# Need help with density parameters of the LambdaCDM model

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Hello,

I would like to have values (or how to calculate the values) of $\Omega_b$ (ordinary matter = baryonic) and $\Omega_{\Lambda}$ (density parameter of the cosmological constant) in relation to, or as a function of, the Hubble constant H at different ages of the universe.

To verify this relationship:

$M_b* \Lambda / 2 \approx 8 Kg/m^{2}$ ( -0.5% compared to 8 = contribution of neutrinos ? )

$M$ for "mass"

Edited by stephaneww
latex

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Are you specifically looking for the density of neutrinos at a given scale factor ? The cosmological constant density stays constant but the matter density (including DM) will scale at $\Omega_m(1+z)^3$ where radiation scales at $\Omega_(1+z)^4$ the cosmological constant remains at the same energy density throughout. It is a constant.

If your goal is specifically the number density of neutrinos the methodology gets rather clunky in calculations using the Maxwell Boltzmann statistics this will correlate its temperature contributions at a given blackbody temperature with the temperature scaling at $1/a$ Though the Stretch column in the calculator in my signature is identical in this ratio as stretch is also the inverse of the scale factor.

Edited by Mordred

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14 minutes ago, Mordred said:

Are you specifically looking for the density of neutrinos at a given scale factor ?

why not after all : it may be interesting.

14 minutes ago, Mordred said:

The cosmological constant density stays constant but the matter density (including DM) will scale at  Ωm(1+z)3 where radiation scales at Ω(1+z)4 the cosmological constant remains at the same energy density throughout. It is a constant.

Thanks a lot.

can we simplify on "short" time scales by saying Mb constant ?

Edited by stephaneww

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Baryonic density will not stay constant but its typically felt the particle number is roughly constant so at short time scales well after nucleosynthesis it would be a reasonable approximation. See chapters 3 and 4 of this article for the Bose-Einstein and Fermi-Dirac /Maxwell Botzmann statistics application. These statistics can calculate the number density of any SM particle with known quantum properties at  given blackbody temperature

http://www.wiese.itp.unibe.ch/lectures/universe.pdf:" Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis.

If I recall correctly he demonstrates how to use this on the neutrino family

Edited by Mordred

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30 minutes ago, Mordred said:

Baryonic density will not stay constant but its typically felt the particle number is roughly constant so at short time scales well after nucleosynthesis it would be a reasonable approximation. See chapters 3 and 4 of this article for the Bose-Einstein and Fermi-Dirac /Maxwell Botzmann statistics application. These statistics can calculate the number density of any SM particle with known quantum properties at  given blackbody temperature ﻿﻿

Perfect. thank you again.

30 minutes ago, Mordred said:

http://www.wiese.itp.unibe.ch/lectures/universe.pdf:" Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis.

If I recall correctly he demonstrates how to use this on the neutrino family

After reading the table of contents and the passage concerning the cosmological parameters, I did not find it. Or else I don't understand.

I have a formula that will go into the speculation section with $\Omega_b$ and $\Omega_\Lambda$ that could be compatible with this:

I'll calculate to see if the numerical values confirm

Thank you again

Edited by stephaneww

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Start with 4.1 Thermodynamical Distributions he later gives baryonic density at $\Omega\le 0.18$ 4.1.13 he discusses the Fermi Dirac statistics but neither here nor there I found an easier way

see equation in the indirect evidence of neutrino background It  is a derivative from the Maxwell Botzmann specific to neutrinos.

Edited by Mordred

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I'm stranded: how to get the values with Planck 2018 please?

edit

I understood : Hubble time is different from age of unoverse

Edited by stephaneww

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I don't think we can at least not without deriving the S-eq matter radiation equality value. I can however see if I can ask Jorrie and Cobert if they are working on the 2018 results for programming the calculator. The primary programmer being Cobert myself and Marcus helped in so far as writing the manuals and adding suggestions and methodologies to make it easier for others to understand. Primary example using Gyrs instead of Mpc. Marcus found people better understood distances in light-years as opposed to parsecs.

