Free-fall Time

and is written as

This leads to the equation motion for a cold gas under a control of the self-gravity is written

(2.14) |

(2.15) |

in which represents the total energy of the pressureless gas element and it is fixed from the initial condition. If the gas is static initially at the distance , the total energy is negative as

(2.17) |

The solutions of equation (2.16) are well known as follows:

- the case of negative energy .
Considering the case that the gas sphere is inflowing ,
equation (2.16) becomes

where we assumed initially at . Using a parameter , the radius of the gas element at some epoch is written

In this case, equation (2.18) reduces to

(2.20)

This corresponds to the closed universe in the cosmic expansion (). - if the energy is equal to zero,
the solution of equation (2.16) is written as

where .