Hubble Parameter as function of time in universe models

In summary, for closed L-CDM universes with a cosmological time scale of 1.5x the critical density, collapse occurs around 100 Gyrs, and for 2x critical density it occurs around 45 Gyrs.
  • #1
timmdeeg
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This graph shows ##H## as a function of time related to the L-CDM model. Do we (@Jorrie) have similar graphs e.g. for ##\Lambda=0##; ##k=-1## critical, ##\Lambda=0##; ##k=0## open, ##\Lambda=0##; ##k=+1## closed?

That would be great, thanks in advance.

1668953461789.png
 
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  • #2
Not precisely, but you can get close enough for all practical purposes by playing around with the input parameters and output options. E.g. ##\Lambda = 0.0000001##, set the output scaling to Normalized, select Chart and set hor and vert scales appropriately:
1669026724844.png

For Open and Closed cases, you play around with ##\Omega##. I have used the http://jorrie.epizy.com/docs/index.html?i=1 version, which has more liberal range limits than the approved Github version.
 
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  • #3
Ah, great, thanks for your advise!

One question, how can I show only one of these curves?
1669049449361.png
 
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  • #4
Just go to 'Column definition and selection'. I usually click 'none' and then select the two or three that I need. The default selections are just to give an idea of how it works.
 
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  • #5
Got it, thanks.
 
  • #6
@Jorrie is there a way for the calculator to show recollapse? I can't seem to get there no matter how I fiddle with the parameters.
 
  • #7
Lightcone8 does not allow for high Omega or very small Lambda, so I'm not sure that collapse can happen with these limitations. Any small Lambda may quickly become dominant again.
As a matter of fact it seems to crash if I set Lambda to 0.001 and Omega to 1.5. Will have to investigate that.

I recall that I have previously simulated a zero lambda situation with collapse on older, less accurate versions, but it will take some searching to find that.
 
  • #8
timmdeeg said:
This graph shows ##H## as a function of time related to the L-CDM model. Do we (@Jorrie) have similar graphs e.g. for ##\Lambda=0##; ##k=-1## critical, ##\Lambda=0##; ##k=0## open, ##\Lambda=0##; ##k=+1## closed?

Bandersnatch said:
@Jorrie is there a way for the calculator to show recollapse?

In a closed matter-only (dust) FLRW univers, parametric expessions for the scale factor ##a## and cosmological time ##t## as functions of conformal time ##η## are (from Ryden)
$$\begin{align}
a\left(\eta\right) &= \frac{1}{2} \frac{\Omega_0}{\Omega_0 - 1} \left( 1 - \cos\eta \right) \\
t\left(\eta\right) &= \frac{1}{2H_0} \frac{\Omega_0}{\left( \Omega_0 - 1 \right)^{3/2}} \left( \eta - \sin\eta \right),
\end{align}$$
with ##0<\eta<2\pi##, and with ##\Omega_0>1## the present density relative to critical density.

The Hubble parameter is given by (with abuse of notation)
$$H\left(\eta\right) = \frac{1}{a} \frac{da}{dt} = \frac{1}{a}\frac{\frac{da}{d\eta}}{\frac{dt}{d\eta}} = \frac{2H_0 \left( \Omega_0 - 1 \right)^{3/2}}{\Omega_0} \frac{\sin\eta}{\left( 1 - \cos\eta \right)^2}.$$
Ii is easy to put ##\eta##, ##t\left(\eta\right)## , and ##H\left(\eta\right)## into three columns of a spreadsheet, and to use these to plot ##H\left(\eta\right)## versus ##t\left(\eta\right)## for ##0<\eta<2\pi##.
 
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  • #9
George Jones said:
Ii is easy to
Yeah, but that requires ME to do some work, instead of somebody else ;)

For those interested, here's the graph for ##\Omega_0=1.5## and ##H_0=67.74##
1669666403563.png

And the spreadsheet

(make a copy if you want to change the parameters)

The behaviour tracks what Jorrie's calc outputs for early periods, so it's probably typed in alright.
The switcheroo towards collapse happens around 100 Gyrs for 1.5x critical density; for 2x density it's about 45 Gyrs; 800 Gyrs for 1.1 - which are the time scales I wanted to get a sense of.
 
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FAQ: Hubble Parameter as function of time in universe models

What is the Hubble parameter?

The Hubble parameter, denoted as H(t), is a measure of the rate at which the universe is expanding at a given time. It is named after the astronomer Edwin Hubble, who first discovered the expansion of the universe.

How does the Hubble parameter change with time?

In most universe models, the Hubble parameter decreases as the universe ages. This is because the expansion of the universe is slowing down due to the effects of gravity. However, in some models, such as the inflationary model, the Hubble parameter may initially increase before decreasing.

What is the significance of the Hubble parameter in cosmology?

The Hubble parameter is a crucial parameter in cosmology as it helps us understand the evolution of the universe. By measuring the Hubble parameter at different times, we can determine the age of the universe, its expansion rate, and the amount of matter and energy it contains.

How is the Hubble parameter calculated in different universe models?

The Hubble parameter is calculated using various methods, including observations of the cosmic microwave background radiation, the redshift of galaxies, and the luminosity and distance of supernovae. Different universe models may use different equations and assumptions to calculate the Hubble parameter.

Can the Hubble parameter change over time?

Yes, the Hubble parameter can change over time in response to different factors, such as the amount and distribution of matter and energy in the universe. It can also be affected by the presence of dark energy, which is thought to be responsible for the current acceleration of the universe's expansion.

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