hubble ultra deep field





“The greatest problem of communication is the illusion that it has been achieved”.  George Bernard Shaw.


3.1   In this study Zirbel’s definition of misconception: “concepts not in agreement with our current understanding of natural science” (Zirbel 2004), will be used, which indicates that any study of misconceptions in cosmology will require previous review of the current paradigms, and also that misconceptions, like paradigms, evolve over time.
The fact that misconceptions evolve may be seen anecdotically by the cosmological constant, which Einstein found himself forced to introduce.  He was laboring under the misconception that the universe was static.  This was not a misconception at the time, since the understanding of natural science before 1920 was that of a static universe.  He felt obliged to adapt his field equations, which had indicated expansion as a natural outcome.   When understanding of natural science developed and observational evidence made expansion of the universe the paradigm, Einstein rapidly withdrew his cosmological constant, with the famous “biggest mistake” comment.  Current understanding, however, is that the universe is accelerating in its expansion due to dark energy, which brings back the need for a term similar to the cosmological constant.  The cosmological constant has progressed from paradigm to misconception and back to paradigm.  This type of flux is typical of the science of cosmology, as it is a science with limited experimental and even observational possibilities, and one of the few disciplines in which the word “untestable” is not necessarily a criticism.  As seen, even the greatest cosmologists may find that today’s paradigm will be tomorrow’s misconception, and possibly even vice versa.  It may merely be a question of time.
In cosmology there are various levels of misconceptions: some start when the student is very young, and hears the expression “Big Bang” for the first time.  Others are introduced in high school when the balloon analogy is used, and still more become ingrained in higher education when the student is asked to master more advanced ideas.  Misconceptions would seem to come in layers, rather like those of an onion. 
There is confirmation of this opinion in the paper by Sadler (1996).  Sadler speaks about the hierarchy of concepts, explaining that ideas in science are often hierarchical in nature.  Students who do not have the foundation stones in place find it very hard to understand new concepts.  These ideas are very resistant to change, and although studies have shown that misconceptions can be changed, conventional courses do little in this aspect.  Sadler explains that:
There is another approach that should be considered, i.e.  choosing concepts on the basis of their structural relationship to learning other concepts.  In this way, concepts that are required for understanding more difficult ideas should be taught first.  When mastery of these ideas is achieved, one can move to harder topics that depend on these prerequisite ideas.
Sadler also finds that courses which cover too much content appear to leave many students with reinforced misconceptions and a decreased ability to answer many astronomical questions.
Some of the principal problems in the field of cosmology are detailed in an article in Scientific American: “Misconceptions about the Big Bang” (Lineweaver and Davis 2005), which was based on a previous paper (Davis and Lineweaver 2003).  In the article the authors talk about various misconceptions on the subject of expansion.  Their article highlights the following six areas of difficulty, where the answers they give have been paraphrased and abbreviated:

1.  What is expansion, anyway?   They explain that Galaxies are not fragments of a Big Bang bomb; rather the space between us and the galaxies is expanding. 
2.  Ubiquitous cosmic traffic jam.  They use this title to describe the Big Bang, and explain that there was no one particular location for the Big Bang, rather it happened everywhere.
3.  Receding faster than light.  Davis and Lineweaver explain that Hubble’s Law predicts that galaxies beyond a certain distance, known as the Hubble distance, recede faster than the speed of light.  People may think that this contradicts Einstein’s special theory of relativity which says that nothing can have a velocity exceeding that of light.  However, Einstein’s theory applies to “normal” velocities – motion through space.  The Hubble flow is not a motion through space; it is a motion of space itself.
4.  Stretching and cooling.  As space expands, light becomes stretched.  As photons travel through expanding space, they lose energy and their temperature decreases.  In this way, the universe cools as it expands.
5.  Running to stay still.  A distant light beam is moving toward us at the speed of light with respect to its local space, but its local space is receding from us faster than the speed of light.  Although the light beam is traveling toward us at the maximum speed possible, it cannot keep up with the stretching of space.  However, since the Hubble constant changes with time, the photons can then find themselves in a region of space that is receding slower than the speed of light.  Thereafter they can approach us.
6.  Is Brooklyn expanding?  No, it isn’t.  Expansion by itself produces no force.  Photon wavelengths expand with the universe because, unlike atoms and cities, photons are not coherent objects whose size has been set by a compromise among forces.  Recently it has been determined that the universe is accelerating its rate of expansion.  If acceleration (of expansion) is constant the planet Earth simply settles into a static equilibrium size slightly larger than the size it would have attained with no acceleration.  However, if acceleration is not constant, it could eventually grow strong enough to tear apart all structures, leading to a “big rip”.  But this rip would occur not because of expansion or acceleration per se but rather because of an accelerating acceleration.

