae, show this acceleration in expansion rate,^[15] indicating the presence of such dark energy.

For a dark-energy–dominated universe, the evolution of the scale factor in theFriedmann–Lemaître–Robertson–Walker metric is easily obtained solving theFriedmann equations:$${\displaystyle a(t)\propto \exp (H_{0}t)}$$Here, the coefficient${\displaystyle H_0}$in the exponential, theHubble constant, is$${\displaystyle H_0=\sqrt{8\pi G{\rho }_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}/3}=\sqrt{\mathrm{\Lambda }/3}.}$$This exponential dependence on time makes the spacetime geometry identical to thede Sitter universe, and only holds for a positive sign of the cosmological constant, which is the case according to the currently accepted value of thecosmological constant, Λ, that is approximately2×10⁻³⁵ s⁻².The current density of theobservable universe is of the order of9.44×10⁻²⁷ kg/m³ and the age of the universe is of the order of 13.8 billion years, or4.358×10¹⁷ s. The Hubble constant,${\displaystyle H_0}$, is ≈70.88 km/s/Mpc (The Hubble time is 13.79 billion years).

• Cosmological principle
• Lambda-CDM model
• Redshift

• Padmanabhan, Thanu (1993).Structure formation in the universe. Cambridge: Cambridge University Press.ISBN 978-0-521-42486-8.
• Spergel, D. N.; et al. (2003)."First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters".Astrophysical Journal Supplement.148 (1):175–194.arXiv:astro-ph/0302209.Bibcode:2003ApJS..148..175S.CiteSeerX 10.1.1.985.6441.doi:10.1086/377226.S2CID 10794058.

• Relation of the scale factor with the cosmological constant and the Hubble constant

• Physical cosmology

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