Cosmological Assumptions

Isotropy and Homogeneity

We will now discuss some of the assumptions that underlie the theory of relativistic cosmology. The main principle that is adopted in mathematically describing the global structure of the universe is called the Cosmological Principle. It is defined as follows.

Assume that there exists a cosmological time scale. Then the Cosmological Principle is formulated in each constant time slice. None of these time slices have privilege points. In this case we say that the time slice is Homogeneous. A spacelike hypersurface (i.e. the time slice) is said to be homogeneous if the geometry admits 3-D space coordinate translations such that the metric is unaffected by those 3-D space translations. Homogeneity requires invariance under transformations like:

 (27.1)

The cosmological Principle also requires that there be no privileged directions on the epoch time slice as well as there being no privileged points. A manifold that has no privileged directions about a point is called Isotropic. Implicit in this statement is the restriction that about the point the manifold is spherically symmetric as well. If we insist on global isotropy then we are constraining the universe manifold to be isotropic about every point. This global constraint can be shown to imply that the universe is homogeneous. The opposite case of this however does not follow. If the universe is globally homogeneous, it is not necessarily globally isotropic. In more technical language the Cosmological Principle can be written as follows:

As stated above the assumption of the Cosmological Principle allows us to develop mathematical models describing the global characteristics of the entire universe. From a phenomenological point of view cosmic homogeneity implies the existence of a smeared out universe at the largest length scales (well over 100 million light-years). Remember 1 light-year is approximately

Isotropy of the universe has good data supporting it. Here are some isotropy determinations from astronomy observations:

1. The visible galaxy distribution is isotropic to within 30%.
2. The Hubble Expansion Constant is isotropicly equal to with 25%.
3. The distribution of radio galaxies is isotropic to within 5%.
4. Cosmic X-Rays are isotropic to within 5%.
5. and, finally, the thing that really nails down the isotropic assumption: the Cosmic Background Radiation which is isotropic to within 2 parts in 100,000.

For more information about the Cosmological Principle and relevant diagrams go to the cosmology.uwinnipeg.ca website and look at the webpage called Isotropy-Homogeneity.

People have studied inhomogeneous and anisotropic solutions of the Einstein Equations in the context of the very early universe since it seems reasonable that the universe would be less symmetrical at the earliest times. Inhomogeneous solutions can be quite globally complicated but anisotropic solutions can be simply described. As an example of an anisotropic metric consider the vacuum solution called the Kasner solution. In c=1 units the line element for this solution looks like the following:

(27.2)

(27.3)

A specific Kasner metric is formed when we pick

The line element is then given by

 (27.4)

It is easy to see that this gravitational field is anisotropic in the z direction. As the universe expands with time in the x and y directions, the universe contracts in the z direction. The entire universe model resembles a planar pancake gravitational distribution as time proceeds.

Weyl's Postulate

In 1923 Hermann Weyl thought about the apparent contradiction in applying a theory like General Relativity, that was set up to be generally covariant, to one particular set of circumstances, that of describing just one universe, our universe. In a universe that is expanding there seems to be a preferred coordinate system, that coordinate system that is comoving with the background expansion flow. Weyl decided that, in the application of GR to a unique symmetrical system like the universe, there must be underlying phenomenologically based postulates that are formulated from local observations. He reasoned that there should be a privileged class of observers that is comoving with the smeared out motion f the galaxies. Weyl postulated a substratum (i.e. a fluid) pervading all of space. In this fluid galaxies move like fundamental particles. A special motion for these galaxies is then assumed.

This postulate implies that there is one, and only one, geodesic passing through each point of spacetime. Hence the matter at any point possesses a unique velocity. The essence of Weyl's Postulate is that we should be able to take the substratum to be represented as a perfect fluid. In reality galaxies will not follow the postulated cosmological motion exactly. There will be random directional motion of the galaxies usually much less than the speed of light (  ). From a global cosmological perspective this velocity is smeared out to be overall negligible and furthermore it will be almost always nonrelativistic. This is in distinction to the velocity of the flow along with the universe's expansion, which can be at quite high speeds. It is, also, always unidirectional (i.e. _ away from you).

Application of the 3 Assumptions to Relativistic Cosmology

In order to describe the universe as a whole even though we have access to only local observations we have to base our cosmology on three main assumptions:

Weyl's Postulate demands that the fluid geodesics be orthogonal to a family of spacelike hypersurfaces. We introduce a set of coordinates . In each spacelike hypersurface corresponding to some time , the 3-D space coordinates of the fluid particle are constant. This is what we mean by saying that we are adopting a comoving coordinate system. We are moving with the expansion flow so that at any given cosmological instant we do not see the expansion motion. We can ensure orthogonality of the spacelike hypersurfaces by adopting the following form for our metric.

(27.5)

With this explicit orthogonality structure we are guaranteed a cosmic interpretation of time.

This global concept of time implies that simultaneity exists along the orthogonal spacelike hypersurface. We define the terms World Map and the World Picture in terms of this cosmic simultaneity.

In a comoving frame the large scale patterns stay the same. As the universe expands we don't see the universe expansion effects. We see our relationship to other parts of the universe stay the same. Consider the situation shown below where points at the endpoints of the triangle pattern always see themselves in the same triangular pattern no matter how much the space between the points has increased.

The Cosmological Principle implies that the 1st triangle must be geometrically similar to the second with respect to time. The cosmological aspect ratio is fixed. This forces the magnification factor due to expansion to be independent of the position of the triangle within the spacelike hypersurface. Mathematically, time can enter into the 3-D metric function  only as a common magnification factor for all 3-D points.

 (27.6)

Note that the magnification factor for the expansion of the universe between any two distinct times  is given by

(27.7)

For this reason the function  is called the Scale Factor. This function must be a real number since we want the spacelike/timelike character of the intervals in the hypersurface to be preserved.

We have placed very strict mathematical constraints on the manifold of points that we wish to model the universe with. We have demanded that the manifold be homogeneous and isotropic in any given time slice and from this arises the constraint that the geometrical arrangement the 3-D space points must be independent of the time variation. What this effectively means is that the curvature at every point in our manifold must be constant. Only with constant curvature, is every point guaranteed to be geometrically equivalent. It is worth remembering here that we are making statements about the universe at the present epoch concerning length scales of the order of 500 million light years and higher.

We have therefore arrived at the conclusion that in our phenomenologically- based cosmological model we must adopt a space of constant 3-D curvature.