how close a fit to LambdaCDM?

[substitutematerials]

Can you guys test drive this heuristic equation relating cosmological redshift (z) to light travel time (t)?

 

 

[math]z=\frac{-ln(1-t)}{\sqrt{1-t^2}}[/math]

 

It seems to mirror LambdaCDM well, but I don't know enough about statistical analysis to really assess it. Light travel time (t) is written with the present equal to zero and the origin of time as 1; multiply by the age of the universe in desired units to get a real time. The equation has no free parameters and is quite simple, so I think its interesting that it seems to fit so well. Useful maybe?

 

I'm generating light-travel times for comparison using Ned Wright's cosmology calculator with default settings. The equation is difficult to solve for light travel time (t) for a given (z), but Mathway can generate roots numerically if the equation is written like this and a value is entered for z:

 

[math]ln\frac{1}{1-t}-z\sqrt{2-2t-(1-t)^2}=0[/math]

 

also here is a graph fitting the equation against some points plotted from the CosmoCalc.

 

Thanks!

 

 

 

[substitutematerials]

Bump

 

Anybody willing to flex their math muscles? Imatfaal, Mordred? Can you generate a comparable fit with a different equation? We can integrate this equation to get co-moving distance (X) also, if that's a more meaningful comparison...

 

[math] X=ct_o(1+\frac{\int_{0}^{t}(z)dt}{t}) [/math]

[Mordred]

Generate a graph then use the lightcone on my signature. Set your number of steps to 100. You will notice that there is further options such as graphing.

 

The lightcone is done in proper distance. I suspect you may or may not see a deviation after Hubble Horizon but thats just from a glance on your last equation.

 

Unfortunately the latex the lightcone generates doesn't post here on this forum. There is something that generates an error specific to the latex format here.

 

The reason I suspect deviations beyond Hubble horizon is that z requires corrections beyond this point.

 

Oh side note the lightcone has a couple of data options. Planck WMAP and combined from the 2012 Planck dataset. It will be closer than the calcs you have been using as it involves those data set parameters.

 

Here is a correction workup I posted in another thread.

 

ets look at the corrections to the redshift formula.

 

First we define a commoving field. This formula though it includes curvature (global) you can set for flat spacetime. A static universe is perfectly flat.

 

[latex]ds^2=c^2dt^2 [\frac {dr^2}{1-kr^2}+r^2 (d\theta^2+sin^2\theta d\phi^2)][/latex]

 

we write [latex](x^0,x^1,x^2,x^3)=(ct,r,\theta,\phi)[/latex]

 

we set the above as

[latex]g_{00}=1,g_{11}=-\frac{R^2(t)}{(1-kr^2)},g_{22}=-R^2 (t)r^2, g_{33}=-R^2 (t)r^2sin^2\theta [/latex]

 

the geodesic equation of the above is

 

[latex]\frac {du^\mu}{d\lambda}+\Gamma^\mu_{\alpha\beta}\mu^\alpha\mu^\beta=0 [/latex]

 

if the particle is massive [latex]\lambda[/latex] can be taken as the proper time s. If it is a photon lambda becomes an affine parameter.

 

So lets look at k=0.

 

we set [latex]d\theta=d\phi=0 [/latex]

 

this leads to

 

[latex]ds^2=c^2t^2-R^2 (t)dr^2=c^2dt^2-dl^2=dt^2 (c^2-v^2)[/latex]

 

where dl is the spatial distance and v=dl/dt is the particle velocity in this commoving frame.

 

Assuming it to be a massive particle of mass "m" [latex]q=m (\frac {dl}{ds})c=(1-\frac {v^2}{c^2})^{\frac{1}{2}}[/latex]

 

from the above a photon emitted at time [latex]t_1[/latex] with frequency [latex]v_1 [/latex] which is observed at point P at time [latex]t_0 [/latex] with frequency [latex]v_0[/latex]

 

with the above equation we get

 

[latex]1+z=\frac {R (t_0)}{R (t_1)}[/latex]

 

Please note were still in commoving coordinates with a static background metric.

 

[latex]z=\frac {v}{c}[/latex] is only true if v is small compared to c.

 

from this we get the Linear portion of Hubbles law

 

[latex]v=cz=c\frac{(t_0-t_1)\dot{R}t_1}{R(t_1)}[/latex]

 

now the above correlation only holds true if v is small. When v is high we depart from the linear relation to Hubbles law.

 

We start hitting the concave curved portion.

 

The departures from the linear relation requires a taylor series expansion of R (t) with the present epoch for this we will also need H_0.

 

note the above line element in the first equation does not use the cosmological constant aka dark energy. This above worked prior to the cosmological constant

 

Now for the departure from the linear portion of Hubbles law.

 

[latex] v=H_Od, v=cz [/latex] when v is small.

 

To this end we expand R (t) about the present epoch t_0.

 

[latex]R (t)=R[(t_0-t)]=R(t_0)-(t_0)-(t_0)\dot {R}(t_0)+\frac {1}{2}(t_0-t)^2\ddot{R}(t_0)...=R (t_0)[1-(t_0-t)H_o-\frac {1}{2}(t_0-t)q_0H^2_0...[/latex]

 

with [latex]q_0=-\frac{\ddot{R}(t_0)R(t_0)}{\dot{R}^2(t_0)}[/latex]

 

q_0 is the deceleration parameter. Sometimes called the acceleration parameter.

 

now in the first circumstances when v is small. A light ray follows

 

[latex]\int_{t_1}^{t^0} c (dt/R (t)=\int_0^{r_1}dr=r_1 [/latex]

 

with the use of this equation and the previous equation we get

 

 

[latex]r=\int^{t_0}_t=\int^{t_0}_t cdt/{(1-R (t_0)[1-(t_0-t)H_0-...]}[/latex]

 

[latex]=cR^{-1}(t_0)[t_0-t+1/2 (t_0-t)^2H_0+...][/latex]

 

here r is the coordinate radius of the galaxy under consideration.

 

Solving the above gives..

 

[latex]t_0-t=\frac {1}{c}-\frac {1}{2}H_0l^2/c^2 [/latex]

 

which leads to the new redshift equation

 

[latex]z=\frac {H_0l^2}{c+\frac {1}{2}(1+q_0)H^2_0l^2/c^2+O (H^3_0l^3)}[/latex]

 

The last equation is the corrected redshift formula when recessive velocity exceeds c..

 

My suspicion is that you will match the linear portion from Hubbles law fairly close, but you will start deviating past Hubble horizon.

 

PS most online calculators don't apply the corrections past Hubble horizon. Though the lightcone calculator is still to good approximation beyond Hubble horizon. It was compared graphically to the lightcones from Lineweaver and Davies.

 

However it isn't 100% accurate either.

 

Good luck and good job in requesting a comparison rather than stating your formulas is correct. Your showing proper methodology.+1

 

Oh I forgot to mention, it took me several years to find the corrections (last equation). Its not something included in the textbooks. Though they all state the cosmological redshift formula is only accurate when recessive velocity is small. I just wish I remember where I found those corrections. It was too long ago and I wrote them down in my notes but forgot to write down the source.

 

(I saved the original paper I got it from on an old phone that died on me and haven't been able to relocate the original paper.) The source is somewhere on arxiv though. I only use peer reviewed sources I trust.

 

Here is the paper we used for developing the lightcone calculator in my signature.

 

http://www.google.ca/url?sa=t&source=web&cd=5&ved=0ahUKEwjG-_D-zJTQAhUC5mMKHcpMCOMQFggpMAQ&url=http%3A%2F%2Fwww.dark-cosmology.dk%2F~tamarad%2Fpapers%2Fthesis_complete.pdf&usg=AFQjCNHLzxKUp15sqgaDF2B8NU6i4xnBdg

 

TM Davies. If you can match up with these lightcones your formulas are reasonably accurate.

Edited November 6, 2016 by Mordred

[substitutematerials]

Thanks Mordred, I'll spend some time with this. As a preliminary measure I added some values from your calculator to my graph, and changed the time of present to reflect the default values (the real present is the time at z=0 from the chart, correct? The Hubble age being the simple 1/Ho?) Just a cursory look suggests that the values in your chart versus my heuristic deviate slightly more between z=10 and z=25 than when compared to Ned's Wright's calculator, but the they re-converge comparably well as we get to higher z values.

[Mordred]

Yeah if I recall we ran into that when we compared to Ned's calc. We opted to compare to Davies. On the top right is your dataset options

 

Stretch equals 1.00 is present age.

 

If you want to contact the developer himself you can find him here. This is the thread that we ran to develop the calculator. Unfortunately the trainer that taught ppl how to use it passed away (Marcus) but the Jorrie still frequents that forum.

 

https://www.physicsforums.com/threads/steps-on-the-way-to-lightcone-cosmological-calculator.634757/

 

Ordinarily I don't like referring to other forums but in this case, the details on the calc usage is best described on this thread.

 

Here is the formulas used in the calculator including reference paper.

 

http://cosmocalc.wikidot.com/advanced-user

 

You can fine tune the calculator by playing with the lower/upper stretch values keeping the number of steps at however many rows you want. This is handy for example in finding the inflection point where we switch from matter dominant to Lambda dominant at roughly 7.3 Gyrs. That thread has an example of doing so.

 

( my involvement was mainly writing the user manual)

Edited November 6, 2016 by Mordred

[substitutematerials]

The lightcone calculator is really great, the table functionality is excellent. It's nice to have the more current data sets as well.

 

The Hubble horizon now is ~13.8 glyr in comoving coordinates, correct? So corresponding to a z of about 1.5. It seems to me that my equation toes the line for longer than that, starting to get kind of bad in the z=8 territory. It's within the right order of magnitude even as far back as 1 second after the big bang though. But again, I'm not a statistician, I can't say if this is an easy to generate coincidence, or perhaps a truly useful equation since it is so simple.

