Redshift z

[quantumtopology]

Scientific Method

 

Confirmation bias

 

Hmmm..., I am aware of the possibility of falling into "confirmation bias" for everyone, but I'm not sure if you bring it up to point out one of the main problems of current cosmology or you are hinting at some specific thing I wrote and you were tired or in a hurry. :)

[modest]

I was thinking that the present discussion could benefit from using the scientific method and avoiding confirmation bias. I’m sure this would go for me as much as anyone. :)

 

~modest

[modest]

To demonstrate numerically (I'm too tired at the moment to plot another diagram), let D_C be comoving distance and D_L be the look-back time,

D

_L

= 1 Gyr, D

_C

= 1.038 Gly

D

_L

= 2 Gyr, D

_C

= 2.156 Gly

D

_L

= 3 Gyr, D

_C

= 3.365 Gly

D

_L

= 4 Gyr, D

_C

= 4.678 Gly

D

_L

= 5 Gyr, D

_C

= 6.109 Gly

D

_L

= 6 Gyr, D

_C

= 7.681 Gly

D

_L

= 7 Gyr, D

_C

= 9.420 Gly

D

_L

= 8 Gyr, D

_C

= 11.367 Gly

D

_L

= 9 Gyr, D

_C

= 13.582 Gly

D

_L

= 10 Gyr, D

_C

= 16.165 Gly

D

_L

= 11 Gyr, D

_C

= 19.293 Gly

D

_L

= 12 Gyr, D

_C

= 23.363 Gly

D

_L

= 13 Gyr, D

_C

= 29.698 Gly

You can see that the change in look-back time decreases at greater comoving distance.

 

Here that plot is:

 

 

and look-back time vs. angular diameter:

 

 

~modest

[quantumtopology]

I was thinking that the present discussion could benefit from using the scientific method and avoiding confirmation bias. I’m sure this would go for me as much as anyone. ;)

 

:)

 

That I take for granted is our common intention, whether we always achieve it is of course arguable.

 

But there is another side to it, at least in my case I maintain a very humble attitude towards this cosmological thing, in the sense that sometimes I feel that pretending to know the essence of the universe s a whole is a task that might be beyond our mental power as humans, so even if I state something affirmatevely, I don't really mean it must be that way, I'm only exploring possibilities.

When I hear many physicist speak about the "first intants of the universe" as if they had been there, I think that's really pretentious.

[coldcreation]

Nice layout above modest. The great value of this dialogue with you is that your arguments compel me to get my ducks in a row, and most importantly, point to areas where I might be going wrong: once again (to paraphrase Hilton Ratcliffe). :hihi:

 

 

Glad I could help :dog:

 

Thanks. Taking short cuts often leads to inaccuracies. But short cuts save time, which is money. :)

 

 

The most obvious question is why D_{(look-back time)} = 13.5 is a horizon. Why can't you see things at D = 14? In an expanding model it makes sense that the universe is only so old. There would be a visible horizon at some point corresponding to the look-back time beyond which the universe was filled with matter too dense for light to get around.

 

Fair question. Again, it's just a short cut to save time. I haven't yet sat down to determine the actual distances that would be observed in a stationary hyperbolic universe.

 

The way I chose to depict the look-back time is based on the observations that make distant SNe Ia appear further away than their redshift would indicate (at least according to the pre-1998 standard model). But that is not all. The idea that distances appear to 'increase' with look-back time, rather than 'decrease' is consistent with the time dilation exhibited by the SNe Ia light curves or rise times. They appear to transpire slower than would be the case in, say, a universe with no spatiotemporal curvature (or without accelerated expansion).

 

Certainly the concept of the universe acceleration "now" as opposed to some other time in the past would be consistent with model C, where distances between spherical shell would appear to become smaller with distance. And certainly too, when plotting the look-back time we have a choice between setting Gly linearly, or redshift z linearly. Both cannot be linear, as that would not agree with observations, but both can be nonlinear. My hunch is that both should be expressed non-linearly, which is exactly what is shown in Figures A.

 

 

Depending of the exact specifications of such a model, distances may turn out to be further away than actually assumed (again, my hunch is that objects would not appear closer). The figure of 13.5 Gly to the horizon is certainly model-based. And yes, it's the wrong model to be basing distances in Figures A. Obviously different models predict different distances. For now, I've accepted the actual distance measurements based on the wide variety of methods described earlier. The point is that, yes, distance will likely differ from the latter, but the shape of the field should remain hyperbolic, i.e., Figure 7A would still hold, in that time dilation and spatial increments would still appear to be greater, larger with distance, but with a new distance to the horizon.

 

First, let's recall that the lookback time tL to an object is the difference between the age t0 of the universe now (at observation) and the age te of the universe at the time the photons were emitted (according to the source). It is used to predict properties of high-redshift objects with evolutionary models. E (z) is the time derivative of the logarithm of the scale factor a (t); the scale factor is proportional to (1 + z), so the product (1 + z) E (z) is proportional to the derivative of z with respect to the lookback time. Source.

 

Keep in mind, too, that the local universe is the reference frame from which we (or any observer) measure distances in the look-back time, and the local geometry of the universe appears Euclidean (this would be true in both hyperbolic and spherical geometries). The further we look out into the past (particularly beyond 2 Gly), the greater is the deviation from linearity, away from the local Euclidean reference frame. We all agree, I think that there is a departure from linearity. The key question is now, whether that departure would best be expressed as being of the type Figure A or C.

 

But your plotted graphs are deceptive in that they seem to portray "greater change" or departure from linearity locally, as opposed to at great distances (where there is a flattening out, or a tendency to return to linearity; 'less change'). Something clearly makes no sense: Locally the universe is flat (like the surface of earth locally), then not flat, then gets flatter again. See what I mean?

 

I'll come back to this in a moment.

 

 

But, if you're suggesting that things are static then I don't understand why D=13.5 would correspond to a horizon.

 

I would suggest that D=13.5 should correspond to z=∞. I think that would fix the problem, but the diagram has D(13.5)=z(10) which clearly seems problematic if you want D=13.5 to be a horizon (regardless if the horizon is an effect of redshift or time dilation).

