Thanks, I think this is related to https://github.com/light-cone-calc/light-cone-calc/issues/6JimJCW said:
Jorrie said:
I think the error is in the UI: we should have ## \Omega_{m,0} = \Omega - \Omega_{\Lambda,0} = 1 - 0.691 = 0.309 ##.JimJCW said:
pbuk said:
I have checked the UI calculation and it seems to be right if radiation density is included, as it should.pbuk said:
Jorrie said:
rholNow = rhocritNow * Ol;
rhol = rholNow;
rhomNow = rhoNow - rhol;
rhom = rhomNow *s*s*s;
rhorNow = rhomNow/s_eq;
rhor = rhorNow *s*s*s*s;Yes, at any time, ##\rho_{0} = \rho_{r,0} + \rho_{m,0} + \rho_{\Lambda,0}##. In the case ##\Omega = 1##, this is subject to the constraint$$\rho_{0} = \rho_\mathrm{critical} = \frac{3 H_0^2 }{8 \pi G},$$pbuk said:
Yes, the same differences occur in Lightcone7, but it is calculated differently than the value in the UI. I will check that part and make it compatible, because I still feel that the value in the UI is correct.pbuk said:
Minor corrections (red) in Calculation.js required to improve the issue:Jorrie said:
George, how do the equations change in the case where ##\Omega \ne 1##?George Jones said:
I am not sure which equations you mean. ##\Omega = \Omega_r + \Omega_m + \Omega_\Lambda## is always true, as is ##\Omega + \Omega_k = 1##. If ##\Omega = 1##, then ##\Omega_k =0##, but if f ##\Omega \neq 1##, then ##\Omega_k \neq 0##.Jorrie said:
I suspect @Jorrie is suffering from the same problem as I am with the confusing terminology. I am working on increasing consistency for my rewrite:George Jones said:
omegaM - density of mass as a fraction of the critical density.omegaLambda - density of the cosmological constant as a fraction of the critical density.omegaRad - density of radiation as a fraction of the critical density.omega - total density of [stuff] as a fraction of the critical density ## \Omega = \Omega_{m} + \Omega_{\Lambda} + \Omega_{rad} ##.omegaK - Do we bother with this?omegaM0 - density of mass now as a fraction of the critical density now.omegaLambda0 - density of the cosmological constant now as a fraction of the critical density now.omegaRad0 - density of radiation now as a fraction of the critical density now.omega0 - total density of [stuff] now as a fraction of the critical density now ## \Omega_{0} = \Omega_{[m,0]} + \Omega_{[\Lambda,0]} + \Omega_{[rad,0]} ##.omegaK0 - curvature now as a fraction of the critical density now (is this actually constant in all models?) ## \Omega_{[K,0]} = 1 - \Omega_{0} ##.Common practice in cosmology is to write "something" in terms of redshift ##z##, i.e., to substitute ##s = 1 + z## for every ##s##. Common cosmological notation is ## [something] = \frac{H\left(z\right)^2}{H_0^2} =: E\left(z\right)^2##. I am not sure why.pbuk said:
Excellent, thank you, so adapting that slightly I can write:George Jones said:
const getESquaredAtS = (s) => {
const s2 = s * s;
return omegaM0 * s2 * s + omegaLambda0 + omegaRad0 * s2 * s2 + OmegaK * s2;
};I think the last term should be ##\Omega_{[K,0]} s^2 ##, because curvature dilates by an inverse square law of the expansion factor.pbuk said:
Yes, ## \LaTeX ## copy and paste error, thanks.Jorrie said:I see in the code piece you have it right...
I might be able to help, but I do not understand the notation. I have figured out "s" for "stretch". To start, I would like to ask about stuff in the following post.Jorrie said:
Does OmegaM represent ##\Omega_{m,0}## or ##\Omega_m \left(t\right)##?Jorrie said:
I do: by eliminating unnecessary conversions I have got the calculations down to this:Jorrie said:
const getParamsAtStretch = (s: number): LcdmModelVariables => {
const eSquared = getESquaredAtStretch(s);
const s2 = s * s;
const h = h0 * Math.sqrt(eSquared);
const omegaM = (omegaM0 * s2 * s) / eSquared;
const omegaLambda = omegaLambda0 / eSquared;
const omegaRad = (omegaRad0 * s2 * s2) / eSquared;
return {
h,
omegaM,
omegaLambda,
omegaRad,
temperature: cmbTemperature * s,
rhoCrit: rhoCrit0 * eSquared,
};
}; for (let i = integrationResults.length - 1; i >= 0; --i) {
const { s, t, dNow: dUnsafe, dPar } = integrationResults[i];
const params = model.getParamsAtStretch(s);
const hGy = params.h * model.kmsmpscToGyr;
// Force dNow to zero at zero redshift.
