What's the nothingness that our universe is expanding into?

[Meron]

We all know that the universe is expanding. What I'm curious about is what it is expanding into.

 

[phinds]

It is not expanding into anything. The universe is everything there is. Either it is spatially finite but unbounded (no edge, no center) or it is infinite (no edge, no center).

 

[Meron]

If the universe is all there is, what was present before the big bang ?

 

[phinds]

If the universe is all there is, what was present before the big bang ?

Unknown. The Big Bang Theory has nothing to say about what came before one Plank time.

 

[Meron]

Interesting :)
Thank you

 

[wabbit]

The answer is "that nothingness is nothing"

 

[Doug Huffman]

@Meron

Try reading some different interpretations of cosmology, in this case particularly Smolin's and Unger's that use principles of Charles Sanders Peirce.

Space is used to describe separation and extent of entities. Where there is only one thing - the "singularity" - then there is no separation or extent. Similarly for time, time is used to describe change and duration. Where there is no change, then there is no time. Another thing and change both come from the singularity.

(You may have a physics significant name-fellow, Matti Meron of UCI as I recall, with whom I used to correspond in the hey-day of Usenet.)

 

[Meron]

:)

 

[slatts]

I am C- caliber in trig and high school physics, and have been taught no math or science beyond them, but I speak 3 languages, am from a city (Miami) where Spanish replaced English for a while, and am descended from an ethnic group (the Irish) whose native language was permanently replaced by English. My impression is that a lot of the distinction between "space" and "nothingness" is shifting, due to the replacement of one "language" (spoken) by another (mathematical). (Notice how I typed "3", instead of "correctly" typing "three"? I did that without thinking about it, because math IS more concise for all kinds of expressions.) From a lot of reading (popularizations by several major physicists, and a bit of philosophy), I've tentatively concluded that it's quite important to notice whether a particular type of "field" can have a zero value or not, in deciding whether portions of it are considered to be at least potentially infinite spatially. It's also rather important to figure out whether diagrams (another potentially concise form of expression) are intended to depict abstractions (and are thereby analogous to, say, bus timetables) or have a closer resemblance to, say, a picture or map: These distinctions get very blurred by instructors trying to retain the attention of bored students, and the blurring has worked its way into the terminology at different times. The blurring has resulted in 21 interpretations of Mach's Principle, one of the ideas that inspired Einstein.

 

[wabbit]

Linguistic differences do not matter much if at all in science.

 

[Doug Huffman]

Perhaps, though, to the scientists. There is a spectrum of interpretations of QM.

 

[slatts]

I agree that linguistic differences should have nothing to do with formulated science, but I'm saying that the conversion of mathematics into spoken language can change the personnel doing the formulation, and consequently change the accuracy or completeness of the science: I don't believe it's clear exactly which of the 21 reported versions of Mach's Principle influenced Einstein, and, if it had been, we might be having less of a problem with, say, singularities than we're having now.

 

[phinds]

I agree that linguistic differences should have nothing to do with formulated science, but I'm saying that the conversion of mathematics into spoken language can change the personnel doing the formulation, and consequently change the accuracy or completeness of the science: I don't believe it's clear exactly which of the 21 reported versions of Mach's Principle influenced Einstein, and, if it had been, we might be having less of a problem with, say, singularities than we're having now.

I don't get how any of that could have had anything to do with singularities. They are just places where the models break down. Are you saying that liguistic differences might have lead to a more complete theory that would not have places where they break down? I find that hard to believe.

 

[wabbit]

I don't believe it's clear exactly which of the 21 reported versions of Mach's Principle influenced Einstein, and, if it had been, we might be having less of a problem with, say, singularities than we're having now.

I find it hard to imagine how you can ascribe a linguistic origin to singularities in GR, and I would be very interested if you could produce evidence for such a claim.

 

[wabbit]

Perhaps, though, to the scientists. There is a spectrum of interpretations of QM.

Indeed but do they depend on which language they are stated in? This can be true for philolosophy or poetry, which can defy translation, but Copenhagen or Everettian or other interpretations of QM do not seem to pose a problem of translation.

 

[Doug Huffman]

No, not of translation but of nuance, subjective QBism on one hand versus a plethora of 'objective' interpretations.