 

At some point in space and time in a geometrically hyperbolic universe light (across the entire electromagnetic spectrum) will cease to arrive at the observer from the horizon. Where and when that point is still need to be determined. I have used 13.5 Gly for convenience. Certainly redshift z would tend to infinity, but where and when, or whether z = ∞ would be attained from the observers perspective depends on the model. Redshift would certainly rise to infinity quickly. The question is would the fact that objects are no longer 'visible' mean that z = ∞, or some arbitrarily large value?

 

For a more complex and fully developed cosmological model of this type, integrating the degree of curvature from the current epoch (z = 0) to (z = ∞) in collaboration with other data, should yield an expression for the geometrical curvature of the global field, from the observer to the horizon (at least).

 

 

 

The striking feature observed here (as in all of the diagram A series) is that the incremental distances of 1.35 Gly 'spherical' shells centered on the observer (at O) appear to increase spatiotemporally the further one gazes into the look-back time, as if stretched, dilated, or both.

 

I have a problem with that assertion. You drew the look-back time quadratically (as you say, "incremental distances of 1.35 Gly... appear to increase") but there is no reason that I can figure for you to have done that. You could just as easily had "incremental distances of 1.35 Gly... appear to decrease" and if you did that it would allow you to say "incremental distances of 1.35 Gly appear to decrease relative to redshift". In other words, between z=1 and z=2 there is quite a bit more look-back time than between z=3 and z=4.

 

It seems to me your assessment would be in contradiction with SNe Ia observations.

 

Though what you write could still be seen as hyperbolic, in the Poincaré model sense, but it would represent a slice through the observable universe, which in the look-back time seems to be dilated increasingly with distance.

 

The idea that the universe is accelerating "now" (and for the past few Gyr) faster than it expanded in the past is deceiving because it would imply a universe that resembles C rather than A: where space is stretched apart more so in the central region of the spherical shells (near the observer), and slows down with distance towards the out edge, appearing more compact near the horizon. The extrapolation of these observations (SNe Ia data) to a cosmic time t would have a structure like C above. But in the look-back time the exact opposite is observed (the only "now" is at the observer). Object appear further away and time appears dilated from our rest-frame (or that of anyone else).

 

 

So the range 9 < z < 10 has a much smaller change in look-back time than 1 < z < 2. Hopefully you see what I mean. It is not the case that incremental distances of look-back time appear to increase relative to redshift. It is rather the opposite.

 

Clearly, both redshift and look-back time would appear nonlinearly to increase with distance. Events appear to take longer in the past relative to the observer's rest-frame. This implies that both spatial and temporal increments and intervals (such as of the outer-most spherical shell) would appear large when viewed from the origin O, i.e., relative to local 'clocks' and 'rods.'

 

So the range 9 < z < 10 has a much larger change in look-back time than 1 < z < 2. And the local universe appears quasi-Euclidean (not stretched out).

 

 

 

To be consistent with your diagram we would need the curve to be a hyperbola. It is possible that you didn't mean that look-back time increases hyperbolically *with respect to redshift*, so the pertinent question would be: relative to what does the change in look-back time increase?

 

The Hubble law is a linear relationship between distance and redshift which assumes that the rate of expansion of the universe is constant. However, when the universe was 'younger,' the expansion rate, and therefore the Hubble 'constant,' was larger than it is today. For more distant galaxies, whose light has been traveling for much longer times, the approximation of constant expansion rate fails, and the Hubble law becomes a non-linear integral relationship dependent on the history of the expansion rate since the emission of the light from the source in question. Observations of the redshift-distance relationship can be used, then, to determine the geometric structure of the universe.

 

For many decades it was believed that the expansion rate was continuously decreasing since the big bang. But recent observations of the redshift-distance relationship using SNe Ia have suggested that in comparatively recent times the expansion rate of the universe has begun to accelerate. Source

 

That can be interpreted geometrically, if we consider the local universe to be quasi-Euclidean out to about 2 Gly, as a departure from linearity with increasing distance, with increasing look-back time. SNe Ia appear, thus, to be dimmer, further away than they would otherwise in a flat, Euclidean universe.

 

The spatiotemporal changes in the look-back time increase (appearing larger) due to hyperbolicity relative to local events and measurements. And this appears to be the case based on a wide variety of data. And, hence, dissimilar to the spacetime manifold represented by figure C. in other words, SNe Ia used as accurate cosmic clocks demonstrate the cosmological time dilation as depicted in Figures A.

 

SNe Ia appear fainter than their local counterparts. Compared with Friedmann models of the universe, the distant SNe are too faint even for a freely coasting “empty” de Sitter universe (Source: Cosmological Implications from Observations of Type Ia Supernovae)

 

This would mean the geometry of the universe has a greater curvature (hyperbolicity) than a empty de Sitter model.

 

Observations of SNe Ia at three epochs in the look-back time conclusively verify the effects of time dilation: temporal changes in the spectra of distant SNe Ia are slower than those of nearby SNe by roughly a factor of 1.36 (Filippenko et al. 1998) Source:

 

We find that the light curves of high-redshift SNe Ia are stretched in a manner consistent with the expansion of space; similarly, their spectra exhibit slower temporal evolution (by a factor of 1 + z) than those of nearby SNe Ia. The luminosity distances of our 16 high-redshift SNe Ia are, on average, 10–15 farther than expected in a low mass-density (ΩM = 0.2) universe with out a cosmological constant. Our analysis strongly supports eternally expanding models with positive cosmological constant and a current acceleration of the expansion. (Filipenko and Riess)

 

 

 

The other observable clearly isn't redshift and you've previously indicated that it isn't galaxy count. It cannot be the case that "incremental distances of 1.35 Gly... appear to increase" relative to itself, so I think we need to put some consideration into this. As it stands, I see no reason to draw the diagram like diagram A rather than diagram B or C.

 

I think it's fair to say that time dilation increase with redshift (i.e., with distance) by at least a factor of (1 + z). See here for example: Time Dilation from Spectral Feature Age Measurements of Type Ia Supernovae. In this study the time dilation factor of (1 + z) is attributed to the expansion of a homogeneous, isotropic universe. In a non-expanding universe that time dilation factor would be attributed to curvature, just a redshift z. So your plotting redshift z and look-back time linearly makes no sense, since it seems to yields greater time dilation at low redshift, and less time dilation at high-z, something inconsistent with observations.

