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The Friedmann equations are a set of equations in physical 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 Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density $ρ$ and pressure $p$ . The equations for negative spatial curvature were given by Friedmann in 1924.^[2]

Assumptions

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the Cosmological Principle; empirically, this is justified on scales larger than ~100 Mpc. The Cosmological Principle implies that the metric of the universe must be of the form:

d s^{2} = a (t)^{2} d s_{3}^{2} - c^{2} d t^{2}

where $d s_{3}^{2}$ is a three dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. The parameter $k$ discussed below takes the value 0, 1, −1 in these three cases respectively. It is this fact that allows us to sensibly speak of a "scale factor", $a (t)$ .

Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute Christoffel symbols, then the Ricci tensor. With the stress-energy tensor for a perfect fluid, we plug them into Einstein field equations and the resulting equations are described below.

The equations

There are two independent Friedmann equations for modeling a homogeneous, isotropic universe. The first is:

\frac{{\dot{a}}^{2} + k c^{2}}{a^{2}} = \frac{8 π G ρ + Λ c^{2}}{3}

which is derived from the 00 component of Einstein's field equations. And the second is:

\frac{\ddot{a}}{a} = - \frac{4 π G}{3} (ρ + \frac{3 p}{c^{2}}) + \frac{Λ c^{2}}{3}

which is derived from the first together with the trace of Einstein's field equations. $H \equiv \frac{\dot{a}}{a}$ is the Hubble parameter, G, Λ, and c are universal constants (G is Newton's gravitational constant, Λ is the cosmological constant, c is the speed of light in vacuum). k is constant throughout a particular solution, but may vary from one solution to another. a, H, ρ, and p are functions of time. $\frac{k}{a^{2}}$ is the spatial curvature in any time-slice of the universe; it is equal to one-sixth of the spatial Ricci curvature scalar R since $R = \frac{6}{c^{2} a^{2}} (\ddot{a} a + {\dot{a}}^{2} + k c^{2})$ in the Friedmann model. We see that in the Friedmann equations, a(t) depends only on ρ, p, Λ, and intrinsic curvature k. It does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for a and k which describe the same physics:

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k = +1, 0 or −1 depending on whether the shape of the universe is a closed 3-sphere, flat (i.e. Euclidean space) or an open 3-hyperboloid, respectively.^[3] If k = +1, then $a$ is the radius of curvature of the universe. If k = 0, then a may be fixed to any arbitrary positive number at one particular time. If k = −1, then (loosely speaking) one can say that i·a is the radius of curvature of the universe.
a is the scale factor which is taken to be 1 at the present time. $k$ is the spatial curvature when $a = 1$ (i.e. today). If the shape of the universe is hyperspherical and $R_{t}$ is the radius of curvature ( $R_{0}$ in the present-day), then $a = R_{t} / R_{0}$ . If $k$ is positive, then the universe is hyperspherical. If $k$ is zero, then the universe is flat. If $k$ is negative, then the universe is hyperbolic.

Using the first equation, the second equation can be re-expressed as

\dot{ρ} = - 3 H (ρ + \frac{p}{c^{2}}),

which eliminates $Λ$ and expresses the conservation of mass-energy $T^{α β}_{; β} = 0$ .

These equations are sometimes simplified by replacing

ρ \to ρ - \frac{Λ c^{2}}{8 π G}

p \to p + \frac{Λ c^{4}}{8 π G}

to give:

H^{2} = {(\frac{\dot{a}}{a})}^{2} = \frac{8 π G}{3} ρ - \frac{k c^{2}}{a^{2}}

\dot{H} + H^{2} = \frac{\ddot{a}}{a} = - \frac{4 π G}{3} (ρ + \frac{3 p}{c^{2}}) .

And the simplified form of the second equation is invariant under this transformation.

The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's 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 Friedmann acceleration equation and reserve the term Friedmann equation for only the first equation.

Density parameter

The density parameter, $Ω$ , is defined as the ratio of the actual (or observed) density $ρ$ to the critical density $ρ_{c}$ of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclidean). In earlier models, which did not include a cosmological constant term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.

