## The Redshift-Distance Relation

### Introduction

The discovery of a relationship between a galaxy's distance and its redshift established that the universe is expanding. And as the distances to farther objects were measured, their redshifts were less than expected, which means in the past, the universe wasn't expanding as fast as it is now.

Currently, there is no single equation to relate distance and redshift that fits the entire range of data. The following equation is proposed as a new redshift-distance relation, which works for measured values and makes predictions to be compared against future measurements.

From this equation, redshift (z) can be found from distance (D) and vice versa, using a constant (L), which is a length of about 26 billion light years.

### Methodology

To help reason about the expansion of space, consider a runner on a track. If the runner maintains a constant speed, the length of each step taken by the runner, and the time between each step, should remain constant throughout the race.

Now consider a race between three such runners beginning at the same speed, each facing a disadvantage.

• Runner 1: Dynamic Track, everything moves away at H0D
• Runner 2: Dynamic Clock, clock speed is c/(c-H0D)
• Runner 3: Dynamic Runner, runs at c-H0D

The three runners should always tie, assuming H0D is equal for all, which you will recognize from Hubble's law, v=H0D.

Runner 1: Dynamic Track
Runner 2: Dynamic Clock
Runner 3: Dynamic Runner

Source Code

Runner 1 on the expanding track will find that the distance between their steps increases. Runners 2 and 3 will find that the time in between their steps increases. Using that data, a z can be calculated for each runner:

 Redshift based on step length Redshift based on step frequency

When the calculated redshifts are compared to the co-moving distance (the distance where the point on the track with that redshift was at the beginning of the race) then we can see that each runner produces the same redshifts. We can then compare those redshifts and distances (green line) to actual data from the universe (yellow dots).

The runners' redshifts increase too quickly with distance to match the observational data. Runner 3, for example, loses the appropriate amount of speed to match the data early on in the race, but loses too much speed as the race goes on. The clock rate of Runner 2 is also increasing too rapidly. To address this, new versions of these runners are made like so:

 Runner 2.1's clock rate Runner 3.1's speed

The new versions of our runners, Runner 2.1 and Runner 3.1, still finish in a tie.

Runner 2.1: Dynamic Clock
Runner 3.1: Dynamic Runner

The redshifts produced by these runners (shown below in purple) don't diverge from the data like the previous group (shown in green).

The term H0/c has units of Mpc-1. If we invert it to c/H0, the result is a length, which we can call L, so L = c/H0.

This allows us to simplify our equations.

 Runner 2.1's clock rate Runner 3.1's speed

The redshifts produced by these runners is a better fit for the data, but it's still not quite right. Out of curiosity, I decided to square the 1+D/L terms, and produced two new runners.

 Runner 2.2's clock rate Runner 3.2's speed

Treating L as a free parameter, through trial and error I found (a bit surprisingly) that L=2c/H0, is a good fit for the data, shown below by the white line. Also shown for comparison are data points (white squares) for the LambdaCDM model, from Wolfram Alpha.

In the redshifts produced by Runner 3, when z=1, 2, 3, and so on, the runner's velocity is v=0.5c, 0.333c, 0.25c, and so on, such that:

And:

Therefore:

Despite being derived from a non-expanding model, this equation predicts the observed z and D of a galaxy in an expanding universe that is accelerating. It is consistent with LambdaCDM where we have observational data, and predicts different redshifts where we don't yet have data, which can be compared with future observations.

### Discussion

Runner 3 is meant to be a helpful tool in thinking about what's going on in a dynamic space (or time) without actually having a dynamic space (or time). A decreasing speed isn't viable as a physical theory because a photon, which the runner represents, cannot slow down.

Runner 2, though, doesn't slow down, avoiding many conflicts with the laws of physics. However, because the redshifts of Runner 2 are caused by a dynamic of time, rather than of space, the size of the universe doesn't change over time, which is in conflict with the big bang theory.

For most cosmologists, the debate on interpreting the redshifts as the expansion of the universe is long settled. It can almost be directly observed that time began around 14 billion years ago. When we look into space and thus back in time, we can see that the most distant galaxies are small and disordered, unlike the galaxies nearby.

