hubble ultra deep field





“Everything should be made as simple as possible,
 but not simpler”.  Albert Einstein.


6.1    From the above studies, some conclusions may be drawn about the parameters which are necessary for effective analogies and visual models of expansion in order to minimize potential misconceptions.  The effectiveness of a model will depend on the intuitiveness of the source concept and the connectivity.  The usefulness of a model will depend on the target concepts addressed and the capacity of the model to integrate hierarchical concepts.  There are therefore two types of bad model.   The first may be efficient – at transmitting misconceptions, and the second may be transmitting the correct concepts – to very few people.  The perfect model would transmit the right target concepts to everybody – an impossible goal, but one which should be striven for.

6.2.1   The source conceptsfor an analogy of expansion should:

  1. Be composed of p-prims, or similar core concepts
  2. Be able to support further, hierarchical concepts
  3. Contain indications of the limitations of the model

6.2.2   The main considerations of connectivity are:

  1. That mapping between the source and target concepts should be intuitive, particularly if no teacher will be present to draw the connections verbally.  This reinforces the need for p-prims as source concepts.
  2. That the connections should be, as far as possible, entertaining.  The student should be intrigued by the model, to encourage him to examine the similarities between the source and target concepts.

6.2.3   The main considerations for target concepts are that the target concepts should not be limited to one area to the exclusion of others.  Either all the target concepts relating to the subject should be included in the model, or it should be an easy matter to extend the model to encompass those concepts which it does not immediately address as hierarchical.  This point is paramount to the acceptation of more advanced concepts and to the model’s success in its goal of transmission without creating misconceptions.   Target concepts may be divided into positive target concepts, which the visual model should transmit as plausible; and negative target concepts, which the visual model should show to be implausible.
In the particular case of a visual model explaining metric expansion of the universe, this study indicates that the following target concepts should be included:

Positive target concepts (Should be plausible):

  1. Space expands with a scale factor which increases with the age of the universe.
  2. Expansion is an initial condition, not a force acting on matter. 
  3. The Cosmic Microwave Background is all around us, we are immersed in it.    
  4. Gravity affects the curvature of space.    
  5. Further hierarchical concepts would be: (i) dark matter, (ii) dark energy, (iii) large scale structure – voids and non-homogeneous expansion and (iv) the tethered galaxy problem.

Negative target concepts (Should be implausible):

  1. Atoms, people, stars, galaxies and anything up to clusters do not join this expansion.
  2. The universe is not expanding into anything else. 
  3. Matter does not have a “velocity” of expansion, as if it were travelling through a static background.
  4. The Big Bang was not an explosion, nor did it take place at some central place.




“For every complex problem there is a solution which is simple, neat,
 and wrong”.  Thomas Henry Huxley.



7.1   EXPANDING BALLOON MODEL   The most significant model used is the expanding balloon model.  This was first used by Sir Arthur Eddington in his book “The Expanding Universe” in 1933, although Brian Greene traces it as far back as a dutch cartoon portraying Dr. de Sitter in 1930 (The Fabric of the Cosmos 2004).
In the balloon analogy the student is invited to imagine a balloon with various small dots drawn on the surface.  The balloon is then blown up, and the galaxies, represented by the small dots on the surface of the balloon, can be seen to be each travelling away from all the others.  Eddington was careful to detail the fallbacks of this analogy, but others have unfortunately not done the same.  This model is shown in the two figures below.

Figures 5 & 6: The balloon analogy

Expanding balloon model 1


expanding balloon model 2

7.2    RAISIN-BREAD ANALOGY   A variant of the expanding balloon model is the raisin-bread analogy.  Here, the raisins in the loaf of bread represent the galaxies, and global expansion of the universe is represented by the expansion of the loaf of bread as it is being cooked in the oven.  William J. Kauffman III in his textbook “Astronomy: The Structure of the universe” (1977) attributes this “recipe” to Dr.  George O. Abell at UCLA.