Zackrisson (2009), in his “Seminar 1: Common Misconceptions about modern Cosmology” identifies five main areas of confusion.  He gives these as the Big Bang, cosmic curvature, expansion of the universe, size of the universe, and distances in cosmology. 
Berman in “Misconceptions on Relativity, Gravitation and Cosmology” (2008) looked through elementary physics textbooks, and found many inclusions of misconceptions.  Amongst these were: “the universe has a centre”, “dark matter and dark energy are the same thing, because of Einstein’s mass-energy relation” and “the universe expands, so it accelerates”.
Davis and Lineweaver (2003) cite 25 examples of confusing comments about expansion from scientific works written by even renowned scientists such as Feynman, Lovell and even Hubble himself.
Peacock also comments on common misconceptions in cosmology in his lecture notes at Caltech (1999).  Here he typifies such misconceptions into four groups: the initial singularity, the origin of the redshift, the nature of the expansion, and the empty universe.
There are also many diametrically opposed statements about the subject in textbooks and popular science literature.  For example, in 1993 Weinberg states: “… space does not expand.  Cosmologists sometimes talk about expanding space, but they should know better.”  And Rees agrees: “Expanding space is a very unhelpful concept.  Think of the universe in a Newtonian way – that is simply in terms of galaxies exploding away from each other” (Chown, New Scientist, 1993).  Yet Carrol and Ostlie in their textbook “An introduction to modern Astrophysics” (2007) say:
… a galaxy’s recessional velocity is not due to its motion through space; instead, the galaxy is being carried along with the surrounding space as the universe expands.   …  In the same manner, a galaxy’s cosmological redshift is produced by the expansion as the wavelength of the light emitted by the galaxy is stretched along with the space through which the light travels.
Indeed, true experts on the subject are not immune to their own mathematical misconceptions.  Harrison (1993) criticizes two of his colleagues for falling into errors of this type.  He says that Sandage remarks that Hubble’s redshift-distance law is valid for all distances, when he should say that the velocity-distance law is valid for all distances.  Harrison goes on to say that Peebles et al tell us that the velocity-distance law requires relativistic correction when the velocity approaches the velocity of light, when this remark is true for the redshift-distance law, but not the velocity-distance law.
Lastly, in their comments about the big bang, NASA in its “Universe 101” page (2010) tells us that in order to avoid misconceptions we should keep in mind the following points:  a) The Big Bang did not occur at a single point in space as an “Explosion” b) It is not logically necessary or sensible to consider the question of what the universe is expanding into.  c) It is beyond the realm of the Big Bang Model to say what gave rise to the Big Bang. 
If we examine the above studies, the one constant factor in all of the papers is expansion.  It forms the basis of the current cosmological paradigm, and is one of the most difficult concepts to explain. 


3.2   The basic misconceptions of the general public would seem to arise from the following areas:


The third problem is a thorny issue, but one which, at this moment, has no particular relevance as a misconception.  The definition of a misconception given above requires that the subject be in disagreement with current understanding of natural science.  Most cosmologists will simply derive the universe backwards to the Planck time and state that nothing can be said about anything previous to this.  Others, such as Penrose (2010) have put forward theories of cyclic universes, where a previous universe existed before the Big Bang.  Current multiverse theory would also indicate that the ex-nihilo premise is unnecessary in modern cosmology. 
The three main misconceptions of the general public are therefore all concerned with global expansion.  The first misconception is that the universe expands from one privileged point, the second that the Big Bang was an explosion, and the third that it is expanding into something else.  All of these points may be incorporated easily into the principal misconceptions for advanced students, enunciated below.


3.3   The principal areas which are leading to misconceptions on the part of more advanced students would seem to be:

The question of superluminal galaxies is a misconception which does not need any visual model to solve.  It is simply a question of definition.  Einstein’s theory of relativity does not allow travel at greater than the speed of light.  However, the intrinsic expansion of space cannot be defined as travel.  In essence this is the solution to the superluminal question.  It is still true that nothing could overtake a light beam.  A verbal analogy could be made by alluding to the fact that each of us is currently travelling at over two million kilometers per hour relative to the CMB, but a traffic policeman can still only fine us if we break the 40 kilometer per hour limit in towns. 



3.4   CONCLUSION   Misconceptions in cosmology arise mainly from the concept of expansion.  On one level students are confused about where the universe expands from, where it expands to, and why it expands at all.  On another level students are uncertain about how it expands, how much of it expands and why we observe a redshift.
The main areas of misconceptions to be treated in this paper are therefore global expansion and the related concepts of redshift and whether or not things expand locally.  According to Zirbel’s definition of misconception cited above, current understanding of these concepts must be examined in order to determine parameters necessary in any visual model which claims to be of aid to the student in his understanding of these topics. 
A paradigm is based on majority opinion in any one scientific area, so we should not expect to find absolute agreement in any bibliographic study.  Cosmology has significantly less than absolute agreement on the subject of expansion, but many of the difficulties are due to different mathematical models used in the calculations, and the different status of the considered observers. 





“Once you can accept the universe as matter that is expanding into nothing that is something, wearing stripes with plaid comes easy”.  Albert Einstein.


