Robertson-Walker

The Cosmological Principle which states that the Universe is always homogeneous and isotropic and leads to the Robertson-Walker metric... 

Here, we shall discuss the implications of cosmological expansion for the searches of a quantum-gravity-induced refractive index and a stochastic effect. We will consider Friedman-Robertson-Walker (FRW) metrics as an appropriate candidate for standard homogeneous and isotropic cosmology. Let R be the FRW scale factor, and a subscript 0 will denote the value at the present era. Ho is the present Hubble expansion parameter, and the deceleration parameter qo is defined in terms of the curvature k of the FRW metric by k ( 2[Pg.588]

We can add the time as the fourth coordinate, to build the equivalent of the Minkowski space-time element. We then get the Robertson-Walker line element after the change of variables f> —> r ... 

The last term corresponds to the constant of integration asociated to the total energy of the sphere (and varies as a2). Its value depends on the initial conditions. Furthermore, it expresses a link between the geometry and the material content of the Universe, which cannot be specified in the Newtonian approach we had and can be justified only within the framework of GR. The form of the above equation is independent of the radius a of the sphere and we shall therefore admit that the equation still holds for the quantity li(t), the constant K being then the constant k which is involved in the Robertson-Walker metric element ... 

As background, we will need to recall the evolution of a Universe governed by the Freedmann-Robertson-Walker (FRW) metric, and of the different components thereof. 

Significant new evidence is the self-similarity of sub-atomic, atomic, biological, planetary and galactic structures, all related to the golden section. The astronomical structures are assumed to trace out the shape of local space. The obvious conclusion is that all of space has a uniform non-zero characteristic curvature, conditioned by the universal constants tt and r. Space, in this sense, is to be interpreted as equivalent to the three-dimensional sub-space of the Robertson-Walker metric [104]. In standard cosmology this sub-space... 

Cosmic structure based on a vacuum interface has been proposed before [49, 7] as a device to rationalize quantum events. To avoid partitioning the universe into regions of opposite chirality the two sides of the interface are joined together with an involution. The one-dimensional analogue is a Mobius strip. Matter on opposite sides of the interface has mutually inverted chirality - matter and anti-matter - but transplantation along the double cover gradually interconverts the two chiral forms. The amounts of matter and anti-matter in such a universe are equal, as required by symmetry, but only one form is observed to predominate in any local environment. Because of the curvature, which is required to close the universe, space itself is chiral, as observed in the structure of the electromagnetic field. This property does not appear in a euclidean Robertson-Walker sub-space. 

Observations of the present universe establish that, on sufficiently large scales, galaxies and clusters of galaxies are distributed homogeneously and they are expanding isotropically. On the assumption that this is true for the large scale universe throughout its evolution (at least back to redshifts 1010, when the universe was a few hundred milliseconds old), the relation between space-time points may be described uniquely by the Robertson - Walker metric... 

Everything discussed so far has been geometrical , relying only on the form of the Robertson-Walker metric. To make further progress in understanding the evolution of the universe, it is necessary to determine the time dependence of the scale factor a(t). Although the scale factor is not an observable, the expansion rate, the Hubble parameter, H = H(t), is. 

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... 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

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... 

The velocity of light in a radial Robertson-Walker direction can be calculated by putting ds = 0, as in Minkowski space, together with angular coordinates d9 and d(p equal to zero, such that... 

Big-bang cosmology is based exclusively on Friedmann solutions, using the Robertson-Walker metric (6.6). The scale factor S t) determines the... 

Inspired by Riemann s ideas William Clifford proposed, as a modification of Newton s universe, a space with constant positive curvature, except for small local variations. This is the most likely precursor of Einstein s closed 3-sphere universe in which matter is modelled as a pressure-less incoherent fluid, or dust, of constant density. The closure ensures constant density in space, that remains constant by defining a time coordinate orthogonal to space. The Robertson-Walker metric is defined on the same principle of constant density in a comoving coordinate system. 

In the Friedmann cosmologies of the GTR, a finite space implies an end to proper time in the future, but this is not required in some nonstandard models. Assuming the Robertson-Walker form of the metric for a homogeneous and isotropic space-time, it is convenient to discuss the future evolution of any such expanding universe in terms of a dimensionless deceleration parameter, defined as ... 

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