 

[slatts]

slatts said: ↑
I agree with Wabbit that linguistic differences should have nothing to do with formulated science, but I'm saying that the conversion of mathematics into spoken language can change the personnel doing the formulation, and consequently change the accuracy or completeness of the science: I don't believe it's clear exactly which of the 21 reported versions of Mach's Principle influenced Einstein, and, if it had been, we might be having less of a problem with, say, singularities than we're having now.

Phinds said:
I don't get how any of that could have had anything to do with singularities. They are just places where the models break down. Are you saying that liguistic differences might have lead to a more complete theory that would not have places where they break down? I find that hard to believe.

Slatts' reply to Phinds is:
In my reply to Meron's post, I was "drawing" an analogy between spoken-and-written "language" per se, and mathematics per se. In my own reply to Wabbit's, I was saying that the conversion of mathematics into spoken/written "language" could, if not done carefully, discourage people like Meron from interesting themselves further in physics, or at least alter the subjects of interest to them. Mach's Principle was attributed to Mach only anecdotally, since he had not given it any notation in physics or math, but Einstein was well enough aware of it that he wrote Mach of some partial confirmation of it which he (E.) was proud of having accomplished experimentally. Since Mach's Principle had to do with the tendency of our arms to rise when we spin around under the stars, it's generally felt to have had to do with a gravitational influence exercised by distant inertial fields. I suspect that Gödel's solution of a rotating universe resulted partly from the well-known knowledge of Einstein's interest in whatever principle was involved. I'm speculating that, if we had known exactly which of the 21 written formulations of Mach's Principle were eventually devised, we might have eliminated singularities from physics years ago, which would have left Hawking free to accomplish even more useful work than he already has, during the time he wasted making bets with Kip Thorne. (This is WAY more speculative than I want to be, but I'm responding to your own reply.)

About the possible disappearance of cosmological singularities from physics (which I hadn't been trying to discuss in this thread), google Ali and Das' "Cosmology From Quantum Potential", written in Dec. 2014. (Not to add to the confusion, I have to mention that, although Ali and Das' reformulation of General Relativity into QM doesn't refer to singularities in particular, their elimination seems--per Bojowald and Rovelli--to be a main aim of Quantum Mechanics, whose control of tiny details does not seem to tolerate the arrival of huge amounts of reality from nowhere very well.)

 

[DaveC426913]

slatts said: ↑

Phinds said:

Slatts' reply to Phinds is:

Dude, use the quote feature in the editor.

 

[slatts]

Linguistic differences do not matter much if at all in science.

I don't get how any of that could have had anything to do with singularities. They are just places where the models break down. Are you saying that liguistic differences might have lead to a more complete theory that would not have places where they break down? I find that hard to believe.

Indeed but do they depend on which language they are stated in? This can be true for philolosophy or poetry, which can defy translation, but Copenhagen or Everettian or other interpretations of QM do not seem to pose a problem of translation.

Thanks, it seemed to work OK this time, although I had to use the .<< edit >> option to add this comment.

slatts said: ↑
I agree that linguistic differences should have nothing to do with formulated science, but I'm saying that the conversion of mathematics into spoken language can change the personnel doing the formulation, and consequently change the accuracy or completeness of the science: I don't believe it's clear exactly which of the 21 reported versions of Mach's Principle influenced Einstein, and, if it had been, we might be having less of a problem with, say, singularities than we're having now.

Phinds said:
I don't get how any of that could have had anything to do with singularities. They are just places where the models break down. Are you saying that liguistic differences might have lead to a more complete theory that would not have places where they break down? I find that hard to believe.
Sorry, I'm basically complaining about imprecision in written language by physicists, and it looks like I'm not being sufficiently precise myself.

Slatts' reply to Phinds is:
In my reply to Meron's post, I was "drawing" an analogy between spoken-and-written "language" per se, and mathematics per se. In my own reply to Wabbit's, I was saying that the conversion of mathematics into spoken/written "language" could, if not done carefully, discourage people like Meron from interesting themselves further in physics, or at least alter the subjects of interest to them. Mach's Principle was attributed to Mach only anecdotally, since he had not given it any notation in physics or math, but Einstein was well enough aware of it that he wrote Mach of some partial confirmation of it which he (E.) was proud of having accomplished experimentally. Since Mach's Principle had to do with the tendency of our arms to rise when we spin around under the stars, it's generally felt to have had to do with a gravitational influence exercised by distant inertial fields. I suspect that Gödel's solution of a rotating universe resulted partly from the well-known knowledge of Einstein's interest in whatever principle was involved. I'm speculating that, if we had known exactly which of the 21 written formulations of Mach's Principle were eventually devised, we might have eliminated singularities from physics years ago, which would have left Hawking free to accomplish even more useful work than he already has, during the time he wasted making bets with Kip Thorne. (This is WAY more speculative than I want to be, but I'm responding to your own reply.)