4.1   INTERPRETATION OF GLOBAL EXPANSION   Global expansion predates the Big Bang.  In fact expansion was the reason the Big Bang theory was formulated.  It is therefore one of the most significant ideas in cosmology, and one which is essential to understanding the subject. 
Global expansion was postulated by Slipher and Hubble as early as 1914, at the beginning of the Great War, to explain observational evidence.  Slipher had studied the wavelength changes of 12 “nebulae” (galaxies), and had found that only one of them (Andromeda) was blueshifted.  The rest were all redshifted.  Slipher was able to continue with his studies, and by 1925 had found that the majority of 40 nebulae he had examined were redshifted.  In 1925, Hubble discovered Cepheid variable stars in M31, and therefore managed to establish, basing his calculations on Henrietta Leavitt’s work, that M31 was an external galaxy.  He then went on to combine his distance measurements with Slipher’s redshift results, and was able to deduce the Hubble relation in 1929.  Hubble found that, in general, the further away a galaxy, the more redshifted it became.  The redshift is proportional to the distance of the galaxy from us.  Hubble (1929) equated redshift with radial velocity, and formulated the velocity-distance law, rather than the redshift-distance law, which was the strict observational result.
At the same time the idea of expansion was being developed theoretically by Einstein and de Sitter.  In 1917 Willem de Sitter had found a solution to Einstein’s equations which described an empty universe containing no matter.  This predicted a radial velocity that increased with the distance from the light’s origin.  As Eddington says, “This was the first hint of an expanding universe.” (1933). Although students may question the validity of considering an empty universe, at that time the only two options were de Sitter’s empty universe (with no matter, but expanding), and Einstein’s static universe (with matter, but no movement).  In fact, de Sitter’s universe is still considered useful today as the end product of accelerated expansion.  In its old age the universe will increasingly tend to a de Sitter universe as the average density of matter approaches zero. 
In 1930, Lemaître sent Eddington a copy of his paper postulating an expanding universe (Neuenschwander 2009).  At the beginning of 1930 Eddington had been worried why there should be only the two solutions of the Einstein field equations, so was receptive to Lemaître’s solution.  By the end of 1930 the Lemaître model of the expanding universe was accepted.  Lemaître had independently rediscovered Friedmann’s 1922 solution.  Robertson at that time also independently rediscovered the same solution and went on to develop the metric with Arthur Walker.  Eddington (1933) explains lucidly that the intermediate solutions of Friedmann and Lemaître are expanding universes, as at one end we have Einstein’s universe with no motion and therefore in equilibrium, then as we proceed along the series, we have model universes showing more and more rapid expansion until we reach de Sitter’s universe at the other end of the series.
Lemaître, in 1931, suggested that the evident expansion in forward time meant that at some point in the past all matter had been concentrated at a point.  He baptized this model the “primeval atom” but it was later misnamed the “Big Bang”.  [Harrison says: “From a purist point of view one cannot but deplore the expression big bang” (Harrison 1993)].  The model was accepted at a special session of the British Association for the Advancement of Science on 31st of October of the same year (Gale 2007).  This was the start of modern cosmology, and marked the approval internationally of global expansion.   
Despite one or two areas of intense research where anomalies have been found, much observational data has been obtained to substantiate the Lambda Cold Dark Matter (hereafter ΛCDM) model which is currently considered the standard of cosmology.  Because it includes the Friedman ̶ Lemaître ̶ Robertson ̶ Walker metric (hereafter reduced to FLRW), expansion is a very important part of this model.  The linearity of Hubble’s law, the concordance of the CMB, the abundance of light elements, the large scale distribution and apparent evolution of galaxies are all in satisfactory agreement with this model. 
However, the epistemological question of what expansion is has created great confusion.  Textbooks speak about “the stretching of space”, or “the stretching of light”, but some speak also about the velocity of recession of the galaxies rather than the expansion of space.  Davis and Lineweaver (2003) find that the question is further complicated by the attempt on the part of teachers to simplify arguments for the student using inexact terminology and incorrect logic.
Historically there has always been great disagreement about the physical cause of Hubble’s law.  Many theories have been put forward to explain the redshift which is observed, apart from radial velocity or the stretching of space.  Einstein’s book “Relativity, The Special and the General Theory” (1916/1961) indicated that light is redshifted as it escapes from gravitational wells, but this effect cannot wholly explain the cosmological redshift seen since the large redshifts documented would have collapsed the source into a singularity had it been solely due to gravitational redshift – and we would therefore be unable to see it at all.  According to Barrow and Tipler in “The Anthropic Cosmological Principle” (1986) Jeans put forward an ingenious theory that the universe was not expanding, rather everything in it was contracting – we are all getting smaller!  Barrow and Tipler say that Steward, Buc and Wold originally proposed the tired light theory, in which redshift is caused by light stretching due to fatigue.  There were also other suggestions, such as a decrease over time of Planck’s constant, and Dirac’s large number hypothesis.  Milne’s much debated theory did not even contain the referent to curved space, or expanding space, because neither of these was operationalizable.  According to the Stanford Encyclopedia of Philosophy (2007), Milne thought that since expanding or curved space could not be observed, it was Euclidean, and that was the end of the matter.
Of all these theories, only three remain consensually valid options.  Gravitational redshift is known to occur, and cannot be entirely ignored.  Doppler redshift will show us the radial part of all peculiar velocity, and is arguably the reason for cosmological redshift in empty universes where the theory of relativity applied is the special, rather than the general.  Metric expansion of space is the third and generally preferred option.  There has been long and hard discussion about the correct interpretation of expansion in recent and not so recent papers, and some of the areas of concern are documented below.
In “Conceptual Problems of the Standard Cosmological Model” Baryshev (2005) defines the space expansion paradigm as being that which gives a proper metric distance of a body with fixed commoving coordinate  as, where is the scale factor at time (t).  The use of such a simple formula in the ΛCDM model really equates expansion to the stretching of space, as the “scale factor” implies, and not as a Doppler shift which many text books mention, to the confusion of students.
Some cosmologists, however, such as Whiting (2004) and Peacock (2008) still suggest that expanding space fails to adequately explain the motion of test particles and should be abandoned.  Whiting says “In fact it seems rather metaphysical to argue whether (on the one hand) two points are actually moving apart, or (on the other) the space between them itself is growing”.   Peacock points out that the idea of “expanding space” is a fallacy, and that the common interpretation of the FLRW metric of saying that galaxies separate “because the space between them expands” leads to confusion. 
Chodorowski, in his paper “Is space really expanding?” (2006) finds that in an empty model the cosmological redshift is a result of the real motion of the source, that is, a Doppler shift.  He points out that there is therefore at least one Friedman model in which expansion of space, in detachment from matter, is an illusion.  He concludes that there is neither absolute space, nor expanding space.  Grøn and Elgarøy (2006) dispute this conclusion in their paper, saying that it is not a correct interpretation of the cosmological redshift for arbitrary universe models.
Francis et al (2007) consider that the concept of expanding space has often been articulated poorly and formulated in contradictory ways.  They show that a consistent description of cosmological dynamics emerges from the idea that the expansion of space is neither more nor less than the increase over time of the distance between observers at rest with respect to the cosmic fluid. 
Clavering (2005) states that in FLRW spacetime models we may construct a commoving frame in which the spacetime of general relativity is treated as expanding whilst matter is, on average, at rest. 
Finally Chodorowski, in a recent paper (2011), comments that the standard interpretation of the cosmological redshift as an effect of the expansion of the universe is obviously correct since:

