Why does this redshift v. light travel time equation work?

[substitute materials]

Why does this redshift versus lighttravel time equation work?

z=-ln(1-t)/sqrt(1-t^2)

where (z) is the cosmological redshift, and (t) is the look-back time where the present equals 0 and the origin of time equals 1.

Ned Wright's cosmology calculator with the default inputs [H_o=69.6, OmegaM=.286, OmegaVac=.714] yields very close values to this equation, especially in the z<2 range that we actually have observational data for. To convert (t) to Giga years as the calculator outputs, just multiply (t) by the age of the universe, 13.721 Gyr in the default calculator setup.

We know (z) is related to scale (a), by
a=1/1+z
So the rest of the values output by the calculator-- co-moving radial distance, angular size distance, and luminosity distance-- are also in agreement with what you would get with the alternate equation after integration over time, multiplication etc.
Why does this work? It doesn't consider mass density, radiation, dark energy, or any free parameter. In the supernova paper, Perlmutter's team notes that the "empty universe" model of Milne generates a nearly best fit graph to their data set, although no equations are included. Is this equation a expression of an emtpy FLRW model? Simply inputing OmegaM=0 and OmegaVac=0 into Ned Wright's calculator does not yield the same relation.
I have a pet theory as to why the equation works, but I won't force it on anybody right now. Mostly I'm eager to see someone else compare the values generated by the equation and the calculator, and get your 2 cents on how close they seem. Coincidence? Could such a shorthand equation be useful to ballpark numbers, even without a explanation?

 

[PeterDonis]

Why does this redshift versus lighttravel time equation work?

Where is this equation coming from?

 

[Chronos]

It's a twisted version of ned wright's cosmological calculator equation.

 

[Chalnoth]

Why does this redshift versus lighttravel time equation work?

z=-ln(1-t)/sqrt(1-t^2)

where (z) is the cosmological redshift, and (t) is the look-back time where the present equals 0 and the origin of time equals 1.

Ned Wright's cosmology calculator with the default inputs [H_o=69.6, OmegaM=.286, OmegaVac=.714] yields very close values to this equation, especially in the z<2 range that we actually have observational data for. To convert (t) to Giga years as the calculator outputs, just multiply (t) by the age of the universe, 13.721 Gyr in the default calculator setup.

We know (z) is related to scale (a), by
a=1/1+z
So the rest of the values output by the calculator-- co-moving radial distance, angular size distance, and luminosity distance-- are also in agreement with what you would get with the alternate equation after integration over time, multiplication etc.
Why does this work? It doesn't consider mass density, radiation, dark energy, or any free parameter. In the supernova paper, Perlmutter's team notes that the "empty universe" model of Milne generates a nearly best fit graph to their data set, although no equations are included. Is this equation a expression of an emtpy FLRW model? Simply inputing OmegaM=0 and OmegaVac=0 into Ned Wright's calculator does not yield the same relation.
I have a pet theory as to why the equation works, but I won't force it on anybody right now. Mostly I'm eager to see someone else compare the values generated by the equation and the calculator, and get your 2 cents on how close they seem. Coincidence? Could such a shorthand equation be useful to ballpark numbers, even without a explanation?

It works sort of okay for the recent universe, but breaks down for the early universe. It's about 30% off at z=10, for example. I'm not sure why that functional form is relatively accurate, even with no prefactors. My bet is it's probably a feature of the fact that dark energy recently became a majority of the energy density of our universe.

 

[substitute materials]

Hi Chalnoth, thanks for crunching numbers. at z=10 I only get a 1.4% difference in light travel times between the equation in the thread and the calculator. Am I doing the math wrong?

 

[Chalnoth]

Hi Chalnoth, thanks for crunching numbers. at z=10 I only get a 1.4% difference in light travel times between the equation in the thread and the calculator. Am I doing the math wrong?

I used the following Google search to do the calculation:
https://www.google.com/search?q=-ln...ome..69i57.15482j0j7&sourceid=chrome&ie=UTF-8

(using light travel time of 13.243 Gyr from the cosmology calculator)

This gives me z=12.85.

 

[Chalnoth]

Maybe you used $t^2$ on the numerator? That does give a more accurate answer for $z=10$, but it does much worse than the equation you wrote at low redshift, and still is pretty far off at high redshift.

 

[substitute materials]

I think I see where we came up with such different discrepancies- I'm working from a given value for z, and then comparing the difference in lookback times- for z= 10, 13.068 Gyr in the novel equation, and 13.243 Gyr for the cosmology calculator, which is only different by 1.32%. Which is the correct way to compare the results do you think? Since (z) is the observable, does it make sense to compare the lookback times? Or is this just forcing the results to look better than they are? Redshift changes a lot with small differences in time once you get that far out.