To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2 atoms per cubic metre.^[4]

Estimated relative distribution for components of the energy density of the Universe. Dark energy dominates the total energy (74%) while dark matter (22%) constitutes most of the mass. Of the remaining baryonic matter (4%), only one tenth is compact. On 21 March 2013, the European-led research team behind the Planck cosmology probe released new data refining these values to 4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy.

A much greater density comes from the unidentified dark matter; both ordinary and dark matter contribute in favor of contraction of the universe. However, the largest part comes from so-called dark energy, which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), the dark energy does not lead to contraction of the universe but rather accelerates its expansion. Therefore, the universe will expand forever.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:

ρ_{c} = \frac{3 H^{2}}{8 π G} .

The density parameter (useful for comparing different cosmological models) is then defined as:

Ω \equiv \frac{ρ}{ρ_{c}} = \frac{8 π G ρ}{3 H^{2}} .

This term originally was used as a means to determine the spatial geometry of the universe, where $ρ_{c}$ is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if $Ω$ is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If $Ω$ is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for $Ω$ in which case this density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the ΛCDM model, there are important components of $Ω$ due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter $k$ is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see. (Similarly, the fact that Earth is approximately flat at the scale of the Netherlands does not imply that the Earth is flat: it only implies that it is much larger than the Netherlands.)

The first Friedmann equation is often seen in terms of the present values of the density parameters, that is^[5]

\frac{H^{2}}{H_{0}^{2}} = Ω_{R} a^{- 4} + Ω_{M} a^{- 3} + Ω_{k} a^{- 2} + Ω_{Λ} .

Here $Ω_{R}$ is the radiation density today (i.e. when $a = 1$ ), $Ω_{M}$ is the matter (dark plus baryonic) density today, $Ω_{k} = 1 - Ω$ is the "spatial curvature density" today, and $Ω_{Λ}$ is the cosmological constant or vacuum density today.

Useful solutions

The Friedmann equations can be solved exactly in presence of a perfect fluid with equation of state

p = w ρ c^{2},

where $p$ is the pressure, $ρ$ is the mass density of the fluid in the comoving frame and $w$ is some constant.

In spatially flat case (k = 0), the solution for the scale factor is

a (t) = a_{0} t^{\frac{2}{3 (w + 1)}}

where $a_{0}$ is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by $w$ is extremely important for cosmology. E.g. $w = 0$ describes a matter-dominated universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as

a (t) \propto t^{2 / 3}

matter-dominated

Another important example is the case of a radiation-dominated universe, i.e., when $w = 1 / 3$ . This leads to

a (t) \propto t^{1 / 2}

radiation dominated

Note that this solution is not valid for domination of the cosmological constant, which corresponds to an $w = - 1$ . In this case the energy density is constant and the scale factor grows exponentially.

Solutions for other values of k can be found at Template:Cite web.

Mixtures

If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then

{\dot{ρ}}_{f} = - 3 H (ρ_{f} + \frac{p_{f}}{c^{2}})

holds separately for each such fluid f. In each case,

{\dot{ρ}}_{f} = - 3 H (ρ_{f} + w_{f} ρ_{f})

from which we get

ρ_{f} \propto a^{- 3 (1 + w_{f})} .

For example, one can form a linear combination of such terms

ρ = A a^{- 3} + B a^{- 4} + C a^{0}

where: A is the density of "dust" (ordinary matter, w = 0) when a = 1; B is the density of radiation (w = 1/3) when a = 1; and C is the density of "dark energy" (w=−1). One then substitutes this into

{(\frac{\dot{a}}{a})}^{2} = \frac{8 π G}{3} ρ - \frac{k c^{2}}{a^{2}}

and solves for a as a function of time.

