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## Novel Redshift Equation

Hi CosmoQuest

I'd like to introduce a theory math first here.

I'm hoping to come away from this with one of 2 things:

1. Evidence that there is no particular significance to this equation. A comparably simple and accurate equation to the one I am presenting, which would suggest that it is just an easy "fit equation", would definitely do the trick.

Or,

2. Help explaining why this equation works as well as it does. I have an explanation which led me to derive the equation, which I can describe in detail, but it is considerably outside the mainstream.

I will be very happy to present my alternative theory here, and hopefully this introductory post isn't too coy or boring; but we don't even need to waste the time on my idea if this equation isn't compelling.

and hey, if you can disabuse me of the notion that this equation is notable, you will save me from getting an embarrassing tattoo.

The equation relates the cosmological redshift to lookback time.

where (z) is the cosmological redshift, and (t) is lookback time where the present equals 0 and the origin equals 1.

The novel equation closely reproduces the relationship of redshift to lookback time in CDM, without any free parameters except the present age of the universe. Because of the relationship of redshift to scale factor, , the cosmological distance scales can also be computed starting from the novel equation. I've included a table of high z supernovas (names, redshifts and peak magnitudes taken from the Perlmutter paper) showing the close fit of this equation, labeled SL, to CDM, using Ned Wright's cosmology calculator with default inputs to generate the CDM values for lookback time, co-moving distance, luminosity distance. Co-moving distance, Luminosity distance and Absolute Magnitude for the SL columns are calculated using standard cosmology equations.

Next is a link to graph plotting the novel equation against two tables spanning a larger range of z values, generated from Ned Wrights CosmoCalc and the Light Cone Calculator.

GRAPH

That's it! Can you help confirm or deny the quantitative efficacy of this equation?

2. Hi substantialmaterials

Welcome to CQ!

I think you will first have to explain to us exactly HOW you got to your new equation.

3. Yes, echoing the previous comment, it seems to fit but if you are wondering whether it is a coincidence or not, we would need to know why you used logs and square roots and the like. Just for example, the price of water per liter is an equation price = volume x unit cost, and the distanced traveled by a train is distance = speed x time, and the equations look just the same, but that doesn't mean that the distance a train travels is somehow related to the unit cost.

But it is an interesting match, so please let us know.

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Originally Posted by substitutematerials

where (z) is the cosmological redshift, and (t) is lookback time where the present equals 0 and the origin equals 1.
The lookback time is a time thus that equation as described is wrong dimensionally. It has ln(time)/ time but z is a number with no units.
But t may be the dimensionless lookback time (tL/tH). But then you include a parameter that needs to be measured - the Hubble time tH) = 1/H0. N.B this is in addition to the present age of the universe that you say that you include.

What you may have is one of the solutions on that page which you may have come across accidently or have a physically valid derivation.

Also: A list of Distance Measures in Cosmology may be helpful.
Last edited by Reality Check; 2017-Mar-14 at 04:02 AM.

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Thanks for taking a look at this guys, Jens and Realitycheck it sounds like you agree that the fit is pretty good. Realitycheck, yes I am using dimensionless lookback time, and so you're quite right- to get a real time out of the equation you need to know the Hubble time through measurement.

To elaborate the mathematical underpinning of the novel equation, it is the definite integral, from the relative cosmic age of the emitter of light to the relative cosmic age of the observer of light, of 1/t expansion:

multiplied by the lorentz factor in which recessional velocity is proportional to lookback time with =1 at t=1, and is treated special relativistically:

Multiply them together and simplify and you get

As you can probably see, this derivation relies on non-standard assumptions. Realitycheck, unless I'm mistaken, the best formula from the standard cosmology for relating redshift to look back time is this:

The above equation is driven by the density of matter, radiation and dark energy, and is quite different from the heuristic I presented. They produce similar results when the above is parameterized according to present convention. Coincidence? Am I describing the Milne universe?

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Originally Posted by substitutematerials
Thanks for taking a look at this guys, Jens and Realitycheck it sounds like you agree that the fit is pretty good.
I actually ignored the fit because your equation looked wrong. What you may have is an coincidental fit from a invalid derivation.
Special relativity is the wrong theory to apply to the expansion of the universe.
How does "recessional velocity is proportional to lookback time" actually give a recessional velocity?