About the possible disappearance of cosmological singularities from physics (which I hadn't been trying to discuss in this thread), google Ali and Das' "Cosmology From Quantum Potential", written in Dec. 2014.

I don't get how any of that could have had anything to do with singularities. They are just places where the models break down. Are you saying that liguistic differences might have lead to a more complete theory that would not have places where they break down? I find that hard to believe

 

[slatts]

Thanks. I'm just realizing that, in my last reply, it had shown my last quotation of myself after my quotation of the other guys and before my quotation of myself. Pretty smart of it.

 

[DaveC426913]

The usual method is this:
Instead of starting with the editor, start with the post(s) you wish to quote.
In the bottom right corner of each post (so posts 10, 13 and 15), there is a [+ QUOTE] button. It's a multi-quote feature.
Click it once on each post. They will accumulate.
Now, at the bottom of the page, where you want to Reply, you'll see under the editor a [INSERT QUOTES] button.
Click that, and your posts will be inserted automatically into the reply. (Don't forget to pare the quoted text down to the essentials.)
Now you can add your comments at-will.

 

[phinds]

I still think you are way off base in thinking that linguistics or different formulations would have had any effect on singularities in our models. I don't think you understand what singularities are. What DO you think they are? How do you think anything you have talked about would have mattered to getting rid of them by formulating better theories than we have now?

 

[slatts]

I still think you are way off base in thinking that linguistics or different formulations would have had any effect on singularities in our models. I don't think you understand what singularities are. What DO you think they are? How do you think anything you have talked about would have mattered to getting rid of them by formulating better theories than we have now?

Like I say, you and I are in a speculative discussion, but I guess it's not a speculative THEORY of PHYSICS, so I guess it's marginally OK per the Forum's rules. (It's absolutely fine with me, and maybe you can advise me on whatever physics terminology I'm using incorrectly.)

I think a singularity is a near-convergence of geodesics at a point in spacetime a little above the Planck scale. Since they're not allow to cross or turn at a sharp angle under its rules, it is the place where relativistic physics does, like you say, break down. As recently as a week or two ago, I had thought, "So what?" After all, for many years the Big Bang had been regarded (mostly because of its improved correlations with chemistry) as a major improvement over Hoyle's steady state theory. As far as physics is concerned, it's my understanding that they are actually similar, and, like inflation (whose sequencing with, or redundance upon, the Big Bang is very unclear to me), both allow for a universe to form with a net expenditure of energy that's at or near zero, through a release of energy from its potential upon the expansion of a gravitational field, which I've heard is negative energy in both its attractive and its repulsive varieties. The Big Bang resolved a very obvious paradox (Olbers') in a very simple way, and I had loved the idea of it for many years.

The problem, which I was largely unaware of until I finished plowing my way through a popularization by Bojowald, is with information theory. As you can see by poor Dave's having to lead me by the hand through a "rather mild" algorithm just now, I'm about as lame on information theory as can be, although I've heard it has a lot to do with entropy. (My only take on entropy is that Boltzmann must've committed suicide out of boredom, but I know it's really big, especially now.) Where this gets into Quantum Mechanics, and why I was suggesting you might like the QM/Relativity synthesis suggested by Ali and Das, is that QM does not allow information to get lost, so, while it's achieving its miraculously accurate (and technologically important) predictions by tracking every path every particle even might take, it's not happy with a whole starload of details suddenly leaving the premises. I know it evaporates in an eon (our time), and that a 2D version of it may remain plastered over the poor star's event horizon meanwhile, but the string theory governing that plastering has about as much fire in it as an ice sculpture of a well-digger's belt buckle floating halfway between us and Andromeda. THAT's why I think you might enjoy Rovelli's "Planck Stars" and the Ali-Das essay.

 

[phinds]

I have no interest in being in a speculative discussion. We are talking past each other.

 

[slatts]

Actually, I forgot to rehash what was speculative (about the effect of imprecise language on the inspiration of prospective physicists), but it was just psychological or literary speculation anyway.

 

[OCR]

Cold and well-diggers... thought I'd heard them all... lol

...has about as much fire in it as an ice sculpture of a well-digger's belt buckle floating halfway between us and Andromeda.