Rescaled Friedmann equation

Set $\tilde{a} = \frac{a}{a_{0}}, ρ_{c} = \frac{3 H_{0}^{2}}{8 π G}, Ω = \frac{ρ}{ρ_{c}}, t = \frac{\tilde{t}}{H_{0}}, Ω_{c} = - \frac{k c^{2}}{H_{0}^{2} a_{0}^{2}}$ , where $a_{0}$ and $H_{0}$ are separately the scale factor and the Hubble parameter today. Then we can have

\frac{1}{2} {(\frac{d \tilde{a}}{d \tilde{t}})}^{2} + U_{e f f} (\tilde{a}) = \frac{1}{2} Ω_{c}

where $U_{eff} (\tilde{a}) = \frac{Ω {\tilde{a}}^{2}}{2}$ . For any form of the effective potential $U_{eff} (\tilde{a})$ , there is an equation of state $p = p (ρ)$ that will produce it.

Notes

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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang). The original Russian manuscript of this paper is preserved in the Ehrenfest archive.
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Template:De icon (English translation in: One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang)
↑ Ray A d'Inverno, Introducing Einstein's Relativity, ISBN 0-19-859686-3.
↑ Rees, M., Just Six Numbers, (2000) Orion Books, London, p. 81, p. 82
↑ http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:astro-ph/0703739

Template:Relativity

ru:Уравнение_Фридмана

[af1922-1] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Template:De icon (English translation in: One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang). The original Russian manuscript of this paper is preserved in the Ehrenfest archive.

[af1924-2] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Template:De icon (English translation in: One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang)

[3] Ray A d'Inverno, Introducing Einstein's Relativity, ISBN 0-19-859686-3.

[4] Rees, M., Just Six Numbers, (2000) Orion Books, London, p. 81, p. 82

[5] ttp://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:astro-ph/0703739

[1]