 

 

 

But, in standard cosmology the diagram would not look like you have drawn it. It would look like diagram C. The most natural thing in ΛCDM against which to compare look-back time is comoving distance. Just like the redshift / look-back time diagram above, a "comoving distance / look-back time diagram" would have positive rather than negative curvature in ΛCDM.

 

Even in the simplest case of an expanding model, the critical model, or the so-called Einstein-de Sitter model (but any expansion model will do), by virtue that objects twice as far recede twice as fast, objects four times as far recede four times as fast, etc, implies that a cross section of an expanding model would resemble Figures A (not B or C). At first glance this redshift-distance relation would appear to be a linear relation of the type B (consistent with Euclidean geometry).

 

In other words, redshift z alone, when interpreted as expansion (or as a curved spacetime phenomenon) is a hyperbolic expression.

 

 

Let's look at this graph below (

Source):

 

 

The ordinate denotes relative age: The present time is given by "1", with nearby galaxies that appear most fully evolved (to us in the present time) having very low redshifts. The exponental drop in the curves (the red curve applies to a Universe with 70% Dark Matter; the blue curve described a Universe without Dark Energy [Cosmological Constant = 0]) shows that the maximum rate of increase in the value of 'z' occurred when the Universe was less than a relative 0.2.

 

 

The redshift of galaxies lying beyond 10 billion light years are observed to rise more rapidly than those observed closer to the present time.

 

 

And in the illustration below; confirmation of the same relation:

 

 

 

 

Note, again, the nonlinearity of the look-back time on the left of the graph. It can be seen that the greatest "change" or departure from linearity (when look-back time is plotted against redshift z) occurs at large distances and exponentially increases away from the observer. This is consistent with the hyperbolicity expressed in Figure 7A (and all the A series).

 

Cosmic distances are generally used to relate redshift values to recessional velocities. When redshifts begin to exceed z = 1, the 'velocity' of the objects in the manifold begin to approach relativistic values. That is, they are ever larger fractions of the speed of light (not ever smaller fractions of light speed). Thus, although the actual speeds continue to increase, the incremental rate of velocity increase itself decreases (slope asymptote approaches 0). This gives rise to a redshift vs recessional speed curve that is like this (Same source as above):

 

 

 

 

This too can be interpreted as a hyperbolic expression for the relationship between recessional velocity (km/s) and reshift z, by virtue of the objects increasing velocity by ever larger fractions of the speed of light.

 

 

To demonstrate numerically (I'm too tired at the moment to plot another diagram), let D_C be comoving distance and D_L be the look-back time, [snip]. You can see that the change in look-back time decreases at greater comoving distance. As D_L approaches the age of the universe D_C approaches 46 Gly (today's horizon in comoving distance)

 

In the look-back time, according to expanding models, galaxies are receding at an increasing rate (as viewed from the observer). And they continue to do so, in the look-back time, until they appear to be receding at the speed of light (as they disappear beyond the horizon). Clearly, that would resemble Figures A (not C) when you make a cross-section slice through the observer rest-frame.

 

In an expanding universe, objects near the horizon 'moving' close to the speed of light will experience deviations from linearity due to a special relativistic time dilation, which in subsequently corrected for by introducing the Lorentz factor γ into the classical Doppler formula. In a curved spacetime interpretation of redshift z, this effect of time dilation is one of general relativity. That would bean that rather plotting the look-back time linearly (as in special relativity) the plot should be expressed non-linearly. The question is whether the nonlinearity should be expressed as figures A or figures C (B in excluded from the start since it is a Euclidean manifold from the observer to the horizon). To answer that question one must turn to empirical observations.

 

Back to the SNe Ia data: A supernova at redshift z will appear to age (1 + z) times more slowly with respect to a local event at z ≈ 0. The prediction of time dilation proportional to (1+z ) is generic to expanding universe models, whether the underlying theory be general relativity (e.g., the FLRW model), special relativity (e.g., the Milne Universe), or Newtonian expansion. See Time Dilation in Type Ia Supernova Spectra at High Redshift.

 

High-redshift Type Ia supernovae are fainter than expected in a decelerating (25% dimmer, or 0.25 mag) or freely coasting universe (15% dimmer), suggesting that the expansion of the universe is accelerating with time. Source: Filippenko

 

For an object of known brightness, the fainter the object the farther away it is and the further back in time you are looking. Source: Perlmutter.

 

 

Figure 1: The Hubble plot: A history of the "size" of the Universe

 

 

This Hubble diagram is shows the "stretching" of the universe as a function of time. As you look farther and farther away, and further back in time, you can find the deviations in the expansion rate. As the universe expands, the wavelengths of the photons travelling to us are 'stretched' exactly proportionately—that is the redshift.

 

In particular, you can think of making a measurement of a supernova explosion at one given time in history. If, for example, you found more redshift at that time than expected from the current expansion rate, that would imply that the expansion was faster in the past and has been slowing down. This would lead you to conclude that there was a higher mass density in the universe.

 

Clearly, the universe appears to be 'stretched' further at high-redshift, than locally (at low-redshift, or at no redshift, at the observer).

 

What this means is that there is a deviations away from a flatness (when Λ = 0). According to Perlmutter: "There would have to be a systematic error as large as almost 50 percent of the brightness of the supernovae for there to be no cosmological constant, if the universe is flat. This is highly unlikely."

 

That is why dark energy (a huge component) had to be introduced. In other words, if one is to judge the departure from linearity away from the flat pre-1998 critical model (ironically called the Einstein-de Sitter: neither of which were flat) without dark energy, the departure from linearity would be huge.

 

Dark energy essentially changes the curvature of the universe, making it either geometrically flat or closed (spherically as in figure C). Without dark energy (i.e., without Lambda-CDM) we would be left with a hyperbolic geometry that resembles figure A.

 

 

If we find more redshift in the past this would imply that the expansion has been slowing down and hence there is more mass in the universe to slow it down. However, we now know that there are other possible cosmological parameters that can work in the other direction; for example, a vacuum energy density can make the universe expand faster. Thus there is a degeneracy here; it is hard to tell apart a situation with more mass or less vacuum energy density.

 

[...]