(Equation 1)                                                                          Equation 1                                                                      

and in all Friedmann cosmological models distances between galaxies grow by the same scale factor.  (In the previous formula  represents the redshift,  the scale factor at time of observation, and the scale factor at time of emission.)
Perhaps Lee has the most useful comment when he says (2004)  “Next time you hear of something strange going on in cosmology remember to think ‘Is this just because of the choice of coordinate system?’” 
Thorne (1995) tells us that the question of which viewpoint (in this case the related concept of curved or flat spacetime) tells the “real truth” is irrelevant for experiments; it is a matter for philosophers to debate, not physicists.  He adds that physicists can and do use the two viewpoints interchangeably when trying to deduce the predictions of general relativity.  He goes on to explain that in relativity there are two accepted paradigms, both with their laws, pictures, and exemplars.  He says that since the laws that underlie the two paradigms are mathematically equivalent, the predictions will be the same.  Thus, physicists become free to use the one that best suits any given situation.  He finishes:  “This freedom carries power.  That is why physicists were not content with Einstein’s curved spacetime paradigm, and have developed the flat spacetime paradigm as a supplement to it.”




Redshift. (z) is defined as:

(Equation 2)   equation 2

which is  the rate of change of the wavelength emitted (λe) and observed (λo).  The paper by Harrison (1993) brings out one of the main problems in understanding redshift.  He asserts that cosmologists generally fail to distinguish between the redshift-distance law proposed by Hubble – a law based on experimental evidence, and the velocity-distance law established later on theoretical grounds.  Both laws are sometimes referred to as Hubble’s Law.  But there is no general proof which exists demonstrating that the two laws are equivalent, and indeed, many astronomers have viewed with reservation the velocity interpretation of extragalactic redshifts.  He concludes that in modern cosmology the universe does not expand in space, but consists of expanding space.  According to Harrison, this picture leads naturally to a distinction between the redshift-distance and the velocity-distance laws.  “In all expanding homogenous and isotropic cosmological models, the linear velocity-distance law is the fundamental relation, valid for all distance, and the linear redshift-distance law is only an approximate relation, valid for small redshifts and distances that are small compared with the Hubble distance”.  He also tells us sternly that expansion redshifts and Doppler redshifts are physically distinct except in Milne’s theory of kinematic relativity which rejects general relativity and the expanding space paradigm.
As Hubble himself commented: 
… the observations as they stand lead to the anomaly of a closed universe, curiously small and dense, and, it may be added, suspiciously young.  On the other hand, if redshifts are not primarily velocity shifts, these characteristics vanish and the observable region appears as a small, homogenous, but insignificant sample of a universe that may extend indefinitely both in space and time.  (1937) 
In the paper “Interpretation of the cosmological metric”, by  Cook and Burns (2008), the authors show that the classically held view (that the FLRW metric of general relativity carries implicitly the assumption that  the cosmological redshift is not a Doppler shift, but is due to a stretching of photon wavelength during propagation in an expanding universe) is not necessarily true.
The standard argument that the cosmic redshift is not a Doppler shift uses the fact that the cosmological redshift for the FLRW metric has the form given in equation 1, which tells us that the redshift depends only upon the scale factor (a) at the time of emission and the time  of reception.  The expansion could both be zero at the time of emission and at the time of reception yet there would still be a cosmological redshift  if there has been some expansion in between these two times.  This implies that such a redshift is not Doppler, and so is usually interpreted as the stretching of the wavelength of light during propagation when the universe is expanding.
Cook and Burns discover that results vary depending on the frame used to develop mathematical models.  If an expanding frame is used then the cause of redshift is wavelength stretching, whilst if a rigid frame is used the cause of redshift is the Doppler effect.  They consider that this, and other, paradoxes can be resolved by discarding the empty universe which is used for both these calculations, and considering a more realistic universe containing both matter and radiation.  Their final conclusions are that there are no true paradoxes because general relativity is a consistent theory.  The different interpretations arise from the different reference frames.  They suggest that the only danger lies in believing that the interpretations in a particular reference frame, such as the conventional interpretations of the FLRW metric, are invariant properties, which they are not. 
In “The kinematic origin of the cosmological redshift”, by Bunn and Hogg (2009) the validity of statements such as “The observed redshift of distant galaxies is not a Doppler shift but is due to the stretching of space” is examined.  Bunn and Hogg feel that the stretching rubber sheet analogy used to introduce many students to cosmology can be compared to the ether in that although it is intuitively appealing it makes no correct testable predictions, and some incorrect ones.  They think that it has no rightful place in the theory.  Bunn and Hogg claim that the most accurate method of labeling cosmic redshift is as Doppler redshift.  Both of these papers suggest that either definition of redshift gives equally correct and physically equivalent descriptions of this cosmic phenomenon. 
There is a further complication of redshift.  Gravitational forces can also create redshift, and these forces must be considered.  Peacock (2008) concludes that the gravitational term is quadratic and so has to be considered when going beyond first-order terms in the Doppler shift, saying “To second order, it is exactly correct to think of the cosmological redshift as a combination of Doppler and gravitational redshifts”.  Grøn and Elgarøy (2006) agree, showing that the cosmic redshift can be separated into a Doppler shift and a gravitational frequency shift if the object is close to the observer cosmically speaking.  They also tell us that this result was first derived by Bondi.  Peacock warns against thinking of high redshift objects as receding at 95% the speed of light, since a non-zero mass density must cause gravitational frequency shifts, which should not be ignored.
In “The Expansion of Space: Free Particle Motion and the Cosmological Redshift” by Whiting (2004) the author explores the meaning of expansion, and states that the cosmological redshift is not an “expansion effect”.  In special cases it can be separated into a kinematic part and a static part, but generally it ought to be considered to be the effect on light of propagation through curved space time.  Whiting concludes, after a long journey through general relativity and Newtonian dynamics, that “it is not at all certain what “the expansion of space” means!”  He derives cosmological redshift as a velocity effect, when observed locally, but warns that General Relativity can only be ignored in an empty, flat universe.  In the case of a spatially flat FLRW universe with no cosmological constant, the dynamic equations can be broken down in this one case into a factor of velocity alone and one of static gravitation alone.  He then finds a more general case using the fact that light propagates along null geodesics, in a space whose curvature enters implicitly through the function a, the scale factor.  Expansion, he finds, of a set of commoving observers, or of “space” itself, is another effect of curvature, not the cause.
Carroll and Ostlie (2007) say that the cosmological redshift is not related to recessional velocity by the Doppler shift equations, which do not include the effects of the expanding curved spacetime of our universe.  They go on to say, however, that astronomers frequently use the Doppler equation

