I have not withdrawn the proposed test. I did get very interested oin the quoted paper because it added a very reasonable third possibility that can also be tested for. That possibility is that:

There are some objects that are physically associated with nearby galaxies (possibly globular clusters) that are acting as strong lenses (~1000 times brightness magnifiers) of distant active galaxies. Under this third possibility, which should be added to the test, quite a few things are explained that have never been explained before:

1. Why quasars have a poor brightness relationship to redshift. Answer - because the amount of brightness magnification has a wide range of possible values.

2. Why quasars are seen to be physically associated with nearby galaxies as found by Arp. Answer - because they are being lensed by objects that are associated with the galaxy.

3. Why quasars can be seen at extremely high redshifts. Answer - because their brightness is being increased approximately 1000 times by lensing.

There is at least one claim by Arp that cannot obviously be explained by the proposal and that is that quasars are seen in matched redshift pairs either side of a galaxy. However that is the only outstanding issue of debate that I can see that is not resolved by the proposal.

Of course that does not make it correct, just an ingenious idea. To be correct it must withstand a variety of additional tests. The probabilities of quasars being seen as that much brighter must be balanced with the number of objects at that distance. There will be a correct distribution of the redshift ratio of the galaxy and quasar according to the optical properties relating to the lensing which can be tested against the sample. It would also be helpful if the actual nature of the lensing objects could be understood well enough to make predictions.
Certainly the distribution of objects in the universe is not random but there is large scale structure. Interestingly, I have been arguing for this fact for decades while people who believed in the big bang told me that the universe was uniform at large scales.

However never mind that. The test fully copes with the non-randomness of both galaxy and quasar distributions. By excluding all cases where the galaxy and quasar redshifts are similar and only including those where the quasar redshift is much greater. Is there any reason to believe that the quasar distribution is an exact copy of the galaxy one but just at 100 times the scale? I think not. In that case, randomizing the pairings is a fully adequate measure to cope with all properties of non-randomness in the samples.
I think that a simple range of redshifts for galaxies is an adequate measure combined with classification as spiral galaxies. It is close by spiral galaxies that Arp claims quasars are ejected from. That is what should be the basis if the test.
Agreed. I am not sure what you mean by outliers, I assume those that are just beyond the magnitude limit or some such factor. I would deal with these by using limits that are well within the catalog sample limits.
If one is doing a test between two alternative cosmologies, then for an honest test the two must be on an equal footing. If you assume that one has only one in a million chance of being right, then a result that would arise only once in one hundred thousand times from the other will be accepted as the most likely choice over a result that agrees 100% with the other. In that case I say the investigator is not scientific but is acting from faith. I have no interest in such investigations or investigators. I am interested in the truth whether or not it agrees with what I think is most likely right now.
I wish to test both Arp and Big Bang as alternatives to explain an analysis yet to be done. That test remains as it was. In case both fail to some extend, it makes sense to now add a third possibility which is the proposal made by Bukhmastova.
When I first proposed the test I was not considering the possibility that quasars were objects that were being strongly lensed by objects associated with nearby galaxies. I do not think that many people have looked at such a proposal seriously. It would reconcile many (but not quite all) of Arp's observations with standard cosmology. Considering how vehemently people, even in this thread, have spoken against Arp, I do not believe that others have thought about this either. But I do think it is well worth thinking about. Isn't science interesting when we explore such possible tests and how we might think about it?

It has taken me a little while to digest these and I have been a little short of time recently. Please correct me if I am wrong, but it seems to me that the lensing being referred to is not in the same category as what Bukhmastova is stating, namely 1000-fold increase in brightness, and caused by very close by objects.

To that extent, although I can see that statistics will be affected, I think that my random re-pairing suggestion would deal with the issues of lensing effects on statistics if there are no real associations between galaxies at low redshift and quasars at high redshift.
I am not claiming Arp vindication. I am proposing to test two rival theories. However Arp has made a prediction about quasar distribution. Before one of the long thin sections of sky was sampled he predicted two peaks in the number of quasars at certain longitudes corresponding to two nearby galaxies near the strip. As I understand it he was quite correct about that.
Well, you may want that, and that is quite reasonable, but it wasn't what I was proposing to test.
I don't think that. However in the quantized redshift revisited thread I did discover that there are times when it is misused. When the redshift is included as a reliable measure of distance to calculate distances between galaxy pairs in 3D and then the result agrees with the big bang, the big bang has actually already been assumed to be correct. When testing between rival theories you cannot assume that one is correct in the analysis except in the case of reductio absurdum.

Let us accept this as correct when testing spatial density. This would not lead to a quasar's brightness having any correlation with a line of sight galaxy's redshift, would it? That is what I propose to test.

rtomes, can you please, in one post, state, simply and clearly, without editorial comment, just what the ATM proposal you intend to present (and defend) is?