[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
-== UGG Noir Luciana ==
+{{Cosmology|cTopic=Expanding universe}}
+[[Image:Aleksandr Fridman.png|thumb|right|200px|[[Alexander Friedmann]]]]
+The '''Friedmann equations''' are a set of [[equation]]s in [[physical cosmology]] that govern the [[metric expansion of space|expansion of space]] in [[Homogeneity (physics)|homogeneous]] and [[isotropy|isotropic]] models of the universe within the context of [[general relativity]]. They were first derived by [[Alexander Alexandrovich Friedmann|Alexander Friedmann]] in 1922<ref name=af1922>{{cite journal | first=A | last=Friedman | authorlink=Alexander Alexandrovich Friedman | title=Über die Krümmung des Raumes | journal=Z. Phys. | volume=10 | year=1922 | issue=1 | pages=377–386 | doi=10.1007/BF01332580|bibcode = 1922ZPhy...10..377F }} {{de icon}} (English translation in: {{cite journal | first=A | last=Friedman | title=On the Curvature of Space | journal=General Relativity and Gravitation | volume=31 | issue=12 | year=1999 | pages= 1991–2000 | bibcode=1999GReGr..31.1991F | doi=10.1023/A:1026751225741}}). The original Russian manuscript of this paper is preserved in the [http://ilorentz.org/history/Friedmann_archive Ehrenfest archive].</ref> from [[Einstein field equations|Einstein's field equations]] of [[gravitation]] for the [[Friedmann–Lemaître–Robertson–Walker metric]] and a [[perfect fluid]] with a given [[density|mass density]] <math>\!\rho</math> and [[pressure]] <math>\!p</math>. The equations for negative spatial curvature were given by Friedmann in 1924.<ref name=af1924>{{cite journal | first=A | last=Friedmann| authorlink=Alexander Alexandrovich Friedman | title=Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes | journal=Z. Phys. | volume=21 | year=1924 | issue=1 | pages=326–332 | doi=10.1007/BF01328280|bibcode = 1924ZPhy...21..326F }} {{de icon}} (English translation in: {{cite journal | first=A | last=Friedmann | title= On the Possibility of a World with Constant Negative Curvature of Space | journal=General Relativity and Gravitation | volume=31 | issue=12 | year=1999 | pages=2001–2008 | bibcode=1999GReGr..31.2001F | doi=10.1023/A:1026755309811}})</ref>
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+==Assumptions==
+{{main|Friedmann–Lemaître–Robertson–Walker metric}}
+The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the [[Cosmological Principle]]; empirically, this is justified on scales larger than ~100 [[Parsec|Mpc]]. The Cosmological Principle implies that the metric of the universe must be of the form:
+:<math> ds^2 = a(t)^2 \, ds_3^2 - c^2 \, dt^2 </math>
+where <math>\! ds_3^2</math> is a three dimensional metric that must be one of '''(a)''' flat space, '''(b)''' a sphere of constant positive curvature or '''(c)''' a hyperbolic space with constant negative curvature. The parameter <math>\! k </math> discussed below takes the value 0, 1, −1 in these three cases respectively. It is this fact that allows us to sensibly speak of a "[[Scale factor (Universe)|scale factor]]", <math>\! a(t) </math>.
-== UGG Bleu ==
+Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute [[Christoffel symbols]], then the [[Ricci tensor]]. With the [[stress-energy tensor]] for a perfect fluid, we plug them into Einstein field equations and the resulting equations are described below.
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+==The equations==
-== Promo UGG ==
+There are two independent Friedmann equations for modeling a homogeneous, isotropic universe. The first is:
+:<math> \frac{\dot{a}^2 + kc^2}{a^2} = \frac{8 \pi G \rho + \Lambda c^2}{3} </math>
+which is derived from the 00 component of [[Einstein field equations|Einstein's field equations]]. And the second is:
+:<math>\frac{\ddot{a}}{a} =  -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}</math>
+which is derived from the first together with the [[Trace (linear algebra)|trace]] of Einstein's field equations. <math>H \equiv \frac{\dot{a}}{a}</math> is the [[Hubble parameter]], ''G'', Λ, and ''c'' are universal constants (''G'' is Newton's [[gravitational constant]], Λ is the [[cosmological constant]], ''c'' is the [[speed of light|speed of light in vacuum]]). ''k'' is constant throughout a particular solution, but may vary from one solution to another. ''a'', ''H'', ρ, and ''p'' are functions of time. <math>k \over a^2</math> is the [[curvature|spatial curvature]] in any time-slice of the universe; it is equal to one-sixth of the spatial [[scalar curvature|Ricci curvature scalar R]] since <math>R = \frac{6}{c^2 a^2}(\ddot{a} a + \dot{a}^2 + kc^2)</math> in the Friedmann model. We see that in the Friedmann equations, a(t) depends only on ρ, ''p'',  Λ, and intrinsic curvature k. It does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for ''a'' and ''k'' which describe the same physics:
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+{{General relativity|cTopic=Equations}}
