Problem for simulation on recollapse model

In summary, the conversation discusses the progress made in the modelisation of the scale factor and the recollapse phenomenon. The speaker has successfully formulated the differential equations and obtained a curve representing the recollapse. However, there may be discrepancies in the results compared to a previous figure, and the speaker is seeking advice on how to resolve this issue.
  • #1
fab13
312
6
Hello,

I post another topic because my first main problem about the modelisation of scale factor by solving numerically friedmann equations is resolved.
However, It remains one problem, that of the recollapse when integration starts at a early age of the universe.

i have reformulated the differential system through the equation :

[tex]H^2(a) = \bigg(\frac{1}{a} \frac{da}{dt}\bigg)^{2} = H_0^2 \left( \frac{\Omega_m}{a^3} + \frac{\Omega_k}{a^2} + \Omega_\Lambda \right)
[/tex]

which gives equation n°1 :

[tex]
\left( \frac{da}{dt} \right)^2 = H_0^2 \left( \frac{\Omega_m}{a} + \Omega_k + \Omega_\Lambda a^{2} \right)
[/tex]

then we have :

[tex] a = \Omega_{m} \bigg( \frac{1}{H_{0}^{2}} \bigg(\frac{da}{dt}\bigg)^{2} - \Omega_{k} - \Omega_{\Lambda} a^{2} \bigg)^{-1} [/tex]​

So, the differential equation can be written in this way :


[tex]

a''=\bigg(\frac{-4 \pi G}{3c^{2}}(\frac{\rho_{0} c^{2}}{a^{3}}+3 p)+\frac{\Lambda}{3}\bigg) \Omega_{m} \bigg( \frac{1}{H_{0}^{2}} \bigg(\frac{da}{dt}\bigg)^{2} - \Omega_{k} - \Omega_{\Lambda} a^{2} \bigg)^{-1} [/tex]​

Finally, i managed to get the curve representing the "recollapse" with [tex]\Omega_{m} > 1 [/tex], [tex] a_{0} = 1 [/tex], and [tex] a'_{0} = H_{0} [/tex] and [tex] \Omega_{\Lambda}=0 [/tex] ( https://www.physicsforums.com/attachment.php?attachmentid=22861&d=1262647111") . The integration begins at [tex] t_{0}=13.7 Gyr [/tex]. The results are validated by the value of the maximum of scale factor before the recollapse :

[tex] a_{max} = a ( - \frac{\Omega_{m}}{\Omega_{k}} )[/tex]

[tex] t_{max} = - \frac{5}{-4} = 1.25 [/tex] on the figure 5​

I would like to get the first part of this curve ( for [tex] t0 << 13.7 Gyr [/tex] ), ie from different initial conditions. I tried with these conditions :

[tex] a_{0} = 0.001 [/tex], and [tex] a'_{0} =H_{0} (5*10^{3}-4)^{1/2} = 70.6824 H_{0} [/tex]

respecting so equation n°1 for the value of [tex] a'_{0} [/tex]. For this case, the Matlab solver begins integration at [tex] t_{0}=380.000 years [/tex], so a redshift z=1000.

I get the https://www.physicsforums.com/attachment.php?attachmentid=22862&d=1262647111". There's well the recollapse design but the maximum is not equal to 1.25 (much more little, about 0.09 ) and the big crunch occurs fastier than on figure 5.

What's the problem, How it is done ? Why i don't get the same recollapse as in figure 5, i mean the same maximum and time "t_final" of big crunch ( [tex] a(t_{final})=0 [/tex].

Thanks a lot.
 
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  • #2




Hello,

Thank you for sharing your progress and results with us. It seems like you have made significant progress in your modelisation of the scale factor and the recollapse phenomenon. It is always exciting to see scientists making progress in their research and tackling challenging problems.

From what I can see, it seems like you have correctly formulated the differential equations and have successfully obtained the curve representing the recollapse. However, it seems like there may be some discrepancies in the values of the maximum scale factor and the time of the big crunch compared to the results in figure 5. This could be due to several factors, such as numerical errors, initial conditions, or assumptions made in the model.

I would suggest double-checking your equations and initial conditions to ensure accuracy. You can also try varying the initial conditions and see how it affects the results. Additionally, you can also compare your results with other models or data to see if they align. This can help identify any potential errors or areas for improvement in your model.

Overall, it seems like you are on the right track and have made significant progress. Keep exploring and refining your model, and I am sure you will be able to resolve any discrepancies and obtain accurate results. Good luck with your research!
 

1. What is the recollapse model?

The recollapse model is a cosmological model that suggests the universe will eventually stop expanding and collapse back in on itself due to the force of gravity.

2. What problems are associated with simulating the recollapse model?

One of the main problems with simulating the recollapse model is the difficulty in accurately modeling the behavior of dark matter, which is a major component of the universe's mass and can greatly affect the rate of collapse.

3. How do scientists simulate the recollapse model?

Scientists use computer simulations that take into account various factors such as the distribution of matter, the expansion rate of the universe, and the effects of dark matter to model the potential recollapse of the universe.

4. What are some potential outcomes of simulating the recollapse model?

Simulations of the recollapse model have shown that the outcome can vary greatly depending on the initial conditions and assumptions used. Some simulations suggest a Big Crunch scenario where the universe collapses into a singularity, while others show a more gradual collapse resulting in a stable, but smaller, universe.

5. How does simulating the recollapse model help us understand the universe?

By simulating the recollapse model, scientists can test different theories and assumptions about the universe's structure and evolution, allowing us to better understand the possibilities and potential outcomes of the universe's fate. It also helps us to refine our understanding of dark matter and its role in the universe's dynamics.

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