Found his latest version for the 2015 Planck results I will adjust my signature to reflect the newer version, sent an email to see if Jorrie is working on the 2018

lol newer version will take me a bit to get used to lmao

Edited by Mordred

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It's okay with:
T H0 : 14,497
T H infinite : 17.51
Omega tot : 0.9989
there's just one tiny little mistake on OmegaM

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kk good email has been sent as I am curious myself if he's going to update this version. It is possible to latex the results here if you look through the small tex output and change tex to latex where required at begin and end or the array commands. Demo

${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{gen}/c&H(t)&rho, kg/m^3 \\ \hline 1090.000&0.000372&0.000627&45.340&0.042&0.0567&21.121&1.561e+6&5.028e-18\\ \hline 339.033&0.002490&0.003948&44.193&0.130&0.1786&10.753&2.477e+5&1.266e-19\\ \hline 104.978&0.015284&0.023444&42.024&0.397&0.5527&5.811&4.172e+4&3.591e-21\\ \hline 32.030&0.090052&0.136169&38.065&1.152&1.6534&3.210&7.182e+3&1.065e-22\\ \hline 9.295&0.521890&0.784445&30.935&3.005&4.6122&1.788&1.247e+3&3.208e-24\\ \hline 2.209&2.976146&4.372203&18.268&5.694&10.8504&1.029&2.237e+2&1.033e-25\\ \hline 0.000&13.799968&14.437487&0.000&0.000&16.5313&1.000&6.774e+1&9.470e-27\\ \hline -0.688&32.967504&17.251750&11.156&35.795&17.2927&2.685&5.669e+1&6.632e-27\\ \hline -0.868&47.865307&17.359119&14.270&108.178&17.3591&6.305&5.634e+1&6.551e-27\\ \hline -0.944&62.796207&17.367429&15.591&279.265&17.3674&14.891&5.631e+1&6.544e-27\\ \hline -0.976&77.729642&17.368136&16.150&683.511&17.3681&35.182&5.631e+1&6.544e-27\\ \hline -0.990&92.663443&17.368093&16.387&1638.657&17.3681&83.127&5.631e+1&6.544e-27\\ \hline \end{array}}$
Edited by Mordred

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I also need need the ratio H/H0 to find OmegaLambda and OmegaM

Edited by stephaneww

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here is the formula

$H = H_0 \left(\Omega_\Lambda + (1-\Omega) (z+1)^2 + \Omega_m (z+1)^3 + \Omega_r (z+1)^4\right)^{0.5}$

other relevant formulas see here

at least those used in the calculator

Edited by Mordred

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12 minutes ago, Mordred said:

here is the formula

H=H0(ΩΛ+(1Ω)(z+1)2+Ωm(z+1)3+Ωr(z+1)4)0.5

oh thank you.

edit ok forgot this question :
$\Omega_m$ is for baryonic matter or all the matter (dark+ordinary)?

I have a new one : where can I find $\Omega_r$ ?

Edited by stephaneww

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use  latex not just tex  as the command this forum requires it fully. Hence on the cosmocalc you need to change tex to latex.

Yes the matter density is both as per the methodology of the FRW metric, for individual particle contributions you need to step into the statistics I mentioned earlier. Recall under the FRW metric its based on the effective equation of states for matter w=0 which encompasses all on relativistic species. Neutrinos has the radiation equation of state however as it is relativistic along with photons.

PS your FRW skills have greatly improved since I first helped you when you first joined well done and keep it up

I've granted a +2 for the considerable improvement

Edited by Mordred

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20 minutes ago, Mordred said:

Recall under the FRW metric its based on the effective equation of states for matter w=0 which encompasses all on relativistic species. Neutrinos has the radiation equation of state however as it is relativistic along with photons.

For the moment, I'm troubled with the equations of state

20 minutes ago, Mordred said:

PS your FRW skills have greatly improved since I first helped you when you first joined well done and keep it up

I've granted a +2 for the considerable improvement

Thanks for these encouragements... but I think I'm doing a little better in cosmology than in FRW: _)

23 minutes ago, stephaneww said:

I have a new one : where can I find Ωr ?

If you have an answer for its current value (planck 2018) I take

Edited by stephaneww

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A simple layman way to think of the equation of state is as an energy density to pressure term ratio, via the $w=\frac{\rho}{p}$. The proofs of each is a good course of study and a good cosmology textbook will include those proofs with the corresponding wavelengths (particularly in the case of radiation).

