Previous lectures in this series have discussed primarily the conventional or `standard' cosmological model. However, the standard model has a number of problems, and so a substantial number of cosmologists (myself included) have devised alternative models in attempts to overcome the deficiencies of the standard model.

The subject of alternative cosmologies is necessarily a controversial one, and this lecture expresses a personal view.

The requirements for a model

A cosmological model requires the following four basic concepts:

Explain or take into account

The concepts of World Picture and World Map were introduced by the British astrophysicist Edward Milne. `World' in this context refers to the observable Universe.

A World Picture is a representation of how an observer sees the Universe. Because of the finite velocity of light, objects seen at distance are seen as they were the light-travel time ago (r = ct). The further away objects are, the younger they are seen. When we look out into the Universe, as we look further out, we are looking further back in time (I like to use the Australian word outback for out in distance and back in time).

A World Map is a theoretical portrayal of the Universe as it would look if the velocity of light were infinite, so that the visible Universe includes all parts of the Universe `at the same time'. Most models (such as the Friedmann-type models) are World Maps. A major difficulty with World Maps is that according to Einstein's relativity, clocks in different parts of the Universe run at different rates.

The majority of models assume that the Universe is homogeneous, ie that matter is spread out uniformly in it. This concept originated from two considerations. Firstly, the Copernican Principle (that the Universe should look much the same everywhere) was thought to imply homogeneity. Secondly, the mathematics of cosmological models is in general greatly simplified by this assumption.

Progress in astronomical observations has revealed that the Universe has, at least as far as we can see, what astronomers call a hierarchical clustering pattern, now (in this second half of the 20th Century) known to mathematicians as a fractal structure. (The mathemathics of fractals was not discovered until the second half of the 20th Century.) Thus, for example,

If the Universe were homogeneous, ie of uniform density, one would expect the number of galaxies within a given radius from us to be proportional to the cube of the radius. This is not so; instead, it appears that the number is approximately proportional to the square of the radius, implying a fractal structure with a `cluster dimension' D = 2.

• Archaic (based on naked-eye astronomy): the Solar System, and bowl of sky with stars attached
• Galactic: our Galaxy only
• Multi-galactic:

  • Friedmann
  • Homogeneous universe: Friedmann-Lemaître-Robertson-Walker models
  • Inhomogeneous: Tolman etc
  • Scalar–tensor
  • Steady-state
  • Static (tired-light): eg Crawford model
  • Kaluza-Klein (more than 3 dimensions of space)
  • Miscellaneous
  • Rost (World Picture)

Friedmann: expansion against gravity

The basic principle of the Friedmann models is that the `particles' of the Universe are flying apart from each other, and that the outward movement is counteracted by gravitation. This principle is expressed by the Friedmann equation, which is based on the law of conservation of energy and states that

Curiously, expansion is thought to be due to expansion of space, opposed by gravitational attraction within existing space.

There are numerous variants of the Friedmann model. The most important are those assuming homogeneity, generally known as Friedmann-Lemaître-Robertson-Walker (FLRW) models. The classical Friedmann model has zero pressure and no cosmological constant.

The characteristic feature of this basic Friedmann model is that matter is distributed isotropically and homogeneously, in a universe (World Map) of equal curvature everywhere. The curvature (and the scale factor) change with time.

Friedmann (1924); Lemaître (1927, 1931); Einstein & de Sitter (1932) (flat space only)

World Map (referred to `inertial frame of cosmic expansion')

Space-time curvature may be positive, zero or negative k = -1 or 0
Riemannian (and Minkowski, for special case k = 0)

Isotropic and homogeneous. Matter is treated as small particles (as in dust).

Einstein's field equations.
The closed variant (k = +1) is believed to be compatible quantitatively with Mach's Principle, the open models not.

Expansion against gravitation (Friedmann equation).
In the closed model (k = +1) the expansion slows to a stop, and the Universe ultimately collapses, may recycle. In the open models, the Universe expands for ever.

