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Topic: Friedmann equations

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	Friedmann biography
		Friedmann and Tamarkin were student leaders of strikes at the school in protest at the government's repressive measures against schools.
		On 13 April 1918 Friedmann was elected an extraordinary professor in the Department of Mathematics and Physics at the University of Perm.
		Friedmann is seen as a profound, independent-minded, and daring thinker who destroys scientific prejudices, myths and dogmas; his intellect sees what others do not see, and will not see what others believe to be obvious but for which there are no grounds in reality.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Friedmann.html (2495 words)

	Friedmann equations - Wikipedia, the free encyclopedia
		They were derived by Alexander Friedmann in 1922 from the Einstein field equations under some assumptions of symmetry appropriate for a cosmological model.
		From his equations, the Friedmann-Lemaître-Robertson-Walker metric was derived for a fluid with a given density and pressure.
		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.
	en.wikipedia.org /wiki/Friedmann_equations (443 words)

	Friedmann Equation
		Friedmann developed it as a relativistic equation in the framework of general relativity, but the description here will be limited to a simplified, non-relativistic version.
		The Friedmann equation which models the expanding universe has a parameter k called the curvature parameter which is indicative of the rate of expansion and whether or not that expansion rate is increasing or decreasing.
		Under this condition, the Friedmann equation can be used to express the critical density of matter in the universe in terms of the current value of the Hubble parameter.
	hyperphysics.phy-astr.gsu.edu /hbase/astro/fried.html (556 words)

	The Uncaused Beginning of the Universe (1988)
		What is important to note about these Friedmann equations is that if p, the density of matter in the universe, is positive, then the right side of the first equation is positive, and this entails that d2a/dt2, the acceleration of the expansion or the deceleration of the contraction, cannot be zero.
		Equations (7) and (8) at most tend to show that acausal laws govern the change of condition of particles, such as the change of particle x's position from q1 to q2.
		The equation describing this state is a quantum tunneling equation, specifically the bounce solution of the Euclidean version of the evolutionary equation of a universe with a closed Robertson-Walker metric.
	www.qsmithwmu.com /the_uncaused_beginning_of_the_universe_(1988).htm (7386 words)

	The Cosmological Models
		One of the reasons to examine the Friedmann Equation is to adjust those free parameters to produce a model that reflects the observed characteristics of the true Universe.
		We begin with the equation of motion, which states that the acceleration of a particle is equal to the gravitational field.
		This equation states that the rate of change of the scaling factor is the sum of a positive gravity term and a negative lambda term.
	mywebpages.comcast.net /bondono2/cosmodel/cosmodel.html (7021 words)

	A Big Bang Cosmological Argument For God's Nonexistence
		Einstein's equation says, in simplified terms, that the geometry (curvature) of spacetime is determined by the distribution of mass and energy in spacetime.
		This equation suggests that if the matter in the universe is sufficiently dense, then the curvature of spacetime will become so great that it eventually curves to a point, as at the tip of a cone.
		The Friedmann solutions tell us that if there is matter evenly distributed throughout the universe, then the universe must be expanding at a decreasing rate or contracting at an increasing rate (except at the instant, if any, at which the expansion stops and changes to a contraction).
	www.infidels.org /library/modern/quentin_smith/bigbang.html (8580 words)

	Is The Universe Static Or Expanding?
		From the interpretation of equation (3) this meant these galaxies were receding at two-fifths of the velocity of light, and an essentially linear relationship was being maintained on the Hubble graph of redshift/distance from (2) or (4) [Couderc, 1960, p.103; Hoyle, 1956, pp.311-312].
		Equation (5) is the end result of a process that Hubble started when he multiplied the redshift z by lightspeed c.
		Friedmann found that these equations describing the behaviour of the cosmos were capable of an infinite number of solutions if Einstein's model of a static universe was abandoned [Friedmann, 1922, p.377].
	www.ldolphin.org /staticu.html (10589 words)

	Alternative forms of the Friedmann equations Text - Physics Forums Library
		equations are about a_t and a_t,t ____often written a-dot and a-doubledot: the first and second time-derivatives of the scale factor a(t).
		The basic constant in that equation is the force c^4/G, or its reciprocal G/c^4, and that force is the definition of F_planck.
		The Friedmann equations which derive from the main GR equation also look a bit simpler in terms of the natural force constant.
	www.physicsforums.com /archive/index.php/t-1423.html (3256 words)

