Friedmann-Lemaitre-Robertson-Walker model

  • #1
Nicole01
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0
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?
 
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  • #2
Nicole01 said:
the equilibrium points of FLRW
An FLRW model doesn't have any equilbrium points. The scale factor is always changing in any FLRW model.
 
  • #3
Nicole01 said:
so that a(t) equals 0?
a(t) is never zero in any FLRW model.
 
  • #4
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
 
  • #5
PeterDonis said:
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.
 
  • #6
Orodruin said:
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.
 
  • #7
Nicole01 said:
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?
 
  • #8
PeterDonis said:
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## ?
 
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  • #9
cianfa72 said:
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
 
  • #10
cianfa72 said:
##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.
 
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FAQ: Friedmann-Lemaitre-Robertson-Walker model

What is the Friedmann-Lemaitre-Robertson-Walker (FLRW) model?

The Friedmann-Lemaitre-Robertson-Walker (FLRW) model is a class of cosmological models that describe a homogeneous, isotropic expanding or contracting universe. It is based on solutions to Einstein's field equations of General Relativity and incorporates the cosmological principle, which states that the universe is homogeneous and isotropic on large scales.

What are the key components of the FLRW metric?

The FLRW metric has several key components: the scale factor \(a(t)\), which describes how the size of the universe changes with time; the curvature parameter \(k\), which can be +1, 0, or -1, corresponding to a closed, flat, or open universe, respectively; and the spatial coordinates, which are usually represented in comoving coordinates. The metric can be written as \(ds^2 = -c^2 dt^2 + a(t)^2 \left( \frac{dr^2}{1-kr^2} + r^2 d\Omega^2 \right)\).

How does the FLRW model describe the expansion of the universe?

The FLRW model describes the expansion of the universe through the scale factor \(a(t)\), which changes over time. The rate of this expansion is governed by the Friedmann equations, which are derived from Einstein's field equations. These equations relate the scale factor to the energy density, pressure, and curvature of the universe. Observations such as the redshift of distant galaxies provide evidence for this expansion.

What are the Friedmann equations?

The Friedmann equations are two differential equations that describe the dynamics of the scale factor \(a(t)\) in the FLRW model. The first Friedmann equation is \(\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{k c^2}{a^2}\), where \(\dot{a}\) is the time derivative of the scale factor, \(G\) is the gravitational constant, \(\rho\) is the energy density, and \(k\) is the curvature parameter. The second Friedmann equation is \(\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3} \left( \rho + \frac{3p}{c^2} \right)\), where \(\ddot{a}\) is the second time derivative of the scale factor and \(p\) is the pressure.

What observational evidence supports the FLRW model?

Several key observations support the FLRW model: the cosmic microwave background radiation (CMB),

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