Expansion of space (Lemaître)

Yes

Finite.

Yes. Current theories of nucleogenesis are compatible with these models

Predicted (Alpher & Gamow) as relic of Big Bang, from end of period when Universe was opaque.

Resolved by expansion of the Universe - there is not enough energy to fill the Universe with radiation at temperature above 3 K.

If k = +1, then q_0 > 0.5, Omega > 1.
If k = 0, then q_0 = 0.5, Omega = 1.

If k= -1, then q_0 < 0.5, Omega < 1.
There is some reasonable evidence for Omega < 1, and some unreliable evidence that q_0 < 0.5. There is no independent evidence about k.

It is possible to estimate the Riemannian curvature very approximately from known data, if one assumes a zero cosmological constant and ignores pressure.

This model, although widely accepted, has never been validated, eg by inserting a full set of known values into the equations. There are major difficulties as listed above. The major difficulty is that asronomical observations have shown that matter in the Universe is arranged in a hierarchical clustering (fractal) pattern, and that gravitational collapse is prevented primarily by orbital or virial motion, while Hubble expansion plays quantitatively a negligible part. Also, the concept of contraction under self-gravitation can only apply to a finite structure: in an infinite universe, there is no centre to contract to, and the gravitational force on any galaxy an any given direction is (on average) balanced by an equal an opposite force in the opposite direction.

Matter has one or more major inhomogeneities, in an overall homogeneous universe. These models can be used to investigate the effects of concentrations of mass on the basic FLRW model. Pros and cons as for FLRW.

In these models, one or more parameters (such as G) are variable) over space and/or time. An example is the Brans-Dicke model, in which the reciprocal of G is a one-component field, the scalar field phi, that is generated by matter in accordance with an additional equation. The name `scalar-tensor' derives from the use of both a scalar and tensors in the field equations. Pros and cons much as for FLRW. None of these models has achieved much acceptance.

Steady state Principle: The Universe is expanding (exponentially), and matter is continually being created to maintain a constant density. This leads to Hoyle's elegant Perfect Cosmological Principle, that the Universe appears much the same to all observers, everywhere, and at all times. (cf the Copernican Cosmological Principle, and Rost's Pluperfect Cosmological Principle (below)).Expansion against gravitation as in the standard Friedmann models.

Hermann Bondi & Thomas Gold (1948)
Fred Hoyle (1948)

World Map.

Flat

Initially thought to isotropic and homogeneous, with matter created uniformly everywhere. Theory later modified, so that matter is created where matter already exists, not in voids, thereby maintaining thehierarchical structure of the Universe.

No inbuilt theory of gravitation.
Probably not compatible quantitatively with Mach's Principle?

Exponential expansion. H is constant, q = -1.
Creation of matter to maintain steady state.
Hoyle says expansion due to newly-created matter forcing existing matter apart. [This would seem to suggest expansion only where matter is, not in voids?]

Expansion of space (Lemaître).

None.

Infinite. Hubble time is finite and constant.

No. This model is not compatible with current theories of nucleogenesis.

Not specifically predicted. Explained as remnant of radiative energy produced by fusion of hydrogren to helium. [Possibly 3 K is the basic temperature of the Universe: there has been an infinite time in which to achieve thermodynamic equilibrium and therefore an isotropic background.]

Unresolved. The Universe is infinite, with an infinite population of stars at various periods of evolution. The observable universe therefore contains an infinity of stars.

k = -1. Otherwise nothing useful.

This model was widely supported for some time, but is no longer thought to be valid. Its major difficulty is lack of a hot primordium, making it incompatible with current theories of nucleogenesis and giving no reason for the 3K CBR.

Static
`Tired-light', Kaluza-Klein
Basic principle is that tidal action on photons in curved geodesics leads to emission of secondary photons.

