The Friedmann Model for the Expansion of the Universe

Written by Frans Nieuwenhout, 16-6-1997

Translated and edited by Inge van de Stadt (12-5-1998)

 

Based on:
* A. Achterberg, Van Oerknal via Niets tot Straling en Stof, een astrofysische inleiding in de kosmologie,
   Epsilon Uitgaven, Utrecht 1994
* A. Achterberg,'Is het oerknalmodel nog te redden',  Zenit, mei 1997, pp. 198-207

Converted to HTML by Gerard Hoogeland (14-5-1998, update 19-11-1998 by Inge van de Stadt)

 

This paper was written for the study group Theoretical Astronomy of the Alkmaar Amateur Astronomical Society, AWSV, (member of the KNVWS). It is an introduction to the basics of cosmology. In response to an article in Zenit a simple Pascal program was written to illustrate the expansion of the Universe.

In the first two sections some fundamental ideas from cosmology are defined to serve as an introduction the expansion model. The most important of these ideas are the Hubble parameter, the Redshift and the Scalefactor. Next the basic equation for a simple model of the expanding universe will be made plausible by the use of the law of energy conservation. Within this Friedmann model the so called 'Cosmological constant' plays a major role. How this Cosmological constant influences the age of the universe is explained in section 5.
 

1. Redshift and Hubble parameter

The redshift for an emitted electromagnetic wave of wavelength le to an observed wavelength lob is equal to their difference in wavelength over the emitted wavelength:

 

(1)

The distance D has been observed to be linearly proportional to the redshift:

 

(2)

where H0 is the Hubble parameter. The subscript 0 means its value at the present time. Its measured value: H0 »  65 km s-1 Mpc-1.
The distance D(t) between two points in the universe with a distance A in comoving coordinates and a scale factor R(t):

 

(3)

At the present time

 

(4)

At other times the distance is given in the following equation:

 

(5)

The velocity V is per definition the time-derivative of the distance:

 

(6)

(7)

(8)

(2)

Where the Hubble parameter H:

 

(10)

By this equation it is clear that the Hubble parameter is not constant with time.
 

2. Scale factor and redshift

The distance a photon travels between emission untill detection is . From the time of emission and the moment of detection the wavelength changes from  to . So the redshift will be:

 

(11)

Using  gives:

 

(12)

So the wavelength is linearly dependent to the scale factor:

 

(13)

The main difference with the classical Doppler effect is that the wavelength of the light keeps changing after emission!
 

3. Dynamics: the Friedmann model

How does the scale factor change with time? Here a classical, that is a non-relativistic derivation is given. It is assumed that the universe is homogeneous with a mass density . Furthermore it is assumed that the contribution of the (gas) pressure is negligible. The starting point for the explanation of the dynamical behaviour of the universe is energy conservation. Choose any sphere of radius . For a galaxy of mass m near the 'surface' of the sphere the sum of its kinetical and gravitational potential energy is constant:

 

(14)

Substituting the Hubble equation and the mass of the sphere gives:

 

(15)

Dividing by and substituting by  gives:

 

(16)

Or,

 

(17)

This is the well known Einstein equation of the Friedmann model. Here k is a constant which corresponds to the average energy per particle. The evolution of the universe depends on the total amount of energy E:

E > 0 and k < 0: open universe, will keep expanding
E = 0 and k = 0: flat universe, expansion will stop when t ® ¥
E < 0 and k > 0: closed universe, expansion is followed by contraction

 A flat universe is only possible at a certain value of the density, the so called critical density which can be found by substituting k = 0 into equation (17):

 

(18)

With the Hubble parameter of 75 km s-1 Mpc-1 the critical density is about 10-26 kg m3
 

4. The Cosmological constant

The solutions of the Friedmann model are dynamical: eternal expansion or expansion followed by contraction. However, the expansion of the Universe was only found by Hubble ten years after the formulation of the Friedmann model. When Friedmann made his solutions public, Einstein reacted by proposing the cosmological constant . It compensates exactly for gravitation and results in a static universe. After it was concluded from the redshift of distant galaxies that the universe as a whole expands, the use of the cosmological constant was abandoned. In recent times there is a 'revival'. First there is a physical phenomenon, the so called vacuum fluctuations, that could justify the use of a constant term in the Einstein equation. Secondly, the introduction of a cosmological constant could account for the age problem of the universe. Simple models of the universe without cosmological constant predict an age of the universe of about 10´ 109 year, which is much younger than the age of the oldest stars, which have been estimated at about 15´ 109 year.