Figure 7:  The Raisin-bread Analogy

The raisin bread analogy

7.3   ANTS-ON-A-RUBBER-SHEET ANALOGY   A further analogy often used is that of ants on a rubber sheet.  Here the ants are only aware of a two dimensional universe – like the famous flatlanders (“Flatland A Romance of Many Dimensions”, Abbott, 1884) – and they see each other receding as the rubber sheet stretches.

Figure 8: The Ants-on-a-rubber-sheet Analogy

the ants on a rubber sheet analogy


7.4    TRIPLE-DECKER MODEL   A refinement of the ants-on-a-rubber-sheet analogy would be the following model, rather more modern, where the three dimensional aspect shows us past, present and future.  This could be called the triple-decker model.

Figure 9: Wikipedia Commons: Universe_expansion.png

triple decker model 

7.5   FUNNEL MODEL   In recent years the “funnel” visual model has become popular in astronomy texts.  Here the Big Bang is portrayed to the left of the diagram, and time evolves to the right.  We can see that the universe is becoming bigger as time passes, although the scale factor is not addressed in this model.  This model is credited to the NASA/WMAP science team, and so is used frequently in modern cosmology. 

Figure 10: Wikipedia commons:

funnel model

7.6   These five models are the main visual models most often used currently in text books and coursework in cosmology.  Although there are many other scientific models in existence, the above models represent the majority of the source domains used for expansion.  The first two are the most used in the teaching of astronomy, although in recent years the last two have become rather more popular.

7.7   CONFORMITY TO PARAMETERS   It is now possible to consider how closely the classical models described above conform to the parameters suggested.
As far as the source concepts are concerned, the balloon, raisin-bread and ants-on-a-rubber-sheet models use simple concepts which may be p-prims, indicating that they will be good at transmitting concepts, and will be efficient if the target concepts are the desired ones.  As will be seen below, they are not.  This provides a reasonable hypothesis that these three models will transmit misconceptions.  The triple-decker and funnel models use more complex source concepts and therefore will not be so efficient at transmitting concepts successfully. 
Connectivity also varies.  Mapping is arguably intuitive in the balloon and raisin-bread models, much less so in the rubber sheet and triple-decker models and unlikely to be intuitive in the funnel model.  The balloon, the ants-on-a-rubber-sheet, and the raisin-bread models could probably be classed as entertaining, the triple-decker and the funnel models less so.


Positive target concepts (Should be plausible)

  1. Space expands with a scale factor.  The target concepts are generally limited to showing that space expands.  The funnel model shows expansion to occur principally in the first 400 years.  A scale factor is possible only in the balloon and the raisin-bread analogies and in both models the connectivity has severe limitations. (For example, in figures 5 and 6 above of the balloon model, the upper parts of the balloon can be seen to expand more than the rest).
  2. Expansion is an initial condition.  The balloon analogy clearly has required much huffing and puffing on the part of somebody to blow it up, and continues to do so.  The stretching of the rubber sheet also requires ongoing force.  The raisin bread needs the oven, and both the funnel and the triple-decker model imply a large explosion powers the expansion.
  3. The cosmic microwave background is all around us.  This concept is arguably hierarchical in all the models, depending on interpretation, but is not plausible without considerable work on the part of the student and some adaptation or extra explanation of the source concepts.
  4. Gravity affects the curvature of space.  None of the models imply that gravity affects the curvature of space.
  5. Further hierarchical concepts such as (i) dark matter, (ii) dark energy, (iii) voids and non homogeneous expansion and (iv) the tethered galaxy problem.  None of the models can be adapted for hierarchical concepts (i), (iii) or (iv).  All can be adapted to the concept of dark energy only by describing it as “more” expansion.



Negative target concepts (Should be implausible)

  1. Atoms, people stars and galaxies do not join this expansion.  None of the models show that clusters do not expand.  The balloon, triple-decker and funnel model in fact strongly support the inference that they do.  The raisin-bread and ants-on-a-rubber-sheet models are better in this respect, but still do not make local expansion particularly implausible since their source concepts consist of generalized expansion, and it is unclear what ants or raisins represent.
  2. The universe is not expanding into anything else.  All the models may be interpreted as expanding into something else, although this is a natural problem in models which aim to show us the entire universe, by definition.
  3. Matter does not have a “velocity” of expansion.  Metric expansion could be considered a hierarchical, if obscure, concept in all the models, although in the funnel model galaxies appear to be almost stationary. 
  4. The Big Bang was not an explosion.  The funnel model and the triple-decker model imply that it was.  The other three models do not suggest a central explosion, although they do suggest expansion from a central place.