(Equation 3)                        doppler equation                                                                                                                                                                        
(where  is the radial velocity), to translate the measured redshift into the radial velocity a galaxy would have if it had a peculiar velocity moving through space instead of a recessional velocity moving along with space. 
In their paper “Expanding Space: The Root of all evil?”  Francis et al, (2007) comment that the use of a variation of the balloon analogy to visualize cosmological redshift is often used.  A wave is sketched on a balloon and as it is blown up the wavelength is seen to increase as the sketch is stretched along with the expansion of the underlying space.  They say that this is generally uncontroversial, but warn that care should be taken in ensuring that the analogy does not mislead.  They claim that the key should be not that the redshift is a gradual process caused by the stretching of the space a photon travels through, but rather that the redshift is caused by the photon being observed in a different frame to that which it is emitted. 
Bunn and Hogg (2009) find that the common belief that the cosmological redshift can only be explained in terms of the stretching of space is based on conflating the properties of a specific coordinate system with properties of space itself, and are of the opinion that this confusion is precisely the opposite of the correct frame of mind in which to understand relativity.  They suggest considering a “Doppler” family, where the infinitesimal frequency shift from each observer to the next is a Doppler shift.  On the other hand, in a “gravitational” family each member is a relative rest to their neighbour at the moment the photon passes by, so that there are no Doppler shifts.  The gravitational family all feel like they are in local gravitational fields, interpreting the shift in the photon’s frequency relative to their neighbour as a gravitational shift.
They are surprised that, since in cosmology, the Doppler family of free fall observers is almost always used, it is a natural choice to use, and it is natural to interpret the cosmological redshift as a Doppler shift.  They go on to explain, however, that there is no fact of the matter about the interpretation of cosmological redshift, that what one concludes depends on one’s coordinate system or method of calculation.  Bunn and Hogg find that in their opinion the natural way to interpret redshift is kinematic, but warn us against confusion between coordinate independent and coordinate dependent statements.
Cook and Burns (2008) agree with this conclusion.  “Observations by a different set of observers (a different reference frame or different spacetime coordinates) can lead to very different but equally correct and physically equivalent descriptions of cosmic phenomena.
Kiang (2003) in “Time, Distance, Velocity, Redshift: a personal guided tour” warns against considering that recession velocity and peculiar velocity are the same thing.  The recession velocity should not be regarded as the property of a source; rather, it should be considered as the property of the point of space in question, whether that point happens to be occupied by a source, a passing photon, or nothing at all.  The peculiar velocity, on the other hand, must have a reference to a material object including photons.
Recently the debate has become even more intense.  In “Eppur si espande” Abramowicz, Bajtlik, Lasota and Moudens (2007) claim to show that, independent of the definition, in a non-expanding space the radar and redshift distances should be the same.  Since they are not, one must conclude that space is expanding.  They call their paper “Eppur si espande”, an adaptation of the quote “Eppur si muove” from Galileo Galilei as he was being tried for heresy against the church for his beliefs.  Their use of this adaptation may be taken as an example of the heated debate which the reason for expansion has caused.
Lewis, Francis, Barnes, Kwan and Berian (2008) examine this point of view and conclude that radar ranging does not require the expansion of space to be a physical phenomenon.  They also state:
… It is a fool’s errand to search for the truth of the existence of expanding space … Any attempts to obtain observations to address the question of whether galaxies are moving through static space or are carried away by the expansion of space are doomed to failure.
Abramowicz et al replied to criticism (Abramowicz, Bajtlik, Lasota and Moudens, 2008) saying categorically “Although the concept of space is not well defined in Einstein’s relativity, one can prove, as we did in “Eppur si espande”, that the statement that the cosmological redshift may be described as a Doppler effect in non-expanding space is false.  Eppur si espande.”  Nevertheless, Francis et al (2007) are of the opinion that this counter-claim is spurious and go on to give some advice.  They say that addressing the issue is important, because the phrase “expansion of space” is in such a wide use that it is no exaggeration to label it the most prominent feature of Big Bang cosmologies.  They finally suggest that the description of the cosmic expansion should be considered a teaching and conceptual aid, rather than a physical theory with an attendant clutch of physical predictions.
Eddington (1933) comments wryly that  “… the more recondite treatment in which the phenomenon (expansion) is presented as a uniform expansion of spherical space, seems to have confused not only the casual reader but some of the experts.”, and it would seem that this comment made nearly a century ago is still valid in today’s cosmology marketplace.


