Friedmann equation

Using the first Friedmann equation (recalling that we neglect the curvature K and the kinetic term in the energy density) gives... 

Using the modified Klein-Gordon equation and then the first Friedmann equation, one gets... 

Assuming that one is in an era where the background equation of state parameter wb is constant, one can derive a relation between II and H2 with the held of the Friedmann equations, so that the three relations above can be translated into... 

The subscript 0 meaning that we consider the corresponding quantities today.) On the other hand, the first Friedmann equation H2 = up leads to... 

The time-evolution of H describes the evolution of the universe. Employing the Robertson-Walker metric in the Einstein equations of General Relativity (relating matter/energy content to geometry) leads to the Friedmann equation... 

The Friedmann equation (eq. 2.9) relates the time-dependence of the scale factor to that of the density. The Einstein equations yield a second relation among these which may be thought of as the surrogate for energy conservation in an expanding universe. 

For matter (non-relativistic matter often called dust ), p p, so that p/po = (ao/a)3. In contrast, for radiation (relativistic particles) p = pi3, so that p/po = (ao/a)4. Another interesting case is that of the energy density and pressure associated with the vacuum (the quantum mechanical vacuum is not empty ). In this case p = —p, so that p = po- This provides a term in the Friedmann equation entirely equivalent to Einstein s cosmological constant A. More generally, for p = wp, p/po = (ao/a)3(1+w Allowing for these three contributions to the total energy density, eq. 2.9 may be rewritten in a convenient dimensionless form... 

To ensure that the physics is described in a manner independent of arbitrary choices, it is useful to introduce the scale change rate, also known as the Hubble parameter, defined as // = (dR/dt)/R. Since H has dimensions of velocity divided by length, 1/// is a characteristic time for evolution of the model. For an expanding empty universe, 1 /H would be the time since the beginning of the expansion. Assuming that the distribution of matter on a large scale can be described in terms of a perfect fluid of total density p and pressure p, the coupled evolution of space-time and matter in the GTR are determined by the Friedmann equations. 

Figure 9 compares Equation 20 with the recent pressure drop flow rate data of Friedmann, Chen, and Gauglitz (5) for a 1 wt% commercial sodium alkyl sulfonate dimer (Chaser SD-1000) stabilized foam in a Berea sandstone. These data are particularly useful because they have been corrected for foam blockage and therefore correctly reflect the flowing bubble regime. The solid line in Figure 9 is best fit according to Equation 20. Unfortunately, neither of the parameters c or 6 is available. Two sets of estimates are shown in Figure 9. When e - 0 (i.e., no surfactant effect) the bubble size is about 30% of a grain diameter. When — 0.1 mm (i.e., a value characteristic of those in Figure 8) the bubble size is about 10 grain diameters. We assert that Equation 20 not only predicts the correct velocity behavior of foam but it does so with reasonable parameter values (23).

It is always risky to identify the origin of an idea. The basic idea of the Big Bang may be identified with the Russian mathematical physicist, Alexander A. Friedmann, who in 1922, armed with Einstein s general relativistic equations, developed the picture of the universe expanding from a point origin. The timing was wrong,... 

Fig. 12. Diagrams showing the pore water isotopic compositions calculated from the direct and indirect temperature constraints and the oxygen isotopic compositions of caicites using the fractionation equation of Friedmann O Neil (1977). As a result of the uncertain temperature constraints for the UMM high trend samples, their composition ranges are probably unrealistic.

The general solution of the field equations (6.4), by using the Robertson-Walker metric, was obtained by Aleksandr Friedmann on substituting (6.6) into (6.4). Details of the procedure are outlined by Narlikar (2002). By considering the matter distribution, as galaxies in space, to be like dust, the... 

Friedmann s expanding-universe solutions of Einstein s equations were assumed to confirm this interpretation. 

The idea that the correlation functions and other statistical moments of the fluid mechanical fields must be recognized as the fundamental characteristics of turbulence was first stated by Keller and Friedmann ([81] see also [8], Sect. 2.2 [112], Chap. 2), who proposed a general method of obtaining the differential equations for the moments of arbitrary order for the description of turbulent flow. The determination of... 

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