Absolutely classic...

Carry on...

 

[PeterDonis]

I think a singularity is a near-convergence of geodesics at a point in spacetime a little above the Planck scale.

No, that's not what a singularity is. Geodesics can converge in a curved spacetime without any problem at all. For example, consider two satellites, both in free-fall circular orbits about the Earth at the same altitude, but with different orbital planes. These satellites both travel on geodesics, and those geodesics will converge (cross) twice per orbit.

The exact definition of a "singularity" in spacetime is somewhat involved, since there are a number of technical conditions involved. But basically, there are two things that can happen in a spacetime that signal that our current theory of spacetime, General Relativity, is reaching its limit:

(1) One or more geodesics can be incomplete: that is, they simply end at a finite value of their affine parameter. For a timelike geodesic, this means someone traveling along it would simply "stop existing" at some finite reading on their clock.

(2) One or more scalars describing the curvature of spacetime can increase without bound; physically, this means that tidal gravity increases without bound, so objects can be stretched or squeezed to an unbounded extent.

Both of these things happen at the "initial singularity" of the idealized FRW spacetimes that are used in cosmology. However, all that really means is that we don't use those idealized spacetimes all the way back to the "initial singularity"; we draw the line somewhere short of that and say that is the current limit of our knowledge.

inflation (whose sequencing with, or redundance upon, the Big Bang is very unclear to me)

It is a matter of terminology whether the term "Big Bang" refers to the state of the universe at the end of the inflation era--a hot, dense plasma that is expanding very rapidly--or whether it includes inflation itself. Physically speaking, inflation happened before the hot, dense, rapidly expanding plasma: what brought that hot, dense, rapidly expanding plasma into being was the rapid conversion of the energy stored in the inflaton field into energy contained in matter and radiation, when the inflaton field underwent a phase transition from the "false vacuum" state (in which it was driving inflation) to the "true vacuum" state, which it has been in ever since. What happened before inflation, or exactly how inflation got started, is still undetermined; there are a number of candidate theories, but we don't have enough evidence to decide which, if any, of them are correct.

both allow for a universe to form with a net expenditure of energy that's at or near zero, through a release of energy from its potential upon the expansion of a gravitational field, which I've heard is negative energy in both its attractive and its repulsive varieties

All of this depends on what you use the term "energy" to refer to. There is a sense in which the "total energy" of the universe is, has been, and will always be zero. With this sense of the term, it is possible to view this net zero energy as an exact balance between the positive energy of matter and radiation and the negative energy of the gravitational field. (Btw, I don't know if this sense of "energy", and the balance between positive energy of matter/radiation and negative energy of gravity, is applicable in the steady-state theory; the "continuous creation" of matter in the steady-state theory is a local process, which is certainly not allowed in our standard model of cosmology--see below.)

However, this sense of the term is useless in practical terms, because it tells us nothing about the dynamics. It tells us nothing about how the expansion got started, or what could start it. It tells us nothing about how the expansion progresses--how the rate of expansion changes as various transitions occur between different types of ordinary matter and radiation being dominant. It tells us nothing about the future of the expansion: whether it will go on forever or not. It doesn't even tell us the most important thing about all of these processes: that locally, they all conserve energy, in the sense that the divergence of the stress-energy tensor (which describes the energy, momentum, pressure, and other stresses at any point in spacetime) is zero. In other words, however we want to define "energy" globally, locally, stress-energy is conserved: you can't create or destroy it, you can only change its form. (This is what is violated by the "continuous creation" of matter in the steady-state theory.)

 

[slatts]

Thanks, this is exactly the sort of thing I'd wanted to find out. I am still just a little shaky on "potential": It seems to me that that word can equally well represent either the energy of a field containing little or no particulate energy or particulate matter, or, provided the theory supporting it confirms its past occupancy by a section of any field at non-zero value and/or predicts its future occupancy by a section of any field at non-zero value, the occupancy of a physical space by a section of a field AT zero value.

Another detail (much less important) that I'd like to verify is whether the release of its potential energy, during the formation or expansion of a gravitational field, can result immediately in the existence of an inflaton field.. (I could easily be wrong about that, but some texts mention the "inflaton"--AKA the inflaton field--in describing the generation of repulsive gravity, and others omit any mention of it.)