4.3   CONCLUSION   The conclusion to be drawn from the above bibliographical study is that there is no absolute consensus at this time amongst the experts, and that there have always been dissenting voices about the physical reason for global expansion.  However, the current paradigm is that of the FLRW scale factor, which is generally taken to mean that space is expanding and that cosmological redshift cannot be regarded as a Doppler shift caused by matter traveling through an unmoving background, although many consider this picture equivalent and useful in some circumstances.
The neatest way to explain expansion is to separate redshift into cosmological and local.  Cosmological redshift occurs on large scales, between objects which are not gravitationally bound.  Local redshift occurs inside bound systems, which are not expanding “with the Hubble flow”.  Local redshift must therefore be purely Doppler or gravitational, whilst cosmological redshift may be considered as metric expansion. 




Mrs. Felix:  Why don’t you do your homework?”
Allen Felix:  “The Universe is expanding.  Everything will fall apart, and we’ll all die.  What’s the point?”
Mrs. Felix:  “We live in Brooklyn.  Brooklyn is not expanding!  Go do your homework!”
(From the film Annie Hall, by Woody Allen)


5.1   INTERPRETATION OF LOCAL EXPANSION   Local expansion, or lack of it, has been known since Hubble’s work.  Willem de Sitter (1931) asked himself if galaxies expand with the universe as early as 1931.  He concluded that they don’t.  Fourteen years later, Einstein and Strauss (1945) were still pondering the influence of expansion on individual stars, coming to the conclusion that “The investigation below yields that the expansion of space has no influence on the structure of the field surrounding an individual star, that it is a static field – if only for an exactly delimited neighbourhood.”  Since then the question of what does and doesn’t expand has been taken up, examined, and dropped, by many researchers.  There is still no definitive agreement about the matter.
Generally, cosmologists believe that gravitationally bound systems will not expand, since they have reached a balance between the various forces acting upon them, and have therefore “overcome” the tendency to expand.  Gravitationally bound systems include you, me, the earth, the sun, our galaxy, and groups of galaxies up to clusters.  These do not, according to the current cosmological paradigm, expand.  Francis et al (2007) consider that there are unambiguous conditions that determine whether an object will be stretched by the expansion of space.  Objects will not expand with the universe when there are sufficient internal forces to maintain the dimensions of the object.
Local expansion is taken to be anything which is gravitationally bound.  In our neighbourhood of the universe this would be the Local Cluster, which would not be expected to join the Hubble flow.  The local Virgo supercluster, of which we are peripheral members, would be expected to join the Hubble flow at this time as it is not gravitationally bound, according to current consensus. 
Sandage, Reindl and Tamman (2010) in a paper published shortly before Sandage’s death on 13th November 2010, determined experimentally the linearity of the cosmic expansion field and found that “The expansion of space is as linear as can be measured after allowance is made for a Virgocentric flow model and for the CMB motion of the Local Supercluster.  Over the range of 200 to 30,000 km/s the value of Ho (the hubble constant) does not change systematically by more than +/- 2.3%.”  They also conclude in their study, interestingly, that the cosmic value of Ho can be found quite locally if sufficient calibrators are available to compensate the locally important peculiar velocities.  Another finding was that “the majority of galaxies within about 3500 km/s share the coherent bulk motion of the Local Supercluster, whereas the galaxies beyond this limit are in rest with respect to the inertial frame of the CMB”.  Here we see that researchers customarily use kilometers per second as a measure of distance, which does not help to clarify the situation as to whether global expansion is a velocity or not.
Araya-Melo, Reisenegger, Meza, van de Weygaert, Dünner and Quintana (2009) find that:
… In the present Universe superclusters are still expanding with the Hubble flow, although at a slightly decelerated rate, or have just started contracting.  Because these structures have not yet fully formed virialized and clearly separated from each other, it is difficult to identify them unambiguously.
However, some experts think that larger structures such as some superclusters are already gravitationally bound.  For example, the Shapley supercluster is identified by Reisenegger, Quintana, Proust and Slezak (2002) as the largest bound structure in the local Universe.  In the same article, Reisenegger et al say that:
Eventually, regions of large enough overdensity can stop their expansion altogether and start recontracting.  Their collapse is then followed by a process of relaxation or “virialization”, after which the resulting object is in an approximate equilibrium state, in which its structure is only occasionally perturbed by merging with other objects.
Not all movement is due to expansion.  There are also peculiar velocities within groups and clusters, within superclusters and filaments, and anomalies in the linearity of the Hubble diagram.  In recent years there has been much speculation about the existence of voids in the large scale structure, and new data has been brought into the general picture.  For example, van de Weygaert, Platen, Tigrak, Hidding, van der Hulst, Aragón-Calvo, Stanonik and van Gorkom (2009) find that underdense regions expand faster than the Hubble flow, and thus expand with respect to the background universe, which means that as voids expand matter is squeezed in between them, and sheets and filaments form the void boundaries.  van de Weygaert and Platen also say (2009) that “voids mark the transition scale at which density perturbations have decoupled from the Hubble flow and contracted into recognizable structural features.”  They find that matter streams out of the void, which means that isolated voids asymptotically evolve towards an underdensity of minus one, pure emptiness.
Other recent studies have examined the peculiar velocities of galaxies in our local universe, and non-uniformities in global expansion, especially in and near to voids. Tully, Shaya, Karchentsev, Courtois, Kocevski, Rizzi and Peel (2007), for example, examined the motion of the Local Sheet and found that it “is participating in the cosmic expansion (though probably somewhat retarded) but simultaneously moving in bulk toward the Vigo Cluster and away from the Local Void.”  They found that the inferred CMB dipole anisotropy of 631 km/s can be decomposed into three almost orthogonal components.  The local component of  is 185 km/sec towards the Virgo Cluster, the second component of 259 km/s is away from the Local void, leaving a third component of 455 km/s towards the Centaurus Cluster.
In the paper “The Influence of the Cosmological Expansion on Local Systems”, by Cooperstock, Faraoni, and Vollick, (1998) the question of whether there is a cutoff at which systems do not partake of the expansion was examined in detail.  They concluded that one would be hard put to justify a particular scale for the onset of expansion, and that it is most reasonable to assume that the expansion does indeed proceed at all scales.  They show that even if the expansion does actually occur at all scales the effects of the cosmological expansion on smaller spatial and temporal scales would be undetectable in general in the foreseeable future, and hence one could just as comfortably hold the view that the expansion occurs strictly on the cosmological scale.  They conclude that it is reasonable to assume that the expansion of the universe affects all scales, but the magnitude of the effect is essentially negligible for local systems, even at the scale of galactic clusters.
Peacock, (2008) comments that the answer to “Does the expansion of the universe cause the Earth and Moon to separate?” is NOT the commonly encountered “it would do, if they weren’t held together by gravity” and in his textbook “Cosmological Physics”, (1999) he also explains that an inability to see that expansion is locally just kinematical lies at the root of one of the worst misconceptions about the Big Bang, and suggests that in the common elementary demonstration of the expansion by means of inflating a balloon, galaxies should be represented by glued-on coins, and not ink drawings – which would expand with the universe.
Francis et al (2007) point out that the student is to be pitied as hybrid explanations treat gravity globally in general relativistic terms and locally as Newtonian, or at best a four force added onto the FLRW metric.  They suggest that a clearer explanation would simply be that on the scales of galaxies the cosmological principle does not hold, even approximately, and the FLRW metric is not valid. 
There is therefore no expansion for the galaxy to overcome, since the metric of the local universe has already been altered by the presence of the mass of the galaxy.  Francis et al add that expansion of space is global but not universal, since the FLRW metric is only a largescale approximation.
Carrol and Ostlie (2007) warn us about assuming that, for example, the orbits of the planets about the sun have been expanding.  They say that gravitationally bound systems do not participate in the universal expansion, and that there is also no compelling evidence that the constants that govern the fundamental laws of physics (such as Newton’s gravitational constant, G) were once different from their present values.  Thus the sizes of atoms, planetary systems, and galaxies have not changed because of the expansion of space.
Eddington, in his small yet still amazingly pertinent book of 1933 had this to say of local expansion:
Within a galaxy the average world-curvature is some thousands of times greater than Lemaìtre’s average for the universe as a whole, and his formulae are inapplicable.  The result is that only the intergalactic distances expand.  The galaxies themselves are unaffected and all lesser systems – star clusters, stars, human observers and their apparatus, atoms – are entirely free from expansion. 