 

 

First of all, almost anyone who has taken a course in cosmology in the previous 25 years will have gotten used to the idea that the curvature of the universe determines its destiny. Thus, a universe that is spatially closed, i.e., one that curves in on itself, will eventually slow to a halt in its expansion and then collapse again; so it will come to an end. On the other hand, a universe that is either flat or curved open will expand forever. This tight relationship between curvature and destiny, however, is only true if we ignore the cosmological constant.

 

With the cosmological constant in the story, all four scenarios are possible: [...][/b]

 

I have chosen to ignore the cosmological constant. If there is no cosmological constant (or if it's value is zero) then the excessive dimness of high-z SNe Ia (making than appear further away than expected) is the signature of hyperbolicity.

 

A spatially closed (geometrically spherical) universe is one that would resemble figure C. But this is not consistent with observations of SNe Ia.

 

Still ignoring dark energy; a spatially flat universe (one that is Euclidean, of the type figure :naughty:, light curves and redshift z displayed by distant SNe Ia would be linearly equivalent to their low-redshift counterparts (i.e., there would be no deviation from linearity). That is not observed.

 

 

The important conclusions can be drawn (again ignoring dark energy): (1) The SNe Ia data is very far from being consistent with a cosmology (pre-Lambda-CDM), that is geometrically Euclidean, "flat." (2) Observations are inconsistent with a universe that is geometrically spherical, "closed." (3) The deviation from linearity observed in SNe Ia data suggests an "open" geometrically hyperbolic universe.

 

 

 

So, you say that your diagram and the data that it represents can be interpreted two ways, but the diagram itself is not clear in what it represents. As far as I can tell, your choice to plot look-back time like diagram A ( where "incremental distances of 1.35 Gly... appear to increase") is arbitrary and doesn't seem to correspond to any data that I know or that you gave.

 

For example, I see you plotted a graph above, of angular diameter distance vs. look-back time. But your plot is misleading. As it turns out, beyond a certain redshift (roughly z = 1.5), the angular diameter distance gets smaller with increasing redshift. But what that means is; an object 'behind' another of the same size, appears larger on the sky, and would therefore have a smaller "angular diameter distance". The fact that one of two objects locate further away that the other of the same intrinsic size would appear larger is a signature of hyperbolicity.

 

In other words, in the currently favoured geometric model of our universe, objects at redshifts greater than about 1.5 appear larger on the sky with increasing redshift. Source: angular size redshift relation. Not only is that a sign of the non-Euclidean nature of the universe, but it is a sign that diagrams A, rather than C, are the more accurate representation of the geometric structure of the physical universe.

 

 

 

It seems to me that we cannot claim the data can be interpreted as static and hyperbolic unless the data is predicted with a static and hyperbolic model.

 

Here is yet another example of the nonlinearity observed with cosmic expansion (same source: nasa.gov):

 

 

 

This relationship describes the redshift in terms of the Scale Factor (referring to the particular time when the light was emitted). Note, from about redshift 2 there is a large change is the scale factor, and this deviation from a linear expansion continues exponentially with increasing redshift z.

 

The plot of z + 1 versus r/R (the distance out to any galaxy ratioed to the distance out to a Universe's edge, set at the Scale Factor R) shows the exponential character of this curve.

 

The scale factor (the radius of the universe) becomes exponentially larger with increasing redshift.

 

That is exactly what happens in Figure 6A. So both the Doppler-like redshift interpretation, and the curved spacetime redshift can claim the data consistent with either expansion of hyperbolicity (respectively): details pending further investigation, of course.

 

While (1) the redshift, according to the standard model (a neo-Newtonian type cosmology), is due to the relative expansion of space (the change in the scale factor to the metric), rather than actual speeding up of increasingly distant galaxies, (2) the general relativistic approach to redshift z, to some extent describes a relative curvature of spacetime, rather than an actual curvature of spacetime found locally. That would be so since each observer is entitled to see the universe as quasi-Euclidean (from wherever they are located) with curvature increasing exponentially (or quadratically) with increasing distance.

 

 

If we could get this same data (as well as other, and preferably directly observable data) by using a static model rather than using standard cosmology then I think saying "both interpretations are valid..." would be quite fine. That, I believe, should be the direction of investigation.

 

True, but for now we're reflecting on the general idea. We don't have the detailed predictions. So all we can do is speculate that both interpretations agree with observations, but only one can be valid. In order to determine which is valid a detailed model is required.

 

For now it seems that in order to compare interpretations we must base the concepts on the observables that we do have at our disposition. That's all we can do.

 

So, while there is speculation involved, the interpretation is not solely based on an idea or a belief, but on some of the physical properties of the observable universe that has already been interpreted and perhaps need to be reinterpreted...

 

 

Otherwise, the idea that the data corresponds is an untested hypothesis and the assertion that it corresponds is an assumption.

 

Obviously the hope would be that the model makes prediction that can be tested against both empirical evidence and the concordance model. One has to begin somewhere!

 

 

 

So, my biggest objection is that we haven't done anything to determine if the above claim is valid or complete rubbish. As far as I can tell, it is right now a hollow claim. The focus of investigation should be on deriving a static model or method (or something) that predicts the observables that are right now being solved with standard cosmology and assumed compatible with some unknown static solution.

 

It sounds like we would need an entire army of physicists to get the job done. Or is this the job of a small, unfunded, group of individuals?

 

 

 

The motion of the objects in the solar system allows the maintenance of stability (for timescales exceeding 10 Gyr).

 

As long as objects in the universe (such as galaxies, or galaxy clusters and superclusters) remain in motion relative to one another, stability can be acquired and maintained, in accord with both classical Newtonian mechanics and Einstein's general theory of relativity. There is no need to extrapolate the problem of stability or instability to the entire universe. If the universe is infinite spatiotemporally it would make even less sense. An infinite universe would have no physical reason to expand or collapse. There is no center, or center of gravity towards which to collapse.

 

If the visible universe had a massive center with pockets of orbiting mass then I'd agree.

 

It is precisely because the universe has no unique massive center that permits equilibrium. If the universe had a massive center with pockets of orbiting mass then I'd agree that the stability of the universe would be in peril.