const dNow = s === 1 ? 0 : dUnsafe;
results.push({
z: s - 1,
a: 1 / s,
s,
t,
dNow,
d: dNow / s,
r: 1 / hGy,
dPar,
vGen: params.h / (s * model.h0),
vNow: dNow * model.h0Gy,
v: (dNow * hGy) / s,
...params,
});
}vGen, it's called XDpar in the LightCone7 code but displayed in the table as ## V_{gen} / c ## and has the descriptionvGen?Yes, good questions, but the work is done moving away from that code now as in my last post. You can see the work in progress at https://github.com/light-cone-calc/light-cone-calc/blob/refactor/src/model.ts.George Jones said:
const rhoCrit0 = rhoConst * h0Seconds * h0Seconds;
const omegaM0 = ((omega - omegaLambda0) * seq) / (seq + 1);
const omegaRad0 = omegaM0 / seq;
const OmegaK0 = 1 - omegaM0 - omegaRad0 - omegaLambda0;##\Omega## or ##\Omega_0##? If ##\Omega_0##, then this is equivalent to taking ##\Omega_{r,0}##, ##\Omega_{m,0}##, and ##\Omega_{\Lambda,0}## as input parameters.pbuk said:
At the risk of outlining what folks already know ... radiation-matter equilibrium happens when ##\rho_r = \rho_m##, or, equivalently, whenpbuk said:
OmegaM represents ##\Omega_{m,0}##. Later on OmegaMatterT represents ##\Omega_m \left(t\right)##George Jones said:
AFAICR, the late Marcus and I were thinking of the worldlines of generic particles that presently cross our Hubble sphere. The variable XDpar appeared in the discussion as the ratio Vthen/Vnow and it was specifically included in the columns for purposes of plotting that interesting wordline mentioned. This is how Vgen[eric] came to be. It shows "dramatically" how a particle could enter the Hubble sphere early on and then be accelerated to cross it again, in the generic case, right now.pbuk said:
Good catch - yes this should be ##\Omega_0##, I will change the variable name.George Jones said:
Many thanks, it's good to have it confirmed.George Jones said:
Jorrie said:
$$\frac{Y_{now}^2}{Y_\infty ^2} = \frac{H_\infty ^2}{H_0^2} = \Omega_{r,0} s^4 + \Omega_{m,0} s^3 + \Omega_{\Lambda,0} + \Omega_{k,0} s^2$$Jorrie said:
is equivalent toJorrie said:
pbuk said:
That's good as that's what I've used and the UI allows entering the parameters to suit different universes!George Jones said:
Great Job! Now we have the UI and the tables agreeing on ##\Omega_m## and also ##\Omega## staying at precisely 1. 00... I have also checked that for ##\Omega \ne 1## the behavior seems to be as expected (and interesting!).pbuk said:
Jorrie said:
Yes you are probably right, however LightCone8 now calculates everything differently as discussed above. By going back to square one on the Physics, aided by @George Jones cross-check to the equations used, I have eliminated the intermediate calculations involving the various densities as below:JimJCW said:
const eSquared = getESquaredAtStretch(s);
const s2 = s * s;
const h = h0 * Math.sqrt(eSquared);
const omegaM = (omegaM0 * s2 * s) / eSquared;
const omegaLambda = omegaLambda0 / eSquared;
const omegaRad = (omegaRad0 * s2 * s2) / eSquared;
return {
h,
omegaM,
omegaLambda,
omegaRad,
omega: omegaM + omegaLambda + omegaRad,
temperature: temperature0 * s,
rhoCrit: rhoCrit0 * eSquared,
};@Jorrie and @pbuk have done tremendous work on LightCone; I haven't done that much. All the stuff I have done in this thread has been performed at coffee shops while "on vacation" at my in-laws' place (11 loud people in the house), and while 3300km from my books. Flying home today.pbuk said:
That's exactly why your contribution has been valuable - it is always important to have eyes on the detail from outside the core team.George Jones said: