Friedmann Equations

 

We now list the necessary mathematical consequences arising from the assumptions.

 

·        Robertson Walker Line Element - Arises from the Cosmological Principle

(29.1)

 

·        Weyl's Postulate - The cosmological substratum should be a perfect fluid (comoving fluid with no fluid particle intersections and no interactions)  This means that we must have the perfect fluid tensor as our energy-momentum source.

(29.2)

 

·        The comoving fluid idea forces us to adopt special coordinates where

 

(29.3)

 

·        We assume that Einstein's General Relativity describes the gravitational evolution of the universe. Hence we get the following field equations.

 

(29.4)

 

 

When combined in the field equations, the above relations lead to only 2 independent evolution equations, called the Friedmann Equations, given as follows:

 

 

(29.5)

 

 

(29.6)

 

We are using units where . The pressure and the energy density are assumed to only be functions of time consistent with the overall isotropy and homogeneity assumption. We require that the energy density always be positive.

 

The two Friedmann equations above can be interpreted as an energy type equation (29.5)and an equation-of-motion type of equation (29.6). It is possible to dig out from these equations the cosmological energy conservation law. Differentiate the first Friedmann equation, multiply it by , and add it to the second Friedmann equation (which in turn has been multiplied by  ). This gives the following equation

(29.7)

Using the first Friedman equation we get this form:

 

(29.8)

 

 

Multiplying through by  we get the following result.

 

 

(29.9)

 

 

We can get a rough interpretation of this equation by making the following argument. Consider a set of fluid particles taking up a volume . Due to the motion of the substratum we should have . We now define the total mass energy in the volume to be . Hence

 

 

(29.10)

 

 

Using this result in (29.9) we obtain the relation

 

(29.11)

 

 

This is just the first law of thermodynamics, which is a direct expression of the conservation of energy. It says that the pressure is doing work in the expansion of the universe.

 

It can be shown that the conservation equation for energy-momentum given by the relation

 

 

(29.12)

 

 

also gives the equation

 

 

(29.13)

.

 

In light of the Bianchi identity constraint on the Einstein tensor this is not surprising since consistency is built in from the form of the field equations.

 

 

Specializing to Zero Pressure

 

 

 

We now consider what the cosmological evolution is like when we restrict ourselves to the  case. This equation of state turns out to be physically reasonable in the late universe. It also has the extra advantage of making the Friedmann equations easier to solve. The pressure  arises from a variety of sources: random star and galaxy motion, heat motion of the molecules, and pressure from radiation. When astronomers doe measurements of the mass energy density of the universe and compare it with the pressure of the universe at the present epoch of approximately 15 billion years they find that the relativistic fluid pressure is significantly less than the mass energy density of the universe.  . This has been the situation ever since the time of the decoupling in the very early universe when time . Hence for times greater than about a million years after the Big Bang it is safe to use the equation of state . Such a universe is said to be a Matter Dominated Universe. The substratum fluid is that of dust. In this case the second of the Friedmann equations (29.6)can be integrated directly. We obtain

 

 

(29.14)

 

where the constant C is a constant of integration. Using the first Friedmann Equation (29.5) we can get the following expression for C.

 

 

 

(29.15)

 

 

This constant represents the energy E of a volume V of the substratum fluid. In the  situation the thermodynamics equation

 

(29.16)

predicts that this energy must be a constant.

 

 

Using the result (29.15) in the first Friedmann Equation we get

 

 

(29.17)

 

This Friedmann evolution equation has exactly the same form as the Newtonian cosmology equation we derived earlier. Various solutions of this equation have been extensively analyzed. Such cosmological models are called Friedmann Models, or FRW models (short for Friedmann-Robertson-Walker). Sometimes the above equation is written in the equivalent form:

 

 

 

(29.18)

 

 

Remember that the k value determines the three dimensional geometry as follows:

 

 

 

 

The Hubble Constant at any particular epoch  is given as

(29.19)

 

 

Hence the above equation (29.18) can be rewritten in terms of the so-called Hubble ‘Constant’ as follows:

 

(29.20)