	Friedmann Equations
		This equation of state turns out to be physically reasonable in the late universe.
		In this case the second of the Friedmann equations (29.6)can be integrated directly.
		Using the first Friedmann Equation (29.5) we can get the following expression for C. This constant represents the energy E of a volume V of the substratum fluid.
	scholar.uwinnipeg.ca /courses/38/4500.6-001/Cosmology/Friedmann-Equations.htm (617 words)

	expantion
		Big Friedmann also has a "k" term in it where k represents the overall spatial curvature (not of space time, merely of space) but they tell us k is zero.
		Now how about the other Friedmann equation, Little Friedmann, the auxilliary controlling a''(t)----the rate of change of the rate of change of the scalefactor.
		BTW these two are simplifications of the original Einstein equation of GR which were derived from the main GR equation by you guessed it a man living in St. Petersburg Russia around 1922 except it was Leningrad then.
	www.physicsforums.com /showthread.php?t=5169 (1228 words)

	The Uncaused Beginning of the Universe
		This suggests that there is some time in the past when all the galactic clusters, or all the materials in these clusters, were arbitrarily close together, and that this time represents the beginning of the universe.
		The field equations can be solved for the universe as a whole if figures reflecting the observed values of the universe are introduced.
		The issue of whether oscillating universes are finite or infinite in respect of the past lost much of its urgency in the middle and late 1960s, with the development of the Hawking-Penrose singularity theorems (Penrose 1965; Hawking 1965, 1966, 1970), which entailed that an inexactly homogeneous and isotropic universe must have a singularity.
	www.infidels.org /library/modern/quentin_smith/uncaused.html (7467 words)

	fried
		Einstein (and most other scientists, for that matter) believed that the universe was static, and he modified his equations by including a “cosmological constant” to keep it so.
		Friedmann made two simple assumptions about the universe: that when viewed at large enough scales, it appears the same both in every direction and from every location.
		Although Friedmann found only this one solution, called a closed universe because the size of the universe is finite, two similar solutions exist.
	www.pbs.org /wnet/hawking/universes/html/univ_fried.html (345 words)

	[No title]
		The principal aim of this Laboratory Project is to write a computer program to solve numerically the Friedmann Equations and investigate their solution as we vary the parameters of our cosmological model.
		Possible Directions To some extent this is an open-ended project: solution of the Friedmann Equations is the basic goal, but there are many different directions in which the project may then proceed.
		Here we simply state the equations, under the approximation that the Universe is matter dominated and its matter content can be treated as a perfect fluid.
	www.astro.gla.ac.uk /honours/labs/Friedmann/friedmann.doc (1856 words)

	Cosmological constant Summary
		Einstein included the term in the equations for general relativity because he was dissatisfied that his equations do not allow for a static universe.
		Moreover, adding the cosmological constant to Einstein's equations does not lead to a universe at equilibrium because the equilibrium is unstable: if the universe expands slightly, then the expansion releases vacuum energy, which causes yet more expansion.
		Ironically, the cosmological constant is still of interest, as observations made in the late 1990s of distance-redshift relations indicate that the expansion of the universe is accelerating.
	www.bookrags.com /Cosmological_constant (1232 words)

	Projects
		The Friedmann equations describe the dynamics of the scale factor in a FRW model-universe.
		When the Friedmann equations have been derived, a brief discussion of the solutions will be given.
		The application of the classical Friedmann equations is limited to the regime where quantum fluctuations of the metric are negligible.
	www.cozmo.dk /cramer/tech (989 words)

	papers
		The field equations and the equations of motions are explicitly given.
		We use two independent data sets to constrain the new density parameter $\Omega_\psi$, which is related to the non-Riemannian structure of the underlying spacetime and supplements the field equations that are very similar to the usual Friedmann equations of general relativity.
		The field equations of the model reduce to a system which is effectively given by the usual Friedmann equations of general relativity, supplied by a correction to the energy density and pressure in form of $\Omega_\psi$, which is related to the non-Riemannian structure of the underlying spacetime.
	folk.uio.no /dirkpu/papers.html (2798 words)

	Observational Cosmology - Unit 2
		This is unnecessarily complex for our purposes, since the dynamical equations can be derived and understood from a Newtonian perspective, and this is the route we will take.
		The lecture and discussion class for this module will also be used to discuss solutions to these equations, and the nature of the cosmological constant.
		Why does the derivation of the Friedmann equation fail in the case of an "island universe", in which the matter density is uniform, but has an edge?
	www.sr.bham.ac.uk /~lrj/obscos/unit2.html (873 words)