7.8   CONCLUSION   None of the models shown above conform to the desired parameters.  All would initially appear to be imperfect visual models.  The balloon model and the raisin-bread model are the most likely to be successful at transmitting concepts, but the concepts they are transmitting do not conform to the parameters.  They probably transmit undesired misconceptions.  The rubber sheet model is entertaining, but less intuitive and does not meet the target concepts.  The triple-decker and the funnel model not only do not meet the target concepts, but also do not fulfil the necessary parameters of connectivity and source concepts and therefore are unlikely to be efficient models.




“Essentially, all models are wrong, but some are useful”.
 George Edward Pelham Box.


8.1   The starting point of Einstein’s general theory of relativity is that gravity – matter – curves space.  This has often been represented through visual models such as the following diagram:


Figure 11: curvature: curved space-time.
Art.  Encyclopædia Britannica Online.  Web.  3 Jul.  2011.

curvature of space time

In such diagrams light is bent towards the object which exerts gravity.  This is a visual model which is familiar, and intuitive.
Gravity here is seen to distort spacetime downward.  What happens to spacetime if expansion is accelerating, if there is a cosmological constant which is also distorting spacetime?  If gravity tends to attract matter, and accelerated expansion tends to disperse it, accelerated expansion has to be considered instinctively to act in an inverse direction to gravity.  Thus gravity and expansion are diametrically opposed to each other – immediately, a landscape of mountains and valleys springs to mind.   
The proposed timeslice model consists simply of showing a developing landscape at three different periods, where expansion is represented by a scale factor, but gravitational collapse is not.   There are three different “timeslices” on the page.  The top one represents a redshift of 3, corresponding to about 10 billion years ago .  The second is scaled to twice the size, representing a redshift of 1, and a time of about 7 billion years ago.  The third shows a redshift of 0, an expansion of four times that of the initial size is shown, and the diagram represents the present.   
In all the diagrams the curvature between set points is divided into equal segments marked in red and blue alternately so that expansion may be measured and seen to be dependent on distance.  The three redshifts of 0, 1 and 3 are also marked, and a scale factor begins at 1 in the oldest diagram and progresses to 4 in the present. 
This model may seem rather curious to those astrophysicists currently working on void studies.  Van de Weygaert et al (2009) point out that in such studies an analogy is frequently drawn between voids and a landscape.  In their vision, however, the valleys represent the voids and the sheets and ridges represent the network of walls filaments and clusters.  This leads to the application of WST, Watershed Transform, which finds its origin in the landscape being flooded by a rising level of water.  In the timeslice model, although the same landscape source concept is used, the voids become the mountains and the walls filaments and clusters become the basins.  This is in order to tie the concept in correctly to gravity being “weight”, and, in a student’s experience, possessing a downward vector.


8.2   The following diagrams show the proposed new model:

Figure 12: the timeslice model

full timeslice model

The model can easily be portrayed by a simple, one-dimensional version where a teacher wishes to draw it himself.  The following diagram demonstrates such a version:

Figure 13: The line version of the timeslice model

line version of the timeslice model


Ho currently could vary between 71.4 and 76.2 according to the latest HST experimental results (Riess et al, 2011).  Using the cosmology calculator (Wright, 2006), together with the concordance model of 0.3 for Ωm, and 0.7 for Ω˄ this gives lookback times rounded to the nearest billion of 7 or 8 for z = 1, and 10 or 11 for z = 3.  The first values have been used in these figures but these values will change according to paradigm.  The scale figures and the redshift are unchanging, however.

For simplicity the scale factor used here is 4 times bigger than the usual R = a(te)/a(to), which would result from equation 1.