If you like I can check my database for some decent articles on the topic as it is fundamental to understanding the FLRW fluid equation and the deceleration equation.

I'm aware your GR isn't strong so in terms of the stress tensor I will try to find a more Newtonian approximation

Edited by Mordred

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On ‎2‎/‎10‎/‎2019 at 8:11 AM, Mordred said:

I don't think we can at least not without deriving the S-eq matter radiation equality value. I can however see if I can ask Jorrie and Cobert if they are working on the 2018 results for programming the calculator. The primary programmer being Cobert myself and Marcus helped in so far as writing the manuals and adding suggestions and methodologies to make it easier for others to understand. Primary example using Gyrs instead of Mpc. Marcus found people better understood distances in light-years as opposed to parsecs.

Found his latest version for the 2015 Planck results I will adjust my signature to reflect the newer version, sent an email to see if Jorrie is working on the 2018

lol newer version will take me a bit to get used to lmao

oh good. it's very good link when I look with options. I use it now before the  2018 planck version

+1

thank you

Edited by stephaneww

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Hello Mordred.

I finally understood the use of the calculator (Planck 2015) and the equality of H with density parameters. But I have not been able to find satisfactory approximations to apply them to the data in the abstract Planck 2018. I have to wait for the update to go further. I will move on to the FLRW and GR equations in the meantime.

PS: I found the button output in Latex and would know how to transform Tex into Latex

edit :

I have find a good approximation but I have a big problem

Edited by stephaneww

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when you copy paste the tex display copy it fully then paste it fully onto the forum post, then edit each instance of tex to the full word latex keeping those terms in the [...]. This will allow you to post the chart onto this forum.

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(base Planck 2015) and :

H : 67.4

Omega L : 6851

Step : 20

I have :

${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&Sc.fctr (a)&S&T (Gy)&D_{hor}(Gly)&H(t)&rho, kg/m^3&Temp(K)&OmegaM&OmegaL&OmegaR \\ \hline 1090.000&0.000917&1091.000&0.000370&0.0567&1.5678e+6&5.1243e-18&2.9784e+3&7.5549e-1&1.2661e-9&2.4451e-1\\ \hline 608.078&0.001642&609.078&0.000972&0.1009&6.1767e+5&7.9532e-19&1.6628e+3&8.4697e-1&8.1576e-9&1.5303e-1\\ \hline 339.033&0.002941&340.033&0.002479&0.1787&2.4879e+5&1.2903e-19&9.2829e+2&9.0838e-1&5.0282e-8&9.1628e-2\\ \hline 188.832&0.005268&189.832&0.006188&0.3153&1.0166e+5&2.1542e-20&5.1824e+2&9.4669e-1&3.0117e-7&5.3311e-2\\ \hline 104.978&0.009436&105.978&0.015217&0.5530&4.1901e+4&3.6600e-21&2.8932e+2&9.6952e-1&1.7726e-6&3.0480e-2\\ \hline 58.165&0.016902&59.165&0.037051&0.9621&1.7360e+4&6.2827e-22&1.6152e+2&9.8275e-1&1.0327e-5&1.7248e-2\\ \hline 32.030&0.030275&33.030&0.089654&1.6549&7.2141e+3&1.0849e-22&9.0173e+1&9.9024e-1&5.9802e-5&9.7027e-3\\ \hline 17.440&0.054230&18.440&0.216104&2.7998&3.0032e+3&1.8801e-23&5.0341e+1&9.9422e-1&3.4507e-4&5.4386e-3\\ \hline 9.295&0.097139&10.295&0.519591&4.6201&1.2522e+3&3.2689e-24&2.8104e+1&9.9498e-1&1.9847e-3&3.0385e-3\\ \hline 4.747&0.173998&5.747&1.245974&7.3303&5.2445e+2&5.7338e-25&1.5690e+1&9.8701e-1&1.1315e-2&1.6828e-3\\ \hline 2.209&0.311671&3.209&2.963870&10.8922&2.2448e+2&1.0505e-25&8.7592e+0&9.3735e-1&6.1759e-2&8.9217e-4\\ \hline 0.791&0.558275&1.791&6.796199&14.4931&1.0647e+2&2.3631e-26&4.8901e+0&7.2506e-1&2.7455e-1&3.8527e-4\\ \hline 0.000&1.000000&1.000&13.795986&16.6666&6.7400e+1&9.4701e-27&2.7300e+0&3.1480e-1&6.8509e-1&9.3386e-5\\ \hline -0.442&1.791233&0.558&23.084440&17.3355&5.7975e+1&7.0067e-27&1.5241e+0&7.4032e-2&9.2595e-1&1.2261e-5\\ \hline -0.662&2.961475&0.338&31.723059&17.4497&5.6279e+1&6.6028e-27&9.2184e-1&1.7384e-2&9.8260e-1&1.7413e-6\\ \hline -0.796&4.896255&0.204&40.497135&17.4966&5.5897e+1&6.5133e-27&5.5757e-1&3.8994e-3&9.9610e-1&2.3625e-7\\ \hline -0.876&8.095059&0.124&49.302428&17.5231&5.5812e+1&6.4936e-27&3.3724e-1&8.6547e-4&9.9913e-1&3.1715e-8\\ \hline -0.925&13.383695&0.075&58.114641&17.5290&5.5793e+1&6.4893e-27&2.0398e-1&1.9163e-4&9.9979e-1&4.2475e-9\\ \hline -0.955&22.127483&0.045&66.928214&17.5304&5.5789e+1&6.4882e-27&1.2338e-1&4.2410e-5&9.9995e-1&5.6857e-10\\ \hline -0.973&36.583733&0.027&75.742301&17.5306&5.5788e+1&6.4880e-27&7.4623e-2&9.3846e-6&9.9998e-1&7.6097e-11\\ \hline -0.983&60.484488&0.017&84.556288&17.5308&5.5788e+1&6.4879e-27&4.5136e-2&2.0766e-6&1.0000e+0&1.0185e-11\\ \hline -0.990&100.000000&0.010&93.370467&17.5307&5.5788e+1&6.4880e-27&2.7300e-2&4.5950e-7&9.9999e-1&1.3631e-12\\ \hline \end{array}}$