In the last ten years however, we have observed massive, dusty galaxies with ordered features and shut down star formation at all distances, well before they are predicted to have appeared[1][2][3][4][5][6][7][8][9].

Mature galaxies in the "early" universe

The distant universe isn't meeting our expectations for when galaxies become massive, dusty, spiraled, barred, disced, bulged, and have shut down star formation:

• A massive stellar bulge in a regularly rotating galaxy 1.2 billion years after the Big Bang[1]
Lelli et al. (1) report a galaxy that had evolved features (both a disk and a bulge) when only 1.2 billion years had elapsed since the Big Bang (∼12.5 billion years ago) (see the figure). This finding suggests that the processes that generate the key features of a mature galaxy arose more rapidly than has been thought.
• A cold, massive, rotating disk galaxy 1.5 billion years after the Big Bang [2]
The existence of such a massive, rotationally supported, cold disk galaxy when the Universe was only 1.5 billion years old favours formation through either cold-mode accretion or mergers, although its large rotational velocity and large content of cold gas remain challenging to reproduce with most numerical simulations.
• A dynamically cold disk galaxy in the early Universe[3]
Interferometric imaging at a spatial resolution of about 60 pc reveals a ratio of rotational-to-random motions of V/σ = 9.7 ± 0.4, which is at least four times larger than expected from any galaxy evolution model at this epoch, but similar to the ratios of spiral galaxies in the local Universe.
• A dominant population of optically invisible massive galaxies in the early Universe [4]
Such a high abundance of massive and dusty galaxies in the early Universe challenges our understanding of massive-galaxy formation.
• Earliest giant galaxies: The birth of monsters [5]
"To complicate things further, if massive galaxies are unexpectedly dustier in the early Universe than astronomers predict then even UltraVISTA wouldn't be able to detect them. If this is indeed the case, the currently-held picture of how galaxies formed in the early Universe may also require a complete overhaul."
• A dusty, normal galaxy in the epoch of reionization [6]
The galaxy is highly evolved: it has a large stellar mass, and is heavily enriched in dust, with a dust-to-gas ratio close to that of the Milky Way. Dusty, evolved galaxies are thus present among the fainter star-forming population at z > 7, in spite of the very short time since they first appeared.
• Some galaxies in the early universe grew up quickly [7]
Fifteen mature galaxies were found at a record-breaking average distance of 12 billion light years, when the universe was just 1.6 billion years old. Their existence at such an early time raises new questions about what forced them to grow up so quickly
• Galaxy Zoo: CANDELS Barred Disks and Bar Fractions [8]
The newly classified galaxies are striking in that they look a lot like those in today's universe, with disks, bars and spiral arms. But theorists predict that these should have taken another 2 billion years to begin to form, so things seem to have been settling down a lot earlier than expected.
• Spitzer Splash Project Dives Deep for Galaxies [9]
"The findings cast doubt on current models of galaxy formation, which struggle to explain how these remote and young galaxies grew so big so fast."

Is the expansion of the universe a fact? This was debated in the 1930's and was very violently interrupted by World War II. This is an unconventional argument to make, but science does not operate in a vacuum. And while the war made science a priority, and produced many giant leaps forward, the science of galactic nebulae was not one of those priorities.

To illustrate how drastic an interruption this was, in December of 1941, Edwin Hubble, the man credited for discovering the expanding universe, announced that he had refuted the expanding universe theory. Soon, he would no longer be looking through telescopes, but instead working in a windtunnel on ballistics research for the war effort.

In the 1950's, a new generation came back to the telescopes, armed with two expanding universe theories: the big bang theory, where the universe was smaller in the past, and the steady-state theory, where expansion of space produces matter, keeping the density of the universe the same.

In the 1960's the discovery of the cosmic microwave background (CMB) led to the acceptance of the big bang theory over the steady state theory. It looked like this:

Big Bang 1.0: Expansion

By the 1970's, it was apparent this theory had some problems (Horizon Problem, Flatness Problem, Magnetic Monopole Problem) which were solved by a period of super-expansion right after the big bang. This period is called cosmic inflation, and it looks like this:

Big Bang 2.0: Inflation

By the 1990's, as mentioned before, the measurements of distances and redshifts showed that the expansion of the universe must be accelerating:

Big Bang 3.0: Acceleration

Today, Big Bang 3.0, also known as LambdaCDM, is famously in conflict with independent measurements[10][11][12][13][14]. A Big Bang 4.0 might be on the way.

