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Observations show that the expansion of the universe is accelerating, such that the velocity at which a distant galaxy recedes from the observer is continuously increasing with time.^[1][2][3] The accelerated expansion of the universe was discovered in 1998 by two independent projects, the Supernova Cosmology Project and the High-Z Supernova Search Team, which used distant type Ia supernovae to measure the acceleration.^[4][5][6] The idea was that as type Ia supernovae have almost the same intrinsic brightness (a standard candle), and since objects that are farther away appear dimmer, the observed brightness of these supernovae can be used to measure the distance to them. The distance can then be compared to the supernovae's cosmological redshift, which measures how much the universe has expanded since the supernova occurred; the Hubble law established that the farther away that an object is, the faster it is receding. The unexpected result was that objects in the universe are moving away from one another at an accelerating rate. Cosmologists at the time expected that recession velocity would always be decelerating, due to the gravitational attraction of the matter in the universe. Three members of these two groups have subsequently been awarded Nobel Prizes for their discovery.^[7] Confirmatory evidence has been found in baryon acoustic oscillations, and in analyses of the clustering of galaxies.

The accelerated expansion of the universe is thought to have begun since the universe entered its dark-energy-dominated era roughly 5 billion years ago.^{[8][notes 1]} Within the framework of general relativity, an accelerated expansion can be accounted for by a positive value of the cosmological constant Λ, equivalent to the presence of a positive vacuum energy, dubbed "dark energy". While there are alternative possible explanations, the description assuming dark energy (positive Λ) is used in the current standard model of cosmology, which also includes cold dark matter (CDM) and is known as the Lambda-CDM model.

Background

In the decades since the detection of cosmic microwave background (CMB) in 1965,^[9] the Big Bang model has become the most accepted model explaining the evolution of our universe. The Friedmann equation defines how the energy in the universe drives its expansion.

${\displaystyle H^{2}={\left({\frac {\dot {a}}{a}}\right)}^{2}={\frac {8{\pi }G}{3}}\rho -{\frac {{\kappa }c^{2}}{a^{2}}}}$

where κ represents the curvature of the universe, a(t) is the scale factor, ρ is the total energy density of the universe, and H is the Hubble parameter.^[10]

We define a critical density

${\displaystyle \rho _{c}={\frac {3H^{2}}{8{\pi }G}}}$

and the density parameter

${\displaystyle \Omega ={\frac {\rho }{\rho _{c}}}}$

We can then rewrite the Hubble parameter as

${\displaystyle H(a)=H_{0}{\sqrt {{\Omega _{k}a^{-2}+\Omega }_{m}a^{-3}+\Omega _{r}a^{-4}+\Omega _{\mathrm {DE} }a^{-3(1+w)}}}}$

where the four currently hypothesized contributors to the energy density of the universe are curvature, matter, radiation and dark energy.^[11] Each of the components decreases with the expansion of the universe (increasing scale factor), except perhaps the dark energy term. It is the values of these cosmological parameters which physicists use to determine the acceleration of the universe.

The acceleration equation describes the evolution of the scale factor with time

${\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4{\pi }G}{3}}\left(\rho +{\frac {3P}{c^{2}}}\right)}$

where the pressure P is defined by the cosmological model chosen. (see explanatory models below)

Physicists at one time were so assured of the deceleration of the universe's expansion that they introduced a so-called deceleration parameter q₀.^[12] Current observations indicate this deceleration parameter is negative.

Relation to inflation

According to the theory of cosmic inflation, the very early universe underwent a period of very rapid, quasi-exponential expansion. While the time-scale for this period of expansion was far shorter than that of the current expansion, this was a period of accelerated expansion with some similarities to the current epoch.

Technical definition

The definition of "accelerating expansion" is that the second time derivative of the cosmic scale factor, ${\displaystyle {\ddot {a}}}$, is positive, which is equivalent to the deceleration parameter, ${\displaystyle q}$, being negative. However, note this does not imply that the Hubble parameter is increasing with time. Since the Hubble parameter is defined as ${\displaystyle H(t)\equiv {\dot {a}}(t)/a(t)}$, it follows from the definitions that the derivative of the Hubble parameter is given by

${\displaystyle {\frac {dH}{dt}}=-H^{2}(1+q)}$

so the Hubble parameter is decreasing with time unless ${\displaystyle q<-1}$. Observations prefer ${\displaystyle q\approx -0.55}$, which implies that ${\displaystyle {\ddot {a}}}$ is positive but ${\displaystyle dH/dt}$ is negative. Essentially, this implies that the cosmic recession velocity of any one particular galaxy is increasing with time, but its velocity/distance ratio is still decreasing; thus different galaxies expanding across a sphere of fixed radius cross the sphere more slowly at later times.

It is seen from above that the case of "zero acceleration/deceleration" corresponds to ${\displaystyle a(t)}$ is a linear function of ${\displaystyle t}$, ${\displaystyle q=0}$, ${\displaystyle {\dot {a}}=const}$, and ${\displaystyle H(t)=1/t}$.

Evidence for acceleration

The rate of expansion of the universe can be analyzed using the magnitude-redshift relationship of astronomical objects using standard candles, or their distance-redshift relationship using standard rulers. Also a factor is the growth of large-scale structure, finding that the observed values of the cosmological parameters are best described by models which include an accelerating expansion.

Supernova observation

In 1998, the first evidence for acceleration came from the observation of Type Ia supernovae, which are exploding white dwarf stars that have exceeded their stability limit. Because they all have similar masses, their intrinsic luminosity can be standardized. Repeated imaging of selected areas of the sky is used to discover the supernovae, then follow-up observations give their peak brightness, which is converted into a quantity known as luminosity distance (see distance measures in cosmology for details).^[13] Spectral lines of their light can be used to determine their redshift.

For supernovae at redshift less than around 0.1, or light travel time less than 10 percent of the age of the universe, this gives a nearly linear distance–redshift relation due to Hubble's law. At larger distances, since the expansion rate of the universe has changed over time, the distance-redshift relation deviates from linearity, and this deviation depends on how the expansion rate has changed over time. The full calculation requires computer integration of the Friedmann equation, but a simple derivation can be given as follows: the redshift z directly gives the cosmic scale factor at the time the supernova exploded.

${\displaystyle a(t)={\frac {1}{1+z}}}$

So a supernova with a measured redshift z = 0.5 implies the universe was 1/1 + 0.5 = 2/3 of its present size when the supernova exploded. In the case of accelerated expansion, ${\displaystyle {\ddot {a}}}$ is positive; therefore, ${\displaystyle {\dot {a}}}$ was smaller in the past than today. Thus, an accelerating universe took a longer time to expand from 2/3 to 1 times its present size, compared to a non-accelerating universe with constant ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Accelerating_universe
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