David Crawford (1987, 1991, 1993, 1995)
(Crawford, David F. (1993) A static stable universe. Astrophusical Journal 410, 488-492.)

Static universe, so World Map = World Picture.
Perfect Cosmological Principle applies.

4-D hypersphere embedded in 5-D (or more) Euclidean space.

Isotropic and homogeneous.

Newtonian.
Probably not compatible quantitatively with Mach's Principle.

Static universe. Particles in motion, centripetal acceleration.

None. (Expansion, as an unrelated phenomenon, not excluded.)

`Tired light' decay of radiation with time (gravitational interaction). Basic principle is that tidal action on photons in curved geodesics leads to emission of secondary photons. `A photon travelling travelling in amedium, which is rare enough for there to be effectively no other interactions, loses energy at a rate proportional to the square root of the local density.'

No.

Infinite.

No. This model is not compatible with current theories of nucleogenesis.

Due to production of secondary photons.

Resolved by finite extent of the Universe, and redshift reducing energy of photons.

Given mass density (rho) only, predicts Hubble parameter, radius, mass.
`Holes' in the redshift distribution due to the effects of high density clouds between us and objects. This offered as an explanation (Crawford 1987a) of observations (Karoji et al, 1976) that galaxies seen through clusters have a larger redshift than those selected away from clusters.

• Universe of infinite age.
• No initial singularity.
• CBR due to emission of secondary photons.
• Perfect Cosmological Principle.
• No `missing mass in galaxies' problem.
• Confirmed predictions:
  • Hubble constant deduced from mass density of Universe.
  • Larger redshift of galaxies seen through clusters.
  • Quasar distribution in accordance with model.

• Hot intergalactic plasma for nucleogenesis.
• Olbers' Paradox resolved by finite Universe and photon redshift.
• Useful predictions.

World Picture.
Principles: (1) As time advances, space expands, clocks run faster, and masses decrease. (2) Fractal distribution of matter. (3) Gravitation quantitatively in accordance with Mach's Principle.

Fred Rost

World Picture; referred to observer's inertia frame. Each observer is in the exact centre of his/her/its observable Universe.

Spacetime in the World Picture is positively curved, closed, consisting of geodesics fanning outwards from observer and joining together again at the cosmic primordium.

Isotropic, fractal.

Quantitative Mach's Principle.

Relativistic.
Expansion is linear in World Picture; exponential in World Map.
Space, time (frequency), mass all change with time; c, h invariant.
Subjective steady state. Pluperfect Cosmological Principle: Universe looks much the same in all directions, everywhere, at all times (apart from local evolution), and at (almost) all scales.

Expansion of space (Lemaître).
Hubble's law predicted (H = cz/DL)

No. Primordium of finite size.

Finite (= Hubble time) in WP; infinite in World Map (ie the Universe is of infinite age, but its age appears finite to us because of the effects of relativity).

Not predicted. Possibly 3 K is the basic temperature of the Universe. Isotropy of CBR may be due to symmetry of World Picture.

Resolved by hierarchical clustering (Charlier's theory), and the finite World Picture.

Data input required: c, G, H_0, D
Predicted: subjective age, radius, mass
Hubble's law (H = cz/DL)

This model is unique in at least two aspects:

I prefer this model to the others mentioned here, because:

1. This model is that of a World Picture, referred to the inertial frame of an observer, and is therefore immediately comparable with observations.
2. The model assumes a fractal structure, in which homogeneity is a special case, and does not assume a necessarily homogeneous Universe on any scale.
3. The age of the Universe is infinite.
4. It offers the greatest range of predictions.
5. The Pluperfect Cosmological principle is the most elegant yet.
6. The model avoids certain problems seen in other models, eg:
   • An initial singularity (Friedmann models)
   • Olbers' Paradox (Steady State model)
   • The inelegance in the Friedmann models of a Universe supposed to be expanding by expansion of space and contracting under gravitation within existing space.