 

 

There is no need for the visible universe to have one "massive center with pockets of orbiting mass". The fact is that there are very many self-gravitating massive objects with pockets of orbiting masses. These are groups of objects that often comprise huge systems, groups of objects that have 'survived' the initial formation processes (i.e., the objects that did not disperse or collapse). So the same process responsible for the maintenance of stability of the solar system—the relationship between, mass, gravitational potential, distance and velocity, but too, angular momentum, resonance ratios, Lagrange points, centrifugal force, and so on—are responsible for the formation and subsequent longevity of the largest systems in the universe.

 

 

The topic of stability is of equal importance as the actual shape or curvature of the universe: the contention that the universe can no longer be considered geometrically Euclidean (flat).

 

The idea that static solutions to the field equations are required of a static model is not justified by empirical evidence, i.e, the notion that the universe is unstable against cosmic expansion or gravitational collapse can no longer be retained.

 

The conclusion would be: There is a good possibility the universe is hyperbolically curved, dynamic yet stationary (i.e., the scale factor does not change with time). But to test the former a detailed model is required. And the latter follows from standard celestial mechanics and general relativity.

 

 

 

CC

[modest]

I believe there is a small issue that is affecting all of your conclusions and it would help if we were on the same page regarding that issue, so instead of replying to things point by point I'd rather address an underlying issue.

 

First though, I should correct,

 

For example, I see you plotted a graph above, of angular diameter distance vs. look-back time. But your plot is misleading. As it turns out, beyond a certain redshift (roughly z = 1.5), the angular diameter distance gets smaller with increasing redshift. But what that means is; an object 'behind' another of the same size, appears larger on the sky, and would therefore have a smaller "angular diameter distance" The fact that one of two objects locate further away that the other of the same intrinsic size would appear larger is a signature of hyperbolicity.

 

According to this graph:

The apparent diameter of objects of equal size gets smaller with distance up until a look-back time of roughly 9.75 (which corresponds to a redshift of 1.689) after which point they get larger. In fact, this:

 

As it turns out, beyond a certain redshift (roughly z = 1.5), the angular diameter distance gets smaller with increasing redshift.

 

is indeed what my diagram shows. It might be more apparent if I add redshift to the graph:

 

 

The angular diameter distance gets smaller above redshift 1.689, hence the angular diameter gets larger above that redshift. This behavior would be an aspect of all expanding models.

 

 

 

The issue that I mentioned in my last post and that I really want to communicate is that one cannot say that distances are non-linear unless they say relative to what those distance are non-linear.

 

Imagine a person who is in a 2 dimensional manifold and they want to know if the manifold is curved. They look around and measure the distance to some objects using a ruler (we'll call this proper distance D_P). They measure the distance to 8 objects (analogous to galaxies):

 

 

The observer is the red dot in the center and the black dots are the objects being measured. The dotted lines represent the distances being measured.

 

From this data alone it is impossible for our 2 dimensional friend to know if he is on a hyperbolic, flat, or spherical surface. It would make no sense for him to say "these distances are less than (or greater than) I expected.

 

To solve the matter he could make two assumptions 1) the manifold is not expanding or shrinking and 2) all of the objects are the same size, x. After doing this he measures the apparent angular diameter of each object, [math]\theta[/math] and uses this formula,

[math]D_A= \frac{x}{\theta}[/math]

to calculate the angular diameter distance (which we'll call D_A) of each object. He now has two distance measures for each object, D_P and D_A:

 

 

From this he can figure out the curvature. He can tell if his manifold is like your diagram A, B, or C. All he has to do is plot his data:

 

 

and derive the fact that this relationship is true:

 

 

Assuming the manifold is not expanding or shrinking he concludes from the data that he lives on a hyperbolic manifold. The diagram directly above claims that at a given distance an object will have a smaller angular diameter (and therefore a larger angular diameter distance) in hyperbolic geometry than euclidean geometry. And, also, that at a given distance an object will have a larger angular diameter (and therefore a smaller angular diameter distance) in spherical geometry than euclidean geometry.

 

In other words, hyperbolic geometry has the effect of making things appear smaller and further away (when measured with angular diameter) than they would in euclidean geometry. This is the opposite of what you said in your last post,

 

The fact that one of two objects locate further away that the other of the same intrinsic size would appear larger is a signature of hyperbolicity.

 

so I feel like I should explain. First, let me give a source backing up the claim,

 

Spatial curvature may either increase or decrease the angular-diameter distance, with a spherical geometry focusing light rays and making objects appear larger, and a hyperbolic geometry having the opposite effect.

angular-diameter distance - not, physical, reciprocity relation

 

Light rays diverge with distance in negatively curved space. So, you might imagine that at X lightyears away there is a large object of diameter D. From earth we consider one degree of angular diameter and ask ourselves if the object of diameter D and distance X is larger than or smaller than that one degree of sky.

 

If space is negatively curved then the lines projected from the 1 degree angle spread out and the object of size X is smaller than those lines measure. In spherical geometry the lines converge and the object is larger than the lines measure. In euclidean geometry the lines are straight and the object fits exactly in the 1 degree triangle. Negative curvature has made the object of diameter X appear smaller in angular diameter as measured from some distance. This might be easy to visualize like so,

 

 

The observer measured the same angular diameter in each case and the white object is the same size in each case, therefore the object appears smallest in the negatively curved case. The angular diameter distance is greatest with negative curvature.

 

So, my main point, in case it got lost, is that one needs two distance measures in order to make any claims about distances increasing or decreasing hyperbolically or spherically. A distance can't change its rate relative to itself or to nothing. That is why I was saying that the data you plotted into diagram 7A was arbitrarily chosen to be like diagram A. You could just as easily plotted it like diagram C. There is not yet any data that you have based that decision on.

 

The incremental amounts of look-back time (the concentric circles) need to increase or decrease relative to something.

 

~modest

[quantumtopology]

So, my main point, in case it got lost, is that one needs two distance measures in order to make any claims about distances increasing or decreasing hyperbolically or spherically.

 

The incremental amounts of look-back time (the concentric circles) need to increase or decrease relative to something

 

I totally agree with this.

