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Дата изменения: Thu May 31 19:32:50 2001 Дата индексирования: Mon Oct 1 19:58:13 2012 Кодировка: Поисковые слова: aurora |
We have to create a framework in which we can relate observationally determined parameters to the large scale charactistics of the Universe:
We assume the Universe to be isotropic and homogenous. In this situation the length of a Geodesic is described by the Robertson-Walker metric:
Where
The gravitational acceleration of the Universe is given by Poisson's Equation:
In the early Universe, pressure from the photon field was very high and the universe acts as an isotropic fluid. This Poisson's equation is modified to
The effective gravitional mass density has been increase due to the pressure (making it hard for structure to form early on). Consider a spherical region with some net gravitational acceleration. The total mass of the sphere is:
As the acceleration on the surface of the sphere goes as GM/rs2 then
Within this sphere there is net energy En = rho*V
From conservation of Energy we have:
The volume and its derivative are:
Solving for p and inserting that into the previous equation yields a relation between only density and its derivatives and the acceleration:
This differential equation can be solved as a non-trivial exercise for the reader to yield
Now let's assume the universe is static: all derivates with respect to rs dissappear and we have
Since mass is positive, then pressure must be negative to solve
this equation. Normal matter doesn't have negative pressure
so Einstein introduces the cosmological constant
to compensate:
Thus we have:
and
This last term shows that the rate of change of R(t) depends on
Suppose K=0 (space is flat) and = 0 and
pressure = 0 then
we have:
Which is solved with:
This is also the solution which satisfies the condition that the mass within a sphere remains constant with time.
Which is only solved if R(t) goes as t2/3. Hence we get
So we have expressions for the critical density and the expansion age of the Universe in terms of the observable Ho.