Formulating these questions has left me more of a fan of math and physics notation than I was a half-hour ago; too bad I can't read a lot of it. (I'm seeing double just from proofreading this series of congealed grunts and moans.) If no one up to entering their configuration space is still minding the store, I won't be surprised, and will check back tomorrow or so. (Move over, Boltzmann...)

 

[PeterDonis]

I am still just a little shaky on "potential"

That word can be used to mean many different things. That's why it's important to use math rather than words when you want to be sure you are clear and unambiguous.

the release of its potential energy, during the formation or expansion of a gravitational field

There's no such thing. Gravitational fields don't "form" or "expand". Spacetime curvature just is. Objects that fall in a gravitational field can be viewed as converting potential energy to kinetic energy, but that's not the same as the field itself "releasing energy"; total energy, counting potential as well as kinetic, is conserved in this process. Nothing is "created"; the local conservation law I described is obeyed.

can result immediately in the existence of an inflaton field

None of the candidates for how inflation got started that I'm aware of look anything like what you're describing. Can you give a reference for what you're talking about?

 

[slatts]

(On p. 292 in Appendix A ("Gravitational Energy") to Guth's 1997 The Inflationary Universe, he uses a drawing showing a spherical "shell" or balloon floating in space, each linked from several points on its surface (by ropes each ending in a coil around a pulley) to the axle of one or another of several dynamos (each mounted on a long rod whose earthly or planetary end is out of the picture) grouped in a ring-shaped formation around that balloon, and describes what I'm calling an expansion of the balloon's gravitational field after its partial deflation. His description is as follows: "...in the shaded region between the original and new positions of the shell, a gravitational field now exists where no field had existed before. [I understand that you're saying he should have said something like "in the shaded region where the density of space had changed"--and I'm guessing you mean changed by a decrease in it--but he didn't.] He immediately continues by saying, "The net effect of this operation is to extract energy, and to create a new region of gravitational field. Thus, energy is released when a gravitational field is created [italics mine]...Since the region began with no gravitational field and hence no energy, the final energy must be negative." Plainly the gravity in this illustration was attractive, but the book was entirely about inflation, and, in a different context within it, on the book's p. 258, he mentions a situation "causing the [false-vacuum] bubble to collapse", not "causing the volume of space to contract" or "causing the density of the true vacuum to increase". (That overall situation is describing inflation induced artificially, and there's no mention of any inflaton field or particle in it..)

While not wanting to seem too off-the-wall, I do see the value of trying to assimilate a viewpoint more comprehensive than Guth's 1997 readers'. Thanks for your help with that.

 

[PeterDonis]

I understand that you're saying he should have said something like "in the shaded region where the density of space had changed"

No, I don't think so. He is saying that new space is being created. In terms of the universe during inflation, he is saying that as the universe inflates, new space is created that didn't exist before.

However, thinking of this as "new space being created" is not really correct relativistically. Strictly speaking, it should just be "the geometry of spacetime is a certain way"--spacetime is shaped like an expanding funnel, so to speak. Spacetime doesn't get "created"; it just is--it's a 4-dimensional geometry that has a certain shape. The "gravitational field" is just a way of referring to the 4-dimensional shape.

In short, I think Guth here is giving a particular interpretation of the theory, which may or may not be a good heuristic to use in trying to understand gravity and energy in general. See below.

The net effect of this operation is to extract energy, and to create a new region of gravitational field. Thus, energy is released when a gravitational field is created [italics mine]...Since the region began with no gravitational field and hence no energy, the final energy must be negative.

Notice that Guth is equivocating here: first he says that a new region is "created", which implies that it didn't exist before; then he says that the region "began with no gravitational field", which implies that it did exist before, just without a gravitational field. But you can't have it both ways. This just shows that you have to be very careful when reading pop science presentations, even when they're written by scientists. Guth is not giving a rigorous scientific description of the theory; he's just giving an analogy which may help with visualization, but has serious limitations.

Notice also that viewing spacetime as a 4-dimensional geometry with a particular shape avoids this difficulty; the shape is what it is, and that's it.

 

[slatts]

I may have looked at one light cone too many, but I do have a question about the funnel analogy. I could understand the funnel with the hyperboloids nested inside it completely IF the funnel itself had curved sides, but it never does. Maybe there's a curvature so subtle that human vision could only detect it in a drawing of the funnel that extended from here to the CMB, but, if so, I've never heard it mentioned.

I'm GUESSING that this MAY have SOMETHING to do with the approach to infinity in calculus. If we could use all the trees on Earth for the paper and write limits on it with digits that could only be read with electron microscopes, would the expenditure in material and effort perhaps correspond to a retreat of the singularity downward in size and backward in time, or am I missing something?

 

[PeterDonis]

I could understand the funnel with the hyperboloids nested inside it completely IF the funnel itself had curved sides, but it never does.

I'm talking about a different "funnel" analogy. If we view the universe spatially as a hypersphere, and drop two spatial dimensions, we can think of it at any instant of time as a circle, and the spacetime of the universe is just a stack of these circles (time is vertical in this analogy). The size of the circles increases as you go upward--that's the expansion of the universe. If the expansion is accelerating, then the rate at which the size of the circles increases upward, is itself increasing, so the stack of circles looks like a funnel with sides that expand outward. It's just an analogy for visualization.

Note that in the other funnel analogy, the one you refer to, the "funnel" itself is not supposed to be part of spacetime; it's just a convenient bit of "scaffolding" to help visualize the model.

If we could use all the trees on Earth for the paper and write limits on it with digits that could only be read with electron microscopes, would the expenditure in material and effort perhaps correspond to a retreat of the singularity downward in size and backward in time

The singularity doesn't "move" in space or time; it's just a particular piece of the 4-d spacetime geometry.

 

[slatts]

I didn't mean that it would move physically; I had thought the conception of it might shift if the approach to infinity didn't have to be delineated with straight sticks laid end-to-end. (It would be something more like pi. I may be trying to say "wave function".)

You see, what I'm trying to say about the singularity is, if it happened to occur at scales whose length factor is so close to the shortest wavelength of light, the termination of the geodesics that would've otherwise continued "down" it may have been a concession to the demand for experimental verification. And, if so, i don't understand how we would be able to make judgments about the nature of whatever lay (in your analogy) "beneath" it. I'm talking about judgments like whether it might be just like "traditional" space (synonymous with traditional "nothingness"), containing non-scalar fields whose values would all be at zero, and might remain there throughout some "time" whose clocks would all have hands that would remain forever motionless to us, but might nevertheless be moving. (I understand that the clocks would have to be pretty tiny: That's more-or-less the point of my inquiry.)

I've googled "scale factor:" dozens of times, and nothing I've been able to comprehend would justify extending the funnel (or stack of discs) analogy until a funnel with curved sides would dwindle into an infinitely thin line that would finally just fade beyond any perceptibility obtainable to us through all the resources in our observable region. IS that the result of the need for experimental verification that distinguishes science from other forms of brain activity, or is there something POSITIVE in the math that would leave that extended analogy markedly less correct than the prevailing one of a straight-sided funnel ending at a singular point?

 

[PeterDonis]

I didn't mean that it would move physically; I had thought the conception of it might shift

Shifting the "conception" won't change the physics. But physically, as I said before, nobody believes the singularity is actually there.

if it happened to occur at scales whose length factor is so close to the shortest wavelength of light

There is no "shortest wavelength of light", at least not in any of our current models. If you're talking about proposed quantum gravity models in which there is no meaning to any length shorter than the Planck length, in those models, "spacetime" itself has no meaning on the Planck scale; it's an emergent phenomenon based on some other kind of physics at the Planck scale (such as strings in string theory or loops in loop quantum gravity).

the termination of the geodesics that would've otherwise continued "down" it may have been a concession to the demand for experimental verification

I'm not sure what you mean here. Nobody is claiming that any geodesics actually "terminate". The point is that the idealized model that contains terminating geodesics (i.e., classical GR in spacetimes with a singularity) can't be right at the point where the geodesics terminate in the model. So we have to find a better model. (Some of the rest of your comments might be attempts to describe a better model, but they don't seem like any of the proposals that I'm aware of.)

I've googled "scale factor:" dozens of times, and nothing I've been able to comprehend would justify extending the funnel (or stack of discs) analogy until a funnel with curved sides would dwindle into an infinitely thin line

Nobody thinks that model is valid at the point of a singularity, so no such justification is needed.

is there something POSITIVE in the math that would leave that extended analogy markedly less correct that the prevailing one of a straight-sided funnel ending at a singular point?

I don't understand. The "straight-sided funnel" is a different model: it models a spacetime whose rate of expansion never changes. We know experimentally that the rate of expansion of our universe has changed, so the "straight-sided funnel" model does not describe it.