 

But I thought we all understood that we were comparing distance according to redshift with distance according to apparent luminosity, and that's were the surprise came with SNeIa (otherwise there wouldn't have any debate or change in the model from CDM to LCDM) ,they appeared as 25% more dim than expected according to redshift distance (and other distance measuring ways)

So for the saddle-shaped universe you just showed, the light emitted by the star is spread on a volume larger than 4pir²: this explains why their luminosity is lower than expected.Wich would be expected in a hyperbolic universe.

 

If I interpreted wrong what we were comparing since the first of CC's diagram, please somebody explain me!!

 

Regards

[coldcreation]

I totally agree with this.

 

But I thought we all understood that we were comparing distance according to redshift with distance according to apparent luminosity, and that's were the surprise came with SNeIa (otherwise there wouldn't have any debate or change in the model from CDM to LCDM) ,they appeared as 25% more dim than expected according to redshift distance (and other distance measuring ways).

 

So for the saddle-shaped universe you just showed, the light emitted by the star is spread on a volume larger than 4pir²: this explains why their luminosity is lower than expected.Wich would be expected in a hyperbolic universe.

 

If I interpreted wrong what we were comparing since the first of CC's diagram, please somebody explain me!!

 

I'm beginning to have doubts now, thanks to modest.

 

But all may be not be lost. Let me explain why in my next post.

 

First, I will plot a diagram of the type C, then we'll take the discussion from there, as whether the diagram would relate to distance observations (e.g., SNe Ia data).

 

Your first choice (diagram C) may have been correct.

 

 

 

CC

[modest]

I totally agree with this.

 

But I thought we all understood that we were comparing distance according to redshift with distance according to apparent luminosity

 

CC was plotting light-travel time. When compared with redshift or luminosity distance it would be like diagram C. The diagrams I've shown demonstrate.

 

and that's were the surprise came with SNeIa (otherwise there wouldn't have any debate or change in the model from CDM to LCDM) ,they appeared as 25% more dim than expected according to redshift distance (and other distance measuring ways)

So for the saddle-shaped universe you just showed, the light emitted by the star is spread on a volume larger than 4pir²: this explains why their luminosity is lower than expected.Wich would be expected in a hyperbolic universe.

 

If I interpreted wrong what we were comparing since the first of CC's diagram, please somebody explain me!!

 

Regards

 

It is true, as I've said, that adding negative curvature decreases luminosity (and increases luminosity distance). It does not follow, however, that "... this explains why their luminosity...".

 

The black curve is the expected cosmology before 1998 with a positive deceleration parameter. The blue curve is what we found from SNe-Ia. It is the current cosmology.

 

 

Adding negative curvature won't get you from the black line to the blue line, so just because negative curvature would have the effect of lifting that black line, that is not what the data shows.

 

~modest

[coldcreation]

It looks like my ducks were not lined up after all . Perhaps diagram C was the correct answer (meaning modest and quantumtopology would have been correct from the get-go, and coldcreation wrong).

 

 

The question now is: does changing the sign of curvature constitute the downfall of the entire foundation upon which redshift z and staticity may be expressed or represented in the physical universe?

 

In other words, could a general relativistically curved non-expanding (stationary) universe exhibit a geometry consistent with a spherical, Riemannian or pseudo-Riemannian manifold (as opposed to Lobachevskian hyperbolicity); where distances would appear to become smaller with greater distance in the look-back time, while retaining the interpretation of redshift z and the time dilation factor; as due to the general curvature of spacetime?

 

 

The apparent U-turn toward positive curvature may be a reasonable response to the problem of a curved spacetime interpretation of redshift z, just as the use of physics is a reasonable response to unphysical aspects of metaphysics. So, another question: is it better to discard a static model that basks in an aura of disappointment, or add an appendage onto it (e.g., curvature with a different sign) which changes the metric properties of the global field? It may still be premature to bring down the entire edifice.

 

I'm not sure yet to what extent changing the sign of curvature signals a need for a wholesale revision, or a major restructuring of the model.

 

 

Perhaps this alternative possibility should be placed out one the table, to see where it leads, if indeed anywhere. The idea, as with expansion models, is that observations must determine the curvature (and yes, too, the details of the model, which in this case remain elusive). All we can do is speculate right now as to which model is in best agreement with the empirical evidence. So let's leave open, then, the idea that curvature could take on two possible forms (pending further investigation, i.e., to see which best matches the empirical data).

 

The Euclidean manifold (B) may turn up later, but for now let's discard it on grounds that we have a universe that possesses a nonzero value for the mass-energy content. Gravity is everywhere present, and in accord with general relativity we adhere to the idea that a Euclidean manifold is untenable; not only due to disagreement with current observations, but it would violate general relativity. That may not please allot of people, but in this model redshift itself is a sign of curvature. In a non-expanding Euclidean manifold there would be no cosmological redshift z (recall, 'tired light' hypotheses are inconsistent with observations).

 

 

We continue the exposé with a schematic diagram that represents an alternative possibility. This reduced dimension portrayal of the manifold is consistent with figure C above. Let's see what happens when redshift z is plotted (somewhat arbitrarily for now) within the look-back time of the observer:

 

 

 

 

 

Figure 1C

Figure 1C represents a cross-section of a non-expanding globally homogenous and isotropic four-dimensional spherically symmetric geometrically curved Riemannian (or pseudo-Riemannian) general relativistic spatiotemporal manifold (i.e., a cross section of the visible universe in reduced dimension).

 

 

 

 

A few remarks:

 

 

• The key difference here, in contrast to Figure 6A (and all the A series), is that spatial distances appear (from the observer rest-frame, O) to become smaller with increasing distance in the look-back time.
   
   
  
• I have not labeled the distance between 'concentric' circles (spherical shells) centered upon the observer (O), but for the sake of argument, let's assume for now that each spherical shell would be less than 2 billion light years apart (or about 0.5 gigaparsec). So the distance to the horizon would be less than 20 Gly from the observer. This would undoubtedly change depending on model specifications.
   
   
  
• Redshift z is plotted more or less in accord with the standard model (out to about z = 1) but diverges exponentially as look-back time tends toward the horizon, where z approaches infinity (thanks Herr modest). This would be analogous to superluminal expansion (where galaxies appear to exceed the speed of light as they disappear beyond the horizon).
   
