Does This Cosmological Model Re-Collapse?

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Discussion Overview

The discussion revolves around a cosmological model characterized by specific density parameters, particularly focusing on whether the model leads to a re-collapse of the universe. Participants are exploring mathematical implications and interpretations of the Friedmann equations in the context of cosmology.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a model with specific density parameters and calculates three values of the scale factor, questioning the significance of the two positive values.
  • Another participant asks for clarification on the mathematical approach being used to analyze the model.
  • There is a suggestion that if the scale factor is normalized time, it may indicate points of collapse and re-collapse, though this interpretation is uncertain.
  • Participants inquire about the definitions of density parameters and the total density of the universe being studied.
  • One participant emphasizes the importance of using the second Friedmann equation to determine the behavior of the Hubble parameter as it approaches zero.
  • Another participant expresses uncertainty about how the derivative of the Hubble parameter relates to identifying the recollapsing universe.
  • There is a challenge regarding the total density parameter, with a participant suggesting it may equal 3.01 instead of 1.
  • A mathematical approach is proposed involving the second derivative test from calculus to analyze the behavior of the scale factor over time.

Areas of Agreement / Disagreement

Participants do not reach consensus on the interpretation of the mathematical results or the definitions of the density parameters. Multiple competing views and uncertainties remain regarding the implications of the calculations and the conditions for re-collapse.

Contextual Notes

Participants express uncertainty about the definitions and implications of the density parameters, as well as the mathematical steps involved in applying the Friedmann equations. There are unresolved questions about the conditions under which the universe may re-collapse.

ajclarke
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Hello.

I have been working through some questions and answers to do with cosmology. One of them asks you to consider a model where:

\Omega_{MO}=3
\Omega_{\Lambda O}=0.01
\Omega_{RO}=0
and asks you to show mathematically that the model re-collapses.

Following through the math, I get three values of a: -14.87,1.51 and 13.36.

Clearly the first can be disregarded and unphysical since a cannot be negative, but I can't decide what's the significance between the second two which allows me to isolate the value corresponding to collapse.

Cheers.
Adam
 
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What 'math' are you following through with?
 
If a is normalized time, then it may have zero diameter 14.87 time units in the past, first collapse 1.51 in the future, and a "recollapse" later. Not sure if that's correct though.
 
What is the definition of \Omega_{s0} for some species s? What is \Omega_{\rm total 0} in the universe you are studying?
 
Make use of the second Friedmann equation to make sure that when H(a) goes to zero, dH/da is negative.
 
zhermes said:
What 'math' are you following through with?

I used the equation for the Hubble Parameter as a function of redshift, then changed this over to be a function of scale factor instead.

cristo said:
What is the definition of \Omega_{s0} for some species s? What is \Omega_{\rm total 0} in the universe you are studying?

\Omega_{total 0} = 1

I don't understand the first bitof the question I'm sorry.

Chalnoth said:
Make use of the second Friedmann equation to make sure that when H(a) goes to zero, dH/da is negative.

I'm uncertain as to how that determines which of the two remaining parameters is the recollapsing universe?
 
ajclarke said:
I'm uncertain as to how that determines which of the two remaining parameters is the recollapsing universe?
If the derivative of the Hubble parameter is negative, then it's recollapsing.
 
Don't you have Ωtotal0 equal to 3.01, instead of unity?
 
Use the second derivative test from elementary calculus. a\left(t\right) has a local maximum at t = t_1 if da/dt \left(t_1 \right) = 0 and d^2 a/dt^2 \left(t_1 \right) < 0. To find d^2 a/dt^2, differentiate the Friedmann equation.
 

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