You do not have a universe containing matter, dark matter or dark energy so maybe a Milne universe but also not the universe we live in.

ETA: Look at the forms of your integral and the mainstream integral. Both are integrals of inverse functions and so natural logarithms will be produced. The mainstream integral has z in it so that should produce a Lorentz like factor.
Last edited by Reality Check; 2017-Mar-14 at 11:18 PM.

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Originally Posted by Reality Check

ETA: Look at the forms of your integral and the mainstream integral. Both are integrals of inverse functions and so natural logarithms will be produced. The mainstream integral has z in it so that should produce a Lorentz like factor.
Thanks for this, the formal similarity in the 2 equations is illuminating, I can see how the natural log would emerge from both, and am trying to understand how the inclusion of z would behave in a lorentz-like manner. However, like the flatness/critical density problem, it seems strange to me that density parameters would be so precisely balanced to yield a curve expressible by much simpler form.

****

So here's my very non-standard theory: The expansion of space is what causes light to propagate- they are synonymous. Light appears to travel 300,000 km/s because the universe grows radially by 300,000 km every second. I am 300,000 km from the space I occupied one second ago, which radiates away from me just as we think of light doing.

Since we know the speed of light is invariant, what I am positing implies an invariant "expansion speed" of space. In fact we measure time definitionally by the expansion of space in this model. The following metaphor of a vinyl record helps to show how this creates an inevitable distortion in the rate of time between different epochs.

On a vinyl record, sound waves are imprinted as grooves embedded in the surface. The grooves play back as sound because of a relative motion, the spinning of the record, relative to the playback needle. In this analogy, light is 'imprinted' on space like the grooves in the record. The relative motion between space and the observer of light is caused by spatial expansion. It would be like a stretchy vinyl record, that plays by expanding rather spinning, dragging the grooves beneath the needle as it grows. It is easy to see how this method of playback would introduce a distortion- by stretching the record, we would be stretching the sound grooves as well- increasing their wavelength and lowering their frequency. It is this effect that results in the cosmological redshift: an intrinsic dilation of time between observers and emitters of light, as function of their relative cosmic age.

So what we have been regarding as the hubble expansion should really be treated as time dilation, and it's evolution is governed by the equation we have been discussing. To understand the evolution of background temperature etc, this time dilation can be converted to constant time, or comoving coordinates,and thus becomes very numerically similar to the present cosmology- which is a universe much bigger than the simple speed of light expansion.

I have some examples of problems this would solve and predictions it would make, but I'll save them for a future post.

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Originally Posted by substitutematerials
So here's my very non-standard theory: The expansion of space is what causes light to propagate- they are synonymous.
Sorry, substitutematerials, but that is not a valid theory.
Maxwell's equations have light as an electromagnetic wave travelling at c (299,792,458 metres per second) in any universe (expanding or contracting).
The expansion of the universe does not expand bound systems, e.g. atoms, you, me, the Earth, the Solar System, galaxies. So the theory predicts that light does not travel from the Sun to us! See Why doesn't the Solar System expand if the whole Universe is expanding?
The expansion of the universe just expands light (red shift). Think about an extremely hypothetical static electromagnetic "wave". If we label a point on this "wave" and a point in front of it then the distance between the points will get bigger - the "wave" will not propagate at all.
The rate of expansion of the universe is measured to vary: Measurement of expansion and change of rate of expansion. Note that the "constant" in Hubble's constant is that the velocity of galaxies increases constantly with distance from us. Then there is dark energy - the measured acceleration of the expansion of the universe.
Last edited by Reality Check; 2017-Mar-16 at 08:36 PM.

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Originally Posted by Reality Check
Sorry, substitutematerials, but that is not a valid theory.
Maxwell's equations have light as an electromagnetic wave travelling at c (299,792,458 metres per second) in any universe (expanding or contracting).
The expansion of the universe does not expand bound systems, e.g. atoms, you, me, the Earth, the Solar System, galaxies. So the theory predicts that light does not travel from the Sun to us! See Why doesn't the Solar System expand if the whole Universe is expanding?
The expansion of the universe just expands light (red shift). Think about an extremely hypothetical static electromagnetic "wave". If we label a point on this "wave" and a point in front of it then the distance between the points will get bigger - the "wave" will not propagate at all.
The rate of expansion of the universe is measured to vary: Measurement of expansion and change of rate of expansion. Note that the "constant" in Hubble's constant is that the velocity of galaxies increases constantly with distance from us. Then there is dark energy - the measured acceleration of the expansion of the universe.
I don't think I am proposing any changes to Maxwell's equations. Just as in relativity, any observer in this model will see light 'traveling' at c regardless of any relative motion or gravitational field. I am, I think, making the claim that only universe possible is one where expansion occurs at c.

The measured rate of change in expansion as defined in CDM (which is a change in time dilation in this model) is accommodated by the equation I provided. The acceleration which necessitated the addition of dark energy emerges naturally from this changing rate of time. That's why the curves match.

The critical change here is that this model redefines the scaling effect measured between observable bodies- which is presently understood to be the expansion of space- to be a dilation of time. Space can expand without changing the distance between objects. This is a substantial redefinition, but can you understand it? Imagine a universe that is finite, with a measurable horizon. When I measure the expansion of space, I am measuring the increase in distance between myself and this horizon, not the distance between myself and any particular object. Gravitationally or otherwise bound objects can and do maintain constant distances. Emitters and observers are light are not at rest relative to the expanding fabric of space, but light is. Can you at least imagine this as a logically coherent alternative? I understand that this is in stark conflict with the extrapolation of General Relativity to a global framework in FLRW space.

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Originally Posted by substitutematerials
I don't think I am proposing any changes to Maxwell's equations.
You are denying Maxwell's equations give electromagnetic waves travelling at c by proposing that they have a speed of the rate of expansion of the universe which is measured to be not c and varies.
The other points are ignored or misunderstood.
• The expansion of the universe does not expand bound systems, e.g. atoms, you, me, the Earth, the Solar System, galaxies. So the theory predicts that light does not travel from the Sun to us!
See Why doesn't the Solar System expand if the whole Universe is expanding?
The billions of galaxies that are outside of the Virgo cluster are not gravitationally bound and obey Hubble's law (look at the graph from that article).
• The expansion of the universe just expands light (red shift). Think about an extremely hypothetical static electromagnetic "wave". If we label a point on this "wave" and a point in front of it then the distance between the points will get bigger - the "wave" will not propagate at all.
• The rate of expansion of the universe is measured to vary: Measurement of expansion and change of rate of expansion. Note that the "constant" in Hubble's constant is that the velocity of galaxies increases constantly with distance from us. Then there is dark energy - the measured acceleration of the expansion of the universe.

This post denies the meaning of expansion in cosmology which is that the distances between points in space (and thus bodies) gets bigger with time!
There is a hint of the mistake of thinking of a universe expanding into something with an example of a finite universe with a visible limit. However, when we measure the expansion of space we measure the distances to objects that we can detect, e.g. galaxies.

A expanding universe is supported by an enormous body of evidence, e.g. What is the evidence for the Big Bang?. An equation almost pulled out of thin air cannot explain this evidence.

So my first formal questions:
IF01: Please calculate the rate of expansion of the [gravitationally bound (thus not expanding) Solar System, e.g. the sized of the orbit of the Earth.
Hint: According to the mainstream it is parts per septillion over the lifetime of the Earth. According to your idea it could be 300,000 km every second, i.e. the light from the Sun can never reach us and we were in the Sun about 8 minutes ago. Or is it the rate of expansion = 0 and so the speed of light may be zero?
IF02: Calculate the speed of light in the Solar System according to your idea?

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Originally Posted by Reality Check

This post denies the meaning of expansion in cosmology which is that the distances between points in space (and thus bodies) gets bigger with time!
Yes! I am denying the standard meaning of expansion in cosmology, this is why I am in the Against the Mainstream forum section. I'm only presenting an alternative, I realize such a radical redefinition has an uphill battle. What I am claiming is that this parenthetical-(and thus bodies)- isn't a priori. Space can expand globally without an increase in distance between particular bodies, and still have local significance. In this model, we detect the expansion of space by measuring the propagation of light, not its cosmological redshift.

Originally Posted by Reality Check

IF01: Please calculate the rate of expansion of the [gravitationally bound (thus not expanding) Solar System, e.g. the sized of the orbit of the Earth.
Hint: According to the mainstream it is parts per septillion over the lifetime of the Earth. According to your idea it could be 300,000 km every second, i.e. the light from the Sun can never reach us and we were in the Sun about 8 minutes ago. Or is it the rate of expansion = 0 and so the speed of light may be zero?
So like I'm saying, the distance between the Earth and sun doesn't have to change even though space is expanding globally. However, the replacement concept for explaining cosmological redshift- time dilation- should in fact be present on the scale of the solar system. This is essentially saying that all clocks speed up over cosmic time, when compared to past clocks. If you look back to my vinyl record analogy, you can see the logical justification for this.

So your question prompts me to present a test for this theory- is there a universal clock acceleration? The effect would be very hard to detect, since it applies to all clocks and is fairly small, but the value of the clock acceleration should equal the Hubble parameter:

If you look at the common units for the Hubble parameter, (km/sec)/megaparsec, you can remove distance to yield inverse time sec^-1 as the SI has done. I am proposing that the correct units are sec/sec^2: an acceleration of proper time.

So how do we look for an effect that changes all clocks uniformly? The best thing I have been able to come up with is to use light delay to compare a clock to its own past rate- to bounce a signal off a distant repeater and look for a discrepancy. If anyone can think of an inexpensive way to set this up with adequate sensitivity, or a different test, I welcome it. The telemetry of the Pioneer spacecraft amounted to such an experiment, and a clock acceleration of the Station Clock was proposed as a highly speculative explanation for the apparent deceleration of the craft, the Pioneer Anomaly.

Because time and distance scales for a human experiment in the solar system are very small compared to the size and age of the universe, H_o can be treated as roughly constant for any practical experiment, although this is an approximation. So we can use the basic form for constant acceleration, Modeling the acceleration in this manner, Anderson et al calculated that the clock acceleration would need be 2.8E-18 sec/sec^2 to explain the anomaly. If we use 72 (km/sec)/mpsc for H_o, the clock acceleration would be 2.3E-18 sec/sec^2, so the Pioneer value is about 22% off. Considering the unintentional, uncontrolled nature of the experiment, especially thermal emission from the craft, this number is quite close.

So in this model, the Pioneer spacecraft appear to slowing down because the clock used to track their telemetry is intrinsically speeding up. This is not a real motion, but rather a fundamental dilation of time, on the scale of the solar system.

This idea makes everybody spit out their coffee, but you hint at it:

Originally Posted by Reality Check

i.e. the light from the Sun can never reach us and we were in the Sun about 8 minutes ago.
We were obviously not inside the sun eight minutes ago- but I'm making the argument that we are now occupying space that was occupied by the sun eight minutes ago. Light didn't travel, but rather was carried by the expansion of space from the sun's former position to our present position. This is an analogous argument to saying, "I am standing where the big bang happened 13.7 billion years ago", which is true in standard cosmology. By claiming that spatial expansion is universal, I am extending this argument to any event in our light cone.

Originally Posted by Reality Check
IF02: Calculate the speed of light in the Solar System according to your idea?
I don't have much to bear on this, except to say that if a second is defined as 300,000 km of spatial expansion, then c=1. I am not proposing a change to the speed of light, I am equating it with a new model of spatial expansion.

12. Originally Posted by substitutematerials
I am, I think, making the claim that only universe possible is one where expansion occurs at c.
I cannot understand what it would mean to say that the universe is expanding at c, or at any fixed speed.

Imagine a universe that is finite, with a measurable horizon. When I measure the expansion of space, I am measuring the increase in distance between myself and this horizon, not the distance between myself and any particular object. Gravitationally or otherwise bound objects can and do maintain constant distances.
If the horizon is moving at c, then presumably it is doing it for all observers. Which means they must all be stationary relative to one another, and the horizon is receding away from them, leaving an ever larger empty space around them. But that is not consistent with what we see.

Or do you mean that the horizon is moving away from us at c, but for distant galaxies (which are receding from us) the horizon is receding more slowly?

13. Originally Posted by substitutematerials
Space can expand globally without an increase in distance between particular bodies
Then what does it mean for space to expand? Expanding space is the increase in distance between bodies.

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Originally Posted by Strange
I cannot understand what it would mean to say that the universe is expanding at c, or at any fixed speed.

If the horizon is moving at c, then presumably it is doing it for all observers. Which means they must all be stationary relative to one another, and the horizon is receding away from them, leaving an ever larger empty space around them. But that is not consistent with what we see.

Or do you mean that the horizon is moving away from us at c, but for distant galaxies (which are receding from us) the horizon is receding more slowly?
Hi Strange, good to see you, thanks for joining in. I think the basic conceptual ask is the same as with the invariance of the speed of light in relativity. All observers measure light to be moving at c, but this does not necessitate that they are stationary relative to one another, right? The discrepancy in reference frames is compensated by time dilation and length contraction.

So we do see distant objects receding more slowly from the horizon from our perspective- i.e. having a recessional velocity that is a fraction of c, but we also see them as time dilated. Any observer at any time would measure the horizon to be receding from themselves at c, and see an object at the horizon as infinitely time dilated.

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Originally Posted by substitutematerials
Yes! I am denying the standard meaning of expansion in cosmology,...
That is not quite what I am saying. You are denying
• The standard meaning of expansion in cosmology (the distance between points getting bigger) and
• The standing meaning of expansion in English (things getting bigger).

A photon going from point A to B is not anything getting bigger. An electromagnetic wave propagating from point A to B is not anything getting bigger. The ATM idea is wrong both in English and cosmology.
There is no longer a Pioneer anomaly (resolved by thermal recoil by several papers from 2012) so your calculation has invalidated your ATM idea.

Second thoughts shows that my IF01/IF02 questions should be that you need to show that a universe expanding at the speed of light will have no measurable effects on the orbit of the Earth. This can be done for the much slower measured expansion of the universe. But looking at the paper suggests that it may be too complex for a formal question.

This still leaves the issue that the measured expansion of the universe is not the speed of light ! To be more exact, the measured redshift of galaxies increases with distance, so their velocities increases up to and past the speed of light: Frequently Asked Questions in Cosmology: Can objects move away from us faster than the speed of light?
This my actual first formal question is:
IF01: Derive the relationship between redshift and distance of galaxies in a universe expanding at the speed of light.
My guess: All of maybe millions of galaxies that we have measured redshift for will have the same redshift according to your ATM idea.

You mention time dilation (as in SR?). Note that this applies to time intervals, not time. So it cannot be arbitrarily plugged into for example red shift derivations. However there is an interesting confirmation of the varying velocity of galaxies - the duration of light curves of type 1a supernova time dilate as expected.
Last edited by Reality Check; 2017-Mar-20 at 12:28 AM.

16. Originally Posted by substitutematerials
So we do see distant objects receding more slowly from the horizon from our perspective- i.e. having a recessional velocity that is a fraction of c, but we also see them as time dilated. Any observer at any time would measure the horizon to be receding from themselves at c, and see an object at the horizon as infinitely time dilated.
So galaxies are moving away from each other. But if "space is expanding at c" then they must all be moving away from each other at c. Which is not what we see. And they would all be time dilated by the same amount.

So what accounts for the observations of Hubble's law?

I don't think I understand what you are trying to say.

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Here's what I'm positing it means for the universe to be expanding at c: The distance from me to the horizon is increasing by 300,000 km every second. One second ago, I was 300,000 km closer to the horizon than I am now, because the universe was smaller by this amount. This is different from conventional ideas of spatial expansion. But it isn't nonsensical.

IF01: Derive the relationship between redshift and distance of galaxies in a universe expanding at the speed of light.
My guess: All of maybe millions of galaxies that we have measured redshift for will have the same redshift according to your ATM idea.

Here you go:

The only 2 numbers we need to compute distances in this system are the measured redshift (z), and the present age of the universe (t_o). First we find the lookback time from the redshift, using the featured equation:

Now we can calculate the most commonly used distance scale, the Co-moving or constant time distance (X). This is given by:

from here Luminosity Distance (Dl) can be computed simply by:

which can then be used to compute absolute magnitude (M) if you have measured apparent magnitude (m).

The (SL) column values in the table I presented in my first post were calculated using these equations.

As an example here's values for z=1 and t_o=13.75E9 years:
t_lb=7.735E9 years, X=9.859E9 lightyears, D_l=19.717E9 lightyears.

I can calculate distances for any value of z that you'd like, but you don't have to take my word for it, please try this functional math yourself.

As for the Pioneer Anomaly, you're right that it is foolish of me to present it as evidence, because it is generally recognized as explained by mundane forces. But this hardly invalidates my idea- it wasn't a controlled experiment looking for such an effect!

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Originally Posted by substitutematerials
Here you go:...
What you imagine about the horizon is irrelevant because we have objects that we can measure the distances to and their velocities (galaxies and their redshifts). Lookback time is irrelevant. Redshift is not magnitude.
IF01: Derive the relationship between redshift and distance of galaxies in a universe expanding at the speed of light.

Mainstream cosmology can derive the empirical Hubble's law from theory. The question is can you derive your "Hubble's law" from your idea? Then the question will be does it match the empirical Hubble's law?

You stated that your idea predicts a value for the Pioneer anomaly. That the anomaly has been explained would be evidence against your ATM idea. But ... I missed that
1. You plugged in the Hubble constant (which varies with distance) rather than the speed of light !
So that prediction was invalid.
2. The anomaly is found using both signals from the spacecraft and the Doppler effect. The latter is not affected by clocks on the spacecraft.
Last edited by Reality Check; 2017-Mar-21 at 04:12 AM.

19. Originally Posted by substitutematerials
Here's what I'm positing it means for the universe to be expanding at c: The distance from me to the horizon is increasing by 300,000 km every second. One second ago, I was 300,000 km closer to the horizon than I am now, because the universe was smaller by this amount. This is different from conventional ideas of spatial expansion. But it isn't nonsensical.

IF01: Derive the relationship between redshift and distance of galaxies in a universe expanding at the speed of light.
My guess: All of maybe millions of galaxies that we have measured redshift for will have the same redshift according to your ATM idea.
Wait, what? How did you come to the conclusion that the universe is expanding at precisely c?

20. And, of course, we can observe galaxies receding at more than c.

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Originally Posted by Reality Check
What you imagine about the horizon is irrelevant because we have objects that we can measure the distances to and their velocities (galaxies and their redshifts). Lookback time is irrelevant. Redshift is not magnitude.
IF01: Derive the relationship between redshift and distance of galaxies in a universe expanding at the speed of light.
Mainstream cosmology can derive the empirical Hubble's law from theory. The question is can you derive your "Hubble's law" from your idea? Then the question will be does it match the empirical Hubble's law?
I think I've answered this question Reality Check. I've provided equations that allow us to relate distance to redshift. I've provided sample values that have good agreement with empirical observation. I'm not clear on what more information you are requesting. In this model, the fundamental relationship is between redshift and relative lookback time. The relationship of distance is redshift is secondary, although the models yield similar values.
Hubble's law

is an approximation- we've observed deviations from it-of the opposite sign from what we expected- and those deviations have required the addition of dark energy. They emerge naturally for the equations I am presenting.

And I'm certainly not claiming Magnitude is redshift! Magnitude is critical to measuring the distance to standard candles, and is a separate observable, so I included it.

Originally Posted by Reality Check
You stated that your idea predicts a value for the Pioneer anomaly. That the anomaly has been explained would be evidence against your ATM idea. But ... I missed that
You plugged in the Hubble constant (which varies with distance) rather than the speed of light !
So that prediction was invalid.
I 'm not claiming that the distance between us and the Pioneers is increasing at the speed of light, this would be absurd as you point out. Nor am I denying the importance of the Hubble parameter. What I am saying is that the Hubble parameter is an acceleration of clocks, which would explain the apparent deceleration of the spacecraft without any real change in speed or forces.
Originally Posted by Reality Check
[*]The anomaly is found using both signals from the spacecraft and the Doppler effect. The latter is not affected by clocks on the spacecraft.[/LIST]
I believe you are incorrect in this point. I've studied the literature in some detail, and I believe that clocks on the spacecraft are not used in the telemetry- the signal from the Station Clock is phase locked by the spacecraft, which function as a mirror for this signal. I used the Hubble parameter because this is the clock acceleration of the Station Clock.

But again, if no one thinks the anomaly is still an anomaly, then I rescind this as evidence. I would continue to make the prediction that a controlled experiment like the one I described will show a clock acceleration though, equal to 1/t_o, and would accept a null result as evidence agains this theory.

Originally Posted by Strange
And, of course, we can observe galaxies receding at more than c.
If you are defining recessional velocity by the difference between comoving distance and light travel distance ((X-d_lt)/t)), this system will also produce a recessional velocity greater than c. If you look at my co-moving coordinates, the universe is larger than would result from a simple speed of light expansion. This is a transformation, but is the most physically meaningful distance scale since it describes the evolution of background temperature and is the one in which we observe a homogenous and isotropic distribution of galaxies.

But it comes out of a universe that grows at c, which creates an intrinsic acceleration of clocks, which can be transformed to yield the larger universe that we see.

In fact, the comoving distance equation can be solved for a lookback time of 1 yielding a finite value, which means this universe I'm describing is finite. The parenthetical portion of the equation solves to 3.9207 for a t of 1, which means that the universe in co-moving coordinates is 3.9207ct, so the radius of the universe at present is 53.9 glyr. If there are observations of objects that are further than this distance, then my model yields a universe that is too small. In the above rubric for recessional velocity, this means an object at the horizon would be receding at 2.9207c. It seems to me that there is controversy about defining recessional velocity anyway- what really matters is distance.

Wait, what? How did you come to the conclusion that the universe is expanding at precisely c?
Hi Abaddon, My model begins by equating the expansion of space with the propagation of light. This is a big change, so hear me out !... When we measure the invariant speed of light, we are actually measuring the speed of spatial expansion, which is being defined as the speed the cosmological horizon is receding from any observer, i.e. how much the universe is getting bigger. So it's no coincidence that the universe is expanding at c, it is by definition. In fact, we measure a second each time the universe has grown by 300,000 km.This is the speed that past moments are connected to present ones. Astronomers often refer to telescopes as time machines, because we only see information from the past, due to the time it takes light to arrive to us. In this alternate model, it is not merely that light is an echo of a past moment, but actually an interaction with that past moment itself, which has expanded to encompass our position. While it feels like a disorienting funhouse mirror way of looking at the universe at first, it makes sense with some of the quantum behaviors of light, like non-locality phenomena such as entanglement. When you observe the photon, it is a interaction with the actual moment of its emission.

If you refer backwards in this topic to my record player analogy, you can see why this would cause an intrinsic change in clock speed as the universe evolves. It is this clock acceleration thats we presently associate with spatial expansion, and can be transformed to coordinates where all galaxies are moving away from each other uniformly. The equation I began with has good numerical agreement with the present cosmology.

22. How does your model account for accelerating expansion rate?

You talk about this "horizon" as if it were some sort of boundary. Is that defining the size of the universe? Is this the same as the cosmological horizon? Is there anything beyond this horizon in your model?

23. Originally Posted by Strange
How does your model account for accelerating expansion rate?
Maybe some sort of unknown energy source? Call it - I don't know, "mystery energy" or "unknown energy" or "dark energy", something like that. It's a perfectly mainstream answer. Of course the model would then have to include that as a variable of some form.

24. Originally Posted by mkline55
Maybe some sort of unknown energy source? Call it - I don't know, "mystery energy" or "unknown energy" or "dark energy", something like that. It's a perfectly mainstream answer. Of course the model would then have to include that as a variable of some form.
You seem to have missed the point. If the expansion is defined by some (arbitrary?) horizon moving "at c" then how can expansion accelerate (as we observe) unless that horizon is now moving at more than c. Which would mean (according to the OP theory) that light now moves at more than c. It doesn't.

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Originally Posted by substitutematerials
I think I've answered this question Reality Check.
That is still wrong, e.g. the Luminosity Distance is not the distance to a galaxy which is the D used in Hubble's law.
IF01: Derive the relationship between redshift and distance of galaxies in a universe expanding at the speed of light.
That will be a function of z (redshift) and D (proper distance) or recessional velocity vr and D (Hubble's law).

What is critical for the standard candles in the cosmic distance ladder is that they allow use to determine absolute magnitude. Their apparent magnitude then gives us their distance. That is irrelevant to you producing a theoretical "Hubble's law" from your idea.
Last edited by Reality Check; Today at 12:09 AM.

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