+*''k'' = +1, 0 or −1 depending on whether the [[shape of the universe]] is a closed [[3-sphere]], flat (i.e. [[Euclidean space]]) or an open 3-[[hyperboloid]], respectively.<ref>Ray A d'Inverno, ''Introducing Einstein's Relativity'', ISBN 0-19-859686-3.</ref> If ''k'' = +1, then <math>a</math> is the radius of curvature of the universe. If ''k'' = 0, then ''a'' may be fixed to any arbitrary positive number at one particular time. If ''k'' = −1, then (loosely speaking) one can say that ''i·a'' is the radius of curvature of the universe.
+*''a'' is the [[Scale factor (Universe)|scale factor]] which is taken to be 1 at the present time. <math>k</math> is the [[curvature|spatial curvature]] when <math>a = 1</math> (i.e. today). If the [[shape of the universe]] is [[Shape of the universe#Spherical universe|hyperspherical]] and <math>R_t</math> is the radius of curvature (<math>R_0</math> in the present-day), then <math>a = R_t/R_0</math>. If <math>k</math> is positive, then the universe is hyperspherical. If <math>k</math> is zero, then the universe is [[Shape of the universe#Flat universe|flat]]. If <math>k</math> is negative, then the universe is [[Shape of the universe#Hyperbolic universe|hyperbolic]].
+Using the first equation, the second equation can be re-expressed as
+:<math>\dot{\rho} = -3 H \left(\rho + \frac{p}{c^2}\right),</math>
+which eliminates <math>\Lambda \!</math> and expresses the conservation of [[mass-energy]] <math> T^{\alpha\beta}{}_{;\beta}  \, = 0</math>.
+These equations are sometimes simplified by replacing
+:<math>\rho \rightarrow \rho - \frac{\Lambda c^2}{8 \pi G}</math>
+:<math>p \rightarrow p + \frac{\Lambda c^4}{8 \pi G}</math>
+to give:
+:<math>H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}</math>
+:<math>\dot{H} + H^2 = \frac{\ddot{a}}{a} = - \frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right).</math>
+And the simplified form of the second equation is invariant under this transformation.
+The Hubble parameter can change over [[time]] if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of [[Hubble's law]]. Applied to a fluid with a given [[equation of state (cosmology)|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 '''Friedmann acceleration equation''' and reserve the term ''Friedmann equation'' for only the first equation.
+==Density parameter==<!--  [[Density parameter]] links to this section  -->
+The density parameter, <math>\Omega</math>, is defined as the ratio of the actual (or observed) density <math>\rho</math> to the critical density <math>\rho_c</math> of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclidean).
+In earlier models, which did not include a [[cosmological constant]] term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.
+To date, the critical density is estimated to be approximately five atoms (of [[monatomic]] [[hydrogen]]) per cubic metre, whereas the average density of [[Baryons#Baryonic_matter|ordinary matter]] in the Universe is believed to be 0.2 atoms per cubic metre.<ref>Rees, M., Just Six Numbers, (2000) Orion Books, London, p. 81, p. 82</ref>[[File:DarkMatterPie.jpg|thumb|right|375px|Estimated relative distribution for components of the energy density of the Universe. [[Dark energy]] dominates the total energy (74%) while [[dark matter]] (22%) constitutes most of the mass.  Of the remaining baryonic matter (4%), only one tenth is compact. On 21 March 2013, the European-led research team behind the [[Planck (spacecraft)|Planck cosmology probe]] released new data refining these values to 4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy.]]
+A much greater density comes from the unidentified [[dark matter]]; both ordinary and dark matter contribute in favor of contraction of the universe. However, the largest part comes from so-called [[dark energy]], which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), the dark energy does not lead to contraction of the universe but rather accelerates its expansion. Therefore, the universe will expand forever.{{Citation needed|date=June 2012}}
+An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:
+:<math>\rho_c = \frac{3 H^2}{8 \pi G}.</math>
+The density parameter (useful for comparing different cosmological models) is then defined as:
+:<math>\Omega \equiv \frac{\rho}{\rho_c} = \frac{8 \pi G\rho}{3 H^2}.</math>
+This term originally was used as a means to determine the [[shape of the universe|spatial geometry]] of the universe, where <math>\rho_c</math> is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if <math>\Omega</math> is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If <math>\Omega</math> is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for <math>\Omega</math> in which case this 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|ΛCDM model]], there are important components of <math>\Omega</math> due to [[baryons]], [[cold dark matter]] and [[dark energy]]. The spatial geometry of the [[universe]] has been measured by the [[WMAP]] spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter <math>k</math> is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see. (Similarly, the fact that Earth is approximately flat at the scale of the Netherlands does not imply that the Earth is flat: it only implies that it is much larger than the Netherlands.)
