Hi, I am trying to resolve a few questions concerning the measurement of cosmological distance and the determination of Hubble’s constant (H=v/d), where (v) is the recessional velocity of the source and (d) is its distance. Various sources quote Hubble’s constant with a confidence in its accuracy of less than 2%: [tex] 70.1 \pm 1.3km/s/mpc = 2.28x10^{-18} \pm 4.23x10^{-20}m/s/m [/tex] http://en.wikipedia.org/wiki/Hubble's_law In practice, I am assuming that this figure is an average of a multitude of measurements associated with objects at various distances, which presumably confirm a value that is essentially constant with distance, but variable with time in an expanding universe? My next assumption is that the velocity has to be determined from its associated redshift (z), which is related to the source and observed wavelength of light received from a given object: [tex] z = \lambda_s - \lambda_o/\lambda_s [/tex] The key problem with this method is determining the source wavelength, which I understand is normally addressed by using a type of star called Cepheids. If the source wavelength is known, non-relativistic velocity is determined from the simple relationship [v=zc], while a relativistic velocity requires the more complex formula: [tex]z = (1+v/c)\gamma – 1 [/tex] However, what is the accuracy of this method? I have seen statements suggesting that for distant objects beyond the Milky Way, the relation is less clear, since the apparent magnitude is affected by spacetime curvature. So, is the Cepheid method the primary method or are other methods, e.g. angular diameter distance, preferred in other circumstances? Presumably, on the accuracy of (H), a cosmic horizon exists at d=c/H, where any source must be receding faster than the speed of light. Based on the excepted value of (H) and the speed of light [c], this would appear to correspond to a radial distance of 13.9 billion light-years. Does this horizon present a barrier to further observation and what is the furthest distance [d] of any accepted measurement associated with (H)? Finally, are there any sources showing (H) against time (t)? Would appreciate any technical clarifications on any of questions raised. Thanks
Great bunch of questions! I should shut up and let Wallace, Marinas, Hellfire, Russ W, Janus and others have a turn. But it's fun to reply where there is so much content. different atomic and molecular species have different wavelengths they absorb and emit light at. the spectrum from a star or galaxy is a long band similar to the rainbow stripe made by a prism and an astronomer can look at it and see PATTERNS of emission and absorption lines. He knows the exact source wavelengths for each line (a sodium, an iron, a hydrogen...) measured in the lab on earth. So he can see how much the patterns have been shifted. the light from a galaxy won't have such sharp lines in its spectrum as the light from an individual star. but galaxies are full of gas clouds which absorb definite wavelengths. so they still manage. there's fascinating technical detail which Wallace could supply and I can't, but that's the general idea Cepheids are a rung in the astronomical distance ladder the construction of the distance ladder is one of the more admirable accomplishments of mankind, there is probably a book about it. Ned Wright probably has a page or two. each lower rung is used to calibrate the rung above it. they start out with simple parallax (trigonometry, like a surveyor) but that only works out to a certain distance because the angles get too small so you use parallax to check and calibrate the next method moving clusters works a bit farther than simple parallax (a textbook would explain, it uses small clusters of a few stars, like the Pleiades, and doppler shift, and angular separation) then there is the H-R diagram which relates a normal star's color-temperature to its brightness. if you know the brightness you can tell the distance (a bright star that looks dim is far away) but obviously to get the H-R in the first place you need more basic distance methods like moving clusters. you need lower rungs to get up to higher ones Cepheids are variable stars whose brightness fluctuates over days and weeks and the rate of fluctuation is proportional to the average brightness. So if you have a way to initially tell the distance to some Cepheids (a lower rung on the ladder) you can figure out their brightness, and thereby determine the proportionality with the fluctuation rate. After that, whenever you see another Cepheid you can measure its fluctuation rate, and from that tell the brightness, and from that tell the distance (because a bright star that looks dim is far away). individual Cepheids can be seen in other galaxies, if the galaxy is not too far away. the relation between how often a Cepheid burps and how bright it is was discovered by a woman named Henrietta who is one of the heroines of astronomy there are a bunch of other handles they have on distance---often called "standard candles". they are things you can recognize which have a known standard brightness (empirically determined by using other distance scales when they occur nearby) astronomers as a group are obsessed with distance and they constantly use each of their distance method to check the others. they compare. they use one method to calibrate and check others. they do huge statistical studies where they count objects at various measured distances just to be sure that the model fits and the estimates are reasonable. any time one distance method gives an answer that is a few percent different from another method, they get excited and intensely debate the possible explanations. one of these disturbances occurred in 1998 where a couple of research teams were using a standard candle called Type 1A Supernovae. A kind of supernova which always explodes with approximately the same brightness and which blazes up and dies down over the course of a few weeks according to a fairly regular timetable. both teams found a discrepancy and their results agreed, we are still living thru the repercussions of that. it is why people talk about dark energy