And, in a separate post, state:

a) exactly what you propose to test; if said test is of "Arp and Big Bang as alternatives", provide sufficient references to papers on these two as to comprehensively, quantitatively, describe them

b) exactly what the test will actually involve; this should be fully quantitative*

c) exactly how you will address the 'seek and ye shall find' shortcoming of 'fishing expeditions'.

* IOW, no more "very different redshifts", or "redshifts are similar", or "strong relationship", or "no significant difference", or "a simple range of redshifts", or "a fully adequate measure", or "well within the catalog sample limits", or ...

Not wishing to revive a now closed thread, the NGC 7603 work was not at all an example of what I suggested.

If the only photons detected by the spectrograph are from the parent and the object 'in' the bridge (whether continuum, emission or absorption lines), then all you learn is that the two objects have different redshifts.

The real test is how does the redshift of lines in the bridge vary along the bridge. If there's a systematic variation of a few tens or hundreds of km/s, then the bridge would seem to be local to the parent, and the high-z object apparently embedded in it would be a chance alignment*.

But - and this is the key part - if the ATM idea does not contain anything on how the redshift of material in such bridges is expected to vary, along the bridge, then this kind of observation is useless - no matter what is found, you have made no progress.

In any case, as rtomes has made clear, this kind of thing is not what his ATM idea (proposal, test) is about, so let's not hijack it ...

* This is just one example, of course.

Redshift is but one of many things to investigate to determine if a bridge is real or not. Actually, if redshift values along the bridge varied from that of the galaxy to that of the object, it would tend to disprove Arp's theory of redshift, and probably require a new theory different than Arp's or mainstream.

Actually, a result of Arp's theory would be that the bridge be either the same redshift as the host galaxy, or the same redshift as the connected object.
An apparent bridge between the objects is not a ATM idea, it is a physical observation that should be analyzed thoroughly. Advancement of knowledge is made by studying a physical observation including testing/analyzing the possibilities to explain it. If a scientist doesn't have a theory to explain the observation, this doesn't mean the observation is useless.
My initial post was meant merely to respond to what I thought was a misleading statement about the scope of Arp's work. It was meant as a one time comment, not a discussion starter.

I already did this in my first post of the thread:
There is no difference between my proposal and the test. The proposal, as stated is to perform a test.
Some of the parameters will need to be different depending on the catalog(s) used to get the data on galaxies and quasars. So I will express things in terms of those parameters:
z_G_min = minimum redshift of galaxies to include in the initial sample.
z_G_max = maximum redshift of galaxies to include in the initial sample.
Galaxy classification to include all types of spirals, exclude all other types.
dist_G_Q = distance limit set for true association of galaxy and quasar in alternative cosmology. This will be around 100 kpc (maximum 150 kpc).
The z limits will need to be set based on probabilities that depend on the particular catalog(s) used because it is desirable that when the alternative cosmological model is considered then the probability of line of sight coincidences must be kept low compared to the actual number of pairs found. Using too low value for z_G_min will cause too many random non-associated pairs.
The proportion of the sky within dist_G_Q of a galaxy at redshift z is:
(dist_G_Q/c/z*H)^2/4/pi
Z can be substituted by z_G_min to check that the number of galaxy-quasar pairs randomly expected is smallish compared to the actual number of pairs.
Because we are testing a correlation versus the galaxy redshift we need a ratio of at least one order of magnitude in the galaxy redshift, but not too much more or we will be sampling galaxies that are too far away for associated quasars to be seen in the alternative model. So z_G_max ~= z_G_min * 10.

For the quasars I think that a simple observed magnitude maximum limit is sufficient as a selection criterion once it is established that the definition of quasar is compatible with that used in Arp (and any other) papers that showed galaxy-quasar associations.
The maximum redshift used for galaxies by the above approach should be significantly less (say at least 25%) than the minimum quasar redshift, so it should satisfy the condition that in the big bang model there is no possibility of real association between the pair.

The magnitude limit for the quasars will depend on the catalog used. But it should be no more than the stated redshift to which the catalog is believed to be reasonably complete. This is to satisfy you, not me. I don't think that such aspects of the catalog will bias the result in any way as they are relatively independent variables. However the reduction of the quasar sample by a brightness limit can also reduce the probability of random line of sight pairs. This may be needed if the catalog(s) are very comprehensive ones. On reflection I think that a redshift limit should not be used as we are plotting the quasar redshift as the dependent variable in our results so that would cause a nasty cutoff that would mess with the statistics of correlation.

After catalog selection but before analysis the number of objects that will be selected needs to be estimated and using the probability of random pairs based on proportion of sky for galaxies and number of quasars in the sample, the free parameters adjusted to reduce the random pairs to less than 1 per galaxy while keeping the quasar sample of a similar order of size to the galaxy one.

Thank you.