   
   
  
• The observer is located at the origin (O). Every other point of the polar coordinate system is in the look-back time (in 360° on the cross section of the manifold, i.e., in all directions), relative to any observer's rest-frame. So an observer located near the horizon would see the universe as if situated at O.
  

 

 

 

From the observer's rest-frame distances appear to become smaller with increasing distance. Time interval appear to slow down with increasing distance from O. This is the relative phenomenon of time dilation. Like the 1917 static de Sitter model, we have a situation where a clock placed at the observer will keep a different time than identical clocks placed elsewhere in the manifold. The timelike intervals depend on distance. The consequence would be that timelike intervals would become smaller for larger distance. In other words (let me see if I can get this right :clock:) clocks would appear to slow down with increasing distance. This is a de Sitter effect in a static universe. (I will come back to this).

 

Interestingly, even though we have switched from a hyperbolic spacetime to a spherical spactime representation (I'm not even sure of that anymore), we still have a de Sitter-like effect operational, but with geometrical properties that resemble Einstein's static model. That would be so since in either geometric structure (hyperbolic or spherical) redshift is a curved spacetime phenomenon (regardless of how the manifold is curved, or regardless of the sign, positive of negative).

 

Interestingly too, there is a 1926 paper (ApJ 64 321) where E. Hubble derives the the radius of curvature of an Einstein static model based on the mass density of nebulae. Hubble's uses the theoretical treatment of Haas (Haas, A. 1924, Introduction to Theoretical Physics, London, Constable & Co.). Even though this displacement toward longer wave-lengths is technically not the same as a de Sitter effect, it is still grounded on a non-expanding world-model. There would be a linear relation with distance over small distances (near the observer) with increasing divergence for larger distances.

 

Indeed, Figure 1C is a general relativistic spacetime manifold not dissimilar to Einstein's 1917 model, i.e. it too has hyperspherical topology and positive spatial curvature, space is neither expanding, contracting, nor flat. The differences are that (1) there is no global instability and (2) no cosmological constant. (3) There is a relationship between redshift z and geometry; a concept which forms the basis for the curved spacetime interpretation for z.

 

The above (redshift z and stability) can be seen as problematic, but they are not without a solution. Briefly, redshift z is not the same as local gravitational redshift, where light is affected as it 'climbs' out of a deep gravitational well. This is a relative effect due the curvature of spacetime as seen from an inertial rest-frame along the geodesic path of the photons prior to arrival. There may still be a missing mass problem, but not of the kind required by local gravitational redshifts, not of the kind required by Lambda-CDM, I suspect. (I will come back to this too).

 

Like the Chronometric model of Segal, I have not yet excluded a quadratic redshift-distance relation (though I have not embraced it either). Prior to the SNe Ia data a quadratic relation (where redshift increases as the square of the distance) had not been observed (using angular diameters and apparent magnitudes of galaxies), but in light of 'new' evidence, this hypothesis should be tested more rigorously.

 

And a quick note on stability: There is no need for a repulsive force to counter the attractive force of gravity since there is no absence of motion. A formal equilibrium is achieved throughout the universe (without balancing gravity and repulsion) in exactly the same way as equilibrium is achieved locally. This is a physical solution that bypasses the entire debate that has transpired in diverse circles for more than three centuries, culminating with the Einstein-de Sitter controversy (1917) and ending with the onset of expansion (1929).

 

The further difference is that the Einstein universe is spatially finite or closed (a three-sphere with a fixed radius r), i.e., with a fixed scale factor (though the model can also be interpreted as infinite spatiotemporally). Figure 1C extends only to the visible horizon, but the universe is considered here to be infinite and without bounds, globally homogenous and isotropic at any given cosmic time. Like the Einstein model, there is no beginning of time, and there is no big bang in the past (there is no expansion).

 

 

 

 

CC

[modest]

Your first choice (diagram C) may have been correct.

 

I agree.

 

Notice the first thing I said when you posted the diagrams:

 

If the lines represent the proper distance then two spaceships at the red galaxy traveling the same speed in opposite directions (one toward green and the other toward blue) would both reach their destinations in the same amount of time. This is the kind of distance cosmologists mean when talking about the shape of the universe and in this case the universe appears flat. Standard cosmology in proper cosmological distance would be option B.

 

If the lines represent the light travel time distance which is to say that light took the same amount of time to get from green to red as it did red to blue (where red to blue happened later than green to red) then I believe C would be the correct rendition.

 

If you're plotting light-travel time (ie look-back time) then diagram C would be correct.

 

But, light travel time is kind of an unnatural choice. Light-travel time mixes space and time. Normally spatial curvature would be talking about proper, comoving distance.

 

Like look-back time, however, the comoving distance is not directly observable.

 

~modest

[quantumtopology]

Interestingly, even though we have switched from a hyperbolic spacetime to a spherical spactime representation (I'm not even sure of that anymore)

 

I wouldn't jump to this conclusion so fast, I'm not at all sure that diagram C is a representation of a spherical universe from our euclidean perspective. Certainly when I chose it I was convinced of the contrary. This deserves a little more of time to be ellucidated.

[modest]

I wouldn't jump to this conclusion so fast, I'm not at all sure that diagram C is a representation of a spherical universe from our euclidean perspective. Certainly when I chose it I was convinced of the contrary. This deserves a little more of time to be ellucidated.

 

I agree.

 

Light travel time is going to end up looking positively curved in just about any situation, and that makes sense if the universe is expanding. But, this doesn't necessarily kill your idea. Firstly, because light travel time is not directly observable, even if standard cosmology renders it like so,

 

 

Who's to say that's right? There is no direct observation that says things are plotted correctly on the y axis of that graph.

 

Unfortunately, now that I think about it, there are other direct observations that would argue very strongly against static, hyperbolic spacetime. For example, quantumtopology posted this link, http://arxiv.org/PS_cache/astro-ph/pdf/9812/9812018v1.pdf, which data fitting a graph like so:

It would be exceedingly difficult to explain how all the data is above the euclidean line rather than below it where it would be expected in a static, hyperbolic universe.

 

Nonetheless, there might still be avenues worth exploring.

 

~modest

[quantumtopology]

Diagram C could be thought as a conformal projection of hyperbolic space from a Euclidean origin.

