244
15 Solutions for the Present Universe
Appendix 3: Some Discarded Cosmological Models
In this chapter we have discussed models that are currently considered viable. We
will mention only briefly some well-known models that are no longer considered
viable.
Static models were first studied, by Einstein and others, before the expansion of
the universe was discovered by Hubble and other astronomers. They are therefore
no longer of physical interest.
Some models are not homogeneous and isotropic. One notable example is that
of Godel, which has a preferred axis and rotates. It is of philosophical and theoretical interest since one must ask “With respect to what can the entire universe
rotate?” (Godel 1949; Adler 1975). The Godel model also has interesting and peculiar causality properties (Hawking 1973). However the model has the fatal flaw of
having no Hubble expansion and is not considered a viable model of the actual
universe.
A steady state model was popular some decades ago, in which the universe
expanded but did not change over time. Spontaneous creation of matter was necessary in this model, and it had no big bang by intent. The discovery of the cosmic
microwave background radiation left over from the big bang greatly reduced interest
in this model and it is no longer in the mainstream of cosmology (Bondi 1948; Hoyle
1948; Liddle 2003).
The de Sitter model has an exponential expansion and no big bang. It is the
asymptotic limit of the LCDM model in the distant future when the scale factor
becomes very large, as we discussed in the preceding sections. We will return to it
as a mathematical guide when we discuss inflation in the very early universe; it is
widely believed that during inflation the universe was dominated by a field or fluid
that behaved much like extremely dense dark energy. As a complete model of the
present universe however it is no longer viable.
Exercises
15.1 Plot the cycloid in (15.11) and show that it looks like the curve in Fig. 15.2.
Look up the name cycloid to see how it relates to the motion of a point on
the edge of a wheel.
15.2 Plot the pseudo-cycloid in (15.12) and show that it looks like the curve in
Fig. 15.2.
15.3 If a galaxy has a redshift of z in a flat matter dominated universe how far
away is it?
15.4 Work out properties of the static model from (15.3), by setting the scale factor
equal to a constant. What values of the curvature parameter k are allowed?
How does the scale factor depend on the density parameters?
15.5 Show explicitly that the static model is not stable. Appendix 1 should be of
help.
15.6 Solve (15.3a) numerically for the LCDM model with a small nonzero k.
15.7 Equation (15.18) can be used to determine the age of the universe in the
LCDM model. Do this by setting the scale factor equal to 1 in (15.18) and
15 Solutions for the Present Universe
Appendix 3: Some Discarded Cosmological Models
In this chapter we have discussed models that are currently considered viable. We
will mention only briefly some well-known models that are no longer considered
viable.
Static models were first studied, by Einstein and others, before the expansion of
the universe was discovered by Hubble and other astronomers. They are therefore
no longer of physical interest.
Some models are not homogeneous and isotropic. One notable example is that
of Godel, which has a preferred axis and rotates. It is of philosophical and theoretical interest since one must ask “With respect to what can the entire universe
rotate?” (Godel 1949; Adler 1975). The Godel model also has interesting and peculiar causality properties (Hawking 1973). However the model has the fatal flaw of
having no Hubble expansion and is not considered a viable model of the actual
universe.
A steady state model was popular some decades ago, in which the universe
expanded but did not change over time. Spontaneous creation of matter was necessary in this model, and it had no big bang by intent. The discovery of the cosmic
microwave background radiation left over from the big bang greatly reduced interest
in this model and it is no longer in the mainstream of cosmology (Bondi 1948; Hoyle
1948; Liddle 2003).
The de Sitter model has an exponential expansion and no big bang. It is the
asymptotic limit of the LCDM model in the distant future when the scale factor
becomes very large, as we discussed in the preceding sections. We will return to it
as a mathematical guide when we discuss inflation in the very early universe; it is
widely believed that during inflation the universe was dominated by a field or fluid
that behaved much like extremely dense dark energy. As a complete model of the
present universe however it is no longer viable.
Exercises
15.1 Plot the cycloid in (15.11) and show that it looks like the curve in Fig. 15.2.
Look up the name cycloid to see how it relates to the motion of a point on
the edge of a wheel.
15.2 Plot the pseudo-cycloid in (15.12) and show that it looks like the curve in
Fig. 15.2.
15.3 If a galaxy has a redshift of z in a flat matter dominated universe how far
away is it?
15.4 Work out properties of the static model from (15.3), by setting the scale factor
equal to a constant. What values of the curvature parameter k are allowed?
How does the scale factor depend on the density parameters?
15.5 Show explicitly that the static model is not stable. Appendix 1 should be of
help.
15.6 Solve (15.3a) numerically for the LCDM model with a small nonzero k.
15.7 Equation (15.18) can be used to determine the age of the universe in the
LCDM model. Do this by setting the scale factor equal to 1 in (15.18) and