5.2   INTERPRETATIONS OF THE TETHERED GALAXY PROBLEM   The Tethered galaxy thought experiment was formulated by a retired chemistry professor, Charles Leffert, who posed the problem to Edward R Harrison, according to an article published by Tim Folger in Discover magazine (1995), “The Ultimate Free Lunch”.  Leffert’s original question was “What happens to two bodies – planets, stars, galaxies, or what have you – in an expanding universe if you join them with a string?”  In “Mining Energy in an expanding Universe” Harrison (1995) found that this problem produced a conflict between global expansion and energy conservation.  In his paper, he concludes that in principle energy can be extracted from the expansion of the universe, and also (tentatively) that in an expanding, spatially unbounded, homogenous universe energy is not conserved.  Regarding this conclusion Martin Rees, Astronomer Royal, commented (Discover magazine, 1995) that the energy gained by the hypothetical winch, comes at the expense of expansion, and that there is therefore no paradox.
From its beginnings as a thought experiment examining conservation of energy on a global level, this problem has in recent years transformed to epitomize the discussion about local expansion, about whether or not something released into the Hubble flow will experience a “force” or not.
A paper by Davis, Lineweaver and Webb (2003) examines the movement of a galaxy which is released into the Hubble Stream from a standstill.  Davis et al found that such a galaxy, instead of joining the Hubble flow immediately and expanding away from the local group, could fall towards the local group, pass through it, and then join the Hubble flow on the other side of the local group in certain circumstances.  Davis et al say “we have shown that an object with a peculiar velocity does rejoin the Hubble flow in eternally expanding universes, but does not feel any force causing it to rejoin the Hubble flow.”  They also derive a result showing that approaching galaxies can be redshifted and receding galaxies can be blueshifted, and say that this is an interesting illustration of the fact that cosmological redshifts are not Doppler shifts.
Peacock (2008) considers a particle (a tethered galaxy) and shows that in an Einstein-de Sitter model, a particle initially at rest with respect to the origin falls towards the origin, passes through it, and asymptotically regains its initial commoving radius on the opposite side of the sky.  Peacock says:
In no sense can “expanding space” be said to have operated … this analysis demonstrates that there is no local effect on particle dynamics from the global expansion of the universe: the tendency to separate is a kinematic initial condition, and once this is removed, all memory of the expansion is lost.
Barnes et al (2006) themselves use a visual model to explain the meaning of an initial kinematic condition.  They suggest that you should imagine yourself and a friend at rest on a large rubber sheet.  (in the dark, to avoid the problem of directly observing spacetime).  Suppose you both observe a glowing ball moving away from you.  “The rubber sheet is being stretched”, you say.  “No it’s not.” replies your friend, “the sheet is still and the ball is rolling away.”  Together, you come up with an ingenious way of finding out who is right.  You take another glowing ball, and drop it onto the sheet a certain distance away.  If the sheet is expanding then we expect it to carry the first ball away; if the sheet is still then the recession of the first ball was due to a kinematical initial condition.  Once this is removed, so is the recession.  They consider this analogy to be appropriate, but they later comment that they believe that some misconceptions amongst cosmologists may have been fostered precisely by this rubber sheet analogy, which they claim here is flawed. 
Whiting (2004) also considers whether galaxies in the Hubble flow move apart because they have a certain initial velocity and an inertia, or because they are carried along in expanding space.  He concludes that, contrary to the expectation of many, a particle removed from the Hubble flow never returns to it.  He believes that there is a qualitative difference to the idea that expansion of space acts like a Newtonian force pushing the galaxies apart.  Whiting finds errors in the Davis, Lineweaver and Webb (2003) investigations, and disagrees with their conclusion that free particles do eventually rejoin the Hubble flow on the far side of the “center”.  He believes that while it is true that in the flat, cosmological-constant situation a nonrelativistic particle disturbed from the Hubble flow in the limit of late times returns to it, he says that such a special case can hardly be taken to show anything general about the nature of spacetime. 
Barnes et al (2006) disagree with the conclusions drawn both by Peacock and by Davis et al about a tethered galaxy rejoining the Hubble flow.  They believe that there may be a misunderstanding of expanding space.  They concur with Peacock in that particles in the Hubble flow do not feel any force, as they are free-falling.  They warn that thinking about expanding space as a viscous force is incorrect. 
Bunn and Hogg (2009) affirm that the common belief that the solar system has a slight tendency to expand due to the Hubble expansion (although this may be thought negligible in practice) is incorrect.  They claim that the tendency to expand due to the stretching of space is nonexistent in the tethered galaxy problem, not merely negligible. 
Finally, Davis et al (2003) tell us that:
Although the galaxy does gradually join the Hubble flow, it does not necessarily recede from us.  In particular, in the currently favored cosmological model, which includes a cosmological constant, the galaxy recedes from the observer as it joins the Hubble flow, but in the previously favored cold dark matter model, the galaxy approaches, passes through the observer, and joins the Hubble flow on the opposite side of the sky.


5.3   CONCLUSION   There is much discussion amongst experts about the reason why local systems do not expand, but there is general agreement that they don’t.  Expansion can be overcome easily by gravity, in which case bound and virialized systems will no longer experience it.  The paradigm to use for teaching purposes is therefore that bound galaxies do not expand relative to each other
Until recently the paradigm has been that the largest bound systems are galaxy clusters, and that galaxy superclusters are not gravitationally bound, and therefore are internally expanding.  Thus space inside clusters of galaxies is not expanding, but space between clusters of galaxies is expanding.  This means that any one galaxy within a cluster will determine a redshift if any galaxy in another cluster is examined, despite some correction being necessary for peculiar velocities of both galaxies.  However galaxies within the same cluster will not be found to be receding from each other, except when redshift is due to normal Doppler shift.  In more recent papers, however there is a tendency to define superclusters as the largest bound structures in the universe, that the real “island universes” caused by accelerated expansion will be superclusters and not clusters [see for example Araya-Melo et al (2009) and Reisenegger et al 2002)]. In any case, the universe has not reached any definitive stage yet, so we may expect that some constructions larger than clusters will eventually virialize and drop out of the Hubble flow before acceleration of the expansion impedes any further gravitational bonding.
Expansion varies according to the matter gradient, with huge voids expanding more than the norm, and expelling the little matter they contain, and the filaments and walls expanding less than the norm, and accreting the matter expelled by the voids.  Expansion therefore is not a completely homogeneous phenomenon.  It is necessary to consider the universe at large scales, and to correct for these peculiar movements in order to confirm the very linear relation which characterizes Hubble’s Law.
As far as this study is concerned, it seems appropriate to take clusters as the boundary, while keeping an open mind about the future.  Certainly the Hubble flow should not be portrayed as happening to all individual galaxies.  Expansion does not occur on a local scale.  Comments in text books such as “all galaxies are receding from us and each other” should not be used.  Firstly, it is incorrect scientifically.  Andromeda and some other galaxies are approaching us, and galaxies in any cluster are also moving with only peculiar motion with respect to an observer in the same cluster.  Secondly the use of the word “receding” is defined by the dictionary as “moving back or backwards” and implies a velocity rather than metric expansion.  Most textbooks try to evade the question of local expansion by stating: “all distant galaxies are receding from each other”, or something similar.  This is also most ambiguous, as no definition of distant is given, and the student may well assume that the adjective distant is applicable to all galaxies.
The tethered galaxy thought experiment has enlivened discussion in recent years, and highlighted important aspects of global expansion.  There seems to be general agreement that consideration of a particle or galaxy dropped into the Hubble flow can show that expansion is an initial kinematic condition and should not be treated as a force acting on matter.  There is also intense speculation about the way a rest galaxy would join global expansion.  It appears that it would not be “swept up” automatically into expansion.  The fate of the galaxy in question may depend on the cosmological model used.  There is some agreement amongst experts that if expansion is decelerating the galaxy would approach the local cluster, and if expansion is accelerating the galaxy would move away from the local cluster.  In both cases it is thought to join the Hubble flow eventually, but on different sides of the cluster.