Newtonian Model of Expanding Universe

It is now an experimental fact that the universe is expanding, and expanding at very close to a rate that would make the universe "flat" or "critical" so that it will expand forever, asymptotically approaching a rest condition at infinite time. Actually, the current information indicates a small acceleration attributed to dark energy, but it is still close enough to the critical density that it makes sense to make ones first model that of a flat or critical universe. To that end, one can model the expansion in terms of a uniform density of particles which interact only gravitationally to create a framework from which to develop more refined models. Those more refined models are often expressed in the Friedman equation.

A spherical shell of mass m which is expanding can be described in terms of its kinetic energy and gravitational potential energy, where that gravitational potential energy is contributed by the mass Mr enclosed by the sphere. The spherical shell of mass has been given an energy E. 一个质量为m的球壳在膨胀时，可以由它的动能和引力势能来描述，其中引力势能由球壳内包含的质量M r贡献。该球壳已被赋予了能量E。 现在已确认一个实验事实，即宇宙正在膨胀，且膨胀速率接近使宇宙变得“平坦”或“临界”以至于宇宙将永远膨胀，渐近地接近无限时间的静止状态。实际上，目前的信息表明存在一小部分加速，归因于暗能量，但这种加速仍接近临界密度，因此合理地将第一个模型设定为平坦或临界宇宙。为此，可以将膨胀建模为粒子的均匀密度，这些粒子仅通过引力相互作用，从而建立一个框架，用于发展更精确的模型。这些更精确的模型通常用弗里德曼方程来表达。

The mechanical energy can be expressed as follows, using the notation of Carroll and Ostlie:

The term on the right is an expression for the total energy of the shell which has been constructed in a form to be of future usefulness. In that expression, c is the speed of light, m is the mass, k is a constant which will be used to represent the "curvature" of the space system, and ϖ is a coordinate whose value represents the chosen mass shell and will remain constant as a "comoving coordinate" to specify that shell. The radius r(t) can then be expressed in terms of a time-dependent scale factor R(t) which specifies the expansion of the space and this comoving coordinate:

Now making use of the Hubble law, we can express the velocity of the expanding shell as

Since the velocity can also be expressed as

Now these terms may be substituted into the energy expression above such that every term is a multiple of the comoving parameter ϖ which may then be canceled out. This suggests that the resulting equations apply to any shell and constitute a description of the expanding space in terms of the expansion factor R(t) and the Hubble parameter H(t). The expansion equation can be put in the two equivalent forms