according to the abstract Planck 2018, $\Omega_b = 0.0492$ and I find $M_b = 1.457*10^{53}kg$
and $\Omega_b / \Omega_m = 0.1563$ for the present time.

For z=0.791 I have a problem determining $\Omega_b$
or it's:  $M_b/M_{total} = 1.457*10^{53}kg/7.559*10^{53}kg=0.1927=\Omega_b$ but then
$\Omega_b / \Omega_m = 0.2658$.

why  $\Omega_b / \Omega_m$ are différents for each time

Edited by stephaneww

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How did you calculate the Baryon density I'm positive you didn't use the Maxwell statistics for it ? You know the Baryon density is only roughly 4% of the total mass density and the dark matter accounts for the larger portion of the matter density.

Baryon density also includes a portion of the radiation density via neutrinos and photons. These are also baryonic particles.

Edited by Mordred

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7 minutes ago, Mordred said:

How did you calculate the Baryon density I'm positive you didn't use the Maxwell statistics for it ?

In abstract $\Omega_b h^2=0.0224$

so $\Omega_b =0.0224/h^2=0.0224/(67.4/100)^2$

Edited by stephaneww

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Yes they give you the value today I don't question that value but how did you calculate your density at Z=0.791. I'm positive you are not familiar enough with the Baryon acoustic oscillations method the Planck paper used to calculate the first value.

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for z=0.791, I used Ned Wright's calculator to determine radius of observable universe, then I find the volume, then I find total mass with the density of your table. I assume that $M_b$ don't change

edit :

1 hour ago, Mordred said:

... I'm positive you didn't use the Maxwell statistics for it ?...

of course

Edited by stephaneww

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wrong assumption mass density of baryons does indeed change with volume the mass density of the 3 neutrino family that Planck universe uses in its calculation will increase at an average $(1-z)^4$ as you go further into the past. Baryon density is directly affected by the change in volume as well as any other reactions to gravity (The value given is the mean average) however baryons can collect into gravity wells lowering the mean average.

As well as any other chemical reactions such as nucleosynthesis processes.

The only at home method I know to calculate the number density of a particle species is via either the Bose-Einstein, Fermi-Dirac or Maxwell Boltzmann statistics. Unless you have access to the Monte Carlo software to handle baryon acoustic oscillations.