	A Babe In the Universe: Einstein's Field Equation
		The Friedmann equations are calculated for a spherical mass distribution of average density, within a larger universe that could be flat or curved.
		I have studied the Einstein equations a little bit, but have found that notation varies greatly from source to source.
		Kappa is another constant that normalises to 8\pi G in the Friedmann equation.
	riofriospacetime.blogspot.com /2006/08/einsteins-field-equation.html (1008 words)

	Friedmann-Robertson-Walker metric - Wikipedia, the free encyclopedia
		The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity and which describes a homogeneous, isotropic expanding/contracting universe.
		Depending on geographical/historical preferences, this may be referred to under the names of a preferred subset of the four scientists Alexander Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker, e.g.
		These equations serve as a first approximation of the standard big bang cosmological model including the current ΛCDM model.
	en.wikipedia.org /wiki/Friedmann-Lema%C3%AEtre-Robertson-Walker (489 words)

	Lecture #14
		Remember all Friedmann cosmologies are decelerating, so let's define a new parameter we can use to describe this deceleration.
		The above equations show that in this case we can define a critical cosmological constant, which is related to curvature.
		Again, from the above equations we see that q=-1 in a de Sitter universe (and that Hubble's constant is truly constant).
	www.astro.virginia.edu /class/spring2003/randall/astr348/lecture14.html (875 words)

	Cosmology Course: General Relativity (Site not responding. Last check: 2007-09-17)
		The combination of the RW metric and Einstein's field equations led to the Friedmann equations, solutions of which were discussed in detail.
		The dynamics of the expansion was studied for special cases (matter and radiation dominated universes) and connected to observable quantities (red shift, density parameter, Hubble constant etc.).
		In the lecture there was only time for some handwaving arguments as to (i) why the metric has to have a particular form and (ii) the derivation of the Friedmann equations from the Einstein equations.
	t8web.lanl.gov /cosmo-course/kat/lect2.html (268 words)

	Helge Kragh, Modern History of Cosmology, Spring 2001
		Copied sources include Friedmann's 1922 paper in English translation, and a general introduction by Einstein of 1920, taken from Munitz, Theories of the Universe.
		Friedmann's paper is highly technical, but much of it can be understood without great difficulties.
		Show that the de Sitter model, too, is among the solutions to the Friedmann equations.
	www.nd.edu /~histast4/exhibits/syllabi/kraghmodhistcosmol.html (2357 words)

	Brane World Cosmology Comes Down to Earth
		In 4D space-time the Friedmann equations describe the evolution of a uniform, homogeneous, and isotropic universe.
		Our French paper, #4, has become highly cited because it derives the equivalent of the Friedmann equation for a 3-brane in higher dimensional space.
		They look like the Einstein equations but they have two additional terms, which arise from the fifth dimension.
	www.sciencewatch.com /july-aug2002/sw_july-aug2002_page6.htm (929 words)

	UTD Physics: Astrophysics, Cosmology, and Relativity
		James is exploring an inverse approach to Einstein’s equations in the presence of various dark energy backgrounds and the use of null (observational) coordinates in this framework
		For example, he is studying Friedmann equations from higher order invariants and their implications for the cosmic acceleration problem.
		Tamara is conducting a study of the effects of various models of dark energy and dark matter on cosmological observables such as the age of the universe and the area distance of the surface of last scattering.
	www.utdallas.edu /~mishak/relativitycosmology.html (579 words)

	Electrodynamics in a Friedmann universe (FRW)
		In this section I generalize the Coulomb and Lorentz gauge of Electrodynamics for a Friedmann universe.
		We afterwards can use Electrodynamics as a model to identify the corresponding gauges in perturbative gravity (see section 4).
		For this task we need the basic equations governing the background, i.e.
	www.mit.edu /~sballmer/physics/dipl/node18.html (106 words)

	Vladimír Skalský
		The FRW equations (1a) and (1b) describe the models of homogeneous and isotropic relativistic universe in the first (linear) approximation.
		These facts make it possible to determine one from the possible solutions of the FRW equations (1a) and (1b), which represents the model of our observed relativistic Universe in the first (linear, Newtonian or classical-mechanical) approximation.
		From these facts, the WMAP measurements, and the relations (1)-(33), it results unambiguously that the RNU model is the only universe model, which describes the properties of our observed relativistic Universe in the first (linear, Newtonian or classical-mechanical) approximation.
	www.euneco.com /cosmology/model.html (1547 words)

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