Interesting questions! We can observe stuff that is beyond the Hubble distance. This is explained in pictures in the Lineweaver SciAm article---I have the link in my signature at the bottom of this post. Really good article. Astronomy teachers at Princeton (and probably other places) use it to help their students understand. It is difficult to see how light from some object receding at c (or greater) could ever reach us but it happens all the time. Much of the light we are now receiving was emitted by objects which were receding faster than c at the time they emitted the light. the reason is related to the change in H over time, which is what you also asked about! I think it shows you are thinking along constructive lines because when you ask two questions the answer to one of them is related to the other question. The reason light can get here from beyond current Hubble distance is because Hubble distance has been increasing for most of history, and is even still increasing (but more slowly than it did in early universe) the current estimate if I remember is that Hubble distance (now approaching 14 Gly) will eventually approach something around 16 Gly asymptotically. what happens is some light is emitted in our direction and it INITIALLY GETS SWEPT BACK and recedes from us (total speed is emitter's recession speed minus c, not as fast as the emitter is receding). it keeps trying to get to us and it keeps getting farther away. but then the Hubble radius itself expands out until it encompasses the light. So now the light is WITHIN the Hubble radius, where things are no longer receding with speed c or greater. Things where the light now is are receding slower than c. So now the light can begin to make some progress. And eventually it gets here! Because the Hubble radius c/H increased at such an extremely rapid rate in the early stages of expansion----by early I mean like when the CMB was emitted by 3000 kelvin gas at the moment space became transparent, say when expansion was only a few hundred thousand years old, not even a million---because in that early era the radius was extending so fast, we are actually now getting light emitted by matter which is now around 46 billion LY away. So the radius of our observable universe is actually around 46 Gly. Astronomers call this the particle horizon. It is how far the matter is that we can now observe and study by the waves that are now arriving from it. What I am telling you is a standard LCDM model number, 46 Gly, the particle horizon, the radius of what is directly observable by the light or other waves coming from it. The most dramatic example of this that I can think of is the CMB itself, you can find the details using one of the cosmology calculators like Ned Wright's (google ned wright) or Morgan's. the CMB was emitted from thin hot gas which at the time of emission was 40 million LY away from us---or the matter which eventually became us---and which was receding at that time at about 57c. Obviously the light would initially have been swept back at about 56c! But it kept coming in our direction, or trying to, and eventually the Hubble radius reached out to where it was, and it finally got here after some roughly roughly 13.7 or 13.9 billion years. Numbers only approximate. And the matter that emitted it is now about 45 Gly from here. So when we observe the CMB we are observing matter that is around 45 Gly away. But which WAS only 40 million lightyears away when it emitted the light. And that matter is about the most distant thing we can observe with presentday instruments. So it makes sense to say, as they do, that the radius of the observable is some 46 Gly. ================= the reason the Hubble radius increases is because it is c/H and because H(t) has been decreasing, rather sharply in early times and more gradually now. that comes from the Friedmann equation and the definition of H as a'/a where a(t) is the scalefactor. Those are technicalities. Acceleration just means that a'(t) is increasing. And even while a' is increasing gradually, the ratio a'/a can decrease----which in fact it is still doing. Using the Friedmann equation the asymptotic limit of H(t) can be estimated you just take the current value, say 71, and you multiply by the square root of whatever you think the dark energy fraction is, say 0.73. Since the Hubble radius is the reciprocal, you get the asymptotic value (what it will be in the far distant future) by taking the current value roughly 14 Gly and DIVIDING by the square root of whatever you think the dark energy fraction is, say 0.73. I'm relying on memory and not re-deriving it at the moment, so I could be wrong. One of the others might correct me on this. Morgan's calculator gives the values of the Hubble parameter H(t) at past epochs, when you put in a redshift. Remember to prime it by saying 0.27 for matter, 0.73 for Lambda, and 71 for Hubble. Ned Wright puts these numbers in as default values instead of making us do it, which is decent of him. This quantitative stuff is to some extent optional.
Sources 4 and 6 on the wikipedia page you link to would be the best places to start ... In particular, the Freedman et al. paper (source 6; one of the most heavily cited papers in modern astronomy) contains a very good overview of the methods they used (there are a lot more than just Cepheids!) plus a quick note on a couple of direct methods, that by-pass the distance ladder ...
Response to #2 Hi Marcus An even better bunch of answers. Many thanks for the informative comments and links, which although I have not fully read, as yet, look very useful. Even followed your link to 'Collected Poems, 1943-2004'. As a complete aside, have you read Byron’s `The Darkness’? Not everybody’s taste, but invokes powerful imaginary. http://englishhistory.net/byron/poems/darkness.html By way of explanation, I recognise that I am at the bottom of a long learning curve with respect to cosmology and at the moment I am just trying to build a framework of general understanding. In this context, your responses were very useful pointers into the underlying detail. As a footnote, I found my way to this forum by coming across your poll about black holes and the universe, which was the subject I was originally looking at, but then realised without understanding cosmology in more detail, I was just leaving myself open to speculation. So thanks again. https://www.physicsforums.com/showthread.php?t=202260 Presumably you are implying that we can identify the spectrum of given atomic and molecular elements by their relative position or pattern to each other. So having identified, say sodium, we know its source wavelength and have a measure of its redshift due to the recessional velocity of the galaxy in question. In principle this would appear to be an accurate method, but can the spectral shift be affected by other factors? However, I probably need to read some of the references before raising too many additional questions, e.g. http://en.wikipedia.org/wiki/Moving_cluster_method http://en.wikipedia.org/wiki/Hertzsprung-Russell_diagram http://en.wikipedia.org/wiki/Cepheid_variable http://en.wikipedia.org/wiki/Type_Ia_supernova I will submit a separate response to your other post #3
Response to #3 Hi Marcus, Again, I know I need to really do some more reading before raising any more questions, but your second response touched on a number of issues that I am still trying to get my head around, i.e. all the various ‘horizons’ in cosmology: Hubble horizon Cosmological horizon Particle horizon ---- Event horizon Collision horizon I mention the black hole event horizon only because it was a topic raised in your poll on the universe as a black hole. It relates to the correlation of the mass density of the universe and the Hubble radius appearing to correspond to the Schwarzschild radius. The collision horizon is not mainstream either, but relates to another `coincidence’ that the mass-density would create a horizon, whereby a photon beyond this point must collide with a mass particle on its way to a central point. It is based on certain assumptions about the cross-sectional area of a proton, but again comes out to approximate the current size of the visible universe. I need to think about what you are saying, but I was not sure about the implication with respect to relativity. When discussing gravitational redshift (GR), people often talk about the wavelength change, but do not always comment on why. One interpretation is that time dilation is the root cause. If so, would a source with an effective recessional velocity [v>c] be subject to time dilation (SR) and any E-M wave emitted, whilst travelling at [c], be infinitely redshifted at source? I have seen one reference that stated that the particle horizon is approximate 3 times the Hubble radius, but have not seen any derivation of its definition. Do you know of any source? I have attached a graph showing (H) against time (t) based on Friedmann’s equation, is there an implication that the compound rate of expansion leading to a definition of physical horizon that differs from the cosmological horizon? Agreed, but presumably all this is really saying is that r=ct where (t) is the age of the universe or the time light has been travelling at [c]? I guess what I was implying in my previous question was that at some point the physical expansion exceeds the speed of light, which then allows an independent particle horizon to be calculated? Last comment because I am probably getting out of my depth here. I was reading about the idea of acoustic peaks the other day and it also contained a reference to .57c. Is there any link with the figure you quote above and is this whole idea accepted within the standard model (LCDM)? http://cmb.as.arizona.edu/~eisenste/acousticpeak/acoustic_physics.html Again, many thanks for your help as I appreciate that some of my questions may seem a ‘bit off the wall’. P.S. I have raised some questions about Friedmann’s equation in a separate thread: https://www.physicsforums.com/showthread.php?t=245886
Response to #4 Nereid, thanks for highlighting the 2 two relevant references within the link, in particular, the Freedmann paper: http://arxiv.org/PS_cache/astro-ph/pdf/0012/0012376v1.pdf
Hubble Space Telescope Results In post #4. Nereid was kind enough to point me in the direction of the following report: Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant: http://arxiv.org/PS_cache/astro-ph/pdf/0012/0012376v1.pdf I have only quickly scanned the 81-page report, but wanted to see if I could confirm the following very, very brief summation of years of work: 1) Cepheid method good to about 30Mpc or 100 million light-years, i.e. < 1% of the Hubble radius. 2) Type 1A Supernovae good to about 400Mpc or 1.3 billion light-years, i.e. 10% of Hubble radius Is this a fair rule-of-thumb assessment of the current state-of-play ?
Re: Hubble Space Telescope Results (bold added) As you have, no doubt, already discovered, when a 'distance' becomes a significant fraction of 14 billion light-years, the relationship between z (redshift), age (of the universe at that z), light-travel time, and much more becomes (cosmological) model-dependent. For example, Ned Wright's excellent online calculator gives a light-travel time of ~10 billion years for an object with an observed z of ~2 (and an age of the universe at this redshift of ~3.5 billion years) ... using 'concordance' parameter values; plug in different parameters and you'll get a different set of values. While most of the distant Ia SNe observed to date have redshifts up to ~1, there have certainly been some with redshifts up to ~2 (just not enough, yet, to robustly constrain cosmological models better than the hundreds observed with lower redshifts). This will very likely change, and quite dramatically, over the next decade or so, as new telescopes start working. Also, I think it's important to note that there are several methods for estimating extra-galactic distances, all of which give results that are consistent with Cepheid and Ia SNe distance estimates, and several of which have great potential for considerably improved reliability and accuracy (for example, check out the water maser cosmology project).
Light emitted from those huge distances,... isn't the frequency of that light extremely low? Can you still call it light?
Re: Response to #2 (bold added) First, there are not many processes that can cause a shift in the position (central wavelength) of a spectral line ... at least, not in environments of interest to those who study the sky (astronomers, etc). Apart from relative motion and the cosmological redshift, there is only a gravitational redshift, which was predicted by Einstein's GR and observed in the lab by Pound and Rebka. Of course, if all you see is a point source, then in principle a range of other mechanisms for creating shifts could be in play; however, in astronomy all (?) extra-galactic point sources (as classes) have associated extended sources whose redshifts are essentially the same. For example, a GRB is a point source, but after it fades its 'parent galaxy' is often seen, and where that galaxy's redshift can be measured, it is similar to that of the GRB (so far anyway). Ditto a quasar or QSO. Second, there are many processes that can result in a broadened spectral line (and some broadening may be asymmetric, so the central wavelength seems to shift a little), but this isn't what you're asking about, is it? Third, the relevant theory of atoms (and more), which has been tested to the highest level of precision of any theory in physics*, allows the possibility of systematic spectral shifts, via changes in the fine structure constant, either by time or space (i.e. this constant is different in different parts of the universe, and/or at different times). It will come as no surprise to learn that firm discovery of such changes would be quite dramatic, suggesting perhaps that one or more of the conservation laws do not apply in at least some places (or times) in the observable universe. So quite a few attempts have been made to constrain any such variations. The net? Depends on how you read the various papers - some claim to have found a clear signal for a tiny, but non-zero, change over cosmological time; others no such change. To give you an idea of how this is done, read this paper by Bahcall et al.; it is a very elegant, simple, robust method (typical of Bahcall). * it's actually the theory underlying atomic theory, QED
Response to #11 Nereid, thanks again for the valuable insights. You essentially list the four basic causes of redshift, i.e. Doppler effect, Relativistic Doppler effect, Expansion of space, Gravitational redshift, but are you alluding to a fifth possibility in connection to the fine structure constant changing with time? I have just downloaded the Bahcall paper, but I have not really had a chance to read it, as yet. No, not really. When I raised the question, prior to your link to the Freedmann paper, I was unsure of the various methods in use or their accuracy. While I understand the general principle of spectral lines and the importance to be able to correlate a number of lines to a given patterns, I didn’t really know whether there were any other factors that had to be taken into consideration, e.g. fine structure constant. While I need to read the paper, is there any speculation as to why the fine structure constant might change with time, e.g. cosmic expansion also causes small changes in atomic orbits?
Cosmological/Particle Horizons? Marcus raised some issues in my mind about a number of horizons in post #3. In particular the cosmological and/or particle horizon, which I am now trying to understand a little better, but believe I may still be making some wrong assumptions. I will present a number of quotes from the following wikipedia page: http://en.wikipedia.org/wiki/Observable_universe To be honest, I was somewhat confused by this statement. Originally, I believed the visible horizon was simply the speed of light times the age of the universe. Where the inverse of the Hubble Constant (H) is said to approximate to the age of universe, although most papers now describe this as a coincidence and not a direct relationship. In post #3, Marcus also alludes to a figure of 45 billion light-years, but describes this as the particle horizon. Are the cosmological and particle horizon just different names for the same thing? Not really sure where this figure comes from? I have attached two graphs, which may be inappropriate, but I wanted to try to illustrate the nature of my questions with a model. It is a model based on the Friedmann equation minus the [tex][k/a^2][/tex] and [tex][\Lambda/3][/tex] terms: [tex]H^2 = \frac{8}{3}\pi G \rho = \frac{8}{3}\pi G \left(\frac{M}{4/3\pi r^3}\right)[/tex] [tex]H = \frac{1}{t} = \sqrt{ \frac{2GM}{r^3}}[/tex] The mass of the universe has been set at [tex]8.5*10^{52}kg[/tex], which roughly corresponds to an often used estimate and leads to a present day value of [H=70.1km/s/mpc]. From these assumptions, a model of an expanding universe is calculated against time. It is believed that this model should be, at least, reflective of a matter-dominated universe existing from 1 billions after the Big Bang to the present-day? See Graph1.jpg: H and radius against the age of the universe The second graph (graph2.jpg) plots the velocity of the expansion, based on the size of universe at the time in question, against the age of the universe. What it seems to suggest is that the physical expansion, due to H, is initially greater than the speed of light, but this slows with the expansion, i.e. as the value of (H) falls. So my question relates to whether this model is, at least, reflective of a basic mechanism by which the physical universe is larger than the visible universe? Would really appreciate any insights as to how the radius of the physical horizon is calculated. Thanks Footnote: this model does not seems to given a value of 46 billion light-years, but this may be the least of its problems.
Re: Cosmological/Particle Horizons? mysearch, I think you'll find the Davis-Linweaver paper http://arxiv.org/abs/astro-ph/0310808 interesting.
Re: Cosmological/Particle Horizons? the figures 45 and 46 are approximate. Let's take it as 46 for discussion sake. The radius of the observable is the same as the particle horizon, essentially by definition, so both are 46 billion LY. But the presentday distance to the material that emitted the CMB, also called the surface of last scattering, is slightly less----say 45.5 or something like that. because with ordinary light we can't see back thru the hot plasma that filled space when expansion was some 400,000 years old. But things like neutrinos and gravity ways CAN get thru, so we can hope to eventually see that last little bit. With light, we see almost to the edge of the observable. With better instruments maybe we see the whole way. But there is no big difference between 45.5 and 46. So the radius of the currently visible, with light, essentially corresponds with the radius of the observable----i.e. the particle horizon distance. You mean the figure 92? 92 is twice 46. Diameter is twice radius. Don't entirely understand the question since why would one need a mechanism to explain the fact that the physical universe is larger than the observable universe (or the visible which is nearly the same)? Do you mean the radius of the physical universe, or the particle horizon? The particle horizon is the same as the radius of the observable universe and has been calculated to be 46 billion LY. The radius of the entire universe is not known. There is no evidence that it has a boundary. It could be finite like the 3D analog of a 2D sphere surface and then it would have a finite circumference. On the other hand it might be infinite. If you would like, I can tell you how to estimate the circumference assuming that the entire thing is spatial finite
Re: Response to #11 As a generalisation, in extra-galactic astronomy, the only causes of 'line shift' worth detailed consideration are the first and third ... for the others: I know there's at least one paper reporting a possible signature of gravitational redshift in highly ionised Fe lines from an AGN; there may be papers reporting relativistic Doppler effect line shifts, but most regimes where this would be detectable don't show line emission (or absorption) in the first place (such as in AGN jets). If there is a systematic change in the fine structure constant, over cosmological time, it is extremely tiny - parts per million, or billion (IIRC), and of no practical relevance to the bread and butter astronomical observations ... I should have mentioned the ISW (Integrated Sachs-Wolfe effect); depending on how you understand this, it is either "Expansion of space, [or] Gravitational redshift", a combo of both, or something different. Of course, it does not show up as a line shift, but I guess in principle it could. Moving away from line shifts to SED shifts (spectral energy distribution, or density), there are many more causes of astronomical interest to list, such as the Sunyaev-Zel'dovich effect ... but I think that would take you way off track, wouldn't it?