Please provide references which contain explicit, detailed, and (preferably) quantitative descriptions of "the rival theories" ("Big Bang" and "Alternative").

Please show, in detail, using pertinent material from those references, how each of the stated "expected outcomes" was derived.

Thank you.

"[...] to include in the initial sample." (emphasis added)
* how many samples will the test involve?
* to what extent will the selection of samples after the initial one be determined by the results of one or more analyses on the initial sample?

"Galaxy classification to include all types of spirals, exclude all other types."
* how will galaxies be classified?
* why are spirals the only galaxies to be included?

"dist_G_Q = distance limit set for true association of galaxy and quasar in alternative cosmology. This will be around 100 kpc (maximum 150 kpc)."
* how will distances be estimated?
* why "around 100 kpc (maximum 150 kpc)"?

"The z limits will need to be set based on probabilities that depend on the particular catalog(s) used [...]"
* how, explicitly and in detail, will "the z limits [...] be set"?
* specifically, to what extent will "the probability of line of sight coincidences" be calculated ab initio?

"it is established that the definition of quasar is compatible with that used in Arp (and any other) papers that showed galaxy-quasar associations"
* how will this be established?

Some minor clarifications:
* what are "c" and "H" in the formula "(dist_G_Q/c/z*H)^2/4/pi"?
* is "Z" a typo for "z" in "Z can be substituted by z_G_min to check [...]"? If not, what is it?

Sorry, I had intended to deal with this in the same post and forgot.

Th initial sample meant the galaxies and quasars eligible by way of magnitude and redshift to be in the sample. The final sample was to include only those that were sufficiently close to each other to qualify as pairs. That means being within (100 kpc or) 150 kpc at the distance of the galaxy.
I propose to use a catalog that has classifications and to accept those. Any classification with "Spiral" in its name would be accepted. The reason is simply that Arp has claimed that quasars are ejected from spiral galaxies along the axis. Therefore that must be the basis of a test for whether he is right or not.
The distance will be based on an accepted estimate of the Hubble constant (say 71 km/s/Mpc) and the redshift of the galaxy. You know the formula. :-)

That distance is a reasonable upper limit for quasars if they are ejected from galaxies. The typical distance is found to be 50 kpc on the projected line of sight. So I was inclined to use 100 kpc, but seeing that other researchers have used 150 kpc, that is also acceptable.
The line of sight coincidences can be set by combining the area of the sky that the chosen distance covers (square degrees or square minutes or whatever) as a fraction of the total sky in the sample with the number of objects that are initially selected. If for example there are 50,000 galaxies and 10,000 quasars initially selected and the region of 150 kpc around each galaxy covers 0.001% of the sky (this will depend on the average redshift so can be modified if need be) in the catalog then we expect about 0.1 quasars per galaxy to be within that region. Such a result is acceptable, being less than 1. That should lead to something of the order of a few thousand pairs, an acceptable result.

Of course I have ignored that even in the big bang there is supposed to be some slight lensing effects. But while this does affect the number of coincidences it does not affect the brightness of the quasar in relation to the galaxy redshift by an amount that could materially alter the scatter diagram in the way that would need to happen if Arp is right.
Maybe some astronomers can help me here.
c is the speed of light
H is the Hubble constant
yes, Z was meant to be z

How, if at all, will the range of distances and redshifts in galaxy clusters (esp rich clusters) be addressed, in estimating the distance of a galaxy, in the samples?

How - explicitly, with formulae and algorithms - will the difference between distance and projected distance be addressed?

Why is "[t]hat distance is a reasonable upper limit for quasars if they are ejected from galaxies"?

In which papers was "[t]he typical distance [...] found to be 50 kpc on the projected line of sight"?

Which "other researchers have used 150 kpc"?

(to be continued)

I am bumping this post, with its questions, because they are essential aspects that need to be addressed before the test(s) can be fully specified (so it seems to me).

This has already been recognised in a later rtomes post:For avoidance of doubt, I am asking for explicit, quantitative statements of the pertinent parts of "the rival theories" ("Big Bang" and "Alternative") - not word salads - and references to source material sufficient for any reader to be able to independently verify the statements.

I am also asking that the derivations of the explicit, quantitative "expected outcomes" be provided.

You may cout me in. I found another redshift that I call "Einstein's universe redshift" since it follows from Einstein's theory of gravitation and shows up in Einstein's stationary universe. This redshift is exponential with distance so it nicely fits the cosmological redshift and it has the same (observationally) acceleration of (apparent in this case) expansion. That's why I think it may replace not only the "cosmological redshift" in your equation for total redshift but it may apply to quasars as well since it shows up in any relatively dense cloud of dust (which I imagine might surround any quasar).

The redshift is predicted by Einstein's theory adjusted to "dynamical friction" of photons (considered "negligible" by the mainstream) through energy conservation. The only adjustment to Einstein's theory turns out to be the introduction of a new tensor, that I call tentati