Vacuum fluctuations is the quantum mechanical phenomenon that in 'empty' space pairs of particles and their anti-particle are continually being created, which destroy each other again after a short time, so that the 'borrowed' energy is released again. By consequence the 'empty' space has also a certain minimum energy content. This vacuum-energy is a property of space, which is not influenced by the overall expansion. The total density in the Einstein equation (17) is the sum of two terms. The first one is due to ordinary matter, whose density decreases with expansion:

 

(19)

where is the density at the present time. It was assumed that the universe is now dominated by matter. In the early universe electromagnetic radiation was the most important contributor to the density. In a universe dominated by radiation, the energy density decreases with R4. This is because of the redshift which makes the energy of the photon decrease with the scale factor according to equation (13). From about 105 year after the big bang matter is the major contributor to the energy-density.

The second term is the constant density due to vacuum fluctuations: . the cosmological constant is defined as a multiple of this vacuum energy:

 

(20)

substituting into the Einstein equation (17) gives:

 

(21)

Dividing by H2 and substitution for the critical density gives the relation between the two density term and the Friedmann constant k:

 

(22)

The densities expressed as a ratio to the critical density are indicated as the parameter Omega:

 
and    

Both Omega density parameters vary with the scale factor. The constant k can be expressed in the present values of the Omega parameters:

 

(23)

The subscript 0 indicates the values at the present time.

Substitution of (10), (19) and (23) into the Einstein equation (22) it is possible to formulate it as a differential equation of the scale factor without the explicit Friedmann constant k:

 

(24)

 This differential equation can be solved numerically starting from 

(You should realise that and that )
 

5. Solutions of Friedmann model
 

It is often assumed that the universe is flat, that is k = 0. Then the sum of the Omega density parameters is exactly 1. In figure 1 the red curve shows how the scalefactor increases with time in a flat universe without cosmological constant. For a Hubble constant of the value of 65 km s1 Mpc-1 the age of the universe is only 10.1´ 109 year. The blue line indicates the present expansion velocity. If it would be constant, that line would show the way the universe expands. In such a case the big bang would have occurred 15.0´ 109 year, at the Hubble time. A more realistic model is given by the green curve. For that solution it was assumed that the present density is 10% of he critical density and that the contribution of the cosmological constant is such that the universe is flat, that is . The resulting age of the universe increases to 19.2´ 109 year.


Figure 1 Scale factor as a function of time by linear expansion (blue), a flat universe without cosmological constant (red) and a flat universe in which the present density of matter is 10% of the critical density (green).

It turns out that the age of the universe is depends strongly on the chosen value of Omega, as can be seen from figure 2. For the smallest possible density of 1% of the critical density, the resulting age is over 30´ 109 year.


Figure 2 Scale factor as a function of time by linear expansion (blue), a flat universe without cosmological constant (red) and a flat universe in which the present density of matter is 1% of the critical density (green).

For a flat universe without cosmological constant the matter density is not enough to stop expansion. For the universe to be closed, the essential condition that the sum of the two Omega terms is larger than 1. Without the cosmological constant, that condition is sufficient: the expansion slows down and eventually turns into contraction as can be seen in figure 3. In this example the present matter density has been put at twice the critical density. Without cosmological constant this gives an age of the universe of only 8.6´ 109 year. The introduction of a cosmological constant in such a case, can also somewhat increase the age of the universe. However, even for small values of Omega lambda the universe will be open. This is illustrated in figure 4. Here is the same value used for the present matter density of twice the critical density, bun now with a present Omega lambda of 0.1. The age only increases marginally to 8.7 year.

On the basis of the simple Friedmann model it can be concluded that an oscillating universe is rather improbable, and that the introduction of the cosmological constant could provide a possible solution for the age problem of the universe. However, the here presented Friedmann model only provides a simplified picture of the expansion of the universe, an therefore it is quite possible that a more profound description would lead to different conclusions.


Figure 3 Scale factor as a function of time for a closed universe with the present matter density twice the critical density, without cosmological constant (green curve).
 


Figure 4 Scale factor as a function of time as in figure 3, but now with a cosmological constant at the present time, WL0 = 0.1