[Mordred]

Hubble horizon is at roughly 4400 Mpc. You can convert that to Gly to get a rough z. I will have to look later but one of the column options should give the Hubble horizon. Glad you like the calc, I find it incredibly handy.

I should mention not all formulas in Cosmology are complex. The FLRW metric is fairly straightforward compared to GR. The advantages a formula has is more than simplicity. Its also its flexibility to define other formulas.

For example this formula is extremely simple and flexible to correlate to more complex formulas.


[latex]1+z=\frac {R (t_0)}{R (t_1)}[/latex]


Whats of greater use though is the formulas involving scale factor.

[latex] H(t)=\frac{\dot{a}(t)}{a(t)}[/latex] for example is further used to calculate proper distance and also temperature thermodynamic relations. There is simple correlations to the scale factor vs temperature evolution. You can see the flexibility in these two articles.

http://arxiv.org/pdf/hep-ph/0004188v1.pdf :"ASTROPHYSICS AND COSMOLOGY"- A compilation of cosmology by Juan Garcıa-Bellido
http://arxiv.org/abs/astro-ph/0409426 An overview of Cosmology Julien Lesgourgues

[latex](\frac{\dot{a}}{a})^2[/latex][latex]=\frac{8\pi G}{3}\frac{\epsilon(t)}{c^2}[/latex][latex]-\frac{kc^2}{R_0^2}\frac{1}{a^2(t)}[/latex]

Another handy article

http://arxiv.org/abs/1302.1498 " The Waters I am Entering No One yet Has Crossed: Alexander Friedman and the Origins of Modern Cosmology" written by Ari Belenkiy

This particular formula is incredibly handy

[latex]H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}[/latex]

Edited November 8, 2016 by Mordred

[Mordred]

 

 

[math] X=ct_o(1+\frac{\int_{0}^{t}(z)dt}{t}) [/math]

 

As it is after work now atm, I can add some details. In particular on Cosmological redshift. There is some handy relations to be familiar with.

first start with commoving coordinates [latex]r_1,\theta, \phi[/latex] photons follow null geodesics [latex]ds^2=0[/latex]

 

thus [latex]cdt=\pm\frac{Rdr}{\sqrt{1-kr^2}}[/latex] key note k in flat geometry =0. The - sign is appropriate as the radius decreases as light approaches us.

this essentially means [latex]\int^{t_0}_{t_1}\frac{cdt}{T(t)}=\int^{r_1}_0\frac{dr}{\sqrt{1-kr^2}}[/latex] For R(t) roughly constant over [latex]\Delta T_0[/latex]

[latex]\frac{c\Delta t_0}{R(t_0)}-\frac{c\Delta t_1}{R(t_1)}=0[/latex]

with c as a constant

[latex]\frac{c\Delta t_0}{c\Delta(t_1)}=\frac{v_0}{v_1}=\frac{\lambda_0}{\lambda_1}=\frac{R(t_0)}{R(t_1}=1+z[/latex]

 

So compare this series of equations with thee one you derived. Further details on the above can be found in "Modern Cosmology" by Max Camenzind page 205

 

(though I changed the commoving coordinates symbols to match those used by wiki and common textbooks)

Edited November 8, 2016 by Mordred

[substitutematerials]

I can mostly wrap my head around this last post, I think I'm basically familiar with those relations. My equations do not allow for any curvature of space (k), they assume flatness apriori as far as can tell.

 

I realized I posted an error in my co-moving equation, it should read:

 

where (t_lb) is lookback time in the units of time used in the speed of light, not (t_o) as is in the earlier post.

 

 

I should mention not all formulas in Cosmology are complex. The FLRW metric is fairly straightforward compared to GR. The advantages a formula has is more than simplicity. Its also its flexibility to define other formulas.

 

 

I agree that the FLRW math is not too onerous. In calling my test equations simple, I'm referring to the lack of free parameters I guess. No need to define the various components of omega to yield a similar redshift /light-travel-time relationship. It could be nothing more than a coincidence though, that's what I hope to determine.

[Mordred]

Yes however lookback time requires those additional parameters to stay accurate. As well as flexible in particular to k values.

 

lets try a key detail.

1) Is the rate of expansion constant over time?

2) What is the importance of those density values in regards to the rate of expansion today as opposed to the rate of expansion then?

3) How does the deceleration equation get involved?

 

(hint number two can be answered from one of the equations I posted)

 

your wondering where your deviations are coming from the above questions will provide your clues.

 

While the formulas above are reasonable approximations. The parameters you mentioned can greatly increase their accuracy. There is a particular variation of the lookback time formula by Hogg's that is far more accurate than the standard look back time formulas. I will post it later when You've had time to consider the above. (Don't forget to look at the z corrections as well)

 

(same hint as per above for question 2) the Hogg's version applies the same equation as per question 1 and 2)

Edited November 9, 2016 by Mordred