+The first Friedmann equation is often seen in terms of the present values of the density parameters, that is<ref>http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:astro-ph/0703739</ref>
+:<math>\frac{H^2}{H_0^2} = \Omega_R a^{-4} + \Omega_M a^{-3} + \Omega_k a^{-2} + \Omega_{\Lambda}.</math>
+Here <math>\Omega_R</math> is the radiation density today (i.e. when <math>a=1</math>), <math>\Omega_M</math> is the matter ([[dark matter|dark]] plus [[baryon]]ic) density today, <math>\Omega_k = 1 - \Omega</math> is the "spatial curvature density" today, and <math>\Omega_\Lambda</math> is the cosmological constant or vacuum density today.
+==Useful solutions==
+The Friedmann equations can be solved exactly in presence of a [[perfect fluid]] with equation of state
+:<math>p=w\rho c^2,</math>
+where <math>p</math> is the [[pressure]], <math>\rho</math> is the mass density of the fluid in the comoving frame and <math>w</math> is some constant.
+In spatially flat case (''k''&nbsp;=&nbsp;0), the solution for the scale factor is
+:<math> a(t)=a_0\,t^{\frac{2}{3(w+1)}} </math>
+where <math>a_0</math> is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by <math>w</math> is extremely important for cosmology. E.g. <math>w=0</math> describes a [[matter-dominated era|matter-dominated]] universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as
+:<math>a(t)\propto t^{2/3}</math> matter-dominated
+Another important example is the case of a [[radiation-dominated era|radiation-dominated]] universe, i.e., when <math>w=1/3</math>. This leads to
+:<math>a(t)\propto t^{1/2}</math> radiation dominated
+Note that this solution is not valid for domination of the cosmological constant, which corresponds to an <math>w=-1</math>. In this case the energy density is constant and the scale factor grows exponentially.
+Solutions for other values of ''k'' can be found at {{cite web|last=Tersic|first=Balsa|title=Lecture Notes on Astrophysics|url=http://nicadd.niu.edu/~bterzic/PHYS652/PHYS652_notes.pdf|accessdate=20 July 2011}}.
+===Mixtures===
+If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then
+:<math>\dot{\rho}_{f} = -3 H \left( \rho_{f} + \frac{p_{f}}{c^2} \right) \,</math>
+holds separately for each such fluid ''f''. In each case,
+:<math>\dot{\rho}_{f} = -3 H \left( \rho_{f} + w_{f} \rho_{f} \right) \,</math>
+from which we get
+:<math>{\rho}_{f} \propto a^{-3 (1 + w_{f})} \,.</math>
+For example, one can form a linear combination of such terms
+:<math>\rho = A a^{-3} + B a^{-4} + C a^{0} \,</math>
+where: ''A'' is the density of "dust" (ordinary matter, ''w''&nbsp;=&nbsp;0) when ''a''&nbsp;=&nbsp;1; ''B'' is the density of radiation (''w''&nbsp;=&nbsp;1/3) when ''a''&nbsp;=&nbsp;1; and ''C'' is the density of "dark energy" (''w''=&minus;1). One then substitutes this into
+:<math>\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2} \,</math>
+and solves for ''a'' as a function of time.
+==Rescaled Friedmann equation==
+Set <math>\tilde{a}=\frac{a}{a_0}, \;\rho_c=\frac{3H_0^2}{8\pi G},\;
+\Omega=\frac{\rho}{\rho_c},\; t=\frac{\tilde{t}}{H_0},\;
+\Omega_c=-\frac{kc^2}{H_0^2 a_0^2}\;</math>, where <math>a_0</math> and <math>H_0</math> are separately the [[Scale factor (Universe)|scale factor]] and the [[Hubble parameter]] today.
+Then we can have
+:<math>\frac{1}{2}\left( \frac{d\tilde{a}}{d\tilde{t}}\right)^2 + U_{\rm eff}(\tilde{a})=\frac{1}{2}\Omega_c</math>
+where <math>U_\textrm{eff}(\tilde{a})=\frac{\Omega\tilde{a}^2}{2}\;</math>. For any form of the effective potential <math>U_\textrm{eff}(\tilde{a})\;</math>, there is an equation of state <math>p=p(\rho)</math> that will produce it.
+==See also==
+* [[Universe]]
+* [[Mathematics of general relativity]]
+* [[Solutions of the Einstein field equations|Solutions of Einstein's field equations]]
+* [[Warm inflation]]
+==Notes==
+<references />
+{{Relativity}}
+{{DEFAULTSORT:Friedmann Equations}}
+[[Category:General relativity]]
+[[Category:Equations]]
+[[ru:Уравнение_Фридмана]]

Whitney immersion theorem: Difference between revisions

Latest revision as of 09:02, 23 March 2013

Contents

Assumptions

The equations

Density parameter

Useful solutions

Mixtures

Rescaled Friedmann equation

See also

Notes

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Whitney immersion theorem: Difference between revisions

Latest revision as of 09:02, 23 March 2013

Assumptions

The equations

Density parameter

Useful solutions

Mixtures

Rescaled Friedmann equation

See also

Notes

Navigation menu

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