Information about Friedmann Equations
cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922[1] from the Einstein field equations for the Friedmann-Lemaître-Robertson-Walker metric and a fluid with a given energy density ρ and pressure 
. The equations are:
is the cosmological constant possibly caused by vacuum energy, 
is the gravitational constant, 
is the speed of light, 
is the scale factor, and 
is the Gaussian curvature when 
(i.e. today). If the shape of the universe is hyperspherical and 
is the radius of curvature (
in the present-day), then 
. Generally, 
is the Gaussian curvature. If is positive, then the universe is hyperspherical. If is zero, then the universe is flat. If is negative, then the universe is hyperbolic. Note that 
and are in general functions of . The Hubble parameter, 
, is the rate of expansion of the universe.
These equations are sometimes simplified by redefining
to give:
The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the energy density, vacuum energy, and curvature). Evaluating the Hubble parameter at the present time yields the Hubble constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.
Some cosmologists call the second of these two equations the acceleration equation and reserve the term Friedmann equation for only the first equation.
, is defined as the ratio of actual (or observed) density to the critical density 
of the Friedmann universe.
An expression for critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the curvature, K, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:
And the expression for the density parameter (useful for comparing different cosmological models) then follows:
This term originally was used as a means to determine the geometry of the field where is the critical density for which the geometry is flat. Assuming a zero vacuum energy density, if is larger than unity, the geometry is closed; the universe will eventually stop expanding, then collapse. If is less than unity, it is open; and the universe expands forever. However, one can also subsume the curvature and vacuum energy terms into a more general expression for in which case this energy density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the Lambda-CDM model, there are important components of due to baryons, cold dark matter and dark energy. The geometry of spacetime has been measured by the WMAP probe to be nearly flat meaning that the curvature parameter κ is zero.
The first Friedmann Equation is often seen in a form with density parameters.
is the radiation density today, 
is the matter (dark plus baryonic) density today, and 
is the cosmological constant or vacuum density today.
,
Ωc=-κ/H0²a0²
where a0 and H0 are separately the scale factor and the Hubble parameter today.
Then we can have
where Ueff(ã)=Ωã²/2. For any form of the effective potential Ueff(ã), there is an equation of state p=p(ρ) that will produce it.
This timeline of cosmological theories and discoveries is a chronological catalog of the evolution of humankind's understanding of the cosmos over the last two-plus millennia.
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)
)
. The equations are:
is the cosmological constant possibly caused by vacuum energy, is the gravitational constant, is the speed of light, is the scale factor, and is the Gaussian curvature when (i.e. today). If the shape of the universe is hyperspherical and is the radius of curvature ( in the present-day), then . Generally, is the Gaussian curvature. If is positive, then the universe is hyperspherical. If is zero, then the universe is flat. If is negative, then the universe is hyperbolic. Note that and are in general functions of . The Hubble parameter, , is the rate of expansion of the universe.
These equations are sometimes simplified by redefining
to give:
The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the energy density, vacuum energy, and curvature). Evaluating the Hubble parameter at the present time yields the Hubble constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.
Some cosmologists call the second of these two equations the acceleration equation and reserve the term Friedmann equation for only the first equation.
The density parameter
The density parameter,, is defined as the ratio of actual (or observed) density to the critical density of the Friedmann universe.
An expression for critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the curvature, K, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:
And the expression for the density parameter (useful for comparing different cosmological models) then follows:
This term originally was used as a means to determine the geometry of the field where
is the critical density for which the geometry is flat. Assuming a zero vacuum energy density, if is larger than unity, the geometry is closed; the universe will eventually stop expanding, then collapse. If is less than unity, it is open; and the universe expands forever. However, one can also subsume the curvature and vacuum energy terms into a more general expression for in which case this energy density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the Lambda-CDM model, there are important components of due to baryons, cold dark matter and dark energy. The geometry of spacetime has been measured by the WMAP probe to be nearly flat meaning that the curvature parameter κ is zero.
The first Friedmann Equation is often seen in a form with density parameters.
is the radiation density today, is the matter (dark plus baryonic) density today, and is the cosmological constant or vacuum density today.
Rescaled Friedmann equation
Set a=ãa0, ρc=3H0²/8πG, ρ=ρcΩ,,
Ωc=-κ/H0²a0²
where a0 and H0 are separately the scale factor and the Hubble parameter today.
Then we can have
where Ueff(ã)=Ωã²/2. For any form of the effective potential Ueff(ã), there is an equation of state p=p(ρ) that will produce it.
References
1. ^ Friedmann, A: Über die Krümmung des Raumes, Z. Phys. 10 (1922), 377-386. (English translation in: Gen. Rel. Grav. 31 (1999), 1991-2000.)
Physical cosmology, as a branch of astronomy, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. Cosmology involves itself with studying the motions of the celestial bodies and the first cause.
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The age of the universe, in physics, is the time elapsed between the Big Bang and the present day. Current observations suggest that this is about 13.7 billion years, with an uncertainty of about +/-200 million years.
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Comoving coordinates
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cosmic microwave background radiation (most often abbreviated CMB but occasionally CMBR, CBR or MBR, also referred to as relic radiation) is a form of electromagnetic radiation discovered in 1965 that fills the entire universe [1].
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In physical cosmology, dark energy is a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of the universe. [1]
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In astrophysics and cosmology, dark matter is hypothetical matter of unknown composition that does not emit or reflect enough electromagnetic radiation to be observed directly, but whose presence can be inferred from gravitational effects on visible matter.
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Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity; it describes a , isotropic expanding or contracting universe.
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In physical cosmology, cosmic inflation is the idea that the nascent universe passed through a phase of exponential expansion that was driven by a negative-pressure vacuum energy density.
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large-scale structure refers to the characterization of observable distributions of matter and light on the largest scales (typically on the order of billions of light-years).
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The metric expansion of space is a key part of science's current understanding of the universe, whereby spacetime itself is described by a metric which changes over time in such a way that the spatial dimensions appear to grow or stretch as the universe gets older.
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In physical cosmology, Big Bang nucleosynthesis (or primordial nucleosynthesis) refers to the production of nuclei other than those of H-1 (i.e. the normal, light isotope of hydrogen, whose nuclei consist of a single proton each) during the early phases of the
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redshift occurs when the electromagnetic radiation, usually visible light, that is emitted from or reflected off an object is shifted toward the (less energetic) red end of the electromagnetic spectrum.
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The shape of the Universe is an informal name for a subject of investigation within physical cosmology. Cosmologists and astronomers describe the geometry of the universe which includes both local geometry and global geometry.
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Structure formation refers to a fundamental problem in physical cosmology. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.7 Gyr ago.
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Physical cosmology
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- Age of the universe
- Big Bang
- Blueshift
- Comoving distance
- Cosmic microwave background
- Dark energy
- Dark matter
- FLRW metric
- Friedmann equations
- Galaxy formation
- Hubble's law
- Inflation
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For a timeline of the cosmos (or universe), see .
This timeline of cosmological theories and discoveries is a chronological catalog of the evolution of humankind's understanding of the cosmos over the last two-plus millennia.
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The ultimate fate of the universe is a topic in physical cosmology. Many possible fates are predicted by rival scientific theories, including futures of both finite and infinite duration.
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Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called "high energy physics"
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Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity.
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Physical cosmology, as a branch of astronomy, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. Cosmology involves itself with studying the motions of the celestial bodies and the first cause.
..... Click the link for more information.
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The metric expansion of space is a key part of science's current understanding of the universe, whereby spacetime itself is described by a metric which changes over time in such a way that the spatial dimensions appear to grow or stretch as the universe gets older.
..... Click the link for more information.
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