From Penrose "Road to reality" talking about the conformal representation of hyperbolic space or Poincare disk.

"Escher has used a particular representation of hyperbolic

geometry in which the entire ‘universe’ of the hyperbolic plane is

‘squashed’ into the interior of a circle in an ordinary Euclidean plane.

The bounding circle represents ‘infinity’ for this hyperbolic universe. We

can see that, in Escher’s picture, the fish appear to get very crowded as they

get close to this bounding circle. But we must think of this as an illusion.

Imagine that you happened to be one of the fish. Then whether you are

situated close to the rim of Escher’s picture or close to its centre, the entire

(hyperbolic) universe will look the same to you. The notion of ‘distance’ in

this geometry does not agree with that of the Euclidean plane in terms of

which it has been represented. As we look down upon Escher’s picture

from our Euclidean perspective, the Wsh near the bounding circle appear to

us to be getting very tiny. But from the ‘hyperbolic’ perspective of the white

or the black fish themselves, they think that they are exactly the same size

and shape as those near the centre. Moreover, although from our outside

Euclidean perspective they appear to get closer and closer to the bounding

circle itself, from their own hyperbolic perspective that boundary always

remains infinitely far away. Neither the bounding circle nor any of the

‘Euclidean’ space outside it has any existence for them. Their entire universe

consists of what to us seems to lie strictly within the circle."

It refers to the Escher drawing with fish , similar to the one showed here in Modest post 439 with bats.After all it wasn't a stereographic projection but the conformal representation, but the projective is quite similar.

There are other representations like the projective one, the hemispheric one ... but all of them resemble diagram C.

[coldcreation]

I wouldn't jump to this conclusion so fast, I'm not at all sure that diagram C is a representation of a spherical universe from our euclidean perspective. Certainly when I chose it I was convinced of the contrary. This deserves a little more of time to be ellucidated.

 

You may be correct. :)

 

 

Let's not forget, gentlemen, we've been considering the simplest case scenarios, where the universe is either hyperbolic of spherical (again, excluding the Euclidean manifold on observational and theoretical grounds).

 

Pictured below is another alternative which is more complex in certain respects than manifolds A, B or C, but certainly directly related to A and C.

 

 

Figure D

A reduced dimension Gaussian spacetime manifold representing the visible universe

 

 

Notice the area directly surrounding the observer (O). If each concentric circle represents 1 Gly, the local universe, out to about 2 Gly, appears quasi-Euclidean. From there, out to about 9 Gly in the look-back time the universe appears spherical, or positively curved in the Reimannian sense. Further, from 9 Gly, the manifold takes on a hyperbolic appearance (negative curvature). Though here the spherical shells (represented by concentric circles) become closer together with increasing distance, the same illustration could be made (with a two-dimensional Gaussian-like function) where the spherical shells appear to become larger with distance.

 

Figure D is a symmetric Gaussian "bell curve" shape that rapidly falls off towards plus or minus infinity. It is a surface of constant curvature. This type of curvature arises by applying exponential functions to a general quadratic function. The value of Gaussian curvature depends only on how distances are measured on the manifold.

 

In the construction of any physical model of the universe, where redshift is due to a curved spacetime phenomenon, this type of geometric curvature would have to be considered a viable option, until ruled out by observations.

 

So my point is that A and C may be slightly too simplistic, especially if we are to consider some type of evolution in the look-back time. The further point to be made is that limiting ones self to the notions of positive and negative curvature flanked on wither side of flat space (-1, 0, +1) seems to limit the scope of possibilities.

 

Sure, first we need a model (if we don't have one already) from which we can draw accurate distance measurements; one that best explains observations. And the conclusions regarding the metric structure of spacetime should follow. But in order to model accurate distances we need a better understanding of the global geometric properties of spacetime, of largest-scale structures and dynamics of the universe, its formation and evolution.

 

Finally, it must be understood the full implications of general relativity, not just as concerns very massive objects or velocities approaching c—or how it can be reconciled with the laws of quantum physics—but in its relation to the physical laws operational both locally and globally, concerning the passage of time, the motion of bodies, the propagation of light and the geometry of spacetime. Then, and only then, will it emerge an accurate representation of the essence of the physical universe and its evolution in time.

 

 

CC

[modest]

From the observer's rest-frame distances appear to become smaller with increasing distance. Time interval appear to slow down with increasing distance from O. This is the relative phenomenon of time dilation. Like the 1917 static de Sitter model, we have a situation where a clock placed at the observer will keep a different time than identical clocks placed elsewhere in the manifold. The timelike intervals depend on distance. The consequence would be that timelike intervals would become smaller for larger distance. In other words (let me see if I can get this right :phones:) clocks would appear to slow down with increasing distance. This is a de Sitter effect in a static universe. (I will come back to this)...

 

I need to think more about this and I'm incredibly short on time this week, but my first thought is that positive curvature provides a repulsive force. What's to keep these galaxies from flying apart? GR says they would want to.

 

The above (redshift z and stability) can be seen as problematic, but they are not without a solution. Briefly, redshift z is not the same as local gravitational redshift, where light is affected as it 'climbs' out of a deep gravitational well. This is a relative effect due the curvature of spacetime as seen from an inertial rest-frame along the geodesic path of the photons prior to arrival.

 

I don't see what would be wrong with that analogy. Positive curvature would be the opposite of a Schwarzschild solution. It'd be the complete opposite of this diagram:

 

-source

 

For any observer, it would be like sitting on a lagrange point. Geodesics converge in every direction. Clocks would be slowed in every direction and light approaching the observer would be redshifted. The problem with your interpretation, as I see it, would be the repulsive 'force'. Any object with some distance from an observer would want to follow an inertial path away from that observer.

 

~modest

[quantumtopology]

We've reached a slightly confusing point :phones: but perhaps that's good :)

Some questions.

 

Modest your graph from post 482 of look-back time versus angular diameter distance is from a expanding universe but in your opinion, does it agree with diagram C?

 

If you didn't know about the data from the CMB pointing so tightly towards a flat space, would you still reject a hyperbolic(expanding) space ?

 

Before we go on, I'd like us to reach an agreement about what diagram C is, would it be easier to consider it only spatial(3D), would it be then representing a hyperbolic view?

 

Regards