Over the decades, some have criticized the big bang theory for the continual need for major, ad hoc changes to fit observations.

Modern Cosmology: Science or Folktale?
In its original form, an expanding Einstein model had an attractive, economic elegance. Alas, it has since run into serious difficulties, which have been cured only by sticking on some ugly bandages: inflation to cover horizon and flatness problems; overwhelming amounts of dark matter to provide internal structure; and dark energy, whatever that might be, to explain the seemingly recent acceleration. A skeptic is entitled to feel that a negative significance, after so much time, effort and trimming, is nothing more than one would expect of a folktale constantly re-edited to fit inconvenient new observations.
Modern Cosmology: Science or Folktale? (2007)

More:

But viable alternatives to the expanding universe theory face considerable obstacles, such as explaining the CMB. If the energy of the CMB is not from the beginning of the universe, what is it from? Is it even related to the redshifts? Without a valid explanation for the CMB, a theory of redshifts, no matter how good it is, is not good enough to topple the expanding universe theory.

### Conclusion

Some readers may not find discussing alternative redshift interpretations to be an effective use of their time. For the reader who wishes to use their time most effectively, having a simple, empirically faithful redshift-distance relation may be useful.

### Sources

1. Lelli, F., et al., A massive stellar bulge in a regularly rotating galaxy 1.2 billion years after the Big Bang. Science Vol. 371, Issue 6530, pp. 713-716 DOI: 10.1126/science.abc1893
2. Neeleman, M., Prochaska, J.X., Kanekar, N. et al. A cold, massive, rotating disk galaxy 1.5 billion years after the Big Bang. Nature 581, 269–272 https://doi.org/10.1038/s41586-020-2276-y 2005.09661 (2020)
3. Rizzo, F., Vegetti, S., Powell, D. et al. A dynamically cold disk galaxy in the early Universe. Nature 584, 201–204 (2020). https://doi.org/10.1038/s41586-020-2572-6
4. Wang, T., Schreiber, C., Elbaz, D. et al. A dominant population of optically invisible massive galaxies in the early Universe. Nature 572, 211–214 (2019). https://doi.org/10.1038/s41586-019-1452-4
5. K. Caputi et al., “Spitzer Bright, UltraVISTA Faint Sources in COSMOS: The Contribution to the Overall Population of Massive Galaxies at z = 3-7”, Astrophysical Journal.
6. Watson, D., Christensen, L., Knudsen, K. et al. A dusty, normal galaxy in the epoch of reionization. Nature 519, 327–330 (2015). https://doi.org/10.1038/nature14164
7. Straatman, Caroline MS, et al. "A substantial population of massive quiescent galaxies at z∼ 4 from ZFOURGE." The Astrophysical Journal Letters 783.1 (2014): L14. https://arxiv.org/abs/1312.4952
8. B. D. Simmons et al. Galaxy Zoo: CANDELS Barred Disks and Bar Fractions. Monthly Notices of the Royal Astronomical Society, 2014 DOI: 10.1093/mnras/stu1817
9. Charles L. Steinhardt et al 2014 ApJL 791 L25 https://iopscience.iop.org/article/10.1088/2041-8205/791/2/L25
10. A. G. Riess et al. "Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics Beyond LambdaCDM" arXiv:1903.07603, (2019).
11. N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters, (2018)
12. A. G. Riess et al., "Cosmic Distances Calibrated to 1% Precision with Gaia EDR3 Parallaxes and Hubble Space Telescope Photometry of 75 Milky Way Cepheids Confirm Tension with LambdaCDM" arXiv:2012.08534 [astro-ph.CO] ApJ., 116, 1009 (2020).
13. Blakeslee, J.P. et al,. "The Hubble Constant from Infrared Surface Brightness Fluctuation Distances" arXiv:2101.02221 (2021)
14. Jones, David, "Crisis in Cosmology: Measuring The Local Value of the Hubble Constant" APS April Meeting 2018 63,4 APR18-2018-000816 (2018)