Newtonian Cosmology

Assume that there exist a finite number, n, of galaxies. These galaxies will be the particles for our Newtonian system. Let galaxy  have mass  and position . We assume a fixed origin at point O. We will assume a cosmological principle that governs the average distribution of the galaxies. We assume that motion about point O must be spherically symmetric. In this case the galaxies, on average, move only radially.

 (26.1)

The kinetic energy for our system of n particles is given by

 (26.2)

Consider two galaxies labelled . The gravitational potential between this pair of galaxies is given by the equation

 (26.3)

For all the particles in our system, the total energy in the form of potential energy can be written as

 (26.4)

We count each pair of galaxies only once. We will assume that there exists a global cosmological force that affects each of the galaxies. We take this force to have the following functional form:

 (26.5)

The constant  is called the cosmological constant. When  this force is pointed radially outward. The force is repulsive relative to the point O. If   the force is attractive relative to the point O.

This force gives rise to the following potential:

 (26.6)

We now write down the total energy in our system of n galaxies.

(26.7)

We will now assume that there is information about the universe (our system of n galaxies) at one of its development times (called an epoch), which we take to be . At any later time the position of the  galaxy is given by

 .

This relation arises out of our assumption that the galaxies are, on average, flowing outward from the origin O in a radial direction. Each galaxy gets affected by the same scale factor, , when it's in the epoch labelled by t.

 (26.9)

This scale factor acts to simply scale up or scale down our Newtonian universe. Since it does not depend on the position coordinate it treats every point ion the manifold exactly the same.

We next write the radial velocity of the galaxies.

 (26.10)

Combining this equation with (26.8) we arrive at the following expression which is independent of the reference epoch.

We define the scale ratio that comes into this equation as follows:

(26.12)

The relation (26.11) is called Hubble's Law and can be written:

This expression is said to be Hubble's Law. In words, this expression can be described as follows. For each epoch in an expanding universe, the radial velocity of recession of a galaxy from a given point is proportional to the distance of the galaxy from that point. The value of   at the given epoch t is called the 'Hubble Constant' for that epoch.

We now derive a cosmological evolution equation from the energy equation (26.7). Substitute equation (26.11) and equation (26.13) into the energy equation

(26.14)

In terms of the parameters defined in terms of the reference epoch at time , we can introduce three reference data constants  to simplify the display of the above equation. This gives:

 (26.15)

Since the cosmological constant is arbitrary at this point we can rewrite the above expression in terms of two new constants A and B.

 (26.16)

This is our cosmological evolution differential equation arising entirely from Newtonian physics.

We now analyze this equation in several cases.

 (26.17)

If the universe is expanding then  will be growing. In this situation the second term in the above equation will be decreasing with time. If we are to keep E constant then the  term must also decrease with time. This means that the universe expansion will slow down as time advances.

(26.18)

The inclusion of this negative definite term will force the derivative term to increase, since it has to compensate by growing to keep E at the same value. Thus the effect of the inclusion of the positive cosmological constant is to drive the universe to expand more quickly. It has become fashionable to call the energy associated with the cosmological constant the ' Dark Energy ' in analogy with the concept of dark matter.

(26.19)

In this case the  term again has to get smaller with time so as to keep E constant.

We now solve the energy equation for  arriving at the following expression:

(26.20)

We now introduce a new symbol, ,  for the universe scale factor so as to put this differential equation into exactly the same form that is most often used in General Relativity.

 (26.21)

where  is a constant. We now multiply the differential equation by  to give

(26.22)

Now we use the  new symbol  in this equation to give

(26.23)

Finally this equation can be written

where

 (26.25)

and

 (26.26) .

It is standard to fix the constant factor  such that

 (26.27)

When , then  is defined arbitrarily. With this definition of  we get three possible values of .

 (26.28)

We constructed our Newtonian cosmological model by starting with a discreet set of particles. It is also possible to derive these equations starting with the continuum situation where a perfect gas is assumed to be the energy-mass source that fills the universe. Equation (26.24) is the same equation as the Friedmann cosmological evolution equation from General Relativistic Cosmology.