Hi , I am very curios abouat some prediction.. electrons closer to the nucleus have bigger speed from what I could deduce from the Bohr radius calculation.. but the Hubble efect predicts that speed gets bigger with the distance. Now i would expect that this two things should be the same , why is this not so? Plz help a noob :)
Response to #14 George: thanks, the Davis-Linweaver paper certainly seems to cover many of the areas of interest. I have raised some related comments in a subsequent response to Marcus.
Response to #15 Marcus: First and foremost: D’Oh - obviously must have contracted horizon blindness due to an overload of horizons. In an attempt to divert attention from this stupidity, I will quote from the Davis-Lineweaver paper, as provided by George Jones (#14): I believe I could be one of the victims of this obvious conspiracy, but hopefully when I have had a chance to assimilate some of the details within the Lineweaver article I will be better able to judge the extent of this conspiracy. However, in the meanwhile I would like to try to clarify what I was trying to do in post #13 via the equation: [1] [tex] H = \frac{1}{t} = \sqrt{ \frac{2GM}{r^3}}[/tex] I was hoping to create a crude model of a matter-dominated universe ranging from +1 billion years through to the present-day. My starting point was the quoted age of the universe, i.e. 13.7 billion years. The reciprocal of this leads to H=71km/s/mpc. This allows the Hubble radius to be defined as c/H=1.29E29m, which allows the associated volume to be calculated. As I have understood things, the only significance of the Hubble radius is that it is the point at which the expansion velocity exceeds the speed of light [c]. I selected a homogeneous density in the order of 9.5E-27kg/m^3 from which I estimated the mass (M) within the Hubble radius volume, e.g. 8.58E52kg. This mass was also selected because it gives the measured value of H via equation (1). As a closed universe, I assumed the mass-density would not change over the period selected, i.e. matter dominated. From equation [1] I plotted H against time. See graph1.jpg in #13 for details. While H varies with time, I assumed that H was linear with distance for any given time. As such, I could plot recession velocity [v] at a given radius against time. By re-arranging equation [1] you can calculate the radius corresponding to H and I assumed that this would allow me to produce a comparative plot of recessional velocity of a volume of space, expanding in time according to H. See graph2.jpg. What this graph suggested to me was that the edge of this volume of space, which today coincides with the Hubble radius, has expanded faster than light until now. If any of the assumptions of this model are valid, it seems to suggest that the universe was initially expanding faster than causality, i.e. ct. However, the model also suggests that as the expansion slows, new sections of the universe will appear within our future light cone, i.e. causality window. I was actually asking how the particle horizon radius was calculated. I don’t know if the answer is in the Lineweaver article, but noted that the figure of 46 billion light-years ignores the issue of inflation. Initially, I didn’t really appreciate whether there was much difference between the physical & particle horizons, but have taken onboard your comments regarding the degree of ambiguity concerning a finite/infinite universe. The way I was starting to consider the particle horizon is as follows: A photon emitted from an atomic transition at point X arrives on Earth after 13.7 billion years. While the distance travelled by the photon is ct, i.e. 13.7 billion lightyears, the distance between the Earth and point X is now much greater than 13.7 billion lightyears due to the subsequent expansion of the universe? Now based on my earlier model, if the photon was too far from the Earth, when emitted, the recessional velocity would be greater than [c] and the photon would still be chasing after us. However, there is also the issue that the recessional velocity appears to slow down over this vast amount of time. If I assume that the photon has travelled 13.7 billion lightyears, does the recessional velocity of point X give rise to the difference between (ct) and the particle horizon, such that: Particle horizon = ct + vt Where t=13.7 billion years, [c] is the speed of light and [v] is the variable recessional velocity of X with time? I accept this is probably too simplistic, but want to know if it basically explained the concept of receiving light from a point far in excess of ct.
Response to #16 Nereid, I have been intrigued by the sheer complexity and depth of this subject. Of course, there is an element of learning to walk before running in my current situation. However, I have very much appreciated your insights into this subject. Thanks