Friedmann-Lemaitre-Robertson-Walker model

[Nicole01]

TL;DR Summary

stability analysis of the FLRW model using the Lyapunov direct function

I am a math student conducting a study on the stability analysis of the FLRW model using the Lyapunov direct function. To do that, I need to find the equilibrium points of FLRW and create a Lyapunov function to carry out the study. Do i find the equilibrium points by setting time derivatives to zero so that a(t) equals 0?

 

[PeterDonis]

the equilibrium points of FLRW

An FLRW model doesn't have any equilbrium points. The scale factor is always changing in any FLRW model.

 

[PeterDonis]

so that a(t) equals 0?

a(t) is never zero in any FLRW model.

 

[Nicole01]

Then how do I conduct stability analysis using lyapunovs function if I don't have equilibrium points to conclude whether the trajectories converge or diverge

 

[Orodruin]

An FLRW model doesn't have any equilbrium points. The scale factor is always changing in any FLRW model.

Making a static universe is literally why Einstein originally introduced the cosmological constant. Then you have pathological cases such as Minkowski space, which is technically on FLRW form.

 

[PeterDonis]

Making a static universe is literally why Einstein originally introduced the cosmological constant. Then you have pathological cases such as Minkowski space, which is technically on FLRW form.

Yes, these are edge cases where one can have a sort of "equilibrium", and at least in the Einstein static universe case, the idea of doing a "stability analysis" makes sense (the result of such an analysis is already known). It does not seem to me that the OP's question was limited to such special cases, however.

 

[PeterDonis]

how do I conduct stability analysis using lyapunovs function if I don't have equilibrium points to conclude whether the trajectories converge or diverge

Um, you can't?

 

[cianfa72]

a(t) is never zero in any FLRW model.

Sorry, $a(t)$ in any FLRW models in standard coordinates is the "scale" factor for the spatial metric on each spacelike hypersurface of constant cosmological time $t$ ?

 

[Ibix]

Sorry, $a(t)$ in any FLRW models in standard coordinates is the "scale" factor for the spatial metric on each spacelike hypersurface of constant cosmological time $t$ ?

Yes

 

[PeterDonis]

$a(t)$ in any FLRW models in standard coordinates is the "scale" factor for the spatial metric on each spacelike hypersurface of constant cosmological time $t$ ?

Yes. And please note that anyone posting in an "A" level thread on this topic should not even need to ask this question. "A" level means you should have a graduate level knowledge of the subject matter, and one of the reasons for that level designation is to avoid having advanced threads cluttered with basic questions.