 

[Chalnoth]

I think I see where we came up with such different discrepancies- I'm working from a given value for z, and then comparing the difference in lookback times- for z= 10, 13.068 Gyr in the novel equation, and 13.243 Gyr for the cosmology calculator, which is only different by 1.32%. Which is the correct way to compare the results do you think? Since (z) is the observable, does it make sense to compare the lookback times? Or is this just forcing the results to look better than they are? Redshift changes a lot with small differences in time once you get that far out.

Lookback time isn't really a measurable quantity. Redshift is.

But yes, because the expansion rate of the universe was much faster early-on, any function of redshift that converges to the current age of the universe will look good for the very early universe using your methodology (how early depends upon how the function converges).

 

[substitute materials]

But yes, because the expansion rate of the universe was much faster early-on, any function of redshift that converges to the current age of the universe will look good for the very early universe using your methodology (how early depends upon how the function converges).

Is this true?

If we take any old function that roughly converges in the near present, say the made up z=2.3(-log(1-t)), it seems to me that it diverges dramatically for the early universe.

We can compare very very far back by evaluating the background temperature, correct? Since the scaling of space is an adiabatic expansion, where T_emission= (z+1) T_observation, or for the CMB,
T=(1090+1)2.73
The original equation in this thread predicts the recombination temperature of 3000 K would have occurred at 590,000 years after T=0, compared to the oft quoted 380,000 years.
Going extremely far back, to just a second after t=0, LamdaCDM has the temperature at around 10 gigakelvins, while this equation yields about 50 gigakelvins. I figured that being within the right order of magnitude so far back was compelling. Sounds like you don't think so? Could you show me another arbitrary function that hews so close?

Thanks for the help, this is fun

 

[Chalnoth]

Is this true?

If we take any old function that roughly converges in the near present, say the made up z=2.3(-log(1-t)), it seems to me that it diverges dramatically for the early universe.

We can compare very very far back by evaluating the background temperature, correct? Since the scaling of space is an adiabatic expansion, where T_emission= (z+1) T_observation, or for the CMB,
T=(1090+1)2.73
The original equation in this thread predicts the recombination temperature of 3000 K would have occurred at 590,000 years after T=0, compared to the oft quoted 380,000 years.

Sure. But now you're comparing the age of the universe at that redshift, rather than the light travel time.

And by the way, by your current measure, your new off-the-cuff approximation does better than the original posted above: for z=1090, I get an age of the universe of 376,000 years from the cosmology calculator. Using your formula, I get 58,500 years, which is off by more than a factor of five, rather than merely the factor of two for this new approximation.

 

[substitute materials]

And by the way, by your current measure, your new off-the-cuff approximation does better than the original posted above: for z=1090, I get an age of the universe of 376,000 years from the cosmology calculator. Using your formula, I get 58,500 years, which is off by more than a factor of five, rather than merely the factor of two for this new approximation.

sorry, I made that confusing. The 590,003 number was from the original equation, can you double check the 58,500 that you got?

Using the off the cuff equation with z=1090 I get a light-travel time of 13.721(1-10^-473) gyr, or effectively an age of 0- extremely wrong by comparison.

Sure. But now you're comparing the age of the universe at that redshift, rather than the light travel time.

sorry,I'm being sloppy switching back and forth.

 

[substitute materials]

Where is this equation coming from?

The way I derived this is pretty dang crackpotty, although I could share it if you'd like. My question for the forum is really about how the results of this equation imitate the standard model. It seems to me that it does very well and could be useful, even without my explanation. I welcome a solid numerical refutation- whether it be another best fit equation that shows this is easy to come up with, or by exposing an error in my comparison of the results. Chalnoth's google search https://www.google.com/search?q=-ln...ome..69i57.15482j0j7&sourceid=chrome&ie=UTF-8 is an easy way to compare my equation to the cosmology calculator by inputing light-travel times. I'm not personally able to solve this equation for t with a known z, but I have a workaround which is to input this reformulation, z\sqrt(2-2t-(1-t)^(2))-ln((1)/(1-t))=0 into the Mathway website, which will take the roots numerically. Input a numerical value for z to get t, then multiply
(t) by the present age of the universe to get lighttravel time.

 

[berkeman]

Thread closed for Moderation...

EDIT -- Thread